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https://arxiv.org/abs/alg-geom/9602016

alg-geom/9602016

Ingo Hadan

Ingo Hadan

Conics Touching a Quartic Surface with 13 Nodes

31 pages, LaTeX 2.09. .tex and .dvi-Files also available at http://www-irm.mathematik.hu-berlin.de/~hadan/papers.html

Motivated by questions occuring in the construction of certain twistor spaces the parameter space of conics tangent to a given quartic is investigated. For a given real quartic surface in complex $\PP ^3$ that has exactly 13 ordinary nodes as singularities this parameter space is studied. In particular, its irreducible components are described.

alg-geom

\section{Introduction} The investigations of this article are motivated by problems arising in the study of twistor spaces over the connected sum of three complex projective planes. In~\cite{kreussler-kurke} it is shown that, under suitable conditions, such a twistor space is a small resolution of a Double Solid branched over a real quartic surface with exactly 13 ordinary nodes. (A Double Solid is a branched double cover of $I\!\!P ^3$; see~\cite{clemens} for many aspects of these varieties. The properties of Double Solids used in this paper are contained in~\cite{kreussler}.) However, it is not known for which quartics and for which resolutions twistor spaces occur. One approach to solving this problem is to study the family of twistor lines. These are smooth rational curves with normal bundle ${\cal O}(1)^{\oplus 2}$ in the total space of the twistor fibration. Those curves will be called ``lines'' in the sequel. The base of the twistor fibration must be contained in the set of real points in the parameter space of all lines in the twistor space. Therefore, it seems to be promising to investigate the parameter space of all lines in small resolutions of Double Solids or at least something which is, hopefully, similar enough to this space. There are some articles on Double Solids e.g. \cite{clemens} or \cite{tichomirov} but there is nothing (as far as known to the author) about lines in Double Solids. In \cite{tichomirov} ``lines'' in Double Solids are studied but Tikhomirov uses an essentially different notion of lines. (His lines are mapped to double tangents of the branch locus -- but cf. Proposition~\ref{geraden-in-DS}.) Furthermore, he assumes the branch locus to be smooth. The following Proposition completes the motivation for the investigations announced in the abstract: \begin{satz} \label{geraden-in-DS} Let $\pi:Z\longrightarrowI\!\!P^3$ be a Double Solid branched over a quartic $B$ that has at most ordinary nodes as singularities. Let $C\subset Z$ be a line (i.e., a smooth rational curve with normal bundle ${\cal O}(1)^{\oplus 2}$) outside the singular set of $Z$. Then the image $\pi(C)$ of $C$ is a conic that has only even order contact with $B$. \end{satz} By this fact the study of the parameter space of all those ``touching conics'' is motivated. In this article we will study the parameter space of touching conics by examining its fibration over $\check{I\!\!P}^3$. (This fibration is given by assigning to each conic the unique plane in which it is contained). In Section~\ref{conics-plane} the generic fibre of this fibration is described. Our approach to the description of the space of touching conics is the detailed study of the {\em reducible} touching conics which are complanar pairs of double tangents at the quartic $B$. Section~\ref{section-dt} is devoted to the examination of the space of double tangents at $B$. These results are used in Section~\ref{sec-tochin-conic} to determine the monodromy of the space \[\mbox{$Y_F$}:=\{(\ell,H)\in \mbox{Grass}(2,4)\times \check{I\!\!P}^3\,|\, \ell \subset H, \ell\;\mbox{is double tangent}\}\] over $\check{I\!\!P}^3 \setminus \Delta$, where $\Delta$ denotes the closed subset of planes $H$ such that $B\cap H$ is singular. As a corollary we get the monodromy of the symmetric product $\left(\mbox{$Y_F$}\stackrel{\mbox{\tiny sym}}{\times}_{\check{I\!\!P}^3}\mbox{$Y_F$}\right) \setminus \!\mbox{Diag}$. The latter can be regarded as the parameter space of reducible touching conics. It serves as a ``frame'' within the parameter space of all touching conics which, finally, is described using the knowledge on this ``frame''. {\bf Acknowledgement:} For their ideas and helpful discussions that essentially influenced my work on this topic, I am most grateful to Bernd Kreu{\ss}ler and Prof.~Kurke. \section{General Preliminaries}\label{gen-prel} First, for completeness, Proposition~\ref{geraden-in-DS} is to be proven, now. {\bf Proof of (\ref{geraden-in-DS}):} Let $\omega_Z$ be the canonical sheaf of $Z$. It holds $\omega_Z \cong \pi^*{\cal O}_{I\!\!P ^3}(-2)$ (cf. \cite{bpv} Lemma~I.17.1) and by adjunction formula we have \hspace*{\mathindent} \begin{tabular}{@{}r@{\quad}c@{\quad}l@{\extracolsep{3em}}l} \rule{0mm}{4ex}$\omega _C$ & $\cong$ & $\omega _Z \otimes \det N_{C|Z}$ & hence\\ \rule{0mm}{4ex}${\cal O}_C(-2)$ & $\cong$ & $\pi ^*{\cal O}_{I\!\!P ^3} (-2) \otimes {\cal O}_C(2)$ &\\ \rule{0mm}{4ex}${\cal O}_C(-4)$ & $\cong$ & $\pi ^*{\cal O}_{I\!\!P ^3} (-2) \otimes {\cal O}_C$ \end{tabular} or for divisors on $C$ \[-4[pt] = \pi ^*(-2H)\cdot [C]\qquad \mbox{($H$ -- a hyperplane section in $I\!\!P ^3$)} \] and therefore \begin{eqnarray*} \pi _*\left(-4 \left[pt\right]\right) & = & \pi _*\left( \pi ^* \left( -2H\right)\cdot \left[C\right]\right) \\ & = &(-2H)\cdot \pi _*\left(\left[C\right]\right) \qquad\qquad\mbox{by projection formula and finally}\\ 2[pt] &=& H \cdot \pi_*\left( \left[C\right]\right). \end{eqnarray*} Thus we get \[ \pi _*\left(\left[C\right]\right) = 2H^2. \] Now, if $\pi(C)$ were of degree one in $I\!\!P^3$ and $\pi |_C: C \rightarrow \pi(C)$ of degree two we would have \[ N_{C|Z} = N_{\pi^{-1}(\pi(C))|Z} \stackrel{!}{\cong} \pi^*N_{\pi(C)|I\!\!P ^3} \cong \pi^*\left( {\cal O}_{\pi(C)}\left( 1\right)^{\oplus 2}\right) \cong {\cal O}_C(2)^{\oplus 2} \] in contradiction to $N_{C|Z} \cong {\cal O}_C(1)^{\oplus 2}$. Therefore $\pi(C)$ is a rational curve of degree two and $\pi|_C:C\rightarrow \pi(C)$ is an isomorphism, i.e., $\pi(C)$ is a smooth conic. Furthermore, $\pi^{-1}(\pi(C))$ must split into two irreducible components if not $\pi(C)\subset B$. Now, let $\pi(C)\not\subset B$, $\mbox{Spec}\,A = U \subset \pi(C)$ a suitable open subset, $f$ the equation of $B$ restricted to $U$, and $A_Z:= \raisebox{0.25ex}{$A[T]$}\big/ \raisebox{-0.25ex}{$(T^2-f)$}$. Then $\mbox{Spec}\, A_Z=\pi^{-1}(U) \subset \pi^{-1}(\pi(C))$ and $\mbox{Spec}\, A_Z$ is reducible if and only if $(T^2-f)$ is reducible in $A[T]$, i.e., if and only if there is an $g\in A$ with $f=g^2$. This proves the proposition. \hspace*{\fill} $\Box$ \subsection{Parameter space of conics}\label{psc} We will construct the parameter space of conics in $I\!\!P ^3$. The space of ``touching conics'' will be contained within this parameter space. Let $\check{I\!\!P} ^3$ be the space of planes in $I\!\!P ^3$ and $\cal S$ the universal subbundle over $\check{I\!\!P} ^3=\mbox{Grass}(3,4)$. Then $P:=I\!\!P (Sym^2 {\cal S}^{\vee})$ is the parameter space of all conics in $I\!\!P ^3$. (Every conic can be given by a plane and a symmetric form of degree two in this plane. On the other hand, every conic determines a unique plane which it sits in and in that plane a symmetric 2-form which is unique up to multiplication by scalars. Even for a double line there is a unique plane in which it is contained. It is determined by the non-reduced subscheme structure of the double line.) The projection $p:P\longrightarrow \check{I\!\!P}^3$ assigns to each conic the unique plane which it is contained in. There is a universal family over $P$, constructed as follows: Let $H:=I\!\!P(p^*{\cal S})$ be the pull-back of the universal plane over $\check{I\!\!P} ^3$, $\tau:H\rightarrow P$ the projection, and ${\cal O}_H(1)$ the relative tautological bundle of $H$ over $P$. Then there is a distinguished section in $(\tau^* {\cal O}_{P|\check{I\!\!P} ^3}(-1))^{\vee} \otimes {\cal O}_H(2)$, for it is \begin{eqnarray*} \left(\tau ^* {\cal O}_{P|\check{I\!\!P} ^3}\left(-1\right)\right)^{\vee} \otimes {\cal O}_H\left(2\right) &=& Hom\left( \tau^* {\cal O}_{P|\check{I\!\!P} ^3}\left(-1\right), \left( {\cal O}_H\left( -1\right) ^{\vee} \right)^{\otimes 2} \right) \\ &=& Hom\left( \tau^*{\cal O}_{P|\check{I\!\!P} ^3}\left(-1\right), Sym^2 \left( {\cal O}_H\left( -1\right) \right) ^{\vee} \right). \end{eqnarray*} and there are canonical injections of vector bundles over $H$: \[ {\cal O}_H(-1)\hookrightarrow \tau^*p^*{\cal S}. \] \[ \tau^* {\cal O}_{P|\check{I\!\!P} ^3}(-1) \hookrightarrow \tau^*p^*(Sym^2 {\cal S}^{\vee}) = \tau^*Sym^2 (p^*{\cal S})^{\vee}. \] The distinguished section is given by the composition \[ \tau^*{\cal O}_{P|\check{I\!\!P} ^3}(-1)\hookrightarrow \tau^*Sym^2 p^*{\cal S}^{\vee} \longrightarrow Sym^2 {\cal O}_H(-1)^{\vee}. \] The universal family over $P$ is the zero locus of this section. The above construction shows that there is a natural projection from the parameter space of ``touching conics'' onto $\check{I\!\!P}^3$, assigning to each conic the plane which it sits in. Section~\ref{conics-plane} is devoted to the study of the fibres of this projection. \subsection{13-nodal quartics} As outlined in the introduction, the focus of the present investigations lies on the study of quartic surfaces with exactly 13 ordinary double points. Those quartics are extensively studied in \cite{kreussler}. The results needed in the sequel are to be summarised here. \begin{satz}\label{quartik13} Every real\/\myfootnote{with respect to the standard real structure of $I\!\!P^3_{\!\!\Bbb C}$} quartic surface $B$ with exactly one real point and 13 ordinary double points can be defined by an equation of the form \begin{eqnarray*} F&=&x_3^2f_2+2x_3L_0L_1L_2 +f_2^2-f_2\left(L_0^2+L_1^2+L_2^2\right) + L_0^2L_1^2+L_0^2L_2^2+L_1^2L_2^2 \\ &=& \frac{1}{4}\left( Q^2 - E_1E_2E_3E_4\right) \\ \end{eqnarray*} where $E_1 = x_3 - L_0 - L_1 - L_2$, $E_2 = x_3 + L_0 + L_1 - L_2$, $E_3 = x_3 + L_0 - L_1 + L_2$, $E_4 = x_3 - L_0 + L_1 + L_2$, $f_2 = x_0^2+x_1^2 +x_2^2$, $Q = 2f_2 +x_3^2 - L_0^2 - L_1^2 - L_2^2$ and $L_j = \sum_{i=0}^2 a_{ij}x_i$ such that $f_2 - L_j^2$ $(j=0,1,2)$ defines three smooth conics with 12 different intersection points. If, moreover, the planes $E_i$ are real then the quadratic forms $f_2 - L_j$ are positive definite. If the forms $L_j$ are mutually linearly independent then $F$ defines a real quartic with exactly 13 ordinary double points and exactly one real point $P= (0:0:0:1)$. Each of the six lines $E_i = E_j = 0$ intersects the quadric $Q$ in two different points which form a conjugate pair of double points of the quartic.\hspace*{\fill}$\Box$ \end{satz} {\bf Remark:} Those quartics $B$ satisfying all the conditions of the above proposition are just the quartics which generically occur in connection with twistor spaces as mentioned in the Introduction (cf. \cite{kreussler} and \cite{kreussler-kurke}). \begin{lemma}\label{branch-sextic} The projection of $B$ (as above -- with real planes $E_i$) from $P$ onto the plane $x_3=0$ defines a double cover of $I\!\!P^2$ which is branched along the sextic $\ti{S} = (f_2 - L_0^2) (f_2 - L_1^2)(f_2 - L_2^2)$ in $I\!\!P ^2$. The conic $f_2=0$ touches $\ti{S}$ in six smooth points.\hspace*{\fill}$\Box$ \end{lemma} \section{Conics touching a plane quartic}\label{conics-plane} In this section\myfootnote{Many ideas of this section are already contained in \cite{salmon}.}, for a given quartic curve $B$ in $I\!\!P^2$, the set of conics that have only even order contact with $B$ (but are different from double lines) is investigated. Those conics will be called {\em touching conics} in the sequel. Throughout this section, let $B\subset I\!\!P ^2$ be an irreducible quartic curve given by a form $F$ of degree four which has at most one ordinary node as its only singularity. Then the following lemma holds. \begin{lemma}\label{Strukt-ber-KS} The set of all touching conics is the union of one-parameter-families. In each family the elements are mutually different. If $B$ is smooth, the families are pairwise disjoint. If $B$ is singular, two families can only intersect in a reducible conic which consists of two lines both containing the singular point of $B$. \end{lemma} \begin{proof} The quadratic form $U$ defines a touching conic if and only if there exist two further quadratic forms $V$ and $W$ such that \[ F = UW - V^2. \] Since \[ UW - V^2 =U(\lambda ^2U+2 \lambda V +W) - (\lambda U + V)^2 \quad \lambda \in \Bbb C, \] by $U$, $W$, and $V$ a whole one-parameter-family of touching conics is given: \begin{equation} \label{epf} \lambda ^2U + 2 \lambda \mu V + \mu ^2 W \quad (\lambda : \mu ) \in I\!\!P ^1 . \end{equation} All conics in such a family are different from each other. Otherwise we would have \[ F= U(\lambda^2 U +2\lambda V + W) - (\lambda U + V)^2 = U(\mu^2 U +2\mu V + W) - (\mu U + V)^2 \] with $\lambda^2 U +2\lambda V + W = \mu^2 U +2\mu V + W$ and $\lambda \ne \mu$. Consequently \[ (\lambda U + V)^2 = (\mu U+ V)^2 \qquad\mbox{hence}\] \[ \lambda U+ V = -(\mu U+V) \quad \mbox{thus} \quad V= \frac{\lambda + \mu}{2} U \qquad\mbox{and finally}\] \[ F=U \left( W-\left( \frac{\lambda +\mu}{2}\right)^2 U\right), \] so that the quartic $B$ would be reducible in contradiction to the assumption. Suppose $U$ is contained in two different families~(\ref{epf}), i.e., there exist $V$, $W$, resp. $V'$ and $W'$ satisfying \[ F= UW -V^2 =UW'-V'^2 \qquad \mbox{and therefore} \] \[ U(W-W') =V^2 - V'^2 =(V+V')(V-V'). \] If $U$ is irreducible it follows $U\,|\,(V+V')$ or $U\,|\,(V-V')$ and consequently $V'= \pm (\lambda U-V)$ and $W'=\lambda ^2U-2\lambda V+W$, i.e., $(U,V,W)$ and $(U,V',W')$ yield the same family. If $U$ is reducible and neither $U\,|\,(V+V')$ nor $U\,|\,(V-V')$ holds, then the intersection of the two lines of $U$ must be a point of $B$ which is necessarily singular. If, finally, $U$ were a double line, $B$ would have an equation of the form $F =L^2W-V^2$ and therefore would have at least two singular points if the line $L=0$ intersects $B$ in two points or a cusp in the intersection point of $B$ with $L=0$. This completes the proof. \end{proof} \begin{lemma} \label{epf-verh} Let $B$ be a quartic with exactly one ordinary node. A conic $U$ consisting of two lines through the node is contained in exactly two one-parameter-families~(\ref{epf}) that intersect in just this conic. \end{lemma} \begin{proof} Let $F= f_1\,x^4 + f_2\,y^4 +\cdots$ be the equation of the quartic $B$ and let the node of $B$ have the coordinates $(0:0:1)$. In suitable coordinates $U$ is given by the equation $xy=0$. By assumption, $F$ is of the form \begin{equation}\label{coeff-comp} F=xyW-V^2 \end{equation} with quadratic forms $V$ and $W$. By comparing the coefficients on both sides of (\ref{coeff-comp}) one finds exactly two one-parameter-families $W_\lambda$ containing the conic $U$. \end{proof} Now, the number of reducible conics in each family~(\ref{epf}) is to be determined. Knowing the number of double tangents at a quartic curve, this permits to count the one-parameter-families of touching conics. \begin{lemma} \label{epf-zerf-el} Every one-parameter-family~(\ref{epf}) contains at least five and at most six reducible elements. The family contains only five reducible conics if and only if it contains a conic which splits into two lines intersecting in a point of $B$ (which is necessarily singular). \end{lemma} \begin{proof} A conic is reducible if and only if the determinant of its matrix of coefficients vanishes. Therefore one has to examine the roots of the polynomial in $\lambda$: $\det (\lambda ^2 U + 2\lambda V + W)$. This polynomial is of sixth degree and, therefore, has at most six roots. For proving the lemma it is necessary to investigate the conditions under which the equation has multiple roots. Since the two triples $(U,V,W)$ and $(\lambda ^2U + 2\lambda V + W, \lambda U + V, U)$ of conics define the same one-parameter-family it is sufficient to study under which conditions $\lambda=0$ is a multiple root. For $\lambda=0$ to be a multiple root, conditions on the coefficients of $U$, $V$, and $W$ are posed. A simple calculation shows that a multiple root corresponds to a reducible conic the lines of which intersect in a point $P\in B$, and that no other conic of the one-parameter-family then can contain $P$. Therefore, at most one root of the considered equation can be of higher multiplicity since $B$ was supposed to have at most one singular point. If the equation had a root of multiplicity greater than two then the quartic $B$, given by the equation $F=UW-V^2$, would have a cusp. This proves the lemma. \end{proof} The following proposition is proved e.g. in \cite{salmon} or \cite{burau}, the smooth case is also treated in \cite{griffiths-harris} Section~4.4. \begin{satz}\label{Anz-DT} Let $B$ be an irreducible quartic curve. If $B$ is smooth then $B$ has 28 double tangents\/\myfootnote{resp. lines that have fourth order contact with $B$}. If $B$ has one ordinary node as its only singularity then $B$ has 22 double tangents. \end{satz} {\bf Remark:} The Pl\"ucker formulas seem to contradict the second part of the Proposition. One must, however, take into account that lines passing through the node and touching the quartic in an other point are not counted by the Pl\"ucker formulas. There are six of those lines as one finds for instance in \cite{salmon}. \begin{satz}\label{Anz-epf} Let $B$ be an irreducible quartic curve. If $B$ is smooth then the set of touching conics splits into 63 mutually disjoint one-parameter-families~(\ref{epf}). If $B$ has an ordinary node as its only singular point then the set of touching conics is divided into 16 families~(\ref{epf}) each of which is disjoint from all other families, and 15 pairs of families that intersect in exactly one conic which consists of two lines intersecting in a point of $B$. \end{satz} \begin{proof} A smooth quartic curve has 28 double tangents, hence, there are 378 pairs of double tangents, i.e., 378 reducible touching conics. Thus, according to Lemma~\ref{Strukt-ber-KS}, there are 63 families~(\ref{epf}) of touching conics and every touching conic is contained in one of these families. In the singular case there are: \begin{enumerate} \renewcommand{\labelenumi}{\alph{enumi})} \item 15 pairs of double tangents intersecting in the node of $B$ \item 96 pairs one line of which contains the node \item 120 pairs no line of which contains the node. \end{enumerate} According to Lemma~\ref{epf-verh} each of the pairs "a)" is contained in two families each of which contains four pairs "c)" (cf. Lemma~\ref{epf-zerf-el}). Thus the pairs "a)" and "c)" spread over 30 families which contain exactly these reducible conics. These families intersect pairwise as stated. The remaining 96 pairs "b)" must be contained in some families~(\ref{epf}), as well. These families do not intersect any other family and each contains six of the 96 pairs "b)" (cf. Lemma~\ref{Strukt-ber-KS} and Lemma~\ref{epf-zerf-el}). Therefore the 96 pairs "b)" generate 16 further families. \end{proof} \begin{lemma}\label{DT-in-EPF} Let $B$ be a smooth quartic curve. \begin{enumerate} \renewcommand{\labelenumi}{\alph{enumi})} \item The double tangents occurring in the reducible conics in one one-parameter-family~(\ref{epf}) are mutually different. \item Let $ab$, $cd$, and $eh$ be reducible conics contained in the same one-parameter-family~(\ref{epf}) ($a$, $b$, $c$, $d$, $e$, and $h$ linear forms). Then the double tangent $e$ does not occur in any reducible conic of that one-parameter-family in which the conics $ac$ and $bd$ are contained. \end{enumerate} \end{lemma} \begin{proof} a) If there were reducible conics $ab$ and $ac$ in the same one-parameter-family then the equation $F$ of $B$ could be written in the form $F = a^2bc - V^2$ with a quadratic form $V$. The intersection points of the conic $V$ with the line $a$ then would be singular points of $F$. b) The quartic $F$ may be written in the form $F = abcd - V^2$ with a quadratic form $V$. Since $eh$ is contained in the one-parameter-family spanned by the conics $ab$ and $cd$ there is a $\lambda \ne 0$ such that $ef = \lambda^2 ab + 2\lambda V +cd$. Now, by writing $F$ in the form $F= ac\cdot bd-V^2$ one finds the one-parameter-family containing $ac$ and $bd$ to be $\mu^2 ac + 2\mu\lambda V +\lambda^2bd$. Suppose $eg$ is contained in the one-parameter-family of $ac$ and $bd$, i.e. there is a $\mu \ne 0$ such that $eg = \mu^2 ac + 2\mu V +bd$. This yields \[ V = \frac{1}{2\lambda}\left( ef - \lambda ^2 ab - cd \right) = \frac{1}{2\mu} \left( eg - \mu^2 ac -bd\right) \] Hence $e(\mu f - \lambda g) = (\lambda b - \mu c)(\lambda \mu a - d)$, i.e. the linear form $e$ divides one of the two forms $(\lambda b - \mu c)$ or $(\lambda \mu a - d)$. Thus the line defined by $e$ contains one of the intersection points of $b$ and $c$ or of $a$ and $d$. In both cases the common point of the three lines is a singular point of $B$. \end{proof} \section{The parameter space of double tangents}\label{section-dt} Let $B \subset I\!\!P^3$ be a quartic surface with ordinary double points as its only singularities which is given by an equation of the form \begin{equation}\label{quartik-equ} g_1 g_3 - g_2^2 = 0 \end{equation} where $g_i$ are homogeneous of degree $i$. Let $(x_0:\ldots :x_4)$ be homogeneous coordinates on $I\!\!P^4$ and let $I\!\!P^3 \subset I\!\!P^4$ as the hyperplane $x_4 = 0$. Then \begin{equation}\label{kubik} x_4^2g_1 + 2x_4 g_2 +g_3 = 0 \end{equation} defines a cubic $K$ in $I\!\!P^4$ with $P:= (0:0:0:0:1)\in K$. Consider the projection from $P$ onto the hyperplane $x_4 = 0$. The extension $\ti{K}\stackrel{\pi}{\longrightarrow} I\!\!P^3$ of this map to the blow-up $\ti{K}$ of $K$ in $P$ is a partial small resolution of the double solid $Z_0$ branched over $B$ (cf. \cite{kreussler-pre}). The $\pi$--fibre of any point $x \in I\!\!P^3$ with $g_1(x)=g_2(x)=g_3(x)=0$ (i.e. the singular points of $B$ in the plane $g_1=0$) is just the strict transform of the line through $P$ and $x$, all other fibres consist of at most two points. In particular, there are exactly six lines through $P$ in $K$ (if the three surfaces $g_i = 0$ intersect properly) namely the six lines that are contracted by $\pi$. The only singularities of $K$ are the preimages of double points of $B$ not contained in the plane $g_1 = 0$. These are ordinary double points, as well. \begin{lemma} $\pi: K\setminus \{P\} \longrightarrow I\!\!P^3$ maps lines in $K\setminus \{P\}$ onto lines that have even intersection with $B$\myfootnote{We will call those lines simply double tangents, i.e. lines with fourth order contact will be called double tangents, too.} or which are contained in $B$. If such a line is contained in $B$ the it passes through a singular point of $B$ which is contained in the plane $g_1=0$. Moreover, such a line has one point of higher order intersection with the cubic $g_3=0$ or the line is contained in this cubic. \end{lemma} \begin{proof} Let \begin{morph} L:&I\!\!P^1 & \lhook\joinrel\longrightarrow & I\!\!P^4 \\ &(s:t) & \longmapsto & (l_0(s,t):\ldots:l_4(s,t)) \end{morph} be the parameter representation of a line $L$ in $I\!\!P^4$, which is contained in $K$. Thus the equation~(\ref{kubik}) restricted to $L$ vanishes, i.e. \[ \left.g_3\right|_L = -l_4^2 \left.g_1\right|_L -2l_4 \left.g_2\right|_L \qquad\mbox{thus} \] \begin{eqnarray*} \left. g_1\right|_L\cdot\left. g_3\right|_L - \left. g_2\right|_L^2 &=& -l_4^2 \left. g_1\right|_L^2 -2l_4 \left. g_1\right|_L\cdot \left. g_2\right|_L - \left. g_2\right|_L^2 \\ &=& - \left( l_4 \left. g_1\right|_L + \left. g_2\right|_L \right)^2 \end{eqnarray*} Hence, equation~(\ref{quartik-equ}) restricted to $\pi(L)$ is a complete square and therefore the image of $L$ in $I\!\!P^3$ is a line with even intersection with $B$ or is contained in $B$. If $\pi(L)$ is contained in $B$ then $L$ is contained in the ramification locus of $\pi$. Hence, the plane spanned by $L$ and $P$ intersects $K$ in a cubic curve which consists of $L$ counting twice and a line through $P$. The lines through $P$ in $K$ are the lines connecting $P$ with the singular points of $B$ in the plane $g_1=0$. Moreover, if $\pi(L)\subset B$ then $(-\l_4\cdot g_1|_L + g_2|_L)$ must vanish and, hence, $g_3|_L = \l_4^2\cdot g_1|_L$. \end{proof} \hspace*{\fill} \begin{minipage}{0.95\textwidth}\sloppy {\bf Remark:} In the case that we are particularly interested in -- $B$ is given by an equation as described in Proposition~\ref{quartik13} and $g_3$ is the product of three of the linear forms $E_i$, say $g_3=E_2E_3E_4$. $Q$ and $E_i$ cannot have common real zeros for there is only one real point on $B$ which is outside the quadric $Q=0$. Therefore the planes $E_i=0$ intersect the quadric $g_2:=Q=0$ along smooth conics. Consequently, there is no line which is contained in $B$ and in the cubic $g_3=0$.\par\smallskip A line $\ell$ (not contained in $g_3=0$) that has a point $P_a$ of higher order intersection with the cubic $g_3=0$ must meet the intersection of two of the three planes the cubic consists of. On the other hand, no common point of three of the four planes $E_i$ is a point of $B$ by Proposition~\ref{quartik13}. In particular, if $\ell\subset B$ then $P_a$ is not contained in the plane $g_1=0$. Hence, if $\ell$ is the image of a line in $K$ which is contained in $B$ then $\ell$ must pass through two singular points of $B$, namely $P_a$ and the the singular point $P_b$ of $B$ in the plane $g_1$ which $\ell$ must pass through by the above lemma. \par\smallskip Therefore, in our case only those lines in $B$ (if any) can appear as images of lines in $K$ that pass through two double points of $B$. But lines through two double points are double tangents unless they are contained in $B$ and hence, lines in $B$ through two double points necessarily appear in the parameter space of double tangents. \end{minipage} \begin{lemma} Every double tangent of $B$ that is not contained in the plane $g_1 =0$ is the image under $\pi$ of a line in $K$. \end{lemma} \begin{proof} Let \begin{morph} L':&I\!\!P^1 & \lhook\joinrel\longrightarrow & I\!\!P^3 \\ &(s:t) & \longmapsto & (l_0(s,t):\ldots:l_3(s,t)) \end{morph} be the parameter representation of a line in $I\!\!P^3$. If $L'$ is contained in the surface $g_3=0$ then there is nothing to show; so, let $L'$ not be contained in $g_3 = 0$. If $L'$ is a double tangent then there exists a quadratic form $q$ on $L'$ such that \[ \left. g_1\right|_{L'}\cdot\left. g_3\right|_{L'} - \left. g_2 \right|_{L'}^2 = - q^2 \qquad\mbox{and therefore} \] \[ \left. g_1\right|_{L'}\cdot\left. g_3\right|_{L'} = \left( \left. g_2\right|_{L'} + q\right)\left( \left. g_2\right|_{L'} - q\right) \] Hence there is a linear form $l_4(s,t)$ on $L'$ such that $l_4 \left. g_1\right|_{L'} = - \left( \left. g_2\right|_{L'} \pm q\right)$. By eventually replacing $q$ with $-q$ we can assume that $l_4 \left. g_1\right|_{L'} = - \left( \left. g_2\right|_{L'} + q\right)$. Then \begin{eqnarray*} \lefteqn{\left. g_1\right|_{L'}\cdot \left( l_4^2 \left. g_1\right|_{L'} + 2l_4 \left. g_2\right|_{L'} + \left. g_3\right|_{L'}\right) =} \hspace{7em}\\[1ex] &=& \left( \left. g_2\right|_{L'} + q\right)^2 +2l_4\left. g_1\right|_{L'}\cdot\left. g_2\right|_{L'} + \left( \left. g_2\right|_{L'} + q\right)\left( \left. g_2\right|_{L'} - q\right)\\ &=& \left( \left. g_2\right|_{L'} + q\right)\left( \left. g_2\right|_{L'} + q + \left. g_2\right|_{L'} - q\right) + 2l_4 \left. g_1\right|_{L'}\cdot\left. g_2\right|_{L'}\\ &=& 2 \left. g_2\right|_{L'}\left(\left. g_2\right|_{L'} + q + l_4\left. g_1\right|_{L'}\right)\\ &=& 0 \end{eqnarray*} Therefore, since $\left. g_1\right|_{L'}\not\equiv 0$, $l_4^2 \left. g_1\right|_{L'} + 2l_4 \left. g_2\right|_{L'} + \left. g_3\right|_{L'}=0$ and hence $(l_1(s,t):\ldots :l_4(s,t))$ ($(s:t) \in I\!\!P^1$) defines a line in $I\!\!P^4$ which is contained in $K$ and the image under $\pi$ of which is $L'$. \end{proof} \begin{lemma}\label{dt-preimage} Let $L$ be a double tangent of $B$ which is not contained in $g_1=0$. If $L'$ and $L''$ are different lines in $K$, which are mapped onto $L$ then $L$ contains one of the points $x$ with $g_1(x)=g_2(x)=g_3(x)=0$. In this case $L'$ and $L''$ are the only lines in $K$ that are mapped to $L$. \end{lemma} \begin{proof} Consider the plane $H$ in $I\!\!P^4$ spanned by $L$ and $P$. Then $H$ is not contained in $K$. If $H$ were contained in $K$ then $L$ would be contained in the locus $g_3 = 0$ in $I\!\!P^3$. $H\subset K$ then implies $g_1=g_2=0$ on $L$. But $L$ was supposed to be not contained in $g_1=0$. Therefore $K\cap H$ is a plane cubic curve that contains $L'$ and $L''$. Hence, this plane cubic must split into three components all of which are lines. One of these lines must contain $P$, but the lines through $P$ are just the lines $\overline{Px}$ with $x \in I\!\!P^3$ and $g_1(x)=g_2(x)=g_3(x)=0$. On the other hand $x$ is in $L$ which proves the lemma. \end{proof} The projection $\pi$ induces a morphism $\bar{\pi}$ between Grassmannians of lines of $I\!\!P^4$ and $I\!\!P^3$: \begin{morph} \bar{\pi} :& \mbox{Grass}(2,5)\setminus \{\mbox{lines through $P$}\}& \longrightarrow & \mbox{Grass}(2,4) \\ &\cup &&\cup \\ &\mbox{Fano}(K)\setminus\{\mbox{lines through $P$}\} & \longrightarrow & \left\{ \parbox{0.35\textwidth}{lines with even intersection with $B$ and lines contained in $B$} \right\} \end{morph} where $\mbox{Fano}(K)$ denotes the set of lines in $K$. The above lemmata suggest that $\bar{\pi}$ defines a birational map onto the set of bitangents of $B$ in $\mbox{Grass}(2,4)$. This will be shown later. \subsection{Lines in a nodal cubic threefold and bisecants of a space curve} Let $K \subset I\!\!P^4$ be a cubic hypersurface with only ordinary double points as singularities. Let \mbox{$P_{\!1}$}{} be one of the double points, $H$ a hyperplane not containing \mbox{$P_{\!1}$}{}, and let $Q'$ be the intersection of the tangent cone of \mbox{$P_{\!1}$}{} with the hyperplane $H$. Finally, let $S = K \cap Q'$. Assume the curve $S$ to have only ordinary double points as singularities. (Under the above assumptions on $K$ this is, in fact, always true, cf. \cite{finkelnberg}.) The singularities of $S$ are just the images of the double points of $K\setminus \{\mbox{$P_{\!1}$}{} \}$. Denote by $p$ the projection $p:K\setminus \{\mbox{$P_{\!1}$}{} \} \longrightarrow H$. Then the following proposition holds: \begin{satz}\label{bisecs-lines} Let $\ell$ be a line in $K$. If $\mbox{$P_{\!1}$}{} \in \ell$ then $\ell$ is a line $\bar{\mbox{$P_{\!1}$}{} x}$ with $x \in S$ and, conversely, every $x \in S$ defines a line in $K$ through \mbox{$P_{\!1}$}{}. A line $\ell$ not containing \mbox{$P_{\!1}$}{} is mapped onto a line $\bar{\ell} = p(\ell) \subset H$ not contained in $Q'$ and either connecting two points of $S$ or being a tangent of $S$ at a smooth point of $S$ or a tangent of $Q'$ at a non-smooth point of $S$. Conversely every such line $\bar{\ell}$ is the image of a line in $K$. \end{satz} \begin{proof}\newcommand{\mbox{$\bar{\ell}$ }}{\mbox{$\bar{\ell}$ }} Here, a sketch of the proof of Finkelnberg (cf. \cite{finkelnberg}) is to be given. The case where $\mbox{$P_{\!1}$} \in \ell$ is obvious. Let \mbox{$\bar{\ell}$ } be an arbitrary line in $H$ and let $V$ be the plane in $I\!\!P^4$ spanned by $\mbox{$\bar{\ell}$ }$ and \mbox{$P_{\!1}$}{}. Noting that $Q' \cong I\!\!P^1 \times I\!\!P^1 \subset I\!\!P^3$ and $S \in \left|{\cal O}_{Q'}(3,3)\right|$ the following cases occur: \begin{itemize} \item {\it $\mbox{$\bar{\ell}$ } \cap S = \emptyset$ and \mbox{$\bar{\ell}$ } intersects $Q'$ transversally.} Then $V \cap K$ is an irreducible plane cubic with an ordinary double point in \mbox{$P_{\!1}$}{}. \item {\it $\mbox{$\bar{\ell}$ } \cap S = \emptyset$ and \mbox{$\bar{\ell}$ } is tangent to $Q'$.} Then $V \cap K$ is an irreducible plane cubic with a cusp in \mbox{$P_{\!1}$}{}. \item {\it $\mbox{$\bar{\ell}$ } \cap S = \{T\}$ and \mbox{$\bar{\ell}$ } intersects $Q'$ transversally.} Then $V \cap K$ consists of the line $\bar{\mbox{$P_{\!1}$}{} T}$ and a conic intersecting $\bar{\mbox{$P_{\!1}$}{} T}$ transversally. The conic must pass through \mbox{$P_{\!1}$}{} and so can not split into two lines since one of the lines were a line through \mbox{$P_{\!1}$}{} and would intersect $S$ in a point different from $T$. If $\bar{\mbox{$P_{\!1}$}{} T}$ were tangent to the conic then $V \cap K$ had a cusp which is impossible since \mbox{$\bar{\ell}$ } intersects $Q'$ in two points. \item {\it $\mbox{$\bar{\ell}$ } \cap S = \{T\}$, $T$ is a non-singular point of $S$ and \mbox{$\bar{\ell}$ } is tangent to $Q'$ but not to $S$.} Then $V \cap K$ consists of a smooth conic through \mbox{$P_{\!1}$}{} and the line $\bar{\mbox{$P_{\!1}$}{} T}$ which is tangent to the conic (by a similar argument as above). \item {\it $\mbox{$\bar{\ell}$ } \cap S = \{T_1,T_2\}$ with $T_1 \ne T_2$ and \mbox{$\bar{\ell}$ } is not contained in $Q'$.} Then $V \cap K$ contains the two lines $\bar{\mbox{$P_{\!1}$}{} T_1}$ and $\bar{\mbox{$P_{\!1}$}{} T_2}$. Even if one or both points $T_i$ are singular points of $S$ none of these lines counts twice since this is only possible if \mbox{$\bar{\ell}$ } is tangent to $Q'$. Therefore $V \cap K$ contains a third line not through \mbox{$P_{\!1}$}{}. \item {\it \mbox{$\bar{\ell}$ } is tangent to $S$ in a smooth point $T$ of $S$.} Then $V \cap K$ splits into the line $\bar{\mbox{$P_{\!1}$}{} T}$ with multiplicity two and a second line that does not pass through any singular point of $K$. (To see that the first line carries multiplicity two move \mbox{$\bar{\ell}$ } a bit.) \item {\it \mbox{$\bar{\ell}$ } is tangent to but not contained in $Q'$ and meets $S$ in a singular point $T$.} Then $V \cap K$ consists of the line $\bar{\mbox{$P_{\!1}$}{} T}$ with multiplicity two (move \mbox{$\bar{\ell}$ } a bit!) and a second line. The second line can not pass through \mbox{$P_{\!1}$}{} since then it would intersect $S$ in a point different from $T$. \item {\it \mbox{$\bar{\ell}$ } is contained in $Q'$.} Then \mbox{$\bar{\ell}$ } meets $S$ in three (not necessaryly different) points. Each of these points is the intersection point of a line through \mbox{$P_{\!1}$}{} in $K$ with $S$, multiple points corresponding to multiple lines. \end{itemize} Therefore, only those lines mentioned in the proposition can be images of lines in $K$ not through \mbox{$P_{\!1}$}{} and each of them determines such a line in $K$. This proves the proposition. \end{proof} \subsection{The Fano scheme of lines on the cubic threefold} {}From now on it becomes convenient to make use of the special type of the quartic $B$ as described in Proposition~\ref{quartik13}: Let $B$ be a real quartic with exactly 13 ordinary double points such that (using the notation of Proposition~\ref{quartik13}) the linear forms $E_1,\ldots , E_4$ are real (i.e. have real coefficients). Take $g_1 = E_1$, $g_3 = E_2E_3E_4$ and $g_2 = Q$ to achieve the form $g_1g_3 - g_2^2$, i.e. $B$ is given by an equation of the form \begin{eqnarray*} F&=&x_3^2f_2+2x_3L_0L_1L_2 +f_2^2-f_2\left(L_0^2+L_1^2+L_2^2\right) + L_0^2L_1^2+L_0^2L_2^2+L_1^2L_2^2 \\ &=& \frac{1}{4}\left( Q^2 - E_1E_2E_3E_4\right) \\ &=& \frac{1}{4}\left( g_2^2 -g_1g_3 \right) \\ \end{eqnarray*} With the notation of the previous paragraph an letting $H\subset I\!\!P^4$ be the plane $x_4=0$, the following lemma holds. \begin{lemma} The curve $S = K \cap Q'$ consists of three components, each a smooth conic in one of the planes $E_i = 0$ ($i=2,3,4$). Each two components intersect in two points. \end{lemma} \begin{proof} First, the equation of $Q'$ is determined: Let $T$ be a point in $H$. Then $T \in Q'$ if and only if the line $\bar{\mbox{$P_{\!1}$}{} T}$ is contained in the tangent cone of $K$ in \mbox{$P_{\!1}$}{} which is the case if and only if this line has third order contact with $K$ in \mbox{$P_{\!1}$}{}. A simple calculation then shows that $T\in Q'$ if and only if $T$ is contained in the zero locus of the quadratic form $\left(x_3 + L_0 +L_1 +L_2\right)^2 - 4f_2$ (notation as in Proposition~\ref{quartik13}). Since $S= K \cap Q' = (K\cap H) \cap Q'$ and since $K \cap H$ is the union of the three planes $E_i = 0$ ($i=2,3,4$), the components of $S$ are the three conics $Q'\cap \{E_i=0\}$ (see also last remark in \cite{kreussler-pre}). These three conics are given by the equations \begin{eqnarray} f_2 - L_2^2 & = & 0 \qquad\mbox{in the plane $E_2=0$} \nonumber\\ f_2 - L_1^2 & = & 0 \qquad\mbox{in the plane $E_3=0$} \label{S-equ}\\ f_2 - L_0^2 & = & 0 \qquad\mbox{in the plane $E_4=0$} \nonumber \end{eqnarray} In the plane $x_3 = 0$, these equations define smooth conics (by Proposition~\ref{quartik13}) and since none of the planes $E_i=0$ contains the point $(0:0:0:1)$ they define smooth conics in the planes $E_i=0$, as well. Another short calculation shows that the quadrics $Q$ and $Q'$ coincide on the lines $E_i = E_j = 0$. Therefore, by the assumptions on the quartic and Proposition~\ref{quartik13}, these lines intersect $Q'$ in two different points. \end{proof} Denote by $\mbox{Bisec}(S)\subset \mbox{Grass}(2,4)$ the closure of the set of bisecants of $S$. By the above lemma, $S$ splits into 3 irreducible components which will be denoted by $S_i,\quad i=1,2,3$. Let $B_{ij} \subset \mbox{Bisec}(S) \quad (1\le i\le j \le 3)$ be the closure of the set of bisecants that connect $S_i$ with $S_j$. $B_{ii}$ then consists of the bisecants and the tangents of $S_i$ whereas $B_{ij}$ contains the lines connecting different points of $S_i$ and $S_j$ together with all tangents at $Q'$ in the to intersection points of $S_i$ and $S_j$. The $B_{ij}$ are the six irreducible components of $\mbox{Bisec}(S)$, for they are irreducible, cover all of $\mbox{Bisec}(S)$ and contain open subsets that are disjoint from all other $B_{ij}$. Now, using Proposition~\ref{bisecs-lines}, morphisms from $B_{ij}$ to $\mbox{Fano}(K)$ are to be constructed which will turn out to induce a birational map between $\mbox{Bisec}(S)$ and $\mbox{Fano}(K)$. Let $\,{\cal U}_{B_{ij}} \subset I\!\!P^3 \times B_{ij}$ be the ''universal line'' over $B_{ij}$: ${\cal U}_{B_{ij}} = \{(x,\ell)\,|\,x \in \ell \}$. Let $C_{B_{ij}}$ be the cone from $\{\mbox{$P_{\!1}$}{}\} \times B_{ij} \subset I\!\!P^4 \times B_{ij}$ over $\,{\cal U}_{B_{ij}}$ (where $I\!\!P^3 \subset I\!\!P^4$ as the hyperplane $x_4=0$): \[ \newlength{\temp} \settowidth{\temp}{$x$ is contained in the plane spanned by $\ell$ and \mbox{$P_{\!1}$}{} in $I\!\!P^4 \times \{\ell\}$} C_{B_{ij}}:= \left\{ (x,\ell) \in I\!\!P^4 \times B_{ij} \left|\; \parbox{0.6\temp}{$x$ is contained in the plane spanned by $\ell$ and \mbox{$P_{\!1}$}{} in $I\!\!P^4 \times \{\ell\}$} \right.\right\} \] Finally let $K_{B_{ij}}:= C_{B_{ij}}\cap \left(K\times B_{ij}\right)$. Every fibre of $K_{B_{ij}}$ over $B_{ij}$ splits into lines at least two of which contain \mbox{$P_{\!1}$}{} (possibly one line counted twice if $\ell \in B_{ij}$ is a tangent at $S$). \[\renewcommand{\arraystretch}{1.5} \begin{array}{ccccl} I\!\!P^3 \times B_{ij} & \subset & I\!\!P^4 \times B_{ij} & \longrightarrow & B_{ij}\\ \cup & & \cup\\ {\cal U}_{B_{ij}} & \subset & C_{B_{ij}} & \supset & \{\mbox{$P_{\!1}$}\}\times B_{ij} = \mbox{vertex of the cone}\\ & & \cup\\ & & K_{B_{ij}} & \supset & G_{B_{ij}}\\ & & \cup\\ \left(\left(S_i \cup S_j \right)\times B_{ij} \right) \cap {\cal U}_{B_{ij}} & \subset & M_{B_{ij}} & = & \mbox{cone over} \quad \left(\left(S_i \cup S_j \right)\times B_{ij} \right)\cap {\cal U}_{B_{ij}} \end{array} \] Consider now $(S_i \cup S_j)\times B_{ij}\cap {\cal U}_{B_{ij}}$ the fibre over $\ell \in B_{ij}$ of which consists of the intersection points of $\ell$ with $S_i \cup S_j$. Let $M_{B_{ij}} \subset I\!\!P^4 \times B_{ij}$ be the cone from $\{\mbox{$P_{\!1}$}{}\} \times B_{ij}$ over $(S_i \cup S_j)\cap {\cal U}_{B_{ij}}$ which by construction is contained in $K_{B_{ij}}$. Since a line $\ell \in B_{ij}$ can meet $S$ in three different points if and only if it meets all three components of $S$ the fibre over $\ell$ consists of one or two lines through \mbox{$P_{\!1}$}{} - one of them possibly counting twice. Let $G_{B_{ij}}$ be the closure in $I\!\!P^4 \times B_{ij}$ of $K_{B_{ij}} \!\setminus M_{B_{ij}}$. By Proposition~\ref{bisecs-lines} $G_{B_{ij}}$ is a family of lines in $I\!\!P^4$ all lying in the cubic $K$. Let $\ell \in B_{ij}$ be an arbitrary line and let $V$ be the plane in $I\!\!P^4$ spanned by $\ell$ and \mbox{$P_{\!1}$}{}. By subtracting $M_{B_{ij}}$ from $K_{B_{ij}}$ just those lines in $K\cap V$ are removed that pass through \mbox{$P_{\!1}$}{} and through the intersection points of $\ell$ with $S_i$ and $S_j$. For each line $\ell \in B_{ij}$ such that $V\cap K$ contains a line not through \mbox{$P_{\!1}$}{} just this line remains in $G_{B_{ij}}$. These $\ell$ are just the lines in $\mbox{Bisec}(S)$ that do not meet all three components of $S$. But also for the lines $\ell$ that are contained in $Q'$ the fibre of $G_{B_{ij}}$ over $\ell$ consists of one single line. $\ell$ must meet all three components of $S$ and, therefore, must be contained in each $B_{ij}$ with $i\ne j$. Subtracting $M_{B_{ij}}$ from $K_{B_{ij}}$ removes just the lines in $V\cap K$ that intersect $S_i$ or $S_j$ leaving the the third line which intersects the third component of $S$ in $G_{B_{ij}}$. Since $G_{B_{ij}}$ is the {\em closure} of $K_{B_{ij}} \!\setminus M_{B_{ij}}$ this construction also yields just one line in the case that $\ell$ meets a singular point of $S$ which is always the intersection of two components of $S$. Since $G_{B_{ij}} \longrightarrow B_{ij}$ is a family of lines in $K$ there is a uniquely determined morphism $B_{ij} \longrightarrow \mbox{Fano}(K)$. The images of all $B_{ij}$ cover $\mbox{Fano}(K)$. For lines in $K$ that do not contain \mbox{$P_{\!1}$}{} this is clear from Proposition~\ref{bisecs-lines}. But also the lines through \mbox{$P_{\!1}$}{} have their representation by an element of one of the $B_{ij}$: Let $T$ be the intersection of such a line with $S$. There are two lines of the rulings of $Q'$ through $T$. Since $S_i \in \left|{\cal O}_{Q'}(1,1)\right|$ for all $i$ non of them can be tangent to $S$. Let $\ell$ be one of these two lines. $\ell$ intersects all three components of $S$ and is therefore contained in each $B_{ij}$ with $i \ne j$. Therefore, we can find $i$ and $j$ such that $T$ is neither contained in $S_i$ nor in $S_j$ and $\ell \in B_{ij}$. (If $T\in S_m \cap S_n$ then e.g. $i=m$ and $j \ne n$ is a fitting choice.) As is clear from the above discussion, the morphism $B_{ij} \longrightarrow \mbox{Fano}(K)$ then maps $\ell \in B_{ij}$ to the line $\bar{\mbox{$P_{\!1}$}{}T}\in \mbox{Fano}(K)$. Now, each $B_{ij}$ contains an open subset (say $U_{ij}$) on which the above morphisms to $\mbox{Fano}(K)$ are injective and such that the images of different $U_{ij}$ do not intersect in $\mbox{Fano}(K)$. (The sets of lines that intersect $S$ in exactly two different nonsingular points will do.) Therefore the morphisms $G_{B_{ij}} \longrightarrow B_{ij}$ induce a birational map $\mbox{Bisec}(S)\relbar\relbar\rightarrow \mbox{Fano}(K)$. \subsection{The components of the space of double tangents} Denote by $\mbox{$Y_0$} \subset \mbox{Grass}(2,4)$ the closed subscheme of double tangents of the quartic $B$. Earlier in this section we constructed a morphism $\mbox{Fano}(K)\setminus \{\mbox{lines through $P$}\} \longrightarrow \mbox{$Y_0$}$. By Lemma~\ref{dt-preimage} this morphism is injective outside the closed subset of lines in $K$ that meet one of the six lines through $P$. As this closed subset is not an irreducible component of $\mbox{Fano}(K)$ (which is clear from the map $\mbox{Bisec}(S) \relbar\relbar\rightarrow \mbox{Fano}(K)$) its complement is open und dense. On the other hand, the image of $\mbox{Fano}(K)\setminus \{\mbox{lines through $P$}\}$ in $\mbox{$Y_0$}$ contains the set of all double tangents outside the plane $g_1=0$. But the double tangents contained in this plane form an irreducible component of their own: Every line in that plane is a double tangent. Thus the set of these double tangents is closed in $\mbox{$Y_0$}$ and of dimension two. On the other hand the dimension of $\mbox{$Y_0$}$ is also two: Consider the variety \[ \mbox{$Y_F$}{}:=\{ (\ell,H) \in \mbox{Grass}(2,4)\times \check{I\!\!P}^3\,|\, \ell\subset H \; \mbox{and}\; \ell\in \mbox{$Y_0$} \} \] which is a subvariety of the flag variety $F(1,2):=\{ (\ell,H) \in \mbox{Grass}(2,4) \times \check{I\!\!P}^3 \,|\, \ell\subset H\}$. This variety is fibred over $\check{I\!\!P}^3$ and the fibre over a general $H\in\check{I\!\!P}^3$ is zero dimensional by Proposition~\ref{Anz-DT}. Hence the dimension of \mbox{$Y_F$}{} is at least three and since every line is contained in a pencil of planes the dimension of \mbox{$Y_0$}{} is at least two. The dimension cannot be greater than two for there is a morphism $\mbox{Fano}(K)\setminus \{\mbox{lines through $P$}\} \longrightarrow \mbox{$Y_0$}$ (which is surjective onto the set of double tangents outside the plane $g_1=0$) and the dimension of $\mbox{Fano}(K)$ is two since it is birationally equivalent to $\mbox{Bisec}(S)$. Therefore \mbox{$Y_0$}{} is the union of two-dimensional varieties and hence is of dimension two. So, the closed subvariety of lines in the plane $g_1=0$ has the same dimension as \mbox{$Y_0$}{} and thus is an irreducible component. This way we have found a rational map $\mbox{Bisec}(S)\relbar\relbar\rightarrow\mbox{$Y_0$}$ which is birational onto those components of \mbox{$Y_0$}{} which are different from the component of lines in the plane $g_1=0$. \mbox{$Y_0$}{} must therefore have seven irreducible components. Four of them consist of all lines in a plane. These are the sets of lines in the planes $E_i = 0$ $(i=1,\ldots,4)$. As the lines in the planes $E_i = 0$ $i=2,3,4$ are contained in $K$ as well as in the hyperplane $x_4 = 0$ they keep fixed under the map $\mbox{Bisec}(S)\relbar\relbar\rightarrow\mbox{$Y_0$}$ and therefore correspond to the lines in the $B_{ii} \subset \mbox{Bisec}(S)$. Thus, the irreducible components of \mbox{$Y_0$}{} are determined: \begin{satz}\label{DT-compon} \mbox{$Y_0$}{} consists of seven irreducible components, namely the four components with all lines in the planes $E_i$ $(i=1,2,3,4)$ and three further components corresponding to the components $B_{ij} \subset \mbox{Bisec}(S)$ with $i\ne j$.\hspace*{\fill}$\Box$ \end{satz} The following proposition will be needed in the next section. \begin{satz}\label{flach} Let $\Delta' \subset \check{I\!\!P}^3$ be the closed set of planes $H$ such that $B\cap H$ is a plane quartic which is not smooth and has more or other singularities than just one ordinary double point. Then $\varphi :\mbox{$Y_F$}|_{\check{I\!\!P}^3 \setminus \Delta'} \longrightarrow \check{I\!\!P}^3 \setminus \Delta'$ is flat. The ramification locus of $\varphi$ is just the set of those $(\ell,H)$ such that $B\cap H$ is a quartic with a node and $\ell$ is a double tangent containing the node. The ramification index in those points is two. Outside the ramification locus $\varphi$ is a smooth 28-fold cover. \end{satz} \begin{proof} \mbox{$Y_F$}{} is contained in the Flag variety $F(1,2)\subset \mbox{Grass}(2,4) \times \check{I\!\!P}^3$ which is a $I\!\!P^2$-bundle over $\check{I\!\!P}^3$. For the proof of the flatness we will determine the Hilbert polynomial of the fibres of $\mbox{$Y_F$}|_{\check{I\!\!P}^3 \setminus \Delta'}$ over $\check{I\!\!P}^3 \setminus \Delta'$ considered as subsets of $\check{I\!\!P}^2$. Let $U\subset \check{I\!\!P}^3$ be a standard open subset (say $U=\{(y_0:\ldots:y_3) \in \check{I\!\!P}^3\,|\, y_3 = -1\}$) such that $F(1,2)|_U\cong U\times \check{I\!\!P}^2$. In $\check{I\!\!P}^2$ choose a standard open set $U' = \{(l_0:l_1:l_2) \in \check{I\!\!P}^2\,|\, l_0 = -1\}$. Then, by substituting $x_3 = x_0y_0+x_1y_2 +x_2y_2$ and $x_0= x_1l_1 + x_2l_2$ in the equation of $B$ we get a map from $U\times U'$ to $I\!\!P^4$ -- the set of binary quartic forms -- which associates to each pair $(\ell,H)\in U\times U'$ the equation of $B$ restricted to $\ell$. If $C\subset I\!\!P^4$ denotes the closed subset parametrising those quartic forms that are complete squares then $\mbox{$Y_F$}|_{U\times U'}$ is the preimage of $C$ under the map $U\times U' \longrightarrow I\!\!P^4$. Therefore, we can get equations for $Y|_{U\times U'}$ (with its reduced scheme structure) by pulling back equations defining $C$. Let $(a:b:c:d:e)\in I\!\!P^4$ correspond to the quartic form $ax^4 + bx^3y + cx^2y^2 + dxy^3 + ey^4$. Obviously we get the equations for $C$ with the reduced scheme structure by eliminating $\xi$, $\eta$, and $\zeta$ from the equations \begin{equation}\label{triv-equ} \renewcommand{\arraystretch}{1.5} \begin{array}{rcl} a & = & \xi^2\\ b & = & 2\xi\eta\\ c & = & 2\xi\zeta-\eta^2\\ d & = & 2\eta\zeta\\ e & = & \zeta^2. \end{array} \end{equation} This is done by the technique of Gr\"obner bases (cf. \cite{cox} for details): One has to compute a Gr\"obner basis for the above equations using the lexicographic monomial order induced by the ordering of variables $(\xi>\eta>\zeta>a>b>c>d>e)$. The equations defining $C\in I\!\!P^4$, then, are those which do not contain $\xi$, $\eta$ or $\zeta$ (cf.~\cite{cox}). The following is the Gr\"obner basis computed by MAPLE~V (the polynomials not containing $\xi$, $\eta$ or $\zeta$ listed last): \begin{equation}\label{GB} \renewcommand{\arraystretch}{1.5} \begin{array}{ccc} \xi ^{2}-a,& 2\,\xi \eta -b,& 2\,\xi \zeta -c+\eta ^{2},\\ \xi b-2\,a\eta ,& -\eta b-4\,\zeta a+2\,\xi c,& \xi d-b\zeta ,\\ 4\,\xi e+\eta d -2\,\zeta c,& b\zeta -\eta c+\eta ^{3},& 4\,a\eta ^{2}-b^{2},\\ b\eta ^{2}-cb+2\,ad,& 2\,c\eta ^{2}-2\,c^{2}+ bd+8\,ea,& d\eta ^{2}-dc+2\,eb,\\ 2\,\eta \zeta -d,& 4\,a\eta c-4\,ab\zeta -b^{2}\eta ,& 2\,a\eta d-b^{2}\zeta ,\\ 4 \,a\zeta d-2\,cb\zeta +\eta bd,& dc\eta -2\,\zeta c^{2}+\zeta bd+8\,ea\zeta ,& \eta d^{2}-2\,\zeta dc+4\,eb\zeta ,\\ 2\,\eta e-\zeta d,& \zeta ^{2}-e\\[3ex] 8\,a^{2}d-4\,cba+b^{3},& cb^{2}+2\,abd-4\,ac^{2}+16\,a^{2}e,& b^{2}d+8\,bea-4\,acd,\\ ad^{2}-eb^{2},& bd^{2}-4\,ceb+8\,ead,& d^{2}c-4\,ec^{2} +2\,ebd+16\,e^{2}a,\\ d^{3}-4\,edc+8\,e^{2}b \end{array} \end{equation} One can easily verify that these polynomials form a Gr\"obner basis with respect to the above monomial order (e.g. by forming S-polynomials and reducing them with respect to the set of polynomials~(\ref{GB})). Even easier is it to check that the equations~(\ref{triv-equ}) and the polynomials~(\ref{GB}) define the same ideal. (The equations~(\ref{triv-equ}) are among the polynomials~(\ref{GB}) and the remaining polynomials are easily reduced to zero using the equations~(\ref{triv-equ}).) Now, let $H\in U\setminus\Delta'$ be a plane in $I\!\!P^3$. For any $(\ell,H)$ in the fibre $\varphi^{-1}(H)$ of \mbox{$Y_F$}{} over $H$ we will compute the length of the local ring ${\cal O}_{\varphi^{-1}(H),(\ell,H)}$. Let \begin{eqnarray*} f & := & a_{44}\,x_0^4 +a_{43}\,x_0^3x_1 +a_{42}\,x_0^2x_1^2 +a_{41}\,x_0x_1^3 +a_{40}\,x_1^4\\ && +x_2\,(a_{33}\,x_0^3+a_{32}\,x_0 ^2x_1 +a_{31}\,x_0x_1^2 +a_{30}\,x_1^3) \\ && +x_2^2(a_{22}\,x_0^2 +a_{21}\,x_0x_1 +a_{20}\,x_1^2)\\ && +x_2^3(a_{11}\,x_0 +a_{10}\,x_1) + a_{0}\,x_2^4 \end{eqnarray*} be the polynomial defining $B\cap H \subset H \cong I\!\!P^2$. Suppose that $\ell \subset I\!\!P^2$ is the line $x_0=0$ contained in $U\times U' \subset F(1,2)$. The equations of \mbox{$Y_F$}{} near $(\ell,H)$ are obtained by substituting $x_0=x_1l_1+x_2l_2$ in $f$ and substituting the resulting coefficients in the seven last polynomials of (\ref{GB}). (The local ring ${\cal O}_{\varphi^{-1}(H),(\ell,H)}$ then is obtained as the factor of $\Bbb C[l_1,l_2]_{(0,0)}$ by the ideal generated by the equations of \mbox{$Y_F$}{} we got in this way.) If $\ell\subset H$ is a double tangent at $B \cap H$ touching $B\cap H$ in the points $(0:1:0)$ and $(0:0:1)$ then the coefficients in $f$ have to satisfy: $a_0=0$, $a_{10}=0$, (since $x=0$ is a tangent at $(0:0:1)$), $a_{40}=0$, $a_{30}=0$ (since $x=0$ is a tangent at $(0:1:0)$), and finally $a_{20}$ must not be zero since otherwise the line $x=0$ would be contained in $B$ (we, thus, can set $a_{20}=1$). Computing the Jacobian of the resulting seven equations in the point $(\ell,H)$ (i.e. for $l_1=l_2=0$) yields the matrix \[ \left( \begin {array}{cc} 0&0\\-4\,a_{41}&0\\0&0\\0&0 \\0&0\\0&-4\,a_{11}\\0&0\end {array}\right) \] Thus, $(\ell,H)$ is a singular point in its fibre if and only if $a_{41}=0$ or $a_{11}=0$, i.e., if and only if $\ell$ touches $B \cap H$ in singular points of $B \cap H$. We only have to compute the ramification index of the points $(\ell,H)$ which are singular in their fibre. Suppose, that the line $\ell$ (given by $x=0$) contains the point $(0:0:1)$ which is assumed to be a singular point of $B \cap H$ and touches $B \cap H$ in the smooth point $(0:1:0)$. In particular we get the condition $a_{11}=0$. We get seven polynomials in $l_1$ and $l_2$, the sixth of which has the form \[ \left( \cdots\; \mbox{terms containing $l_1$ or $l_2$}\;\cdots - 4a_{22} + a_{21}^2 \right) l_2^2. \] But $4a_{22} - a_{21}^2$ must not vanish since otherwise $B\cap H$ would have a cusp in $(0:0:1)$. Thus $(\cdots\; \mbox{terms containing $l_1$ or $l_2$}\;\cdots - 4a_{22} + a_{21}^2)$ is a unit in $\Bbb C[l_1,l_2]_{(0,0)}$. Therefore, this equation can be replaced by $l_2^2$ and the terms containing $l_2^2$ can be cancelled in the other equations without changing the ideal generated by the equations. Now, the second equation has the form \[ \left( \cdots\; \mbox{terms containing $l_1$ or $l_2$}\;\cdots -4\,a_{41} \right) l_1. \] Since $a_{41}$ must not vanish (otherwise $(0:1:0)$ would be singular on $B\cap H$), we can replace this equation by $l_1$ and cancel all terms containing $l_1$ in the other equations. We obtain that the considered ideal in $\Bbb C[l_1,l_2]_{(0,0)}$ is generated by the two elements $l_2^2$ and $l_1$. Hence, if $(\ell,H)$ is of the kind that $\ell$ contains the only node of $B\cap H$ then the local ring of $(\ell,H)$ in its fibre has length two. All other points over $\check{I\!\!P}^3 \setminus \Delta'$ are smooth in their fibres. For $(\ell,H)\in \check{I\!\!P}^3 \setminus \Delta'$ the fibre $\varphi ^{-1}((\ell,H))$ consists, by Proposition~\ref{Anz-DT}, of 28 points if $B\cap H$ is smooth, and of 22 points otherwise. By the Remark following Proposition~\ref{Anz-DT} for a quartic $B\cap H$ with one node, there are six double tangents through the node. The Hilbert polynomial of $\varphi ^{-1}((\ell,H)) \in \check{I\!\!P}^2$, hence, is 28 for $B\cap H$ being smooth and $16+6\cdot 2 = 28$ otherwise. Therefore $\varphi: \mbox{$Y_F$}|_{\check{I\!\!P}^3\setminus \Delta'} \longrightarrow \check{I\!\!P}^3 \setminus \Delta'$ is flat. From the length of the local rings computed above we see that the ramification behaviour is as stated. \end{proof} \section{The parameter space of touching conics}\label{sec-tochin-conic} \subsection{Double tangents and curves in double covers} Let $B \subset I\!\!P ^3$ be a real quartic surface with exactly 13 nodes, given by an equation of the form $E_1E_2E_3E_4 -Q^2$, as described in Proposition~\ref{quartik13}. Let $Z\longrightarrowI\!\!P^3$ be the double cover which is ramified over $B$. It is constructed as follows (cf. \cite{bpv}, I.17.). Denote by $p:L\longrightarrow I\!\!P^3$ the total space of the line bundle ${\cal O}_{I\!\!P^3}(2)$ and by $s\in H^0(I\!\!P^3,{\cal O}_{I\!\!P^3}(4))$ the section who's zero locus is $B$. Finally, let $y\in H^0(L,p^*{\cal O}(2))$ be the tautological section. Then $Z\subset L$ is the zero locus of the section $y^2-p^*(s) \in H^0(L,p^*{\cal O}(4))$. Each of the divisors given by $p^*(E_i)$ splits into two components which are defined by the sections $y^2 \pm \sqrt{-1}p^*(Q)$ of $H^0(L,p^*{\cal O}(2))$. Denote these varieties by $S_i^+$ and $S_i^-$, corresponding to $y^2 + \sqrt{-1}p^*(Q)$ and $y^2 - \sqrt{-1}p^*(Q)$ respectively. Now, let $H\in \check{I\!\!P}^3$ be a general plane. In particular, let $B_H:=B\cap H$ be a smooth quartic curve. The restriction $Z_H$ of $Z$ to $H$ then is the smooth double cover of $H\congI\!\!P^2$ branched along the nonsingular curve $B_H$. Its canonical bundle is \[ K_{Z_H}=p^*{\cal O}_H(-3)\otimes p^*{\cal O}_H(2) = p^*{\cal O}_H(-1), \] which implies that the morphism $p|_H$ is induced by the linear system $|-K_{Z_H}|$. By \cite{griffiths-harris} Chapter~4.4, $Z_H$ is isomorphic to the blow-up of $I\!\!P^2$ in seven points. The 56 $(-1)$-curves are: the seven exceptional divisors $E^i$ $(i=1,\ldots,7)$, the strict transforms of the cubics in $I\!\!P^2$ through all seven points with a node in the $i$-th point $K^i$ $(i=1,\ldots,7)$, the strict transforms of the lines through the $i$-th and the $j$-th point $G^{ij}$ $(1\le i < j\le 7)$, and the strict transforms of the conics through all but the $i$-th and the $j$-th point $C^{ij}$ $(1\le i < j\le 7)$. The projection $p|_H$ maps the 56 $(-1)$-curves onto the 28 double tangents of $B_H$ in such a way that each double tangent has exactly two $(-1)$-curves in its preimage. The restriction of the $S_i^{\pm}$ to $Z_H$ yields eight curves which are denoted by $D_i^{\pm}$. These are mapped onto double tangents of $B_H$ and, hence, are $(-1)$-curves. \begin{lemma} $Z_H$ can be realised as the blow-up of $I\!\!P^2$ in such a way that $D_1^+ = K^7$, $D_1^- = E^7$, $D_2^+=G^{12}$, $D_2^-=C^{12}$, $D_3^+=G^{34}$, $D_3^- = C^{34}$, $D_4^+=G^{56}$, and $D_4^- = C^{56}$. \end{lemma} \begin{proof} All we have to prove is that the $D_i^{\pm}$ intersect in the correct way, i.e., $D_i^+ \cdot D_i^- = 2$ for $i=1,\ldots,4$ and $D_i^+\cdot D_j^+ = D_i^-\cdot D_j^- = 1$ as well as $D_i^+\cdot D_j^- = 0$ for $i\ne j$. {}From the fact that $p|_H$ is induced by the linear system of the anticanonical divisor we deduce that $D_i^+ + D_i^- = -K_{Z_H}$ . Hence \[ 2 = \left(-K_{Z_H}\right)^2 = \left(D_i^+ + D_i^-\right)^2 \] and consequently $D_i^+ \cdot D_i^- = 2$. Next observe that the rational function \[ \frac{y + \sqrt{-1}Q}{(p|_H)^*(E_iE_j)} \] corresponds to the principal divisor $D_k^+ + D_l^+ - D_i^- - D_j^-$, i.e. $[D_k^+ + D_l^+] =[ D_i^- + D_j^-]$ where $\{k,l\} = \{1,\ldots,4\} \setminus \{i,j\}$. This yields \[\renewcommand{\arraystretch}{1.5} \begin{array}{rcccl} \left(D_k^+ + D_l^+\right)^2 & = & \left(D_i^- + D_j^-\right)^2 & = & \left(D_k^+ + D_l^+\right)\left(D_i^- + D_j^-\right)\\ 2D_k^+ D_l^+ - 2 & = & 2D_i^- D_j^- -2 & = & D_k^+ D_i^- + D_k^+ D_j^- + D_l^+ D_i^- + D_l^+ D_j^- \ge 0 \end{array} \] as the product of different effective divisors is always non-negative. Hence $D_i^+\cdot D_j^+ = D_i^-\cdot D_j^- = 1$ since two $(-1)$-curves have intersection product 2 if and only if their sum is an element of $|-K_{Z_H}|$. But then the sum $D_k^+ D_i^- + D_k^+ D_j^- + D_l^+ D_i^- + D_l^+ D_j^-$ must vanish and so $D_i^+\cdot D_j^- = 0$. \end{proof} Consider now ${\cal Z}:= Z\times_{I\!\!P^3} {\cal H}$ where ${\cal H} \subset I\!\!P^3 \times \check{I\!\!P}^3$ is the universal (hyper-)plane. Via ${\cal Z} \rightarrow {\cal H} \rightarrow \check{I\!\!P}^3$, $\cal Z$ is the family of all surfaces $Z_H$, $H\in \check{I\!\!P}^3$. Let $\Delta \subset \check{I\!\!P} ^3$ be the set of all those planes that do not intersect $B$ transversally (i.e., who's intersection with $B$ is not a smooth quartic curve). Then ${\cal Z}|_{\check{I\!\!P}^3 \setminus \Delta} \longrightarrow \check{I\!\!P}^3 \setminus \Delta$ is smooth and proper and therefore, by the Ehresmann--Fibration--Theorem (cf. \cite{lamottke}), locally trivial as a fibration of differentiable manifolds. For any $H_0\in \check{I\!\!P}^3 \setminus \Delta$ the fundamental group $\pi_1(\check{I\!\!P}^3 \setminus \Delta,H_0)$ acts via monodromy on $H^2({\cal Z}|_{H_0},\mbox{$Z\!\!\!Z$}) = H^2(Z_{H_0},\mbox{$Z\!\!\!Z$}) = \mbox{Pic}(Z_{H_0})$ preserving the intersection pairing. In particular the fundamental group acts via monodromy on the set of $(-1)$-curves preserving their intersection behaviour. On the other hand, this fundamental group $\pi_1(\check{I\!\!P}^3 \setminus \Delta,H_0)$ acts via monodromy of the finite unramified cover $\mbox{$Y_F$}|_{\check{I\!\!P}^3\setminus\Delta}\longrightarrow \check{I\!\!P}^3\setminus\Delta$ on the set of double tangents of $B\cap H_0$. (Recall that \mbox{$Y_F$}{} was defined as $\mbox{$Y_F$}:=\{ (\ell,H) \in \mbox{Grass}(2,4)\times \check{I\!\!P}^3\,|\, \ell\in H\,\mbox{and $\ell$ is double tangent at $B$}\}$.) Obviously, this action is the same as the action that is induced from the action on $Z_{H_0}$ by mapping a $(-1)$-curve to its corresponding double tangent. The properties of the monodromy action on the $(-1)$-curves are to be examined in the sequel. First, observe that the eight curves $D_i^{\pm}$ keep fixed under the monodromy action since they are restrictions of the globally defined $S_i^{\pm}$. A pair of $(-1)$-curves lying over the same double tangent is mapped to a pair of $(-1)$ curves that again are mapped onto the same (maybe different) double tangent. This follows from the fact that the sum $[C+C']$ of such a pair $(C,C')$ is equal to the anti-canonical class. which for the $(-1)$-curves is equivalent to $C\cdot C'=2$. The set of $(-1)$-curves that are different from, say, $D_1^{\pm}$ can be split in two monodromy--invariant subsets each containing 27 curves: one subset consisting of all curves $C$ with $C\cdot D_1^- = 1$ and the other containing the curves $C$ satisfying $C\cdot D_1^- = 0$. I.e., these subsets are characterised by the intersection number of their elements with $D_1^- = E^7$ which can take the values $0$ and $1$. As $D_1^-$ is invariant under the monodromy action this condition is invariant and so are the subsets. Each pair $(C,C')$ of $(-1)$-curves with $C\cdot C'=2$ (except the pair $(D_1^+,D_1^-)$) has one member in each of the two sets. Therefore the monodromy action on the $(-1)$ curves is determined by the action on one of the invariant subsets. We choose the set of curves that do not intersect $E^7$. This set with its incidence relations is equivalent to the set of the 27 lines of a smooth cubic surface: We get the correspondence by blowing down $E^7$. The $(-1)$-curves not intersecting $E^7$ are mapped onto the $(-1)$-curves in the blown-down surface which is $I\!\!P^2$ blown-up in six points. \subsection{The lines in a cubic surface} Let $S$ be the cubic surface obtained by blowing down $E^7\subset Z_{H_0}$. Denote the lines in $S$ by $E^i$ $(i=1,\ldots, 6)$ for the images of $E^i\subset Z_{H_0}$, $G^{ij}$ $(1\le i<j\le 6)$ for the images of the corresponding curves $G^{ij}$ in $Z_{H_0}$, and $C^i$ $(i=1,\ldots 6)$ for the images of the curves $C^{i7} \subset Z_{H_0}$. The monodromy action on the $(-1)$-curves of $Z_{H_0}$ induces an action of the fundamental group $\pi_1(\check{I\!\!P}^3\setminus\Delta,H_0)$ on the 27 lines of $S$ and, hence, induces a morphism of $\pi_1(\check{I\!\!P}^3\setminus\Delta,H_0)$ into the group of symmetries of lines in the cubic surface $S$ (i.e. the group of permutations of the 27 lines that respect their incidence relations). Let $G$ be the image of the fundamental group in this symmetry group. Clearly, $G$ leaves the three lines $G^{12}$ ($\,\widehat{=}\, D_2^+$), $G^{34}$ ($\,\widehat{=}\, D_3^+$), and $G^{56}$ ($\,\widehat{=}\, D_4^+$) invariant. To each of them there are exactly 10 lines that intersect this line. Furthermore, to each pair $(C,C')$ of intersecting lines there is exactly one line that intersects them both. Thus, to each of the three lines $G^{12}$, $G^{34}$, and $G^{56}$ there is associated a set of eight lines that intersect this curve and that are different from these three lines. The three sets of eight lines have to be disjoint since the three lines $G^{12}$, $G^{34}$, and $G^{56}$ meet each other and so the unique line that intersects two of them is just the third. \begin{satz}\label{transitiv} $G$ acts transitively on the three $G$--invariant sets of curves associated to the three curves $G^{12}$, $G^{34}$, and $G^{56}$. \end{satz} \begin{proof} The variety \mbox{$Y_F$}{} is naturally fibred over \mbox{$Y_0$}{} -- the parameter space of double tangents. The fibres are isomorphic to $I\!\!P^1$ since each line in $I\!\!P^3$ is contained in a pencil of planes. Therefore the irreducible components of \mbox{$Y_F$}{} are just the preimages of the irreducible components of \mbox{$Y_0$}{}. The projection $\mbox{$Y_F$}{}\longrightarrow\check{I\!\!P}^3$ is a finite cover which is \'etale over $\check{I\!\!P}^3\setminus\Delta$ by Proposition~\ref{flach} and, hence, $\mbox{$Y_F$}|_{\check{I\!\!P}^3 \setminus \Delta}$ is smooth. Therefore its irreducible components are just its arc--connected components in the Euclidian topology. By Proposition~\ref{DT-compon}, \mbox{$Y_F$}{} has seven components. A general fibre $\mbox{$Y_F$}|_H$ over $\check{I\!\!P}^3 \setminus \Delta$ contains 28 points by Proposition~\ref{Anz-DT}, four of them corresponding to the four double tangents in the planes $E_i=0$. (Recall that the closed subset in \mbox{$Y_0$}{} of lines in one of the planes $E_i=0$ were recognised to be irreducible components.) The remaining 24 points belong to double tangents of the other three components of \mbox{$Y_0$}{}. We claim that any fibre $\mbox{$Y_F$}|_H$ over $\check{I\!\!P}^3$ must contain at least one point of each component. For each component of \mbox{$Y_0$}{} there is a plane in $\check{I\!\!P}^3 \setminus \Delta$ which contains a double tangent of this component: For a double tangent $\ell$ that does not pass through one of the singular points of $B$, the pencil of planes containing $\ell$ is not contained in $\Delta \subset \check{I\!\!P}^3$. The set of double tangents through a singular point $p\in B$ is one-dimensional as this set is parametrised by the ramification locus of the projection of $B$ from $p$. On the other hand, each component of \mbox{$Y_0$}{} is two-dimensional and, hence, in any component of \mbox{$Y_0$}{} there is an open set of double tangents $\ell$ not through any of the singular points of $B$. Now, choose a component of \mbox{$Y_0$}{} and a double tangent $\ell$ of this component which does not contain singular points of $B$. Let $H\in \check{I\!\!P}^3 \setminus \Delta$ be a plane containing $\ell$, i.e. $(\ell,H) \in \mbox{$Y_F$}|_{\check{I\!\!P}^3 \setminus \Delta}$. Since $\mbox{$Y_F$}|_{\check{I\!\!P}^3 \setminus \Delta} \longrightarrow \check{I\!\!P}^3 \setminus \Delta$ is an \'etale cover the component of \mbox{$Y_F$}{} containing $(\ell,H)$ dominates $\check{I\!\!P}^3$. Therefore, every plane in $I\!\!P^3$ contains at least one double tangent of each component of \mbox{$Y_0$}{}. As mentioned above, the fundamental group $\pi_1(\check{I\!\!P}^3 \setminus \Delta,H_0)$ acts on the fibre $\mbox{$Y_F$}|_{H_0}$ via monodromy and, as the irreducible components of $\mbox{$Y_F$}|_{\check{I\!\!P}^3 \setminus \Delta}$ are just its connected components, each orbit of the monodromy action contains all those points of the fibre that belong to the same component. On the other hand, the monodromy action on $\mbox{$Y_F$}|_{H_0}$ is obtained from the monodromy action on $(-1)$-curves on $Z|_{H_0}$ by projecting the $(-1)$-curve onto the corresponding double tangent. So the action on the 27 lines of a cubic surface cannot have more than six orbits. (The seventh orbit in $\mbox{$Y_F$}|_{H_0}$ consisting of the point corresponding to the double tangent in the plane $E_1=0$ has no representation among the 27 lines.) The three sets $\{G^{12}\}$ $\{G^{34}\}$ $\{G^{56}\}$ are $G$-invariant and consequently the three $G$--invariant sets of lines intersecting $G^{12}$, $G^{34}$ or $G^{56}$ have to be $G$-orbits. \end{proof} \begin{folg} Let $H$ be a plane in $I\!\!P^3 \setminus \Delta$. There are four components of \mbox{$Y_0$}{} with exactly one double tangent in $H$ -- namely the four components parametrising the lines in the planes $E_i=0$ ($i=1, \ldots, 4$). Of each of the remaining three components of \mbox{$Y_0$}{} there are eight double tangents in $H$. \end{folg} \begin{proof} The 24 double tangents in $H$ which are not contained in one of the four planes $E_i=0$ correspond to the 24 lines in the cubic surface that are not fixed under the action of $G$. These 24 lines in the cubic split into three $G$-orbits, each orbit containing eight of them. The $G$-action on the lines in the cubic is equivalent to the monodromy action on the double tangents in $H$. (The double tangent in the plane $E_1=0$ is fixed by the monodromy.) Hence, the 24 double tangents are spread over the three components in such a way that each component contains eight of them. \end{proof} \begin{satz}\label{generated} $G$ is generated by elements $g$ with the following properties: \begin{itemize} \item $g$ is of order two. \item $g$ leaves at least 15 of the 27 $(-1)$-curves fixed. \end{itemize} In other words: There are at most 6 pairs of lines such that the lines in these pairs are swapped by $g$. \end{satz} \begin{proof} Let $L$ be a general line in $\check{I\!\!P} ^3$. Without loss of generality one can assume that $H_0 \in L$. Then, due to \cite{lamottke} Section~7.4.1, $\pi_1(L\setminus \Delta,H_0) \longrightarrow \pi_1(\check{I\!\!P}^3\setminus \Delta,H_0)$ is surjective and therefore it is sufficient to study the monodromy of paths in $L \setminus \Delta$. The fundamental group $\pi_1(L\setminus \Delta,H_0)$ is generated by the homotopy classes of paths that loop once counter clockwise round one of the points in $L \cap \Delta$. By a sufficiently general choice of $L$, we can achieve that for each $H \in L\cap \Delta$ the plane quartic $B\cap H$ has exactly one ordinary node as its only singularity. For a smooth surface $B$ a generic line in $\check{I\!\!P} ^3$ will do (cf. \cite{lamottke} Section~1.6.4). The proof in \cite{lamottke} works as well in the case of hypersurfaces with isolated singularities. In the proof one only needs to replace the hypersurface by the open set of its regular points in all occurrences. Hence if for a plane $H$ of a generic pencil of planes the quartic curve $B\cap H$ has more or other singularities than one ordinary node then $H$ must contain a singular point of $B$ (which is an ordinary node in our case). But the set of those planes $H$, that contain a singular point of $B$ and for which $B\cap H$ is not a curve with exactly one ordinary node, has at least codimension two in $\check{I\!\!P} ^3$. So we can choose $L$ in such a way that $L$ does not intersect this codimension-2-subset in $\check{I\!\!P}^3$. Now, by Proposition~\ref{flach}, the fibre of \mbox{$Y_F$}{} over any $H \in L\cap \Delta$ contains exactly six ramification points and the ramification index in each of them is two. The monodromy of a loop round $H$ can only interchange two sheets of $\mbox{$Y_F$}|_L$ which meet in one of the ramification points over $H$. Therefore, the monodromy of this loop can swap at most six couples of points in the fibre of \mbox{$Y_F$}{} over $H_0$. Using the correspondence between the monodromy action on $\mbox{$Y_F$}|_{H_0}$ and the monodromy action on the lines in the cubic $S$, this proves the proposition. \end{proof} The group of symmetries of the 27 lines in a cubic surface (i.e. the group of permutations that respect the incidence relations among the lines) has been intensively studied. (Cf.~\cite{henderson}, \cite{segre}, \cite{manin} -- to mention only a few.) The following theorem holds (\cite{manin} Ch.~IV Theorem~1.9): \begin{theorem} The group of symmetries of the 27 lines on a cubic surface is isomorphic to the Weyl--Group \mbox{${\bf E}_6$}.\hspace*{\fill}$\Box$ \end{theorem} In particular, the group $G$ is a subgroup of \mbox{${\bf E}_6$}{}. Using the Propositions~\ref{transitiv} and \ref{generated} we will be able to determine $G \subset \mbox{${\bf E}_6$}$ as a subgroup of \mbox{${\bf E}_6$}{}. For this purpose, we first determine the elements of \mbox{${\bf E}_6$}{} that admit the properties required in Proposition~\ref{generated}. In \cite{manin} (Ch.~IV \S~9) as well as in \cite{swinn} a complete list of the conjugacy classes of \mbox{${\bf E}_6$}{} can be found. Moreover, to each conjugacy class the action of its elements on the 27 lines is described. It turns out that the only conjugacy class who's elements act on the 27 lines as postulated in Proposition~\ref{generated} is the class which is denoted by $C_{16}$ in \cite{manin} and \cite{swinn}. This class contains exactly 36 elements. It is a classical result (cf. e.g. \cite{segre}) that the group of symmetries of the 27 lines in a cubic surface is generated by elements which swap the lines in one of Schl\"afli's 36 ``double six''. A ``double six'' consists of a pair of sextuples of lines such that the lines in each sextuple are mutually skew and each line of one of the tuples intersects exactly five lines of the other tuple. Associating to each line of one tuple the unique line of the other tuple that is skew to this line yields a one-to-one correspondence between the two sextuples of a double six. We identify a double six with the element of \mbox{${\bf E}_6$}{} that exchanges the lines of the sextupels in such a way that each line is swapped with the unique line of the other tuple which is skew to this line. This transformation keeps the other 15 lines fixed. An example of a double six is the pair $[(E^1,\ldots,E^6), (C^1,\ldots,C^6)]$. Obviously, a permutations of the 27 lines corresponding to a double six satisfies the conditions of Proposition~\ref{generated}, i.e. is in the conjugacy class $C_{16}$. As this class contains exactly 36 elements (cf.~\cite{manin} or \cite{swinn}) and as there are exactly 36 double sixes, $C_{16}$ consists just of these double six transformations. Consequently, the group $G$ has its generators among these special transformations. Next, we will give a list of all double six transformations, that leave the three lines $G^{12}$, $G^{34}$, and $G^{56}$ invariant. For this purpose we will use the notation of \cite{manin} and give the transformations in terms of reflections of a root system. The Picard group $\mbox{Pic}(S)$ of a smooth cubic surface $S$ is the free abelian group with generators $[H]$ -- the pull-back of ${\cal O}_{I\!\!P^2}(1)$ under the blow-up morphism $S \longrightarrow I\!\!P^2$ -- and the six classes $[-E^i]$ (where $E^i$ $(i=1,\ldots,6)$ are the exceptional divisors). An element $a[H]-b_1[E^1]-\cdots - b_6[E^6] \in \mbox{Pic}(S)$\label{picard} will be denoted by $(a;b_1,\ldots, b_6)$ in the sequel. The intersection pairing on the Picard group is given by \[ (a;b_1,\ldots,b_6)\cdot (a';b'_1,\ldots,b'_6) = aa' - \sum_{i=1}^6 b_ib'_i. \] Denote by $\omega = -(3;1,1,1,1,1,1)$ the canonical line bundle of the cubic surface. Let $\omega^\perp \subset \mbox{Pic}(S) \otimes \mbox{$I\!\!R$}$ be the orthogonal complement of $\omega$ with respect to the scalar product induced by the intersection pairing. Note that on $\omega^\perp$ the intersection pairing is negative definite (cf.~\cite{manin} Proposition~IV.3.3). There is a root system of type \mbox{${\bf E}_6$}{} in $\omega^{\perp}$ who's reflections are in one-to-one correspondence with the double six transformations. The roots of this root system are the following: \begin{itemize} \item $[E^i] - [E^j] \in \mbox{Pic}(S) \subset \mbox{Pic}(S) \otimes \mbox{$I\!\!R$}$ $(1\le i,j \le 6, i\ne j)$ (30 roots). \item $\pm([H]-[E^i]-[E^j]-[E^k])$, $(1\le i < j < k \le 6)$ (40 roots). \item $\pm (2;1,1,1,1,1,1)$ (2 roots). \end{itemize} For a root $x$ the corresponding reflection $s_x$ is given by \[ s_x(v) = v + (v,x)\,x \] (where the scalar product is the one given by the intersection pairing on $\mbox{Pic}(S)$). The restriction of any reflection to $\mbox{Pic}(S) \subset \mbox{Pic}(S) \otimes\mbox{$I\!\!R$}$ induces an endomorphism of $\mbox{Pic}(S)$ that respects the intersection pairing. In particular it induces a transformation of the 27 lines which is a double six transformation. Now it is easy to find the double six transformations which fix the three lines $G^{12}$, $G^{34}$, and $G^{56}$. These correspond to the roots \begin{equation}\label{roots} \renewcommand{\arraystretch}{1.5} \begin{array}{lll} x_1= (2;1,1,1,1,1,1), & x_2 = (1;1,0,1,0,0,1), & x_3 = (0;-1,1,0,0,0,0), \\ x_4 = (1;1,0,0,1,1,0),& x_5 = (1;1,0,0,1,0,1), & x_6 = (1;0,1,1,0,1,0),\\ x_7 = (0;0,0,-1,1,0,0), & x_8 = (1;0,1,1,0,0,1), & x_9 = (1;0,1,0,1,1,0), \\ x_{10} = (1;0,1,0,1,0,1),& x_{11} = (0;0,0,0,0,-1,1),& x_{12} = (1;1,0,1,0,1,0).\\ \end{array} \end{equation} The corresponding double sixes are: \begin{equation}\label{action} \renewcommand{\arraystretch}{2} \newenvironment{ar}{\renewcommand{\arraystretch}{1.0}% \begin{array}{*{6}{c@{\;}}}}{\end{array}} \begin{array}{ll} x_1 \,\widehat{=}\, \left( \begin{ar} E^1 & E^2 & E^3 & E^4 & E^5 & E^6 \\ C^1 & C^2 & C^3 & C^4 & C^5 & C^6 \end{ar} \right),& x_2 \,\widehat{=}\, \left( \begin{ar} E^1 & E^3 & E^6 & G^{45} & G^{25} & G^{24} \\ G^{36} & G^{16} & G^{13} & C^2 & C^4 & C^5 \end{ar} \right),\\[1ex] x_3 \,\widehat{=}\, \left( \begin{ar} E^1 & C^1 & G^{23} & G^{24} & G^{25} & G^{26}\\ E^2 & C^2 & G^{13} & G^{14} & G^{15} & G^{16} \end{ar} \right),& x_4 \,\widehat{=}\, \left( \begin{ar} E^1 & E^4 & E^5 & G^{36} & G^{26} & G^{23} \\ G^{45} & G^{15} & G^{14} & C^2 & C^3 & C^6 \end{ar} \right),\\[1ex] x_5 \,\widehat{=}\, \left( \begin{ar} E^1 & E^4 & E^6 & G^{35} & G^{25} & G^{23} \\ G^{46} & G^{16} & G^{14} & C^2 & C^3 & C^5 \end{ar} \right),& x_6 \,\widehat{=}\, \left( \begin{ar} E^2 & E^3 & E^5 & G^{46} & G^{16} & G^{14} \\ G^{35} & G^{25} & G^{23} & C^1 & C^4 & C^6 \end{ar} \right),\\[1ex] x_7 \,\widehat{=}\, \left( \begin{ar} E^3 & C^3 & G^{14} & G^{24} & G^{45} & G^{46}\\ E^4 & C^4 & G^{13} & G^{23} & G^{35} & G^{36} \end{ar} \right),& x_8 \,\widehat{=}\, \left( \begin{ar} E^2 & E^3 & E^6 & G^{45} & G^{15} & G^{14} \\ G^{36} & G^{26} & G^{23} & C^1 & C^4 & C^5 \end{ar} \right),\\[1ex] x_9 \,\widehat{=}\, \left( \begin{ar} E^2 & E^4 & E^5 & G^{36} & G^{16} & G^{13} \\ G^{45} & G^{25} & G^{24} & C^1 & C^3 & C^6 \end{ar} \right),& x_{10} \,\widehat{=}\, \left( \begin{ar} E^2 & E^4 & E^6 & G^{35} & G^{15} & G^{13} \\ G^{46} & G^{26} & G^{24} & C^1 & C^3 & C^5 \end{ar} \right),\\[1ex] x_{11} \,\widehat{=}\, \left( \begin{ar} E^5 & C^5 & G^{16} & G^{26} & G^{36} & G^{46}\\ E^6 & C^6 & G^{15} & G^{25} & G^{35} & G^{45} \end{ar} \right),& x_{12} \,\widehat{=}\, \left( \begin{ar} E^1 & E^3 & E^5 & G^{46} & G^{26} & G^{24} \\ G^{35} & G^{15} & G^{13} & C^2 & C^4 & C^6 \end{ar} \right). \end{array} \end{equation} The 12 roots (\ref{roots}) (together with their negatives) form a root system. The group $G'$ which is generated by the double six transformations corresponding to these roots is the Weyl group to this root system. Obviously $G\subset G'$ is a subgroup. To determine $G'$ observe that the four roots $x_3,x_7,x_{11}$ and $x_{12}$ form a basis of the root system (\ref{roots}), i.e. any other root in the system is a linear combination of these four roots and the coefficients are either all non-negative or all non-positive. $G'$ is, thus, uniquely determined by the corresponding Dynkin diagram: \begin{center} \unitlength=0.7pt \begin{picture}(145.00,92.00)(115.00,615.00) \put(125.00,680.00){\line(1,0){50.00}} \put(200.00,680.00){\line(1,0){50.00}} \put(190.00,670.00){\line(0,-1){30.00}} \put(115.00,680.00){\circle*{6}} \put(190.00,680.00){\circle*{6}} \put(260.00,680.00){\circle*{6}} \put(190.00,630.00){\circle*{6}} \put(115.00,700.00){\makebox(0,0)[cc]{$x_3$}} \put(190.00,700.00){\makebox(0,0)[cc]{$x_{12}$}} \put(260.00,700.00){\makebox(0,0)[cc]{$x_{11}$}} \put(190.00,615.00){\makebox(0,0)[cc]{$x_7$}} \end{picture} \end{center} This diagram clearly is the Dynkin diagram of the group $D_4$. The group $D_4$ is isomorphic to the semi-direct product of the permutation group $S_4$ with $(\mbox{$Z\!\!\!Z$}_2)^3$. We are, now, going to show that $G$ must be the whole group $G'$. For this purpose consider the curve $G^{12}$ which is fixed under the group action. As already mentioned, there are ten lines in the cubic $S$ that intersect $G^{12}$. Since for any two intersecting lines in a cubic surface there exists a unique line in this cubic which intersects them both the ten lines which intersect $G^{12}$ are grouped into five pairs of intersecting lines. These are $p_0:=(G^{34},G^{56})$ (which is fixed under the monodromy action) and $p_1:=(G^{46},G^{35})$, $p_2:=(G^{36},G^{45})$, $p_3:=(C^1,E^2)$, $p_4:=(C^2,E^1)$. As the group $G'$ acts on the 27 lines preserving their incidence relations and keeping the line $G^{12}$ fixed, each of the above pairs of intersecting lines must be moved to such a pair by the group action. Using the explicit description (\ref{action}) of the action of $G'$ one observes that any of the transformations corresponding to the $x_i$ interchanges two of the four pairs $p_1,\ldots,p_4$ -- either preserving or reversing the order of the lines in the pairs. As $G\subset G'$ is to act transitively on the lines of the pairs $p_1,\ldots,p_4$ it must, in particular, act transitively on these four pairs. But a transitive subgroup of $S_4$ that is generated by transpositions must be the group $S_4$ itself. (Remember that $G$ is to be generated by a subset of the transformations corresponding to the $x_i$.) Let $x_{i_1}$, $x_{i_2}$, and $x_{i_3}$ correspond to elements of $G$ that act by swapping the pairs $p_1\leftrightarrow p_2$, $p_2\leftrightarrow p_3$, and $p_3\leftrightarrow p_4$ respectively. The subgroup of $G$ generated by these elements is isomorphic to $S_4$. The action of any of its elements does not map one line in a pair $p_i$ onto the other line of this pair. Therefore -- as $G$ is to act transitively on the eight lines -- there is an $x_{i_4}'$ such that the corresponding transformation is in $G$ and is not in the subgroup generated by the $x_{i_j}$ $(j=1,2,3)$. Let $x_{i_4}$ be the transformation that acts on the pairs in the same manner as $x_{i_4}'$ does and which is in the subgroup generated by the $x_{i_j}$ $(j=1,2,3)$. Then the composition of $x_{i_4}$ and $x'_{i_4}$ swaps the lines in the two pairs that are interchanged by the action of $x_{i_4}$ and $x'_{i_4}$. By conjugating $x_{i_4}'$ with the elements of $\left<x_{i_1},x_{i_2},x_{i_3}\right> \subset G$ on can interchange the lines in any two of the four pairs. Hence $G$ contains a subgroup (isomorphic to $(\mbox{$Z\!\!\!Z$}_2)^3$) who's elements interchange the lines in an even number of pairs. This proves that $G$ contains a subgroup which is the semi direct product of $S_4$ and $(\mbox{$Z\!\!\!Z$}_2)^3$ and therefore $G=G'$. Now, that we have a detailed knowledge of the action on the 27 lines (or on the 56 $(-1)$-curves on $Z_{H_0}$) induced by the monodromy, we will examine the induced action on the pairs of (different) lines formed out of these lines. Denote by \mbox{$\cal A$}{}, \mbox{$\cal B$}{}, \mbox{$\cal C$}{} the three orbits consisting of the eight lines intersecting the lines $G^{12}$, $G^{34}$, $G^{56}$ respectively. Then there are the following monodromy invariant subsets in the set of pairs of lines: \begin{itemize} \item The 9 sets $\{G_{12}\}\times\mbox{$\cal A$}$, $\{G_{12}\}\times\mbox{$\cal B$}$, $\{G_{12}\}\times\mbox{$\cal C$}$, $\{G_{34}\}\times\mbox{$\cal A$}$ etc. \item The 3 sets $\{(G_{12},G_{34})\}$, $\{(G_{12},G_{56})\}$, and $\{(G_{34},G_{56})\}$. \item The 3 sets $(\mbox{$\cal A$}\OA)$ (which means the set of pairs consisting of two lines of \mbox{$\cal A$}{}), $(\mbox{$\cal B$}\OB)$, $(\mbox{$\cal C$}\OC)$. \item The 3 sets $(\mbox{$\cal A$}\mbox{$\cal B$})$ (which means the set of pairs consisting of one line of \mbox{$\cal A$}{} and one line of \mbox{$\cal B$}{}), $(\mbox{$\cal A$}\mbox{$\cal C$})$, $(\mbox{$\cal B$}\mbox{$\cal C$})$. \end{itemize} The sets of the first two items are obviously orbits under the monodromy action, whereas the sets of the last two items will turn out to be composite of two orbits. One orbit in the set $(\mbox{$\cal A$}\OA)$ consists of the four pairs $p_1:=(G^{46},G^{35})$, $p_2:=(G^{36},G^{45})$, $p_3:=(C^1,E^2)$, $p_4:=(C^2,E^1)$ which are the four pairs which consist of intersecting lines. We claim that the remaining 24 elements in $(\mbox{$\cal A$}{}\mbox{$\cal A$})$ form an orbit of the monodromy action. Let $(\ell_1,\ell_2)\in \mbox{$\cal A$}$ be a pair of lines which do not intersect. We will calculate the number of elements in the orbit of this pair by finding its stabiliser. Embed $S_4\subset G$ as the subgroup who's elements permute the pairs $p_1,\ldots,p_4$ but leave the order of the lines in the pairs unchanged. If $(\mbox{$Z\!\!\!Z$}_2)^3 \subset G$ is the subgroup of elements that leave the four pairs $p_1,\ldots,p_4$ fixed but swaps the lines in an even number of these pairs then $G$ is the semi-direct product of $S_4$ and $(\mbox{$Z\!\!\!Z$}_2)^3$ (maybe in a different presentation as above). Let the lines $\ell_1$ and $\ell_2$ belong to the pairs $p_{i_1}$ and $p_{i_2}$ respectively. Then the stabiliser of $(\ell_1, \ell_2)$ consists of those elements of $G$ which leave the pairs $p_{i_1}$ and $p_{i_2}$ unchanged (i.e. those which interchange the two other pairs with or without changing the order of the lines in the pairs) together with those elements that interchange $p_{i_1}$ and $p_{i_2}$ in such a way that $\ell_1$ is mapped to $\ell_2$. The stabiliser is, therefore, $(S_2\times S_2) \ltimes \mbox{$Z\!\!\!Z$}_2 \subset S_4 \ltimes (\mbox{$Z\!\!\!Z$}_2)^3 \cong G$. The orbit of $(\ell_1,\ell_2)$, hence must contain $24=192/8$ elements. Applying the same argument to \mbox{$\cal B$}{} and \mbox{$\cal C$}{} yields: \mbox{$\cal A$}{}, \mbox{$\cal B$}{}, and \mbox{$\cal C$}{} each split into two orbits -- one with four and one with 24 elements. In the same manner we will attack the sets $(\mbox{$\cal A$}\mbox{$\cal B$})$, $(\mbox{$\cal A$}\mbox{$\cal C$})$, and $(\mbox{$\cal B$}\mbox{$\cal C$})$ -- e.g. the set $(\mbox{$\cal A$}\mbox{$\cal B$})$ (the two other sets being treated in an analogous way). Consider the line $E^1\in\mbox{$\cal A$}$. The stabiliser in $G$ of $E^1$ consists of exactly those elements which act only on the three pairs different from $(C^2,E^1)$. This subgroup is isomorphic to $S_3 \ltimes (\mbox{$Z\!\!\!Z$}_2)^2$. It is generated by the elements corresponding to the double six transformations $x_6,\ldots,x_{11}$. Using the explicit description (\ref{action}) it is easy to check that the set \mbox{$\cal B$}{} has two orbits under the action of this subgroup -- namely $\{E^3,E^4,G^{25},G^{26}\}$ and $\{C^3,C^4,G^{15},G^{16}\}$. (Note that the second orbit consists of the set of lines in \mbox{$\cal B$}{} that intersect $E^1$.) Thus, any pair $(\ell_1,\ell_2) \in (\mbox{$\cal A$}\mbox{$\cal B$})$ can be moved into a pair $(E^1,\ell)$ via monodromy action and this pair can be moved into each of the pairs $(E^1,\ell')$ with $\ell' \in\{E^3,E^4,G^{25},G^{26}\}$ or $\ell' \in\{C^3,C^4,G^{15},G^{16}\}$ depending on in which set $\ell$ is contained. So $(\mbox{$\cal A$}\mbox{$\cal B$})$ can contain at most two orbits. On the other hand, a pair $(\ell_1,\ell_2)$ of intersecting lines cannot be moved into a pair of non-intersecting lines and vice versa since the monodromy action respects the incidence relations. Hence, $(\mbox{$\cal A$}\mbox{$\cal B$})$ must split in at least two orbits -- one containing pairs of intersecting lines and the other containing pairs of non-intersecting lines. This proves that $(\mbox{$\cal A$}\mbox{$\cal B$})$ (as well as $(\mbox{$\cal A$}\mbox{$\cal C$})$ and $(\mbox{$\cal B$}\mbox{$\cal C$})$) are the composite of two orbits of equal cardinality. Summing up, we have proved the following proposition. \begin{satz}\label{DT-Paare-monod} The set of pairs formed out of the 27 lines has the following orbits under the action of $G$ induced by the monodromy action on the lines. \begin{enumerate} \item Each of the three subsets $(\mbox{$\cal A$}\OA)$, $(\mbox{$\cal B$}\OB)$, $(\mbox{$\cal C$}\OC)$ contains two orbits -- one with four and one with 24 pairs. \item Each of the subsets $(\mbox{$\cal A$}\mbox{$\cal B$})$, $(\mbox{$\cal A$}\mbox{$\cal C$})$, $(\mbox{$\cal B$}\mbox{$\cal C$})$ contains two orbits with 32 pairs -- one orbit with pairs of intersecting lines and one orbit with pairs of non-intersecting lines. \item The 9 sets $\{G_{12}\}\times\mbox{$\cal A$}$, $\{G_{12}\}\times\mbox{$\cal B$}$, $\{G_{12}\}\times\mbox{$\cal C$}$, $\{G_{34}\}\times\mbox{$\cal A$}$ etc. with 8 pairs each and \item the 3 sets $\{(G_{12},G_{34})\}$, $\{(G_{12},G_{56})\}$, and $\{(G_{34},G_{56})\}$ -- are orbits.\hspace*{\fill}$\Box$ \end{enumerate} \end{satz} \subsection{One-parameter-families of touching conics and linear systems} We, now, want to use our knowledge on pairs of lines of a cubic surface and their monodromy to examine the irreducible components of the parameter space of touching conics. For this purpose return to the the double cover $Z$ of $I\!\!P^3$ branched along our quartic $B$. Let $H\subsetI\!\!P^3$ be an arbitrary plane that intersects $B$ transversally and denote by $Z_H$ the restriction of $Z$ to $H$ and by $\pi:Z_H \longrightarrow H$ the induced morphism. (Recall that $Z_H$ is isomorphic to the blow-up of $I\!\!P^2$ in seven points in general position.) We will establish a connection between one-parameter-families of touching conics of $B\cap H$ and certain linear systems in $Z_H$. Thereby the reducible elements of the linear systems will correspond to the reducible touching conics in the one-parameter-families. Let $C_1$ and $C_2$ be two $(-1)$-curves that have different images under $\pi$ (i.e. $\pi(C_1)\ne\pi(C_2)$) which means that they are $(-1)$-curves over different double tangents of $B\cap H$. This is equivalent to $[C_1+C_2] \ne -K_{Z_H}$ where $K_{Z_H}$ denotes the canonical class of $Z_H$. Let $C'_1$ and $C'_2$ be the $(-1)$-curves defined by $[C_i + C'_i] = -K_{Z_H}$. Then these curves intersect as follows: \[ C_1\cdot C_2 = C'_1\cdot C'_2 = 1 - C_1\cdot C'_2 = 1 - C'_1\cdot C_2. \] This follows from \[ 1 = C_1 \cdot (-K_{Z_H}) = C_1\cdot\left( C_2+C'_2\right) = C_1\cdot C_2 + C_1\cdot C'_2 \] and analogous identities. Therefore, by eventually interchanging $C_1$ and $C'_1$, one can achieve that $C_1\cdot C_2 = 1$ (the corresponding double tangents keeping unchanged). \begin{satz} If $C_1$ and $C_2$ are chosen as above with $C_1\cdot C_2 =1$ then the linear system $|C_1+C_2|$ is one-dimensional. Its generic element is a smooth rational curve that by the projection $\pi:Z_H \longrightarrow H$ is mapped onto a touching conic. \end{satz} \begin{proof} For any of the 56 $(-1)$-curves $C$ of $Z_H$ consider the exact sequence \[ 0\longrightarrow {\cal O}_{Z_H}\longrightarrow {\cal O}_{Z_H}(C) \longrightarrow {\cal O}_{C}(C) \longrightarrow 0.\] Noting that $C$ is a smooth rational curve with ${\cal O}_{C}(C)= {\cal O}_{C}(C\cdot C) ={\cal O}_{C}(-1)$ and that $Z_H$ is a smooth rational surface so that $0=h^1({\cal O}_{Z_H}) = h^1({\cal O}_{Z_H}(C))$ we get $h^0({\cal O}_{Z_H}) = h^0({\cal O}_{Z_H}(C)) = 1$. Now, the exact sequence \[ 0\longrightarrow {\cal O}_{Z_H}(C_2) \longrightarrow {\cal O}_{Z_H}(C_1 + C_2) \longrightarrow {\cal O}_{C_1}(C_1 + C_2) \longrightarrow 0 \] and the fact that ${\cal O}_{C_1}(C_1 + C_2) = {\cal O}_{C_1}$ (since $C_1\cdot(C_1+ C_2) = 0$) yield \[ h^0({\cal O}_{Z_H}(C_1 + C_2)) = h^0({\cal O}_{Z_H}(C_2)) + h^0({\cal O}_{C_1}) = 2 \] and, hence, $\dim |C_1+C_2|=1$. $|C_1 +C_2|$ cannot have a fixed component. Since $C_1$ and $C_2$ are irreducible this fixed component would have to be one of these two curves and then the linear system would only contain the divisor $C_1+C_2$ in contradiction to the dimension of the system being 1. Hence, as $(C_1 +C_2)^2=0$, the system cannot have base points at all. By Bertini's Theorem the generic element of $|C_1 +C_2|$ is smooth away from the base locus. As the base locus is empty the generic element of the system is smooth everywhere. Let $C\in |C_1 +C_2|$ be general and $D_1, \ldots, D_n$ be its irreducible components. The $D_i$ cannot intersect each other for any intersection point would be a singular point of $C$. Thus $D_i\cdot D_j = 0$ for $i\ne j$ and consequently \[ 0 = \left( C_1 +C_2\right)^2 = \left( \sum_{i=1}^n D_i\right)^2 = \sum_{i=1}^n D_i^2. \] $|C_1 +C_2|$ has no fixed components -- therefore \[ 0\le D_i\cdot (C_1 +C_2) = D_i^2 \] and hence $D_i^2 = 0$ for all $i$. From the adjunction formula we get \[ g_i := \mbox{genus}(D_i) = \frac{K_{Z_H}\cdot D_i}{2} +1. \] On the other hand $K_{Z_H} \cdot D_i < 0$ since $-K_{Z_H}$ is ample. From $g_i \ge 0$ we then get $K_{Z_H} \cdot D_i = -2$ for all~$i$. But \[ -K_{Z_H} \cdot \left(\sum_{i=1}^n D_i\right) = -K_{Z_H}\cdot \left( C_1 +C_2\right) = 2 \] and hence $n=1$, $C=D_1$, and $\mbox{genus}(C)=0$, i.e. $C$ is a smooth rational curve. Now, let $R\subset Z_H$ be the ramification divisor of the map $Z_{H}\longrightarrow H$: ${\cal O}_{Z_H}(R) = \pi^*({\cal O}_H(2)) = -2K_{Z_H}$. Hence \[ (C_1 +C_2)\cdot [R] = 2\,(C_1 + C_2)\cdot(-K_{Z_H}) = 4 \] and by projection formula \[ 4\,[pt] = \pi_*\left(\left(C_1+C_2\right)\cdot [R]\right) = \pi_*\left(\left(C_1+C_2\right) \cdot \pi^*\left(2\,[\,l\,]\right)\right) = \pi_*\left(C_1+C_2\right)\cdot \left(2\,[\,l\,]\right) \] where $[pt]$ denotes the class of a point and $[\,l\,]$ the class of a line in $H$. Therefore $\pi_*(C_1+C_2)\in |2\,[\,l\,]|$. Let $C\in |C_1+C_2|$ be a general element. If $\pi(C)$ were a line then $C$ were contained in the preimage of a line. As $[C_1+C_2] \ne K_{Z_H}$ there would exist an effective $C'$ such that $[C]+[C'] = -K_{Z_H}$. But then $C'\cdot(-K_{Z_H}) = (-K_{Z_H})^2 - C\cdot(-K_{Z_H}) = 0$ which is impossible as $-K_{Z_H}$ is ample. So the the image of $C$ under $\pi$ must be a smooth conic and $\pi|_C$ is of degree one onto $\pi(C)$. Consider now the preimage $\pi^{-1}(\pi(C))$ in $Z_H$. As $\pi$ is a double cover and $\pi|_C$ is only of degree one, $\pi^{-1}(\pi(C))$ must contain other components than $C$ or $\pi(C)$ must be contained in $B\cap H$. The latter is not possible since $B\cap H$ was supposed to be a smooth quartic curve. Obviously, $\pi^{-1}(\pi(C))$ is an element of $|-2\,K_{Z_H}|$ since $\pi(C)$ is an element of $|{\cal O}_{I\!\!P^2}(2)|$ and $\pi:Z_H\longrightarrow I\!\!P^2\cong H$ is induced by the anticanonical linear system. Therefore, the sum of the other components of $\pi^{-1}(\pi(C))$ must be an element of $|-2\,K_{Z_H} - (C_1 +C_2)| = |C'_1 +C'_2|$. ($C'_i$ was defined to be the $(-1)$-curve in $Z_H$ such that $C_i + C'_i = -K_{Z_H}$.) Let $C'\in |C'_1 + C'_2|$ be the divisor which is complementary to $C$ in $\pi^{-1}(\pi(C))$. Note that $(C'_1 +C'_2)$, as well as $(C_1+C_2)$, is the sum of two $(-1)$-curves with intersection $C'_1 \cdot C'_2 =1$ and consequently the above arguments equally apply to $(C'_1 +C'_2)$. In particular, a general element of $|C'_1 +C'_2|$ is a smooth rational curve which by $\pi$ is mapped onto a smooth conic in $H$. So if $C \in |C_1 +C_2|$ is sufficiently general then $C'$, as well, is a smooth rational curve that is mapped onto a smooth conic. Therefore, $\pi^{-1}(\pi(C))$ splits into two components each of which is a smooth rational curve. Now, one shows just like in the proof of Proposition~\ref{geraden-in-DS} that the preimage in a double cover of a conic in $H$ splits into two components if and only if it has even intersection with the ramification locus $B\cap H \subset H$. Therefore $C$ is mapped onto a touching conic and the proposition is proved. \end{proof} By the above proposition $\pi$ induces a morphism $|C_1+C_2|\cong I\!\!P^1 \longrightarrow I\!\!P^5$ where $I\!\!P^5$ is the parameter space of conics in $H$. This morphism is necessaryly injective as the preimage of $\pi(C)$ consists of $C$ and an element of the linear system $|-K_{Z_H} - [C]|$ which is different from $|C_1 +C_2|$. There is an open subset in $|C_1 +C_2|$ which is mapped into the closed subset of touching conics in $I\!\!P^5$. Therefore any element of $|C_1 +C_2|$ is mapped onto a (maybe reducible) touching conic. Moreover, an element of $|C_1 +C_2|$ is mapped to a reducible conic if and only if it is the sum of two $(-1)$-curves. We, thus, have constructed a correspondence between the linear systems $|C_1 + C_2|$ (with $(-1)$-curves $C_i$ satisfying $C_1 \cdot C_2 =1$) and one-parameter-families of conics in $H$ touching $B\cap H$: For each one parameter family there exist exactly two of these linear systems that are mapped to this family. Let again ${\cal Z}:= Z\times_{I\!\!P^3} {\cal H}$ where ${\cal H} \subset I\!\!P^3 \times \check{I\!\!P}^3$ is the universal (hyper-)plane. We have seen that for any $H_0\in \check{I\!\!P}^3 \setminus \Delta$ the fundamental group $\pi_1(\check{I\!\!P}^3 \setminus \Delta,H_0)$ acts via monodromy of ${\cal Z}|_{\check{I\!\!P}^3\setminus\Delta} \longrightarrow \check{I\!\!P}^3\setminus\Delta$ on $\mbox{Pic}(Z_{H_0})$ preserving the intersection pairing. In particular, the fundamental group acts on the above liner systems. Denote by $\mbox{$X$}'\subset P$ ($P$ -- the parameter space of all conics in $I\!\!P^3$ as constructed in Section~\ref{psc}) the closed subscheme of conics that have only even intersection with $B$ and let $\mbox{$X$}\subset\mbox{$X$}'$ be the union of all irreducible components that do not entirely consist of double lines. From $P$ \mbox{$X$}{} inherits a morphism to $\check{I\!\!P}^3$. By Proposition~\ref{Anz-epf} the fibre of \mbox{$X$}{} over any $H\in \check{I\!\!P}^3\setminus\Delta$ consists of 63 disjoint smooth conic curves in the fibre of $P$ over $H \in \check{I\!\!P}^3$ which is isomorphic to $I\!\!P^5$. The fundamental group $\pi_1(\check{I\!\!P}^3 \setminus \Delta,H_0)$ acts on the set of the 63 one parameter families in the fibre $\mbox{$X$}_{H_0}$ in a natural way by monodromy: Any path $\gamma:[0,1]\longrightarrow \check{I\!\!P}^3 \setminus\Delta$ can be lifted (in a non-unique way) to a path in $\mbox{$X$}|_{\check{I\!\!P}^3\setminus\Delta}$. Though the lift is not unique, the connected component of the fibre $\mbox{$X$}_{\gamma(t)}$ in which the lifted path is contained is well determined. Obviously, the correspondence between the linear systems in $Z_{H_0}$ and the one-parameter-families is compatible with the monodromy action. On the other hand, the orbits of the monodromy in $\mbox{$X$}_{H_0}$ are in bijection with the connected components of $\mbox{$X$}|_{\check{I\!\!P}^3\setminus\Delta}$. But $\mbox{$X$}|_{\check{I\!\!P}^3\setminus\Delta}$ is smooth (since it is flat over $\check{I\!\!P}^3\setminus\Delta$ and has smooth fibres) and therefore the connected components of $\mbox{$X$}|_{\check{I\!\!P}^3\setminus\Delta}$ in the Euclidian topology are its irreducible components. We are mainly interested in the connected components of $\mbox{$X$}|_{\check{I\!\!P}^3\setminus\Delta}$. One way of computing the monodromy orbits would be to list all linear systems $|C_1+C_2|$ with $(-1)$-curves $C_i$ satisfying $C_1\cdot C_2 =1$ and then determining the monodromy orbits of these linear systems using our knowledge on the monodromy of $(-1)$-curves. The monodromy of $\mbox{$X$}|_{\check{I\!\!P}^3\setminus\Delta}$ is then easily calculated. This approach is a bit cumbersome. So we modify this method using our knowledge about one-parameter-families of touching conics. {\sloppy First, we consider the linear system $|G^{12}+G^{34}| = |(2;1,1,1,1,0,0,0)|$ (the elements of $\mbox{Pic}(Z_{H_0})$ are denoted similarly as on page~\pageref{picard}, i.e. $(a;b_1,\ldots,b_7) \in \mbox{Pic}(Z_{H_0})$ denotes the element $a[H] - b_1[E^1] - \cdots - b_7[E^7]$). As the two $(-1)$-curves $G^{12}$ and $G^{34}$ are monodromy invariant the linear system must keep fixed under the monodromy action, as well. This system may be presented as the sum of two $(-1)$-curves in five further ways:} \[ \renewcommand{\arraystretch}{1.5} \begin{array}{rcccccc} (2;1,1,1,1,0,0,0) & = & C^{56}+E^7 & = & C^{57}+E^6 & = & C^{67}+E^5\\ & = & G^{13}+G^{24} & = & G^{14}+G^{23}. \end{array} \] The two pairs $(G^{12},G^{34})$ and $(C^{56},E^7)$ correspond to the pairs of double tangents $(e_2,e_3)$ and $(e_4,e_1)$ where $e_i$ denotes the double tangent given by the equation $E_i=0$ in $H_0$ ($E_i$ being the linear forms appearing in the equation $E_1E_2E_3E_4 -Q^2$ of $B$). The remaining four pairs are those of the set $(\mbox{$\cal A$}\OA)$ when identified with the corresponding lines in a cubic surface. The corresponding one-parameter-family of touching conics, thus, contains four pairs of double tangents all eight double tangents being in the same component of \mbox{$Y_0$}. Denote this component of \mbox{$Y_0$}{} by \mbox{$\cal C$}{}. Analogously (considering the systems $|G^{12}+G^{56}|$ and $|G^{34} + G^{56}|$ respectively) the one-parameter-family of touching conics containing the pairs $(e_2,e_4)$ and $(e_1,e_3)$ contains four pairs formed of double tangents of the component \mbox{$\cal B$}{} of \mbox{$Y_0$}{} and the family containing the pairs $(e_3,e_4)$ and $(e_1,e_2)$ contains four pairs formed of double tangents of the component \mbox{$\cal A$}{}. These three one-parameter-families are the ones which are obvious from the special form of the equation of $B$: $E_1E_2E_3E_4 -Q^2$ is of the form $UW-V^2$ by letting $V=Q$ and letting $U$ be one of $E_1E_2$, $E_1E_3$ or $E_1E_4$. So in any plane $H_0$ we get three one-parameter-families~(\ref{epf}) which we will call the ``obvious'' families. In particular, each of these obvious one-parameter-families contains two reducible conics consisting of the four double tangents $e_1,\ldots,e_4$ and four reducible elements formed of the eight double tangents in $H_0$ that belong to the same component \mbox{$\cal A$}{}, \mbox{$\cal B$}{} or \mbox{$\cal C$}{} of \mbox{$Y_0$}{}. Denote, for simplicity the eight double tangents in $H_0$ of the component \mbox{$\cal A$}{} by $a_1,\ldots,a_8$ and the double tangents of \mbox{$\cal B$}{} and \mbox{$\cal C$}{} by $b_1,\ldots,b_8$ and $c_1,\ldots,c_8$ respectively. We will examine how the pairs formed out of these double tangents can be distributed to the one-parameter-families. By Lemma~\ref{epf-zerf-el} each one-parameter-family of touching conics in $H_0$ contains exactly six reducible conics. From the above arguments we already know the reducible elements of three families (given in terms of pairs of double tangents): \[ \renewcommand{\arraystretch}{1.5} \begin{array}{*{3}{r@{,\hspace{\arraycolsep}}}*{3}{l@{\hspace{\arraycolsep}}}} e_1e_2& e_3e_4& a_1a_2& a_3a_4,&a_5a_6,&a_7a_8 \\ e_1e_3& e_2e_4& b_1b_2&\ldots \\ e_1e_4& e_2e_3& c_1c_2&\ldots \\ \end{array} \] In particular, the equation of the plane quartic $B\cap H_0$ can be written in the form $e_1e_2\cdot a_1a_2 -V^2$ where $V$ is a quadratic form. (Lines and the corresponding linear forms are denoted by the same letter.) By writing this equation as $e_1a_1\cdot e_2a_2 -V^2$ we see that the couples $e_1a_1$ and $e_2a_2$ belong to the same one-parameter-family. By Lemma~\ref{DT-in-EPF}, in this family no further couple of double tangents is contained that has one of the double tangents $e_i$ or $a_i$ as an element. Couples of the form $b_ib_j$ or $c_ic_j$ also must not occur in this family. If, for instance, the couple $b_1b_3$ were in one group together with $e_1a_2$ then the equation of $B\cap H_0$ could be written as $e_1a_2 \cdot b_1b_3 - V'^2 = e_1b_1\cdot a_1b_3 - V'^2$ and thus $e_1b_1$ and $a_1b_3$ would belong to the same family. But this is impossible by the above argument. Hence, only the couples $b_kc_l$ may occur in families together with couples $e_ia_j$. There are exactly 16 families each of which contains two couples $e_ia_j$. On the other hand, there are 64 couples $b_ic_j$ which is just the number of couples needed to complete these 16 families. By the same argument, the couples $a_ib_j$ belong to families which contain two couples of the form $e_kc_l$ and four couples of the form $a_ib_j$ and, analogously, the couples $a_ic_j$ spread over families containing two couples $e_kb_j$ and four couples $a_ic_j$. As we have just seen couples $b_ic_j$ and $b_kc_l$ have to occur in one family. Thus, there is a family containing both $b_ib_k$ and $c_jb_l$. Analogously, there are couples $a_ia_j$ and $b_kb_l$ as well as $a_ia_j$ and $c_kc_l$ in one group. There are $3\cdot({8 \choose 2} -4) = 72$ pairs of the form $a_ia_j$, $b_ib_j$ and $c_ic_j$ which do not occur in the ``obvious'' families. These spread over the remaining 12 one-parameter-families.\myfootnote{In fact, using Lemma~\ref{DT-in-EPF}, one could determine which pairs of double tangents pertain to which one-parameter-family. Those considerations can be found in \cite{salmon}.} In the consequence of these considerations we can determine the connected components of $X|_{\check{I\!\!P}^3 \setminus\Delta}$. Let $\mbox{$Y_F^2$}\subset X|_{\check{I\!\!P}^3 \setminus\Delta}$ be the closed subscheme which parametrises the reducible touching conics. \mbox{$Y_F^2$}{} is naturally isomorphic to the open subset in the symmetric product of \mbox{$Y_F$}{} with itself: \[ \mbox{$Y_F$}\stackrel{\mbox{\tiny sym}}{\times}_{\check{I\!\!P}^3\setminus \Delta} \mbox{$Y_F$} \setminus \mbox{Diag} \;\tilde{\longrightarrow} \;\mbox{$Y_F^2$} \] by simply associating to each pair of complanar double tangents the corresponding reducible touching conic. As any one-parameter-family in any fibre of $X|_{\check{I\!\!P}^3 \setminus\Delta}$ over $\check{I\!\!P}^3\setminus\Delta$ contains reducible conics and as the monodromy action on the set of one-parameter-families in the fibre over $H_0 \in \check{I\!\!P}^3\setminus\Delta$ is independent of the chosen lift of the path one can choose the lift of any path $\gamma\subset \check{I\!\!P}^3\setminus\Delta$ in such a way that the lifted path is contained in \mbox{$Y_F^2$}{}. But the connected components of \mbox{$Y_F^2$}{} are already determined: The following proposition is just a corollary of Proposition~\ref{DT-Paare-monod}. \begin{satz}\label{YY-comp} The connected components of $\mbox{$Y_F^2$}{}\longrightarrow \check{I\!\!P}^3\setminus \Delta$ are the following: \begin{enumerate} \item \label{orb24}Three components with 24 points in each fibre over $\check{I\!\!P}^3 \setminus \Delta$ -- each component corresponding to one orbit (the one with 24 elements) in the sets $(\mbox{$\cal A$}\OA)$, $(\mbox{$\cal B$}\OB)$, and $(\mbox{$\cal C$}\OC)$ of pairs of lines in a cubic. \item \label{orb4}Three components with four points in each fibre -- each component corresponding to the other orbit in the sets $(\mbox{$\cal A$}\OA)$, $(\mbox{$\cal B$}\OB)$, and $(\mbox{$\cal C$}\OC)$ of pairs of lines in a cubic. \item Six components with 32 points in each fibre corresponding to the orbits in the sets $(\mbox{$\cal A$}\mbox{$\cal B$})$, $(\mbox{$\cal A$}\mbox{$\cal C$})$, and $(\mbox{$\cal B$}\mbox{$\cal C$})$. \item \label{orb8}12 components with eight points in each fibre: For each $i=1,\ldots 4$ and each component \mbox{$\cal A$}{}, \mbox{$\cal B$}{} or \mbox{$\cal C$}{} of \mbox{$Y_0$}{} there is an irreducible component of \mbox{$Y_F^2$}{}. The corresponding reducible conics in a plane $H$ consist of the double tangent $e_i$ given by the equation $E_i=0$ in $H$ and one double tangent of the chosen component of \mbox{$Y_0$}{}. \item Six components with just one point in each fibre, namely the six pairs of double tangents $e_ie_j$. \end{enumerate} \end{satz} Consequently, $X|_{\check{I\!\!P}^3 \setminus\Delta}$ has the following connected components: \begin{itemize} \item The three ``obvious'' families in the fibre of $X|_{\check{I\!\!P}^3 \setminus\Delta}$ over $H_0 \in\check{I\!\!P}^3 \setminus\Delta$ are invariant under monodromy as they contain reducible conics consisting of the double tangents $e_ie_j$ which are fixed under monodromy. Each of the three families contains two of them. The other four reducible conics in each family are necessarily the four pairs of double tangents of one of the orbits in item~\ref{orb4}.\ in the above Proposition. \item There are six connected components with eight one-parameter-families in any fibre of $X|_{\check{I\!\!P}^3 \setminus\Delta}$ over $\check{I\!\!P}^3 \setminus\Delta$: All one-parameter-families containing reducible conics of the type $e_ia_j$, $e_ib_j$ or $e_ic_j$ pertain to one of these components. As we have seen, any one-parameter-family that contains those pairs of double tangent contains two of them and four pairs of the form $a_ib_j$, $a_ic_j$, or $b_ic_j$. By Lemma~\ref{DT-in-EPF} and the above discussion, the two pairs $e_ia_j$ and $e_ka_l$ are in the same one-parameter-family only if $i\ne k$ and $j\ne l$ (analogously for $e_ib_j$ and $e_ic_j$). Therefore each component of \mbox{$Y_F^2$}{} in item~\ref{orb8}.\ of Proposition~\ref{YY-comp} intersects a one-parameter-family in a fibre over $\check{I\!\!P}^3 \setminus\Delta$ in at most one point. Consequently, the monodromy orbit of such a one-parameter-family in the fibre of $X|_{\check{I\!\!P}^3 \setminus\Delta}$ over $H_0$ consists of exactly eight families. \item The remaining pairs of double tangents are those of item~\ref{orb24}.\ of Proposition~\ref{YY-comp}. The families containing these reducible conics belong to the same connected component: We have seen that there is a one-parameter-families that contains a pair $a_ia_j$ as well as a pair $b_kb_l$ and a family that contains some $a_ia_j$ together with a $c_kc_l$. So, one can connect the first family with any family containing a pair $a_\bullet a_\bullet$ or a pair $b_\bullet b_\bullet$ by a path lying in \mbox{$Y_F^2$}{} and, analogously, the second family can be connected with any family containing a pair $c_\bullet c_\bullet$. Hence the twelve one-parameter-families belong to the same connected component. \end{itemize} As the connected components of $X|_{\check{I\!\!P}^3 \setminus\Delta}$ are just its irreducible components the following Theorem is proved by the above discussion. \begin{theorem} $X|_{\check{I\!\!P}^3 \setminus\Delta}$ has 10 irreducible components -- namely \begin{itemize} \item three components each with one one-parameter-family in every fibre over $\check{I\!\!P}^3$, \item six components with eight families in every fibre, and \item one component with twelve families in each fibre over $\check{I\!\!P}^3 \setminus\Delta$.\hspace*{\fill}$\Box$ \end{itemize} \end{theorem} The three types of irreducible components differ by the type of reducible conics that they contain. In each one-parameter-family there are reducible conics that contain two, one or none double tangent $e_i$ respectively for the three types. {\bf Remark:} The components of $X$ are not entirely determined by the above theorem. There are at least four components which are contained in $X|_\Delta$ (i.e. over $\Delta \subset \check{I\!\!P}^3$) namely the four sets consisting of all conics in the planes $E_i=0$. All of them have to be touching conics since $B\cap{\{E_i=0\}}$ is a non-reduced conic. But these four components are, conjecturally, all components which are contained in $X|_\Delta$.

9602

alg-geom/9602017

en

https://arxiv.org/abs/alg-geom/9602017

alg-geom/9602017

Nitin Nitsure

Nitin Nitsure

Topology of Conic Bundles - II

5 pages. LaTeX

For conic bundles on a smooth variety (over a field of characteristic $\ne 2$) which degenerate into pairs of distinct lines over geometric points of a smooth divisor, we prove a theorem which relates the Brauer class of the non-degenerate conic on the complement of the divisor to the covering class (Kummer class) of the 2-sheeted cover of the divisor defined by the degenerate conic, via the Gysin homomorphism in etale cohomology. This theorem is the algebro-geometric analogue of a topological result proved earlier.

alg-geom

\subsection*{\hbox{}\hfill{\normalsize\sl #1}\hfill\hbox{}}} \textheight 23truecm \textwidth 15truecm \addtolength{\oddsidemargin}{-1.05truecm} \addtolength{\topmargin}{-1.5truecm} \makeatletter \def\l@section{\@dottedtocline{1}{0em}{1.2em}} \makeatother \begin{document} \title{Topology of Conic Bundles - II} \author{Nitin Nitsure} \date{24 February 1996} \maketitle \centerline{Tata Institute of Fundamental Research, Mumbai 400 005, India.} \centerline{e-mail: [email protected]} \centerline{\sl 1991 Mathematics subject classification : 14F20, 13A20, 12G05} \begin{abstract} \noindent{For} conic bundles on a smooth variety (over a field of characteristic $\ne 2$) which degenerate into pairs of distinct lines over geometric points of a smooth divisor, we prove a theorem which relates the Brauer class of the non-degenerate conic on the complement of the divisor to the covering class (Kummer class) of the 2-sheeted cover of the divisor defined by the degenerate conic, via the Gysin homomorphism in etale cohomology. This theorem is the algebro-geometric analogue of a topological result proved earlier. \end{abstract} \section{Introduction} Let $X$ be a smooth scheme over a field $F$ of characteristic $\ne 2$, and let $C\to X$ be a conic bundle on $X$, whose discriminant defines a smooth divisor $Y\subset X$ with multiplicity $\tau$ (which, if $Y$ is not irreducible, will consist of a positive integer $\tau_i$ for each component $Y_i$ of $Y$). Let $\beta\in H^2(X,\mu_2)$ be the Brauer class of the restriction of $C$ to $X-Y$, which is a $P^1$ fibration which is etale locally trivial (all cohomologies are with respect to etale topology unless otherwise indicated). Suppose that over each geometric point of $Y$, the fiber of $C$ consists of two distinct projective lines meeting at a point. Therefore, the relative Hilbert scheme of lines in $C|Y \to Y$ is a two sheeted finite etale cover $\tilde{Y}\to Y$. Let $\alpha \in H^1(Y,\mu_2)$ be its covering class (`Kummer class'). By smoothness, we have a Gysin homomorphism $H^2(X-Y, \mu_2)\to H^1(Y,\mu_2)$. We prove here the following \begin{theorem} Under the Gysin homomorphism $H^2(X-Y, \mu_2)\to H^1(Y,\mu_2)$, the Brauer class $\beta$ of the $P^1$ fibration on $X-Y$ maps to $\tau\alpha$, where $\tau$ is the vanishing multiplicity of the discriminant and $\alpha$ is the cohomology class of the two sheeted finite etale cover $\tilde{Y}\to Y$. \end{theorem} A topological version of this result was proved in [Ni] for topological conic bundles on manifolds, which is equivalent for complex algebraic conic bundles to the above result. This is because there is a natural isomorphism between etale cohomology with finite constant coefficients (in this case, coefficients $Z\!\!\!Z/(2)$) and the corresponding singular cohomology, which commutes with the two Gysins. The theorem is proved below in two steps. In section 2, we reduce it to proving a purely algebraic lemma (Lemma 2.1 below, which we call as the `main lemma') over a discrete valuation ring. In section 3, we prove the main lemma. \refstepcounter{theorem}\paragraph{Remarks \thetheorem} (1) The main lemma is more transparent than its topological counterpart in [Ni]. (2) After proving the main lemma, enquiries with algebraist colleagues revealed that a more general lemma has already been proved by Colliot-Th\'el\`ene and Ojanguren (see proposition 1.3 in [C-O]). Our proof is more geometric but less general. (3) When the total space of $C$ is nonsingular, it is known that (see [H-N] Proposition 1.4 or [Ne] Theorem 2) $\beta$ is zero only if $\alpha$ is zero. This follows from theorem 1.1 by taking $\tau=1$, though theorem 1.1 makes a stronger statement even in this case. \section{Reduction to Main Lemma} For basic definitions about conic bundles, see for example Newstead [Ne]. It is clearly enough to prove the theorem for each connected component of $X$, so we will assume $X$ to be connected. If $Y=\cup _iY_i$ are the connected components of $Y$, then by replacing $(X,Y)$ by $(X-\cup_{j\ne i}Y_j,Y_i)$ we are reduced to the case where $Y$ also is connected. Hence we can assume that both $X$ and $Y$ are irreducible. Let the conic bundle $C\to X$ be defined via a rank 3 vector bundle $E$ on $X$, together with quadratic form $q$ on $E$ with values in some line bundle $L$ on $X$. The quadratic form $q$ is of rank 3 on $X-Y$ because we have a non-degenerate conic over $X-Y$, and $q$ has rank 2 on $Y$ as we have pairs of distinct lines over geometric points of $Y$. Let $U = Spec R$ be an affine open subscheme of $X$ which intersects $Y$, such that $E$ and $L$ are trivial on $U$, and $Y\cap U$ is defined by a principal ideal $(\pi)$ in $R$. By injectivity of $H^1(Y, \mu_2) \to H^1(Y\cap U, \mu_2)$, it is enough to prove the theorem for $(U,Y\cap U)$ in place of $(X,Y)$. Hence we can assume that the conic bundle is defined by an explicit quadratic form $x^2-ay^2-bz^2$ on the ring $R$, where $a$ is a unit in $R$, while $b\in (\pi)$ vanishes over $Y$. Let $\eta$ be the generic point of $Y$, and let $A$ be the discrete valuation ring $\O_{X,\eta}$. Let $k$ be the function field of $Y$. The morphism $Spec(k) \to Y$ induces an injective homomorphism $H^1(Y,\mu_2)\to H^1(k,\mu_2)$. Hence the theorem follows from the following lemma. \begin{lemma}{\rm (Main Lemma) } Let $F$ be a field of characteristic $\ne 2$. Let $A$ be a discrete valuation ring which is the local ring at the generic point of a smooth divisor in a smooth $F$-variety. Let $K$ be the quotient field of $A$, let $k$ be the residue field, and let $\nu :A-\{ 0\} \to Z\!\!\!Z$ be the discrete valuation. Let $x^2-ay^2-bz^2$ be a quadratic form on $A$, with $a$ a unit in $A$, and $b\ne 0$. Let $(a,b)\in H^2(K,\mu_2)$ be the Brauer class (Hilbert symbol) of the quadratic form on $K$. Let $\ov{a}\in k-\{ 0\}$ be the residue class of $a$, and let $\chi(\ov{a})\in H^1(k,\mu_2)$ be the class of the two sheeted etale cover $k(\ov{a}^{1/2})/k$ (Kummer character). Then under the Gysin homomorphism $H^2(K,\mu_2)\to H^1(k,\mu_2)$, we have (in additive notation) $$(a,b) \mapsto \nu(b) \chi(\ov{a})$$ \end{lemma} \section{Proof of the Main Lemma} In the course of the proof below, we use the following elementary facts which can be found for example in the textbook of Milne [M]. If $S$ is a field or more generally a Henselian local ring, then $Br(S)\to H^2(B, {\underline{GL}}_1)$ is an isomorphism, and provided the characteristic of the residue field is $\ne 2$, the homomorphism $H^2(S,\mu_2)\to H^2(B,{\underline{GL}}_1)$ is injective. Moreover the etale cohomology of $S$ with coefficients in a smooth representable sheaf (for example $\mu_2$ or ${\underline{PGL}}_2$) is isomorphic by restriction to the corresponding etale cohomology of the residue field of $S$. We need two more fact, which are contained in the following two remarks. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem3.1} Let $A$ be a henselian local ring and $B/A$ be a 2-sheeted finite etale cover. If the image of $\gamma\in H^2(A,\mu_2)$ is zero under the composite $$H^2(A,\mu_2) \to H^2(B,\mu_2) \to H^2(L,\mu_2)$$ where $L$ is the quotient field of $B$, then $\gamma$ lies in the image of the canonical (connecting) set map $H^1(A,{\underline{PGL}}_2) \to H^2(A,\mu_2)$ for the following reason. Any generic section of a Brauer-Severi variety on $Spec(B)$ extends to a global section by the valuative criterion of properness and so the map $H^2(B,\mu_2) \to H^2(L,\mu_2)$ is injective, hence $\gamma$ maps to zero under $H^2(A,\mu_2) \to H^2(B,\mu_2)$. Hence $\gamma$ is represented by an element (factor set) of the group cohomology $H^2(Gal(B/A), \mu_2)$. As $Gal(B/A)$ is of order 2, the factor set $\gamma$ defines an Azumaya algebra of rank 2, showing $\gamma$ comes from $H^1(R,{\underline{PGL}}_2)$. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem3.2} Let $K$ be a field of characteristic $\ne 2$, let $a,b\in K-\{ 0\}$ such that $a$ is not a square in $K$, and let $L=K[t]/(t^2-a)$. As the conic in $P^2_K$ defined by $x^2-ay^2-bz^2=0$ has an $L$-rational point, its Brauer class $(a,b)$ is an element of the group cohomology set $H^1(Gal(L/K), PGL_2(L))$. Let $Gal(L/K)=\{ 1,\sigma\}$. A 1-cocycle for $Gal(L/K)$ with coefficients $PGL_2(L)$ therefore consists of an element $g\in PGL_2(L)$ such that $g\sigma(g)=I$ in $PGL_2$. (The corresponding `crossed homomorphism' $Gal(L/K)\to PGL_2(L)$ is defined by $\sigma\mapsto g$.) If $g'\in GL_2(L)$ is an arbitrary lift of $g$, then there must exist some $c\ne 0$ in $L$ such that $g'\sigma(g')=cI$ in $GL_2(L)$ (which implies $c\in K$). In particular it can be seen by using stereographic projection from $(\sqrt{a}, 1,0)\in P^2_K(L)$ that $(a,b)$ is represented in $H^1(Gal(L/K),PGL_2(L))$ by the 1-cocycle defined by $$h=\pmatrix{ & b \cr 1 & \cr }$$ We now prove the main lemma. \paragraph{Proof} If $b$ is a unit in $A$ (that is, $\nu(b)=0$), then the $P^1$ bundle given by $x^2-ay^2-bz^2$ is defined over all of $Spec(A)$, and so by exactness of the Gysin sequence $$H^2(A,\mu_2)\to H^2(K,\mu_2) \to H^1(k,\mu_2)$$ the image of $(a,b)\in H^2(K,\mu_2)$ in $H^1(k,\mu_2)$ is also zero. So we now assume $\ov{b}=0$ (which is anyway the case which interests us). If $\nu(b) = 2n$ is even then as $\chi(\ov{a})$ is 2-torsion we have $\nu(b)\chi(\ov{a}) =0$. Making the change of variable $z'= \pi^nz$ over $K$, we see that $(a,b)=(a,b')\in H^2(K,\mu_2)$ where $b'=b/\pi^{2n}$ . As $\nu(b')=0$, the argument above now completes the proof when $\nu(b)$ is even. When $\nu(b)=2n+1$ is odd, the same change of variables enables us to reduce to the case where $\nu(b)=1$. Hence we assume from now on that $\nu(b)=1$. Note that passing to the completion of $A$ does not affect the residue field and gives a commutative square of the Gysins. Hence it is enough to show the conclusion of the lemma assuming $A$ is complete. If $\ov{a}\in k$ is a square then (by the Henselian property of a complete local ring) $a$ will be a square in $A$ and hence in $K$, and conversely (this uses the discrete valuation) if $a\in A$ is a square in $K$ then it is a square in $A$ and hence $\ov{a}$ is a square in $k$. The conic defined in $P^2_K$ by $x^2-ay^2-bz^2$ will have a $K$ rational point $(a_1, 1, 0)$ whenever $a_1$ is a squareroot of $a$ in $K$, showing its Brauer class over $K$ is zero. Hence if $\chi(\ov{a})=0$ then $(a,b)=0$, and so the lemma holds. Hence we now assume that $\ov{a}$ is not a square in $k$, and therefore $a$ is not a square in $K$ or $A$. (As moreover $\nu(b)=1$, it follows by remark 1.3 (3) that $(a,b)\ne 0$, though this will also follow from the argument below.) Let $B=A[t]/(t-a^2)$ which is a 2-sheeted finite etale cover of $A$, and let $L$ and $l$ be respectively the quotient field and the residue field of $B$. Consider the commutative diagram where the horizantal maps are the two Gysins. $$\matrix{ H^2(K,\mu_2) & \to & H^1(k,\mu_2) \cr \downarrow & & \downarrow \cr H^2(L,\mu_2) & \to & H^1(l,\mu_2) \cr }$$ The kernel of $H^1(k,\mu_2)\to H^1(l,\mu_2)$ is of order two, generated by $\chi(\ov{a})$. As $a$ has a squareroot in $L$, $(a,b)\in Br(K)$ maps to zero in $Br(L)$. Hence to show that $(a,b)\mapsto \chi(\ov{a})$, it is enough to show that the image of $(a,b)$ in $H^1(k,\mu_2)$ is nonzero. By exactness of the Gysin sequence, this will follow if we show that $(a,b)$ does not lie in the image of $H^2(A,\mu_2)\to H^2(K,\mu_2)$. Suppose $(a,b)$ was the image of $\gamma \in H^2(A,\mu_2)$. As $(a,b)$ restricts to zero over $L$, it follows from remark \ref{rem3.1} that $\gamma$ lies in the image of some $\gamma' \in H^1(A,{\underline{PGL}}_2)$ under $H^1(A,{\underline{PGL}}_2)\to H^2(A,\mu_2)$. As $\gamma'$ splits over $B$, it can be regarded as an element of the group cohomology set $H^1(Gal(B/A),PGL_2(B))$, and hence as in remark 3.2, its 1-cocycle is represented by an element $g\in GL_2(B)$ such that $$g\sigma(g) = eI$$ where $Gal(B/A)=\{ 1,\sigma\}$ and $e$ is a unit in $A$. On the other hand, the group cohomology class of $(a,b)\in H^1(Gal(L/K),PGL_2(L))$ can be represented by the matrix $h$ given by remark 3.2. If the cohomology class of $(g)$ were to map to the cohomology class of $(h)$ in $H^1(Gal(L/K),PGL_2(L))$, then by definition of group cohomology there would exist elements $M\in GL_2(L)$ and $0\ne c\in L$ such that $$h = cMg\sigma(M^{-1})$$ Applying $\sigma$, this would give $$h=\sigma(h) = \sigma(c)\sigma(M)\sigma(g)M^{-1}$$ Multiplying the two equations and using $h^2=bI$ and $g\sigma(g)=eI$, we would get $$b/e = Norm(c)$$ But this is impossible as on one hand $e\in A$ is a unit while $b\in A$ has valuation $\nu(b)=1$ so $\nu(b/e)=1$, and on the other hand $\nu$ takes only even values on norms. Hence the image of $(a,b)$ in $H^1(k,\mu_2)$ is nonzero. This completes the proof of the main lemma and hence that of the theorem. \section*{References} [C-O] Colliot-Th\'el\`ene J. L. and Ojanguren M. : `Vari\'et\'es unirationnelles non rationnelles: au-del\`a de l'example d'Artin et Mumford', Inventionnes Math. 97 (1989) 141-158. [H-N] Hirschowitz, A. and Narasimhan, M. S. : `Fibr\'es de 't Hooft speciaux et applications', in {\sl Enumerative geometry and classical algebraic geometry}, Progress in Mathematics 24, Birkhauser, 1982. [M] Milne, J. S. : {\sl \'Etale Cohomology}, Princeton Univ. Press, 1980. [Ne] Newstead, P. E. : `Comparision theorems for conic bundles', Math. Proc. Cambridge Philos. Soc. 90 (1981) 21-31. [Ni] Nitsure, N. : `Topology of conic bundles', J. London Math. Soc. (2) 35 (1987) 18-28. \medskip School of Mathematics, Tata Institute of Fundamental Research, Homo Bhabha Road, Mumbai 400 005. e-mail: [email protected] \end{document} 

9602

alg-geom/9602021

en

https://arxiv.org/abs/alg-geom/9602021

alg-geom/9602021

Chikashi Miyazaki

Chikashi Miyazaki and Wolfgang Vogel

Bounds on cohomology and Castelnuovo-Mumford regularity

LaTeX, 18 pages

The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was introduced by Mumford by generalizing ideas of Castelnuovo. The interest in this concept stems partly from the fact that X is m-regular if and only if for every p \geq 0 the minimal generators of the p-th syzygy module of the defining ideal I of X occur in degree \leq m + p. There are some bounds in the case that X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove our theorems, we need to apply a spectral sequence. We conclude by describing two sharp examples and open problems.

alg-geom

\section{Introduction} Let $X\subseteq{\bf P}_K^N$ be a projective scheme over an algebraically closed field $K$ . We denote by ${\cal I}_X$ the ideal sheaf of $X$. Then $X$ is said to be $m$-regular if $H^i({\bf P}_K^N,{\cal I}_X(m-i))=0$ for all $i\geq 1$ (cf. \cite{Mum}). The Castelnuovo-Mumford regularity $\mbox{reg}(X)$ of $X\subseteq {\bf P}_K^N$, firstly introduced by Mumford by generalizing ideas of Castelnuovo, is the least such integer $m$. The interest in this concept stems partly from the well-known fact that $X$ is $m$-regular if and only if for every $p\geq 0$ the minimal generators of the $p$-th syzygy module of the defining ideal $I$ of $X\subseteq{\bf P}_K^N$ occur in $\mbox{degree}\leq m+p$ (see, e.g., \cite{EG}). There are good bounds in some cases if $X$ is assumed to be smooth (see \cite{BEL} and the references there). Our interest is to consider the case $X$ being locally Cohen-Macaulay and equidimensional. Under this assumption there is a non-negative integer $k$ such that $$(X_0,\cdots,X_N)^k\left[\rixrel{\ell\in{\bf Z}}{\oplus}H^i\left( {\bf P}_K^N,{\cal I}_X(\ell)\right)\right]=0 \mbox{\ for\ }1\leq i\leq\dim X,$$ where ${\bf P}_K^N=\mbox{Proj\ } K[X_0,\cdots,X_N]$. In this case $X\subseteq{\bf P}_K^N$ is called a $k$-Buchsbaum scheme. A refined version is a $(k,r)$-Buchsbaum scheme introduced in \cite{FV}, \cite{HMV}, \cite{HV}: Let $k$ and $r$ be integers with $k\geq 0$ and $1\leq r\leq \dim X$. Then we call $X\subseteq{\bf P}_K^N$ a $(k,r)$-Buchsbaum scheme if, for all $j=0,\cdots,r-1$, $X\cap V$ is a $k$-Buchsbaum scheme for every $(N-j)$-dimensional complete intersection $V$ in ${\bf P}_K^N$ with $\dim(X\cap V)=\dim(X)-j$, (see (2.3) and see also (2.4) for an equivalent definition in case $k=1$.) In recent years upper bounds on the Castelnuovo-Mumford regularity of such a variety $X\subseteq {\bf P}_K^N$ have been given by several authors in terms of $\dim(X)$, $\deg(X)$, $k$ and $r$, (see, e.g., \cite{HMV}, \cite{HM}, \cite{HV}, \cite{NS1}, \cite{NS2}). These bounds are stated as follows: $$\mbox{reg}(X)\leq\left\lceil\frac{\deg(X)-1}{\mbox{codim\ }(X)}\right\rceil+C(k,r,d),$$ where $d=\dim X$, $C(k,r,d)$ is a constant depending on $k$, $r$ and $d$, and $\left\lceil n\right\rceil$ is the smallest integer $\ell\geq n$ for a rational number $n$. In case $X$ is arithmetically Cohen-Macaulay, that is, $k=0$, it is well-known that $C(k,r,d)\leq 1$, (see, e.g., \cite{SV2}, \cite{SV3}). We assume $k\geq 1$. In case $r=1$ it was shown that $C(k,1,d)\leq\left(\begin{array}{c} d+1\\2\end{array}\right)k-d+1$ in \cite{HV}. In \cite{HM} (in a slightly weaker form) and \cite{NS1} it was improved to $C(k,1,d)\leq(2d-1)k-d+1$. Further, a better bound $C(k,1,d)\leq dk$ was obtained in \cite{NS2}. The general case $r\geq 1$ was firstly studied in \cite{HMV}, which was improved in \cite{HV} by showing that $C(k,r,d)\leq(r-1)k+\left(\begin{array}{c} d+2-r\\2\end{array}\right)k-d+1$. The purpose of this paper is to give bounds on $\mbox{reg\ }(X)$ in terms of $\dim (X)$, $\deg (X)$, $k$ and $r$, which improve some of the previous results. In general case we show that $$C(k,r,d)\leq dk-r+1$$ in (3.2). Moreover, in case $k=1$, we show that $$C(1,r,d)\leq\left\lceil\frac{d}{r}\right\rceil.$$ in (3.3) Our methods here is to use a spectral sequence theory for graded modules developed in \cite{M1}, \cite{M2}, \cite{M3} in order to get bounds on the local cohomology and the Castelnuovo-Mumford regularity. Let $R$ be the coordinate ring of $X\subseteq{\bf P}_K^N$ with the homogeneous maximal ideal $\sf m$. Then we define $a_i(R)=\sup\{n;H_{\sf m}^i(R)_n\neq 0\}$ for $i=0,\cdots,d+1$, and $\mbox{reg\ }(R)=\max\{a_i(R)+i;i=0,\cdots,d+1\} $. It is easy to see that $\mbox{reg\ }(X)=\mbox{reg\ }(R)+1$. Among $a_i(R)$'s, $a_{d+1}(R)$, which is called an $a$-invariant of $R$ (cf. \cite{GW}), plays an important role to characterize the graded ring $R$. The results (2.7) and (2.9) on bounds on the cohomology in terms of $a$-invariants are applied to (3.2) and (3.3). In section 2, we consider bounds on the cohomology for graded modules. First we explain a spectral sequence theory in order to get cohomological bounds. Theorem (2.5) is the technical key results of this paper. This theorem gives several important results stated in (2.7), (2.8), (2.9), (2.10), (2.11). In section 3, we describe bounds on the Castelnuovo-Mumford regularity for locally Cohen-Macaulay varieties $X\subseteq{\bf P}_K^N$. Theorem (3.2) and Theorem (3.3) are main results of this section. In section 4, we construct sharp examples of (2.8) and (2.11), and conclude by describing some open problems. \section{Bounds on local cohomology} Throughout this paper let $R=K[R_1]$ be a Noetherian graded $K$-algebra, where $K$ is an infinite field. We denote by $\sf m$ the maximal homogeneous ideal of $R$. Let $M$ be a graded $R$-module. We write $[M]_n$ for the $n$-th graded piece of $M$ and $M(p)$ for the graded module with $[M(p)]_n=[M]_{p+n}$. We set: $$\sup(M)=\sup\{n;[M]_n\neq 0\}.$$ If $M=0$, we set $\sup(M)=-\infty$. Now we assume that $M$ is a finitely generated graded $R$-module with $\dim M=d>0$. Then we set: $$a_i(M)=\sup(H^i_{\sf m}(M)),\qquad i=0,\cdots,d.$$ Also $a_d(M)$ is called the $a$-invariant of $M$ and written as $a(M)$. The Castelnuovo-Mumford regularity of $M$ is defined as follows: $$\mbox{reg\ }(M)=\max\{a_i(M)+i; i=0,\cdots,d\}.$$ In order to state our results we use the following notation: $$\mbox{reg}_n(M)=\max\{a_i(M)+i;i=n,\cdots,d\},$$ where $0\leq n\leq d$. Then we see $\mbox{reg\ }(M)=\mbox{reg}_0(M)=\mbox{reg}_{\mbox{depth\ }(M)} (M)$. Let $X$ be a closed subscheme of ${\bf P}_K^N$. Let $R$ be the coordinate ring of $X$. Then the Castelnuovo-Mumford regularity of $X$ is defined as follows: $$\mbox{reg\ }(X)=\mbox{reg\ }(R)+1.$$ From now on we assume that the graded module $M$ is a generalized Cohen-Macaulay module. In other words $M$ is locally Cohen-Macaulay, that is, $M_{\sf p}$ is Cohen-Macaulay for all non-maximal homogeneous prime ideal $\sf p$ of $R$, and $M$ is equi-dimensional, that is, $\dim\left(R/{\sf p}\right)=d$ for all minimal associated prime ideals $\sf p$ of $M$. \vspace{5mm} \noindent{\bf Definition 2.1.} {\em Let $k$ be a non-negative integer. The graded $R$-module $M$ is called a $k$-Buchsbaum module if ${\sf m}^kH_{\sf m}^i(M)=0$ for all $i<d$.} \vspace{5mm} \noindent{\bf Definition 2.2.} (\cite{FV}, \cite{HMV}, \cite{HV}). {\em Let $k$ and $r$ be integers with $k\geq 0$ and $1\leq r\leq d$. $M$ is called a $(k,r)$-Buchsbaum module if for every s.o.p. $x_1,\cdots,x_d$ of $M$ we have $${\sf m}^kH_{\sf m}^i(M/(x_1,\cdots,x_j)M)=0,$$ for all non-negative integers $i$, $j$ with $j\leq r-1$ and $i+j<d$.} \vspace{5mm} \noindent{\bf Remark.} $(0,d)$-, $(1,d)$-, and $(k,1)$-Buchsbaum modules are the Cohen-Macaulay, Buchsbaum, and $k$-Buchsbaum modules respectively. The $(1,r)$-Buchsbaum module, introduced as $r$-Buchsbaum module in \cite{M1}, was studied also in \cite{M2}, \cite{M3}. \vspace{5mm} \noindent{\bf Remark.} For a fixed integer $r$, $1\leq r\leq d$, every generalized Cohen-Macaulay graded module $M$ has $(k,r)$-Buchsbaum property for $k$ large enough, see \cite{HV}, (2.3). \vspace{5mm} \noindent{\bf Definition 2.3.} (\cite{M3}). {\em Let $r$ and $n$ be integers with $1\leq r\leq n\leq d$. Let $x_1,\cdots,x_n$ be a part of s.o.p. for $M$. We say $x_1,\cdots,x_n$ is $r$-standard, if for any choice $x_{i_1},\cdots,x_{i_\ell}$ $(\ell\leq r-1)$ $$(x_1,\cdots,x_n)H_{\sf m}^j(M/(x_{i_1},\cdots,x_{i_\ell})M)=0$$ for $j+\ell<d$.} \vspace{5mm} \noindent{\bf Remark.} Standard s.o.p. was introduced in several papers, see, e.g., \cite{T}. The $r$-standardness is its generalization. A similar generalization was introduced in \cite{NS1}, (4.2). \vspace{5mm} First we note a characterization of $(1,r)$-Buchsbaum modules in terms of s.o.p. of degree one which improves, e.g., [14], (2.5) of \cite{GM}. \vspace{5mm} \renewcommand{\labelenumi}{\theenumi)} \renewcommand{\theenumi}{\arabic{enumi}} \noindent{\bf Proposition 2.4.}{\em Let $R=K[R_1]$ be a graded ring over an infinite field $K$. Let $\sf m$ be the homogeneous maximal ideal of $R$ generated by $X_0,\cdots,X_N$. Let $M$ be a generalized Cohen-Macaulay graded $R$-module with $\dim(M)=d$. We assume that for all integers $0\leq i_1<\cdots<i_d\leq N$, $X_{i_1},\cdots,X_{i_d}$ is a s.o.p. for $M$. Let $r$ be an integer with $1\leq r\leq d$. Then the following are equivalent: \begin{enumerate} \item $M$ is a $(1,r)$-Buchsbaum module. \item For all integers $0\leq i_1<\cdots<i_d\leq N$, $X_{i_1},\cdots,X_{i_d}$ is an $r$-standard s.o.p. for $M$. \end{enumerate}} \vspace{5mm} \noindent{\bf Proof.} It is clear that 1) implies 2). Now we will show that 2) implies 1) by induction on $r$. It is clear in case $r=1$. We assume that $r>1$. To prove that $M$ is a $(1,r)$-Buchsbaum module we have only to show that, for any part of a s.o.p. $y_1,\cdots,y_r$ for $M$, $y_1,\cdots,y_r$ is $r$-standard, which is equivalent to saying that $\varphi_{y_1\wedge\cdots\wedge y_r}^q(M)$ is a zero map for all $r-1\leq q\leq d-1$ by \cite{M3}, (2.3). (Also see the notation in \cite{M3}.) We set $y_j=\sum_{i=0}^N a_{ji}X_i$ for some $a_{ji}\in R$. Since $X_{i_1},\cdots, X_{i_r}$ is $r$-standard for all $0\leq i_1<\cdots<i_r\leq N$, we see $\varphi_{X_{i_1}\wedge\cdots\wedge X_{i_r}}^q(M)$ is a zero map by \cite{M3}, (2.3). By virtue of \cite{M3}, (3.4), we have that $\varphi_{y_1\wedge\cdots\wedge y_r}^q(M)$ is a zero map. Thus the assertion is proved. \vspace{5mm} The following theorem is the technical key result of this paper. The spectral sequence theory as developed in \cite{M1}, \cite{M2}, \cite{M3} plays an important role to prove it. \vspace{5mm} \noindent{\bf Theorem 2.5.}{\em Let $M$ be a generalized Cohen-Macaulay graded $R$-module with $\dim (M)=d>0$. Let $i$ and $n$ be integers with $0\leq i\leq d-1$ and $1\leq n\leq d$. Let $x_1,\cdots,x_n$ be a part of s.o.p. for $M$ with $\deg(x_j)=e_j\geq 1$, $j=1,\cdots,n$. We set $c_j=\max\{e_{i_1}+\cdots+e_{i_j}; 1\leq i_1<\cdots<i_j\leq n\}$ for $j=1,\cdots,n$. Then we have: \begin{enumerate} \item If $n+i\leq d$, then $$\begin{array}{lrr} \lefteqn{\sup(H_{\sf m}^i(M)/(x_1,\cdots,x_n)H_{\sf m}^i(M))}\\[2mm] & & \leq\max\{a_{i+j}(M)+c_{j+1},a_i(M/(x_1,\cdots,x_n)M);j=1,\cdots,n-1\} {}. \end{array}$$ \item If $n+i>d$, then $$\begin{array}{lrr} \lefteqn{\sup(H_{\sf m}^i(M)/(x_1,\cdots,x_n)H_{\sf m}^i(M))}\\[2mm] & & \leq\max\{a_{i+j}(M)+c_{j+1};j=1,\cdots,d-i\}. \end{array}$$ \end{enumerate} Furthermore let $r$ be an integer with $1\leq r\leq n$. Assume that the sequence $x_1,\cdots,x_n$ is $r$-standard. Then we have: \begin{enumerate} \setcounter{enumi}{2} \item If $n+i\leq d$, then $$a_i(M)\leq\max\{a_{i+j}(M)+c_{j+1},a_i(M/(x_1,\cdots,x_n)M); j=r,\cdots,n-1\}.$$ \item If $n+i>d$, then $$a_i(M)\leq\max\{a_{i+j}(M)+c_{j+1},a(M)+c_{d-i+1};j=r,\cdots,d-i-1\}.$$ \end{enumerate}} \vspace{5mm} In order to prove (2.5) we need to apply the following spectral sequence corresponding to the graded $R$-module $M$: Let $I^\bullet$ be the minimal injective resolution of $M$ in the category of the graded $R$-modules. Let $K_\bullet$ be the Koszul complex $K_\bullet((x_1,\cdots,x_n);R)$ for a part of a s.o.p. $x_1,\cdots,x_n$ for $M$. Then we consider the double complex $B^{\bullet\bullet}=\mbox{Hom}_R (K_\bullet,I^\bullet)$. The filtration $F_t(B^{\bullet\bullet})=\sum_{p\ge q t}B^{p,q}$ gives a spectral sequence $\{E_u^{p,q}\}$ (see, e.g., \cite{God}). Then we have the following isomorphisms: $$E_1^{p,q}\cong K^p((x_1,\cdots,x_n);H_{\sf m}^q(M))$$ and $$H^{p+q}(B^{\bullet\bullet})\cong\left\{\begin{array}{ll} H^{p+q}((x_1,\cdots,x_n);M), & p+q<n\\[2mm] H_{\sf m}^{p+q-n}(M/(x_1,\cdots,x_n)M)(e_1+\cdots+e_n), & p+q\geq n\end{array}\right.$$ for all $p$, $q$, see, e.g., \cite{M3}. Note that the spectral sequence $\{E_u^{p,q}\}$ converges to $H^{p+q}(B^{\bullet\bullet})$. Then we have the following lemma by using the above notation. \vspace{5mm} \noindent{\bf Lemma 2.6.}{\em Let $r$ be an integer with $1\leq r\leq n$. Then the following conditions are equivalent: \begin{enumerate} \item $x_1,\cdots,x_n$ is $r$-standard. \item $d_s^{p,q}:E_s^{p,q}\rightarrow E_s^{p+s,q-s+1}$ is a zero map for all $p$, $q$, $s$ with $q\neq d$, $1\leq s\leq r$. \end{enumerate}} \vspace{5mm} \noindent{\bf Proof.} The lemma follows by induction on $r$ using (3.3) of \cite{M3}. \vspace{5mm} Now let us begin to prove (2.5). \vspace{5mm} \noindent{\bf Proof of (2.5).} Let us consider the spectral sequence $\{E_r^{p,q}\}$ corresponding to the graded $R$-module $M$ discussed above for a part of s.o.p. $x_1,\cdots,x_n$ for $M$. We see that $$E_2^{n,i}\cong H_{\sf m}^i(M)/(x_1,\cdots,x_n)H_{\sf m}^i(M)(c_n)$$ and $$H^{n+i}\cong\left\{\begin{array}{ll} H_{\sf m}^i(M/(x_1,\cdots,x_n)M)(c_n), & n\leq d-i\\ 0, & n>d-i.\end{array}\right.$$ Since $E_1^{p,q}=0$ for $p>n$, we see $E_{\infty}^{p,q}=0$ for $p>n$. Thus we see that the graded $R$-homomorphism $E_\infty^{n,i}\rightarrow H^{n+i}$ is injective and that $E_\infty^{n,i}$ is a quotient of the graded $R$-module $E_2^{n,i}$. First let us prove 1). Take an integer $\ell$ satisfying $\ell>a_{i+j}(M)+c_{j+1}$ for all $j=1,\cdots,n-1$ and $\ell>a_i(M/(x_1,\cdots,x_n)M)$. We want to show $\left[H_{\sf m}^i(M)/ (x_1,\cdots,x_n)\right.$ $\left.H_{\sf m}^i(M)\right]_\ell=0$. Since ${[H^{n+i}]}_{\ell-c_n}={[H_{\sf m}^i (M/(x_1,\cdots,x_n)M)]}_{\ell} = 0$, we have ${[E_\infty^{n,i}]}_{\ell-c_n}=0$. On the other hand we have isomorphisms \begin{eqnarray*} {\left[E_1^{n-j-1,i+j}\right]}_{\ell-c_n} & \cong & {\left[K^{n-j-1}((x_1,\cdots,x_n); H_{\sf m}^{i+j}(M))\right]}_{\ell-c_n}\\ & \cong & {\left[\rixrel{1\leq i_1<\cdots<i_{n-j-1}\leq n}{\oplus} H_{\sf m}^{i+j}(M) (e_{i_1}+\cdots+e_{i_{n-j-1}})\right]}_{\ell-c_n}\\ & \cong & \rixrel{1\leq k_1<\cdots<k_{j+1}\leq n}{\oplus}{\left[H_{\sf m}^{i+j}(M)\right]} _{\ell-(e_{k_1}+\cdots+e_{k_{j+1}})}. \end{eqnarray*} Thus we have ${[E_1^{n-j-1,i+j}]}_{\ell-c_n}=0$ for all $j=1,\cdots,n-1$, because $\ell-(e_{k_1}+\cdots+ e_{k_{j+1}})\geq \ell-c_{j+1}>a_{i+j}(M)$. This leads to the equality ${[E_2^{n,i}]}_{\ell-c_n} ={[E_\infty^{n,i}]}_{\ell-c_n}$. Hence we have ${[E_2^{n,i}]}_{\ell-c_n}=0$. Thus the assertion is proved. Next let us prove 2). Take an integer $\ell$ satisfying $\ell>a_{i+j}(M)+c_{j+1}$ for all $j=1,\cdots,d-i$. Since we have $${[E_1^{n-j-1,i+j}]}_{\ell-c_n}\cong\rixrel{1\leq k_1<\cdots<k_{j+1}\leq n}{\oplus} {[H_{\sf m}^{i+j}(M)]}_{\ell-(e_{k_1}+\cdots+e_{k_{j+1}})}$$ as shown in the proof of 1), we get ${[E_1^{n-j-1,i+j}]}_{\ell-c_n}=0$ for all $j=1,\cdots,d-i$. Thus we have ${[E_2^{n,i}]}_{\ell-c_n}={[E_\infty^{n,i}]}_{\ell-c_n}=0$. Hence the assertion is proved. To prove 3) we take an integer $\ell$ satisfying $\ell>a_{i+j}(M)+c_{j+1}$ for all $j=r,\cdots,n-1$ and $\ell>a_i(M/(x_1,\cdots,x_n)M)$. Since ${[H^{n+i}]}_{\ell-c_n}={[H_{\sf m}^i(M/(x_1,\cdots,x_n) M)]}_\ell=0$, we have ${[E_\infty^{n,i}]}_{\ell-c_n}=0$. Also we see $E_2^{n,i}\cong H_{\sf m}^i (M)(c_n)$, because the sequence $x_1,\cdots,x_n$ is 1-standard. Since $${[E_1^{n-j-1,i+j}]}_{\ell-c_n}\cong \rixrel{1\leq k_1<\cdots<k_{j+1}\leq n}{\oplus} {[H_{\sf m}^{i+j}(M)]}_{\ell-(e_{k_1}+\cdots+e_{k_{j+1}})}.$$ as shown in the proof of 1), we have ${[E_1^{n-j-1,i+j}]}_{\ell-c_n}=0$ for all $j=r,\cdots,n-1$. By using Lemma (2.6), we therefore have ${[E_2^{n,i}]}_{\ell-c_n}={[E_\infty^{n,i}]}_{\ell-c_n}$. Hence we have ${[H_{\sf m}^i(M)]}_\ell=0$. Thus the assertion is proved. Finally let us prove 4). Take an integer $\ell$ satisfying $\ell>a_{i+j}(M)+c_{j+1}$ for all $j=r,\cdots,d-i-1$ and $\ell>a(M)+c_{d-i-1}$. Similarly we have ${[E_\infty^{n,i}]}_{\ell-c_n}=0$, $E_2^{n,i}\cong H_{\sf m}^i(M)(c_n)$, ${[E_1^{n-d+i-1,d}]}_{\ell-c_n}=0$ and ${[E_1^{n-j-1,i+j}]}_{\ell-c_n}=0$ for all $j=r,\cdots,d-i$. By using (2.6) we therefore have ${[E_2^{n,i}]}_{\ell-c_n}={[E_\infty^{n,i}]}_{\ell-c_n}$. Hence we have ${[H_{\sf m}^i(M)]}_\ell =0$. Thus the assertion is proved. This completes the proof of (2.5). \vspace{5mm} The first application of (2.5) is the following theorem which gives bounds on local cohomology. \vspace{5mm} \noindent{\bf Theorem 2.7.}{\em Let $M$ be a generalized Cohen-Macaulay graded $R$-module with $\dim(M)=d>0$. Let $k_1,\cdots,k_{d-1}$ be non-negative integers satisfying that ${\sf m}^{k_j}H_{\sf m}^j(M)=0$ for $j=1,\cdots,d-1$. Let $r$ be an integer with $1\leq r\leq d$. Let $x_1,\cdots,x_d$ be a s.o.p. for $M$ with $\deg(x_j)=1$ for $j=1,\cdots,d$. Assume that the sequence $x_1^{\mu_1},\cdots,x_r^{\mu_r}$ is $r$-standard for some positive integers $\mu_1,\cdots,\mu_r$. Then we have \begin{enumerate} \item $a_i(M)\leq a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r-i)+\sum_{\alpha=i}^{d-r-1}k_\alpha+ \sum_{\alpha=1}^r\mu_\alpha-r$ for $i=1,\cdots,d-r-1$. \item $a_i(M)\leq a_i(M/(x_1,\cdots,x_{d-i})M)+\sum_{\alpha=1}^{d-i}\mu_\alpha-(d-i)$ for $i=d-r,\cdots,d-1$. \item $\mbox{\em reg}_i(M)\leq a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r)+\sum_{\alpha=i}^{d-r-1}k_\alpha +\sum_{\alpha=1}^r\mu_\alpha-r$ for $i=1,\cdots,d-1$. \end{enumerate}} \vspace{5mm} \noindent{\bf Proof.} First we will prove 2). By virtue of (2.5.3) we have $$a_i(M)\leq a_i(M/(x_1^{\mu_1},\cdots,x_{d-i}^{\mu_{d-i}})M)$$ for $i=d-r,\cdots,d-1$, because $x_1^{\mu_1},\cdots,x_{d-i}^{\mu_{d-i}}$ is $(d-i)$-standard. Hence we have $$a_i(M)\leq a_i(M/(x_1,\cdots,x_{d-i})M)+\sum_{\alpha=1}^{d-i}\mu_\alpha-(d-i)$$ for $i=d-r,\cdots,d-1$, by using \cite{NS1}, (6.5). Next we will prove 1). Note that $$a_j(M)\leq\sup(H_{\sf m}^j(M)/{\sf m}H_{\sf m}^j(M))+k_j-1$$ for $j=1,\cdots,d-1$. (See, e.g., \cite{NS2}, (3.3)). By virtue of (2.5.2) we see \begin{eqnarray*} a_i(M) & \leq & \sup(H_{\sf m}^i(M)/(x_1,\cdots,x_d)H_{\sf m}^i(M))+k_i-1\\[2mm] & \leq & \max\{a_{i+j}(M)+j+k_i;j=1,\cdots,d-i\} \end{eqnarray*} for $i=1,\cdots,d-1$. Now we prove the following claim. \vspace{5mm} \noindent{\bf Claim:} Let $i$ be an integer with $1\leq i\leq d-1$. Then we have $$a_i(M)\leq\max\left\{a_j(M)+(j-i)+\sum_{\alpha=i}^{t-1}k_\alpha;j=t, \cdots ,d\right\}$$ for $t=i+1,\cdots,d$. \vspace{5mm} We will show our claim by induction on $t$. The case $t=i+1$ has already shown. In case $a_t(M)=-\infty$, in other words, $k_t=0$, the claim follows immediately from the hypothesis of induction. In case $a_t(M)>-\infty$, in other words, $k_t>0$, we have by the hypothesis of induction \begin{eqnarray*} a_i(M) & \leq & \max\left\{a_j(M)+(j-i)+\sum_{\alpha=i}^{t-1}k_\alpha; j=t,\cdots,d\right\}\\ & \leq & \max\left\{a_{t+s}(M)+s+k_s+(t-i)+\sum_{\alpha=i}^{t-1}k_\alpha,\right.\\ & & \quad \left. a_j(M)+(j-i)+\sum_{\alpha=i}^{t-1}k_\alpha;s=1,\cdots,d-t\mbox{\ and\ }j=t+1,\cdots,d\right\}\\ & = & \max\left\{a_j(M)+(j-i)+\sum_{\alpha=i}^tk_\alpha;j=t+1,\cdots,d \right\}. \end{eqnarray*} Thus the claim is proved. \vspace{5mm} In particular, we have $$a_i(M)\leq\max\left\{a_j(M)+(j-i)+\sum_{\alpha=i}^{d-r-1}k_\alpha;j =d-r, \cdots,d\right\}$$ for $i=1,\cdots,d-r-1$. By using 2), we have \begin{eqnarray*} a_i(M) & \leq & \max\left\{a_j(M/(x_1,\cdots,x_{d-j})M)+(j-i)+\sum_{\alpha =i}^{d-r-1}k_\alpha\right.\\ & & \qquad \left.+\sum_{\alpha=1}^{d-j}\mu_\alpha-(d-j);j=d-r,\cdots,d\right\}\\ & \leq & a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r-i)+\sum_{\alpha=i}^{d-r-1}k_d+\sum_{\alpha =1}^r \mu_\alpha-r \end{eqnarray*} for $i=1,\cdots,d-r-1$. Hence the assertion 1) is proved. The assertion 3) follows immediately from 1) and 2). This completes the proof of (2.7). \vspace{5mm} Our theorem (2.7) has an important corollary which is used in order to prove (3.2). \vspace{5mm} \noindent{\bf Corollary 2.8.}{\em Let $M$ be a generalized Cohen-Macaulay graded $R$-module with $\dim (M)=d>0$. Let $k$ and $r$ be integers with $k\geq 1$ and $1\leq r\leq d$. Assume that $M$ is a $(k,r)$-Buchsbaum module. For $i=1,\cdots,d$, $$\mbox{\em reg}_i(M)\leq a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r)+(d-i)k-r$$ for any part of s.o.p. $x_1,\cdots,x_r$ for $M$ with $\deg(x_j)=1$, $j=1,\cdots,r$.} \vspace{5mm} \noindent{\bf Proof.} Note that $x_1^k,\cdots,x_r^k$ is $r$-standard. By (2.7.3) we have \begin{eqnarray*} \mbox{reg}_i(M) & \leq & a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r)+\sum_{\alpha=i}^{d-r-1}k+ \sum_{\alpha=1}^r k-r\\ & = & a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r)+(d-i)k-r. \end{eqnarray*} \vspace{5mm} \noindent{\bf Remark.} The example (4.2) shows that the bound stated in (2.8) is sharp in case $k=r=1$. \vspace{5mm} Another application of (2.5) is the following result. \vspace{5mm} \noindent{\bf Theorem 2.9.}{\em Let $M$ be a generalized Cohen-Macaulay graded $R$-module with $\dim (M)=d>0$. Let $r$ be an integer with $1\leq r\leq d$. Let $x_1,\cdots,x_d$ be a s.o.p. for $M$ with $\deg(x_j)=1$ for $j=1,\cdots,d$. \begin{enumerate} \item Assume that $x_1,\cdots,x_{d-i}$ is $\min(r,d-i)$-standard for a fixed integer $i=0,\cdots,d-1$. Then we have $$a_i(M)\leq a_i(M/(x_1,\cdots,x_{d-i})M)+\left\lceil\frac{d-i}{r}\right\rceil-1.$$ \item If $x_1,\cdots,x_d$ is $r$-standard, then $$\mbox{\em reg}_i(M)\leq a_i(M/(x_1,\cdots,x_{d-i})M)+i+\left\lceil\frac{d-i}{r}\right\rceil-1$$ for $i=0,\cdots,d-1$. \end{enumerate}} \vspace{5mm} \noindent{\bf Proof.} By (2.5.3) we have \begin{eqnarray*} a_i(M) & \leq & \max\left\{a_j(M)+(j-i)+1, a_i(M/(x_1,\cdots,x_{d-i})M); \right.\\ & & \qquad\left. j=i+r,\cdots,d-1\right\}\\ & \leq & \max\left\{a_j(M)+(j-i)+2,a_i(M/(x_1,\cdots,x_{d-i})M), \right.\\ & & \qquad\left.a_{i+r}(M/(x_1,\cdots,x_{d-r-i})M)+r+1; j=i+2r,\cdots,d-1\right\}\\ & \leq & \max\left\{a_j(M)+(j-i)+2, a_i(M/(x_1,\cdots,x_{d-i})M)+1; \right.\\ & & \qquad\left. j=i+2r,\cdots,d-1\right\}. \end{eqnarray*} By repeating this step we finally have \begin{eqnarray*} a_i(M) & \leq & \max\left\{a_j(M)+(j-i)+\left\lceil\frac{d-i}{r}\right\rceil-1,\right.\\ & & \qquad a_i(M/(x_1,\cdots,x_{d-i})M)+\left\lceil\frac{d-i}{r}\right\rceil-2;\\ & & \qquad \left. j=i+\left(\left\lceil\frac{d-i}{r}\right\rceil-1\right)r,\cdots,d-1\right\}\\ & \leq & a_i(M/(x_1,\cdots,x_{d-i})M)+\left\lceil\frac{d-i}{r}\right\rceil-1 \end{eqnarray*} for $i=0,\cdots,d-1$. Hence the assertion 1) is proved. The assertion 2) is an easy consequence of 1). \vspace{5mm} \noindent{\bf Corollary 2.10.}{\em Let $M$ be a generalized Cohen-Macaulay graded $R$-module with $\dim(M)=d>0$. Let $r$ be an integer with $1\leq r\leq d$. Assume that $M$ is a $(1,r)$-Buchsbaum module. For $i=0,\cdots,d-1$, $$\mbox{\em reg}_i(M)\leq a_{d-r}(M/(x_1,\cdots,x_r)M)+(d-r)+\left\lceil\frac{d-i}{r}\right\rceil-1$$ for any part of s.o.p. $x_1,\cdots, x_r$ with $\deg(x_j)=1$, $j=1,\cdots ,r$.} \vspace{5mm} \noindent{\bf Proof.} It follows immediately from (2.9). \vspace{5mm} \noindent{\bf Remark.} We have assumed that the degree of a s.o.p. $x_1,\cdots,x_d$ is one in (2.7), (2.8), (2.9) and (2.10). We took that assumption for simplicity, although generalized results are similarly proved. \vspace{5mm} Further, we have two more results of Theorem (2.5). \vspace{5mm} \noindent{\bf Corollary 2.11.}{\em Let $M$ be a $k$-Buchsbaum graded $R$-module with $\dim(M)=d>0$. Then we have $$\mbox{\em reg}_i(M)\leq a(M)+d+k(d-i)$$ for $i=1,\cdots,d$.} \vspace{5mm} \noindent{\bf Proof.} It follows immediately from the claim in the proof of (2.7). \vspace{5mm} \noindent{\bf Remark.} Corollary (2.11) was firstly obtained in \cite{NS2}, (3.4). We will show in this paper that the inequality of (2.11) is sharp for all $d$ and $k$, even in the case $M=R$ is an integral domain, see (4.1). \vspace{5mm} \noindent{\bf Proposition 2.12}. {\em Let $M$ be a $(1,r)$-Buchsbaum graded $R$-module with $\dim(M) = d > 0$. Then we have $$\mbox{\em reg}_i(M)\leq a(M)+d+\left\lceil\frac{d-i}{r}\right\rceil$$ for $i=1,\cdots,d$.} \vspace{5mm} \noindent{\bf Proof.} The proof is similar to that of (2.9) by using (2.5.4). \vspace{5mm} \noindent{\bf Remark.} Let us consider in (2.12) the special case that $M$ is Buchsbaum, that is, $r=d$. Then we get for $i\geq 1$ $$\mbox{reg}_i(M)\leq a(M)+d+1.$$ This result can be obtained also by the structure theorem of maximal Buchsbaum modules. Let us explain this approach suggested to us by S. Goto: First we take a polynomial ring $T$ such that $M$ is a maximal $T$-module. By the structure theorem \cite{G}, we see that $M$ as a graded $T$-module is a direct sum of some twistings of $i$-th syzygy modules $E_i$. On the other hand we know that $\mbox{reg\ }(E_i)=i$ and $a(E_i)=i-1$ for $i=1,\cdots,d-1$. Hence we have $$\mbox{reg\ }_i(M)\leq a(M)+d+1$$ for all $i\geq 1$. \section{Bounds on Castelnuovo-Mumford regularity} Let $K$ be an algebraically closed field . Let $X$ be a nondegenerate closed subvariety of ${\bf P}_K^N$ with $\dim(X)=d$. Let $R$ be the coordinate ring of $X$ with $\dim(R)=d+1$. We assume that $X$ is irreducible and reduced, that is, $R$ is an integral domain. Before stating our main results of this section we need the following lemma which is well-known, (see, e.g., \cite{N}, Corollary 2;[20], 4.6(b)). \vspace{5mm} \noindent{\bf Lemma 3.1.}{\em Under the above condition, we have $$a(R)+d+1\leq \left\lceil\frac{\deg(X)-1}{\mbox{\em codim}(X)}\right\rceil$$} \vspace{5mm} The following is our main result of this section, which extends \cite{NS2}, (4.8) and improves \cite{HMV}, (3.1.5) and \cite{HV}, (3.6). \vspace{5mm} \noindent{\bf Theorem 3.2}. {\em Let $X$ be a non-degenerate closed subvariety of ${\bf P}_K^N$ with $\dim(X) = d$ over an algebraically closed field $K$. Let $R$ be the coordinate ring of $X$. Assume that $X$ is a $(k,r)$-Buchsbaum variety for some integer $k$ and $r$ with $k\geq 1$ and $1\leq r\leq d$. Then we have $$\mbox{\em reg\ }(X)\leq\left\lceil\frac{\deg (X)-1}{\mbox{\em codim\ }(X)}\right\rceil+(d+1-\mbox{\em depth\ }(R))k -r+1$$} \vspace{5mm} \noindent{\bf Proof.} It follows immediately from (2.8) and (3.1). \vspace{5mm} \noindent{\bf Remark.} Comparing with the inequality of \cite{HV}, (3.6), we see that our result improves their result in all cases. We have only to check that $$dk-r\leq\left((r-1)+\left(\begin{array}{c}d+2-r\\2\end{array}\right)\right )k-d.$$ Firstly we can easily show that $$d\leq (r-1)+\left(\begin{array}{c}d+2-r\\2\end{array}\right).$$ So we have only to study the case $k=1$, which is left to the readers. \vspace{5mm} The next theorem generalizes known results for quasi-Buchsbaum varieties, see, e.g., \cite{HM}, \cite{NS1}. \vspace{5mm} \noindent{\bf Theorem 3.3}. {\em Let $X$ be a non-degenerate closed subvariety of ${\bf P}_K^N$ with $\dim(X) = d$ over an algebraically closed field $K$. Let $R$ be the coordinate ring of $X$. Assume that $X$ is a $(1,r)$-Buchsbaum variety for some integer $r$ with $1\leq r\leq d$. Then we have $$\mbox{\em reg\ }(X)\leq\left\lceil\frac{\deg(X)-1}{\mbox{\em codim\ }(X)}\right\rceil+ \left\lceil\frac{d+1-\mbox{\em depth\ }(R)}{r}\right\rceil.$$} \vspace{5mm} \noindent{\bf Proof.} It follows immediately from (2.10) and (3.1). \section{Examples and open problems} The purpose of this section is to describe sharp examples of (2.11) and (2.8). Moreover, we conclude with some open problems. Let $X\subseteq{\bf P}_K^N$ be a projective variety with $\dim (X)=d\geq 1$ over an algebraically closed field $K$ of characteristic $0$. Let $R$ be the coordinate ring of $X$. Assume that $X$ is $(k,r)$-Buchsbaum for some integers $k$ and $r$ with $k\geq 1$ and $1\leq r\leq d$. Let $L$ be an $(N-r)$-plane in ${\bf P}_K^N$ with $\dim (X\cap L)=d-r$. Let $R'$ be the coordinate ring of $X\cap L$. Under the above condition, Corollary (2.11), Corollary (2.8) and Theorem (3.2) are stated as follows: $$\begin{array}{rl} \mbox{(2.11)}\qquad & \mbox{reg\ }(X)\leq a(R)+(d+1)+kd+1,\\[5mm] \mbox{(2.8)}\qquad & \mbox{reg\ }(X)\leq a(R')+(d+1-r)+kd-r+1\\[5mm] \mbox{(3.2)}\qquad & \mbox{reg\ }(X)\leq\displaystyle\left\lceil\frac{\mbox{deg}(X)-1}{\mbox{codim\ }(X)}\right\rceil+kd-r+1. \end{array}$$ Note that: $$a(R)+(d+1)\leq a(R')+(d+1-r)\leq\left\lceil\frac{\deg(X)-1}{\mbox{codim\ }(X)}\right\rceil.$$ Hence (3.2) follows from (2.8). Moreover, (2.11) gives the following bound in case $r=1$: ${\rm reg}(X) \leq a(R')+d+kd+1$. But this result is improved in (2.8). Now we will give sharp examples of (2.11) and (2.8) in Example (4.1) and Example (4.2) respectively. These examples are based on ideas of \cite{M1}, (3.4) and \cite{M2}, (3.9). The first example shows that the inequalities of (2.11) are sharp for all $d$ and $k$. \vspace{5mm} \noindent{\bf Example 4.1.} Let $d$ be an integer with $d\geq 1$. Let $Y_j$ be the projective line ${\bf P}_K^1$ over an algebraically closed field $K$ for $j=1,\cdots,d+1$. Let $Y$ be the Segre product of $Y_j$ ($j=1,\cdots,d+1$), that is, $Y=Y_1\times\cdots\times Y_{d+1}$. Let $p_j:Y\rightarrow Y_j$ be the projection for $j=1,\cdots,d+1$. We write an invertible sheaf $p_1^\ast{\cal O}_{{\bf P}^1} (n_1)\otimes\cdots\otimes p_{d+1}^\ast{\cal O}_{{\bf P}^1}(n_{d+1})$ on $Y$ as ${\cal O}_Y(n_1,\cdots, n_{d+1})$ through the isomorphisms $\mbox{Pic\ }(Y)\cong \mbox{Pic\ }(Y_1)\oplus\cdots\oplus \mbox{Pic\ }(Y_{d+1})\cong{\bf Z}^{\oplus d+1}$. On the other hand $Y$ is embedded in the projective space ${\bf P}_K^N$, where $N=2^{d+1}-1$. Then, for any $n\in{\bf Z}$, ${\cal O}_{{\bf P}_K^N} (n)|_Y\cong{\cal O}_Y(n,\cdots,n)$. There exists an irreducible smooth effective divisor $X$ of $Y$ corresponding to the invertible sheaf ${\cal O}_Y(n_1,\cdots,n_{d+1})$ for all positive integers $n_1,\cdots,n_{d+1}$. (See, e.g., \cite{Ha}, p231.) Let $k$ be a positive integer. Let us take integers $n_j=1+(k+1)(j-1)$ for $j=1,\cdots,d+1$. Then we can take a non-singular subvariety $X$ of $Y$ such that the ideal sheaf ${\cal J}_{X/Y}$ is isomorphic to ${\cal O}_Y(-n_1,\cdots,-n_{d+1})$. Let $R$ be the coordinate ring of $X$ in ${\bf P}_K^N$. Then $R$ is an integral domain with $\dim R=d+1$. In order to get $a(R)$ and $a_i(R)$, $i=1,\cdots,d$, we use the following exact sequence and the isomorphisms: $$0\rightarrow H_{\sf m}^{d+1}(R)\rightarrow \rixrel{\ell\in{\bf Z}} {\oplus}H^{d+1}(Y, {\cal J}_{X/Y}(\ell))\rightarrow\rixrel{\ell\in{\bf Z}} {\oplus}H^{d+1}(Y,{\cal O}_Y(\ell)) \rightarrow 0$$ and $$H_{\sf m}^i(R)\cong\rixrel{\ell\in{\bf Z}}{\oplus}H^i(Y,{\cal J}_{X/Y}(\ell)),\quad i\neq 0,\ d+1,$$ because $Y$ is arithmetically Cohen-Macaulay (see, e.g., \cite{M1}, (3.1)). Since we have by K\"{u}nneth's formula: $$H^{d+1}(Y,{\cal J}_{X/Y}(\ell))\cong H^1({\cal O}_{ {\bf P}^1}(\ell-n_1))\otimes\cdots\otimes H^1 ({\cal O}_{{\bf P}^1}(\ell-n_{d+1}))$$ and $$H^{d+1}(Y,{\cal O}_Y(\ell))\cong H^1({\cal O}_{ {\bf P}^1}(\ell))^{\otimes d+1},$$ we see that $H^{d+1}(Y,{\cal J}_{X/Y}(\ell))\neq 0$ if $\ell\leq\min_{1\leq j\leq d+1} (n_j-2)=-1$ and that $H^{d+1}(Y,{\cal O}_Y(\ell))\neq 0$ for $\ell\leq -2$. Thus we have $a(R)=-1$ from the above exact sequence. Next let us take a fixed integer $i$ with $1\leq i\leq d$. Similarly we have by K\"{u}nneth's formula \begin{eqnarray*} H^i(Y,{\cal J}_{X/Y}(\ell)) & \cong & H^0({\cal O}_{ {\bf P}^1}(\ell-n_1))\otimes\cdots\otimes H^0({\cal O}_{{\bf P}^1}(\ell-n_{d-i+1}))\\ & & \ \ \ \otimes H^1({\cal O}_{ {\bf P}^1}(\ell-n_{d-i+2}))\otimes\cdots\otimes H^1({\cal O}_{{\bf P}^1} (\ell-n_{d+1})). \end{eqnarray*} So we see that $H^i(Y,{\cal J}_{X/Y}(\ell))\neq 0$ if $n_{d-i+1}\leq\ell\leq n_{d-i+2}-2$, that is, $(k+1)(d-i+1)-k\leq\ell\leq (k+1)(d-i+1)-1$. Thus we have $a_i(R)=(k+1)(d-i+1)-1$ for $1\leq i\leq d$. Also we have $\mbox{reg}_i(R)=k(d-i+1)+d$ for $1\leq i\leq d$, so $\mbox{reg\ }(R)=(k+1)d$. Further we can easily see that $R$ is a $k$-Buchsbaum ring. Thus we have a non-singular projective $k$-Buchsbaum variety $X$ in ${\bf P}^N$ with $\dim (X)=d$, $a(R)=-1$ and $\mbox{reg\ }(X)=kd+d+1$. Hence this example yields the equality stated in (2.11). \vspace{5mm} The second example shows that the inequalities of (2.8) are sharp in case $k=r=1$. Also it is possible to give examples even in case $r>1$ and $k=1$. In fact we have only to take some higher dimensional projective space instead of ${\bf P}_K^1$ in (4.2). \vspace{5mm} \noindent{\bf Example 4.2.} In Example (4.1), we take $k=1$. Then the quasi-Buchsbaum ring $R$ gives the equality of Corollary (2.8) for all $d$ in case $k=r=1$. In fact, we can easily see that $a(R/hR)=1$ for generic elements $h$ of $R_1$. Then we see that $$\mbox{reg}_i(R)=2 d-i+1$$ and $$a(R/hR)+d+(d+1-i)\cdot 1-1=2 d-i+1$$ for $i=1,\cdots,d$. Hence this example yields the equality stated in (2.8). \vspace{5mm} Finally we conclude by describing some open problems which are arised naturally from our investigation. We take the notation and assumption of the beginning of this section. \vspace{5mm} \noindent{\bf Problem 1.} Generalize (2.11) for $(k,r)$-Buchsbaum varieties for all $r\geq 1$, or construct example of $(k,d)$-Buchsbaum varieties with $\dim (X)=d$ satisfying the equality of (2.11) for all $k$ and $d$. \vspace{5mm} We note that the examples in (4.1) are not $(k,d)$-Buchsbaum for $d\geq 2$ by using arguments of \cite{M1}. \vspace{5mm} \noindent{\bf Problem 2.} Improve the bound stated in (2.8) in order to get sharp examples for all $k$ and $d$ at least in case $r=1$.

9602

alg-geom/9602008

en

https://arxiv.org/abs/alg-geom/9602008

alg-geom/9602008

Bill Oxbury

W.M. Oxbury

Spin Verlinde spaces and Prym theta functions

47 pages, LaTeX

Theta functions of level n on the principally polarised Prym varieties of an algebraic curve are dual to sections of the orthogonal theta line bundle on the moduli space of Spin(n)-bundles over the curve. As a by-product of our computations we also note that when n is odd the pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.

alg-geom

\section{\@startsection {section}{1}{{\bf Z}@}{-3.5ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf} \def\subsection{\@startsection{subsection}{2}{{\bf Z}@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\normalsize\it}} \let\emppsubsection\subsection \newcommand{\numberequationsassubsubsections} \newtheorem{prop}{Proposition}[section] \newtheorem{lemm}[prop]{Lemma} \newtheorem{mumlemm}[prop]{Atiyah-Mumford lemma} \newtheorem{theo}[prop]{Theorem} \newtheorem{cor}[prop]{Corollary} \newtheorem{conj}[prop]{Conjecture} \newtheorem{rem}[prop]{\it Remark} \newtheorem{rems}[prop]{\it Remarks} \newtheorem{ex}[prop]{Example} \begin{document} \title{Spin Verlinde spaces and Prym theta functions} \author{W.M. Oxbury} \date{} \maketitle \centerline{\it Department of Mathematical Sciences} \centerline{\it University of Durham} \bigskip \section{Introduction} The Verlinde formula is a remarkable---and potentially very useful---new tool in the geometry of algebraic curves which is borrowed from conformal field theory. In the first instance it is a trigonometric expression which assigns a natural number $N_l(G,g)$ to data consisting of a semisimple algebraic group $G$, a nonnegative integer $g$ and an auxiliary integer $l \in {\bf Z}$. In physics $N_l(G,g)$ is interpreted as the dimension of the space of conformal blocks at level $l$ in the Wess-Zumino-Witten model of conformal field theory on a compact Riemann surface of genus $g$. In algebraic geometry this `space of conformal blocks' is identified with a vector space of the form $H^0({\cal M}_C(G),{\cal L}^l)$, where ${\cal M}_C(G)$ is the moduli scheme (or stack) of semistable principal $G$-bundles over $C$, and ${\cal L}$ is an ample line bundle on this moduli space generating the Picard group. The feature of the Verlinde formula which motivates this paper is its `numerology'. Namely, when one computes the numbers $N_l(G,g)$ for the classical simple groups one finds that they obey interesting identities which lead one to certain conjectures about the geometry of ${\cal M}_C(G)$. (See \cite{OW}.) In this article we are concerned with identities linking the complex spin groups $G={\rm Spin}_m$ with the configuration of principally polarised Prym varieties associated to the curve $C$ via its unramified double covers. This connection was first observed in \cite{O} for the odd spin groups ${\rm Spin}_{2n+1}$; here we shall give a systematic account of both the odd and even cases. The moduli space ${\cal M}_C(SO_m)$ has, for $m\geq 3$, two connected components labelled by the second Stiefel-Whitney class $w_2$ of the bundles. Each of these components has a $J_2(C)$-Galois cover which is a moduli space for Clifford bundles with fixed spinor norm line bundle $\xi \in {\rm Pic}(C)$, and whose isomorphism class depends only on $\deg \xi $ mod 2. When $\xi = {\cal O}_C$ the Galois cover is precisely ${\cal M}_C({\rm Spin}_m)$ since by definition ${\rm Spin}_m$ is the kernel of the spinor norm; the `sister' space which arises when $\deg \xi$ is odd, we denote by ${\cal M}_C^-({\rm Spin}_m)$. In each case the fibre $J_2(C)$ over ${\cal M}_C(SO_m)$ parametrises liftings of a given $SO_m$-bundle to a Clifford bundle with given spinor norm $\xi$ with $\deg \xi \equiv w_2$ mod 2. Note that for low values of $m$ we recover well-known moduli spaces of vector bundles: for example ${\cal M}_C({\rm Spin}_3) = {\cal SU}_C(2,0)$ and ${\cal M}_C^-({\rm Spin}_3) = {\cal SU}_C(2,1)$ (where ${\cal SU}_C(n,d)$ is the moduli space of rank $n$ vector bundles with fixed determinant of degree $d$); while ${\cal M}_C({\rm Spin}_6) = {\cal SU}_C(4,0)$ and ${\cal M}_C^-({\rm Spin}_6) = {\cal SU}_C(4,2)$. We shall consider the theta line bundle $\Theta({\bf C}^m)$ on these varieties, coming from the standard orthogonal representation ${\bf C}^m$ of the Clifford group, and use the Verlinde formula to count its sections. These preliminary ideas---the spin moduli spaces, theta line bundles on them and the Verlinde calculations---occupy the first five sections of the paper. The central observations on which the paper is based is contained in section \ref{numerology}. This is that the dimension of $H^0({\cal M}_C({\rm Spin}_m),\Theta({\bf C}^m))$ coincides with that of the direct sum of the spaces of {\it even} level $m$ theta functions on all the Prym varieties (including the Jacobian itself); while the dimension of $H^0({\cal M}_C^-({\rm Spin}_m),\Theta({\bf C}^m))$ is equal to that of the direct sum of the spaces of {\it odd} level $m$ theta functions on the Prym varieties. The precise statement (see theorem \ref{numer} and table (\ref{dims})) depends on the parity of $m$: when $m$ is even we have to take theta functions not just on the Prym varieties $P_{\eta}$, $\eta \in J_2(C)\backslash \{{\cal O}\}$, but on the two component abelian subvarieties $$ P_{\eta} \cup P_{\eta}^- = {\rm Nm}^{-1}K_C \subset J^{2g-2}(\cctil_{\eta}), $$ where $\cctil_{\eta} \rightarrow C$ is the double cover corresponding to each $\eta$. There are two remarks to make about this feature of the even spin groups. The first is that it is natural in the sense that, whereas level $m$ theta functions are well-defined on $P_{\eta}$ for all $m$---there is a canonically defined theta divisor ${\Xi_{\eta}}$---on $P_{\eta}^-$ they are well-defined only if $m$ is even. The second is that it corresponds to a direct sum decomposition of $H^0({\cal M}_C({\rm Spin}_m),\Theta({\bf C}^m))$ into two pieces when $m$ is even---apparently the eigenspaces under the involution of the moduli space ${\cal M}_C({\rm Spin}_m)$ corresponding to reflection of the Dynkin diagram. (See section \ref{dynkin}.) In the remaining sections of the paper we make sense of the observations of section \ref{numerology} by constructing homomorphisms: $$ H^0({\cal M}^{\pm}({\rm Spin}_{2n+1}), \Theta({\bf C}^{2n+1}))^{\vee} \rightarrow \sum_{\eta \in J_2}H^0_{\pm}(P_{\eta},({2n+1}){\Xi_{\eta}}) $$ $$ \begin{array}{rcl} H^0({\cal M}^{\pm}({\rm Spin}_{2n}), \Theta({\bf C}^{2n}))^{\vee} &\rightarrow& \displaystyle \sum_{\eta \in J_2}H^0_{\pm}(P_{\eta},2n{\Xi_{\eta}}) \\ &&\ \ \ \ \ \ \displaystyle \oplus \sum_{\eta \not= 0}H^0_{\pm}(P_{\eta}^-,2n{\Xi_{\eta}}).\\ \end{array} $$ Table (\ref{dims}) asserts that each of these maps is {\it between vector spaces of equal dimension}. It is necessary to emphasise that the validity of the right-hand column of the table is dependent on a natural-seeming conjecture \ref{conj} for a Verlinde formula on ${\cal M}^-({\rm Spin}_m)$. When $m=3$ this is the Verlinde formula for rank 2 vector bundles of odd degree due to Thaddeus \cite{T}; further evidence for this `twisted' formula is given in \cite{OW}. It is therefore natural to expect that the above maps are isomorphisms; for $m\geq 5$ this is not known, though we hope to return to the question in a later paper. Cases of low $m$, on the other hand, where the spin moduli spaces can be identified with more familiar moduli spaces of vector bundles, are examined individually in section \ref{numerology}. Finally, when $m$ is odd it is known (see \cite{LS}) that $\Theta({\bf C}^m) = 2{\cal P}$ where the `Pfaffian' line bundle ${\cal P}$ generates the Picard group. In section \ref{pfaffian} we observe---though this remark is independent of the rest of the paper---that the space of sections of this line bundle has a basis labelled by the even theta characteristics of the curve, which directly generalises that constructed by Beauville in \cite{B2} for the case $m=3$. \medskip \noindent {\it Acknowledgements:} In writing this paper the author has benefited greatly from conversations with B. van Geemen, C. Pauly, S. Ramanan and C. Sorger, to all of whom he expresses his gratitude. \section{Moduli spaces of principal bundles on a curve} \label{mod} In this section we shall give a brief account of the moduli spaces of semistable principal bundles over a curve, following [R1], [DN], [KNR]. We begin with a smooth projective complex curve $C$ of genus $g \geq 2$, and a complex connected reductive algebraic group $G$; and we consider algebraic principal $G$-bundles $E\rightarrow C$. Topologically such bundles are classified by the fundamental group of $G$. Just as for vector bundles, one has notions of stability, semistability and S-equivalence for algebraic $G$-bundles, and for stable bundles S-equivalence is the same as isomorphism. (We shall recall in a moment the definition of stability, but it will not be necessary here to define S-equivalence.) The basic result of Ramanathan [R2] is then the following. \begin{theo} Given $C,G$ as above and an element $\gamma \in \pi_1(G)$, there exists a normal irreducible projective variety ${\cal M}(G,\gamma)$ which is a coarse moduli space for families of semistable $G$-bundles of type $\gamma$ on $C$, modulo S-equivalence. \end{theo} Moreover, one has $$ \dim {\cal M}(G,\gamma) = (g-1)\dim G + \dim Z(G), $$ and ${\cal M}(G,\gamma)$ is unirational when $G$ is a simple group [KNR]. The basic construction with principal bundles is the following. If $E$ is a $G$-bundle, and $\rho:G\rightarrow {\rm Aut}(X)$ any left $G$-space, then we can form a bundle $E(X) = E\times_{\rho} X$ with fibre $X$. In case $X=G/P$ is a homogeneous coset space, a section $\sigma: C \rightarrow E(G/P)$ is called a reduction of the strucure group of the bundle to the subgroup $P$. When $P\subset G$ is a maximal parabolic, $E(G/P)\rightarrow C$ can be thought of as a `generalised Grassmannian bundle'. Then by definition, $E$ is {\it semistable} if and only if $$ \deg \sigma^* T^{\rm vert}E(G/P) \geq 0 \qquad \hbox{\sl for all maximal parabolics $P\subset G$,} $$ where $T^{\rm vert}$ denotes the vertical tangent bundle. On the other hand, if $\pi : G' \rightarrow G$ is a group epimorphism then we can view $X=G$ as a left $G'$-space via $\pi$, and so form a $G$-bundle $E=F(G)$ from any $G'$-bundle $F$. $F$ is said to be a {\it lift} of $E$. In particular, if $G'$ is a central extension of $G$ then there is a bijection between maximal parabolics $P\subset G$ and maximal parabolics $P'=\pi^{-1} P\subset G'$, and moreover $F(G'/P') \cong E(G/P)$ if $F$ is any lift of $E$. Consequently: \begin{lemm} \label{1.2} If $E$ is a $G$-bundle and $F$ a lift of $E$ to a central extension of $G$ then $E$ is stable (resp. semistable) if and only if $F$ is. \end{lemm} Finally, of course, we can take for the $G$-space $X$ a finite-dimensional representation $\rho:G\rightarrow GL(V)$, to obtain a vector bundle $E(V)$. In the case when $G = GL_n$ and $V = {\bf C}^n$ is the standard representation, the notions of stability, semistability and S-equivalence are the same for the principal bundle $E$ as for the vector bundle $E(V)$. Thus we shall write ${\cal U}(n,d) = {\cal M}(GL_n,d)$, for $d\in \pi_1(GL_n) \cong {\bf Z}$; this is the moduli space of semistable vector bundles of rank $n$ and degree $d$. Consider now the determinant morphism $$ \det : {\cal U}(n,d) \rightarrow J^d(C). $$ This is a fibration and we shall, as is usual, denote the isomorphism class of the fibre by ${\cal S}{\cal U}(n,d) = {\cal S}{\cal U}_C(n,d) $; or ${\cal S}{\cal U}_C(n)$ when $d=0$. One knows from [DN] that, via det, ${\cal U}(n,d)$ has Picard group $$ {\rm Pic} \ {\cal U}(n,d) \cong {\rm Pic} \ J^d(C) \oplus {\bf Z}\{\Theta_{n,d}\} $$ where $\Theta_{n,d}$ is an ample line bundle on the fibres constructed as follows. It will be convenient, to begin with, to assume that $n$ divides $d$, i.e. that we are dealing with vector bundles of integral slope. Consider first an arbitrary family $F \rightarrow C\times S$ of semistable vector bundles on $C$ with rank $n$, degree $d$ and slope $\mu = d/n \in {\bf Z}$, as above; and we construct a line bundle $\Theta (F) \rightarrow S$, functorial with respect to base change $S' \rightarrow S$, in the following way. Let $\pi : C\times S \rightarrow S$ be the projection. Then (at least Zariski locally) there is a homomorphism of locally free sheaves on $S$, $\phi :K^0 \rightarrow K^1$, having the direct images of $F$ under $\pi$ as kernel and cokernel: \begin{equation} \label{1.3} 0\rightarrow R^0_{\pi}F \rightarrow K^0\map{\phi} K^1\rightarrow R^1_{\pi}F\rightarrow 0. \end{equation} Moreover, the determinant line bundle $$ \textstyle {\rm Det}(F) = (\bigwedge^{\rm top} K^0)^{\vee} \otimes (\bigwedge^{\rm top} K^1) $$ is well-defined and functorial with respect to base change. If $\mu = g-1$ we write $\Theta(F) = {\rm Det}(F)$; and this has a canonical section $\det \phi$, so that in this case $\Theta(F)$ is represented by a canonical Cartier divisor on $S$. Otherwise $\Theta(F)$ is defined to be a suitable twist of ${\rm Det}(F)$ such that $$ \Theta(F\otimes \pi^*L) = \Theta(F) $$ for any line bundle $L\rightarrow S$; i.e. $\Theta$ respects equivalence of families. Now in the case $\mu = g-1$ it is shown in [DN] that the functor $\Theta$ is in fact represented in moduli space by a global Cartier divisor $$ \Theta_{n,n(g-1)} = {\rm Closure}\{{\rm stable}\ V| H^0(V) = H^1(V) \not= 0 \} \subset {\cal U}(n,n(g-1)). $$ In other words $\Theta(F) = f^* \Theta_{n,n(g-1)}$ where $f: S \rightarrow {\cal U}(n,n(g-1))$ is given by the coarse moduli property. For the general case ($\mu \in {\bf Z}$ still) one chooses a line bundle $L\in {\rm Pic}(C)$ with degree chosen so that we get a morphism $$ {\cal U}(n,d) \map{\otimes L} {\cal U}(n,n(g-1)). $$ Now set $$ \Theta_L = (\otimes L)^*\Theta_{n,n(g-1)}. $$ The dependence of $\Theta_L$ on $L$ is then given by (\ref{1.5}) below, which is a consequence of: \begin{lemm} \label{1.4} View $J^0(C)$ as a subgroup of ${\rm Pic}(J^d(C))$ by $L \mapsto \Phi^{-1}\otimes T^*_L\Phi$, where $\Phi$ is any line bundle representing the principal polarisation on $J^d(C)$. Then for any family $F\rightarrow C\times S$ as above, and any $L\in J^0(C)$, we have $$ \Theta(F\otimes pr_C^*L) = {\det}^*(L) \otimes \Theta(F) $$ where $\det : S\rightarrow {\cal U}(n,d) \rightarrow J^d(C)$. \end{lemm} It follows easily from this that when $L,L'$ have the same degree, $\Theta_L$ and $\Theta_{L'}$ are related by: \begin{equation} \label{1.5} \Theta_{L'} = {\det} ^*(L'L^{-1}) \otimes \Theta_L \in {\rm Pic}\ {\cal U}(n,d). \end{equation} We now set $\Theta_{n,d} = \Theta _L$: if $d\not= n(g-1) $ this depends on $L$, {\it but by (\ref{1.5}) its restriction to the fibres of $\det :{\cal U}(n,d) \rightarrow J^d(C)$ is independent of $L$}. \medskip In order to consider bundles of general degree, i.e. non-integral slope, it is necessary to twist by bundles $L$ of higher rank: $$ {\cal U}(n,d) \map{\otimes L} {\cal U}(rn,rn(g-1)), $$ where $L$ has rank $r$. It is easy to check that the necessary and sufficient condition for arranging slope $g-1$ on the right is that: \begin{equation} \label{minr} r\in {n\over {\rm gcd}(n,d)} {\bf Z} \end{equation} The line bundle $\Theta_L$ may now be defined in the same way as above. In this more general situation (\ref{1.5}) becomes: \begin{equation} \label{1.5'} \Theta_{L'} = {\det} ^*(\det L' \otimes \det L^{\vee}) \otimes \Theta_L \in {\rm Pic}\ {\cal U}(n,d), \end{equation} Consequently the restriction of $\Theta _L$ to the fibres of $\det :{\cal U}(n,d) \rightarrow J^d(C)$ is again independent of the choice of $L$ with given rank $r$; and we set $\Theta_{n,d} = \Theta _L$ for any $L$ with $r = {n/ {\rm gcd}(n,d)}$. This is the required generator of the Picard group. Note that if in this construction $L,L'$ are two vector bundles of different ranks $r<r'$ (both satisfying (\ref{minr}), and the degrees of $L$ and $L'$ chosen suitably) then \begin{equation} \label{difr} \Theta_{L'} = \Theta_L^{\otimes r'/r}. \end{equation} \medskip Finally, suppose that we are given a family $E\rightarrow C\times S$ of semistable $G$-bundles, and a representation $\rho:G\rightarrow SL(V)$, where $\dim V =n$. We shall suppose that $\rho$ satisfies the condition: \begin{equation} \label{ss} Z(G)_0 \subset {\rm ker \ } \rho \end{equation} We can form the family of vector bundles $E(V) \rightarrow C\times S$; and by \cite{R2} proposition 2.17 the condition (\ref{ss}) guarantees that these vector bundles are semistable. We thus obtain a theta line bundle $\Theta(E(V))\rightarrow S$, and since $E(V)$ has trivial determinant on the fibres of $\pi:C\times S\rightarrow S$ we deduce from lemma \ref{1.4} the following corollary, which will be needed later: \begin{cor} \label{1.6} For $E\rightarrow C\times S$ and $\rho:G\rightarrow SL(V)$ as above, and for any $L\in J^0(C)$ one has $$ \Theta(E(V)\otimes pr_C^*L) = \Theta(E(V)). $$ \end{cor} Globally $\rho$ satisfying (\ref{ss}) induces a morphism $$ \rho_* : {\cal M}(G,\gamma) \rightarrow {\cal M}(SL_n) \hookrightarrow {\cal U}(n,0) $$ and the functor $E\mapsto \Theta(E(V))$ is represented by the (well-defined) line bundle $$ \Theta(V) := (\rho_*)^* \Theta_{n,0} \in {\rm Pic} \ {\cal M}(G,\gamma). $$ Note that if we let $j: {\cal M}(SL_n) \hookrightarrow {\cal U}(n,0)$ denote the inclusion which identifies ${\cal M}(SL_n)$ with the moduli space of vector bundles of rank $n$ and trivial determinant, via the standard representation ${\bf C}^n$, then by construction $$ \Theta({\bf C}^n) = j^*\Theta_{n,0}. $$ \section{Clifford bundles} \label{clif} Let us consider again the fibration $$ \det :\ {\cal U}_C(n,d) \rightarrow J^d(C); $$ induced, that is, by the determinant homomorphism $GL_n \rightarrow {\bf C}^*$. The fibres of these maps are, up to isomorphism, the $n$ moduli varieties ${\cal S}{\cal U}_C(n,d)$, for $d\in {\bf Z} /n$. In this section we shall describe an alternative generalisation of this situation for $n=2$, obtained by replacing $GL_2$ not by $GL_n$, but by the special Clifford group of a nondegenerate quadratic form. First we need to recall some basic Clifford theory. \subsection{The special Clifford group} Let $Q$ be a nondegenerate quadratic form on a complex vector space $V$ of finite dimension $m$; let $A=A(Q)$ be its Clifford algebra and $A^+$ the even Clifford algebra. Recall that these can be expressed as matrix algebras as follows. If $m=2n$ is even then for any $n$-dimensional isotropic subspace $U\subset V$ one has \begin{equation} \label{evenA} \textstyle A \cong {\rm End} \bigwedge U; \qquad A^+ \cong {\rm End} \bigl( \bigwedge^{\rm even} U \bigr) \oplus {\rm End} \bigl( \bigwedge^{\rm odd} U \bigr). \end{equation} If, on the other hand, $m=2n+1$ is odd then for any direct sum decomposition $V=U \oplus U'\oplus {\bf C}$ where $U,U'$ are $n$-dimensional isotropic subspaces one has \begin{equation} \label{oddA} \textstyle A \cong {\rm End} \bigwedge U \oplus {\rm End} \bigwedge U'; \qquad A^+ \cong {\rm End} \bigwedge U . \end{equation} The `principal involution' of $A$ is $\alpha : x\mapsto -x$ for $x\in V$, i.e. is $\pm 1$ on $A^{\pm}$ respectively. The `principal anti-involution' $\beta$ is the identity on $V$ and reverses the direction of multiplication: $\beta(x_1\ldots x_r) = x_r\ldots x_1$. Then the {\it Clifford group} is $$ C (Q) = \{s\in A^{*} | \alpha (s)Vs^{-1} \subset V\}, $$ where $A^{*}\subset A$ denotes the group of units; and the {\it special Clifford group} is $$ SC(Q) = C(Q) \cap A^+. $$ For $s\in C(Q)$ the transformation $\pi_s: x\mapsto \alpha(s)xs^{-1}$ of $V$ is orthogonal---this is because $C(Q)$ is generated by $x\in V\cap C(Q)$, for which $\pi_x$ is just minus the reflection in the hyperplane $x^{\perp}$. Thus one has a group homomorphism $\pi:C(Q) \rightarrow O(Q)$, which has the following properties. \begin{prop} \label{2.1} \begin{enumerate} \item ${\rm ker \ } \pi = {\bf C}^*$; \item $\pi(C(Q)) = O(Q)$ and $\pi(SC(Q)) = SO(Q)$. \end{enumerate} \end{prop} \begin{cor} \label{2.2} $SC(Q)$ is a connected reductive algebraic group. \end{cor} The {\it spinor norm} is the group homomorphism $$ \begin{array}{rcl} {\rm Nm} \ :\ SC(Q) &\rightarrow& {\bf C}^{*} \\ s &\mapsto& \beta(s)s.\\ \end{array} $$ Equivalently ${\rm Nm} (x_1\ldots x_r) = Q(x_1)\cdots Q(x_r)$ for $x_1,\ldots ,x_r \in V$. Then by definition ${\rm Spin} (Q)= {\rm ker \ } {\rm Nm}$. Note that multiplication by scalars induces a double cover \begin{equation} \label{rescaling} \{\pm(1,1)\} \rightarrow {\bf C}^* \times {\rm Spin}(Q) \rightarrow SC(Q). \end{equation} {}From now on we shall write $C_m$, $SC_m$, ${\rm Spin}_m$ instead of $C(Q)$, $SC(Q)$, ${\rm Spin}(Q)$ when $Q$ is the standard quadratic form on ${\bf C}^m$. Then (using (\ref{2.1})) one has the following commutative diagram of short exact sequences: \begin{equation} \label{2.3} \matrix{ 1&\rightarrow&{\bf C}^{*}&\rightarrow&SC_m&\map{\pi}&SO_m&\rightarrow&1\cr &&&&&&&&\cr &&\uparrow&&\uparrow&&\Vert&&\cr &&&&&&&&\cr 1&\rightarrow&{\bf Z}/2&\rightarrow&{\rm Spin}_m&\rightarrow&SO_m&\rightarrow&1.\cr } \end{equation} \begin{prop} \label{2.4} For $m\geq 3$: \begin{enumerate} \item $SC_m$ has centre $ Z(SC_m) = \cases{{\bf C}^{*} & if $m$ is odd \cr {\bf C}^{*} \times {\bf Z}/2 & if $m$ is even;\cr} $ \item $SC_m$ has fundamental group $\pi_1(SC_m) = {\bf Z}$; and this maps isomorphically to $\pi_1({\bf C}^*) = {\bf Z}$ under the spinor norm. \end{enumerate} \end{prop} {\it Proof.}\ (i) If $m$ is odd the centre of $SC_m$ must be contained in---and hence equal to---the kernel ${\bf C}^*$ of the surjection onto $SO_m$, since the latter has trivial centre. If, on the other hand, $m$ is even, then by the same token $Z(SC_m)$ is contained in $\pi^{-1}(Z(SO_m))$ where $\pi$ denotes the surjection to $SO_m$. In this case $SO_m$ has centre $\{\pm 1\}$. As before everything in $\pi^{-1}(1) = {\bf C}^*$ is central; while if $\{ e_1, \ldots , e_m \} \subset {\bf C}^m$ is any orthonormal basis then the product $e_1 \ldots e_m \in SC_m$ spans $\pi^{-1}(-1) \cong {\bf C}^*$. Since $m$ is even this product anticommutes with each $e_i$, and therefore {\it commutes} with all elements of $A^+$. So $\pi^{-1}\{\pm 1\} \cong {\bf C}^{*} \times {\bf Z}/2$ is contained in and therefore equal to the centre. (ii) From the exact homotopy sequence of the fibration in the upper sequence of (\ref{2.3}), and the vanishing of $\pi_2({\bf C}^*)$, we have a non-split extension $$ \ses{{\bf Z}}{\pi_1(SC_m)}{{\bf Z} /2}. $$ Since the fundamental group of a Lie group is abelian it follows that the only possibility is $\pi_1(SC_m) = {\bf Z}$. The last part now follows from the fact that ${\rm Spin}_m$ is simply-connected. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \subsection{The spin moduli spaces} {}From proposition \ref{2.4} part 2 we see that there is for each $d\in {\bf Z} = \pi_1(SC_m)$ a morphism induced by the spinor norm, and which we shall denote in the same way: $$ {\rm Nm} : {\cal M} (SC_m, d) \rightarrow J^d(C). $$ Moreover, when $m=3$ this is nothing but the determinant morphism for rank 2 vector bundles (see example \ref{clifm=3} below). And just as for rank 2 vector bundles, one has: \begin{prop} \label{2.5} For $d\in {\bf Z}$ and $L\in J^d(C)$, the isomorphism class of the subscheme ${\rm Nm} ^{-1}(L) \subset {\cal M}(SC_m,d)$ depends only on $d$ mod 2. \end{prop} Actually this is essentially trivial and is proved in the same way as for rank 2 vector bundles, once one observes that multiplication of $SC_m$ by its centre (proposition \ref{2.4}) induces a natural generalisation of the tensor product operation of Clifford bundles by line bundles if $m$ is odd, and if $m$ is even by {\it pairs} $(N,\eta)$ where $N$ is a line bundle and $\eta \in H^1(C,{\bf Z}/2) = J_2(C)$. (See also \cite{R}.) We shall write, respectively, $N\otimes E$ and $(N,\eta)\otimes E$ for this product. It follows from the definition of the spinor norm that $$ {\rm Nm}(N\otimes E) = N^2 \otimes {\rm Nm} (E), \quad {\rm resp.} \quad {\rm Nm}((N,\eta)\otimes E) = N^2 \otimes {\rm Nm} (E). $$ So to prove \ref{2.5}: suppose $d=\deg L \equiv d'=\deg L'$ mod 2, and write $L' = N^{2k}\otimes L$ for some $N\in {\rm Pic}^1(C)$, $k\in {\bf Z}$. Then the map ${\cal M}(SC_m,d) \rightarrow {\cal M}(SC_m,d')$ given by $E\mapsto N\otimes E$ (resp. $(N,{\cal O})\otimes E$) restricts to an isomorphism ${\rm Nm}^{-1}(L) \cong {\rm Nm}^{-1}(L')$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} We shall therefore introduce the notation: $$ \begin{array}{rcl} {\cal M}^+({\rm Spin}_m) &=& {\cal M}({\rm Spin}_m) = {\rm Nm}^{-1}({\cal O}_C);\\ {\cal M}^-({\rm Spin}_m) &=& {\rm Nm}^{-1}({\cal O}_C(p)),\\ \end{array} $$ where $p\in C$ is any point of the curve. \begin{rem}\rm \label{w2} The group $SO_m$ has fundamental group ${\bf Z}/2$; so the moduli space of semistable $SO_m$-bundles has two irreducible components ${\cal M}(SO_m,0) = {\cal M}^+(SO_m)$ and ${\cal M}(SO_m,1) = {\cal M}^-(SO_m)$ distinguished by the second Stiefel-Whitney class $w_2$. It is not hard to show (see \cite{O}) that $\deg {\rm Nm}(E) \equiv w_2(E)$ mod 2 and that ${\cal M}^{\pm}({\rm Spin}_m)$ are naturally Galois covers of these components: $$ {\cal M}^{\pm}({\rm Spin}_m) \map{J_2(C)} {\cal M}^{\pm}(SO_m). $$ Note that these maps respect stability---this is a special case of lemma \ref{1.2}. \end{rem} In later sections we shall be interested in the theta line bundle $\Theta({\bf C}^m)$ associated to the orthogonal representation of $SC_m$. Note that by \ref{2.1} and \ref{2.4} the condition (\ref{ss}) is satisfied for $m\geq 3$, so that $\Theta({\bf C}^m)$ is defined everywhere on ${\cal M}(SC_m)$ and hence on both ${\cal M}^{\pm}({\rm Spin}_m)$. \begin{ex} \label{clifm=2} $\bf m = 2.$ \rm Here $SC_2 \cong {\bf C}^* \times {\bf C}^*$, the spinor norm is ${\rm Nm} : (a,b) \mapsto ab$ and the orthogonal representation $SC_2 \rightarrow SO_2 \cong {\bf C}^*$ is $(a,b) \mapsto a/b$. It follows that ${\cal M}(SC_2) \cong {\rm Pic}(C)\times {\rm Pic}(C)$ and each of ${\cal M}^{\pm}({\rm Spin}_2) \cong {\rm Pic}(C)$. $\Theta({\bf C}^2)$ is by definition obtained by pulling back $\Theta_{2,0}$ under $$ \rho_*: {\cal M}(SC_2) = {\rm Pic}(C)\times {\rm Pic}(C) \rightarrow {\cal M}(SL_2). $$ But this map sends a pair of line bundles $(L,N)$ to the vector bundle $LN^{-1}\oplus L^{-1}N$, and this is semistable only if $\deg L = \deg N$. (Note that condition (\ref{ss}) fails for this case!) It follows that the morphism $\rho_*$, and hence $\Theta({\bf C}^2)$, is defined only on ${\rm Pic}(C)\times_{\deg} {\rm Pic}(C)$; and $\Theta({\bf C}^2)$ is thus defined only on the degree 0 component $J(C)$ of ${\cal M}({\rm Spin}_2) = {\rm Pic}(C)$ and is not defined on ${\cal M}^{-}({\rm Spin}_2)$. On the other hand, ${\cal M}({\rm Spin}_2) \rightarrow {\cal M}(SO_2)$ is the squaring map $[2]$ on line bundles, and so one sees that on $J(C)$ the orthogonal theta bundle is: $$ \Theta({\bf C}^2) = [2]^*(2\theta) = 8 \theta, $$ where $\theta$ is the theta divisor on $J(C)$. \end{ex} \begin{ex} \label{clifm=3} $\bf m = 3.$ \rm This is in many ways the most important case. It is well-known that $SC_3$ is, via (\ref{oddA}), equal to the group of units $GL_2 \subset A^+$, and that the spinor norm is the determinant homomorphism so that ${\rm Spin}_3 = SL_2$. (The representation on $SO_3$ is then precisely the action of $GL_2$ on $S^2 \cong {\bf P}_1$ by M\"obius transformations, via stereographic projection.) Thus ${\cal M}({\rm Spin}_3) \cong {\cal SU}_C(2)$ and ${\cal M}^-({\rm Spin}_3) \cong {\cal SU}_C(2,1)$. Let ${\cal L}_d = \Theta_{2,d}$ be the (ample) generators of the Picard groups of these varieties, for $d=0,1$ respectively, as described in the previous section. From the discussion there we can view ${\cal L}_1$ ( $=\Theta_{L'}$ for a suitable rank 2 vector bundle $L'$) as defined on all of ${\cal M}(GL_2)$ while ${\cal L}_0$ ($= \Theta_L$ for a suitable line bundle $L$) is a line bundle defined only on the components of even degree; and by (\ref{difr}) ${\cal L}_1 = {\cal L}_0^2$ on these components. The orthogonal theta bundle is then $\Theta({\bf C}^3) = {\cal L}_1^2 = {\cal L}_0^4$---see (\ref{di2}) below. \end{ex} \begin{ex} \label{clifm=4} $\bf m = 4.$ \rm Again it is very well-known that ${\rm Spin}_4 = SL_2 \times SL_2$. Via (\ref{evenA}) the special Clifford group $SC_4 \subset GL_2 \times GL_2$ is the subgroup consisting of pairs of matrices $(A,B)$ such that $\det A = \det B$; the spinor norm is then the common $2\times 2$ determinant. Thus ${\cal M}(SC_4) = {\cal M}(GL_2) \times_{\det} {\cal M}(GL_2)$ while ${\cal M}^{\pm}({\rm Spin}_4)$ are ${\cal SU}_C(2)\times {\cal SU}_C(2)$ and ${\cal SU}_C(2,1)\times {\cal SU}_C(2,1)$ respectively. The orthogonal representation induces ${\cal M}(SC_4) \rightarrow {\cal M}(SL_4)$ mapping a pair of rank 2 vector bundles $(E,F)$ to $E\otimes F^*$, and from this it follows that the orthogonal theta bundle is (with the notation of the previous example) $$ \Theta ({\bf C}^4) = pr_+^* {\cal L}_1 \otimes pr_-^* {\cal L}_1 = pr_+^* {\cal L}_0^2 \otimes pr_-^* {\cal L}_0^2, $$ where $pr_{\pm}$ denote the respective projections. \end{ex} \begin{ex} \label{clifm=6} $\bf m = 6.$ \rm In this case ${\rm Spin}_6 \cong SL_4$, and one may show that the subgroup (using (\ref{evenA}) once again) $SC_6 \subset GL_4 \times GL_4$ is the image of the homomorphism $$ \begin{array}{rcl} {\bf C}^* \times SL_4 &\rightarrow& GL_4 \times GL_4\\ (\lambda,A) &\mapsto& (\lambda A,\lambda A^{adj,t})\\ \end{array} $$ where $A^{adj,t}$ is the matrix of signed cofactors and $A^{adj} = \det A \times A^{-1} = A^{-1}$. (Compare with (\ref{rescaling}).) Note that projection to the first factor---the first half-spinor representation---induces a double cover $ \{(1,\pm 1)\} \subset SC_6 \map{pr_1} GL_4. $ Moreover, the lift of the determinant function to this double cover has a square root---namely, the spinor norm ${\rm Nm} : (\lambda A,\lambda A^{adj,t}) \mapsto \lambda^2$, where ${\rm Nm}(a)^2 = \det pr_1(a)$ for $a\in SC_6$. At the level of bundles this says we have a commutative diagram $$ \matrix{ {\cal M}(SC_6)&\map{pr_1}&{\cal M}(GL_4)\cr &&\cr {\rm Nm}\downarrow&&\downarrow\det\cr &&\cr {\rm Pic}(C)&\map{[2]}&{\rm Pic}(C)\cr} $$ where each of the horizontal maps has fibre $J_2(C)$. In particular one sees that ${\cal M}({\rm Spin}_6) \cong {\cal SU}_C(4)$ and ${\cal M}^-({\rm Spin}_6) \cong {\cal SU}_C(4,2)$. As in example \ref{clifm=3}, let ${\cal L}_d = \Theta_{2,d}$ be the ample generators of the Picard groups of these varieties, for $d=0,2$ respectively. Then ${\cal L}_2$ ( $=\Theta_{L'}$ for a suitable rank 2 vector bundle $L'$) is defined on all components of ${\cal M}(GL_2)$ of even degree, while ${\cal L}_0$ ($= \Theta_L$ for a suitable line bundle $L$) is defined only on the components of degree $\equiv 0$ mod 4; and by (\ref{difr}) ${\cal L}_2 = {\cal L}_0^2$ on these components. Using (\ref{di2}) from section \ref{vformula} below, the orthogonal theta bundle is then $\Theta({\bf C}^6) = {\cal L}_2 = {\cal L}_0^2$. \end{ex} \section{Orthogonal bundles and theta characteristics} In this section we shall gather together various properties of orthogonal bundles, some possibly well-known, which will be needed later on. \subsection{Isotropic line subbundles} To begin, consider any $SO_m$-bundle $E$ and its associated orthogonal vector bundle $E({\bf C}^m)$. Recall that stability of $E$ is equivalent to the condition that $\deg F <0$ for all {\it isotropic} vector subbundles $F\subset E({\bf C}^m)$. This holds, in particular, if $E({\bf C}^m)$ is stable as a vector bundle. On the other hand, for {\it any} subbundle $F\subset E({\bf C}^m)$, the direct sum with its orthogonal complement fits into an exact sequence $$ \ses{N}{F\oplus F^{\perp}}{M} $$ where $N,M$ are the subbundles generically generated by $F\cap F^{\perp}$ and $F+ F^{\perp}$ respectively. In particular, when $N=0$ this gives rise to an orthogonal splitting $E({\bf C}^m) = F\oplus F^{\perp}$. Applying this idea inductively Ramanan shows (\cite{R} proposition 4.5): \begin{lemm} \label{ramanan} $E$ is a stable $SO_m$-bundle if and only if $E({\bf C}^m)$ is an orthogonal direct sum $E({\bf C}^m)= F_1 \oplus \cdots \oplus F_k$ where the $F_i$ are pairwise non-isomorphic stable vector bundles. In particular $E({\bf C}^m)$ is a stable vector bundle for generic $E\in {\cal M}(SO_m)$. \end{lemm} \noindent (The last assertion here follows from a simple dimension count.) Now suppose that $F\subset E({\bf C}^m)$ is a {\it line} subbundle. Then $N=0$ precisely when $F$ is non-isotropic, and thus $E({\bf C}^m)$ splits in this case, and so fails to be stable as a vector bundle. This shows: \begin{cor} \label{alliso} For generic $E\in {\cal M}(SO_m)$ every line subbundle $F\subset E({\bf C}^m)$ is isotropic. \end{cor} In the next lemma we observe that the bound $\deg F <0$ for isotropic subbundles of $E({\bf C}^m)$---again restricting to line subbundles---can in fact be improved generically: \begin{lemm} \label{nagata} For generic stable $E\in {\cal M}^{\pm}(SO_m)$ every isotropic line subbundle $F\subset E({\bf C}^m)$ satisfies $ \deg F \leq -g+1. $ \end{lemm} {\it Proof.}\ We simply count dimensions of those bundles $E\in {\cal M}(SO_m)$ which can possess an isotropic line subbundle $F\subset E({\bf C}^m)$ with $\deg F = d$. Let $F^{\perp}\subset E({\bf C}^m)$ denote the orthogonal complement: this is a vector bundle of rank $m-1$ on which the quadratic form restricts with rank $m-2$, and the quotient $Q = F^{\perp}/F$ is thus an $SO_{m-2}$-bundle, fitting into an exact diagram: $$ \begin{array}{rcccccccl} &&0&&0&&&&\\ &&\downarrow&&\downarrow&&&&\\ &&&&&&&&\\ &&F&=&F&&&&\\ &&&&&&&&\\ &&\downarrow&&\downarrow&&&&\\ &&&&&&&&\\ 0&\rightarrow&F^{\perp}&\rightarrow&E({\bf C}^m)&\rightarrow&F^{-1}&\rightarrow&0\\ &&&&&&&&\\ &&\downarrow&&\downarrow&&\|&&\\ &&&&&&&&\\ 0&\rightarrow&Q&\rightarrow&(F^{\perp})^{\vee}&\rightarrow&F^{-1}&\rightarrow&0 \\ &&&&&&&&\\ &&\downarrow&&\downarrow&&&&\\ &&0&&0&&&&\\ \end{array} $$ Noting now that $E$ is determined up to isomorphism by $F^{\perp}$ together with its (degenerate) quadratic form (which determines $F$), we see that to construct such a bundle $E$ it is enough to specify the left-hand vertical sequence. For this, $F$ is determined by $g$ parameters; $Q$ by $(g-1)(m-2)(m-3)/2$ parameters; and the extension $F^{\perp}$ by $$ h^1(C,Q^{\vee} \otimes F) -1 = (m-2)(g-1-d) -1 $$ parameters. Consequently, a {\it generic} $E\in {\cal M}(SO_m)$ (in either component of the moduli space) possesses an isotropic line subbundle $F$ of degree $d$ only if the total number of parameters is at least $\dim {\cal M}(SO_m) = (g-1)m(m-1)/2$: $$ \begin{array}{rcl} g+ (g-1)(m-2)(m-3)/2\ \ \ \ \ \ \ &&\\ + (m-2)(g-1-d) -1 &\geq& (g-1)m(m-1)/2,\\ \end{array} $$ which simplifies to $d\leq -g +1$. The lemma follows at once from this. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{rem} \label{rank2case}\rm Note that in the case $m=3$, $E({\bf C}^3) = {\rm ad}\ V$, the bundle of tracefree endomorphisms of a stable rank 2 vector bundle $V$---i.e. that coming from $E$ via the 2-dimensional spin representation. Then there is a one-to-one correspondence between line subbundles $L\subset V$ and isotropic line subbundles $F\subset {\rm ad}\ V$: this is because the quadratic form on ${\rm ad}\ V$ is the Killing form, which for tracefree $2\times 2$ matrices is the determinant. Thus $F$ is isotropic if and only if it consists of nilpotent endomorphisms, and $L$ is then the kernel bundle; and conversely $F = {\rm Hom}(V/L, L) = L^2 \otimes \det V^{\vee}$. If one views $L\subset V$ as determining a cross-section $ l \subset {\bf P} (V)$ of the corresponding ruled surface, then $-\deg F = l\cdot l$ is its self-intersection. So lemma \ref{nagata} says $$ l\cdot l \geq g-1 \qquad \hbox{for any section $ l \subset {\bf P} (V)$,} $$ for a generic rank 2 vector bundle $V$. A well-known result of Nagata \cite{N}, on the other hand, says that {\it every} ruled surface has a section $l$ with self-intersection $ l\cdot l \leq g$. Since, for a given surface, self-intersection of a section is constant ($\equiv \deg V$) mod 2, this implies that the inequality of lemma \ref{nagata} is sharp for $m=3$. \end{rem} \subsection{Conservation of parity} We shall need to make use of the following well-known result of Atiyah \cite{A} and Mumford \cite{M}: \begin{mumlemm} \label{mumlemm} Suppose that $F \rightarrow C$ is a vector bundle admitting a nondegenerate symmetric bilinear form $$ F\otimes F \rightarrow K_C. $$ Then $h^0(C,F)$ is constant modulo 2 under deformation. \end{mumlemm} The case of this which we shall be interested in arises when $F = L\otimes E({\bf C}^m)$ for $L$ a theta characteristic, $L^2 =K$, and $E$ an $SO_m$- (or $SC_m$-) bundle. Let $\vartheta(C) \subset J^{g-1}(C)$ denote the set of theta characteristics. The value of $h^0(C,F)$ mod 2 is in this case given by a calculation of Serre: \begin{prop} \label{2.11} For any $SC_m$-bundle $E\rightarrow C$ and theta characteristic $K^{1\over 2}\in \vartheta(C)$ one has $$ h^0(C,K^{1\over 2} \otimes E({\bf C}^m)) \equiv m h^0(C,K^{1\over 2}) + \deg {\rm Nm}(E) \quad {\rm mod}\ 2. $$ \end{prop} {\it Proof.} By \cite{S}, theorem 2, one has the congruence: $$ h^0(C,K^{1\over 2} \otimes E({\bf C}^m)) \equiv (m+1) h^0(C,K^{1\over 2}) + h^0(C,K^{1\over 2} \otimes w_1(E))+ w_2(E) \quad {\rm mod}\ 2, $$ where $w_1$ and $w_2$ are the Stiefel-Whitney classes. But $w_1(E)$ can be identified with $\det E({\bf C}^m) \in J_2(C) \cong H^1(C,{\bf Z}/2)$, which in our case vanishes since $E$ is a {\it special} Clifford bundle. On the other hand, $w_2(E)\equiv \deg {\rm Nm}(E)$ mod 2 by remark \ref{w2}. So we get the statement in the proposition. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{cor} \label{serre} Suppose that $p: {\widetilde C} \rightarrow C$ is an unramified double cover, and let $\sigma: {\widetilde C}\leftrightarrow {\widetilde C}$ denote the sheet-interchange over $C$. Suppose that $E$ is any $SC_m$-bundle on $C$, and that $L\in {\rm Pic}({\widetilde C})$ satisfies $L\otimes \sigma(L) = K_{{\widetilde C}}$. Then: $$ h^0({\widetilde C},L \otimes p^* E({\bf C}^m)) \equiv m h^0({\widetilde C},L) \quad {\rm mod}\ 2. $$ \end{cor} {\it Proof.}\ $L$ has degree $g({\widetilde C}) -1$, so by the Atiyah-Mumford lemma it suffices to assume that $L$ is the pull-back of a theta characteristic. Then the corollary follows at once from the proposition since $w_2(p^* E)=0$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \subsection{Generically zero results} The main aim of this section is to establish that in each of the results \ref{2.11} and \ref{serre} above, the vanishing (mod 2) of the right-hand side guarantees that $h^0(C,L\otimes E({\bf C}^m)) =0$ for generic stable orthogonal bundle $E$ (theorems \ref{gen0}, \ref{gen0th} and corollary \ref{serregen0}). To this end we shall apply Brill-Noether methods (see, for example, \cite{ACGH}) to the loci: $$ {\cal W}_k = \{ (L,E) | h^0 (C, L\otimes E({\bf C}^m)) \geq k \} \subset J^{g-1}(C) \times {\cal N}^{st}(m) $$ where ${\cal N}^{st}(m) \subset {\cal N}(m)$ is the open set of {\it stable} bundles. Analogously with classical Brill-Noether theory we claim that the Zariski tangent space to ${\cal W}_k$ at a point $(L,E)$ is annihilated by the image of a {\it Petri map} $$ \begin{array}{rcl} \mu : H^0(L\otimes E({\bf C}^m)) \otimes H^0(KL^{-1}\otimes E({\bf C}^m)) &\rightarrow& H^0(K) \oplus H^0(K\otimes E({\bf so}_m))\\ &&\\ &&\cong \Omega^1_{J^{g-1}\times {\cal N}^{st}(m)}|_{(L,E)}.\\ \end{array} $$ Here ${\bf so}_m$ is the Lie algebra of ${\rm Spin}_m$, viewed as the semisimple component of the adjoint representation of $SC_m$. So $E({\bf so}_m)$ is then the vector bundle with fibre ${\bf so}_m$ associated to any Clifford bundle $E$ via this representation; and by standard deformation theory (and this is where we require our bundles to be stable) $H^0(C,K\otimes E({\bf so}_m)) \cong \Omega^1_{{\cal N}^{st}(m)}|_{E}$. The Petri map is defined by \begin{equation} \label{mudef} \mu : s\otimes t \mapsto \< s,t\> \oplus s\wedge t \end{equation} where $\<,\>$ is the symmetric bilinear form on the vector bundle $E({\bf C}^m)$, and we identify ${\bf so}_m \cong \bigwedge^2 {\bf C}^m$. \begin{prop} \label{petri} For $(L,E) \in {\cal W}_k - {\cal W}_{k+1}$ the Zariski tangent space of ${\cal W}_k$ at $(L,E)$ is $ T_{(L,E)}{\cal W}_k = ({\rm image \ } \mu)^{\perp}. $ \end{prop} To prove this we first need: \begin{lemm} Given $(L,E)\in {\rm Pic}(C)\times {\cal N}^{st}(m)$, a section $s\in H^0(C,L\otimes E({\bf C}^m))$, and a tangent vector $\eta \oplus \xi \in H^1(C,{\cal O}) \oplus H^1(C,E({\bf so}_m)) = T_{(L,E)}\bigl( J^{g-1}\times {\cal N}^{st}(m) \bigr)$, $s$ extends to the 1st-order deformation of $L \otimes E({\bf C}^m)$ corresponding to $\eta \oplus \xi$ if and only if $$ \xi s +\eta s=0\in H^1(C,L\otimes E({\bf C}^m)). $$ \end{lemm} {\it Proof.}\ Represent $E$ by transition data $\{h_{\alpha \beta}\}$ with respect to an open cover $\{U_{\alpha}\}$ of $C$, where the $h_{\alpha \beta}$ are holomorphic $SC_m$-valued functions on $U_{\alpha}\cap U_{\beta}$ satisfying the cocycle condition; let $g_{\alpha \beta}$ be the image of $h_{\alpha \beta}$ in $SO_m$---these are then the transition functions for the vector bundle $E({\bf C}^m)$. Likewise represent $L$ by transition data $\{ \phi_{\alpha \beta} \}$ where the $\phi_{\alpha \beta}$ are ${\bf C}^*$-valued functions. Finally, represent the sum $\eta \oplus \xi \in H^1({\cal O}) \oplus H^1({\bf so}_m)$ by a cocycle $\{\eta_{\alpha \beta} \oplus \xi_{\alpha \beta}\}$ (holomorphic Lie algebra-valued functions on the $U_{\alpha}\cap U_{\beta}$). Then the 1st-order deformation corresponding to $\eta \oplus \xi$ is the $SO_m$-bundle on $C\times {\rm Spec \ } {\bf C}[\varepsilon ] /(\varepsilon^2)$ with transition data $$ \begin{array}{rcl} \tilde \phi_{\alpha \beta} &=& (1+\varepsilon \eta_{\alpha \beta})\phi_{\alpha \beta},\\ \tilde g_{\alpha \beta} &=& (1+\varepsilon \xi_{\alpha \beta})g_{\alpha \beta}.\\ \end{array} $$ A section $s\in H^0(C,L\otimes E({\bf C}^m)) $ is now given by a collection $\{s_{\alpha}\}$ of ${\bf C}^m$-valued functions satisfying $$ s_{\alpha} = \phi_{\alpha \beta} g_{\alpha \beta} s_{\beta} $$ on $U_{\alpha}\cap U_{\beta}$; and $s$ extends to the 1st-order deformation $\xi$ provided there exists a 0-cochain $\{s_{\alpha}'\}$ such that $$ \tilde s_{\alpha} = s_{\alpha} + \varepsilon s'_{\alpha} $$ satisfies the cocycle condition $$ \begin{array}{rcl} \tilde s_{\alpha} &=& \tilde \phi_{\alpha \beta} \tilde g_{\alpha \beta} \tilde s_{\beta} \\ &=& \phi_{\alpha \beta} (1+\varepsilon \eta_{\alpha \beta})(1+\varepsilon \xi_{\alpha \beta} )g_{\alpha \beta} (s_{\beta} + \varepsilon s'_{\beta})\\ &=& s_{\alpha} + \varepsilon (\eta_{\alpha \beta} s_{\alpha} +\xi_{\alpha \beta} s_{\alpha} +\phi_{\alpha \beta} g_{\alpha \beta} s'_{\beta} ),\\ \end{array} $$ and hence $ s'_{\alpha} - \phi_{\alpha \beta} g_{\alpha \beta} s'_{\beta} = (\eta_{\alpha \beta} + \xi_{\alpha \beta})s_{\alpha}. $ In other words $s'= \{s'_{\alpha}\}$ has to satisfy $$ \eta s + \xi s = \partial s' \in Z^1(L \otimes E({\bf C}^m)), $$ where $\partial$ is the coboundary operator on 0-cocycles, and $Z^1$ is the group of Cech 1-cocycles. So there exists a solution if and only if $\xi s +\eta s=0$ in cohomology. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} {\it Proof of proposition \ref{petri}.} We write down the Serre duality pairing of $\eta \oplus \xi \in H^1(C,{\cal O}) \oplus H^1(C,E({\bf so}_m))$ with $\mu (s\otimes t)$, for $s\in H^0(C, L\otimes E({\bf C}^m))$ and $t\in H^0(C, KL^{-1}\otimes E({\bf C}^m))$: \begin{equation} \label{pair} \begin{array}{rcl} \bigl(\eta \oplus \xi,\mu (s\otimes t)\bigr)_{\rm Serre} &=&\displaystyle \int_C \bigl(\eta \<s,t\> + {\rm trace}(\xi\ s\wedge t)\bigr)\\ &&\\ &=&\displaystyle \int_C \< \xi s+\eta s,t\>,\\ \end{array} \end{equation} where we have identified $\bigwedge^2 {\bf C}^m$ with ${\bf so}_m$, the space of skew-symmetric $m\times m$ matrices, and used the fact that under this identification one has ${\rm trace}(\xi\ s\wedge t) = \<\xi s,t\>$, as one verifies by an easy calculation. Now for $(L,E) \in {\cal W}_k - {\cal W}_{k+1}$, the Zariski tangent space $T_{(L,E)}{\cal W}_k$ is the linear span of directions $\eta \oplus \xi$ in which {\it all} sections $s$ extend. By the lemma this condition on $\eta \oplus \xi$ is that $\xi s + \eta s =0$ in cohomology for all $s\in H^0(C, L \otimes E({\bf C}^m))$. By (\ref{pair}) this linear span is precisely $({\rm image \ } \mu)^{\perp}$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} We come now to the first main result of this section: \begin{theo} \label{gen0} For generic $(L,E)\in J^{g-1}(C)\times {\cal M}^{\pm}(SO_m)$ one has $H^0(C,L\otimes E({\bf C}^m)) =0$. \end{theo} {\it Proof.}\ We choose a component of $J^{g-1} \times {\cal N}^{st}(m)$, and suppose that $h^0(C,L\otimes E({\bf C}^m))\geq k>0$ everywhere in this component, and generically equal to $k$, for some natural number $k$. We then choose a generic point $(L,E)$ in this component; according to proposition \ref{petri} the Petri map $\mu$ is identically zero here. This means that for arbitrary sections $s\in H^0(C,L\otimes E({\bf C}^m))$ and $t\in H^0(C,KL^{-1}\otimes E({\bf C}^m))$ (and note that both spaces have the same dimension $k$, by Riemann-Roch) we have, on the one hand, $s\wedge t =0$. This implies that $s,t$ generically generate the same line subbundle $F\subset E({\bf C}^m)$; moreover, since $s,t$ are arbitrary we see that $F$ is independent of their choice, and depends only on $L$ and $E$. On the other hand, again by (\ref{mudef}), $\<s,t \>=0$; this implies that $F$ is an {\it isotropic} line subbundle. (Of course, since $E$ is generic this is also forced by corollary \ref{alliso}.) Lemma \ref{nagata} and genericity of $E$ therefore implies that $ \deg F \leq -g+1. $ But by definition of $F$ the spaces of sections $H^0(L\otimes F)$ and $H^0(KL^{-1}\otimes F)$ are both nonzero, which forces $L\otimes F = KL^{-1}\otimes F = {\cal O}$ and hence $L^2 = K$. But this contradicts the genericity of $L$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} We now consider the intersection of the subscheme ${\cal W}_k$ with the fibre $\{K^{1\over 2}\} \times {\cal N}^{st}(m)$, where $K^{1\over 2} \in \vartheta(C)$ is a theta characteristic. (Indeed, theorem \ref{gen0} above also follows---for all cases except ${\cal M}^-(SO_{2n})$---from theorem \ref{gen0th} below.) Fixing $K^{1\over 2}$, one sees that by Serre duality the Petri map now factorises into two maps: $$ \begin{array}{rcl} \mu_J : S^2 H^0(C,K^{1\over 2} \otimes E({\bf C}^m)) & \rightarrow & H^0(C,K),\\ \mu_{{\cal N}}: \bigwedge^2 H^0(C,K^{1\over 2} \otimes E({\bf C}^m)) & \rightarrow & H^0(C,K\otimes\bigwedge^2 E({\bf C}^m))\\ && \cong H^0(C,K\otimes E({\bf so}_m)),\\ \end{array} $$ where $\mu_J : s\otimes t \mapsto \<s,t\>$, and $\mu_{{\cal N}}$ is the natural multiplication map. As an immediate consequence of proposition \ref{petri} we obtain: \begin{cor} \label{thpetri} If $E\in {\cal N}^{st}(m)$ and $K^{1\over 2} \in \vartheta(C)$ satisfy $h^0(C,K^{1\over 2} \otimes E({\bf C}^m)) = k$ then the subscheme ${\cal U}_k = {\cal W}_k |_{\{K^{1/2}\} \times {\cal N}^{st}(m)} \subset {\cal N}^{st}(m)$ has Zariski tangent space $ T_E {\cal U}_k = ({\rm image \ } \mu_{{\cal N}})^{\perp}. $ \end{cor} {}From this follows our second main result: \begin{theo} \label{gen0th} For $m\geq 3$ and for any theta characteristic $K^{1\over 2} \in \vartheta(C)$ we have $h^0(C,K^{1\over 2} \otimes E({\bf C}^m)) = 0$ or 1 for generic $E\in {\cal M}^{\pm}(SO_m)$, where the parity is determined by lemma \ref{2.11}. \end{theo} {\it Proof.}\ We fix our component ${\cal M}^{\pm}(SO_m)$, and suppose that $h^0(C,K^{1\over 2}\otimes E({\bf C}^m))\geq k$ everywhere in this component, generically equal to $k$. We then choose a generic point $E$ in this component; by proposition \ref{thpetri} the Petri map $\mu_{{\cal N}}$ vanishes here. As in the proof of theorem \ref{gen0} this means $s\wedge t =0$ for all sections $s,t \in H^0(C,K^{1\over 2}\otimes E({\bf C}^m))$, so all sections generate the same line subbundle $F\subset E({\bf C}^m)$; in particular $$ H^0(C,K^{1\over 2}\otimes F) = H^0(C,K^{1\over 2}\otimes E({\bf C}^m)). $$ Since $E$ is generic it follows by corollary \ref{alliso} that $F$ is isotropic, and hence by lemma \ref{nagata} that $\deg F \leq -g+1$. But then $\deg K^{1\over 2}\otimes F \leq 0$ and hence $k=h^0(C,K^{1\over 2}\otimes F) \leq 1$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{cor} \label{serregen0} Suppose that $p: {\widetilde C}\rightarrow C$ is an unramified double cover and that $E,L$ is a {\it generic} pair as in corollary \ref{serre}. Then---except possibly in the case of $m$ even, $w_2(E) \equiv 1$---we have $ h^0({\widetilde C},L \otimes p^* E({\bf C}^m)) =0 $ or 1, where the parity is determined by corollary \ref{serre}. \end{cor} {\it Proof.}\ As in the proof of \ref{serre} we take $L= p^* K^{1\over 2}$ to be the pull-back of a theta characteristic. (See also the discussion of section \ref{main}.) Then $$ h^0({\widetilde C},L \otimes p^* E({\bf C}^m)) = h^0(C,K^{1\over 2}\otimes E({\bf C}^m)) + h^0(C,K^{1\over 2}\otimes\eta \otimes E({\bf C}^m)), $$ where $\eta\in J_2(C)$ is the 2-torsion point associated to the covering. If $E$ is generic then by theorem \ref{gen0th} it is possible to choose $K^{1\over 2}$---in all cases except when $m$ is even and $w_2(E)$ is odd---so that the right-hand side is $\leq 1$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{rem}\rm In the case of $m$ even, $w_2(E)$ odd, $h^0({\widetilde C},L \otimes p^* E({\bf C}^m))\geq 2$ for any $L=p^*K^{1\over 2}$; so the above argument fails. Nonetheless, we expect that the result \ref{serregen0} is still true in this case, but its proof requires a refinement of the Brill-Noether analysis of this section. \end{rem} \section{The Verlinde formula} \label{vformula} In this section we shall write down, for the unitary and spin groups, the Verlinde formula which calculates the dimension of the vector spaces $H^0({\cal M}(G),\Theta(V))$. For the derivations of these formulae we refer the reader to \cite{B3}, \cite{OW}. In fact, what one writes down is a natural number $N_l(G)$ depending on the group, on the genus $g$, and on an integer $l$ called the `level'. Then to any representation $V$ of $G$, one associates a level $l=d_V$ such that $\dim H^0({\cal M}(G),\Theta(V)) = N_{d_V}(G)$. \subsection{Preliminaries} For $k\in {\bf N}$ and $r\in {\bf Q}$ we let $$ f_k(r) =4\sin^2(r\pi/k)=(1-\zeta_{k}^r)(1-\zeta_{k}^{-r}) $$ where $\zeta_k = e^{2\pi i /k}$. This satisfies certain obvious identities (we shall usually drop the subscript $k$ for convenience): \begin{lemm} \label{fids1} \item[(i)] $f(r)=f(-r)$; \item[(ii)]$ f(r)=f(k-r)$; \item[(iii)] $ f(k/2)=4$; \item[(iv)] $ f(2r)=f(r)f(k/2-r)$; \item[(v)] $ \prod_{r=1}^{k-1}f(r)=k^2$. \end{lemm} In addition, we shall need the following: \begin{lemm} \label{fids2} \item[(i)] If $k=2n+1$ is odd then $$ \begin{array}{lcl} \prod_{r=1}^{n}f(r) &=& 2n+1,\\ \prod_{r=1}^{2n}f(r)^{[{r\over 2}]} &=& (2n+1)^n.\\ \end{array} $$ \item[(ii)] If $k=2n$ is even then $$ \begin{array}{lcl} \prod_{r=1}^{n-1}f(r) &=& n,\\ \prod_{r=1}^{2n-1}f(r)^{[{r\over 2}]} &=& 2^{n-1} n^n\\ \prod_{r=1}^{2n-1}f(r)^{[{r+1\over 2}]} &=& 2^{n+1} n^n.\\ \end{array} $$ \end{lemm} {\it Proof.}\ (i) The first identity follows at once from parts (ii) and (v) of the previous lemma. For the second, use \ref{fids1} (ii) to write $$ \prod_{r=1}^{2n}f(r)^{[{r\over 2}]} = \prod_{r=1}^{n}f(r)^{[{r\over 2}] + [{2n+1-r\over 2}]} = \prod_{r=1}^{n}f(r)^n = (2n+1)^n. $$ (ii) Again the first identity is an easy consequence of the previous lemma. We shall just give the proof of the third identity, that of the second being being almost the same. The left-hand product can be rewritten, using \ref{fids1} (ii), as: $$ P = f(n)^{[{n+1\over 2}]} \times \prod_{r=1}^{n-1}f(r)^{[{r+1\over 2}] + [{2n-r+1\over 2}]}. $$ We observe that $$ \Bigl[{r+1\over 2}\Bigr] + \Bigl[{2n-r+1\over 2}\Bigr] = \cases{n & if $r$ is even,\cr n+1 & if $r$ is odd.\cr} $$ So suppose first that $n$ is even. Using \ref{fids1} (iii) and $\prod_{r=1}^{n-1}f(r) = n$ we see that $$ \begin{array}{rcl} P &=& 2^n f(1)^{n+1}f(2)^n \cdots f(n-2)^n f(n-1)^{n+1}\\ &=& (2n)^n f(1)f(3) \cdots f(n-1).\\ \end{array} $$ We claim that $f(1)f(3) \cdots f(n-1) = 2$---from which the third identity in \ref{fids2} (ii) follows. To see this, observe first that (using \ref{fids1} (iv)) $$ \begin{array}{rcl} f(2)f(4) \cdots f(n-2) &=& f(n/2)^{-1} \prod_{r=1}^{n-1}f(r) \\ &=& n/2;\\ \end{array} $$ and second that $$ n = f(1) \cdots f(n-1) \times f(2) \cdots f(n-2). $$ The reasoning for $n$ odd is similar. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} In what follows we shall consider sets $U = \{u_1,\ldots ,u_n\}$ of rational numbers; and for such a set we define: $$ \begin{array}{rcl} \Psi_k(U) &=& \displaystyle\prod_{1\le i<j\le n} f(u_i -u_j) \\ &&\\ \Pi_k(U) &=& \displaystyle\prod_{1\le i<j\le n} f(u_i -u_j)f(u_i +u_j) \\ \Phi_k(U) &=& \displaystyle \prod_{1\le i<j\le n} f(u_i -u_j)f(u_i +u_j) \times \prod_{i=1}^n f(u_i) .\\ \end{array} $$ \subsection{Computations} \begin{ex} \label{4.1} $\bf SL_{n}.$ \rm The Verlinde number in this case is: $$ N_l(SL_n) = \sum_U \Bigl( {n(l+n)^{n-1}\over \Psi_{l+n}(U)} \Bigr)^{g-1} $$ where the sum is taken over $U = \{ 0=u_0<u_1<\cdots <u_{n-1}<l+n \}$. Since we are concerned with the spin groups we shall record at this point the result of computing this expression at levels 1 and 2 for ${\rm Spin}_6 = SL_4$: \begin{equation} \label{spin6} \begin{array}{crcl} \bullet & N_1(SL_4) &=& 2^{2g};\\ \bullet & N_2(SL_4) &=& 2^{3g-1}3^{g-1} + 2^{3g-1} + 2^g 3^{g-1}.\\ \end{array} \end{equation} \end{ex} \begin{ex} \label{4.2} $\bf Spin_{2n},\ n \geq 4.$ \rm The Verlinde number at level $l$ is here: $$ N_l({\rm Spin}_{2n}) = \sum_{U\in P_l(2n)} \Bigl( {4k^n \over \Pi_k (U)} \Bigr)^{g-1}, \qquad {\rm where}\ k=l+2n-2; $$ where $P_l = P_l(2n)$ denotes the collection of sets $U=\{u_1<\cdots <u_n\}$ satisfying the following conditions. (Note that this collection is finite provided $n\geq 3$.) \begin{enumerate} \item $ u_i \in {1\over 2}{\bf Z}$; \item $ u_{i+1} -u_{i} \in {\bf Z}$ for $i=1,\ldots n-1$; \item $ u_1 +u_2 >0$; and \item $ u_{n-1} +u_n < k=l+2n-2$. \end{enumerate} It will be convenient to write $P_l = P_l^+ \cup P_l^-$ where $P_l^+$ (resp. $P_l^-$) consists of $U$ with all $u_i \in {\bf Z}$ (resp. $u_i \in {1\over 2}{\bf Z}\backslash {\bf Z}$). Correspondingly the Verlinde number splits up as $N_l = N_l^+ + N_l^-$ where $$ N_l^{\pm}({\rm Spin}_{2n}) = \sum_{U\in P_l^{\pm}(2n)} \Bigl( {4k^n \over \Pi_k (U)} \Bigr)^{g-1}. $$ \begin{rem} \rm $P_l(2n)$ may be viewed as a set of highest weights of irreducible representations of ${\rm Spin}_{2n}$ (namely, those weights in a fundamental chamber for the action of the affine Weyl group of level $l$). Then $P_l^+$ is the subset of `tensor' representations---those which descend to $SO_{2n}$---and $P_l^-$ is the subset of `spinor' representations. The same remark applies to the odd spin groups below. \end{rem} As for $SL_4$ let us note the lowest cases of this formula. (Note, incidentally, that the formulae (\ref{spin2n}) are consistent with ${\rm Spin}_4 = SL_2\times SL_2$ and ${\rm Spin}_6 = SL_4$: for $n=2,3$ respectively $N_l$ coincides with $N_l(SL_2)^2$ and $N_l(SL_4)$.) \begin{equation} \label{spin2n} \begin{array}{clcl} \bullet & N_1({\rm Spin}_{2n}) &=& 2^{2g}\\ \bullet & N_2^+({\rm Spin}_{2n}) &=& (2n)^{g-1}(n-1) + 2^{3g-1}n^{g-1} \\ \bullet & N_2^-({\rm Spin}_{2n}) &=& 2^{3g-1}.\\ \end{array} \end{equation} {\it Proof.}\ We shall just prove the formulae for $N_2^{\pm}$, and leave $N_1$ (which we shall not need) to the reader. We begin by listing the sets over which the summation takes place: $$ \begin{array}{lcl} P_2^+(2n): && \{0,1,\ldots, n-2 ,n\pm 1 \} \\ && \{0,1,\ldots, n-2,n \} \\ && \cdots \\ && \{0,2,\ldots, n-1,n \} \\ && \{\pm 1,2,\ldots, n-1,n \}; \\ &&\\ P_2^-(2n): && \{\pm{1\over 2},{3\over 2}, \ldots, n \pm {1\over 2} \}. \\ \end{array} $$ We recall from \cite{OW} that reflection of the end-points in $0,n$---i.e. the $\pm$ signs in these sets---defines an action of ${\bf Z}/2 \times {\bf Z}/2$ on $P_2(2n)$ under which $\Pi_k(U)$ is easily seen to be invariant. Thus, for example, $P_2^-(2n)$ is a single orbit and so \begin{equation} \label{N2-} N_2^-({\rm Spin}_{2n}) = 4 \times \Bigl({4(2n)^n\over \Pi_{2n}(U)}\Bigr)^{g-1} \end{equation} where $ U = \{{1\over 2},{3\over 2} \ldots,n - {1\over 2} \}. $ We have here $$ \begin{array}{rcl} \Pi_{2n}(U) &=& \displaystyle \prod_{1\leq i<j\leq n } f_{2n}(i-j) f_{2n}(i+j-1) \\ &=& \displaystyle\prod_{r=1}^{2n-1} f_{2n}(r)^{m_r},\\ \end{array} $$ where the multiplicities $m_r$ are to be determined. Namely, $m_r = a_r + b_r$ where $a_r$ is the number of pairs $i<j$ such that $j-i = r$, and $b_r$ is the number of pairs $i<j$ such that $i+j = r+1$. From this we see that $$ \begin{array}{rcl} a_r &=& \cases{n-r & if $r<n$\cr 0 & if $r\geq n$,\cr}\\ b_r &=& \cases{[ {r/ 2}] & if $r<n$\cr [ {(2n-r)/ 2}] & if $r\geq n$;\cr}\\ \end{array} $$ and hence that $$ m_r = n-\Bigl[ {r+1\over 2}\Bigr]. $$ So it follows from \ref{fids1} (v) and \ref{fids2} (ii) that $$ \Pi_{2n}(U) = {(2n)^{2n}\over \prod_{r=1}^{2n-1}f(r)^{[{r+1\over 2}]}} ={1\over 2}(2n)^n ; $$ and hence from (\ref{N2-}) that $N^-_2({\rm Spin}_{2n}) = 2^{3g-1}$. \medskip Let us now turn to $N^+_2({\rm Spin}_{2n})$. For $l=0,1,\ldots ,n$ we shall write $$ U_l = \{ 0,1,\ldots,\widehat l,\ldots ,n\}, $$ i.e. $l$ has been deleted. Thus $P_2^+(2n)$ consists of $U_0,U_1,\ldots,U_n$ together with the reflections of $U_0$ and $U_n$ under the action of ${\bf Z}/2 \times {\bf Z} /2$; and so \begin{equation} \label{N2+} \begin{array}{rcl} N_2^+({\rm Spin}_{2n}) &=& \displaystyle 2 \Bigl( {4(2n)^n \over \Pi_{2n}(U_0)} \Bigr)^{g-1} +2 \Bigl( {4(2n)^n \over \Pi_{2n}(U_n)} \Bigr)^{g-1}\\ &&\\ &&\displaystyle\ \ \ \ \ \ \ +\sum_{l=1}^{n-1}\Bigl( {4(2n)^n \over \Pi_{2n}(U_l)} \Bigr)^{g-1}.\\ \end{array} \end{equation} Now we can write, for each $l$, $\Pi_{2n}(U_l) = N/D_l$ where: $$ \begin{array}{rcl} N &=& \displaystyle \prod_{0\leq i<j\leq n} f(j-i)f(i+j),\\ D_l &=&\displaystyle \prod_{j=l+1}^n f(j-l)f(j+l) \prod_{i=0}^{l-1} f(l-i)f(l+i) \quad {\rm for}\ 1\leq l \leq n-1.\\ \end{array} $$ For $l=0$ and $n$ the denominator takes the slightly simpler forms: $$ D_0 = \prod_{j=1}^n f(j)^2 = 16 n^2, $$ using \ref{fids1} (iii) and \ref{fids2} (ii); and likewise $$ \begin{array}{rcl} D_n &=& \prod_{i=0}^{n-1} f(n-i)f(n+i)\\ &=&\prod_{i=0}^{n-1} f(n-i)^2\\ &=&\prod_{i=1}^{n} f(i)^2\\ &=& 16 n^2.\\ \end{array} $$ To compute the denominator $D_l$ for $1\leq l \leq n-1$: $$ \begin{array}{rclr} D_l &=& f(1) \cdots f(n-l) \times f(2l+1)\cdots f(n+l)&\\ && \times f(1) \cdots f(l) \times f(l) \cdots f(2l -1)&\\ &&&\\ &=&\displaystyle f(n){f(l)\over f(2l)} \prod_{r=1}^{n-1}f(r)^2 {f(n+1)\cdots f(n+l)\over f(n-1)\cdots f(n-l+1)}&\\ &&&\\ &=& 4n^2 {f(l)f(n+l)/ f(2l)}&\hbox{by \ref{fids1} (ii),(iii)}\\ &&&\hbox{and \ref{fids2} (ii);}\\ &=& 4n^2 {f(n+l)/ f(n-l)}&\hbox{using \ref{fids1} (iv);}\\ &=& 4n^2.&\\ \end{array} $$ We next compute the numerator: $$ N = \prod_{r=1}^{2n-1} f(r)^{m_r} $$ where $m_r = a_r + b_r$ and $a_r$ is the number of pairs $i<j$ between $0$ and $n$ such that $j-i =r$, and $b_r$ the number such that $i+j =r$. So: $$ \begin{array}{rcl} a_r &=& \cases{n+1-r & if $r\leq n$\cr 0 & if $r> n$,\cr}\\ b_r &=& \cases{[ {(r+1)/ 2}] & if $r\leq n$\cr [ {(2n-r+1)/ 2}] & if $r> n$;\cr}\\ \end{array} $$ from which we find $$ \begin{array}{rcl} m_r &=& \cases{n+1 -[r/2] & if $r\leq n$\cr n -[r/2] & if $r> n$.\cr}\\ \end{array} $$ Hence $$ N = \prod_{r=1}^{2n-1} f(r)^{n-[r/2]} \times \prod_{r=1}^n f(r) = 4(2n)^{n+1}, $$ using \ref{fids1} (v) and \ref{fids2} (ii). Putting together the above computations we obtain $ \Pi_{2n}(U_l) = (2n)^{n-1}$ for $l=0,n$, and $2^{n+1}n^{n-1}$ for $l=1,\ldots ,n-1$. Substituting into (\ref{N2+}) this gives $$ N^+_2({\rm Spin}_{2n}) = (2n)^{g-1}(n-1) + 2^{3g-1}n^{g-1} $$ as asserted. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \end{ex} \begin{ex} \label{4.3} $\bf Spin_{2n+1}, n \geq 2.$ \rm In this case the Verlinde number is: $$ N_l({\rm Spin}_{2n+1})= \sum_{U\in P_l(2n+1)} \Bigl( {4k^n \over \Phi_k(U)} \Bigr)^{g-1}, \qquad {\rm where}\ k=l+2n-1; $$ where $P_l(2n+1)$ consists of $U=\{0<u_1 <\cdots < u_n \}$ such that \begin{enumerate} \item $ u_i \in {1\over 2}{\bf Z}$; \item $ u_{i+1} -u_{i} \in {\bf Z}$ for $i=1,\ldots n-1$; \item $ u_{n-1} +u_n < k=l+2n-1$. \end{enumerate} As in the even case we shall write $$ N_l^{\pm}({\rm Spin}_{2n+1}) = \sum_{U\in P_l^{\pm}} \Bigl( {4k^n \over \Phi_k (U)} \Bigr)^{g-1}, $$ where $P_l^{\pm}$ denotes the subsets of integral and half-integral $U$ respectively. Here the lowest Verlinde numbers are: \begin{equation} \label{spin2n+1} \begin{array}{clcl} \bullet & N_1({\rm Spin}_{2n+1}) &=& 2^{g-1}(2^g + 1)\\ \bullet & N_2^+({\rm Spin}_{2n+1}) &=& 2^{2g-1} \\ \bullet & N_2^-({\rm Spin}_{2n+1}) &=& (2n+1)^{g-1}(2^{2g-1} + n).\\ \end{array} \end{equation} We shall omit the proof of these formulae, as it is entirely similar to that of (\ref{spin2n}). Moreover they are proved (by a somewhat clumsier method) in \cite{O}. \end{ex} \subsection{Sections of theta bundles} Our interest in the preceding calculations lies in the fact that $N_2({\rm Spin}_m)$ is the dimension of $H^0({\cal M}({\rm Spin}_m),\Theta({\bf C}^m))$, where $\Theta({\bf C}^m)$ is the theta line bundle, defined in section \ref{mod}, for the standard orthogonal representation. To each irreducible representation $V$ of a group $G$ there is associated an integer $d_V$---the {\it height} or {\it Dynkin index} of the representation---for which, for any $r\in {\bf Z}$, one has \begin{equation} \label{verlinde} h^0({\cal M}(G),\Theta(V)^{\otimes r}) = N_{rd_V}(G). \end{equation} (For a useful discussion of the Dynkin index and the proof of (\ref{verlinde}) see \cite{LS}; see also \cite{BL}, \cite{F}, \cite{KNR}.) When $V={\bf C}^m$ is the standard orthogonal representation the height is: \begin{equation} \label{di2} d_V= \cases{ 4 & if $m=3$,\cr 2 & if $m\geq 5$.\cr} \end{equation} What is the analogue of (\ref{verlinde}) for the `twisted' moduli space ${\cal M}^-({\rm Spin}_m)$ defined in section \ref{clif}? The following is a refinement (which deals with even as well as odd $m$) of conjecture (5.2) in \cite{OW}. \begin{conj} \label{conj} For $m\geq 3$, and any representation $SC_m \rightarrow SL(V)$ satisfying (\ref{ss}), $$ h^0({\cal M}^-({\rm Spin}_m),\Theta(V)) = (-1)^m (N_{d_V}^+ -N_{d_V}^-). $$ \end{conj} \begin{ex} $\bf m=3.$ \rm By example \ref{4.1}, $$ N_l({\rm Spin}_3) = N_l(SL_2) = \sum_{j=1}^{l+1} \Bigl( {l+2 \over 2\sin^2{j\pi \over l+2}} \Bigr)^{g-1}. $$ Notice that formally computing \ref{4.3} with $n=1$ (strictly speaking the Verlinde number \ref{4.3} makes sense only for $n\geq 2$) we find (taking $u_0 = 0$): $$ N_l({\rm Spin}_{2n+1})|_{n=1} = \sum_{j=1}^{2l+1} \Bigl( {l+1 \over 4\sin^2{j\pi \over 2l+2}} \Bigr)^{g-1} = N_{2l}(SL_2). $$ Now, by (\ref{verlinde}) and the fact that ${\cal L}_0 = \Theta({\bf C}^2)$ where $d_{{\bf C}^2} =1$, we have $$ h^0({\cal M}({\rm Spin}_3), \Theta(V)) = h^0({\cal SU}_C(2),{\cal L}_0^{d_V}) = N^+_{d_V}(SL_2)+N^-_{d_V}(SL_2). $$ (See example \ref{clifm=3}.) On the other hand, on ${\cal M}^-({\rm Spin}_3) = {\cal SU}_C(2,1)$ we have, given a Clifford representation $V$ for which $d_V$ is even, $\Theta (V) = {\cal L}_0^{d_V} = {\cal L}_1^l$ where $2l = d_V$. Thaddeus's twisted Verlinde formula (\cite{T}, corollary (18)) tells us that $$ \begin{array}{rcl} h^0({\cal SU}_C(2,1),{\cal L}_1^{l}) &=& \displaystyle \sum_{j=1}^{2l+1} (-1)^{j+1}\Bigl( {l+1 \over 4\sin^2{j\pi \over 2l+2}} \Bigr)^{g-1}\\ &&\\ &=& -N^+_{2l}(SL_2)+N^-_{2l}(SL_2);\\ \end{array} $$ or in other words $ {\cal M}^-({\rm Spin}_3),\Theta(V)) = -N^+_{d_V}(SL_2)+N^-_{d_V}(SL_2). $ \end{ex} \begin{ex} $\bf m=6.$ \rm By examples \ref{clifm=6} and \ref{4.1} $$ h^0({\cal M}({\rm Spin}_6), \Theta(V)) = h^0({\cal SU}_C(4),{\cal L}_0^{d_V}) = N^+_{d_V}(SL_4)+N^-_{d_V}(SL_4), $$ where $$ h^0({\cal SU}_C(4),{\cal L}_0^{l}) = N_l(SL_4) = \sum_{U} \Bigl( {4(l+4)^3 \over \Psi_{l+4}(U)} \Bigr)^{g-1}, $$ summed over $U=\{0=u_0<u_1<u_2<u_3\leq l+3 \}\subset {\bf Z}$. On the other hand, ${\cal M}^-({\rm Spin}_6) = {\cal SU}_C(4,2)$ and by the same arguments as in the previous example, the conjecture is in this case equivalent to: $$ \begin{array}{rcl} h^0({\cal SU}_C(4,2),{\cal L}_2^{l}) &=& N^+_{2l}(SL_4) - N^-_{2l}(SL_4)\\ &&\\ &=& \displaystyle \sum_{U} (-1)^{u_2}\Bigl( {4(2l+4)^3 \over \Psi_{2l+4}(U)} \Bigr)^{g-1},\\ \end{array} $$ summed over $U=\{0=u_0<u_1<u_2<u_3\leq 2l+3 \}\subset {\bf Z}$. One can possibly verify this using \cite{BL} theorem 9.4. \end{ex} \section{Numerology} \label{numerology} Our principal aim in this section to make some sense of the Verlinde numbers (\ref{spin6}), (\ref{spin2n}) and (\ref{spin2n+1}) computed in the previous section at level 2---those, that is, associated to the orthogonal representation of the Clifford group. But we begin with a remark, independent of the rest of the paper, which deals with the Verlinde number (\ref{spin2n+1}) at level 1. \subsection{A remark on the generator of ${\rm Pic}\ {\cal M}({\rm Spin}_{2n+1})$} \label{pfaffian} It is known from \cite{LS} that the Picard group of the moduli scheme ${\cal M}({\rm Spin}_{m})$ is infinite cyclic. When $m=2n+1$ is odd the ample generator ${\cal P}$ is constructed as a Pfaffian bundle, i.e. $2{\cal P} = \Theta({\bf C}^{2n+1})$. (When $m=2n$ is even ${\cal P}$ exists only as a Weil divisor class.) Then according to (\ref{spin2n+1}) the space of sections of this line bundle has dimension $$ h^0({\cal M}({\rm Spin}_{2n+1}), {\cal P}) = N_1({\rm Spin}_{2n+1}) = 2^{g-1}(2^g +1). $$ This formula is striking, first because it is independent of $n$ and second because it is the number of even theta characteristics of the curve. In fact it is easy to see how to construct a basis for the linear system $|{\cal P}|$, as follows. Let $\vartheta^+(C)$ denote the set of even theta characteristics, and for each $L \in \vartheta^+(C)$ consider the reduced divisor $$ D_L = \{ E \in {\cal M}({\rm Spin}_{2n+1}) | H^0 (C,L\otimes E({\bf C}^{2n+1})) \not=0\}. $$ By proposition \ref{2.11} the dimension of $H^0 (C,L\otimes E({\bf C}^{2n+1}))$ is always even, and so by theorem \ref{gen0th} $D_L$ is a proper subset of the moduli space. Consequently it is divisor---this is shown in proposition \ref{div} below---and in fact, since we take $D_L$ to be reduced, some multiple $k D_L \in |\Theta({\bf C}^{2n+1})| = |2 {\cal P}|$. Since the dimension of $H^0 (C,L\otimes E({\bf C}^{2n+1}))$ is always even we see that $k$ is at least, and hence equal to, 2. We have therefore constructed a set of divisors on ${\cal M}({\rm Spin}_{2n+1})$: $$ D_L \in | {\cal P} |, \qquad L\in \vartheta^+(C). $$ In the case $n=1$ these were shown by Beauville \cite{B2} to be linearly independent, and hence a basis of the linear system; we expect the same to be true in general. \subsection{Prym varieties} \label{pryms} We recall the following `Verlinde numbers' for principally polarised abelian varieties. Let $(A,\Xi)$ be any principally polarised abelian variety of dimension $g$, where $\Xi$ is a symmetric divisor representing the polarisation; and let $$ H^0(A, m\Xi) = H^0_+(A, m\Xi)\oplus H^0_-(A, m\Xi) $$ be the decomposition into $\pm$-eigenspaces under the canonical involution of $A$. Then by writing down a suitable basis of theta functions one can easily verify that: \begin{equation} \label{abtheta} \dim H^0_{\pm}(A, m\Xi) = \cases{ {(m^g \pm 2^g )/ 2} & if $m\equiv 0$ mod 2,\cr {(m^g \pm 1 )/ 2} & if $m\equiv 1$ mod 2.\cr} \end{equation} Associated to a smooth projective curve $C$ we have a natural configuration of principally polarised {\it Prym varieties}. Let us recall the usual notation (see \cite{ACGH}). For each nonzero half-period $\eta \in J_2(C) \backslash \{{\cal O}\}$ we have an unramified double cover $$p : \cctil_{\eta} \rightarrow C.$$ Writing ${\widetilde J^{2g-2}} = J^{2g-2}(\cctil_{\eta})$ we have $${\rm Nm}_p ^{-1}(K_C) = P_{\eta} \cup P_{\eta}^- \subset {\widetilde J^{2g-2}};$$ where $P_{\eta},\ P_{\eta}^-$ are disjoint translates of the same abelian subvariety, characterised by the condition that for $L\in {\rm Nm}_p^{-1}(K_C)$: \begin{equation} \label{prym+-} h^0(\cctil_{\eta}, L) \equiv \cases{0& mod 2 if $L\in P_{\eta}$,\cr 1& mod 2 if $L\in P_{\eta}^-$.\cr} \end{equation} Then $P_{\eta}$ is called the Prym variety of the covering. We shall denote by ${\Xi_{\eta}}$ the symmetric divisor representing the canonical principal polarisation on $P_{\eta}$, defined by $2 {\Xi_{\eta}} = P_{\eta} \cap \widetilde \Theta$, where $\widetilde \Theta$ is the theta-divisor in ${\widetilde J^{2g-2}}$. We shall allow also $\eta = 0$ by setting $(P_0 ,\Xi_0) = (J^{g-1}(C),\theta)$. Notice that on $P_{\eta}^-$ both the polarisation and the line bundle $2{\Xi_{\eta}}$ are defined, and identify under translation with the same objects on $P_{\eta}$. The distinguished line bundle ${\Xi_{\eta}}$ which represents the polarisation is no longer defined on $P_{\eta}^-$, however. Thus in what follows the line bundle $m{\Xi_{\eta}}$ makes sense on $P_{\eta} ^-$ {\it only if $m$ is even.} \subsection{Spin Verlinde numbers versus Prym Verlinde numbers} We may now formulate the computations of section \ref{vformula} as follows. \begin{theo} \label{numer} \begin{enumerate} \item If $m\geq 6$ is even then $$ \begin{array}{rcl} N_2({\rm Spin}_{m}) = N_2^+ +N_2^- &=& \displaystyle \sum_{\eta\in J_2(C)}h^0_+(P_{\eta},m{\Xi_{\eta}}) + \sum_{\eta \not= 0}h^0_+(P_{\eta}^-,m{\Xi_{\eta}}), \\ N_2^+ -N_2^- &=& \displaystyle\sum_{\eta\in J_2(C)}h^0_-(P_{\eta},m{\Xi_{\eta}}) + \sum_{\eta \not= 0}h^0_-(P_{\eta}^-,m{\Xi_{\eta}}).\\ \end{array} $$ \item If $m\geq 5$ is odd then $$ \begin{array}{rcl} N_2({\rm Spin}_{m}) = N_2^+ +N_2^- &=&\displaystyle \sum_{\eta\in J_2(C)}h^0_+(P_{\eta},m{\Xi_{\eta}}),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ -N_2^+ +N_2^- &=& \displaystyle\sum_{\eta\in J_2(C)}h^0_-(P_{\eta},m{\Xi_{\eta}}).\\ \end{array} $$ \end{enumerate} \end{theo} {\it Proof.}\ Equations (\ref{spin6}), (\ref{spin2n}), (\ref{spin2n+1}) and (\ref{abtheta}). {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{rems}\rm {\it (i)} Of course, when $m$ is even $P_{\eta}$ and $P_{\eta}^-$ are interchangeable as far as the dimensions alone are concerned. But in view of the constructions which follow in the next sections, together with the remarks in section \ref{dynkin}, it will be seen that the right-hand side of part 1 is the correct way to write these identities. {\it (ii)} For $m=3$ there is an analogous statement obtained by replacing $N_2$ by $N_4$. This case has been discussed at length in \cite{OP}. \end{rems} These results imply, via (\ref{verlinde}), (\ref{di2}) and \ref{conj} (assuming the validity of the latter) identities among $h^0$s which we can summarise in the following table. \begin{equation} \label{dims} \begin{array}{|c||l|l|} \hline &&\\ &h^0({\cal M}({\rm Spin}_m),\Theta({\bf C}^m))&h^0({\cal M}^-({\rm Spin}_m),\Theta({\bf C}^m))\\ &&\\ \hline &&\\ {\rm odd}\ m& \displaystyle\sum_{\eta \in J_2} h^0_+(P_{\eta}, m {\Xi_{\eta}})& \displaystyle\sum_{\eta \in J_2} h^0_-(P_{\eta}, m {\Xi_{\eta}})\\ &&\\ \hline &&\\ {\rm even}\ m&\displaystyle\sum_{\eta \in J_2} h^0_+(P_{\eta}, m {\Xi_{\eta}})& \displaystyle\sum_{\eta \in J_2} h^0_-(P_{\eta}, m {\Xi_{\eta}})\\ &&\\ & \displaystyle + \sum_{\eta \in J_2\backslash \{0\}} h^0_+(P_{\eta}^-, m {\Xi_{\eta}}) & \displaystyle + \sum_{\eta \in J_2\backslash \{0\}} h^0_-(P_{\eta}^-, m {\Xi_{\eta}}) \\ &&\\ \hline \end{array} \end{equation} \bigskip Although theorem \ref{numer} is stated only for $m\geq 5$, table (\ref{dims}) is in fact valid for {\it all} $m\in {\bf N}$, as we shall see next by a case-by-case examination. \begin{ex} $\bf m=1.$ \rm We can identify ${\cal M}({\rm Spin}_1)$ with $J_2(C)$; while ${\cal M}^-({\rm Spin}_1)$ is empty. So (\ref{dims}) is trivial in this case. \end{ex} \begin{ex} $\bf m=2.$ \rm {}From example \ref{clifm=2}, ${\cal M}^{\pm}({\rm Spin}_2)$ are both copies of ${\rm Pic}(C)$. However, the orthogonal theta bundle $\Theta({\bf C}^2)$ is defined {\it only} on the degree 0 component $J(C)$ of ${\cal M}({\rm Spin}_2)$, where it can be identified with the line bundle $8\theta$ where $\theta$ is a theta divisor on the Jacobian. So in this case again one may readily check from (\ref{abtheta}) that the identity in (\ref{dims}) holds. But in fact we have more: $$ \begin{array}{rcl} H^0_+(J(C),8\theta) &\cong& \displaystyle \sum_{\eta \in J_2}H^0(P_{\eta},2{\Xi_{\eta}}),\\ H^0_-(J(C),8\theta) &\cong& \displaystyle \sum_{\eta \in J_2\backslash \{0\}}H^0(P_{\eta}^-,2{\Xi_{\eta}}).\\ \end{array} $$ One can see this as follows. Start with the fact that for any symmetric line bundle ${\cal L}$ on an abelian variety $A$ one has $$ H^0(A,{\cal L}^4) \cong \bigoplus_{\alpha\in {\widehat A}_2}H^0(A,{\cal L}\otimes \alpha) $$ where ${\widehat A}_2 = \{\alpha \in {\rm Pic}^0 A | \alpha^2 = {\cal O}_A\}$. This isomorphism is obtained by pulling back sections under the squaring map $[2]:A\rightarrow A$; for each $\alpha$ we have $[2]^*({\cal L}\otimes \alpha) = {\cal L}^4$. Noting that $[2]$ commutes with the involution $[-1] :A\leftrightarrow A$ we see that in fact $ H^0_{\pm}(A,{\cal L}^4) \cong \bigoplus_{\alpha\in {\widehat A}_2}H^0_{\pm}(A,{\cal L}\otimes \alpha). $ If $\theta$ represents a principal polarisation on $A$ then we can identify $A_2 \cong {\widehat A}_2$ by $\eta \mapsto \theta_{\eta} - \theta$ where $\theta_{\eta} = t_{\eta}^*\theta$. In particular, when $A=J(C)$ and ${\cal L} = 2\theta$ we obtain $$ H^0_{\pm}(J,8\theta) \cong \bigoplus_{\eta\in J_2}H^0_{\pm}(J,\theta + \theta_{\eta}). $$ On the other hand $H^0_{\pm}(J,\theta + \theta_{\eta}) \cong H^0(P_{\eta}^{\pm},2{\Xi_{\eta}})$ for $\eta\not=0$. To see this choose $\zeta \in J(C)$ such that $\zeta^2 = \eta$. Then translation by $\zeta$ induces an isomorphism $ t_{\zeta}^* : H^0(J,\theta + \theta_{\eta})\widetilde{\rightarrow} H^0(J,2\theta), $ which is equivariant with respect to $[-1]^*$ acting on the first space and $t_{\eta}^*$ acting on the second. Hence the spaces $H^0_{\pm}(J,\theta + \theta_{\eta})$ identify with the $\pm$-eigenspaces in $H^0(J,2\theta)$ under $t_{\eta}^*$; and these in turn are classically identified with $H^0(P_{\eta}^{\pm},2{\Xi_{\eta}})$ by the Schottky-Jung relations. \end{ex} \begin{ex} $\bf m=3.$ \rm We have seen (see example \ref{clifm=3}) that $\Theta({\bf C}^3)$ restricts to ${\cal L}_0^4 \rightarrow {\cal SU}_C(2)={\cal M}({\rm Spin}_3)$ and to ${\cal L}_1^2 \rightarrow {\cal SU}_C(2,1) = {\cal M}^-({\rm Spin}_3)$---in each case the anticanonical line bundle. The coincidence of dimensions in (\ref{dims}) for this case was the first to be observed, and has been explored in \cite{OP} and \cite{schottky}. \end{ex} \begin{ex} \label{4theta}$\bf m=4.$ \rm By example \ref{clifm=4} we have ${\cal M}({\rm Spin}_4) \cong {\cal SU}_C(2)\times {\cal SU}_C(2)$; while the 4-dimensional orthogonal representation is the tensor product of the 2-dimensional representations of the distinct factors, from which the theta bundle in (\ref{dims}) is $\Theta({\bf C}^4) = pr_+^*{\cal L}_0^2 \otimes pr_-^* {\cal L}_0^2$, where ${\cal L}_0 = \Theta ({\bf C}^2)$ is the ample generator of ${\rm Pic}\ {\cal SU}_C(2) = {\bf Z}$. Thus $$ H^0({\cal M}({\rm Spin}_4), \Theta({\bf C}^4)) \cong \bigotimes^2 H^0({\cal SU}_C(2),{\cal L}_0^2), $$ and since $h^0({\cal SU}_C(2),{\cal L}_0^2) = 2^{g-1}(2^g +1)$ we can at once verify the case $m=4$ of the table. Moreover, just as for the case $m=2$ we can say more: $$ \begin{array}{rcl} S^2 H^0({\cal SU}_C(2),{\cal L}_0^2) &\cong& \displaystyle \sum_{\eta \in J_2}H^0_+(P_{\eta},4{\Xi_{\eta}}),\\ \bigwedge^2 H^0({\cal SU}_C(2),{\cal L}_0^2) &\cong& \displaystyle \sum_{\eta \in J_2\backslash \{0\}}H^0_+(P_{\eta}^-,4{\Xi_{\eta}}).\\ \end{array} $$ As in the case $m=2$ we have written down not just an identity of dimensions, but in fact an isomorphism of vector spaces. In this case it follows easily (for $C$ without vanishing theta-nulls) from the results of \cite{B2}. Similarly ${\cal M}^-({\rm Spin}_4)$ is isomorphic to ${\cal SU}_C(2,1) \times {\cal SU}_C(2,1)$, with theta bundle $\Theta({\bf C}^4) \cong pr_+^*{\cal L}_1 \otimes pr_-^* {\cal L}_1$, where ${\cal L}_1$ is the generator of ${\rm Pic}\ {\cal SU}_C(2,1)$. This time $h^0({\cal SU}_C(2,1),{\cal L}_1^2) = 2^{g-1}(2^g -1)$, the results of Beauville can be applied to give isomorphisms $$ \begin{array}{rcl} S^2 H^0({\cal SU}_C(2,1),{\cal L}_1) &\cong& \displaystyle \sum_{\eta \in J_2}H^0_-(P_{\eta},4{\Xi_{\eta}}),\\ \bigwedge^2 H^0({\cal SU}_C(2,1),{\cal L}_1) &\cong& \displaystyle \sum_{\eta \in J_2\backslash \{0\}}H^0_-(P_{\eta}^-,4{\Xi_{\eta}}),\\ \end{array} $$ and again table (\ref{dims}) is verified in this case. \end{ex} \begin{ex} \label{6theta} $\bf m=6.$ \rm In this case example \ref{clifm=6} identifies $\Theta ({\bf C}^6) \rightarrow {\cal M}^{\pm}({\rm Spin}_6)$ with ${\cal L}_0^2 \rightarrow {\cal SU}_C(4)$ and ${\cal L}_2 \rightarrow {\cal SU}_C(4,2)$ respectively; and table (\ref{dims}) for the case ${\cal M}^+({\rm Spin}_6)$ suggests isomorphisms $$ \begin{array}{rcl} H^0_+({\cal SU}_C(4),{\cal L}_0^2) &\cong& \displaystyle \sum_{\eta \in J_2}H^0_+(P_{\eta},6{\Xi_{\eta}}),\\ H^0_-({\cal SU}_C(4),{\cal L}_0^2) &\cong& \displaystyle \sum_{\eta \in J_2\backslash \{0\}}H^0_+(P_{\eta}^-,6{\Xi_{\eta}});\\ \end{array} $$ for {\it some} $\pm$-decomposition of $H^0({\cal SU}_C(4),{\cal L}_0^2)$. We shall observe in remark \ref{dynkin} in the next section that the involution of ${\cal M}({\rm Spin}_6)$ analogous to the exchange of factors in example \ref{4theta} is the dualising involution $E\mapsto E^{\vee}$ of ${\cal SU}_C(4)$. We therefore conjecture isomorphisms as above where $H^0_{\pm}$ on the left-hand side are the $\pm$-eigenspaces for the dualising involution. For ${\cal M}^-({\rm Spin}_6) = {\cal SU}_C(4,2)$, remark \ref{dynkin} will say that the analogous involution is $E\mapsto E^{\vee}\otimes {\rm Nm} E$, where ${\rm Nm} E$ is the fixed spinor norm, satisfying $({\rm Nm} E)^2 = \det E$. Then we expect, in this case: $$ \begin{array}{rcl} H^0_+({\cal SU}_C(4,2),{\cal L}_2) &\cong& \displaystyle \sum_{\eta \in J_2}H^0_-(P_{\eta},6{\Xi_{\eta}}),\\ H^0_-({\cal SU}_C(4,2),{\cal L}_2) &\cong& \displaystyle \sum_{\eta \in J_2\backslash \{0\}}H^0_-(P_{\eta}^-,6{\Xi_{\eta}}).\\ \end{array} $$ \end{ex} \section{Constructing the homomorphisms: the Jacobian} Our aim in this section and the next is to construct homomorphisms: \begin{equation} \label{oddhomo} H^0({\cal M}^{\pm}({\rm Spin}_{2n+1}), \Theta({\bf C}^{2n+1}))^{\vee} \rightarrow \sum_{\eta \in J_2}H^0_{\pm}(P_{\eta},({2n+1}){\Xi_{\eta}}) \end{equation} \begin{equation} \label{evenhomo} \begin{array}{rcl} H^0({\cal M}^{\pm}({\rm Spin}_{2n}), \Theta({\bf C}^{2n}))^{\vee} &\rightarrow& \displaystyle \sum_{\eta \in J_2}H^0_{\pm}(P_{\eta},2n{\Xi_{\eta}}) \\ &&\ \ \ \ \ \ \displaystyle \oplus \sum_{\eta \not= 0}H^0_{\pm}(P_{\eta}^-,2n{\Xi_{\eta}})\\ \end{array} \end{equation} Table (\ref{dims}) says that in each case the left- and right-hand sides have the same dimension, and so we naturally conjecture that these homomorphisms are isomorphisms, though we shall not prove this here. As remarked in the last section, the first few cases $m=1,2,3,4$ are well understood, and we do have natural isomorphisms (\ref{oddhomo}), (\ref{evenhomo}) in these cases. \subsection{The splitting for even spin groups} \label{dynkin} We shall at this point attempt to explain the splitting $H^0(\Theta({\bf C}^m)) = \sum_{\eta\in J_2} \oplus \sum_{\eta\not=0} $ when $m$ is even. Note that for even $m$ the group ${\rm Spin}_m$ carries a unique nontrivial outer automorphism: $$ \textfont2=\dynkfont\dddnu......\updownarrow $$ This induces an involution $\sigma : {\cal M}(SC_m) \leftrightarrow {\cal M}(SC_m)$ which preserves the spinor norm and hence acts on ${\cal M}({\rm Spin}_m)$ and ${\cal M}^-({\rm Spin}_m)$. (More concretely, the group automorphism is obtained by Clifford conjugation by a vector in ${\bf C}^m$; when $m$ is {\it odd} this is an inner automorphism. $\sigma$ then acts on Clifford bundles by conjugating transition functions with respect to some open cover; when $m$ is odd this action is still defined, of course, but is trivial.) On the other hand this automorphim preserves the orthogonal representation, and hence the isomorphism class of the line bundle $\Theta({\bf C}^m)$; accordingly $H^0(\Theta({\bf C}^m)$ splits into $\pm$-eigenspaces $H^0_+ \oplus H^0_-$ under the action of $\sigma$. For even $m$ we now expect the following refinement of (\ref{evenhomo}): \begin{equation} \label{refine} \begin{array}{rcl} H^0_+({\cal M}^{\pm}({\rm Spin}_m), \Theta({\bf C}^m))^{\vee} &\rightarrow & \displaystyle \sum_{\eta \in J_2}H^0_{\pm}(P_{\eta},m{\Xi_{\eta}}),\\ &&\\ H^0_-({\cal M}^{\pm}({\rm Spin}_m), \Theta({\bf C}^m))^{\vee} &\rightarrow &\displaystyle \sum_{\eta \not= 0}H^0_{\pm}(P_{\eta}^-,m{\Xi_{\eta}}).\\ \end{array} \end{equation} Note that this is exactly what happens in the case $m=4$ (see example \ref{4theta}), where $\sigma$ simply exchanges factors in the products ${\cal M}({\rm Spin}_4) = {\cal SU}_C(2) \times {\cal SU}_C(2)$ and ${\cal M}^-({\rm Spin}_4) = {\cal SU}_C(2,1) \times {\cal SU}_C(2,1)$, so that the summands $H^0_{\pm}$ in (\ref{refine}) are precisely $S^2 H^0$ and $\bigwedge^2 H^0$ appearing in \ref{4theta}. In the case $m=6$---recall example \ref{clifm=6}---we have ${\cal M}({\rm Spin}_6) = {\cal SU}_C(4)$ and ${\cal M}^-({\rm Spin}_6) = {\cal SU}_C(4,2)$. In each case not only the determinant $\det E$ of vector bundles $E$ is fixed, but also a square root ${\rm Nm} E$ of $\det E$. Rank 4 vector bundles come from $SC_6$-bundles via the first half-spinor representation $SC_6 \rightarrow GL_4$; one easily checks that taking instead the second projection to $GL_4$ induces the involution on rank 4 vector bundles $\sigma : E\leftrightarrow E^{\vee}\otimes {\rm Nm} E$. Thus in the case $m=6$ of (\ref{refine}) we expect the situation already described in example \ref{6theta} of the previous section. \subsection{The main construction} \label{main} We shall construct the homomorphisms (\ref{oddhomo}), (\ref{evenhomo}) one summand at a time, concentrating in this section on the summand $\eta =0$, i.e. the projection to Jacobian thetas (see (\ref{s0}) and corollary \ref{par0}). In the next section section we shall construct the remaining projections to the Prym thetas (corollary \ref{pareta}). It will be convenient to denote the two-component variety ${\cal M}({\rm Spin}_m) \cup {\cal M}^-({\rm Spin}_m)$ by ${\cal N}(m)$. Let $\rho: SC_m \rightarrow SL(V)$ be a representation satisfying condition (\ref{ss}) from section \ref{mod}. (We are {\it mainly} concerned with the orthogonal representation $V={\bf C}^m$, and $r=m$ in proposition \ref{div} below.) We consider the subset \begin{equation} \label{ddV} {\cal D}(V) = \{(L,E)|H^0(C,L\otimes E(V)) \not= 0\} \subset J^{g-1}(C)\times {\cal N}(m). \end{equation} It is most important for us that in the case $V = {\bf C}^m$ is the vector representation---by theorem \ref{gen0}---this is a proper subset in each component of the product. For the theta divisor $\theta$ in $J^{g-1}(C)$ and the theta bundle $\Theta(V)$ on ${\cal N}(m)$ we shall abuse notation and use the same symbols to denote also their pull-back to the product $J^{g-1}(C)\times {\cal N}(m)$. \begin{prop} \label{div} If $H^0(C,L\otimes E(V))$ is generically zero---as it is when $V={\bf C}^m$---then ${\cal D}(V)$ is the support of a divisor in $|r\theta + \Theta(V)|$, where $r=\dim V$. \end{prop} {\it Proof.}\ The representation $\rho$ induces a morphism of varieties $$ \alpha : {\cal N}(m) \rightarrow {\cal M}(SC_m) \rightarrow {\cal M}(SL_r); $$ and by definition ${\cal D}(V)$ is (the support of) the pull-back via $$ J^{g-1}(C) \times {\cal N}(m) \map{id \times \alpha} J^{g-1}(C) \times {\cal M}(SL_r) \map{\otimes} {\cal U}(r,r(g-1)) $$ (where, of course, we are identifying ${\cal M}(SL_r)$ with the moduli space of semistable vector bundles of rank $r$ and trivial determinant, and the second map is tensor product of vector bundles) of the canonical theta divisor (see section \ref{mod}) $$ \Theta_{r,r(g-1)} = \{F| H^0( F) \not= 0\} \subset {\cal U}(r,r(g-1)). $$ Since by hypothesis $H^0(C,L\otimes E(V))=0$ generically, it follows that ${\cal D}(V)$ is a well-defined divisor. The proposition now follows from the discussion of section \ref{mod}: first of all, the pull-back of $\Theta_{r,r(g-1)}$ to $J^{g-1}(C) \times {\cal M}(SL_r)$ restricts on a fibre $\{L\} \times {\cal M}(SL_r)$ as the restriction of $\Theta_{r,0}$ from ${\cal U}(r,0)$---this is by definition of $\Theta_{r,0}$---and we have seen that this is just $\Theta({\bf C}^r)$. Hence the pull-back to $J^{g-1}(C) \times {\cal N}(m)$ is $\Theta(V)$ on fibres $\{L\} \times {\cal N}(m)$. On the other hand, restriction to fibres $J^{g-1}(C) \times \{V\}$, for any vector bundle $V$ with rank $r$ and trivial determinant, is well-known to be independent of $V$---see for example \cite{OP} section 3.1. Then it follows from (\ref{difr}) that $\Theta_{r,r(g-1)} $ restricts to $r\theta$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} It follows from the K\"unneth theorem that ${\cal D}(V)$ defines up to scalar a tensor in $$ H^0({\cal N}(m),\Theta(V))\otimes H^0(J^{g-1},r\theta), $$ or equivalently a homomorphism, the projection of (\ref{oddhomo}), (\ref{evenhomo}) at $\eta =0$, \begin{equation} \label{s0} s_0: H^0({\cal N}(m),\Theta(V))^{\vee} \rightarrow H^0(J^{g-1},r\theta). \end{equation} $s_0$ is dual to pull-back of hyperplane sections under the rational map $ f_0 : {\cal N}(m) \rightarrow |r\theta| $ sending $E$ to ${\cal D}(V)|_{J^{g-1}\times \{E\}}$. Now suppose that the vector bundles $E(V)$ are self-dual. This happens when the representation $V$ is symplectic or orthogonal---in particular if $V={\bf C}^m$ or a spin representation. Then it follows easily from Riemann-Roch and Serre duality that the divisors $f_0(E)$ are {\it symmetric}: $$ f_0 : {\cal N}(m) \rightarrow |r\theta|_+ \cup |r\theta|_- $$ where $|r\theta|_{\pm} = {\bf P} H^0_{\pm}(J^{g-1},r\theta)$. \begin{prop} \label{f0} When $V={\bf C}^m$ is the standard orthogonal representation, for $m\geq 3$, $f_0$ respects parity: $ f_0 : {\cal M}^{\pm}({\rm Spin}_m) \rightarrow |m\theta|_{\pm} $ respectively. \end{prop} Before proving this, we need to make a general remark about principally polarised abelian varieties $(A,\Xi)$, where as usual $\Xi$ is a symmetric divisor representing the polarisation. Let $\vartheta(A) = A_2 $ denote the set of 2-torsion points, and let $$ \begin{array}{rcl} \vartheta^{+}(A) &=& \{ x\in A_2 \ |\ {\rm mult}_x \Xi \equiv 0\ \hbox{mod 2} \}\\ \vartheta^{-}(A) &=& \{ x\in A_2 \ |\ {\rm mult}_x \Xi \equiv 1\ \hbox{mod 2} \}\\ \end{array} $$ In the case $(A,\Xi) = (J^{g-1} ,\theta)$ we shall write $\vartheta^{\pm} (J^{g-1}) = \vartheta^{\pm}(C)$. These are the sets of even and odd theta characteristics. In the case of a Prym variety $(P_{\eta} ,{\Xi_{\eta}})$ it is shown in \cite{OP}, proposition 2.3 that we can identify: \begin{equation} \label{Ptheta} \begin{array}{rcr} \vartheta^{\pm} (P_{\eta}) &=& \{\pi^* N = \pi^* (\eta \otimes N) \in {\widetilde J^{2g-2}} \\ &&\hbox{where}\ N, \eta\otimes N \in \vartheta^{\pm}(C)\}.\\ \end{array} \end{equation} On $P_{\eta}^-$ there is again an induced principal polarisation, though, as remarked in section \ref{pryms}, no distinguished theta divisor. Thus we may talk about $\vartheta(P_{\eta}^-)$, whose points are described (using the same methods as in \cite{OP}) by: \begin{equation} \label{P-theta} \begin{array}{rcl} \vartheta (P_{\eta}^-) &=& \{\pi^* N = \pi^* (\eta \otimes N) \in {\widetilde J^{2g-2}} \\ &&\hbox{where}\ N, \eta\otimes N \in \vartheta(C)\ \hbox{have {\it opposite} parity} \}.\\ \end{array} \end{equation} But for $P_{\eta}^-$ the partition into $\vartheta^{\pm}$ is no longer well-defined. \begin{lemm} \label{basepts} \begin{enumerate} \item If $m$ is odd then $|m\Xi|_+$ (resp. $|m\Xi|_-$) is the linear subsystem with base-point set $\vartheta^-(A)$ (resp. $\vartheta^+(A)$). \item If $m$ is even then $|m\Xi|_+$ is base-point free; while $|m\Xi|_-$ is the linear subsystem with base-point set $A_2$. \end{enumerate} \end{lemm} {\it Proof.}\ See \cite{OP} lemma 2.2 (for part 1); or \cite{LB} chapter 4 section 7. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \medskip {\it Proof of proposition \ref{f0}.} We use proposition \ref{2.11}. Suppose first that $m$ is odd: then for $E\in {\cal N}(m)$ and any theta characteristic $L\in \vartheta(C)$ we have $$ h^0(C,L\otimes E({\bf C}^m)) \equiv h^0(C,L) + \deg{\rm Nm} E \quad {\rm mod}\ 2. $$ So if $E\in {\cal M}({\rm Spin}_m)$ then by definition $L\in f_0(E)$ for all {\it odd} theta characteristics; so by the first part of the lemma $f_0(E) \in |m\theta|_+$. Likewise $f_0(E) \in |m\theta|_-$ whenever $E\in {\cal M}^-({\rm Spin}_m)$. If $m$ is even then the same argument shows that ${\cal M}^-({\rm Spin}_m)$ maps into $|m\theta|_-$. ${\cal M}({\rm Spin}_m)$, on the other hand, maps either into $|m\theta|_+$ or $|m\theta|_-$, and to see that it is not the latter it suffices to exhibit a theta characteristic $K^{1\over 2}$ and bundle $E\in {\cal M}({\rm Spin}_m)$ for which $H^0(C,K^{1\over 2}\otimes E({\bf C}^m)) =0$---which is possible by theorem \ref{gen0th}. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{cor} \label{par0} When $V={\bf C}^m$ is the standard orthogonal representation the homomorphism $s_0$ respects parity: $$ s_0 : H^0({\cal M}^{\pm}({\rm Spin}_m),\Theta({\bf C}^m))^{\vee} \rightarrow H^0_{\pm}(J^{g-1}, m\theta) \qquad {\sl respectively.} $$ \end{cor} \section{Constructing the homomorphisms: the Pryms} We next explain how the construction of the previous section may be extended to give the projections to the remaining summands, $\eta \not= 0$, in (\ref{oddhomo}) and (\ref{evenhomo}). These are given in corollary \ref{pareta}, which extends corollary \ref{par0}. To begin, it is necessary to note that semistability of a bundle is preserved under pull-back to the double covers. \begin{lemm} Let $p : {\widetilde C} \rightarrow C$ be any unramified cover of smooth projective curves. Then, if a vector bundle $V\rightarrow C$ is semistable then $p^*V\rightarrow {\widetilde C}$ is semistable. \end{lemm} {\it Proof.}\ By the Narasimhan-Seshadri theorem $V$ is induced from a projective unitary representation of the fundamental group $\pi_1(C)$. Since ${\widetilde C}$ is an unramified cover its fundamental group injects into $\pi_1(C)$, and the restriction of the above representation then induces the pull-back bundle, which is consequently semistable. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{cor} \label{5.5} Let $p : {\widetilde C} \rightarrow C$ be as in the previous lemma, and $E\rightarrow C$ a semistable $G$-bundle. Then $p^*E \rightarrow {\widetilde C}$ is semistable. \end{cor} {\it Proof.}\ The same argument as in the above proof works for $G$-bundles by Ramanathan's generalisation of the Narasimhan-Seshadri theorem \cite{R1}; alternatively apply the lemma to the adjoint bundle ${\rm ad}\ E$: by \cite{R2}, corollary 2.18, semistability of $E$ is equivalent to semistability of ${\rm ad}\ E$ as a vector bundle. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} Let us now return to the double cover $p:\cctil_{\eta} \rightarrow C$. Noting that for a Clifford bundle $E \rightarrow C$ the spinor norm satisfies ${\rm Nm} ( p^*E) = p^* {\rm Nm} ( E)$, and this has even degree, it follows from corollary \ref{5.5} that we obtain a morphism of moduli spaces $$ u = p^*: {\cal N}_C(m) \rightarrow {\cal M}_{\cctil_{\eta}}({\rm Spin}_m). $$ \begin{prop} \label{5.6} For any representation $SC_m \rightarrow SL(V)$ we have $u^*\Theta_{{\cal M}_{\cctil_{\eta}}}(V) = 2\Theta_{{\cal N}_C}(V)$. \end{prop} {\it Proof.} Let $E\rightarrow C\times S$ be an arbitrary family of semistable $SC_m$-bundles, and let $F=E(V)$ be the associated family of vector bundles via the given representation. Let ${\widetilde F} = (p\times {\rm id})^*F$ be the pull-back of the family by the double cover: $$ \matrix{ \cctil_{\eta} \times S&&\map{p\times {\rm id}}&&C\times S\cr &&&&\cr &\hidewidth {\widetilde \pi} \searrow &&\swarrow \pi \hidewidth&\cr &&&&\cr &&S.&&\cr} $$ It is clear from the discussion of section \ref{mod} that to prove the proposition it suffices to show that $$ \Theta({\widetilde F}) = 2 \Theta (F): $$ i.e. the line bundle $\Theta(V) \rightarrow {\cal M}_C(SC_m)$ represents the functor $E\mapsto \Theta (F)$, while the line bundle $u^*\Theta_{{\cal M}_{\cctil_{\eta}}}(V) \rightarrow {\cal M}_C(SC_m)$ represents the functor $E\mapsto \Theta({\widetilde F})$. So to compute $\Theta({\widetilde F})$, first note that by the projection formula applied to $p\times {\rm id}$ we have, for $i=0,1$: $$ \begin{array}{rcl} R^i_{{\widetilde \pi}}({\widetilde F}) &=& R^i_{\pi}(F\otimes p_* {\cal O}_{\cctil_{\eta}}) \\ &=& R^i_{\pi}(F\oplus F\otimes \eta)\\ &=& R^i_{\pi}(F) \oplus R^i_{\pi}(F\otimes \eta).\\ \end{array} $$ If we fit the direct images $R^i_{\pi}(F)$ into an exact sequence (\ref{1.3}), and $R^i_{\pi}(F\otimes \eta)$ into a similar sequence with middle terms ${K^0}' \map{\phi'} {K^1}'$, then we get an exact sequence: $$ 0\rightarrow R^0_{{\widetilde \pi}}{\widetilde F} \rightarrow K^0 \oplus {K^0}' \map{ {\rm diag}(\phi,\phi') } K^1 \oplus {K^1}' \rightarrow R^1_{{\widetilde \pi}}{\widetilde F}\rightarrow 0. $$ It follows at once that $$ {\rm Det}({\widetilde F}) = {\rm Det}(F)\otimes {\rm Det}(F\otimes \eta). $$ But since the bundle $F$ has trivial determinant we can replace Det by $\Theta$ here. And since $\Theta(F\otimes \eta) = \Theta(F)$ by corollary \ref{1.6}, we obtain $\Theta({\widetilde F}) = 2 \Theta(F)$ as required. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} We now consider $$ \widetilde {\cal D}(V) \subset {\widetilde J^{2g-2}} \times {\cal M}_{\cctil_{\eta}}({\rm Spin}_m), $$ denoting the divisor of (\ref{ddV}) and proposition \ref{div} with $C$ replaced by $\cctil_{\eta}$. As a consequence of proposition \ref{5.6}, we see that the pull-back via the map $$ {\rm incl} \times u : (P_{\eta} \cup P_{\eta}^-) \times {\cal N}(m) \rightarrow {\widetilde J^{2g-2}} \times {\cal M}_{\cctil_{\eta}}({\rm Spin}_m), $$ of $\widetilde {\cal D}(V)$ is---{\it provided $H^0(\cctil_{\eta}, L\otimes p^*E(V)) =0$ for generic $(L,E) \in P_{\eta}^{\pm}\times {\cal M}^{\pm}({\rm Spin}_m)$}---a divisor \begin{equation} \label{eeV} {\cal E}_{\eta}(V) \in |2r {\Xi_{\eta}} + 2\Theta_{{\cal N}}|. \end{equation} In the case of the orthogonal representation---except possibly for ${\cal M}^-({\rm Spin}_{2n})$---this is guaranteed by corollary \ref{serregen0}: \begin{prop} \label{gen0eta} Let $V={\bf C}^m$ be the standard orthogonal representation. \begin{enumerate} \item If $m$ is odd and $L\in P_{\eta}^-$ then $H^0(\cctil_{\eta}, L\otimes p^*E(V)) \not=0$ for all $E\in {\cal M}^{\pm}({\rm Spin}_m)$. \item In all other cases---except possibly $E\in {\cal M}^-({\rm Spin}_{2n})$---we have $H^0(\cctil_{\eta}, L\otimes p^*E(V)) =0$ for generic $(L,E) \in P_{\eta}\times {\cal M}^{\pm}({\rm Spin}_m)$ or $P_{\eta}^-\times {\cal M}^{\pm}({\rm Spin}_m)$. \end{enumerate} \end{prop} {\it Proof.}\ Note that by corollary \ref{serre} (applied to $\cctil_{\eta}$), together with (\ref{prym+-}), $h^0(\cctil_{\eta}, L\otimes p^*E({\bf C}^m))$ is odd if $m$ is odd and $L\in P_{\eta}^-$, hence nonzero as asserted. In all other cases, on the other hand, $h^0(\cctil_{\eta}, L\otimes p^*E({\bf C}^m))$ is even. So the proposition is equivalent to \ref{serregen0}. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} The next step is to observe that the divisor thus constructed has multiplicity two: \begin{prop} Suppose that the representation $V$ is orthogonal, i.e. $SC_m \rightarrow SO(V)$ for some invariant quadratic form on $V$. Then, when ${\cal E}_{\eta}(V)$ is a divisor it has multiplicity 2, i.e. ${\cal E}_{\eta}(V) = 2 {\cal D}_{\eta}(V)$ for a divisor ${\cal D}_{\eta}(V) \in |r {\Xi_{\eta}} + \Theta(V)|$. \end{prop} {\it Proof.}\ We are here excluding the case ${\cal N}(m) \times P_{\eta}^-$ for odd $m$. Then we have already observed above that by corollary \ref{serre} and (\ref{prym+-}), $h^0(\cctil_{\eta}, L\otimes p^*E({\bf C}^m))$ is {\it even} for all $(L,E) \in P_{\eta}^{\pm}\times {\cal M}^{\pm}({\rm Spin}_m)$ in the remaining cases. The proposition then follows from the determinantal description of the functor $\Theta$ given in section \ref{mod}. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} Just as for the case $\eta =0$, one now views the divisor ${\cal D}_{\eta}(V)$ as giving a rational map---or, more precisely, a {\it pair} of rational maps if $m$ is even: $$ \begin{array}{rcl} f_{\eta }^{\pm}: {\cal N}(m) & \rightarrow& |m {\Xi_{\eta}} |_{P_{\eta}^{\pm}} \\ E &\mapsto & \{ L \in P_{\eta}^{\pm} | H^0(\cctil_{\eta} , L\otimes p^*E(V)) \not= 0 \}.\\ \end{array} $$ By Riemann-Roch, Serre duality and the fact that $E(V)$ is self-dual, one sees again that $f_{\eta}^{\pm}(E)$ is a {\it symmetric} divisor. This means that each component of ${\cal N}(m)$ maps either into $|m{\Xi_{\eta}} |_{+,P_{\eta}^{\pm}} = {\bf P} H^0_+(P_{\eta}^{\pm} , m {\Xi_{\eta}})$ or into $|m{\Xi_{\eta}} |_{-,P_{\eta}^{\pm}} = {\bf P} H^0_-(P_{\eta}^{\pm} , m {\Xi_{\eta}})$; and the claim is: \begin{prop} \label{pmpm} When $V={\bf C}^m$ is the standard orthogonal representation, for $m\geq 3$, each $f_{\eta} = f_{\eta}^+$ respects parity: $$ f_{\eta}: {\cal M}^{\pm}({\rm Spin}_m) \rightarrow |m{\Xi_{\eta}} |_{\pm,P_{\eta}}; $$ and when $m$ is even $f_{\eta}^-$ respects parity: $$ f_{\eta}^-: {\cal M}^{\pm}({\rm Spin}_m) \rightarrow |m{\Xi_{\eta}} |_{\pm,P_{\eta}^-}. $$ \end{prop} {\it Proof.}\ This uses proposition \ref{2.11} in the same way as the proof of proposition \ref{f0}. We suppose first that $m$ is odd; choose $E\in {\cal N}(m)$ and a theta characteristic $L\in \vartheta(P_{\eta})$. By (\ref{Ptheta}) this means $L= p^*N = p^*(\eta\otimes N)$ for some theta characteristics $N$ and $\eta \otimes N$ {\it of the same parity}. Then we note that $$ h^0(\cctil_{\eta}, L\otimes p^*E({\bf C}^m)) = h^0(C,N\otimes E({\bf C}^m)) + h^0(C,\eta \otimes N\otimes E({\bf C}^m)), $$ and that (by \ref{2.11}) $h^0(C,N\otimes E({\bf C}^m))$ is odd, and hence nonzero, provided $E\in {\cal M}({\rm Spin}_m)$ and $L\in \vartheta^-(P_{\eta})$ {\it or} $E\in {\cal M}^-({\rm Spin}_m)$ and $L\in \vartheta^+(P_{\eta})$. So by part 1 of lemma \ref{basepts} it follows that $f_{\eta}$ maps ${\cal M}^{\pm}({\rm Spin}_m)$ into $|m{\Xi_{\eta}}|_{\pm,P_{\eta}}$ respectively. If $m$ is even then the same argument, using part 2 of lemma \ref{basepts}, shows that ${\cal M}^-({\rm Spin}_m)$ maps under $f_{\eta}$ into $|m{\Xi_{\eta}}|_{-,P_{\eta}}$; and maps under $f_{\eta}^-$ into $|m{\Xi_{\eta}}|_{-,P_{\eta}^-}$ (see (\ref{P-theta})). Finally consider ${\cal M}({\rm Spin}_m) $ for even $m$: for any theta characteristics $N, \eta \otimes N \in \vartheta(C)$, theorem \ref{gen0th} tells us that $h^0(C,N\otimes E({\bf C}^m)) = h^0(C,\eta \otimes N\otimes E({\bf C}^m))=0$ for generic $E\in {\cal M}({\rm Spin}_m)$, so that generically $h^0(\cctil_{\eta}, L\otimes p^*E({\bf C}^m)) = 0$ for any theta characteristic $L \in \vartheta(P_{\eta})$ or $\vartheta(P_{\eta}^-)$. This implies (for each of $P_{\eta}$, $P_{\eta}^-$) that ${\cal M}({\rm Spin}_m) $ does not map into $|m{\Xi_{\eta}}|_-$ and therefore maps into $|m{\Xi_{\eta}}|_+$ as required. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} As a consequence, pull-back of hyperplane sections under each $f_{\eta}^{\pm}$ is dual to a homomorphism (the analogue of (\ref{s0})) \begin{equation} \label{seta} s_{\eta}^{\pm}: H^0({\cal N}(m),\Theta(V))^{\vee} \rightarrow H^0(P_{\eta}^{\pm},r{\Xi_{\eta}}), \end{equation} where $r =\dim V$, and where $s_{\eta}^-$ is defined only for even $m$. Moreover, proposition \ref{pmpm} says that these homomorphisms {\it respect parity}---in other words the analogue of corollary \ref{par0} for this situation is: \begin{cor} \label{pareta} When $V={\bf C}^m$ is the standard orthogonal representation the homomorphisms $s_{\eta}^{\pm}$ respect parity: $$ s_{\eta} = s_{\eta}^+ : H^0({\cal M}^{\pm}({\rm Spin}_m),\Theta({\bf C}^m))^{\vee} \rightarrow H^0_{\pm}(P_{\eta}, m{\Xi_{\eta}}) \qquad {\sl respectively,} $$ and if $m$ is even then additionally: $$ s_{\eta}^- : H^0({\cal M}^{\pm}({\rm Spin}_m),\Theta({\bf C}^m))^{\vee} \rightarrow H^0_{\pm}(P_{\eta}^-, m{\Xi_{\eta}}) \qquad {\sl respectively.} $$ \end{cor}

9602

alg-geom/9602015

en

https://arxiv.org/abs/alg-geom/9602015

alg-geom/9602015

Christoph Lossen

Y.A. Drozd and G.-M. Greuel

Semicontinuity for representations of one-dimensional Cohen-Macaulay rings

LaTeX2e

We show that the number of parameters for CM-modules of prescribed rank is semi-continuous in families of CM rings of Krull dimension 1. This transfers a result of Knoerrer from the commutative to the not necessarily commutative case. For this purpose we introduce the notion of ``dense subrings'' which seems rather technical but, nevertheless, useful. It enables the construction of ``almost versal'' families of modules for a given algebra and the definition of the ``number of parameters''. The semi--continuity implies, in particular, that the set of so-called ``wild algebras'' in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, together with the results of a former paper of the authors the semi-continuity implies that tame is indeed an open property for curve singularities (commutative CM rings). An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or finite dimensional bimodules.

alg-geom

\section*{Introduction}\addcontentsline{toc}{section}{Introduction} ``Algebraic families'' of modules and algebras play an important role in several questions of representation theory. It is often especially useful to know that some ``discrete invariants'' are constant or, at least, are semi--continuous in such families, that is they can change only in ``exceptional points'' which form a family of smaller dimension. Perhaps the best known results in this direction are those of Gabriel \cite{Gab} and Kn\"orrer \cite{Kn}. Gabriel proved that finite representation type is an open condition for finite dimensional algebras (``fat points''), while Kn\"orrer showed that the number of parameters for modules of prescribed rank is semi--continuous in families of commutative Cohen--Macaulay rings of Krull dimension 1 (``curve singularities''). In \cite{DG 2} Kn\"orrer's theorem was used to show that the unimodal singularities of type $T_{pq}$ are of tame Cohen--Macaulay type. Unfortunately, the arguments of \cite{Kn} do not work in the non--commutative case. The aim of this paper is to refine them in such a way that they could be applied to non--commutative Cohen--Macaulay algebras, too. For this purpose we introduce the notion of ``dense subrings'' which seems rather technical but, nevertheless, useful. It enables the construction of ``almost versal'' families of modules for a given algebra (cf.\ Theorem \ref{3.4}) and the definition of the ``number of parameters''. Just as in the commutative case, it is important that the bases of these ``almost versal'' families are projective varieties. Once having this, we are able to prove an analogue of Kn\"orrer's theorem (cf.\ Theorem \ref{4.9}) and a certain variant (cf.\ Theorem 4.11) which turns out to be useful, for instance, to extend the tameness criterion for commutative algebras \cite{DG 2} to the case of characteristic 2. The semicontinuity implies, in particular, that the set of so--called ``wild algebras'' in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, Theorem 4.9, together with the results of \cite{DG 2}, imply that tame is indeed an open property for curve singularities (commutative one-dimensional Cohen--Macaulay rings). An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or elements of finite dimensional bimodules. Though we do not consider here the problem of constructing moduli spaces for Cohen-Macaulay modules (cf. \cite{GP}), we may rephrase the semicontinuity theorem by saying that the dimension of the moduli space for such modules of prescribed rank varies upper semicontinuosly in flat families of Cohen-Macaulay algebras. Likewise, the semicontinuity in other cases (finite-dimensional algebras or bimodules) may be also understood as semicontinuity of the dimension of the corresponding moduli spaces of representations under deformations of the algebra or bimodule. Unfortunately, our results are, just as all known results till now, not sufficient to prove the ``tame is open condition''--conjecture. Nevertheless, the semicontinuity theorem as well as the construction of ``almost versal'' families of modules are not restricted to tame algebras. They have a potentially broader field of applications to classification problems in representation theory. They are a particularly powerful tool if, for a given algebra, the deformation theory of this algebra is sufficiently known and the classification problem for the deformed algebras is easier to solve or even known. The great success of this approach in the commutative case is, of course, also due to the fact that the deformation theory of singularities is a highly developed field. We hope that this paper stimulates further research in the deformation theory for non--commutative Cohen--Macaulay algebras. \newpage \section{Dense subalgebras} \begin{definition}\label{1.1} Let $B$ be a subring of a ring $A$. Call $B$ {\bf dense} in $A$ if any simple $A$--module $U$ is also simple as $B$--$End_A U$--bimodule. \end{definition} {\bf Examples}: \begin{enumerate} \item If $A$ is commutative, then $End_A U = A/Ann\, U$, so any subring $B$ is dense in $A$. \item We can take $A = M_2 ({\Bbb R})$ the ring of $2 \times 2$ real matrices and $B = {\Bbb C}$ (or even $B = {\Bbb Q}(i))$, naturally embedded in is $A$. More generally, let $k$ be some field, $L, K$ and $B$ be its extensions such that $L = KB$ (a composite) and $(L : K) = n$. Then we have a natural embedding $L \to A = M_n(K)$, thus also $B$ is a subalgebra of $A$ and $B$ is dense in $A$. \end{enumerate} \begin{lemma}\label{1.2} Let $D$ be a division algebra over an infinite field $k$, $A = M_n(D)$ and $B \subset A$ a dense subalgebra. Consider $W = M_{n \times m}(D)$ as $A$--module and let $V \subset W$ be a $B$--submodule such that $AV = W$. Suppose that $m = nq + r$ with $0 \le r < n$. Then there exists an automorphism $\sigma$ of $W$ such that $\sigma(V)$ contains the matrices \begin{eqnarray*} E_1 & = & (I\; 0 \ldots 0\; 0'),\;\; E_2 = (0\; I \ldots 0\; 0'), \ldots,\\ E_q & = & (0\; 0 \ldots 0\; I\; 0') \end{eqnarray*} and a matrix of the form $(Y_1\; Y_2 \ldots Y_q\; Y')$. \begin{tabular}{lll} Here & $I$ & denotes the $n \times n$ unit matrix,\\ & $0$ & denotes the $n \times n$ zero matrix, \\ & $0'$& denotes the $n \times r$ zero matrix, \end{tabular} $Y_1, Y_2, \ldots, Y_q$ are some $n \times n$ matrices and $Y'$ is an $n \times r$ matrix of rank $r$ (of course, if $r = 0$, then $0'$ and $Y'$ are empty, so in this case $V$ contains an $A$--basis of $W \simeq q A)$. \end{lemma} {\bf Proof:} Use induction on $m$. First prove the claim in the case $m \le n$. Choose a matrix $X \in V$ of maximal possible rank, say $d$. Then we must show that $d = m$. Suppose that $d < m$. Denote by $\bar{X}_1, \bar{X}_2, \ldots, \bar{X}_m$ the columns of $X$ and let $\bar{X}_1, \bar{X}_2, \ldots, \bar{X}_d$ be linear independent. Then there exists an automorphism $\sigma$ of $W$ such that the last $(m-d)$ columns of $\sigma(X)$ are zero. So we may suppose that $\bar{X}_{d+1} = \ldots = \bar{X}_m = \bar{0}$. Note that $W \simeq mU$ where $U = nD$ is the only simple $A$--module. As $d< n,\; \bar{X}_1, \bar{X}_2, \ldots, \bar{X}_d$ do not span $U$ over $D$. Denote by $V'$ the projection of $V$ onto the $(d + 1)$--st component of $W$ (i.e.\ $V' \subset U$ consists of the first $d + 1$ columns of the matrix from $V$). As $AV = W$, we have $AV' = U$, so $V' \not= 0$. But as $U$ is a simple $B$--$D$--bimodule, $V'D = U$. Therefore, $V$ contains a matrix $Y$ such that its $(d + 1)$--st column $\bar{Y}_{d+1}$ does not lie in $\langle\bar{X}_1,\; \bar{X}_2, \ldots, \bar{X}_d \rangle D$. Choose a $D$--basis of $U$ of the form $\{\bar{X}_1,\; \bar{X}_2, \ldots, \bar{X}_d, \; \bar{Y}_{d+1}, \bar{Z}_{d+2}, \ldots, \bar{Z}_n\}$ and let \[ \bar{Y}_i = \sum^d_{j=1} \bar{X}_i \lambda_{ij} + \bar{Y}_{d+1} \lambda_{i, d+1} + \sum^n_{j=d+2} \bar{Z}_j \lambda_{ij}. \] Again, using an automorphism of $W$, we may suppose that $\lambda_{i, d+1} = 0$ for $i \not= d + 1$. But then rank$(\gamma X + Y) \ge d + 1$ for some $\gamma \in k$, which is a contradiction. Hence, $d = m$. In particular, if $m = n$, there exists an automorphism $\sigma$ of $W$, such that $\sigma(X) = I$. Thus, our claim is proved for $m \le n$. Suppose now that $m > n$ and consider the projection $V'$ of $V$ onto $W' = nU$, the first $n$ components of $W$ (that is the first $n$ columns of each matrix $X \in W)$. As we have proved, there exists an automorphism $\sigma'$ of $W'$ such that $\sigma'(V') \ni I$. We can extend $\sigma'$ to $W$ and thus suppose that $V$ contains a matrix $X$ of the form ($I X'$). Again using an automorphism, we obtain that $X' = 0$, that is $X = E_1$. Now consider the projection $V''$ of $V$ onto $W'' = (m-n)U$, the last components of $W$. Using induction, we may also suppose that the claim is valid for $V''$, thus $V$ contains the matrices of the form: \[ \begin{array}{ll} (X_2 \;\; I \;\; 0 \ldots 0 \;\; 0'),\quad & (X_3\;\; 0\;\; I \ldots 0\;\; 0'),\; \ldots,\\ (X_q\:\; 0 \ldots 0 \;\; I\;\; 0'),\quad & (Y_1\;\; Y_2 \ldots Y_q\;\; Y') \end{array} \] with $rank(Y') = r$. But then, again using an automorphism of $W$, we can make $X_2 = X_3 = \ldots = X_q = 0$, q.e.d. \begin{corollary}\label{1.3} Let $B \subset A$ be a dense subring. Suppose that $A/rad\, A$ is an artinian ring containing an infinite field $k$ in its centre and, moreover, $k \subset B/(B \cap rad\, A)$ (for example $B$ and $A$ are $k$--algebras). Let $V \subset nA$ be a $B$--submodule such that $AV = nA$. Then $V$ contains an $A$--basis of $nA$. \end{corollary} {\bf Proof:} Of course, we may replace $A$ by $A/rad\, A$, so suppose that $A = \prod^s_{i=1} A_i$ with $A_i$ simple artinian. Put $B_i = pr_i\, B,\; V_i = pr_i\, V$, $pr_i$ being the projection from $A$ onto $A_i$. Then $B_i$ is dense in $A_i$ and $A_iV_i = nA_i$. Lemma \ref{1.2} implies that each $V_i$ contains an $A_i$--basis $\{\bar{e}_{ij}| j= 1, \ldots, n\}$ of $nA_i$. Let $e_{ij} \in V$ be such elements that $pr_ie_{ij} = \bar{e}_{ij}\; (j = 1, \ldots, n;\;\; i = 1, \ldots, s)$. Consider in $V$ the elements $e_j(\lambda_1, \ldots, \lambda_s) = \sum^s_{i=1} \lambda_i e_{ij}$ where $\lambda_1, \ldots, \lambda_s \in k$. The sets $T_i = \{(\lambda_1, \ldots, \lambda_s) | pr_i e_j(\lambda_1, \ldots, \lambda_s)$ form a basis of $nA_i\}$, are Zariski--open in $k^s$ and non--empty. As $k$ is infinite, their intersection is also non--empty. But if $(\lambda_1, \ldots, \lambda_s)$ lies in this intersection, then $\{e_j (\lambda_1, \ldots, \lambda_s)|j = 1, 2, \ldots, n\}$ is a basis of $nA$, q.e.d. \begin{definition}\label{1.4} Let $D$ be a skewfield, $A = M_n(D),\;\; U = nD$ the simple $A$--module and \[ {\cal F}:\; U = U_0 \supset U_1 \supset \ldots \supset U_s = \{0\} \] a flag of $D$--subspaces in $U$. Put $A({\cal F}) = \{a \in A \mid a U_i \subset U_i$ for all $i = 0, 1, \ldots, m\}$ and call $A({\cal F})$ a {\bf flag subalgebra} in $A$. If $A = \prod_i A_i$ with $A_i$ simple artinian, call a {\bf flag subalgebra} of $A$ any subalgebra $A'$ of the form $A' = \Pi_i A'_i$ where $A'_i$ is a flag subalgebra in $A_i$ for each $i$. \end{definition} \begin{lemma}\label{1.5} Let $A$ be a semi--simple artinian ring, $A({\cal F})$ a flag subalgebra in $A$ and $B$ a subring of $A({\cal F})$. Then there exists a flag subalgebra $A'$ such that $B \subset A' \subset A({\cal F})$ and $B$ is dense in $A'$. \end{lemma} {\bf Proof}: Obviously, we may suppose $A$ to be simple. Then take a maximal $B$--invariant flag ${\cal F}' \supset {\cal F}$ and put $A' = A({\cal F}')$. \begin{proposition}\label{1.6} Let $A$ be an algebra over a separably closed field $k$, such that $A/rad\,A$ is finite--dimensional over $k$, $B$ a dense subalgebra of $A$ and $K$ a separably generated extension of $k$. Then $B \otimes_k K$ is also dense in $A \otimes_k K$. \end{proposition} {\bf Proof:} Of course, we may suppose $A$ to be simple finite--dimensional, that is $A = M_n(F)$ for some skewfield $F$. As $k$ has no separable extensions, $F$ is really a field \cite{DK}, so it is a pure inseparable extension of $k$. But then $F \otimes_k K$ is again a field, hence $A \otimes_k K$ is simple. Its only simple module is $U \otimes_k K$, where $U$ is the simple $A$--module and $F \otimes_k K$ is its endormorphism ring. But the same observation shows that for any simple $B$--$F$--bimodule $V$ (e.g.\ for $U$) the tensor product $V \otimes_k K$ remains simple, q.e.d. \newpage  \section{Cohen--Macaulay Algebras} We consider here one--dimensional Cohen--Macaulay algebras (not necessarily commutative), also known as orders in semi--simple algebras. \begin{definition}\label{2.1} Call a ring $\Lambda$ a {\bf CM--algebra} (more precisely, {\rm{\bf1}}--{\bf dimensional, analytically reduced Cohen--Macaulay algebra}) if it satisfies the following conditions: \begin{enumerate} \item [(1)] $\Lambda$ is an algebra over a one--dimensional local, commutative, noetherian ring $R$, which is a finitely generated and torsion--free $R$--module. \item [(2)] The completion $\widehat{\Lambda}$ of $\Lambda$ in the ${\frak m}$--adic topology, where ${\frak m}$ is the maximal ideal of $R$, contains no nilpotent ideals. \end{enumerate} \end{definition} It follows from (2) that, in this case, $R$ is a Cohen--Macaulay ring and $\Lambda$ is a maximal Cohen--Macaulay $R$--module. In particular, the ${\frak m}$--adic completion $\widehat{R}$ of $R$ has no nilpotent elements. Denote by $Q$ (respectively $\widehat{Q}$) the total ring of fractions of $R$ (respectively $\widehat{R}$). Then both $Q$ and $\widehat{Q}$ are finite products of fields and $Q\Lambda = Q \otimes_R \Lambda\; (\widehat{Q} \Lambda = \widehat{Q} \otimes_R \Lambda = \widehat{Q} \otimes_{\widehat{R}} \widehat{\Lambda}$) is a semi--simple artinian $Q$--algebra (respectively $\widehat{Q}$--algebra). If $\Gamma$ is a subring of $Q\Lambda$, containing $\Lambda$ and being finitely generated as $\Lambda$--module (or, equivalently, as $R$--module), call it an {\bf overring} of $\Lambda$. Of course, any such overring is also a CM--algebra. If $\Lambda$ has no proper overrings, call it a {\bf maximal} CM--algebra. It is known (cf., e.g., \cite{D1}) that, under condition (1), condition (2) is equivalent to the existence of maximal overrings of $\Lambda$. More precisely, under conditions (1) and (2), the overrings of $\Lambda$ satisfy the ascending chain conditions and any two maximal overrings of $\Lambda$ are conjugate in $Q\Lambda$. Let $\Lambda$ be a CM--algebra. We call a $\Lambda$--module $M$ a $\Lambda$--{\bf lattice} (or a {\bf Cohen--Macaulay}{\boldmath$\Lambda$}{\bf --module}) provided it is a maximal Cohen--Macaulay $R$--module. Denote by CM($\Lambda$) the category of all $\Lambda$--lattices. Any such lattice $M$ embeds naturally into the finitely generated $Q\Lambda$--module $QM = Q\otimes_RM$. So, if $\Gamma$ is an overring of $\Lambda$, the $\Gamma$--module $\Gamma M \subset QM$ is well--defined. The following assertions are rather well--known (for the case when $R$ is a discrete valuation ring, cf. \cite{Rog}; the proofs in the general situation are the same). \begin{proposition}\label{2.2} \begin{itemize} \item[(a)] Any maximal CM--algebra $\Lambda$ is hereditary (that is gl.$\dim\,\Lambda = 1$ or, equivalently, any $\Lambda$--lattice is projective). \item[(b)] Let $\Lambda$ be a maximal CM--algebra and $A$ any flag subalgebra of $\Lambda/rad\,\Lambda$ (cf.\ Definition \ref{1.4}). Then the preimage of $A$ in $\Lambda$ is hereditary and any hereditary CM--algebra can be obtained in this way. \end{itemize} \end{proposition} \begin{corollary}\label{2.3} Let $\Omega$ be a hereditary (e.g.\ maximal) overring of a CM--algebra $\Lambda$. Then there exists a hereditary overring $\Omega'$ such that $\Lambda \subset \Omega' \subset \Omega$ and $\Lambda$ is dense in $\Omega'$. \end{corollary} \begin{proposition}\label{2.4} Suppose that the residue field $k = R/{\frak m}$ is infinite. Let $\Gamma$ be an overring of $\Lambda$ such that $\Lambda$ is dense in $\Gamma$ and $M$ be a Cohen--Macaulay $\Lambda$--module such that $\Gamma M \simeq n \Gamma$. Then $M$ is isomorphic to a module $M'$ such that $n \Lambda \subset M' \subset n \Gamma$. \end{proposition} {\bf Proof:} We may suppose that $M \subset n \Gamma$. By Corollary \ref{1.3} it contains a basis of $n \Gamma$. Then there exists an automorphism $\sigma$ of $n\Gamma$ which maps this basis to the standard one, namely $(1, 0, \ldots, 0),\; (0, 1, 0, \ldots, 0), \ldots, (0, \ldots, 0, 1)$. Therefore, $M' = \sigma(M) \supset n \Lambda$. \begin{proposition}\label{2.5} If $M, M' \subset n\Gamma$ are $\Lambda$--submodules such that $\Gamma M = \Gamma M' = n\Gamma$ (e.g.\ $M$ and $M'$ contain $n\Lambda$), then $M \simeq M'$ if and only if there exists an automorphism $\sigma \in Aut(n\Gamma)$ such that $\sigma(M) = M'$. \end{proposition} The proof is evident. \begin{definition}\label{2.6} For any Cohen--Macaulay $\Lambda$--module $M$ denote $\ell(M)$ the length of the $Q\Lambda$--module $QM$ and call it the {\bf rational length} of $M$. \end{definition} {\bf Remark:} If $\Gamma$ is an overring of $\Lambda$ and $M$ is a Cohen--Macaulay $\Gamma$--module, the rational length of $M$ does not depend on whether we consider $M$ as a $\Lambda$-- or as a $\Gamma$--module. On the other hand, we have to distinguish between $\ell(M)$ and $\ell(\widehat{M})$ where $\widehat{M}$ is the ${\frak m}$--adic completion of $M$. Recall some connections between Cohen--Macaulay modules and their completions. The proofs can be found in \cite{CR} or \cite{Rog} for the case when $R$ is a discrete valuation ring, and they are also valid in the general situation. \begin{proposition}\label{2.7} \begin{enumerate} \item[(a)] $M \simeq N$ if and only if $\widehat{M} \simeq \widehat{N}$. \item[(b)] If $N$ is a Cohen--Macaulay $\widehat{\Lambda}$--module such that $QN \simeq Q\widehat{N}'$ for some Cohen--Macaulay $\Lambda$--module $N'$, then there exists a Cohen--Macaulay $\Lambda$--module $N''$ such that $N \simeq \widehat{N}''$. \item[(c)] If $\widehat{N}$ is isomorphic to a direct summand of $\widehat{M}$, then $N$ is isomorphic to a direct summand of $M$. \end{enumerate} \end{proposition} In the next section we shall use the following simple result. \begin{proposition}\label{2.8} Let $P$ be a projective $\Lambda$--module. Then there exists a projective $\Lambda$--module $P'$ such that $P \oplus P'$ is free of rank $r \le \dim_k(P/rad\, P)$ where $k = R/{\frak m}$. \end{proposition} {\bf Proof:} Due to Proposition \ref{2.7}, we may suppose that $R$ is complete, thus the Krull--Schmidt theorem holds for modules. Let $\Lambda \simeq \oplus^s_{i=1} n_i P_i$, where all $P_i$ are indecomposable and pairwise non--isomorphic. Then $P\simeq \oplus^s_{i=1} m_i P_i$ for some $m_i$. Take $r$ the least integer such that $r n_i \ge m_i$ for all $i$. Then $r\Lambda \simeq P \oplus P'$ for $P' = \oplus^s_{i=1}(rn_i - m_i)P_i$. As $\dim_k(P/rad\,P) = \sum^s_{i=1} m_i \dim_k(P_i/rad\, P_i) \ge m_i$, one obtains $r \le \dim_k (P/rad\, P)$, q.e.d. {\bf Remark:} Obviously, $\dim_k(P/rad\,P) \le \ell(\widehat{P})$, so the last number can also serve as an upper bound for $r$. \newpage  \section{Families of Modules} {}From now on we suppose that our rings are algebras over the field $k = R/{\frak m}$, where $R$ is, as in the preceding paragraph, a Cohen-Macaulay ring. \begin{definition}\label{3.1} Let $X$ be a $k$--scheme, ${\cal O}_X = {\cal O}$ its structure sheaf, $\Lambda$ a CM--algebra (1--dimensional and analytically reduced) and ${\cal M}$ a coherent sheaf on $X$ of $\Lambda \otimes_k {\cal O}$--modules. Call ${\cal M}$ a {\bf family} of Cohen--Macaulay $\Lambda$--modules on $X$ if the following conditions hold: \begin{enumerate} \item[(1)] ${\cal M}$ is $R$--torsion free. \item[(2)]${\cal M}$ is ${\cal O}$--flat. \item[(3)] For each point $x \in X,\; {\cal M}(x) = {\cal M} \otimes_{{\cal O}} k(x)$ is a Cohen--Macaulay $\Lambda(x)$--module, where $\Lambda(x) = \Lambda \otimes_k k(x)$. \end{enumerate} It is easy to see that, under conditions (1) and (2), condition (3) is equivalent to: \begin{enumerate} \item[(3')] For every non--zero divisor $a \in R$, the sheaf ${\cal M}/a{\cal M}$ is also ${\cal O}$--flat. \end{enumerate} \end{definition} We are going to construct some ``almost universal'' families. Let $\Gamma$ be an overring of $\Lambda$ (cf.\ \S 2) and fix some positive integers $n$ and $d$. Put $\Phi = \Gamma/\Lambda$ and consider the Grassmannian $Gr = Gr(n\Phi,d)$, that is the variety of subspaces of codimension $d$ in $n\Phi$. Recall that for every $k$--scheme $X$ the morphisms $X \to Gr$ are in 1--1 correspondence with ${\cal O}$--factormodules of $n\Phi \otimes_k {\cal O}_X$ which are locally free of rank $d$ \cite{Mum}. Consider the subvariety $B = B(n, d; \Lambda, \Gamma)$ of $Gr(n\Phi, d)$ consisting of all $\Lambda$--submodules of $n\Phi$. In other words, the morphisms $X \to B$ are in 1--1 correspondence with $\Lambda \otimes_k {\cal O}_X$--factormodules of $n\Phi\otimes_k {\cal O}_X$ which are locally free over ${\cal O}_X$ of rank $d$. Evidently it is a closed subscheme of $Gr$. Denote by ${\cal F} = {\cal F}(n, d; \Lambda, \Gamma)$ the preimage in $n\Gamma \otimes_k {\cal O}_B$ of the canonical locally free sheaf of corank $d$ on $B$. As $(n\Gamma\otimes_k{\cal O}_B)/{\cal F}$ is flat over ${\cal O}_B$, one can see that ${\cal F}$ is really a family of Cohen--Macaulay $\Lambda$--modules on $B$ having the following universal property (cf.\ \cite{GP}). \begin{proposition}\label{3.2} For any family of Cohen--Macaulay $\Lambda$--modules ${\cal M}$ on a scheme $X$ such that $n\Lambda \otimes_k {\cal O}_X \subset {\cal M} \subset n\Gamma \otimes_k {\cal O}_X$ and $(n\Gamma \otimes_k {\cal O}_X)/{\cal M}$ is locally free over ${\cal O}_X$ of rank $d$, there exists a unique morphism $\varphi : X \to B$ such that ${\cal M} = \varphi^\ast({\cal F})$. \end{proposition} \begin{definition}\label{3.3} Call the families satisfying the conditions of Proposition \ref{3.2} {\bf sandwiched families} with respect to $\Gamma$ of rank $n$ and codimension $d$. In particular, when $X =$ Spec $k$, we have {\bf sandwiched modules} (with respect to $\Gamma$). \end{definition} {}From now on we suppose the ground field $k$ to be algebraically closed. We are going to show that the sandwiched families are, in some sense ``almost versal'', that is any other families can be stably glued from finitely many sandwiched families. Taking into account Corollary \ref{2.3}, this follows from the following result. \begin{theorem}\label{3.4a} Let $\Gamma$ be a hereditary overring of $\Lambda$ such that $\Lambda$ is dense in $\Gamma$. Then, given a family ${\cal M}$ of Cohen--Macaulay $\Lambda$--modules on a reduced $k$--scheme $X$, there exists a descending chain of closed subschemes $X = X_0 \supset X_1 \supset X_2 \supset \cdots \supset X_m = \emptyset$, a set of morphisms $\{\varphi_i : Y_i \longrightarrow B(n_i, d_i;\, \Lambda, \Gamma) \,|\,i = 1, \ldots, m\}$ and a set of projective $\Gamma$--modules $\{P_i\,|\,i = 1, \ldots, m\}$ such that ${\cal M}_{Y_i} \oplus (P_i \otimes_k {\cal O}_{Y_i}) \simeq \varphi^\ast_i {\cal F}(n_i, d_i;\, \Delta, \Gamma)$ where $Y_i = X_{i-1} \backslash X_i$ and $n_i \le \hat{\ell}({\cal M}) = \ell(\widehat{{\cal M}}(x))$ for an arbitrary closed point $x \in X$. \end{theorem} Indeed, we shall establish a more general result, when $\Gamma$ is not necessarily hereditary, but $\Gamma{\cal M}$ is flat over $\Gamma\otimes_k{\cal O}_X$. If $\Gamma$ is hereditary, the last condition becomes superfluous. Since $\Gamma{\cal M}/{\cal M}$ is ${\cal O}_X$--coherent, there exists an open dense subset $U \subseteq X$, on which $\Gamma{\cal M}/{\cal M}$ is flat over ${\cal O}$. Then, as $\Gamma$ is hereditary, it follows from \cite{CE} (Theorem \ref{2.8}) that $\Gamma{\cal M}$ is also $\Gamma\otimes_k{\cal O}$--flat. Moreover, the function $x \mapsto \dim_{k(x)} \Gamma{\cal M}(x)/\mbox{rad}\Gamma{\cal M}(x)$ takes its maximum in some closed point of $X$ and it does not exceed $\hat{\ell}({\cal M})$. Hence, we need only to establish the following fact: \begin{proposition}\label{3.4} Let $\Gamma$ be an overring of $\Lambda$ such that $\Lambda$ is dense in $\Gamma$. Let ${\cal M}$ be a family of Cohen--Macaulay $\Lambda$--modules on a reduced $k$--scheme $X$ such that $\Gamma{\cal M}$ is flat over $\Gamma \otimes_k{\cal O}$. Then there exists an open subscheme $Y \subset X$, a projective $\Gamma$--module $P$ and a morphism $\varphi : Y \to B (n, d; \Lambda, \Gamma)$ for some integers $n$ and $d$ such that the restriction on $Y$ of the family ${\cal M} \oplus (P \otimes_k {\cal O})$ is isomorphic to $\varphi^\ast {\cal F}(n, d; \Lambda, \Gamma$). Moreover, we can choose $n \le \max_g \ell({\cal M}(g))$ where $g$ runs through minimal points of $X$ (that is generic points of its irreducible components). \end{proposition} {\bf Proof:} Of course, we may suppose that $X$ is irreducible. Let $g \in X$ be its generic point. Consider the $\Gamma$--module $\Gamma {\cal M}(g)$. It is finitely generated and flat over $\Gamma(g)$, hence projective \cite{Bou}, (Ch.\ I, \S 2, Ex.\ 15). By Proposition \ref{2.8}, there exists a projective $\Gamma(g)$--module $P'$ such that $\Gamma {\cal M}(g) \oplus P' \simeq n \Gamma(g)$ and we can choose $n \le \dim_{k(g)} (\Gamma {\cal M} (g)/rad\, \Gamma {\cal M}(g))$. If we move to the completions, there is a 1--1--correspondence between projective and semi--simple $\widehat{\Gamma}$--modules and the same is valid for $\widehat{\Gamma}(g)$--modules. But as $k$ is separably closed and $k(g)$ separably generated over $k$, we have seen in the proof of Proposition \ref{1.6} that any simple $\widehat{\Gamma}(g)$--module is of the form $U \otimes_k k(g)$ for some simple $\widehat{\Gamma}$--module $U$. Hence, the same is true for projectives, so $\widehat{P}' \simeq \widehat{P} \otimes_k k(g)$ for some projective $\widehat{\Gamma}$--module $\widehat{P}$. But Proposition \ref{2.7} implies then that $\widehat{P}$ is really a completion of some projective $\Gamma$--module $P$, whence $P' \simeq P \otimes_k k(g)$. Replacing ${\cal M}$ by ${\cal M} \oplus (P \otimes_k {\cal O})$, we may now suppose that $\Gamma {\cal M}(g) \simeq n \Gamma(g)$. But $\Lambda(g)$ is dense in $\Gamma(g)$ by Proposition \ref{1.6}, so we may suppose, using Corollary \ref{1.3}, that ${\cal M}(g)$ contains a basis of $n \Gamma(g)$. By Proposition \ref{2.4}, ${\cal M}(g)$ is isomorphic to a submodule of $n \Gamma(g)$ containing $n \Lambda(g)$. So let $n\Lambda (g) \subset {\cal M}(g) \subset n \Gamma(g)$. Then the same is true on an open subset $Y \subset X$, that is $n \Lambda \otimes {\cal O}_Y \subset {\cal M}_Y \subset n \Gamma \otimes {\cal O}_Y$. Shrinking $Y$, we may also suppose that $(n \Gamma \otimes {\cal O}_Y)/{\cal M}_Y$ is locally free of some rank $d$ (over ${\cal O}_Y$) and it remains to use Proposition \ref{3.2}. An obvious iteration gives us the necessary generalization of Theorem 3.4: \begin{corollary}\label{3.5} Under the conditions of Proposition \ref{3.4} there exists a descending chain of closed subschemes $X = X_0 \supset X_1 \supset X_2 \supset \ldots \supset X_n = \emptyset$, a set of morphisms $\{\varphi_i : Y_i \to B(n_i, d_i, \Lambda, \Gamma) | i = 1, 2, \ldots, n\}$ and a set of projective $\Gamma$--modules $\{P_i | i = 1, 2, \ldots, n\}$ such that ${\cal M}_{Y_i} \oplus (P_i \otimes_k {\cal O}_{Y_i}) \simeq \varphi^\ast_i {\cal F} (n_i, d_i; \Lambda, \Gamma)$ where $Y_i = X_{i-1} \backslash X_i$ and $n_i \le \max \{\dim_{k(x)} (\Gamma {\cal M}(x)/rad\, \Gamma {\cal M}(x)) | x \in X\}$. \end{corollary} Fix now a CM--algebra $\Lambda$ and an overring $\Gamma$. Let $B = B(n, d; \Lambda, \Gamma)$ and ${\cal F} = {\cal F}(n, d; \Lambda, \Gamma)$. Choose a two--sided $\Gamma$--ideal $I \subset rad\,\Lambda$ of finite codimension (over $k$) and put $F = \Gamma/I;\; \bar{\Lambda} = \Lambda/I$. We can identify $Gr(n\Phi, d)$ with the closed subscheme of $Gr(nF, d)$ consisting of all subspaces $V$ containing $n\bar{\Lambda}$. Then $B$ also becomes a closed subvariety of $Gr(nF, d)$. We shall consider the elements of $nF$ as rows of length $n$ with entries from $F$ and identify $Aut(nF)$ with the full linear group $GL(n, F) = G$ acting on $nF$ according to the rule $g \cdot v = vg^{-1}$. Then Proposition \ref{2.5} implies that two subspaces $V, V' \in B$ correspond to isomorphic sandwiched modules if and only if there exists an element $g \in G$ such that $g \cdot V = V'$. Considering the elements of $nF$ as the rows of $n \times n$ matrices, we can identify $nV$ with a subspace in $M_n(F)$. Then we obtain the following: \begin{proposition}\label{3.6} Let $V \in B,\; g \in G$. Then $g \cdot V \in B$ if and only if $g \in G \cap nV$. Hence, $G \cdot V \cap B = (G \cap nV) \cdot V \simeq (G \cap nV)/StV$, where $StV = \{g \in G | gV = V\}$. \end{proposition} As $G$ is open in $M_n(F),\; G \cap nV$ is open in $nV$, hence $\dim(G\cap nV) = \dim nV = n(\gamma n - d)$ where $\gamma = \dim\, F$. Therefore, $\dim(G \cdot V \cap B) = n(\gamma n - d) - \dim\, StV$, whence: \begin{corollary}\label{3.7} For each integer $i$ the set $B_i = \{V \in B | \dim(G\cdot V \cap B) \le i\}$ is closed in $B$. \end{corollary} Put \[ {\rm par}(n, d; \Lambda, \Gamma) = \max_i(\dim B_i - i) \] \mbox{ and} \[ {\rm par}(n; \Lambda, \Gamma)= \max_d {\rm par}(n, d; \Lambda, \Gamma). \] Intuitively, ${\rm par}(n, d; \Lambda, \Gamma)$ is the number of independent parameters defining the isomorphism classes of sandwiched $\Lambda$--modules of rank $n$ and codimension $d$ with respect to $\Gamma$. Corollary \ref{3.5} evidently implies the following result. \begin{corollary}\label{3.7a} Under the conditions of Proposition 3.5, for any closed point $x \in X$ the set $\{y \in X \mid {\cal M}(y) \simeq {\cal M}(x) \otimes_k k(y)\}$ is constructible (that is a finite union of locally closed subsets of $X$) and its dimension is bigger or equal to dim$X -$ {\rm par}$(\widehat{\ell}({\cal M});\, \Delta, \Gamma$). \end{corollary} In particular, this assertion is true for any family of Cohen--Macaulay $\Lambda$--modules if we take for $\Gamma$ an hereditary overring of $\Lambda$ such that $\Lambda$ is dense in $\Gamma$ (which always exists, cf.\ Corollary 2.3). \begin{corollary}\label{3.9} Let $\Gamma$ be any overring of $\Lambda$ and $\Omega$ a hereditary overring of $\Lambda$ such that $\Lambda$ is dense in $\Omega$. Put $\ell_0 = \ell(\widehat{\Lambda})$. Then ${\rm par}(n, \Gamma) \le {\rm par}(\ell_0n, \Omega)$ for all $n$. \end{corollary} Of course, if $\Gamma \subset \Gamma'$ are two overrings of $\Lambda$ and $\dim_k(\Gamma'/\Gamma)= c$, then ${\rm par}(n, d; \Lambda, \Gamma) \le {\rm par}(n, c + d; \Lambda, \Gamma')$, whence ${\rm par}(n; \Lambda, \Gamma) \le {\rm par}(n; \Lambda, \Gamma')$. Put \[ b(n, \Lambda) = \max \{{\rm par}(n; \Lambda, \Gamma)\} \] where $\Gamma$ runs through all overrings of $\Lambda$ (we have actually to look only for maximal ones). Let also $p(n, \Lambda)$ denote the maximal value of $\dim\, X - \dim \{y \in X \mid {\cal M}(y) \simeq {\cal M}(x) \otimes_k k(y)\}$ taken for all families ${\cal M}$ with all possible bases $X$ and for all closed points $x \in X$. \begin{corollary}\label{3.10} Let $\ell_0 = \ell(\widehat{\Lambda})$. Then \[ b(n, \Lambda) \le p(n, \Lambda) \le b (n\ell_0, \Lambda). \] \end{corollary} \newpage  \section{Families of algebras} Now we formulate and prove the semicontinuity statements in two variants: for ``familes of algebras'' (Theorems~\ref{4.7} and \ref{4.9}) and for ``families of generators'' (Theorem~\ref{411}). Again $k$ denotes an algebraically closed field. \begin{definition}\label{4.1} Let $C$ be a reduced algebraic curve over $k$, $\Lambda$ a coherent sheaf of ${\cal O}_C$--algebras, containing no nilpotent ideals and such that for every point $p \in C$, $\Lambda_p$ is maximal Cohen--Macaulay ${\cal O}_{C,p}$--module. Then we call $\Lambda$ a {\bf sheaf of CM--algebras} or just a {\bf CM--algebra on $\bf C$}. \end{definition} If $\Gamma \supset \Lambda$ is another CM--algebra on $C$ and, for each $p \in C$, $\Gamma_p$ is overring of $\Lambda_p$, call $\Gamma$ {\bf an overring} of $\Lambda$. \begin{proposition}\label{4.2} If $\Lambda$ is a CM--algebra on a curve $C$, then, for every point $p \in C$, $\Lambda_p$ is a CM--algebra (in the sense of Definition \ref{2.1}) and, moreover, for almost all points $\Lambda_p$ is maximal. \end{proposition} {\bf Proof:} We only need to prove that $\widehat{\Lambda}_p$ contains no nilpotent ideal. According to \cite{D1}, this is equivalent to the existence of a maximal overring of $\Lambda_p$. Denote by $Z$ the centre of $\Lambda_p$. It is a localization of a finitely generated $k$--algbra, hence, its algebraic closure $\bar{Z}$ in the total quotient ring $Q$ is a finitely generated $Z$--module (cf.\ \cite{Bou}, Ch.V.\ \S 3.2). As before, we consider $\Lambda_p$ embedded in $Q \Lambda_p = Q \otimes_Z \Lambda_p$. Therefore, $\bar{Z}\Lambda_p \subset Q \Lambda_p$ is well--defined. But now $Q\Lambda_p$ is a central, semi--simple, hence, separable $QZ$--algebra, so $\bar{Z}\Lambda_p$ has a maximal overring (cf.\ \cite{CR}). Moreover, as $\bar{Z}_p = Z_p$ for almost all $p \in C$ and $\bar{Z}\Lambda_p$ is maximal for almost all $p$, the same is true also for $\Lambda_p$, q.e.d. Call $\Lambda$ {\bf hereditary} if all $\Lambda_p$ are hereditary (note that, for a general point $g \in C$, $\Lambda_g \simeq Q\Lambda$ is semi--simple). It is well--known (cf.\ \cite{CR}) that one--dimensional CM--algebras can be defined locally: \begin{proposition}\label{4.3} Let $\Lambda$ be a CM--algebra on a curve $C$ and suppose that for each closed point $p \in C$ an overring $\Gamma (p) \supset \Lambda_p$ is given such that $\Gamma(p) = \Lambda_p$ for almost all $p$. Then there exists an overring $\Gamma \supset \Lambda$ such that $\Gamma_p = \Gamma(p)$ for all $p$. \end{proposition} \begin{corollary}\label{4.4} There exists a hereditary overring $\Omega \supset \Lambda$ such that $\Lambda_p$ is dense in $\Omega_p$ for each $p \in C$. \end{corollary} Let now $\Gamma$ be any overring of $\Lambda$. As $\Gamma_p = \Lambda_p$ for almost all $p$, the sum \[ {\rm par}(n; \Lambda, \Gamma) = \sum_{p \in C} {\rm par}(n; \Lambda_p, \Gamma_p) \] is well--defined. \begin{definition}\label{4.5} Let $f : Y \to X$ be a morphism of $k$--schemes and ${\cal L}$ be a coherent sheaf of ${\cal O}_Y$--algebras. Call $({\cal L}, f) = ({\cal L}, f : Y \to X)$ a {\bf family of CM--algebras} with the base $X$ provided the following conditions hold: \begin{enumerate} \item[(1)] $f$ is flat and $f_\ast({\cal L})$ is flat ${\cal O}_X$--module. \item[(2)] $Y(x) = f^{-1}(x)$ is a reduced algebraic curve for each $x \in X$. \item[(3)] ${\cal L}(x)$ is a CM--algebra on $Y(x)$ for each $x \in X$. \end{enumerate} \end{definition} \begin{definition}\label{4.6} Let $({\cal L}, f : Y \to X)$ be a family of CM--algebras with base $X$. A {\bf family of overrings} of $({\cal L}, f)$ is a family $({\cal L}', f)$ (with the same $f$) such that ${\cal L}' \supset {\cal L},\; f_\ast({\cal L}'/{\cal L})$ is ${\cal O}_X$--flat and, for each $x \in X$, ${\cal L}'(x)$ is an overring of ${\cal L}(x)$. \end{definition} Given a family of overrings ${\cal L}' \supset {\cal L}$, we can define the functions on $X$: \begin{eqnarray*} {\rm par}(x,n,d) & := & {\rm par}(x,n,d; {\cal L}, {\cal L}') = {\rm par}(n, d; {\cal L}(x), {\cal L}'(x)),\\ {\rm par}(x, n)& := & {\rm par}(x, n; {\cal L}, {\cal L}') = {\rm par}(n; {\cal L}(x), {\cal L}'(x)). \end{eqnarray*} \begin{theorem}\label{4.7} The functions ${\rm par}(x, n, d)$ and ${\rm par}(x,n)$ are upper-semicontinuous, that is for each integer $i$ and for any $k$--scheme $X$ the sets $X_i(d) = \{x \in X | {\rm par}(x, n, d) \ge i\}$ and $X_i = \{x \in X|{\rm par}(x,n) \ge i\}$ are closed in $X$. \end{theorem} {\bf Proof:} As $X_i = \cup_d X_i(d)$ and since this union is finite, we only need to prove that $X_i(d)$ is closed. Moreover, we may suppose that $X$ is a smooth curve. Let ${\cal N} = {\cal L}'/{\cal L}$. Consider the relative Grassmannian $Gr(n{\cal N}, d) \to X$ and its closed subscheme (over $X$) ${\cal B}(n, d)$ consisting of ${\cal L}$--submodules. Let ${\cal J}$ be the biggest two--sided ${\cal L}'$--ideal contained in ${\cal L}$. Then it is easy to see that ${\cal L}/{\cal J}$ is torsion--free over ${\cal O}_X$, hence, flat. Thus, ${\cal L}'/{\cal J}$ is also flat over ${\cal O}_X$. As in the proof of Proposition \ref{3.6}, identify $Gr(n{\cal N}, d)$ with the closed subscheme of $Gr(n\bar{{\cal L}}', d)$, where $\bar{{\cal L}}' = {\cal L}'/{\cal J}$, and consider the group scheme over $X$, $GL(n, \bar{{\cal L}}')$ acting on the last Grassmannian. The same observations as in the proof of Proposition \ref{3.6} shows that ${\cal B}_j = \{v \in {\cal B}(n, d) \mid \dim\, St\, v \ge j\}$ is closed in ${\cal B}$. As ${\cal B}$ is proper over $X$, its projection $Z_j$ is also closed. But, by definition $X_i = \cup_j X_{ij}$, where $X_{ij} = \{x \in Z_j \mid \dim\, {\cal B}_j(x) \ge i + j\}$ are closed, q.e.d. {\bf Remark:} Suppose that the base of the family $({\cal L}, f)$ is a smooth curve and both ${\cal L}$ and ${\cal L}'$ are Cohen--Macaulay ${\cal O}_Y$--modules. Then $({\cal L}', f)$ is a family of overrings as it follows from \cite{BG} (Example 3.2.5). Moreover, in this case ${\cal O}_Y$ is Cohen--Macaulay itself and $\dim\, Y = 2$. Hence, we are able to construct Cohen--Macaulay ${\cal O}_Y$--modules locally as in the following lemma. \begin{lemma}\label{4.8} Suppose that $Y$ is a reduced 2--dimensional Cohen--Macaulay variety. Let ${\cal M}$ be a Cohen--Macaulay ${\cal O}_Y$--module, $\{y_1, y_2, \ldots, y_m\}$ a set of points of $Y$ of codimension 1 and $N(y_i)$ a finitely generated ${\cal O}_{Y,y_i}$--submodule in ${\cal Q}{\cal M}$ where ${\cal Q}$ is the total quotient ring of ${\cal O}$. Then there exists the Cohen--Macaulay submodule ${\cal N} \subset {\cal Q}{\cal M}$ such that ${\cal N}_{y_i} = N(y_i)$ and ${\cal N}_y = {\cal M}_y$ for all points $y$ of codimension 1, distinct from all $y_i$. \end{lemma} {\bf Proof:} One can easily construct ${\cal N}$ with prescribed localizations as in \cite{Bou} (VII 4.3). Moreover, we may suppose that ${\cal N} = \cap_{codim\,y=1} {\cal N}_y$. But then ${\cal N}$ is Cohen--Macaulay. We can now prove the main result of this paper. Recall that \[ b(n,x) := b(n, {\cal L}(x)) = \max\{ \mbox{par}(n; {\cal L}(x),\, \Gamma\} \] is the maximum number of independent parameters of isomorphism classes of sandwiched ${\cal L}(x)$--modules of rank $n$, which can be thought of as the dimension of the ``moduli space'' of ${\cal L}(x)$--CM--modules of rank $n$. \begin{theorem}\label{4.9} The function $b(n, x) = b(n, {\cal L}(x))$ is upper semi--continuous. \end{theorem} {\bf Proof:} Again we may suppose that $X$ is a smooth curve. Let $g \in X$ be the generic point of $X$ and $\Lambda = {\cal L}(g)$. Find an overring $\Omega \supset \Lambda$ such that $b(n, \Lambda) = {\rm par}(n, \Lambda, \Omega)$. Using Lemma \ref{4.8}, we can construct a family of overrings ${\cal L}' \supset {\cal L}$ with ${\cal L}'(g) = \Omega$. As $b(x) \ge {\rm par}(n, {\cal L}(x), {\cal L}'(x))$ for every $x \in X$, it follows from Proposition \ref{4.7} that the set $\{x \in X \mid b (x) \ge b(g)\}$ is closed. This, of course, proves the theorem. \begin{corollary}\label{4.10} For any family of CM--algebras $({\cal L}, f : Y \to X),$ the set $W({\cal L}) = \{x \in X \mid {\cal L}(x)\mbox{ is wild}\}$ is a countable union of closed subsets of $X$. \end{corollary} (For the definition of tame and wild CM--algebras cf.\ \cite{DG 1}). The proof of this corollary follows from Theorem \ref{4.8} just in the same way as it followed in the commutative case from Kn\"orrer's theorem (cf.\ \cite{DG 2}, Corollary 4.2). Now we consider onother version of the semicontinuity theorem, where algebras are given by parametrized families of generators. Namely, let $X$ be an algebraic $k$--scheme, ${\cal L}$ a family of CM--algebras with the base $X$ and ${\cal I}$ an ideal of ${\cal L}$ such that ${\cal L}/{\cal I}$ is a locally free ${\cal O}_X$--module of finite rank, that is it corresponds to a vector bundle $\pi : F \to X$. The fibres $F(x)$ of $F$ are then finite--dimensional $k(x)$--algebras. Suppose given an algebraic $X$--scheme $f : Y \to X$ and a set of $X$--morphisms $\{\gamma_i : Y \to F \mid i = 1, 2, \ldots, m\}$ (equivalently $Y$--sections of $f^\ast F$). For each point $y \in Y$ denote $A(y)$ the subalgebra of $F(f(y))$ generated by $\{\gamma_1(y),\; \gamma_2(y), \ldots, \gamma_m(y)\}$ and $\Lambda(y)$ the preimage of $A(y)$ in ${\cal L}(y) = {\cal L} \otimes_{{\cal O}_X} k(f(y))$. Then $\Lambda(y)$ is a CM--algebra, thus, given a family of overrings ${\cal L}' \supset {\cal L}$, we may consider, as above, the functions on $Y$: \vspace{-0.5cm} \begin{eqnarray*} p(n,d;\, y) &=& \mbox{par}(n,d;\, \Lambda(y),\; {\cal L}'(y));\\ p(n;\, y) & = &\mbox{par}(n;\, \Lambda(y),\; {\cal L}'(y));\\ b(n,y) & = &b(n, \Lambda(y)). \end{eqnarray*} \begin{theorem}\label{411} In the above situation, the functions $p(n, d;\, y);\; p(n,y)$ and $b(n,y)$ are upper- semicontinuous on $Y$. \end{theorem} {\bf Proof}: Replacing ${\cal L}$ by $f^\ast({\cal L})$, which is a family of CM--algebras on $Y$, we may suppose that $X = Y$ and $f$ is the identity map. Moreover, we may also suppose $X$ to be a smooth curve. As the function $\dim A(y)$ is obviously upper semi--continuous on $Y$, there is an open subset $U \subset Y$ such that $\dim A(y)$ is constant and maximal possible on $U$. Put $d = \dim F(y) - \dim A(y)$. Then we obtain a section $\varphi : U \to \mbox{Gr}(d, F)$ such that $A(y)$ is the subspace of $F(y)$ corresponding to $\varphi(y)$ for each $y \in U$. But as $X$ is a smooth curve and $\mbox{Gr}(d,F)$ is projective over $X$, $\varphi$ can be prolonged to a section $\bar{\varphi} : X \to \mbox{Gr}(d,F)$ (it follows, for example, from \cite{Ha}, Proposition III.9.8). Now $\bar{\varphi}$ gives rise to a subbundle $A' \subset F$ of constant codimension $d$. Denote by $\Lambda'$ its preimage in ${\cal L}$. Note that both conditions \[ \mbox{``}A'(x) \mbox{is a subalgebra of } F(x)\mbox{'' and ``}A'(x) \supset A(x)\mbox{''} \] are evidently closed and hold on $U$. Thus they hold on $X$, that is $\Lambda'(x)$ is a subalgebra of ${\cal L}(x)$ containing $\Lambda(x)$. As ${\cal L}/\Lambda'$ is locally free of finite rank, $\Lambda'$ is really a family of CM--algebras on $X$. Hence, the functions $p'(n,d;\, x),\; p'(n;\, x)$ and $b'(n,x)$ defined just as $p(n,d;\, x),\; p(n;\, x)$ and $b(n,x)$ but using $\Lambda'(x)$ instead of $\Lambda(x)$ are upper semicontinuous. On the other hand we have inequalitites $p(n,d;\, x) \ge p'(n, d;\, x),\; p(n, x) \ge p'(n, x),\; b(n,x) \ge b'(n,x)$ on $X$ and equality on $U$. Therefore, $p(n,d;\, x),\; p(n;\, x)$ and $b(n;x)$ are also upper semicontinuous. To show an application of Theorem~\ref{411}, we extend the criteria of tameness, proved in \cite{DG 2} for the case char $k \not= 2$, to all characteristics. In order to do this, we must first define the singularities $T_{pq}$ in positive characteristic, which are defined for char $k = 0$ as factorrings $(\ast)\hspace{4cm}k[[X,Y]]/(X^p + Y^q + \lambda X^2 Y^2)$. For our purpose it is more convenient to define them using their parametrization given by Schappert \cite{Sch}. Namely, let $\Lambda$ be a local commutative CM--algebra, $\Lambda_0$ its maximal overring. Then $\Lambda_0$ is a direct product of power series rings: \[ \Lambda_0 \simeq k[[t_1]] \times k[[t_2]] \times \dots \times k[[t_s]] \] ($s$ is ``the number of branches'' of $\Lambda$). If $a \in \Lambda$, $a = (a_1, a_2, \dots, a_s)$, with $a_i \in k[[t_i]]$, put {\boldmath$v$}$(a) = (v(a_1), \dots, v(a_2))$, where $v(a_i)$ denotes the usual valuation on the power series (in particular $v(0) = \infty$). Call $\Lambda$ a plane curve singularity if its maximal ideal ${\cal M}$ is generated by two elements: ${\cal M} = (x,y)$. Define the ({\bf valuation}) {\bf type} of $\Lambda$ as the pair ({\boldmath$v$}$(x)$, {\boldmath$v$}$(y)$). \begin{definition}{\rm Let $\Lambda$ be a plane curve singularity. We say that $\Lambda$ is of {\bf type} {\boldmath$T_{pq}$}, where $p, q \in {\Bbb N},\; \frac{1}{p} + \frac{1}{q} \le \frac{1}{2}$, if its valuation type is: \[ \begin{array}{ll} (2, p-2),\; (q-2, 2) & \mbox{for $p,q$ both odd},\\ (1, 1, p-2),\; (\infty, \frac{q}{2} - 1, 2) & \mbox{for $p$ odd, $q$ even},\\ (1, 1, \frac{p}{2} - 1, \infty), \; (\frac{q}{2} - 1, \infty, 1, 1) & \mbox{for $p,q$ both even}. \end{array} \] By \cite{Sch} this definition is equivalent to the equation ($\ast$) if char $k = 0$.} \end{definition} The following theorem was proved in \cite{DG 2} for char $k \not= 2$. Let $\Lambda$ be a local commutative CM--algebra, $\Lambda_0 = k[[t_1]] \times \dots \times k[[t_s]]$ its maximal overring, {\boldmath$m$} = rad$\Lambda$. Denote $t = (t_1, \dots, t_s) \in \Lambda_0$; $\Lambda^\prime = t\Lambda_0 + \Lambda,\; \Lambda^{\prime\prime} = t${\boldmath$m$}$\Lambda_0 + \Lambda$ and $\Lambda^\prime_e = \Lambda^\prime + ke$, where $e \in \Lambda^\prime$ is an idempotent. For each overring $\Gamma \supset \Lambda^\prime$, let $\Gamma/${\boldmath$m$}$\Gamma = L_1 \times \dots \times L_m$, where $L_i$ are local algebras, $d_i = \dim\, L_i$, {\boldmath$d$}$(\Gamma) = (d_1, d_2, \dots, d_m)$ and $d(\Gamma) = d_1 + \dots + d_m$ (the minimal number of generators of $\Gamma$ as $\Lambda$--module). \begin{theorem} If $\Lambda$ is of infinite Cohen--Macaulay type, the following conditions are equivalent: \begin{enumerate} \item[(1)] $\Lambda$ is tame. \item[(2)] $\Lambda$ dominates a plane curve singularity of type $T_{pq}$ for some $p,q$ (that is, $\Lambda$ is isomorphic to an overring of $T_{pq}$). \item[(3)] The following restrictions hold: \begin{enumerate} \item $d(\Lambda_0) \le 4$ and {\boldmath$d$}$(\Lambda_0) \not\in \{(4),\; (3,1),\; (3)\}$, \item $d(\Lambda^\prime) \le 3$ and {\boldmath$d$}$(\Lambda^\prime_e) \not= (3,1)$ for any idempotent $e$, \item if $d(\Lambda_0) = 3$, then $d(\Lambda^{\prime\prime}) \le 2$. \end{enumerate} \end{enumerate} \end{theorem} {\bf Proof}: (1) $\Rightarrow$ (3) and (3) $\Rightarrow$ (2) were proved in \cite{DG 2} and their proofs did not use the restriction char $k \not= 2$. In order to prove (2) $\Rightarrow$ (1), again following \cite{DG 2}, note that the singularity $\Lambda$ of type $T_{pq}$ contains the $\Lambda_0$--ideal $I = b\Lambda_0$, where \[ \begin{array}{ll} b = (t_1^{p+1},\; t_2^{q+1}) & \mbox{for $p,q$ both odd};\\ b = (t_1^{q/2+1},\; t_2^{q/2+1},\; t_3^{p+1}) & \mbox{for $p$ odd, $q$ even};\\ b = (t_1^{q/2+1},\; t_2^{p/2+1},\; t_3^{q/2+1},\; t_4^{p/2+1}) & \mbox{for $p,q$ both even}. \end{array} \] Consider now a new CM--algebra $\Lambda(\lambda),\; \lambda \in k$, containing $I$ and generated modulo $I$ by the following 3 elements: \[ \begin{array}{ll} (\lambda, 1)x,\; (1,\lambda)y,\; xy & \mbox{for $p,q$ both odd},\\ (1,1,\lambda)x,\; (\lambda,\lambda, 1)y,\; xy & \mbox{for $p$ odd, $q$ even},\\ (1, \lambda,1,\lambda)x,\; (\lambda,1, \lambda, 1)y,\, xy & \mbox{for $p,q$ both even}. \end{array} \] If $(p,q) \not\in \{(4,4),\; (3,6)\}$, one can easily check that $\Lambda(\lambda) \simeq \Lambda$ for $\lambda \not= 0$, while $\Lambda(0)$ is a singularity of type $P_{pq}$ in the terminology of \cite{DG 2}, that is generated modulo $\,I \,$ by the elements $\,x_0,y_o \,$ such that {\boldmath$v$}$(x_0)$,{\boldmath$v$}$(y_0)$ are of the form: \[ \begin{array}{ll} (2, \infty),\; (\infty, 2) & \mbox{for $p,q$ both odd},\\ (1, 1, \infty),\; (\infty, \infty, 2) & \mbox{for $p$ odd, $q$ even},\\ (1, 1, \infty, \infty), \; (\infty, \infty, 1, 1) & \mbox{for $p,q$ both even}. \end{array} \] Again, the calculations for $P_{pq}$ in \cite{DG 2} did not use the condition char $k\not= 2$. Hence, they are tame and Theorem 4.11 implies that $\Lambda$ is also tame. The calculation of Dieterich for the remaining case $(p,q) = (3,6)$ or $(p,q) = (4,4)$ (cf.\ \cite{Di 1}, \cite{Di 2}) also did not use any conditions on characteristics. Thus, implication (2) $\Rightarrow$ (1) is completely proved. \newpage \section{Some analogues} Here we give some examples of ``almost versal families'' and semicontinuity theorems for other situations in representation theory. As all the proofs are quite similar (and easier) to those of the preceding sections, we omit them and give only the final formulations of the results analogous to Theorems~\ref{3.4a}, \ref{4.7} and \ref{411}. Although some of the corresponding semicontinuity theorems are known, we hope that the ``unification'' will be useful for these cases too. At least we give new proofs for them. \subsection*{Finite-dimensional algebras} Here, let $A$ be a finite--dimensional algebra over an algebraically closed field $k$. A {\bf family of {\boldmath$A$}-modules\/} parametrized by a $ k $--scheme $ X $ is a sheaf $ {\cal M} $ of $ A\otimes_k{\cal O}_X$--modules, which is coherent and flat over $ {\cal O}_X $. To be in the frame of projective varieties, we can consider first the subvariety $B(P, I, d) \subset Gr(P, d)$, where $P$ is a projective module over a finite--dimensional algebra $A$, $I$ an ideal of $A$ contained in the radical and $B(P, I, d)$ consists of the $A$--submodules $L \subset IP$. $Gr(P,d)$ denotes the Grassmannian of $d$--codimensional subspaces of $P$. Then the canonical sheaf $ {\cal F}={\cal F}(P,I,d) $ on $ B(P,I,d) $ is a family of $ A $--modules and the following result holds. \begin{theorem} \label{5.1} Let $ A $ be a finite-dimensional $ k $-algebra, $ J=rad\, A $ and $ {\cal M} $ a family of $ A $--modules parametrized by a reduced $ k $--scheme $ X $. Then there exists a descending chain of closed subschemes $ X=X_0\supset X_1\supset X_2\supset \dots\supset X_m=\emptyset $ and a set of morphisms $ \{\varphi_i:Y_i\rightarrow B(P_i,J,d_i)\mid i=1,\dots,m\} $ for some projective $ A $--modules $ P_i $ such that $ {\cal M}_{Y_i}\simeq\varphi_i^\ast {\cal F}(P_i,J,d_i) $, where $ Y_i=X_{i-1}\setminus X_i $. Moreover, if $ r= \mbox{ rank } {\cal M} $ (as a locally free sheaf over $ X $), then $ \dim P_i\le rp $, where $ p $ is the maximal dimension of indecomposable projective $ A $--modules. \end{theorem} The group $ G = \mbox{ Aut}_AP $ acts on $ B=B(P,I,d) $ and, as $ I\subset \mbox{ rad } A $, we conclude that $ {\cal F}(x)\simeq{\cal F}(y) $ if and only if $ x $ and $ y $ belong to the same $ G $--orbit. The subsets $ B_i=\{ x\in B | \dim(Gx)\le i \} $ are obviously closed in $ B $. Hence, we can define the {\bf number of parameters\/}: \[ {\rm par}(P,I,d; A) = \max_i(\dim B_i-i) \] and \[ {\rm par}(P,I; A) = \max_d {\rm par}(P,I,d;,A). \] In particular, put \[ {\rm par}(n,d; A)={\rm par}(nA,rad A,d; A)\quad {\rm and}\quad {\rm par}(n; A)={\rm par}(nA,rad A; A). \] Just as for Cohen-Macaulay algebras, these numbers give upper bounds for the number of (independent) parameters of isomorphism classes of $A$--modules of rank $n$ in {\em any\/} family of $ A $--modules. Consider now a {\bf family of algebras} parametrized by a $ k $--scheme $ X $, that is a flat coherent sheaf of $ {\cal O}_X$-algebras $ {\cal A} $. Then we are able to define the following functions on $ X $: \vspace{-0.5cm} \begin{eqnarray*} {\rm par}(x,n,d) &=& {\rm par} (n,d; {\cal A}(x)) ,\\ {\rm par}(x,n) &=& {\rm par} (n; {\cal A}(x)) . \end{eqnarray*} \begin{theorem} \label{5.2} For each family of finite--dimensional $ k $--algebras the functions $ {\rm par}(x,n,d) $ and $ {\rm par}(x,n) $ are upper--semicontinuous. \end{theorem} A version of this theorem was proved by Gei{\ss} \cite{Gei}. Note also that Theorem 5.2 provides a new proof of Gabriel's theorem \cite{Gab} that finite representation type is an open condition. This follows since the Brauer--Thrall conjectures are known to be true for finite dimensional algebras. It is easy to generalize the last theorem to the situation where the algebras are given by ``generators and relations''. Namely, suppose we are given: \begin{itemize} \item a family $ {\cal A} $ of finite--dimensional $ k $-algebras over $X$; \item an algebraic $ X $-scheme $ f:Y\rightarrow X $; \item two sets of $ X $-morphisms $ \{\gamma_i:Y\rightarrow F \mid i=1,\dots,m\} $ and $ \{\rho_j:Y\rightarrow F \mid j=1,\dots,r\} $, where $ F $ is the vector bundle on $ X $ corresponding to the locally free sheaf $ {\cal A} $. \end{itemize} For any point $ y\in Y $, denote by $ I(y) $ the ideal in $ F(f(y)) $ generated by the set $ \{\rho_j(y) | j=1,\dots,r \} $ and by $ A(y) $ the subalgebra of $ F(f(y))/I(y) $ generated by the classes $ \{\gamma_i(y)+I(y) | i=1,\dots,m \} $. Then we can define the functions on $ Y $: \vspace{-0.5cm} \begin{eqnarray*} p(y,n,d) &=& {\rm par} (n,d; A(y)) ,\\ p(y,n) &=& {\rm par} (n; A(y)) . \end{eqnarray*} \begin{theorem} \label{5.3} In the above situation the functions $ p(y,n,d) $ and $ p(y,n) $ are upper-semicontinuous on $ Y $. \end{theorem} \subsection*{Bimodules} Consider now the categories of elements of finite--dimensional bimodules (in the sense of \cite{D2}, although we give here a somewhat different definition). Let $ A $ be a finite--dimensional $ k $--algebra, where $ k $ is again an algebraically closed field, and let $ V $ be a finite--dimensional $ A $--bimodule. The {\bf elements} of $ V $ are, by definition, those of the set $ El(V)=\bigsqcup_{P} V(P) $, where $ P $ runs through all (finitely generated) projective $ A $--modules and $ V(P)=\mbox{ Hom}_A(P,V\otimes_AP) $. Two elements $ u\in V(P) $ and $ u'\in V(P') $ are said to be {\bf isomorphic} if there exists an isomorphism $ p:P\rightarrow P' $ such that $ u'=(1\otimes p)up^{-1} $. Indeed, in \cite{D2} only so--called disjoint bimodules were considered. The bimodule $ V $ is said to be {\bf disjoint} if $ A=A_1\times A_2 $ and $ VA_1=A_2V=0 $. In most applications in representations theory one needs only disjoint bimodules, but non--disjoint ones appear in various ``reduction processes''. To remain in the category of projective varieties, it is convenient to change the problem slightly. Namely, call elements $ u $ and $ u' $ {\bf (projectively) equivalent}, if $ u' $ is isomorphic to $ \lambda u $ for some non--zero $ \lambda\in k $. Obviously, if the bimodule is disjoint, then equivalent elements are isomorphic, but in the non--disjoint case it is not always so. Let $ X $ be a $ k $--scheme and $ {\cal P} $ a flat coherent sheaf of $ A_X $--modules, where $ A_X=A\otimes_k{\cal O}_X $. Put $ V({\cal P})={\cal H}{\it om}_{A_X}({\cal P},V\otimes_A{\cal P}) $. It is a locally free coherent sheaf of $ {\cal O}_X$--modules. Hence, the corresponding projective bundle $ \P_{\cal P}=\P_X(V({\cal P})) $ over $ X $ is defined (cf.\ \cite{Ha}). A {\bf projective family} (or simply {\bf family}, as we do not consider other families here) of elements of the bimodule $ V $ with base $ X $ is, by definition, a section $ \Phi:X\rightarrow \P_{\cal P} $ for some $ {\cal P} $. Note that using projective families we need to consider projective equivalence instead of isomorphism and to exclude zero elements of the bimodule. But this does not essentially differ from the classification problem for the elements of bimodules up to isomorphism. The ``almost universal'' families in this case are more or less evident. Indeed, put, for any projective $ A $--module $ P $, $ B(P)=\P_k(V(P)) $ and $ \tilde P=P\otimes_k{\cal O}_{B(P)} $. Then $ \P_{\tilde P}\simeq B\times B $, where $ B=B(P) $ and the diagonal map $ \Delta_P:B\rightarrow B\times B $ defines a family of elements of $ V $ with the base $ B $. The following result is almost obvious. \begin{theorem}\label{5.4} Let $ A $ be a finite--dimensional $ k $--algebra, $ V $ a finite-dimensional $ A$--bimodule and and $ \Phi:X\rightarrow \P_{\cal P} $ a (projective) family of elements of $ V $. Then there exists a descending chain of closed subschemes $ X=X_0\supset X_1\supset X_2\supset \dots\supset X_m=\emptyset $ and a set of morphisms $ \{\varphi_i:Y_i\rightarrow B(P_i) | i=1,\dots,m\} $ for some projective $ A $--modules $ P_i $ such that $ {\cal P}_{Y_i}\simeq P_i\otimes_k{\cal O}_{Y_i}$. Hence, the restriction of $ \P_{\cal P} $ on $ Y_i= X_{i-1}\backslash X_i $ can be identified with $ Y_i\times B(P_i) $, and, under this identification, $ \Phi_{Y_i}=1\times\varphi_i $. Moreover, $ \dim P_i=rank {\cal P} $ for all $ i $. \end{theorem} The group $ G=\mbox{ Aut}_AP $ acts on $ B=B(P) $ and its orbits are the classes of projective equivalence. Hence, we are again able to define the closed subsets $ B_i=\{ x\in B | \dim(Gx)\le i \} $ and the {\bf number of parameters\/}: \[ {\rm par}(P; A,V) = \max_i(\dim B_i-i) , \] in particular \[ {\rm par}(n; A,V)=par(nA; A,V) . \] Now, given a family of algebras $ {\cal A} $ with base $ X $ and a family of bimodules, that is a coherent sheaf $ {\cal V} $ of $ {\cal A} $--bimodules, flat over $ {\cal O}_X $, we can define the function on $ X $: \[ {\rm par}(x,n) = {\rm par} (n; {\cal A}(x),{\cal V}(x)) . \] \begin{theorem} \label{5.5} For each family of finite--dimensional $ k $--algebras and bimodules the function $ {\rm par}(x,n) $ is upper--semicontinuous. \end{theorem} Of course, one could easily give a version of the last theorem, where the algebras and bimodules are defined by generators and relations, but we leave this obvious generalization to the reader. \subsection*{Remark} In particular, in both cases we can see that the set of wild algebras (or bimodules) in some family is again a countable union of closed subsets. It looks very likely that this set is even closed and, hence, that the set of tame algebras (or bimodules) is open. In order to prove it, one only needs to show that the set of tame algebras (bimodules) is really a countable union of constructible sets (cf.\ \cite{Gab}). If we consider families of commutative CM--algebras, then the set of tame algebras is indeed open. This can be derived from \cite{DG 2} in two ways. The first is to apply the classification of \cite{DG 2} and deformation theory of singularities: the set of singularities which are of finite CM--representation type or which are tame is open in any flat family of singularities. The second is to note that the strict respresentations over the free algebra $k\langle x,y\rangle$ constructed in \cite{DG 2} are of bounded rank. Hence, we can find a common constant $n$ such that a commutative CM--algebra $\Lambda$ is wild if and only if $p(n,\Lambda) > rn$, where $r$ is the rational length of $\Lambda$, which coincides in this case with the number of branches. As $r$ is obviously bounded in any family, we have now only to apply Theorem 4.9. \newpage {\small\addcontentsline{toc}{section}{References}

9310

alg-geom/9310005

en

https://arxiv.org/abs/alg-geom/9310005

alg-geom/9310005

Subhashis Nag

Subhashis Nag and Dennis Sullivan

Teichm\"uller Theory and the Universal Period Mapping via Quantum Calculus and the $H^{1/2}$ Space on the Circle

39 pages (TEX)

The Universal Teichm\"uller Space, $T(1)$, is a universal parameter space for all Riemann surfaces. In earlier work of the first author it was shown that one can canonically associate infinite- dimensional period matrices to the coadjoint orbit manifold $Diff(S^1)/Mobius(S^1)$ -- which resides within $T(1)$ as the (Kirillov-Kostant) submanifold of ``smooth points'' of $T(1)$. We now extend that period mapping $\Pi$ to the entire Universal Teichm\"uller space utilizing the theory of the Sobolev space $H^{1/2}(S^1)$. $\Pi$ is an equivariant injective holomorphic immersion of $T(1)$ into Universal Siegel Space, and we describe the Schottky locus utilizing Connes' ``quantum calculus''. There are connections to string theory. Universal Teichm\"uller Space contains also the separable complex submanifold $T(H_\infty)$ -- the Teichm\"uller space of the universal hyperbolic lamination. Genus-independent constructions like the universal period mapping proceed naturally to live on this completion of the classical Teichm\"uller spaces. We show that $T(H_\infty)$ carries a natural convergent Weil-Petersson pairing.

alg-geom

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\centerline{\bf{TEICHM\"ULLER THEORY AND THE UNIVERSAL PERIOD MAPPING}} \smallskip \centerline{\bf{VIA QUANTUM CALCULUS AND THE $H^{1/2}$ SPACE ON THE CIRCLE}} \medskip \centerline {by} \medskip \centerline{{\bf Subhashis Nag} and {\bf Dennis Sullivan}} \bigskip \nn {\bf Abstract:} Quasisymmetric homeomorphisms of the circle, that arise in the Teichm\"uller theory of Riemann surfaces as boundary values of quasiconfomal diffeomorphisms of the disk, have fractal graphs in general and are consequently not so amenable to usual analytical or calculus procedures. In this paper we make use of the remarkable fact this group $QS(S^{1})$ acts by substitution (i.e., pre-composition) as a family of bounded symplectic operators on the Hilbert space ${\cal H}$=``$H^{1/2}$'' (comprising functions mod constants on $S^1$ possessing a square-integrable half-order derivative). Conversely, and that is also important for our work, quasisymmetric homeomorphisms are actually {\it characterized} amongst homeomorphisms of $S^1$ by the property of preserving the space ${\cal H}$. Interpreting ${\cal H}$ via boundary values as the square-integrable first cohomology of the disk with the cup product symplectic structure, and complex structure provided by the Hodge star, we obtain a universal form of the classical period mapping extending the map of [12] [13] from $Diff(S^{1})/Mobius(S^{1})$ to all of $QS(S^{1})/Mobius(S^{1})$ -- namely to the entire universal Teichm\"uller space, $T(1)$. The target space for the period map is the universal Siegel space of period matrices; that is the space of all the complex structures on ${\cal H}$ that are compatible with the canonical symplectic structure. Using Alain Connes' suggestion of a quantum differential $d^{Q}_{J}f = [J,f]$ -- commutator of the multiplication operator with the complex structure operator -- we obtain in lieu of the problematical classical calculus a quantum calculus for quasisymmetric homeomorphisms. Namely, one has operators $\{h,L\}$, $d\circ\{h,L\}$, $d\circ\{h,J\}$, corresponding to the non-linear classical objects $log (h^\prime)$, ${h'' \over {h'}}dx$, ${1\over 6}Schwarzian(h)dx^{2}$ defined when $h$ is appropriately smooth. Any one of these objects is a quantum measure of the conformal distortion of $h$ in analogy with the classical calculus Beltrami coefficient $\mu$ for a quasiconformal homeomorphism of the disk. Here $L$ is the smoothing operator on the line (or the circle) with kernel $log \vert x-y \vert$, $J$ is the Hilbert transform (which is $d \circ L$ or $L \circ d$), and $\{h,A\}$ means $A$ conjugated by $h$ minus $A$. The period mapping and the quantum calculus are related in several ways. For example, $f$ belongs to ${\cal H}$ if and only if the quantum differential is Hilbert-Schmidt. Also, the complex structures $J$ on ${\cal H}$ lying on the Schottky locus (image of the period map) satisfy a quantum integrability condition $[d^{Q}_{J},J]=0$. Finally, we discuss the Teichm\"uller space of the universal hyperbolic lamination ([20]) as a separable complex submanifold of $T(1)$. The lattice and K\"ahler (Weil-Petersson) metric aspect of the classical period mapping appear by focusing attention on this space. \vfill\eject \baselineskip=14pt \nn {\bf \S 1 - Introduction} The Universal Teichm\"uller Space $T(1)$, which is a universal parameter space for all Riemann surfaces, is a complex Banach manifold that may be defined as the homogeneous space $QS \left({ S^1}\right)/$M\"ob $\left({ S^1}\right)$. Here $QS \left({ S^1}\right)$ denotes the group of all quasisymmetric homeomorphisms of the unit circle $\left({ S^1}\right)$, and M\"ob$\left({ S^1}\right)$ is the three-parameter subgroup of M\"obius transformations of the unit disc (restricted to the boundary circle). There is a remarkable homogeneous K\"ahler complex manifold, $M = \mathop{\rm Diff}\nolimits \left({ S^1}\right)/$ M\"ob $\left({ S^1}\right)$,-- arising from applying the Kirillov-Kostant coadjoint orbit method to the $C^\infty$-diffeomorphism group $\mathop{\rm Diff}\nolimits \left({ S^1}\right)$ of the circle ([22]) - that clearly sits embedded in $T(1)$ (since any smooth diffeomorphism is quasisymmetric). \bigskip In [15] it was proved that the canonical complex-analytic and K\"ahler structures on these two spaces coincide under the natural injection of $M$ into $T(1)$. (The K\"ahler structure on $T(1)$ is formal -- the pairing converges on precisely the $H^{3/2}$ vector fields on the circle.) The relevant complex-analytic and symplectic structures on $M$, (and its close relative $N = \mathop{\rm Diff}\nolimits \left({ S^1}\right)/ \left({ S^1}\right)$), arise from the representation theory of $\mathop{\rm Diff}\nolimits \left({ S^1}\right)$ ; whereas on $T(1)$ the complex structure is dictated by Teichm\"uller theory, and the (formal) K\"ahler metric is Weil-Petersson. Thus, the homogeneous space $M$ is a complex analytic submanifold (not locally closed) in $T(1)$, carrying a canonical K\"ahler metric. \bigskip In subsequent work ([12] [13]) it was shown that one can canonically associate infinite-dimensional period matrices to the smooth points $M$ of $T(1)$. The crucial step in this construction was a faithful representation (Segal [18]) of $\mathop{\rm Diff}\nolimits \left({S^1}\right)$ on the Frechet space $$V = C^\infty \mathop{\rm Maps}\nolimits \left({ S^1, {\bb R} }\right)/{\bb R} ({\mathop{\rm \ the \ constant \ maps \ }\nolimits}) \eqno (1)$$ \nn $\mathop{\rm Diff}\nolimits \left({ S^1}\right)$ acts by pullback on the functions in $V$ as a group of toplinear automorphisms that preserve a basic symplectic form that $V$ carries. \bigskip In order to be able to extend the infinite dimensional period map to the full space $T(1)$, it is necessary to replace $V$ by a suitable ``completed'' space that is invariant under quasisymmetric pullbacks. Moreover, the quasisymmetric homeomorphisms should continue to act as bounded symplectic automorphisms of this extended space. These goals are achieved in the present paper by developing the theory of the Sobolev space on the circle consisting of functions with half-order derivative. This Hilbert space $ H^{1/2}$ = ${\cal H}$, {\it which turns out to be exactly the completion of the pre-Hilbert space $V$}, actually {\it characterizes} quasisymmetric (q.s). homeomorphisms (amongst all homeomorphisms of $S^1$). That fact will be important for our understanding of the period mapping. The symplectic structure, $S$, on $V$ extends to ${\cal H}$ and is preserved by the action of $QS(S^1)$, and indeed we show that this $S$ is the {\it unique} symplectic structure available which is invariant under even the tiny finite-dimensional subgroup M\"ob $\left({ S^1}\right) $ ($\subset QS \left({ S^1}\right) $). \bigskip We utilise several different characterisations of ${\cal H}$ and its complexification. In particular, ${\cal H}$ comprises functions on $S^1$ which are defined quasi-everywhere (i.e., off some set of logarithmic capacity zero); alternatively, they appear as non-tangential limits of harmonic functions of finite Dirichlet energy in the disc. The last-mentioned fact allows us to interpret ${\cal H}$ as the first cohomology space with real coefficients of the unit disc in the Hodge-theoretic sense. That is important for our subsequent discussion of the period mapping as a theory of the variation of $S$-compatible complex structure on this real Hilbert space ${\cal H}$. The fact that quasisymmetric homeomorphisms are the {\it only} ones preserving ${\cal H}$ is necessary in our determination of the universal Schottky locus -- namely the image of $\Pi$. \bigskip We present a section where we discuss quantum calculus on the line (motivated by Alain Connes), the idea being firstly to demonstrate that the $H^{1/2}$ functions have such an interpretation. That then allows us to interpret the universal Siegel space that is the target space for the period mapping as "almost complex structures on the line" and the Teichm\"uller points (i.e., the Schottky locus ) can be interpreted as comprising precisely the subfamily of those complex structures that are {\it integrable}. \bigskip Notice that the fact that capacity zero sets are preserved by quasisymetric transformations -- whereas merely being measure zero is not a q.s.-invariant notion -- goes to exemplify how deeply quasisymmetry is connected to the properties of ${\cal H}$. Other characterisations found below for the complexification ${\bb C} \otimes {\cal H}$ in terms of boundary values for holomorphic and anti-holomorphic functions are of independent importance, and relate to the proof of the uniqueness of the invariant symplectic structure. That proof utilises a pair of irreducible unitary representations from the discrete series for $SL (2, {\bb R})$ and a version of Schur's lemma. \bigskip In universal Teichm\"uller space there resides the separable complex submanifold $T(H_\infty)$ -- the Teichm\"uller space of the universal hyperbolic lamination -- that is exactly the closure of the union of all the classical Teichm\"uller spaces of closed Riemann surfaces in $T(1)$ (see [20]). Genus-independent constructions like the universal period mapping proceed naturally to live on this completed version of the classical Teichm\"uller spaces. We show that $T(H_\infty)$ carries a natural convergent Weil-Petersson pairing. \bigskip We make no great claim to originality in this work. Our purpose is to survey from various different aspects the elegant role of $H^{1/2}$ in universal Teichm\"uller theory, the main goal being to understand the period mapping in the universal context. The Hilbert space ${\cal H}$, its complexification, its symplectic form and its polarizations etc. appear so naturally in what follows that it may not be merely facetious to say that the connection of ${\cal H}$ with Teichm\"uller theory and quasiconformal mappings are not only ``natural'' but almost ``supernatural''. \bigskip \nn {\bf Acknowledgements:} It is our pleasant duty to acknowledge gratefully several stimulating conversations with Graeme Segal, Michel Zinsmeister, M.S. Narasimhan, Alain Connes, Ofer Gabber, Stephen Semmes and Tom Wolff. We heartily thank Michel Zinsmeister for supplying us with his notes on $H^{1/2}$, and for generously permitting us to utilise them in this publication. Also, Graeme Segal suggested the use of Schur's Lemma in order to show the essentially unique nature of the symplectic structure. The ideas on the generalised Jacobi variety (Section 5) arose in conversations with M.S. Narasimhan. \medskip One of us (S.N.) would like to thank very much the IHES (Bures-sur-Yvette) - where most of these results were obtained - as well as the ICTP (Trieste) and the CUNY Graduate Center (New York), for their excellent hospitality during the Autumn/Winter of 1992. He would also like to thank the many intersted participants of the Mathematical Society of Japan International Research Institute on the ``Topology of the moduli space of curves'' (RIMS, Kyoto 1993), for their discussions and useful feedback. \bigskip \nn {\bf \S - 2 The Hilbert space $H^{1/2} $ on the circle and the line.} Let $\Delta$ denote the open unit disc and $U$ the upper half-plane in the plane (${\bb C}$), and $S^1 = \partial \Delta$ be the unit circle. \bigskip Intuitively speaking, the real Hilbert space under concern: $${\cal H} \equiv H^{1/2} \left({ S^1, {\bb R}}\right) /{\bb R} \eqno (2)$$ \nn is the subspace of $L^2 \left({ S^1}\right)$ comprising real functions of mean-value zero on $S^1$ which have a half-order derivative also in $L^2 \left({ S^1}\right)$. Harmonic analysis will tell us that these functions are actually defined off some set of capacity zero (i.e., "quasi-everywhere") on the circle, and that they also appear as the boundary values of real harmonic functions of finite Dirichlet energy in $\Delta$. Our first way (of several) to make this precise is to {\it identify ${\cal H}$ with the sequence space} $$\ell_2^{1/2} = \{ {\mathop{\rm complex \ sequences }\nolimits} \ \ u \equiv (u_1, u_2, u_3, \cdots ): \{ \sqrt{n } \ u_n \} \ {\mathop{\rm is \ square \ summable \ }\nolimits} \}. \eqno (3)$$ \bigskip The identification between (2) and (3) is by showing (see Proof of Theorem 2.1) that the Fourier series $$f \left({ e^{i \theta } }\right) = \sum_{ n = - \infty }^{ \infty } u_n e^{i n \theta} ; {}~~~u_{- n} = \overline u_n, \eqno (4)$$ \nn converges quasi-everywhere and defines a real function of the required type. The norm on ${\cal H}$ and on $\ell_2^{1/2}$ is, of course, the $\ell_2$ norm of $\{ \sqrt{n } \ u_n \}$, i.e., $$ \left\Vert{ f }\right\Vert_{{\cal H}}^{2} = \left\Vert{ u }\right\Vert_{\ell_2^{1/2}}^{2} = 2 \sum_{ n = 1}^{ \infty } n \left\vert{ u_n }\right\vert_{}^{2} .\eqno (5)$$ \bigskip Naturally $\ell_2^{1/2}$ and ${\cal H}$ are isometrically isomorphic separable Hilbert spaces. Note that ${\cal H}$ is a subspace of $L^2 \left({ S^1}\right)$ because $\{ \sqrt{n } \ u_n \}$ in $\ell_2$ implies $\{ \ u_n \} $ itself is in $\ell_2$. \bigskip At the very outset let us note the fundamental fact that the space ${\cal H}$ is evidently closed under {\it Hilbert transform } (``conjugation'' of Fourier series): $$ (Jf) (e^{i \theta}) = - \sum_{ n = - \infty }^{ \infty } i \mathop{\rm sgn}\nolimits (n) u_n e^{i n \theta} .\eqno (6)$$ \bigskip In fact, $J: {\cal H} \rightarrow {\cal H}$ is an isometric isomorphism whose square is the negative identity, and thus $J$ defines a {\it canonical complex structure for ${\cal H}$}. \bigskip \nn {\bf Remark:} In the papers [8],[12] [13] [15], we had made use of the fact that the Hilbert transform defines the almost-complex structure operator for the tangent space of the coadjoint orbit manifolds ($M$ and $N$), as well as for the universal Teichm\"uller space $T(1)$. \bigskip Whenever convenient we will pass to a description of our Hilbert space ${\cal H}$ as functions on the real line, $\bf R$. This is done by simply using the M\"obius transformation of the circle onto the line that is the boundary action of the Riemann mapping ("Cayley transform") of $\Delta$ onto $U$. We thus get an isometrically isomorphic copy, called $H^{1/2}({\bb R})$, of our Hilbert space ${\cal H}$ on the circle defined by taking $f \in {\cal H}$ to correspond to $g \in H^{1/2} ({\bb R})$ where $g = f \circ R, R(z) = {z - i \over z + i} $ being the Riemann mapping. The Hilbert transform complex structure on ${\cal H}$ in this version is then described by the usual singular integral operator on the real line with the "Cauchy kernel" $(x - y)^{-1}$. \bigskip Fundamental for our set up is the dense subspace $V$ in ${\cal H}$ defined by equation (1) in the introduction. At the level of Fourier series, $V$ corresponds to those sequence $\{ u_n \}$ in $\ell_2^{1/2}$ which go to zero more rapidly than $n^{- k}$ for any $k > 0$. This is so because a $C^k$ function has Fourier ceofficients decaying at least as fast as $n^{-k}$. Since trigonometric polynomials are in $V$, it is obvious that $V$ is norm-dense in ${\cal H}$. On $V$ one has the basic symplectic form that we utilised crucially in [12], [13]: $$S: V \times V \rightarrow {\bb R} \eqno (7)$$ \nn given by $$S(f, g) = {1 \over 2 \pi} \int_{ S^1}^{ } f \cdot dg . \eqno (8)$$ \nn This is essentially the signed area of the $(f(e^{i\theta}), g(e^{i\theta}))$ curve in Euclidean plane. On Fourier coefficients this bilinear form becomes $$S(f, g) = 2 \mathop{\rm Im}\nolimits\left({ \sum_{n = 1 }^{\infty} n u_n \overline v_n }\right) = -i\sum_{n = - \infty }^{\infty} n u_n v_{- n } \eqno (9)$$ \nn where $\{ u_n \}$ and $\{ v_n \}$ are respectively the Fourier ceofficients of the (real-valued) functions $f$ and $g$, as in (4). Let us note that the Cauchy-Schwarz inequality applied to (9) shows that {\it this non-degenerate bilinear alternating form extends from $V$ to the full Hilbert space} ${\cal H}$. We will call this extension also $S: {\cal H} \times {\cal H} \rightarrow {\bb R}$. Cauchy-Schwarz asserts: $$ \left\vert{ S(f, g) }\right\vert \leq \left\Vert{ f }\right\Vert \cdot \left\Vert{ g }\right\Vert . \eqno (10)$$ \nn Thus $S$ is a jointly continuous - in fact analytic - map on ${\cal H} \times {\cal H}$. \bigskip The important interconnection between the inner product on ${\cal H}$, the Hilbert-transform complex structure $J$, and the form $S$ is encapsulated in the identity: $$S \left({ f, Jg }\right) = \left\langle{ f, g }\right\rangle, \ {\mathop{\rm for \ all \ }\nolimits} f, g \in {\cal H} \eqno (11)$$ \bigskip {\it We thus see that $V$ itself was naturally a pre-Hilbert space with respect to the canonical inner product arising from its symplectic form and its Hilbert-transform complex structure, and we have just established that the completion of $V$ is nothing other than the Hilbert space ${\cal H}$. Whereas $V$ carried the $C^{\infty}$ theory, beacause it was diffeomorphism invariant, the completed Hilbert space ${\cal H}$ allows us to carry through our constructions for the full Universal Teichm\"uller Space because it indeed is quasisymmetrically invariant.} \bigskip It will be important for us to {\it complexify} our spaces since we need to deal with isotropic subspaces and polarizations. Thus we set $${\bb C} \otimes V \equiv V_{\bb C} = C^\infty \ {\mathop{\rm Maps }\nolimits} \left({ S^1, {\bb C} }\right) / {\bb C} $$ $$ {\bb C} \otimes {\cal H} \equiv {\cal H}_{\bb C} = H^{1/2} \left({ S^1, {\bb C} }\right) / {\bb C} \eqno (12) $$ \nn ${\cal H}_{\bb C}$ is a complex Hilbert space isomorphic to $\ell_2^{1/2} ({\bb C})$ - the latter comprising the Fourier series $$f \left({ e^{i \theta } }\right) = \sum_{n = - \infty }^{ \infty } u_n e^{i n \theta} \ \ , \ \ u_0 = 0 \eqno (13)$$ \nn with $\{ \sqrt{ |n| } \ u_n \}$ being square summable over ${\bb Z} - \{ 0 \}$. Note that the Hermitian inner product on ${\cal H}_{\bb C} $ derived from (5) is given by $$\left\langle{ f, g}\right\rangle = \sum_{ n = - \infty}^{ \infty } |n| u_n \overline v_n. \eqno (14)$$ \nn [This explains why we introduced the factor 2 in the formula (5).] The fundamental {\it orthogonal decomposition} of ${\cal H}_{\bb C}$ is given by $${\cal H}_{\bb C} = W_+ \oplus W_- \eqno (15)$$ \nn where $$W_+ = \{ f \in {\cal H}_{\bb C}: \ {\mathop{\rm all \ negative \ index \ Fourier \ coefficients \ vanish \ }\nolimits} \}$$ \nn and $$\overline W_+ = W_- = \{ f \in {\cal H}_{\bb C}: \ {\mathop{\rm all \ positive \ index \ Fourier \ coefficients \ vanish \ }\nolimits} \}.$$ \bigskip Here we denote by bar the complex anti-linear automorphism of ${\cal H}_{\bb C}$ given by conjugation of complex scalars. \bigskip Let us extend ${\bb C}$-linearly the form $S$ and the operator $J$ to ${\cal H}_{\bb C}$ (and consequently also to $V_{\bb C}$). The complexified $S$ is still given by the right-most formula in (9). Notice that $W_+ $ and $W_-$ can be characterized as precisely the $- i$ {\it and $+ i$ eigenspaces} (respectively) of the ${\bb C}$-linear extension of $J$, the Hilbert transform. Further, each of $W_+ $ and $W_-$ is {\it isotropic} for $S$, i.e., $S(f, g) = 0$, whenever both $f$ and $g$ are from either $W_+$ or $W_-$ (see formula (9)). Moreover, $W_+$ and $W_-$ are {\it positive isotropic} subspaces in the sense that the following identities hold: $$ \left\langle{ f_+, g_+ }\right\rangle_{}^{} = i S \left({ f_+, \overline g_+ }\right), \ for \ f_+, g_+ \in W_+ \eqno (16)$$ \nn and $$ \left\langle{ f_-, g_- }\right\rangle_{}^{} = - i S \left({ f_-, \overline g_- }\right), \ for \ f_-, g_- \in W_- . \eqno (17)$$ \bigskip \nn {\bf Remark:} (16) and (17) show that we could have defined the inner product and norm on ${\cal H}_{\bb C}$ from the symplectic form $S$, by using these relations to {\it define } the inner products on $W_+$ and $W_-$, and declaring $W_+$ to be perpendicular to $W_-$. Thus, for general $f, g \in {\cal H}_{\bb C}$ one has the fundamental identity $$ \left\langle{ f, g }\right\rangle_{}^{} = i S \left({ f_+, \overline g_+ }\right) - i S \left({ f_-, \overline g_- }\right) . \eqno (18)$$ \bigskip \nn We have thus described the Hilbert space structure of ${\cal H}$ simply in terms of the canonical symplectic form it carries and the fundamental decomposition (15). [Here, and henceforth, we will let $f_{\pm}$ denote the projection of $f$ to $W_{\pm}$, etc.]. \bigskip In order to prove the first results of this paper, we have to rely on interpreting the functions in $H^{1/2}$ as boundary values (``traces'') of functions in the disc $\Delta$ that have finite Dirichlet energy, (i.e. the first derivatives are in $L^2 (\Delta)$). We start explaining this material. \bigskip Define the following ``Dirichlet space'' of harmonic functions: $${\cal D} = \{ F : \Delta \rightarrow {\bb R} : F \ {\mathop{\rm is \ harmonic \ }\nolimits}, F(0) = 0, \ \mathop{\rm and}\nolimits \ E(F) < \infty \} \eqno (19)$$ \nn where the energy $E$ of any (complex-valued) $C^1$ map on $\Delta$ is defined as the $L^2 (\Delta) $ norm of $\mathop{\rm grad}\nolimits (F)$ : $$ \left\Vert{ F }\right\Vert_{{\cal D}}^{2} = E(F) = {1\over 2 \pi } \int_{ }^{ }\! \int_{ \Delta}^{ } \left({ \left\vert{ { \partial F \over \partial x } }\right\vert_{}^{2} + \left\vert{ { \partial F \over \partial y } }\right\vert_{}^{2} }\right) dx dy \eqno (20)$$ \nn ${\cal D}$, and its complexification ${\cal D}_{\bb C}$, will be Hilbert spaces with respect to this energy norm. \bigskip We want to identify the space ${\cal D}$ as precisely the space of harmonic functions in $\Delta$ solving the Dirichlet problem for functions in ${\cal H}$. Indeed, the {\it Poisson integral representation allows us to map $P: {\cal H} \rightarrow {\cal D}$ so that $P$ is an isometric isomorphism of Hilbert spaces.} \bigskip To see this let $f (e^{i \theta}) = \displaystyle \sum_{ - \infty }^{ \infty } u_n e^{i n \theta}$ be an arbitrary member of ${\cal H}_{\bb C}$. Then the Dirichlet extension of $f$ into the disc is: $$ F(z) = \sum_{n = - \infty }^{ \infty } u_n r^{|n|} e^{i n \theta } = \left({ \sum_{n = 1 }^{ \infty } u_n z^{n} }\right) + \left({ \sum_{m = 1 }^{ \infty } u_{- m } \overline z^{m} }\right) \eqno (21)$$ \nn where $z = re^{i \theta}$. From the above series one can directly compute the $L^2 (\Delta)$ norms of $F$ and also of $\mathop{\rm grad}\nolimits (F) = (\partial F/ \partial x, \partial F/ \partial y)$. One obtains the following: $$E(F) = {1 \over 2 \pi } \int_{ }^{ }\! \int_{ \Delta}^{ } | \mathop{\rm grad}\nolimits (F) |^2 = \sum_{ - \infty }^{ \infty } | n | |u_n |^2 \equiv \left\Vert{ f }\right\Vert_{{\cal H}}^{2} < \infty \eqno (22)$$ We will require crucially the well-known formula of Douglas (see [2,pg. 36-38]) expressing the above energy of $F$ as the double integral on $S^1$ of the square of the first differences of the boundary values $f$. $$ E(F)= {1\over {16{\pi}^2}} \int_{S^1}^{}\! \int_{S^1}^{} [{(f(e^{i\theta})-f(e^{i\phi}))}/{sin({(\theta - \phi)}/2)}]^2 d\theta d\phi \eqno(23) $$ Transferring to the real line by the M\"obius transform identification of ${\cal H}$ with $H^{1/2}({\bb R})$ as explained before, the above identity becomes as simple as: $$ E(F)= {1\over {4{\pi}^2}} \int_{{\bb R}}^{}\! \int_{{\bb R}}^{} [{(f(x)-f(y))}/{(x-y)}]^2 dxdy= \left\Vert{ f }\right\Vert^{2} \eqno(24) $$ \bigskip \nn Calculating from the series (21), the $L^2$-norm of $F$ itself is: $$ {1 \over 2 \pi } \int_{ }^{ }\! \int_{\Delta }^{ } | F|^2 dx dy = \sum_{ - \infty }^{ \infty } { |u_n |^2 \over (|n| + 1)} \leq E(F) < \infty \eqno (25)$$ \nn (22) shows that indeed {\it Dirichlet extension is isometric from } ${\cal H} $ to ${\cal D}$, whereas (25) shows that the functions in ${\cal D}$ are themselves in $L^2$, so that the {\it the inclusion of ${\cal D} \hookrightarrow L^2 (\Delta)$ is continuous}. (Bounding the $L^2$ norm of $F$ by the $L^2$ norm of its derivatives is a ''Poincar\'e inequality''). \bigskip It is therefore clear that ${\cal D}$ is a subspace of the usual Sobolev space $H^1 (\Delta)$ comprising those functions in $L^2 (\Delta)$ whose first derivatness (in the sense of distributions) are also in $L^2 ( \Delta)$. The theory of function spaces implies (by the ``trace theorems'') that $H^1$ functions lose half a derivative in going to a boundary hyperplane. Thus it is known that the functions in ${\cal D}$ will indeed have boundary values in $H^{1/2}$. See [5] [7] and [21]. \bigskip Moreover, the identity (24) shows that for any $F \in {\cal D}$, the Fourier expansion of the trace on the boundary circle is a Fourier series with $\sum |n| |u_n|^2 < \infty$. But Fourier expansions with coefficients in such a weigted $\ell_2$ space, as in our situation, are known to converge {\it quasi-everywhere} (i.e. off a set of logarithmic capacity zero) on $S^1$. See Zygmund [23, Vol 2, Chap. XIII]. The identification between ${\cal D}$ and ${\cal H}$ (or ${\cal D}_{\bb C}$ and ${\cal H}_{\bb C}$) is now complete. \bigskip It will be necessary for us to identify the $W_\pm$ polarization of ${\cal H}_{\bb C}$ at the level ${\cal D}_{\bb C}$. In fact, let us decompose the harmonic function $F$ of (21) into its holomorphic and anti-holomorphic parts ; these are $F_+ $ and $F_-$, which are (respectively) the two sums bracketed separately on the right hand side of (21). Clearly $F_+$ is a holomorphic function extending $f_+$ (the $W_+$ part of $f$), and $F_-$ is anti-holomorphic extending $f_-$. We are thus led to introduce the following space of holomorphic functions whose derivatives are in $L^2 (\Delta)$: $$ \mathop{\rm Hol}\nolimits_2 (\Delta) = \{ H: \Delta \rightarrow {\bb C}: H \ {\mathop{\rm is \ holomorphic \ }\nolimits}, H(0) = 0 \ \ \mathop{\rm and}\nolimits \ \ \int_{ }^{ }\! \int_{\Delta }^{ } | H' (z) |^2 dx dy < \infty \} .\eqno (26)$$ \bigskip \nn This is a complex Hilbert space with the norm $$ \left\Vert{ H }\right\Vert_{}^{2} = {1 \over 2 \pi } \int_{ }^{ }\! \int_{ \Delta }^{ } | H' (z) |^2 dx dy . \eqno (27)$$ \bigskip \nn If $H(z) = \displaystyle \sum_{ n = 1}^{\infty } u_n z^n$, a computation in polar coordinates (as for (21), (25)) produces $$ \left\Vert{ H }\right\Vert_{}^{2} = \sum_{ n = 1}^{\infty } n \left\vert{ u_n }\right\vert_{}^{2}. \eqno (28)$$ \nn Equations (27) and (28) show that the norm-squared is the Euclidean area of the (possibly multi-sheeted) image of the map $H$. \bigskip We let $\overline {\mathop{\rm Hol}\nolimits_2} (\Delta)$ denote the Hilbert space of antiholomorphic functions conjugate to those in $\mathop{\rm Hol}\nolimits_2 (\Delta)$. The norm is defined by stipulating that the anti-linear isomorphism of $\mathop{\rm Hol}\nolimits_2$ on $\overline {\mathop{\rm Hol}\nolimits_2} $ given by conjugation should be an isometry. The Cauchy-Riemann equation for $F_+$ and $\overline F_-$ imply that $$ \left\vert{ \mathop{\rm grad}\nolimits (F) }\right\vert_{}^{2} = 2 \left\lbrace{ \left\vert{ F'_+ }\right\vert_{}^{2} - \left\vert{ F'_- }\right\vert_{}^{2} }\right\rbrace \eqno (29)$$ \nn and hence $$ \left\Vert{ F_+ }\right\Vert_{}^{2} + \left\Vert{ F_- }\right\Vert_{}^{2} = \left\Vert{ f }\right\Vert_{{\cal H}_{\bb C}}^{2}. \eqno (30)$$ \bigskip Now, the relation between ${\cal D}$ (harmonic functions in $H^1 (\Delta))$ and $\mathop{\rm Hol}\nolimits_2 (\Delta)$ is transparent, so that the holomorphic functions in $\mathop{\rm Hol}\nolimits_2$ will have non-tangential limits quasi-everywhere on $S^1$ - defining a function $W_+$. We thus collect together, for the record, the various representations of our basic Hilbert space: \bigskip \noindent {\bf THEOREM 2.1:} There are {\it canonical isometric isomorphisms} between the following complex Hilbert spaces: \medskip (1) ${\cal H}_{\bb C} = H^{1/2} \left({ S^1, {\bb C} }\right) / {\bb C} = {\bb C} \otimes H^{1/2}({\bb R}) = W_+ \oplus W_-$; \medskip (2) The sequence space $\ell_2^{1/2} ({\bb C})$ (constituting the Fourier coefficients of the above quasi-everywhere defined functions); \medskip (3) ${\cal D}_{\bb C}$, comprising normalized finite-energy harmonic functions (either on $\Delta$ or on the half-plane $U$); [the norm-squared being given by (20) or (22) or (23) or (24)]; \medskip (4) $\mathop{\rm Hol}\nolimits_2 (\Delta) \oplus \overline {\mathop{\rm Hol}\nolimits_2} (\Delta)$. \bigskip \nn Under the canonical identifications, $W_+$ maps to $\mathop{\rm Hol}\nolimits_2 (\Delta)$ and $W_-$ onto $ \overline {\mathop{\rm Hol}\nolimits_2} (\Delta)$. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \noindent {\bf Remark:} One advantage of introducing the full Sobolev space $H^1 (\Delta)$ (rather than only its harmonic subspace ${\cal D}$) is that we may use {\it Dirichlet's principle} to rewrite the norm on ${\cal H}$ as $$ \left\Vert{ f }\right\Vert_{{\cal H}}^{2} = \inf \{ E(F): F \ {\mathop{\rm ranges \ over \ all \ extensions \ to \ }\nolimits} \Delta \ \mathop{\rm of}\nolimits \ f \} \eqno (31)$$ \bigskip \noindent By Dirichlet principle, the infimum is realized by the harmonic extension $P(f) = F$ of (23). In connection with this it is worth pointing out still another formula for the norm : $$ \left\Vert{ f }\right\Vert_{{\cal H}}^{2} = \int_{ S^1 }^{ } F \cdot { \partial F \over \partial n} ds \eqno (32)$$ \nn where $F$ is the harmonic extension to $\Delta$ of $f$. This follows from the well-known Green's identity: $$ \int_{ }^{ }\! \int_{ \Delta}^{ } \left\vert{ \mathop{\rm grad}\nolimits F }\right\vert_{}^{2} = \int_{\Delta}^{ } \! \int_{ }^{ } F (\Delta F) + \int_{ S^1 }^{ } F \cdot {\partial F \over \partial n} \cdot ds \eqno (33)$$ \bigskip \noindent The first term on the right drops out since $F$ is harmonic. Hence (32) follows. The close relation of formula (32) with the symplectic pairing formula (8) should be noted. \bigskip \noindent {\bf Remark:} Since the isomorphisms of the Theorem are all isometric, and because the norm arose from the canonical symplectic structure, (formulas (16), (17), (18)), it is instructive to work out the formulas for the symplectic form $S$ on ${\cal D}_{\bb C}$ and on $\mathop{\rm Hol}\nolimits_2 (\Delta) $. This is left to the interested reader. \nn {\bf \S 3 - Quasisymmetric invariance.} Quasiconformal (q.c.) self-homeomorphisms of the disc $\Delta$ (of the upper half-plane) $U$ are known to extend continuously to the boundary. The action on the boundary circle (respectively, on the real line ${\bb R}$) is called a {\it quasisymmetric} (q.s.) homeomorphism. By [4], $\varphi: {\bb R} \rightarrow {\bb R}$ is quasisymmetric if and only if, for all $x \in {\bb R}$ and all $t > 0$, there exists some $K > 0$ such that $$ {1 \over K} \leq { \varphi (x + t) - \varphi (x) \over \varphi (x) - \varphi (x - t) } \leq K \eqno (34)$$ \bigskip \noindent On the circle this condition for $\varphi: S^1 \rightarrow S^1$ means that $|\varphi(2I)|/|\varphi(I)| \leq K$, where $I$ is any interval on $S^1$ of length less that $\pi$, $2I$ denotes the interval obtained by doubling $I$ keeping the same mid-point, and $| \bullet |$ denotes Lebesgue measure on $S^1$. See [3] [11] and [14] as general references. \bigskip Given any orientation preserving homeomorphism $\varphi: S^1 \rightarrow S^1$, we use it to pullback functions in ${\cal H}$ by precomposition: $$V_\varphi (f) = \varphi^* (f) = f \circ \varphi - {1 \over 2 \pi} \int_{S^1 }^{ } (f \circ \varphi). \eqno (35)$$ \bigskip \noindent [We subtract off the mean value in order that the resulting function also possess zero mean. Since constants pullback to themselves, the operation is well-defined on ${\cal H}$.] \bigskip We prove: \bigskip \noindent {\bf THEOREM 3.1:} $V_\varphi$ maps ${\cal H}$ to itself (i.e., the space ${\cal H} \circ \varphi$ is ${\cal H}$) if and only if $\varphi$ is quasisymmetric. The operator norm of $V_\varphi \leq {\sqrt{K + K^{-1}}}$, whenever $\varphi$ allows a K-quasiconformal extension into the disc. \bigskip \noindent {\bf COROLLARY 3.2:} The group of all quasisymmetric homeomorphism on $S^1, QS \left({ S^1 }\right)$, acts faithfully by bounded toplinear automorphisms on the Hilbert space ${\cal H}$ (and therefore also on ${\cal H}_{\bb C}$). \bigskip \noindent {\bf Proof of sufficiency} Assume $\varphi$ is q.s. on $S^1$, and let $\Phi:\Delta \rightarrow \Delta$ be any quasiconformal extension. Let $f \in {\cal H}$ and suppose $P(f) = F \in {\cal D}$ is its unique harmonic extension into $\Delta$. Clearly $ G \equiv F \circ \Phi$ has boundary values $f \circ \varphi$, the latter being (like $f$) also a continuous function on $S^1$ defined quasi-everywhere. [Here we recall that q.s. homeomorphisms carry capacity zero sets to again such sets, although measure zero sets can become positive measure.] To prove that $f \circ \varphi$ minus its mean value is in ${\cal H}$, it is enough to prove that the Poisson integral of $f \circ \varphi$ again has finite Dirichelet energy. Indeed we will show $$E ({\mathop{\rm harmonic \ extension \ of }\nolimits} \ \varphi^* (f)) \leq 2 \left({ {1 + k^2 \over 1 - k^2} }\right) E (F). \eqno (36)$$ \noindent Here $0 \leq k < 1$ is the q.c. constant for $\Phi$, i.e., $$\left\vert{ \Phi_{\overline z} }\right\vert_{}^{} \leq k \left\vert{ \Phi_{ z} }\right\vert_{}^{} \ \ {\mathop{\rm a.e. \ in }\nolimits} \ \Delta . \eqno (37)$$ \noindent The operator norm of $V_\varphi$ is thus no more that $2^{1/2 } \left({ {\displaystyle 1 + k^2 \over \displaystyle 1 - k^2} }\right)^{1/2 }$. The last expression is equal to the bound quoted in the Theorem, where, as usual, $ K = (1+k)/(1-k)$. \bigskip Towards establishing (36) we prove that the inequality holds with the left side being the energy of the map $G = F \circ \Phi$. Since $G$ is therefore also a finite energy extension of $f \circ \varphi $ to $\Delta$, Dirichlet's principle (see (31) above) implies the required inequality. (Note that since Dirichlet integral is insensitive to adding a constant to a function, the energy of $G$ is the same as the energy of $G-G(0)$.) To compute $E(G)$ we note that $$ \left({ { \partial G \over \partial x } }\right)^2 + \left({ { \partial G \over \partial y } }\right)^2 \leq 2 \left\lbrack{ \left({ { \partial F \over \partial u } }\right)^2 + \left({ { \partial F \over \partial v } }\right)^2}\right\rbrack \left\lbrack{ \left\vert{ \Phi_{ z} }\right\vert_{}^{2} + \left\vert{ \Phi_{\overline z} }\right\vert_{}^{2} }\right\rbrack. \eqno (38)$$ \noindent We have writen $\Phi(x, y) = u(x, y) + iv(x, y)$ and $F = F(u, v)$. (38) follows by straight computation using the chain rule. But notice that the Jacobian of $\Phi$ is $$\mathop{\rm Jac}\nolimits (\Phi) = \left\vert{ \Phi_{ z} }\right\vert_{}^{2} - \left\vert{ \Phi_{\overline z} }\right\vert_{}^{2} . \eqno (39)$$ \noindent By the quasiconformality (37) we therefore get from (38): $$ \left\lbrack{ G_x^2 + G_y^2 }\right\rbrack \leq 2 \left({ { 1 + k^2 \over 1 - k^2} }\right) \left\lbrack{ F_u^2 + F_v^2 }\right\rbrack \mathop{\rm Jac}\nolimits (\Phi) \eqno (40)$$ \nn Using change of variables in the Dirichlet integral we therefore derive $$E(G) \leq 2 \left({ { 1 + k^2 \over 1 - k^2} }\right) E (F) \eqno (41)$$ \nn as desired. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \noindent {\bf Remark:} Since the Dirichlet integral in two dimensions is invariant under conformal mappings, it is not too surprising that it is quasi-invariant under quasiconformal transformations. Such quasi-invariance has been noted before and is applied, for example, in [1] and [16]. \bigskip \noindent {\bf Proof of necessity:} As we mentioned before, the idea of this proof is taken from the notes of M. Zinsmeister. \bigskip Since two-dimensional Dirichlet integrals are conformally invariant, we will pass to the upper half-plane $U$ and its boundary line ${\bb R}$ to aid our presentation. As explained earlier, using the Cayley transform we transfer everything over to the half-plane; the traces on the boundary constitute the space of quasi-everywhere defined functions called $H^{1/2} ({\bb R})$. From the Douglas identity, equation (24), we recall that an {\it equivalent way of expressing the Hilbert space norm on } $H^{1/2} ({\bb R})$ is $$ \left\Vert{ g }\right\Vert_{}^{2} = {1 \over {4 {\pi^2}}} \int_{ }^{ } \!\int_{{\bb R}^2 }^{ } \left\lbrack{ { g(x) - g(y) \over x - y } }\right\rbrack_{}^{2} dx dy, {}~~g \in H^{1/2} ({\bb R}). \eqno (42)$$ \noindent Equation (42) immediately shows that $ \left\Vert{ g }\right\Vert = \left\Vert{ \widetilde g }\right\Vert$ where $\widetilde g(x) = g(ax + b)$ for any real $a (\ne 0) $ and $b$. This will be important. Assume that $\varphi: {\bb R} \rightarrow {\bb R}$ is an orientation preserving homeomorphism such that $V_{\varphi^{-1}}: H^{1/2} ({\bb R}) \rightarrow H^{1/2} ({\bb R})$ is a bounded automorphism. Let us say that the norm of this operator is $M$. Fix a bump function $f \in C_0^\infty ({\bb R})$ such that $f \equiv 1$ on $[- 1, 1] f \equiv 0$ outside $[- 2, 2]$ and $0 \leq f \leq 1$ everywhere. Choose any $c \in {\bb R}$ and any positive $t$. Denote $I_1 = [x - t, x]$ and $I_2 = [x, x + t]$. Set $g(u) = f(au + b)$, choosing $a$ and $b$ so that $g$ is identically $1$ on $I_1$ and zero on $[x + t, \infty)$. By assumption, $g \circ \varphi^{- 1}$ is in $H^{1/2} ({\bb R})$ and $ \left\Vert{ g \circ \varphi^{-1} }\right\Vert_{}^{} \leq M \left\Vert{ g }\right\Vert_{}^{} = M \left\Vert{ f }\right\Vert_{}^{}$. We have $$ \eqalignno{ M \left\Vert{ f }\right\Vert_{}^{} &\geq \int_{ }^{ } \!\int_{{\bb R}^2 }^{ } \left\lbrack{ { g \circ \varphi^{- 1}(u) - g \circ \varphi^{- 1}(v) \over u - v } }\right\rbrack_{}^{2} du dv\cr &\geq \int_{ v = \varphi (x - t) }^{ v = \varphi (x)} \int_{ u = \varphi (x + t) }^{ u = \infty} {1 \over (u - v)^2 } du dv\cr &= \log \left({ 1 + { \varphi (x) - \varphi (x - t) \over \varphi (x + t) - \varphi (x) } }\right) . &(43) \cr}$$ \nn [We utilise the elementary integration $\displaystyle \int_{\gamma}^{ \infty} \displaystyle \int_{ \alpha }^{ \beta} {1 \over (u - v)^2 } du dv = \log \left({ 1 + {\beta - \alpha \over \gamma - \beta } }\right) $, for $\alpha<\beta<\gamma$. ] We thus obtain the result that $${ \varphi (x + t ) - \varphi (x ) \over \varphi (x ) - \varphi (x - t) } \geq { 1 \over e^{M \left\Vert{ f }\right\Vert } - 1 } \eqno (44)$$ \nn for arbitrary real $x$ and positive $t$. By utilising symmetry, namely by shifting the bump to be $1$ over $I_2$ and $0$ for $u \leq x - t$, we get the opposite inequality: $$ { \varphi (x + t ) - \varphi (x ) \over \varphi (x ) - \varphi (x - t) } \leq e^{M \left\Vert{ f }\right\Vert } - 1. \eqno (45)$$ \bigskip \nn The Beurling-Ahlfors condition on $\varphi$ is verified, and we are through. Both the theorem and its corollary are proved. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \nn {\bf \S 4 - The invariant symplectic structure.} The quasisymmetric homeomorphism group, $QS \left({ S^1}\right)$, acts on ${\cal H}$ by precomposition (equation (35)) as bounded operators, {\it preserving the canonical symplectic form} $S: {\cal H} \times {\cal H} \rightarrow {\bb R}$ (introduced in (8), (9), (10)). This is the central fact which we will analyse in this section. It is the crux on which the extension of the period mapping to all of $T(1)$ hinges: \bigskip \nn {\bf PROPOSITION 4.1:} For every $\varphi \in QS \left({ S^1}\right)$, and all $f, g \in {\cal H}$, $$S \left({ \varphi^* (f), \varphi^* (g) }\right) = S (f, g). \eqno (46)$$ \nn Considering the complex linear extension of the action to ${\cal H}_{\bb C}$, one can assert that the only quasisymmetrics which preserve the subspace $W_+ = \mathop{\rm Hol}\nolimits_2 (\Delta)$ are the M\"obius transformations. Then M\"ob $\left({ S^1}\right)$ acts as unitary operators on $W_+$ (and $W_-$). \bigskip Before proving the proposition we would like to point out that this canonical symplectic form enjoys a far stronger invariance property: \bigskip \noindent {\bf LEMMA 4.2:} If $\varphi: S^1 \rightarrow S^1$ is any (say $C^1$) map of winding number (= degree) $k$, then $$S(f \circ \varphi, g \circ \varphi) = kS (f, g) \eqno (47)$$ \nn for arbitrary choice of ($C^1$) functions $f$ and $g$ on the circle. In particular, $S$ is invariant under pullback by all degree one mappings. \bigskip \noindent {\bf Proof:} The proof of (47), starting from (8), is an exercise in calculus. Lift $\varphi$ to the universal cover to obtain $\widetilde \varphi : {\bb R} \rightarrow {\bb R} $; the degree of $\varphi$ being $k (\in {\bb Z})$ implies that $\widetilde \varphi (t + 2 \pi) = \widetilde \varphi (t) + 2 k \pi$. Breaking up $[0, 2 \pi]$ into pieces on which $\widetilde \varphi$ is monotone, and applying the change of variables formula in each piece, produces the result. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \noindent {\bf Proof of Proposition 4.1:} The Lemma shows that (46) is true whenever the quasisymmetric homeomorphism $\varphi$ is at least $C^1$. By Lehto-Virtanen [11, Chapter II, Section 7.4] we know that for arbitrary q.s $\varphi$, there exist real analytic q.s. homeomorphisms $\varphi_m$ (with the same quasisymmetry constant as $\varphi$) that converge uniformly to $\varphi$. An approximation argument, as below, then proves the required result. Let us denote the $n^{\mathop{\rm th}\nolimits}$ Fourier coefficient of a function $f$ on $S^1$ by $F_n (f)$. Recall from equation (9) that $$S(f, g) = - i \sum_{ n = - \infty }^{ \infty } nF_n (f) F_{-n} (g) \eqno (48)$$ \nn for all $f, g$ in ${\cal H}_{\bb C}$. Now since $S$ is continuous it is enough to check (46) on the dense subspace $V$ of smooth functions $f$ and $g$. Therefore assume $f$ and $g$ to be smooth. Since $\varphi_m \rightarrow \varphi $ uniformy it follows that $F_n (f \circ \varphi_m) \rightarrow F_n (f \circ \varphi) $ as $ m \rightarrow \infty$ (for each fixed $n$). Applying the dominated convergence theorem to the sums (48) we immediately see that as $m \rightarrow \infty$, $$ S \left({ \varphi_m^* (f), \varphi_m^* (g) }\right) \rightarrow S \left({ \varphi^* (f), \varphi^* (g) }\right). \eqno (49)$$ \nn Lemma 4.1 says that for each $m$, $S\left({ \varphi_m^* (f), \varphi_m^* (g) }\right) = S(f, g)$. We are through. \bigskip If the action of $\varphi$ on ${\cal H}_{\bb C}$ preserves $W_+$ it is easy to see that $\varphi$ must be the boundary values of some holomorphic map $\Phi: \Delta \rightarrow \Delta$. Since $\varphi$ is a homeomorphism one can see that $\Phi$ is a holomorphic homeomorphism (as explained also in [12, Lemma of Section 1]) - hence a M\"obius transformation. Since every $\varphi$ preserves $S$, and since $S$ induces the inner product on $W_+$ and $W_-$ by (16) (17), we note that such a symplectic transformation preserving $W_+$ must necessarily act unitarily. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \noindent {\bf Remark:} The remarkable invariance property (47) leads us to ask a question that may shed light on the structure of degree $k$ maps of $S^1$ onto itself. Given a vector space $V$ equipped with a bilinear form $S$, one may fix some constant $k (\ne 0)$ and study the family of linear maps $A$ in $\mathop{\rm Hom}\nolimits (V, V) $ such that $$S\left({ A \left({ v_1}\right) , A \left({ v_2}\right) }\right) =k S\left({ v_1, v_2 }\right) \eqno (50)$$ \nn holds for all $v_1, v_2$ in $V$. Of course, the trivial multiplication (by $ \sqrt{ k} $) will be such a map, but we have in Lemma 4.1 a situation where the interesting family of linear maps obtained by degree $k$ pullbacks provide a profusion of examples - precisely when $k$ is an integer. Furthermore, in the situation at hand, we may take $V$ as the space of $C^\infty$ (real or complex) functions on the circle. Then $V$ also carries {\it algebra} structure by pointwise multiplication. The pullbacks by degree $k$ mappings clearly preserve this multiplicative structure (whereas dilatations do not). It is interesting to question whether the only linear maps that preserve the algebra structure and also satisfy the relation (50), (for integer $k$), must necessarily arise from some degree $k$ mapping of $S^1$ on itself. \bigskip Theorem 3.1 and Proposition 4.1 enable us to consider $QS \left({ S^1}\right)$ as a subgroup of the bounded symplectic operators on ${\cal H}$. Since the heart of the matter in extending the period mapping from Witten's homogeneous space $M$ (as in [12], [13]) to $T(1)$ lies in the property of preserving this symplectic form on ${\cal H}$, we prove below that $S$ is indeed the {\it unique} symplectic form that is $\mathop{\rm Diff}\nolimits \left({ S^1}\right)$ or $QS \left({ S^1}\right)$ invariant. It is all the more surprising that the form $S$ is canonically specified by requiring its invariance under simply the 3-parameter subgroup M\"ob $\left({ S^1}\right) (\hookrightarrow \mathop{\rm Diff}\nolimits \left({ S^1}\right) \hookrightarrow QS\left({ S^1}\right)$ ). \bigskip \noindent {\bf THEOREM 4.3:} Let $S \equiv S_1$ be the canonical symplectic form on ${\cal H}$. Suppose $S_2: {\cal H} \times {\cal H} \rightarrow {\bb R}$ is any other continuous bilinear form such that $S_2 ( \varphi^* (f), \varphi^* (g)) = S_2 (f, g)$, for all $f, g$ in ${\cal H}$ whenever $\varphi$ is in M\"ob $\left({ S^1}\right) $. Then $S_2$ is necessarily a real multiple of $S$. Thus every form on ${\cal H}$ that is M\"ob $\left({ S^1}\right) \equiv PSL (2, {\bb R})$ invariant is necessarily non-degenerate (if not identically zero) and remains invariant under the action of the whole of $QS \left({ S^1}\right)$. (Also, it automatically satisfies the even stronger invariance property (47)). \bigskip The proof requires some representation theory. Since this paper is written with complex analysts in mind, we have presented some detail. We start with: \bigskip \noindent {\bf LEMMA 4.4:} The duality induced by canonical form $S_1$ is (the negative of) the Hilbert transform (equation (6)). Thus the map $\Sigma_1$ (induced by $S_1$) from ${\cal H}$ to ${\cal H}^*$ is an invertible isomorphism. \noindent {\bf Proof:} Given the continuous bilinear pairing $S_i: {\cal H} \times {\cal H} \rightarrow {\bb R} (i = 1, 2)$ we are considering the induced ``duality'' maps $$ \Sigma_i: {\cal H} \rightarrow {\cal H}^* \ \ \ \ (i = 1, 2) \eqno (51)$$ \nn which are bounded linear operators defined by: $$ \Sigma_i (g) = S_i (\bullet, g) \ , \ g \in {\cal H}. \eqno (52)$$ \nn since ${\cal H}$ is a Hilbert space, the dual ${\cal H}^*$ is canonically isomorphic to ${\cal H}$ via: $$\lambda(g) = \left\langle{\bullet, g }\right\rangle. \eqno (53)$$ \nn Here $\lambda(g)$ is the linear functional represented by $g \in {\cal H}$. Equation (11) says $S_1 (f, Jg) = \left\langle{ f, g }\right\rangle$ and therefore that $$\Sigma_1 (g) = \lambda (- Jg) \eqno (54)$$ \nn as required. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip The basic tool in proving the Theorem 4.3 is to consider the {\it ``interwining operator''} $$M = \Sigma_1^{-1} \circ \Sigma_2: {\cal H} \rightarrow {\cal H} \eqno (55)$$ \nn which is a bounded linear operator on ${\cal H}$ by the above Lemma. \bigskip \noindent {\bf LEMMA 4.5:} $M$ commutes with every invertible linear operator on ${\cal H}$ that preserves both the forms $S_1$ and $S_2$. \bigskip \noindent {\bf Proof:} $M$ is defined by the identity $S_1 (v, M w) = S_2 (v, w)$. If $T$ preserves boths forms then one has the string of equalities: $$S_1 (Tv, TM w) = S_1 (v, M w) = S_2 (v, w) = S_2 (Tv, T w) = S_1 (Tv, MT w) $$ \bigskip \nn Since $T$ is assumed invertible, this is the same as saying $$ S_1 (v, TM w) = S_1 (v, MT w) , \ \mathop{\rm for}\nolimits \ \mathop{\rm all}\nolimits \ v , w \in {\cal H} \eqno (56)$$ \bigskip \nn But $S_1$ is non-degenerate, namely $\Sigma_1$ was an isomorphism. Therefore (56) implies that $TM \equiv MT$, as desired. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip It is clear that to prove $S_2$ is a real multiple of $S_1$ means that the intertwining operator $M$ has to be simply multiplication by a scalar. This can now be deduced by looking at the complexified representation of M\"ob $\left({ S^1}\right)$ on ${\cal H}_{\bb C}$, which is unitary, and applying Schur's Lemma. \bigskip \noindent {\bf LEMMA 4.6:} The unitary representation of $SL(2, {\bb R})$ on ${\cal H}_{\bb C}$ decomposes into precisely two irreducible pieces - namely on $W_+ $ and $W_-$. In fact these two representations correspond to the two lowest (conjugate) members in the discrete series for $SL (2, {\bb R})$. \bigskip \noindent {\bf Proof:} We refer to [10] or [19] for the list of {\it irreducible unitary representations} of $SL(2, {\bb R})$ that constitute what is called its ``discrete series''. Each of these representations is indexed by an integer $m = \pm 2, \pm 3, \pm 4, \cdots$. For $m \geq 2$ one can write this representation on the $L^2$ space of holomorphic functions in $\Delta$ with the following weighted Poincar\'e measure: $$d \nu_m = (1 - |z|^2)^m {dx dy \over (1 - |z|^2)^2 } \ , \ |z| < 1. \eqno (57)$$ On the Hilbert space $L_{\mathop{\rm Hol}\nolimits}^2 (\Delta, d \nu_m)$ the discrete series representation of $SL (2, {\bb R})$ corresponding to this chosen value of $m$ is given by $\pi_m:SL(2,{\bb R}) \rightarrow \mathop{\rm Aut}\nolimits \left({ L_{\mathop{\rm Hol}\nolimits}^2 (\Delta, d \nu_m)}\right)$, where $$\pi_m (\gamma) (f(z)) = f \left({ {az + b \over cz + d} }\right) (cz + d)^{- m}. \eqno (58)$$ \nn Here, of course, $\gamma \in SL(2, {\bb R})$ corresponds to the $(PSU(1, 1))$ M\"obius transformation $ {\displaystyle az + b \over \displaystyle cz + d} $ on the disc obained as usual by conjugating the $SL(2, {\bb R})$ matrix by the M\"obius isomorphism (Cayley transform) of the upper half-plane onto the disc. For $m \leq - 2$ the anti-holomorphic functions conjugate to those in the above Hilbert spaces need only be used. We claim that the representation given by the operators $V_\varphi$ on $W_+$ (equation (35)), $\varphi \in $ M\"ob $\left({ S^1}\right)$, can be indentified with the $m = 2$ case of the above discrete series of representations of $SL(2, {\bb R})$. Note, M\"ob $\left({ S^1}\right) \equiv PSU(1, 1) \cong SL(2, {\bb R})/(\pm I)$. Recall from Theorem 2.1 that $W_+$ is identifiable as $\mathop{\rm Hol}\nolimits_2 (\Delta)$. The action of $\varphi$ is given on $\mathop{\rm Hol}\nolimits_2$ by : $$V_\varphi (F) = F \circ \varphi - F \circ \varphi (0), ~~F \in \mathop{\rm Hol}\nolimits_2 (\Delta). \eqno (59)$$ \nn But $\mathop{\rm Hol}\nolimits_2$ consists of normalized $(F(0) = 0) $ holomorphic functions in $\Delta$ whose {\it derivative is in} $L^2 (\Delta$, Euclidean measure). From (59), by the chain rule, $$ {d \over dz } V_\varphi (F) = \left({ { dF \over dz } \circ \varphi }\right) \varphi' \eqno (60)$$ \bigskip So we can rewrite the representation on the derivatives of the functions in $\mathop{\rm Hol}\nolimits_2$ by the formula (60) - which coincides with formula (58) for $m = 2$. Indeed $d \nu_2$ is, by (57), simply the Euclidean (Lebesgue) measure on the disc and thus $L_{\mathop{\rm Hol}\nolimits}^2 (\Delta, d \nu_2) \cong \mathop{\rm Hol}\nolimits_2 (\Delta)$. (This last isomorphism being given by sending $F \in \mathop{\rm Hol}\nolimits_2 (\Delta)$ to its derivative.) Our claim is proved. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip It is clear that the representation on the conjugate space will correspond to the $m = - 2$ (highest weight vector of weight $ - 2$) case of the discrete series. In particular, the representations we obtain of M\"ob $\left({ S^1}\right)$ by unitary operators of $W_+ $ and $W_-$ are both {\it irreducible. } \bigskip \noindent {\bf Proof of Theorem 4.3:} By Lemma 4.5, (the ${\bb C}$-linear extension of ) the intertwining operator $M$ commutes with every one of the unitary operators $V_\varphi : {\cal H}_{\bb C} \rightarrow {\cal H}_{\bb C}$ as $\varphi$ varies over M\"ob $\left({ S^1}\right)$. Since $W_+$ and $W_-$ are the only two invariant subspaces for all the $V_\varphi$, as proved above, it follows that $M$ must map $W_+$ either to $W_+$ or to $W_-$. Let us first assume the former case. Then $M$ commutes with all the unitary operators $V_\varphi$ on $W_+$, which we know to be an irreducible representation. Schur's Lemma says that a unitary representation will be irreducible if and only if the only operators that commute with all the operators in the representation are simply the scalars (see [19, page 11]). Since $M$ was a real operator to start with, the scalar must be real. The alternative assumption that $M$ maps $W_+$ to $W_-$ is untenable. In fact, if that were so we could replace $M$ by $M$ followed with complex conjugation. This new $M$ will map $W_+$ to itself and will again commute with all the $V_\varphi$, hence it must be a scalar. Since the original $M$ arose from a real operator this scalar can be seen to be real. But scalar multiplication preserves $W_+ $ - hence the intertwining operator must map $W_+$ (and $W_-$) to itself.\vrule height 0.7em depth 0.2em width 0.5 em \bigskip Our proof is complete. The absolute naturality of the symplectic form thus established will be utilised in understanding the $H^{1/2}$ space as a {\it Hilbertian} space, -- namely a space possessing a fixed symplectic structure but a large family of compatible complex structures. See the following sections. \bigskip \nn {\bf \S 5- The $H^{1/2}$ space as first cohomology:} The Hilbert space $H^{1/2}$, that is the hero of our tale, can be interpreted as the first cohomology space with real coefficients of the "universal Riemann surface" -- namely the unit disc -- in a Hodge-theoretic sense. That will be fundamental for us in explaining the properties of the period mapping on the universal Teichm\"uller space. In fact, in the classical theory of the period mapping, the vector space $H^1(X,{\bb R})$ plays a basic role, $X$ being a closed orientable topological surface of genus $g$ to start with. This real vector space comes equipped with a canonical symplectic structure given by the cup-product pairing, $S$. Now, whenever $X$ has a complex manifold structure, this real space $H^1(X,{\bb R})$ of dimension $2g$ gets endowed with {\it a complex structure $J$ that is compatible with the cup-pairing $S$}. This happens as follows: When $X$ is a Riemann surface, the cohomology space above is precisely the vector space of real harmonic 1-forms on $X$, by the Hodge theorem. Then the {\it complex structure $J$ is the Hodge star operator on the harmonic 1-forms}. The compatibility with the cup form is encoded in the relationships (61) and (62): $$ S(J\alpha, J\beta)= S(\alpha, \beta), {}~~~{\rm for~~ all}~~ \alpha , \beta \in H^1(X,{\bb R}) \eqno(61) $$ \noindent and that, intertwining $S$ and $J$ exactly as in equation (11), $$ S(\alpha, J\beta) = inner ~~ product(\alpha, \beta) \eqno(62) $$ should define a positive definite inner product on $H^1(X,{\bb R})$. [In fact, as we will further describe in Section 7, the Siegel disc of period matrices for genus $g$ is precisely the space of all the $S$-compatible complex structures $J$.] Consequently, the period mapping can be thought of as the variation of the Hodge-star complex structure on the topologically determined symplectic vector space $H^1(X,{\bb R})$. See Sections 7 and 8 below. \bigskip \nn {\bf Remark:} Whenever $X$ has a complex structure, one gets an isomorphism between the real vector space $H^1(X,{\bb R})$ and the $g$ dimensional complex vector space $H^1(X,\cal O)$, where $\cal O$ denotes the sheaf of germs of holomorphic functions. That is so because ${\bb R}$ can be considered as a subsheaf of $\cal O$ and hence there is an induced map on cohomology. It is interesting to check that this natural map is an isomorphism, and that the complex structure so induced on $H^1(X,{\bb R})$ is the same as that given above by the Hodge star. \bigskip For our purposes it therefore becomes relevant to consider, for an {\it arbitrary} Riemann Surface $X$, {\it the Hodge-theoretic first cohomology vector space as the space of $L^2$ (square-integrable) real harmonic 1-forms on $X$}. This real Hilbert space will be denoted ${\cal H}(X)$. Once again, in complete generality, this Hilbert space has a non-degenerate symplectic form $S$ given by the cup (= wedge) product: $$ S(\phi_1, \phi_2) = \int _{}^{}\! \int_{X}^{} \phi_1 \wedge \phi_2 \eqno(63) $$ \nn and the Hodge star is the complex structure $J$ of ${\cal H}(X)$ which is again compatible with $S$ as per (61) and (62). In fact, one verifies that the $L^2$ inner product on ${\cal H}(X)$ is given by the triality relationship (62) -- which is the same as (11). Since in the universal Teichm\"uller theory we deal with the "universal Riemann surface" -- namely the unit disc $\Delta$ -- (being the universal cover of all Riemann surfaces), we require the \bigskip \noindent {\bf PROPOSITION 5.1:} For the disc $\Delta$, the Hilbert space ${\cal H}(\Delta)$ is isometrically isomorphic to the real Hilbert space ${\cal H}$ of Section 2. Under the canonical identification the cup-wedge pairing is the canonical symplectic form $S$ and the Hodge star becomes the Hilbert-transform on ${\cal H}$. \medskip \noindent {\bf Proof:} For every $\phi \in {\cal H}(\Delta)$ there exists a unique real harmonic function $F$ on the disc with $F(0)= 0$ and $dF = \phi$. Clearly then, ${\cal H}(\Delta)$ is isometrically isomorphic to the Dirichlet space ${\cal D}$ of normalized real harmonic functions having finite energy. But in Section 2 we saw that this space is isometrically isomorphic to ${\cal H}$ by passing to the boundary values of $F$ on $S^1$. If $\phi_1 = dF_1$ and $\phi_2 = dF_2$ , then integrating $\phi_1 \wedge \phi_2$ on the disc amounts to, by Stokes' theorem, $$ \int_{}^{}\!\int_{\Delta}^{} dF_1 \wedge dF_2 = \int_{S^1}^{} F_1 dF_2 = S(F_1, F_2) $$ as desired. Finally, let $\phi = udx + vdy$ be a $L^2$ harmonic 1-form with $\phi = dF$. Suppose $G$ is the harmonic conjugate of $F$ with $G(0)=0$. Then $dF + idG$ is a holomorphic 1-form on $\Delta$ with real part $\phi$. It follows that the Hodge star maps $\phi$ to $dG$; hence, under the above canonical identification of ${\cal H}(\Delta)$ with ${\cal H}$, we see that the star operator becomes the Hilbert transform. \vrule height 0.7em depth 0.2em width 0.5 em \noindent {\bf Remark on the generalised Jacobi variety:} The complex torus that is the Jacobi variety of a closed genus $g$ Riemann surface $X$ can be described as the complex vector space $(H^1(X,{\bb R}), {\rm Hodge~~ star})$ modulo the lattice $H_1(X,{\bb Z})$. Indeed, the integral homology group acts on $H^1(X,{\bb R})$ as linear functionals by integration of 1-forms on cycles, and since $H^1(X,{\bb R})$ is a Hilbert space, we may canonically identify the dual space with itself. Thus $H_1(X,{\bb Z})$ appears embedded inside $H^1(X,{\bb R})$, and the quotient is the complex torus that is the classical Jacobi variety of the Riemann surface. But for the same reasons as above, for an {\it arbitrary} Riemann surface $X$, $H_1(X,{\bb Z})$ does sit inside the Hodge-theoretic first cohomology Hilbert space ${\cal H}(X)$. And this last space carries, as we know, the Hodge star complex structure. Thus it makes sense to try to define the generalised Jacobi variety of $X$ as the quotient of this complex Hilbert space by the "discrete subgroup" $H^1(X,{\bb Z})$. For certain classes of open Riemann surfaces that quotient is a reasonable object, and we will report on these matters in future articles. For the unit disc itself then, the generalised Jacobian {\it is} the Hilbert space $H^{1/2}$ = ${\cal H}$ equipped with the Hilbert transform complex structure. \bigskip \nn {\bf \S 6- Quantum calculus and $H^{1/2}$:} \bigskip \nn A.Connes has proposed (see, for example, [6] and Connes' book "Geometrie Non-Commutatif") a "quantum calculus" that associates to a function $f$ an operator that should be considered its quantum derivative -- so that the operator theoretic properties of this $d^{Q}(f)$ capture the smoothness properties of the function. One advantage is that this operator can undergo all the operations of the functional calculus. The fundamental definition in one real dimension is $$ d^{Q}(f)=[J,M_{f}] \eqno(64) $$ \nn where $J$ is the Hilbert transform in one dimension explained in Section 2, and $M_{f}$ stands for (the generally unbounded) operator given by multiplication by $f$. One can think of the quantum derivative as operating (possibly unboundedly) on the Hilbert space $L^{2}(S^1)$ or on other appropriate function spaces. \medskip \nn {\bf Note:} We will also allow quantum derivatives to be taken with respect to other Hilbert-transform like operators; in particular, the Hilbert transform can be replaced by some conjugate of itself by a suitable automorphism of the Hilbert space under concern. In that case we will make explicit the $J$ by writing $d^{Q}_{J}(f)$ for the quantum derivative. See Section 8 for applications. \bigskip As sample results relating the properties of the quantum derivative with the nature of $f$, we quote: $d^{Q}(f)$ is a bounded operator on $L^{2}(S^1)$ {\it if and only if} the function $f$ is of bounded mean oscillation. In fact, the operator norm of the quantum derivative is equivalent to the BMO norm of $f$. Again, $d^{Q}(f)$ is a compact operator on $L^{2}(S^1)$ {\it if and only if} $f$ is in $L^{\infty}(S^1)$ and has vanishing mean oscillation. Also, if $f$ is H\"older, (namely in some H\"older class), then the quantum derivative acts as a compact operator on H\"older. See [6], [6b]. (Note that the union of all the H\"older classes is both quasisymmetrically invariant and Hilbert-transform stable. Moreover, functions that are of bounded variation and H\"older form a quasisymmetrically invariant subspace of $H^{1/2}$.) Similarly, the requirement that $f$ is a member of certain Besov spaces can be encoded in properties of the quantum derivative. \bigskip Our Hilbert space $H^{1/2}({\bb R})$ has a very simple interpretation in these terms: \bigskip \nn {\bf PROPOSITION 6.1:} $f \in H^{1/2}({\bb R})$ if and only if the operator $d^{Q}(f)$ is Hilbert-Schmidt on $L^{2}({\bb R})$ [or on $H^{1/2}({\bb R})$]. The Hilbert-Schmidt norm of the quantum derivative {\it coincides} with the $H^{1/2}$ norm of $f$. \medskip \nn {\bf Proof:} Recall that the Hilbert transform on the real line is given as a singular integral operator with integration kernel $(x-y)^{-1}$. A formal calculation therefore shows that $$ (d^{Q}(f))(g)(x) = \int_{{\bb R}}^{} {{f(x)-f(y)}\over {x-y}} g(y)dy \eqno(65) $$ But the above is an integral operator with kernel $K(x,y)= (f(x)-f(y))/(x-y)$, and such an operator is Hilbert-Schmidt if and only if the kernel is square-integrable over ${\bb R}^{2}$. Utilising now the Douglas identity -- equations (24) or (42) -- we are through. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip Since the Hilbert transform, $J$, is the standard complex structure on the $H^{1/2}$ Hilbert space, and since this last space was shown to allow an action by the quasisymmetric group, $QS({\bb R})$, some further considerations become relevant. Introduce the operator $L$ on 1-forms on the line to functions on the line by: $$ (L\varphi)(x) = \int_{{\bb R}}^{} [log \vert x-y \vert]\varphi (y) dy \eqno(66) $$ One may think of the Hilbert transform $J$ as operating on either the space of functions or on the space of 1-forms (by integrating against the kernel $dx/(x-y)$). Let $d$ as usual denote total derivative (from functions to 1-forms). Then notice that $L$ above is an operator that is essentially a smoothing inverse of the exterior derivative. In fact, $L$ and $d$ are connected to $J$ via the relationships: $$ d \circ L = J_{1-forms}; ~~~ L \circ d = J_{functions} \eqno(67) $$ \bigskip \nn {\bf The Quasisymmetrically deformed operators:} Given any q.s. homeomorphism $h \in QS({\bb R})$ we think of it as producing a q.s. change of structure on the line, and hence we define the corresponding transformed operators, $L^{h}$ and $J^{h}$ by $L^{h}= h \circ L \circ h^{-1}$ and $J^{h}= h \circ J \circ h^{-1}$. ($J$ is being considered on functions in ${\cal H} = H^{1/2}({\bb R})$, as usual.) The q.s homeomorphism (assumed to be say $C^1$ for the deformation on $L$), operates standardly on functions and forms by pullback. Therefore, {\it $J^{h}$ simply stands for the Hilbert transform conjugated by the symplectomorphism $T_{h}$ of ${\cal H}$ achieved by pre-composing by the q.s. homeomorphism $h$.} $J^{h}$ is thus a new complex structure on ${\cal H}$. \smallskip \nn {\bf Note:} The complex structures on ${\cal H}$ of type $J^{h}$ are exactly those that constitute the image of $T(1)$ by the universal period mapping. (See Section 8.) The target manifold, the universal Siegel space, can be thought of as a space of $S$-compatible complex structures on ${\cal H}$. Let us write the perturbation achieved by $h$ on these operators as the "quantum brackets": $$ \{h,L\}=L^{h} - L ;~~~ \{h,J\}=J^{h} - J. \eqno(68) $$ Now, for instance, the operator $d \circ \{h,J\}$ is represented by the kernel $(h \times h)^{*}m - m$ where $m = dx dy/{(x-y)^{2}}$. For $h$ suitably smooth this is simply $d_{y}d_{x}(log [({h(x)-h(y)})/({x-y})])$. It is well known that $(h \times h)^{*}m = m$ when $h$ is a M\"obius transformation. Interestingly, therefore, on the diagonal ($x=y$), this kernel becomes ($1/6$ times) the Schwarzian derivative of $h$ (as a quadratic differential on the line). For the other operators in the table below the kernel computations are even easier. Set $K(x,y) = log [({h(x)-h(y)})/({x-y})]$ for convenience. We have the following table of quantum calculus formulas: $$ \matrix {{\bf Operator} & {\bf Kernel} & {\bf On~~ diagonal} & {\bf Cocycle~~on}~~QS({\bb R}) \cr \{ h,L\} & K(x,y) & log (h^\prime) & function-valued \cr d \circ \{h,L\} & d_{x}K(x,y) & {h'' \over {h'}}dx & 1-form-valued \cr d \circ \{h,J\} & d_{y}d_{x}K(x,y) & {1 \over 6}Schwarzian(h)dx^{2} & quadratic-form-valued} \eqno(69) $$ The point here is that these operators make sense when $h$ is merely quasisymmetric. If $h$ happens to be appropriately smooth, we can restrict the kernels to the diagonal to obtain the respective nonlinear classical derivatives (affine Schwarzian, Schwarzian, etc.) as listed in the table above. \bigskip \nn {\bf Remark:} It is worth pointing out that the central extensions associated to the three cocycles in the horizontal lines of the table above respectively correspond to the subgroups: {\it (i)Translations, (ii)Affine transformations, {\rm and} (iii)Projective (M\"obius) transformations}. \bigskip \nn {\bf \S 7- The universal period mapping on $T(1)$:} \bigskip Having now all the necessary background results behind us, we are finally set to move into the theory of the universal period (or polarisations) map itself. \bigskip The Frechet Lie group, $Diff(S^1)$ operating by pullback (= pre-composition) on smooth functions, had a faithful representation by bounded symplectic operators on the symplectic vector space $V$ of equation (1). That induced the natural map $\Pi$ of the homogeneous space $M=Diff(S^1)/M\ddot{o}b(S^1)$ into Segal's version of the Siegel space of period matrices. In [12] [13] we had shown that this map: $$ \Pi: Diff(S^1)/M\ddot{o}b(S^1) \rightarrow Sp_{0}(V)/U \eqno(70) $$ \nn is {\it equivariant, holomorphic, K\"ahler isometric immersion}, and moreover that it qualifies as a {\it generalised period matrix map} (remembering ([15]) that the domain is a complex submanifold of the universal space of Riemann surfaces $T(1)$). {}From the results of Sections 2, 3, and 4, we know that the full quasisymmetric group, $QS(S^1)$ operates as bounded symplectic operators on the Hilbert space ${\cal H}$ that is the completion of the pre-Hilbert space $V$. The same proof as offered in the articles quoted demonstrates that the subgroup of $QS$ acting unitarily is the M\"obius subgroup. Clearly then we have obtained the {\it extension of } $\Pi$ (also called $\Pi$ to save on nomenclature) {\it to the entire universal Teichm\"uller space}: $$ \Pi: T(1) \rightarrow Sp({\cal H})/U \eqno(71) $$ \bigskip Let us first exhibit the nature of the complex Banach manifold that is the target space of the period map (71). This space, which is the universal Siegel period matrix space, denoted $\cal S_{\infty}$, has several interesting descriptions: \medskip \nn {\bf (a):} $\cal S_{\infty}$= {the space of positive polarizations of the symplectic Hilbert space ${\cal H}$ }. Recall ([12], [13], [18]) that a positive polarization signifies the choice of a closed complex subspace $W$ in ${\cal H}_{{\bb C}}$ such that (i) ${\cal H}_{{\bb C}} = W \oplus \overline W$; (ii) $W$ is $S$-isotropic, namely $S$ vanishes on arbitrary pairs from $W$; and (iii) $iS(w, \overline w)$ defines the square of a norm on $w \in W$. \medskip \nn {\bf (b):} $\cal S_{\infty}$= {the space of $S$-compatible complex structure operators on ${\cal H}$ }. That consists of bounded invertible operators $J$ of ${\cal H}$ onto itself whose square is the negative identity and $J$ is compatible with $S$ in the sense that requirements (61) and (62) are valid. Alternatively, these are the complex structure operators $J$ on ${\cal H}$ such that $H(f,g) = S(f,Jg) + iS(f,g)$ is a positive definite Hermitian form having $S$ as its imaginary part. \medskip \nn {\bf (c):} $\cal S_{\infty}$= {the space of bounded operators $Z$ from $W_{+}$ to $W_{-}$ that satisfy the condition of $S$-symmetry: $S(Z\alpha,\beta)=S(Z\beta,\alpha)$ and are in the unit disc in the sense that $(I-Z \overline Z)$ is positive definite}. The matrix for $Z$ is the "period matrix" of the classical theory. \medskip \nn {\bf (d):} $\cal S_{\infty}$= the homogeneous space $Sp({\cal H})/U$; here $Sp({\cal H})$ denotes all bounded symplectic automorphisms of ${\cal H}$, and $U$ is the unitary subgroup defined as those symplectomorphisms that keep the subspace $W_{+}$ (setwise) invariant. \bigskip Introduce the {\it Grassmannian} $Gr(W_{+}, {\cal H}_{{\bb C}})$ of subspaces of type $W_{+}$ in ${\cal H}_{{\bb C}}$, which is obviously a complex Banach manifold modelled on the Banach space of all bounded operators from $W_{+}$ to $W_{-}$. Clearly, $\cal S_{\infty}$ is embedded in $Gr$ as a complex submanifold. The connections between the above descriptions of the Siegel universal space are transparent: \medskip \nn (a:b) the positive polarizing subspace $W$ is the $-i$-eigenspace of the complex structure operator $J$ (extended to ${\cal H}_{{\bb C}}$ by complex linearity). \medskip \nn (a:c) the positive polarizing subspace $W$ is the graph of the operator $Z$. \medskip \nn (a:d) $Sp({\cal H})$ acts transitively on the set of positive polarizing subspaces. $W_{+}$ is a polarizing subspace, and the isotropy (stabilizer) subgroup thereat is exactly $U$. \bigskip \nn {\bf ${\cal H}$ as a Hilbertian space:} Note that the method (b) above describes the universal Siegel space as a space of Hilbert space structures on the fixed underlying symplectic vector space ${\cal H}$. By the result of Section 4 we know that the symplectic structure on ${\cal H}$ is completely canonical, whereas each choice of $J$ above gives a Hilbert space inner product on the space by intertwining $S$ and $J$ by the fundamental relationship (11) (or (62)). Thus ${\cal H}$ is a {\it "Hilbertian space"}, which signifies a complete topological vector space with a canonical symplectic structure but lots of compatible inner products turning it into a Hilbert space in many ways. \bigskip \nn We come to one of our {\it Main Theorems:} \nn {\bf THEOREM 7.1:} The universal period mapping $\Pi$ is an injective, equivariant, holomorphic immersion between complex Banach manifolds. \nn {\bf Proof:} From our earlier papers [12] [13] we know these facts for $\Pi$ restricted to $M$. The proof of equivariance is the same (and simple). The map is an injection because if we know the subspace $W_{+}$ pulled back by $w_{\mu}$, then we can recover the q.s. homeomorphism $w_{\mu}$. In fact, inside the given subspace look at those functions which map $S^1$ homeomorphically on itself. One sees easily that these must be precisely the M\"obius transformations of the circle pre-composed by $w_{\mu}$. The injectivity (global Torelli theorem) follows. Let us write down the matrix for the symplectomorphism $T$ on ${\cal H}_{{\bb C}}$ obtained by pre-composition by $w_{\mu}$. We will write in the standard orthonormal basis $e^{ik\theta}/k^{1/2}$, $k=1,2,3..$ for $W_{+}$, and the complex conjugates as o.n. basis for $W_{-}$. In ${\cal H}_{\bb C} = W_+ \oplus W_- $ block form, $T$ is given by maps: $A: W_{+} \rightarrow W_{+}$, $B:W_{-} \rightarrow W_{+}$. The conjugates of $A$ and $B$ map $W_{-}$ to $W_{-}$ and $W_{+}$ to $W_{-}$, respectively. The matrix entries for $A=((a_{pq}))$ and $B=((b_{rs}))$ turn out to be: $$ a_{pq}={(2\pi)^{-1}}{p^{1/2}}{q^{-1/2}}\int_{0}^{2\pi} {(w_{\mu}(e^{i\theta}))^{q}}{e^{-ip\theta}}d\theta, ~~p,q \ge 1 $$ $$ b_{rs}={(2\pi)^{-1}}{r^{1/2}}{s^{-1/2}}\int_{0}^{2\pi} {(w_{\mu}(e^{i\theta}))^{-s}}{e^{-ir\theta}}d\theta, ~~r,s \ge 1 $$ Recalling the standard action of symplectomorphisms on the Siegel disc (model {\bf (c)} above), we see that the corresponding operator [=period matrix] $Z$ appearing from the Teichm\"uller point [$\mu$] is given by: $$ \Pi[\mu] = {\overline B}{A^{-1}} $$ The usual proof of finite dimensions shows that for any symplectomorphism $A$ must be invertible -- hence the above explicit formula makes sense. Since the Fourier coefficients appearing in $A$ and $B$ vary only {\it real-analytically} with $\mu$, it may be somewhat surprising that $\Pi$ is actually {\it holomorphic}. In fact, a computation of the first variation of $\Pi$ at the origin of $T(1)$ ( i.e., the derivative map) in the Beltrami direction $\nu$ shows that the following {\it Rauch variational formula} subsists: $$ {(d\Pi([\nu]))_{rs}}={{\pi}^{-1}}{(rs)^{1/2}}\int_{}^{}\int_{\Delta}^{} \nu(z){z^{r+s-2}}dxdy $$ \nn The proof of this formula is as shown for $\Pi$ on the smooth points submanifold $M$ in our earlier papers. The manifest complex linearity of the derivative, i.e., the validity of the Cauchy-Riemann equations, combined with equivariance, demonstrates that $\Pi$ is complex analytic on $T(1)$, as desired. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \nn {\bf Interpretation of $\Pi$ as period map:} Let us take a moment to recall why the map $\Pi$ qualifies as a universal version of the classical genus $g$ period maps. As we had explained in our previous papers, in the light of P.Griffiths' ideas, the classical period map may be thought of as associating to a Teichm\"uller point a positive polarizing subspace of the first cohomology $H^1(X,{\bb R})$. The point is that when $X$ has a complex structure, then the complexified first cohomology decomposes as: $$ H^1(X,{\bb C})= H^{1,0}(X) \oplus H^{0,1}(X) \eqno(72) $$ \nn The period map associates the subspace $H^{1,0}(X)$ -- which is positive polarizing with respect to the cup-product symplectic form -- to the given complex structure on $X$. Of course, $H^{1,0}(X)$ represents the holomorphic 1-forms on $X$, and that is why this is nothing but the usual period mapping. {\it But that is precisely what $\Pi$ is doing in the universal Teichm\"uller space.} Indeed, by the results of Section 5, ${\cal H}$ is the Hodge-theoretic real first cohomology of the disc, with $S$ being the cup-product. The standard complex structure on the unit disc has holomorphic 1-forms that are of the form $dF$ where $F$ is a holomorphic function on $\Delta$ with $F(0)=0$. Thus the boundary values of $F$ will have only positive index Fourier modes -- corresponding therefore to the polarizing subspace $W_{+}$. Now, an arbitrary point of $T(1)$ is described by the choice of a Beltrami differential $\mu$ on $\Delta$ perturbing the complex structure. We are now asking for the holomorphic 1-forms on $\Delta_{\mu}$. Solving the Beltrami equation on $\Delta$ provides us with the $\mu$-conformal quasiconformal self-homeomorphism $w_{\mu}$ of the disc. This $w_{\mu}$ is a holomorphic uniformising coordinate for the disc with the $\mu$ complex structure. The holomorphic 1-forms subspace, $H^{1,0}(\Delta_{\mu})$, should therefore comprise those functions on $S^1$ that are the $W_{+}$ functions {\it precomposed with the boundary values of the q.c. map $w_{\mu}$.} That is exactly the action of $\Pi$ on the Teichm\"uller class of $\mu$. This explains in some detail why $\Pi$ behaves as an infinite dimensional period mapping. \bigskip \nn {\bf Remark:} On Segal's $C^{\infty}$ version of the Siegel space -- constructed using Hilbert-Schmidt operators $Z$, there existed the universal {\it Siegel symplectic metric}, which we studied in [12] [13] and showed to be the same as the Kirillov-Kostant (= Weil-Petersson) metric on $Diff(S^1)/Mob(S^1)$. For the bigger Banach manifold $\cal S_{\infty}$ above, that pairing fails to converge on arbitrary pairs of tangent vectors because the relevant operators are not any more trace-class in general. The difficulties asociated with this matter will be addressed in Section 9 below, and in further work that is in progress. \bigskip \nn {\bf \S 8- The universal Schottky locus and quantum calculus:} \bigskip \nn Our object is to exhibit the image of $\Pi$ in $\cal S_{\infty}$. The result (equation (73)) can be recognized to be a quantum "integrability condition" for complex structures on the circle or the line. \bigskip \nn {\bf PROPOSITION 8.1:} If a positive polarizing subspace $W$ is in the ''universal Schottky locus'', namely if $W$ is in the image of $T(1)$ under the universal period mapping $\Pi$, then $W$ possesses a dense subspace which is {\it multiplication-closed} (i.e., an ``algebra" under pointwise multiplication modulo subtraction of mean-value.) In quantum calculus terminology, this means that $$ [d^{Q}_{J},J] = 0 \eqno(73) $$ where $J$ denotes the $S$-compatible complex structure of ${\cal H}$ whose $-i$-eigenspace is $W$. (Recall the various descriptions of $\cal S_{\infty}$ spelled out in the last section.) \bigskip \nn {\bf Multiplication-closed polarizing subspace:} The notion of being multiplication-closed is well-defined for the relevant subspaces in ${\cal H}_{{\bb C}}$. Let us note that the original polarizing subspace $W_{+}$ contains the dense subspace of holomorphic trigonometric polynomials (with mean zero) which constitute an algebra. Indeed, the identity map of $S^1$ is a member of $W_{+}$, call it $j$, and positive integral powers of $j$ clearly generate $W_{+}$ -- since polynomials in $j$ form a dense subspace therein. Now if $W$ is any other positive polarizing subspace, we know that it is the image of $W_{+}$ under some $T \in Sp({\cal H})$. Thus, $W$ will be multiplication-closed precisely when the image of $j$ by $T$ generates $W$, in the sense that its positive integral powers (minus the mean values) also lie in $W$ (and hence span a dense subspace of $W$). In other words, we are considering $W$ ($\in \cal S_{\infty}$ [description (a)]) to be multiplication-closed provided that the pointwise products of functions from $W$ (minus their mean values) that happen to be $H^{1/2}$ functions actually land up in the subspace $W$ again. Multiplying $f$ and $g$ modulo arbitrary additive constants demonstrates that this notion is well-defined when applied to a subspace. \bigskip \nn {\bf Quantum calculus and equation (73):} We suggest a quantum version of complex structures in one real dimension, and note that the integrable ones correspond to the universal Schottky locus under study. In the spirit of algebraic geometry one takes the real Hilbert space of functions ${\cal H}$ = $H^{1/2}({\bb R})$ as the ``coordinate ring" of the real line. Consequently, a complex structure on ${\bb R}$ will be considered to be a complex structure on this Hilbert space. Since $\cal S_{\infty}$ was a space of (symplectically-compatible) complex structures on ${\cal H}$, we are interpreting $\cal S_{\infty}$ as a space of quantum complex structures on the line (or circle). Amongst the points of the universal Siegel space, those that can be interpreted as the holomorphic function algebra for some complex structure on the circle qualify as the ``integrable'' ones. But $T(1)$ parametrises all the quasisymmetrically related circles, and for each one, the map $\Pi$ associates to that structure the holomorphic function algebra corresponding to it; see the interpretation we provided for $\Pi$ in the last section. It is clear therefore that $\Pi(T(1)$ should be the integrable complex structures. The point is that taking the standard circle as having integrable complex structure, all the other integrable complex structures arise from this one by a $QS$ change of coordinates on the underlying circle. These are the complex structures $J^{h}$ introduced in Section 6 on quantum calculus. The $-i$-eigenspace for $J^{h}$ is interpreted as the algebra of analytic functions on the quantum real line with the $h$-structure. We will see in the proof that (73) encodes just this condition. \bigskip \nn {\bf Proof of Proposition 8.1:} For a point of $T(1)$ represented by a q.s. homeomorphism $\phi$, the period map sends it to the polarizing subspace $W_{\phi} = W_{+} \circ \phi$. But $W_{+}$ was a multiplication-closed subspace, generated by just the identity map $j$ on $S^1$, to start with. Clearly then, $\Pi(\phi) = W_{\phi}$ is also multiplication-closed in the sense explained, and is generated by the image of the generator of $W_{+}$ -- namely by the q.s homeomorphism $\phi$ (as a member of ${\cal H}_{{\bb C}}$). \vrule height 0.7em depth 0.2em width 0.5 em \medskip We suspect that the converse is also true: that the $T(W_{+})$ is such an ''algebra'' subspace for a symplectomorphism $T$ in $Sp({\cal H})$ only when $T$ arises as pullback by a quasisymmetric homeomorphism of the circle. This converse assertion is reminescent of standard theorems in Banach algebras where one proves, for example, that every (conjugation-preserving) algebra automorphism of the algebra $C(X)$ (comprising continuous functions on a compact Hausdorff space $X$) arises from homeomorphisms of $X$. [Remark of Ambar Sengupta.] Owing to the technical hitch that $H^{1/2}$ functions are not in general everywhere defined on the circle, we are as yet unable to find a rigorous proof of this converse. Here is the sketch of an idea for proving the converse. Thus, suppose we are given a subspace $E$ that is multiplication-closed in the sense explained. Now, $Sp({\cal H})$ acts transitively on the set of positive polarizing subspaces. We consider a $T \in Sp({\cal H})$ that maps $W_{+}$ to $E$ preserving the algebra structure (modulo subtracting off mean values as usual). Denote by $j$ the identity function on $S^1$ and let $T(j)=w$ be its image in $E$. Since $j$ is a homeomorphism and $T$ is an invertible real symplectomorphism, one expects that $w$ is also a homeomorphism on $S^1$. (Recall the signed area interpretation of the canonical form (8).) It then follows that the $T$ is nothing other that precomposition by this $w$. That is because: $$ T(j^{m}) = T(j)^{m} - {\rm mean ~ value} = (w(e^{i\theta}))^{m} - {\rm mean ~ value} = j^{m} \circ w - {\rm mean ~ value.} $$ \nn Knowing $T$ to be so on powers of $j$ is sufficient, as polynomials in $j$ are dense in $W_{+}$. Again, since $T$ is the complexification of a real symplectomorphism, seeing the action of $T$ on $W_{+}$ tells us $T$ on all of ${\cal H}_{{\bb C}}$; namely, $T$ is everywhere precomposition by that homeomorphism $w$ of $S^1$. {\it By the necessity part of Theorem 3.1 we see that $w$ must be quasisymmetric}, and hence that the given subspace $E$ is the image under $\Pi$ of the Teichm\"uller point determined by $w$ (i.e., the coset of $w$ in $QS(S^1)/M\ddot{o}b(S^1)$). \bigskip \nn {\it Proof of equation (73)}: Let $J$ be {\it any} $S$-compatible complex structure on ${\cal H}$, namely $J$ is an arbitrary point of $\cal S_{\infty}$ (description (b) of Section 7). Let $J_{0}$ denote the Hilbert transform itself, which is the reference point in the universal Siegel space; therefore $J = T J_{0} T^{-1}$ for some symplectomorphism $T$ in $Sp({\cal H})$. The $-i$-eigenspace for $J_{0}$ is, of course, the reference polarizing subspace $W_{+}$, and the subspace $W$ corresponding to $J$ consists of the functions $(f+i(Jf))$ for all $f$ in ${\cal H}$. Now, the pointwise product of two such typical elements of $W$ gives: $$ (f+i(Jf))(g+i(Jg))= [fg - (Jf)(Jg)] + i[f(Jg) + g(Jf)] $$ \nn In order for $W$ to be multiplication closed the function on the right hand side must also be of the form $(h + i(Jh))$. Namely, for all relevant $f$ and $g$ in the real Hilbert space ${\cal H}$ we must have: $$ J[fg - (Jf)(Jg)] = [f(Jg) + g(Jf)] \eqno(74) $$ Now recall from the concepts introduced in Section 6 that one can associate to functions $f$ their quantum derivative operators $d^{Q}_{J}(f)$ which is the commutator of $J$ with the multiplication operator $M_{f}$ defined by $f$. The quantum derivative is being taken with respect to any Hilbert-transform-like operator $J$ as explained above. But now a short computation demonstrates that equation (74) is the same as saying that: $$ J \circ d^{Q}_{J}(f) = - d^{Q}_{J}(f) $$ operating by $J$ on both sides shows that this is the same as (73). That is as desired. \vrule height 0.7em depth 0.2em width 0.5 em \bigskip \nn {\bf Remark:} For the classical period mapping on the Teichm\"uller spaces $T_g$ there is a way of understanding the Schottky locus in terms of Jacobian theta functions satisfying the nonlinear K-P equations. In a subsequent paper we hope to relate the finite dimensional Schottky solution with the universal solution given above. \bigskip \nn {\bf Remark:} For the extended period-polarizations mapping $\Pi$, the Rauch variational formula that was exhibited in [12], [13], [13a], and here in the proof of Theorem 7.1, continues to hold. \bigskip \nn {\bf \S 9- The Teichm\"uller space of the universal lamination and Weil-Petersson:} The Universal Teichm\"uller Space, $T(1)$=$T(\Delta)$, is a {\it non-separable} complex Banach manifold that contains, as properly embedded complex submanifolds, all the Teichm\"uller spaces, $T_g$, of the classical compact Riemann surfaces of every genus $g$ ($\geq 2$). $T_g$ is $3g-3$ dimensional and appears (in multiple copies) within $T(\Delta)$ as the Teichm\"uller space $T(G)$ of the Fuchsian group $G$ whenever $\Delta/G$ is of genus $g$. The closure of the union of a family of these embedded $T_g$ in $T(\Delta)$ turns out to be a separable complex submanifold of $T(\Delta)$ (modelled on a separable complex Banach space). That submanifold can be identified as being itself the Teichm\"uller space of the "universal hyperbolic lamination" $H_\infty$. We will show that $T(H_{\infty})$ carries a canonical, genus-independent version of the Weil-Petersson metric, thus bringing back into play the K\"ahler structure-preserving aspect of the period mapping theory. \bigskip \noindent {\bf The universal laminated surfaces:} Let us proceed to explain the nature of the (two possible) "universal laminations" and the complex structures on these. Starting from any closed topological surface, $X$, equipped with a base point, consider the inverse (directed) system of all finite sheeted unbranched covering spaces of $X$ by other closed pointed surfaces. The covering projections are all required to be base point preserving, and isomorphic covering spaces are identified. The {\it inverse limit space} of such an inverse system is the "lamination" -- which is the focus of our interest. \medskip \nn {\it The lamination $E_{\infty}$:} Thus, if $X$ has genus one, then, of course, all coverings are also tori, and one obtains as the inverse limit of the tower a certain compact topological space -- every path component of which (the laminating leaves) -- is identifiable with the complex plane. This space $E_{\infty}$ (to be thought of as the "universal Euclidean lamination") is therefore a fiber space over the original torus $X$ with the fiber being a Cantor set. The Cantor set corresponds to all the possible backward strings in the tower with the initial element being the base point of $X$. The total space is compact since it is a closed subset of the product of all the compact objects appearing in the tower. \medskip \nn {\it The lamination $H_{\infty}$:} Starting with an arbitrary $X$ of higher genus clearly produces the {\it same} inverse limit space, denoted $H_{\infty}$, independent of the initial genus. That is because given any two surfaces of genus greater than one, there is always a common covering surface of higher genus. $H_{\infty}$ is our universal hyperbolic lamination, whose Teichm\"uller theory we will consider in this section. For the same reasons as in the case of $E_{\infty}$, this new lamination is also a compact topological space fibering over the base surface $X$ with fiber again a Cantor set. (It is easy to see that in either case the space of backward strings starting from any point in $X$ is an uncountable, compact, perfect, totally-disconnected space -- hence homeomorphic to the Cantor set.) The fibration restricted to each individual leaf (i.e., path component of the lamination) is a universal covering projection. Indeed, notice that the leaves of $H_{\infty}$ (as well as of $E_{\infty}$) must all be simply connected -- since any non-trivial loop on a surface can be unwrapped in a finite cover. [That corresponds to the residual finiteness of the fundamental group of a closed surface.] Indeed, group-theoretically speaking, covering spaces correspond to the subgroups of the fundamental group. Utilising only normal subgroups (namely the regular coverings) would give a cofinal inverse system and therefore the inverse limit would still continue to be the $H_{\infty}$ lamination. This way of interpreting things allows us to see that the transverse Cantor-set fiber actually has a group structure. In fact it is the pro-finite group that is the inverse limit of all the deck-transformation groups corresponding to these normal coverings. \bigskip \noindent {\bf Complex structures :} Let us concentrate on the universal hyperbolic lamination $H_{\infty}$ from now on. For any complex structure on $X$ there is clearly a complex structure induced by pullback on each surface of the inverse system, and therefore $H_{\infty}$ itself inherits a complex structure on each leaf, so that now biholomorphically each leaf is the Poincare hyperbolic plane. If we think of a reference complex structure on $X$, then any new complex structure is recorded by a Beltrami coefficient on $X$, and one obtains by pullback a complex structure on the inverse limit in the sense that each leaf now has a complex structure and the Beltrami coefficients vary continuously from leaf to leaf in the Cantor-set direction. Indeed, the complex structures obtained in the above fashion by pulling back to the inverse limit from a complex structure on any closed surface in the inverse tower, have the special property that the Beltrami coefficients on the leaves are locally constant in the transverse (Cantor) direction. These "locally constant" families of Beltrami coefficients on $H_{\infty}$ comprise the {\it transversely locally constant} (written "TLC") complex structures on the lamination. The generic complex structure on $H_{\infty}$, where all continuously varying Beltrami coefficients in the Cantor-fiber direction are admissible, will be a limit of the TLC subfamily of complex structures. To be precise, a {\it complex structure} on a lamination $L$ is a covering of $L$ bylamination charts (disc) $\times$ (transversal) so that the overlap homeomorphisms are complex analytic on the disc direction. Two complex structures are {\it Teichm\"uller equivalent} whenever they are related to each other by a homeomorphism that is homotopic to the identity through leaf-preserving continuous mappings of $L$. For us $L$ is, of course, $H_{\infty}$. Thus we have defined the set $T(H_{\infty})$. Note that there is a distinguished leaf in our lamination, namely the path component of the point which is the string of all the base points. Call this leaf $l$. Note that all leaves are dense in $H_{\infty}$, in particular $l$ is dense. With respect to the base complex structure the leaf $l$ gets a canonical identification with the hyperbolic unit disc $\Delta$. Hence we have the natural "restriction to $l$" mapping of the Teichm\"uller space of $H_{\infty}$ into the Universal Teichm\"uller space $T(l) = T(1)$. Since the leaf is dense, the complex structure on it records the entire complex structure of the lamination. The above restriction map is therefore actually {\it injective} (see [20]) and therefore describes $T(H_{\infty})$ as an embedded complex analytic submanifold in $T(1)$. Indeed, as we will explain in detail below, $T(H_{\infty})$ embeds as precisely the closure in $T(1)$ of the union of the Teichm\"uller spaces $T(G)$ as $G$ varies over all finite-index subgroups of a fixed cocompact Fuchsian group. These finite dimensional classical Teichm\"uller spaces lying within the separable, infinite-dimensional $T(H_{\infty})$, comprise the TLC points of $T(H_{\infty})$. Alternatively, one may understand the set-up at hand by looking at the direct system of maps between Teichm\"uller spaces that is obviously induced by our inverse system of covering maps. Indeed, each covering map provides an immersion of the Teichm\"uller space of the covered surface into the Teichm\"uller space of the covering surface induced by the standard pullback of complex structure. These immersions are Teichm\"uller metric preserving, and provide a direct system whose direct limit, when completed in the Teichm\"uller metric, gives produces again $T(H_{\infty})$. The direct limit already contains the classical Teichm\"uller spaces of closed Riemann surfaces, and the completion corresponds to taking the closure in $T(1)$. Let us elaborate somewhat more on these various possible embeddings of $T(H_{\infty})$ [ which is to be thought of as the {\it universal Teichm\"uller space of {\bf compact} Riemann surfaces}] within the classical universal Teichm\"uller space $T(\Delta)$. \medskip \noindent {\bf Explicit realizations of $T(H_\infty)$ within the universal Teichm\"uller space:} Start with any cocompact (say torsion-free) Fuchsian group $G$ operating on the unit disc $\Delta$, such that the quotient is a Riemann surface $X$ of {\it arbitrary} genus $g$ greater than one. Considering the inverse limit of the directed system of all unbranched finite-sheeted pointed covering spaces over $X$ gives us a copy of the universal laminated space $H_\infty$ equipped with a complex structure induced from that on $X$. Every such choice of $G$ allows us to embed the separable Teichm\"uller space $T(H_\infty)$ holomorphically in the Bers universal Teichm\"uller space $T(\Delta)$. To fix ideas, let us think of the universal Teichm\"uller space as: $ T(\Delta) = T(1) = QS(S^1)/Mob(S^1)$ as usual. For any Fuchsian group $\Gamma$ define: $$ QS(\Gamma) = \{ w \in QS(S^1): w\Gamma w^{-1}~is~again~a~Mobius~group.\} $$ We say that the quasisymmetric homeomorphisms in $QS(\Gamma)$ are those that are {\it compatible} with $\Gamma$. Then the Teichm\"uller space $T(\Gamma)$ is $QS(\Gamma)/Mob(S^1)$ clearly sits embedded within $T(1)$. [We always think of points of $T(1)$ as left-cosets of the form $Mob(S^1)\circ w$ = $[w]$ for arbitrary quasisymmetric homeomorphism $w$ of the circle.] Having fixed the cocompact Fuchsian group $G$, the Teichm\"uller space $T(H_\infty)$ is now the closure in $T(1)$ of the direct limit of all the Teichm\"uller spaces $T(H)$ {\it as $H$ runs over all the finite-index subgroups of the initial cocompact Fuchsian group $G$}. Since each $T(H)$ is actually embedded injectively within the universal Teichm\"uller space, and since the connecting maps in the directed system are all inclusion maps, we see that the direct limit (which is in general a quotient of the disjoint union) in this situation is nothing other that just the set-theoretic {\it union of all the embedded $T(H)$ as $H$ varies over all finite index subgroups of $G$ }. This union in $T(1)$ constitutes the dense ``TLC" (transversely locally constant) subset of $T(H_\infty)$. Therefore, {\it the TLC subset of this embedded copy of $T(H_\infty)$ comprises the M\"ob-classes of all those QS-homeomorphisms that are compatible with {\it some} finite index subgroup in $G$.} We may call the above realization of $T(H_\infty)$ as {\it `` the $G$-tagged embedding" of $T(H_\infty)$} in $T(1)$. Remark: We see above, that just as the Teichm\"uller space of Riemann surfaces of any genus $p$ have lots of realizations within the universal Teichm\"uller space (corresponding to choices of reference cocompact Fuchsian groups of genus $p$), the Teichm\"uller space of the lamination $H_\infty$ also has many different realizations within $T(1)$. Therefore, in the Bers embedding of $T(1)$, this realization of $T(H_\infty)$ is the intersection of the domain $T(1)$ in the Bers-Nehari Banach space $B(1)$ with the separable Banach subspace that is the inductive (direct) limit of the subspaces $B(H)$ as $H$ varies over all finite index subgroups of the Fuchsian group $G$. (The inductive limt topology will give a complete (Banach) space; see, e.g., Bourbaki's "Topological Vector Spaces".) It is relevant to recall that $B(H)$ comprises the bounded holomorphic quadratic forms for the group $H$. By Tukia's results, the Teichm\"uller space of $H$ is exactly the intersection of the universal Teichm\"uller space with $B(H)$. \nn {\bf Remark:} Indeed one expects that the various $G$-tagged embeddings of $T(H_{\infty}$ must be sitting in general discretely separated from each other in the Universal Teichm\"uller space. There is a result to this effect for the various copies of $T(\Gamma)$, as the base group is varied, due to K.Matsuzaki (preprint -- to appear in Annales Acad. Scient. Fennicae). That should imply a similar discreteness for the family of embeddings of $T(H_{\infty})$ in $T(\Delta)$. It is not hard now to see how many different copies of the Teichm\"uller space of genus $p$ Riemann surfaces appear embedded within the $G$-tagged embedding of $T(H_\infty)$. That corresponds to non-conjugate (in $G$) subgroups of $G$ that are of index $(p-1)/(g-1)$ in $G$. This last is a purely topological question regarding the fundamental group of genus $g$ surfaces. Modular group: One may look at those elements of the full universal modular group $Mod(1)$ [quasisymmetric homeomorphism acting by right translation (i.e., pre-composition) on $T(1)$] that preserve setwise the $G$-tagged embedding of $T(H_\infty)$. Since the modular group $Mod(\Gamma)$ on $T(\Gamma)$ is induced by right translations by those QS-homeomorphisms that are in the normaliser of $\Gamma$: $$ N_{qs}(\Gamma)=\{t \in QS(\Gamma): t\Gamma t^{-1}=\Gamma \} $$ it is not hard to see that only the elements of $Mod(G)$ itself will manage to preserve the $G$-tagged embedding of $T(H_\infty)$. [Query: Can one envisage some limit of the modular groups of the embedded Teichm\"uller spaces as acting on $T(H_\infty)$?] \bigskip \noindent {\bf The Weil-Petersson pairing:} In [20], it has been shown that the tangent (and the cotangent) space at any point of $T(H_{\infty})$ consist of certain holomorphic quadratic differentials on the universal lamination $H_{\infty}$. In fact, the Banach space $B(c)$ of tangent holomorphic quadratic differentials at the Teichm\"uller point represented by the complex structure $c$ on the lamination, consists of holomorphic quadratic differentials on the leaves that vary continuously in the transverse Cantor-fiber direction. Thus locally, in a chart, these objects look like $\varphi(z,\lambda)dz^{2}$ in self-evident notation; ($\lambda$ represents the fiber coordinate). The lamination $H_{\infty}$ also comes equipped with an invariant transverse measure on the Cantor-fibers (invariant with respect to the holonomy action of following the leaves). Call that measure (fixed up to a scale) $d\lambda$. [That measure appears as the limit of (normalized) measures on the fibers above the base point that assign (at each finite Galois covering stage) uniform weights to the points in the fiber.] {}From [20] we have directly therefore our present goal: \medskip \nn {\bf PROPOSITION 9.1:} The Teichm\"uller space $T(H_{\infty})$ is a separable complex Banach manifold in $T(1)$ containing the direct limit of the classical Teichm\"uller spaces as a dense subset. The Weil-Petersson metrics on the classical $T_{g}$, normalized by a factor depending on the genus, fit together and extend to a finite Weil-Petersson inner product on $T(H_{\infty})$ that is defined by the formula: $$ \int_{H_{\infty}}^{} \varphi_{1} \varphi_{2}(Poin)^{-2}dz \wedge d \overline z d\lambda \eqno(75) $$ where $(Poin)$ denotes the Poincare conformal factor for the Poincare metric on the leaves (appearing as usual for all Weil-Petersson formulas). \vrule height 0.7em depth 0.2em width 0.5 em \medskip \nn {\bf Remark on Mostow rigidity for $T(H_{\infty})$:} The quasisymmetric homeomorphism classes comprising this Teichm\"uller space are again very non-smooth, since they appear as limits of the fractal q.s. boundary homeomorphisms corresponding to deformations of co-compact Fuchsian groups. Thus, the transversality proved in [15, Part II] of the finite dimensional Teichm\"uller spaces with the coadjoint orbit homogeneous space $M$ continues to hold for $T(H_{\infty})$. As explained there, that transversality is a form of the Mostow rigidity phenomenon. The formal Weil-Petersson converged on $M$ and coincided with the Kirillov-Kostant metric, but that formal metric fails to give a finite pairing on the tangent spaces to the finite dimensional $T_{g}$. Hence the interest in the above Proposition. \bigskip \nn {\bf \S 10-The Universal Period mapping and the Krichever map:} We make some remarks on the relationship of $\Pi$ with the Krichever mapping on a certain family of Krichever data. This could be useful in developing infinite-dimensional theta functions that go hand-in-hand with our infinite dimensional period matrices. The positive polarizing subspace, $T_{\mu}(W_{+})$, that is assigned by the period mapping $\Pi$ to a point $[\mu]$ of the universal Teichm\"uller space has a close relationship with the Krichever subspace of $L^{2}(S^1)$ that is determined by the Krichever map on certain Krichever data, when $[\mu]$ varies in (say) the Teichm\"uller space of a compact Riemann surface with one puncture (distinguished point). I am grateful to Robert Penner for discussing this matter with me. Recall that in the Krichever mapping one takes a compact Riemann surface $X$, a point $p \in X$, and a local holomorphic coordinate around $p$ to start with (i.e., a member of the ``dressed moduli space''). One also chooses a holomorphic line bundle $L$ over $X$ and a particular trivialization of $L$ over the given ($z$) coordinate patch around $p$. We assume that the $z$ coordinate contains the closed unit disc in the $z$-plane. To such data, the Krichever mapping associates the subspace of $L^{2} (S^1)$ [here $S^1$ is the unit circle in the $z$ coordinate] comprising functions which are restrictions to that circle of holomorphic sections of $L$ over the punctured surface $X-\{p\}$. If we select to work in a Teichm\"uller space $T(g,1)$ of pointed Riemann surfaces of genus $g$, then one may choose $z$ canonically as a certain horocyclic coordinate around the point $p$. Fix $L$ to be the canonical line bundle $T^{*}(X)$ over $X$ (the compact Riemann surface). This has a corresponding trivialization via ``$dz$''. The Krichever image of this data can be considered as a subspace living on the unit horocycle around $p$. That horocycle can be mapped over to the boundary circle of the universal covering disc for $X-\{p\}$ by mapping out by the natural pencil of Poincare geodesics having one endpoint at a parabolic cusp corresponding to $p$. {\it We may now see how to recover the Krichever subspace (for this restricted domain of Krichever data) from the subspace in $H^{1/2}_{{\bb C}}(S^1)$ associated to $(X,p)$ by $\Pi$.} Recall that the functions appearing in the $\Pi$ subspace are the boundary values of the Dirichlet-finite harmonic functions whose derivatives give the holomorphic Abelian differentials of the Riemann surface. Hence, to get Krichever from $\Pi$ one takes Poisson integrals of the functions in the $\Pi$ image, then takes their total derivative in the universal covering disc, and restricts these to the horocycle around $p$ that is sitting inside the universal cover (as a circle tangent to the boundary circle of the Poincare disc). Since Krichever data allows one to create the tau-functions of the $KP$ -hierarchy by the well-known theory of the Sato school (and the Russian school), one may now use the tau-function from the Krichever data to associate a tau (or theta) function to such points of our universal Schottky locus. The search for natural theta functions associated to points of the universal Siegel space $\cal S$, and their possible use in clarifying the relationship between the universal and classical Schottky problems, is a matter of interest that we are pursuing. \bigskip \centerline {{\bf REFERENCES} } \medskip \item{1} L.Ahlfors, Remarks on the Neumann-Poincare integral equation, {\it Pacific J.Math}, 2(1952), 271-280. \medskip \item{2} L.Ahlfors, {\it Conformal Invariants: Topics in Geometric Function Theory}, Mc Graw Hill, \medskip \item{3} L. Ahlfors, {\it Lectures on quasiconformal mappings,} Van Nostrand, (1966). \item{4} A. Beurling and L. Ahlfors, The boundary correspondence for quasiconformal mappings, {\it Acta Math}. 96 (1956), 125-142. \medskip \item{5} A. Beurling and J.Deny, Dirichlet Spaces, {\it Proc. Nat. Acad. Sci.}, 45 (1959), 208-215. \medskip \item{6} A. Connes and D. Sullivan, Quantum calculus on $S^1$ and Teichm\"uller theory, IHES preprint, 1993. \medskip \item{6a} A.Connes, {\it Geometrie Non Commutative}. \medskip \item{6b} R.R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, {\it Annals Math.} 103 (1976),611-635. \medskip \item{7} J. Deny and J.L. Lions, Les espaces de type de Beppo-Levi, {\it Annales Inst. Fourier}, 5 (1953-54), 305-370. \medskip \item{8} F. Gardiner and D. Sullivan, Symmetric structures on a closed curve, {\it American J. Math.} 114 (1992), 683-736. \medskip \item{9} P. Griffiths, Periods of integrals on algebraic manifolds, {\it Bull. American Math. Soc.,}, 75 (1970), 228-296. \medskip \item{10} S. Lang, $SL_2 ({\bb R})$, Springer-Verlag, (1975). \medskip \item{11} O. Lehto and K. Virtanen, {\it Quasiconformal mappings in the plane,} 2nd ed. Springer-Verlag, Berlin, (1973). \medskip \item{12} S. Nag, A period mapping in universal Teichm\"uller space, {\it Bull. American Math. Soc.,} 26, (1992), 280-287. \medskip \item{13} S. Nag, Non-perturbative string theory and the diffeomorphism group of the circle, in {\it ``Topological and Geometrical Methods in Field Theory''} Turku, Finland International Symposium, (eds. J. Mickelsson and O.Pekonen), World Scientific, (1992). \medskip \item{13a} S. Nag, On the tangent space to the universal Teichm\"uller space, {\it Annales Acad. Scient. Fennicae}, vol 18, (1993), (in press). \medskip \item{14} S. Nag, {\it The complex analytic theory of Teichm\"uller spaces,} Wiley-Interscience, (1988). \medskip \item{15} S. Nag and A. Verjovsky, Diff $(S^1)$ and the Teichm\"uller spaces, {\it Commun. Math. Physics}, 130 (1990), 123-138 (Part I by S.N. and A.V. ; Part II by S.N.). \medskip \item{16} M.S. Narasimhan, The type and the Green's kernel of an open Riemann surfaces, {\it Annales Inst. Fourier,} 10 (1960), 285-296. \medskip \item{17} H. Reimann, Ordinary differential equations and quasiconformal mappings, {\it Inventiones Math.} 33 (1976), 247-270. \medskip \item{18} G. Segal, Unitary representations of some infinite dimensional groups, {\it Commun. in Math. Physics} 80 (1981), 301-342. \medskip \item{19} M. Sugiura, {\it Unitary representations and harmonic analysis - an introduction, } 2nd ed., North Holland/Kodansha, (1975) \medskip \item{20} D. Sullivan, Relating the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Milnor Festschrift volume ``Topological methods in modern Mathematics'', (ed. L.Goldberg and A. Phillips), Publish or Perish, 1993, 543-563. \medskip \item{21} H. Triebel, {\it Theory of function spaces,} Birkh\"auser-Verlag, (1983). \medskip \item{22} E. Witten, Coadjoint orbits of the Virasoro group, {\it Commun. in Math. Physics,} 114 (1981), 1-53. \medskip \item{23} A. Zygmund, {\it Trigonometric Series,} Vol 1 and 2, Cambridge Univ. Press (1968). \bigskip \centerline{\bf Authors' addresses} \bigskip $$\vbox{ \offinterlineskip \halign{ # &# \cr \vrule height 12pt depth 5pt width 0pt\cc{ The Institute of Mathematical Sciences} & \cc {City Univ. of New York, Einstein Chair} \cr \vrule height 12pt depth 5pt width 0pt \cc {Madras 600 113, INDIA} &\cc {New York, N.Y. 10036, U.S.A.} \cr \vrule height 12pt depth 5pt width 0pt \cc {and} & \cc {and} \cr \vrule height 12pt depth 5pt width 0pt \cc{I.H.E.S.} & \cc{I.H.E.S.}\cr \vrule height 12pt depth 5pt width 0pt \cc {91440 Bures sur Yvette, FRANCE} & \cc{91440 Bures sur Yvette, FRANCE} \cr }}$$ \end

9310

alg-geom/9310003

en

https://arxiv.org/abs/alg-geom/9310003

alg-geom/9310003

Victor Batyrev

Victor V. Batyrev

Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

43 pages, Latex

We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.

alg-geom

\section{Introduction} \bigskip \hspace*{\parindent} Calabi-Yau $3$-folds caught much attention from theoretical physics because of their connection with the superstring theory \cite{witt}. Physicists discovered a duality for Calabi-Yau $3$-folds which is called {\em Mirror Symmetry} \cite{aspin1,aspin2,aspin3,cand.sch,dixon,gree,lerche,lynker}. This duality defines a correspondence between two topologically different Calabi-Yau $3$-folds $V$ and $V'$ such that the Hodge numbers of $V$ and $V'$ satisfy the relations \begin{equation} h^{1,1}(V) = h^{2,1}(V'), \;\; h^{1,1}(V') = h^{2,1}(V). \label{main.relation} \end{equation} Thousands examples of Calabi-Yau $3$-folds obtained from hypersurfaces in weighted projective spaces have shown a a striking symmetry for possible pairs of integers $(h^{1,1}, h^{2,1})$ relative to the transposition interchanging $h^{1,1}$ and $h^{2,1}$ \cite{cand.sch,kreuzer,schimm}. This fact gives an empirical evidence in favor of the conjectural duality. On the other hand, physicists have proposed some explicit constructions of mirror manifolds \cite{berglund,gree} for several classes of Calabi-Yau $3$-folds obtained from $3$-dimensional hypersurfaces in $4$-dimensional weighted projective spaces ${\bf P}(\omega_0, \ldots, \omega_4)$. The property $(\ref{main.relation})$ for the construction of B. Greene and R. Plesser \cite{gree} was proved by S.-S. Roan \cite{roan1}. \medskip In the paper of P. Candelas, X.C. de la Ossa, P.S. Green and L. Parkes \cite{cand2} the Mirror Symmetry was applied to give predictions for the number of rational curves of various degrees on general quintic $3$-folds in ${\bf P}_4$. For degrees $\leq 3$ these predictions were confirmed by algebraic geometers \cite{eil.st,s.kat}. In \cite{mor1} Morrison has presented a mathematical review of the calculation of P. Candelas et al. \cite{cand2}. Applying analogous method based on a consideration of the Picard-Fuchs equation, he has found in \cite{mor2} similar predictions for the number of rational curves on general members of another families of Calabi-Yau $3$-folds with $h^{1,1} = 1$ constructed as hypersurfaces in weighted projective spaces. Some verifications of these these predictions were obtained by S. Katz in \cite{katz.ver}. The method of P. Candelas et al. was also applied to Calabi-Yau complete intersections in projective spaces by A. Libgober and J. Teitelbaum \cite{libgober} whose calculation gave correct predictions for the number of lines and conics. Analogous results were obtained by physicists A. Font \cite{font}, A. Klemm and S.Theisen \cite{klemm2,klemm3}. \bigskip In this paper we consider families ${\cal F}(\Delta)$ of Calabi-Yau hypersurfaces which are compactifications in $n$-dimensional projective toric varieties ${\bf P}_{\Delta}$ of smooth affine hypersurfaces whose equations have a fixed Newton polyhedron $\Delta$ and sufficiently general coefficients. Our purpose is to describe a general method for constructing of candidates for mirrors of Calabi-Yau hypersurfaces in toric varieties. \medskip It turns out that all results on Calabi-Yau hypersurfaces and complete intersections in weighted and projective spaces, in particular the constructions of mirrors and the computations of predictions for numbers rational curves, can be extended to the case of Calabi-Yau hypersurfaces and complete intersections in toric varieties \cite{batyrev.var,batyrev.quant,bat.cox,bat.straten}. We remark that the toric technique for resolving singularities and computing Hodge numbers of hypersurfaces and complete intersections was first developed by A.G. Khovansky \cite{hov.genus,hov}. For the case of $3$-dimensional varieties with trivial canonical class, toric methods for resolving quotient singularities were first applied by D. Markushevich \cite{mark}, S.-S. Roan and S.-T. Yau \cite{roan0}. \bigskip Let us give an outline of the paper. \medskip Section 2 is devoted to basic terminology and well-known results on toric varieties. In this section we fix our notation for the rest of the paper. We use two definition of toric varieties. These definitions correspond to two kind of combinatorial data, contravariant and covariant ones. \medskip In section 3 we consider general properties of families ${\cal F}(\Delta)$ of hypersurfaces in a toric variety ${\bf P}_{\Delta}$ satisfying some regularity conditions which we call $\Delta$-regularity. These conditions imply that the singularities of hypersurfeace are induced only by singularities of the embient toric variety ${\bf P}_{\Delta}$. The main consequence of this fact is that a resolution of singularities of ${\bf P}_{\Delta}$ immediatelly gives rise to a resolution of singularities of {\em all} $\Delta$-regular hypersurfaces, i.e., we obtain a {\em simultanious resolution} of all members of the family ${\cal F}(\Delta)$. \medskip In section 4 we investigate lattice polyhedra $\Delta$ which give rise to families of Calabi-Yau hypersurfaces in ${\bf P}_{\Delta}$. Such a polyhedra admit a combinatorial characterisation and will be called {\em reflexive polyhedra}. There exists the following crusial observation: \bigskip {\em If $\Delta \subset M_{\bf Q}$ is a reflexive polyhedron, then the corresponding dual polyhedron $\Delta^*$ in dual space $N_{\bf Q}$ is also reflexive.} \bigskip Therefore the set of all $n$-dimensional reflexive polyhedra admits the involution $ \Delta \rightarrow \Delta^*$ which induces the involution between families of Calabi-Yau varieties: \[ {MIR}\, :\, {\cal F}({\Delta}) \rightarrow {\cal F}({\Delta}^*). \] We use the notaion $MIR$ for the involution acting on families of Calabi-Yau hypersurfaces corresponding to reflexive polyhedra in order to stress the following main conjecture: \medskip {\em The combinatorial involution ${\rm Mir} : \Delta \rightarrow \Delta^*$ acting on the set of all reflexive polyhedra of dimension $d$ agrees with the mirror involution on conformal field theories associated to Calabi-Yau varieties from ${\cal F}({\Delta})$ and ${\cal F}({\Delta}^*)$. } \medskip The next purpose of the paper is to give arguments showing that the Calabi-Yau families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$ are good candidates to be mirror symmetric, i.e., the involution $MIR : {\cal F}({\Delta}) \rightarrow {\cal F}({\Delta}^*)$ agrees with properties of the mirror duality in physics. \medskip Let ${\hat Z}$ denotes a {\em maximal projective crepant partial} de\-sin\-gu\-la\-ri\-za\-tion ({\em MPCP}-de\-sin\-gu\-la\-ri\-za\-tion) of a projective Calabi-Yau hypersurface ${\overline Z}$ in ${\bf P}_{\Delta}$. First, using results of Danilov and Khovanski\^i, we show that for $n \geq 4$ the Hodge number $h^{n-2,1}$ of {\em MPCP}-desingularizations of Calabi-Yau hypersurfaces in the family ${\cal F}(\Delta)$ equals the Picard number $h^{1,1}$ of {\em MPCP}-desingularizations of Calabi-Yau hypersurfaces in the family ${\cal F}(\Delta^*)$. As a corollary, we obtain the relation (\ref{main.relation}) predicted by physicists for mirror symmetric Calabi-Yau $3$-folds. Then, we prove that the nonsingular part ${\hat {Z}}$ of $\overline{Z}$ consisting of the union of $Z_f$ with all $(n-2)$-dimensional affine strata corresponding $(n-1)$-dimensional edges of $\Delta$ has {\em always} the Euler characteristic zero. We use this fact in the proof of a simple combinatorial formula for the Euler characteristic of Calabi-Yau $3$-folds in terms of geometry of two $4$-dimensional reflexive polyhedra $\Delta$ and $\Delta^*$. This formula immediately implies that for any pair of dual $4$-dimensional reflexive polyhedra $\Delta$ and $\Delta^*$, Calabi-Yau 3-folds obtained as {\em MPCP}-desingularizations of Calabi-Yau hypersurfaces in ${\cal F}(\Delta)$ and in ${\cal F}(\Delta^*)$ have opposite Euler characteristics. \medskip Section 5 is devoted to relations between our method and other already known methods of explicit constructions of mirrors. First, we calculate the one-parameter mirror family for the family of $(n-1)$-dimensional hypersurfaces of degree $n+1$ in $n$-dimensional projective space and show that our result for quintic therefolds coincides the already known construction of physicists. Next, we investigate the category ${\cal C}_n$ of reflexive pairs $(\Delta, M)$, where $M$ is an integral lattice of rank $n$ and $\Delta$ is an $n$-dimensional reflexive polyhedron with vertices in $M$. Morphisms in the category ${\cal C}_n$ give rise to relations between families of Calabi-Yau hypersurfaces in toric varieties. Namely, the existence of a morphism from $(\Delta_1, M_1)$ to $(\Delta_2, M_2)$ implies that Calabi-Yau hypersurfaces in ${\cal F}(\Delta_1)$ consist of quotients by action of a finite abelian group of Calabi-Yau hypersurfaces in ${\cal F}(\Delta_2)$. We prove that if a reflexive polyhedron $\Delta$ is an $n$-dimensional simplex, then the family ${\cal F}(\Delta)$ consist of deformations of quotiens of Fermat-type hypersurfaces in weighted projective spaces. As a result, we obtain that our method for constructions of mirror candidates coincides with the method of Greene and Plesser \cite{gree}, so that we obtain a generalization of the theorem of S.-S. Roan in \cite{roan1}. \bigskip {\bf Acknowledgements.} The ideas presented in the paper arose in some preliminary form during my visiting Japan in April-June 1991 supported by Japan Association for Mathematical Sciences and T\^ohoku University. I am grateful to P.M.H. Wilson whose lectures on Calabi-Yau $3$-folds in Sendai and Tokyo introduced me to this topic. Geometrical and arithmetical aspects of the theory of higher-dimensional Calabi-Yau varieties turned out to have deep connections with my research interests due to influence of my teachers V.A. Iskovskih and Yu. I. Manin. It is a pleasure to acknowledge helpful recommendations, and remarks concerning preliminary versions of this paper from mathematicians D. Dais, B. Hunt, Y. Kawamata, D. Morrison, T. Oda, S.-S. Roan, D. van Straten, J. Stienstra, and physicists P. Berglund, Ph. Candelas, A. Klemm, J. Louis, R. Schimmrigk, S. Theisen. I am very grateful to D. Morrison, B. Greene and P. Aspinwall who found a serious error in my earlier formulas for the Hodge number $h^{1,1}$ while working on their paper \cite{aspin4}. Correcting these errors, I found an easy proof of the formula for $h^{n-2,1}$. I would like to thank the DFG for the support, and the University of Essen, especially H. Esnault and E. Viehweg, for providing ideal conditions for my work. \bigskip \section{The geometry of toric varieties} \hspace*{\parindent} We follow notations of V. Danilov in \cite{dan1}. Almost all statements of this section are contained in \cite{oda1} and \cite{reid}. \medskip \subsection{Two definitions and basic notations} \hspace*{\parindent} First we fix notations used in the contravariant definition of toric varieties ${\bf P}_{\Delta}$ associated to a lattice polyhedron $\Delta$. \bigskip {$M$} { abelian group of rank $n$; } $\overline{M}$ $ = {\bf Z} \oplus M$; {${\bf T}$} $ = ({\bf C}^*)^n = \{ X = ( X_1 , \ldots, X_n ) \in {\bf C}^n \mid X_1 \cdots X_n \neq 0 \}$ $n$-dimensional algebraic torus ${\bf T}$ over ${\bf C}$; {$M_{\bf Q}$} $= M \otimes {\bf Q}$ the ${\bf Q}$-scalar extension of $M$; {$\overline{M}_{\bf Q}$} $ = {\bf Q} \oplus M_{\bf Q}$; {$\Delta$} a convex $n$-dimensional polyhedron in $M_{\bf Q}$ with integral vertices (i.e., all vertices of $\Delta$ are elements of $M$); {${\rm vol}_M\, \Delta$ (or $d(\Delta)$)} the volume of the polyhedron $\Delta$ relative to the integral lattice $M$, we call it the {\em degree} of the polyhedron $\Delta$ relative to $M$; {$C_{\Delta}$} $ = 0 \cup \{ (x_0, x_1, \ldots, x_n ) \in {\bf Q} \oplus {M}_{\bf Q} \mid ({x_1}/{x_0}, \ldots, {x_n}/{x_0}) \in \Delta, \; x_0 >0 \}$ the $(n+1)$-dimensional convex cone supporting $\Delta$; {$S_{\Delta}$} the the graded subring of ${\bf C} \lbrack X_0, X_1^{\pm 1}, \ldots , X_n^{\pm 1} \rbrack$ with the ${\bf C}$-basis consisting of monomials $X_0^{m_0}X_1^{m_1} \cdots X_n^{m_n}$ such that $(m_0, m_1, \ldots, m_n) \in C_{\Delta}$; {${\bf P}_{\Delta,M}$ (or ${\bf P}_{\Delta}$)} $ = {\rm Proj} \, S_{\Delta}$ be an $n$-dimensional projective toric variety corresponding to the graded ring $S_{\Delta}$; {${\cal O}_{\Delta}(1)$} the ample invertible sheaf on ${\bf P}_{\Delta, M}$ corresponding to the graded $S_{\Delta}$-module $S_{\Delta}(-1)$; {$\Theta$} be an arbitrary $l$-dimensional polyhedral face of $\Delta$ $(l \leq n)$; {${\bf T}_{\Theta}$} the corresponding $l$-dimensional ${\bf T}$-orbit in ${\bf P}_{\Delta}$; {${\bf P}_{\Theta}$} the $l$-dimensional toric variety which is the closure of ${\bf T}_{\Theta}$; {${\bf P}_{\Delta}^{(i)}$} $= \bigcup_{{\rm codim}\, \Theta = i} {\bf P}_{\Theta} = \bigcup_{{\rm codim}\, \Theta \leq i} {\bf T}_{\Theta }$. \bigskip Althought, the definition of toric varieties ${\bf P}_{\Delta}$ associated with integral polyhedra $\Delta$ is very simple, it is not always convenient. Notice that a polyhedron $\Delta$ defines not only the corresponding toric variety ${\bf P}_{\Delta}$, but also the choice of an ample invertible sheaf ${\cal O}_{\Delta}(1)$ on ${\bf P}_{\Delta}$ and an embedding of ${\bf P}_{\Delta}$ into a projective space. In general, there exist infinitely many different ample sheaves on ${\bf P}_{\Delta}$. As a result, there are infinitely many different integral polyhedra defining isomorphic toric varieties (one can take, for example, multiples of $\Delta$ : $k \Delta$, $k = 1,2, \ldots )$. If we want to get a one-to-one correspondence between toric varieties and some combinatorial data, we need covariant definition of toric varieties in terms of fans $\Sigma$ of rational polyhedral cones. This approach to toric varieties gives more possibilities, it allows, for example, to construct affine and quasi-projective toric varieties as well as complete toric varieties which are not quasi-projective. \bigskip {$N$} $= {\rm Hom}\, (M, {\bf Z})$ the dual to $M$ lattice; {$\langle *, * \rangle$}\, : \, $M_{\bf Q} \times N_{\bf Q} \rightarrow {\bf Q} $ the nondegenerate pairing between the $n$-dimensional ${\bf Q}$-spaces $M_{\bf Q}$ and $N_{\bf Q}$; {$\sigma$} an $r$-dimensional $(0 \leq r \leq n)$ convex rational polyhedral cone in $N_{\bf Q}$ having $0 \in N_{\bf Q}$ as vertex; {$\check {\sigma}$} the dual to $\sigma$ $n$-dimensional cone in $M_{\bf Q}$; {${\bf A}_{\sigma, N}$ (or ${\bf A}_{\sigma}$)} $ = {\rm Spec}\, \lbrack \check {\sigma} \cap M \rbrack$ an $n$-dimensional affine toric variety associated with the $r$-dimensional cone $\sigma$; {$N(\sigma)$} minimal $r$-dimensional sublattice of $N$ containing $\sigma \cap N$; {${\bf A}_{\sigma,N(\sigma)}$} the $r$-dimensional affine toric variety corresponding to $\sigma \subset N(\sigma)_{\bf Q}$ (${\bf A}_{N(\sigma)} = {\bf C}^*)^{n-r} \times {\bf A}_{\sigma, N(\sigma)}$); {$\Sigma$} a finite rational polyhedral fan of cones in $N_{\bf Q}$; {$\Sigma^{(i)}$} the set of all $i$-dimensional cones in $\Sigma$; {$\Sigma^{[i]}$} the subfan of $\Sigma$ consisting of all cones $\sigma \in \Sigma$ such that ${\rm dim}\, \sigma \leq i $; {${\bf P}_{\Sigma,N}$ (or ${\bf P}_{\Sigma}$)} $n$-dimensional toric variety obtained by glueing of affine toric varieties ${\bf A}_{\sigma,N}$ where $\sigma \in \Sigma$; {${\bf P}^{[i]}_{\Sigma}$} the open toric subvariety in ${\bf P}_{\Sigma}$ corresponding to $\Sigma^{[i]}$. \bigskip Inspite of the fact that the definition of toric varieties via rational polyhedral fans is more general than via integral polyhedra, we will use both definitions. The choice of a definition in the sequel will depend on questions we are interested in. In one situation, it will be more convenient to describe properties of toric varieties and their subvarieties in terms of integral polyhedra. In another situation, it will be more convenient to use the language of rational polyhedral fans. So it is important to know how one can construct a fan $\Sigma(\Delta)$ from an integral polyhedron $\Delta$, and how one can construct an integral polyhedron $\Delta(\Sigma)$ from a rational polyhedral fan $\Sigma$. The first way $\Delta \Rightarrow \Sigma(\Delta)$ is rather simple: \begin{prop} For every $l$-dimensional face $\Theta \subset \Delta$, define the convex $n$-dimensional cone ${\check {\sigma}}(\Theta) \subset M_{\bf Q}$ consisting of all vectors $\lambda (p - p')$, where $\lambda \in{\bf Q}_{\geq 0}$, $p \in \Delta$, $p' \in \Theta$. Let $\sigma( \Theta) \subset N_{\bf Q}$ be the $(n-l)$-dimensional dual cone relative to ${\check {\sigma}}(\Theta) \subset M_{\bf Q}$. The set $\Sigma(\Delta)$ of all cones $\sigma(\Theta)$, where $\Theta$ runs over all faces of $\Delta$, determines the complete rational polyhedral fan defining the toric variety ${\bf P}_{\Delta}$. \label{def.fan} \end{prop} \bigskip The decomposition of ${\bf P}_{\Delta}$ into a disjoint union of ${\bf T}$-orbits ${\bf T}_{\Theta}$ can be reformulated via cones $\sigma(\Theta)$ in the fan $\Sigma(\Delta) = \Sigma $ as follows. \begin{prop} Let $\Delta$ be an $n$-dimensional $M$-integral polyhedron in $M_{\bf Q}$, $\Sigma = \Sigma(\Delta)$ the corresponding complete rational polyhedral fan in $N_{\bf Q}$. Then {\rm (i)} For any face $\Theta \subset \Delta$, the affine toric variety ${\bf A}_{{\sigma}(\Theta),N}$ is the minimal ${\bf T}$-invariant affine open subset in ${\bf P}_{\Sigma}$ containing ${\bf T}$-orbit ${\bf T}_{\Theta}$. {\rm (ii)} Put ${\bf T}_{\sigma}:= {\bf T}_{\sigma(\Theta)}$. There exists a one-to-one correspondence between $s$-dimensional cones $\sigma \in \Sigma$ and $(n-s)$-dimensional ${\bf T}$-orbits ${\bf T}_{\sigma}$ such that ${\bf T}_{\sigma'}$ is contained in the closure of ${\bf T}_{\sigma}$ if and only if $\sigma$ is a face of $\sigma'$. {\rm (iii)} \[ {\bf P}_{\Sigma}^{[i]} = \bigcup_{{\rm dim}\, \sigma \leq i} {\bf T}_{\sigma} \] is an open ${\bf T}$-invariant subvariety in ${\bf P}_{\Sigma} = {\bf P}_{\Delta}$, and \[ {\bf P}_{\Sigma} \setminus {\bf P}_{\Sigma}^{[i]} = {\bf P}^{(i)}_{\Delta}. \] \label{orbits.fan} \end{prop} \bigskip Let us now consider another direction $\Sigma \Rightarrow \Delta(\Sigma)$. In this case, in order to construct $\Delta(\Sigma)$ it is not sufficient to know only a complete fan $\Sigma$. We need a strictly upper convex support function $h\; :\; N_{\bf Q} \rightarrow {\bf Q}$. \begin{opr} {\rm Let $\Sigma$ be a rational polyhedral fan and let $h\; :\; N_{\bf Q} \rightarrow {\bf Q}$ be a function such that $h$ is linear on any cone $\sigma \subset \Sigma$. In this situation $h$ is called a {\em support function} for the fan $\Sigma$. We call $h$ {\em integral} if $h(N) \subset {\bf Z}$. We call $h$ {\em upper convex} if $h(x + x') \leq h(x) + h(x')$ for all $x, x' \in N_{\bf Q}$. Finally, $h$ is called {\em strictly upper convex} if $h$ is upper convex and for any two distinct $n$-dimensional cones $\sigma$ and $\sigma'$ in $\Sigma$ the restrictions $h_{\sigma}$ and $h_{\sigma'}$ of $h$ on $\sigma$ and $\sigma'$ are different linear functions. } \end{opr} \begin{rem} {\rm By general theory of toric varieties \cite{dan1,oda1}, support functions one-to-one correspond to ${\bf T}$-invariant ${\bf Q}$-Cartier divisors $D_h$ on ${\bf P}_{\Sigma}$. A divisor $D_h$ is a ${\bf T}$-invariant Cartier divisor (or ${\bf T}$-linearized invertible sheaf) if and only if $h$ is integral. $D_h$ is numerically effective if and only if $h$ is upper convex. Strictly upper convex support functions $h$ on a fan $\Sigma$ one-to-one correspond to ${\bf T}$-linearized {\em ample} invertible sheaves ${\cal O}(D_h)$ over ${\bf P}_{\Sigma}$.} \end{rem} The next proposition describes the construction of a polyhedron $\Delta$ from a fan $\Sigma$ supplied with a strictly convex support function $h$. \begin{prop} Let the convex polyhedron $\Delta = \Delta(\Sigma, h)$ to be defined as follows \[ \Delta(\Sigma,h) = \bigcap_{\sigma \in \Sigma^{(n)}} ( -h_{\sigma} + {\check \sigma} ), \] where integral linear functions $h_{\sigma}: N \rightarrow {\bf Z}$ are considered as elements of the lattice $M$. Then one has ${\bf P}_{\Delta} \cong {\bf P}_{\Sigma}$ and ${\cal O}_{\Delta}(1) \cong {\cal O}(D_h)$. \end{prop} We have seen already that the construction of an integral polyhedron $\Delta$ from a fan $\Sigma$ such that ${\bf P}_{\Delta} \cong {\bf P}_{\Sigma}$ is not unique and depends on the choice of an integral strictly upper convex support function $h$. However, there exists an important case when we can make a natural choice of $h$. Indeed, for every ${\bf Q}$-Gorenstein toric variety ${\bf P}_{\Sigma}$ we have the unique support function $h_K$ corresponding to the anticanonical divisor $-K_{\Sigma} = {\bf P}_{\Sigma} \setminus {\bf T}$. \begin{opr} {\rm A toric variety ${\bf P}_{\Sigma}$ is said to be a toric {\em ${\bf Q}$-Fano variety} if the anticanonical support function $h_K$ is strictly upper convex. A toric ${\bf Q}$-Fano variety is called a {\em Gorenstein toric Fano variety} if $h_K$ is integral. } \label{def.fano1} \end{opr} With a toric ${\bf Q}$-Fano variety, one can associate two convex polyhedra: \[ \Delta({\Sigma}, h_K) = \bigcap_{\sigma \in \Sigma^{(n)}} ( -k_{\sigma} + {\check \sigma} ) \] and \[ \Delta^*(\Sigma, h_K) = \{ y \in N_{\bf Q} \mid h_K(y) \leq 1 \}. \] The polyhedra $\Delta({\Sigma}, h_K)$ and $\Delta^*(\Sigma, h_K)$ belong to $M_{\bf Q}$ and $N_{\bf Q}$ respectively. \subsection{Singularities and morphisms of toric varieties} \hspace*{\parindent} \begin{opr} {\rm Let $\sigma \in Sigma$ be an $r$-dimensional cone in $N_{\bf Q}$. We denote by $p_{\sigma}$ the unique ${\bf T}/{\bf T}_{\sigma}$-invariant point on the $r$-dimensional affine toric variety ${\bf A}_{\sigma,N(\sigma)}$ } \end{opr} Since ${\bf A}_{\sigma,N}$ splits into the product \[ ({\bf C}^*)^{n-s} \times {\bf A}_{\sigma, N(\sigma)}, \] open analytical neighbourhoods of any two points $p, p' \in {\bf T}_{\sigma}$ are locally isomorphic. Therefore, it suffices to investigate the structure of toric singularities at points $p_{\sigma}$ where $\sigma$ runs over cones of $\Sigma$. The following two propositions are due to M. Reid \cite{reid}. \begin{prop} Let $n_1, \ldots, n_r \in N$ $(r \geq s)$ be primitive $N$-integral generators of all $1$-dimensional faces of an $s$-dimensional cone $\sigma$. {\rm (i)} the point $p_{\sigma} \in {\bf A}_{\sigma, N(\sigma)}$ is ${\bf Q}$-factorial $($or quasi-smooth$)$ if ond only if the cone $\sigma$ is simplicial, i.e., $r = s$; {\rm (ii)} the point $p_{\sigma} \in {\bf A}_{\sigma, N(\sigma)}$ is ${\bf Q}$-Gorenstein if ond only if the elements $n_1, \ldots, n_r$ are contained in an affine hyperplane \[ H_{\sigma} \; : \; \{ y \in N_{\bf Q} \mid \langle k_{\sigma}, y \rangle = 1 \}, \] for some $k_{\sigma} \in M_{\bf Q}$ $($note that when ${\rm dim}\, \sigma = {\rm dim}\, N$, the element $k_{\sigma}$ is unique if it exists$)$. Moreover, ${\bf A}_{\sigma, N(\sigma)}$ is Gorenstein if and only if $k_{\sigma} \in M$. \label{sing1} \end{prop} \begin{rem} {\rm If ${\bf P}_{\Sigma}$ is a ${\bf Q}$-Gorenstein toric variety, the elements $k_{\sigma} \in M_{\bf Q}$ ($\sigma \in \Sigma^{(n)}$) define together the support function $h_K$ on $N_{\bf Q}$ such that the restriction of $h_K$ on $\sigma \in \Sigma^{(n)}$ is $k_{\sigma}$. The support function $h_K$ corresponds to the anticanonical divisor on ${\bf P}_{\Sigma}$.} \label{anti.des} \end{rem} \begin{prop} Assume that ${\bf A}_{\sigma, N(\sigma)}$ is ${\bf Q}$-Gorenstein {\rm ($see$ \ref{sing1}(ii))}, then {\rm (i)} ${\bf A}_{\sigma, N(\sigma)}$ has at the point $p_{\sigma}$ at most { terminal} singularity if and only if \[ N \cap \sigma \cap \{y \in N_{\bf Q} \mid \langle k_{\sigma}, y \rangle \leq 1 \} = \{ 0, n_1, \ldots, n_r \} ; \] {\rm (ii)} ${\bf A}_{\sigma, N(\sigma)}$ has at the point $p_{\sigma}$ at most { canonical } singularity if and only if \[ N \cap \sigma \cap \{y \in N_{\bf Q} \mid \langle k_{\sigma} , y \rangle < 1 \} = \{ 0 \}. \] \label{sing2} \end{prop} Using \ref{sing1} and \ref{sing2}, we obtain: \begin{coro} Any Gorenstein toric singularity is canonical. \label{gor.can} \end{coro} \begin{opr} {\rm Let $S$ be a $k$-dimensional simplex in ${\bf Q}^n$ $(k \leq n)$ with vertices in ${\bf Z}^n$, $A(S)$ the minimal $k$-dimensional affine ${\bf Q}$-subspace containing $S$. Denote by ${\bf Z}(S)$ the $k$-dimensional lattice $A(S) \cap {\bf Z}^n$. We call $P$ {\em elementary} if $S \cap {\bf Z}(S)$ contains only vertices of $S$. We call $S$ {\em regular} if the degree of $S$ relative to ${\bf Z}(S)$ is $1$.} \end{opr} It is clear that every regular simplex is elementary. The converse is not true in general. However, there exists the following easy lemma. \begin{lem} Every elementary simplex of dimension $\leq 2$ is regular. \label{el.reg} \end{lem} By \ref{sing1} and \ref{sing2}, we obtain: \begin{prop} Let ${\bf P}_{\Sigma}$ be a toric variety with only ${\bf Q}$-Gorenstein singularities, i.e., for any cone $\sigma \in \Sigma$ let the corresponding element $k_{\sigma} \in M_{\bf Q}$ be well-defined {\rm (\ref{sing1}(ii))}. Then {\rm (i)} ${\bf P}_{\Sigma}$ has only ${\bf Q}$-factorial terminal singularities if and only if for every cone $\sigma \in \Sigma$ the polyhedron \[ P_{\sigma} = \sigma \cap \{ y \in N_{\bf Q} \mid \langle k_{\sigma}, y \rangle \leq 1 \} \] is an elementary simplex. {\rm (ii)} ${\bf P}_{\Sigma}$ is smooth if and only if for every cone $\sigma \in \Sigma$ the polyhedron \[ P_{\sigma} = \sigma \cap \{ y \in N_{\bf Q} \mid \langle k_{\sigma}, y \rangle \leq 1 \} \] is a regular simplex. \label{simp} \end{prop} \begin{theo} Let ${\bf P}_{\Sigma}$ be a toric variety. {\rm (i)} ${\bf P}_{\Sigma}$ is smooth. {\rm (ii)} If ${\bf P}_{\Sigma}$ has only terminal singularities, then the open toric subvariety ${\bf P}_{\Sigma}^{[2]}$ is smooth. {\rm (iii)} If ${\bf P}_{\Sigma}$ has only Gorenstein ${\bf Q}$-factorial terminal singularities, then the open toric subvariety ${\bf P}_{\Sigma}^{[3]}$ is smooth. \label{codim2,3} \end{theo} We recall a combinatorial characterization of toric morphisms between toric varieties. \bigskip Let $\phi \; : \; N' \rightarrow N$ be a morphism of lattices, $\Sigma$ a fan in $N_{\bf Q}$, $\Sigma'$ a fan in $N'_{\bf Q}$. Suppose that for each $\sigma' \in \Sigma'$ we can find a $\sigma \in \Sigma$ such that $\varphi(\sigma') \subset \sigma$. In this situation there arises a morphism of toric varieties \[ \tilde {\phi} \; : \; {\bf P}_{\Sigma',N'} \rightarrow {\bf P}_{\Sigma, N}. \] \begin{exam} {Proper birational morphisms.} {\rm Assume that $\varphi$ is an isomorphism of lattices and $\phi(\Sigma')$ is a subdivision of $\Sigma$, i.e., every cone $\sigma \in \Sigma$ is a union of cones of $\varphi(\Sigma')$. In this case $\tilde {\phi}$ is a proper birational morphism. Such a morphism we will use for constructions of desingularizations of toric singularities. } \label{biration} \end{exam} \medskip \begin{opr} {\rm Let $\varphi: W' \rightarrow W$ be a proper birational morphism of normal ${\bf Q}$-Gorenstein algebraic varieties. The morphism $\varphi$ is called {\em crepant} if $\varphi^* K_{W} = K_{W'}$ ($K_W$ and $K_{W'}$ are canonical divisors on $W$ and $W'$ respectively). } \end{opr} \medskip Using the description in \ref{anti.des} of support functions corresponding to anticanonical divisors on ${\bf P}_{\Sigma}$ and ${\bf P}_{\Sigma'}$, we obtain. \begin{prop} A proper birational morphism of ${\bf Q}$-Gorenstein toric varieties \[ \tilde {\phi} \; : \; {\bf P}_{\Sigma'} \rightarrow {\bf P}_{\Sigma} \] is crepant if and only if for every cone $\sigma \in \Sigma^{(n)}$ all $1$-dimensional cones $\sigma'\in \Sigma'$ which are contained in $\sigma$ are generated by primitive integral elements from $N \cap H_{\sigma}$ {\rm ($see$ \ref{sing1}(ii))}. \label{crit.crep} \end{prop} \begin{opr} {\rm Let $\varphi\; : \; W' \rightarrow W$ be a projective birational morphism of normal ${\bf Q}$-Gorenstein algebraic varieties. The morphism $\varphi$ is called a {\em maximal projective crepant partial desingularization} ({\em MPCP-desingularization}) of $W$ if $\varphi$ is crepant and $W'$ has only ${\bf Q}$-factorial terminal singularities.} \end{opr} \medskip Our next purpose is to define some combinatorial notions which we use to construct $MPCP$-desingularizations of Gorenstein toric varieties (see \ref{crep.fano}). \medskip \begin{opr} {\rm Let $A$ be a finite subset in $\Delta \cap {\bf Z}^n$. We call $A$ {\em admissible} if it contains all vertices of the integral polyhedron $\Delta$. } \end{opr} \begin{opr} {\rm Let $A$ be an admissible subset in $\Delta \cap {\bf Z}^n$. By an {\em A-triangulation} of $\Delta \subset {\bf Q}^n$ we mean a finite collection ${\cal T} = \{ \theta \}$ of simplices with vertices in $A$ having the following properties: (i) if $\theta'$ is a face of a simplex $\theta \in {\cal T}$, then $\theta' \in {\cal T}$; (ii) the vertices of each simplex $\theta \in {\cal T}$ lie in $\Delta \cap {\bf Z}^n$; (iii) the intersection of any two simplices $\theta_1, \theta_2 \in {\cal T}$ either is empty or is a common face of both; (iv) $\Delta = \bigcup_{\theta \in {\cal T}} \theta$; (v) every element of $A$ is a vertex of some simplex $\theta \in {\cal T}$. } \label{def.triang} \end{opr} \begin{opr} {\rm An $A$-triangulation ${\cal T}$ of an integral convex polyhedron $\Delta \subset {\bf Q}^n$ is called {\em maximal} if $A = \Delta \cap {\bf Z}^n$. } \end{opr} \begin{rem} {\rm Note that ${\cal T}$ is a maximal triangulation of $\Delta$ if and only if every simplex $\theta \in {\cal T}$ is elementary. Therefore, if ${\cal T}$ is a maximal triangulation of $\Delta$, then for any face $\Theta \subset \Delta$ the induced triangulation of $\Theta$ is also maximal.} \label{rem.max} \end{rem} \medskip Assume that $A$ is admissible. Denote by ${\bf Q}^A$ the ${\bf Q}$-space of functions from $A$ to ${\bf Q}$. Let ${\cal T}$ be a triangulation of $\Delta$. Every element $\alpha \in {\bf Q}^A$ can be uniquely extended to a piecewise linear function $\alpha({\cal T})$ on $\Delta$ such that the restriction of $\alpha({\cal T})$ on every simplex $\theta \in {\cal T}$ is an affine linear function. \begin{opr} {\rm Denote by $C({\cal T})$ the convex cone in ${\bf Q}^A$ consisting of elements $\alpha$ such that $\alpha({\cal T})$ is an upper convex function. We say that a triangulation ${\cal T}$ of $\Delta$ is {\em projective} if the cone $C({\cal T})$ has a nonempty interior. In other words, ${\cal T}$ is projective if and only if there exists a strictly upper convex function $\alpha({\cal T})$. } \label{opr.t.proj} \end{opr} \begin{prop} {\rm \cite{gelf}} Let $\Delta$ be an integral polyhedron. Take an admissible subset $A$ in $\Delta \cap {\bf Z}^n$. Then $\Delta$ admits at least one projective $A$-triangulation, in particular, there exists at least one maximal projective triangulation of $\Delta$. \label{triang.ex} \end{prop} \begin{rem} {\rm In the paper Gelfand, Kapranov, and Zelevinsky \cite{gelf}, it has been used the notion of a {\em regular triangulation of a polyhedron} which is called in this paper a {\em projective triangulation}. The reason for such a change of the terminology we will see in \ref{crep.fano}.} \end{rem} \begin{exam} {\rm {\em Finite morphisms.} Assume that $\phi$ is injective, $\phi(N')$ is a sublattice of finite index in $N$, and $\phi(\Sigma') = \Sigma$. Then $\tilde {\phi}$ is a finite surjective morphism of toric varieties. This morphism induces an \^etale covering of open subsets \[ {\bf P}_{\Sigma'}^{[1]} \rightarrow {\bf P}_{\Sigma}^{[1]} \] with the Galois group ${\rm Coker}\,\lbrack N'\rightarrow N \rbrack$. } \label{final} \end{exam} Recall the following description of the fundamental group of toric varieties. \begin{prop} The fundamental group of a toric variety ${\bf P}_{\Sigma}$ is isomorphic to the quotient of $N$ by sum of all sublattices $N(\sigma)$ where $\sigma$ runs over all cones $\sigma \in \Sigma$. In particular, the fundamental group of the non-singular open toric subvariety ${\bf P}_{\Sigma}^{[1]}$ is isomorphic to the quotient of $N$ by sublattice spanned by all primitive integral generators of $1$-dimensional cones $\sigma \in \Sigma^{(1)}$. \label{fund.group} \end{prop} \bigskip We will use the following characterization of toric Fano varieties in terms of he polyhedra $\Delta({\Sigma}, h_K)$ and $\Delta^*({\Sigma}, h_K)$. \begin{prop} A complete toric variety ${\bf P}_{\Sigma}$ with only ${\bf Q}$-Gorenstein singularities is a ${\bf Q}$-Fano variety {\rm ($see$ \ref{def.fano1})} if and only if $\Delta({\Sigma}, h_K)$ is an $n$-dimensional polyhedron with vertices $-k_{\sigma}$ one-to-one corresponding to $n$-dimensional cones $\sigma \in \Sigma$. In this situation, {\rm (i)} ${\bf P}_{\Sigma}$ is a Fano variety with only Gorenstein singularities if and only if all vertices of $\Delta({\Sigma}, h_K)$ belong to $M$, in particular, ${\bf P}_{\Sigma} = {\bf P}_{\Delta({\Sigma, h_K})}$. {\rm (ii)} ${\bf P}_{\Sigma}$ is a smooth Fano variety if and only if for every $n$-dimensional cone $\sigma \in \Sigma^{(n)}$ the polyhedron $\Delta^*({\Sigma}, h_K) \cap \sigma$ is a regular simplex of dimension $n$. \label{smooth.fano} \end{prop} Now we come to the most important statement which will be used in the sequel. \begin{theo} Let ${\bf P}_{\Sigma}$ be a toric Fano variety with only Gorenstein singularities. Then ${\bf P}_{\Sigma}$ admits at least one MPCP-desingulari\-zation \[ \tilde {\phi} \; : \; {\bf P}_{\Sigma'} \rightarrow {\bf P}_{\Sigma}. \] Moreover, MPCP-desingulari\-zations of ${\bf P}_{\Sigma}$ are defined by maximal projective triangulations of the polyhedron $ \Delta^*(\Sigma, h_K)$, where $h_K$ the integral strictly upper convex support function associated with the anticanonical divisor ${\bf P}_{\Sigma} \setminus {\bf T}$ on ${\bf P}_{\Sigma}$. \label{crep.fano} \end{theo} {\em Proof. } Define a finite subset $A$ in $N$ as follows \[ A = \{ y \in N \mid h_K(y) \leq 1 \} = N \cap \Delta^*({\Sigma}, h_K). \] It is clear that $A$ is an admissible subset of $\Delta^*(\Sigma, h_K)$. By \ref{triang.ex}, there exists at least one projective $A$-triangulation ${\cal T}$ of $\Delta^*(\Sigma, h_K)$. Let $B$ be the boundary of $\Delta^*(\Sigma, h_K)$. For every simplex $\theta \in B$, we construct a convex cone $\sigma_{\theta}$ supporting $\theta$. By definition \ref{def.triang}, the set of all cones $\sigma_{\theta}$ ($\theta \in {\cal T} \cap B$) defines a fan $\Sigma'$ which is a subdivision of $\Sigma$. Since generators of $1$-dimensional cones of $\Sigma$ are exactly elements of $A \cap B$, the morphism ${\bf P}_{\Sigma'} \rightarrow {\bf P}_{\Sigma}$ is crepant (see \ref{crit.crep}). By \ref{opr.t.proj}, there exists a strictly upper convex function $\alpha({\cal T})$. We can also assume that $\alpha({\cal T})$ has zero value at $0 \in N$. Then $\alpha({\cal T})$ defines a strictly convex support function for the fan $\Sigma'$. Thus, ${\bf P}_{\Sigma'}$ is projective. By \ref{simp}(i) and \ref{rem.max}, we obtain that the morphism ${\bf P}_{\Sigma'} \rightarrow {\bf P}_{\Sigma}$ is a $MPCP$-desingulari\-zation. By similar arguments, one can prove that any $MPCP$-desingulari\-zation defines a maximal projective triangulation of $\Delta^*(\Sigma, h_K)$. \bigskip \section{Hypersurfaces in toric varieties} \subsection{Regularity conditions for hypersurfaces} \hspace*{\parindent} A Laurent polynomial $f = f(X)$ is a finite linear combination of elements of $M$ \[ f(X) = \sum c_m X^m \] with complex coefficients $c_m$. The Newton polyhedra $\Delta(f)$ of $f$ is the convex hull in $M_{\bf Q} = M \otimes {\bf Q}$ of all elements $m$ such that $c_m \neq 0$. Every Laurent polynomial $f$ with the Newton polyhedron $\Delta$ defines the affine hypersurface \[ Z_{f, \Delta} = \{ X \in {\bf T} \mid f(X) = 0 \}. \] If we work with a fixed Newton polyhedron $\Delta$, we denote $Z_{f, \Delta}$ simply by $Z_f$. \bigskip Let $\overline{Z}_{f,\Delta}$ be the closure of $Z_{f, \Delta} \subset {\bf T}$ in ${\bf P}_{\Delta}$. For any face $\Theta \subset \Delta$, we put $Z_{f, \Theta} = \overline{Z}_{f, \Delta} \cap {\bf T}_{\Theta}$. So we obtain the induced decomposition into the disjoint union \[ \overline{Z}_{f, \Delta} = \bigcup_{\Theta \subset \Delta} Z_{f, \Theta}. \] \medskip \begin{opr} {\rm Let $L(\Delta)$ be the space of all Laurent polynomials with a fixed Newton polyhedron $\Delta$. A Laurent polynomial $f \in L(\Delta)$ and the corresponding hypersurfaces $Z_{f, \Delta} \subset {\bf T}_{\Delta}$, $\overline{Z}_{f, \Delta} \subset {\bf P}_{\Delta}$ are said to be ${\Delta}$-{\em regular} if for every face $\Theta \subset \Delta$ the affine variety $Z_{f,\Theta}$ is empty or a smooth subvariety of codimension 1 in ${\bf T}_{\Theta}$. Affine varieties $Z_{f,\Theta}$ are called {\em the strata} on $\overline{Z}_{f, \Delta}$ associated with faces $\Theta \subset \Delta$.} \label{d.nondeg} \end{opr} \begin{rem} {\rm Notice that if $f$ is $\Delta$-regular, then ${Z}_{f, \Theta} = \emptyset$ if and only if ${\rm dim}\, \Theta = 0$, i.e., if and only if $\Theta$ is a vertex of $\Delta$. } \label{zero.orb} \end{rem} \bigskip Since the space $L(\Delta)$ can be identified with the space of global sections of the ample sheaf ${\cal O}_{\Delta}(1)$ on ${\bf P}_{\Delta}$, using Bertini theorem, we obtain: \begin{prop} The set of $\Delta$-regular hypersurfaces is a Zariski open subset in $L(\Delta)$. \end{prop} We extend the notion of $\Delta$-regular hypersurfaces in ${\bf P}_{\Delta}$ to the case of hypersurfaces in general toric varieties ${\bf P}_{\Sigma}$ associated with rational polyhedral fans $\Sigma$. \bigskip \begin{opr} {\rm Let $\overline{Z}_{f, \Sigma}$ be the closure in ${\bf P}_{\Sigma}$ of an affine hypersurface $Z_f$ defined by a Laurent polynomial $f$. Consider the induced decomposition into the disjoint union \[ \overline{Z}_{f, \Sigma } = \bigcup_{\sigma \in \Sigma} Z_{f, \sigma}, \] where ${Z}_{f, \sigma } = \overline{Z}_{f, \Sigma } \cap {\bf T}_{\sigma}$. A Laurent polynomial $f$ and the corresponding hypersurfaces $Z_{f} \subset {\bf T}$, $\overline{Z}_{f, \Sigma} \subset {\bf P}_{\Sigma}$ are said to be ${\Sigma}$-{\em regular} if for every $s$-dimensional cone $\sigma \in \Sigma$ the variety $Z_{f,\sigma}$ is empty or a smooth subvariety of codimension 1 in ${\bf T}_{\sigma}$. In other words, $\overline{Z}_{f, \Sigma}$ has only transversal intersections with all ${\bf T}$-orbits ${\bf T}_{\sigma}$ ($\sigma \in \Sigma$). Affine varieties $Z_{f,\sigma}$ are called {\em strata} on $\overline{Z}_{f, \Sigma}$ associated with the cones $\sigma \subset \Sigma$. Denote by $Z_{f, \Sigma}^{[i]}$ the open subvariety in $\overline{Z}_{f, \Sigma}$ defined as follows \[ Z_{f, \Sigma}^{[i]} : = \bigcup_{\sigma \in \Sigma^{[i]}} Z_{f, \sigma} = \overline{Z}_{f, \Sigma} \cap {\bf P}_{\Sigma}^{[i]}. \]} \end{opr} \medskip Let $\sigma$ be an $s$-dimensional cone in $\Sigma$. If we apply the implicit function theorem and the standard criterion of smoothness to the affine hypersurface $Z_{f,\sigma} \subset {\bf T}_{\sigma}$ contained in the open $n$-dimensional affine toric variety \[ {\bf A}_{\sigma, N} \cong ({\bf C}^*)^{n-s} \times {\bf A}_{\sigma, N(\sigma)}, \] then we obtain: \medskip \begin{theo} Small analytical neighbourhoods of points on a $(n -s -1)$-dimensional stratum $Z_{f, \sigma} \subset \overline{Z}_{f, \Sigma}$ are analytically isomorphic to products of a $(s-1)$-dimensional open ball and a small analytical neighbourhood of the point $p_{\sigma}$ on the $(n -s)$-dimensional affine toric variety ${\bf A}_{\sigma, N(\sigma)}$. \label{anal} \end{theo} For the case of $\Delta$-regular hypersurfaces in a toric variety ${\bf P}_{\Delta}$ associated with an integral polyhedron $\Delta$, one gets from \ref{def.fan} the following. \begin{coro} For any $l$-dimensional face $\Theta \subset \Delta$, small analytical neighbourhoods of points on the $(l -1)$-dimensional stratum $Z_{f, \Theta} \subset \overline{Z}_{f, \Delta}$ are analytically isomorphic to products of a $(l-1)$-dimensional open ball and a small analytical neighbourhood of the point $p_{\sigma(\Theta)}$ on the $(n -l)$-dimensional affine toric variety ${\bf A}_{\sigma(\Theta), N(\sigma(\Theta))}$. \label{anal2} \end{coro} Applying \ref{codim2,3}, we also conclude: \begin{coro} For any $\Sigma$-regular hypersurface $\overline{Z}_{f, \Sigma} \subset {\bf P}_{\Sigma}$, the open subset $Z_{f, \Sigma}^{[1]}$ consists of smooth points of $\overline{Z}_{f, \Sigma}$. Moreover, {\rm (i)} $Z_{f, \Sigma}^{[2]}$ consists of smooth points if ${\bf P}_{\Sigma}$ has only terminal singularities. {\rm (ii)} $Z_{f, \Sigma}^{[3]}$ consists of smooth points if ${\bf P}_{\Sigma}$ has only ${\bf Q}$-factorial Gorenstein terminal singularities. {\rm (iii)} $Z_{f, \Sigma}^{[n-1]} = \overline{Z}_{f, \Sigma}$ is smooth if and only if ${\bf P}_{\Sigma}^{[n-1]}$ is smooth. \label{sing.hyp} \end{coro} \bigskip \subsection{Birational and finite morphisms of hypersurfaces} \hspace*{\parindent} \begin{prop} Let $\phi\; : \Sigma' \rightarrow \Sigma$ be a subdivision of a fan $\Sigma$, \[ \tilde {\phi} \; : \; {\bf P}_{\Sigma'} \rightarrow {\bf P}_{\Sigma} \] the corresponding proper birational morphism. Then for any $\Sigma$-regular hypersurface $\overline{Z}_f \subset {\bf P}_{\Sigma}$ the hypersurface $\overline{Z}_{{\tilde {\phi}}^{*}f} \subset {\bf P}_{\Sigma'}$ is $\Sigma'$-regular. \label{subdiv.hyp} \end{prop} {\em Proof. } The statement follows from the fact that for any cone $\sigma' \in \Sigma'$ such that $\phi(\sigma') \subset \sigma \in \Sigma$, one has \[ Z_{{\tilde {\phi}}^* f, \sigma'} \cong Z_{f, \sigma} \times ({\bf C}^*)^{{\rm dim}\, \sigma - {\rm dim}\, \sigma'}. \] \bigskip One can use \ref{sing.hyp} and \ref{subdiv.hyp} in order to construct partial desingularizations of hypersurfaces $\overline{Z}_{f, \Sigma}$. \begin{prop} Let ${\bf P}_{\Sigma}$ be a projective toric variety with only Gorenstein singularities. Assume that \[ \tilde {\phi} \; : \; {\bf P}_{\Sigma'} \rightarrow {\bf P}_{\Sigma} \] is a MPCP-desingularization of ${\bf P}_{\Sigma}$. Then $\overline{Z}_{{\tilde \phi}^*f, \Sigma'}$ is a MPCP-desingularization of $\overline{Z}_{f, \Sigma}$. \label{max.crep} \end{prop} {\em Proof. } By \ref{sing1}, \ref{sing2}, \ref{anal}, $\overline{Z}_{f, \Sigma}$ has at most Gorenstein singularities and $\overline{Z}_{{\tilde \phi}^*f, \Sigma'}$ has at most ${\bf Q}$-factorial terminal singularities. It suffuces now to apply the adjunction formula. \medskip \begin{prop} Let $\phi \; : \; N' \rightarrow N$ be a surjective homomorphism of $n$-dimensional lattices, $\Sigma$ a fan in $N_{\bf Q}$, $\Sigma'$ a fan in $N'_{\bf Q}$. Assume that $\phi(\Sigma') = \Sigma$. Let \[ \tilde {\phi} \; : \; {\bf P}_{\Sigma',N'} \rightarrow {\bf P}_{\Sigma, N} \] be the corresponding finite surjective morphism of toric varieties. Then for any $\Sigma$-regular hypersurface $\overline{Z}_f$ in ${\bf P}_{\Sigma}$ the hypersurface \[ {\tilde {\phi}}^{-1} (\overline{Z}_f) = \overline{Z}_{{\tilde {\phi}}^{*}f} \] is $\Sigma'$-regular. \label{galois} \end{prop} {\em Proof. } It is sufficient to observe that for any cone $\sigma \in \Sigma \cong \Sigma'$ the affine variety ${Z}_{{\tilde {\phi}}^{*}f, \sigma}$ is an \^etale covering of $Z_{f, \sigma}$ whose Galois group is isomorphic to the cokernel of the homomorphism $N'(\sigma) \rightarrow N(\sigma)$ (see \ref{final}). \bigskip \subsection{The Hodge structure of hypersurfaces} \hspace*{\parindent} We are interested now in the calculation of the cohomology groups of $\Delta$-regular hypersurfaces $\overline{Z}_f$ in toric varieties ${\bf P}_{\Delta}$. The main difficulty in this calculation is connected with singularities of $\overline{Z}_f$. So we will try to avoid singularities and to calculate cohomology groups not only of compact compact hypersurfaces $\overline{Z}_f$, but also ones of some naturally defined smooth open subsets in $\overline{Z}_f$. For the last purpose, it is more convenient to use cohomology with compact supports which we denote by $H^i_c(*)$. \medskip First we note that there exists the following analog of the Lefschetz theorem for $\Delta$-regular hypersurfaces proved by Bernstein, Danilov and Khovanski\^i \cite{dan.hov}. \begin{theo} For any open toric subvariety $U \subset {\bf P}_{\Delta}$, the Gysin homomorphism \[ H^i_c ( \overline{Z}_f \cap U) \rightarrow H^{i+2}_c (U) \] is bijective for $i > n -1$ and injective for $i = n - 1$. \label{Lef} \end{theo} Using this theorem, one can often reduce the calculation of cohomology groups $H^i$ (or $H^i_c$) to the "interesting" case $i = n-1$. We consider below several typical examples of such a situation. \begin{exam} {\rm Let $U$ be a smooth open toric subvariety in ${\bf P}_{\Delta}$ (e.g., $U = {\bf T}$). Then $V = U \cap \overline{Z}_f$ is a smooth affine open subset in $\overline{Z}_f$. By general properties of Stein varieties, one has $H^{i}(V) = 0$ for $i > n-1$. Since the calculation of cohomology groups of smooth affine toric varieties is very simple, we obtain a complete information about all cohomology groups except for $i = n-1$ using the following property. } \end{exam} \begin{prop} Let $W$ be a quasi-smooth irreducible $k$-dimensional algebraic variety. Then there exists the Poincare pairing \[ H_c^i (W) \otimes H^{2k -i} (W) \rightarrow H^{2k}_c (W) \cong {\bf C}. \] This pairing is compatible with Hodge structures, where $H^{2k}_c(W)$ is assumed to be a $1$-dimensional ${\bf C}$-space of the Hodge type $(k,k)$. \label{duality} \end{prop} If we take $U = {\bf T}$, then $V = Z_f$. The Euler characteristic of $Z_f$ was calculated by Bernstein, Khovanski\^i and Kushnirenko \cite{kuch}, \cite{hov.genus}. \begin{theo} $ e(Z_f) = \sum_{i \geq 0} (-1)^i {\rm dim}\, H^i(Z_f) = (-1)^{n-1} d_M (\Delta)$. \label{euler} \end{theo} In particular, we obtain: \begin{coro} The dimension of $H^{n-1}(Z_f)$ is equal to $d_M(\Delta) + n -1$. \end{coro} \begin{opr} Let $P$ be a compact convex subset in $M_{\bf Q}$. Denote by $l(P)$ the number of integral points in $P \cap M$, and by $l^*(P)$ the number of integral points in the interior of $P$. \end{opr} There exist general formulas for the Hodge-Deligne numbers $h^{p,q}(Z_f)$ of the mixed Hodge structure in the $(n-1)$-th cohomology group of an arbitrary $\Delta$-regular affine hypersurface in ${\bf T}$ (Danilov and Khovanski\^i \cite{dan.hov}). For the numbers $h^{n-2,1}(Z_f)$ and $h^{n-2,0}(Z_f)$, we get the following (see \cite{dan.hov}, 5.9). \begin{prop} Let ${\rm dim}\, \Delta = n \geq 4$, then \[ h^{n-2,1} (Z_f) + h^{n-2,0} (Z_f) = l^*(2\Delta) - (n+1)l^*(\Delta), \] \[ h^{n-2,0} (Z_f) = \sum_{{\rm codim}\, \Theta =1} l^*(\Theta).\] \label{hd.dh} \end{prop} \bigskip We will use in the sequel the following properties of the numbers $h^{p,q}(H^k_c(Z_f))$ of affine hypersurfaces for cohomology with compact supports (see \cite{dan.hov}): \begin{prop} The Hodge-Deligne numbers $h^{p,q}(H^k_c(Z_f)$ of $\Delta$-regular $(n-1)$-di\-men\-si\-onal affine hypersurfaces $Z_f$ satisfy the properties: {\rm (i)} $h^{p,q}(H^k_c(Z_f) = 0$ for $p \neq q$ and $k > n-1$; {\rm (ii)} $h^{p,q}(H^k_c(Z_f)) = 0$ for $k < n-1$; {\rm (iii)} $h^{p,q}(H^{n-1}_c(Z_f) = 0$ for $p+ q > n-1$. \label{properties.hodge} \end{prop} \bigskip Although we consider the mixed Hodge structure in the cohomology group of the affine hypersurface $Z_f$, we get eventually some information about the Hodge numbers of compactifications of $Z_f$. From general properties of the mixed Hodge structures \cite{deligne}, one obtains: \begin{theo} Let $\overline{Z}$ be a smooth compactification of a smooth affine $(n-1)$-dimensional variety $Z$ such that the complementary set $\overline{Z} \setminus Z$ is a normal crossing divisor. Let \[ j\;: \; Z \hookrightarrow \overline{Z} \] be the corresponding embedding. Denote by \[ j^* \;:\; H^{n-1} (\overline{Z}) \rightarrow H^{n-1} (Z) \] the induced mapping of cohomology groups. Then the weight subspace ${\cal W}_{n-1}H^{n-1}(Z)$ coincides with the image $j^*( H^{n-1}(\overline{Z}))$. In particular, one has the following inequalities between the Hodge numbers of $\overline{Z}$ and the Hodge-Deligne numbers of $Z$: \[ h^{n-1-k,k}(\overline{Z}) \geq h^{n-1-k,k}(H^{n-1}(Z))\;\; (0 \leq k \leq n-1). \] \label{inequal.h} \end{theo} \bigskip \section{Calabi-Yau hypersurfaces in toric varieties} \medskip \subsection{Reflexive polyhedra and reflexive pairs} \hspace*{\parindent} \begin{opr} {\rm If $P$ is an arbitrary compact convex set in $M_{\bf Q}$ containing the zero vector $0 \in M_{\bf Q}$ in its interior, then we call \[ P^* = \{ y \in N_{\bf Q} \mid \langle x, y \rangle \geq -1, \; {\rm for \; all} \; x \in P \}. \] {\em the dual set} relative to $P$. } \label{dual} \end{opr} \bigskip The dual set $P^*$ is a convex compact subset in $N_{\bf Q}$ containing the vector zero $0 \in N_{\bf Q}$ in its interior. Obviously, one has $(P^*)^* = P$. \bigskip \begin{exam} {\rm Let $E$ be a Euclidian $n$-dimensional space, $\langle *, * \rangle$ the corresponding scalar product, \[ P = \{ {x} \mid \langle { x}, { x} \rangle \leq R \} \] the ball of radius $R$. Using the scalar product, we can identify the dual space $E^*$ with $E$. Then the dual set $P^*$ is the ball of radius $1/R$. } \end{exam} \begin{prop} Let $P \subset M_{\bf Q}$ be a convex set containing $0$ in its interior, $C_P \subset \overline{M}_{\bf Q}$ the convex cone supporting $P$, $ \overline{N}_{\bf Q} = {\bf Q} \oplus N_{\bf Q}$ the dual space, $C_{P^*} \subset \overline{N}_{\bf Q}$ the convex cone supporting $P^* \subset N_{\bf Q}$. Then $C_{P^*}$ is the dual cone relative to $C_P$. \label{dual.cones} \end{prop} {\em Proof. } Let $\overline{x} = (x_0, x) \in \overline{M}_{\bf Q}$, $\overline{y} = (y_0, y) \in \overline{M}_{\bf Q}$. Since $x_0$ and $y_0$ are positive, one has \[ \overline{x} \in C_P \Leftrightarrow {x}/{x_0} \in P \;\; {\rm and }\;\; \overline{y} \in C_{P^*} \Leftrightarrow {y}/{y_0} \in P^*. \] Therefore, $\langle \overline{x}, \overline{y} \rangle = x_0y_0 + \langle {x}, {y} \rangle \geq 0 $ if and only if $\langle {x}/{x_0}, {y}/{y_0} \rangle \geq -1$. \bigskip \begin{opr} {\rm Let $H$ be a rational affine hyperplane in $M_{\bf Q}$, $p \in M_{\bf Q}$ an arbitrary integral point. Assume that $H$ is affinely generated by integral points $H \cap M$, i.e., there exists a primitive integral element $l \in N$ such that for some integer $c$ \[ H = \{ x \in M_{\bf Q} \mid \langle x, l \rangle = c \}. \] Then the absolute value $\mid c - \langle p , l \rangle \mid$ is called the {\em integral distance} between $H$ and $p$. } \end{opr} \begin{opr} {\rm Let $M$ be an integral $n$-dimensional lattice in $M_{\bf Q}$, $\Delta$ a convex integral polyhedron in $M_{\bf Q}$ containing the zero $0 \in M_{\bf Q}$ in its interior. The pair $(\Delta, M)$ is called {\em reflexive} if the integral distance between $0$ and all affine hyperplanes generated by $(n -1)$-dimensional faces of $\Delta$ equals 1. If $(\Delta, M)$ is a reflexive pair, then we call $\Delta$ a {\em reflexive polyhedron}.} \label{inver.p} \end{opr} \bigskip The following simple property of reflexive polyhedra is very important. \medskip \begin{theo} Suppose that $(\Delta, M)$ is a reflexive pair. Then $(\Delta^*, N)$ is again a reflexive pair. \end{theo} {\em Proof. } Let $\Theta_1, \ldots , \Theta_k$ be $(n-1)$-dimensional faces of $\Delta$, $H_1, \ldots , H_k$ the corresponding affine hyperplanes. By \ref{inver.p}, there exist integral elements $l_1, \ldots , l_k \in N_{\bf Q}$ such that for all $1 \leq i \leq k$ \[ \Theta_i = \{ x \in \Delta \mid \langle x , l_i \rangle = 1 \},\;\; H_i = \{ x \in M_{\bf Q} \mid \langle x , l_i \rangle = 1 \}. \] Therefore, \[ \Delta = \{ x \in M_{\bf Q} \mid \langle x , l_i \rangle \leq 1 \; (1 \leq i \leq k ) \}. \] So $\Delta^*$ is a convex hull of the integral points $l_1, \ldots , l_k$, i.e., $\Delta^*$ is an integral polyhedron. Let $p_1, \ldots , p_m$ be vertices of $\Delta$. By \ref{dual}, for any $j$ ($1 \leq j \leq m$) \[ \Xi_j = \{ y \in \Delta^* \mid \langle p_j , y \rangle = 1 \} \] is a $(n-1)$-dimensional face of $\Delta^*$. Thus, $\Delta^*$ contains $0 \in N_{\bf Q}$ in its interior, and the integral distance between $0$ and every $(n-1)$-dimensional affine linear subspace generated by $(n-1)$-dimensional faces of $\Delta^*$ equals $1$. \bigskip We can establish the following one-to-one correspondence between faces of the polyhedra $\Delta$ and $\Delta^*$. \begin{prop} Let $\Theta$ be an $l$-dimensional face of an $n$-dimensional reflexive polyhedron $\Delta \subset M_{\bf Q}$, $p_1, \ldots, p_k$ are vertices of $\Theta$, $\Delta^* \in N_{\bf Q}$ the dual reflexive polyhedron. Define the dual to $\Theta$ $(n -l -1)$-dimensional face of $\Delta^*$ as \[ \Theta^* = \{ y \in \Delta^* \mid \langle p_1, y \rangle = \cdots = \langle p_k, y \rangle = - 1 \}. \] Then one gets the one-to-one correspondence $\Theta \leftrightarrow \Theta^*$ between faces of the polyhedra $\Delta$ and $\Delta^*$ reversing the incidence relation of the faces. \label{dual.edge} \end{prop} \begin{opr} {\rm A complex normal irreducible $n$-dimensional projective algebraic variety $W$ with only Gorenstein canonical singularities we call a {\em Calabi-Yau variety} if $W$ has trivial canonical bundle and \[ H^i (W, {\cal O}_W) = 0 \] for $0 < i < n$.} \label{calabi-yau} \end{opr} The next theorem describes the relationship between reflexive polyhedra and Calabi-Yau hypersurfaces. \begin{theo} Let $\Delta$ be an $n$-dimensional integral polyhedron in $M_{\bf Q}$, ${\bf P}_{\Delta}$ the corresponding $n$-dimensional projective toric variety, ${\cal F}(\Delta)$ the family of projective $\Delta$-regular hypersurfaces $\overline{Z}_f$ in ${\bf P}_{\Delta}$. Then the following conditions are equivalent {\rm (i)} the family ${\cal F}(\Delta)$ of $\Delta$-hypersurfaces in ${\bf P}_{\Delta}$ consists of Calabi-Yau varieties with canonical singularities $($see {\rm \ref{calabi-yau}}$);$ {\rm (ii)} the ample invertible sheaf ${\cal O}_{{\Delta}}(1)$ on the toric variety ${\bf P}_{\Delta}$ is anticanonical, i.e., ${\bf P}_{\Delta}$ is a toric Fano variety with Gorenstein singularities; {\rm (iii)} $\Delta$ contains only one integral point $m_0$ in its interior, and $(\Delta-m_0, M)$ is a reflexive pair. \label{equiv} \end{theo} {\em Proof. } Since $\overline{Z}_f$ is an ample Cartier divisor on ${\bf P}_{\Delta}$, (i)$\Rightarrow$(ii) follows from the adjunction formula. The equivalence (ii)$\Leftrightarrow$(iii) follows from \ref{smooth.fano}. Assume that (ii) and (iii) are satisfied. Let us prove (i). By the adjunction formula, it follows from (ii) that every hypersurface $\overline{Z}_f$ has trivial canonical divisor. By the vanishing theorem for arbitrary ample divisors on toric varieties \cite{dan1}, one gets \[ H^i(\overline{Z}_f, {\cal O}_{\overline{Z}_f}) = 0\] for $ 0 < i < n-1$. By \ref{anal2}, singularities of $\overline{Z}_f$ are analytically isomorphic to toric singularities of ${\bf P}_{\Delta}$. Since all singularities of ${\bf P}_{\Delta}$ are Gorenstein, by \ref{gor.can}, they are also canonical. So every $\Delta$-regular hypersurface satisfies \ref{calabi-yau}. \bigskip The next statement follows from definitions of the polyhedra $\Delta(\Sigma, h_K)$ and $\Delta^*(\Sigma, h_K)$. \begin{prop} Let $(\Delta,M)$ be a reflexive pair, $(\Delta^*,N)$ the dual reflexive pair, $\Sigma$ the rational polyhedral fan defining the corresponding Gorenstein toric Fano variety ${\bf P}_{\Delta}$. Then \[ \Delta(\Sigma, h_K) = \Delta, \] \[ \Delta^*(\Sigma, h_K) = \Delta^*. \] In particular, if $\Sigma^*$ is a rational polyhedral fan defining the Gorenstein toric Fano variety ${\bf P}_{\Delta^*}$, then \[ \Delta(\Sigma, h_K) = \Delta^*(\Sigma^*, h_K) \] and \[ \Delta(\Sigma^*, h_K) = \Delta^*(\Sigma, h_K). \] \end{prop} Thus, in order to construct a rational polyhedral fan $\Sigma(\Delta)$ corresponding to a reflexive polyhedron $\Delta$, one can use the following another way: we take the dual reflexive polyhedron $\Delta^*$ and apply the statement \begin{coro} Let $\Delta \subset M_{\bf Q}$ be a reflexive polyhedron, $\Delta^*$ the dual reflexive polyhedron in $N_{\bf Q}$. For every $l$-dimensional face $\Theta$ of $\Delta^*$ define the $(l+1)$-dimensional convex cone $\sigma[\Theta]$ supporting the face $\Theta$ as follows \[ \sigma[\Theta] = \{ \lambda x \in M_{\bf Q} \mid x \in \Theta,\;\; \lambda \in {\bf Q}_{\geq 0} \}. \] Then the set $\Sigma[\Delta^*]$ of all cones $\sigma[\Theta]$ where $\Theta$ runs over all faces of $\Delta^*$ is the complete fan defining the toric Fano variety ${\bf P}_{\Delta}$ associated with $\Delta$. Moreover, every $(l+1)$-dimensional cone $\sigma[\Theta]$ coincides with $\sigma(\Theta^*)$ {\rm ($see$ \ref{def.fan})}, where $\Theta^* \subset \Delta$ is the dual to $\Theta$ $(n-l-1)$-dimensional face of $\Delta$ {\rm ($see$ \ref{dual.edge})}. \end{coro} \bigskip There is the following finiteness theorem for reflexive polyhedra. \begin{theo} There exist up to an unimodular transformation of the lattice $M$ only finitely many reflexive pairs $(\Delta, M)$ of fixed dimension $n$. \end{theo} This statement follows from the finiteness theorem in \cite{bat01}, or from results in \cite{boris,hensley}. \subsection{Singularities and morphisms of Calabi-Yau hypersurfaces} \hspace*{\parindent} Let $\Delta$ be a reflexive polyhedron, $\Delta^*$ the dual reflexive polyhedron. Take a maximal projective triangulation ${\cal T}$ of $\Delta$. It follows from the proof of \ref{crep.fano} that ${\cal T}$ defines a $MPCP$-desingularization \[ \varphi_{\cal T}\; :\; {\hat{\bf P}}_{\Delta} \rightarrow {\bf P}_{\Delta} \] of the Gorenstein toric variety ${\bf P}_{\Delta}$. Let $\overline{Z}_f$ be a $\Delta$-regular Calabi-Yau hypersurface in ${\bf P}_{\Delta}$. Put \[\hat{Z}_f = \varphi^{-1}_{\cal T}(\overline{Z}_f).\] By \ref{max.crep}, \[ \varphi_{\cal T}\; : \; \hat{Z}_f \rightarrow \overline{Z}_f \] is a $MPCP$-desingularization of $\overline{Z}_f$. \begin{opr} {\rm We will call \[ \varphi_{\cal T}\; : \; \hat{Z}_f \rightarrow \overline{Z}_f \] the {\em toroidal MPCP-desingularization of $\overline{Z}_f$ corresponding to a maximal projective triangulation ${\cal T}$ of $\Delta^*$}.} \end{opr} Using \ref{crep.fano} and \ref{codim2,3}(ii), one gets the following. \begin{theo} There exist at least one toroidal MPCP-desingularization $\hat{Z}_f$ of any $\Delta$-regular Calabi-Yau hypersurface in ${\bf P}_{\Delta}$. Such a MPCP-desingularization $\hat{Z}_f$ corresponds to any maximal projective triangulation ${\cal T}$ of the dual polyhedron $\Delta^*$. The codimension of singularities of $\hat{Z}_f$ is always at least $4$. \label{smooth.c} \end{theo} \begin{coro} A toroidal MPCP-desingularization of a projective Calabi-Yau hypersurace $\overline{Z}_f$ associated with a reflexive polyhedron $\Delta$ of dimension $n \leq 4$ is always a smooth Calabi-Yau manifold. \label{smooth.c1} \end{coro} Let ${\cal T}$ be a maximal projective triangulation of $\Delta^*$. For any $l$-dimensional face $\Theta$ of $\Delta$, the restriction of ${\cal T}$ on the dual $(n - l -1)$-dimensional face $\Theta^* \subset \Delta^*$ is a maximal projective triangulation ${\cal T}\mid_{\Theta^*}$ of $\Theta^*$. By \ref{anal}, the analytical decription of singularities along a stratum $Z_{f,\Theta}$ as well as of their $MPCP$-desingularizations reduces to the combinatorial description of a $MPCP$-desingularization of the toric singularity at the unique closed ${\bf T}_{\sigma}$-invariant point $p_{\sigma}$ on the $(n-l)$-dimensional affine toric variety ${\bf A}_{\sigma, N(\sigma)}$, where $\sigma = \sigma[\Theta^*]$. So we introduce the following definition. \begin{opr} {\rm We call the face $\Theta^*$ of the polyhedron $\Delta$ the {\em diagram} of the toric singularity at $p_{\sigma} \in {\bf A}_{\sigma, N(\sigma)}$. A maximal projective triangulation ${\cal T}\mid_{\Theta^*}$ of $\Theta^*$ induced by a maximal projective triangulation ${\cal T}$ of $\Delta^*$ we call {\em a triangulated diagram} of the toric singularity at $p_{\sigma} \in {\bf A}_{\sigma, N(\sigma)}$. } \label{diagram} \end{opr} We have seen already in \ref{anal2} that for any face $\Theta \subset \Delta$ and any $\Delta$-regular Laurent polynomial $f \in L(\Delta)$ local neighbourhoods of points belonging to the same stratum $Z_{f, \Theta}$ are analitically isomorphic. Thus, if a stratum $Z_{f, \Theta}$ consists of singular points of $\overline{Z}_f$, then all these singularities are analitically isomorphic. Our purpose now is to describe singularities along $Z_{f, \Theta}$ and their $MPCP$-desingularizations in terms of triangulated diagrams. \medskip Let ${\cal T}\mid_{\Theta^*}$ be a triangulated diagram. Then we obtain a subdivision $\Sigma({\cal T},\Theta)$ of the cone $\sigma = \sigma[\Theta^*]$ into the union of subcones supporting elementary simplices of the triangulation ${\cal T}\mid_{\Theta^*}$. By \ref{biration}, one has the corresponding projective birational toric morphism \[ \varphi_{{\cal T}, \Theta^*}\; : \;{\bf P}_{\Sigma({\cal T},\Theta)} \rightarrow {\bf A}_{\sigma, N(\sigma)}. \] \begin{theo} For any $l$-dimensional face $\Theta \subset \Delta$ and any closed point $p \in Z_{f, \Theta}$, the fiber $\varphi_{\cal T}^{-1}(p)$ of a MPCP-desingularization $\varphi_{\cal T}$ is isomorphic to the fiber $\varphi_{{\cal T}, \Theta^*}(p_{\sigma})$ of the projective toric morphism $\varphi_{{\cal T}, \Theta^*}$. The number of irreducible $(n-l-1)$-dimensional components of $\varphi_{\cal T}^{-1}(p)$ equals $l^*(\Theta^*)$, i.e., the number of integral points in the interior of $\Theta^*$. Moreover, the Euler characteristic of $\varphi_{\cal T}^{-1}(p)$ equals the number of elementary simplices in the triangulated diagram ${\cal T}\mid_{\Theta^*}$. \label{topol.des} \end{theo} {\em Proof. } Since $ \varphi_{\cal T}\; :\; {\hat{\bf P}}_{\Delta} \rightarrow {\bf P}_{\Delta}$ is a birational toric morphism, the ${\bf T}$-action induces isomorphisms of fibers of $\varphi_{\cal T}$ over closed points of a ${\bf T}$-stratum ${\bf T}_{\Theta}$. Thus, we obtain isomorphisms between fibers of $\varphi_{\cal T}$ over closed points of $Z_{f, \Theta} \subset {\bf T}_{\Theta}$. By \ref{orbits.fan}(i), ${\bf T}_{\Theta}$ is contained in the ${\bf T}$-invariant open subset ${\bf A}_{\sigma,N} \cong {\bf T}_{\Theta} \times {\bf A}_{\sigma, N(\sigma)}$. Thus, we have a commutative diagram \[ \begin{tabular}{ccc} $ \varphi_{\cal T}^{-1}({\bf A}_{\sigma,N})$ & $\rightarrow$ & $ {\bf A}_{\sigma,N}$ \\ $\downarrow$ & & $\downarrow$ \\ $ {\bf P}_{\Sigma({\cal T},\Theta)} $ & $\rightarrow$ & ${\bf A}_{\sigma, N(\sigma)}$ \end{tabular} \] whose vertical maps are divisions by the action of the torus ${\bf T}_{\Theta}$. So the fiber $\varphi_{\cal T}^{-1}(p)$ over any closed point $p \in {\bf T}_{\Theta^*}$ is isomorphic to the fiber $\varphi_{{\cal T}, \Theta^*}^{-1}(p_{\sigma})$ of the projective toric morphism $\varphi_{{\cal T}, \Theta^*}$. Therefore, irreducible divisors of ${\bf P}_{\Sigma({\cal T},\Theta)}$ over $p_{\sigma}$ one-to-one correspond to integral points in the interior of $\Theta^*$. On the other hand, the toric morphism $\varphi_{{\cal T}, \Theta^*}$ has an action the $(n-l)$-dimensional torus ${\bf T}_{\Theta}' = {\rm Ker}\, \lbrack {\bf T} \rightarrow {\bf T}_{\Theta} \rbrack$. Since the closed point $p_{\sigma} \in {\bf A}_{\sigma, N(\sigma)}$ is ${\bf T}_{\Theta}'$-invariant, one has a decomposition of the fiber $\varphi_{{\cal T}, \Theta^*}^{-1}(p_{\sigma})$ into the union of ${\bf T}_{\Theta}'$-orbits. Thus the Euler characteristic of $\varphi_{{\cal T}, \Theta^*}^{-1}(p_{\sigma})$ is the number of zero-dimensional ${\bf T}_{\Theta}'$-orbits. The latter equals the number of $(n-l)$-dimensional cones of $\Sigma({\cal T},\Theta)$, i.e., the number of elementary simplices in the triangulated diagram ${\cal T}\mid_{\Theta^*}$. \medskip \begin{exam} {\rm Let $\Theta$ be an $(n-2)$-dimensional face of a reflexive polyhedron $\Delta$, and let $\Theta^*$ be the dual to $\Theta$ $1$-dimensional face of the dual polyhedron $\Delta^*$. There exists a unique maximal projective triangulation of $\Theta^*$ consisting of $d(\Theta^*)$ elementary, in fact, regular segments. In this case, small analytical neighbourhoods of points on $Z_{f, \Theta}$ are analytically isomorphic to product of $(n-3)$-dimensional open ball and a small analytical neighbourhood of $2$-dimensional double point singularity of type $A_{d(\Theta^*)-1}$. The fiber of $\varphi_{\cal T}$ over any point of $\overline{Z}_{f,\Theta}$ is the Hirzebruch-Jung tree of $l^*(\Theta^*) = d(\Theta^*) -1$ smooth rational curves having an action of ${\bf C}^*$. } \label{duval} \end{exam} \begin{opr} {\rm Let $(\Delta_1, M_1)$ and $(\Delta_2, M_2)$ be two reflexive pairs of equal dimension. {\em A finite morphism} of reflexive pair \[ \phi\; :\; (\Delta_1, M_1) \rightarrow (\Delta_2, M_2) \] is a homomorphism of lattices $\phi\,:\, M_1 \rightarrow M_2$ such that $\phi (\Delta_1) = \Delta_2$. } \label{morph} \end{opr} By \ref{final}, we obtain: \begin{prop} Let \[ \phi\; :\; (\Delta_1, M_1) \rightarrow (\Delta_2, M_2) \] be a finite morphism of reflexive pairs. Then the dual finite morphism \[ \phi^* \; : \; (\Delta_2^*, N_2) \rightarrow (\Delta_1^*, N_1). \] induces a finite surjective morphism \[ \tilde {\phi}^* \; : \; {\bf P}_{\Delta_2,M_2} \rightarrow {\bf P}_{\Delta_1,M_1}. \] Moreover, the restriction \[ \tilde {\phi}^* \; : \; {\bf P}_{\Delta_2,M_2}^{[1]} \rightarrow {\bf P}_{\Delta_1,M_1}^{[1]} \] is an \^etale morphism of degree \[ d_{M_2}(\Delta_2) / d_{M_1}(\Delta_1) = d_{N_1}(\Delta_1^*) / d_{N_2}(\Delta_2^*).\] \label{etale} \end{prop} {\em Proof. } It remains to show that $\tilde {\phi}^* \; : \; {\bf P}_{\Delta_2,M_2}^{[1]} \rightarrow {\bf P}_{\Delta_1,M_1}^{[1]}$ is \^etale. Take a $\Delta_1$-regular Calabi-Yau hypersurface $\overline{Z}_{f} \subset {\bf P}_{\Delta_1,M_1}^{[1]}$. By \ref{subdiv.hyp}, the hypersurface $\overline{Z}_{\tilde{\phi}^*f} = \tilde{\phi}^{-1}(\overline{Z}_f)$ is $\Delta_2$-regular. By \ref{sing.hyp} and \ref{equiv}, two quasi-projective varieties $Z^{[1]}_f$ and ${Z}_{\tilde{\phi}^*f}^{[1]}$ are smooth and have trivial canonical class. Therefore, any finite morphism of these varieties must be \^etale. \subsection{The Hodge number $h^{n-2,1}(\hat{Z}_f)$} \hspace*{\parindent} Let us apply the result of Danilov and Khovanski\^i (see \ref{hd.dh}) to the case of reflexive polyhedra $\Delta$ of dimension $\geq 4$. Using the properties $l^*(2\Delta) = l(\Delta)$, $l^*(\Delta) =1$, we can calculate the Hodge-Deligne number $h^{n-2,1}(Z_f)$ of an affine Calabi-Yau hypersurface $Z_f$ as follows. \begin{theo} Let $\Delta$ be a reflexive $n$-dimensional polyhedron $(n\geq 4)$, then the Hodge-Deligne number $h^{n-2,1}$ of the cohomology group $H^{n-1}(Z_f)$ of any $(n-1)$-dimensional affine $\Delta$-regular Calabi-Yau hypersurface $Z_f$ equals \[ h^{n-2,1} (Z_f) = l(\Delta) - n - 1 - \sum_{{\rm codim}\, \Theta =1 } l^*(\Theta). \] \label{n.def} \end{theo} \bigskip In fact, we can calculate the Hodge-Deligne space $H^{n-2,1}(Z_f)$ itself (not only the dimension $h^{n-2,1}$). \begin{theo} {\rm \cite{batyrev.var}} Let $L^*_1(\Delta)$ be the subspace in $L(\Delta)$ generated by all monomials $X^m$ such that $m$ is an interior integral point on a face $\Theta \subset \Delta$ of codimension $1$. Then the ${\bf C}$-space $H^{n-2,1}(Z_f)$ is canonically isomorphic to the quotient of $L(\Delta)$ by \[ L^*_1(\Delta) + {\bf C}\langle f, f_1, \ldots, f_n \rangle, \] where \[ f_i (X) = X_i \frac{\partial }{\partial X_i} f(X), \;\; (1 \leq i \leq n). \] \end{theo} We want now to calculate for $n \geq 4$ the Hodge number $h^{n-2,1}$ of a {\em MPCP}-de\-sin\-gu\-la\-ri\-za\-tion ${\hat Z}_f$ of the toroidal compactification $\overline{Z}_f$ of $Z_f$. For this purpose, it is convenient to use the notion of the $(p,q)$-{\em Euler characteristic} $e_c^{p,q}$ introduced in \cite{dan.hov}. \begin{opr} {\rm For any complex algebraic variety $V$, $e_c^{p,q}(V)$ is defined as the alternated sum of Hodge-Deligne numbers \[ \sum_{i \geq 0} (-1)^i h^{p,q}(H^i_c(V)). \] } \end{opr} \begin{prop}({\rm see \cite{dan.hov}}) Let $V = V' \times V''$ be a product of two algebraic varieties. Then one has \[ e_c^{p,q}(V) = \sum_{(p'+ p'',q' +q'') = (p,q)} e_c^{p',q'}(V') \cdot e_c^{p'',q''}(V''). \] \label{h.product} \end{prop} \begin{opr} {\rm A stratification of a compact algebraic variety $V$ is a representation of $V$ as a disjoint union of finitely many locally closed smooth subvarieties $\{ V_j \}_{j \in J}$ (which are called {\em strata}) such that for any $j \in J$ the closure of $V_j$ in $V$ is a union of the strata.} \end{opr} The following property follows immediately from long cohomology sequences (see \cite{dan.hov}). \begin{prop} Let $\{ V_j \}_{j \in J}$ be a stratification of $V$. Then \[ e_c^{p,q}(V) = \sum_{j \in J} e_c^{p,q}(V_j). \] \label{addit} \end{prop} Returning to our Calabi-Yau variety $\hat{Z}_f$, we see that $\hat{Z}_f$ is always quasi-smooth. Therefore the cohomology groups $H_c^i(\hat{Z}_f) \cong H^i(\hat{Z}_f)$ have the pure Hodge structure of weight $i$. So, by \ref{addit}, it suffices to calculate the $(n-2,1)$-Euler characteristic \[ e_c^{n-2,1}(\hat{Z}_f) = (-1)^{n-1} h^{n-2,1}(\hat{Z}_f). \] \medskip First, we define a convenient stratification of $\hat{Z}_f$. Let $\varphi_{\cal T} \,: \, \hat{Z}_f \rightarrow \overline{Z}_f$ be the corresponding birational morphism. Then $\hat{Z}_f$ can be represented as a disjoint union \[ \hat{Z}_f = \bigcup_{\Theta \subset \Delta} \varphi^{-1}_{\cal T}(Z_{f, \Theta}). \] On the other hand, all irreducible components of fibers of $\varphi_{\cal T}$ over closed points of $Z_{f, \Theta}$ are toric varieties. Therefore, we can define a stratification of $\varphi^{-1}_{\cal T}(Z_{f, \Theta})$ by smooth affine algebraic varieties which are isomorphic to products ${Z}_{f,\Theta} \times ({\bf C}^*)^k$ some nonnegative integer $k$. As a result, we obtain a stratification of $\hat{Z}_f$ by smooth affine varieties which are isomorphic to products ${Z}_{f,\Theta} \times ({\bf C}^*)^k$ for some face $\Theta \subset \Delta$ and for some nonnegative integer $k$. Second, we note that $(n-2,1)$-Euler characteristic of ${Z}_{f,\Theta} \times ({\bf C}^*)^k$ might be nonzero only in two cases: $\Theta = \Delta$, or ${\rm dim}\, \Theta = n -2$ and $k =1$. The latter follows from \ref{properties.hodge}, \ref{h.product}, and from the observation that the $(p,q)$-Euler characteristic of an algebraic torus $({\bf C}^*)^k$ is nonzero only if $p =q$ . We already know from \ref{n.def} that \[ e_c^{n-2,1}(Z_f) = (-1)^{n-1}(l(\Delta) - n - 1 - \sum_{{\rm codim}\, \Theta =1 } l^*(\Theta)). \] On the other hand, the strata which are isomorphic to ${Z}_{f,\Theta} \times {\bf C}^*$ appear from the fibers of $\varphi_{\cal T}$ over $(n-3)$-dimensional singular affine locally closed subvarieites ${Z}_{f,\Theta} \subset \overline{Z}_f$ having codimension $2$ in $\overline{Z}_f$. By \ref{duval}, a $\varphi^{-1}(Z_{f, \Theta})$ consists of $l^*(\Theta^*) = d(\Theta^*) -1$ irreducible components isomorphic to ${\bf P}_1 \times {Z}_{f,\Theta}$. As a result, for every $(n-2)$-dimensional face $\Theta \subset \Delta$, one obtains $l^*(\Theta^*)$ strata isomorphic to ${Z}_{f,\Theta} \times {\bf C}^*$. On the other hand, one has \[ e_c^{n-2,1}({Z}_{f,\Theta} \times {\bf C}^*) = e_c^{n-3,0}({Z}_{f,\Theta}) \cdot e_c^{1,1}({\bf C}^*). \] It is clear that $e_c^{1,1}({\bf C}^*) =1$. By results of Danilov and Khovanski\^i (see \cite{dan.hov}), one has \[ e_c^{n-3,0}({Z}_{f,\Theta}) = (-1)^{n-3} l^*(\Theta). \] Thus we come to the following result. \begin{theo} For $n \geq 4$, the Hodge number $h^{n-2,1}({\hat Z}_f)$ equals \[ l(\Delta) - n - 1 - \sum_{{\rm codim}\, \Theta =1 } l^*(\Theta) + \sum_{{\rm codim}\, \Theta =2 } l^*(\Theta) \cdot l^*(\Theta^*), \] where $\Theta$ denotes a face of a reflexive $n$-dimensional polyhedron $\Delta$, and $\Theta^*$ denotes the corresponding dual face of the dual reflexive polyhedron $\Delta^*$. \label{n.def.com} \end{theo} \begin{coro} Assume that ${\bf P}_{\Delta}$, or ${\bf P}_{\Delta^*}$ is a smooth toric Fano variety of dimension $n \geq 4$ (see \ref{smooth.fano}(ii)). Then the Hodge-Deligne number $h^{n-2,1}$ of an affine $\Delta$-regular hypersurface $Z_f$ coincides with the Hodge number of a MPCP-desingularization $\hat{Z}_f$ of its toroidal compactification $\overline{Z}_f$. \end{coro} {\em Proof. } If ${\bf P}_{\Delta}$, or ${\bf P}_{\Delta^*}$ is a smooth toric Fano variety, then for any face $\Theta$ of codimension $2$, one has $l^*(\Theta)=0$, or $l^*(\Theta^*)=0$. \bigskip \subsection{The Hodge number $h^{1,1}(\hat{Z}_f)$} \hspace*{\parindent} First we note that the group of principal ${\bf T}$-invariant divisors is isomorphic to the lattice $M$. Applying \ref{crep.fano}, we obtain: \begin{prop} Any MPCP-desingularization $\hat{\bf P}_{\Delta}$ of a toric Fano variety ${\bf P}_{\Delta}$ contains exactly \[ l(\Delta^*) - 1 = {\rm card}\, \{ N \cap \partial \Delta^* \} \] ${\bf T}$-invariant divisors, i.e., the Picard number $\rho(\hat{\bf P}_{\Delta}) = h^{1,1}(\hat{\bf P}_{\Delta})$ equals \[ l(\Delta^*) - n - 1. \] \label{bound.points} \end{prop} \begin{theo} Let $\hat{Z}_f$ be a MPCP-desingularization of a projective $\Delta$-regular Calabi-Yau hypersurface $\overline{Z}_f$, then for $n \geq 4$ the Hodge number $h^{1,1}(\hat{Z}_f)$, or the Picard number of $\hat{Z}_f$, equals \begin{equation} l(\Delta^*) - n - 1 - \sum_{{\rm codim}\, \Theta^* =1 } l^*(\Theta^*) + \sum_{{\rm codim}\, \Theta =2 } l^*(\Theta^*) \cdot l^*(\Theta), \label{for.pic} \end{equation} where $\Theta^*$ denotes a face of the dual to $\Delta$ reflexive $n$-dimensional polyhedron $\Delta^*$, and $\Theta$ denotes the corresponding dual to $\Theta$ face of the reflexive polyhedron $\Delta$. \label{n.pic} \end{theo} {\em Proof. } By \ref{crep.fano}, a $MPCP$-desingularization $\hat{Z}_f$ of a $\Delta$-regular Calabi-Yau hypersurface $\overline{Z}_f$ is induced by a $MPCP$-desingularization $\varphi_{\cal T}\,: \, \hat{\bf P}_{\Delta} \rightarrow {\bf P}_{\Delta}$ of the ambient toric Fano variety ${\bf P}_{\Delta}$. Since $\hat{Z}_f$ has only terminal ${\bf Q}$-factorial singularities, i.e., $\hat{Z}_f$ is quasi-smooth \cite{dan1}, the Hodge structure in cohomology groups of $\hat{Z}_f$ is pure and satisfies the Poincare duality (see \ref{duality}). Therefore, the number $h^{1,1}$ equals the Hodge number $h^{n-2,n-2}$ in the cohomology group $H^{n-2}_c (\hat{Z}_f)$ with compact supports. By the Lefschetz-type theorem (see \ref{Lef}), if $n \geq 4$, then for $i = n-3, n-4$ the Gysin homomorphisms \[ H^i_c(Z_f) \rightarrow H^{i+2}_c ({\bf T}) \] are isomorphisms of Hodge structures with the shifting the Hodge type by $(1,1)$. On the other hand, $H^{2n-1}_c ({\bf T})$ is an $n$-dimensional space having the Hodge type $(n-1,n-1)$, and the space $H^{2n-2}_c ({\bf T})$ has the Hodge type $(n-2,n-2)$. So $H^{2n-3}_c (Z_f)$ is an $n$-dimensional space having the Hodge type $(n-2,n-2)$, and the space $H^{2n-4}_c (Z_f)$ has the Hodge type $(n-3,n-3)$. The complementary set $Y = \hat{Z}_f \setminus Z_f$ is a closed subvariety of $\hat{Z}_f$ of codimension $1$. Consider the corresponding exact sequence of Hodge structures \[ \cdots \rightarrow H_c^{2n-4}(Z_f) \stackrel{\beta_1}{\rightarrow} H_c^{2n-4}(\hat{Z}_f) {\rightarrow} H_c^{2n-4}(Y) {\rightarrow} \\ H_c^{2n-3}(Z_f) \stackrel{\beta_2}{\rightarrow} H_c^{2n-3}(\hat{Z}_f) {\rightarrow} \cdots \] Comparing the Hodge types, we immediately get that $\beta_1$ and $\beta_2$ are zero mappings. Since the space $H_c^{2n-4}(Y)$ does not have subspaces of the Hodge type $(n-1, n-3)$, the Hodge humber $h^{n-2,n-2}$ of $H_c^{2n-4}(\hat{Z}_f)$ equals ${\rm dim}\, H_c^{2n-4} (\hat{Z}_f)$. Thus we get the short exact sequence of cohomology groups of the Hodge type $(n-2, n-2)$ \[ 0 \rightarrow H_c^{2n-4}(\hat{Z}_f) {\rightarrow} H_c^{2n-4}(Y) {\rightarrow} H_c^{2n-3}(Z_f) {\rightarrow} 0.\] It is easy to see that the dimension of $H_c^{2n-4}(Y)$ equals the number of the irreducible components of the $(n-2)$-dimensional complex subvariety $Y$. On the other hand, $Y$ is the intersection of $\hat{Z}_f \subset \hat{\bf P}_{\Delta}$ with the union of all irreducible ${\bf T}$-invariant divisors on the corresponding maximal partial crepant desingularization $\hat{\bf P}_{\Delta}$ of the toric Fano variety ${\bf P}_{\Delta}$. We have seen in \ref{bound.points} that these toric divisors on $\hat{\bf P}_{\Delta}$ are in the one-to-one correspondence to the integral points $\rho \in N \cap \partial \Delta^*$,i.e., we have exactly $l(\Delta^*) -1$ irreducible toric divisors on $\hat{\bf P}_{\Delta}$. Every such a divisor $D_{\rho}$ is the closure of a $(n-1)$-dimensional torus ${\bf T}_{\rho}$ whose lattice of characters consists of elements of $M$ which are orthogonal to $\rho \in N$. It is important to note that $D_{\rho} \cap \hat{Z}_f$ is the closure in $\hat{\bf P}_{\Delta}$ of the affine hypersurface $\varphi_{\cal T}^{-1}(Z_{f,\Theta} \cap {\bf T}_{\rho} \subset {\bf T}_{\rho}$ where $\varphi_{\cal T}({\bf T}_{\rho}) = {\bf T}_{\Theta}$. Note that since $\overline{Z}_f$ does not intersect any $0$-dimensional ${\bf T}$-orbit (\ref{zero.orb}), $\hat{Z}_f$ does not intersect exeptional divisors on $\hat{\bf P}_{\Delta}$ lying over these points. The exceptional divisors lying over ${\bf T}$-invariant points of ${\bf P}_{\Delta}$ correspond to integral points $\rho$ of $N$ in the interiors of $(n-1)$-dimensional faces $\Theta^*$ of $\Delta^*$. So we must consider only \[ l(\Delta^*) -1 - \sum_{{\rm codim}\, \Theta^* =1 } l^*(\Theta^*) \] integral points of $N \cap \partial \Delta$ which are contained in faces of codimension $\geq 2$. If $D_{\rho}$ is an invariant toric divisor on $\hat{\bf P}_{\Delta}$ corresponding to an integral point $\rho$ belonging to the interior of a $(n-2)$-dimensional face $\Theta^*$ of $\Delta^*$, then $D_{\rho} \cap \hat{Z}_f$ consists of $d(\Theta)$ irreducible components whose $\varphi_{\cal T}$-images are $d(\Theta)$ distict points of the zero-dimensional stratum $Z_{f, \Theta} \subset {\bf T}_{\Theta}$. If $D_{\rho}$ is an invariant toric divisor on $\hat{\bf P}_{\Delta}$ corresponding to an integral point belonging to the interior of a face $\Theta^* \subset \Delta^*$ of codimension $\geq 3$, then $D_{\rho} \cap \hat{Z}_f$ is irreducible because ${\bf T}_{\rho} \cap \hat{Z}_f$ is an irreducible affine hypersurface in ${\bf T}_{\rho}$ isomorphic to $Z_{f, \Theta} \times ({\bf C}^*)^{n-1 - {\rm dim}\,\Theta}$. Consequently, the number of irreducible components of $Y$ is \[ \sum_{{\rm codim}\, \Theta^* =2 } d(\Theta)\cdot l^*(\Theta^*) + \] \[ + {\rm number\; of\; integral\; points\; on\; faces}\; \Theta^* \subset \Delta^*, \; {\rm codim}\, \Theta^* \geq 3. \] Since $d(\Theta) = l^*(\Theta) +1$ for any $1$-dimensional face $\Theta \Delta$, we can rewrite the number of the irreducible components of $Y$ as follows \[ {\rm dim}\, H_c^{2n-4}(Y) = l(\Delta^*) - 1 - \sum_{{\rm codim}\, \Theta^* =1 } l^*(\Theta^*) + \sum_{{\rm codim}\, \Theta =2 } l^*(\Theta^*) \cdot l^*(\Theta). \] Since ${\rm dim}\, H_c^{2n-3}(Z_f) = n$, we obtain $(\ref{for.pic})$. \bigskip Applying \ref{n.def.com} and \ref{n.pic}, we conclude. \begin{theo} For any reflexive polyhedron $\Delta$ of dimension $n \geq 4$, the Hodge number $h^{n-1,1}(\hat{Z}_f)$ of a MPCP-desingularization of a $\Delta$-regular Calabi-Yau hypersurface $\overline{Z}_f \subset {\bf P}_{\Delta}$ equals the Picard number $h^{1,1}(\hat{Z}_g)$ of a MPCP-desingularization of a $\Delta^*$-regular projective Calabi-Yau hypersurface $\overline{Z}_g \subset {\bf P}_{\Delta^*}$ corresponding to the dual reflexive polyhedron $\Delta^*$. \label{num.mir} \end{theo} \subsection{Calabi-Yau $3$-folds} \hspace*{\parindent} We begin with the remark that only if ${\rm dim}\, \Delta =4$ both statements \ref{smooth.c1} and \ref{num.mir} hold. In this case, we deal with $3$-dimensional Calabi-Yau hypersurfaces $\overline{Z}_f \subset {\bf P}_{\Delta}$ which admit {\em smooth} $MPCP$-desingularizations $\hat{Z}_f$. Calabi-Yau $3$-folds $\hat{Z}_f$ are of primary interest in theoretical physics. The number \[ \frac{1}{2} e(\hat{Z}_f) = ( h^{1,1}(\hat{Z}_f) - h^{2,1}(\hat{Z}_f)) \] is called {\em the number of generations} in superstring theory \cite{witt}. So it is important to have a simple formula for the Euler characteristic $e(\hat{Z}_f)$. We have already calculated the Hodge numbers $h^{1,1}(\hat{Z}_f)$ and $h^{n-2,1}({Z}_f)$. So we obtain \begin{coro} For any Calabi-Yau $3$-fold $\hat{Z}_f$ defined by a $\Delta$-regular Laurent polynomial $f$ whose Newton polyhedron is a reflexive $4$-dimensional polyhedron $\Delta$, one has the following formulas for the Hodge numbers \[ h^{1,1}(\hat{Z}_f) = l(\Delta^*) - 5 - \sum_{{\rm codim}\, \Theta^* =1 } l^*(\Theta^*) + \sum_{{\rm codim}\, \Theta =2 } l^*(\Theta^*) \cdot l^*(\Theta), \] \[ _+ h^{2,1} (\hat{Z}_f) = l(\Delta) - 5 - \sum_{{\rm codim}\, \Theta =1 } l^*(\Theta) + \sum_{{\rm codim}\, \Theta =2 } l^*(\Theta) \cdot l^*(\Theta^*). \] \label{euler.iso} \end{coro} This implies also. \begin{coro} \[ e(\hat{Z}_f) = (l(\Delta) - l(\Delta^*)) - \; \left( \sum_{ \begin{array}{c} {\scriptstyle {\rm codim}\, \Theta =1} \\ {\scriptstyle \Theta \subset \Delta} \end{array}} l^*(\Theta) - \sum_{\begin{array}{c} {\scriptstyle {\rm codim}\, \Xi =1} \\ {\scriptstyle \Xi \subset \Delta^* } \end{array}} l^*(\Xi)\;\; \right) + \] \[ + \left( \sum_{\begin{array}{c} {\scriptstyle {\rm codim}\, \Theta =2} \\ {\scriptstyle \Theta \subset \Delta} \end{array}} l^*(\Theta) \cdot l^*(\Theta^*) - \sum_{\begin{array}{c}{\scriptstyle {\rm codim}\, \Xi =2} \\ {\scriptstyle \Xi \subset \Delta^*} \end{array}} l^*(\Xi) \cdot l^*(\Xi^*)\; \right). \] \end{coro} \medskip Now we prove a more simple another formula for the Euler characteristic of Calabi-Yau $3$-folds. \begin{theo} Let $\hat{Z}_f$ be a MPCP-desingularization of a $3$-dimensional $\Delta$-regular Calabi-Yau hypersurface associated with a $4$-dimensional reflexive polyhedron $\Delta$. Then \[ e(\hat{Z}_f) = \sum_{\begin{array}{c}{\scriptstyle {\rm dim}\, \Theta =1} \\ {\scriptstyle \Theta \subset \Delta} \end{array}} d(\Theta)d(\Theta^*) - \sum_{\begin{array}{c}{\scriptstyle {\rm dim}\, \Theta =2} \\ {\scriptstyle \Theta \subset \Delta} \end{array}} d(\Theta)d(\Theta^*). \] \label{new.euler} \end{theo} In our proof of theorem \ref{new.euler} we will use one general property of smooth quasi-projective open subsets ${Z}_f^{[1]}$ in $\overline{Z}_f$ consisting of the union of the affine part ${Z}_f$ and all affine strata $Z_{f,\Theta}$, where $\Theta$ runs over all faces of $\Delta$ of codimension 1, i.e., \[{Z}^{[1]}_f = {\bf P}_{\Delta}^{[1]} \cap {\overline{Z}}_f . \] \begin{theo} For arbitrary $n$-dimensional reflexive polyhedron $\Delta$ and $\Delta$-regular Laurent polynomial $f \in L(\Delta)$, the Euler characteristic of the smooth quasi-projective Calabi-Yau variety ${Z}_f^{[1]}$ is always zero. \label{euler.zero} \end{theo} {\em Proof. } Since ${Z}_f^{[1]}$ is smooth, the Euler characteristic of the usual cohomology groups $H^* ({Z}_f^{[1]})$ is zero if and only if the Euler characteristic of the cohomology groups with compact supports $H_c^* ({Z}_f^{[1]})$ is zero (see \ref{duality}). It follows from the long exact sequence of cohomology groups with compact suport that \[ e(H_c^* ({Z}_f^{[1]})) = e(H_c^* ({Z}_f)) + \sum_{{\rm codim \, \Theta} = 1} e(H_c^* ({Z}_{f,\Theta})). \] By \ref{euler}, \[ e(H_c^* ({ Z}_f)) = (-1)^{n-1} d(\Delta), \;\; e(H_c^* ({ Z}_{f,\Theta})) = (-1)^{n-2} d(\Theta). \] Thus, it is sufficient to prove \[ d(\Delta) = \sum_{{\rm codim}\, \Theta =1} d(\Theta). \] The latter follows immediately from the representation of the $n$-dimensional polyhedron $\Delta$ as a union of $n$-dimensional pyramids with vertex $0$ over all $(n-1)$-dimensional faces $\Theta \subset \Delta$. \bigskip {\bf Proof of Theorem \ref{new.euler}}. Let $\varphi_{\cal T}\;:\; \hat{Z}_f \rightarrow \overline{Z}_f$ be a $MPCP$-desingularization. For any face $\Theta \subset \Delta$, let $F_{\Theta}$ denotes $\varphi^{-1}_{\cal T}(Z_{f,\Theta})$. We know that $\varphi$ is an isomorphism over $Z^{[1]}_f$ which is the union of the strata $Z_{f, \Theta}$ (${\rm dim}\, \Theta = 3,4$). By \ref{euler.zero}, $e(Z^{[1]}_f) =0$. Using additivity property of the Euler characteristic, we obtain \[ e(\hat{Z}_f) = \sum_{\begin{array}{c}{\scriptstyle {\rm dim}\, \Theta =1} \\ {\scriptstyle \Theta \subset \Delta} \end{array}} e(F_{\Theta}) + \sum_{\begin{array}{c}{\scriptstyle {\rm dim}\, \Theta =2} \\ {\scriptstyle \Theta \subset \Delta} \end{array}} e(F_{\Theta}). \] Now the statement follows from the following lemma. \begin{lem} Let $\varphi_{\cal T}$ be a MPCP-desingularization as above. Then the Euler characteristic $e(F_{\Theta})$ equals $d(\Theta)d(\Theta^*)$ if ${\rm dim}\, \Theta =1$, and $-d(\Theta)d(\Theta^*)$ if ${\rm dim}\, \Theta =2$. \end{lem} {\em Proof. } If ${\rm dim}\, \Theta =1$, then $Z_{f,\Theta}$ consists of $d(\Theta)$ distinct points. The fiber $\varphi^{-1}_{\cal T}(p)$ over any such a point $p \in Z_{f,\Theta}$ is a union of smooth toric surfaces. By \ref{topol.des}, the Euler characteristic of $\varphi^{-1}_{\cal T}(p)$ equals the number of elementary simplices in the corresponding maximal projective triangulation of $\Theta^*$. Since ${\rm dim}\, \Theta^* =2$, the number of elementary simplices equals $d(\Theta^*)$ (see \ref{el.reg}). Thus, $e(F_{\Theta}) = e(Z_{f,\Theta})e(\varphi^{-1}_{\cal T}(p)) = d(\Theta)d(\Theta^*)$. If ${\rm dim}\, \Theta =2$, then $Z_{f,\Theta}$ is a smooth affine algebraic curve. By \ref{euler}, the Euler characteristic of $Z_{f,\Theta}$ equals $-d(\Theta)$. By \ref{duval}, the fiber $\varphi^{-1}_{\cal T}(p)$ over any point $p \in Z_{f,\Theta}$ is the Hirzebruch-Jung tree of $d(\Theta^*)-1$ smooth rational curves, i.e., the Euler characteristic of $\varphi^{-1}_{\cal T}(p)$ again equals $d(\Theta^*)$. Thus, $e(F_{\Theta}) = -d(Z_{f,\Theta})e(\varphi^{-1}_{\cal T}(p)) = -d(\Theta)d(\Theta^*)$. \medskip Since the duality between $\Delta$ and $\Delta^*$ establishes a one-to-one correspondence between $1$-dimensional (respectively $2$-dimensional) faces of $\Delta$ and $2$-dimensional (respectively $1$-dimensional) faces of $\Delta^*$, we again obtain: \begin{coro} Let $\Delta$ be a reflexive polyhedron of dimension $4$, $\Delta^*$ the dual reflexive polyhedron. Let $\hat{Z}_f$ be a MPCP-desingularization of a $\Delta$-regular Calabi-Yau hypersurface $\overline{Z}_f$ in ${\bf P}_{\Delta}$, $\hat{Z}_g$ be a MPCP-desingularization of a $\Delta^*$-regular Calabi-Yau hypersurface $\overline{Z}_g$ in ${\bf P}_{\Delta^*}$. Then the Euler characteristics of two Calabi-Yau $3$-folds $\hat{Z}_f$ and $\hat{Z}_g$ satisfy the following relation \[ e(\hat{Z}_f) = - e(\hat{Z}_g). \] \label{coro.euler} \end{coro} \medskip \section{Mirror symmetry} \subsection{Mirror candidates for hypersurfaces of degree $n+1$ in ${\bf P}_n$} \hspace*{\parindent} Consider the polyhedron $\Delta_n$ in $M_{\bf Q} \cong {\bf Q}^n$ defined by inequalities \[ x_1 + \ldots + x_n \leq 1, \; x_i \geq -1 \; (1 \leq i \leq n). \] Then $\Delta_n$ is a reflexive polyhedron, and ${\cal F}(\Delta)$ is the family of all hypersurfaces of degree $n+1$ in ${\bf P}_n = {\bf P}_{\Delta_n}$. The polyhedron $\Delta_n$ has $n+1$ $(n-1)$-dimensional faces whose interiors contain exactly $n$ integral points. These $n(n+1)$ integral points form the root system of type $A_n$. The number $l(\Delta_n)$ equals ${ 2n + 1 \choose n }$. Thus the dimension of the moduli space of ${\cal F}({\Delta}_n)$ equals \[ { 2n +1 \choose n } - (n+1)^2.\] The dual polyhedron $\Delta^*_n$ has $n +1$ vertices \[ u_1 =(1,0, \ldots, 0), \ldots , u_n = (0, \ldots, 0, 1), u_{n+1} = (-1, \ldots , -1). \] The corresponding toric Fano variety ${\bf P}_{\Delta_n^*}$ is a singular toric hypersurface ${\bf H}_{n+1}$ of degree $n+1$ in ${\bf P}_{n+1}$ defined by the equation \[ \prod_{i =1}^{n+1} u_i = u_0^{n+1}, \] where $(u_0: \ldots : u_{n+1})$ are homogeneous coordinates in ${\bf P}_{n+1}$. Since the simplex $\Delta_n$ is $(n+1)$-times multiple of $n$-dimensional elementary simplex of degree 1, the degree $d(\Delta_n)$ equals $(n+1)^n$. On the other hand, the dual simplex $\Delta_n^*$ is the union of $n+1$ elementary simplices of degree 1, i.e., $d(\Delta_n^*) = n+1$. There exists a finite morphism of degree $(n+1)^{n-1} = d(\Delta_n)/ d(\Delta_n^*)$ of reflexive pairs \[ \phi \; :\; (\Delta^*_n, N) \rightarrow (\Delta_n, M), \] where $\phi(u_{n+1}) = (-1, \ldots , -1) \in \Delta_n$ and \[ \phi(u_i) = (-1, \ldots, \underbrace{n}_{i}, \ldots, -1 ) \in \Delta_n. \] It is easy to see now that \[ M / \phi(N) \cong ({\bf Z}/(n+1){\bf Z})^{n-1}. \] Let $(v_0 : v_1: \ldots : v_n)$ be the homogeneous coordinates on ${\bf P}_n$. The corresponding to $\phi$ \^etale mapping of smooth quasi-projective toric Fano varieties \[ \tilde {\phi} \;:\; {\bf P}_n^{[1]} \rightarrow {\bf H}_{n+1}^{[1]} \] has the following representation in homogeneous coordinates \[ (v_0 : v_1 : \dots : v_n ) \mapsto (\prod_{i= 0}^n v_i : v_0^{n+1} : v_1^{n+1} : \dots : v_n^{n+1} ) = (u_0 : u_1: \ldots : u_{n+1} ).\] A Calabi-Yau hypersurface $\overline{Z}_f$ in ${\bf H}_{n+1}$ has an equation \[ f(u) = \sum_{i=0}^{n+1} a_i u_i = 0. \] Using \ref{n.def}, it is easy to show that $h^{n-2,1}(Z_f) = 1$. We can also describe the moduli space of the family ${\cal F}(\Delta_n^*)$. Since ${\bf H}_{n+1}$ is invariant under the action of the $n$-dimensional torus \[ {\bf T} = \{ {\bf t}=(t_1, \ldots, t_{n+1}) \in ({\bf C}^*)^{n+1} \mid t_1 \cdots t_{n+1} = 1 \}, \] the equation \[ {\bf t}^*(f(u)) = \sum_{i=0}^{n+1} a_i t_i u_i = 0, \] defines an isomorphic to $\overline{Z}_f$ hypersurface $\overline{Z}_{t^*(f)}$. Moreover, multiplying all coefficients $\{a_i\}\; (0 \leq i \leq n+1)$ by the same non-zero complex number $t_0$, we get also a ${\bf C}^*$-action. Thus, up to the action of the $(n+1)$-dimensional torus ${\bf C}^* \times {\bf T}$ on $n+2$ coefficients $\{ a_i\}$, we get the one-parameter mirror family of Calabi-Yau hypersurfaces in ${\bf H}_{n+1}$ defined by the equation \begin{equation} f_{a}(u) = \sum_{i =1}^d u_i + a u_0 = 0, \label{one.mir} \end{equation} where the number \[ a = a_0 (\prod_{j =1}^{n+1} a_j)^{\frac{-1}{n+1}} \] is uniquely defined up to an $(n +1)$-th root of unity. Using the homogeneous coordinates $\{ v_i \}$ on ${\bf P}_n$, we can transform this equation to the form \begin{equation} {\tilde {\phi}}^*f_{a} = \sum_{i =0}^n v_i^{n+1} + a \prod_{i=0}^n v_i, \label{equ.mir} \end{equation} where $a^{n+1}$ is a canonical parameter of the corresponding subfamily in ${\cal F}(\Delta_n)$ of smooth $(n-1)$-dimensional hypersurfaces in ${\bf P}_n$. \bigskip \begin{theo} The candidate for mirrors relative to the family of all smooth Calabi-Yau hypersurfaces of degree $n+1$ in ${\bf P}_n$ is the one-parameter family ${\cal F}(\Delta^*_n)$ of Calabi-Yau varieties consisting of quotients by the action of the finite abelian group $({\bf Z}/ (n+1){\bf Z})^{n-1}$ of the hypersurfaces defined by the equation $(\ref{equ.mir})$. \label{first.ex} \end{theo} \begin{exam} {\rm If we take $n =4$, then the corresponding finite abelian group is isomorphic to $({\bf Z}/5{\bf Z})^3$ and we come to the mirror family for the family of all $3$-dimensional quintics in ${\bf P}_4$ considered in \cite{cand2}.} \end{exam} \bigskip \subsection{A category of reflexive pairs} \hspace*{\parindent} The set of all reflexive pairs of dimension $n$ forms a category ${\cal C}_n$ whose morphisms are finite morphisms of reflexive pairs (\ref{morph}). The correspondence between dual reflexive pairs defines an involutive functor \[ {\rm Mir}\; : \; {\cal C}_n \rightarrow {\cal C}_n^* \] \[ {\rm Mir}(\Delta, M) = (\Delta^*, N) \] which is an isomorphism of the category ${\cal C}_n$ with the dual category ${\cal C}_n^*$. \bigskip It is natural to describe in ${\cal C}_n$ some morphisms satisfying universal properties. \begin{opr} {\rm Let \[ \phi_0 \; :\; (\Delta_0 , M_0) \rightarrow (\Delta, M) \] be a finite morphism of reflexive pair. The morphism $\phi_0$ is said to be {\em minimal} if for any finite morphism \[ \psi \; :\; (\Delta' , M') \rightarrow (\Delta, M) \] there exists the unique morphism \[ \phi \; :\; (\Delta_0 , M_0) \rightarrow (\Delta', M') \] such that $\phi_0 = \psi \circ \phi$. A reflexive pair in ${\cal C}_n$ is called {\em a minimal reflexive pair}, if the identity morphism of this pair is minimal.} \end{opr} \bigskip Note the following simple property. \begin{prop} Assume that there exist two finite morphisms \[ \phi_1 \; :\; (\Delta_1, M_1) \rightarrow (\Delta_2, M_2) \] and \[ \phi_2 \; :\; (\Delta_2, M_2) \rightarrow (\Delta_1, M_1). \] Then $\phi_1$ and $\phi_2$ are isomorphisms. \label{isom} \end{prop} {\em Proof. } The degrees of $\phi_1$ and $\phi_2$ are positive integers. On the other hand, by \ref{etale}, \[ d(\phi_1) = d_{M_2}(\Delta_2)/ d_{M_1}(\Delta_1), \] \[ d(\phi_2) = d_{M_1}(\Delta_1)/ d_{M_2}(\Delta_2). \] Therefore, $d_{M_1}(\Delta_1) = d_{M_2}(\Delta_2) =1$, i.e., $\phi_1$ and $\phi_2$ are isomorphisms of reflexive pairs. \bigskip \begin{coro} Assume that \[ \phi_0 \; :\; (\Delta_0 , M_0) \rightarrow (\Delta, M) \] is a minimal morphism. Then $(\Delta_0, M_0)$ is a minimal reflexive pair. \end{coro} \begin{coro} For any reflexive pair $(\Delta, M)$ there exists up to an isomorphism at most one minimal pair $(\Delta_0, M_0)$ with a minimal morphism \[ \phi_0 \; : \; (\Delta_0 , M_0) \rightarrow (\Delta, M). \] \end{coro} \bigskip The next proposition completely describes minimal reflexive pairs and the set of all finite morphisms to a fixed reflexive pair $(\Delta, M)$. \medskip \begin{prop} Let $(\Delta, M)$ be a reflexive pair. Denote by $M_{\Delta}$ the sublattice in $M$ generated by vertices of $\Delta$. Let $M'$ be an integral lattice satisfying the condition $M_{\Delta} \subset M' \subset M$. Then $(\Delta, M')$ is also a reflexive pair, and \[ (\Delta, M_{\Delta}) \rightarrow (\Delta, M' ) \] is a minimal morphism. \label{min.mor} \end{prop} The proof immediately follows from the definition of reflexive pair \ref{inver.p}. \medskip \begin{coro} All reflexive pairs having a finite morphism to a fixed reflexive pair $(\Delta, M)$ are isomorphic to $(\Delta, M')$ for some lattice $M'$ such that $M_{\Delta} \subset M' \subset M$. \label{min.pair} \end{coro} \bigskip \begin{opr} {\rm Let \[ \phi^0 \; :\; (\Delta , M) \rightarrow (\Delta^0, M^0) \] be a finite morphism of reflexive pair. The morphism $\phi^0$ is said to be {\em maximal} if for any finite morphism \[ \psi \; :\; (\Delta , M) \rightarrow (\Delta', M') \] there exists the unique morphism \[ \phi \; :\; (\Delta' , M') \rightarrow (\Delta^0, M^0) \] such that $\phi^0 = \phi \circ \psi$. A reflexive pair in ${\cal C}_n$ is called {\em a maximal reflexive pair}, if the identity morphism of this pair is maximal.} \end{opr} \bigskip If we apply the functor ${\rm Mir}$, we get from \ref{min.mor} and \ref{min.pair} the following properties of maximal reflexive pairs. \begin{prop} Let $(\Delta, M)$ be a reflexive pair. Then there exists up to an isomorphism the unique maximal reflexive pair $(\Delta, M^{\Delta})$ having a maximal morphism \[ \phi \; :\; (\Delta , M) \rightarrow (\Delta, M^{\Delta}). \] Moreover, the pair $(\Delta, M^{\Delta}) $ is dual to the minimal pair $(\Delta^*, N_{\Delta^*})$ having the morphism \[ \phi^*\; : \; (\Delta^* , N_{\Delta^*}) \rightarrow (\Delta^*, N) \] as minimal. \end{prop} \bigskip \begin{coro} All reflexive pairs having finite morphisms from a fixed reflexive pair $(\Delta, M)$ are isomorphic to $(\Delta, M')$ for some lattice $M'$ such that $M \subset M' \subset M^{\Delta}$. \end{coro} \bigskip \begin{exam} {\rm Let $(\Delta_n, M)$ and $(\Delta^*_n, N)$ be two reflexive pairs from the previous section. Since the lattice $N$ is generated by vertices of $\Delta_n^*$, the reflexive pair $(\Delta_n^*, N)$ is minimal. Therefore, $(\Delta_n, M)$ is a maximal reflexive pair. } \end{exam} \bigskip \subsection{A Galois correspondence} \hspace*{\parindent} The existence of a finite morphism of reflexive pairs \[ \phi\; : \; (\Delta_1, M_1) \rightarrow (\Delta_2, M_2) \] implies the following main geometric relation between Calabi-Yau hypersurfaces in toric Fano varieties ${\bf P}_{\Delta_1, M_1}$ and ${\bf P}_{\Delta_2, M_2}$. \medskip \begin{theo} The Calabi-Yau hypersurfaces in ${\bf P}_{\Delta_1, M_1}$ are quotients of some Calabi-Yau hypersurfaces in ${\bf P}_{\Delta_2, M_2}$ by the action of the dual to $M_2/ \phi(M_1)$ finite abelian group. \label{quot} \end{theo} {\em Proof. } Consider the dual finite morphism \[ \phi^* \; : \; (\Delta_2^*, N_2) \rightarrow (\Delta_1^*, N_1). \] By \ref{etale}, $\phi^*$ induces a finite \^etale morphism \[ \tilde {\phi}^* \; : \; {\bf P}_{\Delta_2,M_2}^{[1]} \rightarrow {\bf P}_{\Delta_1,M_1}^{[1]} \] of smooth quasi-projective toric Fano varieties. This morphism is defined by the surjective homomorphism of $n$-dimensional algebraic tori \[ {\gamma}_{\phi} \; :\; {\bf T}_{\Delta_2} \rightarrow {\bf T}_{\Delta_1} \] whose kernel is dual to the cokernel of the homomorphism of the groups of characters \[ \phi \; :\; M_1 \rightarrow M_2.\] Therefore, $\tilde {\phi}^*$ is the quotient by the action of the finite abelian group $(M_2/ \phi(M_1))^* = N_1 / \phi^*(N_2)$. The pullback of the anticanonical class of ${\bf P}_{\Delta_1,M_1}^{[1]}$ is the anticanonical class of ${\bf P}_{\Delta_2,M_2}^{[1]}$. Applying \ref{galois}, we obtain that the smooth quasi-projective Calabi-Yau hypersurfaces ${\hat Z}_{f,\Delta_1}$ are \^etale quotients by $N_1 / \phi^*(N_2)$ of some smooth quasi-projective Calabi-Yau hypersurfaces ${\hat Z}_{\tilde {\phi}^*f,\Delta_2}$. \bigskip \begin{coro} The mirror mapping for families of Calabi-Yau hypersurfaces in toric varieties satisfies the following Galois correspondence: If a family ${\cal F}(\Delta_1)$ is a quotient of a family ${\cal F}(\Delta_2 )$ by a finite abelian group ${\cal A}$, then the mirror family ${\cal F}(\Delta_2^*)$ is a quotient of the mirror family ${\cal F}(\Delta_1^*)$ by the dual finite abelian group ${\cal A}^*$. \end{coro} \bigskip \begin{opr} {\rm Let $(\Delta, M)$ be a reflexive pair, $(\Delta^*, N)$ the dual reflexive pair. Denote by $N_{\Delta^*}$ the sublattice in $N$ generated by vertices of $\Delta^*$. The finite abelian group $\pi_1 (\Delta, M) = N/ N_{\Delta^*}$ is called the {\em fundamental group of the pair } $(\Delta, M)$. } \end{opr} \bigskip The fundamental group $\pi_1 (\Delta, M)$ defines a contravariant functor from the category ${\cal C}_n$ to the category of finite abelian groups with injective homomorphisms. \bigskip \begin{prop} Let $(\Delta, M)$ be a reflexive pair, ${\bf P}_{\Delta, M}$ the corresponding toric Fano variety. Then the fundamental group $\pi_1 (\Delta, M)$ is isomorphic to the algebraic $($and topological$)$ fundamental group \[ \pi_1 ({\bf P}^{[1]}_{\Delta} ). \] In particular, the reflexive pair $(\Delta, M)$ is maximal if and only if ${\bf P}^{[1]}_{\Delta}$ is simply connected. \end{prop} {\em Proof. } The statement immediately follows from the description of the fundamental group of toric varieties in \ref{fund.group}. \bigskip \begin{opr} {\rm Let $(\Delta, M)$ be a reflexive pair, $(\Delta^*, N)$ the dual reflexive pair, $(\Delta, M_{\Delta})$ and $(\Delta^*, N_{\Delta^*})$ are minimal pairs, $(\Delta, M^{\Delta})$ and $(\Delta, N^{\Delta^*})$ are maximal pairs. The quotients \[ \pi_1 (\Delta) = N^{\Delta^*} / N_{\Delta^*} \; {\rm and }\; \pi_1 (\Delta^*) = M^{\Delta} / M_{\Delta } \] is called the {\em fundamental groups} of the reflexive polyhedra $\Delta$ and $\Delta^*$ respectively. } \end{opr} \medskip It is clear, $\pi_1 (\Delta)$ and $\pi_1 (\Delta^*)$ are isomorphic dual finite abelian groups. \medskip \begin{opr} {\rm Assume that for a reflexive pair $(\Delta, M)$ there exists an isomorphism between two maximal reflexive pairs \[ \phi\; :\; (\Delta, M^{\Delta}) \rightarrow (\Delta^*, N^{\Delta^*}) .\] Then we call $\Delta$ {\em a selfdual reflexive polyhedron}. } \label{selfdual} \end{opr} \medskip If $\Delta$ is selfdual, then $\Delta$ and $\Delta^*$ must have the same combinatorial type (see \ref{dual.edge}). By \ref{quot}, we obtain. \begin{prop} Let $(\Delta, M)$ be a reflexive pair such that $\Delta$ is selfdual. Then ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$ are quotients respectively by $\pi_1(\Delta, M)$ and $\pi_1(\Delta^*, N)$ of some subfamilies in the family of Calabi-Yau hypersurfaces corresponding to two isomorphic maximal reflexive pairs $(\Delta, M^{\Delta})$ and $(\Delta, N^{\Delta^*})$. Moreover, the order of $\pi_1(\Delta)$ equals to the product of oders of $\pi_1(\Delta, M)$ and $\pi_1(\Delta^*, N)$. \end{prop} \bigskip \subsection{Reflexive simplices} \hspace*{\parindent} In this section we consider Calabi-Yau families ${\cal F}(\Delta)$, where $\Delta$ is a reflexive simplex of dimension $n$. Let $\{ p_0, \ldots, p_n \}$ be vertices of $\Delta$. There exists the unique linear relation among $\{ p_i \}$ \[ \sum_{i = 0}^n w_i p_i = 0, \] where $w_i > 0$ ($0 \leq i \leq n$) are integers and $g.c.d. (w_i ) = 1.$ \begin{opr} {\rm The coefficients $w = \{ w_0, \cdots , w_n \}$ in the above linear relations are called the {\em weights of the reflexive simplex} $\Delta$.} \end{opr} \bigskip Let $\Delta^*$ be the dual reflexive simplex, ${l}_0 , \ldots , {l}_n$ vertices of $\Delta^*$, $p_0, \dots , p_n$ vertices of $\Delta$. By definition \ref{inver.p}, we may assume that \[ \langle p_i , {l}_j \rangle = -1 \; (i \neq j ), \] i.e., that the equation $\langle x , {l}_j \rangle = -1$ defines the affine hyperplane in ${M}_{\bf Q}$ generated by the $(n-1)$-dimensional face of $\Delta$ which does not contain $p_j \in \Delta$. \begin{opr} {\rm The $(n+1)\times(n+1)$-matrix with integral coefficients \[ B(\Delta) = ( b_{ij} ) = ( \langle p_i, {l}_j \rangle ) \] is called {\em the matrix of the reflexive simplex $\Delta$}. } \end{opr} \bigskip \begin{theo} Let $\Delta$ be a reflexive $n$-dimensional simplex. Then {\rm (i)} the matrix $B(\Delta)$ is symmetric and its rank equals $n$; {\rm (ii)} the diagonal coefficients $b_{ii}$ $( 0 \leq i \leq n)$ are positive and satisfy the equation \[\sum_{i =0}^n \frac{1}{b_{ii} + 1} = 1;\] {\rm (iii)} the weights $\{ w_0, \ldots , w_n \}$ of $\Delta$ are the primitive integral solution of the linear homogeneous system with the matrix $B(\Delta)$. Moreover, \[ w_i = \frac{l.c.m. (b_{ii} +1)}{b_{ii} +1}. \] \label{ref.mat} \end{theo} {\em Proof. } The statement (i) follows from the fact that ${\rm rk}\, \{ p_i \} = {\rm rk}\, \{ l_j \} = n $. One gets (ii) by the direct computation of the determinant of $B(\Delta)$ as a function on coefficients $b_{ii}$ ($0 \leq i \leq n$). Finally, (iii) follows from (ii) by checking that \[ \{ 1/(b_{11} +1), \dots , 1/(b_{nn} +1) \} \] is a solution of the linear homogeneous system with the matrix $B(\Delta)$. \bigskip \begin{coro} The matrix $B(\Delta)$ depends only on the weights of $\Delta$. \end{coro} \bigskip Let $\Delta$ be a reflexive simplex with weights $w = \{ w_i \}$. Using the vertices $\{ l_j \}$ of the dual reflexive simplex $\Delta^*$, we can define the homomorphism \[ \iota_{\Delta} \;:\; M \rightarrow {\bf Z}^{n+1},\] where \[ \iota_{\Delta} (m) = (\langle m, {l}_0 \rangle, \ldots , \langle m, {l}_n \rangle ). \] Obviously, $\iota_{\Delta}$ is injective and the image of $\iota_{\Delta}$ is contained in the $n$-dimensional sublattice $M(w)$ in ${\bf Z}^{n+1}$ defined by the equation \[ \sum_{i =0}^n w_i x_i = 0.\] Note that the image $\iota_{\Delta}(p_i)$ is the $i$-th row of $B(\Delta)$. We denote by $\Delta(w)$ the convex hull of the points $\{ \iota_{\Delta}(p_i)\}$ in $M_{\bf Q}(w)$. \medskip \begin{theo} The pair $(\Delta(w), M(w))$ is reflexive and satisfies the following conditions$:$ {\rm (i)} the corresponding to $(\Delta(w), M(w))$ toric Fano variety ${\bf P}_{\Delta(w)}$ is the weighted projective space ${\bf P}(w_0, \cdots, w_n)$; {\rm (ii)} \[ \iota_{\Delta} \; : \; (\Delta, M) \rightarrow (\Delta(w), M(w)) \] is a finite morphism of reflexive pairs; {\rm (iii)} $(\Delta(w), M(w))$ is a maximal reflexive pair. \label{matrix} \end{theo} {\em Proof. } The reflexivity of $(\Delta(w), M(w))$ and the condition (ii) follow immediately from the definition of $\iota_{\Delta}$, since \[ \Delta(w) = \{ (x_0, \ldots, x_n) \in {\bf Q}^{n+1} \mid \sum_{i =0}^n w_i x_i =0, \; x_i \geq -1 \; (0 \leq i \leq n) \}. \] (i) The shifted by $(1, \ldots, 1)$ convex polyhedron \[ \Delta^{(1)}(w) = \Delta(w) + (1, \ldots, 1) \] is the intersection of ${\bf Q}^{n+1}_{\geq 0}$ and the affine hyperplane \[ w_0 x_0 + \cdots + w_n x_n = w_0 + \cdots + w_n = l.c.m.\{ b_{ii} +1 \} = d. \] Therefore, the integral points in $\Delta^{(1)}(w)$ can be identified with all possible monomials of degree $d$ in $n+1$ $w$-weighted independent variables. (iii) Assume that there exists a finite morphism \[ \phi\; :\; (\Delta(w), M(w)) \rightarrow (\Delta', M')\] of reflexive pairs. Obviously, $\Delta'$ must be also a simplex. Since the linear mapping $\phi$ does not change linear relations, $\Delta$ and $\Delta'$ must have the same weights $w = \{w_i \}$. Therefore, by (ii), there exists a finite morphism \[ \iota_{\Delta'} \; : \; (\Delta', M') \rightarrow (\Delta(w), M(w)). \] Therefore, by \ref{isom}, the reflexive pairs $(\Delta(w), M(w))$ and $(\Delta', M')$ are isomorphic. \bigskip Since $B(\Delta) = B(\Delta^*)$ (see \ref{ref.mat} ), we obtain: \begin{coro} Any reflexive simplex $\Delta$ is selfdual. \label{dual.simp} \end{coro} \bigskip \subsection{Quotients of Calabi-Yau hypersurfaces in weighted projective spaces} \hspace*{\parindent} Theorem \ref{matrix} implies \begin{coro} The fundamental group $\pi_1(\Delta)$ of a reflexive simplex $\Delta$ depends only on the weights $w = \{ w_i \}$. This group is dual to the quotient of the lattice $M(w) \subset {\bf Z}^{n+1}$ by the sublattice $M_B(\Delta)$ generated by rows of the matrix $B(\Delta)$. \end{coro} Now we calculate the group $M(w) / M_B(w)$ explicitly. \bigskip Consider two integral sublattices of rang $n+1$ in ${\bf Z}^{n+1}$ \[ \tilde{M} (w) = M(w) \oplus {\bf Z}\langle (1, \ldots, 1) \rangle, \] and \[ \tilde{M}_B (w) = M_B(w) \oplus {\bf Z}\langle (1, \ldots, 1) \rangle. \] Note that $\tilde{M}(w) / \tilde{M}_B(w) \cong M(w) / M_B(w)$. Let $\mu_r$ denote the group of complex $r$-th roots of unity. Put $d_i = b_{ii} +1$ ($ 0 \leq i \leq n$), $d = l.c.m.\{ d_i \}$. The sublattice $\tilde{M}(w)$ is the kernel of the surjective homomorphism \[ \gamma_w \;: \; {\bf Z}^{n+1} \rightarrow \mu_d, \] \[ \gamma_w (a_0, \ldots , a_n) = g^{w_0a_0 + \cdots + w_n a_n}, \] where $g$ is a generator of $\mu_d$. The sublattice $\tilde{M}_B(w)$ is generated by $(1, \ldots, 1)$ and \[ (d_0, 0, \ldots, 0), (0, d_1, \ldots, 0), \dots, (0, \ldots, 0, d_n). \] Therefore, $\tilde{M}_B(w)$ can be represented as the sum of the infinite cyclic group generated by $(1, \ldots, 1)$ and the kernel of the surjective homomorphism \[ \gamma \;: \; {\bf Z}^{n+1} \rightarrow \mu_{d_0} \times \mu_{d_1} \times \cdots \times \mu_{d_n} , \] \[ \gamma (a_0, \ldots , a_n) = g_0^{a_0} g_1^{a_1} \cdots g_n^{a_n}, \] where $g_i$ is a generator of $\mu_{d_i}$ ($ 0 \leq i \leq n$). The order of the element $(1, \ldots, 1)$ modulo ${\rm ker}\, \gamma$ equals $d$. Thus, \[ {\bf Z}^{n+1} / \tilde{M}_B(w) \cong (\mu_{d_0} \times \mu_{d_1} \times \cdots \times \mu_{d_n}) / \mu_d ,\] where the subgroup $\mu_d$ is generated by $g_0g_1 \cdots g_n$. Finally, we obtain. \begin{theo} The fundamental group $\pi_1(\Delta^*) = M(w)/M_B(w)$ of the reflexive simplex $\Delta^*$ with weights $w = \{ w_i \}$ is isomorphic to the kernel of the surjective homomorphism \[ \overline{\gamma}_w \; : \; (\mu_{d_0} \times \mu_{d_1} \times \cdots \times \mu_{d_n}) / \mu_d \rightarrow \mu_d, \] \[ \overline{\gamma}_w ( g_0^{a_0} g_1^{a_1} \cdots g_n^{a_n} ) = g^{w_0a_0 + \cdots + w_n a_n}. \] \label{fund1} \end{theo} By duality, we conclude. \begin{coro} The fundamental group $\pi_1(\Delta)$ of a reflexive simplex $\Delta$ with weights $w$ is isomorphic to the kernel of the surjective homomorphism \[ (\mu_{d_0} \times \mu_{d_1} \times \cdots \times \mu_{d_n}) / \mu_d \rightarrow \mu_d, \] where the homomorphism to $\mu_d$ is the product of complex numbers in $\mu_{d_0}, \mu_{d_1}, \ldots , \mu_{d_n}$, and the embedding of $\mu_d$ in $\mu_{d_0} \times \mu_{d_1} \times \cdots \times \mu_{d_n}$ is defined by \[ g \mapsto ( g^{w_0}, \ldots, g^{w_n}). \] \label{fund2} \end{coro} \begin{coro} The order of $\pi_1(\Delta)$ in the above theorem equals \[ \frac{d_0 d_1 \cdots d_n}{d^2}. \] \end{coro} \bigskip \begin{exam} {\rm Let $(d_0, d_1, \ldots , d_n )$ $(d_i > 0)$ be an integral solution of the equation \begin{equation} \label{weights.proj} \sum_{i =0}^{n} \frac{1}{d_i} = 1. \end{equation} Then the quasi-homogeneous equation \[ v_0^{d_0} + v_1^{d_1} + \cdots + v_n^{d_n} = 0\] defines a $\Delta(w)$-regular Calabi-Yau hypersurface of Fermat-type in the weighted projective space \[ {\bf P}_{\Delta(w)} = {\bf P}(w_0, \ldots, w_n) , \] where \begin{equation} w_i = \frac{l.c.m. (d_i)}{d_i}. \label{new.weights} \end{equation}} \end{exam} \begin{coro} The family ${\cal F}(\Delta(w))$ of Calabi-Yau hypersurfaces in the weighted projective space ${\bf P}_{\Delta(w)}$ consists of deformations of Fermat-type hypersurfaces. If $\Delta$ is a reflexive simplex with weights $w =\{ w_i \}$, then the corresponding family ${\cal F}(\Delta)$ in ${\bf P}_{\Delta}$ consists of quotients of some subfamily in ${\cal F}(\Delta(w))$ by the action of the finite abelian group $\pi_1( \Delta,M)$. \end{coro} If we consider a special case $n =4$, we obtain as a corollary the result of Roan in \cite{roan1}. To prove this, one should use our general result on Calabi-Yau $3$-folds constructed from $4$-dimensional reflexive polyhedra (see \ref{euler.iso}) and apply the following simple statement. \begin{prop} Let $(\Delta, M)$ be a reflexive pair such that $\Delta$ is a $4$-dimensional reflexive simplex with weights $(w_0, \ldots, w_4)$. Then the family ${\cal F}(\Delta)$ consists of quotients by $\pi_1(\Delta, M)$ of Calabi-Yau hypersurfaces in the weighted projective space ${\bf P}(w_0, \ldots, w_4)$ whose equations are invariant under the canonical diagonal action of $\pi_1(\Delta, M)$ on ${\bf P}(w_0, \ldots, w_4)$. \end{prop} \bigskip

9310

alg-geom/9310001

en

https://arxiv.org/abs/alg-geom/9310001

alg-geom/9310001

Lev Borisov

Lev Borisov

Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties

6 pages, Latex file

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.

alg-geom

\section{Introduction} \noindent Mirror Symmetry discovered by physicists for Calabi-Yau manifolds still remains a surprizing puzzle for mathematicians. Some insight on this phenomenon was received from the investigation of Mirror Symmetry for some examples of Calabi-Yau varieties which admit simple birational models embedded in toric varieties. In this context, Calabi-Yau manifolds obtained by the resolution of singularities of complete intersections in toric varieties are the most general examples. In the paper of Batyrev and van Straten \cite{bat.strat}, there was proposed a method for conjectural construction of mirror families for Calabi-Yau complete intersections in toric varieities. Unfortunately, their method fails to provide such a nice duality as it is in the case of hypersurfaces \cite{bat.dual}. The purpose of these notes is to propose a generalized duality which conjecturally gives rise to the mirror involution for complete intersections. I am pleased to thank prof. Batyrev who has edited my original notes. \section{Basic definitions and notations} \noindent Let $M$ and $N = {\rm Hom}(M, {\bf Z})$ be dual free abelian groups of rank $d$, $M_{\bf R}$ and $N_{\bf R}$ be their real scalar extensions and \[ \langle \cdot , \cdot \rangle\; :\; M_{\bf R} \times N_{\bf R} \rightarrow {\bf R} \] be the canonical pairing. For any convex polyhedron $P$ in $M_{\bf R}$ (or in $N_{\bf R})$, we denote its set of vertices by $P^0$. \begin{dfn} {\rm Let $P$ be a $d$-dimensional convex polyhedron in $M_{\bf R}$ such that $P$ contains zero point $0 \in M_{\bf R}$ in its interior. Then \[ P^* = \{ y \in N_{\bf R} \mid \langle x, y \rangle \geq -1 \} \] is called {\em polar}, or {\em dual polyhedron}. } \label{polar} \end{dfn} \begin{dfn} {\rm A convex polyhedron $P$ in $M_{\bf R}$ is called a {\em lattice polyhedron} if $P^0 \subset M \subset M_{\bf R}$. } \end{dfn} \begin{dfn} {\rm (cf. \cite{bat.dual}) Let $\Delta$ be a $d$-dimensional lattice polyhedron in $M_{\bf R}$ such that $\Delta$ contains $0$ in its interior. Then $\Delta$ is called {\em reflexive} if $\Delta^*$ is also a lattice polyhedron.} \end{dfn} \begin{dfn} {\rm Let $P$ be a $d$-dimensional convex polyhedron in $M_{\bf R}$ such that $P$ contains zero point $0 \in M_{\bf R}$ in its interior. We define the $d$-dimensional fan $\Sigma \lbrack P \rbrack$ as the union of the zero-dimensional cone $\{ 0 \}$ together with the set of all cones \[ \sigma\lbrack \theta \rbrack = \{ 0 \} \cup \{ x \in M_{\bf R} \mid \lambda x \in \theta\; \mbox{for some $\lambda \in {\bf R}_{> 0}$} \} \] supporting faces $\theta$ of $P$. } \end{dfn} Next four definitions of this section play the main role in our construction. \begin{dfn} {\rm Let $\Delta \in M_{\bf R}$ be a reflexive polyhedron. Put $E = \{ e_1, \ldots, e_n \} = \Delta^0$. A representation of $E = E_1 \cup \cdots \cup E_r$ as the union of disjoint subsets $E_1, \ldots , E_r$ is called {\em nef-partition of} $E$ if there exist integral convex $\Sigma \lbrack \Delta \rbrack $-piecewise linear functions $\varphi_1, \ldots, \varphi_r $ on $M_{\bf R}$ such that $\varphi_i(e_j) = 1$ if $ e_j \in E_i$, and $\varphi_i(e_j) = 0$ otherwise.} \end{dfn} \begin{rem} {\rm The term {\em nef-partition} is motivated by the fact that such a partition induces a representation of the anticanonical divisor $-K$ on the Gorenstein toric Fano variety ${\bf P}_{\Delta^*}$ as the sum of $r$ Cartier divisors which are nef. } \end{rem} \begin{dfn} {\rm Let $E = E_1 \cup \cdots \cup E_r$ be a nef-partition. Define $r$ convex polyhedra $\Delta_1, \ldots, \Delta_r \subset M_{\bf R}$ as \[ \Delta_i = {\rm Conv}(\{0\} \cup E_i ), \; i =1, \ldots, r. \]} \label{delta} \end{dfn} \begin{rem} {\rm From Definition \ref{delta} we immediately obtain that $\Delta_i \cap \Delta_j = \{0 \}$ if $i \neq j$ and $\Delta = {\rm Conv}( \Delta_1 \cup \cdots \cup \Delta_r )$. } \end{rem} \begin{dfn} {\rm Let $E = E_1 \cup \cdots \cup E_r$ be a nef-partition. Define $r$ convex polyhedra $\nabla_1, \ldots, \nabla_r \subset N_{\bf R}$ as \[ \nabla_i = \{ y \in N_{\bf R} \mid \langle x, y \rangle \geq - \varphi_i(x) \}, \; i =1, \ldots, r. \] } \label{nabla} \end{dfn} \begin{rem} {\rm It is obvious that $\{ 0 \} \in \nabla_1 \cap \cdots \cap \nabla_r$. By Definition \ref{polar}, one has \[ \Delta^* = \{ y \in N_{\bf R} \mid \langle x, y \rangle \geq - \varphi(x) \}, \] where $\varphi = \varphi_1 + \cdots \varphi_r$. Therefore $\nabla_1 \cup \cdots \cup \nabla_r \subset \Delta^*$. Notice that $\nabla_1, \ldots, \nabla_r$ are also lattice polyhedra. This fact follows from the following standard statement.} \label{prop.nabla} \end{rem} \begin{prop} Let $\Sigma$ be any complete fan of cones in $M_{\bf R}$, $\varphi_0$ a convex $\Sigma$-piecewise linear function on $M_{\bf R}$. Then \[ Q_0 = \{ y \in N_{\bf R} \mid \langle x, y \rangle \geq - \varphi_0(x) \} \] is a convex polyhedron whose vertices are restrictions of $\varphi_0$ on cones of maximal dimension of $\Sigma$. \label{bas.lem} \end{prop} \begin{coro} The convex functions $\varphi_1, \ldots, \varphi_r$ have form \[ \varphi_i(x) = - \min_{y \in \nabla_i} \langle x , y \rangle. \] In particular, we have \[ - \min_{ x \in \Delta_j^0,\, y \in \nabla_i^0} \langle x, y \rangle = \delta_{j\,i} \] and \[ \langle \Delta_j, \nabla_i \rangle \geq - \delta_{j\,i}. \] \label{relations} \end{coro} \begin{dfn} {\rm Define the lattice polyhedron $\nabla \in N_{\bf R}$ as \[ \nabla = {\rm Conv}(\nabla_1 \cup \cdots \cup \nabla_r). \]} \end{dfn} \begin{rem} {\rm Remark \ref{prop.nabla} shows that $\nabla \subset \Delta^*$.} \label{inclusion} \end{rem} \section{The combinatorical duality} \begin{prop} $\Delta^* = \nabla_1 + \cdots + \nabla_r$. \end{prop} {\em Proof.} The statement follows from the equality $\sum_i \varphi_i = \varphi$, from Remark \ref{prop.nabla} and Proposition \ref{bas.lem}. \hfill $\Box$ \begin{prop} $\nabla^* = \Delta_1 + \cdots + \Delta_r$. \end{prop} {\em Proof. } Let $x = x_1 + \cdots + x_r$ be a point of $\Delta_1 + \cdots + \Delta_r$ $(x_i \in \Delta_i)$, and $y = \lambda_1 y_1 + \cdots \lambda_r y_r$, $(\lambda_1 + \cdots + \lambda_r = 1,\; \lambda_i \geq 0, \; y_i \in \nabla_i)$ be a point in $\nabla$. By \ref{relations}, \[ \langle x, y \rangle \geq \sum_{i=1}^r \lambda_i \langle x_i, y_i \rangle \geq - \sum_{i=1}^r \lambda_i = -1. \] Hence $\Delta_1 + \cdots + \Delta_r \subset \nabla^*$. Let $y \in (\Delta_1 + \cdots + \Delta_r)^*$. Put \[ \lambda_i = - \min_{ x \in \Delta_i} \langle x, y \rangle. \] Since $0 \in \Delta_i$, all $\lambda_i$ are nonnegative. Since $ \langle \sum_i \Delta_i , y \rangle \geq -1$, we have $\sum_i \lambda_i \leq 1$. Consider the convex function $\varphi_y = \sum_i \lambda_i \varphi_i$. For all $x \in M_{\bf R}$, we have \[ - \varphi_y(x) = \sum_{i=1}^r \lambda_i \varphi_i(x) \leq \langle x, y \rangle . \] By Proposition \ref{bas.lem}, $y$ is contained in the convex hull of all points in $N_{\bf R}$ which are equal to restrictions of $\varphi_y$ on cones of maximal dimension of $\Sigma \lbrack \Delta \rbrack$. By definition of $\varphi_y$, any such a point is a sum $\sum_{i} \lambda_i p_i$ where $p_i \in \nabla_i$. Hence $y \in \nabla$. Thus we have proved that $(\Delta_1 + \cdots + \Delta_r)^* \subset \nabla$. \hfill $\Box$ Since $\nabla$ and $\Delta_1 + \cdots + \Delta_r$ are lattice polyhedra, we obtain: \begin{coro} The polyhedron $\nabla$ is reflexive. \end{coro} \begin{prop} Let $E'= \{e_1', \ldots , e_k' \} = \nabla^0$, $E_i' = \nabla^0_i$ $( i =1, \ldots r)$. Then subsets $E_1', \ldots E_r' \subset E'$ give rise to a nef-partition of $E'$. \end{prop} {\em Proof. } First, we prove that $\nabla_i \cap \nabla_j = \{ 0 \}$ for $i \neq j$. Assume that $e_p' \in \nabla_i \cap \nabla_j$. Using \ref{relations}, we obtain that $e_p'$ has non-negative values at all vertices $e_1, \ldots, e_n$ of $\Delta$. On the other hand, $e_p'$ has zero value at the interior point $0 \in \Delta$. Hence $e_p'$ must be zero. This means that $E_i' \cap E_j' = \emptyset$ for $i \neq j$. Let $e_p'$ be a vertex of $\nabla_i$. We prove that $e_p'$ is also a vertex of $\nabla$. By \ref{relations}, there exists a vertex $e_s \in \Delta_j^0$ such that $\langle e_s, e_p'\rangle = -1$. Moreover, \[ -1 =\min_{y \in \nabla} \langle e_s, y \rangle = \min_{y \in \nabla_i^0} \langle e_s, y \rangle. \] So $e_p'$ is also a vertex of $\nabla$. Define the functions \[ \psi_i \; : \; N_{\bf R} \rightarrow {\bf R},\; i =1, \ldots, r; \] \[ \psi_i(y) = - \min_{x \in \Delta_i} \langle x, y \rangle. \] Obviously, $\psi_1, \ldots, \psi_r$ are convex. By \ref{relations}, $\psi_i(e_p') = 1$ if $e_p' \in \nabla_i$, and $\psi_i(e_p') = 0$ otherwise. We prove that restrictions of $\psi_i$ on cones of $\Sigma \lbrack \nabla \rbrack$ are linear. It is sufficient to consider restrictions of $\psi_i$ on cones $\sigma \lbrack \theta \rbrack$ of maximal dimension where $\theta = \nabla \cap \{y \mid \langle v, y \rangle = -1\}$ is a $(d-1)$-dimensional face of $\nabla$ corresponding to a vertex $v \in \nabla^* = \Delta_1 + \cdots + \Delta_r$. Let $v = v_1 + \cdots + v_i +\cdots + v_r$, where $v_i$ denotes a vertex of $\Delta_i$. If we take another vertex $v_i' \neq v_i$ of $\Delta_i$, then the sum $v = v_1 + \cdots + v_i' +\cdots + v_r$ represents another vertex of $\nabla^*$. Clearly, $\langle v, y \rangle \leq \langle v', y \rangle$ for any $y \in \sigma\lbrack \theta \rbrack$, i.e., $\langle v_i, y \rangle \leq \langle v_i', y \rangle$. Hence the restriction of $\psi_i$ on $\sigma\lbrack \theta \rbrack$ is $- \langle v_i, y \rangle$. \hfill $\Box$ \begin{coro} \[ \Delta_i = \{ x \in m_{\bf R} \mid \langle x, y \rangle \geq - \psi_i(y) \}, \; i =1, \ldots, r. \] \end{coro} \medskip Thus we have proved that the set of reflexive polyhedra with nef-partitions has a natural involution \[ \imath\; : \; (\Delta; E_1, \ldots, E_r) \rightarrow (\nabla; E_1', \ldots, E_r' ). \] On the other hand, every nef-partition of a reflexive polyhedron $\Delta$ defines $r$ base point free linear systems of numerically effective Cartier divisors $\mid D_1 \mid , \ldots, \mid D_r \mid$ such that the sum $D_1 + \ldots + D_r$ is the anticanonical divisor on the Gorenstein toric Fano variety ${\bf P}_{\Delta^*}$. \begin{conj} The duality between nef-partitions of reflexive polyhedra $\Delta$ and $\nabla$ gives rise to pairs of mirror symmetric families of Calabi-Yau complete intersections in Gorenstein toric Fano varieties ${\bf P}_{\Delta^*}$ and ${\bf P}_{\nabla^*}$. \end{conj} \bigskip

9310

alg-geom/9310006

en

https://arxiv.org/abs/alg-geom/9310006

alg-geom/9310006

Rick Miranda

Rick Miranda and Peter F. Stiller

Torsion Sections of Elliptic Surfaces

13 pages, AMS-LaTeX

Given a torsion section of a semistable elliptic surface, we prove equidistribution results for the components of singular fibers which are hit by the section, and for the root of unity (identifying the zero component with ${\Bbb C}$) which is hit by the section in case the section hits the zero component.

alg-geom

\section{Introduction} In this article we discuss torsion sections of semistable elliptic surfaces defined over the complex field $\Bbb C$. Recall that a semistable elliptic surface is a fibration $\pi:X \rightarrow C$, where $X$ is a smooth compact surface, $C$ is a smooth curve, the general fiber of $\pi$ is a smooth curve of genus one, and all the singular fibers of $\pi$ are semistable, that is, all are of type $I_m$ in Kodaira's notation (see [K]). In addition, we assume that the fibration $\pi$ enjoys a section $S_0$; this section defines a zero for a group law in each fiber, making the general fiber an elliptic curve over $\Bbb C$. The Mordell-Weil group of $X$, denoted by $\operatorname{MW}(X)$, is the set of all sections of $\pi$, which is known to form a group under fiber-wise addition; the section $S_0$ is the identity of $\operatorname{MW}(X)$. Note that any section $S \in \operatorname{MW}(X)$ meets one and only one component of each fiber. Now given any section $S \in \operatorname{MW}(X)$, one can ask the following two questions. First, which components of the singular fibers of $X$ does $S$ meet, and second, exactly where in these components does $S$ meet them? When $S$ is a torsion section, the first question was addressed by Miranda in \cite{miranda2}. To describe those results we require some notation. Recall that a singular semistable fiber of type $I_m$ is a cycle of $m$ ${\Bbb P}^1$'s. Suppose that our elliptic fibration $\pi:X \to C$ has $s$ such singular fibers $F_1,\dots,F_s$, with $F_j$ of type $I_{m_j}$. Choose an ``orientation'' of each fiber $F_j$ and write the $m_j$ components of $F_j$ as \[ C^{(j)}_0,C^{(j)}_1,\dots,C^{(j)}_{m_j-1}. \] where the zero section $S_0$ meets only $C^{(j)}_0$ and where for each $k$, $C^{(j)}_k$ meets only $C^{(j)}_{k\pm 1} \mod m_j$. If $m_j = 1$, then $F_j = C^{(j)}_0$ is a nodal rational curve with self-intersection $0$. If $m_j \geq 2$, then each $C^{(j)}_k$ is a smooth rational curve with self-intersection $-2$. Given a section $S$ of $X$, and orientations of all of the singular fibers $F_j$, inducing a labeling of the components as above, we define ``component numbers'' $k_j(S)$ to be the index of the component in the $j^{th}$ fiber $F_j$ which a given section $S$ meets. That is, \[ S \text{ meets } C^{(j)}_{k_j(S)} \text{ in the fiber } F_j. \] This assigns to each section $S$, an $s$-tuple $(k_1(S),\dots,k_s(S))$. The component number $k_j$ can be taken to be defined $\mod m_j$ once the orientation of $F_j$ is chosen; if the orientation is unknown, then $k_j$ is defined $\mod m_j$ only up to sign. Note that $k_j(S_0) = 0$ for every $j$; indeed, after choosing orientations, the assignment of component numbers can be considered a group homomorphism from $\operatorname{MW}(X)$ to $\bigoplus\limits_j \Bbb Z / m_j\Bbb Z$. Now suppose that $S$ is a torsion section of prime order $p$. In this case, since $p\cdot S = S_0$, we must have $p k_j(S) \equiv 0 \mod m_j$ for every $j$. If $p$ does not divide $m_j$, then this forces $k_j(S) = 0$. However if $m_j = p n_j$, then $k_j(S)$ can a priori be any one of the numbers $i n_j$, for $i = 0,\dots, p-1$. This multiple $i$ measures, in some sense, how far around the cycle $S$ is from the zero-section $S_0$ in the $j^{th}$ fiber. Of course, changing the orientation in the fiber $F_j$ will have the effect of changing this multiple $i$ to $p-i$ (if $i \neq 0$; $i=0$ remains unchanged). We are therefore led to a definition of the following quantities. Let $M_i(S)$ denote the number of singular fibers where $k_j(S) = i n_j$ or $k_j(S) = (p-i) n_j$ (weighted by the number $m_j$ of components in the fiber $F_j$), and then divided by the total number $\sum_j m_j$ of components: \[ M_i(S) = { \sum\limits_{j \text{ with } k_j(S) = i n_j \text{ or } (p-i)n_j} m_j \over \sum\limits_j m_j }. \] We may view $M_i$ as a probability, since it is the fraction of the fibers where $S$ meets ``distance $i$'' away from the zero section. The main result of \cite{miranda2} is the following: \begin{theorem} If $S$ is a torsion section of odd prime order $p$, then $M_i(S) = 2p/(p^2-1)$ if $i \neq 0$, and $M_0(S) = 1/(p+1)$. \end{theorem} Notice that, firstly, these fractions are independent of $S$, and secondly, that they are independent of $i$ (for $i$ non-zero). Thus we obtain an ``equidistribution'' property for these component numbers. The proof of the above theorem given in \cite{miranda2} used only basic facts about elliptic surfaces and some intersection theory. In this paper we will address the second of the two questions raised above in a similar spirit. Again let $S$ be a torsion section of order $p$, and consider those fibers where $k_j(S) = 0$, i.e., where $S$ and $S_0$ meet the same component. By the above result, this happens exactly $1/(p+1)$ of the time, where each fiber is weighted by the number $m_j$ of components it has. Each identity component $C_0^{(j)}$ may be naturally identified as a group with $\Bbb C^*$, by sending the two points of intersection with the neighboring components to $0$ and $\infty$, and the point where $S_0$ meets $C_0^{(j)}$ to $1$. (This identification can be made in exactly two ways, corresponding to which node is sent to $0$ and which to $\infty$.) Given such a coordinate choice on $C_0^{(j)}$, the section $S$ of order $p$ will hit $C_0^{(j)}$ at a point whose coordinate is a $p^{th}$ root of unity. Let us denote by $\ell_j(S)$ that integer in $[1,p-1]$ such that $S \cap C_0^{(j)}$ has coordinate\break $\exp(2\pi i \ell_j(S)/p)$. (Since torsion sections can never meet, $\ell_j(S)$ cannot equal $0$.) Note that $\ell_j(S)$ is defined only when $k_j(S)=0$ and may be thought of as being defined modulo $p$; in addition, if we switch the roles of $0$ and $\infty$, we see that $\ell_j(S)$ is replaced by $p - \ell_j(S)$. Thus, combinatorially, we are in a situation identical to the one for the component numbers $k_j$. We will call these numbers $\ell_j(S)$ {\em root of unity numbers} for the section $S$. Now define numbers $R_i(S)$ to be the total number of fibers having $k_j(S)=0$ and having $\ell_j(S) = \pm i \mod p$ (weighted by the number of components $m_j$), and then divided by the total (weighted) number of fibers with $k_j(S)=0$: \[ R_i(S) = { \sum\limits_{j \text{ with } k_j(S)=0 \text{ and } \ell_j(S) = \pm i} m_j \over \sum\limits_{j \text{ with } k_j(S)=0} m_j }. \] The main theorem of this article is an equidistribution property for these fractions $R_i(S)$: namely, if $p$ is odd, $R_i(S) = 2/(p-1)$, independent of $S$ and $i$. These results rely on computations on elliptic modular surfaces. We also give as a by-product the equidistribution for the component numbers $k_j(S)$, both by an explicit computation and via a relationship between the component numbers and the root of unity numbers. Finally we develop an Abel theorem for a singular semistable elliptic curve and use it to compute a Weil pairing on the singular semistable fibers of an elliptic surface. Using this Weil pairing, we may also discover the duality between the component numbers and root of unity numbers. \section{Equidistribution for the roots of unity via the universal property} In this section we will give a proof of the following theorem. \begin{theorem} \label{main_theorem} Fix a prime number $p$, and let $S$ be a section of a smooth semistable elliptic surface which is torsion of order $p$. Then \[ R_1(S) = 1 \text{ if }p=2, \text{ and } R_i(S) = 2/(p-1) \text{ if } p \text{ is odd.} \] \end{theorem} \begin{pf} First note that if $p=2$, then the only possible value for $\ell_j$ is one, so that certainly $R_1(S) = 1$. Similarly, if $p=3$, the only values for $\ell_j$ are $1$ or $2$, which are inverse mod $3$; therefore $R_1(S) = 1$ again. Thus we may assume that $p$ is an odd prime at least five. Second, note that it is immediate to check that if $S$ is a section of $\pi:X \to C$, and if $F:D \to C$ is an onto map of smooth curves, then $S$ induces a section $S'$ on the pull-back surface $\pi':X' \to D$, which is also torsion of order $p$. Moreover, it is easy to see that $R_i(S') = R_i(S)$ for every $i$. If $p \geq 5$, then we may consider the elliptic modular surface $Y_1(p)$ over the modular curve $X_1(p)$, which is defined using the congruence subgroup $\Gamma_1(p)$ of $SL(2,\Bbb Z)$ given by \[ \Gamma_1(p) = \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \;|\; a,d \equiv 1 \mod p, c \equiv 0 \mod p \}. \] Note that $X_1(p)$ is a fine moduli space for elliptic curves with a torsion section of order $p$, and that $Y_1(p) \to X_1(p)$ is the universal family (see \cite{shimura}, [Shd1] or \cite{cox-parry}). This elliptic modular surface also has a natural section $T$ of order $p$, and every elliptic surface with a section $S$ of order $p$ may be obtained via pull-back from this modular surface in such a way that $S$ is the pullback of $T$. By the above remark concerning the constancy of the fraction $R_i$ under pull-back, it suffices to prove that $R_i(T) = 2/(p-1)$, for each $i = 1,\dots, (p-1)/2$; in other words, we need only verify the statement of the theorem for the universal section $T$ of the modular surface $Y_1(p)$ over the modular curve $X_1(p)$. Now the elliptic modular surface has exactly $p-1$ singular fibers, occuring over the cusps of the modular curve $X_1(p)$, of which half are of type $I_1$ and half of type $I_p$ depending on the order of the cusp:\ the total ``weight'' of the singular fibers, which is equal to the Euler number of the modular surface, is $\sum_j m_j = (p^2-1)/2$. For the $I_1$ fibers, the component numbers $k_j(T)$ must be $0$, contributing $[(p-1)/2] / [(p^2-1)/2] = 1/(p+1)$ to the weighted fraction $M_0(T)$. However, by the results of \cite{miranda2} mentioned in the Introduction, the total weighted fraction $M_0(T)$ is $1/(p+1)$, and therefore the component number must be non-zero for all the fibers of type $I_p$. At this point, label the $I_1$ fibers as $F_1,\dots, F_{(p-1)/2}$. Choose an ``orientation'' of each $I_1$ fiber, which in essence means give an isomorphism of its smooth part with $\Bbb C^*$, such that the zero section $T_0$ meets $F_j$ in the point corresponding to the number $1$. Denote by $T_1=T,T_2=2T,\dots,T_{p-1}=(p-1)T$ the non-zero multiples of the universal section $T$. Note that $T_i$ and $T_{p-i}$ meet each $I_1$ fiber $F_j$ in inverse roots of unity; in other words, using the notation of the Introduction, $\ell_j(T_i) = - \ell_j(T_{p-i})$ mod $p$ for every $i$ and $j$. Construct a square matrix $Z$ of size $(p-1)/2$ whose ${ij}^{th}$ entry is the pair of integers $\pm \ell_j(T_i) = \{\ell_j(T_i),\ell_j(T_{p-i})\}$. Now two torsion sections of an elliptic surface never meet, so the sections $\{T_i\}$ are all disjoint. In particular, if we fix an index $j$, in the $j^{th}$ column of this matrix $Z$, we must have $p-1$ distinct integers; since the integers come from $[1,p-1]$, each integer in this range occurs exactly once in each column of $Z$. Next fix an integer $i \in [1,(p-1)/2]$. By the universality of the modular surface, there is an automorphism of the surface fixing the zero section and carrying $T_i$ to $T = T_1$. Therefore, the entries along the $i^{th}$ row must be the same as the entries along the first row, but permuted in some way (indeed, permuted as the automorphism permutes the $I_1$ fibers $F_j$). As noticed above, each integer in $[1,p-1]$ appears exactly once in the first column of $Z$. Hence each integer appears in some row of $Z$. By the above remark, each integer must then appear in the first row, and indeed in every row. This is exactly the statement that $R_i(T) = 2/(p-1)$ in this case. \end{pf} \section{Equidistribution for the roots of unity via explicit computation} In this section we will re-prove Theorem \ref{main_theorem} via an explicit computation. Fix an odd prime $p \geq 5$, and let $\Gamma = \Gamma_1(p)$ be the relevant modular group defined in the proof of the Theorem. Let $\Bbb H$ denote the upper half-plane. Form the semi-direct product $\tilde{\Gamma} = \Gamma \Bbb o \Bbb Z^2$ and recall that $\tilde{\Gamma}$ acts on $\Bbb H \times \Bbb C$ by \[ (\begin{pmatrix} a & b \\ c & d \end{pmatrix},(m,n))\cdot (\tau,w) = ({a\tau + b \over c\tau + d},{1 \over c\tau + d}(w + m\tau + n)). \] Denote the quotient $\Bbb H \times \Bbb C / \tilde{\Gamma}$ by $Y^0_1(p)$, and the quotient $\Bbb H / \Gamma$ by $X^0_1(p)$. The natural map $\pi:Y^0_1(p) \to X^0_1(p)$ induced by sending $(\tau,w)$ to $\tau$ is a smooth elliptic fibration. The curve $X^0_1(p)$ is not compact, but these quotients and the fibration $\pi$ extend to the natural compactifications $\pi:Y_1(p) \to X_1(p)$. The compact curve $X_1(p)$ is obtained by adding $p-1$ points (called cusps) to the open curve $X^0_1(p)$, over which the elliptic fibration has singular curves. These cusps correspond to an equivalence class of rational points on the boundary of the upper half-plane and the vertical point at infinity, $i \infty$. For this modular group $\Gamma_1(p)$, representatives for the cusps can be taken to be the rational numbers \[ {r \over p}, \text{ with }1 \leq r \leq (p-1)/2, \text{ and } {1 \over s} \text{ with }1 \leq s \leq (p-1)/2. \] (See \cite{cox-parry}.) The cusp at $i \infty$ is equivalent to the cusp $1/p$. Over the cusps of the form $r/p$, we have singular fibers for $\pi$ of type $I_1$; over the cusps of the form $1/s$, we have singular fibers of type $I_p$. There are exactly $p$ sections of the map $\pi$, which are induced by letting \[ w(\tau) = {\alpha \over p} \] for $0 \leq \alpha \leq p-1$. (See \cite{shioda}.) The zero-section $T_0$ for $\pi$ is of course defined by $w(\tau) = 0$, while the universal section $T$ can be defined by $w(\tau) = 1/p$. A local coordinate for the modular curve $X_1(p)$ about the point corresponding to the cusp at $\tau = i \infty$ is $q_\infty = \exp(2\pi i \tau)$; the fibers of the modular surface itself may be locally represented near this point as $\Bbb C / (\Bbb Z + \Bbb Z\tau) \cong \Bbb C^* / q_\infty^{\Bbb Z}$. Therefore, if we choose $b,d \in \Bbb Z$ such that $rd-bp=1$, a local coordinate about the point corresponding to the cusp at $\tau = r/p$ is $q_r = \exp(2\pi i (d\tau - b)/(-p\tau + r) )$. This essentially is a translation (via the element $\gamma = \begin{pmatrix} d & -b \\ -p & r \end{pmatrix}$ of $\Gamma_0(p) \subset \operatorname{SL}(2, \Bbb Z)$) from the coordinate system at $\tau = r/p$ to that at $\tau = i\infty$. Notice that when we make this change of coordinates, we can consider the fibers near the cusp $\tau = r/p$ again as $\Bbb C^* / q_r^{\Bbb Z}$, where this time we have $q_r = \exp(2\pi i (d\tau - b)/(-p\tau + r) )$; in other words, locally we are taking the complex plane and dividing by the lattice generated by $1$ and $\gamma\tau$. This is, up to a homothety factor $-p\tau + r$, the lattice generated by $-p\tau + r$ and $d\tau - b$, which is the original lattice in the complex plane with the original variable $w$. Now consider the section defined by $w(\tau) = \alpha/p$; in terms of this new basis for the lattice, we have $w(\tau) = \alpha(d\tau - b) + (\alpha d/p)(-p\tau + r)$. Thus after applying the homothety, and obtaining a new variable $w' = w/(-p\tau + r)$ in the complex plane, we see that this section is definded by $w'(\tau) = (\alpha d/p) + \alpha(\gamma \tau)$. After exponentiating and letting $\tau$ approach $r/p$, we see that the section approaches the point $\exp(2\pi i\alpha d/p)$. This is the root of unity which we have desired to compute. Note that had we used $-\gamma$ instead of $\gamma$ we would have gotten the inverse root of unity and that $d$ is determined only $\bmod\ p$. Using the notation of the Introduction, let us label the $I_1$ fibers of the modular surface as $F_1,\dots,F_{(p-1)/2}$, with $F_r$ corresponding to the cusp at $\tau = r/p$. We have shown that for the section $T_\alpha = \alpha T$, \[ \ell_r(T_\alpha) = \alpha r^{-1}, \] where the value of $r^{-1}$ is taken modulo $p$. This gives an alternate proof of Theorem \ref{main_theorem}, since for any fixed $\alpha$, these values are equidistributed among the nonzero classes mod $p$, up to sign. \section{Equidistribution for the component numbers via explicit computation} The main theorem of \cite{miranda2} concerning the equidistribution of the component numbers can also be proved by appealing to the universal property of the modular surface $\pi:Y_1(p) \to X_1(p)$. Using the notation of the Introduction, one sees that the fractions $M_i(S)$ for a torsion section $S$ of order $p$ remain unchanged under base change. Thus it suffices to check that they have the correct values, namely those given by Theorem~1.1 \ for the universal section $T$ of the modular surface. This we do in this section. As noted in the previous section, the modular surface contains the fibers $F_1,\dots,F_r,\dots,\break F_{(p-1)/2}$ of type $I_1$ over the cusps represented by the rational numbers $r/p$, for $1 \leq r \leq (p-1)/2$. For these fibers, since they have only one component, the component number $k_r(T)$ is zero, contributing $[(p-1)/2]/[(p^2-1)/2] = 1/(p+1)$ to the fraction $M_0(T)$. The rest of the singular fibers of the modular surface will be denoted by $F_{(p+1)/2},\dots,F_{p-1}$ where $F_{p-s}$ will be taken as the fiber over the cusp represented by the rational number $1/s$, for $1 \leq s \leq (p-1)/2$. The component number theorem then follows from the next computation. \begin{proposition} \label{component_number_prop} With the above notation, if $1 \leq s \leq (p-1)/2$, then $k_{p-s}(T) = \pm s$ (the indeterminacy of the sign being due to the choice of orientation of the singular fiber $F_{p-s}$). \end{proposition} \begin{pf} It is convenient to again transport the computation to $\tau = i\infty$ as was done in the previous section. Fix an $s$, with $1 \leq s \leq (p-1)/2$, and let $\gamma = \begin{pmatrix} 1 & 0 \\ -s & 1 \end{pmatrix}$. As $\tau$ approaches $1/s$, $\gamma\tau = \tau/(1-s\tau)$ approaches $i \infty$. Since the fiber $F_{p-s}$ is of type $I_p$, the local model for the fibers of the modular surface near this cusp can be taken to be $\Bbb C^* / q^{p\Bbb Z}$, where $q = \exp(2\pi i [\tau/p(1-s\tau)]) = \exp(2\pi i \gamma\tau/p)$ is the local coordinate on the modular curve $X_1(p)$ near this cusp. The new generators for the lattice with these coordinates are $\tau, (1-s\tau)$ instead of $\tau, 1$. The universal section $T$, which is defined by $w(\tau) = 1/p$, is given in terms of these generators by the formula $w(\tau) = (1/p)(1-s\tau) + (s/p)\tau$. In terms of the new variable $w' = w/(1-s\tau)$, we have $T$ defined by the formula $w'(\tau) = 1/p + s(\gamma\tau/p)$. Exponentiating this so as to use the local coordinate $q$, we see that the universal section $T$ is defined in the general fiber $\Bbb C^* / q^{p\Bbb Z}$ by $\exp(2\pi i [1/p + s(\gamma\tau)/p]) = \zeta_p q^s$, where $\zeta_p = e^{2\pi i/p}$. Now using the toric description of the surface near this cusp, (see \cite[Chapter One, Section 4]{AMRT}), the exponent of $q$ (modulo $p$) in this formula governs which component is hit by the section. In particular, the universal section hits component $s$ in the fiber over the cusp represented by $1/s$. Note that using $-\gamma$ instead of $\gamma$ would result in $-s$ instead of $s$. \end{pf} \section{The Canonical Involution} The matrix $A = \begin{pmatrix} 0 & -1 \\ p & 0 \end{pmatrix}$ normalizes the congruence subgroup $\Gamma_1(p)$, and therefore induces an automorphism of the modular curve $X_1(p)$. Since $A^2$ is a constant matrix, this automorphism (which we will also call $A$) is an involution, called the {\em canonical} involution, which in terms of the variable $\tau$, takes $\tau$ to $-1/p\tau$. The involution $A$ also permutes the cusps, exactly switching the two sets of $(p-1)/2$ cusps; indeed, the rational number $r/p$ is taken by $A$ to the number $-1/r$, which is equivalent to the number $1/r$ under the action of $\Gamma_1(p)$. Thus the $I_1$ cusp represented by $r/p$ is switched via $A$ with the $I_p$ cusp represented by $1/r$. We want to simply remark that under this correspondence, the $I_1$ cusp having $u$ as the root-of-unity number for the universal section $T$ is paired with the $I_p$ cusp having $u^{-1}$ as the component number for $T$. Note that for the $I_1$ fibers, all sections have component number $k_j$ equal to zero, and all non-zero sections have root-of-unity number $\ell_j$ in $G(p) = {(\Bbb Z/p)}^\times/\pm 1$. Moreover, for the $I_p$ fibers, all non-zero sections have component number $k_j$ also in this value group $G(p)$. For a cusp $x$ of type $I_1$, denote the root-of-unity number in $G(p)$ by $\ell_x$; for a cusp $x$ of type $I_p$, denote the component number in $G(p)$ by $k_x$. Then the above statement can be re-phrased as follows. \begin{equation} \label{ALcuspformula} \text{For every cusp } x \text{ of type } I_1,\;\;\; \ell_x = k_{Ax}^{-1} \text{ in } G(p). \end{equation} This is a manifestation of a certain duality between the two notions. This we will explore further in the next sections, via a version of the Weil pairing on singular fibers of elliptic surfaces. Before proceeding to this, we want to make some elementary remarks concerning the modular surface and this involution. As above, denote the modular surface over $X_1(p)$ by $\pi:Y_1(p) \to X_1(p)$. Let $\pi':Y_1(p)' \to X_1(p)$ denote the pull-back of $\pi$ via the involution $A$. This operation exactly switches the fibers over the cusps, so that the fiber of $\pi'$ over an $r/p$ cusp is now of type $I_p$, and the fiber of $\pi'$ over a $1/s$ cusp is of type $I_1$. The universal section $T$ of $\pi$ pulls back to a section $T'$ of $\pi'$. (We note that this surface is the modular surface for the group $\Gamma^1(p)$.) An alternate way of constructing this elliptic fibration $\pi'$ is to take the original modular surface $\pi:Y_1(p) \to X_1(p)$ and divide it, fiber-by-fiber, by the subgroup generated by the universal torsion section $T$. Over the cusps represented by $r/p$, the original modular surface has $I_1$ fibers, which are rational nodal curves; in the quotient there is again a rational nodal curve, but the surface acquires a rational double point of type $A_{p-1}$ at the node, and the minimal resolution of singularities produces a fiber of type $I_p$. Over the cusps represented by $1/p$, we have $I_p$ fibers in the original surface; in the quotient the $p$ components are all identified to a single $I_1$ component. For details concerning this quotient construction, the reader may consult \cite{miranda-persson3}. The section $T'$ in this view comes not from the original section $T$, but from a $p$-section of the modular surface, consisting of a coset of the cyclic subgroup generated by $T$ in the general fiber of $\pi$. (The universal section $T$ and all of its multiples of course descends to the zero-section of $\pi'$.) The formula (\ref{ALcuspformula}) implies that \[ k_x(T') = \ell_x(T)^{-1} \] in the group $G(p)$. \section{The function group of a semistable elliptic curve} In the next section we will develop a version of the Weil pairing on a semistable singular fiber of an elliptic surface (that is, a fiber of type $I_m$). The essential ingredient in the definition of the Weil pairing on a smooth elliptic curve is Abel's theorem; see for example \cite[Chapter III, Section 8]{silverman}. Therefore we require a version of Abel's theorem for a fiber of type $I_m$, and this in turn requires a notion of appropriate rational functions on the degnerate fiber. In this section we describe the set of functions which we will use; Abel's theorem is an immediate consequence of the definition. For a nodal fiber $F$ of type $I_1$, which is irreducible, we have the function field of rational functions on $F$, which may be identified with the field of rational functions on $\Bbb P^1$ which take on the same value at $0$ and $\infty$. The non-zero elements of this field form a multiplicative group, which has as a subgroup ${\cal K}$ the group of functions which are regular and nonzero at the node. This group $\cal K$ comes equipped with a divisor function to the group of divisors $\operatorname{Div}(F^{sm})$ supported on the smooth part $F^{sm}$ of $F$. Since $F^{sm}$ is a group, there is a natural map $\Phi:\operatorname{Div}(F^{sm}) \to F^{sm}$ which takes a formal sum of points of $F^{sm}$ to the actual sum in the group. It is easy to check that Abel's theorem holds: a divisor $D \in \operatorname{Div}(F^{sm})$ is the divisor of a function $f \in {\cal K}$ if and only if $\deg(D) = 0$ and $\Phi(D) = 0$ in the group law of $F^{sm}$. We will now extend this to fibers of type $I_m$ with $m \geq 2$. With this assumption there is no field of functions to employ, since the fiber $F$ is no longer irreducible. Thus we must find a multiplicative group of appropriate functions without the aid of an associated field of rational functions. For this we rely on the existence of a set of coordinates on the components of $F$, which are adapted nicely to both the group law on $F^{sm}$ and to the elliptic surface on which $F$ lies. \begin{definition} Suppose that the singular fiber $F$ (which is assumed to be of type $I_m$) has components $C_0,C_1,\dots,C_{m-1}$, with the zero section $S_0$ meeting $C_0$ and $C_j$ meeting only $C_{j\pm 1}$ for each $j$. Let $u_j$ be an affine coordinate on $C_j$ for each $j$. The set $\{u_j\}$ of coordinates will be called a {\em standard set of affine coordinates on $F$} if the following conditions hold: \begin{itemize} \item[a)] For each $j$, $u_j = 0$ at $C_j \cap C_{j-1}$ and $u_j = \infty$ at $C_j \cap C_{j+1}$. \item[b)] The map $\alpha:\Bbb C^* \times \Bbb Z/m \to F^{sm}$ sending a pair $(t,j)$ to the point $u_j=t$ in component $C_j$ of $F$ is an isomorphism of groups. \item[c)] For each $j$, if we set $v_j = 1/u_j$ to be the affine coordinate on $C_j$ near $u_j = \infty$, then $v_j$ and $u_{j+1}$ extend to coordinate functions $V_j$ and $U_{j+1}$ on the elliptic surface near the point $C_j \cap C_{j+1}$, such that $V_j = 0$ defines $C_{j+1}$ and $U_{j+1} = 0$ defines $C_j$ near this point. \end{itemize} \end{definition} \begin{proposition} Let $F$ be a fiber of type $I_m$ on a smooth elliptic surface $X$. Then a standard set of affine coordinates exist on $F$. Moreover, given the ordering of the components, there are exactly $m$ such standard sets of affine coordinates on $F$. \end{proposition} \begin{pf} The existence of a standard set of affine coordinates on $F$ is an immediate consequence of the local toric description of a smooth semistable elliptic surface near a singular fiber of type $I_m$ given in \cite[Chapter One, Section 4]{AMRT}. Indeed, the standard set of affine coordinates is exactly the set of coordinates on the toric cover described there, which descend nicely to $X$. Note that by $a)$ and $b)$, the coordinate $u_0$ on $C_0$ is determined: it must be $0$ at $C_0\cap C_{m-1}$, $\infty$ at $C_0\cap C_1$, and $1$ at $S_0\cap C_0$. Similarly, each coordinate $u_j$ is determined by the point $u_j=1$, which must be one of the points of $C_j$ of appropriate order. Thus there are exactly $m$ possibilities for $u_1$. By $b)$, once $u_1$ is determined, so is $u_j$. It is easy to check that these $m$ different possibilities all give standard sets (once it is known that one of them does). \end{pf} Note that if $\{u_0,\dots,u_{m-1}\}$ is a standard set of affine coordinates on $F$, then any other standard set is of the form $\{u_j' = \zeta^j u_j\}$ for some $m^{th}$ root of unity $\zeta$. We can now define the group of functions $\cal K$ which plays the role of rational functions on $F$. A bit of notation is useful. Suppose that $g(u)$ is a nonzero rational function of $u$. Firstly, define $n_0(g)$ to be the order of $g$ at $u=0$, and $n_\infty(g)$ to be the order of $g$ at $u=\infty$ (which is the order of $g(1/v)$ at $v=0$). These integers are of course independent of the choice of affine coordinate $u$. Moreover, we have that $g(u)/u^{n_0(g)}$ has a finite value at $u=0$, and $g(u)u^{n_\infty(g)}$ has a finite value at $u=\infty$. Secondly, define $c_0(g)$ to be the value of $g(u)/u^{n_0(g)}$ at $u=0$, and define $c_\infty(g)$ to be the value of $g(u)u^{n_\infty(g)}$ at $u=\infty$. These constants do depend on the choice of coordinate $u$; they are simply the lowest coefficient of the Laurent series for $g$ expanded about $u=0$ and $u=\infty$. If we write \begin{equation} g(u) = \alpha u^\ell \prod_{i=1}^e {(u - \lambda_i)} / \prod_{k=1}^f {(u - \mu_k)}, \end{equation} with $\alpha$, $\lambda_i$ and $\mu_k$ nonzero, and $\ell \in \Bbb Z$, then \begin{eqnarray} n_0(g) &= & \ell\\ n_\infty(g) &=& f - e - \ell \\ c_0(g) &=& \alpha {(-1)}^{e + f} \prod_i \lambda_i / \prod_k \mu_k, \text{ and } \\ c_\infty(g) &=& \alpha. \end{eqnarray} Fix a standard set $\{u_j\}$ of affine coordinates on $F$. Define $\cal K$ to be the set of $m$-tuples of nonzero rational functions $(g_0(u_0), g_1(u_1), \dots, g_{m-1}(u_{m-1}))$ satisfying the following conditions: \begin{itemize} \item[a)] For each $j$, $n_\infty(g_j) + n_0(g_{j+1}) = 0$. \item[b)] For each $j$, $c_\infty(g_j) = c_0(g_{j+1})$. \item[c)] $\sum_{j=0}^{m-1} n_0(g_j) = 0$. \end{itemize} Note that condition $a)$ says that the functions $\{g_j\}$ have opposite orders at the nodes of $F$, and condition $b)$ says that with respect to the standard set of affine coordinates, they have the same leading coefficients in their Laurent series at these nodes. These are local conditions about each of the nodes. The final condition $c)$ is a global condition, saying that the orders sum to zero upon going around the cycle of components. We note that $\cal K$ is a multiplicative group (the operation being defined component-wise). These conditions are motivated by a notion of restriction of functions from the surface to the fiber. Suppose that $G$ is a rational function on $X$. We may uniquely write its divisor, near the fiber $F$, as $\operatorname{div}(G) = H + V$, where $V$ is the ``vertical'' part of the divisor consisting of linear combinations of components of $F$, and $H$ is the ``horizontal'' part of the divisor consisting of linear combinations of multi-sections for the fibration map. If one restricts to the general fiber near $F$, one only sees the contributions from $H$. We want to make the following regularity assumption for the function $G$: \begin{center} $\{*\}$\;No curve appearing in the horizontal part $H$ passes through any node of $F$. \end{center} Under this assumption, we see that the zeroes and poles of $G$, as we approach the singular fiber $F$, survive in the smooth part $F^{sm}$ of $F$. Therefore we have a chance of obtaining a limiting version of Abel's theorem. Take then such a rational function $G$, and let us define a ``restriction'' to the fiber $F$, which will be an element of the group $\cal K$. Fix a standard set of affine coordinates $\{u_j\}$ on $F$, and also assume that we have normalized the base curve so that $\pi = 0$ along $F$. Write $V = \sum_j r_j C_j$ as the vertical part of $\operatorname{div}(G)$. For each $j$, consider the ratio $G/\pi^{r_j}$; this is a rational function on $X$ which does not have a zero or pole identically along $C_j$ (since $\pi$ has a zero of order one along $F$). Restricting this function to $C_j$ gives a nonzero rational function $g_j(u_j)$. \begin{lemma} If $G$ satisfies the regularity condition $\{*\}$, then the $m$-tuple $(g_0,\dots,g_{m-1})$ lies in $\cal K$. \end{lemma} \begin{pf} Near the point $C_j \cap C_{j+1}$, we have local coordinates $U_{j+1}$ and $V_j$ on $X$ as in the definition of a standard set of affine coordinates. The curve $C_j$ is defined locally by $U_{j+1} = 0$, and $C_{j+1}$ by $V_j = 0$. Moreover the fibration map $\pi$ is locally of the form $\pi = U_{j+1} V_j$. Hence near $C_j \cap C_{j+1}$, we may write $G$ as \[ G(U_{j+1},V_j) = U_{j+1}^{r_j} V_j^{r_{j+1}} L(U_{j+1},V_j) \] where the condition $\{*\}$ implies that $L(0,0) \neq 0$. With this notation we have $g_j(v_j) = v_j^{r_{j+1}-r_j} L(0,v_j)$ and $g_{j+1}(u_{j+1}) = u_{j+1}^{r_j-r_{j+1}} L(u_{j+1},0)$. Thus we see that $n_\infty(g_j) = r_{j+1}-r_j$ and $n_0(g_{j+1}) = r_j-r_{j+1}$, proving that condition $a)$ of the definition of $\cal K$ is satisfied. We also have that $c_\infty(g_j) = c_0(g_{j+1}) = L(0,0)$, giving us condition $b)$. Finally, the sum $\sum_j n_0(g_j)$ telescopes to $0$, showing that condition $c)$ holds. \end{pf} {}From this point of view, the definition of $\cal K$, though at first glance rather ad-hoc, is actually quite natural. Motivated by the above, we call $\cal K$ the {\em function group} of the singular fiber $F$. We have a divisor map $\operatorname{div}:{\cal K} \to \operatorname{Div}(F^{sm})$, sending an $m$-tuple $(g_0,\dots,g_{m-1})$ to the formal sum of the zeroes and poles of each $g_j$, throwing away any part of the divisor at the nodes. This map is a group homomorphism. Recall that we also have a natural summation map $\Phi$ from $\operatorname{Div}(F^{sm})$ to $F^{sm}$. Abel's theorem for $F$ can be stated as follows. \begin{theorem} \label{abel} A divisor $D \in \operatorname{Div}(F^{sm})$ is the divisor of an element of $\cal K$ if and only if $\deg(D) = 0$ and $\Phi(D) = 0$. \end{theorem} \begin{pf} Let $(\underline{g}) \in {\cal K}$, and let $D = \operatorname{div}(\underline{g})$. For each $j$, write \[ g_j(u_j) = \alpha_j u_j^{\ell_j} \prod_{i=1}^{e_j} {(u_j - \lambda_i^{(j)})} / \prod_{k=1}^{f_j} {(u_j - \mu_k^{(j)})}. \] {}From the definition of $\cal K$, we must have \begin{eqnarray*} \ell_j - \ell_{j+1} &=& f_j - e_j,\\ \alpha_j &=& \alpha_{j+1} {(-1)}^{e_{j+1} + f_{j+1}} \prod_i \lambda_i^{(j)} / \prod_k \mu_k^{(j)}, \text{ and }\\ \sum_j \ell_j &=& 0. \end{eqnarray*} Summing the first set of equations over $j$, we see that \[ \deg(D) = \sum_j (e_j - f_j) = 0. \] Multiplying the second set of equations over $j$, and applying the above, we have that \[ \prod_j {\prod_i \lambda_i^{(j)} \over \prod_k \mu_k^{(j)} } = 1 \] which shows that the $\Bbb C^*$ part of the group element $\Phi(D)$ is trivial. Finally, to show that the $\Bbb Z/m$ part of $\Phi(D)$ is trivial, we must show that $\sum_j j (e_j - f_j) = 0 \mod m$. Writing $e_j - f_j$ as $\ell_{j+1} - \ell_j$ using the first equation, we see that this sum telescopes to \[ -\ell_1 - \ell_2 - \dots - \ell_{m-1} + (m-1)\ell_0 \] which is $0 \mod m$ by the third equation. This completes the proof of the necessity of the conditions on $D$. We leave the sufficiency, which is equally elementary, to the reader. \end{pf} We note that the only elements $(\underline{g})$ of $\cal K$ which have $\operatorname{div}(\underline{g}) = 0$ are the constant elements, where $g_j = c$ for every $j$ with $c$ being a fixed nonzero complex number. We thus have an exact sequence \[ 0 \to \Bbb C \to {\cal K} \stackrel{\operatorname{div}}{\to} \operatorname{Div}_0(F^{sm}) \stackrel{\Phi}{\to} F^{sm} \to 0 \] where $\operatorname{Div}_0(F^{sm})$ is the group of divisors of degree $0$ on $F^{sm}$. \begin{example} Suppose $F$ is a fiber of type $I_{mk}$, with a standard set of affine coordinates $\{u_j\}$. Fix a primitive $m^{th}$ root of unity $\zeta$, and an integer $\alpha \in [0,m-1]$, and let $p$ be the point of $C_{\alpha k}$ with coordinate $u_{\alpha k} =\zeta$. We note that $p$ is a point of order $m$ in the group law of $F^{sm}$, so that $D = m p - m \underline{0}$ is a divisor on $F^{sm}$ with $\deg(D) = 0$ and $\Phi(D) = \underline{0}$ (where $\underline{0}$ is the origin of the group law, i.e., the point in $C_0$ with coordinate $u_0 = 1$). By Abel's theorem, there is an element $(\underline{g}) \in {\cal K}$ such that $\operatorname{div}(\underline{g}) = D$. If $\alpha = 0$, this element $(\underline{g})$ is \[ g_0 = {(u_0 - \zeta)}^m / {(u_0 - 1)}^m, \;\;\; g_j = 1 \text{ for } j \neq 0. \] If $1 \leq \alpha \leq m-1$, we have \begin{eqnarray*} g_0 &=& u_0^{\alpha}/{(u_0 - 1)}^m, \\ g_j & = & u_j^{\alpha-m} \text{ for }j = 1,\dots,\alpha k - 1, \\ g_{\alpha k} &=& {(-1)}^m u_{\alpha k}^{\alpha-m} {(u_{\alpha k} - \zeta)}^m, \text{ and }\\ g_j &=& {(-1)}^m u_j^{\alpha} \text{ for } j = \alpha k + 1, \dots, mk-1.\\ \end{eqnarray*} \end{example} \section{The limit Weil pairing on a degenerate elliptic curve} The Weil pairing on a smooth elliptic curve $E$ is defined as follows (see \cite[Section III.8]{silverman}). Let $S$ and $T$ be two points of order $m$ on $E$. Choose a rational function $g$ on $E$ with $\operatorname{div}(g) = {[m]}^*(T) - {[m]}^*(\underline{0})$, where $[m]$ denotes multiplication by $m$. Then the Weil pairing $e_m$ on the $m$-torsion points of $E$ is defined by \[ e_m(S,T) = g(X \oplus S) / g(X) \] for any $X \in E$ where both $g(X\oplus S)$ and $g(X)$ are defined and nonzero. (We use $\oplus$ as the group law in $E$ to avoid confusion.) The existence of the rational function $g$ relies solely on Abel's theorem for $E$, as does the fact that $e_m$ has values in the group of $m^{th}$ roots of unity. In the previous section, we developed an Abel theorem for a singular fiber $F$ of type $I_k$ on a smooth elliptic surface, by replacing the notion of the field of rational functions with the limit function group $\cal K$. This allows us to define in the same way a limit Weil pairing on the $m$-torsion points of $F$, which we also denote by $e_m$. In this section we compute it. Since the $e_m$ pairing is isotropic and skew-symmetric, it suffices to compute $e_m(S,T)$ for generators $S$ and $T$ of the group of $m$-torsion points on $F$. Fix an integer $m$; in order that $F$ have $m^2$ $m$-torsion points, $F$ must be of type $I_{mk}$ for some $k$. Since the Weil pairing is formally invariant under base change, we may assume $k=1$ and $F$ is of type $I_m$. Also fix a standard set of affine coordinates $\{u_j\}$ on $F$. Let $\zeta = \exp(2 \pi i/m)$ be a primitive $m^{th}$ root of unity. Let $T$ be the $m$-torsion point of $F$ which is in component $C_0$ having coordinate $u_0 = \zeta$. Let $S$ be the $m$-torsion point of $F$ which is in component $C_1$ having coordinate $u_1 = 1$. These points $S$ and $T$ generate the group of $m$-torsion points of $F$. Let $\nu = \exp(2 \pi i/m^2)$ so that $\nu^m = \zeta$. Consider the point $T' \in F$ in component $C_0$ having coordinate $u_0 = \nu$; note that $m T' = T$, and ${[m]}^*(T) = \sum_R (T' \oplus R)$ where the sum is taken over all $m$-torsion points of $F$. Similarly, ${[m]}^*(\underline{0}) = \sum_R (R)$. The element $(\underline{g}) \in {\cal K}$ such that $\operatorname{div}(\underline{g}) = {[m]}^*(T) - {[m]}^*(\underline{0})$ is defined by \[ g_j(u_j) = \zeta^{-j} {u_j^m - \zeta \over u_j^m - 1}. \] Now choose an $X \in C_0$ with coordinate $u_0 = x$. The point $X\oplus S$ is then the point in $C_1$ with coordinate $x$. Thus \begin{eqnarray*} \underline{g}(X \oplus S) / \underline{g}(X) &=& g_1(x) / g_0(x) \\ &=& \zeta^{-1} {x^m - \zeta \over x^m - 1} / {x^m - \zeta \over x^m - 1} \\ &=& \zeta^{-1}. \end{eqnarray*} This shows that $e_m(S,T) = \zeta^{-1}$. Let $M(t,s) = t T \oplus s S$ be the general point of order $m$ in $F$; note that $M(t,s)$ is in component $C_s$ and has coordinate $u_s = \zeta^t$. Extending the calculation above using the bilinearity and skew-symmetry we obtain the following. \begin{proposition} With the above notation, the limit Weil pairing on $F$ takes the form \[ e_m(M(t_1,s_1),M(t_2,s_2)) = \zeta^{t_1s_2 - t_2s_1}. \] \end{proposition} This limit Weil pairing, although computed using Abel's theorem for the limit function group $\cal K$, is a priori dependent upon the choice of a standard set of affine coordinates on the fiber. However it is easy to see that it is in fact well-defined, independent of this choice. Moreover the limit Weil pairing is indeed the limit of the usual Weil pairing on the nearby smooth fibers of the elliptic surface. This follows from the nature of the element $(\underline{g}) \in {\cal K}$ used in the computation above: each $g_j$ has degree $0$ on the component where it is defined, and hence is the usual restriction of a rational function $G$ on the elliptic surface. This function $G$ can be chosen so that, when restricted to the nearby smooth fibers, it is the function used to define the Weil pairing there. Therefore the limit Weil pairing is the limit of the usual Weil pairing. \section{The root-of-unity and component number relationship via the Weil pairing} Finally, we want to point out that the duality between the root-of-unity results and the component number results, which were mentioned in Section~4 as being related to the canonical involution, can be expressed also in terms of the Weil pairing. We will compute this Weil pairing on the singular fibers using the limit Weil pairing developed in the previous section. First let $W$ be a torsion section of order $m$ of an elliptic surface passing through the zero component $C_0$ of an $I_m$ fiber. Let $\zeta = \exp(2\pi i/m)$ be a primitive $m^{th}$ root of unity. Assume that the $I_m$ fiber is given a standard set of affine coordinates, such that the point $W \cap C_0$ has coordinate $u_0 = \zeta^a$, with $(a,m) = 1$. In the notation of the last section, we have $W = aT$. Let $W^*$ be the set of (local) sections $Z$ such that $e_m(W,Z) = \zeta$. By the computation given in Proposition~7.1 \ we see that such a local section $Z$ is one which passes through component $b$, where $ab \equiv 1 \mod m$. Specializing to the case where $m$ is an odd prime $p$, and using the root-of-unity numbers $\ell$ and the component numbers $k$, we see the following: \begin{equation} \label{W*} \text{If } k_j(W) = 0 \text{ and } \ell_j(W) = a, \text{ then } W^* \text{ is the set of local sections } Z \text{ with } k_j(Z) = a^{-1}. \end{equation} Finally let us return to the modular surface $\pi:Y_1(p) \to X_1(p)$ and the quotient $\pi':Y_1'(p) \to X_1(p)$, as described in Section~4 \ In the above, set $W = T$, the universal section, and fix a cusp $x$ of type $I_1$ on $X_1(p)$. Assume that $\ell_x(T) = a$ for this cusp, that is, $T\cap C_0$ is the point with coordinate $\zeta^a$. The section $T'$ on the quotient is induced by the multisection of $\pi$ which, by (\ref{ALcuspformula}), has $k_x = a^{-1}$. Since the Weil pairing is invariant under isogeny, we see by (\ref{W*}) that $T'$ is exactly the image, locally, of the set $W^*$ of sections pairing with $T$ to give value $\zeta$. Alternatively, we may write \[ T' = \text{ image of } {e_m(T,-)}^{-1}(\zeta). \]

9310

alg-geom/9310007

en

https://arxiv.org/abs/alg-geom/9310007

alg-geom/9310007

Burt Totaro

David Eisenbud, Alyson Reeves, and Burt Totaro

Initial ideals, Veronese subrings, and rates of algebras

24 pages, latex file

We show that high Veronese subrings of any commutative graded ring have a Grobner basis with all relations of degree 2. (The d-th Veronese subring of a ring A_0 + A_1 + A_2 + ... is the ring A_0 + A_d + A_{2d} + ...; ``high'' means we take d sufficiently large, say at least half the regularity of the ideal defining the original ring.) This gives another proof of Backelin's theorem that such Veronese subrings are Koszul algebras (= wonderful rings), i.e., that the minimal resolution of the residue field of such a ring is linear.

alg-geom

\section{ Introduction and motivating results} \label{intro} Given a homogeneous ideal $I \subset S$ it is a matter of both computational and theoretical interest to know how low the degree of $in_>(I)$ can be made by choosing variables and monomial order on $S$ in an appropriate way. In particular, one may ask which ideals $I$ admit {\it quadratic}\ initial ideals; that is, for which $I$ are there choices of variables and order such that $in_>(I)$ can be generated by monomials of degree 2? One of the theoretical reasons for interest in this question is that if $I$ admits a quadratic initial ideal then, by a result of Fr\"oberg and a deformation argument noticed by Kempf and others, $A:=S/I$ is a (homogeneous) Koszul algebra in the sense of Priddy [Pr70]; that is, the residue field $k$ of $A$ admits an $A$-free resolution whose maps are given by matrices of linear forms. Using complex arguments about a lattice of ideals derived from a presentation of $A$ as a quotient of a free noncommutative algebra, Backelin [Ba86] proved that for any graded ring $A$ as above the {\bf $d^{th}$ Veronese subring } $$ A_{(d)} := \bigoplus_{i=1}^{\infty} A_{di},$$ is Koszul for all sufficiently large $d$. Our work started from a request by George Kempf for a simpler proof. In this paper we shall prove the stronger result that $ A_{(d)}$ admits a quadratic initial ideal for all sufficiently large $d$. To make these results more quantitative, we define a measure of the rate of growth of the degrees of the syzygies in a minimal free resolution: \begin{define} Let $A = k \oplus A_1 \oplus A_2 \oplus \cdots$ be a graded ring. For any finitely generated graded $A$-module $M$, set $t_i^A(M) = max\{j\ | \ Tor^A_i(k,M)_j \neq 0\}$, where $Tor_i^A(k,M)_j$ denotes the $j^{th}$ graded piece of $Tor_i^A(k,M)$. The {\bf rate} of $A$ is defined by Backelin [Ba86] to be $$rate(A) = sup \{ (t^A_i (k)-1)/(i-1) | i \geq 2 \}.$$ \end{define} For example, $A$ is Koszul iff $rate(A) = 1$. It turns out that the rate of any graded algebra is finite (see for example [An86]) and Backelin actually proves \begin{thm} \label{thm0} ([Ba86]): $rate(A_{(d)}) \leq max(1, rate(A)/d)$. \end{thm} One should compare this with the rather trivial result (Proposition \ref{mum-prop} below) that if a homogeneous ideal $I$ can be generated by forms of degree $m$, then the ideal $V_d(I)$ defining $A_{(d)}$ can be generated by forms of degree $\leq max (2, \lceil m/d \rceil)$. In the notation introduced above, $\delta(V_d(I)) \leq max (2,\lceil \delta(I)/d \rceil)$. A similar result with $\Delta$ (the minimum, over all choices of variables and of monomial orderings $>$, of the maximum degree of a minimal generator of $in_>(I)$) would lead to a bound on the rate by virtue of Proposition \ref{prop4}. Unfortunately, as we show in Example \ref{counterexample} below, it is {\it not} true that if some initial ideal of $I$ can be generated by monomials of degree $m$ then $V_d(I)$ admits an initial ideal generated by forms of degree $\leq max (2, \lceil m/d \rceil)$. But there {\it is} a replacement for $m$ that makes such a formula true, and this is the main result of this paper. Recall that the {\bf Castelnuovo-Mumford regularity} of $I$ is defined as follows: \begin{define} For $I \subset S$, the {\bf regularity} of $I$ is defined as $$reg(I) = max \{t^S_i (I) - i | i \geq 0 \}.$$ \end{define} Since $t^S_0(I) = \delta(I)\leq reg(I)$, the regularity is $\geq$ the maximal degree of the generators of $I$. One may think of the regularity as a more stable measure of the size of the generators of $I$. Our main result is that we may replace the degree of the generators of $I$ by the regularity and get a bound on the degrees of the initial ideals of Veronese powers: \begin{thm} \label{reg-thm} $$\Delta(V_d(I)) \leq max (2, \lceil reg (I)/d \rceil). $$ In particular, if $d \geq reg( I)/2 $ then $\Delta(V_d(I)) = 2$. \end{thm} In section \ref{multi} we explain how to generalize this result to Segre products of Veronese embeddings. To deduce a version of Backelin's Theorem, one needs a result strengthening the theorem of Fr\"oberg mentioned above. Such a result was stated without proof by Backelin ([Ba86, pp98ff]): \begin{prop} \label{prop4} If $A = S/I$ with $I$ a homogeneous ideal, then $rate(A) \leq \Delta(I) - 1$. In particular, if $\Delta(I) = 2$ then $A$ is Koszul. \end{prop} We will give a simple proof of this proposition (and something more general) in section \ref{resolution} following ideas of Bruns, Herzog, and Vetter [BrHeVe]. Unfortunately the converse of this result is not true: in particular, the algebra $A$ may be Koszul without $I$ admitting a quadratic initial ideal. In section \ref{obstsec}, we formulate another obstruction for an ideal $I\subset S$ to have an quadratic initial ideal. An easily stated part of Theorem \ref{obstruction} is that if $I$ admits a quadratic initial ideal then $I$ contains far more quadrics of low rank than would a generic subspace of quadrics. We may make this quantitative as follows: \begin{cor} \label{cor to Lemma 18} If $I\subset S$ admits a quadratic initial ideal, and $dim (S/I) = n$, then $I$ contains an $m$-dimensional space of quadrics of rank $\leq 2(n+m)-1$ for every $m \leq codim(I)$. \end{cor} The obstruction shows that in certain dimensions, a generic complete intersection of quadrics has no initial ideal $in(I)$ which is generated by quadrics, even though every complete intersection of quadrics is a Koszul algebra. There seems no reason to believe that this obstruction, even with the Koszul condition, is enough to guarantee that an ideal admits a quadratic initial ideal, so we pose as a problem the question raised at the beginning of the introduction : \smallskip {\sl Find necessary and sufficient conditions for an ideal to admit a quadratic initial ideal.} \smallskip A noncommutative analogue of some of the results on initial ideals, in which $T_d$ is replaced by a free noncommutative algebra mapping onto $S$, is given in [Ei]. \section{Initial ideals for Veronese subrings} \label{vero} As above, let $$T_d = k[\{z_{m}\}] \mbox{ where $m$ is a monomial of $S$ of degree $d$},$$ and let $\phi_d: T_d \rightarrow S$ be the map sending $z_{m}$ to $m$. If $J \subset S$ is a homogeneous ideal, let $V_d(J)$ denote the preimage of $J$ in $T_d$. It is easy to see that $V_d(J)$ is generated by the kernel of $\phi_d$ and, for each generator $g$ of $J$ in degree $e$, the preimages of the elements of degree $nd$ in $(x_1,\ldots,x_r)^{nd-e}g$, where $nd$ is the smallest multiple of $n$ that is $\geq e$. These elements have degree $n$ in $T_d$. Since ${\rm ker}(\phi_d)$ is generated by forms of degree 2 it follows that $V_d(J)$ is generated by forms of degree $\leq max(\lceil \delta(J)/d \rceil,2)$. This gives a proof of the following well-known Proposition. \begin{prop} \label{mum-prop} $$\delta(V_d(I)) \leq max(2, \lceil \delta(I)/d \rceil).$$ In particular, if $d \geq \delta(I)/2 $ then $V_d(I)$ is generated by quadrics. \end{prop} This Proposition is mentioned by Mumford in [Mu70] (in a slightly different form), though it is surely much older. We extend the given monomial order on $S$ to a monomial order on $T_d$ as follows: If $a$, $b$ are monomials in $T_d$, then $a > b$ if $\phi(a) > \phi(b)$ or $\phi(a) = \phi(b)$ but $a$ is bigger than $b$ in the reverse lexicographic order: that is, given two monomials in $T_d$ of the same degree having the same image in $S$ we order the factors of each in decreasing order, and take as larger the monomial with the smaller factor in the last place where the two differ. Here the order of the variables $z_{m}$ is defined to be the same as the order in $S$ on the monomials $m$. We first compute the initial ideal of ${\rm ker}(\phi_d)$. \begin{prop} \label{prop8} With notation as above, $in({\rm ker}(\phi_d)) \subset T_d$ is generated by quadratic forms for every $d$. \end{prop} \begin{rem} Barcanescu and Manolache [BaMa82] proved that the Veronese rings are Koszul -- which is a corollary of Proposition \ref{prop8}. \end{rem} {\bf Proof}: The ideal ${\rm ker}(\phi_d)$ is generated by quadratic forms, each a difference of two monomials that go to the same monomial under $\phi_d$. Let $J$ be the monomial ideal generated by the initial terms of these quadratic elements of ${\rm ker}(\phi_d)$. We have $J \subset in({\rm ker}(\phi_d))$, and we wish to prove equality. We will show that distinct monomials of $T_d$ not in $J$ map by $\phi_d$ to distinct monomials of $S$. It will follow that, for each $e$, \begin{eqnarray*} dim S_{de} &\geq &dim (T_d/J)_e \cr &\geq & dim (T_d/(in ({\rm ker}( \phi_d))))_e \cr &=&dim (T_d/({\rm ker}( \phi_d)))_e \cr &=& dim( S_{de}). \end{eqnarray*} Thus $dim (T_d/J)_e = dim (T_d/(in( {\rm ker}( \phi_d))))_e$, and $J = in({\rm ker}( \phi_d))$ as desired. Call the monomials not in $J$ ``standard", and say that a product of monomials of degree $d$ in $S$ is standard if its factors correspond to the factors of a standard monomial in $T_d$. Since $J \subset in({\rm ker}(\phi_d))$, any monomial of $S_{de}$ may be written as a standard product of $e$ monomials of degree $d$. We must show that if $m \in S$ is a monomial of degree $de$, then there is a unique way of writing $m$ as a standard product $m_1\cdots m_e$ of monomials of degree $d$. We claim that this unique product is obtained by writing out the $de$ factors of $m$ in decreasing order $$m = x_1\cdots x_1 x_2\cdots x_2 \cdots x_r\cdots x_r,$$ and taking $m_1$ to be the product of the first $d$ factors, $m_2$ to be the product of the next $d$ factors, and so on. First we prove the claim for a standard product $m_1m_2$ with just two factors. Suppose the sequences of indices of the two factors are $$i_1= (i_{11}\leq \cdots \leq i_{1d}), \; i_2= ( i_{21}\leq \cdots \leq i_{2d})$$ and $m_1 > m_2$. We must show that $ i_{1d} \leq i_{21}$. The product of the monomials $m_1'$ and $m_2 '$ obtained from $m_1$ and $m_2$ by interchanging the factors $x_{ i_{1d}}$ and $x_{ i_{21}}$ represents the same element of $S$. The difference of these products represents a quadratic element of ${\rm ker}(\phi_d)$ . If $ i_{1d} > i_{21}$ then $x_{ i_{1d}} < x_{ i_{21}}$ and thus $m_2' < m_2$. Consequently the leading term of this quadratic element of ${\rm ker}(\phi_d)$ would correspond to $m_1m_2$, and the product $m_1m_2$ would not be standard. Now suppose that $m_1\cdot \cdots \cdot m_e$ is a standard product of degree $de$, with $e$ arbitrary, and $m_1 \geq \cdots \geq m_e$. Let $i_{j1}\leq \cdots \leq \alpha_{jd}$ be the indices of the variables in $m_j$. If $i_{jd}>i_{j+1,1}$ for some $j$ then the product $m_jm_{j+1}$ would not be standard, contradicting the standardness of the entire product. \ \vrule width2mm height3mm \vspace{3mm} \begin{note} Even when $r = 3$ and $d = 2$, the initial terms of the minimal system of generators for the ideal ${\rm ker}(\phi_2)$ do not generate $in_>({\rm ker}(\phi_2))$ under all possible orders $>$. In this case, the ideal is minimally generated by the $2 \times 2$ minors of the symmetric $3 \times 3$ matrix with entries $z_{ij}$, $i \leq j$. There are 29 different initial ideals (depending upon the order chosen), 23 of which are generated entirely in degree 2. The other 6 each require an additional generator of degree three. It would be interesting to characterize the orders $>$ for which the ideal $ in_>({\rm ker} (\phi_d))$ is generated by elements of degree 2. \end{note} Following the proof of Proposition \ref{prop8} we say a monomial of $T_d$ is standard if it is not in $in({\rm ker} (\phi_d))$. Let $\sigma: S \rightarrow T_d$ be the $k$-linear map that takes each monomial to its unique standard representative. Because of the way we have defined the order on $T_d$ we have $in(\sigma(p)) = \sigma(in(p))$. Also, $\sigma$ takes $J$ into $V_d(J)$. As a consequence we have: \begin{lem} \label{lem9} If $J$ is a homogeneous ideal of $S$, then $in(V_d(J))$ is the ideal $K$ generated by $in({\rm ker} (\phi_d))$ and the monomials $\sigma(m)$ for $m \in in(J) \cap (im( \phi_d))$. \end{lem} {\bf Proof}: For each degree $e$ it is clear that $dim_k(T_d/K)_e \leq dim_k(S/J)_{de}$, so it is enough to show that $K \subset in(V_d(J))$. Let $p \in J$ be a form with initial term $m$. Clearly $\sigma(p) \in V_d(J)$, and $in(\sigma(p)) = \sigma(m)$. \ \vrule width2mm height3mm \vspace{3mm} \begin{define} If $m \in S$ is a monomial, then following Eliahou and Kervaire [ElKe90] we write $max(m)$ for the largest index $i$ such that $x_i$ divides $m$. We call a monomial ideal $I$ {\bf (combinatorially) stable} if for every monomial $m \in I$ and $j < max(m)$, the monomial $(x_j/x_{max(m)})m \in I$. \end{define} \begin{thm} \label{char0-thm} If $J$ is a homogeneous ideal of $S$ such that $in(J)$ is combinatorially stable, then $in(V_d(J))$ is generated by $in({\rm ker}(\phi_d))$ and the monomials $\sigma(m)$ where $m$ runs over the minimal generators of $in(J) \cap im (\phi_d)$. Thus if $\delta(in(J)) \leq u$, then $\delta(in(V_d(J))) \leq max(2, \lceil u/d \rceil)$. \end{thm} {\bf Proof}: Let $n \in in(J) \cap im (\phi_d)$. By Lemma \ref{lem9} it suffices to show that $\sigma(n)$ is divisible by some $\sigma(m)$, where $m$ is a minimal generator of $in(J) \cap im (\phi_d)$. Let $m'$ be an element of $in(J) \cap im (\phi_d)$ of minimal degree among those dividing $n$, and write $n = x_{i_1}\cdot \cdots \cdot x_{i_{sd}}$ with $i_1 \leq \cdots \leq i_{sd}$. Say $deg \, m' = de$. Since $in(J)$ is combinatorially stable, it follows that $m := x_{i_1}\cdot \cdots \cdot x_{i_{de}} \in in(J)$, and since $m$ has degree $de$, we have $m \in im(\phi_d)$ as well. If $m$ were not a minimal generator of $in(J) \cap im (\phi_d)$, then some proper divisor of it would be in $in(J) \cap (im (\phi_d))$ and would divide $n$, contradicting our choice of $m'$. As $\sigma(m)$ divides $\sigma(n)$, we are done. \ \vrule width2mm height3mm \vspace{3mm} \begin{ex} 1: The hypothesis of stability cannot be dropped. For example, if $r=3$ and $J = (x_1x_2x_3)$, and we take the lexicographic or reverse lexicographic order on $S$, then the initial ideal $in(V_d(J))$ (defined using the order on $T_d$ we associate to the given order on $S$) requires a cubic generator for all $d$. \end{ex} To obtain Theorem \ref{reg-thm} as given in the introduction, we need to recall the notion of {\bf Castelnuovo-Mumford regularity}. \begin{define} For $I \subset S$, the {\bf regularity} of $I$ is defined as $$reg(I) = max \{t^S_i (I) - i | i \geq 0 \}.$$ \end{define} \begin{note} $t^S_0(I) = \delta(I)\leq reg(I)$. \end{note} Bayer and Stillman [BaSt87] give the following criterion for an ideal to be $m$-regular, assuming (as we do throughout this paper) that the field $k$ is infinite. \begin{thm} [BaSt87] \label{m-reg} Let $I \subset S$ be an ideal generated in degrees $\leq e$. The following conditions are equivalent: \begin{enumerate} \item $I$ is $e$-regular, \item \begin{enumerate} \item For some $j\geq 0$ and for some linear forms $h_1,\ldots,h_j \in S_1$ we have $$((I,h_1,\ldots,h_{i-1}):h_i)_e = (I,h_1,\ldots,h_{i-1})_e $$ for $i=1,\ldots,j$, and \item $$(I,h_1,\ldots,h_j)_e = S_e.$$ \end{enumerate} \item Conditions 2a) and 2b) hold for some $j\geq 0$ and for {\em generic} linear forms $h_1,\ldots ,h_j\in S_1$. \end{enumerate} \end{thm} \begin{note} Let $g$ be generically chosen in the Borel group $B$, the subgroup of $Gl(r)$ consisting of the upper triangular matrices. Then $\langle gx_r,\ldots,gx_{r-j+1} \rangle$ is a generic linear subspace for $I$. Since $gx_i$ is a generic linear form with respect to $(I,gx_r,\ldots,gx_{i+1})$, $x_i$ is a generic linear form with respect to $(g^{-1}I,x_r,\ldots,x_{i+1})$. If $I$ is Borel-fixed, then $g^{-1}I = I$, and hence we can replace $h_1,\ldots,h_j$ by $x_r,\ldots,x_{r-j+1}$ in the statement of Theorem $\ref{m-reg}$ in this case. \end{note} \begin{prop} \label{reg-stab} Let $I \subset S$ be a Borel-fixed monomial ideal generated in degrees $\leq e$. Then $I$ is $e$-regular if and only if $I_e$ is combinatorially stable. \end{prop} {\bf Proof:} By the note following the statement of Theorem \ref{m-reg}, we may replace $h_1,\ldots,h_j$ by $x_r,\ldots,x_{r-j+1}$ in the statement of the theorem. Suppose $I$ is $e$-regular. Then 2b) of Theorem \ref{m-reg} implies that $I$ includes all monomials in $x_1,\ldots,x_{r-j}$ of degree $e$. And 2a) of the same theorem implies that for every monomial $m \in I$ of degree $e$ with $max(m)>r-j$, $I$ also contains $x_k/x_{max(m)} \cdot m$ for every $k$ with $1 \leq k \leq max(m)$. Taken together, these two statements imply $I_e$ is combinatorially stable. Conversely, suppose $I_e$ is combinatorially stable, and let $j$ be the smallest integer such that $I$ contains a power of $x_{r-j}$. It follows that $x_{r-j}^e \in I_e$, and by stability, $(x_1,\ldots,x_{r-j})^e \subset I$, and hence 2b) holds. Let $m \in ((I,x_r,\ldots,x_{i+1}):x_{i})_{e}$ for some $r-j+1 \leq i \leq r$; since $I$ is a monomial ideal, we can assume that $m$ is a monomial in proving 2a). If $m$ is divisible by $x_k$ for some $i+1\leq k \leq r$, then it is clear that $m \in (I,x_r,\ldots,x_{i+1})$. Thus we may assume $m$ is not divisible by $x_{i+1},\ldots,x_r$. Since the monomial $mx_i$ belongs to $(I,x_r,\ldots ,x_{i+1})$, it must belong to $I$. Since it has degree $e+1$, there must be a monomial $m'\in I_e$ and an $l$ such that $mx_i=m'x_l$. Clearly $l\leq i$. Since $m=(x_l/x_i)m'$, combinatorial stability of $I_e$ implies that $m\in I_e$. Thus 2a) of Theorem \ref{m-reg} holds as well, and $I$ is $e$-regular. \ \vrule width2mm height3mm \vspace{3mm} {\bf Proof of Theorem 3:} If $I$ is an ideal in generic coordinates which is $e$-regular, then by Theorem \ref{m-reg}, $in_>(I)$ is generated in degrees $\leq e$, where $<$ is the reverse lexicographic order. By the above proposition, $in(I_e)$ with respect to reverse lexicographic order is combinatorially stable, and hence by Theorem \ref{char0-thm} we have: \begin{eqnarray*} \Delta(V_d(I)) & \leq & \delta(in_{>'}(V_d(gI))) \\ & \leq & max(2, \lceil reg(gI))/d \rceil) \\ & = & max(2, \lceil reg(I)/d \rceil) \end{eqnarray*} (where $g$ is a ``general" choice of coordinates, $>$ is reverse lexicographic order, and $>'$ is the induced order), proving Theorem \ref{reg-thm}. \section{Comments on the main theorem} \label{comments} We have proved that for any homogeneous ideal $I\subset S$, we have $\Delta (V_d(I))\leq max(2,\lceil reg(I)/d \rceil )$. In particular, for suitable coordinates and order on $T_d$, the Veronese ideal $V_d(I)$ has quadratic initial ideal for $d\geq reg(I)/2$, and it follows that the Veronese subring $T_d/V_d(I)\subset S/I$ is Koszul for $d\geq reg(I)/2$. In this section we will estimate the regularity of $I$ in order to bound $\Delta(V_d(I))$ in terms of other invariants of $I$ such as $\Delta (I)$. These results can probably be improved, but we will give an example to show that the most optimistic hopes are false. \begin{thm} \label{delta} Let $r$ be the number of generators of the polynomial ring $S$. For any homogeneous ideal $I\subset S$, $\Delta (V_d(I))\leq max(2,\lceil (r\Delta (I)-r+1)/d \rceil )$. In particular, for suitable coordinates and order on $T_d$, the Veronese ideal $V_d(I)$ has quadratic initial ideal for $d\geq (r\Delta (I)-r+1)/2$. \end{thm} {\bf Proof:} The Taylor resolution [Ta60] gives an upper bound on $reg(I)$, specifically: $$reg(I) \leq r\Delta(I)-r+1.$$ With Theorem \ref{reg-thm}, this gives the result. It is worth mentioning that, by Bayer and Stillman's Theorem \ref{m-reg}, there are actually upper and lower bounds relating $reg(I)$ and $\Delta (I)$: $$\Delta (I) \leq reg(I) \leq r\Delta (I)-r+1.\mbox{ }\ \vrule width2mm height3mm \vspace{3mm} $$ The assumption $d\geq \Delta (I)/2$ is not enough to imply that $V_d(I)$ has quadratic initial ideal, by the example at the end of this section. We do not know the best estimate for $\Delta (V_d(I))$ in terms of $\Delta (I)$. The problem is combinatorial in the sense that it suffices to consider monomial ideals $I$. We have been assuming that the field $k$ is infinite. For arbitrary (in particular, finite) fields $k$, we have a slightly weaker version of Theorem \ref{delta}: there is an order on $T_d$ such that $V_d(I)$ has quadratic initial ideal for $d\geq r\lceil \Delta (I)/2 \rceil$. We omit the proof, which is not too difficult given a definition of the correct order. The ordering which yields this result is defined as follows: \begin{define} For each monomial $m$ in $S$ of degree $d$, we produce a vector $$\nu(m) = (\nu_{11}(m),\nu_{12}(m),\ldots,\nu_{1r}(m),\nu_{21}(m),\ldots, \nu_{2r}(m),\nu_{31}(m)\ldots),$$ where $\nu_{ij}(m) = \cases{ 0 & if $x_j^i | m$ \cr 1 & else}$. The order on monomials in $S$ of degree $d$ is then defined by $m > n$ if $\nu(m) > \nu(n)$ in lexicographic order. We define the order of the variables in $T_d$ using the above order on $S$. Specifically, $z_m > z_n$ if $m > n$ in the order on $S$ defined above. Given this ordering on the variables in $T_d$, let the order on the monomials in $T_d$ be reverse lexicographic order. \ \vrule width2mm height3mm \vspace{3mm} \end{define} For some monomial ideals $I$, we can improve the Taylor bound on the regularity of $I$. First, since one direction of the proof of Proposition \ref{reg-stab} does not use the Borel-fixed hypothesis, we have: \begin{prop} \label{stab-quick} If $in(I)$ is generated in degrees $\leq u$ and $in(I)_u$ is combinatorially stable, then $$reg(I)\leq u.$$ \end{prop} Next we generalize the definition of combinatorial stability. \begin{define} Let $q$ be an integer. A monomial ideal $I$ is {\bf $q$-combinatorially stable}, if for every $m \in I$ and for each $j<max(m)$ there exists an integer $s$ with $1 \leq s \leq q$ such that $x_j^{s}/x_{max(m)}^{s}m \in I$. \end{define} \begin{prop} Let $I$ be an ideal in generic coordinates, and let $e=$ \noindent $\delta(in_>(I))$, where $>$ is reverse lexicographic order. If $I$ is $q$-combinatorially stable, then $reg(I) \leq e+(r-1)(q-1)$. \end{prop} {\bf Proof:} Let $t = e+(r-1)(q-1)$. By Proposition \ref{stab-quick}, we need only show that $J := in(I)_t$ is combinatorially stable. Let $m \in J$. $m = x_1^{b_1+c_1} \cdots x_r^{b_r+c_r}$, where $l := x_1^{b_1} \cdots x_r^{b_r} \in in(I)_e$, and set $n := x_1^{c_1} \cdots x_r^{c_r}$. We have $\sum_{i=1}^r c_i = (r-1)(q-1)$. If $max(m) = max(n)$, then $(x_i/x_{max(m)})m \in I_t$ for all $i$ with $1 \leq i \leq x_{max(m)}$ because $n$ is divisible by $x_{max(m)}$. If $max(m) > max(n)$, then $max(m) = max(l)$, and either there exists some index $k$ such that $c_k \geq q$, or else $c_j = q-1$ for all $j = 1,\ldots,n$. In the first case, we can rewrite $m$ as $l'n'$, where $l' = (x_k^s/x_{max(l)}^s)l$ and $n' = (x_{max(l)}^s/x_k^s) n$, for some $1 \leq s \leq q$. After doing so, $max(m) = max(n)$ and we conclude as before. In the second case, the degree of $x_i$ in $x_i/x_{max(m)}m$ is $b_i + c_i+1$, and $c_i+1 = q$. We may rewrite $(x_i/x_{max(m)})m$ as $(x_i^s/x_{max(l)}^s)ln'$, where $n'= (x_{max(l)}^{s-1}/x_i^{s-1})n$, and $1 \leq s \leq q$ is chosen so that $(x_i^s/x_{max(l)}^s)l \in I$. Thus $(x_i/x_{max(m)})m$ is in $I_t$.\ \vrule width2mm height3mm \vspace{3mm} If $char \ k = 0$, then every ideal $I$ in generic coordinates is Borel-fixed and hence 1-combinatorially stable. In this case, the proposition above yields $reg(I) \leq e$, and in fact, equality holds, as Bayer and Stillman proved in [BaSt87]. In $char \ k = p$, every ideal $I$ in generic coordinates has a $q$-combinatorially stable initial ideal for some $q$ that is a power of $p$ $\leq \delta(in(I))$ [Pa], but even if $q$ is chosen to be as small as possible, $reg(I)$ can be strictly less than $e+(r-1)(q-1)$. An example is the ideal $I = \{a^6,\,a^2b^4,\,a^2c^4,\,b^8,\,c^8\}$, which is 8-combinatorially stable (implying a bound of 22 on the regularity), but has regularity 16. Also, $I$ has a quadratic initial ideal in the Veronese embedding of degree 5, which is strictly less than the degree of 7 given by Theorem \ref{char0-thm}. As noted above, in characteristic 0 and generic coordinates, the regularity of $I$ is equal to $\delta(in(I))$, where the initial ideal is with respect to reverse lexicographic order. In characteristic $p$, we cannot replace $reg(I)$ with $\delta(in(I))$ in the statement of Theorem \ref{reg-thm}, as the following example illustrates. \begin{ex} \label{counterexample} 2: A Borel-fixed ideal $I \subset k[a,b]$, with $char \ k= 2$, $e:= \delta(I) = 6$, such that the algebra $T_3/V_3(I)$ is not Koszul. It follows that the initial ideal of the Veronese embedding of degree $\lceil e/2 \rceil = 3$ is not generated in degree 2 under any order and any generators for the graded algebra $T_3$. In fact the ideal defined below has the same properties for $k$ of characteristic 0, except that it is not Borel-fixed in characteristic 0. \end{ex} Let $I = (a^6, a^2b^4)$, and consider the embedding in degree $3 = \lceil 6/2 \rceil$. Let $A=T_3/V_3(I)$. Thus, in the obvious coordinates $y_i=a^{3-i}b^i$, $$A=k[y_0,y_1,y_2,y_3]/(y_0^2=0,y_0y_2=y_1^2,y_0y_3=y_1y_2, y_1y_3=y_2^2=0).$$ The graded vector space $Tor_3^A(k,k)$ is not entirely in degree 3: it has dimension 26 in degree 3 and dimension 2 in degree 4. So $A$ is not Koszul. In fact, under the induced order used throughout this paper, $in(V_3(I))$ requires 2 cubic generators. However, $in(V_4(I))$ is generated in degree 2. The regularity of $I$ is 9. \section{Resolution of multihomogeneous modules} \label{resolution} Fundamental to the discussion of rates above is the estimate of the rate for a monomial ideal given (without proof) by Backelin in [Ba86]. The case of quadratic monomials follows at once from the more precise result of Fr\"oberg in [Fr75]. Fr\"oberg's result was recently reexamined and reproved by Bruns, Herzog and Vetter [BrHeVe] using a different method. J\"urgen Herzog has pointed out to us that their method actually proves the entire result claimed by Backelin, in a somewhat strengthened form, and we now present this argument. Let $S=k[x_1,\dots,x_r]$ be a polynomial ring over a field $k$. We will regard $S$ as a $Z^r$-graded ring, graded by the monomials. Suppose that $I$ is a monomial ideal of $S$, and set $A := S/I$; the ring $A$ is again $Z^r$-graded. If $M$ is a finitely generated $Z^r$-graded module over $A$, then $M$ has a $Z^r$-graded minimal free resolution over $A$. The vector spaces ${\rm Tor}^A_i(k,M)$ are $Z^r$-graded. For the purpose of bounding degrees it is convenient to turn these multigradings into single gradings. Rather than simply using the total degree, we get a more refined result by defining weights, as follows: Let $w_1,\dots,w_r$ be non-negative real numbers. For any monomial $m = x_1^{\alpha_1}\cdots x_r^{\alpha_r}$ define the {\it weight} of $m$ to be $w(m) = \sum w_i\alpha_i$. Generalizing the definition of $t^A_i(M)$ used above we define $t^A_i(w, M)$ to be the maximal weight, with respect to $w$, of a nonzero vector in $Tor^A_i(k,M)$. We can estimate the $t^A_i(w, M)$ as follows. Given an ordered set $\{g_1,\ldots,g_s\}$ of generators of $M$ we get a filtration $$Ag_1 \subset Ag_1+Ag_2 \subset \cdots \subset Ag_1 + \cdots + Ag_s = M$$ of $M$ with quotients the cyclic modules $A/J_i$ where $J_i = ((g_1,\dots,g_{i-1}):g_i)$. The set of generators $\{g_i\}$ also gives rise to a surjection $\phi$ of a free $A$-module $A^s$ to $M$ sending the $i^{\rm th}$ basis element to $g_i$. It is easy to show that the kernel of $\phi$ has a filtration whose successive quotients are the ideals $J_i$ (see the proof of Theorem \ref{genBackthm}). Thus the weights of the generators of the $J_i$ added to the weights of the $g_i$ give a bound for $t^A_1(w, M)$ (we get a bound and not an exact result because the set of generators for the first syzygy of $M$ produced from sets of generators for the $J_i$ may not be minimal). Moreover, if we have a method for bounding the weights of syzygies of the $J_i$, we may continue this process. The following Lemma provides what we require: \begin{lem} \label{ideallem} Let $S=k[x_1,\dots,x_r]$ be a polynomial ring over a field $k$. Suppose that $I$ is an ideal of $S$, generated by monomials $n_1,\dots,n_t$, and set $A := S/I$. If $J = (m_1,\dots,m_s)\subset A$ is an ideal generated by the images $m_i$ of monomials $m'_i$ of $S$, then the quotient $((m_1,\dots,m_{s-1}):_Am_s)$ is generated by the images in $A$ of divisors of the monomials $m'_1,\ldots,m'_{s-1}$ and proper divisors of the monomials $n_1,\ldots,n_t$. \end{lem} {\bf Proof:} The quotient is the image in $A$ of $((n_1,\dots,n_t,m_1,\dots,m_{s-1}):_S m_s)$ and is thus generated by divisors of the monomials $n_1,\dots,n_t,m_1,\dots,m_{s-1}$. The divisors of the $n_i$ that are not proper go to zero in $A$.\ \vrule width2mm height3mm \vspace{3mm} Using Lemma \ref{ideallem} with the idea above we obtain: \begin{thm} \label{genBackthm} Let $S=k[x_1,\dots,x_r]$ be a polynomial ring over a field $k$, and let $w$ be a weight function on $S$ as above. Suppose that $I$ is an ideal of $S$, generated by monomials, and set $A := S/I$. Let $M$ be a $Z^r$-graded $A$-module with $Z^r$-homogeneous generators $\{g_1, \dots ,g_s\}$, of weights $\leq d$, and set $J_i = (Ag_1+\dots+Ag_{i-1}\,:\,_A\,g_i)$. If the $J_i$ are generated by elements of weight $\leq e$, and both these elements and the proper divisors of the generators of $I$ have weights $\leq f$, then for each integer $i \geq 1$ we have $$ t^A_{i}(w, M) \leq d+e+(i-1)f. $$ \end{thm} {\bf Proof:} We will inductively construct a (not necessarily minimal) free resolution $$ \cdots \, \rightarrow F_2 \rightarrow F_1 \rightarrow F_0 \rightarrow M \rightarrow 0 $$ such that the generators of $F_0$ have weights $\leq d$, the generators of $F_1$ have weights $\leq d+e$, and for $i\geq 2$ the weights of the generators of $F_i$ are $\leq d+e+(i-1)f$. Since the minimal (multigraded) free resolution is a summand of any free resolution, it follows that the weights of the $i^{\rm th}$ free module in the minimal resolution are also $\leq d+e+(i-1)f$, proving the desired inequality. Let $F_0$ be $Z^r$-graded free $A$-module with $s$ generators whose degrees match those of the $g_i$, so that the surjection $\phi_0:\, F_0\rightarrow M$ sending the $i^{\rm th}$ basis vector of $F_0$ to $g_i$ is multihomogeneous. The weights of the generators of $F_0$ are $\leq d$. We will prove by induction on $s$ that the module ${\rm ker}(\phi_0)$ admits a filtration with successive quotients isomorphic, up to a shift in multidegree, to the ideals $J_i$, and that the weight of the generators of this kernel are $\leq d+e$. If we define $F'_0$ to be $F_0$ modulo the first basis element, and define $M'$ by the short exact sequence $$ 0 \rightarrow Ag_1 \rightarrow M \rightarrow M' \rightarrow 0 $$ then by the snake lemma we get a short exact sequence $$ 0 \rightarrow J_1 \rightarrow {\rm ker}(\phi_0) \rightarrow {\rm ker}( \phi'_0) \rightarrow 0,$$ where $\phi'_0:\, F'_0 \rightarrow M'$ is the induced map. By induction ${\rm ker}(\phi'_0) $ has a filtration with quotients $J_2,\dots, J_s$, and generators of weights $\leq d+e$. This gives the desired filtration of ${\rm ker}(\phi_0) $. The weights of the generators of the copy of $J_1$ in the kernel are the weights of the monomials generating the ideal $J_1$ plus the weight of $g_1$, so they are also $\leq d+e$, and we are done. Using this filtration of ${\rm ker}(\phi_0)$, we define a free module $F_1$ whose generators have weights $\leq d+e$ and a map $\phi_1:\, F_1\rightarrow F_0$ sending the generators of $F_1$ to representatives $h_l$ in ${\rm ker}(\phi_0)$ of the generators of the successive quotients $J_i$. We now repeat the argument, replacing $M$ by ${\rm ker}(\phi_0)$ and $\phi_0$ by $\phi_1$. Lemma \ref{ideallem} applied to the ideals $J_i$ implies that the argument works as before if we replace $e$ by $f$: that is, ${\rm ker}(\phi_1)$ has a filtration with successive quotients isomorphic (up to a shift in mult-degree) to ideals with generators of weight $\leq f$. This allows us to construct $F_2$ with generators of degrees $\leq d+e+f$ that map onto generators $h_i$ of ${\rm ker}(\phi_0)$ such that the ideals $(Ah_1+\dots+Ag_{l-1}\,:\,_A\,g_l)$ have generators of weight $\leq f$. We may continue to repeat the argument, using the bound $f$ from the second step on, and constructing the desired resolution. \ \vrule width2mm height3mm \vspace{3mm} In the special case of the resolution of the residue class field $k$, we may take $J_1$ to be the maximal ideal, and we get Backelin's result referred to above: \begin{cor} Let $S=k[x_1,\dots,x_r]$ be a polynomial ring over a field $k$. Suppose that $I$ is an ideal of $S$, generated by monomials of degree $\leq f$, and set $A := S/I$. We have $$ t^A_{i}(k) \leq 1+(i-1)(f-1). $$ \end{cor} Note that Theorem \ref{genBackthm} does {\em not} prove much about the entries of the matrices in even a non-minimal resolution. With the ring $A$ as in the Theorem, it would be interesting to know whether there is a minimal free resolution (say of the residue class field ), with bases for the free modules occurring, such that the entries of the matrices representing the maps of the resolution with respect to the given bases all have low multidegree (or low weight). The rate bound given in the above theorem, together with the requirement that the resolution be minimal (so that each syzygy has weight at least 1) implies that the individual entries of the $i^{th}$ syzygy matrix must have weights bounded by $d+e+(i-1)f-i+1$. We do not know whether there always exists a free resolution with bases --- even non-minimal --- such that the entries appearing in the matrices are all proper divisors of the generators of $I$, or even of the least common multiple of the generators of $I$. \section{Segre Products of Veronese embeddings} \label{multi} The proofs of Proposition \ref{prop8} and Theorem \ref{char0-thm} can be easily generalized to the Segre-Veronese case. Below are the definitions and statements we can make in this case. Let $$S:=k[x_{11},\ldots,x_{1r_1},\ldots,x_{s1},\ldots,x_{sr_s}]$$ be the coordinate ring of $\P {r_1} \times \cdots \times \P {r_s}$, where $x_i = (x_{i1},\ldots,x_{ir_i})$ are the homogeneous coordinates on $\P {r_i}$. And let $$T:=k[\{z_{m}\}, m \mbox{ a monomial of $S$ of multi-degree }(d_1,\ldots,d_s)]$$ be the coordinate ring of $\P N$. \begin{define} $\phi :T \longrightarrow S$ by $\phi(z_{m}) = m = m_1\cdots m_s$, where $m_i = x_{i1}^{\alpha_{i1}} \cdots x_{is}^{\alpha_{is}}$. \end{define} ${\rm ker}(\phi)$ is generated by the quadratic binomials of the form $z_{m}z_{n}-z_{m'}z_{n'},$ where $m\cdot n = m' \cdot n'$ in $S$. As in section \ref{vero}, if $a$, $b$ are monomials in $T$, $a > b$ if $\phi(a) > \phi(b)$, or $\phi(a) = \phi(b)$ and $a>b$ in reverse lexicographic order. \begin{prop} In reverse lexicographic order, the initial terms of the binomials $z_{m}z_{n}-z_{m'}z_{n'}$ generate $in({\rm ker}(\phi))$. \end{prop} \begin{define} The {\bf stabilization} $\{I\}$ of an ideal $I$ is defined to be the ideal generated by $$\{(x_j/x_{max(m)})m \ | m \in I, \ j=1,\ldots,max(m)\}.$$ \end{define} \begin{define} Call a multi-homogeneous monomial ideal $I$ {\it combinatorially stable} if it is combinatorially stable in each set of variables independently. That is, given $$x_1^{\alpha_1}\cdots x_s^{\alpha_s} \in I,$$ we must have $$\{x_1^{\alpha_1}\}\cdot \{x_2^{\alpha_2}\} \cdot \cdots \cdot \{x_s^{\alpha_s}\} \subset I,$$ where $\{x_i^{\alpha_i}\}$ is the set of all monomials necessary for an ideal in $k[x_{i0},\ldots,x_{ir_i}]$ containing $x_i^{\alpha_i}$ to be combinatorially stable, and where the above product is the outer product, i.e. all possible products of elements taken one from each set. \end{define} \begin{thm} If $I$ is a multi-homogeneous ideal whose initial ideal is combinatorially stable, then $in(\sigma(I)) = \sigma(in(I))$ (where the initial terms on the left are computed with respect to the induced order with ties broken by reverse lex). Thus if $in(I)$ is generated in degrees $\leq (u_1,\ldots,u_s)$ ($u_i$ = maximum degree of any generator with respect to the $i^{th}$ set of variables), then $in(V(I))$ is generated in degrees $\leq max(2,\lceil u_1/d_1 \rceil,\ldots,\lceil u_s/d_s \rceil)$. \end{thm} \section{ Another obstruction to having a quadratic initial ideal} \label{obstsec} In this section, we formulate a general obstruction to the existence of a quadratic initial ideal for a given polynomial ideal, beyond the obvious requirement that the ideal must be generated by quadratic polynomials, and even beyond the stronger requirement that the quotient ring must be a Koszul algebra. We use the obstruction to show that in certain dimensions, the ideal of a generic complete intersection of quadrics has no quadratic initial ideal, although every complete intersection of quadrics is a Koszul algebra [BaFr85]. {\bf Note. }In discussing the existence of a quadratic initial ideal for a homogeneous ideal $I$ in a polynomial ring $S=k[x_1,\ldots ,x_r]$, we are asking whether there exists a set of coordinates $x_1',\ldots , x_r'$ (linear combinations of $x_1,\ldots ,x_r\in S_1$) and a monomial order with respect to which $in(I)$ is generated by quadratic polynomials. \begin{thm} \label{obstruction} Let $I$ be a homogeneous ideal in a polynomial ring $S$. Consider the Krull dimensions $r=dim(S)$, $n=dim(S/I)$, $e=r-n$. (Thus, if $n\geq 1$, $n$ is one more than the dimension of the projective variety defined by $I$.) If there are coordinates and a monomial order such that the initial ideal $in(I)$ has quadratic generators, then the ideal $I$ contains $e$ linearly independent quadratic elements of the form: \begin{eqnarray*} c_1x_1^2 &=&x_2L_{1,2}+\cdots +x_{e+n}L_{1,e+n} \\ &\vdots \mbox{ }\\ c_{e}x_e^2 &=& x_{e+1}L_{e,e+1}+\cdots +x_{e+n}L_{e,e+n} \end{eqnarray*} for some basis $x_1,\ldots ,x_{e+n}$ for $S_1$ and some $c_i\in k$ and linear forms $L_{ij}\in S_1$. In particular, $I$ contains an $m$-dimensional space of quadrics of rank $\leq 2(n+m)-1$ for every $m \leq codim(I)$. \end{thm} We recall that the rank of a quadratic form $Q$ over a field $k$ is the rank of a symmetric matrix representing the form. {\bf Proof. }We are given that there is a basis $x_1,\ldots ,x_{e+n}$ for the vector space $S_1$ and an ordering of the $x$-monomials, such that the resulting initial ideal $in(I)$ is generated by $in(I)_2$. We can assume that $x_1>\cdots >x_{e+n}$ in the monomial ordering. We observe that for $i=1,\ldots ,e$, there must be at least $i$ quadratic monomials $x_jx_k$ with $e-i+1 \leq j,k \leq e+n$ which are not allowable. Otherwise the Hilbert series of $S/I$ would be at least equal to the Hilbert series of an algebra $k[x_{e-i+1},\ldots x_{e+n}]/(<i \mbox{ relations})$, so the dimension of $S/I$ would be at least that of the latter ring, which is greater than $n$; this contradicts $dim(S/I)=n$. Thus, for $i=1,\ldots ,e$, there are $i$ monomials $x_jx_k$, $e-i+1\leq j,k\leq e+n$, which are linear combinations of earlier monomials $x_lx_m$. No matter what monomial ordering we are using, at least one of $l$ and $m$ must be $> e-i+1$ in this situation. So, for all $1\leq i\leq e$, $R$ satisfies $i$ independent relations of the form: \begin{eqnarray*} bx_{e-i+1}^2+(a_{e-i+2,1}x_{e-i+2}x_1+\cdots +a_{e-i+2,e+n}x_{e-i+2}x_{e+n}) \cr +(a_{e-i+3,1}x_{e-i+3}x_1+\cdots +a_{e-i+3,e+n}x_{e-i+3}x_{e+n}) +\cdots \quad \quad \cr \quad \quad +(a_{e+n,1}x_{e+n}x_1+\cdots +a_{e+n,e+n}x_{e+n}^2)=0. \end{eqnarray*} This implies the statement of the lemma. \ \vrule width2mm height3mm \vspace{3mm} \begin{cor} Let $k$ be an infinite field, and let $I$ be an ideal in $S=k[x_1,\ldots ,x_{e+n}]$ generated by $e$ generic quadratic forms defined over $k$. We assume that $n\geq 0$. (If $n\geq 1$, $S/I$ is the homogeneous coordinate ring of an $(n-1)$-dimensional complete intersection of quadrics in {\it $\mbox{\bf P} ^{e+n-1}$}.) If $$n< \frac{(e-1)(e-2)}{6}$$ then generic complete intersection ideals $I$ as above do not admit any quadratic initial ideal. \end{cor} {\bf Proof. } Any $n$-dimensional complete intersection of homogeneous quadrics in affine $(e+n)$-space can be described by a point of the Grassmannian of $e$-dimensional subspaces of $S^2V$, $\mbox{Gr}(X_e\subset S^2V)$, where we let $V=S_1$, a vector space of dimension $e+n$ over $k$; conversely, a nonempty open subset of this Grassmannian corresponds to complete intersections. Those $e$-dimensional linear spaces of quadrics which generate an ideal which admits a quadratic initial ideal can, by Lemma \ref{obstruction}, be written in the form: \begin{eqnarray*} c_1x_1^2 &=&x_2L_{1,2}+\cdots +x_{e+n}L_{1,e+n} \\ &\vdots & \quad \quad \quad \quad (*)\\ c_{e}x_e^2 &=& x_{e+1}L_{e,e+1}+\cdots +x_{e+n}L_{e,e+n} \end{eqnarray*} for some basis $x_1,\ldots, x_{e+n}$ for $V$ and some $c_i\in k$ and linear forms $L_{ij}\in V$. We want an upper bound for the dimension of the space of $e$-dimensional linear subspaces of $S^2V$ which can be written in this form. If our bound is less than the dimension of the whole Grassmannian $\mbox{Gr}(X_e\subset S^2V)$, then we will know that generic complete intersections of quadrics in this dimension do not have any quadratic initial ideal. Our dimension estimate (which is not always sharp) is based on the following observation. The basis $x_1,\ldots ,x_{e+n}$ for $V$ is not important, only the flag $\langle x_{e+n},\ldots,x_{e+1}\rangle \subset \langle x_{e+n}, \ldots ,x_{e}\rangle \subset \cdots \subset \langle x_{e+n},\ldots ,x_{1}\rangle =V$. That is, if a linear system of quadrics has the form $(*)$ for one basis $x_1,\ldots ,x_{e+n}$ of $V$, then it has the same form for any basis which gives the same flag $\langle x_{e+1},\ldots,x_{e+n}\rangle \subset\cdots $. For any flag $V_n\subset V_{n+1}\subset \cdots \subset V_{e+n}=V$, we consider an associated flag $W_{a_1}\subset W_{a_2} \subset\cdots\subset W_{a_e}=S^2V$ defined by $$W_{a_i}=(V_{n+i-1}\cdot V)+(V_{n+i}\cdot V_{n+i})\subset S^2V.$$ (That is, in terms of any basis $x_1,\ldots ,x_{e+n}$ adapted to the flag $V_n\subset\cdots $, $W_{a_i}$ is the space of quadrics of the form $c_{e-i+1}x_{e-i+1}^2=x_{e-i+2}L_{e-i+2}+\cdots +x_{e+n}L_{e+n}$, $L_j\in V$.) {\bf Note. }We always use the notation $V_i$, $W_i$, etc.\ to denote $i$-dimensional vector spaces. Then, also associated to any flag $V_n\subset V_{n+1}\subset \cdots \subset V_{e+n}=V$, we can consider the space of flags $X_1\subset \cdots \subset X_e\subset V$ such that $X_i\subset W_{a_i}$ for $i=1,\ldots ,e$. Let $Q_{n,e}$ be the space of flags $V_n\subset V_{n+1}\subset \cdots \subset V_{e+n}=V$ and $X_1\subset \cdots \subset X_e\subset S^2V$ such that $X_i\subset W_{a_i}$ for $i=1,\ldots,e$. Then the image of the map \begin{eqnarray*} Q_{n,e}&\rightarrow &\mbox{Gr}(X_e\subset S^2V) \\ (V_i,X_i) &\mapsto &X_e \end{eqnarray*} contains the set of complete intersections of $e$ homogeneous quadrics in $(e+n)$-space which have a quadratic initial ideal. Moreover, it is easy to compute the dimension of $Q_{n,e}$, which is an iterated projective bundle over the flag manifold $\mbox{Fl} (V_n\subset V_{n+1}\subset\cdots\subset V_{e+n}=V)$: given a flag $V_n\subset\cdots $ and hence the associated subspaces $W_{a_i}$, we first choose the line $X_1\subset W_{a_1}$, then a line $X_2/X_1\subset W_{a_2}/X_1$, and so on. The dimension of the vector space $W_{a_i}$, for $i=1,\ldots,e$, is \begin{eqnarray*} a_i&= &(e+n)+(e+n-1)+\cdots +(e-i-2)+1 \\ &= & (e+n)(e+n+1)/2 -(e-i+1)(e-i+2)/2 +1. \end{eqnarray*} Using this, we compute the dimension of the variety $Q_{n,e}$: \begin{eqnarray*} \mbox{dim }Q_{n,e} &=& \mbox{dim Fl}(V_n\subset\cdots\subset V_{e+n}=V) +\sum_{i=1}^e \mbox{dim }\mbox{\bf P} (W_{a_i}/X_{i-1}) \\ &=& en+\frac{e(e-1)}{2} +\sum_{i=1}^e[\frac{(e+n)(e+n+1)}{2} -\frac{(e-i+1)(e-i+2)}{2}+1-i] \\ &=& en +\frac{e(e-1)}{2}+\frac{e(e+n)(e+n+1)}{2}-(\sum_{j=1}^e \frac{j(j+1)}{2} )+e-\frac{e(e+1)}{2} \\ &=& e(n+\frac{(e+n)(e+n+1)}{2}-\frac{(e+1)(e+2)}{6}). \end{eqnarray*} So \begin{eqnarray*} & &\mbox{dim }Q_{n,e}<\mbox{dim Gr}(X_e\subset S^2V) \\ &\iff &e(n+(e+n)(e+n+1)/2-(e+1)(e+2)/6)<e((e+n)(e+n+1)/2-e) \\ &\iff &n<(e+1)(e+2)/6-e\\ &\iff &n<(e-1)(e-2)/6 \end{eqnarray*} Thus, if $n<(e-1)(e-2)/6$, then the image of the map $Q_{n,e}\rightarrow\mbox{Gr}(X_e\subset S^2V)$ has dimension less than the dimension of $\mbox{Gr}(X_e\subset S^2V)$. Thus for $n<(e-1)(e-2)/6$, the ideal generated by $e$ generic quadratic forms in $e+n$ variables has no quadratic initial ideal. \ \vrule width2mm height3mm \vspace{3mm} \vspace{.2in} For example, the ideal generated by 3 generic quadratic forms in 3 variables has no quadratic initial ideal. Similarly for a generic complete intersection of 5 quadrics in $\mbox{\bf P} ^5$, or a generic complete intersection of 6 quadrics in $\mbox{\bf P} ^7$. (These last examples are smooth curves of genus 129.) Explicitly, say over a field $k$ of characteristic 0, the ideal $$I=(x(x+y),y(y+z),z(z+x))\subset k[x,y,z]=S$$ is a complete intersection of quadrics, and so $S/I$ is a Koszul algebra [BaFr85], but one can check using Lemma \ref{obstruction} that it has no quadratic initial ideal, for any coordinates and any monomial order. (One has to check that no nonzero linear combination of the relations $x(x+y)$, $y(y+z)$, $z(z+x)$ is the square of a linear form, which is easy.)

9310

alg-geom/9310004

en

https://arxiv.org/abs/alg-geom/9310004

alg-geom/9310004

Victor Batyrev

Victor V. Batyrev

Quantum Cohomology Rings of Toric Manifolds

23 pages, Latex

We compute the quantum cohomology ring $H^*_{\varphi}({\bf P}, {\bf C})$ of an arbitrary $d$-dimensional smooth projective toric manifold ${\bf P}_{\Sigma}$ associated with a fan $\Sigma$. The multiplicative structure of $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ depends on the choice of an element $avarphi$ in the ordinary cohomology group $H^2({\bf P}_{\Sigma}, {\bf C})$. There are many properties of the quantum cohomology rings $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ which are supposed to be valid for quantum cohomology rings of K\"ahler manifolds

alg-geom

\section{Introduction} \hspace*{\parindent} The notion of the {\em quantum cohomology ring} of a K\"ahler manifold $V$ naturally appears in the consideration of the so called {\em topological sigma model} associated with $V$ (\cite{witten}, 3a-b). If the canonical line bundle ${\cal K}$ of $V$ is negative, then one recovers the multiplicative structure of the quantum cohomology ring of $V$ from the intersection theory on the moduli space ${\cal I}_{\lambda}$ of holomorphic mappings $f$ of the complex sphere $f\;: \; S^2 \cong {\bf CP}^1 \rightarrow V$ where $\lambda$ is the homology class in $H_2(V, {\bf Z})$ of the image $f({\bf CP}^1)$. If the canonical bundle ${\cal K}$ is trivial, the quantum cohomology ring was considered by Vafa as a important tool for explaining the mirror symmetry for Calabi-Yau manifolds \cite{vafa}. The quantum cohomology ring $QH_{\varphi}(V, {\bf C})$ of a K\"ahler manifold $V$, unlike the ordinary cohomology ring, have the multiplicative structure which depends on the class $\varphi$ of the K\"ahler $(1,1)$-form corresponding to a K\"ahler metric $g$ on $V$. When we rescale the metric $g \rightarrow t g$ and put $t \rightarrow \infty$, the quantum ring "becomes" the classical cohomology ring. For example, for the topological sigma model on the complex projective line ${\bf CP}^1$ itself, the classical cohomology ring is generated by the class $x$ of a K\a"ahler $(1,1)$-form, where $x$ satisfies the quadratic equation \begin{equation} x^2 = 0, \label{first} \end{equation} while the quantum cohomology ring is also generated by $x$, but the equation satisfied by $x$ is different: \begin{equation} x^2 = {\rm exp}(-\int_{\lambda} \varphi), \label{second} \end{equation} $\lambda$ is a non-zero effective $2$-cycle. Similarly, the quantum cohomology ring of $d$-dimensional complex projective space is generated by the element $x$ satisfying the equation \begin{equation} x^{d+1} = {\rm exp}(-\int_{\lambda} \varphi ). \label{third} \end{equation} The main purpose of this paper is to construct and investigate the quantum cohomology ring $QH^*_{\varphi}({\bf P}_{\Sigma},{\bf C})$ of an arbitrary smooth compact $d$-dimensional toric manifold ${\bf P}_{\Sigma}$, where $\varphi$ is an element of the ordinary second cohomology group $H^2({\bf P}_{\Sigma}, {\bf C})$. Since all projective spaces are are toric manifolds, we obtain a generalization of above examples of quantum cohomology rings. According to the physical interpretation, a quantum cohomology ring is a closed operator algebra acting on the fermionic Hilbert space. For example, the equation \ref{third} one should better write as an equations for the linear operator ${\cal X}$ corresponding to the cohomology class $x$: \begin{equation} {\cal X}^{d+1} = {\rm exp}(-\int_{\lambda} \varphi \,) id . \label{four} \end{equation} It is in general more convenient to define quantum rings by polynomial equations among generators. \begin{opr} {\rm Let \[ h(t,x) = \sum_{n \in {\cal N}} c_n(t) x^n \] be a one-parameter family of polynomials in the polynomial ring ${\bf C}\lbrack x \rbrack$, where $x = \{x_i \}_{i \in I}$ is a set of variables indexed by $I$, $t$ is a positive real number, ${\cal N}$ is a fixed finite set of exponents. We say that the polynomial \[ h^{\infty}(x) = \sum_{n \in {\cal N}} c_n^{\infty} x^n \] is the limit of $h(t,x)$ as $t \rightarrow \infty$, if the point $\{ c_n^{\infty} \}_{n \in {\cal N}}$ of the $(\mid {\cal N} \mid -1)$-dimensional complex projective space is the limit of the one-parameter family of points with homogeneous coordinates $\{ c_n(t) \}_{n \in {\cal N}}$. } \end{opr} \begin{opr} {\rm Let $R_{t}$ be a one-parameter family of commutative algebras over {\bf C} with a fixed set of generators $\{ r_i \}$, $t \in {\bf R}_{>0}$. We denote by $J_{t}$ the ideal in ${\bf C} \lbrack x \rbrack$ consisting of all polynomial relations among $\{ r_i \}$, i.e., the kernel of the surjective homomorphism ${\bf C} \lbrack x \rbrack \rightarrow R_t$. We say that the ideal $J^{\infty}$ is the limit of $J_t$ as $t \rightarrow \infty$, if any one-parameter family of polynomials $h(t,x) \in J_t$ has a limit, and $J^{\infty}$ is generated as ${\bf C}$-vector space by all these limits. The ${\bf C}$-algebra \[ R^{\infty} = {\bf C} \lbrack x \rbrack / J^{\infty} \] will be called the {\em limit} of $R_t$. } \end{opr} \begin{rem} {\rm In general, it is not true that if $J^{\infty} = \lim_{t \rightarrow \infty} J_t$, and $J_t$ is generated by a finite set of polynomials $\{ h_1(t,x) \ldots ,h_k(t,x) \}$, then $J^{\infty}$ is generated by the limits $\{ h_1^{\infty}(x), \ldots , h_k^{\infty}(x) \}$. The limit ideal $J^{\infty}$ is generated by the limits $h_i^{\infty}(x)$ {\em only if} the set of polynomials $\{ h_i(t,x) \}$ form a Gr\"obner-type basis for $J_t$. } \end{rem} In this paper, we establish the following basic properties of quantum cohomology rings of toric manifolds: \bigskip {\bf I} : If $\varphi$ is an element in the interior of the K\"ahler cone $K({\bf P}_{\Sigma}) \subset H^2({\bf P}_{\Sigma}, {\bf C})$, then there exists a limit of $QH^*_{t\varphi}({\bf P}_{\Sigma},{\bf C})$ as $t \rightarrow \infty$, and this limit is isomorphic to the ordinary cohomology ring $H^*({\bf P}_{\Sigma},{\bf C})$ (Corollary \ref{kon}). {\bf II} : Assume that two smooth projective toric manifolds ${\bf P}_{\Sigma_1}$ and ${\bf P}_{\Sigma_2}$ are isomorphic in codimension 1, for instance, that ${\bf P}_{\Sigma_1}$ is obtained from ${\bf P}_{\Sigma_2}$ by a flop-type birational transformation. Then the natural isomorphism $H^2({\bf P}_{\Sigma_1}, {\bf C}) \cong H^2({\bf P}_{\Sigma_2}, {\bf C})$ induces the isomorphism between the quantum cohomology rings \[ QH^*_{\varphi}({\bf P}_{\Sigma_1},{\bf C}) \cong QH^*_{\varphi}({\bf P}_{\Sigma_2},{\bf C}) \] (Theorem \ref{flop}). We notice that the ordinary cohomology rings of ${\bf P}_{\Sigma_1}$ and ${\bf P}_{\Sigma_2}$ are not isomorphic in general. {\bf III} : Assume that the first Chern class $c_1({\bf P}_{\Sigma})$ of ${\bf P}_{\Sigma}$ belongs to the closed K\"ahler cone $K({\bf P}_{\Sigma}) \subset H^2({\bf P}_{\Sigma}, {\bf C})$. Then the ring $QH^*_{\varphi}({\bf P}_{\Sigma},{\bf C})$ is isomorphic to the Jacobian ring of a Laurent polynomial $f_{\varphi}(X)$ such that the equation $f_{\varphi}(X) = 0$ defines an affine Calabi-Yau hypersurface $Z_f$ in the $d$-dimensional algebraic torus $({\bf C}^*)^d$ where $Z_f$ is "mirror symmetric" with respect to Calabi-Yau hypersurfaces in ${\bf P}_{\Sigma}$ (Theorem \ref{mirr}). Here by the "mirror symmetry" we mean the correspondence, based on the polar duality {\rm \cite{bat.mir}}, between families of Calabi-Yau hypersurfaces in toric varieties. The properties II and III give a general view on the recent result of P. Aspinwall, B. Greene, and D. Morrison \cite{aspin4} who have shown, for a family of Calabi-Yau $3$-folds $W$ that their quantum cohomology ring $QH^*_{\varphi}(W, {\bf C})$ does not change under a flop-type birational transformation (see also \cite{aspin1,aspin3}). \medskip {\bf IV}: Assume that the first Chern class $c_1({\bf P}_{\Sigma})$ of ${\bf P}_{\Sigma}$ is divisible by $r$, i.e., there exists a an element $h \in H^2({\bf P}_{\Sigma}, {\bf Z})$ such that $c_1({\bf P}_{\Sigma}) = rh$. Then $QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ has a natural ${\bf Z}_r$-grading (Theorem \ref{divis}). We remark that $QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ has no ${\bf Z}$-grading. \medskip The paper is organized as follows. In Sections 2-4, we recall the definition and standard information about toric manifolds. In Section 5, we define the quantum cohomology ring of toric manifolds and prove their properties. In Section 6, we consider examples of the behavior of quantum cohomology rings under elementary birational transformations such as blow-up and flop, we also consider the case of singular toric varieties. In Section 7, we give an combinatorial interpretation of the relation between the quantum cohomology rings and the ordinary cohomology rings. In Section 8, we show that the quantum cohomology ring can be interpreted as a Jacobian ring of some Laurent polynomial. Finally, in Section 9, we prove that our quantum cohomology rings coincide with the quantum cohomology rings defined by $\sigma$-models on toric manifolds. \bigskip {\bf Acknowledgements.} It is a pleasure to acknowledge helpful discussions with Yu. Manin, D. Morrison, Duco van Straten as well as with S. Cecotti and C. Vafa. I would like to express my thanks for hospitality to the Mathematical Sciences Research Institute where this work was conducted and supported in part by the National Science Foundation (DMS-9022140), and the DFG (Forschungsscherpunkt Komplexe Mannigfaltigkeiten). \section{A definition of compact toric manifolds} \hspace*{\parindent} The {\em toric varieties} were considered in full generality in \cite{dan2,oda1}. For the general definition of toric variety which includes affine and quasi-projective toric varieties with singularities, it is more convenient to use the language of schemes. However, for our purposes, it will be sufficient to have a simplified more classical version of the definition for {\em smooth and compact toric varieties} over the complex number field ${\bf C}$. In fact, this approach to compact toric manifolds was first proposed by M. Audin \cite{audin}, and developed by D. Cox \cite{cox}. \bigskip In order to obtain a $d$-dimensional compact toric manifold $V$, we need a combinatorial object $\Sigma$, a {\em complete fan of regular cones}, in a $d$-dimensional vector space over ${\bf R}$. \bigskip Let $N$, $M = {\rm Hom}\, (N, {\bf Z})$ be dual lattices of rank $d$, and $N_{\bf R}$, $M_{\bf R}$ their ${\bf R}$-scalar extensions to $d$-dimensional vector spaces. \begin{opr} {\rm A convex subset $\sigma \subset N_{\bf R}$ is called a {\em regular $k$-dimensional cone} $(k \geq 1)$ if there exist $k$ linearly independent elements $v_1, \ldots, v_k \in N$ such that \[ \sigma = \{ \mu_1 v_1 + \cdots + \mu_k v_k \mid \mu_i \in {\bf R}, \mu_i \geq 0 \}, \] and the set $\{ v_1, \ldots, v_k \}$ is a subset of some ${\bf Z}$-basis of $N$. In this case, we call $v_1, \ldots, v_k \in N$ the {\em integral generators of} $\sigma$. The origin $0 \in N_{\bf R}$ we call the {\em regular $0$-dimensional cone}. By definition, the set of integral generators of this cone is empty.} \end{opr} \begin{opr} {\rm A regular cone $\sigma'$ is called {\em a face} of a regular cone $\sigma$ (we write $\sigma' \prec \sigma)$ if the set of integral generators of $\sigma'$ is a subset of the set of integral generators of $\sigma$. } \label{face} \end{opr} \begin{opr} {\rm A finite system $\Sigma = \{ \sigma_1, \ldots , \sigma_s \}$ of regular cones in $N_{\bf R}$ is called {\em a complete d-dimensional fan} of regular cones, if the following conditions are satisfied: (i) if $\sigma \in \Sigma$ and $\sigma' \prec \sigma$, then $\sigma' \in \Sigma$; (ii) if $\sigma$, $\sigma'$ are in $\Sigma$, then $\sigma \cap \sigma' \prec \sigma$ and $\sigma \cap \sigma' \prec \sigma'$; (iii) $N_{\bf R} = \sigma_1 \cup \cdots \cup \sigma_s$. \\ The set of all $k$-dimensional cones in $\Sigma$ will be denoted by $\Sigma^{(k)}$. } \label{def.fan} \end{opr} \begin{exam} {\rm Choose $d+1$ vectors $ v_1, \ldots, v_{d+1}$ in a $d$-dimensional real space $E$ such that $E$ is spanned by $v_1, \ldots, v_{d+1}$ and there exists the linear relation \[ v_1 + \cdots + v_{d+1} =0. \] Define $N$ to be the lattice in $E$ consisting of all integral linear combinations of $v_1, \ldots, v_{d+1}$. Obviously, $N_{\bf R} = E$. Then any $k$-element subset $I \subset \{ v_1, \ldots, v_{d+1} \}$ $(k \leq d)$ generates a $k$-dimensional regular cone $\sigma(I)$. The set $\Sigma(d)$ consisting of $2^{d+1} -1$ cones $\sigma(I)$ generated by $I$ is a complete $d$-dimensional fan of regular cones. } \label{weig} \end{exam} \begin{opr} {\rm (cf. \cite{bat.class}) Let $\Sigma$ be a complete $d$-dimensional fan of regular cones. Denote by $G(\Sigma) = \{ v_1, \ldots , v_n \}$ the set of all generators of $1$-dimensional cones in $\Sigma$ ($n = {\rm Card}\, \Sigma^{(1)}$). We call a subset ${\cal P} =\{ v_{i_1}, \ldots , v_{i_p} \} \subset G(\Sigma)$ a {\em primitive collection} if $\{v_{i_1}, \ldots , v_{i_p}\}$ is not the set of generators of a $p$-dimensional simplicial cone in $\Sigma$, while for all $k$ $(0 \leq k < p)$ each $k$-element subset of ${\cal P}$ generates a $k$-dimensional cone in $\Sigma$.} \end{opr} \begin{exam} {\rm Let $\Sigma$ be a fan $\Sigma(d)$ from Example \ref{weig}. Then there exists the unique primitive collection ${\cal P} = G(\Sigma(d))$.} \label{ex.prim} \end{exam} \begin{opr} {\rm Let ${\bf C}^n$ be $n$-dimensional affine space over ${\bf C}$ with the set of coordinates $z_1, \ldots, z_n$ which are in the one-to-one correspondence $z_i \leftrightarrow v_i$ with elements of $G(\Sigma)$. Let ${\cal P} =\{ v_{i_1}, \ldots , v_{i_p} \}$ be a primitive collection in $G(\Sigma)$. Denote by ${\bf A}({\cal P})$ the $(n-p)$-dimensional affine subspace in ${\bf C}^n$ defined by the equations \[ z_{i_1} = \cdots = z_{i_p} = 0. \]} \end{opr} \begin{rem} {\rm Since every primitive collection ${\cal P}$ has at least two elements, the codimension of ${\bf A}({\cal P})$ is at least $2$.} \label{cod.2} \end{rem} \begin{opr} {\rm Define the closed algebraic subset $Z(\Sigma)$ in ${\bf C}^n$ as follows \[ Z(\Sigma) = \bigcup_{\cal P} {\bf A}({\cal P}), \] where ${\cal P}$ runs over all primitive collections in $G(\Sigma)$. Put } \[ U(\Sigma) = {\bf C}^n \setminus Z(\Sigma). \] \end{opr} \begin{opr} {\rm Two complete $d$-dimensional fans of regular cones $\Sigma$ and $\Sigma'$ are called {\em combinatorially equivalent} if there exists a bijective mapping $\Sigma \rightarrow \Sigma'$ respecting the face-relation "$\prec$" (see \ref{face}).} \end{opr} \begin{rem} {\rm It is easy to see that the open subset $U(\Sigma) \subset {\bf C}^n$ depends only on the combinatorial structure of $\Sigma$, i.e., for any two combinatorially equivalent fans $\Sigma$ and $\Sigma'$, one has $U(\Sigma) \cong U(\Sigma')$. } \end{rem} \begin{opr} {\rm Let $R(\Sigma)$ be the subgroup in ${\bf Z}^n$ consisting of all lattice vectors $\lambda = (\lambda_1, \ldots, \lambda_n)$ such that $\lambda_1 v_1 + \cdots + \lambda_n v_n = 0$. } \end{opr} Obvioulsy, $R(\Sigma)$ is isomorphic to ${\bf Z}^{n-d}$. \begin{opr} {\rm Let $\Sigma$ be a complete $d$-dimensional fan of regular cones. Define ${\bf D}(\Sigma)$ to be the connected commutative subgroup in $({\bf C}^*)^n$ generated by all one-parameter subgroups \[ a_{\lambda}\;\; : \;\; {\bf C}^* \rightarrow ({\bf C}^*)^n, \;\; \] \[ t \rightarrow (t^{\lambda_1}, \ldots, t^{\lambda_n}) \] where $\lambda \in R(\Sigma)$.} \end{opr} \begin{rem} {\rm Choosing a {\bf Z}-basis in $R(\Sigma)$, one easily obtains an isomorphism between ${\bf D}(\Sigma)$ and $({\bf C}^*)^{n-d}$. } \end{rem} \bigskip Now we are ready to give the definition of the compact toric manifold ${\bf P}_{\Sigma}$ associated with a complete $d$-dimensional fan of regular cones $\Sigma$. \bigskip \begin{opr} {\rm Let $\Sigma$ be a complete $d$-dimensional fan of regular cones. Then quotient \[ {\bf P}_{\Sigma} = U(\Sigma)/{\bf D}(\Sigma) \] is called the {\em compact toric manifold associated with} $\Sigma$. } \end{opr} \begin{exam} {\rm Let $\Sigma$ be a fan $\Sigma(d)$ from Example \ref{weig}. By \ref{ex.prim}, $U(\Sigma(d)) = {\bf C}^{d+1} \setminus \{0 \}$. By the definition of $\Sigma(d)$, the subgroup $R(\Sigma) \subset {\bf Z}^n$ is generated by $(1, \ldots, 1) \in {\bf Z}^{d+1}$. Thus, ${\bf D}(\Sigma) \subset ({\bf C}^*)^n$ consists of the elements $(t, \ldots , t)$, where $t \in {\bf C}^*$. So the toric manifold associated with $\Sigma(d)$ is the ordinary $d$-dimensional projective space.} \end{exam} A priori, it is not obvious that the quotient space ${\bf P}_{\Sigma} = U(\Sigma)/{\bf D}(\Sigma)$ always exists as the space of orbits of the group ${\bf D}(\Sigma)$ acting free on $U(\Sigma)$, and that ${\bf P}_{\Sigma}$ is smooth and compact. However, these facts are easy to check if we take the $d$-dimensional projective space ${\bf P}_{\Sigma(d)}$ as a model example. \bigskip There exists a simple open covering of $U(\Sigma)$ by affine algebraic varieties: \begin{prop} Let $\sigma$ be a $k$-dimensional cone in $\Sigma$ generated by $\{ v_{i_1}, \ldots, v_{i_k} \}$. Define the open subset $U(\sigma) \subset {\bf C}^n$ by the conditions $ z_{j} \neq 0\;\; {\rm for\;\; all}\;\; j \not\in \{ i_1, \ldots , i_k \}$. Then the open subsets $U(\sigma)$ $(\sigma \in \Sigma)$ have the properties: {\rm (i)} \[ U(\Sigma) = \bigcup_{\sigma \in \Sigma} U(\sigma);\] {\rm (ii)} if $\sigma \prec \sigma'$, then $U(\sigma) \subset U(\sigma')$$;$ {\rm (iii)} for any two cone $\sigma_1, \sigma_2 \in \Sigma$, one has $ U(\sigma_1) \cap U(\sigma_2) = U(\sigma_1 \cap \sigma_2)$; in particular, \[U(\Sigma) = \bigcup_{\sigma \in \Sigma^{(d)}} U(\sigma). \] \end{prop} \begin{prop} Let $\sigma$ be a $d$-dimensional cone in $\Sigma^{(d)}$ generated by $\{ v_{i_1}, \ldots, v_{i_d} \} \subset N$. Denote by $ u_{i_1}, \ldots, u_{i_d}$ the dual to $ v_{i_1}, \ldots, v_{i_d}$ ${\bf Z}$-basis of the lattice $M$, i.e, $\langle v_{i_k}, u_{i_l} \rangle = \delta_{k,l}$, where $\langle * , * \rangle\;\;:\;\; N \times M \rightarrow {\bf Z}$ is the canonical pairing between lattices $N$ and $M$. Then the affine open subset $U(\sigma)$ is isomorphic to ${\bf C}^d \times ({\bf C}^*)^{n-d}$, the action of ${\bf D}(\Sigma)$ on $U(\sigma)$ is free, and the space of ${\bf D}(\Sigma)$-orbits is isomorphic to the affine space $U_{\sigma} = {\bf C}^d$ whose coordinate functions $x_1^{\sigma}, \ldots, x_d^{\sigma}$ are $d$ Laurent monomials in $z_1, \ldots, z_n$ $:$ \[ x_1^{\sigma} = z_1^{\langle v_1,u_{i_1} \rangle} \cdots , z_n^{\langle v_n,u_{i_1} \rangle}, \; \ldots \;, x_d^{\sigma} = z_1^{\langle v_1,u_{i_d} \rangle} \cdots z_n^{\langle v_n,u_{i_d} \rangle}. \] \end{prop} The last statement yields a general formula for the local affine coordinates $x_1^{\sigma}, \ldots, x_d^{\sigma}$ of a point $p \in U_{\sigma}$ as functions of its "homogeneous coordinates" $z_1, \ldots, z_n$ (see also \cite{cox}). Compactness of ${\bf P}_{\Sigma}$ follows from the fact that the local polydiscs \[ D_{\sigma} = \{ x \in U_{\sigma}\; : \; \mid x_1^{\sigma} \mid \leq 1, \dots, \mid x_d^{\sigma} \mid \leq 1 \}, \; \; \sigma \in \Sigma^{(d)} \] form a finite compact covering of ${\bf P}_{\Sigma}$. \section{Cohomology of toric manifolds} \hspace*{\parindent} Let $\Sigma$ be a complete $d$-dimensional fan of regular cones. \begin{opr} {\rm A continious function $\varphi\; :\; N_{\bf R} \rightarrow {\bf R}$ is called $\Sigma$-{\em piecewise linear}, if $\sigma$ is a linear function on every cone $\sigma \in \Sigma$.} \end{opr} \begin{rem} {\rm It is clear that any $\Sigma$-piecewise linear function $\varphi$ is uniquely defined by its values on elements $v_i$ of $G(\Sigma)$. So the space of all $\Sigma$-piecewise linear functions $PL(\Sigma)$ is canonically isomorphic to ${\bf R}^n$: $ \varphi \rightarrow (\varphi(v_1), \ldots, \varphi(v_n)).$} \end{rem} \begin{theo} The space $PL(\Sigma)/M_{\bf R}$ of all $\Sigma$\--piece\-wise li\-near func\-tions mo\-dulo the $d$-dimen\-sional sub\-space of glo\-bally li\-near func\-tions on $N_{\bf R}$ is canonically isomorphic to the cohomology space $ H^{2}({\bf P}_{\Sigma}, {\bf R})$. Moreover, the first Chern class $c_1({\bf P}_{\Sigma})$, as an element of $H^{2}({\bf P}_{\Sigma}, {\bf Z})$, is represented by the class of the ${\Sigma}$-piecewise linear function $\alpha_{\Sigma} \in PL(\Sigma)$ such that $\alpha_{\Sigma}(v_1) = \cdots = \alpha_{\Sigma}(v_n) = 1$. \end{theo} \begin{theo} Let $R(\Sigma)_{\bf R}$ be the ${\bf R}$-scalar extension of the abelian group $R(\Sigma)$. Then the space $R(\Sigma)_{\bf R}$ is canonically isomorphic to the homology space $H_{2}({\bf P}_{\Sigma}, {\bf R})$. \end{theo} \begin{opr} {\rm Let $\varphi$ be an element of $PL(\Sigma)$, $\lambda$ an element of $R(\Sigma)_{\bf R}$. Define the {\em degree of $\lambda$ relative to $\varphi$} as \[ {\rm deg}_{\varphi} (\lambda) = \sum_{i =1}^n \lambda_i \varphi(v_i). \] } \label{deg} \end{opr} It is easy to see that for any $\varphi \in M_{\bf R}$ and for any $\lambda \in R(\Sigma)_{\bf R}$, one has ${\rm deg}_{\varphi} (\lambda) = 0$. Moreover, the degree-mapping induces a nondegenerate pairing \[ {\rm deg}\; :\; PL(\Sigma)/M_{\bf R} \times R(\Sigma)_{\bf R} \rightarrow {\bf R} \] which coincides with the canonical intersection pairing \[H^{2}({\bf P}_{\Sigma}, {\bf R}) \times H_2({\bf P}_{\Sigma}, {\bf R}) \rightarrow {\bf R}. \] \begin{opr} {\rm Let ${\bf C}\lbrack z \rbrack$ be the polynomial ring in $n$ variables $z_1, \ldots, z_n$. Denote by $SR(\Sigma)$ the ideal in ${\bf C} \lbrack z \rbrack$ generated by all monomials \[ \prod_{v_j \in {\cal P}} z_j, \] where ${\cal P}$ runs over all primitive collections in $G(\Sigma)$. The ideal $SR(\Sigma)$ is usually called the {\em Stenley-Reisner ideal} of ${\Sigma}$. } \end{opr} \begin{opr} {\rm Let $u_1, \ldots, u_d$ be any ${\bf Z}$-basis of the lattice $M$. Denote by $P(\Sigma)$ the ideal in ${\bf C} \lbrack z \rbrack$ generated by $d$ elements \[ \sum_{i = 1}^{n} \langle v_i , u_1 \rangle z_i, \ldots , \sum_{i = 1}^{n} \langle v_i , u_d \rangle z_i. \]} \end{opr} Obviously, the ideal $P(\Sigma)$ does not depend on the choice of basis of $M$. \begin{theo} The cohomology ring of the compact toric manifold ${\bf P}_{\Sigma}$ is canonically isomorphic to the quotient of ${\bf C} \lbrack z \rbrack$ by the sum of two ideals $P(\Sigma)$ and $SR(\Sigma)$: \[ H^*({\bf P}_{\Sigma}, {\bf C}) \cong {\bf C} \lbrack z \rbrack / (P(\Sigma) + SR(\Sigma)). \] Moreover, the canonical embedding $ H^2({\bf P}_{\Sigma}, {\bf C}) \hookrightarrow H^*({\bf P}_{\Sigma}, {\bf C})$ is induced by the linear mapping \[ PL(\Sigma)\otimes_{\bf R} {\bf C} \rightarrow {\bf C}\lbrack z \rbrack, \;\; \varphi \mapsto \sum_{i =1}^n \varphi_i(v_i) z_i. \] In particular, the first Chern class of ${\bf P}_{\Sigma}$ is represented by the sum $ z_1 + \cdots + z_n$. \label{ord} \end{theo} \begin{exam} {\rm Let ${\bf P}_{\Sigma}$ be $d$-dimensional projective space defined by the fan $\Sigma(d)$ (see \ref{weig}). Then \[ P(\Sigma(d)) = < (z_1 - z_{d+1}), \ldots, (z_d - z_{d+1}) >, \] \[ SR(\Sigma(d)) = < \prod_{i=1}^{d+1} z_i >. \] So we obtain \[ {\bf C} \lbrack z_1, \ldots, z_{d+1} \rbrack / (P(\Sigma(d)) + SR(\Sigma(d)) \cong {\bf C} \lbrack x \rbrack / x^{d+1}. \] } \end{exam} \bigskip \section{Line bundles and K\"ahler classes} \hspace*{\parindent} Let \[ \pi \; : \; U({\Sigma}) \rightarrow {\bf P}_{\Sigma} \] be the canonical projection whose fibers are principal homogeneous spaces of ${\bf D}(\Sigma)$. For any line bundle ${\cal L}$ over ${\bf P}_{\Sigma}$, the pullback $\pi^* {\cal L}$ is a line bundle over $U(\Sigma)$. By \ref{cod.2}, $\pi^* {\cal L}$ is isomorphic to ${\cal O}_{U(\Sigma)}$. Therefore, the Picard group of ${\bf P}_{\Sigma}$ is isomorphic to the group of all ${\bf D}$-linearization of ${\cal O}_{U(\Sigma)}$, or to the group of all characters \[ \chi\; : \; {\bf D}(\Sigma) \rightarrow {\bf C}^*. \] The latter is isomorphic to the group ${\bf Z}^n /M$ where ${\bf Z}^n$ is the group of all ${\Sigma}$-piecewise linear functions $\varphi$ such that $\varphi (N) \subset {\bf Z}$. \begin{prop} Assume that a character $\chi$ is represented by the class of an integral ${\Sigma}$-piecewise linear function $\varphi$. Then the space \[ H^0 ({\bf P}_{\Sigma}, {\cal L}_{\chi} ) \] of global sections of the corresponding line bundle ${\cal L}_{\chi}$, is canonically isomorphic to the space of all polynomials $F(z_1, \ldots, z_n ) \in {\bf C} \lbrack z \rbrack$ satisfying the condition \[ F(t^{\lambda_1}z_1, \ldots, t^{\lambda_n}z_n ) = t^{{\rm deg}_{\varphi} \lambda} F(z_1, \ldots, z_n ) \] \[ {for \;\; all}\;\; \lambda \in R(\Sigma),\; t \in {\bf C}^*. \] The exponentials $(m_1, \ldots, m_n)$ of the monomials satisfying the above condition can be identified with integral points in the convex polyhedron: \[ \Delta_{\varphi} = \{ (x_1, \ldots, x_n) \in {\bf R}^n_{\geq 0} : {\rm deg}_{\varphi} \lambda = \lambda_1 x_1 + \cdots + \lambda_n x_n, \;\; \lambda \in R(\Sigma) \} \] \label{poly1} \end{prop} \begin{opr} {\rm A $\Sigma$-piecewise linear function $\varphi \in PL(\Sigma)$ is called a {\em strictly convex support} function for the fan $\Sigma$, if $\varphi$ satisfies the properties (i) $\varphi$ is an {\em upper convex function}, i.e., \[ \varphi(x) + \varphi(y) \geq \varphi(x +y ); \] (ii) for any two different $d$-dimensional cones $\sigma_1$, $\sigma_2 \in \Sigma$, the restrictions $\varphi {\mid}_{\sigma}$ and $ \varphi {\mid}_{\sigma ' }$ are different linear functions. } \label{support} \end{opr} \begin{prop} If $\varphi$ is a strictly convex support function, then the polyhedron $\Delta_{\varphi}$ is simple $($ i.e., any vertex of $\Delta_{\varphi}$ is contained in $d$-faces of codimension $1$$)$, and the fan $\Sigma$ can be uniquely recovered from $\Delta_{\varphi}$ using the property: \[ \Delta_{\varphi} \cong \{ x \in M_{\bf R}\; : \; \langle v_i, x \rangle \geq -\varphi({v_i}) \}. \] \end{prop} \begin{opr} {\rm Denote by $K(\Sigma)$ the cone in $H^{2}({\bf P}_{\Sigma}, {\bf R}) = PL(\Sigma)/M_{\bf R}$ consisting of the classes of all upper convex ${\Sigma}$-piecewise linear functions $\varphi \in PL(\Sigma)$. We denote by $K^0(\Sigma)$ the interior of $K(\Sigma)$, i.e., the cone consisting of the classes of all strictly convex support functions in $PL(\Sigma)$.} \end{opr} \begin{theo} The open cone $K^0(\Sigma) \subset H^{2}({\bf P}_{\Sigma}, {\bf R})$ consists of classes of K\"ahler $(1,1)$-forms on ${\bf P}_{\Sigma}$, i.e., $K(\Sigma)$ is isomorphic to the closed K\"ahler cone of ${\bf P}_{\Sigma}$. \end{theo} \bigskip Next theorem will play the central role in the sequel. Its statement is contained implicitly in \cite{oda.park,reid}: \bigskip \begin{theo} A $\Sigma$-piecewise linear function $\varphi$ is a strictly convex support function, i.e., $\varphi \in K^0(\Sigma)$, if and only if \[ \varphi(v_{i_1}) + \cdots + \varphi(v_{i_k}) > \varphi(v_{i_1} + \cdots + v_{i_k}) \] for all primitive collections ${\cal P} = \{ v_{i_1}, \ldots , v_{i_k} \}$ in $G(\Sigma)$. \label{crit} \end{theo} \section{Quantum cohomology rings} \hspace*{\parindent} \begin{opr} {\rm Let $\varphi$ be a ${\Sigma}$-piecewise linear function with complex values, or an element of the complexified space $PL(\Sigma)_{\bf C} = PL(\Sigma)\otimes_{\bf R} {\bf C}$. Define the quantum cohomology ring as the quotient of the polynomial ring ${\bf C} \lbrack z \rbrack$ by the sum of ideals $P(\Sigma)$ and $Q_{\varphi}(\Sigma)$: \[ QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C}) : = {\bf C} \lbrack z \rbrack/ (P(\Sigma) + Q_{\varphi}(\Sigma)) \] where $Q_{\varphi}(\Sigma)$ is generated by binomials \[ \exp ( \sum_{i = 1}^n a_i \varphi(v_i)) \prod_{i =1}^n z_j^{a_j} - \exp ( \sum_{j = 1}^n b_j \varphi(v_j)) \prod_{j =1}^n z_j^{b_j} \] running over all possible linear relations \[ \sum_{i = 1}^n a_i v_i = \sum_{j = 1}^n b_j v_j , \] where all coefficients $a_i$ and $b_j$ are non-negative and integral. } \label{def.quant} \end{opr} \begin{opr} {\rm Let ${\cal P} = \{ v_{i_1}, \ldots, v_{i_k} \} \subset G(\Sigma)$ be a primitive collection, $\sigma_{\cal P}$ the minimal cone in $\Sigma$ containing the sum \[ v_{\cal P} = v_{i_1} + \ldots + v_{i_k}, \] $v_{j_1}, \ldots, v_{j_l}$ generators of $\sigma_{\cal P}$. Let $l$ be the dimension of $\sigma_{\cal P}$. By \ref{def.fan}(iii), there exists the unique representation of $v_{\cal P}$ as an integral linear combination of generators $v_{j_1}, \ldots, v_{j_l}$ with positive integral coefficients $c_1, \ldots, c_l$: \[ v_{\cal P} = c_1 v_{j_1} + \cdots + c_l v_{j_l}, \] We put \[ E_{\varphi}({\cal P}) = \exp ( \varphi(v_{i_1} + \ldots + v_{i_k}) - \varphi(v_{i_1}) - \ldots - \varphi(v_{i_k})) \] \[ = \exp ( c_1 \varphi(v_{j_1}) + \cdots + c_l \varphi(v_{j_l}) - \varphi(v_{i_1}) - \ldots - \varphi(v_{i_k})).\]} \end{opr} \begin{theo} Assume that the K\"ahler cone $K(\Sigma)$ has the non-empty interior, i.e., ${\bf P}_{\Sigma}$ is projective. Then the ideal $Q_{\varphi}(\Sigma)$ is generated by the binomials \[ B_{\varphi}({\cal P}) = z_{i_1} \cdots z_{i_k} - E_{\varphi}({\cal P}) z_{j_1}^{c_1} \cdots z_{j_l}^{c_l}, \] where ${\cal P}$ runs over all primitive collections in $G(\Sigma)$. \label{basis} \end{theo} {\em Proof. } We use some ideas from \cite{strum}. Let $\phi$ be an element in $PL(\Sigma)$ representing an interior point of $K(\Sigma)$. Define the weights $\omega_1, \ldots, \omega_n$ of $z_1, \ldots, z_n$ as \[ \omega_i = \phi(v_i)\;\; (1 \leq i \leq n).\] We claim that binomials $B_{\varphi}({\cal P})$ form a reduced Gr\"obner basis for $Q_{\varphi}(\Sigma)$ relative to the weight vector \[ \omega = (\omega_1, \ldots, \omega_n ).\] Notice that the weight of the monomial $z_{i_1} \cdots z_{i_k}$ is greater than the weight of the monomial $z_{j_1}^{c_1} \cdots z_{j_l}^{c_l}$, because \[ \phi(v_{i_1}) + \cdots + \phi(v_{i_k}) > \phi(v_{i_1} + \cdots + v_{i_k}) =c_1 \phi(v_{j_1}) + \cdots c_l \phi(v_{j_l}) \] (Theorem \ref{crit}). So the initial ideal $init_{\omega} \langle B_{\varphi}({\cal P}) \rangle$ of the ideal $\langle B_{\varphi}({\cal P}) \rangle$ generated by $B_{\varphi}({\cal P})$ coincides with the ideal $SR(\Sigma)$. It suffices to show that the initial ideal ${\em init}_{\omega} Q_{\varphi}(\Sigma)$ also equals $SR(\Sigma)$. The latter again follows from Theorem \ref{crit}. \hfill $\Box$ \bigskip \begin{opr} {\rm The tube domain in the complex cohomology space $H^2({\bf P}_{\Sigma}, {\bf C})$: \[ K(\Sigma)_{\bf C} = K(\Sigma) + i H^2({\bf P}_{\Sigma}, {\bf R}) \] we call the {\em complexified K\"ahler cone of } ${\bf P}_{\Sigma}$. } \end{opr} \begin{coro} Let $\varphi$ be an element of $H^2({\bf P}_{\Sigma}, {\bf C})$, $t$ a positive real number. Then all generators $B_{t\varphi}({\cal P})$ of the ideal $Q_{t\varphi}(\Sigma)$ have finite limits as $t \rightarrow \infty$ if and only if $\varphi \in K(\Sigma)_{\bf C}$. Moreover, if $\varphi \in K(\Sigma)_{\bf C}$, then the limit of $ QH^*_{t\varphi}({\bf P}_{\Sigma}, {\bf C}) $ is the ordinary cohomology ring $H^*({\bf P}_{\Sigma}, {\bf C})$. \label{kon} \end{coro} {\em Proof. } Applying Theorem \ref{crit}, we obtain: \[ \lim_{t \rightarrow \infty} B_{t\varphi}({\cal P}) = z_{i_1} \cdots z_{i_k}. \] Thus, \[ \lim_{t \rightarrow \infty} Q_{t\varphi}(\Sigma) = SR(\Sigma). \] By Theorem \ref{ord}, \[ \lim_{t \rightarrow \infty} QH^*_{t\varphi}({\bf P}_{\Sigma}, {\bf C}) = H^*({\bf P}_{\Sigma}, {\bf C}). \] \hfill $\Box$ \bigskip \begin{exam} {\rm Consider the fan $\Sigma(d)$ defining $d$-dimensional projective space (see \ref{weig}). Then we obtain \[ QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C}) \cong {\bf C} \lbrack x \rbrack / (x^{d+1} - \exp ( - {\rm deg}_{ \varphi} \lambda )), \] where $\lambda = (1,\ldots, 1)$ is the generator of $R(\Sigma(d))$. This shows the quantum cohomology ring $QH^*_{\varphi}({\bf CP}^d, {\bf C})$ coincides with the quantum cohomology ring for ${\bf CP}^d$ proposed by physicists. } \end{exam} It is important to remark that the quantum cohomology ring $QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ has no any ${\bf Z}$-grading, but it is possible to define a ${\bf Z}_N$-grading on it. \begin{theo} {\rm Assume that the first Chern class $c_1({\bf P}_{\Sigma})$ is divisible by $r$. Then the ring $QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ has a natural ${\bf Z}_r$-grading.} \label{divis} \end{theo} {\em Proof. } A linear relation \[ \sum_{i = 1}^n a_i v_i = \sum_{j = 1}^n b_j v_j \] gives rise to an element \[ \lambda = (a_1 - b_1, \ldots, a_n - b_n ) \in R(\Sigma). \] By our assumption, \[ {\rm deg}_{\alpha_{\Sigma}} \lambda = \sum_{i = 1}^n a_i - \sum_{j = 1}^n b_j \] is the intersection number of $c_1({\bf P}_{\Sigma})$ and $\lambda \in H_2 ({\bf P}_{\Sigma}, {\bf C})$, i.e., it is divisible by $r$. This means that the binomials \[ \exp (\sum_{i = 1}^n a_i \varphi(v_i)) \prod_{i =1}^n z_j^{a_j} - \exp (\sum_{j = 1}^n b_j \varphi(v_j)) \prod_{j =1}^n z_j^{b_j} \] are ${\bf Z}_r$-homogeneous. \hfill $\Box$ \bigskip Although, the quantum cohomology ring $QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ has no any ${\bf Z}$-grading, it is possible to define a graded version of this quantum cohomology ring over the Laurent polynomial ring ${\bf C} \lbrack z_0, z_0^{-1} \rbrack$. \begin{opr} {\rm Let $\varphi$ be a ${\Sigma}$-piecewise linear function with complex values from the complexified space $PL(\Sigma)_{\bf C} = PL(\Sigma)\otimes_{\bf R} {\bf C}$. Define the quantum cohomology ring \[ QH^*_{\varphi} ({\bf P}_{\Sigma}, {\bf C} \lbrack z_0, z_0^{-1} \rbrack) \] as the quotient of the Laurent polynomial extension ${\bf C} \lbrack z \rbrack \lbrack z_0, z_0^{-1} \rbrack$ by the sum of ideals $Q_{\varphi,z_0}(\Sigma)$ and $P(\Sigma)$: where $Q_{\varphi,z_0}(\Sigma)$ is generated by binomials \[ \exp ( \sum_{i = 1}^n a_i \varphi(v_i)) z_0^{(-\sum_{i = 1}^n a_i)} \prod_{i =1}^n z_i^{a_i} - \exp ( \sum_{j = 1}^n b_j \varphi(v_j)) z_0^{(- \sum_{j = 1}^n b_j)} \prod_{l =1}^n z_j^{b_j} \] running over all possible linear relations \[ \sum_{i = 1}^n a_i v_i = \sum_{j = 1}^n b_j v_j \] with non-negative integer coefficients $a_i$ and $b_j$. } \label{def.quant1} \end{opr} The properties of the ${\bf Z}$-graded quantum cohomology ring \[ QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C} \lbrack z_0, z_0^{-1} \rbrack) \] are analogous to the properties of $QH^*({\bf P}_{\Sigma}, {\bf C})$: \begin{theo} For every binomial $B_{\varphi}({\cal P})$, take the corresponding homogeneous binomial in variables $z_0, z_1, \ldots, z_n$ \[ B_{\varphi,z_0}({\cal P}) = z_{i_1} \cdots z_{i_k} - E_{\varphi}({\cal P}) z_{j_1}^{c_1} \cdots z_{j_l}^{c_l}z_0^{(k - \sum_{s=1}^l c_s)}. \] Then the elements $B_{\varphi,z_0}({\cal P})$ generate the ideal $Q_{\varphi,z_0}(\Sigma)$, and K\"ahler limits of \[ QH^*_{t \varphi}({\bf P}_{\Sigma}, {\bf C} \lbrack z_0, z_0^{-1} \rbrack), \;\; t \rightarrow \infty \] are isomorphic to the Laurent polynomial extension \[ H^*({\bf P}_{\Sigma}, {\bf C}) \lbrack z_0, z_0^{-1} \rbrack \] of the odinary cohomology ring. \end{theo} \bigskip Finally, if the first Chern class of ${\bf P}_{\Sigma}$ belongs to the K\"ahler cone, i.e., $\alpha_{\Sigma} \in PL(\Sigma)$ is upper convex, then it is possible to define the quantum deformations of the cohomology ring of ${\bf P}_{\Sigma}$ over the polynomial ring ${\bf C}\lbrack z_0 \rbrack$. \begin{opr} {\rm Assume that $\alpha_{\Sigma} \in PL(\Sigma)$ is upper convex. We define the quantum cohomology ring \[ QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C} \lbrack z_0 \rbrack) \] over ${\bf C}\lbrack z_0 \rbrack$ as the quotion of the polynomial ring ${\bf C}\lbrack z_0, z_1, \ldots, z_n \rbrack$ by the sum of the ideal $P(\Sigma)\lbrack z_0 \rbrack $ and the ideal \[ {\bf C}\lbrack z_0, z_1, \dots, z_n \rbrack \cap Q_{\varphi,z_0}(\Sigma) \] } \end{opr} \begin{theo} The ideal \[ {\bf C}\lbrack z_0, z_1, \dots, z_n \rbrack \cap Q_{\varphi,z_0}(\Sigma) \] is generated by homogeneous binomials \[ B_{\varphi,z_0}({\cal P}) = z_{i_1} \cdots z_{i_k} - E_{\varphi}({\cal P}) z_{j_1}^{c_1} \cdots z_{j_l}^{c_l}z_0^{(k - \sum_{s=1}^l c_s)} \] where ${\cal P}$ runs over all primitive collections ${\cal P} \subset G(\Sigma)$. $($Notice that convexity of $\alpha_{\Sigma}$ implies $k - \sum_{s=1}^l c_s \geq 0$.$)$ K\"ahler limits of the quantum cohomology ring \[ QH^*_{t \varphi}({\bf P}_{\Sigma}, {\bf C} \lbrack z_0 \rbrack), \;\; t \rightarrow \infty \] are isomorphic to the polynomial extension \[ H^*({\bf P}_{\Sigma}, {\bf C}) \lbrack z_0 \rbrack \] of the odinary cohomology ring. \end{theo} \section{Birational transformations} \hspace*{\parindent} It may look strange that we defined the quantum cohomology rings using infinitely many generators for the ideals $Q_{\varphi}(\Sigma)$ and $Q_{\varphi, z_0}(\Sigma)$, while these ideals have only finite number of generators indexed by primitive collections in $G(\Sigma)$. The reason for that is the following important theorem: \begin{theo} Let $\Sigma_1$ and $\Sigma_2$ be two complete fans of regular cones such that $G(\Sigma_1) = G(\Sigma_2)$, then the quantum cohomology rings $QH^*_{\varphi}({\bf P}_{\Sigma_1}, {\bf C})$ and $QH^*_{\varphi}({\bf P}_{\Sigma_2}, {\bf C})$ are isomorphic. \label{flop} \end{theo} {\em Proof. } Our definitions of quantum cohomology rings does not depend on the combinatiorial structure of the fan $\Sigma$, one needs to know only all lattice vectors $v_1, \ldots, v_n \in G(\Sigma)$, but not the combinatorial structure of the fan $\Sigma$. \hfill $\Box$ \bigskip Since the equality $G(\Sigma_1) = G(\Sigma_2)$ means that two toric varieties ${\bf P}_{\Sigma_1}$ and ${\bf P}_{\Sigma_2}$ are isomorphic in codimension $1$, we obtain \begin{coro} Let ${\bf P}_{\Sigma_1}$ and ${\bf P}_{\Sigma_2}$ be two smooth compact toric manifolds which are isomorphic in codimension $1$, then the rings $QH^*_{\varphi}({\bf P}_1, {\bf C})$ and $QH^*_{\varphi}({\bf P}_2, {\bf C})$ are isomorphic. \end{coro} \begin{exam} {\rm Consider two $3$-dimensional fans $\Sigma_1$ and $\Sigma_2$ in ${\bf R}^3$ such that $G(\Sigma_1) = G(\Sigma_2) = \{ v_1, \ldots, v_6 \}$ where \[ v_1 = (1,0,0),\;v_2 = (0,1,0),\; v_3 = (0,0,1), \] \[ v_4 = (-1,0,0),\;v_5 = (0,-1,0),\; v_6 = (1,1,-1). \] We define the combinatorial structure of $\Sigma_1$ by the primitive collections \[ {\cal P}_1 = \{ v_1, v_4 \}, \;{\cal P}_2 = \{ v_2, v_5 \}, \; {\cal P}_3 = \{ v_3, v_6 \}, \] and the combinatorial structure of $\Sigma_2$ by the primitive collections \[ {\cal P}_1' = \{ v_1, v_4 \}, \;{\cal P}_2' = \{ v_2, v_5 \}, \; {\cal P}_3' = \{ v_1, v_2 \}, \] \[ {\cal P}_4' = \{ v_3, v_5, v_6 \}, \;{\cal P}_5' = \{ v_3, v_4, v_6 \}. \] The flop between two toric manifolds is described by the diagrams: \begin{center} \begin{picture}(360,200) \put(100,150){\makebox(0,0){$\cdot$}} \put(100,50){\makebox(0,0){$\cdot$}} \put(50,100){\makebox(0,0){$\cdot$}} \put(150,100){\makebox(0,0){$\cdot$}} \put(300,150){\makebox(0,0){$\cdot$}} \put(300,50){\makebox(0,0){$\cdot$}} \put(250,100){\makebox(0,0){$\cdot$}} \put(350,100){\makebox(0,0){$\cdot$}} \put(100,150){\makebox(0,0)[b]{$v_3$}} \put(100,50){\makebox(0,0)[t]{$v_6$}} \put(50,100){\makebox(0,0)[r]{$v_1$}} \put(150,100){\makebox(0,0)[l]{$v_2$}} \put(300,150){\makebox(0,0)[b]{$v_3$}} \put(300,50){\makebox(0,0)[t]{$v_6$}} \put(250,100){\makebox(0,0)[r]{$v_1$}} \put(350,100){\makebox(0,0)[l]{$v_2$}} \put(200,100){\makebox(0,0)[b]{$\leftrightarrow$}} \put(50,100){\line(1,1){50}} \put(100,50){\line(1,1){50}} \put(100,150){\line(1,-1){50}} \put(50,100){\line(1,-1){50}} \put(250,100){\line(1,1){50}} \put(300,50){\line(1,1){50}} \put(300,150){\line(1,-1){50}} \put(250,100){\line(1,-1){50}} \put(50,100){\line(1,0){100}} \put(300,50){\line(0,1){100}} \end{picture} \end{center} The ordinary cohomology rings $H^*({\bf P}_{\Sigma_1}, {\bf C})$ and $H^*({\bf P}_{\Sigma_2}, {\bf C})$ are not isomorphic, because their homogeneous ideals of polynomial relations among $z_1, \ldots, z_6$ have different numbers of minimal generators. There exists the polynomial relation in the quantum cohomology ring: \[ \exp (\varphi(v_1) + \varphi(v_2)) z_1 z_2 = \exp (\varphi(v_3) + \varphi(v_6)) z_3 z_6. \] If $\varphi(v_1) + \varphi(v_2) < \varphi(v_3) + \varphi(v_6)$, then we obtain the element $z_3 z_6 \in SR(\Sigma_1)$ as the limit for $t\varphi$, when $t \rightarrow \infty$. On the other hand, if $\varphi(v_1) + \varphi(v_2) > \varphi(v_3) + \varphi(v_6)$, taking the same limit, we obtain $z_1 z_2 \in SR(\Sigma_2)$. } \end{exam} Let us consider another simplest example of birational tranformation. \begin{exam} {\rm The quantum cohomology ring of the $2$-dimensional toric variety $F_1$ which is the blow-up of a point $p$ on ${\bf P}^2$ is isomorphic to the quotient of the polynomial ring ${\bf C} \lbrack x_1, x_2 \rbrack$ by the ideal generated by two binomials \[ x_1(x_1 + x_2) = \exp (-\phi_2);\; x_2^2 = \exp( - \phi_1) x_1, \] where $x_1$ is the class of the $(-1)$-curve $C_1$ on $F_1$, $x_2$ is the class of the fiber $C_2$ of the projection of $F_1$ on ${\bf P}^1$. The numbers $\phi_1$ and $\phi_2$ are respectively degrees of the restriction of the K\"ahler class $\varphi$ on $C_1$ and $C_2$. } \end{exam} \begin{rem} {\rm The definition of the quantum cohomology ring for smooth toric manifolds immediatelly extends to the case of singular toric varieties. However, the ordinary cohomology ring of singular toric varieties is not anymore the K\"ahler limit of the quantum cohomology ring. In some cases, the quantum cohomology ring of singular toric varieties $V$ contains an information about the ordinary cohomology ring of special desingularizations $V'$ of $V$. For instance, if we assume that there exists a projective desingularization $\psi\; : \; V' \rightarrow V$ such that $\psi^*{\cal K}_V = {\cal K}_{V'}$. Then for every K\"ahler class $\varphi \in H^2 (V, {\bf C})$, one has \[ {\rm dim}_{\bf C} H^*_{\varphi} (V, {\bf C}) = {\rm dim}_{\bf C} H^*(V',{\bf C}). \]} \end{rem} \section{Geometric interpretation of quantum cohomology rings} \hspace*{\parindent} The spectra of the quantum cohomology ring ${\rm Spec}\,QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C} )$, and its two polynomial versions \[ {\rm Spec}\,QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C} \lbrack z_0, z_0^{-1} \rbrack), \; {\rm Spec}\,QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C} \lbrack z_0 \rbrack) \] have simple geometric interpretations. \begin{opr} {\rm Denote by $\Pi(\Sigma)$ the $(n-d)$-dimensional affine subspace in ${\bf C}^n$ defined by the ideal $P(\Sigma)$.} \end{opr} \begin{opr} {\rm Choose any isomorphism $N \cong {\bf Z}^d$, so that any element $v \in N$ defines a Laurent monomial $X^v$ in $d$ variables $X_1, \ldots, X_d$. Consider the embedding of the $d$-dimensional torus $T(\Sigma) \cong ({\bf C}^*)^d$ in $({\bf C}^*)^n$: \[ (X_1, \ldots, X_d) \rightarrow (X^{v_1}, \ldots, X^{v_n}). \] Denote by $\Theta(\Sigma)$ the $(n-d)$-dimensional algebraic torus $({\bf C}^*)^n /T(\Sigma)$.} \end{opr} \begin{opr} {\rm Denote by $Exp$ the analytical exponential mapping \[ Exp\; : \; {\cal G} \rightarrow {\bf G} \] where ${\bf G}$ is a complex analytic Lie group, and ${\cal G}$ is its Lie algebra. For example, one has the exponential mapping \[ Exp \;:\; PL(\Sigma)_{\bf C} \rightarrow ({\bf C}^*)^n \] \[ \varphi \mapsto (e^{\varphi(v_1)}, \ldots, e^{\varphi(v_n)}) \] which descends to the exponential mapping \[ Exp \; : \; H^2({\bf P}_{\Sigma}, {\bf C}) \rightarrow \Theta(\Sigma). \] } \end{opr} \begin{prop} The ${\bf T}(\Sigma)$-orbit ${T}_{\varphi}(\Sigma)$ of the point $Exp(\varphi) \in ({\bf C}^*)^n$ is closed, and its ideal is canonically isomorphic to $Q_{\varphi}(\Sigma)$. \end{prop} \begin{coro} The scheme ${\rm Spec}\,QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C} )$ is the scheme-theoretic intersection of the $d$-dimensional subvariety $\overline{T}_{\varphi}(\Sigma) \subset {\bf C}^n$ and the $(n-d)$-dimensional subspace $\Pi(\Sigma)$. \end{coro} \begin{opr} {\rm Let $\tilde{N} = {\bf Z} \oplus N$. For any $v \in N$, define $\tilde{v} \in \tilde{N}$ as $\tilde{v} = (1, v)$. Define the embedding of the $(d+1)$-dimensional torus $T^{\circ}(\Sigma) \cong ({\bf C}^*)^{d+1}$ in $({\bf C}^*)^{n+1}$: \[ (X_0, X_1, \ldots, X_d) \rightarrow (X_0, X^{\tilde{v_1}}, \ldots, X^{\tilde{v_n}}). \]} \label{affine} \end{opr} The quotient $({\bf C}^*)^{n+1}/ T^{\circ}(\Sigma)$ is again isomorphic to $\Theta(\Sigma)$. \begin{prop} The ideal of the ${T}^{\circ}(\Sigma)$-orbit \[ T_{\varphi}^{\circ}(\Sigma) \subset {\bf C}^* \times {\bf C}^n \] of the of the point $(1, {Exp}\,(\varphi)) \in ({\bf C}^*)^{n+1}$ is canonically isomorphic to $Q_{\varphi,z_0}(\Sigma)$. \end{prop} \begin{coro} The scheme ${\rm Spect}\,QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C}\lbrack z_0, z_0^{-1} \rbrack )$ is the scheme-theo\-re\-tic intersection of the $(d+1)$-dimensional subvariety $T_{\varphi}^{\circ}(\Sigma) \subset {\bf C}^* \times {\bf C}^{n}$ and the $(n-d+1)$-dimensional subvariety ${\bf C}^* \times \Pi(\Sigma) \subset {\bf C}^* \times {\bf C}^n$. \end{coro} Similarly, one obtain the geometric interpretation of $QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C}\lbrack z_0 \rbrack )$, when the first Chern class of ${\bf P}_{\Sigma}$ belongs to the K\"ahler cone $K(\Sigma)$. \begin{prop} The scheme ${\rm Spect}\,QH^*_{\varphi}({\bf P}_{\Sigma}, {\bf C}\lbrack z_0 \rbrack )$ is the scheme-theoretic intersection in ${\bf C}^{n+1}$ of the $(d+1)$-dimensional ${T}^{\circ}(\Sigma)$-orbit of the point $(1, Exp(\varphi))$ and the $(n-d+1)$-dimensional affine subspace ${\bf C} \times \Pi(\Sigma) \subset {\bf C}^{n+1}$. \end{prop} The limits of quantum cohomology rings have also geometric interpretations. One obtains, for instance, the spectrum of the ordinary cohomology ring of ${\bf P}_{\Sigma}$ as the scheme-theoretic intersection of the affine subspace $\Pi(\Sigma)$ with a "toric degeneration" of closures of ${\bf T}(\Sigma)$-orbits $\overline{T}_{\varphi}(\Sigma) \subset {\bf C}^n$. Such an interpretation allows to apply methods of M. Kapranov, B. Strumfels, and A. Zelevinsky (see \cite{kap}, Theorem 5.3) to establish connection between vertices of Chow polytope (secondary polyhedron) and K\"ahler limits of quantum cohomology rings. \section{Calabi-Yau hypersurfaces, Jacobian rings and the mirror symmetry} \hspace*{\parindent} Throughout in this section we fix a complete $d$-dimensional fan of regular cones, and we assume that ${\bf P} = {\bf P}_{\Sigma}$ is a toric manifold whose first Chern class belongs to the closed K\"ahler cone $K(\Sigma)$, i.e., $\alpha: = \alpha_{\Sigma}$ is a convex ${\Sigma}$-piecewise linear function. Let $\Delta = \Delta_{\alpha}$, the convex polyhedron in $M_{\bf R}$ (see \ref{poly1}). For any sufficiently general section $S$ of the anticanonical sheaf ${\cal A}$ on ${\bf P}$ represented by homogeneous polynomial $F(z)$, the set $Z = \{ \pi(z) \in {\bf P} \; : \; F(z) = 0 \}$ in ${\bf P}$ is a Calabi-Yau manifold ($c_1({\cal A}) = c_1({\bf P}))$. Since the first Chern class of ${\bf P}$ in the ordinary cohomology ring $H^*({\bf P}, {\bf C})$ is the class of the sum $( z_1 + \cdots + z_n)$, we obtain: \begin{prop} The image of $H^*({\bf P}, {\bf C})$ under the restriction mapping to $H^*(Z, {\bf C})$ is isomorphic to the quotient \[H^*({\bf P}, {\bf C})/ {\rm Ann}( z_1 + \cdots + z_n), \] where ${\rm Ann}( z_1 + \cdots + z_n)$ denotes the annulet of the class of $( z_1 + \cdots + z_n)$ in $H^*({\bf P}, {\bf C})$. \label{image} \end{prop} In general, Proposition \ref{image} allows us to calculate only a part of the ordinary cohomology ring of a Calabi-Yau hypersurface $Z$ in toric variety ${\bf P}$. If the first Chern class of ${\bf P}$ is in the interior of the K\"ahler cone $K(\Sigma)$, then $Z$ is an ample divisor. For $d \geq 4$, by Lefschetz theorem, the restriction mapping $ H^2({\bf P}, {\bf C}) \rightarrow H^2(Z, {\bf C})$ is isomorphism. Thus, using Proposition \ref{image}, we can calculate cup-products of any $(1,1)$-forms on $Z$. \bigskip \begin{opr} {\rm Denote by $\Delta^*$ the convex hull of the set $G(\Sigma)$ of all generators, or equivalently, \[ \Delta^* = \{ v \in N_{\bf R} \mid \alpha(v) \leq 1 \}. \]} \end{opr} \begin{rem} {\rm The polyhedron $\Delta^*$ is dual to $\Delta$ reflexive polyhedron (see \cite{bat.mir}). } \end{rem} \begin{theo} There exists the canonical isomorphism between the quantum cohomology ring \[ QH^*_{\varphi}({\bf P}, {\bf C} ) \] and the Jacobian ring \[ {\bf C} \lbrack X_1^{\pm 1}, \ldots, X_d^{\pm 1} \rbrack / (X_1 \partial f/ \partial X_1, \ldots, X_d \partial f/ \partial X_d ) \] of the Laurent polynomial \[ f_{\varphi}(X) = -1 + \sum_{i =1}^n \exp (\varphi(v_i))^{-1} X^{v_i}. \] This isomorphism is induced by the correspondence \[ z_i \rightarrow X^{v_i}/\exp (\varphi(v_i)) \;\; (1 \leq i \leq n). \] In particular, it maps the first Chern class $(z_1 + \ldots + z_n)$ of ${\bf P}$ to $f_{\varphi}(X) + 1 $. \label{mirr} \end{theo} {\em Proof. } Let \[ {\cal H}\; : \; {\bf C} \lbrack z_1, \ldots, z_n \rbrack \rightarrow {\bf C} \lbrack X_1^{\pm 1}, \ldots , X_d^{\pm 1} \rbrack \] be the homomorphism defined by the correspondence \[ z_i \rightarrow X^{v_i}/\exp (\varphi(v_i)).\] By \ref{def.fan}(iii), ${\cal H}$ is surjective. It is clear that $Q_{\varphi}(\Sigma)$ is the kernel of ${\cal H}$. On the other hand, if we a ${\bf Z}$-basis $\{ u_1, \ldots, u_d \} \subset M$ which establishes isomorphisms $M \cong {\bf Z}^d$ and $N \cong {\bf Z}^d$, we obtain: \[{\cal H}(P(\Sigma)) = < X_1 \partial f/ \partial X_1, \ldots, X_d \partial f/ \partial X_d >. \] \hfill $\Box$ \bigskip \begin{opr} {\rm Let $S_{\Delta^*}$ be the affine coordinate ring of the $T^{\circ}(\Sigma)$-orbit of the point $(1,\ldots, 1) \in {\bf C}^{n+1}$ (see \ref{affine}). } \end{opr} \begin{opr} {\rm For any Laurent polynomial \[ f(X) = a_0 + \sum_{i =1}^n a_i X^{v_i}, \] we define elements \[ F_0, F_1, \ldots, F_d \in S_{\Delta^*} \] as $F_i = \partial X_0 f(X) \partial X_0$, $(0 \leq i \leq d)$. } \end{opr} \begin{rem} {\rm The ring $S_{\Delta^*}$ is a subring of ${\bf C}\lbrack X_0, X_1^{\pm 1}, \ldots, X_d^{\pm 1} \rbrack$. There exists the canonical grading of $S_{\Delta^*}$ by degree of $X_0$. It is easy to see that the correspondence \[ z_0 \rightarrow - X_0, \] \[ z_i \rightarrow X_0X^{v_i}/(\exp ( \varphi(v_i))) \] defines the isomorphism \[ {\bf C} \lbrack z \rbrack / Q_{\varphi}(\Sigma) \cong S_{\Delta^*}. \] This isomorphism maps $(- z_0 + z_1 + \cdots z_n )$ to $F_0$.} \label{iso} \end{rem} \begin{theo} {\rm (\cite{bat.var})} Let \[R_f = S_{\Delta^*} / < F_0, F_1, \ldots, F_d > .\] Then the quotient \[ R_f / {\rm Ann}\,(X_0) \] is isomorphic to the $(d-1)$-weight subspace $W_{d-1}H^{d-1}(Z_f, {\bf C})$ in the cohomology space $H^{d-1}(Z_f, {\bf C})$ of the affine Calabi-Yau hypersurface in $T(\Sigma)$ defined by the Laurent polynomial $f(X)$. \end{theo} For any Laurent polynomial $f(X) = a_0 + \sum_{i=1}^n a_i X^{v_i}$, we can find an element $\varphi \in PL(\Sigma)_{\bf C}$ such that \[ \frac{-a_i}{a_0} = \exp (- \varphi(v_i)). \] A one-parameter family $t \varphi$ in $PL(\Sigma)$ induces the one-parameter family of Laurent polynomials \[ f_t (X) = - 1 + \sum_{i =1}^n \exp (- t \varphi(v_i))X^{v_i}. \] Applying the isomorphism in \ref{iso} and the statement in Theorem \ref{basis}, we obtain the following: \begin{theo} Assume that $\varphi$ is in the interior of the K\"ahler cone $K(\Sigma)$. Then the limit \[ R_{f_t} / {\rm Ann } (X_0) \] is isomorphic to \[H^*({\bf P}, {\bf C})/ {\rm Ann}( z_1 + \cdots + z_n). \] \end{theo} The last statement shows the relation, established in \cite{aspin4}, between the "toric" part of the topological cohomology rings of Calabi-Yau $3$-folds in toric varieties and limits of the multiplicative structure on $(d-1)$-weight part of the Jacobian rings of their "mirrors". \section{Topological sigma models on toric manifolds} So far we have not explained why the ring $H^*_{\varphi} ({\bf P}_{\Sigma}, {\bf C})$ coincides with the quantum cohomology ring corresponding to the topological sigma model on $V$. In this section we want to establish the relations between the ring $H^*_{\varphi} ({\bf P}_{\Sigma}, {\bf C})$ and the quantum cohomology rings considered by physicists. \bigskip In order to apply the general construction of the correlation functions in sigma models (\cite{witten}, 3a ), we need the following information on the structure of the space of holomorphic maps of ${\bf CP}^1$ to a $d$-dimensional toric manifold ${\bf P}_{\Sigma}$. \begin{theo} Let ${\cal I}$ be the moduli space of holomorphic maps $f\;:\; {\bf CP}^1 \rightarrow {\bf P}_{\Sigma}$. The space ${\cal I}$ consists of is infinitely many algebraic varieties ${\cal I}_{\lambda}$ indexed by elements \[ \lambda = (\lambda_1, \ldots, \lambda_n) \in R(\Sigma), \] where the numbers $\lambda_i$ are equal to the intersection numbers ${\rm deg}_{{\bf CP}^1} f^*{\cal O}( Z_i)$ with divisors $Z_i \subset {\bf P}_{\Sigma}$ such that $\pi^{-1}(Z_i)$ is defined by the equation $z_i = 0$ in $U(\Sigma)$. Moreover, if all $\lambda_i \geq 0$, then ${\cal I}_{\lambda}$ is irreducible and the virtual dimension of ${\cal I}_{\lambda}$ equals \[ d_{\lambda} = {\rm dim}_{\bf C} {\cal I}_{\lambda} = d + \sum_{ i =1}^n \lambda_i. \] \end{theo} {\em Proof. } The first statement follows immediatelly from the description of the intersection product on ${\bf P}_{\Sigma}$ (\ref{deg}). Assume now that all $\lambda_i$ are non-negative. This means the that the preimage $f^{-1}(Z_i)$ consists of $\lambda_i$ points including their multiplicities. Let ${\cal F}_{\Sigma}$ be the tangent bundle over ${\bf P}_{\Sigma}$. There exists the generalized Euler exact sequence \[ 0 \rightarrow {\cal O}^{n-d}_{\bf P} \rightarrow {\cal O}_{\bf P}(Z_1) \oplus \cdots \oplus {\cal O}_{\bf P}(Z_n) \rightarrow {\cal F}_{\Sigma} \rightarrow 0. \] Applying $f^*$, we obtain the short exact sequence of vector bundles on ${\bf CP}^1$. \[ 0 \rightarrow {\cal O}^{n-d}_{{\bf CP}^1} \rightarrow {\cal O}_{{\bf CP}^1}({\lambda}_1) \oplus \cdots \oplus {\cal O}_{{\bf CP}^1}({\lambda}_n) \rightarrow f^*{\cal F}_{\Sigma} \rightarrow 0. \] This implies that $h^1({\bf CP}^1,f^*{\cal F}_{\Sigma}) = 0$, and $h^0({\bf CP}^1,f^*{\cal F}_{\Sigma}) = d + \lambda_1 + \cdots + \lambda_n$. The irreducibility of ${\cal I}_{\lambda}$ for $\lambda \geq 0$ follows from the explicit geometrical construction of maps $f \in {\cal I}_{\lambda}$: Choose $n$ polynomials $f_1(t), \ldots, f_n(t)$ such that ${\rm deg}\, f_i(t) = \lambda_i$ $( i =1, \ldots, n)$. If all $\mid \lambda \mid = \lambda_1 + \cdots + \lambda_n$ roots of $\{f_i\}$ are distinct, then these polynomials define the mapping \[ g\; :\; {\bf C} \rightarrow U(\Sigma) \subset {\bf C}^n .\] The composition $\pi \circ g$ extends to the mapping $f$ of ${\bf CP}^1$ to ${\bf P}_{\Sigma}$ whose homology class is $\lambda$. \hfill $\Box$ \bigskip \begin{opr} {\rm Let \[ \Phi \; : \; {\cal I} \times {\bf CP}^1 \rightarrow {\bf P}_{\Sigma} \] be the universal mapping. For every point $x \in {\bf CP}^1$ we denote by $\Phi_x$ the restriction of $\Phi$ to ${\cal I}\times x$. The cohomology classes $z_1 = \lbrack Z_1 \rbrack, \ldots, z_n = \lbrack Z_n \rbrack$ of divisors $Z_1, \dots, Z_n$ on ${\bf P}_{\Sigma}$ in the ordinary cohomology ring $H^*({\bf P}_{\Sigma})$ determine the cohomology classes $W_{z_1}, \ldots, W_{z_n} \in H^*({\cal I})$ which are independent of choice of $x \in {\bf CP}^1$. The element $W_{z_i}$ is the class of the divisor on ${\cal I}$: \[ \{ f \in {\cal I} \mid f(x) \in Z_i \}. \] } \end{opr} The quantum cohomology ring of the sigma model with the target space ${\bf P}_{\Sigma}$ is defined by the intersection numbers \[ (W_{\alpha_1} \cdot W_{\alpha_2} \cdot \cdots \cdot W_{\alpha_k} )_{\cal I} \] on the moduli space ${\cal I}$, where $\Phi_{\alpha} = \Pi_x^*(\alpha)$ \begin{theo} Let ${\bf P}_{\Sigma}$ be a $d$-dimensional toric manifold, $\varphi \in H^2({\bf P}_{\Sigma}, {\bf C})$ a K\"ahler class. Let $\lambda^0 = (\lambda_1^0, \ldots, \lambda_n^0)$ be a non-negative element in $R(\Sigma)$, $\Omega \in H^{2d}({\bf P}_{\Sigma}, {\bf C})$ the fundamental class of the toric manifold ${\bf P}_{\Sigma}$. Then the intersection number on the moduli space ${\cal I}$ \[ (W_{\Omega})\cdot (W_{z_1})^{\lambda_1^0} \cdot(W_{z_2})^{\lambda_2} \cdot \cdots \cdot (W_{z_n})^{\lambda_n^0} \] vanishes for all components ${\cal I}_{\lambda}$ except from $\lambda = \lambda_0$. In the latter case, this number equals \[ \exp (- {\rm deg}_{\varphi} \lambda). \] \end{theo} {\em Proof. } Since the fundamental class $\Omega$ is involved in the considered intersection number, this number is zero for all ${\cal I}_{\lambda}$ such that the rational curves in the class $\lambda$ do not cover a dense Zariski open subset in ${\bf P}_{\Sigma}$. Thus, we must consider only non-negative classes $\lambda$. Moreover, the factors $(W_{z_i})^{\lambda_i^0}$ show that we must consider only those $\lambda = (\lambda_1, \ldots, \lambda_n) \in R(\Sigma)$ such that $\lambda_i \geq \lambda_i^0$, i.e., a mapping $f \in {\cal I}_{\lambda}$ is defined by polynomials $f_1, \ldots, f_n$ such that ${\rm deg}\, f_i \geq \lambda_i$. There is a general principle that non-zero contributions to the intersection product \[ (W_{\alpha_1} \cdot W_{\alpha_2} \cdot \cdots \cdot W_{\alpha_k} )_{\cal I} \] appear only from the components whose virtual ${\bf R}$-dimension is equal to \[ \sum_{i =1} {\rm deg} \, {\alpha_i}. \] In our case, the last number is $d + \lambda_1^0 + \ldots + \lambda_n^0$. Therefore, a non-zero contribution appears only if $\lambda = \lambda^0$. It remains to notice that this contribution equals $\exp(- {\rm deg}_{\varphi} \lambda_0)$. The last statement follows from the observation that the points $f^{-1}(Z_i) \subset {\bf CP}^1$ $( i = 1, \ldots, n)$ define the mapping $f\, : \, {\bf CP}^1 \rightarrow {\bf P}_{\Sigma}$ uniquely up to the action of the $d$-dimensional torus ${\bf T} = {\bf P}_{\Sigma} \setminus (Z_1 \cup \cdots \cup Z_n)$, and the weight of the mapping $f$ in the intersection product is \[ \int_{{\bf CP}^1} f^*(\varphi). \]\hfill $\Box$ \begin{coro} Let ${\cal Z}_i$ be the quantum operator corresponding to the class $\lbrack Z_i \rbrack \in H^2({\bf P}_{\Sigma}, {\bf C})$ $( i =1, \ldots, n)$ considered as an element of the quantum cohomology ring. Then for every non-negative element $\lambda \in R(\Sigma)$, one has the algebraic relation \[ {\cal Z}_1^{\lambda_1} \circ \cdots \circ {\cal Z}_n^{\lambda_n} = \exp (- {\rm deg}_{\varphi} \lambda)\, id. \] \end{coro} It turns out that the polynomial relations of above type are sufficient to recover the quantum cohomology ring $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$: \begin{theo} Let $A_{\varphi}(\Sigma)$ be the quotient of the polynomial ring ${\bf C} \lbrack z \rbrack$ by the sum of two ideals: $P(\Sigma)$ and the ideal generated by all polynomials \[ B_{\lambda} = z_1^{\lambda_1} \cdots z_n^{\lambda_n} - \exp (- {\rm deg}_{\varphi} \lambda) \] where $\lambda$ runs over all non-negative elements of $R(\Sigma)$. Then $A_{\varphi}(\Sigma)$ is isomorphic to $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$. \end{theo} {\em Proof. } Let $B_{\varphi}(\Sigma)$ be the ideal generated by all binomials $B_{\lambda}$. By definition, $B_{\varphi}(\Sigma) \subset Q_{\varphi}(\Sigma)$. So it is sufficient to prove that $Q_{\varphi}(\Sigma) subset B_{\varphi}(\Sigma)$. Let \[ \sum_{i = 1}^n a_i v_i = \sum_{j = 1}^n b_j v_j \] be a linear relation among $v_1, \ldots, v_n$ such that $a_i, \, b_j \geq 0$. Since the set of all nonnegative elements $\lambda =(\lambda_1, \ldots, \lambda_n) \in R(\Sigma)$ $(\lambda_i \geq 0)$ generates a convex cone of maximal dimension in $H^2({\bf P}_{\Sigma}, {\bf C})$, there exist two nonnegative vectors $\lambda$, $\lambda' in R(\Sigma)$ such that \[ \lambda - \lambda' = (\lambda_1 - \lambda_1', \ldots, \lambda_n - \lambda_n') = (a_1 - b_1, \ldots, a_n - b_n ). \] By definition, two binomials $P_{\lambda}$ and $P_{\lambda'}$ are contained in $Q_{\varphi}(\Sigma)$. Hence, the classes of $z_1, \ldots, z_n$ in ${\bf C} \lbrack z \rbrack/ B_{\varphi}(\Sigma)$ are invertible elements. Thus, the class of the binomial \[ \exp ( \sum_{i = 1}^n a_i \varphi(v_i)) \prod_{i =1}^n z_j^{a_j} - \exp ( \sum_{j = 1}^n b_j \varphi(v_j)) \prod_{j =1}^n z_j^{b_j} \] is zero in ${\bf C} \lbrack z \rbrack/ B_{\varphi}(\Sigma)$. Thus, $B_{\varphi}(\Sigma) = Q_{\varphi}(\Sigma)$. \hfill $\Box$.

9203

alg-geom/9203001

en

https://arxiv.org/abs/alg-geom/9203001

alg-geom/9203001

Zube

Severinas Zube

Exceptional linear systems on curves on Enriques surfaces

14 pages, LATEX

The main purpose in this paper is to study the gonality, the Clifford index and the Clifford dimension on linearly equivalent smooth curves on Enriques surfaces. The method is similar to techniques of M.Green $\&$ R.Lazarsfeld and G.Pareschi.

alg-geom

\subsection{ Introduction} In recent years several authors have been led to study the following question : to what extent do linearly equivalent smooth curves on a surface $S$ carry "equally exceptional" linear series? Green and Lazarsfeld investigated the case when $S$ is a K3 surface and proved in [GL] that smooth linearly equivalent curves have the same Clifford index . Clifford index is a natural numerical invariant measuring "exceptional" linear systems on curves . On the other hand Donagi's example shows that smooth linearly equivalent curves on K3 surfaces can have diferent gonality. The question in the case of Del Pezzo surfaces of degree $\geq 2$ has been studied by Pareschi . It follows from the result in [P] that gonality and Clifford index is the same for all smooth linearly equivalent curves, with one exception for gonality , involving curves of genus 3 (see [P] for details ). The purpose of this note is to study the same question in the case of Enriques surfaces. It turns out that the gonality and the Clifford index is not constant for smooth linearly equivalent curves (see section 5.11 ) and in general we obtain the following estimate for the jump of the gonality and the Clifford index in a linear system: \newtheorem{ttt}{Theorem} \begin{ttt} Let C be a smooth irredecible curve of genus g on an Enriques surface S and let gon($|C|$) = min \{gon($C'$) $|$ $C'$ is smooth curve in $|C|$ \}, cliff($|C|$) = min \{cliff($C'$) $|$ $C'$ is smooth curve in $|C'|$ \} be the minimal gonality , Clifford index for $|C|$ . Then for all smooth curves $C'$ in $|C|$ gonality $C'$ is $ \leq gon(|C|) +2$. And if cliff($C'$) = $gon(C') -2$ then Clifford index of $C'$ is $\leq cliff(|C|) +2 $. Moreover, if $gon(|C|) \leq \frac{g-1}{2}$ or $g \geq9$ , then there is a line bundle L on S such that $cliff(C') \leq cliff(L|_{C'} ) +2$ .( see proposition~\ref{PPP} for more precise result ). \end{ttt} \paragraph{Remark}: \\ - It seems that Cliford index always can be computed by gonality, i.e. $cliff(C)$ $= gon(C) -2$ , for all curves contained in Enriques surfaces. (see Theorem 2 and conjecture bellow ). \\ -In the section 5.11 we give examples which show that the gonality for the smooth linearly equivalent cuves is not constant. \\ Another very useful invariant describing "exceptional" curves is the Clifford dimension introduced by Eisinbud , Lange , Martens and Schreyer in [ELMS]. If the curve has the Clifford dimension greater than 1 then it is "special" (see [ELMS] for description of such curves ). For the curves contained in $S$ we have the following result: \newtheorem{cliff}[ttt]{Theorem} \begin{cliff} Let C be a smooth irredecible curve on an Enriques surface S then its Clifford dimension is 1 or greater than 9. ( In fact we prove a somewhat more precise result - see Proposition~\ref{CCC} ). \end{cliff} We conjecture that every smooth irredecible curve $C$ on Enriques surface S has Clifford dimension 1. The paper is organized as follows: In the Section 1.? we introduce notation and collect some preliminary results. In Section 2.? we investigate the bundle $E(C,A)$ - introduced by Lazarsfeld - and obtain the jumping estimate for gonality and prove Theorem 1. Section 3.? is devoted to the proof that a smooth plane curve of the degree $\geq5$ cannot be contained in an Enriques surface . In section 4.? we prove Theorem 2. In last section we explain how to obtain , using results of section 2.? , explicit examples of the smooth linearly equivalent curves of different gonality . This paper was written during my stay at SFB 170 in G$\ddot{o}$ttingen. I am grateful SFB 170 for support and good working condition. \paragraph{Notation}: \\ -We will work over the complex field. \\ -We denote by the same capital letter both a divisor and the line bundle associate to the divisor and we hope that the meaning will be clear from context. \subsection*{Preliminaries} \addtocounter{section}{1} \subsection{E(C,A)} Let $S$ be a regular surface , $C$ a curve contained in $S$, and $A$ a base point free line bundle on $C$. One can associate to the triple $(S,C,A)$ a certain vector bundle on $S$ in a canonical way. We refer the reader to [P], [GL], [T] for details. Let us denote by $F(C,A)$ the vector bundle defined by the sequence: $$0 \to F(C,A) \to H^{0}(A) \otimes O_S \stackrel{ev}{\to}A \to 0, $$ where $A$ is viewed as a sheaf on $S$ and $ev$ is the evaluation map. Dualizing the above sequence we get (see [P]) \begin{equation} 0 \to H^{0}(A)^{*} \otimes O_S \to E(C,A) \to O_{C}(C) \otimes A^ * \to 0 \label{E} \end{equation} here $E(C,A)=F(C,A)^ * = {\cal H}{\sl om}(F(C,A),O_S )$ , $A^* = {\cal H}{\sl om}(A,O_C )$ . To simplify the notation we will omit $(C,A)$, if it is clear from context to which pair $(C,A)$ the bundle $E(C,A)$ is associated. We have \begin{equation} c_1 (E)=C; {~~~} c_2 (E) = deg (A); \end{equation} \begin{equation} rk(E)=h^0 (A);{~~~~} h^i (F(C,A)) =h^{2-i}(E \otimes K_S) = 0{~~~} for {~~~~} i=1,2; \end{equation} where $K_S$ is the canonical bundle of $S$. If $h^0 (O_{C}(C) \otimes A^ * ) > 0$, then $E(C,A)$ is generated by its sections away from a finite set coinciding with the fixed divisor of $O_{C}(C) \otimes A^ * $ Let $s^ \perp $ be a subspase of $H^0 (A)^* $ orthogonal to $ s \in H^0 (A) $ with respect to the natural pairing $$ H^1 (K_C -A) \otimes H^0 (A) = H^0 (A)^* \otimes H^0 (A) \to H^1 (K_C )= \ccc.$$ Then we get another description of the bundle $E(C,A)$: \begin{equation} 0 \to s^ \perp \otimes K_S \to E(C,A) \otimes K_S \to J_ \xi (C+K_S ) \to 0, \label{ej} \end{equation} where $ \xi = (s)_0$ is the zero scheme of the section $s \in H^0 (A) $ (see [T]). \subsection{Enriques surfaces} A smooth irreducible surface $S$, such that $h^1 (O_S )=h^2 (O_S )=0 $ and $2K_S \sim O_S $, is called a Enriques surface. If $C$ is a smooth curve in $S$ , then by adjunction formula one has : \begin{equation} g(C) = \frac{C^2 }{2} +1 . \label{g} \end{equation} Recall that a divisor $D$ on a smooth surface $X$ is said to be $nef$ if $DC \geq 0$ for every curve $C$ on $X$ . The following properties will be used throughout, sometimes without explicit mention: (A)([C,D] Proposition 3.1.6) If $D$ is a $nef$ divisor , then $ \mid D \mid $ has no fixed components , unless $ D \sim 2E+R $, where $ \mid 2E \mid $ is a genus 1 pencil and $R$ a smooth rational curve with $RE=1$ . \label{111} (B)([C,D] Corollary 3.1.3) If $D$ is a $nef$ divisor and $D^2 > 0 $ , then $H^1 (O_S (-D)) = 0$ and $\chi(O(D)) -1 =dim|D| = \frac{D^2 }{2}$. \label{222} (C)([C,D] Proposition 3.1.4) If $ \mid D \mid $ has no fixed components, then one of the following holds: \nonumber \\ (i) $D^2 > 0$ and there exist an irreducible curve $C$ in $ \mid D \mid $. \nonumber \\ (ii) $D^2 =0$ and there exist a genus 1 pencil $ \mid P \mid $ such that $D \sim kP$ for some $k \geq 1 $. \label{333} (D)([C,D]Chapter 4, appendix , corollary 1. and corollary 2.) If $D^2 \geq 6$ and $D$ is $nef$ then $D$ is ample , $2D$ is generated by its global sections, $3D$ is very ample. Let $K^+ = \{D^2 > 0 | D {~}is{~} an {~}effective{~} divisor. \}$. $ K^+ $ is called the positive cone. $ \overline{K^+ } = \{D^2 \geq 0 | D{~} is {~}an{~} effective{~} divisor. \}$ is the closure of the positive cone. The Enriques surface $S$ is called unnodal if there are no smooth (-2) curves contained in $S$. \subsection{Steiner construction} Let $L,M$ be the effective divisors on $S$. Parametrize the two pencils $ \mid L( \lambda ) \mid \\ \subset \mid L \mid $ and $ \mid M( \lambda ) \mid \subset \mid M \mid $ by $ \lambda \in P^1 $, choosing the parametrization so that $L( \lambda ) $ and $ M( \lambda ) $ have no common components for every $ \lambda \in P^1 $. Then the curve : \begin{equation} C= { \cup }_{ \lambda} (L( \lambda ) \cap M( \lambda )) \end{equation} is cleary irreducible if the general curves in $ \mid L( \lambda ) \mid $ and $ \mid M( \lambda ) \mid $ are irreducible. Denote by $ \xi $ (resp. $\eta $) base locus of $ \mid L( \lambda ) \mid $ (resp. $ \mid M( \lambda ) \mid $). Then $ \mid L( \lambda ) \mid = \mid J_{ \xi } (L) \mid , \mid M( \lambda ) \mid = \mid J_{ \xi} (M) \mid $. If $ \xi \cup \eta = P_1 + ... +P_{L^2+M^2}$ consist of different points , then $C$ contain $ \xi \cup \eta $. We have $C \sim L+M $ . Indeed , using our data we can obtain the sequence: \begin{equation} 0 \to \ccc^2 \otimes O_S \stackrel{ s}{ \to} M \oplus L \to { \cal L} \to 0 \end{equation} where $s(( \mu, \nu ) \otimes f ) = s^L _{ \mu / \nu }(f) \oplus s^M _{ \mu / \nu }(f) $ and the map $s^L _{ \mu / \nu } ( \bullet ) {~}$ (resp. $s^M _{ \mu / \nu } ( \bullet ) $ ) is multiplication by $ L( \mu / \nu ) \in|L| $ (resp.$ M( \mu / \nu) \in|M|$), hence is defined by the sequences: \begin{eqnarray} 0 \to O \to L \to O_{L( \mu / \nu )} \to 0 & (s^L _{ \mu / \nu } )\nonumber \\ (resp.{~} 0 \to O \to M \to O_{M( \mu / \nu )} \to 0 & (s^M _{ \mu / \nu }) {~}). \nonumber \end{eqnarray} We see that the sheaf $ \cal L $ has support $C$ and $c_1 (M \oplus L) \sim M+L \sim supp { \cal L} = C$. In the situation described above we will say that $C$ is obtained by the Steiner constrution using $ \mid L( \lambda) \mid $ and $ \mid M( \lambda) \mid $ . Note , that if a curve $C' \in \mid C \mid $ contains the base locus of $ \mid L ( \lambda ) \mid $ , then $C'$ can be obtained by the Steiner construction using $ \mid L ( \lambda ) \mid $ and any another pencil $ \mid M' ( \lambda ) \mid \subset \mid M \mid $ such that $ L ( \lambda ) $ and $ M' ( \lambda ) $ have no fixed components for every $ \lambda \in P^1 $. \subsection{Gonality , Clifford index} Let $C$ be a curve of genus $g$ and and $A$ be a line bundle on $C$ . The Clifford index of $A$ is the integer \[cliff(A) = deg(A) -2r(A) ,\] where $ r(A) = h^0 (A) -1$ is the projective dimension of $|A|$. \\ The Clifford index of $C $ is \[cliff(C) = min \{ cliff(A) {~}|{~}r(A) \geq 1, {~} deg(A) \leq g-1 \}.\] Note that this last definition makes sense only for curves of genus $g \geq 4$. We say that a line bundle $A$ contributes to the Clifford index if it satisfies the inequalities in the above definition and that $A$ computes the Clifford index if $A$ contributes to the Clifford index and $cliff(C) = cliff(A)$. Finaly, we define the Clifford dimension of $C$ as \[ cliffdim(C) = min \{r(A) , A {~}computes {~}the {~}Clifford {~}index{~}of {~}C \}. \] The gonality of $C$, denoted by $gon(C)$, is the minimal degree of a pensil on $C$. Such a definition is nontrivial only for curves of genus $g \geq 3$. Also we say $A$ computes to the gonality if $r(A) = 1$ and $gon(C) = deg(A) $. By Brill-Noether theory one has that \[cliff(C) \leq gon(C) -2 \leq \left[ \frac{g-1}{2} \right] \] and for a general curve both inequalities are equalities. \addtocounter{section}{1} \subsection{Gonality of curves on Enriques surfaces} We will denote (see [C,D] p. 178 ) by \[ \Phi:K^+ \to \bf{Z} _{ \geq 0} \] the function defined by \[ \Phi(C) = inf \{ CE, |2E| {~} is {~}a{~}genus{~} 1 {~}pencil \} . \] \newtheorem{lll}{Lemma }[subsection] \begin{lll} Let $C$ be a smooth curve of genus $g$ on an Enriques surface $S$. Then: \\ (i) $ \Phi(C) \leq \left[ \sqrt{2g-2} \right]$ , where $[l]$ means the integer part of $l$. \\ (ii) if $2 \Phi (C) \leq g-1$ and $ \Phi (C) = CE$ ($ |2E| $ is a genus 1 pencil) , then $cliff(C) \leq cliff(2E|_C ) \leq 2 \Phi(C) - 2$ and $2E|_C$ contributes to the Clifford index. ( by (i) condition $2 \Phi(C) \leq g-1$ is always satisfied if $g \geq 9$). \label{lll} \end{lll} {\bf Proof:} \ \ (i) \ \ is a statment about the Enriques lattice $H^2 (S, \bf{Z})$ and has been proved in [C,D] (Corollary 2.7.1). \\ (ii) Assume $ \Phi (C) = CE $ , where $|2E|$ is a genus 1 pencil. Then we have the sequence: \[ 0 \to O(2E-C) \to O(2E) \to O(2E)|_C \to0 . \] The divisor $2E-C$ cannot be an effective, because $(2E-C)E < 0$ . Therefore $h^0 (2E|_C ) \geq 0 $ and $cliff(C) \leq cliff(2E|_C ) \leq 2 \Phi (C) -2 $, since by our condition $2E|_C$ contributes to the Clifford index.$ \Box$ \newtheorem{PPP}{Proposition}[subsection] \begin{PPP} Let $C$ be a smooth curve on an Enriques surface $S$, and let $A$ be a pencil on $C$ such that $deg(A)=gon(C) \leq \frac{g-1}{2}$. Then (i) There exist an exact sequence: \begin{equation} 0 \to M \to E(C,A) \to L \to 0 , \label{mle} \end{equation} such that M, M-L are line bundles from the positive cone $K^+$ . Moreover the linear systems $\mid L \mid , \mid M \mid $ have no fixed components and $M^2 > LM = deg(A) > L^2 \geq 0$ (if $deg(A) = \frac{g-1}{2} $ then $M-L \in \overline{K^+}$ and $M^2 \geq LM = deg(A) \geq L^2 \geq 0$). (ii) The curve C can be obtained by the Steiner construction for some {~~~~~}$|M( \lambda)| = P^1 \subset \mid M \mid {~~}and {~~}|L( \lambda)| = P^1 \subset \mid L \mid $. (iii) If S is an unnodal Enriques surface , then $ {~~} $ (a) The exact sequence ( \ref{mle}) splits. $ {~~} $ (b) If $gon( \mid C \mid)= gon(C)$ then $L \sim 2E_1 {~} or {~} E_1 + E_2 $, where $ \mid 2E_1 \mid $, \\ $\mid 2E_2 \mid $ are two genus 1 pencils on S. \label{PPP} \end{PPP} {\bf Proof:} \ \ \ 1 Case. \ \ \ $d< \frac{g-1}{2}$. (i). Assume that $d < \frac{g-1}{2} $ , then the vector bundle $E(C,A)$ is not stable in the Bogomolov sense. Indeed $4c_2 (A) = 4deg(A) \leq c_1 (E(C,A) = C^2=2g-2$. By Bogomolov`s theorem [B] we have an exact sequence: \begin{equation} 0 \to M \to E(C,A) \to J_{\xi}(L) \to 0 , \label{new} \end{equation} that such $ (M-L)^2 > 0 $ and $M-L$ is an effective divisor. $E(C,A)$ is a vector bundle genereted by its global sections away from a finet set. Therefore $J_ \xi (L)$ is also generated by its global sections away from a finet set and we obtain that $L^2 \geq 0$ and linear system $ \mid L \mid $ has not fixed components. Claim: \ \ $l( \xi)= length( \xi) = 0$. If $L^2= 0$, then by proposition 3.14 [CD] $L \sim kP$ for some $k \geq 1$. By assumtion, $ deg(A) = kPM+l( \xi) = gon(C)$, but $deg(P \mid _C) =PM \geq deg(A)$, hence we have $l( \xi) = 0, {~}k=1$ . Assume $L^2 > 0$ and $l( \xi) > 0 $, then from exact sequence (~\ref{new} ) we obtain: \begin{eqnarray} h^0 (E(-L)) &=& h^0 (M-L) , \label{0} \\ h^1 (E(-L)) &=& l( \xi) + h^1 (M-L) , \label{H22} \\ h^2 (E(-L)) &=& 0. \end{eqnarray} On the other hand by 1.3 (B) $H^1 (-L) =0$, therefore using (~\ref{E} ) we have: \begin{eqnarray} h^0 (E(-L)) &=& h^0 (C,M \mid _C -A) , \label{13} \\ h^1 (E(-L)) &=& h^1 (C,M \mid _C -A) -2h^2 (-L) , \label{H11} \\ h^2 (E(-L)) &=& 0. \end{eqnarray} By Riemann-Roch on the curve C \begin{eqnarray} h^0 (C,M \mid _C -A) = MC-A-g(C)+1+h^1 (M \mid _C -A) &=& (by{~}~\ref{H11} ,~\ref{H22}{~} and {~} duality){~~} \nonumber \\ =MC - A -( \frac{M^2 + L^2}{2} + ML +1)+1 + \nonumber \\ + l( \xi) +2h^0 (L \otimes K_S )+ h^1 (M-L) &=& (by {~~} R-R {~~} and {~~} ( \ref{new} ) ) \nonumber \\ = M^2 -( \frac{M^2+L^2}{2} + ML +1)+ L^2 +2+ h^1 (M-L) &= & \nonumber \\ = \frac{M^2+L^2}{2} - ML +2+ h^1 (M-L). \label{l} \end{eqnarray} We have $h^2 (M-L) = 0$, hence (~\ref{0}), (~\ref{13} ) and (~\ref{l} ) gives us \begin{equation} \chi (M-L)= \frac{M^2+L^2}{2} - ML +2 \label{17} \end{equation} But by Riemman-Roch theorem $\chi(M-L) = \frac{M^2+L^2}{2} -ML+1$. This contradicts (~\ref{17} ), hence $l( \xi)=0 $ . Now we see that $L$ is nontrivial , because $c_2 (E) = deg(A) = LM$. This implies $(M-L)L > 0$ and $(M-L)M > 0$ , since $ M-L,M \in K^+;L \in \overline{ K^+}$. (ii). By ( \ref{ej}) we have: \begin{equation} 0 \to O_S \to E(C,A) = M \oplus L \to J_ \xi (C ) \to 0, \label{w} \end{equation} where $ \xi = (s)_0 $ is the zero scheme of the section $s \in H^0 (A)$ . Since $H^1 (M) = 0$ we see that the sequence ( \ref{mle} ) is exact on a section level i.e. \[0 \to H^0 (M) \to H^0 (E(C,A)) \to H^0 (L) \to 0 . \] Hence we can write $H^0 (A) \ni s=s_1 \oplus s_2 $ , where $s_1 \in H^0 (L), s_2 \in H^0 (M)$. When $s$ runs through $H^0 (A) $ , $s_1 $ and $s_2$ runs through $|L( \lambda)| = P^1 \subset \mid L \mid$ and $|M( \lambda)| = P_1 \subset \mid M \mid $ respectively. The zero set $ \xi $ consists of points where both sections $s_1 ,s_2$ are zero. This shows that $C$ is obtained by the Steiner construction using $|M( \lambda)| , |L( \lambda)|$ . Now we can see , that the linear system $ \mid M \mid $ has not fixed components, because a fixed part of $ \mid M \mid$ should be also the fixed part of $ \mid C \mid$ , but the curve $C$ is smooth. (iii), \ \ (a). By 1.3 (B) we see that the extention group $Ext^1 (L,M) = H^1 (M-L)=0$, therefore the sequence ( \ref{mle} ) splits. (iii), \ \ (b) . By reducibility lemma 3.2.2 [CD], $L$ is lineary equivalent to a sum of genus 1 curves. Assume $L \sim L_1+L_2$ , where $L_1, L_2$ are effective nontrivial divisors on $S$ such that $L_2$ has no fixed components. Then a curve $C'$ $ \in \mid C \mid $ which is obtained by the Steiner construction using some $|L_2 ( \lambda)| \subset \mid L_2 \mid $ and $|M+L_1 ( \lambda)| \subset \mid M+L_1 \mid$ has gonality $(M+L_1 )L_2 $ . But this number is smaller than $ML= (M' + L)(L_1 + L_2 )= ML_2 + M'L_1 + LL_1$ , where $M'= M-L$. This contradicts our assumtion about minimality of $gon(C)$ . Therefore , there are no such spliting $L$ into two bundles $L_1 ,L_2 $ and we get (iii) , (b). 2 Case. \ \ \ $ d = \frac{g-1}{2}$. Assume $ d = \frac{g-1}{2}$, then by the Riemann-Roch theorem : \begin{equation} \chi (E(C,A),E(C,A)) = 4+c_1 ^2(E) - 4c_2(E) = 4. \nonumber \end{equation} If $E(C,A)$ is $H$-stable in the sense of Mamford-Takemoto ($H$ is an ample divisor on $S$) , then it is simple and $Ext^2 (E,E)^* = Hom(E,E \otimes K_S ) = \ccc $ or 0. Indeed , any nontrivial homomorphism between $E$ and $E \otimes K_S $ should be an isomorphism , because both bundles are stable and have the same determinant. This contadiction shows that $E$ is not $H$-stable for any ample divisor $H$. By 1.3 (D) $C$ is a ample and hence $E$ is not $C$-stable , therefore we have the sequence: \[ 0 \to M \to E(C,A) \to J_{ \xi} (L) \to 0 , \] such that $(M-L)C = M^2 -L^2 \geq 0 $ . Sence $4(ML+l( \xi ))= 4c_2 (E) =c_1 (E) = M^2 + L^2 + 2ML $ we have $ (M-L)^2 =4l( \xi ) \geq 0$ \ i.e. \ $ M-L \in \overline{K^+ }$. And now we can argue as in case 1. above. $ \Box$ {\bf Proof of Theorem 1 } : If $gon(|C|) \leq \frac{g-1}{2}$, then by proposition~\ref{PPP} we are done. On the other hand by Brill-Neother theory we have : \[gon(C) \leq \left[ \frac{g-2}{2} \right] +2 . \Box \] \addtocounter{section}{1} \subsection{Curves of the Clifford dimension 2} It is well known that the curve of Clifford dimension 2 is a smooth plane curve of degree $d \geq 5$ and a line bundle $A$ computing the Clifford index is unique. In this case there is a 1-dimensional family of pencils of degree $d-1$ computing gonality , all obtained by projecting from a point of the curve. \newtheorem{ggg}{Proposition}[subsection] \begin{ggg} An Enriques surface does not contain any smooth plane curve of degree $d \geq5 $ . \end{ggg} {\bf Proof:} \ \ Let $C$ be a curve of degree $ d \geq 6 $ . Recall that for smooth plane curves of degree $d$ by adjunction formula we have $g(C) =\frac{d(d-3)}{2} +1$. By lemma~\ref{lll} (i) \[ \Phi(C) \leq \left[ \sqrt{2g(C)-2} \right] = \left[ \sqrt{d(d-3)} \right] \leq d-2 .\] On the other hand we have \[ \Phi(C) \leq d-2 \leq \frac{d(d-3)}{4} = \frac{g(C) -1}{2}. \] By lemma~\ref{lll} (ii) we obtain \[ cliff(C) =d-4 \leq cliff(2E|_C) \leq 2 \Phi(C) -2 \leq d-4 \] where $\Phi(C) =CE $ and $|2E|$ is a genus 1 pencil . Moreover , $(2E)|_C $ contributes to the Clifford index . This is a contradiction , since we obtain a pencil which computes the Clifford index. If $deg(C)= 5$, then from the exact sequence (~\ref{ej} ) we have : \[h^0 (J_{ \xi} (C+K)) = h^0 (K_C -A) = h^1 (A) = g-d+r(A) = 3, \] where $\xi = (s)_0 $ is zero set of the section $s \in H^0 (A) $ . Therefore we can consider $\xi $ as a divisor on $C$ which is linear equivalent to $A$. A curve from the linear system $D \in |J_A (C+K)|$ cut out a divisor $A+A_D$ on our curve $C$ . The divisor $A+A_D$ is linearly equivalent to the canonical divisor $K_C $, hence $deg(A_D ) = 5$ . $D $ runs through $ |J_A (C+K)|$ and cuts out $A_D$ on the curve $C$, therefore $h^0 (C,A_D) \geq h^0 (C,A) =3$. By the Cliford theorem $deg(A_D ) \geq 2(h^0 (C,A_D )-1)$ , therefore $h^0 (C,A_D ) = 3$. The line bundle $A$ is unique linear systerm of degree 5 and projective dimension 2 , hence we have $A \sim A_D$ . Denote by $C'$ a smooth curve in the linear system $|J_A (C+K)|$ which cuts out the divisor $A+A'$ on our curve $C$ . Then we can consider the divisor $A'$ as a divisor on $C'$. A curve $D' \in |J_A (C+K)|$ cuts out a divisor $A+A_{D'}$ on the curve $C'$ ($A_{D'} \in|A'|$). In the same way as above we get $de! g(A_{D'}) =5 , h^0 (C',A_{D'}) =3$ \addtocounter{section}{1} \subsection{ Curves of the Clifford dimension $r \geq 3$} The curves of Clifford dimension $r \geq3$ are extremly rare . We refer the reader to the preprint [ELMS] for the details. The main result in [ELMS] is: \\ If $C$ has Clifford dimension $r \geq 3$ then one of the following holds: 1. (i) $C$ has genus $g=4r-2$ and Clifford index $2r-3 =[ \frac{g-1}{2}] -1$ . \\ \ \ \ \ (ii) $C$ has a unique line bundle $L$ computing the Clifford index , $L^2$ is the canonical bundle on $C$. 2. If the curve does not satisfy 1. , then $ r \geq10$ and the degree $d$ of the line bundle computing the Clifford index is $ \geq 6r-6$ and its genus is $ \geq 8r-7$. Eisinbud , Lange , Martens and Schreyer conjecture that such a curve does not exists (see [ELMS]). \newtheorem{CCC}{Proposition}[subsection] \begin{CCC} Let $C$ be a smooth curve of Clifford dimension $r \geq 3$ and genus $g$ contained in an Enriques surface $S$. Then $C$ is the curve described by the case 2 above. Moreover $g \geq 2(r-1)^2 +1 \geq 163$ and $2 \left[ \sqrt{2g-2} \right] -2 +2r \geq d $, where $d$ is the degree of a line bundle computing Clifford index. \label{CCC} \end{CCC} {\bf Proof:} \ \ Consider the curve which satisfies 1. above . By lemma~\ref{lll} (ii) we have \[cliff(C) \leq 2 \Phi(C) -2 \leq \frac{g-1}{2} -2 .\] This contradicts the condition $cliff(C) = [ \frac{g-1}{2}] -1 $. For a curve which satisfies 2 lemma~\ref{lll} (i) and (ii) implies that \[ d-2r \leq 2 \Phi(C) -2 \leq 2 \left[ \sqrt{2g-2} \right] -2 .{~~~~~~~} (*) \] But $d \geq 6r-6$ , hence \[ 4r-6 \leq 2[ \sqrt{2g-2}] -2 \] and we get \[ g \geq 2(r-1)^2 +1 \geq 163 . \] Also from (*) we obtain \[d \leq2r-2 + 2 \left[ \sqrt{2g-2} \right] . \Box \] \addtocounter{section}{1} \subsection{ Examples} In this section we will apply proposition~\ref{PPP} to obtain examples of smooth linearly equivalent curves of different gonality. In explicit examples we try to explain the reasons why the gonality is not constant . Assume $S$ is an unnodal Enriques surface. Consider $L = E_1 +E_2 , M = 2L $, where $|2E_1 | , |2E_2 |$ are genus 1 pencils such that $E_1 E_2 = 1$ , then $C \sim M+L$ is very ample by corollary 2 ( appendix , after chapter 4 ) [CD]. If $C$ is a smooth curve in the linear system $|M+L|$ which is obtained by Steiner construction using $P^1 =|L|$ and $|M( \lambda)| \subset |M|$ , then by proposition~\ref{PPP} we see that $gon(C) \leq ML =4$ and actually $gon(C) = 4$. Indeed, if $gon(C) \leq 3$ then by proposition~\ref{PPP} we obtain $M'$ and $L'$ such that $C \sim L'+M' , gon(C) =M'L' > L'^2 \geq 0 $ . Therefore $L'^2 \leq2$ and by (iii)(b) we see that $L \sim E' +E'' $ or $ L=2E'$ for some genus 1 pencils $|2E'| , |2E''| $ such that $E'E''= 1$. But $C \sim 3(E_1 +E_2)$ and $gon(C) = M'L'= CL' - L'^2 \geq 4$ . If $C$ is obtained by Steiner costruction then $C$ contains two base points of pencil $|L|$ . Since $C$ is very ample there is a smooth curve $C' \in |C|$ which does not contain base points of any pencil $|L'| = |E' + E''| $ , because we have at most countable number of such pencils. Now we claim that $gon(C') \geq 5$. Indeed , by proposition~\ref{PPP} (i) , (iii) we have $gon(C') = M'L' \geq 4$ and equality occurs , as we have seen above , only if $L'^2 =2$ and $C'$ is obtained by Steiner construction using $|L'| , |M'( \lambda ) |$ but this is not the case , because $C' $ does not contain base points of linear system $|L'|$. \newtheorem{G}{1.} \begin{G} A general curve in the linear system $|3(E_1 + E_2 ) |$ has gonality $\geq$ 5 and Clifford index $\geq$ 3. There is a linear subsystem $V \subset |C|$ of codimension 2 such that every smooth curve in $V$ has gonality 4 and Clifford index 2. Moreover, gon($|C|$) = 4 \\ \end{G} The subsystem $V \subset |C| $ consists of the curves which contain two base points of the pencil $|L| = |E_1 + E_2 |$ and therefore is obtained by Steiner construction using $|L|$ , $|M|$. Similarly one can obtain the following: \newtheorem{GG}{2.} \begin{GG} A general curve in the linear system $|3E_1 + 4E_2 ) | $ has gonality 6 and Clifford index 4 . There is a linear subsystem $V \subset |C|$ of codimension 2 such that every smooth curve in $V$ has gonality 5 and Clifford index 3. Moreover gon($|C|$) =5. \end{GG} In this case $M=3E_1 + 2E_2$ , $L=E_1 + E_2$. And the curve $C \in |M+L|$ which is obtained by Steiner construction using $|L|$ ,$|M( \lambda) | \subset |M| $ has gonality 5 and Clifford index 3. But $C$ is very ample and therefore a general curve of the linear system $|C|$ does not contain base points of any pencil $|L'|$ . The proof is similar to the proof in the example above and we omit the details. \paragraph{References} : \\ [B] F.A. Bogomolov "Holomorphic tensors and vector bundles on projective varieties" Izvestya of Ac.of Sc. USSR ser. math. v.42 N6 pp.1227-1287. \\ [CD] F. Cossec ; I. Dolgachev "Enriques Surfaces 1", Birkh \"{a}user 1989. \\ [GL] M. Green ; R. Lazarsfeld "Special divisors on curves on K3 surfaces" \ \ \ \ \ \ \ Invent. Math. 89, 357-370, 1989. \\ [ELMS] D. Eisinbud , H. Lange , G. Martens , F.-O. Schreyer "The Clifford dimension of a projective curve ." , preprint. \\ [P] G. Pareschi "Exceptional linear systems on curves on Del Pezzo surfaces" , Math. Ann. 291 , 17-38 (1991) . \\ [T] A.N. Tyurin "Cycles, curves and vector bundles on algebraic surfaces." Duke Math.J. 54, 1-26,(1987). \\ Department of geometry and topology, Faculty of mathematics, Vilnius university, Naugarduko g.24, 2009 Vilnius, Lithuania. \end{document}

9203

alg-geom/9203002

en

https://arxiv.org/abs/alg-geom/9203002

alg-geom/9203002

Motohico Mulase

Yingchen Li and Motohico Mulase

Prym Varieties and Integrable Systems

37 pages in AMS-LaTeX format. Article updated following the published version. Default page size is US Letter

Communications in Analysis and Geometry, Volume 5, (1997) 279--332

A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GL(n) defined on certain infinite-dimensional Grassmannians is studied. It is proved that every finite-dimensional orbit of the action of traceless elements of these commutative Lie algebras is isomorphic to the Prym variety associated with a morphism of algebraic curves. Conversely, it is shown that every Prym variety can be realized as a finite-dimensional orbit of the action of traceless diagonal elements of the formal loop algebra, which defines the multicomponent KP system.

alg-geom

\section{Introduction} \label{sec: intro} \medskip \noindent\textbf{0.1} {From} a geometric point of view, the \emph{Kadomtsev-Petviashvili (KP) equations} are best understood as a set of commuting vector fields, or \emph{flows}, defined on an infinite-dimensional Grassmannian \cite{S}. The Grassmannian $Gr_1(\mu)$ is the set of vector subspaces $W$ of the field $L = \mathbb{C}((z))$ of formal Laurent series in $z$ such that the projection $W\longrarrow \mathbb{C}((z))/\mathbb{C}[[z]]z$ is a Fredholm map of index $\mu$. The commutative algebra $\mathbb{C}[z^{-1}]$ acts on $L$ by multiplication, and hence it induces commuting flows on the Grassmannian. This very simple picture is nothing but the KP system written in the language of infinite-dimensional geometry. A striking fact is that every finite-dimensional orbit (or integral manifold) of these flows is canonically isomorphic to the Jacobian variety of an algebraic curve, and conversely, every Jacobian variety can be realized as a finite-dimensional orbit of the KP flows \cite{M1}. This statement is equivalent to the claim that the KP equations characterize the Riemann theta functions associated with Jacobian varieties \cite{AD}. If one generalizes the above Grassmannian to the Grassmannian $Gr_n(\mu)$ consisting of vector subspaces of $L^{\dsum n}$ with a Fredholm condition, then the formal loop algebra $gl(n,L)$ acts on it. In particular, the Borel subalgebra (one of the maximal commutative subalgebras) of Heisenberg algebras acts on $Gr_n(\mu)$ with the center acting trivially. Let us call the system of vector fields coming from this action the \emph{Heisenberg flows} on $Gr_n(\mu)$. Now one can ask a question: what are the finite-dimensional orbits of these Heisenberg flows, and what kind of geometric objects do they represent? Actually, this question was asked to one of the authors by Professor H.~Morikawa as early as in 1984. In this paper, we give a complete answer to this question. Indeed, we shall prove (see Proposition~\ref{5.1. Proposition} and Theorem~\ref{5.8. Theorem} below) \medskip \begin{thm} \label{A} A finite-dimensional orbit of the Heisenberg flows defined on the Grassmannian of vector valued functions corresponds to a covering morphism of algebraic curves, and the orbit itself is canonically isomorphic to the Jacobian variety of the curve upstairs. Moreover, the action of the traceless elements of the Borel subalgebra (the traceless Heisenberg flows) produces the Prym variety associated with this covering morphism as an orbit. \end{thm} \medskip \begin{rem} The relation between Heisenberg algebras and covering morphisms of algebraic curves was first discovered by Adams and Bergvelt \cite{AB}. \end{rem} \medskip \noindent\textbf{0.2} Right after the publication of works (\cite{AD}, \cite{M1}, \cite{Sh1}) on characterization of Jacobian varieties by means of integrable systems, it has become an important problem to find a similar theory for Prym varieties. We establish in this paper a simple solution of this problem in terms of the \emph{multi-component KP system} defined on a certain quotient space of the Grassmannian of vector valued functions. Classically, Prym varieties associated with degree two coverings of algebraic curves were used by Schottky and Jung in their approach to the Schottky problem. The modern interests in Prym varieties were revived in \cite{Mum1}. Recently, Prym varieties of higher degree coverings have been used in the study of the generalized theta divisors on the moduli spaces of stable vector bundles over an algebraic curve \cite{BNR}, \cite{H}. This direction of research, usually called ``Hitchin's Abelianization Program,'' owes its motivation and methods to finite dimensional integrable systems in the context of symplectic geometry. In the case of infinite dimensional integrable systems, it has been discovered that Prym varieties of ramified double sheeted coverings of curves appear as solutions of the BKP system \cite{DJKM}. Independently, a Prym variety of degree two covering with exactly two ramification points has been observed in the deformation theory of two-dimensional Schr\"odinger operators \cite{No}, \cite{NV}. As far as the authors know, the only Prym varieties so far considered in the context of integrable systems are associated with ramified, double sheeted coverings of algebraic curves. Consequently, the attempts (\cite{Sh2}, \cite{T}) of characterizing Prym varieties in terms of integrable systems are all restricted to these special Prym varieties. Let us define the \emph{Grassmannian quotient} $Z_n(0)$ as the quotient space of $Gr_n(0)$ by the diagonal action of $\big(1 + \mathbb{C}[[z]]z\big)^{\times n}$. The \emph{traceless $n$-component KP system} is defined by the action of the traceless diagonal matrices with entries in $\mathbb{C}[z^{-1}]$ on $Z_n(0)$. Since this system is a special case of the traceless Heisenberg flows, every finite-dimensional orbit of this system is a Prym variety. Conversely, an arbitrary Prym variety associated with a degree $n$ covering morphism of algebraic curves can be realized as a finite-dimensional orbit. Thus we have (see Theorem~\ref{5.14. Theorem} below): \medskip \begin{thm} \label{B} An algebraic variety is isomorphic to the Prym variety associated with a degree $n$ covering of an algebraic curve if and only if it can be realized as a finite-dimensional orbit of the traceless $n$-component KP system defined on the Grassmannian quotient $Z_n(0)$. \end{thm} \medskip \noindent\textbf{0.3} The relation between algebraic geometry and the Grassmannian comes from the cohomology map of \cite{SW}, which assigns injectively a point of $Gr_1(0)$ to a set of geometric data consisting of an algebraic curve and a line bundle together with some local information. This correspondence was enlarged in \cite{M3} to include arbitrary vector bundles on curves. In this paper, we generalize the cohomology functor of \cite{M3} so that we can deal with arbitrary morphisms between algebraic curves. Let ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (n_1, \cdots, n_\ell)$ denote an integral vector consisting of positive integers satisfying that $n = n_1 + \cdots + n_\ell$. \medskip \begin{thm} \label{C} For each ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, the following two categories are equivalent: \begin{enumerate} \item The category $\mathcal{C}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$. An object of this category consists of an arbitrary degree $n$ morphism $f:C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow C_0$ of algebraic curves and an arbitrary vector bundle $\mathcal{F}$ on $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. The curve $C_0$ has a smooth marked point $p$ with a local coordinate $y$ around it. The curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ has $\ell$ $(1\le \ell \le n)$ smooth marked points $\{p_1, \cdots, p_\ell\} = f^{-1}(p)$ with ramification index $n_j$ at each point $p_j$. The curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is further endowed with a local coordinate $y_j$ and a local trivialization of $\mathcal{F}$ around $p_j$. \item The category $\mathcal{S}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$. An object of this category is a triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ consisting of a point $W\in \bigcup_{\mu\in \mathbb{Z}}Gr_n(\mu)$, a ``large'' subalgebra $A_0\subset \mathbb{C}((y))$ for some $y\in \mathbb{C}[[z]]$, and another ``large'' subalgebra $$ A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\subset \bigoplus_{j=1}^\ell \mathbb{C}((y^{1/n_j})) \isom \bigoplus_{j=1}^\ell \mathbb{C}((y_j))\;. $$ In a certain matrix representation as subalgebras of the formal loop algebra $gl\big(n,\mathbb{C}((y))\big)$ acting on the Grassmannian, they satisfy $A_0\subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\cdot W\subset W$. \end{enumerate} \end{thm} \medskip \noindent The precise statement of this theorem is given in Section~\ref{sec: coh}, and its proof is completed in Section~\ref{sec: inverse}. One of the reasons of introducing a category rather than just a set is because we need not only a set-theoretical bijection of objects but also a canonical correspondence of the morphisms in the proof of the claim that every Prym variety can be realized as a finite-dimensional orbit of the traceless multi-component KP system on the Grassmannian quotient. \medskip \noindent\textbf{0.4} The motivation of extending the framework of the original Segal-Wilson construction to include arbitrary vector bundles on curves of \cite{M3} was to establish a complete geometric classification of all the commutative algebras consisting of ordinary differential operators with coefficients in scalar valued functions. If we apply the functor of Theorem~\ref{C} in this direction, then we obtain (see Proposition~\ref{6.14. Proposition} and Theorem~\ref{6.15. Theorem} below): \medskip \begin{thm} \label{D} Every object of the category $\mathcal{C}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ with a smooth curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and a line bundle $\mathcal{F}$ on $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ satisfying the cohomology vanishing condition $$ H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = 0 $$ gives rise to a maximal commutative algebra consisting of ordinary differential operators with coefficients in $n\times n$ matrix valued functions. \end{thm} \medskip\noindent Some examples of commuting matrix ordinary differential operators have been studied before (\cite{G}, \cite{Na}). Grinevich's work is different from ours. In \cite{G} he considers commuting pairs of matrix differential operators. For each commuting pair he constructs a single affine algebraic curve (possibly reducible) in the affine plane and a vector bundle on each of the irreducible components and conversely, given such a collection of algebro-geometric data together with some extra local information he constructs a commuting pair of matrix differential operators. In our case, the purpose is to classify commutative algebras of matrix differential operators. This point of view is more intrinsic than considering commuting pairs because they are particular choices of generators of the algebras. On the algebro-geometric side, we obtain morphisms of two abstract curves (no embeddings) and maps of the corresponding Jacobian varieties. Prym varieties come in very naturally in our picture. Nakayashiki's construction (Appendix of \cite{Na}) is similar to ours, but that corresponds to locally cyclic coverings of curves, i.e.\ a morphism $f:C\longrarrow C_0$ such that there is a point $p\in C_0$ where $f^{-1}(p)$ consists of one point. Since we can use arbitrary coverings of curves, we obtain in this paper a far larger class of totally new examples systematically. As a key step from algebraic geometry of curves and vector bundles to the differential operator algebra with matrix coefficients, we prove the following (see Theorem~\ref{6.5. Theorem} below): \medskip \begin{thm} \label{E} The big-cell of the Grassmannian $Gr_n(0)$ is canonically identified with the group of monic invertible pseudodifferential operators with matrix coefficients. \end{thm} \medskip\noindent Only the case of $n=1$ of this statement was known before. With this identification, we can translate the flows on the Grassmannian associated with an arbitrary commutative subalgebra of the loop algebras into an integrable system of nonlinear partial differential equations. The unique solvability of these systems can be shown by using the generalized Birkhoff decomposition of \cite{M2}. \medskip \noindent\textbf{0.5} This paper is organized as follows. In Section~\ref{sec: 1}, we review some standard facts about Prym varieties. The Heisenberg flows are introduced in Section~\ref{sec: Heis}. Since we do not deal with any central extensions in this paper, we shall not use the Heisenberg algebras in the main text. All we need are the maximal commutative subalgebras of the formal loop algebras. Accordingly, the action of the Borel subalgebras will be replaced by the action of the full maximal commutative algebras defined on certain quotient spaces of the Grassmannian. This turns out to be more natural because of the coordinate-free nature of the flows on the quotient spaces. The two categories we work with are defined in Section~\ref{sec: coh}, where a generalization of the cohomology functor is given. In Section~\ref{sec: inverse}, we give the construction of the geometric data out of the algebraic data consisting of commutative algebras and a point of the Grassmannian. The finite-dimensional orbits of the Heisenberg flows are studied in Section~\ref{sec: Prym}, in which the characterization theorem of Prym varieties is proved. Section~\ref{sec: ODE} is devoted to explaining the relation of the entire theory with the ordinary differential operators with matrix coefficients. The results we obtain in Sections~\ref{sec: coh}, \ref{sec: inverse}, and \ref{sec: ODE} (except for Theorem~\ref{6.15. Theorem}, where we need zero characteristic) hold for an arbitrary field $k$. In Sections~\ref{sec: 1} and \ref{sec: Prym} (except for Proposition~\ref{5.1. Proposition}, which is true for any field), we work with the field $\mathbb{C}$ of complex numbers. \medskip \begin{ack} The authors wish to express their gratitude to the Max-Planck-Institut f\"ur Mathematik for generous financial support and hospitality, without which the entire project would never have taken place. They also thank M.~Bergvelt for sending them the paper \cite{AB} prior to its publication. The earlier version of the current article has been circulated as a Max-Planck-Institut preprint. \end{ack} \bigskip \section{Covering morphisms of curves and Prym varieties} \label{sec: 1} \medskip We begin with defining Prym varieties in the most general setting, and then introduce \emph{locally cyclic coverings} of curves, which play an important role in defining the category of arbitrary covering morphisms of algebraic curves in Section~\ref{sec: coh}. \begin{Def} \label{1.1. Definition} Let $f : C\longrightarrow {C_0}$ be a covering morphism of degree $n$ between smooth algebraic curves $C$ and ${C_0}$, and let $N_f: {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)\longrarrow {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})$ be the norm homomorphism between the Jacobian varieties, which assigns to an element $\sum_q n_q\cdot q\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ its image $\sum_q n_q\cdot f(q)\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})$. This is a surjective homomorphism, and hence the kernel ${\text{\rm{Ker}}}} \def\coker{{\text{\rm{coker}}}(N_f)$ is an abelian subscheme of ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ of dimension $g(C)-g({C_0})$, where $g(C)$ denotes the genus of the curve $C$. We call this kernel the \emph{Prym variety} associated with the morphism $f$, and denote it by $\Prym(f)$. \end{Def} \medskip \begin{rem} \label{{1.2.} Remark} Usually the Prym variety of a covering morphism $f$ is defined to be the connected component of the kernel of the norm homomorphism containing 0. Since any two connected components of ${\text{\rm{Ker}}}} \def\coker{{\text{\rm{coker}}}(N_f)$ are translations of each other in ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$, there is no harm to call the whole kernel the Prym variety. If the pull-back homomorphism $f^{*}: {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})\longrarrow {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ is injective, then the norm homomorphism can be identified with the transpose of $f^{*}$, and hence its kernel is connected. So in this situation, our definition coincides with the usual one. We will give a class of coverings where the pull-back homomorphisms are injective (see Proposition~\ref{1.7. Proposition}). \end{rem} \medskip \begin{rem} \label{{1.3.} Remark} Let ${R}\subset C$ be the ramification divisor of the morphism $f$ of Definition~\ref{1.1. Definition} and $\mathcal{O}_C({R})$ the locally free sheaf associated with ${R}$. Then it can be shown that for any line bundle $\mathcal{L}$ on $C$, we have $N_f(\mathcal{L})=\det(f_*\mathcal{L})\otimes \det\big(f_*\mathcal{O}_C({R})\big)$. Thus up to a translation, the norm homomorphism can be identified with the map assigning the determinant of the direct image to the line bundle on $C$. Therefore, one can talk about the Prym varieties in ${\text{\rm{Pic}}}} \def\Coker{{\text{\rm{Coker}}}} \def\ord{{\text{\rm{ord}}}^d(C)$ for an arbitrary $d$, not just in ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)={\text{\rm{Pic}}}} \def\Coker{{\text{\rm{Coker}}}} \def\ord{{\text{\rm{ord}}}^{0}(C)$. When the curves $C$ and $C_0$ are singular, we replace the Jacobian variety ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ by the generalized Jacobian, which is the connected component of $H^1(C, \mathcal{O}^*_{C})$ containing the structure sheaf. By taking the determinant of the direct image sheaf, we can define a map of the generalized Jacobian of $C$ into $H^1(C_0,\mathcal{O}^*_{C_0})$. The fiber of this map is called the \emph{generalized} Prym variety associated with the morphism $f$. \end{rem} \medskip \begin{rem} \label{{1.4.} Remark} According to our definition (Definition~\ref{1.1. Definition}), the Jacobian variety of an arbitrary algebraic curve $C$ can be viewed as a Prym variety. Indeed, for a nontrivial morphism of $C$ onto $\mathbb{P}^1$, the induced norm homomorphism is the zero-map. Thus the class of Prym varieties contains Jacobians as a subclass. Of course there are infinitely many ways to realize ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ as a Prym variety in this manner. \end{rem} \medskip Let us consider the polarizations of Prym varieties. Let $\Theta_{C}$ and $\Theta_{{C_0}}$ be the Riemann theta divisors on ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ and ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})$, respectively. Then the restriction of $\Theta_{C}$ to $\Prym(f)$ gives an ample divisor $H$ on $\Prym(f)$. This is usually not a principal polarization if $g(C_0)\not= 0$. There is a natural homomorphism $\psi:{\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})\times \Prym(f)\longrarrow {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ which assigns $f^*\mathcal{L}\otimes \mathcal{M}$ to $(\mathcal{L},\mathcal{M})\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})\times \Prym(f)$. This is an isogeny, and the pull-back of $\Theta_{C}$ under this homomorphism is given by $$ \psi^*\mathcal{O}_{{\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)}(\Theta_{C}) \isom \mathcal{O}_{{\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})}(n\Theta_{C_0})\otimes \mathcal{O}_{\Prym(f)}(H)\;. $$ In Section~\ref{sec: coh}, we define a category of covering morphisms of algebraic curves. As a \emph{morphism} between the covering morphisms, we use the following special coverings: \medskip \begin{Def} \label{1.5. Definition} A degree $r$ morphism $\alpha : C\longrarrow {C_0}$ of algebraic curves is said to be a \emph{locally cyclic covering} if there is a point $p\in {C_0}$ such that $\alpha^{*}(p)=r\cdot q$ for some $q\in C$. \end{Def} \medskip \begin{prop} \label{1.6. Proposition} Every smooth projective curve $C$ has infinitely many smooth locally cyclic coverings of an arbitrary degree. \end{prop} \medskip \begin{proof} We use the theory of spectral curves to prove this statement. For a detailed account of spectral curves, we refer to \cite{BNR} and \cite{H}. Let us take a line bundle $\mathcal{L}$ over $C$ of sufficiently large degree. For such $\mathcal{L}$ we can choose sections $s_{i}\in H^{0}(C, \mathcal{L}^i)$, $i$ $=$ 1, 2, $\cdots$, $r$, satisfying the following conditions: \begin{enumerate} \item All $s_{i}$'s have a common zero point, say $p\in C$, i.e., $s_i\in H^0(C, \mathcal{L}^i(-p))$, ${\ } {\ } {\ }$ $i=1,2,\cdots , r$; \item $s_{r}\notin H^{0}(C, \mathcal{L}^{r}(-2p))$. \end{enumerate} Now consider the sheaf $\mathcal{R}$ of symmetric $\mathcal{O}_{C}$-algebras generated by $\mathcal{L}^{-1}$. As an $\mathcal{O}_{C}$-module this algebra can be written as $$ \mathcal{R} = \bigoplus_{i=0}^{\infty }\mathcal{L}^{-i}\;. $$ In order to construct a locally cyclic covering of $C$, we take the ideal $\mathcal{I}_s$ of the algebra $\mathcal{R}$ generated by the image of the sum of the homomorphisms $s_{i}: \mathcal{L}^{-r}\longrarrow \mathcal{L}^{-r+i}$. We define $C_{s}={\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(\mathcal{R}/\mathcal{I}_s)$, where $s = (s_1, s_2, \cdots, s_r)$. Then $C_{s}$ is a spectral curve, and the natural projection $\pi:C_s\longrarrow C$ gives a degree $r$ covering of $C$. For sufficiently general sections $s_{i}$ with properties (1) and (2), we may also assume the following (see \cite{BNR}): \begin{enumerate} \item[(3)] The spectral curve $C_{s}$ is integral, i.e.~reduced and irreducible. \end{enumerate} We claim here that $C_{s}$ is smooth in a neighborhood of the inverse image of $p$. Indeed, let us take a local parameter $y$ of $C$ around $p$ and a local coordinate $x$ in the fiber direction of the total space of the line bundle $\mathcal{L}$. Then the local Jacobian criterion for smoothness in a neighborhood of $\pi^{-1}(p)$ states that the following system $$ \begin{cases} x^{r} +s_{1}(y)x^{r-1}+\cdots +s_{r}(y)=0\\ rx^{r-1}+s_{1}(y)(r-1)x^{r-2} +\cdots +s_{r-1}(y)=0\\ s_{1}(y)'x^{r-1}+ s_{2}(y)'x^{r-2}+\cdots +s_{r}(y)'=0 \end{cases} $$ of equations in $(x, y)$ has no solutions. But this is clearly the case in our situation because of the conditions (1), (2) and (3). Thus we have verified the claim. It is also clear that $\pi^*(p) = r\cdot q$, where $q$ is the point of $C_s$ defined by $x^r = 0$ and $y = 0$. Then by taking the normalization of $C_{s}$ we obtain a smooth locally cyclic covering of $C$. This completes the proof. \end{proof} \medskip \begin{prop} \label{1.7. Proposition} Let $\alpha:C\longrarrow {C_0}$ be a locally cyclic covering of degree $r$. Then the induced homomorphism $\alpha^*:{\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})\longrarrow {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C)$ of Jacobians is injective. In particular, the Prym variety $\Prym(\alp)$ associated with the morphism $\alp$ is connected. \end{prop} \medskip \begin{proof} Let us suppose in contrary that $\mathcal{L}\not\cong {\mathcal{O}}_{{C_0}}$ and $\alpha^{*}\mathcal{L}\cong {\mathcal{O}}_{C}$ for some $\mathcal{L}\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})$. Then by the projection formula we have $ \mathcal{L}\otimes \alpha_{*}{\mathcal{O}}_{C}\cong \alpha_{*}\mathcal{O}_{C}$. Taking determinants on both sides we see that $\mathcal{L}$ is an $r$-torsion point in ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow({C_0})$, i.e.~$\mathcal{L}^r \cong \mathcal{O}_{C_0}$. Let $m$ be the smallest positive integer satisfying that ${\mathcal{L}}^{m} \cong {\mathcal{O}}_{{C_0}}$. Let us consider the spectral curve $$ C' = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big( \bigoplus _{i=0}^{\infty} \mathcal{L}^{-i}/\mathcal{I}_s \big) $$ defined by the line bundle $\mathcal{L}$ and its sections $$ s = (s_1, s_2, \cdots, s_{m-1}, s_m) = (0, 0, \cdots, 0, 1) \in \bigoplus_{i = 1}^m H^{0}({C_0},\mathcal{L}^i)\;. $$ It is easy to verify that $C'$ is an unramified covering of ${C_0}$ of degree $m$. Now we claim that the morphism $\alpha : C\longrarrow {C_0}$ factors through $C'$, but this leads to a contradiction to our assumption that $\alpha$ is a locally cyclic covering. The construction of such a morphism $f:C\longrarrow C'$ over ${C_0}$ amounts to defining an ${\mathcal{O}}_{{C_0}}$-algebra homomorphism \begin{equation} f^{\sharp} :\bigoplus _{i=0}^{\infty}\mathcal{L}^{-i} /\mathcal{I}_s \longrightarrow \alpha_{*}{\mathcal{O}}_{C}\;.\label{1.8} \end{equation} In order to give (\ref{1.8}), it is sufficient to define an ${\mathcal{O}}_{{C_0}}$-module homomorphism $\phi :\mathcal{L}^{-1}\longrarrow \alpha_{*}{\mathcal{O}}_{C}$ such that $\phi ^{\otimes m}: \mathcal{L}^{-m}\cong \mathcal{O}_{C_0}\longrightarrow \alpha _{*}\mathcal{O}_{C}$ is the inclusion map induced by $\alpha$. Since we have $$ H^0(C,\mathcal{O}_C) \isom H^0(C_0,\alpha_*\mathcal{O}_C) \isom H^{0}(C_0, \mathcal{L}\otimes \alpha _{*}\mathcal{O}_{C}) \isom H^{0}(C_0, \mathcal{L}^{m}\otimes \alpha _{*}\mathcal{O}_{C})\;, $$ the existence of the desired $\phi$ is obvious. This completes the proof. \end{proof} \bigskip \section{The Heisenberg flows on the Grassmannian of vector valued functions} \label{sec: Heis} \medskip In this section, we define the Grassmannians of vector valued functions and introduce various vector fields (or flows) on them. Let $k$ be an arbitrary field, $k[[z]]$ the ring of formal power series in one variable $z$ defined over $k$, and $L = k((z))$ the field of fractions of $k[[z]]$. An element of $L$ is a formal Laurent series in $z$ with a pole of finite order. We call $y = y(z)\in L$ an element of \emph{order} $m$ if $y\in k[[z]]z^{-m}\setminus k[[z]]z^{-m+1}$. Consider the infinite-dimensional vector space $V = L^{\dsum n}$ over $k$. It has a natural filtration by the (pole) order $$ \cdots \subset F^{(m-1)}(V) \subset F^{(m)}(V) \subset F^{(m+1)}(V)\subset \cdots\;, $$ where we define \begin{equation} F^{(m)}(V) = \left\{\left. \sum_{j = 0}^\infty a_j z^{-m+j} \; \right|\; a_j\in k^{\dsum n}\right\}\;.\label{2.1} \end{equation} In particular, we have $F^{(m)}(V)\big/F^{(m-1)}(V) \isom k^{\dsum n}$ for all $m\in \mathbb{Z}$. The filtration satisfies $$ \bigcup_{m=-\infty}^\infty F^{(m)}(V) = V\quad{\text{ and }}\quad \bigcap_{m=-\infty}^\infty F^{(m)}(V) = \{0\}\;, $$ and hence it determines a topology in $V$. In Section~\ref{sec: inverse}, we will introduce other filtrations of $V$ in order to define algebraic curves and vector bundles on them. The current filtration (\ref{2.1}) is used only for the purpose of defining the Grassmannian as a pro-algebraic variety (see for example \cite{KSU}). \medskip \begin{Def} \label{2.2. Definition} For every integer $\mu$, the following set of vector subspaces $W$ of $V$ is called the index $\mu$ Grassmannian of vector valued functions of size $n$: $$ Gr_n(\mu) = \{ W\subset V\;|\; \gamma_W {\text{ is Fredholm of index }} \mu\}\;, $$ where $\gamma_W:W\longrarrow V\big/F^{(-1)}(V)$ is the natural projection. \end{Def} \medskip\noindent Let $N_W = \big\{\ord_z(v) \; \big|\; v\in W\big\}$. Then the Fredholm condition implies that $N_W$ is bounded from below and contains all sufficiently large positive integers. But of course, this condition of $N_W$ does not imply the Fredholm property of $\gam_W$ when $n>1$. \medskip \begin{rem} \label{{2.3.} Remark} We have used $F^{(-1)}(V)$ in the above definition as a reference open set for the Fredholm condition. This is because it becomes the natural choice in Section~\ref{sec: ODE} when we deal with the differential operator action on the Grassmannian. {From} purely algebro-geometric point of view, $F^{(0)}(V)$ can also be used (see Remark~\ref{{4.6.} Remark}). \end{rem} \medskip\noindent The \emph{big-cell} $Gr_n^+(0)$ of the Grassmannian of vector valued functions of size $n$ is the set of vector subspaces $W\subset V$ such that $\gamma_W$ is an isomorphism. For every point $W\in Gr_n(\mu)$, the tangent space at $W$ is naturally identified with the space of continuous homomorphism of $W$ into $V/W$: $$ T_WGr_n(\mu) = \Hom_{\text{cont}}(W,V/W)\;. $$ Let us define various vector fields on the Grassmannians. Since the formal loop algebra $gl(n,L)$ acts on $V$, every element $\xi\in gl(n,L)$ defines a homomorphism \begin{equation} W\longrarrow V\overset{\xi}{\longrarrow} V\longrarrow V/W\;,\label{2.4} \end{equation} which we shall denote by ${\Psi}_W(\xi)$. Thus the association $$ Gr_n(\mu)\owns W\longmapsto {\Psi}_W(\xi)\in \Hom_{\text{cont}}(W,V/W) = T_WGr_n(\mu) $$ determines a vector field ${\Psi}(\xi)$ on the Grassmannian. For a subset $\Xi\subset gl(n,L)$, we use the notations ${\Psi}_W(\Xi) = \big\{{\Psi}_W(\xi)\;\big|\;\xi\in\Xi\big\}$ and ${\Psi}(\Xi) = \big\{{\Psi}(\xi)\;\big|\;\xi\in\Xi\big\}$. \medskip \begin{Def} \label{2.5. Definition} A smooth subvariety $X$ of $Gr_n(\mu)$ is said to be an {\it orbit} (or the \emph{integral manifold}) of the flows of ${\Psi}(\Xi)$ if the tangent space $T_WX$ of $X$ at $W$ is equal to ${\Psi}_W(\Xi)$ as a subspace of the whole tangent space $T_WGr_n(\mu)$ for every point $W\in X$. \end{Def} \medskip \begin{rem} \label{{2.6.} Remark} There is a far larger algebra than the loop algebra, the algebra $gl(n,E)$ of \emph{pseudodifferential operators} with matrix coefficients, acting on $V$. We will come back to this point in Section~\ref{sec: ODE}. \end{rem} \medskip Let us choose a monic element \begin{equation} y = z^r + \sum_{m=1}^\infty c_mz^{r+m} \in L \label{2.7} \end{equation} of order $-r$ and consider the following $n\times n$ matrix \begin{equation} h_n(y) = \begin{pmatrix} 0 & & & & 0 & y\\ 1 & 0 & & & & 0\\ & 1 &\ddots\\ & &\ddots& 0\\ & & & 1 & 0\\ & & & & 1 & 0 \end{pmatrix}\label{2.8} \end{equation} satisfying that $h_n(y)^n = y\cdot I_n$, where $I_n$ is the identity matrix of size $n$. We denote by $H_{(n)}(y)$ the algebra generated by $h_n(y)$ over $k((y))$, which is a maximal commutative subalgebra of the formal loop algebra $gl\big(n,k((y))\big)$. Obviously, we have a natural $k((y))$-algebra isomorphism $$ H_{(n)}(y) \isom k((y))[x]/(x^n - y)\isom k((y^{1/n}))\;, $$ where $x$ is an indeterminate. \medskip \begin{Def} \label{2.9. Definition} For every integral vector ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (n_1, n_2, \cdots, n_\ell)$ of positive integers $n_j$ such that $n=n_1 + n_2 + \cdots + n_\ell$ and a monic element $y\in L$ of order $-r$, we define a maximal commutative $k((y))$-subalgebra of $gl\big(n,k((y)\big)$ by $$ H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) = \bigoplus_{j=1}^\ell H_{(n_j)}(y) \isom \bigoplus_{j=1}^\ell k((y^{1/n_j}))\;, $$ where each $H_{(n_j)}(y)$ is embedded by the disjoint principal diagonal blocks: $$ \begin{pmatrix} H_{(n_1)}(y)\\ &H_{(n_2)}(y)\\ &&\ddots\\ &&&H_{(n_\ell)}(y) \end{pmatrix}\;. $$ The algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is called the \emph{maximal commutative algebra of type} ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ associated with the variable $y$. \end{Def} \medskip\noindent As a module over the field $k((y))$, the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ has dimension $n$. \medskip \begin{rem} \label{{2.10.} Remark} The lifting of the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ to the central extension of the formal loop algebra $gl\big(n,k((y))\big)$ is the Heisenberg algebra associated with the conjugacy class of the Weyl group of $gl(n,k)$ determined by the integral vector ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ (\cite{FLM}, \cite{Ka}, \cite{PS}). The word \emph{Heisenberg} in the following definition has its origin in this context. \end{rem} \medskip \begin{Def} \label{2.11. Definition} The set of commutative vector fields ${\Psi}(H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y))$ defined on $Gr_n(\mu)$ is called the \emph{Heisenberg flows} of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (n_1, n_2, \cdots, n_\ell)$ and rank $r$ associated with the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ and the coordinate $y$ of $(\ref{2.7})$. Let $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)_0$ denote the subalgebra of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ consisting of the traceless elements. The system of vector fields ${\Psi}\big(H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)_0\big)$ is called the \emph{traceless Heisenberg flows}. The set of commuting vector fields ${\Psi}\big(k((y))\big)$ on $Gr_n(\mu)$ is called the \emph{$r$-reduced} KP system (or the \emph{$r$-reduction} of the KP system) associated with the coordinate $y$. The usual KP system is defined to be the $1$-reduced KP system with the choice of $y = z$. The Heisenberg flows associated with $H_{(1,\cdots,1)}(z)$ of type $(1, \cdots, 1)$ is called the \emph{$n$-component} KP system. \end{Def} \medskip \begin{rem} \label{{2.12.} Remark} As we shall see in Section~\ref{sec: inverse}, the $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$-action on $V$ is equivalent to the component-wise multiplication of (\ref{4.1}) to (\ref{4.4}). {From} this point of view, the Heisenberg flows of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and rank $r$ are contained in the $\ell$-component KP system. What is important in our presentation as the Heisenberg flows is the new algebro-geometric interpretation of the orbits of these systems defined on the (quotient) Grassmannian which can be seen only through the right choice of the coordinates. \end{rem} \medskip \begin{rem} \label{{2.13.} Remark} The traceless Heisenberg flows of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (2)$ and rank one are known to be equivalent to the BKP system. As we shall see later in this paper, these flows produce the Prym variety associated with a double sheeted covering of algebraic curves with at least one ramification point. This explains why the BKP system is related only with these very special Prym varieties. \end{rem} \medskip \noindent The flows defined above are too large from the geometric point of view. The action of the negative order elements of $gl(n,L)$ should be considered trivial in order to give a direct connection between the orbits of these flows and the Jacobian varieties. Thus it is more convenient to define these flows on certain quotient spaces. So let \begin{equation} H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^- = H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n,k[[y]]y\big) \label{2.14} \end{equation} and define an abelian group \begin{equation} \Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) = {\text{\rm{exp}}}\big(H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^-\big) = I_n + H_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}(y)^-\;.\label{2.15} \end{equation} This group is isomorphic to an affine space, and acts on the Grassmannian without fixed points. This can be verified as follows. Suppose we have $g\cdot W= W$ for some $g =I_n + h\in \Gamma_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ and $W\in Gr_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu)$. Then $h\cdot W\subset W$. Since $h$ is a nonnilpotent element of negative order, by iterating the action of $h$ on $W$, we get a contradiction to the Fredholm condition of $\gam_W$. \medskip \begin{Def} \label{2.16. Definition} The \emph{Grassmannian quotient} of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$ and rank $r$ associated with the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is the quotient space $$ Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y) = Gr_n(\mu)\big/\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\;. $$ We denote by $Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}: Gr_n(\mu)\longrarrow Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu, y)$ the canonical projection. \end{Def} \medskip\noindent Since $\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is an affine space acting on the Grassmannian without fixed points, the affine principal fiber bundle $Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}: Gr_n(\mu)\longrarrow Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu, y)$ is trivial. If the Grassmannian is modeled on a complex Hilbert space, then one can introduce a K\"ahler structure on it, which gives rise to a canonical connection on the principal bundle $Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}$. In that case, there is a standard way of defining vector fields on the Grassmannian quotient by using the connection. In our case, however, since the Grassmannian $Gr_n(\mu)$ is modeled over $k((z))$, we cannot use these technique of infinite-dimensional complex geometry. Because of this reason, instead of defining vector fields on the Grassmannian quotient, we give directly a definition of orbits on $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu, y)$ in the following manner. \medskip \begin{Def} \label{2.17. Definition} A subvariety $\overline{X}$ of the quotient Grassmannian $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ is said to be an \emph{orbit} of the Heisenberg flows associated with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ if the pull-back $Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}^{-1}(\overline{X})$ is an orbit of the Heisenberg flows on the Grassmannian $Gr_n(\mu)$. \end{Def} \medskip\noindent Here, we note that because of the commutativity of the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ and the group $\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$, the Heisenberg flows on the Grassmannian ``descend'' to the Grassmannian quotient. Thus for the flows generated by subalgebras of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$, we can safely talk about the \emph{induced flows} on the Grassmannian quotient. \medskip \begin{Def} \label{2.18. Definition} An orbit $X$ of the vector fields $\Psi(\Xi)$ on the Grassmannian $Gr_n(\mu)$ is said to be of \emph{finite type} if $\overline{X} = Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}(X)$ is a finite-dimensional subvariety of the Grassmannian quotient $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$. \end{Def} \medskip In Section~\ref{sec: Prym}, we study algebraic geometry of finite type orbits of the Heisenberg flows and establish a characterization of Prym varieties in terms of these flows. The actual system of nonlinear partial differential equations corresponding to these vector fields are derived in Section~\ref{sec: ODE}, where the unique solvability of the initial value problem of these nonlinear equations is shown by using a theorem of \cite{M2}. \bigskip \section{The cohomology functor for covering morphisms of algebraic curves} \label{sec: coh} \medskip Krichever \cite{Kr} gave a construction of an exact solution of the entire KP system out of a set of algebro-geometric data consisting of curves and line bundles on them. This construction was formulated as a map of the set of these geometric data into the Grassmannian by Segal and Wilson \cite{SW}. Its generalization to the geometric data containing arbitrary vector bundles on curves was discovered in \cite{M3}. In order to deal with arbitrary covering morphisms of algebraic curves, we have to enlarge the framework of the cohomology functor of \cite{M3}. \medskip \begin{Def} \label{3.1. Definition} A set of \emph{geometric data} of a covering morphism of algebraic curves of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$ and rank $r$ is the collection $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle $$ of the following objects: \begin{enumerate} \item ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (n_1, n_2, \cdots, n_\ell)$ is an integral vector of positive integers $n_j$ such that $n = n_1 + n_2 + \cdots + n_\ell$. \item $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is a reduced algebraic curve defined over $k$, and $\Delta = \{p_1,p_2,\cdots,p_\ell\}$ is a set of $\ell$ smooth rational points of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. \item $\Pi = (\pi_1, \cdots, \pi_\ell)$ consists of a cyclic covering morphism $\pi_j:U_{oj}\longrarrow U_j$ of degree $r$ which maps the formal completion $U_{oj}$ of the affine line $\mathbb{A}^1_k$ along the origin onto the formal completion $U_j$ of the curve ${C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}$ along $p_j$. \item $\mathcal{F}$ is a torsion-free sheaf of rank $r$ defined over $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ satisfying that $$ \mu = \dim_k H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) - \dim_k H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F})\;. $$ \item $\Phi = (\phi_1,\cdots, \phi_\ell)$ consists of an $\mathcal{O}_{U_j}$-module isomorphism $$ \phi_j:\mathcal{F}_{U_j}\overset{\sim}{\longrarrow} \pi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big)\;, $$ where $\mathcal{F}_{U_j}$ is the formal completion of $\mathcal{F}$ along $p_j$. We identify $\phi_j$ and $c_j\cdot \phi_j$ for every nonzero constant $c_j\in k^*$. \item $C_0$ is an integral curve with a marked smooth rational point $p$. \item $f:C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow {C_0}$ is a finite morphism of degree $n$ of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ onto ${C_0}$ such that $f^{-1}(p) = \{p_1,\cdots,p_\ell\}$ with ramification index $n_j$ at each point $p_j$. \item $\pi: U_o\longrarrow U_p$ is a cyclic covering morphism of degree $r$ which maps the formal completion $U_{o}$ of the affine line $\mathbb{A}^1_k$ at the origin onto the formal completion $U_p$ of the curve ${C_0}$ along $p$. \item $\pi_j:U_{oj}\longrarrow U_j$ and the formal completion $f_j:U_j\longrarrow U_p$ of the morphism $f$ at $p_j$ satisfy the commutativity of the diagram $$ \begin{CD} U_{oj} @>{\pi_j}>> U_j\\ @V{\psi_j}VV @VV{f_j}V\\ U_o @>>{\pi}> U_p, \end{CD} $$ where $\psi_j:U_{oj}\longrarrow U_o$ is a cyclic covering of degree $n_j$. \item $\phi:(f_*\mathcal{F})_{U_p}\overset{\sim}{\longrarrow} \pi_*\bigg(\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1) \big)\bigg)$ is an $\big(f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}\big)_{U_p}$-module isomorphism of the sheaves on the formal scheme $U_p$ which is compatible with the datum $\Phi$ upstairs. \end{enumerate} \end{Def} \medskip\noindent Here we note that we have an isomorphism $\psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1) \big) \isom \mathcal{O}_{U_o}(-1)^{\dsum n_j}$ as an $\mathcal{O}_{U_o}$-module. Recall that the original cohomology functor is really a cohomology functor. In order to see what kind of algebraic data come up from our geometric data, let us apply the cohomology functor to them. We choose a coordinate $z$ on the formal scheme $U_o$ and fix it once for all. Then we have $U_o = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[z]]\big)$. Since $\psi_j:U_{oj}\longrarrow U_o$ is a cyclic covering of degree $n_j$, we can identify $U_{oj} = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[z^{1/n_j}]]\big)$ so that $\psi_j$ is given by $z = \big(z^{1/n_j}\big)^{n_j} = z_j^{n_j}$, where $z_j = z^{1/n_j}$ is a coordinate of $U_{oj}$. The morphism $\pi$ determines a coordinate $$ y = z^r + \sum_{m = 1}^\infty c_m z^{r+m} $$ on $U_p$. We also choose a coordinate $y_j = y^{1/n_j}$ of $U_j$ in which the morphism $f_j$ can be written as $y = \big(y^{1/n_j}\big)^{n_j} = y_j^{n_j}$. Out of the geometric data, we can assign a vector subspace $W$ of $V$ by \begin{equation} \begin{aligned} W &= \phi\big(H^0({C_0}\setminus\{p\},f_*\mathcal{F})\big)\\ &\subset H^0\bigg(U_p\setminus\{p\}, \pi_*\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big) \bigg)\\ &= H^0\bigg(U_o\setminus\{o\}, \bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big)\bigg)\\ &\isom H^0\big(U_o\setminus\{o\}, \bigoplus_{j=1}^\ell \mathcal{O}_{U_o}(-1)^{\dsum n_j}\big)\\ &\isom H^0\big(U_o\setminus \{o\}, \mathcal{O}_{U_o}(-1)^{\dsum n}\big) = k((z))^{\dsum n} = V\;. \end{aligned}\label{3.2} \end{equation} Here, we have used the convention of \cite{M3} that $$ \begin{aligned} H^0(C_0\setminus \{p\},\mathcal{O}_{C_0}) &= \varinjlim_m H^0\big(C_0, \mathcal{O}_{C_0}(m\cdot p)\big)\\ H^0(U_o\setminus \{o\},\mathcal{O}_{U_o}) &= \varinjlim_m H^0\big(U_o, \mathcal{O}_{U_o}(m)\big) = k((z))\;, \end{aligned} $$ etc. The coordinate ring of the curve ${C_0}$ determines a scalar diagonal stabilizer algebra \begin{equation} \begin{aligned} A_0 &= \pi^*\big(H^0({C_0}\setminus\{p\},\mathcal{O}_{{C_0}})\big)\\ &\subset \pi^*\big(H^0({U_p}\setminus\{p\},\mathcal{O}_{U_p})\big)\\ &\subset H^0\big(U_o\setminus\{o\},\mathcal{O}_{U_o}\big)\\ &= L \subset gl(n,L) \end{aligned}\label{3.3} \end{equation} satisfying that $A_0\cdot W\subset W$, where $L$ is identified with the set of scalar matrices in $gl(n, L)$. The rank of $W$ over $A_0$ is $r\cdot n$, which is equal to the rank of $f_*\mathcal{F}$. Note that we have also an inclusion $$ A_0 \isom H^0({C_0}\setminus \{p\},\mathcal{O}_{{C_0}})\subset H^0(U_p\setminus \{p\},\mathcal{O}_{U_p}) = k((y)) $$ by the coordinate $y$. As in Section~2 and 3 of \cite{M3}, we can use the formal patching $C_0 = (C_0\setminus \{p\})\cup U_p$ to compute the cohomology group \begin{equation} \begin{aligned} H^1(C_0, \mathcal{O}_{C_0}) &\isom \frac{H^0(U_p\setminus \{p\}, \mathcal{O}_{U_p})}{H^0(C_0\setminus \{p\}, \mathcal{O}_{C_0}) + H^0(U_p,\mathcal{O}_{U_p})}\\ &\isom \frac{k((y))}{A_0 + k[[y]]}\;. \end{aligned} \label{3.4} \end{equation} Thus the cokernel of the projection $\gam_{A_0}:A_0\longrarrow k((y))/k[[y]]$ has finite dimension. The function ring $$ A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} =H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta, \mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) \subset \bigoplus_{j=1}^\ell H^0(U_j\setminus \{p_j\}, \mathcal{O}_{U_j}) $$ also acts on $V$ and satisfies that $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\cdot W\subset W$, because we have a natural injective isomorphism \begin{equation} \begin{aligned} A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta, \mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) &\isom H^0\big(C_0\setminus \{p\}, f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}\big)\\ &\subset H^0\big(U_p\setminus\{p\}, (f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})_{U_p}\big)\\ &= H^0\big(U_p\setminus\{p\}, \bigoplus_{j=1}^\ell f_{j*}\mathcal{O}_{U_j}\big)\\ &=\bigoplus_{j=1}^\ell k((y))\big[ h_{n_j}(y)\big]\\ &= H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\subset gl\big(n,k((y))\big)\;, \end{aligned} \label{3.5} \end{equation} where $h_{n_j}(y)$ is the block matrix of (\ref{2.8}) and $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is the maximal commutative subalgebra of $gl\big(n,k((y))\big)$ of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. In order to see the action of $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ on $W$ more explicitly, we first note that the above isomorphism is given by the identification $y^{1/n_j} = h_{n_j}(y)$. Since the formal completion $\mathcal{F}_{U_j}$ of the vector bundle $\mathcal{F}$ at the point $p_j$ is a free $\mathcal{O}_{U_j}$-module of rank $r$, let us take a basis $\{e_1, e_2, \cdots, e_r\}$ for the free $H^0(U_j,\mathcal{O}_{U_j})$-module $H^0(U_j,\mathcal{F}_{U_j})$. The direct image sheaf $f_{j*}\mathcal{F}_{U_j}$ is a free $\mathcal{O}_{U_p}$-module of rank $n_j\cdot r$, so we can take a basis of sections \begin{equation} \left\{ y^{\alpha/n_j}\tensor e_\beta\right\}_{0\le \alpha< n_j, 1\le\beta\le r} \label{3.6} \end{equation} for the free $H^0(U_p,\mathcal{O}_{U_p})$-module $H^0(U_p,f_{j*}\mathcal{F}_{U_j})$. Since $H^0(U_j,\mathcal{F}_{U_j})$ $=$ $H^0(U_p,f_{j*}\mathcal{F}_{U_j})$, $H^0(U_j,\mathcal{O}_{U_j})$ $=$ $H^0(U_p,f_{j*}\mathcal{O}_{U_j})$ acts on the basis (\ref{3.6}) by the matrix $h_{n_j}(y)\tensor I_r$, where $I_r$ is the identity matrix acting on $\{e_1, e_2, \cdots, e_r\}$. This can be understood by observing that the action of $y^{1/n}$ on the vector $$ (c_0, c_1, \cdots, c_{n-1}) = \sum_{\alpha=0}^{n-1} c_\alpha y^{\alpha/n} $$ is given by the action of the block matrix $h_n(y)$. \medskip \begin{rem} \label{{3.7.} Remark} {From} the above argument, it is clear that the role which our $\pi$ and $\phi$ play is exactly the same as that of the \emph{parabolic structure} of \cite{Mum2}. The advantage of using $\pi$ and $\phi$ rather than the parabolic structure lies in their functoriality. Indeed, the parabolic structure does not transform functorially under morphisms of curves, while our data naturally do (see Definition~\ref{3.14. Definition}). \end{rem} \medskip The algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ has two different presentations in terms of geometry. We have used $$ H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) \isom H^0\big(U_p\setminus\{p\}, (f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})_{U_p}\big) = \bigoplus_{j=1}^\ell k((y))\big[h_{n_j}(y)\big] \subset gl\big(n,k((y))\big) $$ in (\ref{3.5}). In this presentation, an element of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is an $n\times n$ matrix acting on $V\isom H^0\big(U_p\setminus\{p\}, (f_*\mathcal{F})_{U_p}\big)$. The other geometric interpretation is $$ H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\isom H^0\big(U_p\setminus\{p\}, \bigoplus_{j=1}^\ell f_{j*}\mathcal{O}_{U_j}\big)\isom \bigoplus_{j=1}^\ell H^0\big(U_j\setminus\{p_j\},\mathcal{O}_{U_j}\big) =\bigoplus_{j=1}^\ell k((y_j))\;. $$ In this presentation, the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ acts on $$ \begin{aligned} V &\isom H^0\bigg(U_p\setminus\{p\}, \pi_*\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big) \bigg)\\ &\isom \bigoplus_{j=1}^\ell H^0\big(U_{oj}\setminus \{o\}, \mathcal{O}_{U_{oj}}(-1)\big)\\ &= \bigoplus_{j=1}^\ell k((z_j)) \end{aligned} $$ by the component-wise multiplication of $y_j$ to $z_j$. We will come back to this point in (\ref{4.4}). The pull-back through the morphism $f$ gives an embedding $A_0\subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. As an $A_0$-module, $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is torsion-free of rank $n$, because $C_0$ is integral and the morphism $f$ is of degree $n$. Using the formal patching $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta) \cup U_1\cup\cdots\cup U_\ell$, we can compute the cohomology \begin{equation} \begin{aligned} H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) &\isom \frac{\bigoplus_{j=1}^\ell H^0(U_j\setminus \{p_j\}, \mathcal{O}_{U_j})}{H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta,\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) + \bigoplus_{j=1}^\ell H^0(U_j,\mathcal{O}_{U_j})}\\ &\isom \frac{\bigoplus_{j=1}^\ell k((y^{1/n_j}))}{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} + \bigoplus_{j=1}^\ell k[[y^{1/n_j}]]}\\ &\isom \frac{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)}{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} + H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n, k[[y]]\big)}\;. \end{aligned} \label{3.8} \end{equation} This shows that the projection $$ \gam_{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}: A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow \frac{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)}{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n, k[[y]]\big)} $$ has a finite-dimensional cokernel. These discussions motivate the following definition: \medskip \begin{Def} \label{3.9. Definition} A triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ is said to be a set of \emph{algebraic data} of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$, and rank $r$ if the following conditions are satisfied: \begin{enumerate} \item $W$ is a point of the Grassmannian $Gr_n(\mu)$ of index $\mu$ of the vector valued functions of size $n$. \item The type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is an integral vector $(n_1,\cdots, n_\ell)$ consisting of positive integers such that $n = n_1 + \cdots + n_\ell$. \item There is a monic element $y\in L = k((z))$ of order $-r$ such that $A_0$ is a subalgebra of $k((y))$ containing the field $k$. \item The cokernel of the projection $\gam_{A_0}:A_0\longrarrow k((y))/k[[y]]$ has finite dimension. \item $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is a subalgebra of the maximal commutative algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) \subset gl\big(n,k((y))\big)$ of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ such that the projection $$ \gam_{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}:A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow \frac{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)}{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n,k[[y]]\big)} $$ has a finite-dimensional cokernel. \item There is an embedding $A_0\subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ as the scalar diagonal matrices, and as an $A_0$-module (which is automatically torsion-free), $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ has rank $n$ over $A_0$. \item The algebra $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\subset gl\big(n,k((y))\big)$ stabilizes $W\subset V$, i.e.\ $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\cdot W\subset W$. \end{enumerate} \end{Def} \medskip\noindent The homomorphisms $\gam_{A_0}$ and $\gam_{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}$ satisfy the Fredholm condition because (7) implies that they have finite-dimensional kernels. Now we can state \medskip \begin{prop} \label{3.10. Proposition} For every set of geometric data of Definition~\ref{3.1. Definition}, there is a unique set of algebraic data of Definition~\ref{3.9. Definition} having the same type, index and rank. \end{prop} \medskip \begin{proof} We have already constructed the triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ out of the geometric data in (\ref{3.2}), (\ref{3.3}) and (\ref{3.5}) which satisfies all the conditions in Definition~\ref{3.9. Definition} but (1). The only remaining thing we have to show is that the vector subspace $W$ of (\ref{3.2}) is indeed a point of the Grassmannian $Gr_n(\mu)$. To this end, we need to compute the cohomology of $f_*\mathcal{F}$ by using the formal patching $C_0 = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_0)\cup U_p$ (for more detail, see \cite{M3}). Noting the identification $$ \bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big) \isom \mathcal{O}_{U_o}(-1)^{\dsum n} $$ as in (\ref{3.2}), we can show that \begin{equation} \begin{aligned} H^0(C_0,f_*\mathcal{F}) &= H^0(C_0\setminus \{p\},f_*\mathcal{F}) \cap H^0(U_p,f_*\mathcal{F}_{U_p})\\ &\isom W\cap H^0\big(U_p,\pi_*(\mathcal{O}_{U_o}(-1)^{\dsum n})\big)\\ &\isom W\cap H^0\big(U_o,\mathcal{O}_{U_o}(-1)^{\dsum n}\big)\\ &\isom W\cap \big(k[[z]]z\big)^{\dsum n}\\ &= {\text{\rm{Ker}}}} \def\coker{{\text{\rm{coker}}}(\gam_W)\;, \end{aligned} \label{3.11} \end{equation} and \begin{equation} \begin{aligned} H^1(C_0,f_*\mathcal{F}) &\isom \frac{H^0(U_p\setminus \{p\}, f_*\mathcal{F})}{H^0(C_0\setminus \{p\},f_*\mathcal{F}) + H^0(U_p,f_*\mathcal{F}_{U_p})}\\ &\isom \frac{H^0\big(U_p\setminus \{p\}, \pi_*(\mathcal{O}_{U_o}(-1)^{\dsum n})\big)}{W + H^0\big(U_p,\pi_*(\mathcal{O}_{U_o}(-1)^{\dsum n})\big)}\\ &\isom \frac{H^0\big(U_o\setminus \{o\}, \mathcal{O}_{U_o}(-1)^{\dsum n}\big)}{W + H^0\big(U_o,\mathcal{O}_{U_o}(-1)^{\dsum n}\big)}\\ &\isom \frac{k((z))^{\dsum n}}{W + \big(k[[z]]z\big)^{\dsum n}}\\ &=\Coker(\gam_W)\;, \end{aligned} \label{3.12} \end{equation} where $\gam_W$ is the canonical projection of Definition~\ref{2.2. Definition}. Since $f$ is a finite morphism, we have $H^i(C_0,f_*\mathcal{F}) \isom H^i(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \mathcal{F})$. Thus \begin{equation} \mu = \dim_k H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) - \dim_k H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \mathcal{F}) = \dim_k {\text{\rm{Ker}}}} \def\coker{{\text{\rm{coker}}}(\gam_W) - \dim_k \Coker(\gam_W)\;,\label{3.13} \end{equation} which shows that $W$ is indeed a point of $Gr_n(\mu)$. This completes the proof. \end{proof} \medskip \noindent This proposition gives a generalization of the Segal-Wilson map to the case of covering morphisms of algebraic curves. We can make the above map further into a functor, which we shall call the \emph{cohomology functor for covering morphisms}. The categories we use are the following: \medskip \begin{Def} \label{3.14. Definition} The category $\mathcal{C}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of geometric data of a fixed type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ consists of the set of geometric data of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and arbitrary index $\mu$ and rank $r$ as its object. A morphism between two objects $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle $$ of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$ and rank $r$ and $$ \left\langle {f'}:\big( C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta', \Pi', \mathcal{F}', \Phi'\big) \longrarrow \big({C'_0}, p', \pi', {f'}_*\mathcal{F}', \phi'\big)\right\rangle $$ of the same type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu'$ and rank $r'$ is a triple $(\alpha,\beta, \lambda)$ of morphisms satisfying the following conditions: \begin{enumerate} \item $\alpha:C'_0\longrarrow C_0$ is a locally cyclic covering of degree $s$ of the base curves such that $\alpha^{*}(p) = s\cdot p'$, and $\pi$ and $\pi'$ are related by $\pi = \widehat\alpha\circ\pi'$ with the morphism $\widehat\alpha$ of formal schemes induced by $\alpha$. \item $\beta:C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \longrarrow C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is a covering morphism of degree $s$ such that $\Delta' = \beta^{-1}(\Delta)$, and the following diagram $$ \begin{CD} C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} @>{\beta}>> C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\\ @V{{f'}}VV @VV{f}V\\ C'_0 @>>{\alp}> C_0 \end{CD} $$ commutes. \item The morphism $\widehat\beta_j:U'_j\longrarrow U_j$ of formal schemes induced by $\beta$ at each $p'_j$ satisfies $\pi_j = \widehat\beta_j\circ \pi'_j$ and the commutativity of $$ \begin{CD} U_{oj} @>{\pi'_j}>> U'_j @>{\widehat\beta_j}>> U_j\\ @V{\psi_j}VV @V{{f'}_j}VV @V{f_j}VV\\ U_o @>>{\pi'}> U'_{p'} @>>{\widehat\alpha}> U_p. \end{CD} $$ \item $\lam: \beta_*\mathcal{F}'\longrarrow \mathcal{F}$ is an injective $\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}$-module homomorphism such that its completion $\lam_j$ at each point $p_j$ satisfies commutativity of $$ \begin{CD} (\beta_*\mathcal{F}')_{U_j} @>{\lam_j}>> \mathcal{F}_{U_j}\\ @V{\widehat\beta_j(\phi'_j)}V{\wr}V @V{\wr}V{\phi_j}V\\ \widehat\beta_{j*}\pi'_{j*}\mathcal{O}_{U_{oj}}(-1) @= \pi_{j*}\mathcal{O}_{U_{oj}}(-1). \end{CD} $$ In particular, each $\lam_j$ is an isomorphism. \end{enumerate} \end{Def} \medskip \begin{rem} \label{{3.15.} Remark} {From} (3) above, we have $r = s\cdot r'$. The condition (4) above implies that $\mathcal{F}\big/\beta_*\mathcal{F}'$ is a torsion sheaf on $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ whose support does not intersect with $\Delta$. \end{rem} \medskip\noindent One can show by using Proposition~\ref{1.6. Proposition} that there are many nontrivial morphisms among the sets of geometric data with different ranks. \medskip \begin{Def} \label{3.16. Definition} The category $\mathcal{S}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of algebraic data of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ has the stabilizer triples $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ of Definition~\ref{3.9. Definition} of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and arbitrary index $\mu$ and rank $r$ as its objects. Note that for every object $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$, we have the commutative algebras $k((y))$ and $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ associated with it. A morphism between two objects $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ and $(A'_0,A'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},{W'})$ is a triple $(\iota, \epsilon, \omega)$ of injective homomorphisms satisfying the following conditions: \begin{enumerate} \item $\iota:A_0\hookrarrow A'_0$ is an inclusion compatible with the inclusion $k((y))\subset k(({y'}))$ defined by a power series $$ y = y({y'}) = {y'}^s + a_1 {y'}^{s+1} + a_2 {y'}^{s+2} + \cdots\;. $$ \item $\epsilon:A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow A'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is an injective homomorphism satisfying the commutativity of the diagram $$ \begin{CD} A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} @>{\epsilon}>> A'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\\ @VVV @VVV\\ H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) @>>{\mathcal{E}}> H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}({y'}), \end{CD} $$ where the vertical arrows are the inclusion maps, and $$ \mathcal{E}: H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) \isom \bigoplus_{j=1}^\ell k((y^{1/n_j})) \longrarrow \bigoplus_{j=1}^\ell k(({y'}^{1/n_j})) \isom H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}({y'}) $$ is an injective homomorphism defined by the Puiseux expansion $$ y^{1/n_j} = y({y'})^{1/n_j} = {y'}^{s/n_j} + b_1 {y'}^{(s+1)/n_j} + b_2 {y'}^{(s +2)/n_j} + \cdots $$ of $(1)$ for every $n_j$. Note that neither $\epsilon$ nor $\mathcal{E}$ is an inclusion map of subalgebras of $gl(n,L)$. \item $\omega: {W'}\longrarrow W$ is an injective $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$-module homomorphism. We note that ${W'}$ has a natural $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$-module structure by the homomorphism $\epsilon$. As in $(2)$, $\omega$ is not an inclusion map of the vector subspaces of $V$. \end{enumerate} \end{Def} \medskip \begin{thm} \label{3.17. Theorem} There is a fully-faithful functor $$ \kappa_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}:\mathcal{C}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})\overset{\sim}{\longrarrow}\mathcal{S}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}) $$ between the category of geometric data and the category of algebraic data. An object of $\mathcal{C}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of index $\mu$ and rank $r$ corresponds to an object of $\mathcal{S}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of the same index and rank. \end{thm} \medskip \begin{proof} The association of $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ to the geometric data has been done in (\ref{3.2}), (\ref{3.3}), (\ref{3.5}) and Proposition~\ref{3.10. Proposition}. Let $(\alp,\beta,\lam)$ be a morphism between two sets of geometric data as in Definition~\ref{3.14. Definition}. We use the notations $U_j^* = U_j\setminus \{p_j\}$ and $U_p^* = U_p\setminus \{p\}$. The homomorphism $\iota$ is defined by the commutative diagram $$ \begin{CD} A_0 @>{\sim}>> H^0(C_0\setminus \{p\},\mathcal{O}_{C_0}) @>>> H^0(U_p^*,\mathcal{O}_{U_p})\\ @V{\iota}VV @V{\alp^*}VV @V{\widehat{\alp}^*}VV\\ A'_0 @>{\sim}>> H^0(C'_0\setminus \{p'\},\mathcal{O}_{C'_0}) @>>> H^0({U'}_{p'}^*, \mathcal{O}_{{U'}_{p'}}). \end{CD} $$ Similarly, $$ \begin{CD} A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} @>{\sim}>> H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta, \mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) @>>> \bigoplus_{j=1}^\ell H^0(U_j^*,\mathcal{O}_{U_j})\\ @V{\epsilon}VV @V{\beta^*}VV @V{\oplus\widehat{\beta}_j^*}VV\\ A'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} @>{\sim}>> H^0(C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta',\mathcal{O}_{C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) @>>> \bigoplus_{j=1}^\ell H^0({U'}_{j}^*, \mathcal{O}_{{U'}_{j}}) \end{CD} $$ defines the homomorphism $\epsilon$. Finally, $$ \begin{CD} {W'} @= {W'} @>{\omega}>> W\\ @VV{\wr}V @VV{\wr}V @VV{\wr}V\\ H^0(C'_0\setminus \{p'\}, {{f'}}_*\mathcal{F}') @>{\alp_*}>{\sim}> H^0(C_0\setminus \{p\}, f_*\beta_*\mathcal{F}') @>{f_*(\lam)}>> H^0(C_0\setminus \{p\}, f_*\mathcal{F})\\ @V{{f'}^*}V{\wr}V @V{f^*}V{\wr}V @V{f^*}V{\wr}V\\ H^0(C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta', \mathcal{F}') @>{\beta_*}>{\sim}> H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta, \beta_*\mathcal{F}') @>{\lam}>> H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus \Delta, \mathcal{F})\\ @VVV @VVV @VVV\\ \bigoplus_j H^0({U'}_j^*, \mathcal{F}'_{U'_j}) @>{\dsum \widehat{\beta}_{j*}}>{\sim}> \bigoplus_j H^0(U_j^*, \widehat{\beta}_{j*}\mathcal{F}'_{U'_j}) @>{\dsum\lam_j}>> \bigoplus_j H^0(U_j^*, \mathcal{F}_{U_j}) \end{CD} $$ determines the homomorphism $\omega$. In order to establish that the two categories are equivalent, we need the inverse construction. The next section is entirely devoted to the proof of this claim. \end{proof} \medskip The following proposition and its corollary about the geometric data of rank one are crucial when we study geometry of orbits of the Heisenberg flows in Section~\ref{sec: Prym}. \medskip \begin{prop} \label{3.18. Proposition} Suppose we have two sets of geometric data of rank one having exactly the same constituents except for the sheaf isomorphisms $(\Phi ,\phi)$ for one and $(\Phi' ,\phi')$ for the other. Let $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ and $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},{W'})$ be the corresponding algebraic data, where $A_0$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ are common in both of the triples because of the assumption. Then there is an element $g\in \Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ of $(\ref{2.15})$ such that ${W'} = g\cdot W$. \end{prop} \medskip \begin{proof} Recall that $$ \phi:(f_*\mathcal{F})_{U_p}\overset{\sim}{\longrarrow} \pi_*\bigg(\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1) \big)\bigg) $$ is an $\big(f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}\big)_{U_p}$-module isomorphism. Thus, $$ g = \phi'\circ \phi^{-1}: \pi_*\bigg(\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big)\bigg) \overset{\sim}{\longrarrow} \pi_*\bigg(\bigoplus_{j=1}^\ell \psi_{j*} \big(\mathcal{O}_{U_{oj}}(-1)\big)\bigg) $$ is also an $\big(f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}\big)_{U_p}$-module isomorphism. Note that we have identified $\big(f_*\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}\big)_{U_p}$ as a subalgebra of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ in (\ref{3.5}). Indeed, this subalgebra is $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n,k[[y]]\big)$. Therefore, the invertible $n\times n$ matrix $$ g \in k^{*\dsum n} + gl\big(n,k[[y]]y\big) = k^{*\dsum n} + gl\big(n,k[[z]]z\big) $$ commutes with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n,k[[y]]\big)$, where $k^*$ denotes the set of nonzero constants and $k^{*\dsum n}$ the set of invertible constant diagonal matrices. We recall that $k[[z]] = k[[y]]$, because $y$ has order $-1$. The commutativity of $g$ and $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n,k[[y]]\big)$ immediately implies that $g$ commutes with all of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$. But since $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is a \emph{maximal} commutative subalgebra of $gl\big(n,k((y))\big)$, it implies that $g\in \Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$. Here we note that $\phi'_j\circ {\phi_j}^{-1}$ is exactly the $j$-th block of size $n_j\times n_j$ of the $n\times n$ matrix $g$, and that we can normalize the leading term of $\phi'_j\circ {\phi_j}^{-1}$ to be equal to $I_{n_j}$ by the definition (5) of Definition~\ref{3.1. Definition}. Thus the leading term of $g$ can be normalized to $I_n$. {From} the construction of (\ref{3.2}), we have ${W'} = g\cdot W$. This completes the proof. \end{proof} \medskip \begin{cor} \label{3.19. Corollary} The cohomology functor induces a bijective correspondence between the collection of geometric data $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}\big)\right\rangle $$ of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$, and rank one, and the triple of algebraic data $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\overline{W})$ of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$, and rank one satisfying the same conditions of Definition~\ref{3.9. Definition} except that $\overline{W}$ is a point of the Grassmannian quotient $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$. \end{cor} \medskip \begin{proof} Note that the datum $\Phi$ is indeed the block decomposition of the datum of $\phi$. Thus taking the quotient space of the Grassmannian by the group action of $\Gam_n(y)$ exactly corresponds to eliminating the data $\Phi$ and $\phi$ from the set of geometric data of Definition~\ref{3.1. Definition}. \end{proof} \bigskip \section{The inverse construction} \label{sec: inverse} \medskip Let $W\in Gr_n(\mu)$ be a point of the Grassmannian and consider a commutative subalgebra $A$ of $gl(n,L)$ such that $A\cdot W\subset W$. Since the set of vector fields ${\Psi}(A)$ has $W$ as a fixed point, we call such an algebra a commutative \emph{stabilizer} algebra of $W$. In the previous work \cite{M3}, the algebro-geometric structures of arbitrary commutative stabilizers were determined for the case of the Grassmannian $Gr_1(\mu)$ of scalar valued functions. In the context of the current paper, the Grassmannian is enlarged, and consequently there are far larger varieties of commutative stabilizers. However, it is not the purpose of this paper to give the complete geometric classification of arbitrary stabilizers. We restrict ourselves to studying \emph{large} stabilizers in connection with Prym varieties, which will be the central theme of the next section. A stabilizer is said to be large if it corresponds to a finite-dimensional orbit of the Heisenberg flows on the Grassmannian quotient. The goal of this section is to recover the geometric data out of a point of the Grassmannian together with a large stabilizer. Choose an integral vector ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (n_1, n_2, \cdots, n_\ell)$ with $n = n_1 + \cdots + n_\ell$ and a monic element $y$ of order $-r$ as in (\ref{2.7}), and consider the formal loop algebra $gl\big(n,k((y))\big)$ acting on the vector space $V = L^{\dsum n}$. Let us denote $y_j = h_{n_j}(y) = y^{1/n_j}$. We introduce a filtration $$ \cdots \subset H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^{(rm-r)}\subset H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^{(rm)}\subset H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^{(rm+r)}\subset \cdots $$ in the maximal commutative algebra \begin{equation} H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) \isom \bigoplus_{j=1}^\ell k((y))\big[y^{1/n_j}\big] \isom \bigoplus_{j=1}^\ell k((y^{1/n_j})) = \bigoplus_{j=1}^\ell k((y_j))\label{4.1} \end{equation} by defining \begin{equation} H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^{(rm)} = \bigg\{\big(a_1(y_1), \cdots, a_{\ell}(y_{\ell})\big)\; \bigg| \;\max\big[\ord_{y_1}(a_1),\cdots ,\ord_{y_\ell}(a_\ell)\big] \le m\bigg\}\;,\label{4.2} \end{equation} where $\ord_{y_j}(a_j)$ is the order of $a_j(y_j)\in k((y_j))$ with respect to the variable $y_j$. Accordingly, we can introduce a filtration in $V$ which is compatible with the action of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ on $V$. In order to define the new filtration in $V$ geometrically, let us start with $U_o = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[z]]\big)$ and $U_p = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[y]]\big)$. The inclusion $k[[y]]\subset k[[z]]$ given by $y = y(z) = z^r$ $+$ $c_1z^{r+1}$ $+$ $c_2z^{r+2}$ $+ \cdots$ defines a morphism $\pi:U_o\longrarrow U_p$. Let $U_j = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[y_j]]\big)$. The identification $y_j = y^{1/n_j}$ gives a cyclic covering $f_j:U_j\longrarrow U_p$ of degree $n_j$. Correspondingly, the covering $\psi_j:U_{oj}\longrarrow U_o$ of degree $n_j$ of (9) of Definition~\ref{3.1. Definition} is given by $k[[z]]\subset k[[z^{1/n_j}]]$. Thus we have a commutative diagram $$ \begin{CD} k[[z^{1/n_j}]] @<{\pi^*_j}<< k[[y^{1/n_j}]]\\ @A{\psi^*_j}AA @AA{f^*_j}A\\ k[[z]] @<<{\pi^*}< k[[y]] \end{CD} $$ of inclusions, where $\pi^*_j$ is defined by the Puiseux expansion \begin{equation} y_j = y^{1/n_j} = y(z)^{1/n_j} = z^{r/n_j} + a_1z^{(r+1)/n_j} + a_2z^{(r+2)/n_j} + \cdots\label{4.3} \end{equation} of $y(z)$. Recall that in order to distinguish from $U_o = {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[z]]\big)$, we have introduced the notation $U_{oj}$ $=$ ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}\big(k[[z^{1/n_j}]]\big)$ for the cyclic covering of $U_o$. The above diagram corresponds to the geometric diagram of covering morphisms $$ \begin{CD} U_{oj} @>{\pi_j}>> U_j\\ @V{\psi_j}VV @VV{f_j}V\\ U_o @>>{\pi}> U_p. \end{CD} $$ We denote $U^*_o = U_o\setminus \{o\}$, $U_{oj}^* = U_{oj}\setminus \{o\}$, $U_p^*= U_p\setminus \{p\}$, and $U_j^* = U_j\setminus \{p_j\}$ as before. The $k((y))$-algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is identified with the $H^0(U_p^*,\mathcal{O}_{U_p})$-algebra $$ H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) = H^0\big(U_p^*, \bigoplus_{j=1}^\ell f_{j*}\mathcal{O}_{U_j}\big) \isom \bigoplus_{j=1}^\ell H^0(U_j^*,\mathcal{O}_{U_j})\;. $$ Corresponding to this identification, the vector space $V = L^{\dsum n}$ as a module over $L = H^0(U_o^*,\mathcal{O}_{U_o})$ is identified with \begin{equation} V = H^0\bigg(U_o^*,\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big)\bigg) \isom \bigoplus_{j=1}^\ell H^0\big(U_{oj}^*,\mathcal{O}_{U_{oj}}(-1)\big) \isom \bigoplus_{j=1}^\ell k((z^{1/n_j}))\;. \label{4.4} \end{equation} The $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$-module structure of $V$ is given by the pull-back $\bigoplus_{j=1}^\ell \pi_j^*$, which is nothing but the component-wise multiplication of $k((y^{1/n_j}))$ to $k((z^{1/n_j}))$ through (\ref{4.3}) for each $j$. Define a new variable by $z_j = z^{1/n_j}$. We note from (\ref{4.3}) that $y_j= y_j(z^{1/n_j}) = y_j(z_j)$ is of order $-r$ with respect to $z_j$. Now we can introduce a new filtration $$ \cdots \subset V^{(m-1)}\subset V^{(m)}\subset V^{(m+1)}\subset \cdots $$ in $V$ by defining \begin{equation} V^{(m)} = \bigg\{\big(v_1(z_1), \cdots, v_{\ell}(z_{\ell})\big) \in \bigoplus_{j=1}^\ell k((z_j))\; \bigg| \; \max\big[\ord_{z_1}(v_1),\cdots ,\ord_{z_\ell}(v_\ell)\big] \le m\bigg\}\;,\label{4.5} \end{equation} where $\ord_{z_j}(v_j)$ denotes the order of $v_j = v_j(z_j)$ with respect to $z_j$. \medskip \begin{rem} \label{{4.6.} Remark} The filtration (\ref{4.5}) is different from (\ref{2.1}) in general. However, we always have $V^{(0)} = F^{(0)}(V)$ and $V^{(-1)} = F^{(-1)}(V)$. This is one of the reasons why we have chosen $F^{(-1)}(V)$ instead of an arbitrary $F^{(\nu)}(V)$ in the definition of the Grassmannian in Definition~\ref{2.2. Definition}. \end{rem} \medskip It is clear from (\ref{4.2}) and (\ref{4.5}) that $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^{(rm_1)}\cdot V^{(m_2)} \subset V^{(rm_1+m_2)}$, and hence $V$ is a filtered $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$-module. With these preparation, we can state the inverse construction theorem. \medskip \begin{thm} \label{4.7. Theorem} A triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ of algebraic data of Definition~\ref{3.9. Definition} determines a unique set of geometric data $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle\;. $$ \end{thm} \medskip \begin{proof} The proof is divided into four parts. \smallskip \noindent (I) {\sl Construction of the curve $C_0$ and the point $p$:} Let us define $A_0^{(rm)} = A_0\cap k[[y]]y^{-m}$, which consists of elements of $A_0$ of order at most $m$ with respect to the variable $y$. This gives a filtration of $A_{0}$: $$ \cdots \subset A_{0}^{(rm-r)}\subset A_{0}^{(rm)}\subset A_{0}^{(rm+r)} \subset \cdots\;. $$ Using the finite-dimensionality of the cokernel (4) of Definition~\ref{3.9. Definition}, we can show that $A_0$ has an element of order $m$ (with respect to $y$) for every large integer $m\in \mathbb{N}$, i.e.\ \begin{equation} \dim_k A_0^{(rm)}\big/A_0^{(rm-r)} = 1 \quad{\text{ for all }} \quad m>>0\;. \label{4.8} \end{equation} Since $A_0\cdot W\subset W$, the Fredholm condition of $W$ implies that $A_0^{(rm)} = 0$ for all $m<0$. Note that $A_0$ is a subalgebra of a field, and hence it is an integral domain. Therefore, the complete algebraic curve ${C_0} = \Proj(gr A_0)$ defined by the graded algebra $$ gr A_0=\bigoplus_{m=0}^\infty A_0^{(rm)} $$ is integral. We claim that $C_0$ is a one-point completion of the affine curve ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_0)$. In order to prove the claim, let $w$ denote the homogeneous element of degree one given by the image of the element $1\in A_0^{(0)}$ under the inclusion $A_0^{(0)}\subset A_0^{(r)}$. Then the homogeneous localization $(grA_0)_{((w))}$ is isomorphic to $A_{0}$. Thus the principal open subset $D^{+}(w)$ defined by the element $w$ is isomorphic to the affine curve ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_0)$. The complement of ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_0)$ in $C_0$ is the closed subset defined by $(w)$, which is nothing but the projective scheme $$ \Proj\left(\bigoplus_{m=0}^\infty A_0^{(rm)}\big/A_0^{(rm-r)}\right) $$ given by the associated graded algebra of $grA_0$. Take a monic element $a_m\in A_0^{(rm)}\setminus A_0^{(rm-r)}$ for every $m >>0$, whose existence is assured by (\ref{4.8}). Since $a_i\cdot a_j \equiv a_{i+j}$ mod $A_0^{(ri+rj-r)}$, the map $$ \zeta: \bigoplus _{m=0}^\infty A_0^{(rm)}/A_0^{(rm-r)} \longrightarrow k[x]\;, $$ which assigns $x^m$ to each $a_m$ for $m>>0$ and 0 otherwise, is a well-defined homomorphism of graded rings, where $x$ is an indeterminate. In fact, $\zeta$ is an isomorphism in large degrees, and hence we have $$ \Proj\left(\bigoplus_{m=0}^\infty A_0^{(rm)}\big/A_0^{(rm-r)}\right) \isom \Proj \big(k[x]\big) = p\;. $$ This proves the claim. Next we want to show that the added point $p$ is a smooth rational point of $C_0$. To this end, it is sufficient to show that the formal completion of the structure sheaf of $C_0$ along $p$ is isomorphic to a formal power series ring. Let us consider $(grA_0)/({w}^n)$. The degree $m$ homogeneous piece of this ring is given by $A_0^{(rm)}/(w^nA_0^{(rm-rn)})$, which is isomorphic to $ k\cdot a_m\oplus k\cdot a_{m-1}w \oplus \cdots \oplus k\cdot a_{m-n+1}w^{n-1}$ for all $m>n>>0$. {From} this we conclude that $$ gr(A_0)/(w^n)\cong k[x, w]/(w^n) $$ in large degrees for $n>>0$. Therefore, taking the homogeneous localization at the ideal $(w)$, we have $$ \big(gr(A_0)/(w^n)\big)_{((w))}\cong k[w/x]\big/\big((w/x)^n\big) $$ for $n>>0$. Letting $n\rightarrow \infty$ and taking the inverse limit of this inverse system, we see that the formal completion of the structure sheaf of $C_0$ along $p$ is indeed isomorphic to the formal power series ring $k[[w/x]]$. We can also present an affine local neighborhood of the point $p$. Let $a=a(y)\in A_0$ be a monic, nonconstant element with the lowest order. It is unique up to the addition of a constant: $a(y)\mapsto a(y)+c$. This element defines a principal open subset $D^{+}(a)$ corresponding to the ring \begin{equation} \begin{aligned} (grA_0)_{(a)} &= gr A_0 \big[a^{-1}\big]_0\\ &=\{a^{-i}b\;|\;b\in A_0,\; i\ge 0,\; \ord_y(b) - i\cdot\ord_y(a) \le 0\}\\ &\subset k[[y]]\;. \end{aligned}\label{4.9} \end{equation} Since the formal completion of $C_0$ along $p$ coincides with that of $D^+(a)$ at $p$, and since the structure sheaf of the latter is $k[[y]]$ by (\ref{4.9}), we have obtained that $k[[w /x]]= k[[y]]$. Thus $y$ is indeed a formal parameter of the curve $C_0$ at $p$. \smallskip \noindent (II) {\sl Construction of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and $\Delta$:} Since $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\subset H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$, it has a filtration $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)} = A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\cap H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^{(rm)}$ induced by (\ref{4.2}). The Fredholm condition of $W$ again implies that $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)} = 0$ for all $m<0$. So let us define $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = \Proj(gr A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$, where $$ gr A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = \bigoplus_{m=0}^\infty A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\;. $$ This is a complete algebraic curve and has an affine part ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$. The complement $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\setminus {\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ is given by the projective scheme $$ \Proj\left(\bigoplus_{m=0}^\infty A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\big/A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm-r)}\right)\;. $$ The finite-dimensionality (5) of Definition~\ref{3.9. Definition} implies that for every $\ell$-tuple $(\nu_1,\cdots,\nu_\ell)$ of positive integers satisfying that $\nu_j>>0$, the stabilizer algebra $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ has an element of the form $$ \big(a_1(y_1),\;\cdots,\;a_\ell(y_\ell)\big) \in A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \subset \bigoplus_{j=1}^\ell k((y_j)) $$ such that the order of $a_j(y_j)$ with respect to $y_j$ is equal to $\nu_j$ for all $j = 1, \cdots,\ell$. Thus for all sufficiently large integer $m\in\mathbb{N}$, we have an isomorphism $$ A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\big/A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm-r)} \isom k^{\dsum\ell}\;. $$ Actually, by choosing a basis of $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\big/A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm-r)}$ for each $m>>0$, we can prove in the similar way as in the scalar case that the associated graded algebra $\bigoplus _{m=0}^{\infty}A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}/A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm-r)}$ is isomorphic to the graded algebra $\bigoplus _{j=0}^{\ell} k[x_j]$ in sufficiently large degrees, where $x_j$'s are independent variables. The projective scheme of the latter graded algebra is an $\ell$-point scheme. Therefore, the curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is an $\ell$-point completion of the affine curve ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$. Let $$ \Delta = \{p_1,p_2,\cdots,p_\ell\} = \Proj\left(\bigoplus_{m=0}^\infty A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\big/A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm-r)}\right)\;. $$ We have to show that these points are smooth and rational. To this end, we investigate the completion of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ along the subscheme $\{ p_1, p_2, \cdots , p_{\ell}\}$. Let $u$ be the homogeneous element of degree one in $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(r)}$ given by the image of $1\in A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(0)}$ under the inclusion map $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(0)}\subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(r)}$. Then the closed subscheme (the added points) is exactly the one defined by the principal homogeneous ideal $(u)$. We can prove, in a similar way as in (I), that $$ gr(A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})/(u^n)\cong \big(\bigoplus_{j=1}^{\ell}k[x_j]\big)[u]\big/(u^n) \cong \bigoplus_{j=1}^{\ell} \big(k[x_j, u_j]/(u_j^n)\big) $$ in large degrees for $n>>0$, where $x_j$'s and $u_j$'s are independent variables. Letting $n\rightarrow \infty$ and taking the inverse limit, we conclude that the formal completion of the structure sheaf of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ along the subscheme $\{ p_1, p_2, \cdots, p_{\ell}\}$ is isomorphic to the direct sum $\bigoplus_{j=1}^{\ell}k[[u_j/x_j]]$. Thus all of these $\ell$ points are smooth and rational. By considering the adic-completion of the ring $$ (A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})_p =\big\{a^{-i} h\;|\;h\in {A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}^{(rm)},\;i\ge 0,\; m-i\cdot \ord_y(a) \le 0\big\}\;, $$ where $a$ is as in (\ref{4.9}), we can show that $k[[u_j/x_j]] = k[[y_j]]$. So $y_j$ can be viewed as a formal parameter of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ around the point $p_j$. \smallskip \noindent (III) {\sl Construction of the morphism $f$:} The inclusion map $A_0\hookrarrow A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ gives rise to an inclusion \begin{equation} \bigoplus_{q=0}^\infty A_0^{(rq)} \subset \bigoplus_{m=0}^\infty A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\;,\label{4.10} \end{equation} because we have $A_0^{(rq)} \subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}$ for all $m\ge q\cdot \max[n_1,\cdots,n_\ell]$. It defines a finite surjective morphism $f:C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow {C_0}$. Using the formal parameter $y_j$, we know that the morphism $f_j:U_j\longrarrow U_p$ of the formal completion $U_j$ of $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ along $p_j$ induced by $f:C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow C_o$ is indeed the cyclic covering morphism defined by $y = y_j^{n_j}$. Since $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ is a free $k((y))$-module of dimension $n$ and since the algebras $A_0$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ satisfy the Fredholm condition described in (4), (5) and (7) of Definition~\ref{3.9. Definition}, $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is a torsion-free module of rank $n$ over $A_0$. Thus the morphism $f$ has degree $n$. \smallskip \noindent (IV) {\sl Construction of the sheaf $\mathcal{F}$:} We introduce a filtration in $W\subset V$ induced by (\ref{4.5}). The $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$-module structure of $W$ is compatible with the $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y) = \bigoplus_{j=1}^\ell k((y_j))$-action on $V = \bigoplus_{j=1}^\ell k((z_j))$. Note that we have $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm_1)}\cdot W^{(m_2)}\subset W^{(rm_1+m_2)}$, and hence $\bigoplus_{m=-\infty}^\infty W^{(m)}$ is a graded module over $gr A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Let $\mathcal{F}$ be the sheaf corresponding to the shifted graded module $\big(\bigoplus_{m=-\infty}^{\infty}W^{(m)}\big)(-1)$, where this shifting by $-1$ comes from our convention of Definition~\ref{2.2. Definition}. This sheaf is an extension of the sheaf $W^\sim$ defined on the affine curve ${\text{\rm{Spec}}}} \def\proj{{\text{\rm{proj}}}} \def\Proj{{\text{\rm{Proj}}}(A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$. The graded module $\big(\bigoplus_{m=-\infty}^{\infty}W^{(m)}\big)(-1)$ is also a graded module over $gr A_0$ by (\ref{4.10}). It gives rise to a torsion-free sheaf on ${C_0}$, which is nothing but $f_*\mathcal{F}$. Let us define $$ W_p =\big\{a^{-i}w\;|\;w\in W^{(m)},\;i\ge 0,\; m-i\cdot r\cdot \ord_y(a) \le -1\big\}\;, $$ where $a$ is as in (\ref{4.9}). Then $W_p$ is an $(A_0)_p$-module of rank $r\cdot n = r\sum n_j$. The formal completion $(f_*\mathcal{F})_{U_p}$ of $f_*\mathcal{F}$ at the point $p$ is given by the $k[[y]]$-module $W_p\tensor_{(A_0)_p} k[[y]]$, and the isomorphism \begin{equation} W_p\tensor_{(A_0)_p} k[[y]]\isom \bigoplus_{j=1}^\ell k[[z_j]]z_j \label{4.11} \end{equation} gives rise to the sheaf isomorphism $$ \phi:(f_*\mathcal{F})_{U_p}\overset{\sim}{\longrarrow} \pi_*\bigoplus_{j=1}^\ell \psi_{j*}\big(\mathcal{O}_{U_{oj}}(-1)\big) $$ and its diagonal blocks $\Phi = (\phi_1,\cdots,\phi_\ell)$. Since $f_*\mathcal{F}$ has rank $r\cdot n$ over $\mathcal{O}_{C_0}$ from (\ref{4.11}) and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ has rank $n$ over $A_0$, the sheaf $\mathcal{F}$ on $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ must have rank $r$. The cohomology calculation of (\ref{3.11}), (\ref{3.12}) and (\ref{3.13}) shows that the Euler characteristic of $\mathcal{F}$ is equal to $\mu$. Thus we have constructed all of the ingredients of the geometric data of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$, and rank $r$. This completes the proof of Theorem~\ref{4.7. Theorem}. \end{proof} \medskip In order to complete the proof of the categorical equivalence of Theorem~\ref{3.17. Theorem}, we have to construct a triple $(\alpha,\beta,\lam)$ out of the homomorphisms $\iota:A_0\hookrarrow A'_0$, $\epsilon:A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \longrarrow A'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, and $\omega:{W'}\longrarrow W$. Let $s$ be the rank of $A'_0$ as an $A_0$-module. The injection $\iota$ is associated with the inclusion $k((y))\subset k(({y'}))$, and the coordinate $y$ has order $-s$ with respect to ${y'}$. Therefore, we have $r = s\cdot r'$. Recall that the filtration we have introduced in $A_0$ is defined by the order with respect to $y$. The homomorphism $\iota$ induces an injective homomorphism $$ gr A_0 = \bigoplus_{m=0}^\infty A_0^{(rm)}\longrarrow \bigoplus_{m=0}^\infty {A'}_0^{(s\cdot r'm)}\subset \bigoplus_{m=0}^\infty {A'}_0^{(r'm)} = gr {A'}_0\;, $$ which then defines a morphism $\alp:{C'}_0\longrarrow C_0$. Note that the homomorphism $\epsilon$ comes from the inclusion $k((y_j))\subset k(({y'}_j))$ for every $j$. By the Puiseux expansion, we see that every $y_j = y^{1/n_j}$ has order $-s$ as an element of $k(({y'}_j)) = k(({y'}^{1/n_j}))$. Thus we have $$ gr A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = \bigoplus_{m=0}^\infty A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(rm)}\longrarrow \bigoplus_{m=0}^\infty {A'}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(s\cdot r'm)}\subset \bigoplus_{m=0}^\infty {A'}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}^{(r'm)} = gr {A'}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\;, $$ and this homomorphism defines $\beta:{C'}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Finally, the homomorphism $\lam$ can be constructed as follows. Note that $\omega$ gives an inclusion ${W'}^{(m)}\subset W^{(m)}$ as subspaces of $\bigoplus_{j=1}^\ell k((z_j))$ for every $m\in \mathbb{Z}$. Thus we have an inclusion map $$ \bigoplus_{m=-\infty}^\infty {W'}^{(m)} \subset \bigoplus_{m=-\infty}^\infty W^{(m)} \;, $$ which is clearly a $gr A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$-module homomorphism. Thus it induces an injective homomorphism $\lam:\beta_*\mathcal{F}' \longrarrow \mathcal{F}$. One can check that the construction we have given in Section~\ref{sec: inverse} is indeed the inverse of the map we defined in Section~\ref{sec: coh}. Thus we have completed the entire proof of the categorical equivalence Theorem~\ref{3.17. Theorem}. \bigskip \section{A characterization of arbitrary Prym varieties} \label{sec: Prym} \medskip In this section, we study the geometry of finite type orbits of the Heisenberg flows, and establish a simple characterization theorem of arbitrary Prym varieties. Consider the Heisenberg flows associated with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ on the Grassmannian quotient $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ and assume that the flows produce a finite-dimensional orbit at a point $\overline{W}\in Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$. Then this situation corresponds to the geometric data of Definition~\ref{3.1. Definition}: \medskip \begin{prop} \label{5.1. Proposition} Let $W\in Gr_n(\mu)$ be a point of the Grassmannian at which the Heisenberg flows of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and rank $r$ associated with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ generate an orbit of finite type. Then $W$ gives rise to a set of geometric data $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle $$ of type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, index $\mu$, and rank $r$. \end{prop} \medskip \begin{proof} Let $X_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ be the orbit of the Heisenberg flows starting at $W$, and consider the $r$-reduced KP flows associated with $k((y))$. The finite-dimensionality of $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}(X_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ implies that the $r$-reduced KP flows also produce a finite type orbit $X_0$ at $W$. Let $A_0 = \{a\in k((y))\;|\; a\cdot W\subset W\}$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = \{ h\in H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\;|\; h\cdot W\subset W\}$ be the stabilizer subalgebras, which satisfy $A_0\subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. {From} the definition of the vector fields Definition~\ref{2.5. Definition}, an element of $k((y))$ gives the zero tangent vector at $W$ if and only if it is in $A_0$. Similarly, for an element $b\in H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$, ${\Psi}_W(b) = 0$ if and only if $b\in A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Thus the tangent spaces of these orbits are given by $$ T_{W}X_0\isom k((y))\big/A_0 \quad{\text{ and }}\quad T_{W}X_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \isom H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\big/A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\;. $$ Therefore, going down to the Grassmannian quotient, the tangent spaces of $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and $\overline{X}_0 = Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}(X_0)$ are now given by $$ T_{\overline{W}}\overline{X}_0 \isom \frac{k((y))}{A_0 + k((y))\cap gl\big(n, k[[y]]y\big)} = \frac{k((y))}{A_0\dsum k[[y]]y} $$ and $$ T_{\overline{W}}\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \isom \frac{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)}{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} + H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\cap gl\big(n, k[[y]]y\big)} = \frac{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)}{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \dsum H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^-}\;, $$ where $\overline{W} = Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}(W)$, and $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^-$ is defined in (\ref{2.14}). Since both of the above sets are finite-dimensional, the triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ satisfies the cokernel conditions (4) and (5) of Definition~\ref{3.9. Definition}. The rank condition (6) of Definition~\ref{3.9. Definition} is a consequence of the fact that $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ has dimension $n$ over $k((y))$. Therefore, applying the inverse construction of the cohomology functor to the triple, we obtain a set of geometric data. This completes the proof. \end{proof} \medskip \noindent Since $k\subset A_0\subset A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, from (\ref{3.4}) and (\ref{3.8}) we obtain \begin{equation} T_{\overline{W}}\overline{X}_0 \isom\frac{k((y))}{A_0\dsum k[[y]]y} \isom H^1({C_0},\mathcal{O}_{{C_0}}) \label{5.2} \end{equation} and \begin{equation} T_{\overline{W}}\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \isom \frac{H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)}{A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} \dsum H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)^-} \isom H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})\;.\label{5.3} \end{equation} Thus we know that the genera of ${C_0}$ and $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ are equal to the dimension of the orbits $\overline{X}_0$ and $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ on the Grassmannian quotient, respectively. However, we cannot conclude that these orbits are actually Jacobian varieties. The difference of the orbits and the Jacobians lies in the deformation of the data $(\Phi,\phi)$. In order to give a surjective map from the Jacobians to these orbits, we have to eliminate these unwanted information by using Corollary~\ref{3.19. Corollary}. Therefore, in the rest of this section, we have to assume that the point $W\in Gr_n(\mu)$ gives rise to a rank one triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ of algebraic data from the application of the Heisenberg flows associated with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ and an element $y\in L$ of order $-1$. In order to deal with Jacobian varieties, we further assume that the field $k$ is the field $\mathbb{C}$ of complex numbers in what follows in this section. The computation (\ref{5.3}) shows that every element of $H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})$ is represented by \begin{equation} \sum_{j=1}^\ell\sum_{i=-\infty}^\infty t_{ij} y_j^{-i}\in \bigoplus_{j=1}^\ell \mathbb{C}((y_j)) = H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)\;.\label{5.4} \end{equation} The Heisenberg flows at $W$ are given by the equations \begin{equation} \frac{\partial W}{\partial t_{ij}} = y_j^{-i}\cdot W = \big(h_{n_j}(y)\big)^{-i}\cdot W\;, \label{5.5} \end{equation} where $h_{n_j}(y)$ acts on $W$ through the block matrix $$ \begin{pmatrix} 0\\ &\ddots\\ &&h_{n_j}(y)\\ &&&\ddots\\ &&&&0 \end{pmatrix}\;, $$ and the index $i$ runs over all of $\mathbb{Z}$. The formal integration \begin{equation} W(t) = {\text{\rm{exp}}}\left(\sum_{j=1}^\ell\sum_{i=-\infty}^\infty t_{ij} y_j^{-i}\right) \cdot W\label{5.6} \end{equation} of the system (\ref{5.5}) shows that the stabilizers $A_0$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ of $W(t)$ do not deform as $t$ varies, because the exponential factor \begin{equation} e(t) = {\text{\rm{exp}}}\left(\sum_{j=1}^\ell\sum_{i=-\infty}^\infty t_{ij} y_j^{-i}\right)\label{5.7} \end{equation} commutes with the algebra $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$. Note that half of the exponential factor $$ {\text{\rm{exp}}}\left(\sum_{j=1}^\ell\sum_{i=-\infty}^{-1} t_{ij} y_j^{-i}\right) $$ is an element of $\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$. \medskip \begin{thm} \label{5.8. Theorem} Let $y\in L$ be a monic element of order $-1$ and $X_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ a finite type orbit of the Heisenberg flows on $Gr_n(\mu)$ associated with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ starting at $W$. As we have seen in Proposition~\ref{5.1. Proposition}, the orbit $X_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ gives rise to a set of geometric data $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle\;. $$ Then the projection image $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ of this orbit by $Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}:Gr_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu)\longrarrow Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ is canonically isomorphic to the Jacobian variety ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of the curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ with $\overline{W} = Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}(W)$ as its origin. Moreover, the orbit $\overline{X}_0$ of the KP system (written in terms of the variable $y$) defined on the Grassmannian quotient $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ is isomorphic to the deformation space $$ \big\{ \mathcal{N}\tensor f_*\mathcal{F}\;\big|\; \mathcal{N}\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_0)\big\}\;. $$ Thus we have a finite covering ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_0)\longrarrow \overline{X}_0$ of the orbit, which is indeed isomorphic if $f_*\mathcal{F}$ is a general vector bundle on $C_0$. \end{thm} \medskip \begin{proof} Even though the formal integration (\ref{5.6}) is not well-defined as a point of the Grassmannian, we can still apply the same construction of Section~\ref{sec: inverse} to the algebraic data $\big(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},{W}(t)\big)$ understanding that the exponential matrix $e(t)$ of (\ref{5.7}) is an extra factor of degree 0. Of course the curves, points, and the covering morphism $f:C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow C_0$ are the same as before. Therefore, we obtain $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}(t), \Phi(t)\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}(t), \phi(t)\big)\right\rangle\;, $$ where the line bundle $\mathcal{F}(t)$ comes from the $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$-module $W(t)$. We do not need to specify the data $\Phi(t)$ and $\phi(t)$ here, because they will disappear anyway by the trick of Corollary~\ref{3.19. Corollary}. On the curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, the formal expression $e(t)$ makes sense because of the homomorphism $$ {\text{\rm{exp}}}: H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})\owns \sum_{j=1}^\ell\sum_{i=-\infty}^\infty t_{ij} y_j^{-i}\longmapsto \big[e(t)\big]=\mathcal{L}(t)\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})\subset H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{O}^*_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})\;, $$ where $\mathcal{L}(t)$ is the line bundle of degree 0 corresponding to the cohomology class $\big[e(t)\big]\in H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{O}^*_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}})$. Thus the sheaf we obtain from $W(t) = e(t)\cdot W$ is $\mathcal{F}(t) = \mathcal{L}(t)\tensor \mathcal{F}$. Now consider the projection image $\big(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\overline{W}(t)\big)$ of the algebraic data by $Q_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},y}$. Then it corresponds to the data \begin{equation} \left\langle f:\bigg( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{L}(t)\tensor \mathcal{F}\bigg) \longrarrow \bigg({C_0}, p, \pi, f_*\big(\mathcal{L}(t)\tensor \mathcal{F}\big)\bigg)\right\rangle \label{5.9} \end{equation} by Corollary~\ref{3.19. Corollary}. Since ${\text{\rm{exp}}}:H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \mathcal{O}_{C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}}) \longrarrow {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ is surjective, we can define a map assigning (\ref{5.9}) to every point $\mathcal{L}(t)\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of the Jacobian. Through the cohomology functor, it gives indeed the desired identification of ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ and the orbit $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$: $$ {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})\owns \mathcal{L}(t)\longmapsto (\ref{5.9}) \longmapsto \overline{W}(t) \in \overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\;. $$ The KP system in the $y$-variable at $\overline{W}\in Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ is given by the equation $$ \frac{\partial \overline{W}}{\partial s_m} = y^{-m}\cdot \overline{W}\;. $$ The formal integration $$ \overline{W}(s) = {\text{\rm{exp}}}\left( \sum_{m=1}^\infty s_my^{-m}\right) \cdot \overline{W} $$ corresponds to $$ \left\langle f:\bigg( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \big(f^*\mathcal{N}(s)\big)\tensor \mathcal{F}\bigg) \longrarrow \bigg({C_0}, p, \pi, \mathcal{N}(s)\tensor f_*\mathcal{F}\bigg)\right\rangle\;, $$ where $\mathcal{N}(s)\tensor f_*\mathcal{F}$ is the vector bundle corresponding to the $A_0$-module $\overline{W}(s)$. {From} (\ref{5.2}), we have a surjective map of $H^1(C_0,\mathcal{O}_{C_0})$ onto the Jacobian variety ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_0) \subset H^1(C_0,\mathcal{O}^*_{C_0})$ defined by $$ {\text{\rm{exp}}}: H^1(C_0,\mathcal{O}_{C_0})\owns \sum_{m=1}^\infty s_my^{-m} \longmapsto \left[{\text{\rm{exp}}}\bigg(\sum_{m=1}^\infty s_my^{-m}\bigg) \right] = \mathcal{N}(s)\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_0)\;. $$ Thus the orbit $\overline{X}_0$ coincides with the deformation space $\mathcal{N}(s)\tensor f_*\mathcal{F}$, which is covered by ${\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_0)$. The last statement of the theorem follows from a result of \cite{L}. This completes the proof. \end{proof} \medskip\noindent Let $(\eta_1, \cdots, \eta_\ell)$ be the transition function of $\mathcal{F}$ defined on $U_j\setminus \{p_j\}$, where $\eta_j\in \mathbb{C}((y_j))$. Then the family $\mathcal{F}(t)$ of line bundles on $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is given by the transition function $$ \left({\text{\rm{exp}}}\bigg(\sum_{i=1}^\infty t_{i1} y_j^{-i}\bigg)\cdot\eta_1,\; \cdots,\; {\text{\rm{exp}}}\bigg(\sum_{i=1}^\infty t_{i\ell} y_\ell^{-i}\bigg)\cdot\eta_\ell \right)\;, $$ and similarly, the line bundle $\mathcal{L}(t)$ is given by $$ \left({\text{\rm{exp}}}\bigg(\sum_{i=1}^\infty t_{i1} y_j^{-i}\bigg),\; \cdots,\; {\text{\rm{exp}}}\bigg(\sum_{i=1}^\infty t_{i\ell} y_\ell^{-i}\bigg) \right)\;. $$ Here, we note that the nonnegative powers of $y_j=h_{n_j}(y)$ do not contribute to these transition functions. Recall that $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)_0$ denotes the subalgebra of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ consisting of the traceless elements. \medskip \begin{thm} \label{5.10. Theorem} In the same situation as above, the projection image $\overline{X}\subset Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ of the orbit $X$ of the traceless Heisenberg flows ${\Psi}\big(H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)_0\big)$ starting at $\overline{W}$ is canonically isomorphic to the Prym variety associated with the covering morphism $f:C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow {C_0}$. \end{thm} \medskip \begin{proof} Because of Remark~\ref{{1.3.} Remark}, the locus of $\mathcal{L}(t)\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ such that $$ \det\bigg(f_*\big(\mathcal{L}(t) \tensor \mathcal{F}\big)\bigg)= \det(f_*\mathcal{F}) $$ is the Prym variety $\Prym(f)$ associated with the covering morphism $f$. So let us compute the factor \begin{equation} \mathcal{D}(t) = \det\big(f_*(\mathcal{L}(t)\otimes \mathcal{F})\big) \otimes \det(f_*\mathcal{F})^{-1}\;,\label{5.11} \end{equation} which is a line bundle of degree 0 defined on $C_0$. We use the transition function $\eta$ of $f_*\mathcal{F}$ defined on $U_p\setminus \{p\}$ written in terms of the basis (\ref{3.6}). Since $f_*\mathcal{F}(t)$ is defined by the $A_0$-module structure of $W(t) = e(t)\cdot W$, its transition function is given by $$ {\text{\rm{exp}}} \begin{pmatrix} \sum_{i=1}^\infty t_{i1} \big(h_{n_1}(y)\big)^{-i}\\ &\sum_{i=1}^\infty t_{i2} \big(h_{n_2}(y)\big)^{-i}\\ &&\ddots\\ &&&\sum_{i=1}^\infty t_{i\ell} \big(h_{n_\ell}(y)\big)^{-i} \end{pmatrix} \cdot \eta \;, $$ where the $n\times n$ matrix acts on the $y^{\alpha/n_j}$-part of the basis of (\ref{3.6}) in an obvious way. Let us denote the above matrix by $$ T(t) = \begin{pmatrix} \sum_{i=1}^\infty t_{i1} \big(h_{n_1}(y)\big)^{-i}\\ &\sum_{i=1}^\infty t_{i2} \big(h_{n_2}(y)\big)^{-i}\\ &&\ddots\\ &&&\sum_{i=1}^\infty t_{i\ell} \big(h_{n_\ell}(y)\big)^{-i} \end{pmatrix}\;. $$ Then, it is clear that $\mathcal{D}(t) \isom \big[{\text{\rm{exp}}} \;\trace T(t)\big] \in H^1(C_0,\mathcal{O}^*_{C_0})$. {From} this expression, we see that if $\mathcal{L}(t)$ stays on the orbit $\overline{X}$ of the traceless Heisenberg flows, then $\mathcal{D}(t) \isom \mathcal{O}_{C_0}$. Namely, $\overline{X}\subset \Prym(f)$. Conversely, take a point $\overline{W}(t)\in \overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ of the orbit of the Heisenberg flows defined on the quotient Grassmannian $Z_n(\mu,y)$. It corresponds to a unique element $\mathcal{L}(t)\in {\text{\rm{Jac}}}} \def\Prym{{\text{\rm{Prym}}}} \def\trace{{\text{\rm{trace}}}} %Greek letters \def\alp{\alpha} \def\Alp{\Alpha} \def\bet{\beta} \def\gam{{\hbox{\raise1pt\hbox{$\gamma$}}}} \def\del{\delta} \def\Gam{\Gamma} \def\eps{\varepsilon} \def\Del{\Delta} \def\epi{\epsilon} \def\zet{\zeta} \def\tet{\theta} \def\Tet{\Theta} \def\iot{\iota} \def\kap{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\Sig{\Sigma} \def\ome{\omega} \def\Ome{\Omega} \def\rchi{{\hbox{\raise1.5pt\hbox{$\chi$}}}} %regular arrows \def\rarrow{\rightarrow} \def\larrow{\leftarrow} \def\longrarrow{\longrightarrow} \def\longlarrow{\longleftarrow} \def\hookrarrow{\hookrightarrow} \def\hooklarrow{\hookleftarrow(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ by Theorem~\ref{5.8. Theorem}. Now suppose that the factor $\mathcal{D}(t)$ of (\ref{5.11}) is the trivial bundle on $C_0$. Then it implies that $\big[\trace T(t)\big] = 0$ as an element of $H^1(C_0, \mathcal{O}_{C_0})$. In particular, $\trace T(t)$ acts on $\overline{W}$ trivially from (\ref{5.2}). Therefore, $\overline{W}(t)$ is on the orbit of the flows defined by $$ T(t) - I_n\cdot \frac{1}{n}\; \trace T(t)\;, $$ which are clearly traceless. In other words, $\overline{W}(t) \in \overline{X}$. Thus $\Prym(f)\subset \overline{X}$. This completes the proof. \end{proof} \medskip \begin{rem} \label{{5.12.} Remark} Let us observe the case when the curve $C_0$ downstairs happens to be a $\mathbb{P}^1$. First of all, we note that the $r$-reduced KP system associated with $y$ is nothing but the trace part of the Heisenberg flows defined by $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$. Because of the second half statement of Theorem~\ref{5.8. Theorem}, the trace part of the Heisenberg flows acts on the point $\overline{W}\in Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,y)$ trivially. Therefore, the orbit $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ of the entire Heisenberg flows coincides with the orbit $\overline{X}$ of the traceless part of the flows. Of course, this reflects the fact that every Jacobian variety is a Prym variety associated with a covering over $\mathbb{P}^1$. Thus the characterization theorem of Prym varieties we are presenting below contains the characterization of Jacobians of \cite{M1} as a special case. \end{rem} \medskip Now consider the most trivial maximal commutative algebra $H = H_{(1, \cdots, 1)}(z) = \mathbb{C}((z))^{\dsum n}$. We define the group $\Gam_{(1, \cdots, 1)}(z)$ following (\ref{2.15}), and denote by \begin{equation} Z_n(\mu) = Z_{(1, \cdots, 1)}(\mu,z) = Gr_n(\mu)\big/\Gam_{(1, \cdots, 1)}(z)\label{5.13} \end{equation} the corresponding Grassmannian quotient. On this space the algebra $H$ acts, and gives the $n$-component KP system. Let $H_0$ be the traceless subalgebra of $H$, and consider the traceless $n$-component KP system on the Grassmannian quotient $Z_n(\mu)$. \medskip \begin{thm} \label{5.14. Theorem} Every finite-dimensional orbit of the traceless $n$-component KP system defined on the Grassmannian quotient $Z_n(\mu)$ of $(\ref{5.13})$ is canonically isomorphic to a (generalized) Prym variety. Conversely, every Prym variety associated with a degree $n$ covering morphism of smooth curves can be realized in this way. \end{thm} \medskip \begin{proof} The first half part has been already proved. So start with the Prym variety $\Prym(f)$ associated with a degree $n$ covering morphism $f:C\longrarrow C_0$ of smooth curves. Without loss of generality, we can assume that $C_0$ is connected. Choose a point $p$ of $C_0$ outside of the branching locus so that its preimage $f^{-1}(p)$ consists of $n$ distinct points of $C$, and supply the necessary geometric objects to make the situation into the geometric data $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle $$ of Definition~\ref{3.1. Definition} of rank one and type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (1,\cdots,1)$ with $C= C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. The data give rise to a unique triple $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ of algebraic data by the cohomology functor. We can choose $\pi = id$ so that the maximal commutative subalgebra we have here is indeed $H = H_{(1, \cdots, 1)}(z)$. Define $A'_0 = \{a\in \mathbb{C}((z))\;|\; a\cdot W\subset W\}$ and $A'= \{h\in H\;|\; h\cdot W\subset W\}$, which satisfy $A_0\subset A'_0$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\subset A'$, and both have finite codimensions in the larger algebras. {From} the triple of the algebraic data $(A'_0,A',W)$, we obtain a set of geometric data $$ \left\langle {f'}:\big( C'_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta', \Pi', \mathcal{F}', \Phi'\big) \longrarrow \big({C'_0}, p', \pi', {f'}_*\mathcal{F}', \phi'\big) \right\rangle\;. $$ The morphism $(\alp, \beta, id)$ between the two sets of data consists of a morphism $\alpha:C'_0\longrarrow C_0$ of the base curves and $\beta:C'\longrarrow C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Obviously, these morphisms are birational, and hence, they have to be an isomorphism, because $C_0$ and $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ are smooth. Going back to the algebraic data by the cohomology functor, we obtain $A_0 = A'_0$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = A'$. Thus the orbit of the traceless $n$-component KP system starting at $\overline{W}$ defined on the Grassmannian quotient $Z_n(\mu)$ is indeed the Prym variety of the covering morphism $f$. This completes the proof of the characterization theorem. \end{proof} \medskip \begin{rem} \label{{5.15.} Remark} In the above proof, we need the full information of the functor, not just the set-theoretical bijection of the objects. We use a similar argument once again in Theorem~\ref{6.15. Theorem}. \end{rem} \medskip \begin{rem} \label{{5.16.} Remark} The determinant line bundle $DET$ over $Gr_n(0)$ is defined by $$ DET_W = \left(\bigwedge^{\max} {\text{\rm{Ker}}}} \def\coker{{\text{\rm{coker}}}\big(\gam_W\big)\right)^* \bigotimes\bigwedge^{\max} \Coker\big(\gam_W\big)\;. $$ The canonical section of the $DET$ bundle defines the determinant divisor $Y$ of $Gr_n(0)$, whose support is the complement of the big-cell $Gr^+_n(0)$. Note that the action of $\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$ preserves the big-cell. So we can define the \emph{big-cell} of the Grassmannian quotient by $Z^+_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(0, y) = Gr^+_n(0)\big/\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$. The determinant divisor also descends to a divisor $Y/\Gam_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(y)$, which we also call the determinant divisor of the Grassmannian quotient. Consider a point $W\in Gr_n(0)$ at which the Heisenberg flows of rank one produce a finite type orbit $X_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. The geometric data corresponding to this situation consists of a curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ of genus $g = \dim_\mathbb{C} \overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ and a line bundle $\mathcal{F}$ of degree $g - 1$ because of the Riemann-Roch formula $$ \dim_\mathbb{C} H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) - \dim_\mathbb{C} H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = \deg(\mathcal{F}) - r(g-1)\;. $$ Thus we have an \emph{equality} $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = {\text{\rm{Pic}}}} \def\Coker{{\text{\rm{Coker}}}} \def\ord{{\text{\rm{ord}}}^{g-1}(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ from the proof of Theorem~\ref{5.8. Theorem}. The intersection of $\overline{X}_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ with the determinant divisor of $Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(0,y)$ coincides with the theta divisor $\Theta$ which gives the principal polarization of ${\text{\rm{Pic}}}} \def\Coker{{\text{\rm{Coker}}}} \def\ord{{\text{\rm{ord}}}^{g-1}(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$. However, the restriction of this divisor to the Prym variety does not give a principal polarization as we have noted in Section~\ref{sec: 1}. \end{rem} \medskip \begin{rem} \label{{5.17.} Remark} {From} the expression of (\ref{5.9}), we can see that a finite-dimensional orbit of the Heisenberg flows of rank one defined on the Grassmannian quotient gives a family of deformations $f_*\big(\mathcal{L}(t)\tensor \mathcal{F}\big)$ of the vector bundle $f_*\mathcal{F}$ on $C_0$. It is an interesting question to ask what kind of deformations does this family produce. More generally, we can ask the following question: For a given curve and a family of vector bundles on it, can one find a point $W$ of the Grassmannian $Gr_n(\mu)$ and a suitable Heisenberg flows such that the orbit starting from $W$ contains the original family? It is known that for every vector bundle $\mathcal{V}$ of rank $n$ on a smooth curve $C_0$, there is a degree $n$ covering $f:C\longrarrow C_0$ and a line bundle $\mathcal{F}$ on $C$ such that $\mathcal{V}$ is isomorphic to the direct image sheaf $f_*\mathcal{F}$. We can supply suitable local data so that we have a set of geometric data $$ \left\langle f:\big(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}\big) \longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}\big)\right\rangle $$ with $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = C$. Let $(A_0, A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \overline{W})$ be the triple of algebraic data corresponding to the above geometric situation with a point $\overline{W}\in Z_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(\mu,z)$, where $\mu$ is the Euler characteristic of the original bundle $\mathcal{V}$. Now the problem is to compare the family of deformations given by (\ref{5.9}) and the original family. The only thing we can say about this question at the present moment is the following. If the original vector bundle is a general stable bundle, then one can find a set of geometric data and a corresponding point $\overline{W}$ of a Grassmannian quotient such that there is a dominant and generically finite map of a Zariski open subset of the orbit of the Heisenberg flows starting from $\overline{W}$ into the moduli space of stable vector bundles of rank $n$ and degree $\mu + n\big(g(C_0) - 1\big)$ over the curve $C_0$. Note that this statement is just an interpretation of a theorem of \cite{BNR} into our language using Theorem~\ref{5.8. Theorem}. As in the proof of Theorem~\ref{5.14. Theorem}, the Heisenberg flows can be replaced by the $n$-component KP flows if we choose the point $p\in C_0$ away from the branching locus of $f$. Thus one may say that the $n$-component KP system can produce general vector bundles of rank $n$ defined on an arbitrary smooth curve in its orbit. \end{rem} \bigskip \section{Commuting ordinary differential operators with matrix coefficients} \label{sec: ODE} \medskip In this section, we work with an arbitrary field $k$ again. Let us denote by $$ E = \big(k[[x]]\big)((\partial^{-1})) $$ the set of all \hyphenation{pseu-do-dif-fer-en-tial} pseudodifferential operators with coefficients in $k[[x]]$, where $\partial = d/dx$. This is an associative algebra and has a natural filtration $$ E^{(m)} = \big(k[[x]]\big)[[\partial^{-1}]]\cdot \partial^m $$ defined by the \emph{order} of the operators. We can identify $k((z))$ with the set of pseudodifferential operators with constant coefficients by the \emph{Fourier transform} $z$ $=$ $\partial^{-1}$: $$ L = k((z)) = k((\partial^{-1}))\subset E\;. $$ There is also a canonical projection \begin{equation} \rho: E \longrarrow E/Ex \isom k((\partial^{-1})) = L\;,\label{6.1} \end{equation} where $Ex$ is the left-maximal ideal of $E$ generated by $x$. In an explicit form, this projection is given by \begin{equation} \rho : E\owns P = \sum_{m\in \mathbb{Z}} \partial^m\cdot a_m(x) \longmapsto \sum_{m\in\mathbb{Z}} a_m(0)z^{-m}\in L\;.\label{6.2} \end{equation} It is obvious from (\ref{6.1}) that $L$ is a left $E$-module. The action is given by $P\cdot v = P\cdot\rho(Q) = \rho(PQ)$, where $v\in L = E/Ex$ and $Q\in E$ is a representative of the equivalence class such that $\rho(Q) = v$. The well-definedness of this action is easily checked. We also use the notations $$ \begin{cases} D = \big(k[[x]]\big)[\partial]\\ E^{(-1)} = \big(k[[x]]\big)[[\partial^{-1}]]\cdot\partial^{-1}\;, \end{cases} $$ which are the set of linear ordinary differential operators and the set of pseudodifferential operators of negative order, respectively. Note that there is a natural left $\big(k[[x]]\big)$-module direct sum decomposition \begin{equation} E = D\dsum E^{(-1)}\;.\label{6.3} \end{equation} According to this decomposition, we write $P = P^+ \dsum P^-$, $P \in E$, $P^+ \in D$, and $P^-\in E^{(-1)}$. Now consider the matrix algebra $gl(n,E)$ defined over the noncommutative algebra $E$, which is the algebra of pseudodifferential operators with coefficients in matrix valued functions. This algebra acts on our vector space $V = L^{\dsum n} \isom \big(E/Ex\big)^{\dsum n}$ from the left. In particular, every element of $gl(n,E)$ gives rise to a vector field on the Grassmannian $Gr_n(\mu)$ via (\ref{2.4}). The decomposition (\ref{6.3}) induces $$ V = k[z^{-1}]^{\dsum n}\dsum \big(k[[z]]\cdot z\big)^{\dsum n} $$ after the identification $z = \partial^{-1}$, and the base point $k[z^{-1}]^{\dsum n}$ of the Grassmannian $Gr_n(0)$ of index 0 is the residue class of $D^{\dsum n}$ in $E^{\dsum n}$ via the projection $E^{\dsum n} \longrarrow E^{\dsum n}\big/\big(E^{(-1)})^{\dsum n}$. Therefore, the $gl(n,D)$-action on $V$ preserves $k[z^{-1}]^{\dsum n}$. The following proposition shows that the converse is also true: \medskip \begin{prop} \label{6.4. Proposition} A pseudodifferential operator $P\in gl(n,E)$ with matrix coefficients is a differential operator, i.e.~$P\in gl(n,D)$, if and only if $$ P\cdot k[z^{-1}]^{\dsum n}\subset k[z^{-1}]^{\dsum n}\;. $$ \end{prop} \medskip \begin{proof} The case of $n = 1$ of this proposition was established in Lemma~7.2 of \cite{M3}. So let us assume that $P = \big(P_{\mu\nu}\big)\in gl(n,E)$ preserves the base point $k[z^{-1}]^{\dsum n}$. If we apply the matrix $P$ to the vector subspace $$ 0\dsum\cdots\dsum 0\dsum k[z^{-1}]\dsum 0\dsum \cdots \dsum 0 \subset k[z^{-1}]^{\dsum n} $$ with only nonzero entries in the $\nu$-th position, then we know that $P_{\mu\nu}\in E$ stabilizes $k[z^{-1}]$ in $L$. Thus $P_{\mu\nu}$ is a differential operator, i.e.\ $P\in gl(n,D)$. This completes the proof. \end{proof} \medskip Since differential operators preserve the base point of the Grassmannian $Gr_n(0)$, the negative order pseudodifferential operators should give the most part of $Gr_n(0)$. In fact, we have \medskip \begin{thm} \label{6.5. Theorem} Let $S\in gl(n,E)$ be a monic zero-th order pseudodifferential operator of the form \begin{equation} S = I_n + \sum_{m = 1}^\infty s_m(x) \partial^{-m} \;,\label{6.6} \end{equation} where $s_m(x)\in gl\big(n,k[[x]]\big)$. Then the map $$ \sigma : \Sigma\owns S\longmapsto W = S^{-1}\cdot k[z^{-1}]^{\dsum n} \in Gr^+_n(0) $$ gives a bijective correspondence between the set $\Sigma$ of pseudodifferential operators of the form of $(\ref{6.6})$ and the big-cell $Gr_n^+(0)$ of the index 0 Grassmannian. \end{thm} \medskip \begin{proof} Since $S$ is invertible of order 0, we have $S^{-1}\cdot V = V$ and $S^{-1}\cdot V^{(-1)} = V^{(-1)}$, where $V^{(-1)} = F^{(-1)}(V) = \big(k[[z]]z\big)^{\dsum n}$. Thus $V = S^{-1}\cdot k[z^{-1}]^{\dsum n} \dsum V^{(-1)}$, which shows that $\sigma$ maps into the big-cell. The injectivity of $\sigma$ is easy: if $S_1^{-1}\cdot k[z^{-1}]^{\dsum n} = S_2^{-1}\cdot k[z^{-1}]^{\dsum n}$, then $S_1S_2^{-1}\cdot k[z^{-1}]^{\dsum n} = k[z^{-1}]^{\dsum n}$. It means, by Proposition~\ref{6.4. Proposition}, that $S_1S_2^{-1}$ is a differential operator. Since $S_1S_2^{-1}$ has the same form of (\ref{6.6}), the only possibility is that $S_1S_2^{-1} = I_n$, which implies the injectivity of $\sigma$. In order to establish surjectivity, take an arbitrary point $W$ of the big-cell $Gr^+(0)$. We can choose a basis $\big\langle \w_j^\mu\big\rangle_{1\le j\le n, 0\le \mu}$ for the vector space $W$ in the form $$ \w_j^\mu = \e_jz^{-\mu} + \sum_{\nu = 1}^\infty \sum_{i=1}^n \e_iw_{j\nu}^{i\mu}z^\nu\;, $$ where $\e_j$ is the elementary column vector of size $n$ and $w_{j\nu}^{i\mu}\in k$. Our goal is to construct an operator $S\in \Sigma$ such that $S^{-1}\cdot k[z^{-1}]^{\dsum n} = W$. Let us put $S^{-1} = \big(S^i_j\big)_{1\le i,j\le n}$ with $$ S^i_j = \delta^i_j + \sum_{\nu=1}^\infty \partial^{-\nu} \cdot s^i_{j\nu}(x)\;. $$ Since every coefficient $s^i_{j\nu}(x)$ of $S^{-1}$ is a formal power series in $x$, we can construct the operator by induction on the power of $x$. So let us assume that we have constructed $s^i_{j\nu}(x)$ modulo $k[[x]]x^\mu$. We have to introduce one more equation of order $\mu$ in order to determine the coefficient of $x^\mu$ in $s^i_{j\nu}(x)$, which comes from the equation $$ S^{-1}\cdot \e_jz^{-\mu} = {\text{ a linear combination of }} \w_i^{\nu}\;. $$ For the purpose of finding a consistent equation, let us compute the left-hand side by using the projection $\rho$ of (\ref{6.2}): $$ \begin{aligned} S^{-1}\cdot \e_jz^{-\mu} &= \sum_{i=1}^n \e_i\cdot S^i_j\cdot z^{-\mu}\\ &= \e_jz^{-\mu} + \rho\left(\sum_{\nu=1}^\infty\sum_{i=1}^n \partial^{-\nu}\cdot s^i_{j\nu}(x)\e_i\cdot \partial^\mu\right)\\ &= \e_jz^{-\mu} + \rho\left(\sum_{\nu=1}^\infty\sum_{i=1}^n \sum_{m=0}^\mu (-1)^m\binom \mu{m} \e_i\cdot \partial^{\mu-\nu-m} \cdot {s^i_{j\nu}}^{(m)}(x)\right)\\ &= \e_jz^{-\mu} + \sum_{\nu=1}^\infty\sum_{i=1}^n \sum_{m=0}^\mu (-1)^m\binom \mu{m} {s^i_{j\nu}}^{(m)}(0)\cdot \e_i z^{-\mu+\nu+m}\\ &= \e_jz^{-\mu} + \sum_{\alp = 1}^{\mu-1}\sum_{m=0}^\alp \sum_{i=1}^n (-1)^m\binom \mu{m} {s^i_{j\alp - m}}^{(m)}(0)\cdot \e_i z^{-\mu+\alp}\\ &\qquad\qquad + \sum_{m=0}^{\mu-1} \sum_{i=1}^n (-1)^m\binom \mu{m} {s^i_{j\mu - m}}^{(m)}(0)\cdot \e_i \\ &\qquad\qquad + \sum_{\beta=1}^\infty \sum_{m=0}^\mu\sum_{i=1}^n (-1)^m \binom \mu{m} {s^i_{j\beta+\mu - m}}^{(m)}(0) \cdot\e_i z^\beta\;. \end{aligned} $$ \noindent Thus we see that the equation \begin{equation} \begin{aligned} S^{-1}\cdot \e_jz^{-\mu} = \w_j^\mu &+ \sum_{\alp = 1}^{\mu-1}\sum_{m=0}^\alp \sum_{i=1}^n (-1)^m\binom \mu{m} {s^i_{j\alp - m}}^{(m)}(0)\cdot \w_i^{\mu-\alp}\\ &+ \sum_{m=0}^{\mu-1} \sum_{i=1}^n (-1)^m\binom \mu{m} {s^i_{j\mu - m}}^{(m)}(0)\cdot \w_i^0 \end{aligned} \label{6.7} \end{equation} is the identity for the coefficients of $\e_iz^{-\nu}$ for all $i$ and $\nu\ge 0$, and determines $s^i_{j\beta}(0)^{(\mu)}$ uniquely, because the coefficient of $s^i_{j\beta}(0)^{(\mu)}$ in the equation is $(-1)^\mu$. Thus by solving (\ref{6.7}) for all $j$ and $\mu\ge 0$ inductively, we can determine the operator $S$ uniquely, which satisfies the desired property by the construction. This completes the proof. \end{proof} \medskip Using this identification of $Gr^+(0)$ and $\Sigma$, we can translate the Heisenberg flows defined on the big-cell into a system of nonlinear partial differential equations. Since we are not introducing any analytic structures in $\Sigma$, we cannot talk about a Lie group structure in it. However, the exponential map $$ {\text{\rm{exp}}}: gl\big(n,E^{(-1)}\big)\longrarrow I_n + gl\big(n,E^{(-1)}\big) = \Sigma $$ is well-defined and surjective, and hence we can regard $gl\big(n,E^{(-1)}\big)$ as the \emph{Lie algebra} of the infinite-dimensional group $\Sigma$. Symbolically, we have an identification \begin{equation} T_{k[z^{-1}]^{\dsum n}}Gr^+(0) \isom gl\big(n,E^{(-1)}\big) = {\text{Lie}}(\Sigma) = T_{I_n}\Sigma = S^{-1}\cdot T_S\Sigma \label{6.8} \end{equation} for every $S\in \Sigma$. The equation $$ \frac{\partial W(t)}{\partial t_{ij}} = \big(h_{n_j}(y)\big)^{-i}\cdot W(t) $$ is an equation of tangent vectors at the point $W(t)$. We now identify the variable $y$ of (\ref{2.7}) with a pseudodifferential operator \begin{equation} y = \partial^{-r} + \sum_{m=1}^\infty c_m \partial^{-r-m} \label{6.9} \end{equation} with coefficients in $k$. Then the block matrix $h_{n_j}(y)$ of (\ref{5.5}) is identified with an element of $gl(n,E)$. Let $W(t)$ be a solution of (\ref{5.5}) which lies in $Gr^+_n(0)$, where $t = (t_{ij})$. Writing $W(t) = S(t)^{-1}\cdot k[z^{-1}]^{\dsum n}$, the tangent vector of the left-hand side of (\ref{5.5}) is given by $$ \frac{\partial W(t)}{\partial t_{ij}} = \frac{\partial S(t)^{-1}}{\partial t_{ij}}\;, $$ which then gives an element $$ S(t)\cdot \frac{\partial S(t)^{-1}}{\partial t_{ij}} = -\; \frac{\partial S(t)}{\partial t_{ij}} \cdot S(t)^{-1} \in E^{(-1)} $$ by (\ref{6.8}). The tangent vector of the right-hand side of (\ref{5.5}) is $\big(h_{n_j}(y)\big)^{-i}\in \Hom_{\text{cont}}(W,V/W)$, which gives rise to a tangent vector $S(t)\cdot \big(h_{n_j}(y)\big)^{-i}\cdot S(t)^{-1}$ at the base point $k[z^{-1}]^{\dsum n}$ of the big-cell by the diagram $$ \begin{CD} k[z^{-1}]^{\dsum n} @>>> V @>{S\cdot h\cdot S^{-1}}>> V @>>> V\big/k[z^{-1}]^{\dsum n}\\ @V{S^{-1}}VV @V{S^{-1}}VV @VV{S^{-1}}V @VV{S^{-1}}V\\ W @>>> V @>>{h}> V @>>> V/W, \end{CD} $$ where we denote $W = W(t)$, $S = S(t)$ and $h = \big(h_{n_j}(y)\big)^{-i}$. Since the base point is preserved by the differential operators, the equation of the tangent vectors reduces to an equation \begin{equation} \frac{\partial S(t)}{\partial t_{ij}}\cdot S(t)^{-1} = -\bigg(S(t)\cdot \big(h_{n_j}(y)\big)^{-i}\cdot S(t)^{-1}\bigg)^- \label{6.10} \end{equation} in the Lie algebra $gl\big(n,E^{(-1)}\big)$ level, where $(\bullet)^-$ denotes the negative order part of the operator by (\ref{6.3}). We call this equation the \emph{Heisenberg KP system}. Note that the above equation is trivial for negative $i$ because of (\ref{6.9}). In terms of the operator $$ P(t) = S(t)\cdot y^{-1}\cdot I_n\cdot S(t)^{-1}\in gl(n,E) $$ whose leading term is $I_n\cdot \partial^r$, the equation (\ref{6.10}) becomes a more familiar \emph{Lax equation} $$ \frac{\partial P(t)}{\partial t_{ij}} = \left[\bigg(S(t)\cdot \big(h_{n_j}(y)\big)^{-i}\cdot S(t)^{-1}\bigg)^+, \;P(t)\right]\;. $$ In particular, the Heisenberg KP system describes infinitesimal isospectral deformations of the operator $P = P(0)$. Note that if one chooses $y=z=\partial^{-1}$ in (\ref{6.9}), then the above Lax equation for the case of $n=1$ becomes the original KP system. We can solve the initial value problem of the Heisenberg KP system (\ref{6.10}) by the \emph{generalized Birkhoff decomposition} of \cite{M2}: \begin{equation} {\text{\rm{exp}}}\left(\sum_{j=1}^\ell\sum_{i=1}^\infty t_{ij} \big(h_{n_j}(y)\big)^{-i}\right)\cdot S(0)^{-1} = S(t)^{-1}\cdot Y(t)\;,\label{6.11} \end{equation} where $Y(t)$ is an invertible differential operator of infinite order defined in \cite{M2}. In order to see that the $S(t)$ of (\ref{6.11}) gives a solution of (\ref{6.10}), we differentiate the equation (\ref{6.11}) with respect to $t_{ij}$. Then we have $$ S(t)\cdot \big(h_{n_j}(y)\big)^{-i}\cdot S(t)^{-1} = -\;\frac{\partial S(t)}{\partial t_{ij}}\cdot S(t)^{-1} + \frac{\partial Y(t)}{\partial t_{ij}}\cdot Y(t)^{-1}\;, $$ whose negative order terms are nothing but the Heisenberg KP system (\ref{6.10}). It shows that the Heisenberg KP system is a completely integrable system of nonlinear partial differential equations. Now, consider a set of geometric data $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big)\longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle $$ such that $H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = 0$. Then by the cohomology functor of Theorem~\ref{3.17. Theorem}, it gives rise to a triple $(A_0, A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, W)$ satisfying that $W\in Gr^+(0)$. By Theorem~\ref{6.5. Theorem}, there is a monic zero-th order pseudodifferential operator $S$ such that $W = S^{-1}\cdot k[z^{-1}]^{\dsum n}$. Using the identification (\ref{6.9}) of the variable $y$ as the pseudodifferential operator with constant coefficients, we can define two commutative subalgebras of $gl(n,E)$ by \begin{equation} \begin{cases} B_0 = S\cdot A_0\cdot S^{-1} \\ B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = S\cdot A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\cdot S^{-1}\;. \end{cases}\label{6.12} \end{equation} The inclusion relation $A_0\subset k((y))$ gives us $B_0\subset k((P^{-1}))$, where $P =$ $S\cdot y^{-1}\cdot I_n\cdot S^{-1}$ $\in$ $gl(n,E)$. Since $A_0$ and $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ stabilize $W$, we know that $B_0$ and $B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ stabilize $k[z^{-1}]^{\dsum n}$. Therefore, these algebras are commutative algebras of ordinary differential operators with matrix coefficients! \medskip \begin{Def} \label{6.13. Definition} We denote by $\mathcal{C}^+({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},0,r)$ the set of objects $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big)\longrarrow \big({C_0}, p, \pi, f_*\mathcal{F}, \phi\big)\right\rangle $$ of the category $\mathcal{C}({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ of index 0 and rank $r$ such that $$ H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \mathcal{F}) = H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \mathcal{F}) = 0\;. $$ The set of pairs $(B_0,B)$ of commutative algebras satisfying the following conditions is denoted by $\mathcal{D}(n,r)$: \begin{enumerate} \item $k\subset B_0\subset B\subset gl(n,D)$. \item $B_0$ and $B$ are commutative $k$-algebras. \item There is an operator $P\in gl(n,E)$ whose leading term is $I_n\cdot \partial^r$ such that $B_0\subset k((P^{-1}))$. \item The projection map $B_0\longrarrow k((P^{-1}))\big/k[[P^{-1}]]$ is Fredholm. \item $B$ has rank $n$ as a torsion-free module over $B_0$. \end{enumerate} \end{Def} \medskip\noindent Using this definition, we can summarize \medskip \begin{prop} \label{6.14. Proposition} The construction $(\ref{6.12})$ gives a canonical map $$ \chi_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},r}: \mathcal{C}^+({\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},0,r) \longrarrow \mathcal{D}(n,r) $$ for every $r$ and a positive integral vector ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}} = (n_1,\cdots, n_\ell)$ with $n = n_1+\cdots + n_\ell$. \end{prop} \medskip \noindent If the field $k$ is of characteristic zero, then we can construct maximal commutative algebras of ordinary differential operators with coefficients in matrix valued functions as an application of the above proposition. \medskip \begin{thm} \label{6.15. Theorem} Every set $$ \left\langle f:\big( C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}, \Delta, \Pi, \mathcal{F}, \Phi\big)\longrarrow \big({C_0}, p, id, f_*\mathcal{F}, \phi\big)\right\rangle $$ of geometric data with a smooth curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$, $\pi = id$ and a line bundle $\mathcal{F}$ satisfying that $H^0(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = H^1(C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},\mathcal{F}) = 0$ gives rise to a maximal commutative subalgebra $B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\subset gl(n,D)$ by $\chi_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},1}$. \end{thm} \medskip \begin{proof} Let $(B_0,B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}})$ be the image of $\chi_{{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},1}$ applied to the above object, and $(A_0,A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}},W)$ the stabilizer data corresponding to the geometric data. Recall that $B_0 = S\cdot A_0\cdot S^{-1}$, where $S$ is the operator corresponding to $W$. Since $r = 1$ in our case, (\ref{4.8}) implies the existence of an element $a\in A_0$ of the form $$ a = a(z^{-1}) = z^{-m} + c_2 z^{-m+2} + c_3 z^{-m+3} + \cdots \in A_0\subset k((z))\;. $$ We call a pseudodifferential operator $a(\partial)\cdot I_n \in gl(n,E)$ a \emph{normalized} scalar diagonal operator of order $m$ with constant coefficients. Here, we need \end{proof} \medskip \begin{lem} \label{6.16. Lemma} Let $K\in gl(n,E)$ be a normalized scalar diagonal operator of order $m>0$ with constant coefficients and $Q = (Q_{ij})$ an arbitrary element of $gl(n,E)$. If $Q$ and $K$ commute, then every coefficient of $Q$ is a constant matrix. \end{lem} \medskip \begin{proof} Let $K = a(\partial)\cdot I_n$ for some $a(\partial)\in k((\partial^{-1}))$. It is well known that there is a monic zero-th order pseudodifferential operator $S_0\in E$ such that $$ S_0^{-1}\cdot a(\partial)\cdot S_0 = \partial^m\;. $$ Since $a(\partial)$ is a constant coefficient operator, we can show that (see \cite{M3}) $$ S_0^{-1}\cdot k((\partial^{-1})) \cdot S_0 = k((\partial^{-1}))\;. $$ Going back to the matrix case, we have $$ 0 = (S_0\cdot I_n)^{-1}\cdot [Q,K]\cdot (S_0\cdot I_n) = \big[(S_0\cdot I_n)^{-1}\cdot Q \cdot (S_0\cdot I_n),\;\partial^m \cdot I_n\big]\;. $$ In characteristic zero, commutativity with $\partial^m$ implies commutativity with $\partial$. Thus each matrix component $S_0^{-1}\cdot Q_{ij}\cdot S_0$ commutes with $\partial$, and hence $S_0^{-1}\cdot Q_{ij}\cdot S_0$ $\in$ $k((\partial^{-1}))$. Therefore, $Q_{ij}\in k((\partial^{-1}))$. This completes the proof of lemma. \end{proof} \medskip\noindent Now, let $B \supset B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ be a commutative subalgebra of $gl(n,D)$ containing $B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Since $B_0 = S\cdot A_0\cdot S^{-1}$ and $B_0\subset B$, every element of $B$ commutes with $S\cdot a(\partial)\cdot I_n\cdot S^{-1}$. Then by the lemma, we have $$ A = S^{-1}\cdot B\cdot S\subset gl\big(n,k((\partial^{-1}))\big)\;. $$ Note that the algebra $A$ stabilizes $W = S^{-1}\cdot k[z^{-1}]^{\dsum n}$. Since $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(z)$ can be generated by $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ over $k((z)) = k((\partial^{-1}))$, every element of $A$ commutes with $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(z)$. Therefore, we have $A\subset H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(z)$ because of the maximality of $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(z)$. Thus we obtain another triple $(A_0,A,W)$ of stabilizer data of the same type ${\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. The inclusion $A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}\longrarrow A$ gives rise to a birational morphism $\beta:C\longrarrow C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Since we are assuming that the curve $C_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$ is nonsingular, $\beta$ has to be an isomorphism, which then implies that $A = A_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. Therefore, we have $B = B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. This completes the proof of maximality of $B_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}$. \medskip \begin{rem} \label{{6.17.} Remark} There are other maximal commutative subalgebras in $gl(n,D)$ than what we have constructed in Theorem~\ref{6.15. Theorem}. It corresponds to the fact that the algebras $H_{\mathbf{n}}} \def\w{{\mathbf{w}}} \def\e{{\mathbf{e}}} %mathematics operators \def\isom{\cong} \def\tensor{\otimes} \def\dsum{\oplus} \def\Hom{{\text{\rm{Hom}}}(z)$ are not the only maximal commutative subalgebras of the formal loop algebra $gl\big(n, k((z))\big)$. \end{rem} \bigskip \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}

9509

alg-geom/9509004

en

https://arxiv.org/abs/alg-geom/9509004

alg-geom/9509004

Rahul Pandharipande

R. Pandharipande

The Canonical Class of $\overline{M}_{0,n}(P^r,d)$ And Enumerative Geometry

AMSLatex, 13 pages

D. Abramovich found an error in the singularity analysis in the first posting of this paper affecting one formula (Thanks Dan).The error has been corrected. Only the arithmetic genus formula has changed. The geometric genus and all the canonical class formulas are the same.

alg-geom

\section{Summary} Let $\Bbb{C}$ be the field of complex numbers. Let the Severi variety $$S(0,d)\subset \bold P\big(H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))\big)$$ be the quasi-projective locus of irreducible, nodal rational curves. Let $\overline{S}(0,d)$ denote the closure of $S(0,d)$. Let $p_1, \ldots, p_{3d-2}$ be $3d-2$ general points in $\bold P^2$. Consider the subvariety $C_d\subset \overline{S}(0,d)$ corresponding to curves passing through all the points $p_1, \ldots, p_{3d-2}$. $C_d$ is a complete curve in $\bold P \big(H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))\big)$. Let $N_d$ be the degree of $C_d$. $N_d$ is determined by the recursive relation ([K-M], [R-T]): $$N_1=1$$ $$\forall d>1, \ \ \ N_d= \sum_{i+j=d, \ i,j>0} N_iN_j \bigg( i^2j^2 {3d-4 \choose 3i-2} - i^3j {3d-4 \choose 3i-1} \bigg).$$ For $d\geq 3$, $C_d$ is singular. The arithmetic genus $g_d$ of $C_d$ is determined by: $$g_1=0,$$ $$g_2=0,$$ $$2g_d-2= {6d^2+5d-15\over 2d}N_d + {1\over 4d}\sum_{i=1}^{d-1} N_i N_{d-i}\Big(15i^2(d-i)^2 -8di(d-i)-4d\Big) {3d-2 \choose 3i-1}.$$ The last formula holds for $d\geq 3$. The geometric genus $\tilde{g}_{d}$ of the normalization $\tilde{C}_d$ is determined by ($d\geq 1$): $$2\tilde{g}_{d}-2 = -{3d^2-3d+4\over 2d^2} N_d + {1\over 4d^2} \sum_{i=1}^{d-1} N_i N_{d-i} (id-i^2) \Big( (9d+4)i(d-i)-6d^2\Big) {3d-2\choose 3i-1}.$$ These genus formulas are established by adjunction and intersection on Kontsevich's space of stable maps $\barr{M}_{0,n}(\proj^r,d)$. The author thanks D. Abramovich for many useful remarks and for finding an error in the singularity analysis in an earlier version of this paper. \section {The Canonical Class of $\barr{M}_{0,n}(\proj^r,d)$} Let $\barr{M}_{0,n}(\proj^r,d)$ be the coarse moduli space of degree $d$, Kontsevich stable maps from $n$-pointed, genus $0$ curves to $\bold P^r$. Foundational treatments of $\barr{M}_{0,n}(\proj^r,d)$ can be found in [Al], [P1], [K], and [K-M]. Only the case $r\geq 2$ will be considered here. Let $\cal{L}_p$ denote the line bundle obtained on $\barr{M}_{0,n}(\proj^r,d)$ by the $p^{th}$ evaluation map $(1\leq p \leq n)$. Let $\bigtriangleup$ be the set of boundary divisors. Let $\cal{H}$ denote the divisor of maps meeting a fixed codimension 2 linear space of $\bold P^r$. $\cal{H}=0$ if $d=0$. In [P2], it is shown the classes $$\{\cal{L}_p\} \cup \bigtriangleup \cup \{\cal{H}\}$$ span $Pic(\barr{M}_{0,n}(\proj^r,d)) \otimes \Bbb{Q}$. The canonical class of the stack $\overline{\cal{M}}_{0,n}(\bold P^r,d)$ has the following coarse moduli interpretation. $\barr{M}_{0,n}(\proj^r,d)$ is an irreducible variety with finite quotient singularities. When $r\geq 2$, the automorphism-free locus $\overline{M}^*_{0,n}(\bold P^r,d) \subset \barr{M}_{0,n}(\proj^r,d)$ is a nonsingular, fine moduli space with codimension 2 complement except when ([P2]) $$[0,n,r,d]=[0,0,2,2].$$ For $r\geq 2$ and $[0,n,r,d]\neq [0,0,2,2]$, the first Chern class of the cotangent bundle to the moduli space $\overline{M}^*_{0,n}(\bold P^r,d)$ yields the canonical class in $Pic(\barr{M}_{0,n}(\proj^r,d)) \otimes \Bbb{Q}$. Let $P=\{1,2,\ldots, n\}$ be the set of markings ($P$ may be the empty set). The boundary components are in bijective correspondence with data of weighted partitions $(A\cup B, d_A, d_B)$ where: \begin{enumerate} \item[(i.)] $A\cup B$ is a partition of $P$. \item[(ii.)] $d_A+d_B=d$, $d_A \geq 0$, $d_B \geq 0$. \item[(iii.)] If $d_A=0$ (resp. $d_B=0$), then $|A|\geq 2$ (resp. $|B| \geq 2$). \end{enumerate} Define $\cal{D}_{i,j}$ to be the reduced sum of boundary components with $d_A=i$ and $|A|=j$. Note $0\leq i \leq d$ and $0\leq j \leq n$. The divisors $\cal{D}_{0,0}$, $\cal{D}_{0,1}$, $\cal{D}_{d,n-1}$, $\cal{D}_{d,n}$ are equal to 0 by stability. Also, $\cal{D}_{i,j}=\cal{D}_{d-i,n-j}$. Consider first the case $d=0$. $\overline{M}_{0,n}(\bold P^r,0) \stackrel{\sim}{=} \overline{M}_{0,n} \times \bold P^r$. It suffices to determine the canonical class of $\overline{M}_{0,n}$. \begin{pr} The canonical class $\cal{K}_{\barr{M}}$ of $\overline{M}_{0,n}$ is determined in $Pic(\overline{M}_{0,n}) \otimes \Bbb{Q}$ by: \begin{equation} \label{mumkn} \cal{K}_{\barr{M}} = \sum_{j=2}^{[{n\over 2}]} \Big( {j(n-j) \over n-1} -2 \Big) \cal{D}_{0,j}. \end{equation} \end{pr} \noindent The canonical class has a different form in case $d>0$, $n=0$, $r\geq 2$. \begin{pr} The canonical class $\cal{K}_{\barr{M}}$ of $\overline{M}_{0,0}(\bold P^r, d)$ ($d>0, r\geq 2$) is determined in $Pic(\overline{M}_{0,0}(\bold P^r,d) \otimes \Bbb{Q}$ by: \begin{equation} \label{genny} \cal{K}_{\barr{M}}= -{(d+1)(r+1)\over 2d} \cal{H} + \sum_{i=1}^{[{d\over 2}]} \Big( {(r+1)(d-i)i\over 2d}-2 \Big) \cal{D}_{i,0}. \end{equation} \end{pr} \noindent Finally, when $d>0$, $n>0$, $r\geq 2$, the form of the canonical class is the following: \begin{pr} The canonical class of $\cal{K}_{\barr{M}}$ of $\barr{M}_{0,n}(\proj^r,d)$ ($d>0$, $n>0$, $r\geq 2$) is determined in $Pic(\barr{M}_{0,n}(\proj^r,d)) \otimes \Bbb{Q}$ by: \begin{equation} \label{fini} \cal{K}_{\barr{M}}= - {(d+1)(r+1)d-2n \over 2d^2} \cal{H} - \sum_{p=1}^{n} {2\over d}\cal{L}_p \end{equation} $$+\sum_{i=0}^{[{d\over 2}]} \sum_{j=0}^{n} \Big({(r+1)(d-i)di + 2d^2j-4dij+2ni^2 \over 2d^2}-2 \Big) \cal{D}_{i,j}.$$ \end{pr} \noindent Equation (\ref{mumkn}) can be derived from the explicit construction of $\overline{M}_{0,n}$ described in [F-M]. Equations (\ref{mumkn}-\ref{fini}) will be established here via intersections with curves. \section{Computing The Canonical Class} \subsection{Curves in $\barr{M}_{0,n}(\proj^r,d)$} By Proposition (2) of [P2], the canonical projection $$Pic(\barr{M}_{0,n}(\proj^r,d))\otimes \Bbb{Q} \rightarrow Num(\barr{M}_{0,n}(\proj^r,d)) \otimes \Bbb{Q}$$ is an isomorphism. Hence, the canonical class of $\barr{M}_{0,n}(\proj^r,d)$ can be established via intersections with curves. Curves can easily be found in $\barr{M}_{0,n}(\proj^r,d)$. The notation of [P2] is recalled here. Let $C$ be a nonsingular, projective curve. Let $\pi: S=\bold P^1 \times C \rightarrow C$. Select $n$ sections $s_1, \ldots, s_n$ of $\pi$. A point $x\in S$ is an {\em intersection point} if two or more sections contain $x$. Let $\cal{N}$ be a line bundle on $S$ of type $(d,k)$. Let $z_l\in H^0(S, \cal{N})$ $(0\leq l \leq r)$ determine a rational map $\mu: S - \rightarrow \bold P^r$ with simple base points. A point $y\in S$ is a {\em simple base point} of degree $1\leq e\leq d$ if the blow-up of $S$ at $y$ resolves $\mu$ locally at $y$ and the resulting map is of degree $e$ on the exceptional divisor $E_y$. The set of {\em special points} of $S$ is the union of the intersection points and the simple base points. Three conditions are required: \begin{enumerate} \item [(1.)] There is at most one special point in each fiber of $\pi$. \item [(2.)] The sections through each intersection point $x$ have distinct tangent directions at $x$. \item [(3.)] \begin{enumerate} \item[(i.)] $d=0$. No $n-1$ sections pass through a point $x\in S$. \item[(ii.)] $d>0$. If at least $n-1$ sections pass through a point $x\in S$, then $x$ is not a simple base point of degree $d$. \end{enumerate} \end{enumerate} Condition (3.ii) implies there are no simple base points of degree $d$ if $n=0$ or $1$. Let $\overline{S}$ be the blow-up of $S$ at the special points. It is easily seen $\overline{\mu} : \overline{S} \rightarrow \bold P^r$ is Kontsevich stable family of $n$-pointed, genus $0$ curves over $C$. Condition (2) ensures the strict transforms of the sections are disjoint. Condition (3) implies Kontsevich stability. There is a canonical morphism $\lambda:C \rightarrow \barr{M}_{0,n}(\proj^r,d)$. Condition (1) implies $C$ intersects the boundary components transversally. \subsection{$\overline{M}_{0,n}$} Curves in $\overline{M}_{0,n}$ are obtained by the above construction (omitting the map $\mu$). Let $s_1, \ldots, s_n$ be n sections of $\pi:S=\bold P^1 \times C \rightarrow C$ satisfying $(1), (2), (3.i)$. For $1\leq \alpha \leq n$, let $s_\alpha$ be of type $(1,\sigma_\alpha)$ on $S=P^1\times C$. Let $\overline{\pi}:\overline{S} \rightarrow C$ be the blow-up of $S$ as above. Let $s_1, \ldots, s_n$ also denote the transformed sections of $\overline{\pi}$. Let $Q$ denote the points of $C$ lying under the special points of $S$. There is a canonical sequence $$0 \rightarrow R^1\overline{\pi}_*(\omega^*_{\overline{\pi}}(-\sum_{1}^{n} s_\alpha)) \rightarrow \lambda^*(T_{\overline{M}_{0,n}}) \rightarrow \bigoplus_{q\in Q} \Bbb{C}_q \rightarrow 0.$$ (See, for example, [K].) Hence $C\cdot \cal{K}_{\barr{M}}=- deg \big( R^1\overline{\pi}_*(\omega^*_{\overline{\pi}}(-\sum_{1}^{n} s_\alpha)) \big) - C\cdot \sum_{j=2}^{[{n\over 2}]} \cal{D}_{0,j}$. The degree of $R^1\overline{\pi}_*(\omega^*_{\overline{\pi}}(-\sum_{1}^{n} s_\alpha))$ is determined by the Grothendieck-Riemann-Roch formula. Let $x_j$ for $2 \leq j \leq n-2$ be the number of intersection points of $S$ which lie on exactly $j$ sections. If $j\neq n/2$, $C\cdot \cal{D}_{0,j}= x_j+ x_{n-j}$. If $j=n/2$, $C\cdot \cal{D}_{0,j} =x_j$. G-R-R yields: $$ deg \big(R^1\overline{\pi}_* (\omega^*_{\overline{\pi}}(-\sum_{1}^{n} s_\alpha)) \big) =\sum_{1}^{n}2 \sigma_{\alpha} + \sum_{2}^{n-2} (1-j)x_j.$$ By the transverse intersection condition, the following relation must hold: $$\sum_{1}^{n} \sigma_{\alpha} = {1 \over n-1} \sum_{2}^{n-2} {j^2-j\over 2}x_j.$$ Combining equations yields: $$C\cdot \cal{K}_{\barr{M}} = \sum_{2}^{n-2} \Big( j-2 - {j^2-j\over n-1} \big) x_j$$ $$ = \sum_{2}^{n-2} \big({j(n-j)\over n-1} -2 \Big) x_j.$$ Hence both sides of equation (\ref{mumkn}) have the same intersection numbers with $C$. Let $D$ be any nonsingular curve in $\overline{M}_{0,n}$ which intersects the boundary transversely. The universal family over $D$ can be blown-down to a projective bundle $\pi: T \rightarrow D$. The above calculation covers the case where $T=\bold P^1\times D$. The general case (in which T is {\em any} $\bold P^1$-bundle) is identical. Since $A^1(\overline{M}_{0,n})$ is spanned by curves meeting the boundary transversely, Proposition (\ref{mumkn}) is immediate. \subsection{$\overline{M}_{0,0}(\bold P^r,d)$} The case $d>0$, $n=0$, $r\geq 2$ is now considered. Let $\overline{\pi}:\overline{S} \rightarrow C$, $\overline{\mu}:\overline{S} \rightarrow \bold P$ be a family of stable maps as above. There is canonical exact sequence $$0 \rightarrow \overline{\pi}_*(\omega^*_{\overline{\pi}}) \rightarrow \overline{\pi}_*\overline{\mu}^*(T_{\bold P^r}) \rightarrow \lambda^*(T_{\overline{M}_{0,0} (\bold P^r,d)}) \rightarrow \bigoplus_{p\in Q} \Bbb{C}_p \rightarrow 0.$$ Hence $C\cdot \cal{K}_{\barr{M}}= +deg \big( \overline{\pi}_*(\omega^*_{\overline{\pi}}) \big) -deg \big(\overline{\pi}_*\overline{\mu}^*(T_{\bold P^r}))\big) -\sum_{1}^{[{d\over 2}]} \cal{D}_{i,0}$. Let $x_i$ for $1\leq i \leq d-1$ be the number of simple base points of $\mu: S - \rightarrow \bold P^r$ of degree exactly $i$. If $i\neq d/2$, $C\cdot \cal{D}_{i,0}= x_i + x_{d-i}$. If $i=d/2$, $C \cdot \cal{D}_{i.0}=x_i$. Via G-R-R, $$deg\big( \overline{\pi}_*(\omega^*_{\overline{\pi}}) \big) = -\sum_{1}^{d-1} x_i,$$ $$deg\big( \overline{\pi}_*(\overline{\mu}^*(T_{\bold P^r})) \big)= (r+1)(d+1)k - \sum_{1}^{d-1} {(r+1)(i^2+i)\over 2} x_i.$$ Combining equations yields: $$C\cdot \cal{K}_{\barr{M}} = -(r+1)(d+1)k + \sum_{1}^{d-1} \Big( {(r+1)(i^2+i)\over 2}-2 \Big) x_i.$$ Finally $C\cdot \cal{H}$ must be computed: $$C \cdot \cal{H} = 2dk- \sum_{1}^{d-1} i^2 x_i.$$ These equations (plus algebra) verify Proposition (2). As before, the complete proof requires the above calculation for {\em any} $\bold P^1$-bundle $\pi: S \rightarrow C$. Again, the generalization to this case is trivial. \subsection{$\barr{M}_{0,n}(\proj^r,d)$} Finally, the case $d>0$, $n>0$, $r\geq 2$ is considered. Let $\overline{\pi}:\overline{S} \rightarrow C$, $\overline{\mu}:\overline{S} \rightarrow \bold P$ be a family of stable maps as above. Let $s_1, \ldots, s_n$ be n sections of $\pi:S=\bold P^1 \times C \rightarrow C$ satisfying $(1), (2), (3.ii)$. For $1\leq \alpha \leq n$, let $s_\alpha$ be of type $(1,\sigma_\alpha)$ on $S=P^1\times C$. There is a canonical exact sequence $$0 \rightarrow \overline{\pi}_*(\omega^*_{\overline{\pi}}) \rightarrow \overline{\pi}_*(\omega^*_{\overline{\pi}}|_{\sum s_{\alpha}}) \bigoplus \overline{\pi}_*\overline{\mu}^*(T_{\bold P^r}) \rightarrow \lambda^*(T_{\overline{M}_{0,0} (\bold P^r,d)}) \rightarrow \bigoplus_{p\in Q} \Bbb{C}_p \rightarrow 0.$$ Hence $C\cdot \cal{K}_{\barr{M}}= +deg \big( \overline{\pi}_*(\omega^*_{\overline{\pi}}) \big) -(\omega^*_{\overline{\pi}} \cdot \sum_{1}^{n} s_{\alpha}) -deg \big(\overline{\pi}_*\overline{\mu}^*(T_{\bold P^r}))\big) -\sum_{i=1}^{[{d\over 2}]} \sum_{j=0}^{n} \cal{D}_{i,j}$. Let $z_{i,j}$ for $0\leq i \leq d$ and $0\leq j \leq n$ be the number of special points of $S$ that are simple base points of degree exactly $i$ and that lie on exactly $j$ sections. If $i\neq d/2$ or $j\neq n/2$, then $C \cdot \cal{D}_{i,j}= z_{i,j}+ z_{d-i, n-j}$. If $i=d/2$ and $j=n/2$, then $C\cdot \cal{D}_{i,j}=z_{i,j}$. Via G-R-R, $$deg\big( \overline{\pi}_*(\omega^*_{\overline{\pi}}) \big) = -\sum_{i=0}^{d}\sum_{j=0}^{n} z_{i,j},$$ $$deg\big( \overline{\pi}_*(\overline{\mu}^*(T_{\bold P^r})) \big)= (r+1)(d+1)k - \sum_{i=0}^{d}\sum_{j=0}^{n} {(r+1)(i^2+i)\over 2} z_{i,j}.$$ A simple calculation yields: $$\omega^*_{\overline{\pi}} \cdot \sum_{1}^{n} s_{\alpha}= \sum_{1}^{n} 2\sigma_\alpha - \sum_{i=0}^{d} \sum_{j=0}^{n} j z_{i,j}.$$ The two additional intersection numbers are: $$C \cdot \cal{H}= 2dk - \sum_{i=0}^{n} \sum_{j=0}^{n} i^2 z_{i,j},$$ $$C \cdot \sum_{1}^{n} \cal{L}_p = nk+ \sum_{1}^{n}d \sigma_{\alpha} - \sum_{i=0}^{n} \sum_{j=0}^{n} ij z_{i,j}.$$ Now algebra yields the desired equality of intersections that establishes Proposition (3). Again the calculation must be done in case $\pi: S \rightarrow C$ is a $\bold P^1$ bundle. \section{The genus of $C_d$, $\tilde{C}_d$} \subsection{Singularities} Let $C_d\subset \overline{S}(0,d)$ be the dimension $1$ subvariety corresponding to curves passing through $3d-2$ general points $p_1,\ldots, p_{3d-2}$ in $\bold P^2$. Let $\hat{C}_d\subset \overline{M}_{0,0}(\bold P^2, d)$ be the dimension $1$ subvariety corresponding to maps passing through $p_1, \ldots, p_{3d-2}$. The singularities of $C_d$, $\hat{C}_d$ will be analyzed. Let $[\mu]\in \overline{M}_{0,0}(\bold P^2,d)$ correspond to an automorphism-free map with domain $\bold P^1$. There is a normal sequence on $\bold P^1$: $$0 \rightarrow T_{\bold P^1} \stackrel {d\mu}{\rightarrow} \mu^*(T_{\bold P^2}) \rightarrow Norm \rightarrow 0.$$ The tangent space to $\overline{M}_{0,0}(\bold P^2,d)$ is $H^0(\bold P^1, Norm)$. If $\mu$ is an immersion, $Norm \stackrel{\sim}{=} {\cal{O}}_{\bold P^1}(3d-2)$. If $\mu$ is not an immersion $Norm$ will have torsion. A {\em $1$-cuspidal} rational plane curve is an irreducible rational plane curve with nodal singularities except for exactly 1 cusp. If $\mu$ corresponds to a $1$-cuspidal rational curve, then there is a sequence: $$0 \rightarrow \Bbb{C}_p \rightarrow Norm \rightarrow {\cal{O}}_{\bold P^1}(3d-3) \rightarrow 0$$ where $p$ is the point of $\bold P^1$ lying over the cusp. Since $3d-2$ distinct points of $\bold P^1$ always impose independent conditions on $H^0(\bold P^1, {\cal{O}}_{\bold P^1}(3d-2))$ and $H^0(\bold P^1, {\cal{O}}_{\bold P^1}(3d-3))$, Lemma (\ref{t}) has been established: \begin{lm} \label{t} Let $[\mu]\in \hat{C}_d$ be a point corresponding to an irreducible, nodal or $1$-cuspidal rational curve with all singularities distinct from $p_1, \ldots, p_{3d-2}$. $\hat{C}_d$ is nonsingular at $[\mu]$. \end{lm} \noindent The corresponding analysis for $C_d$ is more involved. \begin{lm} \label{tt} Let $x\in C_d$ be a point corresponding to an irreducible, nodal rational curve with nodes distinct from $p_1, \ldots, p_{3d-2}$. $C_d$ is nonsingular at $x$. \end{lm} \begin{pf} Let $X\subset \bold P^2$ be the plane curve corresponding to $x\in C_d$. $S(0,d)$ is nonsingular at $x$ with tangent space $H^0(\tilde{X}, {\cal{O}}_{\bold P^2}(d)-N)$ where $N$ is the divisor of points of $\tilde{X}$ lying over the nodes of $X$. The additional points correspond to $3d-2$ {\em distinct} points of $\tilde{X}$. Since $3d-2$ distinct point on $\bold P^1$ impose $3d-2$ independent linear conditions on sections of ${\cal{O}}_{\bold P^2}(d)-N\stackrel{\sim}{=} {\cal{O}}_{\bold P^1}(3d-2)$, it follows $C_d$ is nonsingular at $x$. \end{pf} Actually, $\overline{M}_{0,0}(\bold P^2,d)$ and $S(0,d)$ are isomorphic on the irreducible, nodal locus. Hence Lemma (\ref{tt}) is unnecessary. \begin{lm} Let $x\in C_d$ be a point corresponding to a $1$-cuspidal rational plane curve with all singularities distinct from $p_1, \ldots, p_{3d-2}$. $C_d$ is cuspidal at $x$. \end{lm} \begin{pf} The versal deformation space of the cusp $Z_0^2+Z_1^3$ is 2 dimensional: $$Z_0^2+Z_1^3+ aZ_1 + b.$$ The locus in the versal deformation space corresponding to equigeneric deformations is determined by the cuspidal curve $4a^3+27b^2=0$. Let $X$ be the plane curve corresponding to $x$. Let $q\in X$ be the cusp. Let $\tilde{X}$ the normalization of $X$. Let $p\in \tilde{X}$ lie over $q$. The nodes of $X$, the points $p_1, \ldots, p_{3d-2}$, and the $2$ dimensional subscheme supported on $q$ and pointing in the direction of the tangent cone of $X$ all together impose independent conditions on the linear system of degree $d$ plane curves. First, this independence will be established. Let $A$ be the subspace of $H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))$ passing through the nodes, points, and the subscheme of length 2. As before, let $N$ denote the divisor of $\tilde{X}$ lying above the nodes. There is a natural left exact sequence obtained by pulling back sections to $\tilde{X}$: $$0 \rightarrow \Bbb{C} \rightarrow A \rightarrow H^0( \tilde{X}, {\cal{O}}_{\bold P^2}(d)-N-p_1-\ldots -p_{3d-2}-3p).$$ By counting conditions, $$dim(A) \geq {(d+1)(d+2)\over 2} - {(d-1)(d-2)\over 2}+1- 3d+2 -2 = 1$$ with equality if only if the conditions are independent. Since $$deg_{\tilde{X}} ({\cal{O}}_{\bold P^2}(d)-N-p_1-\ldots -p_{3d-2}-3p)= d^2- (d-1)(d-2)+2-3d+2-3 = -1,$$ $dim(A)=1$ and the conditions are independent. By the independence result above, the deformations of $X$ parameterized by the linear system of plane curves through the nodes and the points $p_1, \ldots, p_{3d-2}$ surjects on the 2 dimensional versal deformation space of the cusp. The locus of equigeneric deformations of $X$ through the points $p_1, \ldots, p_{3d-2}$ is \'etale locally equivalent to the cusp in the versal deformation space of the cusp. \end{pf} \begin{lm} Let $[\mu]\in \hat{C}_d$ (resp. $x\in C_d$) be a point corresponding to an irreducible, nodal, rational curve with a node at $p_1$ and nodes distinct from $p_2, \ldots, p_{3d-2}$. $\hat{C}_d$ is nodal at $[\mu]$ (resp. $C_d$ is nodal at $x$). \end{lm} \begin{pf} If suffices to prove the result for $\hat{C}_d$. The divisor $D_1 \subset \overline{M}_{0,0}(\bold P^2,d)$ corresponding to curves passing through the point $p_1$ has two nonsingular branches with a normal crossings intersection at $[\mu]$. Let $r,s\in \bold P^1$ lie over $p_1\in \bold P^2$. The two nonsingular branches have the following tangent spaces at $X$: $$H^0(\bold P^1, Norm(-r), \ \ H^0(\bold P^1, Norm(-s)).$$ The remaining $3d-3$ points impose independent conditions on each of these tangent spaces. Etale locally at $[\mu]$, $\hat{C}_d$ is the intersection of the union of linear spaces of dimensions $3d-2$ meeting along a subspace of dimension $3d-3$ with general linear space of codimension $3d-3$. Hence, $\hat{C}_d$ is nodal at $[\mu]$. \end{pf} \begin{lm} \label{ttt} Let $x\in C_d$ be a point corresponding to the union of two irreducible, nodal, rational curves with degrees $i$ and $d-i$ meeting transversely with nodes (including component intersection points) distinct from $p_1, \ldots, p_{3d-2}$. Also assume the components of degrees $i$, $d-i$ contain $3i-1$, $3(d-i)-1$ points respectively. $C_d$ has the singularity type of the coordinate axes at the origin in $\Bbb{C}^{id-i^2}$. \end{lm} \begin{pf} The nodes (including the intersections of the two components of $X$) and the points $p_1, \ldots, p_{3d-2}$ necessarily impose $ (d+1)(d+2)/2$ independent conditions on $H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))$. This independence can be established as follows. Let $\tilde{X}$ be the normalization of $X$ (note $\tilde{X}$ is the disjoint union of two $\bold P^1$'s). Let $A\subset H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))$ be the linear series passing through all the nodes of $X$. There is an exact sequence of vector spaces $$0 \rightarrow \Bbb{C} \rightarrow A \rightarrow H^0(\tilde{X}, {\cal{O}}_{\bold P^2}(d)-N)\rightarrow 0.$$ As before, $N$ is the divisor preimage of the nodes of $X$. Certainly only a $1$ dimensional subspace of $A$ corresponding to the equation of $X$ vanishes on $\tilde{X}$. Surjectivity of the above sequence follows by a dimension count: $$dim(A) \geq {(d+1)(d+2)\over 2} - {(d-1)(d-2)\over 2}-1 = 3d-1,$$ $$h^0(\tilde{X}, {\cal{O}}_{\bold P^2}(d)-N)= d^2-(d-1)(d-2)-2+2=3d-2.$$ The points $p_1, \ldots, p_{3d-2}$ are distinct on $\tilde{X}$ and impose independent conditions on $H^0(\tilde{X}, {\cal{O}}_{\bold P^2}(d)-N)$ by the assumption of their distribution (and the fact $\tilde{X}$ is a disjoint union of $\bold P^1$'s). At $x\in \overline{S}(0,d)$, the closed Severi variety has $id-i^2$ nonsingular branches (one for each intersection point). Let $q\in \bold P^2$ be an intersection point of the two components of $X$. The tangent space $T(q)$ to the branch of $\overline{S}(0,d)$ corresponding to $q$ is simply the linear subspace $T(q)\subset H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))$ of polynomials that vanish at all the nodes of $X$ besides $q$. Let $V\subset H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))$ be the linear subspace of polynomials that vanish at the non-intersection nodes of $X$ and the points $p_1, \ldots, p_{3d-2}$. $C_d$ at $x$ is \'etale locally equivalent to the intersection $$V \cap (T(q_1) \cup T(q_2) \cup \cdots \cup T(q_{id-i^2})).$$ Note $V\stackrel{\sim}{=} \Bbb{C}^{id-i^2}$. Since the nodes of $X$ and the points $p_1, \ldots, p_{3d-2}$ impose independent conditions on $H^0(\bold P^2, {\cal{O}}_{\bold P^2}(d))$, the Lemma is proven. \end{pf} \noindent The last case to be consider is the analogue of Lemma (\ref{ttt}) for $\hat{C}_d$: when $[\mu]\in \hat{C}_d$ corresponds to a map with reducible domain and image satisfying the conditions of (\ref{ttt}). This case can be handled directly. However, it is easier to observe that at such $[\mu]$, $\overline{M}_{0,0}(\bold P^2,d)$ is locally isomorphic to the nonsingular branch in the proof of Lemma (\ref{ttt}) determined by the attaching point of the two components. The singularity analysis then shows $[\mu]\in \hat{C}_d$ is a nonsingular point. For general points $p_1, \ldots, p_{3d-2}$, every point $x\in C_d$, $[\mu]\in \hat{C}_d$ corresponds to exactly one of the three cases covered by Lemmas (1-5). Hence the singularities of $C_d$, $\hat{C}_d$ are established. \label{singa} \subsection{The Arithmetic Genus} Consider the moduli space $\overline{M}_{0,0}(\bold P^2, d)$ for $d\geq 3$ (to avoid $[0,0,r,d]=[0,0,2,2]$). For general points $p_1, \ldots, p_{3d-2}$, the intersection cycle $$\hat{C}_d=\cal{H}_1 \cap \cal{H}_2 \cap \cdots \cap \cal{H}_{3d-2}$$ is a curve in $\overline{M}_{0,0}(\bold P^2, d)$. $\cal{H}_i$ is the divisor of maps passing through the point $p_i$. By the analysis in section (\ref{singa}), $\hat{C}_d$ is nonsingular except for nodes. The nodes occur precisely at the points $[\mu]\in \hat{C}_d$ corresponding to a nodal curve with a node at some $p_i$. Since, for general points, $\hat{C}_d \subset \overline{M}^*_{0,0}(\bold P^2, d)$, the arithmetic genus $\hat{g}_d$ of $\hat{C}_d$ can be determined by the formula for the canonical class and adjunction ($d\geq 3$): $$2\hat{g}_d-2 = \big(\cal{K}_{\barr{M}} + (3d-2) \cal{H}\big)\cdot \cal{H}^{3d-2}.$$ A computation of these intersection numbers in terms of the numbers $N_d$ yields for all $d \geq 3$: $$2\hat{g}_d-2= {(2d-3)(3d+1)\over 2d}N_d + {1\over 4d}\sum_{i=1}^{d-1} N_i N_{d-i}\Big(3i^2(d-i)^2-4di(d-i) \Big) {3d-2 \choose 3i-1}.$$ The natural map $\hat{C}_d \rightarrow C_d$ is a partial desingularization. The arithmetic genus of $C_d$ differs from the arithmetic genus of $\hat{C}_d$ only by the contribution of the singularities of Lemma (3) and (5). Consider first the cusps in $C_d$ determined by Lemma (3). The number of these cusps is exactly the number of $1$-cuspidal, degree $d$, rational curves through $3d-2$ points in the plane. In [P2] it is shown there are $${3d-3\over d}N_d + {1\over 2d} \sum _{i=1}^{d-1} N_iN_{d-i}(3i^2(d-i)^2-2di(d-i)) {3d-2\choose 3i-1}$$ $1$-cuspidal, degree $d$, rational curves through $3d-2$ points. Each cusp contributes $1$ to the arithmetic genus of $C_d$. The singularities of Lemma (5) contribute $${1\over 2} \sum _{i=1}^{d-1} N_iN_{d-i} (i(d-i)-1) {3d-2\choose 3i-1}$$ to the arithmetic genus of $C_d$. The formula for the arithmetic genus of $C_d$ can be deduced by adding these contributions to the formula for $\hat{g}$: $$2g_d-2= {6d^2+5d-15\over 2d}N_d + {1\over 4d}\sum_{i=1}^{d-1} N_i N_{d-i}\Big(15i^2(d-i)^2 -8di(d-i)-4d\Big) {3d-2 \choose 3i-1}.$$ \subsection{The Geometric Genus} The geometric genus, $g(\tilde{C}_d)$ is simple to compute. By Bertini's Theorem, the curve $\tilde{C_d}$ determined in $\overline{M}_{0,3d-2}(\bold P^2,d)$ by $3d-2$ general points is nonsingular and contained in the automorphism-free locus $\overline{M}^*_{0,3d-2}(\bold P^2,d)$. There is sequence of natural maps exhibiting $\tilde{C}_d$ as the normalization of both $\hat{C}_d$ and $C_d$: $$\tilde{C}_d \rightarrow \hat{C}_d \rightarrow C_d.$$ The genus of $\tilde{C}_d$ can be determined by the formula for the canonical class and adjunction: $$2\tilde{g}_d-2 = \big(\cal{K}_{\barr{M}}+2\sum_{1}^{3d-2} c_1(\cal{L}_p)\big) \cdot c^2_1(\cal{L}_1) \cdots c^2_1(\cal{L}_{3d-2})$$ $$= \cal{K}_{\barr{M}} \cdot c^2_1(\cal{L}_1) \cdots c^2_1(\cal{L}_{3d-2}).$$ A computation of these intersection numbers in terms of the numbers $N_d$ yields for all $d\geq 1$: $$2\tilde{g}_{d}-2 = -{3d^2-3d+4\over 2d^2} N_d + {1\over 4d^2} \sum_{i=1}^{d-1} N_i N_{d-i} (id-i^2) \Big( (9d+4)i(d-i)-6d^2\Big) {3d-2\choose 3i-1}.$$ \subsection{The difference $\hat{g}_d- \tilde{g}_d$.} Let $d\geq 3$. The natural map $\tilde{C}_d \rightarrow \hat{C}_d$ is a desingularization. $\hat{C}_d$ has only nodal singularity. The difference, $\hat{g}_d - \tilde{g}_d$, equals the number of nodes of $\hat{C}_d$. Let $M_d$ be the number of irreducible, nodal, rational degree $d$ plane curves with a node at a fixed point and passing through $3d-3$ general point in $\bold P^2$. From the description of the nodes of $\hat{C}_d$, it follows: $$\hat{g}_d- \tilde{g}_d= (3d-2) M_d.$$ Values for low degree $d$ are tabulated below: \begin{tabular}{|l|l|} \hline $d$ & $N_d$ \\ \hline 1 & 1\\ 2 & 1\\ 3 & 12\\ 4 & 620\\ 5 & 87304 \\ 6 & 26312976\\ 7 & 14616808192 \\ 8 & 13525751027392 \\ \hline \end{tabular} \begin{tabular}{|l|l|l|l|l|} \hline $d$& $g_d$& $\hat{g}_d$&$ \tilde{g}_d$ & $M_d$\\ \hline 1 & 0 & 0 & 0 & * \\ 2 & 0 & 0 & 0 & * \\ 3 & 55 & 10 & 3 & 1 \\ 4 & 5447 & 1685 & 725 & 96 \\ 5 & 1059729 & 402261 & 166545& 18132 \\ 6 & 393308785 & 168879025 & 64776625 & 6506400 \\ 7 & 254586817377 & 119342269809 & 42214315809 & 4059366000 \\ 8 & 265975021514145 & 133411753757505 & 43616611944513 & 4081597355136 \\ \hline \end{tabular} \noindent The formula for $M_d$ (for $d\geq 3$) is: $$M_d= {d^2-1\over d^2} N_d - {1\over 4d^2} \sum_{i=1}^{d-1} N_i N_{d-i} (id-i^2) \Big( {(6d+4)i(d-i)-2d^2\over 3d-2} \Big) {3d-2\choose 3i-1}.$$ An alternative method of computing $g_3$ is the following. $\overline{S}(0,3)$ is simply the degree 12 discriminant hypersurface in the linear system of plane cubics. Therefore, $C_3$ is a degree 12 plane curve of arithmetic genus $11\cdot10/2=55$. In fact, $C_3$ has 24 cusp, 28 nodes, and geometric genus 3.

9509

alg-geom/9509009

en

https://arxiv.org/abs/alg-geom/9509009

alg-geom/9509009

V. Batyrev

Victor V. Batyrev and Lev A. Borisov

Mirror duality and string-theoretic Hodge numbers

20 pages, Latex

10.1007/s002220050093

We prove in full generality the mirror duality conjecture for string-theoretic Hodge numbers of Calabi-Yau complete intersections in Gorenstein toric Fano varieties. The proof is based on properties of intersection cohomology

alg-geom

\section{Introduction} The first author has conjectured that the polar duality of reflexive polyhedra induces the mirror involution for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties \cite{bat.dual}. The second author has proposed a more general duality which conjecturally induces the mirror involution for Calabi-Yau {\em complete intersections} in Gorenstein toric Fano varieties \cite{borisov}. The most general form of the combinatorial duality which includes mirror constructions of physicists for rigid Calabi-Yau manifolds was formulated by both authors in \cite{batyrev-borisov1}. The main purpose of our paper is to show that all proposed combinatorial dualities agree with the following Hodge-theoretic property of mirror symmetry predicted by physicists: \bigskip {\em If two {smooth} $n$-dimensional Calabi-Yau manifolds $V$ and $W$ form a {mirror pair}, then their Hodge numbers satisfy the relation \begin{equation} h^{p,q}(V) = h^{n-p,q}(W), \;\;\; 0 \leq p, q \leq n. \label{H1} \end{equation} } \bigskip A verification of this property becomes rather non-trivial if we don't make restrictions on the dimension $n$. The main difficulty is the necessity to work with singular Calabi-Yau varieties whose singularities in general don't admit any crepant desingularization. This difficulty was the motivation for introduction of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$ for singular $V$ \cite{batyrev.dais}. The string-theoretic Hodge numbers $h^{p,q}_{\rm st}(V)$ coincide with the usual Hodge numbers $h^{p,q}(V)$ if $V$ is smooth, and with the usual Hodge numbers of a crepant desingularization $\hat{V}$ of $V$ if such a desingularization exists. Therefore the property (\ref{H1}) must be reformulated as follows: \bigskip {\em Let $(V,W)$ be a mirror pair of singular $n$-dimensional Calabi-Yau varieties. Then the string-theoretic Hodge numbers satisfy the duality: \begin{equation} h^{p,q}_{\rm st}(V) = h^{n-p,q}_{\rm st}(W), \;\;\; 0 \leq p, q \leq n. \label{H2} \end{equation} } \bigskip The string-theoretic Hodge numbers for Gorenstein algebraic varities with toroi\-dal or quotient singularities were introduced and studied in \cite{batyrev.dais}. It was also conjectured in \cite{batyrev.dais} that the conbinatorial construction of mirror pairs of Calabi-Yau complete intersections in Gorenstein toric Fano varieties satisfiies the duality (\ref{H2}). This conjecture has been proved in \cite{batyrev.dais} for mirror pairs of Calabi-Yau hypersurfaces of arbitrary dimension that can be obtained by the Greene-Plesser construction \cite{greene.plesser}. Some other results supporting this conjecture have been obtained in \cite{bat.dual,batyrev-borisov2,roan}. Additional evidence in favor of the conjecture has been received by explicit computations of instanton sums using generalized hypergeometric functions \cite{batyrev.straten,HKTY,klemm-theisen,LT}. \bigskip The paper is organized as follows: \bigskip In Section 2, we introduce some polynomials $B(P; u,v)$ of an Eulerian partially ordered set $P$ using results of Stanley \cite{stanley}. It seems that the polynomial $B(P; u,v)$ have independent interest in combinatorics. For our purposes, their most important property is the relation between $B(P; u,v)$ and $B(P^*; u,v)$, where $P^*$ is the dual to $P$ Eulerian poset (Theorem \ref{duality0}). \bigskip In Section 3, we give an explicit formula for the polynomial $E(Z;u,v)$ which describes the mixed Hodge structure of an affine hypersurface $Z$ in an algebraic torus ${\bf T}$ (Theorem \ref{f-formula}). We remark the following: the explicit formula for $E(Z; 1,1)$ is due to Bernstein, Khovanski\^i and Kushnirensko \cite{khov,kush}; the computation of the polynomial $E(Z; t,1)$ which describes the Hodge filtration on $H^*_c(Z)$ is due to Danilov and Khovanski\^i \cite{danilov.khov} (see also \cite{bat.mixed}); the polynomial $E(Z; t,t)$ which describes the weight filtration on $H^*_c(Z)$ has been computed by Denef and Loeser \cite{DL}. \bigskip In Section 4, we derive an explicit formula for the polynomial $E_{\rm st}(V;u,v)$ where $V$ is a Calabi-Yau complete intersection in a Gorenstein toric Fano variety (Theorem \ref{st.formula}). The coefficients of $E_{\rm st}(V;u,v)$ are equal up to a sign to string-theoretic Hodge numbers of $V$. Since our formula is written in terms of $B$-polynomials as a sum over pairs of lattice points contained in the corresponding pair of dual to each other reflexive Gorenstein cones $C$ and $C^*$, the mirror duality for string-theoretic Hodge numbers becomes immediate consequence of the duality for $B$-polynomials after the transposition $C \leftrightarrow C^*$ (Theorem \ref{duality1}). Following some recent development of ideas of Witten \cite{witten} by Morisson and Plesser \cite{morrison.plesser}, we conjecture that the formula obtained in this paper gives the spectrum of the abelian gauge theory in two dimensions which could be constructed from any pair $(C,C^*)$ of two dual to each other reflexive Gorenstein cones. \bigskip {\bf Acknowledgements.} The first author would like to thank for hospitality the University of Warwick where the paper was finished. \bigskip \section{Combinatorial polynomials of Eulerian posets} Let $P$ be a finite poset (i.e., finite partially ordered set). Recall that the M\"obius function $\mu_P(x,y)$ of a poset $P$ is a unique integer valued function on $P \times P$ such that for every function $f: P \rightarrow A$ with values in an abelian group $A$ the following {\em M\"obius inversion formula} holds: $$ f(y) = \sum_{x \leq y} \mu_P(x,y) g(x), \;\;\; \mbox{\rm where $\;g(y) = \sum_{x \leq y} f(x)$}. $$ {}From now on we always assume that the poset $P$ has a unique minimal element $\hat{0}$, a unique maximal element $\hat{1}$, and that every maximal chain of $P$ has the same length $d$ which will be called the {\em rank of } $P$. For any $x \leq y$ in $P$, define the interval \[ \lbrack x, y \rbrack = \{ z \in P\,:\, x \leq z \leq y \}. \] In particular, we have $P = \lbrack \hat{0}, \hat{1} \rbrack$. Define the rank function $\rho\,:\, P \rightarrow \{0,1,\ldots,d \}$ of $P$ by setting $\rho(x)$ equal to the length of any saturated chain in the interval $\lbrack \hat{0}, x \rbrack$. \begin{dfn} {\rm \cite{stanley} A poset $P$ as above is said to be {\em Eulerian} if for any $x \leq y$ $(x,y \in P)$ we have $$ \mu_P(x,y) = (-1)^{\rho(y) - \rho(x)}. $$} \end{dfn} \begin{rem} {\rm It is easy to see that any interval $\lbrack x, y \rbrack \subset P$ in an Eulerian poset $P$ is again an Eulerian poset with the rank function $\rho(z) - \rho(x)$ for any $z \in \lbrack x, y \rbrack$. If an Eulerian poset $P$ has rank $d$, then the dual poset $P^*$ is again an Eulerian poset with the rank function $\rho^*(x) = d - \rho(x)$. } \end{rem} \begin{exam} {\rm Let $C$ be an $d$-dimensional finite convex polyhedral cone in ${\bf R}^d$ such that $-C \cap C = \{0\} \in {\bf R}^d$. Then the poset $P$ of faces of $C$ satisfies all above assumptions with the maximal element $C$, the minimal element $\{0\}$, and the rank function $\rho$ which is equal to the dimension of the corresponding face. It is easy to show that $P$ is an Eulerian poset of rank $d$.} \end{exam} \begin{dfn} {\rm \cite{stanley} Let $P = \lbrack \hat{0}, \hat{1} \rbrack$ be an Eulerian poset of rank $d$. Define two polynomials $G(P,t)$, $H(P,t) \in {\bf Z} [ t]$ by the following recursive rules: $$ G(P,t) = H(P,t) = 1\;\; \mbox{\rm if $d =0$}; $$ $$ H(P,t) = \sum_{ \hat{0} < x \leq \hat{1}} (t-1)^{\rho(x)-1} G(\lbrack x,\hat{1}\rbrack, t)\;\; (d>0), $$ $$ G(P,t) = \tau_{ < d/2 } \left( (1-t)H(P,t) \right) \;\;( d>0), $$ where $\tau_{ < r }$ denotes the truncation operator ${\bf Z}\lbrack t \rbrack \rightarrow {\bf Z}\lbrack t \rbrack$ which is defined by \[ \tau_{< r} \left( \sum_i a_it^i \right) = \sum_{i < r} a_it^i. \]} \label{GH} \end{dfn} \begin{theo} {\rm \cite{stanley} } Let $P$ be an Eulerian poset of rank $d \geq 1$. Then $$ H(P,t) = t^{d-1} H(P,t^{-1}). $$ \label{symm} \end{theo} \begin{prop} Let $P$ be an Eulerian poset of rank $d \geq 0$. Then $$ t^d G(P,t^{-1}) = \sum_{\hat{0} \leq x \leq \hat{1}} (t-1)^{\rho(x)} G(\lbrack x,\hat{1} \rbrack, t). $$ \label{inv} \end{prop} \noindent {\em Proof.} The case $d=0$ is obvious. Using \ref{symm}, we obtain $$ (t-1) H(P,t) = t^d G(P,t^{-1}) - G(P,t)\;\;(d >0). $$ Now the statement follows from the formula for $H(P,t)$ in \ref{GH}. \hfill $\Box$ \begin{dfn} {\rm Let $P$ be an Eulerian poset of rank $d$. Define the polynomial $B(P; u,v) \in {\bf Z}[ u,v]$ by the following recursive rules: $$ B(P; u,v) = 1\;\; \mbox{\rm if $d =0$}, $$ $$ \sum_{\hat{0} \leq x \leq \hat{1}} B(\lbrack \hat{0}, x \rbrack; u,v) u^{d - \rho(x)} G(\lbrack x , \hat{1}\rbrack, u^{-1}v) = G(P ,uv). $$ } \label{Q} \end{dfn} \begin{exam} {\rm Let $P$ be the boolean algebra of rank $d \geq 1$. Then $G(P,t) = 1$, $H(P, t) = 1 + t + \cdots + t^{d-1}$, and $B(P; u,v) = (1-u)^{d}$. } \end{exam} \begin{exam} {\rm Let $C \subset {\bf R}^3$ be a $3$-dimensional finite convex polyhedral cone with $k$ $1$-dimensional faces ($-C \cap C = \{0\} \in {\bf R}^3$), $P$ the Eulerian poset of faces of $C$. Then $G(P,t) = 1 + (k-3)t$, $H(P, t) = 1 + (k-2)t + (k-2)t^2 + t^3$, and $$B(P; u,v) = 1 - (k- (k-3)v)u + (k -(k-3)v)u^2 - u^3.$$ We notice that $B(P; u,v)$ satisfies the relation $$B(P; u,v) =(-u)^3B(P; u^{-1},v)$$ which is a consequence of the selfduality $P \cong P^*$ and a more general property \ref{duality0}.} \end{exam} \begin{prop} Let $P$ be an Eulerian poset of rank $d> 0$. Then $B(P; u,v)$ has the following properties: {\rm (i)} $B(P; u,1) = (1-u)^d$ and $B(P; 1,v) = 0$; {\rm (ii)} the degree of $B(P; u,v)$ with respect to $v$ is less than $d/2$. \label{degree} \end{prop} \noindent {\em Proof.} The statement (i) follows immediately from \ref{inv} and the recursive definition of $B(P; u,v)$. In order to prove (ii) we use induction on $d$. By assumption, the degree of $B(\lbrack \hat{0}, x \rbrack; u,v)$ with respect to $v$ is less than $\rho(x)/2$. On the other hand, the $v$-degree of $G(\lbrack x , \hat{1}\rbrack; u^{-1}v)$ is less than $(d- \rho(x))/2$ (see \ref{GH}). It remains to apply the recursive formula of \ref{Q}. \hfill $\Box$ \begin{prop} Let $P$ be an Eulerian poset of rank $d$. Then $B$-polynomials of intervals $[\hat{0},x ]$ and $[x, \hat{1}]$ satisfy the following relation: $$ \sum_{ \hat{0} \leq x \leq \hat{1}} B([\hat{0},x ];u^{-1},v^{-1})(uv)^{\rho(x)}(v-u)^{d-\rho(x)} = \sum_{ \hat{0} \leq x \leq \hat{1}} B([x, \hat{1}];u,v)(uv-1)^{\rho(x)}.$$ \label{relation} \end{prop} \noindent {\em Proof}. Let us substitute $u^{-1},v^{-1}$ instead of $u,v$ in the recursive relation \ref{Q}. So we obtain \begin{equation} \sum_{\hat{0} \leq x \leq \hat{1}} B([\hat{0},x];u^{-1},v^{-1}) u^{-d + \rho(x)}G([x, \hat{1}], uv^{-1}) = G(P, u^{-1}v^{-1}). \label{eq1} \end{equation} By \ref{inv}, we have \begin{equation} G(P,u^{-1}v^{-1}) = (uv)^{-d} \sum_{\hat{0} \leq x \leq \hat{1}} (uv-1)^{\rho(x)} G(\lbrack x,\hat{1} \rbrack, uv) \label{eq2} \end{equation} and $$G([x, \hat{1}],uv^{-1}) = \sum_{x \leq y \leq \hat{1}} (u^{-1}v -1)^{\rho(y)-\rho(x)}u^{d-\rho(x)}v^{\rho(x) -d} G(\lbrack y,\hat{1} \rbrack, u^{-1}v)$$ \begin{equation} = \sum_{x \leq y \leq \hat{1}} u^{d-\rho(y)}v^{\rho(x) -d}(v -u)^{\rho(y)-\rho(x)} G(\lbrack y,\hat{1} \rbrack, u^{-1}v). \label{eq3} \end{equation} By \ref{Q}, we also have \begin{equation} G([x, \hat{1}],uv) = \sum_{x \leq y \leq \hat{1}} u^{d - \rho(y)} B([x,y];u,v) G([y,\hat{1}],u^{-1}v). \label{eq4} \end{equation} By substitution (\ref{eq4}) in (\ref{eq2}), and two equations (\ref{eq2}), (\ref{eq3}) in (\ref{eq1}) we obtain: $$ \sum_{ \hat{0} \leq x \leq y \leq \hat{1}} B([\hat{0},x ];u^{-1},v^{-1})u^{\rho(x) - \rho(y)} v^{\rho(x) -d}(v-u)^{\rho(y)-\rho(x)} G([y, \hat{1}], u^{-1}v) $$ \begin{equation} = \sum_{ \hat{0} \leq x \leq y \leq \hat{1}} B([x, y];u,v)u^{-\rho(y)}v^{-d}(uv-1)^{\rho(x)}G([y, \hat{1}], u^{-1}v). \label{eq5} \end{equation} Now we use induction on $d$. It is easy to see that the equation (\ref{eq5}) and the induction hypothesis for $y < \hat{1}$ immediately imply the statement of the proposition. \hfill $\Box$ \begin{prop} The $B$-polynomials are uniquely determined by the relation \ref{relation}, by the property of $v$-degree from \ref{degree}(ii), and by the initial condition $B(P;u,v) =1$ if $d =0$. \label{uniq} \end{prop} \noindent {\em Proof.} Indeed, if we know $B([x,y];u,v)$ for all $\rho(y) - \rho(x) < d$, then we know all terms in \ref{relation} except for $B(P;u,v)$ on the right hand side and $B(P;u^{-1},v^{-1}) (uv)^{d}$ on the left hand side. Because the $v$-degree of $B(P;u,v)$ is less than $d/2$, the possible degrees of monomials with respect to variable $v$ in $B(P;u,v)$ and $B(P;u^{-1},v^{-1})(uv)^d$ do not coincide. This allows us to determine $B(P;u,v)$ uniquely. \hfill $\Box$ \begin{theo} Let $P$ be an Eulerian poset of rank $d$, $P^*$ be the dual Eulerian poset. Then $$ B(P; u,v) = (-u)^d B(P^*;u^{-1},v). $$ \label{duality0} \end{theo} \noindent {\em Proof.} We set $$ {Q}(P; u,v) = (-u)^d B(P^*;u^{-1},v). $$ It is clear that ${Q}(P; u,v) =1$ and $v$-degree of ${Q}(P; u,v)$ is the same as $v$-degree of ${B}(P; u,v)$. By \ref{uniq}, it remains to establish the same recursive relations for ${Q}(P; u,v)$ as for ${B}(P; u,v)$ in \ref{relation}. The last property follows from straightforward computations. Indeed, the equality \begin{equation} \sum_{ \hat{0} \leq x \leq \hat{1}} {Q}([\hat{0},x ];u^{-1},v^{-1})(uv)^{\rho(x)} (v-u)^{d-\rho(x)} = \sum_{ \hat{0}\leq x \leq \hat{1}} {Q}([x, \hat{1}];u,v)(uv-1)^{\rho(x)} \label{rel-dual} \end{equation} is equivalent to the relation \ref{relation} for $B(P^*;u,v^{-1})$: $$ \sum_{ \hat{0}\leq x \leq \hat{1}} {B}([x,\hat{1} ]^*;u^{-1},v)(uv^{-1})^{d-\rho(x)} (v^{-1}-u)^{\rho(x)} $$ $$= \sum_{ \hat{0}\leq x \leq \hat{1}} {B}([\hat{0},x]^*;u,v^{-1})(uv^{-1}-1)^{d-\rho(x)}, $$ because $${Q}([x,\hat{1} ];u,v) =(-u)^{d-\rho(x)} {B}([x,\hat{1} ]^*;u^{-1},v) $$ and $${Q}([\hat{0},x];u^{-1},v^{-1}) =(-u)^{-\rho(x)} {B}([\hat{0},x]^*;u,v^{-1}). $$ \hfill $\Box$ \bigskip \section{E-polynomials of toric hypersurfaces} Let $M$ and $N$ be two free abelian groups of rank $d$ which are dual to each other; i.e., $N = {\rm Hom}(M, {\bf Z})$. We denote by \[ \langle *, * \rangle \;:\; M \times N \rightarrow {\bf Z} \] the canonical bilinear pairing, and by $M_{\bf R}$ (resp. by $N_{\bf R}$) the real scalar extensions of $M$ (resp. of $N$). \begin{dfn} {\rm A subset $C \subset M$ is called a $d$-dimensional rational convex polyhedral cone with vertex $\{0\} \in M$ if there exists a finite set $\{e_1, \ldots, e_k\} \subset M$ such that \[ C = \{ \lambda_1 e_1 + \cdots + \lambda_k e_k \in M_{\bf R} \; : \; \mbox{\rm where }\; \; \lambda_i \in {\bf R}_{\geq 0}\; ( i =1,\ldots,k)\} \] and $-C + C = M_{\bf R}$, $-C \cap C = \{0\} \in M$. } \label{cone} \end{dfn} \begin{rem} {\rm If $C \subset M$ is a $d$-dimensional rational convex polyhedral cone with vertex $\{0\} \in M$, then the dual cone \[ C^* = \{ z \in N_{\bf R} \; : \; \mbox{\rm $\langle e_i, z \rangle \geq 0$ for all $i \in \{ 1, \ldots, k\}$ } \} \] is also a $d$-dimensional rational convex polyhedral cone with vertex $\{0\}$ in the dual space $N_{\bf R}$. Moreover, there exists a canonical bijective correspondence $F \leftrightarrow F^*$ between faces $F \subset C$ and faces $F^* \subset C^*$ $({\rm dim}\, F + {\rm dim}\, F^* = d)$ : \[ F \mapsto F^* = \{ z \in C^*\; : \; \mbox{\rm $\langle z', z \rangle = 0$ for all $z ' \in F$} \} \] which reverses inclusion relation between faces. } \end{rem} Let $P$ be the Eulerian poset of faces of a $d$-dimensional rational convex polyhedral cone $C \subset M_{\bf R}$ with vertex in $\{0\}$. For convenience of notations, we use elements $x \in P$ as indices and denote by $C_x$ the face of $C$ corresponding to $x \in P$, in particular, we have $C_{\hat{0}} = \{0\}$, $C_{\hat{1}} = C$, and $\rho(x) = {\rm dim}\, C_x$. The dual Eulerian poset $P^*$ can be identified with the poset of faces $C_x^*$ of the dual cone $C^* \subset N_{\bf R}$. \begin{dfn} {\rm A $d$-dimensional cone $C$ $(d \geq 1)$ as in \ref{cone} is called {\em Gorenstein} if there exists an element $n_C \in N$ such that $\langle z, n_C \rangle >$ for any nonzero $z \in C$ and all vertices of the $(d-1)$-dimensional convex polyhedron \[ \Delta(C) = \{ z \in C \; : \; \langle z, n_C \rangle =1\} \] belong to $M$. This polyhedron will be called the {\em supporting polyhedron of } $C$. For convinience, we consider $\{0\}$ as a $0$-dimensional Gorenstein cone with the supporting polyhedron $\Delta(\{0\}): = \emptyset$. For any $m \in C \cap M$, we define the {\em degree of} $m$ as $$ {\rm deg}\,m = \langle m, n_C \rangle.$$ } \end{dfn} \begin{rem} {\rm It is clear that any face $C_x$ of a Gorenstein cone is again a Gorenstein cone with the supporting polyhedron \[ \Delta(C_x) = \{ z \in C_x \; : \; \langle z, n_C \rangle =1\}. \]} \end{rem} \bigskip Now we recall standard facts from theory of toric varieties \cite{danilov,fulton,oda} and fix our notations: Let ${\bf P}(C)$ be the $(d-1)$-dimensional projective toric variety associated with a Gorenstein cone $C$. By definition, $$ {\bf P}(C) = {\rm Proj}\, {\bf C} [ C \cap M ] $$ where ${\bf C} [ C \cap M ]$ is a graded semigroup algebra over ${\bf C}$ of lattice points $m \in C \cap M$. Each face $C_x \subset C$ of positive dimension defines an irreducible projective toric subvariety $$ {\bf P}(C_x) = {\rm Proj}\, {\bf C} [ C_x \cap M ] \subset {\bf P}(C) $$ which is a compactification of a $(\rho(x) -1)$-dimensional algebraic torus $$ {\bf T}_x : = {\rm Spec}\,{\bf C} [ M_x ], $$ where $M_x \subset M$ is the subgroup of all lattice points $m \in (-C_x + C_x) \cap M$ such that $\langle m,q \rangle = 0$. Moreover, the multiplicative group low on ${\bf T}_x$ extends to a regular action of ${\bf T}_x$ on ${\bf P}(C_x)$ so that one has the natural stratification $$ {\bf P}(C_x) = \bigcup_{ \hat{0} < y \leq x} {\bf T}_y $$ by ${\bf T}_x$-orbits ${\bf T}_y$. We denote by ${\cal O}_{\bf P}(C)(1)$ the ample tautological sheaf on ${\bf P}(C)$. In particular, lattice points in $\Delta(C)$ can be identified with a torus invariant basis of the space of global sections of ${\cal O}_{\bf P}(C)(1)$. We denote by $\overline{Z}$ the set of zeros of a generic global section of ${\cal O}_{\bf P}(C)(1)$ and set $$ Z_x := \overline{Z} \cap {\bf T}_x\;\; (\hat{0} < x \leq \hat{1}). $$ Thus we have the natural stratification: $$ \overline{Z} = \bigcup_{ \hat{0} < x \leq \hat{1}} Z_x, $$ where each $Z_x$ is a smooth affine hypersurface in ${\bf T}_x$ defined by a generic Laurent polynomial with the Newton polyhedron $\Delta(C_x)$. \begin{dfn} Define two functions $$ S(C_x,t):= (1-t)^{\rho(x)} \sum_{m \in C_x \cap M} t^{{\rm deg}\,m} $$ and $$ T(C_x,t):= (1-t)^{\rho(x)} \sum_{m \in Int(C_x) \cap M} t^{{\rm deg}\,m}, $$ where $Int(C_x)$ denotes the relative interior of $C_x \subset C$. \end{dfn} The following statement is a consequence of the Serre duality (see \cite{danilov.khov,bat.mixed}): \begin{prop} $S(C_x,t)$ and $T(C_x,t)$ are polynomials satisfying the relation $$S(C_x,t) = t^d T(C_x,t^{-1}).$$ \label{ST} \end{prop} \begin{dfn} {\rm \cite{danilov.khov} Let $X$ be a quasi-projective algebraic variety over ${\bf C}$. For each pair of integers $(p,q)$, one defines the following generalization of Euler characteristic: $$ e^{p,q}(X) = \sum_{k} (-1)^{k} h^{p,q}(H_c^k(X)), $$ where $h^{p,q}(H_c^k(X))$ is the dimension of the $(p,q)$-component of the mixed Hodge structure of $H_c^k(X)$ \cite{deligne}. The sum $$E(X; u,v) := \sum_{p,q} e^{p,q}(X) u^p v^q $$ is called {\em $E$-polynomial of $X$}. } \end{dfn} Next statement is also due to Danilov and Khovanski\^i (see \cite{danilov.khov} \S 4 , or another approach in \cite{bat.mixed}): \begin{prop} We set $E(Z_{\hat{0}};t,1): = (t-1)^{-1}$. Then $$ E(Z_x;t,1) = \frac{(t - 1)^{\rho(x)-1} + (-1)^{\rho(x)} S(C_x,t)}{t}$$ for $\rho(x) \geq 0$. \label{s-poly} \end{prop} The purpose of this section is to give an explicit formula for $E$-polynomials of affine hypersurfaces $Z_x \subset {\bf T}_x$. Following the method of Denef and Loeser \cite{DL} combined with ideas of Danilov and Khovanski\^i \cite{danilov.khov}, we compute $E(Z_x;u,v)$ using intersection cohomology introduced by Goresky and MacPherson \cite{GM0}. Recall that intersection cohomology $IH^*(X)$ of a quasiprojective algebraic variety $X$ of pure dimension $n$ over ${\bf C}$ can be defined as hypercohomology of the so called {\em intersection complex} $IC^{\bullet}_X$. Moreover, the intersection complex carries a natural mixed Hodge structure. The weight filtration on the $l$-adic version of intersection cohomology has been introduced and studied by Beilinson, Bernstein, Deligne and Gabber using theory of perverse sheaves \cite{BBD}. The Hodge filtration on intersection cohomology of algebraic varieties over ${\bf C}$ has been introduced by M. Saito using his theory of mixed Hodge modules \cite{saito} (see also \cite{durffe}). \begin{dfn} {\rm Let $X$ be a quasi-projective algebraic variety over ${\bf C}$. For each pair of integers $(p,q)$, one defines the following generalization of Euler characteristic for intersection cohomology: $$ e^{p,q}_{\rm int}(X) = \sum_{k} (-1)^{k} h^{p,q}(IH_c^k(X)), $$ where $h^{p,q}(H_c^k(X))$ is the dimension of the $(p,q)$-component in the mixed Hodge structure of $IH_c^k(X)$. The sum $$E_{\rm int}(X; u,v) := \sum_{p,q} e^{p,q}_{\rm int}(X) u^p v^q $$ is called {\em intersection cohomology $E$-polynomial of $X$}. } \end{dfn} The following statement has been discovered by Bernstein, Khovanski\^i and Mac\-Pherson (two independent proofs are contained in \cite{DL} and \cite{fieseler}): \begin{theo} $$E_{\rm int}({\bf P}(C);u,v)= H(P,uv) =\sum_{\hat{0} < x \leq \hat{1}} (uv-1)^{\rho(x)-1}G([x,\hat{1}],uv).$$ Moreover, the cohomology sheaves ${\cal H}^i(IC_{\bf P(C)}^{\bullet})$ are constant on torus orbits ${\bf T}_x$ and $G([x,\hat{1}],uv)$ is the Poincar{\'e} polynomial describing their dimensions. \label{intersection} \end{theo} \begin{coro} Let $\overline{W} \subset {\bf P}(C)$ be a hypersurface that meets transversally all toric strata ${\bf T}_x \subset {\bf P}(C)$ that it intersects $($$ \overline{W}$ is not necessary ample$)$. Then $$E_{\rm int}(\overline{W};u,v)=\sum_{\hat{0} < x \leq \hat{1}} E(W_x;u,v)G([x,\hat{1}], uv),$$ where $W_x = \overline{W} \cap {\bf T}_x$ $(\hat{0} < x \leq \hat{1})$. \label{e-form} \end{coro} \noindent {\em Proof.} The statement is essentially cointained in \cite{DL}(Lemma 7.7). The key fact is that singularities of $\overline{W}$ along $W_x$ are {\em toroidal } (see \cite{danilov}), i.e. , locally isomorphic to toric singularities which appear on ${\bf P}(C)$. \hfill $\Box$ \bigskip \noindent Applying \ref{s-poly}, we obtain: \begin{coro} $$E_{\rm int}(\overline{Z};t,1)=\sum_{\hat{0} < x \leq \hat{1}} \left( \frac{(t-1)^{\rho(x)-1}+(-1)^{\rho(x)}S(C_x,t)}{t} \right) G([x, \hat{1}], t).$$ \label{s-poly1} \end{coro} \begin{dfn} {\rm Define $H_{\rm Lef}(P,t)$ to be the polynomial of degree $(d-2)$ with the following properties: (i) $H_{\rm Lef}(P,t) = t^{d-2}H_{\rm Lef}(P,t^{-1})$; {(ii)} $ \tau_{\leq (d-2)/2} H_{\rm Lef}(P,t) = \tau_{\leq (d-2)/2} H(P,t)$. } \label{def-lef} \end{dfn} \begin{prop} $$H_{\rm Lef}(P,t) = (1-t)^{-1} (G(P,t)-t^{d-1}G(P,t^{-1})).$$ \label{p-lef} \end{prop} \noindent {\em Proof.} Let us set $$Q(P,t) := (1-t)^{-1} (G(P,t)-t^{d-1}G(P,t^{-1})).$$ We check that the properties \ref{def-lef}(i)-(ii) are satisfied for $Q(P,t)$. Indeed \ref{def-lef}(i) follows immediately from the definiton of $Q(P,t)$. If $$H (P, t) = \sum_{0 \leq i \leq d-1} h_i t^i$$ and $$G (P, t) = h_0 + \sum_{1 \leq i < d/2}(h_i - h_{i-1})t^i,$$ then $$Q(P,t) = h_0 \frac{1 - t^{d-1}}{1-t} + \sum_{1 \leq i < d/2} (h_i- h_{i-1}) \frac{t^i - t^{d-1-i}}{1-t}.$$ This shows (ii) and the fact that $Q(P,t)$ is a polynomial. \hfill $\Box$ \begin{prop} Define $E_{\rm int}^{\rm prim}(\overline{Z};u,v)$ to be the polynomial $$ E_{\rm int}^{\rm prim}(\overline{Z};u,v):= E_{\rm int}(\overline{Z};u,v) - H_{\rm Lef}(P,uv). $$ Then $E_{\rm int}^{\rm prim}(\overline{Z};u,v)$ is a homogeneous polynomial of degree $(d-2)$. \label{sum} \end{prop} \noindent {\em Proof.} By the Lefschetz theorem for intersection cohomology \cite{GM}, we have isomorphisms $$ IH^i({\bf P}(C)) \cong IH^i(\overline{Z}), \; \; (0 \leq i < d-2)$$ and the short exact sequence $$ 0 \rightarrow IH^{d-2}({\bf P}(C)) \rightarrow IH^{d-2}(\overline{Z}) \rightarrow IH^{d-2}_{\rm prim}(\overline{Z}) \rightarrow 0,$$ where $IH^{d-2}_{\rm prim}(\overline{Z})$ denotes the primitive part of intersection cohomology of $\overline{Z}$ in degree $(d-2)$. By purity theorem for intersection cohomology \cite{gabber} (see also \cite{durffe}), the Hodge structure of $IH^{d-2}_{\rm prim}(\overline{Z})$ is pure. On the other hand, it follows from the Poincar{\'e} duality for intersection cohomology that $E_{\rm int}^{\rm prim}(\overline{Z};u,v)$ is the $E$-polynomial of this Hodge structure. \hfill $\Box$ \begin{theo} We set $E(Z_{\hat{0}};u,v):=(uv-1)^{-1}$. Then $E$-polynomials $E(Z_x;u,v)$ of affine toric hypersurfaces satisfy the following resursive relation $$\sum_{\hat{0} \leq x \leq \hat{1}} (E(Z_x;u,v)-(uv)^{-1}(uv-1)^{\rho(x)-1})G([x, \hat{1}],uv)$$ $$=v^{d-2} \sum_{\hat{0} \leq x \leq \hat{1}} (u^{-1}v) (-1)^{\rho(x)} S(C_x,uv^{-1}))G([x, \hat{1}],uv^{-1}).$$ \label{recur} \end{theo} \noindent {\em Proof.} By \ref{s-poly1} and \ref{p-lef}, we have \bigskip \noindent $ E_{\rm int}^{\rm prim}(\overline{Z};t,1) = E_{\rm int}(\overline{Z};t,1) - H_{\rm Lef}(P,t) = $ $$= \sum_{\hat{0} < x \leq \hat{1}} t^{-1}((t-1)^{\rho(x)-1}+(-1)^{\rho(x)}S(C_x,t)) G([x, \hat{1}], t)$$ $$ - (1-t)^{-1} (G(P,t)-t^{d-1}G(P,t^{-1})).$$ Using \ref{inv}, we obtain $$ \sum_{\hat{0} < x \leq \hat{1}} t^{-1}(t-1)^{\rho(x)-1}G([x, \hat{1}], t) = t^{-1}(t-1)^{-1} (t^d G(P,t^{-1}) - G(P,t)). $$ This yields \begin{equation} E_{\rm int}^{\rm prim}(\overline{Z};t,1) = \sum_{\hat{0} \leq x \leq \hat{1}}t^{-1}(-1)^{\rho(x)} S(C_x,t))G([x, \hat{1}],t). \label{mid1} \end{equation} On the other hand, by \ref{e-form} and \ref{p-lef}, we have \bigskip \noindent $E_{\rm int}^{\rm prim}(\overline{Z};u,v) = E_{\rm int}(\overline{Z};u,v) - H_{\rm Lef}(P,uv) $ \\ $$= \sum_{\hat{0} < x \leq \hat{1}} E(Z_x;u,v)G([x, \hat{1}],uv) - (1-uv)^{-1} (G(P,uv)-(uv)^{d-1}G(P,(uv)^{-1})).$$ Using \ref{inv}, we obtain $$ \sum_{\hat{0} \leq x \leq \hat{1}} (uv)^{-1}(uv-1)^{\rho(x)-1}G([x, \hat{1}],uv) = (uv)^{d-1}(uv-1)^{-1}G(P,(uv)^{-1}).$$ This yields \begin{equation} E_{\rm int}^{\rm prim}(\overline{Z};u,v) = \sum_{\hat{0} \leq x \leq \hat{1}} (E(Z_x;u,v)-(uv)^{-1}(uv-1)^{\rho(x)-1})G([x, \hat{1}],uv). \label{mid2} \end{equation} \noindent By \ref{sum}, we have $$E_{\rm int}^{\rm prim}(\overline{Z};u,v) = v^{d-2} E_{\rm int}^{\rm prim}(\overline{Z};uv^{-1},1).$$ It remains to combine (\ref{mid1}) and (\ref{mid2}). \hfill $\Box$ \begin{dfn} {\rm Let $m$ be a lattice point in $C \cap M$. We denote by ${x(m)}$ the minimal element among $x \in P$ such that the face $C_x \subset C$ contains $m$. The interval $[ x(m), \hat{1}] \subset P$ parametrizes the set of all faces of $C$ containing $m$. We identify the dual interval $[ x(m), \hat{1}]^*$ with the Eulerian poset of all faces $C_x^* \subset C$ such that $\langle m, z \rangle= 0$ for all $z \in C_x^*$.} \end{dfn} \begin{theo} Let us set $Z: = Z_{\hat{1}}$. Then there exists the following explicit formula for $E(Z;u,v)$ in terms of $B$-polynomials: $$ E(Z;u,v) = \frac{(uv-1)^{d-1}}{uv} + \frac{(-1)^d}{uv} \sum_{m \in C\cap M} (v-u)^{\rho(x(m))}B([x(m),\hat{1}]^*; u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m}. $$ \label{f-formula} \end{theo} \noindent {\em Proof.} By induction, $E$-polynomials are uniquely determined from the recursive formula \ref{recur}. Therefore, it suffices to show that the functions $$ \frac{(uv-1)^{\rho(x)-1}}{uv} + \frac{(-1)^{\rho(x)}}{uv} \sum_{m \in C_x \cap M} (v-u)^{\rho(x(m))}B([x(m),x]^*; u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m} $$ satisfy the same resursive formula as polynomials $E(Z_x; u,v)$. Indeed, let us substitute these functions instead of $E$-polynomials in the left hand side of \ref{recur} and expand $$(-1)^{\rho(x)}S(C_x,uv^{-1}) = \left(\frac{u}{v} - 1 \right)^{\rho(x)} \sum_{m \in C_x \cap M} \left(\frac{u}{v}\right)^{{\rm deg}\, m}$$ on the right hand side of \ref{recur}. Now we choose a lattice point $m \in C \cap M$, collect terms containing $(u/v)^{{\rm deg}\, m}$ in right and left hand sides, and use the equality (\ref{inv}) $$ \sum_{x(m) \leq x \leq \hat{1}}\left(\frac{u}{v} -1\right)^{\rho(x)} G([x, \hat{1}],uv^{-1}) = \left(\frac{u}{v} -1\right)^{\rho(x(m))} \left(\frac{u}{v}\right)^{d-\rho(x(m))} G([x(m),\hat{1}],u^{-1}v) $$ on the right hand side. By the duality (\ref{duality0}) $$B([x(m),x]^*;u,v) = (-u)^{\rho(x) - \rho(m(x))} B([x(m),x];u^{-1},v),$$ it remains to estabish the recursive relation: $$ \frac{(v-u)^{\rho(x(m))}}{uv} \sum_{x(m) \leq x \leq \hat{1}} (-1)^{\rho(x)}(-u)^{\rho(x) - \rho(m(x))} B([x(m),x];u^{-1},v) G([x,\hat{1}],uv) = $$ $$ = \left(\frac{u}{v} -1\right)^{\rho(x(m))} \frac{v^{d-1}}{u}\left(\frac{u}{v}\right)^{d-\rho(x(m))} G([x(m),\hat{1}], u^{-1}v) $$ which is equivalent to the recursive relation in \ref{Q} after the substitution $u^{-1}$ instead of $u$. \hfill $\Box$ \section{Mirror duality} Let $\overline{M}$ and $\overline{N} = {\rm Hom}(\overline{N}, {\bf Z})$ be dual to each other free abelian groups of rank $\overline{d}$, $\overline{M}_{\bf R}$ and $\overline{N}_{\bf R}$ the real scalar extensions of $\overline{M}$ and $\overline{N}$, $\langle *, *\rangle\;: \; \overline{M} \times \overline{N} \rightarrow {\bf Z}$ the natural pairing. \begin{dfn} {\rm \cite{batyrev-borisov1} Let $C \subset \overline{M}_{\bf R}$ be a $\overline{d}$-dimensional Gorenstein cone. The cone $C$ is called {\em reflexive} if the dual cone $C^* \subset N_{\bf R}$ is also Gorenstein; i.e., there exists a lattice element $m_{C^*} \in M$ such that all vertices of the supporting polyhedron $\Delta(C^*) = \{ z \in C^*\;:\; \langle m_{C^*}, z \rangle =1 \}$ are contained in $M$. In this case, we call $r = \langle m_{C^*}, n_{C} \rangle$ the {\em index} of $C$. } \end{dfn} \begin{dfn} {\rm \cite{bat.dual} Let $M$ be a free abelian group of rank $d$. A $d$-dimensional polyhedron in $M_{\bf R}$ with vertices in $M$ is called {\em reflexive} if it can be identified with a supporting polyhedron of some $(d+1)$-dimensional reflexive Gorenstein cone of index $1$.} \end{dfn} Recall the definition of string-theoretic Hodge numbers of an algebraic variety $X$ with at most Gorenstein toroidal singularities \cite{batyrev.dais}: \begin{dfn} {\rm \cite{batyrev.dais} Let $X = \bigcup_{i \in I} X_i$ be a $k$-dimensional stratified algebraic variety over ${\bf C}$ with at most Gorenstein toroidal singularities such that for any $ i \in I$ the singularities of $X$ along the stratum $X_i$ of codimension $k_i$ are defined by a $k_i$-dimensional finite rational polyhedral cone $\sigma_i$; i.e., $X$ is locally isomorphic to $${\bf C}^{k-k_i} \times U_{\sigma_i}$$ at each point $x \in X_i$ where $U_{\sigma_i}$ is a $k_i$-dimensional affine toric variety which is associated with the cone $\sigma_i$ (see \cite{danilov}). Then the polynomial $$ E_{\rm st}(X;u,v) := \sum_{i \in I} E(X_i;u,v) \cdot S(\sigma_i,uv) $$ is called the {\em string-theoretic E-polynomial of $X$.} If we write $E_{\rm st}(X; u,v)$ in form $$ E_{\rm st}(X;u,v) = \sum_{p,q} a_{p,q} u^{p}v^{q}, $$ then the numbers $h^{p,q}_{\rm st}(X) := (-1)^{p+q}a_{p,q}$ are called the {\em string-theoretic Hodge numbers of $X$.} } \label{st-numbers} \end{dfn} \begin{rem} {\rm Comparing with \ref{intersection} and \ref{e-form}, the definition of the string-theoretic Hodge numbers looks as if there were a complex ${ST}^{\bullet}_X$ whose hypercohomology groups have natural Hodge structure which assumed to be pure if $X$ is compact. We remark that the construction of such a complex ${ST}^{\bullet}_X$ (an analog of the intersection complex) is still an open problem.} \end{rem} Let $V = D_1 \cap \cdots \cap D_r$ be a generic Calabi-Yau complete intersection of $r$ semi-ample divisors $D_1, \ldots, D_r$ in a $d$-dimensional Gorenstein toric Fano variety ${\bf X}$ $(k \geq r)$. According to \cite{batyrev-borisov1}, there exists a $d$-dimensional reflexive polyhedron $\Delta$ and its decompostion into a Minkowski sum $$ \Delta = \Delta_1 + \cdots + \Delta_r,$$ where each lattice polyhedron $\Delta_i$ is the supporting polyhedron for global sections of a semi-ample invertible sheaf ${\cal L}_i \cong {\cal O}_{\bf X}(D_i)$ ($i =1, \ldots, r$). \begin{dfn} {\rm \cite{borisov} Denote by $E_1, \ldots, E_k$ the closures of $(d-1)$-dimensional torus orbits in ${\bf X}$ and set $I := \{ 1, \ldots, k \}$. A decompostion into a Minkowski sum $\Delta = \Delta_1 + \cdots + \Delta_r$ as above is called a {\em nef-partition} if there exists a decomposition of $I$ into a disjoint union of $r$ subsets $I_j \subset I$ $(j =1, \ldots, r)$ such that $$ {\cal O}(D_j) \cong {\cal O}(\sum_{l \in I_j} E_l), \;\; (j =1, \ldots, r)$$.} \label{nef} \end{dfn} Now we put $\overline{M} = {\bf Z}^r \oplus M$, $\overline{d} = d+r$, and define the $\overline{d}$-dimensional cone $C \subset \overline{M}_{\bf R}$ as $$ C:= \{ (\lambda_1, \ldots, \lambda_r, \lambda_1 z_1 + \cdots + \lambda_r z_r) \in \overline{M}_{\bf R}\; :\; \lambda_i \in {\bf R}_{\geq 0}, \; z_i \in \Delta_i, \; i =1, \ldots, r \}. $$ We extend the pairing $\langle \cdot, \cdot \rangle \; : M \times N \rightarrow {\bf Z}$ to the pairing between $\overline{M}$ and $\overline{N} := {\bf Z}^r \oplus N$ by the formula $$\langle (a_1, \ldots, a_r, m), (b_1, \ldots, b_r,n) \rangle = \sum_{i =1}^r a_i b_i + \langle m, n \rangle.$$ \begin{theo} {\rm \cite{borisov,batyrev-borisov1}} Let $\Delta = \Delta_1 + \cdots + \Delta_r $ be a nef-partition. Then it defines canonically a $d$-dimensional reflexive polyhedron $\nabla \subset N_{\bf R}$ and a nef-partition $\nabla = \nabla_1 + \cdots + \nabla_r$ which are uniquely determined by the property that $$ C^*:= \{ (\lambda_1, \ldots, \lambda_r, \lambda_1 z_1 + \cdots + \lambda_r z_r) \in \overline{N}_{\bf R}\; :\; \lambda_i \in {\bf R}_{\geq 0}, \; z_i \in \nabla_i, \; i =1, \ldots, r \} $$ is the dual reflexive Gorenstein cone $C^* \subset \overline{N}_{\bf R}$. \label{nef-partition} \end{theo} \begin{dfn} {\rm \cite{borisov} The nef-partition $\nabla = \nabla_1 + \cdots + \nabla_r$ as in \ref{nef-partition} is called {\em the dual nef-partition}. } \end{dfn} We set $${\bf Y}:= {\bf P}({\cal L}_1\oplus \cdots \oplus {\cal L}_r). $$ Recall the standard construction of the reduction of complete intersection $V \subset {\bf X}$ to a hypersurface $\tilde{V} \subset {\bf Y}$ \cite{batyrev-borisov1}. Let $\pi$ be the canonical projection ${\bf Y} \rightarrow {\bf X}$ and ${\cal O}_{\bf Y}(-1)$ the tautological Grothendieck sheaf on ${\bf Y}$. Since $$\pi_*{\cal O}_{\bf Y}(1) = {\cal L}_1\oplus \cdots \oplus {\cal L}_r,$$ we obtain the isomorphism $$ H^0({\bf Y}, {\cal O}_{\bf Y}(1)) \cong H^0({\bf X}, {\cal L}_1) \oplus \cdots \oplus H^0({\bf X}, {\cal L}_r).$$ Assume that $D_i$ is the set of zeros of a global section $s_i \in H^0({\bf X}, {\cal L}_i)$ $(1 \leq i \leq r)$. We define $\tilde{V}$ as the zero set of the global section $s \in H^0({\bf Y}, {\cal O}_{\bf Y})$ which corresponds to the $r$-tuple $(s_1, \ldots, s_r)$ under above isomorphism. Our main interest is the following standard property (\cite{batyrev-borisov1}): \begin{prop} The restriction of $\pi$ on ${\bf Y} \setminus \tilde{V}$ is a locally trivial ${\bf C}^{r-1}$-bundle in Zariski topology over ${\bf X} \setminus V$. \label{bundle} \end{prop} Let us set $${\bf P} = {\rm Proj}\, \bigoplus_{i \geq 0} H^0({\bf Y}, {\cal O}_{\bf Y}(i)).$$ The following statement is contained in \cite{batyrev-borisov1}: \begin{prop} The tautological sheaf ${\cal O}_{\bf Y}(1)$ is semi-ample and the natural toric morphism $$\alpha \;: \; {\bf Y} \rightarrow {\bf P}$$ is crepant. Moreover, ${\cal O}_{\bf Y}(r)$ is the anticanonical sheaf of ${\bf Y}$, ${\bf P}$ is a Gorenstein toric Fano variety , and $\overline{Z} : = \alpha(\tilde{V})$ is an ample hypersurface in ${\bf P}$. \label{alpha} \end{prop} There is the following explicit formula for $E_{\rm st}(V;u,v)$ in terms of $E_{\rm st}({\bf P};u,v)$ and $E_{\rm st}(\overline{Z};u,v)$: \begin{theo} $$E_{\rm st}(V;u,v) =((uv-1)((uv)^r-1)^{-1})E_{\rm st}({\bf P};u,v) - (uv)^{1-r}E_{\rm st}({\bf P} \setminus \overline{Z};u,v).$$ \label{V-form} \end{theo} \noindent {\em Proof. } Since $V$ is transversal to all toric strata in ${\bf X}$ we have: $$ E_{\rm st}(V;u,v)= E_{\rm st}({\bf X};u,v) - E_{\rm st}({\bf X} \setminus V;u,v).$$ Using the ${\bf CP}^{r-1}$-bundle structure of ${\bf Y}$ over ${\bf X}$, we obtain: $$ E_{\rm st}({\bf X};u,v) = ((uv)^r-1)^{-1}(uv-1)E_{\rm st}({\bf Y};u,v). $$ By \ref{bundle}, we also have $$ E_{st}({\bf X} \setminus V;u,v) = (uv)^{1-r} E_{\rm st}({\bf Y} \setminus \tilde{V};u,v).$$ Since birational crepant toric morphisms do not change string-theoretic Hodge numbers (see \cite{batyrev.dais}), by \ref{alpha}, we conclude $$ E_{\rm st}({\bf Y};u,v)= E_{\rm st}({\bf P};u,v), \;\; E_{\rm st}({\bf Y} \setminus \tilde{V};u,v) = E_{\rm st}({\bf P} \setminus \overline{Z};u,v) .$$ \hfill $\Box$ \begin{dfn} {\rm Let $C \subset \overline{M}_{\bf R}$ be a reflexive Gorenstein cone, $C^* \subset {\overline{N}}_{\bf R}$ the dual reflexive Gorenstein cone. We define $$ \Lambda(C,C^*):= \{ (m,n) \in \overline{M} \oplus \overline{N}\;: \; m \in C, \; n \in C^*,\;\;\mbox{\rm and}\; \; \langle m, n \rangle = 0 \}.$$ } \end{dfn} \begin{dfn} {\rm Let $(m,n)$ be an element of $\Lambda(C,C^*)$. We define the Eulerian poset $P_{(m,n)}$ as the subset of all faces $C_x \subset C$ such that $C_x$ contains $m$ and $\langle z,n \rangle = 0$ for all $z \in C_x$. We denote by $\rho(x^*(n))$ the dimension of the intersection of $C$ with the hyperplane $\langle z,n \rangle = 0$. } \label{n-rho} \end{dfn} \begin{rem} {\rm The dual Eulerian poset $P^*_{(m,n)}$ can be identified with the subset of all faces $C_x^* \subset C^*$ such that $C_x^*$ contains $n$ and $\langle m,z \rangle = 0$ for all $z \in C_x^*$. } \end{rem} \begin{theo} Let us set $\overline{d} = d + r$ and $$A_{(m,n)}(u,v) = \frac{(-1)^{\rho(x^*(n))}}{(uv)^r} (v-u)^{\rho(x(m))}B(P_{(m,n)}^*;u,v)(uv -1)^{\overline{d}-\rho(x^*(n))}. $$ Then $$ E_{\rm st}(V; u,v)= \sum_{(m,n) \in \Lambda(C,C^*)} \left(\frac{u}{v}\right)^{{\rm deg}\,m} A_{(m,n)}(u,v) \left(\frac{1}{uv}\right)^{{\rm deg}\,n} $$ \label{st.formula} \end{theo} \noindent {\em Proof.} By Definition \ref{st-numbers}, $$ E_{\rm st}({\bf P},u,v) = \sum_{\hat{0} < x \leq \hat{1}} (uv - 1)^{\rho(x) -1} S(C_x^*,uv) $$ $$ = \sum_{\hat{0} < x \leq \hat{1}} (uv - 1)^{\rho(x) -1} (uv -1)^{\overline{d} - \rho(x)} T(C_x^*,(uv)^{-1}) $$ $$ = (uv -1)^{\overline{d} -1} \sum_{\hat{0} < x \leq \hat{1}} \left(\sum_{n \in Int(C_x^*) \cap {\overline{N}}} (uv)^{-{\rm deg}\,n} \right) = (uv -1)^{\overline{d}-1} \sum_{n \in \partial C^* \cap {\overline{N}}} (uv)^{-{\rm deg}\,n}, $$ where $\partial C^* = C^* \setminus Int(C^*)$ is the boundary of $C^*$. Since ${\overline{N}} \cap Int(C^*) = p+ {\overline{N}}\cap C^*$ and ${\rm deg}\, p = r$, we conclude: $$ E_{\rm st}({\bf P},u,v) = (1-(uv)^{-r})(uv -1)^{\overline{d}-1} \sum_{n \in C^* \cap {\overline{N}}} (uv)^{-{\rm deg}\,n}$$ $$ = ((uv)^{r}-1)(uv -1)^{\overline{d}-1} \sum_{n \in Int( C^*) \cap {\overline{N}}} (uv)^{-{\rm deg}\,n}. $$ On the other hand, $$ E_{\rm st}({\bf P} \setminus \overline{Z};u,v) = E_{\rm st}({\bf P};u,v) - E_{\rm st}(\overline{Z};u,v). $$ By Definition \ref{st-numbers} and Theorem \ref{f-formula}, $$ E_{\rm st}(\overline{Z};u,v) = \sum_{\hat{0} < x \leq \hat{1}} \left( \frac{(uv-1)^{\rho(x)-1}}{uv} \right) S(C_x^*,uv)$$ $$+ \sum_{\hat{0} < x \leq \hat{1}} \left( \frac{(-1)^{\rho(x)}}{uv} \sum_{m \in C_x \cap {\overline{M}}} (v-u)^{\rho(x(m))}B([x(m),x]^*;u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m}\right) S(C_x^*,uv)$$ $ = (uv)^{-1} E_{\rm st}({\bf P};u,v) + $ $$ + \sum_{\hat{0} < x \leq \hat{1}} \left( \frac{(-1)^{\rho(x)}}{uv} \sum_{m \in C_x \cap {\overline{M}}} (v-u)^{\rho(x(m))}B([x(m),x]^*;u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m} \right) S(C_x^*,uv). $$ By \ref{V-form}, $$E_{\rm st}(V;u,v) = ((uv-1)((uv)^r-1)^{-1} - (uv)^{1-r} + (uv)^{-r}) E_{\rm st}({\bf P},u,v) $$ $$ + \sum_{\hat{0} < x \leq \hat{1}} \left( \frac{(-1)^{\rho(x)}}{(uv)^r} \sum_{m \in C_x \cap {\overline{M}}} (v-u)^{\rho(x(m))}B([x(m),x]^*;u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m} \right) S(C_x^*,uv) $$ $$ = (uv)^{-r}(uv -1)^{\overline{d}} \sum_{n \in Int(C^*) \cap {\overline{N}}} (uv)^{-{\rm deg}\,n} $$ $$ + \sum_{\hat{0} < x \leq \hat{1}} \left( \frac{(-1)^{\rho(x)}}{(uv)^r} \sum_{m \in C_x \cap {\overline{M}}} (v-u)^{\rho(x(m))}B([x(m),x]^*;u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m} \right) S(C_x^*,uv) $$ $$ = \sum_{\hat{0} \leq x \leq \hat{1}} \left( \frac{(-1)^{\rho(x)}}{(uv)^r} \sum_{m \in C_x \cap {\overline{M}}} (v-u)^{\rho(x(m))}B([x(m),x]^*;u,v) \left(\frac{u}{v}\right)^{{\rm deg}\,m} \right) S(C_x^*,uv). $$ It remains to use the formula $$ S(C_x^*,uv)= (uv-1)^{\overline{d} -\rho(x)} \sum_{n \in Int(C_x^*)\cap {\overline{N}}} (uv)^{-{\rm deg}\,n} \;\;\; ( \hat{0} \leq x \leq \hat{1}) $$ and notice that $ \rho(x)= \rho(x^*(n))$ if $n$ is an interior lattice point of $C_x^*$ (see \ref{n-rho}). \hfill $\Box$ \begin{theo} Let $V$ be a $(d-r)$-dimensional Calabi-Yau complete intersection defined by a nef-partition $\Delta = \Delta_1 + \cdots + \Delta_r$, $W$ a $(d-r)$-dimensional Calabi-Yau complete intersection defined by the dual nef-partition $\nabla = \nabla_1 + \cdots + \nabla_r$. Then $$E_{\rm st}(V;u,v) = (-u)^{d-r}E_{\rm st}(W;u^{-1},v),$$ i.e., $$ h^{p,q}_{\rm st}(V) = h^{d-r-p,q}_{\rm st}(W)\;\; 0 \leq p, q \leq d-r. $$ \label{duality1} \end{theo} \noindent {\em Proof.} If we use the duality between two $\overline{d}$-dimensional reflexive Gorenstein cones $C \subset \overline{M}_{\bf R}$ and $C^* \subset \overline{N}_{\bf R}$ \ref{nef-partition}, then the statement of Theorem follows immediatelly from the explicit formula in \ref{st.formula} and from the duality for $B$-polynomials \ref{duality0}.

9509

alg-geom/9509008

en

https://arxiv.org/abs/alg-geom/9509008

alg-geom/9509008

Moriwaki Atsushi

Atsushi Moriwaki

Bogomolov conjecture for curves of genus 2 over function fields

8 pages (with 2 tables). AmSLaTeX ver. 1.2 (with LaTeX2e). In this new version, we remove the assumption of characteristic by remarks of Prof. Liu

In this note, we will show that Bogomolov conjecture holds for a non-isotrivial curve of genus 2 over a function field.

alg-geom

\section{Introduction} \label{intro} Let $k$ be an algebraically closed field, $X$ a smooth projective surface over $k$, $Y$ a smooth projective curve over $k$, and $f : X \to Y$ a generically smooth semistable curve of genus $g \geq 2$ over $Y$. Let $K$ be the function field of $Y$, $\overline{K}$ the algebraic closure of $K$, and $C$ the generic fiber of $f$. Let $j : C(\overline{K}) \to \operatorname{Jac}(C)(\overline{K})$ be a morphism given by $j(x) = (2g-2)x - \omega_C$ and $\Vert\ \Vert_{NT}$ the semi-norm arising from the Neron-Tate height pairing on $\operatorname{Jac}(C)(\overline{K})$. We set \[ B_C(P;r) = \left\{ x \in C(\overline{K}) \mid \Vert j(x) - P \Vert_{NT} \leq r \right\} \] for $P \in \operatorname{Jac}(C)(\overline{K})$ and $r \geq 0$, and \[ r_C(P) = \begin{cases} -\infty & \mbox{if $\#\left(B_C(P;0)\right) = \infty$}, \\ & \\ \sup \left\{ r \geq 0 \mid \#\left(B_C(P;r)\right) < \infty \right\} & \mbox{otherwise}. \end{cases} \] Bogomolov conjecture claims that, if $f$ is non-isotrivial, then $r_C(P)$ is positive for all $P \in \operatorname{Jac}(C)(\overline{K})$. Even to say that $r_C(P) \geq 0$ for all $P \in \operatorname{Jac}(C)(\overline{K})$ is non-trivial because it contains Manin-Mumford conjecture, which was proved by Raynaud. Further, it is well known that the above conjecture is equivalent to say the following. \begin{Conjecture}[Bogomolov conjecture] \label{conj:bogomolov:1} If $f$ is non-isotrivial, then \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) > 0. \] \end{Conjecture} \noindent Moreover, we can think the following effective version of Conjecture~\ref{conj:bogomolov:1}. \begin{Conjecture}[Effective Bogomolov conjecture] \label{conj:bogomolov:2} In Conjecture~\ref{conj:bogomolov:1}, there is an effectively calculated positive number $r_0$ with \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq r_0. \] \end{Conjecture} \noindent In \cite{Mo}, we proved that, if $f$ is non-isotrivial and the stable model of $f : X \to Y$ has only irreducible fibers, then Conjecture~\ref{conj:bogomolov:2} holds. More precisely, \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq \begin{cases} \sqrt{12(g-1)} & \mbox{if $f$ is smooth}, \\ & \\ {\displaystyle \sqrt{\frac{(g-1)^3}{3g(2g+1)}\delta}} & \mbox{otherwise}, \end{cases} \] where $\delta$ is the number of singularities in singular fibers of $f : X \to Y$. In this note, we would like to show the following result. \begin{Theorem} \label{thm:bogomolov:genus:2} If $f$ is non-isotrivial and $g = 2$, then $f$ is not smooth and \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq \sqrt{\frac{2}{135}\delta}. \] \end{Theorem} \section{Notations and ideas} In this section, we use the same notations as in \S\ref{intro}. Let $\omega_{X/Y}^a$ be the dualizing sheaf in the sense of admissible pairing. (For details concerning admissible pairing, see \cite{Zh} or \cite{Mo}.) First we note the following theorem. (cf. \cite[Theorem~5.6]{Zh} or \cite[Corollary~2.3]{Mo}) \begin{Theorem} \label{thm:lower:estimate:r} If $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a > 0$, then \[ \inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq \sqrt{(g-1)(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a}, \] where $(\ \cdot \ )_a$ is the admissible pairing. \end{Theorem} {}From now on, we assume $g = 2$. By the above theorem, in order to get Theorem~\ref{thm:bogomolov:genus:2}, we need to estimate $\left(\omega_{X/Y}^a \cdot \omega_{X/Y}^a \right)_a$. First of all, we can set \addtocounter{Theorem}{1} \begin{equation} \label{eqn:wwa:ww:e} \left(\omega_{X/Y}^a \cdot \omega_{X/Y}^a \right)_a = \left(\omega_{X/Y} \cdot \omega_{X/Y} \right) - \sum_{y \in Y} e_y, \end{equation} where $e_y$ is the number coming from the Green function of $f^{-1}(y)$. This number depends on the configuration of $f^{-1}(y)$. So, let us consider the classification of semistable curves of genus 2. Let $E$ be a semistable curve of genus $2$ over $k$ and $E'$ the stable model of $E$, that is, $E'$ is a curve obtained by contracting all $(-2)$-curves in $E$. It is well known that there are $7$-types of stable curves of genus $2$. Thus, we have the classification of semistable curves of genus $2$ according to type of $E'$ as in Table~\ref{table:classification:semistable:curve:genus:2}. (In Table~\ref{table:classification:semistable:curve:genus:2}, the symbol $A_n$ for a node means that the dual graph of $(-2)$-curves over the node is same as $A_n$ type graph.) The exact value of $e_y$ can be found in Table~\ref{table:d:d:e} and will be calculated in \S\ref{sec:cal:e:y}. Next we need to think an estimation of $\left(\omega_{X/Y} \cdot \omega_{X/Y}\right)$ in terms of type of $f^{-1}(y)$. According to Ueno \cite{U}, there is the canonical section $s$ of \[ H^0(Y, \det(f_*(\omega_{X/Y}))^{10}) \] such that $d_y = \operatorname{ord}_y(s)$ for $y \in Y$ can be exactly calculated under the assumption that $\operatorname{char}(k) \not= 2, 3, 5$. The result can be found in Table~\ref{table:d:d:e}. Prof. Liu points out that by works of T. Saito \cite{Sa} and Q. Liu \cite{Li}, the value $d_y$ in Table~\ref{table:d:d:e} still holds even if $\operatorname{char}(k) \leq 5$. Let $\delta_y$ be the number of singularities in $f^{-1}(y)$. Then, by Noether formula, \[ \deg(\det(f_*(\omega_{X/Y}))) = \frac{\left(\omega_{X/Y} \cdot \omega_{X/Y}\right) + \sum_{y \in Y} \delta_y}{12}. \] On the other hand, by the definition of $d_y$, \[ \sum_{y \in Y} d_y = 10 \deg(\det(f_*(\omega_{X/Y}))). \] Thus, we have \addtocounter{Theorem}{1} \begin{equation} \label{eqn:w:w:eq:d:d} \left(\omega_{X/Y} \cdot \omega_{X/Y}\right) = \sum_{y \in Y}\left(\frac{6}{5}d_y - \delta_y\right). \end{equation} Hence, by (\ref{eqn:wwa:ww:e}) and (\ref{eqn:w:w:eq:d:d}), \addtocounter{Theorem}{1} \begin{equation} \label{eqn:wa:wa:eq:d:d:e} \left(\omega_{X/Y}^a \cdot \omega_{X/Y}^a\right)_a = \sum_{y \in Y}\left(\frac{6}{5}d_y - \delta_y - e_y \right). \end{equation} Therefore, using (\ref{eqn:wa:wa:eq:d:d:e}) and Table~\ref{table:d:d:e}, we have the following theorem. (Note that an inequality \[ \frac{abc}{ab+bc+ca} \leq \frac{a+b+c}{9} \] holds for all positive numbers $a, b, c$.) \begin{Theorem} \label{lower:w:w:admissible:genus:2} If $f$ is non-isotrivial, then $f$ is not smooth and \[ \left(\omega_{X/Y}^a \cdot \omega_{X/Y}^a \right)_a \geq \frac{2}{135}\delta, \] where $\delta = \sum_{y \in Y} \delta_y$. \end{Theorem} Note that non-smoothness of $f$ can be easily derived from the fact that the moduli space $M_2$ of curves of genus $2$ is an affine variety. \section{Calculation of $e_y$} \label{sec:cal:e:y} \setlength{\unitlength}{.5in} Let us start calculations of $e_y$. If the stable model of a fiber is irreducible, $e_y$ is calculated in \cite{Mo}. Thus it is sufficient to calculate $e_y$ for II(a), IV(a,b), VI(a,b,c) and VII(a,b,c). In these cases, the stable model has two irreducible components. Let $f^{-1}(y) = C_1 + \cdots + C_n$ be the irreducible decomposition of $f^{-1}(y)$. We set \[ D_y = \sum_{i=1}^n (\omega_{X/Y} \cdot C_i) v_i, \] where $v_i$ is the vertex in $G_y$ corresponding to $C_i$. Especially, we denote by $P$ and $Q$ corresponding vertexes to stable components. Then, $D_y = P + Q$. Let $\mu$ and $g_{\mu}$ be the measure and the Green function defined by $D_y$. In the same way as in the Proof of Theorem~5.1 in \cite{Mo}, \[ e_y = -g_{\mu}(D_y,D_y) + 4c(G_y,D_y), \] where $c(G_y,D_y)$ is the constant coming from $g_{\mu}$. By the definition of $c(G_y,D_y)$, \[ c(G_y,D_y) = g_{\mu}(P,P) + g_{\mu}(P,D_y). \] Therefore, we have \[ e_y = 7g_{\mu}(P,P) - g_{\mu}(Q,Q) + 2g_{\mu}(P,Q). \] Here claim: \begin{Lemma} $g_{\mu}(P,P) = g_{\mu}(Q,Q)$. In particular, \[ e_y = 6g_{\mu}(P,P) + 2g_{\mu}(P,Q). \] \end{Lemma} {\sl Proof.}\quad By the definition of $c(G_y,D_y)$, \[ c(G_y,D_y) = g_{\mu}(P,P) + g_{\mu}(P,P+Q) = g_{\mu}(Q,Q) + g_{\mu}(Q,P+Q). \] Thus, we can see $g_{\mu}(P,P) = g_{\mu}(Q,Q)$. \QED In the following, we will calculate $e_y$ for each type II(a), IV(a,b), VI(a,b,c) and VII(a,b,c). First we present the dual graph of each type and then show its calculation. \bigskip {\bf Type II(a).} \begin{center} \begin{picture}(6,2) \put(1,1){\circle*{.25}} \put(4,1){\circle*{.25}} \put(1,1){\line(1,0){3}} \put(0.5,1){$P$} \put(4.3,1){$Q$} \put(2.5,1.2){$G$} \put(5,1){$\operatorname{length}(G)=a$} \end{picture} \end{center} In this case, ${\displaystyle \mu= \frac{\delta_P}{2} + \frac{\delta_Q}{2}}$ by \cite[Lemma~3.7]{Zh}. We fix a coordinate $s : G \to [0, a]$ with $s(P) = 0$ and $s(Q) = a$. If we set \[ g(x) = -\frac{s(x)}{2} + \frac{a}{4}, \] then, $\Delta(g) = \delta_P - \mu$ and ${\displaystyle \int_G g \mu = 0}$. Thus, $g(x) = g_{\mu}(P, x)$. Hence \[ g_{\mu}(P, P) = \frac{a}{4}\qquad\mbox{and}\qquad g_{\mu}(P, Q) = -\frac{a}{4}. \] Thus \[ e_y = 6g_{\mu}(P, P) + 2g_{\mu}(P, Q) = a. \] \bigskip {\bf Type IV(a,b).} \begin{center} \begin{picture}(8,2) \put(1,1){\circle*{.25}} \put(3,1){\circle*{.25}} \put(3.55,1){\circle{2}} \put(1,1){\line(1,0){2}} \put(1.0,0.5){$P$} \put(2.8,0.5){$Q$} \put(1.8,1.1){$G_1$} \put(3.5,1.6){$G_2$} \put(4.5,1.25){$\operatorname{length}(G_1) = a$} \put(4.5,0.75){$\operatorname{length}(G_2) = b$} \end{picture} \end{center} We fix coordinates $s : G_1 \to [0,a]$ and $t : G_2 \to [0, b)$ with $s(P) = 0$, $s(Q) = a$ and $t(Q) = 0$. In this case, ${\displaystyle \mu= \frac{\delta_P}{2} + \frac{dt}{2b}}$ by \cite[Lemma~3.7]{Zh}. We set \[ g(x) = \left\{ \begin{array}{ll} {\displaystyle -\frac{s(x)}{2} + \frac{b + 12a}{48}} & \mbox{if $x \in G_1$}, \\ & \\ {\displaystyle \frac{1}{2}\left( \frac{t(x)^2}{2b} - \frac{t(x)}{2} \right) + \frac{b - 12a}{48}} & \mbox{if $x \in G_2$}. \\ \end{array}\right. \] Then, $g$ is continuous, ${\displaystyle \Delta(\rest{g}{G_1}) = \frac{\delta_P}{2} - \frac{\delta_Q}{2}}$, and ${\displaystyle \Delta(\rest{g}{G_2}) = \frac{\delta_Q}{2} - \frac{dt}{2b}}$. Thus, $\Delta(g) = \delta_P - \mu$. Moreover, ${\displaystyle \int_G g \mu = 0}$. Therefore, $g(x) = g_{\mu}(P, x)$. Hence \[ g_{\mu}(P, P) = \frac{b+12a}{48}\qquad\mbox{and}\qquad g_{\mu}(P, Q) = \frac{b-12a}{48}. \] Thus \[ e_y = 6g_{\mu}(P, P) + 2g_{\mu}(P, Q) = a + \frac{b}{6}. \] \bigskip {\bf Type VI(a,b,c).} \begin{center} \begin{picture}(8,2) \put(2,1){\circle*{.25}} \put(4,1){\circle*{.25}} \put(1.45,1){\circle{2}} \put(4.55,1){\circle{2}} \put(2,1){\line(1,0){2}} \put(2.0,0.5){$P$} \put(3.8,0.5){$Q$} \put(2.8,1.1){$G_1$} \put(1.4,1.6){$G_2$} \put(4.5,1.6){$G_3$} \put(5.5,1.5){$\operatorname{length}(G_1) = a$} \put(5.5,1){$\operatorname{length}(G_2) = b$} \put(5.5,0.5){$\operatorname{length}(G_3) = c$} \end{picture} \end{center} We fix coordinates $s : G_1 \to [0,a]$, $t : G_2 \to [0,b)$ and $u : G_3 \to [0,c)$ with $s(P) = 0$, $s(Q) = a$, $t(P) = 0$ and $u(Q) = 0$. In this case, ${\displaystyle \mu= \frac{dt}{2b} + \frac{du}{2c}}$ by \cite[Lemma~3.7]{Zh}. We set \[ g(x) = \left\{ \begin{array}{ll} {\displaystyle \frac{1}{2}\left( \frac{t(x)^2}{2b} - \frac{t(x)}{2} \right) + \frac{b+c+12a}{48}} & \mbox{if $x \in G_2$}, \\ & \\ {\displaystyle -\frac{s(x)}{2} + \frac{b+c+ 12a}{48}} & \mbox{if $x \in G_1$}, \\ & \\ {\displaystyle \frac{1}{2}\left( \frac{u(x)^2}{2c} - \frac{u(x)}{2} \right) + \frac{b+c - 12a}{48}} & \mbox{if $x \in G_3$}. \\ \end{array}\right. \] Then, $g$ is continuous, ${\displaystyle \Delta(\rest{g}{G_1}) = \frac{\delta_P}{2} - \frac{\delta_Q}{2}}$, ${\displaystyle \Delta(\rest{g}{G_2}) = \frac{\delta_P}{2} - \frac{dt}{2b}}$, and ${\displaystyle \Delta(\rest{g}{G_3}) = \frac{\delta_Q}{2} - \frac{du}{2c}}$. Thus, $\Delta(g) = \delta_P - \mu$. Moreover, ${\displaystyle \int_G g \mu = 0}$. Therefore, $g(x) = g_{\mu}(P, x)$. Hence \[ g_{\mu}(P, P) = \frac{b+c+12a}{48}\qquad\mbox{and}\qquad g_{\mu}(P, Q) = \frac{b+c-12a}{48}. \] Thus \[ e_y = 6g_{\mu}(P, P) + 2g_{\mu}(P, Q) = a + \frac{b+c}{6}. \] \bigskip {\bf Type VII(a,b,c).} \begin{center} \begin{picture}(8,2) \put(1,1){\circle*{.25}} \put(4,1){\circle*{.25}} \put(1,1){\line(1,0){3}} \put(2.5,1){\oval(3,1)} \put(0.5,1){$P$} \put(4.3,1){$Q$} \put(2.4,1.58){$G_1$} \put(2.4,1.08){$G_2$} \put(2.4,0.58){$G_3$} \put(5.0,1.5){$\operatorname{length}(G_1) = a$} \put(5.0,1){$\operatorname{length}(G_2) = b$} \put(5.0,0.5){$\operatorname{length}(G_3) = c$} \end{picture} \end{center} We fix coordinates $s : G_1 \to [0,a]$, $t : G_2 \to [0,b]$ and $u : G_3 \to [0,c]$ with $s(P) = 0$, $s(Q) = a$, $t(P) = 0$, $t(Q) = b$, $u(P) = 0$ and $u(Q) = c$. In this case, ${\displaystyle \mu= \frac{ds}{3a} + \frac{dt}{3b} + \frac{du}{3c}}$ by \cite[Lemma~3.7]{Zh}. We set \[ g(x) = \left\{ \begin{array}{ll} {\displaystyle \frac{s(x)^2}{6a} - \left(\frac{1}{6} + \frac{1}{2}\frac{bc}{ab+bc+ca}\right)s(x) + \frac{a+b+c}{108} + \frac{1}{4}\frac{abc}{ab+bc+ca}} & \mbox{if $x \in G_1$}, \\ & \\ {\displaystyle \frac{t(x)^2}{6b} - \left(\frac{1}{6} + \frac{1}{2}\frac{ac}{ab+bc+ca}\right)t(x) + \frac{a+b+c}{108} + \frac{1}{4}\frac{abc}{ab+bc+ca}} & \mbox{if $x \in G_2$}, \\ & \\ {\displaystyle \frac{u(x)^2}{6c} - \left(\frac{1}{6} + \frac{1}{2}\frac{ab}{ab+bc+ca}\right)u(x) + \frac{a+b+c}{108} + \frac{1}{4}\frac{abc}{ab+bc+ca}} & \mbox{if $x \in G_3$}. \\ \end{array}\right. \] Then, $g$ is continuous and $\Delta(g) = \Delta(\rest{g}{G_1})+ \Delta(\rest{g}{G_2})+\Delta(\rest{g}{G_3}) = \delta_P - \mu$. Moreover, ${\displaystyle \int_G g \mu = 0}$. Therefore, $g(x) = g_{\mu}(P, x)$. Hence \[ e_y = 6g_{\mu}(P, P) + 2g_{\mu}(P, Q) = \frac{2}{27}(a+b+c) + \frac{abc}{ab+bc+ca}. \] \bigskip

9509

alg-geom/9509007

en

https://arxiv.org/abs/alg-geom/9509007

alg-geom/9509007

Robert Friedman

Robert Friedman, John W. Morgan

Obstruction bundles, semiregularity, and Seiberg-Witten invariants

AMS-TeX, preprint style

We compare the deformation theory and the analytic structure of the Seiberg-Witten moduli spaces of a K\"ahler surface to the corresponding components of the Hilbert scheme, and show that they are isomorphic. Next we show how to compute the invariant in case the moduli space is smooth but not of the expected dimension, and apply this study to elliptic surfaces. Finally we discuss ruled surfaces, both products and more general ruled surfaces. For product ruled surfaces we relate the infinitesimal structure of the moduli spaces to Brill-Noether theory and compute the invariant in special cases. For more general ruled surfaces, we relate the geometry of the Hilbert scheme to properties of stable bundles and give more general computations.

alg-geom

\section{Introduction} Recently, Seiberg and Witten have introduced new invariants for smooth $4$-manifolds which have led to dramatic progress in understanding the $C^\infty$ properties of algebraic surfaces. Just as with Donaldson theory, the new invariants are computed from a moduli space which, in case the underlying $4$-manifold is a K\"ahler surface $X$, can be identified with a moduli space of holomorphic objects. In Donaldson theory, the holomorphic moduli space is the space of holomorphic structures on a fixed $C^\infty$ complex vector bundle over $X$ satsifying an additional nondegeneracy condition, stability. Such moduli spaces have a rich geometric structure even for very simple K\"ahler surfaces, such as $\Pee ^2$, and seem to become more progressively complicated as the surface becomes more complicated. In Seiberg-Witten theory, the relevant moduli spaces are the spaces of complex curves $D$ on $X$, which are thus parametrized by the Hilbert scheme of $X$, such that $D$ satisfies an additional numerical condition akin to stability. Now the structure of the Hilbert scheme of curves on a smooth surface is an interesting problem in algebraic geometry. However, it turns out for rather trivial reasons involving the Hodge index theorem that the geometric interest of the Seiberg-Witten moduli spaces of a surface $X$ is in a certain sense inversely proportional to the interest in $X$ itself as an abstract surface. Thus for example if $X$ is a minimal surface of general type the Seiberg-Witten moduli spaces are two reduced points corresponding to the trivial (empty) curve. Of course, it is this fact which enables one to prove that the first Chern class of the canonical bundle of a minimal surface of general type is a $C^\infty$ invariant up to sign. At the other extreme, if $X$ is a ruled surface over a curve $C$ of genus at least $2$, then the Seiberg-Witten moduli spaces are connected with the Brill-Noether theory of special divisors on $C$, if $X= \Pee ^1\times C$ is a product ruled surface, and to various interesting questions concerning stable bundles over $C$ in general. Our goal in this paper is to discuss these and other related examples. The outline of this paper is as follows. In Section 1 we construct the Hilbert scheme of a complex surface via $\dbar$ methods. To our knowledge, such a construction has not appeared in the literature. In Section 2 we identify the deformation complex for the Seiberg-Witten equations of a K\"ahler surface in holomorphic terms and show that the Kuranishi model for the Seiberg-Witten equations is the same as the Kuranishi model for the equations defining the Hilbert scheme. In other words, the natural homeomorphism from the Seiberg-Witten moduli space to the Hilbert scheme of ``stable" divisors on $X$ is an isomorphism of real analytic spaces. In Section 3 we discuss how to make computations in case the moduli space is smooth but does not have the expected dimension, using the Euler class of the obstruction bundle. These arguments and various generalizations are well-known to specialists in many different contexts. In Section 4 we apply this study to elliptic surfaces. There is a substantial overlap of the material in Sections 2--4 with the paper of Brussee \cite{2}. The remainder of the paper is concerned with ruled surfaces. We discuss the infinitesimal and analytic structure of the moduli space for product ruled surfaces in Section 5, and then compute the invariant in the special case where the curve involved is a section of the surface (possibly with some fiber components). In Section 6, we deform the surface to a general ruled surface and show that the Hilbert scheme of sections is much better behaved: it is always smooth of the correct dimension. Using this result, we give another computation of the invariants in the case of a section. This computation goes back to Corrado Segre in 1889 \cite{11} and was given a modern proof, for the case of the $0$-dimensional invariant, by Ghione \cite{6}. (Note that Segre considered the case of moduli spaces of sections of arbitrary dimension.) We shall give a quick description of these and related results. These methods generalize to compute the invariant in general homologically; the problem is that it is not known whether, for a general ruled surface, the Hilbert scheme always has the correct dimension. Finally we remark that the computation of the invariant is a special case of the transition formula for Seiberg-Witten invariants for $4$-manifolds with $b_2^+ = 1$. This formula has been computed by the authors, by methods quite reminiscent of those in Section 6, as well as by Li and Liu \cite{9}. Thus our goal in Sections 5 and 6 has been, not so much to compute the invariant (although it is amusing to see the connections with the enumerative calculations of Brill-Noether theory) as it has been to see the relationship between the study of the Seiberg-Witten moduli spaces for ruled surfaces and questions in Brill-Noether theory as well as the theory of rank two stable bundles on curves. \section{1. Structure of the Hilbert scheme.} Let $X$ be an algebraic (or complex) surface, and let $D_0$ be an effective divisor on $X$. We do not assume that $D_0$ is smooth or even reduced. Let $H_{D_0,X}$ be the Hilbert scheme of all effective divisors $D$ on $X$ such that $c_1(\scrO_X(D))= c_1(\scrO_X(D_0))$ in $H^2(X; \Zee)$. As a set, $H_{D_0,X}$ consists of all effective divisors $D$ homologous to $D_0$, i\.e\. algebraically equivalent to $D_0$. There is a morphism $H_{D_0,X}\to \operatorname{Pic}X$ whose fibers are projective spaces. Over $X\times H_{D_0,X}$ there is a tautological divisor $\Cal D$ whose restriction to each slice $X\times \{t\}$ is the divisor $D_t$ on $X$ corresponding to $t$. A general reference for the construction of $H_{D_0,X}$ and its properties is \cite{10}. The infinitesimal structure of $H_{D_0,X}$ is given as follows: from the natural exact sequence $$0 \to \scrO_X \to \scrO_X(D_0) \to \scrO_{D_0}(D_0)\to 0,$$ we have the associated long exact cohomology sequence. The Zariski tangent space to $H_{D_0,X}$ is naturally the space of sections of the normal bundle $H^0(D_0;\scrO_{D_0}(D_0))$. Note that the long exact cohomology sequence gives $$\gather 0 \to H^0(X; \scrO_X(D_0))/H^0(X; \scrO_X) \to H^0(D_0;\scrO_{D_0}(D_0))\\ \to H^1(X; \scrO_X)\to H^1(X; \scrO_X(D_0)). \endgather$$ Here $H^0(X; \scrO_X(D_0))/H^0(X; \scrO_X)$ is the space of sections of $\scrO_X(D_0)$ modulo the line through a nonzero section vanishing along $D_0$, and is thus naturally the tangent space to the linear system $|D_0|$ at $D_0$. The map $H^0(D_0;\scrO_{D_0}(D_0)) \to H^1(X; \scrO_X)$ represents the infinitesimal change in the line bundle $\scrO_X(D_0)$. We let $K_0$ denote the image of $H^0(D_0;\scrO_{D_0}(D_0))$ in $H^1(X; \scrO_X)$, so that $K_0$ is the kernel of the map from $H^1(X; \scrO_X)$ to $H^1(X; \scrO_X(D_0))$ defined by $\sigma _0$. Thus there is an exact sequence $$0 \to H^0(X; \scrO_X(D_0))/H^0(X; \scrO_X) \to H^0(D_0;\scrO_{D_0}(D_0)) \to K_0 \to 0.$$ The obstruction space to the deformation theory of $H_{D_0,X}$ is given as follows: let $K_1$ be the image of $H^1(X; \scrO_X(D_0))$ in $H^0(D_0;\scrO_{D_0}(D_0))$, or in other words the cokernel of the map from $H^1(X; \scrO_X)$ to $H^1(X; \scrO_X(D_0))$ defined by $\sigma _0$. Then $K_1$ is the obstruction space to the functor corresponding to $H_{D_0,X}$. If $K_1=0$, then $H_{D_0,X}$ is scheme-theoretically smooth at $D_0$ of dimension equal to $\dim H^0(D_0;\scrO_{D_0}(D_0))$. (The converse is not necessarily true.) We say that $D_0$ is {\sl semiregular\/} if $K_1=0$, or in other words if the map $H^1(X; \scrO_X(D_0)) \to H^0(D_0;\scrO_{D_0}(D_0))$ is zero. The following theorem was proved by Kodaira-Spencer \cite{7} in the semiregular case (and was claimed by Severi): \theorem{1.1} Let $D_0$ be a curve on $X$. Then the Zariski tangent space $T$ to $H_{D_0,X}$ at $D_0$ fits into an exact sequence $$0 \to H^0(X; \scrO_X(D_0))/H^0(X; \scrO_X) \to T \to K_0 \to 0,$$ where $K_0 = \Ker \{\times \sigma _0\: H^1(X; \scrO_X)\to H^1(X; \scrO_X(D_0))\}$. Locally analytically in a neighborhood of $D_0$, $H_{D_0,X}$ is defined by the vanishing of a convergent power series without constant or linear term from $T$ to $$K_1 = \Coker \{\times \sigma _0\: H^1(X; \scrO_X)\to H^1(X; \scrO_X(D_0))\}.$$ \endstatement To prove Theorem 1.1, one can analyze the deformation theory and obstruction theory for $H_{D_0,X}$ via power series as in \cite{7} and \cite{10}, and apply Schlessinger's theory. Here we give a $C^\infty$ proof of Theorem 1.1. Given $D_0$, let $L_0$ denote the $C^\infty$ complex line bundle defined by $\scrO_X(D_0)$. From this point of view, the scheme $H_{D_0,X}$ is the set (with real analytic structure) of all $C^\infty$ sections of $L_0$ which are complex analytic for some choice of holomorphic structure on $L_0$, modulo the action of the nowhere zero functions acting by multiplication. We fix a given holomorphic structure on $L_0$ with $\dbar$-operator simply denoted by $\dbar$, and a given nonzero holomorphic section $\sigma _0$ of $L_0$ for this holomorphic structure. The equations which say that $s$ is a holomorphic section for some holomorphic structure on $L_0$ read as follows: there exist a $\dbar$-closed $(0,1)$-form $A$ such that $(\dbar +A)(s) = 0$. Thus $H_{D_0,X}$ is the zero set of the function $$F_0\: \Ker \dbar\oplus \Omega ^0(L_0)\subset \Omega ^{0,1}(X) \oplus \Omega ^0(L_0)\to \Omega ^{0,1}(L_0)$$ defined by $$F_0(A,s) = (\dbar +A)( s) = \dbar s + As,$$ modulo the action of $\Cal G_\Cee$, the {\sl complex gauge group\/}, where $\Cal G_\Cee$ is the multiplicative group of nowhere vanishing $C^\infty$ functions on $X$ and $\lambda \in \Cal G_\Cee$ acts on $(A,s)$ via $(A - \dbar \lambda/\lambda, \lambda \cdot s)$. An easy calculation shows that $F_0\circ \lambda = \lambda F_0$. Of course, in order to analyze the equations, we need to pass to an appropriate Sobolev completion of all of these spaces, but we shall leave the details of this standard procedure to the reader. Next we calculate the linearized complex. The space $\Omega ^0(L_0)$ of all $C^\infty$ sections of $L_0$ is a vector space, and we may thus identify the tangent space to $\Omega ^0(L_0)$ at a given section $\sigma _0$ vanishing at $D_0$ with $\Omega ^0(L_0)$ again. The space of all $(0,1)$-conections on $L_0$ is an affine space over $\Omega ^{0,1}(X)$, with origin the $\dbar$-operator corresponding to the given complex structure, and so its tangent space is $\Omega ^{0,1}(X)$. The nowhere zero functions on $X$ may be (locally) identified with $\Omega ^0(X)$, the set of all $C^\infty$ functions on $X$, via the exponential, and the differential at $s= \sigma _0, \lambda = 0$ of $$\lambda \in \Omega ^0(X) \mapsto e^\lambda \cdot s$$ is multiplication by $\sigma _0$: $\lambda \mapsto \lambda \cdot \sigma _0$. Taking the differential of the $\Cal G_\Cee$-action, we obtain a complex $\Cal C_0$: $$0 \to \Omega ^0(X) @>{d_1}>> \Ker \dbar\oplus\Omega ^0(L_0) @>{d_2}>> \Omega ^{0,1}(L_0).$$ Here the map $d_1\:\Omega ^0(X) \to \Ker \dbar\oplus\Omega ^0(L_0)$ sends $\lambda$ to $(-\dbar \lambda,\lambda \cdot \sigma _0)$ and $d_2\:\Ker \dbar\oplus\Omega ^0(L_0) \to \Omega ^{0,1}(L_0)$ sends $(A,s)$ to $\dbar s + A\sigma _0$. However, this complex is not elliptic. Thus, the restriction of $F_1$ to $(\operatorname{Im} d_1)^\perp$ is not Fredholm. To remedy the above problem, consider instead the function $F(A,s) = \pi \circ F_0(A,s)$, where $\pi \: \Omega ^{0,1}(L_0) \to \Ker \dbar$ is orthogonal projection onto the kernel of $\dbar$. (Note that $F$ is not in fact equivariant with respect to the action of $\Cal G_\Cee$.) \lemma{1.2} For $A$ in a neighborhood of zero in $\Ker \dbar$, $F(A,s) = 0$ if and only if $F_0(A,s) = 0$. \endstatement \proof Clearly, if $F_0(A,s) =0$, then $F(A,s) =0$. Conversely, suppose that $F(A,s) = 0$. This says that $F_0(A,s)$ is orthogonal to $\Ker \dbar$, so that $F_0(A,s)= (\dbar +A)(s)=\dbar ^*\gamma$ for some $\gamma \in \Omega ^{0,2}(L_0)$. By hypothesis $\dbar A =0$, so that $(\dbar +A)^2 =0$. Applying $\dbar +A$ to $F_0$, we obtain $(\dbar +A)\dbar ^*\gamma =0$, or in other words $\dbar \dbar ^*\gamma +A\dbar ^*\gamma = 0$. We claim that, in this case, if $A$ lies in some neighborhood of zero, then $\dbar ^*\gamma =0$ and thus $F_0(A,s) = 0$. We may restrict $\gamma $ to $(\Ker \dbar ^*)^\perp = \operatorname{Im}\dbar$, and in this case $\dbar \dbar ^*$ is an isomorphism from $\operatorname{Im}\dbar$ to itself (after taking appropriate completions). Likewise, if $\pi_0$ denotes orthogonal projection from $\Omega ^{0,2}(L_0)$ to $\operatorname{Im}\dbar$, then $\pi _0\circ( \dbar \dbar ^* + A\dbar ^*)$ is a bounded map from $\operatorname{Im}\dbar$ to itself, after taking appropriate completions, which is invertible for $A=0$ and so for $A$ in a neighborhood of zero. It follows that for $A$ in some neighborhood of zero, and for an arbitrary $\gamma \in \Omega ^{0,2}(L_0)$, if $\dbar \dbar ^*\gamma +A\dbar ^*\gamma = 0$, then for $\gamma _0$ the projection of $\gamma $ to $(\Ker \dbar ^*)^\perp$, we have $\dbar ^*\gamma = \dbar ^*\gamma _0$ and $\pi _0\circ( \dbar \dbar ^* + A\dbar ^*)(\gamma _0) = 0$, so that $\gamma _0 = 0$ and $\gamma \in \Ker \dbar ^*$. Hence $\dbar ^*\gamma =0$ and so $F_0(A,s) =0$ as claimed. \endproof The linearization of the equation $F$ and the gauge group action at $(0, \sigma _0)$ gives a complex $\Cal C_1$ defined by the top line of the following commutative diagram: $$\CD 0@>>> \Omega ^0(X) @>{e_1}>> \Ker \dbar \oplus \Omega ^0(L_0) @>{e_2}>> \Ker \dbar @>>> 0\\ @. @| @VVV @VVV @.\\ @. \Omega ^0(X) @>>>\Omega ^{0,1}(X) \oplus \Omega ^0(L_0) @>>> \Omega ^{0,1}(L_0), @. \endCD$$ where the vertical maps are the natural inclusions, and the differentials are given by $e_1 (\lambda) = (-\dbar \lambda, \lambda \sigma _0)$ and $e_2(A,s) = \dbar s + A\sigma _0$. There is a subcomplex $\Cal C'$ defined by $$ \Omega ^0(L_0) @>{\dbar}>> \Ker \{\,\dbar \: \Omega ^{0,1}(L_0)\to \Omega ^{0,2}(L_0)\,\},$$ shifted up a dimension, with differential $\dbar$, and the quotient complex is isomorphic to the complex $\Cal C''$ defined by: $$\Omega ^0(X) @>{\dbar}>> \Ker \{\,\dbar \: \Omega ^{0,1}(X)\to \Omega ^{0,2}(X)\,\}.$$ Thus the deformation complex is elliptic and so the restriction of $F$ to a slice for the $\Cal G_\Cee$ action is Fredholm. Taking the long exact cohomology sequence associated to the exact sequence of complexes $$ 0 \to \Cal C' \to \Cal C_1 \to \Cal C'' \to 0,$$ we see that the cohomology of $\Cal C_1$ fits into the exact sequence $$0 \to H^0(\scrO_X) \to H^0(L_0) \to H^1(\Cal C_1) \to H^1(\scrO_X) \to H^1(L_0) \to H^2(\Cal C_1) \to 0.$$ A routine calculation shows that the induced maps $H^i(\scrO_X) \to H^i(L_0)$ are given by multiplication by $\sigma _0$. Thus $H^1(\Cal C_1)$ satisfies the exact sequence for $T$ given in Theorem 1.1 and $H^2(\Cal C_1) \cong K_1$. This concludes the proof of Theorem 1.1. \qed \medskip The following identifies the quadratic term of the obstruction map: \proposition{1.3} Let $\xi \in H^0(\scrO_{D_0}(D_0))$ be an element of the Zariski tangent space to $H_{D_0,X}$. Then the quadratic term in the obstruction map is equal to $\delta \xi \cup \xi \in H^1(\scrO_{D_0}(D_0))$, where $\delta \: H^0(\scrO_{D_0}(D_0)) \to H^1(\scrO_X)$ is the coboundary map in the natural long exact sequence. \endstatement \proof It is easy to give the quadratic term of the obstruction map using the power series approach of \cite{7}. In terms of the approach outlined here, the quadratic term of $F(A,s)$ is $\pi (A\cdot s)$, where $\dbar A = 0$ and $A\sigma _0 = -\dbar s$. If we identify the class of $(A,s)$ in $H^1(\Cal C_1)$ with an element $\xi \in H^0(\scrO_{D_0}(D_0))$, then it is easy to see that the class of $A$ in $H^{0,1}(X) = H^1(\scrO_X)$ is exactly $\delta \xi$. One then checks that the projection of $A\cdot s$ to $\Ker \dbar$ corresponds to $\delta \xi \cup \xi$. \endproof Note that, if we apply $\delta\: H^1(\scrO_{D_0}(D_0)) \to H^2(\scrO_X)$ to the element $\delta \xi \cup \xi$, we obtain $\delta \xi \cup \delta \xi \in H^2(\scrO_X)$, which is zero since $\delta \xi \cup \delta \xi = -\delta \xi \cup \delta \xi$ as $\scrO_X$ is a sheaf of commutative rings. Thus $\delta \xi \cup \xi$ lies in the image of $H^1(X; \scrO_X(D_0))$ in $H^0(D_0;\scrO_{D_0}(D_0))$. There is also clearly a universal divisor $\Cal D \subset X\times F^{-1}(0)$ defined by the vanishing of $s$. This completes the analytic construction of $H_{D_0,X}$ and the discussion of semiregularity. Note that we have not strictly speaking shown that $\Cal D$ is a divisor on the complex space $X\times H_{D_0,X}$. This would need a discussion of relative $\dbar$-operators similar to, but easier than, the discussion in \cite{4}, Chapter IV, 4.2.3. In other words, we would need to show that $\Cal D$ is a Cartier divisor in the possibly nonreduced complex space $X\times H_{D_0,X}$, which follows by showing that locally on $X\times H_{D_0,X}$, there is a holomorphic embedding of the complex space $X\times H_{D_0,X}$ in $X\times \Cee ^N$ for some $N$ so that $\Cal D$ is locally the restriction of a complex hypersurface. Finally, to identify this construction with the usual construction of $H_{D_0,X}$, and to make a geometric identification of $H_{D_0,X}$ possible, we would have to show that $H_{D_0,X}$ has a universal property. In other words, given a complex space $T$, not necessarily reduced, and a Cartier divisor on $X\times T$, flat over $T$, we need to exhibit a morphism of complex spaces from $T$ to $H_{D_0,X}$. This again can be done along the lines of \cite{4}, Chapter IV. In the cases described in this paper, $H_{D_0,X}$ will be smooth or a union of generically reduced components, and the arguments needed are substantially simpler than the arguments in the general case. The divisor $\Cal D$ is a Cartier divisor and so there is a holomorphic line bundle $\scrO_{X\times H_{D_0,X}}(\Cal D)$ over $X\times H_{D_0,X}$. Slant product with $c_1(\scrO_{X\times H_{D_0,X}}(\Cal D))$ defines a map $H_0(X) \to H^2( H_{D_0,X})$, and we let $\mu$ be the image of the natural generator of $H_0(X; \Zee)$ under this map. Clearly $\mu = \pi _2{}_*c_1(\scrO_{X\times H_{D_0,X}}(\Cal D))= \pi _2{}_*[\Cal D]$. For a fixed $p\in X$, there is the inclusion of the slice $\{p\}\times H_{D_0,X}$ in $X\times H_{D_0,X}$, and clearly $\mu$ is the first Chern class of the line bundle $\scrO_{X\times H_{D_0,X}}(\Cal D)|\{p\}\times H_{D_0,X}$, under the natural identification of $\{p\}\times H_{D_0,X}$ with $H_{D_0,X}$. If $\Cal D$ meets $\{p\}\times H_{D_0,X}$ properly, or in other words if there is no component $\Cal M$ of $ H_{D_0,X}$ such that $p$ lies in every divisor in $\Cal M$, then $\Cal D \cap \{p\}\times H_{D_0,X}$ is a Cartier divisor in $\{p\}\times H_{D_0,X} \cong H_{D_0,X}$, whose support is the set of divisors $D$ such that $p\in D$, and this divisor is a geometric representative for $\mu$. In fact, the divisor $\mu$ is an ample divisor on $ H_{D_0,X}$, which can be shown for example by using the method of Chow schemes described in \cite{10}, Lecture 16, and identifying the numerical equivalence class of $\mu$ up to a positive rational multiple with the natural ample divisor on the Chow scheme. There is another description of the complex line bundle corresponding to $\mu$. For $p\in X$, let $\Cal G_\Cee ^0 \subset \Cal G_\Cee$, the {\sl based gauge group\/}, be the set of $\lambda \in \Cal G_\Cee$ such that $\lambda (p) =1$. Thus the quotient of $\Cal G_\Cee$ by $\Cal G_\Cee^0$ is $\Cee ^*$, and if instead of dividing out $F^{-1}(0)$ by the local action of $\Cal G_\Cee$ we divide out by $\Cal G_\Cee^0$, the result is a $\Cee ^*$-bundle over $H_{D_0,X}$, which thus corresponds to a complex line bundle $\Cal L_0(p)$. We claim that this line bundle has first Chern class equal to $\mu$. First note that there is a universal $C^\infty$ complex line bundle $\Cal L_0$ over $X\times H_{D_0,X}$ whose restriction to the slice $\{p\}\times H_{D_0,X}$ is $\Cal L_0(p)$. Here $\Cal L_0$ is defined as follows: let $\Cal A_\Cee^*(L_0)$ be the set of pairs $(A,s)$ where $A$ is a $(0,1)$-connection on $L_0$ and $s$ is a nonzero section of $L_0$. Then $\Cal G_\Cee$ acts freely on $\Cal A_\Cee^*(L_0)$; let the quotient be denoted $\Cal B_\Cee(L_0)$. Since $\Cal G_\Cee$ also acts as a group of automorphisms of $L_0$, there is a line bundle $\Cal L_0$ over $X\times \Cal B_\Cee(L_0)$ obtained by dividing out $L_0 \times \Cal A_\Cee^*(L_0)$ by the action of $\Cal G_\Cee$. We also denote the restriction of this line bundle to $X\times F^{-1}(0)$ by $\Cal L_0$. So we must identify $\Cal L_0$ with $c_1(\scrO_{X\times H_{D_0,X}}(\Cal D))$ (at least on the reduction of $H_{D_0,X}$). The point is that the tautological section $(s, (A,s))$ of the pullback of $L_0$ to $X\times \Cal A_\Cee^*(L_0)$ is $\Cal G_\Cee$-equivariant and so descends to a section of $\Cal L_0$ over $X\times \Cal B_\Cee(L_0)$. Restricting to $X\times F^{-1}(0)$, we see that $\Cal L_0$ has a section vanishing at $\Cal D$, and this identifies $\Cal L_0$ with $\scrO_{X\times H_{D_0,X}}(\Cal D)$. \section{2. Deformation theory for Seiberg-Witten moduli spaces of K\"ahler surfaces.} In this section we recall the description of Seiberg-Witten moduli spaces for K\"ahler surfaces and compare this description to the discussion of the Hilbert scheme in the previous section. For general references on Seiberg-Witten moduli spaces of K\"ahler surfaces, see \cite{12}, \cite{3}, as well as \cite{5}. For a given metric $g$ on $X$ and Spin${}^c$ structure $\xi$ on $X$ with determinant $L$, the (unperturbed) Seiberg-Witten equations for a pair $(A, \psi)$, where $A$ is a connection on $L$ and $\psi$ is a section of $\Bbb S^+(\xi)$, the plus spinor bundle associated to $\xi$, are $$\align \dirac_A\psi&=0;\\ F_A^+ =q(\psi) &=\psi\otimes \psi^*-\frac{|\psi|^2}{2}\operatorname{Id}. \endalign$$ We let $\Cal M_g(\xi)$ be the corresponding moduli space. In case $X$ is a K\"ahler surface and $g$ is a K\"ahler metric with associated K\"ahler form $\omega$, $\Bbb S^+(\xi) \cong \Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0)$ for a complex line bundle $L_0$, and $\Bbb S^-(\xi) \cong \Omega ^{0,1}(L_0)$. Moreover $L_0 ^{\otimes 2} = L\otimes K_X$, where $L = \det \xi$. We assume that $\omega \cdot c_1(L) \neq 0$, so that there are no reducible solutions, and for simplicity we fix $\omega \cdot c_1(L) <0$. In this case, writing $\psi$ in components $(\alpha, \beta)$, where $\alpha$ is a section of $\Omega ^0(L_0)$ and $\beta$ is a section of $\Omega ^{0,2}(L_0)$, the Seiberg-Witten equations become $$\align F^{0,2} &= \dbar A^{0,1} =\bar\alpha\beta ;\\ (F_A^+)^{1,1} &=\frac{i}{2}(|\alpha|^2-|\beta|^2)\omega ;\\ \bar\partial_A\alpha+\bar\partial^*_A\beta &=0. \endalign$$ Under the assumption that $\alpha \neq 0$, the equations $\dbar A^{0,1} = \bar\alpha\beta$ and $\dbar _A(\bar\partial_A\alpha+\bar\partial^*_A\beta) = 0$ imply that $\beta = 0$, and that $A$ is a $(1,1)$-conection on $L$ \cite{12, 3, 5}. Hence $L$ and $L_0$ have given holomorphic structures, and $\dbar _A\alpha = 0$, so that $\alpha$ is a nonzero holomorphic section of $L_0$. Thus $\alpha$ defines an effective divisor $D$ with $L_0 = \scrO_X(D)$. Taking gauge equivalence defines $\alpha$ up to scalars, or in other words as an element of $|D|$. Thus to each element of $\Cal M_g(\xi)$, there is a well-defined element of $H_{D_0,X}$ for some fixed divisor $D_0$ such that the $C^\infty$ line bundle underlying $\scrO_X(2D_0) \otimes K_X^{-1}$ is $L$. Conversely, to every point of $H_{D_0,X}$ we can associate an irreducible solution of the Seiberg-Witten equations mod gauge equivalence, in other words a point of $\Cal M_g(\xi)$, which essentially follows from a theorem of Kazdan-Warner. It is easy to see that the map from $\Cal M_g(\xi)$ to $H_{D_0,X}$ is a homeomorphism, and we shall show that it is an isomorphism of real analytic spaces in a suitable sense. As in the previous section, we shall pass to Sobolev completions of all of the spaces of $C^\infty$ sections involved without making the choice of completions explicit. We begin by discussing the deformation complex associated to the Seiberg-Witten equations for a K\"ahler surface. For a general Riemannian 4-manifold $X$, at an irreducible solution $(A_0, \psi)$ to the Seiberg-Witten equations, the appropriate deformation complex $\Cal C$ is $$0 \to i\Omega ^0(X;\Ar) @>{\delta _1}>> i\Omega ^1(X;\Ar) \oplus \Bbb S^+(\xi)@>{\delta _2}>> i\Omega ^2_+(X; \Ar) \oplus \Bbb S^-(\xi) \to 0.$$ Here $\Omega ^2_+(X)$ is the space of $C^\infty$ self-dual 2-forms. The differentials are as follows: $\delta _1(\lambda) = (-2d\lambda, \lambda \psi )$ and $$\delta _2(A, \eta) = (d^+A - Dq_{\psi}(\eta), \dirac\eta + \frac12A\cdot \psi).$$ Here $d^+$ is the self-dual part of $d$, $Dq_{\psi}$ is the differential of the quadratic map $q$ in the SW equations, evaluated at $\psi$ on $\psi$, and $\frac12A\cdot \psi$ is the linear term of $\dirac _{A+A_0}\psi $. A calculation shows that $$Dq_\psi(\eta) = \eta \otimes \psi ^* + \psi \otimes \eta ^* - \operatorname{Re} \langle \psi, \eta \rangle \operatorname{Id}.$$ In general, it seems to be somewhat difficult to analyze this complex. In the case of a K\"ahler surface $X$, however, we can give a very explicit description of the cohomology of the deformation complex. First we recall the notation of the previous section: $$\align K_0 &= \Ker \{\times \sigma _0\: H^1(\scrO_X) \to H^1(L_0)\};\\ K_1 &= \operatorname{Coker}\{\times \sigma _0\: H^1(\scrO_X) \to H^1(L_0)\};\\ K_2 &= \Ker \{\times \sigma _0\: H^2(\scrO_X) \to H^2(L_0)\}. \endalign$$ (Note that $\times \sigma _0\: H^2(\scrO_X) \to H^2(L_0) $ is surjective.) \theorem{2.1} Suppose that $X$ is a K\"ahler surface with K\"ahler metric $g$ and associated K\"ahler form $\omega$. Let $\sigma _0$ be a nonzero holomorphic section of $L_0$ corresponding to an irreducible solution $(\sigma _0, 0)$ of the Seiberg-Witten equations. Then the Zariski tangent space $H^1(\Cal C)$ to the Seiberg-Witten moduli space sits in an exact sequence $$0 \to H^0(L_0)/\Cee \sigma _0 \to H^1(\Cal C) \to K_0\to 0$$ and the obstruction space $H^2(\Cal C)$ sits in an exact sequence $$0 \to K_1 \to H^1(L_0)\} \to H^2(\Cal C) \to K_2\to 0.$$ \endstatement \proof The complex $\Cal C$ has the following complex as its symbol complex: $$0 \to i\Omega ^0(X;\Ar) @>{(d,0)}>> i\Omega ^1(X;\Ar) \oplus \Bbb S^+(\xi)@>{(d^+, \dirac)}>> i\Omega ^2_+(X; \Ar) \oplus \Bbb S^-(\xi) \to 0.$$ Thus it is elliptic and its (real) index is the same as the index of the above complex, namely $$1-b_1(X) + b_2^+(X) - 2(h^0(L_0) + h^2(L_0) - h^1(L_0)) = 2\chi (\scrO_X) - 2\chi (L_0).$$ Here we have used the identification of $\dirac$ with $\dbar + \dbar ^*$ up to a factor of $\sqrt{2}$. Note that $\delta _1(\lambda ) = 0$ if and only if $\lambda$ is constant and $\lambda \sigma _0 = 0$. Thus $H^0(\Cal C) = 0$, which just says that the point $(A, (\sigma _0, 0)$ is an irreducible solution to the Seiberg-Witten equations, and we must identify the terms $H^1(\Cal C)$ and $H^2(\Cal C)$. Identify $i\Omega^1(X; \Ar)$ with $\Omega ^{0,1}(X)$, $\Bbb S^+(\xi)$ with $\Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0)$, $i\Omega _+^2(X; \Ar)$ with $i\Omega ^0(X; \Ar)\omega \oplus \Omega ^{0,2}(X)$, and $\Bbb S^-(\xi)$ with $\Omega ^{0,1}(L_0)$. Under these identifications, for $\lambda \in i\Omega ^0(X; \Ar)$, $\delta _1 \lambda = (-\dbar \lambda, \lambda \sigma _0, 0)$ which has image inside $\Omega ^{0,1}(X)\oplus \Omega ^0(L_0)$. Moreover, $\delta _2 (A^{0,1}, \alpha, \beta)$ is given by $$( (\dbar \bar A^{0,1} + \partial A^{0,1})^+ - \operatorname{Re} \langle \sigma _0, \alpha \rangle i\omega, \dbar A^{0,1} -\bar \sigma _0 \beta, \dbar \alpha + \dbar ^*\beta + \frac12A^{0,1} \sigma _0 ).$$ After identifying $i\Omega ^0(X; \Ar)\omega$ with $\Omega ^0(X; \Ar)$ by taking $-i$ times the contraction $\Lambda$ with $\omega$, we can write this as $$\delta _2 (A^{0,1}, \alpha, \beta) = (T_1(A^{0,1}, \alpha), T_2(A^{0,1}, \alpha) + S(\beta)),$$ where (since $\omega \wedge \omega $ is twice the volume form) $$\align T_1(A^{0,1}, \alpha) &= -i(\Lambda\dbar \bar A^{0,1} + \Lambda \partial A^{0,1}) - 2\operatorname{Re} \langle \sigma _0, \alpha \rangle \in \Omega ^0(X; \Ar) ;\\ T_2(A^{0,1}, \alpha) &= (\dbar A^{0,1}, \dbar \alpha + \frac12A^{0,1} \sigma _0) \in \Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0);\\ S(\beta) &= ( -\bar \sigma _0 \beta,\dbar ^*\beta) \in \Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0). \endalign$$ Thus $\delta _2(A^{0,1}, \alpha, \beta) =0$ if and only if $T_1(A^{0,1}, \alpha) = 0$ and $T_2(A^{0,1}, \alpha) + S(\beta) =0$. Next we claim: \lemma{2.2} $T_2(A^{0,1}, \alpha) + S(\beta) =0$ if and only if $\beta = 0$ and $T_2(A^{0,1}, \alpha) =0$. Moreover, $\dbar A^{0,1} =\bar \sigma _0 \beta$ and $\dbar (\dbar \alpha + \dbar ^*\beta + \frac12A^{0,1} \sigma _0) = 0$ if and only if $\beta = 0$ and $\dbar A^{0,1} =0$. \endstatement \proof Clearly, if $\beta = 0$ and $T_2(A^{0,1}, \alpha) =0$, then $T_2(A^{0,1}, \alpha) + S(\beta) =0$. Conversely, suppose that $T_2(A^{0,1}, \alpha) + S(\beta) =0$, or in other words that $\dbar A^{0,1} =\bar \sigma _0 \beta$ and $\dbar \alpha + \dbar ^*\beta + \frac12A^{0,1} \sigma _0=0$. Taking $\dbar$ of the second equation, we find that $$\dbar A^{0,1}\cdot\sigma _0 + \dbar \dbar ^*\beta = |\sigma _0|^2\beta + \dbar \dbar ^*\beta = 0.$$ Taking the inner product with $\beta$ shows that $$\int _X|\sigma _0|^2 |\beta |^2 + \|\dbar ^*\beta\|^2 = 0.$$ Hence $\beta = 0$, and clearly then $T_2(A^{0,1}, \alpha) =0$ as well. The proof of the second assertion is similar. \endproof Now we exhibit an isomorphism from $H^1(\Cal C)$ to $H^1(\Cal C_1)$, where $\Cal C_1$ is the complex defined in the previous section, up to a factor of $2$ (which arises because in our point of view $A^{0,1}$ is a connection on $L_0^{\otimes 2} \otimes K_X$ rather than on $L_0$): $H^1(\Cal C_1)$ is the quotient of $$\{\, (A^{0,1}, \alpha): \dbar A^{0,1} = 0, \dbar \alpha + \frac12A^{0,1}\sigma _0 = 0\,\}$$ by the subgroup of elements of the form $(-2\dbar f, f\sigma_0)$, where $f$ is a {\sl complex\/} valued $C^\infty$ function. To exhibit this isomorphism, given a class in $H^1(\Cal C)$ represented by $(A^{0,1}, \alpha, \beta)$, then, by Lemma 2.2, $\beta = 0$ and $\dbar \alpha + \frac12A^{0,1} \sigma _0 = 0$. Moreover $(A^{0,1}, \alpha)$ is well-defined up to the subgroup of the form $(-2\dbar \lambda, \lambda\sigma_0)$, where $\lambda$ is a purely imaginary $C^\infty$ function. Thus $(A^{0,1}, \alpha)$ defines an element of $H^1(\Cal C_1)$. Conversely, start with a representative $(A^{0,1}, \alpha)$ for an element of $H^1(\Cal C_1)$. Then $(A^{0,1}, \alpha, 0)$ satisfies $T_2(A^{0,1},\alpha) =0$ but not necessarily $T_1(A^{0,1}, \alpha) = 0$. On the other hand, we can change $(A^{0,1}, \alpha)$ by an element of the form $(-2\dbar h, h\sigma _0)$, where $h$ is a {\sl real\/} $C^\infty$ function, without affecting $T_2(A^{0,1},\alpha)$. If we set $\gamma = T_1(A^{0,1}, \alpha)$, then $$T_1(A^{0,1}-2\dbar h, \alpha + h\sigma _0) = \gamma - 2i(\Lambda \dbar \partial h + \lambda \partial \dbar h) -2|\sigma _0|^2 h.$$ {}From the K\"ahler identities, $\Lambda \dbar = -i\partial ^*$ and similarly $\Lambda \partial = i\dbar ^*$. Thus $$i(\Lambda \dbar \partial h + \lambda \partial \dbar h) = 2\operatorname{Re}\dbar ^*\dbar h = -\Delta h,$$ where $\Delta$ is the negative definite Laplacian on $X$, and we seek to solve the equation $$2\Delta h -2|\sigma _0|^2 h = -\gamma.$$ (Note that this equation is the linearized version of the Kazdan-Warner equation used in identifying the Seiberg-Witten moduli space with the Hilbert scheme.) Now we have the following: \lemma{2.3} The operator $\Delta - |\sigma _0|^2$ is an isomorphism from $\Omega ^0(X;\Ar)$ to itself. \endstatement \proof If $\Delta h- |\sigma _0|^2h =0$, then taking the inner product with $h$ we find that $\|dh\|^2 = |\sigma _0|^2h ^2 =0$. Thus $h$ is constant and $|\sigma _0|^2h ^2 =0$, so that $h=0$. Hence the operator $\Delta - |\sigma _0|^2$ is injective. It is an elliptic operator on $\Omega ^0(X; \Ar)$ whose index is the same as the index of the Laplacian on functions, namely zero. Thus it is also surjective. \endproof Thus given the initial representative $(A^{0,1},\alpha)$, there is a unique choice of $h$ such that $T_1(A^{0,1}-2\dbar h, \alpha + h\sigma _0) = 0$. Mapping $(A^{0,1},\alpha)$ to $(A^{0,1}-2\dbar h, \alpha + h\sigma _0, 0)$ then gives a well-defined map from $H^1(\Cal C_1)$ to $H^1(\Cal C)$, and clearly the maps constructed are inverses. We have therefore showed that $H^1(\Cal C) \cong H^1(\Cal C_1)$. By the proof of Theorem 1.1, there is an exact sequence as claimed in the statement of Theorem 2.1. We turn now to the identification of $H^2(\Cal C)$. Given $\gamma \in \Omega ^0(X;\Ar)$, it follows from Lemma 2.3 that we can solve the equation $\Delta h- |\sigma _0|^2h = \gamma$. Thus there exists an $h$ such that $T_1(-2\dbar h, h\sigma _0, 0) = \gamma$, and moreover $(-2\dbar h,h\sigma _0, 0)$ is in the kernel of $T_2+S$. We can therefore identify the cokernel of $\delta _2$ with the cokernel of $$T_2 + S\: \Omega ^{0,1}(X) \oplus \Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0) \to \Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0).$$ Let $\Cal K$ denote the image of $T_2 + S$, so that $\Cal K$ is the set $$\{\, (\dbar A^{0,1} - \bar \sigma _0\beta, \dbar \alpha +\dbar^*\beta + \frac12A^{0,1}\sigma _0): A^{0,1} \in \Omega ^{0,1}(X) , \alpha \in \Omega ^0(L_0), \beta \in \Omega ^{0,2}(L_0)\,\}.$$ First consider the subgroup $0 \oplus \Ker \dbar \subset \Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0)$. For an element in $\Cal K\cap (0 \oplus \Ker \dbar)$ which is the image of $(A^{0,1}, \alpha, \beta)$, we have $\dbar A^{0,1} =\bar \sigma _0 \beta$ and $\dbar (\dbar \alpha + \dbar ^*\beta + \frac12A^{0,1} \sigma _0) = 0$. By Lemma 2.2, this condition is equivalent to $\beta = 0$ and $\dbar A^{0,1} =0$. Thus $$(0 \oplus \Ker \dbar)/\Cal K\cap (0 \oplus \Ker \dbar) \cong \Ker \dbar /\{\,\dbar \alpha +\frac12A^{0,1} \sigma _0: \dbar A^{0,1} =0\,\}.$$ This is clearly the same as $H^1(L_0)/\sigma _0\cdot H^1(\scrO_X)= K_1$. So we have found the subspace of $H^2(\Cal C)$ described in Theorem 2.1. The quotient $K_2'$ of $H^2(\Cal C)$ by $K_1$ is the same as $\Omega ^{0,2}(X) \oplus \operatorname{Im}\dbar/(\operatorname{Id}\oplus \dbar)(\Cal K)$. Now $(\operatorname{Id}\oplus \dbar)(\Cal K)$ is the subgroup $$\{\, (\dbar A^{0,1} - \bar \sigma _0\beta, \dbar \dbar^*\beta + \frac12\dbar A^{0,1}\sigma _0): A^{0,1} \in \Omega ^{0,1}(X) , \beta \in \Omega ^{0,2}(L_0)\,\}.$$ If we consider the projection of this subgroup to the factor $\operatorname{Im}\dbar \subseteq \Omega ^{0,2}(L_0)$, it is surjective since $\dbar \dbar^*$ is an isomorphism on $\operatorname{Im}\dbar$. Thus $K_2'$ is isomorphic to $$\Omega ^{0,2}(X)/\{\, \dbar A^{0,1} - \bar \sigma _0\beta: \dbar \dbar^*\beta + \frac12\dbar A^{0,1}\sigma _0) = 0\,\}.$$ Let $\Cal K' = \{\, \dbar A^{0,1} - \bar \sigma _0\beta: \dbar \dbar^*\beta + \frac12\dbar A^{0,1}\sigma _0) = 0\,\}$. We claim: \lemma{2.4} $\Cal K' \cap \Ker \dbar ^* = \{\, -\bar \sigma _0\beta: \dbar ^*\beta = 0\,\}$. \endstatement \proof Suppose that $$\dbar ^*\dbar A^{0,1} = \dbar ^*\bar \sigma _0\beta; \qquad \dbar \dbar^*\beta =- \frac12\dbar A^{0,1}\sigma _0.$$ Taking the inner product with $A^{0,1}$, we find: $$\gather \|\dbar A^{0,1}\|^2 = \langle \dbar ^*\dbar A^{0,1}, A^{0,1}\rangle = \langle \dbar ^*\bar \sigma _0\beta, A^{0,1}\rangle = \langle \bar \sigma _0\beta, \dbar A^{0,1}\rangle \\= \langle \beta, \sigma _0\dbar A^{0,1}\rangle = -2\langle \beta, \dbar \dbar^*\beta \rangle = -2\|\dbar^*\beta\|^2. \endgather$$ It follows that $\dbar A^{0,1}=0$ and that $\dbar^*\beta =0$, and that $\Cal K' \cap \Ker \dbar ^*$ is as claimed. \endproof Using Lemma 2.4, there is an injection of $\Ker \dbar ^*/(\Cal K' \cap \Ker \dbar ^*)$ into $K_2'$. Now $\Ker \dbar ^*\subseteq \Omega ^{0,2}(X)$ is naturally $H^2(\scrO_X)$, and $\{\, -\bar \sigma _0\beta: \dbar ^*\beta = 0\,\} = \{\, -\sigma _0^*\beta: \dbar ^*\beta = 0\,\}$ is the image of $H^2(L_0)$ under $\sigma _0^*$. The quotient $H^2(\scrO_X)/\sigma _0^*H^2(L_0)$ is isomorphic to the orthogonal complement of $\operatorname{Im}\sigma _0^*$, namely the kernel of multiplication by $\sigma _0$ on $H^2(\scrO_X)$, which we have denoted by $K_2$. So there is an injection of $K_2$ into $K_2'$. The real index of $\Cal C$ is $-\dim _\Ar H^1(\Cal C) + \dim _\Ar H^2(\Cal C)$, and we have shown that one half the real index is at least $$-h^0(L_0)+1 -\dim _\Cee K_0+ \dim _\Cee K_1+\dim_\Cee K_2.$$ Moreover equality holds only if $K_2'$ is exactly equal to $\Ker \dbar ^*/(\Cal K' \cap \Ker \dbar ^*)$, and thus is isomorphic to $K_2 =\Ker \{\times \sigma _0\: H^2(\scrO_X) \to H^2(L_0)\}$. On the other hand, the above alternating sum is the same as $\chi (\scrO_X) - \chi (L_0)$ (take the alternating sum of the dimensions in the cohomology exact sequence associated to multiplying by $\sigma _0$), which as we have seen is one half the real index of $\Cal C$. It follows that $K_2' = \Ker \dbar ^*/(\Cal K' \cap \Ker \dbar ^*)\cong K_2$ and that we have the desired exact sequence for $H^2(\Cal C)$. \endproof \corollary{2.5} If in the above notation $X$ is a K\"ahler surface, $L= K_X^{-1}$ with the \rom{Spin}${}^c$ structure corresponding to the trivial line bundle $L_0$, then the Zariski tangent space is zero-dimensional and the obstruction space is zero. \endstatement \proof In this case $H^0(L_0) = \Cee \sigma _0$, $\times \sigma _0\: H^1(\scrO_X) \to H^1(L_0)$ is an isomorphism, and $\sigma _0H^0(K_X) = H^0(K_X)$. Thus both the Zariski tangent space and the obstruction space are zero. \endproof Next we compare the Kuranishi model of the Seiberg-Witten moduli space to the Kuranishi model of the Hilbert scheme described in the previous section. In what follows we assume that $H^2(\scrO_X) = 0$. In fact, if $X$ is a minimal surface with $H^2(\scrO_X) \neq 0$, then either $X$ is of general type or it is elliptic, a $K3$ surface, or a complex torus. In case $X$ is of general type, the relevant Seiberg-Witten moduli spaces are smooth points corresponding to $\pm K_X$ of the appropriate dimension, and the Kuranishi obstruction space is zero by Corollary 2.5 above. In case $X$ is elliptic, the moduli space need not be of the expected dimension, and the Kuranishi obstruction space need not be zero, but we shall see in the next section that the obstruction map is always identically zero and hence that the map from $\Cal M_g(\xi)$ to $H_{D_0, X}$ is a diffeomorphism between two smooth manifolds. The other cases involve reducible solutions to the Seiberg-Witten equations, and thus are slightly exceptional from our point of view. The case of a nonminimal surface may then be reduced to the minimal case, at least for an open set of K\"ahler metrics; we omit the details. Thus essentially the only interesting case to consider is the case where $H^2(\scrO_X) = 0$. \theorem{2.6} Suppose that $H^2(\scrO_X) = 0$. Then the natural homeomorphism from $\Cal M_g(\xi)$ to $H_{D_0, X}$ is an isomorphism of real analytic spaces. \endstatement \proof We keep the convention that $A$ is a connection on $L = L_0^{\otimes 2}\otimes K_X$, rather than on $L_0$, and that it induces a connection on $L_0$ once we have fixed once and for all a Hermitian connection on $K_X$. Recall that $H_{D_0, X}$ is locally defined as the zeroes of the Fredholm map $F(A,\alpha) = \pi_{\Ker \dbar}(\dbar A + \frac12A\cdot \alpha)$, restricted to a slice of the complex gauge group action on $\Ker \dbar \oplus \Omega ^0(L_0) \subset \Omega ^{0,1}(L_0) \oplus \Omega ^0(L_0)$. As for $\Cal M_g(\xi)$, it is locallly defined by the zero set of the three equations $G = (F_A^+)^{1,1} -\frac{i}{2}(|\alpha|^2-|\beta|^2)\omega$, $\dbar A^{0,1}-\bar\alpha\beta$, and $\bar\partial_A\alpha+\bar\partial^*_A\beta$. Setting the first equation $G$ equal to zero on a slice $S'$ for the real gauge group gives a slice for the complex gauge group: indeed, the differential of $G$ is the map $T_1$ defined in the proof of Theorem 2.1, and Lemma 2.1 shows that, given $(A^{0,1}, \alpha)$, there is a unique real-valued $C^\infty$ function $h$ such that $T_1(A^{0,1}-2\dbar h, \alpha + h\sigma _0) =0$. Thus $$T_1^{-1}(0) \oplus iT\Cal G_\Ar = \Omega ^{0,1}(X) \oplus \Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0).$$ In particular $S' \cap G^{-1}(0)$ is a slice for the complex gauge group in a neighborhood of the origin; denote this slice by $S$. Consider now the remaining two equations. Defining $$\Cal F(A, \alpha, \beta) = (\dbar A^{0,1}-\bar\alpha\beta, \bar\partial_A\alpha+\bar\partial^*_A\beta),$$ we can view $\Cal F$ as a section of the trivial vector bundle over $\Omega ^{0,1}(X) \oplus \Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0)$ with fiber $\Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0)$ whose restriction to the slice $S$ is Fredholm and locally defines $\Cal M_g(\xi)$. By our assumption that $H^2(\scrO_X) = 0$, and since $H^2(\scrO_X)$ surjects onto $H^2(L_0)$, it follows that $\dbar\: \Omega ^{0,1}(L_0) \to \Omega ^{0,2}(L_0)$ is surjective. Hence the natural map $$\Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0) \to \Ker \dbar \oplus \Omega ^{0,2}(X) \oplus \Omega ^{0,2}(L_0)$$ defined by $(\psi, \eta ) \mapsto (\pi _{\Ker \dbar}\eta, \psi, \dbar \eta)$, is an isomorphism. Thus for small $A$, the map $$(\psi, \eta ) \mapsto (\pi _{\Ker \dbar}\eta, \psi, \dbar _A\eta)$$ is again an isomorphism. We may then view this map as an automorphism of the trivial vector bundle over $\Omega ^{0,1}(X) \oplus \Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0)$ (or an appropriate neighborhood of the origin) with fiber $\Omega ^{0,2}(X) \oplus \Omega ^{0,1}(L_0)$. Under this automorphism, $\Cal F$ corresponds to the section $(F_1, F_2)$, where $$\align F_1 &= \pi _{\Ker \dbar}(\dbar _A\alpha + \dbar _A^*\beta);\\ F_2 &= ( \dbar A^{0,1}-\bar\alpha\beta,\dbar _A^2\alpha + \dbar _A\dbar _A^*\beta). \endalign$$ Thus the Kuranishi model for $\Cal F$ on the slice $S$ is the same as that for the pair $(F_1, F_2)$ on $S$. It is easy to check that the differential of the map $F_2$ is the same as the differential of $\Cal F$ followed by $(\Id, \dbar)$. In other words, the cokernel of the differential of $F_2$ is exactly the group $K_2$ of Theorem 1.1, namely the kernel of multiplication from $H^2(\scrO_X)$ to $H^2(L_0)$. Since $H^2(\scrO_X) = 0$ by assumption, the differential of $F_2$ is onto and $F_2^{-1}(0)$ is a smooth submanifold of a neighborhood of the origin in $\Omega ^{0,1}(X) \oplus \Omega ^0(L_0) \oplus \Omega ^{0,2}(L_0)$. Now $F_2 = 0$ if and only if $\dbar A^{0,1}=\bar\alpha\beta$ and $\dbar _A^2\alpha + \dbar _A\dbar _A^*\beta = 0$. As we mentioned earlier, these equations imply that $\beta = 0$ and hence that $\dbar A^{0,1}=0$. Conversely, if $\beta = 0$ and $\dbar A^{0,1} = 0$, then $F_2(A, \alpha, \beta ) = 0$. Solving the equation $\Cal F = 0$ on the slice $S$ is the same then as solving the equation $\pi _{\Ker \dbar}(\dbar _A\alpha) = 0$ on the slice $S\cap (\Ker \dbar \oplus \Omega ^0(L_0))$, at least in a neighborhood of the origin. This is exactly the equation $F(A, \alpha)$ on the slice $S\cap (\Ker \dbar \oplus \Omega ^0(L_0))$ for the complex gauge group. A standard argument (see for example \cite{4}, Chapter 4, proof of Theorem 3.8) shows that this is the same as the usual Kuranishi model for $F$, in other words that this model is isomorphic to the Kuranishi model formed by taking any other slice for the $\Cal G_\Cee$-action. Thus the two Kuranishi models are isomorphic as complex spaces. \endproof \section{3. Obstruction bundles.} Fix an oriented $4$-manifold $X$ with a Riemannian metric $g$ (not necessarily a K\"ahler surface). Let $\Cal A^*(L)$ denote the spaces of pairs $(A, \psi)$, where $A$ is a connection on $L$ and $\psi$ is a nonzero section of $\Bbb S^+(\xi)$ as in the previous section. The real gauge group $\Cal G$ acts on $\Cal A^*(L)$, and we denote the quotient by $\Cal B(L)$. The trivial Hilbert space bundle $i\Omega ^2_+(X; \Ar) \oplus \Bbb S^-(\xi)$ descends to a Hilbert bundle $\Cal H$ over $\Cal B(L)$, and the moduli space $\Cal M_g(\xi)$ is the zero set of the Fredholm section $F(A, \psi)$ defined by the Seiberg-Witten equations. As such $\Cal M_g(\xi)$ has a real analytic structure and in particular a Zariski tangent space. In this section, we are concerned with the following situation: suppose that the space $\Cal M_g(\xi)$ is a smooth compact manifold, not necessarily of the expected dimension. Thus the dimension of the Zariski tangent space of $\Cal M_g(\xi)$ at every point is equal to the dimension of $\Cal M_g(\xi)$ at that point, and these tangent spaces fit together to form the tangent bundle $T\Cal M_g(\xi)$ to $\Cal M_g(\xi)$. Note that the tangent bundle is in fact just $\Ker dF\: T\Cal A^*(L) \to \Cal H$. It follows that the obstruction spaces have locally constant rank on $\Cal M_g(\xi)$ and thus, by standard elliptic theory, fit together to form a vector bundle $\Cal O$ over $\Cal M_g(\xi)$. In case $g$ is a K\"ahler metric on the complex surface $X$, the arguments of the previous section show that the fiber of $\Cal O$ over a point $(A_0, \sigma _0)$ may be canonically identified with the middle cohomology of the elliptic complex $$\Omega ^{0,1} (X; \Ar) \oplus \Omega ^0(L_0) \to \Omega ^{0,2} (X; \Ar) \oplus \Omega ^{0,1}(L_0) \to \Omega ^{0,2}(L_0),$$ where the first map is $(A^{0,1} , \alpha) \mapsto (\dbar A^{0,1}, \dbar \alpha + \frac12A^{0,1}\sigma _0)$ and the second map is $(\varphi, \psi) \mapsto \dbar \psi - \frac12\varphi \cdot \sigma _0)$. Again by a slight modification of the standard theory for the $\dbar$-operator, it follows that the bundle $\Cal O$ is a holomorphic vector bundle over $\Cal M_g(\xi)$. In fact, letting $\frak C$ be the complex $\scrO_{X\times \Cal M_g(\xi)} \to \scrO_{X \times \Cal M_g(\xi)}(\Cal D_0)$, it is easy to see that $\Cal O$ is the $C^\infty$ vector bundle associated to the hyperdirect image $\Ar ^2\pi _2{}_*\frak C \cong R^1\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0)$. To use this information to evaluate Seiberg-Witten invariants, we have the following: \theorem{3.1} In the above notation, suppose that the expected real dimension of $\Cal M_g(\xi)$ is $2d$, and let $\mu$ be the natural class in $H^2(\Cal B(L))$. Then the value of the Seiberg-Witten function on $\xi$ is $\int _{\Cal M_g(\xi)} e(\Cal O)\cup \mu ^d$, where $e(\Cal O)$ is the Euler class of the vector bundle $\Cal O$. In case $X$ is a K\"ahler surface and $\Cal M_g(\xi)$ is equidimensional, this is the same as $\int _{\Cal M_g(\xi)}c_n(\Cal O)\cup \mu ^d$, where $n = \operatorname{rank}\Cal O$. \endstatement In particular we need to calculate $c_n(\Cal O)$: \lemma{3.2} In the above notation, suppose that $X$ is a K\"ahler surface and that $\Cal M_g(\xi)$ is smooth. Then $c_n(\Cal O)$ is the degree $n$ term in $$c(\pi _2{}_!\scrO_{X \times \Cal M_g(\xi)}(\Cal D_0))^{-1}c(T\Cal M_g(\xi)).$$ \endstatement \proof We need to calculate $c_n(\Cal O) = c_n (R^1\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0))$. Now the morphism $\Cal D_0 \to \Cal M_g(\xi)$ has relative dimension one, and so $\pi _2{}_!\scrO_{\Cal D_0}(\Cal D_0)$, which is by definition the alternating sum of the $R^i\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0)$, is just $R^0\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0) - R^1\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0)$. Furthermore, $R^0\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0)$ is just the tangent bundle $T\Cal M_g(\xi)$ to $\Cal M_g(\xi)$. Thus $$c(\Cal O)= c(R^1\pi _2{}_*\scrO_{\Cal D_0}(\Cal D_0)) = c(\pi _2{}_!\scrO_{\Cal D_0}(\Cal D_0))^{-1}c(T\Cal M_g(\xi)).$$ Using the exact sequence $$0 \to \scrO_{X \times \Cal M_g(\xi)} \to \scrO_{X \times \Cal M_g(\xi)}(\Cal D_0) \to \scrO_{\Cal D_0}(\Cal D_0) \to 0,$$ it follows that, in the $K$-theory of $\Cal M_g(\xi)$, $$\pi _2{}_!\scrO_{\Cal D_0}(\Cal D_0) = \pi _2{}_!\scrO_{X \times \Cal M_g(\xi)}(\Cal D_0) - \pi _2{}_!\scrO_{X \times \Cal M_g(\xi)}.$$ Since $\pi _2{}_!\scrO_{X \times \Cal M_g(\xi)}$ is a trivial vector bundle, $$c(\pi _2{}_!\scrO_{\Cal D_0}(\Cal D_0)) = c(\pi _2{}_!\scrO_{X \times \Cal M_g(\xi)}(\Cal D_0)).$$ Putting this together with the above formula for $c(\Cal O)$ gives the statement of (3.2). \endproof \demo{Proof of \rom{(3.1)}} Consider quite generally the following situation: $\Cal H \to \Cal B$ is a Hilbert vector bundle over the connected Hilbert manifold $\Cal B$, and $\sigma$ is a smooth section of $\Cal H$. Let $Z = \sigma ^{-1}(0)$, assumed connected for the sake of simplicity, and suppose that the differential $d\sigma$ is Fredholm of index $e$, at least in a neighborhood of $Z$. Suppose moreover that $Z$ is a smooth compact submanifold of $\Cal B$ of finite dimension $e'$ and that $\Ker d\sigma _z$ has constant rank for every $z\in Z$ and that the corresponding subbundle of $T\Cal B|Z$ is the tangent bundle to $Z$. Define the obstruction bundle $\Cal O\to Z$ by $\Cal O = \Coker (d\sigma |Z)$, of rank $e' - e$. Theorem 3.1 is then a consequence of the following lemma, which implies that the class of a small generic perturbation of $Z$ is the same as the class of a generic section of $\Cal O$, in other words that the Euler class of $\Cal O$ represents the same cohomology class as the Seiberg-Witten class of a generic moduli space. \enddemo \lemma{3.3} In the above situation, suppose that $\sigma _1$ is a small nonlinear Fredholm $C^1$ perturbation of $\sigma$, with $\sigma _1^{-1}(0) = Z_1$, and that $\sigma _1$ is transverse to $0$ in the sense that $d\sigma _1$ is surjective at every point of $Z_1$. Then there exists a section $s$ of $\Cal O \to Z$ which is transverse to $0$ and a small isotopy in $\Cal B$ from $s^{-1}(0)$ to $Z_1$. \endstatement \proof Since $Z$ is compact, standard arguments show that there is a neighborhood $\nu$ of $Z$ in $\Cal B$ which is diffeomorphic to a Hilbert disk bundle over $Z$. Let $\pi \: \nu \to Z$ be the retraction. Over $Z$, there is an orthogonal splitting $\Cal H|Z \cong \operatorname{Im}d\sigma \oplus \Cal O$. Using $\pi$, we can pull this decomposition back to a splitting of $\Cal H|\nu \cong I \oplus \Cal O$. Let $\pi _1 \: \Cal H|\nu \to I$ and $\pi _2 \: \Cal H|\nu \to \Cal O$ be the projections. Consider the composed map $\pi _1\circ \sigma \: \nu \to I$. At $z\in Z$, the differential of this map is just $d\sigma$, and so restricted to a fiber $\pi ^{-1}(z)$, the differential is an isomorphism. It follows that, if $\nu$ is sufficiently small, then $\pi_1\circ \sigma |\pi ^{-1}(z)$ is an open embedding for all $z\in Z$. Now let $\sigma _1$ be a small perturbation of $\sigma$. If $\sigma _1$ is sufficiently close to $\sigma$, then $\sigma _1^{-1}(0) = Z_1 \subseteq \nu$. Consider the map $\pi _1 \circ \sigma _1 \: \nu \to I$. If we restrict this map to a fiber $\pi ^{-1}(z)$ of the map $\pi \: \nu \to Z$, $\pi_1\circ \sigma _1 |\pi ^{-1}(z)$ is close to an open embedding. Thus, as long as $\sigma _1$ is close to $\sigma$, $\pi_1\circ \sigma _1 |\pi ^{-1}(z)$ is also an open embedding. In particular, $(\pi_1\circ \sigma _1)^{-1}(0) = \hat Z_1$ is again a section of $\pi \: \nu \to Z$, and it is close to the zero section $\sigma$. Thus $\hat Z_1$ is isotopic to $Z$ via a small isotopy in $\nu \subseteq \Cal B$. Clearly $Z_1 = \sigma _1^{-1}(0) = (\pi _2 \circ \sigma _1|\hat Z_1)^{-1}(0)$. Moreover $(\pi _2 \circ \sigma _1|\hat Z_1)$ is a section $\hat s$ of the restriction $\Cal O \to \hat Z_1$, and, if $\sigma _1$ is transverse to $0$, then $\hat s$ is also transverse to $0$. Using the isotopy constructed above to identify the section $\hat s$ with a section of $\Cal O\to Z$, we see that we have indeed identified $Z_1$ up to isotopy with a transverse section of $\Cal O \to Z$, as claimed. \endproof There are obvious generalizations of Lemma 3.3, and so of Theorem 3.1. For example, we might only assume that $Z$ is a stratified space with $K$ a compact subset contained in an open subset $U$ of $Z$, such that $U$ is a smooth manifold and $\Ker d\sigma _u = TU_u$ for all $u\in U$. Then a generic small perturbation $\sigma _1$ of $\sigma$ has the property that there exists a neighborhood $N$ of $K$ such that $\sigma _1^{-1}(0) \cap N$ is a smooth manifold isotopic to a transverse section of the obstruction bundle over $N$. For example, we might take for $K$ a subset of the form $\mu _1 \cap \dots \cap \mu _k$, where the $\mu _i$ are generic geometric representatives for the $\mu$-divisor. However, we shall not try to formulate the most general possible result along these lines. \section{4. Elliptic surfaces.} Let $X$ denote an elliptic surface. Suppose that $f$ is the divisor class of a general fiber, the multiple fibers are $F_i$, and that the multiplicity of $F_i$ is $m_i$. Thus $K_X = (p_g-1)f + \sum _i(m_i-1)F_i$. We first consider the much simpler case where $X$ is regular, since this case arises in the smooth classification of elliptic surfaces (see for example \cite{5}, \cite{2}). Then we will discuss the multiplicities for a general elliptic surface. If $X$ is regular, $H^1(\scrO_X) = 0$ and the Seiberg-Witten obstruction space involves only the two terms $H^1(\scrO_X(D_0))$ and $H^2(\scrO_X(D_0))$. The divisor $D_0$ is semiregular if and only if $H^1(\scrO_X(D_0)) = 0$. It follows from (2.3) of \cite{5} that the $D_0$ are exactly the effective divisors which are numerically equivalent to $\frac{1-r}2K_X$, for a rational number $r \leq 1$. In particular $D_0 \cdot K_X = 0$. Another way to describe the $D_0$ is that they are the effective divisors numerically equivalent to a rational multiple of $K_X$ such that $K_X - 2D_0$ has positive fiber degree (is a positive rational multiple of the fiber, or equivalently of $K_X$). A similar statement holds if $X$ is not necessarily assumed to be regular. As $L^2 =0$, the expected dimension of the moduli space is always zero, i\.e\. $X$ is of simple type. We now compute the dimensions of the cohomology groups: \lemma{4.1} Suppose that $X$ is regular, and let $D_0 = af + \sum _ib_iF_i$ with $a\geq 0$ and $0\leq b_i\leq m_i-1$. Then: $$\align h^0(D_0) &= a+1;\\ h^1(D_0) &= \cases 0, & \text{if $a\leq p_g$};\\ a-p_g, &\text{if $a\geq p_g$}. \endcases\\ h^2(D_0) &= \cases p_g-a, & \text{if $a\leq p_g$};\\ 0, &\text{if $a\geq p_g$}. \endcases \endalign$$ \endstatement \proof The statements about $h^0(D_0)$ and $h^2(D_0)$ are clear and the rest follows from Riemann-Roch, since $\chi (\scrO_X(D_0)) = 1+p_g$. \endproof Thus we see that $D_0$ is semiregular if and only if $a\leq p_g$. However, since $X$ is regular and thus $H_{D_0, X}$ is equal to the linear system $|D_0|$, which is a projective space $\Bbb P^a$, $H_{D_0, X}$ is always smooth and the Zariski tangent space is the actual tangent space. To calculate the value of the Seiberg-Witten invariant on $L$, we take the top Chern class of the obstruction bundle. Now the moduli space is $X\times \Pee ^a$, where $\Pee ^a = |D_0|$. Over $X\times \Pee ^a$, there is the incidence divisor $\Cal D$. The obstruction bundle over $\Pee ^a$ has two terms $R^1\pi _2{}_*\scrO_{X\times \Pee ^a}(\Cal D)$ and $R^2\pi _2{}_* \scrO_{X\times \Pee ^a}(\Cal D))$. \proposition{4.2} Suppose in the above notation that $a\leq p_g -1$. Then the multiplicity of the Seiberg-Witten invariant is $(-1)^a\dsize \binom{p_g-1}{a}$. \endstatement \proof By Lemma 4.1, the first term of the obstruction bundle is zero, and we must compute the top Chern class $c_a$ of $R^2\pi _2{}_* \scrO_{X\times \Pee ^a}(\Cal D))$. Next we calculate the class of $\Cal D$ in $\Pic (X\times \Pee ^a)\cong \pi _1^*\Pic X \oplus \Zee \cdot \pi _2^*\scrO_{\Pee ^a}(1)$. Since $\Cal D$ is the incidence divisor, its restriction to the slice $\pi _2^*\{D\}$ is the divisor $D$, whereas its restriction to any slice $\{p\}\times \Pee ^a$ such that $p$ is not in the base locus of $D_0$ is a hyperplane in $\Pee ^a$. Thus $\scrO_{X\times \Pee ^a}(\Cal D) \cong \pi _1^*\scrO_X(D_0) \otimes \pi _2^*\scrO_{\Pee ^a}(1)$. So we have $$R^2\pi _2{}_* \scrO_{X\times \Pee ^a}(\Cal D)) = R^2\pi _2{}_*\left(\pi _1^*\scrO_X(D_0) \otimes \pi _2^*\scrO_{\Pee ^a}(1) \right)= \scrO_{\Pee ^a}(1)^{p_g -a}.$$ Setting $h = c_1(\scrO_{\Pee ^a}(1))$, we want to take the term of degree $a$ in $$\left((1+h)^{p_g-a}\right)^{-1} = (1+h)^{-(p_g-a)}.$$ By the binomial theorem (see below for our conventions on binomial coefficients), this is $\dsize \binom{-(p_g -a)}{a} = (-1)^a\binom{p_g-1}{a}$. \endproof A similar argument shows: \proposition{4.3} Suppose in the above notation that $a\geq p_g$. Then the multiplicity of the Seiberg-Witten invariant is $1$ if $p_g =0$ and is otherwise $0$. \endstatement \proof Since $H^2( D) =0$, by Lemma 4.1, we seek $c_a\left(R^1\pi _2{}_*\scrO_{X\times \Pee ^a}(\Cal D)\right)$. Using the calculation $\scrO_{X\times \Pee ^a}(\Cal D) \cong \pi _1^*\scrO_X(D_0) \otimes \pi _2^*\scrO_{\Pee ^a}(1)$ given above, we need to find $c_a$ of $$R^1\pi _2{}_*\left(\pi _1^*\scrO_X(D_0) \otimes \pi _2^*\scrO_{\Pee ^a}(1)\right) =H^1(D_0) \otimes _\Cee \scrO_{\Pee ^a}(1) = (\scrO_{\Pee ^a}(1))^{a-p_g}.$$ If $p_g =0$, then $c((\scrO_{\Pee ^a}(1))^a) = (1+h)^a$, and thus $c_a((\scrO_{\Pee ^a}(1))^a) = 1$. Otherwise, the multiplicity is $c_a$ of a bundle of rank less than $a$, and so is zero. \endproof \noindent {\bf Remark.} If $p_g =0$, then the multiplicity is always $1$, although we can have $a>0$ if there are more than two multiple fibers. If $p_g >0$ and there are at most two multiple fibers (the case of finite cyclic fundamental group), then it is easy to check that $a\leq p_g-1$. In general however both of the terms in the exact sequence for the obstruction bundle can be nonzero. \medskip Next we turn to elliptic surfaces which are not necessarily regular. To state the result, let us record the following convention on binomial coefficients (made so that the binomial theorem holds): For $m\geq 0$, the binomial coefficient $\dsize\binom{n}{m} =\frac{1}{m!}n(n-1) \cdots (n-m+1)$. Thus it is $1$ if $m=0$, by the usual conventions on the empty product, and it is $0$ if $0\leq n<m$ and $m\neq 0$. Moreover $\dsize\binom{-n}m = (-1)^m\binom{n+m-1}{m}$. For example, $\dsize\binom{-1}{m} = (-1)^m$. With this said, suppose that $X$ is a minimal elliptic surface over a smooth curve of genus $g$ and $D_0 = af + \sum _ib_iF_i$ is a basic class for $X$. Then the value of the Seiberg-Witten invariant on $D_0$ is given by the following formula, due independently to Brussee \cite{2}: \proposition{4.4} Let $\pi \: X \to C$ be a minimal elliptic surface over a smooth curve $C$ of genus $g$ and let $D_0$ be an effective divisor corresponding to a Seiberg-Witten basic class. Suppose that $D_0 = \pi ^*\bold d + \sum _i a_iF_i$, where $\bold d$ is an effective divisor on $C$ of degree $d$, the $F_i$ are the multiple fibers on $X$, of multiplicity $m_i$, and $0 \leq a_i \leq m_i -1$. Then the multiplicity of the Seiberg-Witten invariant is $\dsize (-1)^d\binom{\chi(\scrO_X) + 2g-2}{d}$. \endstatement \proof If $D_0 = \pi ^*\bold d + \sum _i a_iF_i$ as above, there is a natural morphism from $\Sym ^dC = C_d$ to $H_{D_0, X}$ obtained by pulling back the universal divisor on $C\times C_d$ to $X\times C_d$. Slightly tedious arguments left to the reader show that this identifies $H_{D_0, X}$ as a set with $C_d$. To calculate Zariski tangent spaces, it is easy to see that $H^0(\scrO_{D_0}(D_0))$ has dimension $d$ by using the fact that the normal bundle of $F_i$ is torsion of order exactly $m_i$. Thus the dimension of the Zariski tangent space to $H_{D_0, X}$ is equal to the dimension of $H_{D_0, X}=C_d$, and the map $C_d \to H_{D_0, X}$ is an isomorphism. Let $\pi _2 \: X \times C_d \to C_d$ be the second projection, let $p\: X\times C_d \to C_d$ be the map induced by $\pi$, and let $p_2\: C \times C_d \to C_d$ be second projection. Over $C\times C_d$ we have the incidence divisor $\Cal I$ defined by $\Cal I = \{\, (t, \bold d): t\in \operatorname{Supp}\bold d\,\}$. Thus the universal divisor $\Cal D$ on $X \times C_d$ is just $p^*\Cal I$. Let $\psi _2 \: C\times C_d \to C_d$ be projection onto the second factor. By flat base change, $R^i\pi _2 {}_*p^*\scrO_{C\times C_d}(\Cal I) = R^i \psi _2{}_*\scrO_{C\times C_d}(\Cal I)$, and in particular this is zero for $i=2$. Recall that the multiplicity of the Seiberg-Witten invariant is then given by evaluating $c_a(R^1\pi _2{}_*\scrO_{\Cal D}(\Cal D))$ over $C_d$. From the exact sequence $$0 \to \scrO_{X\times C_d} \to \scrO_{X\times C_d}(\Cal D) \to \scrO_{\Cal D}(\Cal D) \to 0,$$ we see that, in the $K$-theory of $C_d$, $$\pi _2{}_!\scrO_{\Cal D}(\Cal D) =\sum _i(-1)^iR^i\pi _2{}_*\scrO_{\Cal D}(\Cal D)= R^0\pi _2{}_*\scrO_{\Cal D}(\Cal D) - R^1\pi _2{}_*\scrO_{\Cal D}(\Cal D)$$ agrees with $\pi _2{}_!\scrO_{X\times C_d}(\Cal D)$ up to the trivial element $\pi _2{}_!\scrO_{X\times C_d}$, and thus they have the same Chern classes. Moreover $$c(R^1\pi _2{}_*\scrO_{\Cal D}(\Cal D)) = c(\pi _2{}_!\scrO_{X\times C_d}(\Cal D))^{-1}c(R^0\pi _2{}_*\scrO_{\Cal D}(\Cal D)).$$ Finally $R^0\pi _2{}_*\scrO_{\Cal D}(\Cal D)$ is just the tangent bundle $T_{C_d}$ to $C_d$. By \cite{1}, p\. 322, $$c(T_{C_d}) = (1+x)^{d+1-g}e^{-\theta /1+x},$$ where $x$ is the class of the divisor $C_{d-1}\subset C_d$ and $\theta$ is the pullback of the theta divisor on $\Pic^d C$ under the natural map. We also have the formula $$\theta ^kx^{d-k} = \frac{g!}{(g-k)!}.$$ To find $c(\pi _2{}_!\scrO_{X\times C_d}(\Cal D))$, we first apply the Grothendieck-Riemann-Roch theorem to find $\ch (\pi _2{}_!\scrO_{X\times C_d}(\Cal D))$: $$\ch (\pi _2{}_!\scrO_{X\times C_d}(\Cal D)) = \pi _2{}_*\left(\ch (\scrO_{X\times C_d}(\Cal D))\pi _1^*\Todd X\right).$$ If $\delta = [\Cal I]$ is the class of $\Cal I$ on $C\times C_d$, then $\ch (\scrO_{X\times C_d}(\Cal D)) = p^*e^\delta$. Moreover $\Todd X = 1 + rf + \chi (\scrO_X)\cdot \text{pt}$ for some rational number $r$. By \cite{1}, p\. 338, $\delta =n[\text{pt}]\otimes 1 + \delta ^{1,1} + 1\otimes x$, where $x$ is the class defined above and $(\delta ^{1,1})^2 = -2[\text{pt}]\otimes \theta$, $(\delta ^{1,1})^2 = (\delta ^{1,1})\cdot [\text{pt}]\otimes 1 = 0$. Since $\pi ^*[\text{pt}] = f$ with $f^2 = 0$, an easy calculation shows that $$\ch (\pi _2{}_!\scrO_{X\times C_d}(\Cal D))= \pi _2{}_*\left(\ch (\scrO_{X\times C_d}(\Cal D))\pi _1^*\Todd X\right) = \chi(\scrO_X) \cdot e^x$$ and thus (setting $\chi(\scrO_X) = \chi$ for brevity) $$c(\pi _2{}_!\scrO_{X\times C_d}(\Cal D)) = (1+x)^\chi.$$ Finally, then, the multiplicity of the Seiberg-Witten invariant is the term of degree $d$ in $$(1+x)^{-\chi}(1+x)^{d+1-g}e^{-\theta /1+x} = (1+x)^{d+1-g-\chi}e^{-\theta /1+x}.$$ Let $N = d+1-g-\chi$. Then $$\align (1+x)^Ne^{-\theta /1+x} &= \sum _i\binom{N}{i}\sum _j\frac{1}{j!}(-1)^j\theta ^j\sum _k\binom{-j}{k}x^{i+k}\\ &= \sum _a\sum _{i+j+k= a} \binom{N}{i}\frac{1}{j!}(-1)^j\binom{-j}{k}\frac{g!}{(g-j)!}\\ &=\sum _a\sum _{i+j+k= a} (-1)^{j+k}\binom{N}{i}\binom{j+k-1}{k}\binom{g}{j}. \endalign$$ Thus the degree $d$ term is $$\sum _j(-1)^j\left(\sum _k(-1)^k\binom{N}{d-j-k}\binom{j+k-1}{k}\right)\binom{g}{j}.$$ To evaluate the term in parentheses above, we have the straightforward combinatorial lemma: \lemma{4.5} We have: $$\sum _{k=0}^a \binom{a+j+e}{a-k}\binom{-j}{k} = \sum _{k=0}^a (-1)^k\binom{a+j+e}{a-k}\binom{j+k-1}{k} = \binom{a+e}{a}.$$ \rom(By our conventions on binomial coefficients, this is $1$ if $a=0$ and is zero for $-a \leq e<0$ and $a\neq 0$.\rom) \endstatement \proof This follows by comparing the coefficient of $t^a$ in the two different power series expansions of $(1+t)^{a+e} = (1+t)^{a+e+j}(1+t)^{-j}$. \endproof Returning to the proof of (4.4), the lemma shows that the term in parentheses is $\dsize \binom{N-j}{N-j-e}$, where $e = 1-g-\chi$ (take $a=d-j$ and $N= a + j + e$). Thus we obtain $$\align \sum _j(-1)^j\binom{N-j}{N-j-e}\binom{g}{j} &= \sum _j(-1)^j\binom{d-j+ 1 -g-\chi}{d-j}\binom{g}{j} \\ &= (-1)^d\sum _j\binom{\chi + g -2}{d-j}\binom{g}{j}, \endalign$$ which is just $(-1)^d$ times the coefficient of $t^d$ in $(1+t)^{\chi + g -2}(1+t)^g=(1+t)^{\chi + 2g -2}$, namely $\dsize (-1)^d \binom{\chi + 2g -2}{d}$, as claimed. \endproof \section{5. Product ruled surfaces.} In this section we shall consider the ruled surfaces $X$ of the form $\Pee ^1\times C$, where $C$ is a curve of genus $g\geq 1$. We shall also always assume that $C$ is a generic curve in the sense of Brill-Noether theory, and shall use \cite{1} as a general reference for the theory of special divisors on curves. Let $\pi _1\: X = \Pee ^1\times C \to \Pee ^1$ be the projection onto the first factor and let $\pi _2\: \Pee ^1\times C \to C$ be the projection onto the second. Let $F_1 = \pi _2^{-1}\{p\}$ be a fiber isomorphic to $\Pee ^1$ and let $F_2 = \pi _1^{-1}\{p\}$ be a fiber isomorphic to $C$. Thus $F_i^2 = 0$ and $F_1\cdot F_2 = 1$. In general we shall refer to a divisor numerically equivalent to $nF_1 + mF_2$ as a divisor {\sl of type\/} $(n,m)$, and similarly for a complex line bundle. Thus for example $K_X$ is of type $(2g-2, -2)$. Hence $K_X^2 = -8(g-1)$. Let $L$ be a line bundle of type $(2a, 2b)$, so that $c_1(L) \equiv K_X \mod 2$. Then $L^2 = 8ab$, and so $L^2 \geq K_X^2$ if and only if $ab \geq 1-g$. Next suppose that $L_0 = (K_X\otimes L)^{1/2}$ has a holomorphic section for some holomorphic structure on $L_0$. As $L_0$ is of type $(g-1+a, b-1)$, we must have $b\geq 1$ and $a\geq 1-g$, and we can write $L_0 = \scrO_X(D_0)$, where $D_0$ is linearly equivalent to $(b-1)\pi _1^*(\text{pt}) + \pi _2^*\bold d$ for some divisor $\bold d$ on $C$ of degree $d = g-1 + a$. Next we turn to the condition that $L\cdot \omega < 0$ for some K\"ahler form $\omega$. The real cohomology classes of K\"ahler metrics are exactly the classes $\omega $ of type $(x,y)$ with $x, y\in \Ar$, and $x,y>0$. Thus $$\omega \cdot L = 2xb + 2ay.$$ Since $b\geq 1$, we must have $a<0$, and it is clear that by choosing $x/y < -a/b$, we can then arrange $\omega \cdot L < 0$. (Note conversely that if $x/y > -a/b$, then $L$ does not correspond to a basic class. Since $-a \leq g-1$ and $b\geq 1$, if we choose $x/y \geq g-1$, then there are no basic classes.) The final conditions on $a$ and $b$ are: $$b \geq 1; \qquad \frac{1-g}b \leq a <0.$$ We note that the expected (complex) dimension of the Seiberg-Witten moduli space is $g-1 +ab$. However, as we shall see, the actual dimension is equal to the expected dimension only for $b=1$. Given a curve $D_0$ of type $(g-1 + a, b-1) = (d,b-1)$, its irreducible components correspond to curves of type $(e_i, c_i)$ with $\sum _ie_i = d$ and $\sum _ic_i = b-1$. For example, if $D_0$ is irreducible, then $D_0$ is simultaneously a cover of $\Pee ^1$ of degree $d$ and a cover of $C$ of degree $b-1$. If $b=1$, then necessarily $D_0$ is a union of $d$ copies of $C$, or more precisely a divisor of the form $\pi _2^*\bold d$ for some divisor $\bold d$ of degree $d$ on $C$. In this case, $H_{D_0, X}$ is just $C_d$, the $d^{\text{th}}$ symmetric product of $C$ with itself. Note that $-K_X$ is a divisor of this type, with $a = 1-g$. In general, for such divisors, we have: \proposition{5.1} In case $c_1(L) = 2aF_1 + 2F_2$, then $H_{D_0, X}= C_d$ and the value of the Seiberg-Witten invariant is $1$. \endstatement \proof We have seen that $H_{D_0, X}= C_d$ as sets. There is an obvious universal divisor on $X\times C_d$ which is the pullback of the universal divisor on $C\times C_d$. Thus there is a morphism from $C_d$ to $H_{D_0, X}$. To see that this morphism is an isomorphism of schemes, it will suffice to show that $H_{D_0, X}$ is smooth of dimension $d$. It is an easy exercise to identify $H^i(\scrO_{D_0}(D_0))$ with $H^i(\Pee ^1; H^0(\scrO_{\bold d})\otimes \scrO_{\Pee ^1}) = H^i(\Pee ^1;\scrO_{\Pee ^1}) \otimes H^0(\scrO_{\bold d})$. This has dimension $d$ for $i=0$ and is zero for $i=1$. Thus $H_{D_0, X}$ is smooth of dimension $d$, and is therefore isomorphic to $C_d$. Note that $X$ is not of simple type if $d>0$. Clearly, for $p\in X$ with $\pi _2(p) = t\in C$, the incidence divisor for $H_{D_0, X}$ and $p\in X$ may be identified with the incidence divisor for $C_d$ and $t\in C$, along with multiplicities. Let $x$ be the class of the divisor in $C_d$. By choosing $d$ distinct points $t_1, \dots, t_d$ and checking that the intersections are transverse, we see that $x^d = 1$. Thus $\mu ^d = 1$ for the Seiberg-Witten moduli space as well, so that the value of the invariant is 1. \endproof For the remainder of this section, we shall mainly be interested in the case $b=2$. In this case every curve $D_0$ of type $(d,1)$ can be written either as $D_0 = D_1 + \pi _2^*\bold d_2$, where $D_1$ is the graph of a map $C\to \Pee ^1$ of degree $d_1$ and $\bold d_2$ is a divisor of degree $d_2 = d-d_1$ on $C$, or $D_0 = \pi _1^*(\text{pt}) + \pi _2^*\bold d$, where $\bold d$ is a divisor of degree $d$ on $C$. More generally, an irreducible divisor $D_0 \subset \Pee ^1 \times C$ of type $(d, b-1)$ which is a section of the line bundle $\pi _1^*\scrO_{\Pee ^1}(b-1) \otimes \pi _2^*\scrO_C(\bold d)$ corresponds to a map $C \to \Sym^{b-1}\Pee ^1 \cong \Pee ^{b-1}$. In this case it is easy to check that the pullback of $\scrO_{\Pee ^{b-1}}(1)$ to $C$ is just $\scrO_C(\bold d)$. Moreover, let $V$ be the smallest linear subspace of $\Cee ^b$ such that $\Pee (V) \subseteq \Pee^{b-1}$ contains the image of $C$. Then $V$ is naturally a quotient of $H^0(\bold d)^*$, of dimension $r+1$, say, corresponding to a linear subseries of $|\bold d|$. Note in particular that we always have $r\leq b-1$. Next suppose that $D_0$ is not necessarily irreducible. Then $D_0$ still corresponds to a linear system $V \subseteq H^0(\bold d)^*$ with $\Pee (V) \subseteq \Sym^{b-1}\Pee ^1 \cong \Pee ^{b-1}$. In fact, if $D_0$ is defined by the section $\sigma _0 \in H^0(\bold b) \otimes H^0(\bold d)$, write $\sigma _0 = \sum _i\alpha _i \otimes \beta _i$, where the $\alpha _i \in H^0(\bold b)$ and the $\beta _i \in H^0(\bold d)$ are linearly independent. For $p\in C$, the morphism $C\to \Sym^{b-1}\Pee ^1$ sends $p$ to $\sum _i\beta _i(p)\alpha _i$, after choosing a coordinate for $\scrO_C(\bold d)$ at $p$. This is well-defined if $p$ is not in the base locus of the span of the $\beta _i$, and extends to a unique morphism $C\to \Sym^{b-1}\Pee ^1$. Now $\sigma _0\in H^0(\bold b) \otimes H^0(\bold d)$ defines a homomorphism $H^0(\bold d)^* \to H^0(\bold b)$ whose image $V$ is spanned by the $\alpha _i$. Thus $V^*\subseteq H^0(\bold d)$ is a linear series. At one extreme, consider divisors $D_0$ of the form $\pi _1^*\bold b + \pi _2^*\bold d$, where $\deg \bold b= b-1$ and $\deg \bold d = d$. In this case, $V$ has dimension 1, the linear series corresponding to $V$ is the single divisor $\bold d$, which is the base locus for the series, and the map $C \to \Sym^{b-1}\Pee ^1 \cong \Pee ^{b-1}$ is constant, with image equal to $\bold b$. In this case, let $\frak M_0 = \Pee ^{b-1}\times C_d$, thinking of this space as parametrizing all divisors of the form $\pi _1^*\bold b + \pi _2^*\bold d$. Then we have an obvious divisor on the product $X\times \frak M_0$, and thus there is an injective morphism $\Pee ^{b-1} \times C_d\to H_{D_0, X}$. In particular, $H_{D_0, X}$ has dimension at least $d+b-1 = g-1 + a + b-1 = g+a+b-2$, whereas the expected dimension of $H_{D_0, X}$ is $g+ab-1$. In this case the difference between $\dim \frak M_0$ and the expected dimension of $H_{D_0, X}$ is $(b-1)(1-a)$. Since $a<0$ we see that the actual dimension is always greater than the expected dimension as long as $b>1$. To see the image of the tangent space to $\frak M_0$ inside the Zariski tangent space of $H_{D_0, X}$, note that the Zariski tangent space of $H_{D_0, X}$ is $H^0(\scrO_{D_0}(D_0))$. In case $D_0 = \pi _1^*\bold b + \pi _2^*\bold d$, there is an map $H^0(\scrO_{\pi _1^*\bold b}) \oplus H^0(\scrO_{\pi _2^*\bold d}) \to H^0(\scrO_{D_0}(D_0))$ given as follows: If we set $E_1 = \pi _1^*\bold b$ and $E_2 = \pi _2^*\bold d$, then this is just the natural map $$(\scrO_X(E_1)/\scrO_X) \oplus (\scrO_X(E_2)/\scrO_X) \to \scrO_X(E_1+E_2)/\scrO_X.$$ Locally for $R = \Cee\{z_1, z_2\}$, this is the same as the map $$R/z_1^{a_1}R \oplus R/z_2^{a_2}R \to R/z_1^{a_1}z_2^{a_2}R$$ defined by $(f,g) \mapsto z_2^{a_2}f+ z_1^{a_1}g$, which is an inclusion since $z_1^{a_1}$ and $z_2^{a_2}$ are relatively prime. Thus the image of the tangent space of $\frak M_0$ is $L_0 = H^0(\scrO_{\pi _1^*\bold b}) \oplus H^0(\scrO_{\pi _2^*\bold d})$, of dimension $b-1 + d$, and the map from $\frak M_0$ to $H_{D_0, X}$ is an immersion. Concerning the structure of $H_{D_0, X}$ at a divisor $D_0$ of the form $\pi _1^*\bold b + \pi _2^*\bold d$, where $\deg \bold b = b-1$ and $\deg \bold d = d = g-1 +a$, we have the following result: \lemma{5.2} The scheme $H_{D_0, X}$ is smooth of dimension $(b-1)(1-a)$ at the divisor $D_0 = \pi _1^*\bold b + \pi _2^*\bold d$ if and only if $h^0(C; \bold d) = 1$, or in other words if and only if $\bold d$ is an effective divisor which does not move in a nontrivial linear system. More generally, if $\dim |\bold d| = r$ and if $T$ is the Zariski tangent space of $H_{D_0, X}$ at $D_0$, then there is an exact sequence $$ 0 \to (H^0(\Pee ^1; \bold b) \otimes H^0(C; \bold d))/\Cee \sigma _0 \to T \to \Ker \{H^1(\scrO_C) \to H^1(\scrO_C(\bold d))\to 0,$$ where the map $H^1(\scrO_C) \to H^1(\scrO_C(\bold d))$ is given by multiplying by the section coresponding to $\bold d$. Thus $$\dim T = b(r+1) -1 + d-r.$$ Finally, the image of $\frak M_0$ is a component of $H_{D_0, X}$. \endstatement \proof First note that $H^0(X; \scrO_X(D_0)) = H^0(\Pee ^1; \scrO_{\Pee ^1}(b-1)) \otimes H^0(C; \bold d)$. Thus if $h^0(C; \bold d) =1$, then every nonzero section of $H^0(X; \scrO_X(D_0))$ is of the form $\pi _1^*\bold b + \pi _2^*\bold d$ for some divisor $\bold b$ of degree $b-1$ on $\Pee ^1$. Hence $h^0(D_0) = b$ and $|D_0|\cong \Pee ^{b-1}$. In general, setting $r+1 = h^0(\bold d)$, $$\dim H^0(\scrO_X(D_0))/\Cee\cdot \sigma _0 = b(r+1) - 1.$$ Now $H^1(\scrO_X) \cong H^0(\scrO_{\Pee ^1}) \otimes H^1(\scrO_C)$. Given a section $\sigma _0 $ of $\scrO_X(D_0)$ of the form $\sigma _1\otimes \sigma _2$, multiplication by $\sigma_0$ is just multiplication by $\sigma _2$ from $H^1(\scrO_C)$ to $H^1(\bold d)$, followed by multiplication by $\sigma _1$. On the other hand, using the exact sequence $$0 \to \scrO_C \to \scrO_C(\bold d) \to \scrO_{\bold d} \to 0,$$ where the map $\scrO_C \to \scrO_C(\bold d)$ is multiplication by $\sigma _2$, it follows that the dimension of the kernel of the map $H^1(\scrO_C)\to H^1(\bold d)$ is $d - r$ (and the dimension of the cokernel is $g-d + r$). So the dimension of the Zariski tangent space is $b(r+1) - 1 + d-r$, whereas the actual dimension of $\frak M_0$ is $b-1+d$. For the generic divisor $\bold d \in C_d$, $r=0$, since $d = g+a-1 <g$. For such a divisor $\bold d$, the map $\frak M_0 \to H_{D_0, X}$ is an embedding near $\bold d$. Thus the image of $\frak M_0$ is a component of $H_{D_0, X}$. Finally $H_{D_0, X}$ is smooth at $\bold d \in \frak M_0$ if and only if $r=0$. \endproof The same argument identifies the obstruction space at $D_0$: \lemma{5.3} Suppose as above that $D_0 = \pi _1^*\bold b + \pi _2^*\bold d$. Then $$\Coker \{\times \sigma _0\: H^1(\scrO_X) \to H^1(\scrO_X(D_0))\}$$ is equal to $H^0(\scrO_{\Pee ^1}(b-1))/\Cee \cdot \sigma _1 \otimes H^1(\bold d)$ and has dimension $(b-1)(g-d + r)$. \qed \endstatement We can put Lemma 5.3 in a more intrinsic global form as follows. The tangent bundle to $\Pee ^{b-1}$ is naturally $\scrO_{\Pee ^{b-1}}(1)^b/\scrO_{\Pee ^{b-1}}$. Over $C\times C_d$ we have the incidence divisor $\Cal I$ defined by $$\Cal I = \{\, (t, \bold d): t\in \operatorname{Supp}\bold d\,\}.$$ Let $\psi _2 \: C\times C_d \to C_d$ be projection onto the second factor. Then using the exact sequence $$0 \to \scrO_{C\times C_d} \to \scrO_{C\times C_d}(\Cal I) \to \scrO_{\Cal I}(\Cal I) \to 0,$$ we have an exact sequence of direct image sheaves by applying $R^i\psi _2{}_*$. Here $\psi _2|\Cal I$ is a $d$-sheeted cover, and in particular it is finite. Moreover $\Cal I$ is a hypersurface in $C\times C_d$ and $\Cal I \cap \psi _2^{-1}(\bold d)$ is identified with the divisor $\bold d$ on $C$. It is easy to see that $R^0\psi _2{}_*\scrO_{\Cal I}(\Cal I) $ is canonically the tangent bundle $T_{C_d}$ of $C_d$. Since it is torsion free, and $R^0\psi _2{}_*\scrO_{C\times C_d} \to R^0\psi _2{}_*\scrO_{C\times C_d}(\Cal I)$ is an isomorphism at a general $\bold d$, $R^0\psi _2{}_*\scrO_{C\times C_d} \cong R^0\psi _2{}_*\scrO_{C\times C_d}(\Cal I)$ and we have an exact sequence $$0 \to T_{C_d} \to R^1\psi _2{}_*\scrO_{C\times C_d} \to R^1\psi _2{}_*\scrO_{C\times C_d}(\Cal I) \to 0.$$ Here $R^1\psi _2{}_*\scrO_{C\times C_d} = H^1(\scrO_C) \times \scrO_{C_d}$ is a trivial bundle of rank $g$ on $C_d$, and we set $\Cal E = R^1\psi _2{}_*\scrO_{C\times C_d}(\Cal I)$. Note that the Chern classes of $\Cal E$ are given by $c(\Cal E) = c(T_{C_d})^{-1}$. \lemma{5.4} Let $p_1\: \Pee ^{b-1} \times C_d \to \Pee ^{b-1}$ be projection onto the first factor and let $p_2\: \Pee ^{b-1} \times C_d \to C_d$ be projection onto the second factor. We then have a map $p_1^*T_{\Pee ^{b-1}} \otimes p_2^*\Cal E \to R^1\pi _2{}_*\scrO_{X\times \frak M_0}(\Cal D)/R^1\pi _2{}_*\scrO_{X\times \frak M_0}$, and it is an isomorphism over those points $(\bold b, \bold d)$ of $\frak M_0$ where $h^0(\bold d) = 1$. \qed \endstatement Let us study the obstruction space for a divisor $D_0$ which is not necessarily in $\frak M_0$. \lemma{5.5} Let $D_0 \subset \Pee ^1 \times C$ be a divisor of type $(d, b-1)$, corresponding to a morphism $C \to \Sym^{b-1}\Pee ^1 \cong \Pee ^{b-1}$. Let $V \subset H^0(\bold d)^*$ be the linear subspace corresponding to the image of $C$. Then the obstruction space for $H_{D_0, X}$ at $D_0$ is zero if and only if the map $C \to \Pee ^{b-1}$ is nondegenerate and the map $$\mu _0 \: V\otimes H^0(K_C - \bold d) \to H^0(K_C)$$ given by cup product is injective, which holds for a generic curve $C$. \endstatement \proof The obstruction space is given as the cokernel of the map $H^1(\scrO_X) \to H^1(\scrO_X(D_0))$ which is multiplication by $\sigma _0$. Here $H^1(\scrO_X) \cong H^0(\scrO_{\Pee ^1}) \otimes H^1(\scrO_C)$ and $H^1(\scrO_X(D_0)) \cong H^0(\scrO_{\Pee ^1}(b-1)) \otimes H^1(\scrO_C(\bold d))$. Let $V$ be the quotient of $H^0(\bold d)^*$ corresponding to the map $\sigma _0 \in H^0(\bold b) \otimes H^0(\bold d) = \Hom (H^0(\bold d)^*, H^0(\bold b))$, and let $\alpha _1, \dots, \alpha _n$ be a basis for $V$ viewed as a subspace of $H^0(\bold b)$. It follows that we can write $\sigma _0 = \sum _i\alpha _i \otimes \beta _i$ for $\beta _i \in H^0(\bold d)$ which are linearly independent. In this case, $\{\beta _i\}$ must also be a basis for $V^* \subseteq H^0(\bold d)$. Multiplication by $\sigma_0$ is equivalent to the map sending $\xi \in H^1(\scrO_C)$ to $\sum _i\alpha _i\otimes (\beta _i\xi)$. This map is surjective if and only if $V = H^0(\scrO_{\Pee ^1}(b-1))$ and the natural map $$H^1(\scrO_C) \to H^0(\scrO_{\Pee ^1}(b-1)) \otimes H^1(\bold d)$$ is surjective. After applying an invertible element of $H^0(\scrO_{\Pee ^1}(b-1))$, we can assume that $\alpha _i = \beta _i^*$, the dual basis to $\beta _i$. In this case multiplication by $\sigma _0$ is easily seen to be the adjoint of the map $\mu _0$. Thus multiplication by $\sigma _0$ is surjective if and only if $V = H^0(\scrO_{\Pee ^1}(b-1))$ and $\mu _0$ is injective. \endproof We note that, in case $\mu _0$ is injective, and in particular for a generic curve $C$, (5.5) identifies the obstruction space as $(H^0(\scrO_{\Pee ^1}(b-1))/V)\otimes H^1(\bold d)$. For example, in case $\sigma _0 = \pi _1^*\bold b+ \pi _2^*\bold d$, $V$ is a line in $H^0(\scrO_{\Pee ^1}(b-1))$ and the obstruction space has dimension $(b-1)(g-d+r)$ as given by Lemma 5.3. In case $b=2$ and $\mu _0$ is injective, the only possibilities are $\dim V =1$ corresponding to $\sigma _0 = \pi _1^*\bold b+ \pi _2^*\bold d$, and $\dim V = 2$, $V = H^0(\scrO_{\Pee ^1}(1))$, corresponding to $D_0 = D_1 + \pi _2^*\bold d'$, where $D_1$ is the graph of a nonconstant map from $C$ to $\Pee ^1$. Thus the obstruction space is necessarily zero in this case, provided that $\mu _0$ is injective. Let us now describe all of the components of $H_{D_0, X}$ in the case $b=2$. In this case $0< -a < (g-1)/2$ and so $d > (g-1)/2$. Now suppose that $D_0 = D_1 + \pi _2^*\bold d'$, where $D_1$ is the graph of a map from $C$ to $\Pee ^1$ of degree $d_1$ and $\bold d'$ has degree $d' = d-d_1$. Thus $D_1$ corresponds to a linear subseries of $|\bold d'|$, which we can write as $\Pee (V^*)$ for some vector space $V$ of dimension two, together with a choice of isomorphism $V \cong \Pee ^1$. In general we can think of $D_0$ as corresponding to a sublinear system of $|\bold d|$ with base points. If $C$ is generic in the Brill-Noether sense, then for there to exist a map from $C$ to $\Pee ^1$ of degree $d_1$, we must have the Brill-Noether number $\rho = 2d_1 -g -2 \geq 0$, in which case the set $G^1_d$ of all linear series of degree $d$ and dimension one has dimension exactly $\rho$. Moreover for a generic curve $C$, if $\rho = 0$ then $G^1_d$ consists of reduced points, whereas if $\rho >0$ then $G^1_d$ is smooth and irreducible of dimension of dimension $\rho$ and the generic linear series in $G^1_d$ is complete. We see then that if $d = (g-1)/2, g/2, (g+1)/2$, then $\frak M_0$ is the unique component of $H_{D_0, X}$. For $d= (g+2)/2$ the components of $H_{D_0, X}$ are $\frak M_0$ together with a number of components isomorphic to $\Pee ^3 = \Pee (H^0(\scrO_{\Pee ^1}(1) \otimes H^0(\bold d))$, one for each $g^1_d$ on $C$. For $d> (g+2)/2$, there are two components, $\frak M_0$ and a smooth component of dimension $\rho + 3 = 2d - g +1$, which is essentially a $\Pee ^3$-bundle over $G^1_d$. Next we discuss the analytic structure of the moduli space in a neighborhood of a singular point in case $b=2$, in other words how the components meet. Recall from Proposition 1.3 that the obstruction map has an intrinsically defined quadratic term given by $\alpha \cup \partial \alpha$. \lemma{5.6} Let $D_0$ be given by the section $\sigma _0 = \sum _i\alpha _i \otimes \beta _i\in H^0(\bold b) \otimes H^0(\bold d)$, where the $\alpha _i$ and $\beta _i$ are linearly independent and $V = \operatorname{span}\{\alpha _i\}$ is a quotient of $H^0(\bold d)^*$ with dual space $V^*\subseteq H^0(\bold d)$. Suppose that $V \neq H^0(\bold d)$ and that the map $\mu _0$ is injective for $\bold d$. Then cup product induces a surjective map $$\gather H^0(\scrO_X(D_0))/\Cee \sigma _0 \otimes \partial (H^0(\scrO_{D_0}(D_0)) \to H^1(\scrO_X(D_0))/\sigma _0H^1(\scrO_X) \\ \cong (H^0(\bold b)/V) \otimes H^1(\bold d). \endgather$$ More precisely, for every $\tau _2 \notin V \subset H^0(\bold d)$, the map $$(H^0(\bold b)\otimes \Cee \tau _2) \otimes \partial (H^0(\scrO_{D_0}(D_0)) \to (H^0(\bold b)/V) \otimes H^1(\bold d)$$ is surjective. \endstatement \proof The image of $\partial \: H^0(\scrO_{D_0}(D_0)) \to H^1(\scrO_X) \cong H^0(\scrO_{\Pee ^1})\otimes H^1(\scrO_C)$ is the set of all $1\otimes \xi$, where $\sum _i\alpha \otimes \xi \beta _i =0$. Since the $\alpha _i$ are linearly independent, this condition is equivalent to the condition that $\xi \beta _i =0$ for all $i$. Cup product of such a class $1\otimes \xi$ with $$\tau _1 \otimes \tau _2 \in H^0(\scrO_X(D_0))\cong H^0(\bold b) \otimes H^0(\bold d)$$ is $\tau _1\otimes (\tau _2\cdot \xi) \in H^1(\scrO_X(D_0))$. For the projection of this map to $$H^1(\scrO_X(D_0))/\sigma _0H^1(\scrO_X) \cong (H^0(\bold b)/V) \otimes H^1(\bold d)$$ to be surjective, it suffices that, setting $$(V^*)^\perp = \{\,\xi \in H^1(\scrO_C): \xi \cdot \beta _i = 0 \text{ for all $i$}\,\},$$ the induced cup product map $H^0(\bold d) \otimes (V^*)^\perp \to H^1(\bold d)$ is surjective. Now by assumption, the adjoint $\mu _0^*$ of the $\mu _0$ map is surjective, where $$\mu _0^*\: H^1(\scrO_C) \to H^1(\bold d) \otimes H^0(\bold d)^*= \Hom (H^0(\bold d), H^1(\bold d)).$$ Since $H^0(\bold d) \neq V$, there exists a $\tau _2 \notin V$. Given $\eta \in H^1(\bold d)$, there exists a linear map $F\: H^0(\bold d) \to H^1(\bold d)$ such that $F(V) = 0$ and $F(\tau _2) = \eta$. The surjectivity of the map $\mu _0^*$ implies that $F$ is given by taking cup product with a $\xi$ such that $\xi \cdot \beta _i = 0$ for all $i$ and $\xi \cdot \tau _2 = \eta$. Thus the image of $H^0(\scrO_X(D_0))/\Cee \sigma _0 \otimes \partial (H^0(\scrO_{D_0}(D_0))$ contains every element of the form $\tau _1 \otimes \eta$, where $\tau _1$ and $\eta$ are arbitrary, and so is all of $(H^0(\bold b)/V) \otimes H^1(\bold d)$. \endproof Using Lemma 5.6, we can describe the local structure of $H_{D_0, X}$ in case $b=2$ near a reducible divisor $\pi _1^*\bold b + \pi _2^*\bold d$, provided that $\bold d$ is generic in the sense that $h^0(\bold d)$ is exactly $2$. \corollary{5.7} With assumptions on the $\mu _0$ map as above, suppose that $b=2$ and that $D_0 = \pi _1^*\bold b + \pi _2^*\bold d$, where $h^0(\bold d) = 2$. Then an analytic neighborhood of $H_{D_0, X}$ near $D_0$ is biholomorphic to a neighborhood of the origin in $L_0 \cup L_1 \subset \Cee ^{d+2}$, where $L_0$ is a hyperplane and $L_1$ is a linear space of dimension $2d-g +1$, not contained in $L_0$. \endstatement \proof The dimension of the Zariski tangent space $T$ to $H_{D_0, X}$ is $2\cdot 2 -1 + d - 1 = d+2$. The dimension of $\Pee ^1 \times C_d$ is $d+1$, and so the image $L_0$ of the tangent space to $\Pee ^1 \times C_d$ at $D_0$ has the expected dimension of a hyperplane in $T$. In fact, we have seen in the discussion prior to Lemma 5.2 that $L_0$ is indeed a hyperplane, defined by the linear form $\ell$, say. Thus if $\Phi\: T \to \Cee ^{g-d+1}$ is the Kuranishi obstruction map, defined in a neighborhood of the origin, then there exists a holomorphic function $f$ with differential $\ell$ such that $\Phi = f\Psi$, and the quadratic term in $\Phi$ is equal to $\ell\cdot d\Psi _0$. The span of $\alpha \cup \partial \alpha$ over all $\alpha$ is thus contained in the image of $d\Psi _0$ and this span is the same, after polarizing, as the image of $\alpha \cup \partial\beta +\beta \cup \partial \alpha$ over all $\alpha, \beta$. Using Lemma 5.6, there exists a choice of $\alpha _i, \beta _i$ with $\partial \alpha _i = 0$ such that the obstruction space is generated by $\alpha _i\cup \partial \beta_i$. Thus $d\Psi _0$ has the same image as the map of Lemma 5.6 and so is surjective. It follows that $\Phi ^{-1}(0) = L_0 \cup \Psi ^{-1}(0)$, where $\Psi ^{-1}(0)$ is a smooth submanifold of $T$ of codimension $g-d+1$. If it does not meet $L_0$ transversally, then $\Ker d\Psi _0 \subseteq L_0$. But $L_0 \cap H^0(\scrO_X(D_0))/\Cee \sigma _0$ is the tangent space to the Segre embedding of $\Pee ^1 \times \Pee ^1$ in $\Pee H^0(\scrO_X(D_0)) = \Pee ^3$. Thus $H^0(\scrO_X(D_0))/\Cee \sigma _0$ is not contained in $L_0$. On the other hand, $H^0(\scrO_X(D_0))/\Cee \sigma _0$ is the tangent space to $\Pee H^0(\scrO_X(D_0))$ and so is unobstructed, so that it must be contained in $\Ker d\Psi _0$. Thus $\Ker d\Psi _0$ is not contained in $L_0$, and so $\Psi ^{-1}(0)$ meets $L_0$ transversally. This concludes the proof. \endproof Finally, for a generic curve $C$, we shall use the description of the components of the moduli space above to make some calculations in case $b=2$ and $d$ is small. We do not need the description of the analytic structure of the moduli space. In this case the moduli space always has the component $\frak M_0= \Pee ^1\times \Sym ^dC$. For $d= g-1/2, g/2, g+1/2$, $\frak M_0$ is the unique component, whereas in general the moduli space is equal to $\frak M_0 \cup \frak M_1$. For $d= g+2/2$, $\frak M_1$ is a union of $k$ copies of $\Pee ^3$, where $k$ is the number of $g^1_d$'s on $C$, and for $d > g+2/2$ $\frak M_1$ is irreducible and smooth, of the expected dimension $\rho +3 = 2d-g +1$. To calulate the value of the Seiberg-Witten invariant, we shall first calculate the contribution from $\frak M_0$ and then the contribution from $\frak M_1$. Note that $\frak M_0$ does not have the expected dimension, which is $2d-g+1$. Moreover the moduli space is in general singular. However it is easy to see that we can choose incidence divisors $\mu _1, \dots, \mu _{2d-g+1}$ which meet properly in the smooth part of the moduli space. Following the procedure of Section 3 (see the comments at the end of the section), we first calculate the top Chern class of the obstruction bundle over $\frak M_0$, which we have seen (Lemma 5.4) is the bundle $p_1^*T_{\Pee ^1} \times p_2^*\Cal E$, at least after cutting down by $\mu ^{2d-g+1}$. Here the $p_i$ are the projections of $\Pee ^1\times \Sym ^dC$ to the $i^{\text{th}}$ factor and $\Cal E$ is the bundle $R^1\psi _2{}_*\scrO_{C\times C_d}(\Cal I)$, of rank $g-d$. The top Chern class of the tensor product of $p_2^*\Cal E$ with the line bundle $p_1^*T_{\Pee ^1}$ is: $c_N(p_1^*T_{\Pee ^1} \times p_2^*\Cal E) = \sum _{i=0}^Np_1^*c_1(T_{\Pee ^1})^ip_2^*c_{N-i}(\Cal E)$. If $h$ is the hyperplane class on $\Pee ^1$, in other words the class of a point, then $p_1^*c_1(T_{\Pee ^1}) = 2p_1^*h$ and $p_1^*c_1(T_{\Pee ^1})^i = 0$ for $i>1$. Thus $$c_N(p_1^*T_{\Pee ^1} \times p_2^*\Cal E) =p_2^*c_N(\Cal E) + 2p_1^*hp_2^*c_{N-1}(\Cal E).$$ By \cite{1}, p\. 322, $c(\Cal E) = c(T_{C_d})^{-1} = (1+x)^{g-1-d}e^{\theta /1+x}$, where $x$ is the class of the divisor $C_{d-1}\subset C_d$ and $\theta$ is the pullback of the theta divisor on $\Pic^d C$ under the natural map, and moreover $\theta ^kx^{d-k} = \frac{g!}{(g-k)!}$. To calculate the Seiberg-Witten invariant, we take $c_N(p_1^*T_{\Pee ^1} \times p_2^*\Cal E)$, where $N = g-d = 1-a$. This gives a class in $H^{2N}( \Pee ^1\times \Sym ^dC)$, and then we further multiply by $\mu ^{d+1-N}$ and evaluate over the fundamental class. On $\Pee ^1\times \Sym ^dC$, it is clear that $\mu = p_1^*h+p_2^*x$, since $(t,p) \in \pi _1^*\{s\} + \pi _2^*\bold d$ if and only if either $t=s$ or $p\in \bold d$, and it is easy to see that the multiplicity is one. Thus we must calculate $$\gather (p_1^* h+p_2^*x)^{d+1-N}p_2^*c_N(\Cal E) + 2p_1^*hp_2^*c_{N-1}(\Cal E))\\ = (2p_1^*h)p_2^*(c_{N-1}(\Cal E)x^{d+1-N} + (d+1-N) p_1^*hp_2^*(c_N(\Cal E)x^{d-N}\\ = 2c_{N-1}(\Cal E)x^{d+1-N} + (d+1-N)c_N(\Cal E)x^{d-N}. \endgather$$ Plugging in for $c(\Cal E)$, we have $$\align c(\Cal E) &= (1+x)^{g-1-d}e^{\theta /1+x} =(1+x)^{N-1}e^{\theta /1+x}\\ &= \sum_{i=0}^{N-1}\binom{N-1}{i}x^i\sum_{j=0}^\infty\frac{1}{j!}\theta ^j\sum_{k=0}^\infty\binom{-j}{k}x^k . \endalign$$ Thus for example the term involving $c_{N-1}(\Cal E)$ becomes $$\align 2&\sum _{i+j+k= N-1}\binom{N-1}{i}\frac{1}{j!}\binom{-j}{k}x^{i+k+d+1-N}\theta ^j\\ =2&\sum _{i+j+k= N-1}\binom{N-1}{i}\frac{1}{j!}\binom{-j}{k}x^{d-j}\theta ^j\\ =2&\sum _{i+j+k= N-1}\binom{N-1}{i}\binom{-j}{k}\frac{1}{j!}\frac{g!}{(g-j)!}\\ =2&\sum _{j=0}^{N-1}\left(\sum _{k=0}^{N-1-j}\binom{N-1}{N-1-j-k}\binom{-j}{k}\right)\binom{N+d}{j}, \endalign$$ where we have used $g=N+d$. Applying Lemma 4.5 with $e=0$ to the inner sum above, where we let $a = N-1-j$ for a fixed $j$, we see that the expression reduces to $\dsize 2\sum _{j=0}^{N-1}\binom{N+d}{j}$. A very similar manipulation with the term $(d+1-N)c_N(\Cal E)x^{d-N}$ gives $$ (d+1-N)c_N(\Cal E)x^{d-N} = (d+1-N)\binom{N+d}{N}.$$ The final contribution for the component $\frak M_0$ is therefore $$2\sum _{j=0}^{N-1}\binom{N+d}{j}+ (d+1-N)\binom{N+d}{N}.$$ Note that $\frak M_0$ is the unique component for $N= d+1, d, d-1$ corresponding to the cases $g = 2d+1, 2d, 2d-1$. Plugging in, we find that the value of the invariant in case $N= d+1$ is $$2\sum _{j=0}^{d}\binom{2d+1}{j}= \sum _{j=0}^{2d+1}\binom{2d+1}{j} = (1+1)^{2d+1} = 2^g.$$ Similar calculations handle the cases $N= d, d-1$, and again give the value $2^g$. For $d= g+2/2,N=d-2, g = 2d-2$, the set $\frak M_1$ consists of $k$ copies of $\Pee ^3$, where $k$ is the number of $g^1_d$'s on $C$. This number has been computed by Castelnuovo \cite{1} p\. 211: it is $$g!\frac1{(g-d+1)!}\frac1{(g-d+2)!} = \frac{(2d-2)!}{(d-1)!\,d!}.$$ In this case, the restriction of $\Cal D$ to each piece $X\times \Pee ^3$ is the incidence divisor, so that $\mu$ restricts to the hyperplane class in each $\Pee ^3$. Thus the final answer is $$2\sum _{j=0}^{d-3}\binom{2d-2}{j} +3\binom{2d-2}{d-2}+ \frac{(2d-2)!}{(d-1)!\,d!}$$ which after a brief manipulation becomes $\dsize\sum _{j=0}^{2d-2}\binom{2d-2}{j} = (1+1)^{2d-2} = 2^g$. Somewhat more involved methods handle the case $N=d-3$, and presumably might be pushed, using excess intersections, to give the general case. However, we shall give a simpler method for the calculation in the next section. \section{6. Deformation to more general ruled surfaces.} In this section, we shall study Seiberg-Witten moduli spaces, or equivalently the Hilbert scheme, for more general ruled surfaces $\Pee (V)$, where $V$ is a general (and in particular stable) rank two bundle over $C$. We shall deal with the case where $\det V$ has even degree, and thus assume that $c_1(V) = 0$. Also, we shall only discuss the case of sections of $V$. However, it will be clear that our methods generalize to handle the case of odd degree as well as more general cases of multisections, and thus suffice for the homological calculations of the invariants in general. We will outline this approach at the end. Throughout this section, we fix a smooth curve $C$ of genus $g\geq 2$. it will not be necessary to assume that $C$ is generic in the Brill-Noether sense. Recall that there is a one-to-one correspondence between irreducible sections $D_0$ of $\Pee (V)$ and line bundles $\lambda$ such that $V\otimes \lambda$ has a nowhere vanishing section, as follows: given a section $D_0$ of $\Pee (V)$, apply $R^i\pi _*$ to the exact sequence $$0 \to \scrO_X \to \scrO_X(D_0) \to \scrO_{D_0}(D_0) \to 0$$ to obtain the exact sequence $$0 \to \scrO_C \to R^0\pi _*\scrO_X(D_0) \to\scrO_{D_0}(D_0) \to 0.$$ Here $R^1\pi _*\scrO_X= 0$ since the fibers are $\Pee ^1$ and we can write $R^0\pi _*\scrO_X(D_0) = V\spcheck \otimes \lambda = V\otimes \lambda$. Note that the normal bundle $\scrO_{D_0}(D_0)$ is naturally $\lambda ^2$. The inverse map sends the section of $V\spcheck \otimes \lambda$ to the homogeneous degree one subvariety of $\Pee(V)$ that it defines. If $D_0$ is not irreducible, then $D_0 = E_0 + \pi ^*\bold e$. In this case the map $\scrO_X \to \scrO_X(D_0)$ factors through $\scrO_X(E_0)$ and the induced map $\scrO_C \to R^0\pi _*\scrO_X(D_0)$ factors as $$\scrO_C \to R^0\pi _*\scrO_X(E_0)\to R^0\pi _*\scrO_X(E_0)\otimes \scrO_C(\bold e) = R^0\pi _*\scrO_X(D_0).$$ Thus $D_0$ still corresponds to a section of $V\spcheck\otimes \lambda$ for an appropriate $\lambda$, but the section vanishes exactly along $\bold e$. Again such a section defines a subvariety of $\Pee (V)$, which is exactly $D_0$ since the section vanishes along $\bold e$. In this way, we can identify $|D_0|$ with $\Pee H^0(V\spcheck \otimes \lambda)$, including the reducible fibers. \proposition{6.1} Let $e$ be an positive integer. For every line bundle $\lambda$ on $C$ of degree $e$, \roster \item"{(i)}" There exist stable bundles $V$ on $C$ together with an exact sequence $$0 \to \lambda ^{-1} \to V \to \lambda \to 0.$$ \item"{(ii)}" For $e< (g-1)/2$, and $V$ a generic stable bundle satisfying \rom{(i)}, if $\mu$ is a line bundle of degree $d\leq e$ and $H^0(V\otimes \mu) \neq 0$, then $\mu = \lambda$ and $H^0(V\otimes \lambda)$ has dimension one. \item"{(iii)}" For $e = (g-1)/2$ and $V$ general, there are exactly $2^g$ distinct $\lambda$ with $H^0(V\otimes \lambda) \neq 0$, and for each such $\lambda$, $\dim H^0(V\otimes \lambda) =1$. Moreover, if $\deg \mu < (g-1)/2$, then $H^0(V\otimes \mu) = 0$. \endroster \endstatement \proof For a line bundle $\lambda$ of degree $e>0$, $\Ext ^1(\lambda, \lambda ^{-1}) = H^1(\lambda ^{-2})$ which has dimension $2e+g-1$, by Riemann-Roch. Let $V$ be a rank two bundle corresponding to an extension class $\xi \in H^1(\lambda ^{-2})$. Suppose that there exists a nonzero map $\mu \to V$, where $\deg \mu = d\leq e$, and that $\mu \neq \lambda$ in case $d=e$. Since $H^0(\mu \otimes \lambda ^{-1})=0$ since $\mu \otimes \lambda ^{-1}$ either has negative degree or has degree zero and is not trivial, there must exist a nonzero section of $\mu \otimes \lambda$ which lifts to a section of $\mu \otimes V$. Note that $(s)$, the divisor of zeroes of $s$, has degree $d+e$, and we can identify the set of all pairs $(\mu, s)$ such that $\mu$ is a line bundle of degree $d$ and $s$ is a nonzero section of $\mu \otimes \lambda$, mod scalars, with $C_{d+e}$. The section $s$ lifts to a section of $\mu \otimes V$ if and only if the coboundary map $\partial (s) = 0$, where $\partial s \in H^1(\mu \otimes \lambda ^{-1})$. Now $\partial s = \xi \cdot s$, the cup product of $s\in H^0(\mu \otimes \lambda)$ with $\xi \in H^1(\lambda ^{-2})$. Consider the exact sequence $$0 \to \lambda ^{-2} @>{\times s}>> \lambda ^{-1}\otimes \mu \to \scrO_{\bold f} \to 0,$$ where $\bold f = (s) \in C_{d+e}$. By assumption, $\partial s = 0$ if and only if $\xi \cdot s = 0$ if and only if the image of $\xi$ in $H^1(\lambda ^{-1}\otimes \mu)$ is zero, if and only if $\xi$ is in the image of $H^0(\scrO_{\bold f})$. By assumption either $\deg (\lambda ^{-1}\otimes \mu) < 0$ or $\deg (\lambda ^{-1}\otimes \mu) = 0$ but $\lambda ^{-1}\otimes \mu$ is not trivial. In either case $H^0(\lambda ^{-1}\otimes \mu) = 0$, and so the image of $H^0(\scrO_{\bold f})$ has dimension $d+e$. Thus the set of possible extension classes $\xi$ for which a given $s$ lifts has dimension $d+e$, and so corresponds to a $\Pee ^{d+e-1} \subseteq \Pee H^1(\lambda ^{-2})= \Pee ^{2e+g-2}$. The set of all $\xi$ for which some $s$ lifts is then the union over all $s$ of a linear subspace of $\Pee ^{2e+g-2}$ of dimension $d+e-1$. Since the set of all $s$ is just $C_{d+e}$, the dimension of the set of all possible $\xi$ is at most $2d+2e-1$. This number is less than $2e+g-2$ exactly when $d < (g-1)/2$. Choosing a bundle $V$ coresponding to an extension class $\xi$ in the complement of this set gives a bundle $V$, written as an extension of $\lambda$ by $\lambda ^{-1}$, such that, if $H^0(V\otimes \mu) \neq 0$ and $\deg \mu \leq \lambda$, then $\mu = \lambda$. In particular, $V$ is stable, proving (i) and (ii), except for the statement that $\dim H^0 (V\otimes \lambda ) = 1$. To see the statement about $\dim H^0(V\otimes \lambda)$, if $s$ is a nonzero section of $\lambda ^2$ which lifts to a section of $V\otimes \lambda$, then arguments similar to those above show that the orthogonal complement of $s\cdot H^0(K_C)$ in $H^1(\lambda ^2)$ has dimension $2e-1$ and so gives a linear space of dimension $2e-2$ inside $\Pee ^{2e+g-2}$. Moreover, the possible $s$ correspond to the case $\mu = \lambda$, and so form a proper subvariety of $C_{d+e} = C_{2e}$. In all, the $\xi$ for which some $s$ lifts, such that the corresponding line bundle $\mu = \lambda$, form a subvariety of $\Pee ^{2e+g-2}$ of dimension at most $4e-3$. Now $4e-3< 2e+g-2$ provided that $e<(g+1)/2$, and thus certainly if $e\leq (g-1)/2$. This establishes the last statement of (ii). The remaining assertion (iii) is a classical formula due to Corrado Segre \cite{11}. It follows from our formulas in the previous section for the zero-dimensional invariant, and will be reproved in more generality shortly. \endproof We now fix a bundle $V$ which will later be assumed generic in an appropriate sense. For each $d$, we consider the following varieties of Brill-Noether type: $$\align W_{1,d}(V) &= \{\, \lambda \in \Pic ^d C: h^0(V\otimes \lambda ) \geq 1\,\};\\ G_{1,d}(V) &= \{\,(s,\lambda): \lambda \in \Pic ^d C, s\in \Pee(H^0(V\otimes \lambda ))\,\}. \endalign$$ Thus there is a natural map $G_{1,d}(V) \to W_{1,d}(V)$. It is also clear that, with $X= \Pee (V)$, $G_{1,d}(V)$ is exactly the Hilbert scheme of $X$ corresponding to sections (possibly reducible) of the appropriate degree and that the map $G_{1,d}(V) \to W_{1,d}(V)$ can be identified with the map from the Hilbert scheme to $\Pic X$. We wish to give another construction of the Hilbert scheme in this context; in other words, we will put another scheme structure on $G_{1,d}(V)$ and then claim that it is in fact the usual one. To do so, we make a construction similar to the usual construction of Brill-Noether theory: fix a divisor $D$ on $C$ of degree $m\gg 0$ such that $h^1(V\otimes \lambda \otimes \scrO_C(D)) = 0$ for all line bundles $\lambda$ of degree $d$. We can assume that $D$ is an effective divisor consisting of reduced points of $C$ if we wish. Consider the restriction sequence $$0 \to V\otimes \lambda \to V\otimes \lambda \otimes \scrO_C(D) \to V\otimes \lambda \otimes \scrO_D(D)\to 0.$$ Taking global sections, there is a map $$\varphi\: H^0(V\otimes \lambda \otimes \scrO_C(D)) \to H^0( V\otimes \lambda \otimes \scrO_D(D)).$$ The first vector space has dimension $2m+2d-2g+2$, by Riemann-Roch, and the second has dimension $2m$, and $\lambda \in W_{1,d}(V)$ if and only if $\varphi$ has a kernel. In this case, the fiber over $\lambda$ in $G_{1,d}(V)$ is just $\Pee (\Ker \varphi) = \Pee(H^0(V\otimes \lambda))$. Globally, let $\Cal P$ be a Poincar\'e line bundle for $C\times \Pic ^dC$, and let $\pi _i$ be the projection of $C\times \Pic ^dC$ to the $i^{\text{th}}$ factor. Set $$\align \Cal E' &= \pi _2{}_*\left(\Cal P\otimes \pi _1^*(V\otimes \scrO_C(D))\right);\\ \Cal E'' &= \pi _2{}_*\left(\Cal P\otimes \pi _1^*(V\otimes \scrO_D(D))\right), \endalign$$ so that there is a natural evaluation map $\Phi \: \Cal E' \to \Cal E''$. Then $W_{1,d}(V)$ is the scheme where $\Phi$ fails to be injective. We can define $G_{1,d}(V)$ similarly: let $p\: \Pee\Cal E' \to \Pic^dC$ be the projection. We have the inclusion of $\scrO_{\Pee\Cal E'}(-1)$ inside $p^*\Cal E'$. Consider the composition $$\scrO_{\Pee\Cal E'}(-1) \to p^*\Cal E' @>{p^*\Phi}>>p^*\Cal E''.$$ If we denote this composition by $\tilde \Phi$, then $\tilde \Phi =0$ at a point $(s, \lambda)$, where $\lambda \in \Pic ^dC$ and $s \in \Pee H^0(V\otimes \lambda \otimes \scrO_C(D))$, if and only if $s$ is in the image of $\Pee H^0(V\otimes \lambda)$. Thus the vanishing of $\tilde \Phi$ defines $G_{1,d}(V)$ as a set inside $\Pee \Cal E'$. Note that $\Pee \Cal E'$ is itself a Hilbert scheme: it is the same as $G_{1, d+m}(V)$, corresponding to the set of all sections of $X$ of degree $2d+2m$. Moreover $G_{1,d}(V)$ is the subset of $\Pee \Cal E'$ consisting exactly of those sections containing $\pi ^*D$. We leave it to the reader to work through the details that the subscheme defined by $\tilde \Phi$ represents the functor corresponding to $G_{1,d}(V)$ (see \cite{1} pp\. 182--184 for the Brill-Noether analogue) and that indeed this identifies $G_{1,d}(V)$ with the Hilbert scheme as schemes. Next suppose that $(s, \lambda)$ is a point of $G_{1,d}(V)$ such that the section $s$ does not vanish. Standard arguments (cf\. \cite{1}, pp\. 185--186) identify the Zariski tangent space to $G_{1,d}(V)$ at the point $(s, \lambda)$ with $H^0(\lambda ^2)= \Hom (\lambda ^{-1}, V/\lambda ^{-1})$, via the exact sequence $$0\to H^0(\scrO_C) \to H^0(V\otimes \lambda) \to H^0(\lambda ^2) \to H^1(\scrO_C) \to H^1(V\otimes \lambda).$$ Moreover the differential of the map from $G_{1,d}(V)$ to $W_{1,d}(V)$ is the obvious map $ H^0(\lambda ^2) \to H^1(\scrO_C)$. Finally, a standard cocycle calculation identifies the obstruction space as $H^1(\lambda ^2)$. Note that, if $\lambda$ corresponds to the section $D_0$ of $X$, then $H^1(\lambda ^2) = H^1(\scrO_{D_0}(D_0))$ is the same obstruction we would have found via the Hilbert scheme, as well it must be since $G_{1,d}(V)$ represents the same functor as the Hilbert scheme. For a general section $s$, suppose that the map $\scrO_C \to V\otimes \lambda$ vanishes along the divisor $\bold e$, so that there is a factorization $$\scrO_C \to V\otimes \lambda \otimes \scrO_C(-\bold e) \to V\otimes \lambda.$$ Let $\lambda _0 = \lambda \otimes \scrO_C(-\bold e)$. Then there is a commutative diagram $$\CD 0 @>>> \scrO_C @>>> V\otimes \lambda \otimes \scrO_C(-\bold e) @>>> \lambda _0^2 @>>> 0\\ @. @| @VVV @VVV @.\\ 0 @>>> \scrO_C @>>> V\otimes \lambda @>>> \lambda _0^2\oplus T @>>> 0. \endCD$$ Here $T$ is a skyscraper sheaf isomorphic to $V\otimes \scrO_{\bold e}$, and so has length $2\deg \bold e = 2e$, say. We can again identify the Zariski tangent space to $G_{1,d}(V)$ at $(s, \lambda)$ with $$\Hom (\lambda ^{-1}, V/\lambda ^{-1}) = H^0 (V\otimes \lambda/s\cdot \scrO_C)=H^0(\lambda_0 ^2\oplus T) = H^0(\lambda_0 ^2) \oplus H^0(T).$$ Moreover the obstruction space is $$\Ext ^1(\lambda ^{-1}, V/\lambda ^{-1}) = H^1(\lambda_0 ^2\oplus T) = H^1(\lambda_0 ^2).$$ This again corresponds to the deformation theory and obstruction theory for the Hilbert scheme: let $D_0 = E_0 + \pi ^*\bold e$, where $E_0$ is an irreducible section of $X$. Apply $R^i\pi _*$ to the exact sequence $$0 \to \scrO_X \to \scrO_X(D_0) \to \scrO_{D_0}(D_0) \to 0,$$ using $R^1\pi _*\scrO_X = R^1\pi _*\scrO_X(D_0) = 0$. We obtain $$0 \to \scrO_C \to V\otimes \lambda \to \pi _*\scrO_{D_0}(D_0) \to 0,$$ and $R^1\pi _*\scrO_{D_0}(D_0) = 0$. Thus $H^i(\scrO_{D_0}(D_0) ) = H^i(\pi _*\scrO_{D_0}(D_0) )= H^i(V\otimes \lambda/s\cdot \scrO_C)$. Summarizing, we have shown the following: \proposition{6.2} Let $(s, \lambda)$ be a point of $G_{1,d}(V)$. Suppose that the section $s$ vanishes exactly along the effective divisor $\bold e$, and set $\lambda _0 = \lambda \otimes \scrO_C(-\bold e)$. Then the Zariski tangent space to $G_{1,d}(V)$ is $H^0(V\otimes \lambda/s\cdot \scrO_C)$, which has dimension $h^0(\lambda _0^2) + 2e$, and the obstruction space is $H^1(\lambda _0^2)$. \qed \endstatement Assuming for simplicity that $s$ does not vanish at any point, and so $\lambda = \lambda _0$ in the above notation, the group $H^1(\lambda ^2)$ arises in yet another way as follows: the universal extension over $\Pee H^1(\lambda ^2)$ of $\lambda$ by $\lambda ^{-1}$ gives rise to a Kodaira-Spencer map from the tangent space of $\Pee H^1(\lambda ^{-2})$ at a nonzero point $\xi \in H^1(\lambda ^{-2})$, namely $H^1(\lambda ^{-2})/\Cee \cdot \xi$, to $H^1(\operatorname{ad}V)$. A diagram chase identifies the cokernel of this map with $H^1(\lambda ^2)$ under the natural map $H^1(\operatorname{ad}V) \to H^1(\lambda ^2)$. Thus $H^1(\lambda ^2)=0$ if and only if the map from the set of extensions to moduli is a submersion at $\xi$. If $d\geq (g-1)/2$, then for a generic choice of $\lambda$ we will indeed have $H^1(\lambda ^2)=0$, and so the map from extensions to moduli will be a submersion where defined. We now show that, for a generic choice of $V$, the Hilbert scheme is always smooth: \proposition{6.3} For a generic stable bundle $V$ with $c_1(V) = 0$ and for all $d$, the Hilbert scheme of sections of $X= \Pee(V)$ of square $2d$ is everywhere smooth of the expected dimension $2d-g+1$. \endstatement \proof Fix a value for $d$ corresponding to the degree of a line subbundle $\lambda$ of $V$. We have seen that, for generic $V$, the space of sections is empty if $d< (g-1)/2$, and if $\lambda$ has degree $>g-1$, then $h^1(\lambda ^2) = 0$ by Serre duality. For $(g-1)/2 \leq d \leq g-1$, let $\Cal P$ be the Poincar\'e line bundle over $C\times \Pic^dC$ and let $\Cal V = R^1\pi _2{}_*\Cal P^{\otimes -2}$. Note that, as $\deg \lambda = d >0$, then $h^0(\lambda ^{\otimes -2}) = 0$ and $h^1(\lambda ^{\otimes -2}) = 2d+g-1$. Hence $\Cal V$ is a vector bundle of rank $2d+g-1$ over $\Pic^dC$ and $\Pee\Cal V$ is a $\Pee ^{2d+g-2}$-bundle over $\Pic^dC$. The space $\Pee\Cal V$ is a moduli space for vector bundles $V$ given as extensions. There is an open subset $\Cal U$ of $\Pee\Cal V$ corresponding to stable bundles, which is nonempty by (i) of (6.1). The remarks prior to the statement of (6.3) imply that the morphism $\Cal U \to \frak M(C)$ is dominant, where $\frak M(C)$ is the moduli space of stable rank two bundles $V$ over $C$ with $c_1(V) = 0$. Let $$\Cal B = \{\, \lambda \in \Pic^dC: h^1(\lambda ^2) \neq 0\,\}.$$ Since $h^1(\lambda ^2) \neq 0$ if and only if $h^0(K_C\otimes \lambda ^{-2}) \neq 0$, $\Cal B$ is the inverse image in $\Pic^dC$ of the set of effective divisors in $\Pic^{2g-2-2d}C$ under the obvious (\'etale) map $\lambda \mapsto K_C\otimes \lambda ^{-2}$. Thus $\Cal B$ has the same dimension as the set of effective divisors in $\Pic^{2g-2-2d}C$, namely $2g-2-2d$. Hence if $p\: \Pee \Cal V \to \Pic ^dC$ is the projection, then $$\dim p^{-1}(\Cal B) = 2g-2 -2d + 2d+g-2 =3g-4.$$ It follows that the image of $p^{-1}(\Cal B)\cap \Cal U$ in $\frak M(C)$ cannot be all of $\frak M(C)$. Thus we can choose a stable bundle $V$ such that, if $\deg \lambda <(g-1)/2$, then there is no nonzero section of $V\otimes \lambda$, and if $\deg \lambda \geq (g-1)/2$ and there is a nowhere vanishing section $s$ of $V\otimes \lambda$, then $ h^1(\lambda ^2) = 0$. It follows that $(s, \lambda)$ is a smooth point of $G_{1,d}(V)$ (or of the Hilbert scheme), and the discussion prior to Proposition 6.2 shows how to extend this to all nonzero sections. Thus the Hilbert scheme of sections is everywhere smooth. \endproof Next we turn to the enumerative geometry of the Hilbert scheme. Since $\scrO_{\Pee\Cal E'}(-1)$ is a line bundle, $\tilde \Phi$ is equivalent to a section of $\scrO_{\Pee\Cal E'}(1)\otimes \Cal E''$ and the class of its zero set, namely $G_{1,d}(V)$, is given by $c_{2m}(\scrO_{\Pee\Cal E'}(1)\otimes \Cal E'')$. If we set $\zeta = c_1(\scrO_{\Pee\Cal E'}(1))$ and use the fact that $\Cal E''$ is a topologically trivial bundle of rank $2m$ \cite{1} p\. 309, then $c_{2m}(\scrO_{\Pee\Cal E'}(1)\otimes \Cal E'')$ is the term of degree $2m$ in $(1+\zeta)^{2m}$, namely $\zeta ^{2m}$. In particular, the class of $W_{1,d}(V)$ is given by $p_*\zeta ^{2m}$, which is the appropriate Segre class of $\Cal E'$. To find it, take $c(\Cal E')^{-1}$. Topologically $\Cal E'$ is two copies of $\pi _2{}_*\left(\Cal P \otimes \pi _1^*\scrO_C(D)\right)$, and using \cite{1} p\. 336, this last bundle has total Chern class $e^{-\theta}$. Thus $c(\Cal E') = e^{-2\theta}$ and $c(\Cal E')^{-1} = e^{2\theta}$. Taking the term of degree $2g-2d-1$, we find \cite{6}: $$[W_{1,d}(V)] = \frac{(2\theta)^{2g-2d-1}}{(2g-2d-1)!}.$$ In particular, when $2d+1 = g$ we obtain $(2\theta)^g/g! = 2^g$, giving the formula of Segre for the case of the zero-dimensional invariant. (In fact, Segre stated the formula for all values of $d$.) To handle the general case, we use: \proposition{6.4} The $\mu$ divisor on $G_{1,d}(V)$ is algebraically equivalent to the restriction of $\zeta = c_1(\scrO_{\Pee\Cal E'}(1))$. \endstatement \corollary{6.5} The value of the Seiberg-Witten invariant is $2^g$. \endstatement \demo{Proof of the corollary} We need to compute $\zeta ^{2m+ 2d-2g+1}$. This is the top Segre class of $\Cal E'$, and by the calculations above it is equal to the degree $g$ term in $e^{2\theta}$, namely $2^g$. \endproof \demo{Proof of \rom{(6.4)}} Keeping our previous notation, note that $\Pee \Cal E'$ is itself a Hilbert scheme, namely the scheme of all sections of $X$ of degree $2d+2m$, and the Hilbert scheme is the subscheme of all sections which are of the form $D_0 + \pi ^*D$. Thus, choosing a point $p\in X$ not lying in $\pi ^*D$, the incidence divisor $\mu (p)$ for the Hilbert scheme is the restriction of the corresponding incidence divisor on $\Pee \Cal E'$. Recall that $\Cal E ' = \pi _2{}_*(\Cal P \otimes \pi _1^*(V\otimes \scrO_C(D)))$. Let $t = \pi (p)$, and suppose that we have chosen the Poincar\'e line bundle $\Cal P$ so that $\Cal P|\{t\}\times \Pic ^dC$ is trivial. Fix the line $\ell \subset V_t$ corresponding to $p\in X$. Identifying the space $(V_t/\ell) \otimes _\Cee \scrO_C(D) _t$ with $\Cee$, there is then a surjection $$\Cal P \otimes \pi _1^*(V\otimes \scrO_C(D)) \to \Cal P \otimes \pi _1^*(V\otimes \scrO_C(D))|\{t\}\times \Pic ^dC \to \Cal P |\{t\}\times \Pic ^dC \cong \scrO_{\Pic ^dC}.$$ Applying $\pi _2{}_*$, we get a map $F\: \Cal E' \to \scrO_{\Pic ^dC}$, which is nonzero if $D$ is sufficiently ample. Clearly $F(s, \lambda) =0$ exactly when $s(t) \in \ell \subset V_t$. In other words, the zero set of $F$ is the incidence divisor $\Cal D$ corresponding to the point $p$. Now $$\align F \in &\Hom (\Cal E', \scrO_{\Pic ^dC}) = H^0(\Pic ^dC; (\Cal E')\spcheck) \\ &= H^0(\Pic ^dC; p_*\scrO_{\Pee \Cal E'}(1)) = H^0(\Pee \Cal E'; \scrO_{\Pee \Cal E'}(1)). \endalign$$ Running through the identifications above, we see that the zero set of $F$ is exactly the zero set of the induced section of $\scrO_{\Pee \Cal E'}(1)$, and a straightforward argument also checks the multiplicity. Thus $\Cal D$ is the zero set of a section of $\scrO_{\Pee \Cal E'}(1)$, and so $[\Cal D] = \zeta$. \endproof We can apply the methods above to handle enumerative questions not directly related to Seiberg-Witten theory. For example, there is the following calculation via the Grothendieck-Riemann-Roch theorem: \theorem{6.6} Suppose that $g$ is even. Then the set of stable rank two bundles $V$ with $\det V=0$ such that there exists a line bundle $\lambda$ of degree $(g-2)/2$ with $h^0(V\otimes \lambda ) \neq 0$ is an irreducible divisor in the moduli space $\frak M(C)$. Its class, at least as a divisor on the moduli functor, is $2^g\Delta$, where $\Delta$ is the first Chern class of the determinant line bundle, which again exists on the moduli functor. \endstatement Finally let us give the general formula for the Seiberg-Witten invariant: \theorem{6.7} Let $X = \Pee ^1 \times C$ be a product ruled surface and let $L$ be a line bundle on $X$ of type $(2a,2b)$, with $b\geq 1$ and $(1-g)/b \leq a <0$. Then for a K\"ahler metric $\omega$ such that $\omega \cdot L < 0$, the value of the Seiberg-Witten invariant on $L$ is $b^g$. \endstatement Note that this formula is a special case of a general transition formula for Seiberg-Witten invariants, which has been established by Li and Liu \cite{9} as well as the authors (unpublished). It would also follow, by copying the arguments above for the case of sections, but working with $\Sym ^mV$ (for $m= b-1$) instead of $V$, if we knew that, for a general ruled surface, the Hilbert scheme was always smooth of the expected dimension. We state this as a conjecture: \medskip \noindent {\bf Conjecture.} Let $X$ be a general ruled surface. Then every component of the Hilbert scheme is smooth of the expected dimension. Here, since $H^2(\scrO_X) = 0$, the expected dimension of the Hilbert scheme at a curve $D$ is $\dsize \frac12(D^2 - D\cdot K_X)$. \medskip Without assuming this conjecture, one can deduce the result from the methods of Section 3 in case the Hilbert scheme is smooth but does not have the expected dimension, and with more work in general, again by reducing it to a homological calculation in a space along the lines of $\Pee \Cal E'$. \Refs \ref \no 1\by E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris \book Geometry of Algebraic Curves volume I \publ Springer Verlag \publaddr New York Berlin Heidelberg Tokyo \yr 1985 \endref \ref \no 2 \by R. Brussee \paper Some $C^\infty$ properties of K\"ahler surfaces \paperinfo Algebraic geometry e-prints 9503004 \endref \ref \no 3\by R. Fintushel, P. Kronheimer, T. Mrowka, R. Stern, and C. Taubes \toappear \endref \ref \no 4\by R. Friedman and J W. Morgan\book Smooth Four-Manifolds and Complex Surfaces, {\rm Ergebnisse der Mathematik und ihrer Grenz\-gebiete 3. Folge} {\bf 27} \publ Springer \publaddr Berlin Heidelberg New York \yr 1994\endref \ref \no 5\bysame \paper Algebraic surfaces and Seiberg-Witten invariants \toappear \endref \ref \no 6 \by F. Ghione \paper Quelques r\'esultats de Corrado Segre sur les surfaces r\'egl\'ees \jour Math. Annalen \vol 255 \yr 1981 \pages 77--95 \endref \ref \no 7\by K. Kodaira and D.C. Spencer \paper A theorem of completeness of characteristic systems of complete continuous systems \jour Amer. J. Math \vol 81 \yr 1959 \pages 477--500 \endref \ref \no 8 \by H. Lange \paper H\"ohere Sekantenvariet\"aten und Vektorb\"undel auf Kurven \jour Manuscripta Math. \vol 52 \yr 1985 \pages 63--80 \endref \ref \no 9 \by T. J. Li and A. Liu \paper General wall crossing formula \paperinfo preprint \endref \ref \no 10\by D. Mumford \book Lectures on Curves on an Algebraic Surface \bookinfo Annals of Mathematics Studies \vol 59\publ Princeton University Press \publaddr Princeton, NJ \yr 1966 \endref \ref \no 11 \by C. Segre \paper Recherches g\'en\'erales sur les courbes et les surfaces r\'egl\'ees alg\'ebriques II \jour Math. Annalen \vol 34 \yr 1889 \pages 1--25 \endref \ref \no 12\by E. Witten \paper Monopoles and four-manifolds \jour Math. Research Letters \vol 1 \yr 1994 \pages 769--796 \endref \endRefs \enddocument

9404

alg-geom/9404014

en

https://arxiv.org/abs/alg-geom/9404014

alg-geom/9404014

Flad

Michel Dubois-Violette and Thierry Masson

Basic cohomology of associative algebras

16 pages, AMS-LaTeX, LPTHE-ORSAY 94/10

Journal of Pure and Applied Algebra n.114 p.39 (1996)

We define a new cohomology for associative algebras which we compute for algebras with units.

alg-geom

\section{Introduction: Definition of the basic cohomology of an associative algebra} Let $\cal A$ be an associative algebra over $\ \Bbb K\ =\Bbb R$ or $\Bbb C$ and let ${\cal A}_{Lie}$ be the underlying Lie algebra (with the commutator as Lie bracket). For each integer $n\in\Bbb N$, let $C^n({\cal A})$ be the vector space of $n$-linear forms on $\cal A$, i.e. $C^n({\cal A})=({\cal A}^{{\otimes}^{n}})^{\ast}$. For $\omega\in C^n({\cal A})$ and $\tau \in C^m({\cal A})$ one defines $\omega . \tau\in C^{n+m}({\cal A})$ by: $$\omega .\tau(A_1,\dots,A_{n+m})=\omega(A_1,\dots,A_n)\tau (A_{n+1},\dots,A_{n+m}),\ \forall A_i\in {\cal A}.$$ Equipped with this product, $C({\cal A})=\displaystyle{\mathop{\oplus}_n} C^n({\cal A})$ becomes an associative graded algebra with unit $(C^0({\cal A})=\ \Bbb K\ )$. One defines a differential $d$ on $C({\cal A})$ by setting for $\omega\in C^n({\cal A})$, $A_i\in {\cal A}$ $$d\omega(A_1,\dots,A_{n+1})=\sum^n_{k=1}(-1)^k\omega(A_1,\dots,A_{k-1},A_kA_{k+1},A_{k+2},\dots,A_{n+1}).$$ Indeed, $d$ is the extension as antiderivation of $C({\cal A})$ of minus the dual of the product of ${\cal A}$ and $d^2=0$ is then equivalent to the associativity of the product of ${\cal A}$. The graded differential algebra $C({\cal A})$ is together with a bimodule $\cal M$ the basic building blocks of the Hochschild complex giving the Hochschild cohomology with value in $\cal M$. Here we do not want to introduce bimodules like $\cal M$. However it is well known, see below, that the cohomology of $C({\cal A})$ is trivial whenever ${\cal A}$ has a unit. Nevertheless, there are two classical cohomologies which can be extracted from the differential algebra $C({\cal A})$, namely the Lie algebra cohomology of ${\cal A}_{Lie}$ and the cyclic cohomology of ${\cal A}$. In fact let ${\cal S}:C({\cal A})\rightarrow C({\cal A})$ and ${\cal C} :C({\cal A})\rightarrow C({\cal A})$ be defined by $$({\cal S}\omega)(A_1,\dots,A_n)=\sum _{\pi\in {\cal S}_{n}}\varepsilon(\pi)\omega (A_{\pi(1)},\dots,A_{\pi(n)})$$ and $$(\cal C\omega)(A_1,\dots,A_n) =\sum_{\gamma\in{\cal C}_{n}}\varepsilon (\gamma) \omega (A_{\gamma(1)},\dots,A_{\gamma(n)})$$ for $\omega\in C^n({\cal A})$, $A_k\in {\cal A}$ and where ${\cal S}_n$ is the group of permutations of $\{1,\dots,n\}$ and ${\cal C}_n$ is the subgroup of cyclic permutations. One has ${\cal S}\circ d=\delta \circ {\cal S}$ where $\delta$ is the Chevalley-Eilenberg differential so $(Im\ {\cal S},\delta)$ is a differential algebra whose cohomology is the Lie algebra cohomology $H({\cal A}_{Lie})$ of the Lie algebra ${{\cal A}}_{Lie}$ [3],[6],[5]. On the other hand, see Lemma 3 in [4] part II, one has ${\cal C}\circ d = b\circ {\cal C}$ where $b$ is the Hochschild differential of $C({{\cal A}},{\cal A}^\ast)$ so $(Im\ {\cal C},b)$ is a complex whose cohomology is the cyclic cohomology $H_\lambda({\cal A})$ of ${\cal A}$ up to a shift $-1$ in degree [4], (it is worth noticing, and this is not accidental, that the same shift occurs in the Loday-Quillen theorem [7]).\\ We want now to point out that there is another natural non-trivial cohomology which may be extracted from the differential algebra $C({\cal A})$. This cohomology is connected with the existence of a canonical operation, in the sense of H. Cartan [2], [5], of the Lie algebra ${\cal A}_{Lie}$ in the graded differential algebra $C({\cal A})$. For $A\in {\cal A}={\cal A}_{Lie}$, define $i_A:C^n({\cal A})\rightarrow C^{n-1}({\cal A})$ by $$i_A\omega(A_1,\dots ,A_{n-1})=\sum^{n-1}_{k=0}(-1)^k \omega (A_1,\dots ,A_k,A,A_{k+1},\dots,A_{n-1})$$ $\forall \omega\in C^n({\cal A})$, $\forall A_i\in {\cal A}$, for $n\geq 1$ and $i_AC^0({\cal A})=0$. For each $A\in {\cal A}$, $i_A$ is an antiderivation of degree $-1$ of $C({\cal A})$ and one has, with $L_A=i_Ad+di_A$, $i_Ai_B+i_Bi_A=0$, $[L_A,i_B]=i_{[A,B]}$, $[L_A,L_B]=L_{[A,B]}$ which are the relations which characterize an operation of ${\cal A}_{Lie}$ in $C({\cal A})$. Notice that then, for $A\in{\cal A}$, the derivation $L_A$ of degree 0 of $C({\cal A})$ is given by $$L_A\omega(A_1,\dots,A_n)=\sum^n_{k=1}\omega (A_1,\dots, [A_k,A],\dots,A_n)$$ for $\omega\in C^n({\cal A})$, $A_i\in {\cal A}$. An element $\omega\in C({\cal A})$ is called {\it horizontal} if $i_A\omega=0$ for any $A\in {\cal A}$, it is called {\it invariant} if $L_A\omega=0$ for any $A\in {\cal A}$ and it is called {\it basic} if it is horizontal and invariant, i.e. if $i_A\omega=0$ and $L_A\omega=0$ for any $A\in {\cal A}$. The set $C_H({\cal A})$ of horizontal elements of ${\cal A}$ is a graded subalgebra of $C({\cal A})$ which is stable by the $L_A$, $A\in {\cal A}$. The set $C_I({\cal A})$ of invariant elements of ${\cal A}$ and the set $C_B({\cal A})$ of basic elements of ${\cal A}$ are two graded differential subalgebras of $C({\cal A})$ $(C_B({\cal A})\subset C_I({\cal A}))$; their cohomologies $H_I({\cal A})$ and $H_B({\cal A})$ are called the invariant cohomology and the basic cohomology of ${\cal A}$. As already claimed, if ${\cal A}$ has a unit then the cohomology $H({\cal A})$ of $C({\cal A})$ is trivial and it turns out that the same is true for the invariant cohomology ; one has the following proposition. \newtheorem{proposition}{PROPOSITION} \begin{proposition} If ${\cal A}$ has a unit, then one has $H^n({\cal A})=0$, $H^n_I({\cal A})=0$ for $n\geq 1$ and $H^0({\cal A})=H^0_I({\cal A})=\ \Bbb K\ $. \end{proposition} \noindent {\bf Proof.} Let $\mbox{\rm 1\hspace {-.6em} l}$ be the unit of ${\cal A}$ and let us define for $n\geq 1$\\ $h:C^n({\cal A})\rightarrow C^{n-1}({\cal A})$ by $h\omega(A_1,\dots,A_{n-1})=-\omega(\mbox{\rm 1\hspace {-.6em} l},A_1,\dots,A_{n-1})$,\\ for $\omega\in C^n({\cal A})$ and $A_i\in {\cal A}$. One has $(dh+hd)\omega=\omega$ and $(L_Ah-hL_A)\omega=0$ for $\omega\in C^n({\cal A})$ and $A\in {\cal A}$. It follows that $h$ is a contracting homotopy for $C^+({\cal A})=\displaystyle{\mathop{\oplus}_{n\geq 1}}C^n({\cal A})$ and for $C^+_I({\cal A}) = \displaystyle{\mathop{\oplus}_{n\geq 1}}C^n_I({\cal A})$, which proves the result.$\square$ The basic cohomology $H_B({\cal A})$ is however non-trivial. In fact it is already non-trivial for ${\cal A}=\ \Bbb K\ $. \begin{proposition} The basic cohomology $H_B(\ \Bbb K\ )$ of $\ \Bbb K\ $ is the free graded commutative algebra with unit generated by an element of degree two;\linebreak[4] $H^{2k}_B(\ \Bbb K\ )=\ \Bbb K\ $, $H^{2k+1}_B(\ \Bbb K\ )=0$ and $H_B(\ \Bbb K\ )$ identifies to the algebra $\ \Bbb K\ [X^2]$ of polynomials in one indeterminate $X^2$ of degree two, $(X^2$ being identified to a non-vanishing element of $H^2_B(\ \Bbb K\ ))$. \end{proposition} \noindent {\bf Proof.} $C(\ \Bbb K\ )$ can be identified to $\ \Bbb K\ [X]$ and coincides with $C_I(\ \Bbb K\ )$ since $L_1=0$. One has $i_1=0$ on the elements of even degrees and $i_1\not= 0$ on the non-vanishing elements of odd degrees.\\ Therefore $C_B(\ \Bbb K\ )=\displaystyle{\mathop{\oplus}_k} C^{2k}(\ \Bbb K\ )=\ \Bbb K\ [X^2]=H_B(\Bbb K).$ $\square$ It is worth noticing here that one has $C^1_B({\cal A})=0$ and therefore $H^1_B({\cal A})=0$ for any associative $\ \Bbb K\ $-algebra ${\cal A}$.\\ In the next section we shall compute $H_B({\cal A})$ for an arbitrary associative $\ \Bbb K\ $-algebra ${\cal A}$ with unit. \section{Computation of the basic cohomology of unital algebras} In this section, ${\cal A}$ is an associative $\ \Bbb K\ $-algebra with a unit denoted by $\mbox{\rm 1\hspace {-.6em} l}$. Let $\cal I^n_S({\cal A}_{Lie})$ denote the space of ad*-invariant homogeneous polynomials of degree $n$ on the underlying Lie algebra ${\cal A}_{Lie}$ of ${\cal A}$. We shall prove the following theorem which generalizes the proposition 2 of \S 1. \begin{theorem} The basic cohomology $H_B({\cal A})$ of ${\cal A}$ identifies with the algebra $\cal I_S({\cal A}_{Lie})$ of invariant polynomials on the Lie algebra ${\cal A}_{Lie}$ where the degree $2n$ is given to the homogeneous polynomials of degree $n$, i.e. $H^{2n}_B({\cal A})\simeq \cal I^n_S({\cal A}_{Lie})$ and $H^{2n+1}_B({\cal A})=0$. In particular, $H_B({\cal A})$ is commutative and graded commutative. \end{theorem} In order to prove this theorem, we shall need some constructions used in equivariant cohomology [1]. Let ${\cal P}^{m,n}$ denote the space of homogeneous polynomial mappings of degree $m$ of ${\cal A}$ in $C^n({\cal A})$. The direct sum $\displaystyle{{\cal P} = \mathop{\oplus}_{m,n}} {\cal P}^{m,n}$ is an associative bigraded algebra in a natural way. One defines the total degree of an element of ${\cal P}^{m,n}$ to be $2m+n$ ; ${\cal P}$ is a graded algebra for the total degree and $C({\cal A})=\displaystyle{\mathop{\oplus}_n}{\cal P}^{0,n}$ is a graded subalgebra of ${\cal P}$. The composition with the differential $d$ of $C({\cal A})$ is a differential, again denoted by $d$, of the graded algebra ${\cal P}$ which extends the differential $d$ of $C({\cal A})$ . One has $d{\cal P}^{m,n}\subset {\cal P}^{m,n+1}$. By using the operation $A\mapsto i_A$, one can define another differential, $\delta$, on ${\cal P}$. Namely if $\omega \in {\cal P}$ is the polynomial mapping $A\mapsto \omega_A$ of ${\cal A}$ in $C({\cal A})$, then $\delta\omega$ is the polynomial mapping $A\mapsto (\delta\omega)_A=i_A\omega_A$ of ${\cal A}$ in $C({\cal A})$. One has $\delta {\cal P}^{m,n}\subset {\cal P}^{m+1,n-1}$ so $\delta$ is of total degree $2-1=1$ and the fact that $\delta$ is an antiderivation satisfying $\delta^2=0$ follows from the fact that, for any $A\in{\cal A}$, $i_A$ is an antiderivation of $C({\cal A})$ satisfying $i^2_A=0$. Notice that $C^n_H({\cal A})$ is the kernel of $\delta \restriction C^n({\cal A})={\cal P}^{0,n}$ $(:{\cal P}^{0,n}\rightarrow {\cal P}^{1,n-1})$.\\ {\it As a vector space}, ${\cal P}^{m,n}$ can be identified to the subspace of elements of $C^{m+n}({\cal A})$ which are symmetric in their $m$ first arguments: For $\omega\in {\cal P}^{m,n}$, $A\mapsto \omega_A$, there is a unique $\xi_\omega \in C^{m+n}({\cal A})$ symmetric in the $m$ first arguments such that $$\omega_A(A_1,\dots , A_n)=\xi_\omega (\underbrace{A,\dots, A}_m, A_1,\dots, A_n),\> \forall A, A_i\in {\cal A}.$$ Let $\cal I^{m,n}$ denote the subspace of ${\cal P}^{m,n}$ consisting of the $\omega\in {\cal P}^{m,n}$ such that $\xi_\omega \in C^{m+n}_I({\cal A})$, (i.e. such that $\xi_\omega$ is invariant). $\cal I = \oplus {\cal I}^{m,n}$ is a graded subalgebra (also a bigraded subalgebra in the obvious sense) of ${\cal P}$ which is stable by $d$ and $\delta$ and, furthermore, $d$ and $\delta$ anticommute on $\cal I$.\\ Notice that one has $\cal I^{m,0}=\cal I^m_S({\cal A}_{Lie})$ and $\cal I^{0,n}=C^n_I({\cal A})$ and that $C^n_B({\cal A})$ is the kernel of $\delta\restriction C^n_I({\cal A})=\cal I^{0,n}$ $(:\cal I^{0,n}\rightarrow \cal I^{1,n-1})$. The algebras ${\cal P}$ and $\cal I$ are bigraded and $d$ and $\delta$ are bihomogeneous, therefore the $d$ and the $\delta$ cohomologies of ${\cal P}$ and $\cal I$ are also bigraded algebras. By using composition with the homotopy $h$ of the proof of proposition 1 and by noticing that $\cal I$ is stable by this composition, one obtains the following generalization of proposition 1. \begin{proposition} One has $H^{m,n}({\cal P},d)=0$, $H^{m,n}(\cal I,d)=0$ for $n\geq 1$ and $H^{m,0}({\cal P},d)={\cal P}^{m,0}$, $H^{m,0}(\cal I,d)=\cal I^{m,0}=\cal I^m_S({\cal A}_{Lie}$). \end{proposition} \noindent Concerning the cohomology of $\delta$ one has the following result \begin{proposition} One has $H^{m,n}({\cal P},\delta)=0$, $H^{m,n}({\cal I},\delta)=0$ for $m\geq 1$ and $H^{0,n}({\cal P},\delta)=C^n_H({\cal A})$, $H^{0,n}({\cal I},\delta)=C^n_B({\cal A})$. \end{proposition} \noindent {\bf Proof.} The last part of the proposition $(m=0)$ is obvious since one has $H^{0,n}({\cal P},\delta)={\rm ker}(\delta\restriction C^n({\cal A}))$ and $H^{0,n}({\cal I},\delta)={\rm ker} (\delta\restriction C^n_I({\cal A}))$. Therefore from now on, assume that one has $m\geq 1$. Define a linear mapping $\ell$ of ${\cal P}$ in itself with $\ell({\cal P}^{m,n})\subset {\cal P}^{m-1,n+1}$ by $$(\ell\omega)_A (A_1,\dots,A_{n+1}) =\frac{d}{dt}\omega_{A+tA_1}(A_2,\dots,A_{n+1})\vert_{t=0}$$ for $ \omega\in {\cal P}^{m,n}$. One has $(\delta\ell + \ell \delta )\omega=m\omega + {\cal H} \omega$ where ${\cal H} \omega$ is given by $$({\cal H}\omega)_A (A_1,\dots,A_n)= \sum^{n+1}_{p=2} (-1)^p \omega_A (A_2,\dots,A_{p-1}, A_1, A_p,\dots,A_n),$$ $(\omega\in {\cal P}^{m,n})$. Notice that if $\omega$ is such that $\omega_A(A_1,\dots,A_n)$ is antisymmetric in $A_1,\dots, A_n$, then ${\cal H}\omega=n\omega$ and therefore $\ell$ gives an homotopy for such $\omega$. The following lemma, which is a combinatorial statement in the algebra of the permutation group, will lead to an homotopy for the general case. The proof of this lemma (which is probably known) will be given in appendix. \begin{lemma} One has on ${\cal P}^{m,n}, \prod^{n-2}_{p=0} ({\cal H} - p\ id)=\prod^{n-1}_{p=0}({\cal H}-p\ id) ={\cal S}$, where ${\cal S}\omega$ is given as before (antisymmetrisation) by $({\cal S} \omega)_A(A_1,\dots,A_n)=\sum_{\pi\in{\cal S}_n} \varepsilon(\pi)\omega_A (A_{\pi(1)},\dots,A_{\pi(n)})$, i.e. $({\cal S}\omega)_A={\cal S}\omega_A$. \end{lemma} Let $\omega\in{\cal P}^{m,n}$ with $m\geq 1$ be such that $\delta\omega=0$. Then $\delta\ell\omega=m\omega+{\cal H}\omega$, so one also have $\delta{\cal H}\omega=0$ and, by induction, $\delta{\cal H}^p\omega=0$ for any integer $p$; i.e. one has $\delta P({\cal H})\omega=0$ for any polynomial $P$. Define $\omega_r\in {\cal P}^{m,n}$, for $r=1,2,\dots,n$, by $\omega_1=\omega,\ \omega_2={\cal H}\omega -(n-2)\omega,\dots,\ \omega_r =\prod^r_{p=2} ({\cal H}-(n-p) id)\omega,\dots,\hfill\\ \omega_n={\cal H}({\cal H} - id)\dots ({\cal H}- (n-2) id)\omega$. One has $\delta\ell\omega_r=m\omega_r+{\cal H}\omega_r=\\ (m+n-r-1)\omega_r+\omega_{r+1}$, i.e. $$\omega_r=\delta\ell\left(\frac{\omega_r}{m+n-(r+1)}\right) - \frac{\omega_{r+1}}{m+n-(r+1)}\raisebox{0.75mm}{,}$$ for $r\leq n-1$. This implies that $$\omega=\delta\ell\left(\sum^{n-1}_{r=1} \frac{(-1)^{r+1}}{\prod^{r+1}_{p=2}(m+n-p)} \omega_r\right) - \frac{(-1)^n}{\prod^n_{p=2}(m+n-p)} \omega_n.$$ On the other hand, it follows from the lemma and the previous discussion (antisymmetry) that $\omega_n=\delta\ell(\frac{1}{m+n}\omega_n)$ and therefore one has an homotopy formula, for $\omega\in{\cal P}^{m,n}$ with $m\geq 1$ satisfying $\delta\omega=0$, of the form $\omega=\delta\delta'\omega$ where $\delta'=\ell\circ Q^{m,n}({\cal H})$ and where the polynomial $Q^{m,n}$ is easily computed from the previous formulae. Since $\ell$ and ${\cal H}$ preserve ${\cal I}$ this achieves the proof of proposition 4.$\square$\\ The proof of the theorem 1 will now follow from $H^{m,n}({\cal I},d)=0$ for $n\geq 1$, $H^{m,n}({\cal I},\delta)=0$ for $m\geq 1$, $$H^{m,0}({\cal I},d)={\cal I}^m_S({\cal A}_{Lie})\ \mbox{and}\ H^{0,n}({\cal I},\delta)=C^n_B({\cal A})$$ by a standard spectral sequence argument in the bicomplex $({\cal I},d,\delta)$.\\ Let $H(\delta\vert d)$ denote the $\delta$-cohomology modulo $d$ of ${\cal I}$, i.e. $$H^{m,n}(\delta\vert d)=Z^{m,n}(\delta\vert d)/B^{m,n}(\delta\vert d)$$ where $Z^{m,n}(\delta\vert d)$ is the space of the $\alpha^{m,n}\in {\cal I}^{m,n}$ for which there is an $\alpha^{m+1,n-2}\in {\cal I}^{m+1,n-2}$ such that $\delta\alpha^{m,n}+d\alpha^{m+1,n-2}=0$ and where $B^{m,n}(\delta\vert d)= \delta {\cal I}^{m-1,n+1}+d{\cal I}^{m,n-1}\ (\subset {\cal I}^{m,n})$. With these notations, one has the following result. \begin{proposition} One has the following isomorphisms:\\ $H^{2p}_B({\cal A})\simeq H^{k,2(p-k)-1}(\delta\vert d) \simeq {\cal I}^p_S({\cal A}_{Lie})$ for $1\leq k \leq p-2$,\\ $H^{2p+1}_B({\cal A})\simeq H^{k,2(p-k)}(\delta\vert d)\simeq 0$ for $1\leq k\leq p-1$, $H^4_B({\cal A})\simeq {\cal I}^2_S({\cal A}_{Lie})$, $H^3_B({\cal A})\simeq 0$ and $H^2_B({\cal A})\simeq {\cal I}^1_S({\cal A}_{Lie})$. \end{proposition} \noindent {\bf Proof.} Let $\alpha^{m,n}\in {\cal I}^{m,n}$ be a $\delta$-cocycle modulo $d$, i.e. there is a $\alpha^{m+1,n-2}\in {\cal I}^{m+1,n-2}$ such that $\delta\alpha^{m,n}+d\alpha^{m+1,n-2}=0$. By applying $\delta$, one obtains $\delta d\alpha^{m+1,n-2}=-d\delta\alpha^{m+1,n-2}=0$, therefore, if $n\geq 4$, there is in view of proposition 3 a $\alpha^{m+2,n-4}\in {\cal I}^{m+2,n-4}$ such that $\delta\alpha^{m+1,n-2}+d\alpha^{m+2,n-4}=0$, which means that $\alpha^{m+1,n-2}$ is also a $\delta$-cocycle modulo $d$. If $\alpha^{m,n}$ is exact, i.e. if there are $\beta^{m-1,n+1}\in {\cal I}^{m-1,n+1}$ and $\beta^{m,n-1}\in {\cal I}^{m,n-1}$ such that $\alpha^{m,n}=\delta\beta^{m-1,n+1}+d\beta^{m,n-1}$, then $d(\alpha^{m+1,n-2}-\delta\beta^{m,n-1})=0$ which implies, again by proposition 3 (since $n-2\geq 2>0$), that there is a $\beta^{m+1,n-3}$ such that $\alpha^{m+1,n-2}=\delta\beta^{m,n-1}+d\beta^{m+1,n-3}$ i.e. $\alpha^{m+1,n-2}$ is also exact. Therefore there is a well defined linear mapping $\partial : H^{m,n}(\delta\vert d)\rightarrow H^{m+1,n-2}(\delta\vert d)$ for $n\geq 4$ such that $\partial [\alpha^{m,n}]=[\alpha^{m+1,n-2}]$. Let now $\alpha^{m+1,n-2}\in {\cal I}^{m+1,n-2}$ be a $\delta$-cocycle modulo $d$, i.e. there is $\alpha^{m+2,n-4}\in {\cal I}^{m+2,n-4}$ such that $\delta\alpha^{m+1,n-2}+d\alpha^{m+2,n-4}=0$. By applying $d$, one obtains $\delta d\alpha^{m+1,n-2}=0$ which implies, in view of proposition 4, that there is a $\alpha^{m,n}\in{\cal I}^{m,n}$ such that $\delta\alpha^{m,n}+d\alpha^{m+1,n-2}=0$. This means that $\partial$ is surjective. Assume that $[\alpha^{m+1,n-2}]=0$ i.e. $\alpha^{m+1,n-2}=\delta\beta^{m,n-1}+d\beta^{m+1,n-3}$ $(\beta\in{\cal I})$ then one has $\delta(\alpha^{m,n}-d\beta^{m,n-1})=0$ which implies that $[\alpha^{m,n}]=0$ if $m\geq 1$ or that $\alpha^{0,n}- d\beta^{0,n-1}\in C^n_B({\cal A})$ if $m=0$, again by proposition 4. {\it Thus} $\partial :H^{m,n}(\delta\vert d)\rightarrow H^{m+1,n-2}(\delta\vert d)$ {\it are isomorphisms for} $n\geq 4$ {\it and} $m\geq 1$ {\it and, for} $m=0$ ($n\geq 4$), $ \partial : H^{0,n}(\delta\vert d) \rightarrow H^{1,n-2}(\delta\vert d)$ {\it is surjective and its kernel is the image of} $C^n_B({\cal A})=H^{0,n}({\cal I},\delta)$ {\it in} $H^{0,n}(\delta\vert d)$.\\ On the other hand, if $\alpha^{0,n}\in {\cal I}^{0,n}$ is a $\delta$-cocycle modulo $d$, i.e. $\delta\alpha^{0,n}+d\alpha^{1,n-2}=0$ then $d\alpha^{0,n}\in C^{n+1}_I({\cal A})$ is a basic cocycle of ${\cal A}$ i.e. $d\alpha^{0,n}\in Z^{n+1}_B({\cal A})$ and if $\alpha^{0,n}$ is exact, i.e. $\alpha^{0,n}=d\beta^{0,n}$ with $\beta^{0,n}\in {\cal I}^{0,n}$, then $d\alpha^{0,n}=0$. Therefore, with obvious notations, one has a linear mapping $d^\sharp:H^{0,n}(\delta\vert d)\rightarrow H^{n+1}_B({\cal A})$, $d^\sharp[\alpha^{0,n}]=[d\alpha^{0,n}]$. If $z^{n+1}\in C^{n+1}_B({\cal A})$ is closed i.e. $z^{n+1}\in Z^{n+1}_B({\cal A})$ then, in view of proposition 1, there is a $\alpha^{0,n}\in C^n_I({\cal A})={\cal I}^{0,n}$ such that $z^{n+1}=d\alpha^{0,n}$; one has $d\delta\alpha^{0,n}=0$, which implies that $\alpha^{0,n}$ is a $\delta$-cocycle modulo $d$ if $n\geq 2$ (by proposition 3). Thus $d^\sharp$ is surjective for $n\geq 2$ and one obviously has $ker(d^\sharp)$ = image of $C^n_B({\cal A})$ in $H^{0,n}(\delta\vert d)$. Applying this for $n\geq 4$ and the previous results, one obtains isomorphisms: $$H^{2p}_B({\cal A})\simeq H^{k,2(p-k)-1}(\delta\vert d)\ \mbox{for}\ 1\leq k\leq p-2$$ and $$H^{2p+1}_B({\cal A})\simeq H^{k,2(p-k)}(\delta\vert d)\ \mbox{for}\ 1\leq k\leq p-1.$$ Thus, to achieve the proof, it remains to show that one has: \begin{description} \item[(i)] $H^{m,2}(\delta\vert d)=0$ for $m\geq 1$ and $H^{0,2}(\delta\vert d)$ = image of $C^2_B({\cal A})$ \item[(ii)] $H^{m,3}(\delta\vert d)\simeq {\cal I}^{m+2}_S({\cal A}_{Lie})$ for $m\geq 1$ and $H^{0,3}(\delta\vert d)$/image of $C^3_B({\cal A})\simeq {\cal I}^2_S({\cal A}_{Lie})$ \item[(iii)] $H^2_B({\cal A})\simeq {\cal I}^1_S({\cal A}_{Lie})$, (remembering that $C^1_B({\cal A})=0$). \end{description} Let $\alpha^{m,2}\in {\cal I}^{m,2}$ be a $\delta$-cocyle modulo $d$ then, (since $d\alpha^{m+1,0}\equiv 0$), $\alpha^{m,2}$ is a $\delta$-cocycle, i.e. $\delta\alpha^{m,2}=0$, which implies, by proposition 4, that $\alpha^{m,2}\in\delta{\cal I}^{m-1,1}$ for $m\geq 1$ and, for $m=0$, $\alpha^{0,2}\in C^2_B({\cal A})=H^{0,2}({\cal I},\delta)$. This proves (i).\\ Let $\alpha^{m,3}\in{\cal I}^{m,3}$ be a $\delta$-cocycle modulo $d$, i.e. there is a $\alpha^{m+1,1}\in {\cal I}^{m+1,1}$ such that $\delta\alpha^{m,3}+d\alpha^{m+1,1}=0$. Then one has $\delta\alpha^{m+1,1}=P^{m+2}\in {\cal I}^{m+2}_S({\cal A}_{Lie})={\cal I}^{m+2,0}$. If $\alpha^{m,3}=\delta\beta^{m-1,4}+d\beta^{m,2}$ for $\beta^{m-1,4}\in {\cal I}^{m-1,4}$ and $\beta^{m,2}\in {\cal I}^{m,2}$, (i.e. if $\alpha^{m,3}$ is exact), one has $d(\alpha^{m+1,1}-\delta\beta^{m,2})=0$ which implies, by proposition 3 and by $d{\cal I}^{m+1,0}=0$, that $\alpha^{m+1,1}=\delta\beta^{m,2}$ and therefore $\delta\alpha^{m+1,1}=P^{m+2}=0$. Thus there is a well defined linear mapping $j:H^{m,3}(\delta\vert d)\rightarrow {\cal I}^{m+2}_S({\cal A}_{Lie})$, ($j([\alpha^{m,3}])=P^{m+2}$). Let $P^{m+2}$ be an arbitrary element of ${\cal I}^{m+2}_S({\cal A}_{Lie})$; then, by proposition 4, there is a $\alpha^{m+1,1}\in{\cal I}^{m+1,1}$ such that $\delta\alpha^{m+1,1}=P^{m+2}$ and, since $dP^{m+2}=0$, one has $\delta d\alpha^{m+1,1}=0$ which implies again by proposition 4 that there is a $\alpha^{m,3}$ such that $\delta\alpha^{m,3}+d\alpha^{m+1,1}=0$. This shows that $j$ is surjective. If $\delta\alpha^{m+1,1}=0$, then, by proposition 4, $\alpha^{m+1,1}=\delta\beta^{m,2}$ and therefore $\delta(\alpha^{m,3}-d\beta^{m,2})=0$ which implies again by proposition 4 that $\alpha^{m,3}=\delta\beta^{m-1,4}+d\beta^{m,2}$ if $m\geq 1$ and, for $m=0$, $\alpha^{0,3}-d\beta^{0,2}\in C^3_B({\cal A})$. This proves (ii).\\ Finally let $z^2\in C^2_I({\cal A})$ be a basic cocycle, i.e. $dz^2=0$ and $\delta z^2=0$, then $z^2=d\alpha^1$ for a unique $\alpha^1\in C^1_I({\cal A})$ (since $dC^0({\cal A})=0$ and by proposition 1). Conversely, if $\alpha^1\in C^1_I({\cal A})$ then $d\alpha^1$ is basic; therefore $H^2_B({\cal A})\simeq C^1_I({\cal A})$ since $C^1_B({\cal A})=0$. But one has canonically $C^1_I({\cal A})={\cal I}^1_S({\cal A}_{Lie}).\square$\\ This proves of course Theorem 1, but it is worth noticing that in the above proof there is also a computation of the $\delta$-cohomology modulo $d$ of ${\cal I}$. \section{Sketch of another approach: Connection with the Lie algebra cohomology} There is another way to study the basic cohomology of ${\cal A}$ which connects it with the Lie algebra cohomology of ${\cal A}_{Lie}$: It is to study the spectral sequence corresponding to the filtration of the differential algebra $C({\cal A})$ associated to the operation $i$ of the Lie algebra ${\cal A}_{Lie}$ in the differential algebra $C({\cal A})$, [5]. This filtration ${\cal F}$ is defined by $${\cal F}^p(C^n({\cal A}))=\{\omega \in C^n({\cal A})\vert i_{A_{1}}\dots i_{A_{n-p+1}} (\omega)=0,\ \forall A_i\in {\cal A}\}$$ for $0\leq p \leq n$ and ${\cal F}^p(C({\cal A}))=\displaystyle{\mathop{\oplus}_{n\geq p}} {\cal F}^p(C^n({\cal A}))$.\\ One has $${\cal F}^0(C({\cal A}))=C({\cal A}),\ {\cal F}^p(C({\cal A})) \cdot {\cal F}^q(C({\cal A})) \subset {\cal F}^{p+q}(C({\cal A}))$$ and $$d{\cal F}^p(C({\cal A})) \subset {\cal F}^p (C({\cal A}))$$ i.e. ${\cal F}$ is a (decreasing) filtration of graded differential algebra. To such a filtration corresponds a convergent spectral sequence $(E_r,d_r)_{r\in \Bbb N}$, where $E_r=\displaystyle{\mathop{\oplus}_{p,q\in \Bbb N}} E^{p,q}_r$ is a bigraded algebra and $d_r$ is a homogeneous differential on $E_r$ of bidegree $(r,1-r)$. The triviality of the cohomology of $C({\cal A})$, (i.e. proposition 1), implies that $E^{p,q}_\infty=0$ for $(p,q)\not=(0,0)$ and $E^{0,0}_\infty= \ \Bbb K\ $. The spectral sequence starts with the graded space $E_0$ associated to the filtration i.e. $E^{p,q}_0={\cal F}^p(C^{p+q}({\cal A}))/{\cal F}^{p+1}(C^{p+q}({\cal A}))$ and $d_0$ is induced by the differential $d$ of $C({\cal A})$. If $\omega\in{\cal F}^p(C^{p+q}({\cal A}))$ then $i_{A_{1}}\dots i_{A_{q}}\omega$ is in $C^p_H({\cal A})$ and is antisymmetric in $A_1,\dots,A_q$. Therefore $(A_1,\dots,A_q)\mapsto i_{A_{1}}\dots i_{A_{q}}\omega$ is a $q$-cochain of the Lie algebra ${\cal A}_{Lie}$ with values in $C^p_H({\cal A})$ for the representation $A\mapsto L_A$ of the Lie algebra ${\cal A}_{Lie}$ in $C^p_H({\cal A})$. This defines a linear map of ${\cal F}^p(C^{p+q}({\cal A}))$ in the space of $q$-cochains of ${\cal A}_{Lie}$ with values in $C^p_H({\cal A})$. The kernel of this map is, by definition, ${\cal F}^{p+1}(C^{p+q}({\cal A}))$. In our case, it is straightforward to show that this map is surjective, i.e. that $E^{p,q}_0$ identifies with the space of $q$-cochains of the Lie algebra ${\cal A}_{Lie}$ with values in the space $C^p_H({\cal A})$ of horizontal elements of $C^p({\cal A})$ and that then, $d_0$ coincides with the Chevalley-Eilenberg differential. Thus $E_1=H(E_0,d_0)$ is the Lie algebra cohomology of ${\cal A}_{Lie}$ with value in $C_H({\cal A})$, $E^{p,q}_1=H^q({\cal A}_{Lie}, C^p_H({\cal A}))$. In particular $E^{0,\ast}_1$ is the ordinary cohomology of ${\cal A}_{Lie}$ (i.e. with value in the trivial representation in $\ \Bbb K\ $) and $E^{\ast,0}_1$ is the space of invariant elements of $C_H({\cal A})$, i.e. the space $C_B({\cal A})$ of basic elements of $C({\cal A})$, $E^{n,0}_1=C^n_B({\cal A})$. Furthermore, on $E^{\ast,0}_1=C_B({\cal A})$, $d_1$ is just the differential $d$ of $C({\cal A})$ restricted to $C_B({\cal A})$. Therefore $E^{\ast,0}_2$ is the basic cohomology $H_B({\cal A})$ of ${\cal A}$, $E^{n,0}_2=H^n_B({\cal A})$. This shows that the spectral sequence connects the basic cohomology of ${\cal A}$ to the Lie algebra cohomology of its underlying Lie algebra ${\cal A}_{Lie}$. The connection between the Lie algebra cohomology of ${\cal A}_{Lie}$ and the ad$^\ast$-invariant polynomials, i.e. $H_B({\cal A})$ in our case, is well known but an interest of the last approach could be to catch the primitive parts. \section*{Appendix: Proof of Lemma 1} Let ${\cal S}_n$ be the group of permutations of $\{1,\dots,n\}$.\\ In the algebra of this group, let us define the antisymmetrisation operator $${\cal S} =\sum_{\pi\in{\cal S}_n}\varepsilon(\pi)\pi$$ and the operators $${\cal H}_{(k)}=\sum_{{\pi\in{\cal S}_n}\atop {\pi^{-1}(k+1)<\dots<\pi^{-1}(n)}}\varepsilon(\pi)\pi$$ for any $1\leq k\leq n$, where $\varepsilon(\pi)$ denotes the signature of the permutation $\pi$.\\ Notice that $${\cal H}_{(n)}={\cal H}_{(n-1)}={\cal S}$$ and one easily shows that $${\cal H}\equiv {\cal H}_{(1)} = \sum^n_{p=1}(-1)^{p+1}\gamma_p$$ where $\gamma_p$ is the permutation $(1,\dots,p,\dots,n)\mapsto (2,\dots,p,1,p+1,\dots,n)$.\\ With these definitions, one has the following result:\\ \noindent{\bf LEMMA} {\it For any} $1\leq k\leq n-1$, $${\cal H}\calh_{(k)} = k {\cal H}_{(k)}+ {\cal H}_{(k+1)}$$ {\bf Proof.}\\ \begin{eqnarray*} {\cal H}\calh_{(k)} & = &\left(\sum^n_{p=1}(-1)^{p+1}\gamma_p\right) \left( \sum_{{\pi\in{\cal S}_n}\atop{\pi^{-1}(k+1)<\dots<\pi^{-1}(n)}} \varepsilon(\pi)\pi\right) \\ & = & \sum^k_{p=1} \sum_{{\pi\in{\cal S}_n}\atop{\pi^{-1}(k+1)<\dots<\pi^{-1}(n)}}(-1)^{p+1}\varepsilon(\pi)\gamma_p\pi\\ & & \mbox{}+ \sum^n_{p=k+1} \sum_{{\pi\in{\cal S}_n}\atop{\pi^{-1}(k+1)<\dots <\pi^{-1}(n)}} (-1)^{p+1}\varepsilon(\pi)\gamma_p\pi \end{eqnarray*} Now, define $\pi'=\gamma_p\pi\in {\cal S}_n$; one has $\varepsilon(\pi')=(-1)^{p+1}\varepsilon(\pi)$.\\ For $p\leq k$, one has $\pi'^{-1}(q)=\pi^{-1}(q)$ for any $k+1\leq q\leq n$.\\ So, in the first summation, for a fixed $p$, the sum over the $\pi\in{\cal S}_n$ such that $\pi^{-1}(k+1)<\dots<\pi^{-1}(n)$ can be replaced by the sum over the $\pi'\in{\cal S}_n$ such that $\pi'^{-1}(k+1)<\dots<\pi'^{-1}(n)$. Thus $$\sum^k_{p=1} \sum_{{\pi\in{\cal S}_n}\atop{\pi^{-1}(k+1)<\dots<\pi^{-1}(n)}} (-1)^{p+1}\varepsilon(\pi)\gamma_p\pi=\sum^k_{p=1} \sum_{{\pi'\in{\cal S}_n}\atop{\pi'^{-1}(k+1)<\dots<\pi'^{-1}(n)}} \varepsilon(\pi')\pi'=k{\cal H}_{(k)}$$ Now, for $p\geq k+1$, one has $\pi'^{-1}(q)=\pi^{-1}(q-1)$ for any $k+2\leq q \leq p$ and $\pi'^{-1}(q)=\pi^{-1}(q)$ for any $p+1\leq q \leq n$. So one has only $$\pi'^{-1}(k+2)<\dots<\pi'^{-1}(n),$$ and in the second summation the sum over $p$ and $\pi$ can be replaced by the sum over the $\pi'\in {\cal S}_n$ such that $\pi'^{-1}(k+2)<\dots<\pi'^{-1}(n).$ Thus $$\sum^n_{p=k+1}\sum_{{\pi\in{\cal S}_n}\atop{\pi^{-1}(k+1)<\dots<\pi^{-1}(n)}} (-1)^{p+1}\varepsilon(\pi)\gamma_p\pi = {\cal H}_{(k+1)}.\ \square$$ By induction, this lemma shows that for any $1\leq k\leq n$ $${\cal H}_{(k)} = \prod^{k-1}_{p=0} ({\cal H} - p\ id)$$ where we recall ${\cal H}\equiv {\cal H}_{(1)}$.\\ So for $k=n$ and $k=n-1$, one has $${\cal H}_{(n)} = \prod^{n-1}_{p=0}({\cal H}- p\ id) = {\cal S}$$ $${\cal H}_{(n-1)} = \prod^{n-2}_{p=0}({\cal H}- p\ id) = {\cal S}$$ Now, notice that the operators ${\cal H}$ and ${\cal S}$ of Lemma 1 are representations of the operators ${\cal H}$ and ${\cal S}$ above in the linear space ${\cal P}^{m,n}$ (in fact only in $C^n({\cal A}))$. This proves Lemma 1.

9404

alg-geom/9404006

en

https://arxiv.org/abs/alg-geom/9404006

alg-geom/9404006

Jeroen Spandaw

K. Hulek and J. Spandaw

Degenerations of abelian surfaces and Hodge structures

35 pages, AMS-LaTeX 1.1 (amstex,amssymb,amscd,theorem), (revised version: only minor TeX-changes)

Schriftenreihe Forschungsschwerpunkt komplexe Mannigfaltigkeiten Heft Nr. 191

In this paper the authors consider a certain toroidal compactification of the moduli space of degenerations of (1,p)-polarized abelian surfaces with (canonical) level structure. Using Hodge theory we give a proof that a degenerate abelian surface associated to a corank 1 boundary point is (almost) completely determined by the boundary point. The crucial ingredient in the proof is the Local Invariant Cycle theorem which relates the variation of Hodge structure to the mixed Hodge structure on the singular surface.

alg-geom

\section{Introduction} The starting point of this note is twofold: In \cite{HKW} the first author together with Kahn and Weintraub constructed and described a torodial compactification of the moduli space of $(1,p)$-polarized abelian surfaces with a (canonical) level structure. Moreover, Mumford's construction was used to associate to each boundary point a degenerate abelian surface. On the other hand Carlson, Cattani and Kaplan gave in \cite{CCK} an interpretation of torodial compactifications of moduli spaces of abelian varieties in terms of mixed Hodge structures. Here we want to discuss a connection between \cite{HKW} and \cite{CCK}. More precisely, we restrict ourselves to corank 1 degenerations in the sense of \cite{HKW}, or equivalently to type II degenerations, i.e. to cycles of elliptic ruled surfaces. Our main result says that a degenerate abelian surface associated to a boundary point is (almost) completely determined by the boundary point (for a precise formulation see theorem \ref{T}). The crucial ingredient in the proof is the Local Invariant Cycle theorem which relates the variation of Hodge structure (VHS) to the mixed Hodge structure (MHS) on the singular surface. In section \ref{section2} we collect some basic facts about semi-stable degenerations of abelian surfaces, some of which are well known. The MHS of a cycle of elliptic ruled surfaces is computed in section \ref{section3}. Section \ref{section4} starts with a review of corank 1 boundary points in moduli spaces of $(1,p)$-polarized abelian surfaces with a level structure. Next the VHS of a specific family associated to such a boundary point is described. Using the Local Invariant Cycle theorem, this and the result from section \ref{section3} give theorem \ref{T}. We would like to thank J. Steenbrink for helpful discussions. Both authors are grateful to the DFG for financial support under grant HU 337/2-4. \section{Preliminaries on degenerations}\label{section2} In this section we want to summarize basic facts on degenerations of abelian surfaces some of which are well known (\cite{FM,P}). \subsection{Cycles of elliptic surfaces} Let $\Delta=\{z\in{\Bbb C}: |z|<1\}$ be the unit disk. We consider proper, flat families $$ \begin{CD} {X}\\ @VV{p}V\\ \Delta \end{CD} $$ with $X_t$ smooth abelian for $t\neq0$. We shall always assume the total space ${X}$ to be smooth and K\"ahler, and the components of the singular fibre $X_0$ to be algebraic. We also assume $X$ to be relatively minimal. If $X_0$ has global normal crossings and no triple points then Persson \cite[proposition~3.3.1]{P} or \cite[p.~11 and 17]{FM} has shown, that $X_0$ is smooth abelian or a cycle of elliptic ruled surfaces, i.e. $$ X_0=Y_1\cup\ldots\cup Y_N. $$ The $Y_i$ are smooth ruled elliptic surfaces. $Y_i$ and $Y_{i+1}$ intersect transversally along a smooth curve which is a section of both $Y_i$ and $Y_{i+1}$. In particular, the $Y_i$ are all ruled surfaces over the same base curve $C$ with two disjoint sections. The situation can be envisaged as in figure~1. \begin{figure} \newsavebox{\mybox} \setlength{\unitlength}{3pt} \begin{center} \begin{picture}(38,76)(10,5) \savebox{\mybox}(22,14){% \begin{picture}(22,14) \put(-2,12){\line(1,0){22}} \put(-2,0){\line(1,0){22}} \multiput(0,-1)(3,0){7}{\line(0,1){14}} \end{picture}} \put(12,62){\usebox{\mybox}} \put(18,74){\circle*{1.5}} \put(18,62){\circle*{1.5}} \put(18,56){\vector(0,1){4}} \put(18,56){\vector(0,-1){0}} \multiput(18,52)(0,1){3}{\circle*{0.1}} \put(18,46){\vector(0,1){4}} \put(18,46){\vector(0,-1){0}} \put(12,32){\usebox{\mybox}} \put(18,44){\circle*{1.5}} \put(18,32){\circle*{1.5}} \put(18,26){\vector(0,1){4}} \put(18,26){\vector(0,-1){0}} \put(12,12){\usebox{\mybox}} \put(18,24){\circle*{1.5}} \put(19.5,25.8){\makebox(0,0)[bl]{$x$}} \put(24,12){\circle*{1.5}} \put(23,10){\makebox(0,0)[tr]{$x{+}s$}} \put(33,11.5){\makebox(0,0)[l]{$C$}} \put(24,8){\vector(0,1){2}} \put(36,8){\oval(24,6)[b]} \put(48,8){\line(0,1){70}} \put(33,78){\oval(30,6)[t]} \put(18,78){\line(0,-1){2}} \savebox{\mybox}{} \end{picture} \end{center} \caption{A cycle of ruled surfaces over $C$ glued with a shift $s$} \end{figure} We take $Y_1$ to be the bottom ruled surface and $Y_n$ the top ruled surface, and call $s$ the gluing parameter or shift of $X_0$. A careful analysis of Persson's proof shows that, if one replaces the hypothesis of global normal crossings by local normal crossings, than the only additional case is the following: $X_0$ is irreducible, its normalization is an elliptic ruled surface with two disjoint sections and $X_0$ arises from this by gluing these two sections with a shift $s$. We shall always refer to this situation (including the case $N=1$) as a cycle of elliptic ruled surfaces. The central fibre of such a degeneration is determined by the following data: \begin{enumerate} \item the number $N$ of components \item the base curve $C$ \item line bundles ${\cal L}_i$ with ${\Bbb P}({\cal O}\oplus{\cal L}_i)\cong \XX{i}$\footnotemark \footnotetext{Strictly speaking for $N=1$ we have to take the normalization.} \item the gluing parameter $s$. \end{enumerate} We first want to show that these data are not independent of each other. Before we can do this, we fix some more notation. First assume that $N\ge 2$. We denote the intersection of $\XX{i}$ and $\XX{i+1}$ by $C_i$. Furthermore we normalize the line bundle ${\cal L}_i$ in such a way that ${\cal O}_{\XX{i}}(C_{i-1})|_{C_{i-1}}={\cal L}_i$ and hence ${\cal O}_{\XX{i}}(C_{i})|_{C_{i}}={\cal L}_i^{-1}$. Under this assumption the line bundle ${\cal L}_i$ is uniquely determined. All indices have to be read modulo $N$. The above discussion also makes sense in case $N=1$, if we replace $\XX{1}$ by its normalization. The next two propositions are special cases of results proved by Persson \cite{P}. \begin{pro} $K_{{X}}={\cal O}_{{X}}$. \end{pro} \begin{pf} Since $K_{{X}}|_{X_t}={\cal O}_{X_t}$ for $t\neq0$ it follows that $$ K_X=\sum_{i=1}^N m_i \XX{i} $$ where the $m_i$ are uniquely defined up to a common summand. After possibly relabelling the components of $X_0$ we can assume $m_1$ to be maximal. Moreover by adding multiples of a fibre of $p$ we may assume that $m_1=0$. This already gives the result for $N=1$. Now assume $N\geq2$. Adjunction gives $$ (K_{{X}}+\XX{i})|_{\XX{i}}=K_{\XX{i}}. $$ Since $$ K_{\XX{i}}=-C_{i-1}-C_{i}+a_i f_{P_i} $$ for a suitable ruling $f_{P_i}$ over a point $P_i\in C$ and $$ \XX{i}|_{\XX{i}} = -C_{i-1}-C_i $$ this implies $$ (m_{i-1}-m_i) C_{i-1} +(m_{i+1}-m_i) C_i -a_i f_{P_i} =0. $$ Since $m_1=0$ we find $$ m_0 C_0 +m_2 C_1 = a_1 f_{P_1}. $$ Since, moreover, all $m_i\le 0$ this shows that $m_0=m_2=a_1=0$. Continuing in this way we find $m_i=a_i=0$ for all $i$. \end{pf} \begin{pro} $\operatorname{(i)}$ All line bundles ${\cal L}_i$ are isomorphic. \noindent $\operatorname{(ii)}$ $\deg {\cal L}_i=0$ for all $i$. \end{pro} \begin{pf} (i) For the normal bundle of $C_i$ in ${X}$ we have \begin{align*} {\cal N}_{C_i/{X}} &= {\cal N}_{C_i/\XX{i}} \oplus {\cal N}_{C_i/\XX{i+1}}\\ &= {\cal L}_i^{-1} \oplus {\cal L}_{i+1}. \end{align*} On the other hand, since $K_{{X}}={\cal O}_{{X}}$ and since $C_i$ is an elliptic curve, we find again by adjunction that $\det {\cal N}_{C_i/{X}}={\cal O}_{C_i}$. This shows (i). (ii) Assume that $\deg{\cal L}_i=m\neq0$. Then $|(C_i)_{\XX{i}}^2|=m$ and $(C_i)_{\XX{i+1}}^2=-(C_i)_{\XX{i}}^2$ for all $i$. But then a topological argument (see \cite[p.~94]{P}) shows that $$ H_1(X_t,{\Bbb Z}) ={\Bbb Z}^3 \oplus {\Bbb Z}_m $$ for general $t$. This contradicts the fact that $X_t$ is abelian. \end{pf} \subsection{Additional structures} We shall now consider further structures on the family $p:X\to\Delta$. First of all we consider degenerations of {\em polarized} abelian surfaces, i.e. we assume that a line bundle ${\cal O}_X(1)$ exists on $X$ such that ${\cal O}_{X_t}(1)={\cal O}(1)|_{X_t}$ for $t\neq0$ is a polarization on the smooth abelian surface $X_t$, and that ${\cal O}_{X_0}(1)$ is ample. We are particulary interested in the case where ${\cal O}_{X_t}(1)$ represents a polarization of type $(1,p)$ where $p\geq3$ is a prime number. We also want to assume that $p:X\to\Delta$ is a degeneration of polarized abelian surfaces with a (canonical) level structure. For the concept of (canonical) level structure on $(1,p)$-polarized abelian surfaces see \cite[I.1]{HKW}. Before we give a formal definition recall the following: Let $Y={\Bbb P}(\cal O\oplus \cal L)$ be a ${\Bbb P}^1$-bundle over an elliptic curve $C$ with $\deg\cal L=0$. Let $Y^0$ be the open part of $Y$ which is given by removing the two sections defined by line bundles $\cal O$, resp. $\cal L$. Then $Y^0$ is a ${\Bbb C}^*$-bundle over the base curve $C$. More precisely $Y^0$ carries the structure of a commutative complex Lie group, and as such it is an extension of the form $$ 1\longrightarrow{\Bbb C}^*\longrightarrow Y^0\longrightarrow C\longrightarrow0. $$ In fact $Y^0$ is a semi-abelian surface of rank 1. \begin{dfn} Let $p:X\to\Delta$ be a degeneration of $(1,p)$-polarized abelian surfaces. We say that this is a degeneration of $(1,p)$-polarized abelian surfaces with a ({\em canonical}) {\em level structure} if the following holds: \noindent (i) There exists an open subset $X^0\subset X$ such that $X^0\to\Delta$ is a family of abelian Lie groups with the following property: $X_t^0=X_t$ for $t\neq0$ and $X_0^0$ is the smooth part of a component of $X_0$ and as such carries the structure of a semi-abelian surface. \noindent (ii) There exists an action of ${\Bbb Z}_p\times{\Bbb Z}_p$ on $X$ over $\Delta$ with the following properties: It leaves ${\cal O}_X(1)$ invariant and defines a (canonical) level structure on $X_t$ for $t\neq0$ (in particular it operates on $X_t$ by translation by elements of order $p$). Moreover, the subgroup of ${\Bbb Z}_p\times{\Bbb Z}_p$ which stabilies $X_0^0$ acts on $X_0^0$ as a subgroup. \end{dfn} The two next results show that the presence of a polarization and a level structure imposes strong conditions on the singular fibre $X_0$. \begin{lem} Let $p:{X}\to\Delta$ be a degeneration of abelian surfaces with $(1,p)$-polarization and a (canonical) level-structure. Then there are only two possibilities: \noindent $\operatorname{(i)}$ The central fibre consists of one component $X_0$. If $\tilde{X_0}$ is its normalization, then ${\cal O}_{\tilde{X_0}}(1)={\cal O}_{\tilde{X_0}}(C_0 + p f_P)$ for a suitable point $P$ in the base curve $C$. \noindent $\operatorname{(ii)}$ The central fibre consists of $p$ components $\XX{i}$, $i=1,\ldots,p$. In this case ${\cal O}_{\XX{i}}(1)={\cal O}_{\XX{i}}(C_{i-1} + f_{P_i})$ for suitable points $P_i$. \end{lem} \begin{pf} We have $$ {\cal O}_{\XX{i}}(1) ={\cal O}_{\XX{i}}(a_i C_{i-1} +b_i f_{P_i}) $$ for suitable integers $a_i$ and $b_i$. Since these line bundles glue together to give a line bundle on $X_0$ it follows immediately that all $b_i$ are equal. We denote this number by $b$. Since ${\cal O}_{X_0}(1)$ is ample, and since we are dealing with a degeneration of abelian surfaces with a $(1,p)$-polarization it follows that \begin{equation}\label{1} 2b\sum_{i=1}^N a_i = 2p \end{equation} with all $a_i>0$. We now consider the action of ${\Bbb Z}_p\times{\Bbb Z}_p$ on the set $\{\XX{1},\ldots,\XX{N}\}$. Let $G$ be the stabilizer of $\XX{1}$. Since $p$ is a prime number, there are three possibilities: (i) $G=\{1\}$. Then $N\ge p^2$ and this contradicts formula~(\ref{1}). (ii) $G={\Bbb Z}_p$. In this case $N\ge p$. It follows from formula~(\ref{1}) that $N=p$ and $a_i=b=1$. (iii) $G={\Bbb Z}_p\times {\Bbb Z}_p$. The group ${\Bbb Z}_p\times {\Bbb Z}_p$ acts on $X_0$ as a group of automorphisms, hence it must leave its singular locus invariant. Since $p$ is an odd prime number the group $G$ must leave the curve $C_0$ invariant. As a subgroup of $X_0^0$ the group $G$ acts by translation on $C_0$. The multiplicative group ${\Bbb C}^\ast$ contains no subgroup isomorphic to ${\Bbb Z}_p\times{\Bbb Z}_p$, hence the group $G$ must contain at least a subgroup ${\Bbb Z}_p$ which acts non-trivially on $C_0$. Since ${\cal O}_X(1)$ is invariant under $G$ it follows that the degree of ${\cal O}_X(1)$ restricted to $C_0$ must be divisible by $p$. I.e. $b$ must be divisible by $p$. By formula~(1) it follows that $b=p$, $N=1$ and $a_1=1$. \end{pf} \begin{rem} \normalshape It is easy to construct degenerations $p:X\to\Delta$ of $(1,p)$-polarized abelian surfaces without a level structure, such that the polarization on different components $Y_i$ is numerically different. \end{rem} In lemma~4 we have seen that the central fibre of a degeneration $p:{X}\to\Delta$ of $(1,p)$-polarized abelian surfaces with a level-structure has either 1 or $p$ components. Recall that $\XX{i}={\Bbb P}({\cal O}\oplus{\cal L}_i)$ and that all ${\cal L}_i$ are isomorphic and of degree 0 (proposition~2). We shall denote this line bundle by ${\cal L}$. Moreover, assume that we have chosen an origin $O$ on the base curve $C$. Then we can consider the shift $s$ as a point on $C$. \begin{pro} Let $p:X\to\Delta$ be a degeneration of $(1,p)$-polarized abelian surfaceswith a (canonical) level structure, and denote the shift of $X_0$ by $s$. Then there are two possibilities: \noindent $\operatorname{(i)}$ $N=1$ and $\cal L={\cal O}_C(ps-pO)$ \noindent $\operatorname{(ii)}$ $N=p$ and there exist a point $s'\in C$ with $s=ps'$ such that $\cal L={\cal O}(s'-O)$. \end{pro} \begin{pf} (i) Assume $N=1$. Then ${\cal O}_{\tilde{X_0}}(1)={\cal O}_{\tilde{X_0}}(C_0+pf_P)$ and hence $$ {\cal O}_{\tilde{X_0}}(1)|_{C_0}={\cal L}\otimes {\cal O}_C(pP),\qquad {\cal O}_{\tilde{X_0}}(1)|_{C_1}={\cal O}_C(pP). $$ Since $C_1$ and $C_0$ are glued with the shift $s$ a necessary and sufficient condition for ${\cal O}_{\tilde{X_0}}(1)$ to descend to a line bundle on $X_0$ is $$ {\cal L}\otimes{\cal O}_C(pP)={\cal O}_C(pP)\otimes{\cal O}_C(s-O)^{\otimes p} $$ which gives the claim. (ii) Assume $N=p$. Then ${\cal O}_{\XX{i}}(1)={\cal O}_{\XX{i}}(C_{i-1}+f_{P_i})$ and hence $$ {\cal O}_{\XX{1}}(1)|_{C_1} ={\cal O}_C(P_1),\qquad {\cal O}_{\XX{2}}(1)|_{C_1}={\cal L}\otimes {\cal O}_C(P_2). $$ From gluing $\XX{1}$ and $\XX{2}$ we obtain $$ {\cal O}_C(P_2)={\cal L}^{-1}\otimes {\cal O}_C(P_1). $$ Continuing in this way, we get $$ {\cal O}_C(P_p)={\cal L}^{-(p-1)}\otimes {\cal O}_C(P_1). $$ Finally gluing $C^p$ and $C_0$ with a shift $s$ gives the condition $$ {\cal L} \otimes {\cal O}_C(P_1) = {\cal L}^{-(p-1)}\otimes {\cal O}_C(P_1)\otimes {\cal O}_C(s-O) $$ i.e. $$ {\cal L}^p = {\cal O}_C(s-O) $$ as claimed. \end{pf} \begin{rem} \normalshape In \cite[part~II]{HKW} a number of explicit examples of degenerations of $(1,p)$-polarized abelian surfaces with a level structure were constructed. It was shown that both types of degenerations which were discussed above actually occur. Moreover, all elliptic curves $C$ and all shifts $s$ can be realised. \end{rem} \section{The mixed Hodge structure on $H^1(X_0)$}\label{section3} Let $X_0=\cup_{i\in{\Bbb Z}_N} Y_i$ be a cycle of elliptic ruled surfaces as in the previous section. In this section we calculate the MHS on $H^1(X_0)$ and show that the base curve $C$ and the shift $s$ can be recovered from it. \subsection{The spectral sequence associated to $X_0$} For technical reasons we want that $X_0$ has global normal crossings. Therefore, we shall first assume $N\geq2$. However, this is not an essential hypothesis (see remark (\ref{R})). The MHS on $H^q(X_0)$ can be computed as follows (see \cite[p.103]{G}). Let $Y^0= \bigsqcup_{i\in\Z_N} Y_i$ and $Y^1=\bigsqcup_{i\in\Z_N} C_i$, where $C_i=Y_i\cap Y_{i+1}$. The maps $\alpha_i$ and $\beta_i$ are defined as the inclusions $$ \alpha_i:C_i\hookrightarrow Y_i\text{ and }\beta_i: C_i\hookrightarrow Y_{i+1}. $$ Consider the double complex \[ A^{pq}=A^q(Y^p) \] of global $C^{\infty}$ differential forms, where $d:A^{pq}\to A^{p,q+1}$ is the exterior derivative and $\delta: A^{0q}=\oplus_{i\in\Z_N} A^q(Y_i) \to A^{1q} =\oplus_{i\in\Z_N} A^q(C_i)$ on $A^q(Y_i)$ is given by $(-\beta_{i-1}^{\ast},\alpha_i^{\ast}):A^q(Y_i)\to A^q(C_{i-1})\oplus A^q(C_i)$. These coboundary maps satisfy the relations $d^2=0$, $d\delta=\delta d$ and $\delta^2=0$. There is a single complex $(A^{\bullet}, D)$ associated to $(A^{\bullet\bullet},d,\delta)$: \[ A^k=\oplus_{p+q=k} A^{pq}\quad\text{$D=(-1)^p d+\delta$ on $A^{pq}$.} \] We call $ {\Bbb H}^{k}:=H_D^{k}(A^{\bullet}) $ the hypercohomology of the double complex. \begin{lem}\label{apq} The hypercohomology ${\Bbb H}^k$ is canonically isomorphic to $H^k(X_0)$. \end{lem} \begin{pf} Let $i_p:Y^p\to X_0$ be the natural map. Let ${\cal A}^{pq}$ be the sheaf on $X_0$ defined by \[ {\cal A}^{pq}(U)=A^q(i_p^{-1}U). \] Set ${\cal A}^k:=\oplus_{p+q=k}{\cal A}^{pq}$ and let $D$ be the sheafified version of the coboundary operator $D$ above. It is shown in \cite[lemma~4.6]{GS} that the complex $({\cal A}^{\bullet},D)$ is an acyclic resolution of the constant sheaf ${\Bbb C}_{X_0}$, hence $H^k(X_0,{\Bbb C})=H^k(\Gamma(X_0,{\cal A}^{\bullet}))=H_D^k(A^{\bullet})={\Bbb H}^k$. \end{pf} There exists a spectral sequence $E_r^{pq}$ with $E_0^{pq}=A^{pq}$ and $d_0=d$. The map $d_1$ is induced by $\delta$ and, more generally, $d_r:E_r^{pq}\to E_r^{p+r,q-r+1}$ is induced by $D$. Notice that $E_1^{pq}=H^q(Y^p)$. Since $A^{pq}=0$ for $p<0$ and $p>1$, the spectral sequence degenerates at $E_2$, i.e. $E_2^{pq}=E_{\infty}^{pq}$. The weight filtration $W_{\bullet}$ on ${\Bbb H}$ is defined to be the filtration induced by $W_k(A^{\bullet})=\oplus_{q\le k} A^{\ast,q}$. By the theory of spectral sequences we have $E_{\infty}^{pq}={\operatorname{Gr}}_{q}^W {\Bbb H}^{p+q}$. Since $A^{pq}=0$ for $p<0$ and $p>1$, this boils down to a commutative diagram with exact rows \[ \begin{CD} 0 @>>> W_0 @>>> H^1(X_0) @>>> {\operatorname{Gr}}_1^W @>>> 0\\ @. @| @| @|\\ 0 @>>> E_{\infty}^{10} @>{i}>> {\Bbb H}^1 @>{\pi}>> E_{\infty}^{01} @>>> 0. \end{CD} \] The map $i$ is induced by the inclusion $A^{10}\to A^{10}\oplus A^{01}$; the map $\pi$ is induced by the projection $A^{10}\oplus A^{01}\to A^{01}$. The Hodge filtration on ${\Bbb H}^k$ is induced by $F^p(A^k)=\oplus_{a+b=k} F^p(A^{a,b})$. In our case we have \[ F^1{\Bbb H}^1={\operatorname{Ker}}(D|_{F^1(Y^0)})={\operatorname{Ker}}(\delta: F^1(Y^0)\to F^1(Y^1)). \] \begin{lem}\label{wf} Let $X_0$ be a cycle of ruled surfaces over a smooth curve $C$. The weight filtration on $H^1(X_0)$ takes the form $$ 0 \subset W_0 \subset W_1=H^1(X_0). $$ Furthermore, $W_0$ is the unique 1-dimensional Hodge structure $T\langle 0 \rangle$ of type $(0,0)$ and ${\operatorname{Gr}}_1^W$ is canonically isomorphic to $H^1(C)$ (as a Hodge structure). \end{lem} \begin{pf} Since $E_1^{pq}=H^q(Y^p)$ we have $W_0=E_{\infty}^{10}=E_2^{10}={\operatorname{Coker}}(\delta: H^0(Y^0)\to H^0(Y^1))$. Now it is easy to see that $$ \begin{CD} H^0(Y^0) @>{\delta}>> H^0(Y^1) @>>> {\Bbb Z} @>>> 0\\ @. @| @AA{S}A\\ @. \oplus H^0(X_i) @= \oplus {\Bbb Z}, \end{CD} $$ where $S$ is the summation map, is commutative and has an exact top row, hence $W_0$ is indeed the trivial Hodge structure $T\langle 0 \rangle$. To show that ${\operatorname{Gr}}_1^W=E_{\infty}^{01}=E_2^{01}={\operatorname{Ker}}(\delta: H^1(Y^0)\to H^1(Y^1))$ is isomorphic to $H^1(C)$, we introduce the following maps: Let $$ \rho_i: Y_i\to C_i\text{ and }\sigma_i: Y_{i+1}\to C_i $$ be the projections and let $\tau_i=\sigma_0 {\scriptstyle\circ} (\alpha_1{\scriptstyle\circ}\sigma_1) {\scriptstyle\circ}(\alpha_2{\scriptstyle\circ}\sigma_2){\scriptstyle\circ}\cdots{\scriptstyle\circ}(\alpha_{i-1}{\scriptstyle\circ}\sigma_{i-1}): Y_i\to C_0=C$. Then one easily shows that the following sequence is exact $$ \begin{CD} 0 @>>> H^1(C) @>{\tau^{\ast}}>> H^1(Y^0) @>{\delta}>> H^1(Y^1), \end{CD} $$ where $\tau^{\ast}:=(\tau_1^\ast,\ldots,\tau_N^\ast): H^1(C) \to H^1(Y^0)=H^1(Y_1)\oplus\cdots\oplus H^1(Y_N)$. \end{pf} \subsection{The base curve and the extension class}\label{subsection22} The mixed Hodge structure $H^1(X_0)$ can be regarded as an extension of the pure Hodge structures $W_0$ and $\operatorname{Gr}^W_1$. We want to show how one can recover the base curve $C$ and the shift $s$ from this extension. In \cite{C} an object $\operatorname{Ext}(\operatorname{Gr}^W_1,W_0)$ was introduced which parametrizes all such extensions. In our case it is an abelian variety, namely the base curve $C$. \begin{lem}\label{lemma14} Let $W_{\bullet}$ be as above. Then there exists a canonical isomorphism between ${\operatorname{Ext}}({\operatorname{Gr}}_1^W,W_0)$ and ${\operatorname{Alb}}(C)$. \end{lem} \begin{pf} For any Hodge structure $H$ we define $J^0 H = H_{{\Bbb C}}/H_{{\Bbb Z}}+ F^0 H$. If $H_1$ and $H_2$ are two Hodge structures, then we can apply this to $H={\operatorname{Hom}}(H_1,H_2)$ and we get ${\operatorname{Ext}}(H_1,H_2)=J^0 {\operatorname{Hom}}(H_1,H_2)$ (see \cite[prop.~2]{C}). Now if $H_1=H^1(X)$, where $X$ is a smooth projective variety and $H_2=T\langle 0 \rangle$, then ${\operatorname{Hom}}(H_1,H_2)$ is the dual Hodge structure of $H^1(X)$ and has weight $-1$. Its Hodge filtration has the form $$ 0=F^1\subset F^0\subset F^{-1}=H^1(X)^\ast, $$ where $F^0(H^1(X)^\ast)=(H^{0,1}(X))^\ast$. Hence $H^1(X)^{\ast}/F^0(H^1(X)^{\ast})=H^{1,0}(X)^{\ast}$ and $J^0(H^1(X)^{\ast})=H^{1,0}(X)^{\ast}/H_1(X,{\Bbb Z})={\operatorname{Alb}}(X)$. \end{pf} In \cite{C} we find the following algorithm to calculate the extension class $e\in J^0{\operatorname{Hom}}(B,A)$ of an extension $$ \begin{CD} 0 @>>> A @>{i}>> H @>{\pi}>> B @>>> 0. \end{CD} $$ \begin{enumerate} \item Choose an integral retraction $r:H \to A$, i.e. a map defined over ${\Bbb Z}$ satisfying $r{\scriptstyle\circ} i ={\bf 1}_A$. \item Define two Hodge filtrations on $A\oplus B$: $$ \tilde{F}_{\infty}^{\bullet}:=(r,\pi)(F^{\bullet}H)\text{ and } \tilde{F}_0^{\bullet}:=F^{\bullet}A \oplus F^{\bullet}B. $$ \item Find a $\psi\in{\operatorname{Hom}}(B,A)_{{\Bbb C}}$ such that $$ \mwp{A}{B} \tilde{F}_0^{\bullet} =\tilde{F}_{\infty}^{\bullet}. $$ \end{enumerate} Then $e=[\psi]\in J^0{\operatorname{Hom}}(B,A)$. Let $X_0$ be a cycle of ruled surfaces over the elliptic curve $C=V/\Lambda$ with shift $s\in {\operatorname{Alb}}(C)$. We now describe the ${\Bbb Z}$-splitting of the exact sequence $0\to W_0\to {\Bbb H}^1\to{\operatorname{Gr}}_1^W\to 0$, as required in the first step of this algorithm, explicitely in terms of $C$ and $s$. Let $\{\lambda_1,\lambda_2\}$ be a ${\Bbb Z}$-basis for $\Lambda$ and let $x_i:V\to{\Bbb R}$ be the corresponding coordinate functions ($i=1$, 2). Let $s_i=x_i(s)$, i.e. \[ s=s_1\lambda_1+s_2\lambda_2. \] (The numbers $s_i$ are defined only up to integers.) Let $\phi\in H^1(C)$. Since $\{dx_1,dx_2\}$ is a basis for $H^1(C)$, we may represent $\phi$ by $\xi_1 dx_1 + \xi_2 dx_2\in A^1(C)$, which we will also denote by $\phi$. Consider \[ \tau^{\ast}\phi:=(\tau_1^{\ast},\ldots,\tau_N^{\ast})\phi\in E_0^{01}. \] Recall that $\tau^{\ast}\phi$ represents an element in $E_{\infty}^{01}$. Set \[ f=(f_1,\ldots,f_N)\in E_0^{10}, \] where \[ f_i=\begin{cases} 0 & 1\le i<N\\ - \xi_1 s_1 - \xi_2 s_2 & i=N \end{cases} \] is a constant function on $C_i$. \begin{lem} The pair $(\tau^{\ast}\phi,f)\in A^{01}\oplus A^{10}$ represents a class in ${\Bbb H}^1$. \end{lem} \begin{pf} First of all, $d(\tau^{\ast}\phi)=0$ since $d\phi=0$. Furthermore, $\delta(\tau^{\ast}\phi)=(0,\ldots,0,(t_{-s}-{\bf 1}_C)^{\ast}\phi)$, where $t_{-s}:x\mapsto x-s: C\to C$. Let $g=\xi_1 x_1+\xi_2 x_2\in{\operatorname{Hom}}_{{\Bbb R}}(V,{\Bbb R})$. Then $\phi=dg$ and $f_N=(t_{-s}-{\bf 1}_C)^{\ast}g$. \end{pf} We write $(\tau^{\ast}\phi,f)=\sigma(\phi)$. By the lemma above, the map $\sigma: H^1(C)\to E_0^{01}\oplus E_0^{10}$ induces a map \[ \bar{\sigma}: {\operatorname{Gr}}_1^W=H^1(C)\to {\Bbb H}^1. \] Since $\pi:{\Bbb H}^1\to {\operatorname{Gr}}_1^W$ is induced by the projection $A^{01}\oplus A^{10}\to A^{01}$, $\pi{\scriptstyle\circ}\bar{\sigma}={\bf 1}_{{\operatorname{Gr}}_1^W}$, i.e. $\bar{\sigma}$ is a splitting over ${\Bbb R}$. In fact, $\bar{\sigma}$ is defined over ${\Bbb Z}$: \begin{lem}\label{fl} The map $\bar{\sigma}$ splits the sequence \[ \begin{CD} 0 @>>> W_0 @>{i}>> {\Bbb H}^1 @>{\pi}>> {\operatorname{Gr}}_1^W @>>> 0 \end{CD} \] over ${\Bbb Z}$. \end{lem} Assuming this for the moment, we show that the extension class is identified with the shift. \begin{cor}\label{cor17} The natural isomorphism ${\operatorname{Ext}}({\operatorname{Gr}}_1^W,W_0)\cong{\operatorname{Alb}}(C)$ identifies the extension class $e=[\psi]$ with the shift $s$. \end{cor} \begin{pf} Let $r:{\Bbb H}^1\to W_0$ be the map such that \[ (r,\pi)=(i,\bar{\sigma})^{-1}:{\Bbb H}^1 \to W_0 \oplus {\operatorname{Gr}}_1^W. \] As before we set \[ \tilde{F}_{\infty}^1=(r,\pi)F^1{\Bbb H}^1\text{ and } \tilde{F}_0^1=F^1(W_0)\oplus F^1 ({\operatorname{Gr}}_1^W) \] in $W_0 \oplus {\operatorname{Gr}}_1^W$. The extension class is represented by a map $\psi: {\operatorname{Gr}}_1^W\to W_0$ such that \[ \begin{pmatrix} {\bf 1}_{W_0} &- \psi\\ 0 & {\bf 1}_{{\operatorname{Gr}}_1^W} \end{pmatrix} \tilde{F}_{\infty}^1=\tilde{F}_0^1. \] Since $\tilde{F}_0^1\cap W_0=F^1(W_0)=0$ we get \[ r(\omega)-\psi(\pi(\omega))=0\quad\text{for all } \omega\in F^1{\Bbb H}^1. \] Now assume that $C={\Bbb C}/({\Bbb Z}\tau+{\Bbb Z})$ and let $\phi=\tau dx_1+dx_2$, where $x_1$, $x_2$ are the coordinates dual to the basis $\lambda_1=\tau$, $\lambda_2=1$. Then $F^1H^1(C)={\Bbb C}\phi$ (see \S4.2). Recall that $(\tau^{\ast}\phi,0)\in A^{01}\oplus A^{10}$ represents an element $\omega\in F^1{\Bbb H}^1$ such that $\pi(\omega)=\phi$. Consider $\sigma(\phi)=(\tau^{\ast}\phi,f)$. Since $i:W_0\to {\Bbb H}^1$ is induced by the inclusion $A^{10}\to A^{01}\oplus A^{10}$, \[ \omega=i(-f)+\sigma(\phi), \] hence \begin{align*} \psi(\tau dx_1 + dx_2) &=\psi(\pi(\omega))\\ & = r(\omega)\\ &=-f \end{align*} in $W_0$. Identifying $W_0$ with ${\Bbb Z}$ as in the proof of lemma~\ref{wf} we find \begin{align*} \psi(\tau dx_1+dx_2) &=- \sum f_i\\ &=-f_N\\ &=\tau s_1 + s_2\\ &=\int_{0}^s \tau dx_1+dx_2. \end{align*} Hence $\psi=\int_0^s$ on $F^1H^1(C)$, i.e. ${\operatorname{Ext}}({\operatorname{Gr}}_1^W,W_0)\cong{\operatorname{Alb}}(C)$ identifies the extension class $e=[\psi]$ with the shift $s$. \end{pf} \begin{rem}\label{R}\normalshape The above result is also true for $N=1$. In this case let $Y_0$ be the normalization of $X_0$. Let $X_0'$ be the cycle of elliptic ruled surfaces consisting of two copies of $Y_0$ and glued with the same shift $s$ as $X_0$. There exists a projection map $X_0'\to X_0$ given by contracting one of the components of $X_0'$. By functoriality, the isomorphism between the vector spaces $H^1(X_0)$ and $H^1(X_0')$ induced by this map is in fact an isomorphism of mixed Hodge structures. Hence we can work with $X_0'$ instead of $X_0$ . \end{rem} \begin{rem}\normalshape Instead of cycles of elliptic ruled surfaces we can also consider cycles of ${\Bbb P}^1$-bundles over an abelian variety $Z$. Again the Hodge filtration takes on the form $0=W_{-1}\subset W_0\subset W_1=H^1(X_0)$ where $W_0$ is the 1-dimensional Hodge structure of type $(0,0)$. As before $\operatorname{Gr}^W_1=H^1(Z)$ and hence $\operatorname{Ext}(\operatorname{Gr}_1^W,W_0)= \operatorname{Alb}(Z)$. The same proof as before also shows that the extension class in $\operatorname{Ext}(\operatorname{Gr}_1^W,W_0)$ again coincides with the shift $s$. \end{rem} \subsection{Proof of lemma~\ref{fl}} Let $c:[0,1]\to X_0$ be the loop described in figure~2 \cite[23.8]{Gre}. \unitlength1cm \begin{figure}\label{f2} \begin{center} \begin{picture}(12,5) \thicklines \put(1,1){\line(1,0){5}} \put(6.4,1){\line(1,0){0.2}} \put(6.9,1){\line(1,0){0.2}} \put(7.4,1){\line(1,0){0.2}} \put(8,1){\line(1,0){3}} \put(1,5){\line(1,0){5}} \put(6.4,5){\line(1,0){0.2}} \put(6.9,5){\line(1,0){0.2}} \put(7.4,5){\line(1,0){0.2}} \put(8,5){\line(1,0){3}} \put(1,1){\line(0,1){4}} \put(3,1){\line(0,1){4}} \put(5,1){\line(0,1){4}} \put(9,1){\line(0,1){4}} \put(11,1){\line(0,1){4}} \thinlines \put(1,2){\line(1,0){5}} \put(6.4,2){\line(1,0){0.2}} \put(6.9,2){\line(1,0){0.2}} \put(7.4,2){\line(1,0){0.2}} \put(8,2){\line(1,0){1}} \put(9,2){\line(2,1){2.0}} \put(9.2,2){\line(1,0){0.1}} \put(9.6,2){\line(1,0){0.1}} \put(10.0,2){\line(1,0){0.1}} \put(10.4,2){\line(1,0){0.1}} \put(10.8,2){\line(1,0){0.1}} \put(1,2){\circle*{0.2}} \put(3,2){\circle*{0.2}} \put(5,2){\circle*{0.2}} \put(9,2){\circle*{0.2}} \put(11,2){\circle*{0.2}} \put(11,3){\circle*{0.2}} \put(0,2.2){\makebox(0.8,0.8)[rb]{$p_0$}} \put(2,2.2){\makebox(0.8,0.8)[rb]{$p_1$}} \put(4,2.2){\makebox(0.8,0.8)[rb]{$p_2$}} \put(8,2.2){\makebox(0.8,0.8)[rb]{$p_{N-1}$}} \put(11.2,3.2){\makebox(0.8,0.8)[lb]{$p_N=p_0$}} \put(11.2,2.2){\makebox(0.8,0.8)[lb]{$p_N+s$}} \put(1,0){\makebox(2,0.8)[t]{$Y_1$}} \put(3,0){\makebox(2,0.8)[t]{$Y_2$}} \put(9,0){\makebox(2,0.8)[t]{$Y_N$}} \end{picture} \caption{The loop $c$} \end{center} \end{figure} Clearly, $H_1(X_0,{\Bbb Z})=H_1(C,{\Bbb Z})\oplus {\Bbb Z} c$. Using the Kronecker product (see \cite{Gre}) the loop $c$ determines a map \[ c: H^1(X_0,{\Bbb Z})\to{\Bbb Z}. \] Since $\bar{\sigma}$ splits the sequence over ${\Bbb R}$, all we have to do is show that $c(\bar{\sigma}(dx_i))\in{\Bbb Z}$ for $i=1$, 2. In fact, we will prove that $c{\scriptstyle\circ}\bar{\sigma}=0$. In order to understand how the loop $c$ operates on ${\Bbb H}^1$, we want to identify ${\Bbb H}^1=H^1(\Gamma(X_0,{\cal A}^{\bullet}))$ with the simplicial cohomology $H_{\nabla}^1(X_0)$. In \cite[p.95]{GM} we find a canonical isomorphism \[ H^k(\Gamma(X_0,{\cal B}^{\bullet}))\to H_{\nabla}^k(X_0), \] induced by integration, where \[ {\cal B}^k:={\operatorname{Ker}}(\delta: {\cal A}^{0k}\to{\cal A}^{1k}). \] One easily checks that the complex $({\cal B}^{\bullet},d)$ is also an acyclic resolution of the constant sheaf ${\Bbb C}_{X_0}$. The inclusions $j_k:{\cal B}^k\hookrightarrow{\cal A}^{0k}\hookrightarrow{\cal A}^k$ commute with the coboundary operators, hence the canonical isomorphism between $H^k(\Gamma(X_0,{\cal B}^{\bullet}))$ and $H^k(\Gamma(X_0,{\cal A}^{\bullet}))$ is induced by $j_k$ (see \cite[5.24]{W}). Let $\phi=\xi_1 dx_1+ \xi_2 dx_2$ and $(\omega,f)=\bar{\sigma}(\phi)$, i.e. \begin{align*} \omega &=(\omega_1,\ldots,\omega_N)\qquad\omega_i=\tau_i^{\ast}\phi\\ f&=(0,\ldots,0,f_N)\qquad f_N=-\xi_1 s_1- \xi_2 s_2 \end{align*} Let $g=(g_1,\ldots,g_N)\in\Gamma(X_0,{\cal A}^0)$ be such that \[ g_i|_{C_j}=\begin{cases} -f_N &(i,j)=(N,0)\\ 0 &\text{otherwise} \end{cases} \] Since $\delta g= -f$, we have that $\omega'=\omega+dg\in \Gamma(X_0,{\cal B}^1)$. Since $(\omega',0)=(\omega,f)+(df,\delta f)$ we have $j_1[\omega']=\bar{\sigma}(\phi)$ in ${\Bbb H}^1$. Now we view the loop $c$ as an element of ${\operatorname{Hom}}_{{\Bbb Z}}(H_{\nabla}^1(X_0,{\Bbb Z}),{\Bbb Z})$. Using the defining properties of $g_i$ and the fact that integrals of $\omega_i$ along the rulings vanish, we get \begin{align*} c(\bar{\sigma}(\phi)) &=\sum_{i=1}^N \int_{p_{i-1}}^{p_i} \omega_i'\\ &=\int_{p_{N-1}}^{p_N} \omega_N'\\ &=g_N(p_N)-g_N(p_{N-1})+\int_{p_N+s}^{p_N} \omega_N\\ &=(s_1\xi_1 +s_2 \xi_2) - (s_1\xi_1 +s_2 \xi_2)\\ &=0. \end{align*} \hfill $\Box$ \section{Variation of Hodge structure}\label{section4} In this section we want to compute the VHS associated to boundary points of the moduli space of $(1,p)$-polarized abelian surfaces with a (canonical) level structure (cf. definition~\ref{D} and theorem~\ref{T}). \subsection{$D$-polarized abelian varieties} Our main reference is \cite{LB}. Fix a type $D$, i.e. an ordered sequence $(d_1,\ldots,d_g)$ of positive integers satisfying $d_i|d_{i+1}$ ($i=1,\ldots,g-1$). We will often write $D={\operatorname{diag}}(d_1,\ldots,d_g)$. Let ${\Bbb H}_g$ be the {\em Siegel space of degree} $g$. To $\tau\in{\Bbb H}_g$ we associate an abelian variety $X_{\tau,D}$ of dimension $g$ and a polarization $E_{\tau,D}$ of type $D$ as follows. First, \[ X_{\tau,D}={\Bbb C}^g/\Lambda_{\tau,D}, \] where \[ \Lambda_{\tau,D}=(\tau, D){\Bbb Z}^{2g}. \] Let $\lambda_i$ ($i=1,\ldots,2g$) be the $i$-th column of $(\tau,D)$. Then we define $E_{\tau,D}$ to be the map $\Lambda\times\Lambda\to{\Bbb Z}$ given by the matrix \[ E_{\tau,D}=\begin{pmatrix} 0 & D\\ -D & 0 \end{pmatrix} \] with respect to the basis $\{\lambda_1,\ldots,\lambda_{2g}\}$ of $\Lambda_{\tau,D}$. For this reason we will refer to this basis as the symplectic basis. As an alternating map $E$ can be regarded as an element in $H^2(X_{\tau,D},{\Bbb Z})$. It is standard knowledge that (up to sign) it is in fact the first Chern class of an ample line bundle on $X_{\tau,D}$, i.e. a polarization. Every $D$-polarized abelian variety is isomorphic to $(X_{\tau,D},E_{\tau,D})$ for some $\tau\in{\Bbb H}_g$. Furthermore, we can construct a \lq\lq universal $D$-polarized abelian variety\rq\rq\ over ${\Bbb H}_g$ as follows: \[ {X}_{D}={\Bbb Z}^{2g}\backslash ({\Bbb C}^g\times{\Bbb H}_g), \] where ${\Bbb Z}^{2g}$ operates on ${\Bbb C}^g\times{\Bbb H}_g$ by \[ l(v,\tau)=(v+(\tau,D)l,\tau) \] for $l\in{\Bbb Z}^{2g}$ and $(v,\tau)\in{\Bbb C}^g\times{\Bbb H}_g$. The projection ${\Bbb C}^g\times{\Bbb H}_g\to{\Bbb H}_g$ induces a projection \[ \pi:{X}_D\to{\Bbb H}_g, \] such that $\pi^{-1}(\tau)=X_{\tau,D}$. $X_{D}$ is a complex manifold and carries a universal polarization ${\cal L}$ (see \cite[lemma~8.7.1]{LB}). We are only interested in $D=(1,p)$, $D=(1)$ and $D=(p)$ and will often omit $D$ if confusion seems unlikely. \subsection{The polarized Hodge structure of $(X_{\tau,D},E_{\tau,D})$} We can describe the Hodge structure of the abelian variety $X_{\tau,D}$ very explicitly. For any abelian variety $X=V/\Lambda$ we have $$ \matrix F^1(H^1(X))=H^{1,0}(X) & \subset & H^1(X,{\Bbb C})\\ \| && \| \\ V^{\ast}={\operatorname{Hom}}_{{\Bbb C}}(V,{\Bbb C}) & \subset & {\operatorname{Hom}}_{{\Bbb R}}(V,{\Bbb C}). \endmatrix $$ Now consider $X=X_{\tau,D}$, let $\{e_1,\ldots,e_g\}$ be the standard basis of $V={\Bbb C}^g$ and let $\{\lambda_1,\ldots,\lambda_{2g}\}$ be the symplectic basis of $\Lambda=\Lambda_{\tau,D}$. Then $\{\lambda_1^{\ast},\ldots,\lambda_{2g}^{\ast}\}$ is a ${\Bbb Z}$-basis of $H^1(X,{\Bbb Z})$ and $\{e_1^{\ast},\ldots,e_g^{\ast}\}$ is a ${\Bbb C}$-basis of $F^1(H^1(X))$. Using the coordinates of $H^1(X,{\Bbb C})=H^1(X,{\Bbb Z})\otimes{\Bbb C}$ determined by $\{\lambda_1^{\ast},\ldots,\lambda_{2g}^{\ast}\}$, $e_i^{\ast}$ is given by the $i$-th row of the period matrix $(\tau,D)$. A polarization $\omega$ on a compact complex manifold $X$ induces a bilinear form $Q$ on $H^n(X)$: $$ Q(\phi,\psi)=(-1)^{n(n-1)/2} \int_{X} \phi\wedge\psi\wedge\omega^{d-n}, $$ where $d=\dim X$. The pair $(H^n(X), Q)$ is a polarized Hodge structure of weight~$n$ (see \cite[p.7]{G}). We now calculate the polarization on $H^1(X_{\tau,D})$ induced by $E_{\tau,D}$. Let $x_i:V\to {\Bbb R}$ ($i=1,\ldots,2g$) be the coordinates with respect to the real basis $\{\lambda_1,\ldots,\lambda_{2g}\}$ of $V$. Then $dx_i$ corresponds to $\lambda_i^{\ast}$ under the natural isomorphism $H_{\hbox{de Rham}}^1(X) \cong {\operatorname{Hom}}(\Lambda,{\Bbb Z})$. Furthermore $E_{\tau,D}\in H^2(X,{\Bbb Z})$ is represented by the 2-form \[ \omega=-\sum_{i=1}^g d_i dx_i\wedge dx_{i+g} \] and \[ \int_X dx_1\wedge dx_{g+1} \wedge\cdots\wedge dx_{g}\wedge dx_{2g} =1 \] (see \cite[lemmas 3.6.4 and~3.6.5]{LB}). It follows that $Q$ is given by the matrix $$ (g-1)! \left(\begin{array}{cc} 0 & -\hat{D}\\ \hat{D} & 0\end{array}\right) \qquad \hat{D}=\Big(\prod_{i=1}^{2g} d_i\Big) D^{-1} $$ with respect to the basis $\{\lambda_1^{\ast},\cdots,\lambda_{2g}^{\ast}\}$ of $H^1(X,{\Bbb Z})={\operatorname{Hom}}(\Lambda,{\Bbb Z})$. \subsection{Boundary points of ${\cal A}^*(1,p)$} We have to recall briefly some facts about compactifications of moduli spaces of abelian surfaces. All relevant details can be found in \cite{HKW}. By $\cal A(1,p)$ we denote the moduli space of $(1,p)$-polarized abelian surfaces with a (canonical) level structure ($p\geq3$, prime). Recall that $$ \cal A(1,p)={\Bbb H}_2/\Gamma_{1,p} $$ where $$ \Gamma_{1,p}=\left\{g\in\operatorname{Sp}(4,{\Bbb Z})\ ;\ g-\mbox{\boldmath$1$}\in \begin{pmatrix} {\Bbb Z}&{\Bbb Z}&{\Bbb Z}&p{\Bbb Z}\\ p{\Bbb Z}&p{\Bbb Z}&p{\Bbb Z}&p^2{\Bbb Z}\\ {\Bbb Z}&{\Bbb Z}&{\Bbb Z}&p{\Bbb Z}\\ {\Bbb Z}&{\Bbb Z}&{\Bbb Z}&p{\Bbb Z} \end{pmatrix}\right\} $$ acts on Siegel space ${\Bbb H}_2$ by $$ g=\begin{pmatrix} A&B\{\Bbb C}&D\end{pmatrix} :\tau\longmapsto(A\tau+B)(C\tau+D)^{-1}. $$ In \cite{HKW} a torodial compactification ${\cal A}^*(1,p)$ of $\cal A(1,p)$ was constructed. To compactify $\cal A(1,p)$ one has to add (non-compact) boundary surfaces (corank 1 boundary points) and boundary curves (corank 2 boundary points). Here we shall restrict ourselves exclusively to the boundary surfaces. These surfaces are indexed by the vertices of the Tits building of $\Gamma_{1,p}$ which correspond to lines $l\subset{\Bbb Q}^4$. According to \cite{HKW} there are two types of boundary surfaces, namely one {\em central boundary surface} $D(l_0)$ and $p(p-1)/2$ {\em peripheral boundary surfaces} $D(l_{(a,b)})$ where $(a,b)\in({\Bbb Z}_p\times{\Bbb Z}_p\setminus\{0\})/(\pm1)$. The group $\operatorname{SL}(2,{\Bbb Z}_p)$ acts on ${\cal A}^*(1,p)$ and permutes the peripheral boundary surfaces of ${\cal A}^*(1,p)$ transitively. Therefore it is enough to consider one of them, namely $D(l_{(0,1)})$. {}From now on let $l=l_0$ or $l_{(0,1)}$. The stabilizer $P(l)$ of $l$ in $\Gamma_{1,p}$ is an extension of the form $$ 1\longrightarrow P'(l)\longrightarrow P(l)\longrightarrow P''(l)\longrightarrow 1 $$ where $P'(l)$ is a rank 1 lattice. The compactification procedure requires that one first takes the partial quotient of ${\Bbb H}_2$ with respect to $P'(l)$. For $l=l_0$ $$ P'(l)=\left\{\left( \begin{array}{c|c} \begin{array}{ccc} && \\ &\mbox{\boldmath$1$}&\\ && \end{array} & \begin{array}{ccc} n&&0\\ && \\ 0&&0\end{array} \\ \hline \begin{array}{ccc} && \\ &0& \\ &&\end{array} & \begin{array}{ccc} && \\ &\mbox{\boldmath$1$}& \\ &&\end{array} \end{array}\right)\ ;\ n\in{\Bbb Z}\right\} $$ and for $l=l_{(0,1)}$: $$ P'(l)=\left\{\left( \begin{array}{c|c} \begin{array}{ccc} && \\ &\mbox{\boldmath$1$}&\\ && \end{array} & \begin{array}{ccc} 0&&0\\ && \\ 0&&np^2\end{array} \\ \hline \begin{array}{ccc} && \\ &0& \\ &&\end{array} & \begin{array}{ccc} && \\ &\mbox{\boldmath$1$}& \\ &&\end{array} \end{array}\right)\ ;\ n\in{\Bbb Z}\right\} $$ The partial quotient map $e(l):{\Bbb H}_2\to{\Bbb H}_2/P'(l)$ is then given by $$ e(l_0):{\Bbb H}_2\longrightarrow{\Bbb C}^*\times{\Bbb C}\times{\Bbb H}_1,\ \begin{pmatrix}\tau_1&\tau_2\\ \tau_2&\tau_3 \end{pmatrix}\longmapsto(e^{2\pi i\tau_1},\tau_2,\tau_3) $$ resp. $$ e(l_{(0,1)}):{\Bbb H}_2\longrightarrow{\Bbb H}_1\times{\Bbb C}\times{\Bbb C}^*,\ \begin{pmatrix}\tau_1&\tau_2\\ \tau_2&\tau_3 \end{pmatrix}\longmapsto(\tau_1,\tau_2,e^{2\pi i\tau_3/p^2}). $$ Here ${\Bbb H}_1$ denotes the usual upper half plane. This maps ${\Bbb H}_2$ to an interior neighbourhood of $\{0\}\times{\Bbb C}\times{\Bbb H}_1$, resp. ${\Bbb H}_1\times{\Bbb C}\times\{0\}$, i.e. the interior of the closure of the image, $X(l)=(\overline{e({\Bbb H}_2)})^o$ is an open neighbourhood of $\{0\}\times{\Bbb C}\times{\Bbb H}_1$, resp. ${\Bbb H}_1\times{\Bbb C}\times\{0\}$. The partial compactification in the direction of $l$ then consists of adding the set $\{0\}\times{\Bbb C}\times{\Bbb H}_1$, resp. ${\Bbb H}_1\times{\Bbb C}\times\{0\}$. There is a natural map $X(l)\to{\cal A}^*(1,p)$ given by dividing out the extended action of $P''(l_0)$ on $X(l)$, which maps $\{0\}\times{\Bbb C}\times{\Bbb H}_1$, resp. ${\Bbb H}_1\times{\Bbb C}\times\{0\}$ to the boundary surface $D(l_0)$, resp. $D(l_{(0,1)})$. \subsection{Two 1-parameter families}\label{subsection34} We consider two 1-parameter families of $(1,p)$-polarized abelian surfaces which are closely related to the central, resp. peripheral boundary components (see proof of theorem~\ref{T}). For $M$ a positive integer we set $$ {\Bbb H}_1(M):=\{\tau\in{\Bbb H}_1\ ;\ \operatorname{Im}\tau>M\} $$ and $$ \Delta^*(M):=\{t\in{\Bbb C}\ ;\ 0<|t|<e^{-2\pi iM}\}. $$ First we fix a pair $(\tau_2,\tau_3)\in{\Bbb C}\times{\Bbb H}_1$. For sufficiently large $M$ we have a map $$ {\Bbb H}_1(M)\longrightarrow{\Bbb H}_2,\ \tau\longmapsto\begin{pmatrix} \tau&\tau_2\\\tau_2&\tau_3\end{pmatrix}. $$ The first 1-parameter family which we want to consider is the pull back of the universal family $\pi:X_D\to{\Bbb H}_2$ to a family $\pi_1:X_D\to{\Bbb H}_1(M)$ via this map. For the second family we fix a pair $(\tau_1,\tau_2)\in{\Bbb H}_1\times{\Bbb C}$ and pull back the universal family $\pi:X_D\to{\Bbb H}_2$ to a family $\pi_2:X_D\to{\Bbb H}_1(M)$ via the map $$ {\Bbb H}_1(M)\longrightarrow{\Bbb H}_2,\ \tau\longmapsto\begin{pmatrix}\tau_1&\tau_2\\ \tau_2&p^2\tau\end{pmatrix} $$ where $M$ is again chosen sufficiently large. We denote the polarized abelian surface associated to $\tau\in{\Bbb H}_1(M)$ by $(X_{\tau},E_{\tau})$. Since in both cases $(X_{\tau},E_{\tau})$ depends only on $\tau\mod{\Bbb Z}$, the family $\pi_i$ is the pull back of a family $p_i: {X}_D\to\Delta^{\ast}(M)$ via the map \[ {\Bbb H}_1(M)\longrightarrow\Delta^*(M),\ \tau\longmapsto t=e^{2\pi i\tau}. \] Consider the maps \[ \tilde{\phi}_1:{\Bbb H}_1(M)\to{\operatorname{Grass}}(2,4),\ \tau\mapsto\sqper{\tau}{\tau_3}, \] and \[ \tilde{\phi}_2: {\Bbb H}_1(M)\to{\operatorname{Grass}}(2,4),\ \tau\mapsto\sqper{\tau_1}{p^2\tau}. \] There image is contained in Griffiths' period domain ${\cal D}_2\subset{\operatorname{Grass}}(2,4)$. The monodromy $T_i$ is defined by $\sideset{^t}{}{\tilde{\phi}}_i(\tau+1)=T_i \sideset{^t}{}{\tilde{\phi}}_i(\tau)$. One easily checks that $T_i=\left(\smallmatrix {\bf 1}_2 & \eta_i\\ 0 & {\bf 1}_2\endsmallmatrix\right)$, where $\eta_1=\left( \smallmatrix 1 & 0\\ 0 & 0 \endsmallmatrix \right)$ and $\eta_2=\left( \smallmatrix 0 & 0\\ 0 & p \endsmallmatrix \right)$ By construction $\tilde{\phi}_i$ descends to a map $\phi_i: \Delta^\ast(M) \to \langle T_i \rangle \backslash {\cal D}_2$: $$ \begin{CD} {\Bbb H}_1 @>{\tilde{\phi_i}}>> {\cal D}_2\\ @V{t}VV @VVV\\ \Delta^\ast(M) @>>{\phi_i}>\langle T_i \rangle \backslash {\cal D}_2 \end{CD} $$ If we identify $H^1(X,{\Bbb C})$ with ${\Bbb C}^4$ using the coordinates induced by the basis $\{\lambda_1^{\ast},\ldots,\lambda_4^{\ast}\}$ of $H^1(X,{\Bbb Z})$), then by the description of the Hodge structure on the first cohomology of an abelian variety given in the previous section, $\phi_i$ is the VHS associated to the family $p_i:{X}_D\to\Delta^{\ast}(M)$. \subsection{The limit mixed Hodge structure}\label{subsection35} The family $p_i:{X}_D\to\Delta^{\ast}(M)$ ($i=1$,2) induces a MHS on $H^1(X_t)$ ($t\in\Delta^{\ast}(M)$ arbitrary but fixed; see \cite[Chapter IV]{G}). Its weight filtration is of the form $0=W_{-1}\subset W_0 \subset W_1 \subset W_2=H^1(X_t)$. It is is given by $$ W_0 = {\operatorname{Im}} N\text{ and } W_1 = {\operatorname{Ker}} N, $$ where $N=\log T=T-\mbox{\boldmath$1$}=\left(\smallmatrix 0 & \eta\\ 0 & 0\endsmallmatrix\right)$. (We have written $T$ instead of $T_i$ etc.) The limit Hodge filtration $F_{\infty}^{\bullet}\subset H^1(X_t)$ is calculated as follows: first define $\tilde{\psi}: {\Bbb H}_1(M) \to \check{{\cal D}}_2\subset{\operatorname{Grass}}(2,4)$ (where $\check{{\cal D}}_2$ is the compact dual of ${\cal D}_2$) by $$ \sideset{^t}{}{\tilde{\psi}}(\tau) =T^{-\tau}\sideset{^t}{}{\tilde{\phi}}(\tau)= (\mbox{\boldmath$1$}-\tau N) \sideset{^t}{}{\tilde{\phi}}(\tau). $$ By construction $\tilde{\psi}(\tau+1)=\tilde{\psi}(\tau)$, hence $\tilde{\psi}$ descends to a map $\psi: \Delta^\ast(M)\to \check{{\cal D}}_2.$ By the work of Griffiths and Schmid $\psi$ extends to $\Delta$ and $F_{\infty}^{\bullet}:=\psi(0)$ together with the monodromy weight filtration defines a MHS on $H^1(X_t)$. {}From now on we will write $e_i$ instead of $\lambda_i^{\ast}$ for the basis vectors of $H^1(X,{\Bbb Z})$. In our case, an immediate calculation shows that $\tilde{\psi}$ is constant and $$ W_0=[e_1],\quad W_1=[e_1, e_2, e_4],\quad F_{\infty}^1=\sqper{0}{\tau_3}. $$ for the family $p_1:{X}_D\to{\Bbb H}_1(M)$ and $$ W_0=[e_2],\quad W_1=[e_1, e_2, e_3],\quad F_{\infty}^1=\sqper{\tau_1}{0}. $$ for the family $p_2:{X}_D\to{\Bbb H}_1(M)$. We shall now restrict our attention to $W_1$. Here the Hodge filtration is given by $$ F_{\infty}^1\cap W_1 = [\tau_2 e_1 + \tau_3 e_2 + pe_4] $$ resp. $$ F_{\infty}^1\cap W_1 = [\tau_1 e_1 + \tau_2 e_2 + e_3]. $$ By abuse of notation we will denote this by $F_{\infty}^1$, too. Finally, recall that all elements of the nilpotent orbit $\{T^{\tau}F_{\infty}^1\}$ of Hodge filtrations on $W_2=H^1(X_t)$ induce the same MHS on $W_1$ (cf. \cite[p.84]{G}). \subsection{Calculation of $\mbox{Gr}_1^W$ and the extension class}\label{subsection36} In the previous section we computed the limit MHS on $H^1(X_t)$ determined by $p_i:X\to\Delta^*(M)$ ($i=1,2$). We now want to compute $\operatorname{Gr}_1^W$ and the extension class. For any integer $n$ and $\tau\in{\Bbb H}_1$ we define $$ X_{\tau,n}=V/\Lambda={\Bbb C} f/({\Bbb Z}\mu_1+{\Bbb Z}\mu_2) $$ where $\mu_1=\tau f$ and $\mu_2=nf$. \begin{pro}\label{P1} $\operatorname{(i)}$ In the central case ($i=1$) $\operatorname{Gr}_1^W=H^1(C)$ where $C=X_{\tau_3,p}$. \noindent $\operatorname{(ii)}$ In the peripheral case ($i=2$) $\operatorname{Gr}_1^W=H^1(C)$ where $C=X_{\tau_1,1}$. \end{pro} \begin{pf} (i) Setting $\mu_1^*=e_2$ and $\mu_2^*=e_4$ we have $$ F^1=[f^*]=[\tau_3\mu_1^*+p\mu_2^*]=[\tau_3e_2+pe_4] $$ and the claim follows immediately from the calculations of \ref{subsection35}. (ii) Setting $\mu_1^*=e_1$ and $\mu_2^*=e_3$ we have $$ F^1=[f^*]=[\tau_1\mu_1^*+\mu_2^*]=[\tau_1e_1+e_3] $$ and the result follows as before. \end{pf} Next we want to calculate the class of the extensions \begin{equation}\label{2} 0\longrightarrow W_0\stackrel{i}{\longrightarrow}W_1\stackrel{\pi}{\longrightarrow} \operatorname{Gr}_1^W\longrightarrow 0. \end{equation} The recipe for finding the extension class of (\ref{2}) demands that we choose an integral retraction $r: W_1\to W_0$ and then transfer $F_{\infty}^1$ to $W_0\oplus {\operatorname{Gr}}_1^W$ via $(r,\pi)$. In the central case we take $r(e_1)=e_1$ and $r(e_2)=r(e_4)=0$ and in the peripheral case $r(e_2)=e_2$ and $r(e_1)=r(e_3)=0$. Then, in both cases, $(r,\pi)={\bf 1}_{{\operatorname{Gr}}_1^W}$, hence $\tilde{F}_{\infty}^1=F_{\infty}^1$. Next we define the trivial extension $$ \tilde{F}_0^1 = F^1(W_0)\oplus F^1({\operatorname{Gr}}_1^W) = F^1({\operatorname{Gr}}_1^W) = [\tau_3 e_2 +pe_4] $$ for the first family and similarly \[ \tilde{F}_0^1 =[\tau_1 e_1 + e_3] \] for the second family. The extension class $e\in J^0{\operatorname{Hom}}({\operatorname{Gr}}_1^W,W_0)$ is represented by any map $\psi: {\operatorname{Gr}}_1^W \to W_0$ such that $$ \mwp{W_0}{{\operatorname{Gr}}_1^W} \tilde{F}_0^1 =\tilde{F}_{\infty}^1. $$ For the first family we can take $\psi=(\tau_2/p) e_4^{\ast}\otimes e_1$ and for the second family $\psi=\tau_2 e_3^{\ast} \otimes e_2$. After identifying $W_0={\Bbb Z}$ and ${\operatorname{Gr}}_1^W=H^1(C)$, where $C$ is as in proposition~\ref{P1}, the extension class $e=[\psi]$ is well defined in \begin{align*} \operatorname{Ext}(\operatorname{Gr}_1^W,W_0) &= J^0 {\operatorname{Hom}}({\operatorname{Gr}}_1^W,W_0)\\ &= H^1(C)^\ast / (H^{0,1}(C))^\ast + H^1(C,{\Bbb Z})^\ast\\ &= H^{1,0}(C)^\ast / H_1(C,{\Bbb Z})\\ &={\operatorname{Alb}}(C) \end{align*} and is represented by $\psi|_{H^{1,0}}$. \begin{pro}\label{P2} Under the identification $\operatorname{Ext}(\operatorname{Gr}_1^W,W_0)=\operatorname{Alb}(C)$ the extension class of (\ref{2}) corresponds to $[\tau_2f]\in\operatorname{Alb}(C)$. \end{pro} \begin{pf} Notice that if $C=V/\Lambda$, then $H^1(C,{\Bbb C})={\operatorname{Hom}}_{{\Bbb R}}(V,{\Bbb C})$ and \[ \begin{CD} H^1(C,{\Bbb C}) @>{\int_{0}^{[v]}}>> {\Bbb C}/{\operatorname{periods}}\\ @| @AAA\\ {\operatorname{Hom}}_{{\Bbb R}}(V,{\Bbb C}) @>{\operatorname{ev}(v)}>> {\Bbb C} \end{CD} \] commutes. The extension class $e\in{\operatorname{Ext}}({\operatorname{Gr}}_1^W,W_0)$ corresponds to $[\tau_2 f]\in{\operatorname{Alb}}(C)$ since \begin{align*} \int_0^{[\tau_2 f]} (\tau_3 e_2 + p e_4) &= \int_0^{[\tau_2 f]} (f^{\ast})\\ &=\tau_2\\ &=\psi(\tau_3 e_2 + p e_4) \end{align*} in the central case and \begin{align*} \int_0^{[\tau_2 f]} (\tau_1 e_1 + e_3) &= \int_0^{[\tau_2 f]} (f^{\ast})\\ &=\tau_2\\ &=\psi(\tau_1 e_2 + e_3) \end{align*} in the peripheral case. \end{pf} \subsection{The number of components}\label{sec47} In this section we will proof the following proposition. \begin{pro}\label{tnoc} Let $\pi\colon X \to \Delta$ be a semi-stable degeneration of $(1,p)$-polarized abelian surfaces with (canonical) level structure. Assume that the central fibre $X_0$ is a cycle of $N$ ruled surfaces. Then the number $N$ is determined by the polarized VHS (PVHS) on $\Delta^\ast$. \end{pro} The polarization on the VHS is induced by the polarization ${\cal L}$ in the following way: For $Y\subset X$ define \[ c_Y = c_1({\cal L}|_Y) \in H^2(Y) \] and \[ Q_Y \colon (a,b) \mapsto a \cup b \cup c_Y\colon H^1(Y) \times H^1(Y) \to H^4(Y). \] We are mostly interested in the case where $Y$ is $X$, $X_0$ or $X_t$. We write $c_0$ for $c_{X_0}$ and $Q_0$ for $Q_{X_0}$. Similarly, we write $c_t$ for $c_{X_t}$ and $Q_t$ for $Q_{X_t}$. The existence of a global level structure is needed to insure that $\deg({\cal L}|_{F_k})$, where $F_k$ is the fibre of $Y_k$, is independent of $k$. Indeed, by lemma 4 of \S2.2 it is always 1. For the rest of the section $\pi\colon X \to \Delta$ is as in proposition \ref{tnoc}. As in \S3.1 we identify $C$ with $C_0=Y_0\cap Y_1$. Let $Y_k$ and $\tau_k\colon Y_k \to C$ be as in \S3.1 and let $i\colon C\hookrightarrow X_0$ and $j_k\colon Y_k \hookrightarrow X_0$ be the inclusion maps. \begin{lem}\label{cpt} Under the isomorphism $(\tau_k^\ast)\colon H^1(C) \to {\operatorname{Gr}}_1^W$, the induced map $(j_k^\ast)\colon W_1 \to {\operatorname{Gr}}_1^W$ corresponds to the map $i^\ast\colon H^1(X_0)\to H^1(C)$ induced by the inclusion $i\colon C\hookrightarrow X_0$. \end{lem} \begin{pf} This follows immediately from $j_k^\ast=\tau_k^\ast i^\ast\colon H^1(X_0) \to H^1(Y_k)$, which in turn follows from the following facts: $\rho_k^\ast$ is the inverse of $\alpha_k^\ast$, $\sigma_k^\ast$ is the inverse of $\beta_k^\ast$, $i=i_0=j_1\beta_0$, and $j_{k+1}\beta_k= j_k\alpha_k$. \end{pf} \begin{pf1} The PVHS on $\Delta^\ast$ determines the bilinear form $Q_t$ on $W_2=H^1(X_t)$. Since the maps on cohomology induced by an inclusion $Y'\subset Y$ are compatible with $Q_Y$ and $Q_{Y'}$, $Q_t$ induces $Q_0$ on $H^1(X_0)=W_1\subset W_2$, i.e., we have a commutative diagram $$ \begin{CD} H^1(X_t) \otimes H^1(X_t) @>{Q_t}>> H^4(X_t)\\ @A{k_t^\ast\times k_t^\ast}AA @AA{k_t^\ast}A\\ H^1(X) \otimes H^1(X) @>{Q}>> H^4(X)\\ @V{k_0^\ast\times k_0^\ast}VV @VV{k_0^\ast}V\\ H^1(X_0) \otimes H^1(X_0) @>{Q_0}>> H^4(X_0). \end{CD} $$ For $t\in\Delta$ let $k_t\colon X_t\hookrightarrow X$ be the inclusion map and let $$ \delta\colon H^4(X_0) \to H^4(X_t) $$ be $k_t^\ast (k_0^\ast)^{-1}$ (recall that $k_0^\ast$ is an isomorphism). Since the monodromy is compatible with $Q_t$, i.e., $ Q_t(Ta,Tb)= Q_t(a,b), $ so is $N=T-{\bf 1}$, i.e. $ Q_t(Na,b)=Q_t(a,Nb). $ Since $W_1=$ Ker $N$ and $W_0=$ Im $N$ $ Q_t(W_1,W_0)=0. $ In other words, there exists a bilinear form $q$ on $H^1(C)$ making the diagram \begin{center} \unitlength1pt \begin{picture}(190,70) \put(0,0){\makebox(90,10){$H^1(C)\times H^1(C)$}} \put(0,50){\makebox(90,10){$H^1(X_0)\times H^1(X_0)$}} \put(130,50){\makebox(60,10){$H^4(X_t)={\Bbb Z}$}} \put(45,40){\vector(0,-1){20}} \put(5,20){\makebox(40,20){$i^\ast\times i^\ast$}} \put(100,55){\vector(1,0){20}} \put(75,20){\vector(2,1){40}} \put(100,18){$q$} \put(100,55){\makebox(20,20){$\delta Q_0$}} \end{picture} \end{center} commute. Furthermore, since $i^\ast\colon H^1(X_0) \to H^1(C)$ is surjective, $q$ is determined by $\epsilon Q_0$ where $$ \varepsilon: H^2(C)\longrightarrow H^4(X_t) $$ is the isomorphism which sends the generator of $H^2(C)$ corresponding to the canonical orientation of $C$ to the same thing on $X_t$. Let $$ \gamma:H^1(X)\times H^1(C)\longrightarrow H^2(C) $$ be the intersection form. We will show in corollary \ref{mcc} that $q=N\varepsilon\gamma$. This means that the data over $\Delta^*$ determine $N$. \end{pf1} \medskip For convenience, we shall first assume that $N\ge2$. The case $N=1$ requires minor modifications and is dealt with later. Since the line bundle determining $Y_k$ has degree $0$, there exists a continuous map $q_k\colon Y_k \to {\Bbb P}^1$ such that $$ (\tau_k,q_k)\colon Y_k \to C \times {\Bbb P}^1 $$ is a homeomorphism. Let $f\in H^2({\Bbb P}^1)$ be the canonical generator, determined by the complex structure. We define $\beta$ by demanding that the diagram $$ \begin{CD} H^2(C) @>{\beta}>> H^4(X_0)\\ @V{({\bf 1},\cdots,{\bf 1})\otimes f}VV @VV{(j_k^\ast)}V\\ \oplus H^2(C)\otimes H^2({\Bbb P}^1) @>>{(\tau_k^\ast \cup q_k^\ast)}> \oplus H^4(Y_k) \end{CD} $$ be commutative. (Notice that $(j_k^\ast)$ is an isomorphism.) \begin{lem}\label{mcl1} The diagram $$ \begin{CD} H^1(X_0)\times H^1(X_0) @>{Q_0}>> H^4(X_0)\\ @V{i^\ast\times i^\ast}VV @AA{\beta}A\\ H^1(C) \times H^1(C) @>>{\gamma}> H^2(C) \end{CD} $$ commutes. \end{lem} \begin{pf} By definition of $\beta$ we have to show that $$ \begin{CD} H^1(X_0)\times H^1(X_0) @>{Q_0}>> H^4(X_0) @>{(j_k^\ast)}>> \oplus H^4(Y_k)\\ @V{i^\ast\times i^\ast}VV @. @AA{(\tau_k^\ast\cup q_k^\ast)}A\\ H^1(C) \times H^1(C) @>>{(\gamma,\ldots,\gamma)}> \oplus H^2(C) @>>{{}\otimes f}> \oplus H^2(C) \otimes H^2({\Bbb P}^1) \end{CD} $$ commutes. In lemma 4 of \S2 we showed that $\deg({\cal L}|_{F_k})=1$ for all $k$, where $F_k$ is the ruling of $Y_k$. Hence we can write $ j_k^\ast (c_0) = \tau_k^\ast (d_k) + q_k^\ast (f) $ for some $d_k\in H^2(C)$. Since $j_k^\ast=\tau_k^\ast i^\ast$ on $H^1(X_0)$ (see proof of lemma \ref{cpt}) $$ j_k^\ast Q_0(a,b) = \tau_k^\ast i^\ast (a\cup b) \cup (\tau_k^\ast (d_k) + q_k^\ast (f)) = \tau_k^\ast i^\ast (a \cup b) \cup q_k^\ast (f), $$ because $i^\ast(a\cup b)\cup d_k\in H^4(C)=0$. \end{pf} The orientations of the components of the singular fibre are compatible with the orientation of the general fibre in the following sense: \begin{lem}\label{mcl2} If $N$ is the number of components of $X_0$, then $\delta\beta=N\epsilon$. \end{lem} \begin{pf} By definition of $\beta$, it suffices to show that the diagram $$ \begin{CD} H^4(X_0) @<{k_0^\ast}<{\sim}< H^4(X)\\ @V{(j_k^\ast)}V{\sim}V @VV{k_t^\ast}V\\ \oplus H^4(Y_k) @>>{({\bf 1},\ldots,{\bf 1})}> H^4(X_t) \end{CD} $$ commutes. The inverse of $k_0^\ast$ is induced by a retraction $r\colon X \to X_0$, which exhibits $X_0$ as a strong deformation retract of $X$ \cite{Cl, P, St}. This retraction restricts to $r_t\colon X_t\to X_0$. Let $Z_k\subset X_t$ be the inverse image of $Y_k$. Then $S_k:=Z_k\cap Z_{k+1}$ is the inverse image of $T_k:=Y_k\cap Y_{k+1}$. Pick any ${\kappa}\in\{1,\ldots,N\}$. Let $Y'=Y_{{\kappa}}$, $Y''=\cup_{k\neq {\kappa}} Y_k$, $Z'=Z_{{\kappa}}$ and $Z''=\cup_{k\neq {\kappa}} Z_k$. Let $S=Z'\cap Z''=S_{{\kappa}-1}\cup S_{{\kappa}}$ and $T=Y'\cap Y''=T_{{\kappa}-1}\cup T_{{\kappa}}$. We will need below that $H^4(Z')=H^4(Z'')=0$. To see this, we use that the retraction $r_t\colon Z_k\to Y_k$ is the real oriented blow up along $T_{k-1}\cup T_k$ \cite[p.~36]{P}. The \lq\lq exceptional divisors\rq\rq\ $S_k=r_t^{-1}(T_k)$ are $S^1$-bundles over $T_k$. They are trivial, because the triple $(Y_k, T_{k-1}, T_k)$ is homeomorphic to $(B\times S^2, B\times\{0\}, B\times\{\infty\})$, where $B=S^1\times S^1$. It follows that the triple $(Z_k,S_{k-1},S_k)$ is homeomorphic to $(B\times S^1\times [0,1], B\times S^1\times \{0\}, B\times S^1\times \{1\})$. In particular, $H^4(Z_k)=0$ and $H^3(Z_k)\to H^3(S_k)$ and $H^3(Z_k)\to H^3(S_{k-1})$ are surjective for all $k$. Mayer-Vietoris now implies that $H^4(Z'')=\oplus_{k\neq {\kappa}} H^4(Z_k)$ and this proves the claim. Consider the commutative diagram $$ \begin{CD} H^4(Y') @<<< H^4(X_0) @>{r_t^\ast}>> H^4(X_t)\\ @A{\sim}AA @AAA @AA{\sim}A\\ H^4(Y',T) @<{\sim}<< H^4(X_0,Y'') @>{\sim}>{r_t^\ast}> H^4(X_t,Z'') \end{CD} $$ The map $H^4(X_0,Y'')\to H^4(X_t,Z'')$ is an isomorphism by Alexander duality since $r_t$ is an homeomorphism $X_0\setminus Y''\to X_t\setminus Z''$ (cf.\ \cite[p.23]{La}). Similarly, $ H^4(X_0,Y'')\to H^4(Y',T)$ is an isomorphism. The left vertical map is an isomorphism because $T$ has topological dimension 2. The right vertical map is surjective because $H^4(Z'')=0$, as we have seen above. But a surjective map form ${\Bbb Z}$ to ${\Bbb Z}$ is automatically injective. It follows from \cite[p.99]{Gre} that the map $H^4(Y')\to H^4(X_0)$ via $H^4(Y',T)$ and $H^4(X_0,Y'')$ composed with the Mayer-Vietoris isomorphism $H^4(Y')\oplus H^4(Y'') \cong H^4(X_0)$ is just the inclusion of $H^4(Y')$ into $H^4(Y')\oplus H^4(Y'')$. We have to show that the composition of this map with $r_t^\ast\colon H^4(X_0)\to H^4(X_t)$ maps the orientation class to the orientation class. As before, we may identify the relative cohomology groups with the cohomology with compact supports of the respective complements. But $r_t\colon X_t\setminus Z''\to Y'\setminus T$ is a complex isomorphism. In particular, it respects the orientation. \end{pf} \begin{cor}\label{mcc} $q=N\epsilon\gamma$. \end{cor} \begin{pf} By lemma \ref{mcl1} $q=\delta\beta\gamma$. Now the result follows immediately from the previous lemma \ref{mcl2}. \end{pf} Finally, we consider the case $N=1$. We proceed as in the case $N\ge 2$. First we have to define $\beta\colon H^2(C)\to H^4(X_0)$. To do this, notice that we have an exact Mayer-Vietoris sequence $$ H^{q-1}(\tilde{C})\to H^q(X_0)\to H^q(\tilde{X_0})\oplus H^q(C)\to H^q(\tilde{C}), $$ where $\tilde{X_0}$ is the normalization of $X_0$ and $\tilde{C}=\pi^{-1}(C)$ (see \cite[p.120]{C}). Hence the normalization map $\pi\colon \tilde{X_0}\to X_0$ induces an isomorphism $H^4(X_0)\to H^4(\tilde{X_0})$. We identify $H^4(X_0)$ with $H^4(\tilde{X_0})$ via $\pi^\ast$ and define $\beta$ by demanding that it maps the orientation class of $C$ to the orientation class of $\tilde{X_0}$. With this definition of $\beta$ and using that the degree $\deg(\pi^\ast({\cal L})|_F)$ of ${\cal L}$ along the fibre $F$ in the normalization is 1 again by lemma 4 of \S2, the diagram in lemma~\ref{mcl1} again commutes. The proof is an easy adaption of the proof given for $N\ge2$ and is left to the reader. To prove lemma \ref{mcl2} for $N=1$, one argues exactly as in the case $N\ge2$ using the commutative diagram $$ \begin{CD} H^4(\tilde{X_0}) @<<< H^4(X_0) @>{r_t^\ast}>> H^4(X_t)\\ @AAA @AAA @AAA\\ H^4(\tilde{X_0},\tilde{C}) @<<< H^4(X_0,C) @>{r_t^\ast}>> H^4(X_t,S), \end{CD} $$ where $S=r_t^{-1}(C)$. (This time, all maps are isomorphisms.) \section{A uniqueness result}\label{section5} Here we combine the results of sections \ref{section3} and \ref{section4} to prove a uniqueness result for degenerate abelian surfaces. In (\ref{subsection34}) we discussed central and peripheral corank 1 boundary points of ${\cal A}^*(1,p)$. Note that the universal family $\pi:X_D\to{\Bbb H}_2$ descends to a family $\pi':X_D\to e({\Bbb H}_2)$. By \cite[part~II.4]{HKW} this family can be extended to a family $\bar\pi:\bar{X}_D\to X(l)$. The fibres of $\bar{X}_D$ over the ''boundary'' $\{0\}\times{\Bbb C}\times{\Bbb H}_1$, resp. ${\Bbb H}_1\times{\Bbb C}\times\{0\}$ are cycles of elliptic ruled surfaces with $N=1$, resp. $N=p$. \begin{dfn}\label{D} Let $[q]\in{\cal A}^*(1,p)$ be a corank 1 boundary point. We say that a surface $X_0$ is a {\em degenerate abelian surface associated to} $[q]$ if the following holds: There exists a 1-parameter degeneration of polarized abelian surfaces with (canonical) level structure $p:X\to\Delta$ as in section 2.2 with central fibre $p^{-1}(0)=X_0$ and an embedding $f:\Delta\to X(l)$ such that: \noindent (i) $f(\Delta)$ meets $\{0\}\times{\Bbb C}\times{\Bbb H}_1$ resp. ${\Bbb H}_1\times{\Bbb C}\times\{0\}$ transversely in the point $q=f(0)$ which is mapped to $[q]$. \noindent (ii) The restriction of $p:X\to\Delta$ to $\Delta^*$ is the pull back of the family $\pi':X_D\to e({\Bbb H}_2)$ via $f$. \end{dfn} \begin{rem}\normalshape The reason why we work with $X(l)$ rather than with the compactification ${\cal A}^*(1,p)$ itself is that the map ${\Bbb H}_2\to\cal A(1,p)$ is branched over two Humbert surfaces $H_1$ and $H_2$. Near these surfaces, resp. their closure in ${\cal A}^*(1,p)$ one has no universal family. \end{rem} \begin{lem}\label{lem25} If $X_0$ is a degeneration with only local normal crossing singularities associated to a corank 1 boundary point $[q]$, then the MHS on $H^1(X_0)$ is determined by this point. \end{lem} \begin{pf} This follows from the Local Invariant Cycle theorem \cite[Ch.~VI]{G} (which also holds when we have local normal crossing rather than global normal crossing) together with the existence of the extended family $\bar{\pi}:\bar{X}_D\to X(l)$: Indeed, compare the family $p:X\to\Delta$ as in definition (\ref{D}) with the family $f^{\ast}\bar{X}_D\to\Delta$. On $\Delta^{\ast}$ these two families agree (they are both the pull back of the universal family), hence they determine the same VHS and thus the same limit MHS on $H^1(X_t)$ ($t\in\Delta^{\ast}$ arbitrary but fixed). But by the Local Invariant Cycle theorem the MHS on the central fibre is determined by this limit MHS at least over ${\Bbb Q}$: more precisely, $H^1(X_0)=W_1(H^1(X_t))=H^1(\bar{\pi}^{-1}(q))$ as ${\Bbb Q}-$MHS. It remains to show that $$ 0\longrightarrow H^1(X)\longrightarrow H^1(X_t)\stackrel{N}{\longrightarrow} H^1(X_t) $$ is exact over ${\Bbb Z}$. For $n\gg0$ the zero locus $Y\subset X$ of a general element of $\Gamma (X,{\cal L}^n)$ is a semi-stable degeneration of the smooth curve $Y_t=Y\cap X_t$. By the Picard-Lefschetz theorem \cite[theorem III.~14.1]{BPV} $$ 0\longrightarrow H^1(Y)\longrightarrow H^1(Y_t)\stackrel{N}{\longrightarrow} H^1(Y_t) $$ is exact over ${\Bbb Z}$. By the Lefschetz hyperplane theorem $H^1(X_t)\longrightarrow H^1(Y_t)$ is injective, hence so is $H^1(X)\longrightarrow H^1(Y)$. We have now reduced our problem to showing that $H^1(Y)/H^1(X)$ is torsion free. Since $H^1(Y)/H^1(X)\hookrightarrow H^2(X,Y)$ it is sufficient to show that $H^2(X,Y)$ or, equivalently $H_1(X,Y)=\text{Coker } (H_1(Y)\longrightarrow H_1(X))$ is torsion free. Since dim ker $N=3$ the central fibre $X_0$ is not smooth abelian. Hence by Persson's theorem $X_0$ is a cycle of elliptic ruled surfaces. Assume that $X_0$ has at least two components. (The remaining case is analogous.) Let $C$ be one of the double curves and $X_i$ one of the components of $X_0$. Then $$ H_1(X)=H_1(X_0)\cong H_1(C)\oplus {\Bbb Z} c $$ where $c\in H_1(X_0)$ is as in 3.3. Let $Y_i=X_i\cap Y$. Then $X_i$ and $Y_i$ are smooth and irreducible. We have a commutative diagram $$ \begin{CD} H_1(C)= @. H_1(X_i) @>>> H_1(X)\\ @. @AAA @AAA\\ @. H_1(Y_i) @>>> H_1(Y). \end{CD} $$ Since $c$ is clearly induced from $Y$ up to $H_1(C)$, it is sufficient to prove that $$ H_1(Y_i)\longrightarrow H_1(X_i) $$ is surjective. But this follows from the Lefschetz hyperplane theorem since $Y_i$ is ample on $X_i$. This shows that $H^1(X_0)=W_1(H^1(X_t))$ depends only on the point$[q]$. \end{pf} \begin{thm}\label{T} Let $X_0$ be a degeneration associated to a corank 1 boundary point $[q]$ of ${\cal A}^*(1,p)$ with only local normal crossing singularities and no triple points. Then $X_0$ is a cycle of ruled surfaces (possibly with only one component) over an elliptic curve $C$. There exists a ${\cal L}\in{\operatorname{Pic}}^0(C)$ such that all components are isomorphic to ${\Bbb P}^1({\cal O}\oplus{\cal L})$. If $q$ is a central boundary point then $X_0$ is completely determined by $q$. If $q$ is peripheral, then the number $N$ of components, the base curve $C$ andthe shift $s$ are uniquely determined by the point $[q]$. The line bundle ${\cal L}\in{\operatorname{Pic}}^0(C)$ determining the elliptic ruled surfaces of the cycle is determined (at least) up to a $p$-torsion point of ${\operatorname{Pic}}^0(C)$. More precisely: \begin{enumerate} \item If $[q]=[\tau_2,\tau_3]$ is a central boundary point, then $N=1$, $C=X_{\tau_3,p}$, $s=[\tau_2]$ and ${\cal L}={\cal O}(s-O)$. \item If $[q]=[\tau_1,\tau_2]$ is a peripheral boundary point, then $N=p$, $C=X_{\tau_1,1}$, $s=[\tau_2]$ and there exists a point $s'\in C$ such that ${\cal L}={\cal O}(s'-O)$. \end{enumerate} \end{thm} \begin{pf} We first note that the pair $(X_{\tau_3,p},[\tau_2])$, resp. $(X_{\tau_1,1},[\tau_2])$ only depends on the point $[q]$ but not on its representative \cite[part~I.3]{HKW}. If $X_0$ has only global normal crossings then Persson's result \cite[proposition 3.3.1]{P} says that $X_0$ is a cycle of elliptic ruled surfaces. The previous lemma asserts that the MHS on $H^1(X_0)$ is determined by $q$. Furthermore, this MHS is the weight 1 part of the limit MHS on $H^1(X_t)$ of the family $p_1$ (resp. $p_2$) defined in \S4.4 in the central (resp. peripheral) case. Indeed, $p_1$ (resp. $p_2$) is the restriction of the universal family to a small disk $\Delta\times (\tau_2,\tau_3)$ (resp. $(\tau_1,\tau_2)\times\Delta$). By lemma~\ref{wf} the base curve $C$ is determined by the equation $H^1(C)= {\operatorname{Gr}}_1^W(H^1(X_0))={\operatorname{Gr}}_1^W(H^1(X_t))$ and this equation was solved in proposition~\ref{P1}. Corollary~\ref{cor17} shows how to recover the shift $s$ from $H^1(X_0)=W_1(H^1(X_t))$ and the result for the families $p_1$ and $p_2$ is given in proposition~\ref{P2}. Applying proposition \ref{tnoc} of \S\ref{sec47} to the family $X$ defining $X_0$ and to the family constructed in \cite{HKW}, one sees that the number of components of $X_0$ is independent of the choice of the disk $\Delta\hookrightarrow X(l)$ (cf.\ proof of lemma \ref{lem25}). But it follows from our calculations in \S4 that the polarization induces $\left(\begin{array}{cc} 0& -1\\1&0\end{array}\right)$ (with respect to the basis $\{e_2,e_4\}\bmod e_1$ of \S\ref{subsection35}) in the central case and $\left(\begin{array}{cc} 0& -p\\p&0\end{array}\right)$ (with respect to the basis $\{e_1,e_3\}\bmod e_2$) in the peripheral case. It follows from proposition \ref{tnoc} of \S\ref{sec47} that the number of components is 1 and $p$ respectively. The statement about the form of ${\cal L}\in{\operatorname{Pic}}^0(C)$ now follows directly from proposition 6 of \S2.2 \end{pf} \begin{rem}\normalshape Note that in the proof of theorem \ref{T} the level structure is not needed to determine $C$ and $s$. \end{rem} \begin{rem} \normalshape It seems reasonable to expect that also in the peripheral case ${\cal L}$ is completely determined by the boundary point $q$. To show this in the spirit of this paper, one probably has to translate the notion of (canonical) level structure to Hodge structures. \end{rem} \begin{rem}\normalshape This result is compatible with \cite{HKW}. Notice that we only used the existence of the degeneration $\bar{X}_D$ as constructed there in order to prove our uniqueness result, but not the precise description of its singular fibres. \end{rem}

9404

alg-geom/9404010

en

https://arxiv.org/abs/alg-geom/9404010

alg-geom/9404010

Robert Friedman

Robert Friedman and Zhenbo Qin

On complex surfaces diffeomorphic to rational surfaces

34 pages, AMS-TeX

10.1007/BF01241123

OSU Math 1995-10

In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.

alg-geom

\section{Introduction} The goal of this paper is to prove the following: \theorem{0.1} Let $X$ be a complex surface of general type. Then $X$ is not diffeomorphic to a rational surface. \endstatement Using the results from [13], we obtain the following corollary, which settles a problem raised by Severi: \corollary{0.2} If $X$ is a complex surface diffeomorphic to a rational surface, then $X$ is a rational surface. Thus, up to deformation equivalence, there is a unique complex structure on the smooth $4$-manifolds $S^2\times S^2$ and $\Pee^2 \# n\overline{\Pee}^2$. \endstatement In addition, as discussed in the book [15], Theorem 0.1 is the last step in the proof of the following, which was conjectured by Van de Ven [37] (see also [14,15]): \corollary{0.3} If $X_1$ and $X_2$ are two diffeomorphic complex surfaces, then $$\kappa (X_1) = \kappa (X_2),$$ where $\kappa (X_i)$ denotes the Kodaira dimension of $X_i$. \endstatement The first major step in proving that every complex surface diffeomorphic to a rational surface is rational was Yau's theorem [40] that every complex surface of the same homotopy type as $\Pee^2$ is biholomorphic to $\Pee^2$. After this case, however, the problem takes on a different character: there do exist nonrational complex surfaces with the same oriented homotopy type as rational surfaces, and the issue is to show that they are not in fact diffeomorphic to rational surfaces. The only known techniques for dealing with this question involve gauge theory and date back to Donaldson's seminal paper [9] on the failure of the $h$-cobordism theorem in dimension 4. In this paper, Donaldson introduced analogues of polynomial invariants for 4-manifolds $M$ with $b_2^+(M) = 1$ and special $SU(2)$-bundles. These invariants depend in an explicit way on a chamber structure in the positive cone in $H^2(M; \Ar)$. Using these invariants, he showed that a certain elliptic surface (the Dolgachev surface with multiple fibers of multiplicities 2 and 3) was not diffeomorphic to a rational surface. In [13], this result was generalized to cover all Dolgachev surfaces and their blowups (the case of minimal Dolgachev surfaces was also treated in [28]) and Donaldson's methods were also used to study self-diffeomorphisms of rational surfaces. The only remaining complex surfaces which are homotopy equivalent (and thus homeomorphic) to rational surfaces are then of general type, and a single example of such surfaces, the Barlow surface, is known to exist [2]. In 1989, Kotschick [18], as well as Okonek-Van de Ven [29], using Donaldson polynomials associated to $SO(3)$-bundles, showed that the Barlow surface was not diffeomorphic to a rational surface. Subsequently Pidstrigach [30] showed that no complex surface of general type which has the same homotopy type as the Barlow surface was diffeomorphic to a rational surface, and Kotschick [20] has outlined an approach to showing that no blowup of such a surface is diffeomorphic to a rational surface. All of these approaches use $SO(3)$-invariants or $SU(2)$-invariants for small values of the (absolute value of) the first Pontrjagin class $p_1$ of the $SO(3)$-bundle, so that the dependence on chamber structure can be controlled in a quite explicit way. In [33], the second author showed that no surface $X$ of general type could be diffeomorphic to $\Pee ^1\times \Pee ^1$ or to $\Bbb F_1$, the blowup of $\Pee^2$ at one point. Here the main tool is the study of $SO(3)$-invariants for large values of $-p_1$, as defined and analyzed in [19] and [21]. These invariants also depend on a chamber structure, in a rather complicated and not very explicitly described fashion. In [34], these methods are used to analyze minimal surfaces $X$ of general type under certain assumptions concerning the nonexistence of rational curves, which are always satisfied if $X$ has the same homotopy type as $\Pee ^1\times \Pee ^1$ or $\Bbb F_1$, by a theorem of Miyaoka on the number of rational curves of negative self-intersection on a minimal surface of general type. The main idea of the proof is to show the following: Let $X$ be a minimal surface of general type, and suppose that $\{E_0, \dots, E_n\}$ is an orthogonal basis for $H^2(X; \Zee)$ with $E_0^2 =1$, $E_i^2 =-1$ for $i\geq 1$, and $[K_X] = 3E_0 -\sum _{i\geq 1}E_i$. Finally suppose that the divisor $E_0 - E_i$ is nef for some $i\geq 1$. Then the class $E_0 - E_i$ cannot be represented by a smoothly embedded 2-sphere. (Actually, in [34], the proof shows that an appropriate Donaldson polynomial is not zero whereas it must be zero if $X$ is diffeomorphic to a rational surface. However, using [26], one can also show that if $E_0 - E_i$ is represented by a smoothly embedded 2-sphere, then the Donaldson polynomial is zero.) At the same time, building on ideas of Donaldson, Pidstrigach and Tyurin [31], using Spin polynomial invariants, showed that no minimal surface of general type is diffeomorphic to a rational surface. We now discuss the contents of this paper and the general strategy for the proof of Theorem 0.1. The bulk of this paper is devoted to giving a new proof of the results of Pidstrigach and Tyurin concerning minimal surfaces $X$. Here our methods apply as well to minimal simply connected algebraic surfaces of general type with $p_g$ arbitrary. Instead of looking at embedded 2-spheres of self-intersection 0 as in [34], we consider those of self-intersection $-1$. We show in fact the following in Theorem 1.10 (which includes a generalization for blowups): \theorem{0.4} Let $X$ be a minimal simply connected algebraic surface of general type, and let $E\in H^2(X; \Zee)$ be a class satisfying $E^2 = -1$, $E\cdot [K_X] = 1$. Then the class $E$ cannot be represented by a smoothly embedded $2$-sphere. \endstatement In particular, if $p_g(X) = 0$, then $X$ cannot be diffeomorphic to a rational surface. The method of proof of Theorem 0.4 is to show that a certain value of a Donaldson polynomial invariant for $X$ is nonzero (Theorem 1.5), while it is a result of Kotschick that if the class $E$ is represented by a smoothly embedded 2-sphere, then the value of the Donaldson polynomial must be zero (Proposition 1.1). In case $p_g(X) =0$, once we have have found a polynomial invariant which distinguishes $X$ from a rational surface, it follows in a straightforward way from the characterization of self-diffeomorphisms of rational surfaces given in [13] that no blowup of $X$ can be diffeomorphic to a rational surface either (see Theorem 1.7). This part of the argument could also be used with the result of Pidstrigach and Tyurin to give a proof of Theorem 0.1. Let us now discuss how to show that certain Donaldson polynomials do not vanish on certain classes. The prototype for such results is the nonvanishing theorem of Donaldson [10]: if $S$ is an algebraic surface with $p_g(S) > 0$ and $H$ is an ample line bundle on $S$, then for all choices of $w$ and all $p \ll 0$, the $SO(3)$-invariant $\gamma _{w,p}(H, \dots, H) \neq 0$. We give a generalization of this result in Theorem 1.4 to certain cases where $H$ is no longer ample, but satisfies: $H^k$ has no base points for $k\gg 0$ and defines a birational morphism from $X$ to a normal surface $\bar X$, and where $p_g(X)$ is also allowed to be zero (for an appropriate choice of chamber). Here we must assume that there is no exceptional curve $C$ such that $H\cdot C=0$, as well as the following additional assumption concerning the singularities of $\bar X$: they should be rational or minimally elliptic in the terminology of [22]. The proof of Theorem 1.4 is a straightforward generalization of Donaldson's original proof, together with methods developed by J\. Li in [23, 24]. Given the generalized nonvanishing theorem, the problem becomes one of constructing divisors $M$ such that $M$ is orthogonal to a class $E$ of square $-1$ and moreover such that $M$ is eventually base point free. (Here we recall that a divisor $M$ is {\sl eventually base point free\/} if the complete linear system $|kM|$ has no base points for all $k\gg 0$.) There are various methods for finding base point free linear systems on an algebraic surface. For example, the well-studied method of Reider [35] implies that, if $X$ is a minimal surface of general type and $D$ is a nef and big divisor on $X$, then $M= K_X+D$ is eventually base point free. There is also a technical generalization of this result due to Kawamata [16]. However, the methods which we shall need are essentially elementary. The general outline of the construction is as follows. Let $E$ be a class of square $-1$ with $K_X \cdot E=1$. It is known that, if $E$ is the class of a smoothly embedded $2$-sphere, then $E$ is of type $(1,1)$ [6]. Thus $K_X+E$ is a divisor orthogonal to $E$. If $K_X+E$ is ample we are done. If $K_X+E$ is nef but not ample, then there exist curves $D$ with $(K_X+E)\cdot D=0$, and the intersection matrix of the set of all such curves is negative definite. Thus we may contract the set of all such curves to obtain a normal surface $X'$. If $X'$ has only rational singularities, then the divisor $K_X+E$ induces a Cartier divisor on $X'$ which is ample, by the Nakai-Moishezon criterion, and so some multiple of $K_X+E$ is base point free. Next suppose that $X'$ has a nonrational singular point $p$ and let $D_1, \dots, D_t$ be the irreducible curves on $X$ mapped to $p$. Then we give a dual form of Artin's criterion [1] for a rational singularity, which says the following: the point $p$ is a nonrational singularity if and only if there exist nonnegative integers $n_i$, with at least one $n_i >0$, such that $(K_X+\sum _in_iD_i)\cdot D_j \geq 0$ for all $j$. Moreover there is a choice of the $n_i$ such that either the inequality is strict for every $j$ or the contraction of the $D_j$ with $n_j \neq 0$ is a minimally elliptic singularity. In this case, provided that $K_X$ is itself nef, it is easy to show that $K_X+\sum _in_iD_i$ is nef and big and eventually base point free, and defines the desired contraction. The remaining case is when $K_X+E$ is not nef. In this case, by considering the curves $D$ with $(K_X+E)\cdot D<0$, it is easy to find a $\Bbb Q$-divisor of the form $K_X+\lambda D$, where $D$ is an irreducible curve and $\lambda \in \Bbb Q^+$, which is nef and big and such that some multiple is eventually base point free, and which is orthogonal to $E$. The details are given in Section 3. These methods can also handle the case of elliptic surfaces (the case where $\kappa (X) =1$), but of course there are more elementary and direct arguments here which prove a more precise result. We have included an appendix giving a proof, due to the first author, R\. Miranda, and J\.W\. Morgan, of a result characterizing the canonical class of a rational surface up to isometry. This result seems to be well-known to specialists but we were unable to find an explicit statement in the literature. It follows from work of Eichler and Kneser on the number of isomorphism classes of indefinite quadratic forms of rank at least 3 within a given genus (see e\.g\. [17]) together with some calculation. However the proof in the appendix is an elementary argument. The methods in this paper are able to rule out the possibility of embedded 2-spheres whose associated class $E$ satisfies $E^2 = -1$, $E\cdot [K_X] = 1$. However, in case $p_g(X) = 0$ and $b_2(X) \geq 3$, there are infinitely many classes $E$ of square $-1$ which satisfy $|E\cdot K_X| \geq 3$. It is natural to hope that these classes also cannot be represented by smoothly embedded 2-spheres. More generally we would like to show that the surface $X$ is strongly minimal in the sense of [15]. Likewise, in case $p_g(X) >0$, we have only dealt with the first case of the ``$(-1)$-curve conjecture" (see [6]). \medskip \noindent {\bf Acknowledgements:} We would like to thank Sheldon Katz, Dieter Kotschick, and Jun Li for valuable help and stimulating discussions. \section{1. Statement of results and overview of the proof} \ssection{1.1. Generalities on $\boldkey S \boldkey O\boldkey (\boldkey 3\boldkey )$-invariants} Let $X$ be a smooth simply connected $4$-manifold with $b_2^+(X) = 1$, and fix an $SO(3)$-bundle $P$ over $X$ with $w_2(P) = w$ and $p_1(P) = p$. Recall that a {\sl wall of type $(w,p)$ for $X$} is a class $\zeta \in H^2(S; \Zee)$ such that $\zeta \equiv w \mod 2$ and $p \leq \zeta ^2 <0$. Let $$\Omega _X = \{x\in H^2(X; \Ar): x^2 >0\}.$$ Let $W^\zeta = \Omega _X \cap (\zeta)^\perp$. A {\sl chamber of type $(w, p)$ for $X$} is a connected component of the set $$\Omega _X - \bigcup\{W^\zeta: \zeta {\text{ is a wall of type $(w,p)$}}\, \}.$$ Let $\Cal C$ be a chamber of type $(w,p)$ for $X$ and let $\gamma_{w,p}(X;\Cal C)$ denote the associated Donaldson polynomial, defined via [19] and [21]. Here $\gamma_{w,p}(X;\Cal C)$ is only defined up to $\pm 1$, depending on the choice of an integral lift for $w$, corresponding to a choice of orientation for the moduli space. The actual choice of sign will not matter, since we shall only care if a certain value of $\gamma_{w,p}(X;\Cal C)$ is nonzero. In the complex case we shall always assume for convenience that the choice has been made so that the orientation of the moduli space agrees with the complex orientation. Via Poincar\'e duality, we shall view $\gamma_{w,p}(X;\Cal C)$ as a function on either homology or cohomology classes. Given a class $M$, we use the notation $\gamma_{w,p}(X;\Cal C) (M^d)$ for the evaluation $\gamma_{w,p}(X;\Cal C) (M, \dots, M)$ on the class $M$ repeated $d$ times, where $d= -p-3$ is the expected dimension of the moduli space. We then have the following vanishing result for $\gamma_{w,p}(\Cal C)$, due to Kotschick [19, (6.13)]: \proposition{1.1} Let $E \in H^2(X; \Bbb Z)$ be the cohomology class of a smoothly embedded $S^2$ in $X$ with $E^2 = -1$. Let $w$ be the second Stiefel-Whitney class of $X$, or more generally any class in $H^2(X, \Zee/2\Zee)$ such that $w\cdot E \neq 0$. Suppose that $M\in H_2(X; \Zee)$ satisfies $M^2 > 0$ and $M \cdot E = 0$. Then, for every chamber $\Cal C$ of type $(w, p)$ such that the wall $W^E$ corresponding to $E$ passes through the interior of $\Cal C$, $$\gamma_{w,p}(X;\Cal C)(M^d) = 0. \qed$$ \endproclaim Note that if $w$ is the second Stiefel-Whitney class of $X$, then $W^E$ is a wall of type $(w,p)$ (and so does not pass through the interior of any chamber) if and only if $E^\perp$ is even. This case arises, for example, if $X$ has the homotopy type of $(S^2\times S^2)\#\overline{\Pee}^2$ and $E$ is the standard generator of $H^2(\overline{\Pee}^2; \Zee) \subseteq H^2(X; \Bbb Z)$. For the proof of Theorem 0.1, the result of (1.1) is sufficient. However, for the slightly more general result of Theorem 1.10, we will also need the following variant of (1.1): \theorem{1.2} Let $E \in H^2(X; \Bbb Z)$ be the cohomology class of a smoothly embedded $S^2$ in $X$ with $E^2 = -1$. Let $w$ be a class in $H^2(X, \Zee/2\Zee)$ such that $w\cdot E \neq 0$. Suppose that $M\in H_2(X; \Zee)$ satisfies $M^2 > 0$ and $M \cdot E = 0$. Then, for every chamber $\Cal C$ of type $(w, p)$ containing $M$ in its closure, $$\gamma_{w,p}(X;\Cal C)(M^d) = 0,$$ unless $p=-5$ and $w$ is the \rom{mod} $2$ reduction of $E$. Thus, except in this last case, $\gamma_{w,p}(X;\Cal C)$ is divisible by $E$. \endproclaim \proof If $W^E$ is not a wall of type $(w,p)$ we are done by (1.1). Otherwise, $E$ defines a wall of type $(w,p)$ containing $M$. Next let us assume that $E^\perp \cap \overline{\Cal C}$ is a codimension one face of the closure $\overline{\Cal C}$ of $\Cal C$. We have an induced decomposition of $X$: $$X = X_0 \# \overline{\Pee}^2.$$ Identify $H_2(X_0; \Bbb Z)$ with the subspace $E^\perp$ of $H_2(X; \Bbb Z)$, and let $\overline{\Cal C}_0 = E^\perp \cap \overline{\Cal C}$. Then $\overline{\Cal C}_0$ is the closure of some chamber ${\Cal C}_0$ of type $(w - e, p + 1)$ on $X_0$, where $e$ is the mod 2 reduction of $E$. Choose a generic Riemannian metric $g_0$ on $X_0$ such that the cohomology class $\omega _0$ of the self-dual harmonic 2-form associated to $g_0$ lies in the interior of $\Cal C_0$. By the results in [39], there is a family of metrics $h_t$ on the connected sum $X_0\# \overline{\Cee P}^2$ which converge in an appropriate sense to $g_0\amalg g_1$, where $g_1$ is the Fubini-Study metric on $\overline{\Pee}^2$, and such that the cohomology classes of the self-dual harmonic 2-forms associated to $h_t$ lie in the interior of $\Cal C$ and converge to $\omega _0$. Standard gluing and compactness arguments (see for example [15], Appendix to Chapter 6) and dimension counts show that the restriction of the invariant $\gamma_{w,p}(X;\Cal C)$ to $H_2(X_0; \Bbb Z)$ vanishes. Consider now the general case where $W^E$ is a wall of type $(w,p)$ and the closure of $\Cal C$ contains $M$ but where $W^E\cap \overline{\Cal C}$ is not necessarily a codimension one face of $\overline{\Cal C}$. Since $W^E$ is a wall of type $(w,p)$ and $M\in E^\perp$, there exists a chamber $\Cal C'$ of type $(w, p)$ whose closure contains $M$ such that $W^E$ is a codimension one face of $\overline{\Cal C}'$. By the previous argument, $\gamma_{w,p}(X;\Cal C')(M^d) =0$ and so it will suffice to show that $$\gamma_{w,p}(X;\Cal C)(M^d) = \gamma_{w,p}(X;\Cal C')(M^d).$$ Note that $\Cal C$ and $\Cal C'$ are separated by finitely many walls of type $(w, p)$ all of which contain the class $M$. Thus, we have a sequence of chambers of type $(w, p)$: $$\Cal C = \Cal C_1, \Cal C_2, \dots, \Cal C_{k - 1}, \Cal C_k = \Cal C'$$ such that for each $i$, $\Cal C_{i - 1}$ and $\Cal C_i$ are separated by a single wall $W_i=W^{\zeta_i}$ of type $(w, p)$ which contains $M$. Since $W_i$ contains $M$, $M \cdot \zeta_i = 0$. By [19, (3.2)(3)] (see also [21]), the difference $\gamma_{w,p}(X;\Cal C_{i - 1}) - \gamma_{w,p}(X;\Cal C_i)$ is divisible by the class $\zeta_i$ except in the case where $p=-5$ and $w$ is the mod 2 reduction of $E$. It follows that, except in this last case, for each $i$, $$\gamma_{w,p}(X;\Cal C_{i - 1})(M^d) = \gamma_{w,p}(X;\Cal C_i)(M^d).$$ Hence $\gamma_{w,p}(X;\Cal C)(M^d) = \gamma_{w,p}(X;\Cal C')(M^d)=0$. \endproof We shall also need the following ``easy" blowup formula: \lemma{1.3} Let $X\#\overline{\Pee}^2 $ be a blowup of $X$, and identify $H_2(X; \Bbb Z)$ with a subspace of $H_2(X\#\overline{\Pee}^2; \Bbb Z)$ in the natural way. Given $w\in H^2(X; \Zee/2\Zee)$, let $\tilde \Cal C$ be a chamber of type $(w, p)$ for $X\#\overline{\Pee}^2$ containing the chamber $\Cal C$ in its closure. Then $$\gamma _{w,p}(X\#\overline{\Pee}^2;\tilde\Cal C) |H_2(X; \Bbb Z) = \pm\gamma_{w,p}(X;\Cal C).$$ \endproclaim \proof Choose a generic Riemannian metric $g$ on $X$ such that the cohomology class $\omega$ of the self-dual harmonic 2-form associated to $g$ lies in the interior of $\Cal C$. We again use the results in [39] to choose a family of metrics $h_t$ on the connected sum $X\# \overline{\Pee}^2$ which converge in an appropriate sense to $g\amalg g'$, where $g'$ is the Fubini-Study metric on $\overline{\Pee}^2$, and such that the cohomology classes of the self-dual harmonic 2-forms associated to $h_t$ lie in the interior of $\tilde\Cal C$ and converge to $\omega$. Standard gluing and compactness arguments (see e\.g\. [15], Chapter 6, proof of Theorem 6.2(i)) now show that the restriction of $\gamma _{w,p}(X\#\overline{\Pee}^2;\tilde\Cal C)$ to $H_2(X; \Bbb Z)$ (with the appropriate orientation conventions) is just $\gamma_{w,p}(X;\Cal C)$. \endproof \ssection{1.2. The case of a minimal $\boldkey X$} In this subsection we shall outline the results to be proved concerning minimal surfaces of general type. One basic tool is a nonvanishing theorem for certain values of the Donaldson polynomial: \theorem{1.4} Let $X$ be a simply connected algebraic surface with $p_g(X) =0$, and let $M$ be a nef and big divisor on $X$ which is eventually base point free. Denote by $\varphi\: X \to \bar X$ the birational morphism defined by $|kM|$ for $k\gg 0$, so that $\bar X$ is a normal projective surface. Suppose that $\bar X$ has only rational or minimally elliptic singularities, and that $\varphi$ does not contract any exceptional curves to points. Let $w\in H^2(X;\Zee/2\Zee)$ be the \rom{mod} $2$ reduction of the class $[K_X]$. Then there exists a constant $A$ depending only on $X$ and $M$ with the following property: For all integers $p\leq A$, let $\Cal C$ be a chamber of type $(w,p)$ containing $M$ in its closure and suppose that $\Cal C$ has nonempty intersection with the ample cone of $X$. Set $d = -p-3$. Then $$\gamma_{w,p}(X;\Cal C)(M^d) >0.$$ \endstatement We shall prove Theorem 1.4 in Section 2, where we shall also recall the salient properties of rational and minimally elliptic singularities. The proof also works in the case where $p_g(X)>0$, in which case $\gamma_{w,p}(X)$ does not depend on the choice of a chamber. We can now state the main result concerning minimal surfaces, which we shall prove in Section 3: \theorem{1.5} Let $X$ be a minimal simply connected algebraic surface of general type, and let $E\in H^2(X; \Zee)$ be a $(1,1)$-class satisfying $E^2=-1$, $E\cdot K_X = 1$. Let $w$ be the \rom{mod} $2$ reduction of $[K_X]$. Then there exist: \roster \item"{(i)}" an integer $p$ and \rom(in case $p_g(X)=0$\rom) a chamber $\Cal C$ of type $(w,p)$ and \item"{(ii)}" a $(1,1)$-class $M\in H^2(X; \Zee)$ \endroster such that $M\cdot E=0$ and $\gamma _{w,p}(X)(M^d) \neq 0$ \rom(or, in case $p_g(X)=0$, $\gamma _{w,p}(X; \Cal C)(M^d) \neq 0$\rom). \endstatement The method of proof of (1.5) will be the following: we will show that there exists an orientation preserving self-diffeomorphism $\psi$ of $X$ with $\psi ^*[K_X] = [K_X]$ and a nef and big divisor $M$ on $X$ such that: \roster \item"{(i)}" $M\cdot \psi ^*E = 0$. \item"{(ii)}" $M$ is eventually base point free, and the corresponding contraction $\varphi\: X \to \bar X$ maps $X$ birationally onto a normal surface $\bar X$ whose only singularities are either rational or minimally elliptic. \endroster Using the naturality of $\gamma _{w,p}(X;\Cal C)$, it suffices to prove (1.5) after replacing $E$ by $\psi ^*E$. In this case, by Theorem 1.4 with $w= [K_X]$, $\gamma _{w,p}(X;\Cal C)(M^d) \neq 0$ for all $p \ll 0$. \corollary{1.6} Let $X$ be a simply connected minimal surface of general type with $p_g(X)=0$. Then there exist \roster \item"{(i)}" a class $w\in H^2(X; \Zee/2\Zee)$; \item"{(ii)}" an integer $p\in \Zee$; \item"{(iii)}" a chamber $\Cal C$ for $X$ of type $(w,p)$, and \item"{(iv)}" a homotopy equivalence $\alpha: X \rightarrow Y$, where $Y$ is either the blowup of ${\Pee}^2$ at $n$ distinct points or $Y=\Pee ^1\times \Pee ^1$, \endroster such that, for $w'= (\alpha ^*)^{-1}(w)$ and $\Cal C' = (\alpha ^*)^{-1}(\Cal C)$, $$\alpha ^*\gamma _{w',p}(Y;\Cal C') \neq \pm\gamma_{w, p}(X; \Cal C).$$ \endproclaim \proof If $X$ is homotopy equivalent to $\Pee ^1\times \Pee ^1$ then the theorem follows from [33]. Otherwise $X$ is oriented homotopy equivalent to $\Pee ^2\# n\overline{\Pee}^2$, for $1\leq n\leq 8$, and we claim that there exists a homotopy equivalence $\alpha \: X \to Y$ such that $\alpha ^*[K_Y] = -[K_X]$. Indeed, every integral isometry $H^2(Y; \Zee) \to H^2(X; \Zee)$ is realized by an oriented homotopy equivalence. Thus it suffices to show that every two characteristic elements of $H^2(Y; \Zee)$ of square $9-n$ are conjugate under the isometry group, which follows from the appendix to this paper. Choosing such a homotopy equivalence $\alpha$, let $e$ be the class of an exceptional curve in $Y$ and let $E = \alpha ^*e$. Then $E^2 =-1$ and $E\cdot [K_X] = 1$. We may now apply Theorem 1.5 to the class $E$, noting that $E$ is a $(1,1)$ class since $p_g(X)=0$. Let $M$ and $\Cal C$ be a divisor and a chamber which satisfy the conclusions of Theorem 1.5 and let $m = (\alpha ^*)^{-1} M$. If $w$ is the mod 2 reduction of $[K_X]$, then $w'$ is the mod two reduction of $[K_Y]$, so that $w'$ is also characteristic. Now $e$ is the class of a smoothly embedded 2-sphere in $Y$ since it is the class of an exceptional curve. Moreover $m\cdot e =0$. By Theorem 1.2, $\gamma_{w', p}(Y;\Cal C')(m^d) = 0$ since $e$ is represented by a smoothly embedded 2-sphere and $w'$ is characteristic. But $\gamma _{w,p}(X;\Cal C)(M^d) \neq 0$ by Theorem 1.5 and the choice of $M$. Thus $\alpha ^*\gamma _{w',p}(Y;\Cal C') \neq \pm\gamma_{w, p}(X; \Cal C)$. \endproof Using the result of Wall [38] that every homotopy self-equivalence from $Y$ to itself is realized by a diffeomorphism, the proof above shows that the conclusions of the corollary hold for {\it every\/} homotopy equivalence $\alpha: X \rightarrow Y$. \ssection{1.3. Reduction to the minimal case} We begin by recalling some terminology and results from [13]. A {\sl good generic rational surface} $Y$ is a rational surface such that $K_Y = -C$ where $C$ is a smooth curve, and such that there does not exist a smooth rational curve on $Y$ with self-intersection $-2$. Every rational surface is diffeomorphic to a good generic rational surface. \theorem{1.7} Let $X$ be a minimal surface of general type and let $\tilde X\to X$ be a blowup of $X$ at $r$ distinct points. Let $E_1', \dots , E_r'$ be the homology classes of the exceptional curves on $\tilde X$. Let $\psi _0\: \tilde X \to \tilde Y$ be a diffeomorphism, where $\tilde Y$ is a good generic rational surface. Then there exist a diffeomorphism $\psi \: \tilde X \to \tilde Y$ and a good generic rational surface $Y$ with the following properties: \roster \item"{(i)}" The surface $\tilde Y$ is the blowup of $Y$ at $r$ distinct points. \item"{(ii)}" If $e_1, \dots, e_r$ are the classes of the exceptional curves in $H^2(\tilde Y; \Bbb Z)$ for the blowup $\tilde Y \to Y$, then possibly after renumbering $\psi^*(e_i) = E_i'$ for all $i$. \item"{(iii)}" Identifying $H^2(X)$ with a subgroup of $H^2(\tilde X)$ and $H^2(Y)$ with a subgroup of $H^2(\tilde Y)$ in the obvious way, we have $\psi^*(H^2(Y)) = H^2(X)$. \endroster Moreover, for every choice of an isometry $\tau$ from $H^2(Y)$ to $H^2(X)$, there exists a choice of a diffeomorphism $\psi$ satisfying \rom{(i)--(iii)} above and such that $\psi ^*|H^2(Y) = \tau$. \endstatement \proof Let $e_i' \in H^2(\tilde Y; \Zee)$ satisfy $\psi_0^*(e_i') = E_i'$. Thus the Poincar\'e dual of $e_i'$ is represented by a smoothly embedded 2-sphere in $\tilde Y$. It follows that reflection $r_{e_i'}$ in $e_i'$ is realized by an orientation-preserving self-diffeomorphism of $\tilde Y$. To see what this says about $e_i'$, we shall recall the following terminology from [13]. Let $\bold H(\tilde Y)$ be the set $\{\, x\in H^2(\tilde Y; \Ar)\mid x^2 =1\,\}$ and let $\Cal K(\tilde Y)\subset H^2(\tilde Y; \Ar)$ be intersection of the closure of the K\"ahler cone of $\tilde Y$ with $\bold H(\tilde Y)$. Then $\Cal K(\tilde Y)$ is a convex subset of $\bold H(\tilde Y)$ whose walls consist of the classes of exceptional curves on $\tilde Y$ together with $[-K_{\tilde Y}]$ if $b_2^-(\tilde Y) \geq 10$, which is confusingly called the {\sl exceptional wall\/} of $\Cal K(\tilde Y)$. Let $\Cal R$ be the group generated by the reflections in the walls of $\Cal K(\tilde Y)$ defined by exceptional classes and define the super $P$-cell $\bold S = \bold S(P)$ by $$\bold S = \bigcup _{\gamma \in \Cal R}\gamma \cdot \Cal K(\tilde Y).$$ By Theorem 10A on p\. 355 of [13], for an integral isometry $\varphi$ of $H^2(\tilde Y; \Ar)$, there exists a diffeomorphism of $\tilde Y$ inducing $\varphi$ if and only if $\varphi (\bold S) = \pm \bold S$. (Here, if $b_2^-(\tilde Y) \leq 9$, $\bold S = \bold H$ and the result reduces to a result of C\.T\.C\. Wall [38].) Note that $\bold H(\tilde Y)$ has two connected components, and reflection $r_e$ in a class $e$ of square $-1$ preserves the set of connected components. Thus if $r_e(\bold S) = \pm \bold S$, then necessarily $r_e(\bold S) = \bold S$. Next we have the following purely algebraic lemma: \lemma{1.8} Let $e$ be a class of square $-1$ in $H^2(\tilde Y; \Zee)$ such that the reflection $r_e$ satisfies $r_e(\bold S) = \bold S$. Then there is an isometry $\varphi$ of $H^2(\tilde Y; \Zee)$ preserving $\bold S$ which sends $e$ to the class of an exceptional curve. \endstatement \proof We first claim that, if $W$ is the wall corresponding to $e$, then $W$ meets the interior of $\bold S$. Indeed, the interior $\operatorname{int}\bold S$ of $\bold S$ is connected, by Corollary 5.5 of [13] p\. 340. If $W$ does not meet $\operatorname{int} \bold S$, then the sets $$\{\, x\in \operatorname{int}\bold S\mid e\cdot x > 0\,\}$$ and $$\{\, x\in \operatorname{int}\bold S\mid e \cdot x < 0\,\}$$ are disjoint open sets covering $\operatorname{int}\bold S$ which are exchanged under the reflection $r_e$. Since at least one is nonempty, they are both nonempty, contradicting the fact that $\operatorname{int} \bold S$ is connected. Thus $W$ must meet $\operatorname{int}\bold S$. Now let $C$ be a chamber for the walls of square $-1$ which has $W$ as a wall. It follows from Lemma 5.3(b) on p\. 339 of [13] that $C\cap \bold S$ is a $P$-cell $P$ and that $W$ defines a wall of $P$ which is not the exceptional wall. By Lemma 5.3(e) of [13], $\bold S$ is the unique super $P$-cell containing $P$, and the reflection group generated by the elements of square $-1$ defining the walls of $P$ acts simply transitively on the $P$-cells in $\bold S$. There is thus an element $\varphi$ in this reflection group which preserves $\bold S$ and sends $P$ to $\Cal K(\tilde Y)$ and $W$ to a wall of $\Cal K(\tilde Y)$ which is not an exceptional wall. It follows that $\varphi (e)$ is the class of an exceptional curve on $\tilde Y$. \endproof Returning to the proof of Theorem 1.7, apply the previous lemma to the reflection in $e_r'$. There is thus an isometry $\varphi$ preserving $\bold S$ such that $\varphi(e_r') = e_r$, where $e_r$ is the class of an exceptional curve on $\tilde Y$. Moreover $\varphi$ is realized by a diffeomorphism. Thus after composing with the diffeomorphism inducing $\varphi$, we can assume that $e_r' = e_r$, or equivalently that $\psi _0^*e_r = [E_r']$. Let $\tilde Y \to \tilde Y_1$ be the blowing down of the exceptional curve whose class is $e_r$. Then $\tilde Y_1$ is again a good generic surface by [13] p\. 312 Lemma 2.3. Since $e_1', \dots, e_{r-1}'$ are orthogonal to $e_r$, they lie in the subset $H^2(\tilde Y_1)$ of $H^2(\tilde Y)$. For $i \neq r$, the reflection in $e_i'$ preserves $W\cap \bold S$, where $W= (e_r)^\perp$. Now $W$ is just $H^2(\tilde Y_1)$ and $\Cal K(\tilde Y) \cap H^2(\tilde Y_1) = \Cal K(\tilde Y_1)$ by [13] p\. 331 Proposition 3.5. The next lemma relates the corresponding super $P$-cells: \lemma{1.9} $W\cap \bold S$ is the super $P$-cell $\bold S_1$ for $\tilde Y_1$ containing $\Cal K(\tilde Y_1)$. \endstatement \proof Trivially $\bold S_1 \subseteq W\cap \bold S$, and both sets are convex subsets with nonempty interiors. If they are not equal, then there is a $P$-cell $P' \subset \bold S_1$ and an exceptional wall of $P'$ which passes through the interior of $\bold S\cap W$. If $\kappa (P')$ is the exceptional wall meeting $\bold S\cap W$, then, by [13] p\. 335 Lemma 4.6, $\kappa (P') - e_r$ is an exceptional wall of $P$ for a well-defined $P$-cell in $\bold S$, and $\kappa (P') - e_r$ must pass through the interior of $\bold S$. This is a contradiction. Hence $\bold S\cap W = \bold S_1$ is a super $P$-cell of $\tilde Y_1$, and we have seen that it contains $\Cal K(\tilde Y_1)$. \endproof Returning to the proof of Theorem 1.7, reflection in $e_{r-1}'$ preserves $\bold S_1$. Applying Lemma 1.8, there is a diffeomorphism of $\tilde Y_1$ which sends $e_{r-1}'$ to the class of an exceptional curve $e_{r-1}$. Of course, there is an induced diffeomorphism of $\tilde Y$ which fixes $e_r$. Now we can clearly proceed by induction on $r$. The above shows that after replacing $\psi _0$ by a diffeomorphism $\psi$ we can find $Y$ as above so that (i) and (ii) of the statement of Theorem 3 hold. Clearly $\psi^*(H^2(Y)) = H^2(X)$. By the theorem of C\.T\.C\. Wall mentioned above, there is a diffeomorphism of $Y$ realizing every integral isometry of $H^2(Y)$. So after further modifying by a diffeomorphism of $Y$, which extends to a diffeomorphism of $\tilde Y$ fixing the classes of the exceptional curves, we can assume that the diffeomorphism $\psi$ restricts to $\tau$ for any given isometry from $H^2(Y)$ to $H^2(X)$. \endproof We can now give a proof of Theorem 0.1: \theorem{0.1} No complex surface of general type is diffeomorphic to a rational surface. \endstatement \proof Suppose that $X$ is a minimal surface of general type and that $\rho \: \tilde X \to X$ is a blowup of $X$ diffeomorphic to a rational surface. We may assume that $\tilde X$ is diffeomorphic via $\psi$ to a good generic rational surface ${\tilde Y}$, and that $\rho '\: {\tilde Y} \to Y$ is a blow up of ${\tilde Y}$ such that $Y$ and $\psi$ satisfy (i)--(iii) of Theorem 1.7. Choose $w, p, \alpha, \Cal C$ for $X$ such that the conclusions of Corollary 1.6 hold, with $\Cal C'$ the corresponding chamber on $Y$, and let $\tilde \Cal C'$ be any chamber for ${\tilde Y}$ containing $\Cal C'$ in its closure. Then $\psi ^*\tilde \Cal C' = \tilde \Cal C$ is a chamber on $\tilde X$ containing $\Cal C$ in its closure. Using the last sentence of Theorem 1.7, we may assume that $\psi ^*|H^2(Y) = \alpha^*$. Thus $\psi ^* (\rho ') ^* = \rho ^*\alpha ^*$. By the functorial properties of Donaldson polynomials, and viewing $H^2(X; \Zee/2\Zee)$ as a subset of $H^2(\tilde X; \Zee/2\Zee)$, and similarly for $\tilde Y$, we have $$\psi ^*\gamma _{w', p}({\tilde Y}, \tilde \Cal C') = \pm\gamma _{\psi ^*w', p}(\tilde X, \tilde \Cal C)= \pm\gamma _{w, p}(\tilde X, \tilde \Cal C).$$ Restricting each side to $\psi ^*H_2(Y) = H_2(X)$, we obtain by repeated application of Lemma 1.3 that $$\alpha ^*\gamma _{w',p}(Y;\Cal C') = \pm\gamma _{w, p}(X;\Cal C).$$ But this contradicts Corollary 1.6. \endproof Using Theorem 1.5, we have the following generalization of Theorem 0.4 in the introduction to the case of nonminimal algebraic surfaces: \theorem{1.10} Let $X$ be a minimal simply connected surface of general type, and let $E\in H^2(X; \Zee)$ satisfy $E^2 = -1$ and $E\cdot K_X = 1$. Let $\tilde X$ be a blowup of $X$. Then, viewing $H^2(X; \Zee)$ as a subset of $H^2(\tilde X; \Zee)$, the class $E$ is not represented by a smoothly embedded $2$-sphere in $\tilde X$. \endstatement \proof Suppose instead that $E$ is represented by a smoothly embedded $2$-sphere. If $p_g(X) >0$, then it follows from the results of [6] that $E$ is a $(1,1)$-class, i\.e\. $E$ lies in the image of $\operatorname{Pic}X$ inside $H^2(X; \Zee)$. Of course, this is automatically true if $p_g(X) = 0$. Next assume that $p_g(X) = 0$. By Theorem 1.5, there exists a $w\in H^2(X; \Zee/2\Zee)$, an integer $p$, and a chamber $\Cal C$ of type $(w,p)$, such that $\gamma _{w,p}(X;\Cal C)(M^d)\neq 0$, where $M$ is a class in the closure of $\Cal C$ and $M\cdot E = 0$. Consider the Donaldson polynomial $\gamma_{w,p}(\tilde X;\tilde \Cal C)$, where we view $w$ as an element of $H^2(\tilde X;\Zee/2\Zee)$ in the natural way and $\tilde \Cal C$ is a chamber of type $(w,p)$ on $\tilde X$ containing $\Cal C$ in its closure. Then $\tilde \Cal C$ also contains $M$ in its closure. Thus, by Theorem 1.2, $\gamma_{w,p}(\tilde X;\tilde \Cal C)(M^d) = 0$. On the other hand, by Lemma 1.3, $\gamma_{w,p}(\tilde X;\tilde \Cal C)(M^d) = \pm \gamma _{w,p}(X;\Cal C)(M^d)\neq 0$. This is a contradiction. The case where $p_g(X) > 0$ is similar. \endproof We also have the following corollary, which works under the assumptions of Theorem 1.10 for surfaces with $p_g>0$: \corollary{1.11} Let $X$ be a simply connected surface of general type with $p_g(X) >0$, not necessarily minimal, and let $E\in H^2(X; \Zee)$ satisfy $E^2 = -1$ and $E\cdot K_X = -1$. Suppose that $E$ is represented by a smoothly embedded $2$-sphere. Then $E$ is the cohomology class associated to an exceptional curve. \endstatement \proof Using [15] and [6], we see that if $E$ is not the cohomology class associated to an exceptional curve, then $E\in H^2(X_{\text{min}}; \Zee)$, where $X_{\text{min}}$ is the minimal model of $X$ and we have the natural inclusion $H^2(X_{\text{min}}; \Zee) \subseteq H^2(X; \Zee)$. We may then apply Theorem 1.10 to conclude that $-E$ cannot be represented by a smoothly embedded $2$-sphere, and thus that $E$ cannot be so represented, a contradiction. \endproof \section{2. A generalized nonvanishing theorem} \ssection{2.1. Statement of the theorem and the first part of the proof} In this section, we shall prove Theorem 1.4. We first recall its statement: \theorem{1.4} Let $X$ be a simply connected algebraic surface with $p_g(X) =0$, and let $M$ be a nef and big divisor on $X$ which is eventually base point free. Denote by $\varphi\: X \to \bar X$ the birational morphism defined by $|kM|$ for $k\gg 0$, so that $\bar X$ is a normal projective surface. Suppose that $\bar X$ has only rational or minimally elliptic singularities, and that $\varphi$ does not contract any exceptional curves to points. Let $w\in H^2(X;\Zee/2\Zee)$ be the \rom{mod} $2$ reduction of the class $[K_X]$. Then there exists a constant $A$ depending only on $X$ and $M$ with the following property: For all integers $p\leq A$, let $\Cal C$ be a chamber of type $(w,p)$ containing $M$ in its closure and suppose that $\Cal C$ has nonempty intersection with the ample cone of $X$. Set $d = -p-3$. Then $$\gamma_{w,p}(X;\Cal C)(M^d) >0.$$ A similar conclusion holds if $p_g(X) >0$. \endstatement \proof We begin by fixing some notation. For $L$ an ample line bundle on $X$, given a divisor $D$ on $X$ and an integer $c$, let $\frak M_L(D, c)$ denote the moduli space of isomorphism classes of $L$-stable rank two holomorphic vector bundles on $X$ with $c_1(V) = D$ and $c_2(V) = c$. Let $w$ be the mod 2 reduction of $D$ and let $p= D^2 - 4c$. Then we also denote $\frak M_L(D, c)$ by $\frak M_L(w, p)$, the moduli space of equivalence classes of $L$-stable rank two holomorphic vector bundles on $X$ corresponding to the choice of $(w,p)$. Here we recall that two vector bundles $V$ and $V'$ are {\sl equivalent\/} if there exists a holomorphic line bundle $F$ such that $V' = V\otimes F$. The invariants $w$ and $p$ only depend on the equivalence class of $V$. Let $\overline{\frak M_L(w, p)}$ denote the Gieseker compactification of $\frak M_L(w, p)$, i\.e\. the Gieseker compactification $\overline{\frak M_L(D, c)}$ of $\frak M_L(D, c)$. Thus $\overline{\frak M_L(w, p)}$ is a projective variety. We now fix a compact neighborhood $\Cal N$ of $M$ inside the positive cone $\Omega _X$ of $X$. Note that, since $M$ is nef, such a neighborhood has nontrivial intersection with the ample cone of $X$. Using a straightforward extension of the theorem of Donaldson [10] on the dimension of the moduli space (see e\.g\. [12] Chapter 8, [32], [24]), there exist constants $A$ and $A'$ such that, for all ample line bundles $L$ such that $c_1(L) \in \Cal N$, the following holds: \roster \item If $p \leq A$, then the moduli space $\overline{\frak M_L(w, p)}$ is good, in other words it is generically reduced of the correct dimension $-p-3$; \item $\frak M_L(w, p)$ is a dense open subset of $\overline{\frak M_L(w, p)}$ and the generic point of $\overline{\frak M_L(w, p)}- \frak M_L(w, p)$ correspond to a torsion free sheaf $V$ such that the length of $V\spcheck{}\spcheck/V$ is one and such that the support of $V\spcheck{}\spcheck/V$ is a generic point of $X$; \item For all $p' \geq A$, the dimension $\dim \frak M_L(w, p') \leq A'$. \endroster We shall need to make one more assumption on the integer $p$. Let $\varphi\: X \to \bar X$ be the contraction morphism associated to $M$. For each connected component $E$ of the set of exceptional fibers of $\varphi$, fix a (possibly nonreduced) curve $Z$ on $X$ whose support is exactly $E$. In practice we shall always take $Z$ to be the fundamental cycle of the singularity, to be defined in Subsection 2.3 below. A slight generalization ([12], Chapter 8) of Donaldson's theorem on the dimension of the moduli space then shows the following: after possibly modifying the constant $A$, \roster \item"(4)" The generic $V\in \frak M_L(w,p)$ satisfies: the natural map $$H^1(X; \operatorname{ad}V) \to H^1(Z; \operatorname{ad}V|Z)$$ is surjective. In other words, the local universal deformation of $V$ is versal when viewed as a deformation of $V|Z$ (keeping the determinant fixed). \endroster We now assume that $p\leq A$. Let $L$ be an ample line bundle which is not separated from $M$ by any wall of type $(w,p)$ (or equivalently of type $(D, c)$), and moreover does not lie on any wall of type $(w,p)$. Thus by assumption, none of the points of $\overline{\frak M_L(D, c)}$ corresponds to a strictly semistable sheaf. Let $C\subset X$ be a smooth curve of genus $g$. Suppose that $C\cdot D = 2a$ is even. Choosing a line bundle $\theta$ of degree $g-1-a$ on $C$, we can form the determinant line bundle $\Cal L(C, \theta)$ on the moduli functor associated to torsion free sheaves corresponding to the values $w$ and $p$ ([15], Chapter 5). Using Proposition 1.7 in [23], this line bundle descends to a line bundle on $\overline{\frak M_L(w, p)}$, which we shall continue to denote by $\Cal L(C, \theta)$. Moreover, by the method of proof of Theorem 2 of [23], the line bundle $\Cal L(C, \theta)$ depends only on the linear equivalence class of $C$, in the sense that if $C$ and $C'$ are linearly equivalent and $\theta '$ is a line bundle of degree $g-1-a$ on $C'$, then $\Cal L(C, \theta) \cong \Cal L(C', \theta')$. Next we shall use the following result, whose proof is deferred to the next subsection: \lemma{2.1} In the above notation, if $k\gg 0$ and $C\in |kM|$ is a smooth curve, then, for all $N\gg 0$, the linear system associated to $\Cal L(C, \theta)^N$ has no base points and defines a generically finite morphism from $\overline{\frak M_L(w, p)}$ to its image. In particular, if $d = \dim \overline{\frak M_L(w, p)}$, then $$c_1(\Cal L(C, \theta))^d > 0.$$ \endstatement It follows by applying an easy adaptation of Theorem 6 in [23] or the results of [25] to the case $p_g(X) =0$ that, since the spaces $\frak M_L(w,p')$ have the expected dimension for an appropriate range of $p'\geq p$, $c_1(\Cal L(C, \theta))^d$ is exactly the value $k^d\gamma_{w,p}(X;\Cal C)(M^d)$. Thus we have proved Theorem 1.4, modulo the proof of Lemma 2.1. This proof will be given below. \enddemo \ssection{2.2. A generalization of a result of Bogomolov} We keep the notation of the preceding subsection. Thus $M$ is a nef and big divisor such that the complete linear system $|k M|$ is base point free whenever $k \gg 0$. Throughout, we shall further assume that $M$ is divisible by $2$ in $\operatorname{Pic}X$. Moreover $w$ and $p$ are now fixed and $L$ is an ample line bundle such that $c_1(L) \in \Cal N$ is not separated from $M$ by a wall of type $(w,p)$ and moreover that $c_1(L)$ does not lie on a wall of type $(w,p)$. In particular the determinant line bundle $\Cal L(C, \theta)$ is defined for all smooth $C$ in $|kM|$ for all $k\gg 0$. We then have the following generalization of a restriction theorem due to Bogomolov [4]: \lemma{2.2} With the above notation, there exists a constant $k_0$ depending only on $w$, $p$, $M$, and $L$, such that for all $k\geq k_0$ and all smooth curves $C\in |kM|$, the following holds: for all $c' \leq c$ and $V \in \frak M_L(D, c')$, either $V|C$ is semistable or there exists a divisor $G$ on $X$, a zero-dimensional subscheme $\Cal Z$ and an exact sequence $$0 \to \scrO_X(G) \to V \to \scrO_X(D-G) \otimes I_{\Cal Z} \to 0,$$ where $2G-D$ defines a wall of type $(w,p)$ containing $M$ and $C \cap \operatorname{Supp}\Cal Z \neq \emptyset$. \endstatement \proof The proof follows closely the original proof of Bogomolov's theorem [4] or [15] Section 5.2. Choose $k_0 \geq -p$ and assume also that there exists a smooth curve $C$ in $|kM|$ for all $k\geq k_0$. Suppose that $V|C$ is not semistable. Then there exists a surjection $V|C \to F$, where $F$ is a line bundle on $C$ with $\deg F=f< (D\cdot C)/2$. Let $W$ be the kernel of the induced surjection $V\to F$. Thus $W$ is locally free and there is an exact sequence $$0 \to W \to V \to F \to 0.$$ A calculation gives $$\align p_1(\operatorname{ad}W) &= p_1(\operatorname{ad}V) + 2D\cdot C + (C)^2 - 4f\\ &> p + k^2(M)^2 \geq p + p^2 \geq 0. \endalign$$ By Bogomolov's inequality, $W$ is unstable with respect to every ample line bundle on $X$. Thus there exists a divisor $G_0$ and an injection $\scrO_X(G_0) \to W$ (which we may assume to have torsion free cokernel) such that $2(L\cdot G_0) > L\cdot (D-C)$, i\.e\. $L \cdot (2G_0 - D +C) > 0$. By hypothesis there is an exact sequence $$0 \to \scrO_X(G_0) \to W \to \scrO_X(-G_0+D-C) \otimes I_{\Cal Z_0}\to 0.$$ Thus $$0< p_1(\operatorname{ad}W) = (2G_0 - D +C)^2 - 4\ell (\Cal Z_0) \leq (2G_0 - D +C)^2.$$ It follows that $(2G_0 - D +C)^2>0$. As $L \cdot (2G_0 - D +C) > 0$ and $(2G_0 - D +C)^2 > 0$, $M \cdot (2G_0 - D +C) \geq 0$ as well, i\.e\. $-(M \cdot (2G_0 - D))\leq k(M)^2$. On the other hand, since $V$ is $L$-stable, $L\cdot (2G_0 - D)< 0$. Since $L$ and $M$ are not separated by any wall of type $(w,p)$, it follows that $M\cdot (2G_0 - D)\leq 0$. Finally using $$\align p_1(\operatorname{ad}W) &= (2G_0 - D +C)^2 - 4\ell (\Cal Z_0) \\ &= p_1(\operatorname{ad}V) + 2D\cdot C + (C)^2 - 4f\\ &> p + k^2(M)^2, \endalign$$ we obtain $$(2G_0 - D)^2 +2k(2G_0 - D)\cdot M > p.$$ Let $m = -(2G_0 - D)\cdot M$. As we have seen above $m\leq kM^2$ and $m\geq 0$. The above inequality can be rewritten as $$ 2km < (2G_0 - D)^2 -p.$$ We claim that $m=0$. Otherwise $$2k < \frac{(2G_0 - D)^2}{m} -\frac{p}{m} .$$ By the Hodge index theorem $(2G_0 - D)^2 M^2 \leq \left[(2G_0 - D)\cdot M\right]^2 = m^2$, so that $(2G_0 - D)^2 \leq m^2/M^2$. Plugging this into the inequality above, using $-p\geq 0$, gives $$2k < \frac{m}{M^2} - \frac{p}{m} \leq k - p,$$ i\.e\. $k< -p$, contradicting our choice of $k$. Thus $m= -(2G_0 - D)\cdot M =0$. Now the inclusions $\scrO_X(G_0) \subset W\subset V$ define an inclusion $\scrO_X(G_0) \subset V$. Thus there is an effective divisor $E$ and an inclusion $\scrO_X(G_0+E) \to V$ with torsion free cokernel. Let $G = G_0 +E$. Thus there is an exact sequence $$0 \to \scrO_X(G) \to V \to \scrO_X(-G+D)\otimes I_{\Cal Z}\to 0.$$ We claim that $(2G-D)\cdot M = 0$. Since $V$ is $L$-stable, $(2G-D)\cdot L < 0$, and since $L$ and $M$ are not separated by a wall of type $(w,p)$, $(2G-D)\cdot M \leq 0$. On the other hand, $$(2G-D)\cdot M = (2G_0 - D) \cdot M + 2(E\cdot M) = -m + 2(E\cdot M)=2(E\cdot M).$$ As $E$ is effective and $M$ is nef, $2(E\cdot M) \geq 0$. Thus $(2G-D)\cdot M = 0$. As $M^2 >0$, we must have $(2G-D)^2<0$. Using $p = (2G-D)^2 -4\ell(\Cal Z)\leq (2G-D)^2$, we see that $2G-D$ is a wall of type $(w,p)$. Finally note that $\operatorname{Supp}\Cal Z\cap C \neq \emptyset$, for otherwise we would have $V|C$ semistable. This concludes the proof of Lemma 2.2. \endproof Returning to the proof of Lemma 2.1, we claim first that, given $k\gg 0$ and $C\in |kM|$, for all $N$ sufficiently large the sections of $\Cal L (C, \theta)^N$ define a base point free linear series on $\overline{\frak M_L(w,p)}$. To see this, we first claim that, for $k \gg 0$, and for a generic $C\in |kM|$, the restriction map $V\mapsto V|C$ defines a rational map $r_C\: \frak M_L(w,p)\dasharrow \frak M(C)$, where $\frak M(C)$ is the moduli space of equivalence classes of semistable rank two bundles on $C$ such that the parity of the determinant is even. It suffices to prove that, for every component $N$ of $\frak M_L(w,p)$ there is one $V\in N$ and one $C \in |kM|$ such that $V|C$ is semistable, for then the same will hold for a Zariski open subset of $|kM|$. Now given $V$, choose a fixed $C_0 \in |kM|$. If $V|C_0$ is not semistable, then by Lemma 2.2 there is an exact sequence $$0 \to \scrO_X(G) \to V \to \scrO_X(-G+D)\otimes I_{\Cal Z} \to 0,$$ where $\Cal Z$ is a zero-dimensional subscheme of $X$ meeting $C_0$. Choosing $C$ to be a curve in $|kM|$ disjoint from $\Cal Z$, which is possible since $|kM|$ is base point free, it follows that the restriction $V|C$ is semistable. For $C$ fixed, let $$\align B_C= \{\,V \in \overline{\frak M_L(w,p)}:&\text{ either $V$ is not locally free over some point of $C$}\\ &\text{ or $V|C$ is not semistable }\,\}. \endalign$$ By the openness of stability and local freeness, the set $B_C$ is a closed subset of $\overline{\frak M_L(w,p)}$ and $r_C$ defines a morphism from $\overline{\frak M_L(w,p)} -B_C$ to $\frak M(C)$. Standard estimates (cf\. [10], [12], [32], [24], [27]) show that, possibly after modifying the constant $A$ introduced at the beginning of the proof of Theorem 1.4, the codimension of $B_C$ is at least two in $\overline{\frak M_L(w,p)}$ provided that $p\leq A$ (where as usual $A$ is independent of $k$ and depends only on $X$ and $M$). Indeed the set of bundles $V$ which fit into an exact sequence $$0 \to \scrO_X(G) \to V \to \scrO_X(D-G)\otimes I_{\Cal Z} \to 0,$$ where $G$ is a divisor such that $(2G-D)\cdot M = 0$, may be parametrized by a scheme of dimension $-\frac34p + O(\sqrt{|p|})$ by e\.g\. [12], Theorem 8.18. Moreover the constant implicit in the notation $O(\sqrt{|p|})$ can be chosen uniformly over $\Cal N$. The case of nonlocally free $V$ is taken care of by assumption (2) in the discussion of the constant $A$: it follows from standard deformation theory (see again [12], [24]) that at a generic point of the locus of nonlocally free sheaves corresponding to the semistable torsion free sheaf $V$ the deformations of $V$ are versal for the local deformations of the singularities of $V$. Thus for a general nonlocally free $V$, $V$ has just one singular point which is at a general point of $X$ and so does not lie on $C$. Thus the set of $V$ which are not locally free at some point of $C$ has codimension at least two (in fact exactly two) in $\overline{\frak M_L(w,p)}$. Let $\Cal L_C$ be the determinant line bundle on $\frak M(C)$ associated to the line bundle $\theta$ (see for instance [15] Chapter 5 Section 2). Then by definition the pullback via $r_C$ of $\Cal L_C$ is the restriction of $\Cal L (C, \theta)$ to $\overline{\frak M_L(w,p)} -B_C$. Since $B_C$ has codimension two, the sections of $\Cal L_C^N$ pull back to sections of $\Cal L (C, \theta)^N$ on $\overline{\frak M_L(w,p)}$. Since $\Cal L_C$ is ample, given $V\in \overline{\frak M_L(w,p)} -B_C$, there exists an $N$ and a section of $\Cal L_C^N$ not vanishing at $r_C(V)$, and thus there is a section of $\Cal L (C, \theta)^N$ not vanishing at $V$. Moreover by [23], for all smooth $C'\in |kM|$ and choice of an appropriate line bundle $\theta '$ on $C'$, there is an isomorphism $\Cal L (C, \theta)^N \cong \Cal L(C', \theta ')^N$. Next we claim that, for every $V\in \overline{\frak M_L(w,p)}$, there exists a $C$ such that $V$ is locally free above $C$ and $V|C$ is semistable. Given $V$, it fails to be locally free at a finite set of points, and its double dual $W$ is again semistable. Thus applying the above to $W$, and again using the fact that $|kM|$ has no base points, we can find $C$ such that $V$ is locally free over $C$ and such that $V|C = W|C$ is semistable. Thus, given $V$, there exists an $N$ and a section of $\Cal L (C, \theta)^N$ which does not vanish at $V$. Since $\overline{\frak M_L(w,p)}$ is of finite type, there exists an $N$ which works for all $V$, so that the linear system corresponding to $\Cal L (C, \theta)^N$ has no base points. Finally we must show that, for $k\gg 0$, the morphism induced by $\Cal L (C, \theta)^N$ is in fact generically finite for $N$ large. We claim that it suffices to show that the restriction of the rational map $r_C$ to $\overline{\frak M_L(w,p)} -B_C$ is generically finite (it is here that we must use the condition on the singularities of $\bar X$ in the statement of Theorem 1.4). Supposing this to be the case, and fixing a $V \in \overline{\frak M_L(w,p)} -B_C$ for which $r_C^{-1}(r_C(V))$ is finite, we consider the intersection of all the divisors in $\Cal L (C, \theta)^N$ containing $V$, where $N$ is chosen so that $\Cal L_C^N$ is very ample. This intersection always contains $V$ and is a subset of $r_C^{-1}(r_C(V)) \cup B_C$. In particular $V$ is an isolated point of the fiber, and so the morphism defined by $\Cal L (C, \theta)^N$ cannot have all fibers of purely positive dimension. Thus it is generically finite. To see that $r_C$ is generically finite, we shall show that, for generic $V$, the restriction map $$r\: H^1(X;\operatorname{ad}V) \to H^1(C;\operatorname{ad}V|C))$$ is injective. The map $r$ is just the differential of the map $r_C$ from $\frak M_L(w,p)$ to $\frak M(C)$ at the point corresponding to $V$, and so if $V$ is generic then $r_C$ is finite. Now the kernel of the map $r$ is a quotient of $H^1(X; \operatorname{ad}V\otimes \scrO_X(-C))$, and we need to find circumstances where this group is zero, at least if $C\in |kM|$ for $k$ sufficiently large. By Serre duality it suffices to show that $H^1(X; \operatorname{ad}V\otimes \scrO_X(C)\otimes K_X)=0$ for $k$ sufficiently large. By applying the Leray spectral sequence to the morphism $\varphi\: X\to \bar X$, it suffices to show that $$H^1(\bar X;R^0\varphi _* (\operatorname{ad}V\otimes \scrO_X(C)\otimes K_X)) =0$$ and that $R^1\varphi _* (\operatorname{ad}V\otimes \scrO_X(C)\otimes K_X)=0$. Now $M$ is the pullback of an ample line bundle $\bar M$ on $\bar X$, and $\scrO_X(C)$ is the pullback of $(\bar M)^{\otimes k}$. Thus for fixed $V$ and $k \gg 0$, $$\align &H^1(\bar X;R^0\varphi _* (\operatorname{ad}V\otimes \scrO_X(C)\otimes K_X)) \\= &H^1(\bar X;R^0\varphi _* (\operatorname{ad}V\otimes K_X)\otimes (\bar M^k)) =0. \endalign$$ Moreover $R^1\varphi _* (\operatorname{ad}V\otimes \scrO_X(C)\otimes K_X)) = R^1\varphi _* (\operatorname{ad}V\otimes K_X)\otimes (\bar M^k)$, so that it is enough to show that $R^1\varphi _* (\operatorname{ad}V\otimes K_X)=0$. By the formal functions theorem, $$R^1\varphi _* (\operatorname{ad}V\otimes K_X) = \varprojlim _mH^1(mZ; \operatorname{ad}V\otimes K_X |mZ),$$ where $Z = \bigcup Z_i$ is the union of the connected components $Z_i$ of the one-dimensional fibers of $\varphi$. Thus it suffices to show that, for all $i$ and all positive integers $m$, $H^1(mZ_i; \operatorname{ad}V\otimes K_X |mZ_i) = 0$. Now by the adjunction formula $\omega _{mZ_i} = K_X\otimes \scrO_X(mZ_i)|mZ_i$, where $\omega _{mZ_i}$ is the dualizing sheaf of the Gorenstein scheme $mZ_i$. Thus $K_X|mZ_i = \scrO_X(-mZ_i) |mZ_i\otimes\omega _{mZ_i}$ and we must show the vanishing of $$H^1(mZ_i; (\operatorname{ad}V\otimes \scrO_X(-mZ_i)) |mZ_i\otimes\omega _{mZ_i}).$$ By Serre duality, it suffices to show that, for all $m>0$, $$H^0(mZ_i; (\operatorname{ad}V\otimes \scrO_X(mZ_i)) |mZ_i)=0.$$ We shall deal with this problem in the next subsection. \medskip \noindent {\bf Remark.} (1) Instead of arguing that the restriction map $r_C$ was generically finite, one could also check that it was generically one-to-one by showing that for generic $V_1$, $V_2$, the restriction map $$H^0(X; Hom (V_1, V_2)) \to H^0(C; Hom (V_1, V_2)|C)$$ is surjective (since then an isomorphism from $V_1|C$ to $V_2|C$ lifts to a nonzero map from $V_1$ to $V_2$, necessarily an isomorphism by stability). In turn this would have amounted to showing that $H^1(X; Hom(V_1, V_2)\otimes \scrO_X(-C))=0$ for generic $V_1$ and $V_2$, and this would have been essentially the same argument. \smallskip (2) Suppose that $\varphi \: X\to \bar X$ is the blowup of a smooth surface $\bar X$ at a point $x$, and that $M$ is the pullback of an ample divisor on $\bar X$. Let $Z\cong\Pee ^1$ be the exceptional curve. In this case, if $c_1(V)\cdot Z$ is odd, say $2a+1$, then the generic behavior for $V|Z$ is $V|Z \cong \scrO_{\Pee ^1}(a)\oplus \scrO_{\Pee ^1}(a+1)$ and the restriction map exhibits $\frak M_L(w,p)$ (generically) as a $\Pee ^1$-bundle over its image (see for instance [5]). Thus the hypothesis that $\varphi$ contracts no exceptional curve is essential. \ssection{2.3. Restriction of stable bundles to certain curves} Let us recall the basic properties of rational and minimally elliptic singularities. Let $x$ be a normal singular point on a complex surface $\bar X$, and let $\varphi \: X \to \bar X$ be the minimal resolution of singularities of $\bar X$. Supppose that $\varphi ^{-1}(x) = \bigcup _iD_i$. The singularity is a {\sl rational\/} singularity if $(R^1\varphi _*\scrO_X)_x = 0$. Equivalently, by [1], $x$ is rational if and only if, for every choice of nonnegative integers $n_i$ such that at least one of the $n_i$ is strictly positive, if we set $Z = \sum _in_iD_i$, the arithmetic genus $p_a(Z)$ of the effective curve $Z$ satisfies $p_a(Z) \leq 0$. Here $p_a(Z) = 1 - \chi (\scrO_Z) = 1 - h^0(\scrO_Z) + h^1(\scrO_Z)\leq h^1(\scrO_Z)$; moreover we have the adjunction formula $$p_a(Z) = 1 + \frac12 (K_X+Z)\cdot Z.$$ Now every minimal resolution of a normal surface singularity $x$ has a {\sl fundamental cycle\/} $Z_0$, which is an effective cycle $Z_0$ supported in the set $\varphi ^{-1}(x)$ and satisfying $Z_0 \cdot D_i \leq 0$ and $Z_0 \cdot D_i < 0$ for some $i$ which is minimal with respect to the above properties. We may find $Z_0$ as follows [22]: start with an arbitrary component $A_1$ of $\varphi ^{-1}(x)$ and set $Z_1= A_1$. Now either $Z_0 = A_1$ or there exists another component $A_2$ with $Z_1\cdot A_2>0$. Set $Z_2 = Z_1 + A_2$ and continue this process. Eventually we reach $Z_k = Z_0$. Such a sequence $A_1, \dots , A_k$ with $Z_i = \sum _{j\leq i}A_j$ and $Z_i\cdot A_{i+1} >0$, $Z_k = Z_0$ is called a {\sl computation sequence}. By a theorem of Artin [1], $x$ is rational if and only if $p_a(Z_0) \leq 0$, where $Z_0$ is the fundamental cycle, if and only if $p_a(Z_0) = 0$. Moreover, if $x$ is a rational singularity, then every component $D_i$ of $\varphi ^{-1}(x)$ is a smooth rational curve, the $D_i$ meet transversally at at most one point, and the dual graph of $\varphi ^{-1}(x)$ is contractible. Next we recall the properties of minimally elliptic singularities [22]. A singularity $x$ is {\sl minimally elliptic\/} if and only if there exists a {\sl minimally elliptic cycle\/} $Z$ for $x$, in other words a cycle $Z= \sum _in_iD_i$ with all $n_i>0$ such that $p_a(Z) = 1$ and $p_a(Z') \leq 0$ for all nonzero effective cycles $Z'<Z$ (i\.e\. such that $Z' = \sum _in_i'D_i$ with $0\leq n_i'\leq n_i$ and $Z'\neq Z$). In this case it follows that $Z=Z_0$ is the fundamental cycle for $x$, and $(K_X+Z_0)\cdot D_i = 0$ for every component $D_i$ of $\varphi ^{-1}(x)$. If $Z_0$ is reduced, i\.e\. if $n_i = 1$ for all $i$, then the possibilities for $x$ are as follows: \roster \item $\varphi ^{-1}(x)$ is an irreducible curve of arithmetic genus one, and thus is either a smooth elliptic curve or a singular rational curve with either a node or a cusp; \item $\varphi ^{-1}(x)= \bigcup _{i=1}^tD_i$ is a cycle of $t\geq 2$ smooth rational curves meeting transversally, i\.e\. $D_i\cdot D_{i+1} =1$, $D_i \cdot D_j \neq 0$ if and only if $i \equiv j \pm 1 \mod t$, except for $t=2$ where $D_1 \cdot D_2 = 2$; \item $\varphi ^{-1}(x)= D_1\cup D_2$, where the $D_i$ are smooth rational, $D_1\cdot D_2 = 2$ and $D_1\cap D_2 $ is a single point (so that $\varphi ^{-1}(x)$ has a tacnode singularity) or $\varphi ^{-1}(x)= D_1\cup D_2 \cup D_3$ where the $D_i$ are smooth rational, $D_i\cdot D_j = 1$ but $D_1\cap D_2 \cap D_3$ is a single point (the three curves meet at a common point). \endroster Here $x$ is called a {\sl simple elliptic singularity\/} in case $\varphi ^{-1}(x)$ is a smooth elliptic curve, a {\sl cusp singularity\/} if $\varphi ^{-1}(x)$ is an irreducible rational curve with a node or a cycle as in (2), and a {\sl triangle singularity\/} in the remaining cases. If $Z_0$ is not reduced, then all components $D_i$ of $\varphi ^{-1}(x)$ are smooth rational curves meeting transversally and the dual graph of $\varphi ^{-1}(x)$ is contractible. With this said, and using the discussion in the previous subsection, we will complete the proof of Theorem 1.4 by showing that $H^0(mZ_i; \operatorname{ad}V\otimes \scrO_X(mZ_i)|mZ_i) = 0$ for all $i$, where $x_1, \dots, x_k$ are the singular points of $\bar X$ and $Z_i$ is an effective cycle with $\operatorname{Supp}Z_i = \varphi ^{-1}(x_i)$. The precise statement is as follows: \theorem{2.3} Let $\varphi \: X \to \bar X$ be a birational morphism from $X$ to a normal projective surface $\bar X$, corresponding to a nef, big, and eventually base point free divisor $M$. Let $w$ be the \rom{mod} $2$ reduction of $[K_X]$, and suppose that \roster \item"{(i)}" $\varphi$ contracts no exceptional curve; in other words, if $E$ is an exceptional curve of the first kind on $X$, then $M\cdot E>0$. \item"{(ii)}" $\bar X$ has only rational and minimally elliptic singularities. \endroster Then there exists a constant $A$ depending only on $p$ and $\Cal N$ with the following property: for every singular point $x$ of $\bar X$, there exists an effective cycle $Z$ with $\operatorname{Supp}Z = \varphi^{-1}(x)$ such that, for all ample line bundles $L$ in $\Cal N$, all $p$ with $p \leq A$, and generic bundles $V$ in $\frak M_L(w,p)$, $$H^0(mZ; \operatorname{ad}V\otimes \scrO_X(mZ)|mZ) = 0$$ for every positive integer $m$. \endstatement The statement of (i) may be rephrased by saying that $X$ is the {\sl minimal resolution\/} of $\bar X$. As $\operatorname{ad}V \subset Hom(V,V)$, it suffices to prove that $H^0(mZ; Hom(V,V)\otimes \scrO_X(mZ)|mZ) = 0$. We will consider the case of rational singularities and minimally elliptic singularities separately. Let us begin with the proof for rational singularities. Let $\varphi ^{-1}(x) = \bigcup _iD_i$, where each $D_i$ is a smooth rational curve. By the assumption (4) of the previous subsection, we can assume that the constant $A$ has been chosen so that $V|D_i$ is a generic bundle over $D_i\cong \Pee ^1$ for every $i$. Thus either there exists an integer $a$ such that $V|D_i \cong \scrO_{\Pee^1}(a) \oplus \scrO_{\Pee^1}(a+1)$, if $w\cdot D_i \neq 0$, or there exists an $a$ such that $V|D_i \cong \scrO_{\Pee^1}(a) \oplus \scrO_{\Pee^1}(a)$, if $w\cdot D_i = 0$. Next, we have the following claim: \claim{2.4} Suppose that $x$ is a rational singularity. Let $\varphi\:X \to \bar X$ be a resolution of $x$. There exist a sequence of curves $B_0, \dots, B_k$, such that $B_i \subseteq \varphi ^{-1}(x)$ for all $i$, with the following property: \roster \item Let $C_i = \sum _{j\leq i}B_i$. Then $B_i \cdot C_i \leq B_i^2 + 1$. \item $C_k = Z_0$, the fundamental cycle of $x$. \endroster \endstatement \proof Since $(K_X+Z_0)\cdot Z_0<0$, there must exist a component $B^{(0)} = D_i$ of $\operatorname{Supp}Z_0 = \varphi ^{-1}(x)$ such that $(K_X+Z_0)\cdot B^{(0)} < 0$. Thus $$Z_0 \cdot B^{(0)} < -K_X\cdot B^{(0)} = (B^{(0)})^2 + 2.$$ Set $Z_1 = Z_0 - B^{(0)}$. Suppose that $Z_1$ is nonzero. Then $Z_1$ is again effective, and by Artin's criterion $p_a(Z_1) \leq 0$. Thus by repeating the above argument there is a $B^{(1)}$ contained in the support of $Z_1$ such that $Z_1 \cdot B^{(1)} < (B^{(1)})^2 + 2$. Continuing, we eventually find $B^{(2)}, \dots, B^{(k)}$ with $B^{(i)}$ contained in the support of $Z_i$, $Z_{i+1} = Z_i -B^{(i)}$ and $Z_k = B^{(k)}$, and such that $Z_i \cdot B^{(i)} < (B^{(i)})^2 +2$. If we now relabel $B^{(i)} = B_{k-i}$, then $Z_i = \sum _{j\leq n-i}B_j$ and the curves $B_0, \dots, B_k$ are as claimed. \endproof Returning to the proof of Theorem 2.3, we first prove that $$H^0(Z_0; Hom(V, V)\otimes \scrO_X(Z_0)|Z_0) = 0.$$ We have the exact sequence $$0 \to \scrO_{C_{i-1}}(C_{i-1}) \to \scrO_{C_i}(C_i) \to \scrO_{B_i}(C_i) \to 0.$$ Tensor this sequence by $Hom (V, V)$. We shall prove by induction that $$H^0(Hom (V, V) \otimes \scrO_{C_i}(C_i)) = 0$$ for all $i$. It suffices to show that $H^0(Hom (V, V) \otimes \scrO_{B_i}(C_i)) = 0$ for all $i$. Now $\scrO_{B_i}(C_i)$ is a line bundle on the smooth rational curve $B_i$. If $V|B_i \cong \scrO_{\Pee^1}(a) \oplus \scrO_{\Pee^1}(a+1)$, then $w\cdot B_i \neq 0$ and so $B_i^2$ is odd. Since $B_i$ is not an exceptional curve, $B_i^2 \leq -3$ and so $B_i \cdot C_i \leq -2$. Thus, as $$Hom (V, V)|B_i = \scrO_{\Pee^1}(-1) \oplus \scrO_{\Pee^1} \oplus \scrO_{\Pee^1} \oplus \scrO_{\Pee^1}(1),$$ we see that $H^0(Hom (V, V)\otimes \scrO_{B_i}(C_i)) = 0$. Likewise if $V|D_i \cong \scrO_{\Pee^1}(a) \oplus \scrO_{\Pee^1}(a)$, then using $B_i \cdot C_i \leq -1$ we again have $H^0(Hom (V, V) \otimes \scrO_{B_i}(C_i)) = 0$. Thus by induction $$H^0(Hom (V, V) \otimes \scrO_{C_k}(C_k)) = H^0(Hom (V, V) \otimes \scrO_{Z_0}(Z_0))=0.$$ The vanishing of $H^0(mZ_0; Hom(V, V)\otimes \scrO_X(mZ_0)|mZ_0)$ is similar, using instead the exact sequence $$0 \to \scrO_{mZ_0+C_{i-1}}(mZ_0 + C_{i-1}) \to \scrO_{mZ_0 +C_i}(mZ_0 + C_i) \to \scrO_{B_i}(mZ_0 + C_i) \to 0.$$ This concludes the proof in the case of a rational singularity. For minimally elliptic singularities, we shall deduce the theorem from the following more general result: \theorem{2.5} Let $\varphi \: X \to \bar X$ be a birational morphism from $X$ to a normal projective surface $\bar X$, corresponding to a nef, big, and eventually base point free divisor $M$. Let $w$ be an arbitrary element of $H^2(X; \Zee/2\Zee)$, and suppose that \roster \item"{(i)}" $\varphi$ contracts no exceptional curve; in other words, if $E$ is an exceptional curve of the first kind on $X$, then $M\cdot E>0$. \item"{(ii)}" If $D$ is a component of $\varphi^{-1}(x)$ such that $w\cdot D \neq 0$, then $Z_0 \cdot D <0$, where $Z_0$ is the fundamental cycle of $\varphi^{-1}(x)$. \endroster Then the conclusions of Theorem \rom{2.3} hold for the moduli space $\frak M_L(w,p)$ for all $p\ll 0$. In particular the conclusions of Theorem \rom{2.3} hold if $\varphi^{-1}(x)$ is irreducible. \endstatement \demo{Proof that \rom{(2.5)} implies \rom{(2.3)}} We must show that every minimally elliptic singularity satisfies the hypotheses of Theorem 2.5(ii), provided that $w$ is the mod 2 reduction of $K_X$. Suppose that $x$ is minimally elliptic and that $w\cdot D \neq 0$. Thus $K_X\cdot D$ is odd. Moreover if $D$ is smooth rational then $D^2\neq -1$ and $K_X\cdot D \geq 0$ so that $K_X\cdot D \geq 1$. Now $(K_X+Z_0)\cdot D = 0$. Thus $Z_0 \cdot D = -(K_X\cdot D)\leq -1$. Likewise if $p_a(D) \neq 0$, so that $D$ is not a smooth rational curve, then $\varphi ^{-1}(x) = D$ is an irreducible curve and (2.3) again follows. \endproof \demo{Proof of Theorem \rom{2.5}} We begin with a lemma on sections of line bundles over effective cycles supported in $\varphi^{-1}(x)$, which generalizes (2.6) of [22]: \lemma{2.6} Let $Z_0$ be the fundamental cycle of $\varphi ^{-1}(x)$ and let $\lambda$ be a line bundle on $Z_0$ such that $\deg (\lambda|D)\leq 0$ for each component $D$ of the support of $Z_0$. Then either $H^0(Z_0; \lambda) =0$ or $\lambda = \scrO_{Z_0}$ and $H^0(Z_0; \lambda) \cong \Cee$. \endstatement \proof Choose a computation sequence for $Z_0$, say $A_1, A_2, \dots, A_k$. Thus, if we set $Z_i = \sum _{j\leq i}A_j$, then $Z_i\cdot A_{i+1} >0$, and $Z_k = Z_0$. Now we have an exact sequence $$0 \to \scrO_{A_{i+1}}(-Z_i)\to \scrO_{Z_{i+1}} \to \scrO_{Z_i} \to 0.$$ Thus $\deg (\scrO_{A_{i+1}}(-Z_i) \otimes \lambda|A_{i+1})<0$. It follows that $H^0(\scrO_{Z_{i+1}}\otimes \lambda) \subseteq H^0(\scrO_{Z_i}\otimes \lambda)$ for all $i$. By induction $\dim H^0(\scrO_{Z_i}\otimes \lambda) \leq 1$ for all $i$, $1\leq i \leq k$. Thus $\dim H^0(Z_0; \lambda) \leq 1$. Moreover, if $\dim H^0(Z_0; \lambda) = 1$, then the natural map $$H^0(\scrO_{Z_{i+1}}\otimes \lambda) \to H^0(\scrO_{Z_i}\otimes \lambda)$$ is an isomorphism for all $i$, and so the induced map $H^0(Z_0; \lambda) \to H^0(A_1; \lambda |A_1)$ is an isomorphism and $\dim H^0(A_1; \lambda |A_1) =1$. Thus $\lambda |A_1$ is trivial and a nonzero section of $H^0(Z_0; \lambda)$ restricts to a generator of $\lambda |A_1$. Since we can begin a computation sequence with an arbitrary choice of $A_1$, we see that a nonzero section $s$ of $H^0(Z_0; \lambda)$ restricts to a nonvanishing section of $H^0(D; \lambda |D)$ for every $D$ in the support of $\varphi ^{-1}(x)$. Thus the map $\scrO_{Z_0} \to \lambda$ defined by $s$ is an isomorphism. \endproof \noindent {\bf Remark.} The lemma is also true if $\lambda$ is allowed to have degree one on some components $D$ of $Z_0$ with $p_a(D)\geq 2$, provided that $\lambda |D$ is general for these components, and a slight variation holds if $\lambda$ is also allowed to have degree one on some components $D$ of $Z_0$ with $p_a(D)=1$. \medskip We next construct a bundle $W$ over $Z_0$ with certain vanishing properties: \lemma{2.7} Suppose that $\varphi \: X \to \bar X$ is the minimal resolution of the normal surface singularity $x$. Let $\mu$ be a line bundle over the scheme $Z_0$. Suppose further that, if $D$ is a component of $\varphi^{-1}(x)$ such that $\deg (\mu |D)$ is odd, then $Z_0 \cdot D <0$, where $Z_0$ is the fundamental cycle of $\varphi^{-1}(x)$. Then there exists a rank two vector bundle $W$ over $Z_0$ with $\det W = \mu$ and such that $$H^0(Z_0; Hom(W,W)\otimes \scrO_X(mZ_0)|Z_0) = 0$$ for every $m\geq 1$. \endstatement \proof Let $\varphi ^{-1}(x) = \bigcup_iD_i$. Then there exists an integer $a_i$ such that $\deg\mu |D_i = 2a_i$ or $2a_i+1$, depending on whether $\deg (\mu| D_i)$ is odd or even. Since $\dim Z_0 = 1$, the natural maps $\operatorname{Pic}Z_0 \to \operatorname{Pic}(Z_0)_{\text{red}} \to \bigoplus _i\operatorname{Pic}D_i$ are surjective. Thus we may choose a line bundle $L_1$ over $Z_0$ such that $\deg (L_1|D_i) = a_i$. It follows that $\mu \otimes L_1^{\otimes - -2}|D_i$ is a line bundle over $D_i$ of degree zero or 1, and if it is of degree 1, then $Z_0\cdot D_i <0$. Hence $\mu \otimes L_1^{\otimes -2}\otimes \scrO_{Z_0}(Z_0)$ has degree at most zero on $D_i$ for every $i$. Set $L_2 = \mu \otimes L_1^{-1}$. Thus $L_1 \otimes L_2 = \mu$ and $\deg (L_2|D_i) = a_i$ or $a_i +1$ depending on whether $\deg (\mu| D_i)$ is even or odd. The line bundle $L_1^{-1}\otimes L_2=\mu \otimes L_1^{\otimes -2}$ thus has degree zero on those components $D_i$ such that $\deg (\mu| D_i)$ is even and 1 on the components $D_i$ such that $\deg (\mu| D_i)$ is odd. Moreover $\deg (L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0}(mZ_0)|D_i) \leq 0$ for every $i$. \claim{2.8} Under the assumptions of \rom{(2.7)}, there exists a nonsplit extension $W$ of $L_2$ by $L_1$ except in the case where $x$ is rational, $\deg (\mu| D_i)$ is odd for at most one $i$, and the multiplicity of $D_i$ in $Z_0$ is one for such $i$, or $\chi (\scrO_{Z_0}) = 0$ and $\deg \mu |D_i$ is even for every $i$. \endstatement \proof A nonsplit extension exists if and only if $h^1(L_2^{-1}\otimes L_1) \neq 0$. Now by the Riemann-Roch theorem applied to $Z_0 = \sum _in_iD_i$, we have $$h^1(Z_0; L_2^{-1}\otimes L_1) = h^0(Z_0; L_2^{-1}\otimes L_1) -\sum _in_i\deg (L_2^{-1}\otimes L_1|D_i) - \chi (\scrO_{Z_0}).$$ Here $\deg (L_2^{-1}\otimes L_1|D_i)= 0$ on those $D_i$ with $\deg (\mu| D_i)$ even and $=-1$ on the $D_i$ with $\deg (\mu| D_i)$ odd. Moreover $h^0(\scrO_{Z_0}) = 1$ by Lemma 2.6 and so $\chi (\scrO_{Z_0})\leq 1$, with $\chi (\scrO_{Z_0}) =1$ if and only if $x$ is rational. Thus $$h^1(Z_0; L_2^{-1}\otimes L_1) \geq \sum \{\, n_i\: \deg (\mu| D_i) \text{ is odd}\,\} - \chi (\scrO_{Z_0}).$$ Hence if $h^1(Z_0; L_2^{-1}\otimes L_1) =0$, then either $x$ is rational, $\deg (\mu| D_i)$ is odd for at most one $i$, and for such $i$ the multiplicity of $D_i$ in $Z_0$ is one, or $\deg (\mu |D_i)$ is even for all $i$ and $\chi (\scrO_{Z_0}) = 0$. \endproof Returning to the proof of (2.7), choose $W$ to be a nonsplit extension of $L_2$ by $L_1$ if such exist, and set $W = L_1 \oplus L_2$ otherwise. To see that $H^0(Z_0; Hom(W,W)\otimes \scrO_X(mZ_0)|Z_0) = 0$, we consider the two exact sequences $$\gather 0 \to L_1 \to W \to L_2 \to 0;\\ 0 \to L_1 \otimes \scrO_{Z_0}(mZ_0)\to W\otimes \scrO_{Z_0}(mZ_0) \to L_2\otimes \scrO_{Z_0}(mZ_0) \to 0. \endgather$$ Clearly $H^0(Z_0; Hom(W,W)\otimes \scrO_X(mZ_0)|Z_0) = 0$ if $$H^0(L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0}(mZ_0)) = H^0(\scrO_{Z_0}(mZ_0)) = H^0(L_2^{-1}\otimes L_1 \otimes \scrO_{Z_0}(mZ_0)) =0.$$ The line bundles $\scrO_{Z_0}(mZ_0)$ and $L_2^{-1}\otimes L_1 \otimes \scrO_{Z_0}(mZ_0)$ have nonpositive degree on each $D_i$ and (since $Z_0 \cdot D_i <0$ for some $i$) have strictly negative degree on at least one component. Thus by Lemma 2.6 $H^0(\scrO_{Z_0}(mZ_0))$ and $H^0(L_2^{-1}\otimes L_1 \otimes \scrO_{Z_0}(mZ_0))$ are both zero. Let us now consider the group $H^0(L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0}(mZ_0))$. By the hypothesis that $Z_0\cdot D_i < 0$ for each $D_i$ such that $\deg (\mu |D_i)$ is odd, the line bundle $L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0}(mZ_0)$ has nonpositive degree on all components $D_i$. Thus by Lemma 2.6 either $H^0(L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0}(mZ_0)) = 0$ or $L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0} (mZ_0) \cong \scrO_{Z_0}$. Clearly this last case is only possible if $m=1$ and $L_1 \cong L_2 \otimes \scrO_{Z_0}(Z_0)$, and if moreover $Z_0 \cdot D_i =0$ if $\deg(\mu |D_i)$ is even and $Z_0 \cdot D_i =-1$ if $\deg(\mu |D_i)$ is odd. As $Z_0\cdot D_i <0$ for at least one $i$, $\deg(\mu |D_i)$ is odd for at least one $i$ as well. In this case, if the nonzero section of $L_1^{-1}\otimes L_2 \otimes \scrO_{Z_0}(Z_0)$ lifts to give a map $L_1\to W\otimes \scrO_{Z_0}(Z_0)$, then the image of $L_1$ in $W\otimes \scrO_{Z_0}(Z_0)$ splits the exact sequence $$0 \to L_1 \otimes \scrO_{Z_0}(Z_0)\to W\otimes \scrO_{Z_0}(Z_0) \to L_2\otimes \scrO_{Z_0}(Z_0) \to 0.$$ Thus $W$ is also a split extension. By Claim 2.8, since $\deg (\mu |D_i)$ is odd for at least one $i$, it must therefore be the case that $x$ is rational, $\deg (\mu| D_i)$ is odd for exactly one $i$, and for such $i$ the multiplicity of $D_i$ in $Z_0$ is one. Moreover $Z_0\cdot D_j \neq 0$ exactly when $j=i$ and in this case $Z_0 \cdot D_i = -1$. But as the multiplicity of $D_i$ in $Z_0$ is 1, we can write $Z_0 = D_i + \sum _{j\neq i}n_jD_j$, and thus $$Z_0^2 = Z_0\cdot D_i = -1.$$ By a theorem of Artin [1], however, $-Z_0^2$ is the multiplicity of the rational singularity $x$. It follows that $x$ is a smooth point and $\varphi$ is the contraction of a generalized exceptional curve, contrary to hypothesis. This concludes the proof of (2.7). \endproof We may now finish the proof of (2.5). Start with a generic vector bundle $V_0 \in \frak M_L(w,p)$ on $X$ satisfying the condition that $H^1(X; \operatorname{ad}V_0)\to H^1(Z_0; \operatorname{ad}V_0|Z_0)$ is surjective. If $\mu = \det V_0|Z_0$, note that, according to the assumptions of (2.5), $\mu$ satisfies the hypotheses of Lemma 2.7. For $V\in \frak M_L(w,p)$, let $$H(mZ_0) = H^0(mZ_0; \operatorname{ad}V\otimes \scrO_X(mZ_0)|mZ_0).$$ Using the exact sequence $$0 \to H((m-1)Z_0) \to H(mZ_0) \to H^0(Z_0;\operatorname{ad}V_0\otimes \scrO_X(mZ_0)|Z_0),$$ we see that it suffices to show that, for a generic $V$, $H^0(Z_0;\operatorname{ad}V\otimes \scrO_X(mZ_0)|Z_0) =0$ for all $m\geq 1$. For a fixed $m$, the condition that $H^0(Z_0;\operatorname{ad}V\otimes \scrO_X(mZ_0)|Z_0) \neq 0$ is a closed condition. Thus since the moduli space cannot be a countable union of proper subvarieties, it will suffice to show that the set of $V$ for which $H^0(Z_0;\operatorname{ad}V\otimes \scrO_X(mZ_0)|Z_0) = 0$ is nonempty for every $m$. Let $\Cal S$ be the germ of the versal deformation of $V_0|Z_0$ keeping $\det V_0|Z_0$ fixed. By the assumption that the map from the germ of the versal deformation of $V_0$ to that of $V_0|Z_0$ is submersive, it will suffice to show that, for each $m\geq 1$, the set of $W\in \Cal S$ such that $H^0(Z_0;\operatorname{ad}W\otimes \scrO_{Z_0}(mZ_0)) = 0$ is nonempty. One natural method for doing so is to exhibit a deformation from $V_0|Z_0$ to the $W$ constructed in the course of Lemma 2.7; roughly speaking this amounts to the claim that the ``moduli space" of vector bundles on the scheme $Z_0$ is connected. Although we shall proceed slightly differently, this is the main idea of the argument. Choose an ample line bundle $\lambda$ on $Z_0$. After passing to some power, we may assume that both $(V_0|Z_0)\otimes \lambda$ and $W\otimes \lambda$ are generated by their global sections. A standard argument shows that, in this case, both $V_0|Z_0$ and $W$ can be written as an extension of $\mu \otimes \lambda$ by $\lambda ^{-1}$: Working with $W$ for example, we must show that there is a map $\lambda ^{-1}\to W$, corresponding to a section of $W\otimes \lambda$, such that the quotient is again a line bundle. It suffices to show that there exists a section $s\in H^0(Z_0; W\otimes \lambda)$ such that, for each $z\in Z_0$, $s(z)\neq 0$ in the fiber of $W\otimes \lambda$ over $z$. Now for $z$ fixed, the set of $s \in H^0(Z_0; W\otimes \lambda)$ such that $s(z) =0$ has codimension two in $H^0(Z_0; W\otimes \lambda)$ since $W\otimes \lambda$ is generated by its global sections. Thus the set of $s \in H^0(Z_0; W\otimes \lambda)$ such that $s(z) =0$ for some $z\in Z_0$ has codimension at least one, and so there exists an $s$ as claimed. Now let $W_0 = \lambda ^{-1} \oplus (\mu \otimes \lambda)$. Let $(\Cal S_0, s_0)$ be the germ of the versal deformation of $W_0$ (with fixed determinant $\mu$). As $Z_0$ has dimension one, $\Cal S_0$ is smooth. Both $V_0|Z_0$ and $W$ correspond to extension classes $\xi, \xi ' \in \operatorname{Ext}^1(\mu \otimes \lambda, \lambda ^{-1})$. Replacing, say, $\xi$ by the class $t\xi, t\in \Cee ^*$, gives an isomorphic bundle. In this way we obtain a family of bundles $\Cal V$ over $Z_0 \times \Cee$, such that the restriction of $\Cal V$ to $Z_0 \times t$ is $V_0|Z_0$ if $t\neq 0$ and is $W_0$ if $t=0$. Hence in the germ $\Cal S_0$ there is a subvariety containing $s_0$ in its closure and consisting of bundles isomorphic to $V_0|Z_0$, and similarly for $W$. As $H^0(Z_0;\operatorname{ad}W\otimes \scrO_{Z_0}(mZ_0)) = 0$, the locus of bundles $U$ in $\Cal S_0$ for which $H^0(Z_0;\operatorname{ad}U\otimes \scrO_{Z_0}(mZ_0)) = 0$ is a dense open subset. Since $\Cal S_0$ is a smooth germ, it follows that there is a small deformation of $V_0|Z_0$ to such a bundle. Thus the generic small deformation $U$ of $V_0|Z_0$ satisfies $H^0(Z_0; \operatorname{ad}U\otimes \scrO_{Z_0}(mZ_0)) = 0$, and so the generic $V\in \frak M_L(w,p)$ has the property that $H^0(Z_0;\operatorname{ad}V\otimes \scrO_X(mZ_0)|Z_0) = 0$ for all $m\geq 1$ as well. As we saw above, this implies the vanishing of $H^0(mZ_0; \operatorname{ad}V\otimes \scrO_X(mZ_0)|mZ_0)$. \endproof \noindent {\bf Remark.} (1) Suppose that $\bar X$ is a singular surface, but that $\varphi \: X \to \bar X$ is not the minimal resolution. We may still define the fundamental cycle $Z_0$ for the resolution $\varphi$. Moreover it is easy to see that $Z_0 \cdot E = 0$ for every component of a generalized exceptional curve contained in $\varphi ^{-1}(x)$. Thus the hypothesis of (ii) of Theorem 2.5 implies that $w\cdot E=0$ for such curves. \smallskip (2) We have only considered contractions of a very special type, and have primarily been interested in the case where $w$ is the mod two reduction of $[K_X]$. However it is natural to ask if the analogues of Theorem 2.3 and 2.5 (and thus Theorem 1.4) holds for more general contractions and choices of $w$, provided of course that no smooth rational curve of self-intersection $-1$ is contracted to a point. Clearly the proof of Theorem 2.5 applies to a much wider class of singularities. Indeed a little work shows that the proof goes over (with some modifications in case there are components of arithmetic genus one) to handle the case where we need only assume condition (ii) of (2.5) for those components $D$ which are smooth rational curves. Another case where it is easy to check that the conclusions of (2.5) hold is where $w$ is arbitrary and the dual graph of the singularity is of type $A_k$. We make the following rather natural conjecture: \medskip \noindent {\bf Conjecture 2.9.} The conclusions of Theorem \rom{1.4} hold for arbitrary choices of $w$ and $\varphi$, provided that $\varphi$ does not contract any exceptional curves of the first kind. \section{3. Nonexistence of embedded $\boldkey 2$-spheres} \ssection{3.1. A base point free theorem} \theorem{3.1} Let $\pi \: X \to X'$ be a birational morphism from the smooth surface $X$ to a normal surface $X'$, not necessarily projective. Suppose that $X$ is a minimal surface of general type, and that $p\in X'$ is an isolated singular point which is a nonrational singularity. Let $\pi ^{-1}(p) = \bigcup _i D_i$. Then: \roster \item"{(i)}" There exist nonnegative integers $n_i$ with $n_i >0$ for at least one $i$ such that $K_X+\sum _in_i D_i$ is nef and big. \item"{(ii)}" Suppose that $q(X) =0$. Then there further exists a choice of $D= \sum _in_iD_i$ satisfying \rom{(i)} with $D$ connected and such that there exists a section of $K_X + D$ which is nowhere vanishing in a neighborhood of $$E=\bigcup\{\,D_j: (K_X + D)\cdot D_j = 0\,\}.$$ In this case either $E= \emptyset$ or $E=\operatorname{Supp}D$ and $D$ is the fundamental cycle of the minimal resolution of a minimally elliptic singularity. \item"{(iii)}" With $D$ satisfying \rom{(i)} and \rom{(ii)}, the linear system $K_X+D$ is eventually base point free. Moreover, if $\varphi \: X \to \bar X$ is the associated contraction, then $\bar X$ is a normal projective surface all of whose singular points are either rational or minimally elliptic. \endroster \endproclaim \proof To prove (i), consider the set of all effective cycles $D = \sum _ia_iD_i$, where the $a_i$ are nonnegative integers, not all zero, and such that $h^1(\scrO_D) \neq 0$. This set is not empty by the definition of a nonrational singularity, and is partially ordered by $\leq$, where $D'\leq D$ if $D-D'$ is effective. Choose a minimal element $D$ in the set. This means that $D = \sum _in_iD_i$ where either $n_i = 1$ for exactly one $i$ and $h^1(\scrO_{D_i}) \neq 0$, or for every irreducible $D_i$ contained in the support of $D$, $D-D_i=D'$ is effective and $h^1(\scrO_{D'})= 0$. If $D''$ is then any nonzero effective cycle with $D''< D$, then there exists an $i$ such that $\scrO_{D-D_i} \to \scrO_{D''}$ is surjective. By a standard argument, $H^1(\scrO_{D-D_i}) \to H^1(\scrO_{D''})$ is surjective and thus $h^1(\scrO_{D''})=0$ for every nonzero effective $D''<D$. Finally note that $D$ is connected, since otherwise we could replace $D$ by some connected component $D_0$ with $h^1(\scrO_{D_0})\neq 0$. Next we claim that $K_X+D$ is nef. Since $K_X$ is nef, it is clear that $(K_X+D)\cdot C \geq 0$ for every irreducible curve $C$ not contained in the support of $D$, and moreover, for such curves $C$, $(K_X+D)\cdot C = 0$ if and only if $C$ is a smooth rational curve of self-intersection $-2$ disjoint from the support of $D$. Next suppose that $D_i$ is a curve in the support of $D$ and consider $(K_X+D)\cdot D_i$. If $D= D_i$ then $K_X+D_i |D_i = \omega _{D_i}$, the dualizing sheaf of $D_i$, and this has degree $2p_a(D_i)-2 \geq 0$ since $p_a(D_i) =h^1(\scrO_{D_i}) > 0$. Otherwise let $D' = D-D_i$ consider the exact sequence $$0 \to \scrO_{D_i}(-D') \to \scrO_D \to \scrO_{D'} \to 0.$$ Thus the natural map $H^1(\scrO_{D_i}(-D')) \to H^1(\scrO_D)$ is surjective since $H^1(\scrO_{D'})=0$, and so $H^1(\scrO_{D_i}(-D'))\neq 0$ as $H^1(\scrO_D)\neq 0$. By duality $H^0(D_i; \omega _{D_i}\otimes \scrO_{D_i}(D'))\neq 0$. On the other hand $\omega _{D_i} = K_X+D_i|D_i$, and so $\deg (K_X+D'+D_i)|D_i = (K_X+D)\cdot D_i \geq 0$; moreover $(K_X+D)\cdot D_i = 0$ only if the divisor class $K_X+D|D_i$ is trivial. Next, $(K_X+D)^2 \geq K_X^2 >0$, so that $K_X+D$ is big. In fact, $$(K_X+D)^2 = K_X\cdot (K_X+D)+(K_X+D)\cdot D\geq K_X^2 + K_X\cdot D \geq K_X^2.$$ Thus $K_X+D$ is big. To see (ii), let $E = \bigcup\{\,D_j\subseteq \operatorname{Supp}D: (K_X + D)\cdot D_j = 0\,\}$. We shall also view $E$ as a reduced divisor. We claim that $\scrO_E(K_X+D) = \scrO_E$. First assume that $E = D$ (and thus in particular that $D$ is reduced); in this case we need to show that $\omega _D = \scrO_D$. By assumption $D$ is connected. Then $\omega _D$ has degree zero on every reduced irreducible component of $D$, and by Serre duality $\chi (\omega _D) = - \chi (\scrO_D) = \frac12(K_X+D)\cdot D=0$. As $h^1(\omega _D) = h^0(\scrO_D) = 1$, $h^0(\omega _D) = 1$ as well. As $\omega _D$ has degree zero on every component of $D$, if $s$ is a section of $\omega _D$, then the restriction of $s$ to every component $D_i$ of $D$ is either identically zero or nowhere vanishing. Thus if $s$ is nonzero, since $D$ is connected, $s$ must be nowhere vanishing. It follows that the map $\scrO_D\to \omega _D$ is surjective and is thus an isomorphism. If $D\neq E$, we apply the argument that showed above that $(K_X+D)\cdot D_i \geq 0$ to each connected component $E_0$ of the divisor $E$, with $D'=D-E_0$, to see that there is a section of $\scrO_{E_0}(K_X+D)$. Since $\scrO_{E_0}(K_X+D)$ has degree zero on each irreducible component of $E_0$, the argument that worked for the case $D=E$ also works in this case. Now let us show that, provided $q(X)=0$, a nowhere zero section of $\scrO_E(K_X+D) = \scrO_E$ lifts to a section of $K_X+D$. It suffices to show that, for every connected component $E_0$ of $E$, a nowhere vanishing section of $\scrO_{E_0}(K_X+D)$ lifts to a section of $K_X+D$. Let $D' = D-E_0$. If $D'=0$ then $D=E=E_0$ and we ask if the map $H^0(\scrO_X(K_X+D)) \to H^0(\scrO_D(K_X+D))$ is surjective. The cokernel of this map lies in $H^1(K_X) = 0$ since $X$ is regular. Otherwise $D' \neq 0$. Beginning with the exact sequence $$0 \to \scrO_{D'}(-E_0)\to \scrO_D \to \scrO_{E_0} \to 0,$$ and tensoring with $\scrO_X(K_X+D)$, we obtain the exact sequence $$0 \to \scrO_{D'}(K_X+D-E_0)\to \scrO_D(K_X+D) \to \scrO_{E_0} \to 0.$$ Now $\scrO_{D'}(K_X+D-E_0) = \scrO_{D'}(K_X+D')= \omega _{D'}$ and by duality $h^0(\omega _{D'}) = h^1(\scrO_{D'}) = 0$. Thus $H^0(\scrO_D(K_X+D))$ includes into $H^0(\scrO_{E_0}) = \Cee$ and so it suffices to prove that $H^0(\scrO_D(K_X+D)) \neq 0$, in which case it has dimension one. On the other hand, using the exact sequence $$0 \to \scrO_X(K_X) \to \scrO_X(K_X+D) \to \scrO_D(K_X+D) \to 0,$$ we see that $h^0(\scrO_D(K_X+D)) \geq h^0(\scrO_X(K_X+D))-p_g(X)$. Since $h^2(K_X+D) = h^0(-D)=0$, the Riemann-Roch theorem implies that $$h^0(\scrO_X(K_X+D)) = h^1(\scrO_X(K_X+D)) + \frac12(K_X+D)\cdot D + 1 + p_g(X).$$ Since all the terms are positive, we see that indeed $h^0(\scrO_X(K_X+D))-p_g(X)\geq 1$, and that $h^0(\scrO_X(K_X+D))-p_g(X)=1$ if and only if $h^1(\scrO_X(K_X+D)) = 0$ and $(K_X+D)\cdot D_i = 0$ for every component $D_i$ contained in the support of $D$. This last condition says exactly that $E= \operatorname{Supp}D$, and thus, as $D$ is connected, that $E_0=E$. We claim that in this last case $D$ is minimally elliptic. Indeed, for every effective divisor $D'$ with $0<D'<D$, we have $$p_a(D') = 1 - h^0(\scrO_{D'}) + h^1(\scrO_{D'})= 1 - h^0(\scrO_{D'})\leq 0.$$ Thus $D$ is the fundamental cycle for the resolution of a minimally elliptic singularity. Finally we prove (iii). The irreducible curves $C$ such that $(K_X+D)\cdot C=0$ are the components $D_i$ of the support of $D$ such that $(K_X+D)\cdot D_i = 0$, as well as smooth rational curves of self-intersection $-2$ disjoint from $\operatorname{Supp}D$. These last contribute rational double points, so that we need only study the $D_i$ such that $(K_X+D)\cdot D_i = 0$. We have seen in (ii) that either there are no such $D_i$, or every $D_i$ in the support of $D$ satisfies $(K_X+D)\cdot D_i = 0$ and the contraction of $D$ is a minimally elliptic singularity. Let $\bar X$ be the normal surface obtained by contracting all the irreducible curves $C$ on $X$ such that $(K_X+D)\cdot C=0$. The line bundle $\scrO_X(K_X+D)$ is trivial in a neighborhood of these curves, either because they correspond to a rational singularity or because we are in the minimally elliptic case and by (ii). So $\scrO_X(K_X+D)$ induces a line bundle on $\bar X$ which is ample, by the Nakai-Moishezon criterion. Thus $|k(K_X+D)|$ is base point free for all $k\gg0$. \endproof \ssection{3.2. Completion of the proof} We now prove Theorem 1.5: \theorem{1.5} Let $X$ be a minimal simply connected algebraic surface of general type, and let $E\in H^2(X; \Zee)$ be a $(1,1)$-class satisfying $E^2=-1$, $E\cdot K_X = 1$. Let $w$ be the \rom{mod} $2$ reduction of $[K_X]$. Then there exist: \roster \item"{(i)}" an integer $p$ and \rom(in case $p_g(X)=0$\rom) a chamber $\Cal C$ of type $(w,p))$ and \item"{(ii)}" a $(1,1)$-class $M\in H^2(X; \Zee)$ \endroster such that $M\cdot E=0$ and $\gamma _{w,p}(X)(M^d) \neq 0$ \rom(or, in case $p_g(X)=0$, $\gamma _{w,p}(X; \Cal C)(M^d) \neq 0$\rom). \endstatement \noindent {\it Proof.} We begin with the following lemma: \lemma{3.2} With $X$ and $E$ as above, there exists an orientation preserving diffeomorphism $\psi \: X \to X$ such that $\psi ^*[K_X] = [K_X]$ and such that $\psi ^*E\cdot [C] \geq 0$ for every smooth rational curve $C$ on $X$ with $C^2 = -2$. \endstatement \proof Let $\Delta = \{[C_1], \dots, [C_k]\}$ be the set of smooth rational curves on $X$ of self-intersection $-2$, and let $r_i\: H^2(X; \Zee) \to H^2(X; \Zee)$ be the reflection about the class $[C_i]$. Then $r_i$ is realized by an orientation-preserving self-diffeomorphism of $X$, $r_i^*[K_X] = [K_X]$, and $r_i$ preserves the image of $\operatorname{Pic}X$ inside $H^2(X; \Zee)$. Let $\Gamma$ be the finite group generated group by the $r_i$. Since the classes $[C_i]$ are linearly independent, the set $$\{\, x\in H^2(X; \Ar)\: x\cdot [C_i] \geq 0\,\}$$ has a nonempty interior. Moreover, if $\Delta ' = \Gamma \cdot \Delta$, and we set $W^\delta = \delta ^\perp$ for $\delta \in \Delta'$, then the connected components of the set $H^2(X; \Ar) - \bigcup _{\delta \in \Delta '}W^\delta$ are the fundamental domains for the action of $\Gamma$ on $H^2(X; \Ar)$. Clearly at least one of these connected components lies inside $\{\, x\in H^2(X; \Ar)\: x\cdot [C_i] \geq 0\,\}$. Thus given $E$ (or indeed an arbitrary element of $H^2(X; \Ar)$), there exists a $\gamma \in \Gamma$ such that $\gamma (E)\cdot [C_i] \geq 0$ for all $i$. As every $\gamma \in \Gamma$ is realized by an orientation preserving self-diffeomorphism $\psi$, this concludes the proof of (3.2). \endproof Thus, to prove Theorem 1.5, it is sufficient by the naturality of the Donaldson polynomials to prove it for every class $E$ satisfying $E^2=-1$, $E\cdot K_X = 1$, and $E\cdot [C] \geq 0$ for every smooth rational curve $C$ on $X$ with $C^2 = -2$. We therefore make this assumption in what follows. Given Theorem 1.4, it therefore suffices to find a nef and big divisor $M$ orthogonal to $E$, which is eventually base point free, such that the contraction morphism defined by $|kM|$ has an image with at worst rational and minimally elliptic singularities (note that, since $X$ is assumed minimal, no exceptional curves can be contracted). Thus we will be done by the following lemma: \lemma{3.3} There exists a nef and big divisor $M$ which is eventually base point free and such that \roster \item $M\cdot E = 0$. \item The contraction $\bar X$ of $X$ defined by $|kM|$ for all $k\gg 0$ has only rational and minimally elliptic singularities. \endroster \endstatement \proof To find $M$ we proceed as follows: consider the divisor $K_X + E=M$. As $K_X\cdot E=1$ and $E^2 = -1$, $M$ is orthogonal to $E$. Moreover $M^2=(K_X+E)^2 = K_X^2 + 1 > 0$. We now consider separately the cases where $M$ is nef and where $M$ is not nef. \medskip \noindent {\bf Case I:} $M= K_X+E$ is nef. Consider the union of all the curves $D$ such that $M \cdot D = 0$. The intersection matrix of the $D$ is negative definite, and so we can contract all the $D$ on $X$ to obtain a normal surface $X'$. If $X'$ has only rational singularities, then $M$ induces an ample divisor on $X'$ and so $M$ itself is eventually base point free. In this case we are done. Otherwise we may apply Theorem 3.1 to find a subset $D_1, \dots, D_t$ of the curves $D$ with $M\cdot D=0$ and positive integers $a_i$ such that the divisor $K_X+\sum _ia_iD_i$ is nef, big, and eventually base point free, and such that the contraction $\bar X$ of $X$ has only rational and minimally elliptic singularities, with exactly one nonrational singularity. Note that $D_i\cdot E = -D_i \cdot K_X \leq 0$, and $D_i \cdot E = 0$ if and only if $D_i \cdot K_X = 0$, or in other words if and only if $D_i$ is a smooth rational curve of self-intersection $-2$. Setting $e = -\sum _ia_i(D_i \cdot E)$, we have $e \geq 0$, and $e=0$ if and only if $D_i \cdot E =0$ for all $i$. But as $\bar X$ has a nonrational singularity, we cannot have $D_i \cdot E =0$ for all $i$, for then all singularities would be rational double points. Thus $e>0$. Now the $\Bbb Q$-divisor $M'= K_X + \frac1{e}\sum _ia_iD_i$ is a rational convex combination of $K_X$ and $K_X+\sum _ia_iD_i$, and $M' \cdot E=0$. Moreover either $M'$ is a strict convex combination of $K_X$ and $K_X+\sum _ia_iD_i$ (if $e>1$) or $M'=K_X+\sum _ia_iD_i$ (if $e=1$). In the second case, $M'$ satisfies (1) of Lemma 3.3, and it is eventually base point free by (iii) of Theorem 3.1. Thus $M'$ satisfies the conclusions of Lemma 3.3. In the first case, $M'$ is nef and big, and the only curves $C$ such that $M' \cdot C=0$ are curves $C$ such that $K_X \cdot C = 0$ and $(K_X+\sum _ia_iD_i )\cdot C=0$. The set of all such curves must therefore be a subset of the set of all smooth rational curves on $X$ with self-intersection $-2$. Hence, if $X''$ denotes the contraction of all the curves $C$ on $X$ such that $M'\cdot C=0$, then $X''$ has only rational singularities and $M'$ induces an ample $\Bbb Q$-divisor on $X''$. Once again some multiple of $M'$ is eventually base point free and (1) and (2) of Lemma 3.3 are satisfied. Thus we have proved the lemma in case $K_X+E$ is nef. \medskip \noindent {\bf Case II:} $M= K_X+E$ is not nef. Let $D$ be an irreducible curve with $M\cdot D<0$. We claim first that in this case $D^2<0$. Indeed, suppose that $D^2 \geq 0$. As $\operatorname{Pic}X\otimes _\Zee \Ar$ has signature $(1, \rho -1)$, the set $$\Cal Q = \{\, x\in \operatorname{Pic}X\otimes _\Zee \Ar\: x^2 \geq 0, x\neq 0\,\}$$ has two connected components, and two classes $x$ and $x'$ are in the same connected component of $\Cal Q$ if and only if $x\cdot x' \geq 0$ (cf\. [13] p\. 320 Lemma 1.1). Now $(K_X+E)\cdot K_X = (K_X+E)^2 = K_X^2 +1 >0$, so that $K_X+E$ and $K_X$ lie in the same connected component of $\Cal Q$. Likewise, if $D^2 \geq 0$, then since $K_X\cdot D \geq 0$, $K_X$ and $D$ lie in the same connected component of $\Cal Q$. Thus $D$ and $K_X+E$ lie in the same connected component of $\Cal Q$, so that $(K_X+E)\cdot D \geq 0$. Conversely, if $M\cdot D<0$, then $D^2<0$. Fix an irreducible curve $D$ with $M\cdot D<0$, and let $d = -E\cdot D >K_X\cdot D \geq 0$. Recall that by assumption $E\cdot D \geq 0$ if $D$ is a smooth rational curve of self-intersection $-2$. If $p_a(D) \geq 1$, then set $M' = K_X+ \frac1{d}D$. Then $M'\cdot E = 0$ by construction. Moreover we claim that $M'$ is nef and big. Indeed $$(M')^2 = (K_X+ \frac1{d}D)^2 = K_X\cdot (K_X+ \frac1{d}D) + \frac1{d}(K_X+ \frac1{d}D)\cdot D.$$ Thus $M'$ is big if it is nef and to see that $M'$ is nef it suffices to show that $M'\cdot D \geq 0$. But $$M'\cdot D = K_X\cdot D + \frac1{d}D^2 = 2p_a(D) - 2 - \left(1- \frac1{d}\right)D^2.$$ As $D^2 < 0$, we see that $M'\cdot D\geq 0$, and $M'\cdot D=0$ if and only if $p_a(D) = 1$ and $d=1$. Suppose that $p_a(D) = 1$ and $M'\cdot D=0$. Using the exact sequence $$0 \to \scrO_X(K_X) \to \scrO_X(K_X+D)\to \omega _D \to 0,$$ and arguments as in the proof of Theorem 3.1, we see that the linear system $M'$ is eventually base point free and that the associated contraction has just rational double points and a minimally elliptic singular point which is the image of $D$. In all other cases, $M'\cdot D >0$, so that the curves orthogonal to $M'$ are smooth rational curves of self-intersection $-2$. Again, some positive multiple of $M'$ is eventually base point free and the contraction has just rational singularities. Thus we may assume that $p_a(D) = 0$ for every irreducible curve $D$ such that $M\cdot D <0$. By assumption $D^2 \neq -1, -2$, so that $D^2 \leq -3$. Thus $d= - -D\cdot E \geq 2$. If either $D^2 \leq -4$ or $D^2 = -3$ and $d \geq 3$, then again let $M' = K_X+ \frac1{d}D$. Thus $M'\cdot E =0$ and $$M'\cdot D = K_X\cdot D + \frac1{d}D^2 = - 2 - \left(1- \frac1{d}\right)D^2\geq 0.$$ Thus $M'$ is nef and big, and some multiple of $M'$ is eventually base point free, and the associated contraction has just rational singularities. The remaining case is where there is a smooth rational curve $D$ on $X$ with self-intersection $-3$ and such that $-D\cdot E = 2$. In this case $K_X \cdot D = 1$, and so $D-E$ is orthogonal to $K_X$. Note that $D-E$ is not numerically trivial since $D$ is not numerically equivalent to $E$. Thus, by the Hodge index theorem $(D-E)^2 <0$. But $$(D-E)^2 = -3 + 4 -1 =0,$$ a contradiction. Thus this last case does not arise. \endproof \section{Appendix: On the canonical class of a rational surface} Let $\Lambda _n$ be a lattice of type $(1,n)$, i.e. a free $\Bbb Z$-module of rank $n+1$, together with a quadratic form $q\: \Lambda _n\to \Bbb Z$, such that there exists an orthogonal basis $\{ e_0, e_1, \dots , e_n\}$ of $\Lambda _n$ with $q(e_0) = 1$ and $q(e_i) = -1$ for all $i>0$. Fix once and for all such a basis. We shall always view $\Lambda _n$ as included in $\Lambda _{n+1}$ in the obvious way. Let $$\kappa _n = 3e_0 - \sum _{i=1}^ne_i.$$ Then $q(\kappa _n) = 9-n$ and $\kappa _n$ is characteristic, i.e. $\kappa _n\cdot\alpha \equiv q(\alpha) \mod 2$ for all $\alpha \in \Lambda _n$. The goal of this appendix is to give a proof, due to the first author, R\. Miranda, and J\.W\. Morgan, of the following: \theorem{A.1} Suppose that $n\leq 8$ and that $\kappa\in \Lambda _n$ is a characteristic vector satisfying $q(\kappa ) = 9-n$. Then there exists an automorphism $\varphi$ of $\Lambda _n$ such that $\varphi (\kappa) = \kappa _n$. A similar statement holds for $n=9$ provided that $\kappa$ is primitive. \endproclaim \demo{Proof} We shall freely use the notation and results of Chapter II of [13] and shall quote the results there by number. For the purposes of the appendix, chamber shall mean a chamber in $\{\,x\in \Lambda _n \otimes \Bbb R \mid x^2 =1\,\}$ for the set of walls defined by the set $\{\, \alpha \in \Lambda _n \mid \alpha ^2 = -1\,\}$. Let $C_n$ be the chamber associated to $\kappa _n$ [13, p\. 329, 2.7(a)]: the oriented walls of $C_n$ are exactly the set $$\{\,\alpha\in \Lambda _n\mid q(\alpha ) = -1,\alpha\cdot \kappa _n =1\,\}.$$ Then $\kappa _n$ lies in the interior of $\Bbb R^+\cdot C_n$, by [13, p\. 329, 2.7(a)]. Similarly $\kappa$ lies in the interior of a set of the form $\Bbb R^+\cdot C$ for some chamber $C$, since $\kappa$ is not orthogonal to any wall (because it is characteristic) and $q(\kappa )> 0$. But the automorphism group of $\Lambda _n$ acts transitively on the chambers, by [13 p. 324]. Hence we may assume that $\kappa \in C_n$. In this case we shall prove that $\kappa = \kappa _n$. We shall refer to $C_n$ as the {\sl fundamental chamber} of $\Lambda _n\otimes _{\Bbb Z}\Bbb R$. Let us record two lemmas about $C_n$. \lemma{A.2} An automorphism $\varphi$ of $\Lambda _n$ fixes $C_n$ if and only if it fixes $\kappa _n$. \endproclaim \demo{Proof} The oriented walls of $C_n$ are precisely the $\alpha \in \Lambda _n$ such that $q(\alpha) = -1$ and $\kappa _n\cdot \alpha =1$. Thus, an automorphism fixing $\kappa _n$ fixes $C_n$. The converse follows from [13, p\. 335, 4.4]. \endproof \lemma{A.3} Let $\alpha = \sum _i\alpha _ie_i$ be an oriented wall of $C_n$, where $e_0, \dots , e_n$ is the standard basis of $\Lambda _n$. After reordering the elements $e_1, \dots, e_n$, let us assume that $$|\alpha _1| \geq |\alpha _2| \geq \dots \geq |\alpha _n|.$$ Then for $n\leq 8$, the possibilities for $(\alpha _0, \dots \alpha _n)$ are as follows \rom(where we omit the $\alpha _i$ which are zero\rom): \roster \item $\alpha _0 = 0, \alpha _1 = 1$; \item $\alpha _0 =1, \alpha _1 = \alpha _2 = -1$ $(n\geq 2)$; \item $\alpha _0 = 2, \alpha _1 = \alpha _2 = \alpha _3 = \alpha _4 = \alpha _5 =-1$ $(n \geq 5)$; \item $\alpha _0 = 3, \alpha _1 = -2, \alpha _2 = \alpha _3 = \alpha _4 = \alpha _5 =\alpha _6 = \alpha _7=-1$ $(n \geq 7)$; \item $\alpha _0 = 4, \alpha _1 = \alpha _2 = \alpha _3 = 2, \alpha _4 = \dots = \alpha _8 = -1$ $(n=8)$; \item $\alpha _0 = 5, \alpha _1 = \dots = \alpha _6 = -2, \alpha _7 = \alpha _8 = -1$ $(n=8)$; \item $\alpha _0 = 6, \alpha _1 = -3, \alpha _2 = \dots = \alpha _8 = -2$ $(n=8)$. \endroster \endproclaim \demo{Proof} This statement is extremely well-known as the characterization of the lines on a del Pezzo surface (see [7], Table 3). We can give a proof as follows. It clearly suffices to prove the result for $n=8$. But for $n=8$, there is a bijection between the $\alpha$ defining an oriented wall of $C_8$ and the elements $\gamma \in \kappa _8^{\perp}$ with $q(\gamma )=-2$. This bijection is given as follows: $\alpha$ defines an oriented wall of $C_8$ if and only if $q(\alpha ) = -1$ and $\kappa _8\cdot \alpha = 1$. Map $\alpha$ to $\alpha - \kappa _8 = \gamma$. Thus, as $q(\kappa _8 ) = 1$, $q(\gamma ) = -2$ and $\gamma \cdot \kappa _8 = 0$. Conversely, if $\gamma \in \kappa _8^{\perp}$ satisfies $q(\gamma )=-2$, then $\gamma + \kappa _8$ defines an oriented wall of $C_8$. Now the number of $\alpha$ listed above, after we are allowed to reorder the $e_i$, is easily seen to be $$8+\binom 82 +\binom 85 +8\cdot 7 +\binom 83 +\binom 82 + 8 = 240.$$ Since this is exactly the number of vectors of square $-2$ in $-E_8$, by e\.g\. [36], we must have enumerated all the possible $\alpha$. \endproof Write $\kappa = \sum _{i=0}^na_ie_i$, where $e_i$ is the standard basis of $\Lambda _n$ given above. Since $\kappa \cdot e_i>0$, $a_i <0$. After reordering the elements $e_1, \dots, e_n$, we may assume that $$|a_1|\geq |a_2| \geq \dots \geq |a_n|.$$ By inspecting the cases in Lemma A.3, for every $\alpha = \sum _i\alpha _ie_i$ not of the form $e_i$, $\alpha _i \leq 0$ for all $i\geq 1$. Given $\alpha = \sum _i\alpha _ie_i$ with $\alpha \neq e_i$ for any $i$, let us call $\alpha$ {\sl well-ordered} if $$|\alpha_1|\geq |\alpha_2| \geq \dots \geq |\alpha_n|.$$ Quite generally, given $\alpha = \alpha _0e_0 + \sum _{i>0}\alpha _ie_i$, we define the {\sl reordering} $r(\alpha)$ of $\alpha$ to be $$r(\alpha ) = \alpha _0e_0 + \sum _{i>0}\alpha _{\sigma (i)}e_i,$$ where $\sigma $ is a permutation of $\{ 1, \dots , n\}$ such that $r(\alpha)$ is well-ordered. Clearly $r(\alpha)$ is independent of the choice of $\sigma$. We then have the following: \claim{A.4} $\kappa \in C_n$ if and only if $\kappa \cdot \alpha >0$ for every well-ordered wall $\alpha$. \endstatement \proof Clearly if $\kappa \in C_n$, then $\kappa \cdot \alpha >0$ for every $\alpha$, well-ordered or not. Conversely, suppose that $\kappa \cdot \alpha >0$ for every well-ordered wall $\alpha$. We claim that $$\alpha \cdot \kappa \geq r(\alpha )\cdot \kappa,\tag{$*$}$$ which clearly implies (A.4) since $r(\alpha)$ is well-ordered. Now $$\alpha \cdot \kappa = \alpha _0 a_0 - \sum _{i>0}\alpha _i a_i.$$ Since $\alpha _i < 0$ and $a_i <0$, ($*$) is easily reduced to the following statement about positive real numbers: if $c_1\geq \dots \geq c_n$ is a sequence of positive real numbers and $d_1, \dots, d_n$ is any sequence of positive real numbers, then a permutation $\sigma$ of $\{1, \dots , n\}$ is such that $\sum _ic_id_{\sigma (i)}$ is maximal exactly when $d_{\sigma (1)} \geq \dots \geq d_{\sigma (n)}$. We leave the proof of this elementary fact to the reader. \endproof Next, we claim the following: \lemma{A.5} View $\Lambda _n\subset \Lambda _{n+1}$. Defining $\kappa _{n+1}$ and $C_{n+1}$ in the natural way for $\Lambda _{n+1}$, suppose that $\kappa \in C_n$. Then $\kappa ' =\kappa - e_{n+1} \in C_{n+1}$. \endproclaim \demo{Proof} We have ordered our basis $\{e_0, \dots, e_n\}$ so that $$|a_1|\geq |a_2| \geq \dots \geq |a_n|.$$ Since $a_i <0$ for all $i$, $|a_n|\geq 1$. Thus the coefficients of $\kappa '$ are also so ordered. Note also that all coefficients of $\kappa '$ are less than zero, so that the inequalities from (1) of Lemma A.3 are automatic. Given any other wall $\alpha '$ of $C_{n+1}$, to verify that $\kappa '\cdot \alpha '> 0$, it suffices to look at $\kappa '\cdot r(\alpha ')$, where $r(\alpha ')$ is the reordering of $\alpha '$. Expressing $\alpha '$ as a linear combination of the standard basis vectors, if some coefficient is zero, then $r(\alpha ') \in \Lambda _n$. Clearly, in this case, viewing $r(\alpha ')$ as an element of $\Lambda _n$, it is a wall of $C_n$. Since then $\kappa ' \cdot r(\alpha ') = \kappa \cdot r(\alpha ')$, we have $\kappa ' \cdot r(\alpha ') >0$ in this case. In the remaining case, $r(\alpha ')$ does not lie in $\Lambda _n$. This can only happen for $n=1,4,6,7$, with $\alpha '$ one of the new types of walls corresponding to the cases (2) --- (7) of Lemma A.3. Thus, the only thing we need to check is that, every time we introduce a new type of wall, we still get the inequalities as needed. Since $r(\alpha ')$ is well-ordered, we can assume that it is in fact one of the walls listed in Lemma A.3. The $n=1$ case simply says that $a_0 > -a_1+1$. However, we can easily solve the equations $a_0^2 - a_1^2 = 8$, $a_1<0$ to get $a_0 = 3, a_1 = - -1$. Since $3> 1+1$, we are done in this case. Next assume that $n =4$. We have $\kappa = a_0e_0 + \sum _{i=1}^4a_ie_i$. We must show that $2a_0> -\sum _{i=1}^4a_i + 1$. We know that $a_0>-a_1-a_2$, hence that $a_0 \geq -a_1-a_2 +1$. Moreover $a_0\geq -a_3-a_4 +1$ since $|a_1|\geq |a_2| \geq |a_3| \geq |a_4|$. Adding gives $2a_0 \geq -\sum _{i=1}^4a_i + 2 = (-\sum _{i=1}^4a_i+ 1) +1$ and therefore $2a_0> -\sum _{i=1}^4a_i + 1$. The case where $n=6$ is similar: we must show that $3a_0 > -2a_1 -\sum _{i=2}^6 a_i +1$. But we know that $2a_0 \geq -\sum _{i=1}^5a_i +1$ and that $a_0 \geq -a_1 -a_6 +1$. Adding gives the desired inequality. For $n=7$, we have three new inequalities to check. The inequality $$4a_0> -2a_1 - 2a_2 - 2a_3 - \sum _{i=4}^7a_i +1$$ follows by adding the inequalities $3a_0 > -2a_1 -\sum _{i=2}^7 a_i$ and $a_0> -a_2 - a_3$. The inequality $$5a_0 > -2\sum _{i=1}^6a_i -a_7 +1$$ follows from adding the inequalities $3a_0 > -2a_1 - \sum _{i=2}^7a_i$ and $2a_0 > -\sum _{i=2}^5a_i$. Likewise, the last inequality $$6a_0 > -3a_1 - 2\sum _{i=2}^7a_i+2$$ follows by adding up the three inequalities $3a_0 > -2a_1 -\sum _{i=2}^7 a_i$, $2a_0 > -\sum _{i=1}^5a_i$, and $a_0 > -a_6-a_7$. Thus we have established the lemma. \endproof \noindent {\it Completion of the proof of Theorem \rom{A.1}.} Begin with $\kappa$. Applying Lemma A.5 and induction, if $n<8$, then the vector $\eta = \kappa - \sum _{j=n+1}^8e_j$ lies in the fundamental chamber of $\Lambda _8$. Moreover $\eta$ is a characteristic vector of square $1$. Thus $\eta ^{\perp} \cong - -E_8$. The same is true for $\kappa _8 = 3e_0 - \sum _{i=1}^8e_i= \kappa _n - \sum _{j=n+1}^8e_j$. Clearly, then, there is an automorphism $\varphi$ of $\Lambda _8$ such that $\eta=\varphi (\kappa _8)$. But both $\eta $ and $\kappa _8$ lie in the fundamental chamber for $\Lambda _8$. Since the automorphism group preserves the chamber structure, the automorphism $\varphi$ must stabilize the fundamental chamber. 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9404

alg-geom/9404013

en

https://arxiv.org/abs/alg-geom/9404013

alg-geom/9404013

Lisa Jeffrey

Lisa C. Jeffrey

Symplectic Forms on Moduli Spaces of Flat Connections on 2-manifolds

12 pages, LaTex version 2.09 This paper will appear in Proceedings of the Georgia International Topology Conference (Athens, GA, August 1993), ed. W. Kazez (International Press). The present version (April 1996) is essentially the version that will appear in print. Several sign errors have been corrected. Some errors have been corrected in the proof of nondegeneracy of the symplectic form (Section 5) on an open dense set in the extended moduli space

Proc. Georgia Intl. Topology Conference (Athens, GA, 1993), W. Kazez, ed., AMS/IP Studies in Advanced Mathematics 2 (1997) (Part 1) 268-281.

Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $\omega$ on \lq twisted' moduli spaces of representations of the fundamental group $\pi$ of a 2-manifold $\Sigma$ (the smooth analogues of ${\rm Hom} (\pi_1(\Sigma), G)/G$) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian $G$-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of $G$ on the extended moduli space.

alg-geom

\section{Introduction} Let $\Sigma$ be a closed oriented 2-manifold of genus $g \ge 2$; the fundamental group of $\Sigma$ will be denoted $\pi$. Let $G$ be a compact connected Lie group with Lie algebra $\lieg$. This paper concerns the moduli space $\mg = {\rm Hom} (\pi, G)/G$ of conjugacy classes of representations of $\pi$ into $G$, and certain more general analogues of $\mg$. The space $\mg$ has an open dense set on which the structure of a smooth symplectic manifold is defined. In addition to the definition we have given in terms of representations of $\pi$, the space $\mg$ has two alternative descriptions. The first of these is the gauge theory description: via the holonomy map, $\mg$ is identified with the space of gauge equivalence classes of flat connections on a trivial principal $G$ bundle over $\Sigma$. The second alternative description appears once one fixes a complex structure on the 2-manifold $\Sigma$, so that $\Sigma$ becomes a Riemann surface; $\mg$ is then identified with the space of equivalence classes of semistable holomorphic $G^{\Bbb C }$ bundles over $\Sigma$. The purpose of this paper is to extend work of Karshon [K] and Weinstein [W] to a more general setting; this paper follows [W] closely and should be read in conjunction with it. These two papers complement the work of Goldman [G]. Goldman originally gave a construction in group cohomology of the symplectic form $\symf$ on the space $\mg$; in order to prove $\symf$ is closed, Goldman used the gauge theory description of the symplectic form. Karshon [K] gave a proof of the closedness of the symplectic form using group cohomology; Weinstein [W] reinterpreted Karshon's construction in the setting of the de Rham-bar bicomplex [B,Sh]. In the present work we extend Weinstein's work to construct symplectic forms on relative character varieties of surface groups, and on `twisted' moduli spaces ${\mbox{$\cal M$}}_\beta$ of bundles on Riemann surfaces (see Section 6 of [AB]) associated to an element $\beta \in Z(G)$. The spaces ${\mbox{$\cal M$}}_\beta$ share many properties with ${\mbox{$\cal M$}}$ but in general have less singularities. Indeed, ${\mbox{$\cal M$}}_\beta$ is smooth when $G=SU(n)$ and $\beta$ is a generator of the center of $SU(n)$: in contrast, even when $G = SU(2)$, the space ${\mbox{$\cal M$}}$ is smooth only in the very special case $g = 2$. We also give a group cohomology construction of symplectic forms on the extended moduli spaces $X$ and $\xb$ [J1]: these are finite dimensional symplectic $G$-spaces from which $\mg$ and ${\mbox{$\cal M$}}_\beta$ may be obtained by symplectic reduction. In [J1], symplectic structures on $\xb$ and $X$ (which are in fact the same as the symplectic forms we recover below) were specified using gauge theoretic techniques. Here we compute the moment maps for the $G$ action on $X$ and $\xb$; up to a normalization factor, these coincide with the moment maps found in [J1]. The purpose of the construction of $\xb$ in [J1] was to exhibit ${\mbox{$\cal M$}}_\beta$ as the result of finite dimensional symplectic reduction (in contrast to the infinite dimensional quotient construction given in [AB]). We make further use of this finite dimensional quotient construction in [JK2], where we extend the techniques of [JK1] (which gives a formula for intersection pairings in the cohomology ring of the symplectic quotient ${\mbox{$\cal M$}}_X$ of a finite dimensional Hamiltonian $G$-space $X$, in terms of the $G$-equivariant cohomology $H^*_G(X)$) to treat the intersection pairings in ${\mbox{$\cal M$}}_\beta$, starting from the equivariant cohomology of $\xb$. By this means we give proofs of formulas (found originally by Witten [Wi] using physical methods) for the intersection pairings in $H^*({\mbox{$\cal M$}}_\beta)$. In this paper, the symplectic forms on $X$ and $\xb$ are constructed explicitly in terms of the Maurer-Cartan form on $G$ and the chain homotopy operator that occurs in the standard proof of the Poincar\'e lemma. This explicit description of the symplectic form will be important in [JK2], where we make use of explicit equivariantly closed differential forms representing the relevant classes in de Rham cohomology. In [J2] we extend the methods of this paper to give explicit representatives in De Rham cohomology for all the generators of the cohomology ring of ${\mbox{$\cal M$}}_\beta$ (one of which is the cohomology class of the symplectic form); for the applications in [JK2], we shall need the de Rham representatives for all the generators. After this work was completed, we received the paper of Huebschmann [H], in which he has obtained similar results independently. {\em Acknowledgement:} We thank A. Weinstein for helpful conversations. { \setcounter{equation}{0} } \section{Group cohomology} Let $\FF = \FF_{2g} $ be the free group on $2g$ generators $x_1, \dots, x_{2g}$. We introduce a relation $R = \prod_{i = 1}^g [x_{2i-1}, x_{2i}], $ where $[a,b] $ denotes the commutator $ab a^{-1} b^{-1} $. The fundamental group $\pi$ of a closed 2-manifold of genus $g$ is then given by $$ \pi = \FF/{\mbox{$\cal R$}}$$ where ${\mbox{$\cal R$}}$ is the normal subgroup generated by $R$. We shall work with Eilenberg-Mac Lane group chains (see [G] section 3.8), and shall denote the differential on the group chain complex by $\pr$. The $p$-chains $C_p(\Gamma)$ on a group $\Gamma$ are ${\Bbb Z }$-linear combinations of elements of $\Gamma^p$. In particular, associated to the relation $R$ there is a distinguished 2-chain $c \in C_2(\FF)$ given by \begin{equation} \label{1.1} c = \sum_{i =1}^{2g} (\partial R/\partial x_i, x_i). \end{equation} (Here, $\partial/\partial x_i$ refers to the differential in the Fox free differential calculus: see [G], sections 3.1-3.3.) Goldman ([G], above Proposition 3.9) shows \begin{equation} \label{1.2} \pr c = 1 - R. \end{equation} { \setcounter{equation}{0} } \section{The de Rham-bar bicomplex} Weinstein [W] introduces a bicomplex $(\cbi,\delta, d)$ whose $p,q$ term is $C^q(G^p)$. The second coboundary is the exterior differential $d$, while the first is the differential $\delta$ appearing in group cohomology: \begin{equation} (\delta \beta ) (g_0, \dots, g_p) = (-1)^{p+1} \beta (g_0, \dots, g_{p-1} ) + \sum_{i = 1}^p (-1)^i \beta (g_0, \dots, g_{i-1} g_{i}, \dots, g_p ) + \beta(g_1, \dots, g_p). \end{equation} Let $\YY$ denote ${\rm Hom} (\FF, G) = G^{2g}. $ Then there is a second bicomplex $(\ctbi, \delta, d)$ whose $p,q$ term is $\Omega^q(\FF^p \times Y) $ $ = \Omega^0 (\FF^p) \otimes \Omega^q(\YY)$. (The differential $\delta$ in the bicomplex $\ctbi$ is the adjoint of the differential $\partial $ in the Eilenberg-Mac Lane group chain complex.) As in [W], the evaluation maps $$E_p: \FF^p \times \YY \to G^p$$ give rise to maps $$E_p^*: \Omega^q(G^p) \to \Omega^0 (\FF^p) \otimes \Omega^q(\YY) $$ which combine to form a map of bicomplexes $E^*: \cbi \to \ctbi$. We recall the following elements of $\cbi$ defined in [W]. Let $\mc \in \Omega^1(G) \otimes \lieg$ denote the (left-invariant) Maurer-Cartan form on $G$, and $\bmc$ the corresponding right-invariant form. Define projection maps $\pi_i: G^p \to G$ ($i = 1, \dots, p$) to the $i$'th copy of $G$, and let $\mc_i = \pi_i^* \mc$ and $\bmc_i = \pi_i^* \bmc$. In terms of this notation, the following are introduced in [W]: \begin{equation} \lambda = \frac{1}{6} \mc \cdot [\mc, \mc] \phantom{a} \in \Omega^3(G), \end{equation} \begin{equation} \Omega = \mc_1 \cdot \bmc_2 \in \Omega^2 (G^2). \end{equation} Given $\eta \in \lieg$, Weinstein also introduces \begin{equation} \theta_\eta = \eta \cdot (\mc + \bmc) \in \Omega^1(G). \end{equation} (Here, $\cdot$ denotes an invariant inner product on $\lieg$.) These forms satisfy the following properties ([W], Lemmas 3.1, 3.3 , 4.1 and 4.4): \begin{prop} \label{p1} We have \begin{equation} \label{pe1} d\lambda = 0; \end{equation} \begin{equation} \label{pe2} d \Omega = \delta \lambda; \end{equation} \begin{equation} \label{pe4} \iota_\teta \lambda = d \theta_\eta; \end{equation} \begin{equation} \label{pe3} \iota_\teta \Omega = - \delta \theta_\eta, \end{equation} where $\teta$ is the vector field generated by $\eta$. \end{prop} { \setcounter{equation}{0} } \section{Two-forms on moduli spaces} Let us now introduce \begin{equation} \label{4.1} \symf = \inpr{c, E^* \Omega} \in \Omega^2(\YY). \end{equation} We then have \begin{prop} \label{p2} In terms of the identification of $\YY = {\rm Hom}(\FF, G) $ with $G^{2g}$, we have \begin{equation} \label{dsymf} d \symf = - \epsr^* \lambda \end{equation} where $\epsr: {\rm Hom}(\FF, G) \to G$ is the map given by evaluation on the element $R$ $\in \FF$: \begin{equation} \label{epsrdef} \epsr (g_1, \dots, g_{2g} ) = \prod_{i = 1}^g [g_{2i-1}, g_{2i}]. \end{equation} \end{prop} \Proof We have $$ d \inpr{c, E^* \Omega} = \inpr{c, d E^* \Omega} $$ $$ \phantom{bbbbb} = \inpr{c, E^* d \Omega} = \inpr{c, E^* \delta \lambda} $$ $$ \phantom{bbbbb} = \inpr{c, \delta E^*\lambda} = \inpr{\partial c, E^* \lambda} $$ $$ \phantom{bbbbb} = \inpr{1 - R, E^* \lambda}, $$ where the last step follows from (\ref{1.2}). $\square$ If $t$ is any element of $G$, define $\YY_t = \epsr^{-1} (t) $ $\subset \YY$. The following is an immediate consequence of Proposition \ref{p2}: \begin{prop} \label{p3} The 2-form $\symf$ restricts on $\YY_t$ to a closed form. \end{prop} \begin{definition} If $t$ is an element of $G$, let $\zt \subset G$ be the centralizer of $t$ in $G$. \end{definition} Notice that $\YY_t$ carries an action of $\zt$ by conjugation. \begin{definition} If $t \in G$, the {\em relative character variety} associated to $t$ is the space ${\mbox{$\cal M$}}_t = \YY_t /\zt$, where $\zt$ acts on $\YY_t$ by conjugation. \end{definition} Some properties of the symplectic geometry of relative character varieties were given in [JW1] and [JW2]. Relative character varieties also arise in algebraic geometry where (under appropriate circumstances) they are identified with moduli spaces of semistable parabolic vector bundles on Riemann surfaces: see [MS] (where the identification between relative character varieties and moduli spaces of parabolic bundles is given) or [Se]. A gauge theory construction of a symplectic form on relative character varieties is given in [J1]: it will follow from the Remark at the end of Section 5 that this is essentially the same as the symplectic form we construct here (i.e., the two are the same under a natural map identifying the relevant Zariski tangent spaces). By extending the calculations in [W], we obtain the following two propositions. \begin{prop} \label{p4} The form $\symf$ is invariant under the action of $G$ on $\YY$ by conjugation. \end{prop} \Proof Let $\eta \in \lieg$; we will show that the Lie derivative of $\symf$ with respect to the vector field $\teta$ generated by $\eta$ is zero, in other words ${\mbox{$\cal L$}}_\teta \symf = (d \iota_\teta + \iota_\teta d) \symf = 0 $. Now \begin{equation} \iota_\teta d \symf = - \iota_\teta \epsr^* \lambda = -\epsr^* \iota_\teta \lambda . \end{equation} Also $d \iota_\teta \symf = d \inpr{c, E^* \iota_\teta \Omega}. $ But $\iota_\teta \Omega = - \delta \theta_\eta $ by (\ref{pe3}), so we get $$ d \iota_\teta \symf = - d \inpr{c, E^* \delta\theta_\eta } $$ $$ \phantom{bbbbb} = - d \inpr{c, \delta E^* \theta_\eta } $$ $$ \phantom{bbbbb} = - d \inpr{\partial c, E^* \theta_\eta} = - d\inpr{1 - R, E^* \theta_\eta} $$ $$ = d \epsr^* \theta_\eta = \epsr^* d \theta_\eta. $$ But by (\ref{pe4}) we have $ d \theta_\eta = \iota_\teta \lambda $, so $ d \iota_\teta \symf = \epsr^* \iota_\teta \lambda$, and ${\mbox{$\cal L$}}_\teta \symf = 0 $. $\square$ \begin{prop} \label{p5} We have the following identification of 1-forms on $G^{2g}$: \begin{equation} \label{4.66} \iota_\teta \symf = \epsr^* \theta_\eta. \end{equation} Thus, the restriction of $\symf$ to $\YY_t$ is horizontal with respect to the action of $\zt$. (In other words, if $\eta \in {\rm Lie} (\zt)$ generates the vector field $\tilde{\eta} $ on $\YY$, then $\iota_{\teta} \symf |_{\YY_t} = 0 $, where $\iota_{\teta} $ denotes the interior product with respect to $\teta$.) \end{prop} \Proof We have $$\iota_\teta \symf = \inpr{c, \iota_\teta E^* \Omega} = \inpr{c, E^* \iota_\teta \Omega} = - \inpr{c, E^* \delta \theta_\eta},$$ $$ \phantom{a} = - \inpr{\partial c, E^* \theta_\eta} = - \inpr{1 - R, E^* \theta_\eta} = \epsr^* \theta_\eta. $$ This form necessarily restricts to zero on the level sets of $\epsr$. $\square$ Propositions \ref{p4} and \ref{p5} imply that the form $\symf$ descends to a 2-form $\bar{\omega}$ on the space ${\mbox{$\cal M$}}_t$, which is closed by Proposition \ref{p3}. Nondegeneracy will be established in Corollary \ref{c5.5}. In particular if $t $ is a central element $\beta \in Z(G)$ then $Z_\beta$ is the full group $G$. If $G = U(n)$ and $\beta$ $= e^{2 \pi i d/n} {\rm diag} (1 , \dots, 1)$ then the space ${\mbox{$\cal M$}}_\beta$ appears in algebraic geometry (see [AB]) as the moduli space of semistable holomorphic vector bundles of rank $n$ and degree $d$. When $\beta$ is as above but $G= SU(n)$, the algebraic geometry interpretation of the space ${\mbox{$\cal M$}}_\beta$ is as the moduli space of semistable holomorphic vector bundles of rank $n$ and degree $d$ with fixed determinant.\footnote{When $G = U(n) $ or $SU(n)$ and $d$ is coprime to $n$, the spaces ${\mbox{$\cal M$}}_\beta$ are smooth manifolds. This is in contrast to the spaces $\mg$, which are singular except in a few very special cases.} The constructions in this section exhibit a group cohomology construction of a symplectic form on the \lq twisted' moduli spaces ${\mbox{$\cal M$}}_\beta$ associated to central elements $\beta$ of $G$. It is shown in [J2] that the form $\bar{\omega}$ is in the cohomology class of (a constant multiple of) the standard generator $f_2$ of $H^*({\mbox{$\cal M$}}_\beta; {\Bbb R })$ (in the notation of Sections 2 and 9 of [AB]). In fact in [J2] we extend the construction which gives rise to the symplectic form $\bar{\omega}$, to give representatives in de Rham cohomology for all the generators of the ring $H^*({\mbox{$\cal M$}}_\beta; {\Bbb R })$ given in [AB]. Our applications in [JK2] will rely at least as heavily on the fact that $\bar{\omega}$ is a de Rham representative of the cohomology class $f_2$ as on its being nondegenerate: in any case, many results of the type we shall invoke for Hamiltonian $G$-manifolds generalize (cf. [KT]) to manifolds where the symplectic form degenerates on a locus of measure $0$. { \setcounter{equation}{0} } \section{The symplectic form on the extended moduli space} Let $\beta$ be an element of the center $Z(G)$. The associated {\em extended moduli space} $X_\beta$ constructed in [J1] may be described as a fibre product \begin{equation} \xb = (\epsr \times \epmb)^{-1} (\diag) \subset G^{2g} \times \lieg. \end{equation} Here, $\diag$ is the diagonal in $G \times G$ and $\epsr: G^{2g} \to G$ was defined above, while $\epm: \lieg \to G$ is the exponential map and $\epmb = \beta \cdot \epm$. The space $\xb$ is equipped with two canonical projection maps $\proj_1: \xb \to G^{2g}$ and $\proj_2: \xb \to \lieg$, for which there is the following commutative diagram: \begin{equation} \begin{array}{lcr} \xb & \stackrel{\proj_2}{\lrar} & \lieg \\ \scriptsize{\proj_1} \downarrow & \phantom{\stackrel{\proj}{\lrar} } & \downarrow \scriptsize{e_\beta} \\ G^{2g} & \stackrel{\epsr }{\lrar} & G\\ \end{array} \end{equation} A straightforward argument using the regular value theorem endows $\xb$ with a smooth structure on an open dense set $\xb^s$ including $\proj_2^{-1}(0)$: see [J1], Proposition 5.4, where an explicit characterization of the singular locus of $\xb$ is given. The space $\xb$ carries an action of the group $G$. A gauge theoretic construction of a $G$-invariant closed 2-form on $\xb$ was given in [J1], and it was shown that this form is nondegenerate on an open dense set in $\xb$ and that the action of $G$ is Hamiltonian where the 2-form is nondegenerate. There is thus an open dense set in $\xb$ which is a smooth finite dimensional Hamiltonian $G$-space such that the space ${\mbox{$\cal M$}}_\beta$ is given by symplectic reduction of this $G$-space at $0$. By extending the techniques of [W], we construct here a $G$-invariant closed 2-form on $\xb$ which is nondegenerate on an open dense set, and show that the moment map $\mu$ is given by a constant multiple of $\proj_2$, as was shown in [J1]. The symplectic form will in fact turn out to be the same as the one constructed using gauge theory (see the Remark at the end of this section): in other words, one of these symplectic forms is the pullback of the other under a natural map. It follows from our construction that the symplectic quotient (at 0) with respect to the action of $G$ on $\xb$ is the twisted moduli space ${\mbox{$\cal M$}}_\beta$ described above. For $t \in G$, the relative character variety ${\mbox{$\cal M$}}_t$ from Section 4 is the symplectic reduction of $X$ at the orbit ${\mbox{$\cal O$}}_\Lambda \subset \lieg$ (under the adjoint action) that corresponds to an element $\Lambda \in \lieg$ for which $\exp (\Lambda) = t$. To construct the symplectic form, we first construct a form $\sigfo \in \Omega^2(\lieg)$ for which \begin{equation} \label{5.1}\epm^* \lambda = d \sigfo. \end{equation} The existence of such a form follows from the following standard result (see e.g. [Wa], Lemma 4.18): \begin{prop} \label{poincare} [Poincar\'e Lemma] \noindent (a) If $\gamma \in \Omega^{p+1} (V)$ where $V$ is a vector space, and $d \gamma = 0 $, then there is a form $\sigf \in \Omega^p(V)$ with $\gamma = d \sigf$. \noindent (b) Denote by $\hop$ the map $\Omega^{p+1} (V) \to \Omega^p(V)$ sending $\gamma$ to $\sigf$. Then $d \hop + \hop d = {\rm id}.$ \end{prop} For $\beta \in \Omega^*(V)$, $\hop \beta$ is given at $v \in V$ by \begin{equation} \label{p.2} (\hop \beta)_v = \int_0^1 F_t^* (\iota_{\bar{v} } \beta) dt \end{equation} where $\bar{v} $ is the vector field on $V$ which takes the constant value $v$, and $F_t$ is the map $V \to V$ given by multiplication by $t$. In our case the form $$\sigma = \hop (\epm^* \lambda) $$ is $G$ -invariant because $\lambda$ is $G$-invariant and $\epm$ is a $G$-equivariant map. We now restrict to the fibre product $\xb \subset G^{2g} \times \lieg$. For $(h, \Lambda) \in \xb$ we have $\epsr(h) = \epmb(\Lambda)$, so if $(H, \zeta) \in T_h G^{2g} \times \lieg$ represents an element in the tangent space to $\xb$, we have \begin{equation} \label{n5.5} \epsr_* H = \epmb_* \zeta. \end{equation} We define a 2-form on $\xb$ by \begin{equation} \tw = \proj_1^* \omega + \proj_2^* \sigfo, \end{equation} where $\omega$ was defined in (\ref{4.1}). We find that $$d (\proj_1^* \omega + \proj_2^* \sigfo) = \proj_1^* d \omega + \proj_2^* d \sigfo $$ so \begin{equation} d \tw (H, \zeta) = - \epsr^* \lambda (H) + d \sigfo (\zeta) \phantom{aaaa} \mbox{(by (\ref{dsymf}))} \end{equation} $$ \phantom{bbbbb} = - \lambda(\epsr_* H) + \lambda (\epm_* \zeta) = - \lambda(\epsr_* H) + \lambda (\epmb_* \zeta) = 0 . $$ (Here, we have used the fact that $\lambda$ is invariant under multiplication by $\beta$, so $\epm^* \lambda = \epmb^* \lambda$.) So we have \begin{prop} The 2-form $\tw$ on $G^{2g} \times \lieg$ restricts on $\xb$ to a closed form. \end{prop} The $G$-invariance of $\tw$ follows because $\sigfo$ and $\omega$ are $G$-invariant and the projection maps $\proj_1$ and $\proj_2$ are $G$-equivariant maps. We may now identify the moment map for the action of $G$ on $\xb$: in other words, we find a function $\mu: X \to \lieg$ such that $\iota_\teta \tw = \eta \cdot d \mu$ where $\teta$ is the vector field on $\xb$ generated by $\eta$ $ \in \lieg$.\footnote{Ordinarily the moment map is specified as a map into $\lieg^*$. Here we have used the inner product to identify $\lieg^*$ with $\lieg$.} First we recall from (\ref{4.66}) that $ (\iota_\teta \omega) = \epsr^* \theta_\eta $. Now since $\sigfo= \hop (\epm^* \lambda) $ $= \hop (\epmb^* \lambda)$, we have $$ \iota_\teta \sigfo = \iota_\teta (\hop \epmb^* \lambda) = - \hop (\iota_\teta \epmb^* \lambda) = - \hop (\epmb^* \iota_\teta \lambda) $$ \begin{equation} \label{5.8} \phantom{bbbbb} = - \hop (d \epmb^* \theta_\eta) \phantom{a} \mbox{(by (\ref{pe4})) }.\end{equation} Combining (\ref{5.8}) with Proposition \ref{poincare} (b) we find \begin{equation} \label{5B} \iota_\teta \sigfo = -\epmb^* \theta_\eta + d (I \epmb^* \theta_\eta) \end{equation} Adding (\ref{4.66}) and (\ref{5B}) we have $$ \iota_\teta (\tw)= d (\hop \epmb^* \theta_\eta) $$ so that a moment map $\mu: \xb \to \lieg$ for the action of $G$ on $\xb$ is given by \begin{equation} \eta \cdot \mu = \hop \epmb^* \theta_\eta. \end{equation} Now \begin{equation} (\hop \epmb^* \theta_\eta)_\Lambda = \int_0^1 F_t^* \Bigl (\epmb^* \theta_\eta (\Lambda) \Bigr ) \phantom{a} \mbox{by (\ref{p.2})} \end{equation} $$ \phantom{bbbbb} = \int_0^1 (\epmb^* \theta_\eta)_{\Lambda t} (\Lambda) = \int_0^1 (\theta_\eta)_{\epmb(\Lambda t)} (\epmb_* \Lambda) $$ $$ = \int_0^1 \eta \cdot (\alpha + \bar{\alpha})_{\epmb(\Lambda t)} (\epmb_* \Lambda) $$ $$ = 2 \eta \cdot \Lambda. $$ Thus we have explicitly specified a moment map for the action of $G$, which is equivariant with respect to the adjoint action of $G$: \begin{prop} \label{p5.3} A moment map for the action of $G$ on $\xb$ is given by the map $\mu = 2 \proj_2: \; (h, \Lambda) \mapsto 2 \Lambda. $ \end{prop} To complete the identification of the form $\tw $ as a symplectic form on an open dense set in $\xb$, one needs the following: \begin{prop} \label{p5.5} The form $\tw$ is a nondegenerate bilinear form on the Zariski tangent space $T_{(h, \Lambda)} \xb$, for any $(h, \Lambda) \in \xb$ for which $(d \epsr)_h$ is surjective. \end{prop} Our proof of this Proposition parallels the gauge theory argument given in [J1] (see Proposition 3.1 of [J1] for the case $G = SU(2)$). This material is treated in [H] (Theorem 4.4 and Section 5) and [K] (Theorem 4): the proof we sketch is essentially the one given by Huebschmann [H], to whom the group cohomology proof of the nondegeneracy of the symplectic form on an open neighbourhood of the zero locus of the moment map in the extended moduli space is due.\footnote{The proof given in [H] applies to a suitable neighbourhood of the zero locus of the moment map in $\xb$ which is contained in $\proj_2^{-1} ({\mbox{$\cal O$}}_{\rm reg})$, where ${\mbox{$\cal O$}}_{\rm reg}$ is the subset of $\lieg$ where the exponential map is regular: the subset $\proj_2^{-1} ({\mbox{$\cal O$}}_{\rm reg})$ is a proper subset of the smooth locus of $\xb$. We have adapted the proof so it applies to the Zariski tangent space $ T_{(h, \Lambda)} \xb$ for all points $ (h, \Lambda) \in \xb$ for which $(d \epsr)_h$ is surjective.} \newcommand{\hl}{ {(h, \Lambda) } } \noindent{\em Proof of Proposition \ref{p5.5}:} To establish nondegeneracy of $\tw$ we proceed as follows. Proposition \ref{p5.3} establishes that if $\teta$ is the vector field associated to the action of $\eta$ on $\xb$, and if $(H, \zeta) \in T_{(h, \Lambda)} \xb$ for $(h, \Lambda) \in \xb$ (in other words, $(d \epsr)_h H = (d e_\beta)_\Lambda \zeta),$ then \begin{equation} \tw_{(h, \Lambda)} (\teta, (H,\zeta) ) = 2 \eta \cdot \zeta. \end{equation} Thus to establish nondegeneracy of $\tw$ at those $\hl$ for which $(d \epsr)_h$ is surjective (see (\ref{n5.5})) it suffices to establish it on the orthocomplement in ${\rm Ker} (d \proj_2) \subset T_\hl \xb$ of the image of the action of $\lieg$. This means we must establish that $\omega$ is nondegenerate restricted to \begin{equation} \frac{ T_\hl \Bigl (\proj_2^{-1} (\Lambda) \subset \xb \Bigr )}{ \{ \teta: \eta \in {\rm Stab} (\Lambda) \} }. \end{equation} \newcommand{\gh}{\lieg_{ h} } \newcommand{\geh}{\lieg_{ \epsr(h)} } \newcommand{\td}{{\delta} } We have commutative diagrams \begin{equation} \label{5.4} \begin{array}{ccccccccc} \phantom{ 0} & \phantom{\lrar} & \phantom{a} \phantom{a} 0 \phantom{a} & \phantom{\lrar} & \phantom{B^1(\FF;} \phantom{a} 0 \phantom{ \gh)} & \phantom{ \lrar} & \phantom{ B^1 ({\Bbb Z };} \phantom{a} 0 \phantom{ \geh)} & \phantom{\lrar} & \phantom{\dots} \\ \phantom{ 0} & \phantom{\lrar} & \phantom{a} \downarrow \phantom{a} & \phantom{\lrar} & \phantom{B^1(\FF;} \downarrow \phantom{ \gh)} & \phantom{ \lrar} & \phantom{ B^1 ({\Bbb Z };} \downarrow \phantom{ \geh)} & \phantom{\lrar} & \phantom{\dots} \\ 0 & \lrar & K_B & \lrar & B^1(\FF; \gh) &\stackrel{\td}{ \lrar} & B^1 ({\Bbb Z }; \geh) & \lrar & \dots \\ \phantom{ 0} & \phantom{\lrar} & \phantom{a} \downarrow \phantom{a} & \phantom{\lrar} & \phantom{B^1(\FF;} \downarrow \phantom{ \gh)} & \phantom{ \lrar} & \phantom{ B^1 ({\Bbb Z };} \downarrow \phantom{ \geh)} & \phantom{\lrar} & \phantom{\dots} \\ 0 & \lrar & K_C & \lrar & C^1(\FF; \gh) & \stackrel{\td}{\lrar} & C^1 ({\Bbb Z }; \geh) & \lrar & \dots \\ \phantom{ 0} & \phantom{\lrar} & \phantom{a} \downarrow \phantom{a} & \phantom{\lrar} & \phantom{B^1(\FF;} \downarrow \phantom{ \gh)} & \phantom{ \lrar} & \phantom{ B^1 ({\Bbb Z };} \downarrow \phantom{ \geh)} & \phantom{\lrar} & \phantom{\dots} \\ 0 & \lrar & K_H& \lrar & H^1(\FF; \gh) & \stackrel{\td}{\lrar} & H^1 ({\Bbb Z }; \geh) & \lrar & \dots \\ \phantom{ 0} & \phantom{\lrar} & \phantom{a} \downarrow \phantom{a} & \phantom{\lrar} & \phantom{B^1(\FF;} \downarrow \phantom{ \gh)} & \phantom{ \lrar} & \phantom{ B^1 ({\Bbb Z };} \downarrow \phantom{ \geh)} & \phantom{\lrar} & \phantom{\dots} \\ \phantom{ 0} & \phantom{\lrar} & \phantom{a} \phantom{a} 0 \phantom{a} & \phantom{\lrar} & \phantom{B^1(\FF;} \phantom{a} \phantom{a} 0 \phantom{ \gh)} & \phantom{ \lrar} & \phantom{ B^1 ({\Bbb Z };} \phantom{a} 0 \phantom{ \geh)} & \phantom{\lrar} & \phantom{\dots} \\ \end{array} \end{equation} in which the columns are short exact sequences. Here, if $\Gamma$ is a discrete group (where $\Gamma = {\Bbb Z }$ or $\Gamma = \FF$) equipped with a representation $\rho: \Gamma \to G$, the notation $C^*(\Gamma; \lieg_\rho)$ refers to the Eilenberg-Mac Lane group cochain complex with coefficients in the $\Gamma$-module $\lieg_\rho$ specified by $\rho$ under the adjoint action of $G$ on $\lieg$. For $\Gamma = \FF$ or $\Gamma = {\Bbb Z }$, we have $C^1(\Gamma; \lieg_\rho) = Z^1(\Gamma; \lieg_\rho). $ The maps $\td$ are induced by $d \epsr$, while $K_B, K_C$ and $ K_H$ are the kernels of the maps $\td$ on $B^1, C^1$ and $ H^1$. Then the vector space $T_\hl (\proj_2^{-1} (\Lambda) ) $ is identified with $K_C \subset C^1 (\FF; \gh) $, while $ { \{ \teta: \eta \in {\rm Stab} (\Lambda) \} }$ is identified with $K_B = K_C \cap B^1 (\FF; \gh)$. Then we see from (\ref{5.4}) that the quotient $K_C/K_B$ is canonically identified with $K_H = {\rm Ker} \Bigl (\td: H^1(\FF; \gh) \to H^1 ({\Bbb Z }; \geh) \, \Bigr ). $ We have the long exact sequence \begin{equation} \label{5.15} 0 \to H^0 (\FF; \gh) \to H^0 ({\Bbb Z }; \geh) \stackrel{\delta^*}{\to} H^1 (\FF,{\Bbb Z } ; \gh) \to H^1 (\FF ; \gh) \to \end{equation} $$ \stackrel{\delta}{\to} H^1 ({\Bbb Z } ; \geh) \to H^2 (\FF,{\Bbb Z } ; \gh) \to 0 . $$ The vector spaces and maps in this sequence satisfy Poincar\'e duality. Furthermore, the pairing $\, \cdot \,: \lieg \otimes \lieg \to {\Bbb R }$ gives rise to a cup product pairing \begin{equation} \label{5.07} H^1(\FF, {\Bbb Z } ; \gh) \otimes H^1(\FF; \gh) \to H^2(\FF, {\Bbb Z }; {\Bbb R }) \cong {\Bbb R },\end{equation} and Poincar\'e duality in (\ref{5.15}) implies that the restriction of this pairing to $\bigl ( H^1(\FF, {\Bbb Z }; \gh)/{\rm Im }(\delta^*) \, \bigr ) \otimes \bigl ({\rm Ker}(\delta) \subset H^1(\FF; \gh) \bigr ) $ is nondegenerate. Now one may show (see for instance [K] Theorem 4 or [H] Theorem 4.4) that this cup product is the restriction of the form $\omega$ to ${\rm Ker} (\delta) \subset H^1(\FF; \gh) $ $\cong T_h \Bigl (\epsr^{-1} \bigl (C_{\epsr(h)} \bigr )/G \Bigr ) $. (Here, $C_{\epsr(h)} $ denotes the conjugacy class of $\epsr(h)$ in $G$.) The nondegeneracy of the pairing arising from the cup product thus completes the proof of nondegeneracy. $\square$ \noindent{\em Remark:} Goldman ([G], proof of Proposition 3.7) has shown that $${\rm Im} (d \epsr)_h = \Bigl ( {\rm Lie } (Stab (h) ) \Bigr )^\perp. $$ Suppose $G = SU(n)$ and $\beta$ is a generator of $Z(G)$. The subset of $\xb$ where $(d \epsr)_h$ is surjective then contains the zero level set of the moment map. Thus the following is an immediate consequence of Proposition \ref{p5.5}: \begin{corollary} \label{c5.5} Let $G = SU(n)$ and suppose $\beta$ is a generator of $Z(G)$. Then $0$ is a regular value of the moment map $\mu = 2 \proj_2$. Further, ${\mbox{$\cal M$}}_\beta$ is a smooth manifold and the form $\bar{\omega}$ on ${\mbox{$\cal M$}}_\beta$ is nondegenerate.\footnote{When zero is a regular value of the moment map on $\xb$, standard arguments establish the smoothness of the reduced space ${\mbox{$\cal M$}}_\beta$.} \end{corollary} \noindent{\em Remark:} At the end of the last Proof, we alluded to the identification of $\omega$ (on quotients of level sets of $\proj_2$) with the bilinear form given by the cup product (\ref{5.07}). It is easy to see that the vector spaces $ H^1(\FF; \gh) $ and $ H^1(\FF, {\Bbb Z } ; \gh) $ (arising from group cohomology with coefficients in the $\FF$-module $\lieg_h$ specified by $h \in {\rm Hom}(\FF, G) $) are the same as the vector spaces $H^1(\Sigma-D^2 ; d_A)$ and $H^1(\Sigma- D^2, \partial D^2 ; d_A)$ arising in the gauge theory description of $\xb$ (cf. Section 2.2 of [J1]). Here, $A$ is a flat connection on the punctured surface $\Sigma - D^2$ whose holonomy gives rise to the representation $h$ of the fundamental group $\FF$. This identification comes from the identification between gauge equivalence classes of flat $G$ connections and conjugacy classes of representations of the fundamental group into $G$, which arises from the map sending a flat connection to the representation given by its holonomy.\footnote{Using the holonomy representation, a homeomorphism from the extended moduli space (as defined gauge theoretically in Section 2.1 of [J1]) to $\xb$ (as defined above) is given in Section 2.3 of [J1]. This map identifies the relevant Zariski tangent spaces and gives rise to the identification of the symplectic forms.} Furthermore, the pairing $ H^1(\FF; \gh) \otimes H^1(\FF, {\Bbb Z } ; \gh) \to {\Bbb R }$ arising from the group cohomology cup product is the same as the pairing $(\alpha, \beta) \mapsto \int_{\Sigma - D^2} \alpha \cdot \wedge \beta $ that gives the symplectic form in the gauge theory description. (Here, $\alpha$ and $\beta$ are $d_A$-closed $\lieg$-valued 1-forms on $\Sigma - D^2$, and $\alpha \cdot \wedge \beta$ is the element of $\Omega^2(\Sigma)$ that arises from the wedge product combined with the pairing $ \lieg \otimes \lieg \to {\Bbb R }$ given by the invariant inner product $\, \cdot \,$.) Hence the symplectic form we have constructed on $\xb$ is in fact the same as the one constructed in [J2] using gauge theory. \vspace{0.2in} {\Large \bf References} [AB] Atiyah, M.F. and Bott, R., The Yang-Mills equations over Riemann surfaces, {\em Phil. Trans. R. Soc. Lond.} {\bf A 308} (1982), 523-615. [B] Bott, R., On the Chern-Weil homomorphism and the continuous cohomology of Lie groups, {\em Advances in Math.} {\bf 11} (1973), 289-303. [G] Goldman, W., The symplectic nature of fundamental groups of surfaces, {\em Advances in Math.} {\bf 54} (1984), 200-225. [H] Huebschmann, J., Symplectic and Poisson structures of certain moduli spaces I, preprint hep-th/9312112 (1993); {\em Duke Math. J.} {\bf 80} (1995) 737-756. [J1] Jeffrey, L.C., Extended moduli spaces of flat connections on Riemann surfaces, {\em Math. Annalen} {\bf 298} (1994), 667-692. [J2] Jeffrey, L.C., Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds, {\em Duke Math. J.} {\bf 77} (1995) 407-429. [JK1] Jeffrey, L.C., Kirwan, F.C., Localization for nonabelian group actions, preprint alg-geom/9307001; {\em Topology} {\bf 34} (1995) 291-327. [JK2] Jeffrey, L.C., Kirwan, F.C., Intersection theory on moduli spaces of holomorphic bundles on a Riemann surface, {\em Elec. Res. Notices AMS} {\bf 1} (No. 2) (1995) 57-71. [JW1] Jeffrey, L.C., Weitsman, J. Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map, {\em Advances in Math.} {\bf 109} (1994), 151-168. [JW2] Jeffrey, L.C., Weitsman, J. Torus actions and the topology and symplectic geometry of flat connections on 2-manifolds, in {\em Topology, Geometry and Field Theory} (Proceedings of the Taniguchi Foundation Symposium on Low Dimensional Topology and Topological Field Theory, Kyoto, January 1993), ed. K. Fukaya, M. Furuta, T. Kohno, D. Kotschick, World Scientific, 1994. [K] Karshon, Y., An algebraic proof for the symplectic structure of moduli space, {\em Proc. Amer. Math. Soc.} {\bf 116} (1992), 591-605. [KT] Karshon, Y., Tolman, S., The moment map and line bundles over presymplectic toric manifolds, {\em J. Diff. Geom.} {\bf 38} (1993), 465-484. [MS] Mehta, V., Seshadri, C.S., Moduli of vector bundles on curves with parabolic structure, {\em Math. Annalen} {\bf 248} (1980), 205-239. [Se] C.S. Seshadri, {\em Fibr\'es Vectoriels sur les Courbes Alg\'ebriques,} {\em Ast\'erisque} {\bf 96} (1982). [Sh] Shulman, H.B., Characteristic classes and foliations, Ph.D. Thesis, University of California, Berkeley (1972). [Wa] Warner, F.W., {\em Foundations of Differentiable Manifolds and Lie Groups}, Springer-Verlag, 1983. [W] Weinstein, A., The symplectic structure on moduli space, in {\em The Floer Memorial Volume}, ed. H. Hofer, C. Taubes, A. Weinstein, E. Zehnder, Birkh\"auser (Progress in Math. {\bf 133}) (1995) 627-635. [Wi] E. Witten, { Two dimensional gauge theories revisited}, preprint hep-th/9204083; {\em J. Geom. Phys.} {\bf 9} (1992), 303-368. \end{document}

9404

alg-geom/9404012

en

https://arxiv.org/abs/alg-geom/9404012

alg-geom/9404012

Lisa Jeffrey

Lisa C. Jeffrey

Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds

20 pages, Latex file

Duke Math. J. 77 (1995) 407-429

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces $BK$ where $K$ is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form. This work solves a problem posed by Weinstein, who gave a corresponding construction (following Karshon and Goldman) of the symplectic forms on these moduli spaces. We also give a corresponding construction of equivariant differential forms on the extended moduli space $X$, which is a finite dimensional symplectic space equipped with a Hamiltonian action of $K$ for which the symplectic reduced space is the moduli space of representations of the 2-manifold fundamental group in $K$. (This paper is in press in Duke Math. J. The substance of the text is unaltered; inor changes and corrections have been made to the file to correspond to the version that will be published.)

alg-geom

\section{Introduction} \label{s1} Let $K$ be a compact connected semisimple Lie group, and let $\beta$ be an element in the center $Z(K)$. If $\Sigma$ is a closed 2-manifold of genus $g \ge 2$ with fundamental group $\fund$, and $\sio = \Sigma - D^2$, then $\pi_1(\sio) = \free$ where $\free $ is the free group on $2g$ generators. There is an associated moduli space of representations ${\mbox{$\cal M$}}_\beta = Y_\beta/K$ where $\yyb = \{ \rho \in {\rm Hom} (\free, K): \rho(R) = \beta \} $ and $K$ acts on ${\rm Hom} (\free, K) $ by conjugation. Here $R$ is the element of $\free $ corresponding to a loop winding once around the boundary $\partial \Sigma_0$. Because $\beta $ is in the center $Z(K)$, each point in the space ${\mbox{$\cal M$}}_\beta$ gives rise to a representation of the fundamental group $\fund$ of the closed surface $\Sigma$ into the group $K_c = K/Z(K)$. The space ${\mbox{$\cal M$}}_\beta$ has two alternative descriptions. Via the holonomy map, ${\mbox{$\cal M$}}_\beta$ may be identified with the space of gauge equivalence classes of flat connections on a principal $K$ bundle over $\sio$, for which the holonomy around the boundary $\partial \sio$ is the element $\beta$. We obtain a second alternative description once we fix a complex structure on $\Sigma$: then ${\mbox{$\cal M$}}_\beta$ becomes identified with a space of semistable holomorphic vector bundles (of prescribed rank and degree) over $\Sigma$. Atiyah and Bott worked in this holomorphic setting, and described the generators of the cohomology ring of ${\mbox{$\cal M$}}_\beta$ in terms of a holomorphic vector bundle $\Bbb U$ over ${\mbox{$\cal M$}}_\beta \times \Sigma$ (the {\em universal bundle}), whose restriction to a point $m \in {\mbox{$\cal M$}}_\beta$ is the holomorphic vector bundle over $\Sigma$ corresponding to $m$. The K\"unneth decomposition of the Chern classes of $\Bbb U$ yields classes in $H^*({\mbox{$\cal M$}}_\beta)$ which are the generators of the cohomology ring. The purpose of the present paper is to give an explicit description of these generators in a representation theoretic setting, using group cohomology. Our starting point is the paper [W] of Weinstein: the construction we present below generalizes Weinstein's construction of the symplectic form on moduli space, whose cohomology class is one of the generators described above. Goldman [G] constructed the symplectic form using group cohomology, but his proof that this form was closed used its gauge theory description. Karshon [K] gave the first group cohomology proof that the symplectic form was closed. In [W], Weinstein interpreted Karshon's construction in terms of the realization of $H^*(BK)$ via the de Rham cohomology of simplicial manifolds, which is due to Bott and Shulman [B1,Sh]. Here we extend Weinstein's treatment to all the generators of $H^*({\mbox{$\cal M$}}_\beta)$;\footnote{In this paper, all cohomology groups are taken with complex coefficients; one could give an equivalent treatment using rational or real coefficients.} this problem was posed in the final section of [W]. The key advance which facilitates our construction is a description of the universal bundle and its base space $\Sigma\times \yyb$ as simplicial manifolds; such a description is possible since $\Sigma$ is a realization of $B \fund$ and $\sio$ a realization of $B \free$. The spaces $EK$ and $BK$ may also be described as simplicial manifolds, and we exhibit the classifying map for the universal bundle as a map of simplicial manifolds. The generators of $H^*({\mbox{$\cal M$}}_\beta)$ are given by pairing the characteristic classes of the universal bundle with homology classes of $\Sigma$ via the slant product; in the present setting the relevant homology classes are classes in the group homology of $\fund$. One feature which we generalize from [W] is the role of equivariant cohomology. For any group $G$, the total space of the classifying bundle $EG$ over the classifying space $BG$ is equipped with an action of $G$ which covers the action of $G$ on $BG$ associated to the adjoint action of $G$ on itself. We construct a universal bundle $\ee$ over $\Sigma \times \yyb$, which is equipped with an action of $K$ covering the action of $K$ on $\Sigma \times \yyb$. The classifying map for $\ee$ is a $K$-equivariant map; our de Rham representatives for the generators of the cohomology ring of ${\mbox{$\cal M$}}_\beta$ (Theorem \ref{t7.1}) are obtained by using the classifying map to pull back {\em equivariant} cohomology classes from the classifying space. (The appropriate classifying space is $B \hatk$ rather than $BK$, where $\hatk = K/Z(K)$.) These equivariant characteristic classes appear naturally in the {\em Cartan model} for equivariant cohomology via an equivariant version of the Chern-Weil map [BGV]; we evaluate invariant polynomials on an equivariant extension of the curvature form of a distinguished invariant connection on the classifying bundle $E \hatk$ over $B \hatk$. Thus we have constructed the \lq grand unified theory' conjectured in the final section of [W], in which extensions of characteristic forms to equivariantly closed forms appear naturally. Finally in Theorem \ref{t8.1} we generalize our construction to give equivariantly closed forms on the {\em extended moduli space} $\xb$, a noncompact Hamiltonian $K$-space which contains $\yyb$ as the zero locus of the moment map, so that ${\mbox{$\cal M$}}_\beta$ may be obtained from $\xb$ by symplectic reduction. The explicit description of the de Rham representatives for the generators (Propositions \ref{p7.*2} and \ref{p8.2}) will be a key step in a future paper on intersection pairings in $H^*({\mbox{$\cal M$}}_\beta)$, where we shall require explicit representatives for the equivariant cohomology classes on $\xb$, to estimate their growth and show they satisfy appropriate boundedness properties as the value of the moment map on $\xb$ tends to infinity. This paper is organized as follows. In Section \ref{s2} we give the representation theory definition of the universal bundle, which is reworked in the setting of simplicial manifolds in Section \ref{s6}. Section \ref{s3} contains background material on simplicial manifolds and the de Rham complex on simplicial manifolds; Section \ref{s4} explains the simplicial realization of the classifying space $BG$ and the Bott-Shulman construction of universal characteristic classes, while Section \ref{s5} gives the generalization of the Bott-Shulman construction to equivariant cohomology. Sections \ref{s7} and \ref{s8} contain our final results: Section \ref{s7} exhibits the generators of the cohomology ring of ${\mbox{$\cal M$}}_\beta$ by pulling back equivariant characteristic classes from $H^*(B \hatk)$ via the classifying map, while Section \ref{s8} gives the generalization to equivariant cohomology classes on the extended moduli space $\xb$ which restrict on $\yyb$ to the classes from Section \ref{s7}. \noindent{\em Acknowledgement:} We thank Alan Weinstein for helpful conversations, and Ezra Getzler for an explanation of the results of [Ge] and their relevance to this paper. { \setcounter{equation}{0} } \section{Definition of the universal bundle} \label{s2} Let $K$ be a compact connected semisimple Lie group, and let $\hatk = K/Z(K)$. We start with the following construction of a family of principal bundles over a closed 2-manifold $\Sigma$ of genus $g \ge 2$ (the universal bundle) which generalizes a construction given in [T]. Let $\tsig \cong D^2$ be the universal cover of $\Sigma$, and let $\fund = \free/{\mbox{$\cal R$}}$ be the fundamental group of $\Sigma$; here, $\free = \free_{2g}$ is the free group on $2g$ generators $x_1, \dots, x_{2g}$, and ${\mbox{$\cal R$}}$ is the normal subgroup generated by the relator $R = \prod_{j = 1}^g x_{2j-1} x_{2j} x_{2j-1}^{-1} x_{2j}^{-1}$. If $\beta$ is a prescribed element of the center $Z(K)$, we let \begin{equation} \label{1.1} \yyb = \{ \rho \in \,{\rm Hom}\,(\FF, K) \, : \, \rho(R) = \beta \}. \end{equation} Define ${\mbox{$\cal M$}}_\beta = \yyb/K$; for some groups $K$ and some choices of $\beta$ (for instance $K = SU(N)$ and $\beta$ a generator of $Z(K)$) the space ${\mbox{$\cal M$}}_\beta$ is a smooth manifold. Instead of working on ${\mbox{$\cal M$}}_\beta$ we shall equivalently work with $K$-equivariant objects on $\yyb$. We form the universal bundle $\pi: \pun \to $ $B = \Sigma \times \yyb$ as follows, with fibre $\hatk = K/Z(K)$. The universal bundle may equivalently be described as a family $\pun$ of flat bundles over $\sio = \Sigma - D^2 $, such that for a point $\rho \in Y_\beta$ (for which the corresponding point $[\rho] \in $ ${\mbox{$\cal M$}}_\beta$ specifies an equivalence class of flat bundles), the bundle $\pun|_{\sio \times \rho} $ is the flat bundle parametrized by $\rho$. We define the bundle $\pun$ by \begin{equation} \label{2.02} \pun = ( \tsig \times \yyb \times \hatk)/\fund \end{equation} where $p \in \fund$ $= \pi_1(\Sigma)$ acts\footnote{We write the action of $K$ on $\hatk$ (by composing the quotient map $K \to \hatk$ with the action of $\hatk$ on itself by left multiplication) as $t \in K: \phantom{a} k \in \hatk \mapsto tk \in \hatk.$} on $ \tsig \times \yyb \times \hatk $ as follows: \begin{equation} \label{1.2} p: \; (\sigma, \rho , k ) \to (\sigma p, \rho, \rho(\tilde{p} )^{-1} k ). \end{equation} Here, $\tilde{p} $ is an element of $\free$ which descends to $p \in \fund$ under the quotient map. Notice that since $\rho(R) = \beta \in Z(K)$, the element $\rho(\tilde{p} ) $ is well defined in $\hatk $ $= K/Z(K)$. The map $\pi: \pun \to B = \Sigma \times \yyb$ is given by $\pi (\sigma, \rho, k) = ([\sigma], \rho)$. Now the bundle $\pun$ is equipped with an action of $K$, which is given by an action on $\tsig \times \yyb \times \hatk$ as follows: \begin{equation} \label{1.3} \ttt \in K: (\sigma, \rho, k) \mapsto (\sigma, \ttt \rho \ttt^{-1}, \ttt k). \end{equation} It is easy to verify that this action is compatible with the action of $\fund$ on $\tsig \times \yyb \times \hatk$, and so it descends to a well defined action on the bundle $\pun$. The $K$-equivariant principal bundle $\pun$ over $\Sigma \times \yyb$ may be thought of as a principal bundle over $\Sigma \times {\mbox{$\cal M$}}_\beta$. { \setcounter{equation}{0} } \section{Simplicial manifolds} \label{s3} Let $\simp^n$ $ = \{ (t_0, \dots, t_n) \in [0,1]^{n+1}: \sum_i t_i = 1 \} $ denote the standard $n$-simplex. A {\em simplicial manifold} $X = \{n \mapsto X(n) \} $ is a contravariant functor from the category of simplices to the category of $C^\infty$ manifolds. In particular, to every nonnegative integer $n$ there is associated a manifold $X(n)$ (corresponding to $\simp^n$), and there are a collection of maps $\epsilon_i: X(n) \to X(n-1) $ (the {\em face maps}) for $0 \le i \le n$ which are functorially associated to the inclusion maps $\epsilon^i: \simp^{n-1} \to \simp^n$ of the $i$'th face. (There are also {\em degeneracy maps} $\delta^*: X(n-1) \to X(n)$ associated to simplicial maps $\delta: \simp^n \to \simp^{n-1}$, but these will not directly concern us.) See for instance [Du,Ge,Se1,Se2] for a more extensive discussion of simplicial manifolds, and Chapter 1 of [M] for background on simplicial objects. There are two versions of the de Rham complex on a simplicial manifold $X$, which appear in the work of Dupont [Du] (Section 2). One version corresponds to the {\em fat realization} $\parallel X \parallel $ of $X$, which is defined as the quotient space of $\coprod_n \simp^n \times X(n)$ by the equivalence relation \begin{equation} (\epsilon^i \times {\rm id}) (t,x) \sim ({\rm id} \times \epsilon_i) (t,x), \phantom{bbbbb} (t, x) \in \simp^{n-1} \times X(n). \end{equation} Following Dupont [Du], we then make the following definition of a $k$-form on $X$: \begin{definition} \label{d3.1} A $k$-form on $X$ is a collection of $k$-forms $\phi^{(n)} \in \Omega^k(\simp^n \times X(n) ) $ satisfying \begin{equation} \label{3.2} ({\rm id} \times \epsilon_i)^* \phi^{(n-1)} = (\epsilon^i \times {\rm id})^* \phi^{(n)} \end{equation} on $\simp^{n-1} \times X(n)$ for all $0 \le i \le n$ and all $n$ $\ge 1$. \end{definition} \begin{definition} The bicomplex $(A^{*, *} (X), d_\simp, d_X) $ is the complex $\bigoplus_{k,l} A^{k,l} (X) $, where $A^{k,l} (X)$ is the vector space spanned by forms $\phi = \{ \phi^{(n)} \} $ on $X$ (in the sense of Definition \ref{d3.1}) for which $\phi^{(n)} $ is a linear combination of forms $ \omega_k \wedge \eta_l$ where $\omega_k \in \Omega^k(\simp^n) $ and $\eta_l \in \Omega^l (X(n) ) $. This bicomplex is equipped with the differentials $d_\simp$ and $d_X$ which are the exterior differentials on $\simp^n$ and $X(n)$ respectively. \end{definition} There is a second bicomplex ${\mbox{$\cal A$}}^{*,*} (X)$ which represents the de Rham complex on the simplicial manifold $X$: \begin{definition} The bicomplex $({\mbox{$\cal A$}}^{*,*} (X), \delta, d)$ is defined by \begin{equation} {\mbox{$\cal A$}}^{k,l} (X) = \Omega^l (X(k) ); \end{equation} we define the differential $\delta$ by \begin{equation} \label{3.01} \delta = \sum_i (-1)^i \epsilon_i^*, \end{equation} while $d= d_X$ is the exterior differential on $X(k)$ for all $k$. \end{definition} Lemma 2.3 of [Du] gives a chain homotopy equivalence between the double complexes $A^{*,*} (X)$ and ${\mbox{$\cal A$}}^{*,*} (X)$. In one direction the map ${\mbox{$\cal I$}}: A^{k,l}(X) \to {\mbox{$\cal A$}}^{k,l} (X) $ giving rise to the chain homotopy is defined by integration over the simplex $\simp^n$: $${\mbox{$\cal I$}}: \phi^{(n)} \in \Omega^*(\simp^n \times X(n)) \mapsto \int_{\simp^n} \phi^{(n)}. $$ Let $G$ be a real Lie group with Lie algebra $\lieg$. Then there is a standard realization (see for example [Se1]) of the classifying space $BG $ and the universal bundle $EG \to BG$ as simplicial manifolds, which is given as follows. (See also [Du], Section 3.) We define simplicial manifolds $\bng$ and $\ng$ by \begin{equation} \label{2.1} \bng(n) = G^{n+1} \end{equation} \begin{equation} \label{2.2} \ng(n) = G^n \end{equation} Here, the face maps $\bar{\epsilon_i}: \bng(n) \to \bng(n-1) $ are as follows: \begin{equation} \label{3.6a} \bar{\epsilon_i} (g_0, \dots, g_n) = (g_0, \dots, \hat{g_i} , \dots, g_n) \phantom{bbbbb} (i = 0, \dots, n) \end{equation} while the face maps $\epsilon_i: \ng(n) \to \ng(n-1) $ are given by \begin{equation} \label{3.6b} \begin{array}{lcr} \epsilon_0 (h_1, \dots, h_n) &=& (h_2, \dots, h_n) \\ \epsilon_i (h_1, \dots, h_n) &=& (h_1, h_2, \dots, h_i h_{i + 1} , \dots, h_n) \phantom{bbbbb} (i = 1, \dots, n-1) \\ \epsilon_n(h_1, \dots, h_n) &=& (h_1, \dots, h_{n - 1} ). \end{array} \end{equation} There is then a map of simplicial manifolds $q: \bng \to \ng$ given by a collection of maps $q_n: \bng(n) \to \ng(n)$, defined by \begin{equation} \label{2.3} q_n (g_0, \dots, g_n) = (g_0 g_1^{-1}, \dots, g_{n-1} g_n^{-1}). \end{equation} The simplicial $G$-bundle $\bng \stackrel{q}{\to} \ng$ is the simplicial realization of the $G$-bundle $EG \to BG$. The action of $G$ on the total space $\bng(n) $ of the principal $G$-bundle $\bng(n) \to \ng(n)$ is given by the action of $G$ on $\bng(n) = G^{n + 1} $ by right multiplication. In the case $X = \ng$, the bicomplex $({\mbox{$\cal A$}}^{*,*} (X), \delta, d)$ is the bicomplex referred to in [W] as the {\em de Rham-bar bicomplex}. Now suppose the group $G$ is acted on by another group $H$, in other words that there is a homomorphism from $H$ to ${\rm Aut} (G)$. Then the simplicial manifold $NG$ is also acted on by $H$. We shall consider the case when $H=G$ acts on $G$ by the adjoint action. In this case the action of $G$ on itself by {\em left} multiplication gives rise to an action of $G$ on $\bng$ which covers the adjoint action of $G$ on $\ng$.\footnote{This action is not to be confused with the action of $G$ on $\bng$ by {\em right} multiplication, which gives $\bng $ its structure of principal $G$-bundle over $\ng$.} Explicitly, the action of $G$ on $\bng(n) $ is given by $$g \in G: (g_0, \dots, g_n) \mapsto (g g_0, \dots, g g_n), $$ while the action on $\ng(n) $ is given by $$ g \in G : (\gamma_1, \dots, \gamma_n) \mapsto (g \gamma_1 g^{-1}, \dots, g \gamma_n g^{-1} ) . $$ Thus for this particular action (the adjoint action of $H = G$ on $G$), we have exhibited the structure of an $H$-equivariant bundle on the simplicial bundle $\bng \to \ng$. The existence of this $H$-equivariant bundle structure will be crucial for what follows: it does not hold for general actions of $H$ on $G$. { \setcounter{equation}{0} } \section{Characteristic classes} \label{s4} In this section, let $G$ be a compact connected Lie group with Lie algebra $\lieg$. In terms of the construction of $\bng$ and $\ng$ as simplicial manifolds, there is a standard construction of generators for $H^*(BG)$ (in other words, of characteristic classes) due to Bott and Shulman [B1,Sh]. According to these references, an invariant polynomial $Q \in S(\liegs)^G$ of degree $r$ gives rise to differential forms $\BPhi_n(Q) \in \Omega^{2r-n} (G^{n+1} ) $ which in fact pull back from differential forms $\Phi_n(Q) \in \Omega^{2r-n}(G^n)$ under the projection maps $q_n$ defined in (\ref{2.3}). These forms are compatible with the face maps of $\bng$ and $\ng$, and fit together to form a closed form of degree $2r$ (in the sense of Definition \ref{d3.1}) on $NG$ (see [Sh] Proposition 10 or [B1] Section 1). These differential forms are constructed as follows (see [Du] Lemma 3.8, [Ge], and [Sh] Section II). We let $\theta$ $\in \Omega^1 (G) \otimes \lieg$ be the (left invariant) Maurer-Cartan form on $G$. Also, let $\bar{\theta} \in \Omega^1(G) \otimes \lieg$ be the corresponding right invariant form: in other words $\theta(g \eta) = \eta= \bar{\theta}(\eta g)$, where $g \in G$ and $\eta\in \lieg$. Defining $\proj_i: G^{n+1} \to G$ ($i = 0, \dots, n$) as the projection on the $i$'th copy of $G$, we denote $\proj_i^* \theta$ by $\theta_i$. Let $t = (t_0, \dots, t_n) \in \simp^n$, where $\simp^n = \{ (t_0, \dots, t_n) \in {\Bbb R }^{n+1}: \phantom{a} $ $ \sum_{i = 0 }^n t_i = 1 \} $ is the standard $n$-simplex. We define $\theta $ $ \in \Omega^1 (\simp^n \times G^{n+1} ) \otimes \lieg$ by \begin{equation} \label{3.1} \theta: t \in \simp^n \mapsto \theta(t) = \sum_{i = 0 }^n t_i \theta_i \in \Omega^1(\simp^n \times G^{n+1} ) \otimes \lieg \end{equation} This is a connection on the principal $G$-bundle $\simp^n \times \bng(n) \to \simp^n \times \ng(n)$. Define $$F_{\theta(t) } = d(\theta(t) ) + {\frac{1}{2} } [\theta(t), \theta(t) ] $$ to be the curvature of the connection $\theta(t)$. We then define \begin{equation} \label{3.101} \BPhi_n(Q) = \int_{\simp^n} Q(F_{\theta(t) } ) \in \Omega^{2r-n} (G^{n+1} ). \end{equation} In other words, $Q(F_{\theta(t)} ) $ represents an element in $A^{*, *} (\bng) $ and $\BPhi_n(Q) $ represents the corresponding element in ${\mbox{$\cal A$}}^{*, *}(\bng)$. The $\BPhi_n(Q)$ give a differential form $\BPhi(Q)$ on the (simplicial) total space $\bng$, which pulls back from a differential form (in the sense of Definition \ref{d3.1}) $\Phi(Q) = \{ \Phi_n(Q) , n = 1, \dots, r\}$ on $\ng$. One may take $\Phi_n(Q) = \sigma_n^* \BPhi_n(Q)$, where $\sigma_n$ are sections of the bundles $q_n: \bng(n) \to \ng(n)$ (\ref{2.3}) given by \begin{equation} \label{4.2a} \sigma_n: (g_1, \dots, g_n) \mapsto (g_1 g_2 \cdots g_n, g_2 \cdots g_n, \dots, g_n, 1 ). \end{equation} Each term in $F_{\theta(t)} $ contains at most one power of $dt_i$, so for $n > r $, $Q(F_{\theta(t)}) $ restricts to $0$ on $\simp^n$ and one finds $\Phi_n(Q) = 0 $ ([Sh], Proposition 10). The map $Q \mapsto \Phi(Q) $ is called the {\em Bott-Shulman map}. By construction $\Phi(Q)$ is closed under the total differential $D$ which is given (on elements of form degree $p$ in $X = \ng$) by $D = \delta + (-1)^p d$.\footnote{We use the following (standard) sign convention for total differentials of double complexes $C^{*,*} $ with differentials $d_1: C^{p,q} \to C^{p+1,q} $ and $d_2: C^{p,q} \to C^{p,q+1} $. If $d_1 d_2 = - d_2 d_1$ then the total differential is $D = d_1 + d_2$. If $d_1 d_2 = d_2 d_1$ -- as is the case here -- then the total differential $D$ on $C^{p,q} $ is $d_1 + (-1)^p d_2$.} In other words, \begin{equation}\label{4.m1} (\delta \pm d) (\Phi_1 (Q) + \Phi_2(Q) + \dots + \Phi_r(Q) ) = 0 \end{equation} or \begin{equation} \label{4.m2} d \Phi_1(Q ) = 0 , \end{equation} \begin{equation} \label{4.m3} \delta \Phi_{j - 1} (Q) = (-1)^{j+1} d \Phi_j(Q ) \in \Omega^{2r-j+1} (G^j) \phantom{a} (1 < j \le r), \end{equation} \begin{equation} \delta \Phi_r(Q) = 0. \end{equation} The Bott-Shulman map gives a collection of differential forms on the simplicial realization $\ng$ of $BG$, which are representatives in de Rham cohomology for the elements of $H^*(BG)$. \noindent{\em Example.} Let $G$ be a compact connected Lie group with Lie algebra $\lieg$, and let \begin{equation} \label{4.001}Q_2 = \langle \cdot, \cdot \rangle \end{equation} be the quadratic form on $\lieg$ determined by an invariant inner product $\langle \cdot, \cdot \rangle$. Then we find that \begin{equation} \label{4.003} \Phi_1 (Q_2) = - \frac{1}{6} \langle \theta, [\theta, \theta] \rangle \;\: {\stackrel{ {\rm def} }{=} } \;\: - \lambda \in \Omega^3(G), \end{equation} \begin{equation} \label{4.004} \Phi_2(Q_2) = \langle \theta_1, \bar{\theta}_2 \rangle \;\: {\stackrel{ {\rm def} }{=} } \;\: \Omega \in \Omega^2(G^2). \end{equation} Here, $\theta_1 = \proj_1^* \theta$ is the (left invariant) Maurer-Cartan form on the first copy of $G$ and $\bar{\theta}_2 = \proj_2^* \bar{\theta} $ is the corresponding right invariant form on the second copy of $G$. { \setcounter{equation}{0} } \section{Equivariant characteristic classes} \label{s5} \noindent{\em 5.1 The equivariant simplicial de Rham complex} Let us now consider the equivariant analogue of the construction given in the previous section. For a general manifold $M$ acted on by a compact connected Lie group $H$, there is an equivariant analogue $\Omega^*_H(M)$ of the de Rham complex (the {\em Cartan model} -- see [BGV], Section 7.1) whose cohomology is the equivariant cohomology $H^*_H(M)$. The complex $\Omega^*_H(M)$ is defined by $$ \Omega^*_H(M) = \Bigl ( \Omega^*(M) \otimes S(\lieh^*) \Bigr )^H. $$ The grading on this complex is given as follows. If $\alpha: \phi \in \lieh \mapsto \alpha(\phi) \in \Omega^*(M)$ is an element of $\Omega^*_H(M)$ which is of degree $r$ as a polynomial in $\phi$ and of degree $p$ as a differential form on $M$, then it is assigned grading $p + 2r$. The differential on $\Omega^*_H(M) $ is given by \begin{equation} \label{5.01} (d_H \alpha) (\phi) = (d - \iota_\phit ) (\alpha( \phi) ) \end{equation} where $\phit$ is the vector field on $M$ generated by $\phi$ and $\iota$ is the interior product. \noindent{\em Remark.} The operators $d$ and $\iota_\phit$ on $\Omega^*_H(M)$ in fact anticommute (since $ d \iota_\phit + \iota_\phit d = L_{\phit} $, and the Lie derivative $L_{\phit} $ vanishes on $\Omega^*_H(M)$.) Thus $\Omega^*_H(M)$ may itself be given a bicomplex structure (see e.g. [H], Section 1); we shall not need this structure here. If a compact connected Lie group $H$ acts on a {\em simplicial} manifold $X$, we may form two double complexes which compute the equivariant cohomology of $X$, by analogy with the double complexes which appeared in Section \ref{s3}. One such complex $(A^{*,*}_H(X), d_\simp, d_H)$ is defined as \begin{equation} A^{*,*}_H(X) = (A^{*,*} (X) \otimes S(\liehs))^H, \end{equation} where the differentials are the exterior differential $d_\simp$ on $\simp^n$ and the Cartan model equivariant exterior differential $d_H$ (\ref{5.01}) on $X(n)$. The other double complex $({\mbox{$\cal A$}}^{*,*}_H(X), \delta, d_H) $ is defined by \begin{equation} {\mbox{$\cal A$}}^{n,q}_H(X) = \Omega^q_H(X(n) ) \end{equation} where the differential $\delta$ is still given by (\ref{3.01}) and the differential $d_H$ is given by (\ref{5.01}). As in Section \ref{s3}, a map from $A^{*,*}_H (X) $ to ${\mbox{$\cal A$}}^{*,*}_H (X)$ is given by integration over $\simp^n$. The cohomologies of the total complexes of $A^{*,*}_H(X) $ and ${\mbox{$\cal A$}}^{*,*}_H(X) $ are isomorphic, since Dupont's construction ([Du], Theorem 2.3) of a chain homotopy equivalence between the two complexes extends to the equivariant case. Applying minor modifications of the proofs in [Ge] (combining Theorem 2.2.3 with the evident extension of Corollary 1.2.3 to simplicial manifolds), one finds that the cohomology of either of these total complexes indeed equals the equivariant cohomology of the simplicial manifold $X$. \noindent{\em 5.2 Equivariant characteristic classes} Now let $X$ be a manifold acted on by a compact connected Lie group $H$, and let $P$ be a principal $G$-bundle over $X$ (for some compact connected Lie group $G$), such that $H$ acts on the total space of $P$ compatibly with its action on $X$. Let $\theta \in \Omega^1(P) \otimes \lieg $ be a connection on $P$, invariant under the action of $H$. We then define the {\em moment} $\mu \in (C^\infty (P) \otimes \lieg)^H \otimes \lieh^* $ (see Section 7.1 of [BGV]) by \begin{equation} \label{4.002} \phi \in \lieh \mapsto \mu(\theta) (\phi) = - \iota_\phit \theta \in \lieg, \end{equation} where $\phit$ is the vector field on $P$ generated by $\phi \in \lieh$. Given an invariant polynomial $Q \in S(\liegs)^G$, we may form the corresponding equivariant characteristic class of $P$, which has a representative in the Cartan model given by ([BGV] Theorem 7.7) \begin{equation} Q (F_\theta + \mu) \in (\Omega^*(P) \otimes S(\lieh^*) )^H. \end{equation} This differential form represents a class in $H^*_H(P) $ which is the pullback of a class in $H^*_H(X)$, the equivariant characteristic class corresponding to the invariant polynomial $Q$. \noindent{\em 5.3 Equivariant version of the Bott-Shulman map} We shall now specialize the construction of Section 5.1 to the case when the simplicial manifold $X$ is $\bng$ or $\ng$, and the group $H$ acts on $\bng$ and $\ng$ compatibly with the projection map from $\bng$ to $\ng$. We require also that the connection $\theta(t)$ (\ref{3.1}) on $\simp^n \times \bng(n)$ be preserved by the action of $H$. This is true, for instance, when $G$ is a quotient of $H$ and $H$ acts via the left action of $G$ on $\bng$ and the adjoint action of $G$ on $\ng$.\footnote{We shall apply this when $H = K$ is compact semisimple and $G = \hatk = K/Z(K)$.} The connection $\theta(t) $ (\ref{3.1}) on $\simp^n \times \bng(n) $ is then invariant under this action of $G$. Hence, according to Section 5.2, an invariant polynomial $Q$ on $\lieg$ of degree $r$ gives rise to an element $$\BPhihh_\simp (Q)= Q\Bigl (F_{\theta(t) } + \mu(\theta(t) ) \Bigr ) \in A^{*,*}_H(\bng), $$ which pulls back from $\Phihh_\simp (Q) \in A^{*,*}_H(\ng).$ Integration over the simplex $\simp^n$ gives an element $$ {\BPhihh}(Q) = \int_{\simp^n} Q\Bigl (F_{\theta(t) } + \mu(\theta(t) ) \Bigr ) \in {\mbox{$\cal A$}}^{*,*}_H(\bng), $$ which pulls back from $\Phihh(Q) \in {\mbox{$\cal A$}}^{*,*}_H(\ng) $ and represents the equivariant characteristic class associated to $Q$ in the Cartan model for $\Omega^*_H(BG)$. We denote the component on $\ng(n)$ by $\Phihh_n(Q) \in \Omega^{2r-n}_H(\ng(n) ) $. In the same way as in (\ref{4.m1}-\ref{4.m3}), we have \begin{equation} \label{5.m1} (\delta \pm d_H) (\Phihh_1 (Q) + \Phihh_2(Q) + \dots \Phihh_r(Q) ) = 0 \end{equation} or \begin{equation} \label{5.m2} d_H \Phihh_1(Q ) = 0 , \end{equation} \begin{equation} \label{5.m3} \delta \Phihh_{j - 1} (Q) = (-1)^{j+1} d_H \Phihh_j(Q ) \in \Omega^{2r-j+1}_H (G^j), \phantom{a} (1 < j \le r) \end{equation} \begin{equation} \delta \Phihh_r(Q) = 0. \end{equation} \noindent{\em Example.} Let $Q_2 $ be as in (\ref{4.001}). The moment $\mu(\theta)$ (corresponding to the Maurer-Cartan form $\theta$ $ \in \Omega^1(G) \otimes \lieg$, which yields the connection $\theta(t)$ on $\bng$ given by (\ref{3.1})) is defined by $\mu(\theta)(\phi) = - \iota_{\tilde{\phi} } \theta$. Now for the action of $H = G $ on $G $ by left multiplication, we have \begin{equation} \label{5.9a} \mu(\theta) (\phi)_g = - {\rm Ad} (g^{-1} ) \phi \end{equation} (in the notation of (\ref{4.002})). This leads to \begin{equation} \Phihh_1(Q_2 ) = \Phi_1(Q_2) - \Theta = - \lambda - \Theta, \end{equation} where \begin{equation} \Theta (\phi) = \langle \phi, \theta + \bar{\theta} \rangle \in \Omega^3_H(G). \end{equation} We also have \begin{equation} \Phihh_2(Q_2 ) = \Phi_2(Q_2) = \Omega = \langle \theta_1, \bar{\theta}_2 \rangle \end{equation} (cf. (\ref{4.004})). This result was obtained by Weinstein ([W], Section 4). (Differences between our formulas and Theorem 4.5 of [W] result from Weinstein's use of the right adjoint action on $NG$ rather than the left adjoint action: see the first sentence of the proof of [W], Lemma 4.1.) { \setcounter{equation}{0} } \section{The classifying map for the universal bundle} \label{s6} In this section, we describe our construction (from Section 2) of the universal bundle $\pun \to B = \Sigma \times \yyb$ in terms of simplicial manifolds. We now take the group $G$ in (\ref{2.1}-\ref{2.2}) to be the fundamental group $\fund$ of the closed 2-manifold $\Sigma$. Recall that $K$ was a compact connected semisimple Lie group and $\hatk = K/Z(K)$. Then we define simplicial manifolds $\ee$, $\bb$ by \begin{equation} \label{5.1} \ee(n) = (\fund^{n+1} \times \yyb \times \hatk )/\fund \end{equation} \begin{equation} \label{5.2} \bb(n) = \fund^n \times \yyb, \end{equation} where $\Pi$ acts by (\ref{1.2}). Again the face maps of $\ee$ are given by (\ref{3.6a}), while those of $\bb$ are given by (\ref{3.6b}). There is a (simplicial) projection map $\pi_n: \ee(n) \to \bb(n) $ given by \begin{equation} \label{5.3} \pi_n (p_0, \dots, p_n, \rho, k) = (p_0 p_1^{-1}, \dots, p_{n-1} p_n^{-1}, \rho). \end{equation} These maps fit together to give a map of simplicial manifolds $\ee \stackrel{\pi}{\lrar} \bb$. The spaces $\ee$ and $\bb$ are equipped with actions of $K$ given by \begin{equation} \label{5.5} \ttt \in K: (p_0, \dots, p_n, \rho, k) \mapsto (p_0, \dots, p_n, \ttt \rho \ttt^{-1}, \ttt k) \end{equation} \begin{equation} \label{5.6} \ttt \in K: (p_1, \dots, p_n, \rho) \mapsto (p_1, \dots, p_n, \ttt \rho \ttt^{-1}) \end{equation} Since $\tsig = E \fund $ and $\Sigma = B \fund$, the simplicial manifolds $\ee$ and $\bb$ (with $K$ actions) together with the $K$-equivariant map $\pi: \ee \to \bb$ are the simplicial realization of the $K$-equivariant bundle $\pun \to B = \Sigma \times \yyb$. \newcommand{\ev}[1]{{ \rm ev}_{ {#1} } } We can also define maps of simplicial manifolds $\psie: \ee \to \bnk$, $ \psib: \bb \to \nk$ for which the diagram \begin{equation} \label{5.4} \begin{array}{lcr} \ee & \stackrel{\psie}{\lrar} & \bnk \\ \downarrow & \phantom{\stackrel{\psie}{\lrar} } & \downarrow \\ \bb & \stackrel{\psib}{\lrar} & \nk \\ \end{array} \end{equation} commutes. We take $$ \psie (n): \ee(n) \to \bnk (n) $$ to be given by \begin{equation} \label{5.05} \psie(n) (p_0, \dots, p_n, \rho, k) = ( \ev{p_0} (\rho) k, \dots, \ev{p_n} (\rho) k) \end{equation} while $\psib(n): \bb(n) \to \nk(n) $ is defined by \begin{equation} \label{5.06} \psib(n) (p_1, \dots, p_n, \rho) = ( \ev{p_1} (\rho) , \dots, \ev{p_n}(\rho) ) \end{equation} Here, $\ev{p} : {\rm Hom} (\FF, K) \to K$ is the evaluation map associated to an element $p \in \FF$, and we have observed that $\yyb$ is contained in $ {\rm Hom} (\FF, K)$. If $p_j = x_{r(j)} $ in terms of the standard generators of $\free$, and we identify $\yyb$ with a subspace of $K^{2g}$ so that a point $\rho \in \yyb$ becomes identified with ${\bf h } = (h_1, \dots, h_{2g} )$ where $h_j = \rho(x_j)$, then in this notation we have \begin{equation} \psie(n) (x_{r(0)}, \dots, x_{r(n)}, {\bf h}, k) = (h_{r(0)}k, \dots, h_{r(n)}k) \end{equation} and similarly \begin{equation} \label{6.*1} \psib(n) (x_{r(1)}, \dots, x_{r(n)}, {\bf h}) = (h_{r(1)}, \dots, h_{r(n)}). \end{equation} Further, the following is easily verified: \begin{prop} The map $\psie$ is equivariant with respect to the actions of $K$, where $\ttt \in K$ acts on $\ee$ by (\ref{5.5}), and on $\bnk$ by left multiplication. The map $\psib$ is $K$-equivariant, where $\ttt \in K$ acts on $\bb$ by (\ref{5.6}), and on $\nk$ by the adjoint action. \end{prop} Thus we see that $\psib$ is the simplicial realization of the classifying map for the bundle $\pun \to \Sigma \times \yyb$ (defined in Section 2), {\em as a $K$-equivariant bundle}. \noindent{\em Remark:} Since $\Sigma = B \fund$ has cohomology only in dimensions $0,1,2$, we shall in fact only be concerned with $\nk(n) $ for $n \le 2$. { \setcounter{equation}{0} } \section{Pullbacks of equivariant characteristic classes} \label{s7} We now use the classifying maps $\psie$ and $\psib$ from Section 6 to pull back the equivariant characteristic classes constructed in Section 5. In this section we restrict to $K=SU(N)$, so $ \hatk = PSU(N)$, and we assume that $\beta$ is a generator of $Z(K)$. In this situation ${\mbox{$\cal M$}}_\beta$ is a smooth manifold. This choice of $K$ and $\beta$ is the case for which the generators of $H^*({\mbox{$\cal M$}}_\beta)$ are given explicitly in Section 9 of [AB]. Let $Q_r $ $(r \ge 2)$ be the invariant polynomial on $ \liek = su(N)$ which corresponds to the $r$'th Chern class. Then according to [AB] (Sections 2 and 9), the generators of the ring $H^*({\mbox{$\cal M$}}_\beta) $ (using rational, real or complex coefficients) are given as follows. We fix generators $c \in H_2 (\Sigma; {\Bbb Z })$ and $\alpha_j \in H_1(\Sigma; {\Bbb Z }) $ $(j = 1, \dots, 2g)$. Let us define a rank $N$ vector bundle $\uu$ over $\Sigma \times \yyb$ (equipped with a $K$ action covering the $K$ action on $\yyb$), whose transition functions are in $SU(N)$ and are lifts of the transition functions of the principal $\hatk$-bundle $\pun$ in (\ref{2.02}) (or $\ee$ in (\ref{5.1})) from $ \hatk = PSU(N)$ to $SU(N)$. (Such a lift exists according to Section 9 of [AB]; moreover, the characteristic classes of the vector bundle $\uu$ defined using such transition functions are independent of the choice of lift.) We then have the following description of a set of generators of $H^*({\mbox{$\cal M$}}_\beta)$ $\cong \hk(\yyb)$: \begin{equation} f_r = (c, \crk(\uu) ) \end{equation} \begin{equation} b_r^j= (\alpha_j, \crk(\uu) ) \end{equation} \begin{equation} a_r= (1, \crk(\uu) ) \end{equation} Here, $\crk$ is the $r$'th equivariant Chern class, and $(\cdot, \cdot ) $ denotes the canonical pairing $H_* (\Sigma ) \otimes \hk(\Sigma \times \yyb) \to \hk(\yyb) \cong H^*(\mb). $ In the simplicial manifold description of the universal bundle (\ref{5.1}-\ref{5.2}), we may represent the generators of $H_*(\Sigma)$ by $c \in H_2 (\fund)$, $\alpha_j \in H_1 (\fund)$ where $H_j (\fund)$ denotes Eilenberg-Mac Lane group homology (see e.g. [Mac]). Thus we find the following \begin{theorem} \label{t7.1} The generators $a_r$, $b_r^j$ and $f_r$ of $\hk(\yyb)$ are given in terms of the map $\Phih$ from Section 5.3 by the following elements $\tar, \tbrj$ and $\tfr$ of $\Omega^*_K(\yyb)$: \begin{equation} \tbrj = \Bigl (\alpha_j, \psib^* \Phih_1 (Q_r) \Bigr ) \end{equation} \begin{equation} \tfr = \Bigl ( c, \psib^* \Phih_2 (Q_r) \Bigr ), \end{equation} while $\tar$ is the image of $Q_r \in S(\lieks)^K $ in $\Omega^*_K(\yyb)$. Here, the classifying map $\psib$ was defined by (\ref{5.06}). \end{theorem} Let us now identify $\yyb$ with a subspace of $K^{2g}$, and write the projection on the $j$'th copy of $K$ as $\pi_j: K^{2g} \to K$. We then have the following result. \begin{prop} \label{p7.*2} The classes $f_r$ and $b_r^j$ are represented in the Cartan model by $K$-equivariant polynomial maps $$ \phi \in \liek \mapsto \tilde{f_r} (\phi), \tilde{b_r^j} (\phi) \in \Omega^*(\yyb). $$ Through our choice of generators $x_1, \dots, x_{2g} $ for $\FF$, we have identified $\yyb$ with a subvariety of $K^{2g}$. Let $\theta$ $\in \Omega^1(K) \otimes \liek$ denote the (left invariant) Maurer-Cartan form. In terms of this notation, each of the forms $\tfr(\phi), \tbrj(\phi) \in \Omega^*(\yyb)$ is given by a linear combination of polynomials in the quantities $\ev{p}^* (\theta) $ and ${\rm Ad}(\ev{p})(\phi)$. Here, $\ev{p}: {\rm Hom} (\FF, K) \to K$ is the evaluation map associated to an element $p \in \FF$, and $p$ ranges over a finite set of elements in $\FF$, while $j = 1, \dots, 2g$. \end{prop} \Proof For any $ \phi \in \liek$, we have \begin{equation} \label{7.3a} \BPhih{2}(Q_r )(\phi) = \sum_{A, B} \kappa_{AB} Q_r\Bigl (\theta_{\alpha_1}, \theta_{\alpha_2}, [\theta_{\alpha_3},\theta_{\alpha_4} ], \dots, [\theta_{\alpha_{2r-2m- 3} },\theta_{\alpha_{2r-2m-2} } ], {\rm Ad} (k_{\beta_1}^{-1} ) \phi, \dots, {\rm Ad} (k_{\beta_m}^{-1} ) \phi \Bigr ) \end{equation} $$ \phantom{aaaaaaa}\in \Omega^{2r-2-2m} (K^3). $$ Here, $A = (\alpha_1, \dots, \alpha_{2r-2-2m})$, $B = (\beta_1, \dots, \beta_m) $ are multi-indices, and $K_{AB} $ are certain constants. Also, $\alpha_i, \beta_i $ $ = 0, 1, 2,$ and $k_\beta$ refers to the $\beta$-th copy of $K$ in $K^3$. (Here, the factors involving $\phi$ result from (\ref{5.9a}).) Thus also by (\ref{4.2a}) we have \begin{equation} \label{7.4c} \Phih_2(Q_r) (\phi) = \sigma_2^* \BPhih{2}(Q_r) (\phi) \in \Omega^{2r-2-2m} (K^2), \end{equation} where $\sigma_2: K^2 \to K^3$ is the map \begin{equation} \label{7.4b} \sigma_2: (k_1, k_2) \mapsto (1, k_1, k_1 k_2) \end{equation} So we have \begin{equation} \label{7.5a} \tfr(\phi) = \Bigl (c, \psib^* \Phih_2(Q_r) (\phi) \Bigr ) \in \Omega^{2r-2-2m} \Bigl ({\rm Hom} (\FF, K) \Bigr ) \end{equation} Finally we use the explicit description of $c \in C_2 (\free)$ ([G], above Proposition 3.1): \begin{equation} \label{7.*1} c = \sum_{j = 1}^{2g} (\frac{\partial R}{\partial x_j} \otimes x_j), \end{equation} where $\partial R/\partial x_j $ $ \in C_1(\free)$ denotes the differential in the Fox free differential calculus (see e.g. [G], Sections 3.1-3.3). We may write $\partial R/\partial x_j = \sum_f a_f f$ where $a_f \in {\Bbb Z }$ and $f$ are certain elements in $\free$ given explicitly by words in the generators $x_1, \dots, x_{2g}$. We have in fact \begin{equation} \frac{\partial R}{\partial x_j} = \gamma_j^0 - \gamma_j^1, \end{equation} where $ \gamma_j^\alpha \in \FF$ is given by \begin{equation} \begin{array}{lcr} \gamma_{2j-1}^0 &=& \prod_{l = 1}^{j-1} [x_{2l-1}, x_{2l} ] \\ \gamma_{2j-1}^1 &=& \gamma_{2j-1}^0 x_{2j-1} x_{2j} x_{2j-1}^{-1} \\ \gamma_{2j}^0 &=& \gamma_{2j-1}^0 x_{2j-1} \\ \gamma_{2j}^1 &=& \gamma_{2j-1}^0 [x_{2j-1}, x_{2j} ] . \end{array} \end{equation} Thus we have from (\ref{7.5a}) \begin{equation} \label{7.5b} \tfr(\phi) = \sum_{j = 1}^{2g} \sum_{\tau = 0 }^1 (-1)^\tau \Bigl (\ev{\gamma_j^\tau} \times \ev{x_j} \Bigr )^* \Phih_2 (Q_r) (\phi) \end{equation} where $\ev{p} : {\rm Hom} (\FF, K) \to K $ is the evaluation map associated to an element $p$ in $\FF$ and one defines $\ev{\gamma_j^\tau} \times \ev{x_j}: {\rm Hom} (\FF, K) \to K^2$. Combining (\ref{7.5b}) with (\ref{7.4c}) we see that \begin{equation} \label{7.5c} \tfr(\phi) = \sum_{j= 1}^{2g} \sum_{\tau = 0}^1 (-1)^\tau \Psi_{j,\tau}^* \BPhih{2}(Q_r)(\phi) , \end{equation} where $\Psi_{j,\tau} = \ev{1} \times \ev{\gamma_j^\tau} \times \ev{\gamma_j^\tau x_j} : $ $ {\rm Hom}(\FF, K) \to K^3$ is the map \begin{equation} \Psi_{j,\tau}: \rho \mapsto (1, \rho(\gamma_j^\tau), \rho(\gamma_j^\tau x_j)). \end{equation} Using the formula (\ref{7.3a}) for $ \BPhih{2}(Q_r) (\phi)$, this gives the desired result. Explicitly, if we decompose the right hand side of (\ref{7.3a}) as $\BPhih{2}(Q_r)(\phi) = \sum_{a} \kappa_a \omega^0_{a}\wedge \omega^1_{a}\wedge $ $ \omega^2_{a} $ for $\omega^j_{a}$ $\in \Omega^*(K_j) $ (where $K_j$ is the $j$-th copy of $K$ in $K^3$), we have \begin{equation} \label{7.6a} (c, \psib^* \Phih_2(Q)(\phi) ) = \sum_{j= 1}^{2g} \sum_{\tau = 0}^1 (-1)^\tau \sum_{a} \kappa_a (\ev{1}^* \omega^0_a) (\ev{\gamma^\tau_j}^* \omega^1_a) (\ev{\gamma^\tau_j x_j}^* \omega^2_a). \end{equation} Let us define elements $z_{j,\alpha}^\tau$ in $\FF $ (for $ \alpha = 0,1,2$ and $\tau = 0, 1$) by $z_{j,0}^\tau = 1, $ $ z_{j,1}^\tau = \gamma^\tau_j, $ $z_{j,2}^\tau = \gamma^\tau_j x_j$. Then we have using (\ref{7.3a}) and (\ref{7.5a}) \begin{equation} \label{7.6b} \tfr (\phi) = \sum_{j = 1}^{2g} \sum_{\tau = 0}^1 (-1)^\tau \sum_{A, B} \kappa_{AB} Q_r \Bigl ( \ev{z_{j,\alpha_1}^\tau }^* \theta, \ev{z_{j,\alpha_2}^\tau }^* \theta, [\ev{z_{j,\alpha_3}^\tau}^* \theta, \ev{z_{j,\alpha_4}^\tau}^* \theta ], \dots, \end{equation} $$ \phantom{aaaaaaa}\dots [ \ev{z_{j,\alpha_{2r-2m-3}}^\tau }^* \theta, \ev{z_{j,\alpha_{2r-2m-2}}^\tau}^* \theta ], {\rm Ad} (\ev{(z_{j,\beta_1}^\tau)^{-1} })( \phi ) , \dots, {\rm Ad} (\ev{(z_{j,\beta_m}^\tau)^{-1} })( \phi ) \Bigr ) \in \Omega^{2r-2-2m}(\yyb). $$ The proof for $\tbrj(\phi) $ is similar but easier. $\square$ \noindent{\em Remark:} If $m: K \times K \to K$ is the multiplication map, and $i: K \to K$ is the inversion map, we have \begin{equation} m^* \theta = \theta_2 + {\rm Ad}(k_2^{-1}) \theta_1, \phantom{bbbbb} (i^* \theta)_k = {\rm Ad}(k^{-1}) \theta. \end{equation} Applying this to (\ref{7.6b}) we get an alternative expression in terms of the generators $x_j$ for $\FF$. { \setcounter{equation}{0} } \section{Extended moduli spaces} \label{s8} The {\em extended moduli space } $\xb$ was defined in [J1]; it is a symplectic space equipped with a Hamiltonian action of $K$, such that the space $\yyb$ treated above embeds in $\xb$ as the zero locus of the moment map. Thus the symplectic reduction of $\xb$ is the space $\mb = \yyb/K$ associated to the central element $\beta$ $\in Z(K)$. The symplectic form on $\xb$ was given a gauge theoretic construction in [J1], and a construction via group cohomology (using techniques similar to the present paper) was given in [J2] and independently in [H]. In this section we shall extend the results of the previous sections to obtain de Rham representatives for classes in $\hk(\xb) $ whose restrictions to $\hk(\yyb)$ are the classes constructed in Section 6. Let us first recall from [J1] or [J2] the definition of the space $\xb$. We had defined an element $ R = \prod_{j = 1}^g x_{2j-1} x_{2j} x_{2j-1}^{-1} x_{2j}^{-1} \in \free $ in terms of the generators $x_j$ of $\free$. The space $\xb$ is defined as a fibre product \begin{equation} \label{8.1} \xb = (\epsr \times \epmb)^{-1} (\diag) \subset K^{2g} \times \liek. \end{equation} Here, ${\rm Hom} (\FF, K)$ has been identified with $K^{2g}$, while $\diag$ is the diagonal in $K \times K$ and $\epsr: K^{2g} \to K$ is the map $\epsr: \rho \in {\rm Hom} (\FF, K) \mapsto \rho(R) . $ The map $\epm: \liek \to K$ is the exponential map, and $\epmb = \beta \cdot \epm$ in terms of the element $\beta \in Z(K)$. The space $\xb$ is equipped with two canonical projection maps $\proj_1: \xb \to K^{2g}$ and $\proj_2: \xb \to \liek$, and there is a commutative diagram \begin{equation} \label{7.2} \begin{array}{lcr} \xb & \stackrel{\proj_2}{\lrar} & \liek \\ \scriptsize{\proj_1} \downarrow & \phantom{\stackrel{\psie}{\lrar} } & \downarrow \scriptsize{e_\beta} \\ K^{2g} & \stackrel{\epsr }{\lrar} & K\\ \end{array} \end{equation} Define $\sio = \Sigma - D^2$. We shall now adapt the techniques from Section \ref{s6} to construct a principal $ \hatk$ bundle $\eex$ over $\sio \times \xb$, equipped with a distinguished homotopy class of trivializations on $\partial \sio \times \xb$, and with an action of $K$ covering the action of $K$ on $\xb$. This $\hatk$ bundle will restrict on $\yyb$ to the $K$-equivariant bundle $\ee$ constructed in Section \ref{s6}; hence its equivariant characteristic classes are the extensions to $\xb$ of the equivariant characteristic classes of the bundle $\ee$, which were used in Section \ref{s7} to construct the generators of the cohomology ring $H^*(\mb)$. We start with a construction of spaces analogous to (\ref{5.1}-\ref{5.2}) and maps analogous to (\ref{5.05}-\ref{5.06}); the spaces (obtained from an action of $\free$ similar to (\ref{1.2})) are \begin{equation} \label{8.01f} \eex(n) \;\: {\stackrel{ {\rm def} }{=} } \;\: ( \free^{n+1} \times \xb \times \hatk)/\free , \end{equation} \begin{equation} \label{8.02f} \bbx(n) \;\: {\stackrel{ {\rm def} }{=} } \;\: \free^n \times \xb. \end{equation} These bundles are pulled back in an obvious way (via the maps $\proj_1: \xb \to K^{2g} $) from bundles \begin{equation} \label{8.01fn} \ceex(n) \;\: {\stackrel{ {\rm def} }{=} } \;\: ( \free^{n+1} \times K^{2g} \times \hatk)/\free , \end{equation} \begin{equation} \label{8.02fn} \cbbx(n) \;\: {\stackrel{ {\rm def} }{=} } \;\: \free^n \times K^{2g}. \end{equation} Denote the canonical maps $\eex \to \ceex$ (resp. $\bbx \to \cbbx$) by $p_E$ (resp. $p_B$): then there is an obvious commutative diagram \begin{equation} \label{7.2*1} \begin{array}{lcr} \eex & \stackrel{p_E}{\lrar} & \ceex \\ \phantom{\scriptsize{\proj_1}} \downarrow & \phantom{\stackrel{\psie}{\lrar} } & \downarrow \phantom{\scriptsize{e_\beta} } \\ \bbx & \stackrel{{p_B}}{\lrar} & \cbbx \\ \end{array} \end{equation} The maps \begin{equation} \label{8.01}\psiex(n): \eex(n) = ( \free^{n+1} \times \xb \times \hatk)/\free \to \hatk^{n+1}, \end{equation} \begin{equation} \label{8.02} \psibx(n) : \bbx(n) = \free^n \times \xb \to \hatk^n \end{equation} are given by $\psiex = \cpsiex \circ p_E$, $\psibx = \cpsibx \circ p_B $, where $$\cpsiex(n): ( \free^{n+1} \times {\rm Hom}(\free, K) \times \hatk)/\free \to \hatk^{n+1}, $$ $$ \cpsibx(n): \free^n \times {\rm Hom}(\free, K) \to \hatk^n$$ are defined (as in (\ref{5.05}-\ref{5.06})) by \begin{equation} \label{8.05p} \cpsiex(n) (p_0, \dots, p_n, \rho, k) = ( \rho(p_0) k, \dots, \rho(p_n) k) \end{equation} \begin{equation} \label{8.06p} \cpsibx(n) (p_1, \dots, p_n, \rho) = ( \rho(p_1) , \dots, \rho(p_n)). \end{equation} These maps give a classifying map for the bundle \begin{equation} \eex = (E \free \times \xb\times \hatk )/\free \to \bbx = B \free \times \xb. \end{equation} In other words, the following diagram commutes: \begin{equation} \label{7.1} \begin{array}{lcr} \eex & \stackrel{\psiex}{\lrar} & \bnk \\ \downarrow & \phantom{\stackrel{\psie}{\lrar} } & \downarrow \\ \bbx & \stackrel{\psibx}{\lrar} & \nk \\ \end{array} \end{equation} Of course, $\sio$ has the homotopy type of $B \free$, so $\eex$ may be regarded as a bundle over $\sio \times \xb$. The action of $K$ on $\eex$ is given by the following action on $E \free \times \xb \times \hatk$: \begin{equation} \label{8.z2} \ttt \in K: \; (e, (\rho,\Lambda), k)\mapsto (e, (\ttt \rho \ttt^{-1},\ttt \Lambda \ttt^{-1}), \ttt k). \end{equation} (Here, $(\rho, \Lambda) \in \xb \subset K^{2g} \times \liek ;$ see (\ref{8.1}).) The maps in the diagram (\ref{7.1}) are $K$-equivariant, where $K$ acts on $\bnk$ by left multiplication and on $\nk$ by the adjoint action. We now construct a trivialization of $\eex$ over $\partial \sio \times \xb$. Consider the map $\psibx(1): \phantom{a} \bbx(1) = $ $\free \times \xb \to \nk(1) = \hatk $ determined by (\ref{7.1}). Its restriction to ${\Bbb Z } \cdot R \times \xb$ (where ${\Bbb Z } \cdot R$ is the abelian subgroup of $\free$ generated by the element $R$) lifts to a map ${\Bbb Z } \cdot R \times \xb \to \liek$, because of the diagram (\ref{7.2}). Since $\liek$ is contractible, this implies there is a canonical trivialization of the bundle $\eex$ over ${\Bbb Z } \cdot R \times \xb$. Now since ${\Bbb Z } \cdot R = {\Bbb Z } = \pi_1 (\partial \sio) $, and in fact the map ${\Bbb Z } \cdot R \to \free$ gives rise to a map $B {\Bbb Z } \to B \free$ which is identified with the map $\partial \sio \to \sio$, we have constructed a trivialization of the bundle $\eex$ over $\partial \sio \times \xb$. Via the classifying map given by (\ref{7.1}), the canonical $K$-invariant connection $\theta(t) $ (\ref{3.1}) on $\bnk \to \nk$ pulls back to a $K$-invariant connection on the bundle $\eex \to \sio \times \xb$. The equivariant characteristic classes of $\eex$ are then given by pulling back the classes\footnote{Recall that $ K$ acts on $\bnk(n) = \hatk^{n+1} $ by left multiplication.} $\Phih(Q_r) $ $\in H^*_K(\nk) $ via the classifying map $\psibx$. Since the restriction of $\psibx$ to $\partial \sio \times \xb$ factors through a map to a contractible space, the classes $\psibx^* \Phih ( Q_r ) $ live in $H^* (\sio, \partial \sio) \otimes \hk(\xb)$. Their restrictions to $(\sio, \partial \sio) \times \yyb$ identify (since $H^*(\sio, \partial \sio) $ $\cong H^*(\Sigma)$) with the classes in $H^*(\Sigma) \otimes \hk(\yyb)$ constructed in Section \ref{s7}. The generators $\tar$, $\tbrj$ and $\tfr$ of $H^*(\mb) \cong \hk(\yyb)$ were obtained in Section \ref{s7} by pairing ${\psib}^* \Phih(Q_r) $ with the generators of $H_*(\Sigma)$ via the slant product. We obtain the equivariant extensions of $\tar, \tbrj $ and $ \tfr$ to $\xb$ by pairing $\psibx^* \Phih(Q_r) $ with the generators of $H_*(\sio, \partial \sio) \cong H_*(\Sigma)$. To construct these equivariant extensions explicitly, we must introduce the equivariant complex $\Omega^*_K(\liek)$, where $ K$ acts on $\liek$ via the adjoint action. Now there is an equivariant map $\hop: \Omega^{*+1}_K (\liek) \to \Omega^*_K (\liek)$ which appears in the usual proof of the Poincar\'e lemma (see e.g. [Wa], Lemma 4.18). The definition is as follows: for any $v \in \liek$ and $\beta \in \Omega^*_K(\liek)$ we define \begin{equation} \label{p.2} (\hop \beta)_v = \int_0^1 F_t^* (\iota_{\bar{v} } \beta ) dt \end{equation} where $F_t: \liek \to \liek$ is the map given by multiplication by $t $ and $\bar{v} $ is the vector field on $\liek $ which takes the constant value $v$. The map $\hop$ is usually defined as a chain homotopy in the ordinary de Rham complex on a vector space; it generalizes to the equivariant de Rham complex on the vector space $\liek$ because the maps $F_t$ are $K$-equivariant maps. More precisely, the map $\hop$ satisfies the property \begin{equation} \label{8.001} \hop d_K + d_K\hop = 1, \end{equation} in other words $\hop$ defines a chain homotopy equivalence (which is in fact equivariant). The formula (\ref{8.001}) follows because $\hop d + d \hop = 1$ and also $\hop \iota_\phit = - \iota_\phit \hop$, where $\iota_\phit $ is the interior product with the vector field $\phit$ generated by the action of an element $\phi \in \liek$. Let us denote the generator of $H_2(\sio, \partial \sio) $ $\cong H_2(\free, {\Bbb Z } \cdot R) $ by $c$ and those of $H_1(\sio, \partial \sio) $ $ \cong H_1(\free, {\Bbb Z } \cdot R) $ by $\alpha_j$ ($j = 1, \dots, 2g$). We then have \begin{theorem} \label{t8.1} Let $K = SU(n)$. The extensions to $\hk(\xb)$ of the generators $\tbrj, \tfr$ of $ \hk(\yyb)$ are represented by equivariant differential forms $\tar(X), \tbrj(X), \tfr(X) \in \Omega^*_K(\xb) $ defined by \begin{equation} \label{8.f} \tfr(X) = \proj_1^*\tfr(X)_1 + \proj_2^* \tfr(X)_2 \end{equation} \begin{equation} \label{8.g} \tbrj(X) = \proj_1^* \tbrj(X)_1 \end{equation} where \begin{equation} \label{8.17a} \tfr(X)_1 = (c, \cpsibx^* \Phih_2 (Q_r) ), \end{equation} \begin{equation} \label{8.17b} \tfr(X)_2 = - \hop \Bigl ( e_\beta^* \Phih_1 (Q_r) \Bigr ), \phantom{bbbbb} \mbox{and} \end{equation} \begin{equation} \label{8.b} \tbrj(X)_1 = (\alpha_j, \cpsibx^* \Phih_1(Q_r) ). \end{equation} The equivariant differential form $\tar(X)$ representing the extension of $\tar$ to $\Omega^*_K(\xb)$ is the image of $Q_r \in S(\lieks)^K $ in $\Omega^*_K(\xb)$. Here, $ K $ acts on $\liek$ by the adjoint action. Also, $\hop: \Omega^{p+1}_K (\liek) \to \Omega^p_K (\liek)$ is the chain homotopy map defined in (\ref{p.2}), and $Q_r$ is the invariant polynomial on $\liek$ corresponding to the $r$th Chern class. \end{theorem} \Proof Let $Q$ be an invariant polynomial of degree $r$ on $\liek$. The class $\Phih (Q ) $ $ = \{ \Phih_n(Q) \} $ $ \in \Omega^*_K(\nk) $ satisfies $ \Phih_1 (Q) \in \Omega^{2r-1}_K(\hatk)$, $ \Phih_2 (Q) \in \Omega^{2r-2}_{\hmac}(\hatk^2)$. Since $\Phih (Q) $ is closed under the total differential $d_{\hmac} \pm \delta$\footnote{Recall the sign convention for total differentials of double complexes introduced in Section 4.} of the complex ${\mbox{$\cal A$}}^{*,*}_{\hmac} (\nk)$, we have \begin{equation} \delta \Phih_1(Q) = - d_{\hmac} \Phih_2 (Q) \in \Omega^{2r-1}_{\hmac}(\hatk^2). \end{equation} Pulling back via the classifying map $\psibx$, we find \begin{equation} d_{\hmac} \Bigl ( \psibx^* \Phih_2 (Q) \Bigr ) = \psibx^*(d_{\hmac} \Phih_2 (Q) ) \end{equation} $$ \phantom{bbbbb} = - \psibx^* \delta \Phih_1 (Q) = - \delta \psibx^* \Phih_1 (Q). $$ We now consider the equivariant complex $\Omega^*_{\hmac}(\liek)$, where $ K$ acts on $\liek$ by the adjoint action. Now $e_\beta^* \Phih_1(Q) \in \Omega^{2r-1}_{\hmac} (\liek) $ is equivariantly closed (by (\ref{5.m2})), so since $H^*_{\hmac}(\liek) $ $\cong H^*(BK)$ is nonzero only in even dimensions, there is an element $\sigma_Q \in \Omega_{\hmac}^{2r-2} (\liek) $ such that $e_\beta^* \Phih_1 (Q) = d_{\hmac} \sigma_Q$. In fact, because of (\ref{8.001}) one may choose \begin{equation} \label{8.z3} \sigma_Q = \hop (e_\beta^* \Phih_1(Q)). \end{equation} The idea of using the operator $\hop$ to construct closed forms is due originally to Weinstein, who suggested this approach in the case $Q = Q_2$. We thus find that $d_{\hmac} (\psibx^* \Phih_2 (Q) ) = -\delta \psibx^* \Phih_1(Q)$. So if $c \in C_2 (\free)$ is the element representing the generator of $H_2(\sio, \partial \sio)$, for which $\partial c = 1-R \in C_1(\free)$ ([G], above Proposition 3.9), then we have \begin{equation} d_{\hmac} \Bigl (c, \psibx^* \Phih_2 (Q) \Bigr ) = -\Bigl (c, \delta \psibx^* \Phih_1 (Q) \Bigr ) \end{equation} $$ \phantom{a} = - (\partial c, \psibx^* \Phih_1 (Q) ) $$ $$ \phantom{a} = -(1-R, \psibx^* \Phih_1 (Q)) $$ $$ \phantom{a} = (\epsr \circ \proj_1)^* \Phih_1 (Q) \in \Omega^*_K(\xb) \phantom{a} (\mbox{cf. (\ref{7.5b})}) $$ $$ \phantom{a} = \proj_2^* (e_\beta^* \Phih_1( Q) ) \phantom{a} \mbox{by (\ref{7.2})}. $$ Thus because of (\ref{8.001}) and (\ref{8.z3}) we easily obtain $$d_{\hmac}(c, \psibx^* \Phih_2 (Q) ) = d_{\hmac} \proj_2^*( \sigma_Q), $$ so that the element $$ \proj_1^* (c, \cpsibx^* \Phih_2 (Q) ) - \proj_2^*( \sigma_Q) $$ is equivariantly closed in $\Omega^{2r-2}_{\hmac}(\xb)$. When $Q = Q_r$ is the invariant polynomial on $\liek = su(N)$ corresponding to the $r$'th Chern class, this element is the extension to $ \Omega^*_K(\xb) $ of the element $\tfr$ described in Section 7. If $\alpha_j$ ($j =1, \dots, 2g$) are the generators of $H_1 (\sio) = H_1 (\free)$, we find that $$d_{\hmac} (\alpha_j, \psibx^* \Phih_1 (Q) ) = 0 $$ since $d_{\hmac} \Phih_1(Q) = 0 $. The element $(\alpha_j, \psibx^* \Phih_1 (Q) )$ represents the extension of the element $\tbrj \in \Omega^{2r-1}_K(\yyb)$ to $\Omega^{2r-1}_K(\xb)$. $\square $ \noindent{\em Remark.} The appearance of a second term in (\ref{8.f}) corresponding to the boundary of $\sio$ should not be surprising. We know a de Rham representative of a differential form $\psibx^* \Phih_2(Q)$ lives in $\Omega^*(\sio) \otimes \Omega^*_K(\xb)$ and is equivariantly closed. Evaluating such a form on $c \in C_2(\free)$ corresponds to integrating it over $\sio$. Now for any Cartesian product of manifolds $S \times Z$ with $\partial Z = 0 $, Stokes' Theorem for integration over the fibre $S$ takes the form \begin{equation} \int_S d_{S \times Z} \alpha - (-1)^{{\rm dim} S} d_Z \int_S \alpha + \int_{\partial S} \alpha = 0. \end{equation} The same is true when $Z$ is a closed manifold acted on by a group $K$ (and $K$ acts trivially on $S$), when the de Rham exterior differential $d$ is replaced by the Cartan model exterior differential $d_K$. Taking $Z = \xb$ and $S = \sio$, we obtain when $\alpha $ is the equivariantly closed differential form $\psibx^* \Phih_2(Q) $, \begin{equation} \label{8.0z1} d_K \int_\sio \psibx^* \Phih_2(Q) = \int_{\partial \sio} \psibx^* \Phih_2(Q). \end{equation} But over $\partial \sio \times \xb$, the classifying map $\psibx$ factors through an equivariant map to a contractible space (see the paragraph after (\ref{8.z2})), so the right hand side of (\ref{8.0z1}) can explicitly be written as a $d_K$-exact form. One choice of such an exact form is the image under $d_K$ of the second term on the right hand side of (\ref{8.f}). \noindent{\em Example.} Let $K = SU(n)$, and let $Q_2$ $ = \langle \cdot , \cdot \rangle $ be the invariant inner product on $\liek$ that gives rise to the second Chern class. The corresponding generator $\tilde{f}_2$ is the cohomology class of the standard symplectic form on ${\mbox{$\cal M$}}_\beta$. In this case, the formula (\ref{8.f}) generalizes the result of [J2], which gives the extension of the symplectic form $\omega$ on ${\mbox{$\cal M$}}_\beta = \yyb/K$ to a closed 2-form $\tilde{\omega}$ on $\xb$ (indeed a symplectic form on a neighbourhood of $\yyb$ in $\xb$). This 2-form is defined by ([J2], (5.5)) \begin{equation} \tilde{\omega} = \proj_1^* \omega - \proj_2^* \sigma, \end{equation} where $\sigma = \sigma_{Q_2} = \hop (e_\beta^* \Phi_1(Q_2) )$ and $$ \Phi_1(Q_2) = -\lambda = -(1/6) \langle \theta, [\theta, \theta] \rangle \in \Omega^3 (\hatk). $$ The 2-form $\omega \in \Omega^2 (K^{2g})$ is defined (see (\ref{4.004})) by \begin{equation} \omega = (c, \psibx^* \Omega) \end{equation} where $\Omega = \Phi_2 (Q_2) = \Phi_2^K(Q_2) = \langle \theta_1, \bar{\theta_2} \rangle \in \Omega^2 (K^2). $ (Here, $\theta_i$ is the left invariant Maurer-Cartan form on the $i$'th copy of $K$, while $\bar{\theta_i} $ is the corresponding right-invariant form.) The corresponding {\em equivariantly} closed 2-form $\bar{\omega} $ on $\xb$ is the element in $\Omega^2_K(\xb) $ given in terms of the Cartan model (for $\phi \in \liek$) by \begin{equation} \bar{\omega} (\phi) = \tilde{\omega} + \langle \mu, \phi \rangle, \end{equation} where $\mu = - 2 \proj_2: \xb \to \liek$ is the moment map for the action of $K$ on $\xb $ ([J2], Proposition 5.4). Indeed we have as a special case of (\ref{8.f}) \begin{equation} \bar{\omega} = \proj_1^* \omega - \proj_2^* \sigma^K \end{equation} where $\sigma^K = \hop e_\beta^*(\Phih_1(Q_2) ). $ Here, $\Phi_1^K(Q_2) = -\lambda - \Theta$, where $\Theta \in \Omega^3_{\hmac}(K) $ is given by $\Theta(\phi) = \langle \phi, \theta + \bar{\theta} \rangle $: see the end of Section 5.3. Finally, we have the following analogue of Proposition \ref{p7.*2}, whose proof is very similar to the proof of that Proposition. \noindent{\em Remark:} The equivariant forms $ \tar(X) $ are (by Theorem \ref{t8.1}) simply the invariant polynomials $Q_r \in S (\lieks)^K$ $\cong H^*_K ({\rm pt})$; this simple observation about the $\tar(X) $ is replaced by the result of Proposition \ref{p8.2} for the equivariant forms $\tbrj(X)$ and $\tfr(X)$. \begin{prop} \label{p8.2}{\bf (a)} {}~In the notation of Theorem \ref{t8.1}, the differential forms $\tfr(X)_1 $ and $\tbrj(X)_1 $ are given by polynomial maps $$ \phi \in \liek \mapsto \tfr(X)_1 (\phi), \tbrj(X)_1 (\phi) \in \Omega^*({\rm Hom} (\FF, K) ). $$ Let $\theta$ $\in \Omega^1(K) \otimes \liek$ denote the (left invariant) Maurer-Cartan form; then each of the forms $\tfr(X)_1 (\phi), \tbrj(X)_1 (\phi) \in \Omega^*( {\rm Hom}(\FF,K) ) $ is given by a linear combination of polynomials in the quantities $\ev{p}^* (\theta) $ and ${\rm Ad}({\ev{p}})(\phi)$. Here, $\ev{p}: {\rm Hom} (\FF, K) \to K$ is the evaluation map associated to an element $p \in \FF$, and $p$ ranges over a finite set of elements in $\FF$, while $j = 1, \dots, 2g$. \noindent{\bf (b)} The differential forms $\tfr(X)_2 (\phi) $ from Theorem \ref{t8.1} are given in terms of a basis for $\Omega^*(\liek) $ (determined by an orthonormal basis for $\frak{t} $ and a choice of roots for the action of $K$ on $\frak{t}^\perp$) by a collection of smooth bounded functions on $\liek$. \end{prop} \vspace{0.2in} {\Large \bf References} [AB] Atiyah, M.F. and Bott, R., The Yang-Mills equations over Riemann surfaces, {\em Phil. Trans. R. Soc. Lond.} {\bf A 308} (1982), 523-615. [BGV] Berline, N., Getzler, E., Vergne, M. {\em Heat Kernels and Dirac Operators}, Springer-Verlag (Grundlehren vol. 298), 1992. [B1] Bott, R., On the Chern-Weil homomorphism and the continuous cohomology of Lie groups, {\em Advances in Math.} {\bf 11} (1973), 289-303. [BSS] Bott, R., Shulman, H., Stasheff, J., On the de Rham theory of certain classifying spaces, {\em Advances in Math.} {\bf 20 } (1976) 43-56. [Du] Dupont, J.L., Simplicial De Rham cohomology and characteristic classes of flat bundles, {\em Topology} {\bf 15} (1976) 233-245. [Ge] Getzler, E., The equivariant Chern character for non-compact Lie groups, {\em Advances in Math.}, to appear. [G] Goldman, W., The symplectic nature of fundamental groups of surfaces, {\em Advances in Math.} {\bf 54} (1984), 200-225. [H] Huebschmann, J., Symplectic forms and Poisson structures of certain moduli spaces, preprint (1993). [J1] Jeffrey, L.C., Extended moduli spaces of flat connections on Riemann surfaces, {\em Math. Annalen} {\bf 298} (1994) 667-692. [J2] Jeffrey, L.C., Symplectic forms on moduli spaces of flat connections on 2-manifolds, to appear in Proceedings of the Georgia International Topology Conference (Athens, GA, 1993), ed. W. Kazez. [JK] Jeffrey, L.C., Kirwan, F.C., Localization for nonabelian group actions, preprint alg-geom/9307001 (1993); {\em Topology} (in press). [JW] Jeffrey, L.C., Weitsman, J. Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map, {\em Advances in Math.} {\bf 109} (1994) 151-168. [K] Karshon, Y., An algebraic proof for the symplectic structure of moduli space, {\em Proc. Amer. Math. Soc.} {\bf 116} (1992) 591-605. [M] May, J.P., {\em Simplicial Objects in Algebraic Topology}, Van Nostrand (1967). [Mac] Mac Lane, S., {\em Homology}, Springer-Verlag (Grundlehren v. 114) (1963). [Se1] Segal, G. Classifying spaces and spectral sequences, {\em Publ. Math. IHES} {\bf 34} (1968) 105-112. [Se2] Segal, G. Categories and cohomology theories, {\em Topology} {\bf 13} (1974) 293-312. [Sh] Shulman, H.B., Characteristic classes and foliations, Ph.D. Thesis, University of California, Berkeley (1972). [T] Thaddeus, M., Conformal field theory and the cohomology of the moduli space of stable bundles, {\em J. Diff. Geo.} {\bf 35} (1992) 131-149. [Wa] Warner, F.W., {\em Foundations of Differentiable Manifolds and Lie Groups}, Springer-Verlag, 1983. [W] Weinstein, A., The symplectic structure on moduli space, to appear in Andreas Floer memorial volume, Birkh\"auser. \end{document}

 
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alg-geom/9401005

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https://arxiv.org/abs/alg-geom/9401005

alg-geom/9401005

E. Looijenga

Eduard Looijenga

Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map

13 pages in AMS-TeX v2.1

This replacement corrects statement and proof of the main result. Also, a section on the universal Abel-Jacobi map has been added.

alg-geom

\section{\global\advance\headnumber by1\global\labelnumber=0{{\the\headnumber}.\ }} \define\label{(\global\advance\labelnumber by1 \the\headnumber .\the\labelnumber )\enspace} \NoBlackBoxes \topmatter \title Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel--Jacobi map \endtitle \rightheadtext{Stable cohomology of mapping class groups} \author Eduard Looijenga \endauthor \abstract The irreducible representations of the complex symplectic group of genus $g$ are indexed by nonincreasing sequences of integers $\lambda =(\lambda _1\ge \lambda_2\ge\cdots)$ with $\lambda _k=0$ for $k>g$. A recent result of N.V.~Ivanov implies that for a given partition $\lambda$, the cohomology group of a given degree of the mapping class group of genus $g$ with values in the representation associated to $\lambda$ is independent of $g$ if $g$ is sufficiently large. We prove that this stable cohomology is the tensor product of the stable cohomology of the mapping class group and a finitely generated graded module over $\Q [c_1,\dots ,c_{|\lambda |}]$, where $\deg (c_i)=2i$ and $|\lambda |=\sum _i\lambda _i$. We describe this module explicitly. In the same sense we determine the stable rational cohomology of the moduli space of compact Riemann surfaces with $s$ given ordered distinct (resp. not necessarily distinct) points as well as the stable cohomology of the universal Abel--Jacobi map. These results take into account mixed Hodge structures. \endabstract \address Faculteit Wiskunde en Informatica, Universiteit Utrecht, PO Box 80.010, 3508 TA Utrecht, The Netherlands \endaddress \email looijenga\@math.ruu.nl \endemail \subjclass Primary 32G15, 20F34; Secondary 55N25, 20C99 \endsubjclass \keywords Stable cohomology, mapping class group, Abel--Jacobi map \endkeywords \endtopmatter \document \head \section Introduction \endhead The mapping class group $\G _{g,r}^s$ can be defined in terms of a compact connected oriented surface $S_g$ of genus $g$ on which are given $s+r$ (numbered) distinct points $(x_i)_{i=1}^{r+s}$: it is then the connected component group of the group of orientation preserving diffeomorphisms of $S_g$ which fix each $x_i$ and are the identity on the tangent space of $S_g$ at $x_i$ for $i=s+1,\dots ,s+r$. It is customary to omit the suffix $r$ resp. $s$ when it is zero. Harer's stability theorem says essentially that $H^k(\G _{g,r}^s;\Z )$ only depends on $s$ if $g$ is large compared to $k$. For a more precise statement it is convenient to make a definition first. There is a natural outer homomorphism $\Gamma ^s_{g,r+1}\to \Gamma ^s_{g+1,r}$ (that is, an orbit of homomorphisms under the inner automorphism group of the target group, so that there is well-defined map on homology) and there is a forgetful homomorphism $\Gamma ^s_{g,r+1}\to \Gamma ^s_{g,r}$. For a coefficient ring $R$ and an integer $g_0\ge 0$, we define $N(g_0;R)$ as the maximal integer $N$ such that both induce isomorphisms on homology with coefficients in $R$ in degree $\le N$ for all $g\ge g_0$ and $s,r\ge 0$. Harer showed in \cite{3} that $N(g;\Z )\ge {1\over 3}g$ and Ivanov \cite{5}, \cite{6} improved this to $N(g;\Z )\ge {1\over 2}g -1$. Recently, Harer proved that $N(g;\Q )\le {2\over 3}g$ and that $N(g;\Q )\ge {2\over 3}g$ is almost true: it holds, provided that we restrict to the mapping class groups with $r\ge 1$ \cite{4}. It is likely that in fact $N(g;\Q )\ge {2\over 3}g-1$. We will be mostly concerned with $N(g;\Q )$ and so we shall write for this number $N(g)$ instead. A consequence of the stability property is that for every integer $s\ge 0$ we have a stable cohomology algebra $H^{\bullet}(\G _{\infty}^s;R)$ (where as before, we omit the superscript $s$ if it is equal to $0$). As the notation suggests, this is indeed the cohomology of a group $\G _{\infty}^s$, namely the group of compactly supported mapping classes of an oriented connected surface of infinite genus relative $s$ given numbered points. (The number of ends of this surface may be arbitrary.) \medskip Consider the symplectic vector space $V_g:=H^1(S_g;\Q )$. Its symplectic form is preserved by the natural action of the mapping class group $\G _g$ on $V_g$ and so any finite dimensional representation $U$ of the algebraic group $Sp(V_g)$ can be regarded as a representation of $\G _g$; in particular we have defined the cohomology groups $H^k(\G _g ;U)$. A basic fact of representation theory is that the isomorphism classes of the irreducible complex representations of $Sp(V_g)$ are in a natural bijective correspondence with $g$-tuples of nonnegative integers $(a_1,\dots ,a_g)$. For instance, the $k$th basis vector $(0,\dots ,0,1,0\dots ,0)$ corresponds to the $k$th exterior power of $V_g$. This result goes back to Weyl, who gave in addition a functorial construction of such a representation inside $\Sym ^{a_1}(\wedge ^1 V_g)\otimes\cdots\otimes\Sym ^{a_g}(\wedge ^g V_g)$. He labeled this representation by the sequence $(a_1+\cdots +a_g,a_2+a_3+\cdots +a_g,\dots ,a_{g-1}+a_g,a_g)$. We will follow his convention, so for us a numerical partition $\lambda=(\lambda _1\ge \lambda _2\ge\cdots )$ with at most $g$ parts (that is, $\lambda _k=0$ for $k>g$) determines an irreducible representation $S_{\langle {\lambda}\rangle}(V_g)$. It follows from a recent result of Ivanov \cite{6} that for fixed $k$ and $\lambda$, the cohomology groups $H^k(\G _{g,1};S_{\langle {\lambda}\rangle}(V_g))$ are independent of $g$ if $k\le N(g)-|\lambda |$, where $|\lambda |:=\sum _i\lambda _i$ is the size of the partition. We shall give an independent proof of this in the undecorated case $r=s=0$ which generalizes in a straightforward manner to the case of arbitrary $r$ and $s$ and we determine these stable cohomology groups as $H^{\bullet}(\G _{\infty} ;\Q)$-modules at the same time. \proclaim {\label Theorem} For every numerical partition $\lambda$ of $s$, there is a graded finitely generated $\Q [c_1,\dots ,c_s]$-module $B^{\bullet}_{\lambda}$ (where $c_i$ has degree $2i$) and a natural homomorphism $$ H^{\bullet}(\G _{\infty};\Q )\otimes B^{\bullet}_{\lambda}\to H^{\bullet}(\G _g; S_{\langle{\lambda}\rangle}(V_g)), $$ which is an isomorphism in degree $\le N(g)-|\lambda |$. \endproclaim To describe $B^{\bullet}_{\lambda}$, denote the coordinates of $s$-space $\Q ^s$ by $u_1,\dots ,u_s$ so that $\Q [u_1,\dots ,u_s]$ is its algebra of regular functions. We grade this algebra by giving each coordinate $u_i$ weight $2$. A {\sl diagonal} of $\Q ^s$ is by definition an intersection of the hyperplanes $u_i=u_j$; this includes the intersection with empty index set, that is, $\Q ^s$ itself. It is clear that a partition $P$ of the set $\{ 1,\dots ,s\}$ determines (and is determined by) a diagonal $\Delta _P$. Notice that the algebra of regular functions $\Q [\Delta _P]$ is a quotient of $\Q [u_1,\dots ,u_s]$ by a graded ideal, so that it inherits a grading. Denote by $l_i(P)$ the number of parts of $P$ of cardinality $i$, and consider $$ \bigoplus _P t^{-s}u^{\codim (P)+2l_1(P)+l_2(P)}\Q [\Delta _P],\tag{1} $$ where $\codim (P)$ is short for $\codim (\Delta _P)$ and $t$ resp. $u$ has formal degree $1$ resp. $2$. (The difference between $t^2$ and $u$ becomes manifest in the Hodge theory: $t$ has Hodge type $(0,0)$, whereas $u$ has Hodge type $(1,1)$.) We regard this as a graded module over $\Q [u_1,\dots ,u_s]$ that has a natural action of the symmetric group $\Sy _s$. We tensorize this module with the signum representation of $\Sy _s$ and denote the resulting graded $\Q [u_1,\dots ,u_s]$-module with $\Sy _s$-action by $B^{\bullet}_s$. Now recall that every numerical partition $\lambda$ of $s$ determines an equivalence class $(\lambda )$ of irreducible representations of $\Sy _s$ and that this gives a bijection between these two sets. For instance, the coarsest partition $(s)$ labels the trivial representation, whereas the finest partition $(1^s)$ corresponds to the signum representation. Passing from a partition $\lambda$ to the conjugate partition $\lambda '$ corresponds to taking the tensor product with the signum representation. In the case at hand we have a decomposition of $B^{\bullet}_s$ into isotypical subspaces: $$ B^{\bullet}_s=\oplus _{\lambda} B^{\bullet}_{\lambda}\otimes (\lambda),\quad \text{with } B^{\bullet}_{\lambda}=\Hom _{\Sy _s}((\lambda ),B^{\bullet}_s). $$ Clearly, this is also a decomposition into graded $\Q [u_1,\dots ,u_s]^{\Sy _s}$-submodules. We identify the latter ring with $\Q [c_1,\dots ,c_s]$, where $c_k$ is the $k$th elementary symmetric function in the $u_i$'s. This completes the description of $B^{\bullet}_{\lambda}$. It follows from the work of M. Saito that $H^{\bullet}(\G _g; S_{\langle{\lambda}\rangle}(V_g))$ carries a natural mixed Hodge structure. It is known that $H^{\bullet}(\G _{\infty};\Q )$ has a natural mixed Hodge structure as well (see \refer{2.5}). We will find that if we put a Hodge structure on $B^{\bullet}_{\lambda}$ by giving its degree $2d-|\lambda |$-part Hodge type $(d,d)$, then the homomorphism of \refer{1.1} is a morphism of mixed Hodge structures. M.\ Pikaart has recently shown that $H^n(\G _{\infty};\Q )$ is pure of weight $n$; this implies that $H^n(\G _g; S_{\la\lambda\ra }(V_g))$ is pure of weight $n+|\lambda |$ in the stable range $n\le N(g)-|\lambda |$. \medskip {\it Example 1.} The $s$-th symmetric power of $V_g$ corresponds to $S_{\langle s\rangle}(V_g)$ and so the stable cohomology of $\Gamma _g$ with values in $\Sym ^s(V_g)$ is the tensor product of $H^{\bullet}(\G _{\infty} )$ with $B^{\bullet}(1^s)$, that is, the isotypical subspace of expression \refer{1} corresponding to the signum representation of $\Sy _s$. Any partition different from the partition into singletons is invariant under a transposition and so will not contribute. This leaves us therefore with the $(1^s)$-isotypical subspace of $t^{-s}u^{2s}\Q [u_1,\dots ,u_s]$. This is the free $\Q [c_1,\dots ,c_s]$-module generated by the element $t^{-s}u^{2s}\prod _{i>j} (u_i-u_j)$, hence is of the form $t^{-s}u^{{1\over 2}s(s+3)}\Q [c_1,\dots ,c_s]$. We find that $$ H^{\bullet}(\G _{\infty };\Sym ^s (V_{\infty}))= t^{-s}u^{{1\over 2}s(s+3)}H^{\bullet}(\G _{\infty };\Q )[c_1,\dots ,c_s]. $$ \smallskip {\it Example 2.} For $g\ge s$, the primitive subspace $\Pr ^s(V_g)$ of the $s$-th exterior power of $V_g$ corresponds to $S_{\langle 1^s\rangle V_g}$ and so the stable cohomology of $\Gamma _g$ with values in $\Pr ^s(V_g)$ is the tensor product of $H^{\bullet}(\G _{\infty};\Q )$ with ${\Sy _s}$-invariant part of \refer{1}. This is naturally written as a sum over the numerical partitions of $s$. Here a numerical partition is best described by means of the exponential notation: $(1^{l_1}2^{l_2}3^{l_3}\cdots )$, where $l_k$ is the number of parts of cardinality $k$ (so that $\sum _k kl_k =s$). Its contribution is then $$ \otimes _{k\ge 1}t^{-l_k k}u^{l_k\max (2, k-1)}\Q [c_1,c_2,\dots ,c_{l_k}]. $$ If we sum over all sequences $(l_1,l_2,\dots )$ of nonnegative integers which become eventually zero, we get $$ \otimes _{k\ge 1}\bigl( \oplus _{l=0}^{\infty} t^{-lk}u^{l\max (2, k-1)}\Q [c_1,c_2,\dots ,c_l]\bigr) . $$ and $B^{\bullet}_{(1^s)}$ is the $t^{-s}$-part of this expression. For instance, $$\align B^{\bullet}_{(1^2)}&= t^{-2}(u^4\Q [c_1,c_2]\oplus u^2\Q [c_1]),\\ B^{\bullet}_{(1^3)})&=t^{-3}(u^6\Q [c_1,c_2,c_3]\oplus u^4\Q [c_1]\otimes\Q [c_1]\oplus u^2\Q [c_1]). \endalign $$ Since $H^1(\G _{\infty};\Q )=0$, it follows in particular that $H^1(\G _{\infty},S_{\langle 1^3\rangle})$ has dimension one. (It is easy to see that $H^1(\G _{\infty};,S_{\langle \lambda\rangle})=0$ for all other numerical partitions $\lambda$.) \medskip\label The cohomology groups of $\G _g$ with values in a symplectic representation have a geometric interpretation as the cohomology of a local system (in fact, of a variation of polarized Hodge structure). The Teichm\"uller space $\Cal{T}_g$ of conformal structures on $S_g$ modulo isotopy is a {\it contractible} complex manifold (of complex dimension $3g-3$, when $g\ge 2$). The action of $\G _g$ on it is properly discontinuous and a subgroup of finite index acts freely. The orbit space $\M _g:=\G _g \backslash \Cal{T}_g$ is naturally interpreted as the coarse moduli space of smooth complex projective curves. Via this interpretation, it gets the structure of a normal quasi-projective variety. It follows from the preceding that $\M _g$ has the rational cohomology of $\G _g$. More generally, if $U$ is a rational representation of $\G _g^s$, then we have a natural isomorphism $$ H^{\bullet}(\G _g;U)\cong H^{\bullet}(\M _g;\Bbb{U}), $$ where $\Bbb{U}$ is the sheaf on $\M _g$ which is the quotient of the trivial local system $\Cal{T}_g\times U\to \Cal{T}_g$ by the (diagonal) action of $\G _g$. This applies in particular to the representations $S_{\lambda}(V_g)$. This representation appears in the cohomology of degree $s:=|\lambda |$ of the configuration space of $s$ numbered (not necessarily distinct) points on $S_g$. So if $\Cu _g^s\to\M _g$ denotes the $s$-fold fiber product of the universal curve, then its $s$th direct image contains the local system (in the orbifold sense) associated to $S_{\lambda}(V_g)$ as a direct summand. By a theorem of Deligne, the Leray spectral sequence of the forgetful map $\Cu _g^s\to M_g$ degenerates at the $E_2$-term and thus the stable cohomology of this representation is realized inside $H^{\bullet}(\Cu ^s_g;\Q )$. This fact will be used in an essential way. \remark {Acknowledgements} I thank Dick Hain for discussions and for suggesting to use Deligne's degeneration theorem. I am also grateful to him for pointing out that the way \refer{1.1} was stated in an earlier version could not be correct. His student S.\ Kabanov obtained related results that should appear soon. I am also indebted to the referee for invaluable comments, in particular for a suggestion for shortening the original proof of \refer{2.3}. \endremark \head \section Stable cohomology of $\M _g^s$ and $\Cu _g^s$ \endhead We first state and prove an immediate consequence of the stability theorems. For $s\ge 1$, the class of a (Dehn) twist about $x_s$ generates an infinite cyclic central subgroup of $\G _{g,r+1}^{s-1}$. The quotient group can be identified with $\G _{g,r}^s$ and so we have a Gysin sequence $$ \cdots\to H^{k-2}(\G _{g,r}^s;\Z ){\buildrel u_s\cup\over\longrightarrow} H^k(\G _{g,r}^s;\Z)\to H^k(\G _{g,r+1}^{s-1};\Z)\to\cdots, $$ where $u_s\in H^2(\G _{g,r}^s;\Z )$ is the first Chern class. Similarly, $x_i$ determines a first Chern class $u_i\in H^2(\G _{g,r}^s;\Z )$ for $i=1,\dots ,s$. These classes are clearly stable and we shall not make any notational distinction between the $u_i$'s and their stable representatives. \proclaim{\label Proposition} The stable cohomology algebra over of the mapping class groups of surfaces with $s$ distinct numbered points is a graded polynomial algebra on the stable cohomology ring of the absolute mapping class groups. More precisely, there is a natural $\Sy _s$-equivariant graded ring homomorphism $$ H^{\bullet}(\G _{\infty};\Z )[u_1,\dots ,u_s]\to H^{\bullet}(\Gamma ^s_{g,r};\Z), $$ which is an isomorphism in degree $\le N(g;\Z )$. In particular, the rational stable cohomology algebra of the mapping class groups of surfaces with a $s$ unlabeled points is a graded polynomial $H^{\bullet}(\G _{\infty};\Q)$-algebra on the elementary symmetric functions $c_1,\dots ,c_s$ of the $u_i$'s. \endproclaim \demo{Proof} The composite of the forgetful maps $\G _{g,r}^{s-1}\to \G _{g,r-1}^s$ and $\G _{g,r-1}^s\to \G _{g,r-1}^{s-1}$ induces an isomorphism on $H^k(-;\Z )$ for large $g$. So in this range the forgetful map induces a surjection $H^k(\Gamma _{g,r-1}^s;\Z )\to H^k(\Gamma _{g,r}^{s-1};\Z )$. If we feed this in the above Gysin sequence, we get short exact sequence $$ 0\to H^{\bullet -2}(\G _{\infty}^s;\Z ){\buildrel u\cup\over\longrightarrow} H^{\bullet}(\G _{\infty} ^s;\Z )\to H^{\bullet}(\G _{\infty}^{s-1};\Z )\to 0 $$ so that $H^{\bullet}(\G _{\infty}^s;\Z )\cong H^{\bullet}(\G _{\infty}^{s-1};\Z )[u]$ as algebra's. The assertion now follows with induction on $s$. \enddemo Let us fix a finite set $X$. We denote by $\Cu ^X_g$ the moduli space of pairs $(C,x )$ where $C$ is a compact Riemann surface of genus $g$ and $x:X\to C$ is a map. Let $j:\M ^X_g\subset \Cu ^X_g$ be the open subset defined by the condition that $x$ be injective. Just as $\M _g$ is a virtual classifying space for $\G _g$, $\M ^X_g$ is one for $\G _g^{|X|}$; in particular, $\G _g^{|X|}$ and $\M ^X_g$ have the same rational cohomology. This enables us to restate \refer{2.1} in more geometric terms. Let $\Cu _g\to\M _g$ be the universal curve and denote by $\theta$ its relative tangent sheaf. For every $i\in X$, the map $(C,x)\mapsto x(i)$ defines a projection of $\Cu ^X_g$ onto the $\Cu _g$; denote by $\theta _i$ the pull-back of $\theta $ under this map. One easily recognizes the first Chern class of $\tau _i|\M ^X_g$ as the first Chern class defined above. So \refer{2.1} implies: \proclaim{\label Proposition} The ring homomorphism $$ H^{\bullet}(\M _g;\Q)[u_i:i\in X]\to H^{\bullet}(\M _g^X;\Q), \quad u_i\mapsto c_1(\theta _i)|\M ^X_g. $$ is an isomorphism in degree $\le N(g)$. \endproclaim We will use this proposition to prove that the rational cohomology of $\Cu ^X_g$ also stabilizes. We begin with attaching to $X$ a graded commutative $\Q [u_i :i\in X]$-algebra. It is convenient to introduce an auxiliary graded commutative $\Q$-algebra $\tilde A_X^{\bullet}$ first. The latter is defined by the following presentation: for each nonempty subset $I$ of $X$, $\tilde A_X^{\bullet}$ has a generator $u_I$ of degree two (we also write $u_i$ for $u_{\{ i\}})$, and these are subject to the relations $u_Iu_J=u_iu_{I\cup J}$ if $i\in I\cap J$. So if $i\in I$, then $u_Iu_I=u_iu_I$. It is then easy to see that the $\Q [u_i:i\in X]$-submodule generated by $u_I$ is defined by the relations $(u_i-u_j)u_I=0$ whenever $i,j\in I$. The monomials $\prod _I u_I^{r_I}$ for which $I$ runs over the members of a partition of $X$ form an additive basis of $\tilde A_X^{\bullet}$. (To make the indexing effective, let us agree that we only allow $r_I$ to be zero if $I$ is a singleton.) We then let $A^{\bullet}_X$ be the $\Q [u_i:i\in X]$-subalgebra of $\tilde A_X^{\bullet}$ generated by the elements $a_I:=u_I^{|I|-1}$, where $I$ runs over the subsets of $X$ with at least two elements. These generators obey the relations $$\align u_ia_I&:=u_ja_I \text{ if } i,j\in I,\\ a_Ia_J&:= u_i^{|I\cap J| -1}a_{I\cup J} \text{ if } i\in I\cap J. \endalign $$ and it is easy to see we thus obtain a presentation of $A^{\bullet}_X$ as a graded commutative $\Q [u_i :i\in X]$-algebra. Notice that as a $\Q [u_i :i\in X]$-algebra, $A^{\bullet}_X$ is already generated by the $a_I$'s with $|I|=2$. For every partition $P$ of $X$ we put $a_P:=\prod _{I\in P; |I|\ge 2} a_I$ (with the convention that $a_P=1$ if $P$ is the partition into singletons). These elements generate $A^{\bullet}_X$ as a $\Q [u_i :i\in X]$-module. In fact, $$ A_X^{\bullet}=\bigoplus _{P|X} \Q [u_I:I\in P]a_P. $$ We give $\tilde A_X^{\bullet}$ a (rather trivial) Hodge structure: its degree $2p$-part has Hodge type $(p,p)$. Clearly, $A_X^{\bullet}$ is then a Hodge substructure. \smallskip Given a partition $P$ of $X$, then the pairs $(C,x: X\to C)$ for which every member of $P$ is contained in a fiber of $x$ define a closed subvariety $i_P:\Cu _g(P)\subset \Cu ^X_g$. Those for which $P$ is the partition defined by $x$ make up a Zariski-open subvariety $\M _g(P)\subset \Cu _g(P)$. Notice that $\Cal{C}_g(P)$ resp. $\M _g(P)$ can be identified with $\Cu ^{X/P}_g$ resp. $\M ^{X/P}_g$ (where $X/P$ stands for quotient of $X$ by the equivalence relation defined by $P$, or rather the set of parts of $P$), and that this is a submanifold in the orbifold sense. For every nonempty $I\subset X$, let $P_I$ be the partition of $X$ whose parts are $I$ and the singletons in $X-I$. So the corresponding orbifold $\Cu _g(P_I)$ has codimension $|I|-1$ in $\Cu _g^X$. \proclaim{\label Theorem} There is an algebra homomorphism $$ \phi _g^X: H^{\bullet}(\M _g;\Q )\otimes A_X^{\bullet}\to H^{\bullet}(\Cu ^X_g;\Q ) $$ that extends the natural homomorphism $H^{\bullet}(\M _g;\Q )\to H^{\bullet}(\Cu ^X_g;\Q )$, sends $1\otimes u_i$ to $c_1(\theta _i)$ and sends $1\otimes a_I$ to the Poincar\'e dual of the class of $\Cu _g(P_I)$. This is an $\Sy _X$-equivariant algebra homomorphism and is also a morphism of mixed Hodge structures. Moreover, $\phi _g^X$ is an isomorphism in degree $\le N(g)$. \endproclaim \demo{Proof} For the first statement we must show that if $I,J\subset X$ have at least two elements, $i\in I\cap J$, $j\in I$ and $P$ is a partition of $X$, then $$ \align i_{P!}(1)&= \prod _{I'\in P; |I'|\ge 2}i_{P_{I'}!}(1),\\ c_1(\theta _i)i_{P_I!}(1)&=c_1(\theta _j)i_{P_I!}(1),\\ i_{P_I!}(1)i_{P_J!}(1)&=c_1(\theta _i)^{|I\cap J|-1} i_{P_{I\cup J}!}(1). \endalign $$ The first identity is geometrically clear and the second follows from the fact that $\theta _i$ and $\theta _j$ have isomorphic restrictions to $\Cu _g(P_I)$. To derive the last identity, we use the following lemma (the proof of which is left to the reader): \enddemo \proclaim{\label Lemma} Let $U$ and $V$ be closed complex submanifolds of a complex manifold $M$ whose intersection $W:=U\cap V$ is also a complex manifold. Suppose that any tangent vector of $M$ which is tangent to both $U$ and $V$ is tangent to $W$. Then $i_{U!}(1)i_{V!}(1)=i_{W!}(e)$, where $e$ is the euler class of the cokernel of the natural monomorphism $\nu _W\to \nu _U|W\oplus\nu _V|W$. \endproclaim \demo{Completion of the proof of \refer{2.3}} We apply the orbifold version of this lemma to $M=\Cu _g^X$, $U=\Cu _g (P_I)$, $V=\Cu _g(P_J)$ so that $W=\Cu _g(P_{I\cup J})$. The desired assertion then follows if we use the fact that the bundles appearing in the monomorphism $\nu _W\to \nu _U|W\oplus\nu _V|W$ are all direct sums of copies of $\theta _i|W$. The second statement of the theorem is clear. To prove the last, let $U_k$ resp. $S_k$ denote the union of the strata $\M _g(P)$ of $\codim \le k$ resp.~$=k$. We prove with induction on $k$ that the homomorphism $$ \bigoplus _{\codim P\le k} H^{\bullet}(\M _g;\Q )\otimes \C [u_I:I\in P]a_P\to H^{\bullet}(U_k;\Q ) $$ is an isomorphism in degree $\le N(g)$. For $k=0$ this is \refer{2.2}. If $k\ge 1$, then consider the Gysin sequence of the pair $(U_k,S_k)$: $$ \cdots\to H^{n-2k}(S_k;\Q )\to H^n(U_k;\Q )\to H^n(U_{k-1};\Q )\to\cdots . $$ In degrees $\le N(g)$ the isomorphism of $\bigoplus _{\codim P\le k} H^{\bullet}(\M _g;\Q )\otimes \C [u_I:I\in P]a_P$ onto $H^n(U_{k-1};\Q )$ factorizes over $H^n(U_k;\Q )$. So the Gysin sequence splits in this range. Since $\bigoplus _{\codim P =k} H^{\bullet}(\M _g;\Q )\otimes \C [u_I:I\in P]$ maps isomorphically onto $H^{\bullet}(S_k;\Q )$ in degree $\le N(g)$, the theorem follows. \enddemo {\it Remark.} For a curve $C$, the image of $A^{\bullet}_X$ in $H^{\bullet}(C^X;\Q )$ is contained in the Hodge ring of $C^X$. If $C$ is general, then it is in fact equal to it. \medskip\label A virtual classifying space of $\Gamma _g^{r+s}$ is the moduli space of $r+s$-pointed curves $\M _g^{r+s}$. It carries $r+s$ relative tangent bundles $\theta _1,\dots ,\theta _{r+s}$ and the total space of the $(\C ^*)^r$-bundle $\M _{g,r}^s\to \M_g^{r+s}$ defined by $\theta _{1+s},\dots ,\theta _{r+s}$ is a virtual classifying space for $\Gamma _{g,r}^s$. The comparison maps that enter in the statement of the stability theorem admit simple descriptions in these algebro-geometric terms and so preserve Hodge structures. This is probably well-known, but since we do not know a reference, we explain this. Clearly, the forgetful homomorphism $\Gamma _{g,r+1}^s\to \Gamma _{g,r}^s$ corresponds to the obvious projection $\M _{g,r+1}^s\to \M _{g,r}^s$. To exhibit the outer homomorphism $\Gamma _{g,r+1}^s\to \Gamma _{g+1,r}^s$, we first note that $\M _g^{r+s+1}\times \M _1^1$ parametrizes a codimension one stratum of the Knudsen--Deligne--Mumford compactification of $\M _{g+1}^{r+s}$: a pair $((C;x_1,\dots ,x_{r+s+1}),(E;O))$ determines a stable $r+s$-pointed genus $(g+1)$-curve by identifying $x_{r+s+1}$ with $O$. The normal bundle of this stratum is just the exterior tensor product of the relative tangent bundles of the factors. Denote the complement of its zero section by $U$ and let $U_e$ be a general fibre of the projection of $U\to \M_1^1$. We can identify $U_e$ with $\M _{g,1}^{r+s}$ and there is a natural restriction homomorphism $$ H^{\bullet}(\M _{g+1}^{r+s};\Q )\to H^{\bullet}(U;\Q )\to H^{\bullet}(U_e;\Q )\cong H^{\bullet}(\M _{g,1}^{r+s};\Q ). $$ The $(\C ^*)^r$-bundles that lie over these spaces determine likewise a homomorphism $H^{\bullet}(\M _{g+1,r}^s;\Q )\to H^{\bullet}(\M _{g,1+r}^s;\Q )$. This is the one we were after. Thus $H^{\bullet}(\G _{\infty };\Q )$ acquires a canonical mixed Hodge structure. The composite map $$ H^{\bullet}(\G _{\infty};\Q )\otimes A_X\to H^{\bullet}(\G _g;\Q )\otimes A_X\cong H^{\bullet}(\M _g;\Q )\otimes A_X\to H^{\bullet}(\Cu ^X _g;\Q ) $$ is evidently a morphism of mixed Hodge structures. \medskip For every $i\in X$ we have a projection $f_i:\Cu _g^X\to \Cu _g^{X-\{ i\}}$ and an obvious inclusion $A^{\bullet}_{X-\{i\}}\hookrightarrow A^{\bullet}_X$ (with image the linear combinations of monomials in which the $u_I$ with $i\in I$ occur with exponent $0$). The two are related: \proclaim{\label Lemma} The map $f_i^*:H^{\bullet}(\Cu _g^{X-\{ i\}};\Q ) \to H^{\bullet}(\Cu _g^X;\Q )$ is covered by the inclusion $A^{\bullet}_{X-\{i\}}\hookrightarrow A^{\bullet}_X$ tensorized with the identity of $H^{\bullet}(\G _{\infty};\Q )$. \endproclaim The proof is straightforward. Denote by $\overline{H}^{\bullet}(\Cu _g^X;\Q )$ the quotient of $H^{\bullet}(\Cu _g^{X};\Q )$ by the subspace spanned by the images of $f_i^*, u_if_i^*$, $i\in X$. \proclaim{\label Corollary} Put $$ A'{}^{\bullet}_X:=\bigoplus _{P|X} (\prod _{\{ i\}\in P} u_i^2)\Q[u_I:I\in P]a_P. $$ Then there is a natural homomorphism of mixed Hodge structures $$ H^{\bullet}(\G _{\infty};\Q )\otimes A'{}^{\bullet}_X\to\overline {H}^{\bullet}(\Cu _g^X;\Q ) $$ which is an isomorphism in degree $\le N(g)$. \endproclaim \head \section The Schur--Weyl functor \endhead \label We continue to denote by $X$ a finite nonempty set. The product $S_g^X$ comes with an obvious $\Sy _X$-action and so there is a resulting action of $\Sy _X$ on $H^{\bullet}(S_g^X;\Q )$. Any diffeomorphism $f$ of $S_g$ induces a diffeomorphism of $S_g^X$ commuting with this action. Thus is obtained an action of the product $\Gamma _g\times \Sy _X$ on $H^{\bullet}(S_g^X;\Q )$. As before, $V_g:=H^1(S_g;\Q )$ and $\symp (V_g)$ denotes its symplectic group. It is easily seen that the $\Gamma _g\times \Sy _X$-action factorizes through one of $\symp (V_g)\times\Sy _X$. Let us write the total cohomology $H^{\bullet}(S_g;\Q )$ as $\Q \oplus V_gt\oplus \Q u$ where $u$ is the canonical class (we assume $g\ge 2$ here) and $t$ shifts the degree by $1$. The K\"unneth rule gives an isomorphism $$ H^{\bullet}(S_g^X;\Q )=\bigoplus _{I,J\subset X;I\cap J=\emptyset} V_g^{\otimes I}t^Iu^J, $$ where $t^I$ should be thought of as a generator of the signum representation of $\Sy _I$ placed in degree $|I|$ and $u^J=\prod _{j\in J}u_j$. The multiplication in the right-hand side obeys the Koszul sign rule and is given by contractions stemming from the symplectic form on $V_g$. If we define $\overline{H}^{\bullet}(S_g^X;\Q )$ as in the relative case, that is, as the quotient of $H^{\bullet}(S_g^X;\Q)$ by the span of the images of $H^{\bullet}(S_g^{X-\{ i\}};\Q), u_iH^{\bullet}(S_g^{X-\{ i\}};\Q)$, $i\in X$, then we see that in terms of the K\"unneth decomposition this reduces to the single summand $V_g^{\otimes X}t^X$. \medskip\label Since $\pi :\Cu _g^X\to \M _g$ is a projective morphism whose total space is an orbifold, it follows from a theorem of Deligne \cite{1} that its Leray spectral sequence degenerates at the $E_2$-term over $\Q$. In other words, $H^{\bullet}(\Cu _g^X;\Q )$ has a canonical decreasing filtration $L^{\bullet}$, the {\it Leray filtration}, such that there is a natural isomorphism $$ \operatorname{Gr}_L^kH^n (\Cu _g^X;\Q )\cong H^k(\M _g;R^{n-k}\pi _* \Q ). $$ The maps $f_i^*$ in \refer{2.6} are strict with respect to the Leray filtrations. So if we combine this with \refer{2.7} we find: \proclaim{\label Corollary} There is a natural graded $\Sy _s$-equivariant map of $H^{\bullet}(\G _{\infty};\Q )$-modules $$ H^{\bullet}(\G _{\infty};\Q )\otimes A'{}^{\bullet}_X \to H^{\bullet}(\G _g; V_g ^{\otimes X})t^X $$ which is an isomorphism in degree $\le N(g)$. \endproclaim We want to decompose $V_g^{\otimes X}$ as a $Sp(V_g)\times\Sy _X$-representation. Following Weyl this is done in two steps. The first step involves Weyl's representation $V_g^{\langle X\rangle}$ whose definition we presently recall. Let $\omega\in V_g\otimes V_g$ correspond to the symplectic form on $V_g$. For every ordered pair $(i,j)\in X$ with $i\not= j$, we have a natural homomorphism $V_g^{\otimes (X-\{i,j\})}\to V_g^{\otimes X}$ defined by placing $\omega$ in the $(i,j)$-slot. Reversing the order gives minus this map. So the natural assertion is that we have a map $$ \bigoplus _{I\subset X; |I|=2} V_g^{\otimes (X -I)}t^I\to V_g^{\otimes X}. $$ It is easy to see that this map is injective; its cokernel is by definition $V_g^{\la X\ra}$. Notice that $V_g^{\langle X\rangle}$ is in a natural way a representation of $\symp (V_g)\times\Sy _X$. The second step is the decomposition of $V_g^{\langle X\rangle}$: Weyl proved the remarkable fact that $$ V_g^{\langle X\rangle}\cong \bigoplus _{\lambda} S_{\langle {\lambda}\rangle}(V_g)\boxtimes (\lambda ), $$ where $\lambda $ runs over the numerical partitions of $|X|$ in at most $g$ parts and $(\lambda )$ denotes the corresponding equivalence class of irreducible representations of $\Sy _X$. In particular, all these irreducible representations appear with multiplicity one. A modern account of the proof and of related results can be found in the book by Fulton and Harris \cite{2}. \proclaim{\label Theorem} If $A''{}^{\bullet}_X\subset A'{}^{\bullet}_X$ is the $\Q [u_i:i\in X]$-submodule defined by $$ A''_X=\bigoplus _{P|X} (\prod _{\{ i\}\in I}u_i^2)(\prod _{I\in P;|I|=2}u_I) \Q [u_I :I\in P]a_P, $$ then there is a natural graded $\Sy _X$-equivariant map of $H^{\bullet}(\G _g,\Q )[u_i:i\in X]$-modules $$ H^{\bullet}(\G _{\infty};\Q )\otimes A''{}^{\bullet}_X \to H^{\bullet}(\G _g; V_g ^{\la X\ra })t^X, $$ and this map is an isomorphism in degree $\le N(g)$. \endproclaim \demo{Proof} If $I\subset X$ is a two-element subset, then the Poincar\'e dual of the hyperdiagonal $\Cu _g(P_I)\subset \Cu _g^X$ is $u_I$. The restriction of this element to the fiber $S_g^X$ is $(\sum _{i\in I}u_i)+\omega t^I$. So the map $$ H^{\bullet -2}(\G _g; V_g ^{\otimes (X-I)})t^{X-I}\to H^{\bullet}(\G _g; V_g ^{\otimes X})t^X $$ defined by multiplication with $\omega t^I$ is covered by the map $H^{\bullet}(\G _{\infty};\Q )\otimes A'{}^{\bullet}_{X-I}\to H^{\bullet}(\G _{\infty};\Q )\otimes A'{}^{\bullet}_X$ which is multiplication by $u_I-\sum_{i\in I}u_i$. The theorem follows. \enddemo \medskip \demo{Proof of \refer{1.1}} We decompose both members of the stable isomorphism \refer{3.4} according to the action of $\Sy _X$. Let $\lambda$ be a numerical partition of $|X|$ and let $\lambda '$ be the conjugate partition. According to Weyl's decomposition theorem, the $(\lambda )$-isotypical component of $H^{\bullet}(\G _g, V_g ^{\la X\ra})$ is $H^{\bullet}(\G _g, S_{\la\lambda\ra}(V_g))$. Note that this is also the isotypical component of type $\lambda '$ of $H^{\bullet}(\G _g; V_g ^{\langle X\rangle})t^X$. On the other hand, it is easily seen that the $(\lambda ')$-isotypical component of $A''{}^{\bullet}_X$ can be identified with $t^sB^{\bullet}(\lambda)$. Since the Leray spectral sequence \refer{3.2} is a spectral sequence of mixed Hodge structures, the homomorphism of \refer{1.1} is actually a morphism of mixed Hodge structures. The theorem follows. \enddemo \head \section Stable cohomology of the universal Abel--Jacobi map \endhead If we give $S_g$ a complex structure, then $S_g$ becomes a compact Riemann surface $C$ of genus $g$, so that we have defined an Abel--Jacobi map $ \Sym ^s(C)\to\Pic ^s(C)$. The induced map on cohomology has been determined by Macdonald: \proclaim{\label Proposition} {\rm (Macdonald \cite{7})} Identify $H^{\bullet}(\Pic ^s(C);\Q )$ with the exterior algebra on $V_g$ so that the Abel--Jacobi map determines an algebra homomorphism $\wedge ^{\bullet}V_g\to H^{\bullet}(\Sym ^s(C);\Q )$. Let $\wedge ^{\bullet}V_g[y]\to H^{\bullet}(\Sym ^s(C);\Q )$ be the extension that sends the indeterminate $y$ of degree two to the sum of the fundamental classes of the factors. Then this map is surjective and its its kernel is the degree $>s$-part of the ideal $\Cal{I}$ in $\wedge ^{\bullet}V_g [y]$ generated by $\{ v\wedge v'-(v.v')y : v,v'\in V_g \}$. \endproclaim If $s>2g-2$, then the Abel--Jacobi map is the projectivization of a vector bundle of rank $s+1-g$ over $\Pic (C)$ and we can interpret the image of $y$ as the first Chern class of the associated line bundle over $\Sym ^s(C)$. Macdonald also expresses the Poincar\'e duals of the diagonals of $C^s$ in terms of this presentation. \smallskip It is our aim to make a corresponding discussion for the universal situation in the stable range. The morphism $\Cu _g\to \M _g$ defines a relative Picard bundle $\Pic (\Cu _g/\M _g)\to \M_g$ (in the orbifold sense) whose connected components are still indexed by the degree: $\Pic ^k(\Cu _g / \M _g)$, $k\in\Z$. The degree $0$-component is called the {\it universal Jacobian} and is also denoted $\J _g$. \proclaim{\label Lemma} For every $k\in\Z$, there is a natural isomorphism $$ H^{\bullet}(\Pic ^k(\Cu _g/\M _g);\Q )\cong H^{\bullet}(\J _g;\Q ). $$ \endproclaim \demo{Proof} The relative canonical sheaf defines a section of $\Pic ^{2g-2}(\Cu _g/\M _g)$. On a suitable Galois cover $\tilde{\M} _g\to \M _g$ (with Galois group $G$, say) this section becomes divisible by $2g-2$ and thus produces a section of $\Pic ^1(\tilde\Cu _g/\tilde{\M } _g)$. This determines an isomorphism $\Pic ^k(\tilde\Cu _g/\tilde\M _g)\cong \Pic ^0(\tilde\Cu _g/\tilde\M _g)$. Although this isomorphism will not in general be $G$-equivariant, it will differ from any $G$-translate by a section of $\Pic ^0(\tilde\Cu _g/\tilde M _g)$ of finite order, and so the induced map on rational cohomology is $G$-equivariant. By passing to the $G$-invariants we obtain an isomorphism $H^{\bullet}(\Pic ^k(\Cu _g/\M _g);\Q )\cong H^{\bullet}(\J _g;\Q )$. One checks that this map does not depend on choices. \enddemo Let $X$ be a finite nonempty set as before. We wish to determine the subalgebra of $\Sy _X$-invariants of $A^{\bullet}_X$, at least stably. Recall that an additive basis of $A^{\bullet}_X$ consists of the set of elements of the form $\prod _{I\in P} u_I^{r_I}$, where $P$ runs over the partitions of $X$ and $r_I\ge |I|-1$. Let us define a partial ordering on the collection of partitions of $X$ by: $P\le Q$ if $P=Q$ or if for the smallest number $k$ such that the $k$-element members of $P$ and $Q$ do not coincide every $k$-element member of $P$ is a $k$-element member of $Q$. This determines a partial ordering on the set of monomials: $\prod _{I\in P} u_I^{r_I}\le\prod _{J\in Q} u_J^{s_J}$ if $P<Q$ or if $P=Q$ and $r_I\le s_I$ for all $I$. The defining relations for $A^{\bullet}_X$ show that a product of two monomials associated to partitions $P$ and $Q$ is a monomial associated to a partition that dominates both $P$ and $Q$. Denote by $S:=\sum _{\sigma\in\Sy _X}\sigma$ the symmetrizer operator (acting on $A_X$). Given a partition $P$ of $X$, then the smallest terms in $\prod _{I\in P} S(u_I^{r_I})$ are monomials associated to a partition that is a $\Sy _X$-translate of $P$. In this expression the part associated to $P$ is the subsum corresponding to the partial symmetrizer $S_P:=\sum _{\sigma\in\Sy _X;\sigma (P)=P}\sigma$. If $l_k$ is the number of members of $P$ with $k$ elements, then the image of $S_P$ can be identified with $$ \Q [c_1,c_2,\dots ,c_{l_1}]\otimes \bigotimes _{k\ge 2;l_k>0} (c_{l_k}^{k-1}\Q [c_1,c_2,\dots ,c_{l_k}]). $$ Here $c_l$ in the tensor factor with index $k$ is to be thought of as the $l$th elementary symmetric function in the $u_I$'s with $I\in P$ and $|I|=k$ (and so has degree $2l$); the appearance of $c_{l_k}^{k-1}$ comes from the condition $r_I\ge |I|-1$. So if we put $$ C^{\bullet}_{\infty}:= \Q [c_1,c_2,\cdots ]\otimes \bigotimes _{k\ge 2}\bigl( \Q\oplus \bigoplus _{l\ge 1}c_l^{k-1}\Q [c_1,c_2,c_3,\dots ,c_l]\bigr) , $$ then we find: \proclaim{\label Corollary} There is natural surjective homomorphism of graded algebra's $C^{\bullet}_{\infty}\to (A^{\bullet}_X)^{\Sy _X}$. Its restriction to the $\Q$-span of all the monomials involving the variables $c^{(k_1)}_{l_1},\dots ,c^{(k_r)}_{l_r}$ (where $c_l^{(k)}$ denotes the variable $c_l$ that occurs in the $k$th tensor power) with $\sum _i k_il_i\le |X|$ is a linear isomorphism. \endproclaim If $Y$ is another finite set with $|Y|\ge |X|$, then any injection $f: X\hookrightarrow Y$ induces an algebra homomorphism $A^{\bullet}_X\to A^{\bullet}_Y$. Since all such injections are in the same $\Sy _Y$-orbit they give rise to the same algebra homomorphism $(A^{\bullet}_X)^{\Sy _X}\to (A^{\bullet}_Y)^{\Sy _Y}$. In particular, $(\tilde A^{\bullet}_X)^{\Sy _X}$ only depends on $|X|$. So if we write $\tilde C^{\bullet}_{|X|}$ for this algebra, then we have a direct system $\cdots\to C^{\bullet}_s\to C^{\bullet}_{s+1}\to\cdots $. The corollary shows that the limit of this direct system can be identified with $C^{\bullet}_{\infty}$. It follows from \refer{2.3} that we have an algebra homomorphism $$ H^{\bullet}(\G _{\infty};\Q )\otimes C^{\bullet}_s\to H^{\bullet}((\Cu _g ^s);\Q )^{\Sy _s}\cong H^{\bullet}((\Cu _g ^s)^{\Sy _s};\Q ) $$ that is an isomorphism in degree $\le N(g)$. In the limit this yields a homomorphism $$ H^{\bullet}(\G _{\infty};\Q )\otimes C^{\bullet}_{\infty}\to H^{\bullet}((\Cu _g ^s)^{\Sy _s};\Q ) $$ that is an isomorphism in degree $\le \min (2s,N(g))$. The image of $c_1\otimes 1\otimes 1\otimes\cdots$ is easily seen to be proportional to the element $y$ appearing in Macdonald's theorem. We put $$ C'{}^{\bullet}_{\infty}:= \Q [c_2,c_3,\cdots ]\otimes\bigotimes _{k\ge 2}\bigl( \Q+\oplus _{l\ge 1}c_l^{k-1}\Q [c_1,c_2,c_3,\dots ,c_l]\bigr) . $$ \proclaim{\label Theorem} The algebra homomorphism above fits in a commutative square of algebra homomorphisms $$ \matrix H^{\bullet}(\G _{\infty};\Q )\otimes C^{\bullet}_{\infty}&\longrightarrow & H^{\bullet}((\Cu _g ^s)^{\Sy _s};\Q )\\ \cup &&\uparrow\\ H^{\bullet}(\G _{\infty};\Q )\otimes C'{}^{\bullet}_{\infty}&\longrightarrow & H^{\bullet}(\Pic ^s(\Cu _g/\M _g);\Q ), \endmatrix $$ in which the right vertical map is induced by the Abel--Jacobi map. The lower horizontal map is an isomorphism in degree $\le \min (s,N(g))$ so that in the limit we have an isomorphism $$ H^{\bullet}(\G _{\infty};\Q )\otimes C'{}^{\bullet}_{\infty}\cong \bigoplus _{s=0}^{\infty} H^{\bullet}(\G _{\infty};\wedge ^s)t^s. $$ \endproclaim \demo{Proof} If we combine Macdonald's theorem with the Leray spectral of sequence of the map $(\Cu _g ^s)^{\Sy _s}\to \M_g$, then we see that the map $$ H^{\bullet}(\Pic ^s(\Cu _g/\M _g);\Q )[y]\to H^{\bullet}((\Cu _g ^s)^{\Sy _s};\Q ) $$ that sends $y$ to $c_1\otimes 1\otimes 1\otimes\cdots $ is an isomorphism in degree $\le s$. The theorem follows from this. \enddemo \Refs \ref\no 1 \by P\. Deligne \paper Th\'eor\`eme de Lefschetz et crit\`eres de d\'eg\'en\'erescence de suites spectrales \jour Inst\. Hautes \'Etudes Sci\. Publ\. Math\.\vol 35 \yr 1968 \pages 259--278 \endref \ref\no 2 \by W.~Fulton and J.~Harris \book Representation theory \bookinfo Graduate Texts in Math. \vol 129 \publ Springer Verlag \publaddr New York \yr 1991 \endref \ref\no 3 \by J.~Harer \paper Stability of the homology of the mapping class groups of orientable surfaces \jour Ann\. of Math\. \vol 121 \yr 1985 \pages 215--249 \endref \ref\no 4 \by J.~Harer \paper Improved stability for the homology of the mapping class groups of surfaces \paperinfo preprint Duke University (1993) \endref \ref\no 5 \by N.V.~Ivanov \paper Complexes of curves and the Teichm\"uller modular group \jour Uspekhi Mat\. Nauk \vol 42 \yr 1987 \lang Russian \pages 110-126 \transl\nofrills English transl. in \jour Russian Math\. Surveys \vol 42 \yr 1987 \pages 55--107 \endref \ref\no 6 \by N.V.~Ivanov \paper On the homology stability for Teichm\"uller modular groups: closed surfaces and twisted coefficients \inbook Mapping class groups and moduli spaces of Riemann surfaces \eds C.F.~B\"odig-heimer and R.M.~Hain \pages 149--194 \bookinfo Contemp\. Math\. \vol 150 \publ AMS \yr 1993 \endref \ref\no 8 \by I.G.~Macdonald \paper Symmetric products of an algebraic curve \jour Topology \vol 1 \pages 319--343 \yr 1962 \endref \endRefs \enddocument \bye

9401

alg-geom/9401008

en

https://arxiv.org/abs/alg-geom/9401008

alg-geom/9401008

Steven Bradlow

Steven B. Bradlow and Oscar Garcia-Prada

Stable triples, equivariant bundles and dimensional reduction

41 pages, LaTeX

This is a resubmission of preprint 9401008 , which has some TeXnical errors introduced by the "reform" procedure (designed to avoid precisely these problems!). The original can be formatted by editing out the messages "%% following line cannot be broken..." at line 1674. This problem has been corrected in the current version. The content of the paper has not been changed in any way. My aplogies for any inconvenience caused. SB

alg-geom

\section{Introduction}\label{intro} The Hitchin-Kobayashi correspondence between stable bundles and solutions to the \linebreak Hermitian-Einstein equations allows one to apply analytic methods to the study of stable bundles. One such analytic technique, which has not yet been much exploited, is that of dimensional reduction. This is a useful tool for studying certain special solutions to partial differential equations; in particular it is useful for studying solutions which are invariant under the action of some symmetry group. When applied to the Hermitian-Einstein equations, it thus provides a way of looking at holomorphic bundle structures which are both stable and invariant under some group action on the bundle, i.e. of looking at {\em equivariant stable bundles}. The main idea in dimensional reduction is the following: Suppose we have a partial differential equation defined on a space which has a symmetry, i.e. which supports some group action. Then by integrating over the group orbits, any solution which is invariant under the group action becomes an object defined in a space of lower dimension than that of the original setting for the general solutions. This lower dimensional space is the orbit space of the group action, and in that space the special solutions to the original equations can be re-interpreted as ordinary solutions to some new set of equations. In particular, suppose that we start with a 4-manifold, $M$, a Lie group $G$\ which acts on it, and a complex vector bundle $E$\ to which this action lifts. In such a situation, there can be equivariant solutions to the Hermitian-Einstein equations, and dimensional reduction can be applied. The orbit space $E/G$\ will be a new bundle over $M/G$, the orbit space of the group action on the original 4-manifold. The equivariant solutions to the original Hermitian-Einstein equations will be solutions to a new set of equations on the bundle $E/G\longrightarrow M/G$. For example, the equations introduced by Hitchin in \cite{H}, namely the Anti-Self-Dual equations on a Riemann surface, can be viewed as arising in this way. A special case of the above situation occurs when the 4-manifold $M$\ is a complex surface, and the orbit space $M/G$\ also admits a complex structure. Solutions to the Hermitian-Einstein equations then correspond to stable holomorphic structures on the bundle $E$. In such a situation, dimensional reduction acquires an extra, holomorphic interpretation. It results in information about the equivariant stable bundles on $M$\ being encoded in a holomorphic interpretation for the dimensionally reduced equations on $M/G$ . These sort of ideas are developed in \cite{GP3}, where they are applied to certain $SU(2)$-equivariant bundles over ${{X\times\dP^1}}$. Here $X$ is a closed Riemann surface and the $SU(2)$-action is trivial on $X$ and the standard one on $\dP^1$. In this case, the equivariant holomorphic bundles over ${{X\times\dP^1}}$\ correspond to holomorphic pairs (i.e. bundles plus prescribed global sections) over $X$. The dimensional reduction of the Hermitian-Einstein equations gives the vortex equations, and the stable equivariant bundles on ${{X\times\dP^1}}$\ correspond (by dimensional reduction) to $\tau$-stable holomorphic pairs on $X$, with $\tau$-stability as defined in \cite{B2} and \cite{GP3}. However not all the $SU(2)$-equivariant holomorphic bundles over ${{X\times\dP^1}}$\ correspond to holomorphic pairs on $X$. In fact those that do form a rather restricted subset of the set of all such equivariant bundles. A very natural relaxation of this restriction leads to a class of equivariant bundles on ${{X\times\dP^1}}$\ which still corresponds to data on the (lower dimensional) space $X$, but not necessarily to holomorphic pairs. What such bundles correspond to is a pair of bundles plus a holomorphic homomorphism between them. We call such data a holomorphic triple. In this paper we undertake a detailed investigation of holomorphic triples over the closed Riemann surface $X$. In particular, we define, in Section 3, a notion of stability for such objects. We explore the relationship between the stability of a triple and the stability of the corresponding equivariant bundle over ${{X\times\dP^1}}$. An important feature of the definition is that, like in the case of holomorphic pairs, it involves a real parameter. This can be traced back to the fact that the definition of stability for a bundle over ${{X\times\dP^1}}$ depends on the polarization (choice of K\"{a}hler\ metric) on ${{X\times\dP^1}}$. We discuss the nature of this parameter, and its influence on the properties of the stable triples. We show for example that {\em In all cases, with one exception, the parameter in the definition of triples stability lies in a bounded interval. The interval is partitioned by a finite set of non-generic values.} Our main result is given in Section 4. Loosely speaking, it is that the stable triples over $X$\ can be considered the dimensional reduction of the stable equivariant bundles over ${{X\times\dP^1}}$. In other words, {\em A holomorphic triple over $X$\ is stable if and only if the corresponding \linebreak $SU(2)$-equivariant extension over ${{X\times\dP^1}}$\ is stable} In \cite{GP3} dimensional reduction is applied to the Hermitian-Einstein equation on equivariant bundles over ${{X\times\dP^1}}$. The result is that on bundles corresponding to triples over $X$, the equivariant solutions correspond to solutions to a pair of {\em Coupled Vortex Equations} on the two bundles in the triple. By combining this result, our dimensional reduction result for stable bundles, and the Hitchin-Kobayashi correspondence, we can thus show {\em There is a Hitchin-Kobayashi correspondence between stability of a triple and existence of solutions to the Coupled Vortex Equations.} This is discussed in Section 5. In Section 6 we discuss the moduli spaces of stable triples. By identifying these as fixed point sets of an $SU(2)$-action on the moduli spaces of stable bundles over ${{X\times\dP^1}}$, we obtain results such as {\em For fixed value of the stability parameter, the moduli space of stable triples is a quasi- projective variety. For generic values of the parameter, and provided the ranks and degrees of the two bundles satisfy a certain coprimality condition, the moduli space is projective} In Section 2 we have collected together the basic definitions and background material that we will need. \section{Background and Preliminaries}\label{background} \subsection{Basic Definitions Let $X$ be a compact Riemann surface. \begin{definition} A holomorphic triple on $X$ is a triple ${(E_1,E_2,\Phi)}$ consisting of two holomorphic vector bundles $E_1$ and $E_2$ on $X$ together with a homomorphism $\Phi:E_2\longrightarrow E_1$, i.e. an element $\Phi\in H^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$. \end{definition} In this paper we will develop a theory of holomorphic triples as objects in their own right. We will also show how they arise from $\su$-equivariant\ holomorphic vector bundles over ${{X\times\dP^1}}$. Let ${SU(2)}$ act on ${{X\times\dP^1}}$ trivially on $X$ and in the standard way on $\dP^1$, that is we regard $\dP^1$ as the homogeneous space ${SU(2)}/U(1)$. Let $F$ be a $C^\infty$ complex vector bundle\ over ${{X\times\dP^1}}$. \begin{definition} The bundle $F$ is said to be ${SU(2)}$-{\em equivariant} if there is an action of ${SU(2)}$ on $F$ covering the action on ${{X\times\dP^1}}$. Similarly a holomorphic vector bundle\ $F$ is ${SU(2)}$-{\em equivariant} if it is $\su$-equivariant\ as a $C^\infty$ bundle and in addition the action of ${SU(2)}$ on $F$ is holomorphic. \end{definition} \subsection{Smooth and Holomorphic Equivariant bundles Our main objective in this section will be the study of $\su$-equivariant\ holomorphic vector bundles\ on ${{X\times\dP^1}}$, however, before addressing this, we shall analyse the much easier problem of classifying the $\su$-equivariant\ $C^\infty$ ones. Let $p$ and $q$ be the projections from ${{X\times\dP^1}}$ to the first and second factors respectively. \begin{prop}\label{equi-vb} Every $\su$-equivariant\ $C^\infty$ vector bundle\ $ F$ over ${{X\times\dP^1}}$ can be equivariantly decomposed, uniquely up to isomorphism, as $$ F=\bigoplus_i {p^\ast} E_i\otimes {q^\ast} H^{\otimes n_i}, $$ \noindent where $E_i$ is a $C^\infty$ vector bundle\ over $X$, $H$ is the $C^\infty$ line bundle over $\dP^1$ with Chern class 1, and $n_i \in {\Bbb Z}$ are all different. \end{prop} {\em Proof}. See \cite[Proposition 3.1]{GP3}. We shall describe now $\su$-invariant\ holomorphic structures on a fixed $\su$-equivariant\ $C^\infty$ vector bundle\ over ${{X\times\dP^1}}$. We shall restrict ourselves, however, to the case which is relevant in connection to holomorphic triples. Let $E_1$ and $E_2$ be $C^\infty$ vector bundles\ on $X$ and let $H$ be as in Proposition \ref{equi-vb}. Consider the $\su$-equivariant\ $C^\infty$ vector bundle\ \begin{equation} F= {p^\ast} E_1\oplus {p^\ast} E_2\otimes {q^\ast} H^{\otimes 2}, \label{equi-f} \end{equation} Note that the total space of ${p^\ast} E_1$ is $E_1\times \dP^1$, and the action of ${SU(2)}$ that we are considering is trivial on $E_1$ and the standard one on $\dP^1$; similarly for ${p^\ast} E_2$. On the other hand, recall that the ${SU(2)}$-equivariant line bundle $H^{\otimes 2}$ over $\dP^1\cong {SU(2)}/U(1)$ corresponds to the one dimensional representations of $U(1)$ given by $e^{i2\alpha}$, i.e. $H^{\otimes 2}={SU(2)}\times_{U(1)}{\Bbb C}$, where $(g,v)\sim (g',v')$ if there is an $e^{i\alpha}\in U(1)$ such that $g'=e^{-i\alpha}g$ and $v'=e^{i2\alpha}v$. The action of ${SU(2)}$ on ${SU(2)}\times{\Bbb C}$, given by $$ \gamma.(g,v)=(\gamma g,v)\;\;\;\;\;\mbox{for}\;\;\;\gamma\in {SU(2)}\;\;\; \mbox{and}\;\;\;(g,v)\in {SU(2)}\times {\Bbb C}, $$ descends to an action on $H^{\otimes 2}$. In order to avoid the introduction of more notation we shall denote a $C^\infty$ vector bundle\ and the same bundle endowed with a holomorphic structure by the same symbol. The distinction will be made explicit unless it is obvious from the context. \subsection{Dimensional Reduction of Bundles \begin{prop}\label{equi-hvb} Every $\su$-equivariant\ holomorphic vector bundle\ $F$ with underlying $\su$-equivariant\ $C^\infty$ structure given by (\ref{equi-f}) is in one-to-one correspondence with a holomorphic extension of the form \begin{equation} {0\lra \ps E_1\lra F\lra \ps E_2\otimes\qs\cod\lra 0}, \label{extension} \end{equation} where $E_1$ and $E_2$ are the bundles over $X$ defining (\ref{equi-f}) equipped with holomorphic structures. Moreover, every such extension is defined by an element $\Phi\in\mathop{{\fam0 Hom}}\nolimits(E_2,E_1)$, and is thus in one-to-one correspondence with the holomorphic triple ${(E_1,E_2,\Phi)}$ on $X$. \end{prop} {\em Proof}. We shall give here only a brief sketch of the proof (see \cite[Proposition 3.9]{GP3} for details). Let $E_1$ and $E_2$ be two holomorphic vector bundles\ over $X$. The extensions over ${{X\times\dP^1}}$ of the form (\ref{extension}) are parametrized by $$ H^1({{X\times\dP^1}}, {p^\ast} (E_1\otimes E_2^\ast)\otimes {q^\ast}{\cal O}(-2)); $$ but this is isomorphic to $$ H^0(X, E_1\otimes E_2^\ast)\otimes H^1(\dP^1,{\cal O}(-2)) \cong H^0(X,E_1\otimes E_2^\ast), $$ by means of the K\"{u}nneth formula, the fact that $H^0(\dP^1,{\cal O}(-2))=0$ and $$H^1(\dP^1,{\cal O}(-2))\cong H^0(\dP^1,{\cal O})^\ast\cong {\Bbb C}\ .$$ Therefore after fixing an element in $H^1(\dP^1,{\cal O}(-2))$, the homomorphism $\Phi$ can be identified with the extension class defining $F$. Certainly, since the action of ${SU(2)}$ on the extension class is trivial (note that this action is induced from the action on $E_1\otimes E_2^\ast$, which is trivial), then the bundle $F$ defined by the triple ${(E_1,E_2,\Phi)}$ is an $\su$-equivariant\ holomorphic vector bundle. One can show with a little bit more work that in fact every $\su$-equivariant\ holomorphic structure on the $\su$-equivariant\ $C^\infty$ bundle (\ref{equi-f}) defines an extension of the form (\ref{extension}). \hfill$\Box$ What this Proposition says is that holomorphic triples over $X$ can be regarded as a ``{\em dimensional reduction}'' of certain $\su$-equivariant\ holomorphic vector bundles\ over ${{X\times\dP^1}}$. \subsection{Subtriples \begin{definition} A triple $T' = {(E_1',E_2',\Phi')}$\ is a subtriple of $T ={(E_1,E_2,\Phi)}$\ if \begin{tabbing} \ \ \ \=(1)\ \= \kill \> {\em (1)} \> $E'_i$\ is a coherent subsheaf of $E_i$, for $i=1,2$\\ \> {\em (2)}\> $\Phi' = \Phi|_{E_2}$, i.e. $\Phi'$\ is the restriction of $\Phi$. \end{tabbing} In other words we have the commutative diagram $$ \begin{array}{ccc} E_2& \stackrel{\Phi}{\longrightarrow} & E_1 \\ \uparrow& & \uparrow\\ E_2'& \stackrel{\Phi'}{\longrightarrow} & E_1'. \end{array} $$ If $E'_1=E'_2=0$, the subtriple is called the trivial subtriple. \end{definition} \noindent {\em Remark.} When studying stability criteria, it will suffice, as usual, to consider saturated subsheaves, that is subsheaves whose quotient sheaves are torsion free. On a Riemann surface these are precisely subbundles. With this definition, subobjects of the triple ${(E_1,E_2,\Phi)}$\ are related to subsheaves of the corresponding $SU(2)$-equivariant bundle $F\longrightarrow {{X\times\dP^1}}$\ in an appropriate way. First note that the correspondence between triples on $X$ and bundles on ${{X\times\dP^1}}$ can be extended more generally to arbitrary coherent sheaves. Namely, if $S_1$ and $S_2$ are two coherent sheaves on $X$ and $\Psi\in \mathop{{\fam0 Hom}}\nolimits( S_2,S_1)$ the triple $( S_1, S_2,\Psi)$ defines a coherent sheaf $U$ over ${{X\times\dP^1}}$. This sheaf is given, as for bundles, as an extension \begin{equation} 0\longrightarrow {p^\ast} S_1\longrightarrow U\longrightarrow{p^\ast} S_2\otimes{\cO(2)}\longrightarrow 0. \label{c-ext} \end{equation} The proof is very much as for the case of bundles, once we have fixed $ S_1$ and $ S_2$, the extensions as (\ref{c-ext}) are parametrized by $$ \mathop{{\fam0 Ext}}\nolimits^1_{{X\times\dP^1}}({p^\ast} S_2\otimes{q^\ast}{\cO(2)},{p^\ast} S_1). $$ But, by the K\"unneth formula for the $\mathop{{\fam0 Ext}}\nolimits$ groups, this group is isomorphic to $$ \mathop{{\fam0 Hom}}\nolimits_X( S_2, S_1)\otimes\mathop{{\fam0 Ext}}\nolimits^1_{\dP^1}({\cal O},{\cO(2)})\oplus \mathop{{\fam0 Ext}}\nolimits^1_X( S_2, S_1)\otimes\mathop{{\fam0 Hom}}\nolimits_{\dP^1}({\cO(2)},{\cal O}). $$ This reduces to $\mathop{{\fam0 Hom}}\nolimits_X(S_2,S_1)$, since $\mathop{{\fam0 Hom}}\nolimits_{\dP^1}({\cO(2)},{\cal O})\cong H^0(\dP^1,{\cal O}(-2))=0$ and $$\mathop{{\fam0 Ext}}\nolimits^1_{\dP^1}({\cal O},{\cO(2)})\cong H^1(\dP^1,{\cal O}(-2))\cong{\Bbb C}\ .$$ \begin{lemma}\label{equi-sub} Let $F\longrightarrow{{X\times\dP^1}}$ be the bundle associated to a triple ${(E_1,E_2,\Phi)}$. Then every $\su$-invariant\ coherent subsheaf $F'\subset F$ is an extension of the form \begin{equation} {0\lra \ps E_1'\lra F'\lra \ps E_2'\otimes\qs\cod\lra 0},\label{s-ext} \end{equation} with $E_1'\subset E_1$ and $E_2'\subset E_2$ coherent subsheaves, making the following diagram commutative $$ \begin{array}{ccccccccc} 0& \longrightarrow & {p^\ast} E_1 &\longrightarrow & F & \longrightarrow & {p^\ast} E_2\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0\\ & &\uparrow & & \uparrow & & \uparrow & & \\ 0& \longrightarrow & {p^\ast} E_1' &\longrightarrow & F' & \longrightarrow & {p^\ast} E_2'\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0. \end{array} $$ Thus $F'$ corresponds to a triple ${(E_1',E_2',\Phi')}$, for $\Phi'\in \mathop{{\fam0 Hom}}\nolimits(E_2',E_1')$. \end{lemma} {\em Proof}. Let $f: F'\rightarrow{p^\ast} E_2\otimes{q^\ast}{\cO(2)}$ be the composition of the injection $ F'\rightarrow F$ with the surjective map $ F\rightarrow{p^\ast} E_2\otimes{q^\ast}{\cO(2)}$. Consider the commutative diagram $$ \begin{array}{ccccccccc} 0& \longrightarrow & {p^\ast} E_1 &\longrightarrow & F & \longrightarrow & {p^\ast} E_2\otimes{q^\ast}{\cO(2)} &\longrightarrow & 0\\ & &\uparrow & & \uparrow & & \uparrow & & \\ 0& \longrightarrow & \mathop{{\fam0 Ker}}\nolimits f &\longrightarrow & F' & \longrightarrow & \mathop{{\fam0 Im}}\nolimits f &\longrightarrow & 0. \end{array} $$ The ${SU(2)}$-invariance of $ F'$ implies that of $\mathop{{\fam0 Ker}}\nolimits f$ and $\mathop{{\fam0 Im}}\nolimits f$. It suffices therefore to show that if $E$ is a holomorphic vector bundle over $X$ and if ${p^\ast} E$ is the pull-back to ${{X\times\dP^1}}$, then every $\su$-invariant\ subsheaf of ${p^\ast} E$ is isomorphic to a sheaf of the form ${p^\ast} E'$ for $ E'$ a subsheaf of $ E$. Indeed, the action of ${SU(2)}$ on ${p^\ast} E$ can be extended to an action of $SL(2,{\Bbb C})$. Let $F'\subset {p^\ast} E$ be a $SL(2,{\Bbb C})$-invariant coherent subsheaf. Consider the action of a subgroup ${\Bbb C}^\ast\subset SL(2,{\Bbb C})$ on $X\times{\Bbb C}\subset{{X\times\dP^1}}$ and let $A=H^0(X,E)$\ be the space of global sections. Clearly $H^0(X\times {\Bbb C},F')\subset A[t]$, that is, $H^0(X\times{\Bbb C}, F')=\bigoplus_{k=0}^N B_k$, where an element of $B_k$ is of the form $st^k$ for $s\in A$. The action of $\alpha\in {\Bbb C}^\ast$ is given by $$ \alpha(st^k)=s\alpha^kt^k. $$ By choosing another subgroup ${\Bbb C}^\ast\subset SL(2,{\Bbb C})$, the $SL(2,{\Bbb C})$-invariance of $F'$ implies that $H^0(X\times{\Bbb C}, F')=B_0$ and hence $F'={p^\ast} E'$ for $E'\subset E$ a coherent subsheaf. \hfill$\Box$ We shall show in the next lemma that the triple associated to $F'\subset F$ is in fact a subtriple of ${(E_1,E_2,\Phi)}$, and conversely, every subtriple of ${(E_1,E_2,\Phi)}$ defines a unique $\su$-invariant\ coherent subsheaf of $F$. \begin{lemma}\label{push-out} Let $ E_1'\subset E_1$ and $ E_2'\subset E_2$ be coherent subsheaves and let $\Phi'\in \mathop{{\fam0 Hom}}\nolimits( E_2', E_1')$. Let $ F'$ be the coherent sheaf over ${{X\times\dP^1}}$ defined by the triple ${(E_1',E_2',\Phi')}$. Then $ F'$ is a subsheaf of $F$ making the diagram $$ \begin{array}{ccccccccc} 0& \longrightarrow & {p^\ast} E_1 &\longrightarrow & F & \longrightarrow & {p^\ast} E_2\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0\\ & &\uparrow & & \uparrow & & \uparrow & & \\ 0& \longrightarrow & {p^\ast} E_1' &\longrightarrow & F' & \longrightarrow & {p^\ast} E_2'\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0, \end{array} $$ commutative if and only if ${(E_1',E_2',\Phi')}$ is a subtriple of ${(E_1,E_2,\Phi)}$. \end{lemma} {\em Proof}. Consider the diagram $$ \mathop{{\fam0 Hom}}\nolimits( E_2', E_1') \stackrel{i}{\longrightarrow}\mathop{{\fam0 Hom}}\nolimits( E_2', E_1)\stackrel{j}{\longleftarrow} \mathop{{\fam0 Hom}}\nolimits( E_2, E_1). $$ To say that ${(E_1',E_2',\Phi')}$ is a subtriple of ${(E_1,E_2,\Phi)}$ is equivalent to saying that $$ i(\Phi')=j(\Phi). $$ Under the isomorphisms $$ \begin{array}{lll} \mathop{{\fam0 Hom}}\nolimits( E_2', E_1')&\cong &\mathop{{\fam0 Ext}}\nolimits^1({p^\ast} E_1',{p^\ast} E_2'\otimes{q^\ast}{\cO(2)})\\ \mathop{{\fam0 Hom}}\nolimits( E_2', E_1)&\cong &\mathop{{\fam0 Ext}}\nolimits^1({p^\ast} E_1,{p^\ast} E_2'\otimes{q^\ast}{\cO(2)})\\ \mathop{{\fam0 Hom}}\nolimits( E_2, E_1)&\cong &\mathop{{\fam0 Ext}}\nolimits^1({p^\ast} E_1,{p^\ast} E_2\otimes{q^\ast}{\cO(2)}), \end{array} $$ $i(\Phi')$ defines an extension $\tilde F^{(i)}$ which makes the following diagram commutative \begin{equation} \begin{array}{ccccccccc} 0& \longrightarrow & {p^\ast} E_1 &\longrightarrow &\tilde F^{(i)} & \longrightarrow & {p^\ast} E_2'\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0\\ & &\uparrow & & \uparrow & & \parallel & & \\ 0& \longrightarrow & {p^\ast} E_1' &\longrightarrow & F' & \longrightarrow & {p^\ast} E_2'\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0. \end{array} \label{push-out1} \end{equation} In particular $ F'$ is a subsheaf of $\tilde F^{(i)}$. On the other hand $j(\Phi)$ defines an extension $\tilde F^{(j)}$ which fits in the following commutative diagram \begin{equation} \begin{array}{ccccccccc} 0& \longrightarrow & {p^\ast} E_1 &\longrightarrow & F & \longrightarrow & {p^\ast} E_2\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0\\ & &\parallel & & \uparrow & & \uparrow & & \\ 0& \longrightarrow & {p^\ast} E_1 &\longrightarrow & \tilde F^{(j)} & \longrightarrow & {p^\ast} E_2'\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0, \end{array} \label{push-out2} \end{equation} and in particular $\tilde F^{(j)}$ is a subsheaf of $ F$. Since $i(\Phi')=j(\Phi)$, $\tilde F^{(i)}\cong \tilde F^{(j)}$ and we can compose the above two diagrams to obtain the desired result. \hfill$\Box$ \subsection{Simple Triples \begin{definition}\label{def-simple}Let \begin{equation} H^0(E_1,E_2,\Phi)=\bigl\{(u,v)\in H^0(\mathop{{\fam0 End}}\nolimits E_1)\oplus H^0(\mathop{{\fam0 End}}\nolimits E_2)\bigm|u\Phi=\Phi v\bigr\}.\label{} \end{equation} We say a holomorphic triple $(E_1,E_2,\Phi)$\ is simple if $H^0(E_1,E_2,\Phi)\simeq{\bf} C$, i.e. if the only elements in $H^0(E_1,E_2,\Phi)$\ are of the form $\lambda({\bf} I_1,{\bf} I_2)$\ where $\lambda$\ is a constant and $({\bf} I_1,{\bf} I_2)$\ denote the identity maps on $E_1$\ and $E_2$. \end{definition} This definition too is dictated by the correspondence between triples on $X$\ and equivariant holomorphic extensions over $X\times\Bbb P^1$: \begin{prop} The triple ${(E_1,E_2,\Phi)}$ is simple if and only if the $\su$-equivariant\ bundle $F$ associated to ${(E_1,E_2,\Phi)}$ is ${SU(2)}$-equivariantly simple. \end{prop} {\em Proof}. The definition of simplicity for an equivariant bundle is the obvious generalization of that for an ordinary holomorphic bundle. Namely, an equivariant bundle is said to be equivariantly simple if it has no other invariant endomorphisms than the constant multiples of the identity. The proof of the Proposition follows from the following lemma. \begin{lemma}\label{equi-hom} Let $T={(E_1,E_2,\Phi)}$ and $T'={(E_1',E_2',\Phi')}$ be two holomorphic triples over $X$ and $F$ and $F'$ be the corresponding $\su$-equivariant\ holomorphic vector bundles over ${{X\times\dP^1}}$. Then, every $\su$-equivariant\ homomorphism $g: F\longrightarrow F'$ induces homomorphisms $u: E_1\longrightarrow E_1'$ and $v:E_2\longrightarrow E_2'$ such that \begin{equation} u\Phi=\Phi' v. \label{commu} \end{equation} Conversely, given morphisms $u$ and $v$ satisfying (\ref{commu}) there exists a unique morphism $g: F\longrightarrow F'$ inducing $u$ and $v$. \end{lemma} {\em Proof}. The map $g$ can be decomposed as $$ g=\left( \begin{array}{cc} g_1 & f_1\\ f_2 & g_2 \end{array} \right), $$ where $g_1:{p^\ast} E_1\longrightarrow{p^\ast} E_1'$, $g_2:{p^\ast} E_2\otimes{q^\ast}{\cO(2)}\longrightarrow {p^\ast} E_2'\otimes{q^\ast}{\cO(2)}$, $f_1:{p^\ast} E_2\otimes{q^\ast}{\cO(2)}\longrightarrow{p^\ast} E_1'$ and $f_2:{p^\ast} E_1\longrightarrow {p^\ast} E_2'\otimes{q^\ast}{\cO(2)}$. By invariance it is very easy to see (cf. \cite[Proposition 3.9]{GP3}) that $g_1={p^\ast} u$,\ $g_2={p^\ast} v$, and $f_1=0=f_2$. Equation (\ref{commu}) follows from the commutativity of the following diagram $$ \begin{array}{ccccccccc} 0& \longrightarrow & \;{p^\ast} E_1 &\longrightarrow & \;F & \longrightarrow & {p^\ast} E_2\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0\\ & &{p^\ast} u{\downarrow} & & g{\downarrow} & & {p^\ast} v{\downarrow} & & \\ 0& \longrightarrow & {p^\ast} E'_1 &\longrightarrow & F' & \longrightarrow & {p^\ast} E_2'\otimes{q^\ast}{\cal O}(2) &\longrightarrow & 0. \end{array} $$ \hfill$\Box$ From this lemma it follows that every $\su$-invariant\ endomorphism of $F$ induces endomorphisms $u: E_1\longrightarrow E_1$ and $v: E_2\longrightarrow E_2$, satisfying $u\Phi=\Phi v$. And conversely, given endomorphisms $u$ and $v$ satisfying $u\Phi=\Phi v$ there exists a unique endomorphism of $F$ inducing $u$ and $v$. \hfill$\Box$ The definition of simplicity for a triple given above is also motivated by a deformation theory description of the ``tangent space'' to the space of triples. We will say more about this in Section \ref{moduli}. \subsection{Reducible Triples A related, but inequivalent, notion to simplicity is that of irreducibility. We make the following definitions. \begin{definition} We say the triple $T={(E_1,E_2,\Phi)}$\ is reducible if there are direct sum decompositions $E_1 = \bigoplus_{i=1}^n E_{1i}$, $E_2 = \bigoplus_{i=1}^n E_{2 i}$, and $\Phi =\bigoplus_{i=1}^n \Phi_i$, such that $\Phi_i\in \mathop{{\fam0 Hom}}\nolimits(E_{2i},E_{1i})$. We adopt the convention that if $E_{2i}=0$\ or $E_{1i}=0$\ for some i, then $\Phi_i$\ is the zero map. With $T_i=(E_{1i},E_{2i},\Phi_i)$, we write $T=\bigoplus_{i=1}^n T_i$. Thus $T$\ is reducible if it has a decomposition as a direct sum of subtriples. If $T$ is not reducible, we say $T$ is irreducible. \end{definition} \begin{prop} If a triple $T={(E_1,E_2,\Phi)}$\ is simple, then it is irreducible. \end{prop} {\em Proof}. Suppose $T$\ is reducible, with $T=\bigoplus_{i=1}^n T_i$. Then we can define $(u,v)\in H^0(E_1,E_2,\Phi)$\ by $u=\bigoplus_{i=1}^n \lambda_i {\bf} I_{1i}$, $v=\bigoplus_{i=1}^n \lambda_i {\bf} I_{2i}$, where for each $i$, $\lambda_i\in {\bf} C$\ and ${\bf} I_{1i}({\bf} I_{2i})$\ is the identity map on $E_{1i}(E_{2i})$. Clearly $T$\ is not simple. \hfill$\Box$ \begin{prop} A holomorphic triple $T={(E_1,E_2,\Phi)}$\ over $X$\ is irreducible if and only if the corresponding SU(2)-equivariant extension $F\longrightarrowX\times\Bbb P^1$\ is equivariantly irreducible, i.e. cannot be decomposed as a sum of SU(2)-equivariant extensions of the form (\ref{s-ext}). \end{prop} {\em Proof}. This follows directly from the relation between subtriples of $T$\ and \linebreak $SU(2)$-equivariant subbundles of $F$ (cf. Lemmas \ref{equi-sub} and \ref{push-out}). \hfill$\Box$ \subsection{Equations for special Metrics Given a holomorphic vector bundle\ over a compact Riemann surface\ there is a natural condition for a Hermitian metric\ on it: that of being projectively flat. By choosing a metric on $X$ one can rewrite this condition in a way which turns out to be the right generalization for higher dimensional manifolds: the {\em Hermitian--Einstein} condition. Since we shall use this notion on $X$ as well as on ${{X\times\dP^1}}$, we shall define it on a compact K\"{a}hler\ manifold of arbitrary dimension $(M,\omega)$. Let $E$ be a holomorphic vector bundle\ over $M$ and $h$ be a Hermitian metric\ on $E$. Recall that there is on $E$ a unique connection compatible with both the metric and the holomorphic structure---the so-called {\em metric connection}. Let $F_h$ be its curvature and $\Lambda F_h$ be the contraction of $F_h$ with the K\"{a}hler\ form $\omega$. $\Lambda F_h$ is hence a smooth section of $\mathop{{\fam0 End}}\nolimits E$. The metric $h$ is said to be Hermitian--Einstein\ with respect to $\omega$ if \begin{equation} \sqrt{-1}\Lambda F_h=\lambda {\bf} I_E, \label{he} \end{equation} where ${\bf} I_E\in \Omega^0(\mathop{{\fam0 End}}\nolimits E)$ is the identity and $\lambda$ is a constant which is determined by integrating the trace of (\ref{he}). Using that the degree of $E$, defined as $$ \deg E=\frac{1}{(m-1)!}\int_M c_1(E)\wedge\omega^{m-1}, $$ where $m$ is the dimension of $M$ and $c_1(E)$ is the first Chern class of $E$, given via Chern--Weil theory by $$ \deg E=\frac{i}{2\pi}\int_M\mathop{{\fam0 Tr}}\nolimits(\Lambda F_h) \frac{\omega^m}{m!}, $$ we obtain $$ \lambda=\frac{2\pi}{\mathop{{\fam0 Vol}}\nolimits M}\frac{\deg E}{\mathop{{\fam0 rank}}\nolimits E}. $$ Coming back to our compact Riemann surface\ $X$, let us choose a metric on $X$ with K\"{a}hler\ form $\omega_X$ and volume normalized to one. Given a holomorphic triple\ ${(E_1,E_2,\Phi)}$ on $X$ it was shown in \cite{GP3} that there are natural equations for metrics on the bundles $E_1$ and $E_2$. These equations, formally similar to the Hermitian--Einstein\ equations, involve in a natural way the endomorphism $\Phi$. If $E_1$ and $E_2$ are endowed with Hermitian metrics one can form smooth sections of \ $\mathop{{\fam0 End}}\nolimits E_1$\ and \ $\mathop{{\fam0 End}}\nolimits E_2$ \ respectively by taking the compositions $\Phi\Phi^\ast$\ and $\Phi^\ast\Phi$. Here $\Phi^\ast$\ is the adjoint of $\Phi$\ with respect to the metrics of $E_1$ and $E_2$. The equations for the metrics $h_1$ and $h_2$ on $E_1$ and $E_2$, respectively, are given by \begin{equation} \left. \begin{array}{l} \sqrt{-1} \Lambda F_{h_1}+\Phi\Phi^\ast=2\pi\tau {\bf} I_{E_1}\\ \sqrt{-1} \Lambda F_{h_2}-\Phi^\ast\Phi=2\pi\tau'{\bf} I_{E_2} \end{array}\right \}, \label{cves} \end{equation} where $\tau$ and $\tau'$ are real parameters. We first observe that, in order to solve (\ref{cves}), the parameters $\tau$ and $\tau'$ must be related. Indeed, by adding the trace of the two equations in (\ref{cves}), and since $\mathop{{\fam0 Tr}}\nolimits(\Phi\Phi^\ast)= \mathop{{\fam0 Tr}}\nolimits(\Phi^\ast\Phi)$, we get $$ \sqrt{-1}\mathop{{\fam0 Tr}}\nolimits(\Lambda F_{h_1})+\sqrt{-1} \mathop{{\fam0 Tr}}\nolimits(\Lambda F_{h_2})=2\pi r_1\tau +2\pi r_2\tau', $$ where $r_1$ and $r_2$ are the ranks of $E_1$ and $E_2$ respectively. By integrating this equation and, since $$ \deg E_1=\frac{\sqrt{-1}}{2\pi}\int_X \mathop{{\fam0 Tr}}\nolimits(\Lambda F_{h_1})\omega \;\;\;\mbox{and}\;\;\; \deg E_2=\frac{\sqrt{-1}}{2\pi}\int_X \mathop{{\fam0 Tr}}\nolimits(\Lambda F_{h_2})\omega, $$ we obtain \begin{equation} r_1\tau+r_2\tau '=\deg E_1+\deg E_2. \label{t-t'} \end{equation} There is therefore just one independent parameter, that we choose to be $\tau$. These equations are called the {\em coupled $\tau$-vortex equations} by analogy with the vortex equations on a single bundle studied in \cite{B1,B2,GP1,GP2}. \subsection{Dimensional Reduction of Equations The coupled vortex equations have similar interpretations to the Hermitian--Einstein\ equation and the vortex equations on a single bundle. They can be interpreted both as the equations satisfied by the minima of a certain gauge-theoretical functional---a generalized Yang--Mills--Higgs-type functional---as well as moment map equations in the sense of symplectic geometry (see \cite[Section 2]{GP3} for details). In fact, the relation between the coupled vortex equations\ and the Hermitian--Einstein\ equation that we shall exploit here is of a more intimate nature. Namely, the coupled vortex equations\ are a dimensional reduction of the Hermitian--Einstein\ equation under the action of ${SU(2)}$ on ${{X\times\dP^1}}$. Of course, in order to talk about the Hermitian--Einstein\ equation on ${{X\times\dP^1}}$ one needs to choose a K\"{a}hler\ metric. We shall consider the one-parameter family of $\su$-invariant\ K\"{a}hler\ metrics with K\"{a}hler\ form $$ {\omega_\sig}=\frac{\sigma}{2}{p^\ast}\omega_X\oplus\omega_{\dP^1}, $$ where $\omega_{\dP^1}$ is the Fubini-Study K\"{a}hler\ form normalized to volume one, and $\sigma\in{\Bbb R}^+$. \begin{prop}\label{dr}Let $T={(E_1,E_2,\Phi)}$ be a holomorphic triple and $F$ be the $\su$-equivariant\ holomorphic bundle over ${{X\times\dP^1}}$ associated to $T$, that is given as an extension \begin{equation} {0\lra \ps E_1\lra F\lra \ps E_2\otimes\qs\cod\lra 0}.\label{extension-a} \end{equation} Suppose that $\tau$ and $\tau'$ are related by (\ref{t-t'}) and let \begin{equation} \sigma=\frac{(r_1+r_2)\tau-(\deg E_1 +\deg E_2)}{r_2}.\label{s-t} \end{equation} Then $E_1$ and $E_2$ admit metrics satisfying the coupled $\tau$-vortex equations\ if and only if $F$ admits an $\su$-invariant\ Hermitian--Einstein\ metric with respect to ${\omega_\sig}$. \end{prop} {\em Proof}. We shall give here just a sketch of the proof (see \cite[Proposition 3.11]{GP3} for details). First one has the following result, which is a special case of the general characterization of an $\su$-invariant\ Hermitian metric on an $\su$-equivariant\ vector bundle over ${{X\times\dP^1}}$. \begin{lemma} Let $h$ be an $\su$-invariant\ Hermitian metric\ on the bundle $F\longrightarrow{{X\times\dP^1}}$ associated to the triple ${(E_1,E_2,\Phi)}$. Then $h$ is of the form \begin{equation} h={p^\ast} h_1\oplus{p^\ast} h_2\otimes{q^\ast} h_2',\label{inv-met} \end{equation} where $h_1$ and $h_2$ are metrics on $E_1$ and $E_2$, respectively, and $h_2'$ is an $\su$-invariant\ metric on ${\cO(2)}$. Conversely, given metrics $h_1$, $h_2$ and $h_2'$ as above, (\ref{inv-met}) defines an $\su$-invariant\ metric on $F$. \end{lemma} Let $F_1$ and $F_2$ be the curvatures of the metric connections of ${p^\ast} h_1$ and ${p^\ast} h_2\otimes {q^\ast} h_2'$ respectively. Then $$ \begin{array}{l} F_1={p^\ast} F_{h_1}\\ F_2={p^\ast} F_{h_2}\otimes 1 + {\bf} I_{E_2}\otimes{q^\ast} F_{h_2'}. \end{array} $$ The curvature of the metric connection corresponding to $h$ is given by $$ F_h= \left( \begin{array}{cc} F_1 - \beta \wedge \beta ^\ast & D' \beta \\ -D''\beta^\ast & F_2 -\beta ^\ast \wedge \beta \end{array} \right), $$ where $\beta\in\Omega^{0,1}({{X\times\dP^1}},{p^\ast}(E_1\otimes E_2^\ast)\otimes{q^\ast}{\cal O}(-2))$ is a representative of the extension class in $H^1({{X\times\dP^1}},{p^\ast}(E_1\otimes E_2^\ast)\otimes{q^\ast}{\cal O}(-2))$ defining (\ref{extension-a}), and $$ D:\Omega^1({{X\times\dP^1}},{p^\ast}(E_1\otimes E_2^\ast)\otimes{q^\ast}{\cal O}(-2))\longrightarrow \Omega^2({{X\times\dP^1}},{p^\ast}(E_1\otimes E_2^\ast)\otimes{q^\ast}{\cal O}(-2)) $$ is built from the metric connections of ${p^\ast} h_1$ and ${p^\ast} h_2\otimes {q^\ast} h_2'$. As explained in Proposition \ref{equi-hvb}, $\beta={p^\ast}\Phi\otimes{q^\ast}\alpha$, where $\alpha\in\Omega^{0,1}(\dP^1,{\cal O}(-2))$ is the unique $\su$-invariant\ representative of the element in $H^1(\dP^1,{\cal O}(-2))$, which has to be fixed in order to associate the extension (\ref{extension-a}) to ${(E_1,E_2,\Phi)}$. One can choose the constant in $H^1(\dP^1,{\cal O}(-2))\cong{\Bbb C}$ such that $\alpha\wedge\alpha^\ast=\frac{1}{\sigma}\omega_{\dP^1}$. Let $\Lambda_\sigma$ be the contraction with the K\"{a}hler\ form ${\omega_\sig}$. A straightforward computation shows that if $\sigma$ is related to $\tau$ by (\ref{s-t}), then $h$ is Hermitian--Einstein\ with respect to ${\omega_\sig}$. That is $$ \sqrt{-1}\Lambda_\sigma F_h=\lambda{\bf} I_F, $$ if and only if $h_1$ and $h_2$ satisfy the coupled $\tau$-vortex equations. We have assumed that if the relation between $\sigma$ and $\tau$ is given by (\ref{s-t}), then $\sigma>0$. However, we will show in Section \ref{stability} that this can actually be derived from the coupled vortex equations. \hfill$\Box$ \noindent{\em Remark}.\ The choice of the K\"{a}hler\ metric on ${{X\times\dP^1}}$ that we have made differs from the one made in \cite{GP3}. There the parameter $\sigma$ is multiplying the metric on $\dP^1$, i.e. ${\omega_\sig}={p^\ast}\omega_X\oplus\sigma{q^\ast}\omega_{\dP^1}$. This, and the fact that the volume of $X$ was not normalized to one, explains why the relation between $\tau$ and $\sigma$ given there is the inverse of (\ref{s-t}). \subsection{Invariant Stability and the Hitchin-Kobayashi Correspondence It is very well-known that the existence of a Hermitian--Einstein\ metric on a holomorphic vector bundle\ is governed by the algebraic-geometric condition of {\em stability}. Recall that a holomorphic vector bundle\ $E$ over a compact K\"{a}hler\ manifold $(M,\omega)$ is said to be {\em stable} if $$ \mu(E')<\mu(E) $$ for every non-trivial coherent subsheaf $E'\subset E$. Where $$ \mu(E')=\frac{\deg E'}{\mathop{{\fam0 rank}}\nolimits E'} $$ is the {\em slope} of $E'$. The precise relation between the Hermitian--Einstein\ condition and stability is given by the so-called {\em Hitchin--Kobayashi correspondence}, proved by Donaldson \cite{D1,D2} in the algebraic case and by Uhlenbeck and Yau \cite{U-Y} for an arbitrary compact K\"{a}hler\ manifold (see also \cite{Ko,L,A-B,N-S}): \begin{thm}\label{h-k} Let $E$ be a holomorphic vector bundle\ over a compact K\"{a}hler\ manifold $(M,\omega)$. Then $E$ admits a Hermitian--Einstein\ metric if and only if $E$ is polystable, that is a direct sum of stable bundles of the same slope. \end{thm} From this theorem and Proposition \ref{dr} we conclude that the existence of solutions to the coupled vortex equations\ must be dictated by the stability of the bundle $F\longrightarrow{{X\times\dP^1}}$ associated to the triple ${(E_1,E_2,\Phi)}$. In fact, since the Hermitian--Einstein\ metric on $F$ is $\su$-invariant, the condition that $F$ has to satisfy is a slightly weaker condition than stability, namely that of {\em invariant stability}. Let $(M,\omega)$ be a compact K\"{a}hler\ manifold and $G$ be a compact Lie group acting on $M$ by isometric biholomorphisms. Let $E$ be a $G$-equivariant holomorphic vector bundle\ over $M$. We say that $E$ is $G$-{\em invariantly stable} if $$ \mu(E')<\mu(E) $$ for every $G$-invariant non-trivial coherent subsheaf $E'\subset E$. The basic relation between $G$-invariant stability and ordinary stability is given by the following theorem (cf. \cite[Theorem 4]{GP2}). \begin{thm} \label{isvs} Let $E$ be a $G$-invariant holomorphic vector bundle as above. Then $E$ is $G$-invariantly stable if and only if $E$ is $G$-indecomposable and is of the form $$ E=\bigoplus _{i=1}^n E_i $$ \noindent where $E_i$ is a stable bundle, which is the transformed of $E_1$ by an element of $G$. \end{thm} As a corollary of Theorems \ref{h-k} and \ref{isvs} one obtains a $G$-invariant version of the Hitchin-Kobayashi correspondence (cf. \cite[Theorems 4 and 5]{GP2}): \begin{thm}\label{inv-h-k} Let $E$ be a $G$-equivariant holomorphic vector bundle\ over a compact K\"{a}hler\ manifold $(M,\omega)$. Then $E$ admits a $G$-invariant Hermitian--Einstein\ metric if and only if $E$ is $G$-invariantly polystable, that is a direct sum of $G$-invariantly stable bundles of the same slope. \end{thm} From Proposition \ref{dr} and Theorem \ref{inv-h-k} we obtain the following existence theorem. \begin{thm}\label{exst} Let $T={(E_1,E_2,\Phi)}$ be a holomorphic triple\ over a compact Riemann surface\ $X$ equipped with a metric. Let $F\longrightarrow{{X\times\dP^1}}$ be the bundle associated to $T$ as above. Let $\sigma$ and $\tau$\ be real parameters related by (\ref{s-t}). Then $E_1$ and $E_2$ admit metrics satisfying the coupled $\tau$-vortex equations\ if and only if $F$ is a ${SU(2)}$-invariantly polystable bundle with respect to the K\"{a}hler\ form ${\omega_\sig}$ defined above. \end{thm} \section{Definition and Properties of Stability for Triples}\label{stability} The existence theorem \ref{exst} gives conditions on the extension $F\longrightarrow{{X\times\dP^1}}$\ for existence of solutions to the coupled vortex equations on ${(E_1,E_2,\Phi)}$. We would like to express these conditions entirely in terms of the data on ${(E_1,E_2,\Phi)}$. Indeed this is one of our primary objectives in this paper. To achieve this, we will need an appropriate notion of stability for a triple. In this section we define such a concept for holomorphic triples, and discuss some properties that follow from the definition. Keeping our earlier notation, we let $E_1$\ and $E_2$\ be holomorphic vector bundles over a Riemann surface $X$. We denote their ranks by $r_1$\ and $r_2$\ respectively, and their degrees by $d_1$\ and $d_2$. We let $\Phi:E_2\longrightarrow E_1$\ be a holomorphic bundle homomorphism, i.e. $\Phi\in H^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$. Our definition of stability for the triple $T={(E_1,E_2,\Phi)}$\ has two equivalent formulations. The first has some advantages when considering the relation between stability and the coupled vortex equations, while the second has the virtue that it is in the style of the definition of parabolic stability, and thus looks more familiar. Both definitions involve a real parameter, with the result that there is a 1-parameter family of stability criteria for triples. This is the same phenomenon as is observed in the case of holomorphic pairs. All our results can be of course be stated in terms of either definition, and for the sake of completeness we will give both versions. \begin{definition}\label{t-stab} Let \ $T'=(E'_1,E'_2,\Phi)$\ be a nontrivial subtriple of \ $(E_1,E_2,\Phi)$, with \ $\mathop{{\fam0 rank}}\nolimits E'_1=r_1'$\ and \ $\mathop{{\fam0 rank}}\nolimits E'_2=r_2'$. For any real $\tau$\ define \begin{equation} \theta_{\tau}(T')=(\mu(E'_1\oplus E'_2)-\tau)-\frac{r_2'}{r_2}\frac{r_1+r_2}{r_1'+r_2'} (\mu(E_1\oplus E_2)-\tau).\label{Theta} \end{equation} The triple $T=(E_1,E_2,\Phi)$ is called $\tau$-stable if $$\theta_{\tau}(T')<0$$ for all nontrivial subtriples $T'=(E'_1,E'_2,\Phi)$. The triple is called $\tau$-semistable if for all subtriples $$\theta_{\tau}(T')\le0.$$ \end{definition} \begin{definition}\label{s-stab} With $\sigma$\ a real number, define the $\sigma$-degree and $\sigma$-slope of a subtriple $T'=(E'_1,E'_2,\Phi)$\ by $$\deg_{\sigma}(T')=\deg(E_1'\oplus E_2')+r_2'\sigma ,$$ and $$\mu_{\sigma}(T')=\frac{\deg_{\sigma}(T')}{r_1'+r_2'}\ .$$ The triple $T=(E_1,E_2,\Phi)$\ is called $\sigma$-stable if for all nontrivial subtriples $T'=(E'_1,E'_2,\Phi)$\ we have $$\mu_{\sigma}(T') < \mu_{\sigma}(T)\ .$$ \end{definition} A straightforward computation shows the equivalence of these two definitions. \begin{prop}\label{tstab-sstab} Fix $\tau$\ and $\sigma$\ such that $$\sigma = \frac{r_1+r_2}{r_2}(\tau-\mu(T))\ ,$$ or equivalently $$\tau = \mu_{\sigma}(T)\ .$$ Then for any subtriple $T'=(E'_1,E'_2,\Phi)$, the following are equivalent: \begin{tabbing} \ \ \ \=(1)\ \= \kill \> {\em (1)}\> $\theta_{\tau}(T')<0\ $,\\ \> {\em (2)} $\mu_{\sigma}(T') < \mu_{\sigma}(T)$. \end{tabbing} That is, the triple is $\tau$-stable if and only if it is $\sigma$-stable. A similar result holds with ``~ $<$\ ~" replaced by ``~ $=$\ ~". \end{prop} \noindent {\em Remark.} There are two special cases where the notion of stability for a triple is especially simple, namely when $\Phi=0$, and when $E_2$\ is a line bundle. \begin{lemma}Suppose that $\Phi=0$. The degenerate holomorphic triple $(E_1,E_2,0)$\ is $\tau$-semistable if and only if $\tau=\mu(E_1)$\ and both bundles are semistable. Such triple cannot be $\tau$-stable. \end{lemma} {\em Proof}. Subtriples of $T=(E_1,E_2,0)$\ are all of the form $T'=(E'_1,E'_2,0)$, with $E'_1$\ and $E'_2$\ being any holomorphic subbundles of $E_1$\ and $E_2$\ respectively. Applying the condition $\theta_{\tau}(T')\le 0$\ to subtriples of the form $T'=(E'_1,0,0)$\ gives \begin{equation} \mu(E'_1)\le \tau,\label{3.2} \end{equation} while applying the condition to subtriples of the form $T'=(E'_1,E_2,0)$\ gives stability \begin{equation} \mu(E_1/E'_1)\ge \tau.\label{3.3} \end{equation} These two inequalities imply \begin{equation} \mu(E_1)\le\tau\le\mu(E_1).\label{3.4} \end{equation} That is, $\tau=\mu(E_1)$, and hence $E_1$\ is a semistable bundle. Similarly, by considering the subtriples $(0,E'_2,0)$\ and $(E_1,E'_2,0)$, we see that $E_2$\ is also semistable. Notice that the inequalities in (\ref{3.2}) and (\ref{3.3}) cannot be made strict without leading to a contradiction in (\ref{3.4}). \hfill$\Box$ \begin{cor}The map $\Phi$\ cannot be identically zero in a $\tau$-stable triple. \end{cor} \begin{lemma}In the case where $E_2=L$\ is a line bundle, i.e. $r_2=1$, the above definition is equivalent to the notion of $\tau$-stability defined in \cite{GP3}. It thus corresponds to the $(\tau-\deg L$)-stability for the holomorphic pair $(E_1\otimes L^*, \Phi)$. \end{lemma} {\em Proof}. In this case there are only two types of subtriple possible, corresponding to $r_2'=0$\ or $r_2'=1$. In the first case the subtriples are of the form $(E'_1,0,0)$, where $E'_1$\ is an arbitrary holomorphic subbundle of $E_1$. The condition $\theta_{\tau}(T')<0$\ then reduces to $$\mu(E'_1)<\tau.$$ In the second case, the subtriples are of the form $(E'_1,E_2,\Phi)$\ where $E'_1$\ is a holomorphic subbundle such that $\Phi(E_2)\subset E'_1$. For such subtriples the condition $\theta_{\tau}(T')<0$\ is equivalent to $$(r_1'+1)\mu(E'_1\oplus E_2)-(r_1+1)\mu(E_1\oplus E_2)-(r_1'-r_1)\tau <0,$$ i.e. $$\mu(E_1/E'_1)>\tau.$$ \hfill$\Box$ Definition \ref{t-stab} can thus be considered a natural extension of the $\tau$-stability for pairs defined in \cite{B2}. For the more general triples which we are considering here however, the number of different possibilities for subtriples is too large to reformulate the definition of $\tau$-stability in the style of \cite{GP3} or \cite{B2}, i.e. in terms of separate slope conditions on the various families of subtriples. The $\tau$-stability\ of a triple does however imply the following conditions on subtriples: \begin{prop}\label{slopes}Let $(E_1,E_2,\Phi)$\ be a $\tau$-stable triple. Let $\tau'$\ be related to $\tau$ by \begin{equation} r_1\tau+r_2\tau'=\deg E_1+\deg E_2.\label{3.5} \end{equation} Then \begin{tabbing} \ \ \ \=(0)\ \= \kill \> {\em (1)}\> $\mu(E_1')<\tau$\ for all holomorphic subbundles $E_1'\subset E_1$,\\ \> {\em (2)}\> $\mu(E_2')<\tau'$\ for all holomorphic subbundles $E_2'\subset E_2$\ such that $E_2'\subset \mathop{{\fam0 Ker}}\nolimits(\Phi),$\\ \> {\em (3)}\> $\mu(E_2'')>\tau'$\ for all holomorphic quotients of $E_2$,\\ \> {\em (4)}\> $\mu(E_1'')>\tau$\ for all holomorphic quotients of $E_1$\ such that $\pi\circ\Phi(E_2)=0$,\\ \> \> where $\pi:E_1\longrightarrow E_1''$\ denotes projection onto the quotient. \end{tabbing} \end{prop} {\em Proof}. These are immediate consequences of the stability condition, i.e. $$\theta_{\tau}(E'_1,E'_2,\Phi')<0,$$ applied to the following special subtriples \begin{tabbing} \ \ \ \=(0)\ \= \kill \> (1) $(E_1',0,\Phi)$,\\ \> (2) $(0,E_2',\Phi)$,\\ \> (3) $(E_1,E'_2,\Phi)$, with $E_2''=E_2/E'_2$,\\ \> (4) $(E'_1,E_2,\Phi)$, with $E_1''=E_1/E'_1$. \end{tabbing} \hfill$\Box$ Notice that (\ref{3.5}) can be expressed as $$\tau' = \mu(E_1\oplus E_2) - \frac{r_1}{r_1+r_2}\sigma\ .$$ An equivalent formulation of Proposition \ref{slopes} is thus \begin{prop}\label{slopes'}Let $T=(E_1,E_2,\Phi)$\ be a $\sigma$-stable triple. Then \begin{tabbing} \ \ \ \=(0)\ \= \kill \> {\em (1)}\> $\mu(E_1')<\mu(T) + \frac{r_2}{r_1+r_2}\sigma$\ for all holomorphic subbundles $E_1'\subset E_1$,\\ \> {\em (2)}\> $\mu(E_2')<\mu(T) - \frac{r_1}{r_1+r_2}\sigma$\ for all holomorphic subbundles $E_2'\subset E_2$\ such that\\ \> \> $E_2'\subset \mathop{{\fam0 Ker}}\nolimits\Phi,$\\ \> {\em (3)}\> $\mu(E_2'')>\mu(T) - \frac{r_1}{r_1+r_2}\sigma$\ for all holomorphic quotients, $E_2''$, of $E_2$,\\ \> {\em (4)}\> $\mu(E_1'')>\mu(T) + \frac{r_2}{r_1+r_2}\sigma$\ for all holomorphic quotients, $E_1''$, of $E_1$\ such that\\ \> \> $\pi\circ\Phi(E_2)=0$, where $\pi:E_1\longrightarrow E_1''$\ denotes projection onto the quotient. \end{tabbing} \end{prop} \subsection{Stable implies simple} An important consequence of stability for holomorphic bundles is that the only automorphisms of a stable bundle are the constant multiples of the identity, i.e. stable bundles are simple. We now show that this remains true in the case of holomorphic triples, where the definition of simplicity is that given in Definition \ref{def-simple}. The key result is the following Proposition. \begin {prop}\label{s-s} Let $(E_1,E_2,\Phi)$\ be a $\tau$-stable holomorphic triple. Let $(u,v)$\ be in $H^0(E_1,E_2,\Phi)$. Either $(u,v)$\ is trivial, or both $u$\ and $v$\ are isomorphisms. \end{prop} {\em Proof}. Suppose that $u$\ and $v$\ are both neither trivial nor isomorphisms. Consider the triples $K=(\mathop{{\fam0 Ker}}\nolimits u,\mathop{{\fam0 Ker}}\nolimits v,\Phi)$\ and $I=(\mathop{{\fam0 Im}}\nolimits u,\mathop{{\fam0 Im}}\nolimits v,\Phi)$, where $\mathop{{\fam0 Ker}}\nolimits$\ and $\mathop{{\fam0 Im}}\nolimits$\ denotes the kernels and images of the maps. Since $u\Phi=\Phi v$, these are both proper subtriples of ${(E_1,E_2,\Phi)}$, and thus the $\tau$-stability condition gives \begin{equation} \theta_{\tau}(K)<0,\label{3.17} \end{equation} and \begin{equation}\theta_{\tau}(I)<0.\label{3.18} \end{equation} We also have the exact sequences $$ 0\longrightarrow\mathop{{\fam0 Ker}}\nolimits u\longrightarrow E_1\longrightarrow\mathop{{\fam0 Im}}\nolimits u\lra0, $$ and $$ 0\longrightarrow\mathop{{\fam0 Ker}}\nolimits v\longrightarrow E_2\longrightarrow\mathop{{\fam0 Im}}\nolimits v\lra0. $$ Let $\sigma_u$\ and $\rho_u$\ denote the ranks of $\mathop{{\fam0 Ker}}\nolimits u$\ and $\mathop{{\fam0 Im}}\nolimits u$, and similarly for $\sigma_v$\ and $\rho_v$. Then from the exact sequences we get \begin{equation} (\sigma_u+\sigma_v)\mu(\mathop{{\fam0 Ker}}\nolimits u\oplus \mathop{{\fam0 Ker}}\nolimits v) + (\rho_u+\rho_v)\mu(\mathop{{\fam0 Im}}\nolimits u\oplus \mathop{{\fam0 Im}}\nolimits v)= (r_1+r_2)\mu(E_1\oplus E_2).\label{3.19} \end{equation} But by definition of $\theta_{\tau}(K)$, $$ \begin{array}{ll} (\sigma_u+\sigma_v)\mu(\mathop{{\fam0 Ker}}\nolimits u\oplus \mathop{{\fam0 Ker}}\nolimits v)&= (\sigma_u+\sigma_v)\theta_{\tau}(K)\\ &+\frac{\sigma_v}{r_2}(r_1+r_2)\mu(E_1\oplus E_2) +(\sigma_u+\sigma_v-\frac{\sigma_v}{r_2}(r_1+r_2))\tau, \end{array} $$ with a similar expression for $(\rho_u+\rho_v)(\mu(\mathop{{\fam0 Im}}\nolimits u\oplus \mathop{{\fam0 Im}}\nolimits v)$. Also, $\sigma_u+\rho_u=r_1$, and $\sigma_v+\rho_v=r_2$. Hence from (\ref{3.19}) we obtain $$r_1\theta_{\tau}(K)+r_2\theta_{\tau}(I)=0.$$ This is incompatible with (\ref{3.17}) and (\ref{3.18}). \hfill$\Box$ \begin{cor}\label{stab-simple} If $(E_1,E_2,\Phi)$\ is $\tau$-stable, then it is simple. \end{cor} {\em Proof}. Let $(u,v)$\ be a nontrivial element in $H^0(E_1,E_2,\Phi)$. By the above Proposition, both $u$\ and $v$\ are isomorphisms. Fix a point $p$\ on the base of the bundles, and let $\lambda$\ be an eigenvalue of $v:E_2|_p\longrightarrow E_2|_p$, i.e. of $v$\ acting on the fibre over $p$. Now define $$\hat u =u-\lambda {\bf} I_1,$$ $$\hat v =v-\lambda {\bf} I_2.$$ Clearly $(\hat u,\hat v)$\ is in $H^0(E_1,E_2,\Phi)$, but since $\hat u$\ is not an isomorphism, it follows from Proposition \ref{s-s} that both are identically zero, i.e. $$(u,v)=\lambda ({\bf} I_1,{\bf} I_2).$$ \hfill$\Box$ We see, in particular, that stable triples are necessarily irreducible. For reducible triples, we can however define a notion of polystability. This will be useful when we consider the relation between stability and the coupled vortex equations. \begin{definition}Let $T={(E_1,E_2,\Phi)}$\ be a reducible triple, with $T=\bigoplus_{i=1}^n T_i$. Suppose that in each summand $T_i=(E_{1i},E_{2i},\Phi_i)$, the map $\Phi_i$\ is non-trivial unless $E_{1i}=0$\ or $E_{2i}=0$. Fix value of $\tau$, and let $\tau'$\ be related to $\tau$\ as in (\ref{3.5}). We say that $T$\ is $\tau$-polystable if for each summand $T_i$ \begin{tabbing} \ \ \ \= (0)\ \= \kill \> {\em (1)}\> if $\Phi_i\ne 0$, then $T_i$ is $\tau$-stable,\\ \> {\em (2)}\> if $E_{1i}=0$, then $E_{2i}$\ is a stable bundle of slope $\tau'$,\\ \> {\em (3)}\> if $E_{2i}=0$, then $E_{1i}$\ is a stable bundle of slope $\tau$. \end{tabbing} \end{definition} \subsection{Duality for triples} Associated to a triple $T={(E_1,E_2,\Phi)}$ there is always a {\em dual triple} $T^*={(E_2^*,E_1^*,\Phi^*)}$, where $\Phi^*$ is the transpose of $\Phi$, i.e. the image of $\Phi$ via the canonical isomorphism $$ \mathop{{\fam0 Hom}}\nolimits(E_2,E_1)\cong \mathop{{\fam0 Hom}}\nolimits(E_1^*,E_2^*). $$ It is reasonable that the stability of $T$ should be related to that of $T^*$. More precisely. \begin{prop}\label{duality} $T={(E_1,E_2,\Phi)}$ is $\tau$-stable\ if and only if $T^*={(E_2^*,E_1^*,\Phi^*)}$ is $(-\tau')$-stable, where $\tau'$ is related to $\tau$ by (\ref{3.5}). Equivalently, $T$\ is $\sigma$-stable if and only if $T^*$\ is $\sigma$-stable. \end{prop} {\em Proof}. Let $T'={(E_1',E_2',\Phi')}$ be a subtriple of $T$. This defines a quotient triple $T''=(E_1'',E_2'',\Phi'')$, where $E_1''=E_1/E_1'$, $E_2''=E_2/E_2'$, and $\Phi''$ is the morphism induced by $\Phi$. $T''^*=(E_2''^*,E_1''^*,\Phi''^*)$ is the desired subtriple of $T^*$. Since one has the isomorphism $T\cong T^{**}$ we can conclude that there is a one-to-one correspondence between subtriples of $T$ and subtriples of $T^*$. It is not difficult to verify that $\theta_\tau(T')<0$ is equivalent to $\theta_{-\tau'}(T''^*)<0$. The equivalence of the $\sigma$-stability for $T$\ and $T^\ast$\ now follows from the fact that if $\tau=\mu_{\sigma}(T)$, then $$-\tau'=-\mu(E_1\oplus E_2)-\frac{r_1}{r_1+r_2}\sigma = -\mu_{\sigma}(T^*)\ .$$ \hfill$\Box$ \subsection{Constraints on the parameters} \begin{prop}\label{l-bound}Let $(E_1,E_2,\Phi)$\ be a $\tau$-stable triple, and let $\tau'$\ be as above. Then \begin{tabbing} \ \ \ \= (0)\ \= \kill \> {\em (1)}\> $\tau >\mu(E_1)$,\\ \> {\em (2)}\> $\tau' <\mu(E_2)$, and\\ \> {\em (3)}\> $\tau -\tau' >0$, \end{tabbing} Equivalently, if $(E_1,E_2,\Phi)$\ is $\sigma$-stable, then \begin{tabbing} \ \ \ \= (0)\ \=\kill \> {\em (1)}\> $\sigma > \mu(E_1)-\mu(E_2)$,\\ \> {\em (2)}\> $\sigma > 0$. \end{tabbing} \end{prop} {\em Proof}. The first two statement follows from cases (1) and (3) in Proposition \ref{slopes} with $E_1'=E_1$, and $E''_2=E_2$\ respectively. To prove the third statement, let $K$\ be the subbundle of $E_2$\ generated by the kernel of $\Phi$, and let $I$\ be the subbundle of $E_1$\ generated by the image of $\Phi$. Since the triple is assumed to be $\tau$-stable, $\Phi$, and therefore $I$, is non-trivial. By (1) in Proposition \ref{slopes} we thus have \begin{equation} \mu(I)<\tau.\label{3.6} \end{equation} But we also have $0\longrightarrow K\longrightarrow E_2\longrightarrow I\longrightarrow 0$, i.e. $I$\ is a quotient of $E_2$. It thus follows from (3) in Proposition \ref{slopes} that \begin{equation} \mu(I)>\tau'.\label{3.7} \end{equation} The bounds on $\sigma$\ can be obtained from those on $\tau$\ by substituting $\tau =\mu_{\sigma}(T)$, and using the fact that $$\sigma =\tau-\tau'$$ if $\tau'$\ is as above. \hfill$\Box$ Part (1) of this proposition gives the lower bound on the allowed range for $\tau$. In almost all cases the rank and degree of $E_1$\ and $E_2$\ also impose an upper bound on $\tau$. In fact \begin{prop}\label{u-bound}Let ${(E_1,E_2,\Phi)}$\ be a triple with $r_1\ne r_2$. If the triple is $\tau$-stable\, then \begin{equation} \tau<\mu(E_1)+\frac{r_2}{|r_1-r_2|}(\mu(E_1)-\mu(E_2))\label{3.8a} \end{equation} Equivalently, if the triple is $\sigma$-stable, then \begin{equation} \sigma < (1+\frac {r_1+r_2}{|r_1-r_2|})(\mu(E_1)-\mu(E_2)).\label{3.8b} \end{equation} \end{prop} {\em Proof}. Let $K=\mathop{{\fam0 Ker}}\nolimits \Phi$ and $I=\mathop{{\fam0 Im}}\nolimits \Phi$. Consider the subtriples $$ T_1=(0, K,\Phi)\;\;\;\;\text{and}\;\;\;\;T_2=(I,E_2,\Phi). $$ Since $r_1\ne r_2$, $\Phi$\ cannot be an isomorphism and at least one of these must be a proper subtriple. Let $r_2'=\mathop{{\fam0 rank}}\nolimits K$, $r_2''=\mathop{{\fam0 rank}}\nolimits I$, $d_2'=\deg K$ and $d_2''=\deg I$. A straightforward computation shows that \begin{eqnarray} \theta_\tau(T_1)<0\Longleftrightarrow& d_2'-r_2'(d_1+d_2)+r_1r_2'\tau<0 \label{3.9}\\ \theta_\tau(T_2)<0\Longleftrightarrow&d_2''-d_1+(r_1-r_2'')\tau<0.\label{3.10} \end{eqnarray} Adding (\ref{3.9}) to $r_2$ times (\ref{3.10}), and noting that $d_2=d_2'+d_2''$ and $r_2=r_2'+r_2''$, we get that \begin{equation} r_2(d_2-d_1)-r_2'(d_1+d_2)+(r_2(r_1-r_2)+r_2'(r_1+r_2))\tau<0. \label{3.11} \end{equation} On the other hand combining (3) in Proposition \ref{l-bound} and (\ref{3.5}) we obtain \begin{equation} d_1+d_2-(r_1+r_2)\tau<0. \label{3.12} \end{equation} Adding (\ref{3.11}) to $r_2'$ times (\ref{3.12}) we get $$ (r_1-r_2)\tau<d_1-d_2. $$ If now $r_1>r_2$, then we get \begin{equation} \tau<\frac{d_1-d_2}{r_1-r_2}\label{3.13} \end{equation} or equivalently $$ \tau<\mu(E_1)+\frac{r_2}{r_1-r_2}(\mu(E_1)-\mu(E_2)). $$ To obtain the bound in the case $r_1<r_2$, note that by Proposition \ref{duality} the $\tau$-stability of ${(E_1,E_2,\Phi)}$ is equivalent to the $(-\tau')$-stability of the dual triple ${(E_2^*,E_1^*,\Phi^*)}$, where $\tau'$ is given by \begin{equation} r_1\tau+r_2\tau'=d_1+d_2. \label{3.14} \end{equation} Hence we can apply (\ref{3.13}) to ${(E_2^*,E_1^*,\Phi^*)}$ to get that $$ -\tau'<\frac{d_1-d_2}{r_2-r_1}, $$ which together with (\ref{3.14}) leads to $$ \tau<\frac{(2r_2-r_1)d_1-r_1d_2}{(r_2-r_1)r_1}, $$ i.e. $$ \tau<\mu(E_1)+\frac{r_2}{r_2-r_1}(\mu(E_1)-\mu(E_2)). $$ \hfill$\Box$ Combining the lower and upper bounds on $\tau$\ (or $\sigma$) we can deduce \begin{cor} If $\mathop{{\fam0 rank}}\nolimits E_1$ and $\mathop{{\fam0 rank}}\nolimits E_2$\ are unequal, then a triple ${(E_1,E_2,\Phi)}$\ cannot be stable unless $\mu(E_2)<\mu(E_1)$. \end{cor} Furthermore, by the proof of Proposition \ref{u-bound} we get the following corollary. \begin{cor}\label{Corollary 3.8}Let ${(E_1,E_2,\Phi)}$\ be $\tau$-stable\, and suppose that $r=s$. If $\Phi$\ is not an isomorphism, then $d_1>d_2$. In particular, in any $\tau$-stable\ triple ${(E_1,E_2,\Phi)}$, the bundle map $\Phi$\ is an isomorphism if and only if $r_1=r_2$\ and $d_1=d_2$. \end{cor} {\em Proof}. The fact that $d_1>d_2$ follows from the inequality $$ (r_1-r_2)\tau<d_1-d_2\ , $$ which applies if $\Phi$\ is not an isomorphism. In particular, if $\Phi$\ is not an isomorphism then $d_1\ne d_2$. Conversely, if $\Phi$\ is an isomorphism, then clearly $r_1=r_2$\ and $d_1=d_2$. \hfill$\Box$ It is when $\Phi$\ is an isomorphism that the range for $\tau$\ can fail to be bounded. For example \begin{prop}\label{Proposition 3.6}Suppose that $E_1$\ and $E_2$\ are both stable bundles of rank $r$ and degree $d$, and that $\Phi:E_2\longrightarrow E_1$\ is non trivial. Then for any $\tau>\mu(E_1)$\ the holomorphic triple $(E_1,E_2,\Phi)$\ is $\tau$-stable. \end{prop} {\em Proof}. Let $\mu(T)=\mu(E_1\oplus E_2)$, and for a subtriple $T'=(E'_1,E'_2,\Phi)$\ set $\mu(T')=\mu(E'_1\oplus E'_2)$. Since $E_1$\ and $E_2$\ are stable and of equal slope, we have $\mu(T')<\mu(T)$\ for all subtriples. Thus $$\theta_{\tau}(T')\le (\mu(T)-\tau)\frac{r_1'-r_2'}{r_1'+r_2'}.$$ Since $\Phi$\ is a nontrivial map between stable bundles of the same rank and degree, it must be a multiple of the identity (cf. \cite{O-S-S}). In particular $\Phi$\ is injective and hence $r_1'-r_2'\ge 0$. Thus $\theta_{\tau}(T')\le 0$. In fact, $\theta_{\tau}(T')<0$\ unless $r_1'=r_2'$. But in that case, we can write $$\theta_{\tau}(T')=r_1'(\mu(E_1')-\mu(E_1))+r_1'(\mu(E_2')-\mu(E_2)),$$ which is strictly negative. \hfill$\Box$ \subsection{Special values for $\tau$\ and $\tau$-semistability}\label{critical-values} In principle $\tau$\ is a continuously varying real parameter. The stability properties of a given triple do not likewise vary continuously, but can change only at certain rational values of $\tau$. This is the same phenomenon as appears in the case of stable pairs. In both cases it is due to the fact that, except for $\tau$\ itself, all numerical quantities in the definition of stability are rational numbers with bounded denominators. In the case of holomorphic pairs, this has the additional consequence that for the generic choice of $\tau$\ there is no distinction between stability and semistablity. This is in contrast to the case of pure bundles, where the notions of stability and semistability coincide only when the rank and degree of the bundle are coprime. The next proposition shows that for a holomorphic triple both the value of $\tau$\ and the greatest common divisor of the rank and degree, are relevant. \begin{prop}\label{critical}Let \ $T=(E_1,E_2,\Phi)$\ be a $\tau$-semistable triple, and let $T'=(E'_1,E'_2,\Phi')$\ be a subtriple such that $\theta_{\tau}(T')=0.$ Then either \begin{equation} r_1r_2'=r_2r_1'\;\; \text{and}\;\; \mu(E'_1\oplus E'_2)=\mu(E_1\oplus E_2),\label{3.15a} \end{equation} or \begin{equation} \frac{r_2(r_1'+r_2')\mu(T')-r_2'(r_1+r_2)\mu(T)}{r_2r_1'-r_1r_2'}= \tau.\label{3. 15b} \end{equation} In particular, if \ $r_1+r_2$\ and $d_1+d_2$\ are coprime, and $\tau$\ is a not rational number with denominator of magnitude less than $r_1r_2$, then all $\tau$-semistable triples are $\tau$-stable. \end{prop} {\em Proof}. From the definition of $\theta_{\tau}$, we see that $\theta_{\tau}(T')=0$\ is equivalent to $$(\mu(E'_1\oplus E'_2)-\frac{r_2'}{r_2}\frac{r_1+r_2}{r_1'+r_2'}\mu(E_1\oplus E_2))= \tau\frac{r_1'r_2-r_1r_2'}{r_1'r_2+r_2'r_2}.$$ If $r_1'r_2-r_1r_2'\ne 0$\ we get (\ref{3.15b}), and if $r_1'r_2-r_1r_2'=0$\ then $$ \frac{r_2'}{r_2}\frac{r_1+r_2}{r_1'+r_2'}=1 $$ and we get (\ref{3.15a}). \hfill$\Box$ Next we compare the stability conditions for a triple and for the two bundles in the triple. \begin{prop}\label{ts-ss}Let $(E_1,E_2,\Phi)$\ be a non-degenerate holomorphic triple. There is an $\epsilon>0$, which depends only on the degrees and ranks of $E_1$\ and $E_2$, and such that for $\mu(E_1)<\tau<\mu(E_1)+\epsilon$\ the following is true: \begin{tabbing} \ \ \ \= (0)\ \= \kill \> {\em (1)}\> If $(E_1,E_2,\Phi)$\ is a $\tau$-stable triple, then both $E_1$\ and $E_2$\ are semistable bundles.\\ \> {\em (2)}\> Conversely, if $E_1$\ and $E_2$\ are stable bundles, then $(E_1,E_2,\Phi)$\ will be a $\tau$-stable\\ \> \> triple for any choice of $\Phi\in H^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$. \end{tabbing} \end{prop} {\em Proof}. For all subbundles $E_1'\subset E_1$\ the slope $\mu(E_1')$\ is a rational number with denominator less than $r_1$. Clearly, if we pick $\epsilon$\ small enough then the interval $(\mu(E_1),\mu(E_1)+\epsilon)$\ contains no rational numbers with denominator less than $r_1$. The condition $\mu(E_1')<\tau$\ is thus equivalent to the condition $\mu(E_1')\le\mu(E_1)$, i.e. to the semistability of $E_1$. Furthermore, as noted above, if $\tau < \mu(E_1)+\epsilon$\ then $\tau' > \mu(E_2)-\frac{r_1}{r_2} \epsilon$. Hence if $\frac{r_1}{r_2}\epsilon$\ is small enough, then the condition $\mu(E_2/E_2')>\tau'$\ for all subbundles $E_2'\subset E_2$\ becomes equivalent to the condition that $\mu(E_2/E_2')\ge\mu(E_2)$. Conversely, suppose $\tau =\mu(E_1)+\delta$\ for some $\delta >0$, and that $\Phi$\ is any section of $H^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$. Then for any subtriple $(E'_1,E'_2,\Phi)$\ we get $$(r_1'+r_2')\theta_{\tau}{(E_1',E_2',\Phi')}=r_1'(\mu(E'_1)-\mu(E_1))+r_2'(\mu(E'_2)- \mu(E_2))+(r_2r_1'-r_1r_2')\delta,$$ where $r_1'=\mathop{{\fam0 rank}}\nolimits E'_1$\ and $r_2'=\mathop{{\fam0 rank}}\nolimits E'_2$. If $E_1$\ and $E_2$\ are stable, and $\delta$\ is small enough, then it follows from this that $\theta_{\tau}{(E_1',E_2',\Phi')}<0$\ for all subtriples. \hfill$\Box$ \section{Main theorem}\label{theorem} In this section we shall show how the stability of a holomorphic triple over $X$ relates to the stability of the associated (SU(2)-equivariant) bundle over $X\times\Bbb P^1$. As in Section \ref{background}, let $F\longrightarrowX\times\Bbb P^1$\ be the extension associated to the triple ${(E_1,E_2,\Phi)}$, i.e. let $F$ be \begin{equation} 0\longrightarrow{p^\ast} E_1\longrightarrow F\longrightarrow {p^\ast} E_2\otimes{q^\ast}{\cal O}(2)\longrightarrow 0, \label{extension-b} \end{equation} where $p$ and $q$ are the projections from ${{X\times\dP^1}}$ to $X$ and $\dP^1$ respectively, and ${\cO(2)}$ is the line bundle of degree 2 over $\dP^1$. To relate the $\tau$-stability of ${(E_1,E_2,\Phi)}$ to the stability of $ F$ we need to consider some K\"{a}hler\ polarization on ${{X\times\dP^1}}$. The parameter $\tau$ will be encoded in this polarization. Let us choose a metric on $X$ with K\"{a}hler\ form $\omega_X$, with volume normalized to one. The metric we shall consider on ${{X\times\dP^1}}$ will be, as in Section \ref{background}, the product of a the metric on $X$ with a coefficient depending on a parameter $\sigma>0$, and the Fubini--Study metric on $\dP^1$ with volume also normalized to one. The \kahler\ form corresponding to this metric depending on the parameter $\sigma$ is \begin{equation} {\omega_\sig}=\frac{\sigma}{2}{p^\ast}\omega_X\oplus {q^\ast}\omega_{\dP^1}.\label{kah-pol} \end{equation} We can now state the main result of this section. \begin{thm}\label{tsvs} Let ${(E_1,E_2,\Phi)}$ be a holomorphic triple over a compact Riemann surface\ $X$. Let $ F$ be the holomorphic bundle over ${{X\times\dP^1}}$ defined by ${(E_1,E_2,\Phi)}$ as in Proposition \ref{equi-hvb}, and let \begin{equation} \sigma(\tau)=\frac{(r_1+r_2)\tau-(\deg E_1 +\deg E_2)}{r_2}.\label{s-t'} \end{equation} Suppose that in ${(E_1,E_2,\Phi)}$\ the two bundles $E_1$\ and $E_2$\ are not isomorphic. Then ${(E_1,E_2,\Phi)}$ is $\tau$-stable\ (equivalently $\sigma$-stable) if and only if $ F$ is stable with respect to ${\omega_\sig}$. In the case that $E_1\cong E_2\cong E$, the triple $(E,E,\Phi)$ is $\tau$-stable (equivalently $\sigma$-stable) if and only if $F$ decomposes as a direct sum $$ F={p^\ast} E\otimes {q^\ast}{\cal O}(1)\oplus {p^\ast} E\otimes {q^\ast}{\cal O}(1), $$ and ${p^\ast} E\otimes {q^\ast}{\cal O}(1)$\ is stable with respect to ${\omega_\sig}$. \end{thm} {\em Proof}. As mentioned in \S\ref{stability} , if $ E_2$ is a line bundle the $\tau$-stability of ${(E_1,E_2,\Phi)}$ is equivalent to the $\tau$-stability of the pair $( E_1\otimes E_2^\ast,\Phi)$ in the sense of Bradlow \cite{B2}. In this case Theorem \ref{tsvs} has been proved in \cite[Theorem 4.6.]{GP3}. The main ideas of that proof extend to the general case in a rather straightforward manner. Recall that the bundle $ F$ associated to ${(E_1,E_2,\Phi)}$ comes equipped with a holomorphic action of ${SU(2)}$. It makes sense therefore to talk about the $\su$-invariant\ stability of $ F$. As explained in \S\ref{background}, this is like ordinary stability, but the slope condition has to be satisfied only for $\su$-invariant\ subsheaves of $ F$. In order to prove the theorem we shall prove first the following slightly weaker result. \begin{prop}\label{tsvis} Let ${(E_1,E_2,\Phi)}$ be a holomorphic triple over a compact Riemann surface\ $X$. Let $ F$ be the holomorphic bundle over ${{X\times\dP^1}}$ defined by ${(E_1,E_2,\Phi)}$ and let $\sigma$ and $\tau$ be related by ({\em \ref{s-t'}}). Then ${(E_1,E_2,\Phi)}$ is $\tau$-stable\ (equivalently $\sigma$-stable) if and only if $ F$ is ${SU(2)}$-invariantly stable with respect to ${\omega_\sig}$. \end{prop} {\em Proof}. We saw in \S\ref{background} (Lemmas \ref{equi-sub} and \ref{push-out}) that there is a one-to-one correspondence between subtriples $T'={(E_1',E_2',\Phi')}$ of $T$ and $\su$-invariant\ coherent subsheaves $F'\subset F$. Moreover, the subsheaf $F'$ defined by $T'$ is an extension of the form \begin{equation} {0\lra \ps E_1'\lra F'\lra \ps E_2'\otimes\qs\cod\lra 0}.\label{subextension} \end{equation} In terms of the parameter $\tau'$ as defined in (\ref{3.5}), the relation between $\sigma$ and $\tau$, given by (\ref{s-t'}), can be rewritten as $$ \sigma=\tau-\tau'. $$ If ${(E_1,E_2,\Phi)}$ is $\tau$-stable\ (equivalently, $\sigma$-stable) it follows from (3) in Proposition \ref{l-bound} that $\sigma$ defined by (\ref{s-t'}) is positive. The slope of $ F'$ with respect to ${\omega_\sig}$ is defined as $$ {\mu_\sig}( F')=\frac{\deg_\sigma F'}{\mathop{{\fam0 rank}}\nolimits F'}, $$ where $\deg_\sigma F'$ is the degree of $F'$, which is given by $$ \deg_\sigma F=\frac{1}{2}\int_{{{X\times\dP^1}}}c_1( F)\wedge{\omega_\sig}. $$ The proposition follows now from the following lemma. \begin{lemma}\label{slope}Let $T'$ be a subtriple of $T$ and $F'$ the corresponding $\su$-invariant\ subsheaf of $F$. Let $\sigma$ be as in Proposition \ref{tsvis}. The following are equivalent \begin{tabbing} \ \ \ \= (0)\ \= \kill \> {\em (1)}\> ${\mu_\sig}( F')<{\mu_\sig}( F)$\\ \> {\em (2)}\> $\theta_\tau(T')<0$\\ \> {\em (3)}\> ${\mu_\sig}(T')<{\mu_\sig}(T)$. \end{tabbing} \end{lemma} {\em Proof}. The equivalence between (2) and (3) corresponds, of course, to the two equivalent definitions of stability for $T$ (cf. Proposition \ref{tstab-sstab}). From (\ref{extension-b}) and (\ref{subextension}) we obtain that $$ {\mu_\sig}( F)=\frac{\deg E_1+\deg E_2+\sigma r_2}{r_1+r_2}, $$ and $$ {\mu_\sig}( F')=\frac{\deg E'_1+\deg E'_2+\sigma r_2'}{r_1'+r_2'}, $$ where $r_1'=\mathop{{\fam0 rank}}\nolimits E_1'$ and $r_2'=\mathop{{\fam0 rank}}\nolimits E_2'$. From Definition \ref{t-stab} we immediately obtain the equivalence between (1) and (3). \hfill$\Box$ \noindent{\em Remark.} As usual, in order for $F$ to be $\su$-invariantly stable\ it is enough to check condition (1) of Lemma \ref{slope} only for saturated $\su$-invariant\ subsheaves, that is $\su$-invariant\ subsheaves $F'$ such that the quotient $F/F'$ is torsion-free. Such a subsheaf $F'$ is a subbundle outside of a set of codimension greater or equal than 2. Hence by ${SU(2)}$-invariance one concludes that $F'$ must be actually a subbundle of $F$ over the whole ${{X\times\dP^1}}$. It is easy to see that the saturation of $F'$ implies that of ${p^\ast} E_1'$ and ${p^\ast} E_2'\otimes{q^\ast}{\cO(2)}$ in (\ref{subextension}), and hence $E_1'\subset E_1$ and $E_2'\subset E_2$ are in fact subbundles. In other words, the one-to-one correspondence between $\su$-invariant\ subsheaves of $F$ and subtriples of $T$ established in Lemma \ref{push-out} sends saturated subsheaves into saturated subtriples. To prove the theorem, we first observe that if $F$ is $\su$-invariantly stable\ then, from Theorem \ref{isvs}, it decomposes as a direct sum \begin{equation} F= F_1\oplus F_2\oplus ...\oplus F_k \label{decomposition} \end{equation} of stable bundles, where $ F_i$ is the transformed by an element of ${SU(2)}$ of a fixed subbundle $ F_1$ of $ F$. For the remaining parts of the theorem, the proof splits into two cases, corresponding to whether $E_1$\ and $E_2$\ are isomorphic (as holomorphic bundles) or not. We treat the non-isomorphic case first. Notice that in this case, the map $\Phi$\ certainly cannot be an isomorphism. Clearly if $F$ is stable it is in particular $\su$-invariantly stable\ and hence by the previous Proposition, the corresponding triple will be $\tau$-stable. Suppose now that ${(E_1,E_2,\Phi)}$ is $\tau$-stable, and that $\Phi$\ is not an isomorphism. Our strategy to prove the stability of $ F$ will be to prove that $ F$ is simple, that is $H^0(\mathop{{\fam0 End}}\nolimits F)\cong {\Bbb C}$, and hence there must be just one summand in the decomposition of $F$ given by (\ref{decomposition}). To compute $H^0(\mathop{{\fam0 End}}\nolimits F)\cong H^0( F\otimes F^\ast)$ let us tensor (\ref{extension-b}) with $ F^\ast$. We obtain the short exact sequence $$ 0\longrightarrow{p^\ast} E_1\otimes F^\ast\longrightarrow F\otimes F^\ast\longrightarrow {p^\ast} E_2\otimes{q^\ast}{\cO(2)}\otimes F^\ast\longrightarrow 0, $$ and the corresponding sequence in cohomology \begin{equation} 0\longrightarrow H^0({p^\ast} E_1\otimes F^\ast)\longrightarrow H^0( F\otimes F^\ast)\longrightarrow H^0({p^\ast} E_2\otimes{q^\ast}{\cO(2)}\otimes F^\ast)\longrightarrow. \label{simple} \end{equation} We first compute $H^0({p^\ast} E_1\otimes F^\ast)$. Dualizing (\ref{extension-b}), tensoring with ${p^\ast} E_1$, and using that $H^0({p^\ast}( E_1\otimes E_2^\ast)\otimes{q^\ast}{\cal O}(-2))=0$, we have the sequence in cohomology \begin{equation} 0\longrightarrow H^0({p^\ast} E_1\otimes F^\ast)\longrightarrow H^0({p^\ast}( E_1\otimes E_1^\ast))\stackrel{g}{\longrightarrow} H^1({p^\ast}( E_1\otimes E_2^\ast)\otimes {q^\ast}{\cal O}(-2)). \label{cs1} \end{equation} By the K\"unneth formula $$ H^0({p^\ast}( E_1\otimes E_1^\ast))\cong H^0( E_1\otimes E_1^\ast) \;\;\;\;\mbox{and}\;\;\;\; H^1({p^\ast} (E_1\otimes E_2^\ast)\otimes{q^\ast}{\cal O}(-2))\cong H^0( E_1\otimes E_2^\ast). $$ Now, thanks to these isomorphisms, $g$ can be interpreted as the map $H^0( E_1\otimes E_1^\ast)\rightarrow H^0( E_1\otimes E_2^\ast)$ defined by $\Phi$, i.e. $g(u)=u\Phi$. Now, from the $\tau$-stability of ${(E_1,E_2,\Phi)}$ one has from Corollary \ref{stab-simple} that ${(E_1,E_2,\Phi)}$\ is simple. Thus $\mathop{{\fam0 Ker}}\nolimits g\cong 0$ and from the exactness of (\ref{cs1}) one obtains \begin{equation} H^0({p^\ast} E_1\otimes F^\ast)=0. \label{van1} \end{equation} To compute $H^0({p^\ast} E_2\otimes {q^\ast}{\cO(2)}\otimes F^\ast)$, we dualize (\ref{extension-b}) and tensor it with ${p^\ast} E_2\otimes{q^\ast}{\cO(2)}$, to get the sequence \begin{equation} 0\longrightarrow H^0({p^\ast}( E_2\otimes E_2^\ast))\longrightarrow H^0({p^\ast} E_2\otimes{q^\ast}{\cO(2)}\otimes F^\ast) \longrightarrow H^0({p^\ast}( E_1^\ast\otimes E_2)\otimes{q^\ast}{\cO(2)}). \label{cs2} \end{equation} \begin{lemma} Let ${(E_1,E_2,\Phi)}$ be $\tau$-stable\ and suppose that $\Phi$ is not an isomorphism, then $H^0( E_1^\ast\otimes E_2)=0$. \label{van2} \end{lemma} {\em Proof}. Suppose that there is a non-zero homomorphism $\Psi: E_1\rightarrow E_2$. Let $u=\Phi\Psi\in H^0(\mathop{{\fam0 End}}\nolimits E_1)$ and $v=\Psi\Phi\in H^0(\mathop{{\fam0 End}}\nolimits E_2)$. Then $u\Phi=\Phi v$ and, since ${(E_1,E_2,\Phi)}$ is simple, we have that $$ u=\lambda {\bf} I_{ E_1}\;\;\;\;\mbox{and}\;\;\;\; v=\lambda {\bf} I_{E_2},\;\;\;\mbox{for} \;\;\;\lambda\in{\Bbb C}. $$ If $\lambda\neq 0$, we easily see that $\Phi$ is an isomorphism, contradicting the assumption of the Lemma. Thus $\lambda=0$ and then $$ \mathop{{\fam0 Im}}\nolimits \Psi\subseteq\mathop{{\fam0 Ker}}\nolimits \Phi \;\;\;\;\;\mbox{and}\;\;\;\;\; \mathop{{\fam0 Im}}\nolimits \Phi\subseteq\mathop{{\fam0 Ker}}\nolimits \Psi. $$ We can therefore consider the subtriples of ${(E_1,E_2,\Phi)}$ $$ T_1=( K, E_2,\Phi)\;\;\;\;\mbox{and}\;\;\;\;T_2=(0, I,\Phi), $$ where $ K=\mathop{{\fam0 Ker}}\nolimits \Psi$ and $ I=\mathop{{\fam0 Im}}\nolimits \Psi$. Let $r_1'=\mathop{{\fam0 rank}}\nolimits K$, $r_1''=\mathop{{\fam0 rank}}\nolimits I$, $d_1'=\deg K$ and $d_1''=\deg I$. Applying the $\tau$-stability condition to $T_1$ and $T_2$ we get the inequalities $$ \begin{array}{l} r_2d_1''-r_1''(d_1+d_2)+r_1r_1''\tau<0\\ d_1'-d_1+(r_1-r_1')\tau<0. \end{array} $$ From this and using that from $$ 0\longrightarrow K\longrightarrow E_1\longrightarrow I\longrightarrow 0, $$ $r_1=r_1'+r_1''$ and $d_1=d_1'+d_1''$, we obtain that $$ \tau<\mu( E_1\oplus E_2), $$ which is equivalent to $\sigma(\tau)<0$, contradicting the $\tau$-stability of ${(E_1,E_2,\Phi)}$. \hfill$\Box$ From the K\"unneth formula and Lemma \ref{van2} we get that $$ H^0({p^\ast}( E_1^\ast\otimes E_2)\otimes{q^\ast}{\cO(2)})\cong H^0( E_1^\ast\otimes E_2)\otimes H^0({\cO(2)})\cong 0, $$ and from (\ref{cs2}) $$ H^0({p^\ast} E_2\otimes{q^\ast}{\cO(2)}\otimes F^\ast)\cong H^0( E_2\otimes E_2^\ast). $$ From this and (\ref{van1}) the first three terms in (\ref{simple}) reduce to $$ 0\longrightarrow H^0( F\otimes F^\ast)\stackrel{i}{\longrightarrow} H^0( E_2\otimes E_2^\ast). $$ Since $ F$ is $\su$-invariantly stable\ then it is ${SU(2)}$-invariantly simple, i.e. the only $\su$-invariant\ endomorphisms are multiples of the identity. Let $\Psi\in H^0( F\otimes F^\ast)$ be a non $\su$-invariant\ endomorphism of $ F$, i.e. $\Psi^g\neq\Psi$ for some $g\in {SU(2)}$. Since $i$ must be compatible with the action of ${SU(2)}$, we get $$ i(\Psi^g)=(i(\Psi))^g. $$ On the other hand, since the action of ${SU(2)}$ on $H^0( E_2\otimes E_2^\ast)$ is trivial $$ (i(\Psi))^g=i(\Psi), $$ hence $ i(\Psi)=i(\Psi^g)$ contradicting the injectivity of $i$. Thus $H^0( F\otimes F^\ast)\cong{\Bbb C}^\ast$, which concludes the proof of our theorem for the case where $E_1$\ and $E_2$\ are not isomorphic. Now suppose that $E_1\cong E_2$. We first prove \begin{lemma}\label{ts-iso} Suppose $E_1\cong E_2$. Then for any $\tau>\mu(E_1)$, the triple $(E_1, E_2,\Phi)$\ is $\tau$- stable if and only if $\Phi$\ is an isomorphism and $E_1$\ is stable. \end{lemma} {\em Proof}. Suppose that the triple is $\tau$-stable. Then (by Corollary 3.8) $\Phi$\ is an isomorphism. Now consider the subtriples of the form $T'=(\Phi(E'_2), E'_2,\Phi)$. These have $r_1'=r_2'$\ and, since $\Phi$\ is an isomorphism, $\mu(T')=\mu(E'_2)$. Hence $\theta_{\tau}(T')=\mu(E'_2)-\mu(E_2)$, and thus the $\tau$-stability of the triple implies the stability of $E_2$. Conversely, if $E_2\cong E_1$\ and both are stable, then all non trivial $\Phi$ are isomorphisms. It now follows as a special case of Proposition 3.6 that the triple $(E_1, E_2,\Phi)$\ is $\tau$-stable. \hfill$\Box$ Suppose that $E_1\cong E_2\cong E$\ and that the bundle $F$ associated to $(E,E,\Phi)$ is of the form \begin{equation} F={p^\ast} E\otimes{q^\ast}{\cal O}(1)\oplus {p^\ast} E\otimes{q^\ast}{\cal O}(1).\label{2-sum} \end{equation} If we assume now that ${p^\ast} E\otimes {q^\ast}{\cal O}(1)$ is stable, then $E$ is also stable and hence $ H^0(E\otimes E^*)\cong{\Bbb C}$. Thus there is only one non-trivial extension class (corresponding to $\Phi=\lambda {\bf} I$). We must now examine this (unique) non-trivial extension $$ 0\longrightarrow {p^\ast} E \longrightarrow F \longrightarrow {p^\ast} E\otimes q^*\cal O(2)\longrightarrow 0. $$ This is of course nothing else but the pull-back to ${{X\times\dP^1}}$ of the non-trivial extension on $\dP^1$ $$ 0\longrightarrow{\cal O}\longrightarrow{\cal O}(1)\oplus{\cal O}(1)\longrightarrow{\cal O}(2)\longrightarrow 0, $$ tensored with ${p^\ast} E$. Thus the action of ${SU(2)}$ permutes the two summands in (\ref{2-sum}) and from Theorem \ref{isvs} we conclude that $F$ is an $\su$-invariantly stable\ bundle. The $\tau$-stability of ${(E_1,E_2,\Phi)}$ follows now from Proposition \ref{tsvis}. Conversely, suppose that ${(E_1,E_2,\Phi)}$ is $\tau$-stable, then from Lemma \ref{ts-iso} we obtain that $E_1\cong E_2\cong E$ is stable and $\Phi$ is hence a non-trivial multiple of the identity. From the above discussion we conclude that $$ F={p^\ast} E\otimes{q^\ast}{\cal O}(1)\oplus {p^\ast} E\otimes{q^\ast}{\cal O}(1). $$ On the other hand, from Proposition \ref{tsvis}, we argue as before that $F$ \ is certainly invariantly stable . Also (\ref{simple}), (\ref{cs1}) and (\ref{van1}) show that we have an exact sequence $$0\longrightarrow H^0(F\otimes F^*)\longrightarrow H^0(p^*E\otimes q^*\cal O(2)\otimes F^\ast).$$ Using that $H^0(E\otimes E^*)\cong {\bf} C$, the exact sequence (\ref{cs2}) becomes, $$0\longrightarrow {\bf} C\longrightarrow H^0(p^*E\otimes q^*\cal O(2)\otimes F^*) \longrightarrow {\bf} C^3.$$ From these two exact sequences, we see that \begin{equation} 1\le h^0(F\otimes F^*)\le h^0(p^*E\otimes q^*\cal O(2)\otimes F^*)\le 4\ . \label{constraint} \end{equation} But since $F$\ is invariantly stable it is given by the direct sum (\ref{decomposition}). In that case, $$H^0(F\otimes F^*)\cong GL(k,{\bf} C)\ ,$$ where $k$\ is the number of stable summands in $F$. It follows that since $h^0(F\otimes F^*)\ne 1$, then $h^0(F\otimes F^*)=k^2-1$\ for some integer $k$. The only possibility consistent with the constraint (\ref{constraint}) is thus $h^0(F\otimes F^*)=3$, i.e. $k=2$. Hence the bundle ${p^\ast} E\otimes{q^\ast}{\cal O}(1)$ in the decomposition of $F$ is stable, which finishes the proof of the Theorem. \hfill$\Box$ Notice that the conclusion of Proposition \ref{tsvis} extends straightforwardly to cover polystable objects. We thus get the following corollary, which will be useful in the next section \begin{thm}\label{ptsvps} Let ${(E_1,E_2,\Phi)}$ be a holomorphic triple and $F$ be the corresponding holomorphic bundle over ${{X\times\dP^1}}$. Let $\tau$\ and $\sigma$\ be related as above. Then ${(E_1,E_2,\Phi)}$ is $\tau$-polystable if and only if $F$ is ${SU(2)}$-invariantly polystable with respect to ${\omega_\sig}$. \end{thm} \section{Relation to vortex equations} In this section we relate the $\tau$-stability of a triple to the existence of solutions to the coupled $\tau$-vortex equations. Using the idea of dimensional reduction this can be viewed as simply a special case of the Hitchin-Kobayashi correspondence between stable bundles and Hermitian-Einstein metrics. Indeed, specializing to the case of SU(2)-equivariant bundles over \ ${{X\times\dP^1}}$\ as in \S \ref{background} (i.e. to the case of the bundles associated with triples over $X$), we have already assembled seen the results we need. From \cite{GP3} (cf. also Theorem \ref{inv-h-k}) we know that the SU(2)-equivariant bundle $F$\ admits a Hermitian-Einstein metric with respect to $\omega_{\sigma}$\ if and only if $F$\ is polystable with respect to $\omega_{\sigma}$. By Theorem \ref{ptsvps} we know that $F$\ is ${SU(2)}$ invariantly polystable with respect to $\omega_{\sigma}$\ if and only if the corresponding triple ${(E_1,E_2,\Phi)}$\ is $\tau$-polystable with $\tau$\ and $\sigma$\ related by (\ref{s-t'}). Finally, by Proposition \ref{dr} there is a bijective correspondence between the SU(2)-equivariant Hermitian-Einstein metrics on $F$ and the solutions to the coupled vortex equations (\ref{cves}) on ${(E_1,E_2,\Phi)}$. Putting all this together, we see that we have filled in three sides of the following ``commutative diagram'' for a holomorphic triple over $X$\ and the corresponding SU(2)-equivariant bundle over ${{X\times\dP^1}}$\ (with K\"{a}hler\ form $\omega_{\sigma}$) \begin{tabular}{cccc} & & & \\ & SPECIAL &\ \ Hitchin-Kobayashi \ \ & HOLOMORPHIC \\ & METRICS & correspondence & INTERPRETATION \\ & & & \\ & SU(2)-invariant Solutions & Th. 2.18 & SU(2)-invariantly \\ On ${{X\times\dP^1}}$ & to Hermitian-Einstein & & Polystable \\ & Equations & & Extensions \\ & & & \\ dim.\ red. &Prop. 2.14 & &Th. 4.6 \\ & & &\\ & Solutions to & & $\tau$-stable\\ On $X$ & Coupled $\tau$-Vortex & & holomorphic\\ & Equations & & Triples\\ \end{tabular} By tracing around the three sides of this diagram which already have arrows filled in, we can fill in the arrows on the fourth side and prove \begin{thm}\label{existence} Let $T=(E_1,E_2,\Phi)$\ be a holomorphic triple. Then the following are equivalent. \ \ \ {\em (1)}\ The bundles support Hermitian metrics $h_1, h_2$\ such that the coupled $\tau$-vortex equations are satisfied, i.e. such that \begin{equation} \sqrt{-1}\Lambda F_{h_1}+\Phi \Phi^*=2\pi\tau{\bf} I_{E_1},\label{4.1a} \end{equation} \begin{equation} \sqrt{-1}\Lambda F_{h_2}-\Phi^*\Phi=2\pi\tau'{\bf} I_{E_2},\label{4.1b} \end{equation} with \begin{equation} r_1\tau+r_2\tau'=\deg E_1 + \deg E_2, \label{4.1c} \end{equation} \ \ \ {\em (2)}\ The triple is $\tau$-polystable. \end{thm} Notice that the statement and conclusion of this theorem make no mention of ${{X\times\dP^1}}$\ or the $\su$-equivariant\ bundle $F$. One might thus expect a more direct proof that does not use dimensional reduction. We will not attempt to prove both directions of the biconditional in the theorem in this way, but the one direction is quite easily seen. That is, one can show how the $\tau$-stability condition can be derived directly as a consequence of the coupled vortex equations. We do this as follows. Let $T'=(E'_1,E'_2,\Phi)$\ be a holomorphic saturated sub-triple of $T$, and let $$E_1=E'_1\oplus E''_1,$$ and $$E_2=E'_2\oplus E''_2$$ be smooth orthogonal splittings of $E_1$\ and $E_2$. With respect to these splittings, we get a block diagonal decomposition of $\sqrt{-1}\Lambda F_{h_1}$, $\sqrt{-1}\Lambda F_{h_2}$as \begin{equation} \sqrt{-1}\Lambda F_{h_i}=\left(\begin{array}{cc} \sqrt{-1}\Lambda F'_i+\Pi_i & \ast\\ \ast& \sqrt{-1}\Lambda F''_i-\Pi_i \end{array}\right),\label{4.2} \end{equation} where $\Pi_i$\ is a positive definite endomorphism coming from the second fundamental form for the inclusion of $E'_i$\ in $E_i$. We also get a decomposition of $\Phi$\ as \begin{equation} \Phi=\left(\begin{array}{cc} \Phi' & \Theta\\ 0 & \Phi'' \end{array}\right).\label{4.3} \end{equation} The coupled vortex equations thus split into equations on the summands of $E_1$\ and $E_2$\ to yield \begin{eqnarray} \sqrt{-1}\Lambda F'_1+\Pi_1+\Phi' (\Phi')^*+\Theta\Theta^* &=2\pi\tau{\bf} I'_1,\label{4.4a}\\ \sqrt{-1}\Lambda F''_1-\Pi_1+\Phi'' (\Phi'')^*&=2\pi\tau{\bf} I''_1,\label{4.4b}\\ \sqrt{-1}\Lambda F'_2+\Pi_2-(\Phi')^* \Phi'&=2\pi\tau'{\bf} I'_2,\label{4.4c}\\ \sqrt{-1}\Lambda F''_2-\Pi_2-(\Phi'')^* \Phi''- \Theta^*\Theta &=2\pi\tau'{\bf} I''_2.\label{4.4d} \end{eqnarray} We now integrate the trace of these equations over the base manifold $X$. We use the notation $$ d'_i=\deg E'_i=\frac{\sqrt{-1}}{2\pi}\int_X \mathop{{\fam0 Tr}}\nolimits(\Lambda F'_i), $$ $$ d''_i=\deg E''_i=\frac{\sqrt{-1}}{2\pi}\int_X \mathop{{\fam0 Tr}}\nolimits(\Lambda F''_i), $$ $$ ||\Pi_i||^2=\frac{1}{2\pi}\int_X \mathop{{\fam0 Tr}}\nolimits(\Pi_i), $$ $$ ||\Theta||^2=\frac{1}{2\pi}\int_X \mathop{{\fam0 Tr}}\nolimits(\Theta\Theta^*), $$ From equations (\ref{4.1a}) and (\ref{4.1c}) we get \begin{equation} d''_1+d''_2 - (||\Pi_1||^2+||\Pi_2||^2)-||\Theta||^2=\tau(r_1-r_1')+\tau'(r_2-r_2'), \label{4.5} \end{equation} and from (\ref{4.1b}) and (\ref{4.4a}) we get \begin{equation} d'_1+d'_2 + (||\Pi_1||^2+||\Pi_2||^2)+||\Theta||^2=\tau r_1'+\tau' r_2'.\label{4.6} \end{equation} Using the fact that $d_i=d'_i+d''_i$, these equations can be combined to give \begin{equation} \deg(E'_1\oplus E'_2)+2(||\Pi_1||^2+||\Pi_2||^2+||\Theta||^2)=r_1'\tau+r_2'\tau'\label{4.7} \end{equation} It now follows by the positivity of the terms $||\Pi_i||^2$\ and $||\Theta||^2$, that \begin{equation} \mu(T')\le \frac{r_1'\tau+r_2'\tau'}{r_1'+r_2'}. \label{4.8} \end{equation} But from equation (\ref{4.1c}) we have that $$\frac{r_1\tau+r_2\tau'}{r_1+r_2} = \mu(T),$$ and by using this to solve for $\tau'$, one sees that (\ref{4.8}) is equivalent to the condition $\theta_{\tau}(T')\le 0$. Furthermore, it follows from (\ref{4.7}) that in order to have $\theta_{\tau}(T')= 0$, one needs $$||\Pi_1||^2=||\Pi_2||^2=||\Theta||^2=0.$$ This means that the bundles split holomorphically as $E_i=E'_i\oplus E''_i$\ and $\Phi=\Phi'\oplus \Phi''$, i.e. the triple $T={(E_1,E_2,\Phi)}$\ splits as $T=(E'_1,E'_2,\Phi')\oplus (E''_1,E''_2,\Phi'')$. Each summand separately supports a solution to the coupled $\tau$-vortex equations. It is possible that in this splitting $\Phi'$\ or $\Phi''$\ is trivial. The bundles in the degenerate subtriple then each supports solutions to the Hermitian-Einstein equations. We have thus proven the following proposition: \begin{prop}Let $T=(E_1,E_2,\Phi)$\ be a holomorphic triple in which the bundles support Hermitian metrics $h_1, h_2$\ such that the coupled $\tau$-vortex equations are satisfied. Then \begin{tabbing} \ \ \ \=(0)\ \= \kill \> {\em (1)}\> $T$\ splits as a direct sum of triples $(E_{1i},E_{2i},\Phi_i)$, i.e. $E_1=\bigoplus E_{1i}$, $E_2=\bigoplus E_{2i}$,\\ \> \> and $\Phi=\bigoplus \Phi_i$,\\ \> {\em (2)}\> each summand $(E_{1i},E_{2i},\Phi_i)$\ is either a $\tau$-stable triple, or $\Phi_i=0$\ and both \\ \> \> bundles are stable. In the latter case, the slope of $E_{1i}$\ is $\tau$\ and the slope of $E_{2i}$\\ \> \> is $\tau'$. \end{tabbing} That is $T$\ is $\tau$-polystable. \end{prop} \section{Moduli spaces}\label{moduli} \subsection{Moduli spaces of stable triples} Recall that two triples $T={(E_1,E_2,\Phi)}$ and $T'={(E_1',E_2',\Phi')}$ are isomorphic if there exist isomorphisms $u:E_1\longrightarrow E_1'$\ and $v:E_2\longrightarrow E_2'$\ making the following diagram commutative $$ \begin{array}{ccc} \; E_2& \stackrel{\Phi}{\longrightarrow} & \; E_1 \\ v{\downarrow}& & u{\downarrow}\\ \; E_2'& \stackrel{\Phi'}{\longrightarrow} & \; E_1'. \end{array} $$ After fixing the topological invariants of our bundles, that is the ranks $r_1$ and $r_2$ and the first Chern classes $d_1$ and $d_2$, let $\glM$ be the set of equivalence classes of holomorphic triples and $\glM_\tau\subset\glM$ be the subset of equivalence classes of $\tau$-stable\ triples. Our goal in this section is to show that $\glM_\tau$ has the structure of an algebraic variety, more precisely: \begin{thm}\label{MODULI} Let $X$ be a compact Riemann surface\ of genus $g$ and let us fix ranks $r_1$ and $r_2$ and degrees $d_1$ and $d_2$. The moduli space of $\tau$-stable\ triples $\glM_\tau$ is a complex analytic space with a natural K\"{a}hler\ structure outside of the singularities. Its dimension at a smooth point is \begin{equation} 1+r_2d_1-r_1d_2+(r_1^2+r_2^2-r_1r_2)(g-1). \label{dimension} \end{equation} The moduli space, $\glM_\tau$ is non-empty if and only if $\tau$ is inside the interval \begin{equation} (\mu(E_1), \mu_{MAX}) \label{interval} \end{equation} where \begin{equation} \mu_{MAX}=\mu(E_1)+\frac{r_2}{|r_1-r_2|}(\mu(E_1)-\mu(E_2)) \end{equation} if $r_1\neq r_2$, and $\mu_{MAX}=\infty$\ if $r_1=r_2$. Moreover $\glM_\tau$ is in general a quasi-projective variety. It is in fact projective if $r_1+r_2$ and $d_1+ d_2$ are coprime and $\tau$ is generic. \end{thm} {\em Proof}. There are several approaches one can take to prove this theorem. One can use standard Kuranishi deformation methods as done in \cite{B-D1,B-D2} for the construction of the moduli spaces of stable pairs. Alternatively one can use geometric invariant theory methods to give an algebraic geometric construction of our moduli spaces, generalizing the construction of the moduli space of stable pairs given in \cite{Be,T}. We will leave these two direct methods for a future occasion and instead will exploit the relation between $\tau$-stable\ triples and equivariant bundles over ${{X\times\dP^1}}$. This method, which is used in \cite{GP3} to construct the moduli spaces of triples when $E_2$ is a line bundle, leads also to an alternative construction of the moduli spaces of stable pairs. Apart from smoothness considerations, which we shall discuss later, the arguments of the proof are the same that those in \cite{GP3}. Let $\sigma$ be related to $\tau$ by (\ref{s-t'}) and let ${\omega_\sig}$ be the K\"{a}hler\ form on ${{X\times\dP^1}}$ defined by (\ref{kah-pol}). Let ${\cal M}_\sigma$ be the moduli space of stable bundles with respect to ${\omega_\sig}$ whose underlying smooth bundle is defined by (\ref{equi-f}). Let us exclude for the moment the case $r_1=r_2$ and $d_1=d_2$. Let $F\longrightarrow{{X\times\dP^1}}$ be the bundle associated to ${(E_1,E_2,\Phi)}$ as in Proposition \ref{equi-hvb}. Theorem \ref{tsvs} says that the correspondence ${(E_1,E_2,\Phi)}\longmapsto F$ defines a map $$ \glM_\tau\longrightarrow{\cal M}_\sigma. $$ The action of ${SU(2)}$ on ${{X\times\dP^1}}$ defined in Section \ref{background} induces an action on ${\cal M}_\sigma$ and, since the bundle $F$ associated to ${(E_1,E_2,\Phi)}$ is $\su$-equivariant\ the image of the above map is contained in ${\cal M}_\sigma^{{SU(2)}}$---the set of fixed points of ${\cal M}_\sigma$ under the ${SU(2)}$ action. As proved in \cite[Proposition 5.3]{GP3} the set ${\cal M}_\sigma^{{SU(2)}}$ can be described as a disjoint union of a finite number of sets $$ {\cal M}_\sigma^{{SU(2)}}=\bigcup_{i\in I}{\cal M}_\sigma^i. $$ The index $I$ ranges over the set of equivalence classes of different smooth $\su$-equivariant\ structures on the smooth bundle $F$ defined by (\ref{equi-f}). Of course the way of writing $F$ in (\ref{equi-f}) already exhibits a particular $\su$-equivariant\ structure, but in principle the bundle $F$ might admit different ones. The set ${\cal M}_\sigma^i$ corresponds to the set of equivalence classes in ${\cal M}_\sigma$ admitting a representative which is $\su$-equivariant\ for the smooth equivariant structure defined by $i\in I$. An equivariant smooth structure defines an action on the space of smooth automorphisms of the bundle $F$ and, as shown in \cite[Theorem 5.6]{GP3} the sets ${\cal M}_\sigma^i$ can be described as the set of equivalence classes of $\su$-equivariant\ holomorphic structures on the underlying smooth $\su$-equivariant\ bundle defined by $i$, modulo $\su$-equivariant\ isomorphisms. Let $i_0$ be the $C^\infty$ $\su$-equivariant\ structure on $F$ defined by (\ref{equi-f}). As shown in Proposition \ref{equi-hvb} there is a one-to-one correspondence \begin{equation} \{\mbox{holomorphic triples}\}\stackrel{1-1}{\longleftrightarrow} \{\mbox{$i_0$-equivariant holomorphic vector bundles}\}. \label{tri-equiv} \end{equation} On the other hand by Lemma \ref{equi-hom} the equivariant homomorphisms between two equivariant holomorphic bundles $F$ and $F'$ corresponding to triples $T$ and $T'$, respectively, are in one-to-one correspondence with the morphisms between $T$ and $T'$. In fact the correspondence (\ref{tri-equiv}) descends to the quotient and thus from Theorem \ref{tsvs} we can identify $\glM_\tau$ with ${\cal M}_\sigma^{i_0}$. The properties of $\glM_\tau$ follow now from standard facts about the more familiar moduli spaces of stable bundles ${\cal M}_\sigma$ \cite{D-K,G,M,Ko}, and more particularly of the fixed-point sets ${\cal M}_\sigma^i$ (see \cite[Theorem 5.6]{GP3} for details). Namely, \begin{thm}\label{fix-points}${\cal M}_\sigma^i$ is a complex analytic variety. A point $[F]\in {\cal M}_\sigma^i$ is non-singular if it is non-singular as a point of ${\cal M}_\sigma$. The tangent space at such a point can be identified with the $\su$-invariant\ part of $H^1({{X\times\dP^1}}, \mathop{{\fam0 End}}\nolimits F)$. ${\cal M}_\sigma^i$ has a natural K\"{a}hler\ structure induced from that of ${\cal M}_\sigma$. Moreover if $\sigma$ is a rational number then ${\cal M}_\sigma^i$ is a quasi-projective variety. \end{thm} {}From this theorem and the identification of $\glM_\tau$ with ${\cal M}_\sigma^{i_0}$ we deduce that $\glM_\tau$ is a complex analytic variety with a K\"{a}hler\ metric outside the singularities. To compute the dimension of the tangent space at a smooth point $[T]$ it suffices to compute the dimension of the $\su$-invariant\ part of $H^1({{X\times\dP^1}},\mathop{{\fam0 End}}\nolimits F)$. This can be done in a similar way to that of \cite[Theorem 5.13]{GP3} to obtain that $$ \mbox{dim}\glM_\tau=1+\chi(E_1\otimes E_2^\ast)-\chi(\mathop{{\fam0 End}}\nolimits E_1)-\chi(\mathop{{\fam0 End}}\nolimits E_2), $$ which by Riemann-Roch yields (\ref{dimension}). We consider now the case $r_1=r_2=r$ and $d_1=d_2=d$. In this case by Lemma \ref{ts-iso} we can identify the moduli space $\glM_\tau$ with the moduli space of stable bundles of rank $r$ and degree $d$ on $X$. The theorem follows now from well-known results about this moduli space \cite{A-B,N-S}. The fact that $\glM_\tau$ is empty outside the interval (\ref{interval}) if $r_1\neq r_2$ and outside $(\mu(E_1),\infty)$ if $r_1=r_2$ follows from Proposition \ref{u-bound}. As explained in Proposition \ref{critical} the non-generic values divide this intervals in subintervals in such a way that the stability properties of a given triple do not change for two values of $\tau$ in the same subinterval. Therefore we can always choose $\tau$ (and hence $\sigma$) to be rational, which by Theorem \ref{fix-points} gives that $\glM_\tau$ is quasi-projective. To show the compactness of $\glM_\tau$ when $r_1+r_2$ and $d_1+d_2$ are coprime and $\tau$ is generic (we are also assuming that $r_1\neq r_2$ or $d_1\neq d_2$) we consider a sequence of points in ${\cal M}_\sigma^{i_0}$. This sequence must converge in $\overline{{\cal M}_\sigma}$---the Uhlenbeck compactification of ${\cal M}_\sigma$. Using ${SU(2)}$-invariance one can see that the limit has to correspond to a polystable element, but by Proposition \ref{critical} this has to be actually stable, that is the limit must be in ${\cal M}_\sigma$ and hence in ${\cal M}_\sigma^{i_0}$ since this is closed. The compactness when $r_1=r_2$ and $d_1=d_2$ follows from the compactness of the moduli space of stable bundles of rank $r$ and degree $d$ when $r$ and $d$ are coprime. The compactness of $\glM_\tau$ can also be obtained (as it is done for pairs in \cite{B-D1}) from the fact that it can be identified with the moduli space of solutions to the coupled vortex equations and these are moment map equations as we shall explain later. \hfill$\Box$ It was shown in \cite[Theorem 5.13]{GP3} that when $E_2$ is a line bundle our moduli spaces are smooth for every value of $\tau$. This does not seem to be the case when $E_2$ is of arbitrary rank. However we can show the following \begin{prop}Let $T={(E_1,E_2,\Phi)}$ be a holomorphic triple such that $\Phi$ is either injective or surjective, then $[T]$ is a smooth point of $\glM_\tau$. \end{prop} {\em Proof}. Let \begin{equation} {0\lra \ps E_1\lra F\lra \ps E_2\otimes\qs\cod\lra 0}, \label{extension-3} \end{equation} be the extension over ${{X\times\dP^1}}$ corresponding to $T$. To prove the smoothness of $\glM_\tau$ at the point $[{(E_1,E_2,\Phi)}]$ it suffices to show that $H^2({{X\times\dP^1}},\mathop{{\fam0 End}}\nolimits F)=0$. Tensoring (\ref{extension-3}) with $F^\ast$ the last terms in the corresponding long exact sequence are \begin{equation} H^2({p^\ast} E_1\otimes F^\ast)\longrightarrow H^2(F\otimes F^\ast)\longrightarrow H^2({p^\ast} E_2\otimes{q^\ast}{\cO(2)}\otimes F^\ast)\longrightarrow 0. \label{2-coh-seq} \end{equation} By Serre duality $$ H^2({p^\ast} E_1\otimes F^\ast)\cong H^0({p^\ast}( E_1^\ast\otimes K)\otimes F)^\ast $$ $$ H^2({p^\ast} E_2\otimes{q^\ast}{\cal O}(2)\otimes F^\ast)\cong H^0({p^\ast} (E_2^\ast\otimes K)\otimes{q^\ast}{\cal O}(-4)\otimes F)^\ast, $$ where $K$ is the canonical line bundle of $X$. It is easy to see that $H^0({p^\ast} (E_2^\ast\otimes K)\otimes{q^\ast}{\cal O}(-4)\otimes F)\cong 0$. To analyse $H^0({p^\ast}( E_1^\ast\otimes K)\otimes F)$ we tensor (\ref{extension-3}) with ${p^\ast}(E_1^\ast\otimes K)\otimes{q^\ast}{\cal O}(-2)$, and since $H^0({p^\ast}(E_1\otimes E_1\otimes K)\otimes{q^\ast}{\cal O}(-2))\cong 0$, we obtain \begin{equation} 0\longrightarrow H^0({p^\ast}(E_1^\ast\otimes K){q^\ast}{\cal O}(-2)\otimes F)\longrightarrow H^0({p^\ast}(E_1^\ast\otimes E_2\otimes K))\stackrel{f}{\longrightarrow} H^1({p^\ast}(E_1\otimes E_1^\ast\otimes K \otimes{\cal O}(-2)).\label{coh-sequence} \end{equation} The map $f$ in the above sequence is essentially the map $$ \begin{array}{ccc} H^0(E_1^\ast\otimes E_2\otimes K)& \stackrel{f}{\longrightarrow}& H^0(E_1\otimes E_1^\ast\otimes K)\\ \Psi&\longmapsto&\Phi\circ\Psi. \end{array} $$ Assume now that $\Phi$ is injective, if we prove that $f$ is also injective, by the exactness of (\ref{coh-sequence}) we would be done. Suppose that $\mathop{{\fam0 Ker}}\nolimits f\neq 0$. This means that there exists a non-zero map $\Psi: E_1\longrightarrow E_2\otimes K$, and since $\Psi\circ\Phi=0$, $\mathop{{\fam0 Im}}\nolimits \Psi$ is a non-trivial subsheaf contained in $\mathop{{\fam0 Ker}}\nolimits \Phi$ contradicting the injectivity. To prove smoothness when $\Phi$ is surjective, we consider the dual triple $T^\ast={(E_2^*,E_1^*,\Phi^*)}$. $\Phi^\ast$ is now injective and the result follows from the fact that $T={(E_1,E_2,\Phi)}$ is a smooth point if and only if $T^\ast={(E_2^*,E_1^*,\Phi^*)}$ is a smooth point. \subsection{Abel--Jacobi maps As shown in Proposition \ref{ts-ss} there is a range for the parameter $\tau$ such that the $\tau$-stability of a triple ${(E_1,E_2,\Phi)}$ implies the semistability of $E_1$ and $E_2$. Let $\glM_0$ be the moduli space of $\tau$-stable\ triples for $\tau$ in such a range. Let $N(r,d)$ be the Seshadri compactification of the moduli space of stable bundles of rank $r$ and degree $d$ over $X$, that is, the space of $S$-equivalence classes of semistable bundles. There are natural `` Abel--Jacobi'' maps $\pi_1$ and $\pi_2$ $$ \begin{array}{ccc} \glM_0 & \stackrel{ \large \pi_2}{\longrightarrow}& N(r_2,d_2)\\ & & \\ \pi_1{\downarrow} & & \\ & & \\ N(r_1,d_1) & & \end{array} $$ defined as $$ \pi_1([{(E_1,E_2,\Phi)}])=[E_1]\;\;\;\mbox{and}\;\;\;\pi_2([{(E_1,E_2,\Phi)}])=[E_2]. $$ We know also from Proposition \ref{ts-ss} that if both $E_1$ and $E_2$ are stable then the intersection of the fibres $\pi_1^{-1}([E_1])$ and $\pi_2^{-1}([E_2])$ can be identified with $\dP(H^0(E_1\otimes E_2^\ast))$. In general, though, this intersection for non-stable points is hard to describe. If $\mu(E_1\otimes E_2^\ast)> 2g-2$, that is if $$ r_2d_1-r_1d_2>r_1r_2(2g-2), $$ where $g$ is the genus of $X$, then $H^1(E_1\otimes E_2^\ast)=0$ for $E_1$ and $E_2$ stable and the projection from $\glM_0$ to $N(r_1,d_1)\times N(r_2,d_2)$ is a fibration on the the stable part. Recall that if $(r_1,d_1)=1$ and $(r_2,d_2)=1$ then stability and semistability coincide and there exist universal bundles $$ {\Bbb E}_1\longrightarrow X\times N(r_1,d_1)\;\;\;\mbox{and}\;\;\;{\Bbb E}_2\longrightarrow X\times N(r_2,d_2). $$ Let us denote by $p_1$, $p_2$ and $\pi$ the projections from $X\times N(r_1,d_1)\times N(r_2,d_2)$ to $X\times N(r_1,d_1)$, $X\times N(r_2,d_2)$, and $N(r_1,d_1)\times N(r_2,d_2)$ respectively. It is clear that $\glM_0$ can be identified with \begin{equation} \dP(\pi_\ast(p_1^\ast{\Bbb E}_1\otimes p_2^\ast {\Bbb E}_2^\ast)).\label{proj} \end{equation} But in the non-coprime situation we have no universal bundles ${\Bbb E}_1$ and ${\Bbb E}_2$ available and the analogue of (\ref{proj}) has to be constructed as a moduli space in its own right. As explained in Theorem \ref{MODULI} the moduli space of $\tau$-stable\ triples is non-empty if and only $\tau$ is in the interval $I=(\mu(E_1),\mu_{MAX})$. We saw in \S \ref{critical-values} that the stability properties of a given triple can change only at certain rational values of $\tau$ (the critical values) which divide $I$ in a finite number of subintervals. The moduli spaces for values of $\tau$ in the same open subinterval are then isomorphic, and they might change only when crossing one of the critical values. We expect that, as in the case of stable pairs \cite{B-D-W, T}, the moduli spaces for consecutive intervals must be related by some sort of flip-type birational transformation. This, as well as the construction of a ``master'' space for triples (cf. \cite{B-D-W}) containing the moduli space of triples for all possible values of $\tau$, will be dealt with in a future paper. \subsection{Vortices} Thanks to our existence theorem the moduli space of stable triples can be interpreted as the moduli space of solutions to the coupled vortex equations. To understand the meaning of this statement one needs to regard the vortex equations as equations for unitary connections instead of equations for metrics. This point of view corresponds to the fact that fixing a holomorphic structure and varying the metric on a vector bundle is equivalent to fixing the metric and varying the holomorphic structure---or the corresponding connection. Recall that the space of unitary connections on a smooth Hermitian vector bundle can be identified with the space of $\overline{\partial}$-operators which in turn corresponds with the space of holomorphic structures on our bundle. Let $E_1$ and $E_2$ be smooth vector bundles over $X$ and $h_1$ and $h_2$ be Hermitian metrics on $E_1$ and $E_2$ respectively. Let ${\cal A}_1$ (resp. ${\cal A}_2$) be the space of unitary connections on $(E_1,h_1)$ (resp. $(E_2,h_2)$). Let $(A_1,A_2,\Phi)\in {\cal A}_1 \times {\cal A}_2\times \Omega^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$. The vortex equations can be regarded as the equations for $(A_1, A_2,\Phi)$ \begin{equation} \left. \begin{array}{l} \overline{\partial}_{A_1\ast A_2}\Phi=0\\ \sqrt{-1} \Lambda F_{A_1}+\Phi\Phi^\ast=2\pi\tau {\bf} I_{E_1}\\ \sqrt{-1} \Lambda F_{A_2}-\Phi^\ast\Phi=2\pi\tau'{\bf} I_{E_2} \end{array}\right \}.\label{c-cves} \end{equation} The connections $A_1$ and $A_2$ induce holomorphic structures on $E_1$ and $E_2$ and the first equation in (\ref{c-cves}) simply says that $\Phi$ must be holomorphic. Let ${\cal G}_1$ and ${\cal G}_2$ be the gauge groups of unitary transformations of $(E_1,h_1)$ and $(E_2,h_2)$ respectively. ${\cal G}_1\times{\cal G}_2$ acts on ${\cal A}_1\times{\cal A}_2\times \Omega^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$ by the rule $$ (g_1,g_2).(A_1, A_2, \Phi)=(g_1A_1g_1^{-1},g_2A_2g_2^{-1},g_1\Phi g_2^{-1}). $$ The action of \ ${\cal G}_1\times{\cal G}_2$\ preserves the equations and the moduli space of {\em coupled $\tau$-vortices} is defined as the space of all solutions to (\ref{c-cves}) modulo this action. The moduli space of vortices can be obtained as a symplectic reduction (see \cite[Section 2.2]{GP3}) in a similar way to the moduli space of Hermitian--Einstein\ connections: ${\cal A}_1\times{\cal A}_2\times\Omega^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1))$ admits a K\"{a}hler\ structure which is preserved by the action of ${\cal G}_1\times{\cal G}_2$. Associated to this action there is a moment map given precisely by \begin{equation} (A_1,A_2,\Phi)\longmapsto (\Lambda F_{A_1}-\sqrt{-1}\Phi\Phi^\ast+2\sqrt{-1}\pi\tau, \Lambda F_{A_2}+\sqrt{-1}\Phi^\ast\Phi+2\sqrt{-1}\pi\tau').\label{mp} \end{equation} Let $\mu$ be this moment map restricted to the subvariety $$ {\cal N}=\{(A_1,A_2,\Phi)\in {\cal A}_1\times{\cal A}_2\times\Omega^0(\mathop{{\fam0 Hom}}\nolimits(E_2,E_1)) \;\; |\;\; \overline{\partial}_{A_1\ast A_2}\Phi=0\}. $$ The moduli space of $\tau$-vortices is then nothing else but the symplectic quotient $$ \mu^{-1}(0,0)/{\cal G}_1\times{\cal G}_2, $$ and Theorem \ref{existence} can be reformulated by saying that there is a one-to-one correspondence $$ \mu^{-1}(0,0)/{\cal G}_1\times{\cal G}_2\stackrel{1-1}{\longleftrightarrow}\glM_\tau. $$ \section{Some generalizations} 1. Although for simplicity we have worked on a Riemann surface, most of our results extend in a straightforward manner to a compact complex manifold of arbitrary dimension. Of course, as in ordinary stability, one needs to choose a K\"{a}hler\ metric in order to define the degree of a coherent sheaf and hence the slopes involved in the definition of $\tau$-stability for a triple. 2. One of our main goals in this paper has been to show that our stability condition for a triple corresponds to the existence of solutions to the coupled vortex equations. This is the main reason for defining our stability criterium only for vector bundles. One can more generally define $\tau$-stability for a triple consisting of two (torsion free) coherent sheaves and a morphism between them. The main results of Sections \ref{stability} and \ref{theorem} go through in this more general situation. 3. A. King \cite{K} has been able to characterize all $\su$-equivariant\ holomorphic vector bundles on ${{X\times\dP^1}}$. Generalizing the results in Section \ref{background}, he has shown that these bundles are in one-to-one correspondence with $(2n-1)$-tuples consisting of $n$ holomorphic vector bundles $E_1$, ..., $E_n$ over $X$ and a chain of morphisms $$ E_n\stackrel{\Phi_{n-1}}{\longrightarrow} E_{n-1}\longrightarrow ...\longrightarrow E_2\stackrel{\Phi_1}{\longrightarrow} E_1. $$ He has defined a stability condition for such a $(2n-1)$-tuple which involves $(n-1)$ parameters and that specializes to our stability condition for a triple when $n=2$. In fact he considers this notion for more general diagrams than the one above. Presumably this stability condition governs, as for triples, the existence of Hermitian metrics on the bundles $E_i$ satisfying some generalized vortex equations naturally associated to the $(2n-1)$-tuple. 4. Our results have also been extended in a different direction \cite{B-GP} to parabolic triples, that is to triples in which the bundles are endowed with parabolic structures. The Higgs field can be either a parabolic morphism or a meromorphic morphism with simple poles at the parabolic points and whose residues respect the parabolic structure in some precise sense. In both cases one can prove a Hitchin-Kobayashi correspondence, although the metrics involved now have singularities at the parabolic points. \vspace{12pt} \noindent {\em Acknowledgements}.\ The authors would like to thank Alastair King and Jun Li for helpful conversations and the following institutions for their hospitality during the course of this project: The Mathematics Institute of the University of Warwick, England; I.H.E.S., Fpance; the Mathelatics Department of UC Berkeley, USA; and C.I.M.A.T., Mexico.

9401

alg-geom/9401002

en

https://arxiv.org/abs/alg-geom/9401002

alg-geom/9401002

Joerg Winkelmann

Holomorphic functions on an algebraic group invariant under a Zariski-dense subgroup

Plain TeX, 9 pages. TeX errors corrected

Let G be complex linear-algebraic group, H a subgroup, which is dense in G in the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1) G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every H-invariant holomorphic function on G is constant. If G=G', furthermore every H-invariant meromorphic or plurisubharmonic function is constant. Finally an example of Margulis is used to show the existence of an algebraic group G with G=G' such that there exists a Zariski-dense discrete subgroup without any semisimple element.

alg-geom

\section{References} \let\c\Cedille \def\(##1\){} \frenchspacing \setbox0=\hbox{99.\enspace}\refindent=\wd0\relax \def\item##1##2:{\goodbreak \hangindent\refindent\hangafter=1\noindent\hbox to\refindent{\hss\refcite{##1}\enspace}\petcap \ignorespaces##2\rm: }} \def\refcite#1{\citeerrortrue\dorefcite#1[]\endcite\ifciteerror\errmessage{Citation error}\fi.} \def\dorefcite#1[#2]#3\endcite{\def\next{#2}% \ifx\next\empty\makecite{#1}{}\else\makecite{#2}{}\fi}% \def\AMS#1{\relax} \catcode`\@=11 \def\proclaim@{\bf}{Examples}{}{\rm}{\proclaim@{\bf}{Examples}{}{\rm}} \catcode`\@=12 \def\Rightarrow{\Rightarrow} \def\mathop{{\cal L}{\it ie}}{\mathop{{\cal L}{\it ie}}} \defG_{\s R}{G_{\s R}} \defcitations{BO,BKO,HO,HM,M1,M2,Mi,T} \titel{Holomorphic Functions on an Algebraic Group \\ Invariant under Zariski-dense Subgroups} \section{Introduction} Let $G$ be a reductive complex linear-algebraic group, $\Gamma$ a subgroup, $\cal O(G)^\Gamma$ the algebra of $\Gamma$-invariant holomorphic functions on $G$. It is known \cite{BO} that $\cal O(G)^\Gamma=\s C$ if $\Gamma$ is dense in $G$ with respect to the algebraic Zariski-topology. We are interested in similar results for non-reductive groups. If $G$ is a complex linear-algebraic group with $G/G'$ non-reductive (where $G'$ denotes the commutator group), then there exists a surjective group morphism $\tau:G\to(\s C,+)$ and $\Gamma=\tau^{-1}(\s Z)$ is a Zariski-dense subgroup of $G$ with $\cal O(G)^\Gamma\simeq\cal O(\s C^*)\ne\s C$. Hence we are led to the question whether the following two properties are equivalent: Let $G$ be a connected complex linear-algebraic group. \item{(i)} $G/G'$ is reductive. \item{(ii)} $\cal O(G)^\Gamma=\s C$ for every Zariski-dense subgroup $\Gamma$. The above argument gave us $(ii)\Rightarrow (i)$ and the result of Barth and Otte \cite{BO} implies the equivalence of $(i)$ and $(ii)$ for $G$ reductive. We will prove that $(i)$ and $(ii)$ are likewise equivalent in the following two cases: \item{a)} $G$ is solvable. \item{b)} The adjoint representation of $S$ on $\mathop{{\cal L}{\it ie}}(U)$ has no zero weight, where $S$ denotes a maximal connected semisimple subgroup of $G$ and $U$ the unipotent radical of $G$. Case $b)$ is equivalent to each of the following two conditions \item{b')} $G/G'$ is reductive and the semisimple elements are dense in $G'$. \item{b'')} $G/G'$ is reductive and $N_{G'}(T)/T$ is finite, where $T$ is a maximal torus in $G'$ and $N_{G'}(T)$ denotes the normalizer of $T$ in $G'$. For instance, if we take $G$ to be a semi-direct product $SL_2(\s C)\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho (\s C^n,+)$ with $\rho:SL_2(\s C)\to GL_n(\s C)$ irreducible, then $G$ fulfills the condition of case $b)$ if and only if $n$ is an even number. The proof for case $a)$ is based on the usual solvable group methods and the structure theorem on holomorphically separable solvmanifolds by Huckleberry and E. Oeljeklaus. The proof for case $b)$ relies on the discussion of semisimple elements of infinite order in such a $\Gamma$. For this reason we conclude the paper with an example of Margulis which implies that $G=SL_2(\s C)\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho (\s C^3,+)$ ($\rho$ irreducible) admits a Zariski-dense discrete subgroup $\Gamma$ such that no element of $\Gamma$ is semisimple. Thus condition $b)$ is really needed in order to find semisimple elements in Zariski-dense subgroups. In consequence, our method does not work for the example of Margulis. However, this only means that we can not prove $\cal O(G)^\Gamma=\s C$ for Margulis' example. We have no knowledge whether there actually exist non-constant holomorphic functions in this case. Finally we discuss invariant meromorphic and plurisubharmonic functions on certain groups. \section{Solvable groups} Here we will discuss solvable groups. First we will develope some auxiliary lemmata. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a connected complex linear-algebraic group such that $G/G'$ is reductive. Then $[G,G']=G'$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} By taking the appropriate quotient, we may assume $[G,G']=\{e\}$. We have to show that this implies $G'=\{e\}$. Now $[G,G']=\{e\}$ means that $G'$ is central, hence $Ad(G)$ factors through $G/G'$. But $G/G'$ is reductive and acts trivially (by conjugation) on both $G/G'$ and $G'$. Due to complete reducibility of representations of reductive groups it follows that $Ad(G)$ is trivial, \ie $G$ is abelian, \ie $G'=\{e\}$. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a connected complex linear-algebraic group, $H\subset G'$ a connected complex Lie subgroup which is normal in $G'$. Then $H$ is algebraic. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $U$ denote the unipotent radical of $G'$. Then $G'/U$ is semisimple. Now $A=(H\cap U)^0$ is algebraic, because every connected complex Lie subgroup of a unipotent group is algebraic. Normality of $H$ implies that $H/(H\cap U)$ is semisimple. Hence $H/A$ is semisimple, too. It follows that $H/A$ is an algebraic subgroup of $G'/A$. Thus $H$ has to be algebraic. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a connected topological group, $H$ a normal subgroup, such that $H\cap G'$ is totally disconnected. Then $H$ is central. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} For each $h\in H$ the set $S_h=\{ghg^{-1}h^{-1}:g\in G\}$ is both totally disconnected and connected and therefore reduces to $\{e\}$. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a connected complex linear-algebraic group, $A\subset G'$ a complex Lie subgroup which is normal in $G$ and Zariski-dense in $G'$. Assume moreover that $[G,G']=G'$. Then $A=G'$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} The connected component $A^0$ of $A$ is algebraic (Lemma 2). Thus $G/A^0$ is again algebraic. Moreover $G'/A^0$ is the commutator group of $G/A^0$. Therefore, by replacing $G$ with $G/A^0$ we may assume that $A$ is totally disconnected. But totally disconnected normal subgroups of connected Lie groups are central (Lemma 3). Since $A$ is Zariski-dense in $G'$, and $[G,G']=G'$, this may occur only for $A=G'=\{e\}$. Thus $A=G'$. \qed \proclaim@{\bf}{Theorem Let $G$ be a connected solvable complex linear-algebraic group, $\Gamma$ a subgroup which is dense in the algebraic Zariski-topology. Assume that $G/G'$ is reductive. Then $\cal O(G)^\Gamma=\s C$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $G/\Gamma \to G/H$ denote the holomorphic reduction, \ie $$H=\{g\in G: f(g)=f(e)\forall f\in\cal O(G)^\Gamma\}.$$ Now $\Gamma\subset H$ normalizes $H^0$. Since $\Gamma$ is Zariski-dense in $G$ and the normalizer of a connected Lie subgroup is necessarily algebraic, it follows that $H^0$ is normal in $G$. Let $A=H^0\cap G'$. This is again a closed normal subgroup in $G$. By a result of Huckleberry and E.\ Oeljeklaus \cite{HO} $H/H^0$ is almost nilpotent (\ie admits a subgroup of finite index which is nilpotent). Let $\Gamma_0$ be a subgroup of finite index in $\Gamma$ with $\Gamma_0/(\Gamma_0\cap H^0)$ nilpotent. By definition this means there exists a number $k$ such that $C^k\Gamma_0\subset H^0$ where $C^k$ denotes the central series. Now $[G,G']=G'$ implies $C^kG=G'$ for all $k\ge 1$. Therefore $C^k\Gamma_0$ is Zariski-dense in $G'$. It follows that $A=H^0\cap G'$ is a closed normal Lie subgroup of $G$ which is Zariski-dense in $G'$. By the preceding lemma it follows that $A=G'$, \ie $G'\subset H$. Now $G/G'$ is assumed to be reductive. Thus the statement of the theorem now follows from the result for reductive groups (\cite{BO}). \qed \section{Groups with many semisimple elements} Here we will prove the following theorem. \proclaim@{\bf}{Theorem Let $G$ be a connected complex linear-algebraic group. Assume that $G/G'$ is reductive and that furthermore one (hence all) of the following equivalent conditions is fulfilled. \item{(1)} $G'$ contains a dense open subset $\Omega$ such that each element in $\Omega$ is semisimple. \item{(2)} For any maximal torus $T$ in $G'$ the quotient $N_{G'}(T)/T$ is finite. \item{(3)} Let $S$ denote a maximal connected semisimple subgroup of $G$ and $U$ the unipotent radical of $G$. Let $\rho:S\to GL(\mathop{{\cal L}{\it ie}} U)$ denote the representation obtained by restriction from the adjoint representation $Ad:G\to GL(\mathop{{\cal L}{\it ie}} G)$. The condition is that all weights of $\rho$ are non-zero. Under these assumptions $\cal O(G)^\Gamma=\s C$ for any Zariski-dense subgroup $\Gamma\subset G$. \endproclaim \proclaim@{\bf}{Examples}{}{\rm} \item{(a)} Let $G$ be a reductive group. Then $G/G'$ is reductive and $G'$ semisimple, hence $N_{G'}(T)/T$ finite for any maximal torus $T\subset G'$. Therefore this theorem is a generalization of the result of Barth and Otte \cite{BO} on redutive groups. \item{(b)} Let $G$ be a parabolic subgroup of a semisimple group $S$. $G/G'$ is obviously reductive. Furthermore a maximal torus $T$ in $G$ is already a maximal torus in $S$. Hence $N_S(T)/T$ is finite. Consequently $N_G(T)/T$ is finite and $G$ fulfills the assumptions of the theorem. \item{(c)} Let $G$ be a semi-direct product of $SL_2(\s C)$ with a unipotent group $U\simeq\s C^n$ induced by an irreducible representation $\xi:SL_2(\s C)\to GL(U)$. Then $G$ fulfills the assumptions of the theorem if and only if $n$ is even. \endproclaim Now we will demonstrate that $(1)$, $(2)$ and $(3)$ are indeed equivalent. The equivalence of $(2)$ and $(3)$ is rather obvious from standard results on algebraic groups. For the equivalence of $(1)$ and $(2)$ we need some elementary facts on semisimple elements in a connected algebraic group $G$. Let $G_s$ denote the set of all semisimple elements in $G$ and $T$ be a maximal torus in $G$. Now $g\in G_s$ iff $g$ is conjugate to an element in $T$. It follows that $G_s$ is the image of the map $\zeta:G\times T\to G_s$ given by $\zeta(g,t)=gtg^{-1}$. In particular $G_s$ is a constructible set. Now a torus contains only countably many algebraic subgroups, hence a generic element $h\in T$ generates a Zariski-dense subgroup of $T$. It follows that for a generic element $h\in T$ the assumption $g\in G$ with $ghg^{-1}\in T$ implies $gTg^{-1}=T$. {}From this it follows that a generic fiber of $\zeta$ has the dimension $\dim N_G(T)$. Therefore the dimension of $G_s=Image(\zeta)$ equals $\dim G-\dim N_G(T)$. Thus we obtained the following lemma, which implies the equivalence of $(1)$ and $(2)$. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a connected linear-algebraic group, $T$ a maximal torus and $G_s$ the set of semisimple elements in $T$. Then $G_s$ is dense in $G$ if and only if $\dim N_G(T)=\dim T$. \endproclaim Next we state some easy consequences of the assumptions of Theorem 2. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be an algebraic group fulfilling the assumptions of Theorem 2 and $\tau:G\to H$ a surjective morphism of algebraic groups. Then $H$ likewise fulfills the assumptions of Theorem 2. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Surjectivity of $\tau$ gives a surjective morphism of algebraic groups from $G/G'$ onto $H/H'$. Therefore $H/H'$ is reductive. The surjectivity of $\tau$ furthermore implies $\tau(G')=H'$. Since morphisms of algebraic groups map semisimple elements to semisimple elements, it follows that $H$ fulfills condition $(1)$. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be an algebraic group fulfilling the assumptions of Theorem 2. Then the center $Z$ of $G$ must be reductive. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Condition $(2)$ implies that $(Z\cap G')^0$ is contained in a maximal torus of $G'$. Since $G/G'$ is reductive, this implies that $Z$ is reductive. \qed The following lemma illuminates why semisimple elements are important for our purposes. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a complex linear-algebraic group, $g\in G$ an element of infinite order, $\Gamma$ the subgroup generated by $g$ and $H$ the Zariski-closure of $\Gamma$. Then $Z=H/\Gamma$ is a Cousin group (hence in particular $\cal O(Z)=\s C$) if $g$ is semisimple; but $Z$ is biholomorphic to some $(\s C^*)^n$ (hence holomorphically separable) if $g$ is not semisimple. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Note that $\bar\Gamma=H$ implies $H=H^0\Gamma$. Hence $H/\Gamma=H^0/(H^0\cap\Gamma)$ is connected. If $g$ is semisimple, the Zariski-closure of $\Gamma $ is reductive and the statement follows from \cite{BO}. If $g$ is not semisimple then $H\simeq(\s C^*)^{n-1}\times\s C$ for some $n\ge 1$ and $g$ is not contained in the maximal torus of $H$. This implies $H/\Gamma\simeq(\s C^*)^n$. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a connected real Lie group, $\Gamma$ a subgroup such that each element $\gamma\in\Gamma$ is of finite order. Then $\Gamma$ is almost abelian and relatively compact in $G$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} If $G$ is abelian, then $G\simeq\s R^k\times(S^1)^n$. In this case $\Gamma\subset(S^1)^n$ and the statement is immediate. Now let us assume that $G$ may be embedded into a complex linear-algebraic group $\tilde G$. Let $H$ denote the (complex-algebraic) Zariski-closure of $\Gamma$ in $\tilde G$. By the theorem of Tits \cite{T} $\Gamma$ is almost solvable, hence $H^0$ is solvable. Now the commutator group of $H^0$ is unipotent and therefore contains no non-trivial element of finite order. Hence $\Gamma\cap H^0$ is abelian, which completes the proof for this case, since we discussed already the abelian case. Finally let us discuss the general case. By the above considerations $Ad(\Gamma_0)$ is contained in an abelian connected compact subgroup $K$ of $Ad(G)$ for some subgroup $\Gamma_0$ of finite index in $\Gamma$. Now $N=(Ad)^{-1}(K)$ is a central extension $1\to Z\to N\to K\to 1$. (where $Z$ is the center of $G$). But complete reducibility of the representations of compact groups implies that this sequence splits on the Lie algebra level. Hence $N$ is abelian and we can complete the proof as before. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a complex linear-algebraic group, $\Gamma$ a Zariski-dense subgroup. Then $\Gamma$ contains a finitely generated subgroup $\Gamma_0$ such that the Zariski-closure of $\Gamma_0$ contains $G'$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Consider all finitely generated subgroups of $\Gamma$ and their Zariski-closure in $G$. There is one such group $\Gamma_0$ for which the dimension of the Zariski-closure $A$ is maximal. Clearly $A$ must contain the connected component of the Zariski-closure for any finitely generated subgroup of $\Gamma$. This implies that $A^0$ is normal in $G$. Furthermore maximality implies that the group $\Gamma/A^0$ contains no element of infinite order. Hence $\Gamma/A^0$ is almost abelian, which implies $G'\subset A$ ($\Gamma/A^0$ is Zariski-dense in $G/A^0$). \qed Caveat: There is no hope for $A=G$, \eg take $G=\s C^*$ and let $\Gamma$ denote the subgroup which consists of all roots of unity. Theorem 2 follows by induction on $dim(G)$ using the following lemma. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a positive-dimensional complex linear-algebraic group fulfilling the assumptions of Theorem 2, $\Gamma$ a Zariski-dense subgroup. Then there exists a positive-dimensional normal algebraic subgroup $A$ with $\cal O(G)^\Gamma\subset\cal O(G)^A$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} If $G$ is abelian, the assumptions imply that $G$ is reductive and $\cal O(G)^\Gamma=\s C$. Otherwise let $H=\{g:f(g)=f(e)\forall f\in\cal O(G)^\Gamma\}$. Now $\Gamma\subset H$, hence $\Gamma$ normalizes $H^0$. The normalizer of a connected Lie subgroup is algebraic, thus $H^0$ is a normal subgroup of $G$. It follows that $(H\cap G')^0$ is a normal algebraic subgroup (Lemma 2). This completes the proof unless $H\cap G'$ is discrete. This leaves the case where $(H\cap G')$ is discrete. Then $H^0$ is contained in the center $Z$ of $G$ (Lemma 3). Let $A$ denote the Zariski-closure of $H^0$. The center is reductive (Lemma 7). It follows that for each $Z$-orbit every $H^0$-invariant functions is already $A$-invariant. Therefore we can restrict to the case where $H$ is discrete. Now $\Gamma$ is discrete and contains a subgroup $\Gamma_0$ which is finitely generated and whose Zariski-closure contains $G'$. By a theorem of Selberg $\Gamma_0$ contains a subgroup of finite index $\Gamma_1$ which is torsion-free. Now let $\Gamma_2=\Gamma_1\cap G'$. Then being Zariski-dense, $\Gamma_2$ must contain a semisimple element of infinite order. Using Lemma 8, this yields a contradiction to the assumption that $H$ is discrete. \qed \section{An example} At a first glance, it seems to be obvious that a Zariski-dense subgroup should contain enough elements of infinite order to generate a subgroup which is still Zariski-dense. However, one has to careful. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G=\s C^*\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}\s C$ with group law $(\lambda,z)\cdot(\mu,w)=(\lambda\mu, z+\lambda w)$ and $\Gamma$ the subgroup generated by the elements $a_n=(e^{2\pi i/n},0)$ ($n\in\s N)$ and $a_0=(1,1)$. Then $\gamma\in G'=\{(1,x):x\in \s C\}$ for any element $\gamma\in\Gamma$ of infinite order, although $\Gamma$ is Zariski-dense in $G$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} It is clear that $\Gamma$ is Zariski-dense in $G$. The other assertion follows from the fact that any element in $G$ is either unipotent or semisimple. Hence every element $g\in G\setminus G'$ is conjugate to an element in $\s C^*\times\{0\}$. \qed \section{Margulis' example} We will use an example of Margulis to demonstrate the following. \proclaim@{\bf}{Proposition There exists a discrete Zariski-dense subgroup $\Gamma$ in $G=SL_2(\s C)\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho(\s C^3,+)$ with $\rho$ irreducible such that $\Gamma$ contains no semisimple element. \endproclaim Thus the condition $G/G'$ reductive is not sufficient to guarantee the existence of semisimple elements in Zariski-dense subgroups. Margulis \cite{M1,M2} constructed his example in order to prove that there exist free non-commutative groups acting on $\s R^n$ properly discontinuous and by affin-linear transformations, thereby contradicting a conjecture of Milnor \cite{Mi}. We will now start with the description of Margulis' example. Let $B$ denote the bilinear form on $\s R^3$ given by $B(x_1,x_2,x_3)=x_1^2+x_2^2-x_3^2$, $W=\{x\in \s R^3:B(x,x)=0\}$ the zero cone and $W^+=\{x\in W:x_3>0\}$ the positive part. Let $S=\{x\in W^+:\abs{x}=1\}$. Let $H$ be the connected component of the isometry group $O(2,1)$ of $B$. (As a Lie group $H$ is isomorphic to $PSL_2(\s R)$.) Let $G_{\s R}=H\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}(\s R^3,+)$ the group of affine-linear transformations on $\s R^3$ whose linear part is in $H$. The following is easy to verify: Let $x^+,x^-$ to different vectors in $S$. Then there exists a unique vector $x^0$ such that $B(x^0,x^0)=1$, $B(x^+,x^0)=0=B(x^-,x^0)$ and $x^0,x^-,x^+$ form a positively oriented basis of the vector space $\s R^3$. Furthermore for any $\lambda\in ]0,1[$ there is an element $g\in H$ (depending on $x^+,x^-\in S$ and $\lambda$) defined as follows: $$ g:ax^0+bx^-+cx^+ \mapsto ax^0+{b\over\lambda}x^-+\lambda c x^+$$ Conversely any non-trivial diagonalizable element $g\in H$ is given in such a way and $x^+$, $x^-$ and $\lambda$ are uniquely determined by $g$. The result of Margulis is the following: Let $x^+,x^-,\tilde x^+,\tilde x^-$ four different points in $S$, $\lambda,\tilde\lambda\in]0,1[$, $v,\tilde v\in\s R^3$ such that $v,x^-,x^+$ resp. $\tilde v,\tilde x^-,\tilde x^+$ forms a positively oriented basis of $\s R^3$. Let $h,\tilde h\in H$ be the elements corresponding to $x^-,x^+,\lambda$ resp. $\tilde x^-,\tilde x^+,\tilde\lambda$ and $g,\tilde g\in G_{\s R}=H\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}\s R^3$ given by $g=(h,v)$, $\tilde g=(\tilde h,\tilde v)$. Then there exists a number $N=N(g,\tilde g)$ such that the elements $g^N$, $\tilde g^N$ generate a (non-commutative) free discrete subgroup $\Gamma\subsetG_{\s R}$ such that the action on $\s R^3$ is properly discontinuous and free. Now an element $g\inG_{\s R}$ is conjugate to an element in $H$ if and only if $g(w)=w$ for some $w\in\s R^3$. Hence no element in $\Gamma$ is conjugate to an element in $H$. In particular no element in semisimple. Furthermore it is clear that $\Gamma$ is Zariski-dense in the complexification $G=SL_2(\s C)\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}\s C^3$ of $G_{\s R}$. \section{Meromorphic functions} \proclaim@{\bf}{Proposition Let $G$ be a connected complex linear-algebraic group with $G=G'$ and an open subset $\Omega$ such that each element in $\Omega$ is semisimple. Let $\Gamma$ be a Zariski-dense subgroup. Then any $\Gamma$-invariant plurisubharmonic or meromorphic function is constant and there exist no $\Gamma$-invariant hypersurface. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} We may assume that $\Gamma$ is closed (in the Hausdorff topology). Since $G=G'$, it follows that $H^0$ is a normal algebraic subgroup for each Zariski-dense subgroup $H$. Therefore we may assume that $\Gamma$ is discrete and furthermore it suffices (by induction on $\dim(G)$ to demonstrate that the functions resp. hypersurfaces are invariant under a positive-dimensional subgroup. Now $G=G'$ implies that $\Gamma$ admits a finitely generated subgroup $\Gamma_0$ which is still Zariski-dense. By the theorem of Selberg $\Gamma_0$ admits a subgroup of finite index $\Gamma_1$ which is torsion-free. Thus $\Gamma_1$ contains a semisimple element of infinite order $\gamma$ which generates a subgroup $I$ whose Zariski-closure $\bar I$ is a torus. $G=G'$ implies that this torus is contained in a connected semisimple subgroup $S$ of $G$. Now known results on subgroups in semisimple groups \cite{HM}\cite{BKO} imply that the functions resp. hypersurfaces are invariant under $\bar I$, which is positive-dimensional. \qed For this result it is essential to require $G=G'$ and not only $G/G'$ reductive. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G=\s C^*\times\s C^*$ and $\Gamma\simeq\s Z$ a (possibly Zariski-dense) discrete subgroup. Then $G$ admits $\Gamma$-invariant non-constant plurisubharmonic and meromorphic functions. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} $G/\Gamma\simeq\s C^2/\Lambda$ with $\Lambda\simeq\s Z^3$. Let $V=<\Lambda>_{\s R}$ the real subvector space of $\s C^2$ spanned by $\Lambda$ and $t:\s C^2\to\s C^2/V\simeq\s R$ a $\s R$-linear map. Then $t^2$ yields a $\Gamma$-invariant plurisubharmonic function on $G$. Let $L=V\cap iV$ and $\gamma\in\Lambda\setminus L$. Let $H=<\gamma>_{\s C}$. Then $H\ne L$, hence $H+L=\s C^2$. It follows that the $H$-orbits in $G/\Gamma$ are closed and induce a fibration $G/\Gamma\to G/H\Gamma$ onto a one-dimensional torus. One-dimensional tori are projective and therefore admit non-constant meromorphic functions. \qed \medskip\sectno=-100 \section{References \item{BO} Barth, W.; Otte, M.: Invariante Holomorphe Funktionen auf reduktiven Liegruppen. \sl Math. Ann. \bf 201\rm, 91--112 (1973) \item{BKO} Berteloot, F.; Oeljeklaus, K.: Invariant Plurisubharmonic Functions and Hypersurfaces on Semisimple Complex Lie Groups. \sl Math. Ann. \bf 281\rm, 513--530 (1988) \item{HO} Huckleberry, A.T.; Oeljeklaus, E.: On holomorphically separable complex solvmanifolds. \sl Ann. Inst. Fourier \bf XXXVI 3\rm, 57--65 (1986) \item{HM} Huckleberry, A.T.; Margulis, G.A.: Invariant analytic hypersurfaces. \sl Invent. Math. \bf 71\rm, 235--240 (1983) \item{M1} Margulis, G.A.: Free totally discontinuous groups of affine transformations. \sl Soviet Math. Doklady \bf 28\rm, 435--439 (1983) \item{M2} Margulis, G.A.: Complete affine locally flat manifolds with free fundamental group. \sl Zapiski Nau\v cn. Sem. Leningrad Otd. Mat. Inst. Steklov \bf 134\rm, 190-205 (1984) [in Russian] \item{Mi} Milnor, J.: On fundamental groups of completely affinely flat manifolds. \sl Adv. Math. \bf 25\rm, 178--187 (1977) \item{T} Tits, J.: Free subgroups in linear groups. \sl J. Algebra \bf 20\rm, 250--270 (1972) \endarticle \\

9401

alg-geom/9401009

en

https://arxiv.org/abs/alg-geom/9401009

alg-geom/9401009

Shelly Cook

Michele Cook

The Connectedness of Space Curve Invariants

18 pages, Latex v2.09

It is a result of Gruson and Peskine that the invariants of a set points in $\ptwo$ in general position are connected. Associated to a space curve there are sequences of invariants which generalize the invariants of points in $\ptwo$. The main result of this paper is to show that the invariants of reduced, irreducible, non-degenerate curves in $\pthree$ also satisfy a connectedness property. This result greatly restricts the kinds of Borel-fixed monomial ideals which can occur as generic initial ideals of such curves and thus gives us more control over their Hilbert functions.

alg-geom

\section*{Introduction} Let $ S = \C[x_1, \dots , x_{n+1}] $ be the set of polynomials in n+1 variables over {\bf C} (usually corresponding to the homogeneous coordinate ring of $\pn$). Let $\succ$ be the reverse lexicographical order on the monomials of $S$. Given a homogeneous ideal $I$ in $S$, $\gi$, the {\bf generic initial ideal} of $I$, is the Borel-fixed monomial ideal associated to $I$. (For details see [B], [BM], [BS] or [Gr].) The generic initial ideal reflects much of the structure of the original ideal, for example, it has the same Hilbert function and the same regularity. Also, many geometric problems can be reduced to combinatorial ones which may be phrased in terms of generic initial ideals. Thus, we would like to have some rules governing the kinds of Borel-fixed monomial ideals which can occur from geometry. The first set of rules are supplied by a result of Gruson and Peskine ([GP]): \begin{theorem}[Gruson-Peskine] Let $\G \subset \ptwo$ be a set of points in general position, with generic initial ideal $\giG = (x_1^s, \ x_1^{s-1}x_2^{\lm_{s-1}}, \ \dots , \ x_1x_2^{\lm_{1}},\ x_2^{\lm_{0}}).$ Then the {\bf invariants}, $\{\lm_{i}\}_{i=0}^{s-1}$, satisfy $$ \lm_{i+1} + 2 \geq \lm_i \geq \lm_{i+1} + 1 \ \ \text{for all} \ i < s-1 $$ and we say that the invariants of $\G$ are {\bf connected}. \end{theorem} Associated to a space curve there are invariants which generalize the invariants of points in $\ptwo$ (see below). The main result of this paper is to show that the invariants of a reduced, irreducible, non-degenerate curves in $\pthree$ also satisfy a connectedness property: Let $C$ be a reduced, irreducible, non-degenerate curve in $\pthree$, As $I_C$ is saturated, the generators of $\gi$ will be of the form $x_1^{i_1}x_2^{i_2}x_3^{f(i_1,i_2)}$. \newline Let $$ s_k = \text{min}\{ i\ | \ f(i,0) \leq k \}, $$ $$ \mu_i(k) = \text{min}\{j\ |\ f(i,j) \leq k \} \ \text{for} \ 0 \leq i \leq s_k - 1. $$ We call $\{\mu_i(k)\}$ the {\bf invariants} of $C$. For $k >> 0$, $\mu_i(k) = \lm_i$, where the $\{\lm_i \}$ are the invariants of a generic hyperplane section, $\G = H \cap C$, of C. The main result of this paper is to prove: \begin{theorem}[The connectness of curve invariants] \label{theorem:connect} Let $C$ be a reduced, irreducible, non-degenerate curve in $\pthree$, then the invariants, $\{\mu_i(j)\}$, of $C$ are such that $$ \mu_{i+1}(k)+2 \geq \mu_i(k) \geq \mu_{i+1}(k)+1 \ \text{for}\ 0 \leq i < s_k-1, $$ and we say that $\{ \mu_i(k) \}$ is {\bf connected}. Furthermore, if $s_k < s_0$, then $\mu_{s_k-1}(k) \leq 2$. \end{theorem} This result greatly restricts the kind of Borel-fixed monomial ideal which can occur as the generic initial ideal of such curves and thus gives us quite a lot of control over their Hilbert functions. The paper is broken into parts as follows: In Section 1, we will prove the Connectedness Theorem. The proof incorporates the ideas used in the proof of the Gruson, Peskine result ([GP], [EP]) and a more differential approach due to Strano ([S]). In Section 2, we will give some further restrictions on the generic initial ideal of a reduced, irreducible, non-degenerate curve in $\pthree$. We will generalize a result of Strano on the effect of certain generators of the generic initial ideal of the curve on the syzygies of the hyperplane section ideal to a larger group of ideals. We will also generalize another result of Strano on complete intersections to curves whose hyperplane sections have invariants of which the first few are like those of a complete intersection. I believe this last result will lead to some interesting results on the gaps of Halphen ([E]). {\it Acknowledgement.} This paper is taken from my dissertation and I would like to take this opportunity to thank my advisor Mark Green, for his insight and patience. \section{Proof of the Connectedness Theorem} \subsection{A pictorial description of monomial ideals} We will first give an description, due to M. Green ([Gr]), of how an ideal $I$ whose generators are monomials in three variables may be described pictorially. Hopefully, this desciption will make subsequent explanations clearer. \newline First, draw a triangle corresponding to the monomials $x_1^ix_2^j$, where $i + j = n$, $0 \leq n \leq n_0$. Let $f(i,j) = \text{min}\{k \ | \ x_1^ix_2^jx_3^k \in I \}$. For each $i,j$, if $f(i, j) = \infty $ (i.e. $x_1^ix_2^jx_3^k \notin I$ for all $k \geq 0$) put a circle in the (i, j) position, if $ 0 < f(i, j) < \infty$ put the number $f(i, j)$ in the (i, j) position, and if $f(i, j) = 0$ put an X in the (i, j) position. \newline Thus the triangle \vspace{5mm} \begin{picture}(270,78)(0,0) \put(150,75){\line(3,-5){42}} \put(150,75){\line(-3,-5){42}} \put(150,5){\line(1,0){42}} \put(150,5){\line(-1,0){42}} \put(150,61){\circle{12}} \put(142.5,49){\circle{12}} \put(157.5,49){\circle{12}} \put(135,37){\circle{12}} \put(150,37){\circle{12}} \put(165,37){\circle{12}} \put(127.5,25){\circle{12}} \put(127.5,25){\makebox(0,0){1}} \put(142.5,25){\circle{12}} \put(142.5,25){\makebox(0,0){1}} \put(157.5,25){\circle{12}} \put(157.5,25){\makebox(0,0){3}} \put(172.5,25){\circle{12}} \put(120,13){\makebox(0,0){X}} \put(135,13){\makebox(0,0){X}} \put(150,13){\makebox(0,0){X}} \put(165,13){\makebox(0,0){X}} \put(180,13){\circle{12}} \put(180,13){\makebox(0,0){1}} \end{picture} corresponds to the ideal $(x_1^3x_3, x_1^2x_2x_3, x_1x_2^2x_3^3, x_1^4, x_1^3x_2, x_1^2x_2^2, x_1x_2^3, x_2^4x_3, x_2^5)$. \newline {\bf Note:} For (i, j) not in the picture, it is assumed that $x_1^ix_2^j \in I$. \newline {\bf Remark:} It is the Borel-fixedness that ensures the step like look. \newline In this example the invariants are as follows: $$ \begin{array}{llll} \mu_0(0) = 5 & \mu_1(0) = 3 & \mu_2(0) = 2 & \mu_3(0) = 1\\ \mu_0(1) = 4 & \mu_1(1) = 3 & \mu_2(1) = 1 \\ \mu_0(2) = 4 & \mu_1(2) = 3 & \mu_2(2) = 1 \\ \mu_0(3) = 4 & \mu_1(3) = 2 & \mu_2(3) = 1 \end{array}$$ and $\mu_i(k) = \mu_i(3)$ for $k \geq 3$. {\bf Note:} The invariants of this monomial ideal are connected. To prove Theorem 1, we will first show in section 1.2, that if there is an $i$ and $j$, with $0 \leq i < s_j-1$, such that $\mu_{i+1}(j) + 2 < \mu_i(j)$, then for generic hyperplanes $H, H'$, the ideal $J = (I|_H:(H'\cap H)^j)$ can be ``split'' so that there exists a homogeneous polynomial $X$ of degree $i+1$ such that $\text{gin}(X)\cap\text{gin}(J) = \text{gin}(X \cap J)$. We will then show in section 1.3, that if $I$ arises as the ideal of a reduced, irreducible, non-degenerate curve $C \in \pthree$, such an $X$ cannot exist. \subsection{Splitting a non-connected ideal} {\bf Definition.} An {\bf elementary move} $e_k$ for $ 1 \leq k \leq n$ is defined by: $$ e_k(x^J) = x^{\hat{J}},$$ where $\hat{J} = (j_1, \dots j_{k-1},j_k + 1, j_{k+1} - 1, j_{k+2}, \dots, j_{n+1})$ and where we adopt the convention that $x^J = 0 $ if $j_m < 0$ for some $m$. One can show that a monomial ideal $I$, is Borel-fixed if and only if for all $x^J \in I$ and for every elementary move $e_k$, $e_k(x^J) \in I$. \begin{theorem}[syzygy configuration] \label{theorem:syzcon} Let I be a Borel-fixed monomial ideal with generators $x^{J_1}, \ \dots , x^{J_N}$. Then the first syzygies of I is generated by $$\{ x_i \otimes x^{J_j} - x^{L_{ij}} \otimes x^{J_{l_{ij}}} \ | \ 1 \leq j \leq N, \ 0 < i < max(J_j), \ \text{min}(L_{ij}) \geq \text{max}(J_{l_{ij}})\}, $$ where $\text{max}(J) = \text{max}\{ i \ |\ j_i > 0 \}$ and $\text{min}(J) = \text{min}\{i | j_i > 0 \}$. \end{theorem} The proof of this is due to Green and may be found in [Gr]. \begin{theorem}\label{theorem:newgen} Let $I$ be a homogeneous ideal generated in degree $\leq n$, with generators $x^{J_1}, \ \dots , x^{J_N}$ of $\gi$ in degree $\leq n$. Then any generator $P$ of $\text{gin}(I)$ in degree $n+1$ is such that $ P \prec x_ix^{J_j}$, for some $J_j$ such that $|J_j| = n$ and $ i < \text{max}(J_j)$. \end{theorem} {\bf Proof.} We will first use the syzygies of $(x^{J_1}, \ \dots , x^{J_N})$ to obtain some new generators of $\text{gin}(I)_{n+1}$ which satisfy the conditions stated above. \newline $(x^{J_1}, \ \dots , x^{J_N})$ is a Borel fixed monomial ideal so by Theorem~\ref{theorem:syzcon}, the syzygies among the $x^{J_j} $ are generated by syzygies of the form $$ \{\ x_k \otimes x^{J_j}- x^{L_{kj}} \otimes x^{J_{l_{kj}}} \ | \ 1 \leq j \leq N, \ 1 \leq k < \text{max}(J_j)\}.$$ By Galligo's Theorem ([Ga]) we may assume, after a possible change of basis, that $\gi = \text{in}(I)$. Let $g_i \in I$ be such that $\text{in}(g_i) = x^{J_i}$ and assume the $g_i$ are monic. Given a syzygy as above, let $h_1 = x_kg_j-x^{L_{kj}}g_{l_{kj}}$, then $\text{in}(h_1) \prec x_kx^{J_j}$. \newline Given $h_i$, if $\text{in}(h_i)= x^{K_{i+1}}x^{J_{j_{i+1}}}$, let $h_{i+1} = h_i - a_{i+1}x^{K_{i+1}}g_{j_{i+1}}$, where $a_{i+1}$ is the leading coefficient of $h_i$, so that $\text{in}(h_{i+1}) \prec \text{in}(h_i) \prec x_kx^{J_j}$. \newline This process must terminate, so for $i$ sufficiently large we get either $h_i = 0$, (in particular this will occur if deg$h_i \leq n$) or $x^{J_j} $ does not divide $\text{in}(h_i)$ for all $1 \leq j \leq N$, in which case $\text{in}(h_i)$ is a new generator of $\gi$ satisfying the conditions stated in the theorem. \newline Now let $P = \text{in}(h) $ be a generator of $\text{gin}(I)_{n+1}$, $h = \sum f_ig_i. $ \newline Let $i_0$ be such that, $\text{in}(f_{i_0}g_{i_0})$ is maximal, and of the maximal $\text{in}(f_{i_0}g_{i_0})$, $g_{i_0}$ has maximal degree, and among those $g_{i_0}$ of the same maximal degree $\text{in}(g_{i_0})$ is minimal. \newline As $P \neq \text{in}(f_{i_0}g_{i_0})$, there exists $i_1$ such that $\text{in}(f_{i_1}g_{i_1}) = \text{in}(f_{i_0}g_{i_0})$. We have picked $i_0$ in such a way that either deg$g_{i_0} > $ deg$g_{i_1}$, or deg$g_{i_0} = $ deg$g_{i_1}$ and $\text{in}(g_{i_0}) \prec \text{in}(g_{i_1})$. \newline We now want to show that $x_i | \text{in}(f_{i_0})$ for some $ i < \text{max}(J_{i_0})$, where $\text{in}(g_{i_0}) = x^{J_{i_0}}.$ {\bf Claim.} Let $x^A, x^B$ be generating monomials of a Borel-fixed monomial ideal, $I$, such that $|A| > |B|$ and $x^Mx^A = x^Nx^B $ for some monomials $x^M$ and $x^N$. Then there exists $x_i | x^M$ such that $i < \text{max}(A)$. {\bf Proof of Claim.} Let $A = (a_1, \dots ,a_s,0, \dots ,0)$, and $B = (b_1, \dots ,b_s,b_{s+1}, \dots ,b_{n+1})$, assume that $s = \text{max}(A)$. Suppose $b_i \leq a_i$ for all $i \leq s$, then we may apply elementary moves to $B$ to get $\hat B $ such that $x^{\hat{B}} \in I$ with $\hat{b_i} \leq a_i$ for all $i$, and as $|\hat{B}| = |B| < |A| $, this would imply $x^A$ cannot be a generator. Therefore there exists $b_i > a_i$ for some $i \leq s$, and if $i < s$ we are done. If $b_i \leq a_i$ for all $i < s$ but $b_s > a_s$ we may again apply elementary moves to $B$ and get the same conclusion. Therefore there exists $b_i > a_i $ for $i < s$. \vspace{1 mm} If deg$g_{i_0} = $ deg$g_{i_1}$ and $\text{in}(g_{i_0}) \prec \text{in}(g_{i_1})$, let $\text{in}(g_{i_0}) = x^A$, $\text{in}(g_{i_1}) = x^B$ where $A = (a_1, a_2, \dots a_{n+1})$ and $B = (b_1, b_2, \dots b_{n+1})$. Then there exists $s$ such that $a_k = b_k $ for all $k > s$ and $a_s > b_s$. As the degrees are the same, there must exist $a_i < b_i$ for some $i < s$ and hence $x_i | \text{in}(f_{i_0})$ for some $ i <s \leq \text{max}(J_{i_0})$. \newline Thus, in either case we have $x_i | \text{in}(f_{i_0})$ with $ i < \text{max}(J_{i_0})$. Consider the syzygy $$ x_i \otimes x^{J_{i_0}}- x^{L_{ii_0}} \otimes x^{J_{l_{ii_0}}}$$ and let $h^*$ be the element of $I$ obtained from this syzygy, as above. \newline $h^* = x_ig_{i_0} - x^{L_{ii_0}}g_{l_{ii_0}} - \sum a_ix^{K_i}g_i$ where $\text{in}(x^{K_i}g_i) \prec \text{in}(x_ig_{i_0})$, and by Theorem~\ref{theorem:syzcon} either $|J_{i_0}| > |J_{l_{ii_0}}|$ or $ \text{in}(g_{l_{ii_0}}) \succ \text{in}(g_{i_0})$. Let $h_1 = h - e_{i_0} h^* $, where $e_{i_0} = a_{i_0}\text{in}(f_{i_0})/x_i$ and $a_{i_0}$ is the leading coefficient of $f_{i_0}$. \newline If deg$e_{i_0} = 0$, $\text{in}(h_1) \preceq \text{max}\{\text{in}(h) ,\text{in}(h^*)\}$. If in$(h) = $ in$(h^*)$, we are done as in$h^* \prec x_ix^{J_{i_0}}$. Therefore we may assume in$(h) \neq $ in$(h^*)$, and so either in$(h_1) = $ in$(h^*)$ or in$(h_1) = $ in$(h)$. In the first case we get $P = \text{in}(h) \prec \text{in}(h^*) \prec x_ix^{J_{i_0}}$, and we are done. \newline If deg$e_{i_0} \geq 1$, $h^* = 0$ and $\text{in}(h_1) = \text{in}(h) = P$. \newline Therefore we are left with the case in$(h_1) = P$, however in adding the multiple of $h^*$, we only added terms which either have initial term less than the term $\text{in}(f_{i_0}g_{i_0})$, or if the term has the same initial term as the $g_{i_0}$ term, corresponding to the $x^{L_{ii_0}}g_{l_{ii_0}}$ term, then either $g_{l_{ii_0}}$ of lesser degree, or $\text{in}(g_{l_{ii_0}}) \succ \text{in}(g_{i_0}) $. Therefore we may proceed by induction and get the required result.\qed Now suppose that $\gi$ is such that there exists $i$ and $j$ such that $$ \mu_{i+1}(j)+2 < \mu_i(j), \ \text{where}\ 0 \leq i < s_j-1. $$ Consider the ideal $J = (I|_H:(H'\cap H)^j)$ for generic hyperplanes $H, H'$. Then $\text{gin}(J) = (\text{gin}(I):x_3^j)$ and $J$ has invariants $\nu_i(k) = \mu_i(k+j)$, and in particular $$ \nu_{i+1}(0)+2 < \nu_i(0) . $$ For example, pictorially we could be in the following situation: \vspace{.5cm} \begin{picture}(300,78)(0,0) \put(0,68){$\text{gin}(I) = $} \put(60,75){\line(3,-5){42}} \put(60,75){\line(-3,-5){42}} \put(60,5){\line(1,0){42}} \put(60,5){\line(-1,0){42}} \put(60,61){\circle{12}} \put(52.5,49){\circle{12}} \put(67.5,49){\circle{12}} \put(45,37){\circle{12}} \put(45,37){\makebox(0,0){4}} \put(60,37){\circle{12}} \put(75,37){\circle{12}} \put(37.5,25){\circle{12}} \put(37.5,25){\makebox(0,0){1}} \put(52.5,25){\circle{12}} \put(52.5,25){\makebox(0,0){1}} \put(67.5,25){\circle{12}} \put(67.5,25){\makebox(0,0){2}} \put(82.5,25){\circle{12}} \put(30,13){\makebox(0,0){X}} \put(45,13){\makebox(0,0){X}} \put(60,13){\makebox(0,0){X}} \put(75,13){\makebox(0,0){X}} \put(90,13){\circle{12}} \put(90,13){\makebox(0,0){3}} \put(120,68){$\text{gin}(I|_H:(H\cap H')^2) = $} \put(250,75){\line(3,-5){42}} \put(250,75){\line(-3,-5){42}} \put(250,5){\line(1,0){42}} \put(250,5){\line(-1,0){42}} \put(250,61){\circle{12}} \put(242.5,49){\circle{12}} \put(257.5,49){\circle{12}} \put(235,37){\circle{12}} \put(235,37){\makebox(0,0){2}} \put(250,37){\circle{12}} \put(265,37){\circle{12}} \put(227.5,25){\makebox(0,0){X}} \put(242.5,25){\makebox(0,0){X}} \put(257.5,25){\makebox(0,0){X}} \put(272.5,25){\circle{12}} \put(220,13){\makebox(0,0){X}} \put(235,13){\makebox(0,0){X}} \put(250,13){\makebox(0,0){X}} \put(265,13){\makebox(0,0){X}} \put(280,13){\circle{12}} \put(280,13){\makebox(0,0){1}} \end{picture} Let $K$ be the ideal generated by elements of degree $\leq i + \nu_{i+1}(0) +2$ in $J$. We want to show that there exists an ideal $K \subseteq \hat{K} \subseteq J$ such that $\text{gin}(\hat{K}) = (x_1^{i+1}) \cap \text{gin}(J)$. \begin{corollary} \label{corollary:ngcor} All elements of $\text{gin}(K) $ are divisible by $x_1^{i+1}$. \end{corollary} {\bf Proof.} Let $x_1^ax_2^bx_3^c \in \text{gin}(K)_d$ for $d \leq i + \nu_{i+1}(0) + 2$. If $a \leq i$ , then by Borel-fixedness $x_1^ix_2^{\nu_{i+1}(0) + 2} \in \text{gin}(K) \subset \text{gin}(J)$, but $x_1^ix_2^{\nu_i(0)} $ is a generator of $\text{gin}(J) $ and so $\nu_{i+1}(0) + 2 \geq \nu_i(0)$. But $ \nu_i(0) > \nu_{i+1}(0)+2 $. \newline Suppose by induction, all elements of $\text{gin}(K)_d $ are divisible by $x_1^{i+1}$ for $d \geq i+\nu_{i+1}(0) + 2$. {\bf Claim.} If $d \geq i+\nu_{i+1}(0) +2 $, then any generator of $\text{gin}(K)_d$ has an $x_3$ term. {\bf Proof of claim.} If $x_1^ax_2^bx_3^c $ a gen\-er\-ator of $\text{gin}(K)_d$, then $a \geq i+1$. Let $a = i+1+j$, we have $x_1^{i+1+j}x_2^{\nu_{i+1+j}(0)} \in \text{gin}(J)$, and $i+1+j + \nu_{i+1+j}(0) \leq i+1 + \nu_{i+1}(0) < d$, hence $b < \nu_{i+1+j}(0)$ and $c > 0$. \vspace{1 mm} Let $P = x_1^ax_2^bx_3^c $ be a generator of $\text{gin}(K)_{d+1}$. If $a \leq i$, then by Borel-fixedness $x_1^ix_2^{d+1-i} \in \text{gin}(K)_{d+1}$, and as $x_1^{i+1} $ divides all elements of degree $\leq d$, $x_1^ix_2^{d+1-i}$ is a generator of $\text{gin}(K)_{d+1}$, and hence by Theorem~\ref{theorem:newgen} $ x_1^ix_2^{d+1-i} \prec x_kx^{J}$ for $x^{J}$ some generator of $\text{gin}(K)_d$ and $k < \text{max}(J)$. However, by the claim $x^{J}$ has an $x_3$ term and thus $x_1^ix_2^{d+1-i} \succ x_kx^{J}$, which is a contradiction. \qed \begin{lemma}\label{lemma:lem} Let $K$ be an ideal in $\C[x_1,x_2,x_3]$ such that $\text{gin}(K) \subseteq (x_1^k)$, with $k \geq 1$ and $k$ maximal. Then, after a possible change of basis, there exists a homogeneous polynomial $X$ such that $in(X) = x_1^k$ and $K = XK'$. \end{lemma} We will need to use the following Proposition to prove Lemma~\ref{lemma:lem} and later, to prove Theorem~\ref{theorem:connect} \begin{proposition}\label{proposition:fam} Let $\{ X_H \}$ be a family of homogeneous polynomials in the polynomial ring $S = \C[x_1, \dots , x_{n+1}]$, (with $n \geq 2$) parametrized by generic hyperplane sections $H= \sum t_ix_i$ in such a way that the coefficients of $X_H$ are differentiable functions in the $t_i$. Suppose that, if we differentiate with respect to $H = \sum t_ix_i$ we get $$x_j\frac{\p X_H}{\p t_i} - x_i\frac{\p X_H}{\p t_j} \in (X_H) \ \text{mod}(H).$$ Then $X_H$ is a constant up to a multiple of $H$, for generic $H$. \end{proposition} {\bf Proof (Green).} Let $Y = X_H|_H$, as $(x_j\frac{\p X_H}{\p t_i} - x_i\frac{\p X_H}{\p t_j})|_H \in (Y)$, we may let $(x_j\frac{\p X_H}{\p t_i} - x_i\frac{\p X_H}{\p t_j})|_H = l_{ij}Y$. $$x_k(x_j\frac{\p X_H}{\p t_i} - x_i\frac{\p X_H}{\p t_j}) - x_j(x_k\frac{\p X_H}{\p t_i} - x_i\frac{\p X_H}{\p t_k}) + x_i(x_k\frac{\p X_H}{\p t_j} - x_j\frac{\p X_H}{\p t_k}) = 0, $$ thus $$ (x_kl_{ij} - x_jl_{ik} + x_il_{jk})Y = 0. $$ Assuming $Y \neq 0$ generically (otherwise we are done), we get $$ x_kl_{ij} - x_jl_{ik} + x_il_{jk} = 0, $$ and hence $$ l_{ij} = \al x_j + \be x_i, \ l_{ik} = \al x_k + \ga x_i, \ l_{jk} = -\be x_k + \ga x_j, $$ for some $\al$, $\be$ and $\ga$. \newline Therefore $$(x_j(\frac{\p X_H}{\p t_i} - \al X_H)- x_i(\frac{\p X_H}{\p t_j} + \be X_H))|_H = 0, $$ and we have $$ \frac{\p X_H}{\p t_i} - \al X_H = x_iU \ \text{mod}(H), \ \frac{\p X_H}{\p t_j} + \be X_H = x_jU \ \text{mod}(H)$$ and similarly $$ \frac{\p X_H}{\p t_k} + \ga X_H = x_kU \ \text{mod}(H). $$ Letting $\al = \al_i$, $\be = -\al_j$, $\ga = -\al_k$ we get $ \frac{\p X_H}{\p t_i} - x_iU = \al_iX_H \ \text{mod}(H) $. We may vary $X_H$ continuously by any multiple of $H$ without changing the hypothesis or claims of the Proposition, so letting $X_H' = X_H - HU$ we get $$ \frac{\p X_H'}{\p t_i} = \frac{\p X_H}{\p t_i} - x_iU - H\frac{\p U}{\p t_i} = \al_iX_H - H\frac{\p U}{\p t_i} \ \text{mod}(H) = \al_iX_H' \ \text{mod}(H) $$ Therefore we may assume that $X_H$ is such that $$ \frac{\p X_H}{\p t_i} = \al_iX_H + HU_i. $$ Differentiating again we get $$\begin{array}{rl} \frac{\p^2X_H}{\p t_j \p t_i} &= \frac{\p \al_i}{\p t_j}X_H + \al_i\frac{\p X_H}{\p t_j} + x_jU_i + H\frac{\p U_i}{\p t_j} \\ &= \frac{\p \al_i}{\p t_j}X_H + \al_i(\al_jX_H + HU_j)+ x_jU_i + H\frac{\p U_i}{\p t_j} \\ \end{array}$$ and $$\begin{array}{rl} \frac{\p^2X_H}{\p t_i \p t_j} &= \frac{\p \al_j}{\p t_i}X_H + \al_j\frac{\p X_H}{\p t_i} + x_iU_j + H\frac{\p U_j}{\p t_i} \\ &= \frac{\p \al_j}{\p t_i}X_H + \al_j(\al_iX_H + HU_i) + x_iU_j + H\frac{\p U_j}{\p t_i}. \end{array}$$ Thus $$ x_jU_i -x_iU_j = (\frac{\p \al_j}{\p t_i} - \frac{\p \al_i}{\p t_j})X_H + H(\al_jU_i - \al_iU_j + \frac{\p U_j}{\p t_i} - \frac{\p U_i}{\p t_j})$$ and so $$( x_k(\frac{\p \al_j}{\p t_i} - \frac{\p \al_i}{\p t_j}) - x_j(\frac{\p \al_k}{\p t_i} - \frac{\p \al_i}{\p t_k}) + x_i(\frac{\p \al_k}{\p t_j} - \frac{\p \al_j}{\p t_k}))X_H |_H = 0.$$ Therefore $\frac{\p \al_j}{\p t_i} = \frac{\p \al_i}{\p t_j}$ for all $i, j$, and hence there exists $\al$ such that $\al_i = \frac{\p \al}{\p t_i}$ for all $i$. \newline Thus we may assume $ \frac{\p X_H}{\p t_i} = \frac{\p \al}{\p t_i}X_H + HU_i. $ Let $X_H' = e^{-\al}X_H$, then $$ \frac{\p X_H'}{\p t_i} = e^{-\al}(\frac{\p X_H}{\p t_i}-\frac{\p \al}{\p t_i}X_H) = 0 \ \text{mod}(H)$$ Therefore we may assume $ \frac{\p X_H}{\p t_i} = HU_i$. \newline Let $ \frac{\p X_H}{\p t_i} = H^kU_i$, with $k \geq 1$. Let $I$ be such that $|I| = k$, then $$ \frac{\p^{k+1}X_H}{\p t^I\p t_i} = k!x^IU_i \ \text{mod}(H).$$ If $I + v_i = I' + v_j$, then $ \frac{\p^{k+1}X_H}{\p t^I\p t_i} = \frac{\p^{k+1}X_H}{\p t^{I'}\p t_j}$ and so $ (x^{I'}U_j - x^IU_i)|_H = 0$. Now, $x^{I'} = x_ix^J$ and $x^I = x_jx^J$ and so $(x_iU_j - x_jU_i)|_H = 0$, therefore $U_i = x_iV + HV_i$ and $ \frac{\p X_H}{\p t_i} = H^k(x_iV + HV_i)$. Let $X_H' = X_H - \frac{1}{k+1}H^{k+1}V$, then $$ \begin{array}{rl} \frac{\p X_H'}{\p t_i} &= \frac{\p X_H}{\p t_i}- H^kx_iV - \frac{1}{k+1}H^{k+1}\frac{\p V}{\p t_i} \\ &= H^{k+1}V_i -\frac{1}{k+1}H^{k+1}\frac{\p V}{\p t_i} \\ &= H^{k+1}W_i. \end{array}$$ Therefore we may assume $\frac{\p X_H}{\p t_i} = 0$ and hence that $X_H$ is a constant up to a multiple of $H$. \qed {\bf Proof of Lemma~\ref{lemma:lem}} \newline $K \subseteq K^{sat}$, therefore $\text{gin}(K) \subseteq \text{gin}(K^{sat})$ and $\text{gin}(K^{sat})$ is generated by $$\{ x_1^ax_2^b | x_1^ax_2^bx_3^c \in \text{gin}(K) \ \text{for some} \ c \geq 0\}.$$ Therefore $\text{gin}(K^{sat}) \subseteq (x_1^k)$ so we may assume $K$ is saturated. \newline Let $H$ be a generic hyperplane section, then $\text{gin}((K|_H)^{sat}) = (x_1^k)$, and hence $(K|_H)^{sat} = (X_H)$. \newline Let $l$ be such that $(K|_H : \text{\bf m}^l) = (K|_H)^{sat} $. Then for all monomials $x^I \in \text{\bf m}^l$, there exists a polynomial $A_{H,I}$, such that $x^IX_H + HA_{H,I} \in K$. Differentiating with respect to $H = \sum t_ix_i$ we get $$ x^I\frac{\p X_H}{\p t_i} + x_iA_{H,I} \in K|_H, $$ and thus $$x_j\frac{\p X_H}{\p t_i} - x_i\frac{\p X_H}{\p t_j} \in (K|_H : \text{\bf m}^l) = (K|_H)^{sat} = (X_H).$$ Thus considering the $X_H$ as elements of $\C[x_1, x_2, x_3]$, the $X_H$ satisfy the hypothesis of Proposition~\ref{proposition:fam} and so $X = X_H|_H$ is independent of $H$. \newline Let $Y \in K$, Then $Y|_H \in K|_H \subseteq (K|_H)^{sat} = (X)$ and $Y = A_HX+HB_H \in K$. Differentiating with respect to $H$ we get $0 =(\frac{\p A_H}{\p t_j}X + x_iB_H)|_H \in K|_H \subseteq (X)$. Therefore $(x_iB_H)|_H \in (X)$ and thus $B_H|_H \in (X)$. Let $B_H = XC_H + HB_H'$, then $Y = XA_H' + H^2B_H' \in (X)$. \newline Let $Y = A_HX+H^kB_H$, with $ k \geq 2$. Differentiating with respect to $H$, $k$ times, we get $$0 =(\frac{\p^k A_H}{\p t^I}X + k!x^IB_H)|_H \in K|_H \subseteq (X).$$ Therefore $(x^IB_H)|_H \in (X)$ and thus $B_H|_H \in (X)$. Letting $B_H = XC_H + HB_H'$, we get $Y = XA_H' + H^{k+1}B_H' \in (X)$. Therefore we may assume $B_H = O$ and hence $Y = XA_H \in (X)$. \qed \noindent Let $K$ be an ideal contained in $J$ maximal with respect to the properties that $K = XK'$ with $\text{in}(X) = x_1^{i+1}$, and $(K)_d = (J)_d$ for $d \leq i+\nu_{i+1}(0) +2$. $K = X \cap J$. \begin{lemma} $\text{gin}(X) \cap \text{gin}(J) \subseteq \text{gin}(X \cap J)$ \end{lemma} {\bf Proof.} By Galligo's Theorem ([Ga]), we may assume $\text{gin}(I) = \text{in}(I)$ for all ideals $I = (X), J$ or $K$. \newline Let $M = x_1^ax_2^bx_3^c \in \text{gin}(X) \cap \text{gin}(J)$, then $a \geq i+1$, and we may write $M = x_1^{a - \al}x_2^{b - \be}x_3^{c - \ga}( x_1^{\al}x_2^{\be}x_3^{\ga})$ where $A = x_1^{\al}x_2^{\be}x_3^{\ga}$ is a generator of $\text{gin}(J)$. If deg$A \leq i + \nu_{i+1}(0) + 2$, then $A \in \text{gin}(X \cap J)$. Suppose deg$A > i + \nu_{i+1}(0) + 2$, and $\al$ is maximal, if $\al < a$ and $\be $ or $\ga \geq 1$, then either $x_1^{\al +1}x_2^{\be -1}x_3^{\ga}$ or $x_1^{\al +1}x_2^{\be }x_3^{\ga - 1} \in \text{gin}(J)$. Then there would exist $B = x_1^{\al'}x_2^{\be'}x_3^{\ga'} $, a generator of $\text{gin}(J)$, dividing this element and $\al' = \al +1 $ otherwise $A$ would not be a generator, and $B | M$ which contradicts the maximality of $\al$. If $\be = \ga = 0$, but then $x_1^{\al}$ is a generator of $J$ and so $\al \leq i + 1 + \nu_{i+1}$. Therefore we may assume $\al = a \geq i+1$. If $\ga \leq 1$ then deg$A \leq i+\nu_{i+1}(0) +2 $ therefore we may assume $\ga \geq 2$. \newline Let's assume $M = x_1^ax_2^bx_3^c \in \text{gin}(X) \cap \text{gin}(J)$, is a generator of $\text{gin}(J)$ with deg$M > i+\nu_{i+1}(0)+2$, $c \geq 2$, and suppose $M \not\in \text{gin}(X \cap J) = \text{gin}(K)$. Pick $M$ satisfying this of minimal degree $m$, and among those of minimal degree, let $M$ be maximal. Pick $f \in J$ with $\text{in}(f) = M$. Let $L$ be the ideal generated by $K$ and $f$. Then $\text{gin}(L)_d = \text{gin}(K)_d$ for $d < m$ and $\text{gin}(L)_m = \text{gin}(K)_m + M$. Then the generators of $\text{gin}(L)$ in degree $\leq m$ form a Borel-fixed monomial ideal and every generator of degree $m$ has an $x_3$ term. Thus as in the proof of Theorem~\ref{theorem:newgen} and Corollary~\ref{corollary:ngcor} all elements of $\text{gin}(L)$ must be divisible by $x_1^{i+1}$. This however contradicts the maximality of $K$, therefore $\text{gin}(X) \cap \text{gin}(J) \subset \text{gin}(X \cap J)$. \qed \subsection{Proof of the Connectedness Theorem} \noindent Suppose $\mu_{i+1}(j) + 2 < \mu_{i}(j)$ for $i < s_j -1$. Consider $J = (I|_H:(H'\cap H)^j)$, for generic hyperplanes $H, H'$ and let $K$ be the ideal contained in $J$ such that $(K)_d = (J)_d$ for $d \leq i+2+\mu_{i+1}(j)$ and $K = XK'$ with $\text{in}(X) = x_1^{i+1}$. \begin{proposition} $X$ is constant up to a multiple of $H$. \end{proposition} {\bf Proof.} \noindent Pick $ p = p(H,H') \in K'$ such that deg$(pX) = m \leq i+1+\mu_{i+1}(j)$ and $(p,X) = 1$. (As $x_1^{i+1}x_2^{\mu_{i+1}(j)} \in \text{gin}(J)_{i+1+\mu_{i+1}(j)} = \text{gin}(K)_{i+1+\mu_{i+1}(j)}$, there exists $p \in K'$ such that in$(p) = x_2^{\mu_{i+1}(j)}$, and so $p$ and $X$ cannot have a common factor.) \newline Then $$ (H')^jpX + HA \in I \ \text{for some} \ A = A(H,H').$$ Keeping $H'$ fixed and letting $H = \sum t_ix_i$ vary and differentiating with respect to the $t_i$ we get $$ (H')^j(\frac{\p p}{\p t_i}X + p\frac{\p X}{\p t_i}) + x_iA \in I_{H}$$ and so $$ x_k(\frac{\p p}{\p t_i}X + p\frac{\p X}{\p t_i}) - x_i(\frac{\p p}{\p t_k}X + p\frac{\p X}{\p t_k}) \in (I_H : (H')^j)_{m+1} = J_{m+1}.$$ $m+1 \leq i+2+\mu_{i+1}(j)$, hence $(J)_{m+1} = (K)_{m+1} \subseteq (X)$. Therefore $p(x_j\frac{\p X}{\p t_i} - x_i\frac{\p X}{\p t_j}) \in (X)$, $(p, X) = 1$ and so $x_j\frac{\p X}{\p t_i} - x_i\frac{\p X}{\p t_j} \in (X)$. Therefore, for generic $H$, $X_H = X$, satisfy the hypothesis of Proposition~\ref{proposition:fam} and hence we may assume $X$ is constant up to a multiple of $H$. \qed {\bf Proof of Theorem~\ref{theorem:connect}} Keeping $H'$ fixed and letting $Y = (H')^jX$, we have for all hyperplanes $H$ and all $p_H =p(H,H') \in K'_{H,H'}$ there exists $A_H = A(H,H')$ such that $$ p_HY + HA_H \in I. $$ Let $\G_H = H \cap C$, then $\G_H \subset V(p_HY) = V(Y) \cup V(p_H)$. If for a generic $H$, there exists $q_H $ a point in $\G_H$ such that $q_H \in V(Y)$, then $S = \{q_H \ | \ q_H \in V(Y) \}$ will be a 1-dimensional space and $S \subset C$, $C$ is reduced and irreducible therefore $S$ is dense in $C$ and hence $ C \subset V(Y)$. However $V(Y) = V((H')^j) \cup V(X)$ and as C is nondegenerate, this would imply $ C \subset V(X)$. However, $i < s_j - 1 \leq s_0 - 1$ and so $i+1 < s_0$. But $s_0$ is the smallest degree of elements of $I$, therefore $C \not\subset V(X)$. Therefore for generic $H, H'$, $ \G_H \subset V(K')$. However the invariants of $(K')^{sat}$ are $\lm_{i+1} > \dots > \lm_{s-1}$ and as $i+1 > 0$, $ \G_H \not\subset V(K')$. This concludes the proof of the main part of Theorem~\ref{theorem:connect}. \vspace{2mm} Now suppose $s_k < s_0$, and $\mu_{s_k-1}(k) \geq 3$. For generic hyperplane sections $H$ and $H'$, consider $(I|_H : (H'\cap H)^k)$, we have $(I|_H : (H'\cap H)^k)_{s_k} = (X_{H,H'})_{s_k}$, and $(I|_H : (H'\cap H)^k)_{s_k+1} = (X_{H,H'})_{s_k+1}$, and so there exists a homogeneous polynomial $A_{H,H'}$ such that $$ (H')^kX_{H,H'} + HA_{H,H'} \in I.$$ Keeping $H'$ fixed and differentiating with respect to $H = \sum t_ix_i$, we get $$ (H')^k(\frac{\p X_{H,H'}}{\p t_i}) + x_iA_{H,H'} \in I_{H}$$ and so $$ (x_j\frac{\p X_{H,H'}}{\p t_i} - x_i\frac{\p X_{H,H'}}{\p t_j}) \in (I|_H : (H'\cap H)^k)_{s_k+1} = (X_{H,H'})_{s_k+1}.$$ Thus by Proposition~\ref{proposition:fam}, $X_{H,H'}$ is constant with respect to $H$. \newline Let $X_{H,H'} = X_{H'}$, Fix $H'$ and let $\G_{H}$ be a generic hyperplane section of $C$. As $ (H')^kX_{H'} + HA_{H,H'} \in I$, $\G_{H} \subset V(H') \cup V(X_{H'})$. But as the points of $\G_{H}$ are in general position, there must exist at least one point of $\G_{H} \in V(X_{H'})$. But varying $H$ as above would imply that $C \subset V(X_{H'})$, and hence that $s_k \geq s_0$, which is a contradiction. This completes the proof of Theorem~\ref{theorem:connect}. \qed \section{Further Results on the Generic Initial Ideal of a Curve} \subsection{Generalized Strano} \noindent This result generalizes a result of Strano ([S]): \noindent {\bf Definition.} If $x_1^ix_2^jx_3^{f(i,j)}$ is a generator of $\text{gin}(I_C)$ with $f(i,j) > 0$, then $x_1^ix_2^jx_3^k$ is a {\bf sporadic zero} for all $0 \leq k < f(i,j)$. \begin{theorem}[Strano] \label{theorem:strano} If C is a reduced irreducible curve and has a sporadic zero in degree m, then $I_{\G}$ has a syzygy in degree $\leq m+2$. \end{theorem} \begin{theorem}[Generalized Strano] \label{theorem:genstr} Let C be a reduced irreducible curve with a sporadic zero $x_1^ix_2^jx_3^{k-a}$ of degree m, such that $x_1^ix_2^jx_3^k$ is a generator of $\text{gin}(I_C)$. Then, for generic hyperplanes $H$ and $H'$, $J = (I_C|_H : (H \cap H')^a)$ has a syzygy in degree $\leq m+2$. \end{theorem} \noindent{\bf Proof.} \noindent $x_1^ix_2^jx_3^{k-a} \in \text{gin}(J)_m$, therefore there exists $F \in (I_C|_H : (H \cap H')^a)_m = (J)_m$ varying differentiably with $H$ and $H'$, and hence for generic $H$ and $H'$ there exists $A$, depending on $H$ and $H'$ such that $$ (H')^aF + HA \in I_C.$$ If we were to keep $H'$ fixed, the coefficients of $F$ and $A$ would be homogeneous polynomials in the coefficients $t_i$ of $H = \sum t_ix_i$. We will choose $F$, which is a bihomogeneous polynomial in $t_i$ and $x_i$, such that the degree of $F$ with respect to $t_i$ is minimal. We will also assume that $\text{gin}(J) = \text{in}(J)$ and that $\text{in}(F) = x_1^ix_2^jx_3^{k-a}$. \newline Differentiating with respect to $H$, keeping $H'$ fixed we get $$ (H')^a\frac{\p F}{\p t_j} + x_jA \in I_C|_H$$ and so $$x_j\frac{\p F}{\p t_i} - x_i\frac{\p F}{\p t_j} \in (J)_{m+1}.$$ Hence $$x_k(x_j\frac{\p F}{\p t_i} - x_i\frac{\p F}{\p t_j}) - x_j(x_k\frac{\p F}{\p t_i} - x_i\frac{\p F}{\p t_k}) + x_i(x_k\frac{\p F}{\p t_j} - x_j\frac{\p F}{\p t_k}) = 0 $$ is a syzygy of $J$ in degree $m+2$. \noindent Suppose $J$ does not have a syzygy in degree $\leq m+2$, then $$x_j\frac{\p F}{\p t_i} - x_i\frac{\p F}{\p t_j} = x_jU_i - x_iU_j \ \text{where} \ U_i \in (J)_m.$$ Rewriting, we get $$x_j(\frac{\p F}{\p t_i}-U_i) - x_i(\frac{\p F}{\p t_j}-U_j) = 0$$ and so $$\frac{\p F}{\p t_i} = U_i + x_iR.$$ Letting $F' = F- HR$ we get $$ \frac{\p F'}{\p t_i} = \frac{\p F}{\p t_i} - x_iR - H\frac{\p R}{\p t_i} = U_i \ \text{mod}(H) \in (J)_m.$$ As we have assumed the degree of $F$ is minimal with respect to $t_i$ we get that $F$ is constant up to a multiple of $H$. Hence, by an argument similar to that of Theorem~\ref{theorem:connect}, $F \in I_C$. This, however, is a contradiction. \qed \noindent {\bf Example.} \newline The following diagram can not correspond to a generic initial ideal of a curve, even though it is connected. \vspace{5mm} \begin{picture}(370,75)(0,7) \put(100,75){\line(3,-5){42}} \put(100,75){\line(-3,-5){42}} \put(100,5){\line(1,0){42}} \put(100,5){\line(-1,0){42}} \put(100,61){\circle{12}} \put(92.5,49){\circle{12}} \put(107.5,49){\circle{12}} \put(85,37){\circle{12}} \put(100,37){\circle{12}} \put(115,37){\circle{12}} \put(77.5,25){\circle{12}} \put(77.5,25){\makebox(0,0){1}} \put(92.5,25){\circle{12}} \put(107.5,25){\circle{12}} \put(122.5,25){\circle{12}} \put(70,13){\makebox(0,0){X}} \put(85,13){\makebox(0,0){X}} \put(100,13){\circle{12}} \put(100,13){\makebox(0,0){1}} \put(115,13){\circle{12}} \put(115,13){\makebox(0,0){2}} \put(130,13){\circle{12}} \end{picture} \noindent We have a sporadic zero in degree $3$, and so by the Theorem, $J = (I_C|_H : (H)^1)$ has a syzygy in degree $\leq 5$. The diagram of $\text{gin}(J)$ is as follows: \begin{picture}(370,75)(0,7) \put(100,75){\line(3,-5){42}} \put(100,75){\line(-3,-5){42}} \put(100,5){\line(1,0){42}} \put(100,5){\line(-1,0){42}} \put(100,61){\circle{12}} \put(92.5,49){\circle{12}} \put(107.5,49){\circle{12}} \put(85,37){\circle{12}} \put(100,37){\circle{12}} \put(115,37){\circle{12}} \put(77.5,25){\makebox(0,0){X}} \put(92.5,25){\circle{12}} \put(107.5,25){\circle{12}} \put(122.5,25){\circle{12}} \put(70,13){\makebox(0,0){X}} \put(85,13){\makebox(0,0){X}} \put(100,13){\makebox(0,0){X}} \put(115,13){\circle{12}} \put(115,13){\makebox(0,0){1}} \put(130,13){\circle{12}} \end{picture} \noindent $J$ has only two generators in degree $\leq 4$, corresponding to $x_1^3$ and $x_1^2x_2^2$ and so if there were a syzygy in degree $\leq 5$ this would imply that we may ``split'' the ideal as in the proof of Theorem~\ref{theorem:connect}. But this would give a contradiction. \vspace{.5cm} \noindent More generally, if $s_k = \text{min}\{i | f(i,0) \leq k\}$ and for $k >> 0 $ $s_0 > s_k $ and $\mu_{s_k-1}(k) = \lambda_{s-1} = 2$, then $f(s-2,3) \leq f(s,0)$. \begin{picture}(370,75)(0,7) \put(100,75){\line(3,-5){42}} \put(100,75){\line(-3,-5){42}} \put(100,5){\line(1,0){42}} \put(100,5){\line(-1,0){42}} \put(92.5,49){\circle{12}} \put(85,37){\circle{12}} \put(100,37){\circle{12}} \put(77.5,25){\circle{12}} \put(77.5,25){\makebox(0,0){a}} \put(92.5,25){\circle{12}} \put(107.5,25){\circle{12}} \put(100,13){\makebox(0,0){b}} \put(100,13){\circle{12}} \put(115,13){\circle{12}} \put(115,13){\makebox(0,0){c}} \end{picture} By connectedness $b \leq a$. If $c > a$, then if we let $J = (I_C|_H : (H)^a)$, then $J$ has a syzygy in degree $\leq s+2$. But again this would imply that we could ``split'' the ideal. \subsection{Complete Intersections and Almost Complete Intersections} The result in this section is inspired by the work of Ellia ([E]) and again generalizes a result of Strano ([S]): \begin{theorem}[Strano] If $C$ is a reduced irreducible curve whose generic hyperplane section has the Hilbert function of a complete intersection of type $(m,n)$, where $m,n > 2$, then $C$ is a complete intersection of type $(m,n)$. \end{theorem} This result follows from connectedness and Theorem~\ref{theorem:strano}. \noindent {\bf Note.} If $\G$ is a set of $d$ points in general position with invariants $\lm_0 > \dots > \lm_{k-1} > 0$ such that $\lm_i = \lm_0 - 2i$ for all $i$. Then $\G$ is a complete intersection of type $(k, d/k)$. (See [Gr].) \begin{proposition} Let C be a reduced, irreducible, non-degenerate curve in $\pthree$, let $\Gamma = C \cap H$ be a generic hyperplane section with invariants $\{\lambda_i\}_{i=0}^{s-1}$. If $\lambda_{s-i} = \lambda_{s-1} +2(i-1)$ for $1 \leq i \leq k$, where $k \geq 3$, then $f(i,j) > 0 $ only if $i < s-k$. \end{proposition} \noindent {\bf Proof.} \newline Let $J = (I|_H : (H \cap H')^j)$ for $j >> 0$, so that $\text{gin}(J) = \text{gin}(I_{\Gamma})$. Let $f$ correspond to $x_1^s$ and let $g$ correspond to $x_1^{s-1}x_2^{\lambda_{s-1}}$, where $f$ and $g$ are in $J$. If $f$ and $g$ have a syzygy in degree $d \leq \lambda_{s-k} + (s-k)$, then generators of $\text{gin}(J)$ in degree $d$ correspond to generators of $J$ and thus we may ``split'' the ideal $J$ as in the proof of Theorem~\ref{theorem:connect}. Therefore $f$ and $g$ have no syzygy in degree $\leq \lambda_{s-k} + (s-k)$. By Theorem~\ref{theorem:strano} or Theorem~\ref{theorem:genstr}, this means that there can be no sporadic zeroes in degree $\leq \lambda_{s-k} + (s-k) -2$. \newline If there is a sporadic zero in degree $\lambda_{s-k} + (s-k) -1 = $ $\lambda_{s-(k-1)} +(s-(k-1)$, then $\mu_{s-(k-1)}(0) > \lambda_{s-(k-1)}$ and $\mu_{s-(k-2)}(0) = \lambda_{s-(k-2)} = \lambda_{s-(k-1)} - 2$, which contradicts the connectedness of the $\{\mu_i(0)\}$. Similarly if $f(s-k, \lambda_{s-k}) > 0$ then $\mu_{s-k}(0) > \lambda_{s-k} = \lambda_{s-(k-1)} + 2 = \mu_{s-(k-1)}(0) + 2$ which again contradicts connectedness. \qed \noindent Thus for the following configuration of a hyperplane section, we can only possibly get a sporadic zero in the (0,6) position. \vspace{5mm} \begin{picture}(370,99)(0,0) \put(140,99){\line(3,-5){56}} \put(140,99){\line(-3,-5){56}} \put(140,5){\line(1,0){56}} \put(140,5){\line(-1,0){56}} \put(140,85){\circle{12}} \put(132.5,73){\circle{12}} \put(147.5,73){\circle{12}} \put(125,61){\circle{12}} \put(140,61){\circle{12}} \put(155,61){\circle{12}} \put(117.5,49){\circle{12}} \put(132.5,49){\circle{12}} \put(147.5,49){\circle{12}} \put(162.5,49){\circle{12}} \put(110,37){\makebox(0,0){X}} \put(125,37){\makebox(0,0){X}} \put(140,37){\circle{12}} \put(155,37){\circle{12}} \put(170,37){\circle{12}} \put(102.5,25){\makebox(0,0){X}} \put(117.5,25){\makebox(0,0){X}} \put(132.5,25){\makebox(0,0){X}} \put(147.5,25){\makebox(0,0){X}} \put(162.5,25){\circle{12}} \put(177.5,25){\circle{12}} \put(95,13){\makebox(0,0){X}} \put(110,13){\makebox(0,0){X}} \put(125,13){\makebox(0,0){X}} \put(140,13){\makebox(0,0){X}} \put(155,13){\makebox(0,0){X}} \put(170,13){\makebox(0,0){X}} \put(185,13){\makebox(0,0){X}} \end{picture} \section*{\bf References} {\bf [B]} D. Bayer, {\it The division algorithm and the Hilbert scheme}, Ph.D. Thesis, Harvard University, Department of Mathematics, June 1982. Order Number 82-22588, University Microfilms International, 300N Zeeb Rd., Ann Arbor, MI 48106. {\bf [BM]} D. Bayer, D. Mumford {\it What can be computed in Algebraic Geometry}, Computational Algebraic Geometry and Commutative Algebra, Symposia Mathematica Volume XXXIV, Cambridge University Press. (1991) pp1-48. {\bf [BS]} D. Bayer, M. Stillman {\it A criterion for detecting m-regularity}, Invent. Math. 87 (1987) pp1-11. {\bf [E]} P. Ellia {\it Sur les lacunes d'Halphen}, Algebraic curves and projective geometry (Trento, 1988), 43-65, Lecture Notes in Math., 1389, Springer, Berlin-New York 1989. {\bf [EP]} P. Ellia, C. Peskine {\it Groupes de points de $\ptwo$: caract\`ere et position uniforme}, Algebriac Geometry (L'Aquila 1988), Lecture Notes in Mathematics, 1417, Springer, Berlin 1990 pp111-116. {\bf [Ga]} A. Galligo {\it A propos du theoreme de preparation de Weierstrass}, Fonctions de Plusieurs Variables Complexes, Lecture Notes in Math., Vol 1974 pp543-579. {\bf [Gr]} M. Green {\it Lecture notes on generic initial ideals}, Unpublished. {\bf [GP]} L. Gruson, C. Peskine {\it Genre des courbes de l'espace projectifs}, Lecture Notes in Mathematics, 687, Springer (1978) pp31-59. {\bf [S]} R. Strano {\it Sulle Sezione Iperpiane Delle Curve}, Rend. Sem. Mat. Fis. Milano 57 (1987) pp125-134. \vspace{.5cm} Michele Cook Department of Mathematics UCLA Los Angeles, CA. 90024 e-mail shelly\verb+@+math.ucla.edu \end{document}

9401

alg-geom/9401003

en

https://arxiv.org/abs/alg-geom/9401003

alg-geom/9401003

Gregory Sankaran

K. Hulek and G.K. Sankaran

The Fundamental Group of Some Siegel Modular Threefolds

9 pages, AMSLaTeX v. 1.1

Let A be the moduli space of (1,p)-polarised abelian surfaces with a level structure, for p an odd prime. Let X be a desingularisation of any algebraic compactification of A. Then X is simply-connected.

alg-geom

\section{Generalities about fundamental groups} The general facts in this section are all well known. \begin{lemma}\label{1:l 1.1} Let $M$ be a connected simply connected real manifold and $G$ a group acting discontinuously on $M$. Let $x\in M$ be a base point. Then the quotient map $\varphi:M\longrightarrow M/G$ induces a map $\psi:G\longrightarrow \pi_1(M/G,\varphi(x))$ which is surjective. \end{lemma} Note that we do not require the action of $G$ to be free or even effective. The map $\psi$ is defined as follows. If $g\in G$ let $\theta_g:[0,1]\longrightarrow M$ be any path in $M$ from $x$ to $g(x):$ that is, $\theta_g(0)=x$ and $\theta_g(1)=g(x)$. Then $\varphi\circ\theta_g:[0,1]\longrightarrow M/G$ is a closed loop in $M/G$, with $\varphi\circ\theta_g(0)=\varphi\circ\theta_g(1)=\varphi(x)$. We put $\psi(g)=[\varphi\circ\theta_g] \in \pi_1(M/G,\varphi(x))$. One checks easily that $\psi$ is well defined and that it is a group homomorphism. It is shown in [G] that $\psi$ is surjective. \begin{lemma}\label{1:l 1.2} Let $M$ be a connected complex manifold and $M_0$ an analytic subvariety. Then the inclusion induces a map $\pi_1(M\setminus M_0,x)\longrightarrow \pi_1(M,x)$ for any base point $x\in M$ which is surjective for $\operatorname{codim }_{ {\Bbb{C}}} M_0\ge 1$ and an isomorphism for $\operatorname{codim }_{ {\Bbb{C}}} M_0 \ge 2$. \end{lemma} \begin{lemma}\label{1:l 1.3} If $X_1$ and $X_2$ are birationally equivalent smooth projective varieties then $\pi_1(X_1)\cong \pi_1(X_2)$. \end{lemma} Proofs of both these lemmas can be found in [HK]. \section{Applications to the case of ${\cal A}_{1,p}$} We fix some notation. $S_2$ is the Siegel upper half-plane of degree 2. We denote by ${\cal A}_{1,p}$ the moduli space of abelian surfaces with a $(1,p)$-polarisation and a level structure, and by ${\cal A}(p^2)$ the moduli space of abelian surfaces of level $p^2$. We let $\Gamma_{1,p}$ and $\Gamma(p^2)$ be the corresponding arithmetic subgroups of $\operatorname{Sp}(4,{\Bbb{Z}})$, so $$ \Gamma_{1,p}=\left\{\gamma\in \operatorname{Sp}(4,{\Bbb{Z}})|\gamma-{\mbox{\boldmath$1$}}\in\left( \begin{array}{cccc} {\Bbb{Z}} & {\Bbb{Z}} & {\Bbb{Z}} & p{\Bbb{Z}}\\ p{\Bbb{Z}} & p{\Bbb{Z}} & p{\Bbb{Z}} & p^2{\Bbb{Z}}\\ {\Bbb{Z}} & {\Bbb{Z}} & {\Bbb{Z}} & p{\Bbb{Z}}\\ {\Bbb{Z}} & {\Bbb{Z}} & {\Bbb{Z}} & p{\Bbb{Z}}\\ \end{array} \right) \right\} $$ (see [HKW1]) and $\Gamma (p^2)$ is the principal congruence subgroup of level $p^2$. We denote by $\Gamma^0_{1,p}$ the arithmetic subgroup of $\operatorname{Sp}(4,{\Bbb{Q}})$ $$ \Gamma_{1,p}^0=\left\{\gamma\in \operatorname{Sp}(4,{\Bbb{Q}})|\gamma\in\left( \begin{array}{cccc} {\Bbb{Z}} & {\Bbb{Z}} & {\Bbb{Z}} & p{\Bbb{Z}}\\ p{\Bbb{Z}} & {\Bbb{Z}} & p{\Bbb{Z}} & p{\Bbb{Z}}\\ {\Bbb{Z}} & {\Bbb{Z}} & {\Bbb{Z}} & p{\Bbb{Z}}\\ {\Bbb{Z}} & \frac 1 p{\Bbb{Z}} & {\Bbb{Z}} & {\Bbb{Z}}\\ \end{array} \right) \right\} $$ (in terms of moduli, this corresponds to keeping the $(1,p)$-polarisation but forgetting the level structure). We denote by ${\cal A}^*_{1,p}$ the toroidal compactification of ${\cal A}_{1,p}$ described in [HKW1], and by ${\cal A}^*(p^2)$ the usual toroidal (or Igusa) compactification of ${\cal A}(p^2)$. Since $\Gamma(p^2)$ is neat, ${\cal A}^*(p^2)$ is nonsingular: indeed, any nonsingular compactification of ${\cal A}(p^2)$ will serve our purpose.\\ Let ${\cal A}'_{1,p}$ be the smooth part of ${\cal A}_{1,p}$. The space ${\cal A}_{1,p}$ is a quotient of $S_2$ by $\Gamma_{1,p}$ and ${\cal A}'_{1,p}$ is the quotient by $\Gamma_{1,p}$ of an open part $S'_2 \subset S_2$. Since the singular locus of ${\cal A}_{1,p}$ has codimension $2,S_2'$ is the complement of an analytic subset of codimension $2$ in $S_2$. In particular, by Lemma 1.2, $S'_2$ is simply connected.\\ {\em Remark. } The quotient map $S'_2 \longrightarrow {\cal A}'_{1,p}=S'_2/\Gamma_{1,p}$ is not unramified. The ramification is in codimension $1$ but the isotropy groups are generated by reflections so the quotient is smooth. If we restrict to the complement of the ramification locus our covering space will not necessarily be simply connected. \begin{proposition}\label{2:p 2.2} Let $f:X \longrightarrow {\cal A}^*_{1,p}$ be a resolution of singularities (so $X$ is smooth and $f$ is a birational morphism which is an isomorphism away from $f^{-1}(\operatorname{Sing} {\cal A}^*_{1,p}))$. Then there are surjective homomorphisms $\Gamma_{1,p} \longrightarrow \pi_1({\cal A}'_{1,p})$ and $\pi_1({\cal A}'_{1,p}) \longrightarrow \pi_1(X)$. \end{proposition} \begin{Proof} The map $\Gamma_{1,p} \longrightarrow \pi_1({\cal A}'_{1,p})$ exists and is surjective by Lemma 1.1. The map $\pi_1({\cal A}'_{1,p})\longrightarrow \pi_1(X)$ comes from Lemma 1.2: we may consider ${\cal A}'_{1,p}$ as a subset of $X$ via $f^{-1}$, which exists on ${\cal A}'_{1,p}$. \end {Proof} We now choose a convenient resolution of singularities to work with. $\Gamma_{1,p}$ is a normal subgroup of $\Gamma^0_{1,p}$ and the quotient (which is isomorphic to $\mbox{SL}(2,{\Bbb{F}}_p)$ as an abstract group) acts on ${\cal A}^*_{1,p}$ (this is shown in [HKW1]). By the general results of Hironaka ([H]) or by an easy explicit construction we can choose $f:X\longrightarrow {\cal A}^*_{1,p}$ to be an equivariant resolution, so that $\Gamma^0_{1,p}/\Gamma_{1,p}$ acts on $X$. Henceforth $X$ will always be such a resolution.\\ Let $\psi:\Gamma_{1,p}\longrightarrow \pi_1(X)$ be the composite of the two maps in Proposition 2.1. Then $\psi$ is surjective. We shall show that $\operatorname{Ker} \psi=\Gamma_{1,p}$ and hence $\pi_1(X)=1$. \section{Level $p^2$} In this section we use results of Kn\"oller ([K]) to prove that $\Gamma(p^2)$, which is a subgroup of $\Gamma_{1,p}$, lies in the kernel of $\psi$.\\ Since $\Gamma(p^2)$ is a normal subgroup of $\Gamma_{1,p}$, there is an action of $\Gamma_{1,p}/\Gamma (p^2)$ on ${\cal A}(p^2)$ and the quotient is ${\cal A}_{1,p}$. The quotient map induces a rational map $h:{\cal A}^*(p^2)--\rightarrow X$, which is a morphism over ${\cal A}'_{1,p}$. We can resolve the singularities of this map so as to get a diagram $$ \unitlength1cm \begin {picture}(5,3) \put(0.5,2){\vector(0,-1){1}} \put(0.8,2){\vector(2,-1){2.1}} \multiput(1.5,0.5)(0.428,0){4}{\line(1,0){0.214}} \put(2.78,0.5){\vector(1,0){0.214}} \put(0.5,2.2){$Y$} \put(0.1,0.4){${\cal A}^*(p^2)$} \put(3.1,0.4){$X$} \put(0.2,1.5){$\sigma$} \put(2.1,0.7){$h$} \put(2.2,1.6){$\tilde{h}$} \end{picture} $$ where $Y$ is smooth, $\sigma$ is an isomorphism over ${\cal A}'(p^2)=h^{-1}({\cal A}'_{1,p})$ and $\tilde h$ is a morphism. In particular we can think of ${\cal A}'(p^2)$ as a subset of $Y$ and ${\cal A}'_{1,p}$ as a subset of $X$, and then $\tilde h|_{{\cal A}'(p^2)}$ is the quotient map ${\cal A}'(p^2)\longrightarrow {\cal A}'_{1,p}$. \begin{theorem}\label{3:t 3.1} {\em (Kn\"oller [K]) } Y is simply connected. \end{theorem} \begin{corollary}\label{3:c 3.2} $\Gamma(p^2)$ is contained in $\operatorname{Ker} \psi$. \end{corollary} \begin{Proof} The quotient map $q:S'_2\longrightarrow {\cal A}'(p^2)$ induces a surjection $\Gamma (p^2)\longrightarrow \pi_1(Y)$. Let $M \in \Gamma(p^2)$, let $x\in S'_2$ be a base point and let $\theta_M$ be a path from $x$ to $M(x)$ in $S'_2$. Then there is a null homotopy $H_M:[0,1]\times[0,1]\longrightarrow Y$ such that $H_M(0,t)=q \theta_M(t)$ and $H_M(1,t)=q(x)\in Y$. So $\tilde{h} H_M$ is a null homotopy of $\tilde{h}q \theta_M$ in $X$. But $[\tilde{h}q \theta_M]$ is the class $\psi(M)\in \pi_1(X)$, since $\tilde{h} q$ is the same as the quotient map $S'_2\longrightarrow {\cal A}'_{1,p}$. \end {Proof} \section{Another element of the kernel} \begin{proposition}\label {4:l 1.1} The element $$ M_0=\left ( \begin{array}{cccc} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right ) $$ of $\Gamma_{1,p}$ is in $\operatorname{Ker }\psi$. \end{proposition} \begin{Proof} We follow the procedure of [K], and choose a path $\theta_{M_0}:[0,1]\longrightarrow S_2$, namely $$ \theta_{M_0}(t)=ic (1-t){\mbox{\boldmath$1$}} + ict M_0({\mbox{\boldmath$1$}}) $$ where $c\in {\Bbb{R}}$ is a positive constant to be specified later. $\theta_{M_0}$ gives rise to a closed loop in ${\cal A}_{1,p}$, starting and finishing at $ic{\mbox{\boldmath$1 $} }$ (so when we specify $c$ we are just choosing a base point).\\ The map $e_{l_0}:S_2\longrightarrow{\Bbb{C}}^*\times {\Bbb{C}} \times S$, corresponding to the central boundary component $D_{l_0}$ of ${\cal A}^*_{1,p}$ (see [HKW1]) is given by $$ e_{l_0}:\left (\begin{array}{cc} \tau_1 & \tau_2\\ \tau_2 & \tau_3 \end{array} \right ) \longmapsto (e^{2\pi i \tau_1}, \tau_2,\tau_3). $$ So $ e_{l_0}\theta_{M_0}(t)=(e^{-2\pi c} e^{-2\pi i t}, 0, ic)$. The homotopy $$ H(s,t)=((se^{2\pi i t}+1-s)e^{-2\pi c}, 0, ic) $$ is evidently a null homotopy of $e_{l_0}\theta_{M_0}$ in ${\Bbb{C}} \times {\Bbb{C}} \times S_1$. As in [K], we let $V$ be the interior of the closure of $S_2/P_{l_0}$ in ${\Bbb{C}} \times {\Bbb{C}} \times S_1:$ then the image of $H$ lies in $V$. But by [HKW2], ${\cal A}^*_{1,p}$ is nonsingular near $D_{l_0}:$ the map $V\longrightarrow {\cal A}^*_{1,p}$ is branched but in some open set $U \subset V$ the branching comes from reflections. If we choose $c$ sufficiently large then the image of $H$ will lie entirely in $U$, and $H$ will therefore give rise to a null homotopy which does not meet the singular locus of ${\cal A}^*_{1,p}$. In particular, the loop in ${\cal A}_{1,p}$ corresponding to $\theta_{M_0}$ actually lies in ${\cal A}'_{1,p}$, and $\theta_{M_0}$ is actually a path in $S'_2$. So $\psi(M_0)$ is trivial (it is even trivial as an element of $\pi_1({\cal A}^*_{1,p}\setminus \mbox{Sing }{\cal A}^*_{1,p}))$. \end{Proof} Now we know that $\operatorname{Ker} \psi$ contains $M_0$ and all of $\Gamma(p^2)$. It is also, of course, a normal subgroup of $\Gamma_{1,p}:$ but because we could choose $X$ to have an action of $\Gamma^0_{1,p}/\Gamma_{1,p}$ it is actually normal as a subgroup of $\Gamma^0_{1,p}$, since $\Gamma^0_{1,p}$ acts on $\pi_1(X)=\Gamma_{1,p}/\operatorname{Ker}\psi$ by conjugation in $\mbox{Sp}(4,{\Bbb{Q}})$. \section{Calculations in $\Gamma _{1,p}$} The remark at the end of section 4, above, shows that to prove the theorem it is enough to check that any normal subgroup of $\Gamma^0_{1,p}$ containing $\Gamma(p^2)$ and $M_0$ must contain $\Gamma_{1,p}$. We shall give a set of generators for $\Gamma_{1,p}$ and show how to produce each one, starting with $M_0$ and $\Gamma(p^2)$ and using multiplication, inversion, and conjugation by elements of $\Gamma^0_{1,p}$. A similar procedure is carried out by Mennicke [M] for principal congruence subgroups of $\mbox{Sp}(2n,{\Bbb{Z}})$. \begin{proposition}\label{5:p1.11} $\Gamma_{1,p}$ is generated by the elements $M_1, M_2, M_3$ and $M_4$ together with the subgroups $j_1 (\operatorname{SL}(2,{\Bbb{Z}}))$ and $j_2(\Gamma_1(p))$, where $$ M_1=\left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 0\\ \end{array}\right ) \qquad M_2=\left ( \begin{array}{cccc} 1 & 0 & 0 & p\\ 0 & 1 & p & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{array}\right ) $$ $$ M_3=\left ( \begin{array}{cccc} 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & -1 & 1 \end{array}\right )\qquad M_4=\left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ -p & 1 & 0 & 0\\ 0 & 0 & 1 & p\\ 0 & 0 & 0 & 1 \end{array}\right ) $$ $$ j_1(\operatorname{SL}(2,{\Bbb{Z}}))=\left\{ \left ( \begin{array}{cccc} a & 0 & b & 0\\ 0 & 1 & 0 & 0\\ c & 0 & d & 0\\ 0 & 0 & 0 & 1\\ \end{array} \right ) |\left ( \begin{array}{cc} a & b\\ c & d \end{array} \right ) \in \operatorname{SL}(2,{\Bbb{Z}}) \right \} $$ $$ j_2(\Gamma_1(p))=\left\{ \left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & a & 0 & pb\\ 0 & 0 & 1 & 0\\ 0 & c/p & 0 & d \end{array} \right ) |\left( \begin{array}{cc} a & b\\ c & d \end{array} \right ) \in \Gamma_1(p) \right \}. $$ \end{proposition} {\em Remark. } Of course one could find a finite set of generations for $\Gamma_{1,p}$ by doing so for $\mbox{SL}(2,{\Bbb{Z}})$ and for $\Gamma_1(p)$. \begin{Proof} It is easier to work, as in Chapter 1 of [HKW1], with $$ \tilde{\Gamma}_{1,p}=\left \{ \gamma \in \mbox{Sp} (\Lambda, {\Bbb{Z}})| \gamma \equiv \left ( \begin{array}{cccc} \ast & \ast & \ast & \ast\\ 0 & 1 & 0 & 0\\ \ast & \ast & \ast & \ast\\ 0 & 0 & 0 & 1 \end{array} \right ) \mbox{mod } p \right \} $$ where $\Lambda$ is the symplectic form $$ \left ( \begin{array}{cc} \phantom{-}0 & \begin{array}{cc} 1 & 0\\ 0 & p \end{array}\\ \begin{array}{cc} -1 & \phantom{-}0\\ \phantom{-}0 & -p \end{array} & 0 \end{array} \right ). $$ $\tilde{\Gamma}_{1,p}$ is an $\mbox{Sp}(4,{\Bbb{Q}})$-conjugate of $\Gamma_{1,p}$. More precisely $$ \tilde{\Gamma}_{1,p}= R {\Gamma}_{1,p} R^{-1} $$ where $$ R=\left ( \begin{array}{cccc} 1 &&&\\ & 1 &&\\ && 1 &\\ &&& p \end{array} \right ). $$ Thus we wish to show that $\tilde{\Gamma}_{1,p}$ is generated by $$ \tilde{M}_1=\left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ p & 0 & 0 & 1\\ \end{array}\right ),\qquad \tilde{M}_2=\left ( \begin{array}{cccc} 1 & 0 & 0 & 1\\ 0 & 1 & p & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{array}\right ) , $$ $$ \tilde{M}_3=\left ( \begin{array}{cccc} 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & -p & 1\\ \end{array}\right ),\qquad \tilde{M}_4=\left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ -p & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\\ \end{array}\right ), $$ $$ \tilde{\jmath}_1(\mbox{SL}(2,{\Bbb{Z}}))=\left \{ \left ( \begin{array}{cccc} a & 0 & b & 0\\ 0 & 1 & 0 & 0\\ c & 0 & d & 0\\ 0 & 0 & 0 & 1\\ \end{array}\right ) | \left ( \begin{array}{cc} a & b\\ c & d \end{array} \right ) \in \mbox{SL}(2,{\Bbb{Z}})\right \} $$ and $$ \tilde{\jmath}_2(\Gamma_1(p))=\left \{ \left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & a & 0 & b\\ 0 & 0 & 1 & 0\\ 0 & c & 0 & d\\ \end{array}\right ) | \left ( \begin{array}{cc} a & b\\ c & d \end{array} \right ) \in \Gamma_1(p) \right \}. $$ Recall that a vector $v \in {\Bbb{Z}}^4$ is short if and only if there exists $w \in {\Bbb{Z}}^4$ such that $v \Lambda ^t w =1:$ otherwise it is long. If $K \in \tilde{\Gamma}_{1,p}$ then the second row of $K$ is certainly long. Suppose the first row is long also. Then, since the first two rows of K span a $\Lambda$-isotropic plane, $K_{14}\equiv 0 \mbox{ mod } p$. But then $$ K\equiv\left ( \begin{array}{cccc} 0 & \ast & 0 & 0\\ 0 & 1 & 0 & 0\\ \ast & \ast & \ast & \ast\\ \ast & \ast & \ast & \ast\\ \end{array}\right ) \mbox{mod } p $$ so $K$ is not invertible mod $p$. But $K \in \mbox{Sp}(4,{\Bbb{Z}})$ and in particular $K$ is invertible.\\ So the first row of $K$ must be short. It is shown in the course of the proof of Proposition 3.38 in [HKW1] that, by multiplying $K$ on the right by a suitable product of $\tilde{M}_i$ and elements of $\tilde{\jmath}_1(\mbox{SL}(2,{\Bbb{Z}}))$, we may assume that the first row of $K$ is $(1, 0, 0, 0)$. Then the symplectic condition gives $$ K=\left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ \ast & a & 0 & c\\ \ast & \ast & 1 & \ast\\ \ast & b & 0 & d \end{array}\right ) $$ for some $\left( \begin{array}{cc} a & b\\ c & d \end{array}\right) \in \Gamma_1 (p)$. Multiplying by $\tilde {\jmath}_2\left( \begin{array}{cc} a & b\\ c & d \end{array}\right)^{-1}$ transforms this into $$ \left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ -np & 1 & 0 & 0\\ \ast & m & 1 & n\\ mp & 0 & 0 & 1 \end{array}\right ) $$ for some integers $m,n$. Multiplying on the right by $\tilde{M}_4^{-n} \tilde{M}_1^{-m}$ reduces to $$ \left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \ast & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right ) $$ which is in $\tilde{\jmath}_1(\mbox{\mbox{SL}}(2,{\Bbb{Z}}))$. This proves the proposition. \end{Proof} Now we produce all the generators using the allowable means described above. \begin{proposition}\label{5:p 5.2} $M_1, M_2, M_3, M_4, j_1({\operatorname{SL}}(2,{\Bbb{Z}}))$ and $j_2(\Gamma_1(p))$ can all be generated from $M_0$ and $\Gamma(p^2)$ by multiplication, inversion, and conjugation in $\Gamma^0_{1,p}$. \end{proposition} \begin{Proof} (i)\quad $M_0=j_1 \left( \begin{array}{cc} 1 & 1\\ 0 & 1 \end{array} \right ) $ and the smallest normal subgroup of $\mbox{SL}(2,{\Bbb{Z}})$ containing $M_0$ is the whole of $\operatorname{SL}(2,{\Bbb{Z}})$, e.g. [B]. Hence we can make $j_1(\operatorname{SL}(2,{\Bbb{Z}}))$.\\ (ii)\quad $ M^{-1}_4 M_0 M_4 M_0^{-1}= \left ( \begin{array}{ccc|c} & {\mbox{\boldmath$1$}} & & \begin{array}{cc} 0 & p\\ p & p^2 \end{array} \\ \hline & 0 \vphantom{\begin{array}{cc} 0 & p\\ p & p^2 \end{array} } & & {\mbox{\boldmath$1$}} \end{array} \right ) $ and $L_1= \left ( \begin{array}{ccc|c} &{\mbox{\boldmath$1$}}& & \begin{array}{cc} 0 & 0\\ 0 & p^2 \end{array} \\ \hline &0\vphantom{\begin{array}{cc} 0 & p\\ p & p^2 \end{array} } & & {\mbox{\boldmath$1$}} \end{array} \right ) \in \Gamma(p^2)$, so \\ $M_2=M_4^{-1} M_0 M_4 M_0^{-1} L_1^{-1}$ and we have made $M_2$.\\ (iii)\quad $ M_1 M_0 M_1^{-1} M_1^{-1} M_0 M_1= \left ( \begin{array}{cccc} 1 & 0 & 2 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & -2 & 0 & 1 \end{array}\right ) $ and since we got $j_1(\operatorname{SL}(2,{\Bbb{Z}}))$ in (i) we may multiply by $j_1\left ( \begin{array}{cc} 1 & -2\\ 0 & 1 \end{array}\right )$ to get $$ \left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & -2 & 0 & 1 \end{array}\right ) $$ Call this matrix $L_2$. Since $p$ is odd we can find integers $\lambda, \mu$ such that $-2\lambda + p^2 \mu=1$. Consider the matrix $L_3= j_2\left ( \begin{array}{cc} 1 & 0\\ p^3 & 1 \end{array} \right ) $, so $L_3\in \Gamma(p^2)$. We then have $$ L^{\lambda}_2 L^{\mu}_3 = \left ( \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 \end{array}\right ) = j_2 \left (\begin{array}{cc} 1 & 0\\ p & 1 \end{array} \right). $$ Call this element $L_4$. We also have $j_2 \left (\begin{array}{cc} 1 & p\\ 0 & 1 \end{array} \right) \in \Gamma (p^2)$, but we have not got all of $j_2(\Gamma_1(p))$ yet. However we need $L_4$ as an auxiliary.\\ (iv)\quad $ L_4 M_1 M_0 M_1^{-1} M_0^{-1}=M_3^{-1}$ so we can now make $M_3.$\\ (v)\quad Put $L_5=j_1\left (\begin{array}{cc} 1 & 0\\ 1 & 1 \end{array} \right ).$ Then $$ L_5^{-1} M_2^{-1} L_5 M_2 L_1^{-1} = M_4 $$ and we have made $M_4$.\\ (vi)\quad $M_3 L_5 L_4 M_3^{-1} L_5^{-1} L_2=M_1^{-1}$, so we can make $M_1$.\\ (vii)\quad It remains to show how to get all of $j_2(\Gamma_1(p))$. So far we have not used the freedom to conjugate by $\Gamma^0_{1,p}$ rather than just by $\Gamma _{1,p}$. Define $$ \Gamma_1'(p^2)=\left\{ Q\in\Gamma_1(p)|Q-{\mbox{\boldmath$1$}}=\left(\begin{array}{cc} p^2{\Bbb{Z}} & p{\Bbb{Z}}\\ p^3{\Bbb{Z}} & p^2{\Bbb{Z}} \end{array}\right) \right\}. $$ Thus $Q\in\Gamma'_1(p^2)$ if and only if $j_2(Q)\in \Gamma(p^2)$. If instead $Q\in \mbox{SL}(2,{\Bbb{Z}})$ then $j_2(Q)\in \Gamma^0_{1,p}$, and we have already generated $j_2(P)$, where $P=\left(\begin{array}{cc} 1 & 0\\ p & 1 \end{array}\right )$, so the problem is now to generate $\Gamma_1(p)$ using $P$, elements of $\Gamma'_1(p^2)$, and conjugation by elements of $\mbox{SL}(2,{\Bbb{Z}})$.\\ A general element of $\Gamma _1(p)$ is of the form $\left( \begin{array}{cc} \lambda p+1 & \alpha p\\ \beta p & \mu p+1 \end{array}\right )$ and since it has determinant equal to 1 $$ (\lambda + \mu)p+(\lambda\mu - \alpha\beta)p^2=0, $$ and in particular $\lambda + \mu\equiv 0 \mbox{ mod } p$. Suppose that $\lambda\equiv 0\mbox{ mod } p$, so that $\mu\equiv 0\mbox{ mod } p$ also. Then $$ P^{-\beta} \left ( \begin{array}{cc} \lambda p+1 & \alpha p\\ \beta p & \mu p+1 \end{array}\right ) = \left (\begin{array}{cc} \lambda p+1 & \alpha p\\ -\beta\lambda p^2 & -\alpha\beta p^2+\mu p+1 \end{array}\right ) $$ which is in $\Gamma_1'(p^2)$. So we can generate any element of $\Gamma_1(p)$ for which $\lambda\equiv 0\mbox{ mod } p$.\\ Suppose then that $(\lambda, p)=1$. If $(\alpha, p)=1$ also then $p|(\lambda-k\alpha)$ for some integer $k$. But now we can conjugate by $\left ( \begin{array}{cc} 1 & 0\\ k & 1 \end{array}\right ) \in \mbox{SL}(2,{\Bbb{Z}})$: $$ \left( \begin{array}{cc} 1 & 0\\ k & 1 \end{array}\right) \left ( \begin{array}{cc} \lambda p+1 & \alpha p\\ \beta p & \mu p+1 \end{array}\right ) \left( \begin{array}{cc} 1 & 0\\ -k & 1 \end{array}\right)= \left ( \begin{array}{cc} (\lambda -k\alpha)p+1 & \alpha p\\ \beta' p & (\mu+k\alpha) p+1 \end{array}\right ) $$ and so we can generate all elements of $\Gamma_1(p)$ for which $\lambda\equiv0\mbox{ mod } p$ or $ \alpha\not\equiv 0 \mbox{ mod } p$. The remaining elements are of the form $$ \left( \begin{array}{cc} \lambda p+1 & \alpha' p^2\\ \ast & \mu p+1 \end{array} \right ) $$ and we multiply by $\left( \begin{array}{cc} 1 & p\\ 0 & 1 \end{array}\right) \in \Gamma'_1(p^2)$: $$ \left( \begin{array}{cc} 1 & p\\ 0 & 1 \end{array}\right) \left( \begin{array}{cc} \lambda p+1 & \alpha' p^2\\ \ast & \mu p+1 \end{array} \right )= \left( \begin{array}{cc} \ast & (\alpha'+\mu) p^2+p\\ \ast & \mu p+1 \end{array} \right ) $$ and $(\alpha'+\mu)p+1\not\equiv 0 \mbox { mod } p$ so this is something we have already generated. \end{Proof \section*{References} \begin{enumerate} \item [{[B]}] J.L. Brenner, The linear homogeneous group III. Ann. Math {\bf71} (1960), 210-223. \item[{[G]}] J. Grosche, \"Uber die Fundamentalgruppen von Quotientr\"aumen Siegelscher Modulgruppen, J. reine angew. Math. {\bf 281} (1976), 53-79. \item[{[HK]}] H. Heidrich \& F.W. Kn\"oller, \"Uber die Fundamentalgruppen Siegelscher Modulvariet\"aten vom Grade 2, Manuscr. Math. {\bf 57} (1987), 249-262. \item[{[H]}] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. Math. {\bf 79} (1964), 109-326. \item[{[HM]}] G. Horrocks \& D. Mumford, A rank 2 vector bundle on ${\Bbb{P}}^4$ with 15,000 symmetries, Topology {\bf 12} (1973), 63-81. \item[{[HKW1]}] K. Hulek, C. Kahn \& S.H. Weintraub, Moduli spaces of abelian surfaces: compactification, degenerations and theta functions, de Gruyter, Berlin 1993. \item[{[HKW2]}] K. Hulek, C. Kahn \& S.H. Weintraub, Singularities of the moduli space of certain Abelian surfaces, Comp. Math. {\bf 79} (1991), 231-253. \item[{[HS]}] K. Hulek, G.K. Sankaran, The Kodaira dimension of certain moduli spaces of Abelian surfaces, Comp. Math., to appear. \item[{[K]}] F.W. Kn\"oller, Die Fundamentalgruppen der Siegelschen Modulvariet\"aten, Abh. Math. Sem. Univ. Hamburg {\bf 57} (1987), 203-213. \item[{[MS]}] N. Manolache \& F. Schreyer, preprint 1993. \item[{[M]}] J. Mennicke, Zur Theorie der Siegelschen Modulgruppe, Math. Annalen {\bf 159} (1965), 115-129. \end{enumerate} \begin{tabular*}{13cm}{l@{\extracolsep{\fill}}l} K. Hulek & G.K. Sankaran\\ Institut f\"ur Mathematik & Department of Pure Mathematics\\ Universit\"at Hannover & and Mathematical Statistics\\ Postfach 6009 & University of Cambridge\\ D-30060 Hannover & Cambridge CB2 1SB\\ Germany & England \end{tabular*} \end{document}

9608

alg-geom/9608034

en

https://arxiv.org/abs/alg-geom/9608034

alg-geom/9608034

Javier Elizondo

E. Javier Elizondo

The ring of global sections of multiples of a line bundle on a toric variety

To appear in Proceedings of the AMS, one figure, 5 pages, Author-supplied DVI file available at: http://calli.matem.unam.mx/investigadores/javier/investigacion.html LaTeX2e Sub-Class: 14C20 14M25

In this article we prove, in a simple way, that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

alg-geom

\section{Introduction} Let us start by stating the result and a few consequences. Let $X$ be an algebraic variety which is complete over a field $K$, and let $D$ be any effective Cartier divisor on $X.$ We denote by ${\cal O}(nD)$ the line bundle associated to $nD,$ with $n \geq 0.$ It is a very interesting problem to know if the ring \begin{equation} \label{ring} R\, := \, \bigoplus_{n\geq 0} \, H^0 \,(X, {\cal O}(nD)) \end{equation} is a finitely generated $K$-algebra. It was O. Zariski who worked out the case of algebraic surfaces in order to solve the Riemann-Roch problem, and gave examples where $R$ is not finitely generated, see \cite{zar-roch}. In this article we prove that the ring $R$ is finitely generated when $X$ is a complete toric variety (perhaps singular). The ring $R$ appears in many interesting problems, for example, if $X$ is a nonsingular projective variety, then the finite generation of $R$ implies that the series \begin{equation} \label{series} \sum_{n \geq 0} \, \dim H^0 (X, {\cal O}(nD)) \, t^n \end{equation} is rational. It was asked by S. D. Cutkosky and V. Srinivas \cite{sri-zar} if this series is rational when $D$ is a nef divisor. Our result gives trivially a positive answer in the case of a complete toric variety, and $D$ any effective Cartier divisor. Furthermore, the rationality of the series also allows us to compute, at least theoretically, the dimension of $ H^0 (X, {\cal O}(nD))$ in terms of $n$. This is the Riemann-Roch problem, see for example \cite{zar-roch} and \cite{sri-zar}. Observe that these dimensions are not given by the Riemann-Roch theorem since the variety $X$ can be singular, and the line bundle ${\cal O}(nD)$ is not necessarily generated by its global sections. We would like to make a final remark. The ring $R$ is a subring of the ring $S \, = \, \bigoplus_{D\geq{0}} \, H^0 (X, {\cal O}(D))$, which is finitely generated in the case of toric varieties. In fact, D. Cox proves in \cite{cox-hom} that the ring $S$ is a polynomial ring graded by the the monoid of effective divisors classes in the Chow group $A_{k-1}(X)$ of $X$, where $k$ is the dimension of $X$. \section{The ring of global sections} In this section we prove that the ring $R$, which was defined by equation (\ref{ring}), is a finitely generated $K$-algebra. Throughout this section $X$ means a complete toric variety over a field $K,$ and $D$ a Cartier divisor in $X.$ The construction of the cone $C_R$, in the proof of the Theorem, is a well know construction, see for example ~\cite{vmh-batyrev}.\ We start by recalling an important lemma that will be use in the proof of the theorem. \begin{lemma}[Gordan] \label{gordan} If $\sigma$ is a strongly convex rational polyhedral cone, and \, ${\sigma}^{\vee}$ its dual, then $S_{\sigma} \, = \, {\sigma}^{\vee} \, \cap \, M$ is a finitely generated semigroup. \end{lemma} Now, we are ready for the main result. \begin{theorem} \label{main} Let $X$ be a complete toric variety, perhaps singular, and let $D$ be an Cartier divisor in $X$. Then the ring $$ R\, := \, \bigoplus_{n\geq 0} \, H^0 (X, {\cal O}(nD)) $$ is finitely generated as a $K$-algebra. \end{theorem} \begin{proof} We first observe that we can consider our divisor $D$ to be $T$-invariant. For the following sequence is exact, see ~\cite[page 116]{dan-tova} $$ 0 \, \longrightarrow \, M \, \longrightarrow \, \Div \, X \, \longrightarrow \, \Pic \,(X) \longrightarrow \, 0 $$ where \, $\Div \, X$ \, is the group of $T$-invariant Cartier divisors and $M := {\Hom}_\field{Z} (N, \field{Z})$ is the dual lattice of $N \cong {\field{Z}}^k$. Then we can write $D$ as \,$D = \, \sum_{i=1}^{s} \, a_i D_i$ with $\{D_i\}$ the set of invariant divisors. We consider the convex rational polyhedron $P_{nD}$ in $M_{\field{Q}} \, := \, M {\bigotimes}_{\field{Z}} \field{Q}$ defined as $$ P_{nD} \, = \, \{u \, \in \, M_{\field{Q}} \, | \, <u,v_i> \, \, \geq \, -n{a_i} \, \mbox{ for all } \, i \} $$ where $v_i$ is the the first element in the lattice appearing in the divisor $D_i$. We know that generators for the space $H^0 (X, {\cal O}(nD) )$ are given by the elements of\, $P_{nD} \, \cap \, M$, which is a finite set because $X$ is a complete variety, this also implies that $P_D$ is a rational convex polytope. Let us embed the k-dimensional {\field{Q}}\,-vector space ${M}_{\field{Q}}$ into the (k+1)-dimensional {\field{Q}}\,-vector space ${M}^{k+1}_{\field{Q}} \, := \, {\field{Z}}^{k+1} \bigotimes_{\field{Z}} \, \field{Q}$ as the hyperplane with equation $x_{k+1} \, = \, 1$, where $(x_1, \ldots , x_{k+1})$ are coordinates for ${M}^{k+1}_{\field{Q}}$. Denote by $C_R$ the ($k+1$)-dimensional cone in ${M}^{k+1}_{\field{Q}}$ generated by the rays starting at the origin and passing through the vertices of $P_D$ (see figure below).\vspace{1cm} \hspace{4cm} \includegraphics{toricfig.eps} \begin{center} {\em The cone \mbox{$C_R$}} \end{center} It follows from the definition that $P_{nD} = n P_{D}$, and this implies that the intersection of the hyperplane $x_{k+1} = n$ with the cone $C_R$ is just the polyhedron $P_{nD}$. In other words, the cone $C_R$ is the cone associated to the ring $R$, in the sense that any element of the ring $R$ is a finite linear combination of integral points of $C_R \cap {\field{Z}}^{k+1}.$ The theorem follows since the semigroup $C_R \cap {\field{Z}}^{k+1}$ is finitely generated by Lemma ~\ref{gordan} (Gordan). \end{proof} \begin{ack} I thank very much V. Srinivas for suggesting me this problem and for many wonderful and fruitful conversations. I also would like to thank A. King for bringing to my attention the article of D. Cox, and to the referee for pointing out the article of V. Batyrev. \end{ack}

9608

alg-geom/9608007

en

https://arxiv.org/abs/alg-geom/9608007

alg-geom/9608007

Yuri G. Prokhorov

Yuri G Prokhorov

On the general elephant conjecture for Mori conic bundles

19 pages LaTeX

Russian Acad. Sci. Sb. Math., 188(11):1665-1685, 1997

Let $f:X\to S$ be an extremal contraction from a threefolds with terminal singularities onto a surface (so called Mori conic bundle). We study some particular cases of such contractions: quotients of usual conic bundles and index two contractions. Assuming Reid's general elephants conjecture we also obtain a rough classification. We present many examples.

alg-geom

\section*{Introduction.} This paper continues our study of extremal contractions from threefolds to surfaces \cite{Pro}, \cite{Pro1}, \cite{Pro2}, \cite{Pro3}. Such contractions occur naturally in birational classification theories of three-dimensional algebraic varieties. We are interested in the local situation. \de{Definition.} Let $(X,C)$ is a germ of three-dimensional normal complex space $X$ along a compact reduced curve $C$ and let $(S,o)$ be a germ of a normal two-dimensional complex space $S$ in $o\in S$. Assume that $X$ has at worst terminal singularities. We say that proper morphism $f:(X,C)\to (S,o)$ is a {\it Mori conic bundle} if \begin{itemize} \item[{\rm (i)}] $f^{-1}(o)_{\mt{red}}=C$, \item[{\rm (ii)}] $f_*\OOO_X=\OOO_S$, \item[{\rm (iii)}] $-K_X$ is $f$-ample. \end{itemize} The first example of Mori conic bundles are conic bundles in the classical sense:\quad $f:(X,C)\to (S,o)$ is called a (usual) {\it conic bundle} if $(S,o)$ is nonsingular and there exists an embedding $i:(X,C)\hookrightarrow \PP^2\times(S,o)$ such that $\OOO_{\PP^2\times S}(X)=\OOO_{\PP^2\times S}(2,0)$. \ede The general question arising very naturally lies in the classification problem of Mori conic bundles. The following two conjectures are interesting for application of the Sarkisov program to study of birational properties of threefolds with structure of conic bundle (see \cite{Iskovskikh}, \cite{Iskovskikh1}). \th{Conjecture I (special case of Reid's general elephants conjecture).} \label{elephant} Let $f: (X,C)\to (S,o)$ be a Mori conic bundle. Then a general member of the anticanonical linear system $|-K_X|$ has only DuVal singularities. \par\endgroup \th{Conjecture II.} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle. Then $(S,o)$ is either nonsingular or a DuVal singularity of type $A_n$. \par\endgroup One can expect that a Mori conic bundle with very singular base is a quotient of a usual conic bundle by a cyclic group (for example it follows from the conjecture \ref{elephant}, see \ref{g.e.}). So this particular case seems to be general. Such quotients are classified in Section \ref{qb}. Surpisingly situation here is not very complicated and there are only three cases. We also study index two Mori conic bundles in Section \ref{s-ind-2}. Assuming conjecture \ref{elephant} we obtain a rough classification of Mori conic bundles in Section \ref{sect-el}. In particular, we will show (see also \cite{Pro}) that the first conjecture implies the second one. We present many examples. The methods are completely elementary. None of the techniques of Mori \cite{Mori-flip} or Shokurov \cite{Shokurov} will be used. Almost all the results of this work were announced in \cite{Pro1} and \cite{Pro2}. \par {\sc Acknowledgments.} I have been working on this problem at Max-Plank Institut f\"ur Matematik in February-April 1995 and at the University of Warwick in November-December 1995. The author would like to thank staffs of these institutes for hospitality. Different aspects of this work were discussed with Professors V.~A.~Iskovskikh, V.~V.~Shokurov and M.~Reid. I am very grateful to them for help and advices. \section{Preliminary results} Throughout this paper a variety is a reduced irreducible complex space. On a normal variety $X$ by $K_X$ we will denote its canonical (Weil) divisor. If $X$ is a variety and $C\subset X$ is its closed subvariety, then $(X,C)$ is a germ of $X$ along $C$. Sometimes we will replace $(X,C)$ by its sufficiently small representative $X$. When we say that a variety $X$ has terminal singularities, it means that singularities are not worse then that, so $X$ can be nonsingular. \par We need some facts about three-dimensional terminal singularities. \de{}\label{terminal} Let $(X,P)$ be a terminal singularity of index $m\ge 1$ and let $\pi:(X\3,P\3 )\to (X,P)$ be the canonical cover. Then $(X\3,P\3 )$ is a terminal singularity of index 1. It is known \cite{Pagoda} that $(X\3,P\3)$ is a hypersurface singularity, i.~e. there exists an $\cyc{m}$-equivariant embedding $(X\3,P\3 )\subset (\CC^4,0)$. \ede \subth{Theorem \cite{Danilov}.} \label{Danilov} If in notations above $(X\3,P\3 )$ is smooth, then it is $\cyc{m}$-isomorphic to $(\CC^3_{x_1,x_2,x_3},0)$ with the action of $\cyc{m}$ by $$ (x_1,x_2,x_3)\longrightarrow (\varepsilon^{a} x_1,\varepsilon^{-a} x_2,\varepsilon^{b} x_3), $$ where $\varepsilon=\exp(2\pi i/m)$, and $a$, $b$ are integers prime to $m$. Conversely every such singularity is terminal. \esubth Such singularity is denoted by $\frac{1}{m}(a,-a,b)$ or $\CC^3/\cyc{m}(a,-a,b)$. \subth{Theorem \cite{Mori-term}, \cite{RYPG}.} \label{cl-term} In notations above assume that $(X\3,P\3)$ is singular, then it is $\cyc{m}$-isomorphic to a hypersurface $\{\phi (x_1,x_2,x_3,x_4)=0\}$ in $(\CC^4_{x_1,x_2,x_3,x_4},0)$, where there are two cases for the action of $\cyc{m}$ \par {\rm (i)} (the main series) $(x_1,x_2,x_3,x_4;\phi)\longrightarrow (\varepsilon^{a} x_1,\varepsilon^{-a} x_2,\varepsilon^{b} x_3,x_4;\phi), $ where $\varepsilon=\exp(2\pi i/m)$, and $a$, $b$ are integer prime to $m$. \par {\rm (ii)} (the exceptional series) $m=4$ and $(x_1,x_2,x_3,x_4;\phi)\to (ix_1,-ix_2,i^{a} x_3,-x_4;-\phi)$, where $a=1$ or $3$. \esubth \subde{Remark.} Terminal singularities of index $>1$ are classified by type of general member $\in |-K_X|$. For example there are four types of index two terminal singularities $cA/2$, $cAx/2$, $cD/2$ and $cE/2$ \cite{RYPG} (we use notations of \cite{KoM}). \esubde \th{Lemma.} \label{st} Let $(X,P)$ be a germ of terminal threefold singularity and let $F\in |-K_{(X,P)}|$ be an element of anticanonical linear system. If $F$ is an irreducible nonsingular surface, then $(X,P)$ is nonsingular. \par\endgroup \subde{Proof.} If $(X,P)$ is of index one, then $F$ is Cartier and $(X,P)$ is nonsingular in this case. So we assume that $(X,P)$ is of index $m>1$. Consider the canonical cover $\pi:(X\3,P\3)\to (X,P)$ and let $F\3:=\pi^{-1}(F)$. Since $\pi$ is \'etale outside $P$ and $F-\{P\}$ is simply connected, the restriction $\pi:F\3\to F$ splits non-trivially, so $F\3=F\3_1+\dots+F\3_m$. But then $P\3$ is the only point of intersection of components $F\3_i$. On the other hand each $F\3_i$ is $\QQ$-Cartier, a contradiction. \qq \esubde It is well known that every DuVal (and, more general, log-terminal) singularity $(F,P)$ is a quotient of a nonsingular germ $(\CC^2,0)$ by a finite group $G$ acting on $\CC^2$ free outside $0$. The order of $G$ is called {\it the topological index } of $(F,P)$. Similar to \ref{st} one can prove the following. \subth{Lemma.}\label{index} Let $(X,P)$ be a germ of a terminal threefold singularity of index $m>1$ and $F\in |-K_{(X,P)}|$ be an anticanonical divisor. Assume that the surface $F$ is reduced, irreducible and the point $(F,P)$ is DuVal of topological index $n$. Then $n$ is divisible by $m$. Moreover if $n=m$, then $(X,P)$ is a cyclic quotient singularity and $(F,P)$ is of type $A_{m-1}$. \esubth Now we present some elementary properties of Mori conic bundles. \th{Theorem \cite{Cut} (see also \ref{non-sing}).} \label{Cut} Let $f:X\to S$ be a Mori conic bundle. Assume that $X$ has only singularities of index $1$. Then $S$ is a nonsingular surface and $f$ is a conic bundle (possibly singular). \par\endgroup The following statement is a consequence of the Kawamata-Viehweg vanishing theorem. \th{Proposition (cf. \cite[(1.2)]{Mori-flip}).} Let $f:X\to S$ be a Mori conic bundle. Then $$ R^if_*\OOO_X=0, \quad i>0. $$ \par\endgroup \subth{Corollary (cf. \cite[(1.3)]{Mori-flip}).} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle. Then \par {\rm (i)} the fiber $C$ is a tree of rational curves, i.~e. $p_a(C_0)=0$ for any one-dimensional subscheme $C_0\subset C$. \par {\rm (ii)} \label{Pic} $\mt{Pic}(X)\simeq H^2(C,\ZZ)\simeq\ZZ^{\rho}$, \quad where $\rho$ is the number of components of $C$. \esubth \de{Construction.} \label{construction} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle. Assume that $(S,o)$ is a singular point. By easy remark in \cite{KoMM} (see also \cite{Ishii}), $(S,o)$ is a quotient singularity. It means that $(S,o)$ is a quotient of nonsingular point $(S',o')$ by a finite group $G$, where the action of $G$ on $S'-o'$ is free. Therefore there exists a faithful representation $G\hookrightarrow GL(T_{o',S'})=GL_2(\CC)$. Let $g:S'\to S$ be the quotient morphism and let $X'$ be the normalization of $X\times_S S'$. We have the following commutative diagram $$ \label{diagram} \begin{array}{ccc} (X',C')&\stackrel{h}{\longrightarrow}&(X,C)\\ \downarrow\lefteqn{\scriptstyle{f'}}&&\downarrow\lefteqn{\scriptstyle{f}}\\ (S',o')&\stackrel{g}{\longrightarrow}&(S,o)\\ \end{array} $$ where $C':=f'^{-1}(o')$. Then $G$ acts on $(X',C')$ and obviously $X=X'/G$, $C=C'/G$. Since the action of $G$ on $S'-o'$ is free, so is the action on $X'-C'$. Therefore $X'$ has only terminal singularities, because $\mt{codim}(C')=2$ (see e.~g. \cite[6.7]{CKM}). It gives us also that the action is free outside a finite number of points $Q_1,\dots,Q_k\in X'$ (for each $Q_i$ its image $P_i=h(Q_i)$ has index $>1$). Since $h:X'\to X$ has no ramification divisors, the anticanonical divisor $-K_{X'}=h^*(-K_X)$ is ample over $S'$. We obtain a new Mori conic bundle $f':(X',C')\to (S',o')$ over a nonsingular base. \par Professor S. Mukai pointed out that actually $X\times_S S'$ is normal, so we can take $X'=X\times_S S'$. \ede \th{Lemma (see \cite{Pro}, \cite{Pro1}, \cite{Ko}).} \label{cyc} In notations above $G$ is cyclic and has at least one fixed point on $X'$. In particular, $(S,o)$ is a cyclic quotient singularity. \par\endgroup \de{Proof.} Since $p_a(C')=0$, it is easy to prove by induction on the number of components of $C$ that $G$ has either a fixed point $Q\in C'$ or an invariant component $C_0'\subset C'$, $C_0'\simeq \PP^1$. But in the second case we have two inclusions $G\subset PGL_2(\CC)$ and $G\subset GL_2(\CC)$, where $G\subset GL_2(\CC)$ contains no quasireflections. By the classification of finite subgroups in $PGL_2(\CC)\simeq SO_3(\CC)$, $G$ is cyclic and has two fixed points on $C_0'$. Therefore in any case $G$ has a fixed point $Q\in X'$. Take a small neighborhood $U\subset X$ of $P:=h(Q)$. We have a surjective map $\pi_1(U-P)\to G$. But $\pi_1(U-P)$ is cyclic because $(X,P)$ is a quotient of a hypersurface singularity by a cyclic group. Whence so is $G$. This proves our claim. \ede \subth{Corollary (cf. \cite{Cut}).} \par {\rm (i)} \label{non-sing} If $X$ has index one, then $(S,o)$ is nonsingular. \par {\rm (ii) } \label{|G|=2} If $X$ has index two, then $(S,o)$ is either nonsingular or DuVal of type $A_1$. \par {\rm (iii) } \label{>n} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle. If $(S,o)$ is a cyclic quotient singularity of type $\frac{1}{n}(a,b)$, then $X$ contains at least one point of index $\ge n$. \esubth \section{Quotients of conic bundles. } \label{qb} By \ref{construction} and by \ref{cyc} every Mori conic bundle $f:(X,C)\to (S,o)$ over a singular base is a quotient of another Mori conic bundle $f':(X',C')\to (S',o')$ with a nonsingular base by a cyclic group. In this section we classify Mori conic bundles $f:(X,C)\to (S,o)$ under the assumption that $X'$ is Gorenstein (and then $f':(X',C')\to (S',o')$ is a conic bundle by \ref{Cut}). First we present several examples. \de{Example}\label{ex1} (Toric example). Let $\PP^1 \times \CC^2 \to\CC^2$ be the standard projection. Define the action of the group $\cyc{n}$ on $\CC^2_{u,v}$ and $\PP^1_{x_0,x_1} \times\CC^2_{u,v}$: $$ (x_0,x_1;u,v)\to (x_0,\varepsilon x_1; \varepsilon^{a} u, \varepsilon^{-a}v), $$ where $\varepsilon =\exp (2\pi i/n)$, $a\in\NN$ and $(n,a)=1$. Denote $X=(\PP^1 \times\CC^2 )/\cyc{n}$, $S=\CC^2 /\cyc{n}$. Then the projection $f:X\to S$ is a Mori conic bundle. The threefold $X$ has on the fiber $f^{-1}(0)$ exactly two terminal points $P_1$, $P_2$ which are cyclic quotients of type $\frac{1}{n}(a,-a,\pm 1)$, the surface $S$ has in 0 a DuVal point of type $A_{n-1}$. \ede \de{Example.}\label{ex3} Consider the following hypersurface in $\PP^2_{x_0,x_1,x_2}\times\CC^2_{u,v}$: $$ X':=\{x_0^2+vx_1^2+\psi(u,v)x_1x_2+ux^2_2=0\}, $$ where $\psi(0,0)=0$. Let $n$ be an odd integer, $n=2q+1$, where $q\in\NN$. Define an action of $\cyc{n}$ on $\PP^2\times\CC^2$ by $$ (x_0,x_1,x_2,u,v)\to (x_0,\varepsilon^{-q}x_1,\varepsilon^{q} x_2,\varepsilon u,\varepsilon^{-1}= v), $$ where $\varepsilon=\exp(2\pi i/n)$. If $\psi(u,v)$ is an invariant, then $\cyc{n}$ acts naturally on $X'$. As in \ref{ex1} the $f:X'/\cyc{n}\to\CC^2/\cyc{n}$ is a Mori conic bundle. The singular locus of $X'/\cyc{n}$ consist of two terminal cyclic quotient points of index $n$. The point $(S,o)$ is DuVal of type $A_{n-1}$. \ede \de{Example.}\label{ex2} Let $X'$ be a hypersurface in $\PP^2 _{x_0,x_1,x_2} \times\CC^2_{u,v}$, defined by the equation $$ x_0^2+x_1^2+x_2^2\phi(u,v)=0, $$ where $\phi(u,v)$ has no multiple factors and contains only monomials of even degree. Denote by $f':X'\to\CC^2$ the natural projection. Then $X'$ has only one singular point $P'=(x_0=x_1=u=v=0)$ on ${f'}^{-1}(0)$. Define the action of $\cyc{2}$ on $X'$ and $\CC^2$ by $$ (x_0,x_1,x_2, u,v)\to (-x_0,x_1,x_2,-u,-v). $$ Let $X=X'/\cyc{2}$, $S=\CC^2/\cyc{2}$. The only fixed point on $X'$ is $P'$ it gives us a unique point $P\in X$ of index two. The variety $X$ has no other singular points. The surface $S$ has a DuVal singularity of type $A_1$ at $0$. There are two cases for $\phi (u,v)$: \par (1) $\mt{mult}_{(0,0)}(\phi)=2$, then $(X,P)$ is terminal of type $cA/2$; \par (2) $\mt{mult}_{(0,0)}(\phi)\ge 4$, then $(X,P)$ is terminal of type $cAx/2$. \ede \th{Theorem.} \label{th1} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle. Assume that $f$ is a quotient of a conic bundle $f':(X',C')\to (\CC^2,0)$ by $\cyc{n}$ such that the action of $\cyc{n}$ on $\CC^2-\{0\}$ is free. Then there exists an analytic isomorphism between $f:(X,C)\to (S,o)$ and one of examples \ref{ex1}, \ref{ex3} or \ref{ex2}. In particular, $(S,o)$ is DuVal of type $A_{n-1}$. \par\endgroup \de{Proof.} By Cartan's lemma \cite{Car}, one can choose coordinates $u, v$ in $\CC^2$ such that the action of $\cyc{m}$ is $(u,v)\longrightarrow (\varepsilon^a u,\varepsilon^b v)$, where $\varepsilon:=\exp (2\pi /m)$, $(a,m)=(b,m)=1$. \par Let $X'_0:=f'^{-1}(0)$ be the scheme-theoretical fiber over $0$. Then $X'_0$ isomorphic to a conic in $\PP^2$ (see \ref{Cut}). There are the following cases. \ede \de{Case I.} \underline{$X'_0$ is a non-degenerate conic.} \label{caI} Then in the analytic situation $X'\simeq\PP^1\times\CC^2$ by the Grauert-Fisher theorem and we may assume that the action of $\cyc{n}$ in some coordinate systems $(x_0,x_1)$ in $\PP^1$ and $(u,v)$ in $\CC^2$ is $$ (x_0,x_1;u,v)\longrightarrow (x_0,\varepsilon x_1;\varepsilon^a u,\varepsilon^b v), \qquad\qquad (a,m)=(b,m)=1, $$ where we may take $\wt(x_0)=0$ because $(x_0,x_1)$ are homogeneous coordinates. There are exactly two fixed points on $X'$: $$ Q_0=\{x_0=u=v=0\},\qquad Q_1=\{x_1=u=v=0\}. $$ They give us two points of index $n$ on $X'/\cyc{n}$ of types $\frac{1}{n}(-1,a,b)$ and $\frac{1}{n}(1,a,b)$, respectively. By \ref{Danilov} these two points are terminal only if $a+b=n$ (recall that $a$, $b$ are defined modulo $n$. We obtain the example \ref{ex1}. \ede \de{Case II. } \underline{The fiber $X'_0$ is reducible.} Then $X'_0=L_1+L_2$ is pair of lines intersecting each other in one point, say $Q$, which is a fixed point. Let $P:=h(Q)$. We fix a generator $s\in\cyc{n}$ and for $\cyc{n}$-semi-invariant $z$ define weight $\wt(z)$ as an integer defined modulo $n$ such that $$ \wt(z)\equiv a\mod m\qquad {\rm iff} \qquad s(z)=\varepsilon^az, $$ where $\varepsilon =\exp{2\pi i/n}$. \par Similar to \ref{caI} consider an $\cyc{n}$-equivariant embedding $X'\subset \PP^2_{x_0,x_1,x_2}\times\CC^2_{u,v}$ such that $x_0,x_1,x_2$ are semi-invariants with $$ \wt(x_0,x_1,x_2;u,v)=(0,p,q;a,b),\qquad (a,n)=(b,n)=1, $$ where we may take $\wt(x_0)=0$ because $(x_0,x_1,x_2)$ are homogeneous coordinates. There are two subcases. \subde{Subcase (i).} \underline{Components of $X_0'$ are $\cyc{n}$-invariant.} \label{non-per} We will derive a contradiction. One can change homogeneous coordinates $x_0,x_1,x_2$ in $\PP^2$ so that $X'_0\subset\PP^2$ is $\{x_0x_1=0\}$. Then $X'$ is given by the equation $$ x_0x_1+\varphi_0(u,v)x_0^2+\varphi_1(u,v)x_0x_1+\varphi_2(u,v)x_0x_2+ \phi(u,v)x_1^2+\psi(u,v)x_1x_2+\zeta(u,v)x^2_2=0, $$ where $\varphi_0(0,0)= \varphi_1(0,0)= \varphi_2(0,0)= \phi(0,0)= \psi(0,0)= \zeta(0,0)=0$. By taking $x_1'=x_1+\varphi_0x_0+\varphi_1x_1+\varphi_2x_2$ we obtain a new equation for $X'$: $$ x_0x_1+\phi(u,v)x_1^2+\psi(u,v)x_1x_2+\zeta(u,v)x^2_2=0, \leqno \abc \label{equation} $$ where $\phi(u,v)$, $\psi(u,v)$, $\zeta(u,v)$ are semi-invariants with suitable weights and $\phi(0,0)=\psi(0,0)=\zeta(0,0)=0$. Then $Q=\{x_0=x_1=u=v=0\}$ and $y_0:=x_0/x_2, y_1=:x_1/x_2, u, v$ are local coordinates in $\PP^2\times\CC^2$ near $Q$. We have $$ (X,P)=\{y_0y_1+\phi(u,v)y_1^2+\psi(u,v)y_1+\zeta(u,v)=0\} /\cyc{n}(-q,p-q,a,b). \leqno \abc \label{pop} $$ The action of $\cyc{n}$ on $X'$ has two more fixed points $Q_1:=\{x_0=x_2=u=v=0\}$ and $Q_2:=\{x_1=x_2=u=v=0\}$. Then $z_0=x_0/x_1, z_2=x_2/x_1, u, v$ are local coordinates in $\PP^2\times\CC^2$ near $Q_1$ and similarly $t_1=x_1/x_0, t_2=x_2/x_0, u, v$ are local coordinates in $\PP^2\times\CC^2$ near $Q_2$. Let $P_i=h(Q_i)$, $i=1,2$. Similar to \ref{pop} $$ \begin{array}{l} (X,P_1)=\{z_0+\phi(u,v)+\psi(u,v)z_2+\zeta(u,v)z_2^2=0\} /\cyc{n}(-p,q-p,a,b)\simeq\\ \CC^3_{z_2,u,v}/\cyc{n}(q-p,a,b),\\ \end{array} \leqno \abc \label{pop1} $$ $$ \begin{array}{l} (X,P_2)=\{t_1+\phi(u,v)t_1^2+\psi(u,v)t_1t_2+\zeta(u,v)t_2^2=0\} /\cyc{n}(p,q,a,b)\simeq\\ \CC^3_{t_2,u,v}/\cyc{n}(q,a,b). \end{array} \leqno \abc \label{pop2} $$ Since the action has a zero-dimension fixed locus, $$ (q,n)=(p-q,n)=1. \leqno \abc \label{=1} $$ By \ref{cl-term} $(X,P)$ a cyclic quotient. It is possible only if $\zeta(u,v)$ contains either $u$ or $v$ terms. Up to permutation of $u, v$ we may assume that $\zeta=u+\dots$. From \ref{equation} we have $\wt(x_0x_1)=\wt(ux_2^2)$. Thus $p=a+2q$. Then $$ (X,P)\simeq \CC^3_{y_0,y_1,v} /\cyc{n}(-q,p-q,b)=\CC^3/\cyc{n}(-q,a+q,b). \leqno \abc \label{popp} $$ We claim that $a+b=0$. Indeed otherwise $n>2$ and from \ref{pop1} we have $b=a+q$ because $p=a+2q$ and $(q,n)=1$. Whence $(X,P)=\CC^3/\cyc{n}(-q,b,b)$. This point cannot be terminal if $n>2$. \par The contradiction shows that $a+b=0$. Point $(X,P)$ is terminal in this case only if $q=b$ (see \ref{popp}). But then $p-q=a+2q-q=0$, a contradiction with \ref{=1}. \esubde \subde{Subcase (ii).} \underline{$\cyc{n}$ permutes components of $X_0'$.} Then $n$ is even, $n=2k$. If $k>1$, then the quotient $X'/\cyc{k}\to\CC^2/\cyc{k}$ is a Mori conic bundle as in \ref{non-per}. We have proved that this is impossible. Therefore $k=1$, $n=2$. Then it is easy to see that $\wt(u)=\wt(v)=1$. We may assume also that the fiber $X'_0\subset\PP^2$ is $\{x_0^2+x_1^2=0\}$. Since $\cyc{2}$ permutes components of $\{x_0^2+x_1^2=0\}$, we may assume that $\cyc{2}$ acts on $x_0$, $x_1$ by $x_0\to -x_0$, $x_1\to x_1$. Then one can change the coordinate system $(x_0,x_1,x_2;u,v)$ such that in $\PP^2\times\CC^2$ the variety $X'$ is given by the equation $$ x_0^2+x_1^2+\phi(u,v)x^2_2=0, \leqno \abc \label{equation1} $$ where $\phi(u,v)$ is an invariant with $\phi(0,0)=0$. It means that $\phi$ contains only monomials of even degree. Then $Q=\{x_0=x_1=u=v=0\}$ is the only fixed point on $C'$ and $y_0:=x_0/x_2, y_1=:x_1/x_2, u, v$ are local coordinates in $\PP^2\times\CC^2$ near $Q$. Whence $$ (X,P)=\{y_0^2+y_1^2+\phi(u,v)=0\}/\cyc{2}. $$ If this point is terminal, then up to permutation of $y_0$, $y_1$ we may assume that $\wt(y_0)=1$, $\wt(y_1)=0$. Thus $\wt(x_2)=0$. Finally singularities of $X'$ are isolated only if $\phi=0$ is a reduced curve. Thus we have the example \ref{ex2}. \esubde \ede \de{Case III.} \underline{$X'_0$ is a double line.} As above consider an $\cyc{n}$-equivariant embedding $X'\subset \PP^2_{x_0,x_1,x_2}\times\CC^2_{u,v}$ such that the action of $\cyc{n}$ is $$ \mt{wt}(x_0,x_1,x_2;u,v)=(0,p,q;a,b),\qquad (a,n)=(b,n)=1. $$ We may assume also that the fiber $X_0'\subset\PP^2$ is $\{x_0^2=0\}$. Then in some semi-invariant coordinate system $(x_0,x_1,x_2;u,v)$ in $\PP^2\times\CC^2$ the variety $X$ is given by the equation $$ x_0^2+\phi(u,v)x_1^2+\psi(u,v)x_1x_2+\zeta(u,v)x^2_2=0, \leqno \ab $$ where $\phi(u,v)$, $\psi(u,v)$, $\zeta(u,v)$ are semi-invariants with $\phi(0,0)=\psi(0,0)=\zeta(0,0)=0$. As above we can take $\wt(x_0)=0$. Denote $p=\wt(x_1)$, $q=\wt(x_2)$. The local coordinate along the fiber $X_0'$ is $x_1/x_2$ and $\wt(x_1/x_2)=p-q$. Since the action of $\cyc{n}$ on $X'$ is free in codimension two, $(p-q,n)=1$. Changing a generator of $\cyc{n}$ we can get $p-q=1$. Fixed points are $Q_1=\{ u=v=0, x_0=x_1=0\}$ and $Q_2=\{u=v=0, x_0=x_2=0\}$. We can take the local coordinates near $Q_1$ and $Q_2$ in $\PP^2\times\CC^2$ as and $(y_0=x_0/x_2,y_1=x_1/x_2,u,v)$ and $(z_0=x_0/x_1,z_2=x_2/x_1,u,v)$, respectively. The point $Q_1\in X'$ gives the singular point $P_1\in X$ of type $$ \{y_0^2+\phi(u,v)y_1^2+\psi(u,v)y_1+\zeta(u,v)=0\}/\cyc{n}(-q,1,a,b). \leqno\abc \label{t1} $$ Similarly, the point $Q_2\in X'$ gives the singular point $P_2\in X$ of type $$ \{z_0^2+\phi(u,v)+\psi(u,v)z_2+\zeta(u,v)z_2^2=0\}/\cyc{n}(-p,-1,a,b). \leqno\abc \label{t2} $$ \subde{} We claim that $(X,P_1)$ and $(X,P_2)$ are from the main series. Indeed assume for example that $(X,P_1)$ is from the special series. Then $n=4$, $q=2$, $p=3$. For $P_2$ all the weights are prime to $n=4$. By \ref{cl-term}, $P_2$ is a cyclic quotient singularity and $X'$ is nonsingular at $Q_2$. It is possible only if $\phi(u,v)$ contains a linear term. Then $\wt(\phi)=\wt(z_0^2)=2$. It gives us $a=\wt(u)=2$ or $b=\wt(v)=2$, a contradiction. \esubde \subde{} Now we claim that $X'$ is nonsingular at $Q_1$ and $Q_2$. As above if $Q_1\in X'$ is singular, then by \ref{cl-term} we have $q=0$, $p=1$. Again by \ref{cl-term}, $P_2$ is a cyclic quotient singularity and $X'$ is nonsingular at $Q_2$. It is possible only if $\phi(u,v)$ contains a linear term. Up to permutation $u, v$ we may assume that $\phi=u+\cdots$. Since $\wt(\phi)=\wt(z_0^2)=-2$, it gives us $a=\wt(u)=-2$. Therefore $n$ is odd and $$ (X,P_1)=\{y_0^2+\phi(u,v)y_1^2+\psi(u,v)y_1+\zeta(u,v)=0\}/ \cyc{n}(0,1,-2,b). \leqno\abc \label{1} $$ $$ \begin{array}{l} (X,P_2)= \{z_0^2+(u+\cdots)+\psi(u,v)z_2+\zeta(u,v)z_2^2=0\}/\cyc{n}(-1,-1,-2,b)\\ \simeq \CC^3_{z_0,z_2,v}/\cyc{n}(-1,-1,b). \end{array} \leqno\abc \label{2} $$ Since $P_2$ is a terminal point and $n$ is odd, from \ref{2} we have $b=1$. But on the other hand from \ref{1} $1+b=0$ or $-2+b=0$, a contradiction. \esubde \subde{} Therefore $X'$ is nonsingular at $Q_1$ and $Q_2$. It follows from \ref{t1} and \ref{t2} that both $\phi(u,v)$ and $\zeta(u,v)$ contain linear terms. Up to permutation of $u,v$ we may assume that $\zeta=u+\dots$. Whence $a=-2q$. Moreover $(X,P_1)=\frac{1}{n}(-q,1,b)$. If $\phi=u+\dots$, then from \ref{t2} $-2p=a=-2+a$, so $n=2$, $a=0$, a contradiction. Therefore $\phi=v+\dots$. It gives us $-2p=b=a-2$ and $(X,P_2)=\frac{1}{n}(-p,-1,a)$. Since $b$ is even, $n$ is odd. Further $a=-2q$, $p=q+1$, $b=-2q-2$. Thus $$ (X,P_1)=\frac{1}{n}(-q,1,-2q-2),\qquad\qquad (X,P_2)=\frac{1}{n}(-q-1,-1,-2q) \leqno \abc \label{ppppp} $$ \esubde \subde{} Now we show that $n=2q+1$. Indeed assume the opposite. Then since $(X,P_2)$ is terminal, from \ref{ppppp} we have either $q+2=0$ or $3q+1=0$. But in the first case $(X,P_1)=\frac{1}{n}(2,1,2)$. This point is terminal only if $n=3$, $q=1$. In the second case $(X,P_1)=\frac{1}{n}(-q,1,-2q-2)$ cannot be terminal. Thus our claim is proved. In particular, we have $a=-2q=1$, $b=-2q-2=-1$. Finally in this case $\wt(x_0,x_1,x_2,u,v)=(0,-q,q;1,-1)$ and by changing coordinates in $\CC^2$ by $u'=\zeta(u,v)$, $v'=\phi(u,v)$ we obtain $$ X\simeq \{x_0^2+vx_1^2+\psi(u,v)x_1x_2+ux^2_2=0\} /\cyc{2q+1}(0,-q,q;1,-1). $$ Therefore $X$ is as in \ref{ex3}. This proves our theorem. \qq \esubde \section{Index two Mori conic bundles.} \label{s-ind-2} In this section index two Mori conic bundles will be investigated. First we consider the case when the base of a Mori conic bundle $f:(X,C)\to (S,o)$ is nonsingular, i.~e. $(S,o)\simeq(\CC^2,0)$. We need some elementary results about extremal neighborhoods \cite{Mori-flip}, \cite{KoM}. Note that if $f:(X,C)\to (S,o)$ is a Mori conic bundle with reducible central fiber $C$, then the Mori cone $\overline{NE}(X/S)$ is generated by extremal rays, because $-K_X$ is ample. Since we consider a germ $(X,C)$, every extremal ray is generated by a component of $C$. On the other hand the dimension of $\overline{NE}(X/S)$ is equal to the number of components of $C$ by \ref{Pic}. Therefore $\overline{NE}(X/S)$ is simplicial and generated by classes of components of $C$. So every irreducible component $C_i\subset C$ gives us (not necessary isolated) extremal neighborhood in the sense of \cite{Mori-flip}. \th{Proposition.} \label{en} Let $(X,C)$ be an extremal neighborhood (not necessary isolated). Assume that $X$ has index two and let $P\in X$ be an index two point. Then \subth{} \cite[(4.6)]{KoM} $P$ is the only point of index two. $C$ has at most three components they all pass through $P$ and they do not intersect elsewhere. \esubth \subth{}\cite[(2.3.2)]{Mori-flip} \label{fu} For every component $C_i\subset C$ one has $(-K_{X}\cdot C_i)=1/2$. \esubth \subth{} \cite[(7.3)]{Mori-flip} A general member $(F,P)\in |-K_{(X,P)}|$ satisfies $F\in |-K_{X}|$, $F\cap C=\{ P\}$ and $(-K_{(X,C_i)}\cdot C_i)=1/2$ for every component $C_i\subset C$. \esubth \par\endgroup The first main result of this section is the following. \th{Theorem.} \label{th3} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle over a nonsingular base surface $(S,o)\simeq (\CC^2,0)$. Assume that $X$ is of index two. Then \esubth \subth{} \label{-h} $X$ contains exactly one index two point $P$, the central fiber $C$ has at most four components they all pass through $P$ and they do not intersect elsewhere. \esubth \subth{} \label{h} There exists a flat elliptic fibration $g:(Y,L)\to (S,o)$ where $(Y,L)$ is a germ of threefold with only isolated Gorenstein terminal singularities along a reducible curve $L$ such that $L=(g^{-1}(o))_{\mt{red}}$ and a general fiber $Y_s:=g^{-1}(s)$, $s\in S$ is a nonsingular elliptic curve. Further $K_Y$ is trivial along $L$ and $g$ factores as $$ g:(Y,L)\stackrel{h}{\longrightarrow} (X,C)\stackrel{f}{\longrightarrow} (S,o), $$ where $h$ is a quotient morphism by $\cyc{2}$ \esubth \subth{} \label{h1} The action of $\cyc{2}$ on $L$ does not interchange components. The locus of $\cyc{2}$-fixed points on $Y$ consists of an isolated point $Q$ such that $h(Q)=P$ and a nonsingular divisor $D\not\ni Q$ such that $D\cap \mt{Sing}(X)=\emptyset$. All the components of $L$ are isomorphic to $\PP^1$. They all pass through $Q$ and they do not intersect elsewhere. \esubth \subth{} \label{-} In notations above we have the following cases for the scheme-theoretical fiber $X_o=f^{-1}(o)$. In each case $C_1,\dots,C_r\simeq\PP^1$ are irreducible components of $C$. \par\noindent\abc\quad \label{1-1-1-1} $X_0\equiv C_1+C_2+C_3+C_4$, \par\noindent\abc\quad \label{1-1-2} $X_0\equiv C_1+C_2+2C_3$, \par\noindent\abc\quad \label{1-3} $X_0\equiv C_1+3C_2$, \par\noindent\abc\quad \label{2-2} $X_0\equiv 2C_1+2C_2$, \par\noindent\abc\quad \label{.4} $X_0\equiv 4C_1$. \esubth \par\endgroup \de{Remark.} Conversely, let $g:(Y,L)\to (S,o)\simeq(\CC^2,0)$ be an elliptic fibration with an action of $\cyc{2}$ such as in \ref{h}-\ref{h1}. If the point $(Y,Q)/\cyc{2}$ is terminal, then $f:(X,C):=(Y,L)/\cyc{2}\to (S,o)$ is a Mori conic bundle of index two. \ede We prove our theorem in several steps. \th{Lemma.} \label{flat} Let $f:X\to S$ be a morphism from a normal threefold with only terminal singularities onto a surface. Assume that all the fibers of $f$ are connected and one-dimensional. Then $f:X\to S$ is flat. \par\endgroup \de{Proof.} Terminal singularities are rational \cite[1-3-6]{KMM} and therefore Cohen-Macaulay \cite{Ke}. Then $f$ is flat by \cite[23.1]{Mat}.\qq \ede \de{} Thus $f:X\to S$ is flat. Let $X_s:=f^{-1}(s)$ be a scheme-theoretical fiber over $s\in S$. Then $(X_o)_{\mt{red}}=C$ and $X_o\equiv \sum n_iC_i$ for some $n_i\in\NN$. Since $X_s\simeq\PP^1$ for general $s\in S$, $(-K_X\cdot X_s)=2$. Thus we have $$ 2=(-K_X\cdot X_s)=(-K_X\cdot X_o)=\sum n_i(-K_X\cdot C_i). \leqno \ab \label{for} $$ It gives us $\sum n_i\le 4$ because $(-K_X\cdot C_i)\in\frac{1}{2}\ZZ$. In particular, $C$ has at most four components. If $C$ is reducible then by \ref{fu} $(-K_X\cdot C_i)=1/2$, so $\sum n_i=4$ and for $X_0$ we have only possibilities as in \ref{1-1-1-1} -- \ref{2-2}. \ede \th{Lemma.} Let $P\in X$ be a point of index 2. Then every component $C_i\subset C$ contains $P$. \par\endgroup \de{Proof.} Assume $C_j\not\ni P$ for some $j$. Then In particular, $C$ is reducible. By \ref{for} we have $\sum_{C_i\ni P}n_i<\sum n_i\le 4$. Let $F\in |-K_{(X,P)}|$ be a general member. Proposition \ref{en} gives us $(F\cdot C_i)=1/2$ for all components $C_i$ passing through $P$. On the other hand $$ 1/2\sum_{C_i\ni P} n_i=\sum_{C_i\ni P} n_i (F\cdot C_i)=(F\cdot X_0) =(F\cdot X_s)\in\NN $$ Thus $1/2\sum_{C_i\ni P} n_i=1$. Therefore $(F\cdot X_0)=(F\cdot X_s)=1$, i.~e. the map $f|_F:F\to S$ is bimeromorphic and finite. Since $S$ is nonsingular $f|_F$ is an isomorphism. So $F$ is also nonsingular. We derive the contradiction with \ref{st}. \qq \ede \th{Lemma.} \label{tor-free} The Weil divisor class group $\mt{Cl}(X)$ has no torsion. \par\endgroup \de{Proof.} If $\xi\in\mt{Cl}(X)$ is a torsion, then $\xi$ defines a cyclic \'etale in codimension 1 cover $X'\to X$ (see e.~g. \cite[(1.11)]{Mori-flip}). The threefold $X'$ is normal and has terminal singularities of indices $\le 2$ only. It follows also that $X'\to X$ is \'etale in codimension 2. Take the Stein factorization $$ \begin{array}{ccc} X'&\longrightarrow&X\\ \downarrow&&\downarrow\\ S'&\longrightarrow&S\\ \end{array} $$ We obtain a new Mori conic bundle $X'\to S'$ and \'etale in codimension one cover $S'\to S$. But $S$ is nonsingular. The contradiction shows that $\mt{Cl}(X)$ is torsion-free. \qq \ede \th{Lemma.} \label{unique} $X$ contains a unique point $P$ of index two. \par\endgroup \de{Proof.} If $C$ is reducible, then $P$ is a unique point of index two on $X$ by \ref{en} and because $p_a(C)=0$. So assume that $C\simeq\PP^1$ and let $P_1,\dots,P_r\in (X,C)$ be all the points of index $2$. \subde{Definition.} Let $X$ be a normal variety and $\mt{Cl}(X)$ be its Weil divisor class group. The subgroup of $\mt{Cl}(X)$ consisting of Weil divisor classes which are $\QQ$-Cartier is called by the semi-Cartier divisor class group. We denote it by $\mt{Cl}^{sc}(X)$. \esubde \subth{Theorem \cite{Pagoda},\cite{Kawamata}.} \label{clsc} Let $(X,P)$ be a germ of 3-dimensional terminal singularity. Then $\mt{Cl}^{sc}(X,P)\simeq\cyc{m}$ and it is generated by the class of $K_{(X,P)}$. \esubth We have the following natural exact sequence (see \cite[(1.8.1)]{Mori-flip}) $$ \begin{array}{ccccccccc} 0&\to&\mt{Pic}(X)&\to&\mt{Cl}^{sc}(X)&\to& \bigoplus_{i=1}^{r}\mt{Cl}^{sc}(X,P_i)&\to&0,\\ & &\| & & & &\| & & \\ & &\ZZ & & & &\bigoplus_{i=1}^{r}\cyc{2} & & \\ \end{array} \leqno \ab \label{unique1} $$ where $\mt{Pic}(X)\simeq\ZZ$ by \ref{Pic} and $\mt{Cl}^{sc}(X,P_i)\simeq\cyc{2}$ by \ref{clsc}. From \ref{tor-free} we get $\mt{Cl}^{sc}(X)\simeq\ZZ$. Hence $r=1$. Thus our lemma is proved. \qq \ede Since $p_a(C)=0$, components of $C$ do not intersect outside $P$. This proves \ref{-h}. \subth{Corollary.} If $C$ is irreducible, then $(-K_X\cdot C)=1/2$. \esubth \subde{Proof.} Suppose that $(-K_X\cdot C)>1/2$ From=20lemma \ref{tor-free} we have $\mt{Cl}^{sc}(X)=\ZZ$. Moreover by \ref{unique1} $\mt{Pic}(X)\subset \mt{Cl}^{sc}(X)$ is a subgroup of index 2. Let $R$ be the ample generator of $\mt{Cl}^{sc}(X)$. Then $(R\cdot C)=1/2$ and $-K_X=kR$, $k\in\ZZ$. On the other hand from \ref{for} we have $(-K_X\cdot C)=1$, $X_o\equiv 2C$ or $(-K_X\cdot C)=2$, $X_o\equiv C$. Whence $-K_X=2R$ or $-K_X=4R$. But then $-K_X$ is Cartier and $X$ has index one, a contradiction. \qq \esubde Therefore $X_0\equiv 4C$, if $C$ is irreducible. This proves \ref{-} \de{Construction.} \label{construction2} Let $B_i$ be a disc that intersects $C_i$ transversally in a general point and let $B=\sum B_i$. Since $(-K_X\cdot C_i)=1/2$ and $\mt{Pic}(X)=\ZZ^r$, $B\in |-2K_X|$. Take a double cover $h:Y\to X$ with ramification divisor $B$. Set $L:=(h^{-1}(C))_{\mt{red}}$ and $D:= (h^{-1}(B))_{\mt{red}}$. We have $$ g:(Y,L)\stackrel{h}{\longrightarrow} (X,C)\stackrel{f}{\longrightarrow} (S,o)\simeq (\CC^2,0). $$ By our construction $Y$ is normal, hence $g:Y\to\CC^2$ has only connected fibers. In a neighborhood of each singular points on $X$ $h$ is \'etale in codimension 1, therefore $Y$ has only terminal singular points (see e.~g. \cite[(3.1)]{Pagoda} or \cite[(6.7)]{CKM}). We have the following equalities for Weil divisors on $Y$ $$ K_{Y}=h^*(K_X)+D,\qquad\qquad D=h^*(-K_X). $$ It gives us $K_{Y}=0$ and $Y$ has only terminal Gorenstein singularities. The morphism $g$ is flat by lemma \ref{flat}. Therefore $g$ is an elliptic fibration. We have proved \ref{h}. It is clear that the locus of $\cyc{2}$-fixed points on $Y$ consists of $D$ and $h^{-1}(P)$. This proves \ref{h1}. Our theorem is proved. \qq \ede Using \cite[(6.2)]{Mori-flip}, \cite[(4.7)]{KoM} and \cite{Pro1} one can improve results of \ref{-}. \subth{Corollary.} In notations and conditions of \ref{th3} we have \par\noindent\abc\quad If $X_0\equiv C_1+C_2+C_3+C_4$, then $P$ is the only singular point of $X$, and $(X,P)$ is of type $cA/2$. \par\noindent\abc\quad If $X_0\equiv C_1+C_2+2C_3$, then $X$ may have one more singular point of index one on $C_3$, and $(X,P)$ is of type $cA/2$. \par\noindent\abc\quad If $X_0\equiv C_1+3C_2$, then $X$ may have one more singular point of index one on $C_3$. \par\noindent\abc\quad If $X_0\equiv 2C_1+2C_2$, then $X$ may have one more singular point of index one on each component $C_i$, $i=1, 2$. \par\noindent\abc\quad If $X_0\equiv 4C_1$, then $X$ may have at most two more singular points of one index 1. \esubth Now we use the construction \ref{construction2} to get examples index two Mori conic bundles. \de{Examples.} In the following examples a Mori conic bundle $f:(X,C)\to (\CC^2,0)$ is constructed as a quotient $X=Y/\cyc{2}\to\CC^2$, where $Y\subset\PP^3\times\CC^2$ is an intersection of two quadrics, $\cyc{2}$ acts on $Y\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ by $$ (x_0,x_1,x_2,x_3;u,v)\longrightarrow (-x_0,-x_1,-x_2,x_3;u,v) $$ and $f$ is induced by the projection of $g:Y\subset\PP^3\times\CC^2\to\CC^2$ on the second factor. We will use the notations $cA/2$, $cAx/2$, $cD/2$, $cE/2$ to distinguish index two terminal singularities \cite{KoM}. \ede \de{Example.} Let $Y\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ is given by the equations $$ \left\{ \begin{array}{l} x_0x_1=(au+bu^2+cuv)x_3^2\\ (x_0+x_1+x_2)x_2=vx_3^2, \end{array}\right. $$ where $a, b, c \in \CC$ are constants. It is easy to check that $Y$ is nonsingular near the central fiber $g^{-1}(0)$ if $a\ne 0$ and has an isolated hypersurface singularity in $u=v=x_0=x_1=x_2=0$ if $a=0$, $c\ne 0$, $b\ne 0$ such that the rank of the quadratic part is equal to $3$. Then the quotient $X:=Y/\cyc{2}\to \CC^2$ is a Mori conic bundle with only one singular point of type $cA/2$ and the central fiber $X_o\equiv C_1+C_2+C_3+C_4$. If $a\ne 0$, then the singular point is a cyclic quotient of type $\frac{1}{2}(1,1,1)$. \ede \de{Example.} Let $Y\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ is given by the equations $$ \left\{ \begin{array}{l} x_0x_1=u(x_0^2+x_1^2+x_2^2-x_3^2)+avx_3^2\\ (x_0+x_1)x_2=vx_3^2+bux_0^2 \end{array}\right. \qquad\qquad a, b \in \CC $$ Then the quotient $X:=Y/\cyc{2}\to \CC^2$ is a Mori conic bundle with the central fiber $X_o\equiv C_1+C_2+2C_3$. $X$ contains exactly one non-Gorenstein point that is of type $\frac{1}{2}(1,1,1)$. If $a=b=0$, then $X$ has also an ordinary double point on $C_3$. \ede \de{Example.} Let $Y\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ is given by the equations $$ \left\{ \begin{array}{l} x_0x_1-x_2^2=ux_3^2\\ x_0x_2=ux_1^2+v(x_2^2+x_3^2) \end{array}\right. $$ Then the quotient $X:=Y/\cyc{2}\to \CC^2$ is a Mori conic bundle with only one singular point of type $\frac{1}{2}(1,1,1)$ and the central fiber $X_o\equiv C_1+3C_2$. \ede The following example shows that all the types of terminal singularities of index two can appear on Mori conic bundles such as in \ref{2-2}. \de{Example.} Let $Y\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ is given by the equations $$ \left\{ \begin{array}{l} x_0x_1=ux_3^2\\ x_2^2=u(x_0^2+x_1^2)+(av+bv^2+cv^3+duv+eu^2v)x_3^2\\ \end{array}\right. $$ where $a,b,c,d,e\in\CC$ are constants. If at least one of $a,b,c$ is non-zero, then $Y$ has near the central fiber $Y_0$ an isolated singularity at $\{x_0=x_1=x_2=u=v=0\}$ (or nonsingular). As above the quotient $X:=Y/\cyc{2}\to \CC^2$ is a Mori conic bundle with the central fiber $X_0\equiv 2C_1+2C_2$ and only one singular point $P:=C_1\cap C_2$. We have the following possibilities: \par $a\ne 0$, then $Y$ is nonsingular, so $P\in X$ is of type $\frac{1}{2}(1,1,1)$, \par $a=0$, $b\ne 0$, then $P\in X$ is of type $cAx/2$, \par $a=b=e=0$, $c\ne 0$, $d\ne 0$, then $P\in X$ is of type $cD/2$, \par $a=b=d=0$, $c\ne 0$, $e\ne 0$, then $P\in X$ is of type $cE/2$. \ede \de{Example.} Let $Y\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ is given by the equations $$ \left\{ \begin{array}{l} x_0^2=ux_2^2+vx_3^2\\ x_1^2=ux_3^2+vx_2^2 \end{array}\right. $$ Then the quotient $X:=Y/\cyc{2}\to \CC^2$ is a Mori conic bundle with irreducible central fiber that contains one singular point of type $\frac{1}{2}(1,1,1)$ and two ordinary double points. \ede Now we investigate index two Mori conic bundles with singular base. \th{Theorem.} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle of index 2 over a surface. Assume that the point $(S,o)$ is singular. Then $(S,o)$ is DuVal of type $A_1$ and $f$ is either as in example \ref{ex2} or as in example \ref{ex1} with $n=2$. \par\endgroup \de{Proof.} We can use construction \ref{construction}. Since $X$ contains only points of indices $\le 2$ by \ref{>n}, $(S,o)$ is a singularity of type $\frac{1}{2}(1,1)=A_1$. We claim that $f':(X',C')\to (\CC^2,0)$ is a conic bundle. It is sufficient to show that $X'$ is Gorenstein \ref{Cut}. Assume the opposite. We remark that $X'$ contains only points of indices $\le 2$ because $X'\to X$ is \'etale in codimension 1. By theorem \ref{th3} $X'$ contains a unique point, denote it by $Q$, of index 2. Then $Q$ is a $\cyc{2}$-fixed point. But then the point $P:=h(Q)$ has index $4$, a contradiction. Therefore $f'$ is a conic bundle and by theorem \ref{th1} $f(X,C)\to (X,C)$ is such as in example \ref{ex1} or example \ref{ex2}. This proves our theorem. \qq \ede \section{The general elephant conjecture.} \label{sect-el} In this section we study Mori conic bundles under the assumption the existence of a good member in $|-K_X|$. \de{Example.} \label{exc3} Let $f:(X,C)\to (S,o)$ be as in example \ref{ex3}. Consider the open subset $U_2=\{ x_2\ne 0\}\subset X$. The local coordinates in $U_2$ $(t_0=x_0/x_2, t_1=x_1/x_2, v)$. Consider also the rational differential $\sigma=(1/t_0)(dt_0\wedge dt_1\wedge dv)$ on $U_2$. Then it is easy to see that $\sigma$ can be extended on $X'$ near $C'$. Since $\sigma$ is $\cyc{n}$-invariant, $\sigma^{-1}$ defines an element $F\in |-K_X|$, the image of $\{x_0=0\}$. It is easy to check that $F$ contains the central fiber $C=(f^{-1}(o))_{\mt{red}}$ and has two singular points of type $A_{n-1}$. \par Similarly one can check that in example \ref{ex1} a general member $F\in |-K_X|$ does not contain $C$ and has two connected components. It contains two singular points of type $A_{n-1}$ on each component. \ede The following theorem improves results of \cite{Pro}. \th{Theorem.} \label{g.e.} Let $f:(X,C)\to (S,o)$ be a Mori conic bundle. Assume that conjecture \ref{elephant} holds. Then we have one of the following: \subth{} \label{part1} $(S,o)$ is nonsingular, \esubth \subth{} \label{part2} $(S,o)$ is DuVal of type $A_1$, \esubth \subth{} \label{imp} $(S,o)$ is DuVal of type $A_3$, in this case $C$ is irreducible, $(X,C)$ has a cyclic quotient singularity $P$ of index $8$ and has no another points of index $>1$ (see \cite{Pro3} for more detailed study of this case). \esubth \subth{} \label{pr} $f:(X,C)\to (S,o)$ is quotient of a nonsingular conic bundle $f':(X',C')\to (S',o')$ with irreducible $C'$ by the group $\cyc{n}$, where $n\ge 3$ and the action $\cyc{n}$ on $(S',o')\simeq (\CC^2,0)$ is free in codimension 1 (i.~e. $f:(X,C)\to (S,o)$ is such as in \ref{ex1} or \ref{ex3}). In particular, $(S,o)$ has type $A_{n-1}$ in this case. \esubth \par\endgroup \de{Proof.} \label{as} Assume that $(S,o)$ is singular and it is not of type $A_1$. We will use the notations of construction \ref{construction}. If $X'$ is of index one, then by \ref{Cut} $(S,o)$ is nonsingular. So we assume that $X'$ is of index $>1$ and show in this case that $f:(X,C)\to (S,o)$ is such as in \ref{imp}. Let $F\in |-K_X|$ be a general member and $F':=h^{-1}(F)$. By our conditions $F$ has only DuVal singularities and since $F'\to F$ is \'etale in codimension 1, so has $F'$ (In particular, $F'$ is irreducible). Since $(-K_X\cdot X_s)=2$, where $X_s$ is a general fiber of $f$, the restriction $f|_F:F\to S$ is generically finite of degree 2. There are the following cases. \ede \de{Case I.} \underline{$F'\cap C'$ is disconnected}. As above since $F'\in |-K_{X'}|$, we see that $f|_{F'}:F'\to S'$ is generically finite of degree 2 and $F'$ has two connected components $F'_1$ and $F'_2$. Let $$ f':F'\stackrel{f'_1}\longrightarrow D'\stackrel{f'_2}\longrightarrow S' $$ be the Stein factorization. Then $f'_1:F'\to D'$ is bimeromorphic and $f'_2:D'\to S'$ is finite of degree 2. In our case $D'$ has exactly two irreducible components $D'_1$ and $D'_2$. Therefore we have $D_1'\simeq D_2'\simeq S'$ and the divisor $D'$ is nonsingular, because so is $S'$. On the other hand by the adjunction formula, $K_{F'}=0$. Whence the morphism $f_1'$ is crepant (i.~e. $K_{F'}={f_1}'^*K_{D'}$). It means that $F'$ is nonsingular and $f'_1=\mt{id}$. By \ref{st}, $X'$ has no points of indices $>1$, a contradiction with our assumption in \ref{as}. \ede \de{Case II.} \underline{$F'\cap C'$ is one point $Q$}. \label{finite} Then $F\cap C=\{ h(Q)\}$ is also one point, say $P$. Therefore the morphism $f|_F:F\to S$ is finite of degree 2 and $P$ is a unique point of index $>1$ and $(F,P)\to (S,o)$ is double cover of isolated singularities. Thus $(S,o)$ is a quotient of DuVal singularity $(F,P)$ by an involution $\tau$. \par Actions of involutions on DuVal singularities were classified by Catanese \cite{Cat}. Recall that $(S,o)$ is of topological index $n\ge 3$ (otherwise we have cases \ref{part1} or \ref{part2} of our theorem). Taking into account that $(S,o)$ is a cyclic quotient from the list in \cite{Cat} we obtain the following \th{Lemma \cite{Cat}.} \label{Cat} Let $(F,P)$ be a germ of DuVal singularity and let $\tau$ be an analytic involution acting on $(F,P)$. Assume that the quotient $(S,o):=(F,P)/\tau$ is a cyclic quotient singularity of type $\frac{1}{n}(a,b)$ with $n\ge 3$ (and $(a,n)=(b,n)=1$). Then there are the following possibilities for $(F,P)\to (S,o)$: $$ \begin{array}{lll} &(F,P)\to (S,o)&n\\ &&\\ \ab\label{(1)}\qquad&E_6\stackrel{2:1}{\longrightarrow}A_2, &n=3,\\ \ab\label{(2)}\qquad&A_{2n-1}\stackrel{2:1}{\longrightarrow}A_{n-1}, &n\ge 1\\ \ab\label{(3)}\qquad&A_{2k}\stackrel{2:1}{\longrightarrow} \frac{1}{2k+1}(k,2k-1),&n=2k+1\\ \ab\label{(4)}\qquad&A_k\stackrel{2:1}{\longrightarrow}A_{2k+1}, &n=2k+1,\\ \ab\label{(5)}\qquad&A_{2k+1}\stackrel{2:1}{\longrightarrow} \frac{1}{4k+4}(2k+1,2k+1),&n=4k+4.\\ \end{array} $$ \par\endgroup The restriction $(F',Q)\to (F,P)$ is \'etale outside $P$ and of degree $n$. Therefore $(F',Q)$ is DuVal. It is easy to see (see e.~g. \cite[4.10]{RYPG}), that $(F,P)$ has no such covers in cases \ref{(4)}, \ref{(5)}. The singularity $(F,P)$ from \ref{(3)} admits only cover by nonsingular $(F',P')$ of degree $n=2k+1$. But then $(X',Q)$ is a nonsingular point by \ref{st}, a contradiction with our assumption in \ref{as}. \par Let $m$ be the index of $(X,P)$, $\pi:(X\3,P\3)\to (X,P)$ be the canonical cover and $F\3:={\pi\3}^{-1}{F}$. As above we have \'etale in codimension one $\cyc{m}$-cover $\pi:(F\3,P\3)\to (F,P)$ of DuVal singularities. Since $\pi$ factores through $(X',Q)$ we have $m\ge n$. \par In case \ref{(1)} $(F,P)=E_6$ by \cite[4.10]{RYPG}, admits only cyclic cover $D_4\stackrel{3:1}{\longrightarrow}E_6$. Then $n=m=3$. Therefore $(X\3,P\3)\simeq (X',P')$ has index 1, a contradiction. \par Finally, consider case \ref{(2)}. Since $(X',P')$ has index $>1$, we have $m>n\ge 3$ and by \cite[4.10]{RYPG} $(F\3,P\3)\stackrel{m:1}{\longrightarrow}(F,P)$ is of type $(\mt{nonsingular})\stackrel{2n:1}{\longrightarrow}A_{2k+1}$, $m=2n$. Then the index of $(X',P')$ is equal to $m/n=2$, $(X\3,P\3)$ is nonsingular, hence $(X',P')$ is a cyclic quotient singularity of type $\frac{1}{2}(1,1,1)$. In particular, $f$ is as in theorem \ref{th3}. Let $X_{o'}'=f'^{-1}(o')$ be the scheme-theoretical fiber of $f'$ over $o'$. Further $P'$ is the only fixed point on $C'$ under the action of $\cyc{n}$. Whence $\cyc{n}$ permutes components $\{C_i'\}$ of $C'$. In particular, the number of components $\ge n\ge 3$ and multiplicities of components in $X_{o'}'$ are the same. Therefore for $f':(X',C')\to (S',o')$ we have the only possibility \ref{1-1-1-1}. Hence we have exactly four components of $X_0'$, $n=4$ and the fiber $X_o'$ is reduced. Then $X'$ is nonsingular outside $P'$. We obtain case \ref{imp} of our theorem. \ede \de{Case III.} \underline{$F'\cap C'$ is one-dimensional and connected}. Then so is $F\cap C$. Let $L:=F'\cap C'$ (with reduced structure). On the surface $F'=h^{-1}(F)$ we have by the adjunction formula that $L$ is Gorenstein and $\omega_L=\OOO_L(L)$, because $\omega_{F'}=\OOO_{F'}$. Since $L$ is contracted by $f'$, the dualizing sheaf $\omega_L$ is anti-ample. Whence $L$ is a reduced conic in $\PP^2$ (see e.~g. \cite{Lip}). \par Let $$ f_F:F\stackrel{f_1}\longrightarrow D\stackrel{f_2}\longrightarrow S $$ be the Stein factorization. Then $f_1:F\to D$ is bimeromorphic and $f_2:D\to S$ is finite of degree 2. By the adjunction formula, $K_F=0$. Therefore the morphism $f_1$ is crepant (i.~e. $K_F=f_1^*K_D$) and $D$ has only DuVal singularities. \par There exists the common minimal resolution $\sigma:\widetilde{F}\to F\to D$. In our case $f_2^{-1}(o)$ consist of only one point $R$. Let $\Gamma=\Gamma(\widetilde{F}/D)$ be a dual graph for $\sigma$. Denote vertices corresponding to components of $h(L)$ (resp. $f_2$-exceptional divisors) by $\bullet$ (resp. $\circ$). Then white vertices form connected subgraphs corresponding singular points of $(F,C)$ and black vertices correspond components of $C$ that contained in $F$. For $f_2:(D,R)\to (S,o)$ we have the same possibilities as for $(F,P)\to (S,o)$ in \ref{Cat}. Let us consider these cases. \subde{Subcase \ref{(1)}.} (i.~e. $(D,R)=E_6$, $(S,o)=A_2$, $n=3$). The group $\cyc{3}$ naturally acts on the curve $L=F'\cap C'$. Since $L$ is a conic, $\cyc{3}$ cannot permute its components. Therefore $\cyc{3}$ has two or three fixed points $Q_i$ on $L$. Then there exists at least two points $P_1,P_2\in X$ of indices $\ge 3$. These points on $F$ have by \ref{index} topological indices $\ge 3$. On the other hand, the graph of the minimal resolution of $(F,P_i)$ must be a white subgraph of $\Gamma$ ($\simeq E_6$). Whence each $(F,P_i)$ is of type $A_2$ and there are only two fixed points on $L$. It is possible only $L$ is irreducible and so is $h(L)$. Thus we have only one possibility for $\Gamma$. $$ \begin{array}{ccccccccc} &&&&\bullet&&&&\\ &&&&|&&&&\\ \circ&\pal&\circ&\pal&\bullet&\pal&\circ&\pal&\circ\\ \end{array} $$ Whence $F$ contains exactly two singular points $P_1$, $P_2$ and they have type $A_2$. The variety $X$ has at these points index 3. Since $h^{-1}(P_i)=Q_i$ by \ref{index}, $X'$ is Gorenstein. We obtain a contradiction with assumptions in \ref{as}. \esubde \subde{Subcase \ref{(2)}} i.~e. $(D,R)=A_{2n-1}$, $(S,o)=A_{n-1}$. If $L$ is irreducible or $\cyc{n}$ doesn't permute components of $L$, then as above $F$ contains at least two points of topological indices $\ge n$. Then $\Gamma$ is $$ \underbrace{\circ\pal\circ\pal\cdots\pal\circ}_{n-1} \pal\bullet\pal\underbrace{\circ\pal\cdots\pal\circ}_{n-1} \leqno \abc \label{lll} $$ Whence $F$ contains exactly two singular points and they have type $A_{n-1}$. By \ref{index} $X'$ is Gorenstein, a contradiction. \par Therefore $L$ is reducible and $\cyc{n}$ interchanges its components. Then $h(L)=F\cap C$ is irreducible, so $\Gamma$ has only one black vertex. Further a $\cyc{n}$-fixed point $Q_1$ on $X'$ gives us the point $P_1\in F$ of topological index $nk$, $k\in\NN$. This point corresponds to a white subgraph in $\Gamma$ of type $A_{nk-1}$. Thus $\Gamma$ again has the form \ref{lll} So $(F,P_1)$ is of type $A_{n-1}$. In particular, by \ref{st} $Q_1$ is nonsingular. The second white subgraph in $\Gamma$ also corresponds to the singular point $P_2\in F$ of type $A_{n-1}$. Then the index of $(X,P_2)$ divides $n$. Since $\cyc{n}$ interchanges two components of $L$, $n=2k$ is even and there are three $\cyc{k}$-fixed points on $L$: $Q_1$, $Q_2$ and $Q_3$, where $h(Q_2)=h(Q_3)=P_2$. Whence the index of $P_2$ is divisible by $k$. By our assumption $(X',C')$ is not Gorenstein, so $(X',Q_2)$, $(X',Q_3)$ have (the same) index $m_0>1$ and $(X',C')$ contains no another points of index $>1$. But then index of $(X,R)$ is $m_0n/2\le n$. Hence $m_0=2$, it means that $f':(X',C')\to (S',o')$ is an index two Mori conic bundle that contains two non-Gorenstein points $Q_2$, $Q_3$, a contradiction with \ref{th3}. \esubde Finally subcases \ref{(3)},\ref{(4)} and \ref{(5)} are impossible as above. \qq \ede \section{An example of Mori conic bundle of index three.} In this section we construct an example of index three Mori conic bundle over a nonsingular base (that is not such as in examples \ref{ex1} or \ref{ex3}). \de{} Let $V\subset\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$ is given by two equations $$ \left\{ \begin{array}{l} a_{1}x_{1}^{3}+a_{2}x_{2}^{3}+ a_{3}x_{1}^{2}x_{2}+a_{4}x_{2}^{2}x_{1}+ a_{5}x_{3}^{2}x_{1}+a_{6}x_{3}^{2}x_{2}=ux_{0}^{3}\\ b_{1}x_{1}^{3}+b_{2}x_{2}^{3}+ b_{3}x_{1}^{2}x_{2}+b_{4}x_{2}^{2}x_{1}+ b_{5}x_{3}^{2}x_{1}+b_{6}x_{3}^{2}x_{2}=vx_{0}^{3}\\ \end{array} \right. $$ where $a_i,b_i\in\CC$ are general constants. Then $V$ is nonsingular and the projection on the second factor $q:V\to\CC^2$ gives us a fibration of curves of degree 9 in $\PP^3_{x_0,x_1,x_2,x_3}\times\CC^2_{u,v}$. The central fiber $q^{-1}(0)$ has exactly nine irreducible components $\Gamma_0,\dots,\Gamma_8$, where $\Gamma_0:=\{u=v=x_1=x_2=0\}$. Define an action of $\cyc{6}$ on $V$ by $$ (x_0,x_1,x_2,x_3;u,v)\longrightarrow (\varepsilon x_0,x_1,x_2,-x_3;u,v), \qquad \varepsilon =\exp (2\pi i/6). $$ First we consider the quotient $p: V\to Y:=V/\cyc{3}$ and the standard projection $g:Y\to\CC^2$. The set of $\cyc{3}$-fixed points is a divisor $R:=\{x_0=0\}\cap V$ and an isolated point $O:=\{x_1=x_2=x_3=u=v=0\}$. The local coordinates on $V$ near $O$ are $(x_1/x_0, x_2/x_0, x_3/x_0)$ with $\cyc{3}$-weights $\wt(x_1/x_0)=\wt(x_2/x_0)=\wt(x_3/x_0)$. Therefore the singular locus of $Y$ consists of one point of type $\frac{1}{3}(1,1,1)$ (denote it by $Q$) that is canonical and Gorenstein. Further by the Hurvitz formula $$ K_V=p^*K_Y+2R, \qquad p^*K_Y=K_V-2R=0. $$ Whence the canonical divisor of $Y$ is trivial and $g:Y\to \CC^2$ is an elliptic fibration. Since $\cyc{3}$ does not permute $\Gamma_0,\dots,\Gamma_8$, the central fiber $g^{-1}(0)$ consists of nine components $L_0,L_1,\dots,L_8$ which are proper transforms of corresponding components $\Gamma_0,\dots,\Gamma_8$ of $q^{-1}(0)$. Now consider the quotient $h:Y\to X:=Y/\cyc{2}$ and the natural morphism $f:X\to\CC^2$. The component $L_0$ is $\cyc{2}$-invariant and $\cyc{2}$ permutes $L_i$, $i=1,\dots,8$ non-trivially. Hence $f^{-1}(0)=g^{-1}(0)/\cyc{2}$ has exactly five components. Denote them by $C_0,\dots,C_4$. To prove that $f:X\to\CC^2$ is a Mori conic bundle, we have to investigate the singular locus of $X$. The set of $\cyc{2}$-fixed points on $V$ is an irreducible divisor $D:=\{x_3=0\}$. Therefore the set of $\cyc{2}$-fixed points on $Y$ is an irreducible Weil divisor $F:=p(D)$ such that $3F$ is Cartier. Moreover $D$ intersects components $\Gamma_0,\dots,\Gamma_8$ transversally. \ede The following is an easy exercise. \th{Lemma.} Let $(Y,Q)$ be a cyclic quotient singularity of type $\frac{1}{3}(1,1,1)$ with action of $\cyc{2}$. Assume that the locus of fixed points of this action is a Weil divisor $F$ such that $3F=0$. Then $(Y,Q)/\cyc{2}$ is terminal of type $\frac{1}{3}(1,1,-1)$. \qq \par\endgroup \de{} By the Hurvitz formula $0=K_Y=h^*(K_X)+F$. Since $(L_i\cdot F)>0$, we have $(h^*(-K_X)\cdot L_i)=(F\cdot L_i)>0$, $i=0,\dots,8$. Hence $(-K_X\cdot C_i)>0$, $i=0,\dots,4$ i.~e. $-K_X$ is relatively $f$-ample (at least near $f^{-1}(0)$. \ede \de{Conclusion.} We get a Mori conic bundle $f:(X,C)\to (\PP^2,0)$ such that its central fiber $C$ has exactly five components and the only singular point $P$ of $X$ is a cyclic quotient singularity of type $\frac{1}{3}(1,1,-1)$. All the components of $C$ pass through $P$ and they do not intersect elsewhere. Note that $h:Y\to X$ is nothing else but Kawamata's double cover trick \cite{Kawamata}. \ede \footnotesize

9608

alg-geom/9608027

en

https://arxiv.org/abs/alg-geom/9608027

alg-geom/9608027

Elizabeth Gasparim

Elizabeth Gasparim

Holomorphic bundles on O(-k) are algebraic

We show that holomorphic bundles on O(-k) for k > 0 are algebraic. We also show holomorphic bundles on O(-1) are trivial outside the zero section.

alg-geom

\section{ Preliminaires } The line bundle on ${\bf P}^1$ given by transition function $z^{k}$ is usually denoted ${\cal O} (-k).$ Since we will be studying bundles over this space, we will denote $ {\cal O} (-k)$ by $M_k$ when we want to view this space as the base of a bundle. We give $M_k$ the following charts $M_k = U \cup V$, for $$U = {\bf C}^2 =\{(z,u)\}$$ $$V = {\bf C}^2 = \{(\xi,v)\}$$ $$U \cap V = ({\bf C} - \{0\}) \times {\bf C}$$ with change of coordinates $$ (\xi,v) = (z^{-1 },z^ku).$$ Since $H^1( {\cal O}(-k), {\cal O}) = 0,$ using the exponential sheaf sequence it follows that $Pic({\cal O}(-k)) = {\bf Z},$ and holomorphic line bundles on $M_k$ are classified by their Chern classes. Therefore it is clear that holomorphic line bundles over $M_k$ are algebraic. We will denote by ${\cal O}^!(j) $ the line bundle on $M_k$ given by transition funcion $z^{-j}.$ If $E$ is a rank $n$ bundle over $M_k,$ then over the zero section (which is a ${\bf P}^1$) $E$ splits as a sum of line bundles by Grothendieck's theorem. Denoting the zero section by $\ell$ it follows that for some integers $j_i$ uniquely determined up to order $E_{\ell} \simeq \oplus_{i = 1}^n {\cal O}(j_i). $ We will show that such $E$ is an algebraic extension of the line bundles $ {\cal O}^!(j_i).$ \section{ Bundles on ${\cal O}(-k)$ are algebraic} \begin{lemma}: Holomorphic bundles on $M_k$ are extensions of line bundles. \end{lemma} \noindent{\bf Proof}: We give the proof for rank two for simplicity. The case for rank $n$ is proved by induction on $n$ using similar calculations. Suppose rank $E$ = 2 and $E_{\ell} \simeq {\cal O}(-j_1) \oplus {\cal O}(-j_2)$ which we may assume to satisfy $j_1 \ge j_2.$ A transition matrix for $E$ from $U$ to $V$ therefore takes the form $$ T = \left(\matrix {z^{j_1} + ua & uc \cr ud & z^{j_2} + ub \cr }\right)$$ where a, b, c, and d are holomorphic functions in $U \cap V.$ We will change coordinates to obtain an upper triangular transition matrix $$ \left(\matrix {z^{j_1} & uc \cr 0 & z^{j_2} \cr }\right),$$ which is equivalent to an extension $$0 \rightarrow {\cal O}^!(-j_1) \rightarrow E \rightarrow {\cal O}^!(-j_2) \rightarrow 0.$$ Our required change of coordinates will be $$\left(\matrix {1 & 0 \cr \eta & 1 \cr }\right) \left(\matrix {z^{j_1} + ua & uc \cr ud & z^{j_2} + ub \cr }\right) \left(\matrix {1 & 0 \cr \xi & 1 \cr }\right)$$ where $\xi$ is a holomorphic function on $U$ and $\eta$ is a holomorphic function on $V$ whose values will be determined in the following calculations. After performing this multiplication, the entry $e(2,1)$ of the resulting matrix is $$ e(2,1)= \eta\, (z^{j_1} + ua) +ud+ [\eta uc+(z^{j_2}+ ub )]\,\xi.$$ We will choose $\xi$ and $\eta $ to make $e(2,1) = 0.$ We write the power series expansions for $\xi$ and $\eta $ as $\xi = \sum_{i = 0} ^\infty \xi_i(z)\, u^i$ and $\eta = \sum_{i = 0} ^\infty \eta_i(z^{-1})\,(z^k u)^i,$ and plug into the expression for $e(2,1) .$ The term independent of $u$ in $e(2,1) $ is $$\eta_0(z^{-1})z^{j_1} + \xi_0(z)z^{j_2}.$$ Since $j_2 - j_1 \le 0$ we may choose $\eta(z^{-1}) = z^{j_2 - j_1}$ and $\xi(z) = 1.$ After these choices $e(2,1)$ is now a multiple of $u.$ Suppose that the coefficients of $\eta$ and $\xi$ have been chosen up to power $u^{n-1}$ so that $e(2,1)$ becomes a multiple of $u^n.$ Then the coefficient of $u^n$ in the expression for $e(2,1)$ is $$ \eta_n\,z^{j_1+kn} + \xi_n\,z^{j_2}+ \Phi$$ where $$\Phi = \sum_{s+i = n}\eta_s\,a_i\,d_n\,z^{sk}+ \sum_{s+i+m = n} \eta_s\,c_i\, \xi_m\,z^{sk} \sum_{m+i = n}\xi_m\,b_i.$$ We separate $\Phi$ into two parts $$ \Phi = \Phi_{>j_2} + \Phi_{\le j_2}$$ where $ \Phi_{>j_2}$ is the part of $\Phi $ containing the powers $z^i $ for $i> j_2$ and $ \Phi_{\le j_2} $ is the part of $\Phi $ containing powers $z^i$ for $i \le j_2.$ We then choose the values of $\eta_n$ and $\xi_n$ as $$\eta_n = z^{-j^1-nk} \Phi_{\le j_2} $$ and $$\xi_n = z^{-j_2} \Phi_{>j_2} .$$ These choices cancel the coefficient of $u^n$ in $e(2,1).$ Induction on $n$ gives $e(2,1)= 0$ And provides a transition matrix of the form $$ T = \left(\matrix {z^{j_1} + ua & uc \cr 0 & z^{j_2} + ub \cr }\right).$$ Now do a similar trick using the change of coordinates $$ \left(\matrix {\eta_1 & 0 \cr 0 & \eta_2 \cr }\right) \left(\matrix {z^{j_1} + ua & uc \cr 0 & z^{j_2} + ub \cr }\right) \left(\matrix {\xi_1 & 0 \cr 0 & \xi_2 \cr }\right)$$ and choose $\xi_1,\, \xi_2,\, \eta_1$ and $\eta_2$ appropriately to obtain a transition matrix of the form $$ T = \left(\matrix {z^{j_1} & uc \cr 0 & z^{j_2} \cr }\right).$$ \hfill\vrule height 3mm width 3mm \begin{theorem}: Holomorphic bundles over $M_k, \,\, k>0$ are algebraic. \end{theorem} \noindent{\bf Proof}: Let $E$ be a holomorphic bundle over $M_k$ whose restriction to the exceptional divisor is $E_{\ell} \simeq \oplus_{i = 1}^n {\cal O}(j_i), $ then $E$ has a transition matrix of the form $$\left(\matrix {z^{j_1} & p_{12} & p_{13} & \cdots \cr 0 & z^{j_2} & p_{23} & p_{24} & \cdots \cr \vdots & & \vdots & \cr 0 & \cdots & 0 & z^{j_{n-1}} & p_{n-1,n} \cr 0 & \cdots & & 0 & z^{j_n} \cr}\right)$$ from $U$ to $V,$ where $p_{ij}$ are polinomials defined on $U \cap V.$ Once again we will give the detailed proof for the case $n = 2.$ The general proof is by induction on $n$ and is essentially the same as for $n = 2$ only notationally uglier.\hfill\vrule height 3mm width 3mm \vspace{5 mm} For the case $n = 2$ we restate the theorem giving the specific form of the polinomial. \begin{theorem}: Let $E$ be a holomorphic rank two vector bundle on $M_k$ whose restriction to the exceptional divisor is $E_{\ell} \simeq {\cal O}(j_1) \oplus {\cal O}(j_2),$ with $j_1 \ge j_2.$ Then $E$ has a transition matrix of the form $$\left(\matrix {z^{j_1} & p \cr 0 & z^{j_2} \cr }\right)$$ from $U$ to $V,$ where the polinomial $p$ is given by $$p = \sum_{i = 1}^{ \left[(j_1 - j_2 -2)/k\right]} \sum_{l = ki+j_2+1}^{j_1-1}p_{il}z^lu^i$$ and $p = 0$ if $j_1< j_2 +2.$ \end{theorem} \noindent {\bf Proof}: Based on the proof of Theorem 2.1 we know that $E$ has a transition matrix of the form $$ \left(\matrix {z^{j_1} & uc \cr 0 & z^{j_2} \cr }\right).$$ We are left with obtaining the form of the polinomial $p,$ for which we perform another set of coordinate changes as follows. $$ \left(\matrix {1 & \eta \cr 0 & 1 \cr }\right) \left(\matrix {z^{j_1} & uc \cr 0 & z^{j_2} \cr }\right) \left(\matrix {1 & \xi \cr 0 & 1 \cr }\right),$$ where the coefficients of $\xi = \sum_{i = 0} ^\infty \xi_i(z)\, u^i$ and $\eta = \sum_{i = 0} ^\infty \eta_i(z^{-1})\,(z^k u)^i,$ will be choosen apropriately in the following steps. After performing this multiplication, the entry $e(1,2)$ of the resulting matrix is $$e(1,2) = z^{j_1}\,\xi +uc + z^{j_2}\,\eta.$$ The term independent of $u$ in the expression for $e(1,2)$ is $z^{j_1}\xi_0(z) + z^{j_2+k} \eta_0(z^{-1}).$ However, we know from the expression for our matrix $T$ (proof of lemma 2.1), that $e(1,2)$ must be a multiple of $u;$ accordingly we choose $\xi_0(z) = \eta_0(z^{-1}) = 0.$ Placing this information into the above equation, we obtain $$e(1,2) = \sum_{n = 1}^\infty (\xi_n(z)z^{j_1} + c_n(z,z^{-1}) + \eta_n(z^{-1})z^ {j_2+kn})\,u^n.$$ Proceeding as we did in the proof of Lemma 2.1, we choose values of $\xi_n$ and $\eta_n$ to cancel as many coefficients of $z$ and $z^{-1}$ as possible. However, here $\xi_n$ appears multiplied by $z^{j_1}$ ( and $\eta_n$ multiplied by $z^{j_2 + kn}$), therefore the optimal choice of coeficients cancels only powers of $z^i$ with $i \ge j_1$ (resp. $z^i$ with $i \le j_2 + kn$). Consequently, $e(1,2)$ is left only with terms in $z^l$ for $j_2+nk < l< j_1 $, and we have the expression $$e(1,2) = \sum_{i = 1}^\infty\sum_{l = nk+j_2+1}^{j_1-1}c_{il}z^lu^i.$$ But $i$ may only vary up to the point where $nk+j_2+1 \leq j_1-1$ and the polynomial $p$ is given by $$p = \sum_{i = 1}^{[(j_1-j_2-2)/k]} \sum_{l = ik+j_2+1}^{j_1-1}p_{il}z^lu^i.$$\hfill \vrule height 3mm width 3mm \section{Triviality outside the zero section} From the previous section we know that bundles on $M_k$ are extensions of line bundles. First we have the following lemma. \begin{lemma}: Line bundles on $M_k$ are trivial outside the zero section. \end{lemma} \noindent{\bf Proof}: A line bundle on $M_k$ can be given by a transition function $z^j$ for some integer $j.$ Then the function given by $z^{k-j}u$ on $U$ and $z^ku$ on $V$ is a global holomorphic section which trivializes the bundle outside the zero section. \hfill\vrule height 3mm width 3mm We now show that the extensions given in Section 2 are trivial outside the zero section. \begin{theorem} Holomorphic vector bundles on ${\cal O}(-1)$ are trivial outside the zero section. \end{theorem} \noindent {\bf Proof}:Let $E$ be a holomorphic bundle on ${\cal O}(-1).$ According to the previous section we know that $E$ is algebraic. Call $F$ the restriction of $E$ to the complement of the zero section, i.e. $F = E|_{\ell^c}.$ Let $\pi : {\cal O}(-1) \rightarrow {\bf C}^2$ be the blow up map. Then $\pi_*(F)$ is an algebraic bundle over ${\bf C}^2 - {0}$ and therefore it extends to a coherent sheaf ${\cal F}$ over ${\bf C}^2.$ Then ${\cal F} ^{**}$ is a reflexive sheaf and as such has singularity set of codimension 3 or more, hence in this case ${\cal F} ^{**}$ is locally free. Moreover, as a bundle on ${\bf C}^2$ it must be holomorphically trivial. But ${\cal F} ^{**}$ restricts to $\pi_*(F)$ on ${\bf C}^2 - {0} ,$ hence $\pi_*(F)$ is trivial and so is $F.$ \hfill\vrule height 3mm width 3mm \begin{corollary} Holomorphic bundles on the blow up of a surface are trivial on a neighborhood of the exceptional divisor minus the exceptional divisor. \end{corollary} \noindent{\bf Proof}: Apply Theorem 3.2 to ${\widetilde {\bf C}^2} = {\cal O}(-1)$.\hfill\vrule height 3mm width 3mm \vspace{7 mm}

9608

alg-geom/9608001

en

https://arxiv.org/abs/alg-geom/9608001

alg-geom/9608001

Alexander I. Suciu

Daniel C. Cohen and Alexander I. Suciu

The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements

27 pages with 7 figures, author-supplied DVI file available at ftp://ftp.math.neu.edu/Pub/faculty/Suciu_Alex/papers/bmono.dvi AMSTeX v 2.1, pictex, edge-vertex-graphs

Commentarii Mathematici Helvetici 72 (1997), no. 2, 285-315.

10.1007/s000140050017

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated ``braided wiring diagram.'' The ensuing presentation of the fundamental group of the complement is shown to be Tietze-I equivalent to the Randell-Arvola presentation. Work of Libgober then implies that the complement of a line arrangement is homotopy equivalent to the 2-complex modeled on either of these presentations. Finally, we prove that the braid monodromy of a line arrangement determines the intersection lattice. Examples of Falk then show that the braid monodromy carries more information than the group of the complement, thereby answering a question of Libgober.

alg-geom

9608

alg-geom/9608021

en

https://arxiv.org/abs/alg-geom/9608021

alg-geom/9608021

Mark De Cataldo

Mark Andrea A. de Cataldo (Washington U. in St. Louis)

Codimension two nonsingular subvarieties of quadrics: scrolls and classification in degree $d\leq 10$

Latex; 20 pages

Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric $\Q{n}$ of dimension $n\geq 5$. We classify such subvarieties when they are scrolls. We also classify them when the degree $d\leq 10$. Both results were known when $n=4$. Keywords: Classification; liaison; low codimension; low degree; quadric; scroll; vector bundle

alg-geom

\section{INTRODUCTION} \label{intr} The paper \ci{ottp5} completes the classification of scrolls as codimension two subvarieties of projective space $\pn{n}$. Ottaviani's proof consists of three parts. First the sectional genus $g$ is exhibited as a function of the degree $d$ of the scroll. The degree $d$ is then bounded from above by the use of Castelnuovo-type bounds for $g$. The final step consists of the construction of varieties with prescribed low invariants which had been accomplished by several authors. In this paper we classify scrolls as codimension two subvarieties of $\Q{n}$; see Theorem \ref{maintm}. The analysis is quite similar to the one of \ci{ottp5} with the following three differences. The first one is that there are fourfolds scrolls on $\Q{6}$. The second difficulty is that the method for bounding the degree of scrolls over surfaces on $\Q{5}$ of \ci{ottp5} is not sufficient; we go around the problem using lemmata \ref{d=6...42} and \ref{d=6812}. Lastly, once we obtain a maximal list of invariants we must construct all the scrolls in question. This is essentially the problem of constructing varieties of low degree and codimension two on $\Q{n}$. We build on the results of \ci{a-s} and \ci{gross} and obtain Theorem \ref{classificationd<12}, i.e. the complete classification in degree $d\leq 10$ and $n\geq 5$. This result highlights the role that some special vector bundles on quadrics play in the construction of subvarieties of quadrics. As a by-pass result of this classification in low degree we are able to construct all scrolls, except for one case: when the degree $d=12$ and the base is a minimal $K3$ surface. We construct an unirational family of these scrolls; see Theorem \ref{esempium}. We do not know whether or not this is the only one. The paper is organized as follows. Section \ref{prel} contains preliminary results. Section \ref{d<=10} contains Theorem \ref{classificationd<12}, Section \ref{secpbundles} contains the main result of this paper, Theorem \ref{maintm}, the proof of which guides the reader through the rest of the section. \smallskip \noindent {\bf Notation and conventions.} Our basic reference is [Ha]. We work over any algebraically closed field of characteristic zero. A quadric $\Q{n}$, here, is a nonsingular hypersurface of degree two in the projective space $\pn{n+1}$. Little or no distinction is made between line bundles, associated sheaves of sections and Cartier divisors. $\lfloor t\rfloor $ denotes the biggest integer smaller than or equal to $t$. $\sim_n$ denotes the numerical equivalence of divisors on a surface. $\odixl{\Q{n}}{1}$ denotes the sheaf $\odixl{\pn{n+1}}{1}_{|\Q{n}}$. If $F$ is a coherent sheaf on $\Q{n}$ and $l$ an integer, then $F(l)$ denotes the sheaf $F\otimes \odixl{\Q{n}}{l}$. \smallskip \noindent {\bf Acknowledgments.} This paper is an expanded and completed version of parts of our dissertation. It is a pleasure to thank our Ph.D. advisor A.J. Sommese, who has suggested to us that we study threefolds on $\Q{5}$. We thank the C.N.R. of the Italian Government and The University of Notre Dame for partial support. We wish to thank K. Chandler and J. Migliore for valuable discussions concerning Proposition \ref{d=6812} and E. Arrondo and G. Ottaviani for useful correspondences. \section{PRELIMINARY MATERIAL} \label{prel} In this section we collect the various results that will be necessary in sections \ref{d<=10} and \ref{secpbundles}. In this section $X$ is a codimension two, nonsingular subvariety of $\Q{n}$, $d$ its degree, $\nb{X}{\Q{n}}$ its normal bundle, $n_i$ the $i^{th}$ Chern class of $\nb{X}{\Q{n}}$, $\iota:X\hookrightarrow \Q{n}$ the embedding, $L$ the restriction of $\odixl{\Q{n}}{1}$ to $X$, $K_X$ the canonical dualizing sheaf of $X$. \subsection{Miscellanea} \label{miscellanea} The cohomology ring of a nonsingular quadric of any dimension is described in \ci{h-p}. Let $h$ be the class of any hyperplane section of $\Q{n}$. We consider the odd dimensional case first: $\Q{2n+1}$. One can describe $H^*(\Q{2n+1},$ $\zed)$ as follows. Let $\Lambda$ be the class of an $n$-dimensional linear space in $\Q{2n+1}.$ The relevant information is, denoting the cup product by ``$\cdot$": \smallskip \noindent $H^{2i+1}(\Q{2n+1},{\Bbb Z})=\{0\}$, $\forall i$;\, $H^{2i}(\Q{2n+1},{\Bbb Z})=$ $\{0\}$, for $i>2n+1$;\, $H^{2i}(\Q{2n+1},{\Bbb Z})=$ ${\Bbb Z}[h^i]$, $i=0,\ldots, n$;\, $H^{2(n+j)}(\Q{2n+1},{\Bbb Z})$ $={\Bbb Z}[\Lambda\cdot h^{j-1}]$, $j=1,\ldots,n+1$;\, $h^{n+1}=2\Lambda$, $h^{2n+1}=2$. \smallskip As to the even dimensional case, we denote by $\Lambda_1$, $\Lambda_2$ the classes of two members of the two rulings of $\Q{2n}$ in $n$-dimensional linear spaces. One has: \smallskip \noindent $H^{2i+1}(\Q{2n},{\Bbb Z})=\{0\}$, $\forall i$;\, $H^{2i}(\Q{2n},{\Bbb Z})=$ $\{0\}$, for $i>2n$;\, $H^{2i}(\Q{2n},{\Bbb Z})=$ ${\Bbb Z}[h^i]$, $i=0,\ldots, n-1$;\, $H^{2i}(\Q{2n},{\Bbb Z})=$ $\zed [\Lambda_1]\bigoplus \zed[\Lambda_2]$;\, $H^{2(n+j)}(\Q{2n+1},{\Bbb Z})$ $={\Bbb Z}[\Lambda_1\cdot h^{j-1}]$ $={\Bbb Z}[\Lambda_2\cdot h^{j-1}]$, $j=1,\ldots,n$;\, $h^{n}=\Lambda_1+\Lambda_2$;\, $h^{2n}=2$;\, $[\Lambda_i] \cdot [\Lambda_j]= \delta_{ij}$, where $\delta_{ij}$ is the Kronecker symbol. \begin{rmk} \label{rmkdiseven} {\rm The above description of the cohomology ring of $\Q{n}$ implies that, for $n\geq 5$, $d$ is an even integer. } \end{rmk} Mumford's {\em self intersection formula} (cf. \ci{fu}, page 103) gives, for $n\geq 5$: \begin{equation} \label{muself} n_2=\frac{1}{2}d L^2. \end{equation} Consider the twisted ideal sheaves $\Il{X}{\Q{n}}{l}:=$ $ \I{X}{\Q{n}}\otimes \odixl{\Q{n}}{l}$. We write the total Chern class of these sheaves as $1+\sum_{i=1}^{n}\gamma_i h^i$. The following is a standard a consequence of \ci{fu}, Theorem and Lemma 15.3. \begin{lm} \label{lmccis} Let $X$ and $\Il{X}{\Q{n}}{l}$ be as above, with $l$ fixed. Assume that $n\geq 5$. Then one has the following relations concerning the Chern classes of \, $\Il{X}{\Q{5}}{l}:$ $$ \gamma_1 = l; \qquad \qquad \gamma_i = \frac{1}{2} (K_X+(5-l)L)^{i-2}\cdot L^{n-i}, \quad \forall i=2,\ldots, n. $$ \end{lm} We now make explicit the {\em Double Point Formul\ae \ } for the embedding $\iota$. The proof is a standard consequence of \ci{fu} Theorem 9.3, once we use (\ref{muself}) and the fact that $n_i=0$ for $i\geq 3$. Denote by $c_i$ the Chern classes of the tangent bundle of $X$. \begin{lm} \label{lmDPF} Let $\iota:X\hookrightarrow \Q{n}$ be as above with $n\geq 5$. Then one has the following relations in the Chow ring of $X$: \begin{equation} \frac{1}{2} d L^2=\frac{1}{2}(n^2 -n +2)L^2 -n c_1\cdot L + c_1^2 - c_2; \label{deg2dpf5} \end{equation} \begin{equation} c_3= \frac{1}{6}(n^3 -3n^2+8n-12)L^3 +\frac{1}{2}(-n^2 +n -2)c_1 L^2+ n(c_1^2-c_2)L +2c_1c_2-c_1^3 ; \label{deg3dpf5} \end{equation} \begin{equation} \label{deg4dpf} c_4= 22L^4 - 24L^3c_1 + 16L^2(c_1^2-c_2) + 12Lc_1c_2 - 6Lc_1^3 -6Lc_3 +2c_1c_3 + (c_1^2-c_2)^2 -c_1^2c_2. \end{equation} \end{lm} For $n=5$ we have: \begin{equation} \label{KL2} K_X\cdot L^2=2(g-1)-2d, \end{equation} \begin{equation} \label{K2L} K_X^2\cdot L=\frac{1}{4}d^2 +\frac{3}{2}d -8(g-1) +6\chi ({\cal O}_S), \end{equation} \begin{equation} \label{K3} K_X^3=-\frac{9}{2}d^2+ \frac{27}{2}d+gd +18(g-1)- 30\chi ({\cal O}_S)-24\chi ({\cal O}_X), \end{equation} \begin{equation} \label{c2L} c_2\cdot L= -\frac{1}{4}d^2 +\frac{5}{2}d +2(g-1)+6 \chi ({\cal O}_S) \end{equation} \begin{equation} \label{c3} c_3=\frac{1}{4}d^2- \frac{1}{2}d -10(g-1)+gd + 24\chi ({\cal O}_S)-30\chi ({\cal O}_X). \end{equation} To prove (\ref{KL2}) we use the genus formula. (\ref{K2L}) follows from \ci{a-s}, Proposition 2.1, after having realized that $K_X^2\cdot L=$ ${K_X}^2_{|S}=$ $(K_S-L_{|S})^2$. The formula for $K_X^3$ follows by ``cutting" (\ref{deg2dpf5}) with $K_X$ and by using the above expressions for $K_X\cdot L^2$, $K_X^2\cdot L$, and the fact that, by Hirzebruch- Riemann-Roch on a threefold, $c_1c_2=24 \chi ({\cal O}_X)$. The proof of (\ref{c2L}) is similar. (\ref{c3}) is obtained from (\ref{deg3dpf5}) by first plugging the expression for $(c_1^2-c_2)$ that one gets form (\ref{deg2dpf5}) and then by plugging the above relations into it. \smallskip Finally we record the expression for the Hilbert polynomial of a threefold $X \subseteq \Q{5}$: \begin{equation} \label{chioxt} \chi ({\cal O}_X (t)) = \frac{1}{6}d\,t^3+[\frac{1}{2}- \frac{1}{2}(g-1)]\,t^2+ [\frac{1}{3}d - \frac{1}{2}(g-1) +\chi ({\cal O}_S)]\,t +\chi ({\cal O}_X), \end{equation} which is an easy consequence of Hirzebruch-Riemann-Roch on a threefold (cf. \ci{ha}, page 437) and of the formul\ae \ above. \begin{fact} \label{unirat} {\rm (Unirationality of the Hilbert scheme) Let $H$ the connected component of the Hilbert scheme of $\Q{n}$ containing the point corresponding to a fixed $X\subseteq \Q{5}$. Denote by $\frak H$ the open subscheme of $H$ corresponding to nonsingular subvarieties. Assume that every subvariety, $X'\in {\frak H}$, admits a resolution of its ideal sheaf of the following form: $$ 0 \to \odix{\Q{n}}^s \to E \to {\cal I}_{X'}(c_1(E)) \to 0, $$ where $E$ is a fixed, locally free sheaf independent of $X'$ and $s$ is a fixed positive integer. \noindent Under the above assumptions $\frak H_{red}$ is integral and unirational. In fact it is enough to observe that the natural rational map ${\Bbb P}(\wedge^s H^0(E)^{\vee}) --> {\frak H}$ is a dominant one. } \end{fact} \medskip In the present context, Lemma 2.3 of \ci{a-s} gives the following: \begin{fact} \label{dimensionhilbert} {\rm (Smoothness and dimension of the Hilbert Scheme) } {\rm If $h^i(\odix{X})=0$, $i\geq 1$, then $h^i({\cal N}_{X,\Q{n}})=0$, $\forall i\geq 1$, $\frak H$ is nonsingular and of dimension $h^0({\cal N}_{X,\Q{n}})$ at $X$. Riemann-Roch on a threefold, $n_1=K_X+5L$, $n_2=(d/2)L^2$, and formul\ae \ (\ref{KL2}), (\ref{K2L}), (\ref{c2L}) give, for $n=5$: $$ \chi ( {\cal N}_{X,\Q{n}})= -\frac{5}{4}d^2 + 10d + 10(g-1) + 5\chi (\odix{S}). $$} \end{fact} \begin{fact} \label{hilbertforci} {\rm (The Hilbert scheme of complete intersections) If $X\subseteq \Q{n}$ is a complete intersection of type $(2,i,j)$ in $\pn{n+1}$, then the corresponding Hilbert scheme $\frak H$ is integral, nonsingular nd rational. \noindent For $i<j$: \begin{eqnarray} \dim {\frak H}=P(n;i,j)\hspace{-0.1in}&:=& [ B(n+1+i,n+1)-B(n+1+i-2,n+1)-1 ] + \nonumber \\ &\ & [ B(n+j,n) - B(n+j-2,n) -1], \nonumber \end{eqnarray} where $B(a,b):=a!/[b!(a-b)!]$, is the usual binomial coefficient. \noindent For $i=j$: $$ \dim {\frak H}=Q(n;i):=2[ B(n+1+i,n+1)-B(n+1+i-2,n+1) -2 ]. $$ } \end{fact} \bigskip The following gives: 1) a method to construct codimension two subvarieties of $\Q{n}$ using vector bundles; 2) a way to reconstruct the ideal sheaf of a codimension two subvariety, $X$, given enough sections of twists of its dualizing sheaf $K_X$. \begin{fact} \label{hartshorne} {\rm The following is a Bertini-type Theorem due to Kleiman; see \ci{kl}. Let $ E,$ $ F$ two vector bundles on $\Q{n}$ of rank $m$ and $m'$ respectively, such that $E^{\vee}\otimes F$ is generated by its global sections. Let $\phi: E \to F$ be an element of $H^0( E^{\vee}\otimes F)$. Define $D_k(\phi)$ to be the closed subscheme of $\Q{n}$ defined, locally, by the vanishing of the $(k+1)\times (k+1)$ minors of a matrix representing $\phi$. For the general $\phi$ and for every $k$: \noindent {\em {\rm a)} either $D_k$ is empty or it has codimension $(m-k)(m'-k)$ and $D_k(\phi)_{sing}\subseteq D_{k-1}(\phi)$; in particular, for $ n< (m-k+1)(m'-k+1)$, $D_k(\phi)$ is nonsingular; \noindent {\rm b)} for $n\geq 5$, assuming that $D_k(\phi)$ has codimension two, $D_k(\phi)$ is connected {\rm (see the remarks following Theorem 2.2 of \ci{so-vdv})}. } \noindent The following fact, proved by Vogelaar, stems from an idea of Serre's; see \ci{o-s-s}, Theorem I.6.4.2 or \ci{a-s} \S 2.3. Let $X\subseteq \Q{n}$ be a local complete intersection of codimension two and $a$ an integer such that the twist $\omega_X(a)$ is generated by $s$ of its global sections. Then we have an exact sequence $$ 0 \to \odix{\Q{n}}^s \to F \to {\cal I}_X(n-a) \to 0, $$ with $F$ locally free. } \end{fact} \subsection{A lifting criterion and bounds for the genera of curves on $\Q{3}$} \label{rothlifting} The following is well known when $\Q{n}$ is replaced by $\pn{n}$, see \ci{boss} for example. The case of $\Q{4}$ is proved in \ci{a-s}, Lemma 6.1. The general case can be proved in the same way. We used it as a tool to prove the finiteness of the number of families of nonsingular codimension two subvarieties of $\Q{5}$ not of general type. See \ci{bounded}, where we prove a more general statement. \begin{pr} \label{roth} Let $X$ be an integral subscheme of degree $d$ and codimension two on $\Q{n}$, $n\geq 4$. Assume that for the general hyperplane section $Y$ of $X$ we have $ h^0({\cal I}_{Y,\Q{n-1}}(\s))\not= 0, $ for some positive integer $\s$ such that $d>2{\s}^2$. Then\, $ h^0({\cal I}_{X,\Q{n}}({\s}))\not= 0. $ \end{pr} \medskip \begin{evidence} \label{1.1} {\rm $C$ is an integral curve lying on a smooth three-dimensional quadric $\Q{3}$, $k$ is a positive integer, $S_k$ is an integral surface in $|\odixl{\Q{3}}{k}|$ containing $C$, $d$ and $g$ are the degree and the geometric genus of $C$, respectively.} \end{evidence} \begin{defin} \label{1.2} {\rm Define $n_0$ and $\epsilon$ when $d>2k(k-1)$ and $\ \theta_0$ and $\epsilon'$ when $d\leq 2k(k-1)$ as follows:} $$ \begin{array}{lrllllcc} &n_0 :=& \lfloor \frac{d-1}{2k}\rfloor+1, \qquad &d \equiv& -\epsilon \quad (mod \ 2k), \quad &0 \leq \epsilon\leq 2k-1;& \\ & & & & & & \\ &{\theta}_0 :=& \lfloor \frac{d-1}{2k}\rfloor +1, \qquad &d \equiv& -\epsilon'\quad (mod\ 2\theta_0),\quad &0 \leq \epsilon'\leq 2\theta_0-1.& \end{array} $$ \end{defin} The following class of curves plays a central role in the understanding of the curves whose genus is the maximum possible. Arithmetically Cohen-Macaulay is denoted by a.C.M.. \begin{defin} \label{1.3} {\rm A curve $C$ as in (\ref{1.1}) is said to be in the class ${\frak S} (d,k)$, if it is nonsingular, projectively normal and linked, in a complete intersection on $\Q{3}$ of type $(k, n_0)$ if $d>2k(k-1)$ (of type $(\theta_0, k)$ if $d\leq 2k(k-1)$), to an {\it a fortiori} a.C.M. curve $D_{\epsilon}$ ($D_{\epsilon'}$, respectively) of degree $\epsilon$ ($\epsilon'$ respectively) lying on a quadric surface hyperplane section of $\Q{3}.$} \end{defin} \begin{pr} \label{1.4} {\em (Cf. \ci{de}.)} Notation as in {\rm (\ref{1.1})} and {\rm Definition \ref{1.2}}. Assume first that $d>2k(k-1)$. Then \noindent {\rm (a)} $$ g-1\leq \pi(d,k) -\Xi $$ where \[ \pi(d,k)= \left\{ \begin{array}{ll} \frac{d^2}{4k} +\frac{1}{2}(k-3)d-\frac{\epsilon^2}{4k}-\epsilon(\frac{k-\epsilon}{2}), & \mbox{if $\ 0\leq \epsilon\leq k,$} \\ \ \\ \frac{d^2}{4k}+\frac{1}{2}(k-3)d-(k-\tilde \epsilon) (\frac{\tilde \epsilon}{2}-\frac{\tilde \epsilon}{4k} + \frac{1}{4}), & \mbox{if $\ k+1\leq \epsilon \leq 2k-1,\ \tilde \epsilon:=\epsilon-k;$} \end{array} \right. \] and \[ \Xi=\Xi(d,k)= \left\{ \begin{array}{ll} 0 & \mbox{ if $\ \epsilon=0,\ 1,\ 2,\ 2k-1,$} \\ \ \\ 1 & \mbox{ if otherwise.} \end{array} \right. \] {\rm (b)} The bound is sharp for $\epsilon =0$, $1,$ $2,$ $3,$ $2k-2,$ $2k-1$. A curve achieves such a maximum possible genus if and only if it is in the class ${\frak S} (d,k)$, except, possibly, the cases $\epsilon=3$, $2k-2$. \medskip \noindent Assume $d\leq 2k(k-1)$. Then statements {\rm a)} and {\rm b)}, with $\pi'(d,k)=\pi(d, \lfloor \frac{d-1}{2k}\rfloor +1)= \pi(d, \theta_0)$ and with $\Xi',$ $\epsilon'$, $(\theta_0,k)$ and $D_{\epsilon'}$ replacing $\Xi,$ $\epsilon,$ $(k,n_0)$ and $D_{\epsilon}$, respectively, hold. \end{pr} \begin{cor} \label{coreasybound} {\em (See \ci{a-s}, Proposition 6.4 for the case $d> 2k(k-1)$.)} Notation as above. Then $$ g-1\leq \frac{d^2}{4k} + \frac{1}{2} (k-3)d. $$ \end{cor} \begin{pr} \label{boundasep} {\em (Cf. \ci{a-s}, Proposition $6.4$.)} Let $C$ be an integral curve in $\Q{3}$, not contained in any surface of degree strictly less than $2k$. Then: $$ g-1\leq \frac{d^2}{2k} +\frac{1}{2}(k-4)d. $$ \end{pr} \subsection{An inequality} \label{useful<} In this section we prove an inequality which is an essential tool for our proof of the classification of scrolls over surfaces on $\Q{5}$. Let $X\subseteq \Q{5}$ be a three dimensional, nonsingular variety, $\s$ be the smallest integer for which there exists a hypersurface $V$ in $|{\cal I}_{X,\Q{5}}(\s)|$ and $\cal N$ the normal bundle of $X$. By the minimality of $\s$, the natural section $\odix{X} \to \check{\cal N} (\s)$ is not the trivial one. The transposed of this section defines the sheaf of ideals of $\odix{X}$ of the singular locus of $V$ restricted to $X$. Let us denote by $\tilde \Sigma$ the associated scheme. We obtain the surjection ${\cal N} \to {\cal I}_{\tilde \Sigma}(\sigma)$. \begin{defin} \label{Divisor} {\rm Let $D$ be the divisorial component of $\tilde \Sigma$, i.e. the unique effective Cartier divisor of $X$ whose sheaf of ideals is the smallest sheaf of principal ideals containing ${\cal I}_{\tilde \Sigma}$; $D$ may be empty. Let $\Sigma$ be the one dimensional component of $\tilde \Sigma$, i.e. the scheme associated with the sheaf of ideals ${\cal I}_{\tilde \Sigma}(D)$; $\Sigma$ is either empty or of pure dimension one.} \end{defin} From the above we get that the following two facts hold. \begin{fact} \label{homdim1} {\rm the sheaf ${\cal I}_{\Sigma}$ is either $\odix{X}$ or it has homological dimension one.} \end{fact} \begin{fact} \label{ISIGMAspanned} {\rm ${\cal I}_{\Sigma}(\s L - D) = {\cal I}_{\tilde \Sigma}(\s L)$; in particular, ${\cal I}_{\Sigma}(\s L - D)$ is generated by global sections since it is a quotient of $\cal N$ which is a quotient sheaf of the globally generated sheaf ${\cal T}_{\Q{5}}$. } \end{fact} \begin{pr} \label{bunchof<>} Let $s_i$, $i=1,$ $2,$ $3$ be the Segre classes of ${\cal I}_{\Sigma}(\s L -D)$. Then\, $ s_1s_2\geq s_3 \geq 0, $ $s_1$ and $s_1^2-s_2$ are represented by effective cycles. Moreover, \begin{equation} \label{55} \chi (\odix{S})\geq \frac{1}{6\s}[(d-12\s )(g-1) + (\frac{1}{4}\s + \frac{3}{2})d^2 -\frac{13}{2}\s d] -\frac{1}{6\s}[\frac{1}{2}dL^2 - (K_X+5L)^2]D. \end{equation} \end{pr} \noindent {\em Proof.} By Fact \ref{ISIGMAspanned} there is a surjection $\odix{X}^{m} \to $${\cal I}_{\Sigma}(\s L -D)$, for some $m$. By Fact \ref{homdim1} the kernel, $F$, of this surjection is locally free. By the definition of Segre classes, $s_i=c_i(\check F)$. The first part of the proposition follows from \ci{boss}, Lemma 5.1. \noindent As to the proof of the last inequality, first we compute the Chern classes $C_i$ of ${\cal I}_{\Sigma}(\s L -D)$ using the following exact sequence which is the Koszul resolution of ${\cal I}_{\Sigma}(\s L -D)$: $$ 0 \to \odixl{X}{K_X -\s L + D} \to {\cal N} \to {\cal I}_{\Sigma}(\s L -D) \to 0; $$ we get \noindent $C_1=\s L -D,$ \noindent $C_2= \frac{1}{2} d L^2 - (K_X+5L)(\s L -D) + (\s L -D)^2,$ \noindent $C_3=- \frac{1}{2}d L^2(K_X +5L)+ \frac{1}{2}dL^2(\s L -D) + (K_X +5L)^2(\s L -D)- 2(K_X +5L )(\s L- D)^2 + (\s L-D)^3.$ \noindent The Segre classes of ${\cal I}_{\Sigma}(\s L -D)$ are $s_1=C_1$, $s_2=C_1^2- C_2$, $s_3= C_1^3- 2C_1C_2 + C_3$. We make explicit these Segre classes using the formul\ae \ for the $C_i$. Then we use (\ref{K2L}) and (\ref{KL2}). We now use the part of the proposition that we have just proved: ({\ref{55}) is $s_3\geq 0$. \blacksquare \subsection{Special vector bundles on quadrics} \label{vbundlesonq} \begin{fact} \label{spinorbundles} {\rm (Spinor Bundles) Here we collect some properties of spinor bundles on quadrics. See \ci{a-ott}. \noindent Let ${\cal S}$ be the spinor bundle on an odd-dimensional quadric and ${\cal S}'$, ${\cal S}''$ be the two spinor bundles on an even dimensional quadric; if $n$ is the dimension of the quadric, the rank of these bundles is $2^{ \lfloor \frac{n-1}{2} \rfloor}$. \noindent For $n=2m+1$ (odd) we have an exact sequence: $$ 0 \to {\cal S} \to \odix{\Q{n}}^{ 2^{m+1} } \to {\cal S}(1) \to 0; $$ for $n=2m$ (even) we have exact sequences: $$ 0 \to S \to \odix{\Q{n}}^{ 2^{m} } \to S(1) \to 0, $$ where $S$ denotes either ${\cal S}'$ or ${\cal S}''$. \noindent For $n=2m+1$ we have ${\cal S}^{\vee} \simeq {\cal S}(1)$. For $n=4m$ we have ${{\cal S}'}^{\vee} \simeq {\cal S}'(1)$ and ${{\cal S}''}^{\vee} \simeq {\cal S}''(1)$; for $n=4m+2$ we have ${{\cal S}'}^{\vee} \simeq {\cal S}''(1)$ and ${{\cal S}''}^{\vee} \simeq {\cal S}'(1)$. Let $i: \Q{2k-1} \hookrightarrow \Q{2k}$ be a nonsingular hyperplane section; then $i^* {\cal S}' \simeq$ $i^* {\cal S}'' \simeq$ $\cal S$. Let $j: \Q{2h} \hookrightarrow \Q{2h+1}$ be a nonsingular hyperplane section; then $j^* {\cal S} \simeq {\cal S}' \oplus {\cal S}''$. \noindent An analogue of Horrocks splitting criterion holds on quadrics; recall that spinor bundles carry no intermediate cohomology: \noindent {\em let $E$ be a vector bundle on $\Q{n}$ then $h^i(E(t))=0$, $0<i<n$, $\forall t \in \zed$ if and only if $E$ splits as the direct sum of line bundles and twists of spinor bundles of $\Q{n}$.} \noindent The Chern polynomial of ${\cal S}(l)$ on $\Q{5}$ is: \begin{eqnarray} c({\cal S}(l)) &=& 1 + (4l-2)h + (6l^2-6l+2)h^2 +(4l^3-6l^2+4l-1)h^3+ \nonumber \\ &+& (l^4-2l^2+2l^2-l)h^4. \nonumber \end{eqnarray} The Chern polynomial of ${\cal S}'(l)$ on $\Q{6}$ is: \begin{eqnarray} c({\cal S}'(l)) &=& 1+ (4l-2 )h+ (6l^{2}-6l+2 )h^{2}+[ (4\,l^{3}-6l^{2}+4l )h^{3} -2{\Lambda_1}] + \nonumber \\ &+&[ ( l^{4} -2l^{3} + 2l^{2})h ^{4} -2l\,{\Lambda_1}h] \nonumber \end{eqnarray} Replacing $\Lambda_1$ by $\Lambda_2$ in the formula above, we get $c({\cal S}''(l))$. } \end{fact} \begin{fact} \label{cayley} {\rm (Cayley bundles) See \ci{ottcayley}. On $\Q{5}$ there is a family of rank two stable vector bundles, called Cayley bundles. Each Cayley bundle $\cal C$ has Chern classes $c_1=-1$, $c_2=1$ and ${\cal C}(2)$ is generated by global sections. Every stable $2$-bundle on $\Q{5}$ with Chern classes $c_1=-1$, $c_2=1$ is a Cayley bundle. Cayley bundles are parameterized by a fine moduli space isomorphic to $\pn{7} \setminus \Q{6}$. A Cayley bundle restricts, on a $\Q{4}$, to a bundle of type $\dot{E}$ which appears in the description of Type 10) of \ci{a-s}, page 44. The Chern polynomial of a ${\cal C}(l)$ is: $c({\cal C}(l))= 1 + (2l-1)h + (l^2-l+1)h^2.$ } \end{fact} \section{CLASSIFICATION FOR $d\leq 10$} \label{d<=10} \subsection{The list} \label{list} In what follows: - $((a,b,c),{ \cal O}(1))$ denotes the polarized pair given by a complete intersection of type $(a,b,c)$ in $\pn{n+1}$ and the restriction of the hyperplane bundle to it; - $(X,L)$ denotes the polarized pair given by a variety $X\subseteq \Q{n}$ and $L:= \odixl{\Q{n}}{1}_{|X}$; if we do not explicitly say the contrary, the embeddings are projectively normal in $\pn{n+1}$; this fact follows from the cohomology of the presentation of the ideal sheaf; - by a {\em presentation} of the ideal sheaf ${\cal I}_X$ we mean an injection of locally free coherent sheaves on $\Q{n}$, $\phi: E \to F$, such that $coker(\phi)\simeq {\cal I}_X(i)$, where $i=c_1(F)-c_1(E)$; we write the presentations so that the integer $i$ is the smallest for which the sheaf $F$ is generated by global sections, so that for that $i$ so will be the sheaf ${\cal I}_{X}(i)$; - $\frak H$ denotes the Hilbert scheme of $\Q{n}$ of a variety fixed by the context; see Fact \ref{unirat}; - $P(n;i,j)$ and $Q(n;i)$ are defined in Fact \ref{hilbertforci}; - a digit ``$\#$)" refers to the type of the surface section as in \ci{a-s}, page 44 (where $d\leq 8$); type $Z^F_{10}$ refers to the paper \ci{gross} - $g,$ $q$ and $p_g$ denote the sectional genus of the embedding line bundle, the irregularity and geometric genus of a surface section, respectively. \begin{tm} \label{classificationd<12} Let $X\subseteq \Q{n}$, $n\geq 5$, a codimension two nonsingular subvariety of degree $d\leq 10$. Then the pair $(X,L)$, a presentation of the ideal of $X$ on $\Q{n}$ and the Hilbert scheme, $\frak H$, of $X$ on $\Q{n}$ are as follows. \medskip \noindent {\rm ($\bullet$)} $d=2$ \smallskip \noindent \underline{\rm Type A):}\quad $((1,1,2), {\cal O}(1))$;\quad $\odixl{\Q{n}}{-1} \to \odix{\Q{5}}^2$;\quad $\frak H$ is integral, nonsingular, rational, of dimension $Q(n;1)$;\quad {\rm 2)};\quad $g=q=p_g=0$. \medskip \noindent {\rm ($\bullet$)} $d=4$ \smallskip \noindent \underline{\rm Type B):}\quad $((1,2,2),{\cal O}(1))$;\quad $\odixl{\Q{n}}{-1} \to \odixl{\Q{n}}{1} \oplus \odix{\Q{n}}$;\quad $\frak H$ is integral, nonsingular, rational and of dimension $P(n;1,2)$;\quad {\rm 6)};\quad $g=1$, $q=p_g=0$. \smallskip \noindent \underline{\rm Type C):}\quad $n=6$, $(\pn{1}\times \pn{3}, {\cal O}(1,1))$;\quad $\odix{\Q{6}}^3 \to {\frak S}(1), $ with ${\frak S}\simeq {\cal S}', {\cal S}''$; \quad $\frak H$ consists of two connected components, which are both nonsingular, integral, unirational and of dimension $15$; \quad {\rm 5)};\quad $g=q=p_g=0$. \smallskip \noindent \underline{\rm Type D):}\quad $n=5$, $({\Bbb P}( {\odixl{\pn{1}}{1}}^2\oplus \odixl{\pn{1}}{2}), \xi)$;\quad $\odix{\Q{6}}^3 \to {\cal S}(1);$ \quad $\frak H$ is integral, nonsingular, unirational and of dimension $15$; \quad {\rm 5)};\quad $g=q=p_g=0$. \medskip \noindent {\rm ($\bullet$)} $d=6$ \smallskip \noindent \underline{\rm Type E):}\quad $((1,2,3),{\cal O}(1))$;\quad $\odixl{\Q{n}}{-1} \to \odixl{\Q{n}}{2}\oplus \odix{\Q{n}}$;\quad ${\frak H}$ is integral, nonsingular, rational and of dimension $P(n;1,3)$;\quad {\rm 12)}; \quad $g=4$, $q=0$, $p_g=1$. \smallskip \noindent \underline{\rm Type F):}\quad $n=5$, $({\Bbb P}( {\cal T}_{\pn{2}}),\xi)$, embedded using a general codimension one linear system ${\frak l}\subseteq| \xi_{ {\cal T}_{\pn{2}} }|$;\quad $\odix{\Q{5}} \to {\cal C}(2)$;\quad $\frak H$ is integral, nonsingular, unirational and of dimension $20$;\quad {\rm 10)};\quad $g=1$, $q=p_g=0$. \smallskip \noindent \underline{\rm Type G):}\quad $n=5$, $f: X \to \pn{1} \times \pn{2}=:Y$ a double cover, branched along a divisor of type $\odixl{Y}{2,2}$, $L\simeq p^* \odixl{ Y }{1,1}$;\quad ${\odixl{\Q{5}}{-1}}^2 \to \odix{\Q{5}}^3$; $\frak H$ is integral, nonsingular, unirational and of dimension $30$;\quad {\rm 11)};\quad $g=2$, $q=p_g=0$. \medskip \noindent {\rm ($\bullet$)} $d=8$ \smallskip \noindent \underline{\rm Type H):}\quad $((1,2,4), {\cal O}(1))$;\quad $\odixl{\Q{n}}{-1} \to \odixl{\Q{n}}{3} \oplus \odix{\Q{n}}$;\quad $\frak H$ is integral, nonsingular, rational and of dimension $P(n;1,4)$;\quad {\rm 20)};\quad $g=9$, $q=0$, $p_g=5$. \smallskip \noindent \underline{\rm Type I):} $((2,2,2), {\cal O}(1) )$;\quad $\odixl{\Q{n}}{-2} \to \odix{\Q{n}}^2$;\quad $\frak H$ is integral, nonsingular, rational and of dimension $Q(n;2)$;\quad {\rm 19)};\quad $g=5$, $q=0$, $p_g=1$. \smallskip \noindent \underline{\rm Type L):} $n=5$, $({\Bbb P}( E), \xi)$, $E$ a rank two vector bundle on $\Q{2}$ as in {\rm \ci{io3}};\quad $\odix{\Q{5}}^4 \to {\cal S}(1)\oplus \odixl{\Q{5}}{1}$;\quad $\frak H$ is integral, nonsingular, unirational and of dimension $35$;\quad {\rm 18)};\quad $g=4$, $q=p_g=0$. \medskip \noindent {\rm ($\bullet$)} $d=10$ \smallskip \smallskip \noindent \underline{\rm Type M):} $((1,2,5), {\cal O}(1) )$;\quad $\odixl{\Q{n}}{-1} \to \odixl{\Q{n}}{4} \oplus \odix{\Q{n}}$;\quad $\frak H$ is integral, nonsingular, rational and of dimension $P(n;1,5)$;\quad $g=16,$ $q=0,$ $p_g=14$. \smallskip \noindent \underline{\rm Type N):}\quad $n=5$, $f_{|K_X+L|}: X \to \pn{1}$ is a fibration with general fiber a Del Pezzo surface $F$, $K_F^2=4$, $K_X=-L+ f^*\odixl{\pn{1}}{1}$;\quad $\odixl{\Q{5}}{-1}^2 \to \odixl{\Q{5}}{1} \oplus \odix{\Q{5}}^2$;\quad $\frak H$ is integral, nonsingular, unirational and of dimension $60$; type $Z_{10}^F$;\quad $g=8$, $q=0$, $p_g=2$. \end{tm} \begin{rmk} {\rm In this remark, by the symbol ${\rm Q} \stackrel{(a,b)}{\sim} {\rm R}$, we mean that {\em every} variety of Type Q) is linked to {\em a} variety of Type R) in a complete intersection of type $(a,b)$ on $\Q{5}$. Using Lemma \ref{pesk} and the presentations of the ideals of the varieties of the above theorem we see that: \noindent ${\rm A} \stackrel{(1,2)}{\sim} {\rm A}$, ${\rm A} \stackrel{(1,3)}{\sim} {\rm B}$, ${\rm A} \stackrel{(1,4)}{\sim} {\rm E}$, ${\rm A} \stackrel{(2,2)}{\sim} {\rm G}$, ${\rm A} \stackrel{(1,5)}{\sim} {\rm H}$, ${\rm A} \stackrel{(1,6)}{\sim} {\rm M}$, ${\rm A} \stackrel{(2,3)}{\sim} {\rm N}$, ${\rm B} \stackrel{(2,2)}{\sim} {\rm B}$, ${\rm B} \stackrel{(2,3)}{\sim} {\rm I}$, ${\rm G} \stackrel{(2,2)}{\sim} {\rm A}$, ${\rm G} \stackrel{(2,3)}{\sim} {\rm G}$, ${\rm G} \stackrel{(2,4)}{\sim} {\rm N}$, ${\rm I} \stackrel{(2,3)}{\sim} {\rm B}$, ${\rm I} \stackrel{(2,4)}{\sim} {\rm I}$, ${\rm I} \stackrel{(3,3)}{\sim} {\rm N}$, ${\rm N} \stackrel{(3,3)}{\sim} {\rm I}$. \noindent The simple details are left to the reader. As for Type F), see Proposition \ref{example}. } \end{rmk} \subsection{The proof} \noindent {\em Proof of Theorem} \ref{classificationd<12}. The degree $d$ is always an even integer by Remark \ref{rmkdiseven}. The statements of the Theorem concerning complete intersections follow from Fact \ref{hilbertforci} and \ci{ha}, III.9.Ex. 9.6. In the sequel, we do not deal with complete intersections. \medskip \noindent {\em Claim. The only nonsingular surfaces on $\Q{4}$ which can be a general hyperplane section of a threefold on $\Q{5}$ of degree $d\leq 10$ are: types} 5), 10), 11) {\em and} 18) {\em from} \ci{a-s} {\em and type} $Z^{10}_F$ {\em from} \ci{gross}. } \medskip \noindent {\em Proof of the Claim.} Let $d\leq 8$. \ci{a-s} page 44 contains the complete list of nonsingular surfaces on a $\Q{4}$ of degree $d\leq 8$. Not all of them can be a general hyperplane section of a threefold on $\Q{5}$. The complete list of linearly normal, nonsingular subvarieties of projective space of degree $d\leq 8$ is given in \ci{io1} and \ci{io3}. We are going to use these results jointly. \noindent Since $H^2(\Q{5},\zed )\simeq \zed <h^2>$, the surface section, $S$, of a degree $d$ threefold $X\subseteq \Q{5}$ has cohomology class $[S]=(d/2) \Lambda_1 + (d/2) \Lambda_2$. This implies that the surfaces of type 1), 3), 4), 7), 8), 13), 14), 15) and 16) of \ci{a-s} page 44 cannot be nonsingular hyperplane sections of any threefold on $\Q{5}$. \noindent Types 2), 6), 12), 19) and 20) are complete intersections and we drop them from the list. \noindent In what follows, assume that $S$ is a surface of a given type and that $X\subseteq \Q{5}$ is a threefold with general surface section $S$. We now exclude types 9) and 17). Type 9). If $X$ existed, a comparison with Ionescu's list \ci{io1} would force $X$ to be a rational scroll over a curve contradicting Proposition \ref{pbundlesovercurves}. Type 17). This type has sectional genus $g=3$ so that, according to Ionescu's list \ci{io3}, $X$ would have to be either a scroll over an elliptic curve, a scroll over $\pn{2}$ or $X$ would have to admit a morphism onto $\pn{1}$ with all fibers quadric surfaces. We exclude the first case because type 17) is simply connected, the second one by Proposition \ref{baseofscroll}. The last one would imply, after having cut (\ref{deg2dpf5}) with a general fiber $F\simeq \Q{2}$, the contradiction $d=6$. \noindent It follows that, except for the case of complete intersections, only the following types are admissible as surface sections of codimension two nonsingular subvarieties of quadrics when $d\leq 8$: \,5), 10), 11), 18). \noindent Let $d=10$. We employ the same technique as above using \ci{gross} and \ci{fa-li} instead of \ci{a-s} and \ci{io1}, \ci{io3}. Looking at the list in \ci{gross} we exclude cases $Z^{10}_A$ and $Z^{10}_B$ since they do not have a balanced cohomology class. Cases $Z^{10}_D$ and $Z^{10}_E$ cannot occur by \ci{fa-li}, since they have sectional genus $g=7$. \noindent The case $C^{10}_A$, where the sectional genus $g=4$ and the irregularity $q=1$ is excluded since, by \ci{fa-li}, we would have $q=0$. The cases $C^{10}_B$ and $C^{10}_C$ are excluded in a similar way. \noindent The case $Z^{10}_C$, which is a rational surface with $g=6$ is excluded as follows. According to \ci{fa-li} there are only two types of threefolds of degree $d=10$ with sectional genus $g=6$; the first is a Mukai manifold (i.e. $K_X=-L$), the second one a scroll over $\pn{2}$. In the former case the surface section would have trivial canonical bundle, contradicting its being rational. The latter case is excluded by Proposition \ref{baseofscroll}. \noindent The proof of the Claim is complete. \medskip \noindent We now show that all the types of the claim occur as nonsingular surface sections of threefolds on $\Q{5}$, that type $5$) is the only one that can occur as a section of a fourfold on $\Q{6}$ and that none of these types can occur as a surface section of any $(n-2)$-fold on $\Q{n}$, for $n\geq 7$. \medskip \noindent {\em The case of Type $5)$.} \noindent Assume that $X\subseteq \Q{6}$ is a fourfold with surface section of type $5$). By Swinnerton-Dyer's classification of varieties of degree $d=4$ (see \ci{io1} for example), $(X,L)$ is of Type C); such a type occurs as a subvariety of $\Q{6}$ as pointed out in Proposition \ref{pbundlesovercurves}. $K_X=\odixl{X}{-2,-4}$, so that $K_X(4)$ is generated by three global sections. Fact \ref{hartshorne} gives us the following exact sequence: \begin{equation} \label{F} 0 \to \odix{\Q{6}}^3 \to F \to {\cal I}_X(2) \to 0, \end{equation} where $F$ is locally free. We want to prove that $F$ is isomorphic to either ${{\cal S}'}_6(1)$ or to ${{\cal S}''}_6(1)$, where the subindeces refer to the fact that that the bundles are the spinor bundles of $\Q{6}$. Consider a general threefold section $T$, which is of type D), and a general surface section $S\subset T$. We have $K_T \simeq K_X(1)\otimes \odix{T}$ and $K_S \simeq K_X(2)\otimes \odix{S}$; there is a canonical identification between $H^0(K_X (4))$, $H^0(K_T(3))$ and $H^0(K_S(2))$ so that the bundles $\cal F_T$ and $\cal F_S$, that we obtain repeating for $T$ and $S$ the construction we have done for $X$ using Fact \ref{hartshorne}, satisfy ${\cal F}_T\simeq F_{|T}$ and ${\cal F}_S\simeq F_{|S}$. \noindent We know, from \ci{a-s}, that, on $\Q{4}$, ${\cal }F_{|S}\simeq {{\cal S}'(1)}_4 \oplus {{\cal S}''}_4(1)$. Recall that spinor bundles have no intermediate cohomology. Let us look at the long cohomology sequences associated with the exact sequences: $$ 0 \to {\cal F}_T(-1+t) \to {\cal F}_T(t) \to {\cal F}_S(t) \to 0, \quad t\in \zed. $$ Firstly we deduce that $H^1({\cal F}_T(-\tau+t))$ surjects onto $ H^1({\cal F}_T(t))$, for every fixed $t$ and every $\tau \geq 0$; Serre Duality and Serre Vanishing imply that $h^1({\cal F}_T(t))=0$, $\forall t$. The vanishing of $h^i({\cal F}_T(t))$ for $2\leq i \leq 4$ are dealt with similarly. We have proved that ${\cal F}_T(t)$ has no intermediate cohomology, so that, by the analogue of Horrocks criterion in Fact \ref{spinorbundles}, ${\cal F}_T$ splits as a direct sum of line bundles and twists of spinor bundles on $\Q{5}$. Since the rank of $\cal S$ is four we see that either ${\cal F}_T\simeq {\cal S}(j)$ or it splits completely as the direct sum of line bundles ${\cal F}_T\simeq \bigoplus_{i=1}^4 \odixl{\Q{5}}{a_i}$. Using the Castelnuovo-Mumford $0$-regularity criterion for global generation (see \ci{a-s}, where it is proved for sheaves on $\Q{4}$; the case of any $\Q{n}$ is analogous) we see that ${\cal F}_T$ is generated by global sections as soon as $h^5({\cal F}_T(-5))=0$ which follows from the cohomology sequence associated with the sequence (\ref{F}) twisted by $-5$ once we observe that $h^4(\Il{T}{\Q{5}}{-3})=3$ and $h^5(\Il{T}{\Q{5}}{-3})=0$. Recall that $1$ is the smallest integer $j$ for which the spinor bundles twisted by $j$ are generated by global sections. It follows that, for the splitting type of ${\cal F}_T$, we have either $j\geq 1$ or $a_i\geq 0$, $\forall i$. We can compute the Chern classes of ${\cal F}_T$ using (\ref{F}), the invariants of \,$T$ and Lemma \ref{lmccis}. Comparing Chern polynomials we deduce that ${\cal F}_T$ cannot split as the direct sum of line bundles, and that, once it is a twist of the spinor bundle ${\cal S}$, $j=1$: ${\cal F}_T \simeq {\cal S}(1)$. \noindent We repeat the argument, replacing $S$ with $T$ and $T$ with $X$, to see that either $F\simeq {\cal S}'(1)$ or $F\simeq {\cal S}''(1)$. \noindent We have proved that every fourfold on $\Q{6}$ with surface section of type 5) is as in Type C) and has the prescribed presentation for its ideal sheaf; we have also proved that {\em every} threefold on $\Q{5}$ with surface section of type 5) is of Type D) and has the prescribed presentation for its ideal sheaf. Conversely, since ${\cal S}(1)$, ${\cal S}'(1)$ and ${\cal S}''(1)$ are globally generated, we use Fact \ref{hartshorne} and our maximal list of varieties of degree $d=4$ to prove that the variety $D_2(\phi)$ is as in C) or D), where $\phi$ is a general element of $H^0(S(1)^3)$ and $S$ one of the three spinor bundles in question. To be precise, Fact \ref{hartshorne} implies that, for a general $\phi$ on $\Q{6}$, $D_1(\phi)$ is either empty or has the expected codimension six and $D_2(\phi)$ will be nonsingular outside $D_1(\phi)$. Porteous' formula, \ci{a-c-g-h}, II.4.2 gives $[D_1(\phi)]= c_3(S(1))^2 - c_2(S(1))c_4(S(1))=0$; the Chern classes of the spinor bundles are listed in \ref{spinorbundles}. It follows that $D_1(\phi)=\emptyset$. \noindent Using facts (\ref{unirat}) and (\ref{dimensionhilbert}) we conclude the proof for type D). To complete the proof for type C) we remark that ${\cal S}'(1)$ distinguishes, via the choice of three general sections, a nonsingular, integral component, say ${\frak H}'$, of $\frak H$. The same is true for ${\cal S}''(1)$ which defines another, distinct, nonsingular component ${\frak H}''$. Since $\frak H$ is nonsingular, ${\frak H}= {\frak H}' \bigsqcup {\frak H}''$. \noindent The dimension of the two components, which are abstractly isomorphic, can be computed using Riemann-Roch and Fact \ref{dimensionhilbert}. The fourfold C) cannot be the hyperplane section of a fivefold; see \ci{io1}. \medskip \noindent {\em The case of Type} 10). \noindent Assume that $X$ is a threefold on $\Q{5}$ whose general surface section, $S$, is of type 10). Since $K_S=-L_{|S}$ and the natural map $Pic(X)\to Pic(S)$ is injective by Lefschetz theorem on hyperplane sections, we have $K_X=-2L$. Looking at \ci{io1} for degree $d=6$ we see that either $(X,L)$ is as in Type F) or $X$ is a scroll over $\Q{2}$; the latter case is not possible by Proposition \ref{baseofscroll}. We have $K_X=-2L$. Fact \ref{hartshorne} yields an extension: $$ 0 \to \odix{\Q{5}} \to F \to {\cal I}_X(3) \to 0, $$ with $F$ locally free of rank two. By \ci{o-s-s}, 2.1.5 one shows that $F(-2)$ is stable. Using Lemma \ref{lmccis} we deduce that, for the Chern classes of $F$, $c_1(F(-2))=-1$ and $c_2(F(-2))=1$. By Fact \ref{cayley} we see that $F(-2)$ is a Cayley bundle. Conversely, \ci{ottcayley}, Theorem 3.7 ensures that the general section of a normalized Cayley bundle twisted by $\odixl{\Q{5}}{2}$ vanishes exactly along a variety of type F). As in \ci{a-ss}, page 209, we see that our scrolls are parameterized by an open dense set, ${\frak U}$ of a projective bundle over the fine moduli space, $\pn{7}\setminus \Q{6}$, of these Cayley bundles. This space is clearly rational and it has dimension 20. \noindent ${\frak U}$ admits a natural morphism onto the Hilbert scheme $\frak H$ of our scrolls. This morphism is one to one. To conclude it is enough to observe that $\frak H$ is nonsingular by Fact \ref{dimensionhilbert}, for then the morphism in question is an isomorphism by Zariski Main Theorem. \noindent Note that, again by \ci{io1}, this threefold is the hyperplane section of only one fourfold, $\pn{2}\times \pn{2}$ embedded via the Segre embedding; this latter can be projected smoothly to $\pn{7}$ but, after this embedding, it does not lie on a smooth quadric $\Q{6}$ by Proposition \ref{nop2inq6}. \medskip \noindent {\em The cases} 11), 18) {\em and} $ Z^{10}_F$. \noindent These cases are analogous to the one of type 5). If the general surface section, $S$, of a threefold $X$ on $\Q{5}$ is of type 11) then there is a morphism with connected fibers $f:S\to \pn{1}$ all fibers of which are conics and $ K_S(1)= f^*\odixl{\pn{1}}{1}$, so that the former sheaf is generated by two global sections. \smallskip \noindent {\em Claim. $K_X(2)$ is generated by its global sections.} Fix any point $x \in X$. Take any nonsingular hyperplane section $S$ of $X$ through $x$; there are plenty of them since the dual variety $\hat X$ does not contain hyperplanes. Kodaira Vanishing implies that $H^0(X, K_X(2))$ surjects onto $H^0(S, K_S(1))$ which in turn generates the stalk of $K_{S}(1)$. The claim follows. \smallskip \noindent A computation analogous to the one of type 5) allows us to conclude that if the threefold in question exists than it is of type G). To prove its existence, we use Fact \ref{hartshorne} for a general morphism $\phi: \odix{\Q{5}}^2 \to \odixl{\Q{5}}{1}^3$. This threefold cannot be the hyperplane section of any fourfold by \ci{io1}. \smallskip \noindent If the type of the general surface section is 18) then we have a morphism $f:S \to \Q{2}$ which is the blowing up of $\Q{2}$ at 10 points and $K_S(2)\simeq f^*\odixl{\Q{2}}{1}$ is generated by four global sections. We argue as above and get Type L). \smallskip \noindent If X is a threefold on $\Q{5}$ with general section $S$ of type $Z^{10}_F$ then the sectional genus $g=8$. By looking at the list in \ci{fa-li}, we see that $X$ is a Del Pezzo fibration $f:X \to Y$ over a curve $Y$ and that $X$ is not the hyperplane section of any fourfold. By looking at the proof of \ci{fa-li} Proposition 4.1 we see that the base of the fibration, $Y$, is a rational curve and that $K_X(1) \simeq f^* \odixl{\pn{1}}{1}$ is generated by two global sections. We argue as above and conclude that the type is N). \blacksquare \section{SCROLLS ON QUADRICS} \label{secpbundles} \subsection{Statement of the main result} \label{main} In this section we classify scrolls as codimension two subvarieties of $\Q{n}$, for $n\geq 5$. A {\em scroll}, here, is a nonsingular subvariety $X\subseteq \Q{n} \subseteq \pn{n+1}$ which admits a surjective morphism $p:X\to Y$ to a lower dimensional variety $Y,$ such that $p$ has equidimensional fibers and the general scheme theoretic fiber is a linear subspace of $\pn{n+1}$ of the appropriate dimension. The case $\dim Y=0$ is the theory of maximal dimensional linear spaces in quadrics, a well known subject; see \ci{h-p}. From now on we assume $\dim Y>0$. \noindent By standard arguments, see \ci{fujitasemipositive} 2.7, we can assume, without loss of generality, that $Y$ is nonsingular and that the polarized pair $(X,L) \simeq (\Bbb P ({\cal E}), \xi_{\cal E})$, where $L$ is the restriction of the hyperplane bundle to $X$ and ${\cal E}:= p_*L$ is an ample, rank $\mu:=\dim X -\dim Y +1$, locally free sheaf generated by its global sections. \begin{rmk} \label{gold} {\rm We assume that $n\geq 5$ since surfaces on $\Q{4}$ which are scrolls over curves have been classified by Goldstein in \ci{go}. They correspond to the surfaces of type 2), 3), 5), and 9) in \ci{a-s}. } \end{rmk} In what follows the Types C), D), F) and I) below refer to Theorem \ref{classificationd<12}; we say that a nonsingular threefold, $X$, on $\Q{5}$ is of Type O), if it has degree $d=12$ and it is a scroll over a minimal $K3$ surface. \begin{tm} \label{maintm} The following is the complete list of \,nonsingular codimension two subvarieties of quadrics $\Q{n}$, $n\geq 5$, which are scrolls. \noindent {\rm Type C)}, $n=6$, $d=4$, scroll over $\pn{1}$ and over $\pn{3}$; \noindent {\rm Type D)}, $n=5$, $d=4$, scroll over $\pn{1}$; \noindent {\rm Type F)}, $n=5$, $d=6$, scroll over $\pn{2}$; \noindent {\rm Type \,L)}, $n=5$, $d=8$, scroll over $\Q{2}$; \noindent {\rm Type O)}, $n=5$, $d=12$, scroll over a minimal $K3$ surface. \end{tm} \noindent {\em Proof.} The proof is the consequence of the lengthy analysis that constitutes the rest of the paper. Here we give the reader directions toward the various relevant statements. \noindent By Fact \ref{n<7} we need to deal only with the cases $n=5,\,6$. \noindent Scrolls over curves are classified by Proposition \ref{pbundlesovercurves}. They correspond to types C) and D). \noindent The are no fourfolds which are scrolls over a surface, by Proposition \ref{nop2inq6}. \noindent The only fourfold which is a scroll over a threefold is of type C), by Proposition \ref{p1bundlesinq6}. \noindent Threefolds which are scrolls over a surface have degree $d=6,\, 8,\, 12$ by Proposition \ref{d=6812} and the base surfaces are as in Proposition \ref{baseofscroll}. The classification in degrees $d=6,\, 8$ is complete; correspondingly we get types F) and L). \noindent For an example and for the general properties of varieties of Type O) see Section \ref{d=12}). \blacksquare \subsection{Preliminary facts} \label{prelfacts} \medskip The Barth-Larsen theorem implies that if $X$ is a nonsingular codimension two subvariety of $\Q{n}$, then the fundamental group $\pi_1(X)$ is trivial for $n\geq 6$, and $Pic(X)\simeq \zed$, generated by the hyperplane bundle, for $n\geq 7$; see \ci{ba}. Since $Pic(X)\simeq \zed$ as soon as $n\geq 7$, we have: \begin{fact} \label{n<7} {\rm There are no codimension two scrolls on $\Q{n}$ for $n\geq 7$ and, for $n=6$, any such is simply connected.} \end{fact} It is therefore enough to study threefolds on $\Q{5}$ which are scrolls over curves and surfaces and fourfolds on $\Q{6}$ which are scrolls over curves, surfaces, and threefolds. \smallskip Let us begin the analysis by fixing some notation. We start with a scroll of degree $d$; let $e_i:=c_i({\cal E})$, $x_i:=c_i(X)$ and $y_i:=c_i(Y)$. Since $p^*$ is injective it is harmless to denote $p^* \alpha$ simply by $\alpha$ while performing computations in the cohomology ring of $X$. \noindent The tautological relation is \begin{equation} \label{tautological} \sum_{i=0}^{\mu} (-1)^i L^{\mu -i}\cdot e_i=0. \end{equation} \noindent Finally, recall the usual exact sequence: \begin{equation} \label{euler} 0\rightarrow \odix{X}\rightarrow p^*({\cal E}^{\lor})\otimes L \rightarrow {\cal T}_X \rightarrow p^*{\cal T}_{Y} \rightarrow 0, \end{equation} which is obtained by pasting together the relative Euler sequence \ci{ha}, II.8.13 and the short exact sequence associated with the epimorphism $dp:{\cal T}_X\to p^*{\cal T}_Y$. \subsection{Scrolls over curves on $\Q{5}$ and on $\Q{6}$} \label{overcurves} The following is proved independently of Theorem \ref{classificationd<12}. \begin{pr} \label{pbundlesovercurves} Let $(X,L)$ be scroll over a nonsingular curve $Y$, on $\Q{n}$. Then $(X,L)$ is one of the following: \noindent {\rm (\ref{pbundlesovercurves}.1)} $ n=6,$ $ ({\Bbb P}_{\pn{1}}(\cal E),\xi_{\cal E})$, ${\cal E}:={{\cal O}_{{\Bbb P}^1}(1)}^4.$ \noindent {\rm (\ref{pbundlesovercurves}.2)} $n=5,$ $({\Bbb P}_{\pn{1}}(\cal E),\xi_{\cal E})$, ${\cal E}:={\cal O}_{{\Bbb P}^1}(2) \oplus {\odixl{\pn{1}}{1}}^2;$ \noindent In particular, in both cases, $d= 4$ and the embedding is projectively normal. \end{pr} \noindent {\em Proof.} Let $F\simeq \pn{n-3}$ be any fiber of the scroll. We cut (\ref{deg2dpf5}) with $F \cdot L^{n-5}$ and solve in $d$. We get $d=4$, so that the structure of $(X,L)$ is given by Theorem 8.10.1 of \ci{be-so-book}. \noindent In both cases it is easy to write down explicit equations for the morphism associated with $|\xi|$; we can check directly that $\xi$ is very ample, that the image lies in a smooth quadric and that the embedding is projectively normal. \blacksquare \begin{rmk} \label{alsoscrolloverthreefold} {\rm Case (\ref{pbundlesovercurves}.1) above is the Segre embedding of $\pn{1}\times\pn{3}$. It is a scroll over a curve if we look at the first projection. If we look at the second projection it is a scroll over $\pn{3}$ with associated vector bundle ${\odixl{\pn{3}}{1}}^{2}$. Case (\ref{pbundlesovercurves}.2) is a general hyperplane section of (\ref{pbundlesovercurves}.1); the natural morphism onto $\pn{3}$ exhibits $X$ as the blow up of $\pn{3}$ along a line.} \end{rmk} \subsection{Threefolds on $\Q{5}$ which are scrolls over surfaces} \label{scrollsoversurfaces} \begin{lm} \label{pbundlesoversurfaces} Let $X\subseteq \Q{5}$ be a codimension two scroll over a surface $Y$. Then either $d=8$ or we have: \noindent $g-1=\frac{1}{8}d(d-6)$, \noindent $\chi (\odix{X})=\chi (\odix{S})=\frac{1}{144}(d^3-18d^2+96d)$, \noindent $e_1^2=\frac{3}{2}d$, \quad $e_2=\frac{d}{2}$, \noindent $K_Y\sim_n \frac{1}{6}(d-12) e_1$. \end{lm} \noindent {\em Proof.} We follow closely a procedure which can be found in \ci{ottp5}. By (\ref{euler}) we get: \noindent $x_1=2L -e_1+y_1$; \noindent $x_2= 2Ly_1 -e_1y_1+y_2$; \noindent $x_3=2y_2$. \noindent We plug the above equalities in (\ref{deg2dpf5}) and (\ref{deg3dpf5}) and get the following two equations: \begin{equation} \label{A} (5-\frac{d}{2})L^2 + L\cdot e_1 -3L\cdot y_1 + e_1^2 + y_1^2 -e_1\cdot y_1 -y_2=0; \end{equation} \begin{equation} \label{B} \frac{d^2}{2}-2d + (d-8)L^2\cdot e_1 - (d-8)L^2\cdot y_1 -4L\cdot e_1^2 - 4L\cdot y_1^2 -2L\cdot y_2 +8L\cdot e_1 \cdot y_1 =0. \end{equation} \noindent We cut (\ref{A}) and the tautological relation with $L,$ $e_1$ and $y_1$ respectively. This way we get six elations which together with (\ref{B}) and the relation $L^3=d$ give a system of eight linear equations in the variables: $v:=(L^3;\ L^2e_1;$ $ L^2y_1;$ $ Le_1^2;$ $ Le_1y_1;$ $ Ly_1^2;$ $ Le_2;$ $ Ly_2)$. The matrix associated with the linear system is: \[M:= \left( \begin{array}{rrrrrrrr} 5-d/2 & 1 & -3 & 1 & -1 & 1 & 0 & -1 \\ 0 & 5-d/2 & 0 & 1 & -3 & 0 & 0 & 0 \\ 0 & 0 & 5-d/2 & 0 & 1 & -3 & 0 & 0 \\ d/2-2 & d-8 & -d+8 & -4 & 8 & -4 & 0 & -2 \\ 1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right) \] and the linear system can be expressed as $Mv^t=(0,0,0,0,0,0,0,d)$. \noindent Since ${\rm det}\, M=72-9d$, the above system of equations has a unique solution if and only if $d\neq 8$. \noindent Let us assume $d\neq 8$. Then the unique solution is: \begin{eqnarray} \label{solutiondnot8} & &\{L^3;\ L^2e_1 ;\ L^2y_1 ;\ Le_1^2 ;\ Le_1y_1; \ Ly_1^2 ;\ Le_2;\ Ly_2 \} = \\ & &\{ d;\ \frac{3d}{2} ;\ \frac{d}{4}(12-d);\ \frac{3d}{2}; \ \frac{d}{4}(12-d);\ \frac{d^3}{24}-d^2 +6d;\ \frac{d}{2}; \ \frac{d^3}{24}-\frac{d^2}{2}+2d\}. \nonumber \end{eqnarray} \noindent We can use (\ref{solutiondnot8}) to compute the genus of a general curve section, C, of $X$. This genus equals the arithmetic genus of the line bundle $e_1$ on $Y$; we get $$ 2(g-1)= -e_1 y_1 + e_1^2=-Le_1y_1 + Le_1^2=\frac{1}{4}d(d-6). $$ An analogous computation gives \begin{equation} \label{chitutti} \chi (\odix{X})=\chi (\odix{Y}) = (1/144)(d^3-18d^2+96d)= \chi (\odix{S}), \end{equation} where the first equality is a standard fact about projective bundles which can be proved using the Leray Spectral sequence and the last one follows from the fact that $S$ is birationally equivalent to $Y$. \noindent To prove that $K_Y$ is numerically equivalent to a rational multiple of $e_1$, we use Hodge index theorem for the surface $Y$: by (\ref{solutiondnot8}), $K_Y^2 e_1^2= (K_Y\cdot e_1)^2$, so that $K_Y \sim_n q\,e_1$ for some rational number $q$ which is straightforward to compute. \blacksquare \begin{lm} \label{d=6...42} Let $X$ be a threefold scroll over a surface on $\Q{5}$. Then $d\leq 42$. Moreover, if a general curve section, $C$, is contained in another quadric hypersurface of \,$\pn{6}$, then $d\leq 12$. \end{lm} \noindent {\em Proof.} By Lemma \ref{pbundlesoversurfaces} we have $g-1=(1/8)d(d-6)$. \noindent Assume that a general curve section $C$ is not contained in any surface, in the corresponding $\Q{3}$, of degree strictly less than $2\cdot 7$. Then Proposition \ref{boundasep} implies $d\leq 42$. \noindent Assume $C$ is contained in a surface ${\cal S}\subseteq \Q{3}$ of degree $2\cdot 6$. Proposition \ref{coreasybound} implies $d\leq 27$. \noindent The same argument repeated for surfaces of degrees $2\cdot 5$, $2\cdot 4$ and $2\cdot 3$ gives $d\leq 18$ in all three cases. \noindent Let us assume that $C$ is contained in a surface of degree $2\cdot 2$ and that $d>8$; by Proposition \ref{roth}, $X$ is contained in another quadric hypersurface of $\pn{6}$. We now prove that, under the above assumptions on $C$, $d\leq 12$. We plug $\s =2$ and the values of $\chi(\odix{S})$ and $g-1$, from Lemma \ref{pbundlesoversurfaces}, in inequality (\ref{55}); we get \begin{equation} \label{pbundles=255} -\frac{1}{288}d(d+6)(d-12)\geq -\frac{1}{12}[ \frac{1}{2}dL^2 - (K_X + 5L)^2]D. \end{equation} By Lemma \ref{pbundlesoversurfaces} we have $$ K_X=-2L +\frac{1}{6}(d-6)e_1. $$ We now plug the above expression for $K_X$ in (\ref{pbundles=255}) using the following relations $e_2=(d/2)f$, $L^2D-Le_1D+e_2D=0$, where $f$ is a fiber of the scroll. After simplifications the result is $$ -d(d+6)(d-12)\geq 12(d+6)Le_1D + d(d+6)(d-12)Df. $$ Since $Le_1D\geq 0$ and $Df\geq 0$ we get $d\leq 12$. Moreover, if $d=12$ then $D$ must be empty. \noindent Finally if $C$ were contained in a surface of degree $2\cdot 1$ then the same would be true for $X$, by Theorem \ref{roth}. But then $X$ would be a scroll on a quadric $\hat\Q{4}$ of $\pn{5}$ with at most one singular point. Weil and Cartier divisors coincide on $\hat\Q{4}$ and $Pic\,(\hat\Q{4})\simeq \zed$ by \ci{ha} II.6 Ex. 6.5. It would follow that $X$ is a complete intersection, a contradiction. \blacksquare \begin{lm} \label{d=6812} Let $X$ be a threefold scroll over a surface on $\Q{5}$. Then $d=6,$ $8$ or $12$. \end{lm} \medskip \noindent {\em Proof.} By Lemma \ref{d=6...42}, $d\leq 42$; by Lemma \ref{pbundlesoversurfaces}, since the invariants there given must be integers we see that the only possibilities for the pairs $(d,g)$ with $d>12$ are $(18,28)$, $(24,55)$, $(30,91)$, $(36,136)$ and $(42,190)$. We prove that the cases $d=18,$ $24,$ $30,$ $36,$ $42$ cannot occur. \noindent Let $C\subseteq \Q{3} \subseteq \pn{4}$ be the general curve section of $X$ and $\G \subseteq \Q{2} \subseteq \pn{3}$ be the general hyperplane section of $C$. We denote by $h_C(i):= h^0( \odixl{\pn{4}}{i} ) - h^0( {\cal I}_{C,\pn{4}}(i))$ the Hilbert function of $C\subseteq \pn{4}$ and by $h_{\G}(i):= h^0( \odixl{\pn{3}}{i} ) - h^0( {\cal I}_{\G,\pn{3}} (i))$ the Hilbert function of $\G \subseteq \pn{3}$. Clearly $h_C(i)\leq h^0(\odixl{C}{i})$, for every $i$. \noindent {\em The case $(18,28)$.} By Riemann-Roch and Serre Duality we have $h^0(\odixl{C}{i})= 18i -27$ for $i\geq 4$; in particular $h^0(\odixl{C}{4})= 45$. $C$ cannot be contained in another quadric of $\pn{4}$, since otherwise, by Proposition \ref{roth} and Lemma \ref{d=6...42}, $d\leq 12$. $C$ cannot be contained in an integral cubic of $\pn{4}$, otherwise, we would get that the genus would be maximal with respect to the bound prescribed by Proposition \ref{1.4} and, since $\epsilon=0$, $C$ would be a complete intersection, forcing $X$ to be one too in view of \ci{ha}, III.9 Ex. 9.6. For the same reason $C$ cannot be contained in an integral quartic of $\pn{4}$. It follows that there are no quartic hypersurfaces containing $C$ except for the ones which are the union of $\Q{3}$ with another quadric; in particular $h_C(4)=55$. We get $55=h_C(4)\leq h^0(\odixl{C}{4})=45$, a contradiction. The case $d=18$ cannot occur. \smallskip \noindent {\em The case} $(24,55)$. As in the previous case we deduce that $C$ is contained in a unique quadric of $\pn{4}$, $C$ is not contained in any integral cubic or quartic of $\pn{4}$. This gives $h_C(4)=55$. As before $h^0(\odixl{C}{5})=66$. By \ci{harrismont} Lemma 3.1 we have $h_C(5) \geq h_C(4) + h_{\G}(5)$ and by \ci{harrismont} Lemma 3.4 we also have that $h_{\G}(5)\geq 16$. It follows that $ 55 + 16 \leq h_C(5) \leq h^0(\odixl{C}{5})=66$, a contradiction. The case $d=24$ cannot occur. \smallskip \noindent {\em The cases $(30,91)$, $(36,136)$ and $(42,190)$}. They are treated as the case $d=24$. In the first case $h^0(\odixl{C}{7})=120$ and the only hypersurfaces of degree seven of $\pn{4}$ which contain $C$, contain $\Q{3}$, so that $h_C(7)=140$, again a contradiction. In the second case $h^0(\odixl{C}{8})=153$ and the only hypersurfaces of degree eight of $\pn{4}$ which contain $C$, contain $\Q{3}$, so that $h_C(8)=289$, again a contradiction. In the last case $h^0(\odixl{C}{10})=231$ and the only hypersurfaces of degree nine of $\pn{4}$ which contain $C$, contain $\Q{3}$, so that $h_C(9)=385$. In particular $h_C(10)>385$, by \ci{harrismont} Lemma 3.1, again a contradiction. \blacksquare \bigskip The proof of the following is independent of Theorem \ref{classificationd<12}. \begin{pr} \label{baseofscroll} Let things be as in Lemma {\rm \ref{d=6812}}. If $d=6$ then $Y\simeq \pn{2}$; if $d=8$ then $Y\simeq \Q{2}$; if $d=12$ then $Y$ is a minimal $K3$ surface. \end{pr} \noindent {\em Proof.} Let $d=6$. By Lemma \ref{pbundlesoversurfaces} $-K_Y$ is ample and $K_Y^2=9$; by the classification of Del Pezzo surfaces we conclude that $Y\simeq \pn{2}$. Let $d=8$. The proof of Proposition \ref{nop2inq6} gives $Y \simeq \Q{2}$. Let $d=12$. By Lemma \ref{pbundlesoversurfaces} $K_Y$ is numerically trivial, so that $Y$ is a minimal model. (\ref{chitutti}) prescribes $\chi ({\cal O}_{Y})=2$, so that, by the Enriques-Kodaira classification, $Y$ is a $K3$ surface. \blacksquare \subsubsection{The case of Type O)} \label{d=12} The purpose of this sections is twofold. First we give an example of a scroll of Type O), making the list of Theorem \ref{maintm} effective. Then we collect information about the arbitrary variety of this type. Let $X$ be of Type O), $\beta_{i,j}:=h^i( \Il{X}{\Q{5}}{j})$ and $\s_i:=h^i(\Il{X}{\Q{5}}{-1}\otimes {\cal S})$. The sheaves $\Psi_i$, first introduced by Kapranov, are defined in \ci{a-ott}. \begin{tm} \label{esempium} Let $ {\cal S}^7 \stackrel{\phi}\to {\Psi}_3 \oplus \odix{\Q{5}}^3 $ be a general morphism. Then, $\phi$ is injective, $X:=D_{27}({\phi})$ is a variety of {\rm Type O)} and we have a resolution of the form $$ 0 \to {\cal S}^7 \stackrel{\phi} \to {\Psi}_3 \oplus \odix{\Q{5}}^3 \to \Il{X}{\Q{5}}{3} \to 0. $$ \end{tm} We will prove this theorem after Proposition \ref{example}. First we determine some properties of the arbitrary variety of Type O). \begin{pr} \label{invd=12} Let $X\subseteq \Q{5}$ be of {\rm Type O)}. \noindent Then: $$ g=10;\, \, K_XL^2=-6; \, \, K^2L=-6;\, \, K_X^3=12; \, \, c_2(X)L=24, \, \, c_1({\cal E})^2=18, \, \, c_2({\cal E})=6. $$ \smallskip \noindent The cohomology of $\odixl{X}{l}$: $$ h^1({\odixl{X}{t}})=0, \, \forall t \in {\zed}; $$ $$ h^2({\odixl{X}{t}})=0, \, \forall t \in {\zed} \, \, except \, \, for \, \, h^2(\odix{X})=1; $$ $$ h^3({\odixl{X}{t}})=0, \, \forall t\geq -1. $$ The following is the Beilinson-Kapranov $E^{p,q}_1$ table for the sheaf $\Il{X}{\Q{5}}{3}$; see {\rm \ci{a-ott}} {\rm Theorem 5.6}. A letter {\rm a} on the left of a vector bundle $B$ means $B$ direct sum with itself {\rm a} times. $$ \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 7{\cal S} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \Psi_3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \beta_{1,2} \Psi_1 & \beta_{1,3} \odix{X} \\ 0 & 0 & 0 & 0 & \beta_{0,2} \Psi_1 & \beta_{0,3} \odix{X}, \end{array} $$ where $\beta_{0,2} - \beta_{1,2} =0$, $\beta_{0,3} - \beta_{1,3}=3$. \noindent Either $\beta_{2,0}=1$ and $\beta_{0,3}=7$ or $\beta_{2,0}=0$ and $ 0 \leq \beta_{1,3} \leq 21$. \noindent If $\beta_{2,0}=1$ then $\Il{X}{\Q{5}}{3}$ can be expressed as the cohomology of a monad of the form {\rm (see \ci{o-s-s} for the definition of monads)}: \begin{equation} \label{II2} 0 \to 7{\cal S} \stackrel{m_1}{\to} \Psi_3 \oplus \Psi_1 \oplus \odixl{\Q{5}}{1} \stackrel{n_1}{\to} 4 \odix{\Q{5}} \to 0 \end{equation} If $\beta_{2,0}=0$ then $\Il{X}{\Q{5}}{3}$ can be expressed as the cohomology of a monad of the form: \begin{equation} \label{I3} 0 \to 7{\cal S} \stackrel{m_2}{\to} \Psi_3 \oplus 3 \odix{\Q{5}}\oplus \beta_{1,3} \odix{\Q{5}} \stackrel{n_2}{\to} \beta_{1,3} \odix{\Q{5}} \to 0. \end{equation} \end{pr} \noindent {\em Proof.} The first list of invariants can be read off from Lemma ref{scrollsoversurfaces} when $d=12$. \noindent As to $h^2(\odixl{X}{t})$ we argue as follows. Via the projection formula and Leray Spectral Sequence, $h^2(\odixl{X}{t})=0$, $\forall t<0$ and for the same reason $h^2(\odix{X})=1$. Since $K_Y$ is trivial, Leray Spectral Sequence and Le Potier's Vanishing Theorem \ci{sh-so} give $h^i(\odixl{X}{1} )=h^i(\odixl{Y}{\cal E})= h^i(\odixl{Y}{K_Y \otimes {\cal E}}) =0$, $\forall $ $i\geq 2$. By Serre Duality and the fact that $L_{|S}(K_S -mL_{|S})=6 - 12m$ we see that $h^2(\odixl{S}{m}) =0$, $\forall m\geq 1$; we conclude for $h^2$ by an easy induction using the sequences \begin{equation} \label{hyperseq} 0 \to \odixl{X}{m-1} \to \odixl{X}{m} \to \odixl{S}{m} \to 0. \end{equation} The vanishings of the $h^3$'s are obvious consequences of Serre Duality. \noindent $h^1(\odixl{X}{t}=0, \, \forall t<0$ by Kodaira vanishing. For $t=0$ the vanishing follows from $h^1(\odix{Y})=0$. Since $X\subseteq \pn{6}$ is linearly normal by a result of Fujita's (cf. \ci{okcod3} , \S 4) and $\chi(\odixl{X}{1})=7$ by Riemann-Roch, we have $h^1(\odixl{X}{1})=0$. To prove the remaining vanishings for $h^1(\odixl{X}{t})$ we argue by induction using the long cohomology sequences associated with the sequences (\ref{hyperseq}), the analogue ones obtained by replacing $X$ and $S$ by $S$ and $C$ (a general curve section of $S$) and observing that the linear systems $|\odixl{C}{t}|$ are non-special for $t\geq 2$. \noindent The Beilinson-Kapranov table is obtained as follows. The vanishings $\beta_{i,j}=0$ for $i=2,\, 3,\, 4,\,5$ and $j=-1,\,0,\, 1,\, 2,\, 3$, except for $\beta_{3,0}=h^0(\odix{X})=1$, are obtained by taking the cohomology of the exact sequences \begin{equation} \label{IOX} 0 \rightarrow \Il{X}{\Q{5}}{l} \rightarrow \odixl{\Q{5}}{l} \rightarrow \odixl{X}{l} \rightarrow 0 \end{equation} and plugging the above values for the cohomology of $\odixl{X}{t}$. For the same reason $\beta_{i,j}=0$ for $i=0, \, 1, \, 2$, $j=-1,\, 0$. $\beta_{0,1}$ is zero because $X$ cannot be degenerate (see the proof of Lemma \ref{d=6...42}). $\beta_{1,1}=0$ since $X\subseteq \pn{6}$ is linearly normal. The relations on the remaining $\beta$'s come from the shape of the Hilbert polynomial $$ \chi (\Il{X}{\Q{5}}{t})={\frac {1}{60}}t^5+{\frac {5}{24}}t^{4} -t^{3}+{\frac {19}{24 }}t^{2}+{\frac {59}{60}}t-1, $$ which vanishes for $t=-1,\, 1,\, 2,$ and has value three for $t=3$. \noindent Because of how this spectral sequence works ($E_{\infty}=E_6$ and $E^{p,q}_{\infty}\simeq \{0\}$ for $p+q\not=0$), we see that $\s_i=0$, for $i=0,\,1,\,2$. \noindent ${\s}_5=0$ by observing the cohomology of (\ref{IOX}) tensored with ${\cal S}$. We use the same sequences, together with Riemann-Roch for $\cal S$ and for ${\cal S}_{|X}$ to get $$ \chi ({\cal S}(t))=\frac{1}{15}t(t+1)(t+2)(t+3)(t+4), $$ and $$ \chi ({\cal S}_{|X} (t))=8t^3 -6t^2 +7; $$ it follows that $ -\s_3 + \s_4=\chi(\I{X}{\Q{5}}\otimes {\cal S}(-1))=7. $ \noindent We now prove that $\s_3 =0$. There is at most one nontrivial differential from $\s_3{\cal S}$, namely the one that hits $E^{-2,0}_4$. On the other hand $E^{-2,0}_4=E^{-2,0}_2=\mbox{Ker}d^{-2,0}_1$. It is enough to show that the last group is trivial. We consider two cases. The former is when $\beta_{0,2}=0$; in this case $\s_3$ is clearly zero. The latter is when $\beta_{0,2}\not=0$. Then $\beta_{0,2}=1$ otherwise $X$ would have $d\leq 8$, the degree of the intersection on $\Q{5}$ of two hypersurfaces of degree four. $\beta_{0,3}=7$ otherwise $X$ would be a complete intersection on $\pn{6}$ of type $(2,2,3)$, a contradiction. By Kapranov's explicit resolution of the diagonal on $\Q{n}\times \Q{n}$, see \ci {a-ott}, we infer that $d^{-2,0}_1$ coincides with the injection $(\Psi_1\simeq)\, {\Omega^1(1)_{\pn{6}}}_{|\Q{5}} \to \odix{\Q{5}}^7$ obtained by restricting the Euler sequence on $\pn{6}$ to $\Q{5}$. It follows that $\s_3=0$ and therefore $\s_4 =7$. \noindent If $\beta_{2,0}\not=0$, we have seen above that $\beta_{2,0}=1$, $\beta_{3,0}=7$ and that $d^{-2,0}_1$ coincides with the injection $(\Psi_1\simeq)\, {\Omega^1(1)_{\pn{6}}}_{|\Q{5}} \to \odix{\Q{5}}^7$ whose cokernel is $\odixl{\Q{5}}{1}$. The statement associated with (\ref{II2}) follows from \ci{a-ott}. \noindent Similarly, we see that the statement associated with (\ref{I3}) holds when $\beta_{2,0}=0$. Since the morphism $n_2$ is trivial on $(3+\beta_{1,3})\odix{\Q{5}}$, the restriction $\nu:={n_2}_{|\Psi_3}$ is surjective. Recall that the rank of $\Psi_3$ is $26$. If $\beta_{1,3}>21$, then the kernel of the map $\nu$ would be a locally free sheaf of rank $r<5$ with fifth Chern class $c_5=c_5(\Psi_3)\not=0$, a contradiction. \blacksquare \bigskip The following is essentially due to Peskine and Szpiro; see \ci{okcod3}, \S1. \begin{lm} \label{pesk} Let $X$ be a codimension two nonsingular subvariety of a nonsingular variety $Z$ of dimension $n\leq 5$ and $L_i$, $i=1,\,2$, two line bundles on $Z$ such that the sheaves $\Il{X}{Z}{L_i}$ are globally generated on $Z$. Let $s_i \in $ $H^0(\Il{X}{Z}{L_i})$ be the choice of two general sections and $V_i$ the two effective divisors associated with the $s_i$. Then $V_1 \cap V_2= X \cup Y,$ as schemes, where $Y$ is nonsingular. \end{lm} We now give a family of examples of degree $d=12$ scrolls on $\Q{5}$. \begin{pr} \label{example} Let $\rho: {\cal C}^{\vee} \to \odixl{\Q{5}}{2}^3$ be a generic morphism. Then $X:=D_1(\rho)$ is a variety of {\rm Type O)} such that $\Il{X}{\Q{5}}{3}$ is generated by global sections; $X$ is linked to a variety $X'$ of \,{\rm Type F)} via the complete intersection of two general elements of $|{\Il{X}{\Q{5}}{3}}|$. \noindent Conversely, if $X\subseteq \Q{5}$ is of {\rm Type O)} and $\Il{X}{\Q{5}}{3}$ is generated by global sections, then $X=D_1(\rho)$ for some $\rho$ as above and $X$ is linked as above to a variety of {\rm Type F)}. \end{pr} \noindent {\em Proof.} For facts about Cayley bundles, see Fact \ref{cayley}. ${\cal C}(2)^3$ is generated by global sections and Fact \ref{hartshorne} implies that $X:=D_1(\rho)$ is a codimension two nonsingular subvariety of $\Q{5}$ and that we have the following exact sequence \begin{equation} \label{alceste} 0 \to {\cal C}^{\vee}(-2) \stackrel{\rho}{\to} \odix{\Q{5}}^3 \to \Il{X}{\Q{5}}{3} \to 0. \end{equation} We compute the total Chern class of \,$\Il{X}{\Q{5}}{3}$ via (\ref{alceste}): $ 1+3\,h+6\,h^{2}+9\,h^{3}+9\,h^{4}$. \noindent We compare it with Lemma \ref{lmccis}\,: \noindent $ \gamma_1=3,$\, $\gamma_2=6=\frac{1}{2}d,$\, $\gamma_3=\frac{1}{2}(K_X+2L)L^2=9,$\, $ \gamma_4= \frac{1}{2}(K_X +2L)^2L=9,$\, $ \gamma_5=\frac{1}{2}(K_X+2L)^3=0,$ \noindent where $L$ denotes $\odixl{\pn{6}}{1}_{|X}$. It follows that $X$ has degree $d=12$. By \ci{be-so-book} Proposition 7.2.2, $K_X+2L$ is generated by global sections since $(X,L)$ cannot be isomorphic to either $(\pn{3}, \odixl{\pn{3}}{1})$, $(\Q{3}, \odixl{\Q{3}}{1})$ or to a scroll over a curve since they all have degree $d=4$ by Proposition \ref{pbundlesovercurves}. The fact that $\gamma_5=0$ implies that $K_X+2L$ cannot be big and the fact that $\gamma_4\not= 0$ implies that the Kodaira dimension $\kappa (K_X+2L)=2$, so that, by \ci{be-so-book} Theorem 7.3.2, $(X,L)$ is an adjunction theoretic scroll over a surface and, by \ci{be-so-book} Proposition 14.1.3, it is actually a scroll in our sense. By Lemma \ref{baseofscroll}, $X$ is a degree $d=12$ scroll over a $K3$ surface. The linking part is proved using Lemma \ref{pesk} to produce a $X'$ of degree $d'=6$ and by observing that the mapping cone construction yields a resolution for $\Il{X'}{\Q{5}}{3}$ which coincides with the one of a variety of Type F). \noindent The converse is proved in a similar way. \blacksquare \bigskip \noindent {\bf Proof of Theorem} {\bf \ref{esempium}}. For the varieties constructed in Proposition \ref{example} we have, by (\ref{alceste}), that $\beta_{2,0}=0$ and $\beta_{0,3}=3$. By looking at the display of the monad (\ref{I3}) with these invariants we get the desired resolution for the ideal sheaves of these varieties. It also follows that, for the generic morphism $\phi$, $\phi$ is injective and $X:=D_{27}({\phi})$ is of Type O) as in the proof of Proposition \ref{example}. \blacksquare \subsection{4-folds which are scrolls on $\Q{6}$} \label{scrollsinq6} \begin{pr} \label{nop2inq6} There are no fourfolds scrolls over surfaces on $\Q{6}$. \end{pr} \noindent {\em Proof.} By contradiction, assume that $X^4$ is such a scroll. Cutting (\ref{deg2dpf5}) with a fiber $F\simeq \pn{2}$, we get $d=8$. We take a general hyperplane section and obtain a scroll, $X$, on $\Q{5}$ so that the previous analysis applies. In fact, if a special fiber, $F$, were isomorphic to $\pn{2}$, then $F_{|F}\simeq \odixl{\pn{2}}{-1}$ so that we would have a contraction morphism $\eta: X\to X'$ and the structural morphism $p:X \to Y$ would factor through $\eta$ violating the upper semicontinuity of the dimension of the fibers. \noindent We solve the linear system contained in the proof of Lemma \ref{scrollsoversurfaces} for $d=8$ and we get that the solutions depend on one additional parameter $t$: \begin{eqnarray} \label{solutiond8} & &\{L^3;L^2e_1 ;L^2y_1 ; Le_1^2 ; Le_1y_1; Ly_1^2 ; Le_2; Ly_2 \} = \\ & &\{ 8; 36-\frac{9}{2}t ;24-3t;36-\frac{9}{2}t; 24-3t; 16-2t;28- \frac{9}{2}t; t \}. \nonumber \end{eqnarray} We observe that $K_Y^2=(4/9)e_1^2$ and that $K_Y\cdot e_1=-(2/3)e_1^2$. Since $e_1$ is ample, the Hodge Index Theorem implies that $3K_Y\sim_n -2e_1$. It follows that $Y$ has to be a Del Pezzo surface. On such a $Y$, numerical and rational equivalence coincide and $3K_Y$ is not divisible by 2 unless $Y$ is a smooth $\Q{2}$. In this case $t=b_2=4$. In particular $\deg e_2 = 10$ and $g=4$. By \ci{io3} this case cannot occur if $\dim X\geq 4$. This contradicts the existence of scrolls over surfaces in $\Q{6}$ of dimension four. \blacksquare \begin{pr} \label{p1bundlesinq6} The only scroll over a threefold on $\Q{6}$ is \, $\pn{1}\times \pn{3}$ embedded with the Segre embedding. \end{pr} \noindent {\em Proof.} The proof runs along the lines of Lemma \ref{pbundlesoversurfaces}. Using (\ref{euler}) we compute the Chern classes of $X$: \noindent $x_1=2L -e_1 +y_1$; \noindent $x_2= 2Ly_1 -e_1y_1 +y_2$; \noindent $x_3= 2Ly_2 - e_1y_2 +y_3$; \noindent $x_4=2Ly_3$. \noindent After having plugged these relations in (\ref{deg2dpf5}), (\ref{deg3dpf5}) and (\ref{deg4dpf}) we get, respectively: \begin{equation} \label{n2} ( \frac{1}{2} d -8)L^2 + L(4y_1-2e_1)-e_1^2+e_1y_1-y_1^2+y_2=0, \end{equation} \begin{eqnarray} \label{n3} & & 8L^3+ L^2(4e_1-8y_1)+L(-2e_1y_1+4y_1^2-4y_2)+ \\ & & e_1^3-e_1^2y_1+e_1y_1^2-e_1y_2-y_1^3 +2y_1y_2-y_3=0, \nonumber \end{eqnarray} \begin{eqnarray} \label{n4} & & 6L^4-8L^3y_1+L^2(4e_1^2-4e_1y_1+8y_1^2-8y_2) + \\ & & L(-2e_1^3+2e_1y_1^2-2e_1y_2-4y_1^3 +8y_1y_2-4y_3)=0. \nonumber \end{eqnarray} We cut the tautological relation and (\ref{n2}) with the following classes: $L^2$, $Le_1$, $Ly_1$, $e_1^2$, $ y_1^2$, $e_1y_1$, $e_2$ and $y_2$; we cut (\ref{n3}) with $L$, $e_1$ and $y_1$. Considering also (\ref{n4}) we get a total of twenty linear equations in the seventeen variables: $L^4$, $L^3e_1$, $L^3y_1$, $L^2e_1^2$, $L^2e_1y_1$, $L^2e_2$, $L^2y_1^2$, $L^2y_2$, $ Le_1^3$, $Le_1^2y_1$, $Le_1e_2$, $Le_1y_1^2$, $Le_1y_2$, $Le_2y_1$, $Ly_1^3$, $Ly_1y_2$ and $Ly_3$. We leave out, on purpose, the condition $L^4=d$. We leave to the reader to check that the resulting linear system has a nontrivial solution only for $d=4$. If $d=4$ we use Theorem \ref{classificationd<12} to conclude. \blacksquare

9608

alg-geom/9608024

en

https://arxiv.org/abs/alg-geom/9608024

alg-geom/9608024

Lucia Caporaso

Lucia Caporaso and Joe Harris

Parameter spaces for curves on surfaces and enumeration of rational curves

77 pages, Tex type: AMSLaTex in LaTex 2.09 compatibility mode, figures are not included in the package, for a copy with pictures write to [email protected]

We answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's formula for rational plane curves of given degree.

alg-geom

\section{Introduction} In this paper we will be concerned with the geometry of families of rational curves on a surface $S$. Let us state the main problem. Let $S$ be a nonsingular, rational surface, and let $D$ be an effective divisor class in $S$. We denote by $|D|$ the set of all effective divisors linearly equivalent to $D$; this is a projective space whose dimension we will denote by $r(D)$. Inside $|D|$, we want to consider the locus of rational curves: we let $$ \tilde V(D) \; = \; \{ [X] \in |D| \ \mbox{such that $X$ is an irreducible, rational curve}\} . $$ This is a locally closed subset of the projective space $|D|$; we let $V(D) \subset |D|$ be its closure. We call $V(D)$ the {\it Severi variety} of rational curves associated to the divisor class $D$ on $S$, and we denote its dimension by $r_0(D)$. We have in general $$ r_0(D) \; \ge \; r(D) - p_a(D) $$ with equality holding in all the cases that we shall study. The particular aspect of the geometry of $V(D)$ of concern to us here is its degree, which we will denote by $N(D)$. This can also be characterized directly: it is the number of irreducible rational curves that are linearly equivalent to $D$ and that pass through $r_0(D)$ general points of $S$. The principal results of this paper will be the computation of $N(D)$ in some cases. For simplicity, we will assume that $N(D)$ is zero if $V(D)$ is empty. \subsection{The general strategy: the cross-ratio method} There are various approaches to the calculation of degrees of Severi varieties (see for example \cite{CH}). The one we take in this paper is based on ideas of Kontsevich and Manin. In their paper \cite{KM} they describe a beautiful formula, found initially by Kontsevich, for the number of plane rational curves of given degree passing through the appropriate number of points (another proof of it was given independently by Ruan and Tian in \cite{RT}). These methods have also been used to give formulas for the degrees of genus 0 Severi varieties on other rational surfaces; see \cite{CM}, \cite{DI} and \cite{KP}. The method that we are going to describe was suggested to us by the ``First Reconstruction Theorem" of \cite{KM}. It is based on the analysis of a one-parameter family of curves: we will consider the family of irreducible rational curves in the linear system $|D|$ passing through $r_0(D) - 1$ general points of $S$. By suitably marking four points on each of these curves, we can (possibly after a base change) associate to the family a cross-ratio function on its base. Moreover, we can do this in such a way that the $N(D)$ curves in the family passing through a given $r_0(D)^{\text{th}}$ point of $S$ are among the zeroes of the cross-ratio function, and all the other zeroes and poles of the cross-ratio function will occur at reducible fibes. Thus, to determine the number of curves in the family passing through the last point, we need to describe the set of its reducible elements (and the multiplicity of the cross-ratio at each). This gives us, in principle, a way of solving the problem recursively. If $D$ and $D'$ are divisor classes on $S$, we say that $D'<D$ if $D-D'$ is effective and non zero. Then finding the number of reducible curves $X$ in $V(D)$ passing through $r_0(D) - 1$ general points involves curves (the components $X_i$ of such curves $X$) in divisor classes $D' < D$, which we may consider known inductively. In fact, in simple cases ($S = \Bbb P^2, \ \Bbb P^1 \times \Bbb P^1$ or $\Bbb F_1$), this works quite smoothly: every union $X = \cup X_i$ of rational curves $X_i \subset S$ is a limit of irreducible rational curves, and we end up with an expression for the number $N(D)$ of rational curves in the linear series $|D|$ through $r_0(D)$ points simply in terms of the numbers $N(D')$ for divisor classes $D' < D$. By contrast, a more delocate situation arises when we consider other ruled surfaces $\Bbb F_n$: here the components $X_i$ of a union $X= \cup X_i$ have to satisfy additional conditions in order for the point $[X] \in |D|$ to be in $V(D)$. In the following chapter, we will analyze exactly this situation, and in the final chapter we will apply the results of this analysis to derive recursive formulas for $N(D)$ on $\Bbb F_n$. Let us describe more precisely the set-up. Fix two irreducible curves $C_3$, $C_4 \subset S$ having positive intersection number with each other and with $D$, and intersecting transversely. Let $q_1,\dots, q_{r_0(D)-1} \in S$ be general points and denote by $H_{q_i}$ the hyperplane in $|D|$ parametrizing curves through $q_i$. Then let $$ \Gamma \; = \; V(D) \cap H_{q_1} \cap \dots \cap H_{q_{r_0(D)-1}} $$ be the corresponding linear section of $V(D)$---equivalently, the closure in $|D|$ of the set of irreducible rational curves passing through $q_1,\dots, q_{r_0(D)-1}$. Now, for a general point $p$ in $S$, we can interpret the degree of $V(D)$ as the number of points $[X] \in \Gamma$ corresponding to curves $X$ that pass through $p$. Let $\cal X \subset \Gamma \times S$ be the universal family over $\Gamma$, that is, the subscheme of $\Gamma \times S$ whose fiber over each point $[X] \in \Gamma$ is simply $X$; let $f : \cal X \to \Gamma$ be the projection. By construction, $f : \cal X \to \Gamma$ is a flat family of curves, whose general fiber is an irreducible rational curve. Next, we introduce a family whose general fiber is the normalization of the corresponding fiber of $\cal X \to \Gamma$. To do this, we first take $\Gamma^\nu \to \Gamma$ the normalization of $\Gamma$, pull the family $\cal X \to \Gamma$ back to $\Gamma^\nu$, and take $\cal X^\nu$ to be the normalization of the total space of this pullback: that is, we set $$ \cal X^\nu \; = \; (\cal X \times_\Gamma \Gamma^\nu)^\nu . $$ The composite map $\cal X^\nu \to \Gamma^\nu$ (which we will again denote by $f$) is then a flat family, with general fiber isomorphic to $\Bbb P^1$. Now, we want endow this family with four sections $$p_i \; : \; \Gamma^\nu \; \longrightarrow \; \cal X^\nu , \ \ \ i=1,2,3,4\ \text{ such that }\ f\circ p_i = id_{\Gamma^\nu}, $$ so as to define a cross-ratio function. To do that, we will take the first two sections to be given by the first two base points $q_1$ and $q_2$, that is, for general $[X] \in \Gamma$ we set $$ p_i([X]) \; = \; ([X], q_i) \quad \text{for} \quad i = 1, 2 ; $$ this then extends to a regular map on all of $\Gamma^\nu$. (The extension follows from the fact that $\Gamma^\nu$ is a smooth curve. The reader may wonder about the points $[X]$ corresponding to the curves in our family with nodes at $q_1$ and $q_2$, where it may at first appear the section $p_i$ can't be well-defined; but in fact $\Gamma$ will have ordinary nodes and those points and the two branches of the normalization $\Gamma^\nu$ over those points exactly correspond to the choices of value for $p_i$.) For the remaining two sections, we want to pick out a point of intersection of each curve $X$ of our family with each of the curves $C_3$ and $C_4$, that is, we want to choose for general $[X] \in \Gamma$, $$ p_j([X]) \; = \; ([X], r_j) \quad \text{with} \quad r_j \in X \cap C_j \quad \text{for} \quad j = 3,4 . $$ This requires another base change. To be precise, let $\pi_2 : \Gamma \times S \to S$ be the projection, and for $j = 3,4$ set $$ {\cal C}_j \; = \; \cal X \cap \pi_2^{-1}(C_j) \; \subset \; \Gamma \times S . $$ The projection ${\cal C}_j \to \Gamma$ will have degree $d_j = (D \cdot C_j)$. We let ${\cal C}_j^0$ be the union of those components of ${\cal C}_j$ of positive degree over $\Gamma$, and finally let $B$ be the normalization of their product over $\Gamma$, that is, $$ B \; = \; ({\cal C}_3^0 \times_\Gamma {\cal C}_4^0)^\nu . $$ Then if we let $$ \cal X ' \; = \; \cal X^\nu \times_{\Gamma^\nu }B\; \longrightarrow \; B $$ be the pullback family, we define two further sections $p_3, p_4 : B \to \cal X "$ by $$ p_j(b) \; = \; (b, \pi_j(b)) $$ for general $b \in B$, where $\pi_j : B \to {\cal C}_j$ is the projection. We now have a family $\cal X ' \to B$ of curves over a smooth base $B$, with general fiber $X'$ isomorphic to $\Bbb P^1$. We may accordingly define the cross-ratio map $$ \phi \; : \; B \; \longrightarrow \; \Bbb P^1 $$ by letting, for general $b \in B$, $\phi(b)$ be the cross-ratio of the four points $p_1(b), p_3(b), p_4(b), p_2(b) \in X' \cong \Bbb P^1$---that is, in terms of any affine coordinate on $X'\cong \Bbb P^1$ we set $$ \phi(b) \; = \; {(p_1 (b)-p_2(b)) (p_3 (b)-p_4(b))\over (p_1 (b)-p_3(b))(p_2 (b)-p_4(b))}. $$ Equivalently, the family $\cal X ' \to B$ will determine a family of stable $4$-pointed rational curves over $B$, and hence a canonical morphism $\tilde\phi$ from $B$ to $\overline{M}_{0,4}$. (It may not be apparent that a further base change is not necessary to arrive at such a family; but this is true and will emerge in the subsequent analysis.) We can then choose an identification $\eta$ of $\overline{M}_{0,4}$ with $\Bbb P ^1$ so that the three points of $\overline{M}_{0,4}$ corresponding to singular (that is, reducible) curves \ \vspace*{1.4in} \noindent \hskip-.3in \special {picture crossratio} \ \noindent map to the points $0, \infty $ and $1$. The composition $\eta \circ \tilde\phi$ is then the map $\phi$ above. Now we want to compute the number of zeroes and poles of $\phi$ to obtain a formula for $N(D)$. Observe now that the cross-ratio function $\phi$ can have a zero or a pole at a point $b \in B$ only under one of two circumstances: if two of the points $p_i(b)$ actually coincide; or if the fiber $X_b$ of $\cal X ' \to B$ over $b$ is reducible. As for the first of these, clearly $p_1(b)$ can never equal $p_2(b)$; and since the points $q_i$ are general, they cannot lie on $C_3$ or $C_4$. The first case will thus occur exactly when $p_3(b) = p_4(b)$, which in turn can happen only when the curve $X_b$ contains one of the points $p$ of intersection of $C_3$ with $C_4$. Conversely, for every curve in the original family $\cal X \to \Gamma$ containing a point $p \in C_3 \cap C_4$, there will be a unique point $b \in B$ lying over $[X] \in \Gamma$ with $p_3(b) = p_4(b)$, namely the point $(p,p) \in {\cal C}_3^0 \times_\Gamma {\cal C}_4^0$ (the fact that this is a smooth point of ${\cal C}_3^0 \times_\Gamma {\cal C}_4^0$, so that there will be a unique point of $B$ over it, follows from the fact that every curve in the original family $\cal X \to \Gamma$ containing a point $p \in C_3 \cap C_4$ will intersect $C_3$ and $C_4$ transversely, which will be a consequence of Proposition \ref{dimensioncount} below). Thus, every curve of our original family passing through a point of $C_3 \cap C_4$ will contribute a zero to the function $\phi$. Once we verify that these are all simple zeroes, we conclude that $\phi$ has a total of $(C_3 \cdot C_4)N(D)$ zeroes at points $b \in B$ corresponding to irreducible curves $X_b$. It remains to describe the zeroes and poles of $\phi$ coming from reducible curves $X$ in the family. Here there is a fundamental difference between the ``simple" cases of surfaces $S = \Bbb P^2$, $\Bbb P^1 \times \Bbb P^1$, or $\Bbb F_1$ and the general surfaces $\Bbb F_n$. The difference is this: for surfaces such as $\Bbb P^2$, the dimension $r_0(D)$ of the family of rational curves in a given linear series $|D|$ is affine linear in the class $D$, without exception: it is given by the formula $$ r_0(D) \; = \; -(K_S \cdot D) - 1 . $$ It follows that for any two effective divisor classes $D_1$, $D_2$ with $D_1 + D_2 = D$ we have $$ r_0(D_1) + r_0(D_2) \; = \; r_0(D) - 1, $$ for three divisor classes $D_1$, $D_2$, $D_3$ with $D_1 +D_2 +D_3 = D$ we have $$ r_0(D_1) + r_0(D_2) + r_0(D_3) \; = \; r_0(D) - 2, $$ and so on. As a consequence, we see that there are no curves $X \in |D|$ with three or more components, all of which are rational, passing through $r_0(D)-1$ general points of $S$; and there are exactly $$ \sum_{D_1+D_2=D}{r_0(D)-1 \choose r_0(D_1)}N(D_1)N(D_2) $$ such curves with two components: for every division of the $r_0(D)-1 = r_0(D_1) + r_0(D_2)$ points $q_i$ into subsets of $r_0(D_1)$ and $r_0(D_2)$, there will be exactly $N(D_1)N(D_2)$ pairs $X_1 + X_2$ with $X_1$ containing the first set and $X_2$ containing the second. By a naive dimension count, moreover, all such curves will be limits of irreducible rational curves. In these cases, then, we can inductively enumerate all reducible elements of our family, and say which contribute to the zeroes and which to the poles of $\phi$; in the end, equating the degrees of the divisors of zeroes and poles of $\phi$ we are led to a recursive formula for the number $N(D)$. For the general $\Bbb F _n$ instead, a dimension count will not be enough to describe all possible types of degenerations occuring in codimension 1. This is due to the presence of the exceptional curve $E$ on $\Bbb F _n$. We shall see that in a family of generically irreducible nodal curves there will be special fibers containing $E$ as a component. The description of these types of degeneration will in fact require a delicate analysis. \subsection{Two sample calculations} To give an idea of what sort of information we need to carry out our project, we will derive formulas for two relatively simple examples. The first of these is the formula found by Kontsevich for the degrees of the Severi varieties of rational plane curves. The second is a direct calculation of the degree of the Severi variety associated to the linear series $|2C|$ on $\Bbb F_2$. We will not justify the assertions made about the types of degenerate fibers occurring in our families, or about the multiplicities of the corresponding zeroes and poles of the cross-ratio function, but we will indicate when such justifications are necessary, and refer to the results of the following chapters. \subsubsection{Kontsevich's formula} The cross-ratio method is well illustrated in the case of $S= \Bbb P^2$. We will derive here Kontsevich's formula for the number $N(d)$ of rational curves passing through $3d-1$ general points in the plane. (Note that the dimension of the variety of rational curves in the linear series $|\cal O_{\Bbb P^2}(d)|$ is $3d-1$. ) To proceed, we take $C_3$ and $C_4$ distinct lines, and choose $q_1,\dots,q_{3d-2}$ general points of the plane. We let $\Gamma \subset |\cal O_{\Bbb P^2}(d)|$ be the closure of the locus of irreducible rational curves of degree $d$ passing through the points $q_i$, and carry out the remaining steps of the construction described above to arrive at a family $\cal X ' \to B$ whose base $B$ is a finite cover of the curve $\Gamma$ of degree $d^2$, with four sections $p_i : B \to \cal X '$ coming from the two base points $q_1$ and $q_2$ and the points of intersection of the curves in the family with the lines $C_3$ and $C_4$. As already explained, our formulas will be obtained by equating the degrees of the divisors of zeroes and poles of the cross-ratio function $\phi$ on $B$. To begin with, the only way $\phi$ can have a zero or pole at a point $b \in B$ corresponding to an irreducible curve $X_b$ is if the curve $X_b$ passes through the point $p = C_3 \cap C_4$ of intersection of the two lines. Now, since the $3d-1$ points $q_1,\ldots,q_{3d-2},p$ are general, there will be exactly $N(d)$ curves $X$ of our original family $\cal X \to \Gamma$ passing through $p$. Moreover, these curves will all correspond to smooth points $[X] \in \Gamma$, so that there will again be exactly $N(d)$ points of $\Gamma^\nu$ corresponding to such curves. Next, since each such curve $X$ intersects $C_3$ and $C_4$ transversely at $p$, the point $(p,p) \in {\cal C}_3^0 \times_\Gamma {\cal C}_4^0$ will be a smooth point of $\Gamma^\nu$ and there will be a unique point of $B$ lying over it. Finally, again as a consequence of the analysis in Chapter 2, each such point of $B$ will be a simple zero of the cross-ratio function; so that in sum we have a total of exactly $N(d)$ zeroes of $\phi$ at points $b \in B$ with $X_b$ irreducible. It remains to count the zeroes and poles of $\phi$ occurring at points corresponding to reducible fibers. We need to make one remark here in general, about the geometry of $\Gamma$ near a point $[X]$ corresponding to a reducible curve $X = X_1 \cup X_2$. If the degree of the components $X_i$ is $d_i$, then each $X_i$ will have ${d_i-1 \choose 2}$ nodes, and in addition there will be $d_1d_2$ transverse points of intersection of $X_1$ with $X_2$; thus $X$ will have $ {d-1 \choose 2} + 1$ nodes in all. Now, as we approach $[X]$ along any branch of $\Gamma$, we see that the limiting positions of the ${d-1 \choose 2}$ nodes of the general curve of our family will consist of the ${d_1-1 \choose 2}$ nodes of $X_1$, the ${d_2-1 \choose 2}$ nodes of $X_2$, and all but one point $r$ of the $d_1d_2$ points of intersection of the two components (so that when we take the normalization of the total space along that branch, what we see is the normalizations of the $X_i$, joined along the one point of each lying over the point $r$). Conversely, if we choose any point $r \in X_1 \cap X_2$, there exist deformations of $X$ smoothing the node $r$, preserving the remaining nodes, and of course continuing to pass through the points $q_1,\dots,q_{3d-2}$. Thus, in sum, we see that the curve $\Gamma$ will have exactly $d_1d_2$ branches at the point $[X]$. With that said, let us proceed to count the remaining zeroes and poles of $\phi$. We have to ask: how many reducible curves $X$ are there in our family with $q_1$ and $q_2$ on one component, and $p_3$ and $p_4$ on the other (by the dimension count made above in general, $X$ can have only two components); and how many are there with $q_1$ and $p_3$ on one component, and $q_2$ and $p_4$ on the other? To start, we can certainly count the number of points $[X] \in \Gamma$ corresponding to reducible curves $X = X_1 \cup X_2$ with $q_1, q_2 \in X_1$: such a curve will consist of a component $X_1$ of degree $d_1$ passing through $q_1, q_2$ and $3d_1-3$ of the other points $q_i$, and a curve $X_2$ passing through the remaining $3d-4-(3d_1-3)=3d_2-1$ of the points $q_3,\dots,q_{3d-2}$. To specify such a curve, then, we have first to break up the set $\{q_3,\dots,q_{3d-2}\}$ into subsets of $3d_1-3$ and $3d_2-1$ points; then choose $X_1$ any one of the $N(d_1)$ rational curves of degree $d_1$ through $q_1$, $q_2$ and the first set, and $X_2$ any one of the $N(d_2)$ rational curves of degree $d_1$ through the second set. The total number of such points on $\Gamma$ is thus $$ N(d_1)N(d_2){3d-4 \choose 3d_1-3} . $$ Next, by the remark above, for each such point $[X] \in \Gamma$, there will be $d_1d_2$ points of $\Gamma^\nu$ lying over it; thus we have $$ N(d_1)N(d_2){3d-4 \choose 3d_1-3}d_1d_2 . $$ such points of $\Gamma^\nu$. Moreover, of the $d^2$ points of $B$ lying over each such point of $\Gamma^\nu$, exactly $d_2^2$ will correspond to triples $([X],p_3,p_4)$ with $p_3, p_4 \in X_2$; so we have a total of $$ N(d_1)N(d_2){3d-4 \choose 3d_1-3}d_1d_2^3 . $$ points of $B$ corresponding to unions $X = X_1 \cup X_2$ of curves of degrees $d_1$ and $d_2$. Finally, we have to check that each of the corresponding points of $B$ is a simple zero of the cross-ratio function $\phi$, which will follow from Lemma~\ref{crossratiomult}. With this verified, we see in sum we have a total of exactly $$ \sum_{d_1+d_2=d} N(d_1)N(d_2){3d-4 \choose 3d_1-3}d_1d_2^3 . $$ zeroes of $\phi$ at points $b \in B$ with $X_b$ reducible. This now accounts for all the zeroes of $\phi$, so that we have $$ \deg \phi \; = \; N(d) + \sum_{d_1+d_2=d} N(d_1)N(d_2){3d-4 \choose 3d_1-3}d_1d_2^3 . $$ The poles of $\phi$ are counted in the same fashion as the zeroes coming from reducible fibers. In fact, the only difference is that now we have to look at curves $X = X_1 \cup X_2$ consisting of a component $X_1$ of degree $d_1$ passing through $q_1$ and $3d_1-2$ of the other points $q_i$, and a curve $X_2$ passing through $q_2$ and the remaining $3d-4-(3d_1-2)=3d_2-2$ of the points $q_3,\dots,q_{3d-2}$. To specify such a curve, then, we have first to break up the set $\{q_3,\dots,q_{3d-2}\}$ into subsets of $3d_1-2$ and $3d_2-2$ points; so that the total number of such points on $\Gamma$ is now $$ N(d_1)N(d_2){3d-4 \choose 3d_1-2} . $$ Again, for each such point $[X] \in \Gamma$, there will be $d_1d_2$ points of $\Gamma^\nu$ lying over it; and now, of the $d^2$ points of $B$ lying over each such point of $\Gamma^\nu$, exactly $d_1d_2$ will correspond to triples $([X],p_3,p_4)$ with $p_3 \in X_1$ and $p_4 \in X_2$; so we have a total of $$ N(d_1)N(d_2){3d-4 \choose 3d_1-2}d_1^2d_2^2 . $$ points of $B$ corresponding to unions $X = X_1 \cup X_2$ of curves of degrees $d_1$ and $d_2$. As before, we have to check that each of the corresponding points of $B$ is a simple pole of $\phi$; once we have done this, since these are all the poles of $\phi$ we conclude that $$ \deg \phi \; = \; \sum_{d_1+d_2=d} N(d_1)N(d_2){3d-4 \choose 3d_1-2}d_1^2d_2^2 . $$ Equating the two values of the degree of $\phi$, we arrive at Kontsevich's formula: $$ N(d) = \sum _{d_1 + d_2 =d} N(d_1)N(d_2)d_1d_2\left[ \left( \begin{array}{c} 3d-4 \\ 3d_1-2 \end{array} \right) d_1d_2 - \left( \begin{array}{c} 3d-4 \\ 3d_1-3 \end{array} \right) d_2^2 \right] . $$ \ \subsubsection{The linear series $|2C|$ on $\Bbb F_2$} \label{2C2} Let $C\in \text{Pic} (\Bbb F _2)$ be as defined in \ref{term}. The linear series $|2C|$ on $\Bbb F_2$ has dimension 8 and arithmetic genus $p_a(2C) = 1$; the Severi variety $V(2C) \subset |2C|$ correspondingly has dimension 7, with general member $X$ a curve with one node. To carry out the calculation, then, we select two general curves $C_3$, $C_4$, which we choose to be linearly equivalent to $C$, and 6 general points $q_1,\dots,q_6 \in S$. We let $\Gamma \subset V(2C)$ be the locus of curves in $V(2C)$ passing through $q_1,\dots,q_6$ and proceed from there as before to construct the family $\cal X ' \to B$, where $B$ will be in this case a $(2C \cdot C_3)(2C \cdot C_4) = 4\cdot 4 = 16$-sheeted cover of the normalization $\Gamma^\nu$ of $\Gamma$. Again, the elements of the family $\cal X \to \Gamma$ passing through the two points of intersection of $C_3$ and $C_4$ each contribute a simple zero to $\phi$, and again these are the only singularities of $\phi$ at points $b \in B$ with $X_b$ irreducible. As for the reducible fibers, we can readily describe those not containing the curve $E$. Since $(2C \cdot E) = 0$ any curve $X$ in the linear system not containing $E$ will be disjoint from it; and since the only curves on $\Bbb F_2$ disjoint from $E$ are those linearly equivalent to a multiple of $C$, any reducible element of $|2C|$ not containing $E$ must be of the form $X = X_1 + X_2$ with $X_1 \sim X_2 \sim C$. Now, the curves of this type form a six-dimensional subvariety of $|2C|$, and it is not hard to see that they are all in $V(2C)$, that is, they are all limits of irreducible singular curves in $|2C|$. In fact, a general such curve $X = X_1 + X_2$ has two nodes, coming from the intersection $X_1 \cap X_2$, each of which may be the limit of the node of a nearby irreducible rational curve. The base $\Gamma$ of our original family will thus have two branches at the point $[X]$ corresponding to such a curve, and the normalization $\Gamma^\nu$ two points lying over $[X]$. We count the number of zeroes and poles of $\phi$ at points $b \in B$ lying over such points as before. A zero arises when the two base points $q_1, q_2$ lie on the same component of $X$, which we will designate $X_1$; thus, to specify such a fiber of $\cal X \to \Gamma$ we have to pick a curve $X_1 \in |C|$ passing through $q_1$, $q_2$ and one point $q_j$ of the four remaining points $q_3,\dots q_6$; and a curve $X_2 \in |C|$ passing through the remaining three of the points $q_3,\dots q_6$. $X$ will thus be determined simply by the choice of $j \in \{3,4,5,6\}$. There will be two points of $\Gamma^\nu$ lying over each such point $[X] \in \Gamma$, as remarked; and a total of $(C \cdot C_3)(C \cdot C_4) = 2 \cdot 2 = 4$ points of $B$ lying over each, corresponding to the choice of $p_3 \in X_2 \cap C_3$ and $p_4 \in X_2 \cap C_4$. Finally, each such point of $B$ will be a simple zero of $\phi$, so we have a total of $$ {4 \choose 1}\cdot 2 \cdot 4 \; = \; 32 $$ zeroes of $\phi$ of this type. The number of poles is described analogously: a pole arises when we have $q_1$ and $p_3$ on one component of $X = X_1 + X_2$ and $q_2$ and $p_4$ on the other; so to specify such a curve $X$ we choose a subset $\{q_i,q_j\} \subset \{q_3,\dots,q_6\}$ and take $X_1$ the (unique) curve in $|C|$ through $q_1$, $q_i$ and $q_j$, and $X_2$ the curve in $|C|$ through $q_2$ and the remaining two points of $\{q_3,\dots,q_6\}$. Again, there are two points of $\Gamma^\nu$ over each such $[X] \in \Gamma$, and four points of $B$ over each of those at which $\phi$ has a pole. These poles are simple, and so we have a total of $$ {4 \choose 2}\cdot 2 \cdot 4 \; = \; 48 $$ poles of $\phi$ of this type. It remains to describe the curves $X$ in our family $\cal X \to \Gamma$ that contain $E$. The curves in the linear series $|2C|$ that contain $E$ form a hyperplane $\Sigma_E = E + |C+2F| \cong \Bbb P^7 \subset |2C| \cong \Bbb P^8$. Thus, even though a general such curve $X = E + X_1$ looks exactly like one of the curves discussed in the last case---two smooth rational components meeting transversely at two points---for purely dimension-theoretic reasons it cannot be a limit of irreducible rational curves in the series $|2C|$. The key question is, then: which curves $X = E + X_1$ are limits of irreducible rational curves? Or, equivalently, what is the intersection $V(2C) \cap H _E$? To answer this, let us see why a general such curve $X = E + X_1$ cannot be such a limit. Suppose we had a one-parameter family $\cal X \subset \Gamma \times \Bbb F_2 \to \Gamma$ of irreducible rational curves specializing to such a curve $X$. After normalizing the base and total space of such a family we would arrive at a family $\cal X^\nu \to \Gamma^\nu$ whose general fiber was smooth, and whose special fiber had one component $\tilde E$ mapping to $E$ and one component $Y$ mapping to $X_1$ via the projection $\pi : \cal X^\nu \to \cal X \to \Bbb F_2$. Since the arithmetic genus of the special fiber must be 0, $\tilde E$ and $Y$ will meet at one point (this will be the point lying over the node of $X$ that is not a limit of nearby curves in the family). \ Now, the inverse image $\pi^{-1}(E) \subset \cal X^\nu$ does not meet the general fiber of $\cal X^\nu \to \Gamma^\nu$; and since it must have pure codimension 1 it can only consist of $\tilde E$ itself: $$ \pi^{-1}(E) \; = \; \tilde E . $$ This is impossible: since $Y$ maps onto $X_1$, it has two points mapping to $E$; but $\pi^{-1}(E) \cap Y = \tilde E \cap Y$ will consist of just one point. Thus $X$ cannot be a limit of irreducible rational curves. This analysis also suggests which reducible curves containing $E$ are such limits. Basically, the contradiction above derived from two hypotheses: that $X_1$ met $E$ transversely in two points; and that $Y$ met $\tilde E$ in just one point. In fact, we will see in the following chapter, as a very special case of Propositions \ref{codimensionone} and \ref{singtotalspace}, that conversely if either of those fails, the curve $X$ is such a limit. In other words, the intersection $V(2C) \cap H_E$ consists of the union of two loci: \ \noindent $\bullet$ \thinspace The locus of curves $X = E + X_1$ with $X_1$ tangent to $E$; and \ \noindent $\bullet$ \thinspace The locus of curves $X = E + X_1 + F$, with $X_1 \sim C+F$ and $F$ a fiber of $\Bbb F_2$. \ In the latter case, the apparent contradiction above is resolved because $Y$ will have two components, each meeting $\tilde E$ at a point, so that the picture of $\cal X^\nu \to \Gamma^\nu$ is \ \vspace*{2.2in} \noindent \special {picture F2type2} \ \noindent In other words, the limiting position of the nodes of nearby fibers of a one-parameter family of irreducible rational curves specializing to $X$ must be the node $X_1 \cap F$ of $X$. In particular, at a point $[X]$ corresponding to such a curve in our original family $\cal X \to \Gamma$, $\Gamma$ will have a single branch, and there will correspondingly be a unique point of $\Gamma^\nu$ lying over $[X] \in \Gamma$. We can at last count the remaining zeroes and poles of $\phi$. First of all, the curves of the form $X = E + X_1$ with $X_1$ tangent to $E$ do not contribute at all: for such a curve, all four points $p_i$ must be distinct points of the component $X_1$. Secondly, since a fiber $F$ can pass through at most one of the two points $q_1, q_2$, the curves of the form $X = E + X_1 + F$, with $X_1 \sim C+F$ and $F$ a fiber of $\Bbb F_2$ will contribute a zero only if the curve $X_1$ contains $q_1$, $q_2$ and three of the four points $q_3,\dots ,q_6$, $F$ is the fiber of $\Bbb F_2$ through the remaining point of $q_3,\dots ,q_6$, and $p_3$ and $p_4$ are chosen to be the unique points of intersection of $F$ with $C_3$ and $C_4$ respectively. There are thus exactly four such points of $\Gamma$, and over each there will be a unique point of $\Gamma^\nu$, and a unique point of $B$ at which $\phi$ has a zero. There is one new wrinkle here: as we will see in Propositions \ref{singtotalspace} and \ref{crossratiomult} below, each of these points is a double zero of $\phi$. Thus we have a total of 8 zeroes of $\phi$ at such points. The poles of $\phi$ at such points are counted similarly. Here there will be exactly two curves $X$ contributing: we could take $F$ the fiber through $q_1$, and $X_1$ the curve in $|C+F|$ through $q_2,\dots,q_6$, and then take $p_3 = F \cap C_3$ and choose $p_4 \in X_1 \cap C_4$; or we could reverse the roles of 1 and 2, and 3 and 4. In each case, there will be a unique point of $\Gamma^\nu$ lying over $[X] \in \Gamma$ and, since $(X_1 \cdot C_3) = (X_1 \cdot C_4) = 3$, three points of $B$ lying over it. Finally, these points are similarly double poles of $\phi$, so we have a total of 12 poles of $\phi$ at such points. The calculation is now complete: equating the number of zeroes and poles of $\phi$, we find that $$ 2 \cdot N(2C) + 36 + 8 \; = \; 48 + 12 $$ and hence $$ N(2C) \; = \; 10 . $$ Actually, there are many (easier) ways to calculate this number. But this example serves to illustrates the questions we must answer in order to apply the cross-ratio method to calculate degrees of rational Severi varieties in general, and also the sort of answer we may find. In the following chapter, we will present a series of results describing exactly what sort of reducible fibers we may expect to find, in codimension $1$, for general linear series $|D|$ on $\Bbb F_n$; and the geometry of the varieties $\Gamma$ and $\cal X$ in a neighborhood of the corresponding points and curves. In the last chapter, we will apply the results of this analysis to derive recursive formulas for the degrees of some Severi varieties on these surfaces. \subsection{Notation and Terminology} \label{term} Our base field will be the field of complex numbers. Let $\Bbb F_n =\Bbb P (\cal O _{\Bbb P ^1}\oplus \cal O _{\Bbb P ^1}(n))$ be a rational ruled surface. The Picard group of $\Bbb F_n$ has rank $2$, and we choose generators as follows: $$ \text{Pic} (\Bbb F_n )= \Bbb Z\cdot C \oplus \Bbb Z \cdot F $$ where $C ^2 = n$, $\ F^2=0$ and $F\cdot C =1$. We denote by $E$ the unique curve of negative self intersection, so that $E^2=-n$ and $E \sim C-nF$. \ Let $D$ be any divisor class on the surface $S = \Bbb F_n$ other than $E$, and let $\underline{m} := (m_1,m_2,\dots,m_k)$ be any sequence of positive integers with $\sum m_i = (D \cdot E)$. We define the locally closed subvariety $\tilde V_{\underline{m}}(D) \subset V(D)$ be the locus of irreducible rational curves $X$ such that, if $\nu : \Bbb P^1 \to X$ is the normalization of $X$, the pullback divisor $$ \nu ^*(E) = \sum m_i\cdot q_i $$ for some collection of distinct points $q_1,\ldots,q_k \in \Bbb P^1$, and we let $V_{\underline{m}}(D) \subset V(D)$ be its closure; for example, as we will see, if $\underline{m} = (1,1,\ldots, 1)$, then $V_{\underline{m}}(D) =V(D)$. When $\underline{m}$ contains a single integer $i$ greater than $1$ (i.e. $\underline{m} = (i,1,1,\ldots ,1)$), we will denote these by $\tilde V_{i}(D) $ and $V_i(D) $ respectively. We set $$r_0^i(D) = \dim (V_i(D))$$ and $$N_i(D) = \deg V_i(D)$$ We have $V(D)$ for $V_1(D)$, $N(D) = N_1(D)$ and $r_0(D) = r_0^1(D)$. We define $N_i(D)$ to be zero if $V_i(D)$ is empty. Similarly, let $\Omega = \{p_1,\ldots,p_k\} \subset E \subset \Bbb F_n$ be any collection of $k$ distinct points. We let $\tilde W^\Omega_{\underline{m}}(D) \subset V(D)$ be the locus of irreducible rational curves $X$ such that, if $\nu : \Bbb P^1 \to X$ is the normalization of $X$, them for some collection of distinct points $q_1,\ldots,q_k \in \Bbb P^1$ we have $$ \nu(q_i) = p_i $$ and $$ \nu ^*(E) = \sum m_i\cdot q_i $$ and again let $W^\Omega_{\underline{m}}(D) \subset V(D)$ be its closure. \subsection{Summary of results} Here we give a list of some known formulas including all the ones that we prove in this paper. These recursions are very similar from a formal point of view. In what follows, we will state them in a way that highlights the analogies. To begin with, fix any rational surface $S$, a divisor class $D$ on it, and two curves $C_3$ and $C_4$. For any pair of divisor classes $D_1$ and $D_2$ we introduce the function $$\gamma (D_1,D_2):=$$ $$N(D_1) N(D_2) \left[ \left( \begin{array}{c} r_0(D) - 3 \\ r_0(D_1)-1 \end{array} \right) (D_1\cdot C_3)(D_2\cdot C_4) - \left( \begin{array}{c} r_0(D) -3 \\ r_0(D_1)-2 \end{array} \right) (D_2\cdot C_3)(D_2\cdot C_4) \right] $$ Using this notation, we state the following results \ \noindent {\bf Recursion for $\Bbb P ^2$} \thinspace (\cite{KM}) {\it Let} $C_3$ {\it and } $C_4$ {\it be two fixed lines in the plane, then} $$N(D) = \sum _{D_1 + D_2 =D} \gamma (D_1,D_2) (D_1 \cdot D_2).$$ \ \noindent {\bf Recursion for $\Bbb P ^1 \times \Bbb P ^1$} \thinspace (\cite{KM}, \cite{DI}, \cite{KP}) {\it Let } $C_3$ {\it and } $C_4$ {\it be two fixed elements of the two distinct rulings, then} $$N(D) = \sum _{D_1 + D_2 =D} \gamma (D_1,D_2) (D_1 \cdot D_2).$$ \ The first new result of this paper is a similar recursion formula for the degrees of Severi varieties of rational curves on the ruled surface $\Bbb F_2$. Note the slightly different form of the recursion: the presence of an extra term not analogous to those in the two preceding formulas is due to the contribution of degenerate curves containing $E$. \ \noindent {\bf Recursion for $\Bbb F_2$} \thinspace (Theorem \ref{F2}) {\it Let } $C_3$ {\it and } $C_4$ {\it be two fixed elements of the class $C$, then} $$2N(D) = \sum _{D_1 + D_2 =D} \gamma (D_1,D_2) (D_1 \cdot D_2)\; + \; 2\sum _{D_1 + D_2 =D-E} \gamma (D_1,D_2) (D_1 \cdot E)(D_2\cdot E).$$ \ Now, we will see that in $\Bbb F_n$, the general reducible curves $X = \cup X_i \in |D|$ that are limits of irreducible rational curves and contain $E$ have the property that each component $X_j$ may have a point of tangency of order $i_j$ with $E$---that is, will belong to $V_{i_j}(D_j)$, where $D_j$ is the divisor class of $X_j$. Accordingly, we shall define later (Section~\ref{Fn}) a generalized version of the number $\gamma (D_1,D_2)$; this will be a function $\gamma _{i_1, i_2,\dots,i_t}(D_1,D_2,\dots,D_t)$ depending recursively on the degrees $N_{i_j}(D_j)$. In these terms, we give a formula expressing the degree $N(D)$ of $V(D)$ on $\Bbb F_n$ in terms of the degrees of the tangential Severi varieties of smaller divisor classes: \noindent {\bf A sample formula for $\Bbb F_n$} \thinspace (Theorem \ref{Fn}) $$ nN(D) = \sum _{D_1 + D_2 = D} (D_1\cdot D_2) \gamma _{1,1}(D_1,D_2) + $$ $$ + \sum _{t=2}^{n} \; \sum _{D_1+D_2+\dots +D_t=D-E} \; \sum _{i_1,\dots,i_t} \; \prod_{j :i_j=1}(E\cdot D_j) \gamma_{i_1,\dots,i_t}(D_1,\dots, D_t) $$ \ The difference here is that in case $n \ge 3$ this does not give a complete recursion: to be able to enumerate rational curves on such surfaces, we would need formulas for the degrees of the ``tangential" Severi varieties as well, that is, we need formulas for $N_i(D)$. The first case for which this occurs is that of $\Bbb F _3$. Very possibly a complete recursion could still be obtained using the cross-ratio method, although the level of difficulty seems to us to get very high. Instead we found a different technique that we successfully applied in a few cases; for example, we obtained a complete set of recursions for the surface $\Bbb F _3$. This different method is the subject of another paper of ours (cf. \cite{CH}); it also is heavily based on the deformation theory results that are developed in the second chapter of this paper. Finally, we obtain a simple closed formula for the class $2C$ on any ruled surface $\Bbb F_n:$ \ \noindent {\bf Closed formula for 2C on $\Bbb F_n$} \thinspace (Theorem \ref{2C}) $$N(2C)=\sum _{k=0}^{n-1}(n-k)^2 {2n+2 \choose k}$$ \newpage \section{Degenerations of rational curves} In this chapter we prove the results on degenerations of rational curves that we will need to obtain our formulas. \subsection{The basic set-up }\label{setup} We start with the complete linear system $|D|$ associated to a divisor class $D$ on the ruled surface $S = \Bbb F_n$, and with the Severi variety $V(D) \subset |D|$. We then choose $ r_0(D) - 1$ general points $q_1,\ldots,q_{r_o (D) -1} \in S$, and let $\Gamma$ be the intersection of $V(D)$ with the linear subspace of curves in $|D|$ passing through $q_1,\ldots, q_{r_o (D) -1}$; we let $\cal X \subset \Gamma \times S$ be the corresponding family of curves over $\Gamma$. Next, we let $\Gamma^\nu \to \Gamma$ be the normalization of the base $\Gamma$, and $$ \cal X^\nu \; = \; \left( \cal X \times_\Gamma \Gamma^\nu \right)^\nu \; \longrightarrow \; \Gamma^\nu $$ the normalization of the pullback of the family to $\Gamma^\nu$, so that $\cal X^\nu \to \Gamma^\nu$ is a family whose general fiber is a smooth rational curve. If $X$ is a fiber of $\cal X \to \Gamma$, the notation $X^\nu$ will be used for a corresponding fiber of the family $\cal X^\nu \to \Gamma^\nu$, which may differ from the normalization of $X$. Then we fix two curves $C_3$ and $C_4$ in $\Bbb F _n$, which will be linearly equivalent to $C$. We need to make a further base change $B \to \Gamma^\nu$, so that the points of intersection of the curves in our family with $C_3$ and $C_4$ become rational over the base. We thus let $B \to \Gamma^\nu$ be any finite cover, unramified at the points $b \in \Gamma^\nu$ with $X^\nu_b$ singular, and let $\cal X' \to B$ be the pullback of the family $\cal X^\nu \to \Gamma^\nu$ to $B$. (By Propositions \ref{dimensioncount} and \ref{codimensionone}, the map $B \to \Gamma^\nu$ introduced in Chapters 1 and 3 in order to define the sections $p_i$ will indeed be unramified at the points of $B$ corresponding to the singular fibers of $\cal X^\nu \to B$.) Because the results of this chapter are all local in the base of our family, however, we will not need to introduce this extra step in the construction. For the remainder of this chapter, accordingly, we will take $B = \Gamma^\nu$; and all of the results of the chapter describing the map $\cal X^\nu \to \Gamma^\nu$ will still hold after the base change $B \to \Gamma^\nu$. Next we introduce the {\it nodal reduction} of the family $\cal X' \to B$. That is to say, after making a base change $\tilde B \to B$ and blowing up the pullback family $\cal X '\times _B \tilde B \to \tilde B$, we arrive at a family $\cal Y \to \tilde B$ such that \ \begin{enumerate} \item $\cal Y \to \tilde B$ is a family all of whose fibers of $\cal Y \to \tilde B$ are reduced curves having only nodes as singularities; \item the total space $\cal Y$ is smooth; \item $\cal Y$ admits a regular birational map $\cal Y \to \cal X' \times_B \tilde B$ over $\tilde B$. \end{enumerate} In fact, most of our concerns with this definition will turn out in the end to be unnecessary: we will see below as a corollary of Propositions \ref{describegamma} and \ref{singtotalspace} that in fact $\cal X' \to B$ is already a family of nodal curves. Thus, in practice, we will not have to make a base change at all at this stage, and $\cal Y$ will be simply the minimal desingularization of $\cal X'$. For this reason (and because $B$ is itself already an arbitrary finite cover of the normalization $\Gamma^\nu$ of our original base $\Gamma$) we will abuse notation slightly and omit the tilde in $\tilde B$, that is, we will speak of the family $\cal Y \to B$. One further remark: in the applications we will have four sections of the family $\cal Y \to B$ and will correspondingly want to consider this as a family of four-pointed nodal curves. For this reason, we may want to make further blow-ups at points where these sections cross. By Propositions \ref{dimensioncount} and \ref{codimensionone}, however, the sections in question will cross only at smooth fibers of $\cal Y \to B$ and so this will not affect our descriptions of the singular fibers of the family. The final construction is one that we will use only in the following chapter, but we mention it here just to have all the definitions in one place. After arriving as above at a family $\cal Y \to B$ of nodal curves with four disjoint sections $p_i$, we may then proceed to blow down ``extraneous" components of fibers $Y$ of $\cal Y \to B$: that is, any component of $Y$ that meets the other components of $Y$ in only one point, and that meets at most one of the sections $p_i$. Iterating this process until there are no extraneous components left, we arrive at what we will call the {\em minimal smooth semistable model} of our family: that is, a family ${\cal Z} \to B$ such that ${\cal Z}$ is smooth, the fibers are nodal, the sections $p_i$ are disjoint and ${\cal Z} \to B$ is minimal with respect to these properties. Note that the special fiber $Z$ of ${\cal Z}$ must be a chain of rational curves $G_0,\dots,G_\ell$ with two of the sections meeting each of the two end components: \vspace*{2.8in} \hskip.6in \special {picture semistable} \ \noindent (the case $\ell = 0$ is simply the case where $Z$ is irreducible). Finally, we can blow down the intermediate components $G_1,\dots,G_{\ell - 1}$ in this chain to arrive at a family ${\cal W} \to B$ of 4-pointed stable curves, called the {\it stable model} of our family. The special fiber of this family will have just two components (or one, if $\ell = 0$), with a singularity of type $A_\ell$ at the point of their intersection. In sum, we have the diagram of families and maps: \vspace*{4in} \hskip.5in \special {picture stablediagram} \ \subsection{The main results from deformation theory} We give here a summary of the main results to be proved in this chapter. \ \noindent $\bullet$ \thinspace The first is Proposition~\ref{dimensioncount} in which we consider the Severi varieties $V(D)$ and $V_{\underline m}(D)$, compute their dimension and describe the geometry of their general point. In particular, we characterize the general fiber of the family $\cal X \to \Gamma$. The results are unsurprising: for example, the general point $[X]$ of $V(D)$ corresponds to a curve $X$ with only nodes as singularities; general points $[X]$, $[X']$ of, respectively, $V(D)$ and $V(D')$ correspond to curves $X$, $X'$ that intersect transversely.. \ \noindent $\bullet$ \thinspace Then, in Proposition \ref{codimensionone}, we study the geometry of the general point of the boundary of $V(D)$. We do that by listing all types of reducible fibers that occur in the family $\cal X \to\Gamma$. This result is not predictable on the basis of a simple dimension count; as we have seen in example \ref{2C2}, in most linear systems $|D|$ on $\Bbb F_n$ the subvariety corresponding to reducible rational curves containing $E$ is larger-dimensional than $V(D)$; so the question of which points of the former lie in the closure of the latter does not have an immediate answer. \ \noindent $\bullet$ \thinspace The third result is Proposition ~\ref{describegamma}, which is specifically about the family $\cal X \to \Gamma$. We describe the geometry of the base $\Gamma$ in a neighborhood of each point $[X] \in \Gamma$ corresponding to a degenerate fiber $X$. In particular, we say how many branches $\Gamma$ has at $[X]$ and say how the nodes of the nearby irreducible fibers approach the singularities of $X$ as we approach $[X]$ along each branch of $\Gamma$. \ \noindent $\bullet$ \thinspace Finally we have Proposition ~\ref{singtotalspace}, describing the singularities of the total space of the families $\cal X \to \Gamma$ and $\cal X^\nu \to \Gamma^\nu$. This will be a crucial ingredient in calculating the multiplicities of zeroes of the cross-ratio function on the base of our family. \ One word of warning is in order. Many of both the statements and proofs of these propositions are just routine verifications of statements easily guessed on the basis of naive dimension counts. At the same time, mixed in with these largely predictable statements are some interesting phenomena . These are described in the second parts of Propositions \ref{codimensionone}, \ref{describegamma} and \ref{singtotalspace}, in which we describe the geometry of the one-parameter families $X \to \Gamma$ and $\cal X^\nu \to \Gamma^\nu$ in a neighborhood of the reducible fibers containing $E$. Near such a curve, the local geometry of the universal family over the Severi variety is, to us, somewhat surprising. \subsection{The geometry of the Severi varieties} Here is the first result about the varieties $V_{\underline m}(D)$ defined in section~\ref{term}. \begin{prop}\label{dimensioncount} Let $|D|$ and $|D'| \ne |E|$ be any linear series on the surface $S = \Bbb F_n$; let $G \subset S$ be any fixed curve not containing $E$ and let $P_1,P_2,\ldots \in S$ be any given finite collection of points. Let $\underline{m} = (m_1,m_2,\dots)$ be any collection of positive integers with $\sum m_i = (D\cdot E)$. \noindent 1. If $V_{\underline{m}}(D) $ is nonempty, then it has pure dimension $$ \dim(V_{\underline{m}}(D)) = -(K_S\cdot D)-1- \sum (m_i - 1) . $$ \noindent 2. A general point $[X]$ of any component of $V_{\underline{m}}(D)$ corresponds to a curve $X \subset S$ having only nodes as singularities, smooth everywhere along $E$, intersecting $G$ transversely and not containing $P_i$ for any $i$. \noindent 3. If $[X]$ and $[X']$ are general points of irreducible components of $V_{\underline{m}}(D)$ and $V_{\underline{m}'}(D')$ respectively, then $X$ and $X'$ intersect transversely, and none of their points of intersection lie on $G$ or $E$. \end{prop} \ \noindent \underbar{Remark}. Many of the techniques necessary to prove this statement are in \cite{H}. In fact, many of these assertions are proved there, but unfortunately with slightly different hypotheses: they are proved first on a general rational surface $S$, but only for $V(D)$, that is, without the tangency condition (Proposition (2.1) of [H]); and then with a single tangency condition, but only with respect to a line in the plane (Lemma (2.4) of [H]). \ \begin{pf} We start with the dimension statement. The assertion that the dimension of $V(D)$ is everywhere equal to $-(K_S\cdot D)-1$ is standard deformation theory (and is well known; c.f. \cite{K}). To see it, observe first that if $[X] \in \tilde V(D)$ is any point and $\nu : X^\nu \to X \subset S$ the normalization of the corresponding curve, the first-order deformations of the map $\nu$ are given by sections of the pullback $\nu^*(T_S)$ of the tangent bundle to $S$. Now, the tangent bundle to the ruled surface $S=\Bbb F_n$ is generated by its global sections everywhere except along $E$; since $X$ doesn't contain $E$, it will likewise be true that the pullback $\nu^*(T_S)$ will be generically generated by its global sections. Since $X^\nu \cong \Bbb P^1$, it follows in turn that $h^1(X^\nu, \nu^*(T_S)) = 0$. The deformations of the map $\nu$ are thus unobstructed, from which it follows that the space of such deformations is smooth of dimension \begin{equation*} \begin{split} h^0(X^\nu, \nu^*(T_S)) &= \deg(\nu^*(T_S)) + 2 \\ &= -(K_S \cdot D) + 2 . \end{split} \end{equation*} If we mod out by automorphisms of the domain $\Bbb P^1$, we see that the space of deformations of the image curve $X \subset S$ as a rational curve has dimension \begin{equation*} \begin{split} h^0(\Bbb P^1, \nu ^*(T_S)) - 3 = -(K_S \cdot D) - 1 . \end{split} \end{equation*} which is the same as the dimension of $T_{[X]}V(D)$. We next establish the \proclaim Claim. The dimension of $\tilde V_{\underline{m}}(D)$, and hence of $V_{\underline{m}}(D)$, is everywhere at least $r_0(D)-\sum (m_i-1)$. To see this, set $l = (D \cdot E)$. Let $[X] \in \tilde V(D)$ be any point, $U$ an analytic neighborhood of $[X]$ in $\tilde V(D)$, $\cal X \subset U \times S \to U$ the universal family of curves over $U$, and $\cal X^\nu$ and $U^\nu$ the normalizations of $\cal X$ and $U$; we may assume that the map $\tau : \cal X^\nu \to U^\nu$ is smooth. Now let $\cal X^\nu_l$ the $l^{\text{th}}$ symmetric fiber product of $\cal X^\nu \to U^\nu$. We then have a map \begin{equation*} \begin{split} \rho \; : \; U \; &\longrightarrow \; \cal X^\nu_l \cr [X] \; &\longmapsto \; \psi_{[X]}^* \nu_{[X]}^*(E) \end{split} \end{equation*} Now, inside the symmetric product $\cal X^\nu_l$, the locus $\Gamma_{\underline{m}}$ of divisors having points of multiplicities $m_i$ or more is irreducible of codimension $\sum (m_i-1)$; since $\tilde V_{\underline{m}}(D) \cap U$ is an open subset of the inverse image $\rho^{-1}(\Gamma_{\underline{m}})$, it follows that it must have dimension at least $\dim(V(D)) - \sum (m_i-1)$ everywhere. \ Note that an analytic neighborhood $U$ of any point of $\tilde V_{\underline{m}}(D)$ admits a map to $E^k$, sending $[X] \in U$ to the images $q_i = \nu(p_i)$; the fibers of this map are analytic open sets in the varieties $W^\Omega_{\underline{m}}(D)$. In particular, we have $$ \dim(V_{\underline{m}}(D)) \; \le \; \dim(W^\Omega_{\underline{m}}(D)) + k $$ so that in order to prove the opposite inequality $\dim(V_{\underline{m}}(D)) \le r_0(D)-\sum (m_i-1)$, it is enough to show that the dimension of the variety $W^\Omega_{\underline{m}}(D)$ is equal to $r_0(D)-\sum m_i$ for any subset $\Omega = \{p_1,\ldots,p_k\} \subset E$. \ To prove the remaining parts of the Proposition requires a tangent space argument. This comes in two parts: first, we will identify the projective tangent space to the space of deformations of a given reduced curve $X$ preserving the geometric genus of $X$; and then the subspaces corresponding to deformations that also preserve singularities other than nodes and/or tangencies with fixed curves. This is the part that is in common with [H], and for the most part we will simply recall here the statements of the relevant results (Theorem~\ref{Zariski} and Lemma~\ref{tangencycondition}). Then, to apply these, we need to estimate the dimension of these subspaces of $|D|$; this is carried out in Lemma~\ref{indcons} and the following argument. We may identify the tangent space to the linear series $|D|$ at $[X]$ with the {\it characteristic series} $$ H^0(X, \cal O_X(X)) = H^0(S, \cal O_S(X))/\Bbb C\tau $$ where $\tau \in H^0(S, \cal O_S(X))$ is the section vanishing along $X$ (this identification is natural up to scalars; more precisely, the tangent space to $\Bbb P(H^0(S, \cal O_S(X)))$ at $[X] = \Bbb C\tau$ is $$ {\text{Hom}}(\Bbb C\tau, H^0(S, \cal O_S(X))/\Bbb C\tau = (\Bbb C\tau)^* \otimes H^0(S, \cal O_S(X))/\Bbb C\tau ). $$ Now suppose that we are given any subvariety $W$ of the linear series $|D|$ on $S$. Let $[X] \in W$ be a general point of $W$. The following theorem of Zariski (\cite{Z}, Theorems 1 and 2) characterizes the tangent space to $W$ at $[X]$: \begin{thm}\label{Zariski} {\bf (Zariski's theorem)} In terms of the identification of the tangent space to the linear series $|\cal O_S(D)|$ at $[X]$ with the characteristic series $H^0(X, \cal O_X(X))$, 1. The tangent space $T_{[X]}W$ is contained in the subspace $H^0(X, {\cal I}(X))$ of $H^0(X, \cal O_X(X))$, where $\cal I \subset \cal O_S$ is the {\em adjoint ideal} of $X$; 2. If $X$ has any singularities other than nodes, then $T_{[X]}W$ is contained in a subspace $H^0(X, {\cal J}(X))$ where ${\cal J} \subsetneqq \cal I$ is an ideal strictly contained in the adjoint ideal. \end{thm} This characterizes the tangent space to $V(D)$ at a general point $[X]$. (If the fact that it does is not clear, it will be after Lemma \ref{indcons} below.) Now, we have to consider the additional information coming from the tangency with $E$. To express this, note first that, if $\nu : X^\nu \to X$ is the normalization of $X$ and ${\cal J} \subset \cal O_{X}$ is any ideal contained in the adjoint ideal of $X$, then the pullback map gives a natural bijection between ideals ${\cal J} \subset {\cal I} \subset \cal O_X$ contained in ${\cal I}$ and ideals $ \nu^*{\cal J}\subset \nu^*{\cal I} \subset \cal O_{X^\nu}$. We will invoke this correspondence implicitly in our notation: if $p \in X^\nu$ is any point, and ${\cal J} \subset \cal O_{X}$ any ideal contained in the adjoint ideal of $X$, we will write ${\cal J}(-mp) \subset \cal O_{X}$ to mean the ideal in $\cal O_X$ whose pullback to $X^\nu$ is $\nu^*{\cal J} \otimes \cal O_{X^\nu}(-mp)$. In these terms, we have the following \begin{lm}\label{tangencycondition} Let $G \subset S$ be any fixed curve and $p \in G$ a smooth point of $G$. Let $W$ be any subvariety of $|D|$. If the general point $[X]$ of $W$ satisfies the condition: there is a point $q \in X^\nu$ such that $\nu(q) = p$ and $$ \text{mult} _q (\nu^*(G)) \; = \; m \, , $$ then the tangent space to $W$ at $[X]$ satisfies $$ T_{[X]}W \; \subset \; H^0(X, {\cal I}(X)(-mp)) . $$ Moreover, if $X$ has any singularities other than nodes, or is singular at the point $p$, we have $$ T_{[X]}W \; \subset \; H^0(X, {\cal J}(X)(-mp)) $$ where ${\cal J} \subsetneqq \cal I$ is an ideal strictly contained in the adjoint ideal. \end{lm} \begin{pf} We will prove the Lemma by applying Zariski's theorem to the proper transform of $X$ on the surface $\tilde S$ obtained by blowing up $S = \Bbb F_n$ a total of $m$ times along the curve $E$. To carry this out, let $S_1 \to S_0$ be the blow-up of $S_0 = S$ at the point $p$, $E_1 \subset S_1$ the exceptional divisor of the blow-up and $p_1 \in E_1$ the point of intersection of $E_1$ with the proper transform of $E$ in $S_1$. Similarly, let $S_2 \to S_1$ be the blow-up of $S_1$ at the point $p_1$, $E_2 \subset S_2$ the exceptional divisor of the blow-up and $p_2 \in E_2$ the point of intersection of $E_2$ with the proper transform of $E$ in $S_1$, and so on, until we arrive at the surface $\tilde S = S_m$; we will denote by $\pi : \tilde S \to S$ the composite of the blow-up maps, by $\tilde X$ the proper transform of $X$ in $\tilde S$ and by $\tilde E_i$ the proper transform of $E_i$ in $\tilde S$; so that the pullback to $\tilde S$ of the divisor $E$ is given by $$ \pi^*E \; = \; \tilde E + \sum i \cdot \tilde E_i \, . $$ We denote by $X'$ the branch of $X$ corresponding to the point $q \in X^\nu$, that is, the image of an analytic neighborhood of $q$ in $X^\nu$, by $\tilde X'$ its proper transform in $\tilde S$, and by $\tilde p$ the point of $\tilde X'$ lying over $p$. Now, let $X_i$ be the proper transform of $X$ in $S_i$, and let $k_i$ be the multiplicity of $X_{i-1}$ at the point $p_{i-1}$; for each $j = 1,\ldots,m$ we will set $$ l_j \; = \; k_1+k_2+\ldots+k_j \, . $$ Thus, for example, we have the equality of divisors $$ \pi^*X \; = \; \tilde X + \sum_{i=1}^m l_i \cdot \tilde E_i \, . $$ Similarly, we let $X_i'$ be the proper transform on $X'$ in $S_i$, $k'_i$ the multplicity of $X'_{i-1}$ at $p_{i-1}$ and $l'_j = k'_1+\ldots k'_j$. Note that $l_j \ge l'_j$ for each $j$; and the requirement that $X'$ have intersection multiplicity $m$ with $E$ at $p$ is equivalent to the assertion that $$ mult_p(X' \cdot E) \; = \; (\pi^*X' \cdot \tilde E) \; = \; l'_m \; = \; m \, , $$ so that we have in particular $l_m \ge m$, with equality if and only if (locally) $X = X'$. We can also write the intersection number $m_p(X' \cdot E)$ as $$ mult_p(X' \cdot E) \; = \; mult_q(\tilde X' \cdot \pi^*E) \; = \; m_q(\tilde X' \cdot (\tilde E + \sum j\cdot \tilde E_j)) $$ so we see that one of three things occurs: either \ $\bullet$ \thinspace $X'$ is smooth, $k_i=1$ for all $i$, and $\tilde X'$ meets the last exceptional divisor $E_m$ transversely; or \ $\bullet$ \thinspace $\tilde X'$ passes through the point $\tilde E_i \cap \tilde E_{i-1}$ for some $i < m$; or \ $\bullet$ \thinspace for some $j < m$, $\tilde X'$ meets the exceptional divisor $\tilde E_j$ at a point other than $\tilde E_j \cap \tilde E_{j-1}$ or $\tilde E_j \cap \tilde E_{j+1}$, and has a point of intersection multiplicity $m/j > 1$ with $\tilde E_j$. \ We now compare the adjoint ideal $\cal I_X$ of $X$ with that of $\tilde X$. The basic fact here is that if $C \subset S$ is any curve on a smooth surface, $p \in C$ a point of multiplicity $k$, and $\tilde C \subset \tilde S$ the proper transform of $C$ in the blow-up $\pi : \tilde S \to S$ of $S$ at $p$, the adjoint ideals of $C$ and $\tilde C$ are related by the formula $$ \pi^*\cal I_C \; = \; \cal I_{\tilde C}(-(m-1)E) $$ where $E$ is the exceptional divisor. Applying this $m$ times to the curve $X$, we have $$ \pi^*\cal I_X \; = \; \cal I_{\tilde X}(-\sum (l_j-j)\tilde E_j)\, . $$ Now, $[X] \in W$ being general, any deformation of $X$ coming from the family $W$ preserves the multiplicities $k_i$, and hence the decomposition $\pi^*X = \tilde X + \sum l_i \tilde E_i$. It also preserves the geometric genus of $\tilde X$, so that identifying the space $H^0(\tilde X, \cal O_{\tilde X}(\tilde X))$ of deformations of $\tilde X \subset \tilde S$ with a subspace of the deformations $H^0(X, \cal O_{X}(X))$ of $X \subset S$ via the pullback map, we have \begin{equation*} \begin{split} T_{[X]}W \; &\subset \; H^0(\tilde X, \cal I_{\tilde X}(\tilde X)) \\ &= \; H^0(\tilde X, (\pi^*\cal I_{\tilde X})(\sum (l_j-j)\tilde E_j)(\pi^*X - \sum l_j \tilde E_j)) \\ &= \; H^0(\tilde X, (\pi^*\cal I_{X})(\pi^*X-\sum j \tilde E_j)) \\ &= \; H^0(\tilde X, \pi^*(\cal I_{X}(X))(-l_mq)) \\ &= \; H^0(X, \cal I_X(-l_mp)) \\ &\subset \; H^0(X, \cal I_X(-mp)) \, . \end{split} \end{equation*} Note that the inclusion in the last line of the above sequence is proper if $X \ne X'$. Now, suppose that $X = X'$ is not smooth at $p$. In this case, as we noted $\tilde X'$ will either be singular at $\tilde p$ or be tangent to $\tilde E_i$ there, or else will pass through the point $\tilde E_i \cap \tilde E_{i-1}$ for some $i$. In the first case, since $\tilde X$ has a unibranch singularity, its deformations correspond to sections of $H^0(\tilde X, {\cal K}(\tilde X))$ for some ideal $\cal K$ strictly contained in the adjoint ideal $\cal I_{\tilde X}$; while in the latter two cases the deformations correspond to sections of $H^0(\tilde X, \cal I_{\tilde X}(\tilde X))$ vanishing at $q$. In either case, the inclusion in the first line of the equation above is strict. Thus $T_{[X]}W \subset H^0(X, \cal I_X(-(m+1)q))$ unless $X$ is smooth at $p$, and the remainder of the statement of the Lemma follows. \end{pf} To conclude the proof of Proposition \ref{dimensioncount} we need one more fact. To state it, let $X \in |D|$ be any irreducible rational curve, $\nu : X^\nu \to X$ the normalization and $p_1,p_2,\ldots \in X^\nu$ any points; suppose that the divisor $\nu^*(E)$ has multiplicity $m_i$ at $p_i$. Let $\cal I \subset \cal O_S$ be the adjoint ideal of $X$, and set $$ {\cal K} \; = \; \cal I(-\sum m_i p_i) \subset \cal O_X $$ Let ${\cal K}'$ be any ideal of index 2 or less in ${\cal K}$---that is, any ideal ${\cal K}' \subset {\cal K}$ with $h^0({\cal K}/{\cal K}') \le 2$, or equivalently an ideal of the form $$ {\cal K}' \; = \; {\cal K}(-q-r) $$ for some pair of points $q, r \in X^\nu$. We will need these ideals ${\cal K}' \subset {\cal K}$ of index 2 in order to see, for example, that a general curve $X \in V(D)$ does not have a node on $E$. In these terms, our result is the \begin{lm}\label{indcons} The ideal $ {\cal K}'$ imposes independent conditions on the linear series $|\cal O_X(X)|$, i.e., $$ h^0(X, {\cal K'}(X)) \; = \; h^0(X, \cal O_X(X)) - \dim_{\Bbb C } (\cal O_X/{\cal K'}) $$ In particular, ${\cal K}$ imposes independent conditions on $|\cal O_X(X)|$, that is, \begin{equation*} h^0(X, {\cal K}(X)) \; = \; r_0(D) - \sum m_i . \end{equation*} \end{lm} \begin{pf} By the adjunction formula we have $$ K_{X^\nu} \; = \; \nu^*( K_S \otimes \cal O_S(X) \otimes {\cal I}) . $$ Thus, $$ \nu^*(\cal O_S(X) \otimes {\cal K}) \; = \; K_{X^\nu} \otimes \nu^*(\cal O_S(-K_S)) \otimes \cal O_{X^\nu}(-\sum m_i p_i) . $$ Now, $\nu^*E - \sum m_i p_i \ge 0$, and on $S = \Bbb F_n$, we have $$ K_S \; = \; \cal O_S(-C-E-2F) $$ so that we have an inequality of divisor classes $$ \nu^*(\cal O_S(X) \otimes {\cal K}) \; \ge \; K_{X^\nu} \otimes \nu^*\cal O_S(C+2F) . $$ Moreover, the divisor class $C+2F$ has intersection number at least 3 with any irreducible curve $X$ not linear equivalent to either $F$ or $E$, so it follows that $$ \deg(\nu^*(\cal O_S(X) \otimes {\cal K})) \; \ge \; -2 + 3 = 1 . $$ Thus $$ \deg(\nu^*(\cal O_S(X) \otimes {\cal K}')) \; \ge -1 . $$ so that $h^1(X^\nu, \nu^*(\cal O_S(X) \otimes {\cal K}'))=0$, and the result follows. \end{pf} We can now complete the proof of Proposition \ref{dimensioncount}. We have already established, in the Claim above, that $$ \dim(\tilde V_{\underline m}(D)) \; \ge \; r_0(D) - \sum (m_i-1) ; $$ but applying Lemmas \ref{tangencycondition} and \ref{indcons} in turn we see that for any subset $\Omega = \{p_1,\ldots,p_k\} \subset E$, \begin{equation*} \begin{split} \dim(\tilde W^\Omega_{\underline m}(D)) \; &\le \; h^0(X, {\cal K}(X)) \\ &= \; r_0(D) - \sum m_i \end{split} \end{equation*} and hence \begin{equation*} \begin{split} \dim(\tilde V_{\underline m}(D)) \; &\le \; \dim(\tilde W^\Omega_{\underline m}(D)) + k \\ &= \; r_0(D) - \sum (m_i-1) \end{split} \end{equation*} so that equality must hold. Moreover, if a general point $[X] \in V_{\underline m}(D)$ corresponded to a curve $X$ with singularities other than nodes, the second inequality above would be strict; so $X$ must be nodal, and smooth at ots points of intersection with $E$. \ We can eliminate all the other possible misbehaviors of our general curve $X$ similarly. If the point $p \in X^\nu$ is mapped to one of the points $P_i$, we would have \begin{equation*} \begin{split} \dim(\tilde V_{\underline m}(D)) \; &\le \; h^0(X, {\cal K}(X)(-p)) \\ &< \; h^0(X, {\cal K}(X)) ; \end{split} \end{equation*} and if the multiplicity of the pullback divisor $\nu^*(G)$ at $p$ were $m > 1$ we would have \begin{equation*} \begin{split} \dim(\tilde V_{\underline m}(D)) \; &\le \; h^0(X, {\cal K}(X)(-(m-1)p)) \\ &< \; h^0(X, {\cal K}(X)) . \end{split} \end{equation*} Suppose next that $X$ had a node on $E$, with branches corresponding to a pair of points $q, r \in X^\nu$ and the branch corresponding to $r$ transverse to $E$. It would follow that $$ h^0(X, {\cal K}(X)(-q-r)) \; = \; h^0(X, {\cal K}(X)) - 1 , $$ since a section of ${\cal K}(X)$ vanishing at $q$ but not at $r$ would correspond to a deformation of $X$ in $\tilde V_{\underline m}(D)$ in which the two branches would meet $E$ in distinct points. Finally, to prove part 3 of Proposition \ref{dimensioncount}, we simply let $X'$ be a general member of the family $\tilde V_{\underline m'}(D')$ and apply the above to $X \in \tilde V_{\underline m}(D)$, including $X'$ in $G$ and its points of intersection with $G$ and $E$ among the points $P_i$. \end{pf} The next Proposition is stated as a characterization of the reducible elements of the one-parameter family $\cal X^\nu \to \Gamma$, but in fact it is a characterization of the codimension one components of the boundary $V(D) \setminus \tilde V(D)$ of $V(D)$. \begin{prop}\label{codimensionone} Let $X \subset S$ be any reducible fiber of the family $\cal X \to \Gamma$. 1. If $X$ does not contain $E$, then $X$ has exactly two irreducible components $X_1$ and $X_2$, with $[X_i]\in V(D_i)$ and $D_1+D_2=D$. Moreover $[X_i]$ is a general point in $V(D_i)$. 2. If $X$ does contain $E$, then $X$ has irreducible components $E$, $X_1,\ldots,X_k$, with $[X_i]\in V(D_i)$ and $E+D_1+\ldots +D_k=D$. Moreover each $X_i$ is general in $V_{m_i}(D_i)$ for some collection $m_1,\dots,m_k$ of positive integers such that $\sum (m_i-1) = n-k$. \end{prop} \ \noindent \underbar{Remark}. Notice that by Proposition~\ref{dimensioncount}, the above result says that if $X$ does not contain $E$, then it has only nodes as singularities. And, if $X$ contains $E$, away from the $k$ points of tangency of $E$ with the curves $X_i$, $X$ has only nodes as singularities. \ \noindent \begin{pf} Assume first that $X$ does not contain $E$. Write the divisor $X$ as a sum $$ X = \sum_{i=1}^k a_i \cdot X_i $$ where $a_i > 0$ and the $X_i$ are irreducible curves in $S$. We claim first that since $[X] \in V(D)$, all the curves $X_i$ must be rational. To see this, take any one-parameter family ${\cal X} \to B$ of irreducible rational curves specializing to $X$. Proceeding as in \ref{setup} we arrive at a family $\cal Y \to B$ of nodal curves, with general fiber $\Bbb P^1$, that admits a regular map $\cal Y \to {\cal X}$. Now, since the fibers of $\cal Y \to B$ are reduced curves of arithmetic genus 0, every component of every fiber of $\cal Y$ must be a rational curve. Thus every component of $X$ is dominated by a rational curve and so must be itself rational. Thus $[X_i] \in V(D_i)$, where $D_i$ are divisor classes such that $\sum a_iD_i = D$. On the other hand, since $X$ is a general member of an $(r_0(D)-1)$-dimensional family, we must have $$ \sum_{i=1}^k r_0(D_i) \; \ge \; r_0(D) - 1 $$ which yields \begin{equation*} \begin{split} \sum_{i=1}^k \bigl( -(K_S \cdot D_i) - 1 \bigr) \; &\ge \; (-K_S \cdot D) - 2 \\ &= \; \sum_{i=1}^ka_i (-K_S \cdot D_i) - 2 . \end{split} \end{equation*} Comparing the two sides, we see that $$ 2 - k - \sum_{i=1}^k(a_i-1) (-K_S \cdot D_i) \; \ge \; 0 . $$ But $(-K_S\cdot D_i)\geq 2$ for any curve $D_i$ on $S$ other than $E$; so we may conclude that all $a_i = 1$ and that $k \le 2$. Moreover, if $k=2$ we have equality in the above inequality, which says that the pair of curves $(X_1, X_2)$ is general in $V(D_1) \times V(D_2)$. We come now to the case where $X$ contains $E$. The first thing we see here is that the dimension-count argument we used above doesn't work: since $$ (-K_S \cdot (X-aE)) \; = \; (-K_S \cdot X) + a(n-2) , $$ the sums $\sum a_iX_i$ of rational curves $X_i \in |D_i|$ may well move in a larger-dimensional family than $X$ itself. The key here is to look at the semistable reduction of a family of curves in $ \tilde V(D)$ specializing to $X$. This will allow us to limit the number of points of intersection of the curves $X_i$ with $E$, that is to say, to show that in fact the $X_i$ belong to $V_{\underline{m}}(D_i)$ for suitable $\underline m$. This replaces the naive bound above on the dimension of the family of such curves $X$ with a stronger one, which turns out to be sharp. Consider then the family $\cal Y \longrightarrow B$ obtained from $\cal X \longrightarrow \Gamma$ as in Section~\ref{setup}. We can thus assume that the total space $\cal Y$ of the family is smooth and every fiber of $\cal Y$ will be a union of smooth rational curves meeting transversely, and whose dual graph is a tree. Now, let $Y$ be the special fiber of $\cal Y \to B$. We decompose $Y$ into two parts: we let $Y_E$ be the union of the irreducible components of $Y$ mapping to $E$, and $Y_R$ the union of the remaining components. Next, we decompose $Y_R$ further into $k$ parts, letting $Y_i$ be the union of the components mapping to $X_i$. Denote the connected components of $Y_E$ by $Z_i$, and for each $i$ let $\alpha_i$ be the degree of the map $\mu|_{Y_i} : Z_i \to E$, so that $\sum \alpha_i = a$. Similarly, let $\{Z_{i,j}\}_j$ be the connected components of $Y_i$ and $\alpha_{i,j}$ the degree of the restriction $\mu|_{Z_{i,j}} : Y_{i,j} \to X_i$, so that $\sum_j \alpha_{i,j} = a_i$. Note that the inverse image of $E$ in $\cal Y$ is given by $$ \pi^{-1}(E) \; = \; Y_E \cup \Gamma_1 \cup \dots \cup \Gamma_b . $$ (Where $\pi : \cal Y \to S$ is, as usual, the natural map.) As we indicated, the essential new aspect of the argument in this case is keeping track of the number of points of intersection of the $X_i$ with $E$. To do this, we note that, over any such point, there will be a point of intersection of a component of $Y_i$ with the inverse image $\pi^{-1}(E)$; which by the expression above for $\pi^{-1}(E)$ will be either a point of intersection of $Y_i$ with $Y_E$ or one of the $b$ points of intersection of the $\Gamma_i$ with $Y$. It thus remains to bound the number $\epsilon$ of points of intersection of $Y_E$ with the remaining parts $Y_i$ of $Y$. This we can do by using the fact that the dual graph of $Y$ is a tree: this says that the number of pairwise points of intersection of the connected components $Z_{i,j}$ of $Y_i$ and the connected components $Z_i$ of $Y_E$ is equal to the total number of all such connected components, minus one. Thus, \begin{equation*} \begin{split} \epsilon \; = \; \#(Y_R \cap Y_E) \; = \; &\#\{ \text{connected components of } Y_E\} \cr + \; &\sum \#\{ \text{connected components of } Y_i\} . \end{split} \end{equation*} Note that the degree $\alpha_i > 0$ on each component $Z_i$ of $Y_E$, so that $$ \#\{ \text{connected components of } Y_E\} \; \le \; a $$ and similarly $$ \#\{ \text{connected components of } Y_i\} \; \le \; a_i . $$ Thus we can deduce in particular that $$ \epsilon \; \le \; a + \sum a_i - 1 . $$ Now, say $X_i \in \tilde V_{{\underline m}^i}(D_i)$ for each $i = 1,\ldots,k$. Let $\nu_i : X_i^\nu \to X_i$ be the normalization map. Choose any irreducible component $X_i^0$ of $Y$ dominating $X_i$ (and hence dominating the normalization $X_i^\nu$), and let $\pi_i : X_i^\nu \to X_i$ be the restriction of $\pi$ to $X_i^\nu$. Trivially, the total number of points of the pullback $\nu_i^*(E)$ of $E$ to $X_i^\nu$ is \begin{equation*} \begin{split} \# \nu_i^*(E) \; &\le \; \#\pi_i^*(E) \\ &= \; \#(X_i^0 \cap Y_E) \end{split} \end{equation*} and hence \begin{equation*} \begin{split} \sum \# \nu_i^*(E) \; &\le \; \sum \#(X_i^0 \cap Y_E) \\ &\le \; \#(Y_R \cap Y_E) \\ &= \; \epsilon \end{split} \end{equation*} with strict inequality if any $a_i > 1$. But the sum of degrees of $E$ on the curves $X_i$ is at least \begin{equation*} \begin{split} \sum \deg(\pi_i^*E) \; &\ge \; \left((\sum X_i) \cdot E\right) \\ &= \; \left( (D - aE - \sum (a_i - 1)D_i ) \cdot E \right) \\ &= \; (D \cdot E) + an - \sum a_i(D_i \cdot E) . \end{split} \end{equation*} Comparing the number of points of the pullbacks of $E$ to the normalizations $X_i^\nu$ with the degrees of these pullbacks, we conclude that there must be multiplicities in these divisors: specifically, the sum $\sum (m_j^i - 1)$ of the multiplicities minus one must be the difference of these numbers, so that \begin{equation*} \begin{split} \sum (m_j^i - 1) \; &\ge \; \sum \deg \pi_i^*(E) - \epsilon - (D \cdot E) \\ &\ge \; (D \cdot E) + an - \sum (a_i - 1)(D_i \cdot E) - a - \sum a_i + 1 - (D \cdot E) \\ &\ge \; a(n-1) - \sum (a_i - 1)(D_i \cdot E) - \sum a_i + 1 . \end{split} \end{equation*} This in turn allows us to bound the number of degrees of freedom of the curves $X_i$: we have \begin{equation*} \begin{split} \sum \dim \tilde V_{{\underline m}^i}(D_i) \; &= \; \sum r_0(D_i) - \sum (m_j^i - 1) \\ &= \; \sum_{i=1}^k \left( (-K_S \cdot D_i) - 1 \right) - \sum (m_j^i - 1) \\ &\le \; \sum (-K_S \cdot D_i) - k - a(n-1) + \sum (a_i - 1)(D_i \cdot E) + \sum a_i - 1 . \end{split} \end{equation*} On the other hand, this must be at least equal to the dimension of $V(D)$ minus one, that is, \begin{equation*} \begin{split} r_0(D) - 1 \; &= \; (-K_S \cdot D) - 2 \\ &= \; a(-K_S \cdot E) + \sum a_i(-K_S \cdot D_i) - 2 \\ &= \; a(n-2) + \sum a_i(-K_S \cdot D_i) - 2 . \end{split} \end{equation*} In the end, then, we must have \begin{equation*} \begin{split} a(n-2) + \sum a_i(-K_S \cdot D_i) - 2 \; &\le \; \sum (-K_S \cdot D_i) - k - a(n-1) \\ &\quad + \sum (a_i - 1)(D_i \cdot E) + \sum a_i - 1 . \end{split} \end{equation*} We can (partially) cancel the $a(n-1)$ and $a(n-2)$ terms, and combine the terms involving $(-K_S \cdot D_i)$ to rewrite this as $$ a + \sum (a_i-1)(-K_S \cdot D_i) - 1 \; \le \; \sum (a_i - 1)(D_i \cdot E) - k + \sum a_i - 1 $$ or, in other words, $$ a + \sum (a_i - 1) \left[ \left( (-K_S - E) \cdot D_i \right) - 1 \right] - 1 \; \le \; 0 . $$ Now, we have already observed that $ -K_S - E \; = \; C + 2F $ meets every curve $X_i$ strictly positively, so that the sum in this last expression is nonnegative. We conclude that $a=1$, and (since any $a_i > 1$ would have led to strict inequality) that all $a_i = 1$. Next, since there is a unique component of $Y$ mapping to each $X_i$, each curve $X_i$ will have at most one point of intersection multiplicity $m > 1$ with $E$. Thus, finally, $X_i$ is a general member of the family $\tilde V_m(D_i)$ for some collection of integers $m_1,\dots,m_k$ with $\sum (m_i - 1) = n-k$, completing the proof of Proposition \ref{codimensionone}. \end{pf} Note that we have not said here that every reducible curve satisfying the conditions of the Proposition in fact lies in the closure of the locus of irreducible rational curves. This is true, and is not hard to see in the case of curves of types (1); but for curves of type (2) it is a deeper fact, and we will require the proof of Proposition~\ref{singtotalspace} to establish it. Having characterized as a set the locus $\Gamma$ of curves in $V(D)$ passing through $q_1,\dots,q_{r_o (D) -1}$, we now turn to a statement about the local geometry of $\Gamma$ around each point. We introduce one bit of terminology here. Let $X$ be a fiber of $\cal X \to \Gamma$; and, in case $\Gamma$ is locally reducible at the point $[X] \in \Gamma$, pick a branch of $\Gamma$ at $[X]$ (that is, a point $b$ of the normalization $\Gamma^\nu$ of $\Gamma$ lying over $[X]$). Let $P$ be a node of $X$. We then make the following \ \noindent {\bf Definition}. If $P$ is a limit of nodes of fibers of $\cal X \to \Gamma$ near $X$ in the chosen branch---that is, if $(P,b)$ is in the closure of the singular locus of the map $\cal X \times_\Gamma (\Gamma^\nu \setminus \{b\}) \to \Gamma^\nu$---we will say that $P$ is an {\it old} node of $X$. If $(P,b)$ is an isolated singular point of the map $X \times_\Gamma (\Gamma^\nu \setminus \{b\}) \to \Gamma^\nu$ we will say that $P$ is a {\it new} node of $X$. \ \noindent Equivalently, $P$ is an old node if the fiber $X^\nu$ of $\cal X^\nu \to \Gamma^\nu$ over $b$ is smooth at the (two) points lying over $P$; if it is a new node, $X^\nu$ will have a single point lying over $P$, which will be a node of $X^\nu$. \ Note that if $P$ is a singular point of $X$ other than a node, the situation is not so black-and-white. For example, if $P$ is an $m$-fold tacnode---that is, if the curve $X$ has two smooth branches at $P$ with contact of order $m$---then a priori, any number $n \le m$ of nodes of nearby fibers may approach $P$ along any branch of $\Gamma$ at $[X]$, with the result that the fiber of $\cal X^\nu \to \Gamma^\nu$ over the corresponding point $b \in \Gamma^\nu$ will have an $(m-n)$-fold tacnode over $P$, or will be smooth over $P$ if $n=m$. (The proof of the relevant case $n=m-1$ will emerge in the proof of Proposition \ref{singtotalspace}.) In these terms, we can state \begin{prop}\label{describegamma} Let $X$ be a reducible fiber of the family $\cal X \to \Gamma$. Keeping the notations and hypotheses of Proposition \ref{codimensionone}, $1$. If $X = X_1 \cup X_2$ does not contain $E$, and $X_1$ and $X_2$ meet at $(D_1 \cdot D_2) = \ell$ points $P_1,\dots,P_\ell$, then in a neighborhood of $[X]$ $\Gamma$ has $\ell$ smooth branches $\Gamma_1 ,\ldots ,\Gamma _{\ell }$; along $\Gamma_i$ the point $P_i$ is new, and all other nodes of $X$ are old. $2a$. If $X = E \cup X_1 \cup \dots \cup X_k$, and $X_i$ meets $E$ transversely in $(D_i \cdot E) = \ell_i$ points $P_{i,1},\dots,P_{i,\ell_i}$, then in a neighborhood of $[X]$ $\Gamma$ consists of $\prod \ell_i$ smooth branches $\Gamma_\alpha = \Gamma_{(\alpha_1,\dots,\alpha_k)}$. Along $\Gamma_\alpha$ the points $P_{1,\alpha_1},\dots,P_{k,\alpha_k}$ are new, and all other nodes of $X$ are old. $2b$. If $X = E \cup X_1 \cup \dots \cup X_k$, and $X_i$ meets $E$ transversely in $(D_i \cdot E) = \ell_i$ points $P_{i,1},\dots,P_{i,\ell_i}$ for $i = 2,\dots,k$, while $D_1$ has a point $P$ of intersection multiplicity $m \ge 2$ with $E$, then in a neighborhood of $[X]$ $\Gamma$ consists of $\prod_{i=2}^k \ell _i$ smooth branches $\Gamma _\alpha = \Gamma_{(\alpha_2,\dots,\alpha_k)}$. Along $\Gamma_\alpha$ the points $P_{2,\alpha_2},\dots,P_{k,\alpha_k}$ are new; all other nodes of $X$ are old; and exactly $m-1$ nodes of nearby fibers will tend to $P$. \end{prop} \noindent \underbar{Remark 1}. The proof of this Proposition will not be complete until the end of the following section. More precisely, we will postpone the proof of the existence and smoothness of the branches of $\Gamma$. Actually, cases $1$ and $2a$ could very well be proved here, but it is more convenient do it later (that is, at the beginning of the proof of Proposition \ref{singtotalspace}). \noindent \underbar{Remark 2}. We believe that an analogous description of the family $\cal X \to \Gamma$ may be given without the assumption that the components of the curve $X$ other than $E$ have altogether at most one point of tangency with $E$, and otherwise intersect $E$ transversely in distinct points. The restricted statement above will suffice for our present purposes. We hope to prove the general statement in the future. \noindent \underbar{Remark 3}. The statement of Proposition \ref{describegamma} can also be expressed in terms of the normalized family $\cal X^\nu \to \Gamma^\nu$, and indeed that is how we will use it in the following chapter. In these terms, the statements are: \ \noindent $1$. \thinspace If $[X]$ is a point of $\Gamma$ corresponding to a curve $X$ in our family not containing $E$, then there will be $(D_1 \cdot D_2) = \ell$ points of $\Gamma^\nu$ lying over $[X]$, corresponding naturally to the nodes of $X$. The fibers of $\cal X^\nu \to \Gamma^\nu$ over these points will be the normalizations of $X$ at all the nodes of $D_1$ and $D_2$ and at all but one of the $\ell$ points of intersection of $D_1$ with $D_2$. \ \noindent $2a$. \thinspace If $X = E + D_1 + \dots + D_k$ contains $E$ and the components $D_i$ intersect $E$ transversely, then the fibers of $\cal X^\nu \to \Gamma^\nu$ over points lying over $[X] \in \Gamma$ are the curves obtained by normalizing $X$ at all nodes of the $D_i$, at all the points of pairwise intersection of the $D_i$, and at all but one of the points of intersection of $E$ with each of the components $D_i$. In other words, the fibers consist of the disjoint union of the normalizations $\tilde D_i$ of the curves $D_i$, each attached to $E$ at one point. \ \noindent $2b$. \thinspace If $X = E + D_1 + \dots + D_k$ as before and one of the components $D_1$ of $X$ has a smooth point $P$ of intersection multiplicity $m \ge 2$ with $E$, then the fibers $X^\nu$ of $\cal X^\nu \to \Gamma^\nu$ corresponding tor $[X] \in \Gamma$ are the curves obtained by normalizing $X$ at all nodes of the $D_i$, at all the points of pairwise intersection of the $D_i$, at all but one of the points of intersection of $E$ with each of the components $D_i$ for $i = 2,\dots,k$, at all the transverse points of intersection of $D_1$ with $E$, and finally taking the partial normalization of $X$ at $P$ having an ordinary node over $P$. (The fact that each fiber of $\cal X^\nu \to \Gamma^\nu$ lying over $X$ has an ordinary node over $P$ follows either from the fact that the $\delta$-invariant of the singularity $P \in X$ is $m$ and that, along each branch, $m-1$ nodes of nearby fibers tend to $P$; or---what is essentially the same thing---the fact that the arithmetic genus of the fibers of $\cal X^\nu \to \Gamma^\nu$ are zero. This will be verified independently in the course of the proof of Proposition \ref{singtotalspace}.) The picture is therefore similar to the preceding case: the fibers consist of the disjoint union of the normalizations $\tilde D_i$ of the curves $D_i$, each attached to $E$ at one point. The one difference is that, while for $i = 2,\dots,k$ the point of attachment of the normalizations $\tilde D_i$ with $E$ can lie over any of the points of intersection of $D_i$ with $E$, the point of intersection of the normalization of $D_1$ with $E$ can only be the point lying over $P$. \ A typical picture of the original curve $X$ and its partial normalization $X^\nu$ is this: \ \vspace*{5.3in} \noindent \special {picture basicdiagram} \ \begin{pf} Consider first of all a reducible curve $X$ in our family that does not contain $E$. By Proposition \ref{codimensionone}, this must be of the form $X = X_1 \cup X_2$ where $X_i$ is a general member of the family $V(D_i)$ with $D_1 +D_2 =D$. In particular, $X_i$ is an irreducible rational curve with $p_a(D_i)$ nodes, and $X_1$ and $X_2$ intersect transversely in $(D_1 \cdot D_2)$ points. Note that $$ p_a(D_i) = \frac{(D_i \cdot D_i) + (D_i \cdot K_S)}{2} + 1 $$ so that the total number of nodes of $X$ will be \begin{equation*} p_a(D_1) + p_a(D_2) + (D_1 \cdot D_2) \; = \; p_a(D) + 1 . \end{equation*} In other words, along any branch of $\Gamma$, all but one of the nodes of $X$ will be limits of nodes of nearby fibers (that is, will be old nodes), while one node of $X$ will be a new node. Note also that not any node of $X$ can be the new node: that must be one of the points of intersection of the two components $X_1$ and $X_2$; otherwise the fiber of the normalization $\cal X^\nu$ would be disconnected. In case $X$ contains $E$, the analogous computation yields that $X$ has $p_a(D)+k$ nodes (or $p_a(D)+k-m$ nodes and one tacnode of order $m$ in case $2b$); hence $X$ has $k$ new nodes (or, $k-1$ in $2b$). Then the analysis in the proof of Proposition \ref{codimensionone} shows that in the normalization of the total space of the family, the corresponding fiber will consist of a curve $\tilde E$ mapping to $E$, plus the normalizations $\tilde X_i$ of the curves $X_i$, each meeting $\tilde E$ in one point and disjoint from each other. In particular, all the nodes of $X$ arising from points of pairwise intersection of the components $X_i$ are old. As for the points of intersection of the components $X_i$ with $E$, there are two cases. First, if a component $X_i$ has a point of contact of order $m > 1$ with $E$, that must be the image of the point $\tilde X_i \cap \tilde E \in \cal X^\nu$; and all the other points of $X_i \cap E$ will be old nodes of $X$ on any branch. On the other hand, if a component $X_i$ intersects $E$ transversely, any one of its points of intersection with $E$ can be a new node. \end{pf} \subsection{Singularities of the total space} We come finally to the fourth result, in which we will describe the singularities of the total space of the normalized family $\cal X^\nu \to \Gamma^\nu$ along a given fiber $X^{\nu }$. (Given a fiber $X$ over $\Gamma$, we will fix a corresponding fiber $X^{\nu }$ throughout.) We will keep a simplified form of the notation introduced in the statement of Proposition \ref{describegamma}: we will denote by $P_1,\dots,P_\ell$ the new nodes of $X$ along $E$, coming from transverse points of intersection of other components of $X$ with $E$; and by $P$ (if it exists) one double point of $X$ other than a node, coming from a point of contact of order $m\geq 2$ of $E$ with another component of $X$. We recall that the nearby fibers of our family are smooth near $P_i$, there will be one point $p_i$ of $\cal X^\nu$ lying over each $P_i$, which will be a node of $X^\nu$, while the nearby fibers have $m-1$ nodes tending to the point $P$, so that the partial normalization $X^\nu \to X$ will again have one point $p$ lying over $P$, and that point will be a node of $X^\nu$. With all this said, we have \begin{prop}\label{singtotalspace} 1. If $X$ does not contain $E$, or if $X$ contains $E$ and the closure of $X \setminus E$ intersects $E$ transversely, then $\cal X^\nu$ is smooth along $X^\nu$. 2. In case $X$ does contain $E$ and the closure of $X \setminus E$ has a point $P$ of intersection multiplicity $m \ge 2$ with $E$, the point $p$ of $X^\nu$ lying over $P$ is a smooth point of $\cal X^\nu$; the other nodes $p_i$ of $X^\nu$ will be singularities of type $A_{m-1}$ of $\cal X^\nu$ . \end{prop} \begin{pf} We start with the first statement, which is by far the easier. Recall that by the two previous propositions $X$, being a general point on a codimension-one locus in $V(D)$, will have $p_a(D)+k$ or $p_a(D)+1$ nodes, depending whether $X$ does or doesn't contain $E$. Of these, $p_a(D)$ will be old nodes and the remaining ones are new nodes; if $E$ is contained in $X$, then the new nodes all lie on $E$. Let $r_1,\dots,r_{p_a(D)}$ be the old nodes of $X$ and let $P$ be any fixed new node. The fiber $X^\nu$ of $\cal X^\nu$ lying over $X$ will be the partial normalization of $X$ at $r_1,\dots,r_{p_a(D)}$, so that $\cal X^\nu$ will certainly be smooth there, and we need only concern ourselves with the point of $\cal X^\nu$ lying over $P$. Consider, in an analytic neighboroohd of $[X]$ in $|D|$, the locus $W$ of curves that pass through the base points $q_1,\dots,q_{r_0(D)-1}$ and that preserve all of the old nodes of $X$. The projective tangent space to $W$ at $[X]$ will be contained in the sub-linear series of $|D|$ of curves passing through the $p_a(D)$ old nodes of $X$ and through $q_1,\dots,q_{r_0(D)-1}$. This gives a total of $r_0(D)-1 + p_a(D)=r(D)-1$ points which, by an argument analogous to the proof of Lemma~\ref{indcons}, impose independent conditions on the linear series $|D|$. We only exhibit the proof in case $E$ is a component of $X$, the other case being similar and easier. Let $\cal H$ be the ideal sheaf of the subscheme of $S$ given by the old nodes $r_1,\dots,r_{p_a(D)}$, and let $\nu :\tilde X\longrightarrow X$ be the normalization map. We have to show that $r_1,\dots,r_{p_a(D)}$ impose independent conditions on $|D|$, which will follow (cf. Lemma~\ref{indcons}) from $$ H^1\bigl( \tilde X, \nu ^* ( \cal O _S(X)\otimes \cal H )\bigr) =0. $$ This, by the adjunction formula, is equivalent to $$ H^0 \bigl( \tilde X , \nu ^* ( K_S \otimes \cal I ) \otimes (\nu ^*\cal H )^{-1}\bigr) =0 $$ where $\cal I$ is the adjoint ideal of $X$. Now notice that the line bundle $\nu ^* (\cal I )\otimes \nu ^* (\cal H )^{-1}$ has degree $-k$ on the component of $\tilde X$ lying over $E$, and degree $-1$ on every other component. Since $ K_S$ has degree $n-2 = k-2$ on $E$ and negative degree on $X_i$, the line bundle $\nu ^* ( K_S \otimes \cal I ) \otimes (\nu ^*\cal H )^{-1}$ cannot have any sections. We conclude that $W$ is smooth of dimension 1. Notice that this completes the proof of Proposition~\ref{describegamma}, parts $1$ and $ 2a$. To anlyze the total space of $\cal X ^\nu $ we consider the map from $W$ to the versal deformation space of the node $(X,P)$. This has nonzero differential because $P$ is not a base point of the linear series of curves passing through $q_1,\dots,q_{r_0(D)-1}$ and through the $p_a(D)$ old nodes of $X$ (to see this, the argument above applied to the ideal sheaf of the union of the old nodes of $X$ and $P$ will work). Thus the family $\cal X^\nu \to \Gamma^\nu$ has local equation $xy-t=0$ near $p$; in particular, it is smooth at $p$. We turn now to the second part, which will occupy us for the remainder of this chapter. We will start by carrying out a global analysis of the family in a neighborhood of the whole fiber $X$, and then proceed to a local analysis around the point $P$ specifically. From the global picture we will establish that, for some integer $\gamma$, the point $P$ will be a singularity of type $A_\gamma$ and the points $P_i$ all singularities of type $A_{\gamma m}$. The local analysis will then show that in fact we have $\gamma=1$. To carry out the global analysis, we use the family $\cal Y \to \Gamma^\nu$ and the map $\pi :\cal Y \to \Bbb F _n$ (cf. section~\ref{setup}), where $\cal Y$ is the minimal desingularization of the surface $\cal X^\nu$. Since the singularities of the fiber of $\cal X^\nu$ are all nodes, the total space $\cal X^\nu$ will have singularities of type $A_{\beta}$ at each; let us say the point $p$ is an $A_\gamma$ singularity of $\cal X^\nu$, and the point $p_i$ an $A_{\gamma_i}$ singularity. When we resolve the singularity at $p$ we get a chain $G_1,\dots,G_{\gamma-1}$ of smooth rational curves; likewise, $p_i$ is replaced by a chain $G_{i,1},\dots,G_{i, \gamma_i-1}$ of smooth rational curves. Denoting the component of $X$ meeting $E$ at $P_i$ by $D_i$ and the component meeting $E$ at $P$ by $D$ (we are not assuming here that these are distinct irreducible components of $X$), we arrive at a picture of the relevant part of the fiber $Y$ of $Y$: \vspace*{3.4in} \noindent \special {picture Fnmult} \ \ (We hope that such a notation will not be too confusing!) We now look at the pull-back of $E$ from $\Bbb F _n$ to $\cal Y$. We can write it as $$ \pi^*(E) = k \cdot E + \sum a_i \cdot G_i + \sum a_{i,j} \cdot G_{i,j} + E' $$ where $E'$ is a curve in $\cal Y$ that meets the fiber $Y$ only along $D_i$ and $D$, with $(E'\cdot D_i) = (E\cdot \pi (D_i)) -1$ and $(E'\cdot D) = (E\cdot \pi (D)) -m$. We can use what we know about the degree of this divisor on the various components of $Y$ to impose conditions on the coefficients $k$, $a_i$ and $a_{i,j}$. First, since $\pi$ maps components $G_i$ and $G_{i,j}$ to points in $\Bbb F_n$, $$ \deg_{G_i}(\pi^*(E)) = \deg_{G_{i,j}}(\pi^*(E)) = 0 . $$ Now, each of the curves $G_i$ and $G_{i,j}$ has self-intersection $-2$; so, setting $a_\gamma = a_{i,\gamma_i} = 0$ and $a_0 = a_{i,0} = k$, we get $$ a_{i-1} - 2a_i + a_{i+1} = 0 $$ for each $i = 1,\dots,\gamma-1$; and similarly $$ a_{i,j-1} - 2a_{i,j} + a_{i,j+1} = 0 $$ for each $j = 1,\dots,\gamma_i-1$---in other words, the sequences $a_0,\dots,a_\gamma$ and $a_{i,0},\dots,a_{i,\gamma_i}$ are arithmetic progressions. On the other hand, the map $\pi$ restricted to the component $D_i$ is transverse to $E$ at $P_i = \pi(p_i)$; so the multiplicity at $p_i$ of the restriction to $D_i$ of the divisor $\pi^*(E) - E'$ is one. This says that $a_{i,\gamma_i-1} = 1$; and similarly $a_{\gamma -1} = m$. Following the arithmetic progression $a_0,\dots,a_\gamma$ up from $D$ to $E$, we arrive at $$ k = \gamma \cdot m $$ and hence $$ \gamma_i = \gamma \cdot m . $$ The proof of the Proposition will be completed once we show that $\gamma =1$, that is, that $p$ is a smooth point of $\cal X ^{\nu }$. Note that this part of the analysis did not rely, except notationally, on the hypothesis that all but one point of intersection of $E$ with the remaining components of $X$ are transverse. If the points $P_i$ were points of intersection multiplicity $m_i$ of $E$ with other components $D_i$ of $X$, we could (always assuming that $m_i-1$ nodes of the general fiber of our family approach $P_i$) carry out the same analysis and deduce that for some integer $k$, the point $p_i$ was a singularity of type $A_{k/m_i}$---loosely speaking, the singularity of $\cal X^\nu$ at $p_i$ is ``inversely proportional" to the order of contact of $D_i$ with $E$ at $P_i$. The remaining question then would be, is the number $k$ as small as possible, that is, the least common multiple of the $m_i$? That is what we will establish with the following local analysis, which does ultimately rely on the hypothesis that all but one of the $m_i$ are one. \subsubsection{The versal deformation space of the tacnode} \label{tacnode} We now carry out the analysis around the point $P$. The versal deformation of $P\in X\subset \Bbb F _n$ has the vector space $\cal O_{\Bbb F_n, P}/{\cal J}$ as base, where ${\cal J}$ is the Jacobian ideal of $X$ at $p$. Choose local coordinates $x,y$ for $\Bbb F_n$ centered at $P$, so that the curve $E$ is given as $y=0$ and the equation of $X$ is $$ y(y+x^m) = y^2 + yx^m = 0 $$ The Jacobian ideal of this polynomial is ${\cal J} = (2y+x^m, yx^{m-1})$. The monomials $y, xy, x^2y,\dots,x^{m-2}y$ and $1, x, x^2,\dots,x^{m-1}$ form a basis for $\cal O_{\Bbb F_n, P}/{\cal J}$, so that we can write down explicitly a versal deformation space: the base $\Delta $ will be an analytic neighborhood of the origin in affine space ${\Bbb A}^{2m-1}$ with coordinates $\alpha_0, \alpha_1,\dots,\alpha_{m-2}$ and $\beta_0, \beta_1,\dots,\beta_{m-1}$, and the deformation space will be the family ${\cal S} \to \Delta$, with ${\cal S} \subset \Delta \times {\Bbb A}^2$, given by the equation $$ y^2 +yx^m + \alpha_0y + \alpha_1xy + \dots + \alpha_{m-2}x^{m-2}y + \beta_0 + \beta_1x + \beta_2x^2 + \dots + \beta_{m-1}x^{m-1} = 0 $$ Inside $\Delta$ we look closely at the closures $\Delta_{m-1}$ and $\Delta_m$ of the loci corresponding to curves with $m-1$ and $m$ nodes, respectively. We have \begin{lm} \label{versal} 1. $\Delta _m$ is given in $\Delta$ by the equations $\beta _0=\ldots =\beta _{m-1} = 0$; in particular it is smooth of dimension $m-1$. 2. $\Delta _{m-1}$ is irreducible of dimension $m$, with $m$ sheets crossing transversely at a general point of $\Delta_m$. \end{lm} \begin{pf} We introduce the {\em discriminant} of the polynomial $f$ above, viewed as a quadratic polynomial in $y$: $$ \delta = \delta_{\alpha,\beta}(x) = (x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_1x + \alpha_0)^2 - 4(\beta_{m-1}x^{m-1} + \dots + \beta_1x + \beta_0) $$ Note that the map $\delta : \Delta \to V$ to the space $V$ of monic polynomials of degree $2m$ in $x$ with vanishing $x^{2m-1}$ term is an isomorphism of $\Delta$ with a neighborhood of the origin in $V$: given an equation \begin{equation*} \begin{split} (x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_1x + \alpha_0)^2 - 4(\beta_{m-1}x^{m-1} + \dots + \beta_1x + \beta_0) \\ =x^{2m} + c_{2m-2}x^{2m-2} + \dots + c_1x + c_0 \end{split} \end{equation*} we can write $$ \alpha_{m-2} = {c_{2m-2} \over 2} $$ $$ \alpha_{m-3} = {c_{2m-3} \over 2} $$ $$ \alpha_{m-4} = {c_{2m-4} - \alpha_{m-2}^2 \over 2} = {4c_{2m-4} - c_{2m-2}^2 \over 8} $$ and so on, recursively expressing the coefficients $\alpha_i$ as polynomials in the coefficients $c_{2m-2}, \dots,c_m$. We can then solve for the $\beta_i$ in terms of the remaining coefficients $c_{m-1},\dots,c_0$, thus obtaining a polynomial inverse to the map $\delta$. Now, since the equation $f$ above for ${\cal S}$ is quadratic in $y$, the fibers of ${\cal S} \to \Delta$ are expressed as double covers of the $x$-line. The discriminant $\delta$ is a polynomial of degree $2m$ in $x$, so that the general fiber of ${\cal S} \to \Delta$, viewed as a double cover of the $x$-axis, will have $2m$ branch points near $P$. To say that any fiber $S_{\alpha,\beta}$ has $m$ nodes is thus tantamount to saying that $\delta_{\alpha,\beta}(x)$ has $m$ double roots--- that $\delta_{\alpha,\beta}(x)$ is the square of a polynomial of degree $m$. The locus of squares being smooth of dimension $m-1$ in $V$, we see that $\Delta_m$ is smooth of dimension $m-1$; indeed, it is given simply by the vanishing $\beta_0 = \dots = \beta_{m-1} = 0$. Similarly, to say that a fiber $S_{\alpha,\beta}$ has $m-1$ nodes amounts to saying that $\delta_{\alpha,\beta}(x)$ has $m-1$ double roots, i.e., that it can be written as a quadratic polynomial in $x$ times the square of a polynomial of degree $m-1$: $$ \delta _{\alpha ,\beta} (x) = (x^{m-1} + \lambda_{m-2}x^{m-2} + \dots + \lambda_1x + \lambda_0)^2(x^2+\mu_1x+\mu_0). $$ The Lemma is then proved. \end{pf} Now we consider the natural map from a suitable analytic neighborhood $W$ of $[X]$ to $\Delta$. To set this up, let $r_1,\dots, r_k$ be the old nodes of $X$; since all the singularities of $X$ other than $P$ are nodes, this will consist of $b$ nodes on $E$ and $k-b$ nodes lying off $E$ where $b = (D\cdot E)$. Since $m-1$ nodes of the general curve of our family tend to $P$, we have $k=p_a(D)-m+1$. Now consider, in an analytic neighborhood of the point $[X] \in |D|$, the locus $W$ of curves passing through the $r_0(D)-1$ assigned points $q_1,\dots,q_{r_0(D)-1}$ and preserving the nodes $r_1,\dots,r_k$ of $X$---that is, such that the restriction of the family of curves $\{D_\lambda\}_{\lambda \in |D|}$ to $W$ is equisingular at each point $r_i$ of $X$. Since this is a total of $r_0(D)-1 + p_a(D)-m+1=r(D)-m$ points and they impose independent conditions on the linear series $|D|$, we see that $W$ is smooth of dimension $m$ at $[X]$. We then get a natural map $\phi : $W$ \to \Delta$ such that $\phi ([X])=0$. We will prove that $\phi $ is an immersion and that the intersection of $\phi (W)$ with $\Delta _{m-1}$ is the union $\Delta _m$ with a smooth curve $\Psi$; moreover $\Psi$ and $\Delta _m$ will have contact of order $m$ at the origin. This will conclude the proof of Proposition~\ref{describegamma}; in fact the original family $\cal X \to \Gamma$ will be the pullback to $W$ of the restriction to $\Psi$ of the versal deformation ${\cal S} \to \Delta$. To illustrate, here is a representation of the simplest case $m=2$. This does not convey the general picture, because $\phi(W) \cap \Delta_{m-1}$ happens to be proper. Also, the picture is inaccurate in at least one respect: the actual surface $\Delta_1$ in the deformation space of a tacnode is also singular along the locus of curves $S_{\alpha,\beta}$ with cusps. \newpage \vspace*{7.3in} \noindent \special {picture deformation} \ \noindent (Note that we see again locally the picture that we have already observed globally in the linear series $|D|$: the closure of the variety $V(D)$ of irreducible rational curves has the expected dimension; but the locus of rational curves has another component of equal or larger dimension.) \ Here is the outline of our argument. First in \ref{phi} we will establish that $\phi$ is an immersion, and identify in part its tangent space. Second, in \ref{example} we treat a special case. We prove the Proposition by direct calculation when $\phi (W)$ is the linear subspace given by equations $\beta _1=\ldots =\beta _{m-1} =0$. The results of \ref{example} also appear in \cite{R}; we include our proof for the sake of completeness. Then we use the action of the automorphism group of the singularity $(X,P)$ (cf. Lemma~\ref{transitivity}) to deduce the statement for any smooth, $m$-dimensional subvariety of $\Delta$ containing $\Delta _m$ whose tangent plane at the origin is not contained in the hyperplane $\beta_0 =0$ . The proof of our Proposition (and of Proposition~\ref{describegamma} ) is then completed in the remaining part of \ref{end}. \ \subsubsection{The deformations coming from $V(D)$} \label{phi} Let $\phi :W\to \Delta$ be as before, denote by $H$ the subspace of $\Delta $ given by $\beta _0=0$. Then we have \begin{lm} \label{firstsinglemma} The map $\phi$ is an immersion; the tangent space to the image at the origin contains the plane $\beta_0 = \dots = \beta_{m-1} = 0$ but is not contained in $H$. \end{lm} \noindent \underbar{Remark}. It is important to note here, and throughout the following argument, that while the loci $\Delta_m$ and $\Delta_{m-1}$ are well-defined subsets of the base $\Delta$ of the deformation space of our tacnode, $H$ is not; it depends on the choice of coordinates. It is well-defined, however, as a hyperplane in the tangent space $X(\Delta)= \cal O_{\Bbb F_n, P}/{\cal J}$ to $\Delta$ at the origin: it corresponds to the quotient ${\bf m}/{\cal J} \subset \cal O_{\Bbb F_n, P}/{\cal J}$ of the maximal ideal ${\bf m} \subset \cal O_{\Bbb F_n, P}$. \ \begin{pf} The projective tangent space to $W$ at the point $[X]$ is the sublinear series of $|D|$ of curves passing through the points $r_1,\dots,r_k$ and $q_1,\dots,q_{r_0(D)-1}$. The kernel of the differential at $[X]$ of the map $\phi$ is thus the vector space of sections of the line bundle ${\cal L} = \cal O_{\Bbb F_n}(D)$ vanishing at $r_1,\dots,r_k$ and $q_1,\dots,q_{r_0(D)-1}$ and lying in the subsheaf ${\cal L} \otimes {\cal J}$, where ${\cal J} \subset \cal O_{\Bbb F_n,P}$ is as before the Jacobian ideal of $[X]$ at $P$. The zero locus of such a section will be a curve in the linear series $|D|$ containing $r_1,\dots,r_k, q_1,\dots,q_{r_0(D)-1}$ and $P$ and so must contain $E$, that is, must be of the form $E+G$ with $G \in |D-E|$. Moreover, from the description above of ${\cal J}$ we see that $G$ must also have contact of order at least $m$ with $E$ at $P$ as well as pass through the $k-b$ nodes of $X$ lying off $E$ and the assigned points $p_1,p_2,q_3,\dots,q_{r_0(D)-1}$. This represents a total of \begin{equation*} \begin{split} m + r_0(D)-1 + p_a(D)-m+1-b&=r(D)-b \\ &= r(D-E)+1 \end{split} \end{equation*} conditions, so we need to show that they are independent to conclude that no such curve exists. But they are also a subset of the adjoint conditions of $X$, hence impose independent conditions on the series $|D+K_{\Bbb F_n}| = |D-C-E-2F|$, and hence on the series $|D-E|$. The remaining statements of the lemma, that the tangent space to the image contains the plane $\beta_0 = \dots = \beta_{m-1} = 0$ but is not contained in the hyperplane $\beta_0 = 0$, follow from the facts that the image contains the subvariety $\Delta_m$ and that not every curve in the linear series $|D|$ containing $r_1,\dots,r_k, q_1,\dots,q_{r_0(D)-1}$ contains $P$ \end{pf} \ \subsubsection{A special case} \label{example} Next, having seen that $\phi(W)$ is a smooth, $m$-dimensional variety of $\Delta$ we will consider the intersection of $\Delta_{m-1}$ with the simplest possible space satisfying the statement of the previous Lemma, the plane $\Lambda$ given by $\beta_1 = \dots = \beta_{m-1} = 0$. We obtain \begin{lm}\label{linearcase} The intersection of $\Delta_{m-1}$ with $\Lambda$ consists of the union of $\Delta _m$ with multiplicity $m$ and a smooth curve $\Psi$ having contact of order $m$ with $\Delta _m$ at the origin. \end{lm} \begin{pf} Restricting to $\Lambda$, we can rewrite the equation of the family more simply as $$ y^2+yx^m + \alpha_0y + \alpha_1xy + \dots + \alpha_{m-2}x^{m-2}y + \beta = 0 $$ and the discriminant as $$ \delta(x) = (x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_1x + \alpha_0)^2 - 4\beta $$ We need now to express the condition that $\delta$ has $m-1$ double roots. One obviously sufficient condition is that $\beta=0$, so that $\delta$ is a square. If we assume $\beta \ne 0$, however, things get more interesting. To see the locus of $(\alpha_0,\dots,\alpha_{m-2},\beta)$ that satisfy this condition, set $$ \nu(x) = x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_1x + \alpha_0 $$ and write $$ \delta(x) = \nu(x)^2-4\beta = (\nu(x)+2\sqrt{\beta})\dot(\nu(x)-2\sqrt{\beta}) . $$ Now, if $\beta \ne 0$, the two factors in this last expression have no common factors; so if their product has $m-1$ double roots, each must have a number of double roots itself: $\nu(x)+2\sqrt{\beta}$ and $\nu(x)-2\sqrt{\beta}$ are polynomials of degree $m$ with a combined total of $m-1$ double roots. In fact, this uniquely characterizes $\nu$ and $\beta$ up to a one-parameter group of automorphisms of $\Bbb P^1$, as we will prove in the following. \begin{lm} \label{sublemma} Let $\gamma$ be a nonzero scalar, and let $m$ be a positive integer. There is a polynomial $\nu(x)$ of degree $m$, monic with no $x^{m-1}$ term, such that 1. if $m$ is odd, the polynomials $\nu(x)+\gamma$ and $\nu(x)-\gamma$ each have $(m-1)/2$ double roots; $\nu(x)$ is unique up to replacing $\nu(x)$ by $-\nu(-x)$; 2. if $m$ is even, $\nu(x)+\gamma$ has $m/2$ double roots and $\nu(x)-\gamma$ has $(m-2)/2$ double roots; in this case $\nu$ is unique. \end{lm} \begin{pf} Suppose that $\nu(x)$ is a polynomial satisfying the conditions of the lemma. Take first the case of $m = 2\ell+1$ odd, and consider the map $\nu : \Bbb P^1 \to \Bbb P^1$ given by $\nu(x)$. This is a map of degree $m$, sending the point $\infty$ to $\infty$, and totally ramified there. In addition the hypotheses assert that over the points $\pm\gamma$ in the target we have $\ell$ ramification points. The point is, this accounts for a total of $(m-1) + 2(\ell-1) = 2m-2$ ramification points, and these are all a map of degree $m$ from $\Bbb P^1$ to $\Bbb P^1$ will have. We have thus specified the covering $\nu$ up to a finite number of coverings, and our principal claim is that in fact we have described $\nu$ uniquely, up to automorphisms of the domain. This is combinatorial. The monodromy permutation $\sigma$ around the point $\infty$ is cyclic, while the the monodromy permutations $\tau$ and $\mu$ around $\gamma$ and $-\gamma$ are each products of $\ell$ disjoint transpositions. Our claim that there is a unique such covering of $\Bbb P^1$ by $\Bbb P^1$ amounts then to the assertion that, up to the action of the symmetric group $\frak{S}_m$ by conjugation, there is a unique pair of permutations $\tau$ and $\mu$, each a product of $\ell$ disjoint transpositions, whose product $\tau \circ \mu$ is cyclic of order $m$. To see this, start with the unique element of the set on which $\tau$ and $\mu$ act that is fixed by $\tau$ , and label it 1. This element cannot also be fixed by $\mu$; give the element exchanged with it by $\mu$ the label 2, and let the element exchanged with 2 by $\tau$ be labelled 3. This also cannot be the fixed point of $\mu$, or else the subset $\{1,2,3\}$ would be fixed by both $\tau$ and $\mu$; let 4 be the element exchanged with it by $\mu$ and 5 the element exchanged with 4 by $\tau$. We can continue in this way until we have exhausted all the elements of the set; and so we see that we can label the elements of the set $\{1,2,\dots,m\}$ so that $$ \mu = (1,2)(3,4)\dots(2l-1,2l) $$ and $$ \tau = (2,3)(4,5)\dots(2l,2l+1) $$ This establishes the uniqueness of the covering $\nu$ up to automorphisms of the domain in case $m$ is odd. The case of $m = 2\ell$ even is similar; we see that we can always label the sheets of $\nu$ so that the monodromy permuations $\tau$ and $\mu$ around $\gamma$ and $-\gamma$ have the form $$ \mu = (1,2)(3,4)\dots(2l-1,2l) $$ and $$ \tau = (2,3)(4,5)\dots(2l-2,2l-1) $$ To complete the proof of Lemma \ref{sublemma}, consider the effect on $\nu$ of automorphisms of the domain. The requirement that $\nu(\infty) = \infty$---that is, that $\nu(x)$ is a polynomial!---restricts us to the group of automorphisms $x \mapsto ax+b$; the requirement that $\nu(x)$ have no $x^{m-1}$ term limits us to automorphisms of the form $x \mapsto ax$; and the fact that $\nu(x)$ is monic says that $a$ must be an $m$-th root of unity. Finally, the fact that the ramification points map to $\pm\gamma$ determines $\nu(x)$ completely in case $m$ is even (where the two branch points $\pm\gamma$ have different multiplicity), and up to the automorphism $x \mapsto -x$ in case $m$ is odd. \end{pf} \ Back to the proof of Lemma \ref{linearcase}. Note first that, by uniqueness, $\nu(x)$ will be even when $m$ is even and odd when $m$ is odd. Note also that if we do not specify the value of $\gamma$ the polynomial $\nu(x)$ will not be unique; we can replace it with $u^m\nu(x/u)$ for any nonzero scalar $u$. Now, suppose first that $m = 2\ell$ is even. Choose $\gamma = 1$, and let $$ \nu(x) = x^m + c_{m-2}x^{m-2} + c_{m-4}x^{m-4} + \dots +c_0 $$ be the polynomial satisfying the conditions of the lemma. Then any collection $(\alpha_0, \alpha_1, \dots,\alpha_{m-2},\beta)$ with $\beta \ne 0$ such that the discriminant $$ \delta(x) = (x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_1x + \alpha_0)^2 - 4\beta $$ has $m-1$ double roots must be of the form \begin{equation*} \begin{split} \alpha_0&= t^\ell \cdot c_0 \\ \alpha_1 &= 0 \\ \alpha_2 &= t^{\ell-1} \cdot c_2 \\ \alpha_3 &= 0 \\ \alpha_4 &= t^{\ell-2} \cdot c_4 \end{split} \end{equation*} and so on, ending with $$ \alpha_{m-2} = t \cdot c_{m-2}; $$ with finally $$ \beta = {t^m \over 4}. $$ This is then a parametric representation of the closure $\Psi$ of the intersection $\Lambda \cap (\Delta_{m-1} \setminus \Delta_m)$. It is obviously a curve; the fact that it is smooth is visible from the coordinate $ \alpha_{m-2} = t \cdot c_{m-2}$; and we see that it has contact of order $m$ with $H$ from the exponent in the expression for $\beta$. Finally, in case $m = 2\ell+1$ is even we get a similar expression. Let $$ \nu(x) = x^m + c_{m-2}x^{m-2} + c_{m-4}x^{m-4} + \dots +c_1x $$ be the polynomial satisfying the conditions of the lemma for $\gamma = 1$. Then any collection $(\alpha_0, \alpha_1, \dots,\alpha_{m-2},\beta)$ with $\beta \ne 0$ such that the discriminant $$ \delta(x) = (x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_1x + \alpha_0)^2 - 4\beta $$ has $m-1$ double roots must be of the form \begin{equation*} \begin{split} \alpha_0 &= 0 \\ \alpha_1 &= t^\ell \cdot c_1 \\ \alpha_2 &= 0 \\ \alpha_3 &= t^{\ell-1} \cdot c_3 \end{split} \end{equation*} and so on, ending with $$ \alpha_{m-2} = t \cdot c_{m-2}; $$ again we have $$ \beta = {t^m \over 4}. $$ So once more we see that $\Psi$ is a smooth curve having contact of order $m$ with $H$ at the origin. \end{pf} \ Let us now prove Proposition \ref{singtotalspace} in this special case. First, in the case $m = 2\ell$ even, the restriction ${\cal S}_\Psi \to \Psi$ of the family ${\cal S} \to \Delta$ to $\Psi$ has equation $$ y^2 + y(x^m + tc_{m-2}x^{m-2} + t^2 c_{m-4}x^{m-4} + \dots + t^{m/2}c_0) + {t^m \over 4}=0 $$ We can think of the total space ${\cal S}_\Psi$ of this family as a double cover of the $(x,t)$-plane, with branch divisor the zero locus of the discriminant $$ \delta = (x^m + tc_{m-2}x^{m-2} + t^2 c_{m-4}x^{m-4} + \dots + t^{m/2}c_0)^2 - t^m $$ By hypothesis, for each value of $t$ the polynomial $\delta$ is the product of the square of a polynomial $g_t(x)$ of degree $m-1$ and a quadratic polynomial $h_t(x)$. Since $\delta $ is even, $g^2$ and $h$ must each be; and given the homogeneity of $\delta $ with respect to $t$ and $x$ we see that we can write $$ \delta = x^2(x^2-\lambda_1t)^2(x^2-\lambda_2t)^2 \dots (x^2-\lambda_{\ell-1}t)^2 \cdot (x^2-\mu t) $$ for suitable constants $\lambda_1,\dots,\lambda_{\ell-1}$ and $\mu$. For example, in case $m=2$, the equation of ${\cal S}_\Psi$ is simply $$ y^2 + y(x^2 + t) + {t^2 \over 4}=0 $$ and the discriminant is just $\delta = x^2(x^2-2t)$. In general, the branch divisor of ${\cal S}_\Psi$ over the $(x,t)$-plane will be simply a union of the $t$-axis, with multiplicity 2; $\ell-1$ parabolas tangent to the $x$-axis at the origin, each with multiplicity 2; and one more parabola tangent to the $x$-axis at the origin and appearing with multiplicity 1. The double cover ${\cal S}_\Psi$ will thus be nodal over the double components of this branch divisor, and smooth elsewhere. Finally, the normalization ${\cal S}_\Psi^\nu$ of the total space ${\cal S}_\Psi$ will be the double cover of the $(x,t)$-plane branched over the single component of multiplicity 1 in the branch divisor; that is, it will have equation $$ y^2 = x^2-\mu t $$ and in particular, since the component $(x^2-\mu t)$ is smooth, ${\cal S}_\Psi^\nu$ will be smooth as well, establishing Proposition \ref{singtotalspace} for this particular family. The picture in case $m = 2\ell+1$ is odd is exactly the same: here ${\cal S}_\Psi$ has equation $$ y^2 + y(x^m + tc_{m-2}x^{m-2} + t^2 c_{m-4}x^{m-4} + \dots + t^{m/2}c_1x) - {t^m \over 4}=0 $$ with discriminant \begin{equation*} \begin{split} \delta &= (x^m + tc_{m-2}x^{m-2} + t^2 c_{m-4}x^{m-4} + \dots + t^{m/2}c_1x)^2 - t^m \\ &= (x^2-\lambda_1t)^2(x^2-\lambda_2t)^2 \dots (x^2-\lambda_{\ell}t)^2 \cdot (x^2-\mu t) \end{split} \end{equation*} for suitable constants $\lambda_1,\dots,\lambda_{\ell}$ and $\mu$. For example, in case $m=3$, the equation of ${\cal S}_\Psi$ will be $$ y^2 + y(x^3 -3tx) - t^3=0 $$ (we are scaling $t$ here to make the coefficients nicer), and the discriminant is just \begin{equation*} \begin{split} \delta &= (x^3 -3tx)^2 +4t^3\\ &=x^6 -6tx^4+9t^2x^2+4t^3 \\ &=(x^2+t)^2(x^2+4t) \end{split} \end{equation*} In general, for $m$ odd the branch divisor of ${\cal S}_\Psi$ over the $(x,t)$-plane will be simply a union of $\ell$ parabolas tangent to the $x$-axis at the origin, each with multiplicity 2; and one more parabola tangent to the $x$-axis at the origin and appearing with multiplicity 1. As before, the normalization ${\cal S}_\Psi^\nu$ of the total space ${\cal S}_\Psi$ will be simply the double cover of the $(x,t)$-plane branched over the single component $(x^2-\mu t)$ of multiplicity 1 in the branch divisor; and as before, since this component is smooth, ${\cal S}_\Psi^\nu$ will be smooth as well, establishing Proposition \ref{singtotalspace} in this case. \end{pf} \subsubsection{The geometry of the locus $\Delta_{m-1}$} \label{end} In order to focus on the essential aspects of the geometry of $\Delta_{m-1}$, and in particular to remove the excess intersection of $\phi(W) \cap \Delta_{m-1}$, we will work on the blow-up $\tau : \tilde\Delta = {\operatorname {Bl}}_{\Delta_m}\Delta \to \Delta$ of $\Delta$ along $\Delta_m$. To express our results, we have to introduce some notation. We will denote by $Z = \tau^{-1}(\Delta_m)$ the exceptional divisor of the blow up, and by $\tilde\Delta_{m-1}$ and $\tilde W$ the proper transforms of $\Delta_{m-1}$ and $\phi(W)$ in $\tilde\Delta$. Our goal will be to describe the intersection $Z_{m-1} : = \tilde\Delta_{m-1} \cap Z$. The fibers of $Z$ over $\Delta_m$ are projective spaces $\Bbb P^{m-1}$with homogeneous coordinates $\beta_0,\dots,\beta_{m-1}$; we will denote the fiber $\tau^{-1}(0)$ of $Z$ over the origin by $\Phi$, by $\Phi_0 \subset \Phi$ the open set given by $\beta_0 \ne 0$, and by $Q$ the point of $\Phi$ with coordinates $[1,0,\dots,0]$ (this is the point of intersection of $\tilde W$ with $\Phi$ in the example above). Note that there is a more intrinsic characterization of the $\Phi$: the tangent space to $\Delta_m$ at the origin is the subspace of $\cal O_{\Bbb F_n, P}/{\cal J}$ of polynomials divisible by $y$, so that $\Phi$---the projectivization of the normal space---is just the space of polynomials in $x$ modulo those vanishing to order $m$ at $P = (0,0)$ and modulo scalars. In these terms, $\Phi_0$ is simply the subspace of polynomials not vanishing at the origin and $Q$ the point corresponding to constants. To study $\Delta_{m-1}$ we use the action of the automorphism group of the deformation space ${\cal S} \to \Delta$. We have many automorphisms of the germ of the singularity $(X, P)$: for example, for any power series $$c(x) = c_1x + c_2x^2 + c_3x^3 + \dots$$ with $c_1 \ne 0$ we can define an automorphism of the germ by \begin{equation*} \begin{split} \gamma_c : (x,y) &\mapsto \bigl(c(x), {c(x)^m \over x^m} \cdot y) \\ &=\bigl(c_1x + c_2x^2 +\dots, (c_1 + c_2x + c_3x^2 + \dots)^m \cdot y \bigr) \\ &=\bigl(c_1x + c_2x^2 +\dots, c_1^my + mc_1^{m-1}c_2xy + (mc_1^{m-1}c_3 + {m(m-1) \over 2}c_1^{m-2}c_2^2)x^2y + \dots \bigr). \end{split} \end{equation*} Let $G$ be the group of automorphisms of the germ $(X, P)$. By the naturality of the versal deformation space, $G$ acts as well on it, that is, $G$ acts equivariantly on $\cal S$ and $\Delta$. Since the action on $\Delta$ preserves the subvariety $\Delta_m$, it lifts to an action on the blow-up $\tilde\Delta$; and since the action on $\Delta$ preserves $\Delta_{m-1}$ the lifted action will preserve $\tilde\Delta_{m-1}$. We can read off from the above expression the action of the automorphism $\gamma_c$ on the tangent space to $\Delta$ at the origin, and thereby on the fiber $\Phi$ of $Z$ over the origin: taking as basis for $X(\Delta) = \cal O_{\Bbb F_n, P}/{\cal J}$ the monomials $1, x, x^2,\dots,x^{m-1}$ and $y, xy, x^2y,\dots,x^{m-2}y$, we can express the relevant part of this action as \begin{equation*} \begin{split} 1 &\mapsto 1 \\ x &\mapsto c_1x + c_2x^2 + \dots + c_{m-1}x^{m-1} \\ x^2 &\mapsto c_1^2x^2 + 2c_1c_2x^3 + \dots \\ &\vdots \\ x^{m-1} &\mapsto c_1^{m-1}x^{m-1} \end{split} \end{equation*} The monomials $x^ky$ are carried into linear combinations of other such monomials; the exact linear combinations will not concern us. The key fact about this action, for our present purposes, follows immediately from the description above: \begin{lm}\label{transitivity} Every orbit of the action of $G$ on $\tilde\Delta$ that intersects $\Phi_0$ contains the point $Q$ in its closure. \end{lm} We are now prepared to state and prove our main lemma on the geometry of $Z_{m-1}$ and $\Delta_{m-1}$. \begin{lm}\label{Zdescription} 1. The fibers of $Z_{m-1}$ over $\Delta_m$ are unions of linear spaces. 2. For any arc $\alpha(t)$ in $\Delta_m$ tending to the origin, the limiting position of the fiber $Z_{\alpha(t)}$ of $Z_{m-1}$ over $\alpha(t)$ is contained in the complement of $\Phi _0$. 3. $\Phi$ itself is contained in (and hence an irreducible component of) $Z_{m-1}$. \end{lm} \begin{pf} The proof is by induction on $m$, using Lemma \ref{linearcase}. First we introduce a natural stratification of the locus $\Delta_m$. Identifying $\Delta_m$ with the space of monic polynomials of degree $m$ in $x$ with no $x^{m-1}$ term, we look at the loci of polynomials with roots of given multiplicity: for any partition $m = m_1 + m_2 + \dots + m_k$ we define the locus $ \Delta\{m_1,\dots,m_k\} \subset \Delta$ by \begin{equation*} \begin{split} \Delta\{m_1&,\dots,m_k\} : = \bigl\{(\alpha_0,\dots,\alpha_{m-2},0,\dots,0) : \\ &x^m + \alpha_{m-2}x^{m-2} + \dots + \alpha_0 = (x-\lambda_1)^{m_1}(x-\lambda_2)^{m_2}\dots(x-\lambda_k)^{m_k} \\ &\qquad \qquad \qquad \text{for some distinct} \; \lambda_1,\dots,\lambda_k \bigr\} \end{split} \end{equation*} Note that the codimension of $\Delta\{m_1,\dots,m_k\}$ in $\Delta_m$ is $\sum (m_\alpha-1)$. Suppose $\alpha$ is any point of $\Delta_m$ other than the origin. Say $\alpha$ lies in the stratum $\Delta\{m_1,\dots,m_k\}$, and write the corresponding polynomial as $$ (x-\lambda_1)^{m_1}(x-\lambda_2)^{m_2}\dots(x-\lambda_k)^{m_k} $$ with $\lambda_1,\dots,\lambda_k$ distinct. The fiber $S_\alpha$ of ${\cal S} \to \Delta$ over $\alpha$ is a reducible curve consisting of two branches, the $x$-axis $(y=0)$ and the curve $y = (x-\lambda_1)^{m_1}(x-\lambda_2)^{m_2}\dots(x-\lambda_k)^{m_k}$, which meet at the $k$ points $r_1=(\lambda_1,0),\dots,r_k=(\lambda_k,0)$ with multiplicities $m_1,\dots,m_k$. Let $\Delta(i)$ be the versal deformation spaces $ \Delta(S_\alpha, r_i)$ of the singular points $r_i \in S_\alpha$. By the openness of versality the natural map $\sigma$ from a neighborhood $U$ of $\alpha$ in $\Delta$ to the product $\prod \Delta(i)$ has surjective differential at $\alpha$ (the fibers are the equisingular deformations of $S_\alpha$, in which only the locations of the points $r_i$ on the $x$-axis vary). Let $\Delta_{m_i-1}$ and $\Delta_{m_i} \subset \Delta(i)$ be the loci in $\Delta(i)$ analogous to $\Delta_{m-1}$ and $\Delta_{m}$ in $\Delta$, that is, the closures of the loci of deformations of the singular points $r_i \in S_\alpha$ with $m_i-1$ and $m_i$ nodes near $r_i$ respectively. Then in the neighborhood $U$ of $\alpha$, we have $$ \Delta_m = \sigma^{-1}\left(\Delta_{m_1} \times \Delta_{m_2} \times \dots \times \Delta_{m_k}\right) $$ and $$ \Delta_{m-1} = \bigcup_{i=1}^k\sigma^{-1}\left(\Delta_{m_1} \times \dots \times \Delta_{m_i-1} \times \dots \times \Delta_{m_k}\right) $$ In other words, the locus $\Delta_{m-1}$ will have $k$ branches in a neighborhood of $\alpha$, each containing $\Delta_m$, along the $i$th of which the fibers of ${\cal S} \to \Delta$ will have $m_j$ nodes tending to $r_j$ for each $j \ne i$ and $m_i-1$ nodes tending to $r_i$. We can use this description to give a more intrinsic characterization of the fiber $Z_\alpha = \tau^{-1}(\alpha)$ of $Z$ over the point $\alpha$, analogous to the one given above for $\Phi$. Briefly, $Z_\alpha$ is the projectivization of the normal space to $\Delta_m$ in $\Delta$ at $\alpha$, which is the product of the normal spaces to the $\Delta_{m_i}$ in $\Delta(i)$ at the origin; this is just the space of polynomials on the $x$-axis modulo those vanishing to order $m_i$ at $r_i$ for each $i$. We may now apply the induction hypothesis to describe, in these terms, the fiber of $Z_{m-1}$over $\alpha$. By the statement of the Lemma for $m = m_i$, the proper transform of the $i$th branch of $\Delta_{m-1}$ will intersect $Z_\alpha$ in the linear subspace of $Z_\alpha$ corresponding to polynomials vanishing to order $m_j$ at $r_j$ for each $j \ne i$; the intersection with $Z_\alpha$ with the proper transform of $\Delta_{m-1}$ itself will be the union of these linear subspaces. This establishes part (1) of the Lemma. Now say that $\alpha(t)$ is any arc in $\Delta_m$ tending to the origin; $\alpha(t)$ will lie in some stratum $\Delta\{m_1,\dots,m_k\}$ for all small $t \ne 0$. As $t$ goes to zero, the singular points $r_i(t)$ of $S_{\alpha(t)}$ approach the point $P$, so that the limiting position of the intersection with $Z_{\alpha(t)}$ of the proper transform of the $i$th branch of $\Delta_{m-1}$ will be simply the linear space of polynomials whose restriction to the $x$-axis vanishes to order $m-m_i$ at $P$; in particular, it is contained in the hyperplane $(\beta_0 = 0) \subset \Phi$ of polynomials vanishing at $P$. We have thus proved parts (1) and (2) of the Lemma, given part (3) for all $m_i < m$. \ Finally, we need to prove for each new value of $m$ that $\Phi$ is contained in (and hence an irreducible component of) $Z_{m-1}$. Now, by Lemma \ref{linearcase}, the point $Q = [1,0,\dots,0] \in \Phi$ lies in $Z_{m-1}$. But we have completely described the closure in $Z_{m-1}$ of the inverse image $\tau^{-1}(\Delta_m \setminus \{0\})$ of the complement of the origin, and $Q$ is not on it. $Q$ must thus lie on an irreducible component of $Z_{m-1}$ not meeting $\tau^{-1}(\Delta_m \setminus \{0\})$, that is to say, an irreducible component of $Z_{m-1}$ contained in $\Phi$; since $Z_{m-1}$ has pure dimension $m-1$, this irreducible component must be $\Phi$ itself. \end{pf} \ For example, here is a picture of $Z_1$ in the case $m=2$. In this case $Z_1$ has only two components, $\Phi$ and a component finite of degree $2$ over $\Delta_2$. \vspace*{3.5in} \noindent \hskip.4in \special {picture blownupdeformation} \ Next, we deduce: \begin{lm} \label{multiplicitym} 1. $\tilde\Delta_{m-1}$ is smooth everywhere along $\Phi_0$ \noindent 2. The intersection multiplicity of $\tilde\Delta_{m-1}$ and $ Z$ along $\Phi$ is $m$. \end{lm} \begin{pf} We use the analysis carried out in Lemma \ref{linearcase}. Let $\tilde\Lambda$ be the proper transform of the linear space $\Lambda$ in $\tilde\Delta$. Since no component of $Z_{m-1}$ other than $\Phi$ passes through $Q$, the only component of the intersection $\tilde\Lambda \cap \tilde\Delta_{m-1}$ containing $Q$ will be the proper transform $\tilde\Psi$ of the curve $\Psi \subset \Delta$ described in Lemma \ref{linearcase}. Since this is smooth, and the intersection $\tilde\Lambda \cap \tilde\Delta_{m-1}$ is proper in a neighborhood of $Q$ ($\tilde\Lambda$ and $ \tilde\Delta_{m-1}$ each have dimension $m$ in the $(2m-1)$-dimensional $\tilde\Delta$, and their intersection is locally a curve) it follows that $ \tilde\Delta_{m-1}$ must be smooth at $Q$. By Lemma \ref{transitivity}, then, it must be smooth at every point of $\Phi_0$. For the second statement, notice that Lemma \ref{linearcase} asserts that this is true when restricted to the proper transform $\tilde\Lambda$, and it follows that it is true on $\tilde\Delta_{m-1}$ \end{pf} \noindent { \it End of the proof of Proposition~\ref{singtotalspace}}. We shall now conclude that the intersection of $\phi (W)$ with $\Delta _{m-1}$ is the union of $\Delta _m$ and a smooth curve $\Psi$, such that $\Psi $ has contact of order $m$ with $\Delta _m$ at the origin. Notice that this will conclude the proof of Proposition~\ref{describegamma} as well. We know from Lemma \ref{firstsinglemma} that $\phi(W)$ is smooth, so that its proper transform $\tilde W$ intersects $Z$ in a section, crossing $\Phi$ at some point $R$; we likewise have from Lemma \ref{firstsinglemma} that $R \in \Phi_0$. $ \tilde\Delta_{m-1}$ is then smooth at $R$. Since the tangent space to $ \tilde\Delta_{m-1}$ at $R$ contains the tangent space to $\Phi$ and the tangent space to $\tilde W$ at $R$ is complementary to the tangent space to $\Phi$, $ \tilde\Delta_{m-1}$ and $\tilde W$ intersect transversely in a smooth curve in a neighborhood of $R$; since that curve is not tangent to $\Phi$ at $R$, its image $\Phi \subset \Delta_{m-1} \cap \phi(W)$ is again a smooth curve. Finally, the intersection number of $\Psi$ with $\Delta_m$ in $\phi(W)$ will be the intersection number of $ \tilde\Delta_{m-1}$, $\tilde W$ and $Z$ at $R$; which by Lemma \ref{multiplicitym} will be $m$. We have thus completed the proof of Proposition \ref{singtotalspace}. Now, the inverse image of $\Psi$ in $W$ is an analytic neighborhood of $\Gamma$; therefore to conclude the proof we need to show that the total space $\cal S _{\Psi }$ of the versal deformation over $\Psi$ is smooth at the point corresponding to $p$. This follows as in the end of \ref{example}. \section{Formulas} Before we prove our formulas, we need a simple result on the order of zeroes and poles of the cross-ratio function $\phi$. \subsection{A remark on the cross-ratio function} Suppose we are given a family $ f : \cal X \to B$ over a smooth one-dimensional base $B$, whose restriction $\tilde f : \tilde {\cal X }= f^{-1}(\tilde B) \to \tilde B$ to the complement $\tilde B = B \setminus \{b_0\}$ of a point $b_0 \in B$ is a family of smooth rational curves; and four sections $p_i : \tilde B \to \cal X$, disjoint over $\tilde B$. We get a map $ \tilde \phi : \tilde B \to \overline{M}_{0,4}$, which then extends over $B$; and the problem is to determine the coefficient of the point $b$ in the pullback via $\tilde \phi$ of the boundary components of $\overline{M}_{0,4}$. To put it another way, the cross-ratio of the four sections $p_1,p_3,p_2,p_4$ defines a rational function on $\tilde B$ and hence on $B$; and we ask simply for the order of zero or pole of this function at $b_0$. We will answer this in terms of any completion of our family to a family of nodal rational curves. Recall first of all the set-up of section~\ref{setup}: we have a resolution of singularities $\cal Y \to B$ of the total space of our family, such that $\cal Y \to B$ is a family of nodal curves and the extensions of the sections $p_i$ to $\cal Y$ are disjoint. We then proceed to blow down ``extraneous" components of $Y$ to arrive at the minimal smooth semistable model of our family: that is, a family ${\cal Z} \to B$ such that ${\cal Z}$ is smooth, the fibers $Z_b$ are nodal, the sections $p_i$ are disjoint and ${\cal Z} \to B$ is minimal with respect to these properties. Finally, we blow down the intermediate components in this chain to arrive at a family ${\cal W} \to B$ of 4-pointed stable curves. The special fiber $W$ of this family will have just two components (or one, if $\ell = 0$), with a singularity of type $A_\ell$ at the point of their intersection. In these terms we prove \begin{lm}\label{crossratiomult} If the sections $p_1$ and $p_2$ (respectively, $p_1$ and $p_3$) meet the same component of $Y$, then the point $b_0$ is a zero (respectively, pole) of multiplicity $\ell$ of the function $\phi$. \end{lm} \begin{pf} We will consider the case where $p_1$ and $p_2$ meet the same component of $W$. Note first that if we blow down the component of $W$ meeting $p_1$ and $p_2$, we arrive at a smooth family, that is (replacing $B$ if necessary by a neighborhood of $b_0$ in $B$), a product $B \times \Bbb P^1$. (Equivalently, we could arrive at this family by blowing down the component of $Z$ meeting $p_1$ and $p_2$, then doing the same thing on the resulting surface, and so on $\ell$ times.) $p_3$ and $p_4$ will remain disjoint from each other in this process, and disjoint from $p_1$ and $p_2$; but $p_1$ and $p_2$ will meet each other with contact of order $\ell$: in other words, we can choose an affine coordinate $z$ on $\Bbb P^1$ and a local coordinate $t$ on $B$ centered around $b_0$ so that the sections $p_i$ are given by $$ p_1(t) = t^\ell; \quad p_2(t) \equiv 0; \quad p_3(t) \equiv 1; \quad {\rm and} \quad p_4(t) \equiv \infty . $$ The cross-ratio function is then $\phi(t) = 1-t^\ell$, which takes on the value 0 with multiplicity $\ell$ at $t=1$ \end{pf} \ \subsection{The recursion for $\Bbb F_2$} Let $D$ be any effective divisor class other than $E$ on the ruled surface $S = \Bbb F_2$. We are going to find a formula for the degree $N(D)$ of the variety $V(D) \subset |D|$. To set this up, we start by choosing as usual $r_0(D) - 1$ general points on $S$, which we label $p_1, p_2, q_3,\dots,q_{r_0(D) - 1}$, and consider the one-parameter family $\cal X \to \Gamma$ of curves $X \in V(D) \subset |D|$ passing through $\{p_1, p_2, q_3,\dots,q_{r_0(D) - 1}\}$. As before, we let $ \Gamma^\nu$ be the normalization of $\Gamma$ and $\cal X^\nu \to \Gamma^\nu$ the normalization of the pullback family. Next, we fix general curves $C_3$ and $C_4 \in |C|$ in the linear series $|C|$, and adopt as usual the convention that we will choose points $p_3$ and $p_4$ on the curves $X$ of our family lying on $C_3$ and $C_4$ respectively. Making the corresponding base change, we arrive at a family $\cal X \to B$; as before, we will denote by $\cal Y$ the minimal desingularization of $\cal X$ and by ${\cal Z} \to B$ the smooth semistable model. Then we calculate the degree of the cross-ratio map $\phi : B \to \overline{M}_{0,4} \cong \Bbb P^1$ in two ways by equating the number of zeroes and poles of $\phi$. We get one contribution to the degree of $\phi^*(0)$ immediately from the curves $X$ in our family that happen to pass through either of the two points of intersection of $C_3$ with $C_4$; this gives a total contribution of $2 \cdot N(D)$ to the degree of $\phi^*(0)$. The remaining zeroes and poles of $\phi$ necessarily correspond to reducible curves in the family $\{X\}$. There are two types of these: those that contain $E$ and those that don't. Consider first a reducible curve $X$ in our family that does not contain $E$. By Proposition \ref{codimensionone}, this must be of the form $X = X_1 + X_2$ where $X_i$ is a general member of the family $V(D_i)$ for some pair of divisor classes $D_1$ and $D_2$ adding up to $D$. In particular, $X_i$ is an irreducible rational curve with $p_a(D_i)$ nodes, and $X_1$ and $X_2$ intersect transversely in $(D_1 \cdot D_2)$ points. Moreover, by Proposition \ref{describegamma}, the curve $\Gamma$ will consist of $(D_1 \cdot D_2)$ smooth branches near the point $[X]$, corresponding to the points of intersection of $X_1$ and $X_2$; thus there are $(D_1 \cdot D_2)$ points in the normalization $\Gamma^\nu$ lying over each such point $[X] \in \Gamma$. How does such a fiber of the family $\cal X \to B$ contribute to the degrees of either $\phi^*(0)$ or $\phi^*(\infty)$? It depends on how the points $p_i$ are distributed. If three or four lie on one component, it does not contribute to either, but if there are two on each it may: for example, if $p_1$ and $p_2$ lie on the same component---say $X_1$---of $X$, and $p_3$ and $p_4$ on the other, we get a zero of $\phi$. Now, as we observed in the proof of Proposition \ref{codimensionone}, each component $X_i$ of $X$ must contain exactly $r_0(D_i)$ of the points $p_1, p_2, q_3,\dots,q_{r_0(D) - 1}$. If $X_1$ is to contain $p_1$ and $p_2$, it will contain $r_0(D_1)-2$ of the points $q_\alpha$, and $X_2$ will contain the remaining $r_0(D)-r_0(D_1)+1 = r_0(D_2)$. Thus, to specify such a fiber, we have first to break the $r_0(D)-3$ points $q_\alpha$ into disjoint sets of $r_0(D_1)-2$ and $r_0(D_2)$. The curve $X_1$ can then be any of the $N(D_1)$ irreducible rational curves in the linear series $|D_1|$ passing through $p_1$, $p_2$ and the first set, while $X_2$ can then be any of the $N(D_2)$ irreducible rational curves in the linear series $|D_2|$ passing through the second set. Altogether, then, we see that there will be $$N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-2}$$ points in $\Gamma$ of this type, and correspondingly $$N(D_1)N(D_2) (D_1 \cdot D_2) \binom{r_0(D)-3}{r_0(D_1)-2}$$ such points in the normalization $\Gamma^\nu$. Finally, if a fiber of $\cal X \to B$ lying over such a point of $\Gamma^\nu$ is to contribute to $\phi^*(0)$, we have to choose $p_3$ and $p_4$ to lie on $X_2$, that is, to be any of the $(D_2 \cdot C)$ points of intersection of $X_2$ with $C_3$ and $C_4$ respectively. There are thus a total of $(D_2 \cdot C)^2$ fibers of $\cal X \to B$ of this type lying over each such point of $\Gamma^\nu$. To complete the calculation of the contribution of fibers of this type to the degree of $\phi^*(0)$, we observe that the fiber of the normalization $\cal X^\nu$ over such a point will have two components, the normalizations of the curves $X_i$, meeting at one point (the point of each lying over the new node). Moreover, by Proposition\ref{singtotalspace}, the total space $\cal X^\nu$ will be smooth at such a point; and it follows by Lemma \ref{crossratiomult} that the corresponding point of $B$ will be a simple zero of $\phi$. In sum, then, fibers of $\cal X \to B$ of this type contribute a total of $$N(D_1)N(D_2) (D_1 \cdot D_2) \binom{r_0(D)-3}{r_0(D_1)-2}(D_2 \cdot C)^2$$ to the degree of $\phi^*(0)$. The contribution of such fibers to the degree of the divisor $\phi^*(\infty)$ is found analogously, the only difference being that, in order to get a pole of the cross-ratio, the points $p_1$ and $p_3$ must lie on one component---say $X_1$---of $X$, while $p_2$ and $p_4$ will lie on the other. Thus, instead of breaking the $r_0(D)-3$ points $q_\alpha$ into subsets of $r_0(D_1)-2$ and $r_0(D_2)$, we divide them into subsets of $r_0(D_1)-1$ and $r_0(D_2)-1$; and instead of $N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-2}$ such points in $\Gamma $ of this type we have $N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-1}$. Similarly, instead of choosing $p_3$ among the $(D_2 \cdot C)$ points of $X_2 \cap C_3$, we choose it among the $(D_1 \cdot C)$ points of $X_1 \cap C_3$; so that instead of $(D_2 \cdot C)^2$ zeroes of the cross-ratio lying over each such point of $\Gamma^\nu$ there will be $(D_1 \cdot C)(D_2 \cdot C)$. Again, each pole of the cross-ratio corresponding to a fiber of this type will have multiplicity one; so the total contribution to the degree of $\phi^*(\infty)$ is $$N(D_1)N(D_2) (D_1 \cdot D_2) \binom{r_0(D)-3}{r_0(D_1)-1}(D_1 \cdot C)(D_2 \cdot C)$$ It remains to add up the number of zeroes and poles of $\phi$ coming from members of our family containing $E$. Proposition \ref{codimensionone} describes all such curves, and the description is particularly simple, given that we are on the surface $\Bbb F_2$. There are only two types: a degenerate member $X$ of our family must consist either of \ \noindent 1. the union of $E$ and an irreducible rational nodal curve $X_1 \in |D-E|$, simply tangent at one point (which will be a smooth point of $X_1$) and meeting transversely elsewhere; \vspace*{1.7in} \hskip.5in \special {picture F2picture1} \noindent or \ \noindent 2. the union of $E$ and two curves $X_i \in |D_i|$, which will correspond to general points of the varieties $V(D_i)$ for some pair of divisor classes $D_1$ and $D_2$ with $D_1 + D_2 = D-E$. In particular, $X_1$ and $X_2$ will intersect each other and $E$ transversely. \vspace*{2.5in} \special {picture F2picture2} \ Now, we can forget about curves of the first type; in fact, since $E$ cannot contain any of the points $p_1,\dots,p_4$, these will be distinct points of $X_1$. Hence the cross-ratio function will not be zero or infinite at such a point of $B$. On the other hand, fibers of the second type may contribute. To see what our family looks like in a neighborhood of such a curve, recall first that by Proposition \ref{describegamma}, as we approach $X$ along any branch of $\Gamma$, all the points of intersection of $X_1$ and $X_2$, as well as all but one of the points of intersection of each curve $X_i$ with $E$, will be old nodes; exactly one of the points of intersection of each $X_i$ with $E$ will be new. The fiber of the normalized family $\cal X^\nu \to \Gamma^\nu$ will thus consist of the normalizations of $X_1$ and $X_2$, each meeting a copy of $E$ in one point and disjoint from each other: \vspace*{3.5in} \hskip.5in \special {picture F2semistable} \noindent Recall also that the total space of $\cal X^\nu$ will be smooth along such a fiber. Again, $E$ can't contain any of the points $p_i$, and if three or four lie on either curve $X_i$ the corresponding point of $B$ will be neither a zero or a pole of $\phi$; but we may get a contribution if two are on each $X_i$. Specifically, if $p_1$ and $p_2$ lie on one component---say $X_1$---and $p_3$ and $p_4$ on the other, we get a zero of $\phi$; while if $p_1$ and $p_3$ lie on a component---again, call this one $X_1$---and $p_2$ and $p_4$ on the other, we get a pole of $\phi$. That said, we can count the number of such fibers exactly as in the preceding case. We do the zeroes first. We begin by specifying a point $[X]$ in $\Gamma$---that is, we break the points $q_\alpha$ into subsets of size $r_0(D_1)-2$ and $r_0(D_2)$ respectively, and choose $X_1$ among the $N(D_1)$ irreducible rational curves in $|D_1|$ through $p_1$, $p_2$ and the first set and $X_2$ among the $N(D_2)$ irreducible rational curves in $|D_2|$ through the second set. Next, a point in $\Gamma^\nu$: we can take any of the $(D_1 \cdot E)(D_2 \cdot E)$ points of $\Gamma^\nu$ lying over $[X] \in \Gamma$. Lastly, we have to choose $p_3$ and $p_4$ among the $(D_2 \cdot C)$ points of intersection of $X_2$ with $C_3$ and $C_4$ respectively. We have, in sum, $$ N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-2}(D_1 \cdot E)(D_2 \cdot E)(D_2 \cdot C)^2 $$ zeroes of $\phi$ of this type. The poles of the cross-ratio coming from such are counted in the same way; the differences being exactly as in the preceding case: in specifying the point $[X] \in \Gamma$ we have to choose a subset of $r_0(D_1)-1$ rather than $r_0(D_1)-2$ of the points $q_\alpha$; and $p_3$ must be chosen among the $(D_1 \cdot C)$ points of $X_1 \cap C_3$. There are thus a total of $$ N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-1}(D_1 \cdot E)(D_2 \cdot E)(D_1 \cdot C)(D_2 \cdot C) $$ poles of this type. There is one important difference between this case and the previous, however: here, the fiber of the normalization $\cal X^\nu \to \Gamma^\nu$ has three components, with the components $X_1$ and $X_2$ containing the points $p_i$ separated by the component $E$. Since by Proposition \ref{singtotalspace} the total space $\cal X^\nu$ is smooth, we see by Lemma \ref{crossratiomult} that such points will be double zeroes and poles of $\phi$. The contribution to the degrees of these divisors coming from fibers of this type is thus twice the number of such fibers. We can now calculate the degree of the divisors $\phi^*(0)$ and $\phi^*(\infty)$. We have \begin{equation*} \begin{split} \deg(&\phi^*(0)) \\ &= 2 \cdot N(D) \\ & \quad + \sum_{D_1+D_2=D \atop D_1, D_2 \ne E} N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-2}(D_1 \cdot D_2) (D_2 \cdot C)^2 \\ & \quad + 2 \cdot \sum_{D_1+D_2=D-E \atop D_1, D_2 \ne E} N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-2}(D_1 \cdot E)(D_2 \cdot E)(D_2 \cdot C)^2 \end{split} \end{equation*} Similarly, \begin{equation*} \begin{split} &\deg(\phi^*(\infty)) \\ & \quad = \sum_{D_1+D_2=D \atop D_1, D_2 \ne E} N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-1}(D_1 \cdot D_2)(D_1 \cdot C)(D_2 \cdot C) \\ & \quad \quad + 2 \cdot \sum_{D_1+D_2=D-E \atop D_1, D_2 \ne E} N(D_1)N(D_2) \binom{r_0(D)-3}{r_0(D_1)-1}(D_1 \cdot E)(D_2 \cdot E)(D_1 \cdot C)(D_2 \cdot C) \end{split} \end{equation*} To express the final result we introduce the notation: \begin{equation*} \begin{split} \gamma (D_1,D_2):=N(D_1) N(D_2) &\biggl[ {r_0(D)-3 \choose r_0(D_1) - 1} (D_1\cdot C)(D_2\cdot C) \\ & \quad - {r_0(D)-3 \choose r_0(D_1) - 2} (D_2\cdot C)^2 \biggr] \end{split} \end{equation*} We now write $\deg(\phi^*(0))=\deg(\phi^*(\infty )))$ and solve the resulting equation for $N(D)$ to arrive at the recursion formula for $N(D)$ on $\Bbb F_2$: \begin{thm} \label{F2} Let $D\in \text{Pic}(\Bbb F _2)$ and let $N(D)$ be the number of irreducible rational curves in the linear series $|D|$ that pass through $r_0(D)$ general points of $\Bbb F _2$; then we have \begin{equation*} \begin{split} N(D) \quad = \qquad &\frac{1}{2} \sum _{D_1+D_2=D \atop D_1, D_2 \ne E} \gamma (D_1,D_2) (D_1 \cdot D_2) \\ & \quad +\sum _{D_1+D_2=D-E \atop D_1, D_2 \ne E} \gamma (D_1,D_2) (D_1 \cdot E)(D_2\cdot E). \end{split} \end{equation*} \end{thm} \subsection{The class $2C$ on $\Bbb F_n$ } We will now analyze the linear series $|2C|$ on the ruled surface $\Bbb F_n$ for any $n$. By restricting ourselves to this linear series we will arrive at a closed-form expression for $N(D)$ rather than a recursion. This is clear: since every linear series $|D|$ on $\Bbb F_n$ with $D < 2C$ that actually contains irreducible curves has arithmetic genus 0, we can say immediately how many degenerate fibers of each type there are in our one-parameter family of curves in $|2C|$. The dimension of the linear series $|2C|$ is $ 3n+2$. The arithmetic genus of the curves in the series is $n-1$, so that the expected dimension of the Severi variety is $r_0(2C) = 2n+3$. This is in fact the actual dimension: any irreducible nodal curve $D \in |2C|$ will be disjoint from $E$ (if it met $E$, it would contain it, having intersection number 0 with it); so that the nodes of $D$ will impose independent conditions on $|2C|$. So, we choose as usual $2n+2$ general points on $\Bbb F_n$, which we label $p_1, p_2, q_3,\dots,q_{2n+2}$ and consider the one-parameter family of curves $X \in |2C|$ passing through $\{p_1, p_2, q_3,\dots,q_{2n+2}\}$; we will denote this family $\cal X \to \Gamma$. As before, we let $ \Gamma^\nu$ be the normalization of $\Gamma$ and $\cal X^\nu \to \Gamma^\nu$ the normalization of the pullback family. Next, we fix general curves $C_3$ and $C_4 \in |C|$ in the linear series $|C|$, and adopt the convention that we will choose points $p_3$ and $p_4$ on the curves $X$ of our family lying on $C_3$ and $C_4$ respectively. Making the corresponding base change, we arrive at a family $\cal X \to B$; as before, we will denote by $\cal Y$ the minimal desingularization of $\cal X$ and by ${\cal Z} \to B$ the smooth semistable model. Now we consider the cross-ratio map $\phi : B \to \overline{M}_{0,4} \cong \Bbb P^1$ as before; we shall obtain a formula for $N(D)$ from $$ \deg \phi ^*(0) = \deg \phi ^*(\infty). $$ Of course, we get one contribution to the degree of $\phi^*(0)$ from the curves in our family that pass through any of the $n$ points of intersection of $C_3$ with $C_4$; this gives a total contribution of $n \cdot N(2C)$ to the degree of $\phi^*(0)$. The remaining zeroes and poles of $\phi$ correspond to reducible curves in the family $\{X\}$. As before we look first at curves that do not contain $E$. They can only be of the form $X = D_1 + D_2$ where $D_1$ and $D_2$ are each linearly equivalent to $C$. Such a fiber of the family $\cal X \to B$ can be either a pole or a zero of $\phi$, depending of course on how the points $p_i$ are distributed. Namely, if $p_1$ and $p_2$ lie on the same component $D_i$ and $p_3$ and $p_4$ on the other, we get a zero. To specify such a fiber, we simply have to break the $2n$ points $q_\alpha$ into disjoint sets of $n-1$ and $n+1$. The two components $D_i$ of the curve $X$ will be the (unique) curve in the series $|C|$ containing $p_1$, $p_2$ and the first subset; and the unique curve in the series $|C|$ containing the second subset. \vspace*{.9in} \noindent \hskip.3in \special {picture FnC+C} \ Next, we have to count the number of points of $B$ lying over each point of $\Gamma$ corresponding to such a curve. Since the general curve of our family has $n-1$ nodes, and the $n$ nodes of $X = D_1 \cup D_2$ impose independent conditions on the series $|2C|$, the curve $\Gamma$ will have $n$ (smooth) branches at the point $[X]$; thus the normalization of $\Gamma$ will have $n$ points lying over $[X]$ (cf. Proposition~\ref{describegamma}). Moreover, for each of these points there will be a point of $B$ for every possible choice of points $p_3$ and $p_4$; these can be any of the $(C \cdot C) = n$ points of intersection of the component $D_i$ not containing $p_1$ or $p_2$ with $C_3$ and $C_4$ respectively. There are thus a total of ${2n \choose n-1} \cdot n \cdot n^2$ fibers of $\cal X \to \Gamma$ of this type. Now, for each such fiber of $\cal X \to \Gamma$, the fiber of $\cal X \to B$ will be simply the normalization of $X$ at the $n-1$ old nodes: \newpage \ \vspace*{.9in} \hskip.3in \special {picture Fnfinal0} In particular, it has just two components and is stable, and as we have seen $\cal X^\nu$ already is smooth at the node of such a fiber. Each such point is thus a simple zero of $\phi$; so the total contribution to the degree of $\phi^*(0)$ of such curves is $$ {2n \choose n-1} \cdot n^3 $$ Similarly, we could have $p_1$ and $p_3$ on the same component $D_i$ and $p_2$ and $p_4$ on the other; in this case, we get a point of $\phi^*(\infty )$. The only difference in this case is that to specify such a fiber, we have to break the $2n$ points $\{q_\alpha\}$ into two disjoint sets of $n$ points apiece. The two components $D_i$ of the curve $X$ will be the (unique) curves in the series $|C|$ containing $p_1$ and the first subset; and the unique curve in the series $|C|$ containing $p_2$ and the second subset. The rest of the analysis is exactly the same---for each such curve, we get $n^3$ points of $B$, each of which is a simple pole of the cross-ratio function $\phi$---so the total contribution to the degree of $\phi^*(\infty)$ of such curves is $$ {2n \choose n} \cdot n^3 $$ \ The remainder of the calculation will be spent evaluating the contributions to the degree of the pullbacks of the boundary components of $\overline{M}_{0,4}$ coming from the curves in our original family containing $E$. As we have indicated, these curves are of $n-1$ types: for each $k = 1,\dots,n-1$ we will have a finite number of curves in our family consisting of the sum of $E$, $k$ fibers $F_1,\dots,F_k$ of $\Bbb F_n \to \Bbb P^1$ and an irreducible curve $D$ linearly equivalent to $C+(n-k)F$, with $D$ having a single point of $(n-k)$-fold intersection with $E$: \newpage \ \vspace*{2.3in} \noindent \special {picture Fn} \ For each of these types, there are a number of possibilities for the distribution of the points $p_1,\dots,p_4$ on the various components. For each such distribution corresponding to points $b$ in the inverse image of a boundary component $\Delta$ of $\overline{M}_{0,4}$, we will consider the number of fibers $X_b$ of that type and the coefficient with which the corresponding points $b \in B$ appear in the divisor $\tilde \phi^*(\Delta)$; in the end we will sum up the contributions to arrive at an expression for $N(2C)$ on $\Bbb F_n$ . \ \ \noindent $\bullet_1$ \thinspace $p_1, p_2 \in D$; $p_3, p_4 \in F_i$. \enspace In such a curve, the fiber components $F_j$ must each contain one of the points $q_\alpha$. To specify such a curve, then, we must first choose a subset of $k$ of the $2n$ points $q_\alpha$ and take $F = \cup F_i$ the unique curve in the linear series $|k\cdot F|$ containing them. Next, we have to single out one of these $k$ points, and label the corresponding fiber $F_i$. At this point $p_3$ and $p_4$ will be determined, as the unique points of intersection of $F_i$ with the curves $C_3$ and $C_4$ respectively. Finally, we choose a curve $D \in |C+(n-k)F|$ passing through the remaining $2n-k$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$. (Note that the ordering of the $k$ points $q_\alpha$ chosen to lie on fibers does not matter; all that counts is which one is chosen to lie on $F_i$.) Now, the linear series $ |C+(n-k)F|$ cuts on the curve $E \cong \Bbb P^1$ the complete linear series $|\cal O_{\Bbb P^1}(n-k)|$. This linear series is parametrized by the space $\Bbb P^{n-k}$ of polynomials of degree $n-k$ on $\Bbb P^1$ modulo scalars; and in that projective space the divisors consisting simply of $n-k$ times a single point---that is, $(n-k)$th powers of linear forms---form a rational normal curve. In the linear series $ |C+(n-k)F|$, then, the locus of curves $D$ having a single point of $(n-k)$-fold intersection with $E$ is a cone over a rational normal curve in $\Bbb P^{n-k}$ (with vertex the subseries $E + |(2n-k)F| \subset |C+(n-k)F|$ of curves containing $E$); in particular, it has degree $n-k$. There are thus exactly $n-k$ curves $D$ linearly equivalent to $|C+(n-k)F|$ passing through $p_1$, $p_2$ and the remaining $2n-k$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$. In sum, the number of fibers $X$ of this type in our family is ${2n \choose k} \cdot k \cdot (n-k)$. It remains to determine, for each such fiber of our family, the coefficient with which the corresponding point $b\in B$ appears in the pullback divisor $\phi^*(0)$. To do this, we need to know the local geometry of the family near $b \in B$; in particular, we need to have the picture of the corresponding fibers of the families $\cal X \to B$ and $Y \to B$. For the first, the only thing we need to know is which of the singular points of the fiber $X$ are limits of nodes of nearby fibers and (in the case of the point of intersection of $D$ with $E$) how many. The answer, as provided in Proposition \ref{singtotalspace}, is that the points of intersection of $D$ with the fibers $F_i$ are all limits of nodes on nearby curves; and the remaining $(n-k-1)$ nodes of the general fiber of the family tend to the point of intersection of $D$ with $E$. When we normalize the total space of the family, then, the curves $D$ and $F_i$ are pulled apart; and the point of intersection of $D$ with $E$ becomes a node, so that the fiber of $\cal X \to B$ over $b$ is \newpage \vspace*{4in} \hskip.8in \special {picture Fnnormal} \ But as we also saw in Proposition \ref{singtotalspace}, $\cal X$ will not be smooth: at the point lying over each point of intersection of $E$ with a fiber $F_i$, $\cal X$ will have a singularity of type $A_{n-k-1}$. Resolving each of these, we arrive at this picture of the fiber of $\cal Y \to B$ over $b$: \newpage \vspace*{3.2in} \noindent \special {picture Fnsmooth} \ Finally, we can blow down the extraneous curves $F_j$ and $G_{j,*}$ for $j \ne i$ to arrive at the picture of the fiber $Z$ of the family $Z \to B$ of semistable 4-pointed curves with smooth total space: \vspace*{2.1in} \noindent \special {picture Fnfinal1} \ Inasmuch as there are $(n-k)$ rational curves in the chain connecting the components $D$ and $F_i$ containing the pairs $\{p_1, p_2\}$ and $\{p_3, p_4\}$, each such fiber represents a point of multiplicity $n-k+1$ in the divisor $\phi^*(0)$. In sum, then, the fibers of this type contribute a total of $$\sum_{k=1}^{n-1}{2n \choose k} \cdot k \cdot (n-k) \cdot (n-k+1)$$ to the degree of $\phi^*(0)$. \ \noindent $\bullet_2$ \thinspace $p_1, p_3 \in D$; $p_2, p_4 \in F_i$ or $p_2, p_4 \in D$; $p_1, p_3 \in F_i$. \enspace In the first of these cases we are simply exchanging the locations of $p_2$ and $p_3$ to obtain a fiber $X$ corresponding to a point $b$ in the inverse image $\phi^*(\infty)$; this will affect the count of the number of such fibers, but not the final configuration on the semistable model with smooth total space, so the multiplicity of each such point in the divisor $\phi^*(0)$ will be as in the preceding case $n-k+1$. The difference here is that, because the fixed point $p_2$ lies on one of the fiber components, we can put the remaining fiber components through only $k-1$ of the points $q_\alpha$; at the same time, we can put the curve $D$ through $p_1$ and the remaining $2n-k+1$. To specify such a curve, then, we must first choose a subset of $k-1$ of the $2n$ points $q_\alpha$ and take $F = \cup F_j$ the unique curve in the linear series $|k\cdot F|$ containing them and $p_2$; the component of $F$ containing $p_2$ we will call $F_i$. As in the preceding case, there will be exactly $n-k$ curves in the linear series $|C+(n-k)F|$ passing through the remaining $2n-k+1$ points $q_\alpha$ and the point $p_1$ and having a point of intersection multiplicity $n-k$ with $E$; the curve $D$ can be any of these. At this point $p_4$ will be determined, as the unique point of intersection of $F_i$ with the curve $C_4$; while $p_3$ can be taken to be any of the $$ (D \cdot C_3) = \bigl( (C+(n-k)F) \cdot C \bigr) = 2n-k $$ \noindent points of intersection of $D$ with $C_3$. The number of fibers $X$ of this type in our family is thus ${2n \choose k-1} \cdot (n-k) \cdot (2n-k)$. As we said, each such fiber $X$ of our family is a pole of order $n-k+1$ of the cross-ratio function $\phi$. Finally, since exchanging $p_1$ with $p_4$ (as in the second case above) yields an identical result, the total contribution of the fibers of these types to the poles of $\phi$ is $$ 2\cdot \sum_{k=1}^{n-1} {2n \choose k-1} \cdot (n-k) \cdot (2n-k) \cdot (n-k+1). $$ \ \noindent $\bullet_3$ \thinspace $p_1, p_2 \in D$; $p_3 \in F_i$ and $p_4 \in F_j$, $i \ne j$. \enspace This case is very similar to the first; again, we have first to select a subset of $k$ of the $2n$ points $q_\alpha$ and take $F = \cup F_i$ the unique curve in the linear series $|k\cdot F|$ containing them. We then have to single out two of these $k$ points, and label the corresponding fibers $F_i$ and $F_j$, which in turn determines the points $p_3 = F_i \cap C_3$ and $p_4 = F_j \cap C_4$. Finally, we take as before $D$ to be any of the $n-k$ curves in $|C+(n-k)F|$ passing through $p_1$ and $p_2$ and the remaining $2n-k$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$. Thus, the number of fibers $X$ of this type in our family is ${2n \choose k} \cdot k \cdot (k-1) \cdot (n-k)$. At this point, we see another difference with the preceding case: here, to arrive at the fiber of the family of semistable 4-pointed curves with smooth total space we blow down the curves $F_m$ and $G_{m,*}$ for all $m$ including $i$ and $j$, to arrive at the simpler fiber: \vspace*{.9in} \noindent \hskip.5in \special {picture Fnfinal2} \ Since this is already semistable, each such fiber represents a simple zero of $\phi$. In sum, then, the fibers of this type contribute a total of $$\sum_{k=1}^{n-1}{2n \choose k} \cdot k \cdot (k-1) \cdot (n-k)$$ to the degree of $\phi^*(0)$. \ \noindent $\bullet_4$ \thinspace $p_1, p_3 \in D$; $p_2 \in F_i$ and $p_4 \in F_j$, $i \ne j$; or $p_2, p_4 \in D$; $p_1 \in F_i$ and $p_3 \in F_j$, $i \ne j$. \enspace This case bears the same relation to the preceding as the second did to the first: we are simply exchanging $p_2$ and $p_3$ (or $p_1$ and $p_4$), so that the fibers $X$ will correspond to poles rather zeroes of $\phi$; the multiplicity will be 1 as in the last case, but the number of such fibers will be different. Take the case $p_1, p_3 \in D$ first. Such a fiber may be specified by choosing first a subset of $k-1$ of the $2n$ points $q_\alpha$ and taking $F = \cup F_i$ the unique curve in the linear series $|k\cdot F|$ containing them and $p_2$; the component containing $p_2$ will will label $F_i$. We then have to single out one of these $k$ points, and label the corresponding fiber $F_j$ , which in turn determines the point $p_4 = F_j \cap C_3$. As before we take $D$ to be any of the $n-k$ curves in $|C+(n-k)F|$ passing through $p_1$ and the remaining $2n-k+1$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$.; the point $p_3$ may be any of the $2n-k$ points of $D \cap C_3$. Thus, the number of fibers $X$ of this type in our family is ${2n \choose k-1} \cdot (k-1) \cdot (n-k) \cdot (2n-k)$. As we said, each such fiber represents a point of multiplicity $1$ in the divisor $\phi^*(\infty )$; and since the case $p_2, p_4 \in D$ contributes an equal number, the fibers of this type contribute a total of $$2 \cdot \sum_{k=1}^{n-1}{2n \choose k-1} \cdot (k-1) \cdot (n-k) \cdot (2n-k) $$ to the degree of $\phi^*(\infty)$. \ \noindent $\bullet_5$ \thinspace $p_3, p_4 \in D$; $p_1 \in F_i$ and $p_2 \in F_j$, $i \ne j$. \enspace This case is also a variant of case (3) above: here we are exchanging both $p_2$ for $p_3$ and $p_1$ for $p_4$. The result is that the fibers $X$ will once more correspond to zeroes of $\phi$, with multiplicity 1 as in the last two cases, but again there will be a different number of such fibers. To evaluate this number, note that this time such a fiber may be specified by choosing first a subset of $k-2$ of the $2n$ points $q_\alpha$ and taking $F = \cup F_i$ the unique curve in the linear series $|k\cdot F|$ containing them and both $p_1$ and $p_2$; $F_i$ will be the component of $F$ containing $p_1$ and $F_j$ the component containing $p_2$. We choose $D$ any of the $n-k$ curves in $|C+(n-k)F|$ passing through the remaining $2n-k+2$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$; and then $p_3$ and $p_4$ may be any of the $2n-k$ points of $D \cap C_3$ and $D \cap C_4$ respectively. Thus, the number of fibers $X$ of this type in our family is ${2n \choose k-2} \cdot (n-k) \cdot (2n-k)^2$; and since each corresponding $b \in B$ is a simple zero of $\phi$, such fibers contribute a total of $$ \sum_{k=1}^{n-1}{2n \choose k-2} \cdot (n-k) \cdot (2n-k)^2 $$ to the degree of $\phi^*(0)$. \ \noindent $\bullet_6$ \thinspace $p_1 \in D$, $p_2 \in F_i$ and $p_3, p_4 \in F_j$, $i \ne j$; or $p_1 \in F_i$, $p_2 \in D$ and $p_3, p_4 \in F_j$, $i \ne j$. \enspace These again give zeroes of $\phi$: to arrive at the semistable model, in the end we will blow down $D$ and the chains $F_m$ and $G_{m,*}$ for all $m \ne j$. Consider first the case $p_1 \in D$, $p_2 \in F_i$. To specify such a fiber we have to select a subset of $k-1$ of the points $q_\alpha$: $F = \cup F_i$ will then be the unique curve in the linear series $|k\cdot F|$ containing them and $p_2$ (and $F_i$ the component of $F$ containing $p_2$). Then we have to choose which of the remaining $k-1$ components of $F$ is to be $F_j$, and take $p_3 = F_j \cap C_3$ and $p_4 = F_j \cap C_4$. Finally, we choose $D$ any of the $n-k$ curves in $|C+(n-k)F|$ passing through $p_1$ and the remaining $2n-k+1$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$; there are thus a total of ${2n \choose k-1} \cdot (k-1) \cdot (n-k)$ such fibers in our family. As for multiplicity, as we said the semistable model with smooth total space is obtained from $Y$ by blowing down $D$ and the chains $F_m$ and $G_{m,*}$ for all $m \ne j$ to arrive at a fiber of the form \vspace*{1.2in} \noindent \special {picture Fnfinal3} \ Since there are $(n-k-1)$ rational curves in the chain connecting the components $E$ and $F_j$ containing the pairs $\{p_1, p_2\}$ and $\{p_3, p_4\}$, each such fiber represents a zero of multiplicity $n-k$ of the function $\phi$. Finally, the case $p_1 \in F_i$, $p_2 \in D$ contributes an equal number; so that the fibers of this type contribute a total of $$2 \cdot \sum_{k=1}^{n-1}{2n \choose k-1} \cdot (k-1) \cdot (n-k)^2$$ to the degree of $\phi^*(0)$. \ \noindent $\bullet_7$ \thinspace $p_1 \in D$, $p_3 \in F_i$ and $p_2, p_4 \in F_j$, $i \ne j$; or $p_2 \in D$, $p_4 \in F_i$, and $p_1, p_3 \in F_j$, $i \ne j$. \enspace These are obtained by exchanging $p_2$ and $p_3$ in the first case immediately above or $p_1$ and $p_4$ in the second (the remaining two possible switches will be considered next), so that the fibers $X$ will correspond to poles rather than zeroes of $\phi$ and will have the same multiplicity $n-k$. We thus simply have to count the number of such fibers, which is straightforward: for example, in the first case ($p_1 \in D$, $p_3 \in F_i$), to specify such a fiber we have to select first a subset of $k-1$ of the points $q_\alpha$ and take $F = \cup F_i$ the unique curve in the linear series $|k\cdot F|$ containing them and $p_2$; $F_j$ will be the component of $F$ containing $p_2$. Then we have to choose which of the remaining $k-1$ components of $F$ is to be $F_i$, and take $p_3 = F_i \cap C_3$ and $p_4 = F_j \cap C_4$. Finally, we choose $D$ any of the $n-k$ curves in $|C+(n-k)F|$ passing through $p_1$ and the remaining $2n-k+1$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$; there are thus a total of ${2n \choose k-1} \cdot (k-1) \cdot (n-k)$ such fibers in our family, so that the fibers of this type contribute a total of $$2 \cdot \sum_{k=1}^{n-1}{2n \choose k-1} \cdot (k-1) \cdot (n-k)^2$$ to the degree of $\phi^*(\infty)$. \ \noindent $\bullet_8$ \thinspace $p_3 \in D$, $p_1 \in F_i$ and $p_2, p_4 \in F_j$, $i \ne j$; or $p_4 \in D$, $p_2 \in F_i$, and $p_1, p_3 \in F_j$, $i \ne j$. \enspace These are the remaining two cases obtained by switching points in case (6) above; as opposed to the immediately preceding case these are obtained by exchanging $p_1$ and $p_4$ in the first case of (6) or $p_2$ and $p_3$ in the second. Thus the fibers will again correspond to poles rather than zeroes of $\phi$ and will again appear with multiplicity $n-k$ in $\phi^*(\infty)$, but the number of such fibers will be different. To compute it, consider the first case ($p_1 \in D$, $p_3 \in F_i$). To specify such a fiber we have to select first a subset of $k-2$ of the points $q_\alpha$ and take $F = \cup F_i$ the unique curve in the linear series $|k\cdot F|$ containing them and both $p_1$ and $p_2$; $F_i$ will be the component of $F$ containing $p_1$ and $F_j$ the component containing $p_2$. This then fixes the point $p_4 = F_j \cap C_4$. We choose $D$ any of the $n-k$ curves in $|C+(n-k)F|$ passing through the remaining $2n-k+2$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$; and $p_3$ can be any of the $2n-k$ points of intersection of $D$ with $C_3$. There are thus a total of ${2n \choose k-2} \cdot (n-k) \cdot (2n-k)$ fibers of this type in our family; so that the fibers of this type contribute a total of $$2 \cdot \sum_{k=1}^{n-1}{2n \choose k-2} \cdot (2n-k) \cdot (n-k)^2$$ to the degree of $\phi^*(\infty)$. \ We come now to the last three cases, those in which none of the four points $p_i$ lie on $D$. The next one is the last to contribute to the degree of $\phi^*(0)$. \ \noindent $\bullet_9$ \thinspace $p_1 \in F_i$, $p_2 \in F_j$ and $p_3, p_4 \in F_\ell$, $i \ne j \ne \ell \ne i$. \enspace To determine a fiber of this type we have to specify first $k-2$ of the points $q_\alpha$, and let $F \in |kF|$ contain $p_1$, $p_2$ and these $k-2$. We then have to pick a component of $F$ among those not containing $p_1$ or $p_2$, and call it $F_\ell$; $p_3$ and $p_4$ will then be the points of intersection of $F_\ell$ with $C_3$ and $C_4$ respectively. As always, $D \in |C+(n-k)F|$ can be any of the $n-k$ curves passing through the remaining $2n-k+2$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$; so there are a total of ${2n \choose k-2} \cdot (k-2) \cdot (n-k) $ such fibers in our family. Finally, for each such fiber, after blowing down $D$ and the chains $F_m$ and $G_{m,*}$ for all $m \ne \ell$ we arrive at the smooth semistable model, whose special fiber has the form \vspace*{2.4in} \noindent \hskip-.5in \special {picture Fnfinal4} There being $n-k-1$ intermediate curves in this chain, each such fiber corresponds to a zero of order $n-k$ of $\phi$; the total contribution of such fibers is thus $$ \sum_{k=1}^{n-1}{2n \choose k-2} \cdot (k-2) \cdot (n-k)^2$$ \ \noindent $\bullet_{10}$ \thinspace $p_1 \in F_i$ $p_3 \in F_j$ and $p_2, p_4 \in F_\ell$; or $p_2 \in F_i$ $p_4 \in F_j$ and $p_1, p_3 \in F_\ell$, $i \ne j \ne \ell \ne i$. \enspace These are the two cases obtained from the preceding by exchanging either $p_2$ and $p_3$ or $p_1$ and $p_4$; each such fiber thus represents a pole of multiplicity $n-k$ of $\phi$. To count the number, consider first the case $p_1 \in F_i$ $p_3 \in F_j$ and $p_2, p_4 \in F_\ell$. To determine such a fiber we specify $k-2$ of the points $q_\alpha$, and let $F \in |kF|$ contain $p_1$, $p_2$ and these $k-2$; the component of $F$ containing $p_1$ we will call $F_i$, and then $p_3$ will be determined as the point $F_i \cap C_3$ . We then have to pick a component of $F$ among those not containing $p_1$ or $p_2$, and call it $F_\ell$; $p_4$ will be the point of intersection of $F_\ell$ with $C_4$. As always, $D \in |C+(n-k)F|$ can be any of the $n-k$ curves passing through the remaining $2n-k+2$ of the points $q_\alpha$ and having a point of $(n-k)$-fold tangency with $E$; so there are a total of ${2n \choose k-2} \cdot (k-2) \cdot (n-k) $ such fibers in our family. Counting both possible exchanges, we see that the total contribution of such fibers to the degree of $\phi^*(\infty)$ is $$ 2 \cdot \sum_{k=1}^{n-1}{2n \choose k-2} \cdot (k-2) \cdot (n-k)^2$$ \ \noindent $\bullet_{11}$ \thinspace $p_1, p_3 \in F_i$ and $p_2, p_4 \in F_j$, $i \ne j$ \enspace This is our final case; note that there is no analogous source of zeroes of $\phi$, since $p_1$ and $p_2$ do not lie on the same fiber of $\Bbb F_n$. First, to count the number: such fibers are determined first by choosing $k-2$ of the points $q_\alpha$, and letting $F$ be the union of fibers containing them and $p_1$ and $p_2$; $F_i$ and $F_j$ will be the fibers containing $p_1$ and $p_2$ and $p_3$ and $p_4$ the points of intersection of these fibers with $C_3$ and $C_4$ respectively, so nothing more need by specified. We only have to choose $D$ among the $n-k$ curves in $|C+(n-k)F|$ passing through the other $2n-k+2$ of the points $q_\alpha$ and meeting $E$ in only one point, so that there are a total of just ${2n \choose k-2} \cdot (n-k) $ such fibers in our family. To arrive at the smooth semistable model near such a fiber, we have to blow down the curve $D$ and the chains $G_{\ell,*}$ for all $\ell \ne i, j$; we arrive at the fiber \vspace*{2.4in} \noindent \hskip-.5in \special {picture Fnfinal5} Since there are $2n-2k-1$ rational curves in the chain connecting $F_i$ and $F_j$, each such fiber gives a pole of multiplicity $2n-2k$; the total contribution of such fibers to the degree of $\phi^*(\infty)$ is thus $$ \sum_{k=1}^{n-1}{2n \choose k-2} \cdot (n-k) \cdot (2n-2k)$$ We are now ready to add up all the contributions to the degrees of $\phi^*(0)$ and $\phi^*(\infty)$, equating the results and solving for $N(2C)$. We have \begin{equation*} \begin{split} \deg(\phi^*(0)) = n \cdot N(2C) + n^3{2n \choose n-1} \\ + \sum_{k=1}^{n-1} (n-k) \biggl[ &{2n \choose k} \bigl( k(n-k+1) + k(k-1) \bigr) \\ + &{2n \choose k-1} \bigl( 2(k-1)(n-k) \bigr) \\ + &{2n \choose k-2} \bigl( (2n-k)^2 + (k-2)(n-k) \bigr) \biggr] . \end{split} \end{equation*} while on the other hand \begin{equation*} \begin{split} \deg(\phi^*(\infty)) = n^3{2n \choose n} \\ + \sum_{k=1}^{n-1} (n-k) \biggl[ &{2n \choose k-1} \bigl( 2(2n-k)(n-k+1) + 2(2n-k)(k-1) + 2(k-1)(n-k) \bigr) \\ + &{2n \choose k-2} \bigl( 2(2n-k)(n-k) +2(k-2)(n-k) + 2(n-k) \bigr) \biggr]. \end{split} \end{equation*} Combining these, we arrive at the expression \begin{equation*} n \cdot N(2C) = n^3 \bigl({2n \choose n} - {2n \choose n-1} \bigr) + S \end{equation*} where \begin{equation*} \begin{split} \!\!\!\!\!\! S = \sum_{k=1}^{n-1} (n-k) \biggl[ &{2n \choose k} \bigl( -k(n-k+1) - k(k-1) \bigr) \\ + &{2n \choose k-1} \bigl( 2(2n-k)(n-k+1) + 2(2n-k)(k-1) + 2(k-1)(n-k)-2(k-1)(n-k) \bigr) \\ + &{2n \choose k-2} \bigl( 2(2n-k)(n-k) +2(k-2)(n-k) + 2(n-k) -(2n-k)^2 - (k-2)(n-k) \bigr) \biggr] . \end{split} \end{equation*} The expression for $S$ may be reduced immediately to \begin{equation*} \begin{split} S = \sum_{k=1}^{n-1} (n-k) \biggl[ &{2n \choose k} \cdot (-kn) \\ + &{2n \choose k-1} \cdot 2(2n-k)n \\ + &{2n \choose k-2} \cdot (-kn) \biggr] . \end{split} \end{equation*} (Note that we can now enlarge the formal limits of summation to include $k=0$ without affecting the sum; this will be convenient in the following calculations.) To reduce this further, we separate it into two terms: we write $S = S' - S''$, where $$ S' = \sum_{k=0}^{n-1} 4n^2(n-k) {2n \choose k-1} $$ and $$ S'' = \sum_{k=0}^{n-1} (n-k) \cdot kn \cdot \biggl[ {2n \choose k} + 2{2n \choose k-1} + {2n \choose k-2} \biggr] . $$ The expression for $S''$ telescopes nicely: since $$ {2n \choose k-1} + {2n \choose k-2} = {2n+1 \choose k-1}, $$ $$ {2n \choose k} + {2n \choose k-1} = {2n+1 \choose k} $$ and $$ {2n+1 \choose k} + {2n+1 \choose k-1} = {2n+2 \choose k}, $$ we have simply $$ S'' = \sum_{k=0}^{n-1} (n-k) \cdot kn \cdot {2n+2 \choose k} $$ As for $S'$, we can combine that with the remaining two terms in the expression for $N(2C)$, and together they simplify. To start with, observe that \begin{equation*} \begin{split} {2n \choose n} - {2n \choose n-1} &= {(2n)! \over n!n!} - {(2n)! \over (n-1)!(n+1)!}\\ &= {(2n)! \over n!(n+1)!} \cdot \bigl( (n+1) - n \bigr) \\ &= {(2n)! \over n!(n+1)!} \\ &= {1 \over n}{2n \choose n-1} \end{split} \end{equation*} so $$ n^3 \bigl({2n \choose n} - {2n \choose n-1} \bigr) = n^2{2n \choose n-1} $$ Now, combining this with the expression for $S'$ above, we have \begin{equation*} \begin{split} n^2{2n \choose n-1} &+ \sum_{k=0}^{n-1} 4n^2(n-k) {2n \choose k-1} \\ &= n^2 \biggl( {2n \choose n-1} + 4{2n \choose n-2} + 8{2n \choose n-3} + \dots + (4n-4){2n \choose 0} \biggr) \end{split} \end{equation*} To reduce this, we use the relation $$ {2n \choose n-1} + {2n \choose n-2} = {2n+1 \choose n-1} $$ to absorb the first term; then 3 times the relation $$ {2n \choose n-2} + {2n \choose n-3} = {2n+1 \choose n-2} $$ to absorb the rest of the second; then 5 times the relation $$ {2n \choose n-3} + {2n \choose n-4} = {2n+1 \choose n-3} $$ to absorb the rest of the third, and so on; ultimately, we arrive at \begin{equation*} \begin{split} n^2{2n \choose n-1} &+ \sum_{k=0}^{n-1} 4n^2(n-k) {2n \choose k-1} \\ &= n^2 \biggl( {2n+1 \choose n-1} + 3{2n+1 \choose n-2} + 5{2n+1 \choose n-3} + \dots + (2n-1){2n+1 \choose 0} \biggr) \end{split} \end{equation*} Now we play the same game again: using the relation $$ {2n+1 \choose n-1} + {2n+1 \choose n-2} = {2n+2 \choose n-1} $$ to absorb the first term, then twice the relation $$ {2n+1 \choose n-2} + {2n+1 \choose n-3} = {2n+2 \choose n-2} $$ to absorb the remainder of the second, and so on, we may re-express this as \begin{equation*} \begin{split} n^2{2n \choose n-1} &+ \sum_{k=0}^{n-1} 4n^2(n-k) {2n \choose k-1} \\ &= n^2 \biggl( {2n+2 \choose n-1} + 2{2n+1 \choose n-2} + 3{2n+1 \choose n-3} + \dots + n{2n+1 \choose 0} \biggr) \\ &= n^2 \sum_{k=0}^{n-1} (n-k) {2n+2 \choose k} \end{split} \end{equation*} Finally, we can combine this and the expression above for $S''$: we have \begin{equation*} \begin{split} n \cdot N(2C) &= n^3 \bigl( {2n \choose n} - {2n \choose n-1} \bigr) + S' - S'' \\ &= n^2 \sum_{k=0}^{n-1} (n-k) {2n+2 \choose k} - n \sum_{k=0}^{n-1} k(n-k) {2n+2 \choose k} \\ &= n \sum_{k=0}^{n-1} (n-k)^2 {2n+2 \choose k}. \end{split} \end{equation*} We have therefore proved the \begin{thm} \label{2C} Let $N(2C)$ be the number of irreducible rational curves in the linear series $|2C|$ on $\Bbb F _n$ passing through $2n+3$ points, then $$ N(2C) = \sum_{k=0}^{n-1} (n-k)^2 {2n+2 \choose k} . $$ \end{thm} For example, on $\Bbb F_2$ we have $$ N(2C) = {6 \choose 1} + 4{6 \choose 0} = 6+4 =10 ; $$ on $\Bbb F_3$ we have $$N(2C) = {8 \choose 2} + 4{8 \choose 1} + 9{8 \choose 0} = 28 + 32 + 9 = 69 ; $$ and on $\Bbb F_4$ we have $$N(2C) = {10 \choose 3} + 4{10 \choose 2} + 9{10 \choose 1} + 16{10 \choose 0}= 120 + 180 + 90 + 16 = 406 $$ and so on. \ We will now show how to arrive at an expression of $N(2C)$ on $\Bbb F_n$ as a coefficient of a simple generating function. We simply write out the sum involved, and then telescope it using the standard binomial relations as before: that is, we write $$ N(2C)= {2n+2 \choose n-1} + 4{2n+2 \choose n-2} + 9{2n+2 \choose n-3} + \dots + n^2{2n+2 \choose 0} $$ and use the relations ${2n+2 \choose n-1} + {2n+2 \choose n-2} = {2n+3 \choose n-1}$, $3{2n+2 \choose n-2} + 3{2n+2 \choose n-3} = 3{2n+3 \choose n-2}$, and so on to rewrite this as \begin{equation*} \begin{split} N(2C) &= {2n+3 \choose n-1} + 3{2n+3 \choose n-2} + 6{2n+3 \choose n-3} + \dots + {n(n+1) \over 2}{2n+3 \choose 0} \\ &= \sum_{k=0}^{n-1} {n-k+1 \choose 2}{2n+3 \choose k}. \end{split} \end{equation*} We can also think of this as the coefficient of $t^n$ in the product of the power series $$ \sum {2n+3 \choose k} t^k = (1+t)^{2n+3} $$ and $$ \sum {\ell+2 \choose 2} t^\ell = {1 \over (1-t)^3} $$ so that we can write $N(2C)$ on $\Bbb F_n$ as the coefficient $$ N(2C) = \bigg[ {(1+t)^{2n+3} \over (1-t)^3} \biggr]_{t^n} . $$ \ \ \subsection{A formula for $\Bbb F _n$} \label{Fn} We conclude our paper with a formula for the general ruled surface $\Bbb F _n$. Here we define the function $\gamma_{i_1,\dots,i_t}(D_{i_1},\dots,D_{i_t})$ giving the contribution to the cross-ratio corresponding to a given decomposition $D = D_1+D_2+\dots+D_t+E$ or $D=D_1+D_2$, with $D_j\in V_{i_j}(D_j)$. Recall that the variety $V_i(D)$ is the closure in $|D|$ of the locus of irreducible rational curves that have a point of contact of order $i$ with the exceptional curve $E$ (cf. \ref{term} ). We define $$ \gamma_{i_1,\dots,i_t}(D_{i_1},\dots,D_{i_t}):= \prod (i_j N_{i_j}(D_j) ) \cdot $$ $$ \cdot \Bigl[ {r_0(D) -3 \choose r_0^{i_1}(D_1) -1,r_0^{i_2}(D_2) -1, r_0^{i_3}(D_3),....} [\sum _{j\geq 3} {(C\cdot D_j) \over i_j}\bigl( {(C\cdot D_1) \over i_1} + {(C\cdot D_2) \over i_2}\bigr) - \sum _{j\geq 3} {(C\cdot D_j)^2 \over i_j}] $$ $$ - {r_0(D) -3 \choose r_0^{i_1}(D_1) -2,r_0^{i_2}(D_2) , r_0^{i_3}(D_3),....} [\sum _{j\geq 2} (C\cdot D_j)^2 ({1 \over i_j }+ {1 \over i_1 }) + {1 \over i_1 }\sum _ {2\leq j<k \leq t}(C\cdot D_j)(C\cdot D_k) ] \Bigr] $$ \noindent In these terms, we can state \begin{thm} \label{Fn} Let $D$ be a divisor on the surface $\Bbb F_n$. Let $N(D)$ be the number of irreducible rational curves in $|D|$ that pass through $r_0(D)$ general points of $\Bbb F_n$. Then $$ nN(D) = $$ $$\sum _{D_1 + D_2 = D} (D_1\cdot D_2) \gamma _{1,1}(D_1,D_2) + $$ $$ + \sum _{t=2}^{n} \; \sum _{D_1+D_2+\dots+D_t=D-E} \; \sum _{i_1,\dots,i_t} \prod _{j :i_j=1}(E\cdot D_{i_j}) \gamma_{i_1,\dots,i_t}(D_{i_1},\dots,D_{i_t}) $$ \end{thm} \ \ \noindent {\bf{Acknowledgments}}. \thinspace Our interest in these questions grew out of conversations with Enrico Arbarello, Ciro Ciliberto and Bill Fulton, to whom we are very grateful. \

9608

alg-geom/9608018

en

https://arxiv.org/abs/alg-geom/9608018

alg-geom/9608018

Trygve Johnsen

Trygve Johnsen

Rank Two Bundles on Algebraic Curves and Decoding of Goppa Codes

An error in (what is now called) Theorem 3.4 has been corrected

We study a connection between two topics: Decoding of Goppa codes arising from an algebraic curve, and rank two extensions of certain line bundles on the curve.

alg-geom

\section{\bf Introduction} \label{intro} Let $C$ be a curve over a finite field $F_q$. Some years ago V. Goppa showed how to produce codes from such a curve (For a survey, see for example Gieseker \cite {G1} ). In this note we will show how socalled syndrome decoding of (the duals of the original ) Goppa codes in an intimate way is connected to the study of rank two bundles, that are extensions of the structure sheaf ${\mathcal O}_C$ and a certain fixed line bundle on $C$ determined by the code in question. In fact, each syndrome corresponds to an extension. Moreover syndromes due to correctable error vectors always correpond to unstable rank two bundles, that is: semistable bundles never correspond to syndromes of correctable errors (Here we call an error vector, or simply error, correctable if its Hamming weight is at most $\frac{d-1}{2}$, where $d$ is the designed minimum distance of the code). For correctable errors the process of error location is translated into finding a certain quotient line bundle of minimal degree (or dually: a line subbundle of maximal degree) of the rank two bundle defined by the syndrome, and then pick the relevant section of this line bundle. If an error is not correctable, there is not necessarily a unique such quotient line bundle of minimal degree. We gain our insight through a certain projective embedding $C \subset {\mathbf P} $, with the property that the columns of the parity check matrix of the Goppa code in question are interpreted as points of the embedded copy of $C$. We study $j$-secant $(j-1)$-planes to $C$, for $j =1,2,...$ . Using the proper definitions of these geometrical objects, we thus obtain a stratification of ${\mathbf P}$, which viewed from one angle is a stratification of syndromes, according to how many errors that have to be made to obtain the syndrome. Viewed from another angle it is a stratification of rank two extensions according to the socalled $s$-invariant. The geometric picture associated to rank two extensions was given very explicitly in Lange and Narasimhan \cite{LN}, and the basic idea was given already in Atiyah \cite{A}. We just remark that the aspects interesting to us carry over to the case of positive characteristic. Then we compare with the picture obtained from the space of syndrome vectors. So far we have not been able to utilize these geometrical observations to make any constructive decoding algorithms. To do so one would have to introduce some kind of "extention arithmetic" to perform decoding. Some explicit considerations along these lines are made in \cite{BC}. For constructive decoding algorithms for Goppa codes in general, see the celebrated Feng and Rao \cite{FR}, or for example Duursma \cite{D}, Justesen et al \cite{JLJH}, Pellikaan \cite{P}, Skorobogatov and Vladut \cite{SV}, or the special issue IEEE Trans. of Info. Theory, Vol. 41, No.6, Nov. 1995. In Section 2 we recall some basic facts concerning algebraic-geometric (Goppa) codes. We also define and make an elementary study of the secant varieties that in a natural way turn up in connection with these codes. In Section 3 we introduce vector bundle language, and in Theorem \ref{mainthm} we present the connection between correctable errors and unstable bundle extensions of rank two. \begin{rem} \label{CIproj} {\rm In Goppa \cite{G2} one describes an algorithm for decoding of Goppa codes from rational curves. There one assumes first of all that at most $t$ errors are made, where $t = [(d-1)/2]$. Then one assumes that exactly $t$ errors are made, and sets up a system of equations to solve the problem of error-location given that this extra assumption holds. If the equations yield no solution, one assumes that $t-1$ errors are made, sets up a new system of equations, and so on. In the end one arrives at a point where one finds a solution since the basic assumption is that at most $t$ errors are made. The coefficient matrices, set up to find the elementary symmetric functions in the parameter values of the error locations are Toeplitz, and in particular symmetric. This process is in many ways reminiscent of describing complete quadrics through various blowing-ups of the space of usual quadrics, which is again a space of symmetric square matrices. Hence, in order to generalize to arbitrary curves, one could describe some kind of analogy to complete quadrics for curves of positive genus. Such a generalization, in terms of an object obtained through various blowing-ups of the secant strata of $C$ inside ${\mathbf P}$, has been given in Bertram \cite{B}, using vector bundle language. It is possible that understanding this or similar objects can give new insight into decoding of Goppa codes.} \end{rem} \section{\bf Definitions and basic facts about Goppa codes} \label{DefBas} A $q$-ary code of length $n$ is a subset of the vector space $F_q^n$, where $F_q$ is a finite field with $q$ elements. A linear code is a linear subspace of $F_q^n$. Let $\overline{F}_q$ be an algebraic closure of $F_q$. Let $C$ be a curve of genus $g$ defined over these fields. Let $D$ and $G$ be divisors on $C$ defined over $F_q$, such that their supports are disjoint. Moreover the support of $D$ consists of $n$ distinct points $P_1,....,P_n$ of degree one, where $n=deg(D)$. For a divisor $M$ defined over $F_q$, denote by $L(M)$ the set of the zero element and those elements $f$ of the function field $F_q(C)$, such that $(f) + M \ge 0$. Denote by $l(M)$ the dimension of $L(M)$ as a vector space over $F_q$. This dimension is the same as the one obtained if we work over $\overline{F}_q$. Following for example van Lint et al \cite{vLvdG} we denote by $C(D,G)$ the code, which is the image of $L(G)$ in $F_q^n$ under the map: $$\phi : f \to\ (f(P_1),.....,f(P_n)).$$ In Pellikaan et al \cite {PSvW}, such a code is called a WAG, which is short for weakly algebraic- geometric code. Moreover one shows there that all linear codes are WAG. By the theorem of Riemann-Roch (which remains valid over finite fields) we have: $$dim (C(D,G)) = l(G) - dim(ker f) = l(G) - l(G - D) = $$ $$ deg(G) + 1 - g + l(K-G) - l(G-D). $$ As usual $K$ denotes a canonical divisor. Set $m=deg(G)$. A WAG is called a SAG (strongly algebraic-geometric code) if the following composite condition is fullfilled: $2g-2< m < n$. For a SAG we observe: $l(K-G) = l(G-D) = 0$, and hence: dim $C(D,G) = m + 1 - g$. By a generator matrix for a code one means a $(k \times n)$-matrix, where the rows constitute a base for the code as a linear space over $F_q$. A generator matrix for $C(D,G)$ is: $$M = \left[\ \vcenter{\halign{$\strut#$\hfill&&\quad\hfill$#$\hfill\cr f_1(P_1)&f_1(P_2)&\cdots &f_1(P_n)\cr \cdots \cr \cdots \cr f_k(P_1)&f_k(P_2)&\cdots &f_k(P_n)\cr}}\ \right], $$ where $k=m+1-g$, and ${f_1,....,f_k}$ is a basis for $L(G)$, both over $F_q$ and over $\overline{F}_q$. By a parity check matrix for a code one means a $((n-k) \times n)$- coefficient matrix of a set of equations cutting out the code as a subspace of $F_q^n$ and of $\overline{F}_q^n$. One denotes by $C^*(D,G)$ the linear code having the matrix $M$ above as parity check matrix. Hence $C^*(D,G)$ is the orthogonal complement of $C(D,G)$ and vice versa. One also says that the codes are dual to each other. For a WAG defined as $C(D,G)$ consider the exact sequence of sheaves on C: $$0 \to\ {\mathcal O}(G-D) \to\ {\mathcal O}(G) \to\ {\mathcal O}(G)/{\mathcal O}(G-D) \to\ 0.$$ The long exact cohomology sequence gives: $$0 \to\ L(G-D) \to\ L(G) \to\ F_q^n \to\ $$ $$H^1(C,{\mathcal O}(G-D)) \to\ H^1(C,{\mathcal O}(G)) \to\ 0.$$ Moreover by Serre duality : $H^1(C,{\mathcal O}(G-D))$ is dual to $L(K+D-G)$, and $H^1(C,{\mathcal O}(G))$ is dual to $L(K-G)$. For a SAG the long exact sequence reduces to: \begin{equation} \label{eq:dual} 0\to\ L(G) \to\ F_q^n \to\ L(K+D-G)^* \to\ 0. \end{equation} Here $K$ can (by Riemann-Roch) be chosen such that $G^* = K+D-G$ has support disjoint from $D$. We can identify the elements of $L(K+D-G)$ with differential forms being zero of order the same as order$(G)$ at the points of the support of $G$, and having at most simple poles at the points of the support of $D$, and no poles elsewhere. Hence we see that evaluating elements of $L(K+D-G)$ can be interpreted as evaluating residues of the differential forms described (after multiplying each value $f(P_i)$ by a non-zero value $Res_{P_i}(\eta )$, where $K=(\eta )$ ). Moreover we have for $f$ in $L(G)$ and $\omega $ a differential form as described: \begin{equation} \label{eq:res} 0 = \Sigma _i Res_{P_i}(f\omega ) = \Sigma _i f(P_i) Res_{P_i}(\omega ). \end{equation} From equations (\ref{eq:dual}) and (\ref{eq:res}) we easily conclude that $C(D,G^*)$ is code equivalent to $C^*(D,G)$. In the original work by Goppa the code obtained from the divisors $D$ and $G$ was $C^*(D,G)$, and it was obtained by means of residues of differential forms. By a Goppa code we will here simply mean a SAG. One easily verifies that $C(D,G^*)$ is a SAG if and only if $C(D,G)$ is so. For a SAG we see that if $ \{ h_1, \cdots ,h_{k^*} \} $ is a basis for $L(G^*)$, then a parity check matrix for $C(D,G)$ is (essentially, after a trivial equivalence operation): $$M^* = \left[\ \vcenter{\halign{$\strut#$\hfill&&\quad\hfill$#$\hfill\cr h_1(P_1)&h_1(P_2)&\cdots &h_1(P_n)\cr \cdots \cr \cdots \cr h_{k^*}(P_1)&h_{k^*}(P_2)&\cdots &h_{k^*}(P_n)\cr}}\ \right], $$ where $k^* = n-k = n-m+g-1$. \begin{rem} \label{columns} {\rm We see that the columns of $M^*$ represent the points of the support of $D$ if $C$ is embedded into ${\mathbf P} = {\mathbf P}^{n-m+g-2}= {\mathbf P} (H^0(C,K+D-G)^*)$ by means of sections of $L(G^*)$.} \end{rem} \medskip For $\bf{w}_1$ and $\bf{w}_2$ in $F_q^n$, let the Hamming distance $d(\bf{w}_1,\bf{w}_2)$ denote the number of coordinate positions, in which $\bf{w}_1$ and $\bf{w}_2$ differ; it is clearly a metric on $F_q^n$. Let the minimum distance of a code be the minimum Hamming distance for any pair of codewords. For a linear code this is easily seen to be the minimum number of non-zero coordinates (minimum weight) for any non-zero codeword. Denote by $d_1$ the minimum distance of the code $C(D,G)$. If a code word has weight $d_1$, then there is a divisor $D_1$ with $D_1 \le D$, and deg($D_1)= n-d_1$, such that $L(G-D_1) \ne 0$, Hence $m-(n-d_1) \ge 0$, that is: $d_1 \ge n-m$. We denote by $d$ the integer $n-m$, which is positive since $C(D,G)$ is a SAG. We call $d$ the designed minimum distance. We also see that $C$ is embedded into ${\mathbf P} (H^0(C,K+H)^*)$, where $d=deg(H)$ (and $H=D-G$). Denote by $t$ the integer $[(d-1)/2]$. We also call $t$ the designed error correcting capacity. Recall the basic fact: \begin{rem} {\rm Let $N$ be the parity check matrix of a code. The minimum distance of the code is equal to $s$ if all choices of $s-1$ columns of $N$ are independent, and some choice of $s$ columns of $N$ are dependent.} \end{rem} \medskip Let $\bf{x}$ be an element (codeword) of $C(D,G) \subset F_q^n$, and assume that $\bf{x}$ is transmitted, and $\bf{y} = \bf{x} + \bf{e}$ is received. The difference $\bf{e}$ is called the error vector. Denote by $S(\bf{y})$ the matrix product $M^*\bf{y}$. Clearly $S(\bf{y})$ is a vector in $F_q^{k^*}$, and $S(\bf{y})$ = $S(\bf{e})$ $=\bf{0}$ if and only if $\bf{y}$ is itself a codeword. $S(\bf{y})$ is called the syndrome vector of $\bf{y}$. We can also interpret $S(\bf{y})$ as a point of ${\mathbf P} = {\mathbf P}^{k^*-1} = {\mathbf P}^{d+g-2}$. For each integer $a$, let the (Hamming) $a$-ball centered at $\bf{x}$ be the set of those $\bf{y}$, such that $d(\bf{x}, \bf{y}) \le$ $a$. The following is immediate from the triangle equality: \begin{rem} The restriction of the map $S$: $F_q^n \to\ F_q^{k^*}$ to any $t$-ball is injective. \end{rem} \medskip \centerline {\bf{Secant varieties} } \medskip Now we view $C$ as any curve defined over the algebraic closure $\overline{F}_q$, and let $C$ be embedded in some projective space over this field. Let $A$ be an effective divisor on $C$, possibly with repeated points. Let $C_j$ be the $j$'th symmetric product of $C$, for $j$=1,2,... . \begin{defn} \label{span} \begin{itemize} \item [(a)] We denote by $Span(A)$ the intersection of all hyperplanes ${\mathcal H}$, such that we have: $\Sigma _iI(Q_i,C \cap\ {\mathcal H})Q_i \ge A$ (Here $I(Q,V_1 \cap\ V_2)$ denotes the usual Bezout intersection number of two varieties of complementary dimension at a point $Q$). \item [(b)]We say that $C$ is $k$-spanned if $dim(Span(A))=j-1$, for all $A$ with $deg(A)=j$, and $j \le k+1$. \item [(c)] We set $Sec_j(C) = \cup Span(A)$, where the union is taken over all $A$ in $C_j$. \item [(d)] For a point $P$ in projective space we set $h(P) = h$ if $P$ is contained in $Sec_h(C) - Sec_{h-1}(C)$. \end{itemize} \end{defn} \begin{prop} \label{emb} Let $C$ be the curve treated in Section 2, defined over $F_q$ and embedded into ${\mathbf P}^{d+g-2}$ by the linear system $K+D-G$ as described. Then we have: \begin{itemize} \item [(a)] $C$ is $(d-2)$-spanned. In particular $C$ is smoothly embedded if $d \ge 3$. \item [(b)] If $h(P) = h \le [(d-1)/2] = t$, then there is a unique effective divisor $A$ with degree at most $h$, such that $P$ is contained in $Span(A)$. \end{itemize} \end{prop} \begin{proof} (a) An easy application of Riemann-Roch. Let $A$ be a divisor of degree $j \le d-1$. Set $H = D-G$. Then $$l(K+H-A) = 2g-2+d-j+1-g +l(A-H) = d+g-1-j = l(K+H) - j,$$ so $A$ imposes $j$ independent conditions on the linear system. (b) Assume $P$ is contained in $Span(A_1) \cap\ Span(A_2)$. If the supports of the divisors $A_1$ and $A_2$ are disjoint, then we have: $Span(A_1+ A_2)$ = the linear span of $Span(A_1) \cup\ Span(A_2)$, so $$dim(Span(A_1+A_2))=1+dim(Span(A_1))+ dim(Span(A_2))-$$ $$ dim((Span(A_1) \cap\ Span(A_2)) \le 1+2(h-1)-1=2h-2.$$ Hence $C$ is not $(2h-1)$-spanned, and thus not $(d-2)$-spanned, since $h \le [(d-1)/2]$. We leave it to the reader to modify the argument if the supports of the two divisors are not disjoint. \end{proof} By abuse of notation (See Remark \ref{columns} above) we denote by $P_i$ column nr. $i$ of the parity matrix $M^*$. Assume that a codeword $\bf{x}$ is transmitted, $\bf{y}$ is received, and that the error vector $\bf{e}$ has weight $h$ with coordinates $e_1,\cdots ,e_h$ in positions $i_1,\cdots ,i_h$ respectively. We have: The syndrome $S(\bf{y})$ = $S(\bf{e})$ = $e_1P_1+ \cdots + e_hP_h$. Interpreting $S(\bf{y})$ as a point of ${\mathbf P}$, we then see that $S(\bf{y})$ is contained in $Sec_h(C)$, and that $h(S(\bf{y})$)$\le h$. Moreover it is clear that if $h \le t$, then $h(S(\bf{y})$)$ = h$, and that the "error divisor" $P_1+\cdots +P_h$ is the unique divisor $A$ of degree at most $h$ over $\overline F_q$, such that $S(\bf{y})$ is contained in $Span(A)$. So, error location amounts to finding such a divisor $A$, given the point $S(\bf{y})$. A priori we know that this divisor consists of distinct points, all of degree 1 defined over $F_q$, and that even the errors $e_1,\cdots ,e_h$ are in $F_q$. \section{\bf Vector bundles of rank two on $C$} \label{Vec2} We continue using the notation from Section 2. The following exposition is to a great extent taken from Lange and Narasimhan \cite{LN} and Bertram \cite{B}. Let $Ext_{{\mathcal O}_C}(H,{\mathcal O}_C)$ be the set of isomorphism classes of exact sequences $$(e): 0 \to\ {\mathcal O}_C \to\ E \to\ H \to 0.$$ The map ${\mathcal O}_C \to\ E$ is denoted by $f$ and the map $E \to\ H$ by $g$. The zero element $(e_0)$ corresponds to the case of a split exact sequence. Here ${\mathcal O}_C$ as usual denotes the structure sheaf on $C$, and $H$ is the fixed line bundle or invertible sheaf $D-G$, see Section 2 (by abuse of notation we do not distinguish between the divisor $H = D-G$, or the invertible sheaf or line bundle, of which the divisor corresponds to a global section). The middle term $E$ is a locally free sheaf, or vector bundle, of rank 2. Standard cohomology theory and Serre duality give: $$Ext_{{\mathcal O}_C}(H,{\mathcal O}_C) = Ext_{{\mathcal O}_C}({\mathcal O}_C,-H) = H^1(C,-H) = H^0(C, K+H)^*.$$ Hence ${\mathbf P}(Ext_{{\mathcal O}_C}(H,{\mathcal O}_C)) = {\mathbf P}(H^0(C, K+H)^*) = {\mathbf P}$. This means that (up to isomorphism and a multiplicative factor) the points of our well-known projective space ${\mathbf P}$ described in Section 2 are identified with extensions as described. \begin{defn} \label{s-inv} Let $E$ be a rank two vector bundle on $C$. \begin{itemize} \item [(a)] Denote by $s(E)$ the integer $$deg(E)-2max(deg(L))=2min(deg(M))- deg(E),$$ where the maximum is taken over all line subbundles $L$ of $E$ and the minimum is taken over the quotient line bundles $M$ of $E$. \item [(b)] $E$ is called stable if $s(E) > 0$; semistable if $s(E) \ge 0$; unstable if $s(E)< 0$. \item [(c)] For an extension $(e)$ as above we set $s((e))=s(E)$, where $E$ is the middle term. An extension is called stable (semistable, unstable) if the middle term E is so. \end{itemize} \end{defn} The definitions of stable and semistable coincide if $deg(E)$ is odd. \medskip For the zero element $(e_o)$ we observe: $s(E) = -d$, and for all non-split extension $(e)$ we have $d \ge s((e)) \ge 2-d$. Moreover, if $M$ is a quotient line bundle of $E$ of minimal degree $(s+d)/2 \ge 1$, with quotient map $h$, then the composition of $h$ and $f$ is non-zero: $$ {\mathcal O}_C \to\ E \to\ M.$$ Hence $M$ is isomorphic to ${\mathcal O}_C(A)$, for an effective divisor $A$ of degree $(s+d)/2$. This implies again that (identifying $(e)$ with its corresponding point of ${\mathbf P}$) the point $(e)$ is contained in the kernel of the map: $$ Ext^1_{{\mathcal O}_C}(H,{\mathcal O}_C) \to\ Ext^1_{{\mathcal O}_C}(H,{\mathcal O}_C(A)),$$ that is, in the kernel of: $$ H^0(C, K+H)^* \to\ H^0(C, K+H-A)^*.$$ This observation has many consequences. First we see that the set of points in ${\mathbf P}$ with $s$-value $2-d$ are precisely those that represent bundles with a quotient bundle of type ${\mathcal O}_C(Q)$, for some point $Q$ on $C$. If we assume that $d \ge 3$, then $C$ is smoothly embedded by Proposition \ref{emb}. Then we can identify $C$ with its embedded image in ${\mathbf P}$, and the point $Q$ on C then corresponds to a bundle extension with a quotient bundle isomorphic to ${\mathcal O}_C(Q)$. Moreover it is clear that the observation above is equivalent to: $(e)$ is contained in $Span(A)$. \begin{rem} \label{spana} {\rm Arguing in a dual way, we get that if (e) is contained in Span(A), then the line bundle corresponding to $H-A$ is a subbundle of E.} \end{rem} Summing up, we now formulate the following result, which is practically identical to Proposition 1.1. of Lange and Narasimhan \cite{LN} (Recall the functions $h$ and $s$ from ${\mathbf P}$ to $\bf{Z}$, introduced in Definition \ref{span}, d.) and Definition \ref{s-inv}, (a), respectively). \begin{prop} \label{s and h} Let $P$ be a point of ${\mathbf P}$. Then $s(P) = 2h(P) - d$. In particular $P$ is a unstable point (semistable, stable) if and only if $h(P)<d/2$ $ (h(P) \ge d/2,$ $ h(P) > d/2)$. \end{prop} We are now able to formulate the main result of the paper: \begin{thm} \label{mainthm} Let $P = S(\bf{y})$ be the syndrome of a received message using the code $C(D,G)$. Then $P$ is the syndrome of some error vector with weight at most the designed error correcting capasity $t = [(d-1)/2]$ only if $P$ is an unstable point. Moreover in that case the process of error location is reduced to finding the error divisor $A$ among global sections of the unique quotient line bundle of degree $h$ of the vector bundle $E(P)$ of rank two, appearing as the middle term in the extension corresponding to $P$. \end{thm} \begin{proof} This follows directly from Proposition \ref{s and h} and the argument above. \end{proof} \begin{rem} \label{iff} {\rm The ``only if'' in the theorem can be replaced by ``if and only if'' if we define the syndrome map over $\overline{F}_q$.} \end{rem} \begin{rem} \label{GIT} {\rm The definitions of stable, semistable, unstable can be viewed as special cases of more general definitions of these concepts in the setting of Geometric Invariant Theory (GIT), which again is an essential tool in building moduli spaces parametrizing various objects. The spaces arise as quotients of various group actions. In order to get quotients with good properties one usually has to disregard certain "bad objects", which are the unstable ones. In our case the relevant construction is that of $\mathcal{SU}$$_2(C)$, the moduli space of isomorphism classes of vector bundles of rank two on $C$ (See Gieseker \cite{Gi}, p. 51-52). This has dimension $4g-3 = g + (3g-3)$ , where the sum decomposition corresponds to $g=dim(Jac(C))$ degrees of freedom to choose a line bundle $H$, and $3g-3= dim (\mathcal{SU}$$_2(C,H))$ degrees of freedom to choose the rank two bundle with determinant $H$ (where $\mathcal{SU}$$_2(C,H)$ = modulo space of rank two bundles with determinant $H$). Moreover, for $\mathcal{SU}$$_2(C,H)$ there are essentially only 2 cases; $deg(H)$ odd and $deg(H)$ even, since tensoring a rank two bundle with a line bundle gives rise to an isomorphism between two such spaces with determinants $H$ with degrees of equal parity. One can show that the natural map: $${\mathbf P} - Sec_t(C) \to\ \mathcal{SU}_2(C,H)$$ is birational if $d = deg(H) = 2g-1$, and that it maps birationally on to the $\theta$-divisor if $d = deg(H) = 2g-2$, where the $\theta$-divisor parametrizes the rank two-bundles with a global section. One observes that the situation in coding theory in a certain way is complementary to that of applying GIT to build $\mathcal{SU}$$_2(C,H)$, since the good(syndrome) points in coding theory are the bad ones for GIT, and vice versa. On the other hand the issue for those who work with moduli spaces is often precisely what to do with the unstable points, so the focal point of the theories are still in a certain sense overlapping. One is for example interested in blowing up various $s$-negative strata of ${\mathbf P}$ to obtain compactifications of ${\mathbf P} - Sec_t(C)$ and $\mathcal{SU}$$_2(C,H)$ with desired properties. See Bertram \cite{Bdiff} and \cite{B}. One could hope that insight in such compactifications could be instrumental in understanding algorithms for decoding of Goppa codes. One can also ask: Is it possible that the minimum distance of $C(D,G)$ exceeds $d$, or weaker: Given a reasonable small integer $k_0$, like 2, 3 or 17; is there a positive limit $s_0$, such that if $s(P) \le s_0 = 2h_0 - d$, then there are at most $k_0$ divisors $A$ (even over $\overline{F}_q$) of degree $h_0$, such that P is contained in $Span(A)$? For practical purposes this would in some situations be almost as good as unique decoding. In vector bundle language one is then interested maximal sublinebundles of $P$ corresponding to divisors of type $H-A$. The question of such maximal subbundles is the main issue in Lange and Narasimhan \cite{LN}.} \end{rem} \begin{example} \label{ratcurv} {\rm Assume $g=0$. Then $C$ is mapped into ${\mathbf P} = {\mathbf P}^{d-2}$ as a curve of degree $d-2$, that is as a rational normal curve. It is well-known that on $C = {\mathbf P}^{1}$ all rank two bundles of degree $d$ split as ${\mathcal O}(a) \oplus\ {\mathcal O}(b)$, with $a+b = d$. If $d$ is odd, we then see the largest possible $s$-value is $-1 = d- 2[(d+1)/2]$. Hence all rank two-bundles are unstable, corresponding to the fact that ${\mathbf P} = Sec_t(C)$ over $F_q$.} \end{example} \centerline {\bf Acknowledgements} \medskip I am grateful to Emma Previato for informing me about the manuscript \cite{BC}, and to the authors of \cite{BC} for correcting a mistake in an old version of my paper. \medskip \centerline{\bf References} \begin{itemize} \bibitem[1]{A} M.~Atiyah, Complex fibre bundles and ruled surfaces, \textit{Proc. London Math. Soc.}, \textbf{5} (1955), 407-434. \bibitem[2]{Bdiff} A.~Bertram, Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space, \textit{Journal of Diff. Geometry }, \textbf{35} (1992), 429-469. \bibitem[3]{B} A.~Bertram, Complete Extensions and their Map to Moduli Space, \textit{London Math. Soc. Lect. Notes Series}, \textbf{179} (1992), 81-91. \bibitem[4]{BC} T.~Bouganis, D. Coles, A Geometric View of Decoding AG Codes, Manuscript, 10 pages (2002). \bibitem[5]{D} I.~Duursma, Algebraic decoding using special divisors, \textit{IEEE Trans. of Info. Theory}, \textbf{39(2)} (1993), 694-698. \bibitem[6]{FR} G.L. ~Feng, T.R.N.~Rao, Decoding Algebraic-Geometric Codes up to the Designed Minimum Distance, \textit{IEEE Trans. of Info. Theory}, \textbf{39(1)} (1993), 37-45. \bibitem[7]{G1} V.D.~Goppa, Algebraic-Geometric codes, \textit{Math. USSR Izvestiya}, \textbf{21} (1983), 75-90. \bibitem[8]{G2} V.D.~Goppa, \textit{Geometry and Codes}, Mathematics and its applications, Soviet series, \textbf{21}, Kluwer Ac.Publ., (1989). \bibitem[9]{Gi} D.~Gieseker, Geometric invariant theory and applications to moduli problems, \textit{Springer Lect. Notes in Math.}, \textbf{996} (1983), 45-63. \bibitem[10]{JLJH} J. ~Justesen, K.J. ~Larsen, H. ~Elbr\"ond Jensen, and T. ~H\"oholdt, Fast decoding of codes from algebraic plane curves, \textit{IEEE Trans. of Info. Theory}, \textbf{38(1)} (1992), 111-119. \bibitem[11]{LN} H.~Lange, M.S.~Narasimhan, Maximal subbundles of rank 2 vector bundles on curve, \textit{Math.Ann.}, \textbf{266} (1983), 55-72. \bibitem[12]{P} R.~Pellikaan, On a decoding algorithm for codes on maximal curves, \textit{IEEE Trans. of Info. Theory}, \textbf{35(6)} (1989), 1228-1232. \bibitem[13]{PSvW} R. ~Pellikaan, B.Z. ~Shen, and G.J.M. ~van Wee Which linear codes are algebraic geometric?, \textit{IEEE Trans. of Info. Theory}, \textbf{37(3)} (1991), 583-682. \bibitem[14]{SV} A.N. ~Skorobogatov and S.G. ~Vladut, On the Decoding of Algebraic-Geometric Codes, \textit{IEEE Trans. of Info. Theory } \textbf{36(5)} (1990), 1051-1060. \bibitem[15]{vLvdG} J. ~van Lint and G. ~van der Geer, \textit{Introduction to Coding Theory and Algebraic Geometry}, DMW Seminar \textbf{12}, Birkhauser (1995). \end{itemize} \end{document}

9608

alg-geom/9608014

en

https://arxiv.org/abs/alg-geom/9608014

alg-geom/9608014

Valery Alexeev

Valery Alexeev and Iku Nakamura

On Mumford's construction of degenerating abelian varieties

Final version, to appear in Tohoku Math. J

We prove that a 1-dimnl family of abelian varieties with an ample sheaf defining principal polarization can be canonically compactified (after a finite base change) to a projective family with an ample sheaf. We show that the central fiber (P,L), which we call an SQAV, has a canonical Cartier theta divisor. We give a combinatorial definition for SQAVs and describe their geometrical properties, in particular compute cohomologies of L^n, n\ge0.

alg-geom

\section*{Introduction} \label{sec:Introduction} \begin{say} Assume that we are given a 1-parameter family of principally polarized abelian varieties with theta divisors. By this we will mean that we are in one of the following situations: \begin{enumerate} \item ${\mathcal R}$ is a complete discrete valuation ring (DVR, for short) with the fraction field $K$, $S=\operatorname{Spec}\, {\mathcal R}$, $\eta=\operatorname{Spec}\, K$ is the generic point, and we have an abelian variety $G_{\eta}$ over $K$ together with an effective ample divisor $\Theta_{\eta}$ defining a principal polarization; or \item we have a projective family $(G,\Theta)$ over a small punctured disk $D_{\varepsilon}^0$. \end{enumerate} In this paper we show that, possibly after a finite ramified base change, the family can be completed in a simple and absolutely canonical manner to a projective family $(P,\Theta)$ with a relatively ample Cartier divisor $\Theta$ over $S$, resp. $D_{\varepsilon}$. Moreover, this construction is stable under further finite base changes. We give a combinatorial description of this family and its central fiber $(P_0,\Theta_0)$ and study their basic properties. In particular, we prove that $P_0$ is reduced and Cohen-Macaulay and that $H^i(P_0,{\mathcal O}(d\Theta_0))$, $d\ge0$ are the same as for an ordinary PPAV (principally polarized abelian variety). \end{say} \begin{say} Existence of such construction has profound consequences for the moduli theory. Indeed, with it one must expect that there exists a canonical compactification $\overline{A}_g$ of the moduli space $A_g$ of PPAVs, similar to the Mumford-Deligne compactification of the moduli space of curves. Without it, one has to believe that there is no single ``best'' geometrically meaningful compactification of $A_g$ and work with the infinitely many toroidal compactifications instead. The moduli implications of our construction are explored in \cite{Alexeev_CMAV}. \end{say} \begin{say} Degenerations of abelian varieties have been studied exhaustively which makes our result somewhat surprising. There is a very complete description of degenerations of polarized abelian varieties of arbitrary degree of polarization over a complete Noetherian domain of arbitrary dimension, with or without an ample line bundle. This description is called Mumford's construction, it was first published in a beautiful short paper \cite{Mumford_AnalyticDegsAVs} and later substantially expanded and improved by Faltings and Chai in \cite{Faltings85,Chai85,FaltingsChai90} (we note a parallel construction of Raynaud which works in the context of rigid analytic geometry). Mumford's construction gives an equivalence of categories $\operatorname{DEG_{pol}}$, resp. $\operatorname{DEG_{ample}}$ of degenerations of polarized abelian varieties, resp. with a line bundle, and the categories $\operatorname{DD_{pol}}$, resp. $\operatorname{DD_{ample}}$ of the ``degeneration data''. \end{say} \begin{say} As an auxiliary tool, Mumford's construction uses {\em relatively complete models.\/} Mumford remarks that such a model ``is neither unique nor canonical'' and that ``in fact, the non-uniqueness of $\widetilde P$ gives one freedom to seek for the most elegant solutions in any particular case''. What we show in this paper is that if one is willing to give up some of the properties of $\widetilde P$ and concentrate on the others, then in fact there is a canonical choice! Here is what we do: \begin{enumerate} \item We only consider the case of a 1-dimensional base $S$. This certainly makes the problem easier but not significantly. In view of the moduli theory one shouldn't expect that a higher-dimensional family can be canonically completed, unless one is in a very special situation, such as for a ``test family'' over a special toric scheme. \item We allow an additional finite ramified base change $S'\to S$, even after one already has the semiabelian reduction. This, again, is perfectly natural from the moduli point of view. \item Most importantly, we do not care where in the central fiber the limit of the zero section of $G_{\eta}$ ends up. Our relatively complete model contains a semiabelian group scheme in many different ways, but the closure of the zero section need not be be contained in any of them. Hence, $P_0$ is a limit of $G_{\eta}$ as an abelian torsor, not as an abelian variety. \item Instead of a section, we pay a very special attention to the limit of the theta divisor $\Theta_{\eta}$, something which was overlooked in the previous constructions. \end{enumerate} For the most part of the paper we work in the algebraic situation, over a complete DVR. The complex-analytic case is entirely analogous, and we explain the differences in Section \ref{sec:Complex-analytic case}. \end{say} \begin{say} Shortly after \cite{Mumford_AnalyticDegsAVs} appeared, a series of works of Namikawa and Nakamura \cite{Namikawa_NewCompBoth, Namikawa_ToroidalDegsAVs, Namikawa_ToroidalDegsAVs2, Namikawa_ToroidalCompSiegel, Nakamura_ModuliSQAVs, Nakamura_CompNeronModel} was published that dealt with the complex-analytic situation. They contain a toric construction for an extended 1-parameter family. This construction is very similar to Mumford's, and the main difference is a substitute for the relatively complete model. One unpleasant property of that substitute is that in dimension $g\ge5$ the central fiber need not be reduced. \end{say} \begin{say} When restricted to the complex-analytic setting, our construction has a lot in common with the Namikawa-Nakamura construction as well. The main difference is again the fact that we use and pay special attention to the theta divisor. Our solution to the problems arising in dimension $g\ge5$ is simple -- a base change. To underline the degree of dependence on the previous work, we call our construction {\em simplified Mumford's construction\/} and we call the central fibers {\em stable quasiabelian varieties,\/} or SQAVs, following Namikawa. We call the pairs $(P_0,\Theta_0)$ {\em stable quasiabelian pairs,\/} or SQAP. \end{say} \begin{say} We note that Namikawa had constructed families ${\mathcal X}^{(2n)}_g$ over the Voronoi compactification $\overline{A}^{\operatorname{VOR}}_{g,1,2n}$ of the moduli space $A_{g,1,2n}$ of PPAVs with a principal level structure of level $2n$, $n\ge3$. The boundary fibers in these families are different for different $n$, and some of them are non-reduced when $g\ge5$. \end{say} \begin{ackno} This research started when the first author was visiting the University of Tokyo and he would like to express his gratitude for the hospitality. We also thank Professors F. Bogomolov, Y.Kawamata, S.Mori and T.Oda for very stimulating conversations. The work of the first author was partially supported by the NSF. The work of the second author was partially supported by the Grant-in-Aid (No. 06452001) for Scientific Research, the Ministry of Education, Science, Sports and Culture, Japan. \end{ackno} \section{Delaunay and Voronoi polyhedral decompositions} \label{sec:Delaunay and Voronoi polyhedral decompositions} \begin{say} The structure of the extended family will be described explicitly in terms of two polyhedral decompositions which we now introduce. \end{say} \begin{notation} $X\simeq{\mathbb Z}^r$ will denote a lattice in a real vector space $X_{\mathbb R}=X\otimes{\mathbb R}$, for a fixed positive integer $r$. $B:X\times X\to{\mathbb R}$ will be a symmetric bilinear form assumed to be positive definite. We denote the norm $\sqrt{B(x,x)}$ by $\|x\|_B$ or simply by $\|x\|$. \end{notation} \begin{defn} \label{defn:Delaunay cell} For an arbitrary $\alpha\inX_{\mathbb R}$ we say that a lattice element $x\in X$ is $\alpha$-nearest if \begin{displaymath} \|x-\alpha\|_B= \min\{ \|x'-\alpha\|_B \quad\big|\, x'\in X \}. \end{displaymath} We define a $B$-Delaunay cell $\sigma$ (or simply a {\em Delaunay cell\/} if $B$ is understood) to be the closed convex hull of all lattice elements which are $\alpha$-nearest for some fixed $\alpha\inX_{\mathbb R}$. Note that for a given Delaunay cell $\sigma$ the element $\alpha$ is uniquely defined only if $\sigma$ has the maximal possible dimension, equal to $r$. In this case $\alpha$ is called the {\em hole \/} of $\sigma$, cf. Section 2.1.2 of the ``encyclopedia of sphere packings and lattices'' \cite{ConwaySloane93}. One should imagine a sphere around the $\alpha$-closest lattice elements (which is known as ``the empty sphere'' because there are no other lattice elements in its interior) with $\alpha$ at the center. Together all the Delaunay cells constitute a locally finite decomposition of $X_{\mathbb R}$ into infinitely many bounded convex polytopes which we call the {\em Delaunay cell decomposition \/} $\operatorname{Del}_B$. \end{defn} \begin{rem} It is clear from the definition that the Delaunay decomposition is invariant under translation by the lattice $X$ and that the 0-dimensional cells are precisely the elements of $X$. \end{rem} \begin{defn} For a given $B$-Delaunay cell $\sigma$ consider all $\alpha\inX_{\mathbb R}$ that define $\sigma$. They themselves form a locally closed bounded convex polytope. We denote the closure of this polytope by $\hat\sigma=V(\sigma)$ and call it the $B$-Voronoi cell or simply the {\em Voronoi cell\/}. The Voronoi cells make up the {\em Voronoi cell decomposition $\operatorname{Vor}_B$\/} of $X_{\mathbb R}$. \end{defn} \begin{say} It is easy to see that the Delaunay and Voronoi cells are dual to each other in the following sense: \end{say} \begin{lem} \label{lem:Voronoi_Delaunay_duality} \begin{numerate} \item For a fixed form $B$ there is a 1-to-1 correspondence between Delaunay and Voronoi cells given by $\hat\sigma=V(\sigma)$, $\sigma=D(\hat\sigma)$. \item $\dim\sigma+\dim\hat\sigma=r$. \item $\sigma\subset\tau$ if and only if $\hat\tau\subset\hat\sigma$. \item For $\sigma=x\in X$ the corresponding Voronoi cell $V(x)$ has the maximal dimension. $V(x)$ is the set of points of $X_{\mathbb R}$ that are at least as close to $x$ as to any other lattice element $x'$. \item For an arbitrary Delaunay cell $\sigma$ the dual Voronoi cell $\hat\sigma$ is the polytope with vertices at holes $\alpha(\tau)$ where $\tau$ goes over all maximal-dimensional Delaunay cells containing $\sigma$. \end{numerate} \end{lem} \begin{rem} With the definition this natural, it is no wonder that Voronoi and Delaunay cells have a myriad of applications in physics, chemistry and even geography (\cite{OkabeBootsSugihara92}, many more references in \cite{ConwaySloane93}) and go by many different names. Some alternative names for Voronoi cells are: Voronoi polytopes, nearest neighbor regions, Dirichlet regions, Brillouin zones, Wigner-Seitz cells. Delaunay cells had been called by various authors Delone cells, Delony cells, $L$-polytopes. Evidently, even the spelling of the last name of Boris Nikolaevich Delone is not agreed upon. A quick computer database search shows that the French variant ``Delaunay'' is prefered in 99\% of all papers, so this is our choice too. \end{rem} \begin{exmp} The Figures 1 and 2 give the only two, up to the action of $SL(2,{\mathbb Z})$, Delaunay decompositions of ${\mathbb Z}^2$. The broken lines show the corresponding Voronoi decompositions. Note that, unlike Delaunay, the Voronoi decompositions may have some continuous moduli. \end{exmp} \begin{say} Here is another way to understand the Delaunay decompositions. \end{say} \begin{lem} \label{lem:paraboloid} Consider an $(r+1)$-dimensional real vector space $X_{\mathbb R}\oplus{\mathbb R}$ with coordinates $(x,x_0)$ and a paraboloid in it defined by the equation \begin{displaymath} x_0=A(x)=B(x,x)/2+lx/2 \end{displaymath} for some $l\inX_{\mathbb R}^*$. Consider the convex hull $Q$ of countably many points on this paraboloid with $x\in X$. This object has a multifaceted shape and projections of the facets onto $X_{\mathbb R}$ are precisely the $B$-Delaunay cells. The equations that cut out the cone at the vertex $(c,A(c))$ are \begin{displaymath} x_0-A(c)\ge B\big( \alpha(\sigma),x\big) +lx/2= dA(\alpha(\sigma))(x) , \end{displaymath} where $\sigma$ goes over all the maximal-dimensional Delaunay cells containing $c$. If $\sigma$ is a maximal-dimensional Delaunay cell, then the interior normal in the dual space $X_{\mathbb R}^*\oplus{\mathbb R}$ to the corresponding facet is $\big(1,-dA(\alpha(\sigma))\big)$. The normal fan to the paraboloid consists of $\{0\}$ and the cones over the shifted Voronoi decomposition $\big(1,-dA(\operatorname{Vor}_B)\big)$. \end{lem} \begin{say} The inequality for $x_0$ does not depend on the cell $\sigma\ni x$ chosen because for any Delaunay vector $v\in\sigma_1\cap\sigma_2$ \begin{displaymath} B(\alpha(\sigma_1),v)=B(v,v)/2=B(\alpha(\sigma_2),v) \end{displaymath} (see \ref{rem:equations_for_hole}). We look at $dA$ here as being a map from $X_{\mathbb R}$ to $X_{\mathbb R}^*$. \end{say} \begin{proof} We claim that the hyperplanes that cut out the facets at the origin are in 1-to-1 correspondence with the maximal-dimensional Delaunay cells containing 0. Let $\sigma\ni0$ be one of such cells with a hole $\alpha$. Then the points $(x,A(x))$ with $x\in\sigma$ lie on the hyperplane $x_0=B(\alpha,x)+lx/2$ and those with $x\notin\sigma$ lie above it. Indeed, \begin{displaymath} A(x)-B(\alpha,x)-lx/2= B(x,x)/2-B(\alpha,x)=(\|\alpha-x\|^2-\|\alpha-0\|^2)/2 \ge0 \end{displaymath} and the equality holds if and only if $x\in\sigma$. The other vertices are checked similarly. \end{proof} \begin{rem}\label{rem:equations_for_hole} As the proof shows, for a maximal-dimensional Delaunay cell the hole $\alpha$ is the unique solution of the system of linear equations \begin{displaymath} 2B(\alpha,x)=B(x,x), \quad x\in\sigma\cap X. \end{displaymath} \end{rem} \begin{defn}\label{defn:eta_xc} We will denote the minimum of the functions $dA(\alpha(\sigma))(x)$ in the above lemma by $\eta(x,c)$. \end{defn} \begin{rem} Note that all $\eta(x,c)$ are integral-valued on $X$ if and only if all $\alpha(\sigma)$ are integral, i.e., belong to $X^*$. \end{rem} \begin{defn}\label{defn:star_Delaunay_primitive} The union of the Delaunay cells containing a lattice vertex $c$ is called the {\em star of Delaunay cells\/} and denoted by $\operatorname{Star}(c)$. For a cell $\sigma\subset\operatorname{Star}(0)$ the nonzero lattice elements $v_1\dots v_m\in\sigma$ are called the Delaunay vectors. In addition, we introduce \begin{displaymath} \operatorname{Cone}(0,\sigma)={\mathbb R}_+v_1+\dots+{\mathbb R}_+v_m, \end{displaymath} a convex cone with the vertex at the origin. By translation we obtain cones $\operatorname{Cone}(c,\sigma)$ with vertices at other lattice elements. The function $\eta(x,c)$ is linear on each $\operatorname{Cone}(c,\sigma)$, $\sigma\subset\operatorname{Star}(c)$. A lattice vector in a cone is called {\em primitive\/} if it cannot be written as a sum of two nonzero lattice vectors in this cone. We will denote by $\operatorname{Prim}$ the union of primitive vectors in Delaunay cells $\sigma$ for all $\sigma\subset\operatorname{Star}(0)$. We say that some elements $x_1\dots x_m$ are {\em cellmates\/} if there exists a (maximal-dimensional) Delaunay cell $\tau\ni0$ such that $x_1\dots x_m\in \operatorname{Cone}(0,\tau)$. \end{defn} \begin{say} We thank S. Zucker for suggesting the term ``cellmates''. \end{say} \begin{defn}\label{defn:generating_cell} We call the cell $\sigma$ {\em generating\/} if its Delaunay vectors contain a basis of $X$ and {\em totally generating\/} if, moreover, the integral combinations $\sum n_iv_i$ of Delaunay vectors with $n_i\ge0$ give all lattice elements of the cone $\operatorname{Cone}(0,\sigma)$. \end{defn} \begin{saynum} In dimensions $g\le4$ all Delaunay cells are totally generating as was proved by Voronoi in a classical series of papers \cite{Voronoi08all}. It was only recently shown that there exist non-generating cells in dimension 5 (see \cite{ErdahlRyshkov87}, p.796). But the following example from \cite{Erdahl92} of a non-generating cell is by far the easiest. \end{saynum} \begin{exmp}\label{exmp:E_8_not_generating} Take $B=E_8$, an even unimodular positive definite quadratic form given by a familiar $8\times8$ matrix with +2 on the main diagonal. By \ref{rem:equations_for_hole} we have \begin{displaymath} E_8(\alpha,v_i)=E_8(v_i,v_i)/2\in{\mathbb Z}, \quad v_i\in\sigma. \end{displaymath} If the Delaunay vectors $v_i$ contained a basis then we would have $\alpha\in X$, since $E_8$ is unimodular. But the holes cannot belong to the lattice by their very definition. In this example each maximal-dimensional Delaunay cell generates a sublattice of index 2 or 3. \end{exmp} \begin{defn}\label{defn:nilpotency} The {\em nilpotency\/} of a Delaunay cell $\sigma$ is the minimal positive integer $n$ such that the lattice generated by the vectors $v_1/n,\dots, v_m/n$ contains $X$. The nilpotency of the Delaunay decomposition is the least common multiple of nilpotencies of its cells. \end{defn} \begin{rem} Existence of non-generating cells is responsible for many combinatorial complications that occur in dimension $g\ge5$. \end{rem} \section{Degeneration data} \label{sec:Degeneration data} \begin{say} The purpose of this section is to recall the Mumford-Faltings-Chai description \cite[III]{FaltingsChai90} of the degenerations of abelian varieties. We will use this description in the next section. We refer the reader to \cite{FaltingsChai90} for basic definitions and facts about polarizations, semiabelian group schemes etc. For easier reference we use the same notation as in \cite{FaltingsChai90} where this is convenient. \end{say} \begin{notation} ${\mathcal R}$ is a Noetherian normal integral domain complete with respect to an ideal $I=\sqrt{I}$, $K$ is the fraction field and $k={\mathcal R}/I$ is the residue ring, $S=\operatorname{Spec}\, {\mathcal R}$, $S_0=\operatorname{Spec}\, k$, and $\eta=\operatorname{Spec}\, K$ is the generic point. ${\mathcal R}$ is assumed to be regular or to be the completion of a normal excellent domain. In our application ${\mathcal R}$ will be a DVR, and $k$ will be the residue field. \end{notation} \begin{defn} The objects of the category $\operatorname{DEG_{ample}}$ are pairs $(G,{\mathcal L})$, where $G$ is a semiabelian group scheme over $S$ with abelian $G_{\eta}$ and ${\mathcal L}$ is an invertible sheaf on $G$ with ample ${\mathcal L}_{\eta}$. The morphisms are group homomorphisms respecting ${\mathcal L}$'s. The objects of the category $\operatorname{DEG_{pol}}$ are pairs $(G,\lambda_{\eta})$ with $G$ as above and with the polarization $\lambda_{\eta}:G_{\eta}\to G^t_{\eta}$, where $G^t_{\eta}$ is the dual abelian variety. The morphisms are group homomorphisms respecting $\lambda$'s. \end{defn} \begin{say} Next, we recall the definition of the degeneration data. We will only need the split case. \end{say} \begin{defn} The degeneration data in the split case consists of the following: \begin{enumerate} \item An abelian scheme $A/S$ of relative dimension $a$, a split torus $T/S$ of relative dimension $r$, $g=a+r$, and a semiabelian group scheme $\widetilde G/S$, \begin{displaymath} 1\to T\to \widetilde G \overset{\pi}{\to} A \to 0. \end{displaymath} This extension is equivalent via negative of pushout to a homomorphism $c:\underline{X}\to A^t$. Here $X$ is a rank $r$ free commutative group, and $\underline{X}=X_S$ is a constant group scheme, the group of characters of $T$. $A^t/S$ is the dual abelian scheme of~$A/S$. \item A rank $r$ free commutative group $Y$ and the constant group scheme $\underline{Y}=Y_S$. \item A homomorphism $c^t:\underline{Y}\to A$. This is equivalent to giving an extension \begin{displaymath} 1\to T^t\to \widetilde G^t \to A^t \to 0, \end{displaymath} where $T^t$ is a torus with the group of characters $\underline{Y}$. \item An injective homomorphism $\phi:Y\to X$ with finite cokernel. \item A homomorphism $\iota: Y_{\eta}\to\widetilde G_{\eta}$ lying over $c^t_{\eta}$. This is equivalent to giving a bilinear section $\tau$ of $(c^t\times c)^* {\mathcal P}^{-1}_{A,\eta}$ on $Y\times X$, in other words a trivialization of the biextension $\tau:1_{Y\times X}\to (c^t\times c)^*{\mathcal P}^{-1}_{A,\eta}$. Here, ${\mathcal P}_{\eta}$ is the Poincare sheaf on $A_{\eta}\times A^t_{\eta}$, which comes with a canonical biextension structure. \item An ample sheaf ${\mathcal M}$ on $A$ inducing a polarization $\lambda_A:A\to A^t$ of $A/S$ such that $\lambda_Ac^t=c\phi$. This is equivalent to giving a $T$-linearized sheaf ${\widetilde {\mathcal L}}=\pi^*{\mathcal M}$ on $\widetilde G$. \item An action of $Y$ on ${\widetilde {\mathcal L}}_{\eta}$ compatible with $\phi$. This is equivalent to a cubical section $\psi$ of $(c^t)^*{\mathcal M}_{\eta}^{-1}$ on $Y$, in other words to a cubical trivialization $\psi:1_{Y}\to(c^t)^*{\mathcal M}_{\eta}^{-1}$, which is compatible with $\tau\circ(\operatorname{id}_Y\times\phi)$. $\psi$ is defined up to a shift by $Y$. \end{enumerate} The trivialization $\tau$ is required to satisfy the following {\em positivity condition}: $\tau(y,\phi y)$ for all $y$ extends to a section of ${\mathcal P}^{-1}$ on $A\underset{S}{\times} A^t$, and it is 0 modulo $I$ if $y\ne0$. The objects of the category $\operatorname{DD_{ample}}$ are the data above, and the morphisms are the homomorphisms of $\widetilde G$'s respecting this data. \end{defn} \begin{defn} Similarly, the objects of the category $\operatorname{DD_{pol}}$ consist of the data as above minus the sheaves ${\mathcal M}$, ${\widetilde {\mathcal L}}$ and the section $\psi$, with the positivity condition again. In addition, one requires the trivialization $\tau$ to be symmetric (in the previous case this was automatic). The morphisms are homomorphisms of $\widetilde G$'s respecting this data. \end{defn} \begin{thm}[Faltings-Chai]\label{thm:Faltings_Chai_equivalence} The categories $\operatorname{DEG_{ample}}$ and $\operatorname{DD_{ample}}$, resp. $\operatorname{DEG_{pol}}$ and $\operatorname{DD_{pol}}$ are equivalent. \end{thm} \begin{say} In the case where $A=0$ and $\widetilde G=T$ is a torus (or, more generally, when $c=c^t=0$) the section $\tau$ is simply a bilinear function $b:Y\times X\to K^*$, and $\psi$ is a function $a:Y\to K^*$ satisfying \begin{enumerate} \item $b(y_1,\phi y_2)=a(y_1+y_2)a(y_1)^{-1}a(y_2)^{-1}$, \item $b(y,\phi y)\in I$ for $y\ne0$. \end{enumerate} This also implies that \begin{enumerate}\setcounter{enumi}{2} \item $b(y_1,\phi y_2) = b(y_2, \phi y_1)$, \item $a(0)=1$, \item $a(y_1+y_2+y_3) a(y_2+y_3)^{-1} a(y_3+y_1)^{-1} a(y_1+y_2)^{-1} a(y_1) a(y_2) a(y_3) =1$. \end{enumerate} \end{say} \begin{say} We will use the following notation. $\widetilde G$ is affine over $A$ and one has $\widetilde G = \operatorname{Spec}\,_A (\oplus_{x\in X} {\mathcal O}_x)$. Each ${\mathcal O}_x$ is an invertible sheaf on $A$, canonically rigidified along the zero section, and one has ${\mathcal O}_x\simeq c(x)$. The pushout of the $T$-torsor $\widetilde G$ over $A$ by $x\in X$ is ${\mathcal O}_{-x}$. The sheaf ${\mathcal M}\otimes{\mathcal O}_x$, rigidified along the zero section, is denoted by ${\mathcal M}_x$. The $Y$-action on $(\oplus_{x\in X}{\mathcal M}_x)\underset{{\mathcal R}}{\otimes}K$ defined by $\psi$ is denoted by $S_y:T^*_{c^t(y)}{\mathcal M}_x \to {\mathcal M}_{x+\phi y,\eta}$. \end{say} \section{Simplified Mumford's construction} \label{sec:Simplified Mumford's construction} \begin{setup} In this section, ${\mathcal R}$ is a DVR, $I=(s)$, and $k$ is the residue field. We will denote the point $S_0$ simply by $0$. We start with an abelian variety $A_{\eta}$ with an effective ample Cartier divisor $\Theta_{\eta}$ defining principal polarization, and ${\mathcal L}_{\eta}={\mathcal O}(\Theta_{\eta})$. Applying the stable reduction theorem (\cite{SGA71,ArtinWinters_StableReduction}) after a finite base change $S'\to S$ we have a semiabelian group scheme $G'/S'$ and an invertible sheaf ${\mathcal L}$ extending $(A_{\eta}',{\mathcal L}_{\eta}')$ such that the toric part $T'_0$ of the central fiber $G'_0$ is split. In order not to crowd notation, we will continue to denote the objects by $S$, $G$, ${\mathcal L}$ etc. We have an object of $\operatorname{DEG_{ample}}$, and, by the previous section, an object of $\operatorname{DD_{ample}}$, i.e., the degeneration data. Since the polarization is principal, $\phi:Y\to X$ is an isomorphism and we can identify $Y$ with $X$. Further, the sheaf ${\mathcal M}$ on $A$ defines a principal polarization. We denote by $\theta_A$ a generator of $H^0(A,{\mathcal M})$. \end{setup} \begin{say} Here is the main object of our study: \end{say} \begin{defn} Consider the graded algebra \begin{displaymath} {\mathcal S}_2=\big(\sum_{d\ge0}(\oplus_{x\in X}{\mathcal O}_x)\otimes {\mathcal M}^d\vartheta^d\big)\underset{R}{\otimes}K, \end{displaymath} where $\vartheta$ is an indeterminate defining the grading. In this algebra consider the subalgebra $_1R$ generated in degree by the ${\mathcal M}_0={\mathcal M}$ and all its $Y$-translates, $S^*_y({\mathcal M}_0)$. This is a locally free graded ${\mathcal O}_A$-algebra. Finally, the algebra $R$ is the saturation of $_1R$ in an obvious sense which will be further explained below. We define the scheme $\widetilde P=\operatorname{Proj}_A R$ and the sheaf ${\widetilde {\mathcal L}}$ on it as ${\mathcal O}(1)$. For each $x\in X=Y$ we have an element $S_x^*(\theta_A)\in H^0(A,{\mathcal M}_x)$ that will be denoted by $\xi_x$. We have a formal power series \begin{displaymath} {\tilde\theta}= \sum_{x\in X} \xi_x. \end{displaymath} \end{defn} \begin{say} We will see that, possibly after another finite base change $S'\to S$, the scheme $\widetilde P$ is a {\em relatively complete model\/} as defined in \cite[III.3]{FaltingsChai90}. Via Mumford's construction, this gives a projective scheme $P/S$ extending $A_{\eta}$. We will see that it naturally comes with a relative Cartier divisor $\Theta$. \end{say} \begin{say} The subalgebra $R$ defines a subalgebra $R'$ in \begin{displaymath} {\mathcal S}_1=\big(\sum_{d\ge0}(\oplus_{x\in X}{\mathcal O}_x)\vartheta^d\big)\underset{R}{\otimes}K. \end{displaymath} One has $\operatorname{Proj} R=\operatorname{Proj} R'$ and ${\widetilde {\mathcal L}}\simeq{\widetilde {\mathcal L}}'\otimes\pi^*{\mathcal M}$. \end{say} \subsection{Case of maximal degeneration} \label{subsec:simplified_max_deg} \mbox{}\smallskip \begin{say} In this case, $A=0$ and $\widetilde G=T=\operatorname{Spec}\, R[w^x \,;\, x\in X]$. $_1R$ is the $R$-subalgebra of $K[\vartheta,w^x \,;\, x\in X]$ generated by $a(x)\vartheta$. We have \begin{displaymath} {\tilde\theta}=\sum_{x\in X}\xi_x = \sum_{x\in X} a(x)w^x \vartheta. \end{displaymath} \end{say} \begin{defn} We define the functions $a_0:Y\to k^*$, $A:Y\to{\mathbb Z}$, $b_0:Y\times X\to k^*$, $B:Y\times X\to{\mathbb Z}$ by setting \begin{displaymath} a(y)=a'(y)s^{A(y)}, \qquad b(y,x)=b'(y,x)s^{B(y,x)} \end{displaymath} with $a'(y),b'(y) \in R\setminus I$ and taking $a_0,b_0$ to be $a',b'$ modulo $I$. \end{defn} \begin{rem} We are using the letter $A$ for two purposes now: to denote an abelian variety, and to denote the integral-valued function above. This should not lead to any confusion since their meanings are very different. \end{rem} \begin{say} Through our identification $\phi:Y{\overset{\sim}{\rightarrow}} X$ the functions $a,A,b$ and $B$ become functions on $X$ and $X\times X$. The functions $a$ and $A$ are quadratic non-homogeneous, the functions $b,B$ are symmetric and they are the homogeneous parts of $a^2,2A$, respectively. We have \begin{displaymath} A(x)=B(x,x)/2+lx/2 \end{displaymath} for some $l\in X^*$. The positivity condition implies that $B$ is positive definite. \end{say} \begin{rem} The function $B:Y\times X\to{\mathbb Z}$ describes the monodromy of the family $G$ and is called the monodromy pairing, cf. \cite[IX.10.4]{SGA71}. \end{rem} \begin{say} Since all $a'(y)$ are invertible in ${\mathcal R}$, the algebra $_1R$ is generated by monomials $\zeta_x=s^{A(x)}w^x\vartheta$, so it is a semigroup algebra. \end{say} \begin{defn} We introduce two lattices $M=X\oplus{\mathbb Z} e_0\simeq {\mathbb Z}^{r+1}$ and its dual $N=X^*\oplus{\mathbb Z} f_0$. \end{defn} \begin{saynum} Each $\zeta_x$ corresponds to a lattice element $(x,A(x))\in M$. These are exactly the vertices of the multifaceted paraboloid $Q$ in Figure~3 which we imagine lying in the hyperplane $(1,M)$ inside ${\mathbb Z}\oplus M$. The extra ${\mathbb Z}$ corresponds to the grading by $\vartheta$. The saturation $R$ of $_1R$ is generated by monomials corresponding to all lattice vectors lying inside $\operatorname{Cone}(Q)$. \end{saynum} \begin{thm}\label{thm:first_structure_thm_family} \begin{numerate} \item $\widetilde P$ is covered by the affine toric schemes $U(c)=\operatorname{Spec}\, R(c)$, $c\in X$, where $R(c)$ is the semigroup algebra corresponding to the cone at the vertex $c\in Q$ of lattice elements \begin{displaymath} \{(x,x_0) \,|\, x_0 \ge \eta(x,c) \} \end{displaymath} ($\eta(x,c)$ is defined in \ref{defn:eta_xc}). \item $R(c)$ is a free ${\mathcal R}$-module with the basis $\zeta_{x,c}=s^{\ulcorner\eta(x,c)\urcorner}w^x$ (here $\ulcorner z\urcorner$ denotes the least integer $\ge z$). \label{numi_basis_Rc} \item All the rings $R(c)$ are isomorphic to each other, and each is finitely generated over ${\mathcal R}$. The scheme $\widetilde P$ is locally of finite type over ${\mathcal R}$. \item $\operatorname{Spec}\, R(c)$ is the affine torus embedding over $S=\operatorname{Spec}\, {\mathcal R}$ corresponding to the cone $\Delta(c)$ over \begin{displaymath} \big(1,-dA(\widehat c) \big) \subset(1,N_0) \subset N, \end{displaymath} where $\widehat c$ is the Voronoi cell dual to $c$. \item $\widetilde P$ is the torus embedding $T_N\operatorname{emb}\Delta$ where $\Delta$ is the fan in $N_{{\mathbb R}}$ consisting of $\{0\}$ and the cones over the shifted Voronoi decomposition $(1,-dA(\operatorname{Vor}_B))$. The morphism $\widetilde P\to S$ is described by the map of fans from $\Delta$ to the half line ${\mathbb R}_{\ge0}f_0$. \item ${\widetilde {\mathcal L}}$ is invertible and ample. \item One has natural compatible actions of $T$ on $\widetilde P$ and of $T\times{\mathbb G}_m$ on ${\widetilde {\mathcal L}}$. \end{numerate} \end{thm} \begin{rem} The reference for torus embeddings over a DVR is \cite[IV,\S 3]{ToroidalEmbeddingsI}. Formally, all computations work the same way as for a toric variety with torus action of $k[s,1/s,w^x \,;\, x\in X]$. \end{rem} \begin{proof} The first part of (i) is simply the description of the standard cover of $\operatorname{Proj}$ by $\operatorname{Spec}\,$'s. The second part, as well as (iv) and (v) follow immediately from lemma \ref{lem:paraboloid}. (ii) and (iii) follow at once from (i). The ring extension $_1R\subset R$ is integral, hence $\operatorname{Proj} R\to \operatorname{Proj}\, _1R$ is well-defined and is finite. The sheaf ${\mathcal O}(1)$ on $\operatorname{Proj}\, _1R$ is invertible and ample, and ${\widetilde {\mathcal L}}$ is its pullback. This gives (vi). The actions in (vii) are defined by the $X$-, resp. $(X\oplus{\mathbb Z})$-gradings. \end{proof} \begin{say} As a consequence, we can apply the standard description in the theory of torus embeddings of the open cover, torus orbits and their closures: \end{say} \begin{thm}\label{thm:toric_descr_family} \begin{numerate} \item For each Delaunay cell $\sigma\in\operatorname{Del}_B$ one has a ring $R(\sigma)$ corresponding to the cone over the dual Voronoi cell $\hat\sigma$. $U(\sigma)$ is open in $\widetilde P$ and $U(\sigma_1)\cap U(\sigma_2)= U(D(\hat\sigma_1\cap \hat\sigma_2))$. \item $R(\sigma)$ is the localization of $R(c)$, $c\in\sigma$, at $\zeta_{d,c}$, $d\in X\cap {\mathbb R}(\sigma-c)$. \item In the central fiber $\widetilde P_0$, the $T_0$-orbits are in 1-to-1 dimension-preserving correspondence with the Delaunay cells $\sigma$. In particular, the irreducible components of $\widetilde P$ correspond to the maximal-dimensional cells. \item The closure $\widetilde V(\sigma)$ of $\operatorname{orb}(\sigma)$ together with the restriction of the line bundle ${\widetilde {\mathcal L}}$ is a projective toric variety over $k$ with a $T_0$-linearized ample line bundle corresponding to the lattice polytope $\sigma$. \item $\widetilde V(\sigma_1)\cap \widetilde V(\sigma_2)= \widetilde V(\sigma_1\cap \sigma_2)$. \item For a maximal-dimensional cell $\sigma$, the multiplicity of $\widetilde V(\sigma)$ in $\widetilde P_0$ is the denominator of $dA(\alpha(\sigma))\in X^*_{{\mathbb Q}}$. \end{numerate} \end{thm} \begin{question} For a maximal-dimensional cell $\sigma$, when is $\widetilde P$ generically reduced at $\widetilde V(\sigma)$? In other words, when is $dA(\alpha(\sigma))$ integral? \end{question} \begin{lem} $dA(\alpha(\sigma))\in X^*$ in any of the following cases: \begin{numerate} \item $\sigma$ is generating. \item $A(x)/n\in{\mathbb Z}$ for every $x\in X$, where $n$ is the nilpotency of $\sigma$. \end{numerate} \end{lem} \begin{proof} (i) is a particular case of (ii), so let us prove the second part. $dA(\alpha)\in X^*$ if and only if $dA(\alpha)(x)\in{\mathbb Z}$ for every $x\in X$. Now let $v_1\dots v_m$ be the Delaunay vectors of $\sigma$. By the definition of the nilpotency in Definition~\ref{defn:nilpotency} we have $x=(1/n) \sum n_iv_i$ for some $n_i\in{\mathbb Z}$. Then \begin{align*} dA(\alpha)(x) &= B(\alpha,x) + lx/2 = \frac{1}{n}\sum n_i\big( B(\alpha,v_i) +lv_i/2 \big)\\ &= \frac{1}{n}\sum n_i \big( B(v_i,v_i)/2 +lv_i/2 \big) = \sum n_i A(v_i)/n \in{\mathbb Z} \end{align*} \end{proof} \begin{saynum} For the central fiber $\widetilde P_0$ to be generically reduced, we need $A(x)$ to be divisible by the nilpotency of the lattice. This certainly holds after a totally ramified base change. Consider the polynomial $z^n-s\in K[z]$. It is irreducible by the Eisenstein criterion. The field extension $K\subset K'=K[z]/(z^n-s)$ has degree $n$ and is totally ramified. The integral closure of $R$ in $K'$ is again a DVR, complete with respect to the maximal ideal $I'=(s')$ (see e.g. \cite[II.3]{Serre79}) and $\operatorname{val}_{s'}(s)=n$, so that $A'(x)=nA(x)$. The following example, very similar to Example \ref{exmp:E_8_not_generating}, shows that this base change is indeed sometimes necessary. \end{saynum} \begin{exmp} Consider the degeneration data $A(x)=E_8(x)/2$, $B(y,x)=E_8(y,x)$. Then $dA=E_8$ and for every hole $\alpha(\sigma)$ we have $dA(\alpha)\notin X^*$. Indeed, otherwise we would have $\alpha\in X$ since $E_8$ is unimodular, and this is impossible by the definition of a hole. In this example every irreducible component of the central fiber $\widetilde P_0$ has multiplicity $2$ or $3$. \end{exmp} \begin{assume}\label{assume:base_change_made} From now on, we assume that the necessary base change has been done, so $dA(\alpha)$ is integral for each hole $\alpha$. This implies that all $\eta(x,c)$ are integral-valued on $X$. \end{assume} \begin{saynum}\label{saynum:Yaction_wP} {\bf $Y$-action on $\widetilde P$.} We are given a canonical $Y$-action on $K[\vartheta,w^x \,;\,x\in X]$ by construction. It is constant on $K$ and sends each generator $\xi_x=a(x)w^x\vartheta$ to another generator $\xi_{x+y}=a(x+y)w^{x+y}\vartheta$. Precisely because $a(x)$ is quadratic, this action extends uniquely to the whole $K[\vartheta,w^x]$. Clearly, the subrings $_1R$ and $R$ are $Y$-invariant, so we have the $Y$-action on $R$ which will be denoted by $S^*_y$. We easily compute: \end{saynum} \begin{eqnarray*} && S^*_y( \frac{a'(x+c)}{a'(x)}\zeta_{x,c} ) = \frac{a'(x+c+y)}{ a'(x+y)} \zeta_{x,c+y}, \\ && S^*_y(\zeta_{x,c} ) = b'(x,y) \zeta_{x+y,c}. \end{eqnarray*} \begin{say} This describes the action $S_y^*:R(c)\to R(c+y)$ and $S_y:\operatorname{Spec}\, R(c+y) \to \operatorname{Spec}\, R(c)$. \end{say} \begin{thm}\label{thm:structure_of_central_fiber} $\widetilde P_0$ is a scheme locally of finite type over $k$. It is covered by the affine schemes $\operatorname{Spec}\, R_0(\sigma)$ of finite type over $k$ for Delaunay cells $\sigma\in\operatorname{Del}_B$, where $R_0(\sigma)=R(\sigma)\underset{{\mathcal R}}{\otimes}k$. The following holds: \begin{numerate} \item $R_0(c)$ is a $k$-vector space with basis $\{\bar\zeta_{x,c}\,;\,x\in X\}$, the multiplication being defined by \begin{displaymath} \bar\zeta_{x_1,c}\dots\bar\zeta_{x_m,c}=\bar\zeta_{x_1+\dots+x_m,c} \end{displaymath} if $x_1\dots x_m$ are cellmates with respect to the $B$-Delaunay decomposition, and $0$ otherwise. \label{enumi:basis_R0} \item For $\sigma\ni c$ the ring $R_0(\sigma)$ is the localization of $R_0(c)$ at $\{\bar\zeta_{d,c}\,;\, d\in X\cap{\mathbb R}(\sigma-c)\}$. \item $U_0(\sigma_1)\cap U_0(\sigma_2)= U_0(D(\hat\sigma_1\cap\hat\sigma_2))$. \item The group $Y$ of periods acts on $\widetilde P_0$ by sending $\operatorname{Spec}\, R_0(c+\phi(y))$ to $\operatorname{Spec}\, R_0(c)$ in the following way: \begin{displaymath} S^*_y(\bar\zeta_{x,c})= b_0(y,x)\, \bar\zeta_{x,c+\phi(y)}. \end{displaymath} \end{numerate} \end{thm} \begin{proof} This follows from \ref{thm:first_structure_thm_family}, \ref{thm:toric_descr_family} and \ref{saynum:Yaction_wP}. Here is a way to see (i) geometrically: each $\zeta_{x_i,c}$ corresponds to a point on a face of the cone of $Q$ at $c$. The sum of several such points lie on a face if and only if they belong to a common face, i.e., if and only if $x_1\dots x_m$ are cellmates. Otherwise, the product corresponds to a point in the interior of the cone and equals $s^n\zeta_{x_1+\dots+x_m,c}$ for some $n>0$. Therefore, it reduces to $0$ modulo $(s)$. \end{proof} \begin{cor}\label{cor:central_fiber_reduced} $\widetilde P_0$ is reduced and geometrically reduced. \end{cor} \begin{lem}\label{lem:theta_n} For each $n$ only finitely many of the elements $\xi_{x+c}\xi_c^{-1}$ in $R_0(c)$ are not zero modulo $I^{n+1}=(s^{n+1})$. For $n=0$ the only ones not zero correspond to the lattice points $x+c\in\operatorname{Star}(\operatorname{Del}_B,c)$, i.e., $x\in\operatorname{Star}(\operatorname{Del}_B,0)$. \end{lem} \begin{proof} Indeed, $\xi_{x+c}\xi_c^{-1}=\zeta_{x,c}s^{B(x,x)/2-B(\alpha,x)}$, and $B$ is positive definite. The second part was proved in Lemma \ref{lem:paraboloid}. \end{proof} \begin{defn} We define the Cartier divisor ${\widetilde\Theta}_0$ on $\widetilde P_0$ by the system of compatible equations $\{{\tilde\theta}/\xi_c \in R_0(c) \}$. Explicitly, \begin{displaymath} {\tilde\theta}/\xi_c = \sum_{x\subset\operatorname{Star}(\operatorname{Del}_B,0)} a_0(x+c)a^{-1}_0(c) \bar \zeta_{x,c}. \end{displaymath} \end{defn} \begin{say} Clearly, ${\tilde\theta}$ defines a $Y$-invariant global section of ${\widetilde {\mathcal L}}_0={\widetilde {\mathcal L}}|_{\widetilde P_0}$, so the divisor ${\widetilde\Theta}_0$ is $Y$-invariant. \end{say} \begin{lem} Once the base change in Assumption \ref{assume:base_change_made} has been made, for any further finite base change $S'\to S$, one has $\widetilde P'\simeq \widetilde P\underset{S}{\times} S'$. In other words, our construction is stable under base change. \end{lem} \begin{proof} This statement is sufficient to check for the semigroup algebras $R(c)$, which is obvious using the basis in \ref{thm:first_structure_thm_family}\ref{numi_basis_Rc}. \end{proof} \subsection{Case of arbitrary abelian part} \label{subsec:Case of principal polarization and arbitrary abelian part:simp} \mbox{}\smallskip \begin{saynum}\label{saynum:descr_wP_nontriv_ab_part} Most of the statements above transfer to the general case without any difficulty. The main difference is that $\widetilde P$ is now fibered over $A$ instead of a point, and each $U(\sigma)$, resp. $U_0(\sigma)$ is an affine scheme over $A$, resp. $A_0$. One easily sees that $\widetilde P$ is isomorphic to the contracted product $\widetilde P^r\overset{T}{\times}\widetilde G$ of an $r$-dimensional scheme $\widetilde P^r$ over $S$ corresponding to the positive definite integral-valued bilinear form $B(x,y)$ on the $r$-dimensional lattice $X$ with $\widetilde G$. Recall that the contracted product is the quotient of $\widetilde P^r\underset{S}{\times}\widetilde G$ by the free action of $T$ with the standard action on the first factor and the opposite action on the second factor. In the same way, $\widetilde P_0 \simeq \widetilde P_0^r\overset{T}{\times}\widetilde G_0$. Moreover, the ${\mathbb G}_m$-torsor $\widetilde {\mathbf L}'$ corresponding to the sheaf ${\widetilde {\mathcal L}}'\simeq {\widetilde {\mathcal L}}\otimes {\mathcal M}^{-1}$ is the contracted product of the ${\mathbb G}_m$-torsor $\widetilde {\mathbf L}^r$ and $\widetilde G\underset{S}{\times}{\mathbb G}_{m,S}$, and similarly for the central fiber. The power series ${\tilde\theta}=\sum\xi_x$ defines a $Y$-invariant section of ${\widetilde {\mathcal L}}$. \end{saynum} \begin{lem}\label{lem:stratification_wP0} $\widetilde P_0$ is a disjoint union of semiabelian varieties $\widetilde G_0(\sigma)$ which are in 1-to-1 correspondence with the Delaunay cells. One has \begin{displaymath} 1\to \operatorname{orb}(\sigma) \to \widetilde G_0(\sigma) \to A \to 0. \end{displaymath} The closure $\widetilde V(\sigma)$ of $\widetilde G_0(\sigma)$ is a projective variety $\widetilde V^r(\sigma)\overset{T}{\times}\widetilde G_0$. For a $0$-dimensional cell $c\in X$ the restriction of ${\tilde\theta}$ to $\widetilde V(c)\simeq A$ is $\xi_x$ and $(\xi_x)=T_{c^t(x)}(\Theta_A)$. In particular, ${\widetilde\Theta}=({\tilde\theta})$ does not contain any of the strata entirely. \end{lem} \begin{proof} The first part follows at once from Theorem \ref{thm:toric_descr_family} by applying the contracted product. The second part is obvious because all the other $\xi_x$, $x\ne c$ are zero on $\widetilde V(c)$. \end{proof} \subsection{Taking the quotient by $Y$} \label{subsec:Taking the quotient by Y} \mbox{}\smallskip \begin{lem} $\widetilde P$ is a relatively complete model as defined in \cite[III.3.1]{FaltingsChai90}. \end{lem} \begin{proof} We do not even recall the fairly long definition of a relatively complete model because most of it formalizes what we already have: a scheme $\widetilde P$ locally of finite type over $R$ with an ample sheaf ${\widetilde {\mathcal L}}$, actions of $Y$ and $T$ etc. There are two additional conditions which we have not described yet. The first one is the completeness condition. It is quite tricky but it is used in \cite{Mumford_AnalyticDegsAVs,FaltingsChai90} only to prove that every irreducible component of $\widetilde P_0$ is proper over $k$. We already know this from \ref{thm:toric_descr_family}. The second condition is that we should have an embedding $\widetilde G\hookrightarrow\widetilde P$. It is sufficient to give such an embedding for the toric case, since then we simply apply the contracted product. In the toric language, $T$ corresponds to a fan in $N_{{\mathbb R}}={\mathbb X}_{{\mathbb R}}^*\oplus{\mathbb R}$ consisting of the ray ${\mathbb R}_{\ge0}f_0$. A map from this fan to the fan $\Delta$, which sends $f_0$ to $\big(1,-dA(\alpha(\sigma))\big)$ for an arbitrary maximal-dimensional Delaunay cell $\sigma$, defines an embedding $T\hookrightarrow \widetilde P^r$. We have used the fact that $dA(\alpha(\sigma))$ is integral here. \end{proof} \begin{rem} Note that the embedding $\widetilde G\hookrightarrow\widetilde P$ defines a section of $\widetilde P$ which has absolutely nothing to do with the zero section $z_{\eta}$ of $A_{\eta}$ and its closure $z$. The embedding $z\hookrightarrow P$ is described by the embedding of fans $({\mathbb Z},{\mathbb R}_{\ge0}f_0)\hookrightarrow (N,\Delta)$, $f_0\mapsto f_0$. From this, we see that $z_0\in\operatorname{orb}(\sigma)$, where $\sigma$ is the ``bottom'' face of the hyperboloid $Q$. It need not be maximal-dimensional. \end{rem} \begin{saynum} We can now apply Mumford's construction as described in \cite{Mumford_AnalyticDegsAVs,FaltingsChai90}. This consists of considering all fattenings $(\widetilde P_n,{\widetilde {\mathcal L}}_n)=(\widetilde P,{\widetilde {\mathcal L}})\underset{R}{\times} R/I^{n+1}$, their quotients $(P_n,{\mathcal L}_n)=(\widetilde P_n,{\widetilde {\mathcal L}}_n)/Y$ and then algebraizing this system to a projective scheme $(P,{\mathcal L})/S$ such that the generic fiber $P_{\eta}$ is abelian and ${\mathcal L}_{\eta}$ defines a principal polarization. By \ref{thm:Faltings_Chai_equivalence}, $(A_{\eta},{\mathcal O}(\Theta_{\eta}))\simeq (P_{\eta},{\mathcal L}_{\eta})$, and since the polarization is principal, this isomorphism is uniquely defined. Thus, we have obtained the extended family. To this construction we will add a theta divisor. By Lemma \ref{lem:theta_n} for each $n$ the power series ${\tilde\theta}$ defines a finite sum in each $R_n(c)$. Hence, we have a compatible system of $Y$-invariant sections of ${\widetilde {\mathcal L}}_n$ that descend to compatible sections of ${\mathcal L}_n$ that algebraize to a section $\theta$ of ${\mathcal L}$. \end{saynum} \begin{say} From Lemma \ref{lem:stratification_wP0} we have: \end{say} \begin{cor}\label{cor:stratification_P0} $P_0$ is a disjoint union of semiabelian varieties $G_0(\bar\sigma)$ which are in 1-to-1 correspondence with the classes of Delaunay cells modulo $Y$-translations. $\Theta_0=(\theta_0)$ does not contain any of the strata entirely. \end{cor} \begin{defn} In the split case, a {\em stable quasiabelian pair}, SQAP for short, over a field $k$ is a pair $(P_0,\Theta_0)$ from our construction. In general, a pair of a reduced projective variety and an ample Cartier divisor over $k$ is called a stable quasiabelian pair if it becomes one after a field extension. We will call $P_0$ itself a stable quasiabelian variety, SQAV for short. \end{defn} \begin{say} Let $\Gamma^2(X^*)$ be the lattice of integral-valued symmetric bilinear forms on $X\times X$. For each Delaunay decomposition $\operatorname{Del}$ let $K(\operatorname{Del})$ be the subgroup of $\Gamma^2(X^*)$ generated by the positive-definite forms $B$ with $\operatorname{Del}_B=\operatorname{Del}$, and set $N(\operatorname{Del})=\Gamma^2(X^*)/K(\operatorname{Del})$. \end{say} \begin{thm} Each SQAP over an algebraically closed field $k$ is uniquely defined by the following data: \begin{enumerate} \item a semiabelian variety $G_0$, $1\to T_0\to G_0 \to A_0\to 0$, \item a principal polarization $\lambda_{A_0}:A_0\to A_0^t$, \item a Delaunay decomposition $\operatorname{Del}$ on $X$, where $X=Y$ is the group of characters of $T_0$, \item a class of bilinear symmetric sections $\tau_0$ of $(c^t_0\times c_0)^*{\mathcal P}^{-1}_{A_0}$ on $X\times X$ modulo the action of the torus $K(\operatorname{Del})\otimes{\mathbb G}_{m,k}$. \end{enumerate} \end{thm} \begin{proof} It is clear from the construction that an SQAP depends only on $A_0,\Theta_0$ and $A,B,\psi_0,\tau_0$. Since we do not care about the origin and the polarization is principal, giving the pair $(A_0,\Theta_0)$ is the same as giving the pair $(A_0,\lambda_{A_0})$. Replacing $\psi$ by $\psi_1$ with the same homogeneous part does not change the isomorphism classes of subalgebras $_1R$ and $R$, since the relations between $S_y({\mathcal M}_0)$ remain the same. The only information we are getting from $B$ is the Delaunay decomposition. Moreover, for a fixed $B$ if we replace the uniformizing parameter $s$ by $\mu s$, $\mu\in R\setminus I$, then the central fiber will not change, but $b(y,x)$ will change to $b(y,x) \mu_0^{B(y,x)}$. Therefore we only have the equivalence class by the $K(\operatorname{Del})\otimes{\mathbb G}_{m,k}$-action. \end{proof} \begin{rem} We can further divide the data above by the finite group of automorphisms of $(A_0,\lambda_{A_0})$ extended by the finite subgroup of $\operatorname{GL}(X)$ preserving $\operatorname{Del}$. The quotient data exactly corresponds to the $k$-points of the Voronoi compactification of $A_g$. Hence, every $k$-point of $\overline{A}_g^{\operatorname{VOR}}$ defines a unique SQAP over $k$. \end{rem} \begin{rem} In \cite{Mumford_KodairaDimSiegel} Mumford considered the first-order degenerations of abelian varieties over ${\mathbb C}$. These are exactly our pairs in the case where the toric part of $\widetilde G$ is 1-dimensional, i.e., $r=1$. \end{rem} \begin{say} In conclusion, we would like to make the following obvious observation. \end{say} \begin{lem} The family $P\to S$ is flat. The family of theta divisors $\Theta\to S$ is also flat. \end{lem} \begin{proof} Indeed, $S$ is integral and regular of dimension 1, and $P$ is reduced and irreducible so the statement follows, e.g., by \cite[III.9.7]{Hartshorne77}. The family of divisors $\Theta\to S$ is flat because $\Theta\cdot P_t$ is defined at every point $t\in S$ (\cite[III.9.8.5]{Hartshorne77}). \end{proof} \section{Further properties of SQAVs} \label{sec:Geometric properties of stable quasiabelian varieties} \begin{say} All statements in this section are stable under field extensions. Therefore without loss of generality we may assume that $k$ is algebraically closed. \end{say} \begin{lem} $P_0$ is Gorenstein. \end{lem} \begin{proof} A Noetherian local ring is Gorenstein if and only if its formal completion is such (\cite[18.3]{Matsumura89}). Therefore, we can check this property on an \'etale cover $\widetilde P_0$ of $P_0$. Moreover, the purely toric case suffices, since $\widetilde P_0$ is a fibration over a smooth variety $A_0$ with a fiber $\widetilde P_0^r$, locally trivial in \'etale topology. Recall that the scheme $\widetilde P$, as any torus embedding, is Cohen-Macaulay. $\widetilde P_0\subset\widetilde P$ is the union of divisors $\widetilde V(\sigma)$ corresponding to the 1-dimensional faces $\Delta(\sigma)$ of the fan $\Delta$ (i.e., to maximal-dimensional Delaunay cells $\sigma$). The following is a basic formula for the dualizing sheaf of a torus embedding: \begin{displaymath} \omega_{\widetilde P}= {\mathcal O}(-\sum \widetilde V(\sigma)) = {\mathcal O}(-\widetilde P_0). \end{displaymath} Since $\widetilde P_0$ is Cartier, $\omega_{\widetilde P}$ is locally free, so $\widetilde P$ is Gorenstein. The scheme $\widetilde P_0$ is then Gorenstein as a subscheme of a Gorenstein scheme that is defined by one regular element. Alternatively, $\widetilde P_0$ is Gorenstein as the complement of the main torus in a toric variety, see \cite[p.126, Ishida's criterion]{Oda_ConvexBodies}. \end{proof} \begin{lem} $\omega_{P_0}\simeq{\mathcal O}_{P_0}$. \end{lem} \begin{proof} As above, we have the canonical isomorphism \begin{displaymath} \omega_{\widetilde P}(\widetilde P_0) \simeq {\mathcal O}_{\widetilde P}, \end{displaymath} which by adjunction gives $\omega_{\widetilde P_0}\simeq{\mathcal O}_{\widetilde P_0}$. Since both sides are invariant under the action of lattice $Y$, this isomorphism descends to $P_0$. \end{proof} \begin{thm} $$h^i(P_0,{\mathcal O})=\binom gi$$. \end{thm} \begin{proof} We want to exploit the fact that $\widetilde P_0$ is built of ``blocks'' $\widetilde V(\sigma)$ and that the cohomologies of each block are easily computable. Indeed, since the fibers of $\widetilde V(\sigma)\to A_0$ are toric varieties $\widetilde V^r(\sigma)$ and $H^i(\widetilde V^r(\sigma),{\mathcal O})=0$ for $i>0$, we have $R^i\pi_*{\mathcal O}_{\widetilde V(\sigma)}=0$ for $i>0$ and $H^i(\widetilde V(\sigma),{\mathcal O})=H^i(A_0,{\mathcal O})$. It is well-known that for an abelian variety $A_0$ these groups have dimension $\binom ai$. The following important sequence for a union of torus orbits is contained in \cite[p.126]{Oda_ConvexBodies}, where it is called Ishida's complex: \begin{displaymath} 0\to {\mathcal O}_{\widetilde P^r_0}\to \oplus_{\dim\sigma=g} {\mathcal O}_{\widetilde V^r(\sigma)} \to \dots \oplus_{\dim\sigma=0} {\mathcal O}_{\widetilde V^r(\sigma)} \to 0. \end{displaymath} The morphisms in this sequence are the restrictions for all pairs $\sigma_1\supset\sigma_2$, taken with $\pm$ depending on a chosen orientation of the cells. By taking the contracted product, we obtain a similar resolution for the sheaf ${\mathcal O}_{\widetilde P_0}$, with $\widetilde V^r(\sigma)$ replaced by $\widetilde V(\sigma)$. Finally, by dividing this resolution by the $Y$-action, we obtain a resolution of ${\mathcal O}_{P_0}$. In this resolution the morphism ${\mathcal O}_{\widetilde V(\bar\sigma_1)} \to {\mathcal O}_{\widetilde V(\bar\sigma_2)}$ is a linear combination of several restriction maps according to the ways representatives of $\bar\sigma_1$ contain representatives of $\sigma_2$. We can now compute $H^i({\mathcal O}_{\widetilde P_0})$ by using the hypercohomologies of the above complex. First, consider the special case where $r=g$ and $a=0$. In this case each $H^0(\widetilde V(\sigma),{\mathcal O})$ is 1-dimensional and the higher cohomologies vanish. Therefore, $H^i(P_0,{\mathcal O})$ are the cohomologies of the complex \begin{displaymath} 0\to \oplus_{\dim\bar\sigma=g} H^0(\widetilde V(\bar\sigma),{\mathcal O}) \to \dots \oplus_{\dim\bar\sigma=0} H^0(\widetilde V(\bar\sigma),{\mathcal O}) \to 0. \end{displaymath} But this complex computes the cellular cohomologies of the cell complex $\operatorname{Del}_B/Y$ whose geometric representation is homeomorphic to ${\mathbb R}^r/{\mathbb Z}^r$. Hence $h^i=\binom ri=\binom gi$. In general, we obtain the spectral sequence $E_1^{pq}=H^p({\mathbb R}^r/{\mathbb Z}^r, H^q(A_0,{\mathcal O})) \Rightarrow H^{p+q}(P_0,{\mathcal O})$ degenerating in degree 1. Therefore, \begin{displaymath} h^i(P_0,{\mathcal O})= \sum_{p+q=i} \binom rp \binom aq = \binom {r+a}i = \binom gi. \end{displaymath} \end{proof} \begin{thm} For every $d>0$ and $i>0$ one has $h^0(P_0,{\mathcal L}_0^d)=d^g$ and $h^i(P_0,{\mathcal L}_0^d)=0$. \end{thm} \begin{proof} Twist the above resolution of ${\mathcal O}_{P_0}$ by ${\mathcal L}$. Once again, the cohomologies of each building block are easy to compute. Indeed, for a toric variety $\widetilde V^r(\sigma)$ higher cohomologies of an ample sheaf vanish. Moreover, since by \ref{thm:toric_descr_family} the pair $(\widetilde V^r(\sigma),{\widetilde {\mathcal L}})$ is the toric variety with a linearized ample sheaf corresponding to the polytope $\sigma\subset X_{{\mathbb R}}$, $H^0(\widetilde V^r(\sigma),{\widetilde {\mathcal L}}^d)$ is canonically the direct sum of 1-dimensional eigenspaces, one for each point $z\in\sigma\cap X$, cf., e.g., \cite[Ch.2]{Oda_ConvexBodies}. Taking into account that higher cohomologies of ample sheaves on abelian varieties vanish, we see that $H^i(\widetilde V(\sigma),{\widetilde {\mathcal L}}^d)=0$ for $i>0$ and $H^0(\widetilde V(\sigma),{\widetilde {\mathcal L}}^d) \simeq H^0(\widetilde V^r(\sigma),{\widetilde {\mathcal L}}^d)\otimes H^0(A_0,{\mathcal M}^d)$. It is well-known that for the abelian variety $A_0$ the latter cohomology space has dimension $d^a$. The above decomposition into eigenspaces extends to the hypercohomologies and we obtain \begin{displaymath} H^i(P_0,{\mathcal L}^d) \simeq H^0(A_0,{\mathcal M}^d) \otimes (\oplus_{\bar z\in X/dX} W^i_{\bar z}), \end{displaymath} where $W^i_{\bar z}$ is the $\bar z$-eigenspace of the $i$-th cohomology of the complex \begin{displaymath} 0\to \oplus_{\dim\bar\sigma=g} H^0(\widetilde V^r(\bar\sigma),{\widetilde {\mathcal L}}^d) \to \dots \oplus_{\dim\bar\sigma=0} H^0(\widetilde V^r(\bar\sigma),{\widetilde {\mathcal L}}^d) \to 0. \end{displaymath} Fix a representative $z\in X/n$ of $\bar z$. Let $\sigma_0$ be the minimal cell containing $z$. There is a 1-to-1 correspondence between the cells $\sigma\ni z$ and the faces of the dual Voronoi cell $\widehat\sigma_0$. Since these are exactly the cells for which the $z$-eigenspace in $H^0(\widetilde V(\sigma),{\widetilde {\mathcal L}}^d)$ are nonzero (and 1-dimensional), we see that $W^i_z$ computes the cellular cohomology $H^i(\widehat\sigma_0,k)$. Since as a topological space $\widehat\sigma_0$ is contractible, $\dim W^0_z=1$ and $W^i_z=0$ for $i>0$. Therefore, $h^i(P_0,{\mathcal L}^d)=0$ and $h^0(P_0,{\mathcal L}^d)=h^0(A_0,{\mathcal M}^d)\cdot |X/dX|=d^ad^r=d^g$. \end{proof} \begin{saynum} As a consequence of this theorem, we can write down explicitly a basis of $H^0(P_0,{\mathcal L}_0^d)$. First, let us do this for the toric case. For each $\bar z\in X/dX$ fix a representative $z\in X/d$ and choose $x_1,\dots,x_d$ with $x_1+\dots+x_d=dz$. Consider a power series \begin{eqnarray*} && {\tilde\theta}'(x_1,\dots,x_d) = \sum_{y\in Y=X} \xi'(x_1+y, \dots, x_d+y), \\ && \xi'(x_1+y, \dots, x_d+y) := \prod_{i=1}^d \xi_{x_i+y}. \end{eqnarray*} This power series is obviously invariant under the $Y$-action on $K[\vartheta,w^x \,|\, x\in X]$ and, under the assumption \ref{assume:base_change_made}, there is a unique nonnegative integer $n=n(d_1,\dots,x_d)$ such that $\xi(x_1+y, \dots, x_d+y)=s^{-n}\xi'(x_1+y, \dots, x_d+y)$ defines a nonzero section of $H^0(\widetilde P_0,{\widetilde {\mathcal L}}_0^d)$, so that ${\tilde\theta}(x_1,\dots,x_d)=s^{-n}{\tilde\theta}'(x_1,\dots,x_d)$ defines a nonzero $Y$-invariant section of $H^0(\widetilde P_0,{\widetilde {\mathcal L}}_0^d)$ which descends to a nonzero section $\theta(x_1,\dots,x_d)$ of $H^0(P_0,{\mathcal L}_0^d)$. Choosing another representatives $z$ and $x_1,\dots,x_d$ changes these sections by multiplicative constants. Therefore, in the cases where we do not care about these constants we will write simply $\xi_z$, ${\tilde\theta}_z$ and $\theta_{\bar z}$. Geometrically, $\xi_z$ is represented by a point on the surface of the multifaceted paraboloid $Q$ of Figure~3 lying over $z$, and ${\tilde\theta}_z$ by the sum of countably many such points lying over all $z+y$, $y\in Y=X$. In general, for any $e\in H^0(A_0,{\mathcal M}^d)$ we repeat the procedure taking instead of $\xi_{x_i+y}$ sections $\xi_{x_i+y}(e)=S^*_{x_i+y}(e)$. This way, after fixing a basis $\{e_1,\dots, e_{d^a}\}$ of $H^0(A_0,{\mathcal M}^d)$ we obtain a basis $\theta_{\bar z}(e_i)$ of $H^0(P_0,{\mathcal L}_0^d)$. \end{saynum} \begin{say} The following is an easy application of Theorem \ref{thm:structure_of_central_fiber}.\ref{enumi:basis_R0}: \end{say} \begin{lem}\label{lem:thetas_explicitly} For each $e\in H^0(A_0,{\mathcal M}^d)$ the following open sets coincide: \begin{displaymath} \{ \xi_z (e) \ne0 \} = \widetilde G_0(\sigma) \cap \pi^{-1} \{ S^*_{dz}(e)\ne0 \}, \end{displaymath} where $\sigma$ is the Delaunay cell containing $z$ in its interior $\sigma^0$. \end{lem} \begin{thm} Let $(P_0,{\mathcal L}_0)$ be an SQAV of dimension $g$. Then the sheaf ${\mathcal O}({\mathcal L}_0^d)$ is very ample if $d\ge 2g+1$. \end{thm} \begin{proof} Consider the toric case first, i.e., assume $r=g$, $a=0$. We need to prove: \begin{numerate} \item Sections $\theta_{\bar z}$, $z\in X/dX$ separate the torus orbits $\operatorname{orb}(\bar \sigma)$. \item They embed every $\operatorname{orb}(\bar \sigma)$. \item This embedding is an immersion at every point. \end{numerate} Let $\sigma$ be a maximal-dimensional cell and pick a point $z\in \sigma^0\cap X/d$ in its interior, which exists by \ref{lem:comb_of_star}. By \ref{lem:thetas_explicitly} $\xi_z$ is nonzero exactly on $\operatorname{orb}(\sigma)$. Therefore, ${\tilde\theta}_z$ is nonzero exactly on the union of $Y$-translates of this orbit, and $\theta_{\bar z}$ is not zero precisely on $\operatorname{orb}(\bar\sigma)$. Thus, we have separated the points of this orbit from all the others. Continuing by induction down the dimension, we get (i). The restriction of each $\theta_{\bar z}$ to $\operatorname{orb}(\bar\sigma)$ is a sum of degree $d$ monomial corresponding to $\bar z$, provided $\bar z\in\bar \sigma$. If $\bar z\in \bar\sigma^0$, there is just one such monomial. Therefore, the condition that suffices for (ii) is that the differences of vectors in $\sigma^0\cap X/d$ generate ${\mathbb R}\sigma\cap X/d$ as a group. This hold by \ref{lem:comb_of_star}.\ref{enumi:differences_generate}. It suffices to prove the immersion condition for the 0-dimensional orbit $p$ corresponding to $0\in X$ only. Indeed, it then holds in an open neighborhood which intersects every other orbit, and due to the torus action everywhere. Moreover, we can work on the \'etale cover $\widetilde P_0$. We have ${\tilde\theta}_0(p)\ne 0$, ${\tilde\theta}_z(p)=0$ for $\bar z\ne0$, and we want to show that ${\tilde\theta}_z/{\tilde\theta}_0$ generate $\mathfrak m/\mathfrak m^2$, where $\mathfrak m$ is the maximal ideal of $R_0(0)$. On the other hand, $\xi_0(p)\ne 0$ as well, so we can consider ${\tilde\theta}_z/\xi_0$ instead. By \ref{thm:structure_of_central_fiber}\ref{enumi:basis_R0}, $\mathfrak m/\mathfrak m^2$ is generated by {\em primitive\/} lattice vectors (see definition \ref{defn:star_Delaunay_primitive}). As an element of $R_0(0)$, ${\tilde\theta}_z/\xi_0$ is the sum with nonzero coefficients of monomials $\bar\zeta_{dz'}$ with $z'\in\operatorname{Star}(0)$ and $\bar z'=\bar z$. Therefore, for (iii) it suffices to have $\operatorname{Prim}/d\subset \operatorname{Star}(0)$ and $(\operatorname{Prim}-\operatorname{Prim})\cap dX = \{0\}$. This follows by \ref{lem:comb_of_star} again. Next, assume that the abelian part $A_0$ is nontrivial. To separate the orbits $\widetilde G_0(\bar\sigma)$ and the points in the orbit, and to see the injectivity, repeat the above arguments with $\xi_z(e_i)$ such that $e_i$ provide an embedding of $A_0$. The immersion is again sufficient to prove at the minimal dimensional stratum. For every $p\in A_0\subset P_0$ for the tangent space we have ${\mathbb T}_{p,P_0}\simeq {\mathbb T}_{p,\widetilde P_0^r}\oplus {\mathbb T}_{p,A_0}$. Hence, if $e_0(p)\ne 0$ then to generate $\mathfrak m/\mathfrak m^2$ it is sufficient to take $\theta_{\bar z}(e_z)/\theta_0(e_0)$ and $\theta_0(e_i)/\theta_0(e_0)$, where $e_z\big(T_{c^t(dz)}(p)\big)\ne 0$ and $e_i/e_0$ generate ${\mathbb T}^*_{p,A_0}$. \end{proof} \begin{lem}\label{lem:comb_of_star} \begin{numerate} \item $\operatorname{Prim}\subset r \operatorname{Star}(0)$. \item Assume $d\ge r+2$. Then for each Delaunay cell $\sigma$ the differences of vectors in $\sigma^0\cap X/d$ generate ${\mathbb R}\sigma\cap X/d$. \label{enumi:differences_generate} \item $\big(\operatorname{Star}(0)-\operatorname{Star}(0)\big)\cap (2+\varepsilon) X= \{0\}$ for any $0<\varepsilon\ll1$. \end{numerate} \end{lem} \begin{proof} Since the restriction of the Delaunay decomposition to ${\mathbb R}\sigma$ is again a Delaunay decomposition, we can assume that $\sigma$ is maximal-dimensional. Let $\sigma\subset\operatorname{Star}(0)$ be a Delaunay cell and let $w\in\operatorname{Cone}(0,\sigma)$ be a primitive lattice vector. Choose arbitrary $r$ linearly independent Delaunay vectors $v_1,\dots,v_r$ with $w\in \operatorname{Cone}(v_1\dots v_r)$ and write $w=\sum p_iv_i$ for some $p_i\in{\mathbb Q}$. Then obviously $p_i\le1$ and $w/r$ belongs to the convex hull of $0,v_1,\dots,v_r$, which is a part of $\operatorname{Star}(0)$. This proves (i). For (ii) note that the vectors $\sum_{i=1}^r v_i$, $(\sum v_i)+v_j$, $j=1\dots r$ and $(\sum v_i)-w$, with $w\ne v_i$ primitive all belong to $\big((r+2)\sigma\big)^0$. For (iii), let $\sigma_1\ne\sigma_2$ be two Delaunay cells in $\operatorname{Star}(0)$. Then for any $y\in X$ the intersection $\sigma_1\cap T_y \sigma_2$ is either $\sigma_1$, or a proper face of $\sigma_1$ or empty. In the first case $y$ must be a Delaunay vector, since both $\sigma_i$ contain $0$. Therefore, $y\notin 2X$. Consequently, for any $y\ne0$, $\operatorname{Star}(0)\cap T_{2y}\operatorname{Star}(0)$ has no interior, and $\operatorname{Star}(0)\cap T_{(2+\varepsilon)y}\operatorname{Star}(0)=\emptyset$. \end{proof} \begin{rem} The bound above is certainly not optimal. However, it seems that a better bound would require going much deeper into the combinatorics of Delaunay cells. In the proof above the only properties we used were that $\operatorname{Del}_B$ is $X$-periodic and that a Delaunay cell does not contain lattice points except its vertices. \end{rem} \section{Additions} \label{sec:Complex-analytic case} \subsection{Complex-analytic case} \label{subsec:Complex-analytic case} \mbox{}\smallskip \begin{say} Everything works the same way as in the algebraic case, only easier. The ring ${\mathcal R}$, resp. $K$ is replaced by the stalk of functions homomorphic, resp. meromorphic in a neighborhood of $0$. A major simplification comes in the construction of the quotient $(\widetilde P,{\widetilde {\mathcal L}})/Y$. Considering the fattenings $(\widetilde P_n,{\widetilde {\mathcal L}}_n)/Y$ and then algebraizing is unnecessary, since the $Y$-action is properly discontinuous in classical topology over a small neighborhood of $0$. Hence, one can take the quotient directly. The combinatorial description of the family and the central fiber and the data for an SQAP remain the same. \end{say} \subsection{Higher degree of polarization} \label{subsec:Higher degree of polarization} \mbox{}\smallskip \begin{say} The formulas in our construction are set up in such a way that we can repeat it for any degree of polarization. The outcome, after a finite base change, is a normal family with reduced central fiber and a relatively ample divisor. However, in this case there are several additional choices to make: \begin{enumerate} \item an embedding ${\mathcal M}_x\hookrightarrow {\mathcal M}_{x,\eta}$ for each nonzero representative of $X/Y$, \item a section $\theta_{A,x}\in H^0(A,{\mathcal M}_x)$ for each representative $x\in X/Y$. \end{enumerate} According to the description of $H^0(A_{\eta},{\mathcal L}_{\eta})$ in \cite[II.5.1]{FaltingsChai90}, this data is equivalent to providing a theta divisor on the generic fiber. This is why there are infinitely many relatively complete models in the case of higher polarization. \end{say} \section{Examples} \label{sec:Examples} \begin{say} Below we list all the SQAPs in dimensions 1 and 2, for illustration purposes. They can already be found in in \cite{Namikawa_NewCompBoth, Namikawa_ToroidalCompSiegel} (over ${\mathbb C}$). \end{say} \subsection{Dimension 1} \label{sec:Dimension 1} \mbox{}\smallskip \begin{say} In this case there is only one Delaunay decomposition of ${\mathbb Z}\subset{\mathbb R}$, so there is only one principally polarized stable quasiabelian pair besides the elliptic curves. The $0$-dimensional Delaunay cells correspond to integers $n$, and $1$-dimensional cells to intervals $[n,n+1]$. By \ref{thm:toric_descr_family} the corresponding toric varieties are projective lines $({\mathbb P}^1,{\mathcal O}(1))$ intersecting at points, and the intersections are transversal by \ref{thm:structure_of_central_fiber}. The theta divisor restricted to each ${\mathbb P}^1$ has to be a section of ${\mathcal O}(1)$, i.e., a point. The quotient $P$ is, obviously, a nodal rational curve. \end{say} \subsection{Dimension 2} \label{sec:Dimension 2} \mbox{}\smallskip \begin{say} Let us look at the case of the maximal degeneration first. There are only two Delaunay decompositions shown on Figures 1,2. In each case the irreducible components are the projective toric varieties described by the lattice polytopes $\sigma$, see \ref{thm:toric_descr_family}. In the first case we have a net of $({\mathbb P}^1\times{\mathbb P}^1,{\mathcal O}(1,1))$'s intersecting transversally at lines which in turn intersect in fours at points. The quotient by ${\mathbb Z}^2$ will have one irreducible component. It is obtained from ${\mathbb P}^1\times{\mathbb P}^1$ by gluing two pairs of zero and infinity sections. Modulo the action of $K(\operatorname{Del})\otimes{\mathbb G}_m$ we have only one parameter $z=b(e_1,e_2)$ and the SQAPs with $z$ and $1/z$ are isomorphic. Therefore, we have a family of SQAPs of this type parameterized by $k^*/{\mathbb Z}_2=k$. The theta divisor is the image of a conic on ${\mathbb P}^1\times{\mathbb P}^1$. It is reduced. It is irreducible unless $z=1$, in which case it is a pair of lines. In the second case \ref{thm:toric_descr_family} we get a net of projective planes $({\mathbb P}^2,{\mathcal O}(1))$ meeting at lines which in turn meet at points. The quotient will have two irreducible component, since there are two non-equivalent maximal-dimensional Delaunay cells modulo the lattice. In this case $K(\operatorname{Del})\otimes{\mathbb G}_m$ is 3-dimensional, so a variety $(P_0,{\mathcal L}_0)$ of this type is unique up to isomorphism. The theta divisor restricted to ${\mathbb P}^2$ has to be a section of ${\mathcal O}(1)$, i.e., a line. Therefore, the theta divisor on $P$ is a union of two rational curves and it is easy to see that they intersect at 3 points, one for each ${\mathbb P}^1$. In other words, the theta divisor in this case is a ``dollar curve''. \end{say} \begin{rem} Note that the lattices in the above two examples are of types $A_1\oplus A_1$ and $A_2$ respectively (see f.e. \cite{ConwaySloane93}). The number 4, resp. 6, of branches meeting at the $0$-dimensional strata has an interesting interpretation in this case. For any lattice of the $A,D,E$-type this is what in the lattice theory called the ``kissing number'' (think of the billiard balls with centers at the lattice elements, each ball is ``kissed'' by 4, resp. 6, other balls). \end{rem} \begin{say} There is only one case for a nontrivial abelian part (besides the smooth abelian surfaces): when $A_0$ is an elliptic curve. This is the simplest case of what Mumford called ``the first order degenerations of abelian varieties'' in \cite{Mumford_KodairaDimSiegel}. Before dividing by the lattice $Y={\mathbb Z}$ we have a locally free fibration over an elliptic curve with a fiber which corresponds to the case of maximal degeneration of dimension 1, i.e,. a chain of projective lines. The group $Y$ acts on this scheme $\widetilde P_0$ by cycling through the chain and at the same time shifting ``sideways'' with respect to the elliptic curve. The theta divisor ${\widetilde\Theta}_0$ on $\widetilde P_0$ is invariant under this shift, so it descends to a divisor on $P_0$. \end{say} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9608

alg-geom/9608016

en

https://arxiv.org/abs/alg-geom/9608016

alg-geom/9608016

Atsushi Nakayashiki

Atsushi Nakayashiki

On the Thomae formula for $Z_N$ curves

Latex file, the constant in Thomae formula is corrected

We shall give an elementary and rigorous proof of the Thomae formula for ${\bf Z}_N$ curves which was discovered by Bershadsky and Radul. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szeg\"{o} kernels and proves the vanishing of zero values of derivatives of theta functions with ${\bf Z}_N$ invariant $1/2N$ characteristics.

alg-geom

\section{Theta function} \par In this paper we mainly follow the notations of the Fay's book \cite{F} which we summarize here. Let $\tau$ be the $g$ by $g$ symmetric matrix whose real part is negative definite. Any element $e\in {\bf C}^g$ is uniquely expressed as \begin{eqnarray} && e=\ch{\delta}{\epsilon} =2\pi i\epsilon+\delta\tau, \label{char} \end{eqnarray} with $\epsilon,\delta\in {\bf R}^g$. Here the vectors $\epsilon,\delta$ etc. are all row vectors. We call $\epsilon,\delta$ the characteristics of $e$. The theta function with characteristics is defined by \begin{eqnarray} \th{\delta}{\epsilon}(z)&=& \sum_{m\in{\bf Z}^g}\exp( {1\over2}(m+\delta)\tau(m+\delta)^{t} +(z+2\pi i\epsilon)(m+\delta)^t ) \nonumber \\ &=&\exp( {1\over2}\delta\tau\delta^t+ (z+2\pi i\epsilon)\delta^t )\theta(z+e), \nonumber \end{eqnarray} where \begin{eqnarray} && \th{0}{0}(z)=\theta(z), \nonumber \end{eqnarray} and $e$ is determined by (\ref{char}). We sometimes use $\theta[e](z)$ instead of $\th{\delta}{\epsilon}(z)$. The transformation property is \begin{eqnarray} && \th{\delta}{\epsilon}(z+2\pi i\lambda+\kappa\tau) \nonumber \\ && = \exp(-{1\over2}\kappa\tau\kappa^t-z\kappa^t+2\pi i(\delta\lambda^t-\epsilon\kappa^t)) \th{\delta}{\epsilon}(z), \nonumber \end{eqnarray} for $\lambda,\kappa\in{\bf Z}^g$. We shall list some of the properties which easily follow from the definitions: \begin{eqnarray} && \th{\delta+m}{\epsilon+n}(z)= \exp(2\pi in\delta^t)\th{\delta}{\epsilon}(z), \label{charprop1} \\ && \th{-\delta}{-\epsilon}(0)= \th{\delta}{\epsilon}(0), \nonumber \\ && {\th{\delta}{\epsilon}(z)\over \th{\delta}{\epsilon}(0)} = {\th{\delta+m}{\epsilon+n}(z)\over \th{\delta+m}{\epsilon+n}(0)}, \nonumber \end{eqnarray} for $m,n\in{\bf Z}^g$. \par Let $C$ be a compact Riemann surface of genus $g$. Let us fix a marking of $C$ \cite{Gun}(\S1). That means, we fix a canonical basis $\{A_i,B_j\}$ of $\pi_1(C)$, a base point $P_0\in C$ and a base point in the universal cover $\tilde{C}$ which lies over $P_0$. We assume that the tails of $A_i,B_j$ are joined to $P_0$. Then we can canonically identify the covering transformation group and the fundamental group $\pi_1(C,P_0)$. We also identify holomorphic $1$-forms on $C$ with holomorphic $1$-forms on $\tilde{C}$ invariant under the action of $\pi_1(C)$. Let us denote by $\pi:\tilde{C}\longrightarrow C$ the projection and by $J(C)$ the Jacobian variety of $C$ which is the set of linear equivalence classes of degree $0$ divisors on $C$. In the following sections we always assume one marking of $C$. Let $\{v_j\}$ be the basis of the normalized holomorphic 1-forms. The normalization is \begin{eqnarray} && \int_{A_j}v_k=2\pi i\delta_{jk}, \nonumber \end{eqnarray} and set \begin{eqnarray} &&\int_{B_j}v_k=\tau_{jk}. \nonumber \end{eqnarray} A flat line bundle on $C$ is described by the character of the fundamental group $\chi:\pi_1(C)\longrightarrow{\bf C}^\ast$, where ${\bf C}^\ast$ is the multiplicative group of non-zero complex numbers. The two representation $\chi_1$ and $\chi_2$ defines a holomorphically equivalent line bundle if and only if there exists an holomorphic 1-form $\omega$ such that \begin{eqnarray} && \chi_1(\gamma)\chi_2(\gamma)^{-1} =\exp(\int_\gamma\omega) \nonumber \end{eqnarray} for any $\gamma\in\pi_1(C)$. Let ${\cal A}$ and ${\cal B}$ be positive divisors of the same degree, say $d$, and set ${\cal A}=\sum_{i=1}^d P_i$, ${\cal B}=\sum_{i=1}^d R_i$. Let us fix points $\tilde{P}_i,\tilde{R}_j$ in $\tilde{C}$ so that they lie over $P_i,R_j$. Let us set \begin{eqnarray} && \int_{{\cal A}}^{{\cal B}}v_i= \sum_{j=1}^d\int_{\tilde{P}_j}^{\tilde{R}_j}v_i, \nonumber \end{eqnarray} where the integration in the right hand side is taken in $\tilde{C}$. Then the flat line bundle corresponding to the degree $0$ divisor ${\cal B}-{\cal A}$ is described by \begin{eqnarray} && \chi(A_i)=1, \qquad \chi(B_i)=\exp(\int_{{\cal A}}^{{\cal B}}v_i). \nonumber \end{eqnarray} Another choice of $\tilde{P}_i,\tilde{R}_j$ gives an equivalent line bundle. We say $\chi$ is unitary if the image is contained in the unitary group $U(1)$. The following proposition is well known and easily proved. \begin{prop} For an isomorphism class of flat line bundles there exists a unique unitary representation $\chi$ which defines the line bundle belonging to that class. \end{prop} \vskip2mm If we take $\delta,\epsilon\in{\bf R}^g$ such that \begin{eqnarray} && (\int_{{\cal A}}^{{\cal B}}v_1,\cdots,\int_{{\cal A}}^{{\cal B}}v_g) =\ch{\delta}{\epsilon} \nonumber \end{eqnarray} as a point on the Jacobian variety of $C$, then the corresponding unitary representation $\tilde{\chi}$ is given by \begin{eqnarray} && \tilde{\chi}(A_j)=\exp(-2\pi i\delta_j), \qquad \tilde{\chi}(B_j)=\exp(2\pi i\epsilon_j). \label{unitary} \end{eqnarray} The multiplicative meromorphic function described by $\tilde{\chi}$ is, for example, given by \begin{eqnarray} && { \th{-\delta}{-\epsilon}(\int_{z_0}^xv-\alpha)\over \theta(\int_{z_0}^xv-\alpha) }, \nonumber \end{eqnarray} where $v$ is the vector of normalized holomorphic 1-forms, $x\in \tilde{C}$, $\alpha\in {\bf C}^g$ and the integration path is taken in $\tilde{C}$. We denote by $\Delta$ the Riemann divisor for our choice of the canonical homology basis which satisfies \begin{eqnarray} && 2\Delta\equiv K_C. \nonumber \end{eqnarray} Here $K_C$ is the divisor class of the canonical bundle of $C$ and $\equiv$ means the linear equivalence. Let $L_0$ be the degree $g-1$ line bundle corresponding to $\Delta$. For a divisor $\alpha$ with degree $0$ let us denote by ${\cal L}_\alpha$ the corresponding flat line bundle and set $L_\alpha={\cal L}_\alpha\otimes L_0$. For a non-singular odd half period $\alpha$ let $h_\alpha$ be the section of $L_\alpha$ which satisfies \begin{eqnarray} && h_\alpha^2(x)= \sum_{j=1}^g {\partial\theta[\alpha]\over \partial z_j}(0)v_j(x). \nonumber \end{eqnarray} Then the prime form is defined by \begin{eqnarray} E(x,y) &=& {\theta[\alpha](y-x)\over h_\alpha(x)h_\alpha(y)}, \nonumber \\ y-x &=& \int_x^y v, \nonumber \end{eqnarray} where $x,y\in \tilde{C}$ and $v=(v_1(x),\cdots,v_g(x))$. Let $\pi_j$ be the projection from $C\times C$ to the $j$-th component and $\delta:C\times C\longrightarrow J(C)$ the map $(x,y)\mapsto y-x$. Then $E(x,y)$ can be considered as a section of the line bundle $\pi_1^\ast L_\alpha\otimes\pi_2^\ast L_\alpha\otimes\delta^\ast\Theta$, where $\Theta$ is the line bundle on $J(C)$ defined by the theta divisor. Let us fix the transformation property of the half differential on $\tilde{C}$ under the action of $\pi_1(C)$ so that the section of $\pi^\ast L_0$ is invariant. This means, in particular, that $E(x,y)$ transforms under the action of $A_i$, $B_i$ in $y$ as \begin{eqnarray} && E(x,y+A_i)=E(x,y), \quad E(x,y+B_i)=\exp(-{\tau_{ii}\over2}-\int_x^y v_i)E(x,y). \nonumber \end{eqnarray} Here we denote the action of $A_i$, $B_i$ in an additive manner. The prime form has the nice expansion as follows. Let $u$ be a local coordinate around $P\in \tilde{C}$. Then the expansion of $E(x,y)$ in $u(y)$ at $u(x)$ takes the form \begin{eqnarray} && E(x,y)\sqrt{du(x)}\sqrt{du(x)} =u(y)-u(x)+O\big((u(y)-u(x))^3\big). \label{primeexp} \end{eqnarray} Since the expansion is of local nature we sometimes use the way of saying that $P\in C$, the local corrdinate $u$ around $P$ and the expansion in $u(y)$ at $u(x)$ etc. \vskip1cm \section{${\bf Z}_N$ curve and ${1\over2N}$ period} \par Let us consider the plane algebraic curve $s^N=f(z)=\prod_{j=1}^{Nm}(z-\lambda_i)$. We compactify it by adding $N$ infinity point $\infty^{(1)},\cdots,\infty^{(N)}$ and denote the compact Riemann surface by $C$. The genus $g$ of $C$ is $g=1/2(N-1)(Nm-2)$. The $N$-cyclic automorphism $\phi$ of $C$ is defined by $\phi:(z,s)\mapsto(z,\omega s)$, where $\omega$ is the $N$-th primitive root of unity. There are $Nm$ branch points $Q_1,\cdots,Q_{Nm}$ whose projection to $z$ coordinate are $\lambda_1,\cdots,\lambda_{Nm}$. The basis of holomorphic 1-forms on $C$ is given by \begin{eqnarray} && w^{(\alpha)}_\beta={z^{\beta-1}dz\over s^{\alpha}} \quad 1\leq\alpha\leq N-1, \quad 1\leq\beta\leq \alpha m-1. \nonumber \end{eqnarray} Let us describe the divisors which we need and their relations. The following lemma is easily proved. \begin{lem} For any $P\in C$ the linear equivalence class $P+\phi(P)+\cdots+\phi^{N-1}(P)$ does not depend on the point $P$. \end{lem} We set \begin{eqnarray} && D\equiv P+\phi(P)+\cdots+\phi^{N-1}(P). \nonumber \end{eqnarray} The following lemma is easily proved. \begin{lem}\label{divrels} The following relations hold. \begin{description} \item[1.] $D\equiv NQ_i\equiv \infty^{(1)}+\cdots+\infty^{(N)}$ for any $i$. \item[2.] $K_C\equiv (L-1)D$, where $L=(N-1)m-1$. \item[3.] $\sum_{j=1}^{Nm}Q_j\equiv mD$. \end{description} \end{lem} \vskip2mm Following Bershadsky-Radul\cite{BR2} we shall describe the important object of our study, the ${\bf Z}_N$ invariant $1/N$ or $1/2N$ periods. Let us consider an ordered partition $\Lambda=(\Lambda_0,\cdots,\Lambda_{N-1})$ of $\{1,2,\cdots, Nm\}$ such that the number $|\Lambda_i|$ of elements of $\Lambda_i$ is equal to $m$ for any $i$. With each $\Lambda$ we associate the divisor class $e_\Lambda$ by \begin{eqnarray} && e_\Lambda \equiv \Lambda_1+2\Lambda_2+\cdots+(N-1)\Lambda_{N-1}-D-\Delta, \nonumber \end{eqnarray} where for a subset $S$ of $\{1,2,\cdots, Nm\}$ we set \begin{eqnarray} && S=\sum_{j\in S}Q_j. \nonumber \end{eqnarray} For a given $\Lambda$ we denote by $\Lambda(j)$ the ordered partition \begin{eqnarray} && \Lambda(j)=(\Lambda_j,\Lambda_{j+1},\cdots,\Lambda_{j-1}). \nonumber \end{eqnarray} Here we consider the index of $\Lambda_j$ by modulo $N$. Then \begin{prop} For any ordered partition $\Lambda$ we have \begin{description} \item[1.] $Ne_\Lambda\equiv0$ for $N$ being even and $2Ne_\Lambda\equiv0$ for $N$ being odd. \item[2.] $e_\Lambda\equiv e_{\Lambda(2)}\equiv\cdots\equiv e_{\Lambda(N)}$. \item[3.] $-e_\Lambda\equiv \Lambda_{N-1}+2\Lambda_{N-2}+\cdots+(N-1)\Lambda_1-D-\Delta$. \end{description} \end{prop} \vskip2mm This proposition is easily proved using Lemma \ref{divrels}. For $\Lambda=(\Lambda_0,\ldots,\Lambda_{N-1})$ we set \begin{eqnarray} \Lambda^{-}=(\Lambda^{-}_0,\ldots,\Lambda^{-}_{N-1})=(\Lambda_0,\Lambda_{N-1},\ldots,\Lambda_1), \nonumber \end{eqnarray} and $\Lambda=\Lambda^{+}$. Let $\theta(z)$ be the theta function associated with our choice of canonical homology basis. Then \begin{prop}\label{nonsing} The $1/2N$ period $e_\Lambda$ is non-singular, that means \begin{eqnarray} && \theta(e_\Lambda)\neq0. \nonumber \end{eqnarray} \end{prop} This proposition was proved in \cite{BR2}. One can find another proof in \cite{G} which is similar to that of \cite{M} in the hyperelliptic case. \vskip1cm \section{Chiral Szeg\"{o} kernel} \par \newtheorem{defi}{Definition} \begin{defi} For $e\in{\bf C}^g$ satisfying $\theta(e)\neq0$ the chiral Szeg\"{o} kernel $R(x,y\vert e)$ is defined by \begin{eqnarray} && R(x,y\vert e)= { \theta[e](y-x)\over\theta[e](0)E(x,y) } \qquad x,y\in \tilde{C}. \nonumber \end{eqnarray} \end{defi} \vskip2mm We remark that $R(x,y\vert e)$ depends only on the image of $e$ to the Jacobian variety $J(C)$. We shall give an algebraic expression for $R(x,y\vert e_\Lambda)$. Let us set \begin{eqnarray} {\cal L}&=&\{-{N-1\over2},-{N-1\over2}+1,\cdots,{N-1\over2}\}, \label{calL} \\ q_l(i)&=&{1-N\over 2N}+\Big\{{l+i+{N-1\over2}\over N}\Big\}, \label{qli} \end{eqnarray} for $l\in{\cal L}$ and $i\in{\bf Z}$. Here $\{a\}=a-[a]$ is the fractional part of $a$, $[a]$ being the Gauss symbol. For an ordered partition $\Lambda=(\Lambda_0,\cdots,\Lambda_{N-1})$ we define the number $k_i, i=1,\cdots,Nm$ by \begin{eqnarray} && i\in\Lambda_j\quad\hbox{if and only if }k_i=j. \label{weight} \end{eqnarray} For each $l\in{\cal L}$ we set \begin{eqnarray} && f_l(x,\Lambda)=\prod_{i=1}^{Nm}(z(x)-\lambda_i)^{q_l(k_i)}\sqrt{dz(x)}. \nonumber \end{eqnarray} The following proposition was found in \cite{BR2}. \begin{prop}\label{bundle} $f_l(x,\Lambda)$ is a meromorphic section of $L_{e_\Lambda}$ whose divisor is \begin{eqnarray} && {\sf div} f_l=\Lambda_{1-j}+2\Lambda_{2-j}+\cdots+(N-1)\Lambda_{-1-j}- \sum_{k=1}^N\infty^{(k)}, \label{divisor} \end{eqnarray} where $l=-(N-1)/2+j$. \end{prop} \vskip2mm Note that the chiral Szeg\"{o} kernel $R(x,y\vert e_\Lambda)$ can be considered as a section of the line bundle $\pi_1^\ast L_{e_\Lambda}\otimes \pi_2^\ast L_{-e_\Lambda}$, where $\pi_i$ is the projection to the $i$-th component of $C\times C$. Let us set \begin{eqnarray} && F(x,y\vert \Lambda)={1\over N} {\sum_{l\in{\cal L}}f_l(x,\Lambda)f_{-l}(y,\Lambda^{-}) \over z(y)-z(x)}. \nonumber \end{eqnarray} Here the choice of the branch of $f_l(x,\Lambda)$ should be specified as in (\ref{branch1}) and (\ref{branch2}). Note that $F(x,y\vert \Lambda)$ and $R(x,y\vert e_\Lambda)$ can be considered as the sections of the same line bundle. Then we have \begin{theo}\label{mainth1} For an ordered partition $\Lambda$ we have \begin{eqnarray} && R(x,y\vert e_\Lambda)=F(x,y\vert \Lambda). \nonumber \end{eqnarray} \end{theo} \vskip2mm As a corollary of this theorem we have the vanishing of the theta derivative constants. \begin{cor}\label{vanish} For any ordered partition $\Lambda$ we have \begin{eqnarray} && {\partial\theta[e_\Lambda]\over\partial z_i}(0)=0 \qquad\hbox{for any $i$}. \label{vanishing} \end{eqnarray} \end{cor} Note that whether the right hand side of (\ref{vanishing}) vanishes or not depends only on the divisor class of $e_\Lambda$ by (\ref{charprop1}). Hence the statement has unambiguously a sense. This curious result is a natural generalization of the hyperelliptic case where $e_\Lambda$ is a non-singular even half period and the corollary is obvious. For general $N$ we do not know whether $\theta[e_\Lambda](z)$ is an even function. \begin{lem}\label{exponent} The following properties hold. \begin{description} \item[1.] $q_l(i)=q_{l^\prime}(i^\prime)$ if $i+l=i^\prime+l^\prime$. \item[2.] $q_l(i+N)=q_l(i)$ for any $i$. \item[3.] $\sum_{i=0}^{N-1}q_l(i)=0$. \end{description} \end{lem} \vskip2mm \noindent Proof. The properties 1 and 2 are obvious. Let us prove 3. Using 1 and 2 we have \begin{eqnarray} && \sum_{i=0}^{N-1}q_l(i)=\sum_{i=0}^{N-1}q_{-{N-1\over2}}(i) =\sum_{i=0}^{N-1}\big({N-1\over2N}+{i\over2}\big)=0. \nonumber \end{eqnarray} $\Box$ \vskip5mm \noindent Proof of Proposition \ref{bundle}. The meromorphy at points except the branch points and $\infty^{(k)}$ is obvious. We can take $t=(z-\lambda_i)^{1/N}$ as a local coordinate around $Q_i$. Then \begin{eqnarray} && (z-\lambda_i)^{q_l(k_i)}\sqrt{dz}=t^{{N-1\over2}+Nq_l(k_i)}\sqrt{Ndt}. \nonumber \end{eqnarray} If we write $l=-(N-1/2)+j$ $(0\leq j\leq N-1)$, we have \begin{eqnarray} && {N-1\over2}+Nq_l(k_i)=N\{{k_i+j\over N}\}. \nonumber \end{eqnarray} At $\infty^{(k)}$ we can take $t=1/z$ as a local coordinate and we have \begin{eqnarray} && f_l(x,\Lambda)={1\over t}\sqrt{dt}(1+O(t)) \nonumber \end{eqnarray} by the property 3 of Lemma \ref{exponent}. Hence $f_l$ is locally meromorphic on $C$ with the divisor (\ref{divisor}). Let us consider $f_l(x,\Lambda)^2$. This is a multi-valued meromorphic 1-form with the divisor \begin{eqnarray} && 2\big(\sum_{k=1}^{N-1}k\Lambda_{k-j} -\sum_{k=1}^N\infty^{(k)}\big)\equiv 2e_{\Lambda(1-j)}+2\Delta\equiv 2e_\Lambda+K_C. \nonumber \end{eqnarray} Hence \begin{eqnarray} && f_l(x,\Lambda)^2\in H^0\big(C, {\cal L}_{e_\Lambda}^{\otimes 2}\otimes \Omega^1_C(-2\sum_{k=1}^{N-1}k\Lambda_{k-j}+ 2\sum_{k=1}^N\infty^{(k)}) \big). \nonumber \end{eqnarray} Since \begin{eqnarray} && {\cal L}_{e_\Lambda}^{\otimes 2}\otimes \Omega^1_C(-2\sum_{k=1}^{N-1}k\Lambda_{k-j}+ 2\sum_{k=1}^N\infty^{(k)})\simeq {\cal O}_C, \nonumber \end{eqnarray} we have \begin{eqnarray} && H^0\big(C, {\cal L}_{e_\Lambda}^{\otimes 2}\otimes \Omega^1_C(-2\sum_{k=1}^{N-1}k\Lambda_{k-j}+ 2\sum_{k=1}^N\infty^{(k)}) \big)={\bf C}f_l(x,\Lambda)^2. \nonumber \end{eqnarray} Note that \begin{eqnarray} && H^0\big(C, L_{e_\Lambda}(-\sum_{k=1}^{N-1}k\Lambda_{k-j}+ \sum_{k=1}^N\infty^{(k)}) \big) \label{cohomology} \end{eqnarray} is one dimensional. Hence $f_l(x,\Lambda)$ can be considered as an element of (\ref{cohomology}). Thus Proposition \ref{bundle} is proved. $\Box$ \begin{lem} The following expression holds : \begin{eqnarray} && f_{-l}(y,\Lambda^{-})=\prod_{i=1}^{Nm}(z(y)-\lambda_i)^{-q_l(k_i)}\sqrt{dz(y)}. \nonumber \end{eqnarray} \end{lem} \vskip2mm \noindent Proof. Recall that \begin{eqnarray} && \Lambda^{-}=(\Lambda^{-}_0,\cdots,\Lambda^{-}_{N-1}), \quad \Lambda^{-}_j=\Lambda_{N-j}. \nonumber \end{eqnarray} Then we have \begin{eqnarray} && f_{-l}(y,\Lambda^{-})= \prod_{i=1}^{Nm}(z(y)-\lambda_i)^{q_{-l}(N-k_i)}\sqrt{dz(y)}. \nonumber \end{eqnarray} Hence it is sufficient to prove \begin{eqnarray} && q_{-l}(N-i)=-q_l(i), \nonumber \end{eqnarray} for any $l$ and $i$. This can be easily checked. $\Box$ \begin{lem} $F(x,y\vert \Lambda)$ is regular outside the diagonal set $\{x=y\}$. \end{lem} \vskip2mm \noindent Proof. A priori we know that $F(x,y\vert \Lambda)$ has poles at most at $z(x)=z(y)$. Hence it is sufficient to prove that $F(x,y\vert \Lambda)$ is regular at $z(x)=z(y)$ and $x\neq y$. If we write $l=-(N-1)/2+j$ we have \begin{eqnarray} q_l(k_i)-q_{-{N-1\over2}}(k_i)= q_{-{N-1\over2}}(j+k_i)-q_{-{N-1\over2}}(k_i) ={j\over N}\hbox{ mod. ${\bf Z}$}. \nonumber \end{eqnarray} Therefore we can set \begin{eqnarray} && q_l(k_i)-q_{-{N-1\over2}}(k_i)={j\over N}+r_{ij}, \quad r_{ij}\in{\bf Z}. \nonumber \end{eqnarray} Let us choose the branch of $f_l(x,\Lambda)$, $f_l(x,\Lambda^{-})$ such that the following equations hold: \begin{eqnarray} f_l(x,\Lambda)&=& (z(x)-\lambda_i)^{r_{ij}}s(x)^j f_{-{N-1\over2}}(x,\Lambda), \label{branch1} \\ f_{-l}(x,\Lambda^{-})&=& (z(x)-\lambda_i)^{-r_{ij}}s(x)^{-j} f_{-{N-1\over2}}(x,\Lambda^{-}). \label{branch2} \end{eqnarray} Then we have \begin{eqnarray} && F(x,y\vert \Lambda) \nonumber \\ &&= f_{-{N-1\over2}}(x,\Lambda) f_{-{N-1\over2}}(y,\Lambda^{-}) \sum_{j=0}^{N-1}\prod_{i=1}^{Nm} \Big( {z(x)-\lambda_i\over z(y)-\lambda_i} \Big)^{r_{ij}} \Big({s(x)\over s(y)}\Big)^j. \nonumber \end{eqnarray} Now in the limit \begin{eqnarray} && z(x)\longrightarrow z(y), \quad s(x)\longrightarrow \omega^r s(y), \quad 1\leq r\leq N-1, \nonumber \end{eqnarray} with $\omega=\exp{2\pi i/N}$ we have \begin{eqnarray} && F(x,y\vert \Lambda)\longrightarrow f_{-{N-1\over2}}(y^{(r)},\Lambda) f_{-{N-1\over2}}(y,\Lambda^{-}) \sum_{j=0}^{N-1}\omega^r=0, \nonumber \end{eqnarray} where $y^{(r)}=(z(y),\omega^rs(y))$. $\Box$ The following lemma is proved by a direct calculation. \begin{lem}\label{expand1} Let $P\in C$ be a non-branch point. We can take $z$ to be a local coordinate around $p$. Then the expansion in $z(y)$ at $z(x)$ takes the form \begin{eqnarray} && F(x,y\vert \Lambda^\pm) \nonumber \\ &&= {\sqrt{dz(x)}\sqrt{dz(y)}\over z(y)-z(x)} \Big[1+ {1\over 2N}\sum_{i,j=1}^{Nm} {q(k_i,k_j)\over(z(x)-\lambda_i)(z(x)-\lambda_j)} (z(y)-z(x))^2+\cdots\Big], \nonumber \end{eqnarray} where $q(i,j)=\sum_{l\in{\cal L}}q_l(i)q_l(j)$. \end{lem} \vskip2mm The following lemma is a consequence of the expansion (\ref{primeexp}) of the prime form $E(x,y)$. \begin{lem}\label{expand3} Under the same conditions of Lemma \ref{expand1} we have \begin{eqnarray} && R(x,y\vert e_\Lambda) \nonumber \\ &&= {\sqrt{dz(x)}\sqrt{dz(y)}\over z(y)-z(x)} \Big[1+ \sum_{i=1}^{g} {\partial\log\theta[e_\Lambda]\over\partial z_i}(0)v_i(x) (z(y)-z(x))+\cdots\Big], \nonumber \end{eqnarray} where $v_i(x)$ means the coefficient of $dz(x)$ in $v_i(x)$. \end{lem} \vskip2mm \noindent Proof of Theorem \ref{mainth1}. Let $\chi$ be the unitary representation corresponding to ${\cal L}_{e_\Lambda}$. If we write \begin{eqnarray} && e_{\Lambda}=\ch{\delta}{\epsilon}, \nonumber \end{eqnarray} then $\chi(A_i)$ and $\chi(B_j)$ are given by (\ref{unitary}) in section 1. The transformation property of $R(x,y\vert e_\Lambda)$ is \begin{eqnarray} && R(x+\gamma_1,y+\gamma_2 \vert e_\Lambda) =\chi(\gamma_1)\chi(\gamma_2)^{-1}R(x,y\vert e_\Lambda), \nonumber \end{eqnarray} for $\gamma_1,\gamma_2\in\pi_1(C)$. On the other hand if we pull $F(x,y\vert \Lambda)$ back to $\tilde{C}\times\tilde{C}$ then \begin{eqnarray} && F(x+\gamma_1,y+\gamma_2 \vert \Lambda) =\chi_1(\gamma_1)\chi_2(\gamma_2)F(x,y\vert \Lambda), \nonumber \end{eqnarray} for some unitary representation $\chi_1$ and $\chi_2$. In fact if $x$ rounds a cycle of $C$ $f_l(x,\Lambda)$ is multiplied by an appropriate $2N$ the root of unity. The same is true for $y$. Let us set \begin{eqnarray} \tilde{F}(x,y\vert \Lambda)= {R(x,y\vert e_\Lambda)\over F(x,y\vert \Lambda)}. \nonumber \end{eqnarray} Then $\tilde{F}(x,y\vert \Lambda)$ is the section of the trivial line bundle and obeys the tensor product of unitary representations of $\pi_1(C)\times\pi_1(C)$. Hence $\tilde{F}(x,y\vert \Lambda)$ is invariant under the action of $\pi_1(C)\times\pi_1(C)$. This means that $R(x,y\vert e_\Lambda)$ and $F(x,y\vert \Lambda)$ have the same transformation property. Therefore the function \begin{eqnarray} && I(x,y)=F(x,y\vert \Lambda)-R(x,y\vert e_\Lambda) \nonumber \end{eqnarray} can be considered as a section of the line bundle $ \pi_1^\ast L_{e_\Lambda}\otimes \pi_2^\ast L_{-e_\Lambda}. $ By Lemma \ref{expand1} and \ref{expand3} we know that $I(x,y)$ is holomorphic except $\cup_{i=1}^{Nm}\{Q_i\}\times \{Q_i\}$. Since $I(x,y)$ is meromorphic on $C\times C$, $I(x,y)$ has no singularity. By Proposition \ref{nonsing}, $e_\Lambda$ is non-singular which means \begin{eqnarray} && H^0(C,L_{e_\Lambda})=0. \nonumber \end{eqnarray} Hence \begin{eqnarray} && H^0(C\times C,\pi_1^\ast L_{e_\Lambda} \otimes \pi_2^\ast L_{-e_\Lambda}) =\pi_1^\ast H^0(C,L_{e_\Lambda})\otimes \pi_2^\ast H^0(C,L_{-e_\Lambda})=0. \nonumber \end{eqnarray} Thus $I(x,y)=0$. $\Box$ \vskip5mm \noindent Proof of Corollary \ref{vanish}. This is a direct consequence of Theorem \ref{mainth1}, Lemma \ref{expand1} and \ref{expand3}. $\Box$ \begin{cor}\label{expand2} Under the same conditions and notations as in Lemma \ref{expand1}, then we have \begin{eqnarray} && R(x,y\vert e_\Lambda)R(x,y\vert -e_\Lambda) \nonumber \\ && ={dz(x)dz(y)\over (z(y)-z(x))^2} \Big[1+ {1\over N}\sum_{i,j=1}^{Nm} {q(k_i,k_j)\over(z(x)-\lambda_i)(z(x)-\lambda_j)} (z(y)-z(x))^2+\cdots\Big], \nonumber \end{eqnarray} \end{cor} \vskip1cm \section{Canonical symmetric differential} \par The canonical symmetric differential $\omega(x,y)$ is defined by the following properties. \vskip3mm \begin{description} \item[1] $\omega(x,y)$ is a meromorphic section of $\pi_1^\ast\Omega_C^1\otimes \pi_2^\ast\Omega_C^1$ on $C\times C$, where $\pi_i$ is the projection to the $i$-th component of $C\times C$. \item[2] $\omega(x,y)$ is holomorphic except the diagonal set $\{x=y\}$ where it has a double pole. For $p\in C$ if we take a local coordinate $u$ around $p$ then the expansion in $u(x)$ at $u(y)$ takes the form \begin{eqnarray} && \omega(x,y)=\Big({1\over(u(x)-u(y))^2}+\hbox{regular}\Big) du(x)du(y). \nonumber \end{eqnarray} \item[3] The $A$ period in $x$ variable is zero: \begin{eqnarray} && \int_{A_j}\omega(x,y)=0 \quad \hbox{for any $j$}. \nonumber \end{eqnarray} \item[4] $\omega(x,y)=\omega(y,x)$. \end{description} \vskip5mm The following proposition is well known. \begin{prop} The canonical differential exists and is unique. \end{prop} In fact there is an analytical description of $\omega(x,y)$ in terms of the theta function (see for example \cite{F} p26, Corollary 2.6) : \begin{eqnarray} && \omega(x,y)=d_xd_y\log E(x,y) =-\sum_{i,j=1}^g{\partial^2\log\theta\over\partial z_i\partial z_j} (y-x-f)v_i(x)v_j(y), \nonumber \end{eqnarray} for any non-singular point $f\in(\Theta)$, where $(\Theta)=(\theta(z)=0)$. The uniqueness can be easily proved using $H^0(C\times C,\pi_1^\ast\Omega_C^1\otimes \pi_2^\ast\Omega_C^1) =\pi_1^\ast H^0(C,\Omega_C^1)\otimes \pi_2^\ast H^0(C,\Omega_C^1)$. There is a remarkable identity due to Fay \cite{F} ( Corollary 2.12 ) connecting the chiral Szeg\"{o} kernel and the canonical symmetric differential. The formula is \begin{eqnarray} && R(x,y\vert e)R(x,y\vert -e) = \omega(x,y)+ \sum_{i,j=1}^g {\partial^2\log\theta[e]\over\partial z_i\partial z_j} (0)v_i(x)v_j(y), \label{remarkable} \end{eqnarray} for any $e\in {\bf C}^g$ such that $\theta(e)\neq0$. For a non-branch point $P\in C$ we can take $z$ as a local coordinate around $P$. Let us define \begin{eqnarray} &&G_z(z)=\lim_{y\rightarrow x} \Big[ \omega(x,y)-{dz(x)dz(y)\over((z(y)-z(x))^2} \Big]. \nonumber \end{eqnarray} It is known\cite{F,HS} that $6G_z$ is the projective connection which satisfies \begin{eqnarray} && 6G_t(t)dt^2=6G_z(z)dz^2+\{z,t\}dt^2 \nonumber \end{eqnarray} for another local coordinate $t$ around $P$, where $\{z,t\}$ is the Schwarzian differential defined by \begin{eqnarray} && \{z,t\}={z^{'''}\over z^{'}}-{3\over2}\Big({z^{''}\over z^{'}}\Big)^2. \nonumber \end{eqnarray} By Corollary \ref{expand2} and (\ref{remarkable}) we have \begin{prop}\label{proj1} \begin{eqnarray} && G_z(z)= {1\over N} \sum_{i,j=1}^{Nm} {q(k_i,k_j) dz(x)^2 \over (z(x)-\lambda_i)(z(x)-\lambda_j)} - \sum_{i,j=1}^g {\partial^2 \log \theta[e_{\Lambda}] \over \partial z_i\partial z_j} (0)v_i(x)v_j(x). \nonumber \end{eqnarray} \end{prop} \vskip2mm As a corollary of this expression we have \begin{cor}\label{projexp1} Let $t=(z-\lambda_i)^{1/N}$ be the local coordinate around the branch point $Q_i$. Then the coefficient of $t^{N-2}dt$ in the Laurent expansion of $G_z(z)$ in $t$ is \begin{eqnarray} && 2N\sum_{j\neq i}{q(k_i,k_j)\over \lambda_i-\lambda_j} -{1\over (N-2)!} \sum_{r,s=1}^g\sum_{\alpha=0}^{N-2} \bic{N-2}{\alpha} {\partial^2 \log \theta[e_{\Lambda}] \over \partial z_r \partial z_s} (0)v_r^{(\alpha)}(Q_i)v_s^{(N-2-\alpha)}(Q_i), \nonumber \end{eqnarray} where $v_r^{(\alpha)}(Q_i)$ is the coefficient of $dt$ in the expansion of $v_k(x)$ in $t$. \end{cor} \vskip1cm \section{Another description of canonical differential} \par Let $P_l^{(l)}(z,w)$ be a polynomial satisfying the conditions \begin{description} \item[1.] $P^{(l)}_l(z,w)=\sum_{j=0}^{lm}P_{lj}^{(l)}(w)(z-w)^j$ with \begin{eqnarray} && P_{l0}^{(l)}(w)=f(w), \quad P_{l1}^{(l)}(w)={l\over N}\sum_{i=1}^{Nm}{f(w)\over w-\lambda_i}. \nonumber \end{eqnarray} \item[2.] $\hbox{deg}_wP_l^{(l)}(z,w)\leq (N-l)m$. \end{description} \vskip2mm The following lemma can be easily proved. \begin{lem} The polynomial $P_l^{(l)}(z,w)$ saisfying the above conditions 1,2 exists for $l=1,\cdots,N-1$. \end{lem} \vskip2mm We set \begin{eqnarray} \xi^{(0)}(x,y)&=&{dz(x)dz(y)\over(z(x)-z(y))^2}, \nonumber \\ \xi^{(l)}(x,y)&=& {P_l^{(l)}(z(x),z(y))dz(x)dz(y)\over s_l(x)s_{N-l}(y)(z(x)-z(y))^2} \quad \hbox{for } l=1,\cdots,N-1, \nonumber \\ \xi(x,y)&=&{1\over N}\sum_{l=0}^{N-1}\xi^{(l)}(x,y), \nonumber \end{eqnarray} where $s_l(x)=s(x)^l$. The condition 1,2 implies that $\xi^{(l)}(x,y)$ is regular on $C\times C$ except $\{z(x)=z(y)\}$. \begin{prop}\label{expxi} \begin{description} \item[1.] $\xi(x,y)$ is holomorphic outside the diagonal set $\{x=y\}$. \item[2.] For a non-branch point $P\in C$ we take $z$ as a local coordinate around $P$. Then the expansion in $z(x)$ at $z(y)$ is \begin{eqnarray} && \xi(x,y)={dz(x)dz(y)\over(z(x)-z(y))^2}+O((z(x)-z(y))^0). \nonumber \end{eqnarray} \end{description} \end{prop} \vskip2mm \noindent Proof. For $y\in C$ let $y^{(r)}=(z(y),\omega^rs(y))$. Suppose that $y$ is not a branch point. Then we can take $z$ as a local coordinate around $y^{(r)}$. By calculation we have the expansion of $\xi^{(l)}(x,y)$ in $z(x)$ at $y^{(r)}$ as \begin{eqnarray} \xi^{(l)}(x,y)&=& \omega^{-rl}\Big[ {1\over (z(x)-z(y))^2} -{l^2\over 2N^2}\Big({d\over dz}\log f(z(y))\Big)^2 -{l\over 2N}{d^2\over dz^2}\log f(z(y)) \nonumber \\ && +{P^{(l)}_{l,2}(z(y))\over f(z(y))} +O\Big((z(x)-z(y))^1\Big)\Big]dz(x)dz(y). \label{expxil} \end{eqnarray} By definition $\xi(x,y)$ is regular except $z(x)=z(y)$. In order to prove the property 1 of the proposition it is sufficient to prove that $\xi(x,y)$ has no singularity at $x=y^{(r)}$ for $1\leq r\leq N-1$. Note that if $y=Q_i$ for some $i$, then $z(x)=z(y)$ is equivalent to $x=y=Q_i$. Hence by the expansion (\ref{expxil}), $\xi(x,y)$ is regular at $x=y^{(r)}$. The property 2 is also obvious from (\ref{expxil}) above. $\Box$ \begin{cor} $\omega(x,y)-\xi(x,y)$ is holomorphic on $C\times C$. \end{cor} \vskip2mm \noindent Proof. By Proposition \ref{expxi}, $\omega(x,y)-\xi(x,y)$ is regular except $\cup_{i=1}^{Nm}(Q_i,Q_i)$. Hence $\omega(x,y)-\xi(x,y)$ is regular everywhere on $C\times C$, since $\omega(x,y)-\xi(x,y)$ is meromorphic on $C\times C$. $\Box$ \vskip1mm By this corollary there exists a set of polynomials $P^{(l)}_k(z,w)$ such that \begin{eqnarray} && \omega(x,y)-\xi(x,y)= \sum_{l=1}^{N-1}\sum_{k=1,k\neq l}^{N-1} {P_k^{(l)}(z(x),z(y))dz(x)dz(y)\over s_k(x)s_{N-l}(y)}, \nonumber \end{eqnarray} where by changing the definition of $P^{(l)}_l(z,w)$ the $k=l$ term can be excluded. The condition for the right hand side to be regular at $z(x)=\infty$ and $z(y)=\infty$ is \begin{eqnarray} && \hbox{deg}_zP^{(l)}_k(z,w)\leq km-2, \quad \hbox{deg}_wP^{(l)}_k(z,w)\leq (N-l)m-2. \nonumber \end{eqnarray} Hence we can write \begin{eqnarray} && P^{(l)}_k(z,w)=\sum_{j=0}^{km-2}P^{(l)}_{kj}(w)(z-w)^j, \nonumber \end{eqnarray} for some polynomials $P^{(l)}_{kj}(w)$. Now by the condition that the $A$ period of $\omega(x,y)$ is zero we have \begin{prop} The following relation holds \begin{eqnarray} && \sum_{l=1}^{N-1}P^{(l)}_{l2}(\lambda_i)= -f^\prime(\lambda_i){\partial\over\partial\lambda_i}\log\det A, \nonumber \end{eqnarray} where $A$ is the $g\times g$ period matrix of non-normalized form: \begin{eqnarray} && A=(\int_{A_i}w^{(\alpha)}_\beta). \nonumber \end{eqnarray} \end{prop} \vskip2mm \noindent Proof. Let us take $t=(z-\lambda_i)^{1/N}$ as a local coordinate around $Q_i$. Then we have \begin{eqnarray} {dz(y)\over s_{N-l}(y)}&=& {N\over\prod_{j\neq i}(\lambda_i-\lambda_j)^{(N-l)/N}} t^{l-1}dt(1+O(t^N)) \quad 1\leq l\leq N-1, \nonumber \\ {dz(y)\over(z(x)-z(y))^2} &=& {N\over (z(x)-\lambda_i)^2} t^{N-1}dt(1+O(t^N)). \nonumber \end{eqnarray} Therefore if we set \begin{eqnarray} && \omega^{(l)}(x)= {1\over N} {P^{(l)}_l(z(x),\lambda_i)dz(x)\over s_l(x)(z(x)-\lambda_i)^2} +\sum_{k=1,k\neq l}^{N-1} {P^{(l)}_k(z(x),\lambda_i)dz(x)\over s_k(x)}, \nonumber \end{eqnarray} then the condition that the coefficients of $dt,tdt,\cdots,t^{N-2}dt$ in the expansion of $\int_{A_j}\omega(x,y)$ vanish is equivalent to \begin{eqnarray} && \int_{A_j}\omega^{(l)}(x)=0 \quad 1\leq l\leq N-1. \label{expomegal} \end{eqnarray} Noting that \begin{eqnarray} && P^{(l)}_l(z,\lambda_i)= {l\over N}f^\prime(\lambda_i)(z-\lambda_i) +\sum_{j=0}^{lm-2}P^{(l)}_{l,j+2}(\lambda_i)(z-\lambda_i)^{j+2}, \nonumber \\ && {\partial\over\partial\lambda_i}{dz\over s_l}= {l\over N}{dz\over s_l(z-\lambda_i)}, \nonumber \end{eqnarray} we see that (\ref{expomegal}) is equivalent to \begin{eqnarray} && {f^\prime(\lambda_i)\over N}{\partial\over\partial\lambda_i} \int_{A_j}{dz\over s_l}+ {1\over N}\sum_{j=0}^{lm-2}P^{(l)}_{l,j+2}(\lambda_i) \int_{A_j}{(z-\lambda_i)^jdz\over s_l} \nonumber \\ && +\sum_{k=1,k\neq l}\sum_{j=0}^{km-2}P^{(l)}_{k,j}(\lambda_i) \int_{A_j}{(z-\lambda_i)^jdz\over s_k}=0. \label{lineareq} \end{eqnarray} We consider (\ref{lineareq}) as a linear equation for the $g$ variables $\{P^{(l)}_{k,r}(\lambda_i)\}$. Solving (\ref{lineareq}) in $P^{(l)}_{l,2}$ by the Cramer's formula and summing up in $l$ we have the statement of the proposition. $\Box$ \vskip2mm The idea of deriving equations of the form (\ref{lineareq}) is due to Bershadsky-Radul\cite{BR1}. By calculations we have \begin{cor}\label{projexp2} The coefficient of $t^{N-2}dt$ in the expansion of $G_z(z)$ in $t=(z-\lambda_i)^{1/N}$ is \begin{eqnarray} && -\mu N\sum_{j=1,j\neq i}^{Nm}{1\over \lambda_i-\lambda_j}- N{\partial\over\partial\lambda_i}\log\det A, \nonumber \end{eqnarray} where \begin{eqnarray} && \mu={(N-1)(2N-1)\over 6N}. \nonumber \end{eqnarray} \end{cor} \vskip1cm \section{Variational formula of period matrix} \par Let us consider the equation \begin{eqnarray} && s_t^N=(z-\lambda_i-t)\prod_{j=1,j\neq i}^{Nm}(z-\lambda_j) \nonumber \end{eqnarray} which is a one parameter deformation of the curve $C$ by a small parameter $t$. We denote the corresponding compact Riemann surface by $C_t$. The notation $s_t$ is different from $s_l=s^l$ in the previous section. We hope that this does not cause any confusion. Let $\bar{\pi}$ be the projection $\bar{\pi}:C\longrightarrow {\bf P}^1$ which maps $(z,s)$ to $z$. We can take a canonical dissection $\{A_i(t),B_j(t)\}$ of $C_t$ such that $\bar{\pi}(A_i(t))$, $\bar{\pi}(B_j(t))$ do not depend on $t$ for $\vert t\vert$ being sufficiently small. The integration of a holomorphic $1$-form on $C_t$ along $A_i(t),B_j(t)$ can be considered as the integration of a multi-valued holomorphic $1$-form on ${\bf P}^1-\{\lambda_1,\cdots,\lambda_{Nm}\}$ along $\bar{\pi}(A_i(t))$, $\bar{\pi}(B_j(t))$. Hence we can think of the integration cycles $A_i(t),B_j(t)$ as if they are independent of $t$. Therefore we simply write $A_i,B_j$ instead of $A_i(t),B_j(t)$ in the calculations in this section. Let $\{v_j(x,t)\}$ be the basis of normailzed holomorphic 1-forms on $C_t$ with respect to $\{A_i(t),B_j(t)\}$. We denote by $\tau(t)=(\tau_{kr}(t))$ the period matrix \begin{eqnarray} && \tau_{kr}(t)=\int_{B_k}v_r(x,t). \nonumber \end{eqnarray} We set \begin{eqnarray} && w^{(\alpha)}_{\beta t}={z^{\beta-1}dz\over s_t^\alpha}. \nonumber \end{eqnarray} We also use our previous notation $v_j(x)=v_j(x,0)$, $\tau_{kr}=\tau_{kr}(0)$, $s=s_0$, $w^{(\alpha)}_\beta=w^{(\alpha)}_{\beta0}$. Our aim in this section is to prove \begin{theo}\label{variation} \begin{eqnarray} && {d\tau_{jk}\over dt}(0)= {1\over N(N-2)!}\sum_{\alpha=0}^{N-2} \bic{N-2}{\alpha}v_j^{(\alpha)}(Q_i)v_k^{(N-2-\alpha)}(Q_i). \nonumber \end{eqnarray} \end{theo} \vskip2mm We define the connection matrix $\sigma$ and $c$ by \begin{eqnarray} && v_j(x)=\sum_{\alpha,\beta}\sigma_{j(\alpha\beta)}w^{(\alpha)}_\beta(x), \quad w^{(\alpha)}_\beta(x)=\sum_{j}c_{(\alpha\beta)j}v_j(x). \label{connection} \end{eqnarray} Let $P\in C$ and $u$ be a local coordinate around $P$. Let $\omega(P;n)$ be the abelian differential of the second kind satisfying the following conditions. \vskip2mm \begin{description} \item[1.] $\omega(P;n)$ is holomorphic except the point $P\in C$ where it has a pole of order $n\geq2$. At $P$ we have the expansion of the form \begin{eqnarray} && \omega(P;n)=-{n-1\over u^n}du(1+O(u^n)). \nonumber \end{eqnarray} \item[2.] $\omega(P;n)$ has zero $A$ periods : \begin{eqnarray} && \int_{A_j}\omega(P;n)=0 \quad\hbox{ for any $j$.} \nonumber \end{eqnarray} \end{description} \vskip2mm The differential $\omega(P;n)$ depends on the choice of the local coordinate $u$. In our case $P=Q_i$ we always take $u=(z-\lambda_i)^{1/N}$ as a local coordinate around $Q_i$. In this sense $\omega(P;n)$ is uniquely determined. It is known that the following relation holds \begin{eqnarray} && \int_{B_j}\omega(P;n)=-{1\over(n-2)!}v_j^{(n-2)}(P), \label{nicerels} \end{eqnarray} where $v_j^{(n-2)}(P)$ is the coefficient of $u^{n-2}du$ in the expansion of $v_j(x)$ in $u$. \begin{lem} If we expand $v_j(x,t)$ as \begin{eqnarray} && v_j(x,t)=v_j(x)+v_{j1}(x)t+\cdots, \label{fexpvt} \end{eqnarray} then we have \begin{eqnarray} && v_{j1}(x)= -\sum_{\alpha,\beta} {\sigma_{j(\alpha\beta)}\lambda_i^{\beta-1}\over \prod_{j\neq i}(\lambda_i-\lambda_j)^{\alpha/N}} \omega(Q_i;\alpha+1). \nonumber \end{eqnarray} \end{lem} \vskip2mm \noindent Proof. We have the expansion \begin{eqnarray} && w^{(\alpha)}_{\beta t}(x)= w^{(\alpha)}_\beta(x)+{\alpha z^{\beta-1}dz\over N(z-\lambda_i)s^\alpha}t +O(t^2), \label{expwt} \end{eqnarray} and the relation \begin{eqnarray} w^{(\alpha)}_{\beta t}(x)= \sum_{j=1}^g\int_{A_j}w^{(\alpha)}_{\beta t}\cdot v_j(x,t). \label{relwv} \end{eqnarray} Substituting the expansions (\ref{fexpvt}) and (\ref{expwt}) into the equation (\ref{relwv}) and comparing the coefficient of $t$ we have \begin{eqnarray} \sum_{j=1}^gv_{j1}(x)\int_{A_j}w^{(\alpha)}_\beta&=& {\alpha\over N}\eta^{(\alpha)}_\beta(x), \nonumber \\ \eta^{(\alpha)}_\beta(x)&=& {z^{\beta-1}dz\over(z-\lambda_i)s^\alpha} -\sum_{j=1}^gv_j(x)\int_{A_j}{z^{\beta-1}dz\over(z-\lambda_i)s^\alpha}. \label{exprels} \end{eqnarray} Then $\eta^{(\alpha)}_\beta(x)$ has the following properties: \vskip2mm \begin{description} \item[1.] $\int_{A_k}\eta^{(\alpha)}_\beta(x)=0$ for any $k=1,\cdots,g$. \item[2.] Taking $u=(z-\lambda_i)^{1/N}$ as a local coordinate around $Q_i$ we have the expansion \begin{eqnarray} && \eta^{(\alpha)}_\beta(x)= {N\lambda_i^{\beta-1}\over f^\prime(\lambda_i)^{\alpha/N}} {du\over u^{\alpha+1}}+O(1). \nonumber \end{eqnarray} \end{description} \vskip2mm Hence we have \begin{eqnarray} && \eta^{(\alpha)}_\beta(x) =-{N\lambda_i^{\beta-1}\over \alpha f^\prime(\lambda_i)^{\alpha/N}} \omega(Q_i;\alpha+1). \label{etacanonical} \end{eqnarray} Since \begin{eqnarray} && \int_{A_j}w^{(\alpha)}_\beta=c_{(\alpha\beta)j}, \nonumber \end{eqnarray} and $\sigma$ is the inverse matrix of $c$, we have the desired result from (\ref{exprels}) and (\ref{etacanonical}). $\Box$ \vskip2mm Now comparing the coefficient of $u^{N-1-\alpha}du$ of the both hand sides of the first equation of (\ref{connection}) we have \begin{eqnarray} && \sum_{\beta=1}^{\alpha m-1}\sigma_{j(\alpha\beta)}\lambda_i^{\beta-1} ={f^\prime(\lambda_i)^{\alpha/N}\over N(N-1-\alpha)!} v_j^{(N-1-\alpha)}(Q_i), \nonumber \end{eqnarray} for $1\leq \alpha\leq N-1$ and thus \begin{eqnarray} v_{j1}(x)= -{1\over N}\sum_{\alpha=1}^{N-1} {1\over (N-1-\alpha)!} v_j^{(N-1-\alpha)}(Q_i)\omega(Q_i;\alpha+1). \nonumber \end{eqnarray} Integrating both hand sides of this equation along the cycle $B_k$ and using the relation (\ref{nicerels}) we obtain \begin{eqnarray} && \int_{B_k}v_{j1}(x)= {1\over N(N-2)!} \sum_{\alpha=0}^{N-2} \bic{N-2}{\alpha}v_j^{(N-2-\alpha)}(Q_i)v_k^{(\alpha)}(Q_i). \nonumber \end{eqnarray} $\Box$ \vskip1cm \section{Thomae formula} \par Now let us prove the generalized Thomae formula. \begin{theo}\label{Thomae} For an ordered partition $\Lambda=(\Lambda_0,\cdots,\Lambda_{N-1})$ we have \begin{eqnarray} && \theta[e_\Lambda](0)^{2N} =C_{\Lambda}(\det A)^{N}\prod_{i<j}(\lambda_i-\lambda_j)^{2Nq(k_i,k_j)+N\mu}, \nonumber \end{eqnarray} where $k_i=j$ for $i\in\Lambda_j$, \begin{eqnarray} && q(i,j)=\sum_{l\in{\cal L}}q_l(i)q_l(j), \quad \mu={(N-1)(2N-1)\over 6N}, \nonumber \end{eqnarray} and $q_l(i)$, ${\cal L}$ are given by (\ref{qli}), (\ref{calL}) in section 3. The complex number $C_{\Lambda}$ does not depend on $\lambda_i$'s. They satisfy $C_{\Lambda}^{2N}=C_{\Lambda^\prime}^{2N}$ for any $\Lambda$, $\Lambda^\prime$. \end{theo} \vskip3mm Since the familly $\{C_t\}$ is locally topologically trivial, we can take a canonical dissection $\{A_i(t), B_j(t)\}$ of $C_t$ such that $A_i(t)$, $B_j(t)$ are continuous in $t$. We assume that $A_i(t),B_j(t)$ does not go through any branch point $Q_i(t)$. We can also define the base points $P_0(t)$ of $C_t$ and $z_0(t)$ of $\tilde{C}_t$ lying over $P_0(t)$ so that they vary continuously in $t$. We identify $Q_i(t)$ with the corresponding point in the fundamental domain on $\tilde{C}_t$ which contains the base point $z_0(t)$. Let $k^{P_0}(t)$ be the vector in ${\bf C}^g$ whose $j$-th component is defined by \begin{eqnarray} && k^{P_0}(t)_j= {2\pi i-\tau_{jj}(t)\over2} +{1\over 2\pi i}\sum_{i\neq j} \int_{A_i(t)}v_i(x,t)\int_{z_0(t)}^xv_j(x,t). \nonumber \end{eqnarray} It is known (see \cite{F}(p8) for example) that \begin{eqnarray} && \Delta-(g-1)P_0(t)=k^{P_0}(t) \nonumber \end{eqnarray} in $J(C)$. Let us define $e_\Lambda(t)$ as an element of ${\bf C}^g$ by \begin{eqnarray} && e_\Lambda(t)= \sum_{j\in\Lambda_1}\int_{z_0(t)}^{Q_j(t)}v(x,t)+\cdots +(N-1)\sum_{j\in\Lambda_{N-1}}\int_{z_0(t)}^{Q_j(t)}v(x,t) -k^{P_0}(t). \nonumber \end{eqnarray} Then $e_\Lambda(t)$ is continuous in $t$. Note that the linear isomorphism ${\bf C}^g\simeq {\bf R}^{2g}$ sending $e$ to its characteristics with respect $\tau(t)$ is analytic in $t$. Therefore if we write \begin{eqnarray} && e_\Lambda(t)=\cht{\delta(t)}{\epsilon(t)}, \nonumber \end{eqnarray} then $\epsilon(t)$ and $\delta(t)$ are continuous in $t$. Since $\epsilon(t)$ and $\delta(t)$ are in $1/2N{\bf Z}^{g}$, they are constant in $t$. Therefore we simply write $\epsilon,\delta$ instead of $\epsilon(t),\delta(t)$. We denote by $\theta_t[e](z)$ the theta function associated with the canonical basis $\{A_i(t),B_j(t)\}$ of $C_t$. We set $\theta[e](z)=\theta_0[e](z)$. Then the function $\theta_t[e_\Lambda(t)](0)$ depends on $t$ only through the period matrix $\tau_{kr}(t)$ since \begin{eqnarray} && \theta_t[e_\Lambda(t)](0)= \sum_{m\in{\bf Z}^g}\exp({1\over2}(m+\delta)\tau(t)(m+\delta)^t +2\pi i\epsilon(m+\delta)^t). \nonumber \end{eqnarray} Using the heat equations \begin{eqnarray} && {\partial^2\theta[e_\Lambda]\over \partial z_k\partial z_r}(z) ={\partial\theta[e_\Lambda]\over\partial\tau_{kr}}(z) \quad(k\neq r), \quad {\partial^2\theta[e_\Lambda]\over \partial z_k^2}(z) =2{\partial\theta[e_\Lambda]\over\partial\tau_{kk}}(z), \nonumber \end{eqnarray} and Lemma \ref{vanish} we have \begin{eqnarray} {\partial\over\partial\lambda_i}\log\theta[e_\Lambda](0)&=& {d\over dt}\log\theta_t[e_\Lambda(t)](0)\Big\vert_{t=0} \nonumber \\ &=&\sum_{1\leq k\leq r\leq g} {\partial \log\theta[e_\Lambda]\over \partial\tau_{kr}}(0) {d\tau_{kr}\over dt}(0) \nonumber \\ &=&{1\over2} \sum_{k,r=1}^g {1\over \theta[e_\Lambda](0)} {\partial^2 \theta[e_\Lambda]\over \partial z_k\partial z_r}(0) {d\tau_{kr}\over dt}(0), \nonumber \\ &=& {1\over2}\sum_{k,r=1}^g {\partial^2 \log\theta[e_\Lambda]\over \partial z_k\partial z_r}(0) {d\tau_{kr}\over dt}(0). \label{logderiv} \end{eqnarray} On the other hand by Corollary \ref{projexp1}, \ref{projexp2} and Theorem \ref{variation} we have \begin{eqnarray} &&-\mu N\sum_{j\neq i}{1\over\lambda_i-\lambda_j} -N{\partial\over\partial\lambda_i}\log\det A \nonumber \\ &&= 2N\sum_{j\neq i}{q(k_i,k_j)\over\lambda_i-\lambda_j} -N\sum_{k,r=1}^g {\partial^2 \log\theta[e_\Lambda]\over \partial z_k\partial z_r}(0) {d\tau_{kr}\over dt}(0). \label{preeq} \end{eqnarray} Substituting (\ref{logderiv}) into (\ref{preeq}) we have \begin{eqnarray} &&{\partial\over\partial\lambda_i}\log\theta[e_\Lambda](0) = {1\over2}{\partial\over\partial\lambda_i}\log\det A +{\mu\over2}\sum_{j\neq i}{1\over\lambda_i-\lambda_j} +\sum_{j\neq i}{q(k_i,k_j)\over\lambda_i-\lambda_j}. \nonumber \end{eqnarray} Hence we have proved the first part of Theorem \ref{Thomae}. \vskip1cm \section{Property of the constant $C_\Lambda$} \par Our aim in this section is to prove the remaining part of Theorem \ref{Thomae}, that is, for any ordered partitions $\Lambda$ and $\Lambda^\prime$ \begin{eqnarray} && C_{\Lambda}^{2N}=C_{\Lambda^\prime}^{2N}. \label{remaining} \end{eqnarray} As in the previous section we identify branch points $Q_i$ with the corresponding points in the fundamental domain in $\tilde{C}$. The key for the proof is the formula of Fay ( \cite{F}, p30, Cor.2.17 ): \begin{eqnarray} && \theta(\sum_{k=1}^gx_k-p-\Delta) =c{\det(v_i(x_j))\over\prod_{i<j}E(x_i,x_j)} {\sigma(p)\over \prod_{k=1}^g\sigma(x_k)} \prod_{k=1}^gE(x_k,p), \nonumber \end{eqnarray} for any $p,x_1,\ldots,x_g\in \tilde{C}$, where $c$ is independent on $p,x_1,\ldots,x_g$ and \begin{eqnarray} && \sigma(p)= \exp(-{1\over2\pi i}\sum_{j=1}^g\int_{A_j}v_j(y)\log E(y,p)). \nonumber \end{eqnarray} Taking ratios for $p=a,b$ equations we have \begin{eqnarray} && {\theta(\sum_{k=1}^gx_k-a-\Delta)\over \theta(\sum_{k=1}^gx_k-b-\Delta)} = {\sigma(a)\over\sigma(b)} \prod_{k=1}^g {E(x_k,a)\over E(x_k,b)}. \nonumber \end{eqnarray} Set $a=Q_i$, $b=Q_j$ $(i\neq j)$ and taking $N$-th power of the both hand sides we obtain \begin{eqnarray} && {\theta(\sum_{k=1}^gx_k-Q_i-\Delta)^N \over \theta(\sum_{k=1}^gx_k-Q_j-\Delta)^N} = \Big({\sigma(Q_i)\over\sigma(Q_j)}\Big)^N \prod_{k=1}^g {E(x_k,Q_i)^N\over E(x_k,Q_j)^N}. \nonumber \end{eqnarray} \noindent Since $NQ_i$ and $NQ_j$ are linearly equivalent, there exists $\lambda(i,j),\kappa(i,j)\in{\bf Z}^g$ such that \begin{eqnarray} && N\int_{z_0}^{Q_i}v(x)-N\int_{z_0}^{Q_j}v(x) =N\int_{Q_i}^{Q_j}v(x)=2\pi i\lambda(i,j)+\kappa(i,j)\tau, \nonumber \\ &&\kappa(j,i)=-\kappa(i,j),\quad \lambda(j,i)=-\lambda(i,j). \nonumber \end{eqnarray} If we set \begin{eqnarray} && f(x)= {E(x,Q_i)^N\over E(x,Q_j)^N}, \nonumber \end{eqnarray} then we have \begin{eqnarray} f(x+A_k)&=&f(x) \nonumber \\ f(x+B_k)&=& \exp(-\sum_{l}\tau_{lk}\kappa(i,j)_l)f(x). \nonumber \end{eqnarray} Hence the function \begin{eqnarray} && \exp(\sum_{l}\int_{z_0}^xv_l(x)\kappa(i,j)_l)f(x) \nonumber \end{eqnarray} can be considered as a single valued function on $C$. Its only zeros are of $N$-th order at $Q_i$ and only poles are of $N$-th order at $Q_j$. Therefore there exists a constant $c_{ij}$ such that \begin{eqnarray} && \exp(\int_{z_0}^xv(x)\kappa(i,j)^t) {E(x,Q_i)^N\over E(x,Q_j)^N} =c_{ij}{z(x)-\lambda_i\over z(x)-\lambda_j}. \nonumber \end{eqnarray} By the property of $\kappa(i,j)$, $c_{ij}$ satisfies \begin{eqnarray} && c_{ij}=c_{ji}^{-1}, \quad c_{ii}=1. \nonumber \end{eqnarray} If we set \begin{eqnarray} && r_{ij}=c_{ij}\Big({\sigma(Q_i)\over\sigma(Q_j)}\Big)^N, \nonumber \\ && w(x\vert i,j)=\int_{z_0}^xv(x)\kappa(i,j)^t, \quad w(\sum_kx_k\vert i,j)=\sum_k w(x_k \vert i,j), \nonumber \end{eqnarray} we have \begin{eqnarray} && \exp(w(\sum_{k=1}^gx_k\vert i,j)) {\theta(\sum_{k=1}^gx_k-Q_i-\Delta)^N \over \theta(\sum_{k=1}^gx_k-Q_j-\Delta)^N} =r_{ij}\prod_{k=1}^g {z(x)-\lambda_i\over z(x)-\lambda_j}, \label{ratio1} \\ && r_{ij}=r_{ji}^{-1}, \quad r_{ii}=1. \nonumber \end{eqnarray} Now let us take an ordered partition $\Lambda=\Lambda^{(1)}=(\Lambda^{(1)}_{0},\ldots,\Lambda^{(1)}_{N-1})$ with \begin{eqnarray} && \Lambda^{(1)}_l=\{i^l_1,\ldots,i^l_m\}, \quad 0\leq l\leq N-1. \nonumber \end{eqnarray} Let us define $\Lambda^{(2)}=(\Lambda^{(2)}_{0},\ldots,\Lambda^{(2)}_{N-1})$ by \begin{eqnarray} && \Lambda^{(2)}_0=\{i^0_1,\ldots,i^0_{m-1},i^{N-1}_m\}, \quad \Lambda^{(2)}_{N-1}=\{i^{N-1}_1,\ldots,i^{N-1}_{m-1},i^{0}_m\}, \nonumber \end{eqnarray} and $\Lambda^{(2)}_l=\Lambda^{(2)}_l(1\leq l\leq N-1)$. If we consider $Q_i$ as $\int_{z_0}^{Q_i}v$, we have the vectors in ${\bf C}^g$: \begin{eqnarray} e_{\Lambda^{(1)}}&=&\Lambda^{(1)}_1+2\Lambda^{(1)}_2+\cdots+ (N-1)(\Lambda^{(1)}_{N-1}\backslash\{i^{N-1}_m\})-Q_{i^{N-1}_m}- k^{P_0}, \nonumber \\ e_{\Lambda^{(2)}}&=&\Lambda^{(2)}_1+2\Lambda^{(2)}_2+\cdots+ (N-1)(\Lambda^{(2)}_{N-1}\backslash\{i^{0}_m\})-Q_{i^{0}_m}-k^{P_0}, \nonumber \\ &=&\Lambda^{(1)}_1+2\Lambda^{(1)}_2+\cdots+ (N-1)(\Lambda^{(1)}_{N-1}\backslash\{i^{N-1}_m\})-Q_{i^{0}_m}-k^{P_0}. \nonumber \end{eqnarray} Putting $i=i^{N-1}_m$, $j=i^0_m$ and \begin{eqnarray} && (x_1,\ldots,x_g)=(\Lambda^{(1)}_1,2\Lambda^{(1)}_2,\ldots, (N-1)(\Lambda^{(1)}_{N-1}\backslash\{i^{N-1}_m\})) \nonumber \end{eqnarray} in (\ref{ratio1}) we have \begin{eqnarray} && \exp(U(\Lambda^{(1)}\vert i^{N-1}_m,i^0_m)) {\theta(e_{\Lambda^{(1)}})^N\over \theta(e_{\Lambda^{(2)}})^N} =r_{i^{N-1}_m,i^0_m} \prod_{r=1}^{N-1} {\prod_{s=1}^{m}}^\prime \Big( {\lambda_{i^r_s}-\lambda_{i^{N-1}_m}\over \lambda_{i^r_s}-\lambda_{i^{0}_m}} \Big)^r, \label{ratio2} \\ && U(\Lambda^{(1)}\vert i^{N-1}_m,i^0_m)= w(\Lambda^{(1)}_1+2\Lambda^{(1)}_2+\cdots+ (N-1)(\Lambda^{(1)}_{N-1}\backslash\{i^{N-1}_m\})\vert i^{N-1}_m,i^0_m). \nonumber \end{eqnarray} Here $\prod'$ means the product for $(r,s)\neq(N-1,m)$. Let us define the elements of ${\bf C}^g$ by \begin{eqnarray} -\bar{e}_{\Lambda^{(1)}}&=& \Lambda^{(1)}_{N-2}+2\Lambda^{(1)}_{N-3}+\cdots+ (N-1)(\Lambda^{(1)}_{0}\backslash\{i^{0}_m\}) -Q_{i^{0}_m}-k^{P_0}, \nonumber \\ -\bar{e}_{\Lambda^{(2)}}&=& \Lambda^{(2)}_{N-2}+2\Lambda^{(2)}_{N-3}+\cdots+ (N-1)(\Lambda^{(2)}_{0}\backslash\{i^{N-1}_m\}) -Q_{i^{N-1}_m}-k^{P_0} \nonumber \\ &=& \Lambda^{(1)}_{N-2}+2\Lambda^{(1)}_{N-3}+\cdots+ (N-1)(\Lambda^{(1)}_{0}\backslash\{i^{0}_m\}) -Q_{i^{N-1}_m}-k^{P_0}, \nonumber \end{eqnarray} where again $Q_i$ denotes $\int_{z_0}^{Q_i}v$. Then if we set $i=i^{0}_m$, $j=i^{N-1}_m$ and \begin{eqnarray} && (x_1,\ldots,x_g)=(\Lambda^{(1)}_{N-2},2\Lambda^{(1)}_{N-3},\ldots, (N-1)(\Lambda^{(1)}_{0}\backslash\{i^{0}_m\})) \nonumber \end{eqnarray} in (\ref{ratio1}) we have \begin{eqnarray} && \exp(U'(\Lambda^{(1)}\vert i^{0}_m,i^{N-1}_m)) {\theta(\bar{e}_{\Lambda^{(1)}})^N\over \theta(\bar{e}_{\Lambda^{(2)}})^N} =r_{i^0_m,i^{N-1}_m} \prod_{r=1}^{N-1} {\prod_{s=1}^{m}}^\prime \Big( {\lambda_{i^{N-1-r}_s}-\lambda_{i^{0}_m}\over \lambda_{i^{N-1-r}_s}-\lambda_{i^{N-1}_m}} \Big)^r, \label{ratio3} \\ && U'(\Lambda^{(1)}\vert i^{0}_m,i^{N-1}_m)= w(\Lambda^{(1)}_{N-2}+2\Lambda^{(1)}_{N-3}+\cdots+ (N-1)(\Lambda^{(1)}_{0}\backslash\{i^{0}_m\})\vert i^{0}_m,i^{N-1}_m). \nonumber \end{eqnarray} Here we have used the property that $\theta(z)$ is an even function of $z$. Multiplying (\ref{ratio2}) and (\ref{ratio3}) we have \begin{eqnarray} && \exp\big(w(-\sum_{l=0}^{N-1}(N-1-2l) \tilde{\Lambda}^{(1)}_l\vert i^{N-1}_m,i^0_m)\big) \Big({ \theta(e_{\Lambda^{(1)}})\theta(\bar{e}_{\Lambda^{(1)}}) \over \theta(e_{\Lambda^{(2)}})\theta(\bar{e}_{\Lambda^{(2)}}) }\Big)^N \nonumber \\ &&=\prod_{r=1}^{N-1}{\prod_{s=1}^{m}}^\prime { (\lambda_{i^r_s}-\lambda_{i^{N-1}_m})^r(\lambda_{i^{N-1-r}_s}-\lambda_{i^{0}_m})^r \over (\lambda_{i^r_s}-\lambda_{i^{0}_m})^r(\lambda_{i^{N-1-r}_s}-\lambda_{i^{N-1}_m})^r }, \label{ratio4} \end{eqnarray} where we set \begin{eqnarray} && \tilde{\Lambda}^{(1)}_0=\Lambda^{(1)}_0\backslash\{i^0_m\}, \quad \tilde{\Lambda}^{(1)}_{N-1}=\Lambda^{(1)}_{N-1}\backslash\{i^{N-1}_m\}, \quad \tilde{\Lambda}^{(1)}_l=\Lambda^{(1)}_l\quad(l\neq 0,N-1). \nonumber \end{eqnarray} Since $\bar{e}_{\Lambda^{(k)}}=e_{\Lambda^{(k)}}$, $k=1,2$ in $J(C)$ we can set \begin{eqnarray} && e_{\Lambda^{(k)}}=\ch{\delta^{(k)}}{\epsilon^{(k)}}= 2\pi i\epsilon^{(k)}+\delta^{(k)}\tau, \quad \delta^{(k)},\epsilon^{(k)}\in{1\over 2N}{\bf Z}^g, \nonumber \\ && \bar{e}_{\Lambda^{(k)}}=e_{\Lambda^{(k)}}+2\pi im^{(k)}+n^{(k)}\tau, \quad m^{(k)},n^{(k)}\in{\bf Z}^g. \nonumber \end{eqnarray} Substituting these equations into (\ref{ratio4}), taking $2N$-th power of both hand sides and using the transformation property of theta functions, we get \begin{eqnarray} && \Big({ \theta[e_{\Lambda^{(1)}}](0) \over \theta[e_{\Lambda^{(2)}}](0) }\Big)^{4N^2} =B \prod_{r=1}^{N-1}{\prod_{s=1}^{m}}^\prime { (\lambda_{i^r_s}-\lambda_{i^{N-1}_m})^{2Nr} (\lambda_{i^{N-1-r}_s}-\lambda_{i^{0}_m})^{2Nr} \over (\lambda_{i^r_s}-\lambda_{i^{0}_m})^{2Nr} (\lambda_{i^{N-1-r}_s}-\lambda_{i^{N-1}_m})^{2Nr} },\label{ratio5} \\ && B= \exp\big( -2n^\prime\tau\kappa^t- N^2\sum_{k=1}^2(-1)^k( n^{(k)}\tau n^{(k)t} +2\delta^{(k)}\tau n^{(k)t} +2\delta^{(k)}\tau \delta^{(k)t}) \big), \nonumber \end{eqnarray} where we set \begin{eqnarray} && Nw(-\sum_{l=0}^{N-1}(N-1-2l)\tilde{\Lambda}^{(1)}_l\vert i^{N-1}_m,i^0_m) =(2\pi im^\prime+n^\prime\tau)\kappa, \nonumber \\ && \kappa=\kappa(i^{N-1}_m,i^0_m). \nonumber \end{eqnarray} We can simplify the right hand side of (\ref{ratio5}) so that there are no common divisor in the numerator and the denominator. The result is \begin{eqnarray} \Big({ \theta[e_{\Lambda^{(1)}}](0) \over \theta[e_{\Lambda^{(2)}}](0) }\Big)^{4N^2}= && B \prod_{r=1}^{N-2}{\prod_{s=1}^{m}} \Big({ \lambda_{i^r_s}-\lambda_{i^{0}_m} \over \lambda_{i^r_s}-\lambda_{i^{N-1}_m} }\Big)^{2N(N-1-2r)} \nonumber \\ && \prod_{s=1}^{m-1} \Big({ (\lambda_{i^{N-1}_s}-\lambda_{i^{N-1}_m}) (\lambda_{i^0_s}-\lambda_{i^{0}_m}) \over (\lambda_{i^0_s}-\lambda_{i^{N-1}_m}) (\lambda_{i^{N-1}_s}-\lambda_{i^{0}_m}) }\Big)^{2N(N-1)}. \label{ratio6} \end{eqnarray} Let us compare this equation with those obtained from the proved part of Theorem \ref{Thomae}. Let $\{k_i\}$, $\{k_i^\prime\}$ correspond to $\Lambda^{(1)}$, $\Lambda^{(2)}$ respectively as in (\ref{weight}). Then by the proved part of the Thomae formula we have \begin{eqnarray} \Big({ \theta[e_{\Lambda^{(1)}}](0) \over \theta[e_{\Lambda^{(2)}}](0) }\Big)^{4N^2} = C\prod_{i<j}(\lambda_i-\lambda_j)^{4N^2(q(k_i,k_j)-q(k^\prime_i,k^\prime_j))}, \label{conseq} \end{eqnarray} where $C=\big(C_{\Lambda^{(1)}}/C_{\Lambda^{(2)}}\big)^{4N^2}$. Note that $4N^2q(k_i,k_j)$ and $4N^2q(k^\prime_i,k^\prime_j)$ are even. Then we have \begin{eqnarray} && \hbox{LHS of (\ref{conseq})} \nonumber \\ && = C \prod_{r=1}^{N-1}{\prod_{s=1}^{m}}^\prime \Big({ \lambda_{i^r_s}-\lambda_{i^{N-1}_m} \over \lambda_{i^r_s}-\lambda_{i^{0}_m} }\Big)^{4N^2q(r,N-1)} \prod_{r=0}^{N-2}{\prod_{s=1}^{m}}^{''} \Big({ \lambda_{i^r_s}-\lambda_{i^{0}_m} \over \lambda_{i^r_s}-\lambda_{i^{N-1}_m} }\Big)^{4N^2q(r,0)} \nonumber \\ && \prod_{s=1}^{m-1} \Big({ (\lambda_{i^{N-1}_s}-\lambda_{i^{0}_m}) (\lambda_{i^{0}_s}-\lambda_{i^{N-1}_m}) \over (\lambda_{i^{N-1}_s}-\lambda_{i^{N-1}_m}) (\lambda_{i^{0}_s}-\lambda_{i^{0}_m}) } \Big)^{4N^2q(0,1)} \nonumber \\ && =C \prod_{r=1}^{N-2}\prod_{s=1}^m \Big({ \lambda_{i^r_s}-\lambda_{i^{0}_m} \over \lambda_{i^r_s}-\lambda_{i^{N-1}_m} }\Big)^{4N^2(q(r,0)-q(r,N-1))} \nonumber \\ && \prod_{s=1}^{m-1} \Big({ (\lambda_{i^{N-1}_s}-\lambda_{i^{N-1}_m}) (\lambda_{i^{0}_s}-\lambda_{i^{0}_m}) \over (\lambda_{i^{N-1}_s}-\lambda_{i^{0}_m}) (\lambda_{i^{0}_s}-\lambda_{i^{N-1}_m}) } \Big)^{4N^2(q(0,0)-q(0,1))}, \label{ratiothoma} \end{eqnarray} where $\prod''$ means the product for $(r,s)\neq(0,m)$. Let us calculate $q_l(r,t)$. By the definition of $q(i,j)$ and Lemma \ref{exponent}, $q(i,j)$ depends only on $\vert i-j\vert$ mod $N$. Hence using $q_l(0)=N/l$ we have \begin{eqnarray} && q(r,t)=q(0,t-r) =\sum_{l\in{\cal L}}q_l(0)q_l(t-r) ={1\over N}\sum_{l\in{\cal L}}lq_l(t-r). \nonumber \end{eqnarray} The following lemma is obtained by a direct calculation. \begin{lem} \begin{eqnarray} \sum_{l\in{\cal L}}lq_l(r)={N^2-1\over12}-{1\over2}r(N-r). \end{eqnarray} \end{lem} \vskip2mm Thus we have \begin{eqnarray} && q(r,t)= {1\over N} \Big({N^2-1\over 12}-{1\over2}(t-r)(N-t+r)\Big). \nonumber \end{eqnarray} In particular \begin{eqnarray} 4N^2\big(q(0,0)-q(0,1)\big)&=&2N(N-1), \nonumber \\ 4N^2\big(q(r,0)-q(r,N-1)\big)&=&2N(N-1-2r). \nonumber \end{eqnarray} Comparing (\ref{ratio6}) and (\ref{ratiothoma}) we have \begin{eqnarray} \Big({C_{\Lambda^{(1)}}\over C_{\Lambda^{(2)}}}\Big)^{2N} =B. \nonumber \end{eqnarray} Let us write \begin{eqnarray} && B=\exp(\sum_{k\leq r}b_{kr}\tau_{kr}). \nonumber \end{eqnarray} Since $C_{\Lambda^{(1)}}$ and $C_{\Lambda^{(2)}}$ do not depend on $\tau$ \begin{eqnarray} && {\partial\over\partial \tau_{kr}}B=b_{kr}B=0, \quad\hbox{for any $k\leq r$}. \nonumber \end{eqnarray} Hence $B=1$. Since any two ordered partitions are transformed to each other by successive exchange of elements of $\Lambda_i$ and $\Lambda_{i+1}$,$i=0,\ldots,N-1$, the equation (\ref{remaining}) are proved. \vskip1cm \section{Examples} \par In this section we shall give examples of Thomae formula for small $N$'s. Recall that \begin{eqnarray} && {\mu\over2}+q(0,j)={N-1\over4}-{j(N-j)\over2N}. \nonumber \\ && q(i,j)=q(0,\vert i-j\vert)=q(0,-\vert i-j\vert). \nonumber \end{eqnarray} We remark that the constants $C_{\Lambda}$ in this section are different from those in Theorem \ref{Thomae} by $\pm1$ times because of the reordering of the difference products. The properties of the constants remain same. \subsection{$N=2$} \par We consider the hyperelliptic curve $s^2=\prod_{j=1}^{2m}(z-\lambda_l)$. Let $\Lambda=(\Lambda_0,\Lambda_1)$ with \begin{eqnarray} && \Lambda_0=\{i_1<\cdots<i_m\}, \quad \Lambda_1=\{j_1<\cdots<j_m\}. \nonumber \end{eqnarray} We have \begin{eqnarray} && q(0,0)={1\over8}, \quad, q(0,1)=-{1\over8}, \quad \mu={1\over4}. \nonumber \end{eqnarray} The Thomae formula is \begin{eqnarray} && \theta[e_\Lambda](0)^4=C_{\Lambda}(\det A)^2 \prod_{k<l}(\lambda_{i_k}-\lambda_{i_l})(\lambda_{j_k}-\lambda_{j_l}). \nonumber \end{eqnarray} This is the original Thomae formula in which case $C_{\Lambda}^2=(2\pi)^{-4(m-1)}$. \subsection{$N=3$} Let $\Lambda=(\Lambda_0,\Lambda_1,\Lambda_2)$. We have \begin{eqnarray} && q(0,0)={2\over9}, \quad q(0,1)=q(0,2)=-{1\over9}, \quad \mu={5\over9}. \nonumber \end{eqnarray} Then \begin{eqnarray} && \theta[e_\Lambda](0)^6=C_{\Lambda}(\det A)^3 \big((\Lambda_0\Lambda_0)(\Lambda_1\Lambda_1)(\Lambda_2\Lambda_2)\big)^3 (\Lambda_0\Lambda_1)(\Lambda_1\Lambda_2)(\Lambda_0\Lambda_2). \nonumber \end{eqnarray} Here if \begin{eqnarray} && \Lambda_i=\{i_1<\cdots<i_m\}, \quad \Lambda_j=\{j_1<\cdots<j_m\}, \nonumber \end{eqnarray} then \begin{eqnarray} && (\Lambda_i\Lambda_i)=\prod_{k<l}(\lambda_{i_k}-\lambda_{i_l}), \quad (\Lambda_i\Lambda_j)=\prod_{k,l=1}^m(\lambda_{i_k}-\lambda_{j_l}). \nonumber \end{eqnarray} Our result shows that $C_{\Lambda}^6$ does not depend on $e_\Lambda$. \subsection{$N=4$} Let $\Lambda=(\Lambda_0,\cdots,\Lambda_3)$. We have \begin{eqnarray} && q(0,0)={5\over16}, \quad q(0,1)=q(0,3)=-{1\over16}, \quad q(0,2)=-{3\over16}, \quad \mu={7\over8}. \nonumber \end{eqnarray} Thomae formula is \begin{eqnarray} \theta[e_\Lambda](0)^8 &=& C_{\Lambda}(\det A)^4 \big((\Lambda_0\Lambda_0)(\Lambda_1\Lambda_1)(\Lambda_2\Lambda_2)(\Lambda_3\Lambda_3)\big)^6 \nonumber \\ && \big((\Lambda_0\Lambda_1)(\Lambda_1\Lambda_2)(\Lambda_2\Lambda_3)(\Lambda_0\Lambda_3)\big)^4 \big((\Lambda_0\Lambda_2)(\Lambda_1\Lambda_3)\big)^2. \nonumber \end{eqnarray} The constant $C_{\Lambda}^{8}$ does not depend on $e_\Lambda$. \subsection{$N=5$} Let $\Lambda=(\Lambda_0,\cdots,\Lambda_4)$. We have \begin{eqnarray} && q(0,0)={2\over5}, \quad q(0,1)=q(0,4)=0, \quad q(0,2)=q(0,3)=-{1\over5}, \quad \mu={6\over5}. \nonumber \end{eqnarray} Thomae formula is \begin{eqnarray} \theta[e_\Lambda](0)^{10} &=& C_{\Lambda}(\det A)^5 \big((\Lambda_0\Lambda_0)(\Lambda_1\Lambda_1)(\Lambda_2\Lambda_2)(\Lambda_3\Lambda_3)(\Lambda_4\Lambda_4)\big)^{10} \nonumber \\ && \big((\Lambda_0\Lambda_1)(\Lambda_1\Lambda_2)(\Lambda_2\Lambda_3)(\Lambda_3\Lambda_4)(\Lambda_0\Lambda_4)\big)^6 \nonumber \\ && \big((\Lambda_0\Lambda_2)(\Lambda_1\Lambda_3)(\Lambda_2\Lambda_4)(\Lambda_0\Lambda_3)(\Lambda_1\Lambda_4)\big)^4. \nonumber \end{eqnarray} The constant $C_{\Lambda}^{10}$ does not depend on $e_\Lambda$. \vskip1cm \section{Concluding Remarks} \par In this paper we have given a rigorous proof of the generalized Thomae formula for ${\bf Z}_N$ curves which was previously discovered by Bershadsky and Radul \cite{BR1,BR2} in the study of a conformal field theory. Here let us make a comment on the related subjects. There are several papers( \cite{G,Far} and references therein) studying the generalization of $\lambda$ function of the elliptic curves to ${\bf Z}_N$ curves by studying the cross ratios of four points on a Riemann surface. In those approaches the only ratios of theta constants appear and Thomae type formula is not used. However in the Smirnov's theta formula for the solutions of $sl_2$ Knizhnik-Zamolodchikov equation on level $0$, Thomae formula is needed. Our strategy to prove the generalized Thomae formula here, which is similar to that of \cite{BR1,BR2}, is the comparison of algebraic and analytic expressions of several quantities. In the hyperelliptic case this can be considered as a part of the more general comparison of algebraic and analytic construction of Jacobian varieties due to Mumford \cite{M}. It will be interesting to study the integrable system associated with ${\bf Z}_N$ curves and to study the generalization of Thomae type formula for spectral curves. In fact Thomae \cite{T1}, Fuchs \cite{F1} derived differential equations satisfied by theta constants with respect to branch points in a more general setting and they could integrate them completely only in the case of hyperelliptic curves. The Thomae formula for ${\bf Z}_N$ curves provides a new example which is integrable. The ${\bf Z}_N$ curves and the $1/2N$ periods in the generalized Thomae formula are related with the Lie algebra $sl_N$ and the weight zero subspace of the tensor products of the vector representation. Hence it is natural to expect that the Thomae type formula has a good description in terms of Lie algebras and their representations. For the evaluation of the constant $C_{\Lambda}$ we need to know the explicit description of canonical cycles of ${\bf Z}_N$ curve. So far we could describe a canonical basis only in the case of $N=3$ (except $N=2$). \vskip4mm \noindent {\Large\hskip4mm Acknowledgement } \vskip4mm \noindent We would like to thank Koji Cho, Norio Iwase, Yasuhiko Yamada for the illuminating discussions.

9608

alg-geom/9608017

en

https://arxiv.org/abs/alg-geom/9608017

alg-geom/9608017

Luca Barbieri-Viale

L. Barbieri-Viale

Balanced Varieties

12 pages, LaTeX

Symp. Algebraic K-Theory and Appl. World Scientific Publ., Singapore, 1999, pag. 298-312 (minor changes)

After the work of Bloch and Srinivas on correspondences and algebraic cycles we begin the study of a birational class of algebraic varieties determined by the property that a multiple of the diagonal is rationally equivalent to a cycle supported on proper subschemes.

alg-geom

\section*{Introduction} One of the main results due to Mumford \cite{MU} is the observation that a complex projective non singular surface having a non zero global holomorphic $2$-form has a non zero Albanese kernel; the result was obtained after Severi's work on algebraic cycles and holomorphic forms. Because of Mumford's result the functor $A_0$ given by $0$-cycles of degree zero modulo rational equivalence is not representable, in contrast with the codimension $1$ case treated by Grothendieck and Murre. These facts lead to the study of those varieties for which the Albanese kernel is zero and to ``weak'' representability.\\ Bloch's proof \cite{Bl1} of the mentioned result by Mumford reveals the motivic (read algebraic) nature of the problem: he observed that the vanishing of the Albanese kernel of a surface $X$ at a sufficiently large base field extension yields the existence of 1-dimensional subschemes $Z_1$ and $Z_2$, $2$-dimensional cycles $\G_1$ and $\G_2$ on $X\times X$, supported on $Z_1\times X$ and $X\times Z_2$ respectively, such that some non-zero multiple of the diagonal is rationally equivalent to $\G_1 +\G_2$ over the base field. Then by investigating the action of the correspondences $\G_1$, $\G_2$ on the trascendental (read not algebraic) part of $\ell$-adic \'etale cohomology one immediately obtains its vanishing.\\ The subsequent paper by Bloch and Srinivas \cite{BlS} generalizes Bloch's argument to varieties of dimension $> 2$ showing its influence on algebraic, Hodge and arithmetic cycles.\\ After all of that one has the temptation of understanding the geometry of those varieties $X$ of pure dimension $n$, for which we assume given proper closed subschemes $Z_1$ and $Z_2$, $n$-dimensional cycles $\G_1$ and $\G_2$ on $X\times X$, supported on $Z_1\times X$ and $X\times Z_2$ respectively, and a positive integer $N$ such that the following equation $$N\D_X = \G_1 +\G_2$$ holds in the Chow group $CH_n(X\times X)$. I like to call such a variety {\it balanced}\, because it appears rather natural to regard $X$ ``balanced'' by $\G_1$ and $\G_2$ on $Z_1$ and $Z_2$ in a motivic way as explained above or as will become more clear in the following.\\ The purpose of this Note is to begin the study (in Section~1) of ``balanced varieties and balanced morphisms'' by showing some basic geometric properties {\it e.g.\/}\ balancing is a birational property, stable under products; I also explain the methods by Bloch and Srinivas \cite{BlS} (in Section~2) obtaining applications in the ``language'' of balanced varieties; I'll sketch some examples (in Section~3) and, finally by the way, I would draw a picture of some expected properties of balanced varieties.\\ I would like to express my best thanks to P.Francia for his warm encouragement, his criticism and his suggestions on some highlights of matters treated herein. \section{Basic geometry of balancing} A variety will be an equidimensional reduced separated scheme of finite type over a fixed base field $k$. A non singular variety will be a regular scheme which is a variety. \B{defi} A balanced variety is a variety $X$ of dimension $n$ over $k$ such that there exist {\it (i)}\, proper closed subschemes $Z_1$ and $Z_2$ in $X$ which are of finite type over $k$, {\it (ii)}\, $n$-dimensional cycles $\G_1$ and $\G_2$ on $X\times_k X$ which are supported on $Z_1\times_k X$ and $X\times_k Z_2$ respectively, and {\it (iii)}\, a positive integer $N$, such that the following equation \B{equation}\label{bileq} N\D_X = \G_1 +\G_2 \E{equation} holds in the Chow group $CH_n(X\times_k X)$, where $\D_X$ is the canonical cycle associated with the diagonal imbedding.\\ In this case we will shortly say that we have a balancing of a variety. We then will say that $X$ is balanced by $\G_1$ and $\G_2$ on $Z_1$ and $Z_2$. We will say that $Z_1$ and $Z_2$ are the balance {\it pans}. The {\it weight}\, of $X$ w.r.t. $Z_1$ and $Z_2$ is the positive integer $\mbox{min} \{\mbox{dim}Z_1,\mbox{dim}Z_2\}$, which we denote by $w(X)$. \E{defi} The basic nice properties of balanced varieties are the following. \B{prop}\label{local} Let $X$ be a variety of dimension $n$. Let $U$ be a Zariski open dense subset of $X$ {\it e.g.\/}\ $X$ integral and $U\subset X$ any non-empty Zariski open. Then $X$ is balanced {\em if and only if}\, $U$ is balanced. \E{prop} \B{proof} Since $U$ contains all generic points we have that dim $Z\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, X-U < n$ whence the diagonal cycle $\D_X$ restricts to the diagonal cycle $\D_U$: so we may restrict the balancing as well. Conversely, if $U$ is balanced we have $N\D_U = \G_1^U+\G_2^U$ where $\G_1^U$ is a cycle supported on $(Z_1\cap U)\times U$ and $\G_2^U$ is a cycle supported on $ U\times (Z_2\cap U)$ for some $Z_1$ and $Z_2$ closed subschemes of $X$. Now we can lift $\G_1^U$ to a cycle $\G_1+\zeta_1$ on $X\times X$ where $\G_1$ is supported on $Z_1\times X$ and $\zeta_1$ is supported on $(Z_1\cap Z)\times Z$; in the same way $\G_2^U$ yields a cycle $\G_2+\zeta_2$ with $\G_2$ supported on $X\times Z_2$ and $\zeta_2$ supported on $Z\times (Z_2\cap Z)$. Since the restriction of $\G_1+\G_2 +\zeta_1+\zeta_2$ to $U\times U$ is $\G_1^U+\G_2^U$ and the restriction of $N\D_X$ is $N\D_U$ we have that $\G_1+\G_2 +\zeta_1+\zeta_2-N\D_X$ is a cycle supported on $Z\times Z$. We then obtain an equation $$N\D_X=\G_1+\G_2+\zeta$$ for some cycle $\zeta$ supported on $Z\times Z$, which is a balancing of $X$.\E{proof} \B{prop}\label{cover} Let $X$ be a balanced variety. Let $f: X\to X'$ be a proper surjective morphism to a variety $X'$ with dim $X$ = dim $X'$. Then $X'$ is balanced. \E{prop} \B{proof} Since $f$ is generically finite we have that the fundamental cycle $[X]$ has proper push-forward $f_*[X]$ = deg($f$)$[X']$. Let denote by $\delta_X$ and $\delta_{X'}$ the diagonal imbeddings of $X$ and $X'$. We clearly have that $\delta_{X'*}\p{\circ}f_*=(f\times f)_*\p{\circ}\delta_{X*}$ whence we have $$(f\times f)_*(\D_X) = \mbox{deg}(f) \D_{X'}$$ by the definition of $\D$. Applying $(f\times f)_*$ to (\ref{bileq}) we have $$N(f\times f)_*(\D_X) = (f\times f)_*(\G_1 + \G_2)$$ so that we obtain $$N\mbox{deg}(f) \D_{X'} = (f\times f)_*(\G_1) + (f\times f)_*(\G_2)$$ Then $X'$ is balanced by $(f\times f)_*(\G_1)$ and $(f\times f)_*(\G_2)$ on $f(Z_1)$ and $f(Z_2)$. \E{proof} We then have: \B{cor}\label{bir} Let $f:X'\mbox{$\stackrel{\pp{-\,-\succ}}{}$} X$ be a proper dominant rational map with dim $X$ = dim $X'$: if $X'$ is balanced then $X$ is balanced. Balancing is a birational property of arbitrary varieties over arbitrary fields. \E{cor} \B{proof} We may indeed restrict $f$ to open dense subsets $U'$ of $X'$ and $U$ of $X$ such that $U'\to U$ is a proper morphism and make use of the Propositions~\ref{local},\ref{cover}. If $X$ is birational to $X'$ then they have isomorphic open dense subsets and we just use Proposition~\ref{local}.\E{proof} \B{prop}\label{prod} Let $X$ be a balanced variety. Then the product of $X$ with any variety is balanced.\E{prop} \B{proof} Let $X'$ be any variety over $k$. We need the following simple Lemma. \B{lemma} Let $X$ and $X'$ be varieties over $k$. Let $$\sigma : X\times X \times X'\times X'\to X\times X' \times X\times X'$$ be the isomorphism given by $\sigma (x_1,x_2,x'_1,x'_2) = (x_1,x'_1,x_2,x'_2)$. Then \B{equation}\label{simde} \sigma_*(\D_X\times \D_{X'}) = \D_{X\times X'} \E{equation} \E{lemma} \B{proof} Because of the functoriality of the exterior product of cycles we have \B{equation}\label{extde} (\delta_X\times\delta_{X'})_*\p{\circ}(-\times \dag) = \delta_{X*}(-)\times\delta_{X'*}(\dag) \E{equation} Moreover we clearly have \B{equation}\label{compde} \sigma\p{\circ}(\delta_X\times\delta_{X'}) = \delta_{X\times X'} \E{equation} So then \B{center} \parbox{7cm}{$\sigma_*(\D_X\times \D_{X'})=$\\ $=\sigma_*(\delta_{X*}[X]\times\delta_{X'*}[X'])=$\hfill by (\ref{extde})\\ $=\sigma_*((\delta_X\times\delta_{X'})_*([X]\times [X']))=$\\$=\sigma_*((\delta_X\times\delta_{X'})_*[X\times X'])=$ \hfill by (\ref{compde})\\ $=(\delta_{X\times X'})_*[X\times X']=$\\ $=\D_{X\times X'}$\\} \E{center} as claimed. \E{proof} Let assume $X$ balanced by $\G_1$ and $\G_2$ on $Z_1$ and $Z_2$ and let (\ref{bileq}) holds. We then have: \B{center} \parbox{7cm}{$N\D_{X\times X'}=$\hfill by (\ref{simde})\\ $=N\sigma_*(\D_X\times \D_{X'})=$\hfill by (\ref{bileq})\\$=\sigma_*((\G_1 +\G_2)\times\D_{X'})=$\\$=\sigma_*(\G_1\times\D_{X'}) +\sigma_*(\G_2\times\D_{X'})$\\} \E{center} where the support of $\sigma_*(\G_1\times\D_{X'})$ is contained in $Z_1\times X'\times X\times X'$ and the support of $\sigma_*(\G_2\times\D_{X'})$ is in $X\times X'\times Z_2\times X$. Then the product $X\times_k X'$ is balanced on $Z_1\times X'$ and $Z_2\times X'$. By symmetry also the product $X'\times_k X$ is balanced. \E{proof} \vspace{0.5cm} We are now going to give another description of the variation of a balancing under birational maps between non-singular varieties. We recall that for any regular algebraic scheme $V$ of pure dimension $n$ and $S\subset V$ a closed subscheme we have a duality isomorphism \B{equation}\label{kdual} H^p_S(V,{\cal K}_p)\by{\simeq} CH_{n-p}(S) \E{equation} where ${\cal K}_p$ is the Zariski sheaf on $V$ associated with Quillen $K$-theory (\cf \cite{BV2}). If we have a cartesian square \B{displaymath}\B{array}{ccc} S' & \hookrightarrow & V' \\ \downarrow & & \downarrow\p{f} \\ S & \hookrightarrow & V \E{array}\E{displaymath} we then have a commutative diagram \B{displaymath}\B{array}{ccc} H^p_S(V,{\cal K}_p) & \to & H^p(V,{\cal K}_p) \\ \downarrow & & \downarrow \ \\ H^p_{S'}(V',{\cal K}_p) & \to & H^p(V',{\cal K}_p) \E{array}\E{displaymath} whence by (\ref{kdual}) the following commutative diagram of Chow groups \B{equation}\label{ciao}\B{array}{ccc} CH_{n-p}(S) & \by{i_*} & CH_{n-p}(V) \\ \downarrow & & \downarrow\p{f^*} \\ CH_{n'-p}(S') & \by{{i'}_*} & CH_{n'-p}(V') \E{array}\E{equation} where $i$ and $i'$ are the imbedding of $S$ in $V$ and $S'$ in $V'$ respectively.\\ Let $f: X'\to X$ be a proper birational morphism between non-singular varieties. For $f: X'\to X$ as above we let consider the cartesian square \B{displaymath}\B{array}{ccc} E_f& \by{\pi} & X \\ \p{e}\downarrow & & \downarrow\p{\delta} \\ X'\times X'& \by{f\times f} & X\times X \E{array}\E{displaymath} where $\delta$ is the diagonal imbedding. Because of (\ref{ciao}) we obtain ($n =$ dim $X =$ dim $X'$) \B{equation}\label{diciao}\B{array}{ccc} CH_{n}(X\times X) & \by{(f\times f)^*} & CH_{n}(X'\times X') \\ \p{\delta_*}\uparrow & & \uparrow\p{e_*} \\ CH_{n}(X) & \by{{\pi}^*} & CH_{n}(E_f) \E{array}\E{equation} and moreover:\\[0.5cm] {\it (i)}\, there is a closed imbedding $i:X'\hookrightarrow E_f$ such that $e\p{\circ}i =\delta_{X'}$ and $\pi\p{\circ}i =f$;\\[0.5cm] {\it (ii)}\, if $Z'=f^{-1}(Z)$ is a closed in $X'$ such that $f:X'-Z'\by{\simeq}X-Z$ then $e(E_f-X')\subset Z'\times Z' - \D_{Z'}$;\\[0.5cm] {\it (iii)}\, $E_f$ can also be regarded as the fiber product of $f$ with itself {\it i.e.\/}\ \B{displaymath}\B{array}{ccc} E_f& \by{\pi '} & X'\\ \p{\pi '}\downarrow & & \downarrow\p{f} \\ X'& \by{f} & X \E{array}\E{displaymath} is a cartesian square, where $f\p{\circ}\pi '=\pi$ and $\pi ':E_f\to X'$ has a section given by $i:X'\hookrightarrow E_f$; we then have a splitting exact sequence \B{equation}\label{split} 0\to CH_n(X')\by{\leftarrow}CH_n(E_f)\to CH_n(E_f-X')\to 0 \E{equation} because of ${\pi}'_*\p{\circ}i_* = 1$.\B{lemma} Notations and hypothesis as above. Then there is a cycle $\zeta_f$ on $X'\times X'$ supported on $Z'\times Z'$ such that the following equation \B{equation}\label{zeta} (f\times f)^*\D_{X}=\D_{X'}+\zeta_f \E{equation} holds in $CH_n(X'\times X')$.\E{lemma} \B{proof} Because of (\ref{diciao}) we have that $$(f\times f)^*\D_{X}=(f\times f)^*(\delta_{X*}[X]) =e_*\pi^*[X]$$ So we are left to show the equation $e_*\pi^*[X]=\D_{X'}+\zeta_f$. The projection ${\pi}'_*$ of $\pi^*[X]\in CH_{n}(E_f)$ is $[X']$ since the composite of $$H^n_X(X\times X,{\cal K}_n)\to H^n_{E_f}(X'\times X',{\cal K}_n) \to H^n_X(X\times X,{\cal K}_n)$$ is the identity, by \cite[Section~7]{BV2} because $f\times f$ is birational, and the composite of $$CH_{n}(E_f)\by{{\pi}'_*}CH_n(X')\by{f_*}CH_n(X)$$ is $\pi_*$ by {\it (iii)}\, above, where $f_*$ is the isomorphism sending $[X']$ to $[X]$ because $f$ is mapping birationally each component of $X$ to a distinct component of $X'$.\\ Moreover by {\it (ii)}\, we have a commutative diagram $$\begin{array}{cc} CH_{n}(E_f-X') \ \ \to \ \ CH_{n}(Z'\times Z' -\D_{Z'}) & {\by{\simeq}}\ \ CH_{n}(Z'\times Z') \\ \p{e_*}\searrow \hspace*{40pt} {\downarrow} &{\downarrow}\\ \ \hspace*{80pt} CH_{n}(X'\times X'-\D_{X'}) & {\leftarrow}\ \ \ CH_{n}(X'\times X') \end{array}$$ since $Z'$ has dimension $\leq n-1$ whence $CH_{n}(Z')=0$. Thus we are able to give a description of $e_*\pi^*[X]$ w.r.t.\/ the splitting (\ref{split}): from the latter diagram we see that the restriction of $\pi^*[X]$ to $CH_{n}(E_f-X')$ ({\it i.e.\/}\ a component of $\pi^*[X]$) has image ({\it i.e.\/}\ $e_*$ of it in $CH_{n}(X'\times X')$ is) a cycle $\zeta_f$ supported on $Z'\times Z'$; the component of $\pi^*[X]$ in $CH_{n}(X')$ ({\it i.e.\/}\ $i_*[X']$) has image $\D_{X'}$ because of {\it (i)}\, {\it i.e.\/}\ $e_*\p{\circ}i_* =\delta_{X'*}$. The claimed formula (\ref{zeta}) is obtained.\E{proof} Assuming $X$ balanced we have: \B{center} \parbox{7cm}{$(f\times f)^*(\G_1 +\G_2)=$\hfill by (\ref{bileq})\\ $=N(f\times f)^*\D_{X}=$\hfill by (\ref{zeta})\\$=N\D_{X'}+N\zeta_f$\\} \E{center} yielding the equation $$N\D_{X'} = (f\times f)^*(\G_1)+(f\times f)^*(\G_2) -N\zeta_f$$ Now, because of (\ref{kdual}) and (\ref{ciao}), we have that the cycle $(f\times f)^*(\G_1)$ is supported on $f^{-1}(Z_1)\times X'$ and $(f\times f)^*(\G_2) -N\zeta_f$ on $X'\times f^{-1}(Z_2)\cup Z'\times Z'$. Then $X'$ is balanced on $f^{-1}(Z_1)$ and $f^{-1}(Z_2)\cup Z'$.\\[0.5cm] {}\hfill In the following any flat morphism will have a relative dimension. \B{defi} Let $p:X\to S$ be a morphism of varieties over $k$ where $S$ is irreducible of dimension $i$. We say that $p:X\to S$ is {\it balanced}\, or that $X$ is balanced {\it over}\, $S$ if $p:X\to S$ is flat of relative dimension $n$ and the equation $\Delta_{X/S}= \G_1+\G_2$ holds in $CH_{n+i}(X\times_SX)$ where $\G_1$ is supported on $Z_1\times_SX$ and $\G_2$ is supported on $X\times_SZ_2$ for some $Z_1$ and $Z_2$ proper closed subschemes of $X$ which are flat over $S$. \E{defi} \B{prop} We have the following basic properties. \B{description} \item[{\sc Base change}] Let $p:X\to S$ be a balanced morphism, $\varphi :S'\to S$ be any flat morphism (but $S'$ irreducible) and $X'= X\times_SS'$. Then $p':X'\to S'$, obtained by base extension, is balanced.\\ \item[{\sc Composition}] If $p:X\to S$ is balanced and $\varphi :S\to T$ is flat ($T$ irreducible) then $\varphi\p{\circ}p :X\to T$ is balanced.\\ \item[{\sc Product}] If $X$ is balanced over $S$ and $Y$ is flat over $S$ then $X\times_SY$ is balanced over $S$. \E{description} \E{prop} \B{proof} A balancing for the base extension $p':X'\to S'$ can be obtained by making use of external products of flat families of cycles because of the equation $\D_{X'/S'}=\D_{X/S}\times_{S}[S']$ (\cf the proof of Proposition~\ref{prod}). The composition $\varphi\p{\circ}p :X\to T$ is known to be flat and the imbedding $j:X\times_SX\hookrightarrow X\times_TX$ is known to be closed, yielding the equation $j_*\D_{X/S}=\D_{X/T}$. Finally: $X\times_SY\to S$ is the composition of the flat morphism $Y\to S$ with the base extension. \E{proof} By the above, if $X$ is balanced over $S$ then $X$ is balanced over the base field. Moreover $X$ can be view as an algebraic family and, for example, we have that $X_s$, the fibre at the generic point $s\in S$, is balanced over $K(S)$ (the function field of $S$). Indeed, let $V$ be any Zariski open neighborhood of the generic point $s$ in $S=\overline{\{s\}}$: by flat base change we have that $X\times_SV\to V$ is balanced. The claim above is obtained by taking the limit over $V$ because we have $$CH_n(X_s)\cong \limdir{V} CH_n(X\times_S V)$$ and $\D_{X_s/K(S)} =\limdir{V}\D_{X\times_S V/V}$.\\ \B{defi}\label{universal} Let $X$ be a variety over a field $k$. We say that $X$ is $K$-balanced if $X_K$ is balanced over the field extension $k\subset K$. If $\Omega$ is a universal domain in the sense of Weil and $X$ is $\Omega$-balanced we say that $X$ is {\it universally}\, balanced. \E{defi} We will see in the next Section, that universally balanced varieties are also balanced over the base field. \section{Actions of a balancing} Let $(H^*,H_*)$ be an appropriate duality theory in the sense of \cite[Sections 6-7]{BV2} {\it e.g.\/}\ De Rham theory or $\ell$-adic \'etale theory. The twisted cohomology functor $H^*(-,\cdot)$ is then equipped with a functorial `cycle class' map $c\ell:H^p(X,{\cal H}^p(\p{p})\to H^{2p}(X,\p{p})$ for $X$ smooth over $k$. An algebraic `${\cal H}$-correspondence' from $X$ to $Y$ is an ${\cal H}$-cohomology class $\alpha\in H^{n+r}(X\times Y, {\cal H}^{n+r}(\p{n+r}))$ ($X$ has pure dimension $n$) and it will be denoted by $\alpha : X\leadsto Y$. As usual $\alpha$ acts on the cohomology groups of non-singular projective varieties $$\alpha_{\sharp}: H^i(X,\p{j})\to H^{i+2r}(Y,\p{j+r})$$ where $\alpha_{\sharp}(-)\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, p_{Y,*}(c\ell(\alpha)\p{\cup} p^*_X(-))$ and $p_{Y}$, $p_{X}$ are the projections of $X\times Y$ on $Y$ and $X$ respectively. Because of our assumptions there is a canonical ring homomorphism from the Chow ring to the ${\cal H}$-cohomology ring (\cf \cite[Sections 5.5 and 6.3]{BV2}). The action of any algebraic cycle on a fixed cohomology theory is given {\it via}\, its ${\cal H}$-cohomology incarnation {\it e.g.\/}\ any cycle algebraically equivalent to zero acts as zero on the singular cohomology of the associated analytic space. In particular any $\alpha : X\leadsto Y$ as above acts on the ${\cal H}$-cohomology groups $$\alpha_{\sharp}: H^p(X,{\cal H}^q(\p{j}))\to H^{p+r}(Y,{\cal H}^{q+r}(\p{j+r}))$$ in a compatible way w.r.t. the coniveau spectral sequences.\\ An easy application of the projection formula (for ${\cal H}$-cohomologies we need \cite[Section 5.4]{BV2}) yields the following useful Lemma. \B{lemma} Let $f: X'\to X$ and $g: Y'\to Y$ be morphisms of non-singular projective varieties. For $\alpha: X'\leadsto Y$ we have \B{equation}\label{uno} (f\times 1_Y)_*(\alpha)_{\sharp} = \alpha_{\sharp} \p{\circ}f^* \E{equation} For $\beta : X\leadsto Y'$ we have \B{equation}\label{due}(1_X\times g)_*(\beta)_{\sharp}= g_*\p{\circ}\beta_{\sharp} \E{equation} \E{lemma} \B{proof} The proof is left as an exercise for the reader. \E{proof} Let now consider a balanced variety $X$ which is non-singular and projective; we are going to make use of (\ref{bileq}) w.r.t. the various cohomology theories. We need to assume resolution of singularities and we let assume that $Z_1$ and $Z_2$ are equidimensional. We then may consider a resolution $Z_1'\to Z_1$ and we have that $\G_1 = (f\times 1_X)_*(\G_1')$ where $\G_1'$ is a cycle on $Z_1'\times X$ of codimension equal to the dimension of $Z_1$ and $f : Z_1'\to X$, so that by (\ref{uno}) we obtain \B{equation} (\G_1)_{\sharp} = (f\times 1_X)_*(\G_1')_{\sharp}= (\G_1')_{\sharp}\p{\circ}f^* \E{equation} since $\G_1': Z_1'\leadsto X$; {\it i.e.\/}\ we have a commutative triangle \B{eqnarray}\label{map1} H^p(X,{\cal H}^q\p{(\cdot)}) & \longby{\G_{1\sharp}} & H^p(X,{\cal H}^q\p{(\cdot)}) \nonumber \\ \searrow & & \nearrow \\ & H^p(Z_1',{\cal H}^q\p{(\cdot)}) &\nonumber \E{eqnarray} We consider as well a resolution of singularities $Z_2'\to Z_2$ and we have $\G_2 = (1_X\times g)_*(\G_2')$ where $\G_2'$ is a cycle on $X\times Z_2'$ whence the degree of the correspondence $\G_2':X\leadsto Z_2'$ is $-\mbox{codim}_X(Z_2)$; by (\ref{due}) we then have \B{equation} (\G_2)_{\sharp} = (1_X\times g)_*(\G_2')_{\sharp} = g_*\p{\circ}(\G_2')_{\sharp} \E{equation} {\it i.e.\/}\ the following triangle \B{eqnarray}\label{map2} H^p(X,{\cal H}^q\p{(\cdot)}) & \longby{\G_{2\sharp}} & H^p(X,{\cal H}^q\p{(\cdot)}) \nonumber \\ \searrow & & \nearrow \\ & H^{p-c}(Z_2',{\cal H}^{q-c}\p{(\cdot-c)}) &\nonumber \E{eqnarray} commutes, where $c=\mbox{codim}_X(Z_2)$. If $Z_2$ (or $Z_1$) is not equidimensional we may consider each smooth component of its resolution $Z_2'$ acting as above. \B{prop} Let consider any cohomology theory $H^*$ as above. Let cd$(k)$ be the ``cohomological dimension'' of the field $k$ {\it i.e.\/}\ we assume that $H^i(U) = 0$ if $i>\mbox{dim} U +\mbox{cd}(k)$ and $U$ is affine. If $X$ is balanced of weight $w$, and either the pans are smooth or can be resolved, then $H^0(X,{\cal H}^q\p{(\cdot)})$ is {\em $N$-torsion} for $q>w+\mbox{cd}(k)$ and some positive integer $N$. \E{prop} \B{proof} Is essentially the same of \cite[Th. 1, ii--iii]{BlS}. By interchanging the pans we may assume that $w=$dim$Z_1$. The action of the cycle $N\D_X$ is the multiplication by $N$ which is given by $\G_{1\sharp}+\G_{2\sharp}$. Beacause of the assumptions and (\ref{map1}) we have $\G_{1\sharp}=0$ since ${\cal H}^q=0$ on $Z_1'$ if $q>w+\mbox{cd}(k)$. Because of (\ref{map2}) $\G_{2\sharp}=0$ since $\mbox{codim}_X(Z_2)>0$. \E{proof} Whenever the sheaves ${\cal H}^*\p{(\cdot)}$ are torsion free we have that $H^0(X,{\cal H}^*\p{(\cdot)})=0$ under the assumptions in the Proposition above. Because of \cite{BV1} (\cf \cite{BV0}, \cite{BV3}) we then have: \B{cor} Let $X$ be balanced over $\C$ of weight $w$ and let assume that Kato's conjecture hold true for function fields. If $q>w$ we then have $$H^0(X,{\cal H}^q(\Z))=0$$ and $$H^0(X,{\cal H}^{q+1}(\Z(q)_{{\cal D}}))=0$$ where ${\cal H}^*(\Z)$ and ${\cal H}^{*}(\Z(\cdot)_{{\cal D}})$ are the Zariski sheaves associated with singular cohomology and Deligne--Beilinson cohomology respectively. \E{cor} As in \cite[Th. 1]{BlS} by the above one obtains the following: for $X$ balanced over $\C$ of weight $\leq 2$ algebraic and homological equivalence coincide for codimension $2$ cycles and for $X$ of weight $\leq 1$ also homological and Abel-Jacobi equivalence coincide in codimension $2$. Indeed the Griffiths group is a quotient of $H^0(X,{\cal H}^3(\Z))$ and the Abel-Jacobi kernel is a quotient of the torsion free group $H^0(X,{\cal H}^{3}(\Z(2)_{{\cal D}}))$ (\cf \cite{BV3} for surfaces).\\ Moreover: \B{prop}\label{forms} Let $X$ (projective, non-singular) be balanced over $\C$ of weight $w$. If $q>w$ then $$H^0(X,\Omega^q_X)=0$$ \E{prop} \B{proof} The previous arguments applies to the twisted cohomology theory given by $F^{\cdot}H^*$ {\it i.e.\/}\ the De Rham filtration, and $F^qH^q(X)=H^0(X,\Omega^q_X)$. Now $\G_{1\sharp}=0$ because $\Omega^q_{Z_1'}=0$ if $q>w=$ dim $Z_1$ and (\ref{map1}). Moreover $\G_{2\sharp}=0$ by $F^{q-c}H^{q-2c}(Z_2')=0$ since $c>0$ {\it i.e.\/}\ the codimension of any component of $Z_2$ cannot be zero. Since $F^{\cdot}H^*$ are real vector spaces we are done. \E{proof} \B{prop}\label{Alba} If $X$ (projective, non-singular) is balanced of weight $w$ over a field $k$, and either the pans are smooth or can be resolved, then there exists a closed subscheme $Y$ of $X$ such that $CH^i(X-Y)_{\Q}=0$ for all $i>w$. For $0$-cycles $Y$ exists of dimension $w$. If, moreover, $k$ is algebraically closed then $A^i(X-Y)=0$ for all $i>w$ and $A_0(X-Y)=0$ as above; finally, in this case, the Albanese kernel of $X$ is contained in the Albanese kernel of the resolved pan of dimension $w$. \E{prop} \B{proof} We may assume that $Z_1$ is irreducible and non-singular of dimension $w$: therefore we obtain that $\G_{1\sharp}=0$ on $CH^i(X)$ if $i>w$, by (\ref{map1}) and $CH^i(Z_1)=0$. Then $\G_{2\sharp}\otimes\Q$ is the isomorphism induced by the multiplication by $N$ on $CH^i(X)_{\Q}$ if $i>w$; then, by (\ref{map2}), the Chow group of $Z_2$ surjects onto $CH^i(X)_{\Q}$ if $i>w$. By taking $Z_2$ as $Y$ we have that $CH^i(X-Y)_{\Q}=0$ as claimed. If $i=\mbox{dim} X$ {\it i.e.\/}\ for $0$-cycles, we then can take $Z_2$ of dimension $w$ and we conclude as above. If $k$ is closed it is well known that $A_0(X)$ is divisible and the Albanese kernel is uniquely divisible; by the same argument we obtain the claimed results. \E{proof} Following Bloch, Srinivas and Jannsen (\cf \cite{Bl1}, \cite{BlS}, \cite[Remark~1.7]{JA}) we have: \B{prop} \label{rap} Let $X$ be a non-singular projective variety over a field $k$. Then $X$ is universally balanced of weight $\leq 1$ if and only if the Chow group of $0$-cycles of degree zero is representable. \E{prop} \B{proof} By the definition, $X$ is universally balanced if $X_{\Omega}$ is balanced over a universal domain $\Omega$. By the Proposition~\ref{Alba} the Albanese kernel $X_{\Omega}$ is contained in the Albanese kernel of (at most) a smooth curve over $\Omega$, which is zero. Conversely, if the Chow group of $0$-cycles is representable then by \cite[Prop.~1.6]{JA} there is (at most) a smooth projective curve over $\Omega$ and a morphism $g:C\to X_{\Omega}$ such that $A_0(X_{\Omega}-g(C))=0$ hence by \cite[Prop.~1]{BlS} $X_{\Omega}$ is balanced of weight $\leq 1$. \E{proof} \B{prop} Let $X$ be a non-singular projective variety over a field $k$. If $X$ is universally balanced then it is balanced. \E{prop} \B{proof} The proof is similar to \cite[Proof of Th.~3.5.(a)-(b)]{JA} which is quite the same of the proof of Proposition~1 of \cite{BlS}.\E{proof} Following Jannsen \cite[\S~3]{JA} we have: \B{cor} If $X$ is universally balanced of weight $\leq w$ then $H^i(X,\p{j})$ (= any twisted cohomology theory in the sense of \cite[\S~3]{JA}) is of coniveau $1$ for $i>w$.\E{cor} Finally, by Proposition~\ref{forms} and Proposition~\ref{rap} we can easily obtain the following. \B{cor} {\em (Mumford-Roitman Theorem)} If $H^0(X,\Omega^q_X)\neq 0$ for some $q>1$ then the Chow group of $0$-cycles of degree zero is not representable. \E{cor} See also \cite[Cor.~3.7]{JA}. \section{Paradigma} We now give some examples.\\ \B{example} The projective space $\P^n_k$ over any field $k$ is balanced of weight $0$ {\it i.e.\/}\ the pans are given by any couple of closed points $x_0$ and $x_1$. In fact we have that $$CH_n(\P^n_k\times \P^n_k) \cong (x_0\times \P^n_k)\Z\bigoplus (\P^n_k\times x_1)\Z$$ \E{example} \B{example} Because of Corollary~\ref{bir} and the above, unirational varieties are balanced over any field. More generally, because of Proposition~\ref{prod}, we have that uniruled varieties are balanced, since are dominated by a product with $\P^1_k$. \E{example} \B{example} If $X$ is a smooth closed subvariety of an abelian variety then $X$ is not balanced over $\C$, since $H^0(X,\Omega^n_X)\neq 0$ for $n=$ dim $X$ and Proposition~\ref{forms}. \E{example} \B{example} The Kummer $3$-fold over $\C$ is balanced of weight $2$ by Bloch and Srinivas \cite{BlS} since $A_0(X)$ is generated by cycles supported on a finite number of surfaces. This is an example of balanced variety which is quite far from ruled varieties. \E{example} \section*{Some remarks and questions} \B{enumerate} \item Of course one might like to define balanced schemes starting from an equidimensional separated scheme over an arbitrary base scheme but then does one have anything to say about it ? Nevertheless one case that sounds interesting is the case of arithmetic schemes {\it i.e.\/}\ regular schemes which are projective and flat over the integers, by mean of arithmetic cycles and correspondences in the sense of H. Gillet and C. Soul\'e. So I expect a parallel theory of ``arithmetically balanced varieties'' by analysing the action of arithmetic correspondences. \item In the language of balanced varieties, Bloch's conjecture give us a numerical criterion for balanced surfaces {\it i.e.\/}\ $p_g=0$ for weight $1$ and $p_g=q=0$ for weight zero. We may expect that the following numerical criterion $$X\mbox{ balanced of weight }w \iff H^0(X,\Omega^q_X) = 0 \mbox{ for } q>w$$ holds true for any projective complex manifold $X$.\\ More in general, for $X$ defined over any field $k$, $X$ will possibly be universally balanced of weight $\leq w$ if and only if any suitable `cohomology theory' will be of coniveau $1$ in degrees $>w$. \item Since the global forms are invariants under a smooth deformation, the previous speculation is suggesting us that the balancing should be a deformation property. We remark that uniruled varieties do have this property, thanks to A. Fujiki and M. Levine (deformation invariance of rationally connected varieties is due to J. Kollar, Y. Miyaoka and S. Mori). \item Let say that a flat family of varieties, parametrized by a nice variety, is balanced if its general member is a balanced variety. Is there a section ? \E{enumerate}

9608

alg-geom/9608020

en

https://arxiv.org/abs/alg-geom/9608020

alg-geom/9608020

Mark De Cataldo

Mark Andrea A. de Cataldo (Washington U. in St. Louis)

A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

Ams-LAtex;13 pages; to appear in Trans. A.M.S

We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\Q{n}$ which are not of general type, for $n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher codimension. The case $n=6$ has been recently settled by Fania-Ottaviani. Keywords: Codimension two, Grassmannians, Lifting, Low codimension, Not of General Type, Polynomial Bound, Quadrics

alg-geom

\section{Introduction} \label{boundednessofthreefoldsnotgentype} There are only finitely many families of codimension two nonsingular subvarieties not of general type of the projective spaces $\pn{n}$, for $n\geq 4$; see \ci{e-p} and \ci{boss1}. More generally, a similar statement holds for the case of higher codimension; see \ci{sc}. In this paper we concentrate on the case of codimension two subvarieties of quadrics. Our main result is Theorem \ref{sigh}: there are only finitely many families of nonsingular codimension two subvarieties not of general type in the quadrics $\Q{n}$, $n=4,$ $5$ or $n\geq 7$. The case $n=4$ is proved in \ci{a-s}, \S6. The case of $n=5$ is at the heart of the paper; the main tools are the semipositivity of the normal bundles of nonsingular subvarieties of quadrics, the Double Point Formula, the Generalized Hodge Index Theorem, bounds for the genus of curves on $\Q{3}$, Proposition \ref{roth} and Corollary \ref{finiteonhyp}. The case $n=6$ is still open$^*$. The case of codimension two with $n\geq 7$ is covered by Theorem \ref{boundednessonqn}, which hinges upon the result of \ci{sc}; it also gives a finiteness result in codimension bigger than two in the same spirit as \ci{sc}. \smallskip The paper is organized as follows. Section \ref{prelim} records, for the reader's convenience, some results used in the paper. A generalization of a lifting criterion of Roth's is contained in section \ref{rothtypelifting}; we shall need the particular case expressed by Proposition \ref{rothpr}. Section \ref{boundednessforn>=7} deals with higher codimensional cases. Sections \ref{chTHFD} and \ref{boundednessonq5} are modeled on \ci{boss1}. Section \ref{chTHFD} contains the lengthy proof of Theorem \ref{polybound} and of its Corollary \ref{finiteonhyp}. Section \ref{boundednessonq5} contains the proof of Theorem \ref{sigh}. \smallskip \noindent {\bf Notation and conventions.} Our basic reference is \cite{ha}. We work over any algebraically closed field of characteristic zero. A quadric $\Q{n}$, here, is a nonsingular hypersurface of degree two in the projective space $\pn{n+1}$. Little or no distinction is made between line bundles, associated sheaves of sections and Cartier divisors; moreover the additive and tensor notation are both used. \smallskip \noindent {\bf Acknowledgments.} It is a pleasure to thank our Ph.D. advisor A.J. Sommese, who has suggested to us to study $3$-folds in $\Q{5}$. \section{Preliminary material} \label{prelim} \begin{pr} \label{easybound} {\rm (Cf. \ci{decastharris} or \ci{thesis}; for the case of $d> 2k(k-1)$ see \ci{a-s}, \S6.)} Let $C$ be an integral curve of degree $d$ contained in an integral surface of degree $2k$ in $\Q{3}$. Then the following bound holds for the genus $g$ of $C$: $$ g-1\leq \frac{d^2}{4k} + \frac{1}{2} (k-3)d. $$ \end{pr} \begin{pr} \label{boundasep} {\rm (Cf. \ci{a-s}, Proposition $6.4$.)} Let $C$ be an integral curve in $\Q{3}$, not contained in any surface in $\Q{3}$ of degree strictly less than $2k$. Then: $$ g-1\leq \frac{d^2}{2k} +\frac{1}{2}(k-4)d. $$ \end{pr} \noindent Let $S$ be a nonsingular surface in $\Q{4}$, $N$ its normal bundle, $C$ a nonsingular hyperplane section of $S$, $g$ its genus, $d$ its degree. Let $V_s \in |{\cal I}_{S,\Q{4}}(s)|$, where $s$ is some positive integer, be an integral hypersurface and $\mu_l:=c_2(N(-l))=$ $(1/2)d^2 +l(l -3)d-2l(g-1)$, $\forall l\in \zed$. \begin{lm} \label{epas} {\rm (Cf. \cite{thesis}, Lemma $2.35$.)} In the above situation: $$ 0\leq \mu_{s}\leq s^2 d. $$ \end{lm} The following proposition follows immediately from Theorem \ref{roth} when the ambient space ${\cal P}^{n+2}$ is chosen to be a quadric $\Q{n+2}$. \begin{pr} \label{rothpr} Let $X$ be an integral subscheme of degree $d$ and codimension two in $\Q{n}$, $n\geq 4$. Assume that for the general hyperplane section $Y$ of $X$ we have $$ h^0(\Q{n-1}, {\cal I}_{Y,\Q{n-1}}(\s))\not= 0, $$ for some positive integer $s$ such that $d>2{\s}^2$. Then $$ h^0(\Q{n}, {\cal I}_{X,\Q{n}}({\s}))\not= 0. $$ \end{pr} \medskip Let $X$ be a degree $d$, nonsingular $3$-fold in $\Q{5}$, $L\simeq {\odixl{\pn{6}}{1}}_{|X}$, $S$ be the surface general hyperplane section of $X$, $C$ the general curve section of $S$ and $g$ the genus of $C$. Using the Double Point Formula (cf. \ci{fu}) for the embedding $X\hookrightarrow \Q{5}$, we get the following formul\ae\ for $K_X\cdot L^2$, $K_X^2\cdot L$, $K_X^3$ as functions of $d$, $g$, $\chi ({\cal O}_X)$, $\chi ({\cal O}_S)$: \begin{equation} \label{KL2} K_X\cdot L^2=2(g-1)-2d, \end{equation} \begin{equation} \label{K2L} K_X^2\cdot L=\frac{1}{4}d^2 +\frac{3}{2}d -8(g-1) +6\chi ({\cal O}_S), \end{equation} \begin{equation} \label{K3} K_X^3=-\frac{9}{4}d^2+\frac{27}{2}d+gd +18(g-1)-30\chi ({\cal O}_S)-24\chi ({\cal O}_X), \end{equation} Finally we record the expression for the Hilbert polynomial of $X$, \begin{eqnarray} \label{chioxt} \chi ({\cal O}_X (t)) = \frac{1}{6}dt^3+[\frac{1}{2}d - \frac{1}{2}(g-1)]t^2+ [\frac{1}{3}d - \frac{1}{2}(g-1) +\chi ({\cal O}_S)]t + \chi ({\cal O}_X). \end{eqnarray} For the details concerning the above formul\ae \ see \ci{thesis}, \S1. \subsection{A Roth-type lifting criterion} \label{rothtypelifting} If the general curve section of a degree $d$ linearly normal surface $S$ in $\pn{4}$ lies on a surface of degree $\s$ in $\pn{3}$, then $S$ lies on some hypersurface of degree $\s$, provided $d>\s^2$ (cf. \cite{ro}). A generalization of this fact to codimension two integral linearly normal subschemes of $\pn{n}$, $n\geq 4$ has been known for some time. In this section we generalize Roth's lifting criterion to a larger class of spaces; see Theorem \ref{roth} and Example \ref{examples}. The proof does not require the concept of linear normality, which was virtually automatic in the case that Roth considered. The proof given below was inspired by \cite{a-s}, Lemma $6.1$. \noindent First we fix some notation. Let ${\cal P}^{n+2}$ be a nonsingular projective variety of dimension $(n+2)$, $n\geq 2$, $L=\odixl{{\cal P}^{n+2}}{1}$ an ample and spanned line bundle on it with $\delta:= L^{n+2}$. Assume that $Pic({\cal P}^{n+2})\simeq \zed [L]$. Let $X^n$ be an integral subscheme of ${\cal P}^{n+2}$ of dimension $n$ and $d:=L^{n}\cdot X$. Denote by ${\cal P}^{i+2}$ the intersection of $(n-i)$ general elements of $|L|$ and by $X^i$ the intersection of the same elements of $|L|$ with $X^n$. \begin{tm} \label{roth} Assume that the natural restriction maps below are surjective $\forall m$: $$ \rho_m:= H^0({\cal P}^{n+2},mL)\to H^0({\cal P}^{n+1}, mL_{|{\cal P}^{n+1}}). $$ If $h^0({\cal I}_{X^{n-1},{\cal P}^{n+1}}(s))\neq 0$ for some $s$ such that $d>\delta s^2$, then $h^0($ ${\cal I}_{X^{n},{\cal P}^{n+2}}(s))$ $\neq 0$. \noindent If $s$ is the minimum such number then $h^0({\cal I}_{X^{n},{\cal P}^{n+2}}(s))=1$. \end{tm} \noindent {\em Proof.} Let us assume that we have proved the theorem for $$s=\sigma:=\min \{t\in {\Bbb N}|\quad h^0 ( {\cal I}_{X^{n-1},{\cal P}^{n+1}} (t) ) \neq 0 \}; $$ we call $\s$ the {\em postulation} of $X^{n-1}$. Then the theorem holds also $\forall s\geq \sigma$. We can thus assume, without loss of generality, that $s=\sigma$. \smallskip \noindent Pick any $V_{\sigma} \in |{\cal I}_{X^{n-1}, {\cal P}^{n+1}}(\s)|$. \medskip \noindent CLAIM. {\em $V_{\sigma}$ is integral}. This follows easily from the minimality of $\sigma$ and the fact that, under our assumptions, $Pic({\cal P}^{n+1})\simeq \zed [L_{|{\cal P}^{n+1}}]$. \medskip \noindent CLAIM. {\em $V_{\sigma}$ is the unique element of $|{\cal I}_{X^{n-1},{\cal P}^{n+1}}(\sigma)|$}. By contradiction, assume that we have two distinct $V^i_{\sigma}$. By the above claim they are both integral. By an easy Bertini-type argument we see that intersecting everything with $n$ general members of $|L|$ we get two distinct integral curves $W^i_{\sigma}\in |\odixl{{\cal P}^2}{\sigma}|$ containing $X^0=\{d$ points$\}$. Since the curves do not have common components we see that $d\leq W^1_{\sigma}\cdot W^2_{\sigma}=\delta {\sigma}^2$; the intersection product here is on ${\cal P}^2$. This is a contradiction and the claim is proved. \smallskip \noindent Let us choose a general line $\ell \subseteq$ $|L|^{\vee}$. Define $\tilde{\cal P}$ to be the blowing up of ${\cal P}^{n+2}$ along the intersection of all the members of $\ell$. Denote by $p$ and $q$ the natural projections to $\ell$ and ${\cal P}^{n+2}$, respectively. By intersecting with general elements of $|L|$ we get the following diagram, where $Y^i$ denotes $q^{-1}X^i$: \medskip $ \begin{array}{lllllllll} \hspace{1cm} Y^0 & \subseteq & Y^1 & \subseteq & \ldots & \subseteq & Y^n & \subseteq & \tilde{\cal P}\stackrel{p}{\longrightarrow} \ell \\ \hspace{1cm} \downarrow & \ & \downarrow & \ & \ & \ & \downarrow & \ & \downarrow q \\ \hspace{1cm} X^0 & \subseteq & X^1 & \subseteq & \ldots & \subseteq & X^n & \subseteq & {\cal P}^{n+2}. \end{array} $ \medskip \noindent We have the following injections, where, for simplicity (and by abuse) of notation, we denote a twist by $q^*\odixl{{\cal P}^{n+2}}{\s}$ simply by a twist by ${\s}$: $$ {\cal I}_{Y^n,\tilde{\cal P}}(\sigma)\to {\cal I}_{Y^{n-1},\tilde{\cal P}}(\sigma)\to \ldots \to {\cal I}_{Y^1,\tilde{\cal P}}(\sigma)\to {\cal I}_{Y^0,\tilde{\cal P}}(\sigma), $$ so that, applying $p_*$, we obtain the following injections: $$ p_*{\cal I}_{Y^n,\tilde{\cal P}}(\sigma)\to p_*{\cal I}_{Y^{n-1},\tilde{\cal P}}(\sigma)\to \ldots \to p_*{\cal I}_{Y^1,\tilde{\cal P}}(\sigma)\to p_*{\cal I}_{Y^0,\tilde{\cal P}}(\sigma). $$ The existence of $V_{\sigma}$, for a general point of $\ell$, ensures that $p_*{\cal I}_{Y^n,\tilde{\cal P}}$ $(\sigma)$ is not the zero sheaf. Since $p$ is dominant and the ideal sheaves ${\cal I}_{Y^i,\tilde{\cal P}}$ are torsion free, we see that the sheaves $p_*{\cal I}_{Y^i,\tilde{\cal P}}(\sigma)$ are torsion free $\forall i$. But $\ell$ is a smooth curve, so that these sheaves are actually locally free. The uniqueness statement, which was shown above, implies that these sheaves are actually line bundles on $\ell$. Since each of the above injections has torsion free cokernel on $\ell$ we deduce that they all are isomorphisms, i.e.: $$ p_*{\cal I}_{Y^n,\tilde{\cal P}}(\sigma)\simeq p_*{\cal I}_{Y^{n-1},\tilde{\cal P}}(\sigma)\simeq \ldots \simeq p_*{\cal I}_{Y^1,\tilde{\cal P}}(\sigma)\simeq p_*{\cal I}_{Y^0,\tilde{\cal P}}(\sigma)\simeq \odixl{\ell}{\tau}, $$ for some $\tau \in \zed$. \noindent By contradiction, assume $\tau<0$. Then $$ 0 = h^0(p_* {\cal I}_{Y^{n-1},\tilde{\cal P}}(\sigma))= h^0({\cal I}_{Y^{n-1},\tilde{\cal P}}(\sigma))= h^0(q_*{\cal I}_{Y^{n-1},\tilde{\cal P}}(\sigma))= h^0({\cal I}_{X^{n-1},{\cal P}^{n+2}}(\sigma)). $$ \noindent By our assumptions we can lift a section defining $V_{\sigma}$ to a non zero element of $H^0($ ${\cal I}_{ X^{n-1},{\cal P}^{n+2} }$ $(\sigma))$. This contradiction proves $\tau \geq 0$. \noindent This proves the first assertion of the theorem. As to the second we need to prove that $\tau =0$. $\tau $ being strictly positive would violate the usual uniqueness. \blacksquare \begin{rmk} {\rm We used the surjectivity of the restriction maps only for $m=\sigma$.} \end{rmk} \begin{rmk} {\rm The cases $({\cal P}^{n+2},L)\cong $ $ (\pn{n+2},$ $\odixl{\pn{n+2}}{1})$, $ (\Q{4},$ $\odixl{\Q{4}}{1})$ seem to be well known. See for example \cite{a-s}, \cite{m-r}, and of course \cite{ro}. However in the case of projective space it seems that the linear normality of $X$ was usually required; after Zak's theorem on tangencies linear normality is automatic, for a nonsingular $X$, unless $ n=4$ and $X$ is the Veronese surface.} \end{rmk} \begin{ex} \label{examples} {\rm The variety ${\cal P}^{n+2}$ can be, for example, a projective space, a nonsingular complete intersection or a Grassmannian; in all these cases $L$ is the hyperplane bundle for the natural embedding. But it can also be chosen to be a Fano variety, of index $r$, with $-K_{\cal P}=rL$, $L$ generated by global sections and $Pic({\cal P})\simeq \zed$ (this is always the case if $r>n/2$), some weighted complete intersections or, more generally, low degree branched coverings of projective spaces \cite{la1} or Grassmannians \cite{ki}. In the last batch of examples, $L$ does not need to be very ample.} \end{ex} The following gives a lifting criterion in any codimension; see \cite{m-r}. Again linear normality is not required. \begin{cor} Let $X^{\nu}$ be an integral subscheme of ${\cal P}^{n+2}$ of dimension $\nu$, $X^{\nu -1}$ the intersection of $X$ with a general member of $|L|$, $\s$ the postulation of $X^{\nu -1}$. Assume that $h^0({\cal I}_{X^{\nu -1}}(\s))=1$ and that $\rho_{\s}$ is surjective. Then $h^0({\cal I}_{X^{\nu}}(\s))=1$. \end{cor} \section{Finiteness on $\Q{n}$, $n\geq 7$} \label{boundednessforn>=7} In this section we remark that, for a nonsingular variety of dimension $\nu \geq \frac{n+3}{2} $ in $\Q{n}$ not of general type, the bound $d\leq 2 n^{n - \nu}$ holds. This gives the finiteness of the corresponding number of families. We thank M. Schneider for pointing out to us that the result of this section could be proved along the lines of his paper \ci{sc}. \begin{tm} \label{boundednessonqn} There are only finitely many components of the Hilbert scheme of $\Q{n}$ corresponding to nonsingular subvarieties of dimension $\nu \geq \frac{n+3}{2} $ which are not of general type. \end{tm} \noindent {\em Proof.} It suffices to bound the degree $d$ of any such $X$. The normal bundle $N$ of $X$ in $\Q{n}$ is generated by global sections. The Proposition of \ci{sc} is valid, on $X$, with ``ample" replaced by ``generated by global sections;" see \ci{fu}, Example 12.1.7. It follows that \begin{equation} \label{schur} c_{n-\nu}(N) \cdot c_1(N)^{2\nu -n} \leq c_1(N)^{\nu}. \end{equation} By the self intersection formula for $X$ on $\Q{n}$ and the structure of the cohomology ring of quadrics we have $c_{n-\nu}(N)=\frac{1}{2}dL^{n -\nu}$. \noindent By \ci{be-so-book}, Theorem 2.3.11 we get that $Pic(X)\simeq \zed [L]$, so that, if $X$ is not of general type, $K_X=eL$, with $e\leq 0$. Adjunction formula gives $c_1(N)=(e+n)L$. By plugging into (\ref{schur}) we get $$ \frac{1}{2}d^2 (e+n)^{2\nu-n} \leq (e+n)^{\nu}d, $$ which gives, after observing that $0 \leq -e \leq \dim X +1 < n $, that $$ d\leq 2 (e+n)^{n-\nu}\leq 2n^{n-\nu}. $$ \blacksquare \section{$3$-folds on a hypersurface of fixed degree} \label{chTHFD} In this section we generalize to the case of $\Q{5}$ the main result of $\S 3$ of \ci{boss1}, which deals with bounds associated with nonsingular $3$-folds contained in a hypersurface of $\pn{5}$. For the analogous result on $\Q{4}$ see \ci{a-s}, Proposition $6.7$. However in both of the above references the result is proved under the assumption that ``$d$ is big enough" with respect to the degree of the hypersurface. Of course this assumption is not a real restriction, since the residual cases are automatically taken care of by the fact that having a bounded degree bounds everything. However, it seems convenient to prove our statements without restrictions. The importance of this bound is more ore less theoretical: it can be used to assert the finiteness of special families of $3$-folds in $\Q{5}$. One should not expect to make an effective use of them and get sharp results. The paper \ci{e-p}, which deals with surfaces in $\pn{4}$, is the original source of the main ideas used in \ci{boss1}, in $\S6$ of \ci{a-s}, and in this section. The theoretical bound given there, for the degree of nonsingular surfaces not of general type in $\pn{4}$, is not an effective one. In the paper \ci{b-f} an effective bound, $d\leq 105$, is proved using initial ideals. \bigskip Let $X$ be a $3$-fold of degree $d$ in $\Q{5}$ contained in an integral hypersurface $V\in |\odixl{\Q{5}}{\s}|$, $S$ a general hyperplane section of $X$, $C$ a general hyperplane section of $S$ and let $g$ be its genus. As a convention, when we write something like ``$+$ l.t. in $\sqrt{d}$," we mean that the coefficients of the lower terms depend only on $\s$. \begin{tm} \label{polybound} Let $X\subseteq V\subseteq\Q{5}$ be as above. There is a degree eight polynomial $P_{\s}(\sqrt d)$, depending only on $\s$ and with positive leading coefficient, such that $$ -\chi(\odix{X})\geq P_{\s}(\sqrt d). $$ \end{tm} \noindent {\em Proof.} \noindent Look at the following three exact sequences. $$ 0\to \odixl{\pn{6}}{t-2} \to \odixl{\pn{6}}{t}\to \odixl{\Q{5}}{t} \to 0, $$ $$ 0\to \odixl{\Q{5}}{t-\s} \to \odixl{\Q{5}}{t}\to \odixl{V}{t} \to 0, $$ $$ 0\to {\cal I}_{X,V}(t) \to \odixl{V}{t}\to \odixl{X}{t} \to 0. $$ One can use the first one to compute $\chi (\odixl{\Q{5}}{t})$, the second one to compute \begin{eqnarray*} \chi (\odixl{V}{t}) &= & \frac{1}{12}\s t^4 + (-\frac{1}{6}\s^2+ \frac{5}{6}\s)t^3 +(\frac{1}{6}\s^3- \frac{5}{4}\s^2+3\s)t^2 \\ & & +(-\frac{1}{12}\s^4+ \frac{5}{6}\s^3- 3\s^2 + \frac{55}{12}\s)t +\frac{1}{60}\s^5 - \frac{5}{24}\s^4 + \s^3 - \frac{55}{24}\s^2 + \frac{149}{60}\s, \end{eqnarray*} finally we use (\ref{chioxt}), $\mu:=\mu_{\s}=$ $\frac{1}{2}d^2+ \s(\s-3)d -2\s(g-1)$ (cf. the notation fixed before Lemma \ref{epas}), and the third short exact sequence to compute \begin{eqnarray*} \chi ({\cal I}_{X,V}(t)) &= & \frac{1}{12}\s t^4 + \frac{1}{6}[(5-\s)\s -d]t^3+ \\ & &[\frac{1}{6}\s^3 -\frac{5}{4}{\s}^2 + 3\s + \frac{1}{4\s}(\frac{d^2}{2} + d\s (\s-3) -\mu ) -\frac{d}{2}]t^2 \\ & & +[ -\frac{1}{12}\s^4 + \frac{5}{6}{\s}^3-3{\s}^2 + \frac{55}{12}\s -\frac{d}{3} \\ & & + \frac{1}{4\s}(\frac{d^2}{2} + d\s (\s-3) -\mu) - \chi (\odix{S})]t \\ & & + \frac{1}{60}\s^5 - \frac{5}{24}\s^4 + \s^3 -\frac{55}{24}\s^2 + \frac{149}{60}\s -\chi (\odix{X}) \\ &=:& Q(t)-\chi (\odix{X}). \end{eqnarray*} It follows that $$ -\chi (\odix{X})=\chi ({\cal I}_{X,V}(t)) - Q(t). $$ Define $$ t_1:=\min\{ t\in {\Bbb N}|\ \ \delta:=2\s t-d>0\ \ {\rm and}\ \ \frac{\delta^2}{2} -\mu -\delta \s (\s-3)>0\}; $$ by \ci{a-s}, page 89: $$ \frac{d}{2\s}\leq t_1 \leq \frac{d}{2\s} + \frac{\sqrt{2d}}{2} + \s $$ By plugging $t_1$ in what above we get, using the above inequalities for $t_1$ and Lemma \ref{epas}: $$ -\chi (\odix{X})=\chi ({\cal I}_{X,V}(t_1)) - Q(t_1)\geq \chi ({\cal I}_{X,V}(t_1))-\frac{1}{64{\s^3}}d^4 + \frac{1}{2\s}\chi (\odix{S})d +\ {\rm l.t.\ in}\ \sqrt{d}; $$ by \ci{a-s}, pages $88-89$, we also know that $$ \chi (\odix{S})\geq \frac{1}{24{\s}^2}d^3 +\ {\rm l.t.\ in}\sqrt{d} $$ so that $$ -\chi (\odix{X})\geq \chi ({\cal I}_{X,V}(t_1)) + \frac{1}{192{\s}^3}d^4 + \ {\rm l.t.\ in}\sqrt{d}. $$ To conclude it is enough to bound conveniently from below $\chi ({\cal I}_{X,V}(t_1))=h^0-h^1+h^2-h^3+ h^4$. This, in turn, can be accomplished by bounding $h^1$ and $h^3$ from above. This is the content of the following technical lemmata. \blacksquare First we fix some notation. By taking general hyperplane sections we obtain the following diagram: $$ \begin{array}{rllllllll} \hspace{1in}\Q{3} & \subseteq & \ldots & \subseteq & \Q{n+1} & \subseteq & \Q{n+2} & \ & \ \\ \cup \ \ & \ & \ & \ & \cup & \ & \cup & \ & \ \\ \tilde{V}^2\to V^2 & \subseteq & \ldots & \subseteq & V^n & \subseteq & V^{n+1} & = & V \\ \cup \ \ & \ & \ & \ & \cup & \ & \cup & \ & \ \\ X^1 & \subseteq & \ldots & \subseteq & X^{n-1} & \subseteq & X^n & = & X \end{array} $$ where $\tilde{V}^2$ is the normalization of $V^2$. \medskip The following lemma is the analogue of \ci{boss1}, Lemma $3.3$. It is proved in the same way using \ci{a-s} lemmata $6.10$, $6.11$ and $6.12$ instead of the lemmata from \ci{e-p} quoted in \ci{boss1}. \begin{lm} \label{uno} Let $X=X^n$ be a degree $d$ nonsingular $n$-dimensional subvariety of $\Q{n+2}$, $n\geq 2$. Assume that $X$ is contained in an integral hypersurface $V=V^{n+1}$ $\in |\odixl{\Q{n+2}}{\s}|$. Define $t_1$ as above. Then there are constants $A_1$, $A_2$, depending only on $\s$, such that $$ \sum_{\nu=t_1}^{\infty} h^1({\cal I}_{X^1,\tilde{V}^2}(\nu)) \leq A_1 \sqrt{d^3} + \ \rm{l.t.\ in }\ \sqrt{d}, $$ and $$ \sum_{\nu=0}^{t_1-1} h^1({\cal I}_{X^1,\tilde{V}^2}(\nu)) \leq A_2 \sqrt{d^5} + \ \rm{l.t.\ in }\ \sqrt{d}. $$ \end{lm} \medskip The next lemma merely generalizes Lemma 3.4 of \ci{boss1}. It should be noted that their proof of it has a flaw since their argument does not work in the case $i=n+1$. However that case is not needed for our (and their) purposes. In any case we easily prove a bound also in that case. \begin{lm} \label{due} Let things be as in the previous lemma. Then $$ h^0({\cal I}_{X,V}(t_1))\leq B_0\sqrt{d^{2n-1}} +\ {\rm l.t. \ in }\ \sqrt{d}; $$ for $i=1,$ $$ h^1({\cal I}_{X,V}(t_1))\leq B_1\sqrt{d^{2n+1}} +\ {\rm l.t. \ in }\ \sqrt{d}, $$ for $i=n-1,\ n$ $$ h^i({\cal I}_{X,V}(t_1))\leq B_i\sqrt{d^{2i+1}} +\ {\rm l.t. \ in }\ \sqrt{d}, $$ and for $i=n+1$ $$ h^{n+1}({\cal I}_{X,V}(t_1))\leq B_{n+1} d^{n+1} +\ {\rm l.t. \ in }\ \sqrt{d}, $$ where the $B_i$'s are positive constants depending only on $\s$. \end{lm} \noindent {\em Proof.} By looking at the following sequences \begin{equation} \label{alce} 0\to {\cal I}_{X^{i},V^{i+1}}(k-1) \to {\cal I}_{X^{i},V^{i+1}}(k) \to {\cal I}_{X^{i-1},V^{i}}(k) \to 0, \end{equation} we deduce \begin{eqnarray*} h^0({\cal I}_{X,V}(t_1)) & \leq & h^0({\cal I}_{X^{n-1},V^{n}}(t_1)) + h^0({\cal I}_{X^{n},V^{n+1}}(t_1-1)) \\ &\leq & \sum_{k=1}^{t_1} h^0({\cal I}_{X^{n-1},V^{n}}(k)) \\ & \vdots & \\ & \leq & \sum_{k=1}^{t_1} \dots \sum_{k=1}^{t_1} h^0({\cal I}_{X^{1},V^{2}}(k)) \qquad \quad (n-1)\ \rm{summands} \\ & \leq & t_1^{n-2} \sum_{k=1}^{t_1} h^0({\cal I}_{X^{1},V^{2}}(k)) \ \leq\ t_1^{n-2} \sum_{k=1}^{t_1} h^0({\cal I}_{X^{1},\tilde{V}^{2}}(k)) \\ & \leq & t_1^{n-2}(A_0\sqrt{d^3} + {\rm l.t \ in }\ \sqrt{d}), \end{eqnarray*} where the last inequality can be found in \ci{a-s}, Lemma 6.15, and $A_0$ depends only on $\s$. Since $t_1\leq \frac{1}{2\s}d + \frac{\sqrt{2d}}{2} + \s$, we have bounded $h^0({\cal I}_{X,V} (t_1))$ as wanted. \smallskip \noindent To bound $h^1$ we argue as above. \noindent For $h^1({\cal I}_{X,V} (t_1))\leq$ $\sum_{k=1}^{t_1}h^1({\cal I}_{X^{n-1},V^n} (k))+$ $h^1({\cal I}_{X^n,V^{n+1}})$, but this last dimension is zero as one can check by looking at the long cohomology sequences associated with the following two exact sequences: \begin{equation} \label{aaa} 0\to {\cal I}_{X^n,\Q{n+2}} \to \odix{\Q{n+2}} \to \odix{X^n} \to 0, \end{equation} \begin{equation} \label{aaaa} 0 \to \odixl{\Q{n+2}}{-\s} \to {\cal I}_{X^n,\Q{n+2}} \to {\cal I}_{X^n,V^{n+1}} \to 0. \end{equation} An easy induction argument, already seen before, using (\ref{alce}) allows us to infer that $$ h^1({\cal I}_{X,V}(t_1))\leq t_1^{n-2} \sum_{k=1}^{t_1}h^1({\cal I}_{X^1,V^2}(k)). $$ To obtain the desired bound on $h^1$ it is enough to prove that: $$ \sum_{k=1}^{t_1}h^1({\cal I}_{X^1,V^2}(k)) \leq F\sqrt{d^5} +\ \rm{l.t. \ in} \sqrt{d}, $$ where $F$ is a constant depending only on $\s$. \noindent This can be proved as follows. The idea is to couple the previous lemma with the cohomology sequences associated with the following exact sequences: $$ 0\to {\cal I}_{X^1,V^2}(k)\to {\cal I}_{X^1,\tilde{V}^2}(k)\to Q(k)\to 0. $$ \noindent Clearly we have $ h^1({\cal I}_{X^1,V^2}(k))\leq h^1({\cal I}_{X^1,\tilde{V}^2}(k))+ h^0(Q(k)) $, $\forall k$, so that, in view of Lemma \ref{uno}, it is enough to prove the following \medskip \noindent CLAIM. $h^0(Q(k))\leq D(k+1)$, $\forall k\geq 0$, {\em where $D$ is a positive constant depending only on $\s$. In particular $\sum_{k=0}^{t_1}h^0(Q(k))\leq (1/2)Dt_1^2+ $ \rm{l.t.} in $t_1$.} \medskip \noindent {\em Proof of the claim.} $Q$ is the structural sheaf of the non-normal locus of $V^2$ twisted by the ideal sheaf of $X^1$. By taking a general hyperplane section we get the following exact sequences $$ 0 \to Q(k-1) \to Q(k) \to Q_{\Gamma}(k) \to 0, $$ where $Q_{\Gamma}(k)\simeq Q_{\Gamma}$ is the structural sheaf of the singular locus of $\Gamma$, a general hyperplane section of $V^2$. As usual $h^0(Q(k))\leq h^0(Q(k-1)) + length(Q_{\Gamma})$, so that $h^0(Q(k))\leq length(Q_{\Gamma})(k+1) + h^0(Q(-1))$. This length is bounded from above by a function of $\s$ only (that is, an irreducible curve of degree $2\s$ on a two dimensional quadric cannot have too many singularities). As to $ h^0(Q(-1))$ one has to exercise caution since the non-normal locus may be non reduced. However by looking at the cohomology sequences associated with (\ref{aaa}) we see that $h^1({\cal I}_{X^1,V^2}(-1))=0$, so that $ h^0(Q(-1))=0$, and the claim is proved. \smallskip \noindent Now we prove the bound for $i=n$. \noindent We start by remarking that for $i=n-1,$ $n,$ $n+1$ and all $k\geq d-1$. $$ h^i({\cal I}_{X,V}(k))=0. $$ If $X^2$ is non degenerate then ${\cal I}_{X^2,\pn{5}}$ is $(d-2)$-regular (cf. \cite{la2}) in the sense of Castelnuovo-Mumford. By looking at the following sequences: $$ 0\to {\cal I}_{X^2,\pn{5}}(-2+k) \to {\cal I}_{X^2,\pn{5}}(k) \to {\cal I}_{X^2,\Q{4}}(k) \to 0, $$ we deduce the vanishings, for $n=2$. An easy inductive argument (cf. \ci{boss1}, page 326) gives the desired vanishings. \noindent If $X^2 \subseteq \pn{5}$ were degenerate, then either $X^2=\pn{2}$, $X^2$ would be a hypersurface of $\pn{3}$ or it would be a nondegenerate surface in $\pn{4}$. In any of these cases we apply the bound for the regularity of the ideal sheaves in \ci{la2} to obtain vanishings for the higher cohomology of ${\cal I}_{X^2,{\Bbb P}^{4}}$ which is easy to lift to the desired vanishings for $X^2$. Again the inductive procedure allows us to conclude. \noindent We have the following chain of inequalities: \begin{eqnarray*} h^n({\cal I}_{X^n,V^{n+1}}(t_1)) & \leq & \sum_{k>t_1}h^{n-1}({\cal I}_{X^{n-1},V^{n+1}}(k)) \leq \sum_{k=1}^{d-4} h^{n-1}({\cal I}_{X^{n-1},V^{n+1}}(k)) \\ & \leq & \ldots \leq \sum_{1}^{d-4} \ldots \ \sum_{1}^{d-4} \sum_{k=1}^{d-4} h^1({\cal I}_{X^1,V^2}(k)) \\ & \leq & (d-4)^{n-2} \sum_{k=1}^{d-4} h^1({\cal I}_{X^1,V^2}(k)) \\ & \leq & B_n\sqrt{d^{2n+1}} + \rm{l.t.\ in} \sqrt{d}, \end{eqnarray*} where $B_n$ depends only on $\s$ and the last inequality follows from Lemma \ref{uno}. \smallskip \noindent The case $i=n-1$ is analogous. \smallskip \noindent Finally the bound for the case $i=n+1$ can be obtained as in the case $i=n$ except for the fact that we end up having to bound $h^2( {\cal I}_{X^1,V^2}(k))$ for $k=1,\ldots,$ $d-4$, and not $h^1$: $$ h^{n+1}({\cal I}_{X,V}(t_1))\leq d^{n-2}(\sum_{k=1}^{d-4}h^2({\cal I}_{X,V}(t_1)). $$ To bound this summand we look at the exact sequences: $$ 0\to {\cal I}_{X^1,V^2}(k) \to \odixl{V^2}{k} \to \odixl{X^1}{k} \to 0, $$ and deduce $$ h^2({\cal I}_{X^1,V^2}(k))\leq h^1(\odixl{X^1}{k}) + h^2(\odixl{V^2}{k}) =h^0(\omega_{X^1}(-k))+h^0(\odixl{X^1}{-3+\s -k}) $$ where the last equality stems from Serre Duality. We are thus left with bounding the two $h^0$ above. The first one can be bounded using Proposition \ref{easybound} on $ h^0(\omega_{X^1})=g(X^1)$: the worst upper bound is of the form $(1/4\s)d^2+$ l.t. in $d$. As to the second $h^0$ its worst upper bound is of the form $(1/2)\s^2$. Adding up for $k=1,\ldots,$ $d-4$ we get that the worst upper bound is $(1/4\s)d^3 +$ l.t. in $d$. \blacksquare \bigskip The following generalizes \ci{boss1}, Corollary $3.1$. \begin{cor} \label{finiteonhyp} Let $\s$ be any positive integer. There are only finitely many components of the Hilbert scheme of $\Q{5}$ corresponding to nonsingular $3$-folds in $\Q{5}$ which are not of general type and are contained in some hypersurface of degree $\s$. \end{cor} \noindent{\em Proof.} It is enough to bound from above the degree of such $3$-folds. Since $\omega_X(-1)$ does not have sections $h^0(\omega_X)\leq h^0(\omega_S)$, where $S$ is any nonsingular hyperplane section of $X$. By the generalized Castelnuovo-type bounds of Harris (cf. \ci{jh}) we have $$ h^0(\omega_S)\leq Ad^3 + \rm{l.t.\ in }\ d, $$ where $A$ is some constant; the Lefschetz hyperplane theorem, coupled with Proposition \ref{easybound}, ensures that $$ h^1(\odix{X})= h^1(\odix{S})\leq h^1(\odix{C})\leq \frac{1}{4\s}d^2 + \rm{l.t.\ in }\ d. $$ It follows that $$ h^0(\omega_S) \geq h^0(\omega_X)= 1+ h^2(\odix{X}) -h^1(\odix{X}) - \chi(\odix{X})\geq \frac{1}{192\s^3}d^4 + \rm{l.t.\ in }\ \sqrt{d}. $$ Comparing the two inequalities for $h^0(\omega_S)$, we conclude that $d$ is bounded. \blacksquare \section{Finiteness for $3$-folds not of general type in $\Q{5}$} \label{boundednessonq5} \begin{pr} \label{generalizedhodge} Let $X$ be a nonsingular $3$-fold in $\Q{5}$ and $k$ a positive integer. Then \begin{equation} \label{genhodge} \chi (\odix{S})\leq \frac{2}{3} \frac{(g-1)^2}{d} - \frac{1}{24}d^2 + \frac{5}{12}d. \end{equation} If $X$ is not of general type then \begin{equation} \label{Xnotgentype-chi} -\chi (\odix{X}) \leq \chi (\odix{S}) + \frac{1}{2}d^2 -2d +2; \end{equation} if $d>2k^2$ and $X$ is not of general type and not contained in any hypersurface of degree strictly less than $2\cdot k$, then \begin{equation} \label{Xnotgentype-chik} -\chi (\odix{X}) \leq \chi (\odix{S}) + \frac{1}{k}d^2 + (k-4)d +2 \end{equation} \end{pr} \noindent {\em Proof.} The first inequality stems from the Generalized Hodge Index Theorem contained in \ci{boss1}: $$ d(K_X^2L)\leq (K_XL^2)^2, $$ we make explicit the left hand side using (\ref{K2L}) and the right hand side using (\ref{KL2}). \noindent For the second one we look at $$ 0 \to K_X(-1) \to K_X \to K_S(-1) \to 0. $$ Since $X$ is not of general type $h^0(K_X(-1))=0$, otherwise $K_X$ would be big, i.e. a $|mK_X|$ would define a birational map. It follows that $h^3(\odix{X})=$ $h^0(K_X)$ $\leq$ $ h^0(K_S(-1))$ $\leq$ $ h^0(K_S)=$ $ h^2(\odix{S})$. \noindent We thus have $$ -\chi (\odix{X})\leq h^1(\odix{X}) + h^3(\odix{X}) \leq \chi (\odix{S}) + 2h^1(\odix{X}), $$ where we have used Lefschetz Theorem on Hyperplane Sections to ensure that $h^1(\odix{X})= h^1(\odix{S})$. $h^1(\odixl{S}{-1})=0$ by Kodaira Vanishing, so that $h^1(\odix{X})= h^1(\odix{S})\leq g$. \noindent If $C$ were contained in a $\pn{3}$ we would use Proposition \ref{easybound} with $k=1$ to conclude. If $C$ were not in any surface of degree strictly less than $2\cdot 2$ we would use Proposition \ref{boundasep} with $k=2$. \noindent The third inequality is proved exactly as the second one by using Proposition \ref{roth} to ensure that a general curve section $C$ is not in any surface of the corresponding $\Q{3}$ of degree strictly less than $2\cdot k$, and Proposition \ref{boundasep} to bound the genus $g$ from above. \blacksquare \begin{pr} \label{s3>=0forN} Let $X$ be a nonsingular $3$-fold in $\Q{5}$. Then $$ 60 \chi (\odix{S}) \geq \frac{3}{2}d^2 - 12d + (d-48)(g-1) + 24 \chi (\odix{X}). $$ \end{pr} \noindent {\em Proof.} Denote by $s_i$ and $n_i$ the Segre and Chern classes respectively of the normal bundle $N$ of $X$ in $\Q{5}$. Since $N$ is generated by global sections we have $s_3\geq 0$. Since $s_3=n_1^3- 2n_1n_2$ we get \noindent $0\leq (K_X+5L)^3 - 2(K_X+5L)\frac{1}{2}dL^2=$ $K^3 + 15K_X^2L + 75K_XL^2 + 125d - d(K_X+5L)L^2.$ \noindent We conclude by (\ref{K3}), (\ref{K2L}) and (\ref{KL2}). \blacksquare \begin{tm} \label{sigh} Let $n=4,$ $5$ or $n\geq 7$. There are only finitely many components of the Hilbert scheme of $\Q{n}$ corresponding to nonsingular $(n-2)$-folds not of general type. \end{tm} \noindent {\em Proof.} By \ci{a-s}, \S6 and Theorem \ref{boundednessonqn} it is enough to consider the case $n=5$. It is enough to bound from above the degree $d$ of such $3$-folds. \noindent Fix a positive integer $k$ and let $d$ be a positive integer such that $d >2k^2.$ Let $X$ be a degree $d$ $3$-fold in $\Q{5}$ not lying on any hypersurface of $\Q{5}$ of degree strictly less than $2\cdot k$; by Proposition \ref{roth}, a general curve section of $X$ does not lie on any surface of the corresponding $\Q{3}$ of degree strictly less than $2\cdot k$. \noindent We couple Proposition \ref{s3>=0forN} and inequality (\ref{Xnotgentype-chik}) of Proposition \ref{generalizedhodge}: $$ 84 \chi (\odix{S}) \geq (\frac{3}{2} - \frac{24}{k})d^2 - ( 24k-84)d -48 + (d-48)(g-1). $$ In what above we plug inequality (\ref{genhodge}) of Proposition \ref{generalizedhodge} and get: $$ \frac{52}{d} (g-1)^2 - \frac{21}{6}d^2 +35d \geq (\frac{3}{2}- \frac{24}{k})d^2 - ( 24k -84)d -48 + (d-48)(g-1). $$ A simple manipulation gives \begin{equation} \label{almost} (g-1)[ \frac{52}{d}(g-1) -d +48] + (\frac{24}{k}-5)d^2 + (24k - 49)d +48 \geq0. \end{equation} Let us now first assume that $g>0$. The aim is to choose $k$ such that the coefficients $\alpha:=(\frac{52}{d}(g-1) -d +48)$ and $\beta:=(\frac{24}{k} -5)$ of $(g-1)$ and $d^2$, respectively, are negative. Once they are negative, since $k$ is fixed, the inequality (\ref{almost}) will force $d$ to be bounded from above. By Proposition \ref{epas} we get \begin{eqnarray*} \frac{52}{d}(g-1) - d + 48 & \leq & \frac{52}{d}[\frac{d^2}{2k} + \frac{1}{2}(k-4)d] -d +48 \\ & = & (\frac{26}{k} -1)d + 26k -56. \end{eqnarray*} Let $k=27$; then $\alpha$ is negative for $d\gg 0$. For the same value of $k$, $\beta$ is negative as well. By what above we infer that $d$ is bounded from above if $g>0$ and $X$ is not in an hypersurface of degree strictly less than $2\cdot 27$. We apply Corollary \ref{finiteonhyp} to see that $d$ is bounded from above for $3$-folds $X$, not of general type, contained in hypersurfaces of degrees less or equal to $2\cdot 27$. This proves the theorem in the case $g>0$. \noindent Assume that $g=0$. Then, by (\ref{almost}): $$ \frac{52}{d} + d -48 + (\frac{24}{k} -5)d^2 + (24k -49)d +48 \geq 0. $$ We argue as above with $k=5$. \blacksquare \bigskip

9608

alg-geom/9608006

en

https://arxiv.org/abs/alg-geom/9608006

alg-geom/9608006

David R. Morrison

David R. Morrison

The Geometry Underlying Mirror Symmetry

26 pages, AmS-LaTeX. Final version, to appear in Proc. European Algebraic Geometry Conference (Warwick, 1996)

New Trends in Algebraic Geometry (K. Hulek, F. Catanese, C. Peters, and M. Reid, eds.), London Math. Soc. Lecture Notes, vol. 264, Cambridge University Press, 1999, pp. 283-310

DUK-M-96-05

The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners. The geometric description---that one Calabi-Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi-Yau manifold---is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometric mirror pairs (in arbitrary dimension) which is independent of those conjectures. We investigate how this new geometric description ought to be related to the mathematical statements which have previously been extracted from mirror symmetry. In particular, we discuss how the moduli spaces of the `mirror' Calabi-Yau manifolds should be related to one another, and how appropriate subspaces of the homology groups of those manifolds could be related. We treat the case of K3 surfaces in some detail.

alg-geom

\section{Quantum mirror symmetry} Moduli spaces which occur in physics often differ somewhat between the classical and quantum versions of the same theory. For example, the essential mathematical data needed to specify the two-dimensional conformal field theory associated to a Calabi--Yau manifold $X$ consists of a Ricci-flat metric $g_{ij}$ on $X$ and an ${\Bbb R}/{\Bbb Z}$-valued harmonic $2$-form $B\in{\cal H}^2(X,{\Bbb R}/{\Bbb Z})$. The classical version of this theory is independent of $B$ and invariant under rescaling the metric; one might thus call the set of all diffeomorphism classes of Ricci-flat metrics of fixed volume on $X$ the ``classical moduli space'' of the theory. The volume of the metric and the $2$-form $B$ must be included in the moduli space once quantum effects are taken into account; in a ``semiclassical approximation'' to the quantum moduli space, one treats the data $(g_{ij},B)$ (modulo diffeomorphism) as providing a complete description of that space. However, a closer analysis of the physical theory reveals that this is indeed only an approximation to the quantum moduli space, with the necessary modifications becoming more and more significant as the volume is decreased. The ultimate source of these modifications---which are of a type referred to as ``nonperturbative'' in physics---is the set of holomorphic curves on $X$ and their moduli spaces. A convenient mathematical way of describing how these modifications work is this: there are certain ``correlation functions'' of the physical theory, which are described near the large volume limit as power series whose coefficients are determined by the numbers of holomorphic $2$-spheres on $X$.\footnote{There are several possible (equivalent) mathematical interpretations which can be given to these correlation functions: they can be interpreted as defining a new ring structure on the cohomology (defining the so-called quantum cohomology ring) or they can be regarded as defining a variation of Hodge structure over the moduli space. We will review this in more detail in section \ref{s:characterizations} below.} The quantum moduli space should then be identified as the natural domain of definition for these correlation functions. To construct it starting from the semiclassical approximation, one first restricts to the open set in which the power series converge, and then extends by analytic continuation to find the complete moduli space.\footnote{There can also be modifications caused by higher genus curves \cite{chiral}, but these are less drastic and are not important for our purposes here.} We refer to this space as the quantum conformal field theory moduli space ${\cal M}_{CFT}(X)$. (When necessary, we use the notation ${\cal M}^{sc}_{CFT}(X)$ to refer to the semiclassical approximation to this space.) A similar story has emerged within the last year concerning the moduli spaces for type IIA and IIB string theories compactified on a Calabi--Yau threefold $X$. The classical low-energy physics derived from these string theories is determined by a quantum conformal field theory, so one might think of the quantum conformal field theory moduli space described above as being a ``classical moduli space'' for these theories. In the semiclassical approximation to the {\it quantum}\/ moduli spaces of these string theories, we encounter additional mathematical data which must be specified. In the case of the IIA theory, the new data consist of a choice of a nonzero complex number (called the ``axion/dilaton expectation value''), together with an ${\Bbb R}/{\Bbb Z}$-valued harmonic $3$-form $C\in {\cal H}^3(X,{\Bbb R}/{\Bbb Z})$. This last object has a familiar mathematical interpretation as a point in the intermediate Jacobian of $X$ (taking a complex structure on $X$ for which the metric is K\"ahler). In the case of the IIB theory on a Calabi--Yau threefold $Y$, the corresponding new data are a choice of nonzero ``axion-dilaton expectation value'' as before, together with what we might call a quantum ${\Bbb R}/{\Bbb Z}$-valued harmonic even class $C\in{\cal H}^{\text{even}}_Q(Y,{\Bbb R}/{\Bbb Z})$. The word ``quantum'' and the subscript ``$Q$'' here refer to the fact that we must use the quantum cohomology lattice rather than the ordinary cohomology lattice in determining when two harmonic $C$'s are equivalent. (The details of this difference are not important here; we refer the interested reader to \cite{udual,mirrorII}.) For both the IIA and IIB theories, a choice of such ``data'' as above can be used to describe a low-energy supergravity theory in four dimensions. Just as in the earlier example, there are additional corrections to the semiclassical description of the moduli space coming from ``nonperturbative effects'' \cite{S,GMS,BBS}, among which are some which go by the name of ``Dirichlet-branes,'' or ``D-branes'' for short. The source of these D-brane corrections differs for the two string theories we are considering: in the type IIA theory, they come from moduli spaces of algebraic cycles on $X$ equipped with flat $U(1)$-bundles, or more generally, from moduli spaces of coherent sheaves on $X$.\footnote{A D-brane in type IIA theory is ordinarily described as a complex submanifold $Z$ together with a flat $U(1)$-bundle on that submanifold; the associated holomorphic line bundle on $Z$ can be extended by zero to give a coherent sheaf on $X$. The {\it arbitrary}\/ coherent sheaves which we consider here correspond to what are called ``bound states of D-branes'' in physics. (This same observation has been independently made by Maxim Kontsevich, and by Jeff Harvey and Greg Moore.)} In the type IIB theory, the D-brane corrections come from (complexified) moduli spaces of so-called supersymmetric $3$-cycles on $Y$, the mathematics of which will be described in the next section. Just as the correlation functions which we could use to determine the structure of the quantum conformal field theory moduli space involved a series expansion with contributions from the holomorphic spheres, the correlation functions in this theory will receive contributions from the coherent sheaves or supersymmetric $3$-cycles, with the precise nature of the contribution arising from an integral over the corresponding moduli space. {\it Quantum mirror symmetry}\/ is the assertion that there should exist pairs of Calabi--Yau threefolds\footnote{There are versions of quantum mirror symmetry which can be formulated in other (low) dimensions, but since these are statements about compactifying ten-dimensional string theories, they cannot be extended to arbitrarily high dimension.} $(X,Y)$ such that the type IIA string theory compactified on $X$ is isomorphic to the type IIB string theory compactified on $Y$; there should be compatible isomorphisms of both the classical and quantum theories. The isomorphism of the classical theories is the statement that the corresponding (quantum-corrected) conformal field theories should be isomorphic. This is the version of mirror symmetry which was translated into mathematical terms some time ago, and leads to the surprising statements relating the quantum cohomology on $X$ to the geometric variation of Hodge structure on $Y$ (and {\it vice versa}\/). On the other hand, the isomorphism of the quantum theories has only recently been explored.\footnote{The speculation some time ago by Donagi and Markman \cite{DonMark} that some sort of Fourier transform should relate the continuous data provided by the intermediate Jacobian to the discrete data provided by the holomorphic curves is closely related to these isomorphisms of quantum theories.} At the semiclassical level, one infers isomorphisms between the intermediate Jacobian of $X$ (the $3$-form discussed above), and an analogue of that intermediate Jacobian in quantum cohomology of $Y$. The full quantum isomorphism would involve properties of the coherent sheaves on $X$, as related to the supersymmetric $3$-cycles on $Y$. In fact, there should be enough correlation functions in the quantum theory to fully measure the structure of the individual moduli spaces of these sheaves and cycles, so we should anticipate that the moduli spaces themselves are isomorphic.\footnote{In the case of coherent sheaves, one should not use the usual moduli spaces from algebraic geometry, but rather some sort of ``virtual fundamental cycle'' on the algebro-geometric moduli space, whose dimension coincides with the ``expected dimension'' of the algebro-geometric moduli space as computed from the Riemann--Roch theorem. When the moduli problem is unobstructed, this virtual fundamental cycle should coincide with the usual fundamental cycle on the algebro-geometric moduli space.} It is this observation which was the key to the Strominger--Yau--Zaslow argument. Strominger, Yau and Zaslow observe that the algebraic $0$-cycles of length one on $X$ (which can be thought of as torsion sheaves supported at a single point) have as their moduli space $X$ itself. According to quantum mirror symmetry, then, there should be a supersymmetric $3$-cycle $M$ on $Y$ with precisely the same moduli space, that is, the moduli space of $M$ should be $X$. Since the complex dimension of the moduli space is three, it follows from a result of McLean \cite{McL} (see the next section) that $b_1(M)=3$. Now as we will explain in the next section, the complexified moduli space $\widehat{X}$ for the supersymmetric $3$-cycles parameterizes both the choice of $3$-cycle $M$ and the choice of a flat $\operatorname{U}(1)$-bundle on $M$. Fixing the cycle but varying the bundle gives a real $3$-torus on $\widehat{X}$ (since $b_1(M)=3$), which turns out to be a supersymmetric cycle on {\it that}\/ space. This is the ``inverse'' mirror transform, based on a cycle $\widetilde{M}$ which is in fact a $3$-torus. Thus, by applying mirror symmetry twice if necessary, we see that we can---without loss of generality---take the original supersymmetric $3$-cycle $M$ to be a $3$-torus. In this case, we say that $Y$ has a {\it supersymmetric $T^3$-fibration}; note that singular fibers must in general be allowed in such fibrations. We have thus obtained the rough geometric characterization of the pair $(X,Y)$ which was stated in the introduction: $X$ should be the moduli space for supersymmetric $3$-tori on $Y$. This characterization is ``rough'' due to technical difficulties involving both the compactifications of these moduli spaces, and the complex structures on them. We will take a different path in section \ref{s:geom} below, and give a precise geometric characterization which sidesteps these issues. This line of argument can be pushed a bit farther, by considering the algebraic $3$-cycle on $X$ in the fundamental class equipped with a flat $U(1)$-bundle (which must be trivial, and corresponds to the coherent sheaf $\O_X$). There is precisely one of these, so we find a moduli space consisting of a single point. Its mirror should then be a supersymmetric $3$-cycle $M'$ with $b_1(M')=0$. Moreover, we should expect quantum mirror symmetry to preserve the intersection theory of the cycles represented by D-branes (up to sign), so since the $0$-cycle and $6$-cycle on $X$ have intersection number one, we should expect $M$ and $M'$ to have intersection number one if $M'$ is oriented properly. In other words, the supersymmetric $T^3$-fibration on $Y$ should have a section,\footnote{The existence of a section is also expected on other grounds: the set of flat $U(1)$-bundles on $M$ has a distinguished element---the trivial bundle. This provides a section for the ``dual'' fibration, and suggests (by a double application of quantum mirror symmetry as above) that the original fibration could have been chosen to have a section, without loss of generality.} and the base of the fibration should satisfy $b_1=0$. The final step in the physics discussion given in \cite{SYZ} is to observe that given a Calabi--Yau threefold with a supersymmetric $T^3$-fibration and a mirror partner, the mirror partner can be recovered by dualizing the tori in the fibration, at least generically. This suggests that by applying an appropriate duality transformation to the path integral---this is known as the ``T-duality argument'' in quantum field theory---one should be able to conclude that mirror symmetry does indeed hold for the corresponding physical theories. Strominger, Yau and Zaslow take the first steps towards constructing such an argument, at appropriate limit points of the moduli space. To complete the argument and extend it to general points in the moduli space, one would need to understand the behavior of the T-duality transformations near the singular fibers; to this end, a detailed mathematical study of the possible singular fibers is needed. Some preliminary information about these singularities can be found in \cite{HL,Br:sub} (see also \cite{GW}). \section{Moduli of special Lagrangian submanifolds}\label{s:SLAG} The structure of the supersymmetric $3$-cycles which played a r\^ole in the previous section was determined in \cite{BBS}, where it was found that they are familiar mathematical objects known as special Lagrangian submanifolds. These are a particular class of submanifolds of Calabi--Yau manifolds first studied by Harvey and Lawson \cite{HL}. We proceed to the definitions. A {\em Calabi-Yau manifold}\/ is a compact connected orientable manifold $Y$ of dimension $2n$ which admits Riemannian metrics whose (global) holonomy is contained in $\operatorname{SU}(n)$. For any such metric, there is a complex structure on the manifold with respect to which the metric is K\"ahler, and a nowhere-vanishing holomorphic $n$-form $\Omega$ (unique up to constant multiple). The complex structure, the $n$-form $\Omega$ and the K\"ahler form $\omega$ are all covariant constant with respect to the Levi--Civita connection of the Riemannian metric. This implies that the metric is Ricci--flat, and that $\Omega\wedge\overline\Omega$ is a constant multiple of $\omega^n$. A {\it special Lagrangian submanifold}\/ of $Y$ is a compact real $n$-manifold $M$ together with an immersion $f:M\to Y$ such that $f^*(\Omega_0)$ coincides with the induced volume form $d\operatorname{vol}_M$ for an appropriate choice of holomorphic $n$-form $\Omega_0$. Equivalently \cite{HL}, one can require that (1) $M$ is a Lagrangian submanifold with respect to the symplectic structure defined by $\omega$, i.e., $f^*(\omega)=0$, and (2) $f^*(\Im\Omega_0)=0$ for an appropriate $\Omega_0$. To state this second condition in a way which does not require that $\Omega_0$ be specified, write an arbitrary holomorphic $n$-form $\Omega$ in the form $\Omega=c\,\Omega_0$, and note that \[ \int_Mf^*(\Omega)=c\int_M f^*(\Omega_0)=c\,(\operatorname{vol} M).\] Thus, the ``appropriate'' $n$-form is given by \[\Omega_0=\frac{(\operatorname{vol} M)\,\Omega}{\int_Mf^*(\Omega)}\] and we can replace condition (2) by \[ \text(2')\qquad f^*\left(\Im\left(\frac{\Omega}{\int_Mf^*(\Omega)} \right)\right)=0.\] (The factor of $\operatorname{vol} M$ is a real constant which can be omitted from this last condition.) Very few explicit examples of special Lagrangian submanifolds are known. (This is largely due to our lack of detailed understanding of the Calabi--Yau metrics themselves.) One interesting class of examples due to Bryant \cite{Br:min} comes from Calabi--Yau manifolds which are complex algebraic varieties defined over the real numbers: the set of real points on the Calabi--Yau manifold is a special Lagrangian submanifold. Another interesting class of examples is the special Lagrangian submanifolds of a K3 surface, which we will discuss in section 6. In general, special Lagrangian submanifolds can be deformed, and there will be a {\it moduli space}\/ which describes the set of all special Lagrangian submanifolds in a given homology class. Given a special Lagrangian $f:M\to Y$ and a deformation of the map $f$, since $f^*(\omega)=0$, the almost-complex structure on $Y$ induces a canonical identification between the normal bundle of $M$ in $Y$ and the tangent bundle of $M$. Thus, the normal vector field defined by the deformation can be identified with a $1$-form on $Y$. The key result concerning the moduli space is due to McLean. \begin{theorem}[McLean \cite{McL}] \quad \begin{enumerate} \item First-order deformations of $f$ are canonically identified with the space of {\em harmonic}\/ $1$-forms on $Y$. \item All first-order deformations of $f:M\to Y$ can be extended to actual deformations. In particular, the moduli space ${\cal M} _{sL}(M,Y)$ of special Lagrangian maps from $M$ to $Y$ is a smooth manifold of dimension $b_1(M)$. \end{enumerate} \end{theorem} \noindent (We have in mind a global structure on ${\cal M}_{sL}(M,Y)$ in which two maps will determine the same point in the moduli space if they differ by a diffeomorphism of $Y$.) McLean also observes that ${\cal M}={\cal M} _{sL}(M,Y)$ admits a natural $n$-form $\Theta$ defined by \[ \Theta(v_1,\dots,v_n)=\int_M\theta_1\wedge\dots\wedge\theta_n\] where $\theta_j$ is the harmonic $1$-form associated to $v_j\in T_{{\cal M} ,f}$. As was implicitly discussed in the last section, the moduli spaces of interest in string theory contain additional pieces of data. To fully account for the ``nonperturbative D-brane effects'' in the physical theory (when $n=3$), the moduli space which we integrate over must include not only the choice of special Lagrangian submanifold, but also a choice of flat $\operatorname{U}(1)$-bundle on that manifold. If we pick a point $b$ on a manifold $M$, then the space of flat $\operatorname{U}(1)$-bundles on $M$ is given by \[\operatorname{Hom}(\pi_1(M,b),\operatorname{U}(1))\cong H^1(M,{\Bbb R})/H^1(M,{\Bbb Z}).\] Thus, if we construct a universal family for our special Lagrangian submanifold problem, i.e., a diagram \[\begin{array}{cc} \phantom{\stackrel{f}{\longrightarrow}\quad} {\cal U} \quad \stackrel{f}{\longrightarrow} & Y\\[3pt] \vcenter{\llap{$\scriptstyle p$}}\big\downarrow&\\[4pt] {\cal M}_{sL}(M,Y)&\\ \end{array}\] with the property that the fibers of $p$ are diffeomorphic to $M$ and $f|_{p^{-1}(m)}$ is the map labeled by $m$, and if $p$ has a section $s:{\cal M}_{sL}(M,Y)\to{\cal U}$, then we can define a moduli space which includes the data of a flat $\operatorname{U}(1)$-bundle by \[{\cal M}_D(M,Y):=R^1p_*{\Bbb R}_{\cal U}/R^1p_*{\Bbb Z}_{\cal U},\] which at each point $m\in{\cal M}_{sL}(M,Y)$ specializes to \[H^1(p^{-1}(m),{\Bbb R})/H^1(p^{-1}(m),{\Bbb Z})\cong \operatorname{Hom}(\pi_1(p^{-1}(m),s(m)),\operatorname{U}(1)).\] (In the case $n=3$, this is the ``D-brane'' moduli space, which motivates our notation.) Note that this space fibers naturally over ${\cal M}_{sL}(M,Y)$, and that there is a section of the fibration, given by the trivial $\operatorname{U}(1)$-bundles. Both the base and the fiber of the fibration ${\cal M}_D(M,Y)\to{\cal M}_{sL}(M,Y)$ have dimension $b_1(M)$, and the fibers are real tori. In fact, we expect from the physics that there will be a family of complex structures on ${\cal M}_D(M,Y)$ making it into a complex manifold of complex dimension $b_1(M)$. The real tori should roughly correspond to subspaces obtained by varying the arguments of the complex variables while keeping their norms fixed. It is expected from the physics that the complex structure should depend on the choice of both a Ricci-flat metric on $Y$ and also on an auxiliary harmonic $2$-form $B$. (This would make ${\cal M}_D(M,Y)$ into a ``complexification'' of the moduli space ${\cal M}_{sL}(M,Y)$ as mentioned in the introduction.) It it not clear at present precisely how those complex structures are to be constructed, although in the case $b_1(M)=n$, a method is sketched in \cite{SYZ} for producing an asymptotic formula for the Ricci-flat metric which would exhibit the desired dependence on $g_{ij}$ and $B$, and the first term in that formula is calculated.\footnote{In the language of \cite{SYZ}, the ``tree-level'' metric on the moduli space is computed, but the instanton corrections to that tree-level metric are left unspecified.} The complex structure could in principle be inferred from the metric if it were known. Motivated by the Strominger--Yau--Zaslow analysis, we now turn our attention to the case in which $M$ is an $n$-torus. The earliest speculations that the special Lagrangian $n$-tori might play a distinguished r\^ole in studying Calabi--Yau manifolds were made by McLean \cite{McL}, who pointed out that if $M=T^n$ then the deformations of $M$ should locally foliate $Y$. (There should be no self-intersections nearby since the harmonic $1$-forms corresponding to the first-order deformations are expected to have no zeros if the metric on the torus is close to being flat.) McLean speculated that---by analogy with the K3 case where such elliptic fibrations are well-understood---if certain degenerations were allowed, the deformations of $M$ might fill out the whole of $Y$. We formulate this as a conjecture (essentially due to McLean). \begin{conjecture}\label{conj:one} Suppose that $f:T^n\to Y$ is a special Lagrangian $n$-torus. Then there is a natural compactification $\overline{{\cal M}}_{sL}(T^n,Y)$ of the moduli space ${\cal M}_{sL}(T^n,Y)$ and a proper map $g:Y\to\overline{{\cal M}}_{sL}(T^n,Y)$ such that \[\begin{array}{ccc} g^{-1}({\cal M}_{sL}(M,Y)) & \hookrightarrow & Y \\[4pt] \vcenter{\llap{$\scriptstyle g$}}\big\downarrow&&\\[4pt] {\cal M}_{sL}(M,Y)&\\ \end{array}\] is a universal family of Lagrangian $n$-tori in the same homology class as $f$. \end{conjecture} \begin{definition} When the properties in conjecture \ref{conj:one} hold, we say that $Y$ has a {\em special Lagrangian $T^n$-fibration}. \end{definition} It is not clear at present what sort of structure should be required of $\overline{{\cal M}}_{sL}(T^n,Y)$: perhaps it should be a manifold with corners,\footnote{This possibility is suggested by the structure of toric varieties, the moment maps for which express certain complex manifolds as $T^n$-fibrations over manifolds with corners (compact convex polyhedra).} or perhaps some more exotic singularities should be allowed in the compactification. We will certainly want to require that the complex structures extend to the compactification, and that the section of the fibration extend to a map $\overline{{\cal M}}_{sL}(T^n,Y)\to Y$. The mirror symmetry analysis of \cite{SYZ} as reviewed in the previous section suggests that the family of dual tori ${\cal M}_D(T^n,Y)$ can also be compactified, resulting in a space which is itself a Calabi--Yau manifold. We formalize this as a conjecture as well. \begin{conjecture}\label{conj:two} The family ${\cal M}_D(T^n,Y)$ of dual tori over ${\cal M}_{sL}(T^n,Y)$ can be compactified to a manifold $X$ with a proper map $\gamma:X\to\overline{{\cal M}}_{sL}(T^n,Y)$, such that $X$ admits metrics with $\operatorname{SU}(n)$ holonomy for which the fibers of $\gamma|_{\gamma^{-1}({\cal M}_{sL}(T^n,Y))}$ are special Lagrangian $n$-tori. Moreover, the fibration $\gamma$ admits a section $\tau:\overline{{\cal M}}_{sL}(T^n,Y)\to X$ such that $\tau({\cal M}_{sL}(T^n,Y))\subset {\cal M}_D(T^n,Y)\subset X$ is the zero-section. \end{conjecture} \noindent It seems likely that for an appropriate holomorphic $n$-form $\Omega_0$ on $X$, the pullback $\tau^*(\Omega_0)$ will coincide with McLean's $n$-form $\Theta$ when restricted to ${\cal M}_{sL}(T^n,Y)$. The most accessible portion of these conjectures would be the following: \begin{subconjecture} The family ${\cal M}_D(T^n,Y)$ of dual tori over ${\cal M}_{sL}(T^n,Y)$ admits complex structures and Ricci-flat K\"ahler metrics. In particular, it has a nowhere vanishing holomorphic $n$-form. \end{subconjecture} Strominger, Yau and Zaslow have obtained some partial results concerning this subconjecture, for which we refer the reader to \cite{SYZ}. It appears, for example, that the construction of the complex structure on the D-brane moduli space should be local around each torus in the torus fibration. \section{Mathematical consequences of mirror symmetry} \label{s:characterizations} There is by now quite a long history of extracting mathematical statements from the physical notion of mirror symmetry. Many of these work in arbitrary dimension, where there is evidence in physics for mirror symmetry among conformal field theories \cite{gp,higherD}.\footnote{In low dimension where a string-theory interpretation is possible, this would become the ``classical'' mirror symmetry which one would also want to extend to a ``quantum'' mirror symmetry if possible.} In this section, we review two of those mathematical statements, presented here as definitions. As the discussion is a bit technical, some readers may prefer to skip to the next section, where we formulate our new definition of {\it geometric mirror pairs}\/ inspired by the Strominger--Yau--Zaslow analysis. Throughout this section, we let $X$ and $Y$ be Calabi--Yau manifolds of dimension $n$. The first prediction one extracts from physics about a mirror pair is a simple equality of Hodge numbers. \begin{definition} We say that the pair $(X,Y)$ {\em passes the topological mirror test}\/ if $h^{n-1,1}(X)=h^{1,1}(Y)$ and $h^{1,1}(X)=h^{n-1,1}(Y)$. \end{definition} Many examples of pairs passing this test are known; indeed, the observation of this ``topological pairing'' in a class of examples was one of the initial pieces of evidence in favor of mirror symmetry \cite{CLS}. Subsequent constructions of Batyrev and Borisov \cite{batyrev:mirror,borisov,BB:dual} show that all Calabi--Yau complete intersections in toric varieties belong to pairs which pass this topological mirror test. For simply-connected Calabi--Yau threefolds, the Hodge numbers $h^{1,1}$ and $h^{n-1,1}$ determine all of the others, but in higher dimension there are more. Na\"{\i}vely one expects to find that $h^{p,q}(X)=h^{n-p,q}(Y)$. However, as was discovered by Batyrev and collaborators \cite{BatD,BB:Hodge}, the proper interpretation of the numerical invariants of the physical theories requires a modified notion of ``string-theoretic Hodge numbers'' $h^{p,q}_{st}$; once this modification has been made, these authors show that $h^{p,q}_{st}(X)=h^{n-p,q}_{st}(Y)$ for the Batyrev--Borisov pairs $(X,Y)$ of complete intersections in toric varieties. The class of pairs for which this modification is needed includes some of those given by the Greene--Plesser construction \cite{gp} for which mirror symmetry of the conformal field theories has been firmly established in physics, so it would appear that this modification is truly necessary for a mathematical interpretation of mirror symmetry. Hopefully, it too will follow from the geometric characterization being formulated in this paper. Going beyond the simple topological properties, a more precise and detailed prediction arises from identifying the quantum cohomology of one Calabi--Yau manifold with the geometric variation of Hodge structure of the mirror partner (in the case that the Calabi--Yau manifolds have no holomorphic $2$-forms). We will discuss this prediction in considerable detail, in order to ensure that this paper has self-contained statements of the conjectures being proposed within it (particularly the ones in sections \ref{s:geom} and \ref{s:filtration} below relating the ``old'' and ``new'' mathematical versions of mirror symmetry). To formulate this precise prediction, let $X$ be a Calabi--Yau manifold with $h^{2,0}(X)=0$, and let $\widetilde{\cal M}^{sc}_{CFT}(X)$ be the moduli space of triples $(g_{ij},B,{\cal J})$ modulo diffeomorphism, where ${\cal J}$ is a complex structure for which the metric $g_{ij}$ is K\"ahler. The map $\widetilde{\cal M}^{sc}_{CFT}(X)\to{\cal M}^{sc}_{CFT}(X)$ is finite-to-one, so this is another good approximation to the conformal field theory moduli space. Moreover, there is a natural map $\widetilde{\cal M}^{sc}_{CFT}(X)\to{\cal M}_{cx}(X)$ to the moduli space of complex structures on $X$, whose fiber over ${\cal J}$ is ${\cal K}_{{\Bbb C}}(X_{{\cal J}})/\operatorname{Aut}(X_{{\cal J}})$, where \[{\cal K}_{{\Bbb C}}(X_{{\cal J}})=\{B+i\,\omega\in {\cal H}^2(X,{\Bbb C}/{\Bbb Z})\ |\ \omega\in{\cal K}_{{\cal J}}\}\] is the {\it complexified K\"ahler cone\footnote{We are following the conventions of \cite{icm} rather than those of \cite{compact,beyond}.} of $X_{{\cal J}}$}\/ (${\cal K}_{{\cal J}}$ being its usual K\"ahler cone), and $\operatorname{Aut}(X_{{\cal J}})$ is the group of holomorphic automorphisms of $X_{{\cal J}}$. The moduli space of complex structures ${\cal M}_{cx}(X)$ has a variation of Hodge structure defined on it which is of geometric origin: roughly speaking, one takes a universal family $\pi:{\cal X}\to {\cal M}_{cx}(X)$ over the moduli space and constructs a variation of Hodge structure on the local system $R^n\pi_*{\Bbb Z}_{\cal X}$ by considering the varying Hodge decomposition of $H^n(X_{{\cal J}},{\Bbb C})$. The local system gives rise to a holomorphic vector bundle ${\cal F}:=(R^n\pi_*{\Bbb Z}_{\cal X})\otimes \O_{{\cal M}_{cx}(X)}$ with a flat connection $\nabla:{\cal F}\to\Omega^1_{{\cal M}_{cx}(X)}\otimes{\cal F}$ (whose flat sections are the sections of the local system), and the varying Hodge decompositions determine the {\it Hodge filtration}\/ \[{\cal F}={\cal F}^0\supset{\cal F}^1\supset\dots\supset{\cal F}^n\supset\{0\},\] a filtration by holomorphic subbundles defined by \[{\cal F}^p|_{{\cal J}}=H^{n,0}(X_{{\cal J}})\oplus\dots\oplus H^{p,n-p}(X_{{\cal J}}),\] which is known to satisfy the {\it Griffiths transversality property}\/ \[\nabla({\cal F}^p)\subset\Omega^1_{{\cal M}_{cx}(X)}\otimes{\cal F}^{p-1}.\] Conversely, given the bundle with flat connection and filtration, the complexified local system $R^n\pi_*{\Bbb C}_{\cal X}$ can be recovered by taking (local) flat sections, and the Hodge structures can be reconstructed from the filtration. The original local system of ${\Bbb Z}$-modules is however additional data, and cannot be recovered from the bundle, connection and filtration alone. The moduli space of complex structures ${\cal M}_{cx}(X)$ can be compactified to a complex space $\overline{\cal M}$, to which the bundles ${\cal F}^p$ and the connection $\nabla$ extend; however, the extended connection $\nabla$ acquires {\it regular singular points}\/ along the boundary ${\cal B}$, which means that it is a map \[\nabla:{\cal F}\to\Omega^1(\log{\cal B})\otimes{\cal F}.\] The residues of $\nabla$ along boundary components describe the {\it monodromy transformations}\/ about those components, the same monodromy which defines the local system. At normal crossings boundary points there is always an associated {\it monodromy weight filtration}, which we take to be a filtration on the homology group $H_n(X)$. The data of the flat connection and the Hodge filtration are encoded in the conformal field theory on $X$ (at least for a sub-Hodge structure containing ${\cal F}^{n-1}$).\footnote{Note that ${\cal F}^n$ appears directly in the conformal field theory, and ${\cal F}^{n-1}/{\cal F}^n$ appears as a class of marginal operators in the conformal field theory. Thus, the conformal field theory contains {\it at least}\/ as much of the Hodge-theoretic data as is described by the smallest sub-Hodge structure containing ${\cal F}^{n-1}$, and quite possibly more.} Since mirror symmetry reverses the r\^oles of base and fiber in the map \[\widetilde{\cal M}^{sc}_{CFT}(X)\to{\cal M}_{cx}(X),\] one of the predictions of mirror symmetry will be an isomorphism between this structure and a similar structure on ${\cal K}_{{\Bbb C}}(X_{{\cal J}})/\operatorname{Aut}(X_{{\cal J}})$. In fact, the conformal field theory naturally encodes a variation of Hodge structure on ${\cal K}_{{\Bbb C}}(X_{{\cal J}})/\operatorname{Aut}(X_{{\cal J}})$. To describe this mathematically, we must choose a {\it framing}, which is a choice of cone \[\sigma={\Bbb R}_+e^1+\dots+{\Bbb R}_+e^r\subset H^2(X,{\Bbb R})\] which is generated by a basis $e^1$, \dots, $e^r$ of $H^2(X,{\Bbb Z})/\text{torsion}$ and whose interior is contained in the K\"ahler cone of $X$. The complexified K\"ahler part of the semiclassical moduli space then contains as an open subset the space \[{\cal M}_A(\sigma):=(H^2(X,{\Bbb R})+i\sigma)/H^2(X,{\Bbb Z}),\] elements of which can be expanded in the form $\sum\left(\frac1{2\pi i}\log q_j\right)e^j$, leading to the alternate description \[{\cal M}_A(\sigma)=\{(q_1,\dots,q_r)\ |\ 0<|q_j|<1\}.\] The desired variation of Hodge structure will be defined on a partial compactification of this space, namely \[\overline{{\cal M}}_A(\sigma):=\{(q_1,\dots,q_r)\ |\ 0\le|q_j|<1\},\] which has a distinguished boundary point $\vec{0}=(0,\dots,0)$. The ingredients we need to define the variation of Hodge structure are the {\it fundamental Gromov--Witten invariants\footnote{These can be defined using techniques from symplectic geometry \cite{Ruan,McDSal,RuanTian} or from algebraic geometry \cite{KontMan,Kont,BehMan,BehFant,Beh,LiTian}.} of $X$}, which are trilinear maps \[\Phi^0_\eta:H^*(X,{\Bbb Q})\oplus H^*(X,{\Bbb Q})\oplus H^*(X,{\Bbb Q})\to{\Bbb Q}.\] Heuristically, when $A$, $B$ and $C$ are integral classes, $\Phi^0_\eta(A,B,C)$ should be the number of generically injective\footnote{We have built the ``multiple cover formula'' \cite{tftrc,manin,voisin} into our definitions.} holomorphic maps $\psi: \C\P^1\to X$ in class $\eta$, such that $\psi(0)\in Z_A$, $\psi(1)\in Z_B$, $\psi(\infty)\in Z_C$ for appropriate cycles $Z_A$, $Z_B$, $Z_C$ Poincar\'e dual to the classes $A$, $B$, $C$, respectively. (The invariants vanish unless $\operatorname{deg} A+\operatorname{deg} B+\operatorname{deg} C=2n$.) {}From these invariants we can define the {\it Gromov--Witten maps} $ \Gamma_\eta:H^k(X)\to H^{k+2}(X) $ by requiring that \[\Gamma_\eta(A)\cdot B|_{[X]}=\frac{\Phi^0_\eta(A,B,C)}{\eta\cdot C}\] for $B\in H^{2n-k-2}(X)$, $C\in H^2(X)$. (This is independent of the choice of $C$.) These invariants are usually assembled into the ``quantum cohomology ring'' of $X$, but here we present this structure in the equivalent form of a variation of Hodge structure over $\overline{{\cal M}}_A(\sigma)$ degenerating along the boundary. To do so, we define a holomorphic vector bundle ${\cal E}:=\left(\bigoplus H^{\ell,\ell}(X)\right)\otimes \O_{\overline{{\cal M}}_A(\sigma)},$ and a flat\footnote{The flatness of this connection is equivalent to the associativity of the product in quantum cohomology.} connection $\nabla:{\cal E}\to\Omega^1_{\overline{\cal M}}(\log{\cal B})\otimes{\cal E}$ with regular singular points along the boundary ${\cal B}= \overline{{\cal M}}_A(\sigma)-{\cal M}_A(\sigma)$ by the formula\footnote{I am indebted to P. Deligne for advice \cite{deligne} which led to this form of the formula.} \[\nabla:=\frac1{2\pi i}\,\left( \sum_{j=1}^r d\mskip0.5mu\log q_j\otimes\operatorname{ad}(e^j)+ \sum_{0\ne\eta\in H_2(X,{\Bbb Z})} d\mskip0.5mu\log\left(\frac1{1-q^{\eta}}\right)\otimes\Gamma_\eta \right)\] where $q^\eta=\prod q_j^{e^j(\eta)}$, and where $\operatorname{ad}(e^j):H^k(X)\to H^{k+2}(X)$ is the adjoint map of the cup product pairing, defined by $\operatorname{ad}(e^j)(A) = e^j\cup A$. We also define a ``Hodge filtration'' \[{\cal E}^p:=\left(\bigoplus_{0\le\ell\le n-p} H^{\ell,\ell}(X)\right)\otimes\O_{\overline{{\cal M}}_A(\sigma)},\] which satisfies $\nabla({\cal E}^p)\subset \Omega^1_{\overline{{\cal M}}}(\log{\cal B}) \otimes{\cal E}^{p-1}.$ This describes a structure we call the {\em framed $A$-variation of Hodge structure with framing $\sigma$}. To be a bit more precise, we should refer to this as a ``formally degenerating variation of Hodge structure,'' since the series used to define $\nabla$ is only formal. (More details about such structures can be found in \cite{parkcity}; cf.~also \cite{deligne}.) There are also some subtleties about passing from a local system of complex vector spaces to a local system of ${\Bbb Z}$-modules which we shall discuss in section \ref{s:filtration} below. The residues of $\nabla$ along the boundary components $q_j=0$ are the adjoint maps $\operatorname{ad}(e^j)$; the corresponding monodromy weight filtration at $\vec0$ is simply \[H_{0,0}(X)\subseteq H_{0,0}(X)\oplus H_{1,1}(X) \subseteq\dots\subseteq(H_{0,0}(X)\oplus\dots\oplus H_{n,n}(X)).\] Under mirror symmetry, this maps to the geometric monodromy weight filtration at an appropriate ``large complex structure limit'' point in $\overline{{\cal M}}_{cx}$ (see \cite{predictions} and references therein). Note that the class of the $0$-cycle is the monodromy-invariant class in $H_{even}(X)$; thus, its mirror $n$-cycle will be the monodromy-invariant class in $H_n(Y)$. Although the choice of a ``framing'' may look unnatural, the relationship between different choices of framing is completely understood \cite{compact} (modulo a conjecture about the action of the automorphism group on the K\"ahler cone). Varying the framing corresponds to varying which boundary point in the moduli space one is looking at, possibly after blowing up the original boundary of the moduli space in order to find an appropriate compactification containing the desired boundary point. We finally come to the definition which contains our precise Hodge-theoretic mirror prediction from physics. \begin{definition} \label{def:HT} Let $X$ and $Y$ be Calabi--Yau manifolds with $h^{2,0}(X)=h^{2,0}(Y)=0$. The pair $(X,Y)$ {\em passes the Hodge-theoretic mirror test}\/ if there exists a partial compactification $\overline{{\cal M}}_{cx}(Y)$ of the complex structure moduli space of $Y$, a neighborhood $U\subset\overline{{\cal M}}_{cx}(Y)$ of a boundary point $P$ of $\overline{{\cal M}}_{cx}(Y)$, a framing $\sigma$ for $H^2(X)$, and a ``mirror map'' $\mu:U\to{\cal M}_A(\sigma)$ mapping $P$ to $\vec0$ such that $\mu^*$ induces an isomorphism between ${\cal E}^{n-1}$ and ${\cal F}^{n-1}$ which extends to an isomorphism between sub-variations of Hodge structure of the $A$-variation of Hodge structure with framing $\sigma$, and the geometric formally degenerating variation of Hodge structure at $P$. \end{definition} The restriction to a sub-variation of Hodge structure (which occurs only when the dimension of the Calabi--Yau manifold is greater than three) seems to be necessary in order to get an integer structure on the local system compatible with the complex variation of Hodge structure. (We will return to this issue in section \ref{s:filtration}.) It seems likely that this is related to the need to pass to ``string-theoretic Hodge numbers,'' which may actually be measuring the Hodge numbers of the appropriate sub-Hodge structures. The property described in the Hodge-theoretic mirror test can be recast in terms of using the limiting variation of Hodge structure on $Y$ to make predictions about enumerative geometry of holomorphic rational curves on $X$. In this sense, there is a great deal of evidence in particular cases (see \cite{predictions,higherD} and the references therein). There are also some specific connections which have been found between the variations of Hodge structure associated to mirror pairs of theories \cite{summing}, as well as a recent theorem \cite{givental} which proves that the expected enumerative properties hold for an important class of Calabi--Yau manifolds. Note that if $(X,Y)$ passes the Hodge-theoretic mirror test in both directions, then it passes the topological mirror test (essentially by definition, since the dimensions of the moduli spaces are given by the Hodge numbers $h^{1,1}$ and $h^{n-1,1}$). \section{Geometric mirror pairs}\label{s:geom} We now wish to translate the Strominger--Yau--Zaslow analysis into a definition of {\it geometric mirror pairs}\/ $(X,Y)$, which we formulate in arbitrary dimension. (As mentioned earlier, the arguments of \cite{SYZ} cannot be applied to conclude that all mirror pairs arise in this way, but it seems reasonable to suppose that a T-duality argument---applied to conformal field theories only---would continue to hold.) The most straightforward such definition would say that $X$ is the compactification of the complexified moduli space of special Lagrangian $n$-tori on $Y$. However, as indicated by our conjectures of section \ref{s:SLAG}, at present we do not have adequate technical control over the compactification to see that it is a Calabi--Yau manifold. So we make instead an indirect definition, motivated by the following observation: if we had such a compactified moduli space $X$, then for generic $x\in X$ there would be a corresponding special Lagrangian $n$-torus $T_x\subset Y$, and we could define an {\em incidence correspondence} \[ Z=\text{closure of}\ \{(x,y)\in X\times Y\ |\ y\in T_x\}. \] By definition, the projection $Z\to X$ would have special Lagrangian $n$-tori as generic fibers. As we saw earlier, from the analysis of \cite{SYZ} it is expected that generic fibers of the other projection $Z\to Y$ will also be special Lagrangian $n$-tori. Furthermore, we should expect that as we vary the metrics on $X$ and on $Y$, the fibrations by special Lagrangian $n$-tori can be deformed along with the metrics. (In fact, it is these dependencies on parameters which should lead to a ``mirror map'' between moduli spaces.) Thus, we will formulate our definition using a {\it family}\/ of correspondences depending on $t\in U$ for some (unspecified) parameter space $U$. \begin{definition} A pair of Calabi--Yau manifolds $(X,Y)$ is a {\em geometric mirror pair}\/ if there is a parameter space $U$ such that for each $t\in U$ there exist \begin{enumerate} \item a correspondence $Z_t\subset(X\times Y)$ which is the closure of a submanifold of dimension $3n$, \item maps $\tau_t:X\to Z_t$ and $\widetilde\tau_t:Y\to Z_t$ which serve as sections for the projection maps $Z_t\to X$ and $Z_t\to Y$, respectively, \item a Ricci-flat metric $g_{ij}(t)$ on $X$ with respect to which generic fibers of the projection map $Z_t\to Y$ are special Lagrangian $n$-tori, and \item a Ricci-flat metric $\widetilde g_{ij}(t)$ on $Y$ with respect to which generic fibers of the projection map $Z_t\to X$ are special Lagrangian $n$-tori. \end{enumerate} Moreover, for generic $z\in Z_t$, the fibers through $z$ of the two projection maps must be canonically dual as tori (with origins specified by $\tau_t$ and $\widetilde \tau_t$). \end{definition} In a somewhat stronger form of the definition, we might require that $U$ be sufficiently large so that the images of the natural maps $U\to{\cal M}_{Ric}(X)$ and $U\to{\cal M}_{Ric}(Y)$ to the moduli spaces of Ricci-flat metrics on $X$ and on $Y$ are open subsets of the respective moduli spaces. It is too much to hope that these maps would be surjective. The best picture we could hope for, in fact, would be a diagram of the form \[ {\cal M}_{Ric}(X) \supseteq U_X \stackrel{\pi_X}{\twoheadleftarrow\joinrel\relbar} U \stackrel{\pi_Y}{\relbar\joinrel\twoheadrightarrow} U_Y \subseteq {\cal M}_{Ric}(Y) \] in which $U_X\subseteq{\cal M}_{Ric}(X)$ and $U_Y\subseteq{\cal M}_{Ric}(Y)$ are open subsets (near certain boundary points in a compactification and contained within the set of metrics for which the semiclassical approximation is valid). The fibers of $\pi_X$ will have dimension $h^{1,1}(X)$, and if the induced map is the mirror map each fiber of $\pi_X$ must essentially be the set of $B$-fields on $X$, i.e., it must be a deformation of the real torus $H^2(X,{\Bbb R}/{\Bbb Z})$. This is compatible with the approximate formula\footnote{The ``tree-level'' formula given in \cite{SYZ} is subject to unspecified instanton corrections.} in \cite{SYZ} for a family of metrics on $Y$, produced by varying the $B$-field on $X$. We expect that geometric mirror symmetry will be related to the earlier mathematical mirror symmetry properties in the following way. \begin{conjecture}\label{conj:three} If $(X,Y)$ is a geometric mirror pair, then the parameter space $U$ and the data in the definition of the geometric mirror pair can be chosen so that \begin{enumerate} \item\label{en:one} $(X,Y)$ passes the topological mirror test\footnote{Part (\ref{en:one}) is a consequence of part (\ref{en:four}) if $h^{2,0}(X)=h^{2,0}(Y)=0$.}, \item $\pi_X:U\to{\cal M}_{Ric}(X)$ lifts to a generically finite map $\widetilde\pi_X:U\to\widetilde U_X\subseteq{\cal M}^{sc}_{CFT}(X)$, \item $\pi_Y:U\to{\cal M}_{Ric}(Y)$ lifts to a generically finite map $\widetilde\pi_Y:U\to\widetilde U_Y\subseteq{\cal M}^{sc}_{CFT}(Y)$, and \item\label{en:four} if $h^{2,0}(X)=h^{2,0}(Y)=0$, then there are boundary points $P\in\overline{\cal M}_{cx}(Y)$, $P'\in\overline{\cal M}_{cx}(X)$ and framings $\sigma$ of $H^2(X)$ and $\sigma'$ of $H^2(Y)$ with partial compactifications $\overline{\widetilde U}_X\subset{\cal M}_A(\sigma)\times\overline{\cal M}_{cx}(X)$ and $\overline{\widetilde U}_Y\subset\overline{\cal M}_{cx}(Y)\times{\cal M}_A(\sigma')$ such that the composite map $(\widetilde\pi_X)_*(\widetilde\pi_Y)^*$ extends to a map $\mu^{-1}\times\mu'$ which consists of mirror maps in both directions (in the sense of definition \ref{def:HT}). In particular, $(X,Y)$ passes the Hodge-theoretic mirror test. \end{enumerate} \end{conjecture} \noindent Even in the case that $h^{2,0}(X)\ne0$, there is an induced map $(\widetilde\pi_X)_*(\widetilde\pi_Y)^*$ which should coincide with the mirror map between the moduli spaces. \bigskip If $X$ has several birational models $X^{(j)}$, then all of the semiclassical moduli spaces ${\cal M}^{sc}_{CFT}(X^{(j)})$ give rise to a common conformal field theory moduli space (see \cite{AGM}, or for a more mathematical account, \cite{beyond}). If we follow a path between the large radius limit points of two of these models, and reinterpret that path in the mirror moduli space, we find a path which leads from one large complex structure limit point of ${\cal M}_{cx}(Y)$ to another. On the other hand, the calculation of \cite{small} shows that the homology class of the torus\footnote{Recall that this is the monodromy-invariant cycle.} in a special Lagrangian $T^n$-fibration does not change when we move from one of these regions of ${\cal M}_{cx}(Y)$ to another. Thus, the moduli space of special Lagrangian $T^n$'s themselves must change as we move from region to region. It will be interesting to investigate precisely how this change comes about. \section{Mirror cohomology and the weight filtration} \label{s:filtration} The ``duality'' transformation which links the two members $X$ and $Y$ of a geometric mirror pair does not induce any obvious relationship between $H^{1,1}(X)$ and $H^{n-1,1}(Y)$, so it may be difficult to imagine how the topological mirror test can be passed by a geometric mirror pair. However, at least for a restricted class of topological cycles, such a relationship {\it can}\/ be found, as part of a more general relationship between certain subspaces of $H_{even}(X)$ and $H_n(Y)$. Fix a special Lagrangian $T^n$-fibration on $Y$ with a special Lagrangian section, and consider $n$-cycles $W\subset Y$ with the property that $W$ is the closure of a submanifold $W_0$ whose intersection with each nonsingular $T^n$ in the fibration is either empty, or a sub-torus of dimension $n-k$ (for some fixed integer $k\le n$). That is, we assume that $W$ can be generically described as a $T^{n-k}$-bundle over a $k$-manifold, with the $T^{n-k}$'s linearly embedded in fibers of the given $T^n$-fibration. We call such $n$-cycles {\it pure}. For any pure $n$-cycle $W\subset Y$, there is a {\it T-dual cycle}\footnote{In physics, when a T-duality transformation is applied to a real torus, a D-brane supported on a sub-torus is mapped to a D-brane supported on the ``dual'' sub-torus (of complementary dimension); this can be mathematically identified as the annihilator. Here, we apply this principle to a family of sub-tori within a family of tori.} $W^\vee\subset X(=\overline{\cal M}_D(T^n,Y))$ defined as the closure of an $n$-manifold $W_0^\vee$ satisfying \[W_0^\vee\cap (T^n)^*= \begin{cases} \text{the annihilator of } W\cap T^n \text{ in } (T^n)^* &\text{if } W\cap T^n\ne\emptyset\\ \emptyset&\text{otherwise} \end{cases} \] for all smooth fibers $(T^n)^*$ in the dual fibration. Since the annihilator of an $(n-k)$-torus is a $k$-torus, we see that $W^\vee$ is generically described as a $T^k$-bundle over a $k$-manifold, and so it defines a class in $H_{2k}(X)$. This is our relationship between the space of pure $n$-cycles on $Y$, and the even homology on $X$. Taking the T-duality statements from physics very literally, we are led to the speculation that pure special Lagrangian $n$-cycles have as their T-duals certain algebraic cycles on $X$; moreover, the moduli spaces containing corresponding cycles should be isomorphic.\footnote{As the referee has pointed out, our ``purity'' condition is probably too strong to be preserved under deformation, but one can hope that all nearby deformations of a (pure) special Lagrangian $n$-cycle are reflected in deformations of the corresponding algebraic cycle.} (Roughly speaking, the $T^k$-fibration on the corresponding algebraic $k$-cycle should be given by holding the norms of some system of complex coordinates on the $k$-cycle fixed, while varying their arguments.) The simplest cases of this statement we have already encountered in the Strominger--Yau--Zaslow discussion: the special Lagrangian $n$-cycles which consist of a single fiber (i.e., $k=0$) are T-dual to the $0$-cycles of length one on $X$, while a special Lagrangian $n$-cycle which is the zero-section of the fibration (i.e., $k=n$) is T-dual to the $2n$-cycle in the fundamental class. This new construction should extend that correspondence between cycles to a broader class (albeit still a somewhat narrow one, since pure cycles are quite special). In fact, the correspondence should be even broader. If we begin with an arbitrary irreducible special Lagrangian $n$-cycle $W$ on $Y$ whose image in ${\cal M}_{sL}(T^n,Y)$ has dimension $k$, then $W$ can be generically described as a bundle of $(n-k)$-manifolds over the image $k$-manifold. The T-dual of such a cycle should be a coherent sheaf ${\cal E}$ on $X$ whose support $Z$ is a complex submanifold of dimension $k$ whose image in ${\cal M}_{sL}(T^n,Y)$ is that same $k$-manifold. Thus, to the homology class of $W$ in $H_n(Y)$ we associate the total homology class in $H_{even}(X)$ of the corresponding coherent sheaf .\footnote{It appears from both the K3 case discussed in the next section, and the analysis of \cite{GHM} that the correct total homology class to use is the Poincar\'e dual of $\operatorname{ch}({\cal E})\sqrt{\operatorname{td} Y}$.} Note that since the support has complex dimension $k$, this total homology class lies in $H_0(X)\oplus H_2(X)\oplus\dots\oplus H_{2k}(X)$. The homology class of the generic fiber of $W$ within $T^n$ should determine the sub-tori whose T-duals would sweep out $Z$; when that homology class is $r$ times a primitive class, the corresponding coherent sheaf should have generic rank $r$ along $Z$. For example, a multi-section of the special Lagrangian $T^n$ fibration which meets the fiber $r$ times should correspond to a coherent sheaf whose support is all of $X$ and whose rank is $r$. We have thus found a mapping from the subspace $H_n^{sL}(Y)$ of $n$-cycles with a special Lagrangian representative, to the subspace $H_{even}^{alg}(Y)$ of homology classes of algebraic cycles (and coherent sheaves). If we consider the Leray filtration on special Lagrangian $n$-cycles on $Y$ \[\S_k:=\{W\in H_n^{sL}(Y)\ |\ \dim(\operatorname{image} W)\le k\},\] then this will map to \[H_0^{alg}(X)\oplus H_2^{alg}(X)\oplus\dots\oplus H_{2k}^{alg}(X)\] (and the pure $n$-cycles on $Y$ will map to homology classes of algebraic cycles on $Y$). But this latter filtration on $H_{even}(X)$ is precisely the monodromy weight filtration of the $A$-variation of Hodge structures on $X$, which should be mirror to the geometric monodromy weight filtration on $Y$!\footnote{This property of the mapping of D-branes has also been observed by Ooguri, Oz and Yin \cite{OOY}.} We are thus led to the following refinement of conjecture \ref{conj:three}. \begin{conjecture}\label{conj:four} If $(X,Y)$ is a geometric mirror pair then there exists a large complex structure limit point $P\in\overline{{\cal M}}_{cx}(Y)$ corresponding to the mirror partner $X$, and a sub-variation of the geometric variation of Hodge structure defined on $H^{sL}_n(Y)^*$ whose monodromy weight filtration at $P$ coincides with the Leray filtration for the special Lagrangian $T^n$-fibration on $Y$. Moreover, under the isomorphism of conjecture \ref{conj:three}, this maps to the sub-variation of the $A$-variation of Hodge structure defined on $H^{alg}_{even}(X)^*$. \end{conjecture} The difficulty in putting an integer structure on the $A$-variation of Hodge structure stems from the fact that $H^{p,p}(X)$ will in general not be generated by its intersection with $H^{2p}(X,{\Bbb Z})$. However, the {\it algebraic}\/ cohomology $H_{even}^{alg}(X)^*$ does not suffer from this problem: its graded pieces are generated by integer $(p,p)$-classes. If conjecture \ref{conj:four} holds, it explains why there is a corresponding sub-variation of the geometric variation of Hodge structure on $Y$, also defined over the integers. We would thus get corresponding local systems over ${\Bbb Z}$ in addition to the isomorphisms of complex variations of Hodge structure. \section{Geometric mirror symmetry for K3 surfaces} The special Lagrangian submanifolds of a K3 surface can be studied directly, thanks to the following fact due to Harvey and Lawson \cite{HL}: given a Ricci-flat metric on a K3 surface $Y$ and a special Lagrangian submanifold $M$, there exists a complex structure on $Y$ with respect to which the metric is K\"ahler, such that $M$ is a complex submanifold of $Y$. This allows us to immediately translate the theory of special Lagrangian $T^2$-fibrations on $Y$ to the standard theory of elliptic fibrations. In this section, we will discuss geometric mirror symmetry for K3 surfaces in some detail. (Some aspects of this case have also been worked out by Gross and Wilson \cite{GW}, who went on to study geometric mirror symmetry for the Voisin-Borcea threefolds of the form $(\mbox{K3}\times T^2)/{\Bbb Z}_2$.) If we fix a cohomology class $\mu\in H^2(Y,{\Bbb Z})$ which is primitive (i.e., $\frac1n\,\mu\not\in H^2(Y,{\Bbb Z})$ for $1<n\in{\Bbb Z}$) and satisfies $\mu\cdot\mu=0$, then for any Ricci-flat metric we can find a compatible complex structure for which $\mu$ has type $(1,1)$ and $\kappa\cdot\mu>0$ ($\kappa$ being the K\"ahler form). The class $\mu$ is then represented by a complex curve, which moves in a one-parameter family, defining the structure of an elliptic fibration. Thus, elliptic fibrations of this sort exist for every Ricci-flat metric on a K3 surface.\footnote{They even exist---although possibly in degenerate form---for the ``orbifold'' metrics which occur at certain limit points of the moduli space: at those points, $\kappa$ is only required to be semi-positive, but by the index theorem $\kappa^\perp$ cannot contain an isotropic vector such as $\mu$, so it is still possible to choose a complex structure such that $\kappa\cdot\mu>0$.} Our conjecture \ref{conj:one} is easy to verify in this case: as is well-known, the base of the elliptic fibration on a K3 surface can be completed to a $2$-sphere, and the resulting map from K3 to $S^2$ is proper. In fact, the possible singular fibers are known very explicitly in this case \cite{kodaira}. To study conjecture \ref{conj:two}, we need to understand the structure of the ``complexified'' moduli space ${\cal M}_D(T^2,Y)$. Since a flat $U(1)$-bundle on an elliptic curve is equivalent to a holomorphic line bundle of degree zero, each point in ${\cal M}_D(T^2,Y)$ has a natural interpretation as such a bundle on some particular fiber of the elliptic fibration. Extending that bundle by zero, we can regard it as a sheaf $\L$ on $Y$, with $\operatorname{supp}({\L})=\operatorname{image}(f)$. We thus identify ${\cal M}_D(T^2,Y)$ as a moduli spaces of such sheaves. Let us briefly recall the facts about the moduli spaces of simple sheaves on K3 surfaces, as worked out by Mukai \cite{Muk:sympl,Muk:bundles}. First, Mukai showed that for any simple sheaf ${\cal E}$ on $Y$, i.e., one without any non-constant endomorphisms, the moduli space $M_{simple}$ is smooth at $[{\cal E}]$ of dimension $\dim\operatorname{Ext}^1({\cal E},{\cal E})=2-\chi({\cal E},{\cal E}).$ (The ``$2$'' in the formula arises from the spaces $\operatorname{Hom}({\cal E},{\cal E})$ and $\operatorname{Ext}^2({\cal E},{\cal E})$, each of which has dimension one, due to the constant endomorphisms in the first case, and their Kodaira--Serre duals in the second case.) Second, Mukai introduced an intersection pairing on $H^{ev}(Y) = H^0(Y) \oplus H^2(Y) \oplus H^4(Y)$ defined by \[(\alpha,\beta,\gamma)\cdot(\alpha',\beta',\gamma')=(\beta\cdot\beta' -\alpha\cdot\gamma'-\gamma\cdot\alpha')|_{[Y]},\] and a slight modification of the usual Chern character $\operatorname{ch}({\cal E})$, defined by \[v({\cal E})=\operatorname{ch}({\cal E})\sqrt{\operatorname{td}(Y)}=((\operatorname{rank}({\cal E}),c_1({\cal E}),\operatorname{rank}({\cal E})+ \frac12(c_1({\cal E})^2-2c_2({\cal E})),\] so that the Riemann--Roch theorem reads \[\chi({\cal E},{\cal F})=v({\cal E})\cdot v({\cal F}).\] In particular, the moduli space ${\cal M}_{simple}(v)$ of simple sheaves with $v({\cal E})=v$ has dimension \[\dim{\cal M}_{simple}(v)=2-\chi({\cal E},{\cal E})=2-v\cdot v.\] In the case of moduli spaces ${\cal M}_{simple}(v)$ of dimension two, Mukai went on to show that whenever the space is compact, it must be a K3 surface. The sheaves $\L$ with support on a curve from our elliptic fibration will have Mukai class $v(\L)=(0,\mu,0)$ for which $v(\L)\cdot v(\L)=\mu\cdot\mu=0$, so the moduli space has dimension two. That is, our moduli space ${\cal M}_D(T^2,Y)$ is contained in ${\cal M}_{simple}(0,\mu,0)$ as an open subset. Our second conjecture will follow if we can show that this latter space is compact, or at least admits a natural compactification. Whether this is true or not could in principle depend on the choice of Ricci-flat metric on $Y$. If we restrict to metrics with the property that $Y$ is algebraic when given the compatible complex structure for which $\mu$ defines an elliptic fibration (this is a dense set within the full moduli space), then techniques of algebraic geometry can be applied to this problem. General results of Simpson \cite{simpson} imply that on an algebraic K3 surface, the set of semistable sheaves with a fixed Mukai vector $v$ forms a projective variety. This applies to our situation with $v=(0,\mu,0)$, and provides the desired compactification. It is to be hoped that compactifications such as this exist even for non-algebraic K3 surfaces. The Mukai class $v=(0,\mu,0)$ should now be mapped under mirror symmetry to the class of a zero-cycle, or the corresponding sheaf $\O_P$; that Mukai class is $(0,0,1)$. In fact, the mirror map known in physics \cite{stringK3} does precisely that: given any primitive isotropic vector $v$ in $H^{ev}(Y)$, there is a mirror map which takes it to the vector $(0,0,1)$. Moreover, it is easy to calculate how this mirror map affects complex structures, by specifying how it affects Hodge structures: if we put a Hodge structure on $H^{ev}(Y)$ in which $H^0$ and $H^4$ have been specified as type $(1,1)$, then the corresponding Hodge structure at the mirror image point has $v^\perp/v$ as its $H^2$. This is {\it precisely}\/ the relationship between Hodge structures on $Y$ and on ${\cal M}_{simple}(v)$ which was found by Mukai \cite{Muk:bundles}! We can thus identify geometric mirror symmetry for K3 surfaces (which associates the moduli spaces of zero-cycles and special Lagrangian $T^2$'s) with the mirror symmetry previously found in physics. It is amusing to note that in establishing this relationship, Mukai used elliptic fibrations and bundles on them in a crucial way. As suggested in the previous section, such a mirror transformation should act on the totality of special Lagrangian $2$-cycles. In fact, it is known that for at least some K3 surfaces, there is a {\it Fourier--Mukai transform}\/ which associates sheaves on ${\cal M}_{simple}(v)$ to sheaves on $Y$ \cite{BBH}. The map between their homology classes is precisely the mirror map.\footnote{One example of this is given by a special Lagrangian section of the $T^2$-fibration, which will map to the class $(1,0,1)$ which is the Mukai vector of the fundamental cycle (i.e., of the structure sheaf) on the mirror.} Thus, proving that there exists such a Fourier--Mukai transform for arbitrary K3 surfaces (even non-algebraic ones) would establish a version of conjecture \ref{conj:four} in this case. \subsection*{Acknowledgments} It is a pleasure to thank Robbert Dijkgraaf, Ron Donagi, Brian Greene, Mark Gross, Paul Horja, Sheldon Katz, Greg Moore, Ronen Plesser, Yiannis Vlassopoulos, Pelham Wilson, Edward Witten, and especially Robert Bryant and Andy Strominger for useful discussions; I also thank Strominger for communicating the results of \cite{SYZ} prior to publication, and the referee for useful remarks on the first version. I am grateful to the Rutgers physics department for hospitality and support during the early stages of this work, to the organizers of the European Algebraic Geometry Conference at the University of Warwick where this work was first presented, and to the Aspen Center for Physics where the writing was completed. This research was partially supported by the National Science Foundation under grant DMS-9401447. \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9608

alg-geom/9608029

en

https://arxiv.org/abs/alg-geom/9608029

alg-geom/9608029

Lisa Jeffrey

Lisa C. Jeffrey (McGill University), Frances C. Kirwan (Oxford University)

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

77 pages, LaTeX version 2.09. This is the text of the revised version which will appear in Annals of Mathematics. An error in Section 2 of the previous version has been corrected

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).

alg-geom

\section{Introduction} Let $n$ and $d$ be coprime positive integers, and define $\mnd$ to be the moduli space of (semi)stable holomorphic vector bundles of rank $n$, degree $d$ and fixed determinant on a compact Riemann surface $\Sigma$. This moduli space is a compact K\"ahler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri 1965 \cite{AB,NS}). The subject of this article is the characterization of the intersection pairings in the cohomology ring\footnote{Throughout this paper all cohomology groups will have complex coefficients, unless specified otherwise.} $H^*(\mnd)$. A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper \cite{AB} on the Yang-Mills equations on Riemann surfaces (where in addition inductive formulas for the Betti numbers of $\mnd$ obtained earlier using number-theoretic methods \cite{DR,HN} were rederived). By Poincar\'{e} duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson \cite{Do} and Thaddeus \cite{T} gave formulas for the intersection pairings between products of these generators in $H^*({\mbox{$\cal M$}}(2,1)) $ (in terms of Bernoulli numbers). Then using physical methods, Witten \cite{tdgr} found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in $H^*(\mnd)$ for general rank $n$. These generalized his (rigorously proved) formulas \cite{qym} for the symplectic volume of $\mnd$: for instance, the symplectic volume of ${\mbox{$\cal M$}}(2,1)$ is given by \begin{equation} \label{svol} {\rm vol}({\mbox{$\cal M$}}(2,1)) = \Bigl (1 - \frac{1}{2^{2g-3}} \Bigr ) \frac{ \zeta(2g-2)}{2^{g-2} \pi^{2g-2}} = \frac{2^{g-1}-2^{2-g}}{(2g-2)!} |B_{2g-2}| \end{equation} where $g$ is the genus of the Riemann surface, $\zeta$ is the Riemann zeta function and $B_{2g-2}$ is a Bernoulli number (see \cite{qym,T,Do}). The purpose of this paper is to obtain a mathematically rigorous proof of Witten's formulas for general rank $n$. Our announcement \cite{JK2} sketched the arguments we shall use, concentrating mainly on the case of rank $n=2$. The proof involves an application of the {\em nonabelian localization principle} \cite{JK1,tdgr}. Let $K$ be a compact connected Lie group with Lie algebra $\liek$, let $(M,\omega)$ be a compact symplectic manifold equipped with a Hamiltonian action of $K$ and suppose that $0$ is a regular value of the moment map $\mu:M \to \lieks$ for this action. One can use equivariant cohomology on $M$ to study the cohomology ring of the reduced space, or symplectic quotient, $\mred = \zloc / K$, which is an orbifold with an induced symplectic form $\omega_0$. In particular it is shown in \cite{Ki1} that there is a natural surjective homomorphism from the equivariant cohomology $H^*_K(M)$ of $M$ to the cohomology $H^*(\mred)$ of the reduced space. For any cohomology class $\eta_0 \in H^*(\mred)$ coming from $\eta \in H^*_K(M)$ via this map, we derived in \cite{JK1} a formula (the residue formula, Theorem 8.1 of \cite{JK1}) for the evaluation $\eta_0[\mred]$ of $\eta_0$ on the fundamental class of $\mred$. This formula involves the data that enter the Duistermaat-Heckman formula \cite{DH}, and its generalization the abelian localization formula \cite{ABMM,BV1,BV2} for the action of a maximal torus $T$ of $K$ on $M$: that is, the set ${\mbox{$\cal F$}}$ of connected components $F$ of the fixed point set $M^T$ of the action of $T$ on $M$, and the equivariant Euler classes $e_F$ of their normal bundles in $M$. Let $\liet$ be the Lie algebra of $T$; then the composition $\mu_T:M \to \liets$ of $\mu:M \to \lieks$ with the natural map from $\lieks$ to $\liets$ is a moment map for the action of $T$ on $M$. In the case when $K=SU(2)$ and the order of the stabilizer in $K$ of a generic point of $\mu^{-1}(0)$ is $n_0$, the residue formula can be expressed in the form \begin{equation} \label{rf2} \eta_0e^{\omega_0}[\mred] = \frac{n_0}{2}{\rm Res}_{X=0} \Bigl( (2X)^2 \sum_{F\in{\mbox{$\cal F$}}_+} h^{\eta}_F(X) dX\Bigr) \end{equation} where the subset ${\mbox{$\cal F$}}_+$ of ${\mbox{$\cal F$}}$ consists of those components $F$ of the fixed point set $M^T$ on which the value taken by the $T$-moment map $\mu_T:M \to \liets \cong {\Bbb R }$ is positive, and for $F\in {\mbox{$\cal F$}}_+$ the inclusion of $F$ in $M$ is denoted by $i_F$ and the meromorphic function $h^{\eta}_F$ of $X\in{\Bbb C }$ is defined by $$h_F^{\eta}(X) = \int_F \frac{i_F^* \eta(X) e^{\overline{\omega}(X)}}{e_F(X)} = e^{\mu_T(F)(X)} \int_F \frac{i_F^*\eta(X) e^{\omega}}{e_F(X)}.$$ when $X\in {\Bbb C }$ has been identified with ${\rm diag}(2\pi i,-2\pi i)X \in \liet\otimes{\Bbb C }$. Here $\overline{\omega}$ is the extension $\omega + \mu$ of the symplectic form $\omega$ on $M$ to an equivariantly closed 2-form, while as before $\omega_0$ denotes the induced symplectic form on $\mred$. Finally ${\rm Res}_{X=0}$ denotes the ordinary residue at $X=0$. The moduli space $\mnd$ was described by Atiyah and Bott \cite{AB} as the symplectic reduction of an infinite dimensional symplectic affine space ${\mbox{$\cal A$}}$ with respect to the action of an infinite dimensional group ${\mbox{$\cal G$}}$ (the {\em gauge group}).\footnote{To obtain his generating functionals, Witten formally applied his version of nonabelian localization to the action of the gauge group on the infinite dimensional space ${\mbox{$\cal A$}}$.} However $\mnd$ can also be exhibited as the symplectic quotient of a finite dimensional symplectic space $M(c)$ by the Hamiltonian action of the finite dimensional group $K=SU(n)$. One characterization of the space $M(c)$ is that it is the symplectic reduction of the infinite dimensional affine space ${\mbox{$\cal A$}}$ by the action of the {\em based} gauge group $\calgo$ (which is the kernel of the evaluation map ${\mbox{$\cal G$}} \to K$ at a prescribed basepoint: see \cite{ext}). Now if a compact group $G$ containing a closed normal subgroup $H$ acts in a Hamiltonian fashion on a symplectic manifold $Y$, then one may \lq\lq reduce in stages'': the space $\mu_H^{-1}(0)/H $ has a residual Hamiltonian action of the quotient group $G/H$ with moment map $\mgh: \mu_H^{-1}(0)/H \to {(\lieg/{\bf h} )}^* $, and $\mu_G^{-1}(0)/G$ is naturally identified as a symplectic manifold with $\mgh^{-1}(0)/(G/H). $ Similarly $M(c)$ has a Hamiltonian action of ${\mbox{$\cal G$}}/{\mbox{$\cal G$}}_0 \cong K$, and the symplectic reduction with respect to this action is identified with the symplectic reduction of ${\mbox{$\cal A$}}$ with respect to the full gauge group ${\mbox{$\cal G$}}$. Our strategy for obtaining Witten's formulas is to apply nonabelian localization to this extended moduli space $M(c)$, which has a much more concrete (and entirely finite-dimensional) characterization described in Section 4 below. Unfortunately technical difficulties arise, because $M(c)$ is both singular and noncompact. The noncompactness of $M(c)$ causes the more serious problems, the most immediate of which is that there are infinitely many components $F$ of the fixed point set $M(c)^T$. These, however, are easy to identify (roughly speaking they correspond to bundles which are direct sums of line bundles), and there are obvious candidates for the equivariant Euler classes of their normal bundles, if the singularities of $M(c)$ are ignored. In the case when $n=2$, for example, a na\"{\i}ve application of the residue formula (\ref{rf2}), with some sleight of hand, would yield $${\rm vol}({\mbox{$\cal M$}}(2,1)) = e^{\omega_0} [{\mbox{$\cal M$}}(2,1)] = (-1)^g {\rm Res}_{X=0} \sum_{j=0}^{\infty} \frac{e^{(2j+1)X}}{2^{g-2}X^{2g-2}}$$ \begin{equation} \label{f3} \phantom{bbbbb} = (-1)^g {\rm Res}_{X=0} \frac{e^{X}}{2^{g-2}X^{2g-2}(1-e^{2X})} = (-1)^{g-1}{\rm Res}_{X=0} \frac{1}{2^{g-1}X^{2g-2}{\rm sinh}(X)}.\end{equation} This does give the correct answer (it agrees with (\ref{svol}) above). However it is far from obvious how this calculation might be justified, since the infinite sum does not converge in a neighbourhood of $0$, where the residue is taken, and indeed the sum of the residues at $0$ of the individual terms in the sum does not converge. These difficulties can be overcome by making use of a different approach to nonabelian localization given recently by Guillemin-Kalkman \cite{GK} and independently by Martin \cite{Ma}. This is made up of two steps: the first is to reduce to the case of a torus action, and the second, when $K=T$ is a torus, is to study the change in the evaluation on the fundamental class of the reduced space $\mu_T^{-1}(\xi)/T$ of the cohomology class induced by $\eta$, as $\xi$ varies in $\liets$. It is in fact an immediate consequence of the residue formula that if $T$ is a maximal torus of $K$ and $\xi\in\liets$ is any regular value sufficiently close to $0$ of the $T$-moment map $\mu_T:M\to \liets$, then the evaluation $\eta_0[\mred]$ of $\eta_0\in H^*(\mred)$ on the fundamental class of $\mred = \zloc/K$ is equal to the evaluation of a related element of $H^*(\mu_T^{-1}(\xi)/T)$ on the fundamental class of the $T$-reduced space $\mu_T^{-1}(\xi)/T$. This was first observed by Guillemin and Kalkman \cite{GK} and by Martin \cite{Ma}, who gave an independent proof which showed that $\eta_0[\mred]$ is also equal to an evaluation on $$\zloc/T = (M_{\liet} \cap \mu_T^{-1}(0))/T$$ where $M_{\liet} = \mu^{-1}(\liet)$. In our situation the space $M_{\liet}$ turns out to be \lq\lq periodic'' in a way which enables us to avoid working with infinite sums except in a very trivial sense. This is done by comparing the results of relating evaluations on $(M_{\liet} \cap \mu_T^{-1}(\xi))/T$ for different values of $\xi$ in two ways: using the periodicity and using Guillemin and Kalkman's arguments, which can be made to work in spite of the noncompactness of $M(c)$. The singularities can be dealt with because $M(c)$ is embedded naturally and equivariantly in a nonsingular space, and integrals over $M(c)$ can be rewritten as integrals over this nonsingular space. In the case when $n=2$ our approach gives expressions for the pairings in $H^*(\mto)$ as residues similar to those in (\ref{f3}) above. When $n>2$ we consider the action of a suitable one-dimensional subgroup $\hat{T}_1$ of $T$, with Lie algebra $\hat{\liet}_1$ say, on the quotient of $\mu^{-1}(\hat{\liet}_1)$ by a subgroup of $T$ whose Lie algebra is a complementary subspace to $\hat{\liet}_1$ in $\liet$. This leads to an inductive formula for the pairings on $H^*(\mnd)$, and thus to expressions for these pairings as iterated residues (see Theorems \ref{mainab} and \ref{t9.6} below, which are the central results of this paper). Witten's formulas, on the other hand, express the pairings as infinite sums over those elements of the weight lattice of $SU(n)$ which lie in the interior of a fundamental Weyl chamber (see Section 2). These infinite sums are difficult to calculate in general, and there is apparently (see \cite{tdgr} Section 5) no direct proof even that they are always zero when the pairings they represent vanish on dimensional grounds. However, thanks to an argument of Szenes (see Proposition \ref{p:sz} below), Witten's formulas can be identified with the iterated residues which appear in our approach. Over the moduli space $\mnd$ there is a natural line bundle ${\mbox{$\cal L$}}$ (the Quillen line bundle \cite{Q}) whose fibre at any point representing a semistable holomorphic bundle $E$ is the determinant line $${\rm det} \bar{\partial} = {\rm det} H^1(\Sigma,E) \otimes {\rm det} H^0(\Sigma,E)^*$$ of the associated $\bar{\partial}$-operator. Our expressions for pairings in $H^*(\mnd)$ as iterated residues, together with the Riemann-Roch formula, lead easily (cf. Section 4 of \cite{Sz}) to a proof of the Verlinde formula for $$\dim H^0(\mnd,{\mbox{$\cal L$}}^k)$$ for positive integers $k$ (proved by Beauville and Laszlo in \cite{BL}, by Faltings in \cite{F}, by Kumar, Narasimhan and Ramanathan in \cite{KNR} and by Tsuchiya, Ueno and Yamada in \cite{TUY}). This paper is organized as follows. In Section 2 we describe the generators for the cohomology ring $H^*(\mnd)$ and Witten's formulas for the intersection pairings among products of these generators. In Section 3 we outline tools from the Cartan model of equivariant cohomology, which will be used in later sections, and the different versions of localization which will be relevant. In Section 4 we recall properties of the extended moduli space $M(c)$, and in Section 5 we construct the equivariant differential forms representing equivariant Poincar\'{e} duals which enable us to rewrite integrals over singular spaces as integrals over ambient nonsingular spaces. Then Section 6 begins the application of nonabelian localization to the extended moduli space, and Section 7 analyses the fixed point sets which arise in this application. Section 8 uses induction to complete the proof of Witten's formulas when the pairings are between cohomology classes of a particular form, Section 9 extends the inductive argument to give formulas for all pairings, and in Section 10 it is shown that these agree with Witten's formulas. Finally as an application Section 11 gives a proof of the Verlinde formula for $\mnd$. We would like to thank the Isaac Newton Institute in Cambridge, the Institute for Advanced Study in Princeton, the Institut Henri Poincar\'{e} and Universit\'{e} Paris VII, the Green-Hurst Institute for Theoretical Physics in Adelaide and the Massachusetts Institute of Technology for their hospitality during crucial phases in the evolution of this paper. We also thank A. Szenes for pointing out an error in an earlier version of the paper: since the original version of this paper was written, Szenes has obtained new results \cite{Sz2} which are closely related to the results given in Section 11 of our paper. { \setcounter{equation}{0} } \section{The cohomology of the moduli space $\mnd$ and Witten's formulas for intersection pairings} \newcommand{\lb}[1]{ {l_{#1} } } \newcommand{\yy}[1]{Y_{#1}} \newcommand{\liner}[1]{L_{#1} } \newcommand{\lambdr}[1]{\Lambda_{#1} } \newcommand{\linestd}{\liner{(\lb{1}, \dots, \lb{n-2} )} } \newcommand{\lambstd}{\lambdr{(\lb{1}, \dots, \lb{n-2} )} } \newcommand{\expsum}[1]{ { (e^{-{#1} } - 1 ) } } \newcommand{\itwopi}{ { 2 \pi i }} \newcommand{\Res}{\res} \newcommand{\indset}{{(l_1, \dots, l_{n-1})} } \newcommand{\indsettwo}{{(l_1, \dots, \dots, l_{n-2})} } \newcommand{\laregint}{ {\lambstd^{reg} } } In order to avoid exceptional cases, we shall assume throughout that the Riemann surface $\Sigma$ has genus $g\geq 2$. A set of generators for the cohomology\footnote{In this paper, all cohomology groups are assumed to be with complex coefficients.} $H^*(\mnd)$ of the moduli space $\mnd$ of stable holomorphic vector bundles of coprime rank $n$ and degree $d$ and fixed determinant on a compact Riemann surface $\Sigma$ of genus $g \ge 2 $ is given in \cite{AB} by Atiyah and Bott. It may be described as follows. There is a universal rank $n$ vector bundle $$ {\Bbb U } \to \Sigma \times \mnd $$ which is unique up to tensor product with the pullback of any holomorphic line bundle on $\mnd$; for definiteness Atiyah and Bott impose an extra normalizing condition which determines the universal bundle up to isomorphism, but this is not crucial to their argument (see \cite{AB}, p. 582). Then by \cite{AB} Proposition 2.20 the following elements of $H^*(\mnd)$ for $2\leq r\leq n$ make up a set of generators: $$ f_r = ([\Sigma], c_r({\Bbb U })), $$ $$ b_r^j = (\alpha_j, c_r({\Bbb U })), $$ $$ a_r = (1, c_r({\Bbb U })). $$ Here, $[\Sigma]$ $ \in H_2(\Sigma)$ and $\alpha_j \in H_1(\Sigma)$ $(j = 1, \dots, 2g)$ form standard bases of $H_2(\Sigma, {\Bbb Z })$ and $H_1(\Sigma, {\Bbb Z })$, and the bracket represents the slant product $H^N(\Sigma \times \mnd) \otimes H_j(\Sigma) \to H^{N-j}(\mnd). $ More generally if $K=SU(n)$ and $Q$ is an invariant polynomial of degree $s$ on its Lie algebra $\liek = su(n)$ then there is an associated element of $H^*(BSU(n))$ and hence an associated element of $H^*(\Sigma \times \mnd)$ which is a characteristic class $Q({\Bbb U })$ of the universal bundle ${\Bbb U }$. Hence the slant product gives rise to classes $$ ([\Sigma],Q({\Bbb U })) \in H^{2s-2}(\mnd), $$ $$ (\alpha_j,Q({\Bbb U })) \in H^{2s-1}(\mnd), $$ and $$ (1,Q({\Bbb U })) \in H^{2s}(\mnd). $$ In particular, letting $\tau_r$ $\in S^r(\lieks)^K$ denote the invariant polynomial associated to the $r$-th Chern class, we recover \begin{equation} \label{1.2} \label{9} f_r = ([\Sigma],\tau_r({\Bbb U })), \end{equation} $$ b_r^j = (\alpha_j,\tau_r ({\Bbb U })) $$ and $$ a_r =(1, \tau_r({\Bbb U })). $$ A special role is played by the invariant polynomial $\tau_2 = - \inpr{\cdot , \cdot}/2 $ on $\liek$ given by the Killing form or invariant inner product. We normalize the inner product as follows for $K = SU(n)$: \begin{equation} \label{1.02} \inpr{X, X} = - {\rm Trace} (X^2)/(4\pi^2). \end{equation} The class $f_2$ associated to $-\inpr{ \cdot, \cdot}/2 $ is the cohomology class of the symplectic form on $\mnd$. As was noted in the introduction, Atiyah and Bott identify $\mnd$ with the symplectic reduction of an infinite dimensional affine space ${\cal A}$ of connections by the action of an infinite dimensional Lie group ${\cal G}$ (the gauge group). They show that associated to this identification there is a natural surjective homomorphism of rings from the equivariant cohomology ring $H^*_{\bar{{\cal G}}}({\cal A})$ to $H^*(\mnd)$, where $\bar{{\cal G}}$ is the quotient of ${\cal G}$ by its central subgroup $S^1$. There is a canonical ${\cal G}$-equivariant universal bundle over $\Sigma \times {\cal A}$, and the slant products of its Chern classes with $1\in H_0(\Sigma)$, $\alpha_j \in H_1(\Sigma)$ for $1\leq j\leq 2g$ and $[\Sigma]\in H_2(\Sigma)$ give generators of $H^*_{{\cal G}}({\cal A})$ which by abuse of notation we shall also call $a_r$, $b_r^j$ and $f_r$. (In fact $H^*_{{\cal G}}({\cal A})$ is freely generated by $a_1,\dots,a_n$, $f_2,\dots,f_n$ and $b_r^j$ for $1<r\leq n$ and $1\leq j\leq 2g$, subject only to the usual commutation relations.) The surjection from ${\cal G}$ to $\bar{{\cal G}}$ induces an inclusion from $H^*_{\bar{{\cal G}}}({\cal A})$ to $H^*_{{\cal G}}({\cal A})$ such that $$H^*_{{\cal G}}({\cal A}) \cong H^*_{\bar{{\cal G}}}({\cal A}) \otimes H^*(BS^1)$$ if we identify $H^*(BS^1)$ with the polynomial subalgebra of $H^*_{{\cal G}}({\cal A})$ generated by $a_1$, and then the generators $a_r$, $f_r$ and $b_r^j$ for $1<r\leq n$ determine generators of $H^*_{\bar{{\cal G}}}({\cal A})$ and thus of $H^*(\mnd)$. These are the generators we shall use in this paper. The normalization condition imposed by Atiyah and Bott corresponds to using the isomorphism $$H^*_{{\cal G}}({\cal A}) \cong H^*_{\bar{{\cal G}}}({\cal A}) \otimes H^*(BS^1)$$ obtained by identifying $H^*(BS^1)$ with the polynomial subalgebra of $H^*_{{\cal G}}({\cal A})$ generated by $2(g-1)a_1 + f_2$; they choose this condition because it has a nice geometrical interpretation in terms of a universal bundle over $\Sigma \times \mnd$. In Sections 4 and 5 of \cite{tdgr}, Witten obtained formulas for generating functionals from which one may extract all intersection pairings $$ \prod_{r = 2}^n a_r^{m_r} f_r^{n_r} \prod_{k_r = 1}^{2g} (b_r^{k_r})^{p_{r, k_r} } [\mnd].$$ Let us begin with pairings of the form \begin{equation} \label{1.00001} \prod_{r = 2}^n a_r^{m_r} \exp f_2 [\mnd]. \end{equation} When $m_r$ is sufficiently small to ensure convergence of the sum, Witten obtains\footnote{In fact $\mnd$ is an $n^{2g}$-fold cover of the space for which Witten computes pairings: this accounts for the factor $n^{2g}$ in our formula (\ref{2.005}). Taking this into account, (\ref{1.1}) follows from a special case of Witten's formula \cite{tdgr} (5.21).} \begin{equation} \label{1.1} \prod_{r = 2}^n a_r^{m_r} \exp f_2 [\mnd] = c^\rho \Gamma (-1)^{n_+ (g-1)} \Biggl ( \sum_{\lambda \in \weightl_{\rm reg} \cap \liet_+ } \frac{c^{- \lambda} \prod_{r = 2}^n \tau_r(\itwopi \lambda) ^{m_r} } {\nusym^{2g-2}(\itwopi \lambda) } \Biggr ), \end{equation} where \begin{equation} \label{2.005} \Gamma = \frac{n^{2g} }{\# \Pi_1(K')} (\frac{ \,{\rm vol}\, (K')}{(2 \pi)^{\dim K'}})^{2g-2} \Bigl ((2 \pi)^{n_+} {\mbox{$\cal D$}}(\rho)\Bigr ) ^{2g-2} = n^g \end{equation} is a universal constant for $K=SU(n)$ and $K' = K/Z(K)$, and the Weyl odd polynomial $\nusym$ on $\liets$ is defined by $$\nusym(X) = \prod_{\gamma > 0 } \gamma(X) $$ where $\gamma$ runs over the positive roots. Here, $\rho$ is half the sum of the positive roots, and $n_+$ $ = n(n-1)/2$ is the number of positive roots. The sum over $\lambda$ in (\ref{1.1}) runs over those elements of the weight lattice $ \weightl$ that are in the interior of the fundamental Weyl chamber.\footnote{The weight lattice $\weightl \subset \liet^*$ is the dual lattice of the integer lattice $\intlat = {\rm Ker} (\exp) $ in $\liet$.} The element \begin{equation} \label{1.p1} c = e^{2 \pi i d/n} {\rm diag}( 1, \dots, 1) \end{equation} is a generator of the centre $Z(K)$ of $K$, so since $\lambda \in \liets$ is in ${\rm Hom} (T, U(1))$, we may evaluate $\lambda$ on $c$ as in (\ref{1.1}): $c^\lambda$ is defined as $\exp \lambda (\tilde{c}) $ where $\tilde{c}$ is any element of the Lie algebra of $T$ such that $\exp \tilde{c} = c$. Note that in fact when $d$ is coprime to $n$ (so that when $n$ is even $d$ is odd) we have $c^{\rho} = (-1)^{n-1}. $ Witten's formula \cite{tdgr} (5.21) covers pairings involving the $f_r$ for $r > 2$ and the $b_r^j$ as well as $f_2$ and the $a_r$. He obtains it by reducing to the special case of pairings of the form (\ref{1.00001}) above (see \cite{tdgr} Section 5, in particular the calculations (5.11) - (5.20)) and then applying \cite{tdgr} (4.74) to this special case. In this special case of pairings of the form (\ref{1.00001}), Witten's formula \cite{tdgr} (5.21) follows from our Theorem \ref{mainab} using Proposition \ref{p:sz} below. Moreover our formula (Theorem \ref{t9.6}) for pairings involving all the generators $a_r$, $b_r^j$ and $f_r$ reduces to the special case just as Witten's does (see Propositions \ref{p9.1} and \ref{p9.2}). Thus Witten's formulas are equivalent to ours, although they look very different (being expressed in terms of infinite sums indexed by dominant weights instead of in terms of iterated residues). For the sake of concreteness it is worth examining the special case when the rank $n = 2$ so that the degree $d$ is odd. In fact, since tensoring by a fixed line bundle of degree $e$ induces a homeomorphism between $\mnd$ and $M(n,d+ne)$, we may assume that $d= 1$. In this case the dominant weights $\lambda$ are just the positive integers. The relevant generators of $H^*(\mto)$ are \begin{equation} f_2 \in H^2(\mto)\end{equation} (which is the cohomology class of the symplectic form on $\mto$) and \begin{equation} a_2 \in H^4(\mto): \end{equation} these arise from the invariant polynomial $\tau_2 = -\inpr{\cdot,\cdot}/2$ by $a_2 = \tau_2(1)$, $f_2 = \tau_2([\Sigma]) $ (see (\ref{1.2})). We find then that the formula (\ref{1.1}) reduces for $m \le g - 2 $ to\footnote{Here, we have identified $-a_2$ with Witten's class $ \Theta$ and $f_2 $ with Witten's class $\omega$.} (\cite{tdgr}, (4.44)) \begin{equation} \label{1.3} a_2^j \exp (f_2 )[{{\mbox{$\cal M$}}}(2,1)] = \frac{ 2^{2g}}{2 (8 \pi^2)^{g-1}} \Biggl ( \sum_{n = 1}^\infty (-1)^{n + 1}\frac{ \pi^{2j} }{ n^{2g-2-2j }} \Biggr ). \end{equation} Thus one obtains the formulas found by Thaddeus in Section 5 of \cite{T} for the intersection pairings $a_2^m f_2^n [\mto]$; these intersection pairings are given by Bernoulli numbers, or equivalently are given in terms of the Riemann zeta function $\zeta(s) = \sum_{n \ge 1} 1/n^s$. As Thaddeus shows in Section 4 of \cite{T}, this is enough to determine all the intersection pairings in the case when the rank $n$ is two, because all the pairings $$ a_2^{m} f_2^{n} \prod_{k = 1}^{2g} (b_2^{k})^{p_{ k} } [{\mbox{$\cal M$}}(2,1)]$$ are zero except those of the form $$ a_2^{m} f_2^{n} b_2^{2i_1-1}b_2^{2i_1 } \ldots b_2^{2i_q-1} b_2^{2i_q } [{\mbox{$\cal M$}}(2,1)]$$ where $m+2n+3q=3g-3$ and $1\leq i_1 < \ldots <i_q \leq g$, and this expression equals the evaluation of $a_2^m f_2^n$ on the corresponding moduli space of rank 2 and degree 1 bundles over a Riemann surface of genus $g-q$ if $q\leq g-2$, and equals 4 if $q=g-1$. \newcommand{\bracearg}[1]{ { [[ #1 ]] } } \newcommand{\tildarg}[1]{ { [[ #1 ]] } } Szenes \cite{Sz} has proved that the expression on the right hand side of (\ref{1.1}) may be rewritten in a particular form. To state the result we must introduce some notation. The Lie algebra $\liet = \liet_n$ of the maximal torus $T$ of $SU(n)$ is $$ \liet = \{ (X_1, \dots, X_n) \in {\Bbb R }^n: X_1 + \dots + X_n = 0 \}. $$ Define coordinates $\yy{j} = e_j(X) = X_j - X_{j +1}$ on $\liet$ for $j = 1, \dots, n-1$. The positive roots of $SU(n)$ are then $\gamma_{jk}(X) = X_j - X_k$ $ = Y_j + \dots + Y_{k-1}$ for $1 \le j < k \le n$. The {\em integer lattice} $\intlat$ of $SU(n)$ is generated by the simple roots $e_j, j = 1, \dots, n - 1$. The dual lattice to $\intlat$ with respect to the inner product $\inpr{ \cdot, \cdot} $ introduced at (\ref{1.02}) is the {\em weight lattice} $\weightl$ $\subset \liet$: in terms of the inner product $\inpr{ \cdot, \cdot} $, it is given by $\weightl = \{ X \in \liet: \; Y_j \in {\Bbb Z } ~\mbox{for $j = 1, \dots, n - 1 $} \}$. We define also $\weightl_{\rm reg} (\liet_n) = \{ X \in \weightl: \; Y_j \ne 0 ~\mbox{ for $ j = 1, \dots, n-1$} $ and $\gamma_{jk} (X) \ne 0 $ for any $j \ne k \}. $ \begin{definition} \label{bracedef} Let $f: \liet \otimes {\Bbb C } \to {\Bbb C } $ be a meromorphic function of the form \begin{equation} \label{fundef} f(X) = g(X) e^{-\gamma (X)} \end{equation} where $\gamma (X) = \gamma_1 Y_1 + \dots + \gamma_{n-1} Y_{n-1}$ for $(\gamma_1, \dots, \gamma_{n-1} )$ $\in {\Bbb R }^{n-1}$. We define $$ \tildarg{\gamma} = (\tildarg{\gamma}_1, \dots, \tildarg{\gamma}_{n-1} )$$ to be the element of ${\Bbb R }^{n-1}$ for which $0 \le \tildarg{\gamma}_j < 1$ for all $j = 1, \dots, n-1$ and $\tildarg{\gamma} = \gamma ~{\rm mod}~ {\Bbb Z }^{n-1}$. (In other words, $\bracearg{\gamma} =$ $\sum_{j = 1}^{n-1} \tildarg{\gamma}_j e_j$ is the unique element of $\liet$ $\cong {\Bbb R }^{n-1}$ which is in the fundamental domain defined by the simple roots for the translation action on $\liet_n$ of the integer lattice, and which is equivalent to $\gamma$ under translation by the integer lattice.) We also define the meromorphic function $\bracearg{f}: \liet \otimes {\Bbb C } \to {\Bbb C } $ by $$ \bracearg{f}(X) = g(X) e^{- \tildarg{\gamma}(X)}. $$ \end{definition} \begin{prop} \label{p:sz}{\bf [Szenes]} Let $f: \liet\otimes {\Bbb C } \to {\Bbb C }$ be defined by $$ f(X) = \frac{\prod_{r = 2}^n \tau_r(X)^{m_r} e^{ - \ct(X) } } {\nusym(X)^{2g-2} }. $$ Provided that the $m_r$ are sufficiently small to ensure convergence of the sum, we have $$ \sum_{\lambda \in \weightl_{reg} (\liet_{n}) \cap \liet_+ } f(\itwopi\lambda) = \frac{1}{n!} \Res_{Y_1 = 0} \dots \Res_{Y_{n-1} = 0 } \Bigl ( \frac{\sum_{w \in W_{n-1}} \bracearg{w(f)}(X)}{\expsum{\yy{n-1} } \dots \expsum{\yy{1}} } \Bigr ), $$ where $W_{n-1} \cong S_{n-1}$ is the Weyl group of $SU(n-1)$ embedded in $SU(n)$ in the standard way using the first $n-1$ coordinates $X_1,\dots,X_{n-1}$. \end{prop} \begin{rem} \label{r2.1} Here, we have introduced coordinates $Y_j = e_j(X)$ on $\liet$ using the simple roots $$\{ e_j: j = 1, \dots n-1 \} $$ of $\liet$, and $ \weightl_{\rm reg}$ denotes the regular part of the weight lattice $\weightl$ (see below). Also, we have introduced the unique element $\tc$ of $\liet$ which satisfies $e^{2 \pi i \tc} = c$ and which belongs to the fundamental domain defined by the simple roots for the translation action on $\tor$ of the integer lattice $\Lambda^I$: this simply means that $\inpr{\tc,X} = \gamma_1 Y_1 + \dots + \gamma_{n-1} Y_{n-1}$ where $0\leq \gamma_j <1$ for $1\leq j\leq n-1$. (In the notation introduced in Definition \ref{bracedef} this says that $ \tc = \bracearg{ ~(d/n, d/n, \dots, -(n-1)d/n) }$.) Also, $\lietpl$ denotes the fundamental Weyl chamber, which is a fundamental domain for the action of the Weyl group on $\liet$. If $g(Y_k, \dots, Y_{n-1})$ is a meromorphic function of $Y_k, \dots, Y_{n-1}$, we interpret $\res_{Y_k = 0 } g(Y_{k}, \dots, Y_{n-1})$ as the ordinary one-variable residue of $g$ regarded as a function of $Y_k$ with $Y_{k+1}, \dots, Y_{n-1}$ held constant. \end{rem} The rest of this section will be devoted to a proof of Proposition \ref{p:sz}. We shall prove the following theorem: \begin{theorem} \label{t2.04} Let $f: \liet_{n} \otimes {\Bbb C } \to {\Bbb C }$ be a meromorphic function of the form $f(X) = g(X) e^{-\gamma (X)}$ where $\gamma (X) = \gamma_1 Y_1 + \dots + \gamma_{n-1} Y_{n-1}$ with $0 \leq \gamma_{n-1} < 1$, and $g(X)$ is a rational function of $X$ with poles only on the zeros of the roots $\gamma_{jk}$ and decaying sufficiently fast at infinity. Then $$\sum_{\lambda \in \weightl_{reg} (\liet_{n})} f(\itwopi\lambda)= \Res_{Y_1 = 0} \dots \Res_{Y_{n-1} = 0 } \Bigl ( \frac{\sum_{w \in W_{n-1}} \bracearg{w(f)}(X) }{\expsum{\yy{n-1} } \dots \expsum{\yy{1}} } \Bigr ) $$ where $W_{n-1}$ is the Weyl group of $SU(n-1)$ embedded in $SU(n)$ using the first $n-1$ coordinates. \end{theorem} \begin{rem} \label{r:intadd} Notice that if $f$ is as in the hypothesis of the Theorem (but here one may omit the hypothesis that $0 \le \gamma_{n-1} <1$) then $$\sum_{\lambda \in \weightl_{reg} (\liet_{n})} f(\itwopi\lambda) = \sum_{\lambda \in \weightl_{reg} (\liet_{n})} \bracearg{f}(\itwopi\lambda), $$ where $\bracearg{f}$ was defined in Definition \ref{bracedef}. \end{rem} \noindent{\em Proof of Proposition \ref{p:sz} given Theorem \ref{t2.04}:} The function \begin{equation} \label{2.fn} f(X) = \frac{\prod_{r = 2}^n \tau_r(X)^{m_r} e^{ - \ct(X) } } {\nusym(X)^{2g-2} } \end{equation} satisfies the hypotheses of the Theorem, provided that the $m_r$ are small enough to ensure convergence of the sum. Notice that if $\lambda \in \weightl_{\rm reg} (\liet_n)$ then $e^{ - \itwopi \ct(\lambda)} = c^{ - \lambda} $ satisfies $c^{- \lambda} = c^{- w\lambda } $ for all elements $w$ of the Weyl group $W$. Thus for this particular $f$ we have that $$\sum_{\lambda \in \weightl_{reg} (\liet_{n})} f(\itwopi\lambda) = n! \sum_{\lambda \in \weightl_{reg} (\liet_{n}) \cap \liet_+ } f(\itwopi\lambda) . $$ So $$ \sum_{\lambda \in \weightl_{reg} (\liet_{n}) \cap \liet_+ } f(\itwopi\lambda) = \frac{1}{n!} \Res_{Y_1 = 0} \dots \Res_{Y_{n-1} = 0 } \Bigl ( \frac{\sum_{w \in W_{n-1}} \bracearg{w(f)}(X)}{\expsum{\yy{n-1} } \dots \expsum{\yy{1}} } \Bigr ) $$ which is the statement of Proposition \ref{p:sz}.\hfill $\square$ It remains to prove Theorem \ref{t2.04}. By induction on $n$ it suffices to prove \begin{lemma} \label{l2.04} Let $f = f_{(n)} : \liet_n \to {\Bbb C }$ be as in the statement of Theorem \ref{t2.04}. Define $f_{(n-1)} : \liet_{n-1} \to {\Bbb C }$ by $$f_{(n-1)} (Y_1, \dots, Y_{n-2}) = \Res_{\yy{n-1} = 0} \frac{f(Y_1, \dots, Y_{n-1})}{e^{- \yy{n-1}} - 1}. $$ Then $$ \sum_{\lambda \in \weightl_{reg} (\liet_{n} )} f_{(n)} (\itwopi \lambda) = \sum_{\lambda \in \weightl_{reg} (\liet_{n-1}) } \sum_{j=1}^{n-1} (q_j f)_{(n-1)} (\itwopi \lambda), $$ where $q_j$ is the element of the Weyl group $W_{n-1} \cong S_{n-1}$ represented by swapping the coordinates $X_j$ and $X_{n-1}$. \end{lemma} \begin{rem} Note that by Remark \ref{r:intadd}, the sum $\sum_{j=1}^{n-1} (q_j f)_{(n-1)} (\itwopi \lambda)$ is equal to $$\sum_{j=1}^{n-1} \bracearg{ (q_j f)_{(n-1)} } (\itwopi \lambda). $$ \end{rem} \begin{rem} Note that the function $ \bracearg{ (q_j f)_{(n-1)} }$ satisfies the hypotheses of Theorem \ref{t2.04}. \end{rem} \noindent{\em Proof of Lemma \ref{l2.04}:} Let $\lb{j} $ $(j = 1, \dots, n - 2) $ be integers such that \begin{equation} \label{2.cond} \lb{j} + \lb{j+1} + \dots + \lb{k} \ne 0 ~\mbox{for any}~ 1 \le j \le k \le n - 2. \end{equation} Define $\liner{(\lb{1}, \dots, \lb{n-2} )}$ to be the line $\{ (\itwopi \lb{1}, \dots, \itwopi \lb{n-2}, \yy{n-1} ): \yy{n-1} \in {\Bbb C } \}$. The condition (\ref{2.cond}) states that all the roots $\gamma_{jk}$ for $1 \le j < k \le n - 1 $ are nonzero on $\liner{(\lb{1}, \dots, \lb{n-2} )}$. Let $f: \liet\otimes {\Bbb C } \to {\Bbb C }$ be a meromorphic function as in the statement of Theorem \ref{t2.04}, having poles only at the zeros of the roots $\gamma_{jk}$. We shall think of $f$ as a function $f(Y_1, \dots, Y_{n-1})$ of the coordinates $Y_1, \dots, Y_{n-1}. $ Define $\laregint$ to be $$\laregint = \{ X \in \linestd: ~Y_j(X) = X_j - X_{j+1} \in \itwopi {\Bbb Z }, $$ $$~\gamma_{jk}(X) \ne 0 ~\mbox{for any $j \ne k $} \}. $$ The sum of all residues of the function $g_{\indsettwo} $ on ${\Bbb C }$ given by $$g_\indsettwo (\yy{n-1}) = \frac{f(\itwopi \lb{1}, \itwopi \lb{2}, \dots, \yy{n-1} ) }{ e^{- \yy{n-1} } - 1 } $$ is zero and these residues occur when $\yy{n-1} \in \itwopi {\Bbb Z }$. Therefore we find that the sum $$ \sum_{ p \in \laregint } f (p) $$ is given by $$ - \sum_{(\itwopi\lb{1}, \dots,\itwopi \lb{n-2}, \itwopi \lb{n-1} ) \in \laregint } \Res_{\yy{n-1} = \itwopi l_{n-1} } \frac{f(\itwopi \lb{1}, \itwopi \lb{2}, \dots, \itwopi \lb{n-2}, \yy{n-1} ) }{ e^{- \yy{n-1} } - 1 } $$ \begin{equation} \label{2.001} = \sum_{j =1}^{n-2} \Res_{\yy{n-1} = -\itwopi (\lb{j} + \dots + \lb{n-2} ) } \frac{f(\itwopi \lb{1}, \dots, \itwopi \lb{n-2}, \yy{n-1} ) }{ e^{- \yy{n-1} } - 1 } \: + \end{equation} $$ \Res_{\yy{n-1} = 0 } \frac{f(\itwopi \lb{1}, \dots, \itwopi \lb{n-2}, \yy{n-1} ) }{ e^{- \yy{n-1} } - 1 }. $$ \newcommand{\lj}[1]{l^{(j)}_{#1} } \begin{prop} \label{p2.3} Let $p_j$ be the point $ X \in \linestd $ for which $Y_{n-1} = - \itwopi (l_j + \dots + l_{n-2})$, or equivalently $ X_n = X_j . $ Then $$ \res_{Y_{n-1} = - \itwopi (l_j + \dots + l_{n-2} ) } \Biggl ( \frac{ f(\itwopi l_1, \dots, \itwopi l_{n-2}, Y_{n-1}) } {e^{-\yy{n-1} } - 1} \Biggr ) $$ $$ = \res_{Y_{n-1} = 0 } \Biggl ( \frac{ q_j(f)\Bigl (\itwopi \lj{1}, \dots, \itwopi \lj{n-2}, Y_{n-1}) \Bigr ) } {e^{-\yy{n-1} } - 1} \Biggr ). $$ Here, we define an involution $q_j: \liet \to \liet$ (for $j = 1, \dots, n-1$) by $$q_j(X_1, \dots, X_j, \dots, X_{n-1}, X_n) = (X_1, \dots, X_{j-1},X_{n-1},X_{j+1}, \dots, X_{n-2},X_{j},X_n),$$ and the integers $\lj{1}, \dots, \lj{n-1} $ are defined by the equation \begin{equation} \label{e:lj} q_j(X)|_{X = (\itwopi l_1, \dots, \itwopi l_{n-2}, Y_{n-1}) } = (\itwopi \lj{1}, \dots, \itwopi \lj{n-2}, \itwopi \lj{n-1} + Y_{n-1}) . \end{equation} \end{prop} \Proof For $j \le n-2$, the involution $q_j$ is given in the coordinates $(Y_1, \dots, Y_{n-1})$ by $q_j: (Y_1, \dots, Y_{n-1}) \mapsto (Y'_1, \dots, Y'_{n-1}) $ where $Y'_k = Y_k$ for $k \ne j-1, j, n-2, n-1$ and \begin{equation} \label{2.009}Y'_{j-1} = Y_{j-1} + \dots + Y_{n-2}, \end{equation} \begin{equation} \label{2.0010} Y'_j = - \sum_{j \le k \le n-2} Y_k, \end{equation} \begin{equation} \label{2.0011} Y'_{n-2} = - \sum_{j \le k \le n-3} Y_k, \end{equation} \begin{equation} \label{2.0012} Y'_{n-1} = Y_j + \dots + Y_{n-1}. \end{equation} For $j = n-1$, $q_j$ is the identity map. Notice that $Y_{n-1}' $ is the only one of the transformed coordinates that involves $Y_{n-1}.$ Notice also that $q_j$ takes $p_j$ to a point where $Y'_{n-1} = 0 . $ We now examine the image of $\linestd$ under $q_j$. The integers $\lj{1}, \dots, \lj{n-1}$ were defined by the equation (\ref{e:lj}): in fact $\lj{k} = l_k$ for $k \ne j-1, j, n-2, n-1$ and \begin{equation} \label{2.0009}\lj{j-1} = l_{j-1} + \dots + l_{n-2}, \end{equation} \begin{equation} \label{2.00010} \lj{j} = - \sum_{j \le k \le n-2} l_k, \end{equation} \begin{equation} \label{2.00011} \lj{n-2} = - \sum_{j \le k \le n-3} l_k, \end{equation} \begin{equation} \label{2.00012} \lj{n-1} = l_j + \dots + l_{n-2}. \end{equation} \newcommand{\yj}{{ Y^{(j)}_{n-1} } } We have that $$ \res_{Y_{n-1} = - \itwopi (l_j + \dots + l_{n-2} ) } \Biggl ( \frac{ f(\itwopi l_1, \dots, \itwopi l_{n-2}, Y_{n-1}) } {e^{-\yy{n-1} } - 1} \Biggr ) = $$ $$ \res_{Y_{n-1} = - \itwopi (l_j + \dots + l_{n-2} ) } \Biggl ( \frac{ f(\itwopi l_1, \dots, \itwopi l_{n-2}, Y_{n-1}) } {e^{-\yy{n-1} - \itwopi (l_j + \dots + l_{n-2}) } - 1} \Biggr ) $$ (because $e^{\itwopi l_k } = 1$ for all $k = j, \dots, n-2$) $$ = \res_{{\yj }= 0 } \Biggl ( \frac{ q_j(f)( \itwopi \lj{1}, \dots, \itwopi \lj{j-1} , \itwopi \lj{j}, \itwopi \lj{j+1}, \dots, \itwopi \lj{n-2}, \yj } {e^{-\yj } - 1} \Biggr ) $$ by the formulas (\ref{2.009} - \ref{2.0012}) where we have defined $\yj = Y_{n-1} + \itwopi (l_j + \dots l_{n-2})$ so that $d\yj = dY_{n-1} $. This completes the proof. \hfill $\square$ \begin{corollary} \label{c2.0} We have $$ \sum_{p \in \laregint} f(p) = $$ $$ \sum_{j = 1}^{n-2} \res_{Y_{n-1} = 0 } \Biggl ( \frac{ q_j(f)( \itwopi \lj{1}, \dots, \itwopi \lj{n-2}, Y_{n-1} ) } {e^{-Y_{n-1} } - 1} \Biggr ) + $$ \begin{equation} \label{2.0055} \Res_{\yy{n-1} = 0 } \frac{f(\itwopi \lb{1}, \dots, \itwopi \lb{n-2}, \yy{n-1} ) }{ e^{- \yy{n-1} } - 1 }, \end{equation} where the integers $\lj{1}, \dots, \lj{n-2} $ were defined by (\ref{2.0009} - \ref{2.00011}). \end{corollary} \Proof This follows by adding the results of Proposition \ref{p2.3} over all $j = 1, \dots, n-1$: on one side this yields the sum on the right hand side of (\ref{2.001}) (which according to (\ref{2.001}) is equal to $\sum_{p \in \laregint} f(p)$), and on the other side yields the sum on the right hand side of (\ref{2.0055}). \hfill $\square$ We shall complete the proof of Lemma \ref{l2.04} by summing the equality given in Corollary \ref{c2.0} over all possible $(l_1, \dots, l_{n-2})$ satisfying (\ref{2.cond}): the proof reduces to the following lemma. \begin{lemma} In the notation of Proposition \ref{p2.3}, $(\lj{1}, \dots, \lj{n-2} ) \in \weightl_{\rm reg} (\liet_{n-1})$. Moreover for any $(l'_1, \dots, l'_{n-2}) \in \weightl_{\rm reg} (\liet_{n-1})$ there is exactly one sequence of integers $(l_1, \dots, l_{n-2})$ satisfying (\ref{2.cond}) such that $$( \lj{1}, \dots, \lj{n-2}) =( l'_1, \dots, l'_{n-2} ).$$ \end{lemma} \Proof This follows immediately from the proof of Proposition \ref{p2.3} and the fact that the restriction of $q_j$ to $\liet_{n-1}$ is given by the action of an element of the Weyl group $W_{n-1}$ and hence maps $ \weightl_{\rm reg} (\liet_{n-1})$ to itself bijectively.\hfill $\square$ This completes the proof of Lemma \ref{l2.04} and hence of Theorem \ref{t2.04} and Proposition \ref{p:sz}. { \setcounter{equation}{0} } \section{Residue formulas and nonabelian localization} Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian action of a compact connected Lie group $K$ with Lie algebra $\liek$. Let $\mu:M \to \lieks$ be a moment map for this action. The $K$-equivariant cohomology with complex coefficients $H^*_K(M)$ of $M$ may be identified with the cohomology of the chain complex \begin{equation} \label{1.001} \Omega^*_K(M) = (S(\lieks) \otimes \Omega^*(M))^K \end{equation} of equivariant differential forms on $M$, equipped with the differential\footnote{This definition of the equivariant cohomology differential differs by a factor of $i$ from that used in \cite{tdgr} but is consistent with that used in \cite{JK1}.} \begin{equation} \label{1.0002} (D\eta)(\xvec) = d(\eta (\xvec) ) - \iota_{\xvec^{\#}} (\eta(\xvec) ) \end{equation} where $\xvec^{\#}$ is the vector field on $M$ generated by the action of $\xvec$ (see Chapter 7 of \cite{BGV}). Here $(\Omega^*(M),d)$ is the de Rham complex of differential forms on $M$ (with complex coefficients), and $S(\lieks)$ denotes the algebra of polynomial functions on the Lie algebra $\liek$ of $K$. An element $\eta \in \Omega^*_K(M)$ may be thought of as a $K$-equivariant polynomial function from $\liek$ to $\Omega^*(M)$, or alternatively as a family of differential forms on $M$ parametrized by $\xvec \in \liek$. The standard definition of degree is used on $\Omega^*(M)$ and degree two is assigned to elements of $\lieks$. In fact as a vector space, though not in general as a ring, when $M$ is a compact symplectic manifold with a Hamiltonian action of $K$ then $H^*_K(M)$ is isomorphic to $H^*(M) \otimes H^*_K$ where $H^*_K=\Omega^*_K({\rm pt}) =S(\lieks)^K$ is the equivariant cohomology of a point (see \cite{Ki1} Proposition 5.8). The map $\Omega^*_K(M) \to \Omega^*_K({\rm pt}) = S(\lieks)^K$ given by integration over $M$ passes to $\hk(M)$. Thus for any $D$-closed element $\eta \in \Omega^*_K(M)$ representing a cohomology class $[\eta]$, there is a corresponding element $\int_M \eta \in \Omega^*_K({\rm pt})$ which depends only on $[\eta]$. The same is true for any $D$-closed element $\eta = \sum_j \eta_j$ which is a formal series of elements $\eta_j$ in $\Omega^j_K(M)$ without polynomial dependence on $\xvec$: we shall in particular consider terms of the form $$\eta (\xvec) e^{ (\bom(\xvec))} $$ where $\eta \in \Omega^*_K(M)$ and $$\bom(\xvec) = \omega + \mu(\xvec) \in \Omega^2_K(M). $$ Here $\mu:M \to \lieks$ is identified in the natural way with a linear function on $\liek$ with values in $\Omega^0(M)$. It follows directly from the definition of a moment map\footnote{We follow the convention that $d\mu(X)=\iota_{X^{\sharp}}\omega$; some authors have $d\mu(X)=-\iota_{X^{\sharp}}\omega$.} that $D \bom = 0 $. If $\xvec$ lies in $\liet$, the Lie algebra of a chosen maximal torus $T$ of $K$, then there is a formula for $\int_M \eta(\xvec)$ (the {\em abelian localization formula} \cite{AB,BGV,BV1,BV2}) which depends only on the fixed point set of $T$ in $M$. It tells us that \begin{equation} \label{1.002} \int_M \eta(\xvec) = \sum_{F \in {\mbox{$\cal F$}}} \int \frac{i_F^* \eta(\xvec)} {e_F(\xvec)} \end{equation} where ${\mbox{$\cal F$}}$ indexes the components $F$ of the fixed point set of $T$ in $M$, the inclusion of $F$ in $M$ is denoted by $i_F$ and $e_F $ $\in H^*_T(M)$ is the equivariant Euler class of the normal bundle to $F$ in $M$. In particular, applying (\ref{1.002}) with $\eta $ replaced by the formal equivariant cohomology class $\eta e^{ \bom}$ we have \begin{equation} \label{1.003} h^\eta(\xvec) \;\: {\stackrel{ {\rm def} }{=} } \;\: \int_M \eta(\xvec)e^{ \bom(\xvec)} = \sum_{F \in {\mbox{$\cal F$}}} h^\eta_F(\xvec), \end{equation} where \begin{equation} \label{1.004} \hfeta(\xvec) = e^{ \mu(F)(\xvec)}\int_F \frac{i_F^* \eta(\xvec) e^{ \omega} }{e_F (\xvec) }. \end{equation} Note that the moment map $\mu$ takes a constant value $\mu(F) \in\liets$ on each $F \in {\mbox{$\cal F$}}$, and that the integral in (\ref{1.004}) is a rational function of $\xvec$. We shall assume throughout that $0$ is a regular value of the moment map $\mu:M \to \lieks$; equivalently the action of $K$ on $\zloc$ has only finite isotropy groups. The reduced space $$\mred = \zloc/K$$ is then a compact symplectic orbifold. The cohomology (with complex coefficients, as always in this paper) $H^*(\mred)$ of this reduced space is naturally isomorphic to the equivariant cohomology $H^*_K(\zloc)$ of $\zloc$, and by Theorem 5.4 of \cite{Ki1} the inclusion of $\zloc$ in $M$ induces a surjection on equivariant cohomology $$\hk(M) \to \hk(\zloc).$$ Composing we obtain a natural surjection $$\Phi: \hk(M) \to H^*(\mred)$$ which we shall denote by $$\eta \mapsto \eta_0.$$ When there is no danger of confusion we shall use the same symbol for $\eta\in\hk(M)$ and any equivariantly closed differential form in $\Omega^*_K(M)$ which represents it. Note that $(\bom)_0\in H^*(\mred)$ is represented by the symplectic form $\omega_0$ induced on $\mred$ by $\omega$. \noindent{\bf Remark} Later we shall be working with not only the reduced space $\xred = \mu^{-1}(0)/K$ with respect to the action of the nonabelian group $K$, but also $\mu^{-1}(0)/T$ and $\mredt{\xi}$ $ = \mu_T^{-1}(\xi)/T$ for regular values $\xi$ of the $T$-moment map $\mu_T$ which is the composition of $\mu$ with restriction from $\lieks$ to $\liets$. We shall use the same notation $\eta_0$ for the image of $\eta$ under the surjective homomorphism $\Phi$ for whichever of the spaces $\mu^{-1}(0)/K, $ $\mu^{-1}(0)/T $ or $\mu_T^{-1}(0)/T $ we are considering, and the notation $\eta_{\xi}$ if we are working with $\mu_T^{-1}(\xi)/T$. It should be clear from the context which version of the map $\Phi$ is being used. \bigskip The main result (the residue formula, Theorem 8.1) of \cite{JK1} gives a formula for the evaluation on the fundamental class $[\mred]\in H_*(\mred)$, or equivalently (if we represent cohomology classes by differential forms) the integral over $\mred$, of the image $\eta_0 e^{\omega_0}$ in $H^*(\mred)$ of any formal equivariant cohomology class on $M$ of the type $\eta e^{\bom}$ where $\eta \in \hk(M)$. \begin{theorem} \label{t4.1}{\bf (Residue formula, \cite{JK1} Theorem 8.1)} Let $\eta \in \hk(M) $ induce $\eta_0 \in H^*(\xred)$. Then \begin{equation} \label{jk81} \eta_0 e^{{}\omega_0} [\xred] = n_0 {C_K} \res \Biggl ( \nusym^2 (X) \sum_{F \in {\mbox{$\cal F$}}} \hfeta(X) [d X] \Biggr ), \end{equation} where the constant\footnote{This constant differs by a factor of $(-1)^s (2\pi)^{s-l} $ from that of \cite{JK1} Theorem 8.1. The reason for the factor of $ (2\pi)^{s-l} $ is that in this paper we shall adopt the convention that weights $\beta \in \liets$ send the integer lattice $\Lambda^I = {\rm Ker}(\exp:\liet \to T)$ to ${\Bbb Z }$ rather than to $2\pi {\Bbb Z }$, and that the roots of $K$ are the nonzero weights of its complexified adjoint action. In \cite{JK1} the roots send $\Lambda^I$ to $2 \pi {\Bbb Z }$. The reason for the factor of $(-1)^s$ is an error in Section 5 of \cite{JK1}. In the last paragraph of p.307 of \cite{JK1} the appropriate form to consider is $\prod_{j=1}^s (\theta^j dz'_j)$, and since 1-forms anticommute this is $(-1)^s/i^s$ times the term in $\exp (idz'(\theta))$ which contributes to the integral (5.4) of \cite{JK1}. The constant also differs by a factor of $i^s$ from that of \cite{JK3} Theorem 3.1, because in that paper the convention adopted on the equivariant cohomology differential is that of \cite{tdgr}, not that of \cite{JK1}.} $C_K$ is defined by \begin{equation} \label{4.001} C_K = \frac{(-1)^{s+n_+}}{ |W| \,{\rm vol}\, (T)}, \end{equation} and $n_0$ is the order of the stabilizer in $K$ of a generic point\footnote{Note that in \cite{JK1} and \cite{JK2} $n_0$ is stated incorrectly to be the order of the subgroup of $K$ which acts trivially on $\zloc$ (i.e. the kernel of the action of $K$ on $\zloc$): see the correction in Section 3 of \cite{JK3}. When $K=T$ is abelian, however, the stabilizer in $K$ of a generic point of $\zloc$ is equal to the kernel of the action of $K$ on $\zloc$. Moreover since the coadjoint action of $T$ on $\liets$ is trivial, when $K=T$ this subgroup acts trivially on the normal bundle to $\zloc$ in $M$ and hence is the kernel of the action of $K$ on $M$.} of $\zloc$. \end{theorem} In this formula $|W|$ is the order of the Weyl group $W$ of $K$, and we have introduced $s = \dim K$ and $l = \dim T$, while $n_+ = (s-l)/2$ is the number of positive roots. The measure $[dX]$ on $\liet$ and volume $ \,{\rm vol}\, (T)$ of $T$ are obtained from the restriction of a fixed invariant inner product on $\liek$, which is used to identify $\lieks$ with $\liek$ throughout. Also, ${\mbox{$\cal F$}}$ denotes the set of components of the fixed point set of $T$, and if $F$ is one of these components then the meromorphic function $\hfeta$ on $\liet \otimes {\Bbb C }$ is defined by (\ref{1.004}). The polynomial $\nusym: \liet \to {\Bbb R }$ is defined by $$\nusym(X) = \prod_{\gamma > 0 } \gamma(X),$$ where $\gamma$ runs over the positive roots of $K$. Note that it would perhaps be more natural to combine $(-1)^{n_+}$ from the constant $C_K$ with $\nusym^2(X)$ and replace them by the product $$\prod_{\gamma} \gamma(X)$$ of all the positive and negative roots of $K$. The formula (\ref{jk81}) was called a residue formula in \cite{JK1} because the quantity $\res$ (whose general definition was given in Section 8 of \cite{JK1}) can be expressed as a multivariable residue\footnote{An alternative definition in terms of iterated 1-variable residues is given in Section 3 of \cite{JK3}.}, whose domain is a class of meromorphic differential forms on $\liet\otimes{\Bbb C }$. It is a linear map, but in order to apply it to individual terms in the residue formula some choices have to be made which do not affect the residue of the whole sum. Once the choices have been made one finds that many of the terms in the sum contribute zero, and the formula can be rewritten as a sum over a certain subset ${\mbox{$\cal F$}}_+$ of the set ${\mbox{$\cal F$}} $ of components of the fixed point set $M^T$. When the rank of $K$ is one and $\liet$ is identified with ${\Bbb R }$, we can take $${\mbox{$\cal F$}}_+ = \{ F \in {\mbox{$\cal F$}}: \mu_T(F) > 0 \}. $$ In this paper we shall be particularly interested in the case where $K$ has rank one, for which the results are as follows. \begin{corollary} \label{c4.2} {\bf \cite{Kalkman,Wu,JK1}} In the situation of Theorem \ref{t4.1}, let $K = U(1)$. Then $$ \eta_0 e^{{}\omega_0} [\xred] = - n_0 \res_{X=0} \Bigl ( \sum_{F \in {\mbox{$\cal F$}}_+} \hfeta(X) \Bigr ) $$ where $n_0$ is the order of the stabilizer in $K$ of a generic point in $\zloc$. Here, the meromorphic function $\hfeta $ on $ {\Bbb C }$ is defined by (\ref{1.004}), and $\res_{X=0} $ denotes the coefficient of $1/X$, where $X \in {\Bbb R }$ has been identified with $2 \pi i X \in \liek$. \end{corollary} \noindent{\bf Remark} The notation $\res_{X=0}$ is intended to indicate the variable $X$ with respect to which the residue is calculated, as well as the point 0 at which the residue is taken, so that, for example, $\res_{-X=0} f(X) = -\res_{X=0} f(X)$. It would perhaps be more natural to use the notation $\res_{X=0} f(X)dX$, but we shall have numerous formulas involving iterated residues of this type which would then become too long and unwieldy. \begin{corollary} \label{c4.3}{\bf(cf. \cite{JK1}, Corollary 8.2)} In the situation of Theorem \ref{t4.1}, let $K = SU(2)$. Then $$ \eta_0 e^{{}\omega_0} [\xred] = \frac{n_0}{2} \res_{X=0} \Bigl ( (2 X)^2 \sum_{F \in {\mbox{$\cal F$}}_+} \hfeta(X) \Bigr ) . $$ Here, $n_0$, $\res_{X=0}$, $\hfeta$ and ${\mbox{$\cal F$}}_+$ are as in Corollary \ref{c4.2}, and $X \in {\Bbb R } $ has been identified with ${\rm diag} (2 \pi i, - 2 \pi i) X \in \liet$. \end{corollary} \begin{rem} \label{omiteo} Note that if the degree of $\eta$ is equal to the dimension of $\mred$ then $$\eta_0 e^{\omega_0} [\mred] = \eta_0 [\mred].$$ Alternatively for $K=U(1)$ or $K=SU(2)$, if we multiply $\omega$ and $\mu$ by a real scalar $\epsilon >0$ and let $\epsilon$ tend to $0$ we obtain $$ \eta_0 [\xred] = - n_0 \res_{X=0} \Bigl ( \sum_{F \in {\mbox{$\cal F$}}_+} \int_F \frac{i_F^*\eta(X)}{e_F(X)} \Bigr ) $$ or $$ \eta_0 [\xred] = \frac{n_0}{2} \res_{X=0} \Bigl ((2X)^2 \sum_{F \in {\mbox{$\cal F$}}_+} \int_F \frac{i_F^*\eta(X)}{e_F(X)} \Bigr ). $$ \end{rem} The results we have stated so far require the symplectic manifold $M$ to be compact, and this condition is not satisfied in the situation in which we would like to apply them (in order to obtain formulas for the intersection pairings in the cohomology of moduli spaces of bundles over compact Riemann surfaces). Luckily there are other related results due to Guillemin and Kalkman \cite{GK}, and independently Martin \cite{Ma}, which as we shall see can be generalized to noncompact symplectic manifolds. Guillemin and Kalkman and Martin have approached the problem of finding a formula for $$\eta_0 [\mred] = \int_\mred \eta_0$$ in terms of data on $M$ localised near $M^T$ in a slightly different way from that described above. As Guillemin and Kalkman observe, it follows immediately from the residue formula that if $\xi \in\liets$ is a regular value of the $T$-moment map $\mu_T:M \to \liets$ which is sufficiently close to $0$ then \begin{equation} \label{rtmt} \eta_0 [\mred] = \frac{(-1)^{n_+}n_0(\nusym^2\eta)_{\xi}} {n_0^T|W|} [\mu_T^{-1}(\xi)/T]\end{equation} where $n_0$ (respectively $n_0^T$) is the order of the stabilizer in $K$ (respectively $T$) of a generic point of $\zloc$ (respectively $\mu_T^{-1}(0)$) and $\mu_T^{-1}(\xi)/T$ is the reduced space for the action of $T$ on $M$ with respect to the shifted moment map $\mu_T - \xi$. Also $(\nusym^2 \eta)_{\xi} \in H^*(\mu_T^{-1}(\xi)/T)$ is the image of $ \nusym^2 \eta$ under the surjection $\Phi:H^*_T(M) \to H^*(\mred)$. Here $\eta\in \hk(M)$ and $\nusym \in S(\liets) = H^*_T$ are regarded as elements of $H^*_T(M)$ via the natural identification of $H^*_K(M)$ with the Weyl invariant part $(H^*_T(M))^W$ of $H^*_T(M)$ and the natural inclusion $H_T^* \to H^*_T(M)$. Martin gives a direct proof of (\ref{rtmt}) without appealing to the residue formula, which shows also that for any $\xi$ sufficiently close to $0$ \begin{equation} \label{rtmt2} \eta_0 [\mred] = \frac{n_0(\nusym \eta)_{\xi}} {n'_0|W|} [\mu^{-1}(\xi)/T]\end{equation} where $n'_0$ is the order of the stabilizer in $T$ of a generic point in $\zloc$, provided that $\mu^{-1}(\xi)/T$ is oriented appropriately. \begin{rem}\label{orient} The symplectic form $\omega$ induces an orientation on $M$, and the induced symplectic forms on $\mred = \zloc / K$ and on $\mu_T^{-1}(\xi)/T$ induce orientations on these quotients. We have made a choice of positive Weyl chamber for $K$ in $\liet$; this determines a Borel subgroup $B$ (containing $T$) of the complexification $G$ of $K$, such that the weights of the adjoint action of $T$ on the quotient $\lieg / \lieb$ of the Lie algebra $\lieg$ of $G$ by the Lie algebra $\lieb$ of $B$ are the positive roots of $K$. We then get an orientation of the flag manifold $K/T$ by identifying it with the complex space $G/B$. Modulo the action of finite isotropy groups we have a fibration $$\zloc/T \to \zloc/K$$ with fibre $K/T$; thus the symplectic orientation of $\zloc/K$ and the orientation of $K/T$ determined by the choice of Weyl chamber induce an orientation of $\zloc /T$. Since 0 is a regular value of $\mu$, if $\xi$ is sufficiently close to 0 there is a homeomorphism from $\zloc/T$ to $\mu^{-1}(\xi)/T$ induced by a $T$-equivariant isotopy of $M$, so we get an induced orientation of $\mu^{-1}(\xi)/T$. This is the orientation of $\mu^{-1}(\xi)/T$ which we shall use. Note that given a positive Weyl chamber we have another choice of orientation on $\mu^{-1}(\xi)/T$ which is compatible with the symplectic orientation on $\mu_T^{-1}(\xi)/T$ and the orientation of the normal bundle to $\mu^{-1}(\xi)/T$ in $\mu_T^{-1}(\xi)/T$ induced by identifying it in the natural way with the kernel of the restriction map $\lieks \to \liets$, thence via the fixed invariant inner product on $\liek$ with $\liek/\liet$ and thus finally with the complex vector space $\lieg/\lieb$ as above. Because we have used the inner product to identify $\liek/\liet$ with its dual here, this orientation differs from the one chosen above by a factor of $(-1)^{n_+}$ where $n_+$ is the number of positive roots. \end{rem} \begin{prop} \label{p:sm} {\bf (Reduction to the abelian case)} {\sc [S. Martin] \cite{Ma}} If $T$ is a maximal torus of $K$ and $K$ acts effectively on $M$, then for any regular value $\xi$ of $\mu_T$ sufficiently close to $0$ we have that $$ \int_{\mu^{-1}(0)/K} (\eta e^{ \bom} )_0 = \frac{n_0}{n'_0|W|} \int_{\mu^{-1}(0)/T} (\nusym \eta e^{ \bom} )_0 = \frac{n_0}{n'_0|W|} \int_{\mu^{-1}(\xi)/T} (\nusym \eta e^{ \bom} )_{\xi} $$ $$ = \frac{(-1)^{n_+}n_0}{n_0^T|W|} \int_{\mu_T^{-1}(\xi)/T} (\nusym^2 \eta e^{ \bom} )_{\xi} $$ where $n_0$ is the order of the stabilizer in $K$ of a generic point of $\zloc$ and $n_0^T$ (respectively $n'_0$) is the order of the stabilizer in $T$ of a generic point of $\mu_T^{-1}(0)$ (respectively $\zloc$). \end{prop} \begin{rem} Note that $(-1)^{n_+}\nusym^2$ is the product of all the roots of $K$, both positive and negative. \end{rem} Martin proves this result by considering the diagram $$\begin{array}{cccccc} & \mu^{-1}(0)/T & \cong & \mu^{-1}(\xi)/T & \hookrightarrow & \mu_T^{-1}(\xi)/T \\ & \downarrow & & & & \\ \mred = & \mu^{-1}(0)/K & & & & \end{array}$$ where the homeomorphism from $\mu^{-1}(0)/T $ to $\mu^{-1}(\xi)/T $ is induced by a $T$-equivariant isotopy of $M$ (for $\xi$ sufficiently close to $0$). For simplicity we shall consider the case when $n_0=n'_0=n_0^T =1$. As before we use a fixed invariant inner product on $\liek$ to identify $\lieks$ with $\liek$, which splits $T$-equivariantly as the direct sum of $\liet$ and its orthogonal complement $\liet^{\perp}$. The projection of $\mu:M \to \lieks\cong\liek$ onto $\liet^{\perp}$ then defines a $T$-equivariant section of the bundle $M \times \liet^{\perp}$ on $M$, which has equivariant Euler class $(-1)^{n_+}\nusym$ if we orient $\liet^{\perp} \cong \liek/\liet$ by identifying it with the dual of the complex vector space $\lieg/\lieb$ as in Remark \ref{orient}. Hence if $\xi$ is a regular value of $\mu_T$ then $\mu^{-1}(\xi)/T$ is a zero-section of the induced orbifold bundle $\mu_T^{-1}(\xi) \times_T \liet^{\perp}$ on $ \mu_T^{-1}(\xi)/T$, whose Euler class is $(-1)^{n_+}\nusym_{\xi}$. Thus under the conventions for orientations described in Remark \ref{orient}, evaluating the restriction to $\mu^{-1}(\xi)/T$ of an element of $H^*(\mu_T^{-1}(\xi))/T$ on the fundamental class $[\mu^{-1}(\xi)/T]$ gives the same result as multiplying by $(-1)^{n_+}\nusym_{\xi}$ and evaluating on the fundamental class $[\mu_T^{-1}(\xi)/T]$. Now Martin observes that since the natural map $$\Pi:\mu^{-1}(0)/T \to \mu^{-1}(0)/K = \mred$$ is a fibration with fibre $K/T$, modulo the action of finite isotropy groups which act trivially on cohomology with complex coefficients, and since the Euler characteristic of $K/T$ is nonzero (in fact it is the order $|W|$ of the Weyl group of $K$), the evaluation of a cohomology class $\eta_0 \in H^*(\mred)$ on $[\mred]$ is given by the evaluation of an associated cohomology class on $[\zloc/T]$. More precisely we have \begin{equation} \label{martin} \eta_0 [\mred] = \frac{e(V)}{|W|} \Pi^*(\eta_0) [\mu^{-1}(0)/T] \end{equation} where $e(V)$ is the Euler class of the vertical subbundle of the tangent bundle to $\mu^{-1}(0)/T$ with respect to the fibration $\Pi$. As this Euler class is induced by $\nusym$ under the orientation conventions of Remark \ref{orient}, this completes the proof. \begin{rem} \label{noncom} \label{leg} In this proof we saw that $\nusym_{\xi}$ is the cohomology class in $H^*(\mu_T^{-1}(\xi)/T)$ which is Poincar\'{e} dual to the homology class represented by $\mu^{-1}(\xi)/T$. Thus $\nusym_{\xi}$ may be represented by a closed differential form on $\mu_T^{-1}(\xi)/T$ with support in an arbitrarily small neighbourhood of $\mu^{-1}(\xi)/T$. If we interpret $\nusym_{\xi}$ in this way, Martin's proof of Proposition \ref{p:sm} is valid even when $M$ is noncompact and has singularities, provided that for $\xi$ near $0$ the subset $\mu^{-1}(\xi)$ is compact and does not meet the singularities of $M$. Note also that $K$ and hence $T$ act with at most finite isotropy groups on a neighbourhood of $\mu^{-1}(0)$ in $\mu_T^{-1}(0)$, and so $\mu_T^{-1}(0)/T$ has at worst orbifold singularities in a neighbourhood of $\mu^{-1}(0)/T$. This means that in Proposition \ref{p:sm} we do not need to perturb the value of the $T$-moment map $\mu_T$ from $0$ to a nearby regular value $\xi$ if, as above, we represent $\nusym_{0}$ by a differential form on $\mu_T^{-1}(0)/T$ with support in a sufficiently small neighbourhood of $\zloc/T$. \end{rem} This result reduces the problem of finding a formula for $\eta_0 [\mred]$ in terms of data on $M$ localized near $M^T$ to the case when $K=T$ is itself a torus. Guillemin and Kalkman, and independently Martin, then follow essentially the same line. This is to consider the change in $$\eta_{\xi} [\mu_T^{-1}(\xi)/T],$$ for fixed $\eta \in H^*_T(M)$, as $\xi$ varies through the regular values of $\mu_T$. This is sufficient, if $M$ is a compact symplectic manifold, because the image $\mu_T(M)$ is bounded, so if $\xi$ is far enough from $0$ then $\mu_T^{-1}(\xi)/T$ is empty and thus $\eta_{\xi} [\mu_T^{-1}(\xi)/T]=0.$ More precisely, the convexity theorem of Atiyah \cite{Aconv} and Guillemin and Sternberg \cite{GSconv} tells us that the image $\mu_T(M)$ is a convex polytope; it is the convex hull in $\liets$ of the set $$\{ \mu_T(F) : F\in{\mbox{$\cal F$}}\}$$ of the images $\mu_T(F)$ (each a single point of $\liets$) of the connected components $F$ of the fixed point set $M^T$. This convex polytope is divided by codimension-one \lq\lq walls'' into subpolytopes, themselves convex hulls of subsets of $\{ \mu_T(F) : F\in{\mbox{$\cal F$}}\}$, whose interiors consist entirely of regular values of $\mu_T$. When $\xi$ varies in the interior of one of these subpolytopes there is no change in $\eta_{\xi} [\mu_T^{-1}(\xi)/T],$ so it suffices to understand what happens as $\xi$ crosses a codimension-one wall. Any such wall is the image $\mu_T(M_1)$ of a connected component $M_1$ of the fixed point set of a circle subgroup $T_1$ of $T$. The quotient group $T/T_1$ acts on $M_1$, which is a symplectic submanifold of $M$, and the restriction of the moment map $\mu_T$ to $M_1$ has an orthogonal decomposition $$\mu_T|_{M_1} = \mu_{T/T_1} \oplus \mu_{T_1}$$ where $\mu_{T/T_1}: M_1 \to (\liet/\liet_1)^*$ is a moment map for the action of $T/T_1$ on $M_1$ and $\mu_{T_1}:M_1 \to \liets_1$ is constant (because $T_1$ acts trivially on $M_1$). If $\xi_1$ is a regular value of $\mu_{T/T_1}$ then we have a reduced space $$(M_1)_{{\rm red}} = \mu_{T/T_1}^{-1}(\xi_1 )/ (T/T_1).$$ Guillemin and Kalkman show that if $T$ acts effectively on $M$ (or equivalently if $n_0^T =1$; see Footnote 9) then, for an appropriate choice of $\xi_1$, the change in $\eta_{\xi} [\mu_T^{-1}(\xi)/T]$ as $\xi$ crosses the wall $\mu_T(M_1)$ can be expressed as $$({\rm res}_{M_1} (\eta))_{\xi_1} [(M_1)_{{\rm red}}]$$ for a certain residue operation (see Footnote 11 below) $${\rm res}_{M_1} : H^*_T(M) \to H^{*-d_1}_{T/T_1}(M_1)$$ where $d_1 = {\rm codim} M_1 -2$. (Of course care is needed here about the direction in which the wall is crossed; this can be resolved by a careful analysis of orientations). By induction on the dimension of $T$ this gives a method for calculating $\eta_{\xi} [\mu_T^{-1}(\xi)/T]$ in terms of data on $M$ localized near $M^T$. It is easiest to see how this version of localization is related to the residue formula of \cite{JK1} in the special case when $K=T=U(1)$. In this case $$\Omega_T^*(M) \cong {\Bbb C }[X] \otimes \Omega^*(M)^T$$ is the tensor product of a polynomial ring in one variable $X$ (representing a coordinate function on the Lie algebra $\liet$) with the algebra of $T$-invariant de Rham forms on $M$. The Guillemin-Kalkman residue operation $${\rm res}_{M_1} : H^*_T(M) \to H^{*-d_1}_{T/T_1}(M_1)$$ is then given in terms of the ordinary residue on ${\Bbb C }$ by $${\rm res}_{M_1} (\eta) = \res_{X=0} \frac{\eta|_{M_1}(X)}{e_{M_1}(X)}$$ where $\eta|_{M_1}(X)$ and the equivariant Euler class $e_{M_1}(X)$ of the normal bundle to $M_1$ in $M$ are regarded as polynomials in $X$ with coefficients in $H^*(M_1)$. More precisely we formally decompose this normal bundle (using the splitting principle if necessary) as a sum of complex line bundles $\nu_j$ on which $T$ acts with nonzero weights $\beta_j \in \liets \cong {\Bbb R }$, and because $c_1(\nu_j) \in H^*(M_1)$ is nilpotent we can express $$\frac{\eta|_{M_1}(X)}{e_{M_1}(X)} = \frac{\eta|_{M_1}(X)}{\prod_j (c_1(\nu_j) + \beta_j X)} = \frac{\eta|_{M_1}(X)}{\prod_j (\beta_j X)} \prod_j \Bigl(1+ \frac{c_1(\nu_j)}{\beta_j X}\Bigr)^{-1}$$ as a finite Laurent series in $X$ with coefficients in $H^*(M_1)$. Then ${\rm res}_{M_1}(\eta)$ is simply the coefficient of $1/X$ in this expression\footnote{When the dimension $l$ of $T$ is greater than one the Guillemin-Kalkman residue operation $${\rm res}_{M_1} : H^*_T(M) \to H^{*-d_1}_{T/T_1}(M_1)$$ is defined in almost exactly the same way, by choosing a coordinate system $X=(X_1,\ldots,X_l)$ on $\liet$ where $X_1$ is a coordinate on $\liet_1$, and taking the coefficient of $1/X_1$ in $\frac{\eta|_{M_1}(X)}{e_{M_1}(X)} $ expanded formally as a Laurent series in $X_1$ with coefficients in ${\Bbb C }[X_2, \ldots, X_l] \otimes \Omega^*(M)^T$.}. Since $T_1 = T$ acts trivially on $M_1$, we have $M_{1,{\rm red}} = M_1$ and $M_1$ is a connected component of the fixed point set $M^T$, i.e. $M_1 \in {\mbox{$\cal F$}}$. Therefore $$({\rm res}_{M_1} (\eta))_{\xi_1} [(M_1)_{{\rm red}}] = \res_{X=0} \int_{M_1} \frac{\eta|_{M_1}(X)}{e_{M_1}(X)}.$$ Of course as $K=T=U(1)$ the convex polytope $\mu_T(M)$ in $\liets \cong {\Bbb R }$ is a closed interval, divided into subintervals by the points $\{\mu_T(F) : F \in {\mbox{$\cal F$}}\}$. Thus the argument of Guillemin and Kalkman just described, amplified by some careful consideration of orientations, tells us that if $\xi >0$ is a regular value of $\mu_T$ and $n_0^T =1$ then the difference $$ \eta_{\xi}[\mu_T^{-1}(\xi)/T] - \eta_0[\mu_T^{-1}(0)/T] $$ can be expressed as \begin{equation} \label{gk} \sum_{M_1 \in {\mbox{$\cal F$}}: 0<\mu_T(M_1)<\xi} {\rm res}_{M_1} (\eta) [M_1] = \res_{X=0} \sum_{F\in{\mbox{$\cal F$}}: 0<\mu_T(F) < \xi} \int_F \frac{i_F^*\eta(X)}{e_F(X)}. \end{equation} If we take $\xi > {\rm sup} (\mu_T(M))$ then this gives the same result as Corollary \ref{c4.2} (cf. Remark \ref{omiteo}). \begin{prop} \label{p:gkm} {\bf ( Dependence of symplectic quotients on parameters)} {\sc Guillemin-Kalkman \cite{GK} ; S. Martin \cite{Ma} } If $K=T = U(1)$ and $n_0^T$ is the order of the stabilizer in $T$ of a generic point of $\mu_T^{-1}(0)$ then\footnote{The convention of Guillemin and Kalkman for the sign of the moment map differs from ours (see Footnote 8). This accounts for a difference in sign between their formula and ours.} $$ \int_{\mu_T^{-1}(\xi_1)/T } (\eta e^{ \bom} )_{\xi_1} - \int_{\mu_T^{-1}(\xi_0)/T } (\eta e^{ \bom} )_{\xi_0} = n_0^T \sum_{F \in {\mbox{$\cal F$}}: \xi_0 < \mu_T(F) < \xi_1} {\rm Res}_{X=0} e^{ \mu_T(F)X} \int_F \frac{\eta(X) e^{ \omega} }{e_F(X) } . $$ where $X \in {\Bbb C }$ has been identified with $2 \pi i X \in \liet \otimes {\Bbb C }$ and $\xi_0 < \xi_1$ are two regular values of the moment map. \end{prop} \begin{rem} \label{arm} As we have already noted these results can be deduced easily from the residue formula of \cite{JK1} when $M$ is a compact symplectic manifold. However the proof of Proposition \ref{p:gkm}, just like that of Proposition \ref{p:sm} (see Remark \ref{noncom}), can be adapted to apply in circumstances when $M$ is not compact and the residue formula of \cite{JK1} is not valid. Indeed, as Guillemin and Kalkman observe, in the case when $K=T=U(1)$ the basis of their argument applies to any compact oriented $U(1)$-manifold $Y$ with boundary such that the action of $T=U(1)$ on the boundary $\partial Y$ is locally free. Let us suppose for simplicity that $T$ acts effectively on $M$ (i.e. that $n_0^T=1$; see Footnote 9) and let $\zeta$ be a $U(1)$-invariant de Rham one-form on $Y-Y^T$ with the property that $\iota_{v}(\zeta) = 1$, where the vector field $v$ is the infinitesimal generator of the $U(1)$-action. Guillemin and Kalkman showed that, at the level of forms, the map $\Phi:H_T^*(Y)\to H^*(\partial Y/T)$ which is the composition of the restriction map from $H_T^*(Y)$ to $H_T^*(\partial Y)$ with the inverse of the canonical isomorphism $H_T^*(\partial Y) \to H^*(\partial Y/T)$ is given by $$\Phi(\eta) = \res_{X=0} \iota_{v}(\frac{\zeta\eta}{X-d\zeta})$$ (see (1.18) of \cite{GK}, noting that Guillemin and Kalkman have a different convention for the equivariant cohomology differential, which accounts for the minus sign). If tubular neighbourhoods $U_1,\ldots,U_N$ of the components $F_1,\ldots,F_N$ of the fixed point set $Y^T$ are removed from $Y$, then Stokes' theorem can be applied to the manifold with boundary $Y-\bigcup_{j=1}^N U_j$ using the formal identity $$D(\frac{\zeta\eta}{X-d\zeta}) = \eta$$ on $Y - \bigcup_{j=1}^N U_j$ to give, after using the fact that $\int_{\partial Y} \alpha = \int_{\partial Y/T} \iota_v(\alpha)$ and taking residues at $X=0$, the formula $$\int_{\partial Y/T} \Phi(\eta) = \res_{X=0} \sum_{j=1}^N \int_{F_j} \frac{\eta|_{F_j}(X)}{e_{F_j}(X)}$$ where $e_{F_j}$ is the equivariant Euler class of the normal bundle to $F_j$ in $Y$. The formula of Proposition \ref{p:gkm} comes directly from this when the manifold with boundary $Y$ is $\mu_T^{-1}[\xi_0,\xi_1]$ for a moment map $\mu_T:M \to \liets\cong{\Bbb R }$ with regular values $\xi_0<\xi_1$, but there is no need for $\mu_T$ to be a moment map or for $M$ to have a symplectic structure for the formula to be valid. It is enough for $\mu_T:M \to {\Bbb R }$ to be a smooth $T$-invariant map with regular values $\xi_0 < \xi_1$ such that $T$ acts freely on the intersections of $\mu_T^{-1}(\xi_0)$ and $\mu_T^{-1}(\xi_1)$ with the support of the equivariant differential form $\eta$. There is also no need to assume that $M$ is compact; it suffices to suppose that $\mu_T:M\to {\Bbb R }$ is a proper map. Indeed, the assumption that $\mu_T$ is proper can itself be weakened; the same proof applies provided only that the intersection of $\mu_T^{-1}[\xi_0,\xi_1]$ with the support of the equivariant differential form $\eta$ is compact. \end{rem} { \setcounter{equation}{0} } \section{Extended moduli spaces} In \cite{ext} certain \lq\lq extended moduli spaces'' of flat connections on a compact Riemann surface with one boundary component are studied. They have natural symplectic structures, and can be used to exhibit the moduli spaces $\mnd$ of interest to us as finite-dimensional symplectic quotients or reduced spaces. Our aim is to obtain Witten's formulas for intersection pairings on $H^*(\mnd)$ by applying nonabelian localization to these extended moduli spaces. They have a gauge-theoretic description (cf. the introduction to this paper), but we shall use a more concrete (and entirely finite dimensional) characterization given in \cite{ext}. The space with which we want to work is defined by \begin{equation} \label{4.1} \mc = (\epsr{K} \times \epc)^{-1} (\bigtriangleup) \subset \,{\rm Hom}\, (\FF, K) \times \liek, \end{equation} where $\FF$ is the free group on $2g$ generators $ \{x_1, \dots, x_{2g} \}$; we identify $\FF$ with the fundamental group of the surface $\Sigma$ with one point removed, in such a way that $x_1, \dots, x_{2g} $ correspond to the generators $\alpha_1, \dots, \alpha_{2g}$ of $H_1(\Sigma,{\bf Z})$ chosen in Section 2. Then $\epsr{K}: \,{\rm Hom}\,(\FF, K) \to K$ is the evaluation map on the relator $r = \prod_{j = 1}^g [x_{j}, x_{j+g}]$ \begin{equation} \label{4.2} \epsr{K} (h_1, \dots, h_{2g} ) = \prod_{j = 1}^g [h_{j} , h_{j+g} ]. \end{equation} The map $\epc: \liek \to K $ is defined by \begin{equation} \label{4.3} \epc(Y) = \cent \exp (Y), \end{equation} where the generator $\cent$ of the centre of $K$ was defined at (\ref{1.p1}) above. The diagonal in $K \times K $ is denoted $\bigtriangleup$. The space $M(c)$ then has canonical projection maps $\proj_1, \proj_2 $ which make the following diagram commute: \begin{equation} \label{4.4} \begin{array}{lcr} \xc & \stackrel{\proj_2}{\lrar} & \liek \\ \scriptsize{\proj_1} \downarrow & \phantom{\stackrel{aaaa}{\lrar} } & \downarrow \scriptsize{e_c} \\ \homfk & \stackrel{\epsr{K} }{\lrar} & K\\ \end{array} \end{equation} In other words, $\xc$ is the fibre product of $\homfk $ and $\liek$ under the maps $\epsr{K}$ and $\epc$. The action of $K$ on $\xc$ is given by the adjoint actions on $K$ and $\liek$. The space $\xc$ has the following properties (see \cite{ext} and \cite{J1}): \begin{prop} \label{p0} {\bf (a)} The space $\xc$ is smooth near all $(h, \Lambda) \in \homfk \times \liek$ for which the linear space $z(h) \cap \ker (d \exp)_\Lambda \ne \{0 \} $. Here, $z(h) $ is the Lie algebra of the stabilizer $Z(h) $ of $h$. {\bf (b) } There is a $K$-invariant 2-form $\omega$ on $\homfk \times \liek$ whose restriction to $\xc$ is closed and which defines a nondegenerate bilinear form on the Zariski tangent space to $\xc$ at every $(h, \Lambda)$ in an open dense subset of $\xc$ containing $\xc \cap (K^{2g} \times \{0\})$. Thus the form $\omega$ gives rise to a symplectic structure on this open subset of $\xc$. {\bf (c) } With respect to the symplectic structure given by the 2-form $\omega$, a moment map $\mu: \xc\to \lieks$ for the action of $K$ on $\xc$ is given by the restriction to $\xc$ of $- \proj_2$, where $\proj_2: \xc \to \liek$ is the projection map to $\liek$ (composed with the canonical isomorphism $\liek \to \lieks$ given by the invariant inner product on $\liek$). {\bf (d) } The space $\xc$ is smooth in a neighbourhood of $\mu^{-1} (0).$ {\bf (e)} The symplectic quotient $\xred = \xc\cap\mu^{-1}(0)/K$ can be naturally identified with $\epsilon_K^{-1}(c)/K = \mnd$. \end{prop} \begin{rem} We shall also use $\mu$ to denote the map $$\mu:K^{2g}\times \liek \to \liek$$ defined by $$\mu(h,\Lambda) = -\Lambda,$$ even though it is only its restriction to $M(c)$ which is a moment map in any obvious sense. That is why we write $\xc\cap\zloc/K$ instead of $\zloc/K$ in (e) above. \end{rem} \begin{rem} \label{r:fibprod} Using our description (\ref{4.4}) of $\xc$ as a fibre product, it is easy to identify the components $F$ of the fixed point set of the action of $T$. We examine the fixed point sets of the action of $T$ on $\homfk$ and $\liek$ and find \begin{equation} \label{4.5} \begin{array}{lcr} \xc^T & \stackrel{\proj_2}{\lrar} & \liet \\ \scriptsize{\proj_1} \downarrow & \phantom{\stackrel{aaaa}{\lrar} } & \downarrow \scriptsize{e_c} \\ {\rm Hom}(\FF, T) & \stackrel{\epsr{K} }{\lrar} & 1 \in T \\ \end{array} \end{equation} (Notice that $\epsr{K}$ sends ${\rm Hom}(\FF, T)$ to $1$ because $T$ is abelian.) Thus \begin{equation} \label{fixed} M(c)^T = {\rm Hom}(\FF, T) \times e_c^{-1}(1) = T^{2g} \times \{ \delta - \tilde{c}: \phantom{a} \delta \in \intlat \subset \liet \} \end{equation} where $\tilde{c} $ is a fixed element of $\liet $ for which $\exp \tilde{c} = c$. (Here, $\intlat$ denotes the integer lattice ${\rm Ker}(\exp) \subset \liet$.) If we ignore the singularities of $M(c)$, this description also enables us to find a plausible candidate for the equivariant Euler class $e_{\fd}$ of the normal bundle of each component $T^{2g} \times (\delta - \tilde{c}) $ in $M(c)^T$ (indexed by $\delta \in \intlat$). This should be simply the equivariant Euler class of the normal bundle to $T^{2g}$ in $K^{2g}$, implying that $e_{\fd}$ is in fact independent of $\delta$ and is given by \begin{equation} \label{33} e_{\fd} (\xvec) = (\prod_{\gamma} \gamma)^g = ((-1)^{n_+ } \nusym(\xvec)^2)^{g}. \end{equation} The symplectic volume of the component $F_\delta $ is independent of $\delta$ (indeed these components are all identified symplectically with $T^{2g}$): we denote the volume of $F_\delta$ by $\int_F e^{\omega}$. The constant value taken by the moment map $\mu_T $ on the component $F = F_\delta$ is given by $\tc - \delta$. \end{rem} We shall need also the following property (proved in \cite{J2}): \begin{prop} \label{abftil} The generating classes $a_r$, $b_r^j$ and $f_r$ ($r = 2, \dots, n$, $j = 1, \dots , 2g$) extend to classes $\tar$, $\tbrj$ and $\tfr$ $ \in H^*_K(\xc)$. \end{prop} Indeed, because of our conventions on the equivariant differential, the construction of \cite{J2} (which will be described at the beginning of Section 9) tells us that the equivariant differential form $\tar \in \Omega^*_K(\xc)$ whose restriction represents the cohomology class $a_r \in H^*(\mnd)$ is $\tau_r(-X)$, where as above $\tau_r \in S^r(\lieks)^K$ $\cong \hk({\rm pt})$ is the invariant polynomial which is associated to the $r$th Chern class (see \cite{J2}). Moreover $\tf_2$ is the extension $\bar{\omega}=\omega + \mu$ of the symplectic form $\omega$ to an equivariantly closed differential form (see \cite{J2} again). Finally we shall need to work with the symplectic subspace $\emtc =\xc \cap \mu^{-1}(\liet)$ of $\xc$, which is no longer acted on by $K$ but is acted on by $T$. The space $\emtc$ has an important periodicity property: \begin{lemma} \label{l4.3} Suppose $\tran $ lies in the integer lattice $\Lambda^I = {\rm Ker} (\exp )$ in $\liet$. Then there is a homeomorphism $s_\tran: K^{2g}\times \liek \to K^{2g} \times \liek$ defined by $$ s_\tran: (h, \Lambda) \mapsto (h, \Lambda + \tran) $$ which restricts to a homeomorphism $s_\tran: \emtc \to \emtc$. \end{lemma} \Proof This is an immediate consequence of the definition of $\emtc$ and the fact that $\exp(\Lambda + \Lambda_0) = \exp(\Lambda)\exp(\Lambda_0)$ when $\Lambda$ and $\Lambda_0$ commute. \hfill $\square$ Let us examine the behaviour of the images in $\hht(\emtc)$ of these extensions $\tar$, $\tbrj$, $\tfr$ $ \in \hk(\mc)$ of the generating classes $a_r, b_r^j, f_r$ (see Proposition \ref{abftil}) under pullback under these homeomorphisms $s_{\Lambda_0}: \emtc \to \emtc$. By abuse of language, we shall refer to these images also as $\tar, \tbrj $ and $\tfr$. We noted above that the classes $\tar$ are the images in $\hk(\mc)$ of the polynomials $\tau_r(-X) \in \hk = S(\lieks)^K$ (cf. (\ref{9})). Moreover (by \cite{J2}, (8.18)) the classes $\tbrj \in \hk(\mc)$ are of the form $\tbrj = \proj_1^* (\tbrj)_1 $ where $ (\tbrj)_1 \in \hk(K^{2g} )$ and $\proj_1: \mc \to K^{2g}$ is the projection in (\ref{4.4}). It follows that $$\stran^* \tbrj = \tbrj$$ and $$\stran^* \tar = \tar. $$ Furthermore we see from (8.30) of \cite{J2} that $\tf_2(X)$ is of the form \begin{equation} \label{8.3} \tf_2(X) = \proj_1^* f_2^1 + \langle \mu,X \rangle \end{equation} where $f_2^1 \in H^*_K(K^{2g})$ and $\mu: \mc \to \liek $ is the moment map (which is the restriction to $\mc$ of minus the projection $K^{2g} \times \liek \to \liek$: see Proposition \ref{p0}). It follows from this that for any $\tran $ in the integer lattice $\Lambda^I$ of $\liet$ (the kernel of the exponential map), \begin{equation} \label{8.4} \stran^* \tf_2(X) = \tf_2(X) - \langle \tran,X \rangle . \end{equation} { \setcounter{equation}{0} } \section{Equivariant Poincar\'{e} duals} We are aiming to apply nonabelian localization to the extended moduli space $M(c)$ defined in the previous section. In order to overcome the problem that $M(c)$ is singular, instead of working with integrals over $M(c)$ of equivariant differential forms, we shall integrate over $K^{2g}\times \liek$ after first multiplying by a suitable equivariantly closed differential form on $K^{2g}\times \liek$ with support near $M(c)$ which can be thought of as representing the equivariant Poincar\'{e} dual to $M(c)$ in $K^{2g}\times \liek$. So we need to construct such an equivariantly closed differential form. \begin{rem} In our earlier article \cite{JK2} covering the case when the bundles have rank $n=2$, we overcame the problem of the singularities of $M(c)$ in a slightly different way, by perturbing the central constant $c\in SU(n)$ to a nearby element of the maximal torus $T$. This method can be generalized to cover the cases when $n>2$, but it seems a little more straightforward to use equivariant Poincar\'{e} duals, so we adopt the latter approach here. \end{rem} \begin{rem} Related constructions of equivariantly closed differential forms representing the Poincar\'e dual to a submanifold appear already in the literature.\footnote{We thank P. Paradan for pointing out that the references cited below contain such constructions.} In Kalkman's paper \cite{Kalkman2} and Mathai-Quillen's paper \cite{MQ}, an equivariantly closed differential form which is rapidly decreasing away from a submanifold and represents the Poincar\'e dual to the submanifold is given: such a form is often referred to as the {\em Thom form}, as the cohomology class it represents is the Thom class of the normal bundle to the submanifold. The forms constructed in \cite{Kalkman2} and \cite{MQ} are not compactly supported: a construction of a compactly supported equivariantly closed form representing the Poincar\'e dual of a submanifold is given in section 2.3 of \cite{DV}. For completeness, in this section we provide a construction of an equivariantly closed form representing the Poincar\'e dual. \end{rem} First we consider the simpler problem of constructing an equivariant Poincar\'{e} dual to the origin in a one dimensional representation $\chi$ of a circle. If we did not need to find a form with support near the origin we could represent the equivariant Poincar\'{e} dual by $\chi$ itself, regarded as an equivariant differential form. However compact support will be important later, so we need to be a little more careful. \begin{lemma} \label{pd1} Let $T=U(1)$ act on ${\Bbb C }$ via a weight $\chi:T \to U(1)$. Then we can find an equivariantly closed differential form $\alpha_{\chi}\in\Omega_T^2({\Bbb C })$ on ${\Bbb C }$ with compact support arbitrarily close to 0, such that $$\int_{{\Bbb C }} \eta \alpha_{\chi} = \eta|_0 \in H^*_T$$ for all equivariantly closed forms $\eta \in \Omega_T^*({\Bbb C })$. Moreover $\alpha_{\chi} \in \chi + D(\Omega^*_T({\Bbb C }))$, so that $\alpha_{\chi}$ represents the same equivariant cohomology class on ${\Bbb C }$ as $\chi$. \end{lemma} \Proof Let $X^{\sharp}$ denote the vector field on ${\Bbb C }$ given by the infinitesimal action of $X\in \liet$. There is a $T$-invariant closed differential 1-form on ${\Bbb C }-\{0\}$, given in polar coordinates $(r,\theta)$ by $\frac{d\theta}{2\pi}$, such that $\iota_{X^{\sharp}}(\frac{d\theta}{2\pi})$ is identically equal to $\chi(X)$ for every $X\in\liet$. We can choose a smooth $T$-invariant function $b:{\Bbb C } \to [0,\infty)$ with support in an arbitrarily small neighbourhood of 0 which is identically equal to 1 on some smaller neighbourhood of 0, and let $$\alpha_{\chi}(X) = \chi(X) + D((1-b)\frac{d\theta}{2\pi}) = \chi(X) + d((1-b)\frac{d\theta}{2\pi}) + (b-1)\chi(X)$$ where $D$ is the equivariant differential defined at (3.2) and $d$ is the ordinary differential. Then $\alpha_{\chi}$ is equivariantly closed and is zero outside the support of $b$. Suppose that $\eta\in\Omega_T^*({\Bbb C })$ is equivariantly closed. We wish to show that $$\int_{{\Bbb C }} \eta \alpha_{\chi} = \eta|_0.$$ First we shall show that the integral $$\int_{{\Bbb C }} \eta \alpha_{\chi} $$ is independent of the choice of the function $b$. If $\rho>0$ is sufficiently small and $R>0$ is sufficiently large, then $b$ is identically equal to 1 on the disc $D_{\rho}$ centre 0 and radius $\rho$, and $b$ is identically equal to 0 outside the disc $D_R$ centre 0 and radius $R$. Then $$\int_{{\Bbb C }} \eta \alpha_{\chi} = \chi\int_{D_{\rho}}\eta + \int_{D_R-D_{\rho}} \eta \alpha_{\chi} .$$ Now $\eta$ is a polynomial function from $\liet$ to the ordinary de Rham complex $\Omega^*({\Bbb C })$, so we can write $$\eta = \eta^{(0)} + \eta^{(1)} + \eta^{(2)}$$ where $\eta^{(j)}$ is a polynomial function from $\liet$ to $\Omega^j({\Bbb C })$ for $j=0,1,2$. Similarly $$\alpha_{\chi} = \alpha_{\chi}^{(0)} + \alpha_{\chi}^{(1)} + \alpha_{\chi}^{(2)}$$ where $\alpha_{\chi}^{(0)}=b\chi$, $ \alpha_{\chi}^{(1)} = 0$ and $\alpha_{\chi}^{(2)} = d((1-b)\frac{d\theta}{2\pi})$. Since $D\eta = d\eta -\iota_{X^{\sharp}}\eta$ is zero, we have $d\eta^{(0)} = \iota_{X^{\sharp}}\eta^{(2)}$. As any 2-form on ${\Bbb C }$ is a $C^{\infty}$ function on ${\Bbb C }$ multiplied by the nowhere vanishing 2-form given in polar coordinates by $\frac{rd\theta dr}{2\pi}$, and since $\iota_{X^{\sharp}}(\frac{rd\theta dr}{2\pi}) = \chi(X) r dr$, it follows that $$\chi(X) \eta^{(2)}(X) = \frac{d\theta}{2\pi} d\eta^{(0)}(X)$$ on ${\Bbb C }-\{0\}$ where $d\theta$ is defined. Hence $$\int_{D_R-D_{\rho}} \eta \alpha_{\chi} = \int_{D_R-D_{\rho}} \eta^{(2)}\alpha^{(0)}_{\chi} + \eta^{(0)}\alpha^{(2)}_{\chi}$$ $$= \int_{D_R-D_{\rho}} b \frac{d\theta}{2\pi} d\eta^{(0)} + \eta^{(0)} d((1-b)\frac{d\theta}{2\pi})$$ $$= -\int_{D_R-D_{\rho}} d(b \eta^{(0)} \frac{d\theta}{2\pi}) $$ $$ = \int_{\partial D_{\rho}} b \eta^{(0)} \frac{d\theta}{2\pi} - \int_{\partial D_{R}} b \eta^{(0)} \frac{d\theta}{2\pi}$$ $$ = \int_{\partial D_{\rho}} \eta^{(0)} \frac{d\theta}{2\pi}$$ by Stokes' theorem, since $b$ is identically one on $\partial D_{\rho}$ and identically zero on $\partial D_R$. It follows that $$\int_{{\Bbb C }} \eta \alpha_{\chi}=\chi \int_{D_{\rho}} \eta + \int_{\partial D_{\rho}} \eta^{(0)} \frac{d\theta}{2\pi}$$ is independent of the choice of $b$. Now $\rho$ can be taken arbitrarily small, and $\chi \int_{D_{\rho}} \eta \to 0$ as $\rho \to 0$. Moreover by continuity, for fixed $X \in \liet$ and any $\epsilon >0$ we can choose $\rho$ so that $\eta^{(0)}$ differs from $\eta^{(0)}|_0 = \eta|_0$ by at most $\epsilon$ on $D_{\rho}$. Then $$|\int_{\partial D_{\rho}} \eta^{(0)} \frac{d\theta}{2\pi} - \eta^{(0)}|_0| = |\int_{\partial D_{\rho}} (\eta^{(0)}-\eta^{(0)}|_0) \frac{d\theta}{2\pi}| \leq \epsilon.$$ Thus $\int_{{\Bbb C }} \eta \alpha_{\chi} - \eta|_0 $ tends to zero as $\rho$ tends to 0. Since $\int_{{\Bbb C }} \eta\alpha_{\chi}$ and $\eta|_0$ are independent of $\rho$ we deduce that $$\int_{{\Bbb C }}\eta \alpha_{\chi} = \eta|_0 $$ as required. $\square$ \begin{lemma} \label{pd1.5} Let $T$ be a torus acting trivially on ${\Bbb R }$. Then we can find an equivariantly closed differential form $\alpha_{0}\in\Omega_T^*({\Bbb R })$ on ${\Bbb R }$ with compact support arbitrarily close to 0, such that $$\int_{{\Bbb R }} \eta \alpha_0 = \eta|_0 \in H^*_T$$ for all equivariantly closed forms $\eta \in \Omega_T^*({\Bbb R })$. \end{lemma} \Proof We have $\Omega_T^*({\Bbb R })=S(\liet^*)\otimes \Omega^*({\Bbb R })$ and $\eta \in S(\liet^*)\otimes \Omega^0({\Bbb R })$ is equivariantly closed if and only if it is constant on ${\Bbb R }$, so we can take $\alpha_{0}$ to be the standard volume form on ${\Bbb R }$ multiplied by any bump function compactly supported near $0$ with unit integral. $\square$ \begin{corollary} \label{pd2} Let $T$ be a torus acting linearly on ${\Bbb C }^n$ with weights $\chi_1,\ldots,\chi_n$ and trivially on ${\Bbb R }^m$. Then we can find an equivariantly closed differential form $\alpha \in \Omega_T^{2n}({\Bbb C }^n\times{\Bbb R }^m)$ on ${\Bbb C }^n\times{\Bbb R }^m$ with compact support arbitrarily close to 0, such that $$\int_{{\Bbb C }^n\times{\Bbb R }^m} \eta \alpha = \eta_0 \in H_T^*$$ for all equivariantly closed forms $\eta\in \Omega_T^*({\Bbb C }^n\times{\Bbb R }^m)$. Moreover if $m=0$ then $\alpha \in \chi_1\ldots\chi_n + D(\Omega_T^*({\Bbb C }^n))$. \end{corollary} \Proof The action of $T$ on the copy of ${\Bbb C }$ in ${\Bbb C }^n$ on which it acts via the weight $\chi_j$ factors through an action of $T/\ker \chi_j \cong U(1)$ (unless $\chi_j = 0$ in which case we can replace $\ker\chi_j$ by any subtorus of $T$ of codimension one). We can construct $\alpha_{\chi_j}\in \Omega_{U(1)}^*({\Bbb C })$ as in Lemma \ref{pd1} and $m$ copies of $\alpha_0$ as in Lemma \ref{pd1.5}, and then define $\alpha$ to be the wedge product of the pullbacks of the $\alpha_{\chi_j}$ and $\alpha_0$ to $\Omega_T^*({\Bbb C }^n\times {\Bbb R }^m)$ via the projections of ${\Bbb C }^n\times{\Bbb R }^m$ to ${\Bbb C }$ and ${\Bbb R }$ and the homomorphisms $T\to U(1)$ induced by the weights $\chi_j$. $\square$ Now we shall relax our assumption that $c$ is a central element of $K$, and assume only that $c \in T$. This will be important later when we apply induction on $n$ (see Remark \ref{induct} below). \begin{corollary} \label{pd3} Let $T$ be the maximal torus of $K=SU(n)$ acting on $K$ by conjugation. If $c \in T$ then we can find a $T$-equivariantly closed differential form $\alpha\in\Omega_T^*(K)$ on $K$ with support arbitrarily close to $c$ such that $$\int_K \eta \alpha = \eta|_c \in H^*_T$$ for all $T$-equivariantly closed differential forms $\eta\in\Omega_T^*(K)$. \end{corollary} \Proof There is a $T$-equivariant diffeomorphism $\phi$ from a $T$-invariant neighbourhood $U$ of 0 in the Lie algebra $\liek$ of $K$ to a $T$-invariant neighbourhood $V$ of $c$ in $K$ given by $$\phi (X) = c \exp(X).$$ By Corollary \ref{pd2} we can find $\tilde{\alpha} \in \Omega_T^*(\liek)$ with arbitrarily small compact support contained in $U$, such that $$\int_{\liek} \eta \tilde{\alpha} = \eta|_0 \in H^*_T$$ for all equivariantly closed forms $\eta\in \Omega^*_T(\liek)$. Then we can define $\alpha $ to be $(\phi^{-1})^*(\tilde{\alpha})$. $\square$ Note that $$M(c)= \Bigl \{ (h_1, \dots, h_{2g}, \Lambda ) \in K^{2g} \times \liek \; : \; \prod_{j = 1}^g h_{2j-1} h_{2j} h_{2j-1}^{-1} h_{2j}^{-1} = c \exp (\Lambda) \Bigr \} $$ can be expressed as $\mc = P^{-1}(c)$ where $P: K^{2g} \times \liek \to K$ is defined by $$ P \Bigl ( h_1, \dots, h_{2g}, \Lambda \Bigr ) = \prod_{j = 1}^g h_{2j-1} h_{2j} h_{2j-1}^{-1} h_{2j}^{-1} \exp (-\Lambda). $$ \begin{prop} \label{defa} If $T$ is the maximal torus of $K=SU(n)$ and $c \in T$ then there is a $T$-equivariantly closed differential form $\alpha\in\Omega^*(K^{2g}\times\liek)$ of degree $n^2 -1$ on $K^{2g}\times \liek$ with support contained in a neighbourhood of $M(c)$ of the form $P^{-1}(V)$ where $V$ is an arbitrarily small neighbourhood of $c$ in $K$, such that $$\int_{K^{2g}\times\liek} \eta\alpha = \int_{M(c)} \eta|_{M(c)} \in H^*_T$$ for any $T$-equivariantly closed form $\eta\in \Omega^*_T(K^{2g}\times\liek)$ for which the intersection of $P^{-1}(\bar{V})$ with the support of $\eta$ is compact. \end{prop} \Proof By Corollary \ref{pd3} we can find a $T$-equivariantly closed differential form $\hat{\alpha} \in \Omega_T^*(K)$ on $K$ with support in $V$ such that $$\int_K \eta \hat{\alpha} = \eta|_c \in H^*_T$$ for all $T$-equivariantly closed forms $\eta\in \Omega^*_T(K)$. Let $\alpha = P^*(\hat{\alpha})$; by the functoriality of the equivariant pushforward map (cf. Section 3 of \cite{ABMM}) this has the properties we want. $\square$ \begin{rem} In fact if $V'$ is any neighbourhood of $c$ in $K$ containing $\bar{V}$ then we have $$\int_{P^{-1}(V')} \eta\alpha = \int_{M(c)} \eta|_{M(c)} \in H^*_T$$ for any $T$-equivariantly closed form $\eta\in \Omega^*_T(P^{-1}(V'))$ on $P^{-1}(V')$ for which the intersection of $P^{-1}(\bar{V})$ with the support of $\eta$ is compact. \end{rem} \begin{rem} As we are going to use Proposition \ref{defa} to convert integrals over $M(c)$ into integrals over $K^{2g}\times \liek$ (or at least over neighbourhoods of $M(c)$ in $K^{2g}\times \liek$ of the form $P^{-1}(V)$ for arbitrarily small neighbourhoods $V$ of $c$ in $K$) we shall need to be able to extend $T$-equivariant cohomology classes $\eta$ on $M(c)$ to $T$-equivariant cohomology classes on neighbourhoods of $M(c)$ in $K^{2g}\times \liek$ of this form $P^{-1}(V)$. This will always be possible by the continuity properties of cohomology (see e.g. \cite{Dold} VIII 6.18) because $\eta$ will always have compact support in $M(c)$; more precisely we will in fact be converting integrals over $M(c)\cap(K^{2g} \times B)$ for compact subsets $B$ of $\liek$ into integrals over $P^{-1}(V)\cap(K^{2g} \times B)$. \end{rem} Note that the centre $Z_n$ of $K=SU(n)$ is a finite group of order $n$ which acts trivially on $K^{2g} \times \liek$. \begin{lemma} \label{l:5.10} Suppose that $c = \diag(c_1,\ldots,c_n) \in T$ is such that the product of no proper subsequence of $c_1,\ldots,c_n$ is 1. Then the quotient $T/Z_n$ of $T$ by the centre $Z_n$ of $K=SU(n)$ acts freely on $P^{-1}(V) \cap \mu^{-1}(0)$ for any sufficiently small $T$-invariant neighbourhood $V$ of $c$ in $K$. \end{lemma} \Proof Suppose that $T/Z_n$ does not act freely on $P^{-1}(V)\cap\mu^{-1}(0)$. Then there exist $t_1,\ldots,t_n\in {\Bbb C }$, not all equal, such that $t_1\dots t_n =1$, and some element $(h,0)=(h_1,\ldots,h_{2g},0)$ of $P^{-1}(V)\cap \mu^{-1}(0)$ fixed by $\diag(t_1,\ldots,t_n)$. Then each $h_j$ is block diagonal with respect to the decomposition of $\{1,\ldots,n\}$ as the union of $\{i:t_i = t_1\}$ and $\{i:t_i \neq t_1\}$, which implies that $$P(h,0) = \matr{A}{0}{0}{B}$$ where $A$ and $B$ are products of commutators and hence satisfy $\det A =1 = \det B$. The result follows. \hfill $\square$ \begin{rem} \label{aa} It follows from this lemma that we can extend the definition of the composition $$\Phi: H^*_T(M(c)) \to H_T^*(M(c) \cap \mu^{-1}(0)) \cong H^*(M(c) \cap \mu^{-1}(0)/T)$$ to $$\Phi: H^*_T(P^{-1}(V)) \to H_T^*(P^{-1}(V) \cap \mu^{-1}(0)) \cong H^*(P^{-1}(V) \cap \mu^{-1}(0)/T).$$ By 1.18 of \cite{GK} (see Remark \ref{arm} above), when $T=U(1)$ is a circle then $\Phi$ is given on the level of forms by $$\Phi(\eta) = \res_{X=0} \iota_v (\frac{\zeta \eta}{X-d\zeta})$$ where the vector field $v$ is the infinitesimal generator of the $U(1)$ action and $\zeta$ is a $U(1)$-invariant differential 1-form on $P^{-1}(V)\cap\mu^{-1}(0)$ such that $\iota_v(\zeta)=1$. (Strictly speaking the residue is an invariant form on $P^{-1}(V)\cap\mu^{-1}(0)$ which descends to a form on $(P^{-1}(V)\cap\mu^{-1}(0))/T$). Thus when $T=U(1)$ we have $$\int_{M(c)\cap\zloc/T} \Phi(\eta) = \int_{M(c)\cap\zloc} \res_{X=0} \frac{\zeta \eta}{X-d\zeta},$$ and it follows that if $\alpha$ is defined as in Proposition \ref{defa} for $n=2$ and $V'$ is any neighbourhood of $c$ in $SU(2)$ containing $\bar{V}$ we have $$ \int_{P^{-1}(V') \cap \mu^{-1}(0)/T} \Phi(\eta\alpha) = \int_{P^{-1}(V')\cap\zloc} \res_{X=0} \frac{\zeta \eta\alpha }{X-d\zeta} $$ $$ = \int_{M(c)\cap\zloc} \res_{X=0} \frac{\zeta \eta}{X-d\zeta} = \int_{M(c) \cap \mu^{-1}(0)/T} \Phi(\eta)$$ for any $T$-equivariantly closed differential form $\eta \in \Omega^*_T(P^{-1}(V'))$ such that the intersection of $P^{-1}(\bar{V})$ with the support of $\eta$ is compact. Here we have used the same notation for $\eta$ and its restriction to $M(c)$. When $n>2$, so that the maximal torus $T$ of $K=SU(n)$ has dimension higher than one, then $\Phi(\eta)$ and $\int_{M(c)\cap\zloc/T} \Phi(\eta)$ are given by similar formulas involving $n-1$ iterated residues (see \cite{GK}). In particular the support of $\Phi(\eta)$ is contained in the image of the support of $\eta$, and $$ \int_{P^{-1}(V') \cap \mu^{-1}(0)/T} \Phi(\eta\alpha) = \int_{M(c) \cap \mu^{-1}(0)/T} \Phi(\eta)$$ for any $T$-equivariantly closed differential form $\eta \in \Omega^*_T(P^{-1}(V'))$ such that the intersection of $P^{-1}(\bar{V})$ with the support of $\eta$ is compact. \end{rem} { \setcounter{equation}{0} } \section{Nonabelian localization applied to extended moduli spaces} Na\"{\i}ve application of the residue formula (Theorem \ref{t4.1}) to the extended moduli space $M(c)$, using (\ref{9}) and Remark \ref{r:fibprod} and ignoring the fact that $M(c)$ is noncompact and has singularities, yields \begin{equation} \label{1.7} \prod_{r = 2}^n a_r^{m_r} \exp ({f_2}) [\mnd] = n C_K \res \Biggl ( \nusym^2(X) (\int_F e^{\omega}) \sum_{\delta \in \intlat} \frac{ \prod_{r = 2}^n \tau_r(-X)^{m_r}e^{(\tc - \delta) (X)} } {((-1)^{n_+} \nusym^{2}(X))^g} \Biggr ) \end{equation} where the constant $C_K$ is defined at (3.7). The main problem with (\ref{1.7}) (related to the noncompactness of $\xc$, which permits the fixed point set $M(c)^T$ to be the union of infinitely many components $F_\delta$) is that the sum over $\delta$ does not converge for $\xvec \in \liet$. In this section we shall instead apply the version of nonabelian localization due to Guillemin-Kalkman and Martin (Propositions 3.6 and 3.9) to $M(c)$, using Remarks \ref{leg} and \ref{arm}; this will lead to a proof that (\ref{1.7}) is true if interpreted appropriately (see Remark \ref{naive}). First we use Proposition \ref{p:sm}. \begin{lemma} \label{l3} Let $|W|=n!$ be the order of the Weyl group $W$ of $K=SU(n)$, and let $c=\diag(e^{2\pi i d/n},\ldots,e^{2\pi i d/n})$ where $d$ is coprime to $n$. If $V$ is a sufficiently small neighbourhood of $c$ in $K$ that the quotient $T/Z_n$ of $T$ by the centre $Z_n$ of $K=SU(n)$ acts freely on $P^{-1}(V) \cap \zloc$ (see Lemma \ref{l:5.10}), then for any $\eta\in\hk(X)$ we have $$ \int_{\mnd} \Phi(\eta e^{ \bom} ) = \frac{1}{|W|} \int_{N(c)} \Phi(\nusym\eta e^{ \bom} ) = \frac{1}{|W|}\int_{N(V)} \Phi(\nusym \eta e^{ \bom}\alpha)$$ where $$N(c)=\xc \cap \mu^{-1}(0)/T $$ for $\mu:K^{2g} \times \liek \to \liek$ given by minus the projection onto $\liek$ and $$N(V) = P^{-1}(V)\cap \zloc /T.$$ Also $\alpha$ is a $T$-equivariantly closed form on $K^{2g}\times \liek$ representing the $T$-equivariant Poincar\'{e} dual to $M(c)$, which is chosen as in Proposition \ref{defa} so that the support of $\alpha$ is contained in $P^{-1}(V)$ and has compact intersection with $\zloc$. \end{lemma} \Proof Since $\mnd = \xc \cap \zloc/K$, we can first identify $\int_{\mnd}\Phi(\eta e^{ \bom} ) $ with $$ \frac{1}{|W|} \int_{N(c)} \Phi(\nusym\eta e^{ \bom} ) $$ via Proposition \ref{p:sm}, whose proof works in this situation even though $M(c)$ is noncompact and singular, because $\mu$ is proper and $M(c)$ is nonsingular in a neighbourhood of $\mu^{-1}(0)$ (see Remark \ref{leg}). Then we use Remark \ref{aa}. \hfill $\square$ Next we need to summarize some conventions on the roots and weights of $SU(n)$. The simple roots $\{e_j: j = 1, \dots, n-1\} $ of $SU(n)$ are elements of $\liets$; in terms of the standard identification of $\liet$ with $\{ (X_1, \dots, X_n ) \in {\Bbb R }^n: \sum_i X_i = 0 \}$ under which $(X_1, \dots, X_n ) \in {\Bbb R }^n$ satisfying $ \sum_i X_i = 0$ corresponds to $X={\rm diag}({2\pi i X_1},\ldots,{2\pi i X_n})\in \liet$, they are given by \begin{equation} \label{6.1} e_j(X) = X_j - X_{j+1}. \end{equation} The dual basis to the basis of simple roots (with respect to the inner product $<\cdot,\cdot>$ defined at (\ref{1.02}) above, which is the usual Euclidean inner product on ${\Bbb R }^n$) is the set of {\em fundamental weights} $w_j \in \liets$ given by \begin{equation} \label{5.fundwts} w_j(X) = X_1 + \dots + X_j. \end{equation} If we use this same inner product to identify $\liets$ with $\liet$, the simple roots become identified with a set of generators $$\he{j}=(0,\ldots,0,1,-1,0,\ldots,0)$$ for the integer lattice $\intlat$ of $\liet$, and the fundamental weights correspond to elements $\hw{j} \in \liet$ given by $$\hw{j} = (1, \dots, 1, 0, \dots, 0 ) - \frac{j}{n}(1, \dots, 1). $$ In particular we have \begin{equation} \hat{w}_{n-1} = \frac{1}{n}(1, \dots, 1, - (n-1) ). \end{equation} Since we shall later apply induction on $n$, it will be convenient to label certain spaces, groups and Lie algebras by the associated value of $n$. In particular the space $M(c )$ will sometimes be denoted by $M_n(c)$, the maximal torus $T$ of $SU(n)$ by $\tor$, its Lie algebra $\liet$ by $\liet_n$, and the map $\Phi$ by $\Phi_n$. We define a one dimensional torus $\tone \cong S^1$ in $SU(n) $ generated by $\he{1}$: it is identified with $S^1 $ via \begin{equation} t \in S^1 \mapsto ( t, t^{-1},1, \dots, 1 ) \in \tone. \end{equation} The (one dimensional) Lie algebra $\lietone$ is spanned by $\he{1}$. Its orthocomplement in $\liet$ is \begin{equation} \lietc = \{ (X_1, \dots, X_n) \in {\Bbb R }^n: X_{1} = X_2, \sum_{j = 1}^{n} X_j = 0 \}. \end{equation} Define $\torc $ to be the torus given by $\exp (\lietc)$: $$ \torc = \{ (t_1,t_{1},t_3 \dots, t_{n-1}, t_{n} ) \in U(1)^{n} : \, \, (t_1)^2(\prod_{j = 3}^{n} t_j ) = 1 \}; $$ then $\torc$ is isomorphic to the maximal torus of $SU(n-1)$ (i.e. $\torc \cong (S^1)^{n-2} $) so this does not conflict with the notation already adopted. \begin{rem} The multiplication map \label{f6.4} $\tone \times \torc \to \tone \torc = \tor $ is a covering map with fibre $\tone \cap \torc = {\Bbb Z }_2 = \{(t,t^{-1},1, \dots, 1 ): \;\; t = t^{-1}\}. $ \end{rem} There is the following decomposition of the ring homomorphism $ \phil$. \begin{prop} \label{stages} For any symplectic manifold $M$ equipped with a Hamiltonian action of $T_n$ such that $\torc$ acts locally freely on $\mu_{\torc}^{-1}(0)$, the symplectic quotient $\mu_{\tor}^{-1}(0)/\tor$ may be identified with the symplectic quotient of $\mu_{\torc}^{-1}(0)/\torc$ by the induced Hamiltonian action of $\tone$. Moreover if in addition $T_n$ acts locally freely on $\mu_{\tor}^{-1}(0)$ then the ring homomorphism $\phil: H^*_\tor(M) \to H^*( \mu_\tor^{-1}(0)/\tor)$ factors as $$ \phil = \phione \circ \phicom$$ where $$\phicom: H^*_{\tor}(M) \to H^*_{\tor}(\mu_{\torc}^{-1}(0)) \cong H^*_{\tone\times T_{n-1}}(\mu_{\torc}^{-1}(0)) \cong H^*_\tone (\mu_\torc^{-1}(0)/\torc) $$ and $$ \phione: H^*_\tone (\mu_\torc^{-1}(0)/\torc ) \to H^*\Bigl((\mu_\torc^{-1}(0) \cap \mu_\tone^{-1}(0)/\torc\times \tone \Bigr ) \cong H^*(\mu_\tor^{-1}(0)/\tor). $$ \end{prop} \Proof The isomorphisms $$ H^*_{\tor}(\mu_{\torc}^{-1}(0)) \cong H^*_{\tone\times T_{n-1}}(\mu_{\torc}^{-1}(0)) \cong H^*_\tone (\mu_\torc^{-1}(0)/\torc) $$ follow from Remark \ref{f6.4} and the fact that the cohomology with complex coefficients of the classifying space of a finite group is trivial. Since $\mu_{\tor}$ is a $\tor$-invariant map, its projection $\mu_{\tone}$ onto $\lietone$ descends to $\mu_{\torc}^{-1}(0)/\torc$ and defines a moment map for the induced $\tone$-action with respect to the induced symplectic structure on $\mu_{\torc}^{-1}(0)/\torc$. The rest then follows from Remark \ref{f6.4} and naturality (cf. \cite{GK}, after (2.9)).\hfill $\square$ \begin{rem} \label{induct} {}From now on, thanks to Lemma \ref{l3}, we shall be working with quotients by $T$ and subgroups of $T$, rather than quotients by $K$. Because of this our arguments will apply to $M(c)$ when $c$ belongs to $T$ but is no longer necessarily a central element of $K$. This will be important later, when we apply induction on $n$ using Proposition \ref{stages}. The only condition we will need to impose on $c\in T$ is that $c = \diag(c_1,\ldots,c_n)$ where the product of no proper subsequence of $(c_1,\ldots,c_n)$ is 1; this is certainly true for our original choice of $c$ when $c_j = e^{2\pi i d/n}$ for all $j$ with $d$ coprime to $n$. So for any $c\in T$, let us define $$M(c) = M_n(c) = P^{-1}(c) = \Bigl \{ (h_1, \dots, h_{2g}, \Lambda ) \in K^{2g} \times \liek \; : \; \prod_{j = 1}^g h_{2j-1} h_{2j} h_{2j-1}^{-1} h_{2j}^{-1} = c \exp (\Lambda) \Bigr \} $$ where $P: K^{2g} \times \liek \to K$ is defined by $$ P \Bigl ( h_1, \dots, h_{2g}, \Lambda \Bigr ) = \prod_{j = 1}^g h_{2j-1} h_{2j} h_{2j-1}^{-1} h_{2j}^{-1} \exp (-\Lambda). $$ Let us also define \begin{equation} \nl(c) = M(c ) \cap \mu^{-1}(0) / \tor \end{equation} and \begin{equation} \nl(V) = P^{-1}(V) \cap \mu^{-1}(0) / \tor \end{equation} where $V$ is a small $T$-invariant neighbourhood of $c$ in $K$. \end{rem} \begin{prop} \label{p7.1} \label{f6.6} Suppose $c = \diag (c_1, \dots, c_n)\in T$ is such that the product of no proper subset of $(c_1,\ldots,c_n)$ is 1. Then the group $\torc/{\Bbb Z }_n$, where ${\Bbb Z }_n$ consists of the identity matrix multiplied by $n$th roots of unity, acts freely on $P^{-1}(V) \cap \mu_{\torc}^{-1} (0)$ for any sufficiently small $T$-invariant neighbourhood $V$ of $c$ in $K$. Hence the quotient $P^{-1}(V) \cap \mu_{\torc}^{-1} (0)/\torc$ is smooth. \end{prop} \Proof The conjugation action of $(t_1, \dots, t_n) \in U(1)^n$ on the space of $n \times n$ matrices sends $$ (A_{ij}) \mapsto (t_i t_j^{-1} A_{ij}). $$ Clearly ${\Bbb Z }_n$ acts trivially. Let us assume that $(h, \Lambda) \in M(c) \cap \mu_{\torc}^{-1}(0) $ is fixed by the action of some element of $\torc$ which is not in ${\Bbb Z }_n$. After rearranging the coordinates $X_3, \ldots, X_n$ if necessary, we may assume that there is some $k$ between $3$ and $n$ such that this element of $\torc$ is of the form $(t_1, t_1, t_3, \ldots, t_{n-1}, t_{n})$ where $t_i = t_{1}$ if and only if $i\leq k$. Then each $h_j $ is block diagonal of the form $$\matr{h_j^1}{0}{0}{h_j^2} $$ where $h_j^1$ is a $k\times k$ matrix and $h_j^2$ is $(n-k)\times(n-k)$. As the determinant of any commutator is one, it follows that $\prod_{j=1}^n [h_{2j-1}, h_{2j}] $ is block diagonal of the form \begin{equation} \label{7.0} \matr{A}{0}{0}{B} \end{equation} where $\det A = \det B = 1$. But $\Lambda$ is also block diagonal of the same form $$\matr{\Lambda_1}{0}{0}{\Lambda_2} ,$$ and since $(h, \Lambda) \in \mu_{\torc}^{-1}(0) $ the diagonal entries of $\Lambda$ are $(2\pi i\lambda, -2\pi i\lambda,0, \dots, 0) $ for some $\lambda \in {\Bbb R }$. Thus as $k\geq 3$ both $\Lambda_1$ and $\Lambda_2$ have trace $0$, so $\det\exp\Lambda_1 = 1 = \det\exp\Lambda_2$. Since $(h, \Lambda) \in M(c)$ it follows that the matrix $A$ must equal $$ \diag (c_{1}, \ldots, c_{k})\exp\Lambda_1, $$ and hence $$c_1\ldots c_k = \det A =1.$$ This contradiction to the hypotheses on $c$ shows that $\torc/{\Bbb Z }_n$ acts freely on $M(c)\cap \mu_{\torc}^{-1}(0)$, and the same argument shows that $\torc/{\Bbb Z }_n$ acts freely on $P^{-1}(V) \cap \mu_{\torc}^{-1}(\hat{V})$ for any sufficiently small $T$-invariant neighbourhood $V$ of $c$ in $K$ and any sufficiently small neighbourhood $\hat{V}$ of 0 in $\liet_{n-1}$. The result follows. \hfill $\square$ \begin{definition}\label{YY} Let us introduce coordinates $$Y_k = e_k(X) = \inpr{\he{k},X} $$ on $\liet$, corresponding to the simple roots $e_k \in \liets$. \end{definition} We are now in a position to exploit Proposition \ref{p:gkm} and Remark \ref{arm}, by using the translation map $s_{\tran} $ defined by Lemma \ref{l4.3}, where $\tran = \he{1}$ lies in the integer lattice $\intlat$ and so satisfies $\exp(\Lambda_0)=1$. \begin{lemma} \label{old5.17} \label{l6.4} Suppose $c = \diag (c_1, \dots, c_n)\in T$ is such that the product of no proper subset of $(c_1,\ldots,c_n)$ is 1. Suppose also that $\eta$ is a polynomial in the $\tar$ and $\tbr$, so that $s_{\he{1}}^*\eta = \eta$. If $V$ is a sufficiently small $T$-invariant neighbourhood of $c$ in $K$ so that $P^{-1}(V) \cap \mu^{-1}(\lietone)/\torc$ is smooth (see Proposition \ref{f6.6}), and if $\nl(V) = P^{-1}(V) \cap \mu^{-1}(0)/\tor$ as before, then $$\int_{\nl(V)} \phil( \eta e^{ \bom} e^{ - Y_1 } \alpha ) = \int_{P^{-1}(V) \cap \mu^{-1}(- \he{1} )/\tor} \phil (\eta e^{ \bom}\alpha ) $$ $$ = \int_{\nl(V)} \phil (\eta e^{ \bom}\alpha ) - n_0 \sum_{F\in{\mbox{$\cal F$}}: -||\he{1}||^2 < \inpr{\he{1},\mu(F)} < 0} \res_{Y_1 = 0 } \int_F \frac{ \phicom ( \eta e^{ \bom}\alpha)} {e_{{F}} } $$ where ${\mbox{$\cal F$}}$ is the set of components of the fixed point set of the action of $\tone$ on $P^{-1}(V)\cap \mu^{-1}(\lietone)/\torc$, and $e_F$ denotes the $\tone$-equivariant Euler class of the normal to $F$ in $P^{-1}(V)\cap \mu^{-1}(\lietone)/\torc$ for any $F\in{\mbox{$\cal F$}}$, while $n_0$ is the order of the subgroup of $\tone/\tone\cap\torc$ that acts trivially on $P^{-1}(V) \cap \mu^{-1}(\lietone)/\torc$. Also $\alpha$ is the $T$-equivariantly closed differential form on $K^{2g}\times \liek$ given by Proposition \ref{defa} which represents the equivariant Poincar\'{e} dual of $M(c)$, chosen so that the support of $\alpha$ is contained in $P^{-1}(V)$. \end{lemma} \Proof Since $\mu^{-1}(\lietone)=K^{2g}\times\lietone$ is contained in $\mu_{\torc}^{-1}(0)$, it follows from Proposition \ref{f6.6} that if $V$ is a sufficiently small $T$-invariant neighbourhood of $c$ in $K$, then $\torc/{\Bbb Z }_n$ acts freely on $P^{-1}(V)\cap \mu^{-1}(\lietone)$ and so the quotient $P^{-1}(V)\cap \mu^{-1}(\lietone)/\torc$ is smooth. Since the restriction of $\mu_{\tone}$ to $\mu^{-1}(\lietone)$ is proper, and the support of $\alpha$ is contained in $P^{-1}(V)$, by Remark \ref{arm} Guillemin and Kalkman's proof of Proposition \ref{p:gkm} can be applied to the $\tone$-invariant function induced by $\mu_{\tone}$ on the smooth manifold $P^{-1}(V)\cap \mu^{-1}(\lietone)/\torc$ and the $\tone$-equivariant form induced by $\eta e^{\bom}\alpha$. In fact since $\tone\cap\torc \cong {\Bbb Z }_2$ acts trivially we can work with the action of $\tone/\tone\cap\torc$ instead of the action of $\tone$ (the Lie algebra and moment map are of course the same). This fits better with the choice of coordinates $Y_k$ defined by the simple roots $\hat{e}_k$ because the simple root $\hat{e}_{1}$ takes $(t,t^{-1},1,\ldots,1)\in\tone$ to $t^2$ and thus induces an isomorphism from $\tone/\tone\cap\torc$ to $S^1$. By combining this with Proposition \ref{stages} we get $$ \int_{P^{-1}(V) \cap \mu^{-1} (0 )/\tor } \phil ( \eta e^{ \bom} \alpha) - \int_{P^{-1}(V) \cap \mu^{-1} (- \he{1} )/\tor } \phil ( \eta e^{ \bom}\alpha) $$ $$ ~~~= n_0 \res_{Y_1 = 0} \sum_{F\in{\mbox{$\cal F$}}: -||\he{1}||^2 < \inpr{\he{1},\mu(F)} < 0} \int_{{F}} \frac{\phicom ( \eta e^{ \bom}\alpha) }{e_{{F}} }. $$ Now note that the restriction of $P:K^{2g}\times\liek \to K$ to $\mu^{-1}(\liet) = K^{2g}\times \liet$ is invariant under the translation $s_{\Lambda_0}$ for $\Lambda_0 \in \Lambda^I$. Therefore by construction the restriction of $\alpha$ to $\mu^{-1}(\liet)$ is also invariant under this translation. Thus by (\ref{8.4}) and Definition \ref{YY} $$\int_{P^{-1}(V) \cap \mu^{-1} (-\he{1})/\tor} \phil (\eta e^{ \bom} \alpha ) = \int_{\nl(V)} \phil \Bigl ( s_{\he{1}}^* (\eta e^{ \bom}\alpha ) \Bigr ) = \int_{\nl(V)} \phil (\eta e^{ \bom} e^{ - Y_1}\alpha ). $$ The result follows. \hfill $\square$ \begin{rem} It will follow from the proof of Proposition \ref{p7.2} below that $n_0=1$ here (see Remark \ref{n0=2}). \end{rem} { \setcounter{equation}{0} } \section{Fixed point sets of the circle action} In this section we shall consider the components $F\in{\mbox{$\cal F$}}$ of the fixed point set of the action of $\tone$ on the quotient $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ (which appeared in Lemma \ref{old5.17}). Since $P^{-1}(c)=M(c)$ and $V$ is an arbitrarily small $T$-invariant neighbourhood of $c$ in $K$, we may assume that every $F\in{\mbox{$\cal F$}}$ contains a component of the fixed point set of the action of $\tone$ on $M(c) \cap\mu^{-1}(\lietone)/\torc$, and each of these components is contained in a unique $F\in{\mbox{$\cal F$}}$. So we shall start by analysing the components of the fixed point set of the action of $\tone$ on $M(c) \cap\mu^{-1}(\lietone)/\torc$. We shall find that they can be described inductively in terms of products of spaces of the form $N(c)$ (see Remark \ref{induct}) for smaller values of $n$. This will enable us to use induction in the next two sections to express the intersection pairings $\int_{\mnd} \Phi(\eta e^{\bom})$ on the moduli spaces $\mnd$ as iterated residues (see Theorem \ref{mainab} and Theorem \ref{t9.6}). \begin{prop} \label{p7.2} Suppose that $c=\diag(c_1,\ldots,c_n)\in SU(n)$ is such that the product of no proper subsequence of $(c_1,\ldots,c_n)$ is 1. Then the components of the fixed \label{fil} point set of the action of $\tone$ on the quotient $(M(c) \cap\mu^{-1}(\lietone))/\torc$ may be described as follows. For any subset $I$ of $\{ 3, \dots, n\}$ let $I_1=I\cup\{1\}$ and let $I_2= \{ 1, \dots, n\} - I_1$. Let $H_I$ be the subgroup of $SU(n)$ given by $$H_I = \{ (a_{ij}) \in SU(n): a_{ij} = 0 ~\mbox{if $(i,j) \in (I_1\times I_2)\cup (I_2\times I_1)$ } \}. $$ Suppose that $\lambda \in {\Bbb R }$ is a solution of $$ e^{-2\pi i \lambda} = c_{i_1} \dots c_{i_{r}} = \prod_{j\in I_1} c_j$$ where $r$ is the number of elements of $I_1=\{i_1,\ldots,i_r\}$, so that $$ e^{2\pi i \lambda} = \prod_{j\in I_2} c_j.$$ Then we have a component of the fixed point set given by $\fisml = \tilde{F}_{I,\lambda}/\torc$ where $$\tilde{F}_{I,\lambda} = M(c) \cap (H_I^{2g} \times \{\lambda \hat{e}_1 \}),$$ and every component is of this form for some subset $I$ of $\{3,\ldots, n\}$ and solution $\lambda$ to the equation above. \end{prop} \Proof Suppose the $\tone$ orbit of a point $ (h_1, \dots, h_{2g}, \Lambda) \in SU(n)^{2g} \times \lietone$ is contained in its orbit under $\torc$. A general element of the $\tone$ orbit of an $n\times n$ matrix $ A = (a_{ij})$ under conjugation looks like $$ \left \lbrack \begin{array}{lcccr} a_{11} & t^2a_{12} & ta_{13} & \dots & t a_{1n} \\ t^{-2}a_{21} & a_{22} & t^{-1}a_{23} & \dots & t^{-1} a_{2n} \\ t^{-1}a_{31} & ta_{32} & a_{33} & \dots & a_{3n} \\ \vdots & \vdots & \vdots & \dots & \vdots \\ t^{-1} a_{n1}& ta_{n2} & a_{n3} & \dots & a_{nn} \end{array}\right \rbrack $$ while a general element of the $\torc$ orbit of $A$ looks like $$ \left \lbrack \begin{array}{lcccr} a_{11} & a_{12} & t_1 t_3^{-1}a_{13} & \dots & t_1 t_n^{-1} a_{1n} \\ a_{21} & a_{22} & t_1 t_3^{-1}a_{23} & \dots & t_1 t_n^{-1} a_{2n} \\ t_3 t_1^{-1} a_{31} & t_3 t_1^{-1}a_{32} & a_{33} & \dots & t_3 t_n^{-1} a_{3n} \\ \vdots & \vdots & \vdots & \dots & \vdots \\ t_n t_1^{-1} a_{n1}&t_n t_1^{-1} a_{n2} & t_n t_3^{-1} a_{n3} &\dots & a_{nn} \end{array}\right \rbrack . $$ For each $t$ there exist $t_1,t_3,\ldots,t_n$ such that these two matrices are equal when $A$ is any of $h_1,\ldots, h_{2g}$ and $\Lambda$. Choose $t\neq t^{-1}$ and let $I$ denote the set of $j$ in $\{ 3, \dots, n\}$ for which $t_1 t_j^{-1} = t$. Similarly, define $J$ to be the set of $j$ in $\{ 3, \dots, n\}$ for which $t_1 t_j^{-1} = t^{-1}$, and let $K = \{ 3, \dots, n\} - I - J.$ Reordering the coordinates one finds that all the $h_j$ and $\Lambda$ are block diagonal where the blocks correspond to $ I \cup \{1\}$, $ J \cup \{2\} $ and $K$. Conversely, if all the $h_j$ are block diagonal of this form and $\Lambda\in\lietone$, then the $\tone$ orbit of $(h_1,\ldots,h_{2g},\Lambda)$ is contained in its $T_{n-1}$ orbit since given any $t\in U(1)$ we can find $(t_1, t_1, t_3, \ldots,t_n)$ in $T_{n-1}$ satisfying $t_1 t_j^{-1} = t$ if $j\in I$ and $t_1 t_j^{-1} = t^{-1}$ if $j \in J.$ We next prove that $K$ is empty. Suppose otherwise; then as the determinant of any commutator is one, $\det \prod_{j=1}^g [h_{2j-1}^{[K]}, h_{2j}^{[K]}] =1 $ (where the superscript $[K]$ denotes the block of the matrix corresponding to $K$). Thus the $K$ block in $c $ also has determinant $1$. This is impossible by the hypothesis on $c$. Suppose now that $(h_1, \dots, h_{2g}, \Lambda) $ $ \in M(c)\cap \mu^{- 1}(\lietone)$ lies in $H_I^{2g} \times \lietone.$ Then $$\Lambda = \lambda \he{1} = 2 \pi i {\rm diag}(\lambda,-\lambda,0,\ldots,0)$$ for some $\lambda \in {\Bbb R }$, so the blocks $\Lambda^{[I_1]}$ and $\Lambda^{[I_2]}$ of $\Lambda$ corresponding to $I_1 = I \cup \{1\}$ and $I_2 = J \cup \{2\}$ satisfy $\det\exp\Lambda^{[I_1]} = e^{2 \pi i \lambda}$ and $\det\exp\Lambda^{[I_2]}= e^{-2 \pi i \lambda}$. But $$\det (\prod_{j=1}^g [h_{2j-1}^{[I_1]}, h_{2j}^{[I_1]}])=1= \det (\prod_{j=1}^g [h_{2j-1}^{[I_2]}, h_{2j}^{[I_2]}]) $$ because the determinant of any commutator is one. It therefore follows from the definition of $M(c)$ that $$ e^{-2\pi i \lambda} = \prod_{j\in I_1} c_j.$$ This is enough to complete the proof. $\square$ \begin{rem} \label{n0=2} The proof of this proposition shows that the elements of $\tone$ which act trivially on the quotient $(M(c) \cap\mu^{-1}(\lietone))/\torc$ are precisely those represented by $t$ satisfying $t=t^{-1}$, i.e. $t=\pm 1$, or equivalently those in $\tone\cap \torc$. Thus the size $n_0$ of the subgroup of $\tone/\tone\cap\torc$ acting trivially on the quotient $M(c) \cap\mu^{-1}(\lietone))/\torc$ is 1 (cf. Lemma \ref{old5.17}). \end{rem} \begin{prop} \label{p7.3} Suppose that $c=\diag(c_1,\ldots,c_n)\in SU(n)$ is such that the product of no proper subsequence of $(c_1,\ldots,c_n)$ is 1. Suppose that $I$ is a subset of $\{3,\ldots,n\}$ with $r-1$ elements where $1\leq r\leq n-1$, and let $I_1 = I \cup \{1\} = \{i_1,\ldots,i_r\}$ and $I_2 = \{1,\ldots,n \}- I_1 = \{i_{r+1},\ldots, i_n\}$. Suppose also that $\lambda \in {\Bbb R }$ is a solution of $$ e^{-2\pi i \lambda} = \prod_{j\in I_1} c_j$$ so that $ e^{2\pi i \lambda} = \prod_{j\in I_2} c_j.$ Let $$ c(I_1,\lambda) = \diag(c^{I,\lambda}_{i_1},\ldots, c^{I,\lambda}_{i_r})$$ and $$ c(I_2, -\lambda) = \diag(c^{I,\lambda}_{i_{r+1}}, \ldots, c^{I,\lambda}_{i_n})$$ where $ c^{I,\lambda}_j = c_j$ if $j \geq 3$, while $ c^{I,\lambda}_{1} = c_{1} e^{2\pi i \lambda}$ and $c^{I,\lambda}_2 = c_2 e^{-2\pi i \lambda}.$ Let $\fisml$ be defined as in Proposition \ref{p7.2}. Then there is a finite to one (in fact $(r(n-r))^{2g}$ to one) surjective smooth map $$ \Psi_{I,\lambda}: (S^1)^{2g} \times N_r (c(I_1,\lambda)) \times N_{n-r} (c(I_2,-\lambda)) \to \fisml. $$ \end{prop} \Proof We define a homomorphism $$\rho_I: S^1 \times SU(r) \times SU(n-r) \to H_I \subset SU(n)$$ given by $$\rho_I : (s,A,B) \mapsto \matr{s^{n-r} A}{0}{0}{s^{-r} B} $$ with respect to the decomposition of $\{1,\ldots,n\}$ as $I_1\cup I_2$. Note that $\rho_I$ restricts to an $r(n-r)$ to one surjective homomorphism $$\rho_I:S^1 \times T_r \times T_{n-r} \to \tor.$$ If $\epsr{r}: SU(r)^{2g} \to SU(r)$ is defined by $ \epsr{r} (h_1, \dots, h_{2g}) = \prod_{j=1}^g [h_{2j-1}, h_{2j}] $ then \begin{equation} \label{7.4} \epsr{n}\Bigl (\rho_I(s_1, A_1, B_1), \dots, \rho_I(s_{2g}, A_{2g}, B_{2g}) \Bigr ) = \matr{\epsr{r} (A_1, \dots, A_{2g}) } {0}{0}{ \epsr{n-r}(B_1, \dots, B_{2g} ) }. \end{equation} Let us define a map $$ \Psi_{I,\lambda}: (S^1)^{2g} \times N_r (c(I_1,\lambda)) \times N_{n-r} (c(I_2,-\lambda)) \to \fisml$$ as the quotient of $$\tpsi_{I,\lambda}: (S^1)^{2g} \times \Bigl (\mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\lambda)) \Bigr ) \times \Bigl (\mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\lambda)) \Bigr ) \to \tilde{F}_{I,\lambda} $$ defined by $$\tpsi_{I,\lambda} \Bigl ( (s_1, \dots, s_{2g}), (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}, 0), (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}, 0) \Bigr ) $$ $$ = \Bigl ( \rho_I (s_1, h_{1}^{[I_1]}, h_{1}^{[I_2]}),\rho_I (s_{2}, h_{2}^{[I_1]}, h_{2}^{[I_2]} ), \dots, \rho_I (s_{2g},h_{2g}^{[I_1]}, h_{2g}^{[I_2]} ), 2\pi i \diag(\lambda, -\lambda,0,\ldots, 0) \Bigr ). $$ Here, $\tilde{F}_{I,\lambda}$ was defined in Proposition \ref{p7.2}. We must check that the image of $\tpsi_{I,\lambda}$ is contained in $\tilde{F}_{I,\lambda}$. We have $$\epsr{r} (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}) = \diag(c^{I,\lambda}_{i_1},\ldots, c^{I,\lambda}_{i_r}) = c(I_1,\lambda)$$ and $$ \epsr{n-r} (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}) =\diag(c^{I,\lambda}_{i_{r+1}}, \ldots, c^{I,\lambda}_{i_n}) = c(I_2,-\lambda). $$ In order to show that $\tpsi_{I,\lambda} \Bigl ( (s_1, \dots, s_{2g}), (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}, 0), (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}, 0) \Bigr ) $ lies in $\tilde{F}_{I,\lambda}$ we need to check that if $\Lambda = 2\pi i \diag(\lambda,-\lambda,0,\ldots,0)$ then $ c \exp ( \Lambda)$ is block diagonal of the form $$ \matr{c(I_1,\lambda) }{0}{0} {c(I_2,-\lambda)} $$ with respect to the decomposition of $\{1,\ldots,n\}$ as $I_1\cup I_2$. This follows by the choice of $c^{I,\lambda}_1,\ldots,c^{I,\lambda}_n$. We must also check that the map $\Psi_{I,\lambda}$ is well defined on the quotient by the action of $T_{r}\times T_{n-r}$: in other words we must check that for any $t=(t_1, \dots,t_n)$ $\in U(1)^n$ satisfying $t_{i_1}, \dots, t_{i_r} = t_{i_{r+1}}, \dots, t_{i_n} = 1 $ so that $t_{I_1}=(t_{i_1} \dots t_{i_r})\in T_r$ and $t_{I_2} =( t_{i_{r+1}} \dots t_{i_n} ) \in T_{n-r}$, we have $$\tpsi_{I,\lambda} \Bigl ( (s_1, \dots, s_{2g}), (t_{I_1}h_1^{[I_1]}(t_{I_1})^{-1}, \dots, t_{I_1}h_{2g}^{[I_1]} (t_{I_1})^{-1}, 0), (t_{I_2}h_1^{[I_2]}(t_{I_2})^{-1}, \dots, t_{I_2}h_{2g}^{[I_2]} (t_{I_2})^{-1}, 0) \Bigr ) $$ \begin{equation} \label{7.009} = \tilde{t} \tpsi_{I,\lambda} \Bigl ( (s_1, \dots, s_{2g}), (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}, 0), (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}, 0) \Bigr ) \end{equation} for some $\tilde{t} = (\tilde{t}_1, \dots, \tilde{t}_n) $ satisfying $\prod_{j = 1}^n \tilde{t}_j = 1$ and $\tilde{t}_{1} = \tilde{t}_2$. For any $s \in U(1)$, we may conjugate all the $h_j^{[I_1]}$ by $s^{n-r}$ and all the $h_j^{[I_2]}$ by $s^{-r} $ without changing the image under $\tpsi_{I,\lambda} $; choosing $s$ so that $s^{n-r} \tilde{t}_{1} = s^{-r} \tilde{t}_2$ we find that the equation (\ref{7.009}) is satisfied for $\tilde{t}_j = t_j s^{n-r} $ (when $j \in I_1$) and $\tilde{t}_j = t_{j} s^{-r} $ (when $j \in I_2$). To show that $\Psi_{I,\lambda}$ is finite-to-one and surjective, suppose that $(h_1,\ldots,h_{2g},\Lambda) \in \tilde{F}_{I,\lambda} $; we must check that a finite (and nonzero) number of $T_r \times T_{n-r}$ orbits in $$(S^1)^{2g} \times \Bigl (\mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\lambda)) \Bigr ) \times \Bigl (\mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\lambda)) \Bigr )$$ map into the $\torc$ orbit of $(h_1,\ldots,h_{2g},\Lambda)$. Now by the definition of $\tilde{F}_{I,\lambda}$ we have $\Lambda = 2 \pi i \diag (\lambda,-\lambda,0,\ldots,0)$ and each $h_j$ is block diagonal of the form $$\matr{\ha{j}}{0}{0}{\hb{j}} $$ with respect to the decomposition of $\{1,\ldots,n\}$ as $I_1\cup I_2$. So $$\tpsi_{I,\lambda} \Bigl ( (s_1, \dots, s_{2g}), (H_1^{[I_1]}, \dots, H_{2g}^{[I_1]}, 0), (H_1^{[I_2]}, \dots, H_{2g}^{[I_2]}, 0) \Bigr ) $$ belongs to the $\torc$ orbit of $(h_1,\ldots,h_{2g},\Lambda)$ if and only if there is some $\tilde{t} = (\tilde{t}_1, \dots, \tilde{t}_n) $ satisfying $\prod_{j = 1}^n \tilde{t}_j = 1$ and $\tilde{t}_{1} = \tilde{t}_2$ such that $$s_j^{n-r} H_j^{[I_1]} = \tilde{t}_{I_1} h_j^{[I_1]}(\tilde{t}_{I_1})^{-1}$$ and $$s_j^{-r} H_j^{[I_2]} = \tilde{t}_{I_2} h_j^{[I_2]}(\tilde{t} _{I_2})^{-1}$$ where $\tilde{t}_{I_1}=(\tilde{t}_{i_1}, \dots ,\tilde{t}_{i_r})\in T_r$ and $\tilde{t}_{I_2} =( \tilde{t}_{i_{r+1}}, \dots, \tilde{t}_{i_n} ) \in T_{n-r}$. Since $\det H_j^{[I_1]} =1 = \det H_j^{[I_2]} $ and $\det h_j^{[I_1]} \det h_j^{[I_2]} =1$, by the argument of the previous paragraph this happens if and only if $(s_j)^{r(n-r)} = \det h_j^{[I_1]}$ and $s_j^{n-r} H_j^{[I_1]}$ is conjugate to $ h_j^{[I_1]}$ and $s_j^{-r} H_j^{[I_2]}$ is conjugate to $h_j^{[I_2]}$. Thus $\Psi_{I,\lambda}$ is surjective and $(r(n-r))^{2g}$ to one. $\square$ \begin{rem} Note that by the definition of $c_j^{I,\lambda}$ (see Proposition \ref{p7.3}) no proper subsequence of $(c^{I,\lambda}_{i_1},\ldots,c^{I,\lambda}_{i_r})$ or $(c^{I,\lambda}_{i_{r+1}},\ldots,c^{I,\lambda}_{i_n})$ has product equal to 1, because the same is true of $(c_1,\ldots,c_n)$. \end{rem} \begin{rem} \label{big} It follows from Proposition \ref{p7.3} that if $\Psi_{I,\lambda}$ is orientation preserving (and we shall see below in Remark \ref{indbasrem} that $\Psi_{I,\lambda}$ takes a natural symplectic orientation on $(S^1)^{2g} \times N_r(c(I_1,\lambda)) \times N_{n-r}(c(I_2,-\lambda))$ to the symplectic orientation induced by $\omega$ on $F_{I,\lambda}$) then \begin{equation} \label{snow} \int_{F_{I,\lambda}} \phicom (\eta e^{ \bom}) = (r(n-r))^{-2g} \int_{(S^1)^{2g} \times N_r(c(I_1,\lambda)) \times N_{n-r}(c(I_2,-\lambda)) } \Psi_{I,\lambda}^* \phicom (\eta e^{ \bom}) \end{equation} where both sides are elements of $H^*_{\tone}$. To be more precise we should replace $\eta e^{ \bom}$ on each side of this equation by its restriction to $\tilde{F}_{I,\lambda}$, and as in Proposition \ref{stages} and Lemma \ref{l6.4} we use the double cover $\tone \times T_{n-1} \to \tor$ to define $$\phicom:H^*_{\tor}(\tilde{F}_{I,\lambda}) \to H^*_{\tone}(\fisml) \cong H^*_{\tone} \otimes H^*(\fisml).$$ Recall from the proof of the last proposition that the homomorphism $$\rho_I: S^1 \times SU(r) \times SU(n-r) \to H_I \subset SU(n)$$ given by $$\rho_I : (s,A,B) \mapsto \matr{s^{n-r} A}{0}{0}{s^{-r} B} $$ with respect to the decomposition of $\{1,\ldots,n\}$ as $I_1\cup I_2$ restricts to an $r(n-r)$ to one surjective homomorphism $$\rho_I:S^1 \times T_r \times T_{n-r} \to \tor.$$ It is easy to check that the inclusions of $\tone$, $T_r$ and $T_{n-r}$ in $T_n$ induce an isomorphism $$\hat{\rho}_r:\tone \times T_r \times T_{n-r} \to \tor$$ such that $\rho_I$ and $\hat{\rho}_r$ have the same restriction to $T_r \times T_{n-r}$. The composition of this restriction with the natural surjection from $\tor$ to $\tor/\tone\cong T_{n-1} /(\tone \cap T_{n-1})$ gives an isomorphism $$T_r \times T_{n-r} \to \tor/\tone\cong T_{n-1} /(\tone \cap T_{n-1}).$$ Moreover the composition of $\rho_I$ with the inverse of $\hat{\rho}_r$ defines a finite (in fact $r(n-r)$ to one) cover $$\nu_I:S^1 \times T_r \times T_{n-r} \to \tone \times T_r \times T_{n-r}$$ which restricts to the identity on $T_r \times T_{n-r}$ and induces a finite cover $\nu_I:S^1 \to \tone$ and isomorphisms on Lie algebras and equivariant cohomology. The argument in the proof of the last proposition to show that the map $\Psi_{I,\lambda}$ is well defined on the quotient by the action of $T_r \times T_{n-r}$ may be rephrased as the statement that the map $$\tpsi_{I,\lambda}: (S^1)^{2g} \times \Bigl (\mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\lambda)) \Bigr ) \times \Bigl (\mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\lambda)) \Bigr ) \to \tilde{F}_{I,\lambda} $$ defined in the proof of Proposition \ref{p7.3} satisfies $$\tpsi_{I,\lambda} \Bigl (t ((s_1, \dots, s_{2g}), (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}, 0), (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}, 0)) \Bigr ) $$ $$ = \rho_I(t) \tpsi_{I,\lambda} \Bigl ( (s_1, \dots, s_{2g}), (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}, 0), (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}, 0) \Bigr ) $$ for all $t \in S^1 \times T_r \times T_{n-r}$, where $S^1$ acts trivially. Thus $\tpsi_{I,\lambda}$ and $\rho_I$ induce $\tpsi_{I,\lambda}^*$ from $H^*_{\tor}(\tilde{F}_{I,\lambda}) $ to $$H^*_{S^1} \otimes H^*((S^1)^{2g}) \otimes H^*_{T_r} (\mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\lambda) )) \otimes H^*_{T_{n-r}} (\mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\lambda)))$$ and $$ \nu_I^* \int_{(S^1)^{2g} \times N_r(c(I_1,\lambda)) \times N_{n-r}(c(I_2,-\lambda)) } \Psi_{I,\lambda}^* \phicom (\eta e^{ \bom}) $$ $$ \label{7.51} = \int_{(S^1)^{2g} \times N_r(c(I_1,\lambda)) \times N_{n-r}(c(I_2,-\lambda)) } (1\otimes\Phi_r \otimes \Phi_{n-r}) \tpsi_{I,\lambda}^*(\eta e^{\bom}), $$ where both sides are elements of $H^*_{S^1}$. Hence by (\ref{snow}), if $Y_1$ is the coordinate on $\lietone$ given by the restriction of $Y_1=X_1-X_2$ on $\liet$ and $Y_1^I$ is the coordinate on the Lie algebra of $S^1$ obtained from $Y_1$ via the isomorphism on Lie algebras induced by $\nu_I:S^1 \to \tone$, then $$ \res_{Y_1 =0} \int_{F_{I,\lambda}} \phicom (\eta e^{ \bom}) = $$ $$(r(n-r))^{-2g} \res_{Y_1^I=0} \int_{(S^1)^{2g} \times N_r(c(I_1,\lambda)) \times N_{n-r}(c(I_2,-\lambda)) } (1\otimes\Phi_r \otimes \Phi_{n-r}) \tpsi_{I,\lambda}^*(\eta e^{\bom}). $$ Since $S^1$ acts trivially, the residue operation $\res_{Y^I_1=0}:H^*_{S^1} \to {\Bbb C }$ can be extended to map $$H^*_{S^1} \otimes H^*((S^1)^{2g}) \otimes H^*_{T_r} \Biggl( \mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\lambda) )\Biggr ) \otimes H^*_{T_{n-r}} \Biggl( \mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\lambda))\Biggr) $$ to $$H^*((S^1)^{2g}) \otimes H^*_{T_r} \Biggl( \mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\lambda) )\Biggr) \otimes H^*_{T_{n-r}} \Biggl( \mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\lambda))\Biggr) $$ so that it commutes with $\Phi_r$ and $\Phi_{n-r}$ and with integration over $N_r(c(I_1,\lambda))$ and integration over $ N_{n-r}(c(I_2,-\lambda))$. In particular by expressing integrals over products as iterated integrals we obtain $$\res_{Y_1 =0} \int_{F_{I,\lambda}} \phicom (\eta e^{ \bom})$$ $$ = (r(n-r))^{-2g} \int_{N_r(c(I_1,\lambda)) } \Phi_r (\res_{Y_1^I=0} \int_{N_{n-r}(c(I_2,-\lambda))} \Phi_{n-r}( \int_{(S^1)^{2g} } \tpsi_{I,\lambda}^*(\eta e^{\bom})))$$ $$=(r(n-r))^{-2g} \int_{N_{n-r}(c(I_2,-\lambda)) } \Phi_{n-r} (\res_{Y_1^I=0} \int_{N_r(c(I_1,\lambda))} \Phi_{r}( \int_{(S^1)^{2g} } \tpsi_{I,\lambda}^*(\eta e^{\bom}))).$$ This will be important when we apply induction later. \end{rem} Recall from Lemma \ref{old5.17} that ${\mbox{$\cal F$}}$ is the set of components of the fixed point set of the action of $\tone$ on the quotient $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$, where $V$ is a sufficiently small $T$-invariant neighbourhood of $c$ in $K$. Every $F\in{\mbox{$\cal F$}}$ contains a component $\fisml$ of the fixed point set of the action of $\tone$ on $M(c) \cap\mu^{-1}(\lietone)/\torc$, and each $\fisml$ is contained in a unique $F\in{\mbox{$\cal F$}}$ (see Proposition \ref{p7.2} for the definition of $\fisml$). For each $I$ and $\lambda$ we now need to understand the normal bundle in $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ to the component $F\in {\mbox{$\cal F$}}$ of the fixed point set which contains $\fisml$. First, we observe that there is the following decomposition: \begin{rem} \label{p5.11} \label{feb} Let $I$ be a subset of $\{3,\ldots,n\}$ with $r-1$ elements where $1\leq r\leq n-1$, let $I_1=I\cup\{1\}$ and let $I_2= \{ 1, \dots, n\} - I_2$. Then $$\nusym_n(X) = \nusym^{[I_1]}_{r}(X) \nusym^{[I_2]}_{n-r}(X) \tau_I(X) $$ where $$\nusym^{[I_1]}_{r}(X) = \prod_{1\leq j<k\leq r } (X_{i_j} - X_{i_k}) $$ is the product of the positive roots of $SU(r)$ embedded in $SU(n)$ via the inclusion of $I_1$ in $\{1,\ldots,n\}$, $$\nusym^{[I_2]}_{n-r}(X) = \prod_{r+1\leq j < k \leq n } (X_{i_j} - X_{i_k}) $$ is the product of the positive roots of $SU(n-r)$ embedded in $SU(n)$ via the inclusion of $I_2$ in $\{1,\ldots,n\}$, and $$\tau_I(X) = \pm \prod_{1\leq j \leq r < k \leq n} ( X_{i_j} - X_{i_k}), $$ where the sign is $+$ or $-$ depending on whether the permutation $$ \left \lbrack \begin{array}{lcccr} 1&2&\dots & n \\ i_1 & i_2 & \dots &i_n \end{array} \right \rbrack $$ is even or odd. Note also that $$(-1)^{r(n-r)} (\tau_I(X))^2 = \prod_{(i,j)\in I_1\times I_2\cup I_2\times I_1} ( X_i - X_j). $$ \end{rem} Now we can find the $\tone$-equivariant Euler class of the normal bundle in $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ to the component $F\in {\mbox{$\cal F$}}$ of the fixed point set which contains $\fisml$. \begin{lemma} \label{p5.13} Let $I$ be a subset of $\{3,\ldots,n\}$ with $r-1$ elements where $1\leq r\leq n-1$, and let $\lambda \in {\Bbb R }$ be a solution of the equation $$ e^{-2\pi i \lambda} = \prod_{j\in I_1} c_j.$$ Then the $\tone$-equivariant Euler class of the normal bundle in $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ to the component $F\in {\mbox{$\cal F$}}$ of the fixed point set of the action of $\tone$ on $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ which contains $\fisml$ is given by $ e_F = $ $(-1)^{r(n-r)g} \Phi_{n-1}(\tau_I^{2g})$. \end{lemma} \Proof The proof of Proposition \ref{p7.2} shows that the component $F\in {\mbox{$\cal F$}}$ of the fixed point set of the action of $\tone$ on $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ which contains $\fisml$ is $$F = P^{-1}(V) \cap (H_I^{2g} \times \lietone)/\torc,$$ whereas $\mu^{-1}(\lietone)=K^{2g}\times\lietone$. The $T$-equivariant Chern roots of the normal bundle to $H_I^{2g}$ in $K^{2g}$ are $X_i-X_j$ for $(i,j) \in I_1\times I_2 \cup I_2 \times I_1$ with multiplicity $g$. The result follows by Remark \ref{p5.11}. \hfill $\square$ \begin{lemma} \label{l5.21} Let $I$ be a subset of $\{3,\ldots,n\}$ with $r-1$ elements where $1\leq r\leq n-1$, let $I_1=I\cup \{1\}$, let $I_2 = \{1,\ldots,n\} - I_1$ and let $\lambda \in {\Bbb R }$ be a solution of the equation $$ e^{-2\pi i \lambda} = \prod_{j\in I_1} c_j.$$ Let $F$ be the component of the fixed point set of the action of $\tone$ on $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ which contains $\fisml$, where $\fisml$ is as defined in Proposition \ref{p7.2}. We then have $$ \int_F \frac{\phicom ( \eta e^{ \bom} \alpha)} {e_F } = (-1)^{r(n-r)(g-1)} \int_{F_{I,\lambda}} \phicom (\frac{ \eta e^{ \bom} }{\tau_I^{2g-1}}) $$ $$= (-1)^{r(n-r)(g-1)} \int_{F_{I,\lambda}} \phicom (\frac{\nusym^{[I_1]}_{r}(X) \nusym^{[I_2]}_{n-r}(X) \eta e^{ \bom} } {\nusym_n(X)\tau_I^{2g-2}}) $$ where $\alpha$ is the $\tor$-equivariant differential form on $K^{2g}\times\liek$ given by Proposition \ref{defa} which is supported near $M(c)$ and represents the equivariant Poincar\'{e} dual of $M(c)$ in $K^{2g}\times\liek$. \label{bag} \end{lemma} \Proof The $\tor$-equivariant differential form $\alpha$ on $K^{2g}\times\liek$ which represents the equivariant Poincar\'{e} dual of $M(c)=P^{-1}(c)$ in $K^{2g}\times\liek$ was defined in Proposition \ref{defa} as a pullback via the map $P:K^{2g}\times\liek \to K$. By using the restriction $P:H_I^{2g}\times\hat{\liet}_1 \to H_I$ we can similarly define a $\tor$-equivariant differential form $\alpha_I$ on $H_I^{2g}\times\hat{\liet}_1$ which represents the equivariant Poincar\'{e} dual of $M(c)\cap(H_I^{2g}\times\hat{\liet}_1)$ in $H_I^{2g}\times\hat{\liet}_1$. The restriction of $\Phi_{n-1}(\alpha_I)$ then represents the Poincar\'{e} dual to $F_{I,\lambda}$ in $F$, provided suitable orientations are chosen. Note that $\{1,\dots,1\}\times \liek$ is transverse to both $M(c)=P^{-1}(c)$ and $\mu^{-1}(0)=K^{2g}\times\{0\}$ in $K^{2g}\times \liek$, and that if $\Lambda\in\liek$ then $$\mu(1,\dots,1,\Lambda) = -\Lambda$$ while $$P(1,\dots,1,\Lambda) = \exp(-\Lambda).$$ {}From the orientation conventions of Remark \ref{orient} it follows that the normal to $P^{-1}(H_I)$ in $K^{2g}\times \liek$ is $\tor$-equivariantly isomorphic to the kernel of the restriction map $\lieks \to {\bf h}_I^*$. Thus the restriction of $(-1)^{r(n-r)}\tau_I \alpha_I$ to $H_I^{2g}\times\hat{\liet}_1$ has compact support near $M(c)$ and locally represents the equivariant Poincar\'{e} dual to $M(c)$ in $K^{2g}\times \liek$, so we can substitute it for $\alpha$ on $H_I^{2g}\times\hat{\liet}_1$ and we can substitute $(-1)^{r(n-r)}\Phi_{n-1}(\tau_I \alpha_I)$ for $\Phi_{n-1}(\alpha)$ on $F$. We have that $e_F = (-1)^{r(n-r)g} \phicom(\tau_I^{2g}) $ by the last lemma. We therefore get $$ \int_{F } \frac{\phicom ( \eta e^{ \bom}\compform ) } {e_F} = (-1)^{r(n-r)(g-1)} \int_{F } \phicom (\frac{ \eta e^{ \bom}\alpha_I } {\tau_I^{2g-1}}) $$ $$ = (-1)^{r(n-r)(g-1)} \int_{F_{I,\lambda}} \phicom (\frac{ \eta e^{ \bom} }{\tau_I^{2g-1}}),$$ and Remark \ref{feb} completes the proof. \hfill $\square$ \begin{rem} \label{fi} The condition for $F\in {\mbox{$\cal F$}}$ to appear in the sum in the statement of Lemma \ref{l6.4} was that \begin{equation} \label{5.54} -|\!|\he{1}|\!|^2 < \inpr{\he{1},\mu(F)} <0. \end{equation} Let $I$ be a subset of $\{3,\ldots,n\}$ with $r-1$ elements where $1\leq r\leq n-1$, and let $\lambda \in {\Bbb R }$ be a solution of the equation $$ e^{-2\pi i \lambda} = \prod_{j\in I_1} c_j.$$ If $F\in {\mbox{$\cal F$}}$ is the component of the fixed point set of the action of $\tone$ on $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ which contains $\fisml$, then $$\mu_{\tone}(F) = \mu_{\tone} (\fisml) = -\lambda \hat{e}_1.$$ We thus find that for each $I$ there is precisely one solution $\lambda\in{\Bbb R }$ to the equation $$ e^{-2\pi i \lambda} = \prod_{j\in I_1} c_j$$ such that the component $F$ of the fixed point set of the action of $\tone$ on $P^{-1}(V) \cap\mu^{-1}(\lietone)/\torc$ which contains $\fisml$ contributes to the sum in Lemma \ref{l6.4}. This solution is $\lambda = \delta_I$ where $\delta_I$ is the non-integer part of $$\frac{i}{2\pi } \log \prod_{j\in I_1} c_j,$$ and so we have $$\mu_{\tone}(F) = - \delta_I \hat{e}_1.$$ (Note that since $\prod_{j\in I_1} c_j$ has modulus 1 but is not equal to 1, the non-integer part of $\frac{i}{2\pi} \log \prod_{j\in I_1} c_j$ is well defined as an element of the open interval $(0,1)$ in ${\Bbb R }$.) We therefore define $$F_I = F_{I,\delta_I},$$ \label{bog} and also $\Psi_I=\Psi_{I,\delta_I}$ and $\tpsi_I=\tpsi_{I,\delta_I}.$ \end{rem} We can now deduce the following result. \begin{prop} \label{crucial} If $\eta(X)$ is a polynomial in the $\tar$ and $\tbr$, so that $s_{\he{1}}^*\eta = \eta$, then $$ \int_{\nl(c)} \phil( \eta e^{\bom} ) - \int_{\nl(c)} \phil (\eta e^{ \bom}e^{- Y_1 } ) = \int_{\nl(c)} \phil \Bigl ((1- e^{- Y_1}) \eta e^{ \bom} \Bigr ) $$ $$= \sum_{1\leq r\leq n-1} \sum_{I\subseteq \{3,\ldots,n\},|I|=r-1} (-1)^{r(n-r)(g-1)} \res_{Y_1 = 0 } \int_{F_I} \phicom (\frac{ \eta e^{ \bom} }{\tau_I^{2g-1}}). $$ \end{prop} \Proof Recall that the coordinates $Y_k = e_k(X) = < \hat{e}_k,X>$ were introduced in Definition \ref{YY}. The result then follows immediately from Lemma \ref{l6.4}, Lemma \ref{bag} and Remark \ref{bog} above, together with Lemma \ref{l3} and Remark \ref{n0=2}. \hfill $\square$ \begin{rem} \label{formal} This proposition is also true for formal equivariant cohomology classes $\eta = \sum_{j=0}^{\infty} \eta_j$ with $\eta_j \in H^j_K(M(c))$, because all but finitely many $\eta_j$ contribute zero to both sides of the equations. \end{rem} \begin{corollary} \label{c6.5} Suppose $\eta$ is a polynomial in the $\tar$ and $\tbr$, so that $s_{\he{1}}^*\eta = \eta$. Then $$ \int_{\nl(c)} \phil (\nusym_n \eta e^{ \bom} ) = \sum_{1\leq r\leq n-1} \sum_{I\subseteq \{3,\ldots,n\},|I|=r-1} (-1)^{r(n-r)(g-1)} \res_{Y_1 = 0 } \int_{F_I} \phicom (\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta e^{ \bom} }{\tau_I^{2g-2}( 1- e^{- Y_1} ) }). $$ \end{corollary} \Proof This follows by applying Remark \ref{p5.11} and Proposition \ref{crucial} with $\eta$ replaced by the formal equivariant cohomology class $\eta \nusym_n / (1 - e^{-Y_1}) $. This is valid by Remark \ref{formal} because $Y_1$ divides $\nusym_n(X)$ and so $ \nusym_n / (1 - e^{-Y_1}) $ can be expressed as a power series in $Y_1$ whose coefficients are polynomials in the other coordinates $Y_2,\ldots, Y_{n-1}$. \hfill $\square$ \bigskip \begin{rem} \label{indbasrem} Recall from the proof of Proposition \ref{p7.3} that $$\tpsi_{I}: (S^1)^{2g} \times \Bigl (\mu_{SU(r)}^{-1}(0) \cap M_r (c(I_1,\delta_I)) \Bigr ) \times \Bigl (\mu_{SU(n-r)}^{-1}(0) \cap M_{n-r} (c(I_2,-\delta_I)) \Bigr ) \to \tilde{F}_{I,\delta_I} $$ is defined for $\delta_I$ as in Remark \ref{fi} by $$\tpsi_{I} \Bigl ( (s_1, \dots, s_{2g}), (h_1^{[I_1]}, \dots, h_{2g}^{[I_1]}, 0), (h_1^{[I_2]}, \dots, h_{2g}^{[I_2]}, 0) \Bigr ) $$ $$ = \Bigl ( (\rho_I (s_1, \ha{1}, \hb{1}),\rho_I (s_{2}, \ha{2}, \hb{2}), \dots, \rho_I (s_{2g}, \ha{2g}, \hb{2g}), 2\pi i \diag(\delta_I,-\delta_I,0,\ldots, 0) \Bigr ) $$ using the map $$\rho_I: S^1 \times SU(r) \times SU(n-r) \to S(U(r) \times U(n-r)) \subset SU(n)$$ given by $$\rho_I : (s,A,B) \mapsto \matr{s^{n-r} A}{0}{0}{s^{-r} B} $$ with respect to the decomposition of $\{1,\ldots,n\}$ as $I_1\cup I_2$, which restricts to an $r(n-r)$ to one surjective homomorphism $$\rho_I:S^1 \times T_r \times T_{n-r} \to \tor.$$ Since $\bom = \omega + \mu$ is constructed using the inner product $<,>$ defined at (2.2) on the Lie algebra $\liek$ of $K=SU(n)$, and since $\rho_I$ embeds the Lie algebras of $S^1$, $SU(r)$ and $SU(n-r)$ as mutually orthogonal subspaces of $\liek$, we have $$\tpsi_I^*(\bom) = \bom_r + \bom_{n-r} + \Omega - \delta_I \hat{e}_{1}$$ for some $\Omega\in H^2((S^1)^{2g})$, where $\bom_r$ and $\bom_{n-r}$ are defined like $\bom$ but with $n$ replaced by $r$ and $n-r$. Thus we have $$\tpsi_{I}^*(\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta e^{ \bom} }{\tau_I^{2g-2}(1 - e^{- Y_1}) }) =e^{ \bom_r + \bom_{n-r} + \Omega - \delta_I Y_1} \tpsi_{I}^*(\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta }{\tau_I^{2g-2}(1 - e^{- Y_1}) }).$$ \end{rem} Since $\tpsi_I^*(\nusym^{[I_1]}_r) = \nusym_r$ and $\tpsi_I^*(\nusym^{[I_2]}_{n-r}) = \nusym_{n-r}$, we can combine this with Corollary \ref{c6.5} and Remark \ref{big} to obtain the result on which is based the inductive proof of Witten's formulas in the next section. \begin{prop} \label{indbas} If $c\in T$ satisfies the conditions of Remark \ref{induct}, and if $\eta(X)$ is a polynomial in the $\tar$ and $\tbr$ so that $s_{\he{1}}^*\eta = \eta$, then $$ \int_{\nl(c)} \phil (\nusym_n \eta e^{ \bom} ) = \sum_{1\leq r\leq n-1} \sum_{I\subseteq \{3,\ldots,n\},|I|=r-1} (-1)^{r(n-r)(g-1)} \res_{Y_1 = 0 } \int_{F_I} \phicom (\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta e^{ \bom} }{\tau_I^{2g-2}(1- e^{- Y_1} ) })$$ where $$\res_{Y_1 = 0 } \int_{F_I} \phicom (\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta e^{ \bom} }{\tau_I^{2g-2}(1- e^{- Y_1}) })$$ is equal to $(r(n-r))^{-2g}$ times $$ \int_{N_r(c(I_1,\delta_I)) } \Phi_r (\nusym_r e^{\bom_r} \res_{Y_1^I=0} \int_{N_{n-r}(c(I_2,-\delta_I))} \Phi_{n-r}( \nusym_{n-r} e^{\bom_{n-r}} \int_{(S^1)^{2g} } e^{\Omega} \tpsi_{I}^*(\frac{ \eta e^{- \delta_I Y_1} }{\tau_I^{2g-2}(1- e^{- Y_1} ) })))$$ and also to $(r(n-r))^{-2g}$ times $$\int_{N_{n-r}(c(I_2,-\delta_I)) } \Phi_{n-r} (\nusym_{n-r} e^{\bom_{n-r}} \res_{Y_1^I=0} \int_{N_r(c(I_1,\delta_I))} \Phi_{r}(\nusym_r e^{\bom_r} \int_{(S^1)^{2g} } e^{\Omega} \tpsi_{I}^*(\frac{ \eta e^{-\delta_I Y_1} }{\tau_I^{2g-2}(1- e^{- Y_1}) }))).$$ Here $c(I_1,\delta_I))$ and $c(I_2,-\delta_I)$ are defined as in Proposition \ref{p7.3} with $\delta_I$ as in Remark \ref{bog} and $\bom_r$, $\bom_{n-r}$ and $\Omega$ as in Remark \ref{indbasrem}. \end{prop} \begin{rem} \label{c} For any $\gamma \in \tor$ a unique $\tilde{\gamma} \in \liet_n$ can be chosen so that $\exp \tilde{\gamma}=\gamma$ and $\tilde{\gamma}$ belongs to the fundamental domain defined by the simple roots for the translation action on $\liet_n$ of the integer lattice $\Lambda^I$ (i.e. $\tilde{\gamma} = \gamma_1\hat{e}_1 + \ldots + \gamma_{n-1}\hat{e}_{n-1}$ with $0\leq \gamma_j <1$ for $1\leq j\leq n-1$). Suppose that $\tilde{c}(I_1,\delta_I)\in\liet_r$ and $\tilde{c}(I_2,-\delta_I)\in\liet_{n-r}$ are chosen in this way in the fundamental domains defined by the simple roots for the translation actions on $\liet_r$ and $\liet_{n-r}$ of their integer lattices, satisfying $$\exp \tilde{c}(I_1,\delta_I) = c(I_1,\delta_I)= \diag(c^{I,\delta_I}_{i_1},\ldots, c^{I,\delta_I}_{i_r})$$ and $$\exp \tilde{c}(I_2,-\delta_I) = c(I_2,-\delta_I) = \diag(c^{I,\delta_I}_{i_{r+1}}, \ldots, c^{I,\delta_I}_{i_n}),$$ where (as in Proposition \ref{p7.3} and Remark \ref{fi}) we define $\delta_I$ to be the non-integer part of $\frac{i}{2\pi}\log \prod_{j\in I_1} c_j$ and let $ c^{I,\delta_I}_j = c_j$ if $j\geq 3$, and $ c^{I,\delta_I}_{1} = c_{1} e^{2\pi i \delta_I}$ and $c^{I,\delta_I}_2 = c_2 e^{-2\pi i \delta_I}.$ In the proof of the main theorem (Theorem \ref{mainab}) of the next section we shall need to consider the elements $w^1_I$ and $w^2_I$ of the subgroup $S_{n-1}$ of the Weyl group $W\cong S_n$ of $SU(n)$ given by the permutations $$ \left \lbrack \begin{array}{lcccr} 1&2&\dots & n \\ i_1 & i_2 & \dots &i_n \end{array} \right \rbrack $$ and $$ \left \lbrack \begin{array}{lcccccr} 1&2&\dots & n-r & n-r+1 & \dots & n \\ i_{r+1} & i_{r+2} & \dots &i_n &i_1 & \dots & i_r \end{array} \right \rbrack, $$ in the cases when $i_r=1$ and $i_{r+1}=2$ and $i_n=n$ and when $i_1=1$ and $i_n=2$ and $i_r=n$ respectively. We will use the fact that if $i_r=1$ and $i_{r+1}= 2$ and $i_n=n$ then $$w^1_I(\tc) = \matr{\tilde{c}(I_1,\delta_I)}{0}{0}{\tilde{c}(I_2,-\delta_I)} + (1 -\delta_I) \hat{e}_{1}, $$ where the block diagonal form is taken with respect to the decomposition of $\{1,\ldots,n\}$ as $ \{1,\ldots,r\}\cup \{r+1,\ldots,n\} $. To see why this is the case, note that $$w^1_I(\tc)(X) = \gamma_1 (X_{i_1} - X_{i_2}) + \dots \gamma_{n-1} (X_{i_{n-1}} - X_{i_n})$$ where $\gamma_k$ is the non-integer part of $\frac{1}{2\pi i} {\rm log} \prod_{j\leq k} c_{i_j},$ so that $\gamma_r = 1 - \delta_I$ and if $k<r$ then $\gamma_k$ is the non-integer part of $\frac{1}{2\pi i} {\rm log} \prod_{j\leq k} c_{i_j}^{I,\delta_I}$ whereas if $k>r$ then $\gamma_k$ is the non-integer part of $$-\delta_I + \frac{1}{2\pi i} {\rm log} \prod_{r < j\leq k} c_{i_j} = \frac{1}{2\pi i} {\rm log} \prod_{r < j\leq k} c_{i_j}^{I,\delta_I}.$$ Similarly if $i_1 =1$ and $i_n =2$ and $i_r = n$ then $$w^2_I(\tc) = \matr{\tilde{c}(I_2,-\delta_I)}{0}{0}{\tilde{c}(I_1,\delta_I)} - \delta_I \hat{e}_{1}$$ where the block diagonal form is taken with respect to the decomposition of $\{1,\ldots,n\}$ as $ \{1,\ldots,n-r\}\cup \{n-r+1,\ldots,n\} $. \end{rem} { \setcounter{equation}{0} } \section{Proof of the iterated residue formula} In this section we shall use induction to prove Witten's formulas in the formulation given in Section 2 (see Proposition \ref{p:sz}) involving iterated residues, for pairings of the form \begin{equation} \prod_{r=2}^n a_r^{m_r} \prod_{k_r=1}^{2g} (b_r^{k_r})^{p_{r,k_r}}\exp (f_2) [\mnd] \end{equation} for nonnegative integers $m_r$ and $p_{r,k}$. The induction is based on Proposition \ref{indbas}. In the next section we shall extend the proof to give formulas for all pairings, and in the following section we shall show that these formulas are equivalent to those of Witten. We are aiming to prove \begin{theorem} \label{mainab} Let $c=\diag\, (e^{2\pi i d/n},\ldots, e^{2\pi i d/n})$ where $d \in\{1,\ldots,n-1\}$ is coprime to $n$, and suppose that $\eta\in H^*_{SU(n)}(M_n(c))$ is a polynomial $Q(\tilde{a}_2,\ldots,\tilde{a}_n, \tilde{b}_2^1,\ldots,\tilde{b}_n^{2g})$ in the equivariant cohomology classes $\tilde{a}_r$ and $\tilde{b}_r^j$ for $2\leq r\leq n$ and $1\leq j\leq 2g$. Then the pairing $Q(a_2,\ldots,a_n,b_2^1,\ldots,b_n^{2g})\exp (f_2) [\mnd]$ is given by $$\int_{\mnd} \Phi (\eta e^{\bom} ) = \frac{(-1)^{n_+(g-1)}}{n!} \res_{Y_{1} = 0} \dots \res_{Y_{n-1} = 0} \Biggl ( \frac{\sum_{w \in W_{n-1}} e^{ \inpr{ \tildarg{w \tc},X} } \int_{\tor^{2g}} \eta e^{ \omega} } {\nusym_n^{2g-2} \prod_{1\leq j \leq n-1} ( \exp (Y_j)-1 ) } \Biggr ), $$ where $n_+ = \frac{1}{2} n(n-1)$ is the number of positive roots of $K=SU(n)$ and $X\in\tor$ has coordinates $Y_1=X_1-X_2,\ldots,Y_{n-1}=X_{n-1}-X_n$ defined by the simple roots, while $W_{n-1} \cong S_{n-1}$ is the Weyl group of $SU(n-1)$ embedded in $SU(n)$ in the standard way using the first $n-1$ coordinates. The element $\tilde{c}$ was defined in Remark \ref{r2.1}: it is the unique element of $\liet_n$ which satisfies $e^{2\pi i\tilde{c}} = c$ and belongs to the fundamental domain defined by the simple roots for the translation action on $\liet_n$ of the integer lattice $\Lambda^I$. Also, the notation $\bracearg{\gamma}$ (introduced in Definition \ref{bracedef}) means the unique element which is in the fundamental domain defined by the simple roots for the translation action on $\liet_n$ of the integer lattice and for which $\bracearg{\gamma}$ is equal to $\gamma$ plus some element of the integer lattice. \end{theorem} \begin{rem} \label{interp} Here the integral $$ \int_{\tor^{2g}} \eta e^{ \omega} $$ is to be interpreted as the integral of the restriction of $\eta e^{ \omega} $ over a connected component $$\tor^{2g} \times \{ \lambda\}$$ (for some $\lambda \in \liet_n$ satisfying $c\exp \lambda = 1$) of the fixed point set of the action of $\tor$ on $M_{n}(c)$. It does not matter which component we choose here, because $\eta$ and $\omega$ are invariant under the translation maps $s_{\Lambda_0}$ defined at Lemma \ref{l4.3} for $\Lambda_0$ in the integer lattice of $\liet_n$. \end{rem} \begin{rem} \label{rem8.3} (a) We can substitute $-X$ for $X$ in Theorem \ref{mainab} to get $$\int_{\mnd} \Phi (\eta e^{\bom} ) = \frac{(-1)^{n_+(g-1)}}{n!} \res_{Y_{1} = 0} \dots \res_{Y_{n-1} = 0} \Biggl ( \frac{\sum_{w \in W_{n-1}} e^{ - \inpr{\tildarg{w\tc},X} } \int_{\tor^{2g}} \eta(-X) e^{ \omega} } {\nusym_n^{2g-2} \prod_{1\leq j \leq n-1} (1 - \exp (-Y_j)) } \Biggr ). $$ \noindent (b) When $\eta$ is a polynomial in $a_2,\dots,a_n$ then $$ \int_{\tor^{2g}} \eta e^{ \omega} = \eta \int_{\tor^{2g}} e^{ \omega} = n^g \eta $$ (see Lemma \ref{l9.3b} below). Since $\tilde{a}_r$ is represented by the polynomial $\tau_r(-X)$ for $2\leq r\leq n$ (see Proposition \ref{abftil} or Section 9 below), this means that, by (a) above, Theorem \ref{mainab} combined with Proposition \ref{p:sz} gives us Witten's formula (2.4). \noindent (c) We can also replace the symplectic form $\omega$ by any nonzero scalar multiple $\epsilon \omega$. Then the moment map $\mu$ is multiplied by the same scalar $\epsilon$, and the proof of Theorem \ref{mainab} yields $$\int_{\mnd} \Phi (\eta e^{\epsilon \bom} ) = \frac{(-1)^{n_+(g-1)}}{n!} \res_{Y_{1} = 0} \dots \res_{Y_{n-1} = 0} \Biggl ( \frac{\sum_{w \in W_{n-1}} e^{ \inpr{\epsilon \tildarg{w\tc},X} } \int_{\tor^{2g}} \eta e^{\epsilon \omega} } {\nusym_n^{2g-2} \prod_{1\leq j \leq n-1} ( \exp (\epsilon Y_j)-1) } \Biggr ). $$ If the degree of $\eta$ is equal to the dimension of $\mnd$ then the left hand side of this equation is equal to $$\int_{\mnd} \Phi (\eta)$$ and hence is independent of $\epsilon$. Thus in this case we can take any nonzero value of $\epsilon$ on the right hand side, or let $\epsilon$ tend to zero, to give alternative formulas for $\int_{\mnd} \Phi (\eta)$. \end{rem} Recall from Lemma \ref{l3} that \begin{equation}\label{oldl5.10} \int_{\mnd} \Phi (\eta e^{\bom} ) = \frac{1}{n!} \int_{\nl(c)} \phil (\nusym_n \eta e^{ \bom} ) .\end{equation} Proposition \ref{indbas} tells us that $\int_{\nl(c)} \phil (\nusym_n \eta e^{ \bom} )$ can be expressed in terms of iterated integrals of the same form for smaller values of $n$, but with $c$ no longer central in $K=SU(n)$. We shall therefore obtain Theorem \ref{mainab} from the following result involving values of $c$ which are not central (cf. Remark \ref{induct}), which will be proved by induction on $n$. \begin{prop} \label{beg} Let $c=\diag(c_1,\ldots,c_n) \in \tor$ be such that the product of no proper subset of $c_1,\ldots,c_n$ is 1. If $\eta(X)$ is a polynomial in the $\tar$ and $\tbr$, so that $s_{\he{l}}^*\eta = \eta$, then $$\int_{\nl(c)} \phil (\nusym_n \eta e^{ \bom} ) = (-1)^{n_+(g-1)} \res_{Y_{1} = 0} \dots \res_{Y_{n-1} = 0} \Bigl ( \frac{\sum_{w \in W_{n-1}} e^{ \inpr{\tildarg{w\tc} ,X} } \int_{\tor^{2g}} \eta e^{ \omega} } {\nusym_n^{2g-2} \prod_{1\leq j \leq n-1} ( \exp (Y_j)-1 ) } \Bigr ), $$ where $W_{n-1} \cong S_{n-1}$ is the Weyl group of $SU(n-1)$, embedded in $SU(n)$ in the standard way using the first $n-1$ coordinates, and $\tc = (\tc_1,\dots,\tc_n) \in \liet_n$ satisfies $e^{2\pi i \tc}=c$ and belongs to the fundamental domain defined by the simple roots for the translation action on $\liet_n$ of the integer lattice $\Lambda^I$. \end{prop} \noindent{\bf Proof of Theorem \ref{mainab} from Proposition \ref{beg}:} Note that when $c = \diag(e^{2\pi i d/n},\ldots, e^{2\pi i d/n})$ we had introduced an element $\tilde{c} $ $\in$ $\liet_n$ (see Remark \ref{r2.1}) which satisfies $e^{2\pi i\tilde{c}} = c$ and belongs to the fundamental domain defined by the simple roots for the translation action on $\tor$ of the integer lattice $\Lambda^I$. Thus Theorem \ref{mainab} follows immediately from (8.2) and Proposition \ref{beg}. \hfill $\square$ \bigskip \noindent {\bf Proof of Proposition \ref{beg}:} The proof is by induction on $n$. When $n=1$ then both $SU(n)$ and the torus $\tor$ are trivial, $\nusym_n = 1$ and both $M_n(c)$ and $N_{n}(c)$ are single points. Thus in this case Proposition \ref{beg} reduces to the tautology $ \eta = \eta$ for any $\eta \in H^*_{SU(1)}(M_1(c))$. Now let us assume that $n>1$ and that the result is true for all smaller values of $n$. By Proposition \ref{indbas} we have $$ \int_{\nl(c)} \phil (\nusym_n \eta e^{ \bom} ) = \sum_{1\leq r\leq n-1} \sum_{I\subseteq \{3,\ldots,n\},|I|=r-1} (-1)^{r(n-r)(g-1)} \res_{Y_1 = 0 } \int_{F_I} \phicom (\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta e^{ \bom} }{\tau_I^{2g-2}(1- e^{- Y_1}) })$$ where $$\res_{Y_1 = 0 } \int_{F_I} \phicom (\frac{\nusym^{[I_1]}_r \nusym^{[I_2]}_{n-r} \eta e^{ \bom} }{\tau_I^{2g-2}(1- e^{ -Y_1}) })$$ is equal to $ (r(n-r))^{-2g}$ times the iterated integral $$\int_{N_r(c(I_1,\delta_I)) } \Phi_r (\nusym_r e^{\bom_r} \res_{Y_1^I=0} \int_{N_{n-r}(c(I_2,-\delta_I))} \Phi_{n-r}(\nusym_{n-r} e^{\bom_{n-r}} \int_{(S^1)^{2g} } e^{\Omega} \tpsi_{I}^*(\frac{ \eta e^{- \delta_I Y_1} }{\tau_I^{2g-2}( 1-e^{ -Y_1}) })))$$ and also to $(r(n-r))^{-2g}$ times the iterated integral $$\int_{N_{n-r}(c(I_2,-\delta_I)) } \Phi_{n-r} ( \nusym_{n-r} e^{\bom_{n-r}} \res_{Y_1^I=0} \int_{N_r(c(I_1,\delta_I))} \Phi_{r}( \nusym_r e^{\bom_r} \int_{(S^1)^{2g} } e^{\Omega} \tpsi_{I}^*(\frac{ \eta e^{- \delta_I Y_1} }{\tau_I^{2g-2}( 1- e^{- Y_1}) }))),$$ for $c(I_1,\delta_I)$ and $c(I_2,-\delta_I)$ defined as in Proposition \ref{p7.3} with $\delta_I$ as in Remark \ref{bog} and $\tau_I$ as in Remark \ref{feb}. Here $\Omega \in H^2((S^1)^{2g})$ satisfies $$\tpsi_{I}^*(\bom) = \bom_r + \bom_{n-r} + \Omega - \delta_I \hat{e}_1$$ as in Remark \ref{indbasrem}. We need to consider separately those $I$ containing $n$ and those for which $n$ is not an element of $I$; first let us suppose that $n$ is not an element of $I$. Note that $$(-1)^{(r(n-r)+ \frac{1}{2}r(r-1) + \frac{1}{2}(n-r)(n-r-1))(g-1)} =(-1)^{\frac{1}{2}n(n-1)(g-1)},$$ and $$\frac{e^{-\delta_I Y_1}}{1-e^{-Y_1}} = \frac{e^{(1-\delta_I) Y_1}}{e^{Y_1}-1} .$$ The finite cover $\rho_I:S^1 \times T_r \times T_{n-r} \to \tor$ is $r(n-r)$ to one, so that it induces an $(r(n-r))^{2g}$ to one surjection from $(S^1)^{2g} \times T_r^{2g} \times T_{n-r}^{2g}$ to $\tor^{2g}$ and we have $$\int_{T_n^{2g}} \eta e^{\omega} = \int_{(S^1)^{2g}\times T_r^{2g} \times T_{n-r}^{2g}} \eta e^{\omega_r + \omega_{n-r} + \Omega}.$$ Moreover this finite cover $\rho_I:S^1 \times T_r \times T_{n-r} \to \tor$ takes the coordinate $Y_1 = X_1 - X_2$ on $\liet$ to the coordinate $Y^I_1$ on the Lie algebra of $S^1$. Since $\tpsi_I^*$ was defined using $\rho_I$ (see Remark \ref{big}), we deduce using Remark \ref{indbasrem} and Remark \ref{p5.11} and induction on $n$ that $(-1)^{r(n-r)(g-1)}$ times the iterated integral $$ \int_{N_r(c(I_1,\delta_I)) } \Phi_r (\nusym_r e^{\bom_r} \res_{Y_1^I=0} \int_{N_{n-r}(c(I_2,-\delta_I))} \Phi_{n-r}(\nusym_{n-r} e^{\bom_{n-r}} \int_{(S^1)^{2g} } e^{\Omega} \tpsi_{I}^*(\frac{ \eta e^{- \delta_I Y_1} }{\tau_I^{2g-2}( 1- e^{- Y_1}) })))$$ equals $(-1)^{n_+(g-1)} (r(n-r))^{2g} $ times the iterated residue $$\res_{X_{i_{1}} - X_{i_{2}}=0} \dots \res_{X_{i_{r-1}} - X_{i_{r}} =0} \res_{X_1-X_2 =0} \res_{X_{i_{r+1}} - X_{i_{r+2}} =0} \dots \res_{X_{i_1} - X_{i_2} =0} $$ $$\sum_{w_1\in W_{r-1}}\sum_{w_2\in W_{n-r-1}} \frac{ e^{ \inpr{\tildarg{ {w}_1\tc(I_1,\delta_I) } ,Y_{I_1}}} e^{ \inpr{ \tildarg{ {w}_2 \tc(I_2,-\delta_I ) } ,Y_{I_2}} } e^{(1-\delta_I) Y_1} \int_{T_{n}^{2g}} \eta e^{\omega} } { \nusym_{n}^{2g-2} ( e^{Y_1}-1) \prod_{j\neq r} ( \exp (X_{i_{j}}-X_{i_{j+1}})-1 ) } \Biggr ) $$ where $Y_{I_1}$ and $Y_{I_2}$ are the projections of $X$ onto the Lie algebras of the maximal tori $T_r$ and $T_{n-r}$ of $SU(r)$ and $SU(n-r)$ embedded in $SU(n)$ via the decomposition of $\{1,\ldots,n\}$ as $I_1 \cup I_2$, and $W_{r-1}$ and $W_{n-r-1}$ are the Weyl groups of $SU(r-1)$ and $SU(n-r-1)$ embedded in $SU(r)$ and $SU(n-r)$ using all but the last coordinates. There is no need to assume that $i_1 < i_2 < \dots <i_r$ and $i_{r+1} < i_{r+2} < \dots < i_n$ here. We simply need that $I_1 = I \cup \{1\} = \{i_1,\ldots,i_r\}$ and $I_2 = \{1,\ldots,n\} - I_1 = \{i_{r+1},\ldots, i_n\}$. So let us assume that $$i_r=1$$ and $$i_{r+1} =2.$$ We are also supposing that $n$ is not an element of $I$ (i.e. that $n\in I_2$) so we may assume in addition that $i_n = n$. Then we can apply the Weyl transformation $w_I^1 \in W_{n-1}$ given by the permutation $$\left \lbrack \begin{array}{ccccc} 1 & \dots & r & \dots & n-1 \\ i_1 & \dots & i_r & \dots & i_{n-1} \end{array} \right \rbrack $$ together with Remark \ref{c} to identify the iterated residue above with $$(-1)^{n_+(g-1)} (r(n-r))^{2g} \res_{Y_{1}=0} \dots \res_{Y_{n-1} =0} \sum_{w_1\in W_{r-1}} \sum_{w_2\in W_{n-r-1}} \frac{ e^{\inpr{\tildarg{w^1_I w_1 w_2(\tc) },X}} \int_{T_{n}^{2g}}\eta e^{\omega} } { \nusym_{n}^{2g-2} \prod_{1\leq j \leq n-1} (1- \exp (-Y_j)) } . $$ When $n\in I$ the argument is similar but we apply induction to $(-1)^{r(n-r)(g-1)}$ times $$ \int_{N_{n-r}(c(I_2,-\delta_I)) } \Phi_{n-r} (\nusym_{n-r} e^{\bom_{n-r}} \res_{Y_1^I=0} \int_{N_{r}(c(I_1,\delta_I))} \Phi_{r}(\nusym_{r} e^{\bom_{r}} \int_{(S^1)^{2g} } e^{\Omega} \tpsi_{I}^*(\frac{ \eta e^{ -\delta_I Y_1} }{\tau_I^{2g-2}(1 - e^{- Y_1}) })))$$ and observe that $$\res_{X_1-X_2=0} \frac{e^{-\delta_I(X_1-X_2)}}{1-e^{-(X_1-X_2)}} = - \res_{X_2-X_1=0} \frac{e^{\delta_I(X_2-X_1)}}{1-e^{X_2-X_1}} $$ (see the Remark after Corollary \ref{c4.2}). As $I_1 = I \cup \{1\} = \{i_1,\ldots,i_r\}$ and $I_2 = \{1,\ldots,n\} - I_1 = \{i_{r+1},\ldots, i_n\}$ and $n \in I$ we can assume that $i_1 =1$, $i_r = n$ and $i_n=2$. Then we use the Weyl transformation $w^2_I \in W_{n-1}$ given by the permutation $$\left \lbrack \begin{array}{ccccccc} 1 & \dots & n-r & n-r+1 & \dots & n-1 \\ i_{r+1} & \dots & i_n & i_1 & \dots & i_{r-1} \end{array} \right \rbrack $$ together with Remark \ref{c} to equate the iterated integral above with $$ \res_{Y_{1}=0} \dots \res_{Y_{n-1} =0} \sum_{w_1\in W_{r-1}} \sum_{w_2\in W_{n-r-1}} \frac{ e^{\inpr{ \tildarg{ w^2_I w_1w_2 (\tc) },X}} \int_{T_{n}^{2g}}\eta e^{\omega} } { \nusym_{n}^{2g-2} \prod_{1\leq j \leq n-1} ( \exp (Y_j)-1) } . $$ Thus it suffices to prove \begin{lemma} For each subset $I$ of $\{3,\ldots,n\}$ with $r-1$ elements, let us fix $i_1,\ldots,i_n$ such that $I\cup \{1\} = \{i_1,\ldots, i_r\}$ and $\{2,\ldots,n\} - I = \{i_{r+1},\ldots, i_n\}$ and also $i_r=1$, $i_{r+1}=2$ and $i_n=n$ (if $n \not\in I$) or $i_1=1$, $i_r=n$ and $i_n=2$ (if $n \in I$). Define permutations $w_I^1$ (for $I$ such that $n \not\in I$) and $w^2_I$ (for $I$ such that $n \in I$) as above. Then as (i) $r$ runs over $\{1,\ldots,n-1\}$, (ii) $w_1$ runs over permutations of $\{1,\ldots,r\}$ fixing $r$, (iii) $w_2$ runs over permutations of $\{r+1,\ldots,n\}$ fixing $n$ and (iv) $I$ runs over subsets of $\{3,\ldots,n\}$ with $r-1$ elements not containing $n$, \noindent the product $w^1_I w_1 w_2 $ runs over the set of permutations $w$ of $\{1,\ldots,n\}$ fixing $n$ such that $$w^{-1}(1)<w^{-1}(2).$$ Moreover if instead of (iv) $I$ runs over subsets of $\{3,\ldots,n\}$ with $r-1$ elements containing $n$, then the product $w^2_I w_1 w_2 $ runs over the set of permutations $w$ of $\{1,\ldots,n\}$ fixing $n$ such that $$ w^{-1}(1)>w^{-1}(2).$$ \end{lemma} \Proof If $w \in W_{n-1}$ satisfies $w^{-1}(1)<w^{-1}(2)$ let $r=w^{-1}(1)$ and $I=\{j:w^{-1}(j)<r\}$. On the other hand if $w \in W_{n-1}$ satisfies $w^{-1}(1) > w^{-1}(2)$ let $r=n-w^{-1}(2)$ and $I=\{j>1:w^{-1}(j) > n-r\} \cup \{n\}$. In each case it is easy to check that there exist unique choices of $w_1$ and $w_2$ such that $w^1_I w_1 w_2 =w$ or $w^2_I w_1 w_2 =w.$ This completes the proof of the lemma and hence of Proposition \ref{beg}. \begin{rem} \label{naive} It is shown in Proposition 3.4 of \cite{JK3} that the multivariable residue (multiplied by the constant $C_K$) of Theorem \ref{t4.1} and formula (\ref{1.7}) can be replaced by the iterated one-variable residue $$\res^+_{Y_{1}=0} \dots \res^+_{Y_{n-1}=0} $$ multiplied by the Jacobian (in this case $1/n$) of the change of coordinates from an orthonormal system to $(Y_1,\ldots,Y_{n-1})$. Here, if $\res_{y=0}g(y)$ denotes the coefficient of $y^{-1}$ in the Laurent expansion about 0 of a meromorphic function $g(y)$ of one complex variable $y$, then $\res^+$ is defined for meromorphic functions of the special form $\sum_{1\leq i\leq s} e^{\lambda_i y} q_i(y)$, where $\lambda_1,\ldots,\lambda_s$ are real numbers and $q_1,\ldots,q_s$ are rational functions of one variable with complex coefficients, by $$\res^+_{y=0} (\sum_{1\leq i\leq s} e^{\lambda_i y} q_i(y)) = \sum_{1\leq i\leq s, \lambda_i>0} \res_{y=0}(e^{\lambda_i y} q_i(y)).$$ Since $$\frac{e^{\gamma y}}{e^{y}-1}$$ can be formally expanded as $$-\sum_{m\in {\Bbb Z }, m+\gamma >0} e^{(m+\gamma)y}$$ when $0 < \gamma<1$, the formula (\ref{1.7}) can be formally rewritten as $$\prod_{r=2}^n a_r^{m_r} \exp(f_2) [\mnd] = \frac{(-1)^{n_+(g-1)}}{n} \res^+_{Y_{1}=0} \dots \res^+_{Y_{n-1}=0} \frac{ e^{\inpr{ \tc,X} } \int_{T_n^{2g}} e^{\omega} \prod_{r=2}^n \tau_r^{m_r}} {\nusym_n^{2g-2} \prod_{1\leq j\leq n-1} (e^{Y_j} -1)}.$$ Moreover the multivariable residue Res is invariant under the action of the Weyl group, as are all the other ingredients of the right hand side of (\ref{1.7}) except for $\tc$. Thus by averaging (\ref{1.7}) over the Weyl group we obtain a special case of Theorem \ref{mainab}. \end{rem} { \setcounter{equation}{0} } \section{Residue formulas for general intersection pairings} \newcommand{\abk}{{\Phi} } \newcommand{\tq}{\tilde{q}} \newcommand{\hattau}{\hat{\tau}} \newcommand{\srj}{{s_r^j} } \newcommand{\ssjg}{{s_s^{j+g} } } \newcommand{\psitexp}{ {\check{\psi}_{X,B}} } \newcommand{\tfq}{{ \tilde{f}_{ (q) } }} \newcommand{\xsharp}{{X^\#}} \newcommand{\free}{\FF} \newcommand{\epmb}{{\epc}} \newcommand{\thetsimp}[1]{\theta^{(#1)} } \newcommand{\spart}[2]{ \frac{\partial}{\partial s_{#1}^{#2} } } \newcommand{\mom}{J} \newcommand{\conn}[1]{ {\Theta^{(#1)} } } \newcommand{\qsgn}{{q}} \newcommand{\qought}{{q_{(o)} }} \newcommand{\tsign}{{\tau }} \newcommand{\Ssign}[1]{{ S^{#1}}} \newcommand{\signr}{ { (-1)^r }} \newcommand{\signs}{ { (-1)^s }} \newcommand{\signq}{**sign?**} \newcommand{\gen}{{\zeta }} \newcommand{\hu}[1]{ \hat{u}_{#1} } \newcommand{\zz}[1]{ Z_{#1} } \newcommand{\pqab}{ (\partial^2 \qsgn)(\hu{a}, \hu{b}) } \newcommand{\pqabx}{ (\partial^2 \qsgn)_X(\hu{a}, \hu{b}) } \newcommand{\simp}{\bigtriangleup} \newcommand{\quest}{**??**} \newcommand{\tarnox}{{\tilde{a}_r} } \newcommand{\tbrnox}{{\tilde{b}_r^j}} \newcommand{\hess}{{H_\liet}} In order to obtain explicit formulas for all the pairings, Witten observes that they can be obtained from those for the $a_r$ and $f_r$ via his formula \cite{tdgr} (5.20). In this section we shall generalize our version of his formula (\cite{tdgr} (4.74), which is our Theorem \ref{mainab} via the results of Section 2) to give formulas for $\int_{\mnd} \abk (\eta e^\bom)$ where $\eta$ is an equivariant cohomology class that does not simply involve the $\tar$ but also involves the $\tbrj $ and the $\tfr$ (see Theorems \ref{t9.5} and \ref{t9.6} below). The key step in the proof is Lemma \ref{l9.2}, combined with the argument used in Sections 5-8 to prove Theorem \ref{mainab}. In the next section we shall see that Theorem \ref{t9.5} yields Witten's formula \cite{tdgr} (5.20). This will follow from certain equations satisfied by the formula given in Theorem \ref{t9.4} (Propositions \ref{p9.1} and \ref{p9.2}). The next lemma (from \cite{J2}) will give an explicit formula for an equivariant cohomology class $\tfr$ on $M(c)$ such that $\abk(\tfr) = f_r$ (cf. Proposition \ref{abftil}). In order to state it, we introduce the following notation. \begin{definition} {\bf (The moment)} If $\theta$ is the Maurer-Cartan form on $K$, the moment $\mom(\theta) \in \Bigl ( \Omega^1 (K) \otimes \lieks \Bigr )^K $ is defined for $X \in \liek$ by \begin{equation} \label{9.09} \mom(\theta) (X)_k = - \iota_{\xsharp} \theta = -\Ad (k^{-1}) X, \end{equation} where $\xsharp$ is the vector field on $K$ given by the left action of $X$ on $K$. \end{definition} \begin{rem} See \cite{BGV}, Chapter 7 for an explanation of the role of the moment in the construction of equivariant characteristic classes, via an equivariant version of Chern-Weil theory. Given a principal bundle over a $K$-manifold equipped with a compatible action of $K$ on the total space of the bundle, the moment $\mom$ plays the same role as the symplectic moment map plays for a principal $U(1)$ bundle ${\mbox{$\cal L$}}$ over a Hamiltonian $K$-manifold with $c_1({\mbox{$\cal L$}}) = [\omega]$ (and with a lift of the action of $K$ to the total space of ${\mbox{$\cal L$}}$). In particular, the appropriate notion of ``equivariant curvature'' is the sum of the usual curvature and the moment $\mom$. \end{rem} In the next few paragraphs we provide a brief outline of the use of the Bott-Shulman construction (see for instance \cite{BSS} and other references given in \cite{J2}) to obtain equivariant differential forms representing the equivariant characteristic classes $\tfr$. This material is summarized from \cite{J2}, which gives a construction of de Rham representatives for equivariant characteristic classes giving rise to the characteristic classes of the universal bundle over $\mnd \times \Sigma$. This was accomplished by regarding this bundle (and the classifying space for it) as {\em simplicial manifolds}. For more details see \cite{J2}. Let $\simp^2 = \{ (t_0, t_1, t_2) \in [0,1]^3: t_0 + t_1 + t_2 = 1 \} $ be the standard 2-simplex. There is a principal $K$-bundle $$ \simp^2 \times K^3 \stackrel{\pi_2}{\longrightarrow} \simp^2 \times K^2 $$ for which the bundle projection $\pi_2: K^3 \to K^2$ is given by $$ \pi_2 (g_0, g_1, g_2) = (g_0 g_1^{-1}, g_1 g_2^{-1}) ~~~\mbox{ (\cite{J2}, (3.9)) }. $$ We define a connection $\conn{2} $ on the total space of this bundle by $$ \conn{2} = \sum_{i = 0}^2 t_i \thetsimp{i} ~~ \in \Omega^1 (\simp^2 \times K^3) \otimes \liek, $$ where $\thetsimp{i}$ $ \in \Omega^1(K^3) \otimes \liek$ is the Maurer-Cartan form on the $i$-th copy of $K$. The curvature $$ F_{\conn{2}} \in \Omega^2 (\simp^2 \times K^3) \otimes \liek $$ of the bundle is \begin{equation} \label{curv} F_{\conn{2}} = \sum_id ( t_i \thetsimp{i}) + [\conn{2}, \conn{2}] . \end{equation} We use this connection and curvature and the Chern-Weil theory of equivariant characteristic classes (see for instance Chapter 7 of \cite{BGV}) to define an equivariant form on the total space $\simp^2 \times K^3$ of the bundle, which represents the equivariant characteristic class associated to $\tau_r$ in equivariant cohomology. We then integrate this equivariant form over the simplex $\simp^2.$ Finally, we may pull this form back to the base space $K^2$ via a section $\sigma_2: K^2 \to K^3 $ given by $$ \sigma_2(k_1, k_2) = (k_1 k_2, k_2, 1) ~~\mbox{(\cite{J2}, (4.3)) } $$ Explicitly, we make the following definition: \begin{definition} Let $\Phi_2^K (\tau_r) = \sigma_2^* \bar{\Phi}_2^K (\tau_r) $ $\in \Omega^{2r-2}_K (K \times K ) $ (see \cite{J2}, above (4.3)) where the section $\sigma_2$ was defined above, and \begin{equation} \label{8.9} \bar{\Phi_2}^K (\tau_r) = \int_{\simp^2} \tau_r (F_{\theta(t)} + \mom(\theta(t) ) ) . \end{equation} \end{definition} Let $\simp^1 = \{ (t_0, t_1) \in [0,1]^2: t_0 + t_1 = 1 \} $ $ \cong [0,1]$ be the standard 1-simplex. We shall perform a similar construction using a principal $K$-bundle $$\simp^1 \times K^2 \stackrel{\pi_1}{\longrightarrow} \simp^1 \times K.$$ The bundle projection $\pi_1: K^2 \to K$ is defined by $$\pi_1(g_0, g_1) = g_0 g_1^{-1}. $$ A section $\sigma_1: K \to K^2 $ of the bundle is given by $\sigma_1(k) = (k,1). $ On the total space $\simp^1 \times K^2$ we define a connection $$\conn{1} = \sum_{i = 0}^1 t_i \thetsimp{i} \in \Omega^1 (\simp^1 \times K^2) \otimes \liek, $$ where $\thetsimp{i}$ $ \in \Omega^1(K^2) \otimes \liek$ is the Maurer-Cartan form on the $i$-th copy of $K$. The definition of the curvature $$F_{\conn{1}} \in \Omega^2 (\simp^1 \times K^2) \otimes \liek$$ is similar to ({\ref{curv}}). As before, we evaluate the invariant polynomial $\tau_r$ on the equivariant curvature and integrate over the simplex $\simp^1$ to get an equivariant form over $K \times K$, and finally we pull this form back to $K$ using the section $\sigma_1$: explicitly, we make the following \begin{definition} We define $$\Phi_1^K(\tau_r) = \sigma_1^* \bar{\Phi}_1^K(\tau_r) \in \Omega^{2r-1}(K),$$ where \begin{equation} \label{9.072} \bar{\Phi}_1^K(\tau_r) = \int_{\simp^1} \tau_r(F_{\theta(t)} + \mom(\theta(t) ) ) \in \Omega^*_K (K \times K). \end{equation} \end{definition} \begin{definition} {\bf (Equivariant chain homotopy)} We define a chain homotopy $$I_K: \Omega^{* +1}_K (\liek) \to \Omega^{* }_K (\liek) $$ as follows: when $v \in \liek $, we have \begin{equation} \label{9.2a} (I_K \beta)_v = \int_0^1 F_t^* (\iota_{\bar{v} } \beta) dt, \end{equation} where $F_t: \liek \to \liek$ is multiplication by $t$ and $\bar{v}$ is the vector field on $\liek$ which takes the constant value $v$. \end{definition} \begin{lemma} \label{l9.1} { \bf(\cite{J2}, Theorem 8.1) } The equivariant cohomology class of the equivariant differential form $$\tfr = \proj_1^* \tfr_1 + \proj_2^* \tfr_2 $$ is a lift of $f_r \in H^{2r-2} (\mnd)$ to $H^{2r-2}_K (M(c))$. Here, the maps $\proj_1$ and $\proj_2$ are the projection maps from $M(c)$ to $K^{2g}$ and $\liek$ defined at (\ref{4.4}). Also, from (\cite{J2}, (7.13)), \begin{equation} \label{9.3} \tfr_1 = \Bigl ( \sum_{j=1}^g (- {\rm ev}_{\gamma_{j}^1} \times {\rm ev}_{x_{j}} + {\rm ev}_{\gamma_{j+g}^0} \times {\rm ev}_{x_{j+g}} \Bigr )^* \Phi_2^K(\tau_r) (X) + \end{equation} $$\Bigl ( \sum_{j=1}^g (- {\rm ev}_{\gamma_{j+g}^1} \times {\rm ev}_{x_{j+g}} + {\rm ev}_{\gamma_{j}^0} \times {\rm ev}_{x_{j}} \Bigr )^* \Phi_2^K(\tau_r) (X) \in \Omega^*_K (K^{2g}) $$ and \begin{equation} \label{9.4} \tfr_2 = - I_K (\epc^* \Phi_1^K (\tau_r)) \in \Omega^*_K(\liek) \end{equation} where $\gamma_j^\alpha$ (for $\alpha = 0,1$ and $j = 1, \dots, 2g$) are certain elements of $\free^{2g}$ (the free group on $2g$ generators $x_1, \dots, x_{2g}$, as in Section 4), whose definition is given in (7.12) of \cite{J2}, and for any $z \in \free^{2g}$, $ {\rm ev}_z: K^{2g} \to K$ denotes the evaluation map on $z$. Here, $\epc: \liek \to K$ is defined by $\epc(\Lambda) = c \exp \Lambda$ where the central element $c = e^{2 \pi i d/n } \diag(1, \dots, 1) $ was defined at (\ref{1.p1}). \end{lemma} By (\ref{9.072}) we have \begin{equation} \label{9.005} \bar{\Phi_1}^K (\tau_r)(-X) = \int_{t \in [0,1]} \tau_r\Bigl (dt (\thetsimp{0} - \thetsimp{1}) + t d\thetsimp{0} + (1-t) d \thetsimp{1} + {\frac{1}{2} } [t \thetsimp{0} + (1-t) \thetsimp{1}, t \thetsimp{0} + (1-t) \thetsimp{1}] + \end{equation} $$ t \Ad(g_0^{-1} ) X + (1-t) \Ad(g_1^{-1})X \Bigr ). $$ Now $$ \bar{\Phi}_1^K (\tau_r)|_{T \times T}(-X) = \int_{t \in [0,1]} \tau_r(dt (\thetsimp{0} - \thetsimp{1}) + X) $$ since $$d \thetsimp{i}+ {\frac{1}{2} } [\thetsimp{i}, \thetsimp{i}] = 0 $$ and the restrictions of $[\thetsimp{i}, \thetsimp{i}] $ to $T$ vanish. Further $$\sigma_1^* \bar{\Phi}_1^K (\tau_r)|_T (-X) = \int_{t \in [0,1]} \tau_r(dt \theta + X), $$ where $\theta$ is the Maurer-Cartan form on $T$. If $\tau_r (Z_1, \dots, Z_{n-1}) = \sum_I (\tau_r)_I Z^I $ where $I = (i_1, \dots, i_{n-1})$ is a multi-index and $Z^I = Z_1^{i_1} \dots Z_{n-1}^{i_{n-1}} $ (in terms of a coordinate system $\{Z_a = \inpr{\hu{a},X}, \: a = 1, \dots, n-1 \} $ on $\liet$, specified by an oriented orthonormal basis $\hu{a}$ for $\liet $ for which $\theta_a, a = 1, \dots, n-1$ are the corresponding components of the Maurer-Cartan form $\theta \in \Omega^1 (T) \otimes \liet$), then we have $$ {\Phi_1^K} (\tau_r)|_{T}(-X) = \sum_I \int_{t \in [0,1]} (\tau_r)_I (dt \theta_1 + Z_1)^{i_1} \dots (dt \theta_{n-1}+ Z_{n-1})^{i_{n-1}} $$ \begin{equation} \label{9.7} = \sum_{a= 1}^{n-1} \theta_a \partial \tau_r /\partial Z_a . \end{equation} \begin{lemma} \label{l9.3} We have for $\Lambda \in \liet$ (in terms of the Maurer-Cartan form $\theta \in \Omega^1(T) \otimes \liet$) that $$I_K (\epc^* \theta)_\Lambda = \Lambda. $$ \end{lemma} \Proof We have $$ I_K (\epc^* \theta)_\Lambda = \int_0^1 F_t^* (\epc^* \theta (\bar{\Lambda} ) )dt = \Lambda $$ since $\epc^* \theta(\bar{\Lambda} ) : \liet \to {\Bbb R } $ is the function with constant value $\Lambda$. \hfill $\square$ Let $q \in S(\lieks)^K$ be an invariant polynomial which is given in terms of the elementary symmetric polynomials $\tau_j$ by \begin{equation} \label{9.0077} q(X) = \tau_2(X) + \sum_{r = 3}^n \delta_r \tau_r(X). \end{equation} The associated element $\tfq $ of $\hk(M(c))$ is defined by \begin{equation} \label{9.001} \tfq = \tf_2 + \sum_{r = 3}^n \delta_r \tf_r. \end{equation} Here, the $\delta_r$ are formal nilpotent parameters: we expand $\exp \tfq$ as a formal power series in the $\delta_r$. We can alternatively regard the $\delta_r$ as real parameters and $\exp \tfq$ as a formal equivariant cohomology class: the integral $$ \int_{\mnd} \abk (\exp \tfq) $$ and the integral appearing in (\ref{9.2}) are well defined and are polynomial functions of the $\delta_j$, since $\int_{\mnd} \abk(\eta) = 0 $ unless $2 {\rm deg}(\eta) = \dim \mnd$. Note that by Lemma \ref{l9.1} we can write $\tfq(X) = \proj_1^* \tfq(X)_1 + \proj_2^* \tfq(X)_2$ where $\tfq(X)_1 \in \Omega^*_K(K^{2g} ) $ and $\tfq(X)_2 \in \Omega^*_K(\liek). $ Then we have \begin{lemma} \label{l9.3a} For $X \in \liet$, the restriction of $\tfq(-X)_2 $ to $\mu^{-1}(\liet)$ is given at $(h_1, \dots, h_{2g}, \Lambda) $ $ \in \mu^{-1} (\liet) \subset K^{2g} \times \liek$ by $$ \tfq(-X)_2|_{\mu^{-1}(\liet)} (h_1, \dots, h_{2g}, \Lambda) = - (d \qsgn)_X (\Lambda). $$ \end{lemma} \Proof $$\tfq(-X)_2|_{\mu^{-1}(\liet)} = - I_K \epmb^* \Phi_1^K (q)(-X) $$ $$ = - I_K (\sum_{a = 1}^{n-1} \theta^a\partial \qsgn /\partial Z^a ) ~~ \mbox{by (\ref{9.7}) } $$ $$ = - (d \qsgn)_X (\Lambda) ~~\mbox{by Lemma \ref{l9.3}}. $$ \hfill $\square$ \begin{lemma} \label{l9.2} Assume that $X \in \liet$. Let $\Lambda = \sum_{a=1}^{n-1} m_a \he{a} \in \intlat$ for $m_a \in {\Bbb Z }$ (where the simple roots $\he{a} $ were defined in (\ref{6.1}))\footnote{Note that the $\he{a}$ are a basis of $\liet$, but not an orthonormal basis.} and let $s_\Lambda$ denote the homeomorphism of $\mtc$ given by Lemma \ref{l4.3}. Then we have that on $\mtc$ \begin{equation} \label{9.007} s_\Lambda^* \tfq(-X) = \tfq(-X) - (d\qsgn)_X (\Lambda),\end{equation} or equivalently $$ s_\Lambda^* \tfq(X) = \tfq(X) + (d\qought)_{X} (\Lambda),$$ where we have introduced the notation $$\qought (X) = q( -X). $$ \end{lemma} \noindent{\em Remark:} This result generalizes (\ref{8.4}). \noindent{\em Proof of Lemma \ref{l9.2}:} Since $\tfq(X) = \proj_1^* \tfq(X)_1 + \proj_2^* \tfq(X)_2$, we need to prove the formula for $s_\Lambda^* \proj_2^* \tfq(X)_2 $ where $\tfq(X)_2 = - I_K \epmb^* \Phi_1^K (q)$ for $\Phi_1^K (q) \in \Omega^*_K(K). $ Lemma \ref{l9.2} then follows from Lemma \ref{l9.3a}. \hfill $\square$ \begin{theorem} \label{t9.4} Suppose $\eta$ is a polynomial in the $\tar$ and $\tbrj$. Let $q \in S(\lieks)^K$. Then for any $X \in \liet$ we have $$ \int_{N_n(V) } \abk \Biggl (\eta e^{\tfq} \Bigl ( e^{ (d \qought)_X (\he{1})} - 1 \Bigr ) \alpha \Biggr ) = - \sum_{F\in{\mbox{$\cal F$}}: -||\he{1}||^2 < \inpr{\he{1},\mu(F)} < 0} \res_{Y_{1} = 0 } \int_F \frac{\phicom ( \eta e^{\tfq} \alpha ) } {e_{ F}} . $$ Here, we sum over the components $F $ of the fixed point set of $\hat{T_1} $ in $P^{-1}(V) \cap \mu^{-1} (\hat{\liet}_1)/T_{n-1} $; the notation is as in the statement of Lemma \ref{old5.17}. The notation $\qought$ was introduced in the statement of Lemma \ref{l9.2}. We have defined the map $\phicom$ in Proposition \ref{stages}, and after (\ref{snow}). \end{theorem} \Proof This follows from the same proof as for Lemma \ref{l6.4}, replacing (\ref{8.4}) by its generalization Lemma \ref{l9.2}. \hfill $\square$ We aim to prove the following result by induction: \begin{theorem} \label{t9.5} {\bf (a)} For the particular $q$ defined in (\ref{9.0077}), we have $$ \int_{N_n(c) } \abk ( e^{\tfq} \nusym_n \eta ) =\frac{(-1)^{n_+(g-1)} }{n!}\sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \frac { \int_{T^{2g}\times \{ -\tildarg{w\tc} \} )}\Bigl ( e^{ \tfq(X) } \eta(X) \Bigr ) }{\nusym(X)^{2g-2} \prod_{j = 1}^{n-1} (\exp -B(-X)_{j} - 1) }, $$ where $\eta$ is a polynomial in the $\tarnox$ and $\tbrnox$ and $B(X)_j = -(d\qsgn)_X (\he{j}) $. Here we have used the fixed invariant inner product on $\liek$ to identify $d\qsgn_X: \liet \to {\Bbb R }$ with an element of $\liet$ and thus define the map $B: \liet \to \liet.$ The notation $\tildarg{\gamma}$ was introduced in Definition \ref{bracedef}. \end{theorem} Substitution of $-X$ for $X$ on the right hand side of the equation in Theorem \ref{t9.5} (a) gives the equivalent formulation \noindent{\bf Theorem \ref{t9.5} (b)} {\em In the notation of Theorem \ref{t9.5} (a) we have $$ \int_{N_n(c) } \abk ( e^{\tfq} \nusym_n \eta ) =\frac{(-1)^{n_+(g-1)} }{n!} \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \frac { \int_{T^{2g}\times \{ - \tildarg{w\tc} \} } \Bigl ( e^{\tfq(-X)} \eta(-X) \Bigr ) }{\nusym(X)^{2g-2} \prod_{j = 1}^{n-1} (1 - \exp -B(X)_{j} ) }. $$ } Finally we may use Lemma \ref{l9.8} and Lemma \ref{l9.9'} (a) where the restrictions to $T^{2g}$ of the equivariant cohomology classes $\tfr$ and $\tbrj$ are expressed in terms of the basis $\zeta_a^j$ for $H^1 (T^{2g})$ (for $a = 1, \dots, n-1$ and $j = 1, \dots, 2g$). We also use Lemma \ref{l9.03}, where the symplectic volume of $T^{2g}$ is calculated. These lemmas enables us to compute $\int_{T^{2g}} e^{ \tfq(-X) } \eta(-X)$ and rephrase Theorem \ref{t9.5} (b) as follows. (Here we have also reformulated the left hand side of Theorem \ref{t9.5} (b) in terms of the pairings on $\mnd$, using Lemma \ref{l3}.) \begin{theorem} \label{t9.6} In the notation of Theorem \ref{t9.5} we have \noindent{\bf (a)} $$ \int_{\mnd} \exp (f_2 + \delta_3 f_3 + \dots + \delta_n f_n) \prod_{r = 2}^n a_r^{m_r} \prod_{k_r = 1}^{2g} (b_r^{k_r})^{p_{r,k_r} } = $$ \begin{equation} \frac{ (-1)^{n_+(g-1)} }{n!} \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \Biggl ( \frac { e^{ d \qsgn_X (\tildarg{w\tc} ) } \Bigl ( \prod_{r = 2}^n \tau_r(X)^{m_r} \Bigr ) }{\nusym(X)^{2g-2} \prod_{j = 1}^{n-1} (1 - \exp -B(X)_{j} ) } \times \end{equation} $$ \int_{T^{2g} } \exp \Bigl \{ - \sum_{a,b} \sum_{j = 1}^g \zeta_a^j \zeta_b^{j + g} \partial^2 \qsgn_X(\hu{a}, \hu{b}) \Bigr \} \prod_{r = 2}^n \prod_{k_r = 1}^{2g} \Bigl ( \sum_{a = 1}^{n-1} (d\tau_r)_X (\hu{a}) \gen_a^{k_r} \Bigr )^{p_{r,k_r}} \Biggr ). $$ \noindent{\bf (b)} In particular we have that $$ \int_{\mnd} \exp (f_2 + \delta_3 f_3 + \dots + \delta_n f_n) \prod_{r = 2}^n a_r^{m_r} = $$ \begin{equation} (-1)^{n_+(g-1)} \frac{n^{g}}{n!} \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \frac { e^{ d \qsgn_X (\tildarg{w\tc} )} \prod_{r = 2}^n \tau_r(X)^{m_r} (\det H_\liet (X))^g }{\nusym(X)^{2g-2} \prod_{j = 1}^{n-1} (1 - \exp - B(X)_{j}) }. \end{equation} \end{theorem} \begin{rem} In the preceding Theorem, we have used the following notation. The $a_r, f_r$ and $b_r^j$ (for $r = 2, \dots, n$ and $j = 1, \dots, 2g$) are generators of the cohomology ring, introduced in Section 2. The $\tau_r$ are the elementary symmetric polynomials, and the $\delta_r$ are formal nilpotent parameters which were introduced in (\ref{9.0077}). The polynomial $q = \tau_2 + \sum_{r = 3}^n \delta_r \tau_r$ was introduced in (\ref{9.0077}). Its derivative $dq_X: \liet \to {\Bbb R }$ is identified with an element of $\liet$ via the inner product $\inpr{\cdot,\cdot}$ on $\liet$, and hence $dq: \liet \to \liet^*$ is identified with a map $B: \liet \to \liet$ (see the statement of Theorem \ref{t9.5}(a)). If $\gamma \in \liet$, the notation $\tildarg{\gamma}$ was introduced in Definition \ref{bracedef}: it is the unique element in the fundamental domain defined by the simple roots for the translation action on $\liet$ of the integer lattice which is equivalent to $\gamma$ under translation by the integer lattice. The $\zeta_a^j$ are derived from the components of the Maurer-Cartan form $\theta \in \Omega^1(T) \otimes \liet$ in terms of an orthonormal basis $\{ \hu{a}, a = 1, \dots, n -1\}$ of $\liet$: they have been identified with a basis of $H^1 (T^{2g})$. (See Definition \ref{zedef} below.) Finally $\det H_\liet(X)$ is the determinant of the Hessian of $q: \liet \to {\Bbb R },$ in terms of the coordinates on $\liet$ given by the orthonormal basis $\{\hu{a} \}$: it is independent of the choice of orthonormal basis. We note that in Theorem \ref{t9.6} the orthonormal basis introduced above could be replaced by a general basis, provided one defines the $\zeta_a^j$ using that basis, and multiplies the Hessian by a factor due to the change of basis: see Remark \ref{r10.1} below. \end{rem} \begin{rem} We can replace $f_2$ by any nonzero constant scalar multiple $\epsilon f_2$ provided we replace the polynomial $q$ by $q^\epsilon$ where $$q^\epsilon(X) = \epsilon \tau_2(X) + \delta_3 \tau_3(X) + \dots + \delta_n\tau_n(X) $$ (cf. Remark \ref{rem8.3} (b)). \end{rem} In order to prove Theorem \ref{t9.5} and hence Theorem \ref{t9.6} we follow the proof of Theorem \ref{mainab} using \begin{lemma} \label{l9.7} Suppose $\eta$ is a polynomial in the $\tar$ and $\tbrj$. Then for any $X \in \liet$ and $q \in S(\lieks)^K$ chosen as in (\ref{9.0077}), we have \begin{equation} \label{9.02} \int_{N_n(V)} \abk \Bigl (\eta \nusym e^{\tfq } \alpha \Bigr ) = - \sum_{F\in{\mbox{$\cal F$}}: -||\he{1}||^2 < \inpr{\he{1},\mu(F)} < 0} \res_{Y_{1} = 0 } \int_F \frac{\phicom ( \nusym \eta(X) e^{\tfq} \alpha ) } {e_{F} \Bigl ( e^{ (d\qought)_X (\he{1})} - 1 \Bigr ) },\end{equation} where the notation is as in the statement of Lemma \ref{old5.17}. \end{lemma} \Proof This follows from Theorem \ref{t9.4} by replacing $\eta $ by \begin{equation} \label{e9.16} \frac{\eta \nusym } {\Bigl ( e^{(d\qought)_X (\he{1})} - 1 \Bigr ) } = \frac{\eta \bar{\nusym } Y_{1} } {\Bigl ( e^{ (d\qought)_X (\he{1}) } - 1 \Bigr ) }, \end{equation} where we have defined $\bar{\nusym} = \nusym/Y_1. $ Notice that $(d\qought)_X (\he{1})$ is divisible by $Y_1$: to see this, we observe that if we define the generating functional $$P(X_1, \dots, X_n) = \prod_{j = 1}^n (1 + t X_j) = \sum_{r = 0}^n \tau_r(X_1, \dots, X_n) t^r $$ (where the $\tau_r$ are the elementary symmetric polynomials) then $$ dP = \left ( (1+ tX_1) tdX_2 + (1 + t X_2) tdX_1 \right) \prod_{j = 3}^n (1 + t X_j) + {\cal P}$$ where ${\cal P}$ is a collection of terms involving $dX_3, \dots, dX_n$. Evaluating $dP$ on $\he{1} = (1,-1, 0, \dots, 0)$ we thus obtain $$ t^2 (-Y_1) \prod_{j=3}^n (1 + tX_j) = \sum_{r = 0 }^n t^r (d\tau_r)_X(\he{1}). $$ It follows that the $(d\tau_r)_X(\he{1})$ (and hence $(d \qought)_X (\he{1}) $) are divisible by $Y_1$. Thus $-(d\qought)_X(\he{1}) = - Y_{1}(1 + {\nu})$ where ${\nu} \in H^*_{T} $ has degree at least $ 1$, so we have $$\Bigl ( e^{ (d \qought)_X (\he{1})} - 1 \Bigr ) = - Y_{1}(1 - \tilde{\nu}) $$ where $\tilde{\nu} = \sum_{j \ge 1} \tilde{\nu}_j $ is a formal sum of classes $\tilde{\nu}_j$ with degree at least $1$ in (a completion of) $H^*_T$. Then the expression $$ \abk \Biggl ( \frac{\eta \nusym } {\Bigl ( e^{ (d\qought)_X (\he{1})} - 1 \Bigr ) } \Biggr ) $$ (which appears on the left hand side of the equation in Theorem \ref{t9.4}) is well defined. On the right hand side, we may replace $$\frac{ Y_{1} } {\Bigl ( e^{+ (d\qought)_X (\he{1})} - 1 \Bigr ) }$$ by $- (1 - \tilde{\nu})^{-1} = -\sum_{s \ge 0 } {\tilde{\nu}}^s. $ \hfill $\square$ We now use Lemma \ref{l9.7} to prove Theorem \ref{t9.5} (a) by induction on $n$. The proof follows the outline of the proof of Theorem \ref{mainab} when $q = q_2$, with the following modifications: \begin{enumerate} \item $e^\bom$ is replaced by $e^{\tfq}$ (and $e^\omega$ replaced by $e^{\tfq - \tilde{f_2}} e^\omega $). \item$(d\qought)_X (\he{1})$ replaces $-Y_1$, so $e^{ (d\qought)_X (\he{1}) } - 1 $ replaces $e^{-\normcon Y_1} - 1$. \item In particular, $-(d \qought)_X (\he{1})$ $ = -B (-X)_1 $ replaces $ Y_1$ in the identity $$ \frac{e^{- \delta_I Y_1} }{1 - e^{-Y_1} } = \frac{e^{(1- \delta_I )Y_1} }{e^{Y_1} - 1 } $$ which is used in the proof of Proposition \ref{beg}. \end{enumerate} We also use the elementary fact that $(d \qought)_X = - (dq)_{-X} = B (-X). $ { \setcounter{equation}{0} }\ \section{Witten's formulas for general intersection pairings} In this section we state and prove Witten's formulas (Propositions \ref{p9.1} and \ref{p9.2} below; cf. \cite{tdgr}, Section 5, in particular the calculations (5.11)-(5.20)), which enabled him to calculate general intersection pairings in terms of those of the form $$\int_{\mnd} \prod_r a_r^{m_r} e^{f_2} . $$ We shall prove these formulas starting from our explicit formulas for the general intersection pairings (see Theorem \ref{t9.5}). Some of the notation in the statement of Propositions \ref{p9.1} and \ref{p9.2} was introduced at the beginning of Section 9. The invariant polynomial $q$ was defined by (\ref{9.0077}). Using the invariant metric on $\liek$, the map $-d\qsgn: \liek \to \lieks$ may be regarded as a map $B = B^{(2)} + \sum_{r \ge 3} \delta_r B^{(r)} :\liek \to \liek$, where we have written $B^{(r)} = -d \tau_r: \liek \to \liek$; we find $B^{(2)} = -d \tau_2 = id: \liek \to \liek$. (Note that we have put $\tau_2(X) = - {\frac{1}{2} } \inpr{X,X}$ in terms of the inner product $\inpr{\cdot,\cdot} $ defined at (\ref{1.02}).) The other maps $B^{(r)}$ are not linear. The Hessian of $-\qsgn$ is $H$; it is a function from $\liek$ to symmetric bilinear forms on $\liek$. If $k, l$ run over an orthonormal basis $\{\hat{v}_k \} $ of $\liek$ then the Hessian at $X$ is the matrix \begin{equation} \label{hessdef} H(X)_{kl} = - (\partial^2 \qsgn)_X (\hat{v}_k, \hat{v}_l) . \end{equation} \begin{rem} \label{r10.1} In most places in Sections 9 and 10, the orthonormal basis $\{\hu{a}\} $ for $\liet$ may be replaced by any basis for $\liet$ (including the basis $\{ \he{a}, a = 1, \dots, n-1 \}$, which is of course not orthonormal), and similarly for the orthonormal basis $\{ \hat{v}_l \} $ for $\liek$. However it is more convenient to define the determinant of the Hessian (given in (\ref{hessdef})) in terms of an {\em orthonormal} basis, since one must otherwise include a normalization factor proportional to the square of the determinant of a matrix whose columns are the basis elements. The second place where it is useful to introduce an orthonormal basis is in the definition of the symplectic form in terms of the generators $\zeta_a^j$ for the cohomology of $T^{2g}$: the symplectic form is defined using the inner product $\inpr{\cdot, \cdot} $ on $\liet$, and the formula (Lemma \ref{l9.7a}) for the restriction of the symplectic form to $T^{2g}$ is cleaner in terms of an orthonormal basis. For these reasons we have chosen to use an orthonormal basis for $\liet$ throughout Sections 9 and 10, although in many specific instances this basis may be replaced by a general basis. In particular in the statement of our main theorem Theorem \ref{t9.6}, it is easy to check that the orthonormal basis may be replaced by a general basis, provided that the $\zeta_a^j$ are also defined using this basis, and that the Hessian is multiplied by the appropriate factor. \end{rem} \newcommand{\winv}{{ { B^{-1} }}} \newcommand{\bwinv}{ (\winv) } We assume the $\delta_r$ are formal {\em nilpotent} parameters: then the invertibility of $B $ is guaranteed. We write $ B^{-1}: \liek \to \liek$ as the inverse of $B$. (If the $\delta_r$ are nilpotent, the inverse of $B$ may be written as a formal power series in the $\delta_r$.) \begin{prop} \label{p9.1} For any invariant polynomial $\tau \in S(\lieks)^K$, the integral \begin{equation} \label{10.1a} \int_{\mnd} \abk ( \tau(-X) \exp \tfq ) \end{equation} is equal to the integral \begin{equation} \label{9.1} \int_{\mnd} \abk \Biggl (\tau(\winv(-X) ) \Bigl ( \det H(\winv(-X)) \Bigr )^{g-1} \Biggr ) \exp f_2 \end{equation} which is of the form that may be calculated by Theorem \ref{mainab}. \end{prop} \begin{prop} \label{p9.2} Let $\tau \in S(\lieks)^K$ be an invariant polynomial, so that $$\tau = \sum_{m_2, \dots, m_n} c_{m_2, \dots, m_n} \prod_{r = 2}^n \tarnox^{m_r}$$ is a polynomial in the $\tilde{a}_2, \dots, \tilde{a}_n$. Let $s_r^j$ be real parameters (for $r = 2, \dots, n$ and $j = 1, \dots, 2g$). Then we have \begin{equation} \label{9.2} \int_{\mnd} \abk \Bigl ( \tau(X) \exp (\sum_{r = 2}^n \sum_{j=1}^{2g} \srj \tbrj ) \exp \tfq \Bigr ) = \int_{\mnd} \abk\Bigl (\tau(X) \exp \hattau(X) \exp \tfq \Bigr ). \end{equation} Here, the invariant polynomial $\hattau$ on $\liek$ is defined (for $X \in \liet$) by $$\hattau(-X) = - \sum_{a,b=1}^{n-1} \sum_{r,s = 2}^{n} \sum_{j=1}^{g} s^j_r s^{j+g}_s (d\tau_r)_X(\hu{a}) (d\tau_s)_X(\hu{b}) (\partial^2 \qsgn )^{-1}_{ab}, $$ where $\{ \hu{a}: a = 1, \dots, n-1 \} $ denotes an oriented orthonormal basis of $\liet$: see (\ref{hattaudef}) for the definition. \end{prop} \begin{rem} Notice that in our conventions on the equivariant cohomology differential and the moment, the construction of \cite{J2} described at the beginning of Section 9 yields $$\tarnox(X) = \tau_r(-X) . $$ Thus $\tau(X) = \sum_{m_2, \dots, m_n} c_{m_2, \dots, m_n} \prod_{r = 2}^n \tau_r(-X)^{m_r}. $ \end{rem} Proposition \ref{p9.1} is proved by comparing Theorem \ref{t9.6} (b) (applied to (\ref{10.1a})) with Theorem \ref{mainab} (applied to (\ref{9.1})). Proposition \ref{p9.2} is obtained by applying Theorem \ref{t9.5} (b) to both sides of (\ref{9.2}) and examining the restrictions to $T^{2g}$ (which are computed in Lemmas \ref{l9.8} and \ref{l9.10}). Propositions \ref{p9.1} and \ref{p9.2} enable us to extract formulas for all pairings, by differentiating the formulas (\ref{9.1}) and (\ref{9.2}) with respect to the parameters $\delta_r$ and $\srj$ and then setting these parameters equal to zero. In fact, for any nonnegative integers $n_r$ (for $r \ge 3$) we have $$ \Biggl (\prod_{r = 3}^n \Bigl ( \frac{\partial}{\partial \delta_r}\Bigr )^{n_r} \int_{\mnd} \abk ( \tau(X) \exp \tfq ) \Biggr )_{\delta_3 = \dots = \delta_n = 0 } = \int_{\mnd} \prod_{r = 3}^n f_r^{n_r} \abk \Bigl ( \tau(X) \Bigr ) \exp f_2 , $$ and likewise for any nonnegative integers $n_r$ (with $n_2 = 0 $) and any choices of $p_{r, j_r} = 0, 1$ we have $$ \Biggl ( \prod_{r = 2}^n \Bigl ( \frac{\partial}{\partial \delta_r} \Bigr)^{n_r} \prod_{j_r = 1}^{2g} \Bigl ( \frac{\partial}{\partial s_{r}^{ j_r} }\Bigr )^{p_{r,j_r} } \int_{\mnd} \abk \Bigl ( \tau(X) \exp (\sum_{r = 2}^n \sum_{j=1}^{2g} \srj \tbrj ) \exp \tfq \Bigr ) \Biggr)_{\delta_r = 0 , ~s_{r,j} = 0} $$ $$ = \int_{\mnd} \abk \Bigl ( \tau(X) \Bigr ) \exp f_2 \prod_{r = 2}^n f_r^{n_r} \prod_{j_r = 1}^{2g} \Bigl ( b_{r}^{j_r} \Bigr )^{p_{r,j_r}} $$ (where the parameters $\delta_r$ and $\srj$ on the left hand side run over $r = 2, \dots, n$ and $j = 1, \dots, 2g$). We can use Proposition \ref{p9.2} to give an explicit formula for pairings of the form \begin{equation} \label{9.31} \int_{\mnd} \Phi \Bigl ( \prod_{r=2}^n \prod_{k_r = 1}^{2g} (\tilde{b}_r^{k_r}(X) )^{p_{r,k_r} } \tau(X) \Bigr ) e^{f_2} \end{equation} where $p_{r,k_r} = 0 $ or $1$. We note that by Proposition \ref{p9.2} this equals \begin{equation} \label{9.32} \prod_{r = 2}^n \prod_{k_r = 1}^{2g} \Bigl ( \frac{\partial}{\partial s_r^{k_r} } \Bigr )^{p_{r,k_r} } \int_{\mnd} \Phi \Bigl ( \tau(X) \exp \hattau(X) \Bigr ) e^{f_2} \mid_{s_r^j = 0 ~\forall ~ r,j} \end{equation} where \begin{equation} \label{9.32a} \hattau(-X) = - \sum_{a, b = 1}^{n-1} \sum_{j = 1}^g \sum_{r,s = 2}^n s_r^j s_s^{j+g} (d \tau_r)_X (\hu{a} ) (d \tau_s)_X (\hu{b} ) (\partial^2 q)^{-1}_{ab} . \end{equation} Here $\hu{a}$ are an oriented orthonormal basis of $\liet$. We introduce $T_{rs}: \liek \to {\Bbb R }$ given by\footnote{Notice that $T_{rs}$ is an invariant polynomial on $\liek$.} $$ T_{rs} (-X) = - \sum_{a, b = 1}^{n-1} (d \tau_r)_X (\hu{a} ) (d \tau_s)_X (\hu{b} )(\partial^2 q)^{-1}_{ab}. $$ Thus we may rewrite (\ref{9.32a}) as \begin{equation} \label{9.33} \hattau(X) = \sum_{r , s = 2}^n T_{rs}(X) (\sum_{j = 1}^g s_r^j s_s^{j+g} ). \end{equation} We observe that in order for the pairing (\ref{9.31}) to be nonzero, one requires $p_{r,j} = 0 $ or $1$ for all $r$ and $j$ (since the $b_r^j$ are of odd degree). Further, in order for the expression (\ref{9.32}) to yield a nonzero answer, we require for each $j = 1, \dots, g$ that $$p_{2,j} + \dots + p_{n,j} = p_{2,j+g} + \dots + p_{n,j+g} = l_j $$ for some $l_j$. We may then rewrite (\ref{9.32}) as \begin{equation} \label{9.34} \Biggl ( \prod_{j = 1}^g \Bigl ( \spart{r_1}{j}\dots \spart{r_{l_j}}{j} \Bigr ) \Bigl ( \spart{s_1}{j+g}\dots \spart{s_{l_j}}{j+g} \Bigr ) \int_{\mnd} \Phi \Bigl ( \tau(X) \exp \hattau(X) \Bigr ) e^{f_2} \Biggr )_{s_r^j = 0 ~\forall ~ r,j}. \end{equation} Because $\hattau$ is quadratic in the $s_r^j$ and we are setting all the $s_r^j$ to zero in the end, for each $j$ we may represent the symbols $\spart{r}{j} $ and $\spart{r}{j+g} $ as 1-valent vertices (labelled by $r$) in a bipartite graph: there must be exactly one edge coming out of each of these vertices, and these edges must connect the symbol $\spart{r}{j} $ with a symbol $\spart{s}{j+g} $ for some $s$. Such bipartite graphs of course correspond to permutations $\sigma_j$ of $\{ 1, \dots, l_j \}$. It follows from (\ref{9.33}) that $$ \spart{r}{j} \spart{s}{j+g} \hattau(X) = T_{rs} (X) ~\mbox{for any $j$} $$ so that \begin{equation} \label{9.35} \prod_{j = 1}^g \Bigl ( \spart{r_1}{j}\dots \spart{r_{l_j} }{j} \Bigr ) \Bigl ( \spart{s_1}{j+g}\dots \spart{s_{l_j}}{j+g} \Bigr ) \int_{\mnd} \Phi \Bigl ( \tau(X) \exp \hattau(X) \Bigr ) e^{f_2} \end{equation} $$ = \int_{\mnd} \Phi \Biggl ( \prod_{j = 1}^g \sum_{ \sigma_j} T_{r_1 s_{\sigma_j(1)} } (X) \dots T_{r_{l_j} s_{\sigma_j(l_j) } } (X) \tau(X) \Biggr) e^{f_2} $$ where we sum over all permutations $\sigma_j$ of $ \{ 1, \dots, l_j \}$. Hence we obtain by Remark \ref{rem8.3} (a) and Lemma \ref{l3} \begin{theorem} \begin{equation} \label{9.36} \int_{\mnd} \prod_{j = 1}^g b_{r_1}^j \dots b_{r_{l_j} }^j b_{s_1}^{j+g} \dots b_{s_{l_j} }^{j +g} \Phi (\tau(X) ) e^{f_2} \end{equation} $$ = \int_{\mnd} \Phi \Biggl ( \prod_{j = 1}^g \sum_{ \sigma_j} T_{r_1 s_{\sigma_j(1)} }(X) \dots T_{r_{l_j} s_{\sigma_j(l_j) } } (X) \tau(X) \Biggr) e^{f_2} $$ which equals $(-1)^{n_+(g-1) } \frac{n^{g}}{n!}$ times the iterated residue $$ \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \frac{ \prod_{j = 1}^g \sum_{\sigma_j} T_{ {r_1} s_{\sigma_j(1) } } (-X) \dots T_{ {r_{l_j} } s_{\sigma_j(l_j) } } (-X) \tau(-X) e^{ - \inpr{\bracearg{w\tc}, X} } }{\nusym^{2g-2}(X) (1 - \exp (-Y_1) ) \cdots ( 1 - \exp (-Y_{n-1} ) ) } . $$ \end{theorem} Let $\hu{a} $ $(a = 1, \dots, n-1)$ denote an oriented orthonormal basis on $\liet$. For $X \in \liet$ define coordinates $\zz{a}$ by $\zz{a} = (X, \hu{a}) $ so that $X = \sum_a \zz{a} \hu{a}.$ Write the Maurer-Cartan form $ \theta$ on $T$ as $\theta = \sum_a \theta_a \hu{a}; $ then the $\theta_a $ form a set of generators of $H^1(T). $ \begin{definition} \label{zedef} A set of generators $\{\zeta^j_a \} $ ($j = 1, \dots, 2g; $ $a = 1, \dots, n-1$) for $H^1(T^{2g})$ is defined by by specifying that $\zeta^j_a = \pi_j^* \theta_a$ where $\pi_j: T^{2g} \to T$ is the projection onto the $j$'th copy of $T$. \end{definition} \begin{lemma} \label{l9.3b} We have $ \int_T \theta_1 \wedge \dots \wedge \theta_{n-1} = \,{\rm vol}\, (T). $ Here, $ \,{\rm vol}\, (T)$ is the Riemannian volume of $T = \liet/\intlat$ in the metric $\langle \cdot , \cdot \rangle$: in other words it is given by $(\det E)^ {\frac{1}{2} } $ $ = \sqrt{n} $ where $E$ is the $(n-1) \times (n-1) $ matrix (known as the Cartan matrix) given by ${E}_{ab} = \inpr{\he{a}, \he{b}} $ in terms of the basis for the integer lattice $\intlat \subset \liet$ over ${\Bbb Z }$ given by the simple roots $\{\he{a} \}$, $a = 1, \dots, n-1$. \end{lemma} \begin{lemma} \label{l9.7a} The restriction of $\tfq(X)_1$ to $T^{2g}$ is given in terms of the generators $\gen_a^j$ of $H^1(T^{2g})$ by $$ \tfq(X)_1 \mid_{T^{2g} } = {\frac{1}{2} } \sum_{a,b} \partial^2 \qsgn (\hu{a}, \hu{b}) \sum_{j = 1}^g (-\gen_a^j \gen_b^{j+g} + \gen_a^{j+g} \gen_b^j ), $$ where $\{ \hu{a} \}$ are an oriented orthonormal basis of $\liet$. \end{lemma} \Proof We need to understand the restriction of $\tfq$ to $T^{2g}$. As in (\ref{9.3}), we have $$ \tfq(X)_1 = \Bigl ( \sum_{j=1}^g (- {\rm ev}_{\gamma_{j}^1} \times {\rm ev}_{x_{j}} + {\rm ev}_{\gamma_{j+g}^0} \times {\rm ev}_{x_{j+g}} \Bigr )^* \Phi_2^K(q) (X) + $$ $$\Bigl ( \sum_{j=1}^g (- {\rm ev}_{\gamma_{j+g}^1} \times {\rm ev}_{x_{j+g}} + {\rm ev}_{\gamma_{j}^0} \times {\rm ev}_{x_{j}} \Bigr )^* \Phi_2^K(q) (X) \in \Omega^*_K (K^{2g}) $$ where (after restricting to $T \times T \times T$) \begin{equation} \label{9.002} \bar{\Phi}_2^K (q) |_{T\times T \times T}(-X) = \int_{(t_0, t_1, t_2) \in \simp^2} \qsgn ( \sum_{k = 0 }^2 dt_k \thetsimp{k} + X) \in \hht(T\times T \times T) \end{equation} (\cite{J2}, above (5.6)) and $\Phi_2^K(q) = \sigma_2^* \bar{\Phi}_2^K(q) $ where $\sigma_2: (g_1, g_2) \mapsto (g_1 g_2, g_2, 1). $ By (\ref{9.002}) we have \begin{equation} \label{9.003} \Phi_2^K(q)|_{T \times T}(-X) = - {\frac{1}{2} } \sum_{a,b} \partial^2 \qsgn(\hu{a}, \hu{b}) \gen_a^1 \gen_b^2 \in \hht (T \times T).\end{equation} For the purposes of evaluation on $T^{2g}$ the generators $\gamma_j^\tau$ in (\ref{9.3}) reduce to $$\gamma_{j}^0 = \gamma_{j+g}^1 = 1,~~ \gamma_{j}^1 = x_{j+g}, ~~\gamma_{j+g}^0 = x_{j}, $$ where $x_1, \dots, x_{2g}$ are the chosen generators of $\free^{2g}$. So we get from (\ref{9.3}) $$ \tfq(X)_1 \mid_{T^{2g} } = \sum_{j=1}^g \Bigl ( -{\rm ev}_{x_{j+g}} \times {\rm ev}_{x_j} + {\rm ev}_{x_{j}} \times {\rm ev}_{x_{j+g} } \Bigr )^* \Phi_2^K(q) (X). $$ We find that $$ \tfq(-X)_1|_{T^{2g}} = - {\frac{1}{2} } \sum_{a,b} \partial^2 \qsgn (\hu{a}, \hu{b}) \sum_{j = 1}^g (\gen_a^j \gen_b^{j+g} - \gen_a^{j+g} \gen_b^j ) = - \sum_{a,b} \partial^2 \qsgn (\hu{a}, \hu{b}) \sum_{j = 1}^g \gen_a^j \gen_b^{j+g}. \square$$ Similarly for $\Lambda \in \liet \subset \liek$ we have $$ \tfq(-X)_2(\Lambda) = - (d\qsgn)_X (\Lambda) $$ (see Lemma \ref{l9.3a}). As a result we see immediately that \begin{lemma} \label{l9.8} Suppose $c \exp \Lambda = 1.$ Then we have $$ \int_{T^{2g} \times \{ \Lambda \} } \exp \tfq (-X) = \int_{T^{2g} \times \{\Lambda \} } \exp \tf_{q_2}(-X) (\det \hess(X))^g, $$ $$ = e^{-(d \qsgn)_X (\Lambda) } \int_{T^{2g} } (\det \hess(X))^g \exp \omega $$ where $\omega$ is the standard symplectic form on $T^{2g}$ and the quadratic form $\hess(X) $ is the Hessian of the restriction of $-\qsgn$ to $\liet$ (evaluated on an oriented {\em orthonormal basis} of $\liet$). In other words, $$\hess(X)_{ab} = -(\partial^2 \qsgn)_X (\hu{a}, \hu{b}) $$ where $\{ \hu{a}: a = 1, \dots, n-1 \} $ is an oriented orthonormal basis for $\liet$. \end{lemma} \Proof This follows by integrating $$ \exp -\sum_{a,b} \partial^2 \qsgn (\hu{a}, \hu{b}) \sum_{j = 1}^g (\gen_a^j \gen_b^{j+g}) $$ over $T^{2g}$. (Notice that $\partial^2 \qsgn (\hu{a}, \hu{b})$ is symmetric in $a$ and $b$.) \hfill $\square$ \begin{lemma} \label{l9.03} We have that $$ \int_{T^{2g} } \exp \omega = n^g. $$ \end{lemma} \Proof This follows from Lemmas \ref{l9.3b} and \ref{l9.7a}. \hfill $\square$ In order to prove Proposition \ref{p9.1}, note that by Theorem \ref{t9.5}(b), we have that $$\int_{N_n(c)} \Phi ( e^{\tfq} \nusym_n \eta) $$ equals $\frac{(-1)^{n_+(g-1) } }{n!}$ times the iterated residue $$ \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \frac{ \int_{T^{2g} \times \{ -\tildarg{w\tc} \} } \Bigl ( e^{ \tfq (-X)} \eta(-X) \Bigr ) }{\nusym(X)^{2g-2} (1 - e^{ -B( X)_{n-1}} ) \dots (1 - e^{ -B(X)_1 }) }. $$ This applies in particular when $\eta (X) = \tau(-X) $ is a linear combination of monomials $\prod_r \tarnox^{m_r}$ in the $\tarnox$ which does not involve the $\tbrnox$; since $\tarnox(X) =\tau_r(-X),$ it is natural to write $\eta(-X) = \tau(X). $ For $\eta$ of this form, the expression above equals $$ \frac{(-1)^{n_+(g-1) } }{n!} \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \frac{ \int_{T^{2g} \times \{ - \tildarg{w \tc} \} } e^{ \tf_{q_2}(-X) } \Bigl (\det \hess(X) \Bigr )^g \tau(X) } {\nusym(X)^{2g-2} (1 - e^{ -B(X)_{n-1} } ) \dots (1 - e^{ -B(X)_1} ) } $$ by Lemma \ref{l9.8}. We now replace $X$ by $\winv(X)$ (where the transformation $\winv:$ $\liek \to \liek$ was defined above Proposition \ref{p9.1}). This change of variables produces a Jacobian $\Bigl ( \det \hess(\winv(X)) \Bigr )^{-1} $. Thus we obtain that \begin{equation} \label{9.004} \int_{N_n(c)} \abk (\nusym e^{\tfq} \eta) = \frac{(-1)^{n_+(g-1) } }{n!} \sum_{w \in W_{n-1} } \res_{Y_{1} = 0 } \dots \res_{Y_{n-1} = 0 } \Biggl ( \Bigl (\det \hess (\winv(X)) \Bigr )^{g -1} \times \end{equation} $$ \frac{ \int_{T^{2g} \times \{ - \tildarg{w \tc} \} } e^{\tf_{q_2} (-\winv(X)) } \tau (\winv(X) ) }{ \nusym^{2g-2}(\winv(X)) (1 - e^{ - Y_{n-1}} ) \dots (1 - e^{ - Y_1}) } \Biggr ). $$ Now we have \begin{lemma} \label{l9.9} $$ \nusym^2 (\winv(X) ) = \nusym^2(X) (\det (-\partial^2 \qsgn)_\lietp )^{-1} , $$ where $ (\partial^2 \qsgn)_\lietp $ denotes the restriction of the symmetric bilinear form $ (\partial^2 \qsgn)_X$ on $\liek$ to a symmetric bilinear form on $\lietp$, which is then identified with a linear map from $\lietp$ to itself using the fixed invariant inner product. \end{lemma} This lemma will be used in establishing Proposition \ref{p9.1} since the Hessian $H$ appearing in that proposition is the Hessian of the ($K$-invariant) function $-q: \liek \to {\Bbb R }$, which is block diagonal with one block being the Hessian $H_{\liet}$ of the restriction of this function to $\liet$ and the other block being $ -(\partial^2 \qsgn)_\lietp $. \Proof We introduce the (orthonormal) basis $X_\gamma, Y_\gamma$ for $\lietp$ corresponding to the positive roots $\gamma$, and a corresponding system of coordinates $x_\gamma, y_\gamma$ on $\lietp$: we have $$ [X_\gamma, X] = \gamma(X) Y_\gamma, ~~~[Y_\gamma, X] = -\gamma(X) X_\gamma.$$ We observe that the map $B$ and its inverse $\winv$ on $\liek$ are $K$-equivariant, and map $\liet$ to $\liet$ and $\lietp$ to $\lietp$. Hence \begin{equation} (d \winv)_X ([X_\gamma, X] ) = [X_\gamma, \winv(X) ] \end{equation} and $$\winv \Biggl ( {\rm Ad} \exp (X_\gamma) ( X ) \Biggr ) = {\rm Ad} \exp (X_\gamma) (\winv (X) ), $$ and similarly for $Y_\gamma$. We find \begin{equation} \frac{\gamma(\winv ( X))}{\gamma(X)} = (d\bwinv_{y_\gamma})_X(Y_\gamma) = (d\bwinv_{x_\gamma})_X(X_\gamma) \end{equation} where $\bwinv_{x_\gamma}, \bwinv_{y_\gamma}:$ $\lietp \to {\Bbb R }$ are the coordinate functions in the directions $ x_\gamma$ and $y_\gamma$. Thus, \begin{equation} \label{6.10} {\mbox{$\cal D$}}^{2}(\winv(X)) = {\mbox{$\cal D$}}^{2}(X) (\det d \bwinv_\perp) \end{equation} $$ \phantom{a} \phantom{a} = {\mbox{$\cal D$}}^{2}(X) (\det -\partial^2 \qsgn)_{\lietp}^{-1} $$ where $d \bwinv_\perp$ is the square matrix of partial derivatives of the $\lietp$ components of $\winv$ in the directions along $\lietp$. This completes the proof of Lemma \ref{l9.9}. \hfill $\square$ Proposition \ref{p9.1} now follows immediately by using (\ref{9.004}) and Lemma \ref{l9.9} to express (\ref{10.1a}) as an iterated residue, and observing that Theorem \ref{mainab} (in the version given by Remark \ref{rem8.3} (a)) yields the same iterated residue for (\ref{9.1}). Let us now consider the proof of Proposition \ref{p9.2}. For the rest of this section let $a = 1, \dots, n-1$ index an oriented {\em orthonormal} basis $\{\hu{a}\}$ of $\liet$. We have $\tbrj = \proj_1^* \tilde{b}_r^{j,1} $ where $ \tilde{b}_r^{j,1} = \ev{x_j}^* \Phi_1^K (\tau_r) $ and $\Phi_1^K (\tau_r) = \sigma_1^* \bar{\Phi_1^K }(\tau_r) $ where $\bar{\Phi_1^K }(\tau_r)$ was defined by (\ref{9.005}). Also, $x_j$ (for $j=1, \dots, 2g$) are our chosen set of generators of $H_1 (\Sigma)$. Theorem \ref{mainab} applies when $\eta(X) = \tau(X) \exp \sum_{j=1}^{2g} \sum_{r \ge 2} s_r^j \tbrj $ for $s_r^j \in {\Bbb C }$ and $\tau \in S(\lieks)^K$. Define $S^j \in S(\lieks)^K$ by $S^j (X) = \sum_{r \ge 2} s_r^j \tau_r(X)$; we then define $\tilde{b}_{S^j}^j$ by $\tilde{b}_{S^j}^j (X) = \sum_{r \ge 2} s_r^j \tbrj$. \begin{lemma} \label{l9.9'} (a) The restriction to $T^{2g}$ of $\tbrnox(-X) $ is $ \sum_{a = 1}^{n-1} (d \tau_r)_X (\hu{a} ) \gen_a^j $ where $\gen_a^j$ (for $a = 1, \dots, {n-1}$ and $j = 1, \dots, 2g$) are the elements of the basis of $H^1 (T^{2g}) $ corresponding to an oriented orthonormal basis $\{ \hu{a} \} $ for $ \liet$. (b) The restriction to $T^{2g}$ of $\tilde{b}_{S^j}^j(-X) $ is $ \sum_{a = 1}^{n-1} (d \Ssign{j})_X (\hu{a} ) \gen_a^j $. \end{lemma} \Proof We have by (8.21) of \cite{J2} that $$\tbrj_1 = \ev{x_j}^* \Phi_1^K (\tau_r). $$ By (\ref{9.7}) we have that $$ \Phi_1^K (\tau_r)|_{T^{2g}}(-X) = \sigma_1^* \bar{\Phi}_1^K (\tau_r)|_{T^{2g}}(-X) = \sum_{a = 1}^{n-1} (d\tau_r)_X (\hu{a}) \theta_a $$ so $$ \tbrnox(-X)_1 \mid_{T^{2g}} = \ev{x_j}^* \sigma_1^* \bar{\Phi}_1^K (\tau_r)|_{T^{2g}}(-X) = \sum_{a = 1}^{n-1} (d\tau_r)_X (\hu{a}) \gen_a^j $$ since the generators $\gen_a^j$ of $H^1(T^{2g}) $ become identified with the components $\theta_a$ of the Maurer-Cartan form on the $j$-th copy of $T$ in $T^{2g}$.\hfill $\square$ \begin{lemma} \label{l9.10} In the notation introduced just before Lemma \ref{l9.9'}, we have \begin{equation} \label{eq10.13} \int_{T^{2g} } \exp \tfq (-X) \exp \sum_{j,r} s_r^j \tbrnox(-X) = \int_{T^{2g} } \exp \tfq(-X) \exp \hattau(X) \end{equation} where $$\hattau(X) = - \sum_{a,b=1}^{n-1} \sum_{j=1}^{g} (d\Ssign{j})_X(\hu{a}) (d\Ssign{j+g})_X(\hu{b}) (\partial^2 \qsgn)^{-1}_{ab} $$ \begin{equation} \label{hattaudef} = - \sum_{a,b=1}^{n-1} \sum_{r,s = 2}^{n} \sum_{j=1}^{g} s^j_r s^{j+g}_s (d\tau_r)_X(\hu{a}) (d\tau_s)_X(\hu{b}) (\partial^2 \qsgn )^{-1}_{ab}. \end{equation} Here, $\{ \hu{a}: a = 1, \dots, n-1 \} $ denotes an oriented orthonormal basis of $\liet$. \end{lemma} \Proof We need to consider the left hand side of (\ref{eq10.13}), which is \begin{equation} \label{9.11} \int_{T^{2g} } \exp \tfq(-X) \exp \sum_{j,r} s_r^j \tbrnox(-X) . \end{equation} By Lemma \ref{l9.7a} the restriction of $\tfq(-X)$ to $T^{2g}$ is $\exp - {\frac{1}{2} } \sum_{a,b} \sum_{j=1}^g(\partial^2 \qsgn )_X(\hu{a}, \hu{b}) (\gen_a^j \gen_b^{j+g} - \gen_a^{j+g} \gen_b^{j} ) $ while $ \exp \sum_{j,r} s_r^j \tbrnox(-X) $ restricts on $T^{2g}$ (by Lemma \ref{l9.9'}) to $$\exp \sum_{r,j} s_r^j \sum_{a = 1}^{n-1} (d \tau_r)_X(\hu{a}) \zeta_a^j. $$ Thus for any given $j= 1, \dots, g$ we must compute the integral \begin{equation} \label{9.14} \int_{T^2} \exp \Bigl ( \sum_{\sigma, \tau} y_\sigma A^{\sigma \tau} y_\tau /2 + \sum_\sigma y_\sigma B_j^\sigma \Bigr ) \end{equation} where $\sigma$ runs over pairs $(a,i)$ for $a = 1, \dots, n-1$ and $i = 0, 1$ (where $i = 0$ corresponds to $j$ and $i = 1$ to $j+g$) and $y_{a,i} = \zeta_a^{j+gi} $. Here, the matrix $A$ is given by \begin{equation} A^{a 0, b1} = -\pqabx = - A^{a 1, b0}; \; \; A^{a 0, b0} = A^{a 1, b1} = 0; \end{equation} thus the Pfaffian of $A$ (whose square is $\det A$) is given by $${\rm Pf} (A) = \det \Bigl (- \partial^2 q|_\liet\Bigr). $$ For $j = 1, \dots, g$ the vector $B_j$ is \begin{equation} \label{9.16} B_j^{a0} = -(\partial \Ssign{j})_X (\hu{a})\, ; \phantom{a} \phantom{a} \phantom{a} B_j^{a1} = -(\partial \Ssign{j+g})_X (\hu{a}). \end{equation} The result is that \begin{equation} \label{9.17} \int_{T^2} \exp \Bigl ( \sum_{\sigma, \tau} y_\sigma A^{\sigma \tau} y_\tau /2 + \sum_\sigma y_\sigma B_j^\sigma \Bigr ) = \int_{T^2} \exp \Bigl ( \sum_{\sigma, \tau} y_\sigma A^{\sigma \tau} y_\tau /2 \Bigr ) \exp ( - B_j^t A^{-1} B_j )/2 \end{equation} $$ = {\rm Pf} (A) \exp ( - B_j^t A^{-1} B_j )/2 , $$ where $B_j^t$ denotes the transpose of the vector $B_j$. Thus we find that (\ref{9.11}) becomes \begin{equation} \label{9.12} \det (-\partial^2 \qsgn |_\liet )^g \exp \Biggl ( \sum_{a,b = 1}^{n-1} \sum_{j = 1}^g (d\Ssign{j} )_X (\hu{a}) (d\Ssign{j+g} )_X (\hu{b}) ( \partial^2 \qsgn )^{-1}_{a,b} \Biggr ), \end{equation} which equals the right hand side of (\ref{eq10.13}). This completes the proof of Lemma \ref{l9.10}.\hfill$\square$ Proposition \ref{p9.2} follows from Theorem \ref{t9.5} once we have shown that $$ \int_{T^{2g}} \exp \tfq(-X) \exp \sum_{r \ge 2} s_r^j \tbrnox(-X) = \int_{T^{2g} } \exp \tfq(-X) \exp \hattau(X)$$ where $\hattau$ is given by (\ref{9.32a}). This is now clear from Lemma \ref{l9.10}. \hfill $\square$ { \setcounter{equation}{0} } \section{The Verlinde formula} \newcommand{\tilc}[1]{ { \tilde{c}_{#1} } } \newcommand{\wtilc}[1]{ { (w\tilde{c})_{#1} } } \newcommand{\bracewtilc}[1]{ { \tildarg{w\tilde{c}}_{#1} } } \newcommand{\dgk}{{ D_{n,d}(g,k) }} \newcommand{\vgk}{{ V_{n,d}(g,k) }} \newcommand{\lineb}{L} \newcommand{\resid}[1]{\res_{#1 = 0 } } \newcommand{\residone}[1]{\res_{#1 = 1 } } \newcommand{\kmod}{r} \newcommand{\gmax}{{\gamma_{\rm max} }} \newcommand{\constq}{ {\frac{1}{n} }} \newcommand{\sol}{ { S_{0 \lambda} }} \newcommand{\ch}{ { \rm ch} } \newcommand{\td}{ { \rm td} } \newcommand{\tg}{{\tilde{\gamma} } } \renewcommand{\ell}{{{\mbox{$\cal L$}}}} The Verlinde formula is a formula for the dimension $\dgk$ of the space of holomorphic sections of powers of $\ell$ , where $\ell$ is a particular line bundle over $\mnd$: it has been proved by Beauville and Laszlo \cite{BL}, Faltings \cite{F}, Kumar, Narasimhan and Ramanathan \cite{KNR} and Tsuchiya, Ueno and Yamada \cite{TUY}. In this section we show how the Verlinde formula follows from our formula (Theorem \ref{mainab}) for intersection pairings in $\mnd$. A line bundle $\ell$ over $\mnd$ may be defined for which $c_1 (\ell) = n f_2$, since $n f_2 \in H^2(\mnd, {\Bbb Z })$ (see \cite{DN}). As described in Section 1, this bundle is the {\em determinant line bundle}. Whenever $k$ is a positive integer divisible by $n$, we then define \begin{equation} \label{10.1} \dgk = \dim H^0(\mnd, \ell^{k/n} ). \end{equation} Let us introduce $$\kmod = k + n;$$ let us also introduce the highest root $\gmax$, which is given by $\gmax(X) = X_n - X_1$ or $\gmax = e_1 + e_2 + \dots + e_{n-1}$. We then make the following definition: \begin{definition} \label{d10.1} The {\em Verlinde function} $\vgk$ is given by $$\vgk = \sum_{\lambda \in \weightl_{\rm reg} \cap \liet_+: \inpr{\lambda, \gmax} < \kmod} \frac{e^{-\itwopi \inpr{ \lambda - \rho ,\tc} } } {(\sol(k) )^{2g-2} } $$ where $\rho$ is half the sum of the positive roots and $$ \sol(k) = \frac{1 }{\sqrt{n} \kmod^{(n-1)/2} } \prod_{\gamma > 0 } 2 \sin \pi \inpr{\gamma,\lambda} /\kmod . $$ \end{definition} (See \cite{GW} (A.44) and \cite{qym} (3.16).) Verlinde's conjecture says that the Verlinde function specifies the dimension of the space of holomorphic sections of $\ell^{k/n}$: \begin{theorem} \label{verl} {\bf (Verlinde's conjecture)} $$ \dgk = \vgk.$$ \end{theorem} We shall show how to extract Verlinde's conjecture from our previous results: an outline of the method we use was given by Szenes \cite{Sz} (Section 4.2). In fact $H^i (\mnd, \ell^m ) = 0 $ for all $i > 0$ and $m> 0$ by an argument using the Kodaira vanishing theorem and the facts that ${\mbox{$\cal L$}}$ is a positive line bundle and the canonical bundle of $\mnd$ is equal to $\ell^{-2} $ (see \cite{Beauv1} Section 5 and Th\'eor\`eme F of \cite{DN}), so $\dgk$ is given for $k>0 $ by the Riemann-Roch formula: \begin{equation} \label{10.2} \dgk = \int_{\mnd }\ch \ell^{k/n} \td \mnd. \end{equation} We use the following results to convert (\ref{10.2}) into a form to which we may apply our previous results. \begin{lemma} \label{p10.1} For any complex manifold $M$ the Todd class of $M$ is given by $$\td(M) = e^{c_1(M)/2} \hat{A} (M) $$ where $c_1(M)$ is the first Chern class of the holomorphic tangent bundle of $M$, and $\hat{A}(M)$ is the $A$-roof genus of $M$. \end{lemma} \Proof See for example \cite{Gilkey}, pages 97-99. \hfill $\square$ \begin{prop} \label{p10.2} We have $$\hat{A} (\mnd) = \abk \Bigl ( \prod_{\gamma > 0} \frac{ \gamma(X)/2 }{\sinh \gamma(X)/2 } \Bigr )^{2g-2}. $$ \end{prop} \Proof This is proved by Newstead\footnote{Newstead writes the details of the proof only for $n=2$ but the same proof yields the result for general $n$.} in \cite{Newstead}. \begin{lemma} \label{p10.3} We have $$c_1(\mnd) = 2 n f_2. $$ \end{lemma} \Proof This is proved in \cite{DN}, Th\'eor\`eme F .\hfill $\square$ Of course the Chern character of $\ell^{k/n}$ is given by $\ch \ell^{k/n} = e^{k f_2}. $ Thus we obtain \begin{corollary} \label{p10.4} The quantity $\dgk$ is given by $$ \dgk = \int_{\mnd} e^{(k + n)f_2} \abk \Bigl ( \prod_{\gamma > 0} \frac{ \gamma(X) }{e^{\gamma(X)/2} - e^{-\gamma(X)/2} } \Bigr )^{2g-2}. $$ \end{corollary} \Proof This follows immediately from (\ref{10.2}), Lemmas \ref{p10.1} and \ref{p10.3} and Proposition \ref{p10.2}.\hfill $\square$ \begin{theorem} \label{t10.5} We have $$\dgk = \frac{(-1)^{n_+(g-1)}}{n!}\sum_{w \in W_{n-1} } \resid{Y_{1}} \dots \resid{Y_{n-1} } \Biggl ( e^{\kmod \inpr{\tildarg{w\tc},X} } \int_{T^{2g}} e^{\kmod \omega} \times $$ \begin{equation} \prod_{\gamma > 0} \Bigl (\frac{ \gamma(X)}{e^{\gamma(X)/2} - e^{-\gamma(X)/2} }\Bigr )^{2g-2} \frac{1 } { \prod_{j = 1}^l (e^{ \kmod Y_j} - 1) \nusym(X)^{2g-2} } \Biggr ).\end{equation} \end{theorem} \Proof This comes straight from Corollary \ref{p10.4} and Theorem \ref{mainab}. Note that because the factor $e^{f_2}$ in the statement of Theorem \ref{mainab} has been replaced by $e^{\kmod f_2} $, it is necessary to replace $ e^{\inpr{\tildarg{w\tc},X} } $ by $e^{\kmod \inpr{\tildarg{w\tc},X} } $, and $e^{Y_j} - 1$ by $e^{\kmod Y_j} - 1$ (cf. Remark \ref{rem8.3} (c) ). \hfill $\square$ We introduce $Z_j = \exp Y_j$. Since for any $w \in W_{n-1}$ we have that $$\tildarg{w\tc} = \bracewtilc{1} \he{1} + \bracewtilc{2} \he{2} + \dots + \bracewtilc{n-1} \he{n-1} $$ (as in the statement of Proposition \ref{p:sz}) with $n \bracewtilc{j} \in {\Bbb Z } $ for all $j$, and $0 \le \bracewtilc{j} < 1$ for all $j$, we obtain $$ e^{ \kmod \inpr{\bracewtilc,X} } = Z_1^{ \bracewtilc{1} \kmod } Z_2^{ \bracewtilc{2} \kmod } \dots Z_{n-1}^{ \bracewtilc{n-1} \kmod}. $$ (Recall that $k$ and $r$ are divisible by $n$ so $e^{ \kmod \inpr{\tc,X} } $ is a well defined single valued function of $Z_1$, $\dots, Z_{n-1}$.) Thus we can equate $\dgk$ with $$ \frac{(-1)^{n_+(g-1) } }{n!} \sum_{w \in W_{n-1} } \residone{Z_{1}} \dots \residone{Z_{n-1} } \Biggl ( \Bigl ( \prod_{j = 1}^{n-1} \frac{1}{Z_j} \Bigr ) \int_{T^{2g}} e^{\kmod \omega} \times $$ $$ \frac{ Z_1^{ \bracewtilc{1} \kmod } Z_2^{ \bracewtilc{2} \kmod} \dots Z_{n-1}^{ \bracewtilc{n-1}\kmod} }{\prod_{\gamma > 0 } (\tg^{1/2}- \tg^{-1/2})^{2g-2} (Z_1^{ \kmod} - 1) \dots (Z_{n-1}^{ \kmod} - 1 ) } \Biggr ) $$ \begin{equation} \label{10.5} = \frac{(-1)^{n-1 + n_+(g-1) } }{n!} \sum_{w \in W_{n-1} } \residone{Z_{1}} \dots \residone{Z_{n-1} } \Biggl ( \Bigl ( \prod_{j = 1}^{n-1} \frac{1}{Z_j} \Bigr ) \times \end{equation} $$ \int_{T^{2g}} e^{\kmod \omega} \frac{ Z_1^{ - \bracewtilc{1} \kmod} Z_2^{ - \bracewtilc{2}\kmod} \dots Z_{n-1}^{ -\bracewtilc{n-1} \kmod} }{\prod_{\gamma > 0 } (\tg^{1/2}- \tg^{-1/2})^{2g-2} (Z_1^{ - \kmod} - 1) \dots (Z_{n-1}^{ - \kmod} - 1 ) } \Biggr ). $$ Here, we have introduced $\tg $ defined (for the root $\gamma = e_r + e_{r+ 1} + \dots + e_{s-1}$) by $$\tg(Z_1, \dots, Z_{n-1}) = Z_r \dots Z_{s-1}.$$ We also have \begin{lemma} \label{p10.6} $$\int_{T^{2}} e^{\omega} = n $$ and hence $$\int_{T^{2g}} e^{r \omega} = r^{(n-1)g} n^g. $$ \end{lemma} \Proof This follows from Lemma \ref{l9.03}.\hfill $\square$ The following may be proved by the same method as in Section 2 (see \cite{Sz}): \begin{prop} \label{p:vsz} Suppose $f$ is the meromorphic function on the complexification $T^{{\Bbb C }}$ of $T$ defined by \begin{equation} \label{11.6} f(Z) = (-1)^{n-1} (-1)^{n_+(g-1)} \kmod^{(n-1)(g-1)} n^{g -1} \frac{Z_1^{-\tilc{1}\kmod} \dots Z_{n-1}^{-\tilc{n-1}\kmod} } { \prod_{\gamma> 0 } (\tg^{1/2} - \tg^{-1/2} )^{2g-2} }. \end{equation} Then we have that \begin{equation}\frac{1}{(n-1)!} \residone{Z_{1}} \dots \residone{Z_{n-1} } \sum_{w \in W_{n-1} } \prod_{j = 1}^{n-1} \Bigl ( \frac{\kmod }{Z_j} \Bigr ) \frac{\bracearg{ w f} (Z)} {\prod_{j = 1}^{n-1} (Z_j^{ - \kmod} - 1 ) } = \sum_{\lambda \in \weightl_{\rm reg} \cap \lietpl : \inpr{\lambda,\gmax} < r} f(\exp \itwopi \lambda/\kmod). \end{equation} Here, $W_{n-1} $ is the permutation group on $\{ 1, \dots, n-1 \} $ which is (isomorphic to) the Weyl group of $SU(n-1)$, and $\bracearg{w f} $ is the function \begin{equation} \label{11.6'} \bracearg{wf}(Z) = (-1)^{n-1} (-1)^{n_+(g-1)} \kmod^{(n-1)(g-1)} n^{g -1} \frac{Z_1^{-\bracewtilc{1}\kmod} \dots Z_{n-1}^{-\bracewtilc{n-1}\kmod} } { \prod_{\gamma> 0 } (\tg^{1/2} - \tg^{-1/2} )^{2g-2} }. \end{equation} \end{prop} \noindent{\em Remark:} Notice that we have $$ \sum_{ \lambda \in \weightl_{\rm reg} \cap \lietpl : \inpr{\lambda,\gmax} < r } f(\exp \itwopi \lambda/r) = \frac{1}{n-1} \sum_{m_j = 1}^{r-1} f\Bigl (e^{ \itwopi ( \sum_j m_j w_j )/r } \Bigr ) . $$ (Here, the $w_j$ are the fundamental weights, which are dual to the simple roots.) The set $\{ X \in \liet: \; $ $ X = \sum_j \lambda_j \he{j}, 0 \le \lambda_j < 1, \; j = 1, \dots, n-1 \} $ is a fundamental domain for the action of the integer lattice $\Lambda^I$ on $\liet$, while the set $\{ X \in \lietpl \subset \liet: \gamma_{\rm max} (X) < 1 \} $ is a fundamental domain for the {\em affine Weyl group} $W_{\rm aff}$ (the semidirect product of the Weyl group and the integer lattice), and $\intlat$ has index $(n-1)! $ (rather than $n!$) in $W_{\rm aff} $ (in other words a fundamental domain for $\intlat$ contains $(n-1)!$ fundamental domains for $W_{\rm aff}$). This difference accounts for the factor $1/(n-1)!$ in Proposition \ref{p:vsz} which replaces the factor $1/n!$ in its analogue Proposition \ref{p:sz}. Applying Proposition \ref{p:vsz} we find (recalling from Section 2 that $(-1)^{n-1} = c^\rho$ when $n$ and $d$ are coprime) that \begin{equation} \label{10.10} \dgk = (-1)^{n_+(g-1)} \kmod^{(n-1)(g-1)} n^{g -1} c^\rho \sum_{\lambda \in \weightl_{\rm reg} \cap \lietpl: \inpr{\lambda, \gmax} < \kmod } \frac{ e^{ - \itwopi \inpr{\tc, \lambda} } } { \prod_{\gamma> 0 } (e^{\itwopi \inpr{\frac{\gamma}{2\kmod}, \lambda} } - e^{-\itwopi \inpr{\frac{\gamma}{2\kmod}, \lambda} } )^{2g-2}}. \end{equation} This gives \begin{equation} \label{10.11} \dgk = (-1)^{n_+(g-1)} r^{(n-1)(g-1)} n^{g-1} \sum_{\lambda \in \weightl_{\rm reg} \cap \lietpl: \inpr{\lambda, \gmax} < \kmod } \frac{ e^{ - \itwopi \inpr{\tc, \lambda - \rho } } } {\prod_{\gamma > 0 } (2 i \sin \pi \inpr{\gamma,\lambda}/\kmod )^{2g-2} } \end{equation} \begin{equation} \label{10.12} ~~~= \kmod^{(n-1)(g-1)} n^{g-1} \sum_{\lambda \in \weightl_{\rm reg} \cap \liet_+: \inpr{\lambda, \gmax} < \kmod } \frac{ e^{ - \itwopi \inpr{\tc, \lambda - \rho } } } {\prod_{\gamma > 0 }\Bigl (2 \sin \frac{\pi \inpr{\gamma,\lambda} }{\kmod }\Bigr )^{2g-2} }. \end{equation} Comparing with Definition \ref{d10.1}, we see that $\dgk = \vgk$. This completes the proof of Theorem \ref{verl}. \hfill $\square$

9608

alg-geom/9608003

en

https://arxiv.org/abs/alg-geom/9608003

alg-geom/9608003

Yuichiro Takeda

A relation between standard conjectures and their arithmetic analogues

AMS-LaTeX file, 10 pages

In this paper we resolve the arithmetic analogues of standard conjectures for an arithmetic variety, which are proposed by Gillet and Soule, into original standard conjectures and similar conjectures for two cycle class groups. One is the homologically trivial cycles. This is well-known as conjectures about the height pairing raised by Beilinson. The other is the cycle class groups consisting of vertical cycles. We show that the conjectures for the above two cycle class groups with Grothendieck's standard conjectures imply their arithmetic analogues.

alg-geom

\section{Introduction} \vskip 1pc Let $X$ be a smooth projective variety of dimension $n$ defined over an algebraically closed field $k$. We denote by $A^p(X)$ the vector space over $\Bbb R$ consisting of algebraic cycles of codimension $p$ modulo homological equivalence. In \cite{grothendieck} Grothendieck conjectured that $A^p(X)$ behaves like complex cohomology. \begin{conj}[Standard conjectures] \ \ \ \ Let $H$ be an ample line bundle on $X$ and $$ L_H:A^p(X)\to A^{p+1}(X) $$ the homomorphism intersecting with the first Chern class $c_1(H)$. Then for $p\leq \frac{n}{2}$, we have the followings: \noindent $\bold A_p(X, H)$: \ $L_H^{n-2p}:A^p(X)\to A^{n-p}(X)$ is an isomorphism. \noindent $\bold H_p(X, H)$: \ For $0\not= x\in A^p(X)$ such that $L_H^{n+1-2p}(x)=0$, $(-1)^p\operatorname{deg} (L_H^{n-2p}(x)x)$ is positive. \end{conj} \vskip 1pc $\bold A_p(X, H)$ is called the hard Lefschetz conjecture and $\bold H_p(X, H)$ is called the Hodge index conjecture. When the characteristic of $k$ is zero, the Hodge index conjecture is already proved. On the other hand, for an arithmetic variety the intersection theory of cycles was established by Arakelov \cite{arakelov} for surfaces and Gillet and Soul\'{e} \cite{GS1} for higher dimensional varieties. It is quite natural to ask whether analogues of standard conjectures hold in this situation. We now explain this. Let $X$ be a regular scheme which is projective and flat over $\Bbb Z$. We assume that the generic fiber $X_{\Bbb Q}$ is smooth over $\Bbb Q$. Such a scheme is called an arithmetic variety. For an arithmetic variety $X$ the arithmetic Chow group $\widehat{CH} ^p(X)$ is defined and the intersection product on $\widehat{CH} ^p(X)_{\Bbb Q}$ is established in \cite{GS1}. We denote by $F_{\infty } $ the complex conjugation on the complex manifold $X(\Bbb C)$ associated with the scheme $X\underset{\Bbb Z}{\otimes }\Bbb C$. A line bundle $H$ on $X$ togather with an $F_{\infty } $-invariant smooth hermitian metric $\Vert \ \Vert $ on the pull back $H_{\Bbb C}$ on $X(\Bbb C)$ is called a hermitian line bundle. For a hermitian line bundle $(H, \Vert \ \Vert )$ on $X$, the arithmetic first Chern class $\hat{c}_1(H, \Mtrc ) \in \widehat{CH} ^1(X)$ is defined. By intersecting with this class we obtain a homomorphism $$ L_{H,\Mtrc } :\widehat{CH} ^p(X)_{\Bbb R}\to \widehat{CH} ^{p+1}(X)_{\Bbb R}. $$ In \cite{GS} Gillet and Soul\'{e} proposed the following conjectures: \begin{conj}[Arithmetic analogues of standard conjectures] \ \ \ \ Let $n$ be the relative dimension of $X$ over $\Bbb Z$ and $H$ an ample line bundle on $X$. Then there exists an $F_{\infty } $-invariant hermitian metric $\Vert \ \Vert $ on $H_{\Bbb C}$ satisfying the followings for $2p\leq n+1$: \noindent $\bold A_p(X, H, \Vert \ \Vert )$: \ $L_{H,\Mtrc } ^{n+1-2p}:\widehat{CH} ^p(X)_{\Bbb R}\to \widehat{CH} ^{n+1-p}(X)_{\Bbb R}$ is an isomorphism. \noindent $\bold H_p(X, H, \Vert \ \Vert )$: \ For $0\not= x\in \widehat{CH} ^p(X)_{\Bbb R}$ such that $L_{H,\Mtrc } ^{n+2-2p}(x)=0$, $(-1)^p\widehat{\DEG} (L_{H,\Mtrc } ^{n+1-2p}(x)x)$ is positive. \end{conj} \vskip 1pc When $n=1$, Conjecture 1.2 is derived from the Hodge index theorem for arithmetic surfaces by Faltings \cite{faltings} and Hriljac \cite{hriljac}. This was extended to higher dimensional varieties by Moriwaki \cite{moriwaki}. He proved $\bold H_1(X, H, \Vert \ \Vert )$ for any arithmetically ample hermitian line bundle $(H, \Vert \ \Vert )$ on an arithmetic variety $X$. In \cite{kunnemann} K\"{u}nnemann showed that the conjectures are deduced from the similar conjectures for Arakelov Chow groups and proved them for projective spaces. The aim of this paper is to resolve Conjecture 1.2 into other well-known conjectures including original standard conjectures. In \cite{faltings, hriljac} for an arithmetic surface $X$ $\bold H_1(X, H, \Vert \ \Vert )$ is proved by the positivity of N\'{e}ron-Tate height pairing of the Jacobian of $X$ and by some arguments on the intersection of cycles of $X$ whose supports do not meet the generic fiber $X_{\Bbb Q}$. In other words, our main results are generalisations of their methods to higher dimensional varieties. As a consequence, all results as mentioned above are obtained, independent of the notion of the arithmetic ampleness. \vskip 2pc \section{Statements of the main results} \vskip 1pc We first recall some basic facts of Arakelov intersection theory. Throughout the paper, $n$ is the relative dimension of an arithmetic variety $X$ over $\Bbb Z$. For an arithmetic variety $X$, we put \begin{gather*} Z^{p,p}(X)=\{\omega ; \text{real closed $(p,p)$-form on $X(\Bbb C)$ with $F_{\infty } ^*\omega =(-1)^p\omega $}\}, \\ H^{p,p}(X)=\{c\in H^{p,p}(X(\Bbb C)); \text{$c$ is real with $F_{\infty } ^*c=(-1)^pc$}\}. \end{gather*} Then we have an exact sequence \begin{align*} CH^{p-1,p}(X)_{\Bbb R}\overset{\rho }{\to }&H^{p-1,p-1}(X)\overset{a} {\to }\widehat{CH} ^p(X)_{\Bbb R}\overset{(\zeta ,\omega )}{\longrightarrow } \\ &CH^p(X)_{\Bbb R}\oplus Z^{p,p}(X)\overset{cl-h}{\to }H^{p,p}(X)\to 0, \end{align*} where the definitions of $CH^{p,p-1}, a, \zeta $ and $\omega $ are seen in \cite[3.3]{GS1}. The map $cl:CH^p(X)_{\Bbb R}\to H^{p,p}(X)$ is the cycle class map and $h$ is the canonical projection map. The map $\rho :CH^{p-1,p}(X)_{\Bbb R}\to H^{p-1,p-1}(X)$ is the regulator map up to constant factor by \cite[Theorem 3.5]{GS1}. We fix a smooth $F_{\infty } $-invariant K\"{a}hler metric $h$ on $X(\Bbb C)$. The pair $\overline{X}=(X, h)$ is called an Arakelov variety. By identifying an element of $H^{p,p}(X)$ with a harmonic $(p,p)$-form with respect to $h$, we can regard $H^{p,p}(X)$ as a subspace of $Z^{p,p}(X)$. We put $CH^p(\overline{X})=\omega ^{-1}(H^{p,p}(X))$ and call it Arakelov Chow group. \begin{assa} \ \ \ \ For $p\geq 0$, the vector space $H^{p,p}(X)$ is spanned by images of $\rho $ and $cl$, that is, $$ H^{p,p}(X)=\operatorname{Im} \rho \oplus \operatorname{Im} cl. $$ \end{assa} \vskip 1pc This is a part of Beilinson conjectures. If Assumption 1 holds for $X$ and for $p-1$, by the definition of the Arakelov Chow group we obtain the following exact sequence: $$ CH^{p-1}(X)_{\Bbb R}\overset{a\cdot cl}{\to }CH^p(\overline{X})_{\Bbb R} \overset{\zeta }{\to }CH^p(X)_{\Bbb R}\to 0. $$ Let $(H, \Vert \ \Vert )$ be a hermitian line bundle on $X$. Suppose that $H$ is ample and that $\Vert \ \Vert $ is a positive metric. Then the first Chern form of $(H, \Vert \ \Vert )$ determines an $F_{\infty } $-invariant K\"{a}hler metric $h$ on $X(\Bbb C)$. Since the product with the K\"{a}hler form respects harmonicity of forms, for the Arakelov variety $\overline{X}=(X, h)$ we can define a homomorphism $$ L_{H,\Mtrc } :CH^p(\overline{X})_{\Bbb R}\to CH^{p+1}(\overline{X})_{\Bbb R}. $$ \begin{conj} \ \ \ \ For an ample line bundle $H$ on $X$, there exists a positive $F_{\infty } $- invariant hermitian metric $\Vert \ \Vert $ on $H_{\Bbb C}$ satisfying the followings for $2p\leq n+1$: \noindent $\bold A\bold A_p(X, H, \Vert \ \Vert )$: \ $L_{H,\Mtrc } ^{n+1-2p}: CH^p(\overline{X}) _{\Bbb R}\to CH^{n+1-p}(\overline{X})_{\Bbb R}$ is an isomorphism. \noindent $\bold A\bold H_p(X, H, \Vert \ \Vert )$: \ For $0\not= x\in CH^p(\overline{X}) _{\Bbb R}$ such that $L_{H,\Mtrc } ^{n+2-2p}(x)=0$, $(-1)^p\widehat{\DEG} (L_{H,\Mtrc } ^{n+1-2p}(x)x)$ is positive. \end{conj} \vskip 1pc \begin{thm} \ \ \ \ $\bold A\bold A_p(X, H, \Vert \ \Vert )$ implies $\bold A_p(X, H, \Vert \ \Vert )$. $\bold A\bold H_p(X, H, \Vert \ \Vert )$ implies $\bold H_p(X, H, \Vert \ \Vert )$. \end{thm} \vskip 1pc Theorem 2.1 was proved by K\"{u}nnemann in \cite{kunnemann}. He also proved Conjecture 2.1 for projective spaces. Recently the author proved Conjecture 2.1 for regular quadric hypersurfaces in \cite{takeda}. From now on, every arithmetic variety $X$ is assumed to be irreducible. Then $X$ is defined over the ring of integers $\cal O_K$ of an algebraic number field $K$ and the generic fiber $X_K$ is geometrically irreducible. The cycle class map $cl:CH^p(X)_{\Bbb R}\to H^{p,p}(X)$ factors through the Chow group of the generic fiber and the restriction map $CH^p(X)\to CH^p(X_K)$ is surjective. Hence the image of $cl$ coincides with the image of $CH^p(X_K)_{\Bbb R}$. We denote it by $A^p(X_K)$. Then the preceding exact sequence yields $$ 0\to A^{p-1}(X_K)\to CH^p(\overline{X})_{\Bbb R}\overset{\zeta }{\to } CH^p(X)_{\Bbb R}\to 0. $$ For an ample line bundle $H$ on $X$, we can consider the hard Lefschetz conjecture and the Hodge index conjecture for $A^p(X_K)$ although $X_K$ is not defined over an algebraically closed field. We denote them by $\bold A_p(X_K, H_K)$ and $\bold H_p(X_K, H_K)$ respectively. For an algebraic closure $\overline{K}$ of $K$, standard conjectures for $(X_{\overline{K}}, H_{\overline{K}})$ imply these for $(X_K, H_K)$. In particular, $\bold H_p(X_K, H_K)$ is true. We put $$ CH_{\operatorname{fin} }^p(X)=\operatorname{Ker} (CH^p(X)\to CH^p(X_K)). $$ For an ample line bundle $H$ on $X$, we can define a homomorphism $$ L_H:CH_{\operatorname{fin} }^p(X)_{\Bbb R}\to CH_{\operatorname{fin} }^{p+1}(X)_{\Bbb R} $$ by intersecting with the first Chern class $c_1(H)$. For $x\in CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ we denote by $\tilde{x}\in CH^p (\overline{X})_{\Bbb R}$ a lifting of $x$. Then we define a pairing $$ \langle \ , \ \rangle :CH_{\operatorname{fin} }^p(X)_{\Bbb R}\otimes CH_{\operatorname{fin} } ^{n+1-p}(X)_{\Bbb R}\to \Bbb R $$ by $\langle x, y\rangle =\widehat{\DEG} (\tilde{x}\tilde{y})$. This definition is independent of the choice of liftings. Here we propose the following conjectures: \begin{conj} \ \ \ \ \noindent $\bold F\bold A_p(X, H)$: \ $L_H^{n+1-2p}:CH_{\operatorname{fin} }^p(X)_{\Bbb R}\to CH_{\operatorname{fin} }^{n+1-p}(X)_{\Bbb R}$ is an isomorphism. \noindent $\bold F\bold H_p(X, H)$: \ For $0\not=x\in CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ such that $L_H^{n+2-2p}(x)=0$, $(-1)^p\langle L_H^{n+1-2p}(x), x\rangle $ is positive. \end{conj} We now construct the height pairing by means of Arakelov intersection theory. We put $$ CH^p(X_K)^0_{\Bbb R}=\operatorname{Ker} (cl:CH^p(X_K)_{\Bbb R}\to H^{p,p}(X)). $$ We need the following hypothesis. \begin{assb} \ \ \ \ For any $x\in CH^p(X_K)^0_{\Bbb R}$, there exists a lifting $\tilde{x} \in CH^p(\overline{X})_{\Bbb R}$ such that $\tilde{x}$ is orthogonal to every vertical cycles. That is to say, it holds that $$ \widehat{\DEG} (\tilde{x}\cdot (C, 0))=0 $$ for any cycle $C$ of codimension $n+1-p$ whose support does not meet the generic fiber $X_K$. \end{assb} \vskip 1pc If $CH^p(X_K)^0_{\Bbb R}$ and $CH^{n+1-p}(X_K)^0_{\Bbb R}$ admit the above assumption, then we can define a pairing $$ \langle \ , \ \rangle :CH^p(X_K)^0_{\Bbb R}\otimes CH^{n+1-p}(X_K)^0 _{\Bbb R}\to \Bbb R $$ by $\langle x, y\rangle =\widehat{\DEG} (\tilde{x} \tilde{y})$, where $\tilde{x}$ and $\tilde{y}$ are liftings which satisfy the above assumption. This definition is independent of the choice of the liftings. Suppose that the pairing $$ \langle \ , \ \rangle :CH_{\operatorname{fin} }^p(X)_{\Bbb R}\otimes CH_{\operatorname{fin} } ^{n+1-p}(X)_{\Bbb R}\to \Bbb R $$ is nondegenerate. For a lifting $\tilde{x}\in CH^p(\overline{X})_{\Bbb R}$ of $x\in CH^p(X_K)^0_{\Bbb R}$, the homomorphism $$ CH_{\operatorname{fin} }^{n+1-p}(X)_{\Bbb R}\to \Bbb R, \ y\mapsto \widehat{\DEG} (\tilde{x}\tilde{y}) $$ is determined independently of the choice of the lifting $\tilde{y}$. Because of the nondegeneracy of the pairing there exists a unique $z\in CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ such that $$ \widehat{\DEG} (\tilde{x}\tilde{y})=\langle z, y\rangle $$ for any $y\in CH_{\operatorname{fin} }^{n+1-p}(X)_{\Bbb R}$. Then for a lifting $\tilde{z}$, $\tilde{x}-\tilde{z}$ is a lifting of $x\in CH^p(X_K)^0_{\Bbb R}$ which satisfies Assumption 2. In the same way we can show that $CH^{n+1-p}(X_K)^0_{\Bbb R}$ also admits Assumption 2 and we can define the height pairing. In particular, if $\bold F\bold A_k(X, H)$ and $\bold F\bold H_k(X, H)$ hold for $k\leq p$, then Assumption 2 holds for $p$ and $n+1-p$. \vskip 1pc {\it Remark 1}: \ We assume that the pairing $$ \langle L_H^{n+1-2p} \ \ , \ \rangle :CH_{\operatorname{fin} }^p(X)_{\Bbb R}\otimes CH_{\operatorname{fin} }^p(X)_{\Bbb R}\to \Bbb R $$ is nondegenerate. Then we can say that $CH^{n+1-p}(X_K)^0_{\Bbb R}$ admits Assumption 2 and that there exists a lifting $\tilde{x}\in CH^p(\overline{X})_{\Bbb R}$ of arbitrary $x\in CH^p(X_K)^0_{\Bbb R}$ such that $L_{H,\Mtrc } ^{n+1-2p}(\tilde{x})$ is orthogonal to any vertical cycles. Therefore we can define the pairing $$ \langle L_{H_K}^{n+1-2p} \ \ , \ \rangle :CH^p(X_K)^0_{\Bbb R}\otimes CH^p(X_K)^0_{\Bbb R}\to \Bbb R $$ although $CH^p(X_K)^0_{\Bbb R}$ may not admit Assumption 2. \vskip 1pc For the height pairing for $X$, Beilinson conjectured an analogues of standard conjectures in \cite[\S 5]{beilinson}. \begin{conj} \ \ \ \ For $2p\leq n+1$, we assume either Assumption 2 for $p$ and $n+1-p$ or nondegeneracy of the pairing of $CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ in Remark 1. Then we have the followings: \noindent $\bold H\bold A_p(X, H)$: \ $L_{H_K}^{n+1-2p}:CH^p(X_K)^0_{\Bbb R}\to CH^{n+1-p}(X_K)^0_{\Bbb R}$ is an isomorphism. \noindent $\bold H\bold H_p(X, H)$: \ For $0\not=x\in CH^p(X_K)^0_{\Bbb R}$ such that $L_{H_K}^{n+2-2p}(x)=0$, $(-1)^p\langle L_{H_K}^{n+1-2p}(x), x\rangle $ is positive. \end{conj} \vskip 1pc {\it Remark 2}: \ We now choose another positive $F_{\infty } $-invariant hermitian metric $\Vert \ \Vert ^{\prime }$ on $H_{\Bbb C}$. Then there exists an $F_{\infty } $-invariant positive real valued function $f$ on $X(\Bbb C)$ such that $\Vert \ \Vert ^{\prime }=f\Vert \ \Vert $. We denote the K\"{a}hler metric associated with $\Vert \ \Vert ^{\prime }$ by $h^{\prime }$ and let $\overline{X}^{\prime }=(X, h^{\prime })$. For $x\in CH_{\operatorname{fin} }^p(X)_{\Bbb R}$, we choose a lifting $\tilde{x}\in CH^p(\overline{X})_{\Bbb R}$ of $x$. Since $\omega (\tilde{x})=0$, $\tilde{x}$ is also contained in the Arakelov Chow group $CH^p(\overline{X}^{\prime })_{\Bbb R}$ of $\overline{X}^{\prime }$. Hence the definition of pairing of $CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ does not depend on the choice of positive metrics on $H_{\Bbb C}$. In the same way it can be shown that the definition of the pairing of $CH^p(X)^0_{\Bbb R}$ and Assumption 2 do not also depend on the choice of metrics. Moreover since \begin{align*} \widehat{\DEG} (\hat{c}_1(H, \Vert \ \Vert ^{\prime }&)^{n+1-2p}\tilde{x}^2)=\widehat{\DEG} ((\hat{c}_1(H, \Mtrc ) -2a(\log f))^{n+1-2p}\tilde{x}^2) \\ &=\widehat{\DEG} (\hat{c}_1(H, \Mtrc ) ^{n+1-2p}\tilde{x}^2-2(n+1-2p)a(\log fc_1(H, \Mtrc ) ^{n-2p} \omega (\tilde{x})^2)) \\ &=\widehat{\DEG} (\hat{c}_1(H, \Mtrc ) ^{n+1-2p}\tilde{x}^2), \end{align*} the conjectures $\bold F\bold H_p(X, H)$ and $\bold H\bold H_p(X, H)$ do not depend on the choice of metrics. \vskip 1pc Here we state our main theorem. \begin{mthm} \ \ \ \ Let $X$ be an arithmetic variety defined over the ring of integers $\cal O_K$ of an algebraic number field $K$. Suppose that the generic fiber $X_K$ is geometrically irreducible. Let $H$ be an ample line bundle on $X$. Given a positive $F_{\infty } $-invariant hermitian metric $\Vert \ \Vert $ on $H_{\Bbb C}$, we define a metric $\Vert \ \Vert _{\sigma } $ by $\Vert \ \Vert _{\sigma } =\exp (\sigma )\Vert \ \Vert $ for $\sigma \in \Bbb R$. \noindent i) \ We assume Assumption 1 for $p-1$ and $n-p$. Then $\bold F\bold A_p(X, H)$, $\bold H\bold A_p(X, H)$, $\bold A_p(X_K, H_K)$ and $\bold A_{p-1}(X_K, H_K)$ imply $\bold A_p(X, H, \Vert \ \Vert _{\sigma } )$ for almost all $\sigma $. \noindent ii) \ We assume Assumption 1 for $p-1$. We suppose that the pairings $$ \langle L_H^{n+1-2p} \ \ , \ \rangle :CH_{\operatorname{fin} }^p(X)_{\Bbb R} \otimes CH_{\operatorname{fin} }^p(X)_{\Bbb R}\to \Bbb R $$ and $$ \langle \ , \ \rangle :CH_{\operatorname{fin} }^{p-1}(X)_{\Bbb R}\otimes CH_{\operatorname{fin} }^{n+2-p}(X)_{\Bbb R}\to \Bbb R $$ are nondegenerate. Then $\bold F\bold H_p(X, H)$, $\bold H\bold H_p(X, H)$ and $\bold A_{p-1}(X_K, H_K)$ imply \linebreak $\bold H_p(X, H, \Vert \ \Vert _{\sigma } )$ for $0 \ll -\sigma $. \end{mthm} \vskip 1pc After arithmetic ampleness of a hermitian line bundle was defined in \cite{zhang}, a finer version of Conjecture 1.2 is proposed. This says that for arithmetically ample hermitian line bundle $(H, \Vert \ \Vert )$ on $X$, $\bold A_p(X, H, \Vert \ \Vert )$ and $\bold H_p(X, H, \Vert \ \Vert )$ hold. For arbitrary hermitian line bundle $(H, \Vert \ \Vert )$ which satisfies the conditions in Main Theorem and for $0\ll -\sigma $, the hermitian line bundle $(H, \Vert \ \Vert _{\sigma } )$ becomes arithmetically ample. But it is difficult to compare the upper bound of $\sigma $ such that $(H, \Vert \ \Vert _{\sigma } )$ is arithmetically ample with that for which ii) of Main Theorem holds. \begin{cor} \ \ \ \ If $X$ is a Grassmannian or a projective smooth toric scheme or a generalized flag scheme, which is a quotient scheme of a split reductive group scheme by a Borel subgroup, defined over a ring of integers $\cal O_K$. Then Conjecture 1.2 holds for any ample line bundle $H$ with a positive metric $\Vert \ \Vert $ and for $0\ll -\sigma $. \end{cor} {\it Proof}: \ \ Since $X$ can be stratified into finite pieces isomorphic to affine spaces over $\cal O_K$, Assumption 1 holds. Since $CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ and $CH^p(X_K)^0_{\Bbb R}$ vanish for all $p$, all assumptions about these are vacuous. For an embedding $\tau :K\to \Bbb C$, we denote by $X_{\tau }$ the complex manifold associated with the scheme $X\underset{\tau }{\otimes }\Bbb C$. Then the cycle class map $cl:CH^p(X_K)_{\Bbb R}\to H^{p,p}(X_{\tau })$ is bijective. So the standard conjectures for $X_K$ are valid. Hence Main Theorem implies Conjecture 2.2. \qed \vskip 1pc \begin{cor} \ \ \ \ Let $X$ be an arithmetic variety. Then the Hodge index conjecture $\bold H_1(X, H, \Vert \ \Vert _{\sigma } )$ of codimension one holds for any ample line bundle $H$ with a positive metric $\Vert \ \Vert $ and for $0\ll -\sigma $. \end{cor} {\it Proof}: \ \ We have only to verify the conditions in Main Theorem for $p=1$. We denote by $\Sigma _K$ the set of all infinite places of $K$. Then $H^{0,0}(X)\simeq \underset{\sigma \in \Sigma _K}{\oplus }\Bbb R$ and $CH^{0,1}(X)\simeq \cal O_K^{\times }$. Hence by Dirichlet unit theorem Assumption 1 for $p=0$ holds. Nondegeneracy of the pairing $\langle L_H^{n-1} \ \ , \ \rangle $ on $CH_{\operatorname{fin} }^1(X)_{\Bbb R}$ and negativity of $\langle L_H^{n-1}x, x\rangle $ for $0\not= x\in CH_{\operatorname{fin} }^1(X)_{\Bbb R}$ have been already proved in \cite[Lemma 1.3]{moriwaki}. Since $CH^0(X)_{\Bbb R}\simeq \Bbb R$ and $CH^{n+1}(X)_{\Bbb R}=0$, we have $CH_{\operatorname{fin} }^0(X)_{\Bbb R}=CH_{\operatorname{fin} }^{n+1}(X)_{\Bbb R}=0$. $\bold H\bold H_1(X, H)$ holds by the positivity of N\'{e}ron-Tate height pairing and $\bold A_0(X_K, H_K)$ is trivial. Hence the proof has completed. \qed \vskip 2pc \section{Proof of the main theorem} \vskip 1pc By Theorem 2.1 we have only to prove $\bold A\bold A_p(X, H, \Vert \ \Vert _{\sigma } )$ and $\bold A\bold H_p(X, H, \Vert \ \Vert _{\sigma } )$. We begin by the proof of i). Since $\bold F\bold A_p(X, H)$, $\bold H\bold A_p(X, H)$ and $\bold A_{p-1}(X_K, H_K)$ hold, $$ L_H^{n+1-2p}:CH^p(X)_{\Bbb R}\to CH^{n+1-p}(X)_{\Bbb R} $$ is surjective and its kernel is isomorphic to $$ A_{\operatorname{prim} }^p(X_K)=\operatorname{Ker} (L_{H_K}^{n+1-2p}:A^p(X_K)\to A^{n+1-p}(X_K)). $$ We put $$ A_{\operatorname{coprim} }^{n-p}(X_K)=\operatorname{Cok} (L_{H_K}^{n+1-2p}:A^{p-1}(X_K)\to A^{n-p}(X_K)) $$ and consider the following diagram: \begin{equation*} \begin{CD} @. 0 @. @. A_{\operatorname{prim} }^p(X_K) @. \\ @. @VVV @. @VVV @. \\ 0 @>>> A^{p-1}(X_K) @>>> CH^p(\overline{X})_{\Bbb R} @>>> CH^p(X)_{\Bbb R} @>>> 0 \\ @. @VV{L_{H_K}^{n+1-2p}}V @VV{L_{H,\MTRC } ^{n+1-2p}}V @VV{L_H^{n+1-2p}}V @. \\ 0 @>>> A^{n-p}(X_K) @>>> CH^{n+1-p}(\overline{X})_{\Bbb R} @>>> CH^{n+1-p}(X)_{\Bbb R} @>>> 0 \\ @. @VVV @. @VVV @. \\ @. A_{\operatorname{coprim} }^{n-p}(X_K) @. @. 0. @. \end{CD} \end{equation*} To show that $L_{H,\MTRC } ^{n+1-2p}$ is bijective, we have only to prove that the edge homomorphism of the above diagram is an isomorphism. Given $x\in A_{\operatorname{prim} }^p(X_K)$, we choose a lifting $\tilde{x}\in CH^p(\overline{X})_{\Bbb R}$ of $x$. Then we have \begin{align*} L_{H,\MTRC } ^{n+1-2p}(\tilde{x})&=\hat{c}_1(H, \Vert \ \Vert _{\sigma } )^{n+1-2p}\tilde{x} \\ &=(\hat{c}_1(H, \Mtrc ) -2a(\sigma ))^{n+1-2p}\tilde{x} \\ &=(\hat{c}_1(H, \Mtrc ) ^{n+1-2p}-2\sigma (n+1-2p)a(c_1(H, \Mtrc ) ^{n-2p}))\tilde{x} \\ &=\hat{c}_1(H, \Mtrc ) ^{n+1-2p}\tilde{x}-2\sigma (n+1-2p)a(c_1(H, \Mtrc ) ^{n-2p}x). \end{align*} Hence if we denote by $F_{\sigma }$ the edge homomorphism of the above diagram for $\sigma $, then we have $$ F_{\sigma }=F_0-2\sigma (n+1-2p)L_H^{n-2p}. $$ $\bold A_p(X_K, H_K)$ and $\bold A_{p-1}(X_K, H_K)$ imply that $L_H^{n-2p}:A_{\operatorname{prim} }^p(X_K)\to A_{\operatorname{coprim} }^{n-p}(X_K)$ is an isomorphism. Hence for all but finitely many $\sigma $, $F_{\sigma }$ is an isomorphism. We turn to the proof of ii). Let $x$ be a primitive element in $CH^p(\overline{X})_{\Bbb R}$ with respect to $L_{H,\Mtrc } $, that is, $L_{H,\Mtrc } ^{n+2-2p}(x)=0$ holds. Then for a closed $(p,p)$-form $\omega $ we have \begin{align*} L_{H,\MTRC } ^{n+2-2p}(x+a(\omega ))&=(\hat{c}_1(H, \Mtrc ) -2a(\sigma ))^{n+2-2p}(x+a(\omega )) \\ &=(\hat{c}_1(H, \Mtrc ) ^{n+2-2p}-2\sigma (n+2-2p)a(c_1(H, \Mtrc ) ^{n+1-2p}))(x+a(\omega )) \\ &=a(c_1(H, \Mtrc ) ^{n+2-2p}\omega -2\sigma (n+2-2p)c_1(H, \Mtrc ) ^{n+1-2p}\omega (x)). \end{align*} Hence $x+a(\omega )\in CH^p(\overline{X})_{\Bbb R}$ is primitive with respect to $L_{H,\MTRC } $ if and only if $$ c_1(H, \Mtrc ) ^{n+2-2p}\omega =2\sigma (n+2-2p)c_1(H, \Mtrc ) ^{n+1-2p}\omega (x). $$ Since $c_1(H, \Mtrc ) ^{n+2-2p}\omega (x)=0$, $\bold A_{p-1}(X_K, H_K)$ implies that $\omega (x)$ is written by $$ \begin{cases} \omega _0(x)+c_1(H, \Mtrc ) \omega _1(x) &\text{if} \ 2p<n+1 \\ c_1(H, \Mtrc ) \omega _1(x) &\text{if} \ 2p=n+1, \end{cases} $$ where each $\omega _i(x)$ is contained in $A_{\operatorname{prim} }^{p-i}(X_K)$. If $2p=n+1$, then we set $\omega _0(x)=0$. By the above equality we have $$ \omega =2\sigma (n+2-2p)\omega _1(x). $$ Hence we can say that the vector spaces consisting of primitive cycles in $CH^p(\overline{X})_{\Bbb R}$ with respect to $L_{H,\MTRC } $ are isomorphic for every $\sigma \in \Bbb R$. We first assume $\omega (x)=0$. Then the restriction of $\zeta (x)\in CH^p(X)_{\Bbb R}$ to the generic fiber $X_K$ is homologically trivial. We denote the restriction of $\zeta (x)$ by $x^{\prime }$. Let $x_1$ be a lifting of $x^{\prime }$ which is orthogonal to $L_H^{n+1-2p} CH_{\operatorname{fin} }^p(X)_{\Bbb R}$ and $x_0=x-x_1$. We suppose that $x_0$ is represented by $(Z, 0)$ and the support of the cycle $Z$ does not meet the generic fiber $X_K$. We identify $x_0$ with the cycle $[Z]\in CH_{\operatorname{fin} }^p(X)_{\Bbb R}$. Since $L_{H,\MTRC } ^{n+2-2p}(x)=0$, we have $$ L_{H,\MTRC } ^{n+2-2p}(x_1)=-L_H^{n+2-2p}(x_0)\in CH_{\operatorname{fin} }^{n+2-p}(X)_{\Bbb R}. $$ Since $\langle \ , \ \rangle :CH_{\operatorname{fin} }^{p-1}(X)_{\Bbb R}\otimes CH_{\operatorname{fin} }^{n+2-p}(X)_{\Bbb R}\to \Bbb R$ is nondegenerate and $L_{H,\MTRC } ^{n+2-2p}(x_1)$ is orthogonal to $CH_{\operatorname{fin} }^{p-1}(X)_{\Bbb R}$, the both sides of the above equality are zero. Hence we have \begin{align*} (-1)^p\widehat{\DEG} (L_{H,\MTRC } ^{n+1-2p}(x)x)&=(-1)^p\langle L_H^{n+1-2p}(x_0), x_0 \rangle \\ & \ +(-1)^p\langle L_{H_K}^{n+1-2p}(x^{\prime }), x^{\prime }\rangle \end{align*} and it is positive by $\bold F\bold H_p(X, H)$ and $\bold H\bold H_p(X, H)$. We next consider when $\omega (x)\not= 0$. Then any primitive cycle $y$ in $CH^p(\overline{X})_{\Bbb R}$ with respect to $L_{H,\MTRC } $ is written by $x+a(\omega )$ where $x$ is a primitive cycle with respect to $L_{H,\Mtrc } $ and $\omega =2\sigma (n+2-2p)\omega _1(x)$. Then we have \begin{align*} L_{H,\MTRC } ^{n+1-2p}(y)y&=(\hat{c}_1(H, \Mtrc ) -2a(\sigma ))^{n+1-2p}(x+a(\omega ))^2 \\ &=(\hat{c}_1(H, \Mtrc ) ^{n+1-2p}-2\sigma (n+1-2p)a(c_1(H, \Mtrc ) ^{n-2p}))(x^2+2a(\omega (x) \omega ))). \end{align*} By substituting the above equality for $\omega $ and $\omega (x)= \omega _0(x)+c_1(H, \Mtrc ) \omega _1(x)$ we have \begin{multline*} \widehat{\DEG} (L_{H,\MTRC } ^{n+1-2p}(y)y)=\widehat{\DEG} (\hat{c}_1(H, \Mtrc ) ^{n+1-2p}x^2)-\sigma (n+1-2p) \operatorname{deg} (c_1(H, \Mtrc ) ^{n-2p}\omega _0(x)^2) \\ +\sigma (n+3-2p)\operatorname{deg} (c_1(H, \Mtrc ) ^{n+2-2p}\omega _1(x)^2). \end{multline*} The Hodge index theorem for $(X_K, H_K)$ implies $$ (-1)^p\operatorname{deg} (c_1(H, \Mtrc ) ^{n-2p}\omega _0(x)^2)>0 $$ and $$ (-1)^{p+1}\operatorname{deg} (c_1(H, \Mtrc ) ^{n+2-2p}\omega _1(x)^2)>0 $$ if $\omega _i(x)\not= 0$. When $2p<n+1$, $\omega (x)\not= 0$ implies $\omega _0(x)\not= 0$ or $\omega _1(x)\not= 0$. Hence for $0 \ll -\sigma $, $(-1)^p\widehat{\DEG} (L_{H,\MTRC } ^{n+1-2p}(x)x)$ is positive. When $2p=n+1$, $\omega _0(x)=0$ and $\omega _1(x)\not= 0$. Hence in this case the same inequality is obtained. \qed \vskip 2pc

9608

alg-geom/9608004

en

https://arxiv.org/abs/alg-geom/9608004

alg-geom/9608004

Pelham Wilson

Mark Gross, P.M.H. Wilson

Mirror Symmetry via 3-tori for a class of Calabi-Yau Threefolds

30 pages, plain TeX, epsf.tex, amssym.tex, 3 figures. Final version

We give an example of the recent proposed mirror construction of Strominger, Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper first considers mirror symmetry for K3 surfaces in light of this construction. We then consider the example of mirror symmetry for Calabi-Yau threefolds of the type considered by Voisin and Borcea, of the form SxE/involution where S is a K3 surface with involution, and E is an elliptic curve. We show how dualizing a family of special Lagrangian real 3-tori does actually produce the mirrors in these examples.

alg-geom

9608

alg-geom/9608011

en

https://arxiv.org/abs/alg-geom/9608011

alg-geom/9608011

Pandharipande Rahul

W. Fulton and R. Pandharipande

Notes on stable maps and quantum cohomology

Latex2e 52 pages. Final version

These are notes from a jointly taught class at the University of Chicago and lectures by the first author in Santa Cruz. Topics covered include: construction of moduli spaces of stable maps, Gromov-Witten invariants, quantum cohomology, and examples. These notes will appear in the proceedings of the 1995 Santa Cruz conference.

alg-geom

\section{{\bf Introduction}} \subsection{Overview} The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields non-trivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let $N_d$ be the number of degree $d$, rational plane curves passing through $3d-1$ general points in $\mathbf P^2$. Since there is a unique line passing through $2$ points, $N_1=1$. The quantum cohomology ring of $\mathbf P^2$ yields the following beautiful associativity relation determining all $N_d$ for $d\geq 2$: $$N_d= \sum_{d_1+d_2=d, \ d_1, d_2 > 0} N_{d_1} N_{d_2} \bigg( d_1^2 d_2^2 \binom{3d-4}{3d_1-2} - d_1^3 d_2 \binom{3d-4}{3d_1-1} \bigg).$$ Similar enumerative formulas are valid on other homogeneous varieties. Viewed from classical enumerative geometry, the quantum ring structure is a complete surprise. The path to quantum cohomology presented here follows the work of Kontsevich and Manin. The approach is algebro-geometric and involves the construction and geometry of a natural compactification of the moduli space of maps. The large and exciting conjectural parts of the subject of quantum cohomology are avoided here. We focus on a part of the story where the proofs are complete. We also make many assumptions that are not strictly necessary, but which simplify the presentation. It should be emphasized that this is in no way a survey of quantum cohomology, or any attempt at evaluating various approaches. In particular, the symplectic point of view is not covered (see [R-T]). Another algebro-geometric approach, using a different compactification, can be found in [L-T 1]. These notes are based on a jointly taught course at the University of Chicago in which our main efforts were aimed at understanding the papers of Kontsevich and Manin. We thank R. Donagi for instigating this course. Thanks are due to D. Abramovich, P. Belorousski, I. Ciocan-Fontanine, C. Faber, T. Graber, S. Kleiman, A. Kresch, C. Procesi, K. Ranestad, H. Tamvakis, J. Thomsen, E. Tj\o{}tta, and A. Vistoli for comments and suggestions. A seminar course at the Mittag-Leffler Institute has led to many improvements. Some related preprints that have appeared since the Santa Cruz conference are pointed out in footnotes. \subsection{Notation} \label{nota} In this exposition, for simplicity, we consider only homology classes of even dimension. To avoid doubling indices, we set, for a complete variety $X$, $$A_d X=H_{2d}(X, \mathbb{Z}), \ \ A^d X = H^{2d}(X, \mathbb{Z}).$$ When $X$ is nonsingular of dimension $n$, identify $A^d X$ with $A_{n-d} X$ by the Poincar\'e duality isomorphism $$A^d X \stackrel{\sim}{\rightarrow} A_{n-d} X, \ \ c \mapsto c \mathbin{\text{\scriptsize$\cap$}} [X].$$ In particular, a closed subvariety $V$ of $X$ of pure codimension $d$ determines classes in $A_{n-d} X$ and $A^d X$ via the duality isomorphism. Both of these classes are denoted by $[V]$. For homogeneous varieties, which are our main concern, the Chow groups coincide with the topological groups. Hence $A_d X$ and $A^d X$ can be taken to be the Chow homology and cohomology groups for homogeneous varieties (see [F]). If $X$ is complete, and $c$ is a class in the ring $A^*X = \bigoplus A^dX$, and $\beta$ is a class in $A_kX$, we denote by $\int_\beta c$ the degree of the class of the zero cycle obtained by evaluating $c_k$ on $\beta$, where $c_k$ is the component of $c$ in $A^kX$. When $V$ is a closed, pure dimensional subvariety of $X$, we write $\int_V c$ instead of $\int_{[V]} c$. We use cup product notation $\mathbin{\text{\scriptsize$\cup$}}$ for the product in $A^*X$. We will be concerned only with varieties over $\mathbb{C}$ since the relevant moduli spaces have not yet been constructed in positive characteristic. Let $[n]$ denote the finite set of integers $\{1,2, \ldots,n\}$. \subsection{{Compactifications of moduli spaces }} \label{rmod} An important feature of quantum cohomology is the use of intersection theory in a space of maps of curves into a variety, rather than in the variety itself. To carry this out, a good compactification of such a space is required. At least when $X$ is sufficiently positive, Kontsevich has constructed such a compactification. We start, in this section, by reviewing some related moduli spaces with similar properties. Kontsevich's space of stable maps will be introduced in section \ref{kontdef}. Algebraic geometers by now have become quite comfortable working with the moduli space $M_g$ of projective nonsingular curves of genus $g$, and its compactification $\barr{M}_g$, whose points correspond to projective, connected, nodal curves of arithmetic genus $g$, satisfying a stability condition (due to Deligne and Mumford) that guarantees the curve has only a finite automorphism group. These moduli spaces are irreducible varieties of dimension $3g - 3$ if $g \geq 2$, smooth if regarded as (Deligne-Mumford) stacks, and with orbifold singularities if regarded as ordinary coarse moduli spaces. Some related spaces have become increasingly important. The moduli space $M_{g,n}$ parametrizes projective nonsingular curves $C$ of genus $g$ together with $n$ distinct marked points $p_1, \ldots , p_n$ on $C$. $M_{g,n}$ has a compactification $\barr{M}_{g,n}$ whose points correspond to projective, connected, nodal curves $C$, together with $n$ distinct, nonsingular, marked points, again with a stability condition equivalent to the finiteness of automorphism groups. $\barr{M}_{g,1}$ is often called the universal curve over $\barr{M}_g$ (although, as coarse moduli spaces, this is a slight abuse of language). The first remarkable feature of the space $\barr{M}_{g,n}$ is that it compactifies $M_{g,n}$ without ever allowing the points to come together. When points on a smooth curve approach each other, in fact, the curve sprouts off one or more components, each isomorphic to the projective line, and the points distribute themselves at smooth points on these new components, in a way that reflects the relative rates of approach. \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.70000cm,0.70000cm> \unitlength=0.70000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 3.888 23.495 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 3.571 23.019 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 3.095 22.860 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 2.301 22.860 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 9.286 22.701 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 10.397 21.749 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 10.238 22.860 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.157}}} at 9.445 21.907 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 6.191 22.860 5.874 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 6.193 22.862 5.870 23.021 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 7.925 23.796 10.624 21.573 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 10.573 23.233 9.049 21.431 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 6.623 21.907 6.677 22.013 6.728 22.111 6.777 22.201 6.823 22.284 6.868 22.360 6.911 22.430 6.993 22.551 7.070 22.650 7.146 22.730 7.222 22.793 7.300 22.843 7.425 22.886 7.496 22.896 7.571 22.900 7.651 22.898 7.733 22.893 7.816 22.886 7.901 22.878 7.986 22.869 8.070 22.862 8.152 22.858 8.231 22.857 8.307 22.861 8.378 22.872 8.503 22.917 8.591 22.975 8.675 23.049 8.759 23.142 8.844 23.256 8.887 23.322 8.931 23.395 8.977 23.475 9.025 23.563 9.074 23.658 9.125 23.761 9.179 23.873 9.235 23.995 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.445 22.860 4.549 22.860 4.604 22.860 4.683 22.860 4.763 22.860 4.763 22.860 4.830 22.948 4.921 23.019 5.020 22.969 5.080 22.860 5.140 22.751 5.239 22.701 5.337 22.751 5.397 22.860 5.458 22.969 5.556 23.019 5.647 22.948 5.715 22.860 5.796 22.854 5.874 22.860 5.953 22.860 6.032 22.860 6.088 22.860 6.191 22.860 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 1.518 21.922 1.572 22.028 1.623 22.126 1.671 22.216 1.718 22.299 1.763 22.375 1.805 22.444 1.887 22.565 1.965 22.665 2.041 22.744 2.117 22.808 2.195 22.858 2.319 22.901 2.390 22.911 2.466 22.914 2.545 22.913 2.627 22.908 2.711 22.901 2.796 22.892 2.880 22.884 2.964 22.877 3.046 22.873 3.126 22.872 3.201 22.876 3.272 22.887 3.397 22.932 3.485 22.990 3.570 23.064 3.654 23.156 3.738 23.271 3.782 23.337 3.826 23.410 3.872 23.490 3.919 23.577 3.968 23.673 4.020 23.776 4.073 23.888 4.130 24.009 / \put{\SetFigFont{8}{9.6}{rm}4} [lB] at 3.651 23.654 \put{\SetFigFont{8}{9.6}{rm}2} [lB] at 2.857 23.019 \put{\SetFigFont{8}{9.6}{rm}1} [lB] at 2.064 23.019 \put{\SetFigFont{8}{9.6}{rm}3} [lB] at 3.334 23.178 \put{\SetFigFont{8}{9.6}{rm}1} [lB] at 9.525 21.590 \put{\SetFigFont{8}{9.6}{rm}2} [lB] at 10.319 21.907 \put{\SetFigFont{8}{9.6}{rm}4} [lB] at 9.207 22.860 \put{\SetFigFont{8}{9.6}{rm}3} [lB] at 10.160 23.019 \linethickness=0pt \putrectangle corners at 1.501 24.026 and 10.649 21.406 \endpicture} \end{center} \vspace{-0pt} The spaces $\barr{M}_{g,n}$ again are smooth stacks, or orbifold coarse moduli spaces, of dimension $3g - 3 + n$, as long as this integer is nonnegative. The case of genus zero plays a prominent role in our story. In this case, $\barr{M}_{0,n}$ is a fine moduli space and a nonsingular variety. A point in $\barr{M}_{0,n}$ corresponds to a curve which is a tree of projective lines meeting transversally, with $n$ distinct, nonsingular, marked points; the stability condition is that each component must have at least three special points, which are either the marked points or the nodes where the component meets the other components. For $n = 3$, of course, $M_{0,3} = \barr{M}_{0,3}$ is a point. For $n = 4$, $M_{0,4}$ parametrizes $4$ distinct marked points on a projective line. Since, up to isomorphism, one can fix the first three of these points, say to be $0$, $1$, and $\infty$, $M_{0,4}$ is isomorphic to $\mathbf P^1 \mathbin{\text{\scriptsize$\setminus$}} \{0,1,\infty\}$. It is not hard to guess what $\barr{M}_{0,4}$ must be. In fact, the three added points are represented by the following three marked curves: \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.50000cm,0.50000cm> \unitlength=0.50000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 2.328 22.464 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 3.397 23.336 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 5.222 23.368 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 6.096 22.511 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 9.938 22.447 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 11.127 23.417 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 12.795 23.417 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 13.760 22.447 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 17.570 22.464 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 18.713 23.385 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 20.428 23.400 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.182}}} at 21.364 22.447 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 1.270 21.590 5.080 24.765 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 24.765 6.985 21.590 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 8.901 21.607 12.711 24.782 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 11.424 24.765 14.599 21.590 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 16.521 21.590 20.331 24.765 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 19.061 24.750 22.236 21.575 / \put{\SetFigFont{9}{10.8}{rm}1} [lB] at 9.542 22.543 \put{\SetFigFont{9}{10.8}{rm}3} [lB] at 10.653 23.448 \put{\SetFigFont{9}{10.8}{rm}2} [lB] at 13.075 23.480 \put{\SetFigFont{9}{10.8}{rm}4} [lB] at 13.822 22.608 \put{\SetFigFont{9}{10.8}{rm}1} [lB] at 17.187 22.591 \put{\SetFigFont{9}{10.8}{rm}4} [lB] at 18.235 23.400 \put{\SetFigFont{9}{10.8}{rm}2} [lB] at 20.633 23.480 \put{\SetFigFont{9}{10.8}{rm}3} [lB] at 21.552 22.559 \put{\SetFigFont{9}{10.8}{rm}1} [lB] at 1.969 22.591 \put{\SetFigFont{9}{10.8}{rm}2} [lB] at 2.985 23.417 \put{\SetFigFont{9}{10.8}{rm}3} [lB] at 5.332 23.544 \put{\SetFigFont{9}{10.8}{rm}4} [lB] at 6.191 22.623 \linethickness=0pt \putrectangle corners at 1.245 24.807 and 22.261 21.550 \endpicture} \end{center} \vspace{-0pt} In general, the closures of the loci of trees of a given combinatorial type are smooth subvarieties of $\barr{M}_{0,n}$, and all such loci meet transversally. There is a divisor $D(A | B)$ in $\barr{M}_{0,n}$ for each partition of $[n]$ into two disjoint sets $A$ and $B$, each with at least two elements. A generic point of $D(A | B)$ is represented by two lines meeting transversally, with points labeled by $A$ and $B$ on each: \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.50000cm,0.50000cm> \unitlength=0.50000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 10.863 20.489 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 11.553 21.241 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 12.759 22.638 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 14.935 23.067 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 15.761 22.686 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 16.538 22.289 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.178}}} at 18.538 21.336 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 10.173 19.685 14.586 24.750 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 12.713 24.147 19.888 20.686 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 10.422 20.688 10.422 20.758 10.374 20.815 10.394 20.925 10.445 21.025 10.520 21.114 10.610 21.195 10.705 21.268 10.797 21.335 10.878 21.396 10.939 21.452 11.030 21.547 11.148 21.664 11.260 21.790 11.333 21.907 11.335 21.958 11.314 21.994 11.350 21.990 11.419 22.005 11.543 22.111 11.607 22.183 11.670 22.263 11.731 22.343 11.788 22.420 11.883 22.543 11.974 22.657 12.030 22.729 12.088 22.804 12.149 22.878 12.209 22.946 12.321 23.048 12.393 23.074 12.474 23.021 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.967 23.552 14.997 23.654 15.109 23.639 15.170 23.645 15.276 23.616 15.396 23.567 15.460 23.536 15.525 23.503 15.591 23.468 15.657 23.432 15.723 23.395 15.787 23.357 15.910 23.286 16.020 23.224 16.112 23.175 16.178 23.142 16.258 23.101 16.348 23.054 16.443 23.005 16.539 22.958 16.631 22.916 16.716 22.882 16.789 22.860 16.857 22.851 16.927 22.858 16.988 22.913 16.988 22.839 17.028 22.767 17.081 22.705 17.180 22.636 17.306 22.567 17.376 22.532 17.450 22.499 17.526 22.465 17.603 22.433 17.680 22.401 17.757 22.371 17.832 22.341 17.903 22.313 17.971 22.287 18.034 22.262 18.140 22.217 18.209 22.188 18.295 22.156 18.392 22.122 18.493 22.084 18.594 22.043 18.689 22.000 18.771 21.954 18.836 21.905 18.900 21.789 18.862 21.709 / \put{\SetFigFont{9}{10.8}{it}A} [lB] at 10.833 22.132 \put{\SetFigFont{9}{10.8}{it}B} [lB] at 17.062 23.042 \linethickness=0pt \putrectangle corners at 10.147 24.776 and 19.914 19.660 \endpicture} \end{center} \vspace{-0pt} It is convenient to allow labeling by finite sets other than $[n]$; we write $\barr{M}_{g,A}$ for the corresponding moduli space where $A$ is the labeling set. Let $B\subset A$ (if $g=0$, then let $|B|\geq 3$). It is a fundamental fact that there is a morphism $\barr{M}_{g,A} \rightarrow \barr{M}_{g,B}$ which ``forgets'' the points marked in $A \mathbin{\text{\scriptsize$\setminus$}} B$. On the open locus $M_{g,n}$ this map is the obvious one, but it is more subtle on the boundary: removing some points may make a component unstable, and such a component must be collapsed. For example, the map from $\barr{M}_{0,5}$ to $\barr{M}_{0,4}$ forgetting the point labeled $5$ sends \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.50000cm,0.50000cm> \unitlength=0.50000cm 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\put{\SetFigFont{9}{10.8}{rm}5} [lB] at 15.064 22.081 \put{\SetFigFont{10}{12.0}{rm}to} [lB] at 17.937 22.272 \linethickness=0pt \putrectangle corners at 8.862 24.807 and 23.518 18.993 \endpicture} \end{center} \vspace{-0pt} \noindent and \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.50000cm,0.50000cm> \unitlength=0.50000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 13.621 22.708 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 14.110 21.099 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 15.608 23.027 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 9.936 20.384 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 11.252 22.081 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 23.834 22.686 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 25.074 21.838 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 21.448 22.126 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.207}}} at 20.185 20.496 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 8.888 19.035 13.348 24.765 / \putrule from 13.348 24.765 to 13.348 24.782 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 11.333 24.304 16.063 21.004 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 19.033 19.018 23.493 24.748 / \putrule from 23.493 24.748 to 23.493 24.765 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 16.284 23.893 13.252 19.971 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 21.507 24.335 26.238 21.035 / \put{\SetFigFont{9}{10.8}{rm}1} [lB] at 9.459 20.400 \put{\SetFigFont{10}{12.0}{rm}to} [lB] at 17.937 22.272 \put{\SetFigFont{9}{10.8}{rm}2} [lB] at 14.285 20.561 \put{\SetFigFont{9}{10.8}{rm}4} [lB] at 15.888 22.765 \put{\SetFigFont{9}{10.8}{rm}5} [lB] at 13.921 22.748 \put{\SetFigFont{9}{10.8}{rm}3} [lB] at 10.888 22.274 \put{\SetFigFont{9}{10.8}{rm}2} [lB] at 24.096 22.862 \put{\SetFigFont{9}{10.8}{rm}4} [lB] at 25.271 21.971 \put{\SetFigFont{9}{10.8}{rm}3} [lB] at 21.112 22.274 \put{\SetFigFont{9}{10.8}{rm}1} [lB] at 19.808 20.591 \linethickness=0pt \putrectangle corners at 8.862 24.807 and 26.264 18.993 \endpicture} \end{center} \vspace{-0pt} \noindent The algebra that shows this is a morphism is carried out in [Kn]. In particular, for any subset $\{i,j,k,l\}$ of four integers in $[n]$, we have a morphism from $\barr{M}_{0,n}$ to $\barr{M}_{0,\{i,j,k,l\}}$. The inverse image of the point $P(i,j \mid k,l)$ \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.224}}} at 6.905 21.368 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.224}}} at 7.889 20.210 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.224}}} at 4.887 21.406 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.224}}} at 3.700 20.201 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 5.074 23.493 8.884 19.048 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.540 19.050 6.985 23.495 / \put{\SetFigFont{9}{10.8}{it}l} [lB] at 8.240 20.398 \put{\SetFigFont{9}{10.8}{it}i} [lB] at 3.270 20.527 \put{\SetFigFont{9}{10.8}{it}j} [lB] at 4.350 21.685 \put{\SetFigFont{9}{10.8}{it}k} [lB] at 7.205 21.668 \linethickness=0pt \putrectangle corners at 2.515 23.520 and 8.909 19.022 \endpicture} \end{center} \vspace{-0pt} \noindent is a divisor on $\barr{M}_{0,n}$. This inverse image is a multiplicity-free sum of divisors $D(A | B)$: the sum is taken over all partitions $A\cup B= [n]$ satisfying $i,j \in A$ and $k,l\in B$. The fact that the three boundary points in $\barr{M}_{0,\{i,j,k,l\}} \cong \mathbf P^1$ are linearly equivalent implies their inverse images in $\barr{M}_{0,n}$ are linearly equivalent as well. Hence, the fundamental relation is obtained: \begin{equation} \label{divisor} \sum_{i,j \in A \ k,l \in B} D(A | B) \, = \; \sum_{i,k \in A \ j,l \in B} D(A | B) \; = \; \sum_{i,l \in A \ j,k \in B} D(A | B) \end{equation} in $A^1(\barr{M}_{0,n})$. Keel [Ke] has shown that the classes of these divisors $D(A | B)$ generate the Chow ring, and that the relations (\ref{divisor}), together with the (geometrically obvious) relations $D(A | B) {\mathbf \cdot} D(A' | B') = 0$ if there are no inclusions among the sets $A$, $B$, $A'$, $B'$, give a complete set of relations. \subsection{The space of stable maps} \label{kontdef} In the remainder of the introduction, the basic ideas and constructions in quantum cohomology are introduced. The goal here is to give a precise overview with no proofs. The ideas introduced here are covered carefully (with proofs) in the main sections of these notes. Let $X$ be a smooth projective variety, and let $\beta$ be an element in $A_1 X$. Let $M_{g,n}(X,\beta)$ be the set of isomorphism classes of pointed maps $(C, p_1, \ldots, p_n, \mu)$ where $C$ is a projective nonsingular curve of genus $g$, the markings $p_1, \ldots , p_n$ are distinct points of $C$, and $\mu$ is a morphism from $C$ to $ X$ satisfying $\mu_*([C]) = \beta$. $(C, p_1, \ldots , p_n, \mu)$ is {\em isomorphic} to $(C', p_1', \ldots, p_n', \mu' )$ if there is a scheme isomorphism $\tau : C \rightarrow C'$ taking $p_i$ to $p_i'$, with $\mu'\circ\tau = \mu$. Of course, if $\beta \neq 0$, $M_{g,n}(X,\beta)$ is empty unless $\beta$ is the class of a curve in $X$. There are also other problems. For example, if $g = 0$, which will be the case of interest to us, $M_{g,n}(X,\beta)$ is empty if $\beta \neq 0$ and $X$ contains no rational curves. To obtain a well-behaved space, one needs to make strong assumptions on $X$. In general, there is a compactification $$ M_{g,n}(X,\beta) \subset \barr{M}_{g,n}(X,\beta) , $$ whose points correspond to stable maps $(C, p_1, \ldots , p_n, \mu)$ where $C$ a projective, connected, nodal curve of arithmetic genus $g$, the markings $p_1, \ldots , p_n$ are distinct nonsingular points of $C$, and $\mu$ is a morphism from $C$ such that $\mu_*([C]) = \beta$. Again, the stability condition (due to Kontsevich) is equivalent to finiteness of automorphisms of the map. Alternatively, $(C, p_1, \ldots, p_n, \mu)$ is a {\em stable} map if both of the following conditions hold for every irreducible component $E\subset C$: \begin{enumerate} \item[(1)] If $E\stackrel {\sim}{=} \mathbf P^1$ and $E$ is mapped to a point by $\mu$, then $E$ must contain at least three special points (either marked points or points where $E$ meets the other components of $C$). \item[(2)] If $E$ has arithmetic genus 1 and $E$ is mapped to a point by $\mu$, then $E$ must contain at least one special point. \end{enumerate} Condition (2) is relevant only in case $g=1$, $n=0$, and $\beta=0$ (in other cases, (2) is automatically satisfied). From conditions (1) and (2), it follows that $\barr{M}_{1,0}(X,0)= \emptyset$. Thus, in practice, (1) is the important condition. When $X$ is a point, so $\beta = 0$, one recovers the pointed moduli space of curves $\barr{M}_{g,n} \stackrel {\sim}{=} \barr{M}_{g,n}(\text{point}, 0)$. When $X\stackrel {\sim}{=} \mathbf P^r$ is a projective space, we write $\barr{M}_{g,n}(\mathbf P^r,d)$ in place of $\barr{M}_{g,n}(\mathbf P^r, \,d[\text{line}])$. The simplest example is $\barr{M}_{0,0}(\mathbf P^r,1)$, which is the Grassmannian $\mathbf G(\mathbf P^1, \mathbf P^r)$. If $n\geq 1$, $\barr{M}_{0,n}(\mathbf P^r,1)$ is a locally trivial fibration over $\mathbf G(\mathbf P^1, \mathbf P^r)$ with the configuration space $\mathbf P^1[n]$ of [F-M] as the fiber. Let us look at the space $\barr{M}_{0,0}(\mathbf P^2,2)$. An open set in this space is the space of nonsingular conics, since to each such conic $D$ there is an isomorphism $\mathbf P^1 \stackrel {\sim} {\rightarrow} D \subset \mathbf P^2$, unique up to equivalence. Singular conics $D$ that are the unions of two lines are similarly the isomorphic image $C \stackrel {\sim} {\rightarrow} D \subset \mathbf P^2$, where $C$ is the union of two projective lines meeting transversally at a point. This gives: \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.765 2.064 24.686 1.945 24.606 1.905 24.527 1.905 24.448 1.905 24.368 1.905 24.304 1.905 24.190 1.905 24.114 1.905 24.026 1.905 23.925 1.905 23.812 1.905 23.697 1.905 23.589 1.905 23.489 1.905 23.396 1.905 23.310 1.905 23.232 1.905 23.161 1.905 23.098 1.900 22.989 1.885 22.900 1.826 22.781 / \plot 1.826 22.781 1.746 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 20.637 2.064 20.717 1.945 20.796 1.905 20.876 1.905 20.955 1.905 21.034 1.905 21.099 1.905 21.213 1.905 21.289 1.905 21.377 1.905 21.477 1.905 21.590 1.905 21.705 1.905 21.813 1.905 21.914 1.905 22.007 1.905 22.092 1.905 22.170 1.905 22.241 1.905 22.304 1.900 22.414 1.885 22.503 1.826 22.622 / \plot 1.826 22.622 1.746 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 10.446 24.750 10.604 24.671 10.724 24.591 10.763 24.512 10.763 24.433 10.763 24.353 10.763 24.289 10.763 24.175 10.763 24.099 10.763 24.011 10.763 23.911 10.763 23.798 10.763 23.682 10.763 23.574 10.763 23.474 10.763 23.381 10.763 23.295 10.763 23.217 10.763 23.147 10.763 23.083 10.768 22.974 10.783 22.885 10.843 22.766 / \plot 10.843 22.766 10.922 22.686 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 10.446 20.623 10.604 20.702 10.724 20.781 10.763 20.861 10.763 20.940 10.763 21.020 10.763 21.084 10.763 21.198 10.763 21.274 10.763 21.362 10.763 21.462 10.763 21.575 10.763 21.691 10.763 21.798 10.763 21.899 10.763 21.992 10.763 22.077 10.763 22.156 10.763 22.226 10.763 22.290 10.768 22.399 10.783 22.488 10.843 22.607 / \plot 10.843 22.607 10.922 22.686 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.605 24.765 14.446 24.686 14.327 24.606 14.287 24.527 14.287 24.448 14.287 24.368 14.287 24.304 14.287 24.190 14.287 24.114 14.287 24.026 14.287 23.925 14.287 23.812 14.287 23.697 14.287 23.589 14.287 23.489 14.287 23.396 14.287 23.310 14.287 23.232 14.287 23.161 14.287 23.098 14.283 22.989 14.268 22.900 14.208 22.781 / \plot 14.208 22.781 14.129 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.605 20.637 14.446 20.717 14.327 20.796 14.287 20.876 14.287 20.955 14.287 21.034 14.287 21.099 14.287 21.213 14.287 21.289 14.287 21.377 14.287 21.477 14.287 21.590 14.287 21.705 14.287 21.813 14.287 21.914 14.287 22.007 14.287 22.092 14.287 22.170 14.287 22.241 14.287 22.304 14.283 22.414 14.268 22.503 14.208 22.622 / \plot 14.208 22.622 14.129 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \circulararc 180.000 degrees from 11.906 22.225 center at 12.541 22.225 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \putrule from 11.906 22.225 to 11.906 23.654 \putrule from 11.906 23.654 to 11.906 23.654 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \putrule from 13.176 22.225 to 13.176 23.654 \putrule from 13.176 23.654 to 13.176 23.654 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 22.860 24.765 23.019 24.686 23.138 24.606 23.178 24.527 23.178 24.448 23.178 24.368 23.178 24.304 23.178 24.190 23.178 24.114 23.178 24.026 23.178 23.925 23.178 23.812 23.178 23.697 23.178 23.589 23.178 23.489 23.178 23.396 23.178 23.310 23.178 23.232 23.178 23.161 23.178 23.098 23.182 22.989 23.197 22.900 23.257 22.781 / \plot 23.257 22.781 23.336 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 22.860 20.637 23.019 20.717 23.138 20.796 23.178 20.876 23.178 20.955 23.178 21.034 23.178 21.099 23.178 21.213 23.178 21.289 23.178 21.377 23.178 21.477 23.178 21.590 23.178 21.705 23.178 21.813 23.178 21.914 23.178 22.007 23.178 22.092 23.178 22.170 23.178 22.241 23.178 22.304 23.182 22.414 23.197 22.503 23.257 22.622 / \plot 23.257 22.622 23.336 22.701 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \ellipticalarc axes ratio 3.493:1.746 360 degrees from 9.842 22.701 center at 6.350 22.701 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 15.875 20.637 22.066 24.765 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 16.510 24.765 20.637 20.796 / \linethickness=0pt \putrectangle corners at 1.729 24.790 and 23.353 20.606 \endpicture} \end{center} \vspace{-0pt} \noindent We also have maps from the same $C$ to $\mathbf P^2$ sending each line in the domain onto the same line in $\mathbf P^2$. To determine this map up to isomorphism, however, the point that is the image of the intersection of the two lines must be specified, so the data for this is a line in $\mathbf P^2$ together with a point on it. Finally, there are maps from $\mathbf P^1$ to a line in the plane that are branched coverings of degree two onto a line in the plane. These are determined by specifying the line together with the two branch points. The added points consist of: \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) 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({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.288}}} at 17.949 22.297 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \put{\makebox(0,0)[l]{\circle*{ 0.288}}} at 19.873 23.288 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.175 21.114 9.366 24.289 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 15.621 21.114 22.130 24.448 / \linethickness=0pt \putrectangle corners at 1.729 24.782 and 23.353 20.606 \endpicture} \end{center} \vspace{-0pt} \noindent Thus, we recover the classical space of complete conics -- but in quite a different realization from the usual one. The same discussion is valid when $\mathbf P^2$ is replaced by $\mathbf P^r$, but this time the space is not the classical space of complete conics in space. The classical space specifies a plane together with a complete conic contained in the plane; Kontsevich's space has blown down all the planes containing a given line. Let $X$ be a complete nonsingular variety with tangent bundle $T_X$. $X$ is {\em convex} if, for every morphism $\mu: \mathbf P^1 \rightarrow X$, \begin{equation} \label{cconnv} H^1(\mathbf P^1,\,\mu^*(T_X)) = 0. \end{equation} Convexity is a very restrictive condition on $X$. The main examples of convex varieties are homogeneous spaces $X=G/P$, where $G$ is a Lie group and $P$ is a parabolic subgroup. Since $G$ acts transitively on $X$, $T_X$ is generated by global sections. Hence, $\mu^*(T_X)$ is globally generated for every morphism of $\mathbf P^1$, and the vanishing (\ref{cconnv}) is obtained. Projective spaces, Grassmannians, smooth quadrics, flag varieties, and products of such varieties are all homogeneous. It is for homogeneous spaces that the theory of quantum cohomology takes its simplest form. The development of quantum cohomology in sections 7--10 is carried out only in the homogeneous case. Other examples of convex varieties include abelian varieties and projective bundles over curves of positive genus. The genus 0 moduli space of stable maps is well-behaved in case $X$ is convex. In this case, $\barr{M}_{0,n}(X,\beta)$ exists as a projective nonsingular stack or orbifold coarse moduli space, containing $M_{0,n}(X,\beta)$ as a dense open subset. When $\barr{M}_{0,n}(X,\beta)$ is nonempty, its dimension is given by $$ \dim\barr{M}_{0,n}(X,\beta) = \dim X + \int_\beta c_1(T_X) + n - 3 . $$ Here, $c_1(T_X)$ is the first Chern class of the tangent bundle to $X$. We assume always that the right side of this equation is nonnegative. In the stack or orbifold sense, this is a compactification with normal crossing divisors. When $X$ is projective space, $\barr{M}_{0,n}(X,d)$ is irreducible. These assertions are Theorems 1--3 in these notes and are established in sections 1--6. We will also write $\barr{M}_{0,A}(X,\beta)$ when the index set is a set $A$ instead of $[n]$. These varieties also have forgetful morphisms $\barr{M}_{0,A}(X,\beta) \rightarrow \barr{M}_{0,B}(X,\beta) $ when $B$ is a subset of $A$. In addition, if $|A|\geq 3$, there are morphisms $\barr{M}_{0,A}(X,\beta) \rightarrow \barr{M}_{0,A}$ that forget the map $\mu$. In both these cases, as before, one must collapse components that become unstable. When $X$ is convex, the spaces $\barr{M}_{0,n}(X, \beta)$ have fundamental boundary divisors analogous to the divisors $D(A |B)$ on $\barr{M}_{0,n}$. Let $n\geq 4$. Let $A\cup B$ be a partition of $[n]$. Let $\beta_1 + \beta_2=\beta$ be a sum in $A_1 X$. There is a divisor in $\barr{M}_{0,n}(X,\beta)$ determined by: \begin{equation} \label{bddeff} D(A,B; \beta_1, \beta_2) = \barr{M}_{0, A\cup\{\bullet\}}(X, \beta_1) \times_X \barr{M}_{0, B\cup \{\bullet\}}(X, \beta_2), \end{equation} $$ D(A,B; \beta_1, \beta_2) \subset \barr{M}_{0,n}(X,\beta).$$ A moduli point in $D(A,B, \beta_1, \beta_2)$ corresponds to a map with a reducible domain $C= C_1 \cup C_2$ where $\mu_*([C_1])= \beta_1$ and $\mu_*([C_2])=\beta_2$. The points labeled by $A$ lie on $C_1$ and points labeled by $B$ lie on $C_2$. 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21.946 20.265 22.014 20.240 22.080 20.213 22.145 20.184 22.270 20.120 22.388 20.048 22.501 19.969 22.607 19.882 22.707 19.787 22.801 19.684 22.889 19.574 22.970 19.455 / \plot 22.970 19.455 23.597 18.479 / \put{\SetFigFont{9}{10.8}{it}A} [lB] at 10.803 22.043 \put{\SetFigFont{9}{10.8}{it}B} [lB] at 22.312 21.630 \put{\SetFigFont{9}{10.8}{it}A} [lB] at 28.884 21.725 \put{\SetFigFont{9}{10.8}{it}B} [lB] at 37.615 21.963 \linethickness=0pt \putrectangle corners at 10.139 23.497 and 38.964 18.057 \endpicture} \end{center} \vspace{-0pt} \noindent Finally, the fiber product in the definition (\ref{bddeff}) corresponds to the condition that the maps must take the same value in $X$ on the marked point $\bullet$ in order to be glued. This fiber product is defined via evaluation maps discussed in the next section. For $i,j,k,l$ distinct in $[n]$, set $$D(i,j \mid k,l) = \sum D(A,B; \beta_1, \beta_2).$$ The sum is over all partitions $A \cup B =[n]$ with $i,j \in A$ and $\ k,l \in B$ and over all classes $\beta_1, \beta_2 \in A_1X$ satisfying $\beta_1+\beta_2=\beta$. Using the projection $\barr{M}_{0,n}(X, \beta) \rightarrow \barr{M}_{0,\{i,j,k,l\}} \stackrel {\sim}{=} \mathbf P^1$, the fundamental linear equivalences \begin{equation} \label{mlinee} D(i,j\mid k,l)= D(i,k\mid j,l)= D(i,l\mid j,k) \end{equation} on $\barr{M}_{0,n}(X, \beta)$ are obtained via pull-back of the the 4-point linear equivalences on $\barr{M}_{0, \{i,j,k,l\}}$ as in (\ref{divisor}). \subsection{Gromov-Witten invariants and quantum cohomology} Let $X$ be a convex variety. For each marked point $1\leq i\leq n$, there is a canonical {\em evaluation map} $$\rho_i:\barr{M}_{0,n}(X,\beta)\rightarrow X$$ defined for $[C, p_1, \ldots, p_n, \mu]$ in $\barr{M}_{0,n}(X, \beta)$ by: $$\rho_i([C, p_1, \ldots, p_n, \mu]) = \mu(p_i).$$ Given classes $\gamma_1, \ldots, \gamma_n$ in $A^*X$, a product is determined in the ring $A^*(\barr{M}_{0,n}(X,\beta))$ by: \begin{equation} \label{iprod} \rho_1^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} \rho_n^*(\gamma_n) \in A^*(\barr{M}_{0,n}(X,\beta)). \end{equation} If $\sum \text{codim}(\gamma_i) = \text{dim} (\barr{M}_{0,n}(X, \beta))$, the product (\ref{iprod}) can be evaluated on the fundamental class of $\barr{M}_{0,n}(X, \beta)$. In this case, the {\em Gromov-Witten invariant} $I_\beta(\gamma_1 \cdots \gamma_n)$ is defined by: \begin{equation} \label{gwitt} I_\beta(\gamma_1 \cdots \gamma_n) = \int_{\barr{M}_{0,n}(X, \beta)} \rho_1^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} \rho_n^*(\gamma_n) . \end{equation} The multiplicative notation in the argument of $I_{\beta}$ is used to indicate $I_{\beta}$ is a symmetric function of the classes $\gamma_1, \ldots, \gamma_n$. Let $X$ be a homogeneous space. Poincar\'e duality and Bertini-type transversality arguments imply a relationship between the Gromov-Witten invariants and enumerative geometry. If $\gamma_i=[V_i]$ for a subvariety $V_i \subset X$, the Gromov-Witten invariant (\ref{gwitt}) equals the number of marked rational curves in $X$ with $i^{th}$ marked point in $V_i$, suitably counted. For example, when $X=\mathbf P^2$, $\beta= d[\text{line}]$, $n=3d-1$, and each $V_i$ is a point, $$N_d= I_{d}( \underbrace{[p] \cdots [p]}_{3d-1}).$$ The Gromov-Witten invariants are used to define the quantum cohomology ring. Associativity of this ring is established as a consequence of the 4-point linear equivalences (\ref{mlinee}). Associativity amounts to many equations among the Gromov-Witten invariants which often lead to a determination of all the invariants in terms of a few basic numbers. Given $\gamma_1, \ldots, \gamma_n \in H^*X$ (not necessarily of even degrees), there are more general Gromov-Witten invariants in $H^* \barr{M}_{0,n}$ defined by $$I^{X}_{0,n,\beta}(\gamma_1 \otimes \cdots \otimes \gamma_n) = \eta_*( \rho_1^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} \rho_n^*(\gamma_n))$$ where $\eta: \barr{M}_{0,n}(X, \beta) \rightarrow \barr{M}_{0,n}$ is the projection. The set of multilinear maps $$\{ I^{X}_{0,n, \beta}: (H^*X)^{\otimes n} \rightarrow H^*\barr{M}_{0,n} \}$$ is called the {\em Tree-Level System of Gromov-Witten Invariants}. We will not need these generalities here. The construction and proofs of the basic properties of $\barr{M}_{0,n}(X,\beta)$ are undertaken in sections 1--6. The theory of Gromov-Witten invariants and quantum cohomology for homogeneous varieties is presented in sections 7--10 with the examples of $\mathbf P^2$, $\mathbf P^3$, and a smooth quadric 3-fold ${\mathbf Q}^3$ worked out in detail. If Theorems 1--3 are taken for granted, sections 1--6 can be skipped. No originality is claimed for these notes except for some aspects of the proofs of Theorems 1--4. Constructions of Kontsevich's moduli space of stable maps can also be found in [A], [K], [B-M]. In [A], a generalization to the case in which the domain is a surface is analyzed. \subsection{Calculation of $N_d$} We end this introduction by sketching how these moduli spaces of maps can be used to calculate the number $N_d$ of degree $d$ rational plane curves passing through $3d-1$ general points in $\mathbf P^2$. The formula (\ref{rrecc}) will be recovered in section 9 from the general quantum cohomology results, but it may be useful now to see a direct proof. For $d=1$, $N_1=1$ is the number of lines through 2 points. $N_d$ is determined for $d\geq 2$ by the {\em recursion formula}: \begin{equation} \label{rrecc} N_d= \sum_{d_1+d_2=d, \ d_1, d_2 > 0} N_{d_1} N_{d_2} \bigg( d_1^2 d_2^2 \binom{3d-4}{3d_1-2} - d_1^3 d_2 \binom{3d-4}{3d_1-1} \bigg). \end{equation} For example, (\ref{rrecc}) yields$^1$ : \footnotetext[1]{The number $N_3=12$ is the classically known number of singular members in a pencil of cubic curves through 8 given points. The number $N_4=620$ was computed by H. Zeuthen in [Z]. Z. Ran reports $N_4=620$ as well as the higher $N_d$ 's can be derived from his formulas in [R1]; see [R2] for a comparison of the two approaches. Some $N_d$'s are also computed in [C-H 2].} $$N_2=1,\ N_3=12, \ N_4=620, \ N_5=87304, \ N_6=26312976, ...$$ The strategy of proof is to utilize the fundamental linear relations (\ref{mlinee}) among boundary components of $\barr{M}_{0,n}(\mathbf P^2,d)$. Intersection of a curve $Y$ in this moduli space with the linear equivalence (\ref{mlinee}) will yield (\ref{rrecc}). We will take $n=3d$ (not $3d-1$) with $d\geq 2$, so $n\geq 6$. Label the marked points by the set $$\{1,2,3, \ldots, n-4, q, r, s, t\}.$$ The forgetful morphism $\barr{M}_{0,n}(\mathbf P^2,d) \rightarrow \overline{M}_{0,\{q,r,s,t\}}$ yields the relations (\ref{mlinee}) on $\barr{M}_{0,n}(\mathbf P^2,d)$: \begin{equation} \label{mlinel} D(q,r\mid s,t) = D(q,s\mid r,t). \end{equation} Recall from section \ref{kontdef}: $$D(q,r \mid s,t) = \sum_{q,r \in A, \ s,t \in B, \ d_1+d_2=d} D(A,B; d_1,d_2).$$ The curve $Y \subset \barr{M}_{0,n}(\mathbf P^2,d)$ is determined by a selection of general points and lines in $\mathbf P^2$. More precisely, let $z_1, \ldots, z_{n-4}, z_s, z_t$ be $n-2$ general points in $\mathbf P^2$ and let $l_{q}, l_r$ be general lines. Let the curve $Y$ be defined by the intersection: $$Y= \rho_1^{-1}(z_1) \cap \cdots \cap \rho_{n-4}^{-1}(z_{n-4}) \cap \rho^{-1}_q(l_q) \cap \rho^{-1}_r(l_r) \cap \rho^{-1}_s(z_s) \cap \rho^{-1}_t(z_t).$$ $\barr{M}_{0,n}(\mathbf P^2,d)$ is a nonsingular, fine moduli space on the open set of automorphism-free maps (see section 1.2). It is not difficult to show the locus of maps with non-trivial automorphisms in $\barr{M}_{0,n}(\mathbf P^2,d)$ is of codimension at least 2 if $(n,d) \neq (0,2)$. Therefore, by Bertini's theorem applied to each evaluation map and the generality of the points and lines, we conclude $Y$ is a nonsingular curve contained in the automorphism-free locus which intersects all the boundary divisors transversally at general points of the boundary. It remains only to compute the intersection of $Y$ with each side of the linear equivalence (\ref{mlinel}). The points of $$Y \cap\ D(A, B; d_1, d_2)$$ correspond bijectively to maps $\mu:C=C_A\cup C_B \rightarrow \mathbf P^2$ satisfying: \begin{enumerate} \item[(a)] $C_A, C_B \stackrel {\sim}{=} \mathbf P^1$ and meet transversally at a point. \item[(b)] The markings of $A$, $B$ lie on $C_A$, $C_B$ respectively. \item[(c)] $\mu_*([C_A])= d_1[\text{line}]$, $\mu_*([C_B])= d_2[\text{line}].$ \item [(d)] $\forall 1\leq i \leq n-4, \ \mu(i)=z_i.$ \item [(e)] $\mu(q)\in l_q$, $\mu(r)\in l_r$, $\mu(s)=z_{s}$, $\mu(t)=z_{t}$. \end{enumerate} Let $q, r \in A$ and $s, t \in B$. $Y\cap\ D(A, B;0,d)$ is nonempty only when $A=\{q,r\}$. In this case, $C_A$ is required to map to the point $l_q \cap l_r$. The restriction $\mu:C_B\rightarrow \mathbf P^2$ must map the $3d-2$ markings on $C_B$ to the $3d-2$ given points, and in addition, $\mu$ maps the point $C_A \cap C_B$ to $l_q \cap l_r$. Therefore, $$\# \ Y\cap\ D (\{q,r\},\{1,\ldots, n-4,s,t\};0,d) = N_d.$$ For $1\leq d_1 \leq d-1$, $Y\cap D(A,B; d_1, d_2)$ is nonempty only when $|A|= 3d_1+1$. There are $\binom{3d-4}{3d_1-1}$ partitions satisfying $q, r\in A$, $s,t\in B$, and $|A|=3d_1+1$. A simple count of maps satisfying (a)-(e) yields $$\# \ Y\cap \ D(A, B; d_1,d_2) = N_{d_1}N_{d_2} d_1^3 d_2$$ for each partition. There are $N_{d_1}$ choices for the image of $C_A$ and $N_{d_2}$ choices for the image of $C_B$. The points labeled $q$ and $r$ map to any of the $d_1$ intersection points of $\mu(C_A)$ with $l_q$ and $l_r$ respectively. Finally, there are $d_1d_2$ choices for the image of the intersection point $C_A \cap C_B$ corresponding to the intersection points of $\mu(C_A) \cap \mu(C_B)\subset \mathbf P^2$. The last case is simple: $Y\cap\ D(A, B;d,0)=\emptyset$. Therefore, $$\#\ Y\cap \ D(q,r \mid s,t)= N_d + \sum_{d_1+d_2=d, \ d_1>0, \ d_2 >0} N_{d_1} N_{d_2} d_1^3 d_2 \binom{3d-4}{3d_1-1}.$$ Now consider the other side of the linear equivalence (\ref{mlinel}). Let the markings now satisfy $q,s \in A$ and $r,t \in B$. $Y\cap \ D(A, B; 0,d)$ and $Y\cap\ (A , B; d,0)$ are both empty. For $1\leq d_1 \leq d-1$, $Y\cap\ (A\cup B, d_1, d_2)$ is nonempty only when $|A|=3d_1$. There are $\binom{3d-4} {3d_1-2}$ such partitions and $$ \# \ Y\cap\ D(A, B; d_1,d_2)= N_{d_1}N_{d_2} d_1^2 d_2^2$$ for each. Therefore, $$ \#\ Y\cap\ D(q,s \mid r,t)= \sum_{d_1+d_2=d, \ d_1>0, \ d_2 >0} N_{d_1} N_{d_2} d_1^2 d_2^2 \binom{3d-4}{3d_1-2}.$$ The linear equivalence (\ref{mlinel}) implies $$\# \ Y\cap\ D(q,r \mid s,t) = \# \ Y\cap\ D(q,s \mid r,t).$$ The recursion (\ref{rrecc}) follows immediately. In the general development of quantum cohomology described in sections 8 and 9, these numerical relations obtained by intersection with the basic linear equivalences arise as ring associativity relations. \section{{\bf Stable maps and their moduli spaces}} \subsection{Definitions} An $n$-pointed, genus $g$, complex, {\em quasi-stable} curve $$(C,\ p_1, \ldots, p_n)$$ is a projective, connected, reduced, (at worst) nodal curve of arithmetic genus $g$ with $n$ distinct, nonsingular, marked points. Let $S$ be an algebraic scheme over $\mathbb{C}$. A {\em family} of $n$-pointed, genus $g$, quasi-stable curves over $S$ is a flat, projective map $\pi:\mathcal{C}\rightarrow S$ with $n$ sections $p_1, \ldots, p_n$ such that each geometric fiber $(\mathcal{C}_s, \ p_1(s), \ldots, p_n(s))$ is an $n$-pointed, genus $g$, quasi-stable curve. Let $X$ be an algebraic scheme over $\mathbb{C}$. A {\em family of maps} over $S$ from $n$-pointed, genus $g$ curves to $X$ consists of the data $(\pi:\mathcal{C}\rightarrow S, \{ p_i \}_{1\leq i \leq n}\ , \mu:\mathcal{C} \rightarrow X)$: \begin{enumerate} \item[(i)] A family of $n$-pointed, genus $g$, quasi-stable curves $\pi: \mathcal{C}\rightarrow S$ with $n$ sections $\{p_1, \ldots, p_n \}$. \item[(ii)] A morphism $\mu: \mathcal{C} \rightarrow X$. \end{enumerate} Two families of maps over $S$, $$(\pi:\mathcal{C}\rightarrow S, \{p_i\}, \mu), \ \ (\pi':\mathcal{C}' \rightarrow S, \{p'_i\}, \mu'),$$ are {\em isomorphic} if there exists a scheme isomorphism $\tau: \mathcal{C}\rightarrow \mathcal{C}'$ satisfying: $\pi= \pi'\circ \tau$, $p'_i= \tau \circ p_i$, $\mu= \mu' \circ \tau$. When $\pi:C \rightarrow \text{Spec}(\mathbb{C})$ is the structure map, $(\pi:C\rightarrow \text{Spec}(\mathbb{C}), \{p_i\}, \mu)$ is written as $(C,\{p_i\}, \mu)$. Let $(C,\{p_i\}, \mu)$ be a map from an $n$-pointed quasi-stable curve to $X$. The {\em special points} of an irreducible component $E\subset C$ are the marked points and the component intersections of $C$ that lie on $E$. The map $(C, \{p_i\}, \mu)$ is {\em stable} if the following conditions hold for every component $E\subset C$: \begin{enumerate} \item[(1)] If $E\stackrel {\sim}{=} \mathbf P^1$ and $E$ is mapped to a point by $\mu$, then $E$ must contain at least three special points. \item[(2)] If $E$ has arithmetic genus 1 and $E$ is mapped to a point by $\mu$, then $E$ must contain at least one special point. \end{enumerate} A family of pointed maps $(\pi:\mathcal{C}\rightarrow S, \{p_i\}, \mu)$ is {\em stable} if the pointed map on each geometric fiber of $\pi$ is stable. If $X= \mathbf P^r$, stability can be expressed in the following manner. Let $\omega_{\mathcal{C}/S}$ denote the relative dualizing sheaf. A flat family of maps $(\pi:\mathcal{C}\rightarrow S,\ \{p_i\}, \mu)$ is {\em stable} if and only if $\omega_{\mathcal{C}/S}(p_1+\ldots+p_n)\otimes \mu^*({\mathcal{O}}_{\mathbf P^r}(3))$ is $\pi$-relatively ample. Let $X$ be an algebraic scheme over $\mathbb{C}$. Let $\beta\in A_1 X$. A map $\mu:C \rightarrow X$ {\em represents} $\beta$ if the $\mu$-push-forward of the fundamental class $[C]$ equals $\beta$. Define a contravariant functor $\barr{\mathcal{M}}_{g,n}(X,\beta)$ from the category of complex algebraic schemes to sets as follows. Let $\barr{\mathcal{M}}_{g,n}(X,\beta)(S)$ be the set of isomorphism classes of stable families over $S$ of maps from $n$-pointed, genus $g$ curves to $X$ representing the class $\beta$. \subsection{Existence} Let $X$ be a projective, algebraic scheme over $\mathbb{C}$. Projective coarse moduli spaces of maps exist for general $g$. In the genus $0$ case, if $X$ is a projective, nonsingular, convex variety, the coarse moduli spaces are normal varieties with finite quotient singularities. \begin{tm} \label{rep} There exists a projective, coarse moduli space $\barr{M}_{g,n}(X,\beta)$. \end{tm} $\barr{M}_{g,n}(X,\beta)$ is a scheme together with a natural transformation of functors $$\phi: \barr{\mathcal{M}}_{g,n}(X,\beta) \rightarrow \mathcal{H}om_{Sch}(*, \barr{M}_{g,n}(X,\beta))$$ satisfying properties: \begin{enumerate} \item[(I)] $\phi(\text{Spec}(\mathbb{C})): \barr{\mathcal{M}} _{g,n}(X,\beta) (\text{Spec}(\mathbb{C})) \rightarrow \mathcal{H}om(\text{Spec}(\mathbb{C}), \barr{M} _{g,n}(X,\beta))$ is a set bijection. \item[(II)] If $Z$ is a scheme and $\psi: \barr{\mathcal{M}}_{g,n}(X,\beta) \rightarrow \mathcal{H}om(*,Z)$ is a natural transformation of functors, then there exists a unique morphism of schemes $$\gamma: \barr{M}_{g,n}(X,\beta) \rightarrow Z$$ such that $\psi= \tilde{\gamma} \circ \phi$. ($\tilde{\gamma}: \mathcal{H}om(*,\barr{M}_{g,n}(X,\beta)) \rightarrow \mathcal{H}om(*,Z)$ is the natural transformation induced by $\gamma$.) \end{enumerate} Let $(C, \{p_i \}, \mu)$ be a map of an $n$-pointed, quasi-stable curve to $X$. An automorphism of the map is an automorphism, $\tau$, of the curve $C$ satisfying $$p_i=\tau(p_i), \ \ \mu=\mu\circ \tau.$$ It is straightforward to check that $(C,\{p_i \}, \mu)$ is stable if and only if $(C,\{p_i \}, \mu)$ has a finite automorphism group. Let $\barr{M}^*_{g,n}(X,\beta)\subset \barr{M}_{g,n}(X,\beta)$ denote the open locus of stable maps with no non-trivial automorphisms. A nonsingular variety $X$ is convex if for every map $\mu:\mathbf P^1\rightarrow X$, $H^1(\mathbf P^1, \mu^*(T_X))=0$ (see section 0.4). The second and third theorems concern the convex, genus $0$ case. \begin{tm} Let $X$ be a projective, nonsingular, convex variety. \begin{enumerate} \item[(i)] $\barr{M}_{0,n}(X,\beta)$ is a normal projective variety of pure dimension $$\text{\em{dim}} (X)+\int_{\beta} c_1(T_X) +n-3.$$ \item[(ii)] $\barr{M}_{0,n}(X,\beta)$ is locally a quotient of a nonsingular variety by a finite group. \item[(iii)] $\barr{M}^*_{0,n}(X,\beta)$ is a nonsingular, fine moduli space (for automorphism-free stable maps) equipped with a universal family. \end{enumerate} \label{t2} \end{tm} \noindent In part (i), $\barr{M}_{0,n}(X,\beta)$ is not claimed in general to be irreducible (or even nonempty). In fact, if the language of stacks is pursued, it can be seen that the moduli problem of stable maps from $n$-pointed, genus $0$ curves to a nonsingular, convex space $X$ determines a complete, nonsingular, algebraic stack. For simplicity, the stack theoretic view is not taken in these notes; the experienced reader will see how to make the required modifications. The {\em boundary} of $\barr{M}_{0,n}(X,\beta)$ is the locus corresponding to reducible domain curves. The boundary of the fine moduli space $\barr{M}_{0,n}$ is a divisor with normal crossings. In the coarse moduli spaces $\barr{M}_{g}$ and $\barr{M}_{g,n}$, the boundary is a divisor with normal crossings modulo a finite group. $\barr{M}_{0,n}(X,\beta)$ has the same boundary singularity type as these moduli spaces of pointed curves. \begin{tm} \label{t33} Let $X$ be a nonsingular, projective, convex variety. The boundary of $\barr{M}_{0,n}(X,\beta)$ is a divisor with normal crossings (up to a finite group quotient). \end{tm} The organization of the construction is as follows. First $\barr{M}_{g,n}(\mathbf P^r,d)$ is explicitly constructed in sections 2--4. If $X\subset \mathbf P^r$ is a closed subscheme, it is not difficult to define a natural, closed subscheme $$\barr{M}_{g,n}(X,d)\subset \barr{M}_{g,n}(\mathbf P^r,d)$$ of maps that factor through $X$. $\barr{M}_{g,n}(X,d)$ is a disjoint union of the spaces $\barr{M}_{g,n}(X,\beta)$ as $\beta$ varies in $A_1 X$. By the universal property, it can be seen that the coarse moduli spaces $\barr{M}_{g,n}(X,\beta)$ do not depend on the projective embedding of $X$ (see section 5). The deformation arguments required to deduce Theorem 2 from the convexity assumption are covered in section 5. The boundary of the space of maps is discussed in section 6. \subsection{Natural structures} \label{introint} The universal property of the moduli space of maps immediately yields geometric structures on $\barr{M}_{g,n}(X,\beta)$. Consider first the marked points. The $n$ marked points induce $n$ canonical evaluation maps $\rho_1, \ldots, \rho_n$ on $\barr{M}_{g,n}(X,\beta)$. For $1\leq i \leq n$, define a natural transformation $$\theta_i: \barr{\mathcal{M}} _{g,n}(X,\beta) \rightarrow \mathcal{H}om(*, X)$$ as follows. Let $\zeta=(\pi:\mathcal{C} \rightarrow S, \ \{p_i\},\ \mu)$ be an element of $\barr{\mathcal{M}}_{g,n}(X,\beta)(S)$. Let $$\theta_i(S)(\zeta)= \mu \circ p_i \in \mathcal{H}om(S,X).$$ $\theta_i$ is easily seen to be a natural transformation. By Theorem \ref{rep}, $\theta_i$ induces a unique morphism of schemes $\rho_i: \barr{M}_{g,n}(X,\beta) \rightarrow X.$ By the universal properties of the moduli spaces $\overline{M}_{g,n}$ of $n$-pointed Deligne-Mumford stable genus $g$ curves (in case $2g-2+n>0$), each element $\zeta\in \barr{\mathcal{M}}_{g,n}(X,\beta)(S)$ naturally yields a morphism $S\rightarrow \overline{M}_{g,n}$ ([Kn]). Therefore, there exist natural forgetful maps $\eta: \barr{M}_{g,n}(X,\beta) \rightarrow \overline{M}_{g,n}$. \section{\bf{Boundedness and a quotient approach}} \subsection{Summary} In this section, the case $X=\mathbf P^r$ will be considered. The boundedness of the moduli problem of pointed stable maps is established. The arguments lead naturally to a quotient approach to the coarse moduli space. To set up the quotient approach, a result on equality loci of families of line bundles is required. \subsection{Equality of line bundles in families} Results on scheme theoretic equality loci are recalled. Let $\pi: \mathcal{C} \rightarrow S$ be a flat family of quasi-stable curves. By the theorems of cohomology and base change (cf. [H]), there is a canonical isomorphism ${\mathcal{O}}_S \stackrel {\sim}{=} \pi_*({\mathcal{O}}_{\mathcal{C}}) $. Hence, for any line bundle $\mathcal{N}$ on $S$, there is a canonical isomorphism $\mathcal{N} \stackrel {\sim}{=} \pi_* \pi^*(\mathcal{N})$. Suppose $\mathcal{L}$ and $\mathcal{M}$ are two line bundles on $\mathcal{C}$. The existence of a line bundle $\mathcal{N}$ on $S$ such that $\mathcal{L} \otimes \mathcal{M}^{-1}\stackrel {\sim}{=} \pi^*(\mathcal{N})$ is equivalent to the joint validity of (a) and (b): \begin{enumerate} \item[(a)] $\pi_*(\mathcal{L} \otimes \mathcal{M}^{-1})$ is locally free. \item[(b)] The canonical map $\pi^*\pi_*(\mathcal{L}\otimes \mathcal{M}^{-1}) \rightarrow \mathcal{L}\otimes \mathcal{M}^{-1}$ is an isomorphism. \end{enumerate} Let $\mathcal{L}_s$ be a line bundle on the geometric fiber $\mathcal{C}_s$ of $\pi$. The {\em multidegree} of $\mathcal{L}_s$ assigns to each irreducible component of $\mathcal{C}_s$ the degree of the restriction of $\mathcal{L}_s$ to that component. \begin{pr} \label{mumford} Let $\mathcal{L}$, $\mathcal{M}$ be line bundles on $\mathcal{C}$ such that the multidegrees of $\mathcal{L}_s$ and $\mathcal{M}_s$ coincide on each geometric fiber $\mathcal{C}_s$. Then, there is a unique closed subscheme $T\rightarrow S$ satisfying the following two properties: \begin{enumerate} \item[(I)] There is a line bundle $\mathcal{N}$ on $T$ such that $\mathcal{L}_T \otimes \mathcal{M}_T^{-1} \stackrel {\sim}{=} \pi^*(\mathcal{N})$. \item[(II)] If $(R \rightarrow S,\ \mathcal{N})$ is a pair of a morphism from $R$ to $S$ and a line bundle on $R$ such that $\mathcal{L}_R \otimes \mathcal{M}^{-1}_R \stackrel {\sim}{=} \pi^*(\mathcal{N})$, then $R\rightarrow S$ factors through $T$. \end{enumerate} \end{pr} \noindent {\em Proof.} The proof of the Theorem of the Cube (II) in [M1] also establishes this proposition. The multidegree condition implies $\mathcal{L}_s \stackrel {\sim}{=} \mathcal{M}_s$ if and only if $h^0(\mathcal{C}_s, \mathcal{L}_s \otimes \mathcal{M}_s^{-1})=1$. The multidegree condition is required for $T$ to be a {\em closed} subscheme. \qed \vspace{+10pt} \subsection{Boundedness} \label{bound} Let $(C, \{p_i\}, \mu)$ be a stable map from an $n$-pointed, genus $g$ curve to $\mathbf P^r$. Let $$\mathcal{L}= \omega_{C}(p_1+\ldots +p_n) \otimes \mu^*({\mathcal{O}} _{\mathbf P^r}(3)).$$ $\mathcal{L}$ is ample on $C$. A simple argument shows there exists an $f=f(g,n,r,d)>0$ such that $\mathcal{L}^f$ is very ample on $C$ and $h^1(C, \mathcal{L}^f)=0$, so $$\text{degree}(\mathcal{L}^f)= f\cdot(2g-2+n+3d) =e,$$ $$h^0(C, \mathcal{L}^f)= e-g+1.$$ Let $W\stackrel {\sim}{=} \mathbb{C}^{e-g+1}$ be a vector space. An isomorphism \begin{equation} \label{choice} W^* \stackrel {\sim} {\rightarrow} H^0(C, \mathcal{L}^f) \end{equation} induces embeddings $\iota:C \hookrightarrow \mathbf P(W)$ and $\gamma: C \hookrightarrow \mathbf P(W) \times \mathbf P^r$ where $\gamma=(\iota, \mu)$. The $n$ sections $\{p_i\}$ yield $n$ points $(\iota \circ p_i, \mu \circ p_i)$ of $\mathbf P(W)\times \mathbf P^r$. Let $H$ be the Hilbert scheme of genus $g$ curves in $\mathbf P(W)\times \mathbf P^r$ of multidegree $(e,d)$. Let $P_i=\mathbf P(W)\times \mathbf P^r$ be the Hilbert scheme of a point in $\mathbf P(W)\times \mathbf P^r$. Via the isomorphism (\ref{choice}), a point in $H\times P_1\times \ldots \times P_n$ is associated to the stable map $(C, \{p_i\}, \mu)$. The locus of points in $H\times P_1\times \ldots \times P_n$ corresponding to stable maps has a natural quasi-projective scheme structure. There is a natural closed incidence subscheme $$ I \subset H \times P_1 \times P_2 \times \ldots \times P_n$$ corresponding to the locus where the $n$ points lie on the curve. There is an open set $U\subset I$ satisfying the following: \begin{enumerate} \item[(i)] The curve $C$ is quasi-stable. \item[(ii)] The natural projection $C \rightarrow \mathbf P(W)$ is a non-degenerate embedding. \item[(iii)] The $n$ points lie in the nonsingular locus of $C$. \item[(iv)] The multidegree of ${\mathcal{O}}_{\mathbf P(W)}(1) \otimes {\mathcal{O}}_{\mathbf P^r}(1)|_{C}$ equals the multidegree of $$\omega^f_C(fp_1+fp_2+ \ldots + fp_n)\otimes {\mathcal{O}}_{\mathbf P^r}(3f+1)|_{C}.$$ \end{enumerate} By Proposition \ref{mumford}, there exists a natural closed subscheme $J\subset U$ where the line bundles of condition (iv) above coincide. $J$ corresponds to the locus of stable maps. The natural $PGL(W)$-action on $\mathbf P(W)\times \mathbf P^r$ yields $PGL(W)$-actions on $H$, $P_i$, $I$, $U$, and $J$. To each stable map from an $n$-pointed, genus $g$ curve to $\mathbf P^r$, we have associated a $PGL(W)$-orbit in $J$. If two stable maps are associated to the same orbit, the two stable maps are isomorphic. The stability condition implies that a stable map has no infinitesimal automorphisms. It follows that the $PGL(W)$-action on $J$ has finite stabilizers. \subsection{Quotients} The moduli space of stable maps is $J/PGL(W)$. It may be possible to construct the quotient $J/PGL(W)$ via Geometric Invariant Theory. Another method will be pursued here. The quotient will be first constructed as a proper, algebraic variety by using auxiliary moduli spaces of pointed curves. Projectivity will then be established via J. Koll\'ar's semipositivity approach. \section{{\bf A rigidification of} $\barr{\mathcal{M}}_{g,n}(\proj^r,d)$} \subsection{Review of Cartier divisors} An effective Cartier divisor $D$ on a scheme $Y$ is a closed subscheme that is locally defined by a non-zero-divisor. An effective Cartier divisor determines a line bundle $\mathcal{L}={\mathcal{O}}(D)$ together with a section $s\in H^0(Y, \mathcal{L})$ locally not a zero-divisor such that $D$ is the subscheme defined by $s=0$. (As an invertible sheaf, ${\mathcal{O}}(D)$ can be constructed as the subsheaf of rational functions with at most simple poles along $D$ with $s$ equal to the function $1$, see [M2].) Conversely, if the pair $(\mathcal{L}, s)$ satisfies: \begin{enumerate} \item[(i)] $\mathcal{L}$ is line bundle on $Y$. \item[(ii)] $s\in H^0(Y,\mathcal{L})$ is a section locally not a zero divisor. \end{enumerate} then the zero scheme of $s$ is an effective Cartier divisor on $Y$. \begin{lm} \label{cart} Let the pairs $(\mathcal{L},s)$ and $(\mathcal{L}', s')$ satisfy (i) and (ii) above. If the two pairs yield the same Cartier divisor, then there exists a unique isomorphism $\mathcal{L} \rightarrow \mathcal{L}'$ taking $s$ to $s'$. \end{lm} \subsection{Definitions} We assume throughout the construction that $r>0$, $d>0$, and $(g,n,r,d)\neq (0,0,1,1)$. If $r=0$, the functor of stable maps to $\mathbf P^0$ is coarsely represented by $\overline{M}_{g,n}$. If $d=0$, the functor $\barr{\mathcal{M}}_{g,n}(\mathbf P^r,0)$ is coarsely represented by $\overline{M}_{g,n}\times \mathbf P^r$ and, $\barr{M}_{0,0}(1,1)$ is easily seen to be $\text{Spec}(\mathbb{C})$. For all other values, the construction of $\barr{M}_{g,n}(X,\beta)$ will be undertaken. Let $\mathbf P^r=\mathbf P(V)$. Then, $V^* =H^0(\mathbf P^r, {\mathcal{O}}_{\mathbf P^r}(1))$. Let $\overline{t}=(t_0, \ldots, t_r)$ span a basis of $V^*$. A {\em $\overline{t}$-rigid stable} family of degree $d$ maps from $n$-pointed, genus $g$ curves to $\mathbf P^r$ consists of the data $$(\pi:\mathcal{C} \rightarrow S,\ \{p_i\}_{1\leq i \leq n}\ ,\ \{q_{i,j}\}_{0\leq i \leq r,\ 1\leq j \leq d}\ ,\ \mu)$$ where: \begin{enumerate} \item[(i)] $(\pi:\mathcal{C} \rightarrow S,\ \{p_i\}, \mu)$ is a stable family of degree $d$ maps from $n$-pointed, genus $g$ curves to $\mathbf P^r$. \item[(ii)] $(\pi:\mathcal{C} \rightarrow S, \ \{p_i \}, \{ q_{i,j} \})$ is a flat, projective family of $n+d(r+1)$-pointed, genus $g$, Deligne-Mumford stable curves with sections $\{p_i \}$ and $\{ q_{i,j} \}$. \item[(iii)] For $0\leq i \leq r$, there is an equality of Cartier divisors $$\mu^*(t_i)= q_{i,1} + q_{i,2} + \ldots + q_{i,d}.$$ \end{enumerate} Condition (iii) implies each fibered map of the family intersects each hyperplane $(t_i)\subset \mathbf P^r$ transversally. 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There are no Deligne-Mumford stable $2$-pointed genus $0$ curves. This is why $(0,0,1,1)$ is avoided. Define a contravariant functor $\barr{\mathcal{M}}_{g,n}(\proj^r,d,\barr{t})$ from the category of complex algebraic schemes to sets as follows. Let $\barr{\mathcal{M}}_{g,n}(\proj^r,d,\barr{t}) (S)$ be the set of isomorphism classes of $\overline{t}$-rigid stable families over $S$ of degree $d$ maps from $n$-pointed, genus $g$ curves to $\mathbf P^r$. Note that the functor $\barr{\mathcal{M}}_{g,n}(\proj^r,d,\barr{t})$ depends only upon the spanning hyperplanes $(t_i)\subset \mathbf P^r$ and not upon the additional $\mathbb{C}^*$-choices in the defining equations $t_i$ of the hyperplanes. Nevertheless, it is natural for the following constructions to consider the equations of the hyperplanes $\overline{t}=(t_0, \ldots, t_r)$. \begin{pr} \label{bartg} There exists a quasi-projective coarse moduli space, $$\barr{M}_{g,n}(\proj^r,d,\barr{t}),$$ and a natural transformation of functors $$\psi: \barr{\mathcal{M}}_{g,n}(\proj^r,d,\barr{t})\rightarrow \mathcal{H}om(*, \barr{M}_{g,n}(\proj^r,d,\barr{t}))$$ satisfying the analogous conditions (I) and (II) of Theorem \ref{rep}. \end{pr} \noindent The genus 0 case is simpler. \begin{pr} \label{barto} $\barr{M}_{0,n}(\proj^r,d,\barr{t})$ represents the functor $\barr{\mathcal{M}}_{0,n}(\proj^r,d,\barr{t})$ and is a nonsingular algebraic variety. \end{pr} \subsection{Proofs} A complete proof of Proposition \ref{barto} will be given. The proof of Proposition \ref{bartg} is almost identical. Remarks indicating the differences will be made. The dependence of the coarse and fine moduli property on the genus in Propositions \ref{bartg} and \ref{barto} is a direct consequence of the fact that $\overline{M}_{g,n}$ is a coarse moduli space for $g>0$ and a fine moduli space for $g=0$. The idea behind the construction is the following. Let $m=n+d(r+1)$. The data of the $\overline{t}$-rigid stable family immediately yields a morphism of the base $S$ to $\overline{M}_{g,m}$. In fact, the image of $S$ lies in a universal, locally closed subscheme of $\overline{M}_{g,m}$. This subscheme is denoted by $B$. The first step of the construction is to identify $B$. The morphism $S\rightarrow B$ does not contain all the data of the $\overline{t}$-rigid stable family. Consider the case in which the base $S$ is a point. The corresponding point in $B$ records the domain curve $C$, the marked points $\{p_i\}$, and the pull-back divisors under $\mu$ of the hyperplanes in $\mathbf P^r$ determined by $\overline{t}$. The map $\mu$ is determined by the pull-back divisors up to the diagonal torus action on $\mathbf P^r$. The torus information is recorded in the total space of $r$ tautological $\mathbb{C}^*$-bundles over $B$. The $\overline{t}$-rigid moduli space is expressed as the total space of these $r$ distinct $\mathbb{C}^*$-bundles over $B$. To canonically construct the universal family over the $\overline{t}$-rigid moduli space, the equations $t_i$ of the hyperplanes are needed. This is why the equations $t_i$ (rather than the spanning hyperplanes $(t_i)$) are explicitly chosen. Proposition \ref{barto} is proved by an explicit construction of $\barr{M}_{0,n}(\proj^r,d,\barr{t})$ together with a universal family of $\overline{t}$-rigid stable maps. Let $\barr{M}_{0,m}$ be the Mumford-Knudsen compactification of the moduli space of $m$-pointed, genus 0 curves. Let $\pi: \mku \rightarrow \barr{M}_{0,m}$ be the universal curve with $m$ sections $\{p_i\}_{1\leq i \leq n}$ and $\{q_{i,j} \} _{0 \leq i \leq r, \ 1 \leq j \leq d}$. Since $\mku$ is nonsingular and the sections are of codimension $1$, there are canonically defined line bundles: $$\mathcal{H}_i = {\mathcal{O}}_{\mku}(q_{i,1} + q_{i,2} + \ldots + q_{i,d}),$$ for $0 \leq i \leq r$. Let $s_i\in H^0(\mku, \mathcal{H}_i)$ be the canonical section representing the Cartier divisor $(q_{i,1}+ q_{i,2}+ \ldots + q_{i,d})$. For any morphism $\gamma:X\rightarrow \barr{M}_{0,m}$, consider the fiber product: \begin{equation*} \begin{CD} X\times_{\barr{M}_{0,m}} \mku @>{\overline{\gamma}}>> \mku \\ @VV{\pi_X}V @VV{\pi}V \\ X @>{\gamma}>> \barr{M}_{0,m} \\ \end{CD} \end{equation*} We call the morphism $\gamma:X \rightarrow \barr{M}_{0,m}$ {\em $\mathcal{H}$-balanced} if \begin{enumerate} \item[(a)] For $1\leq i \leq r$, $\pi_{X*} \overline{\gamma}^*(\mathcal{H}_i \otimes \mathcal{H}_0^{-1})$ is locally free. \item[(b)] For $1\leq i \leq r$, the canonical map $$\pi_X^* \pi_{X*} \overline{\gamma}^*(\mathcal{H}_i \otimes \mathcal{H}_0^{-1}) \rightarrow \overline{\gamma}^*(\mathcal{H}_i \otimes \mathcal{H}_0^{-1})$$ is an isomorphism. \end{enumerate} If $\gamma$ is $\mathcal{H}$-balanced, the line bundles $\overline{\gamma}^{*}(\mathcal{H}_i)$ are isomorphic on the fibers of $\pi_X$. Let $B\subset \barr{M}_{0,m}$ be the universal, locally closed subscheme satisfying the two following properties: \begin{enumerate} \item[(i)] The inclusion $\iota: B \hookrightarrow \barr{M}_{0,m}$ is $\mathcal{H}$-balanced. \item[(ii)] Every $\mathcal{H}$-balanced morphism $\gamma: X \rightarrow \barr{M}_{0,m}$ factors (uniquely) through $B$. \end{enumerate} By Proposition \ref{mumford}, $B$ exists. In fact, $B\subset \barr{M}_{0,m}$ is a Zariski open subscheme. In the $g>0$ case, the above constructions exist over the stacks $\overline{M}_{g,m}$ and $\overline{U}_{g,m}$. $B_{g,m}$ is a locally closed substack of $\overline{M}_{g,m}$ of positive codimension. Let $\mathcal{G}_i=\pi_{B*} \overline{\iota}^*(\mathcal{H}_i\otimes \mathcal{H}_0^{-1})$ for $1\leq i \leq r$. Let $\tau_i: Y_i \rightarrow B$ be the total space of the canonical $\mathbb{C}^*$-bundle associated to $\mathcal{G}_i$. $Y_i$ is the affine bundle associated to $\mathcal{G}_i$ minus the zero section. The pull-back $\tau_i^*(\mathcal{G}_i)$ has a tautological non-vanishing section and hence is canonically trivial. Consider the product $$Y=Y_1 \times_B \times Y_2 \times_B \ldots \times_B Y_r$$ equipped with projections $\rho_i: Y\rightarrow Y_i$ and a morphism $\tau: Y \rightarrow B$. Form the cartesian square: \begin{equation*} \begin{CD} \mathcal{U} @>{\overline{\tau}}>> \mku \\ @VV{\pi_Y}V @VV{\pi}V \\ Y @>{\tau}>> B\subset \barr{M}_{0,m}. \\ \end{CD} \end{equation*} The line bundles $\overline{\tau}^*(\mathcal{H}_i)$ for $1\leq i \leq r$ are canonically isomorphic to $\mathcal{L}=\overline{\tau}^*(\mathcal{H}_0)$ on $\mathcal{U}$ since $$\overline{\tau}^*(\mathcal{H}_i\otimes \mathcal{H}_0^{-1}) \stackrel {\sim}{=} \pi_Y^*\rho_i^* \tau_i^*(\mathcal{G}_i)$$ and $\tau_i^*(\mathcal{G}_i)$ is canonically trivial. Via pull-back and the canonical isomorphisms, $\overline{\tau}^*(s_i)$ canonically corresponds to a section of $\mathcal{L}$. Since these $r+1$ sections do not vanish simultaneously, they define a morphism of $\mu:\mathcal{U} \rightarrow \mathbf P^r$. The canonical method of obtaining $\mu$ is as follows. Define a vector space map $V^* \rightarrow H^0(\mathcal{L})$ by sending $t_i$ to $\overline{\tau}^*(s_i)$. The induced surjection $V^*\otimes {\mathcal{O}} \rightarrow \mathcal{L}$ canonically yields a morphism $$\mu: \mathcal{U}\rightarrow \mathbf P^r.$$ Note that the equations $t_i$ are used to define the morphism $\mu$. The sections $\{p_i \}$, $\{q_{i,j} \}$ pull back to sections of $\pi_Y$. We claim that the family \begin{equation} \label{uufam} (\pi_Y:\mathcal{U} \rightarrow Y, \{p_i\}, \{q_{i,j}\}, \mu) \end{equation} is a universal family of $\overline{t}$-rigid stable maps, so $\barr{M}_{0,n}(\proj^r,d,\barr{t})=Y$. The stability of the family of maps \begin{equation} \label{ufam} (\pi_Y: \mathcal{U} \rightarrow Y, \{p_i\}, \mu) \end{equation} is straightforward. Each fiber $C$ of $\pi_Y$ is an $m$-pointed, genus $0$ stable curve with markings $\{p_i\}$ and $\{q_{i,j}\}$. Let $E\subset C$ be an irreducible component. Suppose $\text{dim}(\mu(E))=0$. By the transversality condition (iii), $E$ has no markings from the sections $\{q_{i,j}\}$. Since $C$ is a stable $m$-pointed curve and no $\{q_{i,j}\}$ markings lie on $E$, $\text{deg}_E(\omega_C(p_1+\ldots +p_n))>0$. Hence, condition (1) in the definition of map stability (section 1.1) holds for $E$. Therefore (\ref{ufam}) is a stable family of maps. By construction, it is a $\overline{t}$-rigid stable family. Finally, it must be shown (\ref{uufam}) is universal. Let \begin{equation} \label{testf} (\pi: \mathcal{C} \rightarrow S, \{p_i \}, \{ q_{i,j} \}, \nu) \end{equation} be a family of $\overline{t}$-rigid stable maps. Since $(\pi:\mathcal{C} \rightarrow S, \{p_i \}, \{q_{i,j}\})$ is a flat family of $m$-pointed, genus $0$ stable curves, there is an induced map $\lambda : S \rightarrow \overline{M}_{0,m}$ such that the pull-back family $S\times_{\barr{M}_{0,m}} \mku$ is canonically isomorphic to $(\pi:\mathcal{C} \rightarrow S, \{p_i\}, \{q_{i,j} \})$. First we show $\lambda$ is $\mathcal{H}_i$-balanced. The pair $(\overline{\lambda}^*(\mathcal{H}_i), \overline{\lambda}^*(s_i))$ yields the Cartier divisor $q_{i,1} + \ldots + q_{i,d}$ on $\mathcal{C}$. The map $\nu$ is induced by a vector space homomorphism $\psi:V^* \rightarrow H^0(\mathcal{C}, \nu^*({\mathcal{O}}_{\mathbf P(V)}(1)))$. Let $z_i= \psi(t_i)$. By condition (iii) of $\overline{t}$-rigid stability, the pair $(\nu^*({\mathcal{O}}_{\mathbf P(V)}(1)), z_i)$ yields the Cartier divisor $q_{i,1} + \ldots + q_{i,d}$ on $\mathcal{C}$. By Lemma \ref{cart}, there are {\em canonical} isomorphisms \begin{equation} \label{caniso} \overline{\lambda}^*(\mathcal{H}_i) \stackrel {\sim}{=} \nu^*({\mathcal{O}}_{\mathbf P(V)}(1)) \end{equation} for all $i$. Hence $\lambda$ is $\mathcal{H}_i$-balanced. By the universal property of $B$, $\lambda$ factors through $B$: $\lambda: S \rightarrow B$. There are canonical isomorphisms \begin{equation} \label{can2} \pi_*(\overline{\lambda}^*(\mathcal{H}_i \otimes \mathcal{H}_0^{-1})) \stackrel {\sim}{=} \lambda^*(\mathcal{G}_i). \end{equation} The canonical isomorphisms (\ref{caniso}) yield canonical sections of $\overline{\lambda}^*(\mathcal{H}_i \otimes \mathcal{H}_0^{-1})$. The canonical isomorphisms (\ref{can2}) then yield nowhere vanishing sections of $\lambda^*(\mathcal{G}_i)$ over $S$. Hence there is a canonical a map $S \rightarrow Y$. It is easily checked the pull-back of the universal family over $Y$ yields a $\overline{t}$-rigid stable family of maps canonically isomorphic to (\ref{testf}). \section{{\bf The construction of} $\barr{M}_{g,n}(\proj^r,d)$} \subsection{Gluing} \label{glu} While a given pointed stable map $\mu: C \rightarrow \mathbf P^r$ may not be rigid for a given basis $\overline{t}$ of $V^*=H^0(\mathbf P^r,{\mathcal{O}}_{\mathbf P^r}(1))$, the map will be rigid (by Bertini's theorem) for some choice of basis. The moduli space $\barr{M}_{g,n}(\proj^r,d)$ is obtained by gluing together quotients of $\barr{M}_{g,n}(\proj^r,d,\barr{t})$ for different choices of bases $\overline{t}$. For notational convenience, set $\barr{M}(\overline{t})= \barr{M}_{g,n}(\proj^r,d,\barr{t})$. We write $(\pi: \mathcal{U} \rightarrow \barr{M}(\overline{t}), \{p_i\}, \{q_{i,j}\}, \mu)$ for the universal family of $\overline{t}$-rigid stable maps in the genus 0 case. If $g>0$, more care is required. Let $\mathfrak{S}_d$ denote the symmetric group on $d$ letters. The group $$G=G_{d,r} = \mathfrak{S}_d \times \ldots \times \mathfrak{S}_d \ \ \ (r+1 \ \text{factors})$$ has a natural action on $\barr{M}(\overline{t})$ obtained by permuting the ordering in each of the $r+1$ sets of sections $\{q_{i,1}, \ldots, q_{i,d}\}$, $0 \leq i \leq r$. For any $\sigma \in G$, the family \begin{equation} \label{permm} (\pi: \mathcal{U} \rightarrow \overline{M}(\overline{t}), \{p_i\}, \{q_{i, \sigma(j)}\}, \mu) \end{equation} is also a $\overline{t}$-rigid family over $\barr{M}(\overline{t})$. By the universal property, the permuted family (\ref{permm}) induces an automorphism of $\barr{M}(\overline{t})$. Since $\barr{M}(\overline{t})$ is quasi-projective and $G$ is finite, there is a quasi-projective quotient scheme $\barr{M}(\overline{t})/G$. Let $\overline{t}$ and $\overline{t}'$ be distinct choices of bases of $V^*$. Let $\mu: \mathcal{U} \rightarrow \mathbf P^r$ be the universal family over $\barr{M}(\overline{t})$. Let $$\barr{M}(\overline{t}, \overline{t}') \subset \barr{M}(\overline{t})$$ denote the open locus over which the divisors $\mu^*(t'_0), \ldots, \mu^*(t'_r)$ are \'etale, disjoint, and disjoint from the sections $\{p_i\}$. The open set $\barr{M}(\overline{t}, \overline{t}')$ is certainly $G$-invariant. Let $\barr{M}(\overline{t}, \overline{t}')/G$ denote the quasi-projective quotient. \begin{pr} \label{bsugg} There is a canonical isomorphism $$\barr{M}(\overline{t}, \overline{t}')/G \stackrel {\sim}{=} \barr{M}(\overline{t}', \overline{t})/G.$$ \end{pr} \noindent {\em Proof.} The divisors $\mu^*(t'_i)$ define an \'etale Galois cover $\mathcal{E}$ of $\barr{M}(\overline{t}, \overline{t}')$ with Galois group $G$ over which a $\overline{t}'$-rigid stable family is defined. The fiber of $\mathcal{E}$ over $(C, \{p_i\}, \{q_{i,j}\}, \mu)$ is the set of orderings $\{q'_{i,j}\}$ of the points mapped by $\mu$ to the hyperplane $(t_i'=0)$. Therefore there is a map \begin{equation} \label{patcher} \mathcal{E} \rightarrow \barr{M}(\overline{t}') \end{equation} which is easily seen be $G$-equivariant for the Galois $G$-action on $\mathcal{E}$ and the $\{q'_{i,j}\}$-permutation $G$-action on the $\barr{M}(\overline{t}')$. Moreover (\ref{patcher}) factors through $\barr{M}(\overline{t}', \overline{t})$. Hence there exists a map of quotients \begin{equation} \label{patch2} \barr{M}(\overline{t}, \overline{t}') \stackrel {\sim}{=} \mathcal{E}/ {\text{Galois}} \rightarrow \barr{M}(\overline{t}', \overline{t})/G. \end{equation} The map (\ref{patch2}) is $G$-invariant for the $\{q_{i,j}\}$-permutation action on $\overline{M}(\overline{t}, \overline{t}')$. Therefore (\ref{patch2}) descends to $\barr{M}(\overline{t}, \overline{t}')/G \rightarrow \barr{M}(\overline{t}', \overline{t})/G$. The inverse is obtained by interchanging $\overline{t}$ and $\overline{t}'$ in the above construction. In fact, there is a natural action of $G \times G$ on $\mathcal{E}$ and canonical isomorphisms $\barr{M}(\overline{t}, \overline{t}')/G \stackrel {\sim}{=} \mathcal{E}/(G\times G) \stackrel {\sim}{=} \barr{M}(\overline{t}', \overline{t})/G$. \qed \vspace{+10pt} In case $g>0$, the coarse moduli spaces $\barr{M}_{g,n}(\proj^r,d,\barr{t})$ do not (in general) have universal families. The permutation action of $G$ can be defined on a Hilbert scheme or a stack and then descended to $\barr{M}_{g,n}(\proj^r,d,\barr{t})$. The open sets $\barr{M}(\overline{t}', \overline{t})$ and $\barr{M}(\overline{t}, \overline{t}')$ are well defined for $g>0$ and still satisfy Proposition \ref{bsugg}. The cocycle conditions on triple intersections are easily established. Hence, the schemes $\barr{M}(\overline{t})/G$ canonically patch together along the open sets $\barr{M}(\overline{t}, \overline{t}')/G$ to form the scheme $\barr{M}_{g,n}(\proj^r,d)$. The results on boundedness show $\barr{M}_{g,n}(\proj^r,d)$ is covered by a finite number of these open sets $\barr{M}(\overline{t})/G$. Hence, $\barr{M}_{g,n}(\proj^r,d)$ is an algebraic scheme of finite type over $\mathbb{C}$. The universal properties of $\barr{M}_{g,n}(\proj^r,d)$ are easily obtained from the universal properties of the moduli spaces of $\overline{t}$-rigid stable maps. \subsection{Separation and completeness} Let $(X,x)$ be a nonsingular, pointed curve. Let $\iota: X\mathbin{\text{\scriptsize$\setminus$}} \{x\}=U\ \hookrightarrow X$. Let \begin{equation} \label{first} (\pi:\mathcal{C} \rightarrow X, \ \{p_i \}, \ \mu) \end{equation} \begin{equation} \label{second} (\pi': \mathcal{C}' \rightarrow X, \ \{p_i'\}, \ \mu') \end{equation} be two families over $X$ of stable maps to $\mathbf P^r=\mathbf P(V)$. \begin{pr} \label{sep} An isomorphism between the families (\ref{first}) and (\ref{second}) over $U$ extends to an isomorphism over $X$. \end{pr} \noindent {\em Proof.} Choose a basis $\overline{t}=(t_0, \ldots, t_r)$ of $V^*$ that intersects the maps $\mu:\mathcal{C}_x \rightarrow \mathbf P^r$ and $\mu':\mathcal{C}'_x \rightarrow \mathbf P^r$ transversally at unmarked, nonsingular points. Since it suffices to prove the isomorphism extends over a local \'etale cover of $(X,x)$, it can be assumed that the Cartier divisors $\mu^*(t_i)$ and ${\mu'}^*(t_i)$ split into sections $\{q_{i,j}\}$ and $\{q'_{i,j}\}$ of $\pi$ and $\pi'$. Then $\mathcal{C}$, $\mathcal{C}'$ are Deligne-Mumford stable $m=n+d(r+1)$ pointed curves. Therefore, by the separation property of the functor of Deligne-Mumford stable pointed curves, there exists an isomorphism (of pointed curves) $\tau: \mathcal{C} \rightarrow \mathcal{C}'$ over $X$. Since $\tau\circ \mu'$ and $\mu$ agree on an open set, $\tau\circ \mu' = \mu$. \qed \vspace{+10pt} \noindent Proposition \ref{sep} and the valuative criterion show $\barr{M}_{g,n}(X, \beta)$ is a separated algebraic scheme. Properness is also established by the valuative criterion. To complete $1$ dimensional families of stable maps, semi-stable reduction techniques for curves are used (as in [K-K-M] and [Ha]). \begin{pr} \label{prop} Let $\mathcal{F}=(\pi:\mathcal{C}\rightarrow U, \ \{p_i\}, \ \mu)$ be a family of stable maps to $\mathbf P^r$. There exists a base change $\gamma:(Y,y) \rightarrow (X,x)$ satisfying: \begin{enumerate} \item[(i)] $\gamma_W: Y\mathbin{\text{\scriptsize$\setminus$}} \{y\}= W \rightarrow U$ is \'etale. \item[(ii)] The pull-back family $\gamma_W^*(\mathcal{F})$ extends to a stable family over $(Y,y)$. \end{enumerate} \end{pr} \noindent {\em Proof.} First, after restriction to a Zariski open subset of $U$, it can be assumed that the fibers $\mathcal{C}_{\xi}$ all have the same number of irreducible components. There may be non-trivial monodromy around the point $x \in X$ in the set of irreducible components of the fibers $\mathcal{C}_{\xi}$. After a base change (possibly ramified at $x$), this monodromy can be made trivial. It can therefore be assumed that $\mathcal{F}$ is a union of stable families $\mathcal{F}_j = (\pi_j:\mathcal{C}_j \rightarrow U, \ \{p^j_i\}, \{p^c_i\}, \ \mu_j )$ where $\pi_j$ is family of {\em irreducible}, nodal, projective curves. The markings $\{p^j_i\}$ are the markings of $\mathcal{C}$ that lie on $\mathcal{C}_j$. The marking $\{p^c_j\}$ correspond to intersections of components in $\mathcal{F}$. It suffices to prove Proposition \ref{prop} separately for each stable family $\mathcal{F}_j$. For technical reasons, it is convenient to consider families of nonsingular curves. After restriction, normalization, and base change of $\mathcal{F}_j$, a family \begin{equation} \label{famm} \tilde{\mathcal{F}}_j = (\tilde{\pi}_j:\tilde{\mathcal{C}}_j \rightarrow U, \{p^j_i\}, \{p^c_i\}, \{p^n_i\}, \tilde{\mu}_j ) \end{equation} can be obtained where $\tilde{\mathcal{F}}_j$ is a family of stable maps of irreducible, {\em nonsingular}, projective curves. The additional markings $\{p^n_i\}$ correspond to the nodes. Consider the nodal locus in $\mathcal{F}_j$. This locus consists of curves and isolated points. Via restriction of $U$ to a Zariski open set, it can be assumed the nodal locus (if non-empty) is of pure dimension $1$. A normalization now separates the sheets along the nodal locus. A base change then may be required to make the separated points $\{p^n_i\}$ sections. If the normalized family $\tilde{\mathcal{F}}_j$ is completed, $\mathcal{F}_j$ can be completed by identifying the nodal markings on $\tilde{\mathcal{F}}_j$. This nodal identification commutes with the map to $\mathbf P^r$. It therefore suffices to prove Proposition \ref{prop} for these normalized families (\ref{famm}). By the above reductions, it suffices to prove Proposition {\ref{prop}} for a family of stable maps of irreducible, nonsingular, projective curves. Let \begin{equation} \label{fammm} (\pi: \mathcal{C} \rightarrow U, \{p_i\}, \mu ) \end{equation} be such a family. Let $\pi:\mathcal{E} \rightarrow X$ be a flat extension of $\pi:\mathcal{C} \rightarrow U$ over the point $x\in X$. After blow-ups in the special fiber of $\mathcal{E}$, it can be assumed the map $\mu: \mathcal{C} \rightarrow \mathbf P^r$ extends to $\mu: \mathcal{E} \rightarrow \mathbf P^r$. By Lemma \ref{ssred} below applied to the flat extension $\pi:\mathcal{E} \rightarrow X$, there exists a base change $\gamma:(Y,y) \rightarrow (X,x)$ and a family of pointed curves $$\pi_Y: \mathcal{C}_Y \rightarrow (Y,y)$$ satisfying conditions (i)$-$(iii) of Lemma \ref{ssred}. Via $\tau : \mathcal{C}_{Y} \rightarrow \mathcal{E}$, ${\mu}$ naturally induces a map $${\mu}_{Y}: {\mathcal{C}}_{Y} \rightarrow \mathbf P^r.$$ The family $({\pi}_{Y}:{\mathcal{C}}_{Y} \rightarrow (Y,y), \{p_i\}, {\mu}_{Y} )$ is certainly an extension of the family over $Y \mathbin{\text{\scriptsize$\setminus$}} \{y\}$ determined by the $\gamma$ pull-back of the stable family (\ref{fammm}). The special fiber is a map of a pointed quasi-stable curve to $\mathbf P^r$. Unfortunately, the special fiber may not be stable. A stable family of maps is produced in two steps. First, unmarked, ${\mu}_{Y}$-collapsed, $-1$-curves in the special fiber are sequentially blow-down. A multiple of the line bundle \begin{equation} \label{lbdle} \omega_{{\pi}_{Y}}( \sum_i p_i) \otimes {\mu}_{Y}^*({\mathcal{O}}_{\mathbf P^r}(3)) \end{equation} is then ${\pi}_{Y}$- relatively basepoint free. Second, as in [Kn], the relative morphism determined by a power of the line bundle (\ref{lbdle}) blows-down the remaining destabilizing $\mathbf P^1$'s to yield a stable extension over $(Y,y)$. \qed \vspace{+10pt} \begin{lm} \label{ssred} Let $\pi_X: \mathcal{S}_X \rightarrow (X,x)$ be a flat, projective family of curves with $l$ sections $s_1, \ldots, s_l$ satisfying the following condition: $\forall \xi \neq x $, $\pi^{-1}(\xi)=\mathcal{C}_{\xi}$ is a projective nonsingular curve with $l$ distinct marked points $s_1(\xi), \ldots, s_l(\xi)$. There exists a base change $\gamma: (Y,y) \rightarrow (X,x)$ \'etale except possibly at $y$ with a family of $l$-pointed curves $\pi_Y: \mathcal{S}_Y \rightarrow (Y,y)$ and a diagram: \begin{equation*} \begin{CD} \mathcal{S}_Y @>{\tau}>> \mathcal{S}_X\\ @VV{\pi_Y}V @VV{\pi_X}V \\ (Y,y) @>{\gamma}>> (X,x) \\ \end{CD} \end{equation*} satisfying the following properties: \begin{enumerate} \item[(i)] $\mathcal{S}_Y$ is a nonsingular surface. $\pi_Y:\mathcal{S}_Y \rightarrow (Y,y)$ is a flat, projective family of $l$-pointed quasi-stable curves. \item[(ii)] For each marking $1\leq i \leq l$, $\tau\circ s_i= s_i\circ \gamma$. \item[(iii)] Over $W=Y\mathbin{\text{\scriptsize$\setminus$}} \{y\}$, there is isomorphism $\mathcal{S}_W \stackrel{\sim}{\rightarrow} \gamma_W^*(\mathcal{S}_U)$, where $U= X \mathbin{\text{\scriptsize$\setminus$}} \{x\}$. The morphism $\tau|_{\mathcal{S}_W}$ is the composition $$\mathcal{S}_W \stackrel{\sim}{\rightarrow} \gamma_{W}^*(\mathcal{S}_U) \rightarrow \mathcal{S}_U$$ where the second map is the natural projection. \end{enumerate} \end{lm} \noindent {\em Proof.} The method is by standard semi-stable reduction (cf. [K-K-M], [Ha]). First, the singularities of $\mathcal{S}_X$ are resolved. Note that all singularities lie in the special fiber. Next, the surface $\mathcal{S}_X$ is blown-up sufficiently to ensure the reduced scheme supported on the special fiber has normal crossing singularities in $\mathcal{S}_X$. The required blow-ups have point centers in the special fiber. Finally, the resulting surface is blown-up further (at points in the special fiber) to ensure the marking sections $s_1, \ldots, s_l$ do not intersect each other and do not pass through nodes of the reduced scheme supported on the special fiber. Let $\hat{\pi}: \hat{\mathcal{S}}_X \rightarrow (X,x)$ be the resulting nonsingular surface. The singularities of the morphism $\hat{\pi}$ are locally of the form $z_1^\alpha z_2^\beta = t$ where $z_1$, $z_2$ are coordinates on $\hat{\mathcal{S}}_X$ and $t$ is a coordinate on $X$. Let $\{\alpha_j, \beta_j\}$ be the set of exponents that occur at the singularities of $\hat{\pi}$. Let $\gamma: (Y,y) \rightarrow (X,x)$ be a base change whose ramification index over $x$ is divisible by all $\alpha_j$ and $\beta_j$. Let $\mathcal{S}_Y$ be the normalization of $\gamma^*(\hat{\mathcal{S}}_X)$. A straightforward local analysis shows the family $\pi_Y: \mathcal{S}_Y\rightarrow (Y,y)$ has an $l$-pointed, reduced, nodal special fiber. The surface $\mathcal{S}_Y$ has singularities of the local form $z_1z_2-t^k$ in the special fiber. Blowing-up $\mathcal{S}_Y$ yields a nonsingular surface with the required properties (i)$-$(iii). \qed \vspace{+10pt} By the valuative criterion, Propositions \ref{sep} and \ref{prop} prove $\barr{M}_{g,n}(\proj^r,d)$ is a separated and proper complex algebraic scheme. \subsection{Projectivity} The projectivity of the proper schemes $\barr{M}_{g,n}(\proj^r,d)$ is established here by a method due to J. Koll\'ar ([Ko1]). Proofs of the projectivity of $\barr{M}_{g,n}(\mathbf P^r, \beta)$ can also be found in [A] and [C]. Koll\'ar constructs ample line bundles on proper spaces via sufficiently nontrivial quotients of semipositive vector bundles. A vector bundle $E$ on an algebraic scheme $S$ is {\em semipositive} if for every morphism of a projective curve $f: C \rightarrow S$, every quotient line bundle of $f^*(E)$ has nonnegative degree on $C$. The first step is a semipositivity lemma. Let \begin{equation} \label{ifam} \mathcal{F}=(\pi: \mathcal{C} \rightarrow S, \ \{p_i\}, \ \mu) \end{equation} be a stable family of maps over $S$ to $\mathbf P^r$. Let $$E_k(\pi)=\pi_*\bigg(\omega^k_{\pi} (\sum_{i=1}^{n} k p_i)\otimes \mu^*({\mathcal{O}}(3k)) \bigg).$$ \begin{lm} \label{semip} $E_k(\pi)$ is a semipositive vector bundle on $S$ for $k\geq 2$. \end{lm} \noindent {\em Proof.} A slight perturbation of the arguments in [Ko1] is required. It suffices to prove semipositivity in case the base is a nonsingular curve $X$. Let $\gamma:Y \rightarrow X$ be a flat base change. By map stability, Serre duality, and the base change theorems, it follows (for $k\geq 2$) $E_k$ commutes with pull-back: $$ E_k(\pi_Y)\stackrel {\sim}{=} \gamma^*(E_k(\pi_X)) $$ where $\pi_Y$ is the pull-back family over $Y$. It therefore suffices to prove semipositivity after base change. Using the methods of section 4.2, it can be assumed (after base change) that $\mathcal{F}$ is a union of component stable families $\mathcal{F}_j = (\pi_j:\mathcal{C}_j \rightarrow X, \ \{p^j_i\}, \{p^c_i\}, \ \mu_j )$ where $\pi_j$ is family of stable maps and the generic element of $\mathcal{F}_j$ is a map of an {\em irreducible}, projective, nodal curve. The notation introduced in the proof of Proposition \ref{prop} is employed. After further base change and normalization of $\mathcal{F}_j$, it can be assumed that \begin{equation} \label{famm2} \tilde{\mathcal{F}}_j = (\tilde{\pi}_j:\tilde{\mathcal{C}}_j \rightarrow X, \ \{p^j_i\}, \{p^c_i\}, \{p^n_i\}, \ \tilde{\mu}_j ) \end{equation} is a family of stable maps where the generic element is a map of an irreducible, projective, {\em nonsingular} curve. A semipositivity result for the family $\tilde{\mathcal{F}}_j$ is first established. Let $H_1$, $H_2$, $H_3 \subset \mathbf P^r$ be general hyperplanes. After base change, it can be assumed $\tilde{\mu}_j^*(H_l)$ is a union of $d$ reduced sections for each $l$. These $3d$ sections are distinct from the sections $\{p^j_i\}$, $\{p^c_i\}$, $\{p^n_i\}$. Therefore, \begin{equation} \label{mess} \omega^k_{\tilde{\pi}_j} \big( \sum k p^j_i+ \sum (k-1) p^c_i + \sum (k-1)p^n_i \big)\otimes \tilde{\mu}_j^*({\mathcal{O}}_{\mathbf P^r}(3k)) \stackrel{\sim}{\rightarrow} \end{equation} $$\omega^k_{\tilde{\pi}_j} (\sum \alpha_q X_q)$$ where $X_q$ are distinct sections of $\tilde{\pi}_j$ and $\alpha_q\leq k$. The surface $\tilde{\mathcal{C}}_j$ has finitely many singularities of the form $z_1z_2-t^{\alpha}$. These singularities are resolved by blow-up, $$\tau: \mathcal{S}_j \rightarrow \tilde {\mathcal{C}_j}.$$ Since the relative dualizing sheaf of the family $\mathcal{S}_j$ is trivial on the exceptional $\mathbf P^1$'s of $\tau$, Lemma \ref{kol} below can be applied to deduce the semipositivity of $F_k(\tilde{\pi}_j)$ for $k\geq 2$ where $$F_k(\tilde{\pi}_j)=\tilde{\pi}_{j*}\bigg(\omega^k_{\tilde{\pi}_j} \big(\sum kp^j_i+\sum (k-1)p^c_i + \sum (k-1)p^n_i\big) \otimes \mu^*({\mathcal{O}}(3k)) \bigg).$$ For $k\geq 2$, the restriction of the line bundle (\ref{mess}) to a fiber of $\tilde{\mathcal{C}}_j$ is equal to \begin{equation} \label{prod} \omega \otimes \omega^{k-1} \big(\sum (k-1)p^j_i+\sum (k-1)p^c_i + \sum (k-1)p^n_i\big)\otimes \end{equation} $$\tilde{\mu}_j^*({\mathcal{O}}_{\mathbf P^r}(3k-3)) \otimes \mu_j^*({\mathcal{O}}_{\mathbf P^r}(3))(\sum_i p^j_i)$$ where $\omega$ is the dualizing sheaf of the fiber. By stability for the family $\tilde{\mathcal{F}}_j$, the product of the middle two factors in (\ref{prod}) is ample for $k\geq 2$. The last factor in (\ref{prod}) is certainly of non-negative degree. By Serre duality, for $k\geq 2$, \begin{equation} \label{van} R^1{\tilde{\pi}}_{j*} \bigg(\omega^k_{\tilde{\pi}_j} \big(\sum kp^j_i+\sum (k-1)p^c_i + \sum (k-1)p^n_i\big) \otimes \mu^*({\mathcal{O}}(3k)) \bigg) =0. \end{equation} The semipositivity of $E_k(\pi)$ will be obtained from the semipositivity of $F_k(\tilde{\pi}_j)$. The $(k-1)$-multiplicities will naturally arise in considering dualizing sheaves on nodal and reducible curves. Let $\tilde{{\pi}}_{\cup_j}: \bigcup_{j} \tilde{\mathcal{C}}_j \rightarrow X$ be the disjoint union of the families $\tilde{\mathcal{F}}_j$. There is natural morphism from the disjoint union to $\mathcal{C}$ $$\rho: \bigcup_{j} \tilde{\mathcal{C}}_j \rightarrow \mathcal{C}$$ obtained by identifying nodal marked points and gluing components along intersection marked points. Consider the natural sequence of sheaves on $\mathcal{C}$: \begin{equation} \label{seqq} 0 \rightarrow \rho_*(\omega_{\tilde{\pi}_{\cup_j}}) \rightarrow \omega_{\pi} \rightarrow K \rightarrow 0. \end{equation} The quotient $K$ is easily identified as $\bigoplus_{p^c_i, p^n_i} {\mathcal{O}}_p$ where the sum is over all nodal and component intersection sections of the family $\mathcal{F}$. Tensoring (\ref{seqq}) with the line bundle $\omega_{\pi}^{k-1}(\sum kp_i) \otimes \mu^*({\mathcal{O}}_{\mathbf P(V)}(3k))$ yields the exact sequence: $$0 \rightarrow \rho_*\bigg(\omega_{\tilde{\pi}_{\cup_j}}^k (\sum kp_i+\sum (k-1)p^c_i + \sum (k-1)p^n_i) \otimes \mu^*({\mathcal{O}}_{\mathbf P(V)}(3k)) \bigg) \rightarrow $$ $$ \omega^k_{\pi}(\sum_i kp_i) \otimes \mu^*({\mathcal{O}}_{\mathbf P(V)}(3k)) \rightarrow \bigoplus_{p^c_i, p^n_i} {\mathcal{O}}_p\otimes \mu^*({\mathcal{O}}_{\mathbf P(V)}(3k)) \rightarrow 0.$$ Certainly $\tilde{\pi}_{\cup_{j}*}= \pi_*\rho_*$. Note the vanishing of $R^1$ determined in (\ref{van}). These facts imply the $\pi$ direct image of the above sequence on $\mathcal{C}$ yields an exact sequence on $X$: $$0 \rightarrow \bigoplus_j F_k(\tilde{\pi}_j) \rightarrow E_k(\pi) \rightarrow \bigoplus_{p^c_i, p^n_i} {\mathcal{O}}_X \otimes \mu^*({\mathcal{O}}_{\mathbf P(V)}(3k)) \rightarrow 0.$$ Finally, since an extension of semipositive bundles is semipositive ([Ko1]), $E_k(\pi)$ is semipositive. \qed \vspace{+10pt} \begin{lm} \label{kol} Let $\pi:\mathcal{S}\rightarrow X$ be a map from a nonsingular projective surface to a nonsingular curve. Assume the general fiber of $\pi$ is nonsingular. Let $X_q$ be a set of distinct sections of $\pi$. Then $$\pi_*(\omega_{\mathcal{S}/X}^k(\sum \alpha_q X_q))$$ is semipositive provided $k\geq 2$ and $\alpha_q\leq k$ for all $q$. \end{lm} \noindent {\em Proof.} This is precisely Proposition 4.7 of [Ko1]. \qed \vspace{+10pt} The second step is the construction of a non-trivial quotient. Let $\mathcal{F}$ be the family (\ref{ifam}). Let $\mathbf P(E_k^*)$ be the projective bundle over $S$ obtained from the subspace projectivization of $E_k^*$. The condition of stability implies there is a canonical $S$-embedding $\iota:\mathcal{C} \rightarrow \mathbf P(E_f^*)$ for some $f=f(d,g,n,r)$ (see section \ref{bound}). The morphism $\mu$ then yields a canonical $S$-embedding: $$\gamma: \mathcal{C} \rightarrow \mathbf P(E_f^*) \times_{\mathbb{C}} \mathbf P^r.$$ The $n$ sections $\{p_i\}$ yield $n$ sections $\{(\iota \circ p_i, \mu \circ p_i) \}$ of $\mathbf P(E_f^*)\times \mathbf P^r$ over $S$. Let $S_i$ denote the subscheme of $\mathbf P(E_f^*)\times \mathbf P^r$ defined by the $i^{th}$ section. Denote the projection of $\mathbf P(E_f^*)\times \mathbf P^r$ to $S$ also by $\pi$. Let $\mathcal{M}={\mathcal{O}}_{\mathbf P(E_f^*)}(1) \otimes {\mathcal{O}}_{\mathbf P^r}(1)$. $\mathcal{M}$ is an $\pi$-relatively ample line bundle. Note $\pi_*(\mathcal{M}^l) \stackrel {\sim}{=} \text{Sym}^l(E_f) \otimes \text{Sym}^l(\mathbb{C}^{r+1}).$ By the stability of semipositivity under symmetric and tensor products ([Ko1]) and Lemma \ref {semip}, $\pi_*(\mathcal{M}^l)$ is semipositive. Fix a choice of $l$ (depending only on $d$, $g$, $n$, and $r$) large enough to ensure \begin{equation} \label{ntq} \pi_*(\mathcal{M}^l) \oplus \ \bigoplus_{i=1}^{n}\pi_*(\mathcal{M}^l) \rightarrow \pi_*(\mathcal{M}^l\otimes {\mathcal{O}}_{\mathcal{C}}) \oplus \ \bigoplus_{i=1}^{n} \pi_*(\mathcal{M}^l\otimes {\mathcal{O}}_{S_i}) \rightarrow 0. \end{equation} Such a choice of $l$ is possible by the boundedness established in section \ref{bound}. Let $Q$ be the quotient in (\ref{ntq}). By boundedness and the vanishing of higher direct images, the quotient $Q$ is a vector bundle for large $l$. The quotient (\ref{ntq}) is nontrivial in the following sense. Let $G=GL$ be the structure group of the bundle $E_f$. $G$ is naturally the structure group of $\pi_*(\mathcal{M}^l)$. Let $W$ be the $G$-representation inducing the bundle $\pi_*(\mathcal{M}^l) \oplus \ \bigoplus_{1}^{n}\pi_*(\mathcal{M}^l)$. Let $q$ be the rank of the quotient bundle of (\ref{ntq}). The quotient sequence (\ref{ntq}) yields a set theoretic classifying map to the Grassmannian: $$\rho: S \rightarrow {\mathbf{Gr}}(q,W ^*)/G.$$ \begin{lm} \label{a} There exists a set theoretic injection $$\delta: \barr{M}_{g,n}(\proj^r,d) \rightarrow {\mathbf{Gr}} (q,W ^*)/G.$$ Let $\lambda: S \rightarrow \barr{M}_{g,n}(\proj^r,d)$ be the map induced by the stable family (\ref{ifam}). There is a (set theoretic) factorization $\rho= \delta \circ \lambda$. \end{lm} \noindent {\em Proof.} For large $l$, the sequence (\ref{ntq}) is equivalent to the data of a Hilbert point in $J$ (see section \ref{bound}). Since the $G$ orbits of $J$ are exactly the stable maps, the lemma follows. \qed \vspace{+10pt} \begin{lm} \label{b} A stable map has a finite number of automorphisms. \end{lm} \noindent {\em Proof.} As simple consequence of the definition of stability, there are no infinitesimal automorphisms. The total number is therefore finite. \qed \vspace{+10pt} Suppose the map to moduli $\lambda:S \rightarrow \barr{M}_{g,n}(\proj^r,d)$ is a generically finite algebraic morphism. Then, in the terminology of [Ko1], Lemmas \ref{a} and \ref{b} show the classifying map $\rho$ is {\em finite} on an open set of $S$. \begin{pr}(Lemma 3.13, [Ko1]) Let the base $S$ of (\ref{ifam}) be a normal projective variety. Suppose the classifying map is finite on an open set of $S$. Then, the top self-intersection number of $\text{Det}(Q)$ on $S$ is positive. \label{dett} \end{pr} If $\barr{M}_{g,n}(\proj^r,d)$ were a fine moduli space equipped with a universal family, $\text{Det}(Q)$ would be well defined and ample (by Proposition \ref{dett} and the Nakai-Moishezon criterion) on $\barr{M}_{g,n}(\proj^r,d)$. Since $\barr{M}_{0,n}(\proj^r,d)$ is expressed locally as a quotient of a fine moduli space by a finite group, it is easily seen $\text{Det}(Q)^k$ is a well defined line bundle on $\barr{M}_{0,n}(\proj^r,d)$ for some sufficiently large $k$. The exponent $k$ is taken to trivialize the $\mathbb{C}^*$-representations at the fixed points. In the higher genus case, $\text{Det}(Q)$ is a well defined line bundle on the Hilbert scheme $J$ or the stack. Since the moduli problem has finite automorphisms, $\text{Det}(Q)^k$ is well defined on the coarse moduli space for some $k$. Since the moduli spaces $\barr{M}_{g,n}(\proj^r,d)$ are not fine, subvarieties are not equipped with stable families. Proposition (\ref{dett}) and the Nakai-Moishezon criterion do not directly establish the ampleness of $\text{Det}(Q)^k$. An alternative approach (due to J. Koll\'ar) is followed. Recall the Hilbert scheme $J$ (of section \ref{bound}) is equipped with a universal family and, therefore, a canonical map $$J\rightarrow \barr{M}_{g,n}(\proj^r,d).$$ Let $X\subset \barr{M}_{g,n}(\proj^r,d)$ be a subvariety. Using $J$ and the finite automorphism property of a stable map, a morphism $Y\rightarrow X$ of algebraic schemes can be constructed satisfying \begin{enumerate} \item[(i)] $Y\rightarrow X$ is finite and surjective. \item[(ii)] $Y$ is equipped with a stable family of maps such that $Y\rightarrow X$ is the corresponding morphism to moduli. \end{enumerate} The existence of $Y\rightarrow X$ is exactly the conclusion of Proposition 2.7 in [Ko1] under slightly different assumptions. Nevertheless, the argument is valid in the present setting. The construction of $Y$ is subtle. First $Y$ is constructed as an algebraic space. Then, a lemma of Artin is used to find an algebraic scheme $Y$. Since $Y$ has a universal family, Proposition \ref{dett} implies $\text{Det}(Q)^k$ has positive top intersection on $Y$ and therefore on $X$. The Nakai-Moishezon criterion can be applied to conclude the ampleness of $\text{Det}(Q)^k$ on $\barr{M}_{g,n}(\proj^r,d)$. \subsection{Automorphisms} We use the notation of sections 3.2 and 4.1. In the genus $0$ case, $\barr{M}(\overline{t})$ is nonsingular. Therefore, the space $\barr{M}_{0,n}(\proj^r,d)$ is locally a quotient of a nonsingular variety by a finite group. \begin{lm} Let $\xi \in \barr{M}(\overline{t})$ be a point at which the $G_{d,r}$ action is not free. Then $\xi$ corresponds to a stable map with nontrivial automorphisms. \end{lm} \noindent {\em Proof.} $G_{d,r}$ acts by isomorphism on the stable maps of the universal family over $\barr{M}( \overline{t})$. The $G_{d,r}$ action is not free at $\xi\in \barr{M}(\overline{t})$ if and only if there exists a $1\neq \gamma\in G_{d,r}$ fixing $\xi$. The element $\gamma$ induces an automorphism of the map corresponding to $\xi$. The automorphism is nontrivial on the marked points $\{q_{i,j}\}$. \qed \vspace{+10pt} Over the automorphism-free locus, the $G_{d,r}$-action on $\barr{M}(\overline{t})$ (and on the universal family over $\barr{M}( \overline{t})$) is free. It follows that the quotient over the automorphism-free locus is a nonsingular quasi-projective variety denoted by $\barr{M}_{0,n}^*(\mathbf P^r,d)$. A universal family over $\barr{M}_{0,n}^*(\mathbf P^r,d)$ is obtained by patching. Theorems \ref{rep} and \ref{t2} have been established in the case $X\stackrel {\sim}{=} \mathbf P^r$. \section{{\bf The construction of} $\barr{M}_{g,n}(X,\beta)$} \subsection{Proof of Theorem \ref{rep}} Let $X$ be a projective algebraic variety. Existence of the coarse moduli space $\barr{M}_{g,n}(X,\beta)$ is established via a projective embedding $\iota: X\hookrightarrow \mathbf P^r$. Let $\iota_*(\beta)$ be $d$ times the class of a line in $\mathbf P^r$. \begin{lm} There exists a natural closed subscheme $$\barr{M}_{g,n}(X,\beta,\overline{t}) \subset \barr{M}_{g,n}(\mathbf P^r,d, \overline{t})$$ satisfying the following property. Let $(\pi:\mathcal{C}\rightarrow{S}, \{p_i\}, \{q_{i,j}\},\mu)$ be a $\overline{t}$-rigid stable family of genus $g$, $n$-pointed, degree $d$ maps to $\mathbf P^r$. Then, the natural morphism $S\rightarrow \barr{M}_{g,n}(\mathbf P^r,d, \overline{t})$ factors through $\barr{M}_{g,n}(X,\beta,\overline{t})$ if and only if $\mu$ factors through $\iota$ and each geometric fiber of $\pi$ is a map to $X$ representing the homology class $\beta\in A_1 X$. \end{lm} \noindent {\em Proof.} The lemma is proved in case $g=0$. If $g>0$, then $\barr{M}_{g,n}(\mathbf P^r,d, \overline{t})$ is not a fine moduli space and the argument is more technical. Let $$(\pi_M: \mathcal{U} \rightarrow \barr{M}_{0,n}(\mathbf P^r,d, \overline{t}), \{p_i\}, \{q_{i,j}\}, \mu)$$ be the universal family over $\barr{M}_{0,n}(\mathbf P^r,d, \overline{t})$. On a genus 0 curve, any vector bundle generated by global sections has no higher cohomology. Therefore, by this cohomology vanishing and the base change theorems, $\pi_{M*}\mu^*({\mathcal{O}}_{\mathbf P^r}(k))$ is a vector bundle for all $k>0$. (This argument must be modified in the $g>0$ case since $\pi_{M*}\mu^*({\mathcal{O}}_{\mathbf P^r}(k))$ need not be a vector bundle even on the Hilbert scheme $J$ or the stack. Nevertheless, it is not hard to define the closed subscheme determined by $X$ on the Hilbert scheme $J$ or the stack and then descend it to the coarse moduli space.) Let $\mathcal{I}_X$ be the ideal sheaf of $X \subset \mathbf P^r$. Let $I_X(k)=H^0(\mathbf P^r, \mathcal{I}_X(k))$. Let $l>>0$ be selected so that $\mathcal{I}_X(l)$ is generated by the global sections $I_X(l)$. These sections $I_X(l)$ yield sections of the vector bundle $\pi_{M*}\mu^*({\mathcal{O}}_{\mathbf P^r}(l))$. Let $Z\subset \barr{M}_{0,n}(\mathbf P^r,d, \overline{t})$ be the scheme-theoretic zero locus of these sections. The restriction of $\mu$ to $\pi_{M}^{-1}(Z)$ factors though $\iota$. Since $Z$ is an algebraic scheme, $Z$ is a finite union of disjoint connected components. The homology class in $A_1(X)=H_2(X, \mathbb{Z})$ represented by a map with moduli point in $Z$ is a deformation invariant of the map. Therefore, the represented homology class is constant on each connected component of $Z$. Let $Z_{\beta}\subset Z$ be the union of components of $Z$ which consist of maps representing the class $\beta\in A_1 X$. Let $\barr{M}_{0,n}(X,\beta,\overline{t})=Z_\beta$. The required properties are easily established. \qed \vspace{+10pt} By the functorial property, $\barr{M}_{g,n}(X,\beta,\overline{t})$ is invariant under the $G_{d,r}\stackrel {\sim}{=} \mathfrak{S}_d\times \cdots \times \mathfrak{S}_d$ action on $\barr{M}_{g,n}(\mathbf P^r,d,\overline{t})$. The quotient $$ \barr{M}_{g,n}(X,\beta,\overline{t})/ G_{d,r}$$ is an open set of $\barr{M}_{g,n}(X,\beta)$. A patching argument identical to the patching argument of section \ref{glu} yields a construction of $\barr{M}_{g,n}(X,\beta)$ as a closed subscheme of $\barr{M}_{g,n}(\proj^r,d)$. The functorial property of $\barr{M}_{g,n}(X,\beta)$ shows the construction is independent of the projective embedding of $X$. Projectivity of $\barr{M}_{g,n}(X,\beta)$ is obtained from the projectivity of $\barr{M}_{g,n}(\proj^r,d)$. This completes the proof of Theorem \ref{rep}. \subsection{Proof of Theorem \ref{t2}} \label{deff} Let $g=0$. Let $X$ be a projective, nonsingular, convex variety. Theorem \ref{t2} is certainly true in case $\beta=0$ since $\barr{M}_{0,n}(X,0)=\barr{M}_{0,n} \times X$. In general, a deformation study is needed to establish Theorem \ref{t2}. By the functorial property, the Zariski tangent space to the scheme $\barr{M}_{0,n}(X,\beta,\overline{t})$ at the point $(C, \{p_i\}, \{q_{i,j}\}, \mu:C\rightarrow X)$ is canonically isomorphic to the space of first order deformations of the pointed $\overline{t}$-stable map $(C, \{p_i\}, \{q_{i,j}\}, \mu:C\rightarrow X)$. The later deformation space corresponds bijectively to the space of first order deformations of the pointed stable map $(C, \{p_i\}, \mu:C\rightarrow X)$. Let $\text{Def}(\mu)$ denote the space of first order deformations of the pointed stable map $(C, \{p_i\}, \mu:C\rightarrow X)$. Consider first the case in which $C\stackrel {\sim}{=} \mathbf P^1$. Let $\text{Def}_R(\mu)$ be the space of first order deformations of $(C, \{p_i\}, \mu:C\rightarrow X)$ with $C$ held rigid. There is an natural exact sequence: $$0\rightarrow H^0(C, T_C) \rightarrow \text{Def}_R(\mu) \rightarrow \text{Def}(\mu)\rightarrow 0.$$ Stability of $\mu$ implies the left map is injective. Let $\text{Hom}(C,X)$ be the quasi-projective scheme of morphisms from $C$ to $X$ representing the class $\beta$. $\text{Hom}(C,X)$ is an open subscheme of the Hilbert scheme of graphs in $C\times X$. The Zariski tangent space to $\text{Hom}(C,X)$ is naturally identified: $$T_{\text{Hom}(C,X)}([\mu]) \stackrel {\sim}{=} H^0(C, \mu^*T_X)$$ (see [Ko2]). There is an exact sequence: $$0 \rightarrow \text{Ker} \rightarrow \text{Def}_R(\mu) \rightarrow H^0(C, \mu^*T_X)\rightarrow 0$$ where $\text{Ker}$ corresponds to the deformations of the markings. Therefore, $\text{dim}_{\mathbb{C}}\text{Ker}=n$. Since $X$ is convex, the above sequences suffice to compute the dimension of $\text{Def}(\mu)$: $$\text{dim}_{\mathbb{C}} \text{Def}(\mu)= \text{dim}(X)+ \int_\beta c_1(T_X) + n-3.$$ The dimension of the tangent space to $\barr{M}_{0,n}(X,\beta,\overline{t})$ is established in case $C\stackrel {\sim}{=} \mathbf P^1$. Before proceeding further, the following deformation result is needed. A proof can be found in [Ko2]. \begin{lm} \label{mapdef} Let $\mathcal{C}/S$ and $\mathcal{X}/S$ be flat, projective schemes over $S$. Let $s\in S$ be a geometric point. Let $C_s$, $X_s$ be the fibers over $s$ and let $f:C_s\rightarrow X_s$ be a morphism. Assume the following conditions are satisfied: \begin{enumerate} \item[(i)] $C_s$ has no embedded points. \item[(ii)] $X_s$ is nonsingular. \item[(iii)] $S$ is equidimensional at $s$. \end{enumerate} Then, the dimension of every component of the quasi-projective variety $\text{Hom}_S(\mathcal{C}, \mathcal{X})$ at the point $[f]$ is at least $$\text{\em{dim}}_{\mathbb{C}}H^0(C_s,f^*T_{X_s})- \text{\em{dim}}_{\mathbb{C}} H^1(C_s, f^*T_{X_s}) + \text{\em{dim}}_s S.$$ \end{lm} Again, let $(C\stackrel {\sim}{=} \mathbf P^1, \{p_i\}, \{q_{i,j}\}, \mu:C\rightarrow X)$ correspond to a point of the space $\barr{M}_{0,n}(X,\beta,\overline{t})$. By Lemma \ref{mapdef} and the convexity of $X$, every component of $\text{Hom}(C,X)$ at $[\mu]$ has dimension at least $\text{dim}_{\mathbb{C}} H^0(C, \mu^*T_X)$. Therefore, every component of $\barr{M}_{0,n}(X,\beta,\overline{t})$ at $[\mu]$ has dimension at least $ \text{dim}(X)+\int_\beta c_1(T_X)+ n-3$. By the previous tangent space computation, it follows $[\mu]$ is a nonsingular point of $\barr{M}_{0,n}(X,\beta,\overline{t})$. Before attacking the reducible case, a lemma is required. \begin{lm} \label{redd} Let $X$ be a nonsingular, projective, convex space. Let $\mu:C \rightarrow X$ be a morphism of a projective, connected, reduced, nodal curve of arithmetic genus 0 to $X$. Then, \begin{equation} \label{kspc} H^1(C, \mu^*T_X)=0. \end{equation} and $\mu^*T_X$ is generated by global sections on $C$. \end{lm} \noindent {\em Proof.} Let $E\subset C$ be an irreducible component of $C$; $E\stackrel {\sim}{=} \mathbf P^1$. Let $$\mu^*T_X|_E\stackrel {\sim}{=} \bigoplus {\mathcal{O}}_{\mathbf P^1}(\alpha_i).$$ Suppose there exists $\alpha_i<0$. The composition of a rational double cover of $E$ with $\mu$ would then violate the convexity of $X$ . It follows that: \begin{equation} \label{condy} \forall i, \ \ \alpha_i\geq 0. \end{equation} We will prove the following statement by induction on the number of components of $C$: \begin{equation} \label{ggll} H^1(C, \mu^* T_X \otimes {\mathcal{O}}_C(-p))=0 \end{equation} for any nonsingular point $p\in C$. Equation (\ref{ggll}) is true by condition (\ref{condy}) when $C\stackrel {\sim}{=} \mathbf P^1$ is irreducible. Assume now $C$ is reducible and $p\in E\stackrel {\sim}{=} \mathbf P^1$. Let $C= C' \cup E$; let $\{p'_1,\ldots, p'_q\}= C'\cap E$. Since $C$ is a tree, $C'$ has exactly $q$ connected components each intersecting $E$ in exactly $1$ point. There is a component sequence: $$ 0 \rightarrow \mu^* T_X|_{C'} \otimes {\mathcal{O}}_{C'}(-\sum_{j=1}^{q}p'_j) \rightarrow \mu^* T_X \otimes {\mathcal{O}}_C(-p) \rightarrow \mu^* T_X |_{E} \otimes {\mathcal{O}}_E(-p) \rightarrow 0.$$ Equation (\ref{ggll}) now follows from the inductive assumptions on $C'$ and $E$. The inductive assumption (\ref{ggll}) is applied to every connected component of $C'$. We now prove $H^1(C, \mu^* T_X)=0$. If $C\stackrel {\sim}{=} \mathbf P^1$, then the lemma is established by condition (\ref{condy}). Assume now $C= C' \cup E$ where $E\stackrel {\sim}{=} \mathbf P^1$. There is a component sequence \begin{equation} \label{igoth} 0 \rightarrow \mu^* T_X|_{C'} \otimes {\mathcal{O}}_{C'}(-\sum_{j=1}^q p'_j) \rightarrow \mu^* T_X \rightarrow \mu^* T_X |_{E} \rightarrow 0. \end{equation} Equation (\ref{kspc}) now follows from (\ref{ggll}) applied to every connected component of $C'$. Finally, an analysis of sequence (\ref{igoth}) also yields the global generation result. $\mu^* T_X|_E$ is generated by global sections by (\ref{condy}). Sequence (\ref{igoth}) is exact on global sections by (\ref{ggll}). Hence $\mu^* T_X$ is generated by global sections on $E$. But, every point of $C$ lies on some component $E\stackrel {\sim}{=} \mathbf P^1$. \qed \vspace{+10pt} In sections 7 and 8, the following related lemma will be required: \begin{lm} \label{george} Let $\mu: \mathbf P^1 \rightarrow X$ be a non-constant morphism to a nonsingular, projective, convex space $X$. Then $\int_{\mu_*[\mathbf P^1]} c_1(T_X) \geq 2$. \end{lm} \noindent {\em Proof.} Since $\mu$ is non-constant, the differential $$d\mu: T_{\mathbf P^1} \rightarrow \mu^*(T_X)$$ is nonzero. Let $s \in H^0(\mathbf P^1,T_{\mathbf P^1})$ be a vector field with two distinct zeros $p_1, p_2 \in \mathbf P^1$. Then, $d\mu(s) \in H^0(\mathbf P^1, \mu^*(T_x))\neq 0$ and $d\mu(s)$ vanishes (at least) at $p_1$ and $p_2$. By the proof of Lemma \ref{redd}, $\mu^*(T_X) \stackrel {\sim}{=} \bigoplus {\mathcal{O}}_{\mathbf P^1}(\alpha_i)$ where $\alpha_i \geq 0$ for all $i$. The existence of $d\mu(s)$ implies that $\alpha_j \geq 2$ for some $j$. \qed \vspace{+10pt} Let $C$ now be a reducible curve. $C$ must be a tree of $\mathbf P^1$'s. Let $q$ be the number of nodes of $C$. Again, let $\text{Def}(\mu)$ be the first order deformation space of the pointed stable map $\mu$. The {\em dual graph} of a pointed curve $C$ of arithmetic genus $0$ consists of vertices and edges corresponding bijectively to the irreducible components and nodes of $C$ respectively. The {\em valence} of a vertex in the dual graph is the numbers of edges incident at that vertex. Let $\text{Def}_G(\mu)\subset \text{Def}(\mu)$ be the first order deformation space of the pointed stable map $\mu$ preserving the dual graph. $\text{Def}_G(\mu)$ is a linear subspace of codimension at most $q$. Let $\text{Def}_G(C)$ be the space of first order deformations of the curve $C$ which preserve the dual graph. A simple calculation yields $$\text{dim}_{\mathbb{C}} \text{Def}_G(C) = \sum_{|\nu|\geq 4} |\nu|-3$$ where the sum is taken over vertices $\nu$ of the dual graph of valence at least 4. The natural linear map $\text{Def}_G(\mu) \rightarrow \text{Def}_G(C)$ is now analyzed. Let $S$ be the nonsingular universal base space of deformations of $C$ preserving the dual graph. Let $\mathcal{C}$ be the universal deformation over $S$. Let $\mathcal{X}=X\times S$. Let $s_0\in S$ correspond to $C$. By Lemmas \ref{mapdef} and \ref{redd}, every component of $\text{Hom}_S(\mathcal{C},\mathcal{X})$ at $[\mu]$ has dimension at least $\text{dim}(X)+\int_{\beta} c_1(T_X)+ \text{dim}(S)$. The tangent space to the fiber of $\text{Hom}_S(\mathcal{C}, \mathcal{X})$ over $s_o$ at $[\mu]$ is canonically $H^0(C, \mu^*T_X)$. The latter space has dimension $\text{dim}(X)+\int_{\beta} c_1(T_X)$. Hence, $\text{Hom}_S(\mathcal{C},\mathcal{X})$ is nonsingular at $[\mu]$ of dimension $\text{dim}(X)+ \int_{\beta} c_1(T_X)+ \text{dim}(S)$ and the projection morphism to $S$ is smooth at $[\mu]$. Therefore, $\text{Def}_G(\mu) \rightarrow \text{Def}_G(C)$ is surjective. The above definitions and results yield a natural exact sequence: $$0\rightarrow \text{Def}_C(\mu) \rightarrow \text{Def}_G(\mu) \rightarrow \text{Def}_G(C)\rightarrow 0$$ where $\text{Def}_C(\mu)$ is the space of first order deformations of the pointed stable map $\mu$ which restrict to the trivial deformation of $C$. As in the case where $C\stackrel {\sim}{=} \mathbf P^1$, $\text{Def}_C(\mu)$ differs from $\text{Def}_R(\mu)$ only by the tangent fields obtained from automorphisms: $$0\rightarrow H^0(C, T^{auto}_C) \rightarrow \text{Def}_R(\mu) \rightarrow \text{Def}_C(\mu) \rightarrow 0.$$ $H^0(C, T^{auto}_C)$ is the space of tangent fields on the components of $C$ that vanish at all the nodes of $C$. Note $H^0(C, T^{auto}_C)= \sum_{|\nu|\leq 3}3-|\nu|$. Finally, there is an exact sequence containing $\text{Def}_R(\mu)$ and the tangent space to $\text{Hom}(C,X)$: $$0 \rightarrow \text{Ker} \rightarrow \text{Def}_R(\mu) \rightarrow H^0(C, \mu^*T_X) \rightarrow 0.$$ From these exact sequences, Lemma \ref{redd}, and some arithmetic, it follows \begin{equation} \label{ooo} \text{dim}_{\mathbb{C}}\text{Def}_G(\mu)= \text{dim}(X)+ \int_{\beta} c_1(T_X) + n-3 -q. \end{equation} Let $\mathcal{C}$ be a smoothing of the reducible curve $C$ over a base $S$ and let $\mathcal{X}=X\times S$. A simple application of Lemma \ref{mapdef} shows that $[\mu]\in \barr{M}_{0,n}(X,\beta,\overline{t})$ lies in the closure of the locus of maps with irreducible domains. Since the irreducible domain locus is pure dimensional of dimension $\text{dim}(X)+\int_{\beta} c_1(T_X)+ n-3$, \begin{equation} \label{tttt} \text{dim}_{\mathbb{C}} \text{Def}(\mu) \geq \text{dim}(X)+\int_{\beta} c_1(T_X)+ n-3. \end{equation} It follows from (\ref{ooo}) and (\ref{tttt}) that $\text{Def}_G(\mu)$ is of maximal codimension $q$ in $\text{Def}(\mu)$ and that the inequality in (\ref{tttt}) is an equality. Since $\text{Def}(\mu)$ is of dimension $\text{dim}(X)+\int_{\beta} c_1(T_X)+ n-3$, $[\mu]$ is a nonsingular point of $\barr{M}_{0,n}(X,\beta,\overline{t})$. Since $\barr{M}_{0,n}(X,\beta,\overline{t})$ is nonsingular of pure dimension $\text{dim}(X)+\int_{\beta} c_1(T_X)+ n-3$, parts (i) and (ii) of Theorem \ref{t2} are established. Part (iii) follows from the corresponding result in the case $X\stackrel {\sim}{=} \mathbf P^r$. \section{{\bf The boundary of} $\barr{M}_{0,n}(X,\beta)$} \subsection{Definitions} \label{bdry} Let $X$ be nonsingular, projective, and convex. Let the genus $g=0$. The {\em boundary} of $\barr{M}_{0,n}(X,\beta)$ is the locus corresponding to reducible domain curves. Boundary properties of the Mumford-Knudsen space $\barr{M}_{0,m}$ (where $m=n+d(r+1)$) are passed to $\barr{M}_{0,n}(\mathbf P^r,d)$ by the local quotient construction. The boundary locus of $\barr{M}_{0,m}$ is a divisor with normal crossings. Since $\barr{M}_{0,n}(\mathbf P^r,d, \overline{t})$ is a product of $\mathbb{C}^*$-bundles over an open set of $\barr{M}_{0,m}$, the boundary locus of $\barr{M}_{0,n}(\mathbf P^r,d, \overline{t})$ is certainly a divisor with normal crossing. $\barr{M}_{0,n}(\mathbf P^r,d)$ is locally the $G_{d,r}$-quotient of $\barr{M}_{0,n}(\mathbf P^r,d, \overline{t})$. The boundary of $\barr{M}_{0,n}(\mathbf P^r,d)$ is therefore a union of subvarieties of pure codimension 1. Over the automorphism-free locus, the boundary of $\barr{M}_{0,n}(\mathbf P^r,d)$ is a divisor with normal crossings. Let $X$ be a nonsingular, projective, convex variety. The corresponding boundary results for $\barr{M}_{0,n}(X,\beta)$ are consequences of the deformation analysis of section \ref{deff}. The boundary locus of $\barr{M}_{0,n}(X,\beta, \overline{t})$ is a divisor with normal crossing singularities. A pointed map $\mu:C\rightarrow X$ such that $C$ has $q$ nodes lies in the intersection of $q$ branches of the boundary. The dimension computation $$\text{dim}_{\mathbb{C}} \text{Def}_G(\mu)= \text{dim}\barr{M}_{0,n}(X,\beta, \overline{t}) - q $$ shows these branches intersect transversally at $[\mu]$. This completes the proof of Theorem 3. In particular $\barr{M}_{0,n}(X,\beta)$ has the same boundary singularity type as $\barr{M}_{g}$ and $\barr{M}_{g,n}$. A class $\beta\in H_2(X,\mathbb{Z})$ is {\em effective} if $\beta$ is represented by some genus $0$ stable map to $X$. If $n=0$, the boundary of $\barr{M}_{0,0}(X, \beta)$ decomposes into a union of divisors which are in bijective correspondence with effective partitions $\beta_1+\beta_2=\beta$. For general $n$, the boundary decomposes into a union of divisors in bijective correspondence with data of weighted partitions $(A,B; \beta_1, \beta_2)$ where \begin{enumerate} \item[(i)] $A \cup B$ is a partition of $[n]=\{1,2, \ldots, n\}$. \item[(ii)] $\beta_1+\beta_2=\beta$, $\beta_1$ and $\beta_2$ are effective . \item[(iii)] If $\beta_1=0$ (resp. $\beta_2=0$), then $|A|\geq 2$ (resp. $|B|\geq 2$). \end{enumerate} $D(A,B; \beta_1, \beta_2)$, the divisor corresponding to the data $(A,B; \beta_1, \beta_2)$, is defined to be the locus of maps $\mu:C_A\cup C_B \rightarrow X$ satisfying the following conditions: \begin{enumerate} \item [(a)] $C$ is a union of two quasi-stable curves $C_A$, $C_B$ of genus $0$ meeting in a point. \item [(b)] The markings of $A$ (resp. $B$) lie on $C_A$ (resp. $C_B$). \item [(c)] The map $\mu_A=\mu|_{C_A}$ (resp. $\mu_B$) represents $\beta_1$ (resp. $\beta_2$). \end{enumerate} The deformation results of section 5 show the locus maps satisfying (a)$-$(c) and $C_A\stackrel {\sim}{=} C_B \stackrel {\sim}{=} \mathbf P^1$ is dense in $D(A,B;\beta_1, \beta_2)$. If $X= \mathbf P^r$, then it is easily seen that $D(A,B; \beta_1, \beta_2)$ is irreducible. In general, we do not claim the divisor $D(A,B; \beta_1, \beta_2)$ is irreducible, although that is the case in all the examples we have seen. \subsection{Boundary divisors} The boundary divisor of $\barr{M}_{0,n}$ corresponding to the marking partition $A\cup B=[n]$ is naturally isomorphic (by gluing) to the product $$\barr{M}_{0,A \cup \{\bullet\}} \times \barr{M}_{0,B \cup \{\bullet\}}.$$ An analogous construction exists for the boundary divisor $D(A, B; \beta_1, \beta_2)$ of the space $\barr{M}_{0,n}(X,\beta)$. Let $K=D(A, B; \beta_1, \beta_2)$ be a boundary divisor of $\barr{M}_{0,n}(X,\beta)$. Let $\barr{M}_A=\barr{M}_{0, A\cup\{\bullet\}}(X,\beta_1)$ and $\barr{M}_B= \barr{M}_{0, B\cup \{\bullet\}}(X, \beta_2)$. Let $e_A: \barr{M}_A \rightarrow X$ and $e_B: \barr{M}_B\rightarrow X$ be the evaluation maps obtained from the additional marking $\bullet$. Let $\tau_A$, $\tau_B$ be the projections of $\barr{M}_A \times \barr{M}_B$ to the first and second factors respectively. Let $\tilde{K}= \barr{M}_A \times_{X} \barr{M}_B$ be the fiber product with respect to the evaluation maps $e_A$, $e_B$. $\tilde{K}\subset \barr{M}_A \times_{\mathbb{C}} \barr{M}_B$ is the closed subvariety $(e_A\times_{\mathbb{C}} e_B)^{-1} (\Delta)$ where $\Delta\subset X \times X$ is the diagonal. Properties of $\tilde{K}$ can be deduced from the local quotient constructions of $\barr{M}_A$ and $\barr{M}_B$. It will be shown that $\tilde{K}$ is a normal projective variety of pure dimension with finite quotient singularities. Let $\barr{M}_A(X,\overline{t}_A)$, $\barr{M}_B(X,\overline{t}_B)$ be the $\overline{t}_A$, $\overline{t}_B$-rigid moduli spaces. $\tilde{K}$ is the $G_A \times G_B$-quotient of the corresponding subvariety $$\tilde{K}(X,\overline{t}_A, \overline{t}_B)\subset\barr{M}_A(X,\overline{t}_A)\times \barr{M}_B(X,\overline{t}_B),$$ $$\tilde{K}(X,\overline{t}_A, \overline{t}_B)=(e_A\times_{\mathbb{C}} e_B)^{-1} (\Delta).$$ The differential of $e_A$ at a point $[\mu]$ of $\barr{M}_A(X,\overline{t}_A)$ is determined in the following manner. The case in which the domain $C\stackrel {\sim}{=}\mathbf P^1$ is irreducible is most straightforward. Then, there are natural linear maps: \begin{equation} \label{hattey} \text{Def}(\mu) \rightarrow H^0(\mu^*T_X/ T_C(-p_\bullet)) \rightarrow T_X({\mu(p_\bullet)}). \end{equation} The first map in (\ref{hattey}) is the natural surjection of $\text{Def}(\mu)$ onto the deformation space of the moduli problem obtained by forgetting all the markings except $\bullet$. The natural fiber evaluation $H^0(\mu^* T_X) \rightarrow T_X({\mu(p_\bullet)})$ is well defined on the space $H^0(\mu^*T_X/ T_C(-p_\bullet))$. This is the second map in (\ref{hattey}). The composition of maps in (\ref{hattey}) is simply the differential of $e_A$ at $[\mu]$. Since $\mu^*T_X$ is generated by global sections by Lemma \ref{redd}, it follows that the differential of $e_A$ is surjective at $[\mu]$. A similar argument shows the differential of $e_A$ is surjective for each $[\mu] \in \barr{M}_A(X,\overline{t}_A)$. The differential of $e_B$ is therefore also surjective. The surjectivity of the differentials of $e_A$ and $e_B$ imply $\tilde{K}(X,\overline{t}_A, \overline{t}_B)$ is nonsingular. Thus $\tilde{K}$ is a normal projective variety of pure dimension with finite quotient singularities. By gluing the universal families over $\barr{M}_A(\overline{t}_A)$ and $\barr{M}_B(\overline{t}_B)$ along the markings $\bullet$, a natural family of Kontsevich stable maps exists over $\tilde{K}(X,\overline{t}_A, \overline{t}_B)$. The induced map $$\tilde{K}(X,\overline{t}_A, \overline{t}_B) \rightarrow K$$ is seen to be $G_A\times G_B$ invariant. Therefore, a natural map $\psi:\tilde{K} \rightarrow K$ is obtained. \begin{lm} Results on the morphism $\psi$ : \label{resy} \begin{enumerate} \item[(i)] If $A\neq \emptyset$ and $B \neq \emptyset$, then $\psi:\tilde{K}\rightarrow K$ is an isomorphism. \item[(ii)] If $A\neq \emptyset$, or $B\neq \emptyset$, or $\beta_A\neq \beta_B$, then $\psi$ is birational. \item[(iii)] If $A=B=\emptyset$ ($n=0$) and $\beta_A=\beta_B= \beta/2$ then $\psi$ is generically $2$ to $1$. \end{enumerate} \end{lm} \noindent {\em Proof.} First part (i) is proven. Let $q_A \in A$ and $q_B \in B$ be fixed markings (whose existence is guaranteed by the assumptions of (i)). Let $\mathcal{L}$ be a very ample line bundle on $X$ against which all degrees of maps are computed. Let $d_A$, $d_B$ be the degrees of $\beta_A$, $\beta_B$ respectively. Let $K=(A\cup B, \beta_A, \beta_B)$. Let $\mu: C\rightarrow \mathbf P^r$ correspond to a moduli point $[\mu] \in K$. Let $C= \bigcup C_i$ be the union of irreducible components. Let $q_A \in C_1$, $q_B\in C_l$ where $1\neq l$ and let $$C_1, C_2, \ldots, C_l$$ be the unique minimal path from $C_1$ to $C_l$ which exists since $C$ is a tree of components. For $1\leq i \leq l-1$, let $x_i=C_i\cap C_{i+1}$. Each node $x_i$ divides $C$ into two connected curves $$C= C_{A,i} \cup C_{B,i}$$ labeled by the points $q_A$, $q_B$. Let $d_i$ be the degree of $\mu$ restricted to $C_{A,i}$. The degrees $d_i$ increase monotonically. Since $[\mu] \in K$, $d_i=d_A$ for some $i$. Let $j$ be the minimal value satisfying $d_j=d_A$. If $d_{j+1} > d_A$, then $\psi^{-1}[\mu]$ is the unique point determined by cutting at the node $x_j$. If $d_{j+1}=d_A$, then the subcurve $$C \ \mathbin{\text{\scriptsize$\setminus$}} \ (C_{A,j} \cup C_{B,j+1})$$ must contain (by stability) a nonempty set of marked points $P_{j+1}$. Let $k$ be maximal index satisfying $d_{j+k}=d_A$. The analogously defined marked point sets $$P_{j+1}, \ldots, P_{j+k}$$ are all nonempty. There must be a index $t$ satisfying $P_{j+t'} \subset A$ for $1\leq t' \leq t$ and $P_{j+t'} \subset B$ for $t < t'\leq k$. $\psi^{-1}[\mu]$ is then the unique point determined by cutting at the node $x_{j+t}$. Therefore, $\psi$ is bijective in case $A$ and $B$ are nonempty. Let $\barr{M}_{0,n}(X,\beta, \overline{t})$ be a locally rigidified moduli space containing the point $[\mu]\in K$. If $|A|, |B| \geq 1$, a similar argument shows the boundary components of $\barr{M}_{0,n}(X,\beta,\overline{t})$ lying over $K$ are {\em disjoint}. Therefore, $K$ is normal. In case $A$ and $B$ are nonempty, $\psi$ is a bijective morphism of normal varieties and hence an isomorphism. Note, for example, that the component $K=D(\emptyset, \emptyset; 2,3)$ of $\barr{M}_{0,0}(\mathbf P^r,5)$ is not normal. $K$ intersects itself along the codimension 2 locus of moduli points $[\mu]$ of the form: $$\mu: C_1\cup C_2 \cup C_3 \rightarrow \mathbf P^r$$ with restricted degrees $d_1=2$, $d_2=1$, $d_3=2$. In this case, $\psi: \tilde{K} \rightarrow K$ is a normalization. Parts (ii) and (iii) follow simply from the defining properties (a)$-$(c) of $K$. \qed \vspace{+10pt} The fundamental relations among the Gromov-Witten invariants will come from the following linear equivalences among boundary components in $\overline{M}_{0,n}(X, \beta)$. \begin{pr} \label{needit} For $i, j, k, l$ distinct in $[n]$, set $$ D(i,j \mid k,l) = \sum D(A,B; \beta_1,\beta_2) , $$ the sum over all partitions such that $i$ and $j$ are in $A$, $k$ and $l$ are in $B$, and $\beta_1$ and $\beta_2$ are effective classes in $A_1X$ such that $\beta_1 + \beta_2 = \beta$. Then, we have the linear equivalence of divisors $$ D(i,j \mid k,l) \sim D(i,l \mid j,k) $$ on $\barr{M}_{0,n}(X,\beta)$. \end{pr} \noindent {\em Proof.} The proof is obtained by examining the map $$ \barr{M}_{0,n}(X,\beta) \rightarrow \barr{M}_{0,n} \rightarrow \barr{M}_{0,\{i,j,k,l\}} \cong \mathbf P^1 , $$ and noting that the divisor $D(i,j \mid k,l)\subset \barr{M}_{0,n}(X,\beta)$ is the multiplicity-free inverse image of the point $D(i,j \mid k,l)\in \barr{M}_{0,\{i,j,k,l\}}$. The deformation methods of section 5 can be used to prove that the inverse image of the point $D(i,j \mid k,l) \in \barr{M}_{0,\{i,j,k,l\}}$ is multiplicity-free. Since points are linearly equivalent on $\mathbf P^1$, the linear equivalence on $\barr{M}_{0,n}(X,\beta)$ is established. \qed \vspace{+10pt} \section{{\bf Gromov-Witten invariants}} In sections 7--10, unless otherwise stated, $X$ will denote a homogeneous variety and the genus $g$ will be zero. Since the the tangent bundle of $X$ is generated by global sections, $X$ is convex. The moduli spaces $\overline{M}_{0,n}(X, \beta)$ are therefore available with the properties proved in sections 1--6. In addition, the cohomology of $X$ has a natural basis of algebraic cycles (classes of Schubert varieties), so $A^i X=H^{2i} X$ can be identified with the Chow group of cycle classes of codimension $i$. The effective classes $\beta$ in $A_1 X$ (see section 6.1) are non-negative linear combinations of the Schubert classes of dimension 1. Each $1$-dimensional Schubert class is represented by an embedding $\mathbf P^1\subset X$. The varieties $\barr{M}_{0,n}(X,\beta)$ come equipped with $n$ morphisms $\rho_1, \ldots , \rho_n$ to $X$, where $\rho_i$ takes the point $[C, p_1, \ldots , p_n, \mu] \in \barr{M}_{0,n}(X,\beta)$ to the point $\mu(p_i)$ in $X$. Given arbitrary classes $\gamma_1, \ldots , \gamma_n$ in $A^*X$, we can construct the cohomology class $${\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n)$$ on $\barr{M}_{0,n}(X,\beta)$, and we can evaluate its homogeneous component of the top codimension on the fundamental class, to produce a number, called a {\em Gromov-Witten invariant}, that we denote by $I_{\beta}(\gamma_1 \cdots \gamma_n)$: \begin{equation} I_{\beta}(\gamma_1 \cdots \gamma_n) \, = \, \int_{\barr{M}_{0,n}(X,\beta)} {\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n) . \end{equation} If the classes $\gamma_i$ are homogeneous, this will be a nonzero number only if the sum of their codimensions is the dimension of $\barr{M}_{0,n}(X,\beta)$, that is, $$\sum \text{codim} (\gamma_i) = \dim X + \int_\beta c_1(T_X) + n - 3.$$ It follows from the definition that $I_{\beta}(\gamma_1 \cdots \gamma_n)$ is invariant under permutations of the classes $\gamma_1, \ldots , \gamma_n$. The conventions of [K-M] require $n\geq 3$. However, it will be convenient for us to take $n\geq 0$. A 0-pointed invariant occurs when the moduli space $\barr{M}_{0,0}(X, \beta)$ is of dimension 0. In this case $I_\beta= \int _{\barr{M}_{0,0}(X, \beta)} 1$. By Lemma \ref{george}, $\barr{M}_{0,0}(X,\beta)$ is of dimension 0 if and only if $\dim (X)=1 $ and $\int_\beta c_1(X)=2$. Hence, for homogeneous varieties, 0-pointed invariants only occur on $X\stackrel {\sim}{=} \mathbf P^1$. In this case, $I_{1}=1$ is the unique 0-pointed invariant. Let $M_{0,n}^*(X, \beta)=M_{0,n}(X, \beta)\cap \overline{M}^*_{0,n}(X,\beta)$. We start with a simple lemma. \begin{lm} \label{freddy} If $n\geq 1$, then $M_{0,n}^*(X, \beta)\subset \barr{M}_{0,n}(X, \beta)$ is a dense open set. \end{lm} \noindent {\em Proof.} If $\beta=0$, then $\overline{M}_{0,n}(X,0)$ is nonempty only if $n\geq 3$. The equality $\barr{M}_{0,n}^*(X, 0)=\barr{M}_{0,n}(X,0)$ is deduced from the corresponding equality for $\barr{M}_{0,n}$. Assume $\beta\neq 0$. By Theorem 3, $M_{0,n}(X, \beta)\subset \barr{M}_{0,n}(X, \beta)$ is a dense open set. Let $(\mathbf P^1, \{p_i\}, \mu)$ be a point in $M_{0,n}(X,\beta)$. It suffices to show that $(\mathbf P^1, \{p_i'\}, \mu)$ is automorphism-free for general points $p_1', \ldots, p_n'\in \mathbf P^1$. The automorphism group $A$ of the unpointed map $\mu: \mathbf P^1 \rightarrow X$ is finite since $\beta \neq 0$. There exists a (nonempty) open set of $\mathbf P^1$ consisting of points with trivial $A$-stabilizers. If $p_1', \ldots, p_n'$ belong to this open subset, the pointed map $(\mathbf P^1, \{p_i'\}, \mu)$ is automorphism-free. \qed \vspace{+10pt} Let $X=G/P$, so $G$ acts transitively on $X$. Let $\Gamma_1, \ldots, \Gamma_n$ be pure dimensional subvarieties of $X$. Let $[\gamma_i]\in A^*X$ be the corresponding classes (see our notational conventions in section \ref{nota}). Assume $$\sum_{i=1}^n \text{codim}(\Gamma_i) = \text{dim}(X)+\int_{\beta}c_1(T_X) +n-3.$$ Let $g\Gamma_i$ denote the $g$-translate of $\Gamma_i$ for $g\in G$. \begin{lm} \label{enuuuu} Let $n\geq 0$. Let $g_1, \ldots, g_n \in G$ be general elements. Then, the scheme theoretic intersection \begin{equation} \label{intttt} \rho_1^{-1}(g_1 \Gamma_1) \cap \cdots \cap \rho_n^{-1}(g_n \Gamma_n) \end{equation} is a finite number of reduced points supported in $M_{0,n}(X, \beta)$ and $$I_{\beta}(\gamma_1 \cdots \gamma_n) = \# \ \rho_1^{-1}(g_1 \Gamma_1) \cap \cdots \cap \rho_n^{-1}(g_n \Gamma_n).$$ \end{lm} \noindent {\em Proof.} If $n=0$, $I_1=1$ on $\mathbf P^1$ is the only case and the lemma holds since $\overline{M}_{0,0}(\mathbf P^1,1)$ is a nonsingular point. Assume $n\geq 1$. $M_{0,n}^*(X, \beta) \subset \overline{M}_{0,n}(X, \beta)$ is a dense open set by Lemma \ref{freddy}. By simple transversality arguments (with respect to the $G$-action), it follows that the intersection (\ref{intttt}) is supported in ${M}_{0,n}^*(X,\beta)$. By Theorem 2, ${M}_{0,n}^*(X,\beta)$ is nonsingular. An application of Kleiman's Bertini theorem ([Kl]) now shows that the intersection (\ref{intttt}) is a finite set of reduced points. To see that the number of points in (\ref{intttt}) agrees with the intersection number, consider the fiber diagram: \begin{equation} \label{jesdia} \begin{CD} \cap_{i=1}^{n} \rho_i^{-1}(g_i \Gamma_i) @>>> \overline{M}\times \prod_{i=1}^{n} g_i \Gamma_i \\ @VVV @VVV \\ \overline{M} @>{\iota}>> \overline{M} \times X^n \\ \end{CD} \end{equation} where $\overline{M}=\overline{M}_{0,n}(X,\beta)$ and $\iota$ is the graph of the morphism $(\rho_1, \ldots, \rho_n)$. From (\ref{jesdia}), one sees that $$\prod_{i=1}^{n}\rho_i^*[g_i \Gamma_i] \cap [\overline{M}] = \iota^*[\overline{M} \times \prod_{i=1}^{n}g_i \Gamma_i]= [\cap_{i=1}^{n} \rho_{i}^{-1}(g_i \Gamma_i)]$$ in $A_0(\overline{M})$, which is the required assertion. \qed \vspace{+10pt} Lemma \ref{enuuuu} relates the Gromov-Witten invariants to enumerative geometry. We see $I_{\beta}(\gamma_1 \cdots \gamma_n)$ equals the number of pointed maps $\mu$ from $\mathbf P^1$ to $X$ representing the class $\beta \in A_1 X$ and satisfying $\mu(p_i) \in g_i \Gamma_i$. We will need three basic properties satisfied by the Gromov-Witten invariants: \vspace{+10pt} (I) $\beta = 0$. \noindent In this case, $\barr{M}_{0,n}(X,\beta) = \barr{M}_{0,n} \times X$, and the mappings $\rho_i$ are all equal to the projection $p$ onto the second factor. Since $${\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n) = p^*(\gamma_1 \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} \gamma_n),$$ \begin{eqnarray*} I_\beta(\gamma_1\cdots \gamma_n) & = & \int_{\barr{M}_{0,n} \times X} p^*(\gamma_1 \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} \gamma_n) \\ & = &\int_{p_*[\barr{M}_{0,n} \times X]} \gamma_1 \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} \gamma_n . \end{eqnarray*} Note that $\overline{M}_{0,n}$ is empty if $0\leq n \leq 2$. If $n > 3$, $p_*[\barr{M}_{0,n} \times X] = 0$, since the fibers of $p$ have positive dimension. The only way the number $I_\beta(\gamma_1 \cdots \gamma_n)$ can be nonzero is when $n = 3$, so that $\barr{M}_{0,n}$ is just a point. In this case, $I_\beta(\gamma_1 {\cdot} \gamma_2 {\cdot} \gamma_3)$ is the classical intersection number $\int_X \gamma_1 \mathbin{\text{\scriptsize$\cup$}} \gamma_2 \mathbin{\text{\scriptsize$\cup$}} \gamma_3$. \vspace{+10pt} (II) $\gamma_1 =1 \in A^0X$. \noindent If $\beta\neq 0$, then the product ${\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n)$ is the pullback of a class on $\barr{M}_{0,n-1}(X,\beta)$ by the map from $\barr{M}_{0,n}(X,\beta)$ to $\barr{M}_{0,n-1}(X,\beta)$ that forgets the first point. Since the fibers of this map have positive dimension, the evaluation $I_\beta(\gamma_1 {\cdots}\gamma_n)$ must vanish. Therefore, by (I), $I_\beta(\gamma_1\cdots \gamma_n)$ vanishes unless $\beta = 0$ and $n = 3$. In this case, $I_0( 1{\cdot} \gamma_2 {\cdot} \gamma_3) = \int_X \gamma_2 \mathbin{\text{\scriptsize$\cup$}} \gamma_3$. \vspace{+10pt} (III) $\gamma_1 \in A^1X$ and $\beta\neq 0$. \noindent In this case, \begin{equation} \label{axxer} I_\beta(\gamma_1\cdots \gamma_n) = \left(\int_\beta \gamma_1\right) \cdot I_\beta(\gamma_2 \cdots \gamma_n). \end{equation} For a map $\mu: C \rightarrow X$ with $\mu_*[C] = \beta$, there are $\left( \int_\beta \gamma_1 \right)$ choices for the point $p_1$ in $C$ to map to a point in $\Gamma_1$, where $\Gamma_1$ is a variety representing $\gamma_1$. Equation (\ref{axxer}) is therefore a consequence of Lemma \ref{enuuuu}. For a formal intersection-theoretic proof of (\ref{axxer}), consider the mapping $$ \psi: \barr{M}_{0,n}(X,\beta) \rightarrow X \times \barr{M}_{0,n-1}(X,\beta) $$ which is the product of $\rho_1$ and the map that forgets the first point. By the K\"unneth formula, we can write $\psi_*[\barr{M}_{0,n}(X,\beta)] = \beta' \times [\barr{M}_{0,n-1}(X,\beta)] + \alpha$, where $\beta'$ is a class in $A_1 X$, and $\alpha$ is some homology class that is supported over a proper closed subset of $\barr{M}_{0,n-1}(X,\beta)$. The class $\beta'$ can be calculated by restricting to what happens over a generic point of $\barr{M}_{0,n-1}(X,\beta)$. Representing such a point by $(C, p_2, \ldots, p_n, \mu)$ with $C\stackrel {\sim}{=} \mathbf P^1$, one sees that the fiber over this point is isomorphic to $C$ and $\beta' = \mu_*[C] = \beta$. Using the projection formula as in (I) and (II), it follows that \begin{eqnarray*} I_\beta(\gamma_1 \cdots \gamma_n) & = & \int_{\beta \times [\barr{M}_{0,n-1}(X,\beta)]} \, \gamma_1 \, \times \, {\rho_2}^*(\gamma_2) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n) \\ & = & \int_\beta \gamma_1 \cdot \int_{\barr{M}_{0,n-1}(X,\beta)} {\rho_2}^*(\gamma_2) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n) , \end{eqnarray*} as asserted. \vspace{+10pt} It should be noted that the generic element of $\overline{M}_{0,0}(X, \beta)$ may not be a {\em birational} map of $\mathbf P^1$ to $X$. This is seen immediately for $X\stackrel {\sim}{=} \mathbf P^1$ where the generic element of $\overline{M}_{0,0}(\mathbf P^1, d)$ is a $d$-fold branched covering of $\mathbf P^1$. This phenomenon occurs in higher dimensions. For example, let $X$ be the complete flag variety $\mathbf{Fl}(\mathbb{C}^3)$ (the space of pairs $(p,l)$ satisfying $p\in l$ where $p$ and $l$ are a point and a line in $\mathbf P^2$). Let $\beta \in A_1 \mathbf{Fl}(\mathbb{C}^3)$ be the class of the curve $\mathbf P^1 \subset \mathbf{Fl}(\mathbb{C}^3)$ determined by all pairs $(p,l)$ for a fixed line $l$. One computes $\int_{\beta} c_1(T_{\mathbf{Fl}(\mathbb{C}^3)})=2$, so the dimension of $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), \beta)$ is $3+2-3=2$ by Theorem 2. Directly, one sees that $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), \beta)$ is isomorphic to the space of lines in $\mathbf P^2$. In particular, $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), \beta)$ has no boundary. As in the case of $\mathbf P^1$, it is seen that every element of $M_{0,0}(\mathbf{Fl}(\mathbb{C}^3), 2\beta)$ corresponds to a double cover of an element of $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), \beta)$. The boundary of $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), 2\beta)$ consists of degenerate double covers. Note also that every element of $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), 2\beta)$ has a nontrivial automorphism. Since the space of image curves of maps in $\barr{M}_{0,0}(\mathbf{Fl}(\mathbb{C}^3), 2\beta)$ is only $2$-dimensional, it follows that all Gromov-Witten invariants of $\mathbf{Fl(\mathbb{C}^3)}$ of the form $I_{2\beta}(\gamma_1\cdots \gamma_n)$ vanish. \section{{\bf Quantum cohomology }} We keep the notation of section 7. Let $T_0 = 1 \in A^0X$, let $T_1, \ldots , T_p$ be a basis of $A^1X$, and let $T_{p+1}, \ldots , T_m$ be a basis for the other cohomology groups. The classes of Schubert varieties form the natural basis for homogeneous varieties. The fundamental numbers counted by the Gromov-Witten invariants are the numbers \begin{equation} \label{idad} N(n_{p+1}, \ldots , n_m; \beta) = I_\beta({T_{p+1}}^{n_{p+1}} \cdots {T_m}^{n_m}) \end{equation} for $n_i\geq 0$. The invariant (\ref{idad}) is nonzero only when $\sum n_i \left(\text{codim} (T_i) - 1 \right) = \dim X + \int_\beta c_1(T_X) \, - \, 3$. In this case, it is the number of pointed rational maps meeting $n_i$ general representatives of $T_i$ for each $i$, $p+1 \leq i \leq m$. Define the numbers $g_{i j}, 0 \leq i, j \leq m$, by the equations \begin{equation} g_{i j} = \int_X T_i \mathbin{\text{\scriptsize$\cup$}} T_j . \end{equation} (If the $T_i$ are the Schubert classes, then for each $i$ there is a unique $j$ such that $g_{ij}\neq 0$. For this $j$, $g_{ij}=1$.) Define $\left( g^{i j}\right)$ to be the inverse matrix to the matrix $\left( g_{i j}\right)$. Equivalently, the class of the diagonal $\Delta$ in $X \times X$ is given by the formula \begin{equation} [\Delta] = \sum_{e\, f} g^{e f} \, T_e \otimes T_f \end{equation} in $A^*(X \times X) = A^*X \otimes A^*X$. The following equations hold: \begin{equation} T_i \mathbin{\text{\scriptsize$\cup$}} T_j = \sum_{e, \, f} \left( \int_X T_i \mathbin{\text{\scriptsize$\cup$}} T_j \mathbin{\text{\scriptsize$\cup$}} T_e \right) g^{e f} T_f = \sum_{e, \, f} I_0(T_i {\cdot} T_j {\cdot} T_e ) g^{e f} T_f . \label{2.3} \end{equation} The idea is to define a ``quantum deformation'' of the cup multiplication of (\ref{2.3}) by allowing nonzero classes $\beta$. Here enters a key idea from physics -- to write down a ``potential function'' that carries all the enumerative information. Define, for a class $\gamma$ in $A^*X$, \begin{equation} \label{2.4} \Phi(\gamma) = \sum_{n \geq 3} \sum_\beta \frac{1}{n!} I_{\beta}(\gamma^n) , \end{equation} where $\gamma^n$ denotes $\gamma \cdots \gamma$ ($n$ times). \begin{lm} \label{uunet} For a given integer $n$, there are only finitely many effective classes $\beta\in A_1 X$ such that $I_\beta(\gamma^n)$ is not zero. \end{lm} \noindent {\em Proof.} Since $X$ is a homogeneous space, the effective classes in $A_1 X$ are the non-negative linear combination of finitely many (nonzero) effective classes $\beta_1, \ldots, \beta_p$. By Lemma \ref{george}, $\int_{\beta_i} c_1(T_X)\geq 2$. Hence, for a given integer $N$, there are only a finite number of effective $\beta$ for which $\int_\beta c_1(T_X) \leq N$. If $I_\beta(\gamma^n)$ is nonzero, then $$\text{dim} \\barr{M}_{0,n}(X, \beta) \leq n \cdot \text{dim}\ X$$ which implies that $\int_\beta c_1(T_X) \leq (n-1) \cdot \text{dim}\ X +3 -n$. \qed \vspace{+10pt} \noindent Let $\gamma = \sum y_i \, T_i$. By Lemma \ref{uunet}, $\Phi(\gamma) = \Phi(y_0, \ldots , y_m)$ becomes a formal power series in $\mathbb Q[[y]] = \mathbb Q[[y_0, \ldots , y_m]]$: \begin{equation} \Phi(y_0, \ldots , y_m) = \sum_{n_0 + \ldots + n_m \geq 3} \sum_\beta I_\beta({T_0}^{n_0} \cdots {T_m}^{n_m} ) \frac{{y_0}^{n_0}}{n_0!} \cdots \frac{{y_m}^{n_m}}{n_m!} \label{2.5} . \end{equation} Define $\Phi_{i j k}$ to be the partial derivative: \begin{equation} \Phi_{i j k} = \frac{\partial^3\Phi}{\partial y_i \, \partial y_j \partial y_k} \; , \; \; 0 \leq i, j, k \leq m . \end{equation} A simple formal calculation, using (\ref{2.5}), gives the following equivalent formula: \begin{equation} \Phi_{i j k} = \sum_{n \geq 0} \sum_\beta \frac{1}{n!} \, I_\beta(\gamma^n \cdot {T_i {\cdot} T_j {\cdot} T_k}) . \label{2.7} \end{equation} Now we define a new ``quantum'' product $*$ by the rule: \begin{equation} T_i \, * \, T_j = \sum_{e, \, f}\Phi_{i j e} \, g^{e f} \, T_f . \label {2.8} \end{equation} The product in (\ref{2.8}) is extended $\mathbb Q[[y]]$-linearly to the $\mathbb Q[[y]]$-module $A^*X\otimes_{\mathbb Z}\mathbb Q[[y]]$, thus making it a $\mathbb Q[[y]]$-algebra. One thing is evident from this remarkable definition: this product is commutative, since the partial derivatives are symmetric in the subscripts. It is less obvious, but not difficult, to see $T_0 = 1$ is a unit for the $*$-product. In fact, it follows from property (I) of section 7, together with (\ref {2.7}), that $$\Phi_{0 j k} \, = \, I_0(T_0 {\cdot} T_j {\cdot} T_k) \, = \, \int_X T_j \mathbin{\text{\scriptsize$\cup$}} T_k \, = \, g_{j k} ,$$ and from this we see that $T_0 * T_j = \sum g_{j e} \, g^{e f} \, T_f = T_j$. The essential point, however, is the associativity: \begin{tm} \label{assss} This definition makes $A^*X\otimes\mathbb Q[[y]]$ into a commutative, associative $\mathbb Q[[y]]$-algebra, with unit $T_0$. \end{tm} We start the proof by writing down what associativity says: $$ (T_i \, * \, T_j) \, * \, T_k \, = \, \sum_{e, \, f} \, \Phi_{i j e} \, g^{e f} \, T_f \, * \, T_k \, = \, \sum_{e, \, f} \, \sum_{c, \, d}\Phi_{i j e} \, g^{e\, f} \, \Phi_{f k c} \, g^{c d} \, T_d , $$ $$ T_i \, * \, (T_j \, * \, T_k) \, = \, \sum_{e, \, f} \, \Phi_{j k e} \, g^{e f} \, T_i \, * \, T_f \, = \, \sum_{e, \, f} \, \sum_{c, \, d}\Phi_{j k e} \, g^{e f} \, \Phi_{i f c} \, g^{c d} \, T_d . $$ Since the matrix $\left( g^{c\, d} \right)$ is nonsingular, the equality of $(T_i \, * \, T_j) \, * \, T_k$ and $T_i \, * \, (T_j \, * \, T_k)$ is equivalent to the equation $$ \sum_{e, \, f}\Phi_{i j e} \, g^{e f} \, \Phi_{f k l} \, = \, \sum_{e, \, f}\Phi_{j k e} \, g^{e f} \, \Phi_{i f l} $$ for all $l$. If we set \begin{equation} \label{rrrr} F(i,j \mid k,l) \, = \, \sum_{e, \, f}\Phi_{i j e} \, g^{e f} \, \Phi_{f k l} , \end{equation} and use the symmetry $\Phi_{i f l} = \Phi_{f i l}$, we see that the associativity is equivalent to the equation \begin{equation} F(i,j \mid k,l) \, = \, F(j,k \mid i,l) . \label {2.9} \end{equation} It follows from (\ref {2.7}) that \begin{equation} F(i,j \mid k,l) \, = \, \sum \frac{1}{n_1! \, n_2!} I_{\beta_1}(\gamma^{n_1} {\cdot} T_i {\cdot} T_j {\cdot} T_e) \, g^{e f} \, I_{\beta_2}(\gamma^{n_2} {\cdot} T_k {\cdot} T_l {\cdot} T_f) , \label {2.10} \end{equation} where the sum is over all nonnegative $n_1$ and $n_2$, over all $\beta_1$ and $\beta_2$ in $A_1X$, and over all $e$ and $f$ from $0$ to $m$. We need the following lemma. Recall from section 6, the divisor $D(A,B;\beta_1, \beta_2)$. In case $A$ and $B$ are nonempty, $$ D(A,B;\beta_1,\beta_2) = \barr{M}_{0,A\cup\{\bullet\}}(X,\beta_1) \, \times_X \, \barr{M}_{0,B\cup\{\bullet\}}(X,\beta_2) . $$ \begin{lm} Let $\iota$ denote the natural inclusion of $D(A,B;\beta_1\beta_2)$ in the Cartesian product $\barr{M}_{0,A\cup\{\bullet\}}(X,\beta_1) \, \times \, \barr{M}_{0,B\cup\{\bullet\}}(X,\beta_2)$, and let $\alpha$ be the embedding of $D(A,B;\beta_1, \beta_2)$ as a divisor in $\barr{M}_{0,n}(X,\beta)$, with $\beta=\beta_1+\beta_2$. Then for any classes $\gamma_1, \ldots , \gamma_n$ in $A^*X$, $$ \iota_* \circ \alpha^* ({\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n)) = $$ $$ \sum_{e, \, f} g^{e f} \left( \prod_{a \in A} {\rho_a}^*(\gamma_a) {\cdot} {\rho_\bullet}^*(T_e) \right) \, \times \, \left( \prod_{b \in B} {\rho_b}^*(\gamma_b) {\cdot} {\rho_\bullet}^*(T_f) \right) . $$ \label{toast} \end{lm} \noindent {\em Proof.} Let $M_1 = \barr{M}_{0,A\cup\{\bullet\}}(X,\beta_1)$, $M_2 = M_{0,B\cup\{\bullet\}}(X,\beta_2)$, $M = \barr{M}_{0,n}(X,\beta)$, and $D = D(A,B;\beta_1\beta_2)$. From the identification of $D$ with $M_1 \, \times_X \, M_2$, we have a commutative diagram, with the right square a fiber square: \begin{equation} \begin{CD} M @<{\alpha}<< D @>{\iota}>> M_1 \times M_2 \\ @V{\rho}VV @V{\eta}VV @VV{\rho'}V \\ X^n @<<{p}< X^{n+1} @>>{\delta}> X^{n+2} \\ \end{CD} \end{equation} Here $\rho$ is the product of the evaluation maps denoted $\rho_i$, $\rho'$ is the product of maps $\rho_i$ and the two others denoted $\rho_\bullet$, $\delta$ is the diagonal embedding that repeats the last factor, and $p$ is the projection that forgets the last factor. Then we have \begin{eqnarray*} \iota_* \circ \alpha^* ({\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n)) & = & \iota_* \circ \alpha^* \circ \rho^* (\gamma_1 \times \cdots \times \gamma_n) \\ & = & \iota_* \circ \eta^* \circ p^*(\gamma_1 \times \ldots \times \gamma_n)\\ & = & \iota_* \circ \eta^*(\gamma_1 \times \ldots \times \gamma_n \times [X]) \\ & = & \rho'{}^* \circ \delta_*(\gamma_1 \times \ldots \times \gamma_n \times [X]) \\ & = & \rho'{}^* (\gamma_1 \times \ldots \times \gamma_n \times [\Delta]) \\ & = & \sum_{e, \, f} \, g^{e f}\rho'{}^*(\gamma_1 \times \ldots \times \gamma_n \times T_e \times T_f) \end{eqnarray*} $$= \sum_{e, \, f} \, g^{e f} \left( \prod_{a \in A} {\rho_a}^*(\gamma_a) {\cdot} {\rho_\bullet}^*(T_e) \right) \, \times \, \left( \prod_{b \in B} {\rho_b}^*(\gamma_b) {\cdot} {\rho_\bullet}^*(T_f) \right) .$$ \qed \vspace{+10pt} Fix $\beta\in A_1 X$ and $\gamma_1, \ldots , \gamma_n \in A^*X$, and fix four distinct integers $q, r, s$, and $t$ in $[n]$. Set \begin{equation} G(q,r \mid s,t) \, = \, \sum g^{e f} I_{\beta_1} \left( \prod_{a \in A} \gamma_a {\cdot} T_e \right) \cdot I_{\beta_2} \left( \prod_{b \in B} \gamma_b {\cdot} T_f \right) , \label{2.11} \end{equation} where the sum is over all partitions of $[n]$ into two sets $A$ and $B$ such that $q$ and $r$ are in $A$ and $s$ and $t$ are in $B$, and over all $\beta_1$ and $\beta_2$ that sum to $\beta$, and over $e$ and $f$ between $0$ and $m$. It follows from Lemma \ref{toast} that $$ G(q,r \mid s,t) = \sum \int_ {D(A,B;\beta_1,\beta_2)} {\rho_1}^*\gamma_1 \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*\gamma_n , $$ the sum over $A$ and $B$ and $\beta_1$ and $\beta_2$ as above. Now Proposition \ref{needit} from section 7 implies \begin{equation} G(q,r \mid s,t) \, = \, G(r,s \mid q,t) . \label{2.12} \end{equation} Apply (\ref{2.12}) in the following case : $$\gamma_i = \gamma, \ \ \ \text{for} \ \ 1 \leq i \leq n - 4, $$ $$\gamma_{n-3} = T_i, \ \ \gamma_{n-2} = T_j, \ \ \gamma_{n-1} = T_k, \ \ \gamma_n = T_l,$$ $$q = n-3, \ \ r = n-2, \ \ s = n-1, \ \ t = n.$$ Then (\ref{2.11}) becomes $$ G(q,r \mid s,t) = \sum \binom {n-4}{n_1 - 2} g^{e f} I_{\beta_1}(\gamma^{n_1-2} {\cdot} T_i {\cdot} T_j {\cdot} T_e) \cdot I_{\beta_2}(\gamma^{n_2-2} {\cdot} T_k {\cdot} T_l {\cdot} T_f) , $$ the sum over $n_1$ and $n_2$, each at least $2$, adding to $n$, and $\beta_1$ and $\beta_2$ adding to $\beta$; the binomial coefficient is the number of partitions $A$ and $B$ for which $A$ has $n_1$ elements, and $B$ has $n_2$ elements. This can be rewritten \begin{equation} G(q,r \mid s,t) = n! \sum \frac{1}{n_1! \, n_2!} g^{ef}I_{\beta_1}(\gamma^{n_1} {\cdot} T_i {\cdot} T_j {\cdot} T_e) \cdot I_{\beta_2}(\gamma^{n_2} {\cdot} T_k {\cdot} T_l {\cdot} T_f) , \label{2.13} \end{equation} the sum over nonnegative $n_1$ and $n_2$ adding to $n-4$, and $\beta_1$ and $\beta_2$ adding to $\beta$. The required equality (\ref{2.9}) then follows immediately from (\ref{2.12}) and (\ref{2.13}), together with (\ref{2.10}). This completes the proof of Theorem \ref{assss}. While the definition of the quantum cohomology ring depends upon a choice of basis $T_0, \ldots, T_m$ of $A^* X$, the rings obtained from different basis choices are canonically isomorphic. The variables $y_0, \ldots, y_m$ should be identified with the dual basis to $T_0, \ldots, T_m$. If $T_0', \ldots, T_m'$ is another basis of $A^* X$ and $T_i'= \sum a_{ij} T_j$ is the change of coordinates, let \begin{equation} \label{duell} y_i = \sum a_{ji}y'_j \end{equation} be the dual coordinate change. Relation (\ref{duell}) yields an isomorphism of $\mathbb Q$-vector spaces $$A^* X \otimes \mathbb Q[[y]] \stackrel {\sim}{=} A^*X \otimes \mathbb Q[[y']].$$ It is easy to check that the quantum products defined respectively on the left and right by the $T$ and $T'$ bases agree with this identification. Let $V$ denote the underlying free abelian group of $A^* X$. Let $\mathbb Q[[V^*]]$ be the completion of the graded polynomial ring $\bigoplus_{i=0}^{\infty} Sym^i(V^*)\otimes \mathbb Q$ at the unique maximal graded ideal. The quantum product defines a canonical ring structure on the free $\mathbb Q[[V^*]]$-module $V \otimes_\mathbb Z \mathbb Q[[V^*]]$. Let $QH^* X = (V \otimes_\mathbb Z \mathbb Q[[V^*]], *)$ denote the quantum cohomology ring. There is a canonical injection of abelian groups $$\iota: A^* X \hookrightarrow QH^* X$$ determined by $\iota(v)=v \otimes 1$ for $v\in V$. The injection $\iota$ is {\em not} compatible with the $\mathbin{\text{\scriptsize$\cup$}}$ and $*$ products. It is worth noting that the quantum cohomology ring $QH^* X$ is {\em not} in general a formal deformation of $A^* X$ over the local ring $\mathbb Q[[V^*]]$. It can be seen directly from the definitions that the $*$-product does not specialize to the $\mathbin{\text{\scriptsize$\cup$}}$-product when the formal parameters are set to 0. At the end of section 9, a presentation of $QH^* \mathbf P^2$ shows explicitly the difference between $A^* \mathbf P^2$ and the specialization of $QH^* \mathbf P^2$. In section 10, a ring deformation of $A^* X$ will be constructed via a smaller quantum cohomology ring. \section{\bf Applications to enumerative geometry} We write the potential function as a sum: $$ \Phi(y_0, \ldots , y_m) = \Phi_{\text{classical}}(y) + \Phi_{\text{quantum}}(y). $$ The classical part has the terms for $\beta=0$: $$ \Phi_{\text{classical}}(y) = \sum_{n_0 + \ldots + n_m = 3} \int_X \left({T_0}^{n_0} \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {T_m}^{n_m}\right) \frac{{y_0}^{n_0}}{n_0!} \cdots \frac{{y_m}^{n_m}}{n_m!} . $$ Since the associativity equations involve only third derivatives, we can modify $\Phi$ by any terms of degree at most 2. Using properties (I)--(III) of section 7, we see that $\Phi_{\text{quantum}}(y)$ can be replaced by $\Gamma(y)$: $$ \Gamma(y) = \sum_{n_{p+1} + \ldots + n_m \geq 0} \sum_{\beta \neq 0} N(n_{p+1}, \ldots , n_m; \beta) \prod_{i = 1}^p e^{\left(\int _\beta T_i\right) y_i} \prod_{i = p+1}^m \frac{{y_i}^{n_i}}{n_i!} , $$ where $N(n_{p+1}, \ldots , n_m; \beta) = I_\beta({T_{p+1}}^{n_{p+1}} \cdots {T_m}^{n_m})$. The partial derivatives of $\Phi_{\text{classical}}$ involve only the numbers $\int_X T_i \mathbin{\text{\scriptsize$\cup$}} T_j \mathbin{\text{\scriptsize$\cup$}} T_k$, while $\Gamma$ involves the interesting enumerative geometry numbers. From this form of $\Gamma$, it is easy to read off its partial derivatives. Let us look again at the projective plane from this point of view. Take the obvious basis: $T_0 = 1$, $T_1$ the class of a line, and $T_2$ the class of a point. Note that $g_{i j}$ is $1$ if $i + j = 2$, and $0$ otherwise, so the same is true for $g^{i j}$. Therefore, $$ T_i \, * \, T_j = \Phi_{i j 0} T_2 + \Phi_{i j 1} T_1 + \Phi_{i j 2} T_0 . $$ For example, \begin{eqnarray*} T_1 \, * \, T_1 & = & T_2 + \Gamma_{111} T_1 + \Gamma_{112} T_0 , \\ T_1 \, * \, T_2 &= & \Gamma_{1 21} T_1 + \Gamma_{122} T_0 ,\\ T_2 \, * \, T_2 & = & \Gamma_{221} T_1 + \Gamma_{222} T_0 . \end{eqnarray*} Therefore, $$ (T_1 \, * \, T_1) \, * \, T_2 = (\Gamma_{221} T_1 + \Gamma_{222} T_0) + \Gamma_{111}(\Gamma_{121} T_1 + \Gamma_{122} T_0) + \Gamma_{112} T_2 , $$ $$ T_1 \, * \, (T_1 \, * \, T_2) = \Gamma_{121} (T_2 + \Gamma_{111} T_1 + \Gamma_{112} T_0) + \Gamma_{122} T_1 . $$ The fact that the coefficients of $T_0$ must be equal in these last two expressions gives the equation: \begin{equation} \Gamma_{222} = {\Gamma_{112}}^2 - \Gamma_{111} \, \Gamma_{122} . \label{2.14} \end{equation} If $\beta = d[\text{line}]$, the number $N(n,\beta)$ is nonzero only when $n = 3d-1$, when it is the number $N_d$ of plane rational curves of degree $d$ passing through $3d-1$ general points. So, $$ \Gamma(y) = \sum_{d \geq 1} N_d e^{dy_1} \frac{{y_2}^{3d-1}}{(3d-1)!} . $$ From this we read off the partial derivatives: \begin{eqnarray*} \Gamma_{222} & = & \sum_{d \geq 2} N_d e^{dy_1} \frac{{y_2}^{3d-4}}{(3d-4)!} \\ \Gamma_{112} & = & \sum_{d \geq 1} d^2 N_d e^{dy_1} \ \frac{{y_2}^{3d-2}}{(3d-2)!} \\ \Gamma_{111} & = & \sum_{d \geq 1} d^3 N_d e^{dy_1} \ \frac{{y_2}^{3d-1}}{(3d-1)!} \\ \Gamma_{122} & = & \sum_{d \geq 1} d N_d e^{dy_1} \ \frac{{y_2}^{3d-3}}{(3d-3)!} . \end{eqnarray*} Therefore, $$ {\Gamma_{112}}^2 = \sum_{d \geq 2} \sum_{d_1 + d_2 = d}{d_1}^2 N_{d_1} {d_2}^2 N_{d_2} e^{dy_1} \frac{{y_2}^{3d-4}}{(3d_1-2)! \ (3d_2-2)!} ,$$ $$\Gamma_{111} \, \Gamma_{122} = \sum_{d \geq 2} \sum_{d_1 + d_2 = d}{d_1}^3 N_{d_1} d_2 N_{d_2} e^{dy_1} \frac{{y_2}^{3d-4}}{(3d_1-1)! \ (3d_2-3)!} . $$ In all these sums, $d_1$ and $d_2$ are positive. Equating the coefficients of $$e^{dy_1} {y_2}^{3d-4}/(3d-4)!,$$ we get the identity $(d\geq 2)$: \begin{equation} N_d = \sum_{d_1 + d_2 = d} N_{d_1} N_{d_2} \left[ {d_1}^2 {d_2}^2 \ \binom {3d-4}{3d_1-2} - {d_1}^3 d_2 \ \binom {3d-4}{3d_1-1} \right] . \label{2.15} \end{equation} Here a binomial coefficient $\binom n m$ is defined to be zero if any of $n$, $m$, or $n - m$ is negative. This is the recursion formula discussed in the introduction. Note that the quantum formalism has removed any necessity to be clever. One simply writes down the associativity equations, and reads off enumerative information. One can organize the information in these associativity equations more systematically as follows (see [DF-I]). Let $F(i,j \mid k,l)$ be defined by (\ref {rrrr}). For $0 \leq i,j,k,l \leq m$, define: \begin{equation*} A(i,j,k,l) = F(i,j\mid k,l)- F(j,k \mid i,l) \end{equation*} $$ = \sum_{e,f} \Phi_{ije}g^{ef}\Phi_{fkl} - \Phi_{jke} g^{ef}\Phi_{fil}.$$ Associativity (Theorem 4) amounts to the equations $A(i,j,k,l)=0$ for all $i,j,k,l$. The symmetry of $\Phi_{ijk}$ in the subscripts and $g^{ef}$ in the superscripts and the basic facts about $\Phi_{0jk}$ imply: \begin{enumerate} \item[(i)] $A(k,j,i,l) = -A(i,j,k,l)$, \item[(ii)] $A(l,k,j,i) = A(i,j,k,l)$, \item[(iii)] $A(i,j,k,l)=0$ if $i=k$ or $j=l$ or if any of the indices $i,j,k,l$ equals 0. \end{enumerate} We consider equations equivalent if they differ by sign. For distinct $i,j,k,l$, the 24 possible equations divide into 3 groups of 8. The equation $A(i,j,k,l)=0$ that says $F(i,j\mid k,l)= F (j,k\mid i,l)$ can be labelled by a duality diagram from topological field theory (see [DF-I]): \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 1.905 24.765 3.175 23.495 / \putrule from 3.175 23.495 to 5.080 23.495 \plot 5.080 23.495 6.350 24.765 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 5.080 23.495 6.350 22.225 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.175 23.495 1.905 22.225 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 13.003 22.528 11.733 21.258 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.268 25.711 12.998 24.441 / \putrule from 12.998 24.441 to 12.998 22.536 \plot 12.998 22.536 14.268 21.266 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 12.990 24.450 11.720 25.720 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 8.871 23.417 8.968 23.534 9.052 23.609 9.129 23.647 9.207 23.654 9.303 23.601 9.368 23.496 9.432 23.391 9.525 23.336 9.599 23.340 9.673 23.373 9.756 23.439 9.855 23.544 / \put{\SetFigFont{10}{12.0}{it}j} [lB] at 11.225 21.004 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 1.446 24.591 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 1.367 21.939 \put{\SetFigFont{10}{12.0}{it}l} [lB] at 6.653 24.575 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 6.636 22.035 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 11.159 25.542 \put{\SetFigFont{10}{12.0}{it}l} [lB] at 14.590 25.624 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 14.556 20.894 \linethickness=0pt \putrectangle corners at 1.367 26.056 and 14.590 20.786 \endpicture} \end{center} \vspace{-0pt} \noindent This diagram corresponds to the equations: $$A(i,j,k,l) = A(j,i,l,k) = A(k,l,i,j) = A(l,k,j,i) = 0$$ $$-A(i,l,k,j) = -A(k,j,i,l) = -A(l,i,j,k) = -A(j,k,l,i) = 0.$$ To obtain the equations, read the labels around the left or right diagram (either clockwise or counterclockwise, but always reading two grouped together at an end first). The other sixteen equations correspond similarly to the diagrams: \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 20.858 24.748 22.128 23.478 / \putrule from 22.128 23.478 to 24.033 23.478 \plot 24.033 23.478 25.303 24.748 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 24.033 23.478 25.303 22.208 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 22.128 23.478 20.858 22.208 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 31.955 22.511 30.685 21.241 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 33.221 25.694 31.951 24.424 / \putrule from 31.951 24.424 to 31.951 22.519 \plot 31.951 22.519 33.221 21.249 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 31.943 24.433 30.673 25.703 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.001 24.748 4.271 23.478 / \putrule from 4.271 23.478 to 6.176 23.478 \plot 6.176 23.478 7.446 24.748 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 6.176 23.478 7.446 22.208 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.271 23.478 3.001 22.208 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.099 22.511 12.829 21.241 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 15.365 25.694 14.095 24.424 / \putrule from 14.095 24.424 to 14.095 22.519 \plot 14.095 22.519 15.365 21.249 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.086 24.433 12.816 25.703 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 27.824 23.400 27.921 23.517 28.004 23.592 28.081 23.630 28.160 23.637 28.255 23.584 28.321 23.479 28.385 23.374 28.478 23.319 28.551 23.323 28.626 23.356 28.709 23.422 28.808 23.527 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 9.967 23.400 10.065 23.517 10.148 23.592 10.225 23.630 10.304 23.637 10.399 23.584 10.464 23.479 10.529 23.374 10.621 23.319 10.695 23.323 10.769 23.356 10.852 23.422 10.952 23.527 / \put{\SetFigFont{9}{10.8}{rm}and } [lB] at 17.340 23.273 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 20.398 24.575 \put{\SetFigFont{10}{12.0}{it}l} [lB] at 25.605 24.558 \put{\SetFigFont{10}{12.0}{it}l} [lB] at 33.543 25.607 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 20.254 21.859 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 25.590 21.905 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 30.048 25.510 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 30.048 21.035 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 33.513 21.006 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 2.542 24.575 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 2.464 21.922 \put{\SetFigFont{10}{12.0}{it}l} [lB] at 7.749 24.558 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 7.732 22.018 \put{\SetFigFont{10}{12.0}{it}l} [lB] at 15.687 25.607 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 15.653 20.877 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 12.368 20.860 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 12.291 25.480 \linethickness=0pt \putrectangle corners at 2.464 26.039 and 33.543 20.758 \endpicture} \end{center} \vspace{-0pt} \noindent In practice, one only needs to write down one equation for each such diagram. When 3 of the 4 labels are distinct, say $i,i,j,k$, there is only 1 equation up to sign (which occurs 8 times). It corresponds to: \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 1.905 24.765 3.175 23.495 / \putrule from 3.175 23.495 to 5.080 23.495 \plot 5.080 23.495 6.350 24.765 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 5.080 23.495 6.350 22.225 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.175 23.495 1.905 22.225 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 13.003 22.528 11.733 21.258 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.268 25.711 12.998 24.441 / \putrule from 12.998 24.441 to 12.998 22.536 \plot 12.998 22.536 14.268 21.266 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 12.990 24.450 11.720 25.720 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 8.871 23.417 8.968 23.534 9.052 23.609 9.129 23.647 9.207 23.654 9.303 23.601 9.368 23.496 9.432 23.391 9.525 23.336 9.599 23.340 9.673 23.373 9.756 23.439 9.855 23.544 / \put{\SetFigFont{10}{12.0}{it}i} [lB] at 1.446 24.591 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 11.159 25.542 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 1.414 21.924 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 6.572 24.606 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 6.716 21.893 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 11.159 21.002 \put{\SetFigFont{10}{12.0}{it}k} [lB] at 14.541 25.512 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 14.624 20.974 \linethickness=0pt \putrectangle corners at 1.414 25.948 and 14.624 20.841 \endpicture} \end{center} \vspace{-0pt} \noindent When two labels are distinct, there is again only 1 equation up to sign (occurring 4 times): \vspace{-0pt} \begin{center} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.40000cm,0.40000cm> \unitlength=0.40000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 1.905 24.765 3.175 23.495 / \putrule from 3.175 23.495 to 5.080 23.495 \plot 5.080 23.495 6.350 24.765 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 5.080 23.495 6.350 22.225 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.175 23.495 1.905 22.225 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 13.003 22.528 11.733 21.258 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 14.268 25.711 12.998 24.441 / \putrule from 12.998 24.441 to 12.998 22.536 \plot 12.998 22.536 14.268 21.266 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 12.990 24.450 11.720 25.720 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 8.871 23.417 8.968 23.534 9.052 23.609 9.129 23.647 9.207 23.654 9.303 23.601 9.368 23.496 9.432 23.391 9.525 23.336 9.599 23.340 9.673 23.373 9.756 23.439 9.855 23.544 / \put{\SetFigFont{10}{12.0}{it}i} [lB] at 1.446 24.591 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 11.159 25.542 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 1.414 21.924 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 6.716 21.893 \put{\SetFigFont{10}{12.0}{it}i} [lB] at 11.159 21.002 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 14.624 20.974 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 6.684 24.591 \put{\SetFigFont{10}{12.0}{it}j} [lB] at 14.592 25.514 \linethickness=0pt \putrectangle corners at 1.414 26.048 and 14.624 20.841 \endpicture} \end{center} \vspace{-0pt} \noindent The symmetry in these diagrams reflects the symmetry in the equations. Taking just one equation for each diagram, one sees that the number $N(m)$ of equations for $\text{rank} (A^*X)=m+1$ is $$ N(m)= 3 \binom {m}{4} + m \binom {m-1}{2}+ \binom{m}{2} = \frac {m(m-1)(m^2-m+2)}{8},$$ so $N(2)=1$, $ N(3)=6$, $N(4)=21$, $N(5)=55$, $N(6)=120$, and $N(7)= 231$. For the complete flag manifold $\mathbf{Fl}(\mathbb{C}^n)$, $m=n!-1$. The number of equations for $\mathbf{Fl}(\mathbb{C}^4)$ is $N(23)=30861$. Let us work this out for the two varieties $X= \mathbf P^3$ and $X= \mathbf{Q}^3$ (a smooth quadric 3-fold), which have very similar classical cohomology rings. Each has a basis : \begin{eqnarray*} T_0 & = & 1,\\ T_1 & = & \text{hyperplane class},\\ T_2 & = & \text{line class}, \\ T_3 & = & \text{point class}. \end{eqnarray*} The difference in the classical product is that $T_1 \mathbin{\text{\scriptsize$\cup$}} T_1= T_2$ for $\mathbf P^3$ but $T_1 \mathbin{\text{\scriptsize$\cup$}} T_1= 2T_2$ for $\mathbf{Q}^3$. Let $c=1$ for $\mathbf P^3$ and $c=2$ for $\mathbf{Q}^3$. The $N(3)=6$ equations are: \begin{align*} \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.25000cm,0.25000cm> \unitlength=0.25000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.486 24.384 \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.507 23.004 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.857 23.971 2.223 23.336 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.606 2.857 23.971 / \putrule from 2.857 23.971 to 3.810 23.971 \plot 3.810 23.971 4.445 24.606 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 23.971 4.445 23.336 / \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.610 24.420 \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.646 22.989 \linethickness=0pt \putrectangle corners at 1.486 24.801 and 4.646 22.894 \endpicture} &\hspace{.4in}& 2 \Gamma_{123}-c\Gamma_{222} & = \Gamma_{111}\Gamma_{222} - \Gamma_{112}\Gamma_{122} \\ \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.25000cm,0.25000cm> \unitlength=0.25000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.507 24.418 \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.496 23.036 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.857 23.971 2.223 23.336 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.606 2.857 23.971 / \putrule from 2.857 23.971 to 3.810 23.971 \plot 3.810 23.971 4.445 24.606 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 23.971 4.445 23.336 / \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.646 22.989 \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 4.650 24.462 \linethickness=0pt \putrectangle corners at 1.496 24.843 and 4.650 22.894 \endpicture} &\hspace{.4in}& \Gamma_{133}-c\Gamma_{223} & = \Gamma_{111}\Gamma_{223} - \Gamma_{113}\Gamma_{122} \\ \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.25000cm,0.25000cm> \unitlength=0.25000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.492 24.416 \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.482 23.036 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.857 23.971 2.223 23.336 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.606 2.857 23.971 / \putrule from 2.857 23.971 to 3.810 23.971 \plot 3.810 23.971 4.445 24.606 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 23.971 4.445 23.336 / \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 4.635 23.036 \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 4.652 24.448 \linethickness=0pt \putrectangle corners at 1.482 24.829 and 4.652 22.940 \endpicture} &\hspace{.4in}& c\Gamma_{233} & = 2\Gamma_{113}\Gamma_{123} - \Gamma_{112}\Gamma_{133} -\Gamma_{111}\Gamma_{233} \\ \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.25000cm,0.25000cm> \unitlength=0.25000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 1.513 23.019 \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.556 24.443 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.857 23.971 2.223 23.336 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.606 2.857 23.971 / \putrule from 2.857 23.971 to 3.810 23.971 \plot 3.810 23.971 4.445 24.606 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 23.971 4.445 23.336 / \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.610 24.420 \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.646 22.989 \linethickness=0pt \putrectangle corners at 1.513 24.824 and 4.646 22.894 \endpicture} &\hspace{.4in}& \Gamma_{233}& = \Gamma_{113}\Gamma_{222} - \Gamma_{112}\Gamma_{223} \\ \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.25000cm,0.25000cm> \unitlength=0.25000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 1.524 22.987 \put{\SetFigFont{6}{7.2}{rm}1} [lB] at 1.501 24.428 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.857 23.971 2.223 23.336 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.606 2.857 23.971 / \putrule from 2.857 23.971 to 3.810 23.971 \plot 3.810 23.971 4.445 24.606 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 23.971 4.445 23.336 / \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.610 24.420 \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 4.604 22.989 \linethickness=0pt \putrectangle corners at 1.501 24.809 and 4.610 22.892 \endpicture} &\hspace{.4in}& \Gamma_{333}& = \Gamma_{123}^2 - \Gamma_{122}\Gamma_{133} +\Gamma_{113}\Gamma_{223}-\Gamma_{112}\Gamma_{233}\\ \font\thinlinefont=cmr5 \begingroup\makeatletter\ifx\SetFigFont\undefined \def#1#2#3#4#5#6}#1#2#3#4#5#6#7\relax{\def#1#2#3#4#5#6}{#1#2#3#4#5#6}}% \expandafter#1#2#3#4#5#6}\fmtname xxxxxx\relax \defsplain{splain}% \ifx#1#2#3#4#5#6}splain \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def#1#2#3#4#5#6}{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter#1#2#3#4#5#6} \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup \mbox{\beginpicture \setcoordinatesystem units <0.25000cm,0.25000cm> \unitlength=0.25000cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 1.507 24.431 \put{\SetFigFont{6}{7.2}{rm}3} [lB] at 1.492 23.019 \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.857 23.971 2.223 23.336 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.223 24.606 2.857 23.971 / \putrule from 2.857 23.971 to 3.810 23.971 \plot 3.810 23.971 4.445 24.606 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 3.810 23.971 4.445 23.336 / \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.610 24.420 \put{\SetFigFont{6}{7.2}{rm}2} [lB] at 4.646 22.989 \linethickness=0pt \putrectangle corners at 1.492 24.812 and 4.646 22.894 \endpicture} &\hspace{.4in}&0 &= \Gamma_{133}\Gamma_{222}-2\Gamma_{123}\Gamma_{223} +\Gamma_{122}\Gamma_{233} \end{align*} The function $\Gamma$ has the form: \begin{equation} \label{dddef} \Gamma= \sum N_{a,b} e^{dy_1} \frac{y_2^a}{a!} \frac{y_3^b}{b!} . \end{equation} For $\mathbf P^3$ the sum in (\ref{dddef}) is over non-negative $a,b$ satisfying $a+2b=4d$, $d\geq 1$. A crucial difference is that for $\mathbf{Q}^3$, the sum in (\ref{dddef}) is over $a+2b=3d$, $d\geq 1$ reflecting the fact that $c_1(T_{\mathbf P^3})=4T_1$ while $c_1(T_{\mathbf{Q}^3})=3T_1$. In each case, $N_{a,b}$ is the number of degree $d$ rational curves in $X$ meeting $a$ general lines and $b$ general points of $X$. Each of the six differential equations above yields a recursion among the $N_{a,b}$: \vspace{+10pt} \noindent (1) For $ a \geq 3,\ b \geq 0$, $\ \ 2d N_{a-2,b+1}-c N_{a,b}=$ $$\sum N_{a_1,b_1}N_{a_2,b_2} \binom{b}{b_1} \bigg( d_1^3 \binom{a-3}{a_1} - d_1^2 d_2 \binom{a-3}{a_1-1} \bigg)$$ \noindent (2) For $ a \geq 2,\ b \geq 1$, $\ \ d N_{a-2,b+1}-c N_{a,b}=$ $$\sum N_{a_1,b_1}N_{a_2,b_2} \binom{a-2}{a_1} \bigg( d_1^3 \binom{b-1}{b_1} - d_1^2 d_2 \binom{b-1}{b_1-1} \bigg)$$ \noindent (3) For $ a \geq 1,\ b \geq 2$, $\ \ c N_{a,b}=$ $$\sum N_{a_1,b_1}N_{a_2,b_2} \bigg(2 d_1^2 d_2 \binom{a-1}{a_1} \binom{b-2}{b_1-1} - d_1^2 d_2 \binom{a-1}{a_1-1} \binom{b-2}{b_1}$$ $$- d_1^3\binom{a-1}{a_1}\binom{b-2}{b_1} \bigg)$$ \noindent (4) For $ a \geq 3,\ b \geq 1$, $\ \ N_{a-2,b+1}=$ $$\sum N_{a_1,b_1}N_{a_2,b_2} d_1^2 \bigg( \binom{a-3}{a_1} \binom{b-1}{b_1-1} - \binom{a-3}{a_1-1} \binom{b-1}{b_1} \bigg)$$ \noindent (5) For $ a \geq 2,\ b \geq 2$, $\ \ N_{a-2,b+1}=$ $$\sum N_{a_1,b_1}N_{a_2,b_2} \bigg( d_1d_2 \binom{a-2}{a_1-1} \binom{b-2}{b_1-1} - d_1 d_2 \binom{a-2}{a_1-2} \binom{b-2}{b_1}$$ $$+ d_1^2\binom{a-2}{a_1}\binom{b-2}{b_1-1} - d_1^2\binom{a-2}{a_1-1}\binom{b-2}{b_1} \bigg)$$ \noindent (6) For $ a \geq 3,\ b \geq 2$, $\ \ 0 =$ $$\sum N_{a_1,b_1}N_{a_2,b_2} d_1 \bigg( \binom{a-3}{a_1} \binom{b-2}{b_1-2} - 2 \binom{a-3}{a_1-1} \binom{b-2}{b_1-1}$$ $$+ \binom{a-3}{a_1-2}\binom{b-2}{b_1} \bigg)$$ In these formulas, the sum is over non-negative $a_1, a_2, b_1, b_2$ satisfying \begin{enumerate} \item[(i)] $a_1+a_2=a$, $b_1+b_2=b$, \item[(ii)] $a+2b=4d$, $a_i+2b_i=4d_i$, $d_i>0$ for $\mathbf P^3$, \item[{}] $a+2b=3d$, $a_i+2b_i=3d_i$, $d_i>0$ for $\mathbf{Q}^3$. \end{enumerate} For $\mathbf P^3$, one starts with the $N_{0,2}=1$ for the number of lines through two points. For $\mathbf{Q}^3$, $N_{1,1}=1$ is not hard to compute directly. In each case, the six recursions are more than enough to solve for all the other $N_{a,b}$. These numbers for $\mathbf P^3$ include the classical results: there are {$N_{4,0}=2$} lines meeting 4 general lines, {$N_{8,0}=92$} conics meeting 8 general lines, and {$N_{12,0}=80160$} twisted cubics meeting 12 general lines. See [DF-I] for more of these numbers$^2$. \footnotetext[2]{The numbers $N_{a,b}$ given in [DF-I] are correct, although their version of equation (6) has a misprint.} For $\mathbf{Q}^3$, computations yield: \begin{tabular}{ll} $(d=1)$ & $ N_{1,1}=1$, $N_{3,0}=1$ \\ $(d=2)$ & $N_{0,3}=1$, $N_{2,2}=1$, $N_{4,1}=2$ , $N_{6,0}=5$ \\ $(d=3)$ & $N_{1,4}=2$, $N_{3,3}=5$, $N_{5,2}=16$, $N_{7,1}=59$, $N_{9,0}=242$ \\ $(d=4)$ & $N_{0,6}=6$, $N_{2,5}=20$, $N_{4,4}=74$, $N_{6,3}=320$, $N_{8,2}=1546$, \\ {} & $N_{10,1}=8148$, $N_{12,0}=46230$ \\ $(d=5)$ & $N_{1,7}=106$, $N_{3,6}=448$, $N_{5,5}=2180$, $N_{7,4}=11910$, \\ & $N_{9,3}=71178$, $N_{11,2}=457788$, $N_{13,1}=3136284$, \\ & $N_{15,0}=22731810$. \end{tabular} The reader is invited to work out the equations for some other simple homogeneous spaces such as $\mathbf P^4$, $\mathbf P^1 \times \mathbf P^1$, $\mathbf{Gr}(2,4)$, or the incidence variety $\mathbf{Fl}(\mathbb{C}^3)$ of points on lines in the plane. For very pleasant excursions along these paths, see [DF-I]. There is a simple method of obtaining a presentation of $QH^* X$ from $\Phi$ and a presentation of $A^*X$. It will be convenient to consider $A^* X_\mathbb Q= H^*(X, \mathbb Q)$, the cohomology ring of $X$ with rational coefficients. Following the notation of section 8, let $QH^* X=(V\otimes _Z \mathbb Q[[V^*]],*)$. There is a canonical embedding: $$\iota_\mathbb Q: A^* X_{\mathbb Q} \hookrightarrow QH^* X$$ of $\mathbb Q$-vector spaces. In the discussion below, $A^* X_{\mathbb Q}$ is viewed as a $\mathbb Q$-subspace of $QH^* X$ via $\iota_\mathbb Q$. The results relating presentations of $A^* X_\mathbb Q$ and $QH^* X$ are established in Propositions \ref{ett} and \ref{tvo}. \begin{pr} \label{ett} Let $z_1,..., z_r$ be homogeneous elements of positive codimension that generate $A^* X_{\mathbb Q}$ as a $\mathbb Q$-algebra. Then, $z_1, \ldots, z_r$ generate $QH^* X$ as a $\mathbb Q[[V^*]]$-algebra. \end{pr} The proof requires a lemma. Note that for $\gamma\in \mathbb Q[[V^*]]$ there is a well-defined constant term $\gamma(0)\in \mathbb Q$. \begin{lm} \label{mygid} Let $T_0, \ldots, T_m$ be any homogeneous $\mathbb Q$-basis of $A^* X_{\mathbb Q}$. Let $w_1, w_2\in A^* X_{\mathbb Q}$ be homogeneous elements. Let $$ w_1 \mathbin{\text{\scriptsize$\cup$}} w_2 = \sum_{k=0}^{m} c_{k}T_k, \ \ c_{k}\in \mathbb Q, $$ $$ w_1 * w_2 = \sum_{k=0}^{m} \gamma_{k} T_k, \ \ \gamma_{k}\in \mathbb Q[[V^*]],$$ be the unique expansions in $A^* X_\mathbb Q$ and $QH^* X$ respectively. \begin{enumerate} \item[(i)] If $\text{\em codim}(T_k)> \text{\em codim}(w_1)+\text{\em codim}(w_2)$, then $\gamma_{k}(0)=0$. \item[(ii)] If $\text{\em codim}(T_k)= \text{\em codim}(w_1)+ \text{\em codim}(w_2)$, then $\gamma_{k}(0)=c_{k}$. \end{enumerate} \end{lm} \noindent {\em Proof.} By linearity of the $*$-product, it can be assumed that $w_1$ and $w_2$ are basis elements $T_i$ and $T_j$ respectively. In the basis $T_0, \ldots, T_m$ of $A^*_\mathbb Q X$, the $*$-product is determined by: $$T_i * T_j= T_i \mathbin{\text{\scriptsize$\cup$}} T_j + \sum_{i=1}^{m} \Gamma_{ijl} g^{lk}T_k $$ where the dual coordinates $y_0, \ldots, y_m$ are taken in $V^*\otimes \mathbb Q$. $\Gamma_{ijl}(0)=\sum_{\beta\neq 0} I_\beta(T_i\cdot T_j \cdot T_l)$. Therefore, if $\Gamma_{ijl}(0) \neq 0$, there must exist a nonzero effective class $\beta\in A_1 X$ such that $$\text{dim} \overline{M}_{0,3}(X, \beta) = \text{codim}(T_i)+ \text{codim}(T_j) +\text{codim}(T_l).$$ Since $X$ is homogeneous, $\int_\beta c_1(X)\geq 2$ by Lemma \ref{george}. By the dimension formula, \begin{equation} \label{arithh} \text{codim}(T_i)+ \text{codim}(T_j) + \text{codim}(T_l) \geq \text{dim}(X)+ 2. \end{equation} Equation (\ref{arithh}) yields $\text{codim}(T_l) \geq \text{dim} (X) - \text{codim}(T_i) -\text{codim}(T_j) +2$. For $g^{lk}$ to be nonzero, it follows that $\text{codim}(T_k) \leq \text{codim}(T_i)+\text{codim}(T_j)-2$. The lemma is proven. \qed \vspace{+10pt} We will apply Lemma \ref{mygid} to products in a basis of $A^* X_\mathbb Q$ consisting of monomials $z^I= z_1^{i_1} \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} z_r^{i_r}$. Let $$z^{*I}= \underbrace{z_1* \cdots *z_1}_{i_1} *\underbrace{z_2*\cdots *z_2}_{i_2}* \cdots *\underbrace{z_r* \cdots *z_r}_{i_r}$$ denote the corresponding monomial in $QH^* X$. Let \begin{equation} \label{monny} \{z^I \ | \ I \in \mathcal{S}\} \end{equation} be a monomial $\mathbb Q$-basis of $A^* X_\mathbb Q$. Choose an ordering of the set $\mathcal{S}$ so that $\text{codim}(z^I) \leq \text{codim} (z^J)$ for $I<J$. Let $$z^{*I}= \sum_{J\in \mathcal{S}} \gamma_{IJ} z^J, \ \ \gamma_{IJ}\in \mathbb Q[[V^*]]$$ be the unique expansion in $QH^* X$. An inductive application of Lemma \ref{mygid} yields: \begin{enumerate} \item[(i)] If $J>I$, then $\gamma_{IJ}(0)=0$. \item[(ii)] $\gamma_{II}(0)=1$. \end{enumerate} Therefore, the matrix $(\gamma_{IJ}(0))$ is invertible over $\mathbb Q$. It follows that the matrix $(\gamma_{IJ})$ is invertible over $\mathbb Q[[V^*]]$. In particular, $\{ z^{*I} \ | \ I \in \mathcal{S}\}$ is a $\mathbb Q[[V^*]]$-basis of $QH^* X$. Proposition \ref{ett} is proved. Let $K$ be the kernel of the surjection $$\phi:\mathbb Q[Z]=\mathbb Q[Z_1, \ldots, Z_r] \rightarrow A^* X_\mathbb Q$$ determined by $\phi(Z_i)=z_i$. Let $K'$ be the kernel of the corresponding surjection $$\phi':\mathbb Q[[V^*]][Z] \rightarrow QH^* X$$ determined by $\phi'(Z_i)=z_i$ Using our choice (\ref{monny}) of monomial basis, there is a method of constructing elements of $K'$ from elements of $K$. Let $f\in K$. The polynomial $f$ is also an element of $\mathbb Q[[V^*]][Z]$. There is a unique expansion: $$\phi'(f)= \sum_{I \in \mathcal{S}} \xi_I z^{*I}, \ \ \xi_I \in \mathbb Q[[V^*]].$$ Then, $f'= f(Z_1, \ldots, ,Z_r)- \sum_{I \in \mathcal{S}} \xi_I Z^I$ is in $K'$. The ideal $K$ is homogeneous provided the degree of $Z_i$ is taken to be the codimension of $z_i$. We need the following fact. \begin{lm} \label{ghead} Let $f\in K$ be homogeneous of degree $d$ and let $I\in \mathcal{S}$. If $\text{\em deg}(Z^I)\geq d$, then $\xi_I(0)=0$. \end{lm} \noindent {\em Proof.} If $d> \text{dim}(X)$, the statement is vacuous. Assume $d\leq \text{dim}(X)$. Let $\phi'(f)= \sum_{I \in \mathcal{S}} \tilde{\xi}_I z^{I}, \ \ \tilde{\xi}_I \in \mathbb Q[[V^*]]$ be the unique expansion. Apply Lemma \ref{mygid} repeatedly to the monomials of $f$ in the basis $\{ z^I\ | \ I \in \mathcal{S}\}$ of $A^* X_\mathbb Q$. It follows that if $\text{deg}(Z^I)\geq d$, then $\tilde{\xi}_I(0)=0$. The change of basis relations (i) and (ii) for the $\mathbb Q[[V^*]]$-basis $\{ z^{*I} \ | \ I \in \mathcal{S}\}$ now imply the lemma. \qed \vspace{+10pt} Now suppose the elements $f_1, \ldots, f_s$ are homogeneous generators of $K$, so $$A^*X_\mathbb Q = \mathbb Q[Z]/(f_1, \ldots,f_s)$$ is a presentation of the cohomology ring. \begin{pr} \label{tvo} The ideal $K'$ is generated by $f_1', \ldots, f_s'$, so $$QH^*X = \mathbb Q[[V^*]][Z]/(f'_1, \ldots, f'_s)$$ is a presentation of the quantum cohomology ring. \end{pr} \noindent {\em Proof.} Since we have a surjection $$\mathbb Q[[V^*]][Z]/ (f_1' \ldots, f_s') \rightarrow QH^*X$$ and $QH^*X$ is a free $\mathbb Q[[V^*]]$-module with basis $\{ z^{*I} \ | \ I \in \mathcal{S}\}$, it suffices to show that the monomials $\{ Z^{I} \ | \ I \in \mathcal{S}\}$ span the $\mathbb Q[[V^*]]$-module on the left. By Nakayama's lemma, it suffices to show that these monomials generate the $\mathbb Q$-vector space \begin{equation} \label{frod} \mathbb Q[[V^*]][Z]/(f_1', \ldots, f_s', \frak{m}), \end{equation} where $\frak{m}\subset \mathbb Q[[V^*]]$ is the maximal ideal. Let $f'_i= f_i- \sum\xi_{iI} Z^I$. Define $\overline{f}'_i \in \mathbb Q[Z]$ by $\overline{f}'_i= f_i- \sum\xi_{iI}(0) Z^I$. The $\mathbb Q$-algebra (\ref{frod}) can be identified with $$\mathbb Q[Z]/(\overline{f}'_1, \ldots, \overline{f}'_s).$$ By Lemma \ref{ghead}, all the terms $\xi_{iI}(0)Z^I$ have strictly lower degree than $f_i$. It is then a simple induction on the degree to see that the same monomials $\{Z^I\}$ that span modulo $(f_1, \ldots, f_s)$ will also span modulo $(\overline{f}'_1, \ldots, \overline{f}'_s)$. \qed \vspace{+10pt} For example, let $X=\mathbf{P}^2$. Let $Z=Z_1$ and let $A^*_\mathbb Q \mathbf{P}^2=\mathbb Q[Z]/Z^3$ be the standard presentation with the monomial basis $1, Z, Z^2$. A presentation of $QH^* \mathbf{P}^2$ is obtained: \begin{equation} \label{pezzx} QH^*\mathbf{P}^2 \stackrel{\sim}{\rightarrow} \mathbb Q[[y_0,y_1, y_2]][Z] /(Z^3- \Gamma_{111}Z^2 -2 \Gamma_{112} Z - \Gamma_{122}) \end{equation} where $\Gamma$ is the quantum potential of $\mathbf{P}^2$. By (\ref{pezzx}) and the determination of $\Gamma$, $$QH^*\mathbf P^2 \otimes_{\mathbb Q[[V^*]]} \mathbb Q[[V^*]]/\frak{m} = \mathbb Q[Z]/(Z^3-1).$$ Note that $QH^* \mathbf P^2$ does not specialize to $A^* \mathbf P^2$. \section{\bf{Variations}} The algebra $QH^* X= A^*X \otimes \mathbb Q [[V^*]]$ may be regarded as the ``big'' quantum cohomology ring. There is also a ``small'' quantum cohomology ring, $QH^*_{s} X$, that incorporates only the $3$-point Gromov-Witten invariants in its product. $QH^*_{s} X$ is obtained by restricting the $*$-product to the formal deformation parameters of the divisor classes. Most computations of quantum cohomology rings have been of this small ring, which is often easier to describe; the small ring is often denoted $QH^* X$. It is simplest to define $QH^*_{s} X$ in the Schubert basis $T_0, \ldots, T_m$. Let \begin{equation} \overline\Phi_{i j k} = \Phi_{i j k}(y_0, y_1, \ldots , y_p, 0, \ldots , 0) = \int_X T_i \mathbin{\text{\scriptsize$\cup$}} T_j \mathbin{\text{\scriptsize$\cup$}} T_k \ + \ \overline\Gamma_{i j k} \label{2.16}. \end{equation} The modified quantum potential $\overline{\Gamma}_{i j k}$ is determined by $$\overline\Gamma_{i j k} = \sum_{n \geq 0} \frac{1}{n!} \sum_{\beta \neq 0} I_\beta(\gamma^ n {\cdot} T_i {\cdot} T_j {\cdot} T_k)$$ where $\gamma = y_1T_1 + \ldots + y_pT_p$. By the divisor property (III) of section 7, \begin{equation} \overline\Gamma_{i j k} = \sum_{\beta \neq 0} I_\beta(T_i {\cdot} T_j {\cdot} T_k) {q_1}^{\int_\beta T_1} \cdots {q_p}^{\int_\beta T_p} , \label{2.17} \end{equation} where $q_i = e^{y_i}$. Note that only $3$-point invariants occur. Let ${\mathbb Z}[q]=[q_1, \ldots, q_p]$. By Theorem 4, the product $$ T_i \, * \, T_j = \sum_{e, \, f} \overline\Phi_{i j e} g^{e f} T_f \, = \, T_i \mathbin{\text{\scriptsize$\cup$}} T_j \ + \ \sum_{e, \, f} \overline\Gamma_{i j e} g^{e f} T_f $$ then makes the ${\mathbb Z}[q]$-module $A^*X \otimes_{\mathbb Z} {\mathbb Z}[q]$ into a commutative, associative ${\mathbb Z}[q]$-algebra with unit $T_0$. From equation (\ref{2.17}), it easily follows that the small quantum cohomology is a deformation of $A^* X$ is the usual sense: $A^* X$ is recovered by setting the variables $q_i=0$. For example, let $X = \mathbf P^r$. Then, $q=q_1$. If $T_i$ is the class of a linear subspace of codimension $i$ and $\beta$ is $d$ times the class of a line, then the number $I_\beta(T_i {\cdot} T_j {\cdot} T_k)$ can be nonzero only if $i + j + k = r + (r \! + \! 1)d$; this can happen only for $d = 0$ or $d = 1$, and in each case the number is $1$. It follows that, \begin{enumerate} \item[(i)] if $i+j \leq r$, then $T_i\, * \, T_j =T_{i+j}$; \item[(ii)] if $r+1 \leq i+j \leq 2r$, then $T_i\, * \, T_j =qT_{i+j-r-1}$. \end{enumerate} Therefore the small quantum cohomology ring is: $$ QH^*_s \mathbf P^r =\mathbb{Z} [ T,\, q ] / (T^{r+1} - q) , $$ where $T = T_1$ is the class of a hyperplane. The following variation of Proposition \ref{tvo} is valid for the small quantum cohomology ring (cf. [S-T]). As before let $z_1, \ldots, z_r$ be homogenous elements of positive codimension that generate $A^* X$. (We use integer coefficients but rational coefficients could be used as well). Let ${\mathbb Z}[Z]= {\mathbb Z}[Z_1, \ldots, Z_r]$, and let $$A^*X = {\mathbb Z}[Z]/ (f_1, \ldots, f_s)$$ be a presentation with arbitrary homogeneous generators $f_1, \ldots, f_s$ for the ideal of relations. Let ${\mathbb Z}[q,Z]={\mathbb Z} [q_1, \ldots, q_p, Z_1, \ldots, Z_r]$. The variables $q_i, Z_j$ are graded by the following degrees: $\text{deg}(q_i)= \int_{\beta_i} c_1(T_X)$ where $\beta_i$ is the class of the Schubert variety dual to $T_i$ and $\text{deg} (Z_j)= \text{codim}(z_j)$. Let $QH_s^* X= A^*X \otimes {\mathbb Z}[q]$ with the quantum product. \begin{pr} \label{tva} Let $f'_1, \ldots, f'_s$ be any homogeneous elements in ${\mathbb Z}[q,Z]$ such that: \begin{enumerate} \item[(i)] $f_i'(0,\ldots,0,Z_1,\ldots, Z_r)= f_i(Z_1, \ldots,Z_r)$ in ${\mathbb Z}[q,Z]$, \item[(ii)] $f'_i(q_1, \ldots, q_p, Z_1, \ldots, Z_r)=0$ in $QH^*_s X$. \end{enumerate} Then, the canonical map \begin{equation} \label{fffff} {\mathbb Z}[q,Z]/ (f'_1, \ldots, f'_s) \rightarrow QH^*_s X \end{equation} is an isomorphism. \end{pr} \noindent {\em Proof.} The proof is by a Nakayama-type induction. As the arguments are similiar to the proof of Proposition \ref{tvo}, we will be brief. The fact that each $q_i$ has positive degree implies the following statement. If $\psi:M\rightarrow N$ is a homogeneous map of finitely generated ${\mathbb Z}[q,Z]$-modules that is surjective modulo the ideal $(q)=(q_1, \ldots, q_p)$, then $\psi$ is surjective. Hence, by (i), the map (\ref{fffff}) is surjective. Similarly, if $\tilde{T}_0, \ldots, \tilde{T}_m$ are homogeneous lifts to ${\mathbb Z}[Z]$ of a basis of $A^* X$, an easy induction shows that their images in ${\mathbb Z}[q,Z]/(f_1', \ldots, f_s')$ generate this ${\mathbb Z}[q]$-module. Since $QH^*_s X$ is free over ${\mathbb Z}$ of rank $m+1$, the map (\ref{fffff}) must be an isomorphism. \qed \vspace{+10pt} A similar calculation, as in [S-T], yields the small quantum cohomology ring of the Grassmannian $X = \mathbf{Gr}(p,n)$ of $p$-dimensional subspaces of $\mathbb C^n$. Let $k = n - p$, let $0 \rightarrow S \rightarrow {{\mathbb C}^n}_X \rightarrow Q \rightarrow 0$ be the universal exact sequence of bundles on $X$, and let $\sigma_i = c_i(Q)$. Set $S_r(\sigma) = \text{det} \left(\sigma_{1+j-i}\right)_{1 \leq i, j \leq r}$, and let $q = q_1$. \begin{pr} The small quantum cohomology ring of $\mathbf{Gr}(p,n)$ is $$ {\mathbb Z}[\sigma_1, \ldots , \sigma_k, q] / \left(S_{p+1}(\sigma), S_{p+2}(\sigma), \ldots , S_{n-1}(\sigma), S_n(\sigma) + (-1)^kq \right) .$$ \end{pr} \noindent {\em Proof.} We use some standard facts about the Grassmannian. In particular, the cohomology has an additive basis of Schubert classes $\sigma_\lambda$, as $\lambda$ varies over partitions with $k \geq \lambda_1 \geq \ldots \geq \lambda_p \geq 0$; $\sigma_\lambda = [\Omega_\lambda]$ is the class of a Schubert variety $$ \Omega_\lambda = \{ L \in X : \dim L \cap V_{k+i-\lambda_i} \geq i\ \text {for}\ 1 \leq i \leq p \} , $$ where $V_1 \subset V_2 \subset \ldots \subset V_n = \mathbb C^n$ is a given flag of subspaces. In $A^*(X)$, $S_r(\sigma)$ represents the $r^{\text th}$ Chern class of $S^\vee$, from which we have $$ A^*(X) = {\mathbb Z}[\sigma_1, \ldots , \sigma_k] / \left( S_{p+1}(\sigma), \ldots , S_n(\sigma)\right) . $$ By Proposition \ref{tva}, it suffices to show that the relations displayed in the proposition are valid in $QH^*_s X$. Since $c_1(T_X) = n \sigma_1$, a number $I_\beta(\gamma_1 {\cdot} \gamma_2 {\cdot} \gamma_3)$ can be nonzero only if the sum of the codimensions of the $\gamma_i$ is equal to $\dim X + nd$, where $\beta$ is $d$ times the class of a line. If $d \geq 1$, all such numbers vanish when $\text{codim}(\gamma_1) + \text{codim} (\gamma_2) < n$. In particular, the relations $S_i(\sigma) = 0$ for $p < i < n$ remain valid in $QH^*_s X$. From the formal identity $$ S_n(\sigma) - \sigma_1\, S_{n-1}(\sigma) + \sigma_2 \, S_{n- 2}(\sigma) - \ldots + (-1)^k\sigma_k \, S_{n-k}(\sigma) = 0 , $$ we therefore have $S_n(\sigma) = (-1)^{k-1}\sigma_k \, S_{n- k}(\sigma)$ in $QH^*_s X$. Since $S_{n-k}(\sigma) = \sigma_{(1^{n-k})}$, the proof will be completed by verifying that $\sigma_k * \sigma_{(1^{n-k})} = q$. Equivalently, when $\beta$ is the class of a line, we must show that $$ I _\beta (\sigma_k , \sigma_{(1^p)}, \sigma_{(k^p)}) = 1 . $$ This is a straightforward calculation. First we have $$ \sigma_k = [\{ L : L \supset A \}] ,\ \sigma_{(1^p)} = [\{ L : L \subset B \}] , \ \sigma_{(k^p)} = [\{ L : L = C \}] , $$ where $A$, $B$, and $C$ are linear subspaces of ${\mathbb C}^n$ of dimensions $1$, $n-1$, and $p$ respectively. It is not hard to verify that any line in $X$ is a Schubert variety of the form $\{ L : U \subset L \subset V\}$, where $U \subset V$ are subspaces of ${\mathbb C}^n$ of dimensions $p-1$ and $p+1$. Such a line will meet the three displayed Schubert varieties only if $V$ contains $A$ and $C$, and $U$ is contained in $B$ and $C$. For $A$, $B$, and $C$ general, there is only one such line, with $U = B \cap C$ and $V$ spanned by $A$ and $C$. \qed \vspace{+10pt} This proposition was proved in another way by Bertram [Ber], where the beginnings of some ``quantum Schubert calculus'' can be found. For the small quantum cohomology ring of a flag manifold, following ideas of Bertram, Givental, and Kim, see [CF]$^3$. \footnotetext[3]{This Schubert calculus is extended to flag manifolds in [F-G-P].} As with the big quantum cohomology ring, the small ring has a basis independent description. Let $\mathbb Z[A_1 X]$ be the group algebra. The small $*$-product is naturally defined on the free $\mathbb Z[A_1 X]$-module $A^* X \otimes_\mathbb Z \mathbb Z[A_1 X]$. If $\beta_1, \ldots, \beta_p$ is a basis of $A_1 X$ consisting of Schubert classes, then the dual Schubert classes $T_1, \ldots, T_p$ satisfy $\int_\beta T_i \geq 0$ for every effective class $\beta$. In this case, the small $*$-product on $A^* X \otimes_\mathbb Z \mathbb Z[A_1 X]$ preserves the $\mathbb Z[q_1, \ldots, q_p]$-submodule: $$A^*X \otimes _\mathbb Z \mathbb Z[q_1, \ldots, q_p] \subset A^* X \otimes_\mathbb Z \mathbb Z[A_1 X].$$ Hence, in the Schubert basis, the small quatum cohomology ring can be taken to be $QH^*_s X= (A^* X \otimes_\mathbb Z \mathbb Z[q_1, \ldots, q_p],*).$ The numbers $I_{\beta}(\gamma_1 \cdots \gamma_n)$ should not be confused with the numbers denoted by the expression $\langle\gamma_1, \ldots, \gamma_n \rangle_{\beta}$ which often occur in discussions of small quantum cohomology rings ([B-D-W], [Ber], [CF]). To define the latter, one {\em fixes} $n$ distinct points $p_1, \ldots, p_n$ in $\mathbf P^1$. Then, $\langle\gamma_1, \ldots, \gamma_n\rangle_{\beta}$ is the number of maps $\mu:\mathbf P^1 \rightarrow X$ satisfying: $\mu_*[\mathbf P^1]=\beta$ and $\mu(p_i)\in \Gamma_i$ for $1 \leq i \leq n$ (where $\Gamma_i$ is a subvariety in general position representing the class $\gamma_i$). For $n=3$, the numbers agree: $I_{\beta}(\gamma_1 \cdot \gamma_2 \cdot \gamma_3) = \langle\gamma_1, \gamma_2, \gamma_3 \rangle_{\beta}$. For $n>3$, the numbers $\langle\gamma_1, \ldots, \gamma_n \rangle_{\beta}$ and $I_{\beta}(\gamma_1 \cdots \gamma_n)$ are solutions to different enumerative problems. In fact, $\langle\gamma_1, \ldots, \gamma_n\rangle_{\beta}$ can be expressed in terms of the $3$-points numbers while $I_{\beta}(\gamma_1 \cdots \gamma_n)$ cannot. For $1 < k < n-1$, $\langle\gamma_1, \ldots, \gamma_n\rangle_{\beta} =$ \begin{equation} \label{ggpp} \sum_{\beta_1+\beta_2=\beta} \sum_{e,f} \langle\gamma_1, \ldots, \gamma_k, T_e\rangle_{\beta_1}g^{ef} \langle T_f,\gamma_{k+1}, \ldots, \gamma_n\rangle_{\beta_2} \end{equation} Equation (\ref{ggpp}) can be seen geometrically by deforming $\mathbf P^1$ to a union of two $\mathbf P^1$'s meeting at a point with $p_1, \ldots, p_k$ going to fixed points on the first line and $p_{k+1}, \ldots, p_n$ going to fixed points on the second. Algebraically, in the small quantum cohomology ring, $$\gamma_1 * \cdots * \gamma_n = \sum_{\beta} \sum_{ e,f} q^{\beta} \langle\gamma_1, \ldots, \gamma_n, T_e\rangle_{\beta} g^{ef} T_f.$$ Equation (\ref{ggpp}) amounts to the associativity of this product. We conclude with a few general remarks to relate the discussion and notation here to that in [K-M 1]. The numbers that we have denoted $I_\beta(\gamma_1 \cdots \gamma_n)$ are part of a more general story. Let $\eta$ denote the forgetful map from $\barr{M}_{0,n}(X,\beta)$ to $\barr{M}_{0,n}$. For any cohomology classes $\gamma_1, \ldots , \gamma_n$ on $X$, one can construct a class \begin{equation} I^X_{0, \, n , \, \beta} (\gamma_1 \otimes \cdots \otimes \gamma_n) = \eta_* \left({\rho_1}^*(\gamma_1) \mathbin{\text{\scriptsize$\cup$}} \cdots \mathbin{\text{\scriptsize$\cup$}} {\rho_n}^*(\gamma_n)\right) \label{2.18} \end{equation} in the cohomology ring $H^*(\barr{M}_{0,n})$. These are called (tree-level, or genus zero) Gromov-Witten classes. The number we denoted $I_\beta(\gamma_1 \cdots \gamma_n)$ is the degree of the zero-dimensional component of this class, which they denote by $ \langle I^X_{0, \, n , \, \beta} \rangle (\gamma_1 \otimes \cdots \otimes \gamma_n)$. The intersections with divisors that we have carried out on $\barr{M}_{0,n}(X,\beta)$ can be carried out with the corresponding divisors on $\barr{M}_{0,n}$; this has the advantage that the intersections take place on a nonsingular variety. One of the main goals of [K-M 1] and especially [K-M 2] is to show how Gromov-Witten classes can be reconstructed from the numbers obtained by evaluating them on the fundamental classes. The idea is that a cohomology class in $H^*(\barr{M}_{0,n})$ is known by evaluating it on the classes of the closures of the strata determined by the combinatorial types of the labeled trees. As we saw and exploited for divisors, these numbers can be expressed in terms of the numbers $I_\beta$ for the pieces making up the tree. Kontsevich and Manin also allow cohomology classes of odd degrees, in which case one has to be careful with signs and the ordering of the terms. For an interesting application to some Fano varieties, see [Bea]. Since the space $H = H^*(X,{\mathbb Q})$ can be identified with its dual by Poincar\'e duality, the maps $I^X_{0, \, n , \, \beta}$ can be regarded as maps \begin{equation} H_*(\barr{M}_{0,n+1}) \rightarrow \text{Hom} (H^{\otimes n} \, , \, H ) .\label{2.19} \end{equation} Both of these, for varying $n$, have a natural operad structure, that on the first coming from all the ways to glue together labeled trees of $\mathbf P^1$'s to form new ones, and the second from all the ways to compose homomorphisms. Remarkably, the associativity (Theorem 4), is equivalent to the assertion that (\ref{2.19}) is a morphism of operads. The structure constants $g^{i j}$ put a metric on the cohomology space $H^*(X,{\mathbb C})$; with coordinates given by the basis for the cohomology, there is a (formal) connection given by the formula $A_{i j}^k = \sum \Phi_{i j e} g^{e k}$. In this formalism of Dubrovin, the associativity translates to the assertion that this is a flat connection. The numbers calculated here are part of a much more ambitious program described in [K-M 1] and [K]. The hope is to extend the story to varieties without the positivity assumptions made here, with some other construction of what should be the fundamental class of $\barr{M}_{g,n}(X,\beta)$. (For varieties whose tangent bundles are not as positive as those considered here, the definition of the potential function $\Phi$ is modified by multiplying the summands in (\ref{2.4}) by $e^{-\int_\beta \omega}$, for a K\"ahler class $\omega$, in the hopes of making the power series converge on some open set of the cohomology space $H$.) Even if this program is carried out, however -- and associativity has been proved by symplectic methods [R-T] in some cases beyond those mentioned here$^4$ -- the interpretation cannot always be in enumerative terms as simple as those we have discussed, cf. [C-M]. \footnotetext[4]{At this conference, J. Li lectured on his work with G. Tian (see [L-T 2]) using cones in vector bundles to construct a ``virtual fundamental class'' to use in place of $[\barr{M}_{g,n}(X, \beta)]$ in case $\barr{M}_{g,n}(X,\beta)$ has the wrong dimension. This approach has been clarified and extended by K. Behrend and B. Fantechi using the language of stacks ([B-F], [B]). Algebraic computations in the non-convex case can be found, for example, in [G], [G-P], [K].} On the other hand, these ideas from quantum cohomology have inspired some recent work in enumerative geometry, even in cases where the associativity formalism does not apply directly, cf. [C-H 1] and [P]$^5$. \footnotetext[5]{Also, [E-K], [K-Q-R].}

9608

alg-geom/9608010

en

https://arxiv.org/abs/alg-geom/9608010

alg-geom/9608010

Guillermo Matera

M. Giusti, J. Heintz, K. H\"agele, J. E. Morais, L. M. Pardo and J. L. Monta\~na

Lower Bounds for diophantine Approximation

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero--dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their {\em polynomial} character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight--line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

alg-geom

\subsection{Lifting fibers by the symbolic Newton-Hensel algorithm}\label{lifting} The idea of using a symbolic adaptation of Newton--Hensel iteration for lifting fibers was introduced in \cite{gihemomorpar} and \cite{gihemopar}. For technical reasons, in these papers it was necessary to use algebraic parameters for the lifting process. We present here a new version of this lifting algorithm in which the use of algebraic numbers is replaced by a certain matrix with integer entries. The whole procedure therefore becomes completely rational. The new lifting process is described in the statement of the next theorem and its proof. \spar Let notations and assumptions be as the same as at the beginning of this section. We fix $1\le i \le n$ and assume for the sake of notational simplicity that the variables \xon are already in Noether position with respect to the variety $V_i$, the variables $\xo {n-i}$ being free. We suppose that the lifting point $P_i$, the coordinates of the ${\rm Z\!\!Z}$-linear form $U_i$ and a geometric solution for the equations of the lifting fiber $V_{P_i}$ are explicitly given. With these conventions we state the main result of this subsection as follows: \begin{thm} \label{newton} There exists a (division-free) straight--line program $\Gamma_i$ in the polynomial ring ${\rm Z\!\!Z}[\xo {n-i}, U_i]$ of size $(id\delta_iL)^{O(1)}$ and non-scalar depth $O((log_2 i+\ell)log_2 \delta_i)$ using as parameters \begin{itemize} \item the coordinates of $P_i$ \item the integers appearing in the geometric solution of the equations of the lifting fiber $V_{P_i}$ and \item the parameters of the input program $\Gamma$ \end{itemize} such that the straight--line program $\Gamma_i$ computes \begin{itemize} \item the minimal polynomial $q_i \in {\rm Z\!\!Z}[X_1,\ldots,X_{n-i},U_i]$ of the primitive element $u_i$ of the ring extension $\Qxo {n-i} \longrightarrow \Q[V_i] = \Qxon /\ifo i$, \item polynomials $\rho^{(i)}_{n-i+1},\ldots,\rho^{(i)}_n \in {\rm Z\!\!Z}[X_1,\ldots,X_{n-i}]$, $\rho:= \prod\limits_{k=n-i+1}^n\rho^{(i)}_k$ and polynomials $v^{(i)}_{r+1},\ldots,v_n^{(i)} \in {\rm Z\!\!Z}[X_1,\ldots,X_{n-i},U_i]$ with $max\{deg_{U_i} v_j^{(i)}\; ; \; r< j \le n\} < \delta_i$ such that \end{itemize} $$\ifo {i}_{\rho}=(q_i(U_i),\rho^{(i)}_{r+1}X_{r+1}- v^{(i)}_{r+1}(U_i),\ldots, \rho^{(i)}_n X_n- v^{(i)}_n(U_i))_{\rho}$$ holds. Without loss of generality we may assume that $\Gamma_i$ represents the coefficients of the polynomials $q_i$ and $v_{r+1}^{(i)},\ldots,v_n^{(i)}$ with respect to $U_i$. \end{thm} \begin{pf} Under our hypotheses, namely that \xon are in Noether position with respect to the variety $V_i$, the variables $\xo {n-i}$ being free, we have the following integral ring extension of reduced rings: $$R_i:=\Q[X_1,\ldots,X_{n-i}] \longrightarrow B_i:=\Q[X_1,\ldots,X_n]/(f_1,\ldots,f_i).$$ Let $P_i =(p_1, \ldots,p_{n-i}) \in {\rm Z\!\!Z}^{n-i}$ be a lifting point of the morphism $\pi_i$ and let $V_{P_i} = \pi_i^{-1}(P_i)$ be its lifting fiber. We have $deg\ V_{P_i}=D_i=rank_{R_i} B_i \le deg\ V_i =\delta_i$. Let $U_i= \lambda_{n-i+1}X_{n-i+1} + \cdots + \lambda_nX_n$$\in{\rm Z\!\!Z}[X_{n-i+1}\klk X_n]$ generate a primitive element of the ring extension $\Q\longrightarrow \Q[V_{P_i}]$ (i.e. $U_i$ separates the points of $V_{P_i}$). The minimal polynomial of the image of $U_i$ in $B_i$ has degree at least the cardinality of the set $U_i(V_{P_i})$. Since the linear form $U_i$ separates the points of $V_{P_i}$, this cardinality is $deg\ V_{P_i}= rank_{R_i} B_i$. We conclude that $U_i$ generates also a primitive element of the ring extension $R_i\longrightarrow B_i$. \spar In the sequel we denote the primitive element generated by the linear form $U_i$ in the ring extensions $\Q\longrightarrow\Q[V_{P_i}]$ and $R_i\longrightarrow B_i$ by the same letter $u_i$.\spar By hypothesis the following data are given explicitly (i.e. by the bit representation of their coefficients): \begin{itemize} \item the primitive minimal equation, say $q\in{\rm Z\!\!Z}[T]$ of the primitive element $u_i$ of $\Q[V_{P_i}]$, \item the parametrization of $V_{P_i}$ by the zeroes of $q$, given by the equations $$X_1-p_1=0, \ldots,X_{n-i}-p_{n-i}=0, q(T) = 0,$$ $$\rho_{n-i+1}X_{n-i+1}-v_{n-i+1}(T)=0, \ldots, \rho_n X_n- v_n(T)=0,$$ \end{itemize} where $v_{n-i+1}\klk v_n$ are polynomials of ${\rm Z\!\!Z}[T]$ of degree strictly less than $D_i=deg\ q$ and $\rho_{n-i+1}\klk \rho_n$ are non-zero integers. Check again Definition \ref{geosolve} to see that the polynomials $\rho_{n-i+1}X_{n-i+1}-v_{n-i+1}(T),\ldots, \rho_n X_n- v_n(T)$ are assumed to be primitive.\spar Let $a \in {\rm Z\!\!Z}$ be the leading coefficient of $q$.\spar We consider now $f_1,\ldots,f_i$ as polynomials in the variables $X_{n-i+1}\klk X_n$, i.e. as elements of the polynomial ring $R_i[X_{n-i+1},\ldots,X_n]$. Let $f:=\ifo i $ and let $$D(f):= \left( {{\partial f_k} \over {\partial X_j}} \right)_{ 1 \le k \le i \atop n-i+1 \le j \le n}$$ be the corresponding Jacobian matrix of $f$ (with respect to the variables $X_{n-i+1}\klk X_n$). Recall that the Newton operator with respect to these variables is defined as $$N_f =\pmatrix{X_{n-i+1}\cr \vdots\cr X_n \cr} - D(f)^{-1} \pmatrix{f_1\cr \vdots\cr f_i \cr}.$$ Let $\kappa$ be a natural number. Using Lemma \ref{lemanewton} we deduce the existence of numerators $g_{n-i+1}, \ldots,g_n$ and a non-zero denominator $h$ in the polynomial ring $R_i[X_{n-i+1},\ldots,X_n]$ such that the following $\kappa$-fold iterated Newton operator has the form: $$N_f^{\kappa}= \pmatrix{{{g_{n-i+1}} \over {h}}\cr \vdots\cr{{g_n} \over {h}} \cr}.$$ Now, let $M_{X_{n-i+1}}, \ldots,M_{X_n}$ be the matrices describing the multiplication tensor of the $\Q$-algebra $\Q[V_{P_i}]$ (recall that by assumption a geometric solution of the polynomial equation system defining $V_{P_i}$ is given and that therefore the matrices $M_{X_{n-i+1}}\klk M_{X_n}$ are known). \spar Let $M$ denote the companion matrix of the polynomial $a^{-1}q(T) \in \Q[T]$. Let $n-i+1\leq j \leq n$. The following identity is immediate: $$\rho_j M_{X_j}= v_j(M).$$ Moreover the matrices $\rho_j a^{D_i -1}M_{X_j}$ have integer entries. Let $\kappa:=1+log_2\delta_i$ and note that $\kappa\geq 1+ log_2 D_i$ holds. Our straight--line program $\Gamma_i$ will execute $\kappa$ Newton steps in a subroutine which we are going to explain now: Let us consider the following column vector of matrices $N_{n-i+1}\klk N_n$ with entries in $\Q(X_1,\ldots,X_{n-i})$: $$ \pmatrix{N_{n-i+1}\cr \vdots\cr N_n \cr} := N_f^{\kappa} ({\underline M})= \pmatrix{{{g_{n-i+1}(X_1,\ldots,X_{n-i}, {\underline M})} \over {h}(X_1,\ldots,X_{n-i}, {\underline M})}\cr \vdots\cr{{g_n(X_1,\ldots,X_{n-i}, {\underline M})} \over {h(X_1,\ldots,X_{n-i}, {\underline M})}} \cr},$$ where ${\underline M}= (M_{X_{n-i+1}}, \ldots,M_{X_n})$ and $g_{n-i+1}\klk g_n$ are the numerator and $h$ the denominator polynomial of Lemma \ref{lemanewton}. Finally, let us consider the matrix $${\cal M}:= U_i(N_{n-i+1},\ldots,N_n)= \lambda_{n-i+1}N_{n-i+1}+ \cdots +\lambda_nN_n.$$ This matrix is a matrix whose entries are rational functions of $\Q(\xo {n-i})$. From the fact that $P_i=(p_1\klk p_{n-i})$ is a lifting point and from the proof of Lemma \ref{lemanewton} one deduces easily that in fact the entries of ${\cal M}$ belong to the local ring $$R_{P_i} := (R_i)_{(X_1-p_1 \klk X_{n-i}-p_{n-i})} = \Q[X_1,\ldots,X_{n-i}]_{(X_1-p_1,\ldots,X_{n-i}-p_{n-i})}.$$ Let $T$ be a new variable. With these notations and assumptions we have the following result: \begin{lem} Let $\chi \in \Q(X_1,\ldots,X_{n-i})[T]$ be the characteristic polynomial of ${\cal M}$ and $m_{u_i} \in R_i[T]$ the minimal integral equation of the primitive element $u_i$ of $B_i$ over $R_i$. Let $\chi(T) = T^{D_i}+ \sum\limits_{k=0}^{D_i -1} a_kT^k$ and $m_{u_i} = T^{D_i}+ \sum\limits_{k=0}^{D_i -1}b_k T^k$ with $a_k, b_k\in\Q(\xo {n-i})$ for $0\leq k\leq D_i-1$. Then all these coefficients satisfy the condition $ord_{P_i}(a_k-b_k) \ge \delta_i+1,$ where $ ord_{P_i}$ denotes the usual order function (additive valuation) of the local ring $R_{P_i}$. \end{lem} \begin{pf} Let $V_{P_i}= \{\xi_1,\ldots,\xi_{D_i} \}$ and fix $1\le l \le D_i$. Let $\zeta_l\!:=U_i(\xi_l)$. Because of the hypothesis made on the lifting fiber $V_{P_i}$ and from Hensel's Lemma (which represents a symbolic version of the Implicit Function Theorem) we deduce that there exist formal power series $R_{n-i+1}^{(l)},\ldots,R_n^{(l)} \in \C[[X_1-p_1,\ldots,X_{n-i}-p_{n-i}]]$ with $R_{n-i+1}^{(l)}(P_i) =\xi_{n-i+1}^{(l)},\ldots,R_n^{(l)}(P_i)=\xi_n^{(l)}$ such that for $R^{(l)}\!:= (X_1-p_1,\ldots,X_{n-i}-p_{n-i}, R_{n-i+1}^{(l)}, \ldots,R_{n}^{(l)})$ the identities \begin{equation} \label{fres0} f_1(R^{(l)})=0,\ldots, f_i(R^{(l)})=0 \end{equation} hold in $\C[[X_1-p_1,\ldots,X_{n-i}-p_{n-i}]]$. Let $u^{(l)}\!:= U_i(R^{(l)}) = \lambda_{n-i+1}R_{n-i+1}^{(l)}\plp \lambda_nR_n^{(l)}$. As shown in \cite{gihemomorpar} the minimal polynomial $m_{u_i}$ of the primitive element $u_i$ verifies in $\C[[X_1-p_1,\ldots,X_{n-i}-p_{n-i}]][T]$ the identity \begin{equation}\label{minimo} m_{u_i}=\prod\limits_{1\le l \le D_i} (T-u^{(l)}). \end{equation} Let us now consider the construction above which produces the matrix ${\cal M} \in (R_{P_i})^{ D_i \times D_i}$ starting with the matrix $M \in \Q^{D_i \times D_i}$ (recall that $M$ is the companion matrix of the polynomial $a^{-1}q(T)\in\Q[T]$). The same construction transforms the $D_i$ distinct eigenvalues of the diagonalizable matrix $M$, namely the values $\zeta_l\!=U_i(\xi_l)$, $1 \le l \le D_i$, into eigenvalues of ${\cal M}$. (Observe that by construction ${\cal M}$ is a rational function of the matrix $M$ and therefore the same rational function applied to any eigenvalue of $M$ produces an eigenvalue of ${\cal M}$). As shown in \cite{gihemomorpar}, in this way we obtain $D_i$ distinct rational functions ${\tilde u}^{(l)} \in \C(X_1,\ldots,X_{n-i})$, which are eigenvalues of ${\cal M}$. Moreover these rational functions are all defined in the point $P_i$ (this means that ${\tilde u}^{(l)} \in\C[X_1\klk X_{n-i}]_{(X_1-p_1\klk X_{n-i}-p_{n-i})}$ holds). For $1\leq l\leq D_i$ the rational functions ${\tilde u}^{(l)}$ can therefore be interpreted as elements of the power series ring $\C[[X_1-p_1,\ldots,X_{n-i}-p_{n-i}]]$ and they satisfy in this ring the condition \begin{equation}\label{eqNr} u^{(l)}-{\tilde u}^{(l)} \in (X_1-p_1,\ldots,X_{n-i}-p_{n-i})^{\delta_i+1}. \end{equation} (For a proof of these congruence relations see \cite{gihemomorpar}). \spar Let us now consider the characteristic polynomial $\chi$ of the matrix ${\cal M}$. Since the coefficients of ${\cal M}$ belong to $R_{P_i} = \Q[X_1\klk X_{n-i}]_{(X_1-p_1,\ldots,X_{n-i}-p_{n-i})}$, the coefficients of $\chi$ do too. Therefore $\chi$ can be interpreted as a polynomial in the variable $T$ with coefficients in the power series ring $\Q[[X_1-p_1\klk X_{n-i}-p_{n-i}]]$. From the fact that the rational functions ${\tilde u}^{(1)}\klk {\tilde u}^{(D_i)}$ represent $D_i$ distinct eigenvalues of ${\cal M}$ we deduce that \begin{equation} \label{chi} \chi(T) = \prod_{1\leq l\leq D_i} (T- {\tilde u}^{(l)}) \end{equation} holds. Let $\sigma_k$ denote the $k$--th elementary symmetric function in $D_i$ arguments. The identity (\ref{chi}) implies that for $0\leq k \leq D_i-1$ we can write the coefficient $a_k$ of $\chi(T)$ as $a_k=(-1)^{D_i - k} \sigma_k({\tilde u}^{(1)},\ldots,{\tilde u}^{(D_i)})$. >From the identity (\ref{minimo}) we deduce that the $k$-th coefficient $b_k$ of the polynomial $m_{u_i}$ satisfies $b_k=(-1)^{D_i - k} \sigma_k( u^{(1)},\ldots,u^{(D_i)})$. From the congruence relations (\ref{eqNr}) and the identities (\ref{minimo}) and (\ref{chi}) we conclude now that for $0\leq k\leq D_i-1$ the congruence relations \begin{equation}\label{eqNoNo} a_k-b_k \in (X_1-p_1,\ldots,X_{n-i}-p_{n-i})^{\delta_i+1} \end{equation} hold in $\Q[[X_1-p_1,\ldots,X_{n-i}-p_{n-i}]]$. This proves the lemma. \qed \end{pf} We continue now the proof of Theorem \ref{newton}. In order to see how to evaluate the coefficients of $\chi$, let us consider $\theta:= det(h({\underline M})) \in R_i$ and the matrix $${\cal M}_1:= \theta {\cal M}.$$ This matrix ${\cal M}_1$ has entries in $R_i$. Now we have the identities $$det(T \cdot Id_{D_i} - {\cal M})= det (T \cdot Id_{D_i} - \theta^{-1} {\cal M}_1) = \theta^{-D_i} det ((\theta T) Id_{D_i} - {\cal M}_1).$$ Let $\phi(T)=T^{D_i}+\phi_{D_i-1}T^{D_i-1}+ \cdots+ \phi_0 \in R_i[T]$ be the characteristic polynomial of ${\cal M}_1$. From the identities above we deduce that for $0\leq k \leq D_i -1$ the $k$-th coefficient of $\chi$ can be written as $$a_k= {{\theta^k \Phi_k} \over {\theta ^{D_i}}}.$$ Executing $\kappa = 1+log_2 \delta_i$ steps in the Newton iteration at the beginning of this proof to produce the entries of the matrix $M$, $\theta$, ${\cal M}$ and finally ${\cal M}_1$ (in this order) and applying Lemma \ref{berk} we produce a straight--line program $\Gamma_i'$ in $\Q(X_1\klk X_{n-i})$ of non-scalar size $O(log_2\delta_i d^3i^6 L)$ and non-scalar depth $O((log_2 i + \ell)log_2\delta_i)$ using as parameters those given by the statement of Theorem \ref{newton} such that $\Gamma_i'$ evaluates the family of polynomials $1, \theta,\ldots, \theta^{D_i}$, $\phi_0,\ldots,\phi_{D_i-1} \in R_i$. \spar Now, applying Strassen's Vermeidung von Divisionen technique (Proposition 23) we obtain a division-free straight--line program $\Gamma_i''$ in $\Qxo {n-i}$ of size $(id\delta_iL)^{O(1)}$, non-scalar depth $O((log_2 i + \ell)log_2\delta_i)$ using as parameters the coordinates $p_1\klk p_{n-i}$ of $P_i$, the coefficients of the linear change of coordinates for the Noether normalization for $V_i$, the rational numbers appearing as coefficients in the geometric solution of the lifting fiber $V_{P_i}$ and the parameters of $\Gamma$ such that $\Gamma_i''$ evaluates for each $0\leq k\leq D_i -1$ the expansion of $a_k$ in $\Q[[X_1-p_1 \klk X_{n-i}-p_{n-i}]]$ up to terms of degree of order $\delta_i +1$. Taking into account the congruence relations (\ref{eqNoNo}) we see that the division--free straight--line program $\Gamma_i''$ evaluates polynomials $g_0,\ldots, g_{D_i-1} \in \Q[X_1,\ldots,X_{n-i}]$ such that $$ b_k -g_k \in (X_1-p_1,\ldots,X_{n-i}-p_{n-i})^{\delta_i+1}$$ holds in $\Q[[X_1-p_1\klk X_{n-i}-p_{n-i}]]$ for any $0\leq k\leq D_i-1$. >From \cite{gihemomorpar} we deduce that the degrees of the coefficients $b_k$ of the minimal polynomial $m_{u_i} \in R_i[T]$ do not exceed $\delta_i$. Putting all this together, we conclude that $$b_k = g_k$$ holds for any $0\leq k\leq D_i-1$. This means that the division-free straight--line program $\Gamma_i''$ computes the coefficients of the polynomial $q_i:= m_{u_i} \in \Q[\xo r, U_i] = R_i[U_i]$. \spar In order to compute the parametrizations $v_{n-i+1}^{(i)}\klk v_n^{(i)} \in\Q[\xo {n-i},T]$ we have to use the same kind of techniques (namely truncated Newton--Hensel iteration) combined with the arguments developed in \cite{krick-pardo1} and applied in \cite{gihemomorpar} and \cite{gihemopar}. Let us be more exact: \spar Let $n-i+1\leq k\leq n$ and let $m_{X_k}\in R_i[X_k]$ be the minimal polynomial of the $R_i$-linear endomorphism given by the image of $X_k$ in the $R_i$-algebra $B_i = \Q[V_i] = \Qxon / \ifo i$. The polynomial $m_{X_k}$ is monic and hence squarefree. We compute the coefficients of the polynomial $m_{X_k}\in R_i[X_k]$ in the same way as before the coefficients of $q_i=m_{u_i}$. So, we have two monic squarefree polynomials $m_{X_k} \in R_i[X_k]$ and $q_i \in R_i[U_i]$. Taking into account that $U_i$ separates the associated primes of the ideal $(m_{X_k}, q_i)$ in $R_i[X_k,U_i]$, we can apply directly \cite[Lemma 26]{krick-pardo1} in order to obtain the parametrization associated to the variable $X_k$. Doing the same for each of the variables involved, namely the variables $X_{n-i+1},\ldots,X_n$, and putting together the corresponding straight--line programs, we obtain a procedure and a straight--line program $\Gamma_i$ of the desired complexity which computes the output of Theorem 28. \qed \end{pf} \subsection{The recursion} \label{liftingpoint} \begin{prop}\label{punto} There exists a division-free arithmetic network of non-scalar size of order $(i d \delta_i L)^{O(1)}$ and non-scalar depth $O((log_2i + \ell)log_2 \delta_i)$ using parameters of logarithmic height bounded by $max \{h, \eta_i, O((log_2i + \ell)log_2 \delta_i)\}$ which takes as input \begin{itemize} \item a Noether normalization for the variety $V_i$, \item a lifting point $P_i$ and \item a geometric solution of the lifting fiber $V_{P_i}$ \end{itemize} and produces as output \begin{itemize} \item a linear change of variables $(X_1,\ldots,X_n) \longrightarrow (Y_1^{(i+1)},\ldots,Y_n^{(i+1)})$ such that the new variables $Y_1^{(i+1)},\ldots,Y_n^{(i+1)}$ are in Noether position with respect to $V_{i+1}$, \item a lifting point $P_{i+1}$ for $V_{i+1}$ and \item a geometric solution for the lifting fiber $V_{P_{i+1}}$. \end{itemize} \end{prop} \begin{pf} The construction of the arithmetic network proceeds in three stages. In the first stage we apply the algorithm underlying Theorem 28. In the second stage we intersect algorithmically the variety $V_i$ with the hypersurfcae $V(f_{i+1})$ in order to produce first a Noether normalization of the variables with respect to the variety $V_{i+1} = V_i \cap V(f_{i+1})$. Then we produce a linear form $U_{i+1}$ representing a primitive element $u_{i+1}$ of the integral ring extension $R_{i+1} \longrightarrow B_{i+1}$ and a straight--line program representing polynomials analogous to the one in the conclusion of Theorem 28. For this purpose we use the algorithm underlying the proof of Proposition 14 in \cite{gihemomorpar}. The only point we have to take care of is that we are working now over the ground field $\Q$ and that we have to take into account the heights of the parameters of $\Q$ we introduce in this straight--line program. \spar We remark that the straight--line program in question has non-scalar size $(id\delta_i L)^{O(1)}$ and non-scalar depth $O(log_2(d\delta_i)+\ell)$ with parameters of logarithmic height bounded by $O(log_2(d\delta_i) + \ell)$. In the third and final stage we consider the polynomial $$J(f_1\ldots,f_{i+1}):= det \left( {{\partial f_k} \over {\partial X_j}} \right)_{ 1 \le k \le i+1 \atop n-i \le j \le n},$$ which is a nonzero divisor modulo the ideal $I_{i+1}=(f_1,\ldots,f_{i+1})$. Let us observe that the polynomial $J(f_1\ldots,f_{i+1})$ can be evaluated by a division-free straight--line program of length $O((i+1)^5+L)$ and depth $O(log_2 (i+1) +\ell)$. \spar Let $\mu \in {\it K}[X_1,\ldots,X_{n-i-1}]$ be the constant term of the characteristic polynomial of the homothety given by $J(f_1\ldots,f_{i+1})$ modulo $I_{i+1}$. Since $J$ represents a nonzero divisor modulo $I_{i+1}$ we conclude that $\mu$ does not vanish. Furthermore we observe that $\mu$ can be evaluated by a division-free straight--line program in $\Q[X_1,\ldots,X_{n-i-1}]$ of length $(i\delta_i L)^{O(1)}$ and depth $O(log_2 d\delta_i + (log_2 i + \ell) log_2 \delta_i)= O((log_2i + \ell)log_2 \delta_i)$, and so does the product $\rho \cdot \mu$, where $\rho= \prod\limits_{n-i+1} ^n \rho_j^{(i)}$ is defined as in the statement of Theorem 28. \spar Using a correct test sequence (see \cite{heschnorr}) we are able to find in sequential time $(i\delta L)^{O(1)}$ and parallel time $O((log_2i + \ell)log_2\delta_i)$ a rational point $P_{i+1} \in {\rm Z\!\!Z}^{n-i-1}$ of logarithmic height bounded by $O((log_2i + \ell)log_2\delta_i)$ which satisfies $(\rho \cdot \mu) (P_{i+1}) \not= 0$. Clearly, $P_{i+1}$ is a lifting point for the variety $V_{i+1}$. In order to obtain a geometric solution of $V_{P_{i+1}}$ with primitive element $u_{i+1}$ induced by the linear form $U_{i+1}$ we have to specialize in the point $P_{i+1}$ the polynomials obtained as output of the second stage (see \cite[Section 3]{gihemomorpar}). By this specialization we obtain the binary representation of the coefficients of certain univariate polynomials in $U_{i+1}$ which represent a geometric solution of the fiber $V_{P_{i+1}}$. Nevertheless it might happen that the height of these coefficients is excessive. In order to control the height of these polynomials we make them primitive. This requires some integer greatest common divisor (gcd) computations which do not modify the asymptotic time complexity of our algorithm. \qed \end{pf} \end{section} \typeout{Section 4} \begin{section}{Lifting Residues and Division modulo a Complete Intersection Ideal}\label{division} This section is dedicated to the proof of Theorem \ref{paso-division}. The outcome is a new trace formula for Gorenstein algebras given by complete intersection ideals. This trace formula makes no reference anymore to a given monomial basis of the algebra. Our trace formula represents an expression which is ``easy-to-evaluate". \begin{subsection}{Trace and Duality} Trace formulas appear in several recent papers treating problems in algorithmic elimination theory. Some of these papers use a trace formula in order to compute a quotient appearing as the result of a division of a given polynomial modulo a given complete intersection ideal (see \cite{figismi,krick-pardo1}). Other papers use trace formulas in order to design algorithms for geometric (or algebraic) solving of zero-dimensional Gorenstein algebras given by complete intersection ideals (\cite{royetal,BeCaRoSz,Cardinal}). The paper \cite{saso} uses a trace formula to obtain an upper bound for the degrees in the Nullstellensatz. \spar However, all these applications of trace formulas require the use of some generating family of monomials of bounded degree which generate the given Gorenstein algebra as a vector space over a suitable field. As a consequence, such trace formulas provide just syntactical complexity or degree bounds and in particular no intrinsic upper complexity bound (as e.g. in Theorem \ref{paso-division}) can be obtained in this way. In this subsection we introduce an alternative trace formula in order to get a maximum benefit from the geometrically and algebraically well suited features of Gorenstein algebras. Let us start with a sketch of the trace theory. For proofs we refer to \cite{kunz}, Appendices E and F. \spar Let $R$ be a ring of polynomials over a given ground field (for our discussion the ground field may be assumed to be $\Q$). Let ${\it K}$ be the quotient field of $R$ and let $R[X_1,\ldots,X_n]$ be the ring of $n$-variate polynomials with coefficients in $R$. Let $f_1,\ldots,f_n$ be a smooth regular sequence of polynomials in the ring $\Rxon$ of degree at most $d$ in the variables $\xon$ generating a radical ideal denoted by $(f_1,\ldots,f_n)$. \spar Consider now the $R$-algebra $B$ given as the quotient of $R[X_1,\ldots,X_n]$ by this ideal: $$B:=R[X_1,\ldots,X_n]/(f_1,\ldots,f_n).$$ We assume that the morphism $R\rightarrow B$ is an integral ring extension representing a Noether normalization of the variety $V\ifon$ defined by the polynomials $\fon$ in a suitable affine space. Thus, $B$ is a free $R$-module of rank bounded by the degree of the variety $V(f_1,\ldots,f_n)$ (this estimation is very coarse but sufficient for our purpose). Moreover, the $R$-algebra $B$ is Gorenstein and the following statements are based on this fact. \spar We consider $B^*:=Hom_R(B,R)$ as a $B$-module by the scalar product $$ B\times B^* \longrightarrow B^*$$ which associates to any $(b,\tau)$ in $B\times B^*$ the $R$-linear map $b\cdot \tau: B\longrightarrow R$ defined by $(b\cdot \tau) (x):= \tau(b x)$ for any element $x$ of $B$. \spar Since the $R$-algebra $B$ is Gorenstein, its dual $B^{*}$ is a free $B$-module of rank one. Any element $\sigma$ of $B^{*}$ which generates $B^{*}$ as $B$-module is called a {\sl trace} of $B$. There are two relevant elements of $B^*$ that we denote by $\Tr$ and $\sigma$. The first one, \Tr, is called the {\em standard trace\/} of $B$ and it is defined in the following way: given $b\in B$, let $\eta_b:B\longrightarrow B$ the $R$-linear map defined by multiplying by $b$ any given element of $B$. The image $\Tr(b)$ under the map $\Tr$ is defined as the ordinary trace of the endomorphism $\eta_b$ of $B$ (note that this definition makes sense since $B$ is a free $R$-module). In order to introduce $\sigma$ (which will be a trace of $B$ in the above sense), we need some additional notations. For any element $g\in\Rxon$ we denote by $\bar{g}$ its image in $B$, i.e. the residue class of $g$ modulo the ideal $\ifon$. Let $Y_1\klk Y_n$ be new variables and let $Y:=(Y_1\klk Y_n)$. Let $1\leq j\leq n$ and let $f_j^{Y}:=f_j(Y_1\klk Y_n)$ be the polynomial of $ \Ryon$ obtained by substituting in $f_j$ the variables $\xon$ by $\yon$. Let us consider the polynomial $$f_j^{Y}-f_j = \sum_{k=1}^n l_{jk}(Y_k-X_k) \in \R[\xon,\yon],$$ where the $l_{jk}$ are polynomials belonging to $R[\xon,\yon]$ having total degree at most $(d-1)$ (observe that the $l_{jk}$ are not uniquely determined by the sequence $\fon$). Let us now consider the determinant $\Delta$ of the matrix $(l_{jk})_{1\leq j,k\leq n}$ which can be written (non uniquely) as $$ \Delta = \sum_m a_m(\xon) b_m(\yon) \in \R[\xon,\yon],$$ with the $a_m$ being elements of $\Rxon$ and the $b_m$ elements of $\Ryon$. \hyphenation{deter-minant} (Observe that it will not be necessary to find the polynomials $a_m$ and $b_m$ alegbraically, we need just their existence for our argumentation.) The polynomial $\Delta$ is called a pseudo-jacobian determinant of the regular sequence \ifon. Observe that the polynomials $a_m$ and $b_m$ can (and will) be chosen to have degrees bounded by {\hbox{$n(d-1)$}} in the variables $\xon$ and $\yon$ respectively. Let $c_m\in\Rxon$ be the polynomial we obtain by substituting in $b_m$ the variables $\yon$ by $\xon$. For $\bar{J}$ the class of the Jacobian determinant $J\ifon$ in $B$ we have the identity $$\bar{J} = \sum_m \bar{a}_m\cdot\bar{c}_m.$$ Moreover the image of the polynomial $\Delta$ in the residue class ring \linebreak $R[\xon,\yon]$ modulo the ideal $(\fon,f_1^Y\klk f_n^Y)$ is independent of the particular choice of the matrix $(l_{kj})_{1\leq k,j\leq n}$. This justifies the name ``pseudo-jacobian" for the polynomial $\Delta$. With these notations there is a unique trace $\sigma \in B^*$ such that the following identity holds in $B$: $$\bar 1 = \sum_m \sigma (\bar{a}_m)\cdot\bar{c}_m.$$ The main property of the trace $\sigma$, known as ``trace formula" (``Tate's trace formula" \cite[Appendix F]{kunz}, \cite{Iversen} being a special case of it) is the following statement: for any $g\in\Rxon$ the identity \begin{equation}\label{eqTrNo} \bar g = \sum_m\sigma(\bar g \cdot \bar{a}_m)\cdot \bar{c}_m \end{equation} holds true in $B$. Let us observe that the polynomial $\sum_m\sigma(\bar g \cdot \bar{a}_m)\cdot c_m \in \Rxon$ underlying the identity (\ref{eqTrNo}) is of degree at most $n(d-1)$ in the variables $\xon$. The main use of this trace formula consists in solving the following problem: \begin{prob}[Lifting of a residual class] \label{problem6} Given a polynomial \linebreak $g\in\Rxon$ of arbitrary degree in $\xon$, find a polynomial $g_1\in\Rxon$ of degree at most $n(d-1)$ in the variables $\xon$ such that $\bar{g}_1 = \bar g$ holds in $B$. \end{prob} As we have seen before, the trace formula (\ref{eqTrNo}) solves Problem \ref{problem6} since it allows us to choose for $g_1$ the polynomial \begin{equation}\label{eqTrNoNo} g_1 := \sum_m\sigma(\bar g \cdot \bar{a}_m)\cdot c_m. \end{equation} However, defining the polynomial $g_1$ by the formula (\ref{eqTrNoNo}) inhibits us from taking advantage of any special ``semantical" features of the $R$-algebra $B$: one ``a priori" needs all monomials of degree at most $n(d-1)$ for the description of the polynomials $c_m$ (and $a_m$). Therefore, we replace the trace formula (\ref{eqTrNo}) by the following alternative one: \begin{prop}[Trace Formula] With the same notations as before, let us consider the free \Rxon-module $\Bxon$ given by extending scalars in $B$ (this means we consider the tensor product $\Bxon := B\otimes_R \Rxon$) and let us also consider the polynomial $\Delta_1\in\Rxon$ given by: $$\Delta_1:=\sum_m\bar{a}_m\cdot c_m \in \Bxon.$$ Then for any $g\in\Rxon$ the following identity holds true in \linebreak $\Rxon$: $$\sum_m\sigma(\bar g \cdot\bar{a}_m)\cdot c_m = \Trt (\bar{J}^{-1} \bar g \cdot \Delta_1).$$ (Here $\Trt:= \Tr \otimes Id_{\Rxon} : \Bxon\longrightarrow \Rxon$ is the standard trace obtained from the standard trace $\Tr: B\longrightarrow R$ by extending scalars). \end{prop} \begin{pf} Let $\Tr: B\longrightarrow R$ be the standard trace of the free $R$-module $B$. Let us recall from \cite[Appendix F]{kunz} that for any $g\in\Rxon$ the identity $$\Tr(\bar{J}^{-1}\bar g) = \sigma(\bar g)$$ holds. From the $\Ryon$-linearity of the map $\Trt:\Byon \longrightarrow \Ryon$ we deduce that any $g\in\Rxon$ satisfies the identities $$\Trt(\bar{J}^{-1}\bar g \Delta_1(\yon)) = \sum_m \Trt(\bar{J}^{-1}\bar g \bar{a}_m\cdot b_m) = \sum_m \Trt(\bar{J}^{-1}\bar g \bar{a}_m) \cdot b_m.$$ In other words we have in $\Ryon$ $$\Trt(\bar{J}^{-1}\bar g \Delta_1(\yon)) = \sum_m \sigma(\bar g \cdot \bar{a}_m) \cdot b_m$$ for any $g\in\Rxon$. Replacing in this identity the variables $\yon$ by $\xon$ we obtain the desired formula $$\Trt(\bar{J}^{-1}\bar g \Delta_1) = \sum_m \sigma(\bar g \cdot a_m) \cdot c_m\ .\mbox{\hskip 3cm \qed}$$ \end{pf} One sees easily that for any $h\in R[\xon,\yon]$, $\Trt(\bar h)$ is simply the standard trace of the image $\bar h$ of $h$ in the $\Ryon$-module $\Byon$. This observation together with Proposition 31 represents our basic tool for the evaluation of formula (\ref{eqTrNoNo}) and hence for the solution of Problem \ref{problem6}. This is the content of the following observations. \spar Let us consider the ${\it K}$-algebra $B' = {\it K}\otimes_R B$ obtained by localizing $B$ in the non-zero elements of $R$. Fix a basis of the finite dimensional ${\it K}$-vectorspace $B'$. Let $M_{X_1} \klk M_{X_n}$ be the matrices of the homotheties $\eta_{X_i}: B' \longrightarrow B'$ with respect to the given basis of $B'$ and let $\Tr$ denote the function which associates to a given matrix its usual trace. With these conventions let $g_1$ be defined as \begin{eqnarray}\label{eqlast41} g_1 & := & \Tr( J(\fon) {(M_{X_1} \klk M_{X_n})}^{-1} \cdot g(M_{X_1} \klk M_{X_n}) \nonumber \\ & & \cdot \Delta(M_{X_1} \klk M_{X_n}, \xon) ) . \end{eqnarray} One easily verifies that $g_1$ belongs to $\Rxon$ and that $\bar g_1 = \bar g$ holds in $B$. \end{subsection} \begin{subsection}{A Division Step} The lifting process presented in the last subsection is now applied to compute the quotient of two polynomials modulo a reduced complete intersection ideal. More precisely, let us consider $f\in\Rxon$ a polynomial which is not a zero-divisor in $B$ and another polynomial $g\in\Rxon$ such that the residual class $\bar f$ divides the residual class $\bar g$ in $B$. The following proposition shows how we can compute a lifting quotient $q\in\Rxon$ for the division of $\bar g$ by $\bar f$ in $B$. \begin{prop}[Division Step]\label{Prop32} Let notations and assumptions be the same as in the previous subsection and let $D$ be the rank of $B$ as free $R$-module. Let be given the following items as input: \begin{itemize} \item a straight--line program $\Gamma'$ of size $L$ and depth $\ell$ representing the polynomials $f, g$ and $\fon$. \item the matrices $M_{X_1}\klk M_{X_n}$ describing the multiplication tensor of $B$ with respect to the given basis of $B'={\it K}\otimes_R B$. \end{itemize} Suppose that $\bar f$ is a non-zero divisor of $\bar B$ and that $\bar f$ divides $\bar g$ in $B$. Then there exists a division-free straight--line program $\Gamma$ in $\Kxon$ of size $L(n d D)^{O(1)}$ and non-scalar depth $O( \ell + log_2 D + log_2n)$ which computes from the entries of the matrices $M_{X_1}\klk M_{X_n}$ and the parameters of $\Gamma'$ a non-zero element ${\theta}$ of $R$, and a polynomial $q$ of $\Rxon$ such that $\theta$ divides $q$ in $\Rxon$ and such that $\bar q \bar f = \bar \theta \bar g$ holds in $B$. \end{prop} \begin{pf} In order to prove this result, let us observe that any basis of $B$ as free $R$-module induces a basis of $\Byon$ as free \Ryon-module. Moreover if $M_{X_i}$ is the matrix of the multiplication by $\bar{X}_i$ in $B$ with respect to a given basis, $M_{X_i}$ represents as well the multiplication by $\bar{X}_i$ in $\Byon$ with respect to the same basis. Next, since the polynomials $f$ and $J\ifon$ are not zero-divisors modulo \ifon, the following matrices are non-singular: $$ F_1 : = f(M_{X_1} \klk M_{X_n}),$$ $$J_1:=det ( {{\partial f_i} \over {\partial X_j}}(M_{X_1} \klk M_{X_n}))_{1\leq i,j\leq n}.$$ Finally, let us denote by $G_1$ and $\Delta_1$ the following two matrices: $$G_1:= g(M_{X_1} \klk M_{X_n}),$$ and $$\Delta_1:=\Delta(M_{X_1} \klk M_{X_n},\yon),$$ where $\Delta$ is the pseudojacobian determinant of $\fon$. Let us remark that the matrices $F_1,J_1$ and $G_1$ have entries in ${\it K}$ while $\Delta_1$ has entries in\linebreak ${\it K}[\yon]$. From formula (\ref{eqlast41}) of the previous subsection we deduce that $q_1 := \Tr(J_1^{-1}\cdot F_1^{-1}\cdot G_1\cdot \Delta_1(\xon))$ is a polynomial of $\Rxon$ which satisfies in $B$ the identity $\bar{q}_1\bar f =\bar g$ in $B$ (by $\Tr$ we denote here the ususal trace of matrices). Finally, let us transpose the adjoint matrices of $F_1$ and $J_1$: $$F_2 := \;\;^t\!Adj(F_1),\mbox{ and } J_2:= \;\;^t\!Adj(J_1).$$ The quotient $q\in\Rxon$ and the non-zero constant ${\theta} \in R$ we are looking for are given as follows: \begin{itemize} \item $q:=\Tr(F_2 \cdot J_2 \cdot G_1\cdot \Delta_1)$ \item ${\theta} := det(F_2)\cdot det(J_2)$ \end{itemize} Clearly, the polynomial $q_1$ is the quotient of $q \over {\theta}$, while $\bar q \bar f = \bar{\theta} \bar g$ holds in $B$. Moreover $q$ can be computed by a {\em division-free\/} straight--line program $\Gamma$ in $\Kxon$ from the entries of $M_{X_1}\klk M_{X_n}$ and the parameters of $\Gamma'$ (note that for the computation of $q_1$ we need divisions). The complexity bounds in the statement of Proposition \ref{Prop32} follow by reconstruction of the straight--line programs that evaluate $f, g, J\ifon, \Delta$ and the determinants involved (cf. Subsection \ref{elementary} above). \spar {\sl Proof of Theorem \ref{paso-division} } \spar In order to prove Theorem \ref{paso-division} we just follow the algorithm underlying the proof of Proposition \ref{Prop32}. We have to add just some comments concerning the matrices $M_{X_1}\klk M_{X_n}$. Let us consider a linear form $u = \lambda_1 X_1 \plp \lambda_nX_n$ (with $\lambda_i\in{\rm Z\!\!Z}$) inducing a primitive element of the zero-dimensional $\Q$-algebra $\Qxon / \ifon$. In this case $R$ will be the field $\Q$. Let $q_u\in{\rm Z\!\!Z}[T]$ be the minimal polynomial of $u$ and $\rho_1X_1-v_1(T)\klk \rho_nX_n-v_n(T)$ the parametrizations of the variety with respect to this primitive element. Let $\alpha$ be the leading coefficient of $q_u$ and $\rho = \prod_{i=1}^n \rho_i$ a discriminant. Then the companion matrix of $q_u$ has the form $$\alpha^{-1} M,$$ where $M$ is a matrix with integer entries. Now, the matrices describing the multiplication tensor of $\Qxon / \ifon$ can be written as $$ M_{X_i} = \rho_i^{-1}\cdot v_i(\alpha^{-1} M)$$ for $1\leq i\leq n$. Taking $g=1$ and $f=f_{n+1}$ in Proposition \ref{Prop32} we obtain ${\theta} \in \Q, {\theta}\neq 0$ and $q\in\Qxon$ such that $$ {\theta} \cdot 1 - q \cdot f_{n+1} \in \ifon$$ holds. Finally, multiplying by appropriate powers of $\alpha$ and $\rho$ we obtain a non-zero integer $a$ and a polynomial $g_{n+1}$ of the form $$ a:=\alpha^N\rho^M\cdot{\theta} \in {\rm Z\!\!Z} \ \ \ \ \ \ \ g_{n+1} := \alpha^N\rho^M q\in \Zxon,$$ such that $a-g_{n+1}f_{n+1}\in \ifon$ holds. The bounds of the Theorem \ref{paso-division} are then obtained from the bounds of Proposition \ref{Prop32}. The bounds for the height of the parameters are obtained simply by choosing an appropriate primitive element $u$ such that $q_u$ and the parametrizations have height equal to the minimal height of the diophantine variety $V:=V\ifon$. \qed \end{pf} \end{subsection} \end{section} \begin{ack} L. M. Pardo wants to thank the \'Ecole Polytechnique X for the invitation and hospitality during his stay in the Fall of 1995, when this paper was conceived. \end{ack} \typeout{References}

9608

alg-geom/9608028

en

https://arxiv.org/abs/alg-geom/9608028

alg-geom/9608028

Dan Edidin

Dan Edidin and William Graham

Algebraic Cuts

Latex2e with amssymb package, 12 pages

Let $X$ be a projective variety with a torus action, which for simplicity we assume to have dimension 1. If $X$ is a smooth complex variety, then the geometric invariant theory quotient $X//G$ can be identifed with the symplectic reduction $X_r$. Lerman introduced a construction (valid for symplectic manifolds) called symplectic cutting, which constructs a manifold $X_c$, such that $X_c$ is the union of $X_r$ and an open subset $X_{>0} \subset X$. Moreover, there is a natural torus action on $X_c$ such that $X_r$ is a component of the fixed locus. Using localization for equivariant cohomology, this construction can be used to study of $X_r$. In this note, we give an algebraic version of this construction valid for projective but possibly singular varieties defined over arbitrary fields. This construction is useful for studying $X_r$ from the point of view of algebraic geometry, using the equivariant intersection theory developed by the authors. At the end of the paper we briefly give an adaptation of Lerman's proof of the Kalkman residue formula and use it to give some formulas for characteristic numbers of quotients by a torus.

alg-geom

\section{Introduction} Let $X$ be a projective variety with a linearized action of an algebraic group $G$. If $X$ is smooth, and the ground field is ${\Bbb C}$, then the geometric invariant theory quotient $X//G$ can be identified with a quotient constructed using symplectic geometry, the ``reduced space'' $X_r$. This result, due to Mumford, Guillemin and Sternberg, connects geometric invariant theory and symplectic geometry. Suppose now that $G = T$ is a torus, which for simplicity we will take to have dimension 1. In \cite{Lerman}, Lerman introduced a construction called symplectic cutting, which constructs a manifold $X_c$ related to $X$, with a $T$-action, and embeds $X_r$ as a component of the fixed point locus $X_c^T$. The complement of $X_r$ in $X_c$ can be identified with an open submanifold of $X$, so the other components of $X_c^T$ are certain components of $X^T$. The components of $X_c^T$ are linked by the localization theorem in equivariant cohomology. Thus, from knowledge of $X^T$ one can (via the cut space) deduce results about $X_r$. For example, Lerman uses cutting to prove a residue formula due to Kalkman, which is closely connected to the localization theorem of Jeffrey-Kirwan-Witten (\cite{G-K}). The purpose of this paper is to present an algebraic version of Lerman's construction, called algebraic cutting, which is valid over arbitrary ground fields and for possibly singular schemes. This is useful in studying $X_r$ from the point of view of algebraic geometry, using the equivariant intersection theory developed in \cite{E-G} in place of equivariant cohomology. For example, Lerman's proof of Kalkman's formula becomes valid for smooth schemes over arbitrary ground fields; at the end of this paper we briefly give an adaptation of that proof. Throughout this paper $T$ will denote a one-dimensional split torus ${\Bbb G}_m$, $X \subset {\bf P}(V)$ a quasiprojective variety with a linearized $T$-action, and $X^s$ the set of stable points for this action. An element of the group $T$ will usually be denoted by $g$. All Chow groups are assumed to have rational coefficients. $A^*_T$ denotes the $T$-equivariant Chow ring of a point; it is isomorphic to the polynomial ring ${\Bbb Q}[t]$. The sign convention for $A^*_T$ is determined as follows. We model $BT$ by $(V - {0})/T = {\bf P}^N$, where $V$ is an $N+1$ dimensional vector space on which $T$ acts with all weights $-1$. The element $t$ represents the hyperplane class. A representation of $T$ is an equivariant vector bundle over a point, and thus has equivariant Chern classes, coming from the induced bundle over $BT$. A 1-dimensional representation with weight $a$ then has equivariant first Chern class $at$. The Chow ring $A^*_{T \times T}$ will be denoted ${\Bbb Q}[t_1,t_2]$. \section{Algebraic cuts} \subsection{Cutting at zero} Let $v_1 , \ldots, v_n$ be a basis of $V$ consisting of weight vectors, of weights $a_1 , \ldots, a_n \in {\Bbb Z}$. If $x \in {\bf P}(V)$ is the line spanned by $\sum x_i v_i$, define (cf. \cite{B-P}) the weights of $x$ (denoted $\Pi(x)$) to be the set of $a_i$ such that $x_i$ is nonzero. As an algebraic analogue of the fact that 0 is a regular value of the moment map, we will assume that for all $x \in X$, $0 \notin \Pi(x)$. Thus, $X$ actually lies in ${\bf P}(W)$, where $W$ is the subspace of $V$ spanned by weight vectors with nonzero weights. By replacing $V$ by $W$ if necessary, we will also assume that $0$ is not a weight of $T$ on $V$. By \cite{B-P} stability with respect to the line bundle ${\cal O}_{{\bf P}(V)}(1)$ can be expressed in terms of the weights: $$X^s = \{x \in X | \Pi(x) \mbox{ contains both positive and negative weights} \}.$$ By assumption, $0$ is not a weight of $T$ on $V$, so $X^{ss} = X^{s}$. Hence if $X$ is projective then the ``reduced space'' $X_r := X_s/T$ is projective. If $X$ is smooth and the ground field is ${\Bbb C}$ then $X_r$ corresponds to the symplectic reduction at zero. Define $$ X_{>0} = \{x \in X | \Pi(x) \mbox{ contains a positive weight} \}. $$ Let $T_{\Delta}$ and $A$ denote the diagonal and antidiagonal copies of $T$ in $T \times T$. Let $(X \times {\Bbb A}^1)^s$ denote the stable points of $X \times {\Bbb A}^1$ with respect to the action of $A$ and the line bundle ${\cal L} = {\cal O}_{{\bf P}(V)}(1) \boxtimes 1$. \begin{defn} \label{d.1} The algebraic cut of $X$ is the quotient $X_c = (X \times {\Bbb A}^1)^s /A$. This scheme has a natural $T$-action, coming from the isomorphism $T \rightarrow (T \times T)/A$, $g \mapsto (g,1) \mbox{ mod }A$. \end{defn} The main results concerning this space are two theorems, which are algebraic versions of Lerman's results. The first says that $X_c$ is projective (if $X$ is) and is assembled out of the pieces $X_{>0}$ and $X_r$. \begin{thm} \label{t.cuts1} $(a)$ If $X$ is projective then $X_c$ is projective. $(b)$ There is a $T$-equivariant open embedding $i_c: X_{>0} \hookrightarrow X_c$ and a $T$-equivariant closed embedding $j_c: X_r \hookrightarrow X_c$ such that $X_c$ is a disjoint union of $X_{>0}$ and $X_r$. (Here $T$ acts trivially on $X_r$.) $(c)$ A neighborhood of $X_r$ in $X_c$ is isomorphic to $X^s \times^T {\Bbb A}^1 \rightarrow X_r$. \end{thm} Proof: To prove (a), we will use the fact that if $Y$ is a projective variety with a linearized $G$-action, then $Y^{ss}/G$ is projective (\cite{GIT}). Since $X \times {\Bbb A}^1$ is not projective we will construct an $A$-equivariant embedding into a projective variety $Y$ with a linearized $A$-action and show that $Y^{ss}$ coincides with $(X \times {\Bbb A}^1)^s$. We do this as follows. For any integer $j$ let ${\Bbb A}^2_{j}$ denote ${\Bbb A}^2$ with weights $0$ and $j$ for the action of $T$. Likewise, set ${\bf P}^1_{j} = {\bf P}(A^2_j)$ with the induced $T$-action. The embedding ${\Bbb A}^1 \hookrightarrow {\bf P}^1_{1}$ given by $z \mapsto [1:z]$ is $T$-equivariant and induces a $T \times T$-invariant embedding $p: X \times {\Bbb A}^1 \hookrightarrow X \times {\bf P}^1_{1}$. Let $a_0, \ldots , a_n$ be the weights of $T$ on $X$. Fix an integer $N$ which is greater than any $a_i$. There is a product action of $T \times T$ on $V \otimes {\Bbb A}^2_{N}$. The weights $T_{\Delta}$ on $V \otimes {\Bbb A}^2_{N}$ are $a_0,\ldots,a_n,a_0+N,\ldots, a_n+N$, and the weights of $A$ are $a_0,\ldots,a_n,a_0-N,\ldots, a_n-N$. Since $N$ is greater than each $a_i$, $0$ is not a weight of $A$ on $V \otimes {\Bbb A}^2_{N}$. Thus, by \cite{B-P} all semi-stable points (with respect to $A$) are stable. Consider the $T \times T$-equivariant finite morphism $\psi : X \times {\bf P}^1_{1} \rightarrow {\bf P}(V \otimes {\Bbb A}^2_{N})$ given by $$ ([x_0: \cdots :x_n], [z_0:z_1]) \mapsto [x_0 z_0^N: \cdots :x_n z_0^N : x_0 z_1^N: \cdots x_n z_1^N]. $$ Let ${\cal M} = \psi^* {\cal O} (1)$. Let ${\bf P}(V \otimes {\Bbb A}^2_{N})^s$ denote the stable points for the action of $A$ with respect to ${\cal O} (1)$ and $(X \times {\bf P}^1_{1})^s$ denote the stable points for the action of $A$ with respect to ${\cal M}$. Since $\psi$ is finite and $X \times {\bf P}^1_{1}$ is proper, by \cite[Theorem 1.19]{GIT}, $(X \times {\bf P}^1_{1})^s = \psi^{-1}({\bf P}(V \otimes {\Bbb A}^2_{N})^s)$. Since 0 is not a weight, all semi-stable points are stable. The stable points in ${\bf P}(V \otimes {\Bbb A}^2_{N})$ are described in \cite{B-P}: if we let $\Pi_A(y)$ denote the $A$-weights of $y \in {\bf P}(V \otimes {\Bbb A}^2_{N})^s$, then $y$ is stable if and only if $\Pi_A(y)$ contains both positive and negative weights. Hence the unstable points $(x,z)$ of $X \times {\bf P}^1_1$ with respect to ${\cal M}$ are of 3 types: $(i)$ $z=[0:1]$, $x$ arbitrary; $(ii)$ $z=[1:0]$, $\Pi(x)$ all positive or all negative; $(iii)$ $z \neq [0:1],[1:0]$, $\Pi(x)$ all negative.\\ In particular, the stable points of $X \times {\bf P}^1_1$ lie in the open set $X \times {\Bbb A}^1 \subset X \times {\bf P}^1$. Restricting to points $(x,w)$ in the open set $X \times {\Bbb A}^1$ we see that there are two types of unstable points: $(ii')$ $w=0$, $\Pi(x)$ all positive or all negative; $(iii')$ $w\neq 0$, $\Pi(x)$ all negative. \medskip The restriction of ${\cal M}$ to $X \times {\Bbb A}^1$ is ${\cal L}$. As in Definition \ref{d.1}, let $(X \times {\Bbb A}^1)^s$ denote the set of stable points relative to to $T$ and ${\cal L}$. By \cite[Prop. 1.18]{GIT} the inclusion $p:X \times {\Bbb A}^1 \subset X \times {\bf P}^1_{1}$ induces a reverse inclusion $$ (X \times {\Bbb A}^1)^s \supset p^{-1}(X \times {\bf P}^1_{1})^s = (X \times {\bf P}^1_{1})^s \cap (X \times {\Bbb A}^1) . $$ However, since $(X \times {\bf P}^1_1)^s \subset X \times {\Bbb A}^1$ this implies $$ (X \times {\Bbb A}^1)^s \supset (X \times {\bf P}^1_{1})^s. $$ To prove that $X_c = (X \times {\Bbb A}^1)^s/A$ is projective it suffices to show that this inclusion is an equality, since the quotient $(X \times {\bf P}^1_{1})^s /T$ is projective. In other words, we must show that all points of type (ii') and (iii') are ${\cal M}$-unstable. We will do this the old fashioned way, and show that any $A$-invariant element of $H^0(X \times {\Bbb A}^1, {\cal L}^d)$ vanishes on points of type (ii') and (iii'). In doing so, we may assume $d >>0$. For $d>>0$, the restriction map $$ H^0({\bf P}(V),{\cal O}(d)) \cong S^d(V^*) \rightarrow H^0(X,{\cal O}(d)), $$ is surjective (here $V^*$ is the dual vector space to $V$). If $\xi \in V^*$ is a positive (resp. negative) weight vector and if $x \in X$ has all positive (resp. negative) weights, then $\xi$ (viewed as a section of ${\cal O}(1)$) vanishes at $x$. Now if $s \in S^d(V^*)$ is a $T$-invariant section, then it is a sum of monomials in weight vectors, with each monomial containing both positive and negative weight vectors. Thus, if $s \in S^d(V^*)^T$ and $\Pi(x)$ consists of all positive or all negative weights, then $s(x) = 0$. Similarly, if $s \in S^d(V^*)$ is a sum of monomials and each monomial contains a negative weight vector (in particular if the weight of $s$ is negative), then $s(x) = 0$ if $\Pi(x)$ has only negative weights. Now if $s \in H^0(X,{\cal O}(d))$ has $T$-weight $k$, and $f_l(w) = w^l \in {\cal O}({\Bbb A}^1)$, then $s \otimes f_l$ has weight $k+l$ for the anti-diagonal torus $A$ (note that $f_l$ has $T$-weight $-l$). Thus, a general $A$-invariant section $\tau$ of $H^0(X \times {\Bbb A}^1, {\cal L}^d)$ is of the form $\tau = \sum \tau_lf_l$, where $\tau_l \in H^0(X,{\cal O}(d))$ has $T$-weight $-l$. We want to show that if $\tau$ is $A$-invariant then $\tau$ vanishes on $(x,w)$ of type $(ii')$ and $(iii')$. On type $(ii')$, we have $$ \tau(x,0) = \sum \tau_l(x) f_l(0) = \tau_0(x). $$ But $\tau_0$ is $T$-invariant, and $(x,0)$ of type $(ii')$ means that $\Pi(x)$ consists of all positive or all negative weights, so by the above remarks, $\tau_0(x) =0$. If $(x,w)$ is of type $(iii')$, then $\Pi(x)$ consists of all negative weights; since the weight of $\tau_i = -i \leq 0$, we have $\tau_i(x) =0$ for all $i$. Hence $\tau$ vanishes on $(x,w)$ of type $(ii')$ and $(iii')$, as desired. This proves (a). We now prove (b) and (c). The embedding $i_c$ is defined to be the quotient by $A$ of the open embedding $$ X_{>0} \times T \hookrightarrow (X \times {\Bbb A}^1)^s; $$ The embedding $j_c$ is defined to be the quotient by $A$ of the embedding $$ J: X^s \times \{ 0 \} \hookrightarrow (X \times {\Bbb A}^1)^s; $$ $J$ is closed since, from the description of $(X \times {\Bbb A}^1)^s$ given in the proof of (a), we have $(X \times {\Bbb A}^1)^s \cap X \times \{ 0 \} = X^s \times \{ 0 \}$. The embedding $i_c$ (resp. $j_c$) is open (resp. closed) since it is the quotient of an open (resp. closed) embedding. This proves (b). A neighborhood of $X^s \times \{ 0 \}$ in $(X \times {\Bbb A}^1)^s$ is given by $X^s \times {\Bbb A}^1$. It follows that a neighborhood of ${X_r}$ in $X_c$ is $T$-equivariantly isomorphic to the fibration $$ (X^s \times {\Bbb A}^1)/A \rightarrow (X^s \times \{ 0 \})/A. $$ This is isomorphic to the fibration $$ X^s \times^T {\Bbb A}^1 \rightarrow X_r, $$ proving (c). $\Box$ \medskip {\bf Remark.} In general $T$ will act with finite stabilizers on $X^s$. If $T$ acts freely on $X^s$ then $X_r$ is regularly embedded with normal bundle $X^s \times^T {\Bbb A}^1 \rightarrow X_r$. We will see in Theorem \ref{t.cuts2} that even if the action of $T$ is not free there is an equivariant pullback and self-intersection formula. \subsection{Cutting at any point} \label{cutany} Instead of considering stable points with respect to the invertible sheaf ${\cal O}_{{\bf P}(V)}(1)$ we can consider stable points with respect to other linearizations. Let $q = a/n$ be a rational number which is not a weight of $\Pi(x)$ for any $x \in X$ (we no longer require that $0$ is not a weight). Then we define $$ X^s(q) = \{x \in X | \Pi(x) \mbox{ contains both weights greater than $q$ and weights less than $q$} \}. $$ Define a $T$-action on $S^n V$ by $g \cdot v^n = g^{-a} (g \cdot v)^n$. Embed ${\bf P}(V)$ in ${\bf P}(S^n V)$ by the Veronese embedding, and let ${\cal M}(q)$ denote the pullback to $X$ of ${\cal O}_{{\bf P}(S^n V)}(1)$. Then (\cite[1.2]{B-P}) $X^s(q)$ is the set of stable points of $X$ with respect to ${\cal M}(q)$. The quotient $X_r(q) = X^s(q)/T$ corresponds (in the symplectic picture) to the reduction of $X$ at $q$. Likewise we can define (in the obvious fashion) the open subscheme $X_{>q}$ of $X$. We let ${\cal L}(q)$ denote the line bundle ${\cal M}(q)\boxtimes 1$ on $X \times {\Bbb A}^1$, and define $X_c(q) = (X \times {\Bbb A}^1)^s /A$, where now stability is defined with respect to ${\cal L}(q)$. The analogue of Theorem \ref{t.cuts1} holds for $X_c(q)$. This can be deduced from Theorem \ref{t.cuts1} by embedding $X$ into ${\bf P}(S^n V)$ as above. \section{Equivariant Intersection theory and algebraic cuts} \subsection{Review of Equivariant Chow groups} In this section we recall the definition and some of the basic properties of equivariant Chow groups \cite{E-G}. For simplicity we will assume that all schemes are quasiprojective and that any group actions are linearized. Let $G$ be a $g$-dimensional group, $X$ an $n$-dimensional scheme and $V$ a representation of $G$ of dimension $l$. Assume that there is an open set $U \subset V$ such that a principal bundle quotient $U \rightarrow U/G$ exist, and that $V-U$ has codimension more than $i$. Let $X_G = (X \times U)/G$. \begin{defn} Set $A_i^G(X) = A_{i+l-g}(X_G)$, where $A_*$ is the usual Chow group. This definition is independent of the choice of $V$ and $U$ as long as $V-U$ has sufficiently high codimension. \end{defn} {\bf Remark:} Because $X \times U \rightarrow X \times^G U$ is a principal $G$-bundle, cycles on $X \times^G U$ exactly correspond to $G$-invariant cycles on $X \times U$. Since we only consider cycles of codimension smaller than the dimension of $X \times (V-U)$, we may in fact view these as $G$-invariant cycles on $X \times V$. Thus every class in $A_i^G(X)$ is represented by a cycle in $Z_{i+l}(X \times V)^G$, where $Z_*(X \times V)^G$ indicates the group of cycles generated by invariant subvarieties. Conversely, any cycle in $Z_{i+l}(X \times V)^G$ determines an equivariant class in $A_i^G(X)$. \medskip The properties of equivariant intersection theory include the following. (1) Functoriality for equivariant maps: proper pushforward, flat pullback, l.c.i pullback, etc. (2) Chern classes of equivariant bundles operate on equivariant Chow groups. (3) If $X$ is smooth, then $\oplus A_*^G(X)$ has a ring structure. (This follows from (1), since the diagonal $X \hookrightarrow X \times X$ is an equivariant regular embedding when $X$ is smooth.) (4) There is a localization theorem for torus actions. \begin{lemma} \label{l.normal} Let $G$ be a group and $X$ a quasi-projective $G$-scheme. Suppose that $X$ is the set of stable points for the linear action of a normal (reductive) subgroup $H \subset G$. Then $A_i^G(X) = A_{i-h}^{G/H}(X/H)$, where $h = \mbox{dim }H$. \end{lemma} Proof. Let $V$ be a representation of $G/H$, and let $U$ be an open set on which $G/H$ acts freely. Then $G$ acts on $X \times U$ with finite stabilizers. Thus, by \cite[Theorem ?]{E-G} $A_*^G(X \times U) = A_*( (X \times U)/G)$. Now $(X \times U)/G = G/H(X \times U)/H$. Since $H$ acts trvially on $U$, $(X \times U)/H = X/H \times U$. Thus $(X \times U)/G = X/H \times^{G/H} U$. Hence if $V - U$ has sufficiently high codimension then $A_*(X \times U) = A_*^{G/H}(X/H)$. Thus, the lemma follows from the homotopy property of equivariant Chow groups. $\Box$ \begin{lemma} Suppose $X$ is a $G$-scheme, $Y$ is an $H$-scheme, and $Q \subset X \times Y$ is $G \times H$-invariant and flat over $X$. Let $\pi: Q \rightarrow X$ be the projection. Then there is a pullback $$ \pi^* : A^G_*(X) \rightarrow A_*^{G \times H}(Q) $$ \end{lemma} Proof. Let $V_G$ be a representation of $G$. Let $p_{Q}: Q \times V_G \rightarrow X \times V_G$ be the projection onto the first and third factors. If $Z \subset (X \times V_G)$ set $\pi^*_Q[Z] = [\pi_{Q}^{-1}Z]$. This is an invariant cycle in $Q \times V_G$ and so defines an element of $A_*^{G \times H}(Q)$. The usual arguments of equivariant theory show that this is well defined. $\Box$ \begin{thm} \label{t.cuts2} Let $X$ be a quasiprojective scheme with a linearized $T$-action and cut scheme $X_c = X_c(q)$ for some $q \in {\Bbb Q}$. Then: (a) There is a pullback of rational equivariant Chow groups $$j_c^*: A_*^T(X_c) \rightarrow A_*^T(X_r)$$ such that $j_{c*} j_c^*$ is multiplication by $t$ under the map $A^1_T \rightarrow A^1_T(X^s) \cong A^1(X_r)$. $(b)$ There is a map $s: A_*^T(X) \rightarrow A_*^T(X_c)$ such that $i^*_c \circ s = i^*$. Here $i$ and $i_c$ denote the embeddings of $X_{>0}$ into $X$ and $X_c$, respectively. $(c)$ If $F \subset X_{>0}^T$, let $i_F$ and $i_{F,c}$ denote the embeddings of $F$ into $X$ and $X_c$ respectively. Then $i_{F,c}^* \circ s = i_F^*$. $(d)$ As maps $A_*^T(X) \rightarrow A_*(X_r)$, we have $ {\cal F} \circ j_c^* \circ s = r^*$, where ${\cal F}$ denotes the forgetful map from equivariant Chow groups to ordinary Chow groups, and $r^*$ is the restriction $A_*^T(X) \rightarrow A_*^T(X_s) \cong A_*(X_r)$. \end{thm} {\bf Remark.} Because $T$ acts trivially on $X_r$, we have $A^*_T(X_r) \cong A_*(X_r) \otimes_{{\Bbb Q}} A^T_*$. Thus, if $\alpha \in A_*^T(X)$, we can write $j_c^* \circ s(\alpha) = \sum \beta_i t^i$. The content of (d) is that $\beta_0 = r(\alpha)$. \medskip Proof: To simplify notation we assume $q = 0$. By the discussion of Section \ref{cutany} this implies the theorem for $q \neq 0$ as well. (a) The $T \times T$ equivariant regular embedding $J: X^s \times \{0\} \hookrightarrow (X \times {\Bbb A}^1)^s$ gives a pullback $J^*$ on $T \times T$-equivariant Chow groups. As noted in the proof of Theorem \ref{t.cuts1}, the map $j_c$ is the quotient of the map $J$ by $A$. We have an isomorphism $T \rightarrow (T \times T) / A$, $g \mapsto (g,1) = (1,g)$ mod $A$. By Lemma \ref{l.normal} we can identify $$ A_*^T(X_c) = A_*^{T \times T}((X \times {\Bbb A}^1)^s) \mbox{ , } A_*^T(X_r) = A_*^{T \times T}(X^s \times \{0\}). $$ Let $j_c^*$ correspond to $J^*$ under this identification. The $T \times T$ equivariant normal bundle to $X^s \times \{0\}$ in $(X \times {\Bbb A}^1)^s$ is the trivial line bundle, where $T \times T$ acts with weight $(0,1)$. Thus $J_*J^*$ is multiplication by $t_2 \in A^*_{T \times T}$. With our identifications, multiplication by either $t_1$ or $t_2$ in $A^*_{T \times T}= {\Bbb Q}[t_1,t_2]$ corresponds to multiplication by $t \in A^*_T = {\Bbb Q}[t]$. Hence $j_{c *}j_c^*$ is multiplication by $t$. This proves (a). (b) We have inclusions $$ X_{>0} \times T \stackrel{I} \rightarrow (X \times {\Bbb A}^1)^s \stackrel{k} \rightarrow X \times {\Bbb A}^1 . $$ The map $i_c$ is the quotient of $I$ by $A$. Consider the following commutative diagram: $$ \begin{array}{cccc} A_*^T(X) \stackrel{\pi^*} \rightarrow A_*^{T \times T}(X \times {\Bbb A}^1) \stackrel{k^*} \rightarrow & A_*^{T \times T}((X \times {\Bbb A}^1)^s) & \stackrel{I^*} \rightarrow & A_*^{T \times T} (X_{>0} \times T) \\ & \small{f} \downarrow & & \small{g} \downarrow \\ & A_*^T(X_c) &\stackrel{i_c^*} \rightarrow & A_*^T( X_{>0} ) \end{array} $$ The vertical maps are from Lemma \ref{l.normal}; here $G = T \times T$, $H = A$, and $G/H = (T \times T)/A \cong T$. We define $$ s: A_*^T(X) \rightarrow A_*^T(X_c) $$ to be $s = f k^* \pi^*$. If $Z$ is a subvariety of $X$, then $$ i_c^* \circ s ([Z]) = g I^* k^* \pi^*[Z] = g[(Z \cap X_{>0}) \times T] . $$ Since $A$ acts freely on $X_{>0} \times T$, $$ g[(Z \cap X_{>0}) \times T] = [Z \cap X_{>0}] = i^*[Z] , $$ proving (b). Part (c) follows from (b). The proof of (d) is similar to (b). $\Box$ \subsection{Kalkman's residue formula} Following \cite{Lerman}, we use algebraic cutting to prove the residue formula of Kalkman. \begin{thm} \label{t.kalkman} Let $X \subset {\bf P}(V)$ be a smooth $n$-dimensional projective variety with a linearized $T$-action. Fix a rational $q$ which is not a weight of $x \in X$. Let $r^*: A^*_T(X) \rightarrow A^*(X_r)$ be as defined above, and let $\alpha \in A^{n-1}_T(X)$. Then $$ \mbox{deg } (r^*(\alpha) \cap [X_r]) = - \sum \mbox{deg }( (\mbox{Res}_{t=0} \frac{i^*_F \alpha}{c^T_{d_F}(N_F X)}) \cap [F]_T) , $$ where the sum is over the components $F$ of $X^T_{>q}$, and $d_F$ is the codimension of $F$. \end{thm} Proof: To simplify notation assume that $q = 0$ in the proof. Recall that $X^T_c = X^T_{>0} \cup X_r$. Each $F \subset X^T_{>0}$ is regularly embedded in $X_c$, so the self-intersection formula applies. Although $X_r$ need not be regularly embedded in $X_c$, a self-intersection formula still holds (Theorem \ref{t.cuts2}(a)). Thus the localization theorem for equivariant Chow groups \cite{E-G2} can be applied just as if $X_r$ were regularly embedded, so \begin{equation} \label{e.kalkman1} s(\alpha) \cap [X_c]_T = \sum_{F \subset X_{>0}} i_{F,c*}( \frac{i^*_{F,c} s(\alpha)}{c^T_{d_F}(N_F X)} \cap [F]_T )+ j_{c*} ( \frac{j_c^*s(\alpha)}{t} \cap [X_r]_T) . \end{equation} This equality holds in $A_*^T(X_c) \otimes_{A_T^*} {\cal Q}$, where ${\cal Q} = {\Bbb Q}[t,t^{-1}]$. Applying Theorem \ref{t.cuts2}(c) we can rewrite this as \begin{equation} \label{e.kalkman2} s(\alpha) \cap [X_c]_T = \sum_{F \subset X^T_{>0}} i_{F,c*} ( \frac{i_F^*\alpha}{c^T_{d_F}(N_F X)} \cap [F]_T) + j_{c*} ( \frac{j_c^* s(\alpha)} {t} \cap [X_r]_T). \end{equation} Let $\pi_Y$ denote the projection of a proper scheme $Y$ to a point and $\pi_{Y*}$ the induced map of equivariant Chow groups. If we apply $\pi_{X_c*}$ to both sides of (\ref{e.kalkman2}), the left side is zero since $\alpha$ has degree $n-1$ and the dimension of $X$ is $n$. Therefore \begin{equation} \label{e.kalkman3} \sum_{F \subset X^T_{>0}} \pi_{F*}( \frac{i_F^*\alpha}{c^T_{d_F}(N_F X)} \cap [F]_T)= - \pi_{X_r*}( \frac{j_c^* s(\alpha)}{t} \cap [X_r]_T). \end{equation} Now, $T$ acts trivially on $F$, so $A^*_T(F) = A^*F \otimes_{{\Bbb Q}} A^*_T$ and $[F]_T = [F] \otimes 1$. The class $$ \frac{i^*_F \alpha}{c^T_{d_F}(N_F X)} \in A^*_T(F) \otimes_{ {\cal Q}} A^*_T = A^*F \otimes_{{\Bbb Q}} {\cal Q} $$ has degree $n-1-d_F = \mbox{dim }F - 1$, so we can write it as $\sum \beta_i t^i$, where $\beta_i \in A^*F$ has degree $\mbox{dim }F - 1 - i$. Thus $$ \frac{i^*_F \alpha}{c^T_{d_F}(N_F X)} \cap [F]_T = (\sum \beta_i \cap [F]) t^i. $$ When we apply $\pi_{F*}$ to this the only term that survives is the term with $t^{-1}$. A similar argument applies to the right side of (\ref{e.kalkman3}). We can therefore rewrite (\ref{e.kalkman3}) as $$ \sum_{F \subset X_{>0}^T} \mbox{deg }( (\mbox{Res}_{t=0} \frac{i^*_F \alpha}{c^T_{d_F}(N_F X)}) \cap [F]) = \mbox{deg } \mbox{Res}_{t=0} \frac{j^* s(\alpha) \cap [X_r]_T}{t}. $$ By the remark after Theorem \ref{t.cuts2}, the right side of this equation is $r^*(\alpha) \cap [X_r]$. $\Box$ \subsection{Characteristic numbers of quotients} Kalkman's residue formula has a number of applications; for instance it can be used to prove the localization theorem of Jeffrey-Kirwan-Witten \cite{G-K}. Another use of the residue formula is to compute some characteristic numbers of quotients of a smooth variety $X$ by a torus. We present some formulas when $T$ acts freely on $X^s(q)$ and the fixed points of $X_{> q}$ are isolated. Let $Y = X^s/T$ and let $\pi: X^s \rightarrow Y$ be the quotient map. Since $T$ acts freely, $Y$ is smooth of dimension $n-1$ and we will use the residue formula to compute the Euler characteristic $\chi(Y) = \mbox{deg }( c_{n-1}(T_Y))$. Since the fiber of $\pi$ is $T$, $T_{X^s/Y}$ is the trivial bundle of rank $1$. There is an exact sequence of equivariant bundles on $X^s$ $$ 0 \rightarrow {\bf 1} \rightarrow T_{X^s} \rightarrow \pi^*{T_{Y}} \rightarrow 0.$$ Hence $r(c_i(T_{X^s})) = c_i(T_{Y})$. Since $T_{X^s}$ is just the restriction of $T_X$ to $X^s$, we can apply the residue formula. If $p \in X_{>q}^T$ is a fixed point let $\alpha_{1}(p), \ldots , \alpha_{n}(p)$ be the weights for the $T$-action on $T_{p}X$. Then $$c^T_{n-1}(T_{X}|_{p}) = t^{n-1} \sum_{i = 1}^{n} \alpha_1(p) \alpha_2(p) \ldots \widehat{\alpha_{i}(p)} \ldots \alpha_n(p)$$ and $c^T_{n}(N_{p}X) = t^n\prod_{i=1}^n \alpha_i(p)$. Thus applying the residue formula we have $$\chi(Y) = - \sum_{p \in X_{>q}^T} \sum_{i = 1}^{n} \frac{1}{\alpha_i(p)}.$$ In a similar manner we can calculate the Todd genus $\chi({\cal O}_Y)$. By Hirzebruch-Riemann-Roch, $\chi({\cal O}_Y)= \mbox{deg }(Td(T_{Y})).$ Using the residue formula to calculate $\mbox{deg }(Td(T_{Y}))$ we obtain $$\chi({\cal O}_Y) = - \sum_{p \in X_{>q}^T} \mbox{res}_{t = 0}\left(\frac{1}{ \prod_{i =1}^{n}(1 - e^{-t \alpha_i(p) })}\right) = - \sum_{p \in X_{>q}^T} \left(\sum_{1 \leq i \neq l \leq n} \frac{\alpha_{l}(p)} {2\alpha_i(p)}\right)$$.

9608

alg-geom/9608002

en

https://arxiv.org/abs/alg-geom/9608002

alg-geom/9608002

Arnaud Beauville

Arnaud Beauville, Yves Laszlo, Christoph Sorger

The Picard group of the moduli of G-bundles on a curve

36 pages, Plain TeX (xypic useful but not compulsory)

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G_2 type (we consider both the coarse moduli space and the moduli stack).

alg-geom

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Card}\nolimits{\mathop{\rm Card}\nolimits} \def\mathop{\rm Tr}\nolimits{\mathop{\rm Tr}\nolimits} \def\mathop{\rm rk\,}\nolimits{\mathop{\rm rk\,}\nolimits} \def\mathop{\rm div\,}\nolimits{\mathop{\rm div\,}\nolimits} \def\mathop{\rm Ad}\nolimits{\mathop{\rm Ad}\nolimits} \def\mathop{\rm Ad}\nolimits{\mathop{\rm Ad}\nolimits} \def\mathop{\rm Res}\nolimits{\mathop{\rm Res}\nolimits} \def\mathop{\rm Lie}\nolimits{\mathop{\rm Lie}\nolimits} \def\mathop{\oalign{lim\cr\hidewidth$\longrightarrow $\hidewidth\cr}}{\mathop{\oalign{lim\cr\hidewidth$\longrightarrow $\hidewidth\cr}}} \def\mathop{\oalign{lim\cr\hidewidth$\longleftarrow $\hidewidth\cr}}{\mathop{\oalign{lim\cr\hidewidth$\longleftarrow $\hidewidth\cr}}} \font\gragrec=cmmib10 \def\hbox{\gragrec \char22}{\hbox{\gragrec \char22}} \font\san=cmssdc10 \def\hbox{\san \char3}{\hbox{\san \char3}} \def\hbox{\san \char83}{\hbox{\san \char83}} \def\sdir_#1^#2{\mathrel{\mathop{\kern0pt\oplus}\limits_{#1}^{#2}}} \input amssym.def \input amssym \vsize = 25truecm \hsize = 16truecm \voffset = -.5truecm \parindent=0cm \baselineskip15pt \overfullrule=0pt \let\bk\backslash \let\lra\longrightarrow \let\ra\rightarrow \font\gros=cmbx12 \font\tgros=cmbx12 at 14pt \newlabel{comp}{1.2 \null\vskip0.5cm \centerline{\tgros The Picard group of the moduli of G-bundles on a curve} \smallskip \centerline{Arnaud {\pc BEAUVILLE}\note{1}{Partially supported by the European HCM project ``Algebraic Geometry in Europe" (AGE).}, Yves {\pc LASZLO}$^1$, Christoph {\pc SORGER}\note{2}{Partially supported by Europroj.}} \vskip 1cm {\bf Introduction} \par\hskip 1truecm\relax This paper is concerned with the moduli space of principal $G$\kern-1.5pt - bundles on an algebraic curve, for $G$ a complex semi-simple group. While the case $G={\bf SL}_r$, which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it became clear that these spaces play an important role in Quantum Field Theory. In particular, if $L$ is a holomorphic line bundle on the moduli space $M_G$, the space $H^0(M_G,L)$ is essentially independent of the curve $X$, and can be naturally identified with what physicists call the {\it space of conformal blocks} associated to the most standard Conformal Field Theory, the so-called WZW-model. This gives a strong motivation to determine the group $\mathop{\rm Pic}\nolimits(M_G)$ of holomorphic line bundles on the moduli space. \par\hskip 1truecm\relax Up to this point we have been rather vague about what we should call the moduli space of $G$\kern-1.5pt - bundles on $X$. Unfortunately there are two possible choices, and both are meaningful. Because $G$\kern-1.5pt - bundles have usually nontrivial automorphisms, the natural solution to the moduli problem is not an algebraic variety, but a slightly more complicated object, the algebraic stack ${\cal M}_G$. This has all the good properties one expects from a moduli space; in particular, a line bundle on ${\cal M}_G$ is the functorial assignment, for every variety $S$ and every $G$\kern-1.5pt - bundle on $X\times S$, of a line bundle on $S$. There is also a more down-to-earth object, the coarse moduli space $M_G$ of semi-stable $G$\kern-1.5pt - bundles; the group $\mathop{\rm Pic}\nolimits(M_G)$ is a subgroup of $\mathop{\rm Pic}\nolimits({\cal M}_G)$, but its geometric meaning is less clear. \par\hskip 1truecm\relax In this paper we determine the groups $\mathop{\rm Pic}\nolimits(M_G)$ and $\mathop{\rm Pic}\nolimits({\cal M}_G)$ for essentially all classical semi-simple groups, i.e.\ of type $A,B,C,D$ and $G_2$. Since the simply-connected case was treated in [L-S] (see also [K-N]), we are mainly concerned with non simply-connected groups. One new difficulty appears: the moduli space is no longer connected, its connected components are naturally indexed by $\pi _1(G)$. Let $\widetilde{G}$ be the universal covering of $G$; for each $\delta\in \pi _1(G)$, we construct a natural ``twisted" moduli stack ${\cal M}_{\widetilde{G}}^\delta$ which dominates ${\cal M}_G^\delta$. (For instance if $G={\bf PGL}_r$, it is the moduli stack of vector bundles on $X$ of rank $r$ and fixed determinant of degree $d$, with $e^{2\pi id/r}=\delta$.) This moduli stack carries in each case a natural line bundle ${\cal D}$, the determinant bundle associated to the standard representation of $\widetilde{G}$. We can now state some of our results; for simplicity we only consider the adjoint groups. \medskip {\bf Theorem}$.-$ {\it Put $\varepsilon _G^{}=1$ if the rank of $G$ is even, $2$ if it is odd. Let $\delta \in \pi _1(G)$. \par\hskip 0.5cm\relax{\rm a)} The torsion subgroup of $\mathop{\rm Pic}\nolimits({\cal M}_G^\delta)$ is isomorphic to $H^1(X,\pi _1(G))$. The torsion-free quotient is infinite cyclic, generated by ${\cal D}^r$ if $G={\bf PGL}_r$, by ${\cal D}^{\varepsilon_G^{}}$ if $G={\bf PSp}_{2l}$ or ${\bf PSO}_{2l}$.} \par\hskip 0.5cm\relax b) {\it The group $\mathop{\rm Pic}\nolimits(M_G^\delta)$ is infinite cyclic, generated by ${\cal D}^{r\varepsilon_G^{}}$ if $G={\bf PGL}_r$, by ${\cal D}^{2\varepsilon_G^{}}$ if $G={\bf PSp}_{2l}$ or ${\bf PSO}_{2l}$.}\note{1} {The statement ``$\mathop{\rm Pic}\nolimits(M_G)$ is generated by ${\cal D}^k$" must be interpreted as ``${\cal D}^k$ descends to $M_G$, and the line bundle on $M_G$ thus obtained generates $\mathop{\rm Pic}\nolimits(M_G)$" -- and similarly for a).} \medskip \par\hskip 1truecm\relax Unfortunately, though our method has some general features, it requires a case-by-case analysis -- in view of the result, this is perhaps unavoidable. An amusing consequence (\S 13) is that the moduli space $M_G$ is {\it not} locally factorial, except when $G$ is simply connected with each simple factor of type $A,C$ or perhaps $E$. However it is always a Gorenstein variety. \vskip1,5cm {\bf Notation}\smallskip \par\hskip 1truecm\relax Throughout this paper we denote by $X$ a smooth projective connected curve over ${\bf C}$; we fix a point $p$ of $X$. We let $G$ be a complex semi-simple group; by a $G$\kern-1.5pt - bundle we always mean a principal bundle with structure group $G$. We denote by ${\cal M}_G$ the moduli stack parameterizing $G$\kern-1.5pt - bundles on $X$, and by $M_G$ the coarse moduli variety of semi-stable $G$\kern-1.5pt - bundles (see \S 7). \vskip2cm \centerline{\gros Part I: The Picard group of the moduli stack} \vskip1cm \section{The stack ${\cal M}_G$} \global\def\currenvir{subsection Our main tool to study $\mathop{\rm Pic}\nolimits({\cal M}_G)$ will be the uniformization theorem of [B-L], [F2] and [L-S], which we now recall. We denote by $LG$ the loop group $G({\bf C}((z)))$, viewed as an ind-scheme over ${\bf C}$, by $L^+G$ the sub-group scheme $G({\bf C}[[z]])$, and by ${\cal Q}_G$ the infinite Grassmannian $LG/L^+G$; it is a direct limit of projective integral varieties ({\it loc.\ cit.}). Finally let $L_XG$ be the sub-ind-group $G({\cal O}(X\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}} p))$ of $LG$. The uniformization theorem defines a canonical isomorphism of stacks $${\cal M}_G\ \mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}\ L_XG\bk {\cal Q}_G\ .$$ \par\hskip 1truecm\relax Let $\widetilde{G}\rightarrow G$ be the universal cover of $G$; its kernel is canonically isomorphic to $\pi_1(G)$. We want to compare the stacks ${\cal M}_G$ and ${\cal M}_{\widetilde{G}}$. \th Lemma \enonce {\rm (i)} The group $\pi_0(LG)$ is canonically isomorphic to $\pi_1(G)$. {\rm (ii)} The quotient map $LG\rightarrow {\cal Q}_G$ induces a bijection $\pi_0(LG)\rightarrow \pi_0({\cal Q}_G)$. Each connected component of ${\cal Q}_G$ is isomorphic to ${\cal Q}_{\widetilde{G}}$. {\rm (iii)} The group $\pi_0(L_XG)$ is canonically isomorphic to $H^1(X,\pi_1(G))$. {\rm (iv)} The group $L_XG$ is contained in the neutral component $(LG)^{\rm o}$ of $LG$. \endth \label{comp} {\it Proof}: Let us first prove (i) when $G$ is simply connected. In that case, there exists a finite family of homomorphisms $x_\alpha:{\bf G}_a\rightarrow G$ such that for any extension $K$ of ${\bf C}$, the subgroups $x_\alpha(K)$ generate $G(K)$ [S1]. Since the ind-group ${\bf G}_a({\bf C}((z)))$ is connected, it follows that $LG$ is connected. \par\hskip 1truecm\relax In the general case, consider the exact sequence $1\ra \pi_1(G)\ra \widetilde{G}\ra G\ra 1$ as an exact sequence of \'etale sheaves on $D^*:=\mathop{\rm Spec} {\bf C}((z))$. Since $H^1(D^*,\widetilde{G})$ is trivial [S2], it gives rise to an exact sequence of ${\bf C}$\kern-1.5pt - groups $$1\ra L\widetilde{G}/\pi_1(G)\lra LG\lra H^1(D^*,\pi_1(G))\ra 1\ .\leqno{(\ref{comp}\ {\it a})}$$ The assertion (i) follows from the connectedness of $L\widetilde{G}$ and the canonical isomorphism $H^1(D^*,\pi_1(G))\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \pi_1(G)$ (Puiseux theorem). \par\hskip 1truecm\relax To prove (ii), we first observe that the group $L^+G$ is connected: for any $\gamma \in L^+G({\bf C})$, the map $ F_\gamma :G\times {\bf A}^1\rightarrow L^+G $ defined by $ F_\gamma (g,t)=g^{-1}\gamma (tz) $ satisfies $F_\gamma (\gamma (0),0)= 1$ and $F_\gamma (1,1)=\gamma$, hence connects $\gamma $ to the origin. Therefore the canonical map $\pi_0(LG)\rightarrow \pi_0(LG/L^+G)$ is bijective. Moreover it follows from (\ref{comp} {\it a}) that $(LG)^{\rm o}$ is isomorphic to $L\widetilde{G}/\pi_1(G)$, which gives (ii). \par\hskip 1truecm\relax Consider now the cohomology exact sequence on $X^*$ associated to the exact sequence $1\ra \pi_1(G)\ra \widetilde{G}\ra G\ra 1$. Since $H^1(X^*,\widetilde{G})$ is trivial [Ha], we get an exact sequence of ${\bf C}$\kern-1.5pt - groups $$1\ra L_X\widetilde{G}/\pi_1(G)\ra L_XG\ra H^1(X^*,\pi_1(G))\ra 1\ .\leqno(\ref{comp}\ {\it b})$$ Since the restriction map $H^1(X,\pi_1(G))\rightarrow H^1(X^*,\pi_1(G))$ is bijective and $L_X\widetilde{G}$ is connected ([L-S], Prop.\ 5.1), we obtain (iii). \par\hskip 1truecm\relax Comparing (\ref{comp}\ {\it a}) and (\ref{comp}\ {\it b}) we see that (iv) is equivalent to saying that the restriction map $H^1(X^*,\pi_1(G))\rightarrow H^1(D^*,\pi_1(G))$ is zero. This follows at once from the commutative diagram of restriction maps $$\diagramme{H^1(X,\pi_1(G)) & \hfl{\sim}{} & H^1(X^*,\pi_1(G))\cr \vfl{}{} & & \vfl {}{} \cr H^1(D,\pi_1(G)) & \hfl{}{} & H^1(D^*,\pi_1(G)) }$$and the vanishing of $H^1(D,\pi_1(G))$.\cqfd \medskip \par\hskip 1truecm\relax For $\delta\in \pi_1(G)$, let us denote by $(LG)^\delta$ the component of $LG$ corresponding to $\delta$ via Prop.\ \ref{comp} (i). \th Proposition \enonce {\rm a)} There is a canonical bijection $\pi_0({\cal M}_G)\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \pi_1(G)$. \par\hskip 1truecm\relax {\rm b)} For $\delta\in \pi_1(G)$, let ${\cal M}_G^\delta$ be the corresponding component of ${\cal M}_G$; let $\zeta $ be any element of $(LG)^\delta({\bf C})$. There is a canonical isomorphism $${\cal M}_G^\delta\ \mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}\ (\zeta^{-1}\,L_XG\,\zeta)\bk {\cal Q}_{\widetilde{G}}\ .$$ \endth \label{Bun_G} {\it Proof}: The first assertion follows from the uniformization theorem and Lemma \ref{comp}, (i), (ii) and (iv). Again by the uniformization theorem, ${\cal M}_G^\delta$ is isomorphic to $L_XG\bk (LG)^\delta/L^+G$; left multiplication by $\zeta^{-1} $ induces an isomorphism of $(LG)^\delta/L^+G$ onto $(LG)^{\rm o}/L^+G={\cal Q}_{\widetilde{G}}$, and therefore an isomorphism of $L_XG \bk (LG)^\delta/L^+G$ onto $(\zeta^{-1}\,L_XG\,\zeta)\bk {\cal Q}_{\widetilde{G}}$.\cqfd \medskip \par\hskip 1truecm\relax Prop.\ \ref{Bun_G} a) assigns to any $G$\kern-1.5pt - bundle $P$ on $X$ an element $\delta$ of $\pi_1(G)$ such that $P$ defines a point of ${\cal M}_G^\delta$; we will refer to $\delta$ as the {\it degree} of $P$. \par\hskip 1truecm\relax We will use Prop.\ \ref{Bun_G} to determine the Picard group of ${\cal M}_G^\delta$; therefore we first need to compute $\mathop{\rm Pic}\nolimits({\cal Q}_{\widetilde{G}})$. We denote by $s$ the number of simple factors of ${\rm Lie}(G)$. \th Lemma \enonce The Picard group of ${\cal Q}_{\widetilde{G}}$ is isomorphic to ${\bf Z}^s$. \endth\label{prod} {\it Proof}: Write $\widetilde{G}$ as a product $\pprod_{i=1}^s\widetilde{G}_i$ of almost simple simply connected groups. Put ${\cal Q}={\cal Q}_{\widetilde{G}}$ and ${\cal Q}_i={\cal Q}_{\widetilde{G}_i}$; the Grassmannian ${\cal Q}$ is isomorphic to $\pprod_{}^{}{\cal Q}_i$. The Picard group of ${\cal Q}_i$ is free of rank $1$ [M]; we denote by ${\cal O}_{{\cal Q}_i}(1)$ its positive generator. The projections ${\cal Q}\rightarrow {\cal Q}_i$ define a group homomorphism $\pprod_{}^{}\mathop{\rm Pic}\nolimits({\cal Q}_i)\rightarrow \mathop{\rm Pic}\nolimits({\cal Q})$; we claim that it is bijective. \par\hskip 1truecm\relax Let ${\cal L}$ be a line bundle on ${\cal Q}$; there are integers $(m_i)$ such that the restriction of ${\cal L}$ to $\{q_1\}\times\ldots\times{\cal Q}_j \times\ldots\times\{q_s\}$, for any $(q_i)\in \pprod_{}^{}{\cal Q}_i$ and any $j$, is isomorphic to ${\cal O}_{{\cal Q}_j}(m_j)$. Then ${\cal L}$ is isomorphic to $\boxtimes_i\,{\cal O}_{{\cal Q}_i}(m_i)$: by writing each ${\cal Q}_i$ as a direct limit of varieties ${\cal Q}_i^{(n)}$, we are reduced to prove that these two line bundles are isomorphic over $\pprod_i^{}{\cal Q}_i^{(n)}$, which follows immediately from the theorem of the square.\cqfd \medskip \par\hskip 1truecm\relax If $A$ is a finite abelian group, we will denote by $\widehat{A}$ its Pontrjagin dual\break $\mathop{\rm Hom}\nolimits(A,{\bf C}^*)$; it is isomorphic (non-canonically) to $A$. \medskip \th Proposition \enonce For $\delta\in \pi_1(G)$, let $q_G^\delta:{\cal Q}_{\widetilde{G}}\rightarrow {\cal M}_G^\delta$ be the canonical projection {\rm (Prop.\ \ref{Bun_G}).} The kernel of the homomorphism $$ (q_G^\delta)^*:\mathop{\rm Pic}\nolimits({\cal M}_G^\delta) \longrightarrow \mathop{\rm Pic}\nolimits({\cal Q}_{\widetilde{G}})\cong {\bf Z}^{s}$$ is canonically isomorphic to $ H^1(X, \pi_1(G)\,\widehat{}\ )$, and its image has finite index. \endth\label{Pic-Bun} {\it Proof}: Since $q_G^\delta$ identifies ${\cal M}_G^\delta$ to the quotient of ${\cal Q}_{\widetilde{G}}$ by $\zeta^{-1}\,L_XG\,\zeta$, line bundles on ${\cal M}_G^\delta$ correspond in a one-to-one way to line bundles on ${\cal Q}_{\widetilde{G}}$ with a $(\zeta^{-1}\,L_XG\,\zeta)$\kern-1.5pt - linearization; in particular, the kernel of $(q_G^\delta)^*$ is canonically isomorphic to the character group $\mathop{\rm Hom}\nolimits(L_XG,{\bf C}^*)$. From the exact sequence (\ref{comp} {\it b}) and the triviality of the character group of $L_X\widetilde{G}$ ([L-S], Cor.\ 5.2) we see that the group $\mathop{\rm Hom}\nolimits(L_XG,{\bf C}^*)$ is isomorphic to $H^1(X, \pi_1(G))\ \widehat{}\ $, which can be identified by duality with $H^1(X, \pi_1(G)\,\widehat{}\ )$. \par\hskip 1truecm\relax Write $\widetilde{G}\cong\pprod_{i=1}^s\widetilde{G}_i$ as in Lemma \ref{prod}. The image of $\pi_1(G)$ under the $i$\kern-1.5pt - th projection $p_i:\widetilde{G}\rightarrow \widetilde{G}_i$ is a central subgroup $A_i$ of $\widetilde{G}_i$; we denote by $G_i$ the quotient $\widetilde{G}_i/A_i$, so that $p_i$ induces a homomorphism $G\rightarrow G_i$. Let $\delta_i$ be the image of $\delta$ in $\pi_1(G_i)$. Choosing a non trivial representation $\rho:G_i\rightarrow {\bf SL}_r $ gives rise to a commutative diagram $$\diagramme{ {\cal Q}_{\widetilde{G}} &\hfl{pr_i}{} &{\cal Q}_{\widetilde{G}_i}&\hfl{}{}& {\cal Q}_{{\bf SL}_r} & \cr \vfl{q_G^\delta}{} & & \vfl{q_{G_i}^{\delta_i}}{} & & \vfl{q^{}_{{\bf SL}_r}}{} &\cr {\cal M}_G^\delta &\hfl{}{} & {\cal M}_{G_i}^{\delta_i} & \hfl{}{} & {\cal M}_{{\bf SL}_r}& \kern-10pt . }$$ The pull back of the determinant bundle ${\cal D}$ on ${\cal M}_{{\bf SL}_r}$ to ${\cal Q}_{{\bf SL}_r}$ is ${\cal O}_{\cal Q}(1)$ [B-L], and the pull back of ${\cal O}_{\cal Q}(1)$ to ${\cal Q}_i:={\cal Q}_{\widetilde{G}_i}$ is ${\cal O}_{{\cal Q}_i}(d_\rho )$ for some integer $d_\rho $ (the Dynkin index of $\rho $, see [L-S]). Therefore $pr_i^*\,{\cal O}_{{\cal Q}_i}(d_\rho )$ belongs to the image of $(q_G^\delta)^*$. It follows that this image has finite index.\cqfd \rem{Remark} In the sequel we will be mostly interested in the case where $G$ is almost simple; then $\pi_1(G)$ is canonically isomorphic to $\hbox{\gragrec \char22}_n$ (the group of $n$\kern-1.5pt - th roots of $1$) or to $\hbox{\gragrec \char22}_2\times \hbox{\gragrec \char22}_2$, and each of these groups is naturally isomorphic to its dual (by choosing $e^{2\pi i/n}$ as generator of $\hbox{\gragrec \char22}_n$). We thus get that the torsion subgroup of $\mathop{\rm Pic}\nolimits({\cal M}_G^\delta)$ is $J_n$ in the first case and $J_2\times J_2$ in the second, where $J_n$ denotes the kernel of the multiplication by $n$ in the Jacobian of $X$. \vskip1cm \section {The twisted moduli stack ${\cal M}_G^\delta$} \label{twist} \global\def\currenvir{subsection \label{M_G^\delta} Proposition \ref{Pic-Bun} takes care of the torsion subgroup of $\mathop{\rm Pic}\nolimits({\cal M}_G^\delta)$; to complete the description of this group we need to determine the image of $(q_G^\delta)^*$, or more precisely to describe geometrically the generators of this image. To do this we will again compare with the simply connected case, by constructing for every $\delta\in \pi_1(G)$ a ``twisted" moduli stack ${\cal M}_{\widetilde{G}}^\delta$ which dominates ${\cal M}_G^\delta$. \par\hskip 1truecm\relax Let $A$ be a central subgroup of $G$, together with an isomorphism $ A\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \pprod_{j=1}^s\hbox{\gragrec \char22}_{r_j}$. Using this isomorphism we identify $A$ to a subgroup of the torus $T=({\bf G}_m)^s$; let $C_AG $ be the quotient of $G\times T$ by the diagonal subgroup $A$. The projection $\partial:C_AG \rightarrow T/A\cong T$ induces a morphism of stacks $\mathop{\rm det}\nolimits:{\cal M}_{C_AG }\rightarrow {\cal M}_{T}$. For each element ${\bf d}=(d_1,\ldots,d_s)$ of ${\bf Z}^s$, let us denote by ${\cal O}_X({\bf d}p)$ the rational point of ${\cal M}_T$ defined by $({\cal O}_X(d_1p),\ldots,{\cal O}_X(d_sp))$. The fiber ${\cal M}^{{\bf d}}_{G,A}$ of $det$ at ${\cal O}_X({\bf d}p)$ depends only, up to a canonical isomorphism, of the class of ${\bf d}$ modulo ${\bf r}=(r_1,\ldots,r_s)$. \par\hskip 1truecm\relax If $S$ is a complex scheme, an object of ${\cal M}^{\bf d}_{G,A}(S)$ is by definition a $C_AG $\kern-1.5pt - bundle $P$ on $X\times S$ together with a $T$\kern-1.5pt - bundle isomorphism of $P\times^{C_AG }T$ with the $T$\kern-1.5pt - bundle associated to ${\cal O}_X({\bf d}p)$. If ${\bf d}=0$, giving such an isomorphism amounts to reduce the structure group of $P$ to $\mathop{\rm Ker}\nolimits\partial=G$: in other words, the stack ${\cal M}^{\bf 0}_{G,A} $ is canonically isomorphic to ${\cal M}_G$. \global\def\currenvir{subsection\label{deg} The projection $p:C_AG \rightarrow G/A$ induces a morphism of stacks\break $\pi:{\cal M}_{G,A}^{\bf d}\rightarrow{\cal M}_{G/A}$. The exact sequence $$1\rightarrow A\longrightarrow C_AG \ \hfl{(p,\partial)}{}\ (G/A)\times T\rightarrow 1$$ gives rise to a cohomology exact sequence $$H^1(X,A)\rightarrow H^1(X,C_AG )\rightarrow H^1(X,G/A)\times H^1(X,T)\rightarrow H^2(X,A)$$ from which we deduce that the degree $\delta\in \pi_1(G)$ of the $\!G$\kern-1.5pt - bundle $\pi(P)$, for $P\in$ ${\cal M}_{G,A}^{\bf d}({\bf C})$, satisfies $\rho(\delta)\,e^{2\pi i{\bf d}/{\bf r}}=1$, where $\rho$ is the natural homomorphism of $\pi_1(G/A)$ onto $A\i ({\bf G}_m)^s$ and $e^{2\pi i{\bf d}/{\bf r}}$ stands for the element $(e^{2\pi id_1/r_1},\ldots ,e^{2\pi id_s/r_s})$ of $ ({\bf G}_m)^s$. We denote by ${\cal M}_{G,A}^\delta$ the open and closed substack $\pi^{-1}({\cal M}_{G/A}^\delta)$ of ${\cal M}_{G,A}^{\bf d}$, where ${\bf d}=(d_1,\ldots,d_s)$ is the unique element of ${\bf Z}^s$ such that $0\leq d_t<r_t$ and $\rho(\delta)\,e^{2\pi i{\bf d}/{\bf r}}=1$ (if $G$ is simply connected, $\rho$ is bijective and ${\cal M}_{G,A}^\delta$ is simply ${\cal M}_{G,A}^{\bf d}$). The induced morphism $\pi:{\cal M}_{G,A}^\delta \rightarrow {\cal M}_{G/A}^\delta$ is surjective. \par\hskip 1truecm\relax We will be mostly interested in the case when $A$ is the center of $G$; then we will denote simply by ${\cal M}_G^\delta$ the stack ${\cal M}^\delta_{G,A}$, for any choice of the isomorphism $ A\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \pprod_{j=1}^s\hbox{\gragrec \char22}_{r_j}$ (up to a canonical isomorphism, the stack ${\cal M}^\delta_{G,A}$ does not depend on this choice). If $\delta$ belongs to $\pi_1(G)\subset\pi_1(G_{\rm ad})$, one gets $\rho(\delta)=1$ hence ${\bf d}=0$: by the above remark, the notation ${\cal M}_{G}^\delta$ is thus coherent with the one introduced in Prop.\ \ref{Bun_G}. \medskip \rem{Examples}\label{ex-M_G^\delta} {\it a}) We take $G=SL_r$, $A=\hbox{\gragrec \char22}_r$. The group $C_AG$ is canonically isomorphic to ${\bf GL}_r$; the stack ${\cal M}_{{\bf SL}_r}^d$ can be identified with the stack of vector bundles $E$ on $X$ with an isomorphism $\hbox{\san \char3}^r E\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal O}_X(dp)$. \par\hskip 1truecm\relax {\it b}) We take for $G$ the group ${\bf O}_{2l}$ or ${\bf Sp}_{2l}$, for $A$ its center, with the unique isomorphism $A\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}\hbox{\gragrec \char22}_2$. The group $C_AG$ is the group $C{\bf O}_{2l}$ or $C{\bf Sp}_{2l}$ of automorphisms of ${\bf C}^{2l}$ respecting the bilinear form up to a (fixed) scalar. The stack ${\cal M}_G^d$ can therefore be viewed as parameterizing vector bundles $E$ on $X$ with a (symmetric or alternate) non-degenerate bilinear form with values in ${\cal O}_X(dp)$. Similarly, the stack ${\cal M}_{{\bf SO}_{2l}}^d$ parameterizes vector bundles $E$ on $X$ with a non-degenerate quadratic form $q:\hbox{\san \char83}^2E\rightarrow {\cal O}_X(dp)$ and an {\it orientation}, i.e.\ an isomorphism $\omega :\mathop{\rm det}\nolimits E\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal O}_X(dlp)$ such that $\omega ^{\otimes2}$ coincides with the quadratic form induced by $q$ on $\mathop{\rm det}\nolimits E$. \par\hskip 1truecm\relax {\it c}) We take $G={\bf Spin}_r$, $A=\hbox{\gragrec \char22}_2$. Then $C_AG$ is the Clifford group and ${\cal M}_{G,A}^{-1}$ is the moduli stack ${\cal M}^-_{{{\bf Spin}_r}}$ considered in [O]. \medskip \global\def\currenvir{subsection\label{unif} Choose any element $\zeta \in (LG_{\rm ad})^\delta({\bf C})$; reasoning as in Prop.\ \ref{Bun_G}, one gets a canonical isomorphism ${\cal M}_{G}^\delta\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}(\zeta^{-1}\,L_X{G}\,\zeta)\bk{\cal Q}_{\widetilde{G}}$ (see also [B-L], 3.6 for the case ${G}={\bf SL}_r$). In particular, the stack ${\cal M}_{G}^\delta$ is connected. Moreover, we see as in the proof of Prop.\ \ref{Pic-Bun} that the torsion subgroup of $\mathop{\rm Pic}\nolimits({\cal M}_{G}^\delta)$ is canonically isomorphic to $H^1(X,\pi_1(G)\,\widehat{}\ )$. \par\hskip 1truecm\relax Let us apply the above construction to the group $\widetilde{G}$, with $A=\pi _1(G)$. Let $\delta\in \pi_1(G)$. {}From the exact sequence (\ref{comp} {\it a}), we see that $\zeta $ is the image of an element of $(L\widetilde{G})^\delta$. Comparing with Prop.\ \ref{Bun_G}, we see that the morphism $q_{G}^\delta:{\cal Q}_{\widetilde{G}}\rightarrow {\cal M}_{G}^\delta$ factors as $$q_{G}^\delta:{\cal Q}_{\widetilde{G}}\qfl{q_{\widetilde{G}}^\delta} {\cal M}_{\widetilde{G}}^\delta \qfl{\pi} {\cal M}_{G}^\delta\ .$$ This shows us the way to determine the group $\mathop{\rm Pic}\nolimits({\cal M}_{G}^\delta)$: we will first compute $\mathop{\rm Pic}\nolimits({\cal M}_{G}^\delta)$ when $G$ is simply connected or $G={\bf SO}_{2l}$, then determine which powers of the generator(s) descend to ${\cal M}_{G}^\delta$. \vskip1cm \section{The Picard group of ${\cal M}_{{\bf PGL}_r}$} \par\hskip 1truecm\relax According to (\ref{Bun_G}), the connected components of ${\cal M}_{{\bf PGL}_r}$ are indexed by the integers $d$ with $0\leq d<r$; the component ${\cal M}_{{\bf PGL}_r}^d$ is dominated by the moduli stack ${\cal M}_{{\bf SL}_r}^d$ parameterizing vector bundles $E$ on $X$ with an isomorphism $\hbox{\san \char3}^rE\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal O}_X(dp)$ (\ref{ex-M_G^\delta} {\it a}). \par\hskip 1truecm\relax Recall that the {\it determinant bundle} ${\cal D}$ on ${\cal M}_{{\bf SL}_r}^d$ is the line bundle\break $\mathop{\rm det}\nolimits R(pr_2)_*({\cal E})$, where ${\cal E}$ is the universal bundle on $X\times {\cal M}_{{\bf SL}_r}^d$. It follows from [B-L], Prop.\ 9.2, that ${\cal D}$ generates $\mathop{\rm Pic}\nolimits( {\cal M}_{{\bf SL}_r}^d)$ and that its inverse image on ${\cal Q}$ generates $\mathop{\rm Pic}\nolimits({\cal Q})$. Therefore our problem is to determine which powers of ${\cal D}$ descend to ${\cal M}^d_{{\bf PGL}_r}$. \th Proposition \enonce The smallest power of ${\cal D}$ which descends to ${\cal M}_{{\bf PGL}_r}^d$ is ${\cal D}^r$. \endth\label{PGL_r} {\it Proof}: Since it preserves the Killing form, the adjoint representation defines a homomorphism ${\rm Ad}:{\bf GL}_r\rightarrow {\bf SO}_{r^2}$. Let $f:{\cal M}_{{\bf SL}_r}^d\rightarrow {\cal M}_{{\bf SO}_{r^2}}$ be the induced morphism of stacks; since ${\rm Ad}$ factors through ${\bf PGL}_r$, $f$ factors through ${\cal M}_{{\bf PGL}_r}^d$. By [L-S], the determinant bundle ${\cal D}_{\bf SO}$ on ${\cal M}_{{\bf SO}_{r^2}}$ admits a square root ${\cal P}$; one has $f^*{\cal D}_{\bf SO}\cong{\cal D}^{2r}$ since the Dynkin index of ${\rm Ad}$ is $2r$, hence $f^*{\cal P}\cong {\cal D}^r$, which implies that ${\cal D}^r$ descends. \par\hskip 1truecm\relax Let $J$ be the Jacobian of $X$, and ${\cal L}$ the Poincar\'e bundle on $X\times J$ whose restriction to $\{p\}\times J$ is trivial. Consider the vector bundles $${\cal F}={\cal L}^{\oplus (r-1)}\oplus {\cal L}^{1-r}(dp)\qquad {\rm and}\qquad {\cal G}= {\cal O}_X^{\oplus (r-1)}\oplus {\cal L}^{-1}(dp)$$ on $X\times J$. We denote by $r^{}_J$ the multiplication by $r$ in $J$, and put $r^{}_{X\times J}=\mathop{\rm Id}\nolimits^{}_X\times r^{}_J$. Since $r_{X\times J}^*\,{\cal L}\cong{\cal L}^r$, one has $r_{X\times J}^*\,{\cal G}\cong {\cal F}\otimes{\cal L}^{-1}$, hence the projective bundles $P({\cal F})$ and $r_{X\times J}^*\,P({\cal G})$ are isomorphic. Therefore we have a commutative\note{1}{By this we always mean 2-commutative, e.g.\ in our case the two functors $\pi\kern 1pt{\scriptstyle\circ}\kern 1pt f$ and $g\kern 1pt{\scriptstyle\circ}\kern 1pt r_J^{}$ are isomorphic.}diagram of stacks $$\diagramme{J & \phfl{f}{} & {\cal M}^d_{{\bf SL}_r} &\cr \vfl{r^{}_J}{} & & \vfl{}{\pi}&\cr J &\phfl{g}{} &{\cal M}_{{\bf PGL}_r}^d &\kern-12pt , }$$ where $f$ and $g$ are the morphisms associated to ${\cal F}$ and $P({\cal G})$ respectively. \par\hskip 1truecm\relax Thus if ${\cal D}^k$ descends to ${\cal M}_{{\bf PGL}_r}^d$, the class of $f^*{\cal D}^k$ in the N\'eron-Severi group $NS(J)$ must be divisible by $r^2$. An easy computation shows that the class of $f^*{\cal D}$ in $NS(J)$ is $r(r-1)$ times the principal polarization; it follows that $r^2$ must divide $kr(r-1)$, which means that $r$ must divide $k$.\cqfd \bigskip \rem{Remark} One can consider more generally the group $G={\bf SL}_r/\hbox{\gragrec \char22}_s$, for each integer $s$ dividing $r$, and the corresponding stacks ${\cal M}_G^d$ for $d\in {r\over s}{\bf Z}$ (mod.$\,r{\bf Z}$). We can prove that {\it the line bundle ${\cal D}^k$ descends to ${\cal M}_G^d$ if and only if $k$ is a multiple of $s/(s,{r\over s})$}. The ``only if" part is proved exactly as above, but the other implication requires some descent theory on stacks which lies beyond the scope of this paper. \vskip1cm \section{The Picard group of ${\cal M}_{{\bf PSp}_{2l}}$} \par\hskip 1truecm\relax According to Prop.\ \ref{Bun_G} the moduli stack ${\cal M}_{{\bf PSp}_{2l}}$ has 2 components ${\cal M}_{{\bf PSp}_{2l}}^d$ $(d=0,1)$; the component ${\cal M}_{{\bf PSp}_{2l}}^d$ is dominated by the algebraic stack ${\cal M}_{{\bf Sp}_{2l}}^d$ parameterizing vector bundles of rank $2l$ on $X$ with a symplectic form $\hbox{\san \char3}^2E\rightarrow {\cal O}_X(dp)$ (Example \ref{ex-M_G^\delta} {\it b}). Let ${\cal D}$ denote the determinant bundle on ${\cal M}_{{\bf Sp}_{2l}}^d$ (i.e.\ the determinant of the cohomology of the universal bundle on $X\times {\cal M}_{{\bf Sp}_{2l}}^d$); it is the pull back of the determinant bundle ${\cal D}_0$ on ${\cal M}_{{\bf SL}_{2l}}^d$ by the $f:{\cal M}_{{\bf Sp}_{2l}}^d\rightarrow {\cal M}_{{\bf SL}_{2l}}^d$ associated to the standard representation. \th Lemma \enonce The group $\mathop{\rm Pic}\nolimits({\cal M}_{{\bf Sp}_{2l}}^d)$ is generated by ${\cal D}$. \endth \label{Sp^d} {\it Proof}: Consider the commutative diagram $$\diagramme{ {\cal Q}_{{\bf Sp}_{2l}}&\hfl{F}{}& {\cal Q}_{{\bf SL}_{2l}}&\cr \vfl{q_{{\bf Sp}_{2l}}^d}{} & & \vfl{}{q_{{\bf SL}_{2l}}^d}&\cr {\cal M}_{{\bf Sp}_{2l}}^d & \hfl{f}{} & {\cal M}_{{\bf SL}_{2l}}^d& \kern-12pt , }$$ where $f$ and $F$ are induced by the embedding ${\bf Sp}_{2l}\rightarrow {\bf SL}_{2l}$, and $q^d_G:{\cal Q}_G\rightarrow {\cal M}_G^d$ is the canonical projection (\ref{unif}). One has ${\cal D}=f^*{\cal D}_0$, $(q_{{\bf SL}_{2l}}^d)^*{\cal D}_0={\cal O}_{{\cal Q}_{ {\bf SL}_{2l}}}(1)$ by [B-L], 5.5, and $F^*{\cal O}_{{\cal Q}_{ {\bf SL}_{2l}}}(1)={\cal O}_{{\cal Q}_{{\bf Sp}_{2l}}}(1)$ since the Dynkin index of the standard representation of ${\bf Sp}_{2l}$ is $1$ ([L-S], Lemma 6.8). It follows that the homomorphism $(q_{{\bf Sp}_{2l}}^d)^*:\mathop{\rm Pic}\nolimits({\cal M}_{ {\bf Sp}_{2l}}^d)\rightarrow \mathop{\rm Pic}\nolimits({\cal Q}_{{\bf Sp}_{2l}})={\bf Z}\,{\cal O}_{\cal Q}(1)$ is surjective. On the other hand, the proof of Prop.\ 6.2 in [L-S] shows that it is injective; our assertion follows.\cqfd \medskip \par\hskip 1truecm\relax In view of the above remarks, Prop.\ \ref{Pic-Bun} and (\ref{unif}) provide us with an exact sequence $$0\rightarrow J_2\rightarrow \mathop{\rm Pic}\nolimits({\cal M}_{{\bf PSp}_{2l}}^d)\qfl{\pi^*} \mathop{\rm Pic}\nolimits({\cal M}_{{\bf Sp}_{2l}}^d)={\bf Z}\,{\cal D}\ ;$$ we now determine the image of $\pi^*$: \th Proposition \enonce The smallest power of ${\cal D}$ which descends to ${\cal M}_{{\bf PSp}_{2l}}^d$ is ${\cal D}$ if $l$ is even, ${\cal D}^2$ if $l$ is odd. \endth \label{PSp} {\it Proof}: The stack ${\cal M}_{{\bf Sp}_{2l}}^d$ parameterizes vector bundles $E$ with a symplectic form $\varphi :\hbox{\san \char3}^2E\rightarrow {\cal O}_X(dp)$ (\ref{ex-M_G^\delta} {\it b}). For such a pair, the form $\hbox{\san \char3}^2\varphi $ defines a quadratic form on $\hbox{\san \char3}^2E$ with values in ${\cal O}_X(2dp)$, hence an ${\cal O}_X$\kern-1.5pt - valued quadratic form on $\hbox{\san \char3}^2E(-dp)$. Put $N=l(2l-1)$; let $f_d:{\cal M}_{{\bf Sp}_{2l}}^d\rightarrow {\cal M}_{{\bf SO}_{N}}$ be the morphism of stacks which associates to $(E,\varphi )$ the pair $(\hbox{\san \char3}^2E(-dp),\hbox{\san \char3}^2\varphi )$. Since the representation $\hbox{\san \char3}^2:{\bf Sp}_{2l}\rightarrow {\bf SO}_N$ factors through ${\bf PSp}_{2l}$, the morphism $f_d$ factors as $$f_d: {\cal M}_{{\bf Sp}_{2l}}^d\rightarrow {\cal M}_{{\bf PSp}_{2l}}^d\rightarrow {\cal M}_{{\bf SO}_{N}}\ .$$ The pull back under $f_d$ of the determinant bundle on ${\cal M}_{{\bf SO}_N}$ is ${\cal D}^{2l-2}$ ($2l-2$ is the Dynkin index of the representation $\hbox{\san \char3}^2$). But we know by [L-S] that this determinant bundle admits a square root, hence ${\cal D}^{l-1}$ descends to ${\cal M}_{{\bf PSp}_{2l}}^d$. On the other hand, the same argument applied to the adjoint representation shows that ${\cal D}^{2l}$ descends (see the proof of Prop.\ \ref{PGL_r}). We conclude that ${\cal D}^2$ descends, and that ${\cal D}$ descends when $l$ is even. \par\hskip 1truecm\relax To prove that ${\cal D}$ does not descend when $l$ is odd, we use the notation of the proof of Prop.\ \ref{PGL_r}, and consider on $X\times J$ the vector bundle ${\cal H}={\cal L}^{\oplus l}\oplus {\cal L}^{-1}(dp)^{\oplus l}$, endowed with the standard hyperbolic alternate form with values in ${\cal O}(dp)$. We see as in {\it loc.\ cit.} that the ${\bf PSp}_{2l}$\kern-1.5pt - bundle associated to ${\cal H}$ descends under the isogeny $2^{}_J$ (observe that ${\cal H}\otimes{\cal L}$ descends, and use the exact sequence $\ 1\rightarrow {\bf G}_m\rightarrow C{\bf Sp}_{2l}\rightarrow $ $\rightarrow {\bf PSp}_{2l}\rightarrow 1$). Therefore the morphism $h:J\rightarrow {\cal M}_{{\bf Sp}_{2l}}^d$ defined by ${\cal H}$ fits in a commutative diagram $$\diagramme{J & \phfl{h}{} & {\cal M}^d_{{\bf Sp}_{2l}} &\cr \vfl{2_J}{} & & \vfl{}{}&\cr J &\phfl{}{} &{\cal M}^d_{{\bf PSp}_{2l}} &\kern-12pt . }$$ Since the class of $f^*{\cal D}$ in $NS(J)$ is $2l$ times the principal polarization, it follows that ${\cal D}$ does not descend.\cqfd \vskip1cm \section{The Picard group of ${\cal M}_{{\bf PSO}_{2l}}$} \global\def\currenvir{subsection Let us consider first the moduli stack ${\cal M}_{{\bf SO}_r}$, for $r\geq 3$. It has two components ${\cal M}_{{\bf SO}_r}^w$, distinguished by the second Stiefel-Whitney class $w\in \hbox{\gragrec \char22}_2$. The Picard group of these stacks is essentially described in [L-S]: to each theta-characteristic $\kappa $ on $X$ is associated a Pfaffian line bundle ${\cal P}_\kappa $ whose square is the determinant bundle ${\cal D}$ (determinant of the cohomology of the universal bundle on $X\times {\cal M}_{{\bf SO}_{r}}^w$); according to Prop.\ \ref{Pic-Bun}, there is a canonical exact sequence $$0\rightarrow J_2\qfl{\lambda } \mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_r}^w) \longrightarrow {\bf Z}\rightarrow 0\ ,$$ where the torsion free quotient is generated by any of the ${\cal P}_\kappa $'s. \medskip \par\hskip 1truecm\relax We can actually be more precise. Let $\theta (X)$ be the subgroup of $\mathop{\rm Pic}\nolimits(X)$ generated by the theta-characteristics; it is an extension of ${\bf Z}$ by $J_2$. \th Proposition \enonce The map $\kappa \mapsto{\cal P}_\kappa $ extends by linearity to an isomorphism ${\cal P}:\theta (X)\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_r}^w)$, which coincides with $\lambda $ on $J_2$. \endth\label{\theta } \par\hskip 1truecm\relax In other words, we have a canonical isomorphism of extensions $$\diagramme{ 0\rightarrow & J_2 & \phfl{}{} & \theta (X) & \phfl{}{}&{\bf Z} &\rightarrow 0 &\cr &\left \| \vbox to 6truemm{}\right. & & \vfl{\cal P}{} & & \left \| \vbox to 6truemm{}\right. &&\cr 0\rightarrow & J_2 & \phfl{\lambda }{} & \mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_{r}}^w) & \phfl{}{}&{\bf Z} &\rightarrow 0 &\kern-12pt . }$$ {\it Proof}: It suffices to prove the formula ${\cal P}_{\kappa\otimes\alpha }={\cal P}_{\kappa}\otimes \lambda (\alpha)$ for any theta-characteristic $\kappa $ and element $\alpha $ of $J_2$. \par\hskip 1truecm\relax Let ${\cal L}$ be the Poincar\'e bundle on $X\times J$, normalized so that its restriction to $\{p\}\times J$ is trivial. Put $d=0$ if $w=1$, $d=1$ if $w=-1$. The vector bundle ${\cal L}(dp)\oplus {\cal L}^{-1}(-dp)\oplus {\cal O}^{r-2}$, with its natural quadratic form and orientation, defines a morphism $g:J\rightarrow {\cal M}_{{\bf SO}_r}^w$. Let us identify $J$ with $\mathop{\rm Pic}\nolimits^{\rm o}(J)$ via the principal polarization. Then the required formula is a consequence of the following two assertions: \par\hskip 0.5cm\relax {\it a}) One has $g^*{\cal P}_{\kappa\otimes\alpha }=(g^*{\cal P}_{\kappa})\otimes \alpha $ for every theta-characteristic $\kappa $ and element $\alpha $ of $J_2$; \par\hskip 0.5cm\relax{\it b}) The map $g^*:\mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_r}^w)_{tors}\rightarrow J_2$ is the inverse isomorphism of $\lambda $. \par\hskip 1truecm\relax Let us prove {\it a}). The line bundle $g^*{\cal P}_\kappa $ is the pfaffian bundle associated to the quadratic bundle ${\cal L}(dp) \oplus {\cal L}^{-1}(-dp)$) and to $\kappa $. Now it follows from the construction in [L-S] that for any vector bundle $E$ on $X\times S$, the pfaffian of the cohomology of $E\oplus (K_X\otimes E^*)$, endowed with the standard hyperbolic form with values in $K_X$, is the determinant of the cohomology of $E$. Because the choice of ${\cal L}$ ensures that the determinant of the cohomology is the same for ${\cal L}$ and ${\cal L}(p)$, we conclude that $g^*{\cal P}_\kappa $ is the determinant of the cohomology of ${\cal L}\otimes\kappa $, i.e.\ the line bundle ${\cal O}_J(\Theta _\kappa )$. Since $\Theta _{\kappa\otimes\alpha }=\Theta _\kappa +\alpha $, the assertion {\it a}) follows. \par\hskip 1truecm\relax Since we already know that $\mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_r}^\pm)_{tors}$ is isomorphic to $J_2$ (Prop.~\ref{Pic-Bun}), {\it a}) implies that $g^*$ is surjective, and therefore bijective. Hence $u=g^*\kern 1pt{\scriptstyle\circ}\kern 1pt \lambda $ is an automorphism of $J_2$. This construction extends to any family of curves $f:{\cal X}\rightarrow {\cal S}$, defining an automorphism of the local system $R^1f_*(\hbox{\gragrec \char22}_2)$ over ${\cal S}$. Since the mono\-dromy group of this local system is the full symplectic group ${\bf Sp}(J_2)$ for the universal family of curves, it follows that $u$ is the identity.\cqfd \medskip \global\def\currenvir{subsection This settles the case of the group ${\bf SO}_r$; let us now consider the group ${\bf PSO}_r$, for $r=2l\geq 4$. The moduli space ${\cal M}_{{\bf PSO}_{2l}}$ has $4$ components, indexed by the center $Z$ of ${\bf Spin}_{2l}$. This group consists of the elements $\{1,-1,\varepsilon ,-\varepsilon \}$ of the Clifford algebra $C({\bf C}^{2l})$, with $\varepsilon ^2=(-1)^l$ ([Bo], Alg\`ebre IX). Each component ${\cal M}_{{\bf PSO}_{2l}}^\delta$, for $\delta\in Z$, is dominated by the algebraic stack ${\cal M}_{{\bf SO}_{2l}}^\delta$ (\ref{M_G^\delta}). For $\delta\in\{\pm 1\}$, this is the same stack as above; the stack ${\cal M}_{{\bf SO}_{2l} }^\varepsilon \cup {\cal M}_{{\bf SO}_{2l}}^{-\varepsilon }$ parameterizes vector bundles with a quadratic form with values in ${\cal O}_X(p)$ and an orientation (\ref{ex-M_G^\delta} {\it b}). Changing the sign of the orientation exchanges the two components ${\cal M}^\varepsilon $ and ${\cal M}^{-\varepsilon }$ (this corresponds to the fact that $\varepsilon $ and $-\varepsilon $ are exchanged by the outer automorphism of ${\bf Spin}(2l)$ defined by conjugation by an odd degree element of the Clifford group). \label{PSO} \medskip \th Lemma \enonce The torsion free quotient of $\mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_{2l}}^{\pm\varepsilon})$ is generated by the determinant bundle ${\cal D}$. \endth {\it Proof}: The same proof as in Lemma \ref{Sp^d} shows that the pull back of ${\cal D}$ by the morphism $q_{{\bf SO}_{2l}}^{\pm\varepsilon }:{\cal Q}_{{\bf\rm Spin}_{2l}}\rightarrow {\cal M}_{{\bf SO}_{2l}}^{\pm\varepsilon}$ is ${\cal O}_{\cal Q}(2)$ (the Dynkin index of the standard representation of ${\bf SO}_{2l}$ is $2$). Therefore it suffices to prove that ${\cal D}$ has no square root in $\mathop{\rm Pic}\nolimits({\cal M}_{{\bf SO}_{2l}}^{\pm\varepsilon})$. \par\hskip 1truecm\relax Let $V$ be a $l$\kern-1.5pt - dimensional vector space; we consider the vector bundle\break $T =(V\otimes_{\bf C}{\cal O}_X)\oplus (V^*\otimes_{\bf C}{\cal O}_X(p))$, with the obvious hyperbolic quadratic form\break $q:\hbox{\san \char83}^2T\rightarrow {\cal O}_X(p)$ and isomorphism $\omega :\mathop{\rm det}\nolimits T \mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal O}_X(lp)$. We choose the sign of $\omega $ so that the triple $T^\varepsilon:=(T,q,\omega ) $ defines a rational point of ${\cal M}_{{\bf SO}_{2l} }^{\varepsilon}$, and put $T^{-\varepsilon }:=(T,q,-\omega )\in {\cal M}_{{\bf SO}_{2l} }^{-\varepsilon}({\bf C})$. The group $G={\bf GL}(V)$ acts on $T$, and this action preserves the quadratic form and the orientation. This defines a morphism $\iota$ of the stack $BG$ classifying $G$\kern-1.5pt - torsors into ${\cal M}_{{\bf SO}_{2l}}^{\pm\varepsilon}$: if $S$ is a ${\bf C}$\kern-1.5pt - scheme and $P$ a $G$\kern-1.5pt - torsor on $S$, one puts $\iota (P)=P\times^GT^{\pm\varepsilon }_S$. \par\hskip 1truecm\relax Recall [L-MB] that the ${\bf C}$\kern-1.5pt - stack $BG$ is the quotient of $\mathop{\rm Spec} {\bf C}$ by the trivial action of $G$; in particular, line bundles on $BG$ correspond in a one-to-one way to $G$\kern-1.5pt - linearizations of the trivial line bundle on $\mathop{\rm Spec} {\bf C}$, that is to characters of $G$. In our situation, the line bundle $\iota ^*{\cal D}$ will correspond to the character of $G$ by which $G$ acts on $\mathop{\rm det}\nolimits R\Gamma (X,T)$. As $G$\kern-1.5pt - modules, we have $$\mathop{\rm det}\nolimits R\Gamma (X,T)\cong\mathop{\rm det}\nolimits R\Gamma (X,V\otimes_{\bf C} {\cal O}_X)\otimes \mathop{\rm det}\nolimits R\Gamma (X,V^*\otimes_{\bf C}{\cal O}_X(p))\ .$$ Now if $L$ is a line bundle on $X$, the $G$\kern-1.5pt - module $\mathop{\rm det}\nolimits R\Gamma (X,V\otimes_{\bf C}L)$ is isomorphic to $\mathop{\rm det}\nolimits(V\otimes H^0(L))\otimes \mathop{\rm det}\nolimits(V\otimes H^1(L))^{-1}=\mathop{\rm det}\nolimits(V)^{\chi (L)}$. We conclude that $\mathop{\rm det}\nolimits R\Gamma (X,T)$ is isomorphic to $\mathop{\rm det}\nolimits(V^*)$, i.e.\ that $\iota ^*{\cal D}$ corresponds to the character $\mathop{\rm det}\nolimits^{-1}:G\rightarrow {\bf C}^*$. Since $\mathop{\rm det}\nolimits$ generates $\mathop{\rm Hom}\nolimits(G,{\bf C}^*)$, our assertion follows.\note{1}{This argument has been shown to us by V. Drinfeld.}\cqfd \th Proposition \enonce Let $\delta\in Z$. The line bundle ${\cal D}$ {\rm (}resp.\ ${\cal D}^2)$ descends on ${\cal M}_{{\bf PSO}_{2l}}^\delta$ if $l$ is even {\rm (}resp.\ odd{\rm );} the corresponding line bundles on ${\cal M}_{{\bf PSO}_{2l}}^\delta$ generate the Picard group. \endth {\it Proof}: We first prove that the Pfaffian bundles ${\cal P}_\kappa $ do not descend to ${\cal M}_{{\bf PSO}_{2l}}^\delta$. If $\delta\in\{\pm\varepsilon \}$, this follows from the above lemma. If $\delta\in\{\pm1\}$, we consider the action of $J_2$ on ${\cal M}_{{\bf SO}_{2l}}^\delta$ deduced from the embedding $\hbox{\gragrec \char22}_2\i {\bf SO}_{2l}$: each element $\alpha \in J_2$ (trivialized at $p$) defines an automorphism -- still denoted $\alpha $ -- of the stack ${\cal M}_{{\bf SO}_{2l}}^\pm$, which maps a quadratic bundle $(E,q,\omega )$ onto $(E\otimes\alpha, q\otimes i_\alpha ,\omega \otimes i_\alpha ^{\otimes l}) $, where $i_\alpha :\alpha ^2\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal O}_X$ is the isomorphism which coincides at $p$ with the square of the given trivialization. \par\hskip 1truecm\relax We claim that $\alpha ^*{\cal P}_\kappa $ {\it is isomorphic to} ${\cal P}_{\kappa \otimes\alpha }$ for every theta-characteristic $\kappa $ and element $\alpha$ of $J_2$. This is easily seen by using the following characterization of ${\cal P}_\kappa $ ([L-S], 7.10): let ${\cal E}$ be the universal bundle on $X\times {\cal M}_{{\bf SO}_{2l}}^\pm$; then the divisor $\Theta _\kappa :=\mathop{\rm div\,}\nolimits Rpr_{2*}({\cal E}\otimes\kappa )$ is divisible by $2$ in $\mathop{\rm Div}\nolimits{\cal M}_{{\bf SO}_{2l}}^\pm$, and ${\cal P}_\kappa $ is the line bundle associated to ${1\over 2}\Theta _\kappa $. By construction $(1_X\times \alpha )^*{\cal E}$ is isomorphic to ${\cal E}\otimes\alpha $, hence $$\alpha ^*\Theta _\kappa =\mathop{\rm div\,}\nolimits Rpr_{2*}((1_X\times \alpha )^*{\cal E}\otimes\kappa )=\mathop{\rm div\,}\nolimits Rpr_{2*}({\cal E}\otimes\alpha \otimes\kappa )=\Theta _{\kappa \otimes\alpha }\ ,$$which implies our claim. Since the map $\kappa \mapsto {\cal P}_\kappa $ is injective (Prop.\ \ref{\theta }), we conclude that ${\cal P}_\kappa $ does not descend. \par\hskip 1truecm\relax The rest of the proof follows closely the symplectic case (Prop.\ \ref{PSp}). For $d=0,1$, the representation $\hbox{\san \char3}^2$ defines a morphism of stacks $g_d:{\cal M}_{{\bf SO}_{2l}}^d\rightarrow {\cal M}_{{\bf SO}_{N}}$, which factors through ${\cal M}_{{\bf PSO}_{2l}}^d$. The pull back under $g_d$ of a square root of the determinant bundle is ${\cal D}^{l-1}$; since ${\cal D}^{2l}$ descends, one concludes that ${\cal D}$ descends when $l$ is even and ${\cal D}^2$ when $l$ is odd. \par\hskip 1truecm\relax To prove that ${\cal D}$ does not descend when $l$ is odd, one considers the quadratic bundle ${\cal H}^\delta$ on $X\times J$ defined by $$\eqalign{ {\cal H}^\delta&={\cal L}^{\oplus l}\oplus({\cal L}^{-1})^{\oplus l}\quad \hbox{if } \delta=1 \cr &= {\cal L}(p)^{\oplus l}\oplus{\cal L}^{-1}(-p))^{\oplus l}\quad \hbox{if } \delta=-1 \cr &= ({\cal L}\oplus{\cal L}^{-1}(p))^{\oplus l}\quad \hbox{if } \delta=\pm\varepsilon \ ,}$$ with the standard hyperbolic quadratic form, and opposite orientations for the cases $\delta=\varepsilon $ and $\delta=-\varepsilon $. \par\hskip 1truecm\relax As above, this gives rise to a commutative diagram $$\diagramme{J & \phfl{h}{} & {\cal M}^\delta_{{\bf SO}_{2l}} &\cr \vfl{2_J}{} & & \vfl{}{}&\cr J &\phfl{}{} &{\cal M}^\delta_{{\bf PSO}_{2l}} &\kern-12pt , }$$from which one deduces that ${\cal D}$ does not descend, since the class of $h^*{\cal D}$ in $NS(J)$ is $2l$ times the principal polarization.\cqfd \vskip2cm \centerline{\gros Part II: The Picard group of the moduli space} \vskip1cm \section{${\bf C}^*$\kern-1.5pt - extension associated to group actions}\label{ext} \par\hskip 1truecm\relax This part is devoted to the Picard group of the moduli space $M_G$. The case of a simply connected group being known, we will consider $M_G$ as a quotient of $M_{\widetilde{G}}$ by the finite group $H^1(X,\pi_1(G))$. Therefore we will first develop some general tools to study the Picard group of a quotient variety. \global\def\currenvir{subsection Let $H$ be a finite group acting on an integral projective variety $M$ over ${\bf C}$ (or, more generally, any variety with $H^0(M,{\cal O}_M^*)={\bf C}^*$), and $L$ a line bundle on $M$ such that $h^*L\cong L$ for all $h\in H$. We associate to this situation a central ${\bf C}^*$\kern-1.5pt - extension $$1\rightarrow {\bf C}^*\longrightarrow {\cal E}(H,L) \qfl{p} H\rightarrow 1 \ ;$$ the group ${\cal E}(H,L)$ consists of pairs $(h,\tilde h)$, where $h$ runs over $H$ and $\tilde h$ is an automorphism of $L$ covering $h$, and $p$ is the first projection. \smallskip \global\def\currenvir{subsection We will need a few elementary properties of these groups: \label{list} \par\hskip 1truecm\relax {\it a}) Let $f:M'\rightarrow M$ be a $H$\kern-1.5pt - equivariant rational map. Pulling back automorphisms induces an isomorphism $f^*:{\cal E}(H,L)\rightarrow {\cal E}(H,f^*L)$. \smallskip \par\hskip 1truecm\relax {\it b}) Recall that the isomorphism classes of central ${\bf C}^*$\kern-1.5pt - extensions of $H$ form a group, canonically isomorphic to $H^2(H,{\bf C}^*)$. Let $r$ be a positive integer. The extension ${\cal E}(H,L^r)$ is isomorphic to the sum of $r$ copies of ${\cal E}(H,L)$; more precisely, the homomorphism $\varphi_r :{\cal E}(H,L)\rightarrow {\cal E}(H,L^r)$ given by $\varphi_r (h,\tilde h)= (h,\tilde h^{\otimes r})$ is a surjective homomorphism, with kernel the group $\hbox{\gragrec \char22}_r$ of $r$\kern-1.5pt - roots of unity, and therefore induces an isomorphism of the push forward of ${\cal E}(H,L)$ by the endomorphism $t\mapsto t^r$ of ${\bf C}^*$ onto ${\cal E}(H,L^r)$. \smallskip \par\hskip 1truecm\relax {\it c}) Let $M'$ be another projective variety, $H'$ a finite group acting on $M'$, $L'$ a line bundle on $M'$ preserved by $H'$. The map $ {\cal E}(H ,L)\times {\cal E}(H' ,L') \rightarrow {\cal E}(H\times H' ,$ $L\boxtimes L')$ given by $((h,\tilde h),(h',\tilde h'))\mapsto (h\times h',\tilde h\,{\scriptstyle\boxtimes}\,\tilde h')$ is a surjective homomorphism, with kernel ${\bf C}^*$ embedded by $t\mapsto(t,t^{-1} )$. \smallskip \par\hskip 1truecm\relax {\it d}) Let $K$ be a normal subgroup of $H$. The group $H/K$ acts on $M/K$; let $L_0$ be a line bundle on $M/K$ preserved by this action, and $L$ the pull back of $L_0$ to $L$. Then the extension ${\cal E}(H ,L)$ is the pull back of ${\cal E}(H/K ,L_0)$ by the projection $H\rightarrow H/K $.\medskip \global\def\currenvir{subsection A $H$\kern-1.5pt - {\it linearization} of $L$ is a section of the extension ${\cal E}(H,L)$. Such a linearization allows us to define, for each point $m$ of $M$, an action of the stabilizer $H_m$ of $m$ in $H$ on the fibre $L_m$; this action is given by a character of $H_m$, denoted by $\chi _m$. \label{char} \par\hskip 1truecm\relax Let $\pi:M\rightarrow M/H$ be the quotient map; if $L_0$ is a line bundle on the quotient $M/H$, the line bundle $L=\pi^*L_0$ has a canonical $H$\kern-1.5pt - linearization. By construction it has the property that at each point $m$ of $M$, the character $\chi _m$ of $H_m$ is {\it trivial}. The converse is true (``Kempf's lemma"), and is actually quite easy to prove in our set-up. Since two $H$\kern-1.5pt - linearizations differ by a character of $H$, we can restate this lemma as follows: assume that $L$ admits a $H$\kern-1.5pt - linearization; then $L$ {\it descends to $M/H$ if and only if there exists a character $\chi $ of $H$ such that $\chi _m=\chi _{|H_m}$ for all} $m\in M$. \par\hskip 1truecm\relax It follows from the above description that the kernel of the homomorphism $\pi^*: \mathop{\rm Pic}\nolimits(M/H)\rightarrow \mathop{\rm Pic}\nolimits(M)$ consists of the $H$\kern-1.5pt - linearizations of ${\cal O}_M$ such that the associated characters $\chi _m$ are trivial, i.e.\ of the characters of $H$ which are trivial on the stabilizers $H_m$. In particular, if the subgroups $H_m$ generate $H$, $\pi^*$ is injective. \global\def\currenvir{subsection \label{produit} Let $M'$ be another projective variety with an action of $H$, and $L'$ a line bundle admitting a $H$\kern-1.5pt - linearization. The $H$\kern-1.5pt - linearizations of $L$ and $L'$ define a $H$\kern-1.5pt - linearization of $L\boxtimes L'$; at each point $(m,m')$, the corresponding character of $H_{(m,m')}=H_m\cap H_{m'}$ is the product of the characters $\chi _m$ of $H_m$ and $\chi' _{m'}$ of $H_{m'}$ associated to the linearizations of $L$ and $L'$. As a consequence, assume that $L$ and $L\boxtimes L'$ descend to $M/H$ and $(M\times M')/H$ respectively, and that $H=\cup_m H_m$; {\it then $L'$ descends to} $M'/H$. \medskip \global\def\currenvir{subsection\label{section-mod} From (\ref{list} {\it b}) we see that {\it the smallest positive integer $n$ such that $L^n$ admits a $H$\kern-1.5pt - linearization is the order of ${\cal E}(H,L)$ in $H^2(H,{\bf C}^*)$}. We want to know which powers of $L^n$ descend to $M/H$. \par\hskip 1truecm\relax Let $r$ be a multiple of $n$. The class $e$ of ${\cal E}(H,L)$ in $H^2(H,{\bf C}^*)$ comes from an element of $H^2(H,\hbox{\gragrec \char22}_r)$, which means that there exists a cocycle $c\in Z^2(H,\hbox{\gragrec \char22}_r)$ representing $e$, or in other words a map $\sigma :H\rightarrow {\cal E}(H,L)$ such that $p\kern 1pt{\scriptstyle\circ}\kern 1pt\sigma ={\rm Id}_H$ and $\sigma (hh')\equiv \sigma (h)\,\sigma (h')$ (mod.$\,\hbox{\gragrec \char22}_r)$ -- let us call such a map a {\it section} ({\it mod.}~$\hbox{\gragrec \char22}_r)$ {\it of} ${\cal E}(H,L)$. Composing $\sigma $ with the homomorphism $\varphi _r: {\cal E}(H,L)\rightarrow {\cal E}(H,L^r)$ (\ref{list} {\it b}) gives a section of the extension ${\cal E}(H,L^r)$, that is a $H$\kern-1.5pt - linearization of $L^r$. \par\hskip 1truecm\relax Let $m$ be a point of $M$. Using this $H$\kern-1.5pt - linearization we get a character $\chi _m$ of $H_m$ (\ref{char}), which can be computed as follows: for $h\in H_m$ the element $\sigma (h)$ acts on $L_m$, and we have $\chi _m(h)=(\sigma (h)_m)^r$. Assume moreover that $h^r=1$ for all $h\in H$; then the element $\sigma (h)^r$ of ${\cal E}(H,L)$ belongs to the center ${\bf C}^*$. Thus $L^r$ {\it endowed with the $H$\kern-1.5pt - linearization deduced from $\sigma $ descends to $M/H$ if and only if $\sigma (h)^r=1$ for all $h$ in} $\cup H_m$. Using \ref{char} we can conclude: \th Proposition \enonce Assume that the order of ${\cal E}(H,L)$ in $H^2(H,{\bf C}^*)$ and of every element of $H$ divides $r$. Let $\sigma :H\rightarrow {\cal E}(H,L)$ be a section {\rm (mod.}\ $\hbox{\gragrec \char22}_r)$. Then $L^r$ descends to $M/H$ if and only if there exists a character $\chi $ of $H$ such that $\sigma (h)^r=\chi (h)$ for all $h\in H$ fixing some point of $M$. \endth \label{desc-prelim} \smallskip \par\hskip 1truecm\relax In the applications we have in mind we will always have $\cup H_m=H$. In this case we get the following condition, which depends only on the extension ${\cal E}(H,L)$ and not on the variety $M$: \th Corollary \enonce Assume that every element of $H$ fixes some point in $M$. Then $L^r$ descends to $M/H$ if and only if the map $h\mapsto \sigma ^r(h)$ from $H$ to ${\bf C}^*$ is a homomorphism. \label{desc-cor} \endth \smallskip \global\def\currenvir{subsection From now on we will assume that the finite group $H$ is {\it abelian}. In that case there is a canonical isomorphism of $H^2(H,{\bf C}^*)$ onto the group ${\rm Alt}(H,{\bf C}^*)$ of bilinear alternate forms on $H$ with values in ${\bf C}^*$ (see e.g.\ [Br], V.6, exer.\ 5) : to a central ${\bf C}^*$\kern-1.5pt - extension $\widetilde{H}\qfl{p} H$ corresponds the form $e$ such that $\ e(p(x) ,p(y ))=$ $xyx^{-1}y ^{-1}\in\mathop{\rm Ker}\nolimits p={\bf C}^*$. Conver\-sely, given $e\in{\rm Alt}(H,{\bf C}^*)$, one defines an extension of $H$ in the following way: choose any bilinear form $\varphi :H\times H\rightarrow {\bf C}^*$ such that $e(\alpha ,\beta )=\varphi (\alpha ,\beta )\varphi (\beta ,\alpha )^{-1}$; take $\widetilde{H}=H\times{\bf C}^*$, with the multiplication law given by $$(\alpha ,t)\,(\beta ,u)=(\alpha +\beta ,tu\,\varphi (\alpha ,\beta ))\ ,$$ the homomorphism $p:\widetilde{H}\rightarrow H$ being given by the first projection. \medskip \global\def\currenvir{subsection Let $r$ be an integer such that $rH=0$. Then the group $H^2(H,{\bf C}^*)\cong$ ${\rm Alt(H,{\bf C}^*)}$ is also annihilated by $r$. Let $e\in {\rm Alt}(H,{\bf C}^*)$; we can choose the bilinear form $\varphi $ with values in $\hbox{\gragrec \char22}_r$. Consider the extension defined as above by $\varphi $. The map $\sigma :H\rightarrow \widetilde{H}$ defined by $\sigma (\alpha )=(\alpha ,1)$ is a section (mod.\ $\hbox{\gragrec \char22}_r)$. An easy computation gives $\sigma (\alpha )^r= \varphi (\alpha ,\alpha )^{{1\over 2}r(r-1)}\in\{1,-1\}$. One has $\sigma ^{2r}(\alpha )=1$, and $\sigma (\alpha )^r=1$ for all $\alpha $ if $r$ is odd. If $r$ is even, the function $\varepsilon :\alpha \mapsto \sigma (\alpha )^r$ is ``quadratic" in the sense that $\varepsilon (\alpha +\beta )=\varepsilon (\alpha )\varepsilon (\beta )\,e(\alpha ,\beta )^{r/2}$. In particular, we see that $\sigma ^r$ is a homomorphism if and only if the alternate form $e^{r/2}$ (with values in $\hbox{\gragrec \char22}_2$) is trivial. In summary: \smallskip \th Proposition \enonce Assume $H$ is commutative, annihilated by $r$; let $e$ be the alternate form associated to ${\cal E}(H,L)$. The line bundle $L^{2r}$ descends to $M/H$; moreover $L^r$ descends, except if $r$ is even and the form $e^{r/2}$ is not trivial. In this last case, if every element of $H$ has some fixed point on $M$, $L^r$ does not descend. \label{desc-prop} \endth \smallskip \rem{ Example} \label{Theta } Let $A$ be an abelian variety of dimension $g\ge 1$, and $\Theta $ a divisor on $A$ defining a principal polarization. Let $A_r$ be the kernel of the multiplication by $r$ in $A$. The group ${\cal E}(A_r,{\cal O}(r\Theta ))$ is the Heisenberg group which plays a fundamental role in Mumford's theory of theta functions; the corresponding alternate form $e_r:A_r\times A_r\rightarrow \hbox{\gragrec \char22}_r$ is the Weil pairing. The group $A_r$ acts on the linear system $|r\Theta |$, and the morphism $A\rightarrow |r\Theta |^*$ associated to this linear system is $A_r$\kern-1.5pt - equivariant; therefore by (\ref{list} {\it a}), the extension ${\cal E}(A_r,{\cal O}_{|r\Theta |^*}(1))$ is isomorphic to ${\cal E}(A_r,{\cal O}(r\Theta ))$. It follows easily that ${\cal E}(A_r,{\cal O}_{|r\Theta |}(1))$ corresponds to the nondegenerate form $e_r^{-1}$. Let $s$ be a positive integer dividing $r$; an easy computation shows that\par \global\def\currenvir{subsection{\it the restriction of $e_r$ to $A_s$ is equal to} $e_s^{r/s}$. \label{restric} \par\hskip 1truecm\relax We conclude from the proposition that:\par -- the line bundle ${\cal O}(2s)$ descends to $|r\Theta |/A_s$; \par -- the line bundle ${\cal O}(s)$ descends to $|r\Theta|/A_s$ if $s$ is odd or $r/s$ is even, but does {\it not} descend if $s$ is even and $r/s$ odd. \vskip1cm \section{The moduli space $M_G$} \global\def\currenvir{subsection\label{coarse} Recall [R1, R2] that a $G$\kern-1.5pt - bundle $P$ on $X$ is {\it semi-stable} (resp.\ {\it stable}) if for every parabolic subgroup $\Pi $, every dominant character $\chi$ of $\Pi $ and every $\Pi $\kern-1.5pt - bundle $P'$ whose associated $G$\kern-1.5pt - bundle is isomorphic to $P$, the line bundle $P'_\chi $ has nonpositive (resp.\ negative) degree. \par\hskip 1truecm\relax Let $\rho:G\rightarrow G^{\prime}$ be a homomorphism of semi-simple groups, and $P$ a $G$\kern-1.5pt - bundle. If $P$ is semi-stable the $G'$\kern-1.5pt - bundle $P_\rho=P\times^G G^{\prime}$ is semi-stable; the converse is true if $\rho $ has finite kernel. In particular $P$ is semi-stable if and only if its adjoint bundle ${\rm Ad}(P)$ is semi-stable. \par\hskip 1truecm\relax We denote by $M_G$ the coarse moduli space of semi-stable principal $G$\kern-1.5pt - bundles on $X$ ({\it loc.\ cit.}). It is a projective normal variety. Let ${\cal M}_G^{ss}$ be the open substack of ${\cal M}_G$ corresponding to semi-stable $G$\kern-1.5pt - bundles; there is a canonical surjective morphism $f:{\cal M}_G^{ss}\rightarrow M_G$. For $\delta\in \pi_1(G)$, $f$ maps the component $({\cal M}_G^{ss})^\delta$ onto the connected component $M_G^\delta$ of $M_G$ parameterizing $G$\kern-1.5pt - bundles of degree $\delta$. \par\hskip 1truecm\relax The definition of (semi-) stability extends to any reductive group $H$: a $H$\kern-1.5pt - bundle $P$ is semi-stable (resp.\ stable) if and only if the $(H/Z^{\rm o})$\kern-1.5pt - bundle $P/Z^{\rm o}$ has the same property, where $Z^{\rm o}$ is the neutral component of the center of $H$. The construction of the moduli space $M_H$ makes sense in this set-up; each component of $M_H$ is normal and projective. \par\hskip 1truecm\relax Let $Z$ be the center of $G$; we choose an isomorphism $Z\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \pprod_{}^{} \hbox{\gragrec \char22}_{r_j}$. Let $\delta\in\pi_1(G_{\rm ad})$. The construction of the ``twisted" moduli stack ${\cal M}_{G}^\delta$ (Section \ref{twist}) obviously makes sense in the framework of coarse moduli spaces. We get a coarse moduli space $M_{G}^\delta$, which parameterizes semi-stable $C_ZG$\kern-1.5pt - bundles with determinant ${\cal O}_X({\bf d}p)$, such that the associated $G_{\rm ad}$\kern-1.5pt - bundle has degree $\delta$, with $\rho (\delta)\,e^{2\pi i{\bf d}/{\bf r}}=1$ (\ref{deg}). The open substack ${\cal M}_{G}^{\delta, ss}$ of ${\cal M}_{G}^\delta$ parameterizing semi-stable bundles maps surjectively onto $M_{G}^\delta$. If $A$ is a central subgroup of $G$, there is a canonical morphism $\pi:M_{G}^\delta\rightarrow M_{G/A}^\delta$ which is a (ramified) Galois covering with Galois group $H^1(X,A)$. The next lemma will allow us to compare the Picard groups of these moduli spaces by applying the results of section \ref{ext}, in particular Prop.\ \ref{desc-prop}, to the action of $H^1(X,A)$ on $M_G^\delta$. \th Lemma \enonce Let $\delta\in \pi _1(G_{\rm ad})$. \par\hskip 1truecm\relax {\rm a)} The moduli space $M_{G}^\delta$ is unirational. \par\hskip 1truecm\relax {\rm b)} Any finite order automorphism of $M_{G}^\delta$ has fixed points. \endth\label{unirat} {\it Proof}: a) The proof in [K-N-R], Cor.\ 6.3, for the untwisted case extends in a straightforward way: by (\ref{unif}) we have a surjective morphism ${\cal Q}_{\widetilde{G}}\rightarrow {\cal M}_{G}^\delta$; so the open subset of ${\cal Q}_{\widetilde{G}}$ parameterizing semi-stable bundles maps surjectively onto $M_G^\delta$. Since ${\cal Q}_{\widetilde{G}}$ is a direct limit of an increasing sequence of generalized Schubert varieties, which are rational, the lemma follows. \par\hskip 1truecm\relax b) This is actually true for any finite order automorphism $g$ of a projective unirational variety $M$. One (rather sophisticated) proof goes as follows: there exists a desingularization $\widetilde{M}$ of $M$ to which $g$ lifts to an automorphism $\tilde g$ [H], necessarily of finite order. Since $H^i(\widetilde{M},{\cal O}_{\widetilde{M}})$ is zero for $i>0$, we deduce from the holomorphic Lefschetz formula that $\tilde g$ has fixed points, hence also $g$.\cqfd \medskip \par\hskip 1truecm\relax Recall that the moduli space $M_G$ is constructed as a {\it good quotient} of a smooth scheme $R$ by a reductive group $\Gamma $ [Se] -- this implies in particular that the closed points of $M_G$ correspond to the closed orbits of $\Gamma $ in $R$. In order to compare the Picard groups of $\mathop{\rm Pic}\nolimits(M_G)$ and $\mathop{\rm Pic}\nolimits({\cal M}_G)$, we will need a more precise result: \smallskip \th Lemma \enonce There exists a presentation of ${\cal M}_{G}^{\delta,ss}$ as a quotient of a smooth scheme $R$ by a reductive group $\Gamma $, such that the moduli space $M_{G}^\delta$ is a good quotient of $R$ by $\Gamma$. \endth\label{comppres} {\it Proof}: We will explain the proof in some detail for the untwisted case, then indicate how to adapt the argument to the general situation. \par\hskip 1truecm\relax We fix a faithful representation $\rho:G\rightarrow {\bf SL}_r$ and an integer $N$ such that for every semi-stable $G$\kern-1.5pt - bundle $P$, the vector bundle $P_{\rho}(Np)$ is generated by its global sections and satisfies $\ H^{1}(X,P _{\rho}(Np))=0$. Let $M=r(N+1-g)$. For any complex scheme $S$, we denote by $\underline{R}_G(S)$ the set of isomorphism classes of pairs $(P ,\alpha)$, where $P $ is a $G$\kern-1.5pt - bundle on $X\times S$ whose restriction to $X\times\{s\}$, for each closed point $s\in S$, is semi-stable, and $\alpha:{\cal{O}}_{S}^M\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} pr_{1*}P _{\rho}(Np)$ an isomorphism. We define in this way a functor $\underline{R}_G$ from the category of ${\bf C}$\kern-1.5pt - schemes to the category of sets; we claim that it is representable by a scheme $R_G$. If $G={\bf SL}_r$, this follows from Grothendieck theory of the Hilbert scheme [G1]. In the general case, we observe that reductions to $G$ of the structure group of a ${\bf SL}_r$\kern-1.5pt - bundle $P$ correspond canonically to global sections of the bundle $P/G$; it follows that $\underline{R}_G$ is isomorphic to the functor of global sections of ${\cal P}/G$, where ${\cal P}$ is the universal ${\bf SL}_r$\kern-1.5pt - bundle on $X\times R_{{\bf SL}_r}$. Again by [G1], this functor is representable by a scheme $R_G$, which is affine over $R_{{\bf SL}_r}$. \par\hskip 1truecm\relax Put $\Gamma ={\bf GL}_M$. The group $\Gamma $ acts on ${\underline R}_G$, and therefore on $R_G$, by the rule $g\cdot(P ,\alpha)=(P ,\alpha g^{-1} )$. This action lifts to the universal $G$\kern-1.5pt - bundle ${\cal{P}}$ over $X\times R_G$ as follows: by construction the universal pair $({\cal{P}},\alpha)$ is isomorphic to $((\mathop{\rm Id}\nolimits\times g)^{*}{\cal{P}},\alpha\kern 1pt{\scriptstyle\circ}\kern 1pt g)$, hence there is an isomorphism of $G$\kern-1.5pt - bundles $\sigma_{g}:(\mathop{\rm Id}\nolimits\times g)^{*}{\cal{P}}\rightarrow {\cal P}$ such that $\alpha \kern 1pt{\scriptstyle\circ}\kern 1pt g^{-1}=pr_{1*}(\sigma_{g,\rho})\kern 1pt{\scriptstyle\circ}\kern 1pt \alpha$. Since $\rho$ is faithful this isomorphism is uniquely determined by $pr_{1*}(\sigma_{g,\rho})$, hence depends only on $g$ and defines the required lifting. \par\hskip 1truecm\relax The $\Gamma $\kern-1.5pt - equivariant morphism $\varphi_{{\cal{P }}}:R_G\rightarrow {\cal{M}}_G$ induces a morphism of stacks $\overline{\varphi}_{{\cal{P}}}:[R_G/\Gamma ]\rightarrow {\cal{M}}_{G}^{ss}$ which is easily seen to be an isomorphism. We also have a $\Gamma $-equivariant morphism $\psi_{\cal P} :R_G\rightarrow M_G$; if there exists a good quotient $R_G//\Gamma $, the universal property of $M_G$ implies that $\psi _{\cal P}$ must induce an isomorphism of this quotient onto $M_G$. The existence of such a good quotient is classical in the case $G={\bf SL}_r$ (possibly after increasing $N$); for general $G$, since the canonical map $R_G\rightarrow R_{{\bf SL}_r}$ is $\Gamma $\kern-1.5pt - equivariant and affine, the existence of a good quotient of $R_{{\bf SL}_r}$ by $\Gamma $ implies the same property for $R_G$ thanks to a lemma of Ramanathan ([R1], lemma 4.1). \par\hskip 1truecm\relax Let us finally consider the twisted case. We choose an embedding of the center $Z$ of $G$ in a torus $T={\bf G}_m^s$, and an embedding $\rho :G\rightarrow \pprod_{i=1}^s {\bf GL}_{r_i}$ such that $\rho (Z)$ is central; we put $S=(\pprod_{}^{} {\bf GL}_{r_i})\times (T/Z)$. The map $(g,t)\mapsto (t^{-1}\rho (g), t\ \hbox{mod.}\ Z)$ of $G\times T$ into $S$ defines an embedding of $C_ZG$ into $S$, which maps the center of $C_ZG $ into the center of $S$, so that a $C_ZG$\kern-1.5pt - bundle $P$ on $X$ is semi-stable if and only if the associated $S$\kern-1.5pt - bundle is semi-stable. We then argue as before, replacing ${\bf SL}_r$ by $S$.\cqfd \smallskip \th Proposition \enonce Assume that the group $G$ is almost simple. The group $\mathop{\rm Pic}\nolimits (M_G^\delta)$ is infinite cyclic, and the homomorphism $\pi^*:\mathop{\rm Pic}\nolimits (M_G^\delta)\rightarrow \mathop{\rm Pic}\nolimits (M_{\widetilde{G}}^\delta)$ is injective. \endth\label{cyc} \par\hskip 1truecm\relax The second assertion follows from Lemma \ref{unirat} b) and (\ref{char}); it is therefore enough to prove the first one when $G$ is simply connected. The proof then is the same as in the untwisted case ([L-S] or [K-N]): since the stack ${\cal M}_G^\delta$ is smooth, the restriction map $\mathop{\rm Pic}\nolimits({\cal M}_G^\delta)\rightarrow \mathop{\rm Pic}\nolimits({\cal M}_G^{\delta,ss})$ is surjective, hence by Prop.\ \ref{Pic-Bun} the group $\mathop{\rm Pic}\nolimits({\cal M}_G^{\delta,ss})$ is cyclic; it remains to prove that the pull back homomorphism $\mathop{\rm Pic}\nolimits(M_G^\delta)\rightarrow \mathop{\rm Pic}\nolimits({\cal M}_G^{\delta,ss})$ is injective. \par\hskip 1truecm\relax We choose a presentation of ${\cal M}_G^{\delta,ss}$ as a quotient of a smooth scheme $R$ by a reductive group $\Gamma $, such that the moduli space ${ M}_G^\delta$ is a good quotient of $R$ by $\Gamma $ (lemma \ref{comppres}); then line bundles on ${\cal M}_G^{\delta,ss}$ (resp.\ on $M_G^\delta$) correspond to line bundles on $R$ with a $\Gamma $\kern-1.5pt - linearization (resp.\ a $\Gamma $\kern-1.5pt - linearization $\sigma$ such that $\sigma (\gamma )_r=1$ for each $(\gamma ,r)\in \Gamma \times R$ such that $\gamma r=r$), hence our assertion.\cqfd \medskip \par\hskip 1truecm\relax In what follows we will identify the group $\mathop{\rm Pic}\nolimits (M_G^\delta)$ with its image in $\mathop{\rm Pic}\nolimits (M_{\widetilde{G}}^\delta)$; our aim will be to find its generator. \vskip1cm \section{The Picard groups of $M_{{\bf Spin}_{r}}$ and $M_{G_2}$} \label{Pic(M_Spin)} \global\def\currenvir{subsection In this section we complete the results of [L-S] in the simply connected case. The cases $G={\bf SL}_r$ or ${\bf Sp}_{2l}$ are dealt with in {\it loc. cit.}. We now consider the case $G={\bf Spin}_r$; we denote by ${\cal D}$ the determinant bundle on $M_{{\bf Spin}_{r}}$ associated to the standard representation $\sigma $ of ${\bf Spin}_{r}$ in ${\bf C}^r$. \th Proposition \enonce Let $r$ be an integer $\geq 7$. The group $\mathop{\rm Pic}\nolimits(M_{{\bf Spin}_r})$ is generated by ${\cal D}$. \endth\label{Pic_Spin} {\it Proof}: Choose a presentation of ${\cal M}_{{\bf Spin}_{r}}^{ss}$ as a quotient of a smooth scheme $R$ by a reductive group $\Gamma $, such that $M_{{\bf Spin}_r}$ is a good quotient of $R$ by $\Gamma $ (lemma \ref{comppres}). Let ${\cal S}$ be the universal ${\bf Spin}_{r}$\kern-1.5pt - bundle on $X\times R$. We fix a theta-characteristic $\kappa $ on $X$; this allows us to define the pfaffian line bundle ${\cal P}_\kappa $ on $R$, which is a square root of $\mathop{\rm det}\nolimits Rpr_{2*}({\cal S}_\sigma \otimes\kappa )$ [L-S]. The action of $\Gamma $ on ${\cal S}$ defines a $\Gamma $\kern-1.5pt - linearization of ${\cal P}_\kappa $. \par\hskip 1truecm\relax By [L-S] we know that the group of $\Gamma $\kern-1.5pt - linearized line bundles on $R$ (isomorphic to $\mathop{\rm Pic}\nolimits({\cal M}_{{\bf Spin}_{r}}^{ss})$) is generated by ${\cal P}_\kappa $, so all we have to prove is that ${\cal P}_\kappa $ itself does not descend to $R//\Gamma $, i.e.\ to exhibit a closed point $s\in R$ whose stabilizer in $\Gamma $ acts nontrivially on the fibre of ${\cal P}_\kappa $ at $s$. If $s$ corresponds to a semi-stable ${\bf Spin}_r$\kern-1.5pt - bundle $P$, its stabilizer is the group $\mathop{\rm Aut}\nolimits(P)$; since the formation of pfaffians commutes with base change, its action on $({\cal P}_\kappa)_s $ is the natural action of $\mathop{\rm Aut}\nolimits(P)$ on $\bigl(\hbox{\san \char3}^{\rm max}H^0(X,P_\sigma \otimes\kappa )\bigr)^{-1}$ [L-S]. \par\hskip 1truecm\relax To construct $P$ we follow [L-S], prop.\ 9.5: we choose a stable ${\bf SO}_4$\kern-1.5pt - bundle $Q$ and a stable ${\bf SO}_{r-4}$\kern-1.5pt - bundle $Q'$ with $w_2(Q)=w_2(Q')=1$. Let $H$ be the subgroup ${\bf SO}_{4}\times {\bf SO}_{r-4}$ of ${\bf SO}_{r}$, and $\widetilde{H}$ its inverse image in ${\bf Spin}_r$. The $H$\kern-1.5pt - bundle $Q\times Q'$ has $w_2=0$ by construction, hence admits a $\widetilde{H}$\kern-1.5pt - structure; we choose one, and take for $P$ the associated ${\bf Spin}_r$\kern-1.5pt - bundle. Let $\gamma $ be a central element of $\widetilde{H}$ lifting the element $(-1,1)$ of $H$. Then $\gamma $ defines an automorphism of $P$, which acts on the associated vector bundle $P_\sigma =Q_\sigma \oplus Q'_\sigma $ as $(-\mathop{\rm Id}\nolimits,\mathop{\rm Id}\nolimits)$ (we use the same letter $\sigma$ to denote the standard representation of all orthogonal group in sight). Therefore $\gamma $ acts on $({\cal P}_\kappa)_s$ by multiplication by $(-1)^{h^0(Q_\sigma \otimes\kappa )}$. But $h^0(Q_\sigma \otimes\kappa )$ is congruent to $w_2(Q)$ (mod.$\,2$) [L-S, 7.10.1], hence our assertion.\cqfd \medskip \rem{Remark} For $r\le6$ the group ${\bf Spin}_r$ is of type A or C, so we already have a complete description of $\mathop{\rm Pic}\nolimits(M_{{\bf Spin}_r})$. It is worth noticing that the above result does not hold for $r\le 6$: using the exceptional isomorphisms one checks easily that $\mathop{\rm Pic}\nolimits(M_{{\bf Spin}_r})$ is generated by a square root of ${\cal D}$ for $r=5$ or $6$ and a fourth root for $r=3$ -- while it is isomorphic to ${\bf Z}^2$ for $r=4$. \medskip \par\hskip 1truecm\relax We now consider the case when $G$ is of type $G_2$. The group $G$ is the automorphism group of the octonion algebra ${\bf O}$ over ${\bf C}$ ([Bo], Alg\`ebre III, App.); in particular it has a natural orthogonal representation $\sigma $ in the $7$\kern-1.5pt - dimensional space ${\bf O}/{\bf C}$. We denote by ${\cal D}$ the determinant bundle on $M_G$ associated to this representation. \th Proposition \enonce The group $\mathop{\rm Pic}\nolimits(M_G)$ is generated by ${\cal D}$. \endth\label{G_2} {\it Proof}: As before we choose a presentation of ${\cal M}_{G}^{ss}$ as a quotient of a smooth scheme $R$ by a reductive group $\Gamma $, such that $M_{G}=R//\Gamma $; choosing a theta-characteristic $\kappa $ on $X$ allows to define a pfaffian line bundle ${\cal P}_\kappa $ on $R$, with a natural $\Gamma $\kern-1.5pt - linearization. By [L-S], thm.\ 1.1, ${\cal P}_\kappa $ generates the group of $\Gamma $\kern-1.5pt - linearized line bundles on $R$; we must again prove that it does not descend to $R//\Gamma $, i.e.\ exhibit a $G$\kern-1.5pt - bundle $P$ such that $\mathop{\rm Aut}\nolimits(P)$ acts nontrivially on $\hbox{\san \char3}^{\rm max}H^0(P_\sigma \otimes\kappa )$. \par\hskip 1truecm\relax Let $V$ be a $3$\kern-1.5pt - dimensional vector space over ${\bf F}_2$. The algebra ${\bf O}$ has a basis $(e_\alpha )_{\alpha \in V}$, with multiplication rule $$e_\alpha \,e_\beta =\varepsilon (\alpha ,\beta )\,e_{\alpha +\beta }\ ,$$for a certain function $\varepsilon :V\times V\rightarrow \{\pm 1\}$. Suppose given a homomorphism $\alpha \mapsto L_\alpha $ of $V$ into $J$. We view $J$ as the moduli space for degree $0$ line bundles with a trivialization at $p$; for each pair $(\alpha ,\beta )$ in $V$ we have a unique isomorphism $u_{\alpha \beta }:L_\alpha \otimes L_\beta \rightarrow L_{\alpha +\beta }$ compatible with these trivializations. We endow the ${\cal O}_X$\kern-1.5pt - module ${\cal A}=\sdir_{\alpha \in V}^{}L_\alpha $ with the algebra structure defined by the map ${\cal A}\otimes{\cal A}\rightarrow {\cal A}$ which coincides with $\varepsilon (\alpha ,\beta )\,u_{\alpha \beta }$ on $L_\alpha \otimes L_\beta $. It is a sheaf of ${\cal O}_X$\kern-1.5pt - algebras, locally isomorphic to ${\cal O}_X\otimes_{\bf C}{\bf O}$. Let $P$ be the associated $G$\kern-1.5pt - bundle (the sections of $P$ over an open subset $U$ of $X$ are algebra isomorphisms of ${\cal O}_U\otimes_{\bf C}{\bf O}$ onto ${\cal A}_{|U}$). Since the pull back of $P$ to any finite covering of $X$ on which the $L_\alpha $'s are trivial is trivial, $P$ is semi-stable. The vector bundle $P_\sigma $ is simply $\sdir_{\alpha \not=0}^{}L_\alpha $. Let $\chi :V\rightarrow \{\pm 1\}$ be a nontrivial character; the diagonal endomorphism $(\chi (\alpha ))_{\alpha \in V}$ of ${\cal A}$ is an algebra automorphism, and therefore defines an automorphism $\iota $ of $P$, which acts on $P_\sigma $ with eigenvalues $(\chi (\alpha ))_{\alpha \not=0}$. Hence $\iota $ acts on $\hbox{\san \char3}^{\rm max}H^0(P_\sigma \otimes\kappa )$ by multiplication by $(-1)^h$, with $h=\sum_{\chi (\alpha )=-1} h^0(L_\alpha \otimes\kappa )$. Since the function $\alpha \mapsto h^0(L_\alpha \otimes\kappa )$ (mod.$\,2$) is quadratic, an easy computation gives that $h$ is even if and only if the image of $\mathop{\rm Ker}\nolimits \chi $ in $J_2$ is totally isotropic with respect to the Weil pairing. Clearly we can choose our map $V\rightarrow J_2$ and the character $\chi $ so that this does not hold; this provides the required example.\cqfd \vskip1cm \section {The Picard group of $M^0_G$} \par\hskip 1truecm\relax In the study of $\mathop{\rm Pic}\nolimits(M_G^\delta)$, contrary to what we found for the moduli stacks, the degree $\delta$ plays an important role. We treat first the degree $0$ case, which is easier. Let us start with the case $A_l$. We recall that the determinant bundle ${\cal D}$ exists on the moduli space $M_{{\bf SL}_r}$, and generates its Picard group. \smallskip \th Proposition \enonce Let $G={\bf SL}_r /\hbox{\gragrec \char22}_s$, with $s$ dividing $r$. {\rm a)} If $s$ is odd or $r/s$ is even, $\mathop{\rm Pic}\nolimits (M_G^0)$ is generated by ${\cal D}^s$. {\rm b)} If $s$ is even and $r/s$ is odd, $\mathop{\rm Pic}\nolimits (M_G^0)$ is generated by ${\cal D}^{2s}$. \par\hskip 1truecm\relax In particular, $\mathop{\rm Pic}\nolimits( M_{{\bf PGL}_r}^0)$ is generated by ${\cal D}^{r}$ if $r$ is odd and by ${\cal D}^{2r}$ if $r$ is even. \endth\label{A0} {\it Proof}: We identify $M_{{\bf SL}_r }$ with the moduli space of semi-stable vector bundles of rank $r$ and trivial determinant on $X$. Let $J^{g-1}$ be the component of the Picard variety of $X$ parameterizing line bundles of degree $g-1$, and $\Theta \i J^{g-1}$ the canonical theta divisor. It is shown in [B-N-R] that for $E$ general in $M_{{\bf SL}_r }$, the condition $H^0(X,E\otimes L)\not=0$ defines a divisor $D(E)$ in $J^{g-1}$ which belongs to the linear system $|r\Theta |$, and that the rational map $D:M_{{\bf SL}_r }\dasharrow |r\Theta |$ thus defined satisfies $D^*{\cal O}(1)={\cal D}$. Using (\ref{list} {\it a}) we deduce that the alternate form associated to ${\cal E}(J_r,{\cal D})$ is the inverse of the Weil pairing $e_r$; its restriction to $J_s$ is $e_s^{-r/s}$ (\ref{restric}). From Prop. \ref{desc-prop}, we conclude that the line bundles ${\cal D}^s$ in case a) and ${\cal D}^{2s}$ in case b) descend to $M_G^0$. \par\hskip 1truecm\relax It remains to prove that these line bundles are indeed in each case generators of $\mathop{\rm Pic}\nolimits (M_G^0)$. Consider first the case $s=r$. Since the extension ${\cal E}(J_r,{\cal D})$ is of order $r$ in $H^2(J_r,{\bf C}^*)$, the smallest power of ${\cal D}$ which admits a $J_r$\kern-1.5pt - linearization is ${\cal D}^r$, so the conclusion follows from Prop.\ \ref{desc-prop}. In the general case, put $M:=M_{{\bf SL}_r }$, and assume that some power ${\cal D}^k$ of ${\cal D}$ descends to $M/J_s$. Observe that $M/J_r$ can be viewed as the quotient of $M/J_s$ by $J_{r/s}$. \par\hskip 1truecm\relax Assume that $r/s$ is even. We know by Prop.\ \ref{desc-prop} that ${\cal D}^{2kr/s}$ descends to $M/J_r$; since $r$ is even, this implies by what we have seen that $2r$ divides $2kr/s$, hence that $k$ is a multiple of $s$. If $r/s$ is odd, then ${\cal D}^{kr/s}$ descends by Prop.\ \ref{desc-prop}, and therefore $k$ is a multiple of $s$ or $2s$ according to the parity of $r$.\cqfd \medskip \global\def\currenvir{subsection We now consider the case of the orthogonal and symplectic group. If $G={\bf SO}_r$ or ${\bf Sp}_r$ ($r$ even), we will denote by ${\cal D}$ the determinant bundle on $M_G$, i.e.\ the pull back of the determinant bundle on $M_{{\bf SL}_r}$ by the morphism associated to the standard representation. We know that the group $\mathop{\rm Pic}\nolimits(M_{{\bf Sp}_{r}})$ is generated by ${\cal D}$ ([L-S], 1.6), and that $\mathop{\rm Pic}\nolimits(M_{{\bf Spin}_{r}})$ is generated by the pull back of ${\cal D}$ (Prop.\ \ref{Pic_Spin}); it follows that the Picard group of each component of $M_{{\bf SO}_r}$ is generated by ${\cal D}$. It remains to consider the groups ${\bf PSp}_{2l} $ and ${\bf PSO}_{2l} $. \th Proposition \enonce Let $G={\bf PSp}_{2l} $ or ${\bf PSO}_{2l}\quad (l\geq 2)$. {\rm a)} If $l$ is even, $\mathop{\rm Pic}\nolimits (M_G^0)$ is generated by ${\cal D}^2$. {\rm b)} If $l$ is odd, $\mathop{\rm Pic}\nolimits (M_G^0)$ is generated by ${\cal D}^4$. \endth \label{BC0} {\it Proof}: The extension ${\cal E}(J_2,{\cal D})$ is the pull back to $J_2$ of the Heisenberg extension of $J_{2l}$, and the corresponding alternate form is $e_2^l$ (\ref{restric}). We deduce from Proposition \ref{desc-prop} that ${\cal D}^{2}$ descends to $M_G^0$ if $l$ is even, and that ${\cal D}^4$ descends but ${\cal D}^2$ does not if $l$ is odd. \par\hskip 1truecm\relax It remains to prove that ${\cal D}$ does not descend when $l$ is even. Let us consider for instance the case of the symplectic group; for every integer $n$, we put $M_n=M_{{\rm\bf Sp}_{2n}}$ and denote by ${\cal D}_n$ the determinant line bundle on $M_n$. Write $l=p+q$, where $p$ and $q$ are odd (e.g.\ $p=1$, $q=l-1)$, and consider the morphism $u:M_p\times M_q\rightarrow M_l$ given by $u((E,\varphi ),(E',\varphi '))=(E\oplus E',\varphi \oplus \varphi ')$. It is $J_2$\kern-1.5pt - equivariant and satisfies $u^*{\cal D}_l={\cal D}_p\boxtimes{\cal D}_q$. The group $J_2\times J_2$ acts on $M_p\times M_q$; from (\ref{list} {\it c}) one deduces that the alternate form $e$ corresponding to the extension ${\cal E}(J_2\times J_2,{\cal D}_p\boxtimes{\cal D}_q)$ is given by $e((\alpha ,\alpha '),(\beta ,\beta '))=e_2(\alpha ,\beta )\,e_2(\alpha ',\beta ')$. If ${\cal D}_l$ descends to $M_l/J_2$, then ${\cal D}_p\boxtimes{\cal D}_q$ descends to $(M_p\times M_q)/J_2$, and we can apply (\ref{list} {\it d}) to the variety $M_p\times M_q$ and the diagonal embedding $J_2\i J_2\times J_2$. We conclude that the form $e$ is the pull back of an alternate form on $J_2$ by the sum map $J_2\times J_2\rightarrow J_2$. This is clearly impossible, which proves that ${\cal D}_l$ does not descend to $M_l/J_2$. \par\hskip 1truecm\relax The same argument applies to the orthogonal groups, except that one has to be careful about the definition of $M_1$: we take it to be the Jacobian of $X$, by associating to a line bundle $\alpha $ on $X$ the vector bundle $\alpha \oplus\alpha ^{-1}$ with the standard isotropic form. Then ${\cal D}_1$ is the line bundle ${\cal O}(2\Theta )$. The alternate form associated to ${\cal E}(J_2,{\cal D}_1)$ is $e_2$, and the rest of the argument applies without any change.\cqfd \medskip \rem{Remark} There remains one case to deal with. When $l$ is even, the center $Z$ of ${\rm\bf Spin}_{2l}$ is isomorphic to $\hbox{\gragrec \char22}_2\times\hbox{\gragrec \char22}_2$, so it contains two subgroups of order $2$ (besides the kernel of the homomorphism ${\bf Spin}_{2l}\rightarrow {\bf SO}_{2l}$). These subgroups are exchanged by the outer automorphisms of ${\bf Spin}_{2l}$, so the corresponding quotient groups are canonically isomorphic; let us denote them by $G$. Since $M_G^0$ dominates $M^0_{{\bf PSO}_{2l}}$, it follows from Prop.\ \ref{BC0} that ${\cal D}^2$ descends to $M_G^0$. If $l$ is not divisible by $4$, one can show that ${\cal D}$ does not descend to $M_G^0$, so $\mathop{\rm Pic}\nolimits(M_G^0)$ {\it is generated by} ${\cal D}^2$. If $l=4$, one sees using the triality automorphism that ${\cal D}$ descends; we do not know what happens for $l=4m$, $m\ge 2$. \vskip1cm \section{The Picard group of $M_{{\bf PGL}_r}^d$} \par\hskip 1truecm\relax In this section we consider the component $M_{{\bf PGL}_r}^d$ of the moduli space $M_{{\bf PGL}_r}$, for $0<d<r$. It is the quotient by $J_r$ of the moduli space $M_{{\bf SL}_r}^d$ of semi-stable vector bundles of rank $r$ and determinant ${\cal O}_X(dp)$. We denote by $\delta$ the g.c.d. of $r$ and $d$. If $A$ is a vector bundle on $X$ of rank $r/\delta$ and degree $(r(g-1)-d)/\delta$ which is general enough, the condition $H^0(X,E\otimes A)\not=0$ defines a Cartier divisor on $M_{{\bf SL}_r}^d$; the associated line bundle ${\cal L}$ (sometimes called the {\it theta line bundle}) is independent of the choice of $A$, and generates $\mathop{\rm Pic}\nolimits(M_{{\bf SL}_r}^d)$ [D-N]. \medskip \th Proposition \enonce The group $\mathop{\rm Pic}\nolimits(M_{{\bf PGL}_r}^d)$ is generated by ${\cal L}^\delta$ if $r$ is odd and by ${\cal L}^{2\delta}$ if $r$ is even. \endth \label{A_d} \par\hskip 1truecm\relax Choose a stable vector bundle $A$ of rank $r/\delta$ and determinant ${\cal O}_X(-{d\over \delta}p)$, and consider the morphism $a:E\mapsto E\otimes A$ of $M_{{\bf SL}_r}^d$ into $M_{{\bf SL}_{r^2/\delta}}^0$. By definition ${\cal L}$ is the pull back of the determinant bundle ${\cal D}$ on the target. The map $a$ is $J_r$ equivariant, hence induces an isomorphism ${\cal E}(J_r,{\cal D})\cong {\cal E}(J_r,{\cal L})$. We have seen in the proof of Prop.\ \ref{A0} that the alternate form associated to ${\cal E}(J_r,{\cal D})$ is $e_r^{-r/\delta}$; hence the smallest power of ${\cal L}$ which descends to $M_{{\bf PGL}_r}^d$ is ${\cal L}^\delta$. Therefore it is enough to prove that ${\cal L}^\delta$ descends to $M_{{\bf PGL}_r}^d$ when $r$ is odd and that ${\cal L}^{2\delta}$ but not ${\cal L}^{\delta}$ descends when $r$ is even. \par\hskip 1truecm\relax We will prove this by reducing to the degree $0$ case with the help of the Hecke correspondence. Let us denote simply by $M$ the moduli space $M_{{\bf SL}_r}^1$ of stable vector bundles of rank $r$ and determinant ${\cal O}_X(p)$ on $X$. There exists a Poincar\'e bundle ${\cal E}$ on $X\times M$; we denote by ${\cal E}_p$ its restriction to $\{p\}\times M$, viewed as a vector bundle on $M$. We fix an integer $h$ with $0<h<r$ and let ${\cal P}={\bf G}_M(h,{\cal E}_p)$ the Grassmann bundle parameterizing rank $h$ locally free quotients of ${\cal E}_p$. A point of ${\cal P}$ can be viewed as a pair $(E,F)$ of vector bundles with $E\in M$, $E(-p)\i F\i E$ and $\mathop{\rm dim}\nolimits (E_p/F_p)=h$. \smallskip \th Lemma \enonce If $E$ is general enough in $M$, for any pair $(E,F)$ in ${\cal P}$ the vector bundle $F$ is semi-stable, and stable if $g\ge3$. \label{slopes} \endth {\it Proof}: We will actually prove a more precise result. If $G$ is a vector bundle on $X$, define the {\it stability degree} $s(G)$ of $G$ as the minimum of the rational numbers $\mu(G'')-\mu(G')$ over all exact sequences $0\rightarrow G'\rightarrow G\rightarrow G''\rightarrow 0$. One has $s(G)\ge0$ if and only if $G$ is semi-stable, $s(G)>0$ if and only if $G$ is stable, and $s(G)=g-1$ when $G$ is a general stable vector bundle [L, Hi]. \par\hskip 1truecm\relax Let $E,F$ be two vector bundles on $X$, with $E(-p)\i F\i E$. The lemma will follow from the inequality $$s(F)\ge s(E)-1$$(note that since $E$ and $F$ play a symmetric role, this implies $|s(E)-s(F)|\le1$). Let $Q_p$ be the sheaf $E/F$ (with support $\{p\}$), and $h$ the dimension of its fibre at $p$. Let $F'$ be a subbundle of $F$, of rank $r'$. From the exact sequence $0\rightarrow F/F'\rightarrow E/F'\rightarrow Q_p\rightarrow 0$ we get $$\mu(F/F')-\mu(F')=\mu(E/F')-{h\over r-r'}-\mu(F')\ge s(E)-{h\over r-r'}\ .$$ Let $K_p:=\mathop{\rm Ker}\nolimits(E_p\rightarrow Q_p)$. The exact sequence $0\rightarrow E(-p)\rightarrow F\rightarrow K_p\rightarrow 0$ induces an exact sequence $0\rightarrow E'\rightarrow F'\rightarrow K_p$, with $E':=F'\cap E(-p)$. Therefore $$\mu(F/F')-\mu(F')\ge\mu (E(-p)/E')-\mu(E')-{r-h\over r'}\ge s(E)-{r-h\over r'}\ .$$ Since one of the two numbers $\displaystyle {h\over r-r'}$ and $\displaystyle {r-h\over r'}$ is $\le1$, we get the required inequality.\cqfd \medskip \par\hskip 1truecm\relax Let us denote by $M'$ the moduli space $M_{{\bf SL}_r}^{1-h}$. Using the lemma we get a diagram \diag \smallskip (``Hecke diagram"), where $q$ (resp.\ $q'$) associates to a pair $(E,F)$ the vector bundle $E$ (resp.\ $F$, provided $F$ is semi-stable). \par\hskip 1truecm\relax Let ${\cal L}$ and ${\cal L}'$ be the theta line bundles on $M$ and $M'$. Let $\delta$ be the g.c.d. of $r$ and $1-h$. \smallskip \th Lemma \enonce One has $K_{\cal P}=q^*{\cal L}^{-1}\otimes q'^*{\cal L}'^{-\delta}$. \label{KP} \endth {\it Proof}: Let $E$ be a general vector bundle in $M$; let us compute the restriction of $q'^*{\cal L}'$ to the fibre $q^{-1}(E)$. On $X\times{\cal P}$ we have a canonical exact sequence $$0\rightarrow {\cal F}\longrightarrow (1_X\times q)^*{\cal E}\longrightarrow (i_p)_*{\cal Q}_p\rightarrow 0\ ,$$ where ${\cal Q}_p$ is the universal quotient bundle of $q^*{\cal E}_p$ on ${\cal P}$ and $i_p$ the embedding of ${\cal P}= \{p\}\times{\cal P}$ in $X\times{\cal P}$. For each point $P=(E,F)$ of ${\cal P}$ this exact sequence gives by restriction to $X\times\{P\}$ the exact sequence $0\rightarrow F\rightarrow E\rightarrow Q_p\rightarrow 0$ defining $P$; in particular, one has ${\cal F}_{X\times\{P\}}=F$, and the map $q':{\cal P}\dasharrow M'$ is the classifying map associated to ${\cal F}$. It follows that $q'^*{\cal L}'$ is the determinant bundle associated to ${\cal F}\otimes A$, where $A$ is a vector bundle of rank $r/\delta$ and appropriate degree. \par\hskip 1truecm\relax Now let $E\in M$; put ${\bf G}=q^{-1}(E)={\bf G}(h,E_p)$, and denote by $\pi:X\times{\bf G}\rightarrow X$ and $\rho :X\times{\bf G}\rightarrow {\bf G}$ the two projections. The restriction of the above exact sequence to $X\times{\bf G}$ gives, after tensor product with $\pi^*A$, an exact sequence$$0\rightarrow {\cal F}\otimes\pi^*A\longrightarrow \pi^*({\cal E}\otimes A)\longrightarrow (i_p)_*{\cal Q}_p^{r/\delta}\rightarrow 0\ ;$$ applying $R\rho _*$ and taking determinants, we obtain $$\mathop{\rm det}\nolimits R\rho _*({\cal F}\otimes \pi^*A)\cong (\mathop{\rm det}\nolimits {\cal Q}_p)^{r/\delta}={\cal O}_{\bf G}(r/\delta)\ .$$ \par\hskip 1truecm\relax The restriction of $K_{\cal P}$ to ${\bf G}$ is $K_{\bf G}={\cal O}_{\bf G}(-r)$; since $\mathop{\rm Pic}\nolimits({\cal P})$ is generated by ${\cal O}_{\cal P}(1)$ and $q^*\mathop{\rm Pic}\nolimits(M)$, one can write $K_{\cal P}=q'^*{\cal L}'^{-\delta}\otimes q^*{\cal L}^a$ for some integer $a$. To compute $a$ we consider the restriction of $q^*{\cal L}$ to a general fibre $q'^{-1}(F)$: by lemma (\ref{slopes}) this fibre can be identified with the Grassmann variety ${\bf G}(r-h,F_p)$, and the same argument as above shows that the restriction of $q^*{\cal L}$ is equal to ${\cal O}_{\bf G}(r)$, that is to the restriction of $K_{\cal P}^{-1}$. This gives $a=-1$, hence the lemma.\cqfd \medskip \par\hskip 1truecm\relax Observe that the group $J_r$ acts in a natural way on ${\cal P}$, by the rule $\alpha \cdot (E,F)=$ $(E\otimes\alpha ,F\otimes\alpha )$; the Hecke diagram is $J_r$\kern-1.5pt - equivariant. \smallskip \th Lemma \enonce Let $s$ be an integer dividing $r$. The canonical bundle $K_{\cal P}$ descends to ${\cal P}/J_s$, except if $s$ is even and $h$ and $r/s$ are odd; in this last case $K_{\cal P}^2$ descends. \endth \label{desc-KP} {\it Proof}: {\it a}) We first prove that $K_M$ descends to $M/J_r$. Let $\pi$ and $\rho $ denote the projections from $X\times M$ onto $X$ and $M$ respectively. By deformation theory, the tangent bundle $T_M$ is canonically isomorphic to $R^1\rho _*({\cal E}nd_0({\cal E}))$, where ${\cal E}nd_0$ denote the sheaf of traceless endomorphisms; it follows that $K_M$ is the inverse of the determinant bundle $\mathop{\rm det}\nolimits R\rho _*({\cal E}nd_0({\cal E}))$. Since ${\cal E}nd_0({\cal E})$ has trivial determinant, this is also equal to $\mathop{\rm det}\nolimits R\rho _*({\cal E}nd_0({\cal E})\otimes \pi^*L)$ for any line bundle $L$ on $X$ (see e.g.\ [B-L], 3.8); therefore $K_M^{-1}$ is the pull back of the generator ${\cal L}$ of $\mathop{\rm Pic}\nolimits(M_{{\bf SL}_{r^2-1}})$ by the morphism $M\rightarrow M_{{\bf SL}_{r^2-1}}$ which maps $E$ to ${\cal E}nd_0(E)$. This morphism factors through the quotient $M/J_r$, hence our assertion. \par\hskip 1truecm\relax {\it b}) Therefore we need only to consider the relative canonical bundle $K_{{\cal P}/M}$, with its canonical $J_r$\kern-1.5pt - linearization. Let $\alpha \in J_r$, and let $P=(E,F)$ be a fixed point of $\alpha $ in ${\cal P}$; we want to compute the tangent map $T_P(\alpha)$ to $\alpha$ at $P$. The vector bundle $E\in M$ is fixed by $\alpha $, and the action of $\alpha $ on the fibre $q^{-1}(E)={\bf G}(E_p)$ is induced by the automorphism $\tilde \alpha $ of $E_p$ obtained from the isomorphism $\varphi_\alpha :E\rightarrow E\otimes\alpha $ (note that $\varphi_\alpha $, hence also $\tilde \alpha $, are uniquely determined up to a scalar, since $E$ is stable). \par\hskip 1truecm\relax Let $0\rightarrow K_p\rightarrow E_p\rightarrow Q_p\rightarrow 0$ be the exact sequence corresponding to $P$. The tangent space to ${\bf G}(E_p)$ at $P$ is canonically isomorphic to $\mathop{\rm Hom}\nolimits(K_p,Q_p)$, hence its determinant is canonically isomorphic to $(\mathop{\rm det}\nolimits E_p)^{-h}(\mathop{\rm det}\nolimits Q_p)^{r}$; we conclude that $\mathop{\rm det}\nolimits T_P(\alpha )$ is equal to $(\mathop{\rm det}\nolimits\tilde \alpha)^h $, where $\tilde\alpha$ is normalized so that $\tilde\alpha^r=1$. \par\hskip 1truecm\relax {\it c}) It remains to compute $\mathop{\rm det}\nolimits\tilde \alpha$. Now the fixed points of $\alpha $ on $M$ are easy to describe [N-R]: let $s$ be the order of $\alpha $, and $\pi:\widetilde{X}\rightarrow X$ the associated \'etale $s$\kern-1.5pt - sheeted covering; a vector bundle $E$ on $X$ satisfies $E\otimes\alpha \cong E$ if and only if it is of the form $\pi_*\widetilde{E}$ for some vector bundle $\widetilde{E}$ on $\widetilde{X}$, of rank $r/s$. To evaluate $\varphi _\alpha $ at $p$, we can trivialize $\widetilde{E}$ in a neighborhood of $\pi^{-1}(p)$: write $\widetilde{E}=\pi^*T$, where $T={\cal O}_X^{r/s}$. Then one has $\pi_*\widetilde{E}=\sdir_{i\in{\bf Z}/s}^{} T\otimes\alpha ^i$, and the isomorphism $\varphi _\alpha $ maps identically $T\otimes\alpha ^i$ onto $(T\otimes\alpha ^{i-1})\otimes\alpha $. It follows that the eigenvalues of $\tilde \alpha $ are the $s$\kern-1.5pt - th roots of $1$, each counted with multiplicity $r/s$. This implies in particular $\mathop{\rm det}\nolimits \tilde \alpha =\zeta ^{r(s-1)/2}$, where $\zeta $ is a primitive $s$\kern-1.5pt - th root of $1$, and therefore $\mathop{\rm det}\nolimits T_P(\alpha )=(-1)^{h(s-1){r\over s}}$. The lemma follows.\cqfd \medskip \global\def\currenvir{subsection{\it Proof of Proposition} \ref{A_d}: \par\hskip 1truecm\relax We first observe that a line bundle $L$ on $M$ descends to $M/J_s$ if and only if its pull back to ${\cal P}$ descends to ${\cal P}/J_s$. In fact, we know by (\ref{list} {\it a}) that $G$\kern-1.5pt - linearizations of $L$ correspond bijectively by pull back to $G$\kern-1.5pt - linearizations of $q^*L$; for $\alpha \in J_s$, any fixed point $E$ of $\alpha $ in $M$ is the image of a point $P\in{\cal P}$ fixed by $\alpha $, so with the notation of (\ref{char}) one has $\chi _E(\alpha )=\chi _P(\alpha )$, which implies our assertion. \par\hskip 1truecm\relax Similarly, a line bundle on $M'$ descends to $M'/J_s$ if and only if its pull back to ${\cal P}$ descends to ${\cal P}/J_s$: what we have to check in order to apply the same argument is that every component of the fixed locus ${\rm Fix}_{M'}(\alpha)$ is dominated by a component of ${\rm Fix}_{{\cal P}}(\alpha)$, and conversely that every component of ${\rm Fix}_{{\cal P}}(\alpha)$ dominates a component of ${\rm Fix}_{M'}(\alpha)$. But this follows easily from the description of the fixed points of $\alpha $ given above (\ref{desc-KP} {\it c}). \par\hskip 1truecm\relax We first consider the case $h=1$. If $r$ is odd, we know from Prop.\ \ref{A0} and lemma \ref{desc-KP} that ${\cal L}'^r$ and $K_{{\cal P}}=q^*{\cal L}^{-1} \otimes q'^*{\cal L}'^{-r}$ descend to ${\cal P}/J_r$; it follows that ${\cal L}$ descends to $M/J_r$. Assume that $r$ is even. Endow $K_{\cal P}$ with its canonical $J_r$\kern-1.5pt - linearization, ${\cal L}^r$ with the $J_r$\kern-1.5pt - linearization defined in (\ref{desc-prop}), and $q^*{\cal L}$ with the $J_r$\kern-1.5pt - linearization deduced from the isomorphism $K_{{\cal P}}\cong q^*{\cal L}^{-1} \otimes q'^*{\cal L}'^{-r}$. Let $\alpha $ be an element of order $r$ in $J_r$, and $P$ a fixed point of $\alpha $ in ${\cal P}$; we know that $\alpha $ acts on $(K_{\cal P})_P$ by multiplication by $-1$ (\ref{desc-KP} {\it c}) and on $(q'^*{\cal L}'^r)_P$ by multiplication by $\varepsilon (\alpha )$ (\ref{desc-prop}), hence it acts on $(q^*{\cal L})_P$ by multiplication by $-\varepsilon (\alpha )$. Since $-\varepsilon (\alpha +\beta )\not=(-\varepsilon (\alpha )\,(-\varepsilon (\beta )))$ when $\alpha $ and $\beta $ are two elements of order $r$ orthogonal for the Weil pairing, we conclude that ${\cal L}$ does not descend, while of course ${\cal L}^2$ descends. \par\hskip 1truecm\relax We now apply the same argument with $h$ arbitrary. If $r$ is odd, $K_{\cal P}$ and $q^*{\cal L}$ descend, hence ${\cal L}'^{\delta}$ descends. If $r$ is even, we get a $J_r$\kern-1.5pt - linearization on $q'^*{\cal L}'^\delta$ such that an element $\alpha$ of order $r$ in $J_r$ acts by multiplication by $(-1)^{h+1}\varepsilon (\alpha )$; again this implies that ${\cal L}'^{\delta}$ does not descend, while ${\cal L}'^{2\delta}$ descends.\cqfd \bigskip \rem{Remark} The methods of this section allow to treat more generally in most cases the group ${\bf SL}_r/\hbox{\gragrec \char22}_s$, for $s$ dividing $r$. We will contend ourselves with an example, which we will need below: the case $G={\bf SL}_{2l}/\hbox{\gragrec \char22}_2$ ($l\ge 1$). The moduli space $M_G$ has two components, namely $M^0_G$ (treated in Prop.\ \ref{A0}) and the quotient $M_G^l$ of $M_{{\bf SL}_{2l}}^l$ by $J_2$. The theta line bundle ${\cal L}$ on $M_{{\bf SL}_{2l}}^l$ is the pull back of the determinant bundle on $M_{{\bf SL}_{4l}}^0$ under the map $E\mapsto E\otimes A$, where $A$ is a stable vector bundle of rank $2$ and degree $-1$. It follows from Prop.\ \ref{A0} that ${\cal L}^2$ descends to $M_G^l$; on the other hand, by Prop.\ \ref{A_d}, ${\cal L}^l$ and therefore ${\cal L}$ do not descend if $l$ is odd. We shall now prove that ${\cal L}$ {\it descends to $M_G^l$ when $l$ is even}. \par\hskip 1truecm\relax Let $\lambda :M_{{\bf SL}_{2l}}^l\rightarrow M_{{\bf SL}_{l(2l-1))}}$ be the morphism $E\mapsto\hbox{\san \char3}^2E(-p)$. One checks easily that the pull back of the determinant bundle ${\cal D}$ on $M_{{\bf SL}(l(2l-1))}$ is ${\cal L}^{l-1}$ (e.g.\ by pulling back to the moduli stack, and using the fact that the Dynkin index of the representation $\hbox{\san \char3}^2$ is $2l-2$). Now $\lambda $ factors through $M_{{\bf SL}_{2l}}^l/J_2$, therefore ${\cal L}^{l-1}$ descends to this quotient. When $l$ is even, this implies that ${\cal L}$ itself descends. \label{M(2l,l)} \vskip1cm \section {The Picard groups of $M_{{\bf PSp}_{2l}}$ and $M_{{\bf PSO}_{2l}}$} \global\def\currenvir{subsection In the case $C_l$, it remains only to consider the component $M_{{\bf PSp}_{2l}}^1$, which is the quotient by $J_2$ of the moduli space $M_{{\bf PSp}_{2l}}^1$ of semi-stable pairs $(E,\varphi )$, where $E$ is a vector bundle of rank $2l$ on $X$ and $\varphi :\hbox{\san \char3}^2E\rightarrow {\cal O}_X(p)$ a non-degenerate alternate form. Let ${\cal L}$ denote the pull back of the theta line bundle by the natural map $M_{{\bf Sp}_{2l}}^1\rightarrow M_{{\bf SL}_{2l}}^1$. \th Proposition \enonce {\rm a)} The group $\mathop{\rm Pic}\nolimits(M_{{\bf Sp}_{2l}}^{1})$ is generated by ${\cal L}$. \par\hskip 1truecm\relax {\rm b)} The group $\mathop{\rm Pic}\nolimits(M_{{\bf PSp}_{2l}}^{1})$ is generated by ${\cal L}$ if $l$ is even, and by ${\cal L}^2$ if $l$ is odd. \endth \label{C} {\it Proof}: By Prop.\ \ref{cyc} to prove a) it suffices to prove that ${\cal L}$ is not divisible. Choose an element $(A,\psi )$ of $M_{{\bf Sp}_{2l-2}}^{1}$, and consider the map $u:M_{{\bf SL}_2}^1\rightarrow M_{{\bf Sp}_{2l}}^{1}$ given by $u(E)=(E,\mathop{\rm det}\nolimits )\oplus (A,\psi )$. The pull back of ${\cal L}$ is the theta line bundle $\Theta $ on $M_{{\bf SL}_2}^1$, hence the assertion a). \par\hskip 1truecm\relax Let us prove b). By Remark \ref{M(2l,l)} we already know that ${\cal L}^2$ descends to $M_{{\bf PSp}_{2l}}^{1}$, and that ${\cal L}$ descends if $l$ is even. Consider the morphism $\mu :M_{{\bf SL}_2}^1\rightarrow M_{{\bf Sp}_{2l}}^{1}$ given by $\mu (E )=(E,\mathop{\rm det}\nolimits)^{\oplus l}$. One has $\mu ^*{\cal L}=\Theta ^l$, so if ${\cal L}$ descends $\Theta ^l$ descends to $M_{{\bf PGL}_2}^1$; by Prop.\ \ref{A_d} this implies that $l$ is even.\cqfd \medskip \global\def\currenvir{subsection Let us consider the group $G={\bf PSO}_{2l}$. The moduli space $M_{G}$ has $4$ components, indexed by the center $\{1,-1,\varepsilon ,-\varepsilon \}$ of ${\bf Spin}_{2l}$ (\ref{PSO}). \par\hskip 1truecm\relax The component $M^1_{{\bf PSO}_{2l}}$ has already been dealt with in Prop.\ \ref{BC0}. The component $M_{{\bf PSO}_{2l}}^{-1}$ is the quotient by the action of $J_2$ of the moduli space $M^{-1}_{{\bf SO}_{2l}}$ of semi-stable quadratic bundles with $w_2=1$. Let ${\cal D}$ denote the determinant bundle on this moduli space. \smallskip \th Proposition \enonce The group $\mathop{\rm Pic}\nolimits(M_{{\bf PSO}_{2l}}^{-1})$ is generated by ${\cal D}^2$ if $l$ is even, by ${\cal D}^4$ if $l$ is odd. \endth {\it Proof}: The same proof as in \ref{BC0} shows that ${\cal D}^2$ descends to $M_{{\bf PSO}_{2l}}^{-1}$ if $l$ is even, and that ${\cal D}^4$ descends but ${\cal D}^2$ does not if $l$ is odd. To prove that ${\cal D}$ does not descend when $l$ is even $\ge 3$, we apply the argument of {\it loc.\ cit.} to the morphism $u:JX\times M_{{\bf SO}_{2l-2}}^{-1}\rightarrow M_{{\bf SO}_{2l}}^{-1}$ deduced from the natural embedding ${\bf SO}_2\times{\bf SO}_{2l-2}\lhook\joinrel\mathrel{\longrightarrow} {\bf SO}_{2l}$ (note that $w_2$ is additive and $w_2(\alpha \oplus\alpha ^{-1})=0$ for $\alpha \in JX$). \par\hskip 1truecm\relax When $l=2$ we consider the morphism $v:M_{{\bf SL}_2}^1\times M_{{\bf SL}_2}^1\longrightarrow M_{{\bf SO}_4}^{-1}$ which associates to a pair $(E,F)$ the vector bundle ${\cal H}om(E,F)$ with the quadratic form defined by the determinant and the orientation deduced from the canonical isomorphism $\mathop{\rm det}\nolimits(E^*\otimes F)\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}$ $(\mathop{\rm det}\nolimits E)^{-2}\otimes(\mathop{\rm det}\nolimits F)^2$. One has $v^*{\cal D}={\cal L} \boxtimes {\cal L}$, where ${\cal L}$ is the theta line bundle on $M_{{\bf SL}_2}^1$. Since ${\cal L}$ does not descend to $M^1_{{\bf PGL}_2}$ (Prop.\ \ref{A_d}), it follows from the commutative diagram $$\diagramme{M_{{\bf SL}_2}^1\times M_{{\bf SL}_2}^1 &\hfl{v}{} &M_{{\bf SO}_4}^{-1} \cr \vfl{}{} && \vfl{}{}\cr M^1_{{\bf PGL}_2}\times M^1_{{\bf PGL}_2}& \hfl{}{} & M_{{\bf PSO}_4}^{-1} }$$that ${\cal D}$ does not descend to $M_{{\bf PSO}_4}^{-1}$.\cqfd \medskip \par\hskip 1truecm\relax We now consider the components $M^{\pm\varepsilon }_{{\bf PSO}_{2l}}$ corresponding to the elements $+\varepsilon $ and $-\varepsilon $ of the center of ${\bf Spin}_{2l}$. Each of these is the quotient by $J_2$ of the moduli space $M^{\pm\varepsilon }_{{\bf SO}_{2l}}$ of semi-stable quadratic bundles $(E,q,\omega )$, where $E$ is a vector bundle of rank $2l$, $q:\hbox{\san \char83}^2E\rightarrow {\cal O}_X(p)$ a quadratic form and $\omega :\mathop{\rm det}\nolimits E\rightarrow {\cal O}_X(lp)$ an isomorphism compatible with $q$; changing the sign of $\omega $ exchanges $M^\varepsilon $ and $M^{-\varepsilon }$ (\ref{PSO}). We denote by ${\cal L}_l$ the pull back of the theta line bundle on $M_{{\bf SL}_{2l}^l}$ under the natural map $M^{\pm\varepsilon }_{{\bf SO}_{2l}}\rightarrow M_{{\bf SL}_{2l}^l}$. \th Proposition \enonce The group $\mathop{\rm Pic}\nolimits(M^{\pm\varepsilon }_{{\bf PSO}_{2l}})$ is generated by ${\cal L}_l$ when $l$ is even, and by ${\cal L}_l^2$ when $l$ is odd. \endth {\it Proof}: We already know that the theta line bundle descends to $M_{{\bf SL}_{2l}}^l/J_2$ when $l$ is even and that its square descends when $l$ is odd (\ref{M(2l,l)}), so we have only to prove that ${\cal L}_l$ does not descend when $l$ is odd. \par\hskip 1truecm\relax Let us first consider the case $l=3$. If $E$ is a vector bundle of rank $4$ and determinant ${\cal O}_X(p)$ on $X$, the bundle $\hbox{\san \char3}^2E$ carries a quadratic form with values in ${\cal O}_X(p)$ (defined by the exterior product) and an orientation. We thus get a morphism $\lambda : M_{{\bf SL}_4}^1\rightarrow M^{\pm\varepsilon }_{{\bf SO}_6}$ such that $\lambda (E\otimes\alpha )=\lambda (E)\otimes\alpha ^2$ for $\alpha \in J_4$. An easy computation shows that $\lambda ^*{\cal L}_3$ is the theta line bundle on $M_{{\bf SL}_4}^1$, which does not descend to $M_{{\bf PGL}_4}^1$ (\ref{A_d}); our assertion follows. \par\hskip 1truecm\relax For $l$ odd $\ge 5$, we consider the morphism $\mu :M^{\pm\varepsilon }_{{\bf SO}_{2l-6}}\times M^{\pm\varepsilon }_{{\bf SO}_6}\longrightarrow M^{\pm\varepsilon }_{{\bf SO}_{2l}}$ deduced from the embedding ${\bf SO}_{2l-6}\times {\bf SO}_6\lhook\joinrel\mathrel{\longrightarrow} {\bf SO}_{2l}$. It is $J_2$\kern-1.5pt - equivariant (with respect to the canonical action of $J_2$ on the spaces $M^{\pm\varepsilon }_{{\bf SO}_{2n}}$, and the diagonal action on the product), and the pull back $\mu ^*{\cal L}_l$ is isomorphic to ${\cal L}_{l-3}\boxtimes {\cal L}_3$. Assume that ${\cal L}_l$ descends to $M^{\pm\varepsilon }_{{\bf PSO}_{2l}}$; since ${\cal L}_{l-3}$ descends, we deduce from \ref{produit} that ${\cal L}_3$ descends, contradicting what we just proved.\cqfd \vskip 1cm \section {Determinantal line bundles} \global\def\currenvir{subsection \label{e_G} We can express the above results in a more suggestive way. Assume $G$ is of type $A,B,C$ or $D$; let $\delta\in \pi_1(G)$. We identify $\mathop{\rm Pic}\nolimits(M_G^\delta)$ to a subgroup of $\mathop{\rm Pic}\nolimits({\cal M}_{\widetilde{G}}^{\delta,ss})$. Let $\sigma $ be the standard representation of $\widetilde{G}$ in ${\bf C}^r$ (for $\widetilde{G}={\bf SL}_r$, ${\bf Spin}_r$ or ${\bf Sp}_r$), and ${\cal D}_\sigma $ the corresponding determinant bundle on ${\cal M}_{\widetilde{G}}^{\delta,ss}$. The results of sections 8 to 11 express the generator of $\mathop{\rm Pic}\nolimits(M_G^\delta)$ as a certain power of ${\cal D}_\sigma$. Using the fact that the pull back to ${\cal M}^{d,ss}_{{\bf SL}_r}$ of the theta line bundle on $M_{{\bf SL}_r }^d$ is $({\cal D}_\sigma )^{r\over (d,r)}$, one finds: \th Proposition \enonce Assume that $G$ is one of the groups ${\bf PGL}_r$, ${\bf PSp}_{2l}$ or ${\bf PSO}_{2l}$. Put $\varepsilon_G^{}=1$ if the rank of $G$ is even, $2$ if it is odd. Let $\delta\in \pi_1(G)$. The group $\mathop{\rm Pic}\nolimits(M_G^\delta)$ is generated by $({\cal D}_\sigma )^{r\varepsilon_G^{}}$ for $G={\bf PGL}_r$, and by $({\cal D}_\sigma)^{2\varepsilon_G^{}}$ for the other groups. \endth \medskip \global\def\currenvir{subsection To produce line bundles on $M_G^\delta$, we have already used the following recipe: to any representation $\rho :G\rightarrow {\bf SL}_N$ we associate the pull back ${\cal D}_\rho $ of the determinant bundle under the morphism $M_G^\delta \rightarrow M_{{\bf SL}_N}$ deduced from $\rho $. These line bundles generate a subgroup $\mathop{\rm Pic}\nolimits_{\rm det}(M_G^\delta)$ of $\mathop{\rm Pic}\nolimits(M_G^\delta)$. We suspect that this subgroup is actually equal to $\mathop{\rm Pic}\nolimits(M_G^\delta)$, i.e.\ that all line bundles on $M_G^\delta$ can be constructed from representations of $G$. We have checked this in some cases: \th Proposition \enonce Assume $G$ is of classical type or of type $G_2$, and either simply connected or adjoint or isomorphic to ${\bf SO}_r$. Then, for every $\delta\in \pi_1(G)$, one has $\mathop{\rm Pic}\nolimits_{\rm det}(M_G^\delta)=\mathop{\rm Pic}\nolimits(M_G^\delta)$. \endth {\it Proof}: The simply connected case, and also the case $G={\bf SO}_r$, follow from [L-S], Prop.\ \ref{Pic_Spin} and \ref{G_2}. \par\hskip 1truecm\relax The other groups are those which appear in the above Proposition; let us denote by $e_G$ the positive integer such that $({\cal D}_\sigma)^{e_G}$ generates $\mathop{\rm Pic}\nolimits(M_G^\delta)$. If $\rho $ is a representation of $G$, with Dynkin index $d_\rho $, the line bundle ${\cal D}_\rho $ on $M_{G}^d$ is isomorphic to $({\cal D}_\sigma) ^{ d_\rho/d_\sigma }$ ($d_\sigma $ is $1$ for the types $A,C$ and $2$ for $B,D$). It follows that $e_G$ divides $d_\rho /d_\sigma$, and that our assertion is equivalent to saying that $e_G$ is the g.c.d.\ of the numbers $d_\rho/d_\sigma$ when $\rho$ runs over the representations of $G$. \par\hskip 1truecm\relax Let us consider the case $G={\bf PGL}_r$. We have $d_{\rm Ad}=2r$, which settles the case $r$ even. If $r$ is odd, consider the representation $\hbox{\san \char83}^2 \otimes \hbox{\san \char3}^{r-2}$ of ${\bf SL}_r$; since $\hbox{\gragrec \char22}_r$ acts trivially, it defines a representation $\rho $ of ${\bf PGL}_r$, whose Dynkin index is $$\eqalign{d_\rho &=d_{\hbox{\san \char83}^2}\,\mathop{\rm dim}\nolimits \hbox{\san \char3}^{r-2}+d_{\hbox{\san \char3}^2}\,\mathop{\rm dim}\nolimits \hbox{\san \char83}^2 \cr &=(r+2){r\choose 2}+(r-2){r+1\choose 2} = r^3-2r\ .\cr}$$ Then $(d_{\rm Ad},d_\rho )=r=e_G$, which proves the result in this case. \par\hskip 1truecm\relax For $G={\bf PSp}_{2l}$, easy computations give $d_{\rm Ad}=2l+2$ and $d_{\hbox{\san \char3}^2}=2l-2$, hence $e_G=(d_{\rm Ad},d_{\hbox{\san \char3}^2})$. For $G={\bf PSO}_{2l}$, one has $d_{\rm Ad}=2(2l-2)$ and $d_{\hbox{\san \char83}^2}=2(2l+2)$, hence $e_G=(d_{\rm Ad},d_{\hbox{\san \char3}^2})/d_\sigma $.\cqfd \medskip \rem{Remark} We can also prove the equality $\mathop{\rm Pic}\nolimits_{\rm det}(M_G^0)=\mathop{\rm Pic}\nolimits(M_G^0)$ for $G={\bf SL}_r/\hbox{\gragrec \char22}_s$ when $s$ and $r/s$ are coprime. Reasoning as above and using Prop. \ref{A0}, we need to prove that the g.c.d.\ of the $d_\rho $'s is $2s$ if $s$ is even, and $s$ if it is odd. We consider the representation $\rho _p=\hbox{\san \char83}^p\otimes \hbox{\san \char3}^{s-p}$ for $1\le p \le s$. Using some nontrivial combinatorics we can prove the relation $\displaystyle \sum_{p=1}^s p\,d_{\rho _p}=(-1)^ss^2$. Since $d_{\rm Ad}=2r$ we find that the g.c.d.\ of the $d_\rho $' divides $(2r,s^2)=s(2{r\over s},s)$, hence our assertion.\cqfd \vskip1cm \section{Local properties of the moduli spaces $M_G$} \def\mathop{\rm Cl}\nolimits{\mathop{\rm Cl}\nolimits} \global\def\currenvir{subsection A $G$\kern-1.5pt - bundle $P $ is called {\it regularly stable} if it is stable and its automorphism group is equal to the center $Z(G)$ of $G$. The open subset $M_G^{\rm reg}$ of $M_G$ corresponding to regularly stable $G$\kern-1.5pt - bundles is smooth, and its complement in $M_G$ is of codimension $\ge 2$, except when $X$ is of genus $2$ and $G$ maps onto ${\bf PGL}_2$: this is seen exactly as the analogous statement for Higgs bundles, which is proved in [F1], thm.\ II.6. In what follows we will assume that we are not in this exceptional case, leaving to the reader to check that our assertions extend by using the explicit description of $M_{{\bf SL}_2}$ in genus $2$. \par\hskip 1truecm\relax Let $i$ be the natural injection of $M_G^{\rm reg}$ into $M_G$. Then the map $i_*$ identifies $\mathop{\rm Pic}\nolimits(M_G^{\rm reg})$ with the Weil divisor class group $\mathop{\rm Cl}\nolimits(M_G)$, that is the group of isomorphism classes of rank $1$ reflexive sheaves on $M_G$ (see [Re], App.\ to \S 1); the restriction map $i^*:\mathop{\rm Pic}\nolimits(M_G)\rightarrow \mathop{\rm Pic}\nolimits(M_G^{\rm reg})$ corresponds to the inclusion $\mathop{\rm Pic}\nolimits(M_G) \i \mathop{\rm Cl}\nolimits(M_G)$. Local factoriality of M$_G$ is equivalent to the equality $\mathop{\rm Pic}\nolimits(M_G) = \mathop{\rm Cl}\nolimits(M_G)$. \par\hskip 1truecm\relax We already know from [D-N] and [L-S] that $M_G$ is locally factorial when $G$ is ${\bf SL}_r$ or ${\bf Sp}_{2l}$. We want to show that these are essentially the only cases where this occurs. \th Proposition \enonce Let $G$ be a simply connected group, containing a factor of type $B_l\ (l\ge 3)$, $D_l\ (l\ge 4)$, $F_4$ or $G_2$. Then $M_G$ is not locally factorial. \endth \par\hskip 1truecm\relax We believe that this still holds if $G$ contains a factor of type $E_l$. This would have the amusing consequence that {\it the semi-simple groups $G$ for which $M_G$ is locally factorial are exactly those which are} special {\it in the sense of Serre, i.e.\ such that all $G$\kern-1.5pt - bundles are locally trivial for the Zariski topology} (see [G2]). \par \smallskip {\it Proof of the Proposition}: We can assume that $G$ is almost simple. Choose a presentation of ${\cal M}_G$ as a quotient of a smooth scheme $R$ by a reductive group $\Gamma $, such that $M_G$ is a good quotient of $R$ by $\Gamma $ (lemma \ref{comppres}). We denote by $\sigma $ the standard representation in ${\bf C}^r$ in case $G={\bf Spin}_r$, in ${\bf C}^7$ if $G$ is of type $G_2$, and the orthogonal representation in ${\bf C}^{26}$ with highest weight $\varpi_4$ if $G$ is of type $F_4$ (we use the standard notation of [Bo], Lie VII). Let ${\cal D}$ be the determinant bundle on $R$ associated to $\sigma $. As in the proof of Prop.\ \ref{Pic_Spin}, the choice of a theta-characteristic $\kappa $ on $X$ allows us to define a square root ${\cal P}_\kappa $ of ${\cal D}$ on $R$, with a canonical $\Gamma $\kern-1.5pt - linearization. We will show that ${\cal P}_\kappa $ descends to the open subset $M_G^{\rm reg}$, but not to $M_G$, thus showing that the restriction map is not surjective. \par\hskip 1truecm\relax The first assertion is clear if $G$ is of type $F_4$ or $G_2$, because then $Z(G)$ is trivial, so $\Gamma $ acts freely on the open subset of $R$ corresponding to regularly stable $G$\kern-1.5pt - bundles. Suppose $G={\bf Spin}_r$; let $Q$ be a $G$\kern-1.5pt - bundle, and $z$ an element of $Z(G)$. The image of $z$ in ${\bf SO}_r$ is either $1$ or possibly $-1$ if $r$ is even; since $h^0(Q_\sigma \otimes\kappa )\equiv rh^0(\kappa )$ (mod.$\,2$) by [L-S], 7.10.1, we conclude that $z$ acts trivially on $\hbox{\san \char3}^{\rm max} H^0(Q_\sigma \otimes\kappa )$, i.e.\ on the fibre of ${\cal P}_\kappa $ at $Q$ (\ref{Pic_Spin}). \par\hskip 1truecm\relax We already know that ${\cal P}_\kappa $ does not descend to $M_G$ when $G={\bf Spin}_r$ (Prop.\ \ref{Pic_Spin}) or $G$ is of type $G_2$ (Prop.\ \ref{G_2}); it remains to show that the class of ${\cal D}$ is not divisible by $2$ in $\mathop{\rm Pic}\nolimits(M_G)$ when $G$ is of type $F_4$. There is a natural inclusion ${\bf Spin}_8\i G$, which induces a morphism $f:M_{{\bf Spin}_8}\rightarrow M_G$. An easy computation gives that the Dynkin index of the restriction to ${\bf Spin}_8$ of the standard representation of $G$ is $6$. Since the Dynkin index of the standard representation of ${\bf Spin}_8$ is $2$, it follows that $f^*{\cal D}$ is isomorphic to ${\cal D}_0^{\otimes 3}$, where ${\cal D}_0$ is the generator $\mathop{\rm Pic}\nolimits(M_{{\bf Spin}_8})$; this implies that ${\cal D}$ is not divisible by $2$ in $\mathop{\rm Pic}\nolimits(M_G)$.\cqfd \medskip \par\hskip 1truecm\relax We now treat the case of a non simply connected group. We start with two lemmas which are certainly well known, but for which we could find no reference: \th Lemma \enonce Let $\pi :\widetilde{Y}\rightarrow Y$ be a ramified Galois covering, with abelian Galois group $A$. If $\pi $ is \'etale in codimension $1$, the variety $Y$ is not locally factorial. \endth\label{factorial} {\it Proof}: Let $Y^{\rm o}$ be an open subset of $Y$ such that $Y\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}} Y^{\rm o}$ has codimension $\ge 2$ and the induced covering $\pi ^{\rm o}:\widetilde{Y}^{\rm o}\rightarrow Y^{\rm o}$ is \'etale. This covering corresponds to a homomorphism $L: \widehat{A}\rightarrow \mathop{\rm Pic}\nolimits(Y^{\rm o})$ such that $\pi _*{\cal O}_{\widetilde{Y}^{\rm o}}=\sdir_{\chi \in \widehat{A}}^{}L(\chi) $. If $Y$ is locally factorial, the restriction map $\mathop{\rm Pic}\nolimits(Y)\rightarrow \mathop{\rm Pic}\nolimits(Y^{\rm o})$ is bijective, so $L$ extends to a homomorphism $ \widehat{A}\rightarrow \mathop{\rm Pic}\nolimits(Y)$ which defines an \'etale covering of $Y$ extending $\pi ^{\rm o}$, and therefore equal to $\pi $. Then $\pi $ is \'etale, contrary to our hypothesis.\cqfd \th Lemma \enonce Let $S$ be a scheme, $H$ an algebraic group, $A$ a closed central subgroup of $H$, $P$ a principal $H$\kern-1.5pt - bundle on $S$. The cokernel of the natural homomorphism $\mathop{\rm Aut}\nolimits(P)\rightarrow \mathop{\rm Aut}\nolimits(P/A)$ is canonically isomorphic to the stabilizer of $P$ in $H^1(X,A)$ {\rm (}for the natural action of $H^1(X,A)$ on $H^1(X,H)\,)$. \endth \label{aut} {\it Proof}: Denote by ${\cal A}ut\,(P)$ the automorphism bundle of the $H$\kern-1.5pt - bundle $P$. We have an exact sequence of groups over $S$ $$1\rightarrow A_S \longrightarrow {\cal A}ut\,(P)\longrightarrow {\cal A}ut\,(P/A)\rightarrow 1$$(to check exactness one may replace $P$ by the trivial $H$\kern-1.5pt - bundle, for which this is clear). The associated cohomology exact sequence reads $$1\rightarrow A \longrightarrow \mathop{\rm Aut}\nolimits(P)\longrightarrow \mathop{\rm Aut}\nolimits(P/A) \longrightarrow H^1(S,A)\qfl{h} H^1(S,{\cal A}ut\,(P))\ .$$ The map $h$ associates to an $A$\kern-1.5pt - bundle $\alpha$ the class of the ${\cal A}ut\,(P)$\kern-1.5pt - bundle $\alpha\times^A{\cal A}ut\,(P)$, which is canonically isomorphic to ${\cal I}som\,(P,\alpha\times^AP)$; the element $h(\alpha)$ is trivial if and only if this last bundle admits a global section, which means exactly that $\alpha\times^AP$ is isomorphic to $P$, hence the lemma.\cqfd \th Proposition \enonce Suppose $G$ is not simpy connected; let $\delta\in \pi _1(G)$. The moduli space $M_G^\delta$ is not locally factorial. \endth {\it Proof}: We first prove that the Galois covering $\pi :M_{\widetilde{G}}^\delta\rightarrow M_G^\delta$ is \'etale above $(M_G^\delta)^{\rm reg}$. We put $A=\pi _1(G)$, and choose an isomorphism $A\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \pprod_{j=1}^s\hbox{\gragrec \char22}_{r_j}$; we use freely the notation of (\ref{M_G^\delta}). We denote by $H$ the group $C_A\widetilde{G}=(\widetilde{G}\times T)/A$. Let $Q\in (M_G^\delta)^{\rm reg}$ and $P$ a point of $M_{\widetilde{G}}^\delta$ above $Q$; we will use the same letters to denote the corresponding bundles. Using the isomorphism $H/A\cong G\times (T/A)$, the condition $\pi(P)=Q$ means that the $(H/A)$\kern-1.5pt - bundle $P/A$ is isomorphic to $Q\times {\cal O}_X({\bf d}p)$. Since $\mathop{\rm Aut}\nolimits(Q)$ is reduced to the center of $G$, the map $\mathop{\rm Aut}\nolimits(P)\rightarrow \mathop{\rm Aut}\nolimits(P/A)$ is surjective; we deduce from lemma \ref{aut} that the stabilizer of $P$ in $H^1(X,A)$ is trivial, i.e.\ $\pi$ is \'etale at $P$. \par\hskip 1truecm\relax It follows that the abelian cover $\pi:M_{\widetilde{G}}^{\delta}\ra M_{G}^{\delta}$ is \'etale in codimension one. Since it is ramified by lemma \ref{unirat}, we conclude from lemma \ref{factorial} that $M_{G}^{\delta}$ is not locally factorial. \medskip \par\hskip 1truecm\relax Finally we observe that, though the moduli space is not locally factorial in most cases, it is always Gorenstein (this is proved in [K-N], thm.\ 2.8, for a simply connected $G$): \th Proposition \enonce The moduli space $M_G$ is Gorenstein. \endth {\it Proof}: We choose again a presentation of ${\cal M}_G$ as a quotient of a smooth scheme $R$ by a reductive group $\Gamma $, such that $M_G$ is a good quotient of $R$ by $\Gamma $ (lemma \ref{comppres}); we denote by ${\cal P}$ the universal bundle on $X\times R$, and by $R^{\rm reg}$ the open subset of $R$ corresponding to regularly stable bundles. Since the center of $G$ is killed by the adjoint representation, the vector bundle $\mathop{\rm Ad}\nolimits({\cal P})$ descends to a vector bundle on $X\times M_G^{\rm reg}$, that we will still denote $\mathop{\rm Ad}\nolimits({\cal P})$. Deformation theory provides an isomorphism $T_{M_G^{\rm reg}} \mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} R^1pr_{2*}(\mathop{\rm Ad}\nolimits {\cal P})$; since $H^0(X,\mathop{\rm Ad}\nolimits(P))=0$ for $P\in M_G^{\rm reg}$, the line bundle $\mathop{\rm det}\nolimits T_{M_G^{\rm reg}}$ is isomorphic to $\mathop{\rm det}\nolimits Rpr_{2*}(\mathop{\rm Ad}\nolimits {\cal P}))$, that is to the restriction to $M_G^{\rm reg}$ of the determinant bundle ${\cal D}_{\rm Ad}$ associated to the adjoint representation. \par\hskip 1truecm\relax Since $M_G$ is Cohen-Macaulay, it admits a dualizing sheaf $\omega $, which is torsion-free and reflexive ([Re], App. of \S 1). The reflexive sheaves $\omega $ and ${\cal D}_{\rm Ad}^{-1}$, which are isomorphic above $M_G^{\rm reg}$, are isomorphic ({\it loc. cit.}), hence $\omega $ is invertible.\cqfd \vskip2cm \centerline{ REFERENCES} \bigskip \baselineskip13pt \def\num#1{\item{\hbox to\parindent{\enskip [#1]\hfill}}} \parindent=1.5cm \num{B-L} A.\ {\pc BEAUVILLE}, Y.\ {\pc LASZLO}: {\sl Conformal blocks and generalized theta functions.} Comm. Math.\ Phys.\ {\bf 164}, 385-419 (1994). \smallskip \num{B-N-R} A.\ {\pc BEAUVILLE}, M.S.\ {\pc NARASIMHAN}, S.\ {\pc RAMANAN}: {\sl Spectral curves and the generalised theta divisor}. J.\ reine angew.\ Math.\ {\bf 398}, 169-179 (1989). \smallskip \num{Bo} N.\ {\pc BOURBAKI}: {\sl El\'ements de Math\'ematique}. Hermann, Paris. \smallskip \num{Br} K.\ {\pc BROWN}: {\sl Cohomology of groups}. GTM {\bf 87}, Springer-Verlag (1982). \smallskip \num{D-N} J.M.\ {\pc DREZET}, M.S.\ {\pc NARASIMHAN}: {\sl Groupe de Picard des vari\'et\'es de modules de fibr\'es semi-stables sur les courbes alg\'ebriques.} Invent.\ math.\ {\bf 97}, 53-94 (1989). \smallskip \num {F 1} G.\ {\pc FALTINGS}: {\sl Stable $G$\kern-1.5pt - bundles and projective connections.} J.\ Algebraic Geometry {\bf 2}, 507-568 (1993). \smallskip \num {F 2} G.\ {\pc FALTINGS}: {\sl A proof for the Verlinde formula.} J.\ Algebraic Geometry {\bf 3}, 347-374 (1994). \smallskip \num {G 1} A.\ {\pc GROTHENDIECK}: {\sl Techniques de construction et d'existence en g\'eom\'etrie alg\'ebrique: les sch\'emas de Hilbert.} S\'em.\ Bourbaki {\bf 221}, 1-28 (1960/61). \smallskip \num {G 2} A.\ {\pc GROTHENDIECK}: {\sl Torsion homologique et sections rationnelles.} S\'emi\-naire Chevalley, $2^e$ ann\'ee, Exp.\ 5. Institut Henri Poincar\'e, Paris (1958). \smallskip \num{Ha} A.\ {\pc HARDER}: {\sl Halbeinfache Gruppenschemata \"uber Dedekindringen}. Invent.\ Math.\ {\bf 4}, 165-191 (1967). \smallskip \num{H} H.\ {\pc HIRONAKA}: {\sl Introduction to the theory of infinitely near singular points}. Memorias de Mat.\ del Inst.\ ``Jorge Juan", Madrid (1974).\smallskip \num{Hi} A.\ {\pc HIRSCHOWITZ}: {\sl Probl\`emes de Brill-Noether en rang sup\'erieur}. C.\ R.\ Acad.\ Sci.\ Paris {\bf 307}, 153-156 (1988). \smallskip \num{K-N} S.\ {\pc KUMAR}, M.S.\ {\pc NARASIMHAN}: {\sl Picard group of the moduli spaces of $G$\kern-1.5pt - bundles}. Preprint alg-geom/9511012.\smallskip \num{K-N-R} S.\ {\pc KUMAR}, M.S.\ {\pc NARASIMHAN}, A.\ {\pc RAMANATHAN}: {\sl Infinite Grassmannians and moduli spaces of $G$\kern-1.5pt - bundles}. Math.\ Annalen {\bf 300}, 41-75 (1994). \smallskip \num{L} H. {\pc LANGE}: {\sl Zur Klassification von Regelmannigfaltigkeiten}. Math.\ Annalen {\bf 262}, 447-459 (1983). \smallskip \num{L-MB} G.\ {\pc LAUMON}, L.\ {\pc MORET-BAILLY}: {\sl Champs alg\'ebriques.} Pr\'epublication Universit\'e Paris-Sud (1992). \smallskip \num{L-S} Y.\ {\pc LASZLO}, Ch.\ {\pc SORGER}: {\sl The line bundles on the moduli of parabolic $G$\kern-1.5pt - bundles over curves and their sections}. Annales de l'ENS, to appear; preprint alg-geom/9507002. \smallskip \num{M} O.\ {\pc MATHIEU}: {\sl Formules de caract\`eres pour les alg\`ebres de Kac-Moody g\'en\'e\-rales.} Ast\'erisque {\bf 159-160} (1988). \smallskip \num{N-R} M.S.\ {\pc NARASIMHAN}, S.\ {\pc RAMANAN}: {\sl Generalized Prym varieties as fixed points}. J.\ of the Indian Math.\ Soc.\ {\bf 39}, 1-19 (1975). \smallskip \num{O} W.M.\ {\pc OXBURY} : {\sl Prym varieties and the moduli of spin bundles}. Preprint, Durham (1995). \smallskip \num{R1} A.\ {\pc RAMANATHAN}: {\sl Stable principal $G$\kern-1.5pt - bundles}. Thesis, Bombay University (1976). \smallskip \num{R2} A.\ {\pc RAMANATHAN}: {\sl Stable principal bundles on a compact Riemann surface}. Math.\ Ann.\ {\bf 213}, 129-152 (1975). \smallskip \num{Re} M.\ {\pc REID}: {\sl Canonical $3$-folds}. Journ\'ees de G\'eom\'etrie alg\'ebrique d'Angers (A. Beauville ed.), 273-310; Sijthoff and Noordhoff (1980). \smallskip \num{Se} C.S.\ {\pc SESHADRI}: {\sl Quotient spaces modulo reductive algebraic groups}. Ann.\ of Math.\ {\bf 95}, 511-556 (1972). \smallskip \num{S1} R.\ {\pc STEINBERG}: {\sl G\'en\'erateurs, relations et rev\^etements de groupes alg\'e\-briques}. Colloque sur la th\'eorie des groupes alg\'ebriques (Bruxelles), 113-127; Gauthier-Villars, Paris (1962). \smallskip \num{S2} R.\ {\pc STEINBERG}: {\sl Regular elements of semisimple algebraic groups}. Publ.\ Math.\ IHES {\bf 25}, 281-312 (1965). \smallskip \def\pc#1{\eightrm#1\sixrm} $$\hbox to 16truecm{\eightrm\vtop{\hbox to 5cm{\hfill A. {\pc BEAUVILLE}, Y. {\pc LASZLO}\hfill} \hbox to 5cm{\hfill DMI -- \'Ecole Normale Sup\'erieure\hfill} \hbox to 5cm{\hfill (URA 759 du CNRS)\hfill} \hbox to 5cm{\hfill 45 rue d'Ulm\hfill} \hbox to 5cm{\hfill F-75230 {\pc PARIS} Cedex 05\hfill}} \vtop{\hbox to 5cm{\hfill Ch. {\pc SORGER}\hfill}\par \hbox to 5cm{\hfill Institut de Math\'ematiques de Jussieu\hfill}\par \hbox to 5cm{\hfill (UMR 9994 du CNRS)\hfill} \hbox to 5cm{\hfill Univ. Paris 7 -- Case Postale 7012\hfill} \hbox to 5cm{\hfill 2 place Jussieu\hfill} \hbox to 5cm{\hfill F-75251 {\pc PARIS} Cedex 05\hfill}}} $$ \end

9608

alg-geom/9608032

en

https://arxiv.org/abs/alg-geom/9608032

alg-geom/9608032

Jun Li

Jun Li and Gang Tian

Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds

LaTex. We updated the bibliography in the revised version

We construct Gromov-Witten invariants of general symplectic manifolds.

alg-geom

\section{Introduction} \maketitle \begin{abstract} In this paper, we first give an intersection theory for moduli problems for nonlinear elliptic operators with certain precompact space of solutions in differential geometry. Then we apply the theory to constructing Gromov-Witten invariants for general symplectic manifolds. \end{abstract} \setcounter{section}{-1} \section{Introduction} \label{sec:0} This paper is a sequel to [LT1]. In [LT1], by using purely algebraic methods, we developped an intersection theory for moduli problems on smooth algebraic varieties over any algebraically closed fields of characteristic zero. An alternative construction was given in [BF]. The key point in [LT1] is the existence of locally free resolutions of tangent complexes involved. In this paper, we apply the same idea to constructing the intersection theory for moduli problems in the differential category. However, the tool will be analytic this time. Given a Banach manifold $B$, a smooth bundle $E\mapsto B$ is Fredholm, if there is a section $s: B \mapsto E$ such that $s^{-1}(0)$ is compact and any linearization of $s$ at any point of $s^{-1}(0)$ is Fredholm of index $r$. Then one can define the determinant line bundle $\det(s)$ over $s^{-1}(0)$. Note that $s:B\mapsto E$ is orientable if $\det (s)$ does. It should be well-known that for any orientable Fredholm bundle $s: B \mapsto E$, one can associate an Euler class $e([s:B\mapsto E])$ in $H_r(B, {\Bbb Z} )$, which depends only on the homotopy class $[s:B\mapsto E]$ of $s: B\mapsto E$. However, its use is very limited. In many problems, such as constructing Donaldson invariants and Gromov-Witten invariants, the zero set $s^{-1}(0)$ is often noncompact, if we insist on smooth Banach manifolds, smooth bundles. For many useful applications, we have to construct Euler classes for spaces, bundles and their sections, which are not necessarily smooth, and prove that they are invariant. In section one, we will give two simple theorems on constructing the Euler classes of so called generalized Fredholm bundles, and more generally, the rational Euler classes of generalized Fredholm orbifold bundles. The main part of this paper is devoted to constructing Gromov-Witten invariants over rational numbers for general symplectic manifolds by establishing the Fredholm properties of the bundle of $(0,1)$-forms over the space of smooth stable maps (cf. section 2). In fact, we have constructed symplectically invariant Euler classes in the space of stable maps. The theory of the Gromov-Witten invariants was first established in a systematical and mathematical way by Ruan and the second author in [RT1], [RT2] for semi-positive symplectic manifolds. They actually constructed the invariants over integers. In [LT1] and [BF], the authors constructed the Gromov-Witten invariants for any algebraic manifolds over any closed fields of characteristic zero. The similar idea was also used by Liu and the second author in proving the Arnold conjecture for nondegenerate Hamiltonian functions on general symplectic manifolds [LiuT]. During the preparation of this paper, we learned that Fukaya and Ono also gave a different construction of the Gromov-Witten invariants and a proof of the Arnold conjecture for nondegenerate Hamiltonian functions for general symplectic manifolds ([FO]). We also learned that one or both of them was also claimed by Hofer and Salamon, and Ruan. Shortly after we finished writting of this paper, we received a preprint [Si] of B. Siebert, in which he gave another different construction of Gromov-Witten invariants for general symplectic manifolds. We believe that our construction can be also used to constructing the Gauge theory invariants, such as Donaldson invariants. It is also interesting to compare the Gromov-Witten invariants constructed here with algebraic ones in [LT1] (cf. [LT2]). We plan to discuss these in forthcoming papers. \tableofcontents \section{Euler classes for Fredholm bundles} In this section, we collect a few simple theorems, which can be proved easily. Let B be a topological space. Recall that a topological bundle $\pi : E \mapsto B$ consists of a continuous map $\pi$ between topological spaces, satisfying: (1) there is a topological subspace $Z$ in $E$ such that $\pi|_Z$ is a homeomorphism from $Z$ onto $B$; (2) For any $x\in B$, the fiber $E_x= \pi^{-1}(x)$ is a vector space with origin at $(\pi|_Z)^{-1}(x)$. A section of $E$ is a map $s: B \mapsto E$ such that $\pi \cdot s$ is the identity map of $B$. Clearly, $Z$ defines a section of $E$, which is usually refered as the 0-section. For any section $s$, its zero locus in $B$ is $s^{-1}(Z)$, which is also denoted by $s^{-1}(0)$. A smooth approximation $(E_i, U_i)$ of $s: B\mapsto E$ consists of an open subset $U_i$ in $B$ and a continuous vector subbundle $E_i$ of finite rank over $U_i$, such that $s^{-1}(E_i)\subset U_i$ is a smooth manifold and $E_i | _{ s^{-1}(E_i)}$ is a smooth bundle over $ s^{-1}(E_i)$ with $s|_{s^{-1}(E_i)}$ smooth . We say that $s: B \mapsto E$ has a weakly smooth structure $\{(E_i, U_i)\}$ of index $r$, if (i) each $(E_i, U_i)$ is a smooth approximation of $s:B \mapsto E$; (ii) $\{U_i\}$ is a covering of $ s^{-1}(0)$; (iii) $s^{-1}(E_i)\subset U_i$ is of dimension $r + {\rm rk }(E_i)$; (iv) For any $x\in s^{-1}(0)\cap U_i\cap U_j$, there is another smooth approximation $E_k \mapsto U_k$ with $x \in U_k$, such that $E_i|_{U_i\cap U_k}$ (resp. $E_j|_{U_j\cap U_k}$) is a subbundle of $E_k|_{U_i\cap U_k}$ (resp. $E_k|_{U_j\cap U_k}$). Given two smooth structures $\{(E_i,U_i)\}$ and $\{(E_j', U_j')\}$ of $s: B\mapsto E$, we say that $\{U_i\}$ is finer than $\{U_j'\}$, if for $x\in s^{-1}(0)\cap U'_j$, there is at least one $(E_i,U_i)$ such that near $x$, (1) $s^{-1}(E_i)\cap s^{-1}(E_j')$ is a smooth submanifold in $U_i$ of dimension $\dim s^{-1}(E_j')$; (2) $E_j'|_{U_i\cap U_j'}$ is a subbundle of $E_i|_{U_i\cap U_j'}$; (3) The restriction $E_j'|_{s^{-1}(E_i)\cap s^{-1}(E_j')}$ is a smooth subbundle of $E_i|_{s^{-1}(E_i)\cap s^{-1}(E_j')}$. We say that $E$ is a generalized Fredholm bundle of index $r$, if there is a continuous section $s: B \mapsto E $ satisfying the followings: \noindent (1) $s^{-1}(0)$ is compact; \noindent (2) $s: B\mapsto E$ has a weakly smooth structure $\{(E_i, U_i)\}$ of index $r$; \noindent (3) There is a finitely dimensional vector space $F$ and a bundle homomorphism $\psi _F: B\times F \mapsto E$, such that for any smooth approximation $(E_i, U_i)$, $\psi_F|_{s^{-1}(E_i)\times F}$ is a smooth map from $s^{-1}(E_i)\times F$ into $E_i$ and transverse to $s$ along $s^{-1}(0)\cap U_i$. Such a section $s$ is called admissible. We call $\{F, (E_i, U_i)\}$ a weakly smooth resolution of $s: B\mapsto E$. Put $W_F = (s-\psi_F)^{-1}(0)\subset B\times F$. Here by abusing the notation, we also regard $s$ as a section of $E$ over $B\times F$. Then $W_F$ is a smooth manifold of dimension $r + {\rm rk}(F)$ near $s^{-1}(0)$, and $s$ lifts to a smooth section $s_F: W_F \mapsto F$, namely, for any $(x, v)\in W_F\subset B\times F$, $s_F(x,v)= v $. Clearly, $s_F^{-1}(0)= s^{-1}(0)$. \begin{rem} One can also define the weakly $C^\ell$-smoothness of $s:B\mapsto E$. We say that $s:B \mapsto E$ is of class $C^\ell$ ($\ell \ge 1$) if any $E_i$ is a $C^\ell$-smooth bundle over a $C^\ell$-smooth manifold $s^{-1}(E_i)$, and $s$, $\psi_F$ are $C^\ell$-smooth along $s^{-1}(E_i)$. \end{rem} \begin{rem} If $\{F', (E_j', U_j')\}$ is another smooth resolution of $s$, we say that $\{F, (E_i, U_i)\}$ is finer than $\{F', (E_j', U_j')\}$, if $F' \subset F$, $\{(E_i, U_i)\}$ is finer than $\{(E_j', U_j')\}$ and $\psi_{F}$ restricts to $\psi_{F'}$ on $F'$. We will identify $\{F',(E_j', U_j')\}$ with $\{F, (E_i, U_i)\}$, if there is another smooth structure $\{F'', (E_k'', U_k'')\}$ finer than $\{F',(E_j', U_j')\}$ and $\{F, (E_i, U_i)\}$. Let $s, s' : B \mapsto E$ be two admissible sections. In the following, unless specified, by $s=s'$, we mean that they are the same as continuous sections and their weakly smooth structures are identical. \end{rem} We say that two generalized Fredholm bundles $s: B \mapsto E$ and $ s': B \mapsto E$ are homotopic to each other, if there is a generalized Fredholm bundle of the form $S: \pi^*_2 E \to [0,1] \times B$ of index $r+1$, such that $S |_{0 \times B} = s$ and $S |_{1 \times B} = s'$, where $\pi_2: [0,1]\times B \mapsto B$ is the natural projection. We denote by $[s : B \to E]$ the equivalence class of generalized Fredholm bundles which are homotopic to $s : B \to E$. We also denote by $r(B,E,s)$ the index of the generalized Fredholm bundle $s: B \mapsto E$. \begin{exam} Let $B$ be a Banach manifold (possibly incomplete). Suppose that $E$ is a vector bundle $E$ over $B$ with a section $s : B \to E$ satisfying: 1) $s^{-1} (0)$ is compact; 2) For any $x \in B$, $L_x (s)$ is Fredholm, where $L_x (s)$ denotes the linearization of $s$ at $x$ with $s(x) =0$, defined as follows: if $\phi : E|_W\mapsto W\times H$ is any local trivialization near $x$, then $$L_x(s) (v) = \phi^{-1} (v (\pi_2\cdot \phi \cdot s)(x)),$$ where $\pi_2$ be the projection from $W\times H$ onto $H$. Since $s(x) =0$, $L_x(s)(v)$ is independent of choices of local trivializations of $E$ near $x$. The index of $L_x (s)$ is independent of $x$ in $B$. Therefore, we can denote this index by $r (B, E, s)$. One can easily show that such a $s: B \mapsto E$ is a generalized Fredholm bundle of index $r(B, E, s)$. \end{exam} Let $s: B\mapsto E$ be a generalized Fredholm bundle. We can define its determinant bundle $\det (s)$ as follows: let $\{F, (E_i, U_i)\}$ be a smooth resolution of $s$, then we define $\det (s)$ to be $\det(TW_F)\otimes \det(F)^{-1}|_{s^{-1}(0)}$. For any smooth approximation $(E_i, U_i)$, $W_F $ is a smooth submanifold in $ U_i \times F$ and its normal bundle can be canonically identified with $E_i|_{W_F}$ by using the differential $d (s- \psi_F)$. It follows that $det(s) $ can be canonically identified with $\det (TU_i\times F)\otimes \det (E_i)^{-1} \otimes \det (F)^{-1}|_{s^{-1}(0)}$, and consequently, $\det (TU_i)\otimes \det (E_i)^{-1}|_{s^{-1}(0)}$. In particular, $\det(s)$ is independent of choices of $\{F, (E_i, U_i)\}$. Moreover, it implies that $\det (s)(x)$ ($x\in s^{-1}(0)$) can be naturally identified with $\det (L_x(s))$, where $L_x(s)$ denotes the linearization of $s$ from $\bigcup \{T_xU_i | x \in U_i\cap s^{-1}(0)\}$ into $E_x$ defined as in Example 1. The determinant $\det (L_x(s))$ is defined in the standard way by using finitely dimensional approximations. We say that $s: B \mapsto E$ is orientable if ${\rm det}(s)$ is orientable, i.e., it admits a nonvanishing section. Clearly, if $s: B\mapsto E$ is orientable, so is any other bundle in $[s:B\mapsto E]$, so we can simply say that $[s:B\mapsto E]$ is an orientable equivalence class. \begin{theorem} For each oriented equivalence class $[s: B\to E]$ of generalized Fredholm bundles, we can assign an oriented Euler class $e([s:B\to E])$ in $ H_r(B, Z)$, where $r=r(B,E,s)$. More precisely, $e([s : B \to E])$ can be represented by an $r$-dimensional manifold. Furthermore, this Euler class satisfies the usual functorial properties for the Euler class of bundles of finite rank. \end{theorem} \vskip 0.1in \noindent {\bf Proof:} First we observe that $s_F^{-1}(0) = s^{-1}(0)$ is compact. Then, by the standard transversality theorem, there is a generic, small section $v: W_F \mapsto F$, such that $(s_F+v)^{-1}(0)$ is a compact submanifold in $B\times F$ of dimension $r$. It has a natural orientation induced by $\det(s)$. We claim that the homology class of $(s_F + v)^{-1}(0)$ is independent of choices of smooth resolutions. Suppose that $\{F', (E_j', U_j')\}$ is another smooth resolution which is finer than $\{F, (E_i, U_i)\}$. Then we have another smooth manifold $W_{F'}$ containing $W_F$ as a submanifold. We may assume that $W_F = W_{F'} \cap B\times F$ and the above $v$ extends to a map from $W_{F'}$ into $F$. Let $F' = F\oplus F'/F$ be a splitting. Write $B\times F'$ as $B\times F\times F'/F$, for any $(x,u_1, u_2)\in W_{F'}\backslash W_{F}$, we have $u_2 \not= 0$. It follows that $(s_{F'} + v)^{-1}(0) = (s_F+v)^{-1}(0)$. Then the claim follows easily. We define $e([s: B\mapsto E])$ to be the homology class in $H_r(B, {\Bbb Z})$, which is represented by $(s_F + v)^{-1}(0)$ in $B\times F$. Here we have used the fact that $B$ is homotopically equivalent to $B\times F$. We can also regard $e([s:B \mapsto E])$ as the intersection class of $W_F$ with $B\times \{0\}$. The class $e([s:B \mapsto E])$ is independent of choices of Fredholm sections $s$ in $[s: B \mapsto E]$. In fact, to prove it, we simply repeat the above arguments for any homotopy $S: B\times [0,1] \mapsto E$ with $S|_0 = s$. One can show that $e([s: B \mapsto E])$ satisfies all functorial properties of the Euler class. So Theorem 1.1 is proved. \begin{rem} Theorem 1.1 still holds even if the assumption (3) on $s: B\mapsto E$ is replaced by \noindent (3)' there are finitely many open subsets $\{V_a\}$ and finitely dimensional vector spaces $F_a$ satisfying: (i) $B = \bigcup _a V_a$; (ii) For each $a$, there is a bundle map $\psi_a: V_a\times F_a \mapsto E|_{V_a}$ which is transverse to $s$ along $s^{-1}(0)$ for any smooth approximations, or equivalently, $(F_a, \psi_a)$ is a weakly smooth resolution of $s|_{V_a}$; (iii) If $\dim F_a \le \dim F_b$, we have that $F_a \subset F_b$ and $$\psi_b |_{(V_a \cap V_b )\times F_a} = \psi _a |_{(V_a \cap V_b )\times F_a}.$$ The proof is not much more difficult than the above one. We can also regard $\{V_a, F_a\}$ as a resolution of $s$. However, when the weakly smooth structure of $B$ admits appropriate cut-off functions, this weakened condition is the same as (3). \end{rem} The assumptions in Theorem 1.1 can be weaken, namely, we do not really need $B$ to be weakly smooth. The following is rather straightforward, if one treats orbifolds like manifolds. Let $B$ be a topological space. We recall that a topological fibration $\pi: E\mapsto B$ is an orbifold bundle, if there is a covering $\{ U_i\}$ of $B$ by open subsets, satisfying: (1) each $U_i$ is of the form $\tilde U_i / \Gamma_i$, where $\Gamma _i$ is a finite group acting on $\tilde U_i$; (2) for each $i$, there is a topological bundle $\tilde E_i \mapsto \tilde U_i$, such that $E|_{U_i}=\tilde E_i/ \Gamma_i$; (3) For any $i, j$, there is a bundle map $\Phi_{ij}$ from $\tilde E_j|_{\pi^{-1}_j(U_i\cap U_j)}$ to $\tilde E_i|_{\pi^{-1}_i(U_i\cap U_j)}$, which descends to the identity map of $E|_{U_i \cap U_j}$, where $\pi_k : \tilde U_k \mapsto U_k$ is the natural projection; (4) For each $x\in \pi_j^{-1}(U_i\cap U_j)$, there is a small neighborhood $U_x$, such that $\Phi_{ij}|_{\pi_j^{-1}(U_x)}$ is an isomorphism from each connected component of $\pi_j^{-1}(U_x)$ onto its image; (5) Each $\Phi_{ij}$ is compatible with actions of $\Gamma_i$ and $\Gamma_j$. Any such a $\pi_i : \tilde U_i\mapsto U_i$ is called a local uniformization of $B$. Note that $\Phi_{ij}\cdot \Phi_{ji}$ may not be an identity. It is only a covering map. We will denote by $\phi_{ij}$ the induced map from $\pi_j^{-1}(U_i\cap U_j)$ to $\pi_i^{-1}(U_i\cap U_j)$. An orbifold section $s: B\mapsto E$ is a continuous map such that for each $i$, $s|_{U_i}$ lifts to a section $s_i$ of $\tilde E_i$ over $\tilde U_i$. Similarly, we can define orbifold bundle homomorphisms, and the zero set $s^{-1}(0)$ in an obvious way. Let $\pi: E\mapsto B$ be a topological orbifold bundle. We say that $E$ is a generalized Fredholm orbifold bundle of index $r$, if there is an orbifold section $s:B\mapsto E$ satisfying: \noindent (1) $s^{-1}(0)$ is compact; \noindent (2) For each local uniformization $\pi_i: \tilde U_i \mapsto U_i$, $s_i$ has a weakly smooth structure of index $r$; \noindent (3) For any $i,j$, $\Phi_{ij}$ respects weakly smooth structures of $s_j: \tilde U_j \mapsto \tilde E_j$ and $s_i: \tilde U_i \mapsto \tilde E_i$; \noindent (4) For each $i$, there is a finitely dimensional vector space $F_i$, on which $\Gamma _i$ acts, and a $\Gamma_i$-equivariant bundle homomorphism $\psi_{i}:\tilde U_i \times F_i\mapsto \tilde E_i$, satisfying: (i) together with the weakly smooth structure, $F_i$ provides a weakly smooth resolution of $s_i$; (ii) For any pair $ i, j$, if $\dim F_j \le \dim F_i$, then there is an injective bundle homomorphism $\theta _{ij}: \pi_j^{-1} (U_i\cap U_j)\times F_j \mapsto \pi_i^{-1} (U_i\cap U_j)\times F_i$, such that $\tilde p _i\cdot \theta_{ij}= \phi_{ij} \cdot \tilde p_j$, where $\tilde p_i$ denotes the natural projection from $\tilde U_i \times F_i$ onto $\tilde U_i$, and $\psi_i \cdot \theta_{ij} = \Phi_{ij}\cdot \psi_j $ on $\pi_j^{-1}(U_i\cap U_j)\times F_j$; (iv) If $\dim F_k \le \dim F_j \le \dim F_i$, then $\theta _{ik}=\theta _{ij}\cdot \theta_{jk}$ over $\pi_k^{-1}(U_i\cap U_j)$; (v) For any $x$ in $U_i\cap U_j$, $\theta_{ij}$ is $\Gamma_x$-equivariant near $\pi^{-1}_j(x)$, where $\Gamma_x$ is the uniformization group of $B$ at $x$. We will also call $\{F_i, \psi_i\}$ a resolution of $s$. Clearly, all $\tilde U_i\times F_i / \Gamma_i$ can be glued together to obtain a topological space $V(F)$. There is a natural projection $p_F: V(F)\mapsto B$. In fact, $V(F)$ is a union of finitely many orbifold bundles, so we may call it an orbifold quasi-bundle. Also, all $\psi_i$ can be put together to form a map $\psi_F$ from $V(F) $ into $E$. Similarly, one can define notions of homotopy equivalence of generalized Fredholm orbifold bundles. One can also compare weakly smooth structures and resolutions of generalized Fredholm orbifold bundles in the same way as we did before. For any generalized Fredholm orbifold bundle $s: B \mapsto E$, we can also associate a determinant orbifold bundle, denoted by $\det (s)$, in the same way as we did before. We say that $s: B \mapsto E$ is orientable if $\det (s) $ does, i.e., $\det (s)$ admits a nonvanishing section. Note that the orientability of generalized Fredholm orbifold bundles is a homotopy invariant. Now we have the following generalization of Theorem 1.1. \begin{theorem} \label{theo:1.2} For each equivalence class $[s: B\to E]$ of generalized Fredholm orbifold bundles, we can assign an oriented Euler class $e([s:B\to E])$ in $H_r(B, {\Bbb Q} )$, where $r$ is the index. Furthermore, this Euler class satisfies the usual functorial properties for the Euler class of bundles of finite rank. \end{theorem} \vskip 0.1in \noindent {\bf Proof:} We will adopt the notations in the above definitions of generalized Fredholm orbifold bundles. As before, we denote by $W_F$ the zero set $(s- \psi_F)^{-1}(0)$. Here again, we regard $s$ as a section of $E$ over $B\times F$ in an obvious way. For each local uniformization $\pi_i : \tilde U_i \mapsto U_i$, $\tilde W_{i}= (s_i - \psi _{i})^{-1}(0)$ is smooth in $\tilde U_i \times F_i$ near $s_i^{-1}(0)$ and of dimension $r + {\rm rk}(F_i)$. Then $W_F$ is obtained by gluing together all $W_i=\tilde W_{i}/ \Gamma_i$. More precisely, for any $\dim F_j \le \dim F_i$, by the above (4), $\tilde W_j \cap \left ( \pi_j^{-1}(U_i) \times F_j\right )$ is locally embedded into $\tilde W_i$ by $\theta_{ij}$, consequently, we can identify $\left (\tilde W_j\cap \pi_j^{-1}(U_i)\times F_j\right )/\Gamma_j$ with a smooth suborbifold, simply denoted by $W_i\cap W_j$ if there is no confusion, in $W_i$. We also denote by $\pi_i$ the projection from $\tilde W_i$ onto $W_i$. Furthermore, $V(F)$ pulls back to an orbifold quasi-bundle $V_0(F)$ over $W_F$. More precisely, $V_0(F)= \bigcup _i V_i$, where each $V_i= \tilde V_i/\Gamma_i$ and $\tilde V_{i} = \tilde W_{i}\times F_i$. As above, if $\dim F_j \le \dim F_i$, $\theta _{ij}$ induces an injective bundle map $$h_{ij}: V_j|_{W_i\cap W_j} \mapsto V_i|_{W_i\cap W_j}.$$ We denote by $F_{ij}$ the orbifold subbundle $h_{ij} (V_j|_{W_i\cap W_j})$. It extends to an orbifold subbundle, still denoted by $F_{ij}$, of $V_i$ over a small neighborhood of $W_i\cap W_j$ in $W_i$. Let $\tilde F_{ij}$ be the lifting of $F_{ij}$ in $\tilde V_i$ over a small neighborhood of $\pi_i^{-1}(W_i\cap W_j)$ in $\tilde W_i$. We define a continuous section $s_F: W_F \mapsto V_0(F)$ as follows: for each $i$, $s_F|_{W_i}$ is descended from the section $$ (x, v) \in \tilde W_i \subset \tilde U_i \times F_i \mapsto v \in F_i.$$ Clearly, $s_F^{-1}(0) = s^{-1}(0)$. Moreover, by the definitions of $s_F$ and $W_F$, we may assume that for some small neighborhood $O$ of $s_F^{-1}(0)$ in $W_F$ and any $\dim F_j \le \dim F_i$, $F_{ij}$ is well defined over $O\cap W_i\cap p_F^{-1}(U_j)$ and $$\left ( s_F|_{W_i}\right )^{-1}(F_{ij}|_{O\cap W_i\cap p_F^{-1}(U_j)}) = \left (s_F|_{W_j}\right )^{-1} (V_j|_{W_i\cap W_j}).$$ To save the notations, we will simply identify $W_F$ with the 0-section in $V_0(F)$. Let $p: V_0(F) \mapsto W_F$ be the natural projection induced by $p_F$. Observe that it induces a homomorphism $\tau : H_*(V_0(F), {\Bbb Q}) \mapsto H_*(B,{\Bbb Q})$. Then we define the Euler class $e([s:B\mapsto E])$ to be $\tau ([W_F \cap G(s_F)])$. Here, $[W_F \cap G(s_F)]$ denotes the intersection class of $W_F$ with the graph $G(s_F)$ of $s_F$ in $V_0(F)$. Using the above properties of $s_F$ and standard arguments, one can show that such an intersection exists in $H_*(V_0(F), {\Bbb Q} )$. Note that $W_F$ and $V_0(F)$ are unions of finitely many orbifolds and $s^{-1}_F(0)$ is compact. For the reader's convenience, we will outline construction of this intersection class by constructing its rational cycle representative. Choose $U_i'$ such that $s^{-1}_F(0)\subset \bigcup _i U_i'$ and its closure $\overline U_i'$ is contained in $U_i$. Put $\tilde W_i' = \tilde W_i \cap \left (\pi_i^{-1}(U_i')\times F_i\right )$ and $W_i'= \tilde W_i'/ \Gamma_i$. Then $s_F^{-1}(0) \subset \bigcup _i W_i'= W_F'$. We also put $\tilde V_i'= \tilde W_i'\times F_i$ and $V_i'=\tilde V_i'/\Gamma_i$. In this proof, by a cocycle $Z'$ of degree $m$ in $p^{-1}(O)$, we mean a union of cycles $Z_i'\subset \overline V_i'\cap p^{-1}(O)$ with its boundary in $\partial \overline V_i'$ and of dimension $\dim F_i + m$, such that $Z_j'\cap \overline V_i'$ is embedded in $Z_i'$ whenever $\dim F_j \le \dim F_i$, and $Z_i'\cap p^{-1}(U) \subset F_{ij}$, where $U$ is some neighborhood of $O\cap \overline W_i' \cap p_F^{-1}(\overline U_j')$. For example, $W_F$ and $G(s_F)$ are cocycles of degree $r$. We say that two cocycles are homologous if they can be deformed to each other through a family of cocycles. Let $m_i$ be the order of the local uniformization group $\Gamma _i$ of $V_i$. Put $m = m_1\cdots m_\ell$. Then $m e([s:B\mapsto E])$ should be the intersection class of $m \,W_F$ with the graph $G(s_F)$ in $V_0(F)$. We will construct an oriented cocycle $Z$ in $p^{-1}(O)$, which is homologous to $m W_F$ in $p^{-1}(O)$, such that $Z$ is transverse to the graph of $s_F$ in $V_0(F)$. We will use the induction for this purpose. Without loss of generality, we may arrange $$\dim F_1 \le \dim F_2 \le \cdots \le \dim F_k \le \cdots.$$ By perturbing $\tilde W_1$ in $\tilde W_1 \times F_1$ and averaging over the action of $\Gamma_1$, we obtain a cycle $\tilde Z_1\subset \tilde V_1=\tilde W_1\times F_1$, satisfying: (i) $\tilde Z_1$ is $\Gamma_1$-invariant; (ii) $\tilde Z_1= m_1 \tilde W_1$ near $\partial \tilde V_1$; (iii) $\tilde Z_1$ is homologous to $m_1 \tilde W_1$ in $\pi_1^{-1}(O)$ with fixed boundary; (iv) $\tilde Z_1$ is transverse to $G(s_{F,1})$ in an neighborhood of $\overline {\tilde W_i'\times F_1}$, where $s_{F,i}$ be the induced section over $\tilde W_{i}$ by $s_F$. We extend $\tilde Z_1 /\Gamma_1$ to a cycle over $\left (\bigcup_{i\ge 2} O\cap W_i\cap p_F^{-1}(U_1)\right )\cup W_1$, such that $\tilde Z_1/\Gamma _1$ is contained in $F_{i1}$ over some neighborhood of $O\cap \overline { W_i'\cap p_F^{-1}(U_1')}$ and coincides with $m_1 W_F$ near each $\left (\partial (O\cap W_i\cap p_F^{-1}(U_1))\right )\cap W_i$. Then we can glue $\tilde Z_1/\Gamma_1$ and $m_1 W_F$ together to form a cocycle $Z_1$ in $p^{-1}(O)$. Now suppose that for $k \ge 1$, we have found cocycles $Z_i $ ($1\le i \le k$) in $p^{-1}(O)$ formed by glueing $m_i Z_{i-1}\backslash p^{-1}(W_i)$ and $\tilde Z_i/\Gamma_i$, where $\tilde Z_i \subset \tilde V_i = \tilde W_i\times F_i$, satisfying: \noindent (i) $\tilde Z_i$ is $\Gamma_i$-invariant and coincides with $\pi^{-1}_i(Z_{i-1})$, where $\pi_i: \tilde V_i \mapsto V_i$ is the projection, near $\partial \tilde V_i$ and $\pi_i^{-1}(\bigcup_{j< i} p^{-1}( W_j))$; \noindent (ii) $\tilde Z_i$ is transverse to the graph $G(s_{F,i})$ in an neighborhood of $\overline {\tilde W_i'\times F_i}$; \noindent (iii) $\tilde Z_i$ is homologous to $\pi_i^{-1}( Z_{i-1})$ in $\tilde V_i$ with boundary fixed. Furthermore, we may assume that for each $l > i$, $Z_i$ is contained in $ F_{li}$ over some neighborhood of $O\cap \overline {W_l'\cap U_i'}$. Now we construct $Z_{k+1}$. Observe that $p|_{Z_k}$ is a branched covering of $Z_k$ over $W_F$ of order $m_1\cdots m_k$. It follows that $\pi_{k+1}^{-1}(Z_k) \subset \tilde V_{k+1}$, counted with multiplicity, is a branched covering of $ W_{k+1}\subset W_F$ of order $m_1\cdots m_{k+1}$. Then, by the standard transversality theorem and the same arguments as we did for $Z_1$, we can have a $\Gamma_{k+1}$-invariant perturbation $\tilde Z_{k+1}$ of $\pi_{k+1}^{-1}(Z_k)$ inside $p^{-1}(O)$, such that all properties for $\tilde Z_i$ ($i\le k$) are satisfied for $\tilde Z_{k+1}$. We define $Z_{k+1}$ to be the glueing of $\tilde Z_{k+1}/ \Gamma_{k+1}$ with $m_{k+1} Z_k$ along the boundary. The method is the same as that in the construction of $Z_1$, so we omit it. The orientation of each $\tilde Z_i$ ($\ge 1$) is naturally induced by that of $\det (s)$, as we did before. Thus by induction, we have constructed an integral cocycle $Z$ in $V_0(F)$ of degree $r$, homologous to $m W_F$ as we wanted. Moreover, one can show that the intersection of $G(s_F)$ with $Z$ is a well-defined cycle in $V_0(F)$. We define $e([s: B\mapsto E])$ to be the homology class in $H_r(B, {\Bbb Q} )$, which is represented by $\frac{1}{m} Z\cap G(s_F)$. We remind the readers that $G(s_F)$ is the graph of $s_F$ in $V_0(F)$. This class is independent of choice of the admissible orbifold section $s$ in $[s: B \mapsto E]$. In fact, to prove it, we simply repeat the above arguments for any homotopy $S: [0,1] \times B \mapsto E$ with $S|_{0\times B} = s$. One can show that $e([s: B \mapsto E])$ satisfies all functorial properties of the Euler class. So Theorem 1.2 is proved. \vskip 0.1in \begin{rem} It is very important to know when $e([s: B \mapsto E])$ is an integer class. Let us stratify $B$ according to the local unformization group, namely, write $B$ as a disjoint union of $B_i$, such that the local unformization group is the same at any points of $B_i$. In fact, each $B_i$ consists of fixed points of local uniformization group of the same type. If $s_i=s|_{B_i}: B_i \mapsto E_i$ is a generalized Fredholm bundle of index $r_i < r$, where $E_i$ is the subbundle of $E$ which consists of fixed points of the local uniformization group, then in the above proof, one can show that $e([s:B\mapsto E])$ is in $H_*(B, {\Bbb Z})$. In the case of Gromov-Witten invariants for rational curves (cf. section 2, 3), if the target manifold is semi-positive, then the above assumptions hold. This explains why the Gromov-Witten invariants in [RT1], [RT2] are integer-valued. If the above assumptions do not hold, in order to get integral Euler classes, one has to use special properties of certain generalized Fredholm orbifold bundles which arise from concrete applications. \end{rem} We believe that Theorem 1.1, 1.2 can be generalized to other singular varieties. However, the resulting Euler class may lie in intersection homology groups. We end this section with a remark. Let us denote formally by $\infty_E$ the rank of $E$ and by $\infty _B$ the dimension of $B$. Then $r=\infty_B - \infty _E$. The Euler class we defined is Poincare dual to the Chern class $c_{\infty _E}(E)$ (formally) of $E$. A natural question is whether or not one can construct reasonable cycles, which are Poincare dual to the lower degree Chern classes $c_{\infty _E -i}(E)$ of $E$, at least under certain assumptions on $[s: B \to E]$. The compactness is the main problem. \section{Gromov-Witten invariants} \label{sec:2} Let $(V, \omega, J)$ be a compact symplectic manifold, where $\omega$ is a symplectic form and $J$ is a compatible almost complex structure, i.e., \begin{displaymath} J^2 = - i d \; , \qquad \omega (J u, J v) = \omega (u, v) \; , \quad \forall u, v \in T V \; . \end{displaymath} Then $g (u, v) = \omega (u, J v)$ defines a Riemannian metric on $V$. Without loss of generality, we may assume that $(V, \omega, J)$ is $C^\infty$-smooth. As usual, if $2g+k \ge 3$, we denote by ${\cal M}_{g,k}$ the moduli space of Riemann surfaces of genus $g$ and with $k$ marked points. Each point of ${\cal M}_{g,k}$ can be presented as \begin{displaymath} \left( \Sigma ; x_1, \ldots, x_k \right) \end{displaymath} where $\Sigma$ is a Riemann surface of genus $g$, $x_1, \ldots, x_k \in \Sigma$ are distinct. We identify $(\Sigma ; x_1, \ldots, x_k)$ with $(\Sigma' ; x'_1, \ldots, x'_k)$, if there is a biholomorphism $f : \Sigma \to \Sigma'$ carrying $x_i$ to $x_i '$. Therefore, ${\cal M}_{0, 3}$ consists of one point. Let $\overline{{\cal M}}_{g, k}$ be the Deligne-Mumford compactification of ${\cal M}_{g,k}$. Then $\overline{{\cal M}}_{g, k}$ consists of all genus $g$ stable curves with $k$ marked points. It is well-known that $\overline{{\cal M}}_{g, k}$ is a K\"{a}hler orbifold (cf. [Mu]). In this section, we will apply Theorem 1.2 to construct the GW-invariant \begin{displaymath} \Psi^V_{(A, g, k)} : H^* (V, {\Bbb Q})^k \times H^* (\overline{{\cal M}}_{g, k}, {\Bbb Q}) \to {\Bbb Q} \; . \end{displaymath} First let us introduce the notion of stable maps. \begin{defn} A stable $C^\ell$-map $(\ell \geq 0)$ with $k$ marked points is a tuple $(f, \Sigma; x_1, \ldots, x_k)$ satisfying: \renewcommand{\theenumi}{\arabic{enumi})} \begin{enumerate} \item \label{item:1} $\Sigma = \bigcup^m_{i = 1} \sum_i$ is a connected curve with normal crossings and $x_1, \ldots, x_k$ are distinct smooth points in $\Sigma$; \item \label{item:2} $f$ is continuous, and each restriction $f |_{\Sigma_i}$ lifts to a $C^\ell$-smooth map from the normalization $\tilde \Sigma_i$ into $V$; \item \label{item:3} If the homology class of $f |_{\Sigma_i}$ is zero in $H_2(V, {\Bbb Q} )$ and $\Sigma _i$ is a smooth rational curve, then $\Sigma_i$ contains at least three of $x_i, \ldots, x_k$ and those points in $\Sing (\Sigma)$, the latter denotes the singular set of $\Sigma$. \end{enumerate} \end{defn} This definition is inspired by the holomorphic stable maps in [KM]. We should also note that $2g+k$ may be less than $3$ in the above definition. Given $(f, \Sigma; x_1, \ldots, x_k)$ as above, let $\Aut (\Sigma; x_1, \ldots, x_k)$ be the automorphism group of $(\Sigma; x_1, \ldots, x_k)$. Note that if $2g + k \ge 3$, $(\Sigma; x_1, \ldots, x_k) \in \overline{M}_{g, k}$ if and only if $\Aut (\Sigma; x_1, \ldots, x_k)$ is finite. Let ${\rm Aut}(f,\Sigma;x_1, \cdots, x_k)$ be the group consisting of all $\sigma $ in $\Aut (\Sigma; x_1, \ldots, x_k)$ such that $f\cdot \sigma = f$. Clearly, if $(f,\Sigma;x_1, \cdots, x_k)$ is stable, then ${\rm Aut}(f,\Sigma;x_1, \cdots, x_k)$ is finite. We say that two stable maps $(f, \Sigma; x_1, \ldots, x_k)$ and $(f', \Sigma'; x'_1, \ldots, x'_k)$ are equivalent if there is a biholomorphism $\sigma : \Sigma \mapsto \Sigma'$ such that $\sigma (x_i) = x'_i$ $ (1 \leq i \leq k)$ and $f' = f \circ \sigma$. We will denote by $[f, \Sigma; x_1, \ldots, x_k]$, usually abbreviated as $[{\cal C}]$, the equivalence class of stable maps equivalent to $(f, \Sigma; x_1, \ldots, x_k)$. Note that in case $\Sigma = \Sigma '$, $\sigma$ is in $\Aut (\Sigma; x_1, \ldots, x_k)$. The genus of a stable map $(f, \Sigma; x_1, \ldots, x_k)$ is defined to the genus of $\Sigma$. Let $\overline {{\cal F}}_A^\ell (V, g, k) $ ($\ell \ge 0$) be the space of equivalence classes $[f, \Sigma; x_1, \ldots, x_k]$ of $C^\ell$ stable maps $ (f, \Sigma; x_1, \ldots, x_k)$ of genus $g$ and with total homology class $A$, which is represented by the image $f(\Sigma)$ in $V$. Clearly, $\overline {{\cal F}}_A^\ell (V, g, k)$ is contained in $\overline {{\cal F}}_A^{\ell'} (V, g, k)$, if $\ell > \ell'$. We will also denote $\overline {{\cal F}}_A^0 (V, g, k)$ by $\overline {{\cal F}}_A (V, g, k)$. For any sequence of stable $C^\ell$-maps $\{(f_i, \Sigma_i; x_{i1}, \cdots, x_{ik})\}$, we say that $(f_i, \Sigma_i; x_{i1}, \cdots, x_{ik})$ converges to $(f_\infty, \Sigma_\infty; x_{\infty 1}, \cdots, x_{\infty k})$ in $C^\ell$-topology, if there are (1) $(\Sigma_i; \{x_{ij}\})$ converges to $(\Sigma_\infty; \{x_{\infty j}\})$ as marked curves; (2) $f_i$ converges to $f_\infty$ in $C^0$-topology on $\Sigma_\infty$; (3) $f_i$ converges to $f_\infty$ in $C^\ell$-topology on any compact subset outside the singular set of $\Sigma_\infty$. Let ${\cal C}_i$ be any sequence of equivalence classes of $C^\ell$-stable maps. We say that $[{\cal C}_i]$ converges to $[{\cal C}_\infty]$, if there are ${\cal C}_i=(f_i, \Sigma_i; x_{i1}, \cdots, x_{ik})$ representing $[{\cal C}_i]$ and converging to a representative ${\cal C}_\infty = (f_\infty, \Sigma_\infty; x_{\infty 1}, \cdots, x_{\infty k})$ of $[{\cal C}_\infty]$ in $C^\ell$-topology. The topology of $\overline {{\cal F}}_A^\ell (V, g,k)$ is given by the sequencial convergence in the above sense. One can easly show that the homotopy class of $\overline {{\cal F}}_A^\ell (V, g, k)$ is independent of $\ell$. We define ${\cal F} _A(V,g,k)$ to be the set of all equivalence classes of stable maps with smooth domain. Put ${\cal F}_A^\ell(V,g,k) = {\cal F}_A(V,g,k)\cap \overline{{\cal F}}_A^\ell(V,g,k)$. \begin{rem} ${\cal F}_A^\ell(V,g,k)$ is basically a family of spaces of maps from Riemann surfaces into $V$. Its topology has been extensively studied in the literature of algebraic topology. Here, one can regard $\overline {\cal F}_A(V,g,k)$ as a partial compactification of ${\cal F}_A(V,g,k)$. This partial compactification seems to have more structures than the original space does. The authors do not know much study on it in the literature. We believe that it deserves more attention. \end{rem} If $2g + k \ge 3$, one can define a natural map $\pi_{g,k}$ from $\overline {{\cal F}}_A (V, g, k) $ onto $\overline {{\cal M}}_{g,k}$ as follows: \begin{displaymath} \pi_{g,k} ( f, \Sigma; x_1, \ldots, x_k) = {\rm Red} (\Sigma; x_1, \ldots, x_k) \end{displaymath} where ${\rm Red} (\Sigma; x_1, \ldots, x_k)$ is the stable reduction of $(\Sigma; x_1, \ldots, x_k)$, which is obtained by contracting all its non-stable irreducible components. Then, we have ${\cal F} _A (V,g,k) = \pi_{g,k}^{-1} ({\cal M} _{g,k})$, moreover, we can describe ${\cal F} _A^\ell (V,g,k)$ locally as follows: given any $[f,\Sigma;x_1,\cdots, x_k]$ in $\pi_{g,k}^{-1}({\cal M}_{g,k})$. Then the automorphism group $\Gamma = {\rm Aut}(\Sigma; x_1,\cdots, x_k)$ is finite. We denote by $\Gamma_0$ its subgroup consisting of automorphisms preserving $f$. Let $W_0$ be a small neighborhood of $(\Sigma;x_1,\cdots, x_k)$ in ${\cal M} _{g,k}$, and $p_{W_0}: \tilde W_0 \mapsto W_0$ be the local uniformization. Note that $\Gamma$ acts on $\tilde W_0$ and $W_0= \tilde W_0 / \Gamma $. One can show that $[f,\Sigma;x_1,\cdots,x_k]$ has a neighborhood of the form $\tilde W_0 \times U/\Gamma _0$, where $U$ is some open neighborhood of $0$ in the space $C^\ell (\Sigma, f^*TV)$ of $f^*TV$-valued $C^\ell$-smooth functions. Note that $\Gamma_0$ acts on $C^\ell(\Sigma, f^*TV)$ naturally. Therefore, ${\cal F}_A^\ell (V,g,k)$ is a Banach orbifold. Without much more difficulty, one can also show that ${\cal F}_A^\ell (V,g,k)$ is a Banach orbifold, even if $2g+k < 3$. However, it seems to be much harder to prove that $\overline {{\cal F}}_A^\ell (V, g,k)$ is smooth. Fortunately, we can avoid it by exploring its weakly smooth structure. Next we define a generalized bundle $E$ over $\overline {{\cal F}}_A^1(V, g,k)$. In the following, we will often denote by ${\cal C}$ a stable map $(f, \Sigma; x_1, \ldots, x_k)$, $f_{{\cal C}}$ the map $f$ and $\Sigma_{\cal C}$ the connected curve $\Sigma$. We define $\wedge ^{0,1}_{{\cal C}}$ as follows: if $\Sigma _{{\cal C}}$ is smooth, then $\wedge ^{0,1}_{{\cal C}}$ consists of all continuous sections $\nu$ in ${\rm Hom }(T\Sigma _{{\cal C}}, f_{{\cal C}}^*TV)$ with $\nu \cdot j_{\cal C} = - J \cdot \nu$, where $j_{\cal C}$ denotes the complex structure on $\Sigma _{\cal C}$. In other words, $\wedge ^{0,1}_{{\cal C}}$ consists of all $f_{{\cal C}}^*TV$-valued (0,1)-forms $\nu$ over $\Sigma _{{\cal C}}$. In general, $\wedge^{0,1}_{\cal C}$ consists of all $f_{{\cal C}}^*TV$-valued (0,1)-form $\nu$ over the normalization of $\Sigma_{\cal C}$, more precisely, if $\Sigma_{\cal C}$ has nodes $q_1, \cdots, q_s$, then $\wedge ^{0,1}_{{\cal C}}$ consists of all $f_{{\cal C}}^*TV$-valued (0,1)-form $\nu$ over ${\rm Reg}(\Sigma _{{\cal C}})$ of $\Sigma _{{\cal C}}$ satisfying: for each $i$, if $D_1$ and $D_2$ are the two local components of $\Sigma_{\cal C}$ near $q_i$, then $\nu |_{D_1}$, $\nu |_{D_2}$ can be extended continuously across $q_i$. Let ${\cal C} = (f, \Sigma; \{x_i\})$ and ${\cal C}'=(f', \Sigma'; \{x_i'\})$ be two equivalent stable maps, and $\sigma $ be the biholomorphism from $\Sigma'$ to $\Sigma$ such that $\sigma (x_i') = x_i$ and $f' = f\cdot \sigma$. For convenience, we sometimes denote ${\cal C}'$ by $\sigma^*({\cal C})$. One can show $$\wedge ^{0,1}_{{\cal C}'} = \sigma ^* \left (\wedge^{0,1}_{{\cal C}}\right ).$$ It follows that $\wedge ^{0,1}_{{\cal C}}$ descends to a space $E_{[{\cal C}]}$ over the equivalence class of ${\cal C}$. We put $E= \bigcup _{[{\cal C}]} E_{[{\cal C}]}$ and equip it with the continuous topology. Then $E$ is a topological fibration over $\overline {{\cal F}}_A^1(V,g,k)$. For simplicity, we will also use $E$ to denote the restriction of $E$ to $\overline {{\cal F}}_A^\ell (V, g,k)$ for any $\ell > 1$. There is a natural section $\Phi([{\cal C}]): \overline {{\cal F}}_A^1(V,g,k) \mapsto E$, i.e., the Cauchy-Riemann equation, defined as follows: for any $C^1$-smooth equivalence class $[{\cal C}]\in \overline {{\cal F}}_A^1(V,g,k) $, we define $\Phi ([{\cal C}])$ to be represented by $$df_{{\cal C}} + J \cdot df_{{\cal C}} \cdot j_{\cal C} \in E_{\cal C},$$ where $j_{\cal C}$ denotes the conformal structure of $\Sigma _{{\cal C}}$. Sometimes, by abusing the notations, we simply write $$\Phi({\cal C}) = df_{{\cal C}} + J \cdot df_{{\cal C}} \cdot j_{\cal C}.$$ Then we have \begin{proposition} For any $\ell \ge 2$, the section $\Phi : \overline {{\cal F}}_A^\ell (V, g, k) \mapsto E$ gives rise to a generalized Fredholm orbifold bundle with the natural orientation and of index $2c_1(V)(A)+2k + (2n-6)(1-g)$. \end{proposition} We will postpone the proof of proposition 2.2 to section 3. Let $\omega'$ be another symplectic form on $V$ and $J'$ be one of its compatible almost complex structure, Recall that $(\omega', J')$ is deformation equivalent to $(\omega, J)$, if there is a smooth family of symplectic forms $\omega _s$ and compatible almost complex structures $J_s$ ($0\le s\le 1$) such that $(\omega _0, J_0)=(\omega , J)$ and $(\omega _1, J_1)=(\omega' , J')$. \begin{proposition} Let $\Phi': \overline {{\cal F}}_A^\ell (V, g, k) \mapsto E$ be the admissible section induced by the Cauchy-Riemann equation of $J'$, where $(\omega', J')$ is given as above. Assume that $(\omega', J')$ is deformation equivalent to $(\omega, J)$. Then $\Phi'$ is homotopic to $\Phi$ as generalized Fredholm orbifold bundles. \end{proposition} The proof of this proposition is identical to that of Proposition 2.2. Using the last two propositions, we can construct symplectic invariants, particularly, the GW-invariants. In the following, if $2g + k < 3$, for convenience, we denote by $\overline {{\cal M}}_{g,k}$ the topological space of one point. Notice that for any $\ell > 0$, $\overline {{\cal F}}_A^\ell (V, g,k)$ is homotopically equivalent to $\overline {{\cal F}}_A(V, g, k)$. Then we can deduce from Theorem 1.2 \begin{theorem} Let $(V, \omega, J)$ be a compact symplectic manifold with compatible almost complex structure. Then for each $g, k$ and $A \in H_2 (V, {\Bbb Z})$, there is a symplectically invariant homomorphism $$\rho^V_{A,g,k} : H^*(\overline {{\cal M}}_{g,k}, {\Bbb Q}) \mapsto H_*({\overline {{\cal F}}}_A (V, g, k), {\Bbb Q}),$$ satisfying: for any $\alpha $, $\beta$ in $H^*(\overline {{\cal M}}_{g,k}, {\Bbb Q})$, $$\rho ^V_{A,g,k} (\alpha \cup \beta ) = \rho ^V_{A,g,k} (\alpha) / \pi_{g,k}^*\beta,$$ where $\pi_{g,k}:{\overline {{\cal F}}}_A (V, g, k) \mapsto \overline {{\cal M}}_{g,k}$ is defined as above. We usually write $\rho^V_{A,g,k} (1)$ as $e_A(V, g,k)$, which is a symplectically invariant class in $$H_{2c_1(V)(A) + 2k + (2n-6) (1-g)}({\overline {{\cal F}}}_A (V, g, k), {\Bbb Q}).$$ Furthermore, if $A=0$, then for any $\beta $ in $\overline {\cal M}_{g,k}$, we have that $\rho ^V_{A,g,k}(\beta)$ takes values in $\tau _*(H_*(\overline {\cal M}_{g,k} \times V, {\Bbb Q}))$, where $\tau : \overline {\cal M}_{g,k} \times V \mapsto \overline F_A(V, g,k)$ is the natural embedding of constant maps. \end{theorem} \vskip 0.1in \noindent {\bf Proof:} By Proposition 2.2, $\Phi: \overline {{\cal F}}_A (V, g, k)\mapsto E$ is a genaralized Fredholm orbifold bundle of index $r$, where $r=2c_1(V)(A) + 2k + (2n-6) (1-g)$. By Proposition 2.3, its homotopy class is independent of choices of $(\omega, J)$. It follows from Theorem 1.2 that there is an Euler class $e([\Phi: \overline {{\cal F}}_A (V, g, k)\mapsto E])$ in $H_{r}({\overline {{\cal F}}}_A (V, g, k), {\Bbb Q})$. Then $\rho ^V_{A,g,k}$ is obtained by taking slant product of this Euler class by cohomological classes in $H^*(\overline {{\cal M}}_{g,k}, {\Bbb Q})$. All the properties can be easily checked. \vskip 0.1in \begin{rem} We conjecture that the invariant $\rho^V_{g,k}$ is integer-valued, i.e., for any $\alpha $ in $H^r(\overline {{\cal M}}_{g,k}, {\Bbb Z})$, $\rho ^V_{g,k} (\alpha)$ is in $H_{2c_1(V)(A) + 2k + 2n (1-g)-r}({\overline {{\cal F}}}_A (V, g, k), {\Bbb Z})$. \end{rem} In order to define the GW-invariants, we observe that there is an evaluation map $$ \begin{array}{rl} ev : \overline {{\cal F}}_A (V, g, k) &\mapsto V^k, \\ ev(f, \Sigma; x_1, \ldots , x_k) &= (f(x_1), \ldots, f(x_k)),\\ \end{array} $$ then we can define the $GW$-invariants \begin{displaymath} \Psi^V_{(A, g, k)} : H^* (V, {\Bbb Q})^k \times H^* (\overline{M}_{g, k}, {\Bbb Q}) \to {\Bbb Q} \; , \end{displaymath} namely, for any $\alpha_1, \ldots, \alpha_k \in H^* (V, {\Bbb Q})$, $\beta \in H^* (\overline{M}_{g, k}, {\Bbb Q})$, \begin{displaymath} \Psi^V_{(A, g, k)} (\beta; \alpha_1, \ldots, \alpha_k) = \ev^* \left( \pi_1^* \alpha_1 \wedge \cdots \wedge \pi^*_k \alpha_k \right) (\rho ^V_{A,g,k}(\beta )) \end{displaymath} \begin{theorem} The $GW$-invariants $\Psi^V_{(A, g, k)}$ are symplectic invariants satisfying the composition law and the basic properties {\rm (cf. [KM], [RT2], [T], [Wi1])}. \end{theorem} \begin{rem} It is believed that $\Psi^V_{(A, g, k)}$ is also integer-valued. In fact, it is true for semi-positive symplectic manifolds (cf. [RT1], [RT2]). \end{rem} \begin{exam} Let $(V, \omega, J)$ be a symplectic manifold as above and $\omega$ be an integer class. Then for any holomorphic curve $C \subset V$, \begin{displaymath} \int_C \omega \geq 1 \end{displaymath} We say that a pseudo-holomorphic map $f : S^2 \to V$ is a line if $\int_{S^2} f^*\omega = 1$. Let $A$ be the homology class of lines, then the moduli space of lines is compact modulo automorphisms of $S^2$. On the other hand, $G = \Aut (S^2)$ acts naturally on $\Map_A (S^2, V)$ and the bundle $\wedge^{0, 1} (T V)$ over $\Map _A(S^2, V)$. Therefore, we have a Fredholm bundle $E$ over $B = \Map_A (S^2, V)_0 / G$, where $\Map_A (S^2, V)_0$ denotes the space of maps which are generically immersive. The Cauchy-Riemann equation descends to a section of $E \to B$. One can show that \begin{displaymath} \Psi^V_{(A, 0, 3)} (\alpha_1, \alpha_2, \alpha_3) = \Bigl(\ev \bigl( \pi^{-1} (e (B, E)) \bigr) \cap (\alpha^*_1 \times \alpha^*_2 \times \alpha^*_3) \Bigr) \hbox{~in~} V^3 \end{displaymath} where $\pi: \Map_A(S^2, V) \mapsto B$ is the natural projection, and $\alpha^*_1 \times \alpha^*_2 \times \alpha^*_3$ are Poincare duals of $\alpha_1, \alpha_2, \alpha_3$. Note that $e (B, E) \in H_r (B, {\Bbb Z})$ for $r = 2 (c_1 (V) \cdot A + n - 3)$. \end{exam} We end up this section with two basic decomposition properties of the symplectic invariant $\rho ^V_{A,g, k}$. Let $\sigma : \overline {\cal M}_{g_1, k_1 +1} \times \overline {\cal M}_{g_2, k_2 +1} \mapsto \overline {\cal M}_{g, k}$, where $g = g_1 + g_2$ and $k = k_1 + k_2$ with $2g_1+k_1\ge 2$, $2g_2+k_2\ge 2$, be the map by glueing the $k_1+1$-th marked point of the first factor to the first marked point of the second factor. We denote by $PD(\sigma)$ the Poincare dual of ${\rm Im}(\sigma)$. The composition law expresses $\rho ^V_{A,g,k} (PD(\sigma))$ in terms of $\rho ^V_{A_1,g_1, k_1+1}$ and $\rho ^V_{A_2,g_2, k_2 +1}$ with $A=A_1+A_2$. Given any decomposition $A= A_1 + A_2$, there is an natural map $$\begin{array}{rl} &p: \overline {\cal F}_{A_1}(V, g_1, k_1 +1)\times \overline {\cal F}_{A_2}(V, g_2, k_2 +1) \mapsto V\times V,\\ &p([h_1, \Sigma_1; x_{1}, \cdots, x_{k_1+1}], [h_2, \Sigma_2; y_1,\cdots, y_{k+2+1}]) = (h_1(x_{k_1+1}), h_2(y_1)).\\ \end{array} $$ Let $\Delta$ be the diagonal in $V\times V$. Then there is an obvious map $\pi$ from $p^{-1}(\Delta)$ onto $\pi_{g,k}^{-1}({\rm Im}(\sigma))$ by identifying $x_{k_1+1}$ with $y_1$. Clearly, $\rho ^V_{A,g,k} (PD(\sigma))$ can be regarded as a class in $H_*(\pi_{g,k}^{-1}({\rm Im}(\sigma)), {\Bbb Q})$. On the other hand, if $\{u_i\}$ is any basis of $H^*(V, {\Bbb Z})$ and $\{u^*_i\}$ is its dual basis, then we have a homology class $$\sum _i \rho ^V_{A_1, g_1, k_1+1}/ ev^* \pi_{k_1 +1}^* u_i \otimes \rho ^V_{g_2, k_2+1}/ ev^*\pi_1^* u_i^*$$ in $H_*(p^{-1}(\Delta), {\Bbb Q})$. The first composition law for $\rho^V_{A,g,k}$ is given by the equation: $$\rho ^V_{A,g,k} (PD(\sigma))= \pi_*\left (\sum _{A=A_1+A_2} \sum _i \rho ^V_{A_1, g_1, k_1+1}/ ev^* \pi_{k_1 +1}^* u_i \otimes \rho ^V_{A_2,g_2, k_2+1}/ ev^*\pi_1^* u_i^* \right ). $$ The second composition law for $\rho ^V_{A,g,k}$ arises from the map $\theta : \overline {\cal M}_{g-1, k+2} \mapsto \overline {\cal M}_{g, k}$, which is obtained by glueing the last two marked points, in a similar way. As above, we define $$\begin{array}{rl} &p: \overline {\cal F}_{A}(V, g-1, k +2) \mapsto V\times V,\\ &p([h, \Sigma; x_{1}, \cdots, x_{k+2}]) = (h(x_{k+1}), h(x_{k+2})).\\ \end{array}$$ We also have the resolution $\pi: p^{-1}(\Delta) \mapsto \pi_{g,k}^{-1}({\rm Im}(\theta))$. Then we have $$\rho ^V_{A,g, k}(PD(\theta)) = \pi_*\left (\sum _i\rho ^V_{A,g-1, k+2}/ \pi_{k+1}^*u_i \wedge \pi^*_{k+2} u_i^*\right ).$$ \section{The proof of Proposition 2.2 and 2.3} \label{sec:3} In the section, we prove Proposition 2.2 in details. The same arguments can be applied to proving Proposition 2.3. We will omit its proof except a few comments at the end of this section. Fix any $\ell \ge 2$. Let $(f, \Sigma; x_1,\ldots, x_k)$ be a stable $C^\ell$-map representing a point in $\overline {{\cal F}}_A^\ell (V, g, k) $. Since the structure of ${\cal F}_A^\ell (V,g,k)$ is clear (cf. section 2), we may assume that $[f, \Sigma; x_1,\ldots, x_k]$ is in $\overline {{\cal F}}_A^\ell (V, g, k) \backslash \overline {{\cal F}}_A^\ell (V, g, k) $. The components of $\Sigma$ can be grouped into two parts: the principal part and the bubbling part. The principal part consists of those components of genus bigger than zero and those rational components, which contain at least three of $x_1, \cdots, x_k$ and the points in ${\rm Sing} (\Sigma )$. Other non-stable rational components consist in the bubbling part. By adding one or two marked points to each bubbling component, we obtain a stable curve $(\Sigma; x_1, \cdots, x_k, z_1,\ldots, z_l)$ in $\overline{{\cal M}}_{g, k +l}$, where $z_1, \ldots, z_l$ are added points. Let $W$ be a small neighborhood of $(\Sigma; \{x_i\},\{ z_j\})$ in $\overline{{\cal M}}_{g, k + l}$, and $\tilde W$ be the uniformization of $W$. Then $W=\tilde W/ \Gamma$, where $\Gamma = {\rm Aut}(\Sigma; \{x_i\}, \{z_j\})$. If $2g + k \ge 3$, we can express $\tilde W$ as follows: by contracting the bubbling part, we obtain the stable reduction $(\Sigma'; y_1, \ldots, y_k)$ of $(\Sigma; x_1, \ldots , x_k)$. Let $W_0$ be a small neighborhood of $(\Sigma'; y_1, \ldots, y_k)$ in $\overline{{\cal M}}_{g,k}$. Let $\tilde W_0$ be the uniformization of $W_0$. Then $W_0=\tilde W_0/ \Gamma$ and $\tilde W = W\times _{W_0}\tilde W_0$. In particular, $\tilde W = W$ is smooth whenever $W_0$ is smooth. Let $\tilde {\cal U} $ be the universal family of curves over $\tilde W$. Clearly, $\tilde {\cal U}$ is smooth. We fix a metric $g$ on $\tilde {\cal U}$. For any two maps $h_1$, $h_2$ from fibers of ${\cal U}$ over $\tilde W$, we define the distance \begin{eqnarray*} d_{\tilde W}(h _1, h_2) = & \sup _{x\in {\rm Dom (h_1)}} \sup _{d_g (y, x) = d_g (x, {\rm Dom (h_2)})} d_V (h_1 (x), h_2(y))\\ +~&\sup _{y\in {\rm Dom (h_2)}} \sup _{d_g (x, y) = d_g (y, {\rm Dom (h_1)})} d_V (h_1 (x), h_2(y)),\\ \end{eqnarray*} where $d_g(\cdot, \cdot )$, $d_V(\cdot , \cdot )$ are distance functions of $g$ and $V$. Let $\Sigma_j$ be any non-stable component of $(\Sigma; x_1, \ldots, x_k)$, then by the definition, the homology class of $f(\Sigma_j)$ is nontrivial. It follows that there is at least one regular value for $f|_{\Sigma_j}$. Therefore, we may choose $z_1, \ldots , z_{l}$, such that for each $i$, $f^{-1} (f(z_i))$ consists of finitely many immersive points. Choose local hypersurfaces $H_1, \ldots, H_{l}$, such that each $H_i$ intersects ${\rm Im}(f)$ transversally at $f(z_i)$. Fix a small $\delta > 0$, we define $$ \begin{array}{rl} \Map _\delta (W) = &\{ (\tilde f, \tilde \Sigma; \{\tilde x_i\},\{ \tilde z_j\})~ |~ (\tilde \Sigma; \{\tilde x_i\}, \{\tilde z_j\}) \in \tilde W, d_{\tilde W} (\tilde f, f) < \delta,\\ &~~~\tilde f ~{\rm is }~C^0 ~{\rm on~}\tilde \Sigma {\rm ~and}~ C^\ell \end{array} $$ We will equip it with the topology: any sequence $(h_a, \Sigma_a; \{x_{ia}\}, \{ z_{ja}\} )$ converges to $(h_\infty, \Sigma_{\infty}; \{x_{i\infty}\}, \{ z_{j\infty}\})$, if $(\Sigma_a; \{x_{ia}\}, \{ z_{ja}\})$ converges to the stable curve $(\Sigma_{\infty}; \{x_{i\infty}\}, \{ z_{j \infty}\})$ in $\tilde W$, and $h_a$ converges to $h_\infty$ in $C^0$-topology everywhere and $C^\ell$-topology outside $\Sing (\Sigma_\infty)$. We denote by $\Sing (\tilde {\cal U})$ the union of singularities of the fibers of $\tilde {\cal U}$ over $\tilde W$. Let $K$ be any compact subset in $\tilde {\cal U} \backslash \Sing(\tilde {\cal U})$ of the form: there is a diffeomorphism $\psi _K: (K\cap \Sigma) \times \tilde W\mapsto K$ such that $\psi_K ((K\cap \Sigma )\times \{t\})$ lies in the fiber of $\tilde {\cal U}$ over $t$. Then we define $$\begin{array}{rl} &\Map _\delta (W, K)\\ =&\{ (\tilde f, \tilde \Sigma; \{\tilde x_i\}, \{\tilde z_j\} )\in \Map_\delta (W)~|~ ||(\tilde f \cdot \psi_K - f)|_{K\cap \Sigma \times \{0\}} ||_{C^\ell } < \delta \}.\\ \end{array} $$ Clearly, each $Map _\delta (W, K)$ is open in $Map _\delta (W)$. By forgetting added marked points, each point in $\Map _\delta (W)$ gives rise to a stable map ${\cal C}$ and consequently, an equivalence class $[{\cal C}]$ in $\overline {{\cal F}}_A^\ell (V, g, k)$. Let $p_W$ be such a projection map into $\overline {{\cal F}}_A^\ell (V, g, k)$. We denote by $\Map _\delta(W_0, K)$ the image of $\Map _\delta (W,K)$ in $\overline {{\cal F}}_A^\ell (V, g, k)$ under the projection $p_W$. Let ${\rm Aut}({\cal C})$ be the automorphism group of the stable map ${\cal C}$. It is a subgroup of $\Gamma$, so it is finite and acts on $\tilde {\cal U}$. Let us denote by $m({\cal C})$ its order. >From now on, $K$ always denotes a compact set in $\tilde {\cal U} \backslash \Sing (\tilde {\cal U})$ containing an open neighborhood of $\bigcup _j f^{-1}(f(z_j))$. Moreover, we may assume that $K$ is invariant under the action of ${\rm Aut}({\cal C})$. \begin{lemma} If $\delta > 0$ is sufficiently small, then the map $p_W|_{\Map _\delta (W,K)}$ is finite-to-one of the order $m({\cal C})$, and $\Map _\delta (W_0, K)$ is an open neighborhood of ${\cal C}$ in $\overline {{\cal F}}_A^\ell (V, g, k)$. Furthermore, there is a canonical action of ${\rm Aut}({\cal C})$ on $\Map_\delta (W,K)$ with the quotient $\Map_\delta (W_0, K)$. In particuar, if ${\cal C}$ has trivial automorphism group, then $p_W|_{\Map _\delta (W,K)}$ is actually one-to-one. \end{lemma} \vskip 0.1in \noindent {\bf Proof:} Suppose that $(h', \Sigma'; \{x_i'\}, \{z_j'\})$ and $(h'', \Sigma''; \{x_i''\}, \{z_j''\})$, which are close to ${\cal C}$, have the same image under the projection $p_W$. Then there is a biholomorphism $\sigma: \Sigma ' \mapsto \Sigma ''$, such that $h' = h''\cdot \sigma$ and $\sigma (x'_i) = x_i''$. Since $h', h''$ are close to $f$, $\sigma $ has to be close to an automorphism of ${\cal C}$. Since $h'(z_j'), h''(z_j'') \in H_j$ for $1\le j \le l$, we have that $\sigma (z'_j)$ and $z_j''$ are close to $f^{-1}(f(z_j))$. Since $f$ is transverse to $H_j$ for each $j$, $p_W$ is finite-to-one of order no more than $ m({\cal C})$. Let us construct the action of ${\rm Aut}({\cal C})$ on $\Map_\delta (W,K)$ with $\Map _\delta (W_0, K)$ as its quotient. In fact, let $\tau \in {\rm Aut}({\cal C})$ and ${\cal C}'=(h', \Sigma'; \{x_i'\}, \{z_j'\})$ in $\Map _\delta (W,K)$. If ${\cal C}'$ is very close to ${\cal C}$, then there is a unique sequence $\{z_{\tau j}\}$ in $\tau (\Sigma')$ such that $h'(\tau ^{-1}(z_{\tau j}) )\in H_j$ and $z_{\tau j}$ is very close to $z_j$. We put $$\tau_*({\cal C}') = (h'\cdot \tau^{-1}, \tau (\Sigma'); \{\tau(x_i')\}, \{ z_{\tau j}\}),$$ then $\tau_*({\cal C}) \in \Map _\delta (W,K)$. Clearly, if $\tau'$ is another one in ${\rm Aut}({\cal C})$, we have that $$(\tau\cdot \tau')_*({\cal C}') = \tau _* (\tau'_*({\cal C}')).$$ It follows that there is an natural action of ${\rm Aut}({\cal C})$ on $\Map _\delta (W,K)$. Clearly, the quotient is $\Map _\delta (W_0,K)$. It also follows that $p_W$ is of order $m({\cal C})$. If $(h, \Sigma'; x_1, \ldots, x_k)$ is a stable map very close to ${\cal C}$, then $h$ is immersive near $z_j$ and there are unique $z_j'$ in $\Sigma '$ near $z_j$, such that $h(z_j') \in H_j$. It follows that $(h, \Sigma'; \{x_i'\}, \{z_j'\})$ is in $\Map _\delta (W, K)$, so $\Map_\delta (W_0,K)$ is a neighborhood of $[{\cal C}]$ in ${\cal F}^\ell _A(V,g,k)$. The lemma is proved. \vskip 0.1in Recall that a $TV$-valued, $(0,1)$-form over the universal family $\tilde {\cal U}$ of curves is a continuous section $\nu$ in $Hom (\pi_1^*T\tilde {\cal U}, \pi^*_2TV)$ satisfying: $\nu \cdot j_{\tilde {\cal U}} = - J\cdot \nu$, where $j_{\tilde {\cal U}}$ denotes the complex structure on $\tilde {\cal U}$. We denote by $\Gamma ^{0,1}_\ell (\tilde {\cal U}, TV)$ the space of such $(0,1)$-forms, which are $C^\ell$ smooth and vanish near ${\rm Sing}(\tilde {\cal U})$. Note that $E|_{\Map_\delta (W_0, K)} \mapsto \Map _\delta (W_0, K)$ lifts to a topological bundle, denoted by $E|_{\Map_\delta (W, K)}$, or simply $E$ if no possible confusions, over $\Map _\delta (W,K)$. In order to prove Proposition 2.2, we need to show that each $E|_{\Map_\delta (W, K)}$ is a generalized Fredholm bundle over $\Map_\delta (W, K)$. Let $\Phi$ be defined by the Cauchy-Riemann equation in section 2. It lifts to a section, still denoted by $\Phi$, of $E$ over $ \Map _\delta (W)$, explicitely, $$\Phi (\tilde f, \tilde \Sigma; \{ \tilde x_i\}, \{\tilde z_j\}) = d\tilde f + J \cdot d\tilde f \cdot j_{\tilde \Sigma}. $$ Let $L_{\tilde f}$ be the linearization of $\Phi$ at $\tilde f$. Then, for any vector field $u$ over $\tilde f (\tilde \Sigma)$, $$L_{\tilde f}(u) = du + J(\tilde f) \cdot du \cdot j_{\tilde \sigma} + \nabla _u J \cdot d\tilde f \cdot j_{\tilde \Sigma}.$$ We denote by $r$ the distance function to the singular set $\Sing (\tilde {\cal U})$ with respect to $g$. For any smooth section $u \in \Gamma ^0(\tilde \Sigma , \tilde f^*TV)$, we define the norm $$||u||_{1,p} = \left ( \int _{\tilde \Sigma} (|u|^p + |\nabla u|^p ) d\mu \right )^{\frac{1}{p}} + \left ( \int _{\tilde \Sigma} r^{-\frac{2(p-2)}{p}} |\nabla u|^2 d\mu \right )^{\frac{1}{2}}, $$ where $p \ge 2$ and $\Gamma ^0(\tilde \Sigma , \tilde f^*TV)$ is the space of continuous sections of $\tilde f^*TV$ over $\tilde \Sigma $, and all norms, covariant derivatives are taken with respect to the metric $g|_{\tilde \Sigma}$. If $\tilde \Sigma$ has more than one components, then $u$ consists of continuous sections of components which have the same value at each node. Then we define $$ \begin{array}{rl} L^{1,p}(\tilde \Sigma, \tilde f^*TV)= \{ u \in \Gamma ^0(\tilde \Sigma , \tilde f^*TV) ~|~ ||u||_{1,p} < \infty \}. \end{array} $$ \begin{lemma} For any $p \ge 2$, there is a uniform constant $c(p)$ such that for any fiber $\tilde \Sigma$ of $\tilde {\cal U}$ over $\tilde W$, and any $u$ in $L^{1,p}(\tilde \Sigma, \tilde f^*TV)$, we have $$||u||_{C^0} \le c(p) ||u||_{1,p}.$$ \end{lemma} \vskip 0.1in \noindent {\bf Proof:} We observe that any small geodesic ball of $\tilde \Sigma$ is uniformly equivalent to an euclidean ball or the union of two euclidean annuli of the same size. Then the lemma follows from the standard Sobolev Embedding Theorem. \vskip 0.1in It follows that $L^{1,p} (\tilde \Sigma, \tilde f^*TV)$ is complete for $p > 2$. On the other hand, for any $v \in {\rm Hom} (T\tilde \Sigma, \tilde f^*TV)$, we define $$||v||_{p} = \left (\int _{\tilde \Sigma} |v|^p d\mu \right )^{\frac{1}{p}} + \left ( \int _{\tilde \Sigma} r ^{-\frac{2(p-2)}{p}} |v|^2 d\mu \right )^{\frac{1}{2}}, $$ where all norms and derivatives are taken with respect to $g|_{\tilde \Sigma}$, too. Then we put $$ L^p (\wedge ^{0,1}(\tilde f^*TV)) =\{ v \in {\rm Hom}(T\tilde \Sigma, \tilde f^*TV) ~|~J\cdot v = - v \cdot j_{\tilde \Sigma},~ ||v||_p < \infty\}. $$ For any $(\tilde f, \tilde \Sigma; \{ \tilde x_i\}, \{\tilde z_j\})$ in $\Map _\delta (W)$, $L_{\tilde f}$ maps the space $L^{1,p}(\tilde \Sigma, \tilde f^*TV)$ into $L^p(\wedge ^{0,1} (\tilde f^*TV))$. Let $L_{\tilde f}^* $ be its adjoint operator with respect to the $L^2$-inner product on $L^2(\wedge ^{0,1} (\tilde f^*TV))$, more explicitly, for any $\tilde f^*TV$-valued (0,1)-form $v$, $$L^*_{\tilde f} v = - e_1(v_1)- e_2(v_2) + B_{\tilde f}( v),$$ where $\{e_1, e_2\}$ is any orthonormal basis of $\tilde \Sigma$ with $j_{\tilde \Sigma} (e_1) = e_2$, $v_i = v (e_i)$ ($i= 1,2$) and $B_{\tilde f}( v)$ is an operator of order $0$, defined by $$2 g_V(u, B_{\tilde f} (v)) = g_V((\nabla _u J )e_2(\tilde f), v_1) - g_V((\nabla _u J)e_1(\tilde f), v_2)$$ for any $u \in L^{1,2} (\tilde \Sigma, \tilde f^*TV)$. We denote by ${\rm Coker}(L_{\tilde f})$ the space of all $v$ in $L^2(\wedge ^{0,1} (\tilde f^*TV))$ such that $L^*_{\tilde f} (v) = 0$. Then by the standard elliptic theory, it is a finite dimensional subspace in $L^p(\wedge ^{0,1} (\tilde f^*TV))$ for any $p$. \begin{lemma} For any $v \in L^p(\wedge ^{0,1} (\tilde f^*TV))$ ($p \ge 2$), there are $v_0 \in {\rm Coker}(L_{\tilde f})$ and $u\in L^{1,p}(\tilde \Sigma, \tilde f^* TV)$, such that $L_{\tilde f} u = v - v_0$. \end{lemma} \vskip 0.1in \noindent {\bf Proof:} By the definition, one can find $u\in L^{1,2}(\tilde \Sigma, \tilde f^* TV)$ and $v_0 \in {\rm Coker}(L_{\tilde f})$, such that $L_{\tilde f} u = v- v_0$. Then the lemma follows from the standard elliptic theory. \vskip 0.1in Let ${\cal C}$ be the fixed holomorphic, stable $C^\ell$-map, in particular, $\Phi ({\cal C}) =0$. For any $v \in \Gamma ^{0,1}_{\ell -1} ({\cal U}, TV)$, we define its restriction $v|_{\tilde {\cal C}}$ to a stable map $\tilde {\cal C}$ as follows: let $\tilde {\cal C} = (\tilde f, \tilde \Sigma; \{ \tilde x_i\}, \{\tilde z_j\})$, then for any $x\in \tilde \Sigma$, we define $$v|_{\tilde {\cal C}} (x) = v (x, \tilde f(x)).$$ Let $S$ be any finitely dimensional subspace in $\Gamma ^{0,1}_{\ell -1} (\tilde {\cal U}, TV)$ ($\ell \ge 2$). We define $$S|_{\cal C} = \{ v|_{\cal C} ~|~ v \in S\}.$$ Then we can define $E_S$ over $\Map _\delta (W, K)$ as follows: for any $\tilde {\cal C}$ in $\Map _\delta (W, K)$, $E_S |_{\tilde {\cal C}} = S|_{\tilde {\cal C}}$. Assume that $\dim S = \dim S|_{\cal C}$. Then if $K$ in $\tilde {\cal U} \backslash {\rm Sing}(\tilde {\cal U})$ is sufficiently large and $\delta$ is sufficiently small, $E_S$ is a bundle of rank $\dim S$ over $\Map _\delta (W, K )$. The following is the main technical result of this section. \begin{proposition} Let $S$ be as above. Suppose that its restriction $S|_{\cal C}$ to ${\cal C}$ is transverse to $L_{f_{\cal C}}$, i.e., if $v_1, \cdots, v_s$ span $S$, then $v_1|_{\cal C} ,\cdots, v_s|_{\cal C}$ and ${\rm Im }(L_{f_{\cal C}})$ generate $L^{1,p}(\wedge ^{0,1}f^*TV)$. Then by shrinking $W$ if necessary, if $\delta $ is sufficiently small and $K$ is sufficiently large, $\Phi ^{-1}(E_S)$ is a smooth submanifold, which contains ${\cal C}$, in $\Map_\delta (W, K)$ and of dimension $2 c_1(V)(A) + 2(n-3)(1-g)+ 2k + \dim S$. Moreover, $E_S \mapsto \Phi^{-1}(E_S)$ is a smooth bundle. \end{proposition} \begin{rem} Suppose that $W$ and $S$ are invariant under the natural action of ${\rm Aut}({\cal C})$. Clearly, there is an induced action of ${\rm Aut}({\cal C})$ on both $\Phi^{-1}(E_S)$ and the total space of the bundle $E_S$ over $\Phi^{-1}(E_S)$. \end{rem} Now let us prove Proposition 3.4. The tool is the Implicit Function Theorem. Let ${\cal C}$ be the stable map $(f, \Sigma; x_1, \cdots, x_k)$ in ${\rm Map }(W)$ as given in Proposition 3.4. We denote by $q_1, \cdots, q_s$ those nodes in $\Sigma$. Recall that $z_1, \cdots, z_l$ be the added points such that $f(z_i) \in H_i$. Fix an $\nu$ in $S$ such that its restriction to $\Sigma_{\cal C}$ is $0$. In fact, for proving Proposition 3.4, we suffice to take $\nu =0$. First we want to construct a family of approximated $(J, \nu_t)$-maps $\tilde f_t$ parametrized by $t \in \tilde W$, where $\nu_t=\nu |_{\Sigma_t}$. Note that a $(J, \nu_t)$-map is a smooth $\tilde f: \Sigma _t\mapsto V$ satisfying the inhomogeneous Cauchy-Riemann equation: $$\Phi(\tilde f)(y) = \nu (y, \tilde f(y)), ~~~~y \in \Sigma_t.$$ For any $q_i$ ($1\le i \le s$), by shrinking $\tilde W$ if necessary, we may choose coordinates $w_{i1}, w_{i2}$, as well as $t$ in $\tilde W$, near ${\cal C}$, such that the fiber $$(\Sigma _t; x_1(t), \cdots, x_k(t), z_1(t), \cdots, z_l(t))$$ of $\tilde {\cal U} $ over $t$ is locally given by the equation $$ w_{i1} w_{i2} = \epsilon _i (t),~~~|w_{i1} | < 1, ~|w_{i2} | < 1, $$ where $\epsilon _i$ is a $C^\infty$-smooth function of $t$. For any $y$ in $\Sigma _t$, if $|w_{i1} (y)| > L \sqrt {|\epsilon (t)|}$ or $|w_{i2} (y)| > L \sqrt {|\epsilon (t)|}$ for all $i$, where $L$ is a large number, then there is a unique $\pi_t(y) $ in $\Sigma = \Sigma _0$ such that $d_g(y, \pi_t(y)) = d_g(y, \Sigma)$. Note that if $y$ is not in the coordinate chart given by $w_{i1}, w_{i2}$, then we simply set $w_{i1}(y) = w_{i2}(y) = \infty$. The following lemma follows from straightforward computations. \begin{lemma} For any $k > 0$, there is a uniform constant $a_k$ such that for any $y$ in $\Sigma _t$ with either $|w_{i1} (y)| > L \sqrt {|\epsilon (t)|}$ or $|w_{i2} (y)| > L \sqrt {|\epsilon (t)|}$ for all $i$, $$|\nabla ^k (\pi_t - id |_{\Sigma_t})| (y) \le a_k \min _i\{ |t|, \frac{|\epsilon_i(t)|}{ d_g(y, q_i)^{k+1}}\},$$ where $\nabla$ denotes the covariant derivative with respect to $g$, and both $\pi_t$, $id$ are regarded as maps from $\Sigma_t$ into $\tilde {\cal U}$. \end{lemma} Let us introduce a complex structure $\tilde J$ on $\tilde {\cal U} \times V$ as follows: for any $u_1 \in T\tilde {\cal U} \subset T(\tilde {\cal U}\times V)$, $$\tilde J (u_1) = j_{\cal U} (u_1) + \nu (j_{\cal U}(u_1));$$ For any $u_2$ in $TV \subset T(\tilde {\cal U}\times V)$, we put $\tilde J (u_2) = J(u_2)$. Define $F: \Sigma \mapsto \tilde {\cal U}\times V$ by assigning $y$ in $\Sigma$ to $(y, f(y))$. We call $F$ the graph map of $f$. One can show that $F$ is $\tilde J$-holomorphic. In fact, for any given $\tilde f: \Sigma _t \mapsto V$, it is a $(J, \nu_t)$-map if and only if its graph map is $\tilde J$-holomorphic (cf. [Gr]). Put $p_i = F(q_i)$. Without loss of generality, we may assume that $$F(\{ w_{i1} w_{i2} = 0 ||w_{i1}| < 1, |w_{i2}| < 1\} )$$ is contained in a coordinate chart $(u_1, \cdots, u_{2N})$ of $\tilde {\cal U} \times V$ near $p_i$. We may further assume that $$ \begin{array}{rl} \tilde J ( \frac{\partial~}{\partial u _i} ) &= \frac{\partial~~}{\partial u_{N+i}} + {\cal O} (|u| ),~\\ \tilde J ( \frac{\partial~~}{\partial u_{N+i}} ) &= - \frac{\partial~}{\partial u_i} + {\cal O} (|u| ),\\ \end{array} $$ where $i = 1,2, \ldots, N$ and $ | u | = \sqrt{\sum_{i=1} ^{2n} |u_i |^2 }$. The curve $F(\Sigma)$ has two components near $p_i$, which intersect transversally there. Then by changing $u_1, \cdots , u_{2N}$ appropriately, we may assume that in complex coordinates $u_1 + \sqrt{-1} u_{N+1}$, ..., $u_N + \sqrt{-1} u_{2N}$, $$ F(w_{i1}, w_{i2}) = (w_{i1}, w_{i2} , 0 ,0,\cdots, 0 ) + {\cal O} ( | w_{1i} | ^2 + |w_{i2}|^2 ). $$ Using this same formula, one can easily extend $F$ to a neighborhood of $q_i$ in ${\cal U}$. \begin{lemma} Let $\pi_2 :\tilde {\cal U} \times V \mapsto V$ be the natural projection. Then there is a uniform constant $a$ such that for any $y$ in $\Sigma_t$ with $\frac{1}{ 2} \le |w_{i1}(y)| \le 1$ or $\frac{1}{2} \le |w_{i2}(y)| \le 1$, $$|\pi_2(F(y)) - f(\pi_t(y))|_{C^2}\le a |\epsilon _i (t)|.$$ \end{lemma} This lemma can be easily proved by straightforward computations. Let $\eta: {\Bbb R}^1 \mapsto {\Bbb R}^1 $ be a cut-off function satisfying: $\eta (x) = 0$ for $|x |\le 1$, $\eta (x) =1$ for $|x| > 2$, and $|\eta ^{(k)}(x) | \le 2^k$. We define $\tilde f_t(y)$, where $y \in \Sigma _t$, as follows: if either $|w_{i1} (y)| > 1$ or $|w_{i2} (y)| > 1$ for all $i$, put $\tilde f_t (y) = f(\pi_t(y))$; If for some $i$, $|w_{i1} (y)| < \frac{1}{2} $ and $|w_{i2} (y)| < \frac{1}{2}$, then we define $\tilde f_t(y) = \pi_2(F(y))$; If $\frac{1}{2} \le |w_{i1} (y)| \le 1$ or $\frac{1}{2} \le |w_{i2} (y)| \le 1$, we define $\tilde f_t(y)$ to be $${\rm exp}_{f(q_i)} \left ( \eta (2 d_g(y, q_i)) {\rm exp}_{f(q_i)}^{-1} f(\pi_t(y)) + (1 - \eta (2 d_g(y, q_i)) ) {\rm exp}_{f(q_i)}^{-1} \pi_2(F(y)) \right ). $$ Since $f$ is continuous at each $q_i$, $\tilde f_t$ is continuous. \begin{lemma} There is a uniform constant $a_f$ such that for any $0 \le k \le \ell$ and $y \in \Sigma _t $, $$ \begin{array}{rl} |\nabla ^k \tilde f_t|(y) &\le a_f \min \{|t|, \frac{|\epsilon(t)|}{d_g(y, q_i)^{k+1}}\},\\ |\nabla ^{k-1}(\Phi (\tilde f_t)- \nu(\cdot, \tilde f_t(\cdot)))|(y) &\le a_f \min _i\{ |t|, \frac{|\epsilon_i(t)|}{d_g(y, q_i)^{k}}\}.\\ \end{array} $$ In particular, $|\nabla \tilde f_t|$ is uniformly bounded. \end{lemma} \vskip 0.1in \noindent {\bf Proof:} By Lemma 3.5 and 3.6, we suffice to prove those estimates near a given node, say $q_i$. Assume that $|w_{i1}|(y), |w_{i2}|(y) < \frac{1}{2}$. Then $\tilde f_t(y) = \pi_2(F(y))$. Let us prove the second estimate. The proof for the first is identical. We omit it. We may assume that $|w_{i1}(y)| \ge |w_{i2}(y)|$. Let $J_0$ be the standard complex structure in the coordinate chart $\{u_1, \cdots u_{2N}\}$, i.e., $$J_0(\frac{\partial }{\partial u_i})=\frac{\partial }{\partial u_{N+i}}, J_0(\frac{\partial }{\partial u_{N+i} })= - \frac{\partial }{\partial u_i},$$ where $i = 1, \cdots, N$. Note that $F$ is holomorphic with respect to $J_0$. Then we can deduce $$ \begin{array}{rl} &\Phi (\tilde f_t)(y) - \nu (y, \tilde f_t(y))\\ = &d\pi_2 \cdot \left (dF + \tilde J \cdot dF\cdot j_{\Sigma_t} \right ) (w_{i1}(y), w_{i2}(y), 0, \cdots,0)\\ =& d\pi_2 \cdot (\tilde J - J_0) \cdot dF \cdot j_{\Sigma_t }(w_{i1}(y), w_{i2}(y), 0, \cdots,0)\\ \le & c |\nabla F| |\tilde J - J_0|(w_{i1}(y), w_{i2}(y), 0, \cdots,0),\\ \end{array} $$ where $c$ is some unfiorm constant. It follows that $$|\Phi (\tilde f_t ) (y)- \nu(y, \tilde f_t(y)| \le \frac{c |\epsilon (t)|} {d_g(y, q_i)}.$$ Similarly, one can deduce other cases of the second estimate from the above identity. \vskip 0.1in For any $t$ small, we denote by $g_t$ the induced metric on $\Sigma_t$ by $g$. Note that $r$ is the distance function from $\Sing (\tilde {\cal U})$ with respect to $g$. For any smooth section $u \in \Gamma ^0(\Sigma _t, \tilde f_t^*TV)$, we recall $$||u||_{1,p} = \left ( \int _{\Sigma_t} (|u|^p + |\nabla u|^p ) d\mu_t \right )^{\frac{1}{p}} + \left ( \int _{\Sigma_t} r^{-\frac{2(p-2)}{p}} |\nabla u|^2 d \mu_t \right )^{\frac{1}{2}}, $$ and $$ \begin{array}{rl} L^{1,p}(\Sigma _t, \tilde f_t^*TV)= \{ u \in \Gamma ^0(\Sigma _t, \tilde f_t^*TV) ~|~ ||u||_{1,p} < \infty \}, \end{array} $$ where $p \ge 2$ and $\Gamma ^0(\Sigma _t, \tilde f_t^*TV)$ is the space of continuous sections of $\tilde f_t ^*TV$ over $\Sigma _t$. If $\Sigma_t$ has more than one components, then $u$ consists of sections which are continuous over each of its components and have the same value at each node. We put $$L^{1,p}= \{ (u,t)~|~u \in L^{1,p}(\Sigma _t, \tilde f_t^*TV)\}. $$ It is a topological bundle over ${\cal U}$. On the other hand, for any $v \in {\rm Hom} (T\Sigma_t,\tilde f_t^*TV)$, we have $$||v||_{p} = \left (\int _{\Sigma_t} |v|^p d\mu_t \right )^{\frac{1}{p}} + \left ( \int _{\Sigma_t} r ^{-\frac{2(p-2)}{p}} |v|^2 d\mu_t \right )^{\frac{1}{2}}, $$ and $$ \begin{array}{rl} L^p (\wedge ^{0,1}(\tilde f_t^*TV)) =\{ v \in {\rm Hom}(T\Sigma_t, \tilde f^*TV) ~|~J\cdot v = - v \cdot j_{\Sigma_t},~ ||v||_p < \infty\}. \end{array} $$ As above, we put $L^p (\wedge ^{0,1}(TV))$ to be the union of all $L^p (\wedge ^{0,1}(\tilde f_t^*TV))$ with $t \in \tilde W$. It is another topological bundle over ${\cal U}$. Furthermore, if $C^\ell_0 (\tilde {\cal U}, TV)$ denotes the space of all $C^\ell$-smooth sections, which vanish near ${\rm Sing}(\tilde {\cal U})$, of $\pi^*_2TV$ over ${\cal U}\times V$, then there is an embedding of $C^\ell_0(\tilde {\cal U}, TV)$ into $L^{1,p}$, where $\pi_2: \tilde {\cal U}\times V \mapsto V$ is the natural projection. Similarly, there is an embedding of $\Gamma^{0,1}_{\ell -1}(\tilde {\cal U}, TV)$ into $L^p(\wedge ^{0,1}TV)$. Note that both $C^\ell _0(\tilde {\cal U}, TV)$ and $\Gamma^{0,1}_{\ell -1}(\tilde {\cal U}, TV)$ are bundles over $\tilde {\cal U}$. By straightforward computations, we can deduce from Lemma 3.7. \begin{lemma} For any $p > 2$, we have $$||\Phi (\tilde f_t )-\nu(\cdot, \tilde f_t(\cdot))||_p \le c |t|^{\frac{1}{2}},$$ where $c$ is a uniform constant. \end{lemma} Next we define a map from $L^{1,p}$ into $L^{p} (\wedge ^{0,1}(TV))$ as follows: for any $(u, t)$ in $L^{1,p}$, $$\Psi (u, t) = \Phi ( exp _{\tilde f_t} u ),$$ where $exp_{\tilde f_t} u$ denotes the function which takes value $exp _{\tilde f_t(x)} u(x)$ at $x$. Clearly, this map $\Psi$ is well-defined, and maps $C^\ell_0 (\tilde {\cal U}, TV)$ into $\Gamma^{0,1}_{\ell -1}(\tilde {\cal U}, TV)$. \begin{rem} Here we have used the fact that $(x,t) \mapsto (x,t, \tilde f_t(x))$ defines a smooth map from $\tilde {\cal U}$ into $\tilde {\cal U}\times V$. \end{rem} Now let us study the linearization $L_t = D_u\Psi$ of $\Psi$ at $(0, t)$: for any $u$, we have $$L_t(u) = du + J(\tilde f_t) \cdot du \cdot j_{\Sigma_t} + \nabla _u J (\tilde f_t) \cdot d \tilde f_t \cdot j_{\Sigma_t}. $$ First we want to establish uniform elliptic estimates for $L_t$. \begin{lemma} There is a uniform constant $c$ such that for any $(u, t)$ in $L^{1,p}$, we have $$||u||_{1,p} \le c ( ||L_t(u) ||_p + ||u||_{1,2}).$$ \end{lemma} \vskip 0.1in \noindent {\bf Proof:} We may assume that $p > 2$, otherwise, the lemma is trivially true. Without loss of generality, we may further assume that $r\le \frac{1}{2}$ if both $w_{i1}$ and $ w_{i2}$ are less than $ \frac{1}{2}$ for some $i$. Let $\eta$ be a cut-off function satisfying: $\eta(x) = 0$ for $|x| \le \frac{1}{4}$, $\eta(x) = 1$ for $|x| > \frac{1}{2}$, and $|\eta'| \le 2$. Put $\tilde u = \eta (r) u$. It vanishes whenever $|w_{i1}|, |w_{i2}| \le \frac{1}{2}$. Moreover, we have $$L_t(\tilde u) = \eta (r) L_t(u) + \eta'(r) \left ( u \,dr + (J(\tilde f_t) u )\,dr\cdot j_{\Sigma_t} \right ).$$ Since $\Sigma _t$ has uniformly bounded geometry in the region where $r \ge \frac{1}{4}$, we can apply the standard $L^p$-estimate for $1^{st}$-order elliptic operators and obtain $$||\tilde u||_{1,p} \le c (||L_t(\tilde u)||_p + ||\tilde u||_{1,2}).$$ Together with the previous identity, we deduce $$||\tilde u||_{1,p} \le c (||L_t(u)||_p + ||u||_p + ||u||_{1,2}).$$ Note that $c$ always denotes a uniform constant, which may depend on $p$. By the Sobolev inequality in dimension two ( $\dim \Sigma_t =2$), we have $||u||_p \le c ||u||_{1,2}$, hence, $$||\tilde u||_{1,p} \le c (||L_t(u)||_p + ||u||_{1,2}).$$ Therefore, we suffice to show that for each $i$, $$ \begin{array}{rl} \left ( \int _{|w_{i1}|, |w_{i2}| \le \frac{1}{2}} |\nabla u |^p d\mu_t \right )^{\frac{1}{p}} &\le c (||L_t(u)||_p + ||u||_{1,2}),\\ \left ( \int _{|w_{i1}|, |w_{i2}| \le \frac{1}{2}} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}}|\nabla u|^2d\mu_t \right )^{\frac{1}{2}} &\le c (||L_t(u)||_p + ||u||_{1,2}).\\ \end{array} $$ Let us first prove the second inequality. Without loss of generality, we assume that $\epsilon _i(t) \not= 0$. Write $w_{i1} = \rho e^{\sqrt{-1} \theta}$, then $w_{i2} = \frac{|\epsilon_i(t)|}{\rho} e^{\sqrt{-1} (\theta + \theta_0)}$, where $\epsilon _i(t) = |\epsilon_i(t)| e^{\sqrt{-1} \theta_0}$. Hence, $|w_{i1}|^2 + |w_{i2}|^2 = \rho ^2 + \frac{|\epsilon_i(t)|^2}{\rho^2}$. Moreover, $|w_{i1}|, |w_{i2}| \le 1$ whenever $|\epsilon_i(t)| \le \rho \le 1$. Put $u_i$ to be zero if either $\rho > 1$ or $\rho < |\epsilon_i(t)|$, and $ (1-\eta (\frac{r}{2})) u$ otherwise. In terms of $\rho$ and $\theta$, we have the following expression: $$L_t(u_i) (\frac{\partial }{\partial \rho}) = \frac{\partial u_i }{\partial \rho} + \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial u_i}{\partial \theta}\right ) +\frac{1}{\rho} (\nabla _{u_i} J )\frac{\partial \tilde f_t}{\partial \theta}.$$ It follows that $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |\frac{\partial u_i }{\partial \rho} + \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial u_i}{\partial \theta}\right )|^2d\mu_t\\ \le & c \left ( ||L_t(u_i)||_p^2 + \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |u_i|^2 |\nabla \tilde f_t|^2 d\mu_t \right )\\ \le & c \left (||L_t(u)||_p^2 + ||u||_{1,2}^2 + \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |u|^2 d\mu_t \right )\\ \end{array} $$ Notice that the integral $$\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p-1}} d\mu_t$$ is bounded by a constant depending only on $p$. However, by the Sobolev Embedding Theorem, we have $$\left (\int _{|w_{i1}|, |w_{i2}| \le 1} |u_i|^{2p} d\mu_t \right )^{\frac{1}{p}} \le c(p) ||u||_{1,2}^2.$$ It follows $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |\frac{\partial u_i }{\partial \rho} + \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial u_i}{\partial \theta}\right )|^2d\mu_t\\ \le & c ( ||L_t(u)||_p^2 + ||u||_{1,2}^2).\\ \end{array} $$ We have $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left ( |\frac{\partial u_i }{\partial \rho}|^2 + \frac{1}{\rho^2} |\frac{\partial u_i}{\partial \theta}|^2\right ) d\mu_t\\ =&\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left (|\frac{\partial u_i }{\partial \rho} + \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial u_i}{\partial \theta}\right )|^2\right .\\ &~~~~~~\left . - 2 \langle \frac{\partial u_i }{\partial \rho}, \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial u_i}{\partial \theta}\right )\rangle \right ) d\mu_t\\ \end{array} $$ Using integration by parts, we derive $$ \begin{array}{rl} &\left | \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \langle \frac{\partial u_i }{\partial \rho}, \frac{1}{\rho} J_0 \left ( \frac{\partial u_i}{\partial \theta}\right )\rangle d \mu_t \right | \\ \le & \frac{p-2}{p}\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} | \langle u_i- a(\rho), \frac{1}{\rho^2} J_0 \left ( \frac{\partial u_i}{\partial \theta}\right )\rangle | d \mu_t,\\ \end{array} $$ where $J_0 = J(q_i)$ and $a(\rho)$ is any function on $\rho$. Using the Poincare inequality on the unit circle and choosing $a(\rho)$ appropriately, we can show that the last integral is no bigger than $$\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \frac{1}{\rho^2} |\frac{\partial u_i}{\partial \theta}|^2 d \mu_t,$$ which is the same as the integral $$ \begin{array}{rl} &\frac{1}{2} \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}}\\ &~~\left ( |\frac{\partial u_i}{\partial \rho}- L_t(u_i)- \frac{1}{\rho} (\nabla _{u_i} J) \frac{\partial \tilde f}{\partial \theta}|^2+ \frac{1}{\rho^2} |\frac{\partial u_i}{\partial \theta}|^2 \right ) d \mu_t\\ \le & c\, (||u_i|_{1,2}^2 + ||L_tu_i||^2_p)\\ &~+ \left ( \frac{1}{2} + \frac{1}{4p} \right ) \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left ( |\frac{\partial u_i}{\partial \rho}) |^2+ \frac{1}{\rho^2} |\frac{\partial u_i}{\partial \theta}|^2 \right ) d \mu_t.\\ \end{array} $$ On the other hand, by the above arguments, one can also show that $$ \begin{array}{rl} &\left | \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \langle \frac{\partial u_i }{\partial \rho}, \frac{1}{\rho} (J-J_0) \left ( \frac{\partial u_i}{\partial \theta}\right )\rangle d \mu_t \right |\\ \le & c ||u_i||_{1, 2} + \frac{1}{2p} \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left ( |\frac{\partial u_i}{\partial \rho} |^2+ \frac{1}{\rho^2} |\frac{\partial u_i}{\partial \theta}|^2 \right ) d \mu_t .\\ \end{array} $$ Combining all the above inequalities, we can deduce the second inequality we wanted. To obtain the first from the second, we decompose the region $\{ |\epsilon_i(t)| \le \rho \le 1$ into subannuli $\{ \delta _j \le \rho \le \delta _{j-1}\}$, where $j=1, \cdots, m$, $\delta_0 =1$, $\delta_m = |\epsilon_i(t)|$ and $1\le \frac{\delta _{j-1}}{\delta_j} < 2$. On each subannulus $\{ \delta _j \le \rho \le \delta _{j-1}\}$, the scaled metric $\delta^{-2}_j g_t$ has bounded geometry, we can apply the standard $L^p$-estimate and obtain $$ \begin{array}{rl} & \int _{ \delta _j \le \rho \le \delta _{j-1}} |\nabla u_i|^p d\mu_t ~~~~~~~~~~~~~~~~\\ \le c& \left ( \int_{ \delta _j \le \rho \le \delta _{j-1}} |L_tu_i|^p d\mu_t + \left (\delta_j^{-\frac{2p-4}{p}} \int_{ \delta _j \le \rho \le \delta _{j-1}} |\nabla u_i|^2 d\mu_t \right )^{\frac{p}{2}}\right ).\\ \end{array} $$ Clearly, the first inequality we wanted follows by suming up these over $j$. The lemma is proved. \vskip 0.1in \begin{lemma} Let $S$ be as in Proposition 3.4 and $t$ sufficiently small. Then for any $p > 2$ and $v$ in $L^p (\wedge ^{0,1} \tilde f_t ^*TV)$, there are $u$ in $L^{1,p}(\tilde f_t^* TV)$ and $v_0$ in $S$, satisfying: $$\begin{array}{rl} & L_t u =v - v_0,\\ &\max \{||u||_{1,p}, ||v_0||_p\} \le c ||v||_p,\\ \end{array} $$ where $c$ is a uniform constant. \end{lemma} \noindent {\bf Proof:} First we prove that there are $u$, $v_0$ such that $L_t u = v -v_0$ for sufficiently small $t$. If not, we can find a sequence $\{t_j\}$ with $\lim t_j =0$ and $v_j$ in ${\rm Coker}(L_{t_j})$, such that each $v_j$ is perpendicular to $S$ with respect to the $L^2$-metric on $L^2 (\wedge ^{0,1} \tilde f_{t_j} ^*TV)$. Note that $v_j \in L^p (\wedge ^{0,1} \tilde f_{t_j} ^*TV)$ for any $p$. We normalize $||v_j||_p = 1$. By using standard elliptic estimates (cf. [GT]), one can easily show that $v_j$ converges to some $v_\infty $ in $L^p (\wedge ^{0,1} \tilde f_{\cal C} ^*TV)$ outside the singular set of $\Sigma _{\cal C}$. Clearly, $v_\infty$ is perpendicular to $S$ and $L_0^*v_\infty =0$, so by our assumptions, $v_\infty = 0$. It follows that for any compact subset $K' \subset {\cal U}$ with $K' \cap {\rm Sing}(\Sigma _{\cal C}) = \emptyset$, we have $$\int _{K' \cap \Sigma_{t_j}} |v_j|^p d\mu_{t_j} + \int _{K' \cap \Sigma_{t_j}} r ^{-\frac{2(p-2)}{p}} |v_j|^2 d\mu_{t_j} \mapsto 0, ~~{\rm as}~j\mapsto \infty. $$ Put $t=t_j$ for any fixed $j$. Let $\epsilon_i (t), w_{i1}, w_{i2}$ be as above, near some node $q_i$ of $ \Sigma _{\cal C}$. As before, without loss of generality, we assume that $\epsilon _i(t) \not= 0$. Write $w_{i1} = \rho e^{\sqrt{-1} \theta}$, then $w_{i2} = \frac{|\epsilon_i(t)|}{\rho} e^{\sqrt{-1} (\theta + \theta_0)}$, where $\epsilon _i(t) = |\epsilon_i(t)| e^{\sqrt{-1} \theta_0}$. Hence, $|w_{i1}|^2 + |w_{i2}|^2 = \rho ^2 + \frac{|\epsilon_i(t)|^2}{\rho^2}$. Moreover, $|w_{i1}|, |w_{i2}| \le 1$ whenever $|\epsilon_i(t)| \le \rho \le 1$. Using $L_{t_j}^* v_j = 0$, we have $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |\frac{\partial v_{j\rho } }{\partial \rho} + \frac{1}{\rho} \frac{\partial v_{j\theta}}{\partial \theta}|^2d\mu_t\\ \le & c \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |v_j|^2 d\mu_t,\\ \end{array} $$ where $v_{j\rho } = v_j (\frac{\partial }{\partial \rho })$ and $v_{j\theta } = v_j (\frac{\partial }{\partial \theta})$. Note that $$v_{j\rho } = J(\tilde f_t) v_{j \theta},~~~ v_{j\theta } = - J(\tilde f_t) v_{j \rho}.$$ It follows that $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |\frac{\partial v_{j\rho } }{\partial \rho} - \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial v_{j \rho }}{\partial \theta}\right )|^2d\mu_t\\ \le & c \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |v_j|^2 d\mu_t.\\ \end{array} $$ We have \begin{eqnarray*} \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left ( |\frac{\partial v_{j\rho} }{\partial \rho}|^2 + \frac{1}{\rho^2} |\frac{\partial v_{j\rho}}{\partial \theta}|^2\right ) d\mu_t\\ =\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left (|\frac{\partial v_{j\rho} }{\partial \rho} - \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial v_{j\rho}}{\partial \theta}\right )|^2\right .\\ \left . + 2 \langle \frac{\partial v_{j\rho} }{\partial \rho}, \frac{1}{\rho} J(\tilde f_t)\left ( \frac{\partial v_{j\rho}}{\partial \theta}\right )\rangle \right ) d\mu_t \end{eqnarray*} Using integration by parts, we derive $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \langle \frac{\partial v_{j\rho} }{\partial \rho}, \frac{1}{\rho} J_0 \left ( \frac{\partial v_{j\rho}}{\partial \theta}\right )\rangle d \mu_t \\ =&\int _{|w_{i1}|=1~{\rm or}~|w_{i2}|=1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \langle v_{j\rho} , \frac{1}{\rho} J_0 \left ( \frac{\partial v_{j\rho}}{\partial \theta}\right )\rangle d \mu_t\\ +& \frac{p-2}{p}\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \langle v_{j\rho}- a(\rho), \frac{1}{\rho^2} J_0 \left ( \frac{\partial v_{j\rho}}{\partial \theta}\right )\rangle d \mu_t,\\ \end{array} $$ where $J_0 = J(q_i)$ and $a(\rho)$ is any function on $\rho$. Using the Poincare inequality on the unit circle and choosing $a(\rho)$ appropriately, we can show that the last integral is no bigger than $$\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \frac{1}{\rho^2} |\frac{\partial v_{j\rho}}{\partial \theta}|^2 d \mu_t.$$ On the other hand, one may assume that for $j$ sufficiently large, $$ \begin{array}{rl} &\int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \langle \frac{\partial v_{j\rho} }{\partial \rho}, \frac{1}{\rho} (J-J_0) \left ( \frac{\partial v_{j\rho}}{\partial \theta}\right )\rangle d \mu_t \\ \le & \frac{1}{2p} \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} \left ( |\frac{\partial v_{j\rho}}{\partial \rho} |^2+ \frac{1}{\rho^2} |\frac{\partial v_{j\rho}}{\partial \theta}|^2 \right ) d \mu_t .\\ \end{array} $$ Combining all above estimates, we have $$\begin{array}{rl} &\lim _{j \to \infty} \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |\nabla v_{j}|^2 d \mu_t\\ =~&\lim_{j \to \infty} \int _{|w_{i1}|, |w_{i2}| \le 1} (|w_{i1}|^2 + |w_{i2}|^2)^{-\frac{p-2}{p}} |v_{j}|^2 d \mu_t\\ =~&0.\\ \end{array} $$ Then one can deduce from this that $\lim_{j\to \infty} ||v_j||_p=0$. A contradiction! Therefore, we have proved the first part. Let us prove the estimate by contradiction. Suppose that it is not true, then there are $u_i$ in $L^{1,p}(\tilde f_{t_i}^*TV)$ and $v_{0i}$ in $S$ satisfying: \noindent (1) $\max\{||u_i||_{1,p},||v_{0i}||_p\}=1$; \noindent (2) $u_i$ are perpedicular to ${\rm Ker}(\pi_S\cdot L_{t_i})$, where $\pi_S$ is the projection onto the orthogonal complement of $S$ in $L^{1,2}(\tilde f_{t_i}^*TV)$; \noindent (3) $\lim_{i\to \infty} ||L_{t_i}u_i + v_{0i}||_p =0$. We may choose $t_i$ such that $\lim _i t_i = t_\infty$ exists. By (1) and the Sobolev Embedding Theorem, we may assume that $u_i$ converges to $u_\infty$ in the $L^{1,2}$-norm. We may further assume that $v_{0i}$ converges to some $v_{0\infty}$. Note that $L_{f} u_\infty = v_{0\infty}$. If $v_{0\infty} \not= 0$, then $u_\infty \not= 0$. Then $u_\infty \in {\rm Ker}(\pi_S\cdot L_{f})$, which is impossible. Therefore, we have $v_{0\infty} =0$. This implies that $\lim ||u_i||_{1,p} = 1$. It follows from Lemma 3.9 that $||u_i||_{1,2}$ is uniformly bounded away from zero. Then one can show that $u_\infty$ is in ${\rm Ker}(\pi_S\cdot L_f)$, a contradiction! The lemma is proved. \vskip 0.1in Let $P$ be a finitely dimensional subspace in $C^\ell_0(\tilde {\cal U}, TV)$. Then for any map $\tilde f: \tilde \Sigma \mapsto V$, where $\tilde \Sigma$ is a fiber of $\tilde {\cal U}$ over $\tilde W$, we define $u|_{\tilde f}$ by $$u|_{\tilde f}(x) = u(x, \tilde f(x)), ~~{\rm for ~any~} x\in \tilde \Sigma,$$ and $$P_{\tilde f} ~=~\{ u|_{\tilde f} ~|~ u \in P\}.$$ We assume that $\dim P = \dim P_f$ and $q_S({\rm Ker}(\pi_S\cdot L_0))= P_f$, where $\pi_S$ is defined in the proof of Lemma 3.10 and $q_S: L^{1,2}(\tilde \Sigma, \tilde f^*TV) \mapsto P_{\tilde f}$ is the projection with respect to the $L^2$-inner product. One can easily deduce from the above lemma the following. \begin{lemma} Let $P$ and $S$ be as above and $t$ be sufficiently small. Then for any $p > 2$, $u_0 \in P_{\tilde f_t}$ and $v$ in $L^p (\wedge ^{0,1} \tilde f_t ^*TV)$, there are unique $u$ in $L^{1,p}(\tilde f_t^* TV)$ and $v_0$ in $S$, satisfying: $$\begin{array}{rl} &q_S(u) =u_0,~~ L_t (u ) = v - v_0,\\ &\max \{||u||_{1,p}, ||v_0||_p\} \le c \max \{||u_0||_{1,p}, ||v||_p\},\\ \end{array} $$ where $c$ is a uniform constant. \end{lemma} \vskip 0.1in \noindent {\bf Proof of Proposition 3.4:} We have the following expansion: $$\Psi (u, t) = \Psi (0, t) + L_t u + H_t (u), $$ where $H_t(u)$ is the term of higher order satisfying: $||H_t(u)||_p \le c ||u||_{C^0} ||u||_{1,p}$ for some uniform constant $c$, which depends only on the derivatives of $J$. By the Sobolev Embedding Theorem, it follows $$||H_t(u)||_p \le c ||u||_{1,p}^2.$$ Also note that $\Psi(0, t) = \Phi (\tilde f_t)$. Consider the map $\Xi: L^{1,p}\times E_S \mapsto L^p(\wedge ^{0,1}TV)\times E_P$, defined by $$\Xi(u,t,v_0) = (\Psi(u,t) + v_0, q_S(u)).$$ Note that $E_P$ is the bundle induced by $P$ over $\tilde W$ with fibers $P_{\tilde f_t}$. The linearization of $\Xi$ at $(0,t,0)$ is the map $$\begin{array}{rl} D\Xi : L^{1,p}(\Sigma_t,\tilde f_t^*TV)\times S_{\tilde f_t} &\mapsto ~L^p(\wedge ^{0,1}\tilde f_t^*TV)\times P_{\tilde f_t},\\ (u,v_0) ~~~&\mapsto ~~(L_t(u) + v_0 , q_S(u)).\\ \end{array} $$ By Lemma 3.11, it is an isomorphism with uniformly bounded inverse. Therefore, by Lemma 3.8 and the Implicit Function Theorem, there is an $\epsilon _0> 0$ such that for any $(0,u_0)\in L^p(\wedge ^{0,1}\tilde f_t^*TV)\times P_{\tilde f_t}$ with $||u_0||_{1,p}< \epsilon_0$ and $d_{\tilde W}(t, 0)< \epsilon_0$, there is a unique $(u,t,v_0)$ satisfying: $$\begin{array}{rl} &\Xi(u,t,v_0)= (0, u_0),~~~\\ &\max \{||u||_{1,p}, ||v_0||_p\} \le c ||u_0||_{1,p},\\ \end{array} $$ where $c$ is some uniform constant. It follows that if $W$ is sufficiently small, the subset $$\{(u,t)\in L^{1,p} | \pi_S\cdot \Psi(u,t)=0, ||u||_{1,p}< \epsilon_0\}$$ is parametrized by $u_0$ in $P$ and $t\in \tilde W$. In particular, it is a smooth manifold of dimension $\dim S + 2c_1(V)(A) +2n(1-g)+2k+2l$. Note that by our choice of $P$, we have $$\dim P = \dim S + 2c_1(V)(A)+ 2n(1-g).$$ We define $Y_{\epsilon_0}(S,W)$ to be $$\{(u,t)\in L^{1,p} | \pi_S\cdot \Psi(u,t)=0, ||u||_{1,p}< \epsilon_0, {\rm exp}_{\tilde f_t(z_j)} u(z_j) \in H_j \},$$ where $z_j$ ($1\le j\le l$) are added points given at the beginning of this section. Then $Y_{\epsilon_0}(S,W)$ is a smooth manifold of dimension $$\dim S + 2c_1(V)(A) + 2n(1-g)+2k.$$ We claim that for $\delta$ sufficiently small and $K$ is sufficiently large, $\Phi^{-1}(E_S)$ is an open set in $Y_{\epsilon_0}(S,W)$. Let $(\tilde f, \tilde \Sigma; \{\tilde x_i\}, \{\tilde z_j\})$ be in $\Phi^{-1}(E_S)$. We denote by $t$ the corresponding point $(\tilde \Sigma; \{\tilde x_i\}, \{\tilde z_j\})$ in $\tilde W$. Using the fact that $d(\tilde f, f) \le \delta$, we can write $\tilde f(x)= {\rm exp}_{\tilde f_t(x)}u(x)$ for some $\tilde f_t^*TV$-valued function $u$. We suffice to show that $u \in L^{1,p}(\Sigma_t, \tilde f_t^*TV)$ and $||u||_{1,p}< \epsilon_0$. It follows from the following lemma. \begin{lemma} For any $p>2$, there is a uniform constant $c$ such that $$\int_{\tilde \Sigma} r^{\frac{2(p-2)}{p}} |u|^p d\mu \le c ||u||_{C^0(K)},$$ where $r$ is the distance function from the set of nodes as we used before. \end{lemma} Write $v_0=\Phi(\tilde f)\in S$. Then $||v_0||_{C^0(K)}\le c \delta$ for some uniform constant $c$. By our choice of $S$, it follows that if $\delta$ is sufficiently small, $||v_0||_{C^1}<<\epsilon_0$. Then Lemma 3.12 can be proved by asymptotic analyses near nodes of $\Sigma$ or the arguments in the proof of Lemma 3.9. Finally, by differentiating $\pi_S\cdot \Phi (\tilde f)=0$ on $t$ and using Lemma 3.11, one can show that $E_S \mapsto \Phi^{-1}(E_S)$ is a smooth bundle and $\Phi|_{\Phi^{-1}(E_S)}$ is a smooth section. This is essentially the smooth dependence of solutions, which are produced by the Implicit Function Theorem, on parameters. Proposition 3.4 is proved. \vskip 0.1in \noindent {\bf Proof of Proposition 2.2:} We first need to construct a covering of $\Phi^{-1}(0)$ by open subsets, which will be parametrized by $[{\cal C} ]= [f, \Sigma; \{x_i\}] \in \Phi^{-1}(0)$, a small number $\delta > 0$, an neighborhood $W_0$ of the stable reduction ${\rm Red}(\Sigma; \{x_i\})$ of $(\Sigma; \{x_i\})$ in $\overline {{\cal M}}_{g,k}$, a compact subset $K$ in the universal family $\tilde {\cal U}$ of curves over $\tilde W$. Here $W$, $\tilde W$ are given as before. We define $$U_\delta ([{\cal C}], W_0, K ) = \Map_\delta (W_0, K),$$ where $\Map_\delta (W_0, K)$ is given in Lemma 3.1. Each $U_\delta ([{\cal C}], W_0, K)$ is of the form $\Map_\delta (W, K) / \Gamma $, where $\Gamma = {\rm Aut}({\cal C})$ and $\Map_\delta (W, K)$ were given as in Lemma 3.1. We put $$\tilde U_\delta ([{\cal C}], W_0, K ) = \Map_\delta (W, K).$$ It is the uniformization of $U_\delta ([{\cal C}], W_0, K)$. Therefore, we have shown that $\overline {{\cal F}}^\ell _A (V, g,k)$ is a topological orbifold. Let $E$ be the space of $TV$-valued $(0,1)$-forms defined in section 2. For each $U_\delta ([{\cal C}], W_0, K)$, as we have already seen, $E$ can be lifted to a topological bundle $E|_{\tilde U_\delta ([{\cal C}], W_0, K)}$ over $\tilde U_\delta ([{\cal C}], W_0, K)$. For the reader's convenience, we recall briefly the definition of this lifted bundle: for any $\tilde {\cal C} = (\tilde f, \tilde \Sigma ;\{\tilde x_i\}, \{\tilde z_j\})$ in $\tilde U_\delta ([{\cal C}], W_0, K)$, the fiber of $E|_{\tilde U_\delta ([{\cal C}], W_0, K)}$ at $\tilde {\cal C}$ consists of all $C^{\ell - 1}$-smooth, $\tilde f^*TV$-valued $(0,1)$-forms on $\tilde \Sigma$. When one passes from $\tilde U_\delta ([{\cal C}], W_0, K)$ to another local uniformization $\tilde U_{\delta'} ([{\cal C}'],W_0',K')$, there is an obvious bundle transition map, which lifts the identity map on $E$, from $E|_{\tilde U_\delta ([{\cal C}], W_0, K)}$ into $E|_{\tilde U_{\delta'} ([{\cal C}'], W_0', K')}$. Moreover, those transition maps satisfy all properties listed in section 1. Therefore, we have a topological orbifold bundle $E$ over $\overline {{\cal F}}^\ell _A (V, g,k)$, which is locally described by those $E|_{\tilde U_\delta ([{\cal C}], W_0, K)}$. The Cauchy-Riemann operator $\Phi$ can be canonically lifted to each local uniformization $\tilde U_\delta ([{\cal C}], W_0, K)$. Now let us check that $\Phi: \overline {{\cal F}}^\ell _A (V, g,k)\mapsto E$ satisfy all properties (1) - (4) in the definition of generalized Fredholm orbifold bundles. All those $U_\delta([{\cal C}], W_0, K)$ cover the moduli space $\Phi^{-1}(0)$ in $\overline {{\cal F}}^\ell _A (V, g,k)$. By the Gromov Compactness Theorem (cf. [Gr], [PW], [Ye], and also [RT1], Proposition 3.1), $\Phi^{-1} (0)$ is compact in $\overline {{\cal F}}_A^\ell (V,g,k)$ ($\ell \ge 2$). For any $S \subset \Gamma^{0,1}({\cal U}, TV)$ with properties stated in Proposition 3.4, we can define a bundle $E_S$ of finite rank as before, where $W_0$, $\delta$ are small and $K$ is big. Moreover, we assume that $S$ is invariant under the action of ${\rm Aut}({\cal C})$. By Proposition 3.4, $(E_S, \Phi^{-1}(E_S))$ is a smooth approximation of $\tilde U_\delta ([{\cal C}], W_0, K)$. Furthermore, $(E_S, \Phi^{-1}(E_S))$ is invariant under the action of ${\rm Aut}({\cal C})$. We denote such a smooth approximation by $$(\tilde E_{\delta, S} ([{\cal C}], W_0, K), \tilde X_{\delta, S} ([{\cal C}], W_0, K)).$$ One can easily show that all the smooth approximations of the form $$(\tilde E_{\delta, S} ([{\cal C}], W_0, K), \tilde X_{\delta, S} ([{\cal C}], W_0, K))$$ are compatible with above transition maps between local uniformizations $\{\tilde U_\delta([{\cal C}], W,K)\}$. Therefore, $\Phi:\overline {{\cal F}}^\ell _A (V, g,k) \mapsto E$ is weakly smooth. Its index can be computed by the Atiyah-Singer Index Theorem and is equal to $2c_1(V)(A) + (2n-3)(1-g) + 2k$. We put $$\begin{array}{rl} &E_{\delta, S} ([{\cal C}], W_0, K)= \tilde E_{\delta, S} ([{\cal C}], W_0, K) / \Gamma,\\ &X_{\delta, S} ([{\cal C}], W_0, K) = \tilde X_{\delta, S} ([{\cal C}], W_0, K)/ \Gamma ,\\ \end{array} $$ where $\Gamma = {\rm Aut}({\cal C})$. We claim that $\Phi^{-1}(0)$ can be covered by finitely many smooth approximations of the form $X_{\delta, S} ([{\cal C}], W_0, K)$. This is the same as saying that for each $[{\cal C}]$ in $\Phi ^{-1}(0)$, there is a small neighborhood $U$ such that for some smooth approximation $X_{\delta, S} ([{\cal C}], W_0, K)$, $$[{\cal C} ]\in U\cap \Phi^{-1}(0) \subset X_{\delta, S} ([{\cal C}], W_0, K).$$ This follows from our construction of $X_{\delta, S} ([{\cal C}], W_0, K)$ and the following lemma. \begin{lemma} Let $\{f_i\}$ be a sequence of $J$-holomorphic maps with fixed homology class $A$, then by taking a subsequence if necessary, we may have that $f_i$ converges to some holomorphic map $f_\infty$, which may be reducible, such that $||f_i||_{1,p}$ is uniformly bounded for any $p > 2$. \end{lemma} \vskip 0.1in \noindent {\bf Proof:} This lemma was in fact essentially proved in [RT1], section 6. It is also true for any sequence of harmonic maps (cf. [CT]). By the Gromov Compactness Theorem, we may assume that $f_i$ converges to $f_\infty$ in the topology of $\overline {\cal F}_A^\ell(V,g,k)$. Then we suffice to show that $|| f_i||_{1,p}$ is uniformly bounded. Let $\Sigma _i$ be the domain of $f_i$ and $q$ be an node of $\Sigma_\infty$, which is the domain of $f_\infty$. Near $q$, $\Sigma _i$ can be locally described by coordinates $w_1, w_2$ with $w_1 w_2 = \epsilon _i$ and $|w_1|, |w_2| \le 1$. Note that $\lim_{i\to \infty} \epsilon _i = 0$. We may assume that $\epsilon_i > 0$ and $f_i(w_1, w_2)$ is very close to $q$. Write $w_1 = s e^{\sqrt{-1} \theta}$, where $\epsilon _i \le s \le 1$. Then the Cauchy-Riemann equation becomes $$\frac{\partial f_i }{\partial s} + \frac{1}{s} J(f_i) \frac{\partial f_i}{\partial \theta} = 0.$$ By the same arguments as in the proof of Lemma 3.9, we can deduce that for any $p > 2$, $$\int _{\epsilon _i \le s \le 1} (|w_1|^2+ |w_2|^2)^{\frac{p-2}{p}} |\nabla f_i|^2 s ds\wedge d\theta \le c_p ,$$ where $c_p$ is a constant depending only on $p$. It follows that $|| f_i||_{1,p}$ is uniformly bounded. The lemma is proved. \vskip 0.1in Now let us construct a resolution $\{F_i, \psi_i\}$ of $\Phi^{-1}(0)$. We cover $\Phi^{-1}(0)$ by smooth approximations $\{(E_{\delta_i, S_i}([{\cal C}_i], W_{0i}, K_i), X_{\delta_i, S_i}([{\cal C}_i], W_{0i}, K_i))\}$, where $1\le i \le m$. For each $i$, we have $$X_{\delta_i, S_i}([{\cal C}_i], W_{0i}, K_i)= \tilde X_{\delta_i, S_i}([{\cal C}_i], W_{0i}, K_i)/ \Gamma_i,$$ where $\Gamma_i$ is the automorphism group of ${\cal C}_i$ and $$\tilde X_{\delta_i, S_i}([{\cal C}_i], W_{0i}, K_i) = \Phi^{-1}(E_{S_i}) \subset \tilde U_{\delta _i}([{\cal C}_i], W_{0i}, K_i)= \Map _{\delta_i}(W_i,K_i).$$ For each $i$, choose $W_{0i}'\subset W_{0i}$ such that if $W_i'$ denotes the corresponding subset in $W_i$, then all the $X_{\delta_i, S_i}([{\cal C}_i], W_{0i}', K_i)$ still cover $\Phi^{-1}(0)$. As we have seen before, any vector $v \in S_i$ induces a section, denoted by $v_s$, of $E_{S_i}$ over $\Map_{\infty}(W_i)$. Let us construct $F_i, \psi_i$ inductively. For $i=1$, we simply define $F_1=S_1$ and $$\begin{array}{rl} \psi_1: &\Map _{\infty }(W_1)\times F_1 \mapsto E_{1},\\ &\psi_1 (\tilde {\cal C}, v) = \eta _1(\Sigma_{\tilde {\cal C}}) v_s(\tilde {\cal C}).\\ \end{array}$$ Here $\eta_i$ is a smooth function on $\tilde W_i$ satisfying: $\eta_i \equiv 1$ on $\tilde W_i'$ and $\eta_i = 0$ near $\partial \tilde W_i$. Suppose that we have defined $F_i, \psi_i$ for $1\le i \le l-1$. Let us define $F_{l}, \psi_{l}$. For each $i < l$ and $v\in F_i$, $\psi_i(v)$ induces a section, say $v_{s,l,i}$, of $\tilde E_l$ over $\tilde U_{\delta_{l}}({\cal C}_{l}, W_{0 l},K_{l})$. Let $F_l$ be the vector space spanned by $S_l$ and $\sigma ^*(v_{s,l,i})$ ($\sigma \in \Gamma_l$, $i< l$). We define $\psi_l(\tilde {\cal C}, v)$ to be $\eta _l v_s(\tilde {\cal C})$ if $v\in S_l$ and $v_{s,l,i}(\sigma(\tilde {\cal C}))$ if $v=\sigma ^*(v_{s,l,i})$. Clearly, $\psi_l$ is $\Gamma_l$-equivariant. Thus we can construct $\{F_i,\psi_i\}_{1\le i\le m}$. The smooth structure of $\Phi: \overline {\cal F}_A^\ell(V,g,k)\mapsto E$ is given by all those smooth approximations $(\tilde E_{\delta, S} ([{\cal C}], W_0, K), \tilde X_{\delta, S} ([{\cal C}], W_0, K))$ satisfying: if ${\cal U}$ (resp. ${\cal U}_i$) is the universal family of curves over $\tilde W$ (resp. $\tilde W_i$) corresponding to $W_0$ (resp. $W_{0i}$), then $S|_{{\cal U}\cap {\cal U}_i}$ contains the image of $\psi_{i}$ for each $i$. One can show that with all these $(\tilde E_{\delta, S} ([{\cal C}], W_0, K), \tilde X_{\delta, S} ([{\cal C}], W_0, K))$, $\{F_i, \psi_i\}$ satisfies all properties required for a smooth resolution. Therefore, $\Phi: \overline {\cal F}_A^\ell(V,g,k)\mapsto E$ is a generalized Fredholm orbifold bundle. Finally, let us give the natural orientation of $\det (\Phi )$ (cf. [R], [RT1]). We first notice that for any ${\cal C}$ representing a point in $\Phi^{-1}(0)$, $\det (\Phi)|_{\cal C}$ can be naturally identified with the determinant of the linear Fredholm operator $L_{\cal C}\Phi$, where $L_{\cal C}\Phi$ denotes the linearization of $\Phi$ at ${\cal C}$. It can be written as $\overline \partial _{\cal C} + B_{\cal C}$, where $\overline \partial _{\cal C}$ is a $J$-linear operator and $B_{\cal C}$ is an operator of zero order. It follows that $L_{\cal C}\Phi$ can connected to $\overline \partial _{\cal C}$ through the canonical path $\{\overline \partial _{\cal C} + t B_{\cal C}\}_{0\le t\le 1}$, so $\det (\Phi )$ is naturally isomorphic to the determinant $\det (\overline \partial )$ of the family of operators $\{\overline \partial _{\cal C}\}_{\cal C}$. However, since each $\overline \partial _{\cal C}$ is J-invariant, there is a natural orientation on $\det (\overline \partial )$. It follows that $\det (\Phi )$ can be naturally oriented. Proposition 2.2 is proved. \begin{rem} In fact, in this concrete case, we can construct the Euler class $e([\Phi: \overline {\cal F}_A^\ell (V,g,k) \mapsto E])$ without using Theorem 1.2. We can use the arguments in the proof of Theorem 1.2 and smooth approximations $$(\tilde E_{\delta, S} ([{\cal C}], W_0, K), \tilde X_{\delta, S} ([{\cal C}], W_0, K))$$ to construct a ${\Bbb Q} $-cycle. This ${\Bbb Q}$-cycle will lie in a finite covering of $\Phi^{-1}(0)$ by finitely dimensional smooth approximations. Here we do need smooth properties of $(\tilde E_{\delta, S} ([{\cal C}], W_0, K), \tilde X_{\delta, S} ([{\cal C}], W_0, K))$ during changes of local uniformizations. \end{rem} \vskip 0.1in \noindent {\bf Proof of Proposition 2.3:} The proof is identical to that of Proposition 2.2. So we omit the details. Let $\Phi: \overline F_A^\ell (V,g,k) \mapsto E$ be the generalized Fredholm orbifold bundle as above, and $\Phi': \overline F_A^\ell (V,g,k) \mapsto E$ is another one induced by the almost complex structure $J'$. Let $\{J_t\}$ be the family of almost complex structures joining $J$ to $J'$. Consider $$\begin{array}{rl} \Psi: [0,1] \times \overline F_A^\ell (V,g,k) & \mapsto E,\\ (t, (f, \Sigma; \{x_i\})) &\mapsto df + J_t(f)\cdot df \cdot j.\\ \end{array} $$ Then $\Psi |_{\{0\}\times \overline F_A^\ell (V,g,k) } = \Phi$ and $\Psi |_{\{1\}\times \overline F_A^\ell (V,g,k) } = \Phi'$. Using the same arguments as above, one can prove that $\Psi: [0,1]\times \overline F_A^\ell (V,g,k) \mapsto E$ is a generalized Fredholm orbifold bundle. Moreover, one can equip this bundle a weakly smooth structure which restricts to the given smooth structures of $\Phi$ and $\Phi'$ along the boundary $\{0, 1\}\times \overline F_A^\ell (V,g,k) $. This follows that $\Phi$ is homotopic to $\Phi'$. That is just what Proposition 2.3 claims. \vskip 0.1in \section{One more example} \label{sec:4} In this section, we consider a simpler example: the Seiberg-Witten invariants of 4-manifolds. The Seiberg-Witten invariants have found many striking applications in the study of 4-dimensional topology (cf. [Wi], [Ta], [KM], [KST]). Here we just give a different approach to defining the Seiberg-Witten invariants, which seems to be of independent interest. We first fix the notation we will use. Let $X$ be a compact oriented smooth 4-manifold and let $c$ be a $\spin^c$ structure on $X$ with the associated $\spin^c$ bundles $W^+$ and $W^-$. Let \begin{displaymath} \rho: \Lambda^+\otimes{\Bbb C}\lra\frak{sl}(W^+) \end{displaymath} be the isomorphism induced by the Clifford multiplication, where $\frak{sl}(W^+)$ is the associated $PSL(2,{\Bbb C})$ bundle of $W^+$. There is also a pairing \begin{displaymath} W^+\times\bar{W}^+\lra\frak{sl}(W^+) \end{displaymath} that is modeled on the map ${\Bbb C}\times\bar{\Bbb C}^2\to\frak{sl}({\Bbb C}^2)$ sending $(v,w)$ to $i(v\bar w^t)_0$, where the subscript means the traceless part. Now the Seiberg-Witten invariants is defined as follows. We first fix a Riemannian metric $g$. Let $L$ be the determinant line bundle of $W^+$ and let ${\cal A} ({\cal L})$ be the space of unitary connections on $L$. Then $A\in{\cal A} ({\cal L})$ induces a Dirac operator $\Gamma(W^+)\to\Gamma(W^-)$. Now let $\tilde{\cal B}$ be the Banach manifold ${\cal A}(L)\times\Gamma(W^+)$ and let $\tilde{\cal E}$ be the constant vector bundle over $\tilde{\cal B}$ with fiber $\Gamma(W^+)\times\Gamma(\frak{sl}(W^+))$. We define a section $\tilde f: \tilde{\cal B}\to\tilde{\cal E}$ via \begin{displaymath} \tilde f(\varphi, A)=(D_A\varphi, \rho( F_A^+)-i\sigma(\varphi,\varphi)). \end{displaymath} Now let $p_0\in X$ be fixed and let ${\cal G}_0=\Map_{p_0}(X,S^1)$ be the pointed gauge group of $L$. Note that ${\cal G}_0$ also acts on $\Gamma(W^+)$ via scalar automorphism of $W^+$. Hence ${\cal G}_0$ acts freely on $\tilde B$ and it lifts to an action on $\tilde {\cal E}$. We let ${\cal B}=\tilde{\cal B}/{\cal G}_0$ and ${\cal E}=\tilde{\cal E}/{\cal G}_0$. Since $\tilde f$ is equivariant under ${\cal G}_0$, $\tilde f$ descends to a section \begin{displaymath} f: {\cal E}\lra{\cal B}\,. \end{displaymath} Note that $f^{-1}(0)$ is compact. Now let ${\cal G}$ be the full gauge group. Then ${\cal G}/{\cal G}_0\cong S^1$ acts on ${\cal E}\to{\cal B}$ and the section $f$ is $S^1$-equivariant as well. The Seiberg-Witten invariants of $X$ is the $S^1$-equivariant version of Euler class of $[f:{\cal B}\to{\cal E}]$ defined in section 1. More precisely, The universal line bundle on ${\cal A}(L)$ descends to a complex line bundle ${\cal L}$ on ${\cal B}$ and the Seiberg-Witten invariants of $X$ is \begin{displaymath} SW: H^2(X,{\Bbb Z})\lra {\Bbb Z} \end{displaymath} defined by \begin{displaymath} SW(L)=<\Euler({\cal E},f)^{S^1}, c_1({\cal L})^k>\,, \end{displaymath} where \begin{displaymath} k={\frac{1}{4}}(c_1(L)^2-(2\chi+3\sigma)), \end{displaymath} is the Fredholm index of $f/S^1: {\cal B}/S^1\to {\cal E}/S^1$. In case the zero locus $f^{-1}(0)$ is disjoint from the fixed point set of $S^1$, which is \begin{displaymath} {\cal B}^{S^1}={\cal A}(L)/{\cal G}_0\times\{0\}\sub {\cal A}(L)\times\Gamma(W^+)/{\cal G}_0, \end{displaymath} then we can work with $[f/S^1:{\cal B}/S^1\to{\cal E}/S^1]$. Let ${\cal B}'=({\cal B}-{\cal B}^{S^1})/S^1$, let ${\cal E}'=({\cal E}|_{{\cal B}'})/S^1$ and let $f'=(f|_{{\cal B}'})/S^1$. Then $[f': {\cal B}'\to {\cal E}']$ is a Fredholm operator as defined in section 1. The Seiberg-Witten invariant then is $$SW(L)=<e[f': {\cal B}'\to {\cal E}'], c_1({\cal L}')^k> $$ where ${\cal L}'$ is the descend of ${\cal L}|_{{\cal B}-{\cal B}^{S^1}}$ to ${\cal B}'$. We now look at the general case. Let \begin{displaymath} {\cal E}|_{\bso}=\oplus_{i=\-\infty}^{\infty}{\cal F}_i \end{displaymath} be the spectral decomposition of the restriction of ${\cal E}$ to $\bso$. Namely, ${\cal F}_i\sub{\cal E}|_{\bso}$ is $S^1$-invariant and the $S^1$ action on ${\cal F}_i$ has weight $i$. Then $f|_{\bso}$ factor through ${\cal F}_0\sub{\cal E}|_{\bso}$. We denote this section by $f_0$. Let $k+1$ be the Fredholm index of $f$ and let $l$ be the Fredholm index of $f_0: T_z\bso \to {\cal F}_{0,z}$. \begin{lemma} Assume $l<0$, then any $S^1$-equivariant Fredholm section $f:{\cal B}\to{\cal E}$ is homotopic to an $S^1$-equivariant Fredholm section $g:{\cal B}\to{\cal E}$ so that $g^{-1}(0)\cap{\cal B}^{S^1}=\emptyset$. \end{lemma} \noindent {\bf Proof:} The proof is straightforward. We first look at the the restriction of $f$ to ${\cal B}^{S^1}$. As we mentioned, it factor through ${\cal F}_0$. Let $h: {\cal B}^{S^1}\to {\cal F}_0$ be this map. Then since $dh$ has negative Fredholm index, by Theorem 1.1, $h$ is homotopic to $\tilde h: {\cal B}^{S^1}\to{\cal F}_0$ so that its vanishing locus is empty. Clearly, for some $S^1$ invariant neighborhood $U$ of ${\cal B}^{S^1}\sub {\cal B}$, we can extend this homotopy, and thus $\tilde h$, within the category of Fredholm operators, to an $S^1$-equivariant $g:{\cal B} \to{\cal E}$ so that the restriction of $g$ to ${\cal B}^{S^1}$ is $\tilde h$ and $g|_{{\cal B}-U}=f|_{{\cal B}-U}$. This proves the Lemma. After having $g$ given by the Lemma, we reduce the situation to when $g^{-1}(0)\cap {\cal B}^{S^1}=\emptyset$. Thus as before, we can define the equivariant Euler class $e[f: {\cal B}^{S^1}\to{\cal E}]^{S^1}$ represented by a smooth $k$ dimensional submanifold in $({\cal B}-{\cal B}^{S^1})/S^1$ to be the class $e[g/S^1: {\cal B}'\to {\cal E}']$. When $l<-1$, then the above argument shows that any two such representatives of $e[f: {\cal B}\to{\cal E}]^{S^1}$ in ${\cal B}'$ are coborbant to each other in ${\cal B}'$. Therefore, they represent a well-defined corbordism class. We now apply the above construction to the Seiberg-Witten invariant, the fixed point set $\bso$ in ${\cal B}$ is ${\cal A}(L)\times\{0\}$. The ${\cal F}_0\sub{\cal E}|_{\bso}$ in this case is the subbundle $\Gamma(\frak{sl}(W^+))$ and the restriction section is $\rho(F_A^+)$, whose Fredholm index is $-b_2^+$. Therefore, when $b_2^+>1$, the Seigerg-Witten invariant is well defined and can be represented by a smooth submanifold in $({\cal B}-\bso)/S^1$.

9608

alg-geom/9608030

en

https://arxiv.org/abs/alg-geom/9608030

alg-geom/9608030

Eleny Ionel

Genus one enumerative invariants in P^n with fixed j invariant

LaTeX, 38 pages

We prove recursive formulas for $\tau_d$, the number of degree $d$ elliptic curves with fixed j-invariant in P^n. We use analysis to relate the classical invariant $\tau_d$ to the genus one perturbed invariant $RT_{1,d}$ defined recently by Ruan and Tian (the later invariant can be computed inductively). By considering a sequence of perturbations converging to zero, we then apply Taubes' Obstruction Bundle method to compute the difference between the two invariants.

alg-geom

\section{Introduction.} \setcounter{equation}{0} A classical problem in enumerative algebraic geometry is to compute the number of degree $d$, genus $g$ holomorphic curves in ${ \Bbbd P}^n$ that pass through a certain number of constraints (points, lines, etc). Let $\sigma_d$ denote the number of degree $d$ rational curves ($g=0$) through appropriate constraints. For example $\sigma_1(pt,pt)=1$ (since 2 points determine a line). The first nontrivial cases were computed around 1875 when Schubert, Halphen, Chasles et al. found $\sigma_2$ for ${ \Bbbd P}^2$ and ${ \Bbbd P}^3$. Later, more low degree examples were computed in ${ \Bbbd P}^2$ and ${ \Bbbd P}^3$, but the progress was slow. Then in 1993 Kontsevich \cite{K} predicted, based on ideas of Witten, that the number $\sigma_d$ of degree $d$ rational curves in ${ \Bbbd P}^2$ through $3d-1$ points satisfies the following recursive relation: \[ \sigma_d=\ma(\sum)_{d_1+d_2=d} \left[ {3d-1\choose 3d_1-1}d_1^2d_2^2- {3d-1\choose 3d_1-2} d_1^3d_2\right] \sigma_{d_1} \sigma_{d_2} \] where $d_i\ne 0$, and $\sigma_1=1$. Ruan-Tian (\cite{rt}, 1994) extended these formulas for $\sigma_d$ in any ${ \Bbbd P}^n$. \medskip When genus $g=1$, the classical problem splits into two totally different problems: one can count (i) elliptic curves with a fixed complex structure, or (ii) elliptic curves with unspecified complex structure (each satisfying the appropriate number of constraints). This paper gives recursive formulas which completely solve the first of these. \medskip Thus our goal is to compute the number $\tau_d$ of degree $d$ elliptic curves in ${ \Bbbd P}^n$ with fixed $j$ invariant. Classically, the progress on this problem has been even slower than on the genus one case. Recently, Pandharipande \cite{rp} found recursive formulas for $\tau_d$ for the 2 dimensional projective space ${ \Bbbd P}^2$ using the Kontsevich moduli space of stable curves. \bigskip We will approach the problem from a different direction, using analysis. Our approach is based on the ideas introduced by Gromov to study symplectic topology. If $(\Sigma,j)$ is a fixed Riemann surface, let \[\{{\vphantom{\ma(\int)_{k}}} \; f:\Sigma\rightarrow { \Bbbd P}^n\; |\; \overline \partial_{J} f=0, \; [f]=d\cdot l\in H_2({ \Bbbd P}^n,{ \Bbbd Z}) \;\}/Aut(\Sigma,j) \] be the moduli space of degree $d$ holomorphic maps $f:\Sigma\rightarrow { \Bbbd P}^n$, modulo the automorphisms of $(\Sigma,j)$. Each constraint, such as the requirement that the image of $f$ passes through a specified point, defines a subset of this moduli space. \smallskip Imposing enough constraints gives a 0-dimensional ``cutdown moduli" space ${\cal M}_d$. To see whether or not it consists of {\em finitely many} points, one looks at its bubble tree compactification $\ov {\cal M}_d$ \cite{pw}. If the constraints cut transversely, then all the boundary strata of $\ov{\cal M}_d$ are at least codimension 1, and thus empty. Unfortunately, transversality fails at multiply-covered maps or at constant maps (called {\em ghosts}), so $\ov{\cal M}_d$ is not a manifold. This was a real problem until 1994, when Ruan and Tian considered the moduli space ${\cal M}_\nu$ of solutions of the perturbed equation: \[ \overline \partial_{J} f=\nu(x,f(x))\] and used marked points instead of moding out by $Aut(\Sigma,j)$. For a generic perturbation $\nu$ the moduli space ${\cal M}_\nu$ is smooth and compact, so it consists of finitely many points that, counted with sign, give an invariant $RT_{d,g}$ (independent of $\nu$). In ${ \Bbbd P}^n$, the genus 0 perturbed invariant $RT_{d,0}$ is equal to the enumerative invariant $\sigma_d$. The perturbed invariants satisfy a degeneration formula that gives not only recursive formulas for the enumerative invariant $\sigma_d$ in ${ \Bbbd P}^n$, but also expresses the higher genus perturbed invariants in terms of the genus zero invariants \cite{rt}. For convenience, these formulas are included in the Appendix. \medskip Unfortunately, when $g=1$, the perturbed invariant $RT_{d,1}$ does not equal the enumerative invariant $\tau_d$. For example, for $d=2$ curves in ${ \Bbbd P}^2$ the Ruan-Tian invariant is $RT_{2,1}=2$ (cf. (\ref{gendeg})), while $\tau_2=0$ (there are no degree 2 elliptic curves in ${ \Bbbd P}^2$). Thus while the Ruan-Tian invariants are readily computable, they differ from the enumerative invariants $\tau_d$. One should seek a formula for the difference between the two invariants. For that, we take the obvious approach: \medskip Start with the genus 1 perturbed invariant $RT_{d,g}$ and consider a sequence of generic perturbations $\nu\rightarrow 0$. A sequence of $(J,\nu)$-holomorphic maps converges either to a holomorphic torus or to a bubble tree whose base is a constant map (ghost base). Proposition \ref{int} shows that the contribution of the $(J,0)$-holomorphic tori is a multiple of $\tau_d$. \medskip We show that the only other contribution comes from bubble trees with ghost base such that the bubble point is equal to the marked point $x_1\in T^2$. To compute this contribution, we use the Taubes' ``Obstruction Bundle" method. Proposition \ref{completion} identifies the moduli space of $(J,\nu)$-holomorphic maps close to a bubble tree with the zero set of a specific section of the obstruction bundle. Studying the leading order term of this section, we are able to compute the corresponding contribution (Proposition \ref{zeros}). Adding both contributions, yields our main analytic result: \newcounter{genthe} \newcounter{gensec} \setcounter{genthe}{\arabic{equation}} \setcounter{gensec}{\arabic{section}} \begin{theorem}\label{gen} Consider the genus 1 enumerative invariant $\tau_d(\beta_1,\dots,\beta_k)$ in ${ \Bbbd P}^n$. Let ${\cal U}_d$ be the $n-1$ dimensional moduli space of 1-marked rational curves of degree $d$ in ${ \Bbbd P}^n$ passing through $\beta_1,\dots,\beta_k$. Let $L\rightarrow {\cal U}_d$ be the relative tangent sheaf, and denote by $\wt L\rightarrow \widetilde{\cal U}_d$ its blow up as in Definition \ref{ltilde}. Then: \[n_j\tau_d(\beta_1,\dots,\beta_k)=RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_k)- \ma(\sum)_{i=0}^{n-1}{n+1\choose i+2} \mbox{\rm ev}^*(H^{n-i-1})c_1^i(\wt L^*)\] where $H^i$ is a codimension $i$ hyperplane in ${ \Bbbd P}^n$, $\mbox{\rm ev}:{\cal U}_d\rightarrow { \Bbbd P}^n$ is the evaluation map corresponding to the special marked point and $n_j=Aut_{x}(j)$ is the order of the group of automorphisms of the complex structure $j$ that fix a point. \end{theorem} Theorem \ref{gen} becomes completely explicit provided we can compute the top power intersections $\mbox{\rm ev}^*(H^{n-i-1})c_1^i(\wt L^*)$. We do this in the second part of the paper, in several steps. For simplicity of notation, let \begin{eqnarray} && x=c_1(L^*) \in H^2({\cal U}_d,{ \Bbbd Z}),\quad \wt x=c_1(\wt L^*) \in H^2(\wt{\cal U}_d,{ \Bbbd Z})\\ &&y=\mbox{\rm ev}^*(H),\quad y\in H^2({\cal U}_d,{ \Bbbd Z})\quad\mbox{or}\quad y\in H^2(\wt{\cal U}_d,{ \Bbbd Z}) \end{eqnarray} depending on the context. In this notation, Theorem \ref{gen} combined with (\ref{gendeg}) becomes: \begin{eqnarray}\label{conden} n_j\tau_d(\; \cdot \;)= \ma(\sum)_{i_1+i_2=n} \sigma_d(H^{i_1},H^{i_2},\; \cdot \;)+ \ma(\sum)_{i=0}^{n-1}{n+1\choose i+2}\;\wt x^i y^{n-1-i}\cdot[\wt{\cal U}_d] \end{eqnarray} Proposition \ref{pwtc1} explains how to get recursive formulas relating $\wt x^i y^j$ to $x^k y^l$ and Proposition \ref{pc1} gives recursive formulas for $x^i y^j$ in terms of the enumerative invariant $\sigma_d$. Finally, the recursive formulas for $\sigma_d$ are known (see \cite{rt}, \cite{K}), so the right hand side of (\ref{conden}) can be recursively computed. \bigskip In the end, we give applications of these formulas. We explicitly work out the formulas expressing the number of degree $d$ elliptic curves passing through generic constraints in ${ \Bbbd P}^2$ and ${ \Bbbd P}^3$ in terms of the rational enumerative invariant $\sigma_d$. For example: \begin{prop}\label{p^3} For $j\ne 0,1728$, the number $\tau_d=\tau_d(p^a,l^b)$ of elliptic curves in ${ \Bbbd P}^3$ with fixed $j$ invariant and passing through $a$ points and $b$ lines (such that $2a+b=4d-1$) is given by: \begin{eqnarray} \tau_d( \cdot )={(d-1)(d-2)\over d}\sigma_d(l,\; \cdot )-{1\over d}\ma(\sum)_{d_1+d_2=d} d_2(2d_1d_2-d)\sigma_{d_1}(l,\; \cdot )\sigma_{d_2}( \cdot ) \end{eqnarray} where $\sigma_d(l,\; \cdot )=\sigma_d(l,p^a,l^b)$ is the number of degree $d$ rational curves in ${ \Bbbd P}^3$ passing through same conditions as $\tau_d$ plus one more line. The sum above is over all decompositions into a degree $d_1$ and a degree $d_2$ component, $d_i\ne 0$, and all possible ways of distributing the constraints $p^a,\;l^b$ on the two components. \end{prop} Using a computer program, one then computes specific invariants: for example, the number of degree 10 tori in ${ \Bbbd P}^3$ with fixed $j$ invariant and passing through 39 lines is: \[ 6\cdot 386805671822029784844530703900638969856 \] when $j\ne 0,1728$. To get $\tau_d$ for $j=0$ or $j=1728$ one simply divides the $\tau_d$ computed for a generic $j$ by 3 or 2 respectively. \medskip \noindent{\bf Acknowledgements.} I would like to thank my advisor Prof. Thomas Parker for introducing me to the subject and for the countless hours of discussions. \section{Analysis} \setcounter{subsection}{0} \setcounter{equation}{0} \setcounter{theorem}{0} \subsection{Setup}\label{Setup} Let $\tau_d$ be the genus one degree $d$ {\em enumerative invariant} (with fixed $j$ invariant) and $\sigma_d$ be the genus zero degree $d$ enumerative invariant in ${ \Bbbd P}^n$. Using analytic methods, we will compute $\tau_d$ by relating it to the perturbed invariant $RT_{d,g}$ introduced by Ruan and Tian \cite{rt}. The later is defined as follows. \smallskip Let $(\Sigma,j)$ be a genus $g$ Riemann surface with a fixed complex structure and $\nu$ an inhomogenous term. A $(J,\nu)$-{\em holomorphic} map is a solution $f:\Sigma\rightarrow{ \Bbbd P}^n$ of the equation \begin{eqnarray}\label{jnu} \overline \partial_{J} f(x)=\nu(x,f(x)). \end{eqnarray} For $2g+l\ge 3$, let $x_1,\dots,x_l$ be fixed marked points on $\Sigma$, and $\alpha_1,\dots,\alpha_l $, $\beta_1,\dots,\beta_k$ be various codimension submanifolds in ${ \Bbbd P}^n$, such that \[{\rm index} \; \overline \partial_{J}= (n+1)d-n(g-1)=\ma(\sum)_{i=1}^l (n-|\alpha_i|)+\ma(\sum)_{i=1}^k (n-1-|\beta_i|)\] For a generic $\nu$, the invariant \[RT_{d,g}(\alpha_1,\dots,\alpha_l\;|\;\beta_1,\dots,\beta_k)\] counts the number of $(J,\nu)$-holomorphic degree $d$ maps $f:\Sigma\rightarrow { \Bbbd P}^n$ that pass through $\beta_1,\dots,\beta_k$ with $f(x_i)\in \alpha_i$ for $i=1,\dots,l$ (for more details see \cite{rt}). \medskip The first part of this paper is devoted to the proof of Theorem \ref{gen}. \medskip \noindent{\large \bf Outline of the Proof of Theorem \ref{gen}.} The proof is done in several steps. The basic idea is to start with the genus 1 perturbed invariant \begin{eqnarray}\label{cond} RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l) \end{eqnarray} and take a sequence of generic perturbations $\nu\rightarrow 0$. Denote by ${\cal M}_{d,1,t\nu}$ the moduli space of $(J,t\nu)$-holomorphic maps satisfying the constraints in (\ref{cond}), and let \begin{eqnarray} {\cal M}^\nu=\ma(\bigcup)_{t\ge 0}{\cal M}_{d,1,t\nu}. \end{eqnarray} As $t\rightarrow 0$, a sequence of $(J,t\nu)$-holomorphic maps converges to a $(J,0)$-holomorphic torus or to a bubble tree (\cite{pw}). Let $\ov {\cal M}^\nu$ denote the bubble tree compactification of ${\cal M}^\nu$ (for details on bubble tree compactifications, see \cite{par}). \medskip Proposition \ref{int} shows that the number of $(J,t\nu)$-holomorphic maps converging to a $J$-holomorphic torus is equal to \[n_j \tau_d(\beta_1,\dots,\beta_k)\] where $n_j=|Aut_{x_1}(j)|$ is the order of the group of automorphisms of the complex structure $j$ that fix the point $x_1$. Namely, \begin{eqnarray} n_j=\left\{\begin{array}{ll} 2 &\mbox{ if } j\ne 0,1728\\ 6&\mbox{ if } j= 0\\ 4&\mbox{ if } j=1728 \end{array}\right.\hskip1in \end{eqnarray} These multiplicities occur because if $f$ is a $J$-holomorphic map, then so is $f\circ \phi$ for any $\phi\in Aut_{x_1}(j)$, but they get perturbed to different $(J,t\nu)$-holomorphic maps. \medskip As $t\rightarrow 0$, there are also a certain number of solutions converging to bubble trees. Because the moduli space of $(J,0)$-holomorphic tori passing through $\beta_1,\dots,\beta_k$ is 0 dimensional, the only bubble trees which occur have a multiply-covered or a ghost base (for these transversality fails, so dimensions jump up). \smallskip A careful dimension count shows that the multiply-covered base strata are still codimension at least one for genus $g=1$ maps in ${ \Bbbd P}^n$. (This is {\em not} true for $g\ge 2$.) But at a ghost base bubble tree the dimension jumps up by $n$ so these strata are $n-1$ dimensional. There are actually 2 such pieces, corresponding to bubble tree where (i) the bubble point is at the marked point $x_1$ and (ii) the bubble point is somewhere else. To make this precise, a digression is necessary to set up some notation. Let \begin{eqnarray}\label{cmo} {\cal M}^0_{d}=\{(f,y_1,\dots,y_k)\;|\; f:S^2\rightarrow { \Bbbd P}^n \mbox{ degree d holomorphic, } f(y_j)\in \beta_j \} \end{eqnarray} be the moduli space of bubble maps, and ${\cal M}_d={\cal M}^0_{d}/G$ be the corresponding moduli space of curves, where $G={\Bbbd PSL}(2,\cx)$. Introduce one special marked point $y\in S^2$ and let \begin{eqnarray}\label{cu} {\cal U}_{d}=\{\; [f,y,y_1,\dots,y_k]\;\;|\;\; [f,y_1,\dots,y_k]\in{\cal M}_d \} \end{eqnarray} be the moduli space of {\em 1-marked curves} and \begin{eqnarray}\label{ev} \mbox{\rm ev}:{\cal U}_d\rightarrow { \Bbbd P}^n,\;\;\;\mbox{\rm ev}([f,y,y_1,\dots,y_k]) =f(y). \end{eqnarray} be the corresponding evaluation map. We will use $f(y)$ to record the image of the ghost base \smallskip For generic constraints $\beta_1,\dots,\beta_k$ the bubble tree compactification of ${\cal U}_d$ is a {\em smooth } ma\-ni\-fold that comes with a {\em natural stratification}, depending on the possible splittings into bubble trees and how the degree $d$ and the constraints $\beta_1,\dots,\beta_k$ distribute on each bubble. \medskip With this, the two ``pieces" of the boundary of $\ov{\cal M}^\nu$ are: \begin{eqnarray}\label{2strata} \{x_1\}\times\ov{\cal U}_d\hskip.2in \mbox{ and}\hskip.2in T^2\times \mbox{\rm ev}^*(\beta_1) \end{eqnarray} The first factor records the bubble point, while the image of the ghost base is encoded in the second factor. For generic constraints, each piece, as well as their intersection, is a smooth manifold, again stratified. \medskip To see which bubble trees with ghost base appear as a limit of perturbed tori, we use the Taubes' Obstruction Bundle. This construction must be performed on the link of each strata. We do this first on the top statum of $\{x_1\}\times\ov{\cal U}_d$, which consists of bubble trees with ghost base and a single bubble. In Section \ref{Approx} we construct a set of approximate maps by gluing in the bubble. The ``gluing data" $[f,y,v]$ at a 1-marked curve $[f,y]$ consists of a nonvanishing vector $v$ tangent to the bubble at the bubble point $y$. Proposition \ref{lg2} shows that the obstruction bundle is then diffeomorphic to $\mbox{\rm ev}^*(T{ \Bbbd P}^n)$. \medskip In Section \ref{Gluing} we try to correct the approximate maps to make them $(J,t\nu)$-holomorphic by pushing them in a direction normal to the kernel of the linearized equation. Those approximate maps that can be corrected to solutions of the equation (\ref{jnu}) are then identified with the zero set of a section $\psi_t$ of the obstruction bundle. Proposition \ref{completion} shows that actually all the solutions of the equation (\ref{jnu}) are obtained this way, i.e. the end of the moduli space of $(J,t\nu)$-holomorphic maps is diffeomorphic to the zero set of the section $\psi_t$. \medskip One might be tempted now to belive that the difference between the two invariants is simply the euler class of the obstruction bundle. But in fact, even in generic conditions, the section $\psi_t$ is {\em not a generic} section of the obstruction bundle. We will see that the obstruction bundle has a nowhere vanishing section, so it has a trivial euler class, while there are examples in which the difference term is certainly not zero. \medskip Now, to understand the zero set of $\psi_t$ it is enough to look at the leading order term of its expansion as $t\rightarrow 0$. By Proposition \ref{psi} this has the form $ df_y(v)+t\bar\nu$ where $\bar\nu$ is the projection of $\nu$ on the obstruction bundle. \medskip The construction described above extends naturally to all the other boundary strata. Each bubble $[f_i,y_i]$ comes with ``gluing data " $[f_i,y_i,v_i]$, consisting of a vector $v_i$ tangent to the bubble at the bubble point $y_i$. But the leading order term of the section $\psi_t$ depends only on the vectors tangent to the {\em first level of nontrivial bubbles}. More precisely, let ${\cal Z}_h\subset \ov{\cal U}_d$ denote the collection of bubble trees for which the image $u=f(y)$ of the ghost base lies on $h$ nontrivial bubbles. Geometrically, the image of a bubble tree in ${\cal Z}_h$ has $h$ components $C_1,\dots,C_h$ that meet at $u$. Let ${\cal W}|_{{\cal Z}_h}\rightarrow {\cal Z}_h$ be the bundle whose fiber is $T_uC_1\oplus\cdots \oplus T_uC_h$. The leading order term of $\psi_t$ on ${\cal Z}_h$ is a section of ${\cal W}$, equal to \[a(f,y,v])+t\bar\nu\ma(=)^{def} df_1(y_1)(v_1)+\dots+df_h(y_h)(v_h)+t\bar\nu\] where $([f_i,y_i,v_i])_{i=1}^h$ is the gluing data corresponding to the bubbles $C_i$, $i=1,\dots,h$. Unfortunately ${\cal W}\rightarrow\ov{\cal U}_d$ is not a vector bundle (its rank is not constant). But if we blow up each strata ${\cal Z}_h$ starting with the bottom one, then the total space of ${\cal W}$ is the same as the total space of $\wt L$, the blow-up of the relative tangent sheaf $L\rightarrow {\cal U}_d$. The leading order term of $\psi_t$ descends as a map $a+t\bar\nu:\wt L\rightarrow \mbox{\rm ev}^*(T{ \Bbbd P}^n)$. Moreover, $\bar \nu$ doesn't vanish on $\mbox{\rm Im}(\cm)=ev_*(\ov{\cal U}_d)$ so it induces a splitting on the restriction \[T{ \Bbbd P}^n/\mbox{\rm Im}(\cm)={ \Bbbd C} \langle \bar\nu\rangle\oplus E.\] Finally, we put all these pieces together in Proposition \ref{zeros} to prove that the number of $(J,\nu)$-holomorphic maps converging as $\nu\rightarrow 0$ to the boundary strata $\{x_1\}\times\ov{\cal U}_d$ is given by the Euler class $c_{n-1}(\mbox{\rm ev}^*(E)\otimes\wt L^*)$. \medskip In Section \ref{Other} we show that the other boundary strata $T^2\times \mbox{\rm ev}^*(\beta_1)$ gives trivial contribution, concluding the proof of the Theorem \ref{gen}. \subsection{The Approximate gluing map}\label{Approx} Let ${\cal U}_d$ be the moduli space of 1-marked rational curves of degree $d$ passing through the conditions $\beta_1,\dots,\beta_k$. In this section we construct a set of approximate maps starting from $ \{x_1\}\times\ov{\cal U}_d $, the first boundary strata in (\ref{2strata}). We will use a: \step{Cutoff function} Fix a smooth cutoff function $\beta$ such that $\beta(r) =0$ for $r\le 1$ and $\beta(r) =1$ for $r\ge 2$. Let $\beta_\lambda (r) = \beta(r/\sqrt \lambda )$. Then $\beta_\lambda$ has the following properties: \begin{eqnarray*} | \beta_\lambda | \le 1\; , \;\;| d \beta_\lambda | \le 2/\sqrt \lambda\;\; \mbox{ and }\;\; d\beta_\lambda \;\;\mbox{ is supported in } \sqrt \lambda \le r \le 2 \sqrt \lambda \end{eqnarray*} \step{The definition of the approximate gluing map on the top stratum} Let ${\cal N}$ denote the top stratum of $\{x_1\}\times\ov{\cal U}_d$. First we need to choose a canonical representative of each bubble curve $[f,y]\in{\cal N}$ (recall that $f(y)$ is the image of the ghost base). Using the $G={\Bbbd PSL}(2,\cx)$ action, we can assume that $y$ is the North pole and $f$ is centered on the vertical axis, which leaves a ${ \Bbbd C}^*\cong S^1\times{ \Bbbd R}_+$ indeterminancy. To break it off, include as gluing data a unit vector tangent to the domain $S^2$ of the bubble at the bubble point $y$. The frame bundle \begin{eqnarray}\label{fr} Fr=\{\; [f,y,u] \;|\; [f,y]\in{\cal U}_d,\; u\in T_yS^2,\;|u|=1\} \end{eqnarray} models the link of ${\cal N}$. The notation $[f,y,u]$ means the equivalence class under the action of $G$ given by: \begin{eqnarray*} g\cdot(f,y,u)=(f\circ g^{-1}, \;g(y),\; g(u) ) \end{eqnarray*} where the compact piece $SO(3)\subset G$ acts on the unit frame $u$ by rotations and the noncompact part acts trivially. \smallskip Fix a nonzero vector $u_1$ tangent to the torus at $x_1$. This determines an identification $T_{x_1}(T^2) \cong { \Bbbd C} $ such that $u_1=1$, giving local coordinates on the torus at $x_1=0$. Similarly, let $u_0$ be a unit vector tangent to the sphere $S^2$ at the north pole and consider the identification \begin{eqnarray}\label{tt2} (T_{x_1}T^2, u_1)\cong (T_NS^2,u_0)\end{eqnarray} that induces natural coordinates on the sphere via the stereographical projection (such that $N=0,\;u_0=1$). These choices of local coordinates on the domain of the bubble tree will be used for the rest of the paper. Fix also a metric on ${ \Bbbd P}^n$ such that we can use normal coordinates up to radius 1. To glue, one needs to make sure that only a small part of the energy of $f$ is concentrated in a neighbourhood of $y$. The convention in \cite{pw} is to rescale $f$ until $\varepsilon_0$ of its energy is distributed in $H_y$, the hemisphere centered at $y$. But since the constructions in the next couple of sections involve quite a few estimates, we prefer to do a different rescaling, that will simplify the analysis. Choose a representative of $[f,y,v]$ such that \begin{eqnarray}\label{uniquerep1} y=0, \;\; u=1,\;\; f \mbox{ centered on the vertical axis } \end{eqnarray} Since on the top strata $[f,y]$ cannot be a ghost, such representative is uniquely determined up to a rescaling factor $r\in{ \Bbbd R}_+$. We will choose this rescaling factor such that moreover \begin{eqnarray}\label{d2f} {\rm max}\{\;|\nabla^2f(z)|,\; |z|\le 1\}\le 2 \end{eqnarray} Note that if the degree of $f$ is not 1, then imposing the extra condition \begin{eqnarray}\label{uniquerep2} {\rm max}\{\;|\nabla^2f(z)|,\; |z|\le 1\}=2 \end{eqnarray} determines uniquely the representative. To see this, choose some representative $\wt f$ as in (\ref{uniquerep1}) and look for a map $f(z)=\wt f(rz)$ satisfying also (\ref{uniquerep2}). The uniqueness comes from the fact that the map $s(r)={\rm max}\{\;|\nabla^2\wt f(z)|,\; |z|\le r\}-{2/r^2}$ is decreasing. \smallskip If the degree of $f$ is 1, (i.e. the image curve is a line), then we could replace (\ref{uniquerep2}) by say $ |df(0)|=1 $ and still have (\ref{d2f}) satisfied. \medskip Finally, the {\em approximate gluing map} \begin{eqnarray}\nonumber &&\gamma_{\varepsilon} :Fr\times (0,\varepsilon) \rightarrow {\cal M} aps(T^2,X) \\ &&\gamma_{\varepsilon}(\;[f,y,u],\;\lambda)={f_\lambda} \end{eqnarray} is constructed as follows: Choose the unique representative of $[f,y,u]$ satisfying (\ref{uniquerep1}) and (\ref{uniquerep2}). The approximate map ${f_\lambda}$ is obtained by gluing to the constant map $f(y)$ defined on $T^2$ the bubble map $f$ rescaled by a factor of $\lambda$ inside a disk $D(0,\sqrt \lambda)\subset T^2$, \[ {f_\lambda} (z) = \beta_ \lambda(|z|) f \l( {\lambda \over z} \r) \] where the multiplication is done in normal coordinates at $f(0)$. Let $Gl=Fr\times (0,\varepsilon)$ denote the set of gluing data. \step{Weighted Norms} On the domain of $f_{\lambda}$ we will use the rescaled metric $g_\lambda=\theta_{\la}^{-2}dzd\bar z$, where \[ \theta_{\la}(z)= (1-\beta_\lambda(z)\;)(\lambda + \lambda^{-1} |z|^2)+\beta_\lambda(z)\] Define \begin{eqnarray*} \| \xi {\|}_{1,p,\lambda}&=& \left( \int |\xi|^p \theta_{\la}^{-2} + | \nabla \xi |^p \theta_{\la}^{p-2} \right)^{1/p} \mbox{ for } \xi \mbox{ vector field along } f_{\lambda} \mbox{ and}\\ \| \eta {\|}_{p,\lambda}&=&\l( \int |\eta |^p \theta_{\la} ^{p-2} \r)^{1/p} \mbox{ for } \eta \mbox{ 1-form along } f_{\lambda} \end{eqnarray*} \noindent The weighted norm of a vector field or 1-form on $f_{\lambda}$ equals its usual norm off $B(0,2\sqrt \lambda)$ and on $ B(0,\sqrt \lambda)$ it is equal with the norm of its pulled back on $S^2$ via a rescaling of factor $\lambda$. The usual Sobolev embeddings hold for this weighted norms with constants independent of $\lambda$. \begin{lemma}{\label{lg1o}} There exists $\varepsilon_{0} > 0$ and constants $C > 0 $ such that for any $p \ge 1$ and $\lambda \; \le \; \varepsilon_{0}$: \begin{equation}\label{fla} \| df_{\lambda} {\|}_{p,\lambda} \; \le \;\;\; C \mbox{ and }\;\;\; \| \overline \partial_{J} f_{\lambda} {\|}_{p,\lambda} \; \le \; C \lambda^ {1/p} \end{equation} \noindent Moreover on the annulus A: $\{\sqrt \lambda \; \le \; |z| \; \le \; 2\sqrt \lambda\}$ we have the following expansion: \begin{equation}{\label{pe1}} \overline \partial_{J} {f_\lambda} = {\sqrt \lambda \over |z|}\; d\beta \cdot d f (y)(u) + \o (\lambda) \end{equation} The estimates are uniform on $Gl\rightarrow {\cal N}$. \end{lemma} \noindent {\bf Proof. } Let $B$ be the disk $|z| \le \sqrt \lambda$. Note that $d {f_\lambda}$ vanishes for $|z| \ge 2\sqrt \lambda$ and by the definition of the weighted norm on $B$, \[ \|d {f_\lambda} \|_{p,\lambda,B} = \|d f \|_{p,B}\] But (\ref{d2f}) implies that \begin{eqnarray}\label{df} {\rm max}\{\;|df(z)|,\; |z|\le 1\}\le 2 \end{eqnarray} In the same time, $ \overline \partial_{J} {f_\lambda}=0$ outside A. Hence we need only to consider what happens in A. But on $A$ \[ |\overline \partial_{J} {f_\lambda} | \; \le \; C | d {f_\lambda} | \; \le \; C( |d \b_\lambda|\;| f | + |\b_\lambda |\;| df |)\; {\lambda\over |z|^2} \; \le \; C{1 \over \sqrt \lambda} \ma( \sup )_{B}|f | +C \; \le \; C \] \noindent since $\ma( \sup )_{B}|f (z)| \; \le \; \sqrt \lambda\; \ma( \sup )_{B}| d f | \; \le \; 2 \sqrt \lambda\;$ in normal coordinates on ${ \Bbbd P}^n$ at $f(y)$. This concludes the first part of the proof. For the second part, notice that on $A$ \[ \overline \partial_{J} {f_\lambda} = \overline \partial_{J} \b_\lambda \cdot f + \b_\lambda \cdot \overline \partial_{J} f = {1\over \sqrt \lambda}\; d \beta \;{z \over |z|}\cdot f \l( {\lambda \over z}\r) \] \noindent since $f $ is holomorphic. But using (\ref{d2f}) in normal coordinates on ${ \Bbbd P}^n$ at $f(y)$ and $y=0$, we get $|f(z)-f(0)-df(0)(z)|\le 2|z|^2$ so \[ f \l({\lambda\over z}\r)={\lambda\over z}\cdot df_y(u)+\o(\lambda)\;\;\mbox{ on }\;A \] Substituting this in the formula for $\overline \partial_{J} {f_\lambda}$ we obtain (\ref{pe1}).\qed \medskip \step{Extending the approximate gluing map} The approximate gluing map extends naturally to the bubble tree compactification $\ov{\cal U}_{d}$ of the moduli space of 1-marked curves. For simplicity, let ${\cal N}$ denote some boundary stratum modeled on a bubble tree $B$ and corresponding to a certain distribution of the degree $d=d_1+\dots+d_m$ on the bubbles. If $[f_i,y_i],\;i=1,\dots,m$ are the bubble curves corresponding to the bubble map $f:B\rightarrow{ \Bbbd P}^n$, then the gluing data $Gl$ is a collection of unit vectors tangent to each sphere in the domain at the corresponding bubble point together with gluing parameters: \begin{eqnarray}\label{gl} Gl=\{\; (\;[f_i,y_i,u_i],\lambda_i\;)_{i=1}^m\;|\; u_i\in T_{y_i}S^2, \;|u_i|\ne 0, \lambda_i\le \varepsilon\} \end{eqnarray} Note that as long as $f_i$ is not a constant map, then we can choose a unique reresentative of $[f_i,y_i,u_i]$ as in (\ref{uniquerep1}), (\ref{uniquerep2}). Then Lemma \ref{lg1o} extends naturally to ${\cal N}$ to give \begin{lemma}\label{lg1} With the notations above, let ${f_\lambda}$ be an approximate gluing map, and $A_1,\dots A_m$ be the corresponding annuli of radii $\lambda_i$ in which the cutoff functions are supported. Then for $\varepsilon$ small enough, there exists a constant $C$ such that: \[\| df_{\lambda} {\|}_{p,\lambda} \; \le \;\;\; C\;,\;\;\;\; \| \overline \partial_{J} f_{\lambda} {\|}_{p,\lambda} \; \le \; C \lambda^ {1/p} \] Moreover, $\overline \partial_{J}{f_\lambda}=0$ except on the annuli $A_i$ that correspond to nontrivial bubbles, where \begin{eqnarray}\label{aprglbd} \overline \partial_{J} {f_\lambda}= -{\sqrt \lambda_i\over |z|}\; d\beta \cdot d f_i (y_i)(u_i) + \o (\lambda_i) \end{eqnarray} The estimates above are uniform on $Gl\rightarrow {\cal N}$. \end{lemma} We will see later that most of the important information is encoded in the first level of nontrivial bubbles. \subsection{The Obstruction Bundle}\label{Obstr} In order to see which of the approximate maps can be corrected to solutions of the equation $\overline \partial_{J} f=\nu$ we need first to understand the behaviour of the linearization of this equation over the space of approximate solutions. \smallskip Recall that transversality fails at a bubble tree with ghost base, so the linearization at such bubble tree is not onto. The cause of that is the ghost base. Thus we start by analysing the ghost maps: Consider the moduli space of holomorphic maps $f:T^2\rightarrow { \Bbbd P}^n$ representing $0\in H_2({ \Bbbd P}^n )$. Obviously, the only such maps are the constant ones (ghosts). If $D_u$ is the linearization of the section $ \overline \partial_{J} : {\cal M} aps(T^2,{ \Bbbd P}^n ) \rightarrow \Lambda^{0,1} $ at $f:T^2\rightarrow { \Bbbd P}^n $, $f(x)=u$ a constant map, then \begin{eqnarray*} \mbox{ index D}_u&=&\mbox{dim }{\rm Ker D}_u- \mbox{dim }{\rm Coker D}_u = c_1(0) + n(1-1)=0 \end{eqnarray*} and \[ {\rm Coker D}_u = \ho(f)\cong T_u{ \Bbbd P}^n\;\;\;\mbox{ (canonically) }\] \noindent since $f^*(T{ \Bbbd P}^n)$ is a trivial bundle, so the elements $\omega\in \ho(f)$ are constant on the torus, i.e. have the form $\omega=Xdz$ for some $X\in T_u{ \Bbbd P}^n$. \bigskip Now if $f:B\rightarrow { \Bbbd P}^n$ is a bubble tree map whose base is a ghost torus $u=f(y)\in { \Bbbd P}^n$, let $D_f$ be the linearization at $f$ of the section $ \overline \partial_{J} : {\cal M} aps(B,{ \Bbbd P}^n ) \rightarrow \Lambda^{0,1}$. Then \begin{eqnarray*} \mbox{ index D}_f&=&\mbox{dim }{\rm Ker D}_f- \mbox{dim }{\rm Coker D}_f =-1 \end{eqnarray*} To describe ${\rm Coker D}_f$ we will use the following: \begin{defn} \label{obstrbd} If $f:B\rightarrow { \Bbbd P}^n$ is as above, let $B_1\subset B$ consist of the domains of all the ghost bubbles with image $f(y)$, $B_2=B -B_1$ and $\wt B\subset B$ denote the first level of bubbles that are not in $B_1$. \end{defn} Then ${\rm Coker D}_f $ is $n$ dimensional, consisting of 1-forms $\omega$ such that \[ \omega=\left\{ \begin{array}{ll} X dz &\mbox{ on } B_1\\ 0&\mbox{ on } B_2 \end{array}\right.\] for some $X\in T_u{ \Bbbd P}^n$. In particular, there is a natural isomorphism \begin{eqnarray}\label{obB} \begin{array}{c} {\rm Coker D} \;\;\cong \;\;\mbox{\rm ev}^*(TP^n)\\ \searrow\;\;\;\;\;\;\;\swarrow\;\;\\ \ov{\cal U}_d \end{array} \end{eqnarray} where $\mbox{\rm ev}:\ov{\cal U}_d \rightarrow { \Bbbd P}^n$ is the evaluation map. Since the moduli space of bubble trees $\ov{{\cal U}_d}$ is compact, there exists a constant $E>0$ such that $D_f D^*_f$ has a zero eigenvalue with multiplicity $n$, and all the other eigenvalues are greater than $2E$ for all $f\in \ov {\cal U}_d$. \bigskip When ${f_\lambda}$ is an approximate map, let $D_\lambda$ be the linearization of \[\overline \partial_{J} : {\cal M} aps(T^2,{ \Bbbd P}^n ) \rightarrow \Lambda^{0,1} \] at ${f_\lambda}$ and $ D^*_\lambda$ its $L^2$-adjoint with respect to the metric $g_\lambda$ on $T^2$. Then $D_\lambda$ is not uniformly invertible. More precisely, \begin{lemma}\label{lg2} For $\lambda > 0$ small, the operator $\Delta_{\lambda}= D_\lambda D^*_\lambda $ has exactly $n$ eigenvalues of order $\sqrt \lambda$ and all the others are greater than E. Moreover, over the set of gluing data $Gl$, the span of low eigenvalues \begin{eqnarray*} \begin{array}{ccc} \lal(f_\lambda) &\hookrightarrow& \Lambda^{0,1}_{low}\\ &&\downarrow \\ &&Gl\end{array} \end{eqnarray*} is a $n$-dimensional vector bundle (called the {\rm Taubes obstruction bundle}), naturally isomorphic to the bundle \[ \mbox{\em{\rm ev}}^* (T{ \Bbbd P}^n )\rightarrow Gl\] where $\mbox{\em{\rm ev}}: Gl \rightarrow{ \Bbbd P}^n $ is the evaluation map. \end{lemma} \noindent {\bf Proof. } The proof is more or less the same as the one Taubes used for the similar result in the context of Donaldson theory, \cite{t2}. For each gluing data in $Gl$, by cutting and pasting eigenvectors we show that the eigenvalues of $\Delta_\lambda=D_\lambda D^*_\lambda$ are $\o (\sqrt \lambda)$ close to those of $\Delta_u=D_u D^*_u$, where $u$ is the point map in the base of the bubble tree. Take for example the top stratum of $\ov{\cal U}_d$. Choose $\{\omega_i,\; i=1,n\}$ a local orthonormal base of ${\rm Coker D}\cong\mbox{\rm ev}^*(T{ \Bbbd P}^n)$ and define \begin{eqnarray}\label{taubes} \omega_{\la}^i (z) = \beta\l({z\over 2\sqrt \lambda}\r) \omega^i(z) \end{eqnarray} A straightforward computation shows that: \begin{eqnarray} \| D^*_\lambda \omega_\lambda \|_{2,\lambda} &\le & \lambda^{1/4} \|\omega_\lambda \|_{2,\lambda}\\ \langle \overline \omega_{\la}^i,\overline \omega_{\la}^j \rangle _{2,\lambda}&=&\delta_{ij}+{\o}(\lambda) \end{eqnarray} The Gramm-Schmidt orthonormalization procedure then provides $n$ eigenvectors $\overline \omega_{\la}^i$ for $\Delta_{\lambda}$ with eigenvalues $\o (\sqrt \lambda)$ such that \[ \overline \omega_{\la}^i =\omega_{\la}^i + \o(\lambda) \] The construction above extends naturally to the other substrata of $\ov{\cal U}_d$. Note that for example when $B_1$ has other components besides $T^2$ then $\overline \omega_{\la}$ is equal to $\omega$ not only on the ghost base, but on all $B_1$ and is extended with 0 starting from the first level of nontrivial bubbles. \medskip An adaptation of Taubes argument from \cite{t1} shows that there are at most $n$ low eigenvalues of $\Delta_\lambda$. Therefore there is a well defined splitting \[\lla(f_\lambda) = \lal(f_\lambda) \oplus \lae(f_\lambda)\] The definition (\ref{taubes}) combined with (\ref{obB}) provides the isomorphism $\Lambda^{0,1}_{low} \cong \mbox{\rm ev}^*(T{ \Bbbd P}^n )$, concluding the proof. \qed \step{The partial right inverse of $D_\lambda$} The restriction of $D_{\lambda} D^{*}_{\lambda} $ to $\Lambda^{0,1}_E$ is invertible (since all its eigenvalues are at least E). Define $ P_\lambda$ to be the composition of the $L^2$-othogonal projection $\Lambda^{0,1} \rightarrow \Lambda^{0,1}_E$ with the operator $ D^*_\lambda ( {D_{\lambda} D^{*}_{\lambda}})^{-1} $ on ${ \Lambda^{0,1}_E}$. Then \begin{eqnarray} P_\lambda : \lla(f_\lambda) \rightarrow \lo(f_\lambda) \end{eqnarray} is the {\em partial right inverse of} $D_\lambda$ and satisfies the uniform estimate: \begin{equation} \|P_\lambda \eta{\|}_{1,p,\lambda} \; \le \; E^{-1} \| \eta {\|}_{p,\lambda} \end{equation} \noindent We will denote by $\pi_{-}^{{f_\lambda}} : \lla(f_\lambda) \rightarrow \lal(f_\lambda)$ the projection onto the fiber of the obstruction bundle. \subsection{The Gluing map}\label{Gluing} The next step is to correct the approximate gluing map to take values in the moduli space ${\cal M}_{t\nu}$ of solutions to the equation \begin{equation}\label{eg0} \overline \partial_{J} f(x)=t\cdot \nu(x,f(x)) \end{equation} \noindent where $\nu$ is generic and fixed and $t$ is a small parameter. If $f_\lambda$ is an approximate map, use the exponential map to write any nearby map in the form $f=\exp_{{f_\lambda}} (\xi) $, for some correction $\xi \in \lo({f_\lambda})$. Let $D_\lambda$ be the linearization of the $\overline \partial_{J}$-section at ${f_\lambda} $ so \begin{equation}\label{del0} \overline \partial_{J} f = \overline \partial_{J} {f_\lambda} + D_\lambda(\xi) + Q_\lambda(\xi) \end{equation} \noindent where $ Q_\lambda$ is quadratic in $\xi$. Similarly, \[\nu(x,f(x))= \nu(x,{f_\lambda}(x)) + d\nu(\xi)+\widetilde Q_\lambda(\xi)\] so equation (\ref{eg0}) can be rewritten as: \begin{eqnarray}\label{eg00} { D_\lambda(\xi)+N_\lambda(\xi,t) = t\nu(x,{f_\lambda}(x)) -\overline \partial_{J} {f_\lambda} } \end{eqnarray} where $N_\lambda(\xi,t)=Q_\lambda(\xi)-td\nu (\xi)-t\widetilde Q_\lambda(\xi)$ is quadratic in $(\xi, t)$. \medskip \noindent The kernel of $D_\lambda$ models the tangent directions to the space of approximate maps, so it is natural to look for a correction in the normal direction. More precisely, we will consider the solutions of (\ref{eg00}) of the form \begin{eqnarray} f=\exp_{f_\lambda} ( P_\lambda \eta)\;\;\; \mbox{ where }\;\;\;\pi_{-}(\eta)=0 \end{eqnarray} Since $ D_\lambda ( P_\lambda(\eta))= \eta$ for such $\eta$, then equation (\ref{eg00}) becomes \begin{equation}\label{eg1} \eta + N_{\lambda,t} ( P_\lambda \eta) = t \nu-\overline \partial_{J} f_\lambda \end{equation} The existence of a solution of (\ref{eg1}) is a standard aplication of the Banach fixed point theorem combined with the estimates in the previous sections. \begin{lemma}\label{l3} There exists a constant $\delta >0$ (independent of $\lambda,t$) such that for $t$ small enough and for any $\alpha \in \lla(f_\lambda) $ so that $\| \alpha {\|}_{p,\lambda} < \delta/2$ the equation: \[ \eta + N_{\lambda,t} ( P_\lambda \eta) = \alpha \] has a unique small solution $\eta \in \lla({f_\lambda})$ with $\| \eta {\|}_{p,\lambda} < \delta $. Moreover, \[\| \eta {\|}_{p,\lambda} < 2 \| \alpha {\|}_{p,\lambda} \] and if $\alpha$ is $C^\infty$, so in $\eta$. \end{lemma} \noindent {\bf Proof. } Apply the contraction principle to the operator \[ T_\lambda : \lla(f_\lambda) \rightarrow \lla(f_\lambda) \] \[ T_\lambda \eta = \alpha - N_{\lambda,t} ( P_\lambda \eta) \] \noindent defined on a small ball centered at 0 in the Banach space $\lla({f_\lambda})$ with the weighted Sobolev norm $L^p_\lambda$. To prove that T is a contraction we note that: \[ \| T_\lambda \eta_1 - T_\lambda \eta_2 {\|}_{p,\lambda} = \| N_{\lambda,t} ( P_\lambda \eta_1)- N_{\lambda,t} ( P_\lambda \eta _2) {\|}_{p,\lambda} \] \noindent and use some estimates of Floer. He proved in \cite{F} that for the quadratic part $Q$ of (\ref{del0}), there exists a constant $C$ depending only on $\| df {\|}_{p,\lambda}$ such that: \begin{eqnarray}\label{floer1} \| Q_f(\xi_1)- Q_f(\xi_2) {\|}_{p,\lambda} \; \le \; C \; (\; \|\xi_1 {\|}_{1,p,\lambda} + \|\xi_2 {\|}_{1,p,\lambda} ) \| \xi_1-\xi_2 {\|}_{1,p,\lambda} \end{eqnarray} \begin{eqnarray} \label{floer2} \| Q_f(\xi) {\|}_{p,\lambda} \; \le \; C \; \|\xi \|_{\infty,\lambda} \cdot \|\xi {\|}_{1,p,\lambda} \end{eqnarray} (Floer's estimates are for the usual Sobolev norm, but the same proof goes through for the weighted norms.) Since $\| d{f_\lambda} {\|}_{p,\lambda}$ is uniformly bounded by Lemma $\ref{lg1}$, the same constant $C$ works for all $ f_\lambda \in Im(\gamma_{\varepsilon})$. Moreover, for $t$ very small the same estimates hold for the nonlinear part $N_{\lambda,t}$. Hence by (\ref{floer1}): \begin{eqnarray*} \| T_\lambda \eta_1 - T_\lambda \eta_2 {\|}_{p,\lambda} & \le& C \;(\; \| P_\lambda \eta_1 {\|}_{1,p,\lambda} + \| P_\lambda \eta_2 {\|}_{1,p,\lambda}) \| P_\lambda (\eta_1-\eta_2) {\|}_{1,p,\lambda} \\ & \le & C/E^2 \;( \;\| \eta_1 {\|}_{p,\lambda} + \| \eta_2 {\|}_{p,\lambda} ) \cdot\| \eta_1-\eta_2 {\|}_{p,\lambda}. \end{eqnarray*} Choosing $\delta < E^2/(4C)$ this implies \[ \| T_\lambda \eta_1 - T_\lambda \eta_2 {\|}_{p,\lambda} \; \le \; 1/2 \;\| \eta_1-\eta_2 {\|}_{p,\lambda} \] for any $\eta_1,\eta_2 \in B(0,\delta)$. Moreover, since $ \| T_\lambda (0) {\|}_{p,\lambda} \le \delta/2 $ then $ T_\lambda :B(0,\delta) \rightarrow B(0,\delta)$ is a contraction. Therefore $T_\lambda$ has a unique fixed point $\eta$ in the ball such that moreover \[ \|\eta {\|}_{p,\lambda} \;\le \;\|T_\lambda\eta -T_\lambda(0){\|}_{p,\lambda} + \|T_\lambda(0){\|}_{p,\lambda}\;\le 1/2 \;\|\eta {\|}_{p,\lambda}+ \| T_\lambda(0){\|}_{p,\lambda}\] \noindent so $\|\eta {\|}_{p,\lambda} \le 2\;\|T_\lambda(0){\|}_{p,\lambda}=2\|\alpha{\|}_{p,\lambda}$. Elliptic regularity implies that $\eta$ is smooth when $\alpha$ is.\qed \begin{cor}\label{defeta} For $t,\lambda$ small enough, equation (\ref{eg1}) has a unique small solution $\|\eta{\|}_{p,\lambda} \; \le \; \delta$. Moreover, \[ \|\eta{\|}_{p,\lambda} \; \le \; C( t|\nu | + \lambda^{1\over p}). \] \end{cor} \noindent {\bf Proof. } Follows immediately from Lemmas \ref{lg1} and \ref{lg2} and the estimate \[ \| \alpha {\|}_{p,\lambda} = \| \;t\nu -\overline \partial_{J} {f_\lambda} {\|}_{p,\lambda} \; \le \; t|\nu| + C \lambda^{1\over p}. \hskip.5in \Box\] \step{The gluing map} Let $ Gl$ be the set of gluing data. The {\em gluing map} is defined by \[ \bar \gamma_{\varepsilon} :Gl \rightarrow {\cal M} aps(T^2, X) \] \[ \bar \gamma_{\varepsilon} ([f,y,u],\lambda) = \bar f_\lambda =\exp_{f_\lambda}(P_\lambda \eta ) \] \noindent where $\eta=\eta(f,y,u,\lambda)$ is the unique solution to the equation (\ref{eg1}) given by Corrolary \ref{defeta}. By construction, $ \bar \gamma_{\varepsilon} $ is a local diffeomorphism onto its image. Moreover, if $\pi_{-}^{f_\lambda}(\eta)=0$ then $\bar {f_\lambda}$ is actually a solution of (\ref{eg0}). \step{The obstruction to gluing} The section \[ \psi_t :Gl \rightarrow \lal(f_\lambda) \;\;\;\mbox { given by } \] \[ \psi_t (f,y,u,\lambda) = \pi_{-}^{f_\lambda}(\eta)= \pi_{-} ^{f_\lambda}( t \nu - \overline \partial_{J} f_\lambda ) - \pi_{-} ^{f_\lambda}( N_{\lambda,t} ( P_\lambda \eta)) \] will be called the {\em obstruction to gluing}. \noindent Let $Z_t =\psi_t^{-1} (0)$ be the zero set of this section. By applying the gluing construction to bubble trees in $Z_t$ we get a subset of the moduli space $ {\cal M}^{t \nu}$. \subsection{Completion of the construction}\label{Compl} We have seen in the previous section that applying the gluing construction to the bubble trees in the zero set $Z_t$ we will get elements of the moduli space ${\cal M}_{d,1,t\nu}$. It is not clear yet why all the elements of this moduli space close enough to the boundary stratum ${\cal N}$ can be obtained by the gluing procedure. The purpose of this section is to clarify this issue. \medskip Recall the construction of the gluing map: Starting with a bubble tree we glue in the bubble to obtain an approximate map ${f_\lambda}$. Then we correct ${f_\lambda}$ by pushing it in a direction normal to the kernel of $D_\lambda$ in order to get an element of the moduli space ${\cal M}^{t\nu}$. The key fact here is that the kernel of the linearization $D_\lambda$ models the tangent space to the approximate maps, and therefore, at least in the linear model, it is enough to look for solutions only in a normal direction. For the construction to be complete though, we need to show that the same thing is true for the nonlinear problem. \medskip More precisely, we will show that for $t$ small, all the elements of the moduli space ${\cal M}_{d,1,t\nu}$ close to the boundary stratum ${\cal N}$ can be reached starting with an approximate map and going out in a normal direction. The proof of the following Theorem is an adaptation of the proof for the same kind of result in the context of Donaldson theory \cite{dk}. It is pretty technical and we include it just for continuity of the presentation. \begin{theorem}\label{completion} The end of the moduli space ${\cal M}_{d,1,t\nu}$ close to the boundary strata ${\cal N}$ is diffeomorphic to the zero set of the section $\psi_t$. More precisely, for $\delta $ and $t $ small enough, there exists an isomorphism \[{\cal M}_{d,1,t\nu} \cap {\bf U}_\delta \cong \psi^{-1}_t(0)\hskip.6in \mbox{where}\] \[{\bf U}_\delta =\{ f:T^2\rightarrow X \mid \exists {f_\lambda} \;s.t.\; f=\exp_{f_\lambda}(\xi) ,\; \| \xi {\|}_{1,2,\lambda} \le \delta,\; \| \overline \partial_{J} f{\|}_{2,\lambda} \le \delta^{3/2} \} \] and $ f_\lambda \in {\rm Im} \gamma_{\varepsilon}$ is some approximate map. \end{theorem} \noindent {\bf Proof. } The proof consists of 2 steps. First, Lemma \ref{nbd} shows that ${\bf U}_\delta$ is actually a neighborhood of ${\cal N}$ in the bubble tree convergence topology. Second, recall that in constructing the section $\psi_t$ we were looking for solutions of the equation (\ref{eg0}) that have the form \begin{equation}\label{form} f=\exp_{{f_\lambda}}( P_\lambda \eta )\;\;\mbox{ for some }\; \|\eta {\|}_{2,\lambda} \; \le \; \delta \end{equation} To prove the Theorem it is enough to show that for $t$ small, all the solutions of the equation (\ref{eg0}) can be written in the form (\ref{form}). This is a consequence of Proposition \ref{prop}. \begin{lemma}\label{nbd} ${\bf U}_\delta\cap \ov{{\cal M}^\nu}$ is a neighborhood of ${\cal N}$ in the bubble tree convergence topology. More precisely, for any $(J,t\nu)$-holomorphic map f close to the boundary strata ${\cal N}$ there exists an approximate map ${f_\lambda}$ such that f can be written in the form \[ f=\exp_{f_\lambda}( \xi )\;\;\mbox{ for some }\; \|\xi {\|}_{1,2,\lambda} \; \le \; \delta \] \end{lemma} \noindent {\bf Proof. } By contradiction, assume there exists a sequence $f_n$ of $(J,t_n\nu)$-holomorphic maps for ${t_n}\rightarrow 0$ such that $f_n$ do not have the required property. By the bubble tree convergence Theorem (\cite{pw}) there exists a bubble tree $f $ such that $f_n\rightarrow f $ uniform on compacts. Moreover, after rescaling the functions $f_n$ by some $\lambda_n$, this becomes a $L^{1,2}$-convergence (cf. \cite{pw}). But this is equivalent to saying that $f_n$ is $L^{1,2,\lambda_n}$ close to $f$. In particular, for $\lambda$ small enough, $f_n$ is $L^{1,2,\lambda_n}$ close to $f_{\lambda_n}$, which gives a contradiction. \qed \begin{prop}\label{prop} For small enough $\delta,t$ any map in ${\bf U}_\delta$ can be represented in the form \[ f=\exp _{f_\lambda} (P_{\lambda} \eta)\;\mbox{ for some }\;f_\lambda \in {\rm Im} \gamma_{\varepsilon}, \;\| \eta {\|}_{2,\lambda} < \delta \;\mbox{ and }\; \pi_{-}^{{f_\lambda}} (\eta)=0 \] \end{prop} \noindent {\bf Proof. } We will use the continuation method. The key fact is that a neighborhood of $f_\lambda $ in $ {\rm Im} \gamma_{\varepsilon} $ is modeled by $ \Lambda^{0}_{low} $ and that the image of $P_\lambda$ spans the normal directions to $ {\rm Im} \gamma_{\varepsilon} $. Let $f \in {\bf U}_\delta$. By definition, there is $f_\lambda \in {\rm Im} \gamma_{\varepsilon}$ such that $ f=\exp_{f_\lambda }\xi $, where $ \| \xi {\|}_{1,2,\lambda}< \delta $. Consider the path $f_s=\exp_{f_ \lambda}( s \xi) $. Let \begin{eqnarray} S =\{ s \in [0,1] \mid \exists f_{\lambda_s} \mbox{ and } \|\eta_s \|_{p,\lambda_s} < \delta \mbox{ such that } f_s= \exp _{f_{\lambda_s}} (P_{\lambda_s} \eta_s) \}. \end{eqnarray} \noindent Note that by definition $f =f_\lambda=\exp_{f_\lambda}(0)$ so $0\in S$. We will show that S is both open and closed and since it is nonempty, $1 \in S $. \medskip \noindent {\bf S is closed}. The only open condition in the definition of S is $\|\eta_s \|_{p,\lambda_s} < \delta $. But since \begin{eqnarray}\nonumber \overline \partial_{J} f_s &=& \overline \partial_{J} f_{\lambda_s} + D_{\lambda_s}(P_{\lambda_s}\eta_s ) + N_{\lambda}(P_{\lambda_s}\eta_s) \hskip.3in \mbox{then}\\ \nonumber \eta_s&=& \overline \partial_{J} f_s - \overline \partial_{J} f_{\lambda_s} - N_{\lambda_s}(P_{\lambda_s}\eta_s) \hskip.5in \mbox{so}\\ \nonumber \| \eta_s {\|}_{2,\lambda} &\le& \| \overline \partial_{J} f_s {\|}_{2,\lambda} + \| \overline \partial_{J} f_{\lambda_s} {\|}_{2,\lambda} + {C \over E^2} \| \eta_s {\|}_{2,\lambda}^2 \\ \label{etas} &\le & \| \overline \partial_{J} f_s {\|}_{2,\lambda} + C \sqrt \lambda + C \| \eta_s {\|}_{2,\lambda} ^2 \end{eqnarray} We need to estimate $\| \overline \partial_{J} f_s {\|}_{2,\lambda}$. But \begin{eqnarray}\nonumber &&\overline \partial_{J} f_s = \overline \partial_{J} f_{\lambda} + s D_{\lambda}(\xi ) +N_{\lambda} (s\xi) \hskip.2in\mbox{and}\\ \nonumber &&\overline \partial_{J} f_1 = \overline \partial_{J} f_{\lambda} + D_{\lambda}(\xi ) +N_{\lambda} (\xi) \hskip.2in\mbox{so}\\ \label{delfs2} &&\overline \partial_{J} f_s = s\overline \partial_{J} f_{1}+(1-s)\overline \partial_{J} f_{\lambda}+N_{\lambda} (s\xi) - sN_{\lambda} (\xi) \end{eqnarray} \noindent The relation (\ref{delfs2}) combined with the estimate (\ref{floer2}) gives $\;\|N_{\lambda} (\xi) {\|}_{2,\lambda} \le C \;\| \xi {\|}_{1,2,\lambda}^2 $ so \[ \| \overline \partial_{J} f_s {\|}_{2,\lambda} \; \le \; \| \overline \partial_{J} f_1 {\|}_{2,\lambda} + \| \overline \partial_{J} f_\lambda {\|}_{2,\lambda} + 2\;C \; \|\xi{\|}_{1,2,\lambda}^2 \; \le \; \sqrt \lambda +\delta^{3/2} + C\; \delta^2\] \noindent Therefore for $\lambda<<\delta$, \begin{equation}\label{delfs} \| \overline \partial_{J} f_s {\|}_{2,\lambda} \; \le \; 2\;C\;\delta^{3/2} \end{equation} Using (\ref{delfs}) in (\ref{etas}) we get \[ \| \eta_s {\|}_{2,\lambda} \; \le \; 2\;C\;\delta^{3/2} + C \sqrt \lambda + C \| \eta_s {\|}_{2,\lambda} ^2 \] \noindent For small $ \lambda \; \le \; \delta^{ 3}$ , the constraint $ \| \eta_s {\|}_{2,\lambda} < \delta $ implies $ \| \eta_s {\|}_{2,\lambda} < \delta/2 $ so it is a closed condition too. \qed \medskip \noindent {\bf S is open}. Assume that $s_0 \in S$, i.e. there exists an approximate map $f_{\lambda_0}$ such that $f_{s_0}=\exp _{f_{\lambda_0}}(P_{\lambda_0}(\eta_0))$. We will show that $s\in S$ for $s$ sufficiently close to $s_0$. For that we need to find an approximate map $f_{\lambda_s}$ and an $\eta_s \in \Lambda^{0}_{E}$ such that: \begin{equation}\label{long} f_s=\exp_{{f_\lambda}}(s\xi) =\; \exp _{f_{\lambda_s}} (P_{\lambda_s} \eta_s) \end{equation} It is enough to prove that the linearization of the equation (\ref{long}) is onto at $s_0$. First we prove that: \begin{lemma}\label{neigh} A small neighborhood ${\cal N}_\delta$ of $f_ \lambda $ in ${\rm Im} \gamma_{\varepsilon}$ is modelled by $ \Lambda^{0}_{low} $. More precisely, there is a well defined map $ g: \Lambda^{0}_{low} \rightarrow \Lambda^{0,1}_E$ such that any approximate map $f \in {\rm Im} \gamma_{\varepsilon}$ has the form $f=\exp _{f_ \lambda } (\zeta+ P_\lambda g( \zeta))$ for some $\zeta \in \Lambda^{0}_{low}$, $ \| \zeta {\|}_{1,2,\lambda} \; \le \; \delta$. \end{lemma} \noindent {\bf Proof. } The first statement is an immediate consequence of the way we constructed the approximate maps. For the second part, notice that any $f \in {\rm Im} \gamma_{\varepsilon} $ close to $f_\lambda$ can be written in the form $f=\exp _{f_ \lambda } (\chi)$, with $\chi$ small. Let $ \chi = \zeta + P_ \lambda \eta $ be the orthogonal decomposition of $\chi$ in $\Lambda^{0}_{low} \oplus \Lambda^{0}_{E} $, where $\eta \in \Lambda^{0,1}_E$ (recall that $P_\lambda: \Lambda^{0,1}_E \rightarrow \Lambda^{0}_{E} $ is an isomorphism). Using the same techniques as in Section \ref{Gluing} we can prove that for any $\zeta\in \Lambda^{0}_{low}$ there exists a unique solution $\eta=g(\zeta)$ to the equation \[ \eta + N_\lambda(P_\lambda\eta) = \overline \partial_{J} f\] \noindent which concludes the proof of Lemma. \qed \bigskip Since the notations are becoming cumbersome, we will illustrate for simplicity the case $s_0=0$. The general case follows similarly. Using Lemma \ref{neigh} we can regard the equation (\ref{long}) as an equation in $(\zeta,\eta) \in \Lambda^{0}_{low} \oplus \Lambda^{0,1}_E $. More precisely, for a fixed $s$ small, we need to find $\zeta \in \Lambda^{0}_{low}$ and $\eta \in \Lambda^{0}_{E}$ such that the approximate map $f= \exp _ {f_ \lambda } ( \zeta+ P_\lambda g( \zeta) )$ solves the equation: \begin{equation}\label{ec0} \exp_f (P_ f \eta) = \exp_{f_\lambda} (s \xi) \end{equation} The linearization of the equation (\ref{ec0}) at $(0,\eta)$ is ${\bf D} : \Lambda^{0}_{low} \oplus \Lambda^{0,1}_E \rightarrow \Lambda^{0}$, \[ {\bf D}_{(0,\eta)} ( {z}, {n} ) = {z} + P_\lambda \nabla g ({z}) + P_ \lambda {n} + \Pi ( {z},\eta )\] \noindent where $\Pi ( {z},\eta )$ is the derivative of $ P_\lambda \eta $ with respect to $ f_\lambda $. \bigskip Our goal is to show that the operator $ {\bf D}_{(0,\eta)}$ is an isomorphism in some appropriate norms on $ \Lambda^{0}_{low} \oplus \Lambda^{0,1}_E $ and $ \Lambda^{0}$. \begin{defn} On $ \Lambda^{0}_{low} \oplus \Lambda^{0,1}_E $ and $\Lambda^{0}$ define the following norms: \begin{eqnarray*} \| ({z},{n}) \|_{B_1}& =&\| {z} {\|}_{1,2,\lambda} + \| {n} + \nabla g ({z}) {\|}_{2,\lambda} \;\;\mbox{ for any } \;\;({z},{n}) \in \Lambda^{0}_{low}\oplus\Lambda^{0,1}_E\\ \| \xi \|_{B_2}& =& \| D_\lambda \xi {\|}_{2,\lambda} \quad \mbox{for any} \quad \xi \in \Lambda^{0} \end{eqnarray*} \end{defn} \noindent Consider the operator $ {\bf T}: \Lambda^{0}_{low} \oplus \Lambda^{0,1}_E \rightarrow \Lambda^{0}$ given by $ T( {z}, {n}) = z+ P_ \lambda ( {n} + \nabla g ({z}) )$. Then ${\bf T}$ is continuous, since \begin{eqnarray*} \| {\bf T}( {z}, {n}) \|_{B_2}& =& \| D_\lambda {z} + {n} + \nabla g ({z}) {\|}_{2,\lambda} \; \le \; \| D_\lambda {z} {\|}_{2,\lambda} + \| {n} + \nabla g ({z}) {\|}_{2,\lambda} \\ &\le& C \lambda^ {1/4} \| {z} {\|}_{1,2,\lambda} + \| {n} + \nabla g ({z}) {\|}_{2,\lambda}\le \|({z},{n}) \|_{B_1} \end{eqnarray*} \noindent for $ \lambda$ small enough. Recall that the low eigenvalues of $D_\lambda$ are of order $\lambda^{1/4}$, and thus $\| D_\lambda {z} {\|}_{2,\lambda} \; \le \; \lambda^ {1/4}\| {z} {\|}_{1,2,\lambda}$ on $\Lambda^{0}_{low}$. \begin{lemma}\label{l10} For $\lambda, \delta $ small enough ${\bf T}$ is invertible, with the operator norm of the inverse uniformly bounded $\| {\bf T}^{-1} \| \; \le \; C_T$ (independent of $\lambda, \delta $). \end{lemma} \noindent {\bf Proof. } Let $\alpha = {z} + P_\lambda( {n} + \nabla g ({z}) ) $. We need to estimate $ \| {z} {\|}_{1,2,\lambda} $ and $\| {n} + \nabla g ({z}) {\|}_{p,\lambda}$ in terms of $ \| \alpha \|_{B_2}$. Since $D_ \lambda \alpha =D_ \lambda {z} + {n}+ \nabla g ({z}) $ then \begin{eqnarray*} \| {n}+ \nabla g ({z}) {\|}_{2,\lambda} & \le& \| \alpha \|_{B_2} + \| D_ \lambda {z} {\|}_{2,\lambda} \; \le \; \| \alpha \|_{B_2} +C \lambda^ {1/4} \| \zeta {\|}_{1,2,\lambda} \\ & \le & \| \alpha \|_{B_2} + C \lambda^ {1/4} \;\| \alpha - P_\lambda ({n}+ \nabla g ({z}) ) {\|}_{1,2,\lambda} \\ & \le & \| \alpha \|_{B_2} + C \lambda^ {1/4}\; \| \alpha \|_{B_2} + C \lambda^ {1/4}\;\| {n}+ \nabla g ({z}) {\|}_{2,\lambda} \end{eqnarray*} So for $\lambda$ small we get the uniform estimate $ \| {n} + \nabla g ({z}) {\|}_{2,\lambda} \; \le \; C_1 \| \alpha \|_{B_2}$. This gives \begin{eqnarray*} \| {z} {\|}_{1,2,\lambda} & =&\| \alpha - P_\lambda ({n} + \nabla g ({z}) ) {\|}_{1,2,\lambda} \; \le \; \| \alpha \|_{B_2} + C \| {n} + \nabla g ({z}) {\|}_{2,\lambda} \\ &\le & C_2 \| \alpha \|_{B_2} \end{eqnarray*} thus \begin{eqnarray*} \| ({z},{n}) \|_{B_1} \; \le \; C_T \| {\bf T}({z},{n}) \|_{B_2} \end{eqnarray*} So ${\bf T}$ is an injective linear operator. But by construction ${\rm inded} ({\bf T})= 0$ thus ${\bf T}$ is invertible, with $\| {\bf T}^{-1} \| \; \le \; C_T$ (independent of $\lambda, \delta $).\qed \begin{lemma} For ${z}$ small, $ \| \Pi ( {z}, \eta) \|_{B_2} \; \le \; C \| \eta {\|}_{2,\lambda} \| ({z}, 0 ) \|_{B_1}$. \end{lemma} \noindent {\bf Proof. } By differentiating the relation $D_f P_f \eta = \eta $ with respect to $f$ at $f_\lambda$ we get \begin{eqnarray*} \partial D_f ( P_ \lambda \eta)({z}) + D_ f (\Pi ( {z}, \eta)) =0 \hskip.2in\mbox{ so}\\ \| \;D_ \lambda (\Pi ( {z}, \eta)) \; {\|}_{2,\lambda} = \| \;\partial D_f ( P_ \lambda \eta)({z})\; {\|}_{2,\lambda} . \end{eqnarray*} Using the expansion of \[ D_f \xi = {1\over 2} \l( \nabla \xi + J(f) \circ \nabla \xi \circ j \r) + {1\over 8} N_f ( \partial_J f , \xi ) \] (cf. \cite{MS}) it is easy to check that \[ \|\; \partial D_f ( P_ \lambda \eta)({z}) \; {\|}_{2,\lambda} \; \le \; C \| {z} \|_{\infty,\lambda} \| P_ \lambda \eta {\|}_{1,2,\lambda} \] uniformly in a neighborhood of $ f_ \lambda $. Therefore \begin{eqnarray*} \| \Pi ( {z}, \eta) \|_{B_2} &=& \|D_ f (\Pi ( {z}, \eta))\; {\|}_{2,\lambda} \; \le \; C \|{z} \|_{\infty,\lambda} \| P_ \lambda \eta {\|}_{1,2,\lambda} \\ &\le& C \| {z} {\|}_{1,2,\lambda} \| \eta {\|}_{2,\lambda} = C \| {z} \|_{B_2} \| \eta {\|}_{2,\lambda}.\hskip.3in \Box \end{eqnarray*} \noindent If we choose $\delta$ small enough then for $ \| \eta {\|}_{2,\lambda} < \delta$, \begin{eqnarray*} \| \Pi ( {z}, \eta) \|_{B_2} \; \le \; C_T/2 \; \|({z},{n}) \|_{B_2} \end{eqnarray*} where $C_T$ is the constant in Lemma $\ref{l10}$ so $ {\bf D}_{(0,\eta)} ( {z}, {n} ) = {\bf T} ( {z}, {n} )+ \Pi ( {z},{n} ) $ is still invertible. This concludes the proof of Proposition \ref{prop}. \qed \subsection{The leading order term of the obstruction $\psi_t$ for $t$ small} \label{Psi} Next step is to identify the leading order term of the section $\psi_t$ as $t\ra0$. Let ${\cal N}$ denote some stratum of $\ov{\cal U}_d$ and $Gl\rightarrow {\cal N}$ denote the gluing data as in (\ref{gl}). For the sake of the gluing construction, the gluing data has to be defined on the domain of the bubble tree. But we will see in a moment that the important information is encoded in the image curves. Introduce first some notation: If $u_i\in T_{y_i}S^2$ is a unit frame and $ \lambda_i$ is the gluing parameter, let \[ v_i=\lambda_i\cdot u_i \in T_{y_i} S^2,\;\; (v_i\ne 0)\;\;\;\; \mbox{ denote the gluing data.}\] \begin{defn} For any $[f,y,v]\in Gl$, such that $f:B\rightarrow { \Bbbd P}^n$ is an element of ${\cal N}$, let $([f_i,y_i,v_i])_{i=1}^m$ be the bubble maps together with the gluing data and let $u$ be the image of the ghost base (so $u=f_j(y_j)$ for all $j\in\wt B$). Set \begin{eqnarray} \label{da}a([f,y,v])&=& \ma(\sum)_{i=1}^n \ma(\sum)_{j\in \wt B} \langle \;df_j(y_j)(v_j) \;,\; X_i\;\rangle \omega_i \\ \label{dbarn} \bar \nu(x)&= & \ma(\sum)_{i=1}^n \int_ {T^2} \langle \;\nu(z, u) \;,\;\omega_i(z)\;\rangle \omega_i \end{eqnarray} \noindent where $\{\omega_i=X_idz,\; i=1,n\}$ is an orthonormal base of $\ho(u)$, $X_i\in T_u{ \Bbbd P}^n$ and $\wt B$ is as in Definition \ref{obstrbd}. \end{defn} Note that $a$ depends {\em only} on the gluing data on the first level $\wt B$ of essential bubbles, and $\bar\nu$ depends only on the image of the ghost base. Then \begin{lemma}{\label{lg4}} Using the notation above, let ${f_\lambda}$ be an approximate gluing map. Then for $t$ and $|\lambda|=\sqrt{\lambda_1^2+\dots\lambda_i^2}$ small enough, \begin{eqnarray}\label{prn} \pi_{-} ^{f_\lambda}( \nu )&=& \bar \nu(u) +\o(|\lambda|) \\ \label{prfl} \pi_{-} ^{f_\lambda}( \overline \partial_{J} f_\lambda )&=& a([f,y,v]) + \o(|\lambda|^{3/2}). \end{eqnarray} and the section $\psi_t$ has the form \begin{eqnarray}\label{psi} \psi_t([f,y,v]) =t \bar\nu(u)+ a([f,y,v]) + \o ( |\lambda| ^{5/4}+ t \sqrt{|\lambda|} + t^2). \end{eqnarray} The estimates above are uniform on ${\cal N}$. \end{lemma} \noindent {\bf Proof. } For the first 2 relations, it is enough to check them on components. Assume for simplicity that $\wt B$ consists of a single bubble $[f,y,v]$. If $\omega=X dz$ is an element of the base for $H^{0,1}$, let $\overline \omega_{\la}$ be the element of the local orthonormal frame for $\lal(f_\lambda) $ provided by Lemma $\ref{lg2}$. Then \begin{eqnarray*} & |\langle \nu ,\omega_{\la}-\overline \omega_{\la} \rg_{\vphantom{I}_\lambda} | \; \le \; \|\nu \|_\infty \| \omega_{\la}-\overline \omega_{\la} {\|}_{2,\lambda} \; \le \; C \lambda& \quad \mbox{ so }\\ &\langle \nu,\overline \omega_{\la} \rg_{\vphantom{I}_\lambda}= \langle \nu,\omega_{\la} \rg_{\vphantom{I}_\lambda}+\o (\lambda)& \end{eqnarray*} On the other hand, using the definition of $\omega_{\la}$ and the fact that $\langle\;,\;\rg_{\vphantom{I}_\lambda}$ is the usual inner product on $T^2$ off a small ball we get \begin{eqnarray*} \langle \nu,\omega_{\la} \rg_{\vphantom{I}_\lambda}&=& \ma(\int)_{|z|>\sqrt \lambda} \langle \nu(z,f(y)), \omega \rangle = \ma(\int)_ {T^2} \langle \nu(z,f(y)), \omega \rangle + \o(\lambda)\;\;\mbox{ so}\\ \langle \nu,\overline \omega_{\la} \rg_{\vphantom{I}_\lambda}&=& \ma(\int)_ {T^2} \langle \nu(z,f(y)), \omega \rangle + \o(\lambda) \end{eqnarray*} which gives (\ref{prn}). Similarly, \[ |\langle \overline \partial_{J} f_\lambda ,\omega_{\la}-\overline \omega_{\la} \rg_{\vphantom{I}_\lambda} | \; \le \; \| \overline \partial_{J} f_\lambda {\|}_{2,\lambda} \| \omega_{\la}-\overline \omega_{\la} {\|}_{2,\lambda} \; \le \; C \lambda^{1/2} \lambda \; \le \; C \lambda^{3/2} \] \noindent and using the estimate (\ref{pe1}) and the definition of $\omega_{\la}$ we get \begin{eqnarray*} \langle \overline \partial_{J} f_\lambda ,\omega_{\la} \rg_{\vphantom{I}_\lambda} &=& \ma(\int)_{\sqrt \lambda \; \le \; |z| \; \le \; 2\sqrt \lambda} {\sqrt \lambda \over |z|}\; d \beta \langle d f (y)(u), X \rangle + \o(\lambda^2)\\ \\ &=& \lambda \langle d f (y)(u), X \rangle + \o(\lambda^2) \end{eqnarray*} Combine the previous 2 relations we get \begin{eqnarray*} \langle \overline \partial_{J} f_\lambda ,\overline \omega_{\la} \rg_{\vphantom{I}_\lambda} = \langle d f (y)(\lambda u), X \rangle + \o(\lambda^{3/2}) \langle d f (y)(v), X\rangle +\o(\lambda^{3/2}) \end{eqnarray*} which implies (\ref{prfl}). \smallskip The general case when $B$ has more bubbles follows in a similar maner using the relation (\ref{aprglbd}) and the fact that $\omega$ is 0 pass the first level of nontrivial bubbles. \smallskip Finally, the relation (\ref{psi}) is a consequence of (\ref{fla}) provided we have an estimate of the the quadratic part. For that use (\ref{floer2}) to get \begin{eqnarray*} \langle\; N_\lambda (P_\lambda \eta)\;,\; \overline \omega_{\la} \rg_{\vphantom{I}_\lambda} & \le & \| N_\lambda (P_\lambda \eta)\; \|_{4/3,\lambda} \;\| \overline \omega_{\la} \|_{4,\lambda} \; \le \; C \| \eta \|_{2,\lambda} \;\| \eta \|_{4/3,\lambda} \;\| \omega \|_{4} \\ & \le & \o( |\lambda|^{1/2} +t)\; \o (|\lambda|^{3/4}+t). \end{eqnarray*} \noindent Thus the quadratic part is $\o( |\lambda| ^{5/4}+t\sqrt{|\lambda|}+ t^2)$. \qed \bigskip \step{ The definition of $\wt L\rightarrow \wt{\cal U}_d$ } {}From this point on, since we are going to look at the leading order term, it will become easier if we forget part of the gluing data. We have already observed that the map $a$ depends only on the gluing data on the first level $\wt B$ of essential bubbles. Moreover, if we denote by \begin{eqnarray}\label{w} w=\ma(\sum)_{j\in \wt B} df_j(y_j)(v_j)\in T_u{ \Bbbd P}^n \end{eqnarray} then the map $a$ and the linear part $\wt\psi_t$ of $\psi_t$ become respectively \begin{eqnarray}\label{ao} a(w)&=&\ma(\sum)_{i=1}^n\langle w, X_i\rangle \omega_i\\ \label{psitdo} \wt\psi_t(w)&=&t \bar\nu(u)+ a(w) \end{eqnarray} Introduce a space ${\cal W}$ together with a projection $\pi:{\cal W}\rightarrow\ov {\cal U}_d$ such that the fiber of $\pi$ at a 1-marked curve (possibly with more components) is the span of the tangent planes to all the image bubbles that meet at the marked point. By definition $w\in {\cal W}$ so (\ref{w}) defines a projection $p:Gl\rightarrow {\cal W}$. Note though that $\pi:{\cal W}\rightarrow\ov{\cal U}_d$ is not a vector bundle, and that ${\cal W}$ it is equal to the {\em relative tangent bundle} $L\rightarrow {\cal U}_d$ on the top strata of $\ov{\cal U}_d$. \medskip Here is a more precise description of ${\cal W}$. Stratify $\ov{\cal U}_d$ by letting ${\cal Z}_h$ be the union of all boundary strata such that the image of the marked point is on $h$ nontrivial bubbles, i.e. \begin{eqnarray}\label{cz} {\cal Z}_h=\{ f:B\rightarrow{ \Bbbd P}^n \;|\; \wt B \mbox{ has } h \mbox{ elements }\} \end{eqnarray} Each $\ov{\cal Z}_h$ is a variety with normal crossings. For transversality arguments we need to use the moduli space $\widehat{\cal Z}_h$ obtained from ${\cal Z}_h$ by collapsing all the ghost bubbles up to the first level of essential bubbles. Note that $\widetilde{\cal Z}_{2}\supset\widetilde{\cal Z}_3\supset\dots$, and the natural projection \[q:{\cal Z}_h\rightarrow\widehat{\cal Z}_h\] has fiber ${\cal U}_{0,h}={\cal M}_{0,h+1}$, the moduli space of $h+1$ marked points on the sphere. Moreover, \begin{eqnarray} {\rm dim}\; {\cal Z}_h= n-h-1\quad \mbox{ and } \quad {\rm dim}\; \widehat{\cal Z}_h= n-2h+1 \end{eqnarray} In particular, ${\cal Z}_h\ne\emptyset$ only for $h\le [{n+1\over 2}]$. \medskip Let $L_i$ be the pullback of the relative tangent sheaf to the $i$'th factor of $\widehat{\cal Z}_h$. When the constraints $\beta_1,\dots,\beta_k$ are in generic position, the fibers of $L_1,\dots,L_h$ over a point in $\widehat{\cal Z}_h$ are linearly independent subspaces of ${ \Bbbd P}^n$. This is because linear dependence imposes $n+1-h$ conditions, and $\widehat{\cal Z}_h$ is only $n-2h+1$ dimensional. So on ${\cal Z}_h$ \begin{eqnarray}\label{cw} {\cal W}|_{{\cal Z}_h}= q^*(L_1\oplus\dots\oplus L_h) \end{eqnarray} \addtocounter{theorem}{1 Since not all the gluing parameters can be zero, a dimension count argument similar to the one above shows that $w$ defined by (\ref{w}) is an element of ${\cal W}-\{0\}$, the space nonzero vectors in ${\cal W}$, thus $p:Gl\rightarrow {\cal W}-\{0\}$. \smallskip Note that ${\cal W}|_{{\cal Z}_h}$ is nothing but the normal bundle of ${\cal Z}_h$ in $\ov{\cal U}_d$, for any $2\le h\le [{n+1\over 2}]$. This observation allows us to get a line bundle out of ${\cal W}$ as follows: \begin{defn}\label{ltilde} Let $N=[{n+1\over 2}]$. Blow up $\ov{\cal U}_d$ along ${\cal Z}_{N}$ (the bottom strata), then blow up the proper transform of ${\cal Z}_{N-1}$ and so on, all the way up to blowing up the proper transform of ${\cal Z}_2$ and denote by \[\rho:\wt{\cal U}_d\rightarrow{\cal U}_d\] the resulting manifold. Similarly, after the first blow up, extend $L$ over the exceptional divisor $E_{N}$ as the universal line bundle over ${ \Bbbd P}(N_{{\cal U}_d}{{\cal Z}_N})$, the projectivization of the normal bundle of ${\cal Z}_N$, and so on. Let $\wt L\rightarrow \wt {\cal U}_d$ denote the blow up of $L$ constructed above. \end{defn} By definition, the total space of $\wt L \rightarrow \wt{\cal U}_d$ is the same as $\rho^*({\cal W})$. From now on, we will make this identification. \medskip Both the map $a$ and the linear part $\wt\psi_t$ of $\psi_t$ pull back to $\wt L-\{0\}$ as \begin{eqnarray}\label{a} a(w)&=&\ma(\sum)_{i=1}^n\langle w, X_i\rangle X_i\\ \label{psitd} \wt\psi_t(w)&=&t \bar\nu(\pi(w))+ a(w) \end{eqnarray} where $\pi:\wt L\rightarrow { \Bbbd P}^n$ is the composition $\wt L\rightarrow \wt{\cal U}_d\ma(\longrightarrow)^{ev} { \Bbbd P}^n$. For simplicity of notation, we have also denoted by $\mbox{\rm ev}:\wt{\cal U}_d\rightarrow { \Bbbd P}^n$ the composition $\wt{\cal U}_d \ma(\rightarrow)^\rho {\cal U}_d\ma(\longrightarrow)^{ev}{ \Bbbd P}^n$. Note that by definition, $a$ is a linear map but $\wt\psi_t$ is not, and we have the following diagramm: \unitlength=1mm \linethickness{0.4pt} \begin{center} \begin{picture}(95,15) \put(20,12){\makebox(0,0)[cc]{$\wt L-\{0\}$ }} \put(50,12){\makebox(0,0)[cc]{$\mbox{\rm ev}^*(T{ \Bbbd P}^n)$}} \put(72,12){\makebox(0,0)[cc]{$T{ \Bbbd P}^n$}} \put(35,0){\makebox(0,0)[cc]{$\widetilde{\cal U}_d$}} \put(72,0){\makebox(0,0)[cc]{${ \Bbbd P}^n$}} \put(28,12){\vector(1,0){12}} \put(34,15){\makebox(0,0)[cc]{$a,\;\wt \psi_t$ }} \put(23,9){\vector(3,-2){9}} \put(47,9){\vector(-3,-2){9}} \put(70,9){\vector(0,-1){6}} \put(72,3){\vector(0,1){6}} \put(74,6){\makebox(0,0)[cc]{$\bar\nu$}} \put(40,0){\vector(1,0){26}} \put(55,2){\makebox(0,0)[cc]{$\mbox{\rm ev}$}} \end{picture} \end{center} \begin{prop}\label{zeropsit} As $t\rightarrow 0$ the zero set of the section $\psi_t$ is homotopic to the zero set of its leading order term \[\wt\psi_t:\wt L-\{0\}\rightarrow \mbox{\rm ev}^*(T{ \Bbbd P}^n)\] \end{prop} \p In generic conditions and for $t$ small enough the zero sets of both sections \[\psi_t:Gl\rightarrow\mbox{\rm ev}^*(T{ \Bbbd P}^n)\;\;\;\mbox{ and }\;\;\; \rho^*p_*(\psi_t):\wt L-\{0\}\rightarrow \mbox{\rm ev}^*(T{ \Bbbd P}^n)\] consist of points lying on the top stratum of ${\cal U}_d$ and $\widetilde{\cal U}_d$ respectively. But on the top stratum, the projection $pr:Gl\rightarrow\wt L-\{0\}$ is an isomorphism, thus the two zero sets are diffeomorphic for $t$ small. Note that (\ref{psi}) gives \[ p_*(\psi_t(w)) =t \bar\nu(u)+ a(w) + \o ( |w| ^{5/4}+ t \sqrt{|w|} + t^2)\] Finally, Lemma \ref{nonvan} gives that $w=\o(t)$ on the zero set of $\psi_t$, so \[ p_*(\psi_t(w)) =t \bar\nu(u)+ a(w) + \o ( t^{5/4})\] giving the desired homotopy as $t\rightarrow 0$. \qed \begin{lemma}\label{nonvan} The linear map $a:\wt L-\{0\}\rightarrow \mbox{\rm ev}^*(T{ \Bbbd P}^n)$ defined in (\ref{a}) has no zeros when the constraints $\beta_1,\dots,\beta_l$ are in a generic position, thus there exists $C>0$ such that \begin{eqnarray}\label{aw} |a(w)|\ge C|w|\end{eqnarray} Moreover, there exists a uniform constant $C$ on $\wt L-\{0\}$ such that the zero set of $\psi_t$ is contained in $|w|\le Ct$. \end{lemma} \noindent {\bf Proof. } First part is a standard transversality argument and dimension count. Note that $a$ induces a map \begin{eqnarray*} a\otimes id:\wt L\otimes \wt L^* & \rightarrow & \mbox{\rm ev}^*(T{ \Bbbd P}^n)\otimes \wt L^* \;\;\;\;\;\; \mbox{ i.e.}\\ a\otimes id:\ov {\cal U}_d\times { \Bbbd C}&\rightarrow & \mbox{\rm ev}^*(T{ \Bbbd P}^n)\otimes \wt L^* \end{eqnarray*} Because of the ${ \Bbbd C}^*$-equivariance of $a$, the zero set of $a:\wt L-\{0\}\rightarrow \mbox{\rm ev}^*(T{ \Bbbd P}^n)$ is the same as the zero set of the section \begin{eqnarray*} \wt a: \wt{\cal U}_d&\rightarrow& ev^*(T{ \Bbbd P}^n)\otimes \wt L^* \\ \wt a(x)&=&(a\otimes id) (x, 1) \end{eqnarray*} If the constraints $\beta_1,\dots,\beta_k$ are in generic position, then $\wt a$ is transverse to the zero set of $\mbox{\rm ev}^*(T{ \Bbbd P}^n)$. But the base $\wt {\cal U}_d$ is only $n-1$ dimensional, while the fiber is $n$ dimensional, so generically $\wt a$ and thus $a$ has no zeros. For the second part, note that on the zero set of $p_*(\psi_t)$ \begin{eqnarray*} 0=p_*(\psi_t)&=& a(w)+t\bar\nu(u)+\o(|w|^{5/4} +|w|^{1/2} \;t +t^2 ) \;\;\;\mbox{ so }\\ a(w)&=&- t\bar\nu(u)-\o(|w|^{5/4} +|w|^{1/2} \;t +t^2 ) \end{eqnarray*} which combined with (\ref{aw}) gives \begin{eqnarray*} C|w|\le |a(w)|&\le& t|\bar\nu(u)|+ \wt C(|w|^{5/4} +|w|^{1/2} \;t +t^2) \;\;\;\mbox{ i.e. }\\ |w|(C-\wt C|w|^{1/4})& \le &Ct \end{eqnarray*} For $t$ and $w$ small, the left hand side is positive, completing the proof. \qed \subsection{The enumerative invariant $\tau_d$}\label{Tau} Next step is to find the zero set of the leading order term of $\psi_t$. As a warm-up we will discuss first the limit case $t=0$. The constructions described in the previous sections apply equally in this case, giving: \begin{prop}\label{psit0} Let ${\cal N}$ be a ghost base boundary stratum of $\ov{\cal U}_d$. Then the moduli space of $J$-holomorphic tori close to ${\cal N}$ is isomorphic to the zero set of a section in the obstruction bundle over the space of gluing data \[ \psi([f_i,y_i,v_i]_{i=1}^m) = a([f_i,y_i,v_i]]_{i=1}^m) +\o ( |\lambda| ^{5/4})\] where a is defined by (\ref{da}). Moreover, for generic constraints $\beta_1,\dots,\beta_l$, the number of $J$-holomorphic tori that define the enumerative invariant \[ \tau_d(\beta_1,\dots,\beta_l)\] is finite, and the moduli space of these holomorphic tori is at a positive distance from the ghost base boundary strata of the bubble tree compactification. \end{prop} \noindent {\bf Proof. } For the second part, note that $\psi$ and $\lambda^{-1}\psi$ have the same zero set, so as $\lambda\rightarrow 0$ the limit of the end of the moduli space of $J$-holomorphic tori is modeled by the zero set of the section $a$. But we have seen that generically $a$ has no zeros, and thus there are no $J$-holomorphic tori in a small neighborhood of that boundary stratum.\qed \bigskip Now we can now evaluate the contribution from the interior: \begin{prop}\label{int} For $t$ small, the number of $(J,t\nu)$-holomorphic maps that satisfy the constraints in the definition of $RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l)$ and are close to some $(J,0)$-holomorphic torus is equal to \[ n_j \tau_d(\beta_1,\dots,\beta_l)\] where $n_j=|Aut_{x_1}(j)|$ is the order of the group of automorphisms of the complex structure $j$ that fix the point $x_1$. \end{prop} \noindent {\bf Proof. } Recall that $RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l)$ counts the number of solutions of the equation \[\overline \partial_{J} f(x)=\nu(x,f(x))\] such that $f(x_1)\in \beta_1$ and $f$ passes through $\beta_2,\dots,\beta_l$. A generic path of perturbations converging to 0 provides a cobordism ${\cal M}^\nu$ to the solutions of the equation \[\overline \partial_{J} f(x)=0\] such that $f(x_1)\in \beta_1$ and $f$ passes through $\beta_2,\dots,\beta_l$. A $(J,0)$-holomorphic torus $f:T^2\rightarrow { \Bbbd P}^n$ is a smooth point of this cobordism, i.e. all the intersections are transversal and the cokernel $H^{0,1}(T^2, f^*(T{ \Bbbd P}^n))$ vanishes (since $f^*(T{ \Bbbd P}^n)$ is a positive bundle for the standard complex structure). \smallskip But the invariant $\tau_d(\beta_1,\dots,\beta_l)$ counts the number of such solutions mod the automorphism group of $j$. Imposing the condition $f(x_1)\in \beta_1$ reduces the stabilizer to just $Aut_{x_1}(j)$. \qed \medskip \addtocounter{theorem}{1 Note that the pertubed invariant counts the number of $(J,\nu)$- holomorphic maps with sign. This sign is determined by the spectral flow of the linearization $D_f$ to $\overline \partial_{J}$. In the limit, when $\nu=0$, we have $D_f=\overline \partial_{J}$ thus all $(J,0)$-tori have a positive sign. This agrees with the way they were counted classically to obtain $\tau_d$. \medskip \begin{lemma}\label{nonvan2} For generic $\nu$ the section $\bar\nu:\mbox{\rm ev}_*(\ov{\cal U}_d)\rightarrow T{ \Bbbd P}^n $ defined by (\ref{dbarn}) has no zeros. \end{lemma} \noindent {\bf Proof. } For generic $\nu$, the section $\bar\nu$ is transverse to the zero section. But the fiber of $T{ \Bbbd P}^n$ is $n$ dimensional, and the base $\mbox{\rm Im}(\cm)=\mbox{\rm ev}_*(\ov{\cal U}_d)$ is only $n-1$ dimensional, so $\bar\nu$ has no zeros generically.\qed \medskip \addtocounter{theorem}{1 The zeros $u\in{ \Bbbd P}^n$ of $\bar\nu$ give the location of the point maps $u$ that can be perturbed away to get genus one $(J,\nu)$-holomorphic maps representing $0\in H_2({ \Bbbd P}^n)$. Since index=0 then generically $\bar \nu$ has finitely many zeros. But $\mbox{\rm Im}(\cm)$ is a codimension 1 subvariety in ${ \Bbbd P}^n$ that doesn't depend on $\nu$. Then we can choose $\nu$ generic so that its zeros do not lie in $\mbox{\rm Im}(\cm)$, and thus $\bar \nu(f(y)) \ne 0$ for any $[f,y]\in\ov{\cal U}_d$. \medskip Moreover, Lemma \ref{zeropsit} showed that as $t\rightarrow 0$ the zero set $Z_t$ of $\psi_t$ is homotopic to the zero set $Z_0$ of the map \begin{eqnarray*} \psi_0: \wt L-\{0\}\rightarrow \mbox{\rm ev}^*(T{ \Bbbd P}^n)\\ \psi_0(w)=\bar \nu(\pi(w)) + a(w) \end{eqnarray*} where $a,\;\bar\nu$ are defined in (\ref{a}), (\ref{dbarn}) and $\pi:\wt L\rightarrow { \Bbbd P}^n$ is the composition $\wt L\rightarrow\wt{\cal U}_d\ma(\rightarrow)^{\mbox{\rm ev}} { \Bbbd P}^n$. We have made a change of variables $w\rightarrow w/t$. \medskip Next we identify the zero set $Z_0$. Since $\bar \nu(u)\ne 0$ on $\mbox{\rm Im}(\cm)$ then it induces a splitting of the obstruction bundle: \begin{eqnarray}\label{e} T{ \Bbbd P}^n\vert_{\mbox{\rm Im}(\cm) }={ \Bbbd C}<\bar \nu> \oplus \;E \end{eqnarray} \noindent where $E$ is an $n-1$ dimensional bundle, so \begin{eqnarray}\label{splitev} \mbox{\rm ev}^*(T{ \Bbbd P}^n) ={ \Bbbd C}<\bar \nu> \oplus\; \mbox{\rm ev}^* E \end{eqnarray} \begin{lemma}\label{etil} The number of zeros (counted with multiplicity) of $\psi_0$ is equal to \[ c_{n-1}(\mbox{\rm ev}^*(E)\otimes \wt L^*) \] \end{lemma} \noindent {\bf Proof. } Using (\ref{splitev}) map $ \psi_0:\wt L-\{0\}\rightarrow\mbox{\rm ev}^*(T{ \Bbbd P}^n) $ splits as \begin{eqnarray} \psi_1(w)=&\bar \nu(\pi(w))\;\; +& a_1(w) \\ \psi_2(w)=& &a_2(w) \end{eqnarray} where $a_i$ denote the projections of $a(w)$. The map $a_2:\wt L-\{0\}\rightarrow\mbox{\rm ev}^*(E)$ is ${ \Bbbd C}^*$-equivariant, so tensored with the identity on $\wt L^*$ induces a ${ \Bbbd C}^*$-equivariant map \[ \bar a_2 :\wt{\cal U}_d\times{ \Bbbd C}^*\rightarrow \mbox{\rm ev}^*(E)\otimes \wt L^*\] that has the same zero set as $a_2$. Let \begin{eqnarray*} \wt a_2:\wt{\cal U}_d\rightarrow\mbox{\rm ev}^*(E)\otimes \wt L^* \;\;\;\;\mbox{ given by }\;\;\; \wt a_2(x)=\bar a_2(x,1) \end{eqnarray*} Then the zero set of $a_2$ is equal to $Z(\wt a_2)\times { \Bbbd C}^*$. To find the zero set of $\psi_0$, for any $(x,v)\in Z(\wt a_2)\times { \Bbbd C}^*$ solve the equation \[ 0=\psi_1(x,v)=\bar \nu(x)+ a_1(x,v)=\bar \nu(x)+v\cdot a_1(x,1)\] Note that $a_1\ne 0$ on $Z(a_2)$ since $a$ has no zeros, so for any $x\in Z(\wt a_2)$ there exists a unique $v\in { \Bbbd C}^*$ such that \[ -\bar \nu(x)=v\cdot a_1(x,1)\] This implies that there exists an isomorphism between the zero set of $\psi_0$ and the zero set of $\wt a_2$. To complete the proof, note that for generic $\nu$ the section $\wt a_2$ is transversal to the zero section of $\mbox{\rm ev}^*(E)\otimes \wt L^*$, so its zero set is given by the Euler class of $\mbox{\rm ev}^*(E)\otimes \wt L^*$.\qed \medskip Finally, we can compute the boundary contribution: \begin{prop}\label{zeros} For $t$ small, the number of $(J,t\nu)$-holomorphic maps that satisfy the constraints in the definition of $RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l)$ and are close to the boundary strata $\{x_1\}\times\ov{\cal U}_d$ is equal to \[ \ma(\sum)_{i=0}^{n-1} {n+1\choose i+2}\mbox{\rm ev}^*(H^{n-1-i})\cdot c_1^i(\wt L^*)\] where $\wt L$ is the blow up of the relative tangent sheaf $L$ as in Definition \ref{ltilde}. \end{prop} \noindent {\bf Proof. } As we have seen previously, the moduli space of $(J,t\nu)$-holomorphic maps that satisfy the constraints in the definition of $RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l)$ and are close to the boundary strata $\{x_1\}\times\ov{\cal U}$ is diffeomorphic to the zero set of the section $\psi_0$. Using Lemma \ref{etil}, the later is equal to \[ c_{n-1}(\mbox{\rm ev}^*(E)\otimes \wt L^*) =\ma(\sum)_{i=0}^{n-1} \mbox{\rm ev}^*(c_{n-i-1}(E)\;)\cdot c_1^i(\wt L^*)\] But by definition $c_i(E)= c_{i}(T{ \Bbbd P}^n)={n+1\choose i}H^{i}$, completing the proof. \qed \subsection{The other contribution}\label{Other} In the previous sections we have described in great length the gluing construction corresponding to the strata $\{x_1\}\times \ov{\cal U}_d$, that consists of a ghost base and a bubble at the marked point $x_1$. Finally, it is the time to sketch the gluing construction corresponding to other boundary stratum $T^2\times\mbox{\rm ev}^*(\beta_1)$, and to explain why it does not give any contribution. \begin{prop}\label{other} For $t$ small, the number of $(J,t\nu)$-holomorphic maps that satisfy the constraints in the definition of $RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l)$ and are close to the boundary strata $T^2\times\mbox{\rm ev}^*(\beta_1)$ is equal to 0. \end{prop} \noindent {\bf Proof. } Construct first the space of approximate maps. The only difference from the gluing construction decribed in Section \ref{Approx} is that we need to allow the bubble point $x\in T^2$ to vary. Since the tangent bundle of the torus is trivial, choose an isomorphism \[ T T^2\cong T^2\times{ \Bbbd C}\] which gives an identification $T _xT^2\cong { \Bbbd C}$ for all $x\in T^2$ (providing local coordinates on $T^2$). The set of gluing data will then be modeled on: \[ T^2\times Fr\times (0,\varepsilon)\] where \[Fr=\{\; [f,y,u] \;|\; [f,y]\in \mbox{\rm ev}^*(\beta_1),\; u\in T_yS^2\;|u|=1\}\] is the restriction of the frame bundle over ${\cal U}_d$ defined by (\ref{fr}). \medskip To glue, use the unit frame $u\in TyS^2$ to identify $T_xT^2 \cong T_yS^2$ which will induce natural coordinates on the sphere via the stereographic projection. \medskip Then all the constructions decribed in Sections \ref{Approx}-\ref{Tau} extend to this case. Since the holomorphic 1-form $\omega\in H^{0,1}(T^2,{ \Bbbd C})$ is constant along the torus, then the isomorphism between the obstruction bundle and $\mbox{\rm ev}^*(T{ \Bbbd P}^n)$ is independent of the bubble point, so \[ \begin{array}{ccc} \;\;\;H^{0,1}\cong\rho^*\mbox{\rm ev}^*(T{ \Bbbd P}^n)&&\mbox{\rm ev}^*(T{ \Bbbd P}^n)\\ \searrow\;\;\;\;\;\swarrow\;\;\;\;\;\;\;&& \downarrow\\ T^2\times\mbox{\rm ev}^*(\beta_1)\;\;\;&\ma(\longrightarrow)^\rho& \mbox{\rm ev}^*(\beta_1) \end{array}\] Moreover, the linear part of the section $\psi_t$ that models the end of the moduli space is also independent of the bubble point. But a dimension count shows that the zero set of a $T^2$-equivariant section in the obstruction bundle must be empty generically. \qed \setcounter{subsection}{0} \setcounter{equation}{0} \setcounter{theorem}{0} \section{Computations}\label{Comp} In this second part of the paper we explain how one can compute the top power intersections $c_1^i(\wt L^*)\mbox{\rm ev}^*(H^{n-1-i})$ involved in Theorem \ref{gen}. The program is simple: first we find recursive formulas for the top intersections $c_1^i( L^*)\mbox{\rm ev}^*(H^{n-1-i})$ (Proposition \ref{pc1}), where $L$ is the relative tangent sheaf of ${\cal U}_d$, an object well known to the algebraic geometers. Next we can exploit the fact that $\wt L$ is a blow up of $L$ to compute its corresponding top intersections. Unfortunately, the notation becomes quickly pretty complicated if we insist on keeping track of all the information, so we chose to indicate at each step only the new changes, leaving out the data that stays the same. \step{Notations} If $\beta_0,\dots,\beta_k$ are various codimension constraints let \[ {\cal U}_d(\beta_0\; ;\; \beta_1,\dots,\beta_k)= \mbox{\rm ev}^*(\beta_0)\; [\;{\cal U}_d(\; ;\beta_1,\dots,\beta_k)\;] \] denote the moduli space of 1-marked cuves in ${ \Bbbd P}^n$ passing through $\beta_0,\dots,\beta_k$, such that the special marked point is on $\beta_0$ and let \[{\cal M}_d(\beta_0,\beta_1,\dots,\beta_k) \] denote the corresponding moduli space of curves (in which we forget the special marked point). \medskip In particular, let ${\cal U}_d= {\cal U}_d(\; ;\beta_1,\dots,\beta_k)$ be the moduli space of 1-marked curves that appears in Theorem \ref{gen}. If $i,j\ge 0$ are such that $i+j$= dim ${\cal U}_d$ then let \begin{eqnarray} \phi_d(i,j\;|\;\beta_1,\dots,\beta_k)= c_1^i(L^*)\;\mbox{\rm ev}^*(H^j)\;[\;{\cal U}_d\;] \end{eqnarray} denote the top intersection. Moreover, if $\wt {\cal U}_d$ is the blow-up ${\cal U}_d$ as in Definition \ref{ltilde}, let \begin{eqnarray}\label{xy} x=c_1(L^*) \in H^2({\cal U}_d,{ \Bbbd Z}),\;\;\wt x=c_1(\wt L^*) \in H^2(\wt{\cal U}_d,{ \Bbbd Z}), \;\; y=\mbox{\rm ev}^*(H) \end{eqnarray} where $y\in H^2({\cal U}_d,{ \Bbbd Z})$ or $y\in H^2(\wt {\cal U}_d,{ \Bbbd Z})$ depending on the context. Note that \begin{eqnarray} \phi_d(i,j\;|\; \cdot \;)=x^i y^j\;[\;{\cal U}_d\;]=x^i\;[{\cal U}_d(H^j; \cdot )] \end{eqnarray} Using the notation above and the degeneration formula (\ref{gendeg}), Theorem \ref{gen} becomes \begin{eqnarray}\label{bau} n_j\tau_d(\cdot)=\ma(\sum)_{i_1+i_2=n}\sigma_d(H^{i_1},H^{i_2},\cdot)- \ma(\sum)_{i=0}^{n-1}{n+1\choose i+2}\wt x^{i}y^{n-1-i}[\wt{\cal U}_d] \end{eqnarray} \addtocounter{theorem}{1\label{steps} To compute a particular value for $\tau_d$ in ${ \Bbbd P}^n$ one should use a computer program based on the following four steps: \begin{enumerate} \item Find $\sigma_d$ using the recursive formula (\ref{spheres}) \item Find $\phi_d(i,j\; |\; \cdot)=c_1^i(L^*)\mbox{\rm ev}^*(H^j)[{\cal U}_d]$ using the recursive formulas of Proposition \ref{pc1}. \item Find recursive formulas for $\wt x^i\cdot y^j=c_1^i(\wt L^*)\mbox{\rm ev}^*(H^j)[\wt {\cal U}_d]$ as outlined in Proposition \ref{pwtc1}. \item Finally, use (\ref{bau}) to get $\tau_d$. \end{enumerate} \subsection{Recursive formulas for $c_1^i(L^*)\mbox{\rm ev}^*(H^{j}) $ } Let ${\cal U}_d$ be some $r$-dimensional moduli space of 1-marked curves of degree $d$ through some constraints $\beta_1,\dots,\beta_k$ (not necessarily the same as in Theorem \ref{gen}) and let $L\rightarrow {\cal U}_d$ be its relative tangent sheaf. In this section we give recursive formulas for top intersections \[ \phi_d(i,j\; |\; \cdot )=c_1^i(L^*)\mbox{\rm ev}^*(H^j)[{\cal U}_d] \] where $i+j=r$ and the constaints $\beta_1,\dots,\beta_k$ are dropped from the notation. \begin{prop}\label{pc1} For every $r$-dimensional moduli space ${\cal U}_d$ of any degree $d\ge 1$, there are the following recursive relations for the top intersections: \begin{eqnarray}\label{c1lini} \phi_d(0,j\;|\; \cdot )&=&\sigma_d(H^j,\; \cdot ) \\ \nonumber \phi_d(i+1,j\;|\; \cdot )&=&-{2\over d}\phi_d(i,j+1\;|\; \cdot \;)+ {1\over d^2}\phi_d(i,j\;|\;H^2,\; \cdot \;)\\ \nonumber &+&\ma(\sum)_{d_1+d_2=d\atop i_1+i_2=n}{d_2^2 \over d^2}\; \phi_{d_1}(i,j\;|H^{i_1}\;,\; \cdot \;)\cdot\sigma_{d_2}(H^{i_2},\; \cdot \;)\nonumber\\ \label{c1li} &+&\ma(\sum)_{d_1+d_2=d\atop i_1+i_2=n+j}{d_2^2 \over d^2} \phi_{d_1}(i-1,i_1\;|\; \cdot \;)\cdot\sigma_{d_2}(H^{i_2},\; \cdot \;) \end{eqnarray} for any $i\ge 0$, where the sums above are over all possible distributions of the constraints $\beta_1,\dots,\beta_k$ on the two factors and $d_1,d_2\ne 0$. When $i=0$, the last term in (\ref{c1li}) is missing. \end{prop} \noindent {\bf Proof. } The first relation follows by definition, and provides the initial step of the recursion. The second one requires more work. In what follows, we will identify a cohomology class like $c_1(L)$ with a divisor representing it. Then: \begin{lemma}\label{lc1l} On ${\cal U}_d$, we have the following relation: \begin{eqnarray}\label{c1l} c_1(L^*)={1\over d^2} {\cal H}-{2\over d}\mbox{\rm ev}^*(H) +{1\over d^2} \ma(\sum)_{d_1+d_2=d}d_2^2 {\cal M}_{d_1,d_2} \end{eqnarray} where ${\cal H}$ denotes the extra condition that the curve passes through $H^2$, and $ {\cal M}_{d_1, d_2}$ denotes the boundary stratum corresponding to the splittings in a degree $d_1$ 1-marked curve and a degree $d_2$ curve, for $d_i\ne 0$ (for all possible distributions of the constraints $\beta_1,\dots,\beta_k$ on the two components). \end{lemma} \noindent {\bf Proof. } Fix 2 hyperplanes in generic position in ${ \Bbbd P}^n$. Each curve in ${\cal U}_d$ intersects a hyperplane in $d$ points. Then the moduli space $Y=\mbox{\rm ev}^*_{k+1}(H)\cap\mbox{\rm ev}^*_{k+2}(H)$ of 1-marked curves passing through $\beta_1,\dots,\beta_k,H,H$ is a $d^2$ fold cover of ${\cal U}_d$: \[ \pi:Y \rightarrow {\cal U}_d, \hskip.3in [f,y_1,\dots,y_k,a,b\;;y]\rightarrow [f,y_1,\dots,y_k\;;y]\] Define the section \[ s([f,y_1,\cdots,y_k,a,b\;;y])={(a-b)dy\over (y-a)(y-b)}\] Then $s$ is a section in the relative cotangent sheaf $L^*$, and it extends to the compactification $\ov{{\cal U}_d}$. As $a$ and $b$ are getting closer together, the section $s$ converges to 0. Thus its zero set is the sum of the divisors $\{a=b\}$ and ${\cal M}(y\; ; \;a,b)$, where ${\cal M}(y\; ;\;a,b)$ is the sum of all boundary strata corresponding to splittings into a degree $d_1$ 1-marked bubble and a degree $d_2$ bubble containing $a,b$ for $d=d_1+d_2$. Note that $d_i\ne 0$. The infinity divisor is $\{y=a\}+\{y=b\}$. Thus \[\pi^*( c_1(L^*))=\; \{a=b\}+ {\cal M}(y\; ; \;a,b)-\{y=a\}-\{y=b\}\] Note that \[ d^2 c_1(L^*)=\pi_*\pi^*( c_1(L^*))\] When projecting down to $\ov {\cal U}_d$, the divisor $\{a=b\}$ becomes ${\cal H}$, and the divisors $ \{y=a\}$, $ \{y=b\}$ become each $d\cdot\mbox{\rm ev}^*(H)$. The rest amounts to summing over all codimension 1 boundary strata. The boundary strata ${\cal M}_{d_1, d_2}$ appears with coefficient $d_2^2$ in $\pi_*({\cal M}(y,a\; ; \;b))$. Combining all the pieces together completes the proof of Lemma. \qed \medskip \addtocounter{theorem}{1 We could have chosen any 2 marked points out of the already existent ones, and then express $c_1(L)$ in terms of them. But then this expression would not look independent of choice. Nevertheless, with some work, one can actually see that all these divisors are homotopic. We have chosen to introduce 2 new marked points to avoid this issue. \medskip \addtocounter{theorem}{1 Note that the base locus of the line bundle $L$ is exactly the union of the divisors ${\cal Z}_h$. Doing the intersection theory in the blow up along the base locus is the same as considering the excess intersection (see \cite{ful}). \bigskip Relation (\ref{c1l}) provides the basic relation for proving (\ref{c1li}): \[ c_1^{i+1}(L^*)=-{2\over d}c_1^i(L^*)\cdot \mbox{\rm ev}^*(H) + {1\over d^2}c_1^i(L^*)\cdot {\cal H}+ \ma(\sum)_{d_1+d_2=d}{d_2^2 \over d^2}\; c_1^i(L^*)\cdot {\cal M}_{d_1,d_2} \] so taking a cup product with $\mbox{\rm ev}^*(H^j)$ we get: \begin{eqnarray}\label{c1ini} \phi_d(i+1,j\;|\; \cdot )&=&-{2\over d}\phi_d(i,j+1\;|\; \cdot )+ {1\over d^2}\phi_d(i,j\;|\;H^2,\; \cdot )\\ \nonumber &+&\ma(\sum)_{d_1+d_2=d}{d_2^2 \over d^2}\; c_1^i(L^*)\mbox{\rm ev}^*(H^j)\cdot {\cal M}_{d_1,d_2} \end{eqnarray} Next, we need to understand the restriction of $L^*$ to the boundary stratum ${\cal M}_{d_1,d_2}$. Let \[p:{\cal M}_{d_1,d_2}\rightarrow {\cal U}_{d_1}\] be the projection on the first component (the one that contains the special marked point $y$). If $A,B$ are the 2 special points of ${\cal M}_{d_1, d_2}$ (where the 2 components meet), let \[\mbox{\rm ev}_A\times\mbox{\rm ev}_B:{\cal U}_{d_1}\times {\cal M}_{d_2}\rightarrow { \Bbbd P}^n\times { \Bbbd P}^n\] be the corresponding evaluation map. Then by definition \begin{eqnarray}\label{cm12} {\cal M}_{d_1,d_2}=(\mbox{\rm ev}_A\times\mbox{\rm ev}_B )^*([\Delta]) \end{eqnarray} where $\Delta$ is the diagonal of ${ \Bbbd P}^n\times{ \Bbbd P}^n$. Moreover, it is known that as divisors, \begin{eqnarray}\label{c1la} c_1(L^*)/{\cal M}_{d_1,d_2} =p^* c_1(L^*_A)+\{ y=A\} \end{eqnarray} where $L_A=L_{|{\cal U}_{d_1}}$ is the relative tangent sheaf of ${\cal U}_{d_1}$. Next step is to find \begin{eqnarray}\label{c1cmd1d2} c_1^i(L^*)/{\cal M}_{d_1,d_2}=\ma(\sum)_{l=0}^i {i\choose l} \;p^* c_1^{i-l}(L^*_A)\cdot(\{ y=A\})^l \end{eqnarray} For the self intersection of the divisor $\{ y=A\}$ note that its normal bundle $N$ inside ${\cal M}_{d_1,d_2}$ is nothing but $p^*(L_A)/\{ y=A\}$, so for $l>0$, \[ (\{ y=A\})^l= c_1(N)^{l-1}=(-1)^{l-1}p^*c_1^{l-1}(L_A^*) \cdot [\{y=A\}] \] Substituting in (\ref{c1cmd1d2}) and after some algebraic manipulations we get: \begin{eqnarray}\label{pof} c_1^i(L^*)\cdot[{\cal M}_{d_1,d_2}] =p^* c_1^i(L^*_A)+p^*c_1^{i-1}(L_A^*)\cdot [\{ y=A\}] \end{eqnarray} We will do the intersection theory inside ${\cal U}_{d_1}\times{\cal M}_{d_2}$. The relation (\ref{cm12}) combined with $[\Delta]=\ma(\sum)_{i_1+i_2=n}H^{i_1}\times H^{i_2}$ gives \begin{eqnarray*} \mbox{\rm ev}^*(H^j)\cdot[{\cal M}_{d_1,d_2}]&=&\ma(\sum)_{i_1+i_2=n} {\cal U}_{d_1}(H^j\; ;\;H^{i_1},\; \cdot \;)\times {\cal M}_{d_2}(H^{i_2},\; \cdot \;)\\ \mbox{\rm ev}^*(H^j)\cdot[\{ y=A\}]&=&\ma(\sum)_{i_1+i_2=n+j} {\cal U}_{d_1}(H^{i_1}\;;\; \cdot \;)\times {\cal M}_{d_2}(H^{i_2},\; \cdot \;) \end{eqnarray*} where we sum over all possible distributions of the constraints on the two components. The relations above imply \begin{eqnarray} \nonumber &&\hskip-.6in \mbox{\rm ev}^*(H^j)\cdot p^*c_1^i(L^*_A)\cdot [{\cal M}_{d_1,d_2}]\\ \nonumber &&=\ma(\sum)_{i_1+i_2=n}(c_1^i(L_A)\cdot {\cal U}_{d_1}(H^j\; ;\;H^{i_1},\; \cdot \;)\;)\times {\cal M}_{d_2}(H^{i_2},\; \cdot \;)\\ &&=\ma(\sum)_{i_1+i_2=n} \phi_{d_1}(i,j\; |\;H^{i_1},\; \cdot \;)\cdot \sigma_{d_2}(H^{i_2},\; \cdot \;) \end{eqnarray} \begin{eqnarray}\nonumber &&\hskip-.6in\mbox{\rm ev}^*(H^j)\cdot p^*c_1^{i-1}(L_A^*)\cdot[\{ y=y_A\}] \\&&=\ma(\sum)_{i_1+i_2=n+j} \nonumber (c_1^{i-1}(L_A)\cdot{\cal U}_{d_1}(H^{i_1}\; ;\; \cdot \;)\;)\times {\cal M}_{d_2}(H^{i_2},\; \cdot \;)\\ &&=\ma(\sum)_{i_1+i_2=n+j} \phi_{d_1}(i-1,i_1\; |\; \cdot \;)\cdot \sigma_{d_2}(H^{i_2},\; \cdot \;) \end{eqnarray} Substituting these relations in (\ref{c1ini}) using (\ref{pof}) we get (\ref{c1li}), which concludes the proof of Proposition \ref{pc1}. \qed \subsection{Recursive formulas for $c_1^i(\wt L^*)\cdot\mbox{\rm ev}^*(H^{j})$ } Next step is to express the top intersections involving the first Chern class of $\wt L$, the blow up of $L$, in terms of the top intersections involving the first Chern class of $L$. The program for such kind of computations is very nicely outlined in \cite{ful}, which we will follow closely. Although recursive formulas can be found for any $n$, the more strata we need to blow up, the longer and more complicated looking these formulas become. For simplicity of the presentation, in this section we will give only the general principles of the algorithm, without working out completely the recursive formulas. In the next section we will use this algorithm to obtain recursive formulas for small values of $n$ (i.e. $n\le 4$). \medskip Let ${\cal U}_d={\cal U}_d(\; ;\beta_1,\dots,\beta_k)$ the some $r$-dimensional moduli space of 1-marked curves. Recall the construction of $\widetilde{\cal U}_d$: starting with ${\cal U}_d$, we first blow up along ${\cal Z}_{N}$, then we blow up the proper transform of ${\cal Z}_{N-1}$ and so on, up to blowing up the proper transform of ${\cal Z}_2$. Since $\wt L$ extends as the blow up of $L$ then \begin{eqnarray} c_1(\wt L)= c_1(L)+\ma(\sum)_{h=2}^{N} E_h \end{eqnarray} where $E_h$ is the exceptional divisor corresponding to the proper transform of ${\cal Z}_h$. \begin{prop}\label{pwtc1} Using the notations above, the top intersections \begin{eqnarray} c_1^i(\wt L^*)\mbox{\rm ev}^*(H^{j}) \end{eqnarray} on $\wt{\cal U}_d$ can be recursively expressed in terms of the top intersections \begin{eqnarray*} \phi_d(k,l\;|\;\cdot\;)=c_1^k(L^*)\cdot\mbox{\rm ev}^*(H^{l}) \end{eqnarray*} on possibly lower dimensional moduli spaces ${\cal U}_{d'}$. \end{prop} \noindent {\bf Proof. } The idea is of course to do inductively one blow up at a time. Although the fact there exist such recursive formulas is not that hard to see, writting them down becomes pretty complicated very quickly. So we explain why such formulas exist, leaving their derivation for later. By definition, \begin{eqnarray*} \wt x=c_1(\wt L^*)=x-\ma(\sum)_{h=2}^{N} E_h \end{eqnarray*} Let \begin{eqnarray} x(h)=x-\ma(\sum)_{l=h}^N E_l \end{eqnarray} so $\wt x(N+1)=x$ and $\wt x(2)=\wt x$. Using $x(h)=x(h+1)-E_h$, and expanding, \begin{eqnarray}\nonumber x(h)^i\cdot y^{j}&=&x(h+1)^{i}\cdot y^{j}+ \ma(\sum)_{l=1}^i{i\choose l}x(h+1)^{i-l} y^{j}(-1)^l\; E_h^l \\ &=&x(h+1)^{i} y^{j}- \ma(\sum)_{l=h}^i{i\choose l}x(h+1)^{i-l} y^{j} s_{l-h}(N_{\wt {\cal U}_d}\widetilde{\cal Z}_h) [\widetilde{\cal Z}_h] \label{wtxiyj} \end{eqnarray} The last equality is a consequence of the following \begin{fact}(cf. \cite{ful}) Assume $X\subset Y$ regular imbedding of codimension $a$, where dim $Y$=r. Let $\pi:\wt Y\rightarrow Y$ be the blow up of $Y$ along $X$ and $E={ \Bbbd P}(N_YX)$ be the exceptional curve. For any $\alpha\in H^{r-l}(Y)$, the top intersection \begin{eqnarray} (\pi^*(\alpha)\cup E^l)\cap[\wt Y]=(-1)^{l-1} (\alpha\cup s_{l-a}(N_YX))\cap[Y] \end{eqnarray} as integers, where $l\ge 1$ and $s(N)$ is the segree class of the normal bundle $N$. \end{fact} Next we need to understand how $x(h+1)$ or equivalently $E_m$ restricts to $\widetilde{\cal Z}_h$, and also we need to find $s(N_{\wt{\cal U}_d}\widetilde{\cal Z}_d)$. First, we find: \step{The normal bundle of ${\cal Z}_h$ in ${\cal U}_d$} Recall that ${\cal Z}_h$ consists of bubble trees with $h$ essential components meeting at the image of the ghost base. So in particular, ${\cal Z}_h$ has components indexed by the different distributions of the degree on the $h$ bubbles: \begin{eqnarray}\label{ovcz} {\cal Z}_{d_1,\dots,d_h}=\mbox{\rm ev}_0^*([\Delta])\subset \ov{\cal M}_{0,h+1}\times {\cal U}_{d_1}\times\dots\times {\cal U}_{d_h} \end{eqnarray} where $d_i\ne 0$ for $i=1,\dots,h$, $\Delta$ is the small diagonal in $({ \Bbbd P}^n)^h$ and \begin{eqnarray} &&\mbox{\rm ev}_0: {\cal U}_{d_1}\times\dots\times {\cal U}_{d_h}\rightarrow ({ \Bbbd P}^n)^h\\ &&ev_0([f_1,y_1],\dots,[f_h,y_h])=(f_1(y_1),\dots,f_h(y_h)) \end{eqnarray} is the evaluation map. Then \begin{eqnarray} {\cal Z}_h={1\over h!}\ma(\bigcup)_{d_1+\dots+d_h=d}{\cal Z}_{d_1,\dots,d_h} \end{eqnarray} where the factor of $h!$ comes from the action of the symmetric group that permutes the order of the $h$ bubbles (yielding the same bubble tree). Let \begin{eqnarray*} \ov {\cal M}_{0,h+1}\times {\cal U}_{d_1}\times\dots\times {\cal U}_{d_h}& \ma(\longrightarrow)^{p_i} &{\cal U}_{d_i} \end{eqnarray*} be the projection and $L_i$ be the relative tangent sheaf of the $i$'th factor. It is easy to check that \begin{lemma}\label{normalbd} Using the notations above, the normal bundle $N_{ {\cal Z}_h} {\cal U}_d$ of ${\cal Z}_h$ in ${\cal U}_d$ is isomorphic to \begin{eqnarray}\label{normbd} p_1^*L_1\oplus \dots\oplus p_h^* L_h \quad \mbox{on each component} \end{eqnarray} so \begin{eqnarray} s(N_{ {\cal Z}_h} {\cal U}_d)={1\over (1-x_1)\cdot \dots\cdot (1-x_h)} \end{eqnarray} where $x_i=p_i^*c_1(L_i^*)$. \end{lemma} \medskip \addtocounter{theorem}{1 One word of caution: so far we have defined $x_i$ on each component ${\cal Z}_{d_1,\dots,d_h}$ of ${\cal Z}_h$, but these definitions {\em do not match} on the intersection of two components. Nevertheless, after we blow up ${\cal U}_d$ as in Definiton \ref{ltilde}, all the components of ${\cal Z}_h$ become disjoint, so doing the intersection theory in the blow up allows up to treat each component separately as if they were disjoint. \medskip Next, use the following \begin{fact}(cf. \cite{ful}) Assume $X,\;Y\subset Z$ are regular imbeddings. Let $\wt Z=Bl_X Z$ be the blow up of $Z$ along $X$ and $\wt Y=Bl_{X\cap Y} Y$ be the proper transform of $Y$. Denote by $E={ \Bbbd P}(N_Z X)$ the exceptional curve in $\wt Z$, and let $F={ \Bbbd P}(N_Y {(X\cap Y})$ be the exceptional curve in $\wt Y$. Then: \begin{eqnarray} E\cap \wt Y&=&F\\ N_{\wt Z} \wt Y&\cong& \pi^*(N_Z Y) \otimes {\cal O}(-F)\quad \mbox{ so } \\ \label{spN} s_p(N_{\wt Z} \wt Y)&=&\ma(\sum)_{i=0}^{p-a} {a+p\choose a+i} \;s_i(N_Z Y)\; F^{p-i} \end{eqnarray} where $a=$rank $N_Z Y$. \end{fact} So \begin{eqnarray} E_l\cap \widetilde{\cal Z}_h=E_{h,l} \end{eqnarray} is the exceptional curve in the blow up of ${\cal Z}_h$ along the proper transform of ${\cal Z}_h\cap{\cal Z}_l$ and \begin{eqnarray} N_{\wt {\cal U}_d} \widetilde{\cal Z}_h&\cong& \pi^*(N_{{\cal U}_d} {\cal Z}_h) \otimes {\cal O}\l(-\ma(\sum)_{l=h+1}^N E_{h,l}\r) \end{eqnarray} Note that ${\cal Z}_h$ has less dimensions than ${\cal U}_d$ and it is stratified by subvarieties ${\cal Z}_h\cap{\cal Z}_l$ for $l\ge h+1$ the same way ${\cal U}_d$ is stratified by the sets ${\cal Z}_l$. Thus we can repeat the same construction. \medskip Inductively, the intersection theory takes place inside a strata of form ${\cal Z}_{h_1}\cap\dots\cap {\cal Z}_{h_l}$. Now, it is not that easy to list and parametrize all the possible bubble tree configurations in this intersection. But a closer look reveals that although the combinatorics involved is complicated, all these intersections have components of the form \begin{eqnarray}\label{piece} ev_0^*([\Delta])\subset{\cal Z}\times {\cal U}_{d_1}\times \dots\times {\cal U}_{d_m} \end{eqnarray} where ${\cal Z}$ is some substrata of $\ov {\cal M}_{0,m+1}$, and $\Delta$ is the small diagonal in $({ \Bbbd P}^n)^m$. Such component comes in with a coeficient of one over the order of the subgroup of permutations that preserve the same bubble tree configuration. The normal bundle of (\ref{piece}) has the same form as in Lemma \ref{normalbd}, and using the arguments outlined above we are inductively decreasing either the number of exceptional curves or the dimension of the moduli space we do the intersection theory over. In either case, the process terminates in finite time, reducing the top intersections $\wt x^i y^j$ on $\wt{\cal U}_d$ we started with to sums of top intersections of the form \begin{eqnarray}\label{mess} x_1^{i_1}\dots x_m^{i_m} y^j \quad \mbox{ on }\quad \mbox{\rm ev}_0(\Delta)\subset {\cal Z}\times {\cal U}_{d_1}\times \dots\times {\cal U}_{d_m} \end{eqnarray} Finally, let \begin{eqnarray*} \pi:{\cal Z}\times {\cal U}_{d_1}\times \dots\times {\cal U}_{d_m}\rightarrow {\cal U}_{d_1}\times \dots\times {\cal U}_{d_m} \end{eqnarray*} be the projection. Since all the classes in (\ref{mess}) are pull-backs by $\pi$ then the top intersection (\ref{mess}) vanishes unless ${\cal Z}\subset {\cal M}_{0,m+1}$ is 0 dimensional. When ${\cal Z}$ is 0-dimensional then using the decomposition of the diagonal \begin{eqnarray*} [\Delta]=\ma(\sum)_{i_1+\dots+i_m=n}H^{i_1}\times\dots\times H^{i_m} \end{eqnarray*} and letting $y_j=\mbox{\rm ev}^*(H)$ on the $j$'th factor we get \begin{eqnarray} \mbox{\rm ev}_0^*([\Delta])&=&\ma(\sum)_{j_1+\dots+j_m=n} y_1^{j_1}\dots y_m^{j_m} [{\cal U}_{d_1}\times\dots\times{\cal U}_{d_m}]\quad \mbox{ and}\\ y^j \mbox{\rm ev}_0^*([\Delta])&=&\ma(\sum)_{j_1+\dots+j_m=n+j} y_1^{j_1}\dots y_m^{j_m}[{\cal U}_{d_1}\times\dots\times {\cal U}_{d_m}] \end{eqnarray} so (\ref{mess}) is equal to \begin{eqnarray} \ma(\sum)_{j_1+\dots+j_m=n+j} (\;x_1^{i_1}y_1^{j_1}[{\cal U}_{d_1}]\;)\times\dots \times (\;x_m^{i_m}y_m^{j_m}[{\cal U}_{d_m}]\;)\\ \label{splity} =\ma(\sum)_{i_1+\dots+i_m=n+j} \phi_{d_1}(i_1,j_1)\cdot\dots\cdot \phi_{d_m}(i_m,j_m) \end{eqnarray} giving Proposition \ref{pwtc1}. \qed \section{Applications to ${ \Bbbd P}^n, \;n\le 4$} Finally, we apply the inductive algorithm described in the previous section to obtain recursive formulas for the elliptic enumerative invariant $\tau_d$ in ${ \Bbbd P}^n$, for $n\le 4$. In this case the story is quite simple, since $N=\left[ {n+1\over 2}\right]\le 2$, so we need to blow up at most one strata, ${\cal Z}_2$. \step{Explicite formulas for $c_1^i(\wt L^*)\cdot\mbox{\rm ev}^*(H^{j})$} Note that for $n=2$ there is nothing to blow up, so \begin{eqnarray}\label{ln2} \wt L=L \quad \mbox{ for }\quad n=2 \end{eqnarray} A litlle more work gives: \begin{lemma}\label{l34} Using the notations in Theorem \ref{gen}, when $n=3,\;4$, we have the following relations: \begin{eqnarray}\label{x1y} &&\hskip-.3in c_1(\wt L)\cdot \mbox{\rm ev}^*(H^{n-2})=\phi_d(1,n-2\;|\;\cdot\;)\\ \label{x2y} &&\hskip-.3in c_1^2(\wt L)\cdot \mbox{\rm ev}^*(H^{n-3})=\phi_d(2,n-3\;|\;\cdot\;)- \hskip-.2in\ma(\sum)_{d_1+d_2=d\atop i_1+i_2=2n-3}\hskip-.1in \sigma_{d_1}(H^{i_1})\cdot \sigma_{d_2}(H^{i_2})\\ \label{x3y} &&\hskip-.3in c_1^3(\wt L)=\phi_d(3,0\;|\;\cdot\;)- \ma(\sum)_{d_1+d_2=d\atop i_1+i_2=4} \phi_{d_1}(1,i_1\;|\;\cdot\;)\cdot \sigma_{d_2}(H^{i_2}) \mbox{ for }n=4 \end{eqnarray} where the sums above are over all possible distributions of the constraints on the two components and $d_i\ne 0$. \end{lemma} \noindent {\bf Proof. } When $n=3,\; 4$ we need to blow up only ${\cal Z}_2$. Use (\ref{wtxiyj}) to get: \begin{eqnarray}\nonumber \wt x^i y^{n-1-i}\;[\wt{\cal U}_d]&=& x^{i}y^{n-1-i}\;[{\cal U}_d] \\ \label{blxy} &-&\ma(\sum)_{l=2}^i {i\choose l}x^{i-l}s_{l-2}(N_{{\cal Z}_2}) y^{n-1-i} \; [{\cal Z}_2] \end{eqnarray} Note that for $i=1$ the sum in (\ref{blxy}) is indexed by the empty set, thus giving (\ref{x1y}). When $i=2$ the sum reduces to: \begin{eqnarray*} x^{0}\cdot s_{0}(N_{{\cal Z}_2})\cdot y^{n-3} \; [{\cal Z}_2]= y^{n-3} \; [{\cal Z}_2]= {1\over 2}\ma(\sum)_{d_1+d_2=d\atop i_1+i_2=2n-3} \sigma_{d_1}(H^{i_1}) \cdot \sigma_{d_2}(H^{i_2}) \end{eqnarray*} by (\ref{splity}), giving (\ref{x2y}). When $n=4$ and $i=3$ then the sum in (\ref{blxy}) becomes \begin{eqnarray}\label{43} (3 x\cdot s_0(N_{{\cal U}_d}{{\cal Z}_2})+s_1(N_{{\cal U}_d}{{\cal Z}_2}))[{\cal Z}_2] \end{eqnarray} But note that $x|_{[{\cal Z}_2]}=0$ and Lemma \ref{normalbd} gives \begin{eqnarray*} s(N_{{\cal U}_d}{{\cal Z}_2})={1\over (1-x_1)(1-x_2)}\quad \mbox{ thus } \quad s_1(N_{{\cal U}_d}{{\cal Z}_2}))=x_1+x_2 \end{eqnarray*} So (\ref{43}) becomes \begin{eqnarray*} (x_1+x_2)\;[{\cal Z}_2]&=& {1\over 2}\ma(\sum)_{d_1+d_2=d\atop j_1+j_2=n}(x_1+x_2)y_1^{j_1}y_2^{j_2} \;[ {\cal U}_{d_1}\times{\cal U}_{d_2}]\\ &=&\ma(\sum)_{d_1+d_2=d\atop j_1+j_2=n}(x_1y_1^{j_1} [{\cal U}_{d_1}])\cdot (y_2^{j_2} [{\cal U}_{d_2}])= \ma(\sum)_{d_1+d_2=d\atop j_1+j_2=n} \phi_{d_1}(1,j_1\;|\;\cdot\; )\cdot\sigma_{d_2}(H^{j_2},\;\cdot\;) \end{eqnarray*} using again (\ref{splity}). \qed \medskip Now we can prove for example that: \begin{prop}\label{p^2} The number $\tau_d(p^{3d-1})$ of degree $d$ elliptic curves in ${ \Bbbd P}^2$ with fixed $j$ invariant and passing though $3d-1$ points is \begin{eqnarray} \tau_d(p^{3d-1})={2\over n_j}{d-1\choose 2}\sigma_d(p^{3d-1}) \end{eqnarray} where $\sigma_d$ is the number of rational curves through $3d$ points, and $n_j$ is the order of the group of automorphisms of the complex structure $j$ fixing a point. \end{prop} \noindent {\bf Proof. } For $n=2$, relation (\ref{conden}) combined with (\ref{ln2}) gives: \begin{eqnarray}\label{relp2} n_j \tau_d(p^{3d-1})= \sigma_d( l,l,p^{3d-1})-3\mbox{\rm ev}^*(H)-c_1( L^*) \end{eqnarray} where $L\rightarrow {\cal U}_d$ is the relative tangent sheaf over the moduli space of 1-marked rational curves of degree $d$ passing through $3d-1$. The moduli space ${\cal M}_d$ of unmarked curves is $n-2=0$ dimensional, consisting of $\sigma_d(p^{3d-1})$ curves. Using (\ref{pc1}) (or easier by inspection) \[ c_1(L^*)=-{2\over d}\sigma_d(l,p^{3d-1})=-2\sigma_d(p^{3d-1})\] \[\mbox{\rm ev}^*(H)=\sigma_d(l,p^{3d-1})=d\sigma_d(p^{3d-1})\;\;\;\mbox{ and }\;\;\; \sigma_d(l,l,p^{3d-1})=d^2 \sigma_d(p^{3d-1})\] So plugging them back in (\ref{relp2}) we obtain \[ \tau_d(p^{3d-1})={1\over n_j}(d^2-3d+2)\;\sigma_d\] which gives (\ref{p^2}). \qed \medskip In particular, \vskip-.5in \begin{eqnarray} \tau_d(p^{3d-1})= \left\{\begin{array}{ll} {d-1\choose 2}\sigma_d& \mbox{ if } j\ne 0, 1728\\ \\ {1\over 2} {d-1\choose 2}\sigma_d& \mbox{ if } j= 0\\ \\ {1\over 3} {d-1\choose 2}\sigma_d& \mbox{ if } j=1728 \end{array}\right. \end{eqnarray} This formula was recently obtained by Panharipande \cite{rp} using different methods. \medskip \noindent Next we can prove that: \begin{prop}\label{compp3} The number $\tau_d=\tau_d(p^a,l^b)$ of elliptic curves in ${ \Bbbd P}^3$ with fixed $j$ invariant and passing through $a$ points and $b$ lines (such that $2a+b=4d-1$) is given by: \begin{eqnarray}\label{formp3} \tau_d={2(d-1)(d-2)\over dn_j}\sigma_d(l)-{2\over dn_j}\ma(\sum)_{d_1+d_2=d} d_2(2d_1d_2-d)\sigma_{d_1}(l)\sigma_{d_2} \end{eqnarray} where $\sigma_d(l)=\sigma_d(p^a,l^b,\;l)$ is the number of degree d rational curves in ${ \Bbbd P}^3$ passing through same conditions as $\tau_d$ plus one more line. By the term $\sigma_{d_1}(l)\sigma_{d_2}$ we understand the sum over all decompositions into a degree $d_1$ and a degree $d_2$ bubble such that the constraints are distributed in all possible ways on the bubbles, and $d_i\ne 0$. \end{prop} \noindent {\bf Proof. } When $n=3$, Theorem \ref{gen} gives: \begin{eqnarray}\nonumber n_j\tau_d(p^a, l^b) &=&\ma(\sum)_{i_1+i_2=3} \sigma_d(H^{i_1},H^{i_2},p^a,l^b )- 6\mbox{\rm ev}^*(H^2)\\ \label{3ci} &-&4\mbox{\rm ev}^*(H) c_1(\wt L^*)-c_1^2(\wt L^*) \end{eqnarray} The moduli space ${\cal M}_d$ of degree $d$ unmarked curves passing through $a$ points and $b$ lines is $n-2=1$ dimensional, with a finite number of bubble trees in the boundary. Then Proposition \ref{compp3} is a consequence of (\ref{3ci}) and the following \begin{lemma} In ${ \Bbbd P}^3$, \begin{eqnarray*} &&\ma(\sum)_{i_1+i_2=3} \sigma_d(H^{i_1},H^{i_2},p^a,l^b )=2d \cdot\sigma_d(p^a,l^{b+1}) \\ &&\mbox{\rm ev}^*(H^2)=\sigma_d(p^a,l^{b+1}) \\ &&\mbox{\rm ev}^*(H)\cdot c_1(\wt L^*)=\mbox{\rm ev}^*(H)\cdot c_1( L^*)= -{1\over d}\sigma_d(l)+ {1\over d}\ma(\sum)_{d_1+d_2=d}d_1d_2^2 \sigma_{d_1}(l)\sigma_{d_2}\\ &&c_1(L^*)^2=-\hskip-.1in \ma(\sum)_{d_1+d_2=d}\hskip-.1in d_2\sigma_{d_1}(l)\sigma_{d_2}\quad\mbox{and}\quad c_1(\wt L^*)^2=-2\hskip-.1in \ma(\sum)_{d_1+d_2=d}\hskip-.1in d_2\sigma_{d_1}(l)\sigma_{d_2} \end{eqnarray*} \end{lemma} \noindent {\bf Proof. } The relations above follow either by definition, or by aplying several times (\ref{c1l}) combined with (\ref{x1y}) or (\ref{x2y}) (and of course some simple algebraic manipulations). \medskip \addtocounter{theorem}{1 If we distribute the constraints in Proposition \ref{compp3} in all possible ways, formula (\ref{formp3}) becomes: \begin{eqnarray}\label{3cf} &\tau_d(p^a,l^b)={2(d-1)(d-2)\over n_jd}\sigma_d(p^a,l^{b+1})& \\ \nonumber &+{2\over n_jd }\ma(\sum)_{d_1=1}^{d-1}\ma(\sum)_{a_1=0}^a \ma(\sum)_{b_1=0}^b {a\choose a_1}{b\choose b_1} d_2(2d_1d_2-d)\;\sigma_{d_1}(p^{a_1},l^{b_1+1})\cdot \sigma_{d_2}(p^{a_2},l^{b_2})& \end{eqnarray} where in the sum above $d_1+d_2=d$, $a_1+a_2=a$ and $b_1+b_2=b$. \step{\bf Example 1} Using a computer program based on (\ref{3cf}) and the recursive formulas (\ref{spheres}) for $\sigma_d$, one recovers for example that in ${ \Bbbd P}^3$ all the degree 2 elliptic invariants are 0 (fact known for a very long time) but also one gets new examples, like: \[ \begin{tabular}{c|ccccc} & $\tau_3(l^{11})$ && $\tau_5(p,l^{17})$ && $\tau_6(p^{11},l)$ \cr \hline $j\ne 0,1728$ & $6\cdot 25920$ && $6\cdot 15856790593536$ && $6\cdot 13260$ \cr $j=0$& $2\cdot 25920$ && $2\cdot 15856790593536$ && $2 \cdot 13260 $ \cr $j=1728$ & $3\cdot 25920$ && $3\cdot 15856790593536$ && $3\cdot 13260 $ \end{tabular} \] \step{\bf Example 2} Similarly, when $n=4$, one can use a computer program based on the four steps described in the Section \ref{Comp} to get for example: \[ \begin{tabular}{c|ccccc} & $\tau_3(5H^2,3l,p)$ && $\tau_3(12H^2,l)$ &&$\tau_3(14H^2)$\cr \hline $j\ne 0,1728$ & $6\cdot 42 $ && $6\cdot202680$&& $6\cdot 1305640$ \cr $j=0$& $2\cdot 42$ && $2\cdot 202680$ && $2\cdot 1305640 $ \cr $j=1728$ & $3\cdot 42$ && $3\cdot 202680$&& $3\cdot 1305640 $ \end{tabular} \] \medskip \addtocounter{theorem}{1 Unfortunately, the number of steps involved in computing the elliptic invariant $\tau_d$ in ${ \Bbbd P}^n$ increases extremely fast with $n$. For example, one can write down the recursive formulas for $n=5,\;6$ that do not look that complicated (we need to blow up only two strata, ${\cal Z}_3$ and ${\cal Z}_2$). But the amount of time necessary to run the corresponding program is too long to produce interesting examples. \section{Appendix} \renewcommand{\theequation}{A.\arabic{equation}} \setcounter{equation}{0} The genus zero perturbed invariant and the genus zero enumerative invariant are equal in ${ \Bbbd P}^n$ (cf. \cite{rt}), i.e. \begin{eqnarray}\label{si=phi} \sigma_d(H^{j_1},H^{j_2},\dots, H^{j_k})= RT_{d,0}(H^{j_1},H^{j_2},H^{j_3}|H^{j_4},\dots, H^{j_k}) \end{eqnarray} Consequences of Ruan-Tian degeneration formula are: \begin{eqnarray}\label{gendeg} RT_{d,1}(\beta_1\;|\;\beta_2,\dots,\beta_l)= \ma(\sum)_{i_1+i_2=n} \sigma_d(H^{i_1},H^{i_2},\beta_1,\dots,\beta_l) \end{eqnarray} and that $\sigma_d$ in ${ \Bbbd P}^n$ satisfies the following recursive formula: for $j_1\ge j_2\ge\dots\ge j_k\ge 2$, \begin{eqnarray}\label{spheres}\nonumber \sigma_d(H^{j_1},H^{j_2},H^{j_3})= d \sigma_d(H^{j_1+j_3-1},H^{j_2})-d \sigma_d(H^{j_1+j_2},H^{j_3-1}) -\sigma_d(H^{j_1},H^{j_2+1},H^{j_3-1})\\ \ma(\sum)_{d_1=1}^{d-1}\ma(\sum)_{i=0}^n {\vphantom{\ma(\int)_{k}}}\sigma_{d_1}(H^{j_1},H^{j_2},H^i)\sigma_{d_2}(H^{j_3-1},H^{n-i}) \ma(\sum)_{d_1=1}^{d-1}\ma(\sum)_{i=0}^n \sigma_{d_1}(H^{j_1},H^{j_3-1},H^i)\sigma_{d_2}(H^{j_2},H^{n-i})\hskip.1in \end{eqnarray} where $\sigma_d(H^{j_1},H^{j_2},H^{j_3})= \sigma_d(H^{j_1},H^{j_2},H^{j_3},H^{j_4},\dots,H^{j_k})$ and the conditions $H^{j_4},\dots,H^{j_k}$ are distributed in the right hand side in all possible ways. Note that $\sigma_1(pt, pt)=1$ gives the initial step of the recursion.

9608

alg-geom/9608012

en

https://arxiv.org/abs/alg-geom/9608012

alg-geom/9608012

Valery Alexeev

Valery Alexeev

Compactified jacobians

AMSLaTeX 1.2/LaTeX2e with epic.sty, eepic.sty, Postscript file is also available at http://domovoy.math.uga.edu/preprints , a few misprints corrected

Let J be the jacobian of a reduced projective curve C with nodes only. 1) We give a simple and natural definition for its many compactifications and show the connection with various other definitions appearing in the literature. 2) Among all compactifications we choose one canonical, and define a theta divisor on it. 3) We give two very explicit and simple descriptions of a stratification of this canonical compactification into homogeneous spaces over J.

alg-geom

\section{Introduction} \label{sec:Introduction} \begin{saynum} Let $C$ be a reduced projective curve over a field $k$ such that $C_{\bar k}$ has only nodes as singularities. The jacobian $\operatorname{Pic}^0\,C$ is a semiabelian variety over $k$ which parameterizes invertible sheaves on $C$ of degree 0 on each irreducible component. It need not be proper. The problem of finding a good compactification for it goes back at least to the work of Igusa \cite{Igusa_SystemsOfJacobians} and the notes of Mumford and Mayer \cite{Mumford_BoundaryPoints,Mayer69}. For an irreducible curve the answer was given already by D'Souza in \cite{DSouza79}. Altman, Kleiman and others extended this work to the families of irreducible curves with more general (for example, nonplanar) singularities in a series of works \cite{AltmanKleiman_CompJac,AltmanKleiman_CompJac2, AltmanIarrobinoKleiman_IrreducibilityCompJac, KleimanKleppe81,Rego_CompactifiedJacobian}, and more recently \cite{Soucaris94,Esteves95}. \end{saynum} \begin{saynum} In the case when $C$ is reducible the situation is more complicated. In a classical paper \cite{OdaSeshadri79} Oda and Seshadri constructed a family of compactified jacobians $\operatorname{Jac}_{\phi}$ parameterized by an element $\phi$ of a certain real vector space. The construction is very general and covers a lot of cases. At the same time it poses a question of giving a more natural definition for $\operatorname{Jac}_{\phi}$ and explaining where exactly the multitude of answers comes from. A related paper is \cite{Seshadri82}. \end{saynum} \begin{rem} It is important to note that the term ``compactified jacobian'' is a misnomer. Most of the varieties $\operatorname{Jac}$ discussed here do not naturally contain $\operatorname{Pic}^0(C)$. This becomes especially clear when one works over a nonclosed field or with families. Instead, there is always an action of $\operatorname{Pic}^0(C)$ and $\operatorname{Jac}$ is stratified into locally closed subschemes so that every stratum is a homogeneous space over $\operatorname{Pic}^0(C)$ and the maximal-dimensional strata are principal homogeneous spaces. However, we will use the term since it is widely accepted. \end{rem} \begin{saynum} The work of Simpson \cite{Simpson94a} on the moduli of coherent sheaves on projective schemes implies, as a very special case of a much more general situation, a natural definition of the compactified jacobian $\operatorname{Jac}_{d,L}(C)$ which depends on an integer $d$, the degree, and on an ample invertible sheaf $L$ on $C$, the polarization. The definition is functorial and therefore also works for families. It turns out that $\operatorname{Jac}_{d,L}(C)$ and Oda-Seshadri's $\operatorname{Jac}_{\phi}(C)$ coincide and there is a simple formula for $\phi$ as a function of $d$ and $L$. An immediate corollary of this is that they are all reduced and Cohen-Macaulay schemes. \end{saynum} \begin{saynum} In the case when the curve $C$ is stable, we can further narrow the choices by using for $L$ the dualizing sheaf $\omega_C$. Then for every $d\in{\mathbb Z}$ the schemes $\operatorname{Jac}_{d,\omega_C}/\operatorname{Aut}(C)$ can be put in a family over the moduli space $\overline{M}_g$, where $g$ is the arithmetical genus of the curve $C$. This is the result of the work \cite{Pandharipande94} of Pandharipande (he also considers sheaves of rank $\ge2$). A yet another family $\overline{P}_d\to\overline{M}_g$ for $d\ge10(2g-2)$ was earlier constructed by Caporaso in \cite{Caporaso94} as the compactification of the universal jacobian. The interpretation of the fiber $\overline{P}_d(C)$ over $[C]\in\overline{M}_g$ is in terms of invertible sheaves on certain semistable curves that have $C$ as a stable model. Pandharipande shows that Caporaso's construction of $\overline{P}_d$ is equivalent to his. \end{saynum} \begin{saynum} Another approach is to look at a one-parameter family of smooth curves $C_t$ degenerating to $C$ and try to find a limit of the family of ``jacobians'' $\operatorname{Jac}_d(C_t)=\operatorname{Pic}^d(C_t)$, perhaps after a finite ramified base change. In the complex analytic situation Namikawa \cite{Namikawa_ToroidalDegsAVs} constructed infinitely many toroidal degenerations of principally polarized abelian varieties that depend on polyhedral decompositions. We note a related work \cite{Kajiwara93} where the compactified jacobians corresponding to polyhedral decompositions appear in the context of log geometry (under the restriction that the irreducible components of the curve $C$ are nonsingular). Among various polyhedral decompositions Namikawa explicitly distinguished one called the Voronoi decomposition (and the Delaunay decomposition dual to it). \end{saynum} \begin{saynum} The degeneration corresponding to the Delaunay and Voronoi decompositions also appears in \cite{AlexeevNakamura96} as a result of the ``simplified Mumford's construction''. There it is shown that a family of principally polarized abelian varieties with theta divisors over spectrum of a complete DVR has the canonical limit (perhaps after making a finite ramified base change first). This limit was called a stable quasiabelian variety (SQAV), and when considered as a pair $(P,B)$ with the theta divisor -- a stable quasiabelian pair (SQAP). This poses a question of whether the SQAP which appears as the limit of jacobians is one of the compactified jacobians $\operatorname{Jac}_{\phi}$, $\operatorname{Jac}_{d,L}(C)$ above, and if yes, then which one. \end{saynum} \begin{saynum} As explained in \cite{Alexeev_CMAV}, an SQAV corresponding to a smooth curve $C$ coincides with $\operatorname{Pic}^{g-1}(C)$ and not with $\operatorname{Pic}^0(C)$. This gives a motivation to look at the case $d=g-1$ more closely. \end{saynum} \begin{saynum} We show that precisely for one degree, $d=g-1$, the scheme $\operatorname{Jac}_{d,L}(C)$ does not depend on the polarization $L$, so one can simply write $\operatorname{Jac}_{g-1}$. For this reason we call it the {\em canonical compactified jacobian.\/} We show that $\operatorname{Jac}_{g-1}$ possesses a natural ample sheaf with a natural section which we call the theta divisor $\Theta_C$. We give a very simple explicit combinatorial description of the stratification of $\operatorname{Jac}_{g-1}$ and the restrictions of $\Theta_C$ on each stratum. The description goes in terms of the orientations on complete subgraphs of the dual graph $\Gamma(C)$ and invertible sheaves on the partial normalizations of $C$ of multidegrees that correspond to these orientations. \end{saynum} \begin{saynum} By considering a degenerating family $C_t\rightsquigarrow C$ of curves and the corresponding degenerating family $\operatorname{Pic}^{g-1}(C_t) \rightsquigarrow \operatorname{Jac}_{g-1}(C)$ one can show that the canonical compactified jacobian is an SQAV (we do not include this argument here). Therefore, the functor associating to each stable curve $C$ its canonical compactified jacobian $(\operatorname{Jac}_{g-1},\Theta)$ should define a map from the Deligne-Mumford compactification $\overline{M}_g$ to the complete moduli space of SQAVs if the latter exists. \end{saynum} \begin{saynum} Finally, an SQAV was defined in \cite{AlexeevNakamura96,Alexeev_CMAV} in terms of some explicit combinatorial data. We give the corresponding data for a curve $C$ (about a half of this description can already be found in \cite[\S18]{Namikawa_NewComp2} and \cite[9.D]{Namikawa_ToroidalCompSiegel} where it is attributed to Mumford). Further, we explain how this second description is related to the previous one. \end{saynum} \begin{saynum} Although the main definitions and results including \ref{defn:Jac_functor} and \ref{thm_Simpson} hold over arbitrary base, for most of the paper we will be working over an algebraically closed field for simplicity. \end{saynum} \begin{ack} I would like to thank Profs. Yu.I. Manin, T. Oda and R. Smith for very useful conversations. It was Prof. Oda who kindly explained me the relation between his and Seshadri's parameter $\phi$ and the polarizations (in the case of degree 0). Of course, it is completely my fault if I didn't get it right. \end{ack} \section{Definition of $\operatorname{Jac}_{d,L}$} \label{sec:Definition of oJd} \begin{say} In this section I give the definition for compactified jacobians which I feel is the easiest and the most natural, and formulate the main existence theorem, due to Simpson. To take the bull by the horns, here it is: \end{say} \begin{defn}\label{defn:cj} For every integer $d$ and a polarization $L$ on $C$, the ``compactified jacobian'' $\operatorname{Jac}_{d,L}$ is the coarse moduli space of semistable w.r.t. $L$ admissible sheaves on $C$ of degree $d$ up to the $\operatorname{gr}$-equivalence. \end{defn} \begin{say} Let us now explain the terms used in this definition. The {\em polarization\/} $L$ is an ample invertible sheaf. By {\em admissible\/} (for our purposes) sheaf we mean a coherent ${\mathcal O}_C$-module $F$ \begin{enumerate} \item which is of rank 1, i.e. it is invertible on a dense open subset of $C$, and \item such that for every $x\in C$ the stalk $F_{x,C}$ is of depth 1. Equivalently, any nonzero subsheaf of $F$ has support of dimension 1. \end{enumerate} The latter condition is what Seshadri \cite{Seshadri82} calls a depth 1 sheaf and what Simpson \cite{Simpson94a} calls a purely dimensional sheaf. \end{say} \smallskip \begin{saynum}\label{saynum:admissible_sheaves} As well known (see f.e. \cite{Seshadri82}) admissible sheaves have a very simple description: \begin{enumerate} \item if $x$ is nonsingular, $F_{x,C}\simeq {\mathcal O}_{x,C}$ \item if $x$ is a node, $F_{x,C}$ is either ${\mathcal O}_{x,C}$ or the maximal ideal $m_{x,C}$. In the latter case $F$ is isomorphic to $\pi(x)_*F(x)$ where $\pi(x)$ is a partial normalization of $C$ at $x$ and $F(x)=\pi(x)^*F/\text{torsion}$. \end{enumerate} Moreover, if we are interested in the depth 1 sheaves that have rank 0 or 1 at every generic point, we need to add the sheaves such that \begin{enumerate}\setcounter{enumi}{2} \item $F_{x,C}=0$ \item if $x$ is a node lying on two irreducible components $C_1$ and $C_2$ with the inclusions $i_k:C_k\to C$, then $F_{x,C}=i_{k*}{\mathcal O}_{C_k}$, $k=1$ or $2$. \end{enumerate} \end{saynum} \begin{lem}\label{lem:admissible_sheaves} For each admissible sheaf $F$ on $C$ denote by $\pi': C'\to C$ the partial normalization of $C$ at the nodes where $F$ is not invertible and by $F'=\pi{'}^*F/tors$. Then $F=\pi'_*F'$. Therefore, every admissible sheaf $F$ on $C$ can be identified with a unique invertible sheaf $F'$ on a unique partial normalization $C'$ of $C$. \end{lem} \begin{proof} Well known, see f.e. \cite{Seshadri82}. \end{proof} \begin{defn}\label{defn:degree} For an admissible sheaf $F$ on $C$ the degree is defined as \begin{displaymath} \deg F= \chi(F) -\chi({\mathcal O}_C) =\chi(F) +g-1, \end{displaymath} where $g$ is the arithmetical genus of $C$. \end{defn} \begin{rem} Note that $\deg F'$ if defined on $C'$ itself is $\deg F$ minus the number of nodes where $F$ is not invertible. \end{rem} \begin{defn} Let $\underline\lambda=(\lambda_1\dots\lambda_s)$ be the multidegree of $L$, and $\underline{r}=(r_1\dots r_s)$ be the multirank of a depth 1 sheaf $F$. The Seshadri slope is \begin{displaymath} \mu_L(F)= \frac{\chi(F)}{\sum \lambda_i r_i} \end{displaymath} \end{defn} \begin{defn} A depth 1 sheaf $F$ on $C$ is said to be stable (resp. semistable) w.r.t. the polarization $L$ if for any nonzero subsheaf $E\subset F$ one has \begin{displaymath} \mu_L(E) < \mu_L(F) \end{displaymath} (resp $\le$). \end{defn} \begin{rem} This definition, due to Seshadri \cite{Seshadri82}, is a particular case of a much more general one given by Simpson in \cite{Simpson94a} that applies to a pure-dimensional sheaf on any projective scheme whatsoever. \end{rem} \begin{lem}\label{lem:criterion_stability_for_admissible_sheaves} A depth 1 sheaf is (semi)stable iff the inequality $\mu_L(E)<$(resp. $\le$) $\mu_L(F)$ is satisfied for finitely many subsheaves of the form $F_D=i_*(i^*F/tors)$ for every subcurve $D\subset C$, where $i:D\to C$ is the inclusion morphism. \end{lem} \begin{proof} Indeed, for any depth 1 subsheaf $E$ with support $D$ one has $E\subset F_D$ and $\mu_L(E)\le\mu_L(F_D)$. \end{proof} \begin{rem} This leads to a series of simple inequalities some of which will be considered in the next sections. Therefore, knowing the multidegree of $F$ and the set of nodes where $F$ is not locally free, it is easy to say whether $F$ is (semi)stable or not. \end{rem} \begin{say} According to the general theory, for every depth 1 sheaf $F$ there exists a Harder-Narasimhan filtration \begin{displaymath} 0=F_0\subset F_1 \subset\dots\subset F_k=F \end{displaymath} with strictly decreasing slopes and semistable quotients $F_i/F_{i+1}$. If $F$ is semistable, then there is a similar (Jordan-Holder) filtration with stable $F_i/F_{i+1}$ which is not unique. However, the graded object $\operatorname{gr}(F)=\oplus F_i/F_{i+1}$ is uniquely defined. \end{say} \begin{defn} Two semistable depth 1 sheaves $F$ and $F'$ are said to be $\operatorname{gr}${\em-equivalent\/} if $\operatorname{gr}(F)\simeq\operatorname{gr}(F')$. \end{defn} \begin{say} Now all the ingredients of the definition \ref{defn:cj} have been introduced. \end{say} \smallskip \begin{say} To put this into a functorial perspective, consider a projective morphism of schemes $\pi:C\times S\to S$ whose every geometric fiber is a reduced curve with nodes only as singularities and a relatively ample sheaf $L$ on $C$. We will say that a coherent sheaf $F$ on $C$ is admissible (resp. stable, resp. semistable) if so is its restriction to every geometric fiber of $\pi$. We say that two sheaves are equivalent (resp. $\operatorname{gr}$-equivalent) if the restrictions on the geometric fibers are isomorphic (resp. $\operatorname{gr}$-equivalent). Now define the moduli functor $Jac_{d,L}(C/S):Schemes\to Sets$ in the following way: \end{say} \begin{defn}\label{defn:Jac_functor} For any scheme $S'$, $Jac^-_{d,L}(C/S)(S')$ is the set of semistable admissible sheaves on $C'=C\underset{S}{\times}S'/S'$ up to the $\operatorname{gr}$-equivalence. The functor $Jac^-_{d,L}(C/S)$ itself is not necessarily a sheaf for the fppf (faithfully flat of finite presentation) topology and therefore cannot be representable even if all the sheaves are stable. This happens basically for the same reason why the functor $\operatorname{Pic}\,S'$ is not a sheaf: one needs a rigidification to kill the infinite ($={\mathbb G}_m$) group of automorphisms. When the smooth locus of $C/S$ has a section, one can use the rigidified version. Or we can follow the same path as for the relative Picard functor, i.e. define $Jac_{d,L}(C/S)$ to be the fppf-sheafification of $Jac^-_{d,L}(C/S)$. \end{defn} \begin{thm}[Simpson]\label{thm_Simpson} The functor $Jac_{d,L}(C/S)$ is coarsely represented by a projective scheme $\operatorname{Jac}_{d,L}(C/S)$. \end{thm} \begin{proof} This may be proved by the same methods as in \cite{Simpson94a} and basically is a very special case of \cite[1.21]{Simpson94a}. Simpson works over $\Bbb C$ but in our particular situation there is no need for this. The hardest question involved, the boundedness, is basically obvious. Alternatively, in \ref{thm:Jacs_are_the_same} we will show that our (semi)stability condition is equivalent to the one that was used by Oda and Seshadri, so in the case $S=\operatorname{Spec}\bar k$ the theorem follows by \cite[12.14]{OdaSeshadri79}. \end{proof} \section{Basic definitions and notations} \label{sec:Basic definitions and notations} \begin{say} The purpose of this section is to fix some notations common to the following sections and to introduce the basic examples of curves on which the later descriptions will be illustrated. \end{say} \begin{notation} \begin{enumerate} \item To any curve $C$ we can associate the unoriented graph $\Gamma(C)$ by assigning a vertex to each irreducible component $C_i$ and an edge to every node. We do not assume $C$ to be connected, so the graph need not be connected either. \item $\pi:\widetilde C\to C$ denotes the normalization of $C$. \item $C_i$ are the irreducible components of $C$, $\widetilde C_i$ are their normalizations. \item $g_i=p_a(C_i)$, $\tilde g_i=p_a(\widetilde C_i)$. \item $h(C)=h(\Gamma(C))$ is the cyclotomic number -- the number of independent loops in $\Gamma$, i.e. the rank of $H_1(\Gamma(C))$ when $\Gamma$ is considered as a cell complex. \end{enumerate} \end{notation} {\bf Six simple examples.} \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \put(0,0){\arc{80}{5.1025}{6.1492}} \put(0,50){\arc{80}{0.1308}{1.1775}} \put(70,25){\circle{4}} \put(80,25){\circle{4}} \drawline(72,25)(78,25) \end{picture} \begin{example}\label{example1} $C=C_1\cup C_2$ intersecting at one point with both $C_i$ smooth. $\tilde g_i=g_i$, $h(C)=0$. \end{example} \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \spline(10,45)(17,37)(21,28)(24,15)(25,5) \spline(10,30)(17,37)(20,49) \spline(5,20)(21,28)(30,35) \spline(10,10)(24,15)(40,15) \spline(40,49)(50,40)(60,37) \spline(53,48)(50,40)(55,30) \put(67,25){\circle{4}} \put(80,25){\circle{4}} \put(90,35){\circle{4}} \put(90,15){\circle{4}} \drawline(69,25)(78,25) \drawline(81.5,26.5)(88.5,33.5) \drawline(81.5,23.5)(88.5,16.5) \put(95,45){\circle{4}} \put(105,35){\circle{4}} \drawline(96.5,43.5)(103.5,36.5) \end{picture} \begin{example}\label{example2} The generalization of the previous example is a curve whose dual graph is a forest. Still $\tilde g_i=g_i$ and $h(C)=0$. \end{example} \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \spline(20,10)(30,20)(40,30)(30,42)(20,30)(30,20)(40,10) \put(90,15){\circle{4}} \put(90,22.5){\arc{15}{1.8555}{7.5647}} \end{picture} \begin{example}\label{example3} An irreducible curve with one node. $\tilde g=g-1$, $h(C)=1$. \end{example} \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \spline(30,35)(26,40)(30,45)(34,40)(30,35) \spline(30,35)(34,28)(30,24) \spline(30,35)(26,28)(30,24) \spline(30,24)(34,18)(30,12)(27,7) \spline(30,24)(26,18)(30,12)(33,7) \put(90,15){\circle{4}} \put(90,20){\arc{10}{1.9327}{7.4871}} \put(90,22.5){\arc{15}{1.8454}{7.5744}} \put(90,25){\arc{20}{1.8019}{7.6179}} \end{picture} \begin{example}\label{example4} The generalization of that is an irreducible curve with $n$ nodes. $\tilde g=g-n$, $h(C)=n$. \end{example} \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \spline(20,15)(27.5,12)(35,15)(37,20)(35,25)(30,27)(25,30)(20,35) (27.5,43)(35,40) \drawline(27.5,49)(27.5,5) \put(70,25){\circle{4}} \put(90,25){\circle{4}} \drawline(72,25)(88,25) \spline(71.5,26.5)(80,32)(88.5,26.5) \spline(71.5,23.5)(80,18)(88.5,23.5) \end{picture} \begin{example}\label{example5} The dollar sign curve. $\tilde g_i=g_i$, $h(C)=2$. \end{example} \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \spline(30,35)(26,40)(30,45)(33,49) \spline(30,35)(34,40)(30,45)(27,49) \spline(30,35)(34,28)(30,24) \spline(30,35)(26,28)(30,24) \spline(30,24)(34,17)(30,12)(27,7) \spline(30,24)(26,17)(30,12)(33,7) \put(70,25){\circle{4}} \put(90,25){\circle{4}} \spline(71.5,26.5)(80,32)(88.5,26.5) \spline(71.5,23.5)(80,18)(88.5,23.5) \spline(71.5,26.5)(80,37)(88.5,26.5) \spline(71.5,23.5)(80,13)(88.5,23.5) \end{picture} \begin{example}\label{example6} The generalization of the dollar sign curve is a curve $C=C_1\cup C_2$ with both $C_i$ smooth and intersecting at $n$ points. $\tilde g_i=g_i$, $h(C)=n-1$. \end{example} \section{Comparison with Oda-Seshadri's compactified jacobians} \label{sec:Comparison with Oda-Seshadri's compactified jacobians} \begin{say} We would like to compare the compactified jacobians introduced in section \ref{sec:Definition of oJd} with those appearing in the classical paper \cite{OdaSeshadri79} of Oda and Seshadri. \end{say} \begin{saynum} The $\operatorname{Jac}_{\phi}$ in \cite{OdaSeshadri79} are constructed using GIT as the moduli spaces of $\phi$-semistable admissible sheaves. Here $\phi$ is an element of a certain real vector space $\partial C_1(\Gamma,{\mathbb R})$ (without loss of generality one can assume that $\phi\in\partial C_1(\Gamma,{\mathbb Q})$). $\Gamma$ is, as in the previous section, the dual graph of $C$ and $C_0,C_1,H_0,H_1,C^0,C^1,H^0$ and $H^1$ are the associated to it chain and (co)homology groups. \end{saynum} \begin{saynum} In \cite{OdaSeshadri79} the main object of interest is the depth 1 sheaves of degree 0. Oda and Seshadri give the combinatorial definition of a $\phi$-stable (resp. semistable) sheaf and introduce the $\phi$-equivalence relation. The main result then is that there exists a reduced scheme $\operatorname{Jac}_{\phi}$ which coarsely represents the functor of $\phi$-semistable sheaves up to $\phi$-equivalence. This is then applied to compactify $\operatorname{Pic}^0\,C$. However, for any depth 1 sheaf of arbitrary degree $d$ one can relate the $\phi$-(semi)stability and equivalence with $(d,L)$-(semi)stability and equivalence. Then whatever is proved for $\operatorname{Jac}_{\phi}$ immediately applies to $\operatorname{Jac}_{d,L}$. Here is the precise connection. \end{saynum} \begin{thm}\label{thm:Jacs_are_the_same} Let $\underline\lambda=(\lambda_i)$ and $\underline\omega=(\lambda_i)$ be the multidegrees of the polarization $L$ and of the dualizing sheaf $\omega_C$ respectively, and $\lambda=\sum\lambda_i$ and $\omega=\sum\omega_i=2g-2$ be the total degrees. Pick arbitrary integers $d_i$ with $\sum d_i=d$ and sufficiently large integers $\tilde{n}_i$. Define $\phi=(\phi_i)\in\partial C_1({\mathbb Q})$ to be a solution of the following system of linear equations \begin{displaymath} (\lambda_i/\lambda)(d-\omega/2)= d_i-\omega_i/2+\tilde{n}_i+\phi_i \end{displaymath} ($\phi$ is only defined up to a shift by a lattice). Then an admissible sheaf of degree $d$ is (semi)stable w.r.t. $L$ iff it is $\phi$-(semi)stable. Two semistable w.r.t. $L$ sheaves are $\operatorname{gr}$-equivalent iff they are $\phi$-equivalent. \end{thm} \begin{proof} This can be extracted from \cite[\S11]{OdaSeshadri79} directly, particularly from the account on pp.52-53. \end{proof} \begin{cor} Every $\operatorname{Jac}_{d,L}$ is isomorphic to one of $\operatorname{Jac}_{\phi}$ and vice versa. \end{cor} \begin{cor} Every $\operatorname{Jac}_{d,L}$ is reduced and Cohen-Macaulay. \end{cor} \begin{proof} Indeed, $\operatorname{Jac}_{\phi}$ is reduced by \cite[11.4]{OdaSeshadri79}. Moreover, the proof shows (pp.60-62) that $\operatorname{Jac}_{\phi}$ is a good GIT quotient of a certain scheme $R$ and there exists an open subscheme $Y\subset R\times {\mathbb P}(E^*)$ such that the projection $R\to Y$ is surjective, and $Y$ is formally smooth over a Hilbert scheme $H$ which is open in a quotient by the symmetric group of $C\times\dots\times C$. Therefore, $H$ is CM, an so is $Y$, and so is $R$, and so is $\operatorname{Jac}_{\phi}$. \end{proof} \section{Description of $\operatorname{Jac}_{g-1}$} \label{sec:first description} \begin{lem} $\operatorname{Jac}_{g-1,L}(C)$ does not depend on the polarization $L$. \end{lem} \begin{proof} Indeed, by definition \ref{defn:degree} the degree $d=g-1$ iff $\chi(F)=0$. Then for any $E\subset F$ the inequality \begin{displaymath} \mu_L(E)\le \,(\text{resp. }<) \,\mu_L(F) \end{displaymath} is equivalent to \begin{displaymath} \chi(E)\le \,(\text{resp. }<) \,0 \end{displaymath} \end{proof} \begin{say} For this reason we will call $\operatorname{Jac}_{g-1}(C)$ the {\em canonical compactified jacobian.} \end{say} \begin{defn} A subgraph $\Gamma'\subset\Gamma$ is said to be {\em generating\/} if $\operatorname{vertices}(\Gamma)=\operatorname{vertices}(\Gamma')$. Every such subgraph corresponds to a partial normalization of $C$ at the nodes $\Gamma-\Gamma'$. We denote this partial normalization by $\pi(\Gamma'):C(\Gamma')\to C$. Note in particular that $C(\Gamma)=C$ and that $\tilde C$ is $C(\Gamma')$ where $\Gamma'$ has all the vertices of $\Gamma$ but no edges at all. \end{defn} \begin{defn} A subgraph $\Gamma'\subset\Gamma$ is said to be {\em complete\/} if $\operatorname{vertices}(\Gamma')\subset \operatorname{vertices}(\Gamma)$ and $\operatorname{edges}(\Gamma')$ are precisely the edges of $\Gamma$ lying inside $\Gamma'$. These graphs correspond to subcurves $D\subset C$. Often we identify such subcurves $D$ with the corresponding subgraphs. \end{defn} \begin{defn} A {\em multidegree\/} of a graph $\Gamma$ is a set $\underline{d}=(d_i)$ of integers for every vertex $C_i$ of $\Gamma$. {\em We will always assume that\/} \begin{displaymath} \sum d_i = g-1 \end{displaymath} A {\em normalization of multidegree\/} $\underline{d}$ is a set of integers $\underline{e}=(e_i)$ defined by \begin{displaymath} e_i=d_i-(\tilde g_i-1). \end{displaymath} It will be called {\em the normalized multidegree.\/} Note that we can use multidegrees $\underline{d}$ and normalized multidegrees $\underline{e}$ interchangeably. Note that $\sum e_i$ equals the number of edges of $\Gamma$. For a subcurve $D\subset C$, i.e. a complete subgraph $\Gamma'\subset\Gamma$, we set \begin{displaymath} d_D=\sum_{C_i\subset D} d_i, \quad e_D=\sum_{C_i\subset D} e_i \end{displaymath} \end{defn} \begin{defnprop}\label{defnprop} A normalized multidegree $\underline{e}$ is called {\em semistable\/} (resp. {\em stable\/}) if any of the following equivalent conditions hold: \begin{enumerate} \item \begin{displaymath} |e_D-\#edges(D)-\frac{1}{2}D(C-D)|\le \frac{1}{2}D(C-D) \end{displaymath} for every subcurve $D\subset C$. Here $D(C-D)$ is the number of points in $D\cap\overline{(C\setminus D)}$ (resp. $<$). \item \begin{displaymath} e_D\le \#edges(D)+D(C-D) \end{displaymath} (resp.$<$). \item there exists an orientation of the graph $\Gamma$ such that $e_i$ equals the number of arrows pointing at $C_i$ (resp. in addition there is no proper subcurve $D\subset C$ such that all arrows between $D$ and $C-D$ go in one direction). \end{enumerate} In this case the multidegree $(\underline d)$ is also called (semi)stable. \end{defnprop} \begin{proof} The implication (i)$\Rightarrow$(ii) is clear and the inverse is obtained by looking at $D'=C-D$. (iii) obviously implies (ii). To prove the implication (ii)$\Rightarrow$(iii) first assume that the normalized multidegree $\underline{e}$ of the graph of $C$ is strictly semistable, i.e. there exists a subcurve $D\subset C$ for which the equality holds. Then consider separately the following multidegrees on $D$ and $C-D$. On $C-D$ simply take the restriction of $\underline{e}$. On $D$, however, for every vertex $C_i$ take $e_i'=e_i$ minus the number of edges between $C_i$ and $C-D$. Then it is easy to show that the two multidegrees thus obtained are semistable. Therefore, the orientations on $D$ and $C-D$ exist by the induction on the number of vertices. To complete the orientation of $C$, orient all the edges between $D$ and $C-D$ to point at $D$. In general, starting with a semistable multidegree as in (ii) we can fix an arbitrary vertex $C_{i_0}$ and change the degrees of $C_{i_0}$ and the neighboring vertices = curves $C_j$ by 1 to make the multidegree strictly semistable, thus reducing to the previous case. Hence, we get an orientation for the modified multidegree. The orientation for the original multidegree is then obtained by reversing the orientations of edges $(i_0,j)$. \end{proof} \begin{say} The third condition of the above definition is the easiest to check. We will call an orientation satisfying (iii) semistable (resp. stable). Note that different orientations may well produce the same multidegree. \end{say} \smallskip \begin{saynum} This is how the above combinatorial definitions relate to the (semi) stability of admissible sheaves on $C$. By lemma \ref{lem:admissible_sheaves} every admissible sheaf $F$ on $C$ can be identified with a unique invertible sheaf $F'$ on a unique partial normalization $C'=C(\Gamma')$ of $C$. Denote by $(\underline d')$ (resp. $(\underline e')$) the corresponding (resp. normalized) multidegrees on $C'$. Then \end{saynum} \begin{lem} If $\deg F=g-1$, then for $(\underline d')$ one has $\sum d'_i=g'-1$. \end{lem} \begin{proof} Obvious. \end{proof} \begin{lem} \begin{enumerate} \item $F$ is semistable iff $(\underline e')$ is semistable. \item $F$ is stable iff $(\underline e')$ is stable and the graphs $\Gamma$ and $\Gamma'$ have the same number of connected components. \end{enumerate} \end{lem} \begin{proof} Follows easily from \ref{defnprop} and \ref{lem:criterion_stability_for_admissible_sheaves}. \end{proof} \begin{say} We can now describe the points of $\operatorname{Jac}_{g-1}(C)$ as follows \end{say} \begin{thm}\label{thm:1description} \begin{enumerate} \item $\operatorname{Jac}_{g-1}(C)$ has a natural stratification into homogeneous spaces over $\operatorname{Pic}^0(C)$. Each stratum corresponds in a 1-to-1 way to a stable multidegree $\underline{d'}$ (resp. stable normalized multidegree $\underline{e'}$) on a generating subgraph $\Gamma'\subset\Gamma$. The $k$-points of this stratum can be identified with $k$-points of $\operatorname{Pic}_{\underline{d'}}(C(\Gamma'))$, i.e. with invertible sheaves on $C(\Gamma')$ of multidegree $\underline{d'}$. The codimension of this stratum equals $h(\Gamma)-h(\Gamma')$. \item There is a natural Cartier divisor $\Theta$ on $\operatorname{Jac}_{g-1}(C)$. Under the above identification, the restriction of $\Theta$ on each stratum corresponds to the sheaves $L$ with $h^0(L)>0$. \end{enumerate} \end{thm} \begin{say} To illustrate this theorem, let us see what happens in our basic examples. \end{say} \setcounter{example}{0} \begin{example} Graph $\Gamma$ doesn't have any stable multidegrees: take $D$ to be one of the vertices. The only possibility then is $\Gamma'$ which is a disjoint union of two vertices and the multidegree $\underline{e}=(0,0)$, i.e. $\underline{d}=(\tilde g_1-1,\tilde g_2-1)=(g_1-1,g_2-1)$. The graph $\Gamma'$ corresponds to the normalization $\widetilde X=X_1\bigsqcup X_2$ and \begin{displaymath} \operatorname{Jac}_{g-1}(C)=\operatorname{Pic}^{g_1-1}(C_1)\oplus\operatorname{Pic}^{g_2-1}(C_2) \end{displaymath} \end{example} \begin{example} Once again, a forest doesn't have any stable orientations unless all the brunches, i.e. edges, are cut. So, there is only one normalized multidegree $\underline e'=(0,\dots,0)$ for a generating subgraph corresponding to the normalization $\widetilde C$ and \begin{displaymath} \operatorname{Jac}_{g-1}(C)=\oplus_i\operatorname{Pic}^{g_i-1}(C_i) \end{displaymath} \end{example} \begin{example} The stable orientations are \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \put(15,15){\circle{4}} \put(15,22.5){\arc{15}{1.8555}{7.5647}} \drawline(16.75,15.25)(19,17.5) \drawline(16.75,15.5)(19.5,15.5) \put(35,15){\circle{4}} \end{picture} The first corresponds to $\operatorname{Pic}^{\tilde g-1+1}(C)=\operatorname{Pic}^{\tilde g}(C)$ and the second -- to $\operatorname{Pic}^{\tilde g-1}(\widetilde X)$. \end{example} \begin{example} There are $2^n$ subgraphs $\Gamma'$: each edge is either included in $\Gamma'$ or it's not. Each graph with $k$ edges obviously defines the multidegree $(\underline d')=(k)$. Therefore there are $\binom n{n-k}=\binom nk$ strata of codimension $n-k$ each corresponding to $\operatorname{Pic}^{g'-1}(C')$. \end{example} \begin{example}[Dollar sign curve] The possible generating subgraphs are: \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \put(15,25){\circle{4}} \put(25,25){\circle{4}} \drawline(17,25)(23,25) \spline(16.5,26.5)(20,30)(23.5,26.5) \spline(16.5,23.5)(20,20)(23.5,23.5) \put(40,25){\circle{4}} \put(50,25){\circle{4}} \spline(41.5,26.5)(45,30)(48.5,26.5) \spline(41.5,23.5)(45,20)(48.5,23.5) \put(40,40){\circle{4}} \put(50,40){\circle{4}} \drawline(42,40)(48,40) \spline(41.5,38.5)(45,35)(48.5,38.5) \put(40,10){\circle{4}} \put(50,10){\circle{4}} \drawline(42,10)(48,10) \spline(41.5,11.5)(45,15)(48.5,11.5) \put(65,40){\circle{4}} \put(75,40){\circle{4}} \spline(66.5,41.5)(70,45)(73.5,41.5) \put(65,25){\circle{4}} \put(75,25){\circle{4}} \drawline(67,25)(73,25) \put(65,10){\circle{4}} \put(75,10){\circle{4}} \spline(66.5,8.5)(70,5)(73.5,8.5) \put(90,25){\circle{4}} \put(100,25){\circle{4}} \end{picture} It is very easy to list all stable orientations and the corresponding stable multidegrees. Here are some of them: \setlength{\unitlength}{0.1cm} \begin{picture}(110,55)(0,0) \thicklines \put(25,25){\circle{4}} \put(40,25){\circle{4}} \drawline(27,25)(38,25) \drawline(38,25)(36,26) \drawline(38,25)(36,24) \spline(26.5,26.5)(32.5,32)(38.5,26.5) \drawline(38.5,26.5)(38,29) \drawline(38.5,26.5)(36,27) \spline(26.5,23.5)(32.5,18)(38.5,23.5) \drawline(26.5,23.5)(27,21) \drawline(26.5,23.5)(29,23) \put(70,25){\circle{4}} \put(85,25){\circle{4}} \spline(71.5,26.5)(77.5,32)(83.5,26.5) \drawline(83.5,26.5)(83,29) \drawline(83.5,26.5)(81,27) \spline(71.5,23.5)(77.5,18)(83.5,23.5) \drawline(71.5,23.5)(72,21) \drawline(71.5,23.5)(74,23) \end{picture} Here is the complete list: \begin{enumerate} \item For the graph $\Gamma$ itself there are two multidegrees $(2,1)$ and $(1,2)$ corresponding to invertible sheaves of multidegree $(\tilde g_1+1,\tilde g_2)=(\tilde g-1,\tilde g_2-1)+(2,1)$ and $(\tilde g_1,\tilde g_2+1)=(\tilde g-1,\tilde g_2-1)+(1,2)$ on $C$. \item In the second column, for each graph we have a unique stable multidegree $(1,1)$. Hence, there are 3 strata of codimension 1 corresponding to invertible sheaves of multidegree $(\tilde g_1,\tilde g_2)=(\tilde g-1,\tilde g_2-1)+(1,1)$. \item In the third column there are no stable orientations -- the graphs are trees. \item From the last column we get the normalized multidegree $(0,0)$ which corresponds to the invertible sheaves on the normalization $\widetilde C$ of $C$ of multidegree $(\tilde g_1-1,\tilde g_2-1)=(\tilde g-1,\tilde g_2-1)+(0,0)$. \end{enumerate} \end{example} \begin{example} This is an exercise no harder then the previous five. Any subgraph $\Gamma'$ has at least one stable orientation with one exception: when $\Gamma'$ contains only one edge. For each such subgraph with $k$ edges the number of possible stable multidegrees is $k-1$. Therefore, there are $\binom nk(k-1)$ strata of codimension $n-k$ for $k>0$ and one stratum for $k=0$. \end{example} \begin{proof}[Proof of \ref{thm:1description}] The proof of (i) follows immediately from \ref{lem:admissible_sheaves} and \ref{lem:criterion_stability_for_admissible_sheaves}. In each $\operatorname{gr}$-equivalence class of strictly semistable sheaves we can choose the one with the minimal graph $\Gamma'$ and it will be stable in our definition. Next, we have to show the existence of a natural line bundle with a natural section on $\operatorname{Jac}_{g-1}(C)$. To define the divisor $\Theta$ in a way similar to how it was done in \cite{Soucaris94,Esteves95} for irreducible $C$. Consider any family $\pi:C\times S\to S$ and an admissible sheaf $F$ of degree $g-1$ on $C\times S$. Then there is a natural line bundle $L(F)$ on $S$ defined as \begin{displaymath} L(F)= (\det R\pi_*F)^{-1} \end{displaymath} see \cite{KnudsenMumford76} for its definition. If we replace $F$ by $F\otimes\pi^*E$, $L(F)$ will be replaced by $L(F)\otimes E^{-\chi(F_t)}$. When $d=g-1$, $\chi=0$ which means that $L(F)$ will not change, so it is universally defined. Moreover, two $\operatorname{gr}$-equivalent families of semistable sheaves produce the same $L(F)$. The latter follows from the fact that if \begin{displaymath} 0\to F'\to F\to F''\to 0 \end{displaymath} is an exact sequence, then $\det R\pi_*F=(\det R\pi_*F')\otimes (\det R\pi_*F'')$, so only the stable factors are important. $\operatorname{Jac}_{d,L}$ and $\operatorname{Jac}_{\phi}$ are constructed using GIT as a quotient of the Grothendieck's $Quot$-schemes. By the universality $L(F)$ descends to $\operatorname{Jac}_{g-1}$. Now fix a point $c\in C$ and consider a universal family $F$ of invertible sheaves of degree $g'-1$ and multidegree $\underline d'$ over $\operatorname{Pic}^{g'-1}(C')$, where $C'=C(\Gamma')$ is any of the partial normalizations of $C$. When does the formula \begin{displaymath} \Theta=\{s\in\operatorname{Pic}^{g'-1}(C') \,|\, h^0(F_s)>0\} \end{displaymath} define a divisor? The answer is given by a theorem of Beauville \cite[2.1]{Beauville_PrymSchottky}: it is exactly when the multidegree $\underline d'$ is semistable using the part (iii) of Definition-Proposition \ref{defnprop}. It is also easy to show directly that if two sheaves of degree $g'-1$ are semistable and $\operatorname{gr}$-equivalent, then $h^0(F_1)\ne0$ iff $h^0(F_2)\ne0$. $\Theta$ provides a section of $(\det R\pi_*F)^{-1}$. \end{proof} \begin{rem} From the above proof we have a yet another characterization of semistable admissible sheaves in degree $g-1$: they have the multidegrees for which the usual definition of the theta-divisor actually gives a divisor. \end{rem} \section{An SQAV corresponding to a curve} \label{sec:Jac second description} \begin{saynum} An SQAV was defined in \cite{AlexeevNakamura96} explicitly starting from the following combinatorial data: \begin{enumerate} \item a lattice $X\simeq{\mathbb Z}^{g'}$ (and a lattice $Y$ isomorphic to it via $\phi:Y{\overset{\sim}{\rightarrow}} X$). \item a symmetric positive definite bilinear form $B:X\times X\to{\mathbb Z}$. \item an abelian variety $A_0$ of dimension $g''$, $g'+g''=g$, with a principal polarization given by an ample sheaf ${\mathcal M}_0$. \item a homomorphism $c_0:X\to A_0^t(k)$ (and a dual homomorphism $c_0^t:Y\to A_0(k)$) defining a semiabelian variety $G_0$ (and a dual semiabelian variety $G^t$). \item a trivialization of the biextension $\tau_0:1_{X\times X}=1_{Y\times X}\to (c^t\times c)^*{\mathcal P}_{A_0}^{-1}$, where ${\mathcal P}_{A_0}$ is the Poincare bundle. \end{enumerate} When the abelian part $A_0$ is trivial, $\tau_0$ becomes simply a bilinear symmetric function $b_0:X\times X\to k^*$. \end{saynum} \begin{saynum} We now would like to explain how to associate this data to a curve $C$. Part of this description can already be found in \cite[\S18]{Namikawa_NewComp2} and \cite[9.D]{Namikawa_ToroidalCompSiegel} where it is attributed to Mumford. \begin{enumerate} \item $X=H_1(\Gamma(C),{\mathbb Z})$. \item By choosing arbitrarily an orientation on $\Gamma$, we get a natural embedding of $X$ in a free abelian group $C_1(\Gamma(C),{\mathbb Z})=\oplus{\mathbb Z} e_j$, each $e_j$ corresponds to an edge of $\Gamma$. The form $B$ is the restriction to $H_1$ of the standard Euclidean form on $C_1$. \item an abelian variety $A_0$ is $\operatorname{Pic}^0(\widetilde C)$. Instead of line bundle on $A_0$ we consider $B_0=Pic^{g-1}(\widetilde C)$ and the natural line bundle $M_0$ on it defines by the theta divisor. A choice of an isomorphism $A_0\to B_0$ doesn't matter. \item every element of $H_1(\Gamma)$ defines a cycle of multidegree $(0,\dots,0)$ on $\widetilde C$, i.e. an element of $A_0^t$. This gives the homomorphism $c$. \item Finally, the map $\tau$ is the most interesting part. The quick answer is that $\tau$ is given by a ``generalized crossratio''. \end{enumerate} \end{saynum} \begin{saynum} Let $f,g$ be two meromorphic functions on a smooth projective curve $X$ with disjoint divisors. Then, defining \begin{displaymath} (f,g)=f(div\, g)= \prod_{x\in C} f(x)^{v_x(g)}, \end{displaymath} one has $(f,g)=(g,f)$ according to A.Weil. For $f=(z-a)/(z-b)$, $g=(z-c)/(z-d)$ this is nothing but the usual crossratio. In \cite[XVII]{SGA4} Deligne showed that to arbitrary two invertible sheaves $L,M$ on $X$ and their meromorphic sections $f,g$ with disjoint divisors one can associate an element $(f,g)$ of a certain one-dimensional vector space $(L,M)$. These one-dimensional vector spaces are bilinear and symmetric in $L,M$ (in the case of degree 0 they form a symmetric biextension of $\operatorname{Pic}^0\times\operatorname{Pic}^0$) and $(f,g)=(g,f)$ if $\deg M\cdot\deg L$ is even and $=-(g,f)$ otherwise. We will call this pairing Deligne symbol. A very nice summary of its properties can be found in \cite{BeilinsonManin86}. \end{saynum} \begin{saynum} Now for every two distinct elements $e_k,e_l$ of the standard basis in $C_1(\Gamma)$ we have two divisors of the total degree 0 on $\widetilde C=\cup \widetilde C_i$. This defines a one-dimensional vector space $V_{k,l}$ and an element $(e_k,e_l)$ in it. If $k=l$, we still have a vector space $V_{k,k}$ but $(e_k,e_k)$ is undefined. {\em We define it arbitrarily.\/} In particular, restricting this to $X\times X\subset C_1\times C_1$, we obtain a pairing on $X\times X$ with the values in a certain collection of one-dimensional vector spaces. Because every element of $H_1$ has degree 0 on each irreducible component $\widetilde C_i$, this pairing is symmetric. It can be checked that these one-dimensional vector spaces are the fibers of $(c^t\times c)^*{\mathcal P}_{A_0}^{-1}$, where ${\mathcal P}_{A_0}$ is the Poincare bundle (= Weil biextension) on $A_0\times A_0$. This defines the trivialization $\tau_0$. In the case $A_0=0$, i.e. when all $C_i$ are rational, $\tau_0=b_0$ is a product of the usual cross ratios. \end{saynum} \begin{saynum}\label{saynum:independence_of_the_choice} Our definition seemingly depends on a choice of $(e_k,e_k)$. However, by \cite{AlexeevNakamura96} an SQAV depends only on the residue class of $\tau_0$ modulo the following equivalence relation. $\tau_0$ can be replaced by \begin{displaymath} \tau'_0(x,y)=\tau_0(x,y)\cdot c^{B_1(x,y)} \end{displaymath} for any $c\in k$ and any symmetric positive bilinear form $B_1$ defining the same Delaunay decomposition as $B$ (for the definitions of the Delaunay decompositions, see \cite{AlexeevNakamura96} or \cite{OdaSeshadri79}). The independence of the choice of $(e_k,e_k)$ now follows because on $C_1({\mathbb R})$ the standard Euclidean form and the form $\sum \lambda_iz_i^2$ for any $\lambda_i>0$ define the same Delaunay decomposition. The Delaunay cells are the standard cubes and their faces. Because, as one can easily show (\cite[3.2]{OdaSeshadri79} or \cite[\S18]{Namikawa_ToroidalCompSiegel}) $C_1(\Gamma(C),{\mathbb Z})\cap H_1(\Gamma(C),{\mathbb R})=H_1(\Gamma(C),{\mathbb Z})$, the Delaunay decomposition of $X\otimes\bR=H_1({\mathbb R})$ is the intersection of this standard Delaunay decomposition with $H_1({\mathbb R})$. Therefore, every cell has the following simple description. For each $1\le i\le\dim C_1$ we choose an integer $n_i$ and two numbers $a_i,b_i$ with either $a_i=b_i=n_i$ or $a_i=n_i$ and $b_i=n_i+1$. Then we obtain the cell $\sigma$ in $C_1({\mathbb R})$ defined by the inequalities \begin{displaymath} a_i \le z_i \le b_i \end{displaymath} and the cell in $\sigma\cap H_1({\mathbb R})$ in $H_1({\mathbb R})$ (it may be empty). \end{saynum} \setcounter{example}{3} \begin{example} In this case $H_1=C_1$ and we have the standard Euclidean space ${\mathbb R}^g\supset{\mathbb Z}^g$. The Delaunay cells are standard cubes and their faces. Modulo the translation by ${\mathbb Z}^g$ there are exactly $\binom nk$ such cells of codimension $k$. These numbers are the same as in section \ref{sec:first description}. Further assume that the curve $C$ is rational for simplicity. Then the symmetric bilinear form $b_0$ is defined by $n(n-1)/2$ crossratios $(e_i,e_j)$, $i< j$. \end{example} \setcounter{example}{4} \begin{example} In this case $H_1(\Gamma(C),{\mathbb Z})\subset C_1(\Gamma(C),{\mathbb Z})$ is the hyperplane $\{x_1+x_2+x_3=0\}$. The Delaunay decomposition is the decomposition of ${\mathbb R}^2$ into unilateral triangles. Modulo the translations there are two cells of dimension 2, 3 cells of dimension 1 and 1 cell of dimension 0. These numbers are the same as in section \ref{sec:first description}. This SQAV does not depend on the form $\tau_0$ as all the $3=\dim S^2(H_1)$ choices are killed by the $3=\dim C_1$ choices for $(e_k,e_k)$. \end{example} \setcounter{example}{5} \begin{example} In this case $H_1(\Gamma(C),{\mathbb Z})\subset C_1(\Gamma(C),{\mathbb Z})$ is the hyperplane $\{x_1+...+x_n=0\}$. The lattice is the standard lattice $A_n$. It can be checked that the number of $k$-dimensional cells is given by the same formula as in the section \ref{sec:first description}. \end{example} \begin{saynum} \cite{AlexeevNakamura96} gives a stratification of an SQAV into locally closed subschemes which are homogeneous spaces over a semiabelian variety. A stratum of dimension $n$ corresponds to a Delaunay cell of dimension $n-\dim A_0$. On the other hand, in section \ref{sec:first description} we have given a similar description for $\operatorname{Jac}_{g-1}$ and the semiabelian variety is $\operatorname{Pic}^0 C$. In all the above examples the numbers of strata of each dimension in both descriptions are the same. We now would like to relate the two descriptions explicitly. \end{saynum} \begin{saynum} Consider an arbitrary orientation of the generating subgraph $\Gamma'\subset\Gamma$. By \ref{defnprop} it corresponds to a semistable multidegree $\underline d'$ of the graph $\Gamma(C)$. Now, depending on whether the edge $e_i$ is oriented the ``right'' way (the same that we used defining $H_1$), the ``wrong'' way, or not present at all, choose $a_i=0,b_i=1$, $a_i=-1,b_i=0$ or $a_i=b_i=0$. This gives a Delaunay cell $\sigma$ of $C_1({\mathbb R})$ and the Delaunay cell $\sigma\cap H_1({\mathbb R})$ of $H_1({\mathbb R})$ as in \ref{saynum:independence_of_the_choice}. Moreover, the orientation is stable iff \begin{displaymath} \dim\sigma=\dim\sigma\cap H_1({\mathbb R}) \end{displaymath} \end{saynum} \begin{proof} The above is Oda and Seshadri's description of the stratification of $\operatorname{Jac}_{\phi}$ for the case $\phi=\partial e(J)/2$, in which case the Namikawa-Delaunay decomposition of \cite{OdaSeshadri79} coincides with the Delaunay decomposition we have described above. Therefore, everything follows from \cite{OdaSeshadri79} and the following lemma \end{proof} \begin{lem} $\operatorname{Jac}_{g-1}$ corresponds to the choice $\phi=\partial e(J)/2$ in the notations of \cite{OdaSeshadri79}. \end{lem} \begin{proof} Follows directly from \ref{thm:Jacs_are_the_same}. \end{proof} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9608

alg-geom/9608013

en

https://arxiv.org/abs/alg-geom/9608013

alg-geom/9608013

Valery Alexeev

Valery Alexeev

Log canonical singularities and complete moduli of stable pairs

AMSLaTeX 1.2/LaTeX2e, Postscript file is also available at http://domovoy.math.uga.edu/preprints

1) Assuming log Minimal Model Conjecture, we give a construction of a complete moduli space of stable log pairs of arbitrary dimension generalizing directly the space M_{g,n} of pointed stable curves. Each stable pair has semi log canonical singularities. 2) We prove that a stable quasiabelian pair, defined by author and I.Nakamura as the limit of abelian varieties with theta divisors, has semi log canonical singularities.

alg-geom

\section{Introduction} \begin{saynum} This paper consists of two parts. In the first part, assuming the log Minimal Model Program (which is currently only known to be true in $\dim\le3$), we construct the complete moduli of ``stable pairs'' $(X,B)$ of projective schemes with divisors that generalize the moduli space of $n$-pointed stable curves $M_{g,n}$ to arbitrary dimension. The construction itself is a direct generalization of that of \cite{Alexeev_Mgn} where it was given in the case of surfaces, and is based in part on ideas from \cite{KollarShepherdBarron88,Kollar_ProjCompleteModuli,Viehweg95}. \end{saynum} \begin{saynum} In the second part of the paper we study the singularities of stable quasiabelian varieties and stable quasiabelian pairs $(X,B)$ that appear in \cite{AlexeevNakamura96} as limits of abelian varieties. We show that the singularities are semi log canonical. This implies, via Koll\'ar's Ampleness Lemma, that over $\Bbb C$ if there exists a compactification of the moduli space $A_g$ of principally polarized abelian varieties by stable quasiabelian pairs, then it is in fact projective. We give more examples of situations where log canonical singularities appear naturally in connection with complete moduli problems. One of them is the minimal and toroidal compactifications of quotients $D/\Gamma$ of bounded symmetric domains by arithmetic groups. We point out the fact, which could be obvious to specialists had they known the definitions, that they all have log canonical singularities and that the minimal (=Baily-Borel) compactification is the log canonical model of any toroidal compactification when the ``boundary'' divisor $B$ is correctly defined. \end{saynum} \begin{ack} I would like to thank Professors V. Batyrev, I. Nakamura, Y. Kawamata, J. Koll\'ar, S. Mori and R. Varley for very helpful conversations. \end{ack} \section{Definitions for singularities} \label{sec:Definitions for singularities} \begin{defn} Let $X$ be a normal variety (not necessarily irreducible) defined over an algebraically closed field $k$ of any characteristic, and let $B_1\dots B_m$ be distinct reduced divisors on $X$. Denote $\sum B_j$ by $B$. Let $i:U\hookrightarrow X$ be the inclusion of the nonsingular part and denote \begin{displaymath} {\mathcal O}(N(K+B))= i_*{\mathcal O}(N(K_U+B\vert_U)), \end{displaymath} where ${\mathcal O}(K_U)$ is the canonical sheaf, the top exterior power of $\Omega^1_U$, and $N$ is an integer. One says that the pair $(X,B)$ has {\em log canonical singularities\/} if \begin{enumerate} \item ${\mathcal O}(N(K+B))$ is invertible for some $N>0$ (one then says that $K+B$ is ${\mathbb Q}$-Cartier). \item for any birational morphism from a normal scheme $f:Y\to X$ one has \begin{displaymath} f_*{\mathcal O}_Y\big( N(K_Y+f^{-1}B+\sum E_i) \big) = {\mathcal O}_X\big(N(K_X+B)\big), \end{displaymath} where $E_i$ are exceptional divisors of $f$. \end{enumerate} \end{defn} \begin{rem} The above definition can be formulated also for the case of a divisor $B=\sum b_jB_j$ with rational coefficients $b_j$ by requiring $N$ to be divisible enough. \end{rem} \begin{saynum} An equivalent way would be to use {\em log codiscrepancies\/} -- the coefficients $a_i$ appearing in the following natural formula: \begin{displaymath} f^*(K_X+B)=K_Y+f^{-1}B+\sum a_iE_i \end{displaymath} The log codiscrepancies depend only on the divisors $E_i$ themselves, i.e. the corresponding discrete valuations of the function field, and not on the model $Y$ chosen. Indeed, every two models $Y_1$ and $Y_2$ are comparable since they are both dominated by a third normal variety $Y_3$ -- take for example the component of the normalization of $Y_1\underset{Y}{\times}Y_2$ which dominates $Y$. \end{saynum} \begin{defn} The singularities are log canonical if all log codiscrepancies are $\le1$. They are {\em log terminal\/} if $a_i<1$, {\em klt\/} if $a_i<1$ and $b_j<1$. And they are {\em canonical (resp. terminal)\/} if $B$ is empty and $a_i\le0$ (resp. $a_i<0$). \end{defn} \begin{rem} In the above definition one usually assumes $Y$ to be non-singular, and then one needs the embedded resolution of singularities and hence characteristic $0$. This does not appear to be necessary. Still, without the resolution of singularities the situation becomes somewhat cumbersome. For example, it is not absolutely obvious that the next definition is equivalent to, or even implies, the previous one (this {\em is} obvious with resolution of singularities). \end{rem} \begin{defn} Let $(X,B)$ be as above. We say that this pair has {\em pre log canonical singularities\/} if there exists a proper birational morphism from a nonsingular variety $f:Y\to X$ such that \begin{enumerate} \item ${\mathcal O}(N(K+B))$ is invertible for some $N>0$. \item the exceptional set of $f$ is a union of divisors $E_i$. \item $\cup f^{-1}B_j\cup E_i$ has normal crossings. \item $f_*{\mathcal O}_Y(N(K_Y+f^{-1}B+\sum E_i)) = {\mathcal O}_X(N(K_X+B))$. \end{enumerate} \end{defn} \begin{say} Another important class of singularities is the following. \end{say} \begin{defn}\label{defn:semi_log_canonical} Let $X$ be a reduced variety (not necessarily irreducible) defined over an algebraically closed field $k$ of any characteristic, and let $B_1\dots B_m$ be distinct reduced divisors on $X$, denote $\sum B_j$ by $B$. In addition, assume that $X$ is quasi-projective over $k$. Let $i:U\hookrightarrow X$ be the union of the open locus of Gorenstein points of $X$ not contained in $B$ and the nonsingular locus of $X$. Denote \begin{displaymath} {\mathcal O}(N(K+B))= i_*{\mathcal O}(N(K_U+B\vert_U)), \end{displaymath} where $N$ is an integer, and ${\mathcal O}(K_U)$ is the restriction of the dualizing sheaf $\omega_{\overline{U}}$ of a projective closure of $U$ (\cite[III.7]{Hartshorne77}). One says that the pair $(X,B)$ has {\em semi log canonical singularities\/} if \begin{enumerate} \item $X$ satisfies the Serre's condition $S_2$. \item $X$ is Gorenstein in codimension $1$. \item none of the irreducible components of $B_j$ is contained in the singular locus of $X$. \item the closed subscheme $\operatorname{cond}(\nu)$ of $X^{\nu}$ corresponding to the conductor of normalization $\nu:X^{\nu}\to X$ is a union of reduced divisors. \item ${\mathcal O}(N(K+B))$ is invertible for some $N>0$. \item the pair $(X^{\nu},\nu^{-1}B+\operatorname{cond}(\nu))$ has log canonical singularities. \end{enumerate} \end{defn} \begin{say} In the same way as above, one can define pre semi log canonical singularities. \end{say} \begin{rem} We note a certain lack of symmetry in the definitions of ${\mathcal O}(N(K+B))$ for log and semi log canonical cases. However, they coincide if $X$ is both normal and quasi-projective. \end{rem} \begin{defn} Under the assumptions above, we will say that $K+B$ is ample if the sheaf ${\mathcal O}_X(N(K+B))$, equivalently ${\mathcal O}_{X^{\nu}}(N(K+B+\operatorname{cond}(\nu)))$, is an ample invertible sheaf for some integer $N>0$. In this case the pair $(X,B)$ is called the {\em log canonical model}. \end{defn} \begin{say} Let us now try to see what is the most general situation where the previous definitions still work. The main thing to understand is the canonical sheaf. The rest transfers over in a pretty straightforward way. \end{say} \smallskip \begin{say} Let us fix a regular Noetherian scheme $S$ (for example spectrum of $\Bbb Z$ or a DVR) and consider a reduced scheme $X$ flat and of finite type over $S$. Let us assume that $\pi:X\to S$ is smooth in codimension 1. Denoting by $i:U\to X$ the embedding of this smooth locus, we can set \begin{displaymath} {\mathcal O}_X(K_{X/S})=i_* {\mathcal O}_X(K_{U/S}), \end{displaymath} where ${\mathcal O}_X(K_{U/S})$ is the top exterior power of $\Omega^1_{U/S}$. We can now define (pre) log canonical singularities of a pair $(X,B=\sum B_j)$, where $B_j\subset X$ are closed codimension 1 subschemes of $X$, by copying the definitions from section \ref{sec:Moduli of stable pairs in general}. In particular, for log canonical singularities we require $X$ to be normal. For (pre) semi log canonical we need to assume that $X/S$ is quasi-projective and that the normalization $X^{\nu}/S$ is smooth in codimension 1. As one can see, in these definitions we use the regular scheme $S$ only as ``the beginning of coordinates'', something to start measuring from. \end{say} \medskip \begin{say} Let us push the limits even little further. Clearly, the definition of (pre) log canonical singularities is stable under \'etale maps. Therefore, they transfer directly to algebraic spaces and algebraic stacks. If $R_X\overset{\to}{\to} U_X$ is an equivalence relation or a groupoid defining $X$, and $R_{B_j},U_{B_j}$ are the closed subschemes corresponding to $B_j$, then we say that the pair $(X,B)$ has (pre) log canonical singularities if the same holds for $(U_X,U_B)$. \end{say} \section{Moduli of stable pairs in general} \label{sec:Moduli of stable pairs in general} \begin{say} The purpose of this section is to describe a construction of complete and projective moduli spaces for stable $n$-dimensional pairs which generalize the usual moduli of stable $n$-pointed curves. This will be done assuming a series of conjectures the main of which is the log Minimal Model Program in dimension $n+1$. These conjectures are theorems only when $n+1=3$, so only in the case of surfaces the results are not hypothetical, and this case was considered in detail in \cite{Alexeev_Mgn}. Where possible, we work in general context, over a fixed base scheme. The bulk of this material, however, applies only to the case of an algebraically closed field of characteristic $0$ because of the Minimal Model Program. We would like to point out that the general framework of what is described here has already been essentially understood in \cite{KollarShepherdBarron88}, \cite{Kollar_ProjCompleteModuli} and \cite{Alexeev_Mgn}. Many important ideas also come from \cite{Viehweg95}. \end{say} \medskip \begin{say} We first remind what a stable $n$-pointed curve is. \end{say} \begin{defn} A stable $n$-pointed curve over an algebraically closed field is a collection $(C;P_1\dots P_m)$, where \begin{enumerate} \setcounter{enumi}{-1} \renewcommand{\theenumi}{(\arabic{enumi})} \item $C$ is a connected projective curve and $P_1\dots P_m$ are points on $C$. \item (condition on singularities) $C$ is reduced and has nodes only, and $P_1\dots P_m$ all lie in the nonsingular part. \item (numerical condition) for every smooth rational curve $E\subset C$, $E$ has at least 3 special points: one of $P_i$ or the nodes; and for every smooth elliptic curve or a rational curve with one node $E\subset C$, $E$ has at least 1 special point. \end{enumerate} A stable $n$-pointed curve over a scheme $S$ is a flat projective morphism $\pi:(C;P_1\dots P_m)\to S$, with $P_i\subset C$ closed subschemes and each $P_i\to S$ also flat, whose every geometric fiber is a stable $n$-pointed curve over a field $k=\bar k$. \end{defn} \begin{say} The moduli stack of stable $n$-pointed curves is proper, and it is coarsely represented by a projective scheme $M_{g,n}$, see \cite{Knudsen83}. \end{say} \medskip \begin{say} {\bf Question.} What is the analog of this in higher dimensions? \medskip One definitely has to consider a collection consisting of a connected projective scheme $X$ plus $m$ closed subschemes. We have two basic choices: they could be points or divisors. Here, we choose divisors: $B_1\dots B_m$. The numerical condition (2) above can be reformulated by saying ``$K_C+\sum P_i$ is ample''. We can now directly transfer this to dimension $n$ if we understand what $K+B=K_X+\sum B_j$ is. Finally, the condition on the singularities. This is the trickiest of the three. The answer comes from the log Minimal Model Program theory: the singularities of $(X,B)$ have to be semi log canonical. \end{say} \medskip \begin{say} We are now ready to introduce our main object. \end{say} \begin{defn} A stable pair over an algebraically closed field is a collection $(X;B_1\dots B_m)$, where \begin{enumerate}\setcounter{enumi}{-1} \renewcommand{\theenumi}{(\arabic{enumi})} \setcounter{enumi}{-1} \item $X$ is a connected projective not necessarily irreducible variety and $B_1\dots B_m$ are reduced divisors on $X$. \item (condition on singularities) the pair $(X,B)$ has semi log canonical singularities. \item (numerical condition) $K+B$ is ample. \end{enumerate} A stable pair over a scheme $S$ of level $N$ is a flat projective morphism $\pi:(X;B_1\dots B_m;{\mathcal L})\to S$, with $B_i\subset X$ closed subschemes, each $B_i\to S$ also flat and ${\mathcal L}$ an invertible sheaf on $X$, whose every geometric fiber is a stable pair over a field $k=\bar k$ and such that the restriction of ${\mathcal L}$ on each geometric fiber coincides with ${\mathcal O}(N(K+B))$. We say that two pairs $(X_1,B_1;{\mathcal L}_1)$ and $(X_2,B_2;{\mathcal L}_2)$ are isomorphic if there exists an isomorphism of $(X_1,B_1)$ and $(X_2,B_2)$ over $S$ that induces a fiber-wise isomorphism of ${\mathcal L}_1$ and ${\mathcal L}_2$. \end{defn} \begin{conj}[Boundedness Conjecture] For every positive rational number $C$ there exist \begin{enumerate} \item a positive integer $N>0$ with the property that for every stable $n$-dimensional stable pair $(X,B)$ with $(K+B)^n=C$ the sheaf ${\mathcal O}(N(K+B))$ is invertible. \item a scheme $S$ of finite type over the base scheme and a flat projective family $(X;B_1\dots B_m)$ whose geometric fibers include all stable $n$-dimensional pairs of level $N$ with $(K+B)^n=C$. \end{enumerate} \end{conj} \begin{say} This has been shown to be true only in dimension 2 (\cite{Alexeev_Boundedness}) and trivially in dimension 1. \end{say} \begin{defn} We now fix a rational number $C$ and an integer $N$ as above and define the functor \begin{displaymath} {\mathcal M}^N_C(S)=\left\{ \begin{aligned} \text{stable $n$-dimensional pairs over }S \\ \text{of level $N$ with } (K+B)^n=C \end{aligned} \right\}/\simeq \end{displaymath} and the moduli stack by the same formula but without dividing by isomorphisms, and by giving ${\mathcal M}^N_C(S)$ the groupoid structure in a natural way. \end{defn} \begin{say} There are other possible definitions for the moduli functor, see f.e \cite{Alexeev_Mgn}. \end{say} \medskip \begin{say} At this point we can choose a certain scheme in a product of Hilbert schemes with the universal family that contains all interesting for us stable pairs. The next step is to separate the stable pairs from wrong fibers, and for this we need to know that our functor is locally closed in the following sense. For every flat projective family $(X,B)\to S$ there exist locally closed subschemes $S_l\subset S$ with the following universal property: \begin{itemize} \item A morphism of schemes $T\to S$ factors through $\coprod S_l$ iff $(X,{\mathcal L})\underset{S}{\times}T \to T$ belongs to ${\mathcal M}^N_C(T)$. \end{itemize} \medskip For our functor this property follows from the following conjecture of Shokurov (\cite{Shokurov91}). \end{say} \begin{conj}[Inversion of log Adjunction] \label{conj:inversion of log adjunction} Let $(X,B)\to S$ be a flat $1$-dimensional family. Assume that there exists an invertible sheaf ${\mathcal L}$ on $X$ whose restriction on each fiber coincides with ${\mathcal O}_X(N(K_X+B))$ as in definition \ref{defn:semi_log_canonical}. Then the $S_2$-fication of the pair $(X_0,B_0)$ has semi log canonical singularities iff the pair $(X,B+X_0)$ has semi log canonical singularities in a neighborhood of $X_0$. \end{conj} \begin{rem} $X_0$ is $S_2$ iff $X$ is $S_3$. In many cases the varieties are Cohen-Macaulay, so taking the $S_2$-fication is unnecessary. \end{rem} \begin{saynum}\label{saynum:inversion_of_log_adjunction} One direction of this conjecture (going from the family to the central fiber) is easy and the proof for the case when $X$ is irreducible can be found in \cite[ch.17]{FAAT}. The general case can be easily deduced from that by taking the normalization. The same reference contains the proof of the opposite direction (the inversion) assuming the log Minimal Model Program in dimension $n+1$. It also contains several special cases where it can be proved without log MMP, using the Kawamata-Viehweg vanishing theorem only. \end{saynum} \medskip \begin{saynum} The inversion of adjunction conjecture implies that if the sheaves ${\mathcal O}_X(N(K_X+B))$ are locally free and can be put together in a flat family then the semi log canonical property is stable under generizations. Indeed, it follows from the definition that for a general fiber $X_t$ the pair $(X,B+X_0+X_t)$ is still semi log canonical. Then $(X_t,B_t)$ is semi log canonical by the easy direction of log adjunction. The question when exactly the sheaves ${\mathcal O}_X(N(K_X+B))$ can be put together in a flat family is rather delicate. It follows from a technical result of Koll\'ar, see f.e. \cite{Alexeev_Mgn}. \end{saynum} \medskip \begin{say} At this point we can pick a sub-family in our universal family that contains exactly our stable pairs. What remains is to take a quotient by the pre equivalence relation (or a groupoid) which is given by the action of the projective linear group $PGL$. This groupoid is easily seen to be flat. It also has a quasifinite stabilizer because stable pairs have finite automorphism groups by \cite{Iitaka82}. The next separateness property implies that the stabilizer is in fact finite. In this situation the quotient exists as a separated algebraic space. Nowadays, there are several convenient references for this statement, for example \cite{Kollar_QuotSpaces} and \cite{MoriKeel95}. As a result, one obtains a coarse moduli space $M_C^N$ as a separated algebraic space of finite type, and we are already working over an algebraically closed field $k$ of $\chr0$ since we used the log MMP. \end{say} \begin{thm} Let us assume the inversion of log adjunction conjecture. Let $(X',B')\to S\setminus 0$ be a $1$-dimensional family without the central fiber which is a stable pair over $S\setminus 0$. Then it can be completed to a stable pair over $S$ in no more than one way up to an isomorphism. \end{thm} \begin{proof} Let $(X,B)\to S$ be one such completion. By the inversion of log adjunction we know that $(X,B+X_0)$ is semi log canonical, possibly after shrinking $S$. Assume first that the scheme $X$ is irreducible, so that the singularities are in fact canonical. Then for any proper birational morphism from a normal variety $f:Y\to X$ and for every positive integer $d$ we have by definition \begin{displaymath} f_*{\mathcal O}_Y\big( dN(K_Y+f^{-1}B+f^{-1}X_0+\sum E_i) \big) = {\mathcal O}_X\big(dN(K_X+B+X_0)\big) \end{displaymath} Here the following three circumstances are important: \begin{enumerate}\renewcommand{\theenumi}{(\arabic{enumi})} \item $f^{-1}X_0+\sum E_i$ is in fact the central fiber of $Y$ with the reduced structure. \item the divisor $X_0$ is relatively trivial. \item the divisor $K+B$ is relatively ample, so that the family $(X,B)\to S$ can be computed as a $\operatorname{Proj}$ of a big graded ring of relative sections of ${\mathcal O}(dN(K+B))$. \end{enumerate} As a result of this, we obtain \begin{displaymath} X=\operatorname{Proj}_{d\ge0}\oplus\pi_*{\mathcal O}_Y(dN(K_Y+f^{-1}B+Y_{0,red})), \end{displaymath} where $\pi$ denotes the morphism $Y\to S$. But this means that the family $(X,B)$ can be uniquely reconstructed from $(Y,f^{-1}B)$. Now, given two families $(X_1,B_1)$ and $(X_2,B_2)$. we can find a normal variety $Y$ which dominates both of them. By uniqueness, we have a canonical isomorphism $(X_1,B_1)\to(X_2,B_2)$. This completes the case when $X$ is irreducible. In general, the above argument shows the uniqueness of $(X^{\nu},B+\operatorname{cond}(\nu))$, and $(X,B)$ is uniquely recoverable from that. \end{proof} \begin{say} Next, we would like to prove that this algebraic space is in fact proper. For this, we have to check the corresponding property for our functor ${\mathcal M}_C^N$. \end{say} \medskip \begin{saynum} The pair $(X,B)$ above is the log canonical model of $(Y,f^{-1}B+Y_{0,red})$. So, the argument actually followed from the uniqueness of the log canonical model. Vice versa, assume that we have the log Minimal Model available. Start with arbitrary compactification $(X,B)\to S$ of a stable pair over $S\setminus0$. Take the normalization. For each irreducible component apply the Semistable Reduction Theorem (of course, $\operatorname{char} 0$ is necessary for that) to obtain, after a finite ramified base change and resolution of singularities, a family with the reduced central fiber such that the irreducible components of the central fiber, $\operatorname{cond}(\nu)$, exceptional divisors of resolution and $B_j$ intersect transversally. Note that it is possible to choose the same base change that works for every irreducible component. And then just find the log canonical model applying log MMP. In fact, we don't need all the results of log MMP but only the following conjecture and only in the $1$-dimensional semistable case. After that, glue the irreducible components back together. That will be the desired family over a finite ramified cover of $S$. This proves that our functor and the moduli space are proper. \end{saynum} \begin{conj}[Existence of log Canonical Model] Let $\pi:(Y,B)\to S$ be a projective morphism and assume that \begin{enumerate} \item the singularities of $(Y,B)$ are log canonical. \item restriction of ${\mathcal O}_Y(N(K+B)$ on each generic fiber is big (contains an ample divisor). \end{enumerate} Then the ring of ${\mathcal O}_S$-modules \begin{displaymath} \oplus_{d\ge0}\pi_*{\mathcal O}_Y(dN(K_Y+B)) \end{displaymath} is finitely generated. \end{conj} \begin{saynum}\label{saynum:ampleness_lemma} The last step is to show that the moduli space $M_C^N$ is projective. This follows by the Koll\'ar's Ampleness Lemma, see \cite{Kollar_ProjCompleteModuli}. The input data for this statement is \begin{enumerate} \item $M$ has to be a proper algebraic space of finite type over an algebraically closed field field $k$ of characteristic $0$. \item On a finite cover of $M$ there has to exist a projective polarized family $(X,B)$ whose every fiber has semi log canonical singularities (Koll\'ar considered the case $B=\emptyset$ but the generalization to the case of reduced $B$ is immediate). For example, this happens when $M$ is a coarse moduli space for some functor of polarized varieties, as in our case. \item The polarization has to be functorial, i.e. compatible with base changes. In our case, the polarization ${\mathcal O}(N(K_{X/S}+B))$ has this property. \end{enumerate} \end{saynum} \section{Examples of log canonical singularities} \label{sec:Examples of log canonical singularities} \begin{say} The following examples should be in any introductory article on log canonical singularities but surprisingly they aren't. \end{say} \begin{lem}\label{lem:torus_embeddings_are_slc} Let $X=T_Nemb(\Delta)$ be a torus embedding over a field $k$ defined by a rational partial polyhedral cone decomposition and $B=\sum B_j$ be the sum of divisors corresponding to the $1$-dimensional faces of the fan $\Delta$. Then the pair $(X,B)$ has pre log canonical singularities, and $B$ (i.e. the pair $(B,0)$) has pre semi log canonical singularities. \end{lem} \begin{proof} The basic formula of the theory of torus embeddings for the canonical sheaf is \begin{displaymath} \omega_X(B)\simeq {\mathcal O}_X \end{displaymath} Every torus embedding has a toric resolution of singularities $f:Y\to X$ such that $f^{-1}B\cup E_i$ has normal crossings, where $E_i$ are the exceptional divisors of $f$. Here $f^{-1}B\cup E_i$ is the union of divisors corresponding to $1$-dimensional faces of the fan of $Y$. Therefore, \begin{displaymath} f^*{\mathcal O}(K_X+B)\simeq f^*{\mathcal O}_X \simeq {\mathcal O}_Y \simeq {\mathcal O}(K_Y+f^{-1}B+\sum E_i) \end{displaymath} and the singularities of the pair $(X,B)$ are pre log canonical. The normalization $B^{\nu}$ of $B$ is a disjoint union of torus embeddings, and $\operatorname{cond}(\nu)$ is again the union of divisors corresponding to the $1$-dimensional faces. This shows that $B$ has pre semi log canonical singularities. \end{proof} \begin{say} Therefore, every time when toric geometry is used, log canonical singularities show up. One of such situations is the following theorem of Mumford \cite[3.4,4.2]{Mumford_HirzebruchProportionality}. \end{say} \begin{thm} Let $\Gamma$ be a neat arithmetic group acting on a bounded symmetric complex domain $D$. Let $(D/\Gamma)^*$ be the Baily-Borel compactification of $D/\Gamma$ and $\overline{D/\Gamma}$ be any of the toroidal compactifications. Denote the boundaries of these compactifications by $\Delta^*$, $\overline{\Delta}$ respectively. Then \begin{align*} (D/\Gamma)^* &= \operatorname{Proj}_{d\ge0} H^0\big( d(K_{(D/\Gamma)^*}+\Delta^*) \big) \\ &= \operatorname{Proj}_{d\ge0} H^0\big( d(K_{\overline{D/\Gamma}}+\overline{\Delta}) \big) \end{align*} \end{thm} \begin{cor} $((D/\Gamma)^*,\Delta^*)$ is the log canonical model of $(\overline{D/\Gamma},\overline{\Delta})$, and they both have log canonical singularities. \end{cor} \begin{say} The above formula in fact is one of the {\em definitions\/} of a log canonical model. We remind that a group $\Gamma$ is called neat if eigenvalues of each element of $\Gamma$ generate a torsion-free subgroup of ${\mathbb C}^*$. The quotient space $D/\Gamma$ by a neat group is nonsingular. What about the general case? It is easy, all one has to do is use the Hurwitz formula (cf. \cite[3.16]{Kollar_SingsPairs}). \medskip Every arithmetic group contains a neat subgroup $\Gamma_0\subset\Gamma$ of finite index. Let $D_j$ be the irreducible ramification divisors of $\overline{D/\Gamma_0}\to\overline{D/\Gamma}$ on $\overline{D/\Gamma}$ with ramification indices $n_j$. Then we immediately obtain the following \end{say} \begin{thm} \begin{align*} (D/\Gamma)^* &= \operatorname{Proj}_{d\ge0} H^0\big( d(K_{(D/\Gamma)^*}+\Delta^* +\sum(1-1/n_j)D_j) \big) \\ &= \operatorname{Proj}_{d\ge0} H^0\big( d(K_{\overline{D/\Gamma}}+\overline{\Delta} +\sum(1-1/n_j)D^*_j) \big) \end{align*} \end{thm} \begin{cor} $((D/\Gamma)^*,\Delta^*+\sum(1-1/n_j)D^*_j)$ is the log canonical model of $(\overline{D/\Gamma},\overline{\Delta}+\sum(1-1/n_j)D_j)$, and they both have log canonical singularities. \end{cor} \begin{exmp} The compactification $\overline{A}_1={\mathbb P}^1$ of the moduli space $A_1$ of elliptic curves does not have log general type: \begin{displaymath} \deg(K_{{\mathbb P}^1}+P_{\infty})= -2+1<0, \end{displaymath} so it is not a log canonical model of anything. However, the sum becomes positive when one adds the terms $(1-1/n_i)P_i$ corresponding to the elliptic curves with automorphisms. This answers the footnote of Mumford appearing on the same page as theorem \cite[4.2]{Mumford_HirzebruchProportionality}. \end{exmp} \begin{say} Another situation is the stable quasiabelian varieties and pairs appearing as the limits of abelian varieties. We refer the reader to \cite{AlexeevNakamura96,Alexeev_CMAV} for their definition. The very construction for them is toric, so not surprisingly we have \end{say} \begin{lem}\label{lem:sqavs_are_slc} Let $P_0$ is a SQAV. Then $P_0$ has pre semi log canonical singularities. \end{lem} \begin{proof} By construction (\cite{AlexeevNakamura96}) there exists an \'etale map $\widetilde P_0\to P_0$, and $\widetilde P_0$ is a union of divisors in a torus embedding $\widetilde P$ corresponding to the $1$-dimensional faces of the fan. The statement now follows from \ref{lem:torus_embeddings_are_slc}. \end{proof} \begin{say} $P_0$ in \cite{AlexeevNakamura96} appears as a central fiber of a one-dimensional degenerating normal family $P/S$ of abelian varieties. Over ${\mathbb C}$, $P$ is a quotient of a torus embedding (which is locally of finite type) by a group ${\mathbb Z}^g$ acting freely in the classic topology. \end{say} \begin{lem} The family $P$ itself has log canonical singularities. \end{lem} \begin{proof} Indeed, the general fiber of $P/S$ is smooth, so all the ``bad'' discrepancies lie over the central fiber. By \ref{lem:torus_embeddings_are_slc} the pair $(P,P_0)$ has log canonical singularities, i.e. the corresponding discrepancies are $a_i\le1$. But the discrepancies of $(P,0)$ have to be less than $a_i$ by at least the multiplicities of $f^*P_0$ along the exceptional divisors. Since $P_0$ is Cartier, these multiplicities are at $\ge1$ and the discrepancies of $P$ are $\le0$. \end{proof} \begin{say} In the principally polarized case a SQAV by \cite{AlexeevNakamura96} comes with a natural theta divisor $\Theta$. \end{say} \begin{rem} An easy generalization of the last lemma is that a pair $(P_0,\varepsilon\Theta_0)$ has semi log canonical singularities for $\varepsilon\ll1$ in $\chr0$. For this one simply has to notice that $\Theta$ does not entirely contain any of the strata of $P_0$: \cite[3.28]{AlexeevNakamura96}. A more interesting is the following. \end{rem} \begin{thm} \label{thm:sqaps_are_slc} A principally polarized stable quasiabelian pair $(P_0,\Theta_0)$ over ${\mathbb C}$ has semi log canonical singularities. \end{thm} \begin{proof} For the abelian varieties this result is a theorem of Koll\'ar \cite{Kollar_ShafMapsnPlurigenera}. The present proof is the adaptation of the proof of that theorem to our situation. By \cite{AlexeevNakamura96} every stable quasiabelian pair appears as the central fiber in a $1$-dimensional family $\pi:(P,\Theta)\to S=D_{\varepsilon}$ with abelian general fiber over a small disk. We denote by $I$ the ideal defining $0\in D_{\varepsilon}$. If we prove that the pair $(P,\Theta+P_0)$ has log canonical singularities then we would be done by the easy direction of the ``inversion of log adjunction theorem'' (see \cite[ch.17]{FAAT} or \ref{conj:inversion of log adjunction}). The locus $Z$ of non-log canonical singularities of $(P,\Theta+P_0)$ coincides with the locus of non-log terminal singularities of the pair $(P,(1-\varepsilon)(\Theta+P_0))$ for $0<\varepsilon\ll1$. We will apply the Kawamata-Viehweg vanishing theorem in the following Nadel's form (see f.e. \cite[2.16]{Kollar_SingsPairs}): \begin{thm} Let $X$ be a normal and proper variety and $N$ a line bundle on $X$. Assume that $N\equiv K_X+\Delta+M$, where $M$ is nef and big ${\mathbb Q}$-Cartier divisor and $\Delta$ effective ${\mathbb Q}$-Cartier divisor with coefficients $<1$. Then there is an ideal sheaf $J\subset{\mathcal O}_X$ such that \begin{displaymath} \operatorname{Supp}({\mathcal O}_X/J)=\{ x\in X \,|\, (X,\Delta) \text{ is not log terminal at } x \} \end{displaymath} \end{thm} We will apply this theorem in the relative situation to the proper morphism $\pi:P\to S$. We have \begin{displaymath} K_P+\Theta+P_0=K_P+(1-\varepsilon)(\Theta+P_0)+ \varepsilon(\Theta+P_0), \end{displaymath} $K_P,P_0$ are relatively trivial and $\Theta$ is relatively ample. Therefore, by the above there exists an ideal $J\subset{\mathcal O}_P$ supporting the locus $Z$ where the pair $(P,\Theta+P_0)$ is not log canonical, and $R^1\pi_*J(\Theta+P_0)=0$. Therefore, the following map is surjective \begin{displaymath} \pi_*{\mathcal O}_P(K_P+\Theta+P_0) \underset{\phi}{\to} \pi_*{\mathcal O}_Z(K_P+\Theta+P_0) \end{displaymath} Since in the nonsingular case the statement holds by Kollar's theorem, after shrinking $S$ the support of $Z_{red}$ will be contained in the central fiber. Therefore $Z$ is a closed complex-analytic subspace of $P_n=P\underset{R}{\times}R/I^n$ for some $n\ge0$, where $R$ is the ring of germs of analytic functions at $0$. Moreover, $Z$ is a closed subspace of the theta divisor $\Theta$. Indeed, as in the proof of lemma \ref{lem:sqavs_are_slc}, theorem \ref{lem:torus_embeddings_are_slc} the pair $(P,P_0)$ is log canonical, therefore the pair $(P,(1-\varepsilon)P_0)$ is log terminal. By theorem 4.6 of \cite{AlexeevNakamura96} we have $H^0({\mathcal O}_{P_0}(\Theta_0))=1$ and $H^i({\mathcal O}_{P_0}(\Theta_0))=0$ for $i>0$. This implies $R^i\pi_*{\mathcal O}_P(\Theta)=0$ for $i>0$ and $\pi_*{\mathcal O}_P(\Theta)={\mathcal O}_S$. We have ${\mathcal O}(K_P)\simeq{\mathcal O}(P_0)\simeq{\mathcal O}$, so ${\mathcal O}(K_P+\Theta+P_0)\simeq {\mathcal O}(\Theta)$. Since $Z$ is a closed subspace of $\Theta$, $\phi$ has to be the zero map. On the other hand, $\pi_*{\mathcal O}_Z(\Theta)\ne0$ for any proper subspace $Z\subset P_n$. In the nonsingular case this is concluded by the semi-continuity argument and the fact that the abelian variety acts transitively by translations. In our situation, there is the action of a semiabelian group $G/S$, and although it is not transitive, still the intersection of translations $g(\Theta)$ by sections $g\in G$ is empty: \cite[3.28]{AlexeevNakamura96}, and this implies $\pi_*{\mathcal O}_Z(\Theta)\ne0$. \end{proof} \begin{cor} Over ${\mathbb C}$, if the compactification of the moduli space $A_g$ by the pairs $(P_0,\Theta_0)$ exists, it is projective. \end{cor} \begin{proof} This follows by applying the Ampleness Lemma of Koll\'ar, cf. \ref{saynum:ampleness_lemma}. \end{proof} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9608

alg-geom/9608022

en

https://arxiv.org/abs/alg-geom/9608022

alg-geom/9608022

Mark De Cataldo

Mark Andrea A. de Cataldo (Washington U. in St. Louis)

Some adjunction-theoretic properties of codimension two nonsingular subvarieties of quadrics

Latex; 16 pages

We make precise the structure of the first two reduction morphisms associated with codimension two nonsingular subvarieties of quadrics $\Q{n}$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed to be not of log-general-type. Keywords: Adjunction Theory, classification, codimension two, conic bundles, low codimension, non log-general-type, quadric, reduction, special variety

alg-geom

\section{INTRODUCTION} \label{intr} Because of the Barth-Larsen Theorem and the Double Point Formula, low codimensional embeddings in projective space are special in many respects. Inspired by the study of the special adjunction-theoretic properties of threefolds in $\pn{5}$ contained in \ci{be-sc-soams}, in this note we study the similar properties for codimension two nonsingular subvarieties of quadrics $\Q{n}$, $n\geq 5$. As it turns out, by analogy with the results of \ci{be-sc-soams}, the reduction morphisms associated with these varieties are almost always isomorphisms; see Theorem \ref{secondreduction}. We give a coarse classification Theorem for the varieties for which the second reduction morphism is not defined, the so-called varieties {\it not of log-general-type}; see Theorem \ref{k+2lnotspanned}, Theorem \ref{k+2lspannednotbig} and Theorem \ref{k+nu-2lnotnefbig}. To prove the latter one we need to analyze the case of Del Pezzo fibrations and, in the same way as in the paper \ci{boss2}, the case of conic bundles on $\Q{5}$; see sections \ref{fanofibrations} and \ref{quadricfibration}, respectively. \smallskip \noindent {\bf Notation and conventions.} Our basic reference is [Ha]. We work over any algebraically closed field of characteristic zero. A quadric $\Q{n}$, here, is a nonsingular hypersurface of degree two in the projective space $\pn{n+1}$. Little or no distinction is made between line bundles, associated sheaves of sections and Cartier divisors. \noindent By {\em scroll} we mean a variety $X \subseteq \pn{N}$, for which $(X,\odixl{\pn{N}}{1}_{|X}) \simeq ({\Bbb P}_Y({\cal E}), \xi_{\cal E})$, where $\cal E$ is a vector bundle on a nonsingular variety $Y$. An adjunction-theoretic scroll (see \ci{be-fa-soadjtp}) is not, in general, a scroll; we denote them by {\em a.t. scrolls}. \smallskip \noindent {\bf Acknowledgments.} This paper is an expanded and completed version of parts of our dissertation. It is a pleasure to thank our Ph.D. advisor A.J. Sommese, who has suggested to us that we study threefolds on $\Q{5}$. We thank the C.N.R. of the Italian Government and The University of Notre Dame for partial support. \section{PRELIMINARY MATERIAL} Let $\iota: X \hookrightarrow \Q{n}$ be the embedding of a degree $d$ nonsingular subvariety of codimension two of $\Q{n}$; let $L$ denote the line bundle $\iota ^*\odixl{\Q{n}}{1}$, $g$ the genus of the curve $C$ obtained by intersecting $(n-3)$ general elements of $|L|$. Denote by $x_i$ the Chern classes of the tangent bundle of $X$ and by $n_i$ the ones of the normal bundle $\nb{X}{\Q{n}}$; by adjunction $K_X=-nL +n_1$ and by the self-intersection formula $n_2=(1/2)dL^2$. The following formul\ae\, which hold in the Chow ring of $X$ for $n\geq 5$, are obtained using the Double Point Formul\ae \ (see \ci{fu}) for $\iota$. \begin{equation} n_2=\frac{1}{2}(n^2 -n +2)L^2 -n x_1\cdot L + x_1^2 - x_2; \label{deg2dpf} \end{equation} \begin{equation} \frac{1}{6}(n^3 -3n^2+8n-12)L^3 +\frac{1}{2}(-n^2 +n -2)x_1 L^2+ n(x_1^2-x_2)L +2x_1x_2-x_1^3 -x_3=0. \label{deg3dpf} \end{equation} \noindent The following formula for surfaces $X$ on $\Q{4}$ with balanced cohomology class can be found in \ci{a-s}. \begin{equation} \label{deg2dpf4} 2K_X^2=\frac{1}{2}\,d^2-3d-8(g-1) +12\chi ({\cal O}_X). \end{equation} In the case of $n=5$, using the above formulae we can express $K_X\cdot L^2$, $K_X^2\cdot L$, $K_X^3$, $x_2 \cdot L$ and $x_3$ as functions of $d$, $g$, $\chi ({\cal O}_X)$, $\chi ({\cal O}_S)$; for example, omitting the dots from now on: \begin{equation} \label{KL2} K_X L^2=2(g-1)-2d, \end{equation} \begin{equation} \label{K2L} K_X^2 L=\frac{1}{4}d^2 + \frac{3}{2}d -8(g-1) +6\chi ({\cal O}_S), \end{equation} \begin{equation} \label{K3} K_X^3=-\frac{9}{4}d^2+ \frac{27}{2}d+gd +18(g-1)- 30\chi ({\cal O}_S)-24\chi ({\cal O}_X). \end{equation} \bigskip \begin{pr} \label{s3>=0forN} Let $X$ be a nonsingular threefold on $\Q{5}$. Then $$ 60 \chi (\odix{S}) \geq \frac{3}{2}d^2 - 12d + (d-48)(g-1) + 24 \chi (\odix{X}) $$ and $$ \chi (\odix{S})\leq \frac{2}{3} \frac{(g-1)^2}{d} - \frac{1}{24}d^2 + \frac{5}{12}d. $$ \end{pr} \noindent {\em Proof.} Denote by $s_i$ and $n_i$ the Segre and Chern classes respectively of the normal bundle $\cal N$ of $X$ in $\Q{5}$. Since $\cal N$ is generated by global sections, we have $s_3\geq 0$. Since $s_3=n_1^3- 2n_1n_2$, we get $$ 0\leq (K_X+5L)^3 - 2(K_X+5L)\frac{1}{2}dL^2= K^3 + 15K_X^2L + 75K_XL^2 + 125d - d(K_X+5L)L^2. $$ The first inequality follows from (\ref{K3}), (\ref{K2L}) and (\ref{KL2}). \noindent We use the Generalized Hodge Index Theorem of \ci{boss1} (see also \ci{be-bi-so}): $$ d(K_X^2L)\leq (K_XL^2)^2 $$ and we make explicit the left hand side using (\ref{K2L}) and the right hand side using (\ref{KL2}). The second inequality follows. \blacksquare \medskip In what follows: - $((a,b,c),{ \cal O}(1))$ denotes the polarized pair given by a complete intersection of type $(a,b,c)$ in $\pn{n+1}$ and the restriction of the hyperplane bundle to it; - $(X,L)$ denotes the polarized pair given by a variety $X\subseteq \Q{n}$ and $L:= \odixl{\Q{n}}{1}_{|X}$; - $g,$ $q$ and $p_g$ denote the sectional genus of the embedding line bundle, the irregularity and geometric genus of a surface section, respectively. \begin{rmk} \label{degeven} Let $X\subseteq \Q{n}$, $n\geq 5$, be any subvariety. Then the degree $d$ of $X$ is even. This follows from the fact that the cohomology class of $[X]$ equals the class $(1/2)\, d\, [\Q{n-2}]$ in $H^4(\Q{n}, \zed)$. \end{rmk} \begin{pr} \label{classificationd<12} {\rm (Cf. \ci{decascroll})} Let $X\subseteq \Q{n}$, $n\geq 5$, a codimension two nonsingular subvariety of degree $d\leq 10$. Then the pair $(X,L)$ is one of the ones below. \smallskip \noindent \underline{\rm Type A):} $d=2$, $((1,1,2), {\cal O}(1))$; $g=q=p_g=0$. \noindent \underline{\rm Type B):} $d=4$, $((1,2,2),{\cal O}(1))$; $g=1$, $q=p_g=0$. \noindent \underline{\rm Type C):} $d=4$, $n=6$, $(\pn{1}\times \pn{3}, {\cal O}(1,1))$; $g=q=p_g=0$. \noindent \underline{\rm Type D):} $d=4$, $n=5$, $({\Bbb P}( {\odixl{\pn{1}}{1}}^2\oplus \odixl{\pn{1}}{2}), \xi)$; \quad {\rm 5)}; $g=q=p_g=0$. \noindent \underline{\rm Type E):} $d=6$, $((1,2,3),{\cal O}(1))$; \quad $g=4$, $q=0$, $p_g=1$. \smallskip \noindent \underline{\rm Type F):} $d=6$, $n=5$, $({\Bbb P}( {\cal T}_{\pn{2}}),\xi)$, embedded using a general codimension one linear system ${\frak l}\subseteq| \xi_{ {\cal T}_{\pn{2}} }|$; $g=1$, $q=p_g=0$. \smallskip \noindent \underline{\rm Type G):} $d=6$ $n=5$, $f: X \to \pn{1} \times \pn{2}=:Y$ a double cover, branched along a divisor of type $\odixl{Y}{2,2}$, $L\simeq p^* \odixl{ Y }{1,1}$; $g=2$, $q=p_g=0$. \noindent \underline{\rm Type H):} $d=8$, $((1,2,4), {\cal O}(1))$; \quad $g=9$, $q=0$, $p_g=5$. \smallskip \noindent \underline{\rm Type I):} $d=8$, $((2,2,2), {\cal O}(1) )$; $g=5$, $q=0$, $p_g=1$. \smallskip \noindent \underline{\rm Type L):} $d=8$, $n=5$, $({\Bbb P}( E), \xi)$, $E$ a rank two vector bundle on $\Q{2}$ as in {\rm \ci{io3}}; $g=4$, $q=p_g=0$. \noindent \underline{\rm Type M):} $d=10$, $((1,2,5), {\cal O}(1) )$; $g=16,$ $q=0,$ $p_g=14$. \noindent \underline{\rm Type N):} $d=10$, $n=5$, $f_{|K_X+L|}: X \to \pn{1}$ is a fibration with general fiber a Del Pezzo surface $F$, $K_F^2=4$, $K_X=-L+ f^*\odixl{\pn{1}}{1}$; $g=8$, $q=0$, $p_g=2$. \end{pr} We say that a nonsingular threefold $X$ on $\Q{5}$ is of Type O), if it has degree $d=12$ and it is a scroll over a minimal $K3$ surface. Such a threefold exists. See \ci{decascroll}. \begin{pr} \label{maintm} {\rm (Cf. \ci{decascroll})} The following is the complete list of \,nonsingular codimension two subvarieties of quadrics $\Q{n}$, $n\geq 5$, which are scrolls. \noindent {\rm Type C)}, $n=6$, $d=4$, scroll over $\pn{1}$ and over $\pn{3}$; \noindent {\rm Type D)}, $n=5$, $d=4$, scroll over $\pn{1}$; \noindent {\rm Type F)}, $n=5$, $d=6$, scroll over $\pn{2}$; \noindent {\rm Type \,L)}, $n=5$, $d=8$, scroll over $\Q{2}$; \noindent {\rm Type O)}, $n=5$, $d=12$, scroll over a minimal $K3$ surface. \end{pr} \begin{pr} \label{coreasybound} {\rm (Cf. \ci{decacurves}, or \ci{a-s} for the case $d> 2k(k-1)$.) } Let $C\subseteq \Q{3}$ be an integral curve of degree $d$ and geometric genus $g$. Assume that $C$ is contained in a surface of $\Q{3}$ of degree $2k$. Then $$ g-1\leq \frac{d^2}{4k} + \frac{1}{2} (k-3)d. $$ \end{pr} \begin{pr} \label{boundasep} {\rm (Cf. \ci{a-s}, Proposition $6.4$.)} Let $C$ be an integral curve in $\Q{3}$, not contained in any surface of $\Q{3}$ of degree strictly less than $2k$. Then: $$ g-1\leq \frac{d^2}{2k} +\frac{1}{2}(k-4)d. $$ \end{pr} \noindent Let $S$ be a nonsingular surface on $\Q{4}$, ${\cal N}$ its normal bundle, $\sigma$ its postulation, $C$ a nonsingular hyperplane section of $S$, $g$ its genus, $d$ its degree. Let $s$ be a positive integer, $V_s \in |{\cal I}_{S,\Q{4}}(s)|$ be integral and $\mu_l:=c_2({\cal N}(-l))=$ $(1/2)d^2 +l(l -3)d-2l(g-1)$, $\forall l\in \zed$. \begin{lm} \label{epas} In the above situation: \quad $ 0\leq \mu_{s}\leq s^2 d. $ \end{lm} \noindent {\em Proof.} The left hand side inequality is just Proposition \ref{coreasybound} above. To prove the right hand side we first assume $s=\s$. Using \ci{a-s}, Lemma 6.8 we conclude (from here on the hypothesis $d>2\s^2$ was not used there) in the case at hand. \noindent Now, for the general case, let $s=\s + t$, where $t$ is a non-negative integer. Then, as it is easily checked, $\mu_s=\mu_{\s}+ \s td + t(\s+t-3)d -2t(g-1)$. We conclude by what proved for $\mu_{\s}$ and by the obvious $g\geq 0$. \blacksquare \begin{rmk} \label{barthlarsen} {\rm Let $X$ be a nonsingular codimension two subvariety of $\Q{n}$. As a consequence of the Barth-Larsen Theorem (see \ci{ba}), we have that: if $n\geq 6$, then the fundamental group $\pi_1(X)$ is trivial; if $n\geq 7$, then $Pic(X)\simeq \zed$, generated by the hyperplane bundle, so that $X$ does not carry any nontrivial morphisms. } \end{rmk} \medskip The following fact is well known when $\Q{n}$ is replaced by $\pn{n}$, see \ci{boss1} for example. The case of $\Q{4}$ is proved in \ci{a-s}, Lemma 6.1. The general case can be proved in the same way. See \ci{bounded}, where we prove a more general statement. We used this ``lifting" criterion as a tool to prove the finiteness of the number of families of nonsingular threefolds on $\Q{5}$ not of general type; see Proposition \ref{sigh} below. \begin{pr} \label{roth} {\rm (Cf. \ci{bounded})} Let $X$ be an integral subscheme of degree $d$ and codimension two on $\Q{n}$, $n\geq 4$. Assume that for the general hyperplane section $Y$ of $X$ we have $ h^0({\cal I}_{Y,\Q{n-1}}(\s))$ $\not= 0, $ for some positive integer $\s$ such that $d>2{\s}^2$. Then\, $ h^0({\cal I}_{X,\Q{n}}({\s}))\not= 0. $ \end{pr} \begin{pr} \label{sigh} {\rm (Cf. \ci{bounded})} Let $n=4,$ $5$ or $n\geq 7$. There are only finitely many components of the Hilbert scheme of $\Q{n}$ corresponding to nonsingular $(n-2)$-folds not of general type. \end{pr} \section{THE STRUCTURE OF THE REDUC\-TION MOR\-PH\-IS\-MS} \label{dpfand divisors} In this section we give, by a systematic use of the double point formul\ae, a precise description of the reduction morphisms associated with codimension two subvarieties of quadrics $\Q{n}$, $n\geq 5$. We apply these formul\ae \ also to the case of divisorial contractions of extremal rays on threefolds on $\Q{5}$. For the language and results of Adjunction Theory, which we are going to use freely for the rest of this note, we refer the reader to \ci{be-fa-soadjtp} and to \ci{be-so}. \smallskip Let $\nu :=n-2$. \begin{lm} \label{numericaldpf} Let $X$ be a codimension two nonsingular subvariety of $\Q{n}$, $n\geq 5$. \noindent Let $D$ be a divisor on $X$ with $(D,\odixl{D}{D})\simeq (\pn{\nu-1}, \odixl{\pn{\nu -1}}{-1})$ and $(K_X+(\nu-1)L)_{|D}\simeq \odix{D}$; then $n=5,$ $6$ and $d=10$. \noindent Let $n=5$. Then we have the following list of possible degrees according to whether $X$ contains a divisor of the given form $(D,\odixl{D}{D})$ with $(K_X+(\nu-2)L)_{|D}\simeq \odix{D}$: \smallskip \noindent {\rm (\ref{numericaldpf}.1)} if $(D,\odixl{D}{D}) \simeq ( \pn{2}, \odixl{ \pn{2} }{-2})$, then $d=20$; \noindent {\rm (\ref{numericaldpf}.2)} if $(D, \odixl{D}{D}) \simeq (\pn{2}, \odixl{\pn{2}}{-1})$, then $d=14$; \noindent {\rm (\ref{numericaldpf}.3)} if $(D ,\odixl{D}{D}) \simeq ( \tilde{ {\Bbb F}}_2, G)$, where $2G=K_D$, then $d=14$; \noindent {\rm (\ref{numericaldpf}.4)} $(D,\odixl{D}{D}) \simeq ( {\Bbb F}_0, G)$, where $2G=K_D$, then $d=14$; \noindent {\rm (\ref{numericaldpf}.5)} the case in which $D$ has two components as in {\rm \ci{be-fa-soadjtp}, Theorem 0.2.1, case b5)}, cannot occur; \noindent {\rm (\ref{numericaldpf}.6)} the case $(D ,\odixl{D}{D}) \simeq ( {\Bbb F}_1, -E-f)$ cannot occur. \noindent {\rm (\ref{numericaldpf}.7)} the cases in which $D$ is as in either \, {\rm a)}, or {\rm b)} of \, {\rm \ci{be-so}}, {\rm Theorem 2.3} cannot occur. \smallskip \noindent Let $n=6$. Assume $X$ contains a surface $\cal S$ such that ${\cal S}\simeq \pn{2}$, $L_{|{\cal S}} \simeq \odixl{\pn{2}}{1}$ and such that the normal bundle ${\cal N}_{{\cal S},X} \simeq {\cal T}^*_{ \pn{2} } (1)$. Then $d=14$. \end{lm} \noindent {\em Proof.} For $n=5$ the proof is the same as the one of \ci{be-sc-soams}, Proposition 1.1, using (\ref{deg2dpf}) in the place of (0.8) of the quoted paper. For $n=6$ we compute all the relevant Chern classes by using (\ref{deg2dpf}), the Euler sequence for ${\cal S}\simeq \pn{2}$ and the exact sequence $$ 0\to {\cal T}_{\cal S} \to {{\cal T}_X}_{|{\cal S}} \to {\cal N}_{{\cal S},X} \to 0. $$ \blacksquare \begin{tm} \label{secondreduction} {\rm (\bf{Structure of the reduction morphisms})} Let $X$ be a nonsingular codimension two subvariety of $\Q{n}$, $n\geq 5$. \noindent Assume that $(X,L)$ admits a first reduction $(X',L')$. Then the first reduction morphism is an isomorphism: $(X,L)\simeq (X',L')$. \noindent Assume that $(X,L)$ admits, in addition, a second reduction $(X'',L'')$. We have: \noindent if \ $n=5$ and $d\not= 14,$ $20$, then $(X,L)=(X',L')$ and the second reduction map $\varphi: X'\to X''$ is the blowing up on a nonsingular $X''$ of a disjoint union of nonsingular integral curves; \noindent if $n=6$ and $d\not= 14$, then $(X,L)=(X',L')$ and the second reduction map $\varphi: X'\to X''$ is the blowing up on a nonsingular $X''$ of a disjoint union of nonsingular integral curves. If in addition $d\not= 16,$ $22$, then the second reduction morphism is an isomorphism: $(X,L)\simeq$ $(X',L')\simeq$ $(X'',L'')$; \noindent if $n\geq 7$, then $(X,L)\simeq $$(X',L')\simeq$$(X'',L'')$. \end{tm} \noindent {\em Proof.} Once $K_X+(n-1)L $ is nef and big, i.e. out of the lists of Theorem \ref{k+2lnotspanned} and Theorem \ref{k+2lspannednotbig}, it fails to be ample only if the first reduction is not an isomorphism; in turn, that happens if and only if $X$ contains some exceptional divisors of the first kind. By Proposition \ref{numericaldpf} this happens only if $d=10$. By Proposition \ref{classificationd<12} the type is either M) or N); neither of them contains an exceptional divisor of the first kind. It follows that if the first reduction exists, then $(X,L)\simeq (X',L')$. \noindent The statements concerning the second reduction morphism can be proved as follows. For $n=5,$ we use Theorem 0.2.1 of \ci{be-fa-soadjtp} coupled with Proposition \ref{numericaldpf}. \noindent For $n=6$ we use Theorem 0.2.2 of \ci{be-fa-soadjtp} and then we take a general hyperplane section and reduce to the case $n=5$, with the difference that now case b2) of Theorem 0.2.1 of \ci{be-fa-soadjtp} does not occur. The case of the blowing up of curves yields $d=16,\,22$, as we now show. Since $X\simeq X'$ we cut (\ref{deg2dpf}) with $F\simeq \pn{2}$, a general fiber of the blowing up. Define $a$ to be the positive integer such that $L_{|F}\simeq \odixl{\pn{2}}{a}$. Since $\nb{F}{X}\simeq$ $\odix{\pn{2}} \oplus$ $\odixl{\pn{2}}{-1}$ and ${K_X}_{|F} \simeq$ $\odixl{\pn{2}}{-2}$ we get $$ (16-d/2)a^2=12a -4. $$ Since $a>0$ we see that $d\leq 30$. The only integer solutions to the relation above are $(d,a)=$ $(16,1)$ and $(22,2)$. This concludes the proof for $n=6$. \noindent Finally, for $n\geq 7$ we use Remark \ref{barthlarsen}. \blacksquare \medskip We now describe Mori contractions for threefolds on $\Q{5}$. \begin{lm} \label{divisorialmoricontractionsd=} Let $X$ be a nonsingular threefold in $\Q{5}$. Let $D$ be an integral divisor on $X$. We have: \smallskip \noindent {\rm (\ref{divisorialmoricontractionsd=}.1)} if $(D,\odixl{D}{D}) \simeq (\pn{2}, \odixl{\pn{2}}{-1})$, then either $d=10$ and $L_{|D}\simeq \odixl{\pn{2}}{1}$, or $d=14$ and $L_{|D}\simeq \odixl{\pn{2}}{2}$; \noindent {\rm (\ref{divisorialmoricontractionsd=}.2)} if $(D,\odixl{D}{D}) \simeq (\pn{2}, \odixl{\pn{2}}{-2})$, then either $d=8$ and $L_{|D}\simeq \odixl{\pn{2}}{1}$, or $d=16$ and $L_{|D}\simeq \odixl{\pn{2}}{2}$; \noindent {\rm (\ref{divisorialmoricontractionsd=}.3)} if $(D,\odixl{D}{D}) \simeq ( {\Bbb F}_0,G)$, then $d\leq 20$; \noindent {\rm (\ref{divisorialmoricontractionsd=}.4)} if $(D,\odixl{D}{D}) \simeq ( \tilde{\Bbb F}_2,G)$, then $d=14$ and $L_D=-G$. \end{lm} \noindent {\em Proof.} The proof is the same as the one of \ci{be-sc-soams}, Proposition 1.1, using (\ref{deg2dpf}) in the place of (0.8) of the quoted paper. \blacksquare \begin{pr} \label{moricontrd>=22} {\rm (\bf{Structure of Mori contractions})} Let $X$ be a nonsingular threefold embedded in $\Q{5}$ with $d\geq 22$ and $K_X$ not nef. Let $\rho: X\to Y$ be the contraction of any extremal ray on $X$. Then $Y$ is nonsingular and either $\rho$ is birational and the blowing up of an integral nonsingular curve on $Y$ or $\rho$ is a conic bundle in the sense of Mori Theory. In particular, if $d\gg 0$, then only the former case can occur. \end{pr} \noindent {\em Proof.} The proof is the same as the one of \ci{be-sc-soams}, Corollary 1.2, using (\ref{deg2dpf}) in the place of (0.8) of the quoted paper. As for the last statement, if $\dim Y \leq 2$, then $X$ is not of general type and we apply Proposition \ref{sigh} \blacksquare \medskip The following conjecture is due to Beltrametti, Schneider and Sommese in the case of $3$-folds on $\pn{5}$. It seems a fairly natural question in view of Proposition \ref{moricontrd>=22}. \begin{conj} \label{conjbescso} There is an integer $d_0$ such that every threefold on $\Q{5}$ of degree $d\geq d_0$ is a minimal model. \end{conj} \section{VARIETIES NOT OF LOG-GENERAL-TYPE} \label{secadjointbundles12} \noindent In this section we give a coarse classification of varieties as in the title. We still make free use of the language of Adjunction Theory. \smallskip Let $(X,L)$ be a degree $d$, $\nu$-dimensional nonsingular subvariety of $\Q{n}$ endowed with its embedding line bundle $L$. The ``Types" we shall consider correspond to the ones of Propositions \ref{classificationd<12} and \ref{maintm}. \smallskip We start by observing that $K_X+(\dim X-1)L$ is spanned by its global sections (spanned for short) except for three varieties. \begin{tm} \label{k+2lnotspanned} Let $(X,L)$ be as above. Then $K_X+(\nu -1)L$ is spanned unless $(X,L)$ is one of the three pairs {\rm A)}, {\rm C)} or {\rm D)}. In particular, $d\leq 4$. \end{tm} \noindent{\em Proof.} By the list on \ci{be-so} page 381, and by the fact that there are no codimension two linear subspaces on ${\Q{n}},$ $\forall n\geq 5$, we need to analyze the a.t. scroll over a curve case only. By flatness an a.t. scroll over a curve is a scroll. The result follows from Theorem \ref{maintm}. \blacksquare \medskip Now we classify those pairs for which $K_X+(\nu -1)L$ is spanned, but for which $\kappa (K_X+(\nu -1)L)<\nu$. \begin{tm} \label{k+2lspannednotbig} Let $(X,L)$ be as above. Assume that $K_X+(\nu -1)L$ is spanned, i.e. $(X,L)$ is not as in {\rm Theorem \ref{k+2lnotspanned}}, but that it is not big. Then $(X,L)$ is one of the following pairs: \noindent {\rm (\ref{k+2lspannednotbig}.1) (Del Pezzo variety):} {\rm Type B)}; {\rm Type F)}; \noindent {\rm (\ref{k+2lspannednotbig}.2) (Quadric Bundle over a curve):} {\rm Type G)}; \noindent {\rm (\ref{k+2lspannednotbig}.3) (A.t. scroll over a surface):} {\rm Type L)}; {\rm Type O)}. \noindent In particular, $d\leq 12$. \end{tm} \noindent {\em Proof.} Let $K_X+(\nu -1)L$ be as in the Theorem, then by \ci{be-so} page 381 $(X,L)$ is either a Del Pezzo variety, a quadric bundle or an a.t. scroll over a surface. \noindent Let us assume that $(X,L)$ is a Del Pezzo variety. By slicing with $(\dim X-2)$ general hyperplanes we get a surface in $\Q{4}$ with $K_S=-L_{|S}$. Since $S$ is Del Pezzo we get $\chi (\odix{S})=g(L)=1$. We plug these values in (\ref{deg2dpf4}) and get: $$ d^2-10d+24=0. $$ It follows that either $d=4$ or $d=6$. The conclusion follows from Proposition \ref{classificationd<12}. \noindent Let us assume that $(X,L)$ is a quadric bundle. Let $F\simeq \Q{n-3}$ be a general fiber of the quadric fibration. Dotting (\ref{deg2dpf}) with $F$ we get $d=6$. We conclude using Proposition \ref{classificationd<12}. \noindent Let us assume that $(X,L)$ is an a.t. scroll over a surface. By \ci{be-so-book}, Proposition 14.1.3 $(X,L)$ is an ordinary scroll with $\kappa (K_X + (n-1)L)=2$. We conclude by comparing with Proposition \ref{maintm}. \blacksquare \medskip Now we deal with the line bundle $K_X+(\nu -2)L$. First we exclude the presence of some special pairs. \begin{lm} \label{noveronese} Let $(X,L)$ be as above, then $(X,L)$ cannot be isomorphic to any of the three pairs $(\pn{4}, \odixl{\pn{4}}{2})$, $(\pn{3},\odixl{\pn{3}}{3})$ and $(\Q{3},\odixl{\Q{3}}{2})$. Moreover, there are no Veronese bundles $(X',L')$ associated with a pair $(X,L)$ on $\Q{5}$. \end{lm} \noindent {\em Proof.} By contradiction assume that $(X,L)\simeq (\pn{4}, \odixl{\pn{4}}{2})$. We intesect two general members of $|L|$ and get a nonsingular surface section $(S,L_{|S})$ which is embedded in $\Q{4}$ with $d=16$, $g=1$ and $\chi(\odix{S})=1$. This contradicts (\ref{deg2dpf4}). We exclude the case in which $(X,L)\simeq (\Q{3},\odixl{\Q{3}}{2})$ in a similar way. \noindent The possibility $(X,L)\simeq (\pn{3},\odixl{\pn{3}}{3})$ is ruled out by Remark \ref{degeven}. \noindent Let us assume that $(X,L)$ is a pair for which $(X',L')$ exists and is a Veronese bundle with associated morphism $p:X \to Y$; in particular $n=5$. By Theorem \ref{secondreduction} $(X,L)\simeq (X',L')$. Dotting (\ref{deg2dpf}) with a general fiber $F$ we get $d=10$. Since for some ample line bundle $\cal L$ on $Y$ $2K_X+3L=p^*{\cal L}$, we have the following relation on a general surface section $S$ of $X$: $$ L_{|S}=-2K_S + {\cal L}_{|S}, $$ which ``squared" gives $d=10 \equiv 0$ $mod(4)$, a contradiction. \blacksquare \begin{tm} \label{k+nu-2lnotnefbig} Assume that we are not on the lists of {\em Theorems \ref{k+2lnotspanned}} and {\em \ref{k+2lspannednotbig}} so that $(X,L)\simeq (X',L')$. If $K_X+(\nu -2)L$ is not nef and big then $(X,L)$ is one of the following pairs: \noindent {\rm (\ref{k+nu-2lnotnefbig}.1) (Mukai variety):} {\rm Type E)}; {\rm Type I)}; \noindent {\rm (\ref{k+nu-2lnotnefbig}.2) (Del Pezzo fibration over a curve):} either {\rm Type N)}, $d=10$ or as in {\rm (\ref{3dimdelpezzofibr}.2)}, $d=12$; \noindent {\rm (\ref{k+nu-2lnotnefbig}.3) (Quadric bundle over a surface):} $n=5,\,6$, a flat quadric bundle over a nonsingular surface: if $n=6$, then $d=12$ and if $n=5$, then either $d\leq 18$ or $d=44$. \noindent {\rm (\ref{k+nu-2lnotnefbig}.4) (A.t. scroll over a threefold):} $n=6$, the scroll map is not flat and $d$ is either $14$ or $20$. \end{tm} \noindent {\em Proof.} Let $K_X+(\nu -1)L$ be as in the Theorem, then by \ci{be-so} page 381-2 and Lemma \ref{noveronese}, $(X,L)$ is either a Mukai variety, a Del Pezzo fibration over a curve, a quadric bundle over a surface or an a.t. scroll of dimension $\nu \geq 4$ over a normal threefold. \noindent Let us assume that $(X,L)$ is a Mukai variety. By slicing to a surface section $S$ we find that $K_S=\odix{S}$, and since $X$ is simply connected it follows that $\pi_1(S)$ is trivial as well; $S$ is thus a $K3$ surface. Using (\ref{deg2dpf4}) we get, using $\chi (\odix{S})=2$, $2(g-1)=d$, that either $d=6$ or $d=8$; accordingly $g=4,\,5$, respectively. The conclusion, in this case, follows from Proposition \ref{classificationd<12}. \smallskip \noindent We deal with the case of Del Pezzo fibrations over a curve in Lemma \ref{delpezzofibrationandkodaira} and Proposition \ref{3dimdelpezzofibr} \smallskip \noindent We now deal with quadric bundles over surfaces. Again, $n=5,\, 6$, by Remark \ref{barthlarsen}. \noindent Let $n=5$ and assume, by contradiction, that there is a divisorial fiber $F$ of the quadric bundle map $p:X \to Y$. Then $F$ is as in \ci{be-so} Theorem 2.3. This contradicts case (\ref{numericaldpf}.7) of Lemma \ref{numericaldpf}. It follows that all the fibers of $p$ are equidimensional. By Theorem \ref{conbundle} it follows that $p$ is a quadric fibration in the sense of section \ref{quadricfibration}. The statement follows form Proposition \ref{conicbundleonsurface} and Remark \ref{maple}. \noindent Let $n=6$. $(X,L)$ is a quadric bundle over a surface, $p:X \to Y$, so is its general hyperplane section. By what proved for the case $n=5$ the base surface $Y$ is nonsingular and by Corollary \ref{liftflat} we deduce that $p$ is flat. If we cut (\ref{deg2dpf}) with a general fiber of $p$ we get $d=12$. \noindent Case (\ref{k+nu-2lnotnefbig}.3) follows. \smallskip \noindent Finally case (\ref{k+nu-2lnotnefbig}.4) follows from Proposition \ref{maintm} which ensures us of the absence, on $\Q{6}$, of adjunction theoretic scrolls over threefolds for which the map $p$ is flat: for if $p$ were flat then $Y$ would be nonsingular by \ci{mats} Theorem $23.7$ and then $X$ would be a projective bundle, a contradiction. If one of these scrolls occurs, since $p$ is not flat and -$K_X$ is $p$-ample, Lemma \ref{conbundle} and \ci{mats}, Theorem $23.1$ ensures there must be a fiber $F$ such that either $F$ contains a divisor or, by \ci{be-so-book}, 14.1.4, $F$ is a surface $\cal S$ as in Proposition \ref{numericaldpf}. In the latter case we get $d=14$. In the former, by slicing with a general hyperplane section, we get a threefold $\tilde{X}$ together with the morphism $\tilde{p}:=p_{|{\tilde X}}:{\tilde X} \to Y$, where $Y$ is the base of the scroll. $\tilde{p}$ is the second reduction morphism for $(\tilde{X}, L_{|{\tilde X}})$, so that the result follows by looking at the divisorial fibers of $\tilde{p}$ and Lemma \ref{numericaldpf}. \blacksquare. \section{FIBRATIONS OVER CUR\-VES WI\-TH GE\-NE\-RAL FI\-BER A DEL PEZ\-ZO MA\-NI\-FOLD} \label{fanofibrations} In this section we study codimension two nonsingular subvarieties of $\Q{n}$, $n\geq 5$, which admit a morphism $f:X\to Y$, with connected fibers, onto a nonsingular curve $Y$, such that the line bundle $K_X + (n-4) L$ is trivial on the general fiber. The general fiber will thus be a nonsingular (adjunction-theoretic) Del Pezzo variety of the appropriate dimension $n-3$. By Remark \ref{barthlarsen} we have $n=5,$ $6$. The following lemma ensures that these fibrations coincide with the Del Pezzo fibrations over curves of Adjunction Theory. \begin{lm} \label{delpezzofibrationandkodaira} Let $X$ be a fibration as above. Then $K_X+(n-1)L$ is ample and $\kappa (K_X+(n-2)L)=\kappa (S)=1$. \end{lm} \noindent {\em Proof.} \noindent Without loss of generality we may assume that $n=5$, for otherwise we cut with a general hyperplane section to the three dimensional case and it is easy to show that if the statements we want to prove hold for the threefold hyperplane section of $X$, then they also hold for $X$. \noindent The generic fiber of $f$ is a nonsingular Del Pezzo surface $F$. Since $K_X+L$ is trivial on the fibers we define $$ \Delta:=L^2\cdot F=L_{|F}^2=K_F^2. $$ Cut (\ref{deg2dpf}) with $F$, using the facts that ${K_X}_{|F}=K_F$ and that $x_2\cdot F=12- \Delta$. We get $$ \Delta=\frac{24}{16-d}. $$ Since $F$ is a Del Pezzo surface and $L$ is very ample, we get $3\leq \Delta\leq 9$. Since $\Delta$ is an integer we have only the following possibilities: \begin{equation} \label{deltaandd} (\Delta, d)= (3,8),\ (4,10), \ (6,12). \end{equation} Using the above invariants, and the lists of Adjunction Theory, it is easy to show that $K_X+(n-1)L$ is ample and that $\kappa(K_X+(n-2)L)=0,$ $1$. By Theorem \ref{k+nu-2lnotnefbig} the case $K_X=-(n-2)L$ cannot occur, since these manifolds do not carry any nontrivial fibration. It follows that $K_X+2L$ is ample, $\kappa(K_X+L)=1$ and, by adjunction, $\kappa(S)=1$. \blacksquare \medskip We need the following facts. \begin{fact} \label{relvan} {\rm Let $f:X\to Y$ be as above. By relative vanishing we have $h^i(\odix{X})=h^i(\odix{Y})$, $\forall i$. } \end{fact} \begin{fact} \label{socrelle} $g(Y)=q(S)$, $2g-2-d=(p_g(S) + q(S) -1)\Delta$; moreover the elliptic fibration $S\to Y$ has no multiple fibers. \end{fact} \noindent The assertion on $g(Y)=q(S)$ follows from Lefschetz Theorem on hyperplane sections, $q(S)=h^1(\odix{X})$, and from Fact \ref{relvan}; the other assertion follows from \ci{socrelle}, 0.5.1. \begin{fact} \label{notinp4} $ S\not \subseteq \pn{4} $. \end{fact} \noindent To prove this, assume that $S\subseteq {\pn{4}}$. We use jointly the double point formula for surfaces on $\pn{4}$, see \ci{ha}, page 434, and (\ref{deg2dpf4}) to compute the values of $g$ and $\chi (\odix(S))$ to conclude that, $d=8,$ $10$ would yield noninteger values, a contradiction, and that if $d=12$ then $g=25$, and $\chi (\odix{S})=13$; this system of invariants is inconsistent by Fact \ref{socrelle}. This proves the assertion. \begin{pr} \label{3dimdelpezzofibr} Let $X\subseteq \Q{n}$, $n\geq 5$, be a nonsingular, codimension two subvariety which admits a fibration $f:X\to Y$ in Del Pezzo manifolds onto a nonsingular curve $Y$; in particular $(K_X+(\dim X -2)L)_{|F}\simeq \odix{F}$, $F$ a general fiber. \noindent Then $Y\simeq \pn{1}$ and either $(X,L)$ is of {\rm Type N)} or only the following systems of invariants is possible: \noindent {\rm (\ref{3dimdelpezzofibr}.1)} $n=5,$ $d=12$, $K_F^2=6$, $g=10$, $p_g(S)=2$, $q(S)=0$, $h^i(\odix{X})=0,\ \forall i>0$. \end{pr} \noindent {\em Proof.} By the proof of Lemma \ref{delpezzofibrationandkodaira} and by the knowledge of degree $d=8,\,10$ varieties stemming from Proposition \ref{classificationd<12}, we only need to rule out the case $d=8$ and make precise the invariants in the case $d=12$. Moreover, by the same lemma, $\kappa (S)=1$. \noindent First let $n=5$. \noindent Now we determine the invariants in the case $d=12$. \noindent We apply formula (\ref{deg2dpf4}) in the case $d=12$. We get \begin{equation} \label{chi} 2(g-1)-3\chi (\odix{S})=9. \end{equation} By Fact \ref{notinp4} and by Proposition \ref{roth} we are in the position to apply the Castelnuovo bound for curves on $\pn{4}$, which gives $g\leq 13$. \noindent (\ref{chi}) implies that $\chi (\odix{S})$ is not a non-negative integer, unless $(g,\chi (\odix{S}))=$ $(7,1)$, $(10,3)$, $(13,5)$. We can rule out the cases: $d=12$ and $(g,\chi(\odix{S}))=$$(7,1)$, $(13,5)$ using Fact \ref{socrelle} which gives $g-7=3(p_g + q -1)$; this last equality together with the given values of $\chi(\odix{S})$ and $g$ gives a non-integer value for $q$, a contradiction. It follows that if $d=12$, then $(g,\chi (\odix{S}))=$ $(10,3)$. To compute the values of $p_g$ and $q$ we use again Fact \ref{socrelle} which gives the number $p_g+q$. Since we know $\chi(\odix{S})$ we get the values of $p_g$ and $q$. \noindent Since $g=q$ we see that $Y\simeq \pn{1}$. The assertions on $h^i(\odix{X})$ follow from Fact \ref{relvan}. \noindent The proposition is thus proved for $n=5$. \noindent Let $n=6$, the only remaining case, by Barth-Larsen Theorem. By slicing with a general hyperplane we get a threefold with a fibration onto a curve whose general fiber is a Del Pezzo manifold so that the above analysis applies. The only difference is that the case $d=10$ does not occur by Proposition \ref{classificationd<12}. \noindent Now we prove that also the case $d=12$ does not occur. \noindent The general fiber of $f$ is a Del Pezzo threefold with $K_F=-2L_{|F}$ and $L_{|F}^3=6$. By explicit classification, see \ci{fujbook}, page 72, either $F\simeq \pn{1}\times \pn{1} \times \pn{1}$ or $F\simeq {\Bbb P}({\cal T}_{\pn{2}})$. In both cases formula (\ref{deg3dpf}) dotted with $F$ gives $x_3\cdot F=x_3(F)=24$. But in the former case $x_3(F)=8$, in the latter $x_3(F)=6$. \blacksquare \subsection{MORE UPPER BOUNDS} This section is not needed for Theorem \ref{k+nu-2lnotnefbig}. \medskip We now give an upper bound for the degree of codimension two, nonsingular subvarieties of $\Q{n}$, $n\geq 5$, which admit a morphism onto a curve such that the a general fiber is a Fano variety. By Barth-Larsen Theorem we need to worry only about the cases $n=5,$ $6$. \begin{pr} \label{boundforfanofibrations} Let $X\subseteq \Q{n}$ a nonsingular subvariety of codimension two and degree $d$ which admits a morphism onto a curve such that the general fiber is a Fano variety. \noindent If $n=5$, then $d\leq 20$. \noindent If $n=6$, then $d\leq 30$. \end{pr} \noindent {\em Proof.} Let $n=5$ and ${\cal L}:=L_{|F}$. Assume that $d\geq 22$. \noindent We cut (\ref{deg2dpf}) with a fiber, $F$, and obtain, on $F$: $$ (11-d/2){\cal L}^2 + 5K_F{\cal L} + K_F^2 -c_2(F)=0. $$ Since $c_2(F)=12- K_F^2$, we get: \begin{equation} \label{=ondelpezzo} (d/2-11){\cal L}^2 + 2K_F^2 -12 + 5K_F {\cal L}=0. \end{equation} Now we use $K_F^2\leq 9$ to get \begin{equation} \label{fibrdelpezzosurfaces} (d/2 -11){\cal L}^2\leq 6 + 5K_F{\cal L}. \end{equation} Since $K_F{\cal L}\leq -1$, we see that either $d=22$, or $d=24$ and ${\cal L}^2=- K_F{\cal L}=1$. In the latter case $F\simeq \pn{2}$ and the Hodge Index Theorem on the surface $F$ says that $K_{\pn{2}}^2=1$, a contradiction. In the former case we use (\ref{=ondelpezzo}): $$ 2K_F^2-12 + 5K_F{\cal L}=0, $$ which gives a contradiction for each value $K_F^2=1,\ldots,$ $9$. It follows that $d\leq 20$. \noindent The proof of the statement for $n=6$ is analogous to the proof of Proposition \ref{fanod<22}, where we use (\ref{deg2dpf}) with $n=6$ cut with the cycle $K_X\cdot F$. \blacksquare \medskip In the same spirit we give an upper bound on the degree of Fano threefolds on $\Q{5}$. \begin{pr} \label{fanod<22} Let $X\subseteq \Q{5}$ be a nonsingular Fano threefold. Then $d\leq 20$. \end{pr} \noindent {\em Proof.} (Cf. \ci{be-sc-soams}, Corollary 1.2.) We cut (\ref{deg2dpf}) with $K_X$ and get, using the fact that $x_1x_2=24 \chi (\odix{X})=24$: $$ (11-d/2)L^2K_X + 5LK_X^2 + K_X^3 + 24=0. $$ Let $$ \lambda:= LK_X^2, \qquad 2\mu:=- L^2K_X=-2g+2+2d; $$ clearly $\lambda$ and $\mu$ are positive integers and the above becomes: \begin{equation} \label{fanolm} (d-22)\mu + 5\mu \lambda + 24 = -K_X^3. \end{equation} By the Generalized Hodge Index Theorem, see \ci{boss1}, we get $(-K_X^3)(-K_XL^2)\leq$ $ (K_X^2L)^2$, or \begin{equation} \label{hodgeonfano} (-K_X)^3(2\mu)\leq \lambda^2. \end{equation} By combining (\ref{fanolm}) and (\ref{hodgeonfano}) we get \begin{equation} \label{l2m} \lambda^2 - 10\mu \lambda - [2(d-22)\mu^2 + 48\mu]\geq 0. \end{equation} If we solve the above in $\lambda$ we get either $\lambda <0$, a contradiction, or $\lambda > 10 \mu$. This implies, in turn, that $\lambda \geq 11$. Since, by the classification of Fano threefolds, $-K_X^3\leq 64$, (\ref{fanolm}) becomes $$ (d-22)\mu +55 + 24\leq 64, $$ a contradiction for $d\geq 22$. \blacksquare \section{QUADRIC FIBRATIONS} \label{quadricfibration} In this section the term ``quadric bundle" is to be intended in the sense of Adjunction Theory. The term ``quadric fibration" is introduced below. By {\em quadric fibration} we mean a nonsingular projective variety $X\subseteq {\Bbb P}$, of dimension $x$, together with a fibration $p:X\to Y$ onto a ({\em a fortiori}) nonsingular variety $Y$ of positive dimension $y$, {\em all} of which fibers are quadrics, not necessarily integral, of the appropriate dimension $(x-y)$. One has non integral fibers only if the relative dimension is one. The case $\dim Y=0$ is trivial. In virtue of Remark \ref{barthlarsen} we have: \begin{fact} {\em There are no codimension two quadric fibrations on $\Q{n}$, for $n\geq 7$ and, for $n=6$, any such is simply connected.} \end{fact} \medskip We restrict ourselves to the case of $n\geq 5$. We begin by fixing some notation and establishing some simple facts. \noindent Let $L$ denote the restriction to $X$ of the hyperplane bundle. The sheaf ${\cal E} : =p_* L$ is locally free on $Y$ of rank $(x-y +2)$. It is easy to check that $\cal E$ is generated by its global sections. The surjection $p^*p_*L\to L$ defines an embedding: $X \hookrightarrow {\Bbb P}(\cal E)$, where $L={\xi_{\cal E}}_{|X}$ and $X$ is defined by a nonzero section of the line bundle $ 2\xi - \pi^* {\cal M}$, for some ${\cal M}\in Pic (Y)$, where $\pi:{\Bbb P}({\cal E})\to Y$ is the bundle projection. The following gives a sufficient condition for a general hyperplane section of $X$ to be a quadric fibration over $Y$. It is a well known ``counting dimensions" argument. \begin{lm} \label{countingdimensions} Let $X\to Y$ be a quadric fibration as above. Assume $2y< x+2$. Then a general hyperplane section $X'$ of $X$ is a quadric fibration over $Y$ via $p_{|X'}:X'\to Y$. \end{lm} \noindent {\em Proof.} Since $\cal E$ is generated by global sections and, by assumption $rank({\cal E})>y$, a general section of it does not vanish on $Y$. Such a section will define, {\em for every} $y\in Y$, a hyperplane $\Lambda_y$ of the corresponding fiber $\pi^{-1}(y)\subseteq {\Bbb P}({\cal E})$. In the case in which the quadrics $p^{-1}(y)$ were integral $\forall y\in Y$, we would be done. This is, in general, not true. However, the singular quadrics of the fibration are parameterized by a proper closed subset $D$ of $Y$ with $\dim D \leq (y-1)$. The hyperplanes of $\Bbb P$ which contain the reduced part, $\Sigma\simeq \pn{x-y}$, of one of the components of one non integral quadric of the fibration form a linear space of dimension $(\dim {\Bbb P}-x+y-1)$ contained in ${\Bbb P}^{\vee}$. The space of these bad hyperplanes is of dimension at most $(\dim D + \dim {\Bbb P} -x +y-1)\leq$ $\dim {\Bbb P}-x+2y-2<$ $\dim {\Bbb P}^{\vee}$. It follows that the general section of $\cal E$ gives a hyperplane section of $X$ which cuts {\em every} quadric of the fibration in a quadric of dimension one less. \blacksquare \medskip \begin{pr} \label{quadricfibrationoncurve} There are no quadric fibrations over curves on $\Q{6}$. The only quadric fibrations over curves on $\Q{5}$ are of {\rm Type G)}. If there is a quadric fibration over a surface on $\Q{6}$, hen it has degree $d=12$. \end{pr} \noindent {\em Proof.} As to quadric fibrations over curves, we cut (\ref{deg2dpf}) with a nonsingular fiber $F\simeq \Q{n-3}$, we get $d=6$. We conclude by comparing with Proposition \ref{classificationd<12}. \noindent As to quadric fibrations over a surface we cut (\ref{deg2dpf}) with a nonsingular fiber $F\simeq \Q{n-4}$ and get $d=12$. \blacksquare \medskip The following proposition and remark describe the situation for threefolds quadric bundles over surfaces. \begin{pr} \label{conicbundleonsurface} Let $X\subseteq \Q{5}$ be a threefold quadric fibration {\rm (conic bundle)} over a surface $Y$. Then either $d\leq 98$ or $X$ is contained in a hypersurface $V\in |\odixl{\Q{5}}{3}|$ and $d\leq 276$. \end{pr} \noindent {\em Proof.} We denote the Chern classes of $X$ and $Y$ by $x_i$ and $b_i$, respectively. We omit the symbol ``$p^*$" for ease of notation. We follow closely the paper \ci{boss2}. First we introduce the following entities and we report from \ci{boss2}, for the reader's convenience, the relations among them which are essential to the computations below (one warning: some of the equalities are only numerical equalities): $\cal M$ was defined at the beginning of the section; $D\in |2e_1 -3{\cal M}|$, it is called the {\em discriminant divisor}; its points correspond to the singular fibers of $p$; $2R\subseteq Y$ the {\em branching divisor} associated with a general hyperplane section, $S$, of $X$, which, in view of Lemma \ref{countingdimensions}, is a cyclic double cover of $Y$; $e_1= 3R-D$; ${\cal M}=2R-D$; $x_1=L +b_1 - R;$ $x_2=L^2 +L\cdot( b_1 -2R +D) + ( -2R^2 - R\cdot b_1+ D\cdot R + b_2 + e_2) ;$ $x_3= 2b_2 -D^2 + Db_1;$ $L\cdot W\cdot W'=2W\cdot W'$, for every pair of divisors $W$ and $W'$ on $Y$; $L^2\cdot W= (4R-D)\cdot W$; $e_2=\frac{1}{2}(12R^2 +D^2 -7DR -d)$. Now we plug the above values of $x_1$ and $x_2$ for $x_1$ and $x_2$ in (\ref{deg2dpf}): \begin{equation} \label{deg2cb} (6-\frac{d}{2})L^2 -4L b_1+5L R+ b_1^2 -b_1R -LD + 3R^2-D R -b_2 -e_2=0. \end{equation} Next we equate the expression above for $x_3$ to the one of (\ref{deg3dpf}), using again the above expressions for $x_1$ and $x_2$: \begin{equation} \label{deg3cb} -(2d+10)b_1R + 2dR^2 + (\frac{d}{2}+4)Db_1 + D^2 -10b_2 + 2b_1^2 -(\frac{d}{2}+5)DR -d(\frac{d}{2} -13)=0. \end{equation} Now we set $$ x:= b_1^2 \quad {\rm and} \quad y:=DR, $$ we cut $(\ref{deg2cb})$ with $R$, $-b_1$, $D$ and $L$, respectively, so that we obtain four linear equations to which we add (\ref{deg3cb}), after having substituted in $x$ and $y$. The result is the following linear system of equations: \begin{equation} \label{systcb} Mv^t=c^t, \end{equation} where $ \hspace{1in} M:= \left( \begin{array}{rrrrr} -8 & 34-2d & 0 & 0 & 0 \\ 2d-34 & 0 & -\frac{d}{2}+8 & 0 & 0 \\ 0 & 0 & -8 & \frac{d}{2}-8 & 0 \\ -18 & 14 & +4 & 0 & -2 \\ -2d-10 & 2d & \frac{d}{2} +4 & 1 & -10 \end{array} \right), $ \smallskip \noindent $v:=(\ b_1R,\ R^2,\ Db_1,\ D^2,\ b_2\ )$ \noindent and \noindent $c:=(\ (8-\frac{d}{2})y,\ -8x,\ (2d-34)y,\ 2x+ 4y + d(\frac{d}{2}-7),\ -2x + (\frac{d}{2} +5)y + d(\frac{d}{2}-13)\ )$. \medskip \noindent Since $P:=-\frac{1}{2}det M=3d^3-27d^2 -1520d +18976>0$, $\forall d>0$, we can solve the above system (\ref{systcb}) and obtain the unique solution: \medskip \noindent $b_1R = -\frac{1}{2}[(-128 d^2 +4480d-39168)x+(2d^3 -111d^2+2020d -12096)y+$ $( 2d^5 -120d^4 +2678d^3 -26304d^2+95744d)]/P,$ \medskip \noindent $R^2 = \frac{1}{4}[(-1024d+18432)x +(3d^3 -8d^2 -2112d+23552)y + (16d^4 -688d^3 +9728d^2 -45056d)]/P, $ \medskip \noindent $b_1D = -2[(-152d^2 +4440d-32128)x + (2d^3 -113d^2 +2099d -12852)y + ( 2d^5 -122d^4 +2766d^3 -27574d^2 +101728d)]/P$, \medskip \noindent $D^2 =-4[(-1216d +16064)x + (-3d^3 +46d^2 +893 d -13736)y + (16d^4 -720 d^3 +10608d^2 -50864d)]/P, $ \medskip \noindent $b_2 =\frac{1}{4}[(12d^3+20d^2 -3648d+13952)x+(d^3 -30d^2 +152d+960)y + (d^5 -27d^4 +274d^3 -4448d^2 +46016d)]/P $. \medskip \noindent Since $\cal E$ is generated by global sections and $D$ is effective we see that $e_2\geq 0$, $e_1D\geq 0$. Also, \ci{boss2}, Lemma 2.9 gives $y=DR\geq 0$. We can make explicit $e_2$ and $e_1$ by the formul\ae \ given at the beginning of this proof and deduce: \begin{eqnarray*} \label{ineqcb} DR &= & y \geq 0, \\ e_2 \cdot P & = & (896d-4480)x - (\frac{19}{2}d^2 -366d+3616)y- \\ & & (\frac{19}{2}d^4 -\frac{843}{2}d^3 + 5864d^2 -24656d)\geq 0, \\ e_1D \cdot P &= & -(4864d-64256)x- (3d^3 -103d^2 + 988d -1984)y+ \\ & & (64d^4 -2880d^3 + 42432d^2 -203456d)\geq 0. \end{eqnarray*} These three inequalities define a region of the plane $(x,y)$. It is straightforward to check that the two lines $e_2=0$ and $e_1D=0$ have slopes $a$ and $b$ whose sign does not change with $d$ if $d\geq 20$. One can check easily that $a>0$ and $b<0$. The intersection of the first line above with the $x$-axis is $$ (x_1,0)_{e_2}=(\frac{(19/2)d^4 - (843/)2d^3+5864d^2 - 24656d}{896d-4480},0); $$ the intersection of the second line with the $x$-axis is $$ (x_2,0)_{e_1D}=(\frac{64d^4 -2880d^3 + 42432d^2 -203456d}{4864d-64256},0). $$ One can check, that, since $d\geq 20$, $x_1<x_2$. The region we are interested in is a triangle with vertices $(x_1,0)_{e_2}$, $(x_2,0)_{e_1D}$ and $(x_3,y_3)_{(e_2=0)\cap (e_1D=0)}.$ \noindent Now we compute the genus of a general curve section, $C$, of $X$. By adjunction $x_1\cdot L^2=2d+2-2g$, so that by what above: \begin{eqnarray*} g-1 & =& \frac{d}{2} -2b_1R+ \frac{Db_1}{2} + 2R^2 -\frac{DR}{2} \\ &= & -2b1R + \frac{Db_1}{2} +2R^2 - \frac{y}{2} +\frac{d}{2} \\ &=&[\,(24d^2-472d+2176)x+( (23/2)d^2 -375d +3044)y + \\ & & ((23/2)d^4 -(891/2)d^3 +5374 d^2 -19024d)\,]\,/\,P. \end{eqnarray*} Again it is not difficult to check that the absolute value of the slope of the above line is bigger than $|b|$. It follows easily that the maximum possible value for $g-1$ in our region is achieved at $(x_2,0)_{e_1D}$, while the minimum is at $(x_1,0)_{e_2}$. We thus get \begin{equation} \label{superbound} \frac{19d^3-187d^2+416d}{224d-1120} \leq g-1 \leq \frac{4d^3-77d^2+321d}{38d-502}. \end{equation} Assume that $C$ is not contained in any surface of $\Q{3}$ of degree strictly less than $2\cdot 11$. Then by (\ref{boundasep}) and by the left hand side inequality of (\ref{superbound}), we get $$ \frac{19d^3-187d^2+416d}{224d-1120} \leq \frac{d^2}{22}+\frac{7}{2}d, $$ which, remembering that $d$ is even and that we are assuming $d\geq 20$, implies $ d\leq 98$. \noindent Assume that $C$ is contained in a surface of degree $2k$, with $k=10,9,\ldots, 3$. By Corollary (\ref{coreasybound}) we infer: $$ \frac{19d^3-187d^2+416d}{224d-1120} \leq \frac{d^2}{4k}+\frac{k-3}{2}d, $$ which implies, as above, that for $k=10, 9, \ldots, 3$, $d\leq 64,$ $58$ $54,$ $48,$ $44,$ $40,$ $40$ and $276$, respectively. \noindent Finally, assume that $C$ is contained in a surface of degree four or two. Using the right hand side inequality of (\ref{superbound}) and Lemma \ref{epas} we get $d\leq 42$ and $d\leq 16 $, respectively. Actually in the last case we get a contradiction, since we are assuming $d\geq 20$. \noindent Finally if $C$ is in a surface of degree six, then $X$ is in a hypersurface of degree six in $\Q{5}$, provided, $d>18$ (cf. Proposition \ref{roth}). \blacksquare \begin{rmk} \label{maple} {\rm We have checked with a Maple routine which are the possible degrees of a threefold on $\Q{5}$ which is a quadric fibration over a surface. For $d\geq 20$ we have imposed the following restrictions on the triples $(d,x,y)$: \noindent 1) $20 \leq d \leq 276$; \noindent 2) for every fixed $d$ as above $(x,y)$ must belong to the triangle of the proof of Proposition \ref{conicbundleonsurface}; \noindent 3) $b_1R,$ $ R^2,$ $ b_1D,$ $D^2,$ $b_2$, $g-1$, $\chi ({\cal O}(Y)$ and $\chi ({\cal O} (S))$ must be integers; \noindent 4) $(g-1)$ must satisfy inequality (\ref{superbound}) and the bound of Theorem 2.3 in \ci{gross}; \noindent 5) $\chi ({\cal O}(S))$ must satisfy the two inequalities of Proposition \ref{s3>=0forN}; \noindent 6) various inequalities stemming from the Hodge Index Theorem on $Y$ as, for example, $(K_YR)^2\geq K_Y^2 R^2$; \noindent 7) if $d> 98$ then $g-1 \leq (1/12)d^2$, see Proposition \ref{coreasybound}; \noindent The result is that the only possible degree, for $d\geq 20$ is $d=44$. \noindent By taking double covers of the four scrolls of \ci{ottp5}, we see that there are flat conic bundles over surfaces for $d=6,\, 12,\, 14,\, 18$. We do not know whether the case $d=44$ occurs. } \end{rmk} \subsection{DIGRESSION} \label{digressione} In the course of the proof of Theorem \ref{k+nu-2lnotnefbig} we used the fact, due to Besana \ci{besana}, that the base of an adjunction theoretic quadric bundle over a surface is nonsingular. The following is a result with a similar flavor. It is probably well known. \begin{lm} \label{conbundle} Let $X$ a nonsingular projective variety of dimension $n$, $p:X\to Y$ a morphism onto a normal projective variety $Y$ of dimension $n-1$ such that all fibers have the same dimension, the general scheme theoretic fiber over a closed point is isomorphic to a conic and $-K_X$ is $p$-ample. Then all the scheme-theoretic fibers are isomorphic to conics, $p$ is flat and $Y$ is nonsingular. \end{lm} \noindent {\em Proof.} The proof is the same as the one of \ci{mori} Lemma 3.25. The only necessary changes are the following: a) replace the line bundle $H$ of \ci{mori}, by a pull-back $p^*A$ of any ample line bundle $A$ on $Y$ and use Kleiman criterion of ampleness to obtain the result analogue to the last assertion of \ci{mori} Lemma; b) replace\ci{mori} Lemma 3.12 by \ci{ando} Lemma 1.5. \blacksquare \begin{cor} \label{liftflat} Let $X$ be a nonsingular projective variety together with a morphism $p:X\to Y$, where $Y$ is a normal variety of dimension $m$. Let $D_i$, $i=1,\ldots, n-m-1$ be divisors on $X$ such that they intersect transversally; denote by $X'$ their intersection. Assume that $p_{|X'}:X'\to Y$ satisfies the hypothesis of {\rm Lemma \ref{conbundle}}. Then $p$ is flat and $Y$ is nonsingular. \end{cor} \noindent {\em Proof.} By the lemma, $p_{|X'}$ is flat. We can ``lift" this flatness to $p$ by virtue of \ci{mats}, Corollary to Theorem 22.5. As above the flatness of $p_{|X'}$ (or of $p$) implies the nonsingularity of $Y$. \blacksquare \begin{cor} \label{conics} Let $X$ a nonsingular projective variety of dimension $n$, $p:X\to Y$ a morphism onto a normal projective variety $Y$ of dimension $n-1$ such that all fibers have the same dimension. If the general fiber of $p$ is actually embeddable as conics with respect to an embedding of $X$, then all scheme theoretic fibers are actually embedded conics, $p$ is flat, $Y$ is nonsingular and $-K_X$ is $p$-ample. \end{cor} \noindent {\em Proof.} We argue as in the proof of the lemma with the simplifications due to the fact that a flat deformation of a conic in projective space is still a conic. The assertion on $-K_X$ follows by observing that, if $L$ denotes the line bundle with which we embed $X$, $K_X+L$ is a pull-back from $Y$. \blacksquare \begin{rmk} {\rm The assumption $-K_X$ is $p$-ample is essentialin the lemma, as the blow up of a $\pn{1}$ bundle over a curve at two distinct points on a fiber shows. Moreover, the above Lemma does not follow directly from \ci{mori} or \ci{ando}, since there are conic bundles which structural morphism is not a Mori contraction. Finally, the above theorem is certainly false if one has $\dim X= \dim Y$. It is a purely local question: consider the quotient of $\Bbb A^2$ by the involution $(x,y)\to (-x,-y)$. } \end{rmk}

9211

alg-geom/9211001

en

https://arxiv.org/abs/alg-geom/9211001

alg-geom/9211001

Daniel Huybrechts

Daniel Huybrechts and Manfred Lehn

Stable pairs on curves and surfaces

37 pages, LaTeX Version 2.09, MPI-92-093

We describe stability conditions for pairs consisting of a coherent sheaf and a homomorphism to a fixed coherent sheaf on a projective variety. The corresponding moduli spaces are constructed for pairs on curves and surfaces. We consider two examples. The fixed sheaf is the structure sheaf or is a vector bundle on a divisor, i.e. Higgs pairs or framed bundles, resp. (unencoded version)

alg-geom

\section{Moduli spaces of stable pairs} Throughout this paper we fix the following notations: $X$ is an irreducible, nonsingular, projective variety of dimension $e$ over an algebraically closed field $k$ of characteristic zero, embedded by a very ample line bundle ${\cal O}_X(1)$. The canonical line bundle is denoted by ${\cal K}_X$. If ${\cal E}$ is a coherent ${\cal O}_X$-module, then $\chi_{{\cal E}}(n):=\chi({\cal E}\otimes{\cal O}_X(n))$ denotes its Hilbert polynomial, ${\rm T}({\cal E})$ its torsion submodule and $\det\,{\cal E}$ its determinant line bundle. The degree of ${\cal E}$, $\deg\,{\cal E}$, is the integral number $c_1(\det\,{\cal E}).H^{e-1}$, where $H\in|{\cal O}_X(1)|$ is a hyperplane section. $\chi$ will always be a polynomial with rational coefficients which has the form $$\chi(z)=\deg\,X\cdot r\cdot\frac{z^e}{e!}+(d-\frac{\deg\,{\cal K}_X}{2}\cdot r)\cdot z^{e-1}+\hbox{\rm Terms of lower order in $z$}.$$ If $\chi=\chi_{{\cal E}}$, then $r={\rm rk}{\cal E}$ and $d=\deg{\cal E}$. Finally, let ${\cal E}_0$ be a fixed coherent ${\cal O}_X$-module. By a \em pair \em we will always mean a pair $({\cal E},\alpha)$ consisting of a coherent ${\cal O}_X$-module ${\cal E}$ with Hilbert polynomial $\chi_{{\cal E}}=\chi$ and a nontrivial homomorphism $\alpha:{\cal E}\rightarrow{\cal E}_0$. We write ${\ke_{\alpha}}$ for ${\rm Ker }\,\alpha$. In the next section we define the notion of semistability for such pairs with respect to an additional parameter $\delta$. To simplify the notations and to be able to treat stability and semistability simultaneously, we employ the following short-hand: Whenever in a statement the word \em (semi)stable \em occurs together with a relation symbol in brackets, say $(\leq)$, the latter should be read as $\leq$ in the semistable case and as $<$ in the stable case. An inequality $p\,(\leq)\,p'$ between polynomials means, that $p(n)\,(\leq)\,p'(n)$ for large integers $n$. If $p$ is a polynomial then $\Delta p(n):=p(n)-p(n-1)$ is the difference polynomial. We proceed as follows: In section 1.1 we define semistability for pairs and formulate the moduli problem. In section 1.2 boundedness results for semistable pairs on curves and surfaces are obtained. Moreover, a close relation between semistability and sectional semistability is established. The notion of sectional stability naturally appears by way of constructing moduli spaces for pairs. This is done in section 1.3 leading to the existence theorem \ref{mainthm}. Section 1.4 is devoted to an invariant theoretical analysis of the construction in 1.3 and the proof of the main technical proposition \ref{prop}.\\ The reader who is familiar with the papers of Gieseker and Maruyama (\cite{g2}, \cite{Ma}) will notice that many of our arguments are generalizations of their techniques. \subsection{Stable pairs and the moduli problem}\label{Kapitel11} Let $\delta$ be a polynomial with rational coefficients such that $\delta>0$, i.\ e.\ $\delta(n)>0$ for all $n\gg 0$. We write $\delta(z)=\sum_{\nu}\delta_{e-\nu}z^{\nu}$. \begin{definition}\label{stabdef} A pair $({\cal E},\alpha)$ is called \em(semi)stable \em (with respect to $\delta$), if the following two conditions are satisfied: \begin{itemize}\item[(1)] ${\rm rk}{\cal E}\.\chi_{{\cal G}}\,(\leq)\,{\rm rk}{\cal G}\.(\chi_{{\cal E}}-\delta)$ for all nontrivial submodules ${\cal G}\subseteq{\ke_{\alpha}}$. \item[(2)] ${\rm rk}{\cal E}\.\chi_{{\cal G}}\,(\leq)\,{\rm rk}{\cal G}\.(\chi_{{\cal E}}-\delta)+{\rm rk}{\cal E}\.\delta$ for all nontrivial submodules ${\cal G}\,(\subseteq)\,{\cal E}$. \end{itemize}\end{definition} If no confusion can arise, we omit $\delta$ in the notations. Note that a stable pair a fortiori is semistable. \begin{lemma}\label{firstobs} Suppose $({\cal E},\alpha)$ is a semistable pair, then: \begin{itemize} \item[i)] ${\ke_{\alpha}}$ is torsion free. $h^0({\cal G})\leq h^0({\rm T}({\cal E}_0))$ for all submodules ${\cal G}\subseteq{\rm T}({\cal E})$. \item[ii)] Unless $\alpha$ is injective, $\delta$ is a polynomial of degree smaller than $d$. \end{itemize}\end{lemma} {\it Proof:} ad i): If ${\cal G}\subset{\ke_{\alpha}}$ is torsion, then ${\rm rk}{\cal G}=0$. Condition (1) then shows $\chi_{{\cal G}}=0$, hence ${\cal G}=0$. Thus $\alpha$ embeds the torsion of ${\cal E}$ into the torsion of ${\cal E}_0$. This gives the second assertion. ad ii): Assume ${\ke_{\alpha}}$ is nontrivial. By i) ${\ke_{\alpha}}$ is torsion free of positive rank, and condition (1) implies $\delta/{\rm rk}{\cal E}\,\leq\,(\chi_{{\cal E}}/{\rm rk}{\cal E}-\chi_{{\ke_{\alpha}}}/{\rm rk}{\ke_{\alpha}})$. The two fractions in the brackets are polynomials with the same leading coefficients. This shows $\deg\delta<e$.\hspace*{\fill}\hbox{$\Box$} Thus if $\deg\delta\geq e$, then $\alpha$ must needs be an injective homomorphism, and isomorphism classes of semistable pairs correspond to submodules of ${\cal E}_0$ with fixed Hilbert polynomial. Note that condition (2) of the definition above is automatically satisfied. So in this case all pairs are in fact stable and parametrized by the projective quotient scheme ${\rm Quot }^{\chi_{{\cal E}_0}-\chi_{{\cal E}}}_{X/{\cal E}_0}$. For that reason we assume henceforth that $\delta$ has the form $$\delta(z)=\delta_1z^{e-1}+\delta_2z^{e-2}+\cdots+\delta_e.$$ \begin{definition}\label{mustab} A pair $({\cal E},\alpha)$ is called $\mu$-(semi)stable (with respect to $\delta_1$), if the following two conditions are satisfied: \begin{itemize} \item[(1)] ${\rm rk}{\cal E}\.\deg{\cal G}\,(\leq)\,{\rm rk}{\cal G}\.(\deg{\cal E}-\delta_1)$ for all nontrivial submodules ${\cal G}\subseteq{\ke_{\alpha}}$. \item[(2)] ${\rm rk}{\cal E}\.\deg{\cal G}\,(\leq)\,{\rm rk}{\cal G}\.(\deg{\cal E}-\delta_1)+{\rm rk}{\cal E}\.\delta_1$ for all nontrivial submodules ${\cal G}\subseteq{\cal E}$ with ${\rm rk}{\cal G}<{\rm rk}{\cal E}$. \end{itemize}\end{definition} As in the theory of stable sheaves there are immediate implications for pairs $({\cal E},\alpha)$: \begin{center} $\mu$-stable $\Rightarrow$ stable $\Rightarrow$ semistable $\Rightarrow$ $\mu$-semistable \end{center} A family of pairs parametrized by a noetherian scheme $T$ consists of a coherent ${\cal O}_{T\times X}$-module ${\cal E}$, which is flat over $T$, and a homomorphism $\alpha:{\cal E}\rightarrow p_X^*{\cal E}_0$. If $t$ is a point of $T$, let $X_t$ denote the fibre $X\times{\rm Spec}\,k(t)$, ${\cal E}_t$ and $\alpha_t$ the restrictions of ${\cal E}$ and $\alpha$ to $X_t$. A homomorphism of pairs $\Phi:({\cal E},\alpha)\rightarrow({\cal E}',\alpha')$ is a homomorphism $\Phi:{\cal E}\rightarrow{\cal E}'$ which commutes with $\alpha$ and $\alpha'$, i.\ e.\ $\alpha'\circ\Phi=\alpha$. The correspondence $$T\mapsto\{\hbox{Isomorphism classes of families of (semi)stable pairs parametrized by $T$}\}$$ defines a setvalued contravariant functor $\underline{\cal M}_\delta^{(s)s}(\chi,{\cal E}_0)$ on the category of noetherian $k$-schemes of finite type. We will prove that for $\dim X\leq 2$ there is a fine moduli space for $\underline {\cal M}_\delta^s (\chi,{\cal E}_0)$. It is compactified by equivalence classes of semistable pairs (\ref{mainthm}). \subsection{Boundedness and sectional stability}\label{Kapitel12} In section \ref{Kapitel13} we will construct moduli spaces of stable pairs by means of geometric invariant theory. The stability property needed in this construction differs slightly from the one given in \ref{stabdef} in refering to the number of global sections rather than to the Euler characteristic of a submodule of ${\cal E}$. In this section we compare the different notions and prove that semistable pairs form bounded families, if the variety $X$ is a curve or a surface. \begin{definition}\label{secstadef} Let $\bar\delta$ be a positive rational number. A pair $({\cal E},\alpha)$ is called \em sectional (semi)\-stable \em (with respect to $\bar\delta$), if ${\ke_{\alpha}}$ is torsionfree and there is a subspace $V\subseteq H^0({\cal E})$ of dimension $\chi({\cal E})$ such that the following conditions are satisfied: \begin{itemize} \item[(1)] ${\rm rk}{\cal E}\.\dim(H^0({\cal G})\cap V)\,(\leq)\,{\rm rk}{\cal G}\.(\chi({\cal E})-\bar\delta)$ for all nontrivial submodules ${\cal G}\subseteq{\ke_{\alpha}}$. \item[(2)] ${\rm rk}{\cal E}\cdot\dim(H^0({\cal G})\cap V)\,(\leq)\,{\rm rk}{\cal G}\cdot(\chi({\cal E})-\bar\delta)+{\rm rk}{\cal E}\cdot\bar\delta$ for all nontrivial submodules ${\cal G}\,(\subseteq)\,{\cal E}$. \end{itemize} \end{definition} We begin with the case of a curve. In this case $\delta$ is a rational number, and the Hilbert polynomial of any ${\cal O}_X$-module ${\cal G}$ depends on ${\rm rk}{\cal G}$ and $\deg{\cal G}$ only. Moreover, the polynomials occuring in the inequalities of definition \ref{stabdef} are linear and have the same leading coefficients. Therefore the Hilbert polynomials $\chi_{{\cal G}}$ can throughout be replaced by the Euler characteristics $\chi({\cal G})$ without changing the essence of the definition. \begin{theorem}\label{boundcurve} Let $X$ be a smooth curve of genus $g$. Assume that $d>r\.(2g-1)+\delta$. \begin{itemize}\item[i)] If $({\cal E},\alpha)$ is semistable or sectional semistable, then ${\cal E}$ is globally generated and $h^1({\cal E})=0$. \item[ii)] $({\cal E},\alpha)$ is a (semi)stable pair if and only if it is sectional (semi)stable.\end{itemize} \end{theorem} {\it Proof:} ad i): On a smooth curve $X$ there is a split short exact sequence $$\ses{{\rm T}({\cal E})}{{\cal E}}{\bar{\cal E}}$$ with locally free $\bar{\cal E}$ for any coherent ${\cal O}_X$-module ${\cal E}$. Now $H^1({\cal E})=H^1(\bar{\cal E})$, and ${\cal E}$ is globally generated if and only if $\bar{\cal E}$ is globally generated. A glance at the short exact sequence $$\ses{\bar{\cal E}(-x)}{\bar{\cal E}}{\bar{\cal E}\otimes{\cal O}_x}$$ for some closed point $x\in X$ shows that the vanishing of $H^1(\bar{\cal E}(-x))$ for all $x\in X$ is a sufficient criterion for both $H^1({\cal E})=0$ and the global generation of ${\cal E}$. If $H^1(\bar{\cal E}(-x))\neq0$, then there is a nontrivial homomorphism $\varphi:\bar{\cal E}\rightarrow{\cal K}_X(x)$. Let ${\cal G}:={\rm T}({\cal E})+{\rm Ker }\varphi$, so that there is a short exact sequence $$\ses{{\cal G}}{{\cal E}}{{\cal K}_X(x-C)}$$ with some effective divisor $C$ on $X$. From this sequence we get $$\chi({\cal G})\geq\chi({\cal E})-\chi({\cal K}_X(x))\quad{\rm and}\quad h^0({\cal G})\geq h^0({\cal E})-h^0({\cal K}_X(x)).$$ On the other hand, $$\chi({\cal G})\leq\frac{{\rm rk}{\cal E}-1}{{\rm rk}{\cal E}}\chi({\cal E})+\frac{\delta}{{\rm rk}{\cal E}},$$ if $({\cal E},\alpha)$ is semistable, and $$\dim(V\cap H^0({\cal G}))\leq\frac{{\rm rk}{\cal E}-1}{{\rm rk}{\cal E}}\chi({\cal E}) +\frac{\delta}{{\rm rk}{\cal E}}$$ for some vector space $V\subseteq H^0({\cal E})$ of dimension $\chi({\cal E})$, if $({\cal E},\alpha)$ is sectional semistable. In the first case we get $\chi({\cal E})\leq{\rm rk}{\cal E}\.\chi({\cal K}_X(x))+\delta$. And in the second case one has \begin{eqnarray*}h^0({\cal E})-h^0({\cal K}_X(x))\leq h^0({\cal G})&\leq& \dim(H^0({\cal G})\cap V)+(h^0({\cal E})-\dim V)\\ &\leq&\frac{{\rm rk}{\cal E}-1}{{\rm rk}{\cal E}}\.\chi({\cal E})+\frac{\delta}{{\rm rk}{\cal E}}+(h^0({\cal E})- \chi({\cal E})) \end{eqnarray*} So in any case we end up with $\deg{\cal E}\leq{\rm rk}{\cal E}\.(2g-1)+\delta$ contradicting the assumption of the theorem. ad ii): By part i) we have $\chi({\cal E})=h^0({\cal E})$, $V=H^0({\cal E})$ and, of course, $\chi({\cal G})\leq h^0({\cal G})$ for any submodule ${\cal G}\subseteq{\cal E}$. Hence sectional (semi)stability implies (semi)stability at once. Conversely, assume that $({\cal E},\alpha)$ is a (semi)stable pair. If for a submodule ${\cal G}$ we have $h^1({\cal G})=0$, then $h^0({\cal G})=\chi({\cal G})$ and there is nothing to show. (This applies in particular when ${\rm rk}{\cal G}=0$). Hence assume $h^1({\cal G})\neq0$. As above this leads to a short exact sequence $$\ses{{\cal G}'}{{\cal G}}{{\cal K}_X(-C)}$$ with ${\rm rk}{\cal G}'={\rm rk}{\cal G}-1$ and some effective divisor $C$ on $X$, so that $h^0({\cal G}')\geq h^0({\cal G})-g$. By induction we may assume that $$h^0({\cal G}')\,(\leq)\,\frac{{\rm rk}{\cal G}-1}{{\rm rk}{\cal E}}(h^0({\cal E})-\delta)+\varepsilon\.\delta$$ with $\varepsilon=0$ if ${\cal G}\subseteq{\ke_{\alpha}}$ and $\varepsilon=1$ if ${\cal G}\,(\subseteq)\,{\cal E}$. Combining these inequalities we get $$h^0({\cal G})\,(\leq)\,\frac{{\rm rk}{\cal G}}{{\rm rk}{\cal E}}(h^0({\cal E})-\delta)+ \varepsilon\.\delta+(g-\frac{h^0({\cal E})}{{\rm rk}{\cal E}}+\frac{\delta}{{\rm rk}{\cal E}}).$$ Since $h^0({\cal E})=\chi({\cal E})=\deg{\cal E}+(1-g){\rm rk}{\cal E}>g\.{\rm rk}{\cal E}+\delta$, we are done.\hspace*{\fill}\hbox{$\Box$} \begin{corollary} Suppose $X$ is a curve. The set of isomorphism classes of ${\cal O}_X$-modules occuring in semistable pairs is bounded.\hspace*{\fill}\hbox{$\Box$} \end{corollary} Before we pass on to surfaces recall the following criterion due to Kleiman which we will use several times: \begin{theorem}[Boundedness criterion of Kleiman]\label{boundcrit} Suppose $\chi$ is a polynomial\\ and $K$ an integer. If $T$ is a set of ${\cal O}_X$-modules ${\cal F}$ such that $\chi_{{\cal F}}=\chi$ and $$h^0(X,{\cal F}|_{H_1\cap\ldots\cap H_i})\leq K\qquad\forall\,i=0,\ldots,e,$$ for a ${\cal F}$-regular sequence of hyperplane sections $H_1,\ldots,H_e$, then $T$ is bounded.\end{theorem} {\it Proof:} \cite[Thm 1.13]{k1}\hspace*{\fill}\hbox{$\Box$} We introduce the following notation: For integers $\rho$ and $\varepsilon$ let $P(\rho,\varepsilon)$ be the polynomial $$P(\rho,\varepsilon,z):=\frac{\rho}{r}(\chi(z)-\delta(z))+\varepsilon\.\delta(z).$$ If ${\cal G}\subseteq{\cal E}$ is a submodule, let $\varepsilon({\cal G})=0$ or $1$ depending on wether ${\cal G}\subseteq{\ke_{\alpha}}$ or not. Then the stability conditions can be conveniently reformulated: \begin{itemize} \item[-] $({\cal E},\alpha)$ is (semi)stable if and only if $\chi_{{\cal G}}\,(\leq)\,P({\rm rk}{\cal G},\varepsilon({\cal G}))$ for all nontrivial submodules ${\cal G}\,(\subseteq)\,{\cal E}$. \item[-] $({\cal E},\alpha)$ is $\mu$-(semi)stable if and only if $\Delta\chi_{{\cal G}}\,(\leq)\,\Delta P({\rm rk}{\cal G},\varepsilon({\cal G}))$ for all nontrivial submodules ${\cal G}\subseteq{\cal E}$ with ${\rm rk}{\cal G}<{\rm rk}{\cal E}$. \item[-] $({\cal E},\alpha)$ is sectional (semi)stable if and only if ${\rm T}({\ke_{\alpha}})=0$ and there is a subspace $V\subseteq H^0({\cal E})$ of dimension $\chi({\cal E})$ such that $\dim(V\cap H^0({\cal G}))\,(\leq)\,P({\rm rk}{\cal G},\varepsilon({\cal G}),0)$ for all nontrivial submodules ${\cal G}\,(\subseteq)\,{\cal E}$. \end{itemize} \begin{lemma}\label{n0} Suppose $X$ is a surface. There is an integer $n_0<0$, depending on $X$, ${\cal O}_X(1)$ and $P$ only, such that $\Delta\chi_{{\cal O}_X(-n_0)}>\Delta P(1,\varepsilon)$ for $\varepsilon=0,1$. \end{lemma} {\it Proof:} As polynomials in $\nu$ the expressions $\Delta\chi_{{\cal O}_X}(\nu-n)$ and $\Delta P(1,\varepsilon,\nu)$ are both linear and have the same positive leading coefficient. Hence for very negative numbers $n$ one has $\Delta\chi_{{\cal O}_X}(\nu-n)>\Delta P(1,\varepsilon,\nu)$.\hspace*{\fill}\hbox{$\Box$} The following technical lemma is an adaptation of \cite[Lemma 1.2]{g2}. Unfortunately, we cannot apply Gieseker's lemma directly because it treats torsion free modules only, even though the necessary modifications are minor. \begin{lemma}\label{techlem} Suppose $X$ is a surface. Let $Q$ be a positive integer. Then there are integers $N$ and $M$, depending on $X,{\cal O}_X(1),P$ and $Q$, such that if $\varepsilon\in\{0,1\}$ and if ${\cal F}$ is an ${\cal O}_X$-module of rank $r'\leq r$ with the properties $h^0({\rm T}({\cal F}))\leq Q$ and $\Delta\chi_{{\cal G}}\leq\Delta P({\rm rk}{\cal G},\varepsilon)$ for all nontrivial submodules ${\cal G}\subseteq{\cal F}$, then either \begin{itemize}\item[] $h^0({\cal F}(n))<P(r',\varepsilon,n)$ for all $n\geq N$, \end{itemize} or the following assertions hold: \begin{itemize} \item[(1)] $\Delta\chi_{{\cal F}}=\Delta P(r',\varepsilon)$, \item[(2)] $h^2({\cal F}(n))=0$ for all $n\geq N$, \item[(3)] $h^0({\cal F}(n_0)|_H)\leq M$ for some ${\cal F}$-regular hyperplane section $H$, \item[(4)] if $h^1({\cal F}(n_0))\leq Q$, then $h^1({\cal F}(n))=0$ for all $n\geq N$. \end{itemize}\end{lemma} {\it Proof:} Let ${\cal F}$ be an ${\cal O}_X$-module satisfying the assumptions of the lemma. For every integer $n$ let ${\cal H}_n'$ denote the image of the evaluation map $H^0({\cal F}(n))\otimes{\cal O}_X\rightarrow{\cal F}(n)$ and ${\cal S}_n'$ the quotient ${\cal F}(n)/{\cal H}'_n$. Let ${\cal H}_n$ be the kernel of the epimorphism $${\cal F}(n)\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow ({\cal S}_n'/{\rm T}({\cal S}_n'))=:{\cal S}_n.$$ Then ${\cal H}_n$ is characterized by the following properties: $H^0({\cal H}_n)=H^0({\cal F}(n))$, ${\cal F}(n)/{\cal H}_n$ is torsion free and ${\cal H}_n$ is minimal with these two properties. Obviously ${\cal H}'_{n-1}(1)\subseteq{\cal H}'_n$ and therefore also ${\cal H}_{n-1}(1)\subseteq{\cal H}_n$. Moreover, being a submodule of the torsion free module ${\cal F}(n-1)/{\cal H}_{n-1}$ the quotient ${\cal H}_n(-1)/{\cal H}_{n-1}$ is itself torsion free. In particular either ${\cal H}_{n-1}={\cal H}_n(-1)$ or ${\rm rk}{\cal H}_{n-1}<{\rm rk}{\cal H}_n$. Let $n_1<\ldots<n_k$ be the indices with ${\rm rk}{\cal H}_{n_i-1}<{\rm rk}{\cal H}_{n_i}$. (If ${\cal F}$ is torsion, then ${\cal H}_n={\cal F}(n)$ for all $n$. Let $k=0$ in this case). By Serre's Theorem ${\cal H}_{n_k}={\cal F}(n_k)$ and $k\leq r'$. Let $s\in H^0{\cal F}(n)$ be a nonzero section. Then either $s$ is a torsion element or induces an injection ${\cal O}_X(-n)\rightarrow{\cal F}$. In the latter case one has $\Delta\chi_{{\cal O}_X(-n)}\leq\Delta P(1,\varepsilon)$. This is impossible for $n\leq n_0$. It follows that $$h^0({\cal F}(n_0))=h^0({\rm T}({\cal F})(n_0))\leq h^0({\rm T}({\cal F}))\leq Q$$ and that ${\cal H}_{n_0}={\rm T}({\cal F})(n_0)$. In particular $n_0<n_1$ if $r'>0$. A generic hyperplane section $H\in|{\cal O}_X(1)|$ has the following properties:\begin{itemize} \item[a)] $H$ is a smooth curve (of genus $g=1+\deg{\cal K}_X/2)$. \item[b)] $H$ is ${\cal H}_n$-regular for all integers $n$. \item[c)] ${\cal H}_n|_H$ is globally generated at the generic point of $H$ for all integers $n$. \end{itemize} (a) is just Bertini's Theorem. For (b) it is enough to consider the sheaves ${\cal H}_{n_i}$, $i=0,\ldots,k$. $H$ must not contain any of the finitely many associated points of the modules ${\cal H}_{n_i}$ in the scheme $X$. But this is an open condition. ${\cal H}_n$ is globally generated outside the support of ${\rm T}({\cal S}_n)$, so for (c) it is sufficient that in addition $H$ should not contain any of the associated points of the ${\rm T}({\cal S}_{n_i})$. Hence for a generic hyperplane section $H$ there are short exact sequences $$\ses{{\cal H}_n(-1)}{{\cal H}_n}{{\cal H}_n|_H},$$ $$\ses{{\cal O}_H^{r_n}}{{\cal H}_n|_H}{Q_n},$$ where $r_n={\rm rk}{\cal H}_n$ and $Q_n$ is an ${\cal O}_H$-torsion module. From the second sequence one deduces estimates $$h^1({\cal H}_n|_H)\leq r_n\.g\qquad{\rm and}\qquad h^1({\cal H}_n(\ell)|_H)=0,$$ if $\deg(K_H-\ell H)<0$, i.\ e.\ if $\ell>(2g-2)/H^2$. In particular we get for all integers $n$ with $n_i+(2g-2)/H^2<n<n_{i+1}$: $$h^1({\cal H}_n|_H)=h^1({\cal H}_{n_i}(n-n_i)|_H)=0.$$ This leads to the inequalities $$\begin{array}{rcl} h^0({\cal F}(n))-h^0({\cal F}(n-1))&=&h^0({\cal H}_n)-h^0({\cal H}_n(-1))\\ &\leq&h^0({\cal H}_n|_H)=\chi({\cal H}_n|_H)+h^1({\cal H}_n|_H) \end{array}$$ and, summing up, $$ h^0({\cal F}(n))-h^0({\cal F}(n_0)\leq \sum_{\nu=n_0+1}^n\chi({\cal H}_{\nu}|_H)+\sum_{\rho=1}^{r_n} \rho g((2g-2)/H^2+1).$$ Let $K:=Q+{r'+1\choose2}g((2g-2)/H^2+1)$. Then $$h^0({\cal F}(n))\leq K+\sum_{\nu=n_0+1}^n\chi({\cal H}_{\nu}|_H)$$ for all integers $n\geq n_0$. Suppose $n_0\leq\nu<n_k$. Then $r_{\nu}<r'$. Since ${\cal H}_{\nu}(-\nu)$ is a submodule of ${\cal F}$, $$ \chi({\cal H}_{\nu}|_H)=\Delta\chi_{{\cal H}_{\nu}(-\nu)}(\nu) \leq\Delta P(r_{\nu},\varepsilon,\nu)$$ Now \begin{eqnarray*} \Delta P(r_{\nu},\varepsilon,\nu)-\Delta P(r',\varepsilon,\nu)&=&(r_{\nu}-r')\.(\deg X\.\nu+d/r+(1-g)-\delta_1)\\ &\leq&-(\deg X\.\nu+C), \end{eqnarray*} where $C$ is a constant depending on $r$, $d$, $n_0$, $\deg X$ and $g$. For $\nu\geq n_k$ one has ${\cal H}_{\nu}={\cal F}(\nu)$ so that $\chi({\cal H}_{\nu}|_H)=\Delta\chi_{{\cal F}}(\nu)$. Let $m(n)=\min\{n,n_k-1\}$. Then the following inequality holds for all $n\geq n_0$: \begin{eqnarray*} h^0({\cal F}(n))-\sum_{\nu=n_0+1}^{n}\Delta P(r',\varepsilon,\nu) &\leq& K-\sum_{\nu=n_0+1}^{m(n)}\{\deg X\.\nu+C\}\\ &&-\sum_{\nu=m(n)+1}^{n}\{\Delta P(r',\varepsilon,\nu)-\Delta\chi_{{\cal F}}(\nu)\}. \end{eqnarray*} Note that the summands of the second sum of the right hand side are all equal to some nonnegative constant $C'$, (and that by convention the sum is 0 if $n<n_k$). Let $f$ be the polynomial $$f(z):=\deg X({z+1\choose 2}-{n_0+1\choose 2})+C\.(z-n_0)-K-P(r',\varepsilon,n_0).$$ Then for $n\geq n_0$: $$h^0({\cal F}(n))-P(r',\varepsilon,n)\leq-f(m(n))-C'\.(n-m(n))$$ There is an integer $N_1>n_0$ such that $f(\nu)>0$ for all $\nu\geq N_1$. Assume $N>N_1$. If $n_k-1\geq N_1$ then for all $n\geq N$ one has $m(n)\geq N_1$, hence $f(m(n))>0$ and $h^0({\cal F}(n))<P(r',\varepsilon,n)$. Hence we can restrict to the case that $n_k$ is uniformly bounded by $N_1$. Let $G:=\max\{-f(n)|n_0\leq n\leq N_1\}$. Suppose $C'>0$. There are positive integers $T,T'$ with $T'$ depending on $X, P$ and $r$ only, such that $C'=T/T'$. Choose an integer $N_2>\max\{N_1,G\.T'+N_1\}$. Assume $N>N_2$. Then for all $n\geq N$ $$h^0({\cal F}(n))-P(r',\varepsilon,n)\leq-f(n_k-1)-(n+1-n_k)\.C'\leq G-(N_2-N_1)\.C'<0.$$ Again we can restrict to the case $C'=0$. But this gives (1). Let $N_3=\lceil N_2+(2g-2)/H^2+1\rceil$ and assume $N>N_3$. Then for all $n\geq N$, one has $n>n_k+(2g-2)/H^2$ so that $h^1({\cal H}_n|_H)=h^1({\cal F}(n)|_H)=0$. In particular $$H^2({\cal F}(n))=H^2({\cal F}(n+1))=H^2({\cal F}(n+2))=\ldots,$$ and these cohomology groups must vanish for $n\gg 0$, hence already for $n\geq N$. This is assertion (2). Moreover, $$h^0({\cal F}(n_0)|_H)\leq h^0({\cal F}(N_3)|_H)=\chi({\cal F}(N_3)|_H)=\Delta\chi_{{\cal F}}(N_3)= \Delta P(r',\varepsilon,N_3)$$ according to (1). Let $M:=\max\{\lceil \Delta P(r',\varepsilon,N_3)\rceil|0\leq r'\leq r\}$. Then (3) holds. It remains to prove (4). Since ${\cal F}(N_3)={\cal H}_{N_3}$, there are short exact sequences $$\ses{{\cal O}_H(\nu-N_3)^{\oplus r'}}{{\cal F}(\nu)|_H}{Q_{N_3}}$$ for all $\nu=n_0,\ldots,N_3$. Hence $$h^1({\cal F}(\nu)|_H)\leq r'\.h^1({\cal O}_H(\nu-N_3))\leq r'\.h^1({\cal O}_H(n_0-N_3))$$ and $$h^2({\cal F}(n_0))\leq h^2({\cal F}(N_3))+\sum_{\nu=n_0+1}^{N_3}h^1({\cal F}(\nu)|_H)\leq r(N_3-n_0)\.h^1({\cal O}_X(n_0-N_3))$$ is uniformly bounded. Since by assumption $h^1({\cal F}(n_0))\leq Q$ and $h^0({\cal F}(n_0))\leq Q$, the Euler characteristic $\chi({\cal F}(n_0))$ lies in a finite set of integers. By (1) $\Delta\chi_{{\cal F}}$ is given. Hence $\chi_{{\cal F}}$ lies in a finite set of polynomials. Using (3) and criterion \ref{boundcrit} we conclude that the set of modules ${\cal F}$ we are left with is bounded. Therefore there is a constant $N_4>N_3$ such that $h^1({\cal F}(n))=0$ if $n\geq N_4$. The lemma holds, if we choose any $N>N_4$.\hspace*{\fill}\hbox{$\Box$} An immediate consequence of this lemma is the following boundedness result: \begin{corollary}\label{bound} Suppose $X$ is a surface. The set of isomorphism classes of ${\cal O}_X$-modules ${\cal E}$ which occur in $\mu$-semistable pairs $({\cal E},\alpha)$ with ${\rm T}({\ke_{\alpha}})=0$ is bounded. \end{corollary} {\it Proof:} Apply lemma \ref{techlem} with $Q=h^0({\cal E}_0)$. The proof of the lemma shows that $h^0({\cal E}(n_0))\leq Q$. By Serre's theorem $h^0({\cal E}(n))=\chi_{{\cal E}}(n)=P(r,1,n))$ for all large enough numbers $n$, so the second alternative of the lemma holds. Part (3) then states: $h^0({\cal E}(n_0)|_H)\leq M$ for some ${\cal E}$-regular hyperplane section $H$ and some constant $M$ which is independent of ${\cal E}$. Therefore the Kleiman criterion applies to the set of modules ${\cal E}(n_0)$ with the constant $K:=\max\{Q,M,r\.\deg X\}$.\hspace*{\fill}\hbox{$\Box$} As a consequence of the corollary there is an integer $\hat N$ such that ${\cal E}(n)$ is globally generated and $h^i({\cal E}(n))=0$ for all $i>0$, $n\geq\hat N$ and for all ${\cal O}_X$-modules ${\cal E}$ satisfying the hypotheses of the corollary. Note that according to lemma $\ref{firstobs}$ among these all the modules occuring in semistable pairs can be found. After these preparations we can prove the equivalent to theorem \ref{boundcurve} in the surface case: \begin{theorem}\label{boundsurf} Suppose $X$ is a surface. There is an integer $N$ depending on $X$, ${\cal O}_X(1)$, $h^0({\cal E}_0)$ and $P$, such that \begin{itemize} \item[i)] if $({\cal E},\alpha)$ is (semi)stable (with respect to $\delta$) then $({\cal E}(n),\alpha(n))$ is sectional (semi)\-stable (with respect to $\delta(n)$) for all $n\geq N$, and \item[ii)] if $({\cal E}(n),\alpha(n))$ is sectional (semi)stable for some $n\geq N$, then $({\cal E},\alpha)$ is (semi)stable. \end{itemize} \end{theorem} {\it Proof:} By the boundedness result \ref{bound} the dimension of $H^1({\cal E}(n_0))$ is uniformly bounded for all ${\cal E}$ satisfying the hypotheses of the corollary. Let $Q:=h^0({\cal E}_0)+\max\{h^1({\cal E}(n_0))\}$. Let $N$ be the number obtained by applying lemma \ref{techlem}. Without loss of generality $N>\hat N$. ad i): Suppose $({\cal E},\alpha)$ is (semi)stable. Apply lemma \ref{techlem} to ${\cal E}$. Since by Serre's theorem $h^0({\cal E}(n))=\chi({\cal E}(n))=P(r,1,n)$ for all sufficiently large $n$, the second alternative of the lemma holds and shows $h^1({\cal E}(n))=h^2({\cal E}(n))=0$ for $n\geq N$. Hence $V:=H^0({\cal E}(n))$ has dimension $\chi(n)$. Now let ${\cal F}$ be a submodule of ${\cal E}$. Then either $h^0({\cal F}(n))<P({\rm rk}{\cal F},\varepsilon({\cal F}),n)$ for all $n\geq N$, in which case we are done, or we have $\Delta\chi_{{\cal F}}=\Delta({\rm rk}{\cal F},\varepsilon({\cal F}))$. Let ${\cal E}'={\ke_{\alpha}}$ if ${\cal F}\subseteq{\ke_{\alpha}}$ and ${\cal E}'={\cal E}$ else. Let ${\cal S}:={\cal E}'/{\cal F}$, $\bar{\cal S}:={\cal S}/{\rm T}({\cal S})$ and let $\bar{\cal F}$ be the kernel of the epimorphism ${\cal E}'\rightarrow\bar{\cal S}$. Then ${\rm rk}{\cal F}={\rm rk}\bar{\cal F}$, $\varepsilon({\cal F})=\varepsilon(\bar{\cal F})$, and we must have $\Delta\chi_{\bar{\cal F}}=\Delta P({\rm rk}{\cal F},\varepsilon({\cal F}))=\Delta\chi_{{\cal F}}$. Hence $\bar{\cal F}/{\cal F}={\rm T}({\cal S})$ has zero-dimensional support. There is a short exact sequence $$\ses{\bar{\cal F}(n_0)}{{\cal E}'(n_0)}{\bar{\cal S}(n_0)}.$$ Now $\bar{\cal S}(n_0)$ cannot have global sections. For otherwise there is a submodule in $\bar{\cal S}$ isomorphic to ${\cal O}_X(-n_0)$. Let ${\cal G}$ be its preimage in ${\cal E}'$. Then $$\Delta\chi_{{\cal G}}=\Delta\chi_{\bar{\cal F}}+\Delta\chi_{{\cal O}_X(-n_0)}\leq \Delta P({\rm rk}{\cal F}+1,\varepsilon({\cal F}))=\Delta P({\rm rk}{\cal F},\varepsilon({\cal F}))+\Delta P(1,0)$$ contradicting lemma \ref{n0}. But this shows that \begin{eqnarray*} h^1(\bar{\cal F}(n_0))\leq h^1({\cal E}'(n_0)) &\leq&h^1({\cal E}(n_0))+h^0({\cal E}(n_0)/{\cal E}'(n_0))\\ &\leq&h^1({\cal E}(n_0))+h^0({\cal E}_0(n_0))\leq Q. \end{eqnarray*} By part (4) of lemma \ref{techlem} we now conclude that $$h^0({\cal F}(n))\leq h^0(\bar{\cal F}(n))=\chi(\bar{\cal F}(n))\,(\leq)\,P({\rm rk}{\cal F},\varepsilon({\cal F}))$$ for all $n\geq N$ if $\bar{\cal F}\,(\subseteq)\,{\cal E}$. Only the case $\bar{\cal F}={\cal E}$ for stable pairs needs special attention: In this case one has $h^0({\cal F}(n))<h^0({\cal E}(n))$, because ${\cal F}$ is a proper submodule of ${\cal E}$ and ${\cal E}(n)$ is globally generated for all $n\geq N$. Hence (semi)stability implies sectional (semi)stability for all $n\geq N$. ad ii) Suppose $({\cal E}(n),\alpha(n))$ is sectional (semi)stable for some $n\geq N$. Assume that there exists a submodule ${\cal F}\subseteq{\cal E}$ with $\Delta\chi_{{\cal F}}>\Delta P({\rm rk}{\cal F}, \varepsilon({\cal F}))$. If such a module exists at all, we may assume that it is maximal with this property among the submodules of ${\cal E}$. Let ${\cal S}={\cal E}/{\cal F}$. The maximality of ${\cal F}$ implies that ${\cal S}$ is torsion free if $\varepsilon({\cal F})=1$ and that $\alpha$ embeds ${\rm T}({\cal S})$ into ${\rm T}({\cal E}_0)$ if $\varepsilon({\cal F})=0$. Hence $h^0({\rm T}({\cal S}))\leq Q$. Suppose ${\cal G}$ is any submodule of ${\cal S}$. Let ${\cal F}'$ be the preimage of ${\cal G}$ under the map ${\cal E}\rightarrow {\cal S}$. Then $$\Delta\chi_{{\cal G}}+\Delta\chi_{{\cal F}}=\Delta\chi_{{\cal F}'}\leq\Delta P({\rm rk}{\cal F}+{\rm rk}{\cal G},1)=\Delta P({\rm rk}{\cal F},\varepsilon({\cal F}))+\Delta({\rm rk}{\cal G},1-\varepsilon({\cal F})).$$ The inequality in the middle of this line is infered from the maximality of ${\cal F}$. Hence $$\Delta\chi_{{\cal G}}\leq\Delta({\rm rk}{\cal G},1-\varepsilon({\cal F}))+ \{\Delta P({\rm rk}{\cal F},\varepsilon({\cal F}))-\Delta\chi_{{\cal F}}\} <\Delta P({\rm rk}{\cal G},1-\varepsilon({\cal F})).$$ Therefore we can apply lemma \ref{techlem} to the module ${\cal S}$ with $\varepsilon=1-\varepsilon({\cal F})$. But we did assume that $({\cal E}(n),\alpha(n))$ was sectional semistable. Hence there exists a vector space $V\subseteq H^0({\cal E}(n))$ of dimension $\chi(n)$ such that $$\dim(V\cap H^0({\cal F}(n)))\leq P({\rm rk}{\cal F},\varepsilon({\cal F}),n)$$ and $$h^0({\cal S}(n))\geq\dim V-\dim(V\cap H^0({\cal F}(n))) \geq\chi(n)-P({\rm rk}{\cal F},\varepsilon({\cal F}),n) =P({\rm rk}{\cal S},1-\varepsilon({\cal F}),n)$$ This excludes the first alternative of the lemma, and we get $\Delta\chi_{{\cal S}}=\Delta({\rm rk}{\cal S},1-\varepsilon({\cal F}))$ and equivalently $\Delta\chi_{{\cal F}}=\Delta({\rm rk}{\cal F},\varepsilon({\cal F}))$, which contradicts the original assumption. Thus we have proven that $\Delta\chi_{{\cal F}}\leq\Delta P({\rm rk}{\cal F},\varepsilon({\cal F}))$. But this means that ${\cal E}$ satisfies the hypotheses of corollary \ref{bound}. By the remark following the corollary we have $h^0({\cal E}(\nu))=\chi({\cal E}(\nu))$ for all $\nu\geq N$ since $N\geq\hat N$, so that necessarily $V=H^0({\cal E}(n))$. Applying lemma \ref{techlem} to ${\cal F}$ we see that either \begin{itemize}\item[] $h^0({\cal F}(\nu))<P({\rm rk}{\cal F},\varepsilon({\cal F}),\nu)$ for all $\nu\geq N$, in particular $\chi_{{\cal F}}<P({\rm rk}{\cal F},\varepsilon({\cal F}))$,\end{itemize} or \begin{itemize}\item[] $h^2({\cal F}(\nu))=0$ for all $\nu\geq N$ and hence $$\chi_{{\cal F}}(n)=h^0({\cal F}(n))-h^1({\cal F}(n))\leq h^0({\cal F}(n))\,(\leq)\,P({\rm rk}{\cal F},\varepsilon({\cal F}),n),$$ which together with $\Delta\chi_{{\cal F}}=\Delta P({\rm rk}{\cal F},\varepsilon({\cal F}))$ implies $\chi_{{\cal F}}\,(\leq)\,P({\rm rk}{\cal F},\varepsilon({\cal F}))$. \end{itemize} This finishes the proof.\hspace*{\fill}\hbox{$\Box$} \subsection{The basic construction}\label{Kapitel13} Let $X$ be a curve or a surface. By the results of the previous section the set of modules ${\cal E}$ with fixed Hilbert polynomial $\chi$ that occur in semistable pairs is bounded. In particular, there is a projective open and closed part $A$ of the Picard scheme ${\rm Pic }(X)$ such that $[\det{\cal E}]\in A$ for all ${\cal E}$ in semistable pairs. Let ${\cal L}\in{\rm Pic }(A\times X)$ be a universal line bundle. Then there is an integer $N$ such that for all $n\geq N$ the following conditions are simultaneously satisfied: \begin{itemize} \item[-] $0<\delta(n)<\chi(n)$. \item[-] ${\cal E}$ is globally generated and $h^i({\cal E}(n))=0$ for all $i>0$ and for all ${\cal E}$ in semistable pairs. \item[-] $({\cal E},\alpha)$ is (semi)stable (with respect to $\delta$) if and only if $({\cal E}(n),\alpha(n))$ is sectional (semi)\-stable (with respect to $\delta(n)$). \item[-] If $p_A,p_X$ denote the projection maps from $A\times X$ to $A$ and $X$, respectively, then $R^ip_{A*}({\cal L}\otimes p_X^*{\cal O}_X(n))=0$ for all $i>0$, ${\cal U}_n:=p_{A*}({\cal L}\otimes p_X^*{\cal O}_X(n))$ is locally free and $p_A^*({\cal U}_n)\otimes p_X^*{\cal O}_X(-n)\rightarrow{\cal L}$ is surjective. \end{itemize} By twisting the pairs $({\cal E},\alpha)$ with ${\cal O}_X(n)$ for sufficiently large $n$ we can always assume that the assertions above hold for $N=0$. We make this assumption for the rest of this section and write $p:=\chi(0)$ and $\bar\delta:=\delta(0)$. Let $V$ be a vector space of dimension $p$ and let $V_X=V\otimes_k {\cal O}_X$. Quotient modules of $V_X$ with Hilbert polynomial $\chi$ are parametrized by a projective scheme ${\rm Quot }^{\chi}_{X/V_X}$ (\cite[3.1.]{Gr}). On the product ${\rm Quot }^{\chi}_{X/V_X}\times X$ there is a universal quotient ${\tilde q}:p_X^*V_X\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\tilde\ke}$. Forming the determinant bundle of ${\tilde\ke}$ induces a morphism $$det:{\rm Quot }_{X/V_X}^{\chi}\longrightarrow{\rm Pic }(X)$$ so that $\det{\tilde\ke}=(det\times{\rm id }_X)^*({\cal L})\otimes p_{Quot}^*({\cal M})$ for some line bundle ${\cal M}\in{\rm Pic }({\rm Quot }^{\chi}_{X/X_V})$. Let $Q$ denote the preimage of $A$ under the map $det$. We use the same symbols for the universal quotient and its restriction to $Q\times X$. Further let $P:=\IP({\rm Hom }(V,H^0({\cal E}_0))^{\rm v})$. Again there is a universal homomorphism ${\tilde a}:(V\otimes_kH^0({\cal E}_0)^{\rm v})\otimes{\cal O}_P\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow {\cal O}_P(1)$. For sufficiently high $n$ the direct image sheaf ${\cal H}:=p_{Q*}({\rm Ker }\,{\tilde q}\otimes p_X^*{\cal O}_X(n))$ is locally free and the canonical homomorphism $$\beta:p_Q^*{\cal H}\rightarrow{\rm Ker }\,{\tilde q}\otimes p_X^*{\cal O}_X(n)$$ is surjective, so that there is an exact sequence $$p_Q^*{\cal H}\otimes p_X^*{\cal O}_X(-n)\rpfeil{10}{\tilde\beta}p_X^*V_X\rpfeil{10}{{\tilde q}}{\tilde\ke} \rpfeil{10}{}0.$$ $\tilde\beta$ induces a homomorphism of ${\cal O}_Q$-modules $$\gamma:{\cal H}\otimes_kH^0({\cal E}_0(n))^{\rm v}\rightarrow{\cal O}_Q\otimes_k(V\otimes_k H^0({\cal E}_0)^{\rm v}).$$ Let ${\cal I}$ be the ideal in the symmetric algebra ${\cal S}^*(V\otimes_kH^0({\cal E}_0)^{\rm v})\otimes_k{\cal O}_Q$ which is generated by the image of $\gamma$ and let $B\subset P\times Q$ be the corresponding closed subscheme. Let $\pi_P:B\rightarrow P$ and $\pi_Q:B\rightarrow Q$ be the projection maps and let ${\cal O}_B(1):=\pi_P^*{\cal O}_P(1)$. This scheme $B$ is the starting point for the construction of the moduli space for semistable pairs. We introduce the following notations: Let $$q_B:=(\pi_Q\times{\rm id }_X)^*{\tilde q}:V\otimes{\cal O}_{B\times X}\rightarrow{\cal E}_B:=(\pi_Q\times{\rm id }_X)^*{\tilde\ke}$$ and $$a_B:=\pi_P^*{\tilde a}:(V\otimes H^0({\cal E}_0)^{\rm v})\otimes{\cal O}_B\rightarrow{\cal O}_B(1).$$ By definition of $B$ an arbitrary morphism $h:T\rightarrow P\times Q$ factors through the closed immersion $B\rightarrow P\times Q$ if and only if the pull-back under $h$ of the composition $p_P^*{\tilde a}\circ p_Q^*\gamma$ is the zero map. This is equivalent to saying that the pull-back under $h\times{\rm id }_X$ of the induced homomorphism $V\otimes{\cal O}_{P\times Q\times X}\rightarrow p_P^*{\cal O}_P(1)\otimes p_X^*{\cal E}_0$ factors through $V\times{\cal O}_{T\times X}\rightarrow(h\times{\rm id }_X)^*{\tilde\ke}$. This applies in particular to $B$ itself. Let $\alpha_B:{\cal E}_B\rightarrow p_B^*{\cal O}_B(1)\otimes p_X^*{\cal E}_0$ be the induced homomorphism. \begin{lemma} (i) There is an open subscheme $Q^0$ of $Q$ such that $u$ is a point in $Q^0$ if and only if $h^i({\tilde\ke}_u)=0$ for all $i>0$ and the homomorphism $\tilde q_u:V_X\otimes k(u)\rightarrow{\tilde\ke}_u$ induces an isomorphism on the spaces of global sections. (ii) Let $({\cal E},\alpha)$ be a flat family of pairs parametrized by a noetherian $k$-scheme $T$. Then there is an open subscheme $S\subseteq T$ such that ${\rm Ker }(\alpha_t)$ is torsionfree for a geometric point $t$ of $T$ if and only if $t$ is a point of $S$. \end{lemma} {\it Proof:} (i) By semicontinuity of $h^i$ there is an open subscheme of $Q$ of points $u$ for which the higher cohomology groups of ${\tilde\ke}_u$ vanish. For those points $h^0({\tilde\ke}_u)=p$ and hence $H^0(q_u)$ is an isomorphism if and only if $h^0({\rm Ker }\,q_u)=0$, which again is an open condition for $u$. (ii) For $n$ large enough there is a locally free ${\cal O}_T$-module ${\cal G}$ and a surjection\\${\cal G}\otimes{\cal O}_X(-n)\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal E}^{\rm v}$ and dually an inclusion $\beta':{\cal E}^{\rm vv}\rightarrow{\cal G}^{\rm v}\otimes{\cal O}_X(n)$. Note that there is an open subscheme $O$ of $T\times X$ which meets every fibre $X_t$ and for which the restriction ${\cal E}|_O$ is locally free, so that in particular $\vartheta:{\cal E}\rightarrow {\cal E}^{\rm vv}$ is an isomorphism when restricted to $O$. If we let $\beta=\beta'\circ\vartheta$, then the kernel of $\beta_t:{\cal E}_t\rightarrow {\cal G}^{\rm v}(t)\otimes{\cal O}_X(n)$ is precisely the torsion part of ${\cal E}_t$. Hence the kernel of $\gamma_t:=(\alpha_t,\beta_t):{\cal E}_t\rightarrow{\cal E}_0\oplus({\cal G} ^{\rm v}(t)\otimes{\cal O}_X(n))$ is the torsion submodule of ${\rm Ker }(\alpha_t)$. It is enough to show that the points $t$ with ${\rm Ker }(\gamma_t)=0$ form an open set. But this is \cite[Cor IV 11.1.2]{EGA}.\hspace*{\fill}\hbox{$\Box$} Let $S$ be the open subscheme of $B$ which according to the lemma belongs to the family $({\cal E}_B,\alpha_B)$, and let $B^0=S\cap(P\times Q^0)$. The algebraic group ${\rm SL}(V)$ acts naturally on $Q$ and $P$ from the right. On closed points $[q:V\otimes{\cal O}_X\rightarrow{\cal E}]$ and $[a:V\rightarrow H^0({\cal E}_0)]$ this action is given by $[q]\cdot g=[q\circ(g\otimes{\rm id }_{{\cal O}_X})]$ and $[a]\cdot g=[a\circ g]$. \begin{lemma} $B^0$ is invariant under the diagonal action of ${\rm SL}(V)$ on $P\times Q$.\end{lemma} {\it Proof:} This is clear from the characterization of $B$ as the subscheme of points $([q],[a])$ for which there is a commuting diagram $$\begin{array}{ccc} V\otimes{\cal O}_X&\rpfeil{15}{q}&{\cal E}\\ \spfeil{12}{a}&&\spfeil{12}{\alpha}\\ H^0({\cal E}_0)\otimes{\cal O}_X&\rpfeil{15}{ev}&{\cal E}_0. \end{array}$$\hspace*{\fill}\hbox{$\Box$} $B^0$ has the following local universal property: \begin{lemma}\label{locunivprop} Suppose $T$ is a noetherian $k$-scheme parametrizing a flat family $({\cal E},\alpha)$ of semistable pairs on $X$. Then there is an open covering $T=\bigcup T_i$ and for each $T_i$ a morphism $h_i:T_i\rightarrow B^0$ and a nowhere vanishing section $s_i$ in $h_i^*{\cal O}_B(1)$ such that the pair $({\cal E},\alpha)|_{T_i}$ is isomorphic to the pair $((f_i\times{\rm id }_X)^*{\cal E}_B, (f_i\times{\rm id }_X)^*(\alpha_B)/s_i)$. \end{lemma} {\it Proof:} Let $T$ be a noetherian scheme and $({\cal E},\alpha)$ a flat family of semistable pairs on $X$ parametrized by $T$. According to the remarks in the first paragraph of this section the direct image sheaf $p_{T*}{\cal E}$ is locally free of rank $p$ \cite[Thm 12.8]{h1}. Hence locally on $T$ there are trivializations $V\otimes{\cal O}_T\rightarrow p_{T*}{\cal E}$, which lead to quotient maps $q:V\otimes{\cal O}_{T\times X}\rightarrow {\cal E}$. By the universal property of $Q$ there is a $k$-morphism $f:T\rightarrow Q$ and a uniquely determined isomorphism $\Phi:(f\times{\rm id }_X)^*{\tilde\ke}\rightarrow{\cal E}$ such that $\Phi\circ(f\times{\rm id }_X)^*{\tilde q}=q$. Moreover, the composition $$V\otimes{\cal O}_{T\times X}\rpfeil{10}{q}{\cal E}\rpfeil{10}{\alpha}p_X^*{\cal E}_0$$ determines a homomorphism $a:V\otimes{\cal O}_T\rightarrow H^0({\cal E}_0)\otimes{\cal O}_T$. By the universal property of $P$ there is a morphism $g:T\rightarrow P$ and a uniquely determined nowhere vanishing section $s$ in $g^*{\cal O}_P(1)$ such that $a=g^*{\tilde a}/s$. It is clear from the construction that $h:=(f,g):T\rightarrow P\times Q$ factors through $B^0$. $\Phi^{-1}$ is an isomorphism from ${\cal E}$ to $(f\times{\rm id }_X)^*{\tilde\ke}=(h\times{\rm id }_X)^*{\cal E}_B$, and $\alpha\circ\Phi=(h\times{\rm id }_X)^*(\alpha_B)/s$.\hspace*{\fill}\hbox{$\Box$} If $h:T\rightarrow B$ and $g:T\rightarrow{\rm SL}(V)$ are morphisms let $h\cdot g$ denote the composition $T\rpfeil{12}{(h,g)}B\times{\rm SL}(V)\rightarrow B$, where the last map is the induced group action of ${\rm SL}(V)$ on $B$. \begin{lemma}\label{induce} Suppose $T$ is a noetherian $k$-scheme and $h=(f,g):T\rightarrow B^0\subset P\times Q$ a $k$-morphism. $h$ induces (locally) isomorphism classes of families of pairs. If $g:T\rightarrow{\rm SL}(V)$ is a morphism, then the families induced by $h$ and $h\cdot g$ are isomorphic. Conversely, if $h_1$ and $h_2$ induce isomorphic families parametrized by $T$, then there is an etale morphism $c:T'\rightarrow T$ and a morphism $g':T'\rightarrow{\rm SL}(V)$ such that the morphisms $(h_1\circ c)\cdot g'$ and $(h_2\circ c)$ are equal. \end{lemma} {\it Proof:} Let $h:T\rightarrow B^0$ be a $k$-morphism. Applying $(h\times{\rm id }_X)^*$ to ${\cal E}_B$ and $\alpha_B$ induces a family ${\cal E}_T$ and a homomorphism $\alpha_T:{\cal E}_T\rightarrow h^*({\cal O}_B(1))\otimes p_X^*{\cal E}_0$. Locally there are nowhere vanishing sections in $h^*{\cal O}_B(1)$. Dividing $\alpha_T$ by any of these sections defines families of pairs. Two such sections differ by a section in ${\cal O}_T^*$. But this sheaf embeds into the sheaf of automorphisms of ${\cal E}_T$. Hence the families induced by different sections are isomorphic. The second statement is clear. For the third assume that $h_1$ and $h_2$ are morphisms such that for $i=1,2$ there are nowhere vanishing sections $s_i\in H^0(T,h_i^* {\cal O}_B(1))$. Let $${\cal E}_i:=(h_i\times{\rm id }_X)^*{\cal E}_B, \quad q_i:=(h_i\times{\rm id }_X)^*q_B\quad\hbox{and}\quad \alpha_i:=(h_i\times{\rm id }_X)^*\alpha_B/s_i.$$ Assume that there is an isomorphism $\Phi:({\cal E}_1,\alpha_1)\rightarrow({\cal E}_2,\alpha_2)$ of pairs. The quotient maps $q_i$ induce isomorphisms $\bar q_i:V\otimes{\cal O}_T\rightarrow p_{T*}{\cal E}_i$ because of the definition of $B^0$ (\cite[Thm 12.11]{h1}). The composition $\bar q_2^{-1}\circ p_{T*}\Phi\circ \bar q_1$ corresponds to a morphism $g:T\rightarrow {\rm GL}(V)$. Define morphisms $c$ and $\ell$ by the fibre product diagram $$\begin{array}{ccc} T'&\rpfeil{20}{c}&T\\ \spfeil{}{\ell}&&\spfeil{}{\det(g)}\\ {\rm\bf G}_m&\rpfeil{20}{p^{th}\,\,power}&{\rm\bf G}_m \end{array}$$ and let $g':=(g\circ c)/\ell:T'\rightarrow{\rm SL}(V)$. It is easy to check that $(h_1\circ c)\.g'=(h_2\circ c)$. \hspace*{\fill}\hbox{$\Box$} ${\tilde q}$ induces a homomorphism $\Lambda^r({\cal O}_Q\otimes V_X)\rightarrow\det{\tilde\ke}=(det\times{\rm id })^*({\cal L})\otimes p_Q^*{\cal M}$ and hence a homomorphism $\Lambda^rV\otimes_k{\cal M}^{\rm v}\rightarrow det^*{\cal U}_0=det^*p_{A*}{\cal L}$ (\cite{Ma}). This finally leads to morphisms $T:Q\rightarrow P':=\IP({\cal H} om(\Lambda^rV,{\cal U}_0)^{\rm v})$ and $\tau:=(\pi_P,T):B\rightarrow P\times P'$. \begin{lemma} ${\rm SL}(V)$ acts naturally on $P'$ from the right, $T$ and $\tau$ are equivariant morphisms with respect to this action.\hspace*{\fill}\hbox{$\Box$} \end{lemma} We can choose a very ample line bundle ${\cal N}$ on $A$ such that ${\cal N}':={\cal O}_{P'}(1)\otimes p_A^*{\cal N}$ is very ample on $P'$. For any positive numbers $\nu,\nu'$ the line bundle ${\cal O}_P(\nu)\otimes({\cal N}')^{\otimes\nu'}$ is very ample on $P\times P'$ and inherits a canonical linearization with respect to the ${\rm SL}(V)$-action \cite[1.4,1.6]{m10}. Choose $\nu$ and $\nu'$such that $\nu/\nu'=r\bar\delta/(p-\bar\delta)$. Let $Z^{(s)s}\subseteq P\times P'$ be the open subscheme of (semi)stable points with respect to this linearization. Here \em stable \em means \em properly stable \em in the sense of Mumford. \begin{theorem}\label{mainresult} The open subscheme $B^{(s)s}=B^0\cap\tau^{-1}(Z^{(s)s})$ of $B$ has the following property: A morphism $h:T\rightarrow B^0$ induces families of (semi)stable pairs in the sense of lemma \ref{induce} if and only if $h$ factors through $B^{(s)s}$. The restriction of $\tau$ to $B^{ss}$ is a finite morphism $\tau^{ss}:B^{ss}\rightarrow Z^{ss}$. \end{theorem} For the proof we need a stability criterion for $\tau([a],[q])$, and we need it in slightly greater generality. But before this, note that if $q:V_X\rightarrow{\cal E}$ defines a point $[q]$ in $Q(k)$, then the fibre of the projective bundle $P'$ through the point $T([q])$ is isomorphic to $P'':=\IP(Hom(\Lambda^rV,H^0(\det{\cal E}))^{\rm v})$, and $\tau([a],[q])$ is a (semi)stable point in $P\times P'$ if and only if it is (semi)stable point in $P\times P''$ with respect to the canonical linearization of ${\cal O}_P(\nu)\otimes{\cal O}_{P''}(\nu')$ (\cite[4.12]{Ma}). In particular, the choice of ${\cal N}$ is of no consequence for the definition of $Z^{(s)s}$. \begin{proposition}\label{prop} Let $({\cal E},\alpha)$ be a pair with $\det{\cal E} \in A$ and torsionfree ${\ke_{\alpha}}$. Suppose there is a generically surjective homomorphism $q:V_X\rightarrow{\cal E}$ such that $q\circ\alpha\neq 0$. Let $T:\Lambda^rV\rightarrow H^0(\det{\cal E})$ and $a:V\rightarrow H^0({\cal E}_0)$ be the derived homomorphisms. Then ${([a],[T])}$ is a (semi)stable point in $P\times P''$ with respect to the given linearization if and only if $q$ injects $V$ into $H^0({\cal E})$ and $({\cal E},\alpha)$ is sectional (semi)stable with respect to $\bar\delta$. \end{proposition} The proof of this proposition is postponed to the next section. \em Proof of theorem \ref{mainresult}: \em Pairs $({\cal E},\alpha)$ that correspond to points $([a],[q])$ in $B^0$ satisfy the hypotheses of the proposition, for $q$ is surjective, $H^0q$ isomorphic and ${\ke_{\alpha}}$ torsionfree. Hence by proposition \ref{prop} and theorems \ref{boundcurve} and \ref{boundsurf} $({\cal E},\alpha)$ is (semi)stable if and only if $\tau([a],[q])$ is a (semi)stable point. This proves the first assertion of the theorem. In order to show that $\tau^{ss}:=\tau|_{B^{ss}}$ is a finite morphism it is enough to show that $\tau^{ss}$ is proper and injective (\cite[IV 8.11.1]{EGA}). This will be done in two steps: \begin{proposition}\label{properness} $\tau^{ss}$ is a proper morphism. \end{proposition} {\it Proof:} Using the valuation criterion it suffices to show the following: Let $\bar C=Spec\,R$ be a nonsingular affine curve, $c_0\in\bar C$ a closed point defined by a local parameter $t\in R$ and $C$ the open complement of $c_0$. Suppose we are given a commutative diagram $$\begin{array}{ccc}C&\rpfeil{15}{h}&B^{ss}\\ \spfeil{}{\iota}&&\spfeil{}{\tau^{ss}}\\ \bar C&\rpfeil{15}{m}&Z^{ss}. \end{array}$$ We must show that (at least locally near $c_0$) there is a lift $\bar h:\bar C\rightarrow B^{ss}$ of $m$ extending $\bar h$. Making $C$ smaller if necessary we may assume that $h$ induces homomorphisms $${\cal O}_C\otimes V_X\rpfeil{12}{q}{\cal F}\rpfeil{12}{\alpha}{\cal O}_C\otimes{\cal E}_0,$$ so that $({\cal F},\alpha)$ is a flat family of semistable pairs. Using Serre's theorem one can find a locally free ${\cal O}_X$-module ${\cal H}$ and an epimorphism ${\cal O}_C\otimes{\cal H}^{\rm v}\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal F}^{\rm v}$. The kernel of the dual homomorphism $\beta:{\cal F}\rightarrow{\cal O}_C\otimes{\cal H}$ is the torsion submodule ${\rm T}({\cal F})$. Since ${\rm Ker }\,\alpha$ and ${\rm Im}\,\alpha$ are $C$-flat, $({\rm Ker }\,\alpha)_c\subset ({\rm Ker }\,\alpha_c)$. Since the kernel of the restriction of $\alpha$ to any fibre $X\times c$, $c\in C$, is torsion free by lemma \ref{firstobs}, ${\rm Ker }\,\alpha$ is also torsion free. Therefore $$(\alpha,\beta):{\cal F}\rightarrow{\cal O}_C\otimes({\cal E}_0\oplus{\cal H})$$ is injective. There are integers $a,b$ such that the composition $${\cal O}_C\otimes V_X\rpfeil{15}{q}{\cal F}\rpfeil{20}{(t^a\alpha,t^b\beta)}{\cal O}_C\otimes ({\cal E}_0\oplus{\cal H})$$ extends to a homomorphism $$\lambda:{\cal O}_{\bar C}\otimes V_X\longrightarrow{\cal O}_{\bar C}\otimes({\cal E}_0\oplus {\cal H})$$ which is nontrivial in each component when restricted to the special fibre $X_{c_0}$. Let $\bar{\cal F}$ be the maximal submodule of ${\cal O}_{\bar C}\otimes({\cal E}_0\oplus{\cal H})$ with the properties \begin{center}$\bar{\cal F}|_{C\times X}={\cal F}$, ${\rm Im}\,\lambda\subseteq \bar{\cal F}$ and $\dim{\rm Supp}(\bar{\cal F}/{\rm Im}\,\lambda)<e$; \end{center} and let $\bar\alpha:\bar{\cal F}\rightarrow{\cal O}_{\bar C}\otimes{\cal E}_0$ be the projection map. Then $\bar{\cal F}$ is $\bar C$-flat, $(\bar{\cal F},\bar\alpha)|_{C\times X}\cong({\cal F},\alpha)$ and $q_{c_0}:V_X \rightarrow{\cal F}_{c_0}$ is generically surjective. Moreover $\bar\alpha_{c_0}$ is nonzero and ${\rm Ker }\,\bar\alpha_{c_0}$ is torsion free. For assume that ${\rm T}({\rm Ker }\,\bar\alpha_{c_0})\neq0$ and let $\tilde{\cal F}$ be the kernel of the composite epimorphism $$\bar{\cal F}\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal F}_{c_0}\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal F}_{c_0}/{\rm T}({\rm Ker }\,\bar\alpha_{c_0}).$$ Then there is a short exact sequence $$\ses{\bar{\cal F}}{\tilde{\cal F}}{{\rm T}({\rm Ker }\,\bar\alpha_{c_0})}.$$ By construction $\bar\alpha$ extends to $\tilde\alpha:\tilde{\cal F}\rightarrow {\cal O}_{\bar C}\otimes{\cal E}_0$. Since ${\cal H}$ is normal and the codimension of ${\rm Supp}\,{\rm T}({\rm Ker }\,\alpha)$ in $\bar C\times X$ is greater than 1, $\bar\beta$ also extends to a homomorphism $\tilde\beta:\tilde{\cal F}\rightarrow{\cal O}_{\bar C}\otimes {\cal H}$. Finally $(\tilde\alpha,\tilde\beta):\tilde{\cal F}\rightarrow{\cal O}_{\bar C}\otimes({\cal E}_0\oplus{\cal H})$ is injective, contradicting the maximality of $\bar{\cal F}$. Hence indeed ${\rm T}({\rm Ker }\,\bar\alpha_{c_0})=0$. Since $q_{c_0}$ is generically surjective, ${\rm Ker }\,\bar\alpha_{c_0}$ torsionfree and $\bar \alpha_{c_0}\circ q_{c_0}\neq 0$, we can apply proposition \ref{prop} to the pair $(\bar{\cal F}_{c_0},\bar\alpha_{c_0})$. By assumption on the map $m$ the induced point in $P\times P'$ is semistable, hence $H^0q_{c_0}$ is injective and $({\cal F}_0,\alpha_0)$ sectional semistable. But then necessarily ${\cal F}_{c_0}$ is globally generated, $H^0q_{c_0}$ isomorphic and $q_{c_0}$ surjective. This shows that $h$ extends to a morphism $\bar h:\bar C\rightarrow B$ with $\bar h(c_0)\in B^{ss}$. \hspace*{\fill}\hbox{$\Box$} \begin{proposition}\label{monomorphism} $\tau^{ss}$ is injective. \end{proposition} {\it Proof:} Assume that for $i=1,2$ there are closed points $([a_i:V\to H^0({\cal E}_0)], [q_i:V_X\to{\cal E}_i])$ with the same image under $\tau$. We may assume that $a_1=a_2$ and $\det\,{\cal E}_1=\det\,{\cal E}_2$. Then there is an open subscheme $\emptyset\not=U\subset X$ such that ${{\cal E}_1}_{|U}$, ${{\cal E}_2}_{|U}$ are locally free and are in fact isomorphic as quotients of ${V_X}_{|U}$. Then ${\cal E}_1/{\rm T}({\cal E}_1)$ and ${\cal E}_2/{\rm T}({\cal E}_2)$ are isomorphic as quotients of $V_X$ via a map $\Phi:{\cal E}_1/{\rm T}( {\cal E}_1)\to{\cal E}_2/{\rm T}({\cal E}_2)$ (\cite{Ma}, lemma 4.8). The kernels of the induced homomorphisms $\alpha_i:{\cal E}_i\to{\cal E}_0$ are torsionfree, so that the natural map ${\cal E}_i\to{\cal E}_0\oplus{\cal E}_i/{\rm T}({\cal E}_i)$ are injective. The diagram $$\begin{array}{ccccc} V_X&\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow&{\cal E}_1&\longrightarrow&{\cal E}_0\oplus{\cal E}_1/{\rm T}({\cal E}_1)\\ \|&&&&\spfeil{1}{{\rm id }+\Phi}\\ V_X&\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow&{\cal E}_2&\longrightarrow&{\cal E}_0\oplus{\cal E}_2/{\rm T}({\cal E}_2) \end{array}$$ commutes and shows ${\cal E}_1$ and ${\cal E}_2$ are isomorphic as quotients of $V_X$.\hspace*{\fill}\hbox{$\Box$} This completes the proof of theorem \ref{mainresult} up to the proof of proposition \ref{prop}.\hspace*{\fill}\hbox{$\Box$} \begin{theorem}\label{mainthm} Assume that $X$ is a smooth projective variety of dimension one or two. Then there is a projective $k$-scheme ${\cal M}_\delta^{ss}(\chi,{\cal E}_0)$ and a natural transformation $$\varphi:\underline{\cal M}_\delta^{ss}(\chi,{\cal E}_0)\longrightarrow{\rm Hom }_{{\rm Spec}\, k}( {}~~~,{\cal M}_\delta ^{ss}(\chi,{\cal E}_0))~~,$$ such that $\varphi$ is surjective on rational points and ${\cal M}_\delta^{ss}(\chi,{\cal E}_0)$ is minimal with this property. Moreover, there is an open subscheme ${\cal M}_\delta^s (\chi,{\cal E}_0)\subset{\cal M}_\delta^{ss}(\chi,{\cal E}_0)$ such that $\varphi$ induces an isomorphism of subfunctors$$\underline{\cal M}_\delta^s(\chi,{\cal E}_0)\stackrel{\cong}{\longrightarrow} {\rm Hom }_{{\rm Spec}\, k}(~~~~,{\cal M}_\delta^s(\chi,{\cal E}_0))~~,$$ i.e. ${\cal M}_\delta^s(\chi,{\cal E}_0)$ is a fine moduli space for all stable pairs.\end{theorem} {\it Proof:} By \cite[1.10]{m10} and \cite{g2} there is a projective $k$-scheme ${\cal M}^{ss}$ and a morphism $\rho:B^{ss}\longrightarrow{\cal M}^{ss}$ which is a good quotient for the ${\rm SL}(V)$-action on $B^{ss}$. By lemma \ref{induce} and theorem \ref{mainresult} any family of semistable pairs parametrized by $T$ induces morphisms $T_i\to B^{ss}$ for an appropriate open covering $T=\bigcup T_i$ such that the composition with $\rho$ glue to a well-defined morphism $T\to{\cal M}^{ss}$. This establishes a natural transformation $$\varphi:\underline{\cal M}_\delta^{ss}(\chi,{\cal E}_0)\longrightarrow{\rm Hom }_{{\rm Spec}\, k}(~ {}~,{\cal M}^{ss})~~.$$ If $\psi:\underline{\cal M}^{ss}\longrightarrow{\rm Hom }(~~,N)$ is a similar transformation, then the family $({\cal E}_B,\alpha_B)|_{B^{ss}}$ induces an ${\rm SL}(V)$-invariant morphism $B^{ss}\longrightarrow N$, which therefore factors through a morphism ${\cal M}^{ss}\longrightarrow N$. Moreover there is an open subscheme ${\cal M}^s\subset{\cal M}^{ss}$ such that $B^s=\rho^{-1}({\cal M}^s)$ and $\rho|_{B^{s}}:B^s\longrightarrow{\cal M}^s$ is a geometric quotient. In order to see that the family $({\cal E}_B,\alpha_B)|_{B^s}$ descends to give a universal pair on ${\cal M}^s$ it is enough to show that the stable pairs have no automorphism besides the identity. But assume that $\Phi\not={\rm id }$ is an automorphism of a stable pair $({\cal E},\alpha)$, i.e. $\Phi:{\cal E}\stackrel{\cong}{\longrightarrow}{\cal E}$ and $\alpha\circ\Phi=\alpha$. Then $\psi=\Phi-{\rm id }$ is a nontrivial homomorphism from ${\cal E}$ to ${\ke_{\alpha}}$. Apply the stability conditions to ${\rm Ker }\psi\subset{\cal E}$ and ${\rm Im}\psi\subset{\ke_{\alpha}}$ to get$${\rm rk}\,{\cal E}\cdot\chi_{{\rm Ker }\psi}<{\rm rk}({\rm Ker }\psi)(\chi_{\cal E}-\delta)+\delta \cdot{\rm rk}\,{\cal E}$$ and $${\rm rk}\,{\cal E}\cdot\chi_{{\rm Im}\psi}<{\rm rk}({\rm Im}\psi)(\chi_{\cal E}-\delta)~.$$ Summing up and using $\chi_{{\rm Im}\psi}+\chi_{{\rm Ker }\psi}=\chi_{\cal E}$ and ${\rm rk}({\rm Im}\psi) +{\rm rk}({\rm Ker }\psi)={\rm rk}{\cal E}$ we get the contradiction $\chi_{\cal E}<\chi_{\cal E}$.\hspace*{\fill}\hbox{$\Box$} \subsection{Geometric stability conditions}\label{Kapitel14} In this section we prove proposition \ref{prop}. Let $q:V_X\rightarrow{\cal E}$ and $\alpha:{\cal E}\rightarrow{\cal E}_0$ be homomorphisms of ${\cal O}_X$-modules. To these data we can associate vector space homomorphisms $T:\Lambda^rV\rightarrow H^0(\det\,{\cal E})$ and $a:V\rightarrow H^0({\cal E}_0)$. If $q$ is generically surjective, then $T$ is nontrivial, and if $\alpha\circ q\neq 0$, then $a$ is nontrivial. Let ${([a],[T])}$ denote the corresponding closed point in $P\times P''$ (notations as in section \ref{Kapitel13}). The group ${\rm SL}(V)$ acts on $P\times P''$ by $${([a],[T])}\cdot g=([a\circ g], [T\circ\Lambda^rg]).$$ We want to investigate the stability properties of ${([a],[T])}$ with respect to an ${\rm SL}(V)$-linearization of the very ample line bundle ${\cal O}_{P\times P''}(\nu,\nu')$, where $\nu,\nu'$ are positive integers. These stability properties depend on the ratio $\eta:=\nu/\nu'$ only. We will make use of the Hilbert criterion to decide about (semi)\-stability. Let $\lambda:{\rm\bf G}_m\rightarrow{\rm SL}(V)$ be a 1-parameter subgroup, i.\ e.\ a nontrivial group homomorphism. There is a basis $v_1,\ldots, v_p$ of $V$ such that ${\rm\bf G}_m$ acts on $V$ via $\lambda$ with weights $\gamma_1,\ldots,\gamma_p\in\IZ$: $$\lambda(u)\.v_i=u^{\gamma_i}\.v_i\qquad\hbox{for all }u\in{\rm\bf G}_m(k).$$ Reordering the $v_i$ if necessary we may assume that $\gamma_1\leq\ldots\leq\gamma_p$, $\sum\gamma_i=0$, since ${\rm det}\lambda=1$, and $\gamma_1<\gamma_p$, since $\lambda\neq 1$. For any multiindex $I=(i_1,\ldots,i_r)$ with $1\leq i_1<\ldots<i_r\leq p$ let $v_I=v_{i_1}\wedge\ldots\wedge v_{i_r}$ and $\gamma_I=\gamma_{i_1}+\cdots+\gamma_{i_r}$. The vectors $v_I$ form a basis of $\Lambda^rV$, and ${\rm SL}(V)$ acts with weights $\gamma_I$ with respect to this basis. $T(v_I)\neq0$ if and only if the sections $q(v_{i_1}),\ldots,q(v_{i_r})$ are generically linearly independent, i.\ e.\ generate ${\cal E}$ generically. Now let $$\mu=\mu([a],\lambda):=-{\rm min}\{\gamma_i|a(v_i)\neq 0\}.$$ $$\mu'=\mu([T],\lambda):=-{\rm min}\{\gamma_I|T(v_I)\neq 0\}$$ \begin{lemma}[Hilbert criterion] ${([a],[T])}$ is a (semi)stable point in $P\times P''$ with respect to ${\cal O}(\nu,\nu')$ if and only if $\hat\mu:=\eta\.\mu+\mu'(\geq)0$ for all 1-parameter subgroups $\lambda$. \end{lemma} {\it Proof:} \cite[Thm 2.1.]{m10}\hspace*{\fill}\hbox{$\Box$} For any linear subspace $W\subset V$ let ${\cal E}_{(W)}\subset{\cal E}$ be the submodule which is characterized by the properties : ${\cal E}/{\cal E}_{(W)}$ is torsionfree and ${\cal E}_{(W)}$ is generically generated by $q(W\otimes{\cal O}_X)$. In particular, let ${\cal E}_{(i)}={\cal E}_{(\langle v_1,\ldots,v_i\rangle)}$, $i=0,\ldots,p$ for a given basis $v_1,\ldots,v_p$. Then there is a filtration $${\rm T}({\cal E})={\cal E}_{(0)}\subset{\cal E}_{(1)}\subset\ldots\subset{\cal E}_{(p-1)}\subset {\cal E}_{(p)}={\cal E}.$$ Since ${\cal E}_{(i)}/{\cal E}_{(i-1)}$ is torsionfree, one has either ${\cal E}_{(i)}={\cal E}_{(i-1)}$ or ${\rm rk}{\cal E}_{(i)}>{\rm rk}{\cal E}_{(i-1)}$. Consequently, there are integers $1\leq k_1<\ldots<k_r\leq p$ marking the points, where the rank jumps, i.\ e.\ $k_{\rho}$ is minimal with ${\rm rk}{\cal E}_{(k_{\rho})}=\rho$. Let $K$ denote the multiindex $(k_1,\ldots,k_r)$. If $I$ is any multiindex as above, let $i_0=0$ and $i_{r+1}=p+1$ for notational convenience. \begin{lemma} $\mu'=-\gamma_K$.\end{lemma} {\it Proof:} By construction $T(v_K)\neq 0$. We must show that $\gamma_K\leq\gamma_I$ for every multiindex $I$ with $T(v_I)\neq 0$. For any $I$ and any $t\in\{1,\ldots,r\}$ we let ${\cal E}_{I,t}={\cal E}_{\langle v_{i_1},\ldots,v_{i_t}\rangle)}$. Now suppose $T(v_I)\neq 0$. Let $\ell={\rm max}\{\lambda|k_t=i_t\quad\forall t<\lambda\}$. If $\ell\geq r+1$, then $I=K$ and we are done. We will procede by descending induction on $\ell$. By definition of $K$, we have $k_{\ell}<i_{\ell}$. Define ${\cal E}_{I,t}'={\cal E}_{(\langle v_{k_1},\ldots,v_{k_{\ell}},v_{i_{\ell}},\ldots v_{i_t}\rangle)}$ for $t=\ell,\ldots p$. Then ${\cal E}_{I,t}\subset{\cal E}'_{I,t}$, and $t\leq{\rm rk}\,{\cal E}'_{I,t}\leq t+1$. Let $m={\rm min}\{t|{\rm rk}\,{\cal E}'_{I,t}=t,\ell \leq t\leq p\}$. Now define a multiindex $$I'=(k_1,\ldots,k_{\ell},i_{\ell},\ldots,i_{m-1},i_{m+1},\ldots,i_p).$$ (If $m=\ell$, drop the $i_{\ell},\ldots,i_{m-1}$ part; if $m=p$, drop the $i_{m+1},\ldots,i_p$ part.) Then we have $T(v_{I'})\neq 0$, and $\gamma_{I'}\leq\gamma_I$ by monotony of $I$ and $\gamma$. Moreover, $I'$ and $K$ agree at least in the first $\ell$ entries. Thus by induction $\gamma_K\leq\gamma_{I'}\leq\gamma_I$.\hspace*{\fill}\hbox{$\Box$} Let $\ell:=\min\{i|a(v_i)\neq0\}$. Obviously $\mu=-\gamma_{\ell}$, so that $\hat\mu=-\gamma_K-\eta\.\gamma_{\ell}$. Now $\ell$ and $K$ depend on the basis $v_1,\ldots,v_p$ only, and $\mu$ is a linear function of $\gamma$ for fixed $\ell$ and $K$. Using these notations, the Hilbert criterion can be expressed as follows: \begin{lemma} ${([a],[T])}$ is a (semi)stable point if and only if $$\min_{\hbox{\em\scriptsize bases of $V$}}\min_{\gamma}-(\gamma_K+\eta\.\gamma_{\ell})\quad (\geq)\quad0.$$\hspace*{\fill}\hbox{$\Box$} \end{lemma} We begin with minimizing over the set of all weight vectors $\gamma$. This is the cone spanned by the special weight vectors $$\gamma^{(i)}=(\underbrace{i-p,\ldots,i-p}_{i},\underbrace{i,\ldots,i}_ {p-i})$$ for $i=1,\ldots,p-1$. For any weight vector $\gamma$ can be expressed as $\gamma=\sum_{i=1}^{p-1}c_i\gamma^{(i)}$ with nonnegative rational coefficients $c_i=(\gamma_{i+1}-\gamma_i)/p$. In order to check (semi)stability for a given point it is enough to show $\hat\mu(\geq)0$ for each of these basis vectors. Let $\delta_i=1$ or $0$ if $\ell\leq i$ or $>i$, respectively. Evaluating $\hat\mu$ on $\gamma^{(i)}$ we get numbers $$\mu^{(i)}=p\.(\max\{j|k_j\leq i\}+\eta\.\delta_i)-i\.(r+\eta).$$ If $i$ increases, $\mu^{(i)}$ decreases unless $i$ equals $\ell$ or any of the numbers $k_j$, in which case $\mu^{(i)}$ might jump. The critical values of $i$ therefore are $\ell-1$ and $k_j-1$, $j=1,\ldots,r$, and the corresponding critical values of $\mu^{(i)}$ are: \begin{tabular}{ll} $p\.(j-1)-(k_j-1)\.(r+\eta)$& if $1\leq j\leq r$, $1<k_j\leq\ell$,\\ $p\.(j-1)-(\ell-1)\.(r+\eta)$& if $1\leq j\leq r+1$, $k_{j-1}<\ell\leq k_j$, $1<\ell$,\\ $p\.(j-1+\eta)-(k_j-1)\.(r+\eta)$& if $1\leq j\leq r$, $\ell<k_j$. \end{tabular} If we put $\ell_j=\min\{k_j,\ell\}$, then the conditions imposed by these values of $\hat\mu$ can be comprised as follows: \begin{tabular}{lll} (1)& $0\,(\leq)\,p\.(j-1)-(\ell_j-1)\.(r+\eta)$ &if $1\leq j\leq r+1$, $1<\ell_j$\\ (2)& $0\,(\leq)\,p\.(j-1+\eta)-(k_j-1)\.(r+\eta)$ &if $1\leq j\leq r$. \end{tabular} In the next step one should minimize these terms over all bases of $V$. But in fact, the relevant information is not the used basis itself but the flag of subspaces of $V$ which it generates. The stability criterion takes the following form: \begin{lemma}\label{1form} ${([a],[T])}$ is a (semi)stable point if and only if \begin{itemize} \item[1)] $\dim W\.(r+\eta)\,(\leq)\,p\.{\rm rk}\,{\cal E}_{(W)}$ for all subspaces $0\neq W\subseteq{\rm Ker }\,a$. \item[2)] $\dim W\.(r+\eta)\,(\leq)\,p\.({\rm rk}\,{\cal E}_{(W)}+\eta)$ for all subspaces $0\neq W\subseteq V$ with ${\rm rk}\,{\cal E}_{(W)}\,(\leq)\,r$. \end{itemize}\hspace*{\fill}\hbox{$\Box$}\end{lemma} We give the stability criterion still another form, shifting our attention from subspaces of $V$ to submodules of ${\cal E}$: \begin{lemma}\label{2form} ${([a],[T])}$ is a (semi)stable point if and only if \begin{itemize} \item[(0)] $H^0q$ is an injective map. \item[(1)] $V\cap H^0{\cal F}=0$ or $\dim(V\cap H^0{\cal F})\.(r+\eta)\,(\leq)\, p\.{\rm rk}{\cal F}$ for all submodules ${\cal F}\subseteq{\rm Ker }\alpha$. \item[(2)] $\dim(V\cap H^0{\cal F})\.(r+\eta)\,(\leq)\,p\.({\rm rk}{\cal F}+\eta)$ for all submodules ${\cal F}\subseteq{\cal E}$ with \newline ${\rm rk}{\cal F}\,(\leq)\,{\rm rk}{\cal E}$. \end{itemize}\end{lemma} {\it Proof:} If ${([a],[T])}$ is semistable, let $W:={\rm Ker } H^0q$. Then $W\subseteq{\rm Ker }\,a$. {}From the lemma above it follows that $\dim W\leq p/(r+\eta)\.{\rm rk}\,{\cal E}_{(W)}=0$. Hence (0) is a necessary condition. It is to show that the conditions(1) and (2) of lemma \ref{1form} and of lemma \ref{2form} are equivalent. Suppose we are given a submodule ${\cal F}\subseteq{\cal E}$. Let $W:=V\cap H^0{\cal F}$. Then $q(W\otimes{\cal O}_X)\subseteq{\cal F}$ and ${\rm rk}\,{\cal F}={\rm rk}\,{\cal E}_{(W)}$. Moreover, if ${\cal F}\subseteq{\ke_{\alpha}}$, then $W\subseteq{\rm Ker }\,a$. Now either $W=0$ or \ref{1form} applies and gives \ref{2form}. Conversely, if $W\subseteq V$ is given, let ${\cal F}:=q(W\otimes{\cal O}_X)$. Then $W\subseteq V\cap H^0{\cal F}$ and ${\rm rk}\,{\cal E}_{(W)}={\rm rk}\,{\cal F}$. Again, if $W\subseteq{\rm Ker }\,a$, then ${\cal F}\subseteq{\ke_{\alpha}}$. Hence \ref{2form} implies \ref{1form}. Finally, we replace $\eta$ by a more suitable parameter: $$\bar\delta=\frac{p\.\eta}{r+\eta}\qquad\eta=\frac{r\.\bar\delta} {p-\bar\delta}.$$ Since $\eta$ was a positive rational number, $\bar\delta$ is confined to the open interval $(0,p)$, which of course tallies with the data of the previous section. The following theorem, which differs from proposition \ref{prop} only in the choice of words, summarizes the discussion of this section: \begin{theorem} If in addition to the global assumptions of this section ${\ke_{\alpha}}$ is torsionfree, then ${([a],[T])}$ is a (semi)stable point of $P\times P''$ if and only if the following conditions are satisfied: \begin{itemize} \item[-] $H^0q$ is an injective homomorphism. \item[-] $({\cal E},\alpha)$ is sectional stable with respect to $\bar\delta$. \end{itemize}\end{theorem} {\it Proof:} If ${\ke_{\alpha}}$ is torsionfree then every nontrivial submodule of ${\ke_{\alpha}}$ has positive rank. Hence condition (1) in \ref{2form} can be replaced by \begin{itemize}\item[(1')] $dim(V\cap H^0{\cal F})\.(r+\eta)(\leq)p\.{\rm rk}\,{\cal F}$ for all submodules ${\cal F}\subseteq{\ke_{\alpha}}$. \end{itemize} As a result of replacing $\eta$ by $\bar\delta$ in (1') and \ref{2form}(0),(2) one obtains the defintion of sectional (semi)stability.\hspace*{\fill}\hbox{$\Box$} \section{Applications} This chapter is organized as follows. In \ref{Kapitel21} we show that the existence of semistable pairs gives an upper bound for $\delta$. Rationality conditions on $\delta$ imply the equivalence of semistability and stability. If $\delta$ varies within certain regions the semistability conditions remain unchanged. This is formulated and specified for the rank two case. \ref{Kapitel22} deals with Higgs pairs. Again we concentrate on the rank two case. We make the first step to generalize the diagrams of Bertram and Thaddeus to algebraic surfaces. The restriction of $\mu-$stable vector bundles~ on an algebraic surface to a curve of high degree induces an immersion of the moduli space~ of vector bundles~ on the surface into the moduli space~ of vector bundles~ on the curve. The understanding of this process is important, e.g. for the computation of Donaldson polynomials and for the study of the geometry of the moduli space on the surface (\cite{T}). With the help of a restriction theorem for $\mu-$stable pairs $({\cal E},\alpha:{\cal E}\to{\cal O})$ we construct an approximation of this immersion, which will hopefully shed some light on the relation between the original moduli spaces. It is remarkable that the limit of any approximation is independent of the polarization. In \ref{Kapitel23} we first compare our stability~ for ${\cal E}_0={\cal O}_D^{\oplus r}$, where $D$ is a divisor on a curve, with the notion of Seshadri of stable sheaves with level structure along a divisor(\cite{Se}). We will have a closer look at the moduli space~ of rank two sheaves of degree 0 with a level structure at a single point. Furthermore certain results from \ref{Kapitel22} are reconsidered in the case of ${\cal E}_0$ being a vector bundle on a divisor. \subsection{Numerical properties of $\delta$}\label{Kapitel21} Let $X$ be a smooth projective variety with an ample divisor $H$, ${\cal E}_0$ a coherent ${\cal O}_X-$module and $\delta$ a positive rational polynomial of degree $\dim X-1$ with leading coefficient $\delta_1\geq0$. \begin{lemma}\label{rest}Assume $({\cal E},\alpha)$ is a semistable pair such that ${\ke_{\alpha}}\not=0$. Then $$\delta(\leq)\chi_{\cal E}-\frac{{\rm rk}{\cal E}}{{\rm rk}{\ke_{\alpha}}}(\chi_{\cal E}-\chi_{{\cal E}_0})~~.$$ If ${\cal E}_0\cong{\cal O}_X$ and ${\rm rk}{\cal E}>1$, then $$\delta(\leq)\frac{{\rm rk}{\cal E}\.\chi_ {{\cal O}_X}-\chi_{\cal E}}{{\rm rk}{\cal E}-1}$$ and in particular $$\delta_{1}(\leq)-\frac{\deg{\cal E}}{{\rm rk}{\cal E}-1}~~.$$ If ${\cal E}_0$ is torsion, then $$\delta(\leq)\chi_{{\cal E}_0}$$ and in particular $\delta_1(\leq)\deg{{\cal E}_0}$.\end{lemma} {\it Proof:} The first inequality follows immediately from the stability condition i). If ${\cal E}_0\cong{\cal O}_X$ use $\chi_{{\ke_{\alpha}}}=\chi_{\cal E}-\chi_{{\rm Im}\alpha}\geq\chi_{\cal E}-\chi_{{\cal E}_0}$ and ${\rm rk}{\ke_{\alpha}}={\rm rk}{\cal E}-1$.\hspace*{\fill}\hbox{$\Box$} It is much more convenient to work with $\mu-$stability~ only. In fact for the general $\delta$ one can achieve that every semistable pair is $\mu-$stable. \begin{lemma}\label{ratio} There exists a discrete set of rationals $0\leq...<\eta_i<\eta_{i+1} <...$ including $0$, such that for $\delta_1\in (\eta_i,\eta_{i+1}) $ every semistable pair with respect to $\delta$ is in fact $\mu-$stable and the $\mu-$stability conditions depend only on $i$.\end{lemma} {\it Proof:} Define $\{\eta_i\}:=[0,-{d}/({r-1}))\cap\{({ar-sd}) /({r-s})|a,s\in\hbox{\sym \char '132},~0\leq s<r\}$. If $\delta_1\in(\eta_i,\eta_{i+1})$, then the right hand sides of the $\mu-$ semistability conditions $\deg{\cal G}\leq{sd}/{r}- \delta_1{s}/{r}$ and $\deg{\cal G}\leq{sd}/{r}+\delta_1({r-s})/{r}$ are not integer ($s= {\rm rk}{\cal G}$). Therefore $\mu-$semistability and $\mu-$stability coincide. Moreover, the integral parts of the right hand sides depend only on $i$, i.e. for two different choices of $\delta_1$ in the intervall $(\eta_i,\eta_{i+1})$ the $\mu-$stability conditions are the same.\hspace*{\fill}\hbox{$\Box$} More explicit results can be achieved in special cases: \begin{proposition}\label{indforlb}For $r=2$ and ${\cal E}_0\in Pic(X)$ and $\delta_1\in(\eta_i,\eta_i+2)$,where $\eta_i:= \max\{0,2i+d\}$ with $i\in\hbox{\sym \char '132}$, every semistable pair is $\mu-$stable. The stability in this region does not depend on $\delta$.\end{proposition} {\it Proof:} For ${\cal E}_0\in Pic(X)$ all semistable pairs $({\cal E},\alpha)$ have torsionfree ${\cal E}$ and ${\rm rk}{\ke_{\alpha}}=1$. In particular the stability~ conditions concern rank one subsheaves only. Now $\delta_1 \in(\eta_i,\eta_i+2)$ is equivalent to $-1-i<{d}/{2}-{\delta_1}/{2}<-i$, $i+d-1<{d}/ {2}+{\delta_1}/{2}<i+d$ and $\delta_1>0$.\hspace*{\fill}\hbox{$\Box$} As the last numerical criterion we mention \begin{lemma} Assume $\delta_1<\min_{0\leq s<r}\{(r-sd)/({r-s}) +{r}({r-s})[{sd}/{r}]\}$. \begin{itemize} \item[i)] Then every sheaf ${\cal E}$ in a semistable pair $({\cal E},\alpha)$ without torsion in dimension zero is torsionfree and $\mu-$semistable. \item[ii)] If ${\cal E}$ is torsionfree and $\mu-$semistable and $\alpha:{\cal E}\to{\cal E}_0$ a nontrivial homomorphism such that ${\ke_{\alpha}}$ does not contain a destabilizing subsheaf, then $({\cal E},\alpha)$ is $\mu-$stable. \end{itemize}\end{lemma} {\it Proof:} The condition on $\delta_1$ is equivalent to either of the two conditions: $[sd/r,sd/r+\delta_1(r-s)/r)\cap\hbox{\sym \char '132}=\emptyset$ for $0\leq s<r$. $[sd/r-\delta_1/r,sd/r)\cap\hbox{\sym \char '132}=\emptyset$ for $0<s\leq r$.\hspace*{\fill}\hbox{$\Box$} \subsection{Higgs pairs in dimension one and two}\label{Kapitel22} A Higgs pair in this context is a vector bundle~ together with a global section. (This notion should not be confused with a Higgs field as a section $\theta\in H^0({\cal E} nd{\cal E}\otimes\Omega^1_X)$ with $\theta\wedge\theta=0$!) Instead of considering a global section we prefer to work with a homomorphism from the dualized bundle to the structure sheaf. These objects will be called pairs as in the general context. First we remind of the situation in the curve case, which was motivation for us to go on. \begin{definition} Let $C$ be a smooth curve. As introduced in \ref{Kapitel13} ${\cal M}_\delta^{ss}(d,2,{\cal O})$(resp. ${\cal M}_\delta^{ss}({\cal Q},2,{\cal O})$) denotes the moduli space of semistable pairs $({\cal E},\alpha:{\cal E} \to{\cal O})$ with respect to $\delta$, where ${\cal E}$ is a rank two sheaf of degree $d$ (with determinant ${\cal Q}$).\end{definition} \begin{remark}\label{torsfree}Notice, that $\delta$ is just a number and that a sheaf occuring in a semistable pair is always torsionfree and hence a vector bundle. Moreover the stability conditions reduce to $\deg({\ke_{\alpha}})\leq{d}/{2}-{\delta}/{2}$ and $\deg({\cal G})\leq{d}/{2}+{\delta}/{2}$ for all line bundles ${\cal G}\subset{\cal E}$.\end{remark} For the following we assume $d<0$. \begin{definition}\label{UC}$U_{C,i}(d):={\cal M}_\delta^{ss}(d,2,{\cal O})$ and $SU_{C,i}({\cal Q}):={\cal M}_\delta^{ss}({\cal Q},2,{\cal O})$, where $\delta\in( max\{0,2i+d\},2i+d+2)$.\end{definition} Note that according to proposition \ref{indforlb} the spaces $U_{C,i}(d)$ and $SU_{C,i}({\cal Q})$ do not depend on the choice of $\delta$ \begin{proposition}(M. Thaddeus) $U_{C,i}(d)$ and $SU_{C,i}({\cal Q})$ are projective fine moduli spaces. Every semistable pair is automatically stable.\end{proposition} {\it Proof:} \cite{Th} or \ref{mainthm}\hspace*{\fill}\hbox{$\Box$} \begin{proposition}\begin{itemize} \item[i)] For $i\geq-d$ the moduli spaces~ $U_{C,i}(d)$ are empty. \item[ii)] For $i=\lfloor-{d}/{2}-1\rfloor+1$ there are morphisms $$U_{C,i}(d)\longrightarrow U(d)$$ and$$SU_{C,i}({\cal Q})\longrightarrow SU({\cal Q})~~,$$where $U(d)$ and $SU({\cal Q})$ are the moduli spaces~ of semistable vector bundles~ of degree $d$ and determinante ${\cal Q}$, resp. The fibre over a stable bundle ${\cal E}$ is isomorphic to $\hbox{\sym \char '120}(H^0({\cal E}^{\rm v})^{\rm v})$. In particular they are projective bundles for $0\gg d\equiv1(2)$. \item[iii)] A pair $({\cal E},\alpha)$ lies in $SU_{C,-d-1}({\cal Q})$ if and only if there is a nonsplitting exact sequence of the form $$\sesq{ {\cal Q}}{}{{\cal E}}{\alpha}{{\cal O}}~.$$Thus $SU_{C,-d-1}\cong\hbox{\sym \char '120}({{\rm Ext}}^1({\cal O}, {\cal Q})^{\rm v})$.\end{itemize} \end{proposition} {\it Proof:} i) and ii) follow from the general criteria. A similar result as iii) holds in the surface case. We give the proof there.\hspace*{\fill}\hbox{$\Box$} The following picture illustrates the situation:$$\begin{array}{clll} SU_{C,\lfloor-{d}/{2}-1\rfloor+1}({\cal Q})&SU_{C,\lfloor-{d}/{2}-1\rfloor+2}( {\cal Q})&.... &SU_{C,-d-1}({\cal Q})\cong\hbox{\sym \char '120}({{\rm Ext}}^1({\cal O},{\cal Q})^{\rm v})\\ \downarrow&&&\\ SU({\cal Q})&&& \end{array}$$ M. Thaddeus is able 'to resolve the picture' by a sequence of blowing ups and downs. In particular all the spaces $SU_{C,i}$ are rational. This process makes it possible to trace a generalized theta divisor on $SU_{C,i}$ to a certain section of ${\cal O}(k)$ on $\hbox{\sym \char '120}(H ^1({\cal Q}))$. This method is used in \cite{Th} to give a proof of the Verlinde formula. We go on to proceed in a similar way in the case of a surface. Let $X$ be an algebraic surface with an ample divisor $H$. Now ${\cal M}_\delta^{ss}(d,c_2,2,{\cal O})$ ($ {\cal M}_\delta^{ss}({\cal Q},c_2,2,{\cal O})$) denotes the moduli space of semistable pairs $({\cal E},\alpha:{\cal E}\to{\cal O}_X)$ with respect to $\delta$, where ${\cal E}$ is a rank two sheaf of degree $d$ ($:=c_1.H$) (with determinant ${\cal Q}$) and second Chern class $c_2$. For the existence of such pairs it is necessary that $\delta$ be linear with nonnegative leading coefficient $\delta_1$. As in \ref{torsfree} a sheaf occuring in a semistable pair is torsionfree and the stability conditions are $$\chi_{\cal G}(\leq)\frac{\chi_{\cal E}}{2}-\frac{\delta}{2}$$ for all rank one subsheaves ${\cal G}\subset{\ke_{\alpha}}$ and $$\chi_{\cal G}(\leq)\frac{\chi_{\cal E}}{2}+\frac{\delta}{2} $$for all rank one subsheaves ${\cal G}\subset{\cal E}$. \begin{definition}\label{UCX}For $\delta$ such that $\delta_1 \in(\max\{0,2i+d\},2i+d+2)$ we define $U_i:={\cal M}_\delta^{ss}(d,c_2,2,{\cal O})$ and $SU_i:={\cal M}_\delta^{ss}({\cal Q},c_2,2,{\cal O})$.\end{definition} Again, note that according to \ref{indforlb} the definition does not depend on the choice of $\delta$. \begin{corollary} $U_i$ and $SU_i$ are projective fine moduli spaces. Every semistable pair is $\mu-$stable.\end{corollary} {\it Proof:} It follows immediately fom \ref{mainthm} and section \ref{Kapitel21}.\hspace*{\fill}\hbox{$\Box$} \begin{proposition}If $({\cal E},\alpha)$ is a $\mu-$semistable pair with respect to $\delta$, then $4c_2({\cal E})-c_1^2({\cal E})\geq-{\delta_1}/({4H^2})$.\end{proposition} {\it Proof:} If $({\cal E},\alpha)$ is a $\mu-$semistable pair the homomorphism $\alpha$ can be extended to a homomorphism ${\cal E}^{{\rm vv}}\to {\cal O}$ and the resulting pair is still $\mu-$semistable with $c_1({\cal E}^{{\rm vv}})=c_1({\cal E})$ and $c_2({\cal E}^{{\rm vv}})\leq c_2({\cal E})$. Thus it is enough to prove the inequality for locally free pairs. If ${\cal E}$ itself is a $\mu-$semistable bundle the Bogomolov inequality says $4c_2-c_1^2\geq0$. If ${\cal E}$ is not $\mu-$semistable, then there is an exact sequence $$\ses{{\cal L}_1}{{\cal E}}{{\cal L}_2\otimes I_{Z}}~~,$$where $I_Z$ is the ideal sheaf of a zero dimensional subscheme and ${\cal L}_1$ and ${\cal L}_2$ are line bundles with ${\deg{\cal E}}/{2}< \deg{\cal L}_1\leq{\deg{\cal E}}/{2}+({1}/{2})\delta_1$ and ${\deg{\cal E}}/{2}-({1}/{2}) \delta_1\leq\deg{\cal L}_2<{\deg{\cal E}}/{2}$. Using $c_2({\cal E})=c_1({\cal L}_1)c_1({\cal L}_2)+l(Z)\geq c_1( {\cal L}_1)c_1({\cal L}_2)=({1}/{4})\{(c_1({\cal L}_1)+c_1({\cal L}_2))^2-(c_1({\cal L}_1 )-c_1({\cal L}_2))^2\}=({1}/{4}) c_1^2({\cal E})-\frac{1}{4}(c_1({\cal L}_1)-c_1({\cal L}_2))^2$ and Hodge index theorem, which gives $(c_1({\cal L}_1)-c_1({\cal L}_2))^2\leq ({(\deg{\cal L}_1-\deg{\cal L}_2)^2})/{H^2}$ we infer the claimed inequality. Notice, that for $\delta_1\to0$ the inequality converges to the usual Bogomolov inequality.\hspace*{\fill}\hbox{$\Box$} \begin{proposition}\begin{itemize} \item[i)] For $i\geq-d$ the moduli spaces~ $U_i$ and $SU_i$ are empty. \item[ii)] If $i=\lfloor-{d}/{2}-1\rfloor+1$, then every pair $({\cal E},\alpha)\in U_i$ has a $\mu-$semistable ${\cal E}$. There is rational map $U_i\to U(c_1,c_2)$ (the moduli space~ of semistable, torsionfree sheaves), which is a morphism for $d\equiv1(2)$. The image of the rational map contains all $\mu-$stable sheaves ${\cal E}$ with ${{\rm Hom({\cal E},{\cal O})}}\not=0$. The fibre over such a point is $\hbox{\sym \char '120}({{\rm Hom}}({\cal E},{\cal O})^{\rm v})$. \item[iii)] Every pair $({\cal E},\alpha)\in SU_{-d-1}$ sits in an nontrivial extension of the form $$\sesq{I_{Z_1}\otimes{\cal Q}}{}{{\cal E}}{\alpha}{I_{Z_2}}~~,$$where $I_{Z_i}$ are the ideal sheaves of certain zero dimensional subscheme. In the case $Z_1=\emptyset$, e.g. ${\cal E}$ is locally free, every such extension gives in turn a stable pair $({\cal E},\alpha)\in SU_{-d-1}$.\end{itemize}\end{proposition} {\it Proof:} {\it i)} and {\it ii)} follow again from \ref{Kapitel21} If $({\cal E},\alpha)\in SU_{-d-1}$, then $\deg{\ke_{\alpha}}< d+{1}/{2}$, which is equivalent to $ \deg({\rm Im}\alpha)>-{1}/{2}$. Since ${\rm Im}\alpha\subset{\cal O}$ it follows ${\rm Im} \alpha= I_{Z_2}$. A splitting of the induced exact sequence would lead to the contradiction $0\leq\deg I_{Z_2} \leq-{1}/{2}$. Let $({\cal E},\alpha)$ be given by a sequence with $ Z_1=\emptyset$. For ${\cal G}\subset{\ke_{\alpha}}$ one gets the required inequality $\deg{\cal G}\leq d<d+{1}/{2}$. If ${\cal G}\subset{\cal E}$ and ${\cal G}\not\subset{\ke_{\alpha}}$, the sheaf ${\cal G}$ has the form ${\cal G}= I_{Z_3}\subset I_{Z_2}$. Without restriction we can assume that ${\cal E}/{\cal G}$ is torsionfree. Since ${\cal E}/{\cal G}$ is an extension of $ I_{Z_2}/I_{Z_3}$ by ${\cal Q}$ and ${{\rm Ext}}^1(I_{Z_2}/I_{Z_3},{\cal Q})=0$, ${\cal G}$ in fact equals $I_{Z_2}$ and therefore defines a splitting of the sequence.\hspace*{\fill}\hbox{$\Box$} \begin{corollary}The set of all pairs $({\cal E},\alpha)\in SU_{-d-1}$ with ${\ke_{\alpha}}$ locally free, which in particular contains all locally free pairs, forms a projective scheme over $Hilb^{c_2}(X)$ with fibre over $[Z]\in Hilb^{c_2}(X)$ isomorphic to $\hbox{\sym \char '120}({{\rm Ext}}^1 (I_Z,{\cal Q})^{\rm v})$. \end{corollary} {\it Proof:} If $({\cal E},\alpha)$ is a universal family over $SU_{-d-1}\times X$, then the set of points $t\in SU_{-d-1}$ with $l(({\rm coker }\,\alpha)_t)$ maximal is closed. It is easy to see that $ ({\rm coker }\,\alpha)_t\cong{\rm coker }(\alpha_t)$ and that $l({\rm coker }(\alpha_t))$ is maximal, i.e. is equal to $c_2$ if $ {\rm Ker }(\alpha_t)$ is locally free. Therefore the set of all pairs with locally free kernel ${\ke_{\alpha}}$ is closed and ${\cal O}/{\rm Im}\alpha$ induces the claimed morphism to $Hilb^{c_2}(X)$.\hspace*{\fill}\hbox{$\Box$} \begin{corollary}\label{indepofpol} The moduli space of all locally free pairs $({\cal E},\alpha)\in SU_{-d-1}$ does not depend on the polarization of $X$.\end{corollary} \begin{remark}{\it i)} Bradlow introduced in \cite{Br} the notion of $\phi-$stability~ with respect to a parameter $\tau$. If we set $\delta_1=-d+({\tau}/{2\pi})vol(X)$ ($d$ is the degree of ${\cal E}$) both notions coincide, i.e. a pair $({\cal E},\alpha:{\cal E}\to{\cal O})$ with a locally free ${\cal E}$ is $\mu-$stable in our sense if and only if $({\cal E}^{\rm v},\phi=\alpha^{\rm v}\in H^0({\cal E}^{\rm v}))$ is $\phi-$stable with respect to the parameter $\tau$ in Bradlow's sense. He proves a Kobayashi-Hitchin correspondence in this situation, i.e. he shows: $({\cal E},\alpha)$ is $\mu-$stable ( or a sum of a $\mu-$stable pair with $\mu-$stable bundles) if and only if the vortex equation has a solution, i.e. there exists a hermitian metric $H$ on ${\cal E}^{\rm v}$, such that $$\Lambda_\omega F_H+\tau\frac{i}{2}id=\frac{i}{2}\phi\otimes\phi^{*_H}~~.$$ $F_H$ is the curvature of the metric connection on ${\cal E}^{\rm v}$, $\omega$ is a fixed K\"ahler form and $\Lambda_\omega$ is the adjoint of $\wedge\omega$. Now,if $({\cal E}, \alpha)\in SU_{-d-1}$ one can take $\delta$ near to $-d$. That corresponds to $\tau\to0$. Although \ref{indepofpol} shows that $SU_{-d-1}$ is independent of the polarization $H$, i.e. of the Hodge metric, for us there is no obvious reason in the analytical equation.\\ {\it ii)} In \cite{Rei} the space $SU_{-d-1}$ is stratified and equipped with certain line bundles. These objects Reider calls Jacobians of rank two alluding to a Torelli kind theorem for algebraic surfaces. \end{remark} In order to study the restriction of $\mu-$stable vector bundles~ to curves of high degree it could be usefull to study the restriction of $\mu-$stable pairs to those curves. As a generalization of a result of Bogomolov we prove \begin{theorem}\label{restofpairs} For fixed $c_1,c_2,\delta$ and $H$ there exists a constant $n_0$, such that for $n\geq n_0$ and any smooth curve $C\in|nH|$ the restriction of every locally free, $\mu-$stable pair to $C$ is a $\mu-$stable pair on the curve with respect to $n\delta_1$.\end{theorem} {\it Proof:} If ${\cal E}$ is locally free the kernel ${\ke_{\alpha}}$ is a line bundle. In particular the restriction of the injection ${\ke_{\alpha}}\subset{\cal E}$ to a curve remains injective. Thus $({\ke_{\alpha}})_C={\rm Ker }(\alpha_C)$. Since $\deg({\ke_{\alpha}})_C=n\deg{\ke_{\alpha}}$, the two inequalities $\deg{\ke_{\alpha}}<{\deg{\cal E}}/{2}-{\delta_1}/{2}$ and $\deg({{\ke_{\alpha}}})_C<{\deg{\cal E}_C}/{2}-{n\delta_1}/{2}$ are equivalent. Thus the first of the stability conditions on $C$ is always satisfied. In order to prove the second we proceed in two steps.\\ i) By Bogomolov's result (\cite{Bo}) there is a constant $n_0$, such that the restriction of a $\mu-$stable vector bundle~ to a smooth curve $C\in|nH|$ for $n\geq n_0$ is stable. Since the inequality $\deg{\cal G}<{\deg{\cal E}_C}/{2}+{n\delta_1}/{2}$ for a line bundle ${\cal G}\subset{\cal E}_C$ is weaker than the stability condition on ${\cal E}_C$, the theorem follows immediately from Bogomolov's result for all $\mu-$stable pairs $({\cal E},\alpha)$, where ${\cal E}$ is a $\mu-$stable vector bundle.\\ ii) Therefore it remains to prove the theorem for pairs with ${\cal E}$ not $\mu-$stable. Any such vector bundle~ is an extension of ${\cal L}_2\otimes I_Z$ by ${\cal L}_1$, where ${\cal L}_1$ and ${\cal L}_2$ are line bundles with ${\deg{\cal E}}/{2}\leq\deg{\cal L}_1<{\deg{\cal E}}/{2}+({1}/{2}) \delta_1$. $I_Z$ is as usual the ideal sheaf of a zero dimensional subscheme. If $C\in|nH|$ is a curve with $C\cap Z=\emptyset$, then the restriction of the extension to $C$ induces the exact sequence $$\ses{({\cal L}_1)_C}{{\cal E}}{({\cal L}_2)_C}~~.$$ If ${\cal G}\subset{\cal E}_C$ is a line bundle, then either ${\cal G}\subset({\cal L}_1)_C$ or ${\cal G}\subset({\cal L}_2)_C$. This implies $\deg{\cal G}\leq\deg({\cal L}_1)_C =n\deg{\cal L}_1<{\deg{\cal E}_C}/{2}+({1}/{2})n\delta_1$ or $\deg{\cal G}\leq\deg({\cal L}_2)_C= n\deg{\cal E}-n\deg{\cal L}_1\leq{\deg{\cal E}_C}/{2}$. Hence $({\cal E}_C,\alpha_C)$ is stable. If $C\cap Z\not=\emptyset$ we only get a sequence of the form $$\ses{({\cal L}_1)_C(Z.C)}{{\cal E}_C} {({\cal L}_2)_C(-Z.C)}$$Notice, that ${\cal O}_C(-Z.C)\cong(I_Z\otimes{\cal O}_C)/{\rm T}(I_Z\otimes{\cal O}_C)$. As above $\deg{\cal G}\leq\deg({\cal L}_1)_C+\deg(Z.C)\leq\deg({\cal L}_1)_C+l(Z)$ or $\deg{\cal G}\leq{ \deg({\cal E}_C)}/{2} $ for every line bundle ${\cal G}\subset{\cal E}_C$. If $\deg{\cal L}_1+{l(Z)}/{n}<{\deg{\cal E}}/{2}+ {\delta_1}/{2}$, then $({\cal E}_C,\alpha_C)$ is stable. There exists a positive number $\varepsilon$ depending only on the degree, $\delta$ and $H$, such that $\deg{\cal L}_1\leq{\deg{\cal E}}/{2}+{\delta_1}/{2}-\varepsilon$. Thus it suffices to bound $l(Z)$ by $n_0\varepsilon$. That is done by the following computation. $l(Z)=c_2-c_1({\cal L}_1) c_1({\cal L}_2)=c_2-{c_1^2}/{4}+({1}/{4})(c_1({\cal L}_1)-c_1({\cal L}_2))^2\leq c_2-{c_1^2}/ {4} +{\delta_1^2}/({4H^2})$. Thus $n_0>({1}/{\varepsilon})(c_2-{c_1^2}/{4}+ {\delta_1^2}/({4H^2}))$ satisfies $l(Z)<n_0\varepsilon$.\hspace*{\fill}\hbox{$\Box$} With the notation of \ref{UC} and \ref{UCX} one proves \begin{corollary} For fixed $c_1,c_2$ and $H$ there exists a number $n_0$, such that for every smooth curve $C\in|nH|$ for $n_0\leq n\equiv1(2)$ and every $i$ with $\lfloor{-d}/{2}-1\rfloor+1 \leq i\leq-d-1$ the restriction of pairs gives an injective immersion, i.e. an injective morphism with injective tangent map:$$U_i^f\to U_{C,in+({n-1})/{2}}$$\end{corollary} (The superscript denotes the subset of all locally free pairs)\\ {\it Proof:} The technical problem here is, that the constant $n_0$ in the last theorem depends on $\delta$ and not only on $i$. Therefore we fix for every $i$ a very special $\delta$, namely $\delta_1=2i+d+1$. Since we only consider finitely many $i$'s there is an $n_0$, such that the restriction gives a morphism $U_i^f\to U_{C,in+({n-1})/{2}}$. Here we use $n\equiv1(2)$. Since the occuring family of vector bundles~ is bounded one can choose $n_0$, such that $H^k(X,{\cal H} om({\cal E},{\cal E}')(-nH))=0$ ($k=0,1$) and $H^0({\cal E}^{\rm v}(-nH))=0$ for $n\geq n_0$ and all vector bundles~ ${\cal E}$ and ${\cal E}'$ occuring in a pair in one of the moduli spaces~ $U_i$. Thus $({\cal E},\alpha)_C \cong({\cal E}',\alpha')_C$ if and only if ${\cal E}\cong{\cal E}'$ and $\alpha$ maps to $\alpha'$ under this isomorphism, i.e. the restriction morphism is injective. A standard argument in deformation theory shows that the Zariski tangent space of $U_i^f$ at $({\cal E},\alpha)$ is isomorphic to the hypercohomology $\hbox{\sym \char '110}^1({\cal E} nd{\cal E}^{\rm v}\to{\cal E}^{\rm v})$ of the indicated complex which is given by $\varphi\mapsto\varphi (\alpha^{\rm v})$ (\cite{We}). Analogously, the Zariski tangent space of $U_{C,j}$ at $({\cal E}_C,\alpha_C)$ is isomorphic to the hypercohomology $\hbox{\sym \char '110}^1({\cal E} nd{\cal E}_C^{\rm v}\to{\cal E}_C^{\rm v})$. The Zariski tangent map is described by the restriction of hypercohomology classes. Both hypercohomology groups sit in exact sequences of the form$$...\to H^0({\cal E}^{\rm v})\to\hbox{\sym \char '110}^1({\cal E} nd{\cal E}^{\rm v}\to{\cal E}^{\rm v})\to H^1({\cal E} nd{\cal E})\to...$$ and$$...\to H^0({\cal E}_C^{\rm v})\to\hbox{\sym \char '110}^1({\cal E} nd{\cal E}_C^{\rm v}\to{\cal E}_C^{\rm v})\to H^1({\cal E} nd{\cal E}_C)\to...~~~,$$resp. By our assumptions the restrictions $H^0({\cal E}^{\rm v})\to H^0({\cal E}_C^{\rm v})$ and $H^1({\cal E} nd{\cal E}^{\rm v})\to H^1({\cal E} nd{\cal E}_C) $ are injective. Hence the Zariski tangent map of the restriction of stable pairs is injective, too.\hspace*{\fill}\hbox{$\Box$} We remark that neither the starting nor the end point of the series of moduli spaces~ on the surface is sent to the corresponding point of the series moduli spaces~ on the curve. A slight generalization of the theorem allows to restrict $\mu-$stable pairs to a stable pair on a curve $C\in|nH|$ with respect to the parameter $n\delta_1+c$, where $c$ is a constant depending only on $\delta_1,c_1,c_2$ and $H$. \subsection{Framed bundles and level structures}\label{Kapitel23} In this paragraph we consider pairs of rank $r$, where ${\cal E}_0\cong{\cal O}_D^{\oplus r}$ or more generally where ${\cal E}_0$ is a vector bundle~ of rank $r$ on a divisor $D$.\\ We start with pairs on a curve. In this case $D$ is a finite sum of points. As far as we know, Seshadri was the first to consider and to construct moduli spaces~ for such pairs. In \cite{Se} they were called sheaves with a level structure. The general stability conditions as developped in this paper and specialized to this case present a slight generalization of Seshadris stability concept in terms of the parameter $\delta$, which in \cite{Se} is always $l(D)$. The geometric invariant theory which Seshadri used to construct the moduli spaces~ differs from the one in \ref{Kapitel13}. In \cite{Se} a point $[{\cal O}^{\oplus N}\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal E}]$ of the Quotscheme is mapped to a point $$([{\cal O}^{\oplus N}(x_1)\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow {\cal E}(x_1)],..., [{\cal O}^{\oplus N}(x_n)\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal E}(x_n)])$$ in the product of Grassmannians (the $x_i$ are sufficiently many generic points. The conditions for a point in this product to be semistable in the sense of geometric invariant theory translate into the semistability properties for pair. However, to generalize the construction to the higher dimensional case one has to map the Quotscheme into a different projective space as in \ref{Kapitel13} and study the stability conditions there. \begin{lemma}If the genus of the curve is at least 2, there exists a semistable pair of rank $r$ and degree $d$ with respect to $({\cal O}_D^{\oplus r},\delta)$ if and only if $0<\delta\leq r\cdot l(D)$.\end{lemma} {\it Proof:} The 'only if' part was proven in \ref{rest}, since $r\cdot l(D)=h^0({\cal E}_0)$. For the 'if' direction we pick a stable vector bundle~ ${\cal E}$ of rank $r$ and degree $d$ and an isomorphism $\alpha:{\cal E}_D\cong{\cal O}_D^{\oplus r}$. The induced pair is semistable.\hspace*{\fill}\hbox{$\Box$} \begin{corollary} The moduli spaces~ ${\cal M}_\delta^{ss}(d,r,{\cal O}_D^{\oplus r})$ of semistable pairs with $0<\delta\leq r\cdot l(D)$ exist as projective schemes of generic dimension $r^2(g-1)+r^2\cdot l(D)$ \end{corollary}(cp. \cite{Se}, III.5., there is a misprint in the dimension formula in \cite{Se})\\ There are two new features in the theory of pairs compared with the moduli spaces~ of vector bundles. First, to compactify one really has to use sheaves with torsion supported on $D$. Secondly, the set of semistable pairs which are not stable may have only codimension 2, whereas the set of semistable vector bundles~ which are not stable is at least $2g-3$ codimensional in the moduli space~ of all semistable vector bundles. To give an example we describe the moduli space ${\cal M}_1^{ss}(0,2,k(P)^{ \oplus 2})$ of sheaves of rank two and degree zero with a level structure at a reduced point $P\in X$ with $\delta=1$. Here we try to compute the S-equivalence in geometric terms, which is not clear to us in the general context.\\ The stability conditions say\begin{itemize} \item[i)] $\deg{\cal G}(\leq)-\frac{1}{2}$ for all rank one subsheaves ${\cal G}\subset{\ke_{\alpha}}={\rm Ker }\alpha$. \item[ii)] $\deg{\cal G}(\leq)\frac{1}{2}$ for all rank one subsheaves ${\cal F}\subset{\cal E}$. \item[iii)] $l({\cal E}/{\ke_{\alpha}})(\geq)1$ \item[iv)] $l({\rm T}({\cal E}))(\leq)1$ \item[v)] $\alpha$ is injective on the torsion ${\rm T}({\cal E})$.\end{itemize} Therefore the sheaves ${\cal E}$ occuring in semistable pairs in ${\cal M}_1^{ss}(0,2,k(P)^{\oplus 2})$ are either locally free or of the form ${\cal F}\oplus k(P)$ with ${\cal F}$ locally free.\\ First we classify all pairs $({\cal E},\alpha)$ with locally free ${\cal E}$. By ii) such a bundle ${\cal E}$ has to be semistable as a bundle. If ${\cal E}$ is a stable bundle, then every pair $({\cal E},\alpha)$ with an arbitrary $\alpha\not=0$ is semistable and is stable if and only if ${\rm rk}(\alpha)=2$, i.e. $\alpha(P)$ is bijective. If ${\cal E}$ is only semistable there are two cases to consider: Either a) ${\cal E}\cong{\cal L}_1\oplus{\cal L}_2$, where ${\cal L}_1$ and ${\cal L}_2$ are line bundles of degree $0$ or b) ${\cal E}$ is given as a nontrivial extension of two such line bundles.\\ a) If ${\cal L}_1\cong{\cal L}_2$ then $({\cal E},\alpha)$ is semistable if and only if $\alpha$ is bijective. If ${\cal L}_1\not\cong{\cal L}_2$ then $({\cal E},\alpha)$ is semistable if and only if none of the restrictions $\alpha_{|{\cal L}_i (P)}$ is trivial. \\ If ${\cal L}_1\cong{\cal L}_2$ and $\alpha$ bijective, the pair $({\cal E},\alpha)$ is in fact stable, since $l({\cal E}/{\ke_{\alpha}})=2>1$. If $\alpha$ is only of rank one we can always find an inclusion ${\cal L}_1\subset{\cal L}_1\oplus{\cal L}_1$ with ${\cal L}_1={\rm Ker }(\alpha_{|{\cal L}_1})$, which contradicts i). For ${\cal L}_1\not\cong{\cal L}_2$ one has to consider line bundles ${\cal L}\subset{\cal L}_1\oplus{\cal L}_2$ of degree zero with ${\cal L}={\rm Ker }(\alpha_{|{\cal L}})$, because that is the only possibility to contradict i). But such a line bundle has to be isomorphic to one of the summands with the natural inclusion. Therefore the stability condition is equivalent to $\alpha_{|{\cal L}_i}\not=0$.\\ b) If ${\cal E}$ is a nonsplitting extension $$\ses{{\cal L}_1}{{\cal E}}{{\cal L}_2}$$ a pair $({\cal E},\alpha)$ is semistable iff ${\cal L}_1\not={\rm Ker }(\alpha_{|{\cal L}_1})$. That is, since every line bundle ${\cal L}\subset{\cal E}$ of degree $0$ either defines a splitting of the sequence or maps isomorphically to ${\cal L}_1$.\\ The next step is to determine all semistable pairs $({\cal F}\oplus k(P),\alpha)$. Here we claim, that such a pair is semistable iff ${\cal F}$ is stable and $\alpha_{|k(P)}$ is injective. Let $({\cal F}\oplus k(P),\alpha)$ be semistable and ${\cal L}\subset{\cal F}$ a line bundle. Then, since ${\cal L}\oplus k(P)\subset{\cal F}\oplus k(P)$, the semistability conditions for the pair give $\deg({\cal L}\oplus k(P))\leq{1}/{2}$, i.e. $\deg{\cal L}\leq-{1}/{2}={\deg{\cal F}}/{2}$. Let now ${\cal F}$ be a stable bundle. If ${\cal L}$ is a rank one subsheaf of ${\cal F}\oplus k(P)$, then it either injectively injects into ${\cal F}$ or has torsion part $k(P)$ and therefore satisfies the required inequality.\\ Next we look at the isomorphism classes of stable pairs. If ${\cal E}$ is a stable bundle, two pairs $({\cal E},\alpha)$ and $({\cal E},\alpha')$ are isomorphic if and only if $\alpha$ and $\alpha'$ differ by a scalar. For ${\cal E}$ of the form ${\cal L}_1\oplus{\cal L}_2$ the automorphism group of ${\cal E}$ is either $\hbox{\sym \char '103}^*\times\hbox{\sym \char '103}^*$ for ${\cal L}_1\not\cong{\cal L}_2$ or ${\rm GL}(2)$ for ${\cal L}_1\cong{\cal L}_2$. In the first case the set of isomorphism classes of stable pairs for fixed ${\cal E}$ is isomorphic to ${\rm PGL}(2)/\{{\beta\,0\choose0\, \gamma}|\beta,\gamma\in\hbox{\sym \char '103}^*\}$. In the latter case all stable pairs are isomorphic for fixed ${\cal E}$, they all define the same point in the moduli space. If ${\cal E}$ is given by a nonsplitting exact sequence$$\ses{{\cal L}_1}{{\cal E}}{{\cal L}_2}$$ the automorphism group is either $\hbox{\sym \char '103}^*$ for ${\cal L}_1\not \cong {\cal L}_2$ or $\{{\beta\,\gamma\choose0\,\beta}|\beta\in\hbox{\sym \char '103}^*,\gamma\in\hbox{\sym \char '103}\}$ for ${\cal L}_1\cong{\cal L}_2$. Therefore every such extension induces either a ${\rm PGL}(2)-$family of stable pairs in the moduli space~ or a ${\rm PGL}(2)/\{{\beta\, \gamma\choose0\,\beta}|\beta\in\hbox{\sym \char '103}^*,\gamma\in\hbox{\sym \char '103}\}-$family of stable pairs in the moduli space.\\ In order to describe the S-equivalence we claim, that the orbit of a pair $({\cal E},\alpha)$ is closed if and only if either the pair is stable, i.e. ${\cal E}$ is a semistable vector bundle~ and $\alpha$ of rank two, or ${\cal E}$ is of the form ${\cal F}\oplus k(P)$ with a stable vector bundle~ ${\cal F}$ of degree $-1$ and $\alpha_{|{\cal F}}=0$.\\ If ${\cal E}$ is locally free and $\alpha$ of rank one there is an extension of the form $$\ses{{\ke_{\alpha}}}{{\cal E}}{k(P)}~~.$$If $\psi\in{\rm Ext}^1(k(P),{\ke_{\alpha}})$ denotes the extension class one can easily construct a family of pairs over $\hbox{\sym \char '103}\cdot\psi$, which gives the pair $({\cal E},\alpha)$ outside $0$ and $({\ke_{\alpha}}\oplus k(P),\alpha \cdot{\rm pr}_{k(P)})$ on the special fibre, where ${\rm pr}_{k(P)}$ is the projection to $k(P)$. Obviously this pair is again semistable. If $({\cal F}\oplus k(P),\alpha)$ is a semistable pair with $\alpha=(\alpha_1,\alpha_2)$, the pair $({\cal F}\oplus k(P),(t\cdot\alpha_1,\alpha_2))$ converges constantly to a pair with $ \alpha_{|{\cal F}}=0$ for $t\to 0$. In order to prove the claim it is therefore enough to show that the orbit of such a pair is closed. If there were a family parametrized by a curve with a point $O$, which outside $O$ were isomorphic to a fixed semistable pair $({\cal F}\oplus k(P),\alpha)$ with $\alpha_{|{\cal F}}=0$ and over this point $O$ isomorphic to another pair of this kind, the family of the kernels would give a family of stable bundles, which would be constant for all points except $O$. Since the stable bundles are separated, it has to be constant everywhere. Finally, using the constance of the images of the maps $\alpha$ outside the point $O$ one concludes that the family of pairs is constant.\\ If ${\cal M}_1^s(0,2,k(P)^{\oplus 2})$ denotes the subset of all stable pairs we summarize the results in the following proposition \begin{proposition}\begin{itemize} \item[i)] ${\cal M}_1^{ss}(0,2,k(P)^{\oplus 2})\setminus {\cal M}_1^s(0,2,k(P)^{\oplus 2})\cong \hbox{\sym \char '120}_1\times U(-1,2)$, where $U(-1,2)$ is the moduli space~ of stable rank two vector bundles~ of degree $-1$. \item[ii)] There is a morphism ${\cal M}_1^s(0,2,k(P)^{\oplus 2})\to U(0,2)$, which is a ${\rm PGL}(2) -$fibre bundle over $U(0,2)^s$ and whose fibre over a point $[{\cal L}_1\oplus{\cal L}_2]\in U(0,2)\setminus U(0,2)^s$ is isomorphic to $${\rm PGL}(2)/\{{\beta\,0\choose0\,\gamma}| \beta,\gamma\in\hbox{\sym \char '103}^*\} \cup\{{\rm PGL}(2)\times\hbox{\sym \char '120}({\rm Ext}^1({\cal L}_1,{\cal L}_2)^{\rm v})\}$$ for ${\cal L}_1\not\cong{\cal L}_2$ and isomorphic to $$\{pt\}\cup\{{\rm PGL}(2)/\{{\beta\,\gamma\choose0\,\beta} |\beta\in\hbox{\sym \char '103}^*,\gamma\in\hbox{\sym \char '103}^*\}\times\hbox{\sym \char '120}({\rm Ext}^1({\cal L}_1,{\cal L}_2)^{\rm v})\}$$ for ${\cal L}_1\cong{\cal L}_2$.\end{itemize}\end{proposition} {\it Proof:} The isomorphism in {\it i)} is given by $({\cal F}\oplus k(P),\alpha)\mapsto (\alpha(k(P)),{\cal F})$. The morphism in {\it ii)} is induced by the universality property of the moduli space.\hspace*{\fill}\hbox{$\Box$} In particular the dimension of ${\cal M}_1^{ss}(0,2,k(P)^{\oplus2})$ is $4g$ ($g$ is the genus of the curve) and the dimension of ${\cal M}_1^{ss}(0,2,k(P)^{\oplus2})\setminus{\cal M}_1^s(0,2,k(P)^{\oplus2})$ is $4g-2$. Thus the codimension is two, independently of the genus. Finally we want to study the situation in the two dimensional case. Let $X$ be a surface with an effective divisor $C$ and ${\cal E}_0$ be a vector bundle~ of rank $r$ on $C$. A framing of a vector bundle~ ${\cal E}$ of rank $r$ on $X$ along $C$ in the strong sense as introduced in \cite{l4} is an isomorphism $\alpha:{\cal E}_C\cong{\cal E}_0$. In \cite{l4} the question of the existence of moduli spaces~ for such pairs $({\cal E},\alpha)$ was asked ($\alpha$ denotes the isomorphism as well as the composition of this isomorphism with the surjection ${\cal E}\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow{\cal E}_C$). In fact, under additional conditions, fine moduli spaces~ for such framed bundles were constructed as algebraic spaces. These additional conditions are: $C$ is good and ${\cal E}_0$ is simplifying. If $C=\sum b_i C_i$ with prime divisors $C_i$ and $b_i>0$ $C$ is called good if there exist nonnegative integers $a_i$, such that $\sum a_i C_i$ is big and nef. The vector bundle~ ${\cal E}_0$ is called simplifying if for two framed bundles ${\cal E}$ and ${\cal E}'$ the group $H^0(X,{\cal H} om({\cal E},{\cal E}')(-C))$ vanishes. At the first glance it is surprising that there are no further stability conditions for such pairs. However, in many situations the general stability conditions of chapter one are hidden behind the concept of framed bundles.\\ \begin{definition} For $0<s<r$ the number $\nu_s({\cal E}_0,C_i)$ is defined as the maximum of ${\deg({\cal F})}/{s}-{\deg({{\cal E}_0}|_{C_i})}/{r}$, where ${\cal F}\subset{{\cal E}_0}|_{C_i}$ is a vector bundle~ of rank $s$.\end{definition} In the following we assume, that there are nonnegative integers $a_i$, s.t. $H=\sum a_i C_i$ is ample. This is equivalent to saying that $X\setminus C$ is affine. \begin{proposition} If $\delta_1$ is positive with $$\max_{0<s<r}\{{r\cdot s}/({r-s})\sum a_i\nu_s({\cal E}_0 ,C_i)\}<\delta_1<(r-1)(C.H)~,$$ then every vector bundle~ ${\cal E}$ of rank $r$ together with an isomorphism $\alpha:{\cal E}_C\cong{\cal E}_0$ forms a $\mu-$stable pair $({\cal E},\alpha)$.\end{proposition} {\it Proof:} The $\mu-$stability for such pairs is defined by the following two inequalities:\\ {\it i)} ${\deg{\cal G}}/{{\rm rk}{\cal G}}<{\deg{\cal E}}/{r}-{\delta_1}/{r}$ for every vector bundle~ ${\cal G}\subset{\ke_{\alpha}}$ with $0<{\rm rk}{\cal G}<r$ and\\ {\it ii)} ${\deg{\cal G}}/{{\rm rk}{\cal G}}<{\deg{\cal E}}/{r}+\delta_1({r-{\rm rk}{\cal G}})/( {r\cdot{\rm rk}{\cal G}})$ for every vector bundle~ ${\cal G}\subset{\cal E}$ with $0<{\rm rk}{\cal G}<r$.\\ We first check {\it ii)}. It is enough to consider vector bundles~ ${\cal G}$, s.t. the quotient ${\cal E}/{\cal G}$ is torsionfree. In particular we can assume, that ${\cal G}_{C_i}\to{\cal E}_{C_i}$ is injective. Then we conclude ${\deg{\cal G}}/{{\rm rk}{\cal G}}={c_1({\cal G}).H}/{{\rm rk}{\cal G}}= \sum a_i{\deg({\cal G}_{C_i})}/{{\rm rk} {\cal G}}\leq\sum a_i({\deg({\cal E}_0)_{C_i}}/{r}+\nu_{{\rm rk}{\cal G}}({\cal E}_0,C_i))= {\deg{\cal E}}/{r}+\sum a_i\nu _{{\rm rk}{\cal G}}({\cal E}_0,C_i)<{\deg{\cal E}}/{r}+(({r-{\rm rk}{\cal G}})/{r\cdot{\rm rk}{\cal G}})\delta_1$. To prove {\it i)} one uses ${\ke_{\alpha}}={\cal E}(-C)$ and {\it ii)}: For ${\cal G}\subset{\ke_{\alpha}}$ the inequality {\it ii)} applied to ${\cal G}(C)\subset{\cal E}$ implies ${\deg{\cal G}}/{{\rm rk}{\cal G}}+{C.H}={\deg{\cal G}(C)}/ {{\rm rk}{\cal G}}<{\deg{\cal E}}/{r}+(({r-{\rm rk}{\cal G}})/{r\cdot{\rm rk}{\cal G}})\delta_1$. Therefore $\delta_1<(r-1)C.H$ suffices to give {\it i)}.\hspace*{\fill}\hbox{$\Box$} \begin{corollary}\label{framedareproj} For $\max_{0<s<r}\{{r\cdot s}/({r-s})\sum a_i\nu_s({\cal E}_0 ,C_i)\}<(r-1)(C.H)$ and $C$, such that there exists an effective, ample divisor $H$, whose support is contained in $C$, the moduli spaces ${\cal M}^{fr}_{X/C/{\cal E}_0/\chi}$ of framed vector bundles~ are quasi-projective.\end{corollary} {\it Proof:} These moduli spaces~ are in fact open subsets of the $\mu-$stable part of the moduli space~ of all semistable pairs $({\cal E},\alpha)$.\hspace*{\fill}\hbox{$\Box$} There is a special interest in the case ${\cal E}_0\cong{\cal O}_C^{\oplus r}$, since the corresponding moduli spaces~ are in fact invariants of the affine surface $X\setminus C$ (\cite{L2}). In this case all the numbers $\nu_s({\cal E}_0,C_i)$ vanish. Therefore a trivially framed bundle gives a $\mu-$stable pair $({\cal E},\alpha)$ with respect to every $\delta_1<(r-1)C.H$.\\ In (\cite{l4},2.1.5.) a sufficient condition for a bundle ${\cal E}_0$ to be simplifying is proven: If ${\rm Hom}({\cal E}_0,{\cal E}_0(-kC))=0$ for all $k>0$, then ${\cal E}_0$ is simplifying. We remark that at least in the rank two case this condition is closely related to the numerical condition we gave. It is possible to make the condition finer, because in the definition of the numbers $\nu_{C_i}$ it is sufficient to take the maximum over those bundles, which actually live on $X$. \newpage

9711

alg-geom/9711028

en

https://arxiv.org/abs/alg-geom/9711028

alg-geom/9711028

Frederic Han

F.Han

On moduli spaces of 4- or 5-instanton bundles

Latex, 49 pages

We study the scheme of multi-jumping lines of an $n$-instanton bundle mainly for $n\leq 5$. We apply it to prove the irreducibility and smoothness of the moduli space of 5-instanton. Some particular situations with higher $c_2$ are also studied.

alg-geom

\section{Some properties of $S$ for $c_{\bf{2}}\bf{\leq 5}$} \subsection{A link between trisecant lines to $S$ and jumping plane} We'd like to study here the lines $d$ of ${\rm I} \! {\rm P}_5$ which meet $M$ in length 3 or more. Those lines have to be included in the Grassmannian G, and each of them can be identified to a plane pencil of lines of ${\rm I} \! {\rm P}_3$. Let $h$ be the $\beta $-plane containing $d$, and $H$ the associated plane of ${\rm I} \! {\rm P}_3$. As $R^2q_{*}$ is zero, we have an isomorphism between the restriction of $R^1q_{*}p^{*}E$ to $h$, and $R^1q_{*}p^{*}E_H$, where $% E_H$ is the restriction of $E$ to $H$, and also between $R^1q_{*}p^{*}E_H$ restricted to a line of $h$ and the sheaf constructed analogously when blowing up the associated point of $H$. (So the projections over $H$ and ${\rm I} \! {\rm P}_1$ will be still denoted by $p$ and $q$). \begin{proposition} \label{pinceaux}For $c_2\leq 5$, let $d$ be a line of ${\rm I} \! {\rm P}_5$ meeting $M$ in a 0 dimensional scheme of length 3 or more not containing jumping lines of order 3 or more, then the plane $H$ associated to $d$ is not a stable plane for $E$. \end{proposition} Let $d$ be such a line, and assume that $H$ is stable (ie that $E_H$ is stable), and blow up $H$ at the point $D$ included in all the lines represented by $d$. The resolution of this blow up is: \begin{center} $0\longrightarrow {\cal O}_{d\times H}(-\tau -\sigma )\longrightarrow {\cal O}_{d\times H}\longrightarrow {\cal O}_{\widetilde{H}}\longrightarrow 0$ \end{center} Twisting it by $p^*E_H$ gives using the functor $q_*$: \begin{center} $0\rightarrow q_*p^*E_H\rightarrow H^1E_H(-1)\otimes {\cal O}_d (-1) \longrightarrow H^1E_H\otimes {\cal O}_d \rightarrow R^1q_*p^*E_H \rightarrow 0$ \end{center} The length induced by the scheme of multi-jumping lines on $d$ is $c_1(R^1q_*p^*E_H)$ and is at least 3 by hypothesis. The sheaf $q_*p^*E_H$ is locally free of rank $-h^1E_H+h^1E_H(-1)=2$ over $d={\rm I} \! {\rm P}_1$. So it has to split in ${\cal O}_d(-a) \oplus {\cal O}_d (-b)$ with $a,b>0$ and $a+b=h^1E_H(-1)-c_1(R^1q_*p^*E_H)$. But $h^1E_H(-1)$ is $c_2$, so it gives already a contradiction for $c_2\leq 4$. For $c_2=5$, one has necessarily $a=b=1$, so $h^0(q_*p^*E_H(\sigma))$ which is also $h^0({\cal J}_D \otimes E_H(1))$ is 2. Those 2 sections $s_1$ and $s_2$ are thus proportional on the conic $Z_{s_1\wedge s_2}$ which has to be singular in $D$ because this point is in the first Fitting ideal of $2{\cal O}_H \rightarrow E_H(1)$. If $Z_{s_1\wedge s_2}$ is made of 2 distinct lines $d_1$ and $d_2$, we have the following exact sequence because both $s_1$ and $s_2$ vanish at $D$: \begin{center} $0\longrightarrow 2{\cal O}_H\stackrel{(s_1,s_2)}{\longrightarrow }% E_H(1)\longrightarrow {\cal O}_{d_1}(\alpha)\oplus {\cal O}_{d_2}(\beta)\longrightarrow 0$ \end{center} Computing Euler-Poincar\'e characteristics gives $\alpha+\beta=-3$, so one of them is less or equal to $-2$, thus one of the $d_i$ is a \prefix{k\geq 3}jumping line containing $D$ which contradicts the hypothesis. It is the same when $Z_{s_1\wedge s_2}$ is a double line $d_1$. We have the exact sequence: \begin{center} $0\longrightarrow 2{\cal O}_H\stackrel{(s_1,s_2)}{\longrightarrow } E_H(1)\longrightarrow {\cal L} \longrightarrow 0$ \end{center} where ${\cal L}$ is a sheave of rank 1 on the plane double structure of $d_1$ (in fact ${\cal L}$ is not locally free at $D$). We have the linking sequence: \begin{center} $0\longrightarrow {\cal O}_{d_1}(\alpha)\longrightarrow {\cal L} \longrightarrow {\cal O}_{d_1}(\beta)\longrightarrow 0$ \end{center} once again $\alpha+\beta=-3$, so either $h^0{\cal O}_{d_1}(\beta)<h^1{\cal O}_{d_1}(\alpha)$, or $h^1{\cal O}_{d_1}(\beta) \neq 0$, but in both cases $R^1q_*p^*{\cal L}_{(d_1)}$ is non zero, so $d_1$ would be a \prefix{k\geq 3}jumping line containing $D$ as previously. \bigskip We can't hope such a result in a jumping plane (ie semi-stable but not stable), but we will show in \ref{pinceauxsauteurfini} that the problems arise only at finitely many points of the plane. \begin{lemma} \label{secantes} Let $F$ be a bundle over ${\rm I} \! {\rm P}_n$ with $c_1F=0$. If $F(k)$ has a section of vanishing locus $Z$, then for any $i\geq k-1$, the scheme of jumping lines of order at least $i+2$ is isomorphic to the scheme of lines at least \prefix{i+k+2}secant to $Z$. \end{lemma} We deduce from the following sequence using the standard construction: \begin{center} $0\longrightarrow {\cal O}_{{\rm I} \! {\rm P}_n} \longrightarrow F(k)\longrightarrow {\cal J}_Z(2k) \longrightarrow 0$ \end{center} an isomorphism $R^1q_*p^*F(i) \simeq R^1q_*p^*{\cal J}_Z(k+i)$ when $i\geq k-1$. And the supports of those sheaves are by definition (Cf [G-P]) the above schemes. \begin{lemma}\label{pinceauxsauteurfini} For $c_2\leq 5$, any $\beta -$plane associated to a jumping plane contain only a finite number of lines of ${\rm I} \! {\rm P}_5$ meeting $M$ in a scheme of length 3 or more made only of 2-jumping lines. \end{lemma} If $H$ is a jumping plane, we can take $k=0$ in lemma \ref{secantes} and study the bisecant lines to $Z$. To prove the lemma it is enough to show that for any point $N$ not in $Z$ with no \prefix{k\geq 3}jumping lines through $N$, the line $N ^{\rm v}$ is not trisecant to $M$ where $N ^{\rm v}$ is the lines of $H$ containing $N$. So let $N$ be such a point, and study the bisecant lines to $Z$ in $N ^{\rm v}$ by blowing up $H$ at $N$. One has from the standard construction the exact sequence because $N\not\in Z$: \begin{center} $0\rightarrow q_*p^*{\cal J}_Z \rightarrow H^1{\cal J}_Z(-1) \otimes {\cal O}_{N ^{\rm v}}(-1) \longrightarrow H^1{\cal J}_Z \otimes {\cal O}_{N ^{\rm v}} \rightarrow R^1q_*p^*{\cal J}_Z\rightarrow 0$ \end{center} where $q_*p^*{\cal J}_Z$ is isomorphic to ${\cal O}_{{\rm I} \! {\rm P}_1}(-a)$ for some $a>0$ with $a=\deg Z-c_1(R^1q_*p^*{\cal J}_Z)$ because there are only a finite number of bisecant to $Z$ through $N$ ($N\not\in Z$). So if $Z$ has at least 3 bisecant (with multiplicity) through $N$, then $a\leq 2$ for $c_2\leq 5$. If $a=1$, then $Z$ is included in a line containing $N$ which must be a $c_2$-jumping line, and it conflicts with the hypothesis. So only the case $a=2$, and $c_2=5$ is remaining. But then, the section of $q_*p^*{\cal J}_Z(2\sigma)$ would give a conic $C$ singular in $N$ and containing $Z$. We have the linking sequence where $d_1$ may be equal to $d_2$: \begin{center} $0 \longrightarrow {\cal O}_{d_1}(-1) \longrightarrow {\cal O}_C \longrightarrow {\cal O}_{d_2} \longrightarrow 0$ \end{center} As previously $Z$ can't be included in a line so it cuts $d_2$ and $h^0({\cal J}_Z \otimes{\cal O}_{d_2})=0$. Twisting the previous sequence by ${\cal J}_Z$ gives when computing Euler-Poincar\'e's characteristics $h^1({\cal J}_Z \otimes{\cal O}_{d_1}(-1))+h^1({\cal J}_Z \otimes{\cal O}_{d_2})=4$ because $Z$ is included in $C$. So one of the $d_i$ would be a trisecant line to $Z$, thus it would be a 3-jumping line through $N$ which contradicts the hypothesis and gives the lemma. \begin{proposition} \label{infinitebis}Any plane $H$ containing infinitely many multi-jumping lines is a jumping plane. In that situation, the support of $R^1q_*p^*E$ induce on the $\beta $-plane associated to $H$ a reduce line with some possible embedded points.\\When $c_2=5$, there is a 3-jumping line in this pencil.\\When $c_2=4$ there are 4 embedded points. \end{proposition} Let's first notice that $H$ can't be a stable plane. So assume that it is stable, then, for $c_2 \leq 5$, the bundle $E_H(1)$ has a section vanishing on some scheme $Z$ of length at most 6. We showed in the proof of \ref{pinceauxsauteurfini} the link between multi-jumping lines in $H$ and trisecant lines to $Z$, but if $Z$ had infinitely many trisecant lines (necessarily passing through a same point of $Z$), then on the blow up of $H$ at this point, the sheaf $p^*E(\tau-3x)$ would have a section (where $x=\tau-\sigma$ is the exceptional divisor) which is not possible because it has a negative $c_2$. So $H$ is a jumping plane and the section of $E_H$ has infinitely many bisecant lines (necessarily containing a same point). Let's blow up $H$ at this point, then $p^*E_H$ has a section vanishing 2 times on the exceptional divisor $x$. The residual scheme of $2x$ is empty when $c_2=4$ and a single point when $c_2=5$. The line of $H$ containing this point and the point blown up is thus a trisecant line to the vanishing of the section of $E_H$, so it is a 3-jumping line. Although it is not required by the following, it is interesting to understand more this situation. For example, when $c_2=4$ we can understand the scheme structure of multi-jumping lines in this plane. Let $s$ be the section of $E_H$, $Z$ its vanishing locus and ${\cal J}_Z$ its ideal. In fact $Z$ is the complete intersection of 2 conics singular at the same point. Let $I$ be the point/line incidence variety in $H ^{\rm v} \times H$, and pull back on $I$ the resolution of ${\cal J}_Z$: \begin{center} $R^1q_*{\cal O}_I(-4\tau )\longrightarrow 2R^1q_*{\cal O}_I(-2\tau ) \longrightarrow R^1q_*p^* {\cal J}_Z \longrightarrow 0$ \end{center} to obtain by relative duality: \begin{center} $q_*{\cal O}_I(2\tau +\sigma ) ^{\rm v} \longrightarrow 2q_* {\cal O}_I(-\sigma ) \longrightarrow R^1q_*p^* {\cal J}_Z \longrightarrow 0$ \end{center} As $I= {\rm I} \! {\rm P}(\Omega _{H ^{\rm v}}(2) ^{\rm v})$ with $0\rightarrow \Omega _{H ^{\rm v}}(2) \rightarrow 3 {\cal O}_{H ^{\rm v}}(1) \rightarrow {\cal O}_{H ^{\rm v}}(2) \rightarrow 0$, one has: \hbox{$q_* {\cal O}_I(k\tau )=Sym_k(\Omega _{H ^{\rm v}}(2))$}, and we are reduced to study the degeneracy locus of a map: $2{\cal O}_{H ^{\rm v}} \longrightarrow Sym_2(\Omega _{H ^{\rm v}}(2))$. This locus contain a reduced line because through a general point $P$ of $H$ there is no conic singular in $P$ containing $Z$ so $a\geq 3$ in the proof of \ref{pinceauxsauteurfini}. So the residual scheme of this line in the degeneracy locus has the good dimension, so we can compute its class with the method of the appendix \ref{appendiceresiduel}, which is simpler here because the excess is a Cartier divisor. So this locus is given by: \begin{center} $c_2(Sym_2(\Omega _{H ^{\rm v}}(2)))-h.c_1(Sym_2(\Omega _{H ^{\rm v}}(2)))+h^2=4h^2$ \end{center} where $h$ is the hyperplane class of $H ^{\rm v}$. The geometric interpretation of this residual locus should be that it is made of twice the lines which occur as doubled lines in the pencil of singular conics containing $Z$. So the ideal of the multi-jumping lines of $E_H$ should look like $(y^2,yx^2(x-1)^2)$ in $H ^{\rm v}$. \subsection{Some preliminary results} In the last section we studied the link between trisecant lines to $S$ and jumping planes. We can notice that some hypothetical bad singularities of $S$ give rise to vector bundles with a line in ${\rm I} \! {\rm P}_3 ^{\rm v}$ of jumping planes. Unfortunately, such bundles exist, for example when the instanton has a $c_2$-jumping line (Cf [S]). A well understanding of those bundle would give many shortcuts, particularly when $c_2=4$, but the following example shows that some may still be unknown. \begin{example} There are 4-instanton with a 2-jumping line such that every plane containing this line is a jumping plane. \end{example} Let's construct this bundle from an elliptic curve $C$ and two degree 2 invertible sheaves ${\cal L}$ and ${\cal L}^{\prime}$ such that $({\cal L} ^{\rm v} \otimes {\cal L}^{\prime})^{\otimes 4}={\cal O}_C$, where 4 is the smallest integer with this property. Let $\Sigma\stackrel{\pi }{\rightarrow }C$ be the ruled surface ${\rm I} \! {\rm P}({\cal L} \oplus {\cal L}^{\prime})$ and $\overline{\Sigma}$ its quartic image in ${\rm I} \! {\rm P}_3={\rm I} \! {\rm P} (H^0({\cal L}) \oplus H^0({\cal L}^{\prime}))$. Let $D=\pi^*({\cal L} ^{\rm v})^{\otimes 4} \otimes {\cal O}_{\Sigma}(4)$. The linear system $|D|$ is \mbox{$h^0(({\cal L} ^{\rm v})^{\otimes 4}\otimes Sym_4({\cal L}\oplus {\cal L}^{\prime}))-1$} dimensional which is 1 due to the relation between ${\cal L}$ and ${\cal L}^{\prime}$. So define the bundle $E$ by the following exact sequence: \begin{center} $0\longrightarrow E(-2)\longrightarrow 2{\cal O}_{{\rm I} \! {\rm P}_3} \longrightarrow \phi _{*} (\pi ^{*}({\cal L} ^{\rm v})^{\otimes 4}\otimes {{\cal O}}_{\Sigma}(4))\longrightarrow 0$ \end{center} where $\phi$ is the morphism from $\Sigma$ to ${\rm I} \! {\rm P}_3$, and where the dimension of $|D|$ proves that $E$ is an instanton. Let $s$ and $s^{\prime}$ be the only sections of $\pi ^{*}{\cal L} ^{\rm v} \otimes {\cal O}_{\Sigma}(1)$ and of $\pi ^{*}{\cal L} ^{\prime \rm v}\otimes {\cal O}_{\Sigma}(1)$. The linear system $|D|$ is made of curves of equation $\lambda s^4+ \mu s^{\prime4}$ with generic element a degree 8 smooth elliptic curve section of $E(2)$. Furthermore, when $\lambda $ or $\mu$ vanishes, this element of $|D|$ provides to the double line of $\overline{\Sigma}$ associated to $s$ or $s^{\prime}$ a multiple structure. Every plane containing one of those lines (notes $d$ and $d^{\prime}$) must be a jumping plane, and every ruling of $\overline{\Sigma}$ is a 4-secant to those section of $E(2)$, so they are 2-jumping lines. In any plane containing $d$ or $d^{\prime}$, the jumping section vanishes on a scheme of ideal $(y^2,x(x-1))$ where $y$ is the equation of $d$ or $d^{\prime}$. So the lines $d$ or $d^{\prime}$ must be 2-jumping lines, and those sections of $E(2)$ induce on the exceptional divisor of ${\rm I} \! {\rm P}_3$ blown up in $d$ or $d^{\prime}$ a curve of bidegree $(2,2)$. \begin{remark} \label{dtemultiple}For every $c_2$, if there is a line $\delta $,at least 1-jumping for some instanton $E$ such that ${\cal J} _\delta ^k\otimes E(k)$ has a section, then $\delta $ is a $k$-jumping line, and for $k \geq 2$ this section is irreducible. \end{remark} Let's first notice that if ${\cal J} _\delta^k\otimes E(k)$ has a section when $\delta$ is a jumping line, then this section is set-theoretically $\delta$. Indeed, consider the blowing up $\widetilde{{\rm I} \! {\rm P}_3^{\prime}}$ of ${\rm I} \! {\rm P}_3$ along $\delta$, and denote by $p^{\prime}$ et $q^{\prime}$ the projections of $\widetilde{{\rm I} \! {\rm P}_3^{\prime}}$ on ${\rm I} \! {\rm P}_3$ et ${\rm I} \! {\rm P}_1$. The restriction of the section of $p^{\prime *}E(k\sigma )$ to the exceptional divisor is a section of ${\cal O}_{{\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1}(k,a) \oplus {\cal O}_{{\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1}(k,-a)$ where $\delta$ is a $a$-jumping line. As $a>0$, this section is in fact a section of ${\cal O}_{{\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1}(k,a)$, so it has no embedded nor isolated points. So the section of $p^{\prime *}E(k\sigma )$ don't have irreducible components which meet the exceptional divisor, but the section of ${\cal J}^k_{\delta} \otimes E(k)$ has to be connected in ${\rm I} \! {\rm P}_3$ for $k\geq 2$ because $h^1E(-2)=0$. So this section must be set-theoretically $\delta$ when $k\geq 2$. On another hand, the class in the Chow ring of $\widetilde{{\rm I} \! {\rm P}_3^{\prime}}$ of a divisor of the exceptional ${\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1$ of bidegree $(k,a)$ is: $k\tau ^2+(a-k)\tau\sigma $, but this curve is included in a section of $p^{\prime *}E(k\sigma )$ of class $c_2\tau^2$ which gives $a=k$. \bigskip The main problem encountered to classify those bundle comes from the fact that the section of $p^{\prime *}E(k\sigma )$ may have a multiple structure not included in the exceptional divisor as in the previous example. So we will not try to classify all those bundle, but we will need the following: \begin{lemma} \label{dteplansaut}When $c_2=4$ and $h^0E(1)=0$, if there is a \prefix{k\leq 2}jumping line $d$, such that every plane containing $d$ is a jumping one, then ${\cal J}_d ^{\otimes 2}\otimes E(2)$ has a section \end{lemma} Still blow up ${\rm I} \! {\rm P}_3$ along $d$, the class of the exceptional divisor $x={\rm I} \! {\rm P}_1 \times d$ is $\tau - \sigma$, and we have the exact sequence: \begin{center} $0\longrightarrow {\cal O}_{{\rm I} \! {\rm P}_3\times{\rm I} \! {\rm P}_1}(-\tau -\sigma )\longrightarrow {\cal O}_{{\rm I} \! {\rm P}_3\times {\rm I} \! {\rm P}_1}\longrightarrow {\cal O}_{\widetilde{{\rm I} \! {\rm P}_3^{\prime}}}\longrightarrow 0$ \end{center} which gives when twisted by $p^{\prime *}E(\tau )$ using the functor $q_*^{\prime }$ the following exact sequence because $h^0E(1)=0$. \begin{center} $0\rightarrow \!q_{*}^{\prime }p^{\prime *}E(\tau )\rightarrow \!H^1E\otimes {\cal O}_{{\rm I} \! {\rm P}\!_1}(-1) \kercoker{K} \!\!H^1E(1)\otimes {\cal O}_{{\rm I} \! {\rm P}\!_1}\rightarrow \!R^1q_{*}^{\prime }p^{\prime *}E(\tau )\rightarrow \!0$ \end{center} The sheaf $R^1q_{*}^{\prime }p^{\prime *}E(\tau )$ is locally free of rank 1 over ${\rm I} \! {\rm P}_1$ because every plane containing $d$ is a jumping one and $c_2=4$. On another hand, the resolution of $x$ in $\widetilde{{\rm I} \! {\rm P}_3^{\prime}}$ gives the exact sequence: \begin{center} $0\rightarrow p^{\prime *}E(\sigma )\rightarrow p^{\prime *}E(\tau )\rightarrow p^{\prime *}E_x(\tau )\rightarrow 0$ \end{center} But $p^{\prime *}E_x(\tau )\simeq {\cal O}_{{\rm I} \! {\rm P}\!_1\times d}(0,-a+1)\oplus {\cal O}_{{\rm I} \! {\rm P}\!_1\times d}(0,a+1)$ with $a\leq 2$, and the map \linebreak\mbox{$R^1q_{*}^{\prime }p^{\prime *}E(\sigma )\rightarrow R^1q_{*}^{\prime }p^{\prime *}E(\tau )$} is a surjection, so $R^1q_{*}^{\prime }p^{\prime *}E(\tau )\simeq {\cal O}_{{\rm I} \! {\rm P}_1}(b)$ with \mbox{$b\geq 1$} due to the surjection of $H^1E\otimes {\cal O}_{{\rm I} \! {\rm P}_1}$ onto $R^1q_{*}^{\prime }p^{\prime *}E$. But $h^1(K)=0$ so \mbox{$h^1(q_{*}^{\prime }p^{\prime *}E(\tau ))\leq 2$} and as $q_{*}^{\prime }p^{\prime *}E(\tau )$ is locally free of rank 3, we can deduce that $h^0(q_{*}^{\prime }p^{\prime *}E(\tau +\sigma ))\neq 0$. So we have a section of ${\cal J}_d\otimes E(2)$ whose restriction to every plane containing $d$ is proportional to the jumping section because the jumping section don't have a vanishing locus included in a line ( $d$ is not $c_2$-jumping). So in every plane containing $d$, the section of ${\cal J}_d E(2)$ must vanish on a conic which has to be twice $d$, so it gives a section of ${\cal J}_d^{\otimes 2}E(2)$. \begin{example} \label{alphaun}There are bundles $E$ with $h^1E(-2)\mbox{mod }2=1$, $c_2=4$ having a line congruence of multi-jumping lines. \end{example} Take 2 skew lines and put on each of them a quadruple structure made by the complete intersection of 2 quadrics singular along those lines. So the disjoint union of these two elliptic quartics gives a section of $E(2)$ where $E$ has $c_2=4$ and $h^1E(-2)\mbox{mod }2=1$ according to [C]. Every line meeting those 2 lines is 4-secant to this section which gives a $(1,1)$ congruence of 2-jumping lines. \begin{lemma} \label{t'hooft}For every $c_2$, the t'Hooft bundles (ie $h^0E(1)\neq 0$) have a 1-dimensional scheme of multi-jumping lines. \end{lemma} We can remark that using deformation theory around such a bundle Brun and Hirshowitz (Cf [B-H]) proved that any instanton in some open subset of the irreducible component of $I_n$ containing the t'Hooft bundles has a smooth curve as scheme of multi-jumping lines. This is much stronger but unfortunately the t'Hooft bundles are not in this open set because they always have a 3-jumping line when $n \geq 3$. ( Take a quadrisecant to the section of $E(1)$ which is made of $n+1$ disjoint lines Cf [H]) \smallskip So, let $s$ be a section of $E(1)$ with vanishing locus $Z$. For any $k\in {\rm I} \! {\rm N}$, according to the \ref{secantes}, the scheme of \prefix{k+2}jumping lines is isomorphic to the scheme of \prefix{k+3}secant lines to $Z$. The lemma is thus immediate when the lines making $Z$ are reduced. If it is not the case, a 2-parameter family of trisecant to $Z$ may arise from 2 kinds of situations: a) There is a line $d$ such that the scheme induce on $d$ by $Z$ has a congruence of bisecant lines which also meet another line $d^{\prime}$ of $Z$. This congruence must then be set-theoretically the lines meeting $d$ and $d^{\prime}$, so for any plane $H$ containing $d^{\prime}$, the lines passing through $d\cap H$ are bisecant to $Z$ at this point. The line $d$ would then be equipped by $Z$ of a multiple structure doubled in every plane containing $d$, which is impossible because those planes would be unstable. b) The scheme induce by $Z$ on some line $d$ (noted $Z_d$) has a 2 -dimensional family of trisecant lines which can't lye in a same plane. Every plane containing $d$ is a jumping one, because the restriction of the section of $E(1)$ to this plane is vanishing on $d$, and has to contain infinitely many multi-jumping lines by hypothesis. As those lines are bisecant to the jumping section which is 0 dimensional, those lines must form a pencil through a point of $d$ by hypothesis. So $d$ is a multi-jumping line which contradicts the \ref{dtemultiple} because $s$ is a section of ${\cal J}_d \otimes E(1)$. \bigskip We can now reformulate using the lemma \ref{t'hooft} the theorem of Coanda (Cf [Co]) in a form which will be used at many times in the following: \begin{theorem} \label{coanda} {\rm \bf (Coanda)} For any $c_2$, if an instanton has not a 1-dimensional family of multi-jumping lines, then it has at most a 1-dimensional family of jumping planes. \end{theorem} Indeed, Coanda showed that any bundle with a 2-dimensional family of jumping planes is either a special t'Hooft bundle which has its multi-jumping lines in good dimension according to the \ref{t'hooft}, or another kind of bundles which are not an instanton. In fact, the last one are in the ''big family'' of Barth-Hulek (Cf [Ba-Hu]). \subsection{A bound on the bidegree of $S$} Let $(\alpha, \beta)$ be the bidegree of $S$, where $\alpha$ (resp $\beta$) is the number of lines of the congruence $S$ passing through a general point of ${\rm I} \! {\rm P}_3$ (ie in a $\alpha$-plane), (resp in a general plane of ${\rm I} \! {\rm P}_3$ (ie in a $\beta$-plane)). \begin{proposition} \label{basdeg}For any $c_2$, if an instanton has at most a 2-dimensional scheme of multi-jumping lines , and if $h^0({\cal J} _P^{\otimes c_2-4} \otimes E(c_2-4))$ is zero for a general point of ${\rm I} \! {\rm P}_3$, then $\alpha \leq 2c_2-6$.\\ Similarly, any stable plane $H$ without \prefix{c_2} and \prefix{c_2-1}jumping lines and such that $h^0({\cal J} _P^{\otimes k}\otimes E_H(k))$ is zero when $P$ is general in $H$ for $0\leq k\leq c_2-4$, then the $\beta$-plane $h$ associated to $H$ cuts the scheme of multi-jumping lines in length at most $2c_2-6$. So if those hypothesis are satified for the general plane then $\beta \leq 2c_2-6$. \end{proposition} {\bf Remark:} The above hypothesis about $h^0({\cal J} _P^{\otimes k}\otimes E_H(k))$ are always satisfied when $c_2=4$ or $5$ because one has in any stable plane $H$ without \prefix{c_2} and \prefix{c_2-1}jumping lines: $h^0(E_H(1))\leq 2$. Furthermore, we will show for those $c_2$, under the assumption of the existence of $S$, that the general member of any 2-dimensional family of planes is stable from the theorem of Coanda stated in the \ref{coanda}, and don't contain \prefix{c_2} or \prefix{c_2-1}jumping lines according to the \ref{maxsauteuses}, \ref{fini4}, and \ref{courbetris}. \bigskip Start first with the bound of $\alpha$, so take an $\alpha$-plane $p$ cutting $S$ in length $\alpha$ and denote by $P$ its associated point of ${\rm I} \! {\rm P}_3$. The standard construction associated to the blow up of ${\rm I} \! {\rm P}_3$ at $P$ gives the following exact sequence above the exceptional divisor $p$: \begin{center} $0\longrightarrow q_*p^*E \longrightarrow H^1E(-1)\otimes ({\cal O}_p \oplus {\cal O}_p(-1)) \longrightarrow H^1E\otimes {\cal O}_p \longrightarrow R^1q_*p^*E \longrightarrow 0$ \end{center} The sheaf $q_*p^*E$ is thus a second local syzygy, so it has a 3-codimensional singular locus hence it is a vector bundle denoted by $F$ in the following. For every instanton, one has $h^1E(-1)=c_2$ and $h^1E=2c_2-2$. So $\alpha$ is given by $\chi(R^1q_*p^*E)=c_2-2+\chi(F)$. The stability of $E$ implies that $F$ has no sections, but it is also locally free of first Chern class $-c_2$, so $\chi(F)=-h^1F+h^0F(c_2-3)$. Take a ${\rm I} \! {\rm P}_1$ in $p$ without any multi-jumping lines. The restriction of this sequence to this ${\rm I} \! {\rm P}_1$ gives an injection of $F_{{\rm I} \! {\rm P}_1}$ in $c_2{\cal O}_{{\rm I} \! {\rm P}_1} \oplus c_2{\cal O}_{{\rm I} \! {\rm P}_1}(-1)$, so the bundle $F_{{\rm I} \! {\rm P}_1}$ has to split into ${\cal O}_{{\rm I} \! {\rm P}_1}(-a) \oplus {\cal O}_{{\rm I} \! {\rm P}_1}(-b)$ with $a+b=c_2$ because this line avoids the multi-jumping lines by hypothesis. We can bound $h^0F_{{\rm I} \! {\rm P}_1}(c_2-3)$ by $c_2-4$ except when $a=0$ or $1$ (which may happen, for example if $E$ had infinitely many jumping planes). But in the cases $a=0$ or $1$, one has $h^0F_{{\rm I} \! {\rm P}_1}(c_2-3)=c_2-2-a$, and as we can assume that $P$ is not on the vanishing of a section of $E(1)$ because for $c_2\geq 2$ $h^0E(1)\leq 2$, we have $h^0F(1)=0$, so there is an injection of $H^0F_{{\rm I} \! {\rm P}_1}(1)$ into $H^1F$. Then in the cases $a=0$ or $1$, one has $h^1F \geq 2-a$. But $h^0F(c_2-4)$ is zero by hypothesis so one has \linebreak$h^0F(c_2-3) \leq h^0F_{{\rm I} \! {\rm P}_1}(c_2-3)$ thus $\chi(F)$ is also bounded by $c_2-4$ in the cases $a=0$ or $1$, which gives $\alpha \leq 2c_2-6$ \bigskip Let's now take care of the bound of $\beta$. Take a stable plane $H$ without \prefix{c_2} and \prefix{c_2-1}jumping lines, and denote by $\beta^{\prime}$ the length of the intersection of the $\beta$-plane $h$ with the scheme of multi-jumping lines $M$. Consider now the incidence variety $I\subset H ^{\rm v} \times H$. The resolution of $I$ twisted by $p^*E_H$ gives the following exact sequence: \begin{center} $0\longrightarrow q_*p^*E_H \longrightarrow H^1E_H(-1)\otimes {\cal O}_{H ^{\rm v}}(-1) \longrightarrow H^1E_H\otimes {\cal O}_{H ^{\rm v}}\longrightarrow R^1q_*p^*E_H \longrightarrow 0$ \end{center} One has $h^1E_H(-1)=c_2$ and $h^1E_H=c_2-2$, so \mbox{$\beta^{\prime}=\chi(R^1q_*p^*E_H) = c_2-2+\chi (F)$}, where this time $F$ is $q_*p^*E_H$ which is still locally free of rank 2 and $c_1F=-c_2$, so we have $\chi (F)=-h^1F+h^0F(c_2-3)$. As $H$ is stable without \prefix{c_2} and \prefix{c_2-1}jumping lines the vanishing locus of a section of $E_H(1)$ is at most on one conic from the \ref{secantes}, so we have $h^0E_H(1) \leq 2$, and we can take a point $P$ of $H$ which is neither in a multi-jumping line nor in the vanishing locus of some section of $E_H(1)$. The line $p\subset H ^{\rm v}$ associated to $P$ don't contain any multi-jumping lines, so we have an injection: $0\rightarrow F_p \longrightarrow c_2{\cal O}_p(-1)$ and $F_p={\cal O}_{{\rm I} \! {\rm P}_1}(-a)\oplus {\cal O}_{{\rm I} \! {\rm P}_1}(-b)$ with $a+b=c_2$. But $P$ is not in the vanishing locus of a section of $E_H(1)$ so $a$ and $b$ are at least 2 then $h^0F_p(c_2-3) \leq c_2-4$. By hypothesis, $h^0F_p(k)=0$ for $0 \leq k \leq c_2-4$, so for those $k$ we have by induction that all the $h^0F(k)$ are zero using the sequence of restriction of $F(k)$ to $p$. In particular we have $h^0F(c_2-4)=0$ so $h^0F(c_2-3)$ is bounded by $h^0F_p(c_2-3) \leq c_2-4$, which gives $\beta^{\prime} \leq 2c_2-6$. Furthermore, the general plane is stable according to [Ba1], so if it satisfies the conditions on $h^0({\cal J}_P^{\otimes k}\otimes E_H(k))$ then we have $\beta=\beta^{\prime}\leq 2c_2-6$. \nolinebreak \raise-0.1em\hbox{$\Box$} \section{``No'' k-jumping lines on $S$ when $k\geq 3$}\label{fini3sautet+} The aim of this section is to prove that $S$ is made of 2-jumping lines except at a finite number of points. Let's first take care of some extremal cases: \subsection{Some too particular jumping phenomena} We want to prove here that under the hypothesis of the existence of $S$ with $c_2\leq 5$, there is at most a \prefix{c_2-1-k}dimensional family of k-jumping lines for $3 \leq k \leq 5$. \begin{lemma} \label{trisautplan}For $c_2\leq 5$ and $i=0$ or $1$, the support of $R^1q_*p^*E(1+i)$ meets any $\beta $-plane $h$ which doesn't have \prefix{k\geq 4+i}jumping lines in a scheme of length at most $c_2-3-i$. \end{lemma} Let's first get rid of the case of a stable plane $H$. For $c_2\leq 5$, we can pick a section of $E_H(1)$, and note $Z$ its vanishing locus. One has $h^1E_H(1)=h^1{\cal J}_Z(2)$ which is at most 1 because $\deg Z\leq 6$ and $Z$ has no \prefix{5+i}secant because there is no \prefix{4+i}jumping lines in $H$. So $R^1q_*p^*E_H(1)$ is the cokernel of some map $(c_2-2){\cal O}_h(-1) \rightarrow {\cal O}_h$, and its support is 0-dimensional according to the \ref{infinitebis}, so it has length 0 or 1. Similarly, if there is a 4-jumping line in $H$, then $R^1q_*p^*E_H(2)$ is the cokernel of some map $2{\cal O}_h(-1) \rightarrow {\cal O}_h$ , so the support of this sheaf has length 1. If $H$ is a jumping plane, then denote by $Z$ the vanishing locus of the section of $E_H$. So we have to study the \prefix{3+i}secant lines to $Z$ with $\deg Z =c_2$, so it is the same problem as the study of the \prefix{2+i}jumping lines of a stable bundle over $H$ with $c_2\leq 4$. So the result for $i=0$ is a deduced from \ref{basdeg}, and to obtain $i=1$, remark that we have shown above that there is at most one 3-jumping line in a stable plane when $c_2 \leq 4$, so there is at most one 4-jumping line in any plane when $c_2 \leq 5$. \nolinebreak \raise-0.1em\hbox{$\Box$} \bigskip So we can obtain the following: \begin{lemma} The jumping lines of order at least 3 of a 4- or 5-instanton make an at most 1-dimensional scheme. \end{lemma} Indeed, if $E$ has a line congruence of \prefix{k\geq 3}jumping lines of bidegree $(\alpha,\beta)$, then $\alpha \leq 1$ and $\beta \leq 1$ because we have shown above that when $c_2\leq 5$ there was no two 3-jumping lines in a stable plane, and there is at most a one parameter family of jumping planes from Coanda's theorem stated in \ref{coanda}. But the congruence $(0,1)$, $(1,0)$, $(1,1)$ contain lines (Cf [R]), and for those $c_2$ it is impossible to have a plane pencil of 3-jumping lines according to the \ref{infinitebis}. \begin{remark} \label{maxsauteuses}When $c_2\leq 5$, if $E$ has a $c_2$-jumping line $d$,then its multi-jumping lines are 3-codimensional in $G$. \end{remark} Let's first recall that the instanton property imply directly that any plane is semi-stable. Furthermore, any semi-stable plane containing a $c_2$-jumping line doesn't contain another multi-jumping line because the vanishing locus $Z$ of a section of $E_H(1)$ or of $E_H$ is included in $d$ because $d$ is \prefix{\deg Z}secant to $Z$, and the multi-jumping lines are at least 2-secant to $Z$. But the lines of ${\rm I} \! {\rm P}_3$ meeting $d$ is a hypersurface of $G$, so it would cut $S$ in infinitely many points and we could find a multi-jumping line in the same plane as $d$ which is impossible. \bigskip So we will assume in the following that $E$ has no \prefix{c_2}jumping lines. \begin{proposition} \label{fini4}The assumption of the existence of $S$ implies that $E$ has at most a finite number of 4-jumping lines when $c_2=5$. \end{proposition} Assume here that there is a 4-jumping line $q$. Consider the hypersurface of $S$ made of the lines of the congruence meeting $q$ (i.e: $T_qG\cap S$). But, when $c_2=5$, any plane $H$ containing $q$ and another multi-jumping line can't be stable from the \ref{secantes}, because if $H$ was stable, the vanishing locus of a section of $E_H(1)$ would be a scheme of degree 6 with $q$ as 5-secant line, so it couldn't have another trisecant. We had also showed in the \ref{infinitebis} that any plane containing $q$ has necessarily a finite number of multi-jumping lines, so the curve $T_qG\cap S$ can't be a plane curve, thus any plane containing $q$ must be a jumping plane, and if there was infinitely many such lines $q$, there would exist a 2-dimensional family of jumping planes, and we could conclude with Coanda's theorem as in the \ref{coanda}. \subsection{Curves of 3-jumping lines} One wants here to prove that there is no curve of \prefix{k\geq 3}jumping lines lying on $S$. When $c_2=4$ this is exactly the method of Coanda, but there are many more problems when $c_2=5$ because the ruled surface used may have a singular locus. Let's first consider the case of a curve of 3-jumping lines. \begin{proposition} \label{courbetris}No 4-instanton with $h^0E(1)=0$ has a curve of jumping lines of order exactly 3.\\When $c_2=5$, if $E$ has a surface $S$ of multi-jumping lines, then there is no curve of 3-jumping lines on $S$. \end{proposition} Assume that there is such a curve of 3-jumping lines (ie without 4-jumping lines), and take $\Gamma$ be an integral curve made only of 3-jumping lines. Denote by $\Sigma$ the ruled surface associated to $\Gamma$, and $\widetilde{\Sigma } \stackrel{\pi }{\longrightarrow }\Sigma $ its smooth model. \begin{itemize} \item When $c_2=4$ \end{itemize} We showed in the \ref{trisautplan} that two 3-jumping lines can't cut one another, so according to Coanda's result (Cf [Co]), the curve $\Gamma$ is either a regulus of some quadric $Q$ of ${\rm I} \! {\rm P}_3$, or the tangent lines to a skew cubic curve, so in both cases, one has $\widetilde{\Sigma }\simeq {\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1$. Let's now apply the method used by Coanda in [Co]. As there is no 4-jumping lines in $\Gamma$, we have the exact sequence for some divisor $A$ of degree $a$ in Num($\Sigma $): \begin{center} $0 \longrightarrow {\cal O}_{\widetilde{\Sigma }}(-A,3)\longrightarrow \pi ^*E_\Sigma \longrightarrow {\cal O}_{\widetilde{\Sigma }}(A,-3) \longrightarrow 0$ \end{center} which gives $c_2(\pi ^*E_\Sigma )=d.c_2=2a3$, where $d=\deg \Sigma$ is 2 or 4. So it gives a contradiction because here $c_2=4$. \begin{itemize} \item When $c_2=5$ \end{itemize} We also have such an exact sequence, but this time $\widetilde{\Sigma }= {\rm I} \! {\rm P}_{\widetilde{\Gamma }}(F ^{\rm v})$ where $\widetilde{\Gamma }$ is the normalization of $\Gamma$, and $F$ is a rank 2 vector bundle having a section (denote by ${\cal L}$ its cokernel), and such that $h^0F(l)=0$ for any invertible sheaf $l$of negative degree. The surjection from $F$ to ${\cal L}$ gives a section $C_0$ of $\widetilde{\Gamma }$ into $\widetilde{\Sigma }$. Let $e=-\deg F$, the intersection form in \mbox{Num$\widetilde{\Sigma }$} is given by $\left( \begin{array}{cc} 0 & 1 \\ 1 & -e \end{array} \right) $ in the basis $(f,C_0)$ where $f$ is the class of a fiber, so we have the relation $5d=6a+9e$ which proves that $d$ is a multiple of 3. On another hand, the dual ruled surface $\Sigma ^{\rm v}$ has no triple points because there is at most two 3-jumping lines in a same plane from the \ref{trisautplan}, so we have the relation from [K]: \begin{equation} \label{nopointtriple} [(d-2)(d-3)-6g](d-4)=0 \end{equation} because $\widetilde{\Sigma }$ and $\widetilde{\Sigma ^{\rm v} }$ have same degree and genus. So when $d=3$ we can work as in [Co] because it gives $g=0$, and $\widetilde{\Sigma }={\rm I} \! {\rm P}({\cal O}_{{\rm I} \! {\rm P}_1}\oplus {\cal O}_{{\rm I} \! {\rm P}_1}(1))$, so $e=1$ and $a=1$, then we have:\\ \centerline{$\pi ^* {\cal O}_\Sigma (1) \stackunder{Num}{\sim } {\cal O}_{\widetilde{\Sigma }}(2,1)$, and $h^2(\pi^* E_\Sigma (-2))=h^0({\cal O}_{{\rm I} \! {\rm P}_1}(-a+4+2g-2-e)\otimes Sym_3F)$}\\ which imply $h^2E_{\Sigma}(-2)\geq 1$ when taking sections in the following sequence where $C^{\prime}$ is at most 1-dimensional. \begin{center} $0 \longrightarrow E_\Sigma (-2)\longrightarrow \pi _*\pi ^*E_\Sigma (-2)\longrightarrow E_\Sigma (-2)\otimes \omega _{C^{\prime}} \longrightarrow 0$ \end{center} The resolution of $\Sigma$ in ${\rm I} \! {\rm P}_3$ twisted by $E(-2)$ gives then $h^3E(-5)\geq 1$, so $h^0E(1)\geq 1$ which contradicts the hypothesis. Thus $d$ is a multiple of 3 and $d\geq 6$. We will now need the following 3 lemmas before going on proving the proposition \ref{courbetris} \begin{lemma} A generic ruling of $\Sigma$ can't meet points of multiplicity 3 or more of $\Sigma$ \end{lemma} If the opposite was true, there would be a point $P$ of $\Sigma$ included in 3 distinct ruling of $\Sigma$. Those 3 lines would give 3 triples points of the pentic curve representing the jumping lines through $P$. This pentic must then be reducible, and those triple points have to lie in a same line which contradicts the fact that a plane has at most two 3-jumping lines (Cf \ref{trisautplan}). \begin{lemma} \label{lieudoublereduit}Any generic ruling of $\Sigma $ meets only reduced components of the double locus of $\Sigma $ \end{lemma} Assume that the double locus of $\Sigma$ has a reducible component meeting all the rulings. So any point of this component is a point $P$ of the double locus where the tangent cone $C_P\Sigma$ is a double plane, so this plane must contain the 2 rulings passing through $P$. On another hand, there are pinch points on $\Sigma$ because the number $2d+4(d-1)$ (Cf [K]) is not zero when $d$ is multiple of 3 and solution of \ref{nopointtriple}. So let $P^{\prime}$ be a pinch point of $\Sigma$, then there is through $P^{\prime}$ a ruling $d$ of $\Sigma$ such that the tangent plane to $\Sigma$ at any point of $d$ is a same plane $H$. This means that the tangent space to $\Gamma$ at $d$ is included in the $\beta$-plane $h$ associated to $H$. So $\Gamma\cap h_{\{d\}}$ has length at least 2. But under the above hypothesis we can find another ruling of $\Sigma$ in $H$. Indeed, by assumption $d$ meets a non reduced component of the double locus in some point $P$, so there is another ruling $d^{\prime}$ of $\Sigma$ through $P$ and we showed that $C_P\Sigma$ was set-theoretically the plane containing $d$ and $d^{\prime}$, but as the tangent space is constant along $d$, $H$ has to be included in $C_P\Sigma$ so $d^{\prime}$ is also in $H$ and $h\cap \Gamma$ has length at least 3 which contradicts the \ref{trisautplan}. \begin{lemma} \label{nplanssaut}A generic ruling of $\Sigma $ meets at least 4 other distinct rulings of $\Sigma$. \end{lemma} a) One have first to proves that a generic ruling of $\Sigma$ meets the double locus in distinct points. According to the \ref{lieudoublereduit}, the only case where this could be not satisfied is when the generic ruling of $\Sigma$ is tangent to some component of the double locus. That is the case where $\Sigma$ is a developable surface. The cone situation is immediate for the lemma \ref{nplanssaut}, so we will first consider the case where $\Sigma$ is a the developable surface of tangent lines to some curve $C$. For any point $P$ of $C$, the osculating plane $H_P$ to $C$ must coincide with the tangent plane to $\Sigma$ at any point of the ruling $d_P$ which is tangent to $C$ at $P$. Furthermore, the tangent space to $\Gamma$ at $d_P$ is made of the pencil of lines of $H_P$ containing $P$, so it is included in the $\beta$-plane associated to $H_P$. We can also remark that for any point $P^{\prime}$ of $C$ distinct from $P$, the planes $H_P$ and $H_{P^{\prime}}$ are distinct, otherwise we the $\beta$-plane associated would cut $\Gamma$ in length 4 which contradicts the \ref{trisautplan}. So we can translate this by the fact that the dual surface $\Sigma ^{\rm v}$ is a developable surface of tangent lines to a smooth curve $C ^{\rm v}$. Consider the projection of $C ^{\rm v}$ to ${\rm I} \! {\rm P}_2$ from a point of ${\rm I} \! {\rm P}_3 ^{\rm v}$ not in $\Sigma ^{\rm v}$. The degree of $\Gamma$ is also the degree of the dual curve of this projection, which is according to Pl\"ucker's formula $d=2(\deg C ^{\rm v} -1)+2g$ because this projection don't have cusp. So when $\Sigma$ is developable, the case $d=6, g=2$ is not possible , then $d\geq 9$. Let $t$ be a general point of $\Gamma$, then the contact between $T_tG$ and $\Gamma$ can't be of order 6 or more, otherwise the osculating plane to $\Gamma$ at $t$ would be included in $G$, so there would exist a $\beta$-plane meeting $G$ in length 3 or more which contradicts the \ref{trisautplan}. But the contact between $T_tG$ and $\Gamma$ must be even (Cf [Co] lemma 6), so $T_tG\cap \Gamma-\{t\}$ is made of $d-4$ distinct points because in general $t$ is not bitangent to $C$ because $\Gamma$ is reduced. So, when $\Sigma$ is developable a generic ruling of $\Sigma$ meets at least 5 other distinct ones. b) So we can assume that $\Sigma$ is not developable, and using the \ref{lieudoublereduit}, that a generic ruling of $\Sigma$ meets the double locus in distinct points. One have thus to compute the number of such points. But as $\Sigma$ is not developable, in general a point $t$ of $\Gamma$ is such that the intersection of $T_tG$ with $\Gamma-\{t\}$ has length $d-2$, which gives the lemma because $d\geq 6$. \nolinebreak \raise-0.1em\hbox{$\Box$} \bigskip We are now ready to continue the proof of \ref{courbetris}. We obtained in the \ref{nplanssaut} that through any ruling $t$ of $\Sigma$ there are at least 4 distinct planes containing another ruling of $\Sigma$. So according to the \ref{trisautplan}, we have at least 4 jumping planes through $t$. We can assume that there are only a finite number of jumping planes through $t$ when $t$ is general because the bundle $E$ don't have a 2-dimensional family of jumping planes (\ref{coanda}). Let's blow up ${\rm I} \! {\rm P}_3$ along $t$ and denote by $p^{\prime}$ and $q^{\prime}$ the projections over ${\rm I} \! {\rm P}_3$ and ${\rm I} \! {\rm P}_1$. We have from the standard construction the following sequence: \begin{center} $0\rightarrow \!q_{*}^{\prime }p^{\prime *}E(\tau )\rightarrow \!H^1E\otimes {\cal O}_{{\rm I} \! {\rm P}\!_1}(-1) \kercoker{K} H^1E(1)\otimes {\cal O}_{{\rm I} \! {\rm P}\!_1}\rightarrow \!R^1q_{*}^{\prime }p^{\prime *}E(\tau )\rightarrow \!0$ \end{center} where $h^1E=8,h^1E(1)=7$, and where $R^1q_*^{\prime }p^{\prime *}E(\tau)$ is not locally free at the jumping planes but has rank 1 because in a stable plane $H$ containing the 3-jumping line $t$, the 6 points of the vanishing locus of a section of $E_H(1)$ must lie on a conic because $t$ is 4-secant to this locus. The sheaf $q_*^{\prime }p^{\prime *}E(\tau)$ is thus locally free of rank 2, and we have: $h^0(q_*^{\prime }p^{\prime *}E(\tau))=0$, $h^1(K)=0$, $h^1(q_*^{\prime }p^{\prime *}E(\tau ))=h^0(K)$, et $h^0(R^1q_*^{\prime } p^{\prime *} E(\tau ))=7-h^1(q_*^{\prime } p^{\prime *}E(\tau ))$. If $h^1(q_*^{\prime }p^{\prime *}E(\tau ))$ is 0 or 1 then $q_*^{\prime } p^{\prime *}E(\tau )= {\cal O}_{{\rm I} \! {\rm P}_1}(-1) \oplus {\cal O}_{{\rm I} \! {\rm P}_1} (-1\ or\ -2)$, so $[q_*^{\prime }p^{\prime *}E(\tau )](1)$ would have a section and ${\cal J}_t\otimes E(2)$ also. But this phenomena must occurred for a generic $t$, so we have $h^0E(2) \geq 2$, but the vanishing locus of those sections of $E(2)$ would then be in a quartic surface, which must contain every 3-jumping line because those lines are 5-secant to those vanishing locus. But it contradicts $\deg \Sigma \geq 6$. So we have $h^0(R^1q_*^{\prime }p^{\prime *}E(\tau )\leq 5$. On another hand $R^2q_*^{\prime }p^{\prime *}E(\tau )$ is zero from Grauert's theorem, so we have by base change for all $y$ in ${\rm I} \! {\rm P}_1$:\\ $R^1q_*^{\prime }p^{\prime *}E(\tau )_{\{y\}}\simeq H^1E_Y(1)$ where $Y$ is the plane associated to $y$. So we have the exact sequence: \begin{center} $0\longrightarrow \stackunder{k\ times}{\oplus }\setlength{\unitlength}{0.01em \longrightarrow R^1q_*^{\prime } p^{\prime *}E(\tau )\longrightarrow R^1q_*^{\prime }p^{\prime *}E(\tau ) ^{\rm vv} \longrightarrow 0$ \end{center} where $k$ is the number of jumping planes through $t$, so $k\geq 4$ from the \ref{nplanssaut}. But $R^1q_*^{\prime }p^{\prime *}E(\tau ) ^{\rm vv} \simeq {\cal O}_{{\rm I} \! {\rm P}_1}(l)$ with $l\geq 0$, so necessarily $l=0,k=4$. Then $d=6$ so $g=2$ from the formula \ref{nopointtriple}. We also have $h^0(R^1q_*^{\prime }p^{\prime *}E(1))=5$ and $q_*^{\prime} p^{\prime *}E(1) \simeq {\cal O}_{{\rm I} \! {\rm P}_1}(-2) \oplus {\cal O}_{{\rm I} \! {\rm P}_1}(-2)$, so $h^0{\cal J}_t^2\otimes E(3) \geq 2$. Take $s$ and $s^{\prime}$ 2 sections of ${\cal J}_t^2\otimes E(3)$. Those sections are proportional on a sextic surface containing the 4 jumping planes $H_i$ passing through $t$ because the restriction of $s$ and $s^{\prime}$ to $H_i$ are multiple of $t^2$ and $E_{H_i}(1)$ can't have a section not proportional to the section of $E_{H_i}$ because there are no 5-jumping lines according to the \ref{maxsauteuses}. So denote by $Q$ the quadric surface such that $Z_{s\wedge s^{\prime}}=\bigcup\limits_{i=1}^k H_i \cup Q$. First notice that for a general plane $H$ containing $t$, the sections $s$ and $s^{\prime}$ gives, after the division by $t^2$, 2 sections of $E_H(1)$ which are proportional on a conic which must be $Q\cap H$. Furthermore, this conic must contain $t$ because $t$ is 4-secant to the vanishing locus of any section of $E_H(1)$. Denote by $d_H$ the line such that $t \cup d_H=Q \cap H$. Next, assume that $\Gamma$ is drawn on $S$ as in the hypothesis of \ref{courbetris}. The curve $T_tG\cap S$ is made of multi-jumping lines meeting $t$, and is an hyperplane section of $S$. We want here to understand this curve by cutting it with planes containing $t$. First remark that there is no irreducible component of $T_tG\cap S$ included in some $\beta$-plane containing $t$, otherwise its associated plane would be a jumping one according to the \ref{infinitebis}, so it would be one of the $H_i$, but the $H_i$ contain two 3-jumping lines, so they contain only a finite number of multi-jumping lines using again the \ref{infinitebis}. So the points of $T_tG\cap S$ which represent lines lying in a stable plane $H$ containing $t$ make a dense subset of $T_tG\cap S$. Now we can notice that in a general plane $H$ containing $t$, the only possible multi-jumping lines are trisecant to any vanishing locus of a section of $E_H(1)$, so they must be $t$ or $d_H$. But $d_H$ is always in $Q$, so the curve $T_tG\cap S$ must be the rulings of $Q$, and it would imply $\deg S=2$, then the curve $\Gamma$ of degree 6 and genus 2 could not lie on $S$ because $h^0{\cal O}_{\Gamma}(1)=6$, and it gives the proposition \ref{courbetris}. \bigskip We'd like now to enlarge the result \ref{courbetris} to the case where the previous curve $\Gamma$ contains 4-jumping lines when $c_2=5$. \begin{proposition} \label{sauteuses4}When $c_2=5$, the surface $S$ don't contain a curve of jumping lines of order 3 in general with some 4-jumping lines \end{proposition} So we assume that there is such a curve, and let $\Gamma$ be an irreducible curve of jumping lines of order 3 in general with at least a 4-jumping line $q$. Denote by $\Sigma$ the ruled surface associated to $\Gamma$. First notice that $\chi(E(2))=0$, and the existence of $q$ implies $h^1E(2)\neq 0$, so $E(2)$ has a section $s$ vanishing on some degree 9 curve $Z$. Furthermore, any 3-jumping lines is 5-secant to $Z$. The part of the proof of the \ref{courbetris} bounding the number of jumping planes containing a general ruling $t$ of $\Sigma$ is still valid in this situation, so we still have $\deg \Gamma \leq 6$. a) Now notice that $\Gamma$ can't be a conic. In fact this is the worst case because the bundles of the family \ref{famille} have such a conic of 3-jumping lines with four 4-jumping lines. So the required contradiction need to use the existence of $S$. In that case $\Sigma$ is a quadric so we have: \begin{center} $0\longrightarrow {\cal O}_{{\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1}(-a,3) \longrightarrow E_{\Sigma} \longrightarrow {\cal O}_{{\rm I} \! {\rm P}_1 \times {\rm I} \! {\rm P}_1}(a,-3) \longrightarrow \setlength{\unitlength}{0.01em ^n\longrightarrow 0$ \end{center} which gives $10=6a+n$ where $n$ is the number of 4-jumping lines on $\Gamma$. So $a$ would be 0 or 1 then $h^0E_{\Sigma}(2)\geq 12$. The resolution of the quadric twisted by $E(2)$ gives the sequence: \begin{center} $0 \longrightarrow H^0E(2)\longrightarrow H^0E_{\Sigma}(2)\longrightarrow H^1E$ \end{center} So \mbox{$h^0E(2)\geq 4$}, and we can solve this case with the \ref{multisautfamille}. \bigskip For the other possible degrees of $\Gamma$, the situation is easier because none of the ruling of $\Sigma$ can meet $q$ because in any plane $H$ containing $q$, this line is according to the \ref{secantes} \prefix{\deg Z-1}secant to the 0-dimensional vanishing locus $Z$ of a section of $E_H$ or of $E_H(1)$, and any ruling of $\Sigma$ is at least 3-secant to $Z$. So if $\Sigma$ has a double locus, it must meet $q$ in a pinch point, so $q$ is a torsal line. In other words, there is a line $L$ included in $G$ meeting $\Gamma$ in length at least 2 around $q$. So, the support of the sheaf $R^1q_{*}p^{*}E(\tau )_L$ must contain twice the point $q$. So this sheaf can't be equal to its restriction at the reduced point $q$, but it has rank 2 at $q$, so $R^1q_{*}p^{*}E(\tau )_L\geq 3$ which is not possible around a 4-jumping line, because the plane associated to $L$ can't be stable, hence according to the \ref{secantes}, it is for $c_2=5$ the same problem as the study of the \ref{pinceaux} for $c_2=4$ around a 3-jumping line, and the proof of \ref{pinceaux} was still valid for $c_2=4$ in this situation. If $\Sigma$ has no singular locus, then as it can't be a quadric, it is the tangential surface of a skew cubic, hence all its lines are torsals, so we can conclude as previously. \nolinebreak \raise-0.1em\hbox{$\Box$} We also have the following: \begin{corollary} \label{c2=5deg}For $c_2=5$, any integral curve in $G$ made only of 2-jumping lines has a degree multiple of 4. The surface $S_{red}$ has a degree multiple of 4. \end{corollary} Indeed, any integral curve of degree $d$ made only of 2-jumping lines gives using the previous notations, a ruled surface such that $5d=4a+4e$. But the results \ref{maxsauteuses}, \ref{fini4}, \ref{courbetris}, and \ref{sauteuses4} imply the general hyperplane section of $S_{red}$ don't contain any \prefix{k\geq3}jumping lines. \section{Trisecant lines to $S$ and applications} In the first section we obtained an interpretation of some trisecant lines to $S$. The key application of this interpretation is the folowing proposition: \subsection{Trisecant lines to the general hyperplane section of $S$} \begin{proposition} \label{c2=5tris} When $c_2\leq 5$, the general hyperplane section of $S_{red}$ has no trisecant lines. \end{proposition} Indeed, on one hand the generic hyperplane section of $S_{red}$ contains no \linebreak\prefix{k\geq 3}jumping lines according to the section \ref{fini3sautet+}, and on the other hand, the lines of ${\rm I} \! {\rm P}_5$ which are in a hyperplane make a 2-codimensional subscheme of the Grassmann manifold $G(1,5)$. So if the general hyperplane section of $S_{red}$ had a trisecant line then $S_{red}$ would have a 2-parameter family of trisecant lines meeting $S$ in 2-jumping lines. But any of those trisecant lines would give a jumping plane according to the \ref{pinceaux}, and as there is only a finite number of such trisecant lines in a same jumping $\beta$-plane according to the \ref{pinceauxsauteurfini}, we would have a 2 parameter family of jumping planes which would contradict the \ref{coanda}. \nolinebreak \raise-0.1em\hbox{$\Box$} \subsection{The cases where $S$ is ruled}\label{reglee} \begin{itemize} \item Let's first consider the case where $S$ is a ruled surface when $c_2=5$. \end{itemize} Every ruling of $S$ must contain a 3-jumping line according to the \ref{infinitebis}, but there is at most a finite number of 3-jumping lines on $S$ from the \ref{courbetris} and the \ref{sauteuses4}. So $S$ must be a cone with vertex a 3-jumping line $t$ , and this cone is in the ${\rm I} \! {\rm P}_4$ constructed by projectivisation of the tangent space $T_tG$. According to the \ref{infinitebis}, any plane contain at most one pencil of multi-jumping lines, so $S_{red}$ must have bidegree $(1,\beta)$, and the bound on $\beta$ of the \ref{basdeg} and the \ref{c2=5deg} imply that $\beta=3$. But the projection from $t$ of this cone gives a curve of bidegree $(1,3)$ in the quadric obtained by projection of $T_tG\cap G$. So there are many planes containing 3 distinct rulings of $S_{red}$, then the generic hyperplane section of $S_{red}$ would have trisecant lines which would contradict the \ref{c2=5tris}. \begin{itemize} \item So assume here that $S$ is a ruled surface and that $c_2=4$. \end{itemize} Denote by $C$ the reduced curve described by the center of the pencils of lines associated to the rulings of $S$. \begin{itemize} \item[-] If $\deg C\geq 3$ \end{itemize} A general hyperplane $H$ meeting $C$ in distinct points contain a line of the congruence $S$ passing through each point of $C\cap H$. As $\beta \leq 2$ and $\deg C\geq 3$, those lines must be bisecant to $C$, and the other bisecant to $C$ are not in the congruence $S$. Hence $S$ is a join between 2 components of $C_1$ and $C_2$ of $C$ with $\deg C_1=1$ because $S$ must be ruled. So every plane $H$ containing $C_1$ contain a pencil of multi-jumping lines as in the \ref{infinitebis} which is centered at a point of $C_2\cap H$. So those plane can't be stable and according to the \ref{dteplansaut} ${\cal J}^2_{C_1}\otimes E(2)$ has a section. Furthermore, this section vanishes at least with multiplicity 4 on $C_1$ and also on $C_2$, but it has degree 8 and must be connected, then $S$ would be a $\beta$-plane which is not possible. \begin{itemize} \item[-] If $\deg C=1$ \end{itemize} Then $S$ is a cone with vertex the point representing $C$, and once again every plane containing $C$ would contain infinitely many multi-jumping lines, hence from the \ref{dteplansaut} ${\cal J}^2_{C}\otimes E(2)$ has a section $s_0$ vanishing at least with multiplicity 4 on $C$ and on the curve $C^{\prime}$, where $C^{\prime}$ is the curve obtained by the center of the pencils of multi-jumping lines which are in planes containing $C$. So we must have $\deg C^{\prime} =\alpha =1$ and the connexity of the vanishing locus imply that $C=C^{\prime}$. So, in any plane $H$ containing $C$, the bundle $E_H$ has a section $s_H$ of type \ref{infinitebis} such that the connected component of $Z_{s_H}$ is on $C$, and any section $\sigma$ of $E_H(1)$ is proportional to $s_H$, otherwise its vanishing locus $Z_{\sigma}$ would be in the line $Z_{\sigma \wedge s_H}$ which would be a 4-jumping line, but it would contradict the \ref{infinitebis}. Therefore, every section of ${\cal J}_C \otimes E(2)$ must be a section of ${\cal J}_C^2 \otimes E(2)$. Furthermore, if we blow up ${\rm I} \! {\rm P}_3$ along $C$, the bundle $q^{\prime}_*p^{\prime *}E$ has rank 1, so $h^0{\cal J}_C^2 \otimes E(2)=1$. This proves that the sections of $E(2)$ have no base curves, and the number of base points is the number of residual points in the intersection of the 3 quartics of $H^0{\cal J}_{Z_{s_0}}(4)$ which is 0 from [Fu] p155. So from Bertini's theorem, there is a smooth section $s$ of $E(2)$, such that $Z_s$ is also connected from $h^1E(-2)=0$, and $Z_s \cap C=\emptyset$. The quartic surface $Z_{s_0 \wedge s}$ must cut any plane $H$ containing $C$ in twice $C$, and a conic which must be singular in some $P_H\in C$ because it contains $Z_{s_H}$. Furthermore, this conic doesn't contain $C$ because $Z_s\cap C= \emptyset$ and the lines through $P_H$ are $0$- or 4-secant to $Z_s$. So this quartic is a ruled surface with $C$ as directrix, but $C$ is in its singular locus so there are rulings of $Z_{s_0 \wedge s}$ through $P_H$ which are not in $H$. So $C$ must be a triple curve of $Z_{s_0 \wedge s}$, which imply that this quartic has a rational basis. But it contradicts the formula of Segre (Cf [G-P2]) applied to $Z_s$ because it is a degree $8$ elliptic curve 4-secant to the rulings of $Z_{s_0 \wedge s}$. \begin{itemize} \item[-] When $\deg C=2$ \end{itemize} The plane $H$ containing the conic $C$ must then contain infinitely many multi-jumping lines which have to contain a same point $O$ according to the \ref{infinitebis}. But the lines of the congruence are secant to $C$, so only one of them contain a fixed point $P$ of $H-\{C,O\}$, then $\alpha =1$. We had already eliminated the cases of the congruences $(1,1)$ ($\deg C=1$), and the congruences $(1,2)$ are the joins between $C$ and a line $d$ cutting $C$ in 1 point (Cf [R]). As previously, every plane containing $d$ would be of type \ref{infinitebis}, and the section of ${\cal J}_d^2 \otimes E(2)$ would vanish on $d\cup C$ at least with multiplicity 4, which is not possible because it has degree 8. \nolinebreak \raise-0.1em\hbox{$\Box$} \subsection{``No'' trisecant cases} The aim of this section is to get rid of the remaining cases. \begin{lemma}\label{degS=4} For $c_2=4$ and 5, the congruence $S_{red}$ can't have degree 4. \end{lemma} The degree 4 surfaces are classified in [S-D], and as we already eliminate the cases of ruled surfaces in \S\ref{reglee}, only the complete intersection of 2 quadrics in ${\rm I} \! {\rm P}_4$ and the Veronese are remaining. \begin{itemize} \item For $c_2=4$ \end{itemize} The Veronese has bidegree $(3,1)$ or $(1,3)$ so it would contradict the \ref{basdeg}. If $S$ is the complete intersection of $G$ with another quadric and some ${\rm I} \! {\rm P}_4$ then $S$ has bidegree $(2,2)$ in G (Cf [A-S]). Furthermore, in that case $S$ is locally complete intersection, so we can compute from the \ref{exresiduel} that the scheme of multi-jumping lines $M$ is made of $S$ with a non empty residual curve $C$. There is a 2 dimensional family of $\beta$-planes which meet $C$, so from the \ref{coanda} its general element must be a stable plane, then it must cut $S$ in good dimension according to the \ref{infinitebis}. But those $\beta$-planes will contain $\beta +1=3$ multi-jumping lines (with multiplicity) which is not possible when $c_2=4$ (Cf \ref{basdeg}). We can remark that it is still true when $C$ is drawn on $S$. Indeed, one has the exact sequence: \begin{center} $0 \longrightarrow {\cal L} \longrightarrow {\cal O}_M \longrightarrow {\cal O}_S \longrightarrow 0$ \end{center} where $C$ is the support of ${\cal L}$. Restrict it to a stable $\beta$-plane $h$ which meet $C$ to have: \begin{center} ${\cal T}or _1({\cal O}_S,{\cal O}_h) \longrightarrow {\cal L} \otimes {\cal O}_h \longrightarrow {\cal O}_{M\cap h} \longrightarrow {\cal O}_{S\cap h} \longrightarrow 0$ \end{center} As $h$ is stable, it cuts $S$ in dimension 0, so $Tor_1({\cal O}_S,{\cal O}_h)={\cal J}_S \cap {\cal J}_h /{\cal J}_S {\cal J}_h=0$, which still gives length${\cal O}_{M\cap h}\geq 3$. \begin{itemize} \item When $c_2=5$ \end{itemize} The complete intersection of 2 quadrics in ${\rm I} \! {\rm P}_4$ is a Del-Pezzo surface isomorphic to ${\rm I} \! {\rm P}_2$ blown up in 5 points embedded by $3L-E_1-...-E_5$. According to the \S\ref{fini3sautet+}, the surface $S$ contains at most a finite number of \prefix{k\geq 3}jumping lines, so we can find on $S$ some cubic curves made only of 2-jumping lines, which contradicts the \ref{c2=5deg}. Similarly, in the Veronese case, there would have many conics made only of 2-jumping lines which also contradicts the \ref{c2=5deg}. \nolinebreak \raise-0.1em\hbox{$\Box$} \begin{proposition} \label{c2=5courbe} For $c_2=4$ or $5$, the scheme of multi-jumping lines of a mathematical instanton is a curve in $G$. \end{proposition} In fact the case $c_2=4$ is already done because $S$ can't have degree 4 (\ref{degS=4}), and the bound of its bidegree of the \ref{basdeg} imply that $\alpha =1$ or $\beta =1$. But those congruences are ruled (Cf [R]) and we can conclude with the \S\ref{reglee}. \bigskip For $c_2=5$, let's first prove the following: \begin{lemma} The set of points $p$ of $S$ such that $\dim T_pS_{red}=4$ is finite. \end{lemma} Assume that there are infinitely many such $p$, then the general hyperplane section contains one of them, so in general, the surface $S_{red}$ has no trisecant through $p$ (Cf \ref{c2=5tris}), so the curve $T_tG \cap S_{red}$ must be an union of lines containing $p$, and as $S$ is not ruled (Cf \S\ref{reglee}), it can happen only at finitely many $p$ which proves the lemma. \nolinebreak \raise-0.1em\hbox{$\Box$} \bigskip So a general hyperplane section $\Gamma$ of $S_{red}$ is locally complete intersection with at most finitely many $p$ such that $\dim T_p\Gamma=2$. According to the \ref{c2=5tris}, the curve $\Gamma $ has no trisecant, and we can adapt LeBarz's formula of [LeBa2]. In fact, when projecting $\Gamma $ to ${\rm I} \! {\rm P}_2$, the singularities of $\Gamma $ won't gives embedded points, so they have to be removed from the contribution of the apparent double points. Hence the number of trisecant to $\Gamma$ is $(d-4)((d-2)(d-3)-6\pi)$ where $\pi$ is the arithmetic genus of $\Gamma $. So we have from the \ref{c2=5tris} and \ref{degS=4} that $(d-2)(d-3)-6\pi=0$. Thus $d-1$ is not a multiple of 3, and then $\Gamma $ is locally complete intersection with maximal genus in ${\rm I} \! {\rm P}_4$, because if $\epsilon$ is the remainder of the division of $d-1$ by 3, and $m=\left[ \frac{d-1}3\right] $, then $\frac{(d-2)(d-3)}6 =\frac{\epsilon ^2-3\epsilon +2}6+ \frac{3m(m-1)}2+ m\epsilon $ because $\epsilon\neq 0 $. According to the \ref{degS=4} and the \ref{c2=5deg}, we have $\deg \Gamma =8$, and as it has maximal genus in ${\rm I} \! {\rm P}_4$, it must be a canonical curve, and it can't be trigonal (ie have a $g^1_3$) because it has no trisecant. So according to Riemann-Roch $\Gamma$ hence $S$ must be the complete intersection of 3 quadrics. So $S$ is the Kummer K-3 surface and has bidegree $(4,4)$ (Cf [A-S]). We can compute from the \ref{exresiduel} that the scheme of multi-jumping lines has a residual curve because $S$ is locally complete intersection, and we construct as in the \ref{degS=4} when $c_2=4$ a 2 parameter family of $\beta$-planes containing 5 multi-jumping lines (with multiplicity), which contradicts the \ref{coanda} or the \ref{basdeg}. \nolinebreak \raise-0.1em\hbox{$\Box$} \section{Applications to moduli spaces}\label{applications} \subsection{The construction of Ellingsrud and Stromme} Let us recall here the construction made in [E-S], and show how the previous results could be used to study I$_n$. Let $N$ be a point such that there exists an instanton $E$ which has only a one dimensional family of jumping lines through $N$ which are all of order $1$. Denote by $U_N$ the subscheme of I$_n$ made of instantons which don't have a multi-jumping line through $N$ and which have a non jumping line through $N$. The result \ref{c2=5courbe} and Grauert-Mullich's theorem implies that the $(U_N)_{N\in {\rm I} \! {\rm P}_3}$ cover $I_n$ for $n=4$ or $5$. To describe $U_N$, first blow up ${\rm I} \! {\rm P}_3$ at $N$, denote by $Y$ the plane parameterizing the lines containing $N$, $p$ and $q$ the projections on ${\rm I} \! {\rm P} _3$ and $Y$, and $\tau=p^*{{\cal O}_{\proj_3}}(1)$, $\sigma=q^*{\cal O}_Y(1)$, and by $V=H^0({\cal O}_Y(1))$. Let $H$ be a $\setlength{\unitlength}{0.01em $-vector space of dimension $n$. ($H$ will stand for $H^2E(-3)$). A point $E$ of $U_N$ is characterized (after the choice of an isomorphism: $H \simeq H^2E(-3)$) by the following data where $F$ is naturally $q_*p^*E$ and $\theta (1)$ is $R^1q_*p^*E(-1)$: \begin{center} $\left\{ \begin{array}{cllc} 1) & & m\in V\otimes S_2H ^{\rm v} & \\ 2) & \text{a surjection: }& 2{\cal O}_{{\rm I} \! {\rm P}(V ^{\rm v})}\rightarrow \theta (2)\rightarrow 0 & \text{(denote by } F \text{ its kernel)} \\ 3) & \text{and a surjection: }& q^{*}F ^{\rm v}\rightarrow q^{*}\theta (\sigma +\tau )\rightarrow 0 & \end{array} \right. $ \end{center} In fact, the symmetric map $m$ is just the restriction to the $\alpha $-plane associated to $N$ of the $2n+2$ rank element of $\stackrel{2}{\Lambda }H^0({\cal O}% _{{\rm I} \! {\rm P}\!_3}(1))\otimes S_2H ^{\rm v}$ of Tjurin, (Cf [T2] or [LeP]) and it can be obtained from the following sequence coming from Beilinson's spectral sequence. \begin{equation} \label{reseau} 0\longrightarrow H\otimes {\cal O}_Y\stackrel{m}{ \longrightarrow }H ^{\rm v}\otimes {\cal O}_Y(1)\longrightarrow \theta (2)\longrightarrow 0 \end{equation} We have also to recall from [E-S] that the data 3) exists only under the condition that the restriction of $F$ to $C$ splits. To be more precise, the restriction of the data 2) to $C$ gives a sequence \begin{equation} \label{Fc} 0 \longrightarrow \theta ^{\rm v}(-1) \longrightarrow F_C \longrightarrow \theta ^{\rm v}(-2)\longrightarrow0 \end{equation} which has to split for the existence of the data 3). This is the main difficulty of this description of $U_N$. So let's first globalize data 2). The group $P=^{GLH}/_{\pm{}Id}$ acts on $H$ by $^t p.h.p$, and it also acts on the previous data according to the exact sequence (\ref{reseau}). Furthermore, 2 elements of $U_N$ are isomorphic if and only if they are in the same orbit of this action. Our reference about properties and existence of the moduli space of those theta characteristic will be [So]. A net of quadrics will be called semi-stable (stable) if and only if, for all non zero totally isotropic (for the net) subspace $L$ we have: \hbox{$\dim L+\dim L^{\bot }\leq \dim H$} ($< \dim H$). Let $\Theta $ be the quotient \hbox{$(V\otimes S_2H ^{\rm v})^{ss}/P$}, and $\Theta ^s_{lf}$ be the stable points which represent locally free sheaves, and $\Theta _0$ be the dense open subset of $\Theta $ made of isomorphic classes of $\theta $-characteristic which have a smooth support. The space $\Theta $ is irreducible of dimension $\frac{n(n+3)}2$, and normal with rational singularities (Cf [So] th\'eor\`eme 0.5), and $\Theta ^s_{lf}$ is smooth (Cf [So] th\'eor\`eme 0.4) because when $\theta$ is locally free, the notion of stable nets coincide with the stability of $\theta $ in the sheaf meaning (without the locally free assumption, the last notion is stronger). By construction, any element of $U_N$ has a non-jumping line through $N$. This implies that the associated net is semi-stable. Furthermore by definition of $U_N$, the $\theta$-characteristic considered are locally free. So we have a morphism: $$I_n \supset U_N \longrightarrow \Theta _{lf}$$ The key of the construction is to understand the fiber of this morphism. We have thus to globalize the data 2) and then, to show that for $n \leq 11$ this map is dominant, so we will have to understand under what condition the data 3) exists. \bigskip Define by $G^{\prime }_{\theta}=G\big(2,H^0(\theta (2))\big)$ the Grassmann manifold of 2-dimensional subspace of $H^0(\theta (2))$, and $G^{\prime \prime }_{\theta}$ be the open subset of $G^{\prime}_{\theta}$ made of pairs of sections of $\theta (2)$ with disjoint zeros. Denote by $K_{\theta}$ the tautological subbundle of ${\cal O}_{G^{\prime \prime}_{\theta}} \otimes H^0(\theta(2))$. The existence condition of the data 3) was already identified in [E-S] \S{4.2} as the vanishing locus of a map: $$\delta_{\theta} :{\cal O}_{G^{\prime \prime }_{\theta}}\longrightarrow H^1({\cal O}% _C(1))\otimes \stackrel{2}{\Lambda }K ^{\rm v}_{\theta}$$ obtained by the following way: \bigskip We pick from data 2) the exact sequence: \begin{center} $0\longrightarrow F\longrightarrow {\cal O}_{{\rm I} \! {\rm P}(V ^{\rm v}% )}\boxtimes K_{\theta}\longrightarrow \theta (2)\boxtimes {\cal O}_{G^{\prime \prime }_{\theta}}\longrightarrow 0$ \end{center} whose restriction to $C\times{}G^{\prime \prime }$ (where $C$ is $\theta$ 's support) gives: \begin{center} $0\longrightarrow {\cal O}_C(1)\boxtimes {\cal O}_{G^{\prime \prime }_{\theta}}\longrightarrow F(\theta (2)\boxtimes {\cal O}_{G^{\prime \prime }_{\theta}})\longrightarrow {\cal O}_C\boxtimes \stackrel{2}{\Lambda }% K_{\theta}\longrightarrow 0$ \end{center} Pushing down this sequence twisted by ${\cal O}_C\boxtimes \stackrel{2}{\Lambda }K ^{\rm v}_{\theta}$ with the second projection of $C\times{}G^{\prime \prime }_{\theta}$ direct image's functor, we obtain the boundary $\delta_{\theta} $. \bigskip Denote by $Z_{\delta_{\theta}} $ the vanishing locus of $\delta_{\theta} $. Then, according to [E-S], the fiber $U_{N,\theta}$ is the product of $Z_{\delta_{\theta}} $ by ${\rm I} \! {\rm P}_4$. $$I_n \supset U_N \longrightarrow \Theta_{lf} \hspace{2cm} \forall \theta \in \Theta_{lf},U_{N,\theta}={\rm I} \! {\rm P}_4 \times Z_{\delta_{\theta}}$$ Let $\Theta _0$ be the subspace of $\Theta $ made of isomorphic classes of $\theta$-characteristic with smooth support. Ellingsrud and Stromme had also remarked ([E-S] \S{4.2}) that for any $\theta$ in $\Theta_0$, the bundle $\stackrel{2}{\Lambda }K_{\theta} ^{\rm v}$ generates $Pic G_{\theta} ^{\prime \prime }$, and that for $n \leq 11$ $\Theta _0$ is included in $U_N$ 's image. This image is thus irreducible, normal and $\frac{n(n+3)}2$ dimensional. Furthermore, if we denote by $U_N ^s$ the elements of $U_N$ which have a stable image in $\Theta $, we obtain that the image of $U_N ^s$ is irreducible and smooth. So we need to prove the following lemma: \begin{lemma} \label{recstable}For $n=4$ and $5$, the $U_N^s$ make a covering of I$_n$ when $N$ travels through ${\rm I} \! {\rm P}_3$ \end{lemma} We have to understand the non stable nets which could occur in the image of $U_N$. Let's recall from [So] that $\theta $ is not stable if there exists a totally isotropic (for the net $m$ associated to $\theta$) subspace $L\subset H$ such that $\dim L+\dim L^{\perp }\geq n$. As there exists a smooth quadric in the net, $L$ has to be at most 2 dimensional. The case $\dim L=1$ has already been avoided in [E-S] \S{5.10} using Barth's condition $\alpha 2$ of [Ba3]. So we have just to study the case $\dim L=2$. If there exists such a 2-dimensional space, then all the quadrics of the net can be written in the following way: $\sum_{i=1}^{i=3}Y_i\left( \begin{array}{cl} \begin{picture}(80,100)(0,20) \put(15,7){0} \put(0,-35){\line(1,0){80}} \put(0,100){\line(1,0){130}} \put(80,20){\line(1,0){50}} \put(80,-35){\line(0,1){55}} \put(0,-35){\line(0,1){135}} \put(130,20){\line(0,1){80}} \end{picture} & \hfill ^{^\tau A_i} \\ _{A_i}\hfill & B_i \end{array} \right) $ where $A_i$ is a $2\times{}2$ matrix, and where $B_i$ is a $2\times{}2$ matrix in case $n=4$ and a $3\times{}3$ matrix in case $n=5$. Their determinant must then be a multiple of $(\det ( \sum_{i=1}^{i=3} Y_iA_i))^2$. The curve of jumping lines of any preimage of this net must contain this double conic. Let $ A_N=$\hbox{\tiny $\left(\begin{array}{cc} a_N & b_N \\ c_N & d_N \end{array}\right)$}$=\sum_{i=1}^{i=3}Y_iA_i$, and first prove the following lemma before continuing the \ref{recstable}: \begin{lemma}\label{conicsing} If the conic $det(A_N)$ is singular in $p$, then either $p$ represents a multi-jumping line or $N$ is in the vanishing locus of a section of $E_H$ for some jumping plane $H$ \end{lemma} The first case is obtained when $a_N,b_N,c_N,d_N$ represent lines containing a same point $p$, and the other cases are reduced to the 2 cases where $a_N,b_N,d_N$ or $b_N,c_N,d_N$ is a basis of $V$, because we are allowed to change the basis of $L$. Computing the remaining coefficient in this basis, and using that $A_N$ is singular, we are reduced by changing the basis of $L$ to the case where $a_N$ and $c_N$ are proportional. But in this case, $a_N$ divides all the $(n-1)\times (n-1)$ minors except one. It means that the line $a_N$ cuts the multi-jumping lines in length $n-1$. The computation of \ref{pinceaux} also proves that this is not possible in a stable plane, and the \ref{pinceauxsauteurfini} that it is impossible in a jumping plane $H$ where $N$ is not in the vanishing locus of the section of $E_H$, because there is no jumping line of order $n$ through $N$ when $a_N,b_N,d_N$ or $b_N,c_N,d_N$ is a basis of $V$. This proves the lemma \ref{conicsing}. \nolinebreak \raise-0.1em\hbox{$\Box$} \bigskip To obtain the \ref{recstable}, we have now to prove that there exists no instanton such that for every $N$ in ${\rm I} \! {\rm P} _3$ the net has such a 2 dimensional totally isotropic space $L_N$. If there was such a bundle, then, its hypersurface of jumping line would contain a double quadratic complex. Let $Q$ be this quadratic complex, and $\Sigma$ be the set of $N\in {\rm I} \! {\rm P}_3$ such that the $\alpha$-plane $\alpha_N$ doesn't cut $Q$ in a smooth conic, and let $\Sigma^{\prime}$ be the set of singular rays (i.e: the points of $G$ which are singularities of $Q\cap\alpha_N$ for some $N$). As $\Sigma$ is the degeneracy locus of a square matrix, it is at least 2 dimensional. We can show that for any $Q$, the set $\Sigma^{\prime}$ is at least 2 dimensional too: let $\sum_{i=0}^5 x_i^2=0$ and $\sum_{i=0}^5 k_ix_i^2=0$ be the equations of $G$ and $Q$, and consider the third quadric $Q^{\prime}: \sum_{i=0}^5 k_i^2x_i^2=0$, then $ G\cap Q\cap Q^{\prime} \subset \Sigma^{\prime}$ because $(x_i)\in G\cap Q\cap Q^{\prime} $ implies that $(k_ix_i)$ is a line meeting the line $(x_i)$ in some point $N$, and for any $(z_i)$ in the $\alpha$-plane $\alpha_N$, the equation $(z_i)$ cut $(k_ix_i)$ which is $\sum_{i=0}^5 k_i x_iz_i=0$ means also $(z_i)\in T_{(x_i)}Q$. Then we can conclude using the lemma \ref{conicsing}, proposition \ref{c2=5courbe}, and Coanda's theorem ([Co]), remarking in the cases where $N$ is on the jumping section of $E_H$ for some jumping-plane $H$, that $E$ can't be a special t'Hooft bundle, otherwise there would exist a $n$-jumping line through $N$ which as been excluded in the demonstration of \ref{conicsing}. \nolinebreak \raise-0.1em\hbox{$\Box$} \subsection{Description of the boundary $\delta_{\theta}$} We'd like now to understand more explicitly the fiber of $Z_{\delta_{\theta}}$ over some $\theta $. In other words, we need a description of the splitting condition of the restriction of $F$ to $C$. The following will consist of 2 descriptions of this condition. The first one is explicit in function of the net of quadrics, and enables to prove that the fiber $Z_{\delta_{\theta }}$ is the complete intersection of $G^{\prime \prime}_{\theta}$ with some hyperplanes This could have been conjectured from Ellingsrud and Stromme viewpoint because of the map $\delta$ of the previous subsection, but we can't conclude directly from this because there is some torsion in $Pic G^{\prime\prime}_{\theta}$ . Furthermore, we will need a second description to understand the ramification of this morphism in term of vector bundle and to get informations on I$_n$. \begin{itemize} \item First description \end{itemize} Let $K$ be the 2-dimensional vector space generated by 2 sections of $\theta (2)$ with disjoint zeros. Let $F$ be the vector bundle over the exceptional plane $Y= {\rm I} \! {\rm P} (V ^{\rm v})$ defined by the exact sequence: \begin{equation}\label{F} 0 \longrightarrow F \longrightarrow K \otimes {\cal O}_Y \longrightarrow \theta (2) \longrightarrow 0 \end{equation} Let $S$ be the kernel of the map $H ^{\rm v} \otimes \theta (3) \rightarrow \theta ^2(4)$ obtained when twisting by $\theta (2)$ the exact sequence (\ref{reseau}) of the net of quadric. As ${\cal E}xt^1(\theta (2),{\cal O}_Y)=\theta (1)$, we can compute the kernel of the restriction of (\ref{F}) to $C$ by dualizing (\ref{F}). This yields to $Tor_1(\theta (2),\theta (2))={\cal O}_C(1)$. We can now deduce from (\ref{reseau}) the exact sequence: $$0\longrightarrow {\cal O}_C(1)\longrightarrow H\otimes \theta (2)\longrightarrow H ^{\rm v}\otimes \theta (3)\longrightarrow \theta ^2(4)\longrightarrow 0$$ Taking there global sections, we get the following diagram where $\Sigma $ is by definition the cokernel of \hbox{$H\otimes H^0(\theta (2))\rightarrow H ^{\rm v}\otimes H^0(\theta (3))$}. \begin{center} {\begin{picture}(3000,1100) \setsqparms[1`1`0`1;800`300] \putsquare(1100,550)[H\otimes H^0(\theta (2))`\phantom{H^0(S)}`H\otimes H^0(\theta (2))`\phantom{H^{{\rm v}}\otimes H^0(\theta (3))};`\wr``b] \setsqparms[1`1`1`1;600`300] \putsquare(1900,550)[H^0(S)`H^1({\cal O}_C(1))`H^{{\rm v}}\otimes H^0(\theta (3))`\Sigma;```] \setsqparms[0`1`1`1;600`300] \putsquare(1900,250)[``H^0(\theta ^2(4))`H^0(\theta ^2(4));```\sim] \putmorphism(400,850)(1,0)[{H^0({\cal O}_C(1))}`\phantom{H\otimes H^0(\theta (2))}`]{700}1a \putmorphism(0,850)(1,0)[0`\phantom{H^0({\cal O}_C(1))}`]{400}1a \putmorphism(2500,850)(1,0)[\phantom{H^1({\cal O}_C(1))}`0`]{400}1a \putmorphism(2500,550)(1,0)[\phantom{\Sigma}`0`]{400}1a \putmorphism(1900,250)(0,-1)[`0`]{250}1a \putmorphism(2500,250)(0,-1)[`0`]{250}1a \putmorphism(1900,1100)(0,-1)[0``]{250}1a \putmorphism(2500,1100)(0,-1)[0``]{250}1a \end{picture}} \end{center} On the other hand, we can compute the following diagram displaying the sequence (\ref{F}) horizontally and the sequence (\ref{reseau}) vertically: \begin{center} {\begin{picture}(2200,1100)(0,0) \setsqparms[1`1`1`1;600`300] \putsquare(500,550)[F\otimes H`K\otimes H\otimes{\cal O}_Y`F\otimes H^{{\rm v}}(1)`K\otimes H^{{\rm v}}(1);```] \setsqparms[1`0`1`1;600`300] \putsquare(1100,550)[\phantom{K\otimes H\otimes{\cal O}_Y}`H\otimes \theta(2)`\phantom{K\otimes H^{{\rm v}}(1)}`H^{{\rm v}}\otimes \theta(3);```] \setsqparms[0`1`1`1;600`300] \putsquare(500,250)[``F\theta(2)`K\otimes \theta(2);```] \setsqparms[0`0`1`1;600`300] \putsquare(1100,250)[``\phantom{K\otimes \theta(2)}`\theta^2(4);```] \putmorphism(0,850)(1,0)[0`\phantom{F\otimes H}`]{500}1a \putmorphism(0,550)(1,0)[0`\phantom{F\otimes H^{{\rm v}}(1)}`]{500}1a \putmorphism(1700,850)(1,0)[\phantom{H\otimes\theta(2)}`0`]{500}1a \putmorphism(1700,550)(1,0)[\phantom{H^{{\rm v}}\otimes\theta(3)}`0`]{500}1a \putmorphism(1700,250)(1,0)[\phantom{\theta^2(4)}`0`]{500}1a \putmorphism(1100,250)(0,-1)[`0`]{250}1a \putmorphism(1100,1100)(0,-1)[0``]{250}1a \putmorphism(1700,250)(0,-1)[`0`]{250}1a \end{picture}} \end{center} We obtain by passing there to global sections the following: \begin{center} {\begin{picture}(2600,1600) \setsqparms[1`1`1`1;800`400] \putsquare(500,250)[H^{{\rm v}}\otimes V\otimes K`H^0(\theta (3))\otimes H^{{\rm v}}`H^0(\theta (2))\otimes K`\Sigma ;a`c``] \putsquare(1300,250)[\phantom{H^0(\theta (3))\otimes H^{{\rm v}}}`H^1(F(1))\otimes H^{{\rm v}}`\phantom{\Sigma}`H^1(F\theta (2));```f] \putsquare(500,650)[H\otimes K`H^0(\theta (2))\otimes H`\phantom{H^{\rm v}\otimes V\otimes K}`\phantom{H^0(\theta (3))\otimes H^{{\rm v}}};``b`] \setsqparms[1`1`1`0;800`400] \putsquare(1300,650)[\phantom{H^0(\theta (2))\otimes H}`H^1(F)\otimes H``;```] \putmorphism(2100,650)(1,0)[\phantom{H^1(F(1))\otimes H^{{\rm v}}}`0`]{500}1a \putmorphism(2100,1050)(1,0)[\phantom{H^1(F)\otimes H}`0`]{500}1a \putmorphism(500,250)(0,-1)[`0`]{250}1a \putmorphism(1300,250)(0,-1)[`0`]{250}1a \putmorphism(500,1350)(0,-1)[0``]{300}1a \putmorphism(2100,1350)(0,-1)[H^0(F\theta (2))``]{300}1a \putmorphism(2100,1600)(0,-1)[0``]{250}1a \end{picture}} \end{center} The splitting condition of sequence (\ref{Fc}) is just $f(g(H^0({\cal O}_C)))=0$ , where $f$ and $g$ are constructed in the following diagram obtained by taking global sections in the sequence (\ref{F}) twisted by $\theta (2)$. \begin{center} {\begin{picture}(2500,550) \setsqparms[1`1`1`1;600`300] \putsquare(1000,0)[H^0 F\theta(2)`H^0{\cal O}_C`H^0 F\theta(2)`K\otimes H^0\theta(2); `\wr`g`] \setsqparms[1`0`1`1;600`300] \putsquare(1600,0)[\phantom{H^0{\cal O}_C}`H^1({\cal O}_C(1))`\phantom{K\otimes H^0\theta(2)}`\Sigma;```f] \putmorphism(1600,550)(0,-1)[0``]{250}1a \putmorphism(2200,550)(0,-1)[0``]{250}1a \putmorphism(0,300)(1,0)[0`\phantom{H^0({\cal O}_C(1))}`]{400}1a \putmorphism(400,300)(1,0)[H^0({\cal O}_C(1))`\phantom{H^0 F\theta(2)}`]{600}1a \end{picture}} \end{center} So, the condition that the 2-dimensional vector space $K\subset H^0(\theta (2))$ gives an instanton, which is also the vanishing of $H^0({\cal O}_C)$ in $\Sigma $, can be translated if $W$ denote the preimage by $c$ of $g(H^0({\cal O}_C))$ in $H ^{\rm v} \otimes V \otimes K$, by the condition: $a(W)\subset {\rm Im}b$. \bigskip We will now identify this condition with an explicit map \hbox{$\stackrel{2}{\Lambda }H^0(\theta (2)) \stackrel{\beta}{\longrightarrow} H^1({\cal O}_C(1))$}. To construct $\beta $, let's first consider the Eagon-Northcott complexes associated to the sequence (\ref{reseau}) twisted by $-1$. This gives for the second symmetric power the following sequence: $$ 0\rightarrow (\stackrel{2}{\Lambda}H)(-2) \rightarrow H ^{\rm v} \otimes H(-1) \kercoker{A ^{\rm v}} S_2H ^{\rm v} \rightarrow \theta ^2(2) \rightarrow 0$$ We obtain by dualizing it the 2 following short exact sequences: \begin{center} $0 \rightarrow S_2H \otimes {\cal O}_Y\rightarrow A \rightarrow {\cal O} _C(1) \rightarrow 0$ and $0\rightarrow A \rightarrow H ^{\rm v} (1) \otimes H \stackrel{e}{\rightarrow} (\stackrel{2}{\Lambda}H) ^{\rm v} (2) \rightarrow 0$ \end{center} Let $S_2H \otimes {\cal O}_Y \stackrel{d}{\rightarrow} H ^{\rm v} (1) \otimes H $ be the composition of the 2 previous injections. Then $d$ and $e$ are illustrated in the following diagram not exact in the middle: \begin{center} {\begin{picture}(2500,300) \setsqparms[1`1`1`1;800`300] \putsquare(500,0)[S_2 H\otimes{\cal O}_Y`H\otimes H^{{\rm v}}(1)`S_2 \left(H^{\rm v}(1)\right)`H^{{\rm v}}(1)\otimes H^{{\rm v}}(1);d`S_2 m`m\otimes 1`] \setsqparms[1`0`1`1;800`300] \putsquare(1300,0)[\phantom{H\otimes H^{{\rm v}}(1)}`(\buildrel{2}\over{\Lambda}H^{{\rm v}})(2)`\phantom{H^{{\rm v}}(1)\otimes H^{{\rm v}}(1)}`(\buildrel{2}\over{\Lambda}H^{{\rm v}})(2);e``\wr`] \putmorphism(0,0)(1,0)[0`\phantom{S_2 H^{\rm v}(1) \otimes{\cal O}_Y}`]{500}1a \putmorphism(0,300)(1,0)[0`\phantom{S_2 H\otimes{\cal O}_Y}`]{500}1a \putmorphism(2100,0)(1,0)[\phantom{(\buildrel{2}\over{\Lambda}H^{\rm v})(2)}`0`]{500}1a \putmorphism(2100,300)(1,0)[\phantom{(\buildrel{2}\over{\Lambda}H^{{\rm v}})(2)}`0`]{500}1a \end{picture} } \end{center} The display of the monad $(d,e)$ made by the first line is: \begin{center} \begin{picture}(2000,1100) \setsqparms[1`1`1`1;600`300] \putsquare(1100,250)[H\otimes H^{\rm v}(1)`B`(\buildrel{2}\over{\Lambda}% H^{\rm v})(2)`(\buildrel{2}\over{\Lambda}H^{\rm v})(2);`e``\sim ] \setsqparms[1`1`1`1;600`300] \putsquare(500,550)[S_2 H \otimes {\cal O}_Y`A`S_2 H \otimes{\cal O}_Y`\phantom{H\otimes H^{\rm v}(1)};`\wr ``d] \setsqparms[1`0`1`0;600`300] \putsquare(1100,550)[\phantom{A}`{\cal O}_C(1)`\phantom{H\otimes H^{\rm v}(1)}`\phantom{B};```] \putmorphism(0,550)(1,0)[0`\phantom{S_2 H \otimes{\cal O}_Y}`]{500}1a \putmorphism(0,850)(1,0)[0`\phantom{S_2 H \otimes{\cal O}_Y}`]{500}1a \putmorphism(1700,550)(1,0)[\phantom{B}`0`]{300}1a \putmorphism(1700,850)(1,0)[\phantom{{\cal O}_C(1)}`0`]{300}1a \putmorphism(1100,250)(0,1)[\phantom{(\buildrel{2}\over{\Lambda}H^{\rm v})(2)}`0`]{250}{1}l \putmorphism(1700,250)(0,1)[\phantom{(\buildrel{2}\over{\Lambda}H^{\rm v})(2)}`0`]{250}{1}l \putmorphism(1100,1100)(0,1)[0`\phantom{A}`]{250}1l \putmorphism(1700,1100)(0,1)[0`\phantom{{\cal O}_C(1)}`]{250}1l \end{picture} \end{center} The last column of this display gives when passing to global sections a boundary map \hbox{$\stackrel{2}{\Lambda }H ^{\rm v}\otimes S_2V \stackrel{\beta_0}{\rightarrow} H^1({\cal O}_C(1))$} which vanishes on the image of the map \hbox{$H\otimes H ^{\rm v} \otimes V \stackrel{b^{\prime }}{\rightarrow } \stackrel{2}{\Lambda }H ^{\rm v} \otimes S_2V$} obtained from $e$ by taking global sections. Using now the surjection of (\ref{reseau}) we obtain the commutative diagram: \begin{center} \begin{picture}(800,300) \setsqparms[1`1`1`1;800`300] \putsquare(0,0)[H\otimes H^{\rm v}\otimes V`H^{\rm v}\otimes H^{\rm v}\otimes S_2 V`H\otimes H^0 \theta (2)`H^{\rm v}\otimes H^0 \theta (3);b^{\prime}```b] \end{picture} \end{center} which identify the conditions $a^{\prime }(W)\subset Im b^{\prime}$ and $a(W) \subset Im b$, where $a^{\prime }$ is defined in the diagram below. As we had already identified the splitting condition with $a(W) \subset Im b$, we can from the following diagram construct a map $\beta$ giving the corollary \ref{scindage}. \begin{center} {\begin{picture}(2000,550) \setsqparms[1`-1`-1`1;800`300] \putsquare(0,0)[H\otimes H^{{\rm v}}\otimes V`\buildrel{2}\over{\Lambda}H^{{\rm v}}\otimes S_2 V`H\otimes H^{{\rm v}}\otimes V`\buildrel{2}\over{\Lambda}(H^{{\rm v}}\otimes V);b^{\prime }`\wr`a^{\prime }`] \setsqparms[1`0`-1`1;800`300] \putsquare(800,0)[\phantom{\buildrel{2}\over{\Lambda}H^{{\rm v}}\otimes S_2 V}`H^1 {\cal O}_C (1)`\phantom{\buildrel{2}\over{\Lambda}(H^{\rm v}\otimes V)}`\buildrel{2}\over{\Lambda}H^0\theta(2);\beta _0``\beta`c] \put(800,350){\vector(0,1){150}} \put(775,525){0} \putmorphism(1600,0)(1,0)[\phantom{\buildrel{2}\over{\Lambda}H^0\theta(2)}`0`]{500}1a \putmorphism(1600,300)(1,0)[\phantom{H^1 {\cal O}_C(1)}`0`]{500}1a \end{picture} } \end{center} \begin{corollary} \label{scindage}The splitting condition of the sequence (\ref{Fc}) is given by the vanishing of a surjective map $\stackrel{2}{% \Lambda }(H^0\theta (2))\stackrel{\beta }{\rightarrow }H^1({\cal O}_C(1))$ \end{corollary} This gives the following description of $Z_{\delta_{\theta}}$ where we denote by $Z_s$ the vanishing locus of a section $s$ of $\theta (2)$: $$Z_{\delta_{\theta}}={\rm I} \! {\rm P}(\{s\wedge s^{\prime }|s,s^{\prime }\in H^0\theta (2),\;Z_s \cap Z_{s^{\prime}}= \emptyset \;\text{and}\;\beta (s\wedge s^{\prime })=0\})$$ \begin{itemize} \item Second description \end{itemize} Let $E$ be an $n$-instanton, $\theta $ its associated $\theta $-characteristic, and $s$, $s^{\prime}$ be 2 sections of $\theta (2)$ associated to $E$. The purpose of this second description is to understand the possible singularities of $Z_{\delta_{\theta}}$ at $s\wedge s^{\prime}$ in terms of the vector bundle $E$. This part will be divided in 3 steps. First, we will construct a new map $\beta_E $, then we'll have to identify the vanishing locus of $\beta$ and $\beta_E$, and in the third step, this will enable us to understand the particular bundles that give singularities of $Z_{\delta_{\theta}}$. \bigskip 1) We will construct here a map \hbox{$\beta_E:\ H^0\theta (2) \oplus H^0\theta (2) \rightarrow H^1 {\cal O}_C(1)$}. Let's take the exact sequence of restriction to the exceptional divisor $x=\tau-\sigma$ twisted by $E(\tau)$, where $\tau=p^* {{\cal O}_{\proj_3}}(1)$ and $\sigma= q^* {\cal O}_Y(1)$. \begin{center} $0\longrightarrow p^{*}E(\sigma )\longrightarrow p^{*}E(\tau )\longrightarrow p^{*}E(\tau )_{\mid x}\longrightarrow 0$ \end{center} apply it to the functor $q_*$ to get the following exact sequence where we recall that $F=q_*p^*E$, and that the 2 dimensional vector space in the right of the sequence is naturally the fiber of $E$ at $N$. \begin{equation} \label{Etau} 0\longrightarrow F(1)\longrightarrow q_{*}p^{*}E(\tau) \longrightarrow 2 {\cal O}_Y \longrightarrow 0 \end{equation} Let $2H^0\theta (2)\stackrel{\delta_E}{\rightarrow} H^1F\theta (3)$ be the boundary map obtained when taking global sections in (\ref{Etau}) twisted by $\theta (2)$. Taking global sections in (\ref{Fc}) twisted by $\theta (3)$ gives us a map $H^1F\theta (3)\rightarrow H^1{\cal O}_C(1)$. Define $\beta _E$ as the composition of $\delta_E$ and this map: $$ \beta_E:\ 2H^0\theta (2) \stackrel{\delta_E}{\longrightarrow} H^1F\theta (3) \longrightarrow H^1({\cal O}_C(1))$$ \bigskip 2) To link $\beta$ and $\beta _E$, we will prove here that for every surjection \hbox{$2{\cal O}_Y \stackrel{(\sigma ,\sigma ^{\prime})}{ \longrightarrow } \theta (2)\rightarrow 0$} where $\sigma ,\sigma ^{\prime }$ are such that $\beta(\sigma\wedge \sigma ^{\prime })=0$, then $\beta _E(-\sigma^{\prime },\sigma)=0$. For all that, apply to the sequence \ref{Etau} the functor $Hom(*,\theta (2))$. It gives a map $Hom(2{\cal O}_Y,\theta (2)) \rightarrow Hom(q_{*}p^{*}E(\tau ),\theta (2))$, which enable to pull back any $\phi: 2{\cal O}_Y \stackrel{(\sigma ,\sigma ^{\prime})}{ \longrightarrow } \theta (2)\rightarrow 0$ by a map $q_*p^*E(\tau)\stackrel{\psi}{\rightarrow} \theta(2)$ making the following diagram commutative:\\ \vbox{\begin{equation}\label{phi} {\begin{picture}(1600,550)(0,550) \setsqparms[1`-1`0`1;500`300] \putsquare(300,250)[F (1)`\phantom{q_* p^* E(\tau )}`F(1)`\phantom{N};`\wr ``] \setsqparms[1`-1`-1`1;500`300] \putsquare(800,250)[q_* p^* E(\tau)`2{\cal O}_{{\rm I} \! {\rm P}_2}`N`F^{\prime };```] \setsqparms[1`-1`-1`0;500`300] \putsquare(800,550)[\theta (2)`\theta (2)`\phantom{mmmm}`\phantom{mmm};\sim `\psi`\phi`] \setsqparms[0`-1`-1`0;500`250] \putsquare(800,850)[0`0`\phantom{mm}`\phantom{mm};```] \putmorphism(0,250)(1,0)[0`\phantom{F(1)}`]{300}1a \putmorphism(0,550)(1,0)[0`\phantom{F(1)}`]{300}1a \putmorphism(1300,850)(1,0)[\phantom{\theta (2)}`0`]{300}1a \putmorphism(1300,250)(1,0)[\phantom{F^{\prime }}`0`]{300}1a \putmorphism(1300,550)(1,0)[\phantom{2{\cal O}_C}`0`]{300}1a \setsqparms[0`-1`-1`0;500`250] \putsquare(800,0)[``0`0;```] \end{picture} } \end{equation} \vskip 5.5em} where $N$ and $F^{\prime}$ are by definition the kernel of $\psi $ and $\phi$, and where the middle line is the sequence \ref{Etau}. Restricting (\ref{phi}) to $C$ gives the 2 following diagrams where the kernel of the restriction of $\psi$ to $C$ is noted $M$:\\ \vbox{ \begin{equation}\label{phiC1} {\begin{picture}(1600,550)(0,550) \setsqparms[1`-1`0`1;500`300] \putsquare(300,250)[F_C (1)`\phantom{q_* p^* E(\tau )_C}`F_C (1)`\phantom{M};`\wr ``] \setsqparms[1`-1`-1`1;500`300] \putsquare(800,250)[q_* p^* E(\tau)_C`2{\cal O}_{{\rm I} \! {\rm P}_2}`M`\theta ^{\rm v}(-2);```] \setsqparms[1`-1`-1`0;500`300] \putsquare(800,550)[\theta (2)`\theta (2)`\phantom{mmmm}`\phantom{mmm};\sim `\psi`\phi`] \setsqparms[0`-1`-1`0;500`250] \putsquare(800,850)[0`0`\phantom{mm}`\phantom{mm};```] \putmorphism(0,250)(1,0)[0`\phantom{F_C(1)}`]{300}1a \putmorphism(0,550)(1,0)[0`\phantom{F_C(1)}`]{300}1a \putmorphism(1300,850)(1,0)[\phantom{\theta (2)}`0`]{300}1a \putmorphism(1300,250)(1,0)[\phantom{\theta ^{\rm v}(-2)}`0`]{300}1a \putmorphism(1300,550)(1,0)[\phantom{2{\cal O}_C}`0`]{300}1a \setsqparms[0`-1`-1`0;500`250] \putsquare(800,0)[``0`0;```] \end{picture} } \end{equation} \vskip 6em}\\ \vbox{ \begin{equation}\label{phiC2} {\begin{picture}(1600,550)(0,550) \setsqparms[1`-1`-1`1;500`300] \putsquare(300,550)[F_C (1)`M`F_C (1)`N_C;`\wr ``] \setsqparms[1`0`-1`1;500`300] \putsquare(800,550)[\phantom{M}`\theta ^{\rm v}(-2)`\phantom{N_C}`F^{\prime }_C;```] \setsqparms[0`-1`-1`1;500`300] \putsquare(800,250)[\phantom{N_C}`\phantom{F^{\prime }_C}`\theta^{\rm v}(-1)`\theta^{\rm v}(-1);```\sim] \setsqparms[0`-1`-1`0;500`250] \putsquare(800,850)[0`0`\phantom{M}`\phantom{\theta ^{\rm v}(-2)};```] \putmorphism(0,550)(1,0)[0`\phantom{F_C(1)}`]{300}1a \putmorphism(0,850)(1,0)[0`\phantom{F_C(1)}`]{300}{1}a \putmorphism(1300,850)(1,0)[\phantom{\theta ^{\rm v}(-2)}`0`]{300}1a \putmorphism(1300,550)(1,0)[\phantom{F^{\prime }_C}`0`]{300}1a \setsqparms[0`-1`-1`0;500`250] \putsquare(800,0)[``0`0;```] \end{picture} } \end{equation} \vskip 5.5em} Construct a map $0 \rightarrow \theta ^{\rm v} \stackrel{i}{\rightarrow}M$ by composition of the injection $0 \rightarrow \theta ^{\rm v} \rightarrow F_C(1)$ of sequence (\ref{Fc}) with the injection of $F_C(1)$ in $M$ of the diagram (\ref{phiC2}). Denote by $M^{\prime}$ the cokernel of $i$, and by $E^{\prime}$ the cokernel of the injection $0 \rightarrow \theta ^{\rm v} \rightarrow q_*p^*E(\tau)_C$ obtained by composing $i$ with the injection of $M$ in $q_*p^*E(\tau)_C$ of the diagram (\ref{phiC1}). Now using the fact that we have an injection of the cokernel of (\ref{Fc}) in $M^{\prime}$ and in $E^{\prime}$, we obtain from (\ref{phiC1}) the following diagram:\\ \vbox{ \begin{equation}\label{Mprime} {\begin{picture}(1600,550)(0,550) \setsqparms[1`-1`0`1;500`300] \putsquare(300,250)[\theta ^{\rm v}(-1)`E^{\prime }`\theta ^{\rm v}(-1)`M^{\prime };`\wr ``] \setsqparms[1`-1`-1`1;500`300] \putsquare(800,250)[\phantom{E^{\prime }}`2{\cal O}_{{\rm I} \! {\rm P}_2}`\phantom{M^{\prime }}`\theta ^{\rm v}(-2);```] \setsqparms[1`-1`-1`0;500`300] \putsquare(800,550)[\theta (2)`\theta (2)`\phantom{E^{\prime }}`\phantom{2{\cal O}_{{\rm I} \! {\rm P}_2}};\sim ``\phi`] \setsqparms[0`-1`-1`0;500`250] \putsquare(800,850)[0`0`\phantom{mm}`\phantom{mm};```] \putmorphism(0,250)(1,0)[0`\phantom{\theta ^{\rm v}(-1)}`]{300}1a \putmorphism(0,550)(1,0)[0`\phantom{\theta ^{\rm v}(-1)}`]{300}1a \putmorphism(1300,250)(1,0)[\phantom{\theta ^{\rm v}(-2)}`0`]{300}1a \putmorphism(1300,550)(1,0)[\phantom{2{\cal O}_C}`0`]{300}1a \setsqparms[0`-1`-1`0;500`250] \putsquare(800,0)[``0`0;```] \end{picture} } \end{equation} \vskip 5.5em} We can now notice that $\beta _E$ is just the boundary obtained when taking global section in the middle line of (\ref{Mprime}) twisted by $\theta(2)$. As the right column of (\ref{Mprime}) twisted by $\theta (2)$ gives an injection on $H^0({\cal O}_C)$ in $2H^0(\theta(2))$ of image $(-\sigma^{\prime}, \sigma)$, we obtain that $\beta_E (-\sigma^{\prime}, \sigma)=0$ if and only if $M^{\prime}$ splits. But $F_C$ split because $\beta(\sigma^{\prime}\wedge \sigma)=0$ by hypothesis, so we have after the choice of a section $\alpha$ of (\ref{Fc}) the diagram: \begin{center} {\begin{picture}(1550,1100) \setsqparms[1`-1`-1`1;500`300] \putsquare(300,550)[\theta ^{\rm v}`M`F_C (1)`N_C;i```] \setsqparms[1`-1`-1`1;450`300] \putsquare(800,550)[\phantom{M}`M^{\prime }`\phantom{N_C}`F^{\prime }_C;```] \setsqparms[0`-1`-1`1;500`300] \putsquare(300,250)[\phantom{F_C (1)}`\phantom{N_C}`\theta ^{\rm v}(-1)`\theta ^{\rm v}(-1);`\alpha ``\sim] \setsqparms[0`-1`-1`0;500`250] \putsquare(300,850)[0`0`\phantom{mm}`\phantom{mm};```] \putmorphism(0,550)(1,0)[0`\phantom{F_C (1)}`]{300}1a \putmorphism(0,850)(1,0)[0`\phantom{\theta ^{\rm v}}`]{300}1a \putmorphism(1250,550)(1,0)[\phantom{F^{\prime }_C}`0`]{300}1a \putmorphism(1250,850)(1,0)[\phantom{M^{\prime }}`0`]{300}1a \setsqparms[0`-1`-1`0;500`250] \putsquare(300,0)[``0`0;```] \end{picture} } \end{center} which proves that $M^{\prime }\simeq F_C^{\prime }$, and that $\beta$ and $\beta_E$ have the same vanishing locus. \nolinebreak \raise-0.1em\hbox{$\Box$} \bigskip 3) Identification of $Z_{\delta_{\theta}}$ 's singularities.\\ Let $E$ be an instanton, and $s\wedge s^{\prime }$ its associated element of $Z_{\delta_{\theta}}$. Denote by $K$ the vector space $Vect(s,s^{\prime})\subset H^0(\theta(2))$. The bundle $E$ gives a singularity of $G(2,H^0(\theta(2)))\cap \ker\beta$ if and only if the Zariski tangent space to $G(2,H^0(\theta(2)))$ at $s\wedge s^{\prime }$ is included in one of the hyperplanes given by $\beta $. In other words, it means that $K$ is in the kernel of one of the skew map given by $\beta $. As $H^1({\cal O}_C(1))=S_{n-4}V ^{\rm v}$, it means that there is some $p\in S_{n-4}V$ such that $K\otimes p$ is in the kernel of \hbox{$K \otimes S_{n-4}V \stackrel{\beta_E}{\rightarrow} H^0(\theta(2)) ^{\rm v} $}. But we can understand this kernel explicitly with the following diagram, where the first 2 lines are obtained from the sequences (\ref{F}) and (\ref{Fc}), and the last from sequence (\ref{Etau}) twisted by ${\cal O}_Y(n-4)$:\\ \vbox{ \begin{equation}\label{diagsing} {\begin{picture}(3620,425)(0,425) {\footnotesize \setsqparms[1`-1`-1`1;600`300] \putsquare(2700,550)[H^2 F(-3)`H^2 (2{\cal O}_C (-3))`H^1(F_C (n-3))`(H^0 \theta(2))^{\rm v};```] \setsqparms[1`-1`0`1;700`300] \putsquare(2000,550)[H^1 \theta(-1)`\phantom{H^2 F(-3)}`H^1 \theta^{\rm v}(n-4)`\phantom{H^1(F_C (n-3))};`\wr ``] \putmorphism(-200,250)(1,0)[0`H^0 (q_* p^* E(n-3)\sigma)`]{500}1a \putmorphism(300,250)(1,0)[\phantom{H^0 (q_* p^* E(n-3)\sigma)}`H^0 (q_* p^* E(\tau +(n-4)\sigma))`]{950}1a \putmorphism(1250,250)(1,0)[\phantom{H^0 (q_* p^* E(\tau +(n-4)\sigma))}`K \otimes S_{n-4}V`]{800}1a \putmorphism(2050,250)(1,0)[\phantom{K \otimes S_{n-4}V}`H^1 (F(n-3))`\delta_E]{650}1a \putmorphism(2700,250)(0,1)[\phantom{MM}`0`]{250}{-1}l \putmorphism(2700,550)(0,1)[\phantom{MM}`\phantom{MM}`]{300}{-1}l \putmorphism(1500,850)(1,0)[0`\phantom{H^1 \theta(-1)}`]{500}{1}a \putmorphism(3300,550)(1,0)[\phantom{(H^0 \theta(2))^{\rm v}}`0`]{320}{1}a \putmorphism(1500,550)(1,0)[0`\phantom{H^1 \theta^{\rm v}(n-4)}`]{500}{1}a \bezier{0}(2050,300)(2050,400)(2300,400) \put(2300,400){\line(1,0){750}} \bezier{0}(3050,400)(3300,400)(3300,500) \put(3300,500){\vector(0,1){20}} \put(3150,370){$\beta_E$} } \end{picture} } \end{equation} \vskip 3.0em} This proves that the maps $\beta_E$ and $\delta _E$ have the same kernel, which is nothing else than the cokernel of $H^0(q_{*}p^{*}E(n-3)\sigma )\stackrel{j}{\rightarrow} H^0(q_{*}p^{*}E(\tau +(n-4)\sigma ))$. \begin{corollary}\label{ramification} The ramification locus of the map $U_n^s \rightarrow \Theta^s_{lf}$ is made of special t'Hooft bundle for $c_2=4$.\\ For $c_2=5$, there are 3 possibilities. If the Zariski tangent space of $Z_{\delta_{\theta}}$ is 15 (respectively 16) dimensional, then this point arise from a t'Hooft bundle (respectively special t'Hooft). If it is 14 dimensional, then the bundle twisted by 2 have 2 sections vanishing at $N$. \end{corollary} {\bf NB:} For $c_2=5$, a bundle such that $h^0{\cal J}_N \otimes E(2)\geq 2$ doesn't necessarily give a singularity of $Z_{\delta_{\theta}}$. The result \ref{ramification} is clear for $c_2=4$ because we have in this situation $h^0(q_*p^*E(\tau)) \geq 2$. When $c_2=5$, any singularity of $Z_{\delta_{\theta}}$ is such that $K \otimes p$ is in the cokernel of $j$ for some $p\in V$, so $h^0(q_*p^*E(\tau +\sigma)) \geq 2$. Although this will be enough to prove the smoothness of I$_5$, we will go further to understand the ramification, and also because it seems it is not enough to obtain the connexity. The cases of a 15 or 16 dimensional Zariski tangent space, have the following interpretation in term of the cokernel of $H^0q_*p^*E(\sigma) \stackrel{j^{\prime}}{\rightarrow} H^0q_*p^*E(\tau)$: Take global sections in (\ref{Etau}) to obtain like in the diagram (\ref{diagsing}) a map $\beta ^{\prime }:K \rightarrow H^1\theta ^{\rm v}(-1) = H^0(\theta (3)) ^{\rm v} $ whose kernel is the cokernel of $j^{\prime}$. In fact $\beta$ is the composition of $\beta^{\prime}$ with the injection \hbox{$H^0(\theta(3)) ^{\rm v} \rightarrow (H^0(\theta(2))\otimes V) ^{\rm v}$}. But the Zariski tangent space of $Z_{\delta_{\theta}}$ at $s \wedge s^{\prime}$ is 15 (resp 16) dimensional if and only if there is a 2 (resp 3) dimensional subspace $W$ of $V$ such that the map \mbox{$K\otimes H^0(\theta (2))\otimes W\rightarrow \setlength{\unitlength}{0.01em $} induced by $\beta $ is zero. -If dim$W=3$, then $\beta^{\prime}=0$ and thus $h^0(q_*p^*E(\tau))=2$. -If dim$W=2$, then we have to find an element $s_0 \in K$ such that the map $\beta^{\prime}(s_0): H^0(\theta(3)) \rightarrow \setlength{\unitlength}{0.01em$ is zero. Let $(p,q)$ be a basis of $W$, and compute the dimension of \hbox{$\{H^0(\theta (2))\otimes p\}\cap \{H^0(\theta (2))\otimes q\}$} in $H^0(\theta(3))$. Indeed, as \hbox{$H^0\theta(3)\otimes K \stackrel{\beta^{\prime}}{\rightarrow} \setlength{\unitlength}{0.01em $} is zero on the image of $H^0(\theta(2)) \otimes W \otimes K$ in $H^0(\theta(3)) \otimes K$, if $\{H^0(\theta (2))\otimes p\}+\{H^0(\theta (2))\otimes q\}$ is 0 or 1 codimensional in $H^0\theta(3)$, then there is such an $s_0$. So assume that it is not the case, then $\dim \{H^0(\theta (2))\otimes p\}\cap \{H^0(\theta (2))\otimes q\}\geq 7$, and this intersection would contain 2 elements independent relatively to $H^0(\theta(1))\otimes pq$. So there would exist 2 sections $s,\sigma$ of $\theta (2)$ (independent relatively to $H^0(\theta(1))\otimes q$), and $s^{\prime},\sigma^{\prime}$ (independent relatively to $H^0(\theta(1))\otimes p$) such that $s.p=s^{\prime}.q$ and $\sigma.p=\sigma^{\prime}.q$. As those sections are not in $H^0(\theta(1))\otimes pq$, the lines $p,q$ have to cut one another in a point $P\in C$, and the 4 remaining points of $q\cap C$ have to be zeros of $s$ and $\sigma$. As $\theta(2)$ is generated by its global sections, those vanishing in a given point are always an hyperplane of $H^0(\theta(2))$. As the intersection of the 5 hyperplanes associated to $q\cap C$ is $H^0(\theta(1))\otimes q$, the intersection of 4 of them can't be 7 dimensional, which contradicts the hypothesis. \begin{remark}\label{ngone} For $c_2=n$, if there is a section of $\theta (2)$ which is in the cokernel of $H^0q_*p^*E(\sigma) \stackrel{j^{\prime}}{\rightarrow} H^0q_*p^*E(\tau)$, then this section vanishes on the vertices of the complete $(n+1)$-gone (inscribed in the curve of jumping lines) obtained from the lines through $N$ which are bisecant to the zero locus of the associated section of $E(1)$ \end{remark} Let $E$ be an $n$-instanton such that there is a section $s$ of $\theta (2)$ in the cokernel of $j^{\prime}$. Let $\Im $ be the ideal of the vanishing locus of the section of $E(1)$ coming from $s$. The bundle $F=q_*p^*E$ is also $q_*p^*\Im(\tau)$, and we can understand $s$ with the following diagram, where the first column is the natural evaluation: \hbox{$ (q_{*}p^{*}E) \otimes (q_{*} {\cal O}_{ \widetilde{{\rm I} \! {\rm P}}\!_3} (\tau))\rightarrow q_{*} p^{*}E(\tau )$}. \begin{center} {\begin{picture}(2200,900) \setsqparms[1`-1`-1`1;900`300] \putsquare(800,0)[q_* p^* E(\tau )`q_{*}p^{*}\Im (2\tau )`F\oplus F(1)`q_{*}p^{*}\Im (\tau )\otimes q_{*}({\cal O}_{\widetilde{{\rm I} \! {\rm P}}\!_3}(\tau ));```\sim ] \setsqparms[0`-1`-1`1;900`300] \putsquare(800,600)[0`0`\theta (2)`L;```] \put(800,375){\vector(0,1){150}} \put(1700,375){\vector(0,1){150}} \putmorphism(300,300)(1,0)[{\cal O}_Y`\phantom{q_* p^* E(\tau )}`]{500}1a \putmorphism(1700,300)(1,0)[\phantom{q_{*}p^{*}\Im (2\tau )}`0`]{500}1a \putmorphism(0,300)(1,0)[0`\phantom{{\cal O}_Y}`]{300}1a \put(400,400){\vector(2,1){250}} \put(500,500){$s$} \end{picture}} \end{center} As $R^1q_*p^*\Im(2\tau) =0$ because there are no multi-jumping lines through $N$, we have for any line $D$ containing $N$ a surjection from $q_*p^*\Im (2\tau)_{\{d\}}$ onto $H^0(\Im_D(2\tau))$ where $\Im_D=\Im \otimes {\cal O}_D$. But the map \hbox{$q_{*}p^{*}\Im (\tau )_{\{d\}}\otimes q_{*}({\cal O}_{\widetilde{{\rm I} \! {\rm P}}\!_3}(\tau )) \rightarrow H^0(\Im _D(2\tau)) $} has to be zero if $D$ is bisecant to the scheme defined by $\Im$, thus the support of $L$ must contain the points corresponding to those lines. Computing the degree, we can conclude that $s$ vanish on the $(n+1)$-gone whose vertices are the projections from $N$ of the previous bisecant. \begin{corollary} When $c_2=4$ the fiber $Z_{\delta_{\theta}}$ is singular if and only if the support of $\theta $ is a Lur\"oth quartic. In this situation, the singularity $s\wedge s^{\prime}$ is such that the zeros of $s$ and $s^{\prime}$ give the pencil of complete pentagons inscribed on the quartic. \end{corollary} \subsection{The normality condition when $c_2 =5$; applications} We want here to understand when $Z_{\delta_{\theta}}$ is not regular in codimension 1. When $c_2=5$, the fiber $Z_{\delta_{\theta}}$ is an open subset of the Grassmannian $G(2,10)$ cut by 3 hyperplanes. Let $A_i$ be the skew forms associated to those hyperplanes, and $a_i$ be their skew linear maps. The key result of \ref{ramification} and \ref{ngone} is that any $s$ in the kernel of all the $a_i$ gives a $(n+1)$-gone inscribed in $\theta$'s support. So we want here to prove the following: \begin{proposition} \label{fibrenonorm}When $c_2=5$, if $G(2,H^0\theta(2)) \cap \ker \beta$ is singular in codimension 1 or have an excess dimension, then the $a_i$ have at least a 4-dimensional common kernel \end{proposition} Both singularity in codimension 1 and excess dimension imply that there is a 12 dimensional subscheme $S$ of $G(2,H^0\theta(2)) \cap \ker\beta$ such that the intersection is not transverse at any points of $S$. So, any $s\wedge s^{\prime }$ of $S$ is necessarily in some $\ker (\sum_{i=1}^{i=3}\lambda_i a_i)$, then $S$ is in $\bigcup\limits_{\lambda _i}G(2,\ker (\sum_{i=1}^{i=3}\lambda _ia_i))$ . As $S$ is 12 dimensional, at least one of those Grassmann manifold is of dimension 10 or more. So this Grassmann manifold has to be at least 12 dimensional, and for this value of $(\lambda _1,\lambda _2,\lambda _3)$ one have $\dim \ker(\sum_{i=1}^{i=3}\lambda _ia_i)=8$ (none of those maps are zero according to the \ref{scindage}). We can assume that this occur for $(1,0,0)$, so $\dim \ker a_1=8$. This yields to the following discussion: a) There is no $(\lambda _1,\lambda _2,\lambda _3) \neq (1,0,0)$ such that $\dim \ker \sum_{i=1}^{i=3}\lambda _ia_i)=8$. Then the union of those Grassmann manifolds is the union of $G(2,\ker a_1)$ with something of dimension at most 8+2. So we must have $G(2,\ker a_1)=S$, but it implies that all the $A_i$ are zero on $\stackrel{2}{\wedge }\ker a_1$. The $a_i$ have then at least a 4 dimensional common kernel. b) Otherwise, we can assume that $\dim \ker a_2=8$. If $a_3$ had also an 8 dimensional kernel, then the $a_i$ would easily have a 4 dimensional common kernel, so let's assume that if $\lambda_3 \neq 0$ then $\dim \ker (\sum_{i=1}^{i=3}\lambda _ia_i) \leq 6$. So \hbox{$\dim \bigcup\limits_{\lambda _i,\lambda_3\neq 0}G(2,\ker (\sum_{i=1}^{i=3}\lambda _ia_i))\leq 10$}, then $S\subset \bigcup\limits_{\lambda _i}G(2,\ker(\sum_{i=1}^{i=2}\lambda _ia_i))$. So $S\cap G(2,\ker a_1)$ is at most 1 codimensional in $G(2,\ker a_1)$, and it is in the vanishing of $A_{2_{|\stackrel{2}{\wedge }\ker a_1}}$ and of $A_{3_{|\stackrel{2}{\wedge }\ker a_1}}$. So those 2 skew forms have to be proportional on $\ker a_1$, then we can assume that $A_{3_{|\stackrel{2}{\wedge }\ker a_1}}=0$ so the $a_i$ have again a 4 dimensional common kernel as claimed. \begin{theorem} \label{irred5} The moduli space I$_5$ is irreducible of dimension 37. \end{theorem} Let's first show that for a general $\theta$ in the image of $U_N^s$, then $Z_{\delta_{\theta}}$ is irreducible. As $Z_{\delta_{\theta}}$ is open in $G^{\prime}_{\theta}\cap \ker \beta$,it is enough to show that $G^{\prime}_{\theta} \cap \ker \beta$ is irreducible, where $G^{\prime}_{\theta}=G(2,H^0(\theta(2)))$. But any degeneracy locus of $3{\cal O}_{G^{\prime}_{\theta}} \rightarrow {\cal O}_{G^{\prime}_{\theta}}(1)$ is connected (Cf [A-C-G-H] p311), and complete intersection. So it satisfies Serre's condition (S2), and in our situation, the \ref{fibrenonorm} shows that if it is not regular in codimension 1 (R1) or if it is excess dimensional, then the support of $\theta$ is a Darboux pentic according to \ref{ngone} (in fact it would be 4 times Darboux if this had a sense!). Anyway $\theta$ can't be general in $\Theta$. Then if $\theta$ is not Darboux, $G^{\prime}_{\theta}\cap \ker \beta$ is normal, connected and 13-dimensional, so it is irreducible and $Z_{\delta_{\theta}}$ too. But we proved in the \ref{recstable} that the $(U_N^s)_{N \in {\rm I} \! {\rm P}_3}$ cover $I_n$. Furthermore, the basis of the fibration $U_N^s\rightarrow \Theta ^s_{lf}$ is irreducible and smooth according to [So], and the fiber is ${\rm I} \! {\rm P}_4 \times Z_{\delta_{\theta}}$ which is irreducible, normal and 17 dimensional when $\theta$ is not Darboux. Furthermore, for any $\theta$ in the image of $U_N^s$, $Z_{\delta_{\theta}}$ is at most 15 dimensional because $G^{\prime}_{\theta}$ is not include in a hyperplane. As $U_N$ is open in I$_5$, all its irreducible components have dimension at least 37. But the Darboux $\theta$ are 3 codimensional in $\Theta^s_{lf}$ which is 20 dimensional, so the preimage of the Darboux $\theta$ is too small to make an irreducible component of $U_N^s$. Then $U_N^s$ has to be irreducible, and 37 dimensional. We can conclude using the \ref{recstable} that it is also the case for I$_5$. \begin{theorem} \label{lissite} The moduli space of mathematical instanton with $c_2=5$ is smooth \end{theorem} As $U_N^s$ is a fibration over $\Theta^s_{lf}$ which is smooth, any bundle which is not in the ramification of this morphism is a smooth point of I$_5$. So we have to check that a bundle $E$ which is in all the ramifications of $U_N^s \rightarrow \Theta^s_{lf}$ when $N$ fills ${\rm I} \! {\rm P}_3$ is a smooth point of I$_5$. According to the \ref{ramification} we must have $h^0({\cal J}_NE(2))\geq 2$ for every $N$. So $h^0E(2) \geq 4$, and either $E$ is a t'Hooft bundle, or $E$ belongs to the family described in the \ref{famille}, and both are smooth points of the moduli space (Cf \ref{lisse}). \subsection{Residual class}\label{appendiceresiduel} Let $E$ be an $n$-instanton, and assume that $E$ has a 2 parameter family $S$ of multi-jumping lines. Furthermore, we assume here that $S$ is irreducible, and that the residual scheme of $S$ in the scheme of multi-jumping lines $M$, is a curve (may be empty) not drawn on $S$. Eventually assume that $S_{red}$ is locally complete intersection.\\ \begin{picture}(800,0)(-60,800) \setsqparms[1`1`1`1;400`400] \putsquare(0,0)[\widetilde{S_{red}}`\widetilde{G}`S_{red}`G;j`g`f`i] \setsqparms[0`0`1`0;400`400] \putsquare(0,400)[`{{\rm I} \! {\rm P}(H^1E ^{\rm v}\otimes {\cal O}_{\widetilde{G}})}`\phantom{\widetilde{G}}`;``\pi`] \end{picture} \hfill \begin{minipage}[t]{10.5cm} In order to define and compute the class of $C$ in the chow ring of $G$ (graduated by the dimension), we will have to work in the blow up $\widetilde{G}$ of $G$ along $S_{red}$. Let $\widetilde{S_{red}}$ be the exceptional divisor, and $x$ be the class of $\widetilde{S_{red}}$ in $A_3\widetilde{G}$, we have: \hfill {$A_k\widetilde{G}=(A_k\widetilde{S_{red}}\oplus A_kG)/\alpha (A_{k}S_{red})$}\hfill\phantom{.}\\ where $\forall y\in A_{k}S_{red}$, $\alpha (y)=(c_1(g^{*}N/{\cal O}_N(-1)\cap g^{*}y,-i_{*}y)$, and where $N$ is the normal bundle of $S_{red}$ in $G$. Let's define:\hfill $\zeta=c_1(j^*{\cal O}_{\widetilde{G}} (\widetilde{S_{red}}) =c_1({{\cal O}}_N(-1))$ \hfill \phantom{.} \end{minipage} The multiplicative structure of $A_*\widetilde{G}$ is given by the following rules:\\ \begin{center} $\left\{ \begin{array}{lcl} f^{*}\gamma .f^{*}\gamma ^{\prime } & = & f^{*}\gamma \gamma ^{\prime } \\ j_{*}\widetilde{\sigma }.j_{*}\widetilde{\sigma ^{\prime }} & = &j_{*}(\zeta \widetilde{\sigma }\widetilde{\sigma ^{\prime }}) \\ f^{*}\gamma .j_{*}\widetilde{\sigma } & = & j_{*}((g^{*}i^{*}\gamma ).\widetilde{% \sigma }) \end{array} \right. $ \end{center} where $\gamma \in A_{*}G$ and $\widetilde{\sigma },\widetilde{\sigma ^{\prime }}\in A_{*}\widetilde{S_{red}}$. So we have: $x^2=j_{*}\zeta =(g^{*}c_1N,-i_{*}[S_{red}])$, because $\zeta =-c_1(g^{*}N/% {\cal O}\!_N(-1))+g^{*}c_1N$ and similarly $x^3=j_{*}\zeta ^2=((c_1N)^2-c_2(N),-i_{*}(c_1(N))$, and \begin{center} $A_{*}\widetilde{S_{red}}=A_{*}S_{red}[\zeta ]/(\zeta ^2-c_1(N)\zeta +c_2(N)) $. \end{center} One has the exact sequence: $H^1E(-1)\otimes Q ^{\rm v} \stackrel{\sigma }{ \longrightarrow }H^1E\otimes {{\cal O}}_G \rightarrow R^1q_{*}p^{*}E\rightarrow 0$, where $Q$ is the tautological quotient bundle of $H^0({\cal O}_{{\rm I} \! {\rm P}\!_3}(1)) ^{\rm v} \otimes {\cal O}_G$. According to [G-P], the scheme of multi-jumping lines $M$ is the locus where $\sigma$ has rank at most $2n-2$. Denote by $U$ the universal subbundle of ${\rm I} \! {\rm P}(H^1E ^{\rm v} \otimes {\cal O}-{\widetilde{G}})$ and $\pi$ the projection on $\widetilde{G}$. The degeneracy locus of $f^*\sigma $ is equal to the one of $f^*\sigma ^{\rm v}$, and we can compute $f^*[M]$ in function of the vanishing locus $Z_s$ of some section $s$ of $U ^{\rm v} \otimes \pi^{*}(H^1E ^{\rm v} \otimes {\cal O}_{\widetilde{G}})$ arising from the sequence: \begin{center} $0\longrightarrow U\longrightarrow \pi ^{*}(H^1E ^{\rm v} \otimes {\cal O}_{ \widetilde{G}}) \stackrel{\pi ^{*}f^{*}\sigma ^{\rm v}}{\longrightarrow }\pi ^{*}(H^1E(-1) \otimes f^{*}Q ^{\rm v}) ^{\rm v}$ \end{center} Indeed, one has $f^{*}M=\pi _{*}(Z_s)$, and $s$ vanishes on the divisor $\pi^*\widetilde{S}$ so it gives a regular section $s^{\prime}$ of $U ^{\rm v} \otimes \pi ^{*}(H^1E ^{\rm v} \otimes {\cal O}_{\widetilde{G}}) (-mx)$ where $S=mS_{red}$ in $A_*G$. Computing this class as in [Fu] Ex 14.4 gives: \begin{center} $Z_{s^{\prime }}=\sum\limits_{i=0}^{2n}(-1)^ic_{2n-i}(U ^{\rm v} \otimes \pi ^{*}f^{*}F)(\pi ^{*}mx)^i$, where $F=H^1E(-1) ^{\rm v} \otimes Q$. \end{center} According to Josefiak-Lascoux-Pragacz, [Fu]Ex14.2.2, one has: \begin{center} $\pi _{*}(c_{2n-i}(U ^{\rm v} \otimes \pi^{*}f^{*}F)) =c_{2n-i-(2n-2)+1} (F-H^1E ^{\rm v} \otimes {\cal O}_{\widetilde{G}}))=c_{3-i}(F)$ \end{center} But the Chern polynomial of $F$ is: \begin{center} $c_Y(F)=1+(nt)Y+[\binom{n+1}2t^2-nu]Y^2+[2\binom{n+1}3tu]Y^3$ \end{center} where $t$ is the class of a hyperplane section of $G$, and $u$ is represented by the lines in a plane. (Let's recall that the Chow ring of $G$ is generated by $t$ and $u$ with the relations: $t^3=2tu$; $u^2=t^2u$. Furthermore, $c_2Q=t^2-u$ and $[S]=\alpha (t^2-u) +\beta u$. So we have: $\pi _{*}(Z_{s^{\prime }})=\sum\limits_{i=0}^{2n}c_{3-i}(F)(-mx)^i=\sum\limits_{i=0}^3c_{3-i}(F)(-mx)^i$\\ $\pi _{*}(Z_{s^{\prime}})=\big( m^3[c_2N-(c_1N)^2]+m^2(c_1N.n.i^{*}t)-m.i^{*}[\binom{n+1}2t^2-nu],$\hfill \phantom{.}\\ \phantom{.}\hfill $2\binom{n+1}3tu-nm^2t.i_{*}[S_{red}]+m^3i_{*}c_1N \big)$\\ \begin{example} \label{exresiduel} If $S$ is a smooth congruence of bidegree $(\alpha,\beta)$, then the residual class is: $\bigl(\frac{\alpha ^2+\beta ^2-(n^2-7n+13)\beta -(n^2-5n+13)\alpha +(2\pi -2)(2n-12)-12\chi ({\cal O}_S)} 2\overline{p},$\\ \phantom{.}\hfill${{[2\binom{n+1}3-\!\!(n-3)(\alpha \!+\!\beta )\!+\!2\pi\!\! -\!\!2]tu \bigr)} }$\\ where $\pi$ is the genus of a hyperplane section of $S$, and $\overline{p}$ is the class of a point in $A_0S$. \end{example} NB: the term of the right is still valid when $S$ is just locally complete intersection, and that is the non vanishing of this term in the required situations that had been used in the \ref{c2=5courbe}. Let $c_i\Omega $ be the Chern classes of the cotangent bundle of $S$. If $C$ is a general hyperplane section of $S$, the normal bundle of $C$ is $N_C=N_{|C} \oplus {\cal O}_C(1)$, where $N$ is still the normal bundle of $S$. But $c_1N_C=c_1(\Omega_G ^{\rm v})_{| C}+c_1(\omega _C)=4(\alpha +\beta )+2\pi -2$, so we have: $t.c_1N=[3(\alpha +\beta )+2\pi -2].\overline{p}$. On the other hand, one has $c_1N^2-c_2N=[5(\alpha +\beta )+8(\pi -1)+c_2\Omega ].\overline{p}$, which gives the formula using Hirzebruch-Riemann-Roch theorem and the following relation satisfied by every smooth congruence (Cf [A-S]): \begin{center} $\alpha ^2+\beta ^2=3(\alpha +\beta )+4(2\pi -2)+2(c_1\Omega )^2-12\chi ({\cal O}_S)$ \end{center} \subsection{The family of $n$-instantons with $h^0(E(2)) \geq 4$} \begin{proposition}\label{famille} Let $E$ be any $n$-instanton with $n \geq 5$, $h^0(E(1))=0$ and \hbox{$h^0 (E(2)) \geq 4$}, then $h^0(E(2))=4$ and there is a section of $E(2)$ vanishing on the union of $2$ curves of arithmetic genus $0$ cutting each other in length 2. Furthermore, one of these curves has to be a skew cubic (not necessarily integral), and the other one may be chosen smooth and has bidegree $(1,n)$ in a smooth fixed quadric. \end{proposition} We can construct from the hypothesis $h^0 (E(2)) \geq 4$ a map \hbox{$4 {\cal O} _{{\rm I} \! {\rm P}_3} \rightarrow E(2)$} whose kernel and cokernel will be denoted by $E^{\prime}(-2)$ and ${\cal L}$. So we have the exact sequence: $$0 \longrightarrow E^{\prime}(-2)\longrightarrow 4 {\cal O} _{{\rm I} \! {\rm P}_3} \longrightarrow E(2) \longrightarrow {\cal L}\longrightarrow 0$$ The key is that the support of ${\cal L}$ has to contain a quadric surface. So we have to eliminate in the following the other cases. \bigskip \noindent\underline{First case:} dim $supp {\cal L} \leq 1$ \begin{lemma} \label{c2E'} If dim $supp {\cal L} \leq 1$, then we have $c_2(E^{\prime})=8-n-d_{\cal L}$, where $d_{\cal L}$ is the degree of the sheaf ${\cal E}xt ^2({\cal L},{\cal O}_{{\rm I} \! {\rm P}_3})$. \end{lemma} NB: In this case $d_{\cal L}$ is also the degree of ${\cal L}$, but we'd like to keep the definition of $d_{\cal L}$ in the other cases. Let $a$ be such that $E ^{\prime \rm v}(-a)$ has a section. Choosing one enables to build the following diagram where the middle line is obtained by dualizing the previous exact sequence, and where $X$ is the vanishing locus of the chosen section ($X$ could be empty):\\ \vbox{ \begin{equation}\label{diag1} \begin{picture}(2550,500)(0,500) \setsqparms[1`1`1`1;650`250] \putsquare(800,250)[4{{\cal O}_{\proj_3}} (-2)`E ^{\prime \rm v}` k{{\cal O}_{\proj_3}}(-2)`{\mathcal{J}}_X (-a);```] \setsqparms[1`0`1`1;600`250] \putsquare(1450,250)[\phantom{E ^{\prime \rm v}}`{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} (-2)) `\phantom{{\mathcal{J}}_X (-a)}`A;```] \setsqparms[1`1`1`0;650`250] \putsquare(800,500)[(4-k){{\cal O}_{\proj_3}} (-2)`{{\cal O}_{\proj_3}} (a)`\phantom{4{{\cal O}_{\proj_3}} (-2)}`\phantom{E ^{\prime \rm v}};```] \setsqparms[1`0`1`0;600`250] \putsquare(1450,500)[\phantom{{{\cal O}_{\proj_3}} (a)}`B`\phantom{E ^{\prime \rm v}}` \phantom{{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} (-2))};```] \putmorphism(0,500)(1,0)[0`E(-4)`]{300}1a \putmorphism(300,500)(1,0)[\phantom{E(-4)}`\phantom{4{{\cal O}_{\proj_3}} (-2)}`]{500}1a \putmorphism(2050,500)(1,0)[\phantom{{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} (-2))}`0`]{550}1a \putmorphism(800,250)(0,1)[`0`]{250}1l \putmorphism(1450,250)(0,1)[`0`]{250}1l \putmorphism(2050,750)(1,0)[\phantom{B}`0`]{550}1a \putmorphism(2050,250)(1,0)[\phantom{A}`0`]{550}1a \putmorphism(2050,250)(0,1)[`0`]{250}1l \putmorphism(800,1000)(0,1)[0``]{250}1l \putmorphism(1450,1000)(0,1)[0``]{250}1l \end{picture} \end{equation} \vskip 5em} We can first take $a=-2$. So there arise $k=3$ quartic surfaces containing $X$, and $B$ has to vanish. The kernel of the last line of (\ref{diag1}) is in this case $E(-4)$. Choose 2 of those quartics, and denote by $\Gamma $ their complete intersection, and by ${\cal J} _{X|\Gamma}$ the ideal of $X$ in $\Gamma$. The last line of (\ref{diag1}) and the resolution of $\Gamma$ give in the following a section $s$ of $E(2)$ whose vanishing locus will be noted $Z$.\\ \vbox{ \begin{equation}\label{diag2} \begin{picture}(2150,550)(50,500) \setsqparms[1`1`1`1;500`250] \putsquare(300,250)[E(-2)`3{{\cal O}_{\proj_3}} `{\mathcal{J}}_Z `{{\cal O}_{\proj_3}};```] \setsqparms[1`1`1`0;500`250] \putsquare(300,500)[{{\cal O}_{\proj_3}}(-4)`2{{\cal O}_{\proj_3}}`\phantom{E(-2)} `\phantom{3{{\cal O}_{\proj_3}}};`s``] \setsqparms[1`0`1`1;500`250] \putsquare(800,250)[\phantom{3{{\cal O}_{\proj_3}}}`{\mathcal{J}}_X (4)` \phantom{{{\cal O}_{\proj_3}}}` {\mathcal{J}}_{X|\Gamma} (4);```] \setsqparms[1`0`1`0;500`250] \putsquare(800,500)[\phantom{2{{\cal O}_{\proj_3}}}`{\mathcal{J}}_{\Gamma} (4)` ` ; ` ` ` ] \setsqparms[1`0`1`1;600`250] \putsquare(1300,250)[\phantom{{\mathcal{J}}_{\Gamma} (4)}`{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} ) ` \phantom{{\mathcal{J}}_{X|\Gamma} (4)}`{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} );``\wr`] \setsqparms[1`0`1`0;600`250] \putsquare(1300,500)[\phantom{{\mathcal{J}}_{\Gamma} (4)}`0``;```] \putmorphism(-50,250)(1,0)[0`\phantom{{\mathcal{J}}_Z }`]{300}1a \putmorphism(-50,500)(1,0)[0`\phantom{E(-2)}`]{350}1a \putmorphism(-50,750)(1,0)[0`\phantom{{{\cal O}_{\proj_3}} (-4)}`]{350}1a \putmorphism(1900,250)(1,0)[\phantom{{\cal E}xt ^2({ \mathcal{L}} ,{{\cal O}_{\proj_3}} )}` 0`]{450}1a \putmorphism(1900,500)(1,0)[\phantom{{\cal E}xt ^2({ \mathcal{L}} ,{{\cal O}_{\proj_3}} )}` 0`]{450}1a \putmorphism(300,250)(0,1)[`0`]{250}1l \putmorphism(800,250)(0,1)[`0`]{250}1l \putmorphism(1300,250)(0,1)[`0`]{250}1l \putmorphism(1900,250)(0,1)[`0`]{250}1l \putmorphism(300,1000)(0,1)[0``]{250}1l \putmorphism(800,1000)(0,1)[0``]{250}1l \putmorphism(1300,1000)(0,1)[0``]{250}1l \end{picture} \end{equation} \vskip 5em} but the third column of (\ref{diag2}) proves that $X$ is linked by the 2 quartics to the support of ${\cal J} _{X|\Gamma}$, and the last line implies that this support is the union of $Z$ with the 1-dimensional part of \hbox{${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$} 's support counted rank of ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$ times. So it proves lemma \ref{c2E'}. \nolinebreak \raise-0.1em\hbox{$\Box$} We can remark now that the degeneracy class of \hbox{$4{{\cal O}_{\proj_3}} \rightarrow E(2)$} is negative when $n \geq 5$, so the support of ${\cal L}$ has to contain at least a one dimensional component. So we have from lemma \ref{c2E'} $c_2 E^{\prime}\leq 2$. \bigskip Let's use again the diagram (\ref{diag1}), but this time with the biggest $a$ such that $E ^{\prime \rm v} (-a)$ has a section. We will now study the possible cases recalling that $A$'s support is at most 1 dimensional. \begin{itemize} \item If $k=2$ Then the kernel of the first line of (\ref{diag1}) is ${{\cal O}_{\proj_3}} (-a-4)$, and it is injected into $E(-4)$. So it gives a section of $E(a)$, and by the hypothesis made on $E$ and $a$, we have $a\geq 2$, and then $X$ is empty because $A$'s support is at most 1 dimensional. So the middle column of (\ref{diag1}) splits. (in this situation we will say in the following $E ^{\prime \rm v}$ splits). \item If $E ^{\prime \rm v}$ splits Then the map from $B$ to ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}}(-2))$ of (\ref{diag1}) is an injection, so $B$'s support has also to be at most 1 dimensional. The only possible cases are thus: \begin{itemize} \item $k=1,a=2,X=\emptyset$, but it implies $E ^{\prime \rm v}= {{\cal O}_{\proj_3}} (2) \oplus {{\cal O}_{\proj_3}} (-2)$, so it contradicts the independence of the 4 chosen sections of $E(2)$. \item $k=2$, then $A$ and $B$'s supports are 1 dimensional of degree $(a-2)^2-c_2(E^{\prime})-a^2$ and $(a+2)^2$. As $E ^{\prime \rm v}$ is splited, the degree of ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$ is the sum of $A$ and $B$'s degree. Using $c_2 E^{\prime}\leq 3-d_{\cal L}$, we obtain the contradiction: $d_{\cal L}\geq a^2+5+d_{\cal L}$. \item $k=3,a=-2$ but it is impossible because $E ^{\prime \rm v} $ is reflexive and \mbox{$c_1(E ^{\prime \rm v})=0$}; $c_2(E ^{\prime \rm v} ) \leq 2$, so $E ^{\prime \rm v} (1)$ must have a section. \end{itemize} So the only remaining case is: \item $k\geq 3,X\neq\emptyset$ The curve $X$ can't be in 3 independent planes, so $a \geq 0$. In other words, it means that $E^{\prime}$ is semi-stable. \begin{itemize} \item If $a=0$, then $X$ has degree $c_2(E^{\prime})$, which must be 1 or 2. So there is a plane $H$ having a curve in its intersection with $X$. This intersection is in fact the union of a curve of degree $b=1$ or $2$ with a scheme $X^{\prime}$ which is the vanishing of a section of $E_H(-b)$. The last line of (\ref{diag1}) has kernel $E(-4)$ because $k\geq 3$, so it gives the following exact sequence: % $$0\longrightarrow E_H(-4) \longrightarrow k {\cal O}_H(-2) \longrightarrow {\mathcal{J}}_{X^{\prime}}(-b)\longrightarrow A^{\prime }\longrightarrow 0$$ % But $X^{\prime}$ has degree $c_2(E^{\prime})+b^2\geq 2$, so it lies in only one line, thus those 3 sections have to be proportional. Then we would have $E_H(-4)=2{\cal O}_H(-2)$, which is impossible because an instanton doesn't have unstable planes. \item If $a=-1$, (i.e: $E ^{\prime \rm v} $ stable) Then we have $c_2(E ^{\prime \rm v} )>0$, and on the other hand, ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$ has a non empty 1 dimensional component, so we have $5\leq n \leq 7$ and $0<c_2(E ^{\prime \rm v} ) \leq 2$, and $d_{\cal L} \leq 2$. Take this time a general plane $H$ such that $E_H$ and $E ^{\prime \rm v} _H$ are stable, and $d_{{\cal L}}=h^0{\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$. Restrict to $H$ the second line of diagram (\ref{diag1}) to obtain: $$ 0\rightarrow E_H(-4)\rightarrow 4 {\cal O} _H(-2) \kercoker{N} E ^{\prime \rm v}_H \rightarrow {\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}}(-2)) _H \rightarrow 0$$ But $h^1E ^{\prime \rm v}_H=0$ because $c_2(E ^{\prime \rm v} ) \leq 2$, and $h^0N(1)=h^1E_H(-3)=h^0E_H=0$, so we have \hbox{$h^0{\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}}(-1)) _H \geq h^0E ^{\prime \rm v} _H(1)$}, which is greater or equal to 4, so it contradicts $d_{\cal L} \leq 2$. \end{itemize} \end{itemize} \bigskip \noindent\underline{Second case:} The support of ${\cal L}$ contain a 2 dimensional part which is a plane $H$. We have the following sequence, where this time $c_1(E ^{\prime \rm v})=-1$. $$0 \longrightarrow E^{\prime}(-2)\longrightarrow 4 {\cal O} _{{\rm I} \! {\rm P}_3} \kercoker{M} E(2) \longrightarrow {\cal L}\longrightarrow 0$$ As previously, we obtain the following diagram:\\ \vbox{ \begin{equation}\label{diag1b} \begin{picture}(2600,550)(0,500) \setsqparms[1`1`1`1;650`250] \putsquare(800,250)[4{{\cal O}_{\proj_3}} (-2)`E ^{\prime \rm v}` k{{\cal O}_{\proj_3}}(-2)`{\mathcal{J}}_X (-a-1);```] \setsqparms[1`0`1`1;600`250] \putsquare(1450,250)[\phantom{E ^{\prime \rm v}}`{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} (-2)) `\phantom{{\mathcal{J}}_X (-a-1)}`A;```] \setsqparms[1`1`1`0;650`250] \putsquare(800,500)[(4-k){{\cal O}_{\proj_3}} (-2)`{{\cal O}_{\proj_3}} (a)`\phantom{4{{\cal O}_{\proj_3}} (-2)}`\phantom{E ^{\prime \rm v}};```] \setsqparms[1`0`1`0;600`250] \putsquare(1450,500)[\phantom{{{\cal O}_{\proj_3}} (a)}`B`\phantom{E ^{\prime \rm v}}` \phantom{{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} (-2))};```] \putmorphism(-100,500)(1,0)[0`M ^{\rm v}(-2)`]{350}1a \putmorphism(250,500)(1,0)[\phantom{M ^{\rm v}(-2)}`\phantom{4{{\cal O}_{\proj_3}} (-2)}`]{550}1a \putmorphism(2050,500)(1,0)[\phantom{{\cal E}xt ^2({\cal L} ,{{\cal O}_{\proj_3}} (-2))}`0`]{550}1a \putmorphism(800,250)(0,1)[`0`]{250}1l \putmorphism(1450,250)(0,1)[`0`]{250}1l \putmorphism(2050,750)(1,0)[\phantom{B}`0`]{550}1a \putmorphism(2050,250)(1,0)[\phantom{A}`0`]{550}1a \putmorphism(2050,250)(0,1)[`0`]{250}1l \putmorphism(800,1000)(0,1)[0``]{250}1l \putmorphism(1450,1000)(0,1)[0``]{250}1l \end{picture} \end{equation} \vskip 5em} Working as in the first case, we prove that the union of a section of $E ^{\prime \rm v} (2)$ with ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$'s support (counted rank of ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}})$ times) is linked by 2 cubic surfaces (sections of $\stackrel{2}{\Lambda}(E ^{\prime \rm v}(2))$) to a section of $M ^{\rm v}(3)$. Furthermore, this section is linked by a section of $E(2)$ to a (possibly empty) curve of $H$. But this plane curve is of degree at most 2 because $E_H$ is necessarily semi-stable. Then, $c_2(M ^{\rm v}(3))=c_2(M ^{\rm v})\geq n+2$ ($n\geq 5$), so we have $d_{\cal L}+c_2(E ^{\prime \rm v})\leq 0$ and $d_{\cal L}\leq 2$. But as $c_1(E ^{\prime \rm v})=-1$ and $c_2(E ^{\prime \rm v})\leq 0$, we must have $h^0(E ^{\prime \rm v})\neq 0$, we can assume $a\geq 0$ in diagram (\ref{diag1b}). \begin{itemize} \item If $X=\emptyset$ Then $E ^{\prime \rm v}$ has to split, so $B$ injects itself in ${\cal E}xt ^2({\cal L},{{\cal O}_{\proj_3}} (-2))$, thus $B$'s support has to be at most 1 dimensional. As $A$ has also a 1 dimensional support (at most), we have only the following cases: \begin{itemize} \item $k=1,a=1$, so $E^{\prime}={{\cal O}_{\proj_3}}(-1)\oplus {{\cal O}_{\proj_3}}(2)$, but it conflicts with the independence of the 4 chosen sections of $E(2)$. \item $k=2$, then $B$ has a 1 dimensional support of degree \hbox{$(2+a)^2\geq 4$}, but it conflicts with $d_{\cal L}\leq 2$. \end{itemize} \item If $X\neq\emptyset$ The facts that $A$ has an at most 1 dimensional support, and that $a\geq 0$ imply $k=2, X={\rm I} \! {\rm P}_1 ,a=0$. Then $X$ must have degree $c_2(E ^{\prime \rm v})$, which contradicts $c_2(E ^{\prime \rm v})\leq 0$. \end{itemize} \noindent\underline{The effective situation:} We can conclude from the previous cases that there is a quadric $Q$ in ${\cal L}$'s support because it can't contain a cubic surface as there is no curve in 3 independent planes. So a section $C$ of $E(2)$ have a component $C_2$ in the quadric $Q$, and another one $C_1$, which has to be skew because $h^0E(1)=0$, and which lies on 3 quadrics. This curve $C_1$ must then be a cubic curve of arithmetic genus 0. Let's remark that $C_2$ has degree $n+1 \geq 6$, so $Q$ is normal because $C$ can't have plane components of degree 3 or more otherwise there would be an unstable planes, which is not possible for an instanton. One has first to check that $C_1$ and $C_2$ have a 0-dimensional intersection to get the \ref{famille}. The cubics arising when the section of $E(2)$ moves, are the vanishing locus of sections of $M ^{\rm v}(2)$, which is reflexive of rank 2. Take a normal quadric $Q^{\prime}$ containing $C_1$, then it must contain a pencil of those cubics say $\lambda C_1+\mu C^{\prime}_1$. But if the associated pencil of sections of $E(2)$ was such that $C_1 \cap C_2$ and $C^\prime_1 \cap C^\prime_2$ were 1 dimensional, then \hbox{$C_1 \cap C^{\prime}_1$} would have a 1 dimensional component, because $C_1 \cap C_2$ and $C^\prime_1 \cap C^\prime_2$ lies in the fixed curve $Q \cap Q^{\prime}$, and this 1-dimensional component of $C_1 \cap C^{\prime}_1$ would be in the singular locus of $Q^{\prime}$ which contradicts its normality. So there is a section of $E(2)$ vanishing on $C=C_1 \cup C_2$, where $C_1$ is a skew cubic of arithmetic genus 0 and $\dim C_1 \cap C_2=0$ . On the other hand, we have $\omega_C={\cal O}_C$, and the exact sequence of liaison (i=1 or 2): $$0\longrightarrow \omega_{C_i} \longrightarrow {\cal O}_C \longrightarrow {\cal O}_{C_{3-i}} \longrightarrow 0$$ gives when i=1 by restriction to $C_1$ that $C_1 \cap C_2$ has length 2, and when i=2 by restriction to $C_2$ that $C_2$ has arithmetic genus 0. But a quadric cone can't have curves of arithmetic genus 0 and degree greater or equal to 4, so $Q$ is smooth and $C_2$ has bidegree $(1,n)$ in $Q$. We'd like now to prove the smoothness of a general $C_2$. Denote by $h$ the 3-dimensional subspace of $|C_2|$ induced by the 4 sections of $E(2)$. The base points of $h$ must be in the singular locus of $Q\cup Q^{\prime}$ for any quadric $Q^{\prime}$ containing $C_1$, hence those points are on $C_1$. But we showed that $C_1 \cap C_2$ was 0-dimensional, so the set of base points of $h$ is at most finite. Furthermore, if $C_2$ is singular in some point $P$, then it must contain the ruling of bidegree $(0,1)$ passing through $P$, so this ruling would be a base curve of $h$ if the general curve was singular at $P$. Hence, the generic element of $h$ is smooth and irreducible of bidegree $(1,n)$, and this is the proposition \ref{famille}. \nolinebreak \raise-0.1em\hbox{$\Box$} \begin{proposition}\label{multisautfamille} Let $E$ be a $n$-instanton with $n \geq 5$, $h^0(E(1))=0$ and \hbox{$h^0 (E(2)) \geq 4$}, then $E$ has only a 1-dimensional scheme of multi-jumping lines. \end{proposition} The bundle $E$ belongs by hypothesis to the family \ref{famille}, so denote by $Q$ the quadric which is the support of the cokernel of the map given by the 4 sections to $E(2)$. Let $s$ be a section of $E(2)$ and $Z_s=C_{2,s}\cup C_{1,s}$ its vanishing locus, where $C_{1,s}$ is the rational cubic and $C_{2,s}$ is the curve of bidegree $(1,n)$ in $Q$. If a line $d$ is a multi-jumping line then $E_d(2)={\cal O}_d(a+2)\oplus {\cal O}_d(-a+2)$ with $a\geq 2$. So a multi-jumping line $d$ meets $Z_s$ if and only if $d\cap Z_s$ has length at least 4. So if $E$ has a 2 parameter family of multi-jumping lines, then infinitely many of them would be 4-secant to $Z_s$ and up to a change of the section $s$, we can assume that infinitely many are not in $Q$. So those lines are 2-secant to $C_{2,s}$ and 2-secant to $C_{1,s}$ because they are not in $Q$ and $C_{1,s}$ don't have trisecant. Denote by $\Sigma$ the ruled surface of ${\rm I} \! {\rm P}_3$ made with those lines. a) If $C_{1,s}$ is smooth.\\ Then the basis of $\Sigma$ is a curve $\Gamma$ on the Veronese surface of bisecant lines to $C_{1,s}$, and $\Sigma$ must contain $C_2$ because $C_2$ is irreducible. But we have a morphism of degree 2 from $C_2$ onto the basis of $\Sigma$ hence $\Gamma$ is smooth and rational, so $\deg \Sigma=2$ or $\deg \Sigma=4$, but $\Sigma$ contains $C_1\cup C_2$ so $\deg \Sigma \geq 4$ because $h^0E(1)=0$, and all the quartics containing $C_1\cup C_2$ are some $Q\cup Q^{\prime}$, where $Q^{\prime}$ is a quadric containing $C_1$, so $\Sigma$ can't be a quartic containing $C_1\cup C_2$ and this case is impossible. b) If $C_{1,s}$ is made of 3 lines containing a same point $N$ independent of $s$, then $C_{2,s}$ must contain $N$ because $Z_s$ is locally complete intersection. As $C_{2,s}$ is smooth at $N$, any line through $N$ meets $Z_s$ in length 2 around $N$, and $N\in Q$, so a line through $N$ don't meet $C_{1,s}$ in another point, and it can't meet $C_{2,s}$ in 2 points distinct of $N$, so it can't be 4-secant to $C_{2,s}\cap C_{1,s}$. There must then exist a plane $H$ containing 2 of the lines of $C_{1,s}$ and infinitely many multi-jumping lines meeting $C_{2,s}$. As $h^0E_H(-1)=0$, $C_{2,s}\cap H$ must be 0-dimensional, and there would be a point $P$ in $C_{2,s}\cap H$ such that every line of $H$ through $P$ is bisecant to $C_{2,s}$. Hence $H$ is tangent to $Q$ at $P$, but one of the ruling of $Q$ through $P$ would be in $H$ and would be only 1 secant to $C_{2,s}$ because it has bidegree $(1,n)$, which contradicts the definition of $P$. \nolinebreak \raise-0.1em\hbox{$\Box$} \begin{proposition}\label{lisse} Let $E$ be a $n$-instanton with $n \geq 5$, $h^0(E(1))=0$ and \hbox{$h^0 (E(2)) \geq 4$}, then $E$ is a smooth point of the moduli space I$_n$. \end{proposition} From classical theory (cf [LeP]), one has to prove that $h^2(E\otimes E)=0$, and if $C$ is the vanishing locus of a section of $E(2)$, then $h^1(E_C(2))=0$ implies this result. As $E_C(2)$ is just the normal bundle of $C$, we are considering the problem of the vanishing of $h^1(N_C)$, so we will solve it as it was done for smoothing questions in [H-H]. \bigskip Let's recall from \ref{famille} that $C=C_1\cup C_2 $ where the $C_i$ have zero arithmetic genus, and where $C_1$ is a skew cubic and $C_2$ is on a smooth quadric $Q$. We want first to prove the vanishing of $h^1(N_{C_i})$. As $C_2$ has bidegree $(1,n)$ on the smooth quadric $Q$, we have $h^1(N_{C_2})=0$. (Cf for example the proof of prop 5.4 $\alpha$ in [H-H]). This will be true for $C_1$ even if it is 3 concurrent lines. Indeed $C_1$ is the vanishing of a section of $R(1)$ where $R$ is a rank 2 reflexive sheaf with $(c_1,c_2,c_3)=(0,2,4)$, and the following exact sequences: \begin{center} $0 \rightarrow 2{{\cal O}_{\proj_3}}(-1)\rightarrow 4{{\cal O}_{\proj_3}} \rightarrow R(1)\rightarrow 0$ \&\nolinebreak $0 \rightarrow 2{{\cal O}_{\proj_3}}(-3)\rightarrow3{{\cal O}_{\proj_3}}(-2)\rightarrow {\cal J}_{C_1}\rightarrow 0$ \end{center} imply that $h^1(R(1))=0$, $h^2({\cal J}_{C_1}R(1))=0$ thus $h^1(R_{C_1}(1))=0$, but \hbox{$R_{C_1}(1)=N_{C_1}$}. So both $h^1N_{C_i}$ are zero, and we can now deduce from it that $h^1N_C$ is also zero. We have the following sequence of liaison: $$0\longrightarrow {\cal O}_C \longrightarrow {\cal O}_{C_1}\oplus {\cal O}_{C_2} \longrightarrow {\cal O}_{C_1 \cap C_2} \longrightarrow 0$$ giving when twisted by $E(2)$: $$0\longrightarrow N_C \longrightarrow E_{C_1}(2)\oplus E_{C_2}(2) \longrightarrow E_{C_1 \cap C_2}(2) \longrightarrow 0$$ Furthermore, the inclusion ${\mathcal{J}}_C \subset {\mathcal{J}}_{C_i}$ gives a map \hbox{$N_{C_i} \rightarrow N_C$}. Then we have the following exact sequence defining ${\cal L}_i$ and $M_i$: $$ ? \longrightarrow N_{C_i} \kercoker{M_i} E_{C_i}(2) \longrightarrow {\cal L}_i \longrightarrow 0$$ As $h^1(N_{C_i})=0$ implies $h^1(M_i)=0$, we have $ \left\{\begin{array}{l} h^1(E_{C_i}(2))=0\\ H^0(E_{C_i}(2))\rightarrow H^0({\cal L}_i) \rightarrow 0 \end{array}\right.$, which gives $h^1(N_C)=0$ when taking global sections in the following diagram. \begin{center} \begin{picture}(2400,500) \setsqparms[1`1`1`1;800`250] \putsquare(300,250)[N_{C_1}\oplus N_{C_2}`E_{C_1}(2)\oplus E_{C_2}(2)`N_C`E_{C_1}(2)\oplus E_{C_2}(2);``\wr`] \setsqparms[1`0`1`1;800`250] \putsquare(1100,250)[\phantom{E_{C_1}(2)\oplus E_{C_2}(2)}`{\cal L}_1 \oplus {\cal L}_2 `\phantom{E_{C_1}(2) \oplus E_{C_2}(2)} `E_{C_1 \cap C_2}(2) ;```] \putmorphism(0,250)(1,0)[0`\phantom{N_C}`]{300}1a \putmorphism(1900,250)(1,0)[\phantom{E_{C_1 \cap C_2}(2)}`0`]{400}1a \putmorphism(1900,500)(1,0)[\phantom{{\cal L}_1 \oplus {\cal L}_2} `0`]{400}1a \putmorphism(300,250)(0,1)[`0`]{250}1l \putmorphism(1900,250)(0,1)[`0`]{250}1l \end{picture} \end{center} We can now conclude that $E$ is a smooth point of the moduli space because $N_C=E_C(2)$, and the previous vanishing gives $h^2(E\otimes E)=0$ using the sequences: \begin{center} $0 \rightarrow E(-2)\rightarrow E\otimes E\rightarrow {\cal J}_C E(2)\rightarrow 0$ and $0 \rightarrow {\cal J}_C E(2)\rightarrow E(2)\rightarrow N_C\rightarrow 0$ \end{center} \subsection{A $\theta$-characteristic on the curve of multi-jumping lines of an $n$-instanton} Assume here that $E$ is an instanton vector bundle with second Chern class $n$. The aim of this part is to study the scheme of multi-jumping lines of $E$ when it satisfies the properties expected from its determinantal structure. For instance, according to [B-H], the general member of the irreducible component of the moduli space containing the t'Hooft bundles satisfy those properties. \begin{proposition} \label{w2}If an $n$-instanton has only \prefix{k\leq 2}jumping lines, and if its scheme of multi-jumping lines is a curve $\Gamma $ in $G$, then we have:\\\centerline{$(R^1q_{*}p^{*}E)^{% \otimes 4}={\cal O}_\Gamma (n);\omega _\Gamma ^o=(R^1q_{*}p^{*}E)^{\otimes 2}(n-4);(\omega _\Gamma ^o)^{\otimes 2}=% {\cal O}_\Gamma (3n-8)$}\\where $\omega _\Gamma ^o$ is a dualizing sheaf on $\Gamma $. \end{proposition} In fact this result is just an analogous of the proposition 1.5 of [E-S]. Let $K$ and $Q$ be the tautological bundles on $G$ such that we have the sequence: \begin{center} $0\longrightarrow K\longrightarrow H^0({\cal O}_{{\rm I} \! {\rm P}_3}(1)) ^{\rm v}\otimes {\cal O}_G\longrightarrow Q\longrightarrow 0$ \end{center} The points/lines incidence variety of ${\rm I} \! {\rm P}_3$ is $F={\rm I} \! {\rm P}_G(Q ^{\rm v})$, so one has the exact sequence, where $q$ still be the projection from $F$ to $G$: \begin{center} $0\longrightarrow {\cal O}_F(-\tau)\longrightarrow q^*Q ^{\rm v} \longrightarrow (\omega_{F/G}(\tau )) ^{\rm v} \longrightarrow 0$ \end{center} We'd like here to make a relative (over $G$) Beilinson's construction. So consider the resolution of the diagonal $\Delta$ of $F\stackunder{G}{\times}F$: \begin{center} $0\longrightarrow {\cal O}_F(-\tau )\stackunder{G}{\boxtimes }\omega _{F/G}(\tau )\longrightarrow {\cal O}_{F\stackunder{G}{\times}% F}\longrightarrow {\cal O}_\Delta \longrightarrow 0$ \end{center} which gives when twisted by $p^{*}E(\tau )\stackunder{G}{ \boxtimes} {\cal O}_F$ the following spectral sequence which stops in $E_2^{p,q}$ and ends to $p^{*}E(\tau )$ in degree 0 and 0 in the other degrees. \begin{center} {\begin{picture}(2000,500)(-300,0) \setsqparms[1`0`0`1;1500`300] \putsquare(0,0)[{q^{*}(R^1q_{*}p^{*}E)\otimes {\cal O}_F(\sigma -\tau )}`{q^{*}(R^1q_{*}p^{*}E(\tau ))}`{q^{*}q_{*}p^{*}E\otimes {\cal O}_F(\sigma -\tau )}`{q^{*}q_{*}p^{*}E(\tau )};```d] \putmorphism(1500,0)(1,0)[\phantom{q^{*}q_{*}p^{*}E(\tau)}``p]{500}1a \putmorphism(1500,600)(0,1)[`\phantom{M}`q]{300}{-1}r \put(1500,70){\line(0,1){150}} \end{picture}} \end{center} By assumption $E$ has no \prefix{k\geq 3}jumping lines, so $R^1q_{*}p^{*}E(\tau )=0$, and we have the following surjection where $M$ denotes its kernel. \begin{center} $0\longrightarrow M\longrightarrow p^{*}E(\tau )\longrightarrow q^{*}(R^1q_{*}p^{*}E)\otimes {\cal O}_F(\sigma -\tau )\longrightarrow 0$ \end{center} Denote by $h$ and $K$ the restrictions of $q$ and $q^{*}(R^1q_{*}p^{*}E)\otimes {\cal O}_F(\sigma -\tau )$ to $q^{-1}(\Gamma)$. The sheaf $R^1q_{*}p^{*}E$ is locally free over $\Gamma$, so $h_{*}K=R^1q_{*}p^{*}E\otimes h_{*}{\cal O}_F(\sigma -\tau )=0$, and we have: $h_{*}M_{| q^{-1}(\Gamma )}=h_{*}(p^{*}E(\tau )_{|q^{-1} (\Gamma )})$. Furthermore, as $K$ is locally free over $\Gamma$ and $\stackrel{2}{\Lambda }E=0$, one has $M_{| q^{-1}(\Gamma )}=K ^{\rm v}(2\tau )$, so $h_{*}(p^{*}E(\tau )_{| q^{-1}(\Gamma )})=(R^1q_{*}p^{*}E) ^{\rm v} \otimes h_{*}{\cal O}_F(3\tau -\sigma )$. But it means that $h_{*}(p^{*}E(\tau )_{| q^{-1}(\Gamma )})\otimes R^1q_{*}p^{*}E={\cal O}_G(-1)\otimes Sym_3Q_{| \Gamma }$, and using that $h_{*}(p^{*}E(\tau )_{| q^{-1}(\Gamma )})$ is locally free of rank 4, we found the relation: $(R^1q_{*}p^{*}E)^{\otimes 4}\otimes \det (h_{*}(p^{*}E(\tau )_{| q^{-1}(\Gamma )}))={\cal O}_\Gamma (2)$. But in fact $q_*p^*E(\tau)$ is locally free over $G$ by base change, so $\det(h_*(p^*E(\tau)_{|q^{-1}(\Gamma)}))= {\cal O}_\Gamma (2-n)$, because $c_1(q_{*}(p^{*}E(\tau )))=2-n$ from Riemann-Roch over $F$. So we have: \begin{center} $(R^1q_{*}p^{*}E)^{\otimes 4}={\cal O}_\Gamma (n)$ \end{center} On another hand, the following resolution of ${\cal O}_F$ in $G\times {\rm I} \! {\rm P}_3$: \begin{center} $0\longrightarrow {\cal O}_{G\times {\rm I} \! {\rm P}_3}(-\sigma -2\tau )\longrightarrow Q ^{\rm v} \otimes {\cal O}_{{\rm I} \! {\rm P}_3}(-\tau )\longrightarrow {\cal O}_{G\times {\rm I} \! {\rm P}_3}\longrightarrow {\cal O}_F\longrightarrow 0$ \end{center} gives when twisted by $p^*E$ the following exact sequence deduced from the Leray spectral sequence: \begin{center} $H^1(E(-1))\otimes Q ^{\rm v} \stackrel{\phi }{\longrightarrow } H^1(E)\otimes {\cal O}_G\longrightarrow R^1q_{*}p^{*}E\longrightarrow 0$ \end{center} The Eagon-Northcott complexes $(E_i)$ associated to $\phi$ gives resolutions of $M_i$, where we have because $\Gamma$ is a curve: $M_0={\cal O}_\Gamma $, $M_1=R^1q_{*}p^{*}E$, $M_i=Sym_iM_1$, and furthermore: \begin{center} $\omega _\Gamma ^o=M_2\otimes \omega _G\otimes \det (H^1(E)\otimes {\cal O}_G) \otimes (H^1(E(-1))\otimes Q ^{\rm v}) ^{\rm v}$ \end{center} But $\omega _G={\cal O}_G(-4)$, and $\det (H^1(E)\otimes {\cal O}_G) \otimes (H^1(E(-1)) \otimes Q ^{\rm v}) ^{\rm v}={\cal O}_G(n)$, so we have: \begin{center} $\omega _\Gamma ^o=(R^1q_{*}p^{*}E)^{\otimes 2}(n-4)$ ans $(\omega _\Gamma ^o)^{\otimes 2}={\cal O}_\Gamma (3n-8)$ \end{center} \begin{remark} $\theta =(R^1q_{*}p^{*}E)\otimes \omega _\Gamma ^o(2-n)$ is a $\theta$-characteristic of $\Gamma $.\\If $n=2n^{\prime }$, then so is $\theta=(R^1q_{*}p^{*}E)(n^{\prime }-2)$. \end{remark} \begin{proposition} If the scheme of multi-jumping lines $\Gamma $ of an $n$-instanton is a curve in $G$ without \prefix{k\geq 3}jumping lines, then:\\ \centerline{$R^1q_{*}p^{*}E(-\tau )_{| \Gamma }\simeq (R^1q_{*}p^{*}E)\otimes Q_{| \Gamma }(-\sigma )$}\\Furthermore, if $\Gamma $ is smooth, then its normal bundle in $G$ is:\\\centerline{$N_{\Gamma ,G}=Sym_2Q\otimes \omega _\Gamma (3-n)$} \end{proposition} This proposition is also due to a relative Beilinson's construction, but this time the resolution of the diagonal is twisted by $p^{*}E\stackunder{G}{\boxtimes }{\cal O}_F$, so we have the spectral sequence: \begin{center} {\begin{picture}(2000,500)(-400,0) \setsqparms[1`0`0`1;1500`300] \putsquare(0,0)[{q^{*}(R^1q_{*}p^{*}E(- \tau ))\otimes {\cal O}_F(\sigma -\tau )}`{q^{*}(R^1q_{*}p^{*}E)}`{q^{*}(q_{*}p^{*}E(- \tau ))\otimes {\cal O}_F(\sigma -\tau )}`{q^{*}q_{*}p^{*}E};d```d'] \putmorphism(1500,0)(1,0)[\phantom{q^{*}q_{*}p^{*}E(\tau)}``p]{500}1a \putmorphism(1500,600)(0,1)[`\phantom{M}`q]{300}{-1}r \put(1500,70){\line(0,1){150}} \end{picture}} \end{center} which ends with $0\rightarrow K\rightarrow p^{*}E\rightarrow N\rightarrow 0$ where $N$ is the kernel of $d$ and $K$ the cokernel of $d^{\prime}$. It gives the exact sequences (*): \begin{center} $0\longrightarrow q^{*}(q_{*}p^{*}E(-\tau ))\otimes {\cal O}_F(\sigma -\tau )\longrightarrow q^{*}(q_{*}p^{*}E)\kercoker{K} p^{*}E \longrightarrow N \rightarrow 0$ $0\longrightarrow N(\tau -\sigma )\longrightarrow q^{*}(R^1q_{*}p^{*}E(-\tau ))\longrightarrow q^{*}(R^1q_{*}p^{*}E)(\tau -\sigma )\longrightarrow 0$ \end{center} Let's restrict this last sequence to $q^{-1}(\Gamma)$, then we can apply the projection formula to $q^{*}(R^1q_{*}p^{*}E(-\tau ))_{| q^{-1}\Gamma}$ and to $q^{*}(R^1q_{*}p^{*}E)(\tau -\sigma )_{| q^{-1}\Gamma }$ because they are locally free, and we obtain the exact sequence: \begin{center} $R^1q_{*}p^{*}E(-\tau )_{| \Gamma }\longrightarrow R^1q_{*}p^{*}E\otimes q_{*}{\cal O}_\Gamma (\tau -\sigma )\longrightarrow R^1q_{*}N_{| q^{-1}\Gamma }(\tau -\sigma )$ \end{center} On another hand the restriction to $q^{-1}\Gamma$ of the first sequence of (*) shows that $R^1q_{*}K_{| \Gamma }(\tau )=0$, so that $R^1q_{*}N_{| q^{-1}\Gamma }(\tau )\simeq R^1q_{*}p^{*}E_\Gamma (\tau )=0$ because $E$ has no \prefix{k \geq 3}jumping lines. So we have a surjection from $R^1q_{*}p^{*}E(-\tau )_{| \Gamma }$ to $R^1q_{*}p^{*}E\otimes Q(-1)$ which is in fact an isomorphism because those 2 sheaves are locally free of rank 2 on $\Gamma$. Furthermore, the curve $\Gamma$ is given by the first Fitting ideal of the following map $\phi$: \begin{center} $H^2(E(-3)\otimes {\cal O}_G(-1)\stackrel{\phi }{\longrightarrow }% H^1(E(-1))\otimes {\cal O}_G \stackrel{j }{\longrightarrow} R^1q_{*}p^{*}E(-\tau )\longrightarrow 0$ \end{center} where $\phi$ is symetric with respect to Serre's duality. Assume now that $\Gamma$ is smooth, and denote by ${\cal L}_{\Gamma}$ the restriction of $R^1q_{*}p^{*}E(-1)$ to $\Gamma$. We just need now, to conclude the proof, to recall the results of Tjurin (Cf [T1]) which give a description of the normal bundle. Let's consider the hypernet of quadrics classically associated to an instanton, and denote by $V=(H^0{\cal O}_{{\rm I} \! {\rm P}_3}(1)) ^{\rm v}$ and $H=H^2E(-3)$. The hypernet gives an inclusion ${\rm I} \! {\rm P}(\stackrel{2}{\Lambda } V) \subset {\rm I} \! {\rm P}(Sym_2H ^{\rm v})$, and denote by $D_2$ the surface made of the quadrics of the hypernet of rank at most $n-2$, assuming here that $G$ and ${\rm I} \! {\rm P}(\stackrel{2}{\Lambda } V)$ cut transversally the stratification of ${\rm I} \! {\rm P}(Sym_2H ^{\rm v})$ by the rank of the quadrics. The second symetric power of the restriction of $j$ to $\Gamma$ gives a surjection $Sym_2H ^{\rm v} \otimes {\cal O}_\Gamma \stackrel{s_2j_{| \Gamma }}{\rightarrow }Sym_2{\cal L}_\Gamma \rightarrow 0$. Denote by $K$ the kernel of $s_2j_{| \Gamma }$, then the fiber ${\rm I} \! {\rm P}_(K_q)$ over some $q\in \Gamma$ is just made of the quadrics of ${\rm I} \! {\rm P}(H)$ which contain the singular locus of $q$. This is also according to [T1] or [T3]\S2 lemma 1.1, the projective tangent space in $q$ to the locus of rank at most n-2 quadrics. The above hypothesis of transversality implies that $\stackrel{2}{\Lambda }V\cap K_q$ is 3 dimensional. Similarly, composing the inclusion $(\stackrel{2}{\Lambda }V)\otimes {\cal O}_\Gamma \subset Sym_2H^{*}\otimes {\cal O}_\Gamma $ with $s_{2j_\Gamma}$ gives the sequence: \begin{center} $0\longrightarrow K^{\prime }\longrightarrow (\stackrel{2}{\Lambda }% V)\otimes {\cal O}_\Gamma \longrightarrow Sym_2{\cal L}_\Gamma \longrightarrow 0$ \end{center} where ${\rm I} \! {\rm P}(K_q^{\prime })$ is the projective tangent space to $D_2$ at a point $q$ of $\Gamma$. So we have the commutative diagram of [T1] restricted to $\Gamma$: \begin{center} {\begin{picture}(2200,1000) \setsqparms[1`-1`-1`1;700`250] \putsquare(400,250)[{K^{\prime }}`{\buildrel\hbox{\tiny2}\over{\Lambda} V\otimes {\cal O}_\Gamma }`{{\cal O}_\Gamma (-1)}`{{\cal O}_\Gamma (-1)};```] \putsquare(400,500)[{T_{D_2}(-1)_{\mid \Gamma }}`{T_{{\rm I} \! {\rm P}\!_5}(-1)_{\mid \Gamma }}`\phantom{K^{\prime }}`\phantom{\buildrel\hbox{\tiny2}\over{\Lambda} V\otimes {\cal O}_\Gamma };```] \putsquare(1100,500)[\phantom{T_{{\rm I} \! {\rm P}\!_5}(-1)_{\mid \Gamma }}`{N_{D_2,{\rm I} \! {\rm P}\!_5}(-1)_{\mid \Gamma }}`\phantom{\buildrel\hbox{\tiny2}\over{\Lambda} V\otimes {\cal O}_\Gamma }`{Sym_2{\cal L}_\Gamma};```] \putmorphism(0,250)(1,0)[{0}`\phantom{{\cal O}_\Gamma (-1)}`]{400}1l \putmorphism(0,500)(1,0)[{0}`\phantom{K^{\prime }}`]{400}1l \putmorphism(0,750)(1,0)[{0}`\phantom{T_{D_2}(-1)_{\mid \Gamma }}`]{400}1l \putmorphism(1100,250)(1,0)[\phantom{{\cal O}_\Gamma (-1)}`{0}`]{700}1l \putmorphism(1800,500)(1,0)[\phantom{Sym_2{\cal L}_\Gamma }`{0}`]{500}1l \putmorphism(1800,750)(1,0)[\phantom{N_{D_2,{\rm I} \! {\rm P}\!_5}(-1)_{\mid \Gamma }}`{0}`]{500}1l \putmorphism(1100,1000)(0,1)[{0}`\phantom{T_{{\rm I} \! {\rm P}\!_5}(-1)_{\mid \Gamma }}`]{250}{-1}m \putmorphism(1800,1000)(0,1)[{0}`\phantom{N_{D_2,{\rm I} \! {\rm P}\!_5}(-1)_{\mid \Gamma }}`]{250}{-1}m \putmorphism(400,250)(0,1)[\phantom{{\cal O}_\Gamma (-1)}`{0}`]{250}{-1}m \putmorphism(1100,250)(0,1)[\phantom{{\cal O}_\Gamma (-1)}`{0}`]{250}{-1}m \putmorphism(1800,500)(0,1)[\phantom{Sym_2{\cal L}_\Gamma}`\phantom{0}`]{250 }{-1}m \putmorphism(400,1000)(0,1)[{0}`\phantom{T_{D_2}(-1)_{\mid \Gamma }}`]{250}{-1}m \end{picture} } \end{center} where the first two columns are the Euler relative exact sequences over $\Gamma$ and \mbox{${\rm I} \! {\rm P}(\stackrel{2}{\Lambda }V)$}. But $\Gamma= D_2 \cap G$, so we have $N_{\Gamma,G}(-1)=N_{D_2,{\rm I} \! {\rm P}_5}(-1)_{|\Gamma}$, then \linebreak$N_{\Gamma ,G}\simeq Sym_2(R^1q_{*}p^{*}E(-\tau )_{| \Gamma })\otimes {\cal O}_\Gamma (1)$, which gives with the previous results: \begin{center} $N_{\Gamma ,G}\simeq Sym_2Q\otimes \omega _\Gamma (3-n)$ \end{center}

9711

alg-geom/9711031

en

https://arxiv.org/abs/alg-geom/9711031

alg-geom/9711031

Jim Bryan

Jim Bryan and Naichung Conan Leung

The enumerative geometry of K3 surfaces and modular forms

24 pages, LaTeX2e with eepic macros

We prove the conjectures of Yau-Zaslow and Gottsche concerning the number curves on K3 surfaces. Specifically, let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C| for any g and n satisfying C^2=2g+2n-2. When g=0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Gottsche has generalized their conjecture to arbitrary g in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to P^2 blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus g constrained to g points are also given in terms of quasi-modular forms.

alg-geom

\section{Introduction} Let $X$ be a $K3$ surface and $C$ be a holomorphic curve in $X$ representing a primitive homology class. For any $g$ and $n$ satisfying $C\cdot C=2g+2n-2,$ we define an invariant $N_{g}(n)$ which counts the number of curves of geometric genus $g$ with $n$ nodes passing through $g$ generic points in $X$ in the linear system $\left| C\right| .$ For each $g,$ consider the generating function \[ F_g\left( q\right) =\sum_{n=0}^\infty N_{g}(n)q^n. \] Our main theorem gives explicit formulas for $F_{g}$ in terms of quasi-modular forms: \begin{theorem}[Main Theorem]\label{thm: main thm} For any $g,$ we have \begin{eqnarray*} F_g\left( q\right) &=&\left(\sum_{k=1}^{\infty }k(\sum_{d|k}d)q^{k-1} \right)^{g} \prod _{m=1}^{\infty }(1-q^{m})^{-24}\\ &=&\left(\frac{d}{dq}G_{2}(q) \right)^{g}\frac{q}{\Delta (q)}. \end{eqnarray*} \end{theorem} So for example, we have \begin{eqnarray*} F_{0}&=&1+ 24q+ 324q^{2}+3200q^{3}+\cdots \\ F_{1}&=&1+ 30q+ 480q^{2}+5460q^{3}+\cdots \\ F_{2}&=&1+36q+672q^{2}+8728q^{3}+\cdots \\ F_{3}&=&1+42q+900q^{2}+13220q^{3}+\cdots . \end{eqnarray*} If we write $q=e^{2\pi i\tau }$ then $\Delta \left( \tau \right) =q\prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{24}=\eta \left( \tau \right) ^{24}$ is a modular form of weight $12$ where $\eta \left( \tau \right) $ is the Dedekind $\eta $ function. In particular, for any $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in SL\left( 2,\mathbf{Z}\right) $ and $\operatorname{Im} (\tau) >0,$ we have \[ \Delta \left( \frac{a\tau +b}{c\tau +d}\right) =\left( c\tau +d\right) ^{12}\Delta \left( \tau \right) . \] $G_{2}(q)$ is the Eisenstein series $$ G_{2}(q)=\frac{-1}{24}+\sum_{k=1}^{\infty }\sigma (k)q^{k} $$ where $\sigma (k)=\sum_{d|k}d$. $G_{2}$ and its derivatives are quasi-modular forms \cite{Gott-conj}. One crucial property for us is the fact that $\prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{-1}$ is the generating function of the partition function $p\left( d\right) $. Namely \begin{eqnarray*} \prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{-1} &=&\sum_{d=0}^{\infty }p\left( d\right) q^{d} \\ &=&\allowbreak 1+q+2q^{2}+3q^{3}+5q^{4}+7q^{5}+ \\ &&11q^{6}+15q^{7}+22q^{8}+\mathcal{O}\left( q^{9}\right). \end{eqnarray*} When $g=0,$ our main theorem proves the formula predicted by Yau and Zaslow \cite{Y-Z} for primitive classes. For $g\geq 0$, G\"ottsche has recently conjectured a very general set of formulas for the number of curves on algebraic surfaces \cite{Gott-conj} and Theorem \ref{thm: main thm} proves his conjecture for primitive classes in $K3$ surfaces. Yau and Zaslow give a beautiful, though indirect, argument that would be a complete proof for the $g=0$ case if one could control the complexity of singularities that can occur for curves in a complete linear system. Beauville \cite{Beau}, Chen \cite{Chen}, and Fantechi-G\"ottsche-van~Straten \cite{F-G-vS} have partial results along this line. We shall use a completely different argument by studying Gro\-mov-Wit\-ten invariants for the twistor family of symplectic structures on a $K3$ surface. We learned the twistor family approach from Li and Liu \cite{Li-Liu} who studied the Seiberg-Witten theory for families and obtained interesting results. In the case of a hyperk\"ahler $K3$, the twistor family is the unit sphere in the space of self-dual harmonic 2-forms. The idea of extending the moduli space of pseudo-holomorphic curves by including the family of non-degenerate, norm 1, self-dual, harmonic 2-forms goes back to Donaldson \cite{Do}. He pointed out that in order to have the theory of pseudo-holomorphic curves on a 4-manifold more closely mimic the theory of divisors on a projective surface, one should include this family. One key point in the proof of our main theorem is the use of the large diffeomorphism group of a $K3$ surface to move $C$ to a particular class $S+\left( g+n\right) F$ on an elliptic $K3$ surface with section $S$ and fiber $F$ which has $24$ nodal fibers. Inside the linear system $\left| S+\left( g+n\right) F\right| $, we can completely understand the moduli space of stable maps and really {\em count} the invariants $N_{g}(n)$, reducing the calculation to the computation of ``local contributions'' by multiple covers. The contribution of multiple covers of the smooth fibers is responsible for the $dG_{2}/dq$ term and the contribution from multiple covers of the nodal fibers is related to the partition function $p\left( d\right)$. The computation for the multiple covers of nodal fibers requires an obstruction bundle computation. This is done by splitting the moduli space into components and then identifying each component with a moduli-obstruction problem arising from the Gromov-Witten invariants of a certain blow-up of $\P ^{2}$. This ``matching'' technique allows us to use known properties of the Gromov-Witten invariants of $\P ^{2}$ blown-up, specifically their invariance under Cremona transformations, to show that the contribution of each component is always 0 or 1. The partition function then arises combinatorially in a somewhat unusual way (see lemma \ref{lem: no. of 1-admissable seqs is p(a)}). These computations occupy section \ref{sec: analysis and local computations}. These invariants $N_{g}(n)$ are all enumerative (Theorem \ref{thm: the invariants are enumerative}). We can also apply our method to other elliptic surfaces. We blow up $\mathbf{P} ^{2}$ at nine distinct points and call the resulting algebraic surface $Y$. We consider the Gro\-mov-Wit\-ten invariant $N_{g}^{Y}(C)$ which counts the number of curves of geometric genus $g$ passing through $g$ generic points in a fixed class $C$. We show that any class $C\in H_{2}(Y)$ whose genus $g$ invariants require exactly $g$ point constraints is related to a class of the form $C_{n}=\left( g+n\right) \left[ 3h-\sum_{i=1}^{9}e_{i}\right] +e_{9}$ by a Cremona transform. Here $h$ is the pullback to $Y$ of the hyperplane class in $\mathbf{P}^{2}$ and $e_{1},...,e_{9}$ are the exceptional curves in $Y$. We obtain the following: \begin{theorem}\label{thm: E1 case} Let $Y$ be the rational elliptic surface; for fixed $g$ let $C_{n}$ be the class $S+(g+n)F$ where $F$ is the fiber and $S$ is any section. Let $N^{Y}_{g}(C_{n})$ denote the number of genus $g$ curves in the class of $C_{n}$ passing through $g$ generic points. Then \begin{eqnarray*} \sum_{n=0}^{\infty }N_{g}^{Y}( C_{n}) q^{n}&=&\left(\sum_{k=1}^{\infty }k(\sum_{d|k}d)q^{k-1} \right)^{g}\prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{-12}\\ &=&\left(\frac{d}{dq}G_{2}(q) \right)^{g}\left(\frac{q}{\Delta (q)} \right)^{1/2 }. \end{eqnarray*} \end{theorem} Note that the Euler characteristic of $Y$ is 12 while the Euler characteristic of $K3$ is 24. For us the relevant manifestation of this fact is that elliptically fibered $K3$ surfaces have (generally) 24 nodal fibers while rational elliptic surfaces have (generally) 12 nodal fibers. Since $N_{g}^{Y}( C_{n})$ is an ordinary Gro\-mov-Wit\-ten invariant (without family), it is an invariant for the deformation class of the symplectic structure on $Y$. In particular, $N_{g}^{Y}( C_{n})$ is independent of the locations of those blow up points in $\mathbf{P}^{2}$ and it is left invariant by Cremona transforms. For the genus zero case, the invariants were obtained by G\"{o}ttsche and Pandharipande \cite{Go-Pa} where they computed the quantum cohomology for $\mathbf{P}^{2}$ blown up at arbitrary number of points using the associativity law. Their numbers are in terms of two complicated recursive formulas and it is not obvious that the numbers that correspond to $N_{0}^{Y}( C_{n})$ can be put together to form modular forms, but Theorem \ref{thm: E1 case} can be verified term by term for $g=0$ using their recursion relations. The foundations on which our calculations rest have been developed by Li and Tian (\cite{Li-Tian3}\cite{Li-Tian2}\cite{Li-Tian}). They construct the virtual fundamental cycle of the moduli space of stable maps both symplectically and algebraically and they show that the two contructions coincide in the projective case. Ionel and Parker have a different approach to computing $N_0^Y(C_n)$ that does not rely on \cite{Li-Tian3}. Although our methods are completely different from those of Yau and Zaslow, for the sake of completeness we ouline their beautiful argument for counting rational curves with $n$ nodes. Choose a smooth curve $C$ in the $K3$ surface $ X$ with $C\cdot C=2n-2.$ By adjunction formula, the genus of $C$ equals $n.$ One can show that $C$ moves in a complete linear system of dimension $n$ using the Riemann-Roch theorem and a vanishing theorem. That is $\left| C\right| \cong \mathbf{P}^{n}.$ Imposing a node will put one constraint on the linear system $\left| C\right| $. Therefore, by imposing $n$ nodes, one expects to see a finite number of rational curves with $n$ nodes. Define $N_{0}(n)$ to be this number. Now look at the compactified universal Jacobian $\pi :\mathcal{\bar{J }}\rightarrow \left| C\right| $ for this linear system (c.f. Bershadsky, Sadov, and Vafa \cite{vafa}). It is a smooth hyperk\"{a}hler manifold of dimension $2n.$ If one assumes that each member in the linear system $\left| C\right| $ has at most nodal singularities, then one can argue that for any $C^{\prime }\in \left| C\right| $ the Euler characteristic of $\pi ^{-1}\left( C^{\prime }\right) $ is always zero unless $C^{\prime }$ is a rational curve with $n$ nodes. In the latter case, the Euler characteristic of $\pi ^{-1}\left( C^{\prime }\right) $ equals one. One concludes that $\chi \left( \mathcal{ \bar{J}}\right) =N_{0}(n).$ One the other hand, $\mathcal{\bar{J}}$ is birational to the Hilbert scheme $ \mathcal{H}_{n}$ of $n$ points in $X,$ which is again another smooth hyperk\"{a}hler manifold. Using a result of Batyrev \cite{Bat} which states that compact, birationally equivalent, projective, Calabi-Yau manifolds have the same Betti numbers, one can conclude that $N_{0}(n)=\chi \left(\mathcal{H}_{n}\right)$. Then one uses the result of G\"{o}ttsche \cite{Gott}, who used Deligne's answer to the Weil conjecture to compute (among other things) the Euler characteristic of $\mathcal{H}_{n}:$ \[ \sum_{n=0}^{\infty }\chi \left( \mathcal{H}_{n}\right) q^{n}=\prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{-24}. \] Finally, combining these results, one obtains $$F_{0}=\sum_{n=0}^{\infty}N_{0}(n)q^{n}=\prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{-24}.$$ To turn this argument into a proof one must address the problems of what kind of singularities can occur in $\left| C\right| $ and how the compactified Jacobian contributes to the Euler characteristic of $\mathcal{\bar{J}}$. Work on these issues has been started by Beauville \cite{Beau}, Chen \cite{Chen}, and Fantechi-G\"ottsche-van~Straten \cite{F-G-vS}. Our proof circumvents these problems by using symplectic geometry and Gro\-mov-Wit\-ten invariants for families of symplectic structures. Our method is also more direct than the Yau-Zaslow argument, avoiding the characteristic $p$ methods employed by G\"ottsche and Batyrev. We end this introduction with some speculation into the meaning of our results. Ordinary Gro\-mov-Wit\-ten invariants give rise to the quantum cohomology ring. There may be a corresponding structure in the context of Gro\-mov-Wit\-ten invariants for families. The computation of Theorem \ref{thm: main thm} shows that there is structure amongst these invariants and suggests that there should be an interesting theory of quantum cohomology that encodes it. The ordinary quantum cohomology ring of $X$ gives a Frobenius structure on $H^{*}(X;\mathbf{C}) $ and the (generalized) mirror conjecture states that this Frobenius structure is equivalent to a Frobenius structure arising from some sort of ``mirror object'' (see Givental \cite{Giv}). In the case of a Calabi-Yau 3-fold, the mirror object is another Calabi-Yau 3-fold and the Frobenius structure arises from its variation of Hodge structure. Theorem \ref{thm: main thm} shows that the Gro\-mov-Wit\-ten invariants for $K3$ with its twistor family can be expressed in terms of quasi-modular forms. If there is a quantum cohomology theory associated to the Gro\-mov-Wit\-ten invariants for families and a corresponding mirror conjecture, then our theorem should provide clues as to what the ``mirror object'' of $K3$ with its twistor family should be. This paper is organized as follows. In section \ref{sec: inv of families} we define invariants for families of symplectic structures; in section \ref{sec: twistor families} we discuss twistor families and define $N_{g}(n)$; in section \ref{sec: computation of Ngn} we compute $N_{g}(n)$ and prove our main theorem; in section \ref{sec: analysis and local computations} we analyze the moduli spaces and compute local contributions; in section \ref{sec: counting on E1} we apply similar techniques to $\mathbf{P}^{2}$ blown up at nine points. The authors are pleased to acknowledge helpful conversations with A. Bertram, A. Givental, L. G\"ottsche, E. Ionel, A. Liu, P. Lu, D. Maclagan, D. McKinnon, T. Parker, S. Schleimer, C. Taubes, A. Todorov, and S.-T. Yau. The authors especially thank L. G\"ottsche for sharing early versions of his conjecture with us and for providing many other valuable communications. We would like to thank R. Pandharipande and L. G\"ottsche for sending us their Maple program to verify our results. Additionally we thank the Park City Mathematics Institute for support and providing a stimulating environment where part of this work was carried out. \section{Invariants of families of symplectic structures}\label{sec: inv of families} In this section, we introduce an invariant for a family of symplectic structures $\omega _{B}:B\rightarrow \Omega _{sympl}^{2}\left( X\right) $ on a compact manifold $X.$ Here $B$ is an oriented, compact manifold and $\omega _{B}$ is a smooth map into the space of symplectic forms $\Omega _{sympl}^{2}\left( X\right) .$ This invariant is a direct generalization of the Gro\-mov-Wit\-ten invariants. Roughly speaking, it counts the number of maps $u:\Sigma \to X$ which are holomorphic with respect to some almost complex structure in a generic family compatible with $\omega _{B}.$ Kronheimer \cite{Kr} and Li and Liu \cite{Li-Liu} have also studied invariants for families of symplectic structures and obtained interesting results. In their paper on Gro\-mov-Wit\-ten invariants for general symplectic manifolds, Li and Tian \cite{Li-Tian} setup a general framework for constructing invariants. Their results are easy to adapt to our setting but we remark that in the case of interest, the full Li-Tian machinery is not needed and the techniques of Ruan-Tian \cite{Ruan-Tian} would suffice. This is because 2-dimension families of symplectic structures on a 4-manifold behave like the semi-positive case for the ordinary invariants, {\em i.e. } the (perturbed) moduli spaces are compactified by strata of codimension at least 2. The definition of Gro\-mov-Wit\-ten invariants for families is also contained as part of the very general approach of Ruan \cite{Ruan}. We employ the Li-Tian machinery because they are also able to relate their symplectic constructions to their purely algebraic ones (\cite{Li-Tian2} and \cite{Li-Tian3}). Let $X$ be a compact smooth manifold. Suppose that $\omega _{B}$ is a smooth family of symplectic structures on $X$ parameterized by an oriented, compact manifold $ B. $ Let $J_{B}:B\rightarrow \mathcal{J}\left( X\right) $ be a smooth family of almost complex structures on $X$ such that $J_{t}=J_{B}\left( t\right) $ is compatible with $\omega _{t}=\omega _{B}\left( t\right) $ for any $t\in B.$ In particular, $g_{t}=\omega _{t}\left( \cdot ,J_{t}\cdot \right) $ is a family of Riemannian metrics on $X.$ It is not difficult to see that $J_{B}$ always exists and is unique up to homotopy. This follows from the fact that the space of all almost complex structures compatible with a fixed symplectic form is contractible. Given $X$ and $\omega _{B}$ as above, we shall define the GW-invariant for family as a homomorphism: \[ \Psi _{\left( A,g,k\right) }^{(X,\omega _{B})}:\bigoplus_{i=1}^{k}H^{a_{i}}\left( X,\mathbf{Q}\right) \bigoplus H^{b}\left( \overline{\mathcal{M}}_{g,k},\mathbf{Q}\right) \rightarrow \mathbf{Q}, \] with \begin{equation}\label{eqn: dimension formula} \sum_{i=1}^{k}a_{i}+b=2c_{1}\left( X\right) \left( A\right) +2k+\dim B+\left( \dim X -6\right) \left( 1-g\right) . \end{equation} Here $A\in H_{2}\left( X,\mathbf{Z} \right) $ and $\overline{\mathcal{M}}_{g,k}$ is the Deligne-Mumford compactification of the moduli space of Riemann surfaces of genus $g$ with $ k $ distinct marked points (define $\overline{\mathcal{M}}_{g,k}$ to be a point if $2g+k<3$). For any particular symplectic structure $\omega _{t}$ and the corresponding almost complex structure $J_{t},$ Li and Tian define a section $\Phi _{t}$ of $E\rightarrow \overline{\mathcal{F}_{A}}$ $\left( X,g,k\right) $ (we recall the definition below) which is equivalent to the Cauchy-Riemann operator. These sections depend on $t\in B$ smoothly so that we have a section $\Phi $ of $E\rightarrow \overline{ \mathcal{F}}_{A}$ $\left( X,g,k\right) $ $\times B.$ Let us first recall their notations: A stable map with $k$ marked points is a tuple $\left( f,\Sigma ;x_{1},...x_{k}\right) $ satisfying: \begin{enumerate} \item [{\bf (i)}] $\Sigma =\bigcup\limits_{i=1}^{m}\Sigma _{i}$ is a connected normal crossing projective curve and $x_{i}$'s are distinct smooth points on $\Sigma$, \item [{\bf (ii)}] $f$ is continuous and $f|_{\Sigma _{i}}$ can be lifted to a smooth map on the normalization of $\Sigma _{i}$, and \item [{\bf (iii)}] if $\Sigma _{i}$ is a smooth rational curve such that $f\left( \Sigma _{i}\right) $ represents a trivial homology class in $H_{2}\left( X,\mathbf{Q}\right) ,$ then the cardinality of $ \Sigma _{i}\bigcap \left( \left\{ x_{1},...,x_{k}\right\} \bigcup S\left( \Sigma \right) \right) $ is at least three where $S\left( \Sigma \right) $ is the singular set of $\Sigma.$ \end{enumerate} Two stable maps $\left( f,\Sigma ;x_{1},...x_{k}\right) $ and $\left( f^{\prime },\Sigma ;x_{1}^{\prime },...x_{k}^{\prime }\right) $ are equivalent if there is a biholomorphism $\sigma :\Sigma \rightarrow \Sigma ^{\prime }$ such that $\sigma \left( x_{i}\right) =x_{i}^{\prime }$ for $ 1\leq i\leq k$ and $f^{\prime }=f\circ \sigma .$ We denote the space of equivalent classes of stable maps of genus $g$ with $k$ marked points and with total homology class $A$ by $\overline{\mathcal{F}}_{A}$ $\left( X,g,k\right) $ and the subspace consisting of equivalent classes of stable maps with smooth domain by $\mathcal{F}_{A}\left( X,g,k\right) .$ The topology of $\overline{\mathcal{F}}_{A}$ $\left( X,g,k\right) $ can be defined by sequential convergence. Next they introduced a generalized bundle $E$ over $\overline{\mathcal{F}}_{A}$ $\left( X,g,k\right) $ as follows: For any $\left[ \left( f,\Sigma ;x_{1},...x_{k}\right) \right] \in \overline{ \mathcal{F}}_{A}$ $\left( X,g,k\right) ,$ the fiber of $E$ consists of all $ f^{*}TX$-valued $\left( 0,1\right) $-forms over the normalization of $\Sigma . $ Equiped with the continuous topology, $E$ is a generalized bundle over $ \overline{\mathcal{F}}_{A}$ $\left( X,g,k\right) $ in the sense of Li and Tian. For each $t\in B,$ there is a section of $E$ given by the Cauchy-Riemann operator defined by $J_{t}:$ Namely, for any $\left[ \left( f,\Sigma ;x_{1},...,x_{k}\right) \right] \in \overline{\mathcal{F}}_{A}$ $\left( X,g,k\right) ,$ we have $\Phi _{t}\left( f,\Sigma ;x_{1},...x_{k}\right) =df+J_{t}\circ df\circ j_{\Sigma }$ where $j_{\Sigma }$ is the complex structure of $\Sigma .$ Putting different $t\in B$ together, we have a section $\Phi $ of $E$ over $\overline{\mathcal{F}}_{A}$ $\left(X,g,k\right) \times B$ given by \[ \Phi \left( \left[ \left( f,\Sigma ;x_{1},...x_{k}\right) \right] ,t\right) =df+J_{t}\circ df\circ j_{\Sigma }. \] The following theorems are easy adaptations of those in of Li and Tian found in \cite{Li-Tian}: \begin{theorem} The section $\Phi $ gives rise to a generalized Fredholm orbifold bundle with the natural orientation and of index $$2c_{1}( X) \left[ A\right] +2k+\dim B+\left( \dim X -6\right) \left( 1-g\right). $$ \end{theorem} \begin{theorem} Let $\omega _{B}$ and $\omega _{B}^{\prime }$ are two families of symplectic structures on $X$ parameterized by $B$. Suppose that they are equivalent to each other under deformations for families. Let $J_{B}$ and $J_{B}^{\prime }$ be two families of almost complex structures on $X$ compatible with corresponding symplectic structures. Suppose that $\Phi $ and $\Phi ^{\prime }$ are the corresponding section of $ E$ over $\overline{\mathcal{F}}_{A}$ $\left( X,g,k\right) \times B.$ Then $ \Phi $ and $\Phi ^{\prime }$ are homotopic to each other as generalized Fredholm orbifold bundles. \end{theorem} Using the main theorem of Li and Tian in their paper, there is an Euler class $e\left( \left[ \Phi :\overline{\mathcal{F}}_{A}\left( X,g,k\right) \times B\rightarrow E\right] \right) $ in $H_{r}\left( \overline{\mathcal{F} _{A}}\left( X,g,k\right) \times B,\mathbf{Q}\right) $ with $r=2c_{1}\left( X\right) \left[ A\right] +2k+\dim B+\left( \dim X -6\right) \left( 1-g\right)$. This class is called the {\em virtual fundamental cycle} of the moduli space of holomorphic stable maps $\mathcal{M}=\mathcal{M} (X,\omega _{B},g,k,C)$. We denote it by $[\mathcal{M}]^{vir}$. To define the invariant for $\omega _{B},$ we consider the following two maps. First we have the evaluation map $ev:\overline{\mathcal{F}}_{A}\left( X,g,k\right) \times B\rightarrow X^{k}$ : \[ ev\left( \left( f,\Sigma ;x_{1},...,x_{k}\right) ,t\right) =\left( f\left( x_{1}\right) ,...,f\left( x_{k}\right) \right) . \] and second we have the forgetful map $\pi _{g,k}:\overline{\mathcal{F}}_{A} \left( X,g,k\right) \times B\rightarrow \overline{\mathcal{M}}_{g,k}:$ \[ \pi _{g,k}\left( \left( f,\Sigma ;x_{1},...,x_{k}\right) ,t\right) =red\left( \Sigma ;x_{1},...,x_{k}\right) . \] Here $red\left( \Sigma ;x_{1},...,x_{k}\right) $ is the stable reduction of $ \left( \Sigma ;x_{1},...,x_{k}\right) $ that is obtained by contracting all of its non-stable irreducible components. Now we can define the invariants \[ \Psi _{\left( A,g,k\right) }^{(X,\omega _{B})}:H^{*}\left( X,\mathbf{Q}\right) ^{k}\times H^{*}\left( \overline{\mathcal{M}}_{g,k},\mathbf{Q}\right) \rightarrow \mathbf{Q}. \] by \[ \Psi _{\left( A,g,k\right) }^{(X,\omega _{B})}\left(\alpha _{1},...,\alpha _{k};\beta \right) =\left( ev^{*}\left( \pi _{1}^{*}\alpha _{1}\wedge ...\wedge \pi _{k}^{*}\alpha _{k}\right) \cup \pi _{g,k}^{*}\left( \beta \right) \right) [\mathcal{M}]^{vir} \] for any $\alpha _{1},...,\alpha _{k}\in H^{*}\left( X,\mathbf{Q}\right) $ and $ \beta \in H^{*}\left( \overline{\mathcal{M}}_{g,k},\mathbf{Q}\right) .$ \begin{theorem} $\Psi _{\left( A,g,k\right) }^{(X,\omega _{B})}$ is an invariant of the deformation class of the family of symplectic structures $\omega _{B}.$ \end{theorem} If $(\hat{\alpha }_{1},\ldots,\hat{\alpha }_{k})$ are geometric cycles in $X$ that are Poincar\'e dual to $(\alpha _{1},\ldots,\alpha _{k})$ and $\hat{\beta }$ is a cycle in $\overline{\mathcal{M}}_{g,k}$ dual to $\beta $, then $\Psi ^{(X,\omega _{B})}_{(A,g,k)}(\alpha _{1},\ldots,\alpha _{k};\beta )$ counts the number of stable maps $f:\Sigma _{g}\to X$ so that \begin{enumerate} \item $f(\Sigma _{g})$ represents the class $A$, \item $f$ is $J_{t}$-holomorphic for some $t\in B$, \item $f(x_{i})$ lies on $\hat{\alpha }_{i}$, and \item the stable reduction of $\Sigma _{g}$ lies in $\hat{\beta }\subset \overline{\mathcal{M}}_{g,k}$. \end{enumerate} One is usually interested in $\hat{\beta }=\overline{\mathcal{M}}_{g,k}$, or sometimes $\hat{\beta }=\text{pt.} \in \overline{\mathcal{M}}_{g,k}$. \section{Twistor families of $K3$ surfaces and the definition of $N_{g}(n)$}\label{sec: twistor families} In this section we collect some general facts about $K3$ surfaces and their twistor families. We show that every twistor family is deformation equivalent and we define $N_{g}(n)$ in terms of the Gro\-mov-Wit\-ten invariant for this family. We show that when $X$ is projective and $|C|$ has only reduced and irreducible curves, $N_{g}(n)$ coincides with the enumerative count defined by algebraic geometers (see \cite{F-G-vS}). The results of the section are summarized in Definition \ref{def: definition of Ng,n}. A $K3$ surface is a simply-connected, compact, complex surface $X$ with $c_{1}(X)=0$. For a general reference on $K3$ surfaces and twistor families we refer the reader to \cite{BPV} or \cite{Bess}. Any pair of $K3$ surfaces are deformation equivalent and hence diffeomorphic. A {\em marking} of a $K3$ surface $X$ is an identification of $(H^{2}(X;\znums),Q_{X})$ with the fixed unimodular form $Q=-2E_{8}\oplus 3\left(\begin{smallmatrix} 0&1\\ 1&0 \end{smallmatrix} \right)$. The space of marked $K3$ surfaces forms a connected, 20 complex dimensional moduli space. The complex structure on a marked $K3$ surface $X$ is determined by how the line $H^{0,2}(X)$ sits in $Q\otimes \cnums $. To make this precise, define the {\em period} $\Omega _{X}$ of $X$ to be the element of the {\em period domain} $$\mathcal{D} =\{\Omega \in \P (Q\otimes \cnums ): \left\langle \Omega ,\overline{\Omega }\right\rangle =0,\left\langle \Omega ,\Omega \right\rangle>0 \}$$ given by the image of $H^{0,2}(X)$ under the marking. The Torelli theorem states that $\mathcal{D}$ is the moduli space of marked $K3$ surfaces, {\em i.e. } every marked $K3$ surface corresponds uniquely to its period point in $\mathcal{D}$ and every $\Omega \in \mathcal{D}$ is the period point of some $K3$ surface. For a fixed $K3$ surface $X$ with period $\Omega _{X} $, a class $\omega _{X}\in Q\otimes \cnums $ is a K\"ahler class for $(X,\Omega _{X})$ if and only if $\left\langle \omega _{X},\Omega _{X}\right\rangle=0$, $\left\langle \omega _{X},\overline{\Omega}_{X}\right\rangle=0$, $\omega _{X}=\overline{\omega}_{X}$, and $\left\langle \omega _{X},\omega _{X}\right\rangle>0$. For any K\"ahler $K3$ surface $(X,\Omega _{X},\omega _{X})$ there is a unique hyperk\"ahler metric by Yau's proof of the Calabi conjecture \cite{Yau}. A hyperk\"ahler metric $g$ determines a 2-sphere worth of K\"ahler structures, namely the unit sphere in the space $\mathcal{H}^{2}_{+,g}$ of self-dual harmonic forms. We can describe the corresponding 2-sphere of period points as follows. Consider the projective plane spanned by $\left\langle \Omega _{X},\overline{\Omega }_{X}, \omega _{X}\right\rangle$ in $\P (Q\otimes \cnums) $. Since $\left\langle \Omega _{X},\overline{\Omega }_{X}, \omega _{X}\right\rangle$ spans $\mathcal{H}^{2}_{+,g}\otimes \cnums$, the intersection of this projective plane and the period domain is the quadric determined by $\left\langle \Omega ,\overline{\Omega }\right\rangle=0$. This is a smooth plane quadric and hence a 2-sphere. This 2-sphere of complex structures together with the corresponding 2-sphere of K\"ahler structures we call a {\em twistor family}. We will use the notations $(J_{T},\omega_{T})$ to refer to a twistor family and $J_{t}$, $\omega_{t}$ for $t\in T$ to refer to individual members. The following proposition was explained to us by Andrei Todorov: \begin{prop}\label{prop: one curve in class of C in twistor family} Let $(X,\Omega _{X},\omega _{X})$ be a marked, K\"ahler $K3$ surface and $(J_{T},\omega_{T})$ the corresponding twistor family. Let $C\in H^{2}(X;\znums )$ be a class of square $C^{2}\geq -2$. Then there is exactly one member $t\in T$ for which there is a $J_{t}$-holomorphic curve in the class of $C$. \end{prop} {\em Proof:} A class $C$ of square $-2$ or larger admits a holomorphic curve if and only if $C\in H^{1,1}(X;\znums )$ and $C$ pairs positively with the K\"ahler class. Since $C$ is a real class, $C\in H^{1,1}$ if and only if $\left\langle C,\Omega_{X}\right\rangle=0$. This equation determines a hyperplane in $\P (H^{2}_{+})$ and so meets the twistor space in 2 points $\pm \Omega_{0}$ (since the twistor space is a quadric). Then exactly one of $\pm\Omega_{0}$ will have its corresponding K\"ahler class pair positively with $C$.\qed We next show that every twistor family is the same up to deformation. \begin{proposition}\label{prop: every twistor family is homotopic} Let $X_{1}$ and $X_{2}$ be two K\"ahler $K3$ surfaces, then the corresponding twistor families $T_{0}$ and $T_{1}$ are deformation equivalent. \end{proposition} {\em Proof:} The moduli space of $K3$ surfaces is connected and the space of hyperk\"ahler structures for a fixed $K3$ surface is contractible (it is the K\"ahler cone). Therefore, the space parameterizing hyperk\"ahler $K3$ surfaces $(X,\Omega_{X},\omega _{X})$ is also connected. We can thus find a path $(X_{s},\Omega _{s},\omega _{s})$, $s\in [0,1]$ connecting $X_{0}$ to $X_{1}$ where the twistor family of $\omega _{i}$ is $T_{i}$ for $i=0,1$. By then associating to each hyperk\"ahler structure $\omega _{s}$ its twistor family $T_{s}$, we obtain a continuous deformation of $T_{0}$ to $T_{1}$. \qed From this proposition we see that the Gro\-mov-Wit\-ten invariants for a twistor family are independent of the choice of a twistor family. We can thus write unambiguously $$\Psi ^{(K3,\omega _{T})}_{(C,g,k)}:H^{*}(K3;\znums )^{k}\otimes H^{*}(\overline{\mathcal{M}}_{g,k})\to \qnums .$$ We are primarily interested in the invariants that count stable maps without fixing the complex structure on the domain. That is, the invariants obtained using the Poincar\'e dual of the fundamental class of $\overline{\mathcal{M}}_{g,k}$. It is enough to consider those constraints that come from the generator of $H^{4}(K3,\znums )$; these count curves passing through fixed generic points. The invariants with the constraint that the $k$th point lies on a fixed generic cycle dual to an element $\beta \in H^{2}(K3)$ can be computed in terms of the invariants for $k-1$ constraints and the pairing $\beta \cdot C$. For this reason, constraining the invariants by elements of $H^{2}(K3)$ is uninteresting, and of course elements of $H^{0}(K3)$ provide no constraints at all. We can thus simplify notation by defining $$\Psi(C,g,k)\equiv \Psi ^{(K3,\omega _{T})}_{(C,g,k)}(PD(x_{1}),\ldots,PD(x_{k});PD([\overline{\mathcal{M}}_{g,k} ])). $$ An important observation about the twistor family is the following. \begin{prop}\label{prop: diffeos preserve T} If $f:K3\to K3$ is an orientation preserving diffeomorphism, then the pullback family $f^{*}(\omega _{T})$ is deformation equivalent to $\omega _{T}$; thus $$ \Psi (C,g,k) =\Psi (f_{*}(C),g,k). $$ \end{prop} {\em Proof:} Let $\omega _{T}$ be the twistor family associated to a hyperk\"ahler metric $g$. Then $f^{*}(\omega _{T})$ is the twistor family associated to the hyperk\"ahler metric $f^{*}(g)$ and so Proposition \ref{prop: every twistor family is homotopic} they are deformation equivalent. \qed The $K3$ surface has a big diffeomorphism group in the sense of Friedman and Morgan \cite{Fr-Mo}, which means that every automorphism of the lattice $Q_X$ which preserves spinor norm can be realized by an orientation preserving diffeomorphism. In particular, one can take any primitive class $C\in H_2(K3;\znums)$ to any other primitive class with the same square via an orientation preserving diffeomorphism. We are now in a position to define $N_{g}(n)$. By the adjunction formula, a holomorphic curve of genus $g$ with $n$ nodes will be in a class $C$ with square $C^{2}=2(g+n)-2$. \begin{definition}\label{def: definition of Ng,n} Let $C$ be any primitive class with $C^{2}=2(g+n)-2$. We define the number $N_{g}(n)$ by \begin{eqnarray*} N_{g}(n)&=&\Psi (C,g,g)\\ &=&\Psi ^{(K3,\omega_T)}_{(C,g,g)} (PD(x_{1}),\ldots,PD(x_{g});PD([\overline{\mathcal{M}}_{g,k}])). \end{eqnarray*} By Proposition \ref{prop: diffeos preserve T} , $N_{g}(n)$ is independent of the choice of the primitive class $C$. By Proposition \ref{prop: every twistor family is homotopic}, $N_{g}(n)$ is independent of the choice of twistor family. Finally, in the case of a projective $K3$ surface with an effective divisor in the class of $C$, Proposition \ref{prop: one curve in class of C in twistor family} shows that $N_{g}(n)$ counts holomorphic maps $f:D\to X$ of genus $g$ curves to $X$ with image in $|C|$ and passing through $g$ generic points. \end{definition} \begin{thm}\label{thm: the invariants are enumerative} If $X$ is generic among those $K3$ surfaces admitting a curve in the class $[C]$, then invariant $N_{g}(n)$ is enumerative. \end{thm} {\sc Proof:}\footnote{This argument is due to Lothar G\"ottsche. We are grateful to him for showing it to us.} The assumption that the $K3$ surface $X$ is generic among those admitting a curve in the class of $C$ guarantees that the primitive class $C$ generates the Picard group. Suppose that the invariant $\Psi (C,g,g)$ differs from the actual count of curves $\Sigma \in |C|$ of genus $g$ passing through $g$ general points (curves counted with multiplicities if the they are not all nodal as in \cite{F-G-vS}). Then there is an extra map $D\to X$ of a curve of arithmetic genus $g$ that has some contracted components and the rest of the map is generically injective with irreducible image. Let $C_{1}$ be a contracted component. Since the $g$ marked points have to go to $g$ distinct points on $X$, $C_{1}$ can have at most 1 marked point. By stability then, either the geometric genus $g(C_{1})$ is larger than 0 or $C_{1}$ intersects the rest of $D$ in at least 2 points. Since the image of $D$ is irreducible, the contracted components cannot all be genus 0 unless the dual graph of $D$ is not a tree. Thus either $D$ has a contracted component of genus greater than 0 or the dual graph of $D$ is not a tree. In either case, the geometric genus of the image is smaller than the arithmetic genus of $D$ and thus the image is a curve of genus less than $g$ passing through $g$ points. This does not occur for $g$ generic points by a dimension count. \begin{rem}\label{rem: our method only is for primitive C} The conjectured formula of Yau and Zaslow applies to non-primitive classes as well. The above definition could be made for arbitrary classes $C$, but {\it a priori} $N_{g}(n)$ would also depend on the divisibility of $C$. Our method of computing $N_{g}(n)$ only applies to primitive classes, so the Yau-Zaslow conjecture remains open for the non-primitive classes. \end{rem} \section{Computation of $N_{g}(n)$}\label{sec: computation of Ngn} To compute $N_{g}(n)$ we are free to choose any family of symplectic structures deformation equivalent to the twistor family and any primitive class $C$ with $C^{2}=2(g+n)-2$. Let $X$ be an elliptically fibered $K3$ surface with a section and 24 nodal singular fibers $N_{1},\ldots,N_{24}$. Endow $X$ with a hyperk\"ahler metric and let $(\omega _{T},J_{T})$ be the corresponding twistor family. Let $S$ denote the section and $F$ the class of the fiber so that $F^{2}=0$, $F\cdot S=1$, and $S^{2} =-2$. Let $C$ be the class $S+(n+g)F$ and fix $g$ generic points $x_{1},\ldots,x_{g}$ not on $S$ that lie on $g$ distinct smooth fibers which we label $F_{1},\ldots, F_{g}$ (see the following illustration). \vskip 15pt \setlength{\unitlength}{0.00083333in} \begingroup\makeatletter\ifx\SetFigFont\undefined \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup {\renewcommand{\dashlinestretch}{30} \begin{picture}(3928,2183)(0,-10) 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\put(401,2121){\makebox(0,0)[lb]{\smash{$N_1$}}} \put(1115,2121){\makebox(0,0)[lb]{\smash{$F_1$}}} \put(2154,2121){\makebox(0,0)[lb]{\smash{$F_g$}}} \put(2965,2121){\makebox(0,0)[lb]{\smash{$N_{24}$}}} \put(3809,1407){\makebox(0,0)[lb]{\smash{$X$}}} \put(3842,11){\makebox(0,0)[lb]{\smash{$\mathbf{P}^1$}}} \put(1505,1472){\makebox(0,0)[lb]{\smash{$\cdots$}}} \put(3257,855){\makebox(0,0)[lb]{\smash{$S$}}} \end{picture} } \vskip 15pt Recall from Definition \ref{def: definition of Ng,n} that $$N_{g}(n)=\Psi (C,g,g). $$ This counts the number of stable maps of geometric genus $g$, in the class of $C$, whose image contains $x_{1},\ldots,x_{g}$, and which are holomorphic for some $J_{t}$, $t\in T$. The space of such maps forms a compact moduli space of virtual dimension 0 which we denote by $\mathcal{M}_{C,g}$. By Proposition \ref{prop: one curve in class of C in twistor family}, there is a unique $t\in T$ so that there are $J_{t}$-holomorphic curves in the class of $C$. This $J_{t}$ must be the original elliptically fibered complex structure. Thus $\mathcal{M}_{C,g}$ consists of stable holomorphic maps whose image are in the linear system $|S+(n+g)F|$ and contain the points $x_{1},\ldots,x_{g}$. Because of the elliptic fibration, the linear system $|C|$ is easy to analyze. The dimension of $|S+(n+g)F|$ is $n+g$ and consists solely of reducible curves which are each a union of the section and $(n+g)$ (not necessarily distinct) fibers. Since the image contains the points $x_{1},\ldots,x_{g}$, it contains the corresponding smooth fibers $F_{1},\ldots,F_{g}$. The image of a map in $\mathcal{M}_{C,g}$ must therefore be the union of the section $S$, the $g$ fibers $F_{1},\ldots,F_{g}$, and some number of nodal fibers (possibly counted with multiplicity). We summarize this discussion in the following \begin{prop} Let $\mathcal{M}_{C,g}$ be the moduli space of stable maps of genus $g$ in the class of $C=S+(g+n)F$ passing through the points $x_{1},\ldots,x_{g}$. Let $\pi :\mathcal{M}_{C,g}\to \mathbf{P}(H^{0}(X,C)) $ be the natural projection onto the linear system $|C|$. Then $\operatorname{Im} (\pi )$ is a finite number of points labeled by the vectors $\mathbf{a}=(a_{1},\ldots,a_{24})$ and $\mathbf{b}=(b_{1},\ldots,b_{g})$ where $a_{j}\geq 0$, $b_{i}\geq 1$, and $\sum a_{j}+\sum{b_{i}}=n+g$. The corresponding divisor in $|C|$ is $$S+\sum_{i=1}^{g}b_{i}F_{i}+\sum_{j=1}^{24}a_{j}N_{j} $$ where $F_{i}$ is the smooth fiber containing $x_{i}$ and $N_{1},\ldots,N_{24}$ are the nodal fibers. \end{prop} The proposition implies that $\mathcal{M}_{C,g}$ is the disjoint union of components $\mathcal{M}_{\mathbf{a},\mathbf{b}}$ labeled by the vectors $\mathbf{a}$ and $\mathbf{b}$. In section \ref{sec: analysis and local computations} we analyze the moduli spaces $\mathcal{M}_{\mathbf{a},\mathbf{b}}$ in detail. The main result of that section (Theorem \ref{thm: contribution of Mab}) is that the contribution to $N_{g}(n)$ from $ \mathcal{M}_{\mathbf{a},\mathbf{b}}$ is the product of the local contributions: $$\prod_{i=1}^{g}b_{i}\sigma (b_{i})\prod_{j=1}^{24}p\left( a_{j}\right) .$$ Our main theorem follows from this and some manipulations with the generating functions. Recall that the generating function of the partition function $p(l)$ is $\prod_{m=1}^{\infty }(1-q^{m})^{-1}$. Recall also that $\mathbf{a}$ is a $24$-tuple of integers with $a_{j}\geq 0$ and $\mathbf{b}$ is a $g$-tuple of integers with $b_{i}\geq 1$ and $|\mathbf{a}|+|\mathbf{b}|=\sum_{j=1}^{24}a_{j}+\sum_{i=1}^{g} b_{i}=n+g$. We compute: \begin{eqnarray*} N_{g}(n)q^{n}&=& \left(\sum_{\begin{smallmatrix} \mathbf{a},\mathbf{b}\\ |\mathbf{a}|+|\mathbf{b}|=n+g\end{smallmatrix}}\prod_{i=1}^{g}b_{i}\sigma (b_{i})\prod_{j=1}^{24}p(a_{j}) \right)q^{n}\\ &=&\sum_{k=0}^{n}\sum_{\begin{smallmatrix} |\mathbf{a}|=n-k\\ |\mathbf{b}|=g+k \end{smallmatrix}}\prod_{i=1}^{g}b_{i}\sigma (b_{i})q^{b_{i}-1}\prod_{j=1}^{24}p(a_{j}) q^{a_{j}}. \end{eqnarray*} Summing over $n$: \begin{eqnarray*} \sum_{n=0}^{\infty }N_{g}(n)q^{n}&=&\sum_{n=0}^{\infty }\sum_{k=0}^{n}\left(\sum_{|\mathbf{b}|=g+k}\prod_{i=1}^{g}b_{i}\sigma (b_{i})q^{b_{i}-1} \right)\left(\sum_{|\mathbf{a}|=n-k}\prod_{j=1}^{24}p(a_{j})q^{a_{j}} \right)\\ &=&\sum_{k=0}^{\infty }\sum_{n=k}^{\infty }\left(\sum_{|\mathbf{b}|=g+k}\prod_{i=1}^{g}b_{i}\sigma (b_{i})q^{b_{i}-1} \right)\left(\sum_{|\mathbf{a}|=n-k}\prod_{j=1}^{24}p(a_{j})q^{a_{j}} \right)\\ &=&\left(\sum_{|\mathbf{b}|\geq g} \prod_{i=1}^{g}b_{i}\sigma (b_{i})q^{b_{i}-1}\right)\left(\sum_{|\mathbf{a}|\geq 0}\prod_{j=1}^{24}p(a_{j})q^{a_{j}} \right)\\ &=&\prod_{i=1}^{g}\left(\sum_{b_{i}=1}^{\infty }b_{i}\sigma (b_{i})q^{b_{i}-1} \right) \prod_{j=1}^{24}\left(\sum_{a_{j}=0}^{\infty }p(a_{j}) q^{a_{j}} \right)\\ &=&\left(\sum_{b=1}^{\infty }b\sigma (b)q^{b-1} \right)^{g}\prod_{m=1}^{\infty }(1-q^{m})^{-24}\\ &=&\left(\frac{d}{dq}G_{2}(q) \right)^{g}\frac{q}{\Delta (q)}. \end{eqnarray*} This proves our main theorem. \section{Analysis of moduli spaces and local contributions} \label{sec: analysis and local computations} The main goal of this section is to compute the contribution of the component $\mathcal{M}_{\mathbf{a},\mathbf{b}}$ to the invariant $N_{g}(n)$. Our strategy is simple in essence. We show that the moduli space can be written as a product of various other moduli spaces and that the tan\-gent-ob\-struct\-ion complex splits into factors that pull back from tan\-gent-ob\-struct\-ion complexes on the other moduli spaces. We then show that those individual moduli-obstruction problems have many components, each of which can be identified with moduli-obstruction problems arising for the Gro\-mov-Wit\-ten invariants of $\P ^{2}$ blown up multiple times. These contributions can then be determined by elementary properties of the Gro\-mov-Wit\-ten invariants on blow ups of $\P ^{2}$. Using Cremona transformations, these contributions can be shown to all either vanish or be equivalent to the number of straight lines between two points (one). The computation then follows from straight forward combinatorics. This section is somewhat notationally heavy so to help the reader navigate we summarize the notation used. We use $\mathcal{M} _{C}$ for the full moduli space of stable genus $g$ maps to $X$ in the class of $C$; $\mathcal{M} _{C,g}$ denotes the subset of $\mathcal{M} _{C}$ where the curves pass through the $g$ points $x_{1},\ldots,x_{g}$. $\mathcal{M} _{C,g}$ breaks into components $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ indexed by vectors $\mathbf{a}=(a_{1},\ldots,a_{24})$ and $\mathbf{b}=(b_{1},\ldots,b_{g})$ determining the image of the map. These components break into further components indexed by ``data'' $\Lambda (\mathbf{b})$, $\lambda (\mathbf{b})$, and $s(\mathbf{a})$ (Theorem \ref{thm: components of Mab}). To prove Theorem \ref{thm: components of Mab}, it is convenient to introduce $\mathcal{M}_{a}$ which denotes the moduli space of genus 0 stable maps to $X$ with image $S+aN$ where $N$ is any fixed nodal fiber. The moduli space $\mathcal{M} _{a}$ breaks into components because of the possibility of ``jumping'' behavior at the node of $N$. This behavior is encoded by certain kinds of sequences $\{s_{n} \}$ (we call {\em admissible}) and hence the components of $\mathcal{M} _{a}$ are indexed by such sequences. We denote those components by $\mathcal{M} _{\{s_{n} \}}$. We compute the contribution of $\mathcal{M} _{\{s_{n} \}}$ by ``matching'' its tangent-obstruction complex with a tangent-obstruction complex on a moduli space of stable maps to a blow up of $\P ^{2}$. This blow up is denoted $\til{P}$ and the relevant moduli space is denoted $\mathcal{M} ^{\til{P}}_{\{s_{n} \}}$. Ultimately, we show that the contribution of each component to the invariant is either 0 or 1; those components that contribute 1 are those for which the relavant admissable sequences have a special property (we call such sequences {\em 1-admissable}). The contribution of $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ is then obtained by counting how many possibilites there are for the data $\Lambda (\mathbf{b})$, $\lambda (\mathbf{b})$, and $s(\mathbf{a})$ that have only 1-admissable sequences. \subsection{Components of $\mathcal{M} _{\mathbf{a},\mathbf{b}} $} We begin by identifying the connected components of $\mathcal{M}_{\mathbf{a},\mathbf{b}}$. Call a sequence $\{s_{n} \}$ {\em admissible} if each $s_{n}$ is a positive integer and the index $n$ runs from some non-positive integer through some non-negative integer (the sequence could consist solely of $\{s_{0} \}$ for example). Write $|s|$ for $\sum_{n}s_{n}$. \begin{thm}\label{thm: components of Mab} The connected components of $\mathcal{M}_{\mathbf{a},\mathbf{b}}$ are indexed by the following data. For each $a_{i}\in \mathbf{a}$ assign an admissible sequence $\{s_{n}(a_{i}) \}$ such that $|s(a_{i})|=a_{i}$. For each $b_{j}\in \mathbf{b}$ assign a sublattice $\Lambda (b_{j})\subset \znums \oplus \znums $ of index $b_{j}$ and an element $\lambda (b_{j})$ of the set $\{1,2,\ldots,b_{j} \}$. \end{thm} \begin{rem}\label{rem: number of lattices is sigma} The number of sublattices of $\znums \oplus \znums $ of index $b$ is classically known and is given by $\sigma (b)=\sum_{d|b}d$. Thus we see that the number of possible choices of data assigned to $\mathbf{b}$ is $\prod_{i=1}^{g}b_{i}\sigma (b_{i})$. \end{rem} Let $f\in \mathcal{M} _{\mathbf{a},\mathbf{b}}$. Since the image of $f:D\to X$ is $S+\sum_{i}b_{i}F_{i}+\sum_{j}a_{j}N_{j}$ and is reducible, $D$ must be reducible and its components must group into the set of components mapping to $S$, $F_{1},\ldots,F_{g}$, and $N_{1},\ldots,N_{24}$. Since the components mapping to each $F_{i}$ must have geometric genus at least 1 and the total geometric genus of $D$ is $g$, the components of $D$ mapping to $F_{i}$ must each be genus 1 and all other components of $D$ are rational. Furthermore, the dual graph of $D$ is a tree and the $g$ marked points are on the $g$ elliptic components which we call $G_{1},\ldots,G_{g}$. Then since $f$ is a stable map, the image of all ghost components of $D$ must lie in the nodal fibers $N_{1},\ldots,N_{24}$. We denote the component of $D$ mapping isomorphically onto $S$ also by $S$. So far then, we can describe the domain $D$ as a rational curve $S$ that has attached to it $g$ marked elliptic curves $G_{1},\ldots,G_{g}$ and 24 components $D_{1},\ldots,D_{24}$ that are either empty (if $a_{i}=0$) or a tree of rational components. Furthermore, $f|_{G_{i}}:G_{i}\to F_{i}$ is a degree $b_{i}$ map preserving the intersection with $S$ and sending the marked point to $x_{i}$ and $f|D_{j}:D_{j}\to N_{j}$ has total degree $a_{j}$. Using the intersection with $S$ as an origin for $G_{i}$ and $F_{i}$, we can identify the number of distinct possibilities for the map $f:G_{i}\to F_{i}$ with the number of degree $b_{i}$ homomorphisms onto a fixed elliptic curve $F_{i}$. This is precisely the number of index $b_{i}$ sublattices of $\znums \oplus \znums $. Additionally, since $x_{i}$ has $b_{i}$ preimages under $f$ (the $x_{i}$'s are chosen generically), there are $b_{i}$ choices for the location of the marked point on $G_{i}$ for each homomorphism $f:G_{i}\to F_{i}$. Thus the data associated to $\mathbf{b}$ in the theorem completely determines $f$ restricted to $G_{1},\ldots,G_{g}$. Now $f|_{S}$ is determined and so we can reconstruct $f$ completely from $f|_{N_{1}},\ldots,f|_{N_{24}}$ and the data $\Lambda (b_{i})$, $\lambda (b_{i})$. It follows that the subset of $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ with fixed data for $\mathbf{b}$ is the product of the moduli spaces $\prod_{j=1}^{24}\mathcal{M} _{[a_{j}]_{j},0}$, where $[c]_{j}$ denotes the 24-tuple $(0,\ldots,c,\ldots,0)$ with $c$ in the $i$th slot and zeros elsewhere. The connected components of $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ are in one to one correspondence with the data $\Lambda (b_{i})$, $\lambda (b_{i})$, and $s(a_{j})$ and Theorem \ref{thm: components of Mab} is proved provided we can show that the connected components of $\mathcal{M} _{[a],0}$ are in one to one correspondence with admissible sequences $s$ of magnitude $|s|=a$. We state this as a \begin{lem}\label{lem: components of nodal multiple covers are admissable seqs} Let $\mathcal{M} _{a}$ be the moduli space of stable, genus 0 maps to $X$ with image $S+aN$ for any fixed nodal fiber $N$. Then $\mathcal{M} _{a}$ is a disjoint union $\coprod _{\{s_{n} \}}\mathcal{M} _{\{s_{n} \}}$ of spaces $\mathcal{M} _{\{s_{n} \}}$ labeled by admissible sequences $\{s_{n} \}$ with $|s|=\sum_{n}s_{n}=a$. \end{lem} {\sc Proof: } Let $\Sigma(a)$ be a genus 0 nodal curve consisting of a linear chain of $2a+1$ smooth components $\Sigma _{-a},\ldots,\Sigma _{a}$ with an additional component $\Sigma _{*}$ meeting $\Sigma _{0}$ (so $\Sigma_{n} \cap\Sigma _{m}= \emptyset $ unless $|n-m|= 1$ and $\Sigma _{*}\cap \Sigma _{n}=\emptyset $ unless $n=0$). Fix a map of $\Sigma(a) $ to $X$ with image $S\cup N$ in the following way. Map $\Sigma _{*}$ to $S$ with degree 1 and map each $\Sigma _{n}$ to $N$ with degree one. Require that a neighborhood of each singular point $\Sigma _{n}\cap \Sigma _{n+1}$ is mapped biholomorphically onto its image with $\Sigma _{n}\cap \Sigma _{n+1}$ mapping to the nodal point of $N$. Let $\{s_{n} \}$ be an admissible sequence with $|s|=a$. Since the index $n$ of the sequence cannot be smaller than $-a$ or larger than $a$, we can extend $\{s_{n} \}$ to a sequence $s_{-a},\ldots,s_{a}$ by setting $s_{n}=0$ for those not previously defined. Define $\mathcal{M} _{\{s_{n} \}}$ to be the moduli space of genus 0 stable maps {\em to} $\Sigma(a) $ in the class $$ \Sigma _{*} +\sum_{n=-a}^{a}s_{n}\Sigma _{n}. $$ By composition with the fixed map from $\Sigma(a) $ to $X$, we get a stable map in $\mathcal{M} _{a}$ from each map in $\mathcal{M} _{\{s_{n} \}}$. To prove the lemma we need to show\footnote{To prove the lemma as stated we also need to show that $\mathcal{M} _{\{s_{n} \}}$ is connected. This is not hard, but since we never actually use this part of the result, we will leave its proof to the reader.} that every map in $\mathcal{M} _{a}$ factors uniquely in this way through a map in $\mathcal{M} _{\{s_{n} \}}$ for some admissible $\{s_{n} \}$. The following figure illustrates some of the phenomenon that can occur. The numbers on components of $D$ indicate the degree of $f$ on that component. 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_{a}$. The graph is a tree with one special vertex $v_{*}$ (the component mapping to $S$) whose valence is 1. Every other vertex $v$ is marked with a non-negative integer $l_{v}$ (the degree of the component associated to $v$) such that the sum of the $l_{v}$'s is $a$. Vertices with a marking of 0 (ghost components) must have valence at least three (stability). We mark the edges in the following way. Each edge corresponds to a nodal singularity in $D$ and if the node is not mapped to the nodal point in $N$, we do not mark the edge. The remaining edges are marked with either a pair of the letters $A$ or $B$, a single letter of $A$ or $B$, or nothing as follows. Label the two branches near the node in $N$ by $A$ and $B$. Three things can then happen for an edge corresponding to a node in $D$ that gets mapped to the node in $N$.\begin{enumerate} \item If the edge connects two ghost components, do not mark the edge. \item If the edge connects one ghost component with one non-ghost component, mark the edge with an $A$ or $B$ depending on whether the non-ghost component is mapped (locally) to the $A$ branch or the $B$ branch. \item Finally, if the edge connects two non-ghost components, then mark the edge with two of the letters $A$ or $B$, one near each of the vertices, according to which branch that corresponding component maps to (locally). Note that all the combinations $AB$, $BA$, $AA$, and $BB$ can occur. \end{enumerate} The markings on our graph now tell us how and when the map ``jumps'' branches. Jumping from $A$ to $B$ will correspond to moving from $\Sigma _{n}$ to $\Sigma _{n+1}$ in the factored map. To determine which component $\Sigma _{n}$ a component of $D$ gets mapped to, we count how many ``jumps'' occur between the corresponding vertex $v$ and the central vertex $v_{*}$. For every non-ghost component vertex $v$ assign its index $n_{v}$ by traveling from $v_{*}$ to $v$ in the graph and counting $+1$ for each $AB$ pair passed through, $-1$ for each $BA$ pair, and $0$ for each $AA$ or $BB$ pair. We can now uniquely factor $f:D\to X$ through the fixed map $\Sigma (a)\to X$. The component of $D$ corresponding to a vertex $v$ gets mapped to $\Sigma _{n_{v}}$. The factorization is unique since away from the $AB$ or $BA$ jumps, $f$ factors uniquely through the normalization of $N$. \qed The marked dual graph for the previously illustrated example is below. \vskip 15pt \setlength{\unitlength}{0.00083333in} \begingroup\makeatletter\ifx\SetFigFont\undefined \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup {\renewcommand{\dashlinestretch}{30} \begin{picture}(4789,2063)(0,-10) \path(48,986)(873,647)(1600,647) (2570,986)(2327,1809) \path(1600,647)(1648,65) \path(2570,986)(3394,986) \path(3394,986)(4364,1374) \path(3443,1858)(3394,986)(3539,211) \put(0,1083){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(4,-1)}}}}} \put(97,792){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}A}}}}} \put(582,647){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}B}}}}} \put(825,744){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(1,0)}}}}} \put(1552,744){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}0}}}}} \put(1745,17){\makebox(0,0)[lb]{\smash{{{\SetFigFont{9}{8.4}{rm}$v_*$}}}}} \put(2521,840){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(2,0)}}}}} \put(2909,1034){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}A}}}}} \put(2521,1180){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}A}}}}} \put(2425,1568){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}B}}}}} \put(2279,1858){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(1,1)}}}}} \put(3345,1955){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(2,0)}}}}} \put(4461,1374){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(1,1)}}}}} \put(3636,162){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}(1,1)}}}}} \put(3491,889){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}0}}}}} \put(3491,1568){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}A}}}}} \put(4025,1083){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}B}}}}} \put(3539,453){\makebox(0,0)[lb]{\smash{{{\SetFigFont{7}{8.4}{rm}B}}}}} \end{picture} } \vskip 15pt Here we've marked the vertices with $(l_{v},n_{v})$ so in this example $s_{-1}=4$, $s_{0}=5$, and $s_{1}=3$. \subsection{The tangent-obstruction complex} To compute the contribution of $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ to $N_{g}(n)=\Psi (C,g,g)$, we recall the definition of the invariant. Let $\mathcal{M} _{C}$ be the moduli space of $g$-marked, genus $g$ stable maps to $X$ (with its twistor family) in the class of $C$. Let $\mathcal{M} _{C,g}\subset \mathcal{M} _{C}$ denote the restriction to those maps that send the $i$th marked point to the point $x_{i}\in X$; the virtual dimension of $\mathcal{M} _{C,g}$ is 0. By definition, $N_{g}(n) $ is the evaluation of $ev^{*}_{1}([x_{1}])\wedge \cdots \wedge ev^{*}_{g}([x_{g}])$ on $[\mathcal{M} _{C}]^{vir}$ which is the same as the class $[\mathcal{M} _{C,g}]^{vir}$. Li and Tian construct the class $[\mathcal{M} _{C,g}]^{vir}$ in the symplectic category in \cite{Li-Tian} and in the algebraic category in \cite{Li-Tian2}. In \cite{Li-Tian3} they show the classes coincide when the symplectic manifold is algebraic. This enables us to compute purely algebro-geometrically although we need the machinery of the symplectic category to define the invariant. The data of the algebraic construction of $[\mathcal{M} _{C,g}]^{vir}$ is the tan\-gent-ob\-struct\-ion complex $[T^{1}\to T^{2}]$. It is a complex of sheaves over $\mathcal{M} _{C,g}$ whose stalks over a map $f:D\to X$ fit into an exact sequence (c.f. \cite{Gr-Pa}) $$ \begin{CD} 0@>>>\operatorname{Aut}(D)@>>>H^{0}(D,f^{*}(TX))@>>>T^{1}@>>>\\ @>>>\operatorname{Def}(D)\oplus T_{t_{0} }B@>>>H^{1}(D,f^{*}(TX))@>>>T^{2}@>>>0 \end{CD} $$ where $\operatorname{Aut}(D)$ is the space of infinitesimal automorphisms of the domain, $\operatorname{Def}(D)$ is the space of infinitesimal deformations of the domain, and $T_{t_{0} }B$ is the tangent space of the family of K\"ahler structures at $t_{0}$ (formally the pull back by $J_{B}:B\to \mathcal{J} $ of the tangent space of $\mathcal{J}$ restricted to $t_{0}\in B$). The sequence can be informally interpreted as follows. The tangent space to $\mathcal{M} _{C,g}$ at a map $f:D\to X$ contains those vector fields along the map modulo those that come from vector fields on the domain. The tangent space also contains infinitesimal deformations of the complex structure on the domain and infinitesimal deformations of the complex structure of the range in the family, however, those infinitesimal deformations are obstructed from being actual deformations if they have non-zero image in $H^{1}(D,f^{*}(TX))$. If the ranks of $T^{i}$ remain constant over the moduli space, then the moduli space is a smooth orbifold and $T^{2}$ is a smooth orbi-bundle. In this case, the virtual fundamental class is simply the Euler class of $T^{2}$. This is, in fact, the case for $\mathcal{M} _{C,g}$. Since we have identified the components of $\mathcal{M} _{C,g}$ with various products of the moduli spaces $\mathcal{M} _{\{s_{n} \}}$ which are smooth orbifolds ($\mathcal{M} _{\{s_{n} \}}$ is smooth since it is a moduli space of stable maps to a range consisting of convex varieties meeting at general points). In the rest of this section we show how the bundle $T^{2}$ can be expressed as a sum of bundles pulling back from the factors of the product. The exact sequence shows that when (a component of) the moduli space of stable maps all have the same image, the tangent obstruction complex only depends upon the restriction of the tangent bundle to $X$ to the image of $f$ (and not on the rest of $X$). Breaking our moduli space into connected components, we have $$[\mathcal{M} _{C,g}]^{vir}=\sum_{\mathbf{a},\mathbf{b}}\sum_{s,\Lambda , \lambda} [\prod_{i=1}^{24}\mathcal{M} _{s(a_{i})}]^{vir}.$$ We wish to determine $[\prod_{i=1}^{24}\mathcal{M} _{s(a_{i})}]^{vir}$ in terms of tan\-gent-ob\-struct\-ion complexes on the $\mathcal{M} _{s(a_{i})}$'s. To specify a tan\-gent-ob\-struct\-ion complex for $\mathcal{M} _{s(a_{i})}$ it is enough to define a rank 2 bundle $T$ on $\Sigma(a_{i}) $ and use the above sequence to define $T^{1}$ and $T^{2}$. Let $T$ be the bundle on $\Sigma(a_{i}) $ that restricted to each component $\Sigma _{i}$ is $T\Sigma _{i}\oplus \mathcal{O}_{\Sigma _{i}}(-2)$ and restricted to $\Sigma _{*}$ is $T\Sigma _{*}\oplus \mathcal{O}_{\Sigma _{*}}(-1)$. Call the resulting tan\-gent-ob\-struct\-ion complex $[T^{1}\to T^{2}]_{s(a_{i})}$. Note that the bundle obtained by pulling back $TX$ by the fixed map $\Sigma(a_{i}) \to S\cup N_{i}$ is isomorphic to $T\otimes \mathcal{O}_{\Sigma _{*}}(1)$. In this subsection we show: \begin{lem}\label{lem: t-o complex is product of t-o complexs from fibers} The tan\-gent-ob\-struct\-ion complex defining $[\prod_{i=1}^{24}\mathcal{M} _{s(a_{i})}]^{vir}$ is isomorphic to the direct sum of tan\-gent-ob\-struct\-ion complexes $[T^{1}\to T^{2}]_{s(a_{i})}$. \end{lem} In the case at hand, $B$ is the hyperk\"ahler family $S(\mathcal{H}^{2}_{+,g})$ and so the tangent space to $t_{0}$ is the space perpendicular to $\omega _{t_{0}}$ in $\mathcal{H}^{2}_{+,g}$ which can be canonically identified with the space of holomorphic 2-forms. Thus $T_{t_{0}}B$ is canonically\footnote{There is a question how to orient the twistor family which corresponds to whether we identify the perpendicular to $\omega _{0}$ with $H^{0}(X,\mathcal{O})$ or $H^{0}(X,\mathcal{O})^{*}$. We choose the convention that makes the intersection of the twistor family with the set of projective $K3$'s positive.} $H^{0}(X,K)^{*}\cong H^{2}(X,\mathcal{O})$. It is convienent to use the exact sequence $$0\to \mathcal{O}\to \mathcal{O}(S)\to N_{S}\to 0 $$ to identify $H^{2}(X,\mathcal{O})$ with $H^{1}(X,N_{S})$ where $N_{S} $ is the normal bundle to $S$ in $X$. Since $TS$ is positive we can in fact identify $T_{t_{0}}B$ with $H^{1}(S,TX|_{S})$. We know that infinitesimal deformations of $f$ in the direction of the twistor family are obstructed (Proposition \ref{prop: one curve in class of C in twistor family}), and we wish to determine the image of $T_{t_{0}}B\to H^{1}(f^{*}(TX))$. Let $D'$ be the union of the components $D_{1},\ldots,D_{24}$ and $G_{1},\ldots,G_{g}$ in $D$ so that $D=D'\cup S$. Twisting the (partial) normalization sequence $$0\to \mathcal{O}_{D}\to \mathcal{O}_{S}\oplus \mathcal{O}_{D'}\to \mathcal{O}_{S\cap D'}\to 0 $$ by $f^{*}(TX)$ and taking cohomology, we see that $H^{1}(D,f^{*}(TX))$ surjects onto $H^{1}(S,f^{*}(TX))$. The composition $$T_{t_{0}}B\cong H^{1}(S,f^{*}(TX))\to H^{1}(D,f^{*}(TX))\to H^{1}(S,f^{*}(TX)) $$ is the identity. We can thus rewrite the sequence for $T^{1}$ and $T^{2}$ as \begin{eqnarray}\label{eqn: sequence for t-o complex without familiy term} 0\to \operatorname{Aut}(D)\to & H^{0}(D,f^{*}(TX))&\to T^{1}\to \\ \nonumber \to \operatorname{Def}(D) \to & H^{1}(D,f^{*}(TX))/H^{1}(S,f^{*}(TX))&\to T^{2}\to 0 . \end{eqnarray} This identifies our tan\-gent-ob\-struct\-ion complex with an equivalent tan\-gent-ob\-struct\-ion complex without a family: since the tan\-gent-ob\-struct\-ion complex only depends upon the restriction of $TX$ to the image of $f$, we can define a new tan\-gent-ob\-struct\-ion complex over $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ by dropping the dependence on the family and twisting the normal bundle of $S$ by $\mathcal{O}(1)$ so that $N_{S}=\mathcal{O}_{S}(-1)$. The above exact sequence shows that the resulting complex is isomorphic to $[T^{1}\to T^{2}]$ since the effect of changing the normal bundle of $S$ from $\mathcal{O}(-2)$ to $\mathcal{O}(-1)$ is solely to replace $H^{1}(D,f^{*}(TX))$ by $H^{1}(D,f^{*}(TX))/H^{1}(S,\mathcal{O}(-2))$. \begin{rem}\label{rem: obstr argumnet applies to E(1) also} The discussion of this section also applies to the rational elliptic surface where we consider ordinary Gro\-mov-Wit\-ten invariants (no family) but the normal bundle to the section is already degree $-1$. \end{rem} To complete the proof of the lemma we now use the (partial) normalization sequence $$0\to \mathcal{O}_{D}\to \mathcal{O}_{S}\oplus _{i} \mathcal{O}_{G_{i}}\oplus _{j} \mathcal{O}_{D_{j}}\to \oplus _{i} \mathcal{O}_{S\cap G_{i}}\oplus _{j} \mathcal{O}_{S\cap D_{j}}\to 0 $$ twisted by $f^{*}(TX)$. With care one can use the isomorphisms obtained from the cohomology sequence to see that the Sequence \ref{eqn: sequence for t-o complex without familiy term} is a direct sum of the sequences defining $[T^{1}\to T^{2}]_{s(a_{i})}$. The dependence on the $G_{j}$ components goes away: the $TX|_{S\cap G_{i}}$ term arising from the normalization sequence cancels with the infinitesimal deformation of $D$ that smooths the intersection of $S$ and $G_{i}$ and the new automorphism of $S$ that occurs when the intersection with $G_{i}$ is removed. Finally, we use the fixed maps $\Sigma(a_{i}) \to S\cup N_{i}$ to identify the $H^{*}(D,f^{*}(TX))$ terms with those terms $H^{*}(D,\til{f}^{*}(T))$ coming from maps $\til{f}:D\to \Sigma(a_{i}) $ in $\mathcal{M}_{s(a_{i})}$.\qed \subsection{Computations via blow-ups on $\P ^{2}$} In the previous subsection, we showed that the virtual fundamental cycle of a component $\prod_{i=1}^{24}\mathcal{M} _{s(a_{i})}$ (for any fixed $s(\mathbf{a})$, $\Lambda (\mathbf{b})$, and $\lambda (\mathbf{b})$) is given by the product of virtual fundamental cycles on $\mathcal{M} _{s(a_{i})}$ defined by the tan\-gent-ob\-struct\-ion complex $[T^{1}\to T^{2}]_{s(a_{i})}$. In this subsection, we will realize the moduli-obstruction problem $(\mathcal{M} _{s(a_{i})},[T^{1}\to T^{2}]_{s(a_{i})})$ as one coming from $\til{P}$, a certain blow-up of $\P ^{2}$ at $2a+3$ points. The homology classes of $\til{P}$ will have a diagonal basis $h$, $e_{-a-1},\ldots,e_{a+1}$ where $h^{2}=1$ and $e_{n}^{2}=-1$. We construct $\til{P}$ as follows. Begin with a linear $\cnums ^{*}$ action on $\P ^{2}$ fixing a line $H$ and a point $p$. Choose three points $p_{-}$, $p_{0}$, and $p_{+}$ on $H$ and blow them up to obtain three exceptional curves $E_{-1}$, $E_{0}$, and $E_{1}$ representing classes $e_{-1}$, $e_{0}$, and $e_{1}$. The proper transform of $H$ is a $(-2)$-curve $\Sigma_{0} $ in the class $h-e_{-1}-e_{0}-e_{1}$. The $\cnums ^{*}$ action extends to this blow-up acting with two fixed points on each of the curves $E_{-1}$, $E_{0}$, and $E_{1}$, namely the intersection with $\Sigma_{0} $ and one other. Blow-up the fixed points on $E_{-1}$ and $E_{1}$ that are not the ones on $\Sigma_{0} $ to obtain two new exceptional curves $E_{-2}$ and $E_{2}$ in the classes $e_{-2}$ and $e_{2}$. Let $\Sigma _{-1}$ and $\Sigma _{1}$ be the proper transforms of $E_{-1}$ and $E_{1}$ and note that they are $(-2)$-spheres in the classes $e_{-1}-e_{-2}$ and $e_{1}-e_{2}$ respectively. The $\cnums ^{*}$ action extends to this blow-up and we can repeat the procedure $a-1$ additional times to obtain $\til{P}$. $\til{P}$ contains $2a+1$ $(-2)$-spheres, namely $\Sigma _{-a},\ldots,\Sigma _{a}$ which represent the classes $$[\Sigma _{n}] =\begin{cases} e_{n}-e_{n+1}&\text{ if $0<n\leq a$,}\\ h-e_{0}-e_{-1}-e_{1}&\text{ if $a=0$,}\\ e_{n}-e_{n-1}&\text{ if $-a\leq n<0$.} \end{cases}$$ We rename the $(-1)$-spheres $E_{0}$, $E_{a+1}$, and $E_{-a-1}$ by $\Sigma _{*}$, $\Sigma _{a+1}$, and $\Sigma _{-a-1}$ and it is a straight forward computation to check that the classes $[\Sigma _{*}],[\Sigma _{-a-1}],\ldots,[\Sigma _{a+1}]$ form an integral basis for $H_{2}(\til{P};\znums )$. The configuration $\Sigma _{*}+\sum_{n=-a}^{a}\Sigma _{n}$ is (as our notation suggests) biholomorphic to $\Sigma (a)$. Furthermore, $T\til{P}|_{\Sigma (a)}$ is isomorphic to the bundle $T$ defining $[T^{1}\to T^{2}]_{s(a)}$. This will allow us to realize our obstruction problem as an ordinary Gromov-Witten invariant: \begin{lem}\label{lem: obstr problem is same as certain GW inv in blown up P2} $[\mathcal{M} _{s(a)}]^{vir}$ is the same as the (ordinary) genus 0 Gro\-mov-Wit\-ten invariant of $\til{P} $ in the class $$ [\Sigma _{*}]+\sum_{n=-a}^{a}s_{n}[\Sigma _{n}] . $$ \end{lem} {\sc Proof:} This follows immediately if we can show that {\em all} the rational curves in the above homology class lie in the configuration $\Sigma(a) $. Note that the curves $\Sigma _{*},\Sigma _{-a-1},\ldots,\Sigma _{a+1}$ are preserved by the $\cnums ^{*}$ action and the only other curves preserved are the proper transforms of lines through the fixed point $p$. We call these additional lines $\Sigma ^{+}$, $\Sigma ^{-}$, $\Sigma ^{0}$, and $\Sigma ^{t}$ which are the proper transforms of the lines $\overline{pp_{+}}$, $\overline{pp_{-}}$, $\overline{pp_{0}}$, and $\overline{pp_{t}}$ where $p_{t} $ is any point on $H$ that is not $p_{+}$, $p_{-}$, or $p_{0}$. See the figure: \vskip 1.25in \setlength{\unitlength}{0.00083333in} \begingroup\makeatletter\ifx\SetFigFont\undefined \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \fi\endgroup {\renewcommand{\dashlinestretch}{30} \begin{picture}(6733,4347)(0,-10) \put(49.077,2698.116){\arc{412.042}{4.5117}{7.6590}} \put(88.536,2373.307){\arc{413.025}{4.5198}{7.6518}} \put(170.500,2049.500){\arc{413.021}{4.5150}{7.6566}} \put(291.500,1401.500){\arc{413.021}{4.5150}{7.6566}} \put(372.808,1076.961){\arc{414.198}{4.5140}{7.6537}} \put(453.035,752.810){\arc{413.812}{4.5226}{7.6547}} \put(5802.849,2698.228){\arc{411.851}{1.7606}{4.9136}} \put(5763.578,2373.383){\arc{413.219}{1.7734}{4.9093}} \put(5681.500,2049.500){\arc{413.021}{1.7682}{4.9098}} \put(5560.500,1401.500){\arc{413.021}{1.7682}{4.9098}} \put(5479.115,1077.077){\arc{414.002}{1.7658}{4.9112}} \put(5399.077,752.885){\arc{414.003}{1.7706}{4.9065}} \put(4108.585,991.708){\arc{991.136}{2.1999}{3.9794}} \path(3210,4320)(130,2739) \path(616,753)(5236,753) \path(2642,4320)(5722,2739) \path(5236,753)(616,753) \path(2886,4320)(3777,1199) \path(2966,4320)(1912,428) \path(2716,686)(2841,186) \path(2774.209,576.859)(2716.000,686.000)(2716.000,562.307) \path(3091,4186)(3925,4270) \path(3207.390,4227.874)(3091.000,4186.000)(3213.402,4168.177) \put(2237,1239){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma ^{t}=h$}}}}} \put(3720,915){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{*}=e_{0}$}}}}} \put(5925,2739){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{a+1}=e_{a+1}$}}}}} \put(5844,2334){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{a}=e_{a}-e_{a+1}$}}}}} \put(5764,2009){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{a-1}=e_{a-1}-e_{a}$}}}}} \put(5844,1725){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\vdots$}}}}} \put(5641,1401){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{3}=e_{3}-e_{4}$}}}}} \put(5561,1036){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{2}=e_{2}-e_{3}$}}}}} \put(5520,712){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{1}=e_{1}-e_{2}$}}}}} \put(697,820){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{-1}=e_{-1}-e_{-2}$}}}}} \put(576,1442){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{-3}=e_{-3}-e_{-4}$}}}}} \put(616,1117){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{-2}=e_{-2}-e_{-3}$}}}}} \put(414,2009){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{-a+1}=e_{-a+1}-e_{-a}$}}}}} \put(373,2293){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{-a}=e_{-a}-e_{-a-1}$}}}}} \put(332,2576){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{-a-1}=e_{-a-1}$}}}}} \put(535,1766){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\vdots$}}}}} \put(10,3752){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma ^{-}=h-e_{-1}-\cdots -e_{-a-1}$}}}}} \put(3494,2617){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma ^{0}=h-e_{0}$}}}}} \put(3940,3752){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma ^{+}=h-e_{1}-\cdots -e_{a+1}$}}}}} \put(4023,4272){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$p$}}}}} \put(2885,19){\makebox(0,0)[lb]{\smash{{{\SetFigFont{10}{8.4}{rm}$\Sigma _{0}=h-e_{-1}-e_{0}-e_{1}$}}}}} \end{picture} } We express the classes of $\Sigma ^{+}$, $\Sigma ^{-}$, $\Sigma ^{0}$ and $\Sigma ^{t}$ in the two homology bases $\{h,e_{-a-1},\ldots,e_{a+1} \}$ and $\{[\Sigma _{*}],[\Sigma _{-a-1}],\ldots,[\Sigma _{a+1}] \}$ as follows: \begin{eqnarray*} \left[ \Sigma ^{t}\right] &=&h=\left[ \Sigma _{*}\right] +\sum_{n=-a-1}^{a+1}\left[ \Sigma _{n}\right] \\ \left[ \Sigma ^{0} \right] &=&h-e_{0}=\sum_{n=-a-1}^{a+1}\left[ \Sigma _{n}\right] \\ \left[ \Sigma ^{+} \right] &=&h-e_{1}-\cdots -e_{a+1}\\ &=&\left[ \Sigma _{*}\right] +\sum_{n=-a-1}^{a+1}\left[ \Sigma _{n}\right] -\sum_{n=1}^{a+1}n\left[ \Sigma _{n}\right] \\ \left[ \Sigma ^{-}\right] &=&h-e_{-1}-\cdots -e_{-a-1}\\ &=&\left[ \Sigma _{*}\right] +\sum_{n=-a-1}^{a+1}\left[ \Sigma _{n}\right] -\sum_{n=1}^{a+1}n\left[ \Sigma _{-n}\right] . \end{eqnarray*} Since $\cnums ^{*}$ acts on $\til{P}$ we get an $\cnums ^{*}$-action on the moduli space $\mathcal{M} ^{\til{P}}_{\{s_{n} \}}$ of genus 0 stable maps to $\til{P}$ in the class $[\Sigma _{*}]+\sum_{n=-a}^{a}s_{n}[\Sigma _{n}]$. We first show that the maps in the fixed point set of $\mathcal{M} ^{\til{P}}_{\{s_{n} \}}$ must have image $\Sigma _{*}+\sum_{n=-a}^{a}s_{n}\Sigma _{n}$. This is essentially for homological reasons: the image of a map in the fixed point set of $\mathcal{M} ^{\til{P}}_{\{s_{n} \}}$ must be of the form $$c_{*}\Sigma _{*}+\sum_{n=-a-1}^{a+1}c_{n}\Sigma _{n}+\sum_{t} c^{t}\Sigma ^{t}+c^{+}\Sigma ^{+}+c^{-}\Sigma ^{-} +c^{0}\Sigma ^{0}$$ for non-negative coefficients given by the $c$'s. Since $[\Sigma _{*}],[\Sigma _{-a-1}],\ldots,[\Sigma _{a+1}]$ form a basis we have \begin{eqnarray*} c_{*}+\sum_{t}c^{t}+c^{+}+c^{-}&=&1,\\ c_{n}+c^{0}+\sum_{t}c^{t}+c^{+}+c^{-}-|n|c^{\operatorname{sign}(n)}&= &\begin{cases} s_{n}&|n|\leq a,\\ 0&|n|=a+1. \end{cases} \end{eqnarray*} The first equation implies that exactly one of $c_{*}$, $c^{t}$, $c^{+}$, or $c^{-}$ is 1 (for some $t$) and the rest are 0. Suppose that $c^{+}=1$; then $c^{-}=0$ and letting $n=-a-1$ in the second equation leads to a contradiction and so we have $c^{+}=0$. A similiar argument shows $c^{-}=0$ and then summing the second equation over $n$ leads to $$\left(\sum_{n=-a-1}^{a+1}c_{n} \right)+(2a+3)(c^{0}+\sum_{t}c^{t})=a $$ which implies that $c^{0}=c^{t}=0$. Thus $c_{*}=1$ and $c_{n}=s_{n}$. Finally, suppose $f\in \mathcal{M} ^{\til{P}}_{\{s_{n} \}}$ is not a fixed point of the $\cnums ^{*}$ action. Then the limit of the action of $\lambda \in \cnums ^{*}$ on $f$ as $\lambda \to 0$ must be fixed an hence has image $\Sigma _{*}+\sum_{n=-a}^{a}s_{n}\Sigma _{n}$. But then the limit of the action as $\lambda \to \infty $ must also be fixed and its image must contain the point $p$ which is a contradiction. Hence every $f\in \mathcal{M} ^{\til{P}}_{\{s_{n} \}}$ is fixed by $\cnums ^{*}$ and so has image $\Sigma _{*}+\sum_{n=-a}^{a}s_{n}\Sigma _{n}$. \qed Now $\til{P}$ is deformation equivalent to the blow-up of $\P ^{2}$ at $2a+3$ generic points and so the invariant for the class $[\Sigma _{*}]+\sum_{n=-a}^{a}s_{n}[\Sigma _{n}]$ can be computed using elementary properties of the invariants for blow-ups of $\P ^{2}$. We follow the notation of \cite{Go-Pa} and recall some of the properties of the invariant. We write $N{(d;\alpha _{1},\ldots)}$ for the genus 0 Gro\-mov-Wit\-ten invariant in the class $dh-\sum_{i}\alpha _{i}e_{i}$ . Here we are not being very picky about the indexing set for the exceptional classes since the invariant is the same under reordering. In the notation $N{(d;\alpha _{1},\ldots)}$ it is implicit that if the moduli space of genus 0 maps in the class $(d;\alpha _{1},\ldots)$ is positive dimensional then we impose the proper number of point constraints and if the dimension is negative the invariant is zero. Also, we drop any $\alpha =0$ terms from the notation so that $N(d;\alpha _{1},\ldots,a_{l},0,\ldots,0)=N(d;\alpha _{1},\ldots,\alpha _{l})$. The invariants satisfy the following properties: \begin{enumerate} \item $N{(d;\alpha _{1},\ldots)}=0$ if any $\alpha <0$ unless $d=0$, $\alpha _{i}=0$ for all $i$ except $i_{0}$ and $\alpha _{i_{0}}=-1$. In the latter case the invariant is 1. \item $N{(d;\alpha _{1},\ldots,\alpha _{l},1)}=N{(d;\alpha _{1},\ldots,\alpha _{l} )}$. \item $N{(d;\alpha _{1},\ldots,\alpha _{l})}=N{(d;\alpha _{\sigma (1)},\ldots,\alpha _{\sigma (l)})}$ for any permutation $\sigma $. \item $N{(d;\alpha _{1},\ldots,\alpha _{l})}$ is invariant under the Cremona transformation which takes the class $$(d;\alpha _{1},\alpha _{2},\alpha _{3},\ldots)$$ to the class $$(2d-\alpha _{1}-\alpha _{2}-\alpha _{3};d-\alpha _{2}-\alpha _{3},d-\alpha _{1}-\alpha _{3},d-\alpha _{1}-\alpha _{2},\ldots).$$ \item $N{(1)}=1$. \end{enumerate} Ordering the exceptional classes in $\til{P}$ by $e_{0},e_{1},e_{-1},e_{2},e_{-2},\ldots$ and rewriting the class $\Sigma _{*}+\sum s_{n}\Sigma _{n}$ in this basis, we can express can express the contribution of $\mathcal{M} _{s(a)}$ as $$ [\mathcal{M}_{s(a)}]^{vir}= N{(s_{0};s_{0}-1,s_{0}-s_{1},s_{0}-s_{-1},s_{1}-s_{2},s_{-1}-s_{-2}, \ldots,s_{-a+1}-s_{-a})}. $$ We call an admissible sequence $\{s_{n} \}$ {\em 1-admissible} if $s_{\pm n\pm 1}$ is either $s_{\pm n}$ or $s_{\pm n}-1$ for all $n$. \begin{lem}\label{lem: contribution is 1 for 1-admissable, 0 otherwise} $[\mathcal{M} _{s(a)}]^{vir}=1$ if $s(a)$ is a 1-admissible sequence and $[\mathcal{M} _{s(a)}]^{vir}=0$ otherwise. \end{lem} {\sc Proof:} Suppose that $[\mathcal{M}_{s(a)}]^{vir}\neq 0$. Since $s_{0}>0$, all the other terms in $(s_{0};s_{0}-1,s_{0}-s_{1},\ldots)$ must be non-negative by property 1. Thus $s_{\pm n\pm 1}\geq s_{\pm n}$ for all $n$. Now by permuting and performing the Cremona transformation, we get \begin{eqnarray*} [\mathcal{M}_{s(a)}]^{vir}&=&N{(s_{0};s_{0}-1,s_{0}-s_{1},s_{\pm n}-s_{\pm n\pm 1},\ldots)}\\ &=&N(1+s_{1}+s_{\pm n\pm 1}-s_{\pm n};s_{1}-s_{\pm n}+s_{\pm n\pm 1}, \\ & &\quad 1+s_{\pm n\pm 1}-s_{\pm n},s_{1}+1-s_{0},\ldots). \end{eqnarray*} Now since $s_{\pm n}\leq s_{1}$, we have $1+s_{1}+s_{\pm n\pm 1}-s_{\pm n}>0$ and so $1+s_{\pm n\pm 1}-s_{\pm n}\geq 0$ which combined with $s_{\pm n\pm 1}\leq s_{\pm n}$ yields $$s_{\pm n\pm 1}\leq s_{\pm n}\leq s_{\pm n\pm 1} $$ and so $s$ is 1-admissible. Suppose then that $s$ is 1-admissible. Then except for the first two terms, the class $(s_{0};s_{0}-1,s_{0}-s_{1},\ldots)$ consists of $0$'s and $1$'s. Thus $[\mathcal{M}_{s(a)}]^{vir}=N{(s_{0};s_{0}-1)}$. Finally, since $N{(s_{0};s_{0}-1)}=N{(s_{0};s_{0}-1,1,1)}$ we can apply Cremona to get $$N{(s_{0};s_{0}-1)}=N{(s_{0}-1;s_{0}-2)} $$ and so by induction $$[\mathcal{M}_{s(a)}]^{vir}=N{(s_{0};s_{0}-1)}=N{(1)}=1 $$ and the lemma is proved.\qed \begin{lem}\label{lem: no. of 1-admissable seqs is p(a)} The number of 1-admissible sequences $s$ with $|s|=a$ is the number of partitions of $a$, $p(a)$. \end{lem} {\sc Proof:}\footnote{We are grateful to D. Maclagan and S. Schleimer for help with this and other combinatorial difficulties.} The number of partitions, $p(a)$, is given by the number of Young diagrams of size $a$. There is a bijective correspondence between 1-admissible sequences and Young diagrams. Given a Young diagram define an 1-admissible sequence $\{s_{n} \}$ by setting $s_{0}$ equal to the number of blocks on the diagonal, $s_{1}$ equal to the number of blocks on the first lower diagonal, $s_{2}$ equal to the number of blocks on the second lower diagonal, and so on, doing the same for $s_{-1}$, $s_{-2}$,\dots with the upper diagonals. It is easily seen that this defines a bijection.\qed Summarizing the results of this section we have: \begin{thm}\label{thm: contribution of Mab} Since every component of $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ contributes either 0 or 1 to $N_{g}(n)$, the overall contribution of $\mathcal{M} _{\mathbf{a},\mathbf{b}}$ is the sum over all the connected components whose contribution is 1. It is thus the sum of all choices of data $\Lambda (\mathbf{b})$ and $\lambda (\mathbf{b})$, and those choices of $s(\mathbf{a})$ that are 1-admissible. For each $a_{j}\in \mathbf{a}$ we have $p(a_{j})$ choices of a 1-admissible sequence $s(a_{j})$ and for each $b_{i}\in \mathbf{b}$ we have $b_{j}\sigma (b_{i})$ choices for the data $\Lambda (b_{i})$ and $\lambda (b_{i})$. Thus the total contribution is: $$[\mathcal{M} _{\mathbf{a}, \mathbf{b}}]^{vir}=\prod_{j=1}^{24}p(a_{j})\prod_{i=1}^{g}b_{i}\sigma (b_{i}). $$ \end{thm} \section{Counting curves on the rational elliptic surfaces}\label{sec: counting on E1} Let $Y$ be the blow up of $\mathbf{P}^{2}$ at nine distinct points. In this section we apply our degeneration method and our local calculations to compute a certain set of Gro\-mov-Wit\-ten invariants of $Y$. We compute the genus $g$ invariants for all classes such that the invariants require exactly $g$ constraints. There is a canonical symplectic form $\omega $ (unique up to deformation equivalence) on $Y$ determined by the blow up of the Fubini-Study form on $\mathbf{P}^{2}.$ If we arrange these nine blow up points lying on a pencil of cubic elliptic curves in $\mathbf{P}^{2},$ then $Y$ has the structure of an elliptic surface with fiber class $F$ representing these elliptic curves in $H_{2}\left( Y,\mathbf{Z}\right) $ and the nine exceptional curves $e_{1},e_{2},...,e_{9}$ are all sections of this elliptic fibration. If $h$ represents the homology class of the strict transform of the hyperplane in $\mathbf{P}^{2},$ then we have $F=3h-e_{1}-...-e_{9}$. In fact $H^{2}\left( Y,\mathbf{Z}\right) $ is generated by $e_{1},...,e_{9}$ and $h$. We abbreviate the class $dh-a_{1}e_{1}-\cdots -a_{9}e_{9}$ by $(d;a_{1},\ldots,a_{9})$. Now we pick any of these sections, $e_{9}$ say, and consider the class $C_{n}= e_{9}+( g+n) F=(3(n+g);g+n,\ldots,g+n,g+n-1)$. It is easy to check that the complete linear system $|C_{n}|$ has dimension $g+n.$ We write $N_{g}^{Y}(C ) $ the Gro\-mov-Wit\-ten invariant for $\left( Y,\omega \right) $ which counts the number of curves of geometric genus $g$ representing the homology class $C$ and passing through $g$ points. {\em i.e. } we define $$N_{g}^{Y}(C)=\Psi ^{Y}_{(C,g,g)}(PD(x_{1}),\ldots,PD(x_{g});PD(\overline{\mathcal{M} }_{g,g})) . $$ We show that the numbers $N_{g}^{Y}(C_{n})$ contain all the genus $g$ Gro\-mov-Wit\-ten invariants that are constrained to exactly $g$ points. This was observed by G\"ottsche who explained the following argument to us: For $N_{g}^{Y}(C)$ to be well defined (see Equation \ref{eqn: dimension formula}) we need $4g=2c_{1}(Y)\cdot C+2g-2(1-g)$, {\em i.e. } $F\cdot C=1$. Now the Gro\-mov-Wit\-ten invariants do not change when $C\mapsto C'$ is induced by a permutation of the exceptional classes $e_{i}$ or a Cremona transform (see \cite{Go-Pa}). Recall that the Cremona transform takes a class $(d;a_{1},\ldots,a_{9})$ to the class $$ (2d-a_{1}-a_{2}-a_{3};d-a_{2}-a_{3}, d-a_{1}-a_{3},d-a_{1}-a_{2},a_{4},\ldots,a_{9}). $$ \begin{lem}\label{lem: lothar's lemma} Let $C\in H_{2}(Y;\znums )$ be a class so that the moduli space of genus $g$ maps has formal dimension $g$. Then the class $C$ can be transformed by a sequence of Cremona transforms and permutations of the $e_{i}$'s to a class of the form $e_{9}+(g+n)F=C_{n}$. \end{lem} {\em Proof:} By permuting the $E_{i}$'s we may assume that $a_{1}\geq a_{2}\geq \cdots \geq a_{9}$. Then the condition $F\cdot C=1$ is equivalent to $3d-1=\sum_{i}a_{i}$ so that $a_{1}+a_{2}+a_{3}\geq d$ with equality if and only if $C=(3i,i,i,i,i,i,i,i,i,i-1)=e_{9}+iF$ for some $i=n+g$. If the equality is strict then we can apply a Cremona transform to obtain $C'=(e,b_{1},\ldots,b_{9})$ with $e<d$. The result follows by descending induction on $d$.\qed The methods of Sections \ref{sec: computation of Ngn} and \ref{sec: analysis and local computations} apply to these invariants (see remark \ref{rem: obstr argumnet applies to E(1) also}). Note that the elliptic fibration of $Y$ has (generically) 12 nodal fibers rather than 24. We get the same formula as in the $K3$ case with the 24 replace by 12. \begin{theorem}\label{thm: E1 calculation} For any $g\geq 0,$ we have \begin{eqnarray*} \sum_{n=0}^{\infty }N_{g}^{Y}(C_{n}) q^{n} &=&\left(\sum_{b=1}^{\infty }b\sigma (b)q^{b-1} \right)^{g}\prod_{m=1}^{\infty }\left( 1-q^{m}\right) ^{-12} \\ &=&\left(\frac{d}{dq}G_{2}(q) \right)^{g}\left(\frac{q}{\Delta (q)} \right)^{1/2} \end{eqnarray*} \end{theorem} When the genus $g$ equals zero, these numbers are computed by G\"{o}ttsche and Pandharipande \cite{Go-Pa}. In fact, they obtain all genus zero Gro\-mov-Wit\-ten invariants for $\mathbf{P}^{2}$ blown up at arbitrary number of points in terms of two rather complicated recursive formulas. Theorem \ref{thm: E1 calculation} can be verified term by term for $g=0$ using the recurrence relations, although the computer calculation becomes extremely lengthy quickly. We know of no way of obtaining the genus 0 closed form of Theorem \ref{thm: E1 calculation} directly from the recurrence relations.

9711

alg-geom/9711029

en

https://arxiv.org/abs/alg-geom/9711029

alg-geom/9711029

Terutake Abe

Terutake Abe (Johns Hopkins University)

Classification of Exceptional Complements: Elliptic Curve Case

LaTeX2e, 14 pages, with 24 postscript figures

We classify the log del Pezzo surface (S,B) of rank 1 with no 1-,2-,3-,4-, or 6-complements with the additional condition that B has one irreducible component C which is an elliptic curve, and that C has the coefficient b in B with (1/n)floor((n+1)b)=1 for n=1,2,3,4, and 6.

alg-geom

\subsection{The dual graph of a configuration} In the following we use the language of graphs to talk about the the configuration of curves. The dual graph of a configuration of curves is a (weighted-multi) graph where we have a vertex for each curve and an n-ple edge for each intersection point with multiplicity $n$ between two curves. Each vertex has a weight $\in {\mathbb Z}$ which is the self-intersection number of the curve. Graphically, we use $\bullet$ (``b(lack)-vertex'') to represent exceptional curves with self-intersection number $\le -2$, $\circ$ (``w(hite)-vertex'') for $(-1)$-curves, and squares for curves with non-negative self-intersection. The weight of a vertex is shown by a number next to each vertex, and multiplicity of an edge by the number of lines joining the two end vertices. ``Blow up of an edge'' means the transformation of the graph reflecting the blow up of the corresponding point, that is, introduce a new white vertex, decrease the multiplicity of edge by 1, decrease the weight of the both end vertices of the edge by 1, and join them to the new white vertex by a simple edge. ``Blow up of a vertex'' reflects the blow up of a point on the curve outside the intersection with neighboring curves: introduce a white vertex, decrease the weight of the vertex by 1, and join it to the new white vertex by a simple edge. Blow up of a complete subgraph of any cardinality $k$ can be defined in the same way. \subsection{Types of Singularities on $C$} \begin{lemma} \begin{itemize} \item[(i)] The singularity of $C$ is at worst a node, and it is outside $Sing(S) \cup Supp(B_1)$. \item[(ii)] At most one component $C_i$ of $B_1$ passes through each point $P\in C$. If $P$ in a smooth point of $S$, then the intersection is normal ,with one possible exception where $C_i$ has coefficient $\frac12$ and has a simple tangency with $C$ at a smooth point $P$ of $S$. \item[(iii)] Singularity $P$ of $S$ on $C$ is a cyclic quotient singularity, i.e. log terminal singularities with resolution graph ${\mathbb A}_n$ (a chain), where $C$ meets one end curve of the chain normally. If another component $C_i$ passes through $P$, then it meets the other end curve normally. \end{itemize} \end{lemma} \proof Note that $K+D$, as defined above, is log canonical by the existence of local complements ([Sh1, Cor.5.9.]). Then all the statements follow from the classification of surface log canonical singularities ([Ka],or [Al2]) and $\frac17$-log terminal condition. For example, for (iii), if we had a type ${\mathbb D}_n$ singularity, (case (6) in [Ka, Th.9.6]) we would have a log decrepancy $\le \frac17$. Note also that the exception in (ii) is the only case where $K+D$ is not log terminal at $P$ ([Sh2, Prop.5.2]). \qed As is well known, the singularities mentioned above are isomorphic, analytically, to the origin 0 in the quotient of ${\mathbb C}^2$ by the action of cyclic group $\mu_m$ of order $m$, where the generator $\varepsilon = e^{\frac{2\pi i}{m}}$ acts by $$ (z_1, z_2) \mapsto (\varepsilon^{-k}\cdot{z_1}, \varepsilon\cdot z_2), \text{ where } 1 \le k \le m \text{ and } gcd(m,k)=1.$$ The minimal resolution of such a singularity has a chain of rational curves $E_1,E_2,\cdots,E_r)$ as its exceptional locus, and the co continued fraction expansion $$\frac{m}{m-k} = w_1 - \cfrac{1}{w_2- \cfrac{1}{w_3-\dotsb}}$$ \noindent give their self intersection numbers ({\it cf.} for example, [Ful]). We call such a singularity $P$ type $[m,k]$. We extend this correspondence to incorporate the information on the component $C_i$ that passes through $P$ ({\it cf.} [Sh1,Cor.3.10], [Sh2,Lemma 2.22]). Namely, if the component $C_i$ has the standard coefficient $\frac{d-1}{d}$ and the singularity $P$ has type $(m^\prime, k^\prime)$, we represent it by the pair $(m,k)=(dm^\prime,dk^\prime)$. The ``dual graph'' of the minimal resolution of this singularity is: \begin{figure}[h] \begin{center} \includegraphics{logterm.eps} \caption {} \end{center} \end{figure} \noindent Generalizing the notation of [KM], we may denote the same singularity by $(\underline{w_1},w_2, \cdots, w_r)_d$ with the underline indecating the curve meeting $C$. This singularity has the minimal log discrepancy $$mld(P,K+B)=a(E_1)=\tfrac{1+(m-k)(1-b)}{m} \text{ }(\le \tfrac{1+\frac17(m-k)}{m}), $$ where $m = d \cdot \text{(index of $P$)}$. Also we denote the co-discrepancy, or the coefficient, of $P$ by $e(P,K+B)=1-mld(P,K+B)$. Now the $\frac17$-log terminal condition $$\tfrac{1+\frac17(m-k)}{m} > \tfrac17$$ is equivalent to $k < 7$. Therefore the possible singularities on $C$ are put into 21 $=6(6+1)/2$ (infinite) series according to the pair $(m (\text{mod }k),k)$ with $1\le k\le 6$. This will be convenient later on. \section {Elliptic Curve Case} Now we start the classification of the case $p_a(C) = 1$. Thus, $C\in S$ is a smooth curve of genus 1 or a rational curve with one node. We call it the ``elliptic curve case''. \begin{lemma} In the ``elliptic curve case'', the condition (EX2) is equivalent to the condition that $(S,B)$ has log-singularities on $C$. That is, either $S$ has singularities on $C$, or $B$ has components other than $C$ (which intersect $C$ since $\rho(S)=1$). \end{lemma} \proof If $(S,B)$ is smooth on $C$ , then $(K+f^*(D)).C=(K+C).C = 0$ on $\uppermin{S}$, so $K+D=K+C\sim 0$ on $S$ and (EX2) is not satisfied. In fact $K+B=K+bC$ has a 1-complement. On the other hand if $(S,B)$ has a singularity on $C$, then $(K+f^*(D)).C>(K+C).C=0$ so we have $K+D>0$ on $S$, which implies (EX2). \qed The case when $S$ is a cone (${\mathbb P}^2 \text{ or } {\mathbb Q}_m$) has been classified elsewhere and from it we have only one case with $C$=elliptic: $S={\mathbb F}_2, C=$double section,$B_1=\frac12C_2$ where $C_2$ is a generator of the cone. Then $C\equiv 2H \equiv -K, C_2\equiv \frac12H$. So $K+\frac67C+\frac47C_2\equiv 0$ and $K+B$ has 7-complement $= 0$. It also has the trivial 8-complement: $K+\frac78 C+ \frac12 C_2 \equiv 0$ (This is the entry $\sharp$1 in the table at the end). From now on we assume $S$ is not a cone. \begin{lemma} $C^2 \ge 3$ on $S^{\text{\rm min}}$. If $(S,B)$ has two singularities on $C$, then $C^2 \ge 6$. On the other hand, the minimum log discrepancy of the singularity $P$ on $C$ with respect to $K+B$ (hence also with respect to $K+bC$) is at least $1-(C^2/7)$. \end{lemma} \proof Because $-(K+B)$ is nef, \begin{align*} 0\geq (K+B).C &= K_{S^{\text{min}}} +bC +\Sigma b_iC_i + \Sigma d_jE_j \cdot C\\ &\geq -(1-b)C^2 + \Sigma b_i + \Sigma_{P}(1-mld(P))\\ &\geq -(1-b)C^2 + min\{b_i, 1-mld(P)\} \\ &\geq -\tfrac17 C^2 + \tfrac37. \end{align*} Note that, because of Lemma 1.1, $1-mld(P)=d_j$ for the exceptional curve $E_j$ meeting $C$. The last inequality holds because we have at least one nonzero $b_i$ or $1-mld(P)$ by Lemma 2.1 and the minimum nonzero value for $b_i$ is $\frac12$, that for $1-mld(P)$ is $\frac12 \cdot \frac67 = \frac37$, the latter being attained when $P$ in duVal of type ${\mathbb A}_1$. Therefore, $C^2 \geq 3$. By the same calculation, if there are two singularities on $C$ we have $0\geq -\frac17 C^2 + \frac67$. On the other hand, the second inequality in particular implies that $1-mld(P) \le (1-b)C^2 \le \tfrac17 C^2$, whence the second assertion. $\qed$ \subsection{Reduction to ${\mathbb F}_2$} We need the following \begin{lemma} Let $E$ be a $(-1)$-curve on $\uppermin{S}$. Then on its image $f_*(E)$, $S$ has either at least two singularities, or one singularity that is not log-terminal for $K+E$. \end{lemma} \proof If , on $E$, $S$ had at most one singularity $P$ that is log-terminal for $K+E$, i.e. a cyclic quotient singularity such that $E$ meets one end curve $E_1$ of the chain of the resolution, then we would have $$ (f_*(E))^2=E.f^*f_*(E)=E.(E+(1-a(E_1))E_1)=-1 + (1 -a(E_1)) < 0$$ which is absurd since $\rho(S)=1$. \qed \medskip Now we can prove the \begin{lemma} We can always obtain ${\mathbb F}_2$ as a smooth model of $S$ (and $C$ as a double section). \end{lemma} \proof $p_a(C)=1$ means that after reconstruction, $C$ is either a cubic in ${\mathbb P}^2$, curve of bidegree $(2,2)$ on ${\mathbb F}_0$, or a double section of ${\mathbb F}_2$. Suppose $S^\prime$ is ${\mathbb P}^2$ and $C$ is a cubic, since there are no irreducible curves with arithmetic genus 1 on ${\mathbb F}_m$, with $m\ge 3$. If $g: S^{\text{min}} \to {\mathbb P}^2$ contracts two or more exceptioncal curves to a point $P \in {\mathbb P}^2$, then we can choose different contractions to get $S^\prime = {\mathbb F}_2$. Therefore we may assume that we have only one exceptional curve for $g$ over each center $P \in {\mathbb P}^2$, and we shall derive a contradiction. Since all the curves contracted by $g$ are $(-1)$-curves on $S^{\text{min}}$, no exceptional curve $E_i$ for $f$ are contracted and all of them are present on ${\mathbb P}^2$ as divisors. Thus we have an inequality $$ 0 \geq deg(K_{{\mathbb P}^2}+bC+B_1^\prime) = -3+\tfrac67\cdot 3 + deg(B_1^\prime) = -\tfrac37 + deg(B_1^\prime) $$ Therefore, since the coefficients are standard, no component $C_i$ other than $C$ are present on ${\mathbb P}^2$. And we have \begin{equation} \Sigma d_j \le \Sigma d_j\cdot deg(E_j) = deg(B_1^\prime) \le \tfrac37 \tag{$*$} \end{equation} We have the two possibilities: (1) $B$ has at least one component, say $C_2$, other than $C$. Then by the above, $C_2$ must be contracted on ${\mathbb P}^2$ and is a $(-1)$-curve on $S^{\text{min}}$. Therefore, by Lemma 2.3, $S$ must have either at least two singularities on $C_2$, or a singularity that is not log-terminal for $K+C_2$. In the former case, then, we would have $\Sigma d_j \ge (\frac12 + \frac12)b_2 \ge \frac12 > \frac37$, contradicting ($*$). In the latter case, we have an exceptional curve $E$ with $a(E,K+C_2)\le 0$. Then because $a(E, K+b_2C_2)$ is a linear function of $b_2$ and we also have $a(E,K+0\cdot C_2)=a(E,K) \le 1$, we have $a(E,K+b_2 C)\le 1-b_2$. Thus $d_2 = 1-a(E,K+B) \ge 1-a(E,K+b_2C_2) \ge b_2 \ge \frac12 > \frac37$, again a contradiction to ($*$). (2) $B$ has no other components than $C$, i.e. $B=bC$, and $S$ has a singularity on $C$. Then ($*$) implies that and we have $K+B > 0$ except in the following case: $S$ has only one duVal singularity $P$ of type ${\mathbb A}_1$ on $C$, the exceptional curve $E_1$ of the resolution of $P$ is a line on ${\mathbb P}^2$, $b=\frac67$, and $B_1^\prime$ has no other component than $E_1$, so that $K_{{\mathbb P}^2}+ B^\prime = K + \frac67 C + \frac37 E_1 \sim 0$. In particular all the singularities on $S$ are duVal so $$f^*(K+B)=K_{\uppermin{S}}+\uppermin{B} = K+\frac67 C+ \frac37 E_1$$ Also, the triviality of $K+B$ means that pull back $g^*$ is crepant so that the above is also equal to $g^*(K+B^\prime).$ On the other hand, since $E_1.C = 3$ on ${\mathbb P}^2$ and $E_1.C=1$ on $S^{\text{min}}$, two of the intersection points of $E_1$ and $C$ has to be blown up on $S^{\text{min}}$. The exceptional curve $E$ for the first of such blowups would have the coefficient $\frac67 + \frac37 - 1 = \frac27$ in $K_{S^{\text{min}}}+B^{\text{min}} = g^*(K+B^\prime)$. Contradicting the explicit form of $\uppermin{B}$ given above. If we have a model $S^\prime = {\mathbb F}_0$, then we have had at least one contraction of $(-1)$-curve so we can get $S^\prime = {\mathbb P}^2$ by choosing other contractions, and we are reduced to the previous case. \qed Therefore we have a ${\mathbb P}^1$-fibration $p:S^{\text{min}}\to {\mathbb F}_2\to {\mathbb P}^1$. Now our strategy for the classification is to start from ${\mathbb F}_2 = S^\prime$, make blow ups to construct $S^{\text{min}}$, choose $B^{\text{min}}$ on it so that resulting $(S,B)$ would have singularities on $C$ ($\Leftrightarrow$ (EX2) by Lemma 2.1) and would satisfy (EX1),(EX3), and (EX4). The conditions (EX1) and (EX4) implies that the number of $(-n)$-curves, $n\ge 2$, on $S^{\text{min}}$ must equal $\rho(S^{\text{min}})-1$, and they are all exceptional for the resolution $f$. These curves are either in the fibres of $p$, or they are not, i.e. they are (multi-) sections of $p$. As for the number of curves of each type, we have the following: \begin{lemma} {\rm (\lbrack Zhang, Lemma 1.5 \rbrack)} We have \begin{eqnarray*} r & = & \sharp\text{\{Exceptional curves $E_i$'s of the resolution f that are not in the fibres of $p$\}} - \text{ 1 } \\ & = & \sharp\text{\{$(-1)$-curves on $S^{\text{min}}$ that are in the fibres of $p$.\}} \\ & & -\ \ \sharp\text{\{Singular fibres of p\}} \end{eqnarray*} \end{lemma} \proof Add ($2 + \sharp\{E_i$'s that are in the fibres of $p$\}) to both sides, and we get two expressions for $\rho(S^{\text{min}})$.$\qed$\ \subsection{The search for exceptions} \begin{case} $r=0$, i.e. minimal section $\Sigma$ on ${\mathbb F}_2$ is the only $E_i$ with $p(E_i) = {\mathbb P}^1$. \end{case} Then there is only one $(-1)$-curve in each singular fibre of $p$. Therefore on each fibre $F$ modified we have to have initially two blow ups at the same point $P$. Suppose $C^2 = w$ before the modification, then according as the intersection multiplicity $i=I(P;F\cap C)=2,1,$ or $0$, i.e. according as $P =$ tangency of $F$ and $C$, normal intersection of $F$ and $C$, or $P \in F \setminus (F\cap C)$, we get one of the three dual graphs in the Figure 2 below. \begin{figure}[h] \begin{center} \begin{tabular}{p{1.0in}p{0.5in}p{1.0in}p{0.5in}p{1.0in}} \includegraphics{blowup1.eps} & \hspace{0.5in} & \includegraphics{blowup2.eps} & \hspace{0.5in} & \includegraphics{blowup3.eps} \\ \mbox{(I)} & & \mbox{(II)} & & \mbox{(III)} \end{tabular} \caption{} \end{center} \end{figure} In the figure the b-vertex at the bottom is the minimal section $\Sigma \in {\mathbb F}_2$. In the case (III), the curve C and neighboring $(-2)$-curve ($=F$) have either two normal intersections, one simple tangency, or $C$ has a node on $F$. Case(III) gives a non log canonical point ({\it cf.} Lemma 1.1(i)) and is excluded. Case(II) gives one example with trivial complement (entry $\sharp2$ in the table at the end): \begin{align*} S &= \text{Gorenstein del Pezzo surface with singularities ${\mathbb A}_1+{\mathbb A}_2$}, \\ C &= \text{elliptic curve through ${\mathbb A}_1$ and ${\mathbb A}_2$ points},\\ K+&B = K+\frac67C\equiv 0 \\ 7(K_{S^{\text{min}}}+ & B^{\text{min}})= 7(K + \frac67C + \frac37E_1 + \frac47E_2 + \frac27E_2) \sim 0 \end{align*} \noindent (Following [MZ], we denote the Gorenstein del Pezzo surfaces of rank 1 by its singularity type, for example, $S(A_1+A_2)$ for the surface above, and their resolution by e.g. $\tilde{S}(A_1 + A_2)$.) Since we already have $K+ B \equiv 0$, if we make any more blow ups (which have to be on the unique $(-1)$-curve) or add other components to $B$, we would have $K+B > 0$ and $(S,B)$ will violate (EX1). So we need not consider this case any longer. Thus we are left with case (I), i.e. two initial blow ups at the ramification point of $C \to {\mathbb P}^1$ (tangency of $C$ and a fibre). In particular, in all the remaining cases, $C^2 \le 6$, because $C^2=8$ on ${\mathbb F}_2$. This implies that a smooth fibre $F$ cannot be a component of $B_1^\prime$, because if it were, we would have $0 \ge (K+bC+B_1).C \ge -(1-b)C^2 + \frac12 F.C \ge -\frac67+\frac12\times 2 = \frac17$, a contradiction. Therefore only singularities on $C$ are those coming from the intersection of $C$ and the singular fibres. After (I), we can only blow up a point on the unique $(-1)$-curve on each fibre: otherwise we would introduce more than one $(-1)$-curves in a fibre, violating $r=0$. There are two types of such blow ups. One is the blow ups of the intersection of $C$ and the $(-1)$-curve, (blow up of the edge between the white vertex and $C$) which decrease $C^2$. The other is the blow ups of a point of $(-1)$-curve outside $C$. We start from the first type of blow ups and get the resolutions of Gorenstein log del Pezzos of rank 1 with $K^2 = C^2 \ge 3$ (Lemma 2.2): $$\begin{array}{lllllll} \tilde{S}(A_1 + A_2)&\longrightarrow& \tilde{S}(A_4) &\longrightarrow& \tilde{S}(D_5) &\longrightarrow& \tilde{S}(E_6)\\ &\searrow& &\searrow& & &\\ & &\tilde{S}(2A_1+A_3) &\longrightarrow& \tilde{S}(A_1+A_5).& & \end{array}$$ Each ``$\longrightarrow$'' represents one blow up, and each ``$\searrow$'' two blow ups on a new fibre . Then, starting from one of these, we make the second type of blow ups, which decrease the minimal log discrepancy of $S$, until either (EX1) or (EX3) is violated (see below). The Gorenstein rank 1 surfaces listed above are the image of $\uppermin{S}$ under the morphism $\phi_{|C|}$ defined by the linear system $|C|$ on it. We denote it by $S_C$, and its resolution (one of the above) by $\tilde{S_C}$. Note that $C$ meets every $(-1)$-curve $E$ on $S_C$ since $C\sim -K_{S_C}$ and $-K.E=1$. Consider blow ups on one fibre starting at one such $E$. By Lemma 2.3, on $E$, $S$ has either at least two singularity or one singularity that is not log-terminal for $K+E$. That is, on $\tilde{S_C}$, either $E$ meets at least two trees $T_1,T_2$ of b-vertices, or one tree $T_3$ that gives non-log-terminal point for $K+E$. Now consider the transformation of the subgraph consisting or $C$, $E$, and trees of b-vertices $T_i$ meeting $E$ on $S_C$. It should always contain a unique w-vertex. If we blow up the vertex $E$, i.e. blow up a point on $E$ other than the intersection points with neighboring exceptional curves, then after the transformation $C$ would meet the b-vertex $E$ in the black graph $T_1 - E - T_2$ or $E - T_3$. Either of these would contracts to a non-log-terminal point on $S$ for $K+C$, contradicting Lemma 1.1. (For an example of the first situation, consider blow up of the white vertex in the configuration (I) in the Figure 2 above. For the second, consider the same in the configuration of the table $\sharp 9$.) Therefore the first blow up has to be at the intersection point of $E$ and one of the neighboring b-vertices, i.e. blow up of the edge joining $E$ and one of its neighbors. The same argument, repeated for the new white vertex $E_1$ at each stage, shows that successive blow ups also must be at the edge joining $E_1$ and a neighboring b-vertex, because the trees now meeting $E_1$ are even bigger than those that met $E$. Thus, by induction, we see that the full inverse image of $E$ is of the form $E - T - E_1 - T^\prime$, where $T$ and $T^\prime$ are chains of b-vertices ($T$, or $T^\prime$ may be a part of a larger tree. And $E$ may meet another tree $T^{\prime\prime}$ in which case $T$ should be empty --- Remember that $C$ meets $E$), and $E_1$ is a w-vertex. The blow up described above either increases the weight of an end vertex of $T$ next to $E_1$, or adds one $(-2)$-curve $E_1$ to it, depending on which side of $E_1$ we blow up. Either of such tranformations (those which preserve log-terminal property), if repeated infinitely many times, make the log-discrepancy with respect to $K+bC$ of the resulting singularity on $C$ monotonically decrease toward $1-b \le \tfrac17$. Hence by Lemma 2.2, after finite number of steps, (EX3) will be violated (or perhaps, (EX1) may be violated first). Therefore this procedure of successive blowups must terminate. We can now refine the lemma 2.2 as follows: If $C^2 < 6$ then we have only one singularity by Lemma 2.2. But on the other hand, if $C^2 \ge 5$ we can have only one singular fibre, which means that in every case we have only one singularity of $(S,B)$ on $C$. (EX1) restricts the possible types of singularities $[m,k]$ on $C$ as follows: \begin{eqnarray*} 0\geq K+bC+B^\prime \cdot C &=& -(1-b)C^2+\tfrac{(k-1)+b(m-k)}{m}\\ &=& \frac17C^2+\tfrac{(k-1)+\frac67(m-k)}{m},\\ &or&\\ (6-C^2)m&\leq& 7-k. \end{eqnarray*} In this way, we find that there are 20 possible $S$'s , with a few different $B$'s for some of the $S$'s. These are summarized in the table below. \begin{case} $r=1$, i.e. we have one exceptional curve, say $E$, other than $\Sigma$ that is a section of ${\mathbb P}^1$-fibration $p$. \end{case} Thus, exactly one fibre contains two $(-1)$- curves in it. If we modify at any other fibre it has to start like (I) of the Figure 2 (two blow ups at the tangency with the fibre) because (II) and (III) have been elimineted. In particular each time we blow up on a new fibre we decrease $C^2$ by at least 2. \begin{claim} Any exceptional curve $E$ that is a (mutli-)section of $p$ is in fact a 1-section that is disjoint from $\Sigma$. \end{claim} \proof Let $E$ be a (multi-)section, and $d=\text{mult}_E(B_1^\prime)$. Then if $F$ is a fibre of $p$, we have $$0 \ge (K+B^\prime).F \ge (K+bC+dE).F \le -2+\tfrac67\cdot 2 + d.$$ Hence $d \le \frac27 < \frac37$. So $E$ cannot intersect $C$ on $\uppermin{S}$. Therefore all the intersection point of $C$ and $E$ have to be blown up on $\uppermin{S}$. If $E$ is not a 1-section disjoint from $\Sigma$, then we have $C.E\ge 6$ on ${\mathbb F}_2$. So we would have $C^2 \le 8 - 6 =2$ on $\uppermin{S}$ which is impossible according to the lemma 2.2. This proves the claim. So let $E$ be a simple section disjoint from $\Sigma$. Then $E.C=4$. Suppose $E$ intersects $C$ at one point with multiplicity 4. Then after four blowups at this point we get $\tilde{S}(A_1+A_3)$ (the configuration of the table $\sharp 17$, with a different choice of fibration), which has already been studied in the case 1 above. If $E$ intersects $C$ at two points with multuplicity 3 and 1 respectively, then by the above observation we have at least $3+2=5$ blow ups on $C$, which gives $\tilde{S}(3A_2)$, with $C$ passing through three $(-1)$-curves joining three ${\mathbb A}_2$ points. Since $C^2=3$ by Lemma 2.2 $C$ can have at worst ${\mathbb A}_1$ (=``type [2,1]'') point on it, but that cannot be attained: We could at best choose $B_1=\frac12 C_2$ where $C_2=$ (image of one of the $(-1)$-curves meeting $C$) and thus get type [2,2] point on $C$, which is worse than ${\mathbb A}_1 = type [2,1]$. If $E$ intersects $C$ at more than 3 points, then we have at least six blow ups on $C$ thus $C^2 \le 2$, which is impossible by Lemma 2.2. Finally, if $E$ intersects $C$ at two points with multiplicity 2 each, we would have two $(-1)$-curves in each fibre, and this violates (EX4). Thus, we get no new examples from case 2. \begin{case} $r\ge 2$, i.e. we have at least two exceptional curves, say, $E_1$ and $E_2$, other than $\Sigma$, that are sections of $p$. \end{case} $E_i$ are simple sections. Then because $C.E_i=4$ and $E_1.E_2=2$, we must have at least $4+4-2=6$ blow ups on $C$ in order to separate $E_i$'s from $C$. Then $C^2 \le 2$ and by Lemma 2.2, this is impossible. \qed It turns out that in every case $K+B$ has a 7-complement. Moreover, we can choose $g$ so that in every case $B_1^\prime$ has only one component which is a fibre of ${\mathbb F}_2$. \bigskip \noindent{\bf Table} Thus we get the following table. Here, \begin{itemize} \item The first column shows the configuration on $S^{\text{min}}$ of the exceptional curve $E_i$'s, $(-1)$-curves, and the components of $B$. `$\circ$' denote $(-1)$-curve, `$\bullet$' are the $E_i$'s with self intersection number ($\le -2$) attached, with `$\leftarrow$' indicating (one possible) $\Sigma \subset{\mathbb F}_2$ after a suitable sequence of contractions of $(-1)$-curves. Squares are curves with non negative self intersection. \item The second column gives the fractional part $B_1$ of the boundary $B$, or rather, of $D$. \item The third column gives the number $(\tfrac67 \le)\hspace{0.1in} max\{b|K+bC+B_1\le 0\}\hspace{0.1in} (<1)$ \item The fourth column gives an example of $n$-complements. \item The last column lists numerical relations between some relevant divisors on S, with $H$ being the generator of $Pic(S)$. \end{itemize} Note that we can compute intersection numbers on $S^{\text{min}}$ using the crepant pullbacks, and a divisor on $S$ is Cartier iff its crepant pullback is Cartier, i.e. iff it is integral ({\it cf.}[Sakai]). The table is organized according to $S_C$, the image of $\uppermin{S}$ under the morphism defined by the linear system $|C|$. \bigskip (1) $S=S(A-1)={\mathbb Q}_2$ (=quadratic cone $\subset {\mathbb P}^3$, $S_C$ = its Veronese image) \bigskip \tablehead \centering{1} & \includefigure{0} & \centering{$\frac12C_2$} & \centering{$\dfrac78$} & \begin{minipage}{1.4in}$\bullet$ 7-compl.$=0$\\ ($K+\frac67C+\frac47C_2\equiv0$)\\ $\bullet$ trivial 8-compl. \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$\begin{array}{ll} -K&\equiv C \equiv 2H \\ C_2&\equiv\frac12H \end{array} $$ \end{center} \end{minipage} \\ \hline \end{tabular} \bigskip (2) $S_C=S_7$ (= a del Pezzo with degree 7) \bigskip \settablewidth \centering{2} & \includefigure{1} & \centering{$0$}& \centering{$\dfrac67$}& \begin{minipage}{1.4in}\centering{trivial 7-compl.\\ ($K+\frac67C\equiv 0$)} \end{minipage} & \begin{minipage}{1.2in}\begin{center} $$\begin{array}{ll} -K& \equiv H \\ C & \equiv \frac76H \end{array} $$ \end{center} \end {minipage} \\ \hline \end{tabular} \newpage \par \bigskip (3) $S_C = S(A_1+A_2)$ \bigskip \tablehead \centering{ 3 }& & \centering{$\frac12C_2$} & \centering{$\dfrac9{10}$} & \begin{minipage}{1.4in}\centering{trivial 10-compl.\\ ($K+\frac9{10}C+\frac12C_2\equiv0$)}\end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv C \equiv H\\ C_2&\equiv\frac16H \end{array} $$ \end{center} \end{minipage} \\ \cline{3-5} & \includefigure{2} & \centering{$\frac23C_2$} & \centering{$\dfrac89$} & \begin{minipage}{1.4in}\centering{ trivial 9-compl.\\($K+\frac89C+\frac23C_2\equiv0$)} \end{minipage} & \\ \cline{3-5} & & \begin{center}$\frac34C_2$\end{center} & \begin{center}$\dfrac78$\end{center} & \begin{center}trivial 8-compl. \\($K+\frac78C+\frac34C_2\equiv0$)\end{center} & \\ \cline{3-5} & & \begin{center}$\frac45C_2$\end{center} & \begin{center}$\dfrac{13}{15}$\end{center} & \begin{center}7-compl.=0\\($K+\frac67C+\frac67C_2\equiv0$)\end{center} & \\ \cline{3-5} & & \begin{center}$\frac56C_2$\end{center} & \begin{center}$\dfrac{31}{36}$\end{center} & \begin{center}7-compl.=0\\($K+\frac67C+\frac67C_2\equiv0$) \end{center} & \\ \hline \centering{ 4 }& \includefigure{3} & \centering{$0$ } & \centering{$\dfrac89$} & \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$C_2$\\($K+\frac67C+\frac27C_2\equiv0$)\\ \bigskip $\bullet$ trivial 9-compl.} \end{minipage}& \begin{minipage}{1.2in \centering{ $ -K \equiv\frac{8}{12}H$\\ $ C \equiv\frac{9}{12}H$\\ $ C_2 \equiv\frac1{12}H$} \end{minipage} \\\hline \centering{ 5 }& \includefigure{4} & \centering{$0$ } & \centering{$\dfrac9{10}$} & \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$3C_2$\\($K+\frac67C+\frac37C_2\equiv0$)\\ \bigskip $\bullet$ trivial 10-compl.}\end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{9}{15}H\\ C &\equiv\frac{10}{15}H\\ C_2&\equiv\frac1{15}H \end{array} $$ \end{center} \end{minipage} \\\hline \centering{ 6 }& \includefigure{5} & \centering{$0$ } & \centering{$\dfrac78$} & \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$2C_2$\\($K+\frac67C+\frac27C_2\equiv0$)\\ \bigskip $\bullet$ trivial 8-compl.}\end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{14}{40}H\\ C &\equiv\frac{16}{40}H\\ C_2&\equiv\frac1{40}H \end{array} $$ \end{center} \end{minipage} \\\hline \end{tabular} \newpage \tablehead \centering{ 7 }& \includefigure{6} & \centering{$0$ } & \centering{$\dfrac{13}{15}$} & \begin{minipage}{1.4in}\centering{ 7-compl.=$C_2$\\($K+\frac67C+\frac17C_2\equiv0$) } \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{13}{35}H\\ C &\equiv\frac{15}{35}H\\ C_2&\equiv\frac1{35}H \end{array} $$ \end{center} \end{minipage} \\\hline \centering{ 8 }& \includefigure{7} & \centering{$0$ } & \centering{$\dfrac{19}{22}$} & \centering{7-compl.=$C_2$\\($K+\frac67C+\frac17C_2\equiv0$)} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{19}{77}H\\ C &\equiv\frac{22}{77}H\\ C_2&\equiv\frac1{77}H \end{array} $$ \end{center} \end{minipage} \\\hline \end{tabular} \par \bigskip (4) $S_C=S(A_4)$ \bigskip \begin{tabular}{|p{0.2in}|p{1.2in}|p{0.8in}|p{0.5in}|p{1.4in}|p{1.2in}|} \hline \centering{ 9 }& \includefigure{8} & \centering{$\frac12C_2$ } & \centering{$\dfrac{9}{10}$} & \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$C_2$\\($K+\frac67C+\frac57C_2\equiv0$)\\ \bigskip $\bullet$ trivial 10-compl.}\end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv C \equiv H\\ C_2&\equiv\frac15H \end{array} $$ \end{center} \end{minipage} \\\cline{3-5} & & \begin{center}$\frac23C_2$ \end{center} & \begin{center}$\dfrac{13}{15}$\end{center} & \begin{center}7-compl.=0\\($K+\frac67C+\frac57C_2\equiv0$)\end{center}& \\\hline \centering{ 10 } & \includefigure{9} & \centering{$0$ } & \centering{$\dfrac{10}{11}$} & \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$4C_2$\\($K+\frac67C+\frac47C_2\equiv0$)\\ \bigskip $\bullet$ trivial 11-compl.} \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv \frac{10}{22}H\\ C &\equiv \frac{11}{22}H\\ C_2&\equiv\frac1{22}H \end{array} $$ \end{center} \end{minipage} \\\cline{3-5} & & \begin{center}$\frac12C_2$ \end{center} & \begin{center}$\dfrac{19}{22}$\end{center} & \begin{center}7-compl.=0\\($K+\frac67C+\frac47C_2\equiv0$)\end{center}& \\\hline \centering{ 11 }& \includefigure{10} & \centering{$0$ } & \centering{$\dfrac{15}{17}$} & \begin{minipage}{1.4in} \centering{7-compl.=$3C_2$\\($K+\frac67C+\frac37C_2\equiv0$)} \end{minipage} & \begin{minipage}{1.2in} $$ \begin{array}{ll} -K &\equiv\frac{15}{3\cdot 17}H\\ C &\equiv\frac{17}{3\cdot17}H\\ C_2&\equiv\frac1{3\cdot17}H \end{array} $$ \end{minipage} \\ \hline \end{tabular} \newpage \tablehead \centering{12}& \includefigure{11} & \centering{$0$} & \centering{$\dfrac78$} & \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$2C_2$\\($K+\frac67C+\frac27C_2\equiv0$)\\ \bigskip $\bullet$ trivial 8-compl.} \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{14}{48}H\\ C &\equiv\frac{16}{48}H\\ C_2&\equiv\frac1{48} \end{array} $$ \end{center} \end{minipage} \\ \hline \centering {13}& \includefigure{12} & \centering{$0$} & \centering{$\dfrac{20}{23}$}& \begin{minipage}{1.4in} \centering{7-compl.=$2C_2$\\($K+\frac67C+\frac27C_2\equiv0$)} \end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{20}{4\cdot23}H\\ C &\equiv\frac14H\\ C_2&\equiv\frac1{4\cdot23}H \end{array} $$ \end{center} \end{minipage} \\ \hline \centering {14}& \includefigure{13} & \centering{$0$} & \centering{$\dfrac{25}{29}$}& \begin{minipage}{1.4in} \centering{7-compl.=$C_2$\\($K+\frac67C+\frac17C_2\equiv0$)} \end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac{25}{5\cdot29}H\\ C &\equiv\frac15H\\ C_2&\equiv\frac1{5\cdot29}H \end{array} $$ \end{center} \end{minipage} \\ \hline \end {tabular} \par \smallskip (5) $S_C=S(D_5)$ \smallskip \begin{tabular}{|p{0.2in}|p{1.2in}|p{0.8in}|p{0.5in}|p{1.4in}|p{1.2in}|} \hline \centering {15}& \includefigure{14}& \centering{$\frac12 C_2$} & \centering{$\dfrac78$}& \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=0\\($K+\frac67C+\frac47C_2\equiv0$)\\ \bigskip $\bullet$ trivial 8-compl.} \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv C \equiv H\\ C_2&\equiv\frac14H \end{array} $$ \end{center} \end{minipage} \\\hline \centering {16}& \includefigure{15} & \centering{$0$} & \centering{$\dfrac{8}{9}$}& \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=$2C_2$\\($K+\frac67C+\frac27C_2\equiv0$)\\ \bigskip $\bullet$ trivial 9-compl.} \end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv \frac8{18}H\\ C &\equiv \frac9{18}H\\ C_2&\equiv\frac1{18}H \end{array} $$ \end{center} \end{minipage} \\\hline \end {tabular} \par \bigskip (6) $S_C=S(A_3+2A_1)$ \bigskip \begin{tabular}{|p{0.2in}|p{1.2in}|p{0.8in}|p{0.5in}|p{1.4in}|p{1.2in}|} \hline \centering {17}& \includefigure{16} & \centering{$\frac12 C_2$} & \centering{$\dfrac{7}{8}$}& \begin{minipage}{1.4in} \centering{$\bullet$ 7-compl.=0)\\($K+\frac67C+\frac47C_2\equiv0$)\\ \bigskip $\bullet$ trivial 8-compl.}\end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv C \equiv H\\ C_2&\equiv\frac14H \end{array} $$ \end{center} \end{minipage} \\ \hline \centering {18}& \includefigure{17} & \centering{$0$} & \centering{$\dfrac{6}{7}$}& \begin{minipage}{1.4in} \centering{trivial 7-compl.\\($K+\frac67C\equiv0$)} \end{minipage} & \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv \frac{12}{42}H\\ C &\equiv \frac{14}{42}H\\ C_2&\equiv\frac3{42}H\\ C_3&\equiv\frac2{42}H \end{array} $$ \end{center} \end{minipage} \\ \hline \end{tabular} \newpage \par\bigskip (7) $S_C=S(E_6)$ \bigskip \begin{tabular}{|p{0.2in}|p{1.2in}|p{0.8in}|p{0.5in}|p{1.4in}|p{1.2in}|} \hline \centering {19}& \includefigure{18}& \centering{$0$} & \centering{$\dfrac{6}{7}$}& \begin{minipage}{1.4in} \centering{trivial 7-compl.\\($K+\frac67C\equiv0$)} \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac6{14}H\\ C&\equiv\frac7{14}H \\ C_2&\equiv\frac1{14}H \end{array} $$ \end{center} \end{minipage} \\ \hline \end {tabular} \par \bigskip (8) $S_C=S(A_5+A_1)$ \bigskip \begin{tabular}{|p{0.2in}|p{1.2in}|p{0.8in}|p{0.5in}|p{1.4in}|p{1.2in}|} \hline \centering {20}& \includefigure{19}& \centering{$0$} & \centering{$\dfrac{6}{7}$}& \begin{minipage}{1.4in} \centering{trivial 7-compl.\\($K+\frac67C\equiv0$)} \end{minipage}& \begin{minipage}{1.2in} \begin{center} $$ \begin{array}{ll} -K &\equiv\frac6{14}H\\ C&\equiv\frac7{14}H \\ C_2&\equiv \frac2{14}H\\ C_3&\equiv \frac1{14}H \end{array} $$ \end{center} \end{minipage} \\ \hline \end{tabular} \newline \newline

9711

alg-geom/9711002

en

https://arxiv.org/abs/alg-geom/9711002

alg-geom/9711002

Jan Stienstra

Jan Stienstra

Resonant Hypergeometric Systems and Mirror Symmetry

37 pages Latex2e; one picture; submitted for publication in the proceedings of the Taniguchi Symposium 1997 "Integrable Systems and Algebraic Geometry"

The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a function $\Psi$ with values in the finite dimensional vector space $R_{A,T}\otimes C$. Applications of these results in the context of toric Mirror Symmetry are described. Building on work of Batyrev we show that the relative cohomology module of a certain hypersurface in a torus is a GKZ hypergeometric $D$-module which over an appropriate domain is isomorphic to the trivial $D$-module $R_{A,T}\otimes O_T$, where $O_T$ is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Gamma-series. As a result one finds the periods of a holomorphic differential form of degree $d$ on a $d$-dimensional Calabi-Yau manifold, needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov we interpret the ring $\cR_{\sA,\gT}$ as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.

alg-geom

\section*{Introduction I} A GKZ hypergeometric system \cite{gkz1} depends on four parameters: two positive integers $N$ and $n$, a set $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ of vectors in ${\mathbb Z }^n$ and a vector $\beta$ in ${\mathbb C }^{n}$. The standard assumptions \cite{gkz1} are \ \begin{condition}\label{gkzcond} \footnote{ ${}\;\;\;{\mathbb Z }^n,{\mathbb R }^n,{\mathbb C }^n$ resp. ${\mathbb Z }^{n\vee},{\mathbb R }^{n\vee},{\mathbb C }^{n\vee}$ denote spaces of column vectors resp. row vectors.}\vspace{-7mm} \begin{eqnarray} \label{eq:aspan} && {\mathfrak a }_1,\ldots,{\mathfrak a }_N \textrm{ generate a rank }n \textrm{ sub-lattice } {\mathbb M } \textrm{ in }{\mathbb Z }^n \\ \label{eq:hyperplane} && \exists\;{\mathfrak a }_0^\vee\in{\mathbb Z }^{n\vee} \textrm{ such that } {\mathfrak a }_0^\vee\cdot{\mathfrak a }_i \,=\, 1 \;\;(i=1,\ldots,N) \end{eqnarray} \end{condition} The GKZ system with these parameters is the following system of partial differential equations for functions $\Phi$ on a torus with coordinates $v_1,\ldots,v_N$: \begin{eqnarray} \label{eq:gkzlin} \left(-\beta+ \sum_{j=1}^N\:{\mathfrak a }_j\,v_j\frac{\partial}{\partial v_j}\right)\;\Phi &=&0\\ \label{eq:gkzmon} \left(\prod_{\ell_j>0} \left[\frac{\partial}{\partial v_j}\right]^{\ell_j} \:-\:\prod_{\ell_j<0} \left[\frac{\partial}{\partial v_j}\right]^{-\ell_j} \right)\Phi&=&0\hspace{3mm}\textrm{ for } \ell\in{\mathbb L } \end{eqnarray} where (\ref{eq:gkzlin}) is in fact a system of $n$ equations and \begin{equation}\label{eq:L} {\mathbb L }\::=\:\{\;\ell=(\ell_1,\ldots,\ell_N)^t\in{\mathbb Z }^N\:|\: \ell_1{\mathfrak a }_1+\ldots+\ell_N{\mathfrak a }_N\,=\,0\;\}\,. \end{equation} Some of the above data are displayed in the following short exact sequence in which ${\mathcal A }$ denotes the linear map ${\mathcal A }\,:\;{\mathbb Z }^N\rightarrow{\mathbb Z }^n$, ${\mathcal A }(\lambda)\:=\:\lambda_1{\mathfrak a }_1+\ldots+\lambda_N{\mathfrak a }_N$. \begin{equation}\label{eq:ses} 0\rightarrow{\mathbb L }\longrightarrow{\mathbb Z }^N\stackrel{{\mathcal A }}{\longrightarrow} {\mathbb M }\rightarrow 0 \end{equation} \emph{We are going to construct solutions for GKZ systems with $\beta \in {\mathbb M }\,.$ Of special interest for applications to mirror symmetry are the cases $\beta=0$ and $\beta=-{\mathfrak a }_0$ with ${\mathfrak a }_0$ as in the definition of reflexive Gorenstein cone (definition \ref{gorco}).} The idea is as follows. Gel'fand-Kapranov-Zelevinskii \cite{gkz1} give solutions for (\ref{eq:gkzlin})-(\ref{eq:gkzmon}) in the form of so-called $\Gamma$-series \begin{equation}\label{eq:gaser} \sum_{\ell\in{\mathbb L }}\;\prod_{j=1}^N \frac{\:v_j^{\gamma_j+\ell_j}}{\Gamma (\gamma_j+\ell_j+1)} \end{equation} $\Gamma$ is the usual $\Gamma$-function, $\ell=(\ell_1,\ldots,\ell_N)^t\in{\mathbb L }\subset{\mathbb Z }^N$. The series depends on an additional parameter $\gamma=(\gamma_1,\ldots,\gamma_N)^t\in{\mathbb C }^N$ which must satisfy \begin{equation} \label{eq:Agabe} \gamma_1{\mathfrak a }_1+\ldots+\gamma_N{\mathfrak a }_N\,=\,\beta \end{equation} Allowing the obvious formal rules for differentiating such $\Gamma$-series one sees that the functional equations of the $\Gamma$-function guarantee that (\ref{eq:gaser}) satisfies the differential equations (\ref{eq:gkzmon}) and that condition (\ref{eq:Agabe}) on $\gamma$ takes care of (\ref{eq:gkzlin}). \emph{The issue is to interpret the $\Gamma$-series (\ref{eq:gaser}) as a function on some domain.} In order that (\ref{eq:gaser}) can be realized as a function $\gamma$ must satisfy more conditions. Gel'fand-Kapranov-Zelevinskii obtain convenient conditions from a triangulation ${\mathcal T }\!$ of the convex hull of $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$. However, if $\beta$ is in ${\mathbb M }$ and the triangulation has more than one maximal simplex, the vectors $\gamma$ which satisfy these extra conditions do not provide enough $\Gamma$-series solutions for the GKZ system. This phenomenon is called \emph{resonance} \cite{gkz1}. An extreme case of resonance, in which all $\Gamma$-series coincide, occurs when $\beta$ is in ${\mathbb M }$ and ${\mathcal T }\!$ is unimodular. \begin{definition} (cf. {} \cite{stur}) A triangulation is called \emph{unimodular} if all its maximal simplices have volume $1\,;$ the volume of a maximal simplex $\mathrm{ conv}\,\{{\mathfrak a }_{i_1},\ldots,{\mathfrak a }_{i_n}\}$ is defined as $|\det({\mathfrak a }_{i_1},\ldots,{\mathfrak a }_{i_n})|\,.$ \end{definition} \ To get around the resonance problem for $\beta \in {\mathbb M }$ we proceed as follows. Fixing a solution $\gamma^\circ\in{\mathbb Z }^N$ for equation (\ref{eq:Agabe}) we write the general solution of (\ref{eq:Agabe}) as $\gamma=\gamma^\circ+{\mathsf c }$ with ${\mathsf c }=(c_1,\ldots,c_N)^t$ such that \begin{equation} \label{eq:linc0} c_1{\mathfrak a }_1+\ldots+c_N{\mathfrak a }_N=0 \end{equation} and note $\gamma+{\mathbb L }={\mathsf c }+{\mathcal A }^{-1}(\beta)\,.$ Thus (\ref{eq:gaser}) becomes $\sum_{\lambda\in{\mathcal A }^{-1}(\beta)}\hspace{2mm}\prod_{j=1}^N \frac{\:v_j^{c_j+\lambda_j}}{\Gamma (c_j+\lambda_j+1)}\:.$ Multiplying this by $\prod_{j=1}^N\Gamma (c_j+1)$ we obtain \begin{equation}\label{eq:form1} \Phi_{{\mathcal T }\!,\beta}({\mathsf v })\::=\:\sum_{\lambda\in{\mathcal A }^{-1}(\beta)}\hspace{2mm} Q_\lambda({\mathsf c })\;\cdot\;\prod_{j=1}^N v_j^{\lambda_j}\;\cdot\;\prod_{j=1}^N v_j^{c_j} \end{equation} where \begin{equation}\label{eq:bico1} Q_\lambda({\mathsf c })\::=\: \frac{\prod_{\lambda_j<0}\prod_{k=0}^{-\lambda_j-1}(c_j-k)} {\prod_{\lambda_j>0}\prod_{k=1}^{\lambda_j}(c_j+k)}\,. \end{equation} The key observation is that (\ref{eq:bico1}) and (\ref{eq:form1}) also make sense when $c_1,\ldots,c_N$ are taken from a ${\mathbb Q }$-algebra in which they are nilpotent. The expression $v_j^{c_j}$ can still be interpreted as $\exp(c_j\log v_j)$. \ \begin{definition}\label{ring} Let ${\mathsf A }=(a_{ij})$ denote the $n\times N$-matrix with columns ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$. For a regular triangulation ${\mathcal T }\!$ (cf. $\S$ \ref{triangs}) of the polytope $\Delta\,:=\mathrm{ conv}\,\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ we define: \begin{equation}\label{eq:ring} {\mathcal R }_{{\mathsf A },{\mathcal T }\!}\::=\:\modquot{{\mathbb Z }[D^{-1}][C_1,\ldots,C_N]}{{\mathcal J }} \end{equation} where ${\mathcal J }$ is the ideal generated by the linear forms \begin{equation} \label{eq:linrel} a_{i1}C_1+\ldots+a_{iN}C_N\hspace{3mm}\textrm{ for } i=1,\ldots,n \end{equation} and by the monomials \begin{equation}\label{eq:srrel} C_{i_1}\cdot\ldots\cdot C_{i_s} \hspace{4mm}\textrm{ with } \hspace{4mm} \mathrm{ conv}\,\{{\mathfrak a }_{i_1},\ldots,{\mathfrak a }_{i_s}\} \hspace{4mm}\textrm{ not a simplex in }{\mathcal T }\!\, ; \end{equation} $D$ is the product of the volumes of the maximal simplices of ${\mathcal T }\!\,.$ We write $c_i$ for the image of $C_i$ in ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\,.$ \end{definition} \ In theorem \ref{dims} we show that ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is a free ${\mathbb Z }[D^{-1}]$-module with rank equal to the number of maximal simplices in the triangulation. This implies that $c_1,\ldots,c_N$ are nilpotent and hence $$ Q_\lambda({\mathsf c })\in{\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q } $$ \begin{theorem}\label{values}\nonumber With this interpretation of $Q_\lambda({\mathsf c })$ the function $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$\\ defined by (\ref{eq:form1}) takes values in the \emph{ algebra } ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C }\,.${\hfill$\boxtimes$} \end{theorem} \ The domain of definition of the function $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$ is discussed hereafter. Relation (\ref{eq:linrel}) ensures that this function $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$ satisfies the differential equations (\ref{eq:gkzlin}); it automatically satisfies (\ref{eq:gkzmon}). Relation (\ref{eq:srrel}) ensures that the series expansion for $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$ only contains $\lambda$'s (i.e. $Q_\lambda({\mathsf c })\neq 0$) which satisfy \begin{equation}\label{eq:support} {\mathcal A }\,\lambda=\beta\hspace{4mm}\textrm{and}\hspace{4mm} \mathrm{ conv}\,\{{\mathfrak a }_i\:|\:\lambda_i<0\}\hspace{3mm} \textrm{is a simplex in the triangulation } {\mathcal T }\!\,. \end{equation} This is important for determining a domain of definition for $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$. As we tried to distinguish a kind of regular behavior for the $\lambda$'s which satisfy (\ref{eq:support}), we were led to triangulations for which the intersection of the maximal simplices is not empty. We call \begin{equation}\label{eq:introcore} \mathrm{ core}\,{\mathcal T }\!\::=\textrm{ intersection of the maximal simplices of }{\mathcal T }\! \end{equation} the core of the triangulation ${\mathcal T }\!$. We use the short notation $i\in\mathrm{ core}\,{\mathcal T }\!$ for ${\mathfrak a }_i\in\mathrm{ core}\,{\mathcal T }\!$. The following result is corollary \ref{corid} in section \ref{coresection}. \ \begin{theorem}\label{hascore} Assume $\mathrm{ core}\,{\mathcal T }\!\neq\emptyset$ and $\beta=\sum_{i\in\mathrm{ core}\,{\mathcal T }\!} m_i{\mathfrak a }_i$ with all $m_i<0$. Then the function $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$ takes values in the \emph{ principal ideal } $c_{\mathrm{ core}\,}{\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C }$ where $$ c_{\mathrm{ core}\,}\::=\:\prod_{i\in\mathrm{ core}\,{\mathcal T }\!} c_i $$ Multiplication by $c_{\mathrm{ core}\,}$ on ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ induces a linear isomorphism \begin{equation}\label{eq:coriso} \modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}} \stackrel{\simeq}{\longrightarrow}c_{\mathrm{ core}\,}{\mathcal R }_{{\mathsf A },{\mathcal T }\!} \end{equation} Thus one can also say that the function $\Phi_{{\mathcal T }\!,\beta}({\mathsf v })$ takes values in the algebra $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}\otimes{\mathbb C }\,.$ {\hfill$\boxtimes$} \end{theorem} \ By composing $\Phi_{{\mathcal T }\!,\beta}$ with a linear map ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\rightarrow{\mathbb C }$ one obtains a ${\mathbb C }$-multi-valued function which satisfies the system of differential equations (\ref{eq:gkzlin})-(\ref{eq:gkzmon}). When $\beta=0$ and ${\mathcal T }\!$ is unimodular all solutions of (\ref{eq:gkzlin})-(\ref{eq:gkzmon}) can be obtained in this way; see theorem \ref{isosol}. For $\beta\neq 0$ not all solutions of (\ref{eq:gkzlin})-(\ref{eq:gkzmon}) can be obtained in this way. Yet what we need for mirror symmetry are the solutions which can be obtained in this way for appropriate $\beta$ and ${\mathcal T }\!$; see theorem \ref{mainthm}. Our proof of this theorem makes essential use of the relation: \begin{equation}\label{eq:recur} \frac{\partial}{\partial v_i}\Phi_{{\mathcal T }\!,\beta}({\mathsf v })\:=\: \Phi_{{\mathcal T }\!,\beta-{\mathfrak a }_i}({\mathsf v })\,. \end{equation} which follows imediately from the formulas (\ref{eq:form1}) and (\ref{eq:bico1}). \begin{remark}\textup{ The ideal generated by the monomials in (\ref{eq:srrel}) is known as the \emph{Stanley-Reisner ideal} and has been defined for finite simplicial complexes in general \cite{stan}. It is well-known \cite{dani,ful,oda} that the cohomology ring of a toric variety constructed from a complete simplicial fan has a presentation by generators and relations as in (\ref{eq:linrel})-(\ref{eq:srrel}). Unimodular triangulations whose core is not empty and is not contained in the boundary of $\Delta$, give rise to such toric varieties and in that case ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is indeed the cohomology ring of a toric variety; see theorem \ref{protor}. However not all triangulations to which the present discussion applies are of this kind. For instance for the triangulation ${\mathcal T }\!_5$ in figure 1 we find ${\mathcal R }_{{\mathsf A },{\mathcal T }\!_5}= {\mathbb Z }[c_1,c_2,c_5]/(c_1^2,\,c_2^2,\,c_5^2,\,c_1c_2,\,c_2c_5)$. An element like $c_2$ which annihilates the whole degree $1$ part of ${\mathcal R }_{{\mathsf A },{\mathcal T }\!_5}$ can not exist in the cohomology of a toric variety.} \end{remark} \begin{remark}\textup{ Our method for solving GKZ systems in the resonant case evolved directly from the $\Gamma$-series of Gel'fand-Kapranov-Zelevinskii. In hindsight it can also be viewed as a variation on the classical method of Frobenius \cite{frob}. The latter would view $\gamma_1,\ldots,\gamma_N$ in (\ref{eq:gaser}) or $c_1,\ldots,c_N$ in (\ref{eq:form1}) as variables with a restriction given by (\ref{eq:Agabe}) or (\ref{eq:linc0}); then differentiate (repeatedly if necessary) with respect to these variables and set $\gamma=(\gamma_1,\ldots,\gamma_N)$ in the derivatives equal to its special value $\gamma^\circ$, c.q. set $c_1=\ldots=c_N=0$, to obtain solutions for (\ref{eq:gkzlin})-(\ref{eq:gkzmon}). Frobenius \cite{frob} considered only functions in one variable. In the case with more variables one also needs a good bookkeeping device for the linear relations between the solutions of the differential equations. The rings ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ resp. $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}$ are such a bookkeeping devices. Hosono-Klemm-Theisen-Yau have applied Frobenius' method directly in the situation of the Picard-Fuchs equations of certain families of Calabi-Yau threefolds; see \cite{HKTY} formulas (4.9) and (4.10). In their work the cohomology ring of the mirror Calabi-Yau threefold plays a similar role of bookkeeper; in fact $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}$ is the cohomology ring of the mirror Calabi-Yau manifold . The way in which we arrive at our result looks quite different from that in \cite{HKTY} $\S 4$. Moreover the formulation in op. cit. is restricted to the situation of Calabi-Yau threefolds.} \end{remark} \begin{remark}\textup{ Some of our $\Phi_{{\mathcal T }\!,\beta}$'s are similar to expressions presented by Givental in \cite{giv} theorems 3 and 4; more specifically, $\vec{g}_l$ in \cite{giv} thm. 4 is a special case of $\Phi_{{\mathcal T }\!,\beta}$ in our theorem \ref{hascore} with $\beta=-\sum_{i\in\mathrm{ core}\,{\mathcal T }\!}{\mathfrak a }_i$, whereas in \cite{giv} thm. 3 there is a difference in that the input data are not subject to (\ref{eq:hyperplane}) in condition \ref{gkzcond}. The algebra $H$ in \cite{giv} thm. 3 is the cohomology algebra of a toric variety while our ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ for appropriate ${\mathcal T }\!$ is also the cohomology algebra of a toric variety. The algebra $H$ in \cite{giv} thm. 4 is the algebra $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}$ in our theorem \ref{hascore}. } \end{remark} $$ $$ For a proper treatment of the logarithms which appear in (\ref{eq:form1}) we set \begin{equation}\label{eq:logcoord} \begin{array}{lcl} v_j&:=&\exp({2\pi{\mathsf i}\, } z_j) \hspace{4mm} (j=1,\ldots,N)\\ {\mathsf z }&:=&(z_1,\ldots,z_N)\:\in\: {\mathbb C }^{N\vee}\\ {\mathsf c }&:=&\,(c_1,\ldots,c_N)^t\in{\mathcal R }_{{\mathsf A },{\mathcal T }\!}\,\otimes{\mathbb Z }^N\,; \end{array} \end{equation} by (\ref{eq:linc0}) $\; {\mathsf c }$ lies in fact in ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\,\otimes{\mathbb L }\,.$ Instead of (\ref{eq:form1}) we now consider \begin{equation}\label{eq:form2} \Psi_{{\mathcal T }\!,\beta}({\mathsf z })\::=\: \sum_{\lambda\in{\mathcal A }^{-1}(\beta)}\hspace{2mm} Q_\lambda({\mathsf c }){\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot\lambda}\cdot{\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot{\mathsf c }}\,. \end{equation} Note that ${\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot{\mathsf c }}$ is just a polynomial, but $\sum_{\lambda\in{\mathcal A }^{-1}(\beta)}\; Q_\lambda({\mathsf c }){\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot\lambda}$ is really a series. In section \ref{solutions} we analyse the convergence of this series and give a domain ${\mathcal V }_{\mathcal T }\!$ in ${\mathbb C }^{N\vee}$ on which the function $\Psi_{{\mathcal T }\!,\beta}$ is defined; see theorem \ref{defdom}. The domain ${\mathcal V }_{\mathcal T }\!$ is invariant under translations by elements of ${\mathbb Z }^{N\vee}$ and by elements of ${\mathbb M }_{\mathbb C }^\vee\::=\:\Hom{{\mathbb M }}{{\mathbb C }}\subset{\mathbb C }^{N\vee}$. {}From (\ref{eq:form2}) one immediately sees \begin{eqnarray}\label{eq:monodromy} \Psi_{{\mathcal T }\!,\beta}({\mathsf z }+\mu)&=&{\mathsf e }^{{2\pi{\mathsf i}\, }\mu\,\cdot{\mathsf c }}\cdot \Psi_{{\mathcal T }\!,\beta}({\mathsf z })\hspace{5mm}\forall \:\mu\in{\mathbb Z }^{N\vee} \\ \label{eq:character} \Psi_{{\mathcal T }\!,\beta}({\mathsf z }+{\mathsf m })&=&{\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf m }\,\cdot\beta}\cdot \Psi_{{\mathcal T }\!,\beta}({\mathsf z }) \hspace{5mm}\forall \:{\mathsf m }\in{\mathbb M }_{\mathbb C }^\vee \,. \end{eqnarray} The functional equation (\ref{eq:monodromy}) gives the monodromy of $\Phi_{{\mathcal T }\!,\beta}$, when viewed as a multivalued function on $\modquot{{\mathcal V }_{\mathcal T }\!}{{\mathbb Z }^{N\vee}}$ with values in the vector space ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C }$. Because of (\ref{eq:linrel}) elements of ${\mathbb M }_{\mathbb Z }^\vee\::=\:\Hom{{\mathbb M }}{{\mathbb Z }}$ give trivial monodromy and the actual monodromy comes from ${\mathbb L }_{\mathbb Z }^\vee\::=\:\Hom{{\mathbb L }}{{\mathbb Z }}$. As ${\mathbb M }_{\mathbb Z }^\vee$ acts trivially, the translation action of ${\mathbb M }_{\mathbb C }^\vee$ descends to an action of the torus $\modquot{{\mathbb M }_{\mathbb C }^\vee}{{\mathbb M }_{\mathbb Z }^\vee}=\Hom{{\mathbb M }}{{\mathbb C }^\ast}$. The functional equation (\ref{eq:character}), whose infinitesimal analogues are the differential equations (\ref{eq:gkzlin}), means that $\Psi_{{\mathcal T }\!,\beta}$ is an eigenfunction with character $\beta$. If one wants an invariant function for $\beta\neq 0$ one must replace the range of values of $\Psi_{{\mathcal T }\!,\beta}$ by $\modquot{({\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C })}{{\mathbb C }^{\,\ast}}$, the orbit space for the natural ${\mathbb C }^{\,\ast}$-action on the vector space ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C }$. On a possibly slightly smaller domain of definition the invariant function even takes values in the projective space ${\mathbb P }({\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C })$. The ${\mathbb M }_{\mathbb C }^\vee$-invariant function $\Psi_{{\mathcal T }\!,\beta}\bmod{\mathbb C }^{\,\ast}$ is defined on the domain ${\mathbb L }_{\mathbb R }^\vee+\sqrt{-1} {\mathcal B }_{\mathcal T }\!$ in ${\mathbb L }_{\mathbb C }^\vee$; cf. formula (\ref{eq:domain4}). The (multivalued) function $\Phi_{{\mathcal T }\!,\beta}\bmod{\mathbb C }^{\,\ast}$ is defined on a domain in the torus $\Hom{{\mathbb L }}{{\mathbb C }^{\,\ast}}$. For a good overall picture it is appropriate to point out here that the pointed secondary fan (the construction of which is recalled in section \ref{secfan}) defines a toric variety which compactifies the torus $\Hom{{\mathbb L }}{{\mathbb C }^{\,\ast}}$. To each regular triangulation of $\Delta$ corresponds a special point in the boundary of this compactification. The domain of definition of $\Phi_{{\mathcal T }\!,\beta}\bmod{\mathbb C }^{\,\ast}$ is the intersection of the torus $\Hom{{\mathbb L }}{{\mathbb C }^{\,\ast}}$ and a neighborhood of the special point corresponding to ${\mathcal T }\!$; see the end of section \ref{solutions}. \begin{example}\textup{ Let ${\mathfrak a }_1,\ldots,{\mathfrak a }_6$ be the columns of the following matrix ${\mathsf A }$:} \end{example} \begin{picture}(300,80)(-90,0) \put(-75,35){\makebox(100,50){${\mathsf A }=\left( \begin{array}{rrrrrr} 1&1&1&1&1&1\\ 0&1&-1&0&1&0\\ 1&1&0&0&0&-1 \end{array} \right)$}} \put(-75,-5){\makebox(100,50){${\mathsf B }=\left( \begin{array}{rrrrrr} 1&0&0&-2&0&1\\ 0&1&1&-3&0&1\\ 0&0&1&-2&1&0 \end{array} \right)$}} \put(70,0){\makebox(50,70){$\Delta=$}} \put(145,10){\line(1,1){25}} \put(145,10){\line(-1,1){25}} \put(145,60){\line(1,0){25}} \put(145,60){\line(-1,-1){25}} \put(170,35){\line(0,1){25}} \put(145,10){\circle*{5}} \put(120,35){\circle*{5}} \put(145,35){\circle*{5}} \put(170,35){\circle*{5}} \put(145,60){\circle*{5}} \put(170,60){\circle*{5}} \put(133,53){\makebox(10,20){${\mathfrak a }_1$}} \put(172,53){\makebox(10,20){${\mathfrak a }_2$}} \put(115,33){\makebox(10,20){${\mathfrak a }_3$}} \put(143,33){\makebox(10,20){${\mathfrak a }_4$}} \put(172,31){\makebox(10,20){${\mathfrak a }_5$}} \put(149,0){\makebox(10,20){${\mathfrak a }_6$}} \end{picture} \noindent These satisfy conditions (\ref{eq:aspan}) and (\ref{eq:hyperplane}) with ${\mathbb M }={\mathbb Z }^3$ and ${\mathfrak a }_0^\vee=(1,0,0)$. Figure 1 shows all regular triangulations of the polytope $\Delta$, with two triangulations joined by an edge iff the corresponding cones in the pointed secondary fan are adjacent. \setlength{\unitlength}{.7pt} \begin{picture}(250,340)(-50,-40) \put(50,-30){\makebox(160,35){\textbf{figure 1}}} \put(-15,-20){\makebox(60,20){${\mathcal T }\!_4$}} \put(185,-20){\makebox(60,20){${\mathcal T }\!_1$}} \put(265,20){\makebox(60,20){${\mathcal T }\!_2$}} \put(-15,229){\makebox(60,20){${\mathcal T }\!_9$}} \put(75,269){\makebox(60,20){${\mathcal T }\!_8$}} \put(265,269){\makebox(60,20){${\mathcal T }\!_7$}} \put(90,65){\makebox(60,20){${\mathcal T }\!_3$}} \put(95,182){\makebox(60,20){${\mathcal T }\!_{10}$}} \put(210,125){\makebox(60,20){${\mathcal T }\!_5$}} \put(200,228){\makebox(60,20){${\mathcal T }\!_6$}} \thicklines \put(15,34){\line(0,1){160}} \put(33,15){\line(1,0){162}} \put(36,23){\line(2,1){15}} \put(64,37){\line(2,1){15}} \put(34,215){\line(1,0){82}} \put(32,223){\line(2,1){51}} 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\put(230,135){\line(-1,1){15}} \put(255,220){\line(1,1){15}} \put(255,220){\line(-1,1){15}} \put(255,250){\line(1,0){15}} \put(255,250){\line(-1,-1){15}} \put(270,235){\line(0,1){15}} \put(255,220){\circle*{3}} \put(240,235){\circle*{3}} \put(255,235){\circle*{3}} \put(270,235){\circle*{3}} \put(255,250){\circle*{3}} \put(270,250){\circle*{3}} \put(270,235){\line(-1,0){30}} \put(270,235){\line(-1,1){15}} \end{picture} The columns of the matrix ${\mathsf B }^t$ constitute a ${\mathbb Z }$-basis for ${\mathbb L }$ by means of which one can identify ${\mathbb L }$ with ${\mathbb Z }^3$ and ${\mathbb L }_{\mathbb R }^\vee$ with ${\mathbb R }^{3\vee}$. The rows ${\mathfrak b }_1,\ldots,{\mathfrak b }_6$ of the matrix ${\mathsf B }^t$ are then identified with the images of the standard basis vectors under the projection ${\mathbb R }^{6\vee}\rightarrow{\mathbb L }_{\mathbb R }^\vee$, dual to the inclusion ${\mathbb L }\subset{\mathbb Z }^6$. Thus one finds $\ell_j={\mathfrak b }_j\cdot\ell$ for every $\ell\in{\mathbb L }_{\mathbb R }\simeq{\mathbb R }^3$ and (\ref{eq:support}) becomes a condition on the signs of ${\mathfrak b }_1\cdot\ell+\gamma^\circ_1,\ldots, {\mathfrak b }_6\cdot\ell+\gamma^\circ_6\,.$ The signs give a vector in $\,\{-1,0,+1\}^6\,$ These sign vectors correspond exactly to the various strata in the stratification of ${\mathbb R }^3$ induced by the six planes ${\mathfrak b }_j\cdot{\mathsf x }+\gamma^\circ_j=0$ ($j=1,\ldots,6$). Figure 2 shows the zonotope spanned by ${\mathfrak b }_1,\ldots,{\mathfrak b }_6$. The $3-j$-dimensional faces of this zonotope correspond bijectively with the $j$-dimensional strata in the stratification for $\gamma^\circ=0$. The stratum with sign vector $(s_1,\ldots,s_6)$ corresponds with the face whose centre is $s_1{\mathfrak b }_1+s_2{\mathfrak b }_2+s_3{\mathfrak b }_3+s_4{\mathfrak b }_4+s_5{\mathfrak b }_5+s_6{\mathfrak b }_6\,.$ The vertices $1$--$14$ (resp. $15$--$28$) of the zonotope have sign vectors $(s_1,\ldots,s_6)$ (resp. $-(s_1,\ldots,s_6)$) as given in table 1. The sign vectors of all faces of the zonotope give all possible signs for $\ell=(\ell_1,\ldots,\ell_6)\in{\mathbb L }\,.$ Thus by comparing this with (\ref{eq:support}) one can see for every triangulation ${\mathcal T }\!$ what types of terms are involved in the series of $\Psi_{{\mathcal T }\!,0}$. For example for triangulation ${\mathcal T }\!_1$ the series of $\Psi_{{\mathcal T }\!_1\!,0}$ involves precisely those $\ell\in{\mathbb L }$ whose sign vector corresponds to a face of the zonotope containing at least one of the vertices $1$, $2$, $3$ or $4$. \begin{picture}(250,250)(-50,-270) \put(50,-240){\makebox(250,250){$ \begin{array}{cc} \epsfig{file=zonotope.eps,height=61mm} & \begin{array}{rrrrrrrl} 1 &+ &+ &+ &- &+ &+ & 15\\ 2 &- &+ &+ &- &+ &+ & 16\\ 3 &+ &- &+ &- &+ &+ & 17\\ 4 &+ &+ &+ &- &- &+ & 18\\ 5 &- &+ &+ &- &- &+ & 19\\ 6 &- &+ &+ &- &+ &- & 20\\ 7 &+ &- &- &- &+ &+ & 21\\ 8 &+ &- &+ &- &+ &- & 22\\ 9 &+ &+ &- &- &- &+ & 23\\ 10 &- &+ &- &- &- &+ & 24\\ 11 &- &+ &+ &+ &- &+ & 25\\ 12 &- &+ &+ &- &- &- & 26\\ 13 &- &- &+ &- &+ &- & 27\\ 14 &- &+ &+ &+ &+ &- & 28 \end{array} \\ & \\ \textbf{figure 2}&\textbf{table 1} \end{array} $}} \end{picture} $$ $$ The series $\Psi_{{\mathcal T }\!_1\!,-{\mathfrak a }_4}$ involves the same $\ell$'s with exception of $\ell=0$ (which corresponds to the $3$-dimensional the zonotope itself). Using the Pochhammer symbol notation $(x)_m:=x\,(x+1)\cdot\ldots\cdot(x+m-1)$ we have \begin{eqnarray*} \Psi_{{\mathcal T }\!_1\!,-{\mathfrak a }_4}&=& c_4\:{\mathsf e }^{-2\pi i z_4}\: T_1^{c_1}T_2^{c_2}T_5^{c_5}\times\\ &\times&\hspace{-2mm}\left\{\hspace{2mm} \sum_{p,q,r\geq 0}(-1)^q \frac{(2c_1+3c_2+2c_5+1)_{2p+3q+2r}}{(c_1)_p(c_2)_q (c_5)_r(c_1+c_2)_{p+q}(c_2+c_5)_{q+r}} T_1^pT_2^qT_5^r \right.\\ &&\hspace{5mm}-c_1\sum_{r\geq 0,-q\leq p<0}(-1)^{q+p} \frac{(2p+3q+2r)!(-p-1)!}{q!r!(p+q)!(q+r)!} T_1^{p}T_2^{q}T_5^{r} \\ &&\hspace{5mm}-c_5\sum_{p\geq 0,-q\leq r<0}(-1)^{q+r} \frac{(2p+3q+2r)!(-r-1)!}{p!q!(p+q)!(q+r)!} T_1^{p}T_2^{q}T_5^{r} \\ &&\left.\hspace{5mm}-c_2 \sum_{{-p\leq q<0\,,-r\leq q<0}} \frac{(2p+3q+2r)!(-q-1)!}{p!r!(p+q)!(q+r)!} T_1^{p}T_2^{q}T_5^{r}\hspace{2mm} \right\} \end{eqnarray*} where $$ T_1:={\mathsf e }^{2\pi i(z_1-2z_4+z_6)}\;,\hspace{3mm} T_2:={\mathsf e }^{2\pi i(z_2+z_3-3z_4+z_6)}\;,\hspace{3mm} T_5:={\mathsf e }^{2\pi i(z_3-2z_4+z_5)} $$ $$ c_4=-2c_1-3c_2-2c_5 $$ and $$ {\mathcal R }_{{\mathsf A },{\mathcal T }\!_1}=\modquot{{\mathbb Z }[c_1,c_2,c_5]}{(c_1^2-c_2^2,\, c_1^2-c_5^2,\,c_1^2+c_1c_2,\,c_1^2+c_2c_5,\,c_1c_5)}\,. $$ Note that $c_4c_1=c_4c_2=c_4c_5$. One may therefore simplify the expression for $\Psi_{{\mathcal T }\!_1\!,-{\mathfrak a }_4}$ and replace $c_2$ and $c_5$ by $c_1$. \section{Regular triangulations and the pointed secondary fan} \label{regtrifan} In this section we review some results about regular triangulations and about the pointed secondary fan, essentially following \cite{bfs}. One may take as a definition of {\it regular triangulations} that these are the triangulations produced by the construction in this section; see in particular proposition \ref{protri}. \subsection{Regular triangulations} \label{triangs} We start from a set of vectors $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ in ${\mathbb Z }^n$ satisfying condition \ref{gkzcond}. Let $\Delta=\mathrm{ conv}\,\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ denote the convex hull of this set of points in ${\mathbb R }^n$. We are interested in triangulations of $\Delta$ such that all vertices are among the marked points ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$. The notation can be conveniently simplified by referring to a simplex $\mathrm{ conv}\,\{{\mathfrak a }_{i_1},\ldots,{\mathfrak a }_{i_m}\}$ by just the index set $\{i_1,\ldots,i_m\}$. We will allways take the indices in increasing order. If ${\mathcal T }\!$ is a triangulation, we write ${\mathcal T }\!^m$ for the set of simplices with $m$ vertices. A triangulation is completely determined by its set of maximal simplices ${\mathcal T }\!^n$. For the construction of a regular triangulation we take an $N$-tuple of positive real numbers ${\mathsf d }=(d_1,\ldots,d_N)$ and consider the polytope \begin{equation}\label{eq:pd} {\mathcal P }_{\mathsf d }:=\mathrm{ conv}\,\{0,d_1^{-1}{\mathfrak a }_1,\ldots,d_N^{-1}{\mathfrak a }_N\}\:\subset\:{\mathbb R }^n\,. \end{equation} Consider a subset $I=\{i_1,\ldots,i_{n}\}$ of $\{1,\ldots,N\}$ for which ${\mathfrak a }_{i_1},\ldots,{\mathfrak a }_{i_{n}}$ are linearly independent. The affine hyperplane through $d_{i_1}^{-1}{\mathfrak a }_{i_1},\ldots,d_{i_{n}}^{-1}{\mathfrak a }_{i_{n}}$ is given by the equation $D_{{\mathsf d },I}({\mathsf x })=0$ with \begin{equation}\label{eq:faceq} D_{{\mathsf d },I}({\mathsf x })\::=\: \det\left(\begin{array}{cccc} d_{i_1}^{-1}{\mathfrak a }_{i_1}&\ldots&d_{i_{n}}^{-1}{\mathfrak a }_{i_{n}}&{\mathsf x }\\ 1&\ldots&1&1\end{array}\right) \end{equation} Write $I^\ast:=\{1,\ldots,N\}\setminus I$. Then $\{d_{i_1}^{-1}{\mathfrak a }_{i_1},\ldots,d_{i_{n}}^{-1}{\mathfrak a }_{i_{n}}\}$ lies in a codimension $1$ face of ${\mathcal P }_{\mathsf d }$ if and only if for all $j\in I^\ast$: \begin{equation}\label{eq:face} D_{{\mathsf d },I}(d_j^{-1}{\mathfrak a }_j)\cdot D_{{\mathsf d },I}(0)\geq 0 \end{equation} This face is a simplex with vertices $d_{i_1}^{-1}{\mathfrak a }_{i_1},\ldots,d_{i_{n}}^{-1}{\mathfrak a }_{i_{n}}$ iff $D_{{\mathsf d },I}(d_j^{-1}{\mathfrak a }_j)\neq 0$ for every $j\in I^\ast$. Thus if ${\mathsf d }$ does not lie on any hyperplane in ${\mathbb R }^N$ given by the vanishing of $D_{{\mathsf d },I}(d_j^{-1}{\mathfrak a }_j)$ for some $I$ and $j$ with $j\not\in I$, then all faces of ${\mathcal P }_{\mathsf d }$ opposite to the vertex $0$ are simplicial. In this case the projection with center $0$ projects the boundary of ${\mathcal P }_{\mathsf d }$ onto \emph{a triangulation ${\mathcal T }\!$ of $\Delta$.} The maximal simplices of ${\mathcal T }\!$ are those $I=\{i_1,\ldots,i_{n}\}$ for which $D_{{\mathsf d },I}(d_j^{-1}{\mathfrak a }_j)\cdot D_{{\mathsf d },I}(0)>0$ holds for every $j\in I^\ast\,.$ \ Let ${\mathsf A }=(a_{ij})$ denote the $n\times N$-matrix with columns ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$. The triangulation obviously depends only on ${\mathsf d }$ modulo the row space of ${\mathsf A }\,.$ Let us reformulate the above construction accordingly. Take ${\mathbb L }=\ker{\mathsf A }\subset{\mathbb Z }^N$ as in (\ref{eq:L}). Assumption (\ref{eq:hyperplane}) implies $\ell_1+\ldots+\ell_N\:=\:0$ for every $\ell=(\ell_1,\ldots,\ell_N)^t\in{\mathbb L }\,.$ Take an ${(N-n)}\times N$-matrix ${\mathsf B }$ with entries in ${\mathbb Z }$ such that columns of ${\mathsf B }^t$ constitute a basis for ${\mathbb L }\,.$ Let $w\in{\mathbb R }^{N-n}$. Then there exists a row vector of positive real numbers ${\mathsf d }=(d_1,\ldots,d_N)$ such that $w\:=\:{\mathsf B }{\mathsf d }^t$. Take the matrices $$ \widetilde{{\mathsf A }}\::=\;\left(\begin{array}{c|c} {\mathsf A } & 0\\ \hrulefill&\hrulefill\\ {\mathsf d } &1\end{array}\right) \hspace{5mm}\textrm{ and}\hspace{5mm} \widetilde{{\mathsf B }}\::=\:\left(\begin{array}{c|c}{\mathsf B }&-w\end{array}\right)\,. $$ Denote by $\widetilde{{\mathsf A }}_K$ (resp. $\widetilde{{\mathsf B }}_K$) the submatrix of $\widetilde{{\mathsf A }}$ (resp. $\widetilde{{\mathsf B }}$) composed of the entries with column index in a subset $K$ of $\{1,\ldots,N+1\}$. Since ${\mathrm{ rank}\,\,}\widetilde{{\mathsf A }}\,=\,n+1\:,$ ${\mathrm{ rank}\,\,}\widetilde{{\mathsf B }}\,=\,{N-n}$ and $\widetilde{{\mathsf A }}\cdot\widetilde{{\mathsf B }}^t\:=\:0$ there is a non-zero $r\in{\mathbb Q }$ such that for every $J\subset\{1,\ldots,N+1\}$ of cardinality $n+1$ and $J^\prime=\{1,\ldots,N+1\}\setminus J$ $$ \det(\widetilde{{\mathsf A }}_J)\,=\, (-1)^{\sum_{j\in J} j}\;r\,\det(\widetilde{{\mathsf B }}_{J^\prime}) $$ One sees that (\ref{eq:face}) is equivalent to \begin{equation}\label{eq:om} (-1)^{\sharp \{h\in I^\ast\:|\: h> j\}} \det\left(\left(\begin{array}{c|c}{\mathsf B }_{I^\ast\setminus\{j\}} &w\end{array}\right)\right) \cdot\det\left({\mathsf B }_{I^\ast}\right)\;\geq\;0\,; \end{equation} here ${\mathsf B }_{I^\ast}$ resp. ${\mathsf B }_{I^\ast\setminus\{j\}}$ is the submatrix of ${\mathsf B }$ consisting of the entries with column index in $I^\ast$ resp. $I^\ast\setminus\{j\}\,.$ \emph{Thus the triangulation ${\mathcal T }\!$ can also be constructed from (\ref{eq:om}).} \subsection{The pointed secondary fan}\label{secfan} For a more intrinsic formulation which does not refer to a choice of a basis for ${\mathbb L }$ we consider the ${(N-n)}$-dimensional real vector space ${\mathbb L }_{\mathbb R }^\vee\::=\:\Hom{{\mathbb L }}{{\mathbb R }}\,.$ Let ${\mathfrak b }_1,\ldots,{\mathfrak b }_N\in{\mathbb L }_{\mathbb R }^\vee$ be the images of the standard basis vectors of ${\mathbb R }^{N\vee}$ under the surjection ${\mathbb R }^{N\vee}\twoheadrightarrow{\mathbb L }_{\mathbb R }^\vee$ dual to the inclusion ${\mathbb L }\hookrightarrow{\mathbb Z }^N$. Let ${\mathfrak B }$ (resp. ${\mathfrak D }$) be the collection of those subsets $J$ of $\{1,\ldots,N\}$ of cardinality ${N-n}$ (resp. ${N-n}-1$) for which the vectors ${\mathfrak b }_j$ ($j\in J$) are linearly independent. For $K=\{k_1,\ldots,k_{N-n}\}\in{\mathfrak B }$ and $J=\{j_1,\ldots,j_{{N-n}-1}\}\in{\mathfrak D }$ we write \begin{eqnarray*} {\mathcal C }_K&:=&\{t_1{\mathfrak b }_{k_1}+\ldots+t_{N-n}{\mathfrak b }_{k_{N-n}}\in{\mathbb L }_{\mathbb R }^\vee\;|\; t_1,\ldots,t_{N-n}\in{\mathbb R }_{\geq 0}\;\}\\ H_J&:=&\{t_1{\mathfrak b }_{j_1}+\ldots+t_{{N-n}-1}{\mathfrak b }_{j_{{N-n}-1}}\in{\mathbb L }_{\mathbb R }^\vee\;|\; t_1,\ldots,t_{{N-n}-1}\in{\mathbb R }\;\}\,. \end{eqnarray*} Choosing a basis for ${\mathbb L }$ as before one can identify ${\mathbb L }_{\mathbb R }^\vee$ with ${\mathbb R }^{{N-n}\vee}$ and ${\mathfrak b }_1,\ldots,{\mathfrak b }_N$ with the rows of matrix ${\mathsf B }^t$. The inequality (\ref{eq:om}) becomes equivalent to the statement $w\in{\mathcal C }_{I^\ast}\,.$ The condition $D_{{\mathsf d },I}(d_j^{-1}{\mathfrak a }_j)\neq 0$ for the left hand factor in (\ref{eq:face}) becomes equivalent to $w\not\in H_J$ for $J=\{1,\ldots,N+1\}\setminus(I\cup\{j\})$. Thus the preceding discussion shows: \ \begin{proposition}\label{protri} \textup{ (cf. \cite{bfs} lemma 4.3.)} For $w\in{\mathbb L }_{\mathbb R }^\vee\setminus\bigcup_{J\in{\mathfrak D }}H_J$ the set $$ {\mathcal T }\!^n\::=\:\{\,I\:|\:I^\ast\in{\mathfrak B }\textrm{ and }w\in{\mathcal C }_{I^\ast}\,\} $$ is the set of maximal simplices of a regular triangulation ${\mathcal T }\!$ of $\Delta\,$.\\ \textup{(Recall the notation $I^\ast:=\{1,\ldots,N\}\setminus I\,.$)} {\hfill$\boxtimes$} \end{proposition} \ If ${\mathcal T }\!$ is a regular triangulation of $\Delta$ write \begin{equation}\label{eq:ct} {\mathcal C }_{{\mathcal T }\!}\:=\:\bigcap_{I\in{\mathcal T }\!^n} {\mathcal C }_{I^\ast}\,. \end{equation} Then every $w\in{\mathcal C }_{{\mathcal T }\!}\setminus\bigcup_{J\in{\mathfrak D }}H_J$ leads by the above construction to the same triangulation ${\mathcal T }\!$. The cones ${\mathcal C }_{\mathcal T }\!$ one obtains in this way from all regular triangulations of $\Delta$ constitute the collection of maximal cones of a complete fan in ${\mathbb L }_{\mathbb R }^\vee\,.$ This fan is called \emph{ the pointed secondary fan}. \ \begin{remark}\label{delzant}\textup{ The dual (or polar) set of ${\mathcal P }_{\mathsf d }$ in (\ref{eq:pd}) is (e.g. \cite{bat1} def.4.1.1, \cite{ful} p.24) \begin{equation}\label{eq:dupo} {\mathcal P }_{\mathsf d }^\vee\::=\:\{\;{\mathsf y }\in{\mathbb R }^{n\vee}\:|\;{\mathsf y }\cdot{\mathsf x }\geq -1 \;\textrm{ for all }\;{\mathsf x }\in{\mathcal P }_{\mathsf d }\;\} \end{equation} It is the intersection of half-spaces given by the inequalities $$ {\mathsf y }\cdot{\mathfrak a }_i+d_i\geq 0\hspace{5mm} (i=1,\ldots,N) $$ ${\mathcal P }_{\mathsf d }^\vee$ is an unbounded polyhedron. Its vertices correspond with the codimension $1$ faces of ${\mathcal P }_{\mathsf d }$ which do not contain $0$.} \textup{ Adding to ${\mathsf d }$ an element of the row space of matrix ${\mathsf A }$ amounts to just a translation of the polyhedron ${\mathcal P }_{\mathsf d }^\vee$ in ${\mathbb R }^{n\vee}$. If ${\mathsf d }$ gives rise to a unimodular triangulation, then ${\mathcal P }_{\mathsf d }^\vee$ is an (unbounded) \emph{Delzant polyhedron} in the sense of \cite{guillemin} p.8. Thus, by the constructions in \cite{guillemin}} a point in the real cone ${\mathcal C }_{{\mathcal T }\!}$ for a unimodular regular triangulation ${\mathcal T }\!$ can be interpreted as a parameter for the symplectic structure of a toric variety. In view of formula (\ref{eq:domain4}) this applies in particular to the imaginary part of the variable ${\mathsf z }$ in (\ref{eq:form2}). \end{remark} \section{The ring ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\,.$} \label{thering} \begin{theorem}\label{dims} Consider the ring ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ as in definition \ref{ring}. \begin{enumerate} \item ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is a free ${\mathbb Z }[D^{-1}]$-module of rank $\sharp\,{\mathcal T }\!^n\,.$ \item ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is a graded ring. Let ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(k)}$ denote its homogeneous component of degree $k$. Then the Poincar\'e series of ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is: $$ \sum_{k\geqslant 0} \left(\mathrm{ rank}\,\,{\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(k)}\right)\,\tau^k\:=\: \sum_{m=0}^{n}\sharp\, (\,{\mathcal T }\!^m\,)\,\tau^m(1-\tau)^{n-m} $$ where $\sharp\, (\,{\mathcal T }\!^m\,)\,=$ the number of simplices with $m$ vertices; $\sharp\, (\,{\mathcal T }\!^0\,)=1$ by convention. In particular $$ {\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(k)}=0\hspace{4mm}\textrm{for}\hspace{3mm} k\geqn\,. $$ \item $\{c_I\:|\:I\in{\mathcal T }\!^n\}$ is a ${\mathbb Z }[D^{-1}]$-basis for ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\,.$ \hspace{3mm} (cf. formula (\ref{eq:cbas})) \end{enumerate} \end{theorem} The \textbf{proof of theorem \ref{dims}} closely follows the proofs of Danilov ( \cite{dani} $\S$ 10) and Fulton ( \cite{ful} $\S$ 5.2) for the analogous presentation of the Chow ring of a complete simplicial toric variety. We include a proof here in order check that it needs no reference to algebraic cycles and also works when the simplicial complex is homeomorphic to a ball instead of a sphere as in \cite{dani,ful}. For the construction of a basis for ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ we choose a vector $\xi$ in $\Delta$ which should be linearly independent from every $n-1$-tuple of vectors in $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$. If $I$ is a maximal simplex, then $\{{\mathfrak a }_i\}_{i\in I}$ is a basis of ${\mathbb R }^n$ and $\xi\,=\,\sum_{i\in I} x_i{\mathfrak a }_i$ with all $x_i\neq 0\,.$ We define \begin{eqnarray} \label{eq:imin} I^-\:&:=&\:\{\,i\in I\:|\: x_i<0\,\}\:, \\ c_I\:&:=&\:\prod_{i\in I^-}\;c_i \:.\label{eq:cbas} \end{eqnarray} \ Let ${\mathcal T }\!$ be associated with ${\mathsf d }=(d_1,\ldots,d_N)$ as in section \ref{triangs}. For $I\in{\mathcal T }\!^n$ let $p_I$ be the positive real number such that $p_I\xi$ lies in the affine hyperplane through the points $d_{i}^{-1}{\mathfrak a }_{i}$ with $i\in I$, i.e. $D_{{\mathsf d },I}(p_I\xi)=0\,.$ We may assume that ${\mathsf d }$ is chosen such that $p_{I_1}\neq p_{I_2}$ whenever $I_1\neq I_2\,;$ indeed, for $I_1\neq I_2$ the equality $p_{I_1}= p_{I_2}$ amounts to a non-trivial linear equation for $d_1,\ldots,d_N\,.$ As in \cite{ful} we define a total ordering on ${\mathcal T }\!^n$: \begin{equation}\label{eq:ord} I_1<I_2 \hspace{4mm}\textrm{ iff }\hspace{4mm} p_{I_1}<p_{I_2}\,. \end{equation} \begin{lemma}\label{grot}\textup{ (cf. \cite{ful} p.101($\ast$))} If $I_1^-\subset I_2$ then $I_1\leq I_2$ . \end{lemma} \textbf{proof:} By definition of $p_{I_1}$ there exist $s_j\in{\mathbb R }$ such that $ p_{I_1}\xi\:=\sum_{j\in I_1} s_j d_j^{-1} {\mathfrak a }_j $ and $ 1\:=\sum_{j\in I_1} s_j \:. $ If $I_1\neq I_2$ and $I_1^-\subset I_2$ then $s_j>0$ for every $j\in I_1\setminus I_2\,.$ Using this and (\ref{eq:face}) for $I_2$ one checks: $D_{{\mathsf d },I_2}(p_{I_1}\xi)\cdot D_{{\mathsf d },I_2}(0)>0\,.$ This shows that $0$ and $p_{I_1}\xi$ lie on the same side of the affine hyperplane through the points $d_i^{-1}{\mathfrak a }_i$ with $i\in I_2\,.$ Hence: $p_{I_1}<p_{I_2}\,.$ {\hfill$\boxtimes$} \begin{lemma}\label{tussen}\textup{ (cf. \cite{ful} p.102)} Let $J$ be a simplex in ${\mathcal T }\!$. Then: $I^-\:\subset\:J\:\subset\:I$ where $I:=\min\{I^\prime\in{\mathcal T }\!^n\:|\:J\subset I^\prime\}\,.$ \end{lemma} \textbf{proof:} The conclusion is clear if $I=J$. So assume $I\neq J$ and take $i\in I\setminus J$. Then $I\setminus\{i\}$ is a codim $1$ simplex in the triangulation, which either is contained in the boundary of $\Delta$ or is contained in another maximal simplex $I^\prime\neq I$. If $I\setminus\{i\}$ is a boundary simplex, then $\xi$ and ${\mathfrak a }_i$ are on the same side of the linear hyperplane in ${\mathbb R }^n$ spanned be the vectors ${\mathfrak a }_j$ with $j\in I\setminus\{i\}\,.$ This implies $x_i>0$ in the expansion $\xi\,=\,\sum_{j\in I} x_j{\mathfrak a }_j$. So $i\not\in I^-$. If $I\setminus\{i\}$ is contained in a maximal simplex $I^\prime\neq I$, then $J\subset I^\prime$ and hence $I<I^\prime$. Now look at the two expansions $\xi\,=x_i{\mathfrak a }_i+\,\sum_{j\in I\cap I^\prime} x_j{\mathfrak a }_j$ and $\xi\,=\,y_k{\mathfrak a }_k+\,\sum_{j\in I\cap I^\prime} y_j{\mathfrak a }_j$ where $\{k\}=I^\prime\setminus (I\cap I^\prime)$. Then $y_k<0$ because $I^{\prime -}\not\subset I$ by the preceding lemma. On the other hand, $x_i$ and $y_k$ have different signs because ${\mathfrak a }_i$ and ${\mathfrak a }_k$ lie on different sides of the linear hyperplane spanned by the vectors ${\mathfrak a }_j$ with $j\in I\cap I^\prime$. We see $x_i>0$ and $i\not\in I^-$. Conclusion: $I^-\subset J$. {\hfill$\boxtimes$} \begin{proposition}\label{generate} The elements $c_I$ $(\,I\in{\mathcal T }\!^n\,)$ generate ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ as a ${\mathbb Z }[D^{-1}]$-module. \end{proposition} \textbf{proof:} ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is linearly generated by monomials in $c_1,\ldots,c_N\,.$ For one $I_0\in{\mathcal T }\!^n$ we have $I_0^-=\emptyset\,,$ hence $c_{I_0}=1\,.$ Therefore we only need to show that for every $j$ and every $I_1$ the product $c_j\cdot c_{I_1}$ can be written as a linear combination of $c_I$'s. If $j\in I_1$ one can use the linear relations (\ref{eq:linrel}) to express every $c_i$ with $i\in I_1$ as a ${\mathbb Z }[D^{-1}]$-linear combination of $c_k$'s with $k\not\in I_1\,.$ Since this works for $c_j$ in particular, the problem can be reduced to showing that a monomial of the form $\prod_{i\in J} c_i$ with $J$ a simplex of the triangulation, can be written as a linear combination of $c_I$'s. Given such a $J$ take $I_J\in{\mathcal T }\!^n$ such that $I_J^-\subset J\subset I_J\,;$ see lemma \ref{tussen}. If $J=I_J^-$, then $\prod_{i\in J} c_i\,=\,c_{I_{\!J}}$ and we are done. If $J\neq I_J^-$ take $m\in J\setminus I_J^-$ and use the linear relations (\ref{eq:linrel}) to rewrite $c_m$ as a ${\mathbb Z }[D^{-1}]$-linear combination of $c_k$'s with $k\not\in I_J\,.$ This leads to an expression for $\prod_{i\in J} c_i$ as a ${\mathbb Z }[D^{-1}]$-linear combination of monomials of the form $\prod_{i\in K} c_i$ with $K$ a simplex of the triangulation and $I_J^-\subsetneq K\,.$ Given such a $K$ take $I_K\in{\mathcal T }\!^n$ such that $I_K^-\subset K\subset I_K\,.$ Then, according to lemma \ref{grot}, $I_J<I_K\,.$ We proceed by induction.{\hfill$\boxtimes$} \ Next we follow Danilov's arguments in \cite{dani} remark 3.8 to prove \begin{equation}\label{eq:poinc} \sum_{k\geqslant 0} \left(\dim _{{\mathbb Q }} \,{\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(k)}\otimes{\mathbb Q }\right)\,\tau^k\:=\: \sum_{m=0}^{n}\sharp\, (\,{\mathcal T }\!^m\,)\,\tau^m(1-\tau)^{n-m} \end{equation} We have added a few references of which \cite{munk} is most relevant because it deals with a triangulation of a polytope, while \cite{dani} deals with a triangulation of a sphere. In \cite{stan,munk} the Stanley-Reisner ring ${\mathbb Q }[{\mathcal T }\!\,]$ of the simplicial complex ${\mathcal T }\!$ over the field ${\mathbb Q }$ is defined as the quotient of the polynomial ring ${\mathbb Q }[C_1,\ldots,C_N]$ modulo the ideal generated by the monomials (\ref{eq:srrel}). ${\mathbb Q }[{\mathcal T }\!]$ is a Cohen-Macaulay ring of Krull dimension $n$; see \cite{munk} thm.2.2 and \cite{stan} thm 1.3. By definition \ref{ring} there is a natural homomorphism ${\mathbb Q }[{\mathcal T }\!]\rightarrow{\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$ with kernel generated by the $n$ elements $\alpha_i:=a_{i1}C_1+a_{i2}C_2+\ldots+a_{iN}C_N$. By proposition \ref{generate} the ring ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$ is a finite dimensional ${\mathbb Q }$-vector space and hence has Krull dimension $0$. It also follows from proposition \ref{generate} that ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$ and ${\mathbb Q }[{\mathcal T }\!]$ are local rings. We can now apply \cite{matsumura} thm.16.B and see that $\alpha_1,\ldots,\alpha_n$ is a regular sequence. As pointed out in \cite{dani} remark 3.8b this implies that the Poincar\'e series of ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$ is equal to $(1-\lambda)^n$ times the Poincar\'e series of ${\mathbb Q }[{\mathcal T }\!\,]$. The latter is $\sum_{m=0}^n\,\sharp\,({\mathcal T }\!^m)\,\lambda^m(1-\lambda)^{-m}$ by \cite{stan} thm. 1.4 (where it is called Hilbert series). Formula (\ref{eq:poinc}) follows. We see that $\dim_{\mathbb Q } {\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }=\sharp\,{\mathcal T }\!^n$ and hence that the elements $c_I$ $(I\in{\mathcal T }\!^n)$ are linearly independent over ${\mathbb Q }$. \emph{This completes the proof of theorem \ref{dims}}{\hfill$\boxtimes$} \ \begin{corollary}\label{allvert} If ${\mathcal T }\!$ is unimodular, then \begin{enumerate} \item ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is a free ${\mathbb Z }$-module with rank equal to $\mathrm{ vol}\,\Delta$. \item $\Delta\,\cap\,{\mathbb Z }^n \:=\:{\mathcal T }\!^1\:=\: \{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ \item there is an isomorphism ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(1)}\stackrel{\sim}{\rightarrow} {\mathbb L }_{\mathbb Z }^\vee $ such that $ c_j\leftrightarrow{\mathfrak b }_j$ ($j=1,\ldots,N$) \end{enumerate} \end{corollary} \textbf{proof:} \textbf{(i)} immediately follows from theorem \ref{dims}. \\ \textbf{(ii)} Assume that there is a lattice point in $\Delta$ which is not a vertex of ${\mathcal T }\!\,.$ This point lies in some maximal simplex and gives rise to a decomposition of this simplex into at least two integral simplices. This contradicts the assumption. \\ \textbf{(iii)} Because of (ii) all monomials in (\ref{eq:srrel}) have degree $\geq 2$. Consequently, ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(1)}$ is just the quotient of ${\mathbb Z }\,C_1\oplus\ldots\oplus{\mathbb Z }\,C_N$ modulo the span of the linear forms in (\ref{eq:linrel}). This quotient is ${\mathbb L }_{\mathbb Z }^\vee$. {\hfill$\boxtimes$} \section{A domain of definition for the function $\Psi_{{\mathcal T }\!,\beta}\,.$} \label{solutions} We first investigate for which $\lambda$'s one possibly has $Q_\lambda({\mathsf c })\neq 0$ in ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }\,.$ For $I\in{\mathcal T }\!^n$ let ${\mathsf A }_I$ denote the $n\timesn$-submatrix of ${\mathsf A }$ with columns ${\mathfrak a }_i\;(i\in I)\,.$ By ${\mathsf p }_I$ we denote the $N\times N$-matrix whose entries with row index not in $I$ are all $0$ and whose $n\times N$-submatrix of entries with row index in $I$ is ${\mathsf A }_I^{-1}{\mathsf A }\,.$ This ${\mathsf p }_I$ is an idempotent linear operator on ${\mathbb R }^N\,.$ Now define: \begin{equation}\label{eq:project} {\mathfrak P }_{\mathcal T }\!\::=\:\mathrm{ conv}\,\{{\mathsf p }_I\:|\:I\in{\mathcal T }\!^n\} \hspace{4mm}\textrm{in}\hspace{3mm}\textrm{Mat}_{N\times N}({\mathbb R })\,. \end{equation} The image of the idempotent operator ${\mathsf 1 }-{\mathsf p }_I$ is ${\mathbb L }_{\mathbb R }\,.$ Therefore all elements of ${\mathsf 1 }-{\mathfrak P }_{\mathcal T }\!=\mathrm{ conv}\,\{{\mathsf 1 }-{\mathsf p }_I\:|\:I\in{\mathcal T }\!^n\}$ are idempotent operators on ${\mathbb R }^N$ with image ${\mathbb L }_{\mathbb R }\,.$ Hence all elements of ${\mathfrak P }_{\mathcal T }\!$ are also idempotent operators on ${\mathbb R }^N$. For every $\lambda\in{\mathbb Z }^N$ one has the polytope ${\mathfrak P }_{\mathcal T }\!(\lambda)$ which is the convex hull of $\{{\mathsf p }_I(\lambda)\:|\:I\in{\mathcal T }\!^n\}$ in ${\mathbb R }^N\,.$ This obviously depends only on $\lambda\bmod {\mathbb L }\,.$ \ \begin{lemma}\label{support} If $\lambda\in{\mathbb Z }^N$ is such that $Q_\lambda({\mathsf c })\neq 0$ in ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$, then $\lambda$ lies in the set ${\mathfrak P }_{\mathcal T }\!(\lambda)\:+\:\,{\mathcal C }_{\mathcal T }\!^\vee$. Here ${\mathcal C }_{\mathcal T }\!^\vee$ is the dual of the cone ${\mathcal C }_{{\mathcal T }\!}$ defined in (\ref{eq:ct}): \begin{equation}\label{eq:duco} {\mathcal C }_{{\mathcal T }\!}^\vee\::=\:\{\ell\in{\mathbb L }_{\mathbb R }\;|\;\omega\cdot\ell\geq 0 \;\textrm{ for all }\;\omega\in{\mathcal C }_{\mathcal T }\!\;\}\,. \end{equation} \end{lemma} \textbf{proof:} If $Q_\lambda({\mathsf c })\neq 0$ then $\{i\,|\:\lambda_i<0\}$ is contained in some maximal simplex $I$ of ${\mathcal T }\!\,.$ Let $\ell=({\mathsf 1 }-{\mathsf p }_I)(\lambda)\,.$ Then $\ell=(\ell_1,\ldots,\ell_N)\in{\mathbb L }_{\mathbb R }$ and ${\mathfrak b }_j\cdot\ell=\ell_j=\lambda_j\geq 0$ for all $j\in I^\ast\,.$ This shows $({\mathsf 1 }-{\mathsf p }_I)(\lambda)\in{\mathcal C }_{I^\ast}^\vee\subset{\mathcal C }_{\mathcal T }\!^\vee\,.$ {\hfill$\boxtimes$} \ \begin{lemma}\label{estimate} The coefficients in the power series expansion \begin{equation}\label{eq:qx} \frac{\prod_{\lambda_j<0}\prod_{k=0}^{-\lambda_j-1}(k+x_j)} {\prod_{\lambda_j>0}\prod_{k=1}^{\lambda_j}(k-x_j)} \,=\,\sum_{m_1,\ldots,m_N\geq 0} K_{m_1,\ldots,m_N} x_1^{m_1}\cdot\ldots\cdot x_N^{m_N} \end{equation} satisfy \begin{equation}\label{eq:schat} 0\leq K_{m_1,\ldots,m_N}\,\leq\, N^{\parallel\lambda\parallel} \cdot2^{\parallel m\parallel+N}\cdot N!\cdot (\max(1,N-\deg\lambda))! \end{equation} with $\parallel m\parallel:=\sum_{i=1}^N m_i$ and $\parallel\lambda\parallel:=\sum_{i=1}^N |\lambda_i|$ and $\deg\lambda:=\sum_{i=1}^N \lambda_i={\mathfrak a }_0^\vee\cdot\beta\,.$ \end{lemma} \textbf{proof:} Clearly $K_{m_1,\ldots,m_N}\geq 0$. Clearly also $2^{-\parallel m\parallel}K_{m_1,\ldots,m_N}$ is less than the value of the left hand side at $x_1=\ldots=x_N\,=\,\frac{1}{2}\,.$ Therefore $$ K_{m_1,\ldots,m_N}\:<\: 2^{\parallel m\parallel +S}\cdot\frac{\prod_{\lambda_j<0}\,(-\lambda_j)!}{ \prod_{\lambda_j>0}\,(\lambda_j-1)!} \:\leq\: 2^{\parallel m\parallel+S}\cdot\frac{P!}{(R-S)!}\cdot S^{R-S} $$ where $P\,=\,-\sum_{\lambda_i<0}\lambda_i$ and $R\,=\,\sum_{\lambda_i>0}\lambda_i$ and $S\,=\,\sharp\{i\,|\,\lambda_i>0\}\,.$ If $P\leq R-S$ then $\frac{P!}{(R-S)!}\leq 1$. If $P>R-S$ then $\frac{P!}{(R-S)!}\,\leq\, 2^P\cdot (P-R+S)!$. Combining these estimates one arrives at (\ref{eq:schat}). {\hfill$\boxtimes$} \ The sum of the series $\sum_{\lambda\in{\mathcal A }^{-1}(\beta)}\; Q_\lambda({\mathsf c }){\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot\lambda}$ in formula (\ref{eq:form2}) should be computed as a limit for $L\rightarrow\infty$ of partial sums $\Sigma_L$ taking only terms with $\parallel\lambda\parallel\:\leq L\,.$ These sums only involve $\lambda$'s with $Q_\lambda({\mathsf c })\neq 0\,.$ According to lemma \ref{support} such a $\lambda$ is of the form $\lambda=\tilde{\lambda}+\ell$ with $\ell\in{\mathcal C }_{\mathcal T }\!^\vee$ and with $\tilde{\lambda}$ contained in a compact polytope which only depends on $\beta$. Therefore $\parallel \lambda\parallel\leq\parallel \ell\parallel +$ some constant which only depends on $\beta$. Since $$ Q_\lambda({\mathsf c })=(-1)^{\sharp\{i\,|\,\lambda_i<0\}} \sum_{m_1,\ldots,m_N\geq 0,\,\parallel m\parallel\leq n} (-1)^{\parallel m\parallel} K_{m_1,\ldots,m_N} c_1^{m_1}\cdot\ldots\cdot c_N^{m_N} $$ lemma \ref{estimate} shows that the coordinates of $Q_\lambda({\mathsf c })$ with respect to a basis of the vector space ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$ are less than $N^{\parallel\ell\parallel}$ times some constant which only depends on $\beta\,.$ Thus one sees that the limit of the partial sums exists if the imaginary part $\Im\,{\mathsf z }$ of ${\mathsf z }$ satisfies \begin{equation}\label{eq:domain1} \Im\,{\mathsf z }\cdot\ell\:>\: \frac{\log N}{2\pi}\parallel\ell\parallel \hspace{4mm}\textrm{ for all } \ell\in{\mathcal C }_{\mathcal T }\!^\vee \end{equation} Let $p:{\mathbb R }^{N\vee}\rightarrow{\mathbb L }_{\mathbb R }^\vee$ denote the canonical projection. If $b\in{\mathbb L }_{\mathbb R }^\vee$ is any vector which satisfies \begin{equation}\label{eq:bdom} b\cdot\ell\:>\:\frac{\log N}{2\pi}\parallel\ell\parallel \hspace{4mm}\textrm{ for all } \ell\in{\mathcal C }_{\mathcal T }\!^\vee\,, \end{equation} then $b\in{\mathcal C }_{\mathcal T }\!$ and every ${\mathsf z }$ with the property $ p\,(\Im\,{\mathsf z })\in b+{\mathcal C }_{\mathcal T }\!$ satisfies (\ref{eq:domain1}). Let us therefore define \begin{equation}\label{eq:domain4} {\mathcal B }_{\mathcal T }\!\::=\: \bigcup_{b\;\textrm{ s.t. (\ref{eq:bdom})}}\; (\:b\:+\:{\mathcal C }_{\mathcal T }\!\:) \end{equation} The above discussion proves: \ \begin{theorem}\label{defdom} Formula (\ref{eq:form2}): $$ \Psi_{{\mathcal T }\!,\beta}({\mathsf z })\::=\: \sum_{\lambda\in{\mathcal A }^{-1}(\beta)}\hspace{2mm} Q_\lambda({\mathsf c }){\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot\lambda}\cdot{\mathsf e }^{{2\pi{\mathsf i}\, }{\mathsf z }\cdot{\mathsf c }} $$ defines a function with values in ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\,\otimes{\mathbb C }$ on the domain \begin{equation}\label{eq:defdom} {\mathcal V }_{\mathcal T }\!\::=\{{\mathsf z }\in{\mathbb C }^{N\vee}\;|\; p\,(\Im\,{\mathsf z })\in{\mathcal B }_{\mathcal T }\!\;\}\,. \end{equation} {\hfill$\boxtimes$} \end{theorem} \ In order to have a more global geometric picture of where the domain of definition of the function $\Psi_{{\mathcal T }\!,\beta}$ is situated we give a brief description of \emph{the toric variety associated with the pointed secondary fan.} The pointed secondary fan is a complete fan of strongly convex polyhedral cones which are generated by vectors from the lattice ${\mathbb L }_{\mathbb Z }^\vee\,.$ By the general theory of toric varieties \cite{ful,oda} this lattice-fan pair gives rise to a toric variety. In the case of ${\mathbb L }_{\mathbb Z } ^\vee$ and the pointed secondary fan the general construction reads as follows. For each regular triangulation ${\mathcal T }\!$ one has the cone ${\mathcal C }_{\mathcal T }\!$ in the secondary fan and one considers the monoid ring ${\mathbb Z }[{\mathbb L }_{\mathcal T }\!]$ of the sub-monoid ${\mathbb L }_{\mathcal T }\!$ of ${\mathbb L }\,:$ \begin{equation}\label{eq:lt} {\mathbb L }_{\mathcal T }\!\::={\mathbb L }\cap{\mathcal C }_{\mathcal T }\!^\vee\:=\: \{\:\ell\in{\mathbb L }\:|\: \omega\cdot\ell\geq 0\hspace{3mm} \textrm{ for all }\omega\in{\mathcal C }_{\mathcal T }\!\:\}\,. \end{equation} The affine schemes ${\mathcal U }_{\mathcal T }\!\::=\:{\mathrm{ spec}\,\:}{\mathbb Z }[{\mathbb L }_{\mathcal T }\!]$ for the various triangulations naturally glue together to form the toric variety for the pointed secondary fan. A complex point of ${\mathcal U }_{\mathcal T }\!$ is just a homomorphism from the additive monoid ${\mathbb L }_{\mathcal T }\!$ to the multiplicative monoid ${\mathbb C }\,.$ There is a special point in ${\mathcal U }_{\mathcal T }\!$, namely the homomorphism sending $0\in{\mathbb L }_{\mathcal T }\!$ to $1$ and all other elements of ${\mathbb L }_{\mathcal T }\!$ to $0$. A disc of radius $r\,,$ $0<r<1\,,$ about this special point consists of homomorphisms ${\mathbb L }_{\mathcal T }\!\rightarrow{\mathbb C }$ with image contained in the disc of radius $r$ in ${\mathbb C }\,.$ A vector ${\mathsf z }\in{\mathbb C }^{N\vee}$ defines the homomorphism $$ {\mathbb L }\rightarrow{\mathbb C }^\ast\,,\;\;\ell\mapsto {\mathsf e }^{2\pi i {\mathsf z }\cdot\ell} $$ and hence a point of the toric variety. The point lies in the disc of radius $r<1$ about the special point corresponding to a regular triangulation ${\mathcal T }\!$ iff $\Im{\mathsf z }\cdot\ell>-\frac{\log r}{2\pi}$ holds for every $\ell\in{\mathbb L }_{\mathcal T }\!$. It suffices of course to require this only for a set of generators of ${\mathbb L }_{\mathcal T }\!$. If $b$ is in ${\mathcal C }_{\mathcal T }\!$ and $K$ is such that $K>b\cdot\ell$ for all $\ell$ from a set of generators of ${\mathbb L }_{\mathcal T }\!$, then the set $\{\:{\mathsf z }\in{\mathbb C }^{N\vee} \;|\; p\,(\Im\,{\mathsf z })\in b+{\mathcal C }_{\mathcal T }\!\:\}$ contains the intersection of the disc of radius $\exp(-2\pi K)$ with the torus $\Hom{{\mathbb L }}{{\mathbb C }^\ast}\,.$ \emph{This shows that the domain of definition of the function $\Psi_{{\mathcal T }\!,\beta}$ is situated about the special point associated with ${\mathcal T }\!$ in the toric variety of the pointed secondary fan.} \section{The special case $\beta=0$} \label{betanul} The function $\Psi_{{\mathcal T }\!,0}$ is invariant under the action of ${\mathbb M }_{\mathbb C }^\vee$; see (\ref{eq:character}). So it is in fact a function on the domain ${\mathbb L }_{\mathbb R }^\vee\:+\:\sqrt{-1} {\mathcal B }_{\mathcal T }\!$ in ${\mathbb L }_{\mathbb C }^\vee\,.$ For $F\in\Hom{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{{\mathbb C }}$ we have the ${\mathbb C }$-valued function $F\Psi_{{\mathcal T }\!,0}$ on ${\mathbb L }_{\mathbb R }^\vee\:+\:\sqrt{-1} {\mathcal B }_{\mathcal T }\!\,.$ \ \begin{lemma}\label{locsys} If $F\Psi_{{\mathcal T }\!,0}$ is the $0$-function on ${\mathbb L }_{\mathbb R }^\vee\:+\:\sqrt{-1} {\mathcal B }_{\mathcal T }\!$ then $F\,=\,0$. \end{lemma} \textbf{proof:} By lemma \ref{support} the series $\Psi_{{\mathcal T }\!,0}$ involves only $\lambda$'s in ${\mathbb L }\cap{\mathcal C }_{\mathcal T }\!^\vee$ and $\lambda=0$ is really present with $Q_0({\mathsf c })=1$. Moreover ${\mathcal B }_{\mathcal T }\!$ is contained in the interior of ${\mathcal C }_{\mathcal T }\!\,.$ Therefore, if $F\Psi_{{\mathcal T }\!,0}\,=\,0\,,$ then the polynomial function $$ \sum_{m_1,\ldots,m_N\geq 0} \frac{({2\pi{\mathsf i}\, })^{m_1+\ldots+m_N}}{m_1!\cdot\ldots\cdot m_N!}\cdot F(c_1^{m_1}\cdot\ldots\cdot c_N^{m_N})\cdot z_1^{m_1}\cdot\ldots\cdot z_N^{m_N} $$ is bounded on an unbounded open domain in ${\mathbb C }^N$. So, this is the zero polynomial. Therefore $F(c_1^{m_1}\cdot\ldots\cdot c_N^{m_N})=0$ for all $m_1,\ldots,m_N\geq 0\,.$ {\hfill$\boxtimes$} \ \begin{theorem}\label{isosol} If $\beta=0$ and ${\mathcal T }\!$ is unimodular, then there is an isomorphism: $$ \Hom{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{{\mathbb C }}\stackrel{\sim}{\rightarrow} \textbf{solution space of (\ref{eq:gkzlin})-(\ref{eq:gkzmon})}\;, \hspace{5mm}F\mapsto F\Phi_{{\mathcal T }\!,0} \;. $$ \end{theorem} \textbf{proof:} Lemma \ref{locsys} shows that the map is injective. {}From corollary \ref{allvert} we know $\dim\Hom{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{{\mathbb C }}\,=\,\mathrm{ vol}\,\Delta$. Since the triangulation ${\mathcal T }\!$ is unimodular, the proof of \cite{stur} prop.13.15 shows that the normality condition for the correction in \cite{gkz1a} to \cite{gkz1} thm. 5 is satisfied. Therefore the number of linearly independent solutions of the GKZ system (\ref{eq:gkzlin})-(\ref{eq:gkzmon}) at a generic point equals $\mathrm{ vol}\,\Delta$. {\hfill$\boxtimes$} \section{Triangulations with non-empty core} \label{coresection} The intersection of all maximal simplices in a regular triangulation ${\mathcal T }\!$ of $\Delta$ is a remarkable structure. We call it the core of ${\mathcal T }\!$. It is a simplex in the triangulation ${\mathcal T }\!\,.$ Since we identify simplices with their index sets, we view $\mathrm{ core}\,{\mathcal T }\!$ also as a subset of $\{1,\ldots,N\}\,.$ \ \begin{definition}\label{tcore} \hspace{5mm}$\displaystyle{\mathrm{ core}\,{\mathcal T }\!\::=\:\bigcap_{I\in{\mathcal T }\!^n}\,I}$ \end{definition} \ \begin{lemma}\label{corebound} A simplex which does not contain $\mathrm{ core}\,{\mathcal T }\!$ lies in the boundary of $\Delta\,.$ \end{lemma} \textbf{proof:} It suffices to prove this for simplices of the form $I\setminus \{j\}$ with $I\in{\mathcal T }\!^n$ and $j\in\mathrm{ core}\,{\mathcal T }\!\,.$ Since every maximal simplex contains $j$, $I$ is the only maximal simplex which contains $I\setminus \{j\}\,.$ Therefore $I\setminus \{j\}$ lies in the boundary of $\Delta\,.$ {\hfill$\boxtimes$} \ \begin{lemma}\label{lcore} \hspace{5mm}$\displaystyle{\mathrm{ core}\,{\mathcal T }\!\:=\:\{j\:|\:\ell_j\leq 0 \; \textrm{ for all }\;\ell\in{\mathcal C }_{{\mathcal T }\!}^\vee\:\} } $ \end{lemma} \textbf{proof:} $\supset\,:$ assume $j\not\in\mathrm{ core}\,{\mathcal T }\!$, say $j\not\in I$ for some $I\in{\mathcal T }\!^n$. Then there is a relation ${\mathfrak a }_j-\sum_{i\in I} x_i{\mathfrak a }_i\,=\,0\,;$ whence an $\ell\in{\mathbb L }$ with $\ell_j>0$ and $\{i\,|\,\ell_i<0\}\subset I$. As in the proof of lemma \ref{support} this implies $\ell\in{\mathcal C }_{{\mathcal T }\!}^\vee\,.$ $\subset\,:$ assume $j\in\mathrm{ core}\,{\mathcal T }\!\,.$ First consider an $\ell\in{\mathbb L }_{\mathbb R }$ such that $\{i\,|\,\ell_i<0\}$ is a simplex. Let $L=\sum_{\ell_i>0} \ell_i=\sum_{\ell_i<0} -\ell_i\,.$ The relation in (\ref{eq:L}) can be rewritten as \begin{equation}\label{eq:convell} \sum_{\ell_i>0} \frac{\ell_i}{L}{\mathfrak a }_i \:=\: \sum_{\ell_i<0} \frac{-\ell_i}{L}{\mathfrak a }_i \end{equation} Suppose $\ell_j>0$. Then the simplex $\{i\,|\,\ell_i<0\}$ lies in a boundary face of $\Delta\,.$ Take a linear functional $F$ whose restriction to $\Delta$ attains its maximum exactly on this face. Evaluate $F$ on both sides of (\ref{eq:convell}). The value on the right hand side is $\max F$, but on the left hand side it is $<\max F$, because $F({\mathfrak a }_j)<\max F$ and $\ell_j>0$. Contradiction! Therefore we conclude: $\ell_j\leq 0$ if $\ell$ is such that $\{i\,|\,\ell_i<0\}$ is a simplex. From the constructions in section \ref{secfan} one sees that $\{i\,|\,\ell_i<0\}$ is a simplex if and only if $\ell\in{\mathcal C }_{I^\ast}^\vee$ for some $I\in{\mathcal T }\!^n$; note: $\ell_j={\mathfrak b }_j\cdot\ell$. Since ${\mathcal C }_{{\mathcal T }\!}^\vee$ is the Minkowski sum of the cones ${\mathcal C }_{I^\ast}^\vee$ with $I\in{\mathcal T }\!^n$ we finally get: $\ell_j\leq 0$ for every $\ell\,\in{\mathcal C }_{{\mathcal T }\!}^\vee\,.$ {\hfill$\boxtimes$} \ \begin{definition} \begin{equation}\label{eq:ccore} c_{\mathrm{ core}\,}\::=\:\prod_{i\in\mathrm{ core}\,{\mathcal T }\!} c_i \end{equation} \end{definition} \ \begin{corollary} \label{coreideal} If $\lambda$ is such that ${\mathcal A }\lambda\,=\,\sum_{i\in\mathrm{ core}\,{\mathcal T }\!} m_i {\mathfrak a }_i$ with all $m_i<0$ then $\lambda_i<0$ for every $i\in\mathrm{ core}\,{\mathcal T }\!$ and hence $$ Q_\lambda({\mathsf c })\,\in\,c_{\mathrm{ core}\,}{\mathcal R }_{{\mathsf A },{\mathcal T }\!} $$ \end{corollary} \textbf{proof:} Let $\mu=(\mu_1,\ldots,\mu_N)$ be defined by $\mu_i=m_i$ for $i\in\mathrm{ core}\,{\mathcal T }\!$ and $\mu_i=0$ for $i\not\in\mathrm{ core}\,{\mathcal T }\!$. Then ${\mathfrak P }_{\mathcal T }\!(\lambda)={\mathfrak P }_{\mathcal T }\!(\mu)$ in lemma \ref{support}. {}From the definitions one sees immediately that ${\mathfrak P }_{\mathcal T }\!(\mu)=\{\mu\}\,.$ The result now follows from lemmas \ref{support} and \ref{lcore}. {\hfill$\boxtimes$} \ \begin{corollary}\label{corid} If $\mathrm{ core}\,{\mathcal T }\!$ is not empty and $\beta\,=\,\sum_{i\in\mathrm{ core}\,{\mathcal T }\!} m_i {\mathfrak a }_i\;$ with all $\; m_i<0\;$ then the function $\Psi_{{\mathcal T }\!,\beta}$ takes values in the ideal $c_{\mathrm{ core}\,}{\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb C }\,.$ {\hfill$\boxtimes$} \end{corollary} \ \begin{theorem}\label{corsol} If $\mathrm{ core}\,{\mathcal T }\!$ is not empty and $\beta\,=\,\sum_{i\in\mathrm{ core}\,{\mathcal T }\!} m_i {\mathfrak a }_i\;$ with all $\; m_i<0\;$ then the linear map $$ \Hom{c_{\mathrm{ core}\,}{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{{\mathbb C }}\longrightarrow \textbf{solution space of (\ref{eq:gkzlin})-(\ref{eq:gkzmon})}\;, \hspace{5mm}F\mapsto F\Phi_{{\mathcal T }\!,\beta} $$ is injective. \end{theorem} \textbf{proof:} {}From lemma \ref{support} and the proof of corollary \ref{coreideal} one sees that the series $\Psi_{{\mathcal T }\!,\beta}$ involves only $\lambda$'s in $\mu+({\mathbb L }\cap{\mathcal C }_{\mathcal T }\!^\vee)$ and that $\lambda=\mu$ is really present: $$ Q_\mu({\mathsf c })=c_{\mathrm{ core}\,}\cdot U\hspace{4mm}\textrm{with}\hspace{4mm} U\::=\:\prod_{i\in\mathrm{ core}\,{\mathcal T }\!} \prod_{k=1}^{-m_i-1}(c_i-k)\,. $$ The rest of the proof is analogous to the proof of lemma \ref{locsys}. In particular, if $F\Psi_{{\mathcal T }\!,\beta}$ is the $0$-function on ${\mathcal V }_{\mathcal T }\!$, then $F(c_{\mathrm{ core}\,}\cdot U\cdot c_1^{n_1}\cdot\ldots\cdot c_N^{n_N})=0$ for all $n_1,\ldots,n_N\geq 0\,.$ The desired result now follows because $U$ is invertible in the ring ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes{\mathbb Q }$. {\hfill$\boxtimes$} \ \ \ \begin{center}\textbf{PART II}\end{center} \section*{Introduction II} One aspect of the mirror symmetry phenomenon (cf. \cite{bundel1,bundel2}) is that (generalized) Calabi-Yau manifolds seem to come in pairs $(X,Y)$ with the geometries of $X$ and $Y$ related in a beautifully intricate way. On one side of the mirror - usually called \emph{the B-side} - it is the geometry of complex structure, of periods of a holomorphic differential form, of variations of Hodge structure. On the other side - \emph{the A-side} - it is the geometry of symplectic structure, of algebraic cycles and of enumerative questions about curves on the manifold. Batyrev \cite{bat1} showed that behind many examples of the mirror symmetry phenomenon one can see a simple combinatorial duality. Batyrev and Borisov gave a generalization of this combinatorial duality and formulated a \emph{mirror symmetry conjecture for generalized Calabi-Yau manifolds} in arbitrary dimension ( \cite{babo} 2.17). The fundamental combinatorial structure is a \emph{reflexive Gorenstein cone}. \begin{definition}\label{gorco} \textup{( \cite{babo} definitions 2.1-2.8.)} A cone $\Lambda$ in ${\mathbb R }^n$ is called a \emph{Gorenstein cone} if it is generated, i.e. \begin{equation}\label{eq:lambda} \Lambda={\mathbb R }_{\geq 0}{\mathfrak a }_1\,+\ldots+\,{\mathbb R }_{\geq 0}{\mathfrak a }_N\,, \end{equation} by a finite set $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}\subset{\mathbb Z }^n$ which satisfies condition \ref{gkzcond}. It is called a \emph{reflexive Gorenstein cone} if both $\Lambda$ and its dual $\Lambda^\vee$ are Gorenstein cones, \begin{equation}\label{eq:dula} \Lambda^\vee\::=\:\{\:{\mathsf y }\in{\mathbb R }^{n\vee}\:|\: \forall\,{\mathsf x }\in\Lambda\,:\:{\mathsf y }\cdot{\mathsf x }\,\geq\,0\:\}\,, \end{equation} i.e. there should also exist a vector ${\mathfrak a }_0\in{\mathbb Z }^n$ and a set $\{{\mathfrak a }_1^\vee,\ldots,{\mathfrak a }_{N^\prime}^\vee\}\subset{\mathbb Z }^{n\vee}$ of generators for $\Lambda^\vee$ such that ${\mathfrak a }_i^\vee\cdot{\mathfrak a }_0=1$ for $i=1,\ldots,N^\prime$. The vectors ${\mathfrak a }_0^\vee$ and ${\mathfrak a }_0$ are uniquely determined by $\Lambda$. The integer ${\mathfrak a }_0^\vee\cdot{\mathfrak a }_0$ is called \emph{ the index} of $\Lambda\,.$ \end{definition} \ For a reflexive Gorenstein cone one has one new datum in addition to the data for GKZ systems; namely ${\mathfrak a }_0$. It has the very important property \begin{equation}\label{eq:alint} \textrm{interior}(\Lambda)\cap{\mathbb Z }^n\,=\,{\mathfrak a }_0+\Lambda\,. \end{equation} \ \emph{Our aim is to show that in the case of a mirror pair $(X,Y)$ associated with a reflexive Gorenstein cone $\Lambda$ and a unimodular regular triangulation ${\mathcal T }\!$ whose core is not empty and is not contained in the boundary of $\Delta\,,$ the periods of a holomorphic differential form on $X$ are given by the function $\Phi_{{\mathcal T }\!,-{\mathfrak a }_0}$ which takes values in the ring $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}\otimes{\mathbb C }$ and that the ring $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}$ is isomorphic with a subring of the Chow ring of $Y$.} \ This project naturally has a B-side and an A-side which we develop separately in Part II B and Part II A. Our method puts some natural restrictions on the generality. For Part II B we must eventually assume that there is a unimodular triangulation ${\mathcal T }\!$ of the polytope \begin{equation}\label{eq:delta} \Delta\::=\:\mathrm{ conv}\,\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}= \{{\mathsf x }\in\Lambda\:|\:{\mathfrak a }_0^\vee\cdot{\mathsf x }\,=\,1\:\}\,. \end{equation} This restriction which comes from the use of theorem \ref{isosol}, also implies \begin{equation}\label{eq:a0z} {\mathfrak a }_0\in{\mathbb Z }_{\geq 0}{\mathfrak a }_1+\ldots+{\mathbb Z }_{\geq 0}{\mathfrak a }_N\,. \end{equation} So, $\Phi_{{\mathcal T }\!,-{\mathfrak a }_0}$ is defined in Part I. For Part II A we must additionally assume that the core of ${\mathcal T }\!$ is not empty and is not contained in the boundary of $\Delta\,.$ \begin{center}\textbf{PART II B}\end{center} \section*{Introduction II B} For a Gorenstein cone $\Lambda$ we denote the monoid algebra ${\mathbb C }[\Lambda\cap{\mathbb Z }^n]$ by ${\mathcal S }_\Lambda$ and view it as a subalgebra of the algebra ${\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]$ by identifying ${\mathsf m }=(m_1,\ldots,m_n)^t\in\Lambda\cap{\mathbb Z }^n$ with the Laurent monomial ${\mathsf u }^{\mathsf m }\,:=$ $u_1^{m_1}\cdot\ldots\cdot u_n^{m_n}$. For ${\mathsf m }\in{\mathbb Z }^n$ we put $\deg{\mathsf u }^{\mathsf m }\::=\,\deg{\mathsf m }\::=\,{\mathfrak a }_0^\vee\cdot{\mathsf m }\,.$ Thus ${\mathcal S }_\Lambda$ becomes a graded ring. The scheme ${\mathbb P }_\Lambda\::=\:\textrm{Proj }{\mathcal S }_\Lambda$ is a projective toric variety. If $\Lambda$ is a reflexive Gorenstein cone, the zero set in ${\mathbb P }_\Lambda$ of a global section of ${\mathcal O }_{{\mathbb P }_\Lambda}(1)$ is called a \emph{generalized Calabi-Yau manifold} of dimension $n-2$ ( \cite{babo} 2.15). The toric variety ${\mathbb P }_\Lambda$ is a compactification of the $n-1$-dimensional torus \begin{equation}\label{eq:torus2} {\mathbb T }\::=\:\modquot{\widetilde{\mathbb T }}{({\mathbb Z }{\mathfrak a }_0^\vee\otimes{\mathbb C }^{\,\ast})} \end{equation} where \begin{equation}\label{eq:torus1} \widetilde{\mathbb T }\::=\:\Hom{{\mathbb Z }^n}{{\mathbb C }^{\,\ast}} \:=\:{\mathbb Z }^{n\vee}\otimes{\mathbb C }^{\,\ast} \end{equation} is the $n$-dimensional torus of ${\mathbb C }$-points of $\textrm{Spec }{\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]\,.$ A global section of ${\mathcal O }_{{\mathbb P }_\Lambda}(1)$ is given by a Laurent polynomial \begin{equation}\label{eq:laurent} {\mathsf s }=\sum_{{\mathsf m }\in\Delta\cap{\mathbb Z }^n} v_{\mathsf m } {\mathsf u }^{\mathsf m }\,. \end{equation} with $\Delta$ as in (\ref{eq:delta}). As in \cite{bat2} we assume from now on \ \begin{condition}\label{aisall} \hspace{10mm}$\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}=\Delta\cap{\mathbb Z }^n$ \end{condition} \ The Laurent polynomial ${\mathsf s }$ gives a function on $\widetilde{\mathbb T }$ which is homogeneous of degree $1$ for the action of ${\mathbb Z }{\mathfrak a }_0^\vee\otimes{\mathbb C }^{\,\ast}\,.$ Let \begin{equation}\label{eq:zs} {\mathsf Z }_{\mathsf s }\::=\{\textrm{ zero locus of } \;{\mathsf s }\:\}\subset{\mathbb T } \end{equation} Over the complementary set ${\mathbb T }\setminus{\mathsf Z }_{\mathsf s }$ there is a section of $\widetilde{\mathbb T }\rightarrow{\mathbb T }$ which identifies ${\mathbb T }\setminus{\mathsf Z }_{\mathsf s }$ with the zero set $\widetilde{\mathsf Z }_{{\mathsf s }-1}$ of ${\mathsf s }-1$ in $\widetilde{\mathbb T }$: \begin{equation}\label{eq:varieties} {\mathbb T }\setminus{\mathsf Z }_{\mathsf s }\:\simeq\:\widetilde{\mathsf Z }_{{\mathsf s }-1}\subset\widetilde{\mathbb T } \end{equation} One may say that according to Batyrev \cite{bat2} \emph{the geometry on the B-side of mirror symmetry is encoded in the weight $n$ part ${\mathcal W }_n H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })$ of the Variation of Mixed Hodge Structure of $H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })$}; the variation comes from varying the coefficients $v_{\mathsf m }$ in (\ref{eq:laurent}). \begin{remark}\textup{ One usually formulates Mirror Symmetry with on the B-side the Variation of Hodge Structure on the $d$-th cohomology of a $d$-dimensional Calabi-Yau manifold. For a CY hypersurface in a toric variety the Poincar\'e residue mapping gives an isomorphism with the $d+1$-st cohomology of the hypersurface complement, at least on the primitive parts (see {} \cite{bat2} prop.5.3). For a CY complete intersection of codimension $>1$ in a toric variety one needs besides the Poincar\'e residue mapping also corollary 3.4 and remark 3.5 in \cite{babo} to relate the CYCI's cohomology to the cohomology of the complement of a generalized Calabi-Yau hypersurface in a toric variety, i.e. to the situation we are studying in this paper. Our investigations do however also allow on this B-side of the mirror generalized Calabi-Yau hypersurfaces which are not related to CY complete intersections, although on the other A-side we do eventually want a Calabi-Yau complete intersection (see \cite{babo} \S 5 for an example of mirror symmetry with such an asymmetry between the two sides). } \end{remark} \ In \cite{bat2} Batyrev described the weight and Hodge filtrations of this Variation of Mixed Hodge Structure (VMHS) in terms of the combinatorics of $\Lambda\,.$ In particular, ${\mathcal W }_n H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })$ corresponds with the ideal ${\mathbb C }[\textrm{interior}(\Lambda)\cap{\mathbb Z }^n]$ in ${\mathcal S }_\Lambda\,.$ If $\Lambda$ is a reflexive Gorenstein cone of index $\kappa\,,$ this ideal is the principal ideal generated by ${\mathsf u }^{{\mathfrak a }_0}$ (cf. (\ref{eq:alint})) and the part of weight $n$ and Hodge type $(n-\kappa,\kappa)$ has dimension $1\,.$ Batyrev \cite{bat2} also showed that the periods of the rational $(n-1)$-form \begin{equation}\label{eq:ratform} \omega_\mu\::=\:\frac{{\mathsf u }^\mu}{{\mathsf s }^{\deg\mu}}\, \frac{dt_2}{t_2}\wedge\ldots\wedge\frac{dt_{n}}{t_{n}} \end{equation} ($\mu\in\Lambda\cap{\mathbb Z }^n\,,\; t_2,\ldots,t_{n}$ coordinates on ${\mathbb T }$ ) as functions of the coefficients $v_{\mathsf m }$ satisfy a GKZ system of differential equations (\ref{eq:gkzlin})-(\ref{eq:gkzmon}) with parameters $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ and $\beta=-\mu\,.$ However, \emph{ not all solutions of this system are ${\mathbb C }$-linear combinations of the periods of $\omega_\mu\,.$ Theorem \ref{mainthm} shows precisely which solutions of this system are ${\mathbb C }$-linear combinations of the periods of $\omega_\mu$ in case $\mu\in{\mathbb Z }_{\geq 0}{\mathfrak a }_1+\ldots+{\mathbb Z }_{\geq 0}{\mathfrak a }_N\,.$ } The key point of our method is to study the VMHS on $H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\,.$ This has the advantage that \emph{if ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$ generate ${\mathbb Z }^n$, then $H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})$ is a hypergeometric ${\mathcal D }$-module} as in \cite{gkz1} with parameters $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ and $\beta=0\,;$ see theorem \ref{hygeoDmod}. \ If ${\mathsf s }$ is $\Lambda$-regular (cf. definition \ref{regular}) there is an exact sequence of mixed Hodge structures \begin{equation}\label{eq:exseq} 0\rightarrow H^{n-1}(\widetilde{\mathbb T })\rightarrow H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})\rightarrow H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\rightarrow H^{n}(\widetilde{\mathbb T })\rightarrow 0 \end{equation} The left hand $0$ results from a theorem of Bernstein-Danilov-Khovanskii \cite{dk,bat2}. On the right we used $ H^{n}(\widetilde{\mathsf Z }_{{\mathsf s }-1})=0$ because $\widetilde{\mathsf Z }_{{\mathsf s }-1}$ is an affine variety of dimension $n-1\,.$ Writing as usual ${\mathbb Q }(m)$ for the $1$-dimensional ${\mathbb Q }$-Hodge structure which is purely of weight $-2m$ and Hodge type $(-m,-m)$ one has \begin{equation}\label{eq:hodgetate} H^{n-1}(\widetilde{\mathbb T })\simeq{\mathbb Q }^n\otimes{\mathbb Q }(1-n)\:,\hspace{5mm} H^{n}(\widetilde{\mathbb T })\simeq{\mathbb Q }(-n)\,. \end{equation} Morphisms of mixed Hodge structures are strictly compatible with the weight filtrations ( \cite{del} thm. 2.3.5). Thus the sequence (\ref{eq:exseq}) in combination with (\ref{eq:varieties}) gives the isomorphisms \begin{equation}\label{eq:isos} {\mathcal W }_i H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })\stackrel{\simeq}{\longrightarrow} {\mathcal W }_i H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})\stackrel{\simeq}{\longrightarrow} {\mathcal W }_i H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1}) \end{equation} for $ i \leq 2n-3\,.$ In particular if $n\geq 3\,,$ the weight $n$ part relevant for the geometry on the B-side of mirror symmetry will get a complete and simple description by our analysis of the GKZ hypergeometric ${\mathcal D }$-module $H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\,.$ \ \begin{remark}\textup{ Though it plays no role in this paper I want to point out that there is an interesting relation with recent work of Deninger \cite{den}. The group $G$ of diagonal $n\timesn$-matrices with entries $\pm 1$ acts naturally on $\widetilde{\mathbb T }=\Hom{{\mathbb Z }^n}{{\mathbb C }^{\,\ast}}\,.$ {}From the inclusion $\imath: \widetilde{\mathsf Z }_{{\mathsf s }-1} \hookrightarrow\widetilde{\mathbb T }$ one gets the $G$-equivariant map $G\times\widetilde{\mathsf Z }_{{\mathsf s }-1} \rightarrow\widetilde{\mathbb T }\;,$ $\; (g,z)\mapsto g\cdot\imath (z)$. Corresponding to this map there is an exact sequence of mixed Hodge structures with $G$-action analogous to (\ref{eq:exseq}). Taking isotypical parts for the character $\det: G\rightarrow \{\pm 1\}$ and using $H^{n-1}(\widetilde{\mathbb T })(\det)=0\:,$ $H^{n}(\widetilde{\mathbb T })(\det)\stackrel{\simeq}{\rightarrow}H^{n}(\widetilde{\mathbb T })$ and $H^{n-1}(G\times\widetilde{\mathsf Z }_{{\mathsf s }-1})(\det)\stackrel{\simeq}{\rightarrow} H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})$ one finds the short exact sequence \begin{equation}\label{eq:ses1} 0\rightarrow H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})\rightarrow H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,(G\times\widetilde{\mathsf Z }_{{\mathsf s }-1}))(\det)\rightarrow H^{n}(\widetilde{\mathbb T })\rightarrow 0 \end{equation} see \cite{den} (12). In \cite{den} remark 2.4 Deninger sketches how the extension (\ref{eq:ses1}) comes from a Steinberg symbol in the group $K_n(\widetilde{\mathsf Z }_{{\mathsf s }-1})$ in the algebraic $K$-theory of $\widetilde{\mathsf Z }_{{\mathsf s }-1}$; in our coordinates (see remark \ref{homocoord3}) this Steinberg symbol reads \begin{equation}\label{eq:symbol} \{u_1,u_2,\ldots,u_n\}\:\in\:K_n\left( \modquot{{\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]}{({\mathsf s }-1)}\right) \end{equation} The exact sequence (\ref{eq:exseq}) decomposes into two short exact sequences \begin{eqnarray} \label{eq:ses2} 0\rightarrow H^{n-1}(\widetilde{\mathbb T })\rightarrow H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1}) \rightarrow PH^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})\rightarrow 0 \\ \label{eq:ses3} 0\rightarrow PH^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})\rightarrow H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\rightarrow H^{n}(\widetilde{\mathbb T })\rightarrow 0 \end{eqnarray} which define the \emph{primitive part} of cohomology ( \cite{bat2} def. 3.13). The relation between the various cohomology groups is best displayed in the following commutative diagram with injective horizontal and surjective vertical arrows: \begin{equation}\label{eq:diagram} \begin{array}{ccccc} H^{n-1}(\widetilde{\mathbb T })&\rightarrow&H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})& \rightarrow&H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,(G\times\widetilde{\mathsf Z }_{{\mathsf s }-1}))(\det)\\ &&\downarrow&&\downarrow\\ &&PH^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1})&\rightarrow&H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\\ &&&&\downarrow\\&&&&H^{n}(\widetilde{\mathbb T }) \end{array} \end{equation} With varying coefficients $v_{\mathsf m }$ the story plays in the category of Variations of Mixed Hodge Structures. With coefficients $v_{\mathsf m }$ fixed in some number field the story plays in a category of Mixed Motives. A challenge for further research is to combine these stories and our results on hypergeometric systems.} \end{remark} \section{VMHS associated with a Gorenstein cone} \label{vmhs} In this section we prove theorem \ref{hygeoDmod}. This result is essentially implicitly contained in \cite{bat2}. Our proof is mainly a review of constructions and results in \cite{bat2}. Shifting emphasis from the polytope $\Delta$ to the cone $\Lambda$ we write ${\mathcal S }_\Lambda$ (instead of $S_\Delta$ as in \cite{bat2}) for the monoid algebra ${\mathbb C }[\Lambda\cap{\mathbb Z }^n]$ viewed as a subalgebra of ${\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]\,.$ The grading is given by $\deg{\mathsf u }^{\mathsf m }\,=\,{\mathfrak a }_0^\vee\cdot{\mathsf m }$ for ${\mathsf m }\in{\mathbb Z }^n$. A homogeneous element ${\mathsf s }$ of degree $1$ in ${\mathcal S }_\Lambda$ is a Laurent polynomial as in (\ref{eq:laurent}): \begin{equation}\label{eq:laurent2} {\mathsf s }=\sum_{i=1}^N\;v_i\,{\mathsf u }^{{\mathfrak a }_i} \end{equation} with coefficients $v_i\in{\mathbb C }\,.$ Let $\widetilde{\mathbb T }$, ${\mathbb T }$, ${\mathsf Z }_{\mathsf s }$ and $\widetilde{\mathsf Z }_{{\mathsf s }-1}$ be as in (\ref{eq:torus2})-(\ref{eq:varieties}). \ \begin{remark}\label{homocoord1}\textup{ When comparing with \cite{bat2} one should keep in mind that in op.cit. $n$ is the dimension of the polytope $\Delta$ whereas here $n$ is the dimension of the cone $\Lambda$ and the polytope $\Delta$ has dimension $n-1\,.$ Also one has to make the following change of coordinates on ${\mathbb Z }^n$ and ${\mathbb Z }^{n\vee}$. The idempotent $n\timesn$-matrix ${\mathfrak a }_1\cdot{\mathfrak a }_0^\vee$ gives rise to a direct sum decomposition ${\mathbb Z }^{n\vee}={\mathbb Z }{\mathfrak a }_0^\vee\oplus \,\Xi$ and thus to a basis $\{{\mathfrak a }_0^\vee,\,\alpha_2,\ldots,\,\alpha_{n}\}$ for ${\mathbb Z }^{n\vee}$. The coordinate change on ${\mathbb Z }^n$ amounts to multiplying vectors in ${\mathbb Z }^n$ by the matrix $M=(m_{ij})$ with rows ${\mathfrak a }_0^\vee,\,\alpha_2,\ldots,\,\alpha_{n}\,.$ In particular, in the new coordinates ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$ all have first coordinate $1\,.$} \textup{ The above coordinate change also induces a change of coordinates on $\widetilde{\mathbb T }\,:$ $u_j=\prod_{i=1}^n t_i^{m_{ij}}$. The map $\widetilde{\mathbb T }\rightarrow{\mathbb T }$ is then just omitting the coordinate $t_1\,.$ In $t$-coordinates ${\mathsf s }$ takes the form $t_1\cdot f$ where $f$ is a Laurent polynomial in the variables $t_2,\ldots,t_n\,.$ Thus ${\mathsf s }$ corresponds with $F_0$ and ${\mathsf s }-1$ with $F$ in \cite{bat2} def. 4.1. } \end{remark} \begin{remark} \label{homocoord3}\textup{ When comparing with \cite{den} one sees again a shift of dimensions from $n$ in op. cit. to $n-1$ here; $T^n$ with coordinates $t_1,\ldots,t_n$ in op. cit. is our ${\mathbb T }$ with coordinates $t_2,\ldots,t_n\,.$ The polynomial $P$ of op. cit. and our ${\mathsf s }$ are related by ${\mathsf s }=t_1\cdot P$. The identification of ${\mathbb T }\setminus{\mathsf Z }_{\mathsf s }$ with $\widetilde{\mathsf Z }_{{\mathsf s }-1}$ now gives for the Steinberg symbols $\{P,t_2,\ldots,t_n\}\,=\,-\{t_1,t_2,\ldots,t_n\} \,=\,\{u_1,u_2,\ldots,u_n\}$ if the coordinates are ordered such that $\det M=-1\,.$ } \end{remark} \ Before we can state Batyrev's results we need some definitions/notations. \\ \cite{bat2} def. 2.8 defines an ascending sequence of homogeneous ideals in ${\mathcal S }_\Lambda$: \begin{equation}\label{eq:idealweight} I_\Delta^{(0)}\subset I_\Delta^{(1)}\subset\ldots\subset I_\Delta^{(n)}\subset I_\Delta^{(n+1)} \end{equation} where $I_\Delta^{(k)}$ is generated by the elements ${\mathsf u }^{\mathsf m }$ with ${\mathsf m }$ in $\Lambda\cap{\mathbb Z }^n$ but not in any codimension $k$ face of $\Lambda\,;$ in particular \begin{equation}\label{eq:weightfilt} I_\Delta^{(0)}=0\:,\hspace{3mm} I_\Delta^{(1)}={\mathbb C }[\mathrm{interior}(\Lambda)\cap{\mathbb Z }^n]\:,\hspace{3mm} I_\Delta^{(n)}={\mathcal S }_\Lambda^+\:,\hspace{3mm} I_\Delta^{(n+1)}={\mathcal S }_\Lambda \end{equation} ${\mathcal S }_\Lambda^+$ is the ideal in ${\mathcal S }_\Lambda$ generated by the monomials of degree $>0\,.$ \\ \cite{bat2} p.379 defines a descending sequence of ${\mathbb C }$-vector spaces in ${\mathcal S }_\Lambda$: \begin{equation}\label{eq:hodgefilt} \ldots\supset{\mathcal E }^{-k}\supset{\mathcal E }^{-k+1}\supset\ldots \supset{\mathcal E }^{-1}\supset{\mathcal E }^0\supset{\mathcal E }^1=0 \end{equation} where ${\mathcal E }^{-k}$ is spanned by the monomials ${\mathsf u }^{\mathsf m }$ with $\deg{\mathsf u }^{\mathsf m }\leq k\,.$ \\ \cite{bat2} def. 7.2 defines the differential operators \begin{equation}\label{eq:difops} D_i\::=\:u_i\frac{\partial}{\partial u_i}\,+\, u_i\frac{\partial {\mathsf s }}{\partial u_i}\;,\hspace{5mm} (i=1,\ldots,n) \end{equation} These operate on ${\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]\,,$ preserving ${\mathcal S }_\Lambda$ and ${\mathcal S }_\Lambda^+\,.$ \\ \cite{bat2} thm. 4.8 can be used as a definition: \begin{definition}\label{regular} ${\mathsf s }$ is said to be \emph{$\Lambda$-regular} if $\displaystyle{u_1\frac{\partial {\mathsf s }}{\partial u_1},\: u_2\frac{\partial {\mathsf s }}{\partial u_2}\ldots,\: u_n\frac{\partial {\mathsf s }}{\partial u_n}}$ is a \emph{regular sequence} in ${\mathcal S }_\Lambda\,.$ \end{definition} \begin{theorem}\label{TZ} \textup{(summary of results in \cite{bat2})}\\ If ${\mathsf s }$ is $\Lambda$-regular, then there is a commutative diagram \begin{equation} \label{eq:HTZ} \begin{array}{ccccc} \modquot{{\mathcal S }_\Lambda^+}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda^+}& \stackrel{\simeq}{\rightarrow} & H^{n-1}(\widetilde{\mathsf Z }_{{\mathsf s }-1}) & \stackrel{\simeq}{\rightarrow} & H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })\\[.5em] \downarrow&&\downarrow&&\\[.5em] \modquot{{\mathcal S }_\Lambda}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda}& \stackrel{\simeq}{\rightarrow} & H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})&& \end{array} \end{equation} in which the horizontal arrows are isomorphisms. These isomorphisms restrict to the following isomorphisms relating (\ref{eq:weightfilt}) and (\ref{eq:hodgefilt}) with the weight and Hodge filtrations on $ H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })$ and $ H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1}) \,.$\\ For $k=-1,0,1,\ldots,n,n+1\,:$ \[ \begin{array}{lcl} \textrm{image } I_\Delta^{(k)} \textrm{ in } \modquot{{\mathcal S }_\Lambda^+}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda^+} &\stackrel{\simeq}{\rightarrow}& {\mathcal W }_{k+n-1} H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s }) \\[.5em] \textrm{image } {\mathcal E }^{-k}\cap {\mathcal S }_\Lambda^+ \textrm{ in } \modquot{{\mathcal S }_\Lambda^+}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda^+} &\stackrel{\simeq}{\rightarrow}& {\mathcal F }^{n-k}H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s }) \\[.5em] \textrm{image } I_\Delta^{(k)} \textrm{ in } \modquot{{\mathcal S }_\Lambda}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda} &\stackrel{\simeq}{\rightarrow}& {\mathcal W }_{k+n-1} H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1}) \\[.5em] \textrm{image } {\mathcal E }^{-k}\textrm{ in } \modquot{{\mathcal S }_\Lambda}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda} &\stackrel{\simeq}{\rightarrow}& {\mathcal F }^{n-k} H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1}) \end{array} \] \end{theorem} \textbf{proof:} The statements for $H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })$ are theorems 7.13, 8.1 and 8.2 in \cite{bat2}. The statements about $ H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})$ can also be derived with the methods of op. cit., as follows. Recall that $H^\ast(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})$ is the cohomology of the cone of the natural map of DeRham complexes $\Omega_{\widetilde{\mathbb T }}^\bullet\rightarrow \Omega_{\widetilde{\mathsf Z }_{{\mathsf s }-1}}^\bullet$ and that this cone complex is in degrees $i$ and $i+1$ \begin{equation}\label{eq:relcomplex} \begin{array}{rcccl} \ldots\rightarrow & \Omega_{\widetilde{\mathbb T }}^i\oplus\Omega_{\widetilde{\mathsf Z }_{{\mathsf s }-1}}^{i-1} &\longrightarrow& \Omega_{\widetilde{\mathbb T }}^{i+1}\oplus\Omega_{\widetilde{\mathsf Z }_{{\mathsf s }-1}}^i &\rightarrow\ldots\\[.7em] &(\omega_1,\omega_2)&\mapsto& (-d\omega_1,d\omega_2+\omega_1|_{\widetilde{\mathsf Z }_{{\mathsf s }-1}})& \end{array} \end{equation} A basis for the ${\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]$-module $\Omega_{\widetilde{\mathbb T }}^\bullet$ is given by the forms $\frac{du_{i_1}}{u_{i_1}}\wedge\ldots\wedge\frac{du_{i_r}}{u_{i_r}}$. Let $\Omega_{\widetilde{\mathbb T },0}^\bullet$ denote the subgroup of $\Omega_{\widetilde{\mathbb T }}^\bullet$ consisting of the linear combinations of the basic forms with coefficients in ${\mathbb C }\,.$ The standard differential $d$ on $\Omega_{\widetilde{\mathbb T }}^\bullet$ is $0$ on $\Omega_{\widetilde{\mathbb T },0}^\bullet$. The inclusion of complexes $\Omega_{\widetilde{\mathbb T },0}^\bullet\hookrightarrow \Omega_{\widetilde{\mathbb T }}^\bullet$ is a quasi-isomorphism. So in (\ref{eq:relcomplex}) we may replace $\Omega_{\widetilde{\mathbb T }}^\bullet$ by $\Omega_{\widetilde{\mathbb T },0}^\bullet$. For the proof of {} \cite{bat2} thm.7.13 Batyrev uses the ${\mathbb C }$-linear map ${\mathcal R }:\,{\mathcal S }_\Lambda^+\rightarrow\Omega_{\widetilde{\mathsf Z }_{{\mathsf s }-1}}^{n-1}\:,$ $\;{\mathcal R }({\mathsf u }^{\mathsf m }):=(-1)^{\deg{\mathsf m }-1}(\deg{\mathsf m }-1)!\,{\mathsf u }^{\mathsf m } \frac{dt_2}{t_2}\wedge\ldots\wedge\frac{dt_n}{t_n}$ (cf. remark \ref{homocoord1} for the $t$-coordinates). Let us extend this to a ${\mathbb C }$-linear map ${\mathcal R }:\,{\mathcal S }_\Lambda\rightarrow\Omega_{\widetilde{\mathbb T },0}^n \oplus\Omega_{\widetilde{\mathsf Z }_{{\mathsf s }-1}}^{n-1}$ by setting ${\mathcal R }(1)=\left( \frac{dt_1}{t_1}\wedge\ldots\wedge\frac{dt_n}{t_n}\,,\:0\right)\,.$ This induces a surjective linear map $ {\mathcal S }_\Lambda\longrightarrow H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1}) $ with $\sum_{i=1}^n D_i {\mathcal S }_\Lambda^+$ in its kernel. Note $\displaystyle{D_i(1)=u_i\frac{\partial{\mathsf s }}{\partial u_i}}\,.$ A direct calculation shows for $i=1,\ldots,n$: $$ (-1)^{i-1}{\mathcal R }(t_i\frac{\partial{\mathsf s }}{\partial t_i}) \,=\, d\left(\frac{dt_1}{t_1}\wedge\ldots\wedge \widehat{\frac{dt_i}{t_i}}\wedge\ldots\wedge \frac{dt_n}{t_n}\,,\:0\right) $$ in $\Omega_{\widetilde{\mathbb T },0}^n \oplus\Omega_{\widetilde{\mathsf Z }_{{\mathsf s }-1}}^{n-1}\,.$ Therefore ${\mathcal R }$ induces a surjective linear map $$ \modquot{{\mathcal S }_\Lambda}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda} \rightarrow H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\,. $$ A simple dimension count now shows that this is in fact an isomorphism. The statements about the Hodge filtration and the weight filtration on $ H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})$ follow from the corresponding statements for $H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })$ and from (\ref{eq:hodgetate}). {\hfill$\boxtimes$} \ The \emph{principal ${\mathsf A }$-determinant} of Gel'fand-Kapranov-Zelevinskii \cite{gkz2} is a polynomial $E_{{\mathsf A }} (v_1,\ldots,v_N)\in {\mathbb Z }[v_1,\ldots,v_N]$ such that (see \cite{bat2} prop. 4.16): \begin{equation}\label{eq:disc} {\mathsf s } \textrm{ is } \Lambda\textrm{-regular} \hspace{5mm}\Longleftrightarrow \hspace{5mm} E_{{\mathsf A }} (v_1,\ldots,v_N)\neq 0 \end{equation} \emph{Now we want to vary the coefficients $v_i$ in (\ref{eq:laurent2}) and work over the ring} \begin{equation}\label{eq:basering} {\mathbb C }[{\mathsf v }]\::=\:{\mathbb C }[v_1,\ldots,v_N,E_{{\mathsf A }}^{-1}]\,. \end{equation} Let $\Omega^\bullet$ resp. $\widetilde{\Omega}^\bullet$ denote the DeRham complex of ${\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]\otimes{\mathbb C }[{\mathsf v }]$ relative to ${\mathbb C }[{\mathsf v }]$ resp. relative to ${\mathbb C }\,.$ Define on these complexes a new differential \begin{eqnarray} \nonumber &&\delta: \Omega^i\rightarrow\Omega^{i+1} \textrm{ resp. } \widetilde{\Omega}^i\rightarrow \widetilde{\Omega}^{i+1} \\ \label{eq:newd} &&\delta\omega\::=\: d\omega+ d{\mathsf s }\wedge\omega \end{eqnarray} where $d$ is the ordinary differential on DeRham complexes. As a basis for the ${\mathbb C }[u_1^{\pm 1},\ldots,u_n^{\pm 1}]\otimes{\mathbb C }[{\mathsf v }]$-module $\Omega^1$ (resp. $\widetilde{\Omega}^1$ ) we take $\frac{du_1}{u_1},\ldots,\frac{du_n}{u_n}$ (resp. $\frac{du_1}{u_1},\ldots,\frac{du_n}{u_n},dv_1,\ldots,dv_N$) and extend it by taking wedge products to a basis for $\Omega^\bullet$ (resp. $\widetilde{\Omega}^\bullet$ ). Let $\Omega_\Lambda^\bullet$ (resp. $\Omega_{\Lambda^+}^\bullet$ ) denote the subgroups of $\Omega^\bullet$ consisting of the linear combinations of the given basic forms with coefficients in ${\mathcal S }_\Lambda\otimes{\mathbb C }[{\mathsf v }]$ (resp. ${\mathcal S }_\Lambda^+\otimes{\mathbb C }[{\mathsf v }]$ ). Define $\widetilde{\Omega}_\Lambda^\bullet$ (resp. $\widetilde{\Omega}_{\Lambda^+}^\bullet$ ) in the same way as subgroups of $\widetilde{\Omega}^\bullet\,.$ The differential $\delta$ (\ref{eq:newd}) preserves these subgroups. Thus we get the two complexes \begin{eqnarray*} (\Omega_\Lambda^\bullet,\delta)&:&\hspace{3mm} \Omega_\Lambda^0\,\stackrel{\delta}{\rightarrow} \Omega_\Lambda^1\,\stackrel{\delta}{\rightarrow} \ldots\stackrel{\delta}{\rightarrow} \Omega_\Lambda^{n-1}\,\stackrel{\delta}{\rightarrow} \Omega_\Lambda^n \\ (\widetilde{\Omega}_\Lambda^\bullet,\delta)&:&\hspace{3mm} \widetilde{\Omega}_\Lambda^0\,\stackrel{\delta}{\rightarrow} \widetilde{\Omega}_\Lambda^1\,\stackrel{\delta}{\rightarrow} \ldots\stackrel{\delta}{\rightarrow} \widetilde{\Omega}_\Lambda^{n-1}\,\stackrel{\delta}{\rightarrow} \widetilde{\Omega}_\Lambda^{n} \stackrel{\delta}{\rightarrow} \widetilde{\Omega}_\Lambda^{n+1}\,\stackrel{\delta}{\rightarrow} \ldots\stackrel{\delta}{\rightarrow}\widetilde{\Omega}_\Lambda^{N+n} \end{eqnarray*} Then \begin{equation} \label{eq:dmod1} H^n(\Omega_\Lambda^\bullet,\delta)\:=\: \left(\modquot{{\mathcal S }_\Lambda}{\sum_{i=1}^n D_i {\mathcal S }_\Lambda}\right) \otimes{\mathbb C }[{\mathsf v }] \end{equation} The Gauss-Manin connection \begin{equation} \label{eq:gaussmanin} \nabla\,:\;H^n(\Omega_\Lambda^\bullet,\delta)\rightarrow H^n(\Omega_\Lambda^\bullet,\delta)\otimes\Omega^1_{{\mathbb C }[{\mathsf v }]/\,{\mathbb C }} \end{equation} on this module is described by the Katz-Oda construction (cf. \cite{katz} \S 1.4) as follows. Lift the given $\xi\in H^n(\Omega_\Lambda^\bullet,\delta)$ to an element $\tilde{\xi}$ in $\widetilde{\Omega}_\Lambda^{n}$. Then $\nabla\xi$ is the cohomology class of $\delta\tilde{\xi}\in\widetilde{\Omega}_\Lambda^{n+1}$ in $H^n(\Omega_\Lambda^\bullet,\delta)\otimes\Omega^1_{{\mathbb C }[{\mathsf v }]/\,{\mathbb C }}\,.$ Having $\nabla\xi$ one defines $\frac{\partial}{\partial v_j}\xi\in H^n(\Omega_\Lambda^\bullet,\delta)$ by \begin{equation}\label{eq:deriv} \nabla\xi\:=\:\sum_{j=1}^N\: \left(\frac{\partial}{\partial v_j}\xi\right)\,\otimes\,dv_j \end{equation} In particular for $\mu\in\Lambda\cap{\mathbb Z }^n$ and \begin{equation}\label{eq:ximu} \xi_\mu\::=\:\textrm{cohomology class of } {\mathsf u }^\mu\cdot\frac{du_1}{u_1}\wedge\ldots\wedge\frac{du_n}{u_n}\; \in H^n(\Omega_\Lambda^\bullet,\delta) \end{equation} we find \begin{eqnarray}\label{eq:dximu1} \frac{\partial}{\partial v_j}\xi_\mu&=&\textrm{cohomology class of } {\mathsf u }^{{\mathfrak a }_j+\mu}\cdot\frac{du_1}{u_1}\wedge\ldots\wedge\frac{du_n}{u_n} \\ \label{eq:dximu2} &=& \xi_{\mu+{\mathfrak a }_j} \end{eqnarray} The form $\xi_\mu$ for $\mu\neq 0$ corresponds via (\ref{eq:dmod1}) and {} \cite{bat2} thm.7.13 with the form $\omega_\mu$ in (\ref{eq:ratform}); more precisely $\xi_\mu$ is the cohomology class of $\omega_\mu$ modulo $H^{n-1}(\widetilde{\mathbb T })$. \ \begin{corollary}\label{gkzximu} \begin{eqnarray} \label{eq:gkzxilin} \left(\mu+ \sum_{j=1}^N\:{\mathfrak a }_j\,v_j\frac{\partial}{\partial v_j}\right)\;\xi_\mu &=&0 \\ \label{eq:gkzximon} \left(\prod_{\ell_j>0} \left[\frac{\partial}{\partial v_j}\right]^{\ell_j} \:-\:\prod_{\ell_j<0} \left[\frac{\partial}{\partial v_j}\right]^{-\ell_j} \right)\xi_\mu&=&0\hspace{3mm}\textrm{ for } \ell\in{\mathbb L } \end{eqnarray} \end{corollary} \textbf{proof:} On the level of differential forms in the complex $(\Omega_\Lambda^\bullet,\delta)$ the $i$-th equation of (\ref{eq:gkzxilin}) reads \begin{eqnarray*} &&\left(\mu_i+ \sum_{j=1}^N\:a_{ij}\,v_j\frac{\partial}{\partial v_j}\right) {\mathsf u }^\mu\cdot\frac{du_1}{u_1}\wedge\ldots\wedge\frac{du_n}{u_n}\;= \\ &&=\; \delta\left((-1)^{i-1}{\mathsf u }^\mu \frac{du_1}{u_1}\wedge\ldots\wedge\frac{du_{i-1}}{u_{i-1}}\wedge \frac{du_{i+1}}{u_{i+1}}\wedge\ldots\wedge\frac{du_n}{u_n}\right) \end{eqnarray*} (\ref{eq:gkzximon}) follows immediately from (\ref{eq:dximu1}). {\hfill$\boxtimes$} \ \begin{remark}\textup{ We have essentially repeated the proof of \cite{bat2} thm. 14.2. There is however a small difference: Batyrev uses coefficients in ${\mathcal S }_\Lambda^+$ where we are using coefficients in ${\mathcal S }_\Lambda\,.$ His differential equations hold for $H^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s }) =H^n(\Omega_{\Lambda^+}^\bullet,\delta)$ whereas ours only hold in the primitive part $PH^{n-1}({\mathbb T }\setminus{\mathsf Z }_{\mathsf s })\,.$ On the other hand we can also treat $\xi_0$. The following theorem shows that this gives an important advantage.} \end{remark} \begin{theorem}\label{hygeoDmod} If $\Lambda\cap{\mathbb Z }^n\:=\:{\mathbb Z }_{\geq 0}{\mathfrak a }_1+\ldots+{\mathbb Z }_{\geq 0}{\mathfrak a }_N\,,$ then $\xi_0$ generates $H^n(\Omega_\Lambda^\bullet,\delta)$ as a module over the ring ${\mathcal D }\::=\:{\mathbb C }[v_1,\ldots,v_N,E_{{\mathsf A }}^{-1}, \frac{\partial}{\partial v_1},\ldots,\frac{\partial}{\partial v_N}]\,.$ The annihilator of $\xi_0$ in \rule{0mm}{3mm}${\mathcal D }$ is the left ideal generated by the differential operators \[ \sum_{j=1}^N\:a_{ij}\,v_j\frac{\partial}{\partial v_j} \hspace{4mm}\textrm{and} \hspace{4mm} \prod_{\ell_j>0} \left[\frac{\partial}{\partial v_j}\right]^{\ell_j} \:-\:\prod_{\ell_j<0} \left[\frac{\partial}{\partial v_j}\right]^{-\ell_j} \] with $1\leq i\leqn$ and $\ell\in{\mathbb L }\,.$ In other words, $ H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})= H^n(\Omega_\Lambda^\bullet,\delta)$ is the \emph{hypergeometric ${\mathcal D }$-module in the sense of {} \cite{gkz1}} \S 2.1 with parameters $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ and $\beta=0\,.$ \end{theorem} \textbf{proof:} Let ${\mathcal M }_0$ denote the hypergeometric ${\mathcal D }$-module with parameters $\beta=0$ and $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ as in \cite{gkz1} section 2.1. By corollary \ref{gkzximu} and formula (\ref{eq:dximu2}) we have a surjective homomorphism of ${\mathcal D }$-modules ${\mathcal M }_0\rightarrow H^n(\Omega_\Lambda^\bullet,\delta)\,.$ The filtration of ${\mathcal D }$ by the order of differential operators induces an ascending filtration on ${\mathcal M }_0$ and $H^n(\Omega_\Lambda^\bullet,\delta)\,.$ It suffices to prove that the above surjection induces an isomorphism for the associated graded modules. According to {} \cite{gkz1} prop.3 $gr\,{\mathcal M }_0$ is isomorphic to the quotient of the ring ${\mathbb C }[x_1,\ldots,x_N]\otimes{\mathbb C }[{\mathsf v }]$ by the ideal generated by the linear forms $\sum_{j=1}^N\,a_{ij}x_j$ for $i=1,\ldots,n$ and by the polynomials $\prod_{\ell_j>0} x_j^{\ell_j}\,-\,\prod_{\ell_j<0} x_j^{-\ell_j}$ with $\ell\in{\mathbb L }\,.$ Via the substitution homorphism $x_j\mapsto {\mathsf u }^{{\mathfrak a }_j}$ this quotient ring is isomorphic to the quotient of the ring ${\mathcal S }_\Lambda\otimes{\mathbb C }[{\mathsf v }]$ by the ideal generated by $\displaystyle{u_1\frac{\partial {\mathsf s }}{\partial u_1},\: u_2\frac{\partial {\mathsf s }}{\partial u_2}\,,\ldots,\: u_n\frac{\partial {\mathsf s }}{\partial u_n}}\,.$ Using (\ref{eq:dximu1}), (\ref{eq:dmod1}) and (\ref{eq:difops}) one checks that the latter quotient ring is isomorphic to $gr\,H^n(\Omega_\Lambda^\bullet,\delta)\,.$ {\hfill$\boxtimes$} \ \begin{center}\textbf{PART II A}\end{center} \section*{Introduction II A} In this Part II A we give our results the flavor of Mirror Symmetry by showing that for a regular triangulation ${\mathcal T }\!$ which satisfies conditions (\ref{eq:core1}), (\ref{eq:core2}), (\ref{eq:vol1}), the ring ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is the cohomology ring of a toric variety constructed somehow from the dual Gorenstein cone $\Lambda^\vee$ and that the ring $ \modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}$ is a subring of the Chow ring of a Calabi-Yau complete intersection in that toric variety; more precisely the subring is the image of the Chow ring of the ambient toric variety. We construct several toric varieties which are also used in \cite{babo}. As we want to promote the use of triangulations we give a construction of these toric varieties as a quotient of an open part of ${\mathbb C }^d$ ($d$ an appropriate dimension) by a torus. The torus is related to ${\mathbb L }$ and the open part is given by the triangulation ${\mathcal T }\!$. Such a construction of toric varieties is well known (see for instance \cite{guillemin}). \section{Triangulations with non-empty core and completely split reflexive Gorenstein cones.} \label{gorenstein} \begin{proposition}\label{corecs} Assume that ${\mathcal T }\!$ satisfies the following three conditions \begin{eqnarray} &&\mathrm{ core}\,{\mathcal T }\!\textrm{ is not empty and } \mathrm{ core}\,{\mathcal T }\!\,=\,\{1,\ldots,\kappa\}\label{eq:core1}\\ &&\mathrm{ core}\,{\mathcal T }\!\textrm{ is not contained in the boundary of } \Delta \label{eq:core2}\\ &&{\mathcal T }\! \textrm{ is unimodular}\label{eq:vol1} \end{eqnarray} Then $\Lambda\::=\:{\mathbb R }_{\geq 0}{\mathfrak a }_1\,+\ldots+\,{\mathbb R }_{\geq 0}{\mathfrak a }_N$ is a reflexive Gorenstein cone of index $\kappa$ and the dual cone $\Lambda^\vee$ is \emph{completely split} in the sense of {} \cite{babo} definition 3.9. \end{proposition} \textbf{proof:} By lemma \ref{corebound} and hypotheses (\ref{eq:core1}) and (\ref{eq:core2}) the $(n-2)$-dimensional simplices in the boundary of $\Delta$ are precisely the simplices $I\setminus\{i\}$ with $I\in{\mathcal T }\!^n$ and $i=1,\ldots,\kappa$. It follows that the dual cone $\Lambda^\vee$ is generated by the set of row vectors $\{{\mathfrak a }_{I,i}^\vee\;|\;I\in{\mathcal T }\!^n,\;i=1,\ldots,\kappa\:\}$ where \[ {\mathfrak a }_{I,i}^\vee\;:=\;\textrm{ the } i\textrm{-th row of the matrix } {\mathsf A }_I^{-1} \] Hypothesis (\ref{eq:vol1}) implies ${\mathfrak a }_{I,i}^\vee\in{\mathbb Z }^{n\vee}$ for all $I,i\,.$ By construction \begin{equation}\label{eq:splitting} {\mathfrak a }_{I,i}^\vee\cdot{\mathfrak a }_j\:=\: \left\{\begin{array}{ll}\geq 0 & \textrm{ for } j=1,\ldots,N\\ 1 & \textrm{ if } j=i\\ 0 & \textrm{ if } 1\leq j\leq\kappa\,,\;j\neq i \end{array}\right. \end{equation} So if we take \begin{equation}\label{eq:a0} {\mathfrak a }_0\::=\:{\mathfrak a }_1+\ldots+{\mathfrak a }_\kappa\;\in{\mathbb R }^n \end{equation} then $$ {\mathfrak a }_{I,i}^\vee\cdot{\mathfrak a }_0\,=\,1\hspace{5mm}\textrm{for}\hspace{3mm} I\in{\mathcal T }\!^n,\;i=1,\ldots,\kappa\,. $$ This shows that $\Lambda^\vee$ is a Gorenstein cone. Hence $\Lambda$ is a reflexive Gorenstein cone with index ${\mathfrak a }_0^\vee\cdot{\mathfrak a }_0=\kappa\,.$ Every element of $\Lambda^\vee$ can be written as $\sum_{I,i}\:s_{I,i}{\mathfrak a }_{I,i}^\vee$ with all $s_{I,i}\in{\mathbb R }_{\geq 0}\,.$ Such a sum can be rearranged as $\sum_{i=1}^\kappa t_i \alpha_i$ with $t_i=\sum_I\,s_{I,i}$ and $\alpha_i\in\square_i$ where \begin{equation}\label{eq:boxi} \square_i\::=\,\mathrm{ conv}\,\{{\mathfrak a }_{I,i}^\vee\:|\:I\in{\mathcal T }\!^n\:\}\,. \end{equation} $\square_i$ is a lattice polytope in the $(n-\kappa)$-dimensional affine subspace of ${\mathbb R }^{n\vee}$ given by the equations $\xi\cdot{\mathfrak a }_i=1$ and $\xi\cdot{\mathfrak a }_j=0$ if $1\leq j\leq\kappa\,,\;j\neq i$ (cf. (\ref{eq:splitting})). This shows that $\Lambda^\vee$ is a \emph{completely split} reflexive Gorenstein cone of index $\kappa$ in the sense of {} \cite{babo} definition 3.9. Note that the dimension of $\square_i$ equals $n-2$ minus the dimension of the minimal face of $\Delta$ which contains $\{{\mathfrak a }_j\:|\:j\in\mathrm{ core}\,{\mathcal T }\!\setminus\{i\}\;\}\,.$ {\hfill$\boxtimes$} \section{Triangulations and toric varieties} \label{tritorvar} \emph{We assume from now on that ${\mathcal T }\!$ satisfies the conditions (\ref{eq:core1}), (\ref{eq:core2}), (\ref{eq:vol1}).} Take some $I_0\in{\mathcal T }\!^n$ and consider the matrix $(u_{ij})\,:=\,{\mathsf A } _{I_0}^{-1}{\mathsf A }$. Then in definition \ref{ring} the linear forms \begin{equation}\label{eq:linrelu} u_{i1}C_1+u_{i2}C_2+\ldots+u_{iN}C_N\hspace{5mm}(i=1,\ldots,n) \end{equation} together with the monomials in (\ref{eq:srrel}) give another system of generators for the ideal ${\mathcal J }$. The corresponding relations in ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ for $i=1,\ldots,\kappa$ express $c_1,\ldots,c_\kappa$ as linear combinations of $c_{\kappa+1},\ldots,c_N$. The relations for $i=\kappa+1,\ldots,n$ do not involve $c_1,\ldots,c_\kappa\,.$ Also the monomials in (\ref{eq:srrel}) do not involve $C_1,\ldots,C_\kappa\,.$ Let ${\mathfrak u }_{\kappa+1},\ldots,{\mathfrak u }_N\,\in{\mathbb R }^{n-\kappa}$ be the columns of the matrix $(u_{ij})_{\kappa<i\leqn,\kappa<j\leq N}$. There is a simplicial fan ${\mathcal F }^\prime$ in ${\mathbb R }^{n-\kappa}$ given by the cones \begin{equation} \label{eq:profan} {\mathbb R }_{\geq 0}{\mathfrak u }_{i_1}+\ldots+{\mathbb R }_{\geq 0}{\mathfrak u }_{i_s}\hspace{5mm} \textrm{ with }i_1,\ldots,i_s > \kappa \textrm{ and }\{i_1,\ldots,i_s\}\in{\mathcal T }\! \end{equation} i.e. the index set is a simplex in the triangulation ${\mathcal T }\!$. The fan ${\mathcal F }^\prime$ is complete iff $0\in{\mathbb R }^{n-\kappa}$ is a linear combination with positive coefficients of the vectors ${\mathfrak u }_{\kappa+1},\ldots,{\mathfrak u }_N$. This is equivalent to condition (\ref{eq:core2}). Condition (\ref{eq:vol1}) implies that ${\mathcal F }^\prime$ is a fan of regular simplicial cones, i.e. its maximal cones are spanned by a basis of ${\mathbb Z }^{n-\kappa}$. Combining these considerations with \cite{dani} thm.10.8 or \cite{ful} prop.p.106 we find: \begin{theorem}\label{protor} If the triangulation ${\mathcal T }\!$ satisfies conditions (\ref{eq:core1}), (\ref{eq:core2}), (\ref{eq:vol1}), then ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is isomorphic to the cohomology ring $H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })$ of the $(n-\kappa)$-dimensional smooth projective toric variety ${\mathbb P }_{\mathcal T }\!$ associated with the fan ${\mathcal F }^\prime$ (see definition \ref{ptor}); more precisely: $$ {\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(m)}\simeq H^{2m}({\mathbb P }_{\mathcal T }\!,{\mathbb Z })\;,\hspace{5mm} m=0,1,\ldots,n-\kappa\,. $$ and ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(m)}=0$ for $m>n-\kappa\,.${\hfill$\boxtimes$} \end{theorem} \ There is much more geometry in those three conditions than was used for theorem \ref{protor}. Consider in ${\mathbb R }^n$ the fan ${\mathcal F }$ consisting of the cones \begin{equation} \label{eq:fan} {\mathbb R }_{\geq 0}{\mathfrak a }_{i_1}+\ldots+{\mathbb R }_{\geq 0}{\mathfrak a }_{i_s}\;\;,\hspace{5mm} \{i_1,\ldots,i_s\}\in{\mathcal T }\!\,. \end{equation} The standard constructions produce a toric variety ${\mathbb E }_{\,{\mathcal T }\!}$ from this fan. We recall the construction of the toric variety ${\mathbb E }_{\,{\mathcal T }\!}$ as a quotient of an open part of ${\mathbb C }^N$ by the torus ${\mathbb L }\otimes{\mathbb C }^{\,\ast}$. This torus appears here because ${\mathbb L }$ is the lattice of linear relations between the vectors ${\mathfrak a }_1,\ldots,{\mathfrak a }_N\,;$ by condition (\ref{eq:vol1}) and corollary \ref{allvert} these are exactly the generators of the $1$-dim cones of the fan ${\mathcal F }\,.$ Take ${\mathbb C }^N$ with coordinates $x_1,\dots,x_N$ and define \begin{eqnarray} {\mathbb C }^N_I&\::=\:&\{\,(x_1,\dots,x_N)\in{\mathbb C }^N\;|\;x_j\neq 0\textrm{ if } j\not\in I\:\}\hspace{3mm}\textrm{ for }\hspace{3mm}I\in{\mathcal T }\!^n \nonumber \\ \label{eq:cnt} {\mathbb C }^N_{\mathcal T }\!&\::=\:&\bigcup_{I\in{\mathcal T }\!^n}\:{\mathbb C }^N_I \end{eqnarray} The torus ${\mathbb C }^{\,\ast N}$ acts on ${\mathbb C }^N$ via coordinatewise multiplication. The inclusion ${\mathbb L }\subset{\mathbb Z }^N$ induces an inclusion of tori ${\mathbb L }\otimes{\mathbb C }^{\,\ast}\subset{\mathbb C }^{\,\ast N}$. Thus ${\mathbb L }\otimes{\mathbb C }^{\,\ast}$ acts on ${\mathbb C }^N\,.$ For $\ell=(\ell_1,\ldots,\ell_N)\in{\mathbb L }\,,\;t\in{\mathbb C }^{\,\ast}$ the element $\ell\otimes t$ acts as \begin{equation}\label{eq:action} (\ell\otimes t)\cdot(x_1,\dots,x_N)\::=\: (t^{\ell_1}x_1,\ldots,t^{\ell_N}x_N) \end{equation} \begin{definition}\label{etor}\hspace{5mm} $\displaystyle{{\mathbb E }_{\,{\mathcal T }\!}\::=\: \modquot{{\mathbb C }^N_{\mathcal T }\!}{{\mathbb L }\otimes{\mathbb C }^{\,\ast}}\,.}$ \end{definition} \ Take an ${(N-n)}\times N$-matrix ${\mathsf B }$ with entries in ${\mathbb Z }$ such that the columns of ${\mathsf B }^t$ constitute a basis for ${\mathbb L }\,.$ For $I\subset\{1,\ldots,N\}$ we denote by ${\mathsf A }_I$ (resp. ${\mathsf B }_{I^\ast}$) the submatrix of ${\mathsf A }$ (resp. ${\mathsf B }$) composed of the entries with column index in $I$ (resp. in $I^\ast\::=\,\{1,\ldots,N\}\setminus I$ ). Consider $I=\{i_1,\ldots,i_n\}\in{\mathcal T }\!^n$. Then $\det({\mathsf B }_{I^\ast})\,=\,\pm \det({\mathsf A }_I)\,=\,\pm 1$ by condition (\ref{eq:vol1}). So ${\mathsf B }_{I^\ast}$ is invertible over ${\mathbb Z }\,.$ From this one easily sees that there is an isomorphism \begin{eqnarray}\label{eq:chart} {\mathbb C }^n&\stackrel{\simeq}{\longrightarrow}& \modquot{{\mathbb C }^N_I}{{\mathbb L }\otimes{\mathbb C }^{\,\ast}}\\ \nonumber (y_1,\ldots,y_n)&\mapsto& (x_1,\dots,x_N) \textrm{ with } x_j=\left\{\begin{array}{ll} y_t&\textrm{ if } j=i_t\in I\\ 1&\textrm{ if } j\not\in I \end{array}\right. \end{eqnarray} Hence ${\mathbb E }_{\,{\mathcal T }\!}$ is a smooth toric variety. The torus $\modquot{{\mathbb C }^{\ast N}}{{\mathbb L }\otimes{\mathbb C }^{\,\ast}}={\mathbb M }\otimes{\mathbb C }^{\,\ast}$ acts on ${\mathbb E }_{\,{\mathcal T }\!}$ and the variety ${\mathbb E }_{\,{\mathcal T }\!}$ contains ${\mathbb M }\otimes{\mathbb C }^{\,\ast}$ as a dense open subset. One constructs in the same way the toric variety ${\mathbb P }_{\mathcal T }\!$ from the fan ${\mathcal F }^\prime$ (see (\ref{eq:profan})). Now the lattice of linear relations between the generators ${\mathfrak u }_{\kappa+1},\ldots,{\mathfrak u }_N$ of the $1$-dimensional cones of the fan ${\mathcal F }^\prime$ is the image of the composite map ${\mathbb L }\hookrightarrow{\mathbb Z }^N\twoheadrightarrow {\mathbb Z }^{N-\kappa}$. This map ${\mathbb L }\rightarrow {\mathbb Z }^{N-\kappa}$ is also injective. Take ${\mathbb C }^{N-\kappa}$ with coordinates $x_{\kappa+1},\dots,x_N$ and define \begin{eqnarray} {\mathbb C }^{N-\kappa}_I&\::=\:&\{\,(x_{\kappa+1},\dots,x_N)\in{\mathbb C }^N\;|\;x_j\neq 0\textrm{ if } j\not\in I\:\}\hspace{3mm}\textrm{ for }\hspace{3mm}I\in{\mathcal T }\!^n \nonumber \\ {\mathbb C }^{N-\kappa}_{\mathcal T }\!&\::=\:&\bigcup_{I\in{\mathcal T }\!^n}\:{\mathbb C }^{N-\kappa}_I \end{eqnarray} ${\mathbb L }\otimes{\mathbb C }^{\,\ast}$ is a subtorus of ${\mathbb C }^{\,\ast {N-\kappa}}$ and acts accordingly; i.e. as in (\ref{eq:action}) using only the coordinates with index $>\kappa\,.$ \ \begin{definition}\label{ptor}\hspace{5mm} $\displaystyle{{\mathbb P }_{\mathcal T }\!\::=\: \modquot{{\mathbb C }^{N-\kappa}_{\mathcal T }\!}{{\mathbb L }\otimes{\mathbb C }^{\,\ast}}\,.}$ \end{definition} \ ${\mathbb P }_{\mathcal T }\!$ is a smooth projective toric variety: smooth for the same reason as ${\mathbb E }_{\,{\mathcal T }\!}$ and projective because the fan ${\mathcal F }^\prime$ is complete. Projection onto the last $N-\kappa$ coordinates induces a surjective morphism \begin{equation}\label{eq:morphism} \pi\,:\;{\mathbb E }_{\,{\mathcal T }\!}\rightarrow{\mathbb P }_{\mathcal T }\! \end{equation} As (\ref{eq:cnt}) puts no restriction on the coordinates $x_1,\ldots,x_\kappa\,,$ the fibers of $\pi$ are complex vector spaces of dimension $\kappa\,;$ more precisely, (\ref{eq:chart}) gives a trivialization \[ \modquot{{\mathbb C }^N_I}{{\mathbb L }\otimes{\mathbb C }^{\,\ast}}\;\simeq\;{\mathbb C }^n\;\simeq\; {\mathbb C }^\kappa\times{\mathbb C }^{n-\kappa}\;\simeq\; {\mathbb C }^\kappa\times\left(\modquot{{\mathbb C }^{N-\kappa}_I}{{\mathbb L }\otimes{\mathbb C }^{\,\ast}} \right) \] Thus: \begin{proposition} ${\mathbb E }_{\,{\mathcal T }\!}$ has the structure of a vector bundle of rank $\kappa$ over ${\mathbb P }_{\mathcal T }\!\,.$ {\hfill$\boxtimes$} \end{proposition} \ The dual vector bundle ${\mathbb E }_{\,{\mathcal T }\!}^\vee\rightarrow{\mathbb P }_{\mathcal T }\!$ can be constructed as \begin{equation}\label{eq:edual} {\mathbb E }_{\,{\mathcal T }\!}^\vee\::=\: \modquot{{\mathbb C }^N_{\mathcal T }\!}{({\mathbb L }\otimes{\mathbb C }^{\,\ast})^\prime} \end{equation} with ${\mathbb C }^N_{\mathcal T }\!$ as in definition \ref{etor}, but with the action of ${\mathbb L }\otimes{\mathbb C }^{\,\ast}$ slightly modified from (\ref{eq:action}): the element $\ell\otimes t$ now acts as \begin{equation}\label{eq:duaction} (\ell\otimes t)\cdot^\prime\,(x_1,\dots,x_N)\::=\: (t^{-\ell_1}x_1,\ldots,t^{-\ell_\kappa}x_\kappa, t^{\ell_{\kappa+1}}x_{\kappa+1},\ldots,t^{\ell_N}x_N) \end{equation} For the sake of completeness we also describe the construction of the bundle of projective spaces ${\mathbb P }{\mathbb E }_{\,{\mathcal T }\!}\rightarrow{\mathbb P }_{\mathcal T }\!$ associated with the vector bundle ${\mathbb E }_{\,{\mathcal T }\!}\rightarrow{\mathbb P }_{\mathcal T }\!\,.$ Take as before ${\mathbb C }^N$ with coordinates $x_1,\dots,x_N\,.$ Define for $i\in\mathrm{ core}\,{\mathcal T }\!$ and $I\in{\mathcal T }\!^n$ \begin{eqnarray} {\mathbb C }^N_{i,I}&\::=\:&\{\,(x_1,\dots,x_N)\in{\mathbb C }^N\;|\;x_i\neq 0 \textrm{ and }x_j\neq 0\textrm{ if } j\not\in I\:\} \nonumber \\ \label{eq:cnpt} {\mathbb C }^N_{{\mathcal T }\!\circ}&\::=\:&\bigcup_{i\in\mathrm{ core}\,{\mathcal T }\!\,,\:I\in{\mathcal T }\!^n}\:{\mathbb C }^N_{i,I} \end{eqnarray} Write ${\mathsf k }:=(k_1,\ldots,k_N)^t$ with $k_j=1$ if $j\in\mathrm{ core}\,{\mathcal T }\!$ resp. $k_j=0$ if $j\not\in\mathrm{ core}\,{\mathcal T }\!\,,$ i.e. ${\mathsf k }=(1,\ldots,1,0,\ldots,0)^t$. Clearly ${\mathsf k }\not\in{\mathbb L }\,.$ Hence ${\mathbb Z }\cdot{\mathsf k }\oplus{\mathbb L }\subset{\mathbb Z }^N$ and $({\mathbb Z }\cdot{\mathsf k }\oplus{\mathbb L })\otimes{\mathbb C }^{\,\ast}\:\subset\:{\mathbb C }^{\,\ast N}$. Then \begin{equation}\label{eq:pxtor} {\mathbb P }{\mathbb E }_{\,{\mathcal T }\!}\::=\: \modquot{{\mathbb C }^N_{{\mathcal T }\!\circ}}{({\mathbb Z }\cdot{\mathsf k }\oplus{\mathbb L })\otimes{\mathbb C }^{\,\ast}}\,. \end{equation} with the morphism ${\mathbb P }{\mathbb E }_{\,{\mathcal T }\!}\rightarrow{\mathbb P }_{\mathcal T }\!$ induced from projection onto the last $N-\kappa$ coordinates. There are two kinds of codim $1$ simplices in the triangulation ${\mathcal T }\!\,:$ those which do contain $\mathrm{ core}\,{\mathcal T }\!$ and those which do not. Those which do not contain $\mathrm{ core}\,{\mathcal T }\!$ are precisely the ones of the form $I\setminus\{i\}$ with $I\in{\mathcal T }\!^n$ and $i\in\mathrm{ core}\,{\mathcal T }\!\,.$ Notice the relation with (\ref{eq:cnpt}). The codim $1$ simplices which do not contain $\mathrm{ core}\,{\mathcal T }\!$ constitute a triangulation of the boundary of $\Delta\,.$ Let as in (\ref{eq:a0}) \[ {\mathfrak a }_0\::=\:{\mathfrak a }_1+\ldots+{\mathfrak a }_\kappa\,. \] Then ${\mathbb Z }\cdot{\mathsf k }\oplus{\mathbb L }\subset{\mathbb Z }^N$ is precisely the lattice of linear relations between the vectors ${\mathfrak a }_1-\frac{1}{\kappa}{\mathfrak a }_0,\:{\mathfrak a }_2-\frac{1}{\kappa}{\mathfrak a }_0,\ldots, {\mathfrak a }_N-\frac{1}{\kappa}{\mathfrak a }_0\,.$ Thus we see: \ \begin{proposition} ${\mathbb P }{\mathbb E }_{\,{\mathcal T }\!}$ is the $(n-1)$-dimensional smooth projective toric variety associated with the lattice ${\mathbb Z } ({\mathfrak a }_1-\frac{1}{\kappa}{\mathfrak a }_0)+\ldots+{\mathbb Z }({\mathfrak a }_N-\frac{1}{\kappa}{\mathfrak a }_0)$ and the fan consisting the cones with apex $0$ over the simplices of the triangulation of the boundary of $-\frac{1}{\kappa}{\mathfrak a }_0+\Delta$ induced by ${\mathcal T }\!\,.$ {\hfill$\boxtimes$} \end{proposition} \section{Calabi-Yau complete intersections in toric varieties} \label{cicy} According to proposition \ref{corecs} conditions (\ref{eq:core1}), (\ref{eq:core2}), (\ref{eq:vol1}) imply that $\Lambda^\vee$ is a completely split reflexive Gorenstein cone. In \cite{babo} Batyrev and Borisov relate this splitting property to complete intersections in toric varieties. Formulated in our present context this relation is as follows. A (global) section of ${\mathbb E }_{\,{\mathcal T }\!}^\vee\rightarrow{\mathbb P }_{\mathcal T }\!$ is given by polynomials $P_i(x_{\kappa+1},\ldots,x_N)$ $(i=1,\ldots,\kappa)$ which satisfy the homogeneity condition \begin{equation} \label{eq:homogeneous} P_i(t^{\ell_{\kappa+1}}\cdot x_{\kappa+1},\ldots, t^{\ell_N}\cdot x_N)\:=\:t^{-\ell_i}\cdot P_i(x_{\kappa+1},\ldots,x_N) \end{equation} for every $ t\in{\mathbb C }^{\,\ast} $ and $\ell=(\ell_1,\ldots,\ell_N)^t\in{\mathbb L }\,.$ The vector bundle is a direct sum of line bundles and the polynomial $P_i$ gives a section of the $i$-th line bundle. The polynomial $P_i$ is a linear combination of monomials $x_{\kappa+1}^{m_{\kappa+1}}\cdot\ldots\cdot x_N^{m_N}$ such that \[ \ell_{\kappa+1}m_{\kappa+1}+\ldots+\ell_N m_N\,=\,-\ell_i \hspace{4mm}\textrm{ for all }\ell=(\ell_1,\ldots,\ell_N)\in{\mathbb L }\,. \] These monomials correspond bijectively to the elements $(m_1,\ldots,m_N)$ in the row space of matrix ${\mathsf A }$ which satisfy $m_i=1$, $m_j=0$ if $1\leq j\leq\kappa, j\neq i$ and $m_j\geq 0$ if $j>\kappa\,.$ Equivalently, these monomials correspond bijectively to the elements ${\mathsf w }\in{\mathbb Z }^{n\vee}$ which satisfy \begin{equation}\label{eq:splitting2} {\mathsf w }\cdot{\mathfrak a }_j\:=\: \left\{\begin{array}{ll}\geq 0 & \textrm{ for } j=1,\ldots,N\\ 1 & \textrm{ if } j=i\\ 0 & \textrm{ if } 1\leq j\leq\kappa\,,\;j\neq i \end{array}\right. \end{equation} So the monomials in the polynomial $P_i$ correspond bijectively to the integral lattice points in the polytope $\square_i$; see (\ref{eq:boxi}). The zero locus of the section of ${\mathbb E }_{\,{\mathcal T }\!}^\vee\rightarrow{\mathbb P }_{\mathcal T }\!$ corresponding to the polynomials $P_i(x_{\kappa+1},\ldots,x_N)$ $(i=1,\ldots,\kappa)$ is clearly the complete intersection in ${\mathbb P }_{\mathcal T }\!$ with (homogeneous) equations \begin{equation}\label{eq:cyci} P_i(x_{\kappa+1},\ldots,x_N)=0\hspace{5mm} (i=1,\ldots,\kappa) \end{equation} If the coefficients of these polynomials satisfy a $\Lambda^\vee$-regularity condition, then this complete intersection is a Calabi-Yau variety $Y$ of dimension $n-2\kappa\,.$ The ring ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is isomorphic to the cohomology ring of the toric variety ${\mathbb P }_{\mathcal T }\!\,.$ The elements $-c_1,\ldots,-c_{\kappa}$ are the Chern classes of the hypersurfaces associated with the polynomials $P_1,\ldots,P_{\kappa}$. With as before $c_{\mathrm{ core}\,}=c_1\cdot\ldots\cdot c_{\kappa}$, the ring $\modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}$ is isomorphic to the image of $H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })$ in $H^\ast(Y,{\mathbb Z })$. \begin{center}\section*{Conclusions}\end{center} Consider the map ${\mathsf v }:\,{\mathbb C }^{N\vee}\rightarrow{\mathbb C }^{N\vee}\:,$ $\;{\mathsf v }(z_1,\ldots,z_N)\::=\: ({\mathsf e }^{2\pi i\,z_1},\ldots,{\mathsf e }^{2\pi i\,z_N})\,.$ According to \cite{gkz2} p.304 cor.1.7 there is a vector $b\in{\mathcal C }_{\mathcal T }\!$ such that \begin{equation}\label{eq:discno} E_{{\mathsf A }} ({\mathsf v }({\mathsf z }))\neq 0\hspace{4mm}\textrm{for all} \hspace{3mm} {\mathsf z }\in{\mathbb C }^{N\vee} \hspace{3mm}\textrm{such that}\hspace{3mm} p\,(\Im\,{\mathsf z })\in b+{\mathcal C }_{\mathcal T }\!\,; \end{equation} here $p:{\mathbb R }^{N\vee}\rightarrow{\mathbb L }_{\mathbb R }^\vee$ denotes the surjection dual to the inclusion ${\mathbb L }\hookrightarrow{\mathbb Z }^n$. This shows how one can replace the domain of definition ${\mathcal V }_{\mathcal T }\!$ of the functions $\Psi_{{\mathcal T }\!,\beta}$ (cf. (\ref{eq:defdom})) by a slightly smaller domain ${\mathcal V }^\prime_{\mathcal T }\!$ such that on ${\mathsf v }({\mathcal V }^\prime_{\mathcal T }\!)$ the function $E_{{\mathsf A }} $ is nowhere zero. The ${\mathcal D }$-module $ H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})$ is therefore defined on ${\mathsf v }({\mathcal V }^\prime_{\mathcal T }\!)$; cf. theorem \ref{hygeoDmod}. Its pullback to ${\mathcal V }^\prime_{\mathcal T }\!$ is the ${\mathcal D }_{\mathcal T }\!$-module $H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\otimes {\mathcal O }_{\mathcal T }\!\,,$ where ${\mathcal O }_{\mathcal T }\!$ denotes the ring of holomorphic functions on ${\mathcal V }^\prime_{\mathcal T }\!$ and ${\mathcal D }_{\mathcal T }\!$ denotes the corresponding ring of differential operators. The functions $\Psi_{{\mathcal T }\!,\beta}$ are also defined on the domain ${\mathcal V }^\prime_{\mathcal T }\!$ and $$ \Psi_{{\mathcal T }\!,\beta}\:\in\: {\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes {\mathcal O }_{\mathcal T }\!\,. $$ ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes {\mathcal O }_{\mathcal T }\!$ is a ${\mathcal D }_{\mathcal T }\!$-module with ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ as its group of horizontal sections. The following theorem summarizes the results of this paper: \ \begin{theorem}\label{mainthm} Let $\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ be a finite subset of ${\mathbb Z }^n$ which satisfies condition \ref{gkzcond}. Let $\Lambda:={\mathbb R }_{\geq 0}{\mathfrak a }_1+\ldots+{\mathbb R }_{\geq 0}{\mathfrak a }_N$ be the associated Gorenstein cone and $\Delta:=\mathrm{ conv}\, \{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}\,.$ \begin{enumerate} \item If there exists a unimodular regular triangulation of $\Delta$, then condition \ref{aisall} is satisfied and ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$ generate ${\mathbb Z }^n$, i.e. \begin{equation}\label{eq:unimodex} \Delta\cap{\mathbb Z }^n=\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}\hspace{4mm}\textrm{and} \hspace{4mm}{\mathbb M }={\mathbb Z }^n\,. \end{equation} \item For every unimodular regular triangulation ${\mathcal T }\!$ there is an isomorphism of ${\mathcal D }_{\mathcal T }\!$-modules on ${\mathcal V }^\prime_{\mathcal T }\!$ : \begin{equation}\label{eq:isodmod} H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\otimes {\mathcal O }_{\mathcal T }\!\;\simeq\; {\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes {\mathcal O }_{\mathcal T }\! \end{equation} through which $\xi_0$ corresponds with $\Psi_{{\mathcal T }\!,0}$. More generally $\xi_\mu$ corresponds with $\Psi_{{\mathcal T }\!,-\mu}$ if $\mu\in\Lambda\cap{\mathbb Z }^n\,.$ \item In particular if $\Lambda$ is a reflexive Gorenstein cone of index $\kappa$ and ${\mathcal T }\!$ is a unimodular regular triangulation, then ${\mathcal W }_n H^n(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\otimes {\mathcal O }_{\mathcal T }\!$ is generated as a ${\mathcal D }_{\mathcal T }\!$-module by $\xi_{{\mathfrak a }_0}$ and corresponds via (\ref{eq:isodmod}) with the sub-${\mathcal D }_{\mathcal T }\!$-module of ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes {\mathcal O }_{\mathcal T }\!$ generated by $\Psi_{{\mathcal T }\!,-{\mathfrak a }_0}\,.$ Moreover $\xi_{{\mathfrak a }_0}$ has weight $n$ and Hodge type $(n-\kappa,\kappa)$. \item If $\Lambda$ is a reflexive Gorenstein cone and ${\mathcal T }\!$ is a unimodular regular triangulation with non-empty core, then (\ref{eq:isodmod}) induces an isomorphism \begin{equation}\label{eq:isodmodw} \begin{array}{rcl} {\mathcal W }_n H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\otimes {\mathcal O }_{\mathcal T }\! &\simeq& c_{\mathrm{ core}\,}{\mathcal R }_{{\mathsf A },{\mathcal T }\!}\otimes {\mathcal O }_{\mathcal T }\! \\[.6em] &\simeq& \modquot{{\mathcal R }_{{\mathsf A },{\mathcal T }\!}}{\mathrm{ Ann}\, c_{\mathrm{ core}\,}}\:\otimes {\mathcal O }_{\mathcal T }\! \end{array} \end{equation} \item Now assume ${\mathcal T }\!$ satisfies conditions (\ref{eq:core1}), (\ref{eq:core2}), (\ref{eq:vol1}), i.e. ${\mathcal T }\!$ is a unimodular regular triangulation whose core is not empty and is not contained in the boundary of $\Delta\,.$ Then \begin{enumerate} \item $\Lambda$ is a reflexive Gorenstein cone. \item ${\mathcal R }_{{\mathsf A },{\mathcal T }\!}$ is isomorphic to the cohomology ring $H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })$ of the $(n-\kappa)$-dimensional smooth projective toric variety ${\mathbb P }_{\mathcal T }\!$: \begin{equation}\label{eq:rattor} {\mathcal R }_{{\mathsf A },{\mathcal T }\!}^{(m)}\simeq H^{2m}({\mathbb P }_{\mathcal T }\!,{\mathbb Z })\;,\hspace{5mm} m=0,1,\ldots,n-\kappa \end{equation} and in particular for $m=1$: ${\mathbb L }_{\mathbb Z }^\vee\simeq\textrm{Pic}({\mathbb P }_{\mathcal T }\!)\,.$ \item $c_{\mathrm{ core}\,}=c_\kappa({\mathbb E }_{\,{\mathcal T }\!})\,,$ the top Chern class of the vectorbundle ${\mathbb E }_{\,{\mathcal T }\!}\,.$ \item The zero locus of a general section of the dual vector bundle ${\mathbb E }_{\,{\mathcal T }\!}^\vee$ is an $n-2\kappa$-dimensional Calabi-Yau complete intersection in ${\mathbb P }_{\mathcal T }\!$. \item \begin{eqnarray} H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\otimes {\mathcal O }_{\mathcal T }\!&\simeq& H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })\otimes {\mathcal O }_{\mathcal T }\! \\ {\mathcal W }_n H^{n}(\widetilde{\mathbb T }\,\mathrm{ rel }\,\widetilde{\mathsf Z }_{{\mathsf s }-1})\otimes {\mathcal O }_{\mathcal T }\! &\simeq& c_\kappa({\mathbb E }_{\,{\mathcal T }\!})\,H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })\otimes {\mathcal O }_{\mathcal T }\! \end{eqnarray} \item The monodromy representation is isomorphic to the representation of $\textrm{Pic}({\mathbb P }_{\mathcal T }\!)$ on $H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })$ ( resp. on $c_\kappa({\mathbb E }_{\,{\mathcal T }\!})\,H^\ast({\mathbb P }_{\mathcal T }\!,{\mathbb Z })$ ) in which the Chern class $c_1({\mathcal L })$ of a line bundle ${\mathcal L }$ acts as multiplication by $\exp(c_1({\mathcal L }))\,.$ \end{enumerate} \end{enumerate} \end{theorem} \textbf{Proof:} \textbf{(i)}: corollary \ref{allvert}. \textbf{(ii)}: theorems \ref{isosol} and \ref{hygeoDmod}, formulas (\ref{eq:recur}) and (\ref{eq:dximu2}). \textbf{(iii)}: formulas (\ref{eq:alint}), (\ref{eq:a0z}), (\ref{eq:weightfilt}) and theorem \ref{TZ}. \textbf{(iv)}: corollary \ref{corid} and theorem \ref{corsol}. \textbf{(v}\textit{a}\textbf{)}: proposition \ref{corecs}. \textbf{(v}\textit{b}\textbf{)}: theorem \ref{protor} and corollary \ref{allvert}. \textbf{(v}\textit{c}\textbf{)}: section \ref{cicy}. \textbf{(v}\textit{d}\textbf{)}: section \ref{cicy}. \textbf{(v}\textit{e}\textbf{)}: (\ref{eq:isodmod}) and (\ref{eq:isodmodw}). \textbf{(v}\textit{f}\textbf{)}: formula (\ref{eq:monodromy}). {\hfill$\boxtimes$} \ All cases which have on the A-side of mirror symmetry a smooth complete intersection Calabi-Yau variety in a smooth projective toric variety, are covered by this theorem. Indeed, a smooth projective toric variety ${\mathbb P }$ of dimension $d$ can be constructed from a complete simplicial fan in which every maximal cone is generated by a basis of the lattice ${\mathbb Z }^d$. Let ${\mathsf u }_1,\ldots,{\mathsf u }_p\in{\mathbb Z }^d$ be the generators of the $1$-dimensional cones in the fan and let $$ \overline{{\mathbb L }}:=\{(m_1,\ldots,m_p)\in{\mathbb Z }^p\:|\:m_1{\mathsf u }_1+\ldots m_p{\mathsf u }_p=0\:\} $$ The toric variety ${\mathbb P }$ can also be obtained as the quotient of a certain open part of ${\mathbb C }^p$ by the action of the subtorus $\overline{{\mathbb L }}\otimes{\mathbb C }^\ast$ of $({\mathbb C }^\ast)^p$. The Calabi-Yau complete intersection $Y$ of codimension $\kappa$ in ${\mathbb P }$ is the common zero locus of polynomials $P_1,\ldots,P_\kappa$ which are homogeneous for the action of $\overline{{\mathbb L }}\otimes{\mathbb C }^\ast$. The homogeneity of $P_i$ is given by a character of this torus, i.e. by a linear map $\chi_i:\overline{{\mathbb L }}\rightarrow{\mathbb Z }\,.$ Now set $N=p+\kappa$ and $n=d+\kappa\,.$ Let $$ {\mathbb L }:=\{(-\chi_1({\mathsf m }),\ldots,-\chi_\kappa({\mathsf m }),m_1,\ldots,m_p) \in{\mathbb Z }^N\:|\:{\mathsf m }=(m_1,\ldots,m_p)\in\overline{{\mathbb L }}\:\}\,. $$ Then ${\mathbb L }$ has rank $N-n$. The Calabi-Yau condition for $Y$ implies $\ell_1+\ldots+\ell_N=0$ for every $\ell=(\ell_1,\ldots,\ell_N)\in{\mathbb L }$. Let ${\mathsf B }$ be an $(N-n)\times N$-matrix with entries in ${\mathbb Z }$ such that the columns of ${\mathsf B }^t$ constitute a basis for ${\mathbb L }\,.$ Let ${\mathsf A }$ be an $n\times N$-matrix of rank $n$ with entries in ${\mathbb Z }$ such that ${\mathsf A }\cdot{\mathsf B }^t=0\,.$ Then the columns ${\mathfrak a }_1,\ldots,{\mathfrak a }_N$ of ${\mathsf A }$ satisfy condition \ref{gkzcond}. One obtains a regular triangulation of $\Delta:=\mathrm{ conv}\,\{{\mathfrak a }_1,\ldots,{\mathfrak a }_N\}$ which satisfies the three conditions (\ref{eq:core1}), (\ref{eq:core2}), (\ref{eq:vol1}), by taking as its maximal simplices all $\mathrm{ conv}\,\{{\mathfrak a }_1,\ldots,{\mathfrak a }_\kappa,{\mathfrak a }_{\kappa+i_1},\ldots,{\mathfrak a }_{\kappa+i_d}\}$ for which ${\mathsf u }_{i_1},\ldots,{\mathsf u }_{i_d}$ span a maximal cone in the fan defining ${\mathbb P }$. \subsection*{Acknowledgments} I want to thank the Taniguchi Foundation and the organizers of the Taniguchi Workshop ``Special Differential Equations'' 1991 for inviting me to this interesting workshop where I first learnt about GKZ hypergeometric functions and regular triangulations. Special thanks are for Bruce Hunt, who kindly pointed out that the example in my talk \cite{sti} at the workshop reminded him very much of the examples in Batyrev's talk a few weeks earlier in Kaiserslautern. I want to thank the Japan Society for the Promotion of Science for a JSPS Invitation Fellowship in November-December 1996 and Kobe University for support for a visit in July 1997. The stimulating atmosphere I experienced during these two visits to Kobe was very important for finishing this work. I also want to thank the organizers of various workshops - in particular the Taniguchi Symposium ``Integrable Systems and Algebraic Geometry'' 1997 - to which I was invited during these visits. I am very grateful to Masa-Hiko Saito for arranging as my host both visits and for the pleasant co-operation.

9711

alg-geom/9711030

en

https://arxiv.org/abs/alg-geom/9711030

alg-geom/9711030

Vicente Munoz Velazquez

Vicente Mu\~noz

Quantum cohomology of the moduli space of stable bundles over a Riemann surface

15 pages, LaTeX2e, no figures

We determine the quantum cohomology of the moduli space of odd degree rank two stable vector bundles over a Riemann surface $\Sigma$ of any genus. This work together with dg-ga/9710029 prove that this quantum cohomology is isomorphic to the instanton Floer cohomology of the three manifold $\Sigma \times S^1$. (Note: There is some overlap with the previous paper: alg-geom/9711013).

alg-geom

\section{Introduction} \label{sec:intro} Let $\S$ be a Riemann surface of genus $g \geq 2$ and let $M_{\Sigma}$ denote the moduli space of flat $SO(3)$-connections with nontrivial second Stiefel-Whitney class $w_2$. This is a smooth symplectic manifold of dimension $6g-6$. Alternatively, we can consider $\S$ as a smooth complex curve of genus $g$. Fix a line bundle $\L$ on $\S$ of degree $1$, then $M_{\Sigma}$ is the moduli space of rank two stable vector bundles on $\S$ with determinant $\L$, which is a smooth complex variety of complex dimension $3g-3$. The symplectic deformation class of $M_{\Sigma}$ only depends on $g$ and not on the particular complex structure on $\S$. The manifold $X=M_{\Sigma}$ is a positive symplectic manifold with $\pi_2(X)={\Bbb Z}$. For such a manifold $X$, its quantum cohomology, $QH^*(X)$, is well-defined (see~\cite{Ruan}~\cite{RT}~\cite{McDuff}~\cite{Piunikhin}). As vector spaces, $QH^*(X)=H^*(X)$ (rational coefficients are understood), but the multiplicative structure is different. Let $A$ denote the positive generator of $\pi_2(X)$, i.e. the generator such that the symplectic form evaluated on $A$ is positive. Let $N=c_1(X)[A] \in {\Bbb Z}_{>0}$. Then there is a natural ${\Bbb Z}/2N{\Bbb Z}$-grading for $QH^*(X)$, which comes from reducing the ${\Bbb Z}$-grading of $H^*(X)$. (For the case $X=M_{\Sigma}$, $N=2$, so $QH^*(M_{\Sigma})$ is ${\Bbb Z}/4{\Bbb Z}$-graded). The ring structure of $QH^*(X)$, called quantum multiplication, is a deformation of the usual cup product for $H^*(X)$. For $\a \in H^p(X)$, $\b \in H^q(X)$, we define the quantum product of $\a$ and $\b$ as $$ \a \cdot \b =\sum_{d \geq 0} \P_{dA}(\a, \b), $$ where $\P_{dA}(\a, \b) \in H^{p+q-2Nd} (X)$ is given by $<\P_{dA}(\a, \b),\gamma>=\Psi^X_{dA}(\a, \b, \gamma)$, the Gromov-Witten invariant, for all $\gamma \in H^{\dim X-p-q+2Nd} (X)$. One has $\P_0(\a, \b)=\a \cup \b$. The other terms are the quantum correction terms and they all live in lower degree parts of the cohomology groups. It is a fact~\cite{RT} that the quantum product gives an associative and graded commutative ring structure. To define the Gromov-Witten invariant, let $J$ be a generic almost complex structure compatible with the symplectic form. Then for every $2$-homology class $dA$, $d \in {\Bbb Z}$, there is a moduli space ${\cal M}_{dA}$ of pseudoholomorphic rational curves (with respect to $J$) $f : {\Bbb P}^1 \rightarrow X$ with $f_*[{\Bbb P}^1] = dA$. Note that ${\cal M}_0=X$ and that ${\cal M}_{dA}$ is empty for $d<0$. For $d \geq 0$, the dimension of ${\cal M}_{dA}$ is $\dim X +2Nd$. This moduli space ${\cal M}_{dA}$ admits a natural compactification, $\overline{{\cal M}}_{dA}$, called the Gromov-Uhlenbeck compactification~\cite{Ruan}~\cite[section 3]{RT}. Consider now $r \geq 3$ different points $\seq{P}{1}{r} \in {{\Bbb P}}^1$. Then we have defined an evaluation map $ev: {\cal M}_{dA} \rightarrow X^r$ by $f \mapsto (f(P_1), \ldots, f(P_r))$. This map extends to $\overline{{\cal M}}_{dA}$ and its image, $ev(\overline{{\cal M}}_{dA})$, is a pseudo-cycle~\cite{RT}. So for $\a_i \in H^{p_i}(M_{\Sigma})$, $1 \leq i \leq r$, with $p_1+ \cdots +p_r = \dim X + 2Nd$, we choose generic cycles $A_i$, $1 \leq i \leq r$, representatives of their Poincar\'e duals, and set \begin{equation} \label{eqn:qu1} \Psi_{dA}^X(\seq{\a}{1}{r}) = <A_1\times\cdots\times A_r, [ev(\overline{{\cal M}}_{dA})]>= \# ev_{P_1}^*(A_1) \cap \cdots \cap ev_{P_r}^*(A_r), \end{equation} where $\#$ denotes count of points (with signs) and $ev_{P_i}: {\cal M}_{dA} \rightarrow X$, $f \mapsto f(P_i)$. This is a well-defined number and independent of the particular cycles. Also, as the manifold $X$ is positive, $\text{coker} L_f=H^1({\Bbb P}^1, f^* c_1(X))=0$, for all $f \in {\cal M}_{dA}$ (see~\cite{Ruan} for definition of $L_f$). By~\cite{Ruan} the complex structure of $X$ is generic and we can use it to compute the Gromov-Witten invariants. Also for $r \geq 2$, let $\a_i \in H^{p_i}(M_{\Sigma})$, $1 \leq i \leq r$, then $$ \a_1 \cdots \a_r =\sum_{d \geq 0} \P_{dA}(\a_1, \ldots ,\a_r), $$ where the correction terms $\P_{dA}(\a_1, \ldots ,\a_r) \in H^{p_1+\cdots +p_r - 2 N d}(X)$ are determined by $<\P_{dA}(\a_1, \ldots ,\a_r), \gamma> = \Psi_{dA}^X(\a_1, \ldots ,\a_r, \gamma)$, for any $\gamma \in H^{\dim X +2Nd-(p_1+\cdots +p_r)}(X)$. Returning to our manifold $X=M_{\Sigma}$, there is a classical conjecture relating the quantum cohomology $QH^*(M_{\Sigma})$ and the instanton Floer cohomology of the three manifold $\Sigma \times {\Bbb S}^1$, $HF^*(\Sigma \times {\Bbb S}^1)$ (see~\cite{Floer}). In~\cite{Vafa} a presentation of $QH^*(M_{\Sigma})$ was given using physical methods, and in~\cite{Floer} it was proved that such a presentation was a presentation of $HF^*(\Sigma \times {\Bbb S}^1)$ indeed. Here we determine a presentation of $QH^*(M_{\Sigma})$ and prove the isomorphism $QH^*(M_{\Sigma}) \cong HF^*(\Sigma \times {\Bbb S}^1)$. Siebert and Tian have an alternative program~\cite{Siebert} to find the presentation of $QH^*(M_{\Sigma})$, which goes through proving a recursion formula for the Gromov-Witten invariants of $M_{\Sigma}$ in terms of the genus $g$. The paper is organised as follows. In section~\ref{sec:ordinary} we review the ordinary cohomology ring of $M_{\Sigma}$. In section~\ref{sec:curves} the moduli space of lines (rational curves representing $A$) in $M_{\Sigma}$ is described. This makes possible to compute the Gromov-Witten invariants $\Psi_A^{M_{\Sigma}}$, which determine the first quantum correction terms of the quantum products in $QH^*(M_{\Sigma})$. Section~\ref{sec:GW-inv} is devoted to this task. In~\cite{D1} Donaldson uses this information alone to determine $QH^*(M_{\Sigma})$ in the case of genus $g=2$. It is somehow natural to expect that this idea can be developed in the general case $g \geq 3$. In section~\ref{sec:quantum} we give an explicit presentation of $QH^*(M_{\Sigma})$ for $g \geq 3$ (theorem~\ref{thm:main}), concluding the proof of $QH^*(M_{\Sigma}) \cong HF^*(\Sigma \times {\Bbb S}^1)$ (corollary~\ref{cor:21}). The two main ingredients that we make use of are the $\text{Sp}\, (2g,{\Bbb Z})$-decomposition of $H^*(M_{\Sigma})$ under the action of the mapping class group (not ignoring the non-invariant part as it was customary) and a recursion similar to that in~\cite{Siebert} (lemma~\ref{lem:17}). The difference with~\cite{Siebert} lies in the fact that we fix the genus, so that we do not need to compare the Gromov-Witten invariants for moduli spaces of Riemann surfaces of different genus. Finally in section~\ref{sec:6} we discuss the cases $g=1$ and $g=2$, which are slightly different. \noindent {\em Acknowledgements:\/} The author is greatly indebted to the two sources of inspirations who have made possible this work: Prof.\ Simon Donaldson, for his encouragement and invaluable help, and Bernd Siebert and Gang Tian, for helpful conversations and for providing with a copy of their preprint~\cite{Siebert}, which was very enlightening. \section{Classical cohomology ring of $M_{\Sigma}$} \label{sec:ordinary} Let us recall the known description of the homology of $M_{\Sigma}$~\cite{King}~\cite{ST}~\cite{Floer}. Let ${\cal U} \rightarrow \S \times M_{\Sigma}$ be the universal bundle and consider the K\"unneth decomposition of \begin{equation} c_2(\text{End}_0 \, {\cal U})=2 [\S] \otimes \a + 4 \psi -\b, \label{eqn:qu2} \end{equation} with $\psi=\sum \gamma_i \otimes \psi_i$, where $\{\seq{\gamma}{1}{2g}\}$ is a symplectic basis of $H^1(\S;{\Bbb Z})$ with $\gamma_i \gamma_{i+g}=[\S]$ for $1 \leq i \leq g$ (also $\{\gamma^{\#}_i\}$ will denote the dual basis for $H_1(\S;{\Bbb Z})$). Here we can suppose without loss of generality that $c_1({\cal U})=\L + \a$ (see~\cite{ST}). In terms of the map $\mu: H_*(\S) \to H^{4-*}(M_{\Sigma})$, given by $\mu(a)= -{1 \over 4} \, p_1({\frak g}_{{\cal U}}) /a $ (here ${\frak g}_{{\cal U}} \to \S \times M_{\Sigma}$ is the associated universal $SO(3)$-bundle, and $p_1({\frak g}_{{\cal U}}) \in H^4(\S\xM_{\Sigma})$ its first Pontrjagin class), we have $$ \left\{ \begin{array}{l} \a= 2\, \mu(\S) \in H^2 \\ \psi_i= \mu (\gamma_i^{\#}) \in H^3, \qquad 1\leq i \leq 2g \\ \b= - 4 \, \mu(x) \in H^4 \end{array} \right. $$ where $x \in H_0(\S)$ is the class of the point, and $H^i=H^i(M_{\Sigma})$. These elements generate $H^*(M_{\Sigma})$ as a ring~\cite{King}~\cite{Thaddeus}, and $\a$ is the positive generator of $H^2(M_{\Sigma};{\Bbb Z})$. We can rephrase this as saying that there exists an epimorphism \begin{equation} \AA(\S)= {\Bbb Q}[\a,\b ]\otimes \L(\seq{\psi}{1}{2g}) \twoheadrightarrow H^*(M_{\Sigma}) \label{eqn:qu3} \end{equation} (the notation $\AA(\S)$ follows that of Kronheimer and Mrowka~\cite{KM}, although it is slightly different). Recall that $\deg(\a)=2$, $\deg(\b)=4$ and $\deg(\psi_i)=3$. The mapping class group $\text{Diff}(\S)$ acts on $H^*(M_{\Sigma})$, with the action factoring through the action of $\text{Sp}\, (2g,{\Bbb Z})$ on $\{\psi_i\}$. The invariant part, $H_I^*(M_{\Sigma})$, is generated by $\a$, $\b$ and $\gamma=-2 \sum_{i=0}^g \psi_i\psi_{i+g}$. Then there is an epimorphism \begin{equation} {\Bbb Q}[\a,\b,\gamma ] \twoheadrightarrow H^*_I(M_{\Sigma}) \label{eqn:qu4} \end{equation} which allows us to write $$ H_I^*(M_{\Sigma})= {\Bbb Q} [\a, \b, \gamma]/I_g, $$ where $I_g$ is the ideal of relations satisfied by $\a$, $\b$ and $\gamma$. {}From~\cite{ST}, a basis for $H_I^*(M_{\Sigma})$ is given by the monomials $\a^a\b^b\gamma^c$, with $a+b+c<g$. For $0 \leq k \leq g$, the primitive component of $\L^k H^3$ is $$ \L_0^k H^3 = \ker (\gamma^{g-k+1} : \L^k H^3 \rightarrow \L^{2g-k+2} H^3). $$ The spaces $\L^k_0 H^3$ are irreducible $\text{Sp}\, (2g,{\Bbb Z})$-modules, i.e. the transforms of any nonzero element of $\L^k_0 H^3$ under $\text{Sp}\, (2g,{\Bbb Z})$ generate the whole of it. The description of the ideals $I_g$ and the cohomology ring $H^*(M_{\Sigma})$ is given in the following \begin{prop}[\cite{ST}~\cite{King}] \label{prop:1} Define $q^1_0=1$, $q^2_0=0$, $q^3_0=0$ and then recursively, for all $r \geq 1$, $$ \left\{ \begin{array}{l} q_{r+1}^1 = \a q_r^1 + r^2 q_r^2 \\ q_{r+1}^2 = (\b+(-1)^{r+1}8) q_r^1 + {2r \over r+1} q_r^3 \\ q_{r+1}^3 = \gamma q_r^1 \end{array} \right. $$ Then $I_g=(q^1_g,q^2_g,q^3_g) \subset {\Bbb Q}[\a,\b,\gamma]$, for all $g \geq 1$. Note that $\deg(q_g^1)=2g$, $\deg(q_r^2)=2g+2$ and $\deg(q_g^3)=2g+4$. Moreover the $\text{Sp}\, (2g,{\Bbb Z})$-decomposition of $H^*(M_{\Sigma})$ is \begin{equation} H^*(M_{\Sigma})= \bigoplus_{k=0}^{g-1} \L_0^k H^3 \otimes {\Bbb Q} [\a, \b, \gamma]/I_{g-k}. \label{eqn:antes} \end{equation} \end{prop} This proposition allows us to find a basis for $H^*(M_{\Sigma})$ as follows. Let $\{x_i^{(k)}\}_{i \in B_k}$ be a basis of $\L_0^k H^3$, $0 \leq k \leq g-1$. Then \begin{equation} \{x^{(k)}_i\a^a\b^b\gamma^c/ k=0,1,\ldots, g-1, \> a+b+c < g-k, \> i \in B_k \} \label{eqn:antes2} \end{equation} is a basis for $H^*(M_{\Sigma})$. If we set \begin{equation} x^{(k)}_0= \psi_1 \psi_2 \cdots \psi_k \in\L_0^k H^3, \label{eqn:antes3} \end{equation} then proposition~\ref{prop:1} says that a complete set of relations satisfied in $H^*(M_{\Sigma})$ are $x_0^{(k)} q^i_{g-k}$, $i=1,2,3$, $0 \leq k \leq g$, and the $\text{Sp}\, (2g,{\Bbb Z})$ transforms of these. \section{Holomorphic lines in $M_{\Sigma}$} \label{sec:curves} In order to compute the Gromov-Witten invariants $\Psi_A^{M_{\Sigma}}$, we need to describe the space of lines, i.e. rational curves in $M_{\Sigma}$ representing the generator $A \in H_2(M_{\Sigma};{\Bbb Z})$, $$ {\cal M}_A=\{f: {\Bbb P}^1 \rightarrow M_{\Sigma} / \text{$f$ holomorphic, } f_*[{\Bbb P}^1]=A\}. $$ Let us fix some notation. Let $J$ denote the Jacobian variety of $\S$ parametrising line bundles of degree $0$ and let ${\cal L} \rightarrow \S\times J$ be the universal line bundle. If $\{\gamma_i\}$ is the basis of $H^1(\S)$ introduced in section~\ref{sec:ordinary} then $c_1({{\cal L}}) = \sum \gamma_i \otimes \phi_i \in H^1(\S) \otimes H^1(J)$, where $\{\phi_i\}$ is a symplectic basis for $H^1(J)$. Thus $c_1({{\cal L}})^2 = -2 [\S] \otimes \o \in H^2(\S) \otimes H^2(J)$, where $\o=\sum_{i=1}^g \phi_i \wedge \phi_{i+g}$ is the natural symplectic form for $J$. Consider now the algebraic surface $S=\Sigma \times {\Bbb P}^1$. It has irregularity $q=g \geq 2$, geometric genus $p_g=0$ and canonical bundle $K \equiv -2\S+(2g-2){{\Bbb P}}^1$. Recall that $\L$ is a fixed line bundle of degree $1$ on $\S$. Fix the line bundle $L=\L \otimes {\cal O}_{{\Bbb P}^1}(1)$ on $S$ (we omit all pull-backs) with $c_1= c_1(L) \equiv {\Bbb P}^1+\S$, and put $c_2=1$. The ample cone of $S$ is $\{a{\Bbb P}^1 +b \S \, / \, a,b >0 \}$. Let $H_0$ be a polarisation close to ${\Bbb P}^1$ in the ample cone and $H$ be a polarisation close to $\S$, i.e. $H= \S +t{\Bbb P}^1$ with $t$ small. We wish to study the moduli space ${\frak M}= {\frak M}_H(c_1,c_2)$ of $H$-stable bundles over $S$ with Chern classes $c_1$ and $c_2$. \begin{prop} \label{prop:2} ${\frak M}$ can be described as a bundle ${\Bbb P}^{2g-1} \rightarrow {\frak M}= {\Bbb P}({\cal E}_{\zeta}^{\vee}) \rightarrow J$, where ${\cal E}_{\zeta}$ is a bundle on $J$ with $\text{ch}\: {\cal E}_{\zeta} = 2g + 8 \o$. So ${\frak M}$ is compact, smooth and of the expected dimension $6g-2$. The universal bundle ${\cal V} \rightarrow S \times {\frak M}$ is given by $$ 0 \rightarrow {\cal O}_{{\Bbb P}^1}(1) \otimes {\cal L} \otimes \l \rightarrow {\cal V} \rightarrow \L \otimes {\cal L}^{-1} \rightarrow 0, $$ where $\l$ is the tautological line bundle for ${\frak M}$. \end{prop} \begin{pf} For the polarisation $H_0$, the moduli space of $H_0$-stable bundles with Chern classes $c_1, c_2$ is empty by~\cite{Qin}. Now for $p_1=-4c_2+c_1^2=-2$ there is only one wall, determined by $\zeta \equiv -{\Bbb P}^1 +\S$ (here we fix $\zeta= 2\S- L=\S-c_1(\L)$ as a divisor), so the moduli space of $H$-stable bundles with Chern classes $c_1, c_2$ is obtained by crossing the wall as described in~\cite{wall}. First, note that the results in~\cite{wall} use the hypothesis of $-K$ being effective, but the arguments work equally well with the weaker assumption of $\zeta$ being a good wall~\cite[remark 1]{wall} (see also~\cite{Gottsche} for the case of $q=0$). In our case, $\zeta \equiv -{\Bbb P}^1+\S$ is a good wall (i.e. $\pm \zeta+K$ are both not effective) with $l_{\zeta}=0$. Now with the notations of~\cite{wall}, $F$ is a divisor such that $2F-L \equiv \zeta$, e.g. $F=\S$. Also ${\cal F} \rightarrow S \times J$ is the universal bundle parametrising divisors homologically equivalent to $F$, i.e. ${\cal F}={\cal L} \otimes {\cal O}_{{\Bbb P}^1}(1)$. Let $\pi: S \times J \rightarrow J$ be the projection. Then ${\frak M}=E_{\zeta}= {\Bbb P}({\cal E}_{\zeta}^{\vee})$, where $$ {\cal E}_{\zeta} = {{\cal E}}\text{xt}^1_{\pi}({\cal O}(L-{\cal F}),{\cal O}({\cal F}))= R^1\pi_* ({\cal O}(\zeta )\otimes {\cal L}^2). $$ Actually ${\frak M}$ is exactly the set of bundles $E$ that can be written as extensions $$ \exseq{{\cal O}_{{\Bbb P}^1}(1) \otimes L}{E}{\L \otimes L^{-1}} $$ for a line bundle $L$ of degree $0$. The Chern character is computed in~\cite[section 3]{wall} to be $\text{ch}\: {\cal E}_{\zeta} = 2g + e_{K-2\zeta}$, where $e_{\a}= -2 ({\Bbb P}^1 \cdot \a) \o$ (the class $\S$ defined in~\cite[lemma 11]{wall} is ${\Bbb P}^1$ in our case). Finally, the description of the universal bundle follows from~\cite[theorem 10]{wall}. \end{pf} \begin{prop} \label{prop:3} There is a well defined map ${\cal M}_A \rightarrow {\frak M}$. \end{prop} \begin{pf} Every line $f:{{\Bbb P}}^1 \rightarrow M_{\Sigma}$ gives a bundle $E=(\text{id}_{\S}\times f)^*{\cal U}$ over $\Sigma \times {\Bbb P}^1$ by pulling-back the universal bundle ${{\cal U}} \rightarrow \S \times M_{\Sigma}$. Then for any $t \in {\Bbb P}^1$, the bundle $E|_{\S \times t}$ is defined by $f(t)$. Now, by equation~\eqref{eqn:qu2}, $p_1(E)= p_1({\cal U})[\S \times A]= -2\a [A] =-2$. Since $c_1(E) = (\text{id}_{\S}\times f)^*c_1({\cal U})=\L +\S$, it must be $c_2=1$. To see that $E$ is $H$-stable, consider any sub-line bundle $L \hookrightarrow E$ with $c_1(L) \equiv a{\Bbb P}^1 +b\S$. Restricting to any $\S \times t \subset \Sigma \times {\Bbb P}^1$ and using the stability of $E|_{\S \times t}$, one gets $a \leq 0$. Then $c_1(L) \cdot \S < {c_1(E)\cdot \S \over 2}$, which yields the $H$-stability of $E$ (recall that $H$ is close to $\S$). So $E \in {\frak M}$. \end{pf} Now define $N$ as the set of extensions on $\S$ of the form \begin{equation} 0 \rightarrow L \rightarrow E \rightarrow \L \otimes L^{-1} \rightarrow 0, \label{eqn:qu5} \end{equation} for $L$ a line bundle of degree $0$. Then the groups $\text{Ext}^1( \L \otimes L^{-1}, L) =H^1(L^2 \otimes \L^{-1})= H^0(L^{-2} \otimes \L \otimes K)$ are of constant dimension $g$. Moreover $H^0(L^{2} \otimes \L^{-1})=0$, so the moduli space $N$ which parametrises extensions like~\eqref{eqn:qu5} is given as $N={\Bbb P}({\cal E}^{\vee})$, where ${\cal E}= {{\cal E}}{\hbox{xt}}^1_p(\L \otimes {{\cal L}}^{-1}, {{\cal L}})=R^1 p_*({{\cal L}}^{2} \otimes \L^{-1})$, $p:\S\times J \rightarrow J$ the projection. Then we have a fibration ${{\Bbb P}}^{g-1} \rightarrow N={\Bbb P}({\cal E}^{\vee}) \rightarrow J$. The Chern character of ${\cal E}$ is \begin{equation} \begin{array}{rcl} \text{ch}\:({\cal E}) &=& \text{ch}\:(R^1 p_*({{\cal L}}^{2}\otimes \L ^{-1})) = - \text{ch}\:(p_{!}( {{\cal L}}^2 \otimes \L ^{-1}) ) = \\ &=& -p_*( (\text{ch}\:\,{{\cal L}})^2 \> (\text{ch}\:\,\L)^{-1} \> \text{Todd} \> T_{\S}) = \\ &=& -p_*((1+c_1({{\cal L}})+{1 \over 2}c_1({{\cal L}})^2)^2(1-\L)(1-{1 \over 2}K)) = \\ &=& -p_*(1-{1 \over 2}K+2c_1({{\cal L}}) -4 \o \otimes [\S]-\L) = g+4\o. \end{array} \label{eqn:qu6} \end{equation} It is easy to check that all the bundles in $N$ are stable, so there is a well-defined map $$ i: N \rightarrow M_{\Sigma}. $$ Now we wish to construct the space of lines in $N$. Note that $\pi_2(N)=\pi_2({\Bbb P}^{g-1})={\Bbb Z}$, as there are no rational curves in $J$. Let $L \in \pi_2(N)$ be the positive generator. We want to describe $$ {\cal N}_L=\{f: {\Bbb P}^1 \rightarrow N / \text{$f$ holomorphic, } f_*[{\Bbb P}^1]=L\}. $$ For the projective space ${\Bbb P}^n$, the space $H_1$ of lines in ${\Bbb P}^n$ is the set of algebraic maps $f: {{\Bbb P}}^1 \rightarrow {{\Bbb P}}^n$ of degree $1$. Such an $f$ has the form $f[ x_0, x_1 ]=[x_0 u_0 +x_1 u_1]$, $[x_0,x_1] \in {\Bbb P}^1$, where $u_0$, $u_1$ are linearly independent vectors in ${\Bbb C}^{n+1}$. So $$ H_1= {\Bbb P}(\{(u_0, u_1)/ \text{$u_0$, $u_1$ are linearly independent} \}) \subset {\Bbb P}(({\Bbb C}^{\,n+1} \oplus {\Bbb C}^{\,n+1})^{\vee}) ={\Bbb P}^{2n+1}. $$ The complement of $H_1$ is the image of ${\Bbb P}^n \times {\Bbb P}^1 \hookrightarrow {\Bbb P}^{2n+1}$, $([u] , [x_0, x_1]) \mapsto [x_0 u, x_1 u]$, which is a smooth $n$-codimensional algebraic subvariety. So ${\cal N}_L$ can be described as the fibration \begin{equation} \begin{array}{ccccc} H_1 &\rightarrow & {\cal N}_L & \rightarrow& J \\ \bigcap & & \bigcap & & \| \\ {\Bbb P}^{2g-1} &\rightarrow & {\Bbb P}(({\cal E} \oplus {\cal E})^{\vee}) & \rightarrow & J \end{array} \label{eqn:qu7} \end{equation} \begin{rem} \label{rem:4} Note that ${\cal E}_{\zeta}= R^1\pi_* ({\cal O}(\zeta )\otimes {\cal L}^2)= R^1\pi_*({\cal O}_{{\Bbb P}^1}(1) \otimes {{\cal L}}^{2} \otimes \L^{-1})= H^0 ({\cal O}_{{\Bbb P}^1}(1)) \otimes R^1p_*( {{\cal L}}^{2} \otimes \L^{-1}) = H^0 ({\cal O}_{{\Bbb P}^1}(1)) \otimes {\cal E} \cong {\cal E} \oplus {\cal E}$. So ${\frak M}={\Bbb P}(({\cal E} \oplus {\cal E})^{\vee})$, canonically. \end{rem} \begin{prop} \label{prop:5} The map $i:N \rightarrow M_{\Sigma}$ induces a map $i_*: {\cal N}_L \rightarrow {\cal M}_A$. The composition ${\cal N}_L \rightarrow {\cal M}_A \rightarrow {\frak M}$ is the natural inclusion of~\eqref{eqn:qu7}. \end{prop} \begin{pf} The first assertion is clear as $i$ is a holomorphic map. For the second, consider the universal sheaf on $\S\times N$, \begin{equation} 0 \rightarrow {{\cal L}} \otimes U \rightarrow {{\Bbb E}} \rightarrow \L \otimes {\cal L}^{-1} \rightarrow 0, \label{eqn:qu8} \end{equation} where $U={\cal O}_N(1)$ is the tautological bundle of the fibre bundle ${\Bbb P}^{g-1} \rightarrow N \rightarrow J$. Any element in ${\cal N}_L$ is a line ${{\Bbb P}}^1 \hookrightarrow N$, which must lie inside a single fibre ${\Bbb P}^{g-1}$. Restricting~\eqref{eqn:qu8} to this line, we have an extension $$ 0 \rightarrow L \otimes {{\cal O}}_{{{\Bbb P}}^1}(1) \rightarrow E \rightarrow \L \otimes L^{-1} \rightarrow 0 $$ on $S=\Sigma \times {\Bbb P}^1$, which is the image of the given element in ${\frak M}$ (here $L$ is the line bundle corresponding to the fibre in which ${{\Bbb P}}^1$ sits). Now it is easy to check that the map ${\cal N}_L \rightarrow {\frak M}$ is the inclusion of~\eqref{eqn:qu7}. \end{pf} \begin{cor} \label{cor:6} $i_*$ is an isomorphism. \end{cor} \begin{pf} By proposition~\ref{prop:5}, $i_*$ has to be an open immersion. The group $PGL(2,{\Bbb C})$ acts on both spaces ${\cal N}_L$ and ${\cal M}_A$, and $i_*$ is equivariant. The quotient ${\cal N}_L/PGL(2,{\Bbb C})$ is compact, being a fibration over the Jacobian with all the fibres the Grassmannian $\text{Gr}\,({\Bbb C}^{\,2}, {\Bbb C}^{\,g-1})$, hence irreducible. As a consequence $i_*$ is an isomorphism. \end{pf} \begin{rem} \label{rem:7} Notice that the lines in $M_{\Sigma}$ are all contained in the image of $N$, which is of dimension $4g-2$ against $6g-6=\dim M_{\Sigma}$. They do not fill all of $M_{\Sigma}$ as one would naively expect. \end{rem} \section{Computation of $\Psi_A^{M_{\Sigma}}$} \label{sec:GW-inv} The manifold $N$ is positive with $\pi_2(N)={\Bbb Z}$ and $L \in \pi_2(N)$ is the positive generator. Under the map $i: N \rightarrow M_{\Sigma}$, we have $i_* L=A$. Now $\dim N= 4g-2$ and $c_1(N)[L]=c_1({\Bbb P}^{g-1})[L]=g$. So quantum cohomology of $N$, $QH^*(N)$, is well-defined and ${\Bbb Z}/2g{\Bbb Z}$-graded. {}From corollary~\ref{cor:6}, it is straightforward to prove \begin{lem} \label{lem:8} For any $\a_i \in H^{p_i}(M_{\Sigma})$, $1 \leq i \leq r$, such that $p_1 + \cdots + p_r =6g-2$, it is $\Psi_A^{M_{\Sigma}}(\seq{\a}{1}{r})= \Psi_L^N(\seq{i^*\a}{1}{r})$. $\quad \Box$ \end{lem} It is therefore important to know the Gromov-Witten invariants of $N$, i.e. its quantum cohomology. {}From the universal bundle~\eqref{eqn:qu8}, we can read the first Pontrjagin class $p_1( {{\frak g}}_{{\Bbb E}}) = -8 [\S]\otimes\o + h^2- 2 [\S]\otimes h + 4h\cdot c_1({{\cal L}}) \in H^4(\S\times N)$, where $h=c_1(U)$ is the hyperplane class. So on $N$ we have \begin{equation} \left\{ \begin{array}{l} \a=2 \mu(\S) = 4\o + h \\ \psi_i=\mu(\gamma_i^{\#}) = - h \cdot \phi_i \\ \b= -4 \mu(x) = h^2 \end{array} \right. \label{eqn:qu9} \end{equation} Let us remark here that $h^2$ denotes ordinary cup product in $H^*(N)$, a fact which will prove useful later. Now let us compute the quantum cohomology ring of $N$. The cohomology of $J$ is $H^*(J)= \L H_1$, where $H_1=H_1(\S)$. Now the fibre bundle description ${\Bbb P}^{2g-1} \rightarrow N={\Bbb P}({\cal E}^{\vee}) \rightarrow J$ implies that the usual cohomology of $N$ is $H^*(N)= \L H_1 [h]/<h^g+c_1h^{g-1}+ \cdots +c_g=0>$, where $c_i=c_i({\cal E})={4^i \over i!} \o^i$, from~\eqref{eqn:qu6}. As the quantum cohomology has the same generators as the usual cohomology and the relations are a deformation of the usual relations~\cite{ST2}, it must be $h^g+c_1h^{g-1}+ \cdots +c_g=r$ in $QH^*(N)$, with $r \in {\Bbb Q}$. As in~\cite[example 8.5]{RT}, $r$ can be computed to be $1$. So \begin{equation} QH^*(N)= \L H_1 [h]/<h^g+c_1h^{g-1}+ \cdots +c_g=1> . \label{eqn:qu10} \end{equation} \begin{lem} \label{lem:9} For any $s \in H^{2g-2i}(J)$, $0 \leq i \leq g$, denote by $s \in H^{2g-2i}(N)$ its pull-back to $N$ under the natural projection. Then the quantum product $h^{2g-1+i} s$ in $QH^*(N)$ has component in $H^{4g-2}(N)$ equal to ${(-8)^i\over i!} \o^i \wedge s$ (the natural isomorphism $H^{4g-2}(N) \cong H^{2g}(J)$ is understood). \end{lem} \begin{pf} First note that for $s_1,s_2 \in H^*(J)$ such that their cup product in $J$ is $s_1s_2=0$, then the quantum product $s_1s_2 \in QH^*(N)$ vanishes. This is so since every rational line in $N$ is contained in a fibre of ${\Bbb P}^{2g-1} \rightarrow J \rightarrow N$. Next recall that $h^{g-1+i}s$ has component in $H^{4g-2}(N)$ equal to $s_i({\cal E}) \wedge s={(-4)^i\over i!} \o^i \wedge s$. Then multiply the standard relation~\eqref{eqn:qu10} by $h^{g-1+i}s$ and work by induction on $i$. For $i=0$ we get $h^{2g-1}s=h^{g-1}s$ and the assertion is obvious. For $i>0$, $$ h^{2g-1+i}s +h^{2g-2+i}c_1s+ \cdots +h^{2g-1}c_is=h^{g-1+i}s. $$ So the component of $h^{2g-1+i}s$ in $H^{4g-2}(N)$ is $$ - \sum_{j=1}^i {(-8)^{i-j}\over (i-j)!} \o^{i-j} c_j s + {(-4)^i\over i!} \o^i s = {(-8)^i\over i!} \o^i s - \sum_{j=0}^i {(-8)^{i-j}\over (i-j)!}{4^j\over j!} \o^i s + {(-4)^i\over i!} \o^i s = {(-8)^i\over i!} \o^i s. $$ \end{pf} \begin{lem} \label{lem:10} Suppose $g>2$. Let $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ have degree $6g-2$. Then $$ \Psi^N_L(\a, \stackrel{(a)}{\ldots}, \a, \b , \stackrel{(b)}{\ldots} ,\b,\psi_{i_1}, \ldots, \psi_{i_r})= <(4\o + X)^a(X^2)^b \phi_{i_1}\cdots\phi_{i_r} X^r, [J]>, $$ evaluated on $J$, where $X^{2g-1+i}={(-8)^i \over i!} \o^i \in H^*(J)$. \end{lem} \begin{pf} By definition the left hand side is the component in $H^{4g-2}(N)$ of the quantum product $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in QH^*(N)$. From~\eqref{eqn:qu9}, this quantum product is $(4\o + h)^a (h^2)^b (-h\phi_{i_1})\cdots(-h\phi_{i_r})$, upon noting that when $g>2$, $\b=h^2$ as a quantum product as there are no quantum corrections because of the degree. Note that $r$ is even, so the statement of the lemma follows from lemma~\ref{lem:9}. \end{pf} Now we are in the position of relating the Gromov-Witten invariants $\Psi_A^{M_{\Sigma}}$ with the Donaldson invariants for $S=\S\times {\Bbb P}^1$ (for definition of Donaldson invariants see~\cite{DK}~\cite{KM}). \begin{thm} \label{thm:11} Suppose $g>2$. Let $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ have degree $6g-2$. Then $$ \Psi^{M_{\Sigma}}_A(\a, \stackrel{(a)}{\ldots}, \a, \b , \stackrel{(b)}{\ldots} ,\b,\psi_{i_1}, \ldots, \psi_{i_r})= (-1)^{g-1} D^{c_1}_{S, H} ((2\S)^a(-4 \text{pt})^b \gamma_{i_1}^{\#} \cdots\gamma_{i_r}^{\#}), $$ where $D^{c_1}_{S, H}$ stands for the Donaldson invariant of $S=\S\times{\Bbb P}^1$ with $w=c_1$ and polarisation $H$. \end{thm} \begin{pf} By definition, the right hand side is $\epsilon_S(c_1) <\a^a\b^b\psi_{i_1}\cdots\psi_{i_r}, [{\frak M}]>$, where $\a =2\mu(\S) \in H^2({\frak M})$, $\b =-4\mu(x) \in H^4({\frak M})$, $\psi_i =\mu(\gamma_i^{\#}) \in H^3({\frak M})$. Here the factor $\epsilon_S(c_1)=(-1)^{K_S c_1 +c_1^2 \over 2}=(-1)^{g-1}$ compares the complex orientation of ${\frak M}$ and its natural orientation as a moduli space of anti-self-dual connections~\cite{DK}. By~\cite[theorem 10]{wall}, this is worked out to be $(-1)^{g-1} <(4\o + X)^a(X^2)^b \phi_{i_1}\cdots\phi_{i_r} X^r,[J]>$, where $X^{2g-1+i}=s_i({\cal E}_{\zeta})={(-8)^i \over i!} \o^i$. Thus the theorem follows from lemmas~\ref{lem:8} and~\ref{lem:10}. \end{pf} \begin{rem} \label{rem:12} The formula in theorem~\ref{thm:11} is not right for $g=2$, as in such case, the quantum product $h^2 \in QH^*(N)$ differs from $\b$ by a quantum correction. \end{rem} \begin{rem} \label{rem:13} Suppose $g \geq 2$ and let $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ have degree $6g-6$. Then \begin{eqnarray*} \Psi^{M_{\Sigma}}_0(\a, \stackrel{(a)}{\ldots}, \a, \b , \stackrel{(b)}{\ldots} ,\b,\psi_{i_1}, \ldots, \psi_{i_r}) &=& \epsilon_S({\Bbb P}^1) < \a^a\b^b\psi_{i_1}\cdots\psi_{i_r},[M_{\Sigma}]> = \\ &=& - D^{{\Bbb P}^1}_{S, H} ((2\S)^a(-4 \text{pt})^b \gamma_{i_1}^{\#} \cdots\gamma_{i_r}^{\#}), \end{eqnarray*} as the moduli space of anti-self-dual connections on $S$ of dimension $6g-6$ is $M_{\Sigma}$. \end{rem} \section{Quantum cohomology of $M_{\Sigma}$} \label{sec:quantum} It is natural to ask to what extent the first quantum correction determines the full structure of the quantum cohomology of $M_{\Sigma}$. In~\cite{D1}, Donaldson finds the first quantum correction for $M_{\Sigma}$ when the genus of $\S$ is $g=2$ and proves that this is enough to find the quantum product. Now it is our intention to show how the Gromov-Witten invariants $\Psi^{M_{\Sigma}}_A$ determine completely $QH^*(M_{\Sigma})$. First we check an interesting fact. \begin{lem} \label{lem:14} Let $g \geq 3$. Then $\gamma =-2 \sum \psi_i\psi_{i+g}$ as elements in $QH^*(M_{\Sigma})$ (i.e. using the quantum product in the right hand side). \end{lem} \begin{pf} Let $\hat \gamma =-2 \sum \psi_i\psi_{i+g} \in QH^*(M_{\Sigma})$. In principle, it is $\hat \gamma=\gamma + s\a$, for some $s \in {\Bbb Q}$. Let us show that $s=0$. Multiplying by $\a^{3g-4}$, we have $\hat \gamma\a^{3g-4} =\gamma\a^{3g-4} +s \a^{3g-3}$. Considering the component in $H^{6g-6}(M_{\Sigma})$ and using lemma~\ref{lem:8}, we have $$ -2 \sum \Psi_L^N(\a, \stackrel{(3g-4)}{\ldots}, \a, \psi_i, \psi_{i+g})= \Psi_L^N(\a, \stackrel{(3g-4)}{\ldots}, \a, \gamma) + s <\a^{3g-3},[M_{\Sigma}]>. $$ Now, in $N$, $\gamma$ is the cup product $-2 \sum \phi_i\phi_{i+g} h^2$. It is easy to check that this coincides with the quantum product $-2 \sum \phi_i\phi_{i+g} h^2$. For $g >3$ it is evident because of the degree. For $g=3$ there might be a quantum correction in $H^0(N)$, but this is $-2 \sum \Psi_L^N(\phi_i,\phi_{i+g}, h,h,\text{pt})=0$ (since lines are contained in the fibres). Now lemma~\ref{lem:10} and its proof imply that $-2 \sum \Psi_L^N(\a, \ldots, \a, \psi_i, \psi_{i+g})=\Psi_L^N(\a, \ldots, \a, \gamma)$, so $s=0$. \end{pf} We are pursuing to prove an isomorphism between $QH^*(M_{\Sigma})$ and $HF^*(\Sigma \times {\Bbb S}^1)$, the instanton Floer homology of the three manifold $\Sigma \times {\Bbb S}^1$. First recall the main result contained in~\cite{Floer}. \begin{thm}[\cite{Floer}] \label{thm:15} Define $R^1_0=1$, $R^2_0=0$, $R^3_0=0$ and then recursively, for all $r \geq 1$, $$ \left\{ \begin{array}{l} R_{r+1}^1 = \a R_r^1 + r^2 R_r^2 \\ R_{r+1}^2 = (\b+(-1)^{r+1}8) R_r^1 + {2r \over r+1} R_r^3 \\ R_{r+1}^3 = \gamma R_r^1 \end{array} \right. $$ Put $I'_r=(R^1_r,R^2_r,R^3_r) \subset {\Bbb Q}[\a,\b,\gamma]$, $r \geq 0$. Then the $\text{Sp}\, (2g,{\Bbb Z})$-decomposition of $HF^*(\Sigma \times {\Bbb S}^1)$ is $$ HF^*(\Sigma \times {\Bbb S}^1)= \bigoplus_{k=0}^{g-1} \L_0^k H^3 \otimes {\Bbb Q} [\a, \b, \gamma]/I'_{g-k}. $$ \end{thm} The elements $R_r^1$, $R_r^2$ and $R_r^3$ are deformations graded mod $4$ of $q_r^1$, $q_r^2$ and $q_r^3$, respectively. This means that we can write \begin{equation} \label{eqn:qu11} R_r^i= \sum_{j \geq 0} R_{r,j}^i, \end{equation} where $\deg(R_{r,j}^i)=\deg(q_r^i)-4j$, $j \geq 0$, and $R_{r,0}^i=q_r^i$. In the case of $QH^*(M_{\Sigma})$ we shall have \begin{prop} \label{prop:16} The $\text{Sp}\, (2g,{\Bbb Z})$-decomposition of $QH^*(M_{\Sigma})$ is $$ QH^*(M_{\Sigma})= \bigoplus_{k=0}^{g-1} \L_0^k H^3 \otimes {\Bbb Q} [\a, \b, \gamma]/J_{g-k}, $$ where $J_r$ is generated by three elements $Q_r^1$, $Q_r^2$ and $Q_r^3$, which are deformations graded mod $4$ of $q_r^1$, $q_r^2$ and $q_r^3$, respectively. \end{prop} \begin{pf} The action of $\text{Sp}\, (2g,{\Bbb Z})$ on $M_{\Sigma}$ being symplectic (see~\cite[section 3.1]{Siebert}), we have an epimorphism of rings (with $\text{Sp}\, (2g,{\Bbb Z})$-actions) like in~\eqref{eqn:qu3} $$ \AA(\S) \twoheadrightarrow QH^*(M_{\Sigma}). $$ This induces an epimorphism on the invariant parts $$ {\Bbb Q}[\a,\b,\gamma ] \twoheadrightarrow QH^*_I(M_{\Sigma}), $$ where $\gamma=-2 \sum_{i=0}^g \psi_i\psi_{i+g}$ (see lemma~\ref{lem:14}). Therefore we have maps \begin{equation} \L_0^k (\seq{\psi}{1}{2g}) \otimes {\Bbb Q}[\a,\b,\gamma ] \rightarrow QH^*(M_{\Sigma}). \label{eqn:qu12} \end{equation} Let $V_k$ be the image of the map~\eqref{eqn:qu12}. As $\L_0^k H^3$, $0 \leq k \leq g-1$, are inequivalent irreducible $\text{Sp}\, (2g,{\Bbb Z})$-modules, the subspaces $V_k$ are pairwise orthogonal. On the other hand, the existence of the basis~\eqref{eqn:antes2} of $H^*(M_{\Sigma})$ and the results in~\cite{ST2} imply that $\{x^{(k)}_i\a^a\b^b\gamma^c/ k=0,1,\ldots, g-1, \> a+b+c < g-k, \> i \in B_k \}$ (where quantum products are now understood) is a basis of $QH^*(M_{\Sigma})$. So the subspaces $V_k$ generate $QH^*(M_{\Sigma})$, i.e. \begin{equation} QH^*(M_{\Sigma})= \bigoplus_{k=0}^{g-1} V_k. \label{eqn:qu13} \end{equation} Actually this decomposition coincides with the decomposition~\eqref{eqn:antes}. This is proved by giving a definition of $V_k$ independent of the ring structure (cup product or quantum product). For instance, say that $V_k$ is the space generated by elements which are orthogonal to $V_0$, $\ldots$, $V_{k-1}$ and such that the $\text{Sp}\, (2g,{\Bbb Z})$-module generated by them have dimension equal to $\dim \L_0^k H^3$. Our second purpose is to describe the kernel of~\eqref{eqn:qu12}, i.e. the relations satisfied by the elements of $\L_0^k H^3$, $\a$, $\b$ and $\gamma$. The results in~\cite{ST2} imply that we only need to write the relations of $V_k \subset H^*(M_{\Sigma})$ in terms of the quantum product. Fix $k$, and recall $x^{(k)}_0=\psi_1\psi_2\cdots \psi_k \in \L_0^k H^3$ from~\eqref{eqn:antes3}. By section~\ref{sec:ordinary} the relations in $V_k$ are given by $x^{(k)}_0 q_{g-k}^i$, $i=1,2,3$, and its $\text{Sp}\, (2g,{\Bbb Z})$-transforms. We rewrite these relations in terms of the quantum product, using the basis~\eqref{eqn:antes2}, as \begin{equation} x^{(k)}_0 q_{g-k}^i= \sum_{a+b+c<g-k} x_{abc} \a^a\b^b\gamma^c \in QH^*(M_{\Sigma}) \label{eqn:qu14} \end{equation} where $x_{abc} \in \L^k_0 H^3$, and the monomials in the right hand side have degree strictly less that the degree of the left hand side. Now we want to prove that $x_{abc}$ are all multiples of $x^{(k)}_0$. Suppose not. Then it is easy to see that there exists $\phi \in \text{Sp}\, (2g,{\Bbb Z})$ satisfying $\phi( x^{(k)}_0)= x^{(k)}_0$ and $\phi(x_{abc}) \neq x_{abc}$. Consider~\eqref{eqn:qu14} minus its transform under $\phi$. This is a relation between the elements of the basis of $V_k$, which is impossible. Therefore~\eqref{eqn:qu14} can be rewritten as $$ x^{(k)}_0 (q_{g-k}^i+ Q_{g-k,1}^i + Q_{g-k,2}^i +\cdots )=0, $$ where $\deg Q_{g-k,j}^i= \deg q_{g-k}^i- 4j$, $j\geq 1$. This finishes the proof. \end{pf} \begin{lem} \label{lem:17} $\gamma J_k \subset J_{k+1} \subset J_k$, for $k=0,1,\ldots, g-1$. \end{lem} \begin{pf} Let $f\in J_k \subset {\Bbb Q}[\a,\b,\gamma]$. By definition (proposition~\ref{prop:16}) this means that the quantum product $\psi_1 \cdots \psi_{g-k} f=0$. Using the action of $\text{Sp}\, (2g,{\Bbb Z})$ we have $\psi_1 \cdots \psi_{g-k-1} \psi_i f=0$, for $g-k \leq i \leq g$. Thus $\psi_1 \cdots \psi_{g-(k+1)} \gamma f=0$, i.e. $\gamma f \in J_{k+1}$. For the second inclusion, let $f\in J_{k+1}$. Then $\psi_1 \cdots \psi_{g-(k+1)} f=0$ and hence $\psi_1 \cdots \psi_{g-k} f=0$, i.e. $f \in J_k$. \end{pf} \begin{prop} \label{prop:18} There are numbers $c_r, d_r \in {\Bbb Q}$, $1 \leq r \leq g-1$, such that for $0 \leq r \leq g-1$ it is $$ \left\{ \begin{array}{l} Q_{r+1}^1 = \a Q_r^1 + r^2 Q_r^2 \\ Q_{r+1}^2 = (\b+c_{r+1}) Q_r^1 + {2r \over r+1} Q_r^3 \\ Q_{r+1}^3 = \gamma Q_r^1 + d_{r+1} Q_r^2 \end{array} \right. $$ \end{prop} \begin{pf} Completely analogous to the proof of~\cite[theorem 10]{Floer}. \end{pf} \begin{prop} \label{prop:19} For all $1 \leq r \leq g$, $c_r= (-1)^{r+g+1}\, 8$ and $d_r=0$. \end{prop} \begin{pf} We write $R_{g-k}^i= \sum_{j \geq 0} R_{g-k,j}^i$ and $Q_{g-k}^i= \sum_{j \geq 0} Q_{g-k,j}^i$, as in~\eqref{eqn:qu11}, for $0 \leq k \leq g-1$. Then $R_{g-k,0}^i=Q_{g-k,0}^i=q_{g-k}^i$. The coefficients $c_r$ and $d_r$ are determined by the first correction term $Q_{r,1}^i$ of $Q_r^i$. By the definition of $R^i_r$ in theorem~\ref{thm:15}, we only need to check that $R_{g-k,1}^i=(-1)^g Q_{g-k,1}^i$, for $i=1,2,3$, $0 \leq k \leq g-1$. Fix $i$ and $k$. Recall $x_0^{(k)}=\psi_1 \cdots \psi_k \in \L_0^k H^3$. By theorem~\ref{thm:15}, $x_0^{(k)} R^i_{g-k} =0 \in HF^*(\Sigma \times {\Bbb S}^1)$. Pick an arbitrary $f= \a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ of degree $6g-2-\deg( x_0^{(k)} q^i_{g-k})$. In $HF^*(\Sigma \times {\Bbb S}^1)$ the pairing $<x_0^{(k)} R^i_{g-k},f>=0$, i.e. $D^{(w,\S)}_{S,H} (\bar x_0^{(k)} \bar R^i_{g-k} \bar f)=0$, where $\bar R^i_{g-k}=R^i_{g-k}(2\S,-4 x,-2\sum \gamma^{\#}_i\gamma^{\#}_{g+i})$, and analogously for $\bar f$ and $\bar x_0^{(k)}$ (for the notation $D^{(w,\S)}$ see~\cite{Floer}). This means that $$ D^{{\Bbb P}^1}_{S,H} (\bar x_0^{(k)} \bar R^i_{g-k,1} \bar f) + D^{c_1}_{S,H} (\bar x_0^{(k)} \bar R^i_{g-k,0} \bar f)=0. $$ From theorem~\ref{thm:11} and remark~\ref{rem:13} we have that the component in $H^{6g-6}(M_{\Sigma})$ of the quantum product $- x_0^{(k)}R^i_{g-k,1} f + (-1)^{g-1} x_0^{(k)}R^i_{g-k,0} f \in QH^*(M_{\Sigma})$ vanishes, i.e. \begin{equation} -< x_0^{(k)}R^i_{g-k,1}, f> + (-1)^{g-1}< x_0^{(k)}R^i_{g-k,0} ,f>=0 \label{eqn:main1} \end{equation} in $QH^*(M_{\Sigma})$. On the other hand, proposition~\ref{prop:16} says that $x_0^{(k)} Q^i_{g-k} =0 \in QH^*(M_{\Sigma})$. Multiplying by $f$, $x_0^{(k)}Q_{g-k}^i f =0$, so the component in $H^{6g-6}(M_{\Sigma})$ of the quantum product $x_0^{(k)}Q^i_{g-k,1} f + x_0^{(k)}Q^i_{g-k,0} f \in QH^*(M_{\Sigma})$ is zero. Thus \begin{equation} < x_0^{(k)}Q^i_{g-k,1}, f> + < x_0^{(k)}Q^i_{g-k,0} ,f>=0 \label{eqn:main2} \end{equation} in $QH^*(M_{\Sigma})$. Equations~\eqref{eqn:main1} and~\eqref{eqn:main2} imply together that \begin{equation} < x_0^{(k)}Q^i_{g-k,1}, f>= (-1)^g < x_0^{(k)}R^i_{g-k,1}, f>, \label{eqn:main3} \end{equation} for any $f \in \AA(\S)$ of degree $6g-2-\deg( x_0^{(k)}q^i_{g-k})= 6g-6-\deg( x_0^{(k)} Q^i_{g-k,1})$. As we are considering the pairing on classes of complementary degree, equation~\eqref{eqn:main3} holds in $H^*(M_{\Sigma})$ as well. By section~\ref{sec:ordinary}, $Q^i_{g-k,1} \equiv (-1)^g R^i_{g-k,1} \pmod{I_{g-k}}$. Considering the degrees, it must be $Q^1_{g-k,1} =(-1)^g R^1_{g-k,1}$ and $Q^2_{g-k,1} =(-1)^g R^2_{g-k,1}$. For $i=3$, the difference $Q^3_{g-k,1} -(-1)^g R^3_{g-k,1}$ is a multiple of $q^1_{g-k}$. The vanishing of the coefficient of $\a^{g-k}$ for both $Q_{g-k}^3$ and $R_{g-k}^3$ (see theorem~\ref{thm:15} and equation~\eqref{eqn:qu14}) implies $Q^3_{g-k,1} =(-1)^g R^3_{g-k,1}$. \end{pf} Putting all together we have proved the following \begin{thm} \label{thm:main} The quantum cohomology of $M_{\Sigma}$, for $\S$ a Riemann surface of genus $g \geq 3$, has a presentation $$ QH^*(M_{\Sigma})= \bigoplus_{k=0}^{g-1} \L_0^k H^3 \otimes {\Bbb Q}[\a,\b,\gamma] /J_{g-k}. $$ where $J_r=(Q^1_r, Q^2_r,Q^3_r)$ and $Q^i_r$ are defined recursively by setting $Q^1_0=1$, $Q^2_0=0$, $Q^3_0=0$ and putting for all $r \geq 0$ $$ \left\{ \begin{array}{l} Q_{r+1}^1 = \a Q_r^1 + r^2 Q_r^2 \\ Q_{r+1}^2 = (\b+(-1)^{r+g+1}8) Q_r^1 + {2r \over r+1} Q_r^3 \\ Q_{r+1}^3 = \gamma Q_r^1 \end{array} \right. $$ \end{thm} \begin{cor} \label{cor:21} Let $\S$ be a Riemann surface of genus $g \geq 3$. Then there is an isomorphism $$ QH^*(M_{\Sigma}) \stackrel{\simeq}{\ar} HF^*(\Sigma \times {\Bbb S}^1). $$ For $g$ even, the isomorphism sends $(\a,\b,\gamma) \mapsto (\a,\b,\gamma)$. For $g$ odd, the isomorphism sends $(\a,\b,\gamma) \mapsto (\sqrt{-1}\, \a,-\b,-\sqrt{-1}\, \gamma)$. \end{cor} \begin{pf} This is a consequence of the descriptions of $QH^*(M_{\Sigma})$ and $HF^*(\Sigma \times {\Bbb S}^1)$ in theorem~\ref{thm:main} and theorem~\ref{thm:15}, respectively. \end{pf} \begin{rem} \label{rem:22} Alternatively, we can say that for any $g \geq 1$ there is an isomorphism $QH^*(M_{\Sigma}) \stackrel{\simeq}{\ar} HF^*(\Sigma \times {\Bbb S}^1)$, taking $(\a,\b,\gamma) \mapsto (\sqrt{-1}^{\,g}\a,\sqrt{-1}^{\,2g}\b,\sqrt{-1}^{\,3g}\gamma)$. \end{rem} \section{The cases $g=1$ and $g=2$} \label{sec:6} Let us review the cases of genus $g=1$ and $g=2$ in the view of theorem~\ref{thm:main}. These cases are somehow atypical, as the generators precise the introduction of quantum corrections, a fact already noted in~\cite{Vafa}. \begin{ex} \label{ex:23} Let $\S$ be a Riemann surface of genus $g=1$. Then $M_{\Sigma}$ is a point and we can write $$ QH^*(M_{\Sigma})= {\Bbb Q}[\a,\hat \b,\gamma] /(\a,\hat \b+8,\gamma), $$ where we have defined $\hat \b=\b-8$. This agrees with theorem~\ref{thm:main} but with corrected generators. Again $QH^*(M_{\Sigma}) \stackrel{\simeq}{\ar} HF^*(\Sigma \times {\Bbb S}^1)$, where $(\a,\hat \b,\gamma) \mapsto (\sqrt{-1}\, \a,-\b,-\sqrt{-1}\, \gamma)$. \end{ex} \begin{ex} \label{ex:24} Let $\S$ be a Riemann surface of genus $g=2$. The quantum cohomology ring $QH^*(M_{\Sigma})$ has been computed by Donaldson~\cite{D1}, using an explicit description of $M_{\Sigma}$ as the intersection of two quadrics in ${\Bbb P}^5$. Let $h_2$, $h_4$ and $h_6$ be the integral generators of $QH^2(M_{\Sigma})$, $QH^4(M_{\Sigma})$ and $QH^6(M_{\Sigma})$, respectively. Then, with our notations, $\a=h_2$, $\b = -4 h_4$ and $\gamma = 4 h_6$ (see~\cite{Vafa}). Define $\hat \gamma =-2 \sum \psi_i\psi_{i+g} \in QH^*(M_{\Sigma})$. The computations in~\cite{D1} yield $\hat \gamma =\gamma -4 \a$ (compare with lemma~\ref{lem:14}). Put $\hat \b=\b + 4$. It is now easy to check that the relations found in~\cite{D1} can be translated to $$ QH^*(M_{\Sigma})= \left( H^3 \otimes {\Bbb Q}[\a,\hat \b,\hat \gamma] /(\a,\hat \b-8,\hat \gamma) \right) \oplus {\Bbb Q}[\a,\hat \b,\hat \gamma] /(Q_2^1,Q_2^2,Q_2^3), $$ where $Q_2^1= \a^2+\hat \b-8$, $Q_2^2= (\hat \b+8) \a +\hat \gamma$ and $Q_2^3= \a \hat \gamma$ (defined exactly as in theorem~\ref{thm:main}, but with corrected generators). Now $QH^*(M_{\Sigma}) \stackrel{\simeq}{\ar} HF^*(\Sigma \times {\Bbb S}^1)$, where $(\a,\hat \b,\hat \gamma) \mapsto (\sqrt{-1}\, \a,-\b,-\sqrt{-1}\, \gamma)$. \end{ex} The artificially introduced definition of $\hat \b$ is due to the same phenomenon which causes the failure of lemma~\ref{lem:10} for $g=2$, i.e. the quantum product $h^2$ differs from $\b$ in~\eqref{eqn:qu9} (defined with the cup product) because of a quantum correction in $QH^*(N)$ which appears when $g=2$.

9711

alg-geom/9711019

en

https://arxiv.org/abs/alg-geom/9711019

alg-geom/9711019

Victor V. Batyrev

Stringy Hodge numbers and Virasoro algebra

10 pages, AMSLaTeX

Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was discovered by Libgober and Wood that the product of the Chern classes $c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has been used by Eguchi, Jinzenji and Xiong in their approach to the quantum cohomology of $X$ via a representation of the Virasoro algebra with the central charge $c_n(X)$. In this paper we define for singular varieties $X$ a rational number $c_{st}^{1,n-1}(X)$ which is a stringy version of the number $c_1c_{n-1}$ for smooth $n$-folds. We show that the number $c_{st}^{1,n-1}(X)$ can be expressed in the same way using the stringy Hodge numbers of $X$. Our results provides an evidence for the existence of an approach to quantum cohomology of singular varieties $X$ via a representation of the Virasoro algebra whose central charge is the rational number $e_{st}(X)$ which equals the stringy Euler number of $X$.

alg-geom

\section{Introduction} Let $X$ be an arbitrary smooth projective variety of dimension $n$. The $E$-polynomial of $X$ is defined as \[ E(X; u,v):= \sum_{p,q} (-1)^{p+q} h^{p,q}(X) u^p v^q \] where $h^{p,q}(X)= dim\, H^q(X, \Omega^p_X)$ are Hodge numbers of $X$. Using the Hirzebruch-Riemann-Roch theorem, Libgober and Wood \cite{LW} has proved the following equality (see also the papers of Borisov \cite{LB1} and Salamon \cite{S}): \begin{theo} \[ \frac{d^2}{d u^2}{E}_{\rm st}(X; u,1)|_{u =1} = \frac{3n^2 - 5n}{12} c_n(X) + \frac{c_1(X)c_{n-1}(X)}{6}. \] \label{d-rel} \end{theo} \noindent By Poincar{\'e} duality for $X$, one immediatelly obtains \cite{LB1,S}: \begin{coro} Let $X$ be an arbitrary smooth $n$-dimensional projective variety. Then $c_1(X)c_{n-1}(X)$ can be expressed via the Hodge numbers of $X$ using the following equality \[ \sum_{p,q} (-1)^{p+q} h^{p,q}(X) \left(p - \frac{n}{2} \right)^2 = \frac{n}{12}c_n(X) + \frac{1}{6} c_1(X)c_{n-1}(X), \] where \[ c_n(X) = \sum_{p,q} (-1)^{p+q}h^{p,q}(X) \] is the Euler number of $X$. \label{vir-rel} \end{coro} \begin{coro} Let $X$ be an arbitrary smooth $n$-dimensional projective variety with $c_1(X) =0$. Then the Hodge numbers of $X$ satisfy the following equation \[ \sum_{p,q} (-1)^{p+q} h^{p,q}(X) \left(p - \frac{n}{2} \right)^2 = \frac{n}{12} \sum_{p,q} (-1)^{p+q}h^{p,q}(X), \] In particular, for hyper-K\"ahler manifolds $X$ this equation reduces to \[ 2 \sum_{j =1}^{2n} (-1)^j (3j^2 - n) b_{2n-j}(X) = nb_{2n}(X), \] where \[ b_i(X) = \sum_{p+q=i} h^{p,q}(X) \] is $i$-th Betti number of $X$. \label{cy-rel} \end{coro} \begin{rem} {\rm We that if $X$ is a $K3$-surface, then the relation \ref{cy-rel} is equivalent to the equality $c_2(X) =24$. For smooth Calabi-Yau $4$-folds $X$ the relation \ref{cy-rel} has been observed by Sethi, Vafa, and Witten \cite{W2} (it is equivalent to the equality \[ c_4(X) = 6( 8 - h^{1,1}(X) + h^{2,1}(X) - h^{3,1}(X)), \] if $h^{1,0}(X) = h^{2,0}(X) = h^{3,0}(X) =0$). } \end{rem} There are a lot of examples of Calabi-Yau varieties $X$ having at worst Gorenstein canonical singularities which are hypersurfaces and complete intersections in Gorenstein toric Fano varieties \cite{BA,BB0}. It has been shown in \cite{BD} that for all these examples of singular Calabi-Yau varieties $X$ one can define so called {\em stringy Hodge numbers} $h^{p,q}_{\rm st}(X)$. Moreover, the stringy Hodge numbers of Calabi-Yau complete intersections in Gorenstein toric varieties agree with the topological mirror duality test \cite{BB}. It was a natural question posed in \cite{B1}, whether one has the same identity for stringy Hodge numbers of singular Calabi-Yau varieties as for usual Hodge numbers of smooth Calabi-Yau manifolds, i.e. \begin{equation} \sum_{p,q} (-1)^{p+q} h^{p,q}_{\rm st}(X) \left(p - \frac{n}{2} \right)^2 = \frac{n}{12} \sum_{p,q} (-1)^{p+q}h^{p,q}_{\rm st}(X) = \frac{n}{12} e_{\rm st}(X). \label{st-rel} \end{equation} \noindent The purpose of this paper is to show that the formula (\ref{st-rel}) holds true. Moreover, one can define a rational number $e_{\rm st}^{1, n-1}(X)$ which is a stringy version $c_1(X)c_{n-1}(X)$ such that the stringy analog of \ref{vir-rel} \begin{equation} \sum_{p,q} (-1)^{p+q} h^{p,q}_{\rm st}(X) \left(p - \frac{n}{2} \right)^2 = \frac{n}{12}e_{\rm st}(X) + \frac{1}{6} c_{\rm st}^{1,n-1}(X) \label{st-rel2} \end{equation} holds true provided the stringy Hodge numbers of $X$ exist. \section{Stringy Hodge numbers} Recall our general approach to the notion of stringy Hodge numbers $h^{p,q}_{\rm st}(X)$ for projective algebraic varieties $X$ with canonical singularites (see \cite{B1}). Our main definition in \cite{B1} can be reformulated as follows: \begin{dfn} {\rm Let $X$ be an arbitrary $n$-dimensional projective variety with at worst log-terminal singularites, $\rho\, : \, Y \to X$ a resolution of singularities whose exceptional locus $D$ is a divisors with normally crossing components $D_1, \ldots, D_r$. We set $I: = \{1, \ldots, r \}$ and $D_J: = \bigcap_{j \in J} D_j$ for all $J \subset I$. Define the {\bf stringy $E$-function} of $X$ to be \[ E_{\rm st} (X; u,v) := \sum_{J \subset I} E(D_J; u,v) \prod_{j \in J} \left(\frac{uv-1}{(uv)^{a_j+1} -1} -1 \right), \] where the rational numbers $a_1, \ldots, a_r$ are determined by the equality \[ K_Y = \rho^*K_X + \sum_{i =1}^r a_i D_i. \] Then the {\bf stringy Euler number} of $X$ is defined as \[ e_{\rm st}(X) := \lim_{u,v \to 1} E_{\rm st} (X; u,v) = \sum_{J \subset I} c_{n - |J|}(D_J) \prod_{j \in J} \left(\frac{-a_j}{a_j+1} \right), \] where $c_{n-|J|}(D_J)$ is the Euler number of $D_J$ (we set $ c_{n-|J|}(D_J)=0$ if $D_J$ is empty). } \label{def-e} \end{dfn} \begin{dfn} {\rm Let $X$ be an arbitrary $n$-dimensional projective variety with at worst Gorenstein canonical singularites. We say that {\bf stringy Hodge numbers of $X$ exist}, if $E_{\rm st}(X; u,v)$ is a polynomial, i.e., \[ E_{\rm st}(X; u,v) = \sum_{p,q} a_{p,q}(X)u^pv^q .\] Under the assumption that $E_{\rm st}(X; u,v)$ is a polynomial, we define the {\bf stringy Hodge numbers} $h^{p,q}_{\rm st}(X)$ to be $(-1)^{p+q}a_{p,q}$. } \end{dfn} \begin{rem} {\rm In the above definitions, the condition that $X$ has at worst log-terminal singularities means that $a_i > -1$ for all $i \in I$; the condition that $X$ has at worst Gorenstein canonical singularities is equivalent for $a_i$ to be nonnegative integers for all $i \in I$ (see \cite{KMM}).} \end{rem} The following statement has been proved in \cite{B1}: \begin{theo} Let $X$ be an arbitrary $n$-dimensional projective variety with at worst Gorenstein canonical singularites. Assume that stringy Hodge numbers of $X$ exist. Then they have the following properties: {\rm (i)} $h^{0,0}_{\rm st}(X)= h^{n,n}_{\rm st}(X) =1$; {\rm (ii)} $h^{p,q}_{\rm st}(X)=h^{n-p,n-q}_{\rm st}(X)$ and $h^{p,q}_{\rm st}(X)=h^{q,p}_{\rm st}(X)$ $\forall p,q$; {\rm (iii)} $h^{p,q}_{\rm st}(X) = 0$ $\forall p,q >n$. \label{st-pr} \end{theo} \section{The number $c_{\rm st}^{1,n-1}(X)$} \begin{dfn} {\rm Let $X$ be an arbitrary $n$-dimensional projective variety $X$ having at worst log-terminal singularities and $\rho\;:\; Y \to X$ is a desingularization with normally crossing irreducible components $D_1,\ldots, D_r$ of the exceptional locus. We define the number \[ c_{\rm st}^{1, n-1}(X): = \sum_{J \subset I} \rho^*c_1(X) c_{n-|J|-1}(D_J) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right), \] where $\rho^*c_1(X) c_{n-|J|-1}(D_J)$ is considered as the intersection number of the $1$-cycle $c_{n-|J|-1}(D_J) \in A_1(D_J)$ with the $\rho$-pullback of the class of the anticanonical ${\Bbb Q}$-divisor of $X$.} \label{def-c1} \end{dfn} \begin{rem} {\rm It is not clear a priori that the number $c_{\rm st}^{1, n-1}(X)$ in the above the definition does not depend on the choice of a desingularization $\rho$. Later we shall see that it is the case.} \end{rem} \begin{prop} For any smooth $n$-dimensional projective variety $V$, one has \[ \frac{d}{d u} {E}(V;u,1)|_{u =1} = \frac{n}{2} c_n(V). \] \label{f-1} \end{prop} \noindent {\em Proof.} By definition of $E$-polynomials, we have \[ \frac{d}{d u} {E}(V;u,1)|_{u =1} = \sum_{p,q} p(-1)^{p+q}h^{p,q}(V). \] The Poincar{\'e} duality $h^{p,q}(V) = h^{n-p,n-q}(V)$ $\forall p,q$ implies that \[ \sum_{p,q} \left(p - \frac{n}{2} \right) (-1)^{p+q}h^{p,q}(V) = 0. \] Hence, \[ \sum_{p,q} p(-1)^{p+q}h^{p,q}(V) = \frac{n}{2} \sum_{p,q} (-1)^{p+q}h^{p,q}(V) =\frac{n}{2} c_n(V). \] \hfill $\Box$ \begin{prop} For any $n$-dimensional projective variety $X$ having at worst log-terminal singularities, one has \[ \frac{d}{d u} {E}_{\rm st}(X;u,1)|_{u =1} = \frac{n}{2} e_{\rm st}(X). \] \label{f-2} \end{prop} \noindent {\em Proof.} By definition \ref{def-e}, we have \[ E_{\rm st} (X; u,1) = \sum_{J \subset I} E(D_J; u,1) \prod_{j \in J} \left( \frac{u -1}{u^{a_j +1} -1} -1 \right). \] Applying \ref{f-1} to every smooth submanifold $D_J \subset Y$, we obtain \[ \frac{d}{d u} {E}_{\rm st}(X;u, 1)|_{u =1} = \sum_{J \subset I} \frac{(n-|J|)}{2} c_{n-|J|}(D_J) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) + \] \[ + \sum_{J \subset I} \frac{|J|}{2} c_{n-|J|}(D_J) \prod_{j \in J} \left( \frac{-a_j}{(a_j +1)} \right) = \frac{n}{2} \sum_{J \subset I} c_{n-|J|}(D_J) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) = \frac{n}{2}e_{\rm st}(X). \] \hfill $\Box$ \begin{prop} Let $V$ be a smooth projective algebraic variety of dimension $n$ and $W \subset V$ a smooth irreducible divisor on $V$ or empty divisor $($the latter means that ${\cal O}_V(W) \cong {\cal O}_V)$. Then \[ c_1({\cal O}_V(W)) c_{n-1}(V) = c_{n-1}(W) + c_1({\cal O}_W(W)) c_{n-2}(W), \] where $c_{n-1}(W)$ is considered to be zero if $W = \emptyset$. \label{div} \end{prop} \noindent {\em Proof.} Consider the short exact sequnce \[ 0 \to T_W \to T_V|_W \to {\cal O}_W(W) \to 0, \] where $T_W$ and $T_V$ are tangent shaves on $W$ and $V$. It gives the following the relation betwen Chern polynomials \[ (1 + c_1({\cal O}_W(W) t) ( 1 + c_1(D)t + c_2(D)t^2 + \cdots + c_{n-1}(D)t^{n-1}) = \] \[ = 1 + c_1( T_V|_W)t + c_2( T_V|_W)t^2 + c_{n-1}( T_V|_W) t^{n-1} ). \] Comparing the coefficients by $t^{n-1}$ and using $c_{n-1}( T_V|_W) = c_1({\cal O}_V(W)) c_{n-1}(V)$, we come to the required equality. \hfill $\Box$ \begin{coro} Let $Y$ be a smooth projective variety, $D_1, \ldots, D_r$ smooth irreducible divisors with normal crossings, $I: = \{1, \ldots, r \}$. Then for all $J \subset I$ and for all $j \in J$ one has \[ c_1({\cal O}_{D_{J \setminus \{j \} }}(D_j)) c_{n-|J|}( D_{J \setminus \{j \} }) - c_{n-|J|}( D_{J}) = c_1({\cal O}_{D_J}(D_j))c_{n-|J|-1}(D_J), \] where $D_J$ is the complete intersection $\bigcap_{j \in J} D_j$. \label{rec} \end{coro} \noindent {\em Proof.} One sets in \ref{div} $V:= D_{J \setminus \{j \}}$ and $W:= D_{J}$. \hfill $\Box$ \begin{prop} Let $\rho\; : \; Y \to X$ be a desingularization as in \ref{def-c1}. Then \[ \sum_{J \subset I} c_1(D_J) c_{n-|J|-1}(D_J) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) = c_{\rm st}^{1, n-1}(X) +\] \[ + \sum_{J \subset I} \left( \sum_{j \in J} (a_{j} +1)c_{n-|J|}(D_J) \right)\prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) . \] \label{relat-div} \end{prop} \noindent {\em Proof.} Using the formula \[ c_1(Y) = \rho^*c_1(X) + \sum_{i \in I} -a_i c_1({\cal O}_Y(D_i)) \] and the adjunction formula for every complete intersection $D_J$ $(J \subset I)$, we obtain \[ c_1(D_J) = \rho^*c_1(X)|_{D_J} + \sum_{j \in J} (-a_j-1)c_1({\cal O}_{D_J}(D_j)) + \sum_{j \in I \setminus J} (-a_j)c_1({\cal O}_{D_J}(D_j)). \] Therefore \begin{equation} \sum_{J \subset I} c_1(D_J) c_{n-|J|-1}(D_J) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) = c_{\rm st}^{1, n-1}(X) + \label{eq3} \end{equation} \[ + \left( \sum_{j \in J} (-a_j-1)c_1({\cal O}_{D_J}(D_j)) c_{n-|J|-1}(D_J) \right) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) +\] \[ + \left( \sum_{j \in I \setminus J} (-a_j)c_1({\cal O}_{D_J}(D_j)) c_{n-|J|-1}(D_J) \right) \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) . \] Using \ref{rec}, we obtain \begin{equation} \sum_{j \in J} (-a_j-1)c_1({\cal O}_{D_J}(D_j)) c_{n-|J|-1}(D_J) = \label{eq4} \end{equation} \[ = \sum_{j \in J} (-a_j-1) \left(c_1({\cal O}_{D_{J \setminus \{j \} }}(D_j)) c_{n-|J|}( D_{J \setminus \{j \} }) - c_{n-|J|}( D_{J}) \right). \] By substitution (\ref{eq4}) to (\ref{eq3}), we come to the required equality. \hfill $\Box$ \begin{theo} Let $X$ be an arbitrary $n$-dimensional projective variety variety with at worst log-terminal singularities. Then \[ \frac{d^2}{d u^2}{E}_{\rm st}(X; u,1)|_{u =1} = \frac{3n^2 - 5n}{12} e_{\rm st}(X) + \frac{1}{6}c_{\rm st}^{1,n}(X). \] \label{main} \end{theo} \noindent {\em Proof.} Using the equalities \[ \frac{d}{d u}\left( \frac{u-1}{u^{a+1} -1} -1 \right)_{u=1} = \frac{-a}{2(a+1)}, \;\; \frac{d^2}{d u^2} \left( \frac{u-1}{u^{a+1} -1} -1 \right)_{u=1} = \frac{a(a+2)}{6(a+1)} \] together with the identities in \ref{d-rel} and \ref{f-1} for every submanifold $D_J \subset Y$, we obtain \[ \frac{d^2}{d u^2} {E}_{\rm st}(X; u,1)|_{u =1} = \sum_{J \subset I} \frac{c_1(D_J)c_{n - |J|-1}(D_J)}{6} \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) + \] \[ + c_{n-|J|}(D_J) \frac{3(n-|J|)^2 - 5(n-|J|)}{12} \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) + \] \[ + \sum_{J \subset I} \frac{(n-|J|)|J|c_{n-|J|}(D_J)}{2} \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) + \] \[ + \sum_{J \subset I} \frac{c_{n-|J|}(D_J)(|J|-1)|J|}{4} \prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right) + \] \[ + \sum_{J \subset I} \frac{c_{n-|J|}(D_J)(-\sum_{j \in J} (a_j +2) )}{6} \prod_{j \in J} \left( \frac{-a_j}{a_j+1} \right). \] By \ref{relat-div}, the first term of the above equals \[ \frac{1}{6} c_{\rm st}^{1, n-1}(X) + \frac{1}{6} \sum_{J \subset I} \left( \sum_{j \in J} (a_{j} +1)c_{n-|J|}(D_J) \right)\prod_{j \in J} \left( \frac{-a_j}{a_j +1} \right). \] Now the required statement follows from the equality \[ \frac{ \sum_{j \in J} (a_{j} +1)}{6} + \frac{3(n-|J|)^2 - 5(n-|J|)}{12} + \frac{(n-|J|)|J|}{2} + \] \[ + \frac{(|J|-1)|J|}{4} + \frac{-\sum_{j \in J} (a_j +2) }{6} = \frac{3n^2 - 5n}{12}. \] \hfill $\Box$ \begin{coro} The number $c_{\rm st}^{1,n}(X)$ does not depend on the choice of the desingularization $\rho\;: \; Y \to X$. \end{coro} \noindent {\em Proof.} By \ref{f-2} and \ref{main}, $c_{\rm st}^{1,n}(X)$ can be computed in terms of derivatives of the stringy $E$-function of $X$. But the stringy $E$-function does not depend on the choice of a desingularization \cite{B1}. \hfill $\Box$ \begin{coro} Let $X$ be a projective variety with at worst Gorenstein canonical singularities. Assume that the stringy Hodge numbers of $X$ exist. Then \[ \sum_{p,q} (-1)^{p+q} h^{p,q}_{\rm st}(X) \left(p - \frac{n}{2} \right)^2 = \frac{n}{12}e_{\rm st}(X) + \frac{1}{6} c_{\rm st}^{1,n-1}(X). \] \end{coro} \noindent {\em Proof.} The equality follows immediately from \ref{main} using the properties of the stringy Hodge numbers \ref{st-pr}. \hfill $\Box$ \begin{coro} If the canonical class of $X$ is numerically trivial, then $c_{\rm st}^{1,n}(X) =0$. In particular, for Calabi-Yau varieties with at worst Gorenstein canonical singularities we have \[ \frac{d^2}{d u^2}{E}_{\rm st}(X; u,1)|_{u =1} = \frac{3n^2 - 5n}{12} e_{\rm st}(X), \] and therefore stringy Hodge numbers of $X$ satisfy the identity $($\ref{st-rel}$)$ provided these stringy numbers exist. \end{coro} \section{Virasoro Algebra} Recall that the Virasoro algebra with the central charge $c$ consists of operators $L_n$ $(m \in {\Bbb Z})$ satisfying the relations \[ [L_n, L_m] = (n-m)L_{n+m} + c\frac{n^3-n}{12}\delta_{n+m,0} \;\;n, m \in {\Bbb Z}. \] For arbitrary compact K\"ahler manifold $X$, Eguchi et. al have proposed in \cite{EHX,EMX} a new approach to its quantum cohomology and to its Gromov-Witten invariants for all genera $g$ using so called the Virasoro condition: \[ L_n Z = 0, \forall n \geq -1, \] where \[ Z = {\rm exp}\, F = {\rm exp}\left( \sum_{g \geq 0} \lambda^{2g-2}F_g \right) \] is the partition function of the topological $\sigma$-model with the target space $X$ and $F_g$ the free energy function corresponding to the genus $g$. In this approach, the central charge $c$ acts as the multiplication by $c_n(X)$. Moreover, all Virasoro operators $L_n$ can be explicitly written in terms of elements of a basis of the cohomology of $X$, their gravitational descendants and the action of $c_1(X)$ on the cohomology by the multiplication. In particular the commutator relation \[ [L_1, L_{-1}] = 2 L_0 \] implies the precisely the identity of Libgober and Wood in the form \[ \sum_{p,q}(-1)^{p+q} h^{p,q}(X) \left( \frac{n+1}{2} - p \right) \left(p - \frac{n-1}{2} \right) = \frac{1}{6} \left( \frac{3-n}{2} c_n(X) - c_1(X)c_{n-1}(X) \right). \] Now let $X$ be a projective algebraic variety with at worst log-terminal singularities. We conjecture that there exists an analogous approach to the quantum cohomology as well as to the Gromov-Witten invariants of $X$ for all genera using the Virasoro algebra in such a way that for any resolution of singularities $\rho\; : \; Y \to X$ the corresponding Virasoro operators can be explicitely computed via the numbers $a_i$ appearing in the formula \[ K_X = \rho^*K_X + \sum_{i =1}^r a_i D_i \] and bases in cohomology of all complete intersections $D_J$ together with the multiplicative actions of $c_1(D_J)$ in them. We consider our main result \ref{main} as an evidence in favor of this conjecture.

9711

alg-geom/9711013

en

https://arxiv.org/abs/alg-geom/9711013

alg-geom/9711013

Vicente Munoz Velazquez

Vicente Mu\~noz

First quantum correction for the moduli space of stable bundles over a Riemann surface

13 pages, LaTeX2e, no figures

We compute some Gromov-Witten invariants of the moduli space of odd degree rank two stable vector bundles over a Riemann surface of any genus. Next we find the first correction term for the quantum product of this moduli space and hence get the two leading terms of the relations satisfied by the natural generators of its quantum cohomology. Finally, we use this information to get a full description of the quantum cohomology of the moduli space when the genus of the Riemann surface is 3.

alg-geom

\section{Introduction} \label{sec:intro} Let $\S$ be a Riemann surface of genus $g \geq 2$ and let $M_{\Sigma}$ denote the moduli space of flat $SO(3)$-connections with nontrivial second Stiefel-Whitney class $w_2$. This is a smooth symplectic manifold of dimension $6g-6$. Alternatively, we can consider $\S$ as a smooth complex curve of genus $g$. Fix a line bundle $\L$ on $\S$ of degree $1$, then $M_{\Sigma}$ is the moduli space of rank two stable vector bundles on $\S$ with determinant $\L$, which is a smooth complex variety of complex dimension $3g-3$. The symplectic deformation class of $M_{\Sigma}$ only depends on $g$ and not on the particular complex structure on $\S$. The manifold $X=M_{\Sigma}$ is a positive symplectic manifold with $\pi_2(X)={\Bbb Z}$. For such a manifold $X$, its quantum cohomology, $QH^*(X)$, is well-defined (see~\cite{Ruan}~\cite{RT}~\cite{McDuff}~\cite{Piunikhin}). As vector spaces, $QH^*(X)=H^*(X)$ (rational coefficients are understood), but the multiplicative structure is different. Let $A$ denote the positive generator of $\pi_2(X)$, i.e. the generator such that the symplectic form evaluated on $A$ is positive. Let $N=c_1(X)[A] \in {\Bbb Z}_{>0}$. Then there is a natural ${\Bbb Z}/2N{\Bbb Z}$-grading for $QH^*(X)$, which comes from reducing the ${\Bbb Z}$-grading of $H^*(X)$. (For the case $X=M_{\Sigma}$, $N=2$, so $QH^*(M_{\Sigma})$ is ${\Bbb Z}/4{\Bbb Z}$-graded). The ring structure of $QH^*(X)$, called quantum multiplication, is a deformation of the usual cup product for $H^*(X)$. For $\a \in H^p(X)$, $\b \in H^q(X)$, we define the quantum product of $\a$ and $\b$ as $$ \a \cdot \b =\sum_{d \geq 0} \P_{dA}(\a, \b), $$ where $\P_{dA}(\a, \b) \in H^{p+q-2Nd} (X)$ is given by $<\P_{dA}(\a, \b),\gamma>=\Psi^X_{dA}(\a, \b, \gamma)$, the Gromov-Witten invariant, for all $\gamma \in H^{\dim X-p-q+2Nd} (X)$. One has $\P_0(\a, \b)=\a \cup \b$. The other terms are the correction terms and all live in lower degree parts of the cohomology groups. It is a fact~\cite{RT} that the quantum product gives an associative and graded commutative ring structure. To define the Gromov-Witten invariant, let $J$ be a generic almost complex structure compatible with the symplectic form. Then for every $2$-homology class $dA$, $d \in {\Bbb Z}$, there is a moduli space ${\cal M}_{dA}$ of pseudoholomorphic rational curves (with respect to $J$) $f : {\Bbb P}^1 \rightarrow X$ with $f_*[{\Bbb P}^1] = dA$. Note that ${\cal M}_0=X$ and that ${\cal M}_{dA}$ is empty for $d<0$. For $d \geq 0$, the dimension of ${\cal M}_{dA}$ is $\dim X +2Nd$. This moduli space ${\cal M}_{dA}$ admits a natural compactification, $\overline{{\cal M}}_{dA}$, called the Gromov-Uhlenbeck compactification~\cite{Ruan}~\cite[section 3]{RT}. Consider now $r \geq 3$ different points $\seq{P}{1}{r} \in {{\Bbb P}}^1$. Then we have defined an evaluation map $ev: {\cal M}_{dA} \rightarrow X^r$ by $f \mapsto (f(P_1), \ldots, f(P_r))$. This map extends to $\overline{{\cal M}}_{dA}$ and its image, $ev(\overline{{\cal M}}_{dA})$, is a pseudo-cycle~\cite{RT}. So for $\a_i \in H^{p_i}(M_{\Sigma})$, $1 \leq i \leq r$, with $p_1+ \cdots +p_r = \dim X + 2Nd$, we choose generic cycles $A_i$, $1 \leq i \leq r$, representatives of their Poincar\'e duals, and set \begin{equation} \label{eqn:qu1} \Psi_{dA}^X(\seq{\a}{1}{r}) = <A_1\times\cdots\times A_r, [ev(\overline{{\cal M}}_{dA})]>= \# ev_{P_1}^*(A_1) \cap \cdots \cap ev_{P_r}^*(A_r), \end{equation} where $\#$ denotes count of points (with signs) and $ev_{P_i}: {\cal M}_{dA} \rightarrow X$, $f \mapsto f(P_i)$. This is a well-defined number and independent of the particular cycles. Also, as the manifold $X$ is positive, $\text{coker} L_f=H^1({\Bbb P}^1, f^* c_1(X))=0$, for all $f \in {\cal M}_{dA}$. By~\cite{Ruan} the complex structure of $X$ is generic and we can use it to compute the Gromov-Witten invariants. Also for $r \geq 2$, let $\a_i \in H^{p_i}(M_{\Sigma})$, $1 \leq i \leq r$, then $$ \a_1 \cdots \a_r =\sum_{d \geq 0} \P_{dA}(\a_1, \ldots ,\a_r), $$ where the correction terms $\P_{dA}(\a_1, \ldots ,\a_r) \in H^{p_1+\cdots +p_r - 2 N d}(X)$ are determined by $<\P_{dA}(\a_1, \ldots ,\a_r), \gamma> = \Psi_{dA}^X(\a_1, \ldots ,\a_r, \gamma)$, for any $\gamma \in H^{\dim X +2Nd-(p_1+\cdots +p_r)}(X)$. Returning to our manifold $X=M_{\Sigma}$, there is a classical conjecture relating the quantum cohomology $QH^*(M_{\Sigma})$ and the instanton Floer cohomology of $\Sigma \times {\Bbb S}^1$, $HF^*(\Sigma \times {\Bbb S}^1)$ (see~\cite{Floer}). In~\cite{Vafa} a presentation of $QH^*(M_{\Sigma})$ was given using physical methods, and in~\cite{Floer} it was proved that such a presentation was a presentation of $HF^*(\Sigma \times {\Bbb S}^1)$ indeed. Siebert and Tian have a program~\cite{Siebert} to find the presentation of $QH^*(M_{\Sigma})$, which goes through proving a recursion formula for the Gromov-Witten invariants of $M_{\Sigma}$ in terms of the genus $g$. This will complete the proof of the conjectural isomorphism $QH^*(M_{\Sigma}) \cong HF^*(\Sigma \times {\Bbb S}^1)$. The purpose of this paper is two-fold. On the one hand we aim to compute the Gromov-Witten invariants $\Psi_A^{M_{\Sigma}}$, relating them to the Donaldson invariants for the algebraic surface $S=\S\times{\Bbb P}^1$. This gives the first correction term of the quantum product of $M_{\Sigma}$ and hence the first two leading terms of the relations satisfied by the generators of $QH^*(M_{\Sigma})$. In particular we obtain the coefficient $c_g$ of~\cite[section 3.3]{Siebert}. On the other hand we infer a full presentation of $QH^*(M_{\Sigma})$ when the genus of $\S$ is $g=3$ (recall that the case of genus $g=2$ was worked out by Donaldson~\cite{D1}). This is the starting point of the induction in~\cite{Siebert}. The paper is organised as follows. In section~\ref{sec:ordinary} we review the ordinary cohomology ring of $M_{\Sigma}$. In section~\ref{sec:curves} the moduli space of lines in $M_{\Sigma}$ is described, in order to compute the corresponding Gromov-Witten invariants in section~\ref{sec:GW-inv}. In section~\ref{sec:quantum} we study the quantum cohomology of $M_{\Sigma}$ and finally determine it completely in the case of genus $g=3$ in section~\ref{sec:g3}. \noindent {\em Acknowledgements:\/} I want to thank my D. Phil.\ supervisor Simon Donaldson, for his encouragement and invaluable help. Also I am very grateful to Bernd Siebert and Gang Tian for helpful conversations and for letting me have a copy of their preprint~\cite{Siebert}. \section{Classical cohomology ring of $M_{\Sigma}$} \label{sec:ordinary} Let us recall the known description of the homology of $M_{\Sigma}$~\cite{King}~\cite{ST}~\cite{Floer}. Let ${\cal U} \rightarrow \S \times M_{\Sigma}$ be the universal bundle and consider the K\"unneth decomposition of \begin{equation} c_2(\text{End}_0 \, {\cal U})=2 [\S] \otimes \a + 4 \psi -\b, \label{eqn:qu2} \end{equation} with $\psi=\sum \gamma_i \otimes \psi_i$, where $\{\seq{\gamma}{1}{2g}\}$ is a symplectic basis of $H^1(\S;{\Bbb Z})$ with $\gamma_i \gamma_{i+g}=[\S]$ for $1 \leq i \leq g$ (also $\{\gamma^{\#}_i\}$ will denote the dual basis for $H_1(\S;{\Bbb Z})$). Here we can suppose without loss of generality that $c_1({\cal U})=\L + \a$ (see~\cite{ST}). In terms of the map $\mu: H_*(\S) \to H^{4-*}(M_{\Sigma})$, given by $\mu(a)= -{1 \over 4} \, p_1({\frak g}_{{\cal U}}) /a $ (here ${\frak g}_{{\cal U}} \to \S \times M_{\Sigma}$ is the associated universal $SO(3)$-bundle, and $p_1({\frak g}_{{\cal U}}) \in H^4(\S\xM_{\Sigma})$ its first Pontrjagin class), we have $$ \left\{ \begin{array}{l} \a= 2\, \mu(\S) \in H^2 \\ \psi_i= \mu (\gamma_i^{\#}) \in H^3, \qquad 1\leq i \leq 2g \\ \b= - 4 \, \mu(x) \in H^4 \end{array} \right. $$ where $x \in H_0(\S)$ is the class of the point, and $H^i=H^i(M_{\Sigma})$. These elements generate $H^*(M_{\Sigma})$ as a ring~\cite{King}~\cite{Thaddeus}, and $\a$ is the positive generator of $H^2(M_{\Sigma};{\Bbb Z})$. We can rephrase this as saying that there exists an epimorphism \begin{equation} \AA(\S)= {\Bbb Q}[\a,\b ]\otimes \L(\seq{\psi}{1}{2g}) \twoheadrightarrow H^*(M_{\Sigma}) \label{eqn:qu3} \end{equation} (the notation $\AA(\S)$ follows that of Kronheimer and Mrowka~\cite{KM}). The mapping class group $\text{Diff}(\S)$ acts on $H^*(M_{\Sigma})$, with the action factoring through the action of $\text{Sp}\, (2g,{\Bbb Z})$ on $\{\psi_i\}$. The invariant part, $H_I^*(M_{\Sigma})$, is generated by $\a$, $\b$ and $\gamma=-2 \sum_{i=0}^g \psi_i\psi_{i+g}$. Then there is an epimorphism \begin{equation} {\Bbb Q}[\a,\b,\gamma ] \twoheadrightarrow H^*_I(M_{\Sigma}) \label{eqn:qu4} \end{equation} which allows us to write $$ H_I^*(M_{\Sigma})= {\Bbb Q} [\a, \b, \gamma]/I_g, $$ where $I_g$ is the ideal of relations satisfied by $\a$, $\b$ and $\gamma$. Recall that $\deg(\a)=2$, $\deg(\b)=4$ and $\deg(\gamma)=6$. From~\cite{ST}, a basis for $H_I^*(M_{\Sigma})$ is given by the monomials $\a^a\b^b\gamma^c$, with $a+b+c<g$. For $0 \leq k \leq g$, the primitive component of $\L^k H^3$ is $$ \L_0^k H^3 = \ker (\gamma^{g-k+1} : \L^k H^3 \rightarrow \L^{2g-k+2} H^3). $$ Then the $\text{Sp}\, (2g,{\Bbb Z})$-decomposition of $H^*(M_{\Sigma})$ is~\cite{King} $$ H^*(M_{\Sigma})= \bigoplus_{k=0}^g \L_0^k H^3 \otimes {\Bbb Q} [\a, \b, \gamma]/I_{g-k}. $$ \begin{prop}[\cite{ST}] \label{prop:1} For $g=1$, let $q_1^1=\a$, $q_1^2=\b$, $q_1^3=\gamma$. Define recursively, for $g \geq 1$, $$ \left\{ \begin{array}{l} q_{g+1}^1= \a q_g^1 +g^2 q_g^2 \\ q_{g+1}^2 = \b q_g^1 + {2g \over g+1} q_g^3 \\ q_{g+1}^3 = \gamma q_g^1 \end{array} \right. $$ Then $I_g=(q_g^1, q_g^2, q_g^3)$, for all $g \geq 1$. Note that $\deg(q_g^1)=2g$, $\deg(q_g^2)=2g+2$ and $\deg(q_g^3)=2g+4$. \end{prop} \section{Holomorphic lines in $M_{\Sigma}$} \label{sec:curves} In order to compute the Gromov-Witten invariants $\Psi_A^{M_{\Sigma}}$, we need to describe the space of lines, i.e. rational curves in $M_{\Sigma}$ representing the generator $A \in H_2(M_{\Sigma};{\Bbb Z})$, $$ {\cal M}_A=\{f: {\Bbb P}^1 \rightarrow M_{\Sigma} / \text{$f$ holomorphic, } f_*[{\Bbb P}^1]=A\}. $$ Let us fix some notation. Let $J$ denote the Jacobian variety of $\S$ parametrising line bundles of degree $0$ and let ${\cal L} \rightarrow \S\times J$ be the universal line bundle. If $\{\gamma_i\}$ is the basis of $H^1(\S)$ introduced in section~\ref{sec:ordinary} then $c_1({{\cal L}}) = \sum \gamma_i \otimes \phi_i \in H^1(\S) \otimes H^1(J)$, where $\{\phi_i\}$ is a symplectic basis for $H^1(J)$. Thus $c_1({{\cal L}})^2 = -2 [\S] \otimes \o \in H^2(\S) \otimes H^2(J)$, where $\o=\sum_{i=1}^g \phi_i \wedge \phi_{i+g}$ is the natural symplectic form for $J$. Consider now the algebraic surface $S=\Sigma \times {\Bbb P}^1$. It has irregularity $q=g \geq 2$, geometric genus $p_g=0$ and canonical bundle $K \equiv -2\S+(2g-2){{\Bbb P}}^1$. Recall that $\L$ is a fixed line bundle of degree $1$ on $\S$. Fix the line bundle $L=\L \otimes {\cal O}_{{\Bbb P}^1}(1)$ on $S$ (we omit all pull-backs) with $c_1= c_1(L) \equiv {\Bbb P}^1+\S$, and put $c_2=1$. The ample cone of $S$ is $\{a{\Bbb P}^1 +b \S \, / \, a,b >0 \}$. Let $H_0$ be a polarisation close to ${\Bbb P}^1$ in the ample cone and $H$ be a polarisation close to $\S$, i.e. $H= \S +t{\Bbb P}^1$ with $t$ small. We wish to study the moduli space ${\frak M}= {\frak M}_H(c_1,c_2)$ of $H$-stable bundles over $S$ with Chern classes $c_1$ and $c_2$. \begin{prop} \label{prop:2} ${\frak M}$ can be described as a bundle ${\Bbb P}^{2g-1} \rightarrow {\frak M}= {\Bbb P}({\cal E}_{\zeta}^{\vee}) \rightarrow J$, where ${\cal E}_{\zeta}$ is a bundle on $J$ with $\text{ch}\: {\cal E}_{\zeta} = 2g + 8 \o$. So ${\frak M}$ is compact, smooth and of the expected dimension $6g-2$. The universal bundle ${\cal V} \rightarrow S \times {\frak M}$ is given by $$ 0 \rightarrow {\cal O}_{{\Bbb P}^1}(1) \otimes {\cal L} \otimes \l \rightarrow {\cal V} \rightarrow \L \otimes {\cal L}^{-1} \rightarrow 0, $$ where $\l$ is the tautological line bundle for ${\frak M}$. \end{prop} \begin{pf} For the polarisation $H_0$, the moduli space of $H_0$-stable bundles with Chern classes $c_1, c_2$ is empty by~\cite{Qin}. Now for $p_1=-4c_2+c_1^2=-2$ there is only one wall, determined by $\zeta \equiv -{\Bbb P}^1 +\S$ (here we fix $\zeta= 2\S- L=\S-c_1(\L)$ as a divisor), so the moduli space of $H$-stable bundles with Chern classes $c_1, c_2$ is obtained by crossing the wall as described in~\cite{wall}. First, note that the results in~\cite{wall} use the hypothesis of $-K$ being effective, but the arguments work equally well with the weaker assumption of $\zeta$ being a good wall~\cite[remark 1]{wall} (see also~\cite{Gottsche} for the case of $q=0$). In our case, $\zeta \equiv -{\Bbb P}^1+\S$ is a good wall (i.e. $\pm \zeta+K$ are both not effective) with $l_{\zeta}=0$. Now with the notations of~\cite{wall}, $F$ is a divisor such that $2F-L \equiv \zeta$, i.e. $F=\S$. Also ${\cal F} \rightarrow S \times J$ is the universal bundle parametrising divisors homologically equivalent to $F$, i.e. ${\cal F}={\cal L} \otimes {\cal O}_{{\Bbb P}^1}(1)$. Let $\pi: S \times J \rightarrow J$ be the projection. Then ${\frak M}=E_{\zeta}= {\Bbb P}({\cal E}_{\zeta}^{\vee})$, where $$ {\cal E}_{\zeta} = {{\cal E}}\text{xt}^1_{\pi}({\cal O}(L-{\cal F}),{\cal O}({\cal F}))= R^1\pi_* ({\cal O}(\zeta )\otimes {\cal L}^2). $$ Actually ${\frak M}$ is exactly the set of bundles $E$ that can be written as extensions $$ \exseq{{\cal O}_{{\Bbb P}^1}(1) \otimes L}{E}{\L \otimes L^{-1}} $$ for a line bundle $L$ of degree $0$. The Chern character is~\cite[section 3]{wall} $\text{ch}\: {\cal E}_{\zeta} = 2g + e_{K-2\zeta}$, where $e_{\a}= -2 ({\Bbb P}^1 \cdot \a) \o$ (the class $\S$ defined in~\cite[lemma 11]{wall} is ${\Bbb P}^1$ in our case). Finally, the description of the universal bundle follows from~\cite[theorem 10]{wall}. \end{pf} \begin{prop} \label{prop:3} There is a well defined map ${\cal M}_A \rightarrow {\frak M}$. \end{prop} \begin{pf} Every line $f:{{\Bbb P}}^1 \rightarrow M_{\Sigma}$ gives a bundle $E=(\text{id}_{\S}\times f)^*{\cal U}$ over $\Sigma \times {\Bbb P}^1$ by pulling-back the universal bundle ${{\cal U}} \rightarrow \S \times M_{\Sigma}$. Then for any $t \in {\Bbb P}^1$, the bundle $E|_{\S \times t}$ is defined by $f(t)$. Now, by equation~\eqref{eqn:qu2}, $p_1(E)= p_1({\cal U})[\S \times A]= -2\a [A] =-2$. Since $c_1(E) = (\text{id}_{\S}\times f)^*c_1({\cal U})=\L +\S$, it must be $c_2=1$. To see that $E$ is $H$-stable, consider any sub-line bundle $L \hookrightarrow E$ with $c_1(L) \equiv a{\Bbb P}^1 +b\S$. Restricting to any $\S \times t \subset \Sigma \times {\Bbb P}^1$ and using the stability of $E|_{\S \times t}$, one gets $a \leq 0$. Then $c_1(L) \cdot \S < {c_1(E)\cdot \S \over 2}$, which yields the $H$-stability of $E$ (recall that $H$ is close to $\S$). So $E \in {\frak M}$. \end{pf} Now define $N$ as the set of extensions on $\S$ of the form \begin{equation} 0 \rightarrow L \rightarrow E \rightarrow \L \otimes L^{-1} \rightarrow 0, \label{eqn:qu5} \end{equation} for $L$ a line bundle of degree $0$. Then the groups $\text{Ext}^1( \L \otimes L^{-1}, L) =H^1(L^2 \otimes \L^{-1})= H^0(L^{-2} \otimes \L \otimes K)$ are of constant dimension $g$. Moreover $H^0(L^{2} \otimes \L^{-1})=0$, so the moduli space $N$ which parametrises extensions like~\eqref{eqn:qu1} is given as $N={\Bbb P}({\cal E}^{\vee})$, where ${\cal E}= {{\cal E}}{\hbox{xt}}^1_p(\L \otimes {{\cal L}}^{-1}, {{\cal L}})=R^1 p_*({{\cal L}}^{2} \otimes \L^{-1})$, $p:\S\times J \rightarrow J$ the projection. Then we have a fibration ${{\Bbb P}}^{g-1} \rightarrow N={\Bbb P}({\cal E}^{\vee}) \rightarrow J$. The Chern character of ${\cal E}$ is \begin{equation} \begin{array}{rcl} \text{ch}\:({\cal E}) &=& \text{ch}\:(R^1 p_*({{\cal L}}^{2}\otimes \L ^{-1})) = - \text{ch}\:(p_{!}( {{\cal L}}^2 \otimes \L ^{-1}) ) = \\ &=& -p_*( (\text{ch}\:\,{{\cal L}})^2 \> (\text{ch}\:\,\L)^{-1} \> \text{Todd} \> T_{\S}) = \\ &=& -p_*((1+c_1({{\cal L}})+{1 \over 2}c_1({{\cal L}})^2)^2(1-\L)(1-{1 \over 2}K)) = \\ &=& -p_*(1-{1 \over 2}K+2c_1({{\cal L}}) -4 \o \otimes [\S]-\L) = g+4\o. \end{array} \label{eqn:qu6} \end{equation} It is easy to check that all the bundles in $N$ are stable, so there is a well-defined map $$ i: N \rightarrow M_{\Sigma}. $$ Now we wish to construct the space of lines in $N$. Note that $\pi_2(N)=\pi_2({\Bbb P}^{g-1})={\Bbb Z}$, as there are no rational curves in $J$. Let $L \in \pi_2(N)$ be the positive generator. We want to describe $$ {\cal N}_L=\{f: {\Bbb P}^1 \rightarrow N / \text{$f$ holomorphic, } f_*[{\Bbb P}^1]=L\}. $$ For the projective space ${\Bbb P}^n$, the space of lines $H_1$ is the set of algebraic maps $f: {{\Bbb P}}^1 \rightarrow {{\Bbb P}}^n$ of degree $1$. Such an $f$ has the form $f[ x_0, x_1 ]=[x_0 u_0 +x_1 u_1]$, $[x_0,x_1] \in {\Bbb P}^1$, where $u_0$, $u_1$ are linearly independent vectors in ${\Bbb C}^{n+1}$. So $$ H_1= {\Bbb P}(\{(u_0, u_1)/ \text{$u_0$, $u_1$ are linearly independent} \}) \subset {\Bbb P}(({\Bbb C}^{\,n+1} \oplus {\Bbb C}^{\,n+1})^{\vee}) ={\Bbb P}^{2n+1}. $$ The complement of $H_1$ is the image of ${\Bbb P}^n \times {\Bbb P}^1 \hookrightarrow {\Bbb P}^{2n+1}$, $([u] , [x_0, x_1]) \mapsto [x_0 u, x_1 u]$, which is a smooth $n$-codimensional algebraic subvariety. Now ${\cal N}_L$ can be described as \begin{equation} \begin{array}{ccccc} H_1 &\rightarrow & {\cal N}_L & \rightarrow& J \\ \bigcap & & \bigcap & & \| \\ {\Bbb P}^{2g-1} &\rightarrow & {\Bbb P}(({\cal E} \oplus {\cal E})^{\vee}) & \rightarrow & J \end{array} \label{eqn:qu7} \end{equation} \begin{rem} \label{rem:4} Note that ${\cal E}_{\zeta}= R^1\pi_* ({\cal O}(\zeta )\otimes {\cal L}^2)= R^1\pi_*({\cal O}_{{\Bbb P}^1}(1) \otimes {{\cal L}}^{2} \otimes \L^{-1})= H^0 ({\cal O}_{{\Bbb P}^1}(1)) \otimes R^1p_*( {{\cal L}}^{2} \otimes \L^{-1}) = H^0 ({\cal O}_{{\Bbb P}^1}(1)) \otimes {\cal E} \cong {\cal E} \oplus {\cal E}$. So ${\frak M}={\Bbb P}(({\cal E} \oplus {\cal E})^{\vee})$, canonically. \end{rem} \begin{prop} \label{prop:5} The map $i:N \rightarrow M_{\Sigma}$ induces a map $i_*: {\cal N}_L \rightarrow {\cal M}_A$. The composition ${\cal N}_L \rightarrow {\cal M}_A \rightarrow {\frak M}$ is the natural inclusion of~\eqref{eqn:qu7}. \end{prop} \begin{pf} The first assertion is clear as $i$ is a holomorphic map. For the second, consider the universal sheaf on $\S\times N$, \begin{equation} 0 \rightarrow {{\cal L}} \otimes U \rightarrow {{\Bbb E}} \rightarrow \L \otimes {\cal L}^{-1} \rightarrow 0, \label{eqn:qu8} \end{equation} where $U={\cal O}_N(1)$ is the tautological bundle of the fibre bundle ${\Bbb P}^{g-1} \rightarrow N \rightarrow J$. Any element in ${\cal N}_L$ is a line ${{\Bbb P}}^1 \hookrightarrow N$, which must lie inside a single fibre ${\Bbb P}^{g-1}$. Restricting~\eqref{eqn:qu8} to this line, we have an extension $$ 0 \rightarrow L \otimes {{\cal O}}_{{{\Bbb P}}^1}(1) \rightarrow E \rightarrow \L \otimes L^{-1} \rightarrow 0 $$ on $S=\Sigma \times {\Bbb P}^1$, which is the image of the given element in ${\frak M}$ (here $L$ is the line bundle corresponding to the fibre in which ${{\Bbb P}}^1$ sits). Now it is easy to check that the map ${\cal N}_L \rightarrow {\frak M}$ is the inclusion of~\eqref{eqn:qu7}. \end{pf} \begin{cor} \label{cor:6} $i_*$ is an isomorphism. \end{cor} \begin{pf} By proposition~\ref{prop:5}, $i_*$ has to be an open immersion. The group $PGL(2,{\Bbb C})$ acts on both spaces ${\cal N}_L$ and ${\cal M}_A$, and $i_*$ is equivariant. The quotient ${\cal N}_L/PGL(2,{\Bbb C})$ is compact, being a fibration over the Jacobian with all the fibres the Grassmannian $\text{Gr}\,({\Bbb C}^{\,2}, {\Bbb C}^{\,g-1})$, hence irreducible. As a consequence $i_*$ is an isomorphism. \end{pf} \begin{rem} \label{rem:7} Notice that the lines in $M_{\Sigma}$ are all contained in the image of $N$, which is of dimension $4g-2$ against $6g-6=\dim M_{\Sigma}$. They do not fill all of $M_{\Sigma}$ as one would naively expect. \end{rem} \section{Computation of $\Psi_A^{M_{\Sigma}}$} \label{sec:GW-inv} The manifold $N$ is positive with $\pi_2(N)={\Bbb Z}$ and $L \in \pi_2(N)$ is the positive generator. Under the map $i: N \rightarrow M_{\Sigma}$, we have $i_* L=A$. Now $\dim N= 4g-2$ and $c_1(N)[L]=c_1({\Bbb P}^{g-1})[L]=g$. So quantum cohomology of $N$, $QH^*(N)$, is well-defined and ${\Bbb Z}/2g{\Bbb Z}$-graded. {}From corollary~\ref{cor:6}, it is straightforward to prove \begin{lem} \label{lem:8} For any $\a_i \in H^{p_i}(M_{\Sigma})$, $1 \leq i \leq r$, such that $p_1 + \cdots + p_r =6g-2$, it is $\Psi_A^{M_{\Sigma}}(\seq{\a}{1}{r})= \Psi_L^N(\seq{i^*\a}{1}{r})$. $\quad \Box$ \end{lem} It is therefore important to know the Gromov-Witten invariants of $N$, i.e. its quantum cohomology. {}From the universal bundle~\eqref{eqn:qu8}, we can read the first Pontrjagin class $p_1( {{\frak g}}_{{\Bbb E}}) = -8 [\S]\otimes\o + h^2- 2 [\S]\otimes h + 4h\cdot c_1({{\cal L}}) \in H^4(\S\times N)$, where $h=c_1(U)$ is the hyperplane class. So on $N$ we have \begin{equation} \left\{ \begin{array}{l} \a=2 \mu(\S) = 4\o + h \\ \psi_i=\mu(\gamma_i^{\#}) = - h \cdot \phi_i \\ \b= -4 \mu(x) = h^2 \end{array} \right. \label{eqn:qu9} \end{equation} Let us remark that $h^2$ denotes ordinary cup product in $H^*(N)$, a fact which will prove useful later. Now let us compute the quantum cohomology ring of $N$. The cohomology of $J$ is $H^*(J)= \L H_1$, where $H_1=H_1(\S)$. Now the fibre bundle description ${\Bbb P}^{2g-1} \rightarrow N={\Bbb P}({\cal E}^{\vee}) \rightarrow J$ implies that the usual cohomology of $N$ is $H^*(N)= \L H_1 [h]/<h^g+c_1h^{g-1}+ \cdots +c_g=0>$, where $c_i=c_i({\cal E})={4^i \over i!} \o^i$, from~\eqref{eqn:qu6}. As the quantum cohomology has the same generators as the usual cohomology and the relations are a deformation of the usual relations~\cite{ST2}, it must be $h^g+c_1h^{g-1}+ \cdots +c_g=r$ in $QH^*(N)$, with $r \in {\Bbb Q}$. As in~\cite[example 8.5]{RT}, $r$ can be computed to be $1$. So \begin{equation} QH^*(N)= \L H_1 [h]/<h^g+c_1h^{g-1}+ \cdots +c_g=1> . \label{eqn:qu10} \end{equation} \begin{lem} \label{lem:9} For any $s \in H^{2g-2i}(J)$, $0 \leq i \leq g$, denote by $s \in H^{2g-2i}(N)$ its pull-back to $N$ under the natural projection. Then the quantum product $h^{2g-1+i} s$ in $QH^*(N)$ has component in $H^{4g-2}(N)$ equal to ${(-8)^i\over i!} \o^i \wedge s$. \end{lem} \begin{pf} First note that for $s_1,s_2 \in H^*(J)$ such that their cup product in $J$ is $s_1s_2=0$, then the quantum product $s_1s_2 \in QH^*(N)$ vanishes. This is so since every rational line in $N$ is contained in a fibre of ${\Bbb P}^{2g-1} \rightarrow J \rightarrow N$. Next note that $h^{g-1+i}s$ has component in $H^{4g-2}(N)$ equal to $s_i({\cal E}) \wedge s={(-4)^i\over i!} \o^i \wedge s$. Then multiply the standard relation~\eqref{eqn:qu9} by $h^{g-1+i}$ and work by induction on $i$. For $i=0$ we get $h^{2g-1}s=h^{g-1}s$ and the assertion is obvious. For $i>0$, $$ h^{2g-1+i}s +h^{2g-2+i}c_1s+ \cdots +h^{2g-1}c_is=h^{g-1+i}s. $$ So the component of $h^{2g-1+i}s$ in $H^{4g-2}(N)$ is $$ - \sum_{j=1}^i {(-8)^{i-j}\over (i-j)!} \o^{i-j} c_j s + {(-4)^i\over i!} \o^i s = {(-8)^i\over i!} \o^i s - \sum_{j=0}^i {(-8)^{i-j}\over (i-j)!}{4^j\over j!} \o^i s + {(-4)^i\over i!} \o^i s = {(-8)^i\over i!} \o^i s. $$ \end{pf} \begin{lem} \label{lem:10} Suppose $g>2$. Let $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ have degree $6g-2$. Then $$ \Psi^N_L(\a, \stackrel{(a)}{\ldots}, \a, \b , \stackrel{(b)}{\ldots} ,\b,\psi_{i_1}, \ldots, \psi_{i_r})= <(4\o + X)^a(X^2)^b \phi_{i_1}\cdots\phi_{i_r} X^r, [J]>, $$ evaluated on $J$, where $X^{2g-1+i}={(-8)^i \over i!} \o^i \in H^*(J)$. \end{lem} \begin{pf} By definition the left hand side is the component in $H^{4g-2}(N)$ of the quantum product $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in QH^*(N)$. From~\eqref{eqn:qu9}, this quantum product is $(4\o + h)^a (h^2)^b (-h\phi_{i_1})\cdots(-h\phi_{i_r})$, upon noting that when $g>2$, $\b=h^2$ as a quantum product as there are no quantum corrections because of the degree. Note that $r$ is even, so the statement of the lemma follows from lemma~\ref{lem:9}. \end{pf} Now we are in the position of relating the Gromov-Witten invariants $\Psi_A^{M_{\Sigma}}$ with the Donaldson invariants for $S=\S\times {\Bbb P}^1$ (for definition of Donaldson invariants see~\cite{DK}~\cite{KM}). \begin{thm} \label{thm:11} Suppose $g>2$. Let $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ have degree $6g-2$. Then $$ \Psi^{M_{\Sigma}}_A(\a, \stackrel{(a)}{\ldots}, \a, \b , \stackrel{(b)}{\ldots} ,\b,\psi_{i_1}, \ldots, \psi_{i_r})= (-1)^{g-1} D^{c_1}_{S, H} ((2\S)^a(-4 \text{pt})^b \gamma_{i_1}^{\#} \cdots\gamma_{i_r}^{\#}), $$ where $D^{c_1}_{S, H}$ stands for the Donaldson invariant of $S=\S\times{\Bbb P}^1$ with $w=c_1$ and polarisation $H$. \end{thm} \begin{pf} By definition, the right hand side is $\epsilon_S(c_1) <\a^a\b^b\psi_{i_1}\cdots\psi_{i_r}, [{\frak M}]>$, where $\a =2\mu(\S) \in H^2({\frak M})$, $\b =-4\mu(x) \in H^4({\frak M})$, $\psi_i =\mu(\gamma_i^{\#}) \in H^3({\frak M})$. Here the factor $\epsilon_S(c_1)=(-1)^{K_S c_1 +c_1^2 \over 2}=(-1)^{g-1}$ compares the complex orientation of ${\frak M}$ and its natural orientation as a moduli space of anti-self-dual connections~\cite{DK}. By~\cite[theorem 10]{wall}, this is worked out to be $(-1)^{g-1} <(4\o + X)^a(X^2)^b \phi_{i_1}\cdots\phi_{i_r} X^r,[J]>$, where $X^{2g-1+i}=s_i({\cal E}_{\zeta})={(-8)^i \over i!} \o^i$. Thus the theorem follows from lemmas~\ref{lem:8} and~\ref{lem:10}. \end{pf} \begin{rem} \label{rem:12} The formula in theorem~\ref{thm:11} is not right for $g=2$, as in such case, the quantum product $h^2 \in QH^*(N)$ differs from $\b$ by a quantum correction. \end{rem} \begin{rem} \label{rem:13} Suppose $g \geq 2$ and let $\a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ have degree $6g-6$. Then \begin{eqnarray*} \Psi^{M_{\Sigma}}_0(\a, \stackrel{(a)}{\ldots}, \a, \b , \stackrel{(b)}{\ldots} ,\b,\psi_{i_1}, \ldots, \psi_{i_r}) &=& \epsilon_S({\Bbb P}^1) < \a^a\b^b\psi_{i_1}\cdots\psi_{i_r},[M_{\Sigma}]> = \\ &=& - D^{{\Bbb P}^1}_{S, H} ((2\S)^a(-4 \text{pt})^b \gamma_{i_1}^{\#} \cdots\gamma_{i_r}^{\#}), \end{eqnarray*} as the moduli space of anti-self-dual connections on $S$ of dimension $6g-6$ is $M_{\Sigma}$. \end{rem} \section{Quantum cohomology ring of $M_{\Sigma}$} \label{sec:quantum} The action of the mapping class group on $M_{\Sigma}$ is symplectic, so the quantum product restricts to the invariant part of the cohomology, thus having defined $QH^*_I(M_{\Sigma})$ (see~\cite[section 3.1]{Siebert}). To give a description of it, let us define \begin{equation} \left\{ \begin{array}{l} \hat{\alpha}=\a \\ \hat{\beta}=\b + r_g \\ \hat{\gamma} =-2 \sum \psi_i\psi_{i+g} \end{array} \right. \label{eqn:qu12} \end{equation} where $r_g \in {\Bbb Q}$, $g \geq 1$ (to be determined shortly), and $\hat{\gamma}$ is given using the quantum product. It might happen that $\hat{\gamma}=\gamma +s_g \a$, $s_g \in {\Bbb Q}$. The need of introducing this quantum corrections in the generators was noticed already in~\cite{Vafa}. As a consequence of the description of $H_I^*(M_{\Sigma})$ given in section~\ref{sec:ordinary}, the results in~\cite{ST2} imply that there is a presentation $$ QH^*_I(M_{\Sigma})= {\Bbb Q} [\hat{\alpha}, \hat{\beta}, \hat{\gamma}]/J_g, $$ where the ideal $J_g$ is generated by three elements $Q_g^1$, $Q_g^2$ and $Q_g^3$ which are deformations graded mod $4$ of $q_g^1$, $q_g^2$ and $q_g^3$, respectively (see~\cite{Siebert}). This means that $J_g=(Q^1_g,Q^2_g,Q^3_g)$ for \begin{equation} \label{eqn:qu11} Q_g^i= \sum_{j \geq 0} Q_{g,j}^i, \end{equation} where $\deg(Q_{g,j}^i)=\deg(q_g^i)-4j$, $j \geq 0$, and $Q_{g,0}^i=q_g^i$. There is still one source of possible ambiguity coming from adding a scalar multiple of $Q_g^1$ to $Q_g^3$. To avoid this, we require the coefficient of $\hat{\alpha}^g$ in $Q_g^3$ to be zero. Recall the main result from~\cite{Floer}. \begin{thm}[\cite{Floer}] \label{thm:14} Define $R^1_0=1$, $R^2_0=0$, $R^3_0=0$ and then recursively, for all $g \geq 1$, $$ \left\{ \begin{array}{l} R_{g+1}^1 = \a R_g^1 + g^2 R_g^2 \\ R_{g+1}^2 = (\b+(-1)^{g+1}8) R_g^1 + {2g \over g+1} R_g^3 \\ R_{g+1}^3 = \gamma R_g^1 \end{array} \right. $$ Then the invariant part of the instanton Floer cohomology of $\Sigma \times {\Bbb S}^1$ is $HF^*(\Sigma \times {\Bbb S}^1)_I ={\Bbb Q}[\a,\b,\gamma]/(R^1_g,R^2_g,R^3_g)$. $R^1_g$, $R^2_g$, $R^3_g$ are uniquely determined by the conditions that the leading term of $R^i_g$ is $q^i_g$, $i=1,2,3$ and that the coefficient of $\a^g$ in $R^3_g$ is zero. \end{thm} Accounting for the difference in signs between theorem~\ref{thm:11} and remark~\ref{rem:13}, the prospective generators of $J_g$ are defined as follows. \begin{defn} \label{def:15} For $i=1,2,3$ and $g \geq 1$, set $$ \hat{R}^i_g(\hat{\alpha},\hat{\beta},\hat{\gamma})= (\sqrt{-1})^{-\deg q^i_g} R^i_g(\sqrt{-1}^{\,g}\hat{\alpha},\sqrt{-1}^{\,2g}\hat{\beta}, \sqrt{-1}^{\,3g}\hat{\gamma}), $$ i.e. when $g$ is even, $\hat{R}^i_g=R^i_g$ and when $g$ is odd, $\hat{R}^i_g$ is obtained from $R^i_g$ by changing the sign of the homogeneous components of degrees $\deg(q^i_g)-4-8j$, $j\geq 0$. \end{defn} Thus we expect that $Q_g^i=\hat{R}_g^i$, $i=1,2,3$ (i.e. $J_g=(\hat R_g^1,\hat{R}_g^2,\hat{R}_g^3)$), for $g \geq 1$ (see~\cite{Vafa}~\cite{Siebert}). Let us review the known cases. \begin{ex} \label{ex:16} For $g=1$, we set $r_1=-8$, so that $\hat{\alpha}=\a$, $\hat{\beta}=\b - 8$, $\hat{\gamma} =\gamma$. Then the ideal of relations is generated by $\hat{R}^1_1= \hat{\alpha}$, $\hat{R}^2_1=\hat{\beta}+ 8$ and $\hat{R}^3_1=\hat \gamma$. The correction $r_1=-8$ is arranged in such a way that things work. \end{ex} \begin{ex} \label{ex:17} For $g=2$, the quantum cohomology ring $QH^*(M_{\Sigma})$ has been computed by Donaldson~\cite{D1}, using an explicit description of $M_{\Sigma}$ as the intersection of two quadrics in ${\Bbb P}^5$. Let $h_2$, $h_4$ and $h_6$ be the integral generators of $QH^2(M_{\Sigma})$, $QH^4(M_{\Sigma})$ and $QH^6(M_{\Sigma})$, respectively. Then, with our notations, $\a=h_2$, $\b = -4 h_4$ and $\gamma = 4 h_6$ (see~\cite{Vafa}). The computations in~\cite{D1} yield $\hat{\gamma} =\gamma -4 \a$. Now we set $r_2=4$, i.e. $\hat{\beta}=\b + 4$. It is now easy to check that the relations found in~\cite{D1} can be translated to the relations $\hat{R}_2^1= \hat{\alpha}^2+\hat{\beta}-8$, $\hat{R}_2^2= (\hat{\beta}+8) \hat{\alpha} +\hat{\gamma}$ and $\hat{R}_2^3= \hat{\alpha}\hat{\gamma}$ for $QH_I^*(M_{\Sigma})$. The artificially introduced term $r_2=4$ is due to the same phenomenon which causes the failure of lemma~\ref{lem:10} for $g=2$, i.e. the quantum product $h^2$ differs from $\b$ in~\eqref{eqn:qu9} (defined with the cup product) because of a quantum correction in $QH^*(N)$ which appears when $g=2$. \end{ex} In the general case we have \begin{thm} \label{thm:main} Let $g \geq 3$. Put $r_g=0$, so that $\hat{\alpha}=\a$ and $\hat{\beta}=\b$. Then we can write $$ \left\{ \begin{array}{l} Q^1_g = \hat{R}^1_g +f^1_g \\ Q^2_g = \hat{R}^2_g +f^2_g \\ Q^3_g = \hat{R}^3_g +f^3_g \end{array} \right. $$ where $\deg(f^i_g) \leq \deg(q^i_g) -8$, $i=1,2,3$. \end{thm} \begin{pf} As in equation~\eqref{eqn:qu11}, we can write $\hat{R}_g^i= \sum_{j \geq 0} \hat{R}_{g,j}^i$, where $\deg(\hat{R}_{g,j}^i)=\deg(q_g^i)-4j$, $j \geq 0$ (and analogously for $R_g^i$). Clearly, $\hat{R}_{g,0}^i=q_g^i= Q_{g,0}^i$, for $i=1,2,3$. We want to check that $Q_{g,1}^i=\hat{R}_{g,1}^i= (-1)^g R_{g,1}^i$. Pick any $z= \a^a\b^b\psi_{i_1}\cdots\psi_{i_r} \in \AA(\S)$ of degree $6g-2-\deg(q^i_g)$. By theorem~\ref{thm:14}, $\phi^w(\S \times D^2,R^i_g)=0$ (see~\cite{Floer} for notations), so $D^{(w,\S)}_{S,H} (R^i_g z)=0$, i.e. $$ D^{{\Bbb P}^1}_{S,H} (R^i_{g,1} z) + D^{c_1}_{S,H} (R^i_{g,0} z)=0. $$ From theorem~\ref{thm:11} and remark~\ref{rem:13} this is translated as the component in $H^{6g-6}(M_{\Sigma})$ of the quantum product $-R^i_{g,1} z + (-1)^{g-1}R^i_{g,0} z \in QH^*(M_{\Sigma})$ vanishing. On the other hand, by definition $Q_g^i z =0 \in QH^*(M_{\Sigma})$, so the component in $H^{6g-6}(M_{\Sigma})$ of the quantum product $Q^i_{g,1} z + Q^i_{g,0} z \in QH^*(M_{\Sigma})$ is zero. Thus $<Q^i_{g,1}, z>= <(-1)^g R^i_{g,1}, z>$, for any $z$ of degree $6g-2-\deg(q^i_g)$, and hence $Q^i_{g,1} \equiv (-1)^g R^i_{g,1} \pmod{I_g}$ ($I_g$ is the ideal defined in section~\ref{sec:ordinary}). This gives the required equality $Q_{g,1}^i=\hat{R}_{g,1}^i$ (in the case $i=3$ we have to use the vanishing of the coefficient of $\hat{\alpha}^g$ for both $Q_g^3$ and $R_g^3$). \end{pf} \section{The case of genus $g=3$} \label{sec:g3} It is natural to ask to what extent the first quantum correction determine the full structure of the quantum cohomology of $M_{\Sigma}$. In~\cite{D1}, Donaldson finds the first quantum correction for $M_{\Sigma}$ when the genus of $\S$ is $g=2$ and proves that this determines the quantum product. In this section we are going to check that this also happens for $g=3$, finding thus the quantum cohomology of the moduli space of stable bundles over a Riemann surface of genus $g=3$. \begin{prop} \label{prop:19} Let $\S$ have genus $g=3$. Then $Q^1_3=\hat{R}^1_3$, $Q^2_3=\hat{R}^2_3$ and $Q^3_3=\hat{R}^3_3$, i.e. $$ QH^*_I(M_{\Sigma}) = {\Bbb Q}[\hat{\alpha},\hat{\beta},\hat{\gamma}]/(\hat{R}^1_3, \hat{R}^2_3, \hat{R}^3_3). $$ \end{prop} \begin{pf} Theorem~\ref{thm:main} says that $$ \left\{ \begin{array}{l} Q^1_g = \hat{R}^1_g = \hat{\alpha}(\hat{\alpha}^2 +\hat{\beta} +8) + 4(\hat{\alpha}\hat{\beta} -8\hat{\alpha} +\hat{\gamma}) \\ Q^2_g = \hat{R}^2_g +x = (\hat{\beta}+8)(\hat{\alpha}^2 +\hat{\beta} +8) +{4 \over 3}\hat{\alpha}\hat{\gamma} +x \\ Q^3_g = \hat{R}^3_g +y \hat{\alpha} = \hat{\gamma} (\hat{\alpha}^2 +\hat{\beta} +8)+y \hat{\alpha} \end{array} \right. $$ where $x, y \in {\Bbb Q}$. The main tool that we shall use is the nilpotency of $\hat{\gamma}$. Actually, from its definition~\eqref{eqn:qu12}, $\hat{\gamma}^4=0$ in $QH^*(M_{\Sigma})$. Suppose $y \neq 0$. Then the third relation implies $\hat{\alpha}^4=0$. Multiplying the first relation by $\hat{\alpha}$ we get $5\hat{\alpha}^2\hat{\beta} + 24\hat{\alpha}^2 +4 \hat{\alpha}\hat{\gamma}=0$. This is a relation of degree $2g+2=8$. Thus it must be a multiple of $Q^2_g$, which is not. This contradiction implies $y=0$. Let us see $x =0$. It is easy to check that $$ \hat{\gamma}^3= {\hat{\gamma}^2 \over 4} \hat{R}^1_3 -{3\hat{\gamma}(\hat{\beta}-8) \over 4}\hat{R}^2_3 + {3(\hat{\beta}+8)(\hat{\beta}-8) -\hat{\alpha}\hat{\gamma} \over 4} \hat{R}^3_3. $$ The relations above imply that $\hat{\gamma}^3= -{3 \over 4} \hat{\gamma}(\hat{\beta}-8) x$ in $QH^*(M_{\Sigma})$. So $\hat{\gamma}^2(\hat{\beta}-8)=0$. Also, the third relation (with $y=0$) gives $\hat{\gamma}\hat{\alpha}^2 = -\hat{\gamma}(\hat{\beta}+8)$. Thus $\hat{\gamma}^2\hat{\alpha}^2=-16 \hat{\gamma}^2$ and $\hat{\gamma}^2\hat{\beta}= 8\hat{\gamma}^2$. Multiply the first relation by $\hat{\gamma}^2$ to get $\hat{\gamma}^3=0$. As $\hat{\gamma}(\hat{\beta}-8) \neq 0$ (because this is not a multiple of the only relation of degree $2g=6$), it must be $x=0$. \end{pf} \begin{cor} \label{cor:20} Let $\S$ be a Riemann surface of genus $g=3$. Then $$ QH^*(M_{\Sigma})= \bigoplus_{k=0}^{g-1} \L_0^k H^3 \otimes {\Bbb Q} [\hat{\alpha}, \hat{\beta}, \hat{\gamma}]/\hat{I}_{g-k} $$ where we put $\hat{I}_r=( R^1_r(\sqrt{-1}^{\,g}\hat{\alpha},\sqrt{-1}^{\,2g}\hat{\beta},\sqrt{-1}^{\,3g}\hat{\gamma}), R^2_r(\sqrt{-1}^{\,g}\hat{\alpha},\sqrt{-1}^{\,2g}\hat{\beta},\sqrt{-1}^{\,3g}\hat{\gamma}), \allowbreak R^3_r(\sqrt{-1}^{\,g}\hat{\alpha},\sqrt{-1}^{\,2g}\hat{\beta},\sqrt{-1}^{\,3g}\hat{\gamma}))$, $1 \leq r \leq g$. \end{cor} \begin{pf} This is an easy consequence of the former proposition and~\cite[lemma 7]{Floer}, noting that the hypothesis only has to be checked up to $g=3$. One has to be careful with the exponents of $\sqrt{-1}$ everywhere. \end{pf}

9711

alg-geom/9711014

en

https://arxiv.org/abs/alg-geom/9711014

alg-geom/9711014

Parusinski Adam

Adam Parusinski

Topological Triviality of $\mu$-constant Deformations of Type f(x) + tg(x)

AMS-LaTeX, 7 pages

Universite d'Angers prepublication no. 48

We show that every $\mu$-constant family of isolated hypersurface singularities of type f(x) + tg(x), where t is a parameter, is topologically trivial. In the proof we construct explicitely a vector field trivializing the family. The proof uses only the curve selection lemma and hence, for an appropriately translated statement, also works over the reals. Some applications to the study of singularities at infinity of complex polynomials are given.

alg-geom

\section{{\L}ojasiewicz-type Inequalities}\label{Loj} \bigskip \begin{thm}\label{loj} Let $F:(\mathbb{K}^n,0) \to (\mathbb{K},0)$, $g:(\mathbb{K}^n,0) \to (\mathbb{K},0)$ be two germs of $\mathbb{K}$-analytic functions. Then there exists a real constant $C$ such that for $p\in F ^{-1} (0)$ and sufficiently close to the origin \begin{equation} |g(p)| \le C |p| \inf_{\eta \in \K} ( |\eta \grad F(p) + \grad g(p) |) . \end{equation} \end{thm}\medskip \begin{proof} Suppose that this is not the case. Then, by the curve selection lemma, there exist a real analytic curve $p(s)$ and a function $\eta (s)$, $s\in [0, \varepsilon)$, such that \item {(a)} $p(0)=0$; \item {(b)} $F(p(s)) \equiv 0$, and hence $dF(p(s)) \Dot p(s) \equiv 0$; \begin{equation*} |g(p(s))| \gg |p(s)| |\eta (s) \grad F(p(s)) + \grad g(p(s)) |, \qquad \text { as } \quad s\to 0 . \tag {c} \end{equation*} Since $p(0)=0$ and $g(p(0))=0$ we have asymptotically as $s\to 0$ \begin{equation*} s |\Dot p(s)| \sim | p(s)| \qquad \text {and } \quad s \frac d {ds} g(p(s)) \sim g(p(s)) . \end{equation*} But \begin{equation*} \frac d {ds} g(p(s)) = dg(p(s)) \Dot p(s) = (\eta(s) dF + dg) (p(s)) \Dot p(s) . \end{equation*} Hence \begin{equation*} |g(p(s))| \sim | s \frac d {ds} g(p(s))| \lesssim s |\Dot p(s)| |\eta(s) \grad F(p(s)) + \grad g(p(s)) |, \end{equation*} which contradicts (c). This ends the proof. \end{proof} \bigskip \begin{rem} By standard argument, based again on the curve selection lemma, (1.1) implies that in a neighbourhood of the origin \begin{equation} |g(p)| \ll \inf_{\eta \in \K} ( |\eta \grad F(p) + \grad g(p) |) \qquad \text {as } p \to g^{-1} (0). \end{equation} Hence, by \cite{loj}, there exists a constant $\alpha>0$ such that \begin{equation} |g(p)| \le |g(p)|^\alpha \inf_{\eta \in \K} ( |\eta \grad F(p) + \grad g(p) |) \end{equation} which is a variation of the second {\L}ojasiewicz Inequality \cite{loj}[$\S$18]. \end{rem} \bigskip \begin{cor}\label{ag} Let $F(x,t)= f(x) +t g(x)$, where $f,g:(\mathbb{K}^n,0) \to (\mathbb{K},0)$ are germs of $\mathbb{K}$-analytic functions and $t\in \mathbb{K}$. Then the following conditions are equivalent: \medskip \item {(i)} \centerline {$|g(x)| \ll |\grad F(x,t)| \qquad \text {as } \quad (x,t) \to (0,0) .$} \smallskip \item {(ii)} \centerline {$\quad |g(x)| \ll \inf_\lambda \{|\grad F(x,t)+\lambda \grad g(x) |\}$ } as $(x,t)\in F^{-1} (0)$ and $(x,t) \to (0,0)$. \end{cor}\medskip \begin{proof} (i) $\Longrightarrow$ (ii) follows immediately from theorem \ref{loj}. Indeed, $\grad F(x,t) = (\gradx f +t \gradx g, g)$, so (i) is equivalent to $|g(x)| \ll |\gradx f +t \gradx g|$ as $(x,t) \to (0,0)$. Hence $|g(x)| \ll |\grad F(x,t) + \lambda \grad g(x) |$ for $\lambda$ small enough, say $\lambda < \varepsilon$. For $\lambda \ge \varepsilon$ the required inequality follows from theorem \ref{loj} by setting $\mu = \lambda^{-1}$. Suppose now that (i) fails. Then there exists a real analytic curve $(x(s), t(s))\to (0,0)$ such that $|g(x(s))| \ge C |\grad F(x(s),t(s))| \gg |F(x(s),t(s))|$. This follows $f(x(s))/g(x(s)) \to 0$ as $s\to 0$. Set $\tilde t(s) = - f(x(s))/g(x(s))$. Then $(x(s), \tilde t(s))\to (0,0)$, $F(x(s), \tilde t(s))\equiv 0$ and $g(x(s)) \ge C |\grad F(x,t)+\lambda (s) \grad g(x)|$ for $\lambda (s) = t(s) - \tilde t(s)$, which contradicts (ii). \end{proof} \bigskip \begin{rem} (ii) of corollary \ref{ag} is equivalent to the following: suppose that $(x,t)\in F^{-1} (0)$ and $(x,t) \to (0,0)$ then \begin{equation*} \varphi(x) = |g(x)|/ (\inf_\lambda \{|\gradx f(x) + \lambda \gradx g(x)|\}) \to 0 . \end{equation*} Note that, though $\varphi(x)$ does not depend on $t$, the existence of $t=t_x$ such that $(x,t)\in F^{-1} (0)$, $(x,t) \to (0,0)$, gives a nontrivial condition on $x$. If $\gradx f$ and $\gradx g$ are independent then \begin{equation} \inf_\lambda \{|\gradx f(x) + \lambda \gradx g(x)|^2 = \frac {|\gradx f|^2 |\gradx g|^2 - |<\gradx f, \gradx g>|^2} {|\gradx g|^2}, \end{equation} and the minimum is attained at $\lambda = - \frac {<\gradx f, \gradx g>} {|\gradx g|^2}$. Hence (ii) of corollary \ref{ag} is equivalent to \begin{equation} \frac {|g(x)|^2 |\gradx g|^2} {\Delta } \to 0 , \end{equation} where $\Delta = |\gradx f|^2 |\gradx g|^2 - |<\gradx g,\gradx f>|^2$. \end{rem} \bigskip \bigskip \section{Topological Triviality}\label{Triv}\bigskip \begin{proof} [Proof of theorem \ref{triv}] First we show how to trivialize the zero set $X = F^{-1} (0)$. For this we construct a vector field $\v$ on $X$ whose $t$-coordinate is $\partial_t$ and which restricted to the t-axis is precisely $\partial_t$. Moreover, for the reason we explain below, we require that $\v$ is tangent to the levels of $g$. If one looks for such a vector field of the form \begin{equation*} \v = \partial_t + a \gradx f + b \gradx g \end{equation*} the condition of tangency to the levels of $F$ and $g$ impose that on $X'= X\setminus g^{-1} (0)$ \begin{equation} \v = \partial_t - {\frac {\bar g |\gradx g|^2} \Delta} \gradx f + {\frac {\bar g <\gradx f,\gradx g>} \Delta} \gradx g , \end{equation} where again $\Delta = |\gradx f|^2 |\gradx g|^2 - |<\gradx g,\gradx f>|^2$. We claim that \begin{equation} |\v - \partial_t| \to 0 \qquad \text {as } \quad (x,t) \to g^{-1} (0) , F(x,t)=0 . \end{equation} It suffices to show it on any real analytic curve $(x(s),t(s))\to (x_0,t_0) \in g^{-1} (0)\cap F^{-1}(0)$. We may assume $(x_0,t_0) =(0,0)$. Since $|\v - \partial_t|^2 = \frac {|g(x)|^2 |\gradx g|^2} {\Delta }$, (2.2) follows from Corollary \ref{ag} and (1.5). To show that the flow of $\v$ is continuous we use $g$ as a control function. First note that, by (1.4), $X' =X \setminus g^{-1} (0)$ is nonsingular. By (1.8), $\{\Delta =0\} \subset \{g=0\}$, and hence $\v$ is smooth on $X'$. Moreover the integral curves of $\v|_{X'}$ cannot fall on $Y=X\cap g^{-1} (0)$ since $\v$ is tangent to the levels of $g$. Extend $\v$ on $Y$ by setting $\v|_Y \equiv \partial_t$. Let $\Phi(p,s)$ be the flow of $\v$ and let $p_0 = (x_0,t_0)\in Y$, $p_1 = (x_1,t_1) \in X'$. By (2.2) \begin{equation} |\v(\Phi (p_0,s)) - \v(\Phi (p_1,s))| \le \varepsilon (g(\Phi (p_1,s))) = \varepsilon (g(p_1)) , \end{equation} and $\varepsilon (\eta)\to 0$ as $\eta \to 0$. Hence \begin{equation} |(\Phi (p_0,s)) - (\Phi (p_1,s))| \le |p_0-p_1| + |s| \varepsilon (g(p_1)) . \end{equation} This shows the continuity of $\Phi$ and ends the first part of the proof of theorem, the triviality of $X$ along the $t$-axis. To trivialize $F$ as a function we use the Kuo vector field \cite{kuo} \begin{equation} \Vec {\mathbf{w}} = \partial_t - {\frac {\bar g} {|\gradx F|^2} } \gradx F . \end{equation} $\Vec {\mathbf{w}}$ is tangent to the levels of $F$ and hence, by (0.1), one may show that the flow of $\Vec {\mathbf{w}}$ is continuous in the complement of $X=F^{-1} (0)$. Unfortunately this argument does not work on $X$ and the integral curves of $\Vec {\mathbf{w}}|_X$ may disappear at the singular locus of $F$. To overcome this difficulty we "glue" $\v$ and $\Vec {\mathbf{w}}$. Fix a sufficiently small neighbourhood $\mathcal U$ af the origin in $\mathbb{K}^n \times \mathbb{K}$. Let $\mathcal V_1$ be a neighbourhood of $X'=X\setminus Y$ in $\mathcal U \setminus Y$ such that (1.5) still holds on $\mathcal V_1$. Fix a partition of unity $\rho_1, \rho_2$ subordonated to the covering $\mathcal V_1,\mathcal V _2 = \mathcal U\setminus X$ of $\mathcal U \setminus Y$ and put \begin{equation*} \Vec {\mathbf{V}} = \rho_1 \v + \rho_2 \Vec {\mathbf{w}} . \end{equation*} Clearly $\Vec {\mathbf{V}}$ is tangent to the levels of $F$ on $\mathcal U \setminus Y$ and to the levels of $g$ on $X \setminus Y$. Hence its flow is continuous on $\mathcal U \setminus Y$. Moreover \begin{equation*} |\Vec {\mathbf{V}} - \partial_t| \le \rho_1 |\v- \partial_t| + \rho_2 |\Vec {\mathbf{w}}- \partial_t| , \end{equation*} so it goes to $0$ as $(x,t)\to Y$. Now the continuity of the flow of $\Vec {\mathbf{V}}$ follows by the same argument as the continuity of the flow of $\v$. This ends the proof of theorem. \end{proof} \medskip \begin{cor}\label{mu-constant} Let $f,g:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ be such that $F(x,t)= f(x) +t g(x)$ is a $\mu$-constant family of isolated singularites. Then $F$ is topologically trivial. \end{cor}\smallskip \begin{proof} Let $L$ denote the t-axis. By \cite{le-saito} $\mu$-constancy is equivalent to \begin{equation} |\partial_t F| \ll |\grad F(x,t)| , \end{equation} as $(x,t) \to L $. Hence, by theorem \ref {triv} it suffices to show (2.6) for $(x,t)\to (x_0,t_0)$, where $g(x_0)=0$ and $x_0\ne 0$. But this is obvious since, by our hypothesis, $\gradx F(x_0,t_0) \ne 0$. \end{proof} \bigskip \begin{rem} Suppose that $F(x,t)= f(x) +t g(x)$ is a family of isolated singularities and that in a neighbourhood of the origin \begin{equation} |g(x)| \le C |x| |\grad F(x,t)| . \end{equation} Then, as shown in \cite{kuo}, the vector field $\Vec {\mathbf{w}}$ of (2.5) trivializes $F$ along t-axis. The flow of $\Vec {\mathbf{w}}$ is continuous since $\rho (x,t) = |x|$ changes slowly along the trajectories of $\Vec {\mathbf{w}}$. That is, for a trajectory $(x(s), s)$ \begin{equation} |\frac {d\rho }{ds}| \le C \rho , \end{equation} which implies \begin{equation} \rho (x(0),0) e^{-C|s|} \le |\rho (x(s),s)| \le \rho (x(0),0) e^{C|s|}, \end{equation} see \cite{kuo} for the details. It is not difficult to see that (2.8) and (2.9) are also satisfied on the trajectories of vector field $\v$ of (2.1), provided (2.7) holds. Indeed, the proof of theorem \ref{loj} actually gives \begin{equation} |g(p)| \le C \inf_{\eta \in \K} (|x| |\eta \gradx F(p) +\grad g(p)| + |t|| \eta \partial F/\partial t (p)|) . \end{equation} This allows one to show that (2.7) is equivalent to \begin{equation*} |g(x)| \le C |x| \inf_\lambda \{|\grad F(x,t)+\lambda \grad g(x) |\} \quad \text { on } F^{-1} (0). \end{equation*} Hence $|\v(x,t) - \partial_t| \le C |x|$ which easily implies (2.8) and (2.9). \end{rem} \vskip20pt \bigskip \section{Singularities at infinity of complex polynomials}\label{infty} \bigskip Let $f(x_1,\ldots ,x_n)$ be a complex polynomial of degree $d$. Let $\tilde f(x_0, x_1, \ldots, x_n)$ denote the homogenization of $f$. Set \begin{equation*} X = \{ (x,t)\in \mathbb P ^n \times \mathbb{C} | \, F(x,t) = \widetilde f(x) - tx_0^d =0\} \end{equation*} and let $\overline f:X\to \mathbb{C}$ be induced by the projection on the second factor. Then $\overline f$ is the family of projective closures of fibres of $f$. Let $H_{\infty}= \{x_0=0\} \subset \mathbb P ^n$ be the hyperplane at infinity and let $X_{\infty}=X\cap (H_\infty \times \mathbb{C})$. $\overline f$ can be used to trivialize a non-proper function $f$ as follows. Fix $t_0\in \mathbb{C}$. If $\overline f$ is topologically trivial over a neighbourhood of $t_0$ and the trivialization preserves $X_{\infty}$, then clearly $f$ is topologically trivial over the same neighbourhood of $t_0$. Thus one may use Stratification Theory in order to trivialize $f$. On the other hand, by a simple argument, see for instance \cite{parus1}, the following condition on the asymptotic behaviour of $\grad f$ at infinity assures the topological triviality of $f$ over a neighbourhood of $t_0$: $\grad f$ does not vanish on $f^{-1} (t_0)$ and \begin{equation} |x| |\grad f(x)| \ge \delta > 0 , \end{equation} for all $x$ such that $|x|\to \infty$ and $f(x) \to t_0$. Following \cite{pham} the condition (3.1) is called Malgrange's Condition. These two approaches are related by the following result. \bigskip \begin{thm}\label{banach} \cite{parus2, tibar} \par Let $p_0\in X_\infty$, $t_0=f(p_0)$. Then the following conditions are equivalent: \item {(i)} $\overline f$ has no vanishing cycles in a neighbourhood of $p_0$. \item {(ii)} (3.1) holds as $(x,f(x))\to p_0$ in $\mathbb P^n\times \mathbb{C}$. \end{thm} We show below that the proofs of theorem \ref{banach} given in \cite{parus2, tibar} can be considerably simplified using a version of corollary \ref{ag}. For this we first sketch the main steps of the proof of \cite{parus2}. We work locally at $p_0\in X_\infty$. We may assume that $p_0 = ((0:0:\ldots :0:1),0) \in \mathbb P ^n\times \mathbb{C} $, so that $y_0 = x_n^{-1}$, $y_i = x_i/ x_n$ for $i=1, \ldots, n-1$, and $t$, form a local system of coordinates at $p_0$. In this new coordinate system $X$ is defined by \begin{equation*} F(y_0, y_1, \ldots, y_{n-1},t) = \tilde f(y_0, y_1, \ldots, y_{n-1},1) - t y_0^d = 0 . \end{equation*} It is easy to see that (ii) of theorem \ref{banach} is equivalent to \begin{equation} |\partial F/\partial t (p)| \le C |\partial F/\partial y_1, \ldots, \partial F/\partial y_{n-1})(p)| \, , \end{equation} for $p\in X$ and close to $p_0$. On the other hand, less trivial but standard arguments, see e.g. \cite{parus2}[Definition-Proposition 1], show that (i) of theorem \ref{banach} is equivalent to the fact that $(p_0,dt)$ does not belong to the characteristic variety of $X$, which is equivalent to \begin{equation} |\partial F/\partial t (p)| \le C | \partial F/\partial y_0, \ldots, \partial F/\partial y_{n-1})(p)| \, , \end{equation} in a neighbourhood of $p_0$. The proofs of (3.2) $\Longleftrightarrow$ (3.3) given in \cite{parus2,tibar} are based on the results of \cite{BMM}. But it may be easily shown using theorem \ref{loj}. Indeed, take $g(y) = y_0^d$. Then \begin{equation*} \inf_\lambda \{|\grad F(y,t)+\lambda \grad g(y) |\} = |\partial F/\partial y_1, \ldots, \partial F/\partial y_{n-1})(p)| , \end{equation*} and (3.2) $\Longleftrightarrow$ (3.3) is a variation of corollary \ref{ag} with the same proof. It seems to be particularly surprising that unlike the previous proofs the one presented above does not use the complex structure at all. \bigskip

9711

alg-geom/9711008

en

https://arxiv.org/abs/alg-geom/9711008

alg-geom/9711008

Victor V. Batyrev

Stringy Hodge numbers of varieties with Gorenstein canonical singularities

26 pages, AMSLaTeX, to appear in the Proceedings of Taniguchi Symposium 1997,"Integrable Systems and Algebraic Geometry, Kobe/Kyoto"

We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we propose a general method to define stringy Hodge numbers for projective algebraic varieties with at worst Gorenstein canonical singularities. This allows us to formulate the topological mirror duality test for arbitrary Calabi-Yau varieties with canonical singularities. In Appendix we explain non-Archimedian integrals over spaces of arcs. We need these integrals for the proof of the main technical statement used in the definition of stringy Hodge numbers.

alg-geom

\section{Introduction} Topological field theories associated with Calabi-Yau manifolds predict the following Hodge-theoretic property of mirror manifolds \cite{W}: \bigskip {\em If two smooth $d$-dimensional Calabi-Yau manifolds $V$ and $V^*$ form a mirror pair, then their Hodge numbers must satisfy the relations \begin{equation} h^{p,q}(V) = h^{d-p,q}(V^*), \;\; 0 \leq p, q \leq d. \label{d-rel} \end{equation} In particular, one has the following equality for the Euler numbers \begin{equation} e(V) = (-1)^d e(V^*). \end{equation} } The relation (\ref{d-rel}) is known as a {\em topological mirror symmetry test} \cite{M}. The physicists' discovery of mirror symmetry has been supported by a lot of examples of mirror pairs $(V,V^*)$ consisting of Calabi-Yau varieties with Gorenstein abelian quotient singularities \cite{GP}. Some further examples of mirror pairs $(V,V^*)$ consisting of Calabi-Yau varieties with Gorenstein toroidal singularities have been proposed in \cite{BA,BS,BB0,LB}. We want to stress that a mathematical formulation of the {topological mirror symmetry test} for Calabi-Yau varieties with singularities is a priori not clear, because (\ref{d-rel}) does not hold for usual Hodge numbers of $V$ and $V^*$ if at least one of these Calabi-Yau varieties is not smooth. Let us illustrate this problem using some simplest examples: \begin{exam} {\rm Let $V \subset {\Bbb P}^4$ be a Fermat quintic 3-fold defined by the equation $f(z)= \sum_{i =0}^4 z_i^5 =0$. Physicists predict that the mirror of $V$ is isomorphic to the quotient $V^*= V/G$, where $G \cong ({\Bbb Z}/5Z)^3$ is maximal diagonal subgroup of $PSL(5,{\Bbb C})$ which acts trivially on the polynomial $f$. It is easy to check that \[ 1= h^{2,1}(V^*) = h^{2,1}(V^*) = h^{1,1}(V)= h^{2,2}(V). \] However one has \[ 1 = h^{1,1}(V^*) = h^{2,2}(V^*) \neq h^{2,1}(V) = h^{2,1}(V) =101. \] In order to obtain the required duality (\ref{d-rel}), one needs to construct a smooth Calabi-Yau $3$-fold ${\widehat{V}}^*$ by resolving singularities of $V^*$. It is possible to show that the Hodge numbers $h^{1,1}({\widehat{V}}^*)= h^{2,2}({\widehat{V}}^*)= 101$ do not depend on the choice of such a desingularization (see \cite{B} for more general result).} \end{exam} \begin{exam} {\rm Let $V \subset {\Bbb P}^5$ be a Fermat octic 4-fold defined by the equation $f(z)= z_0^4 + z_1^4 + z_2^8 + z_3^8 + z_4^8 + z_5^8 =0$. Then physicists predict that the mirror of $V$ is isomorphic to the quotient $V^*= V/G$, where $G$ is maximal diagonal subgroup of $PSL(6,{\Bbb C})$ which acts trivially on the polynomial $f$. By resolving abelian quotient singularities on $V^*$, one obtains a smooth Calabi-Yau $4$-fold ${\widehat{V}}^*$ with the Hodge numbers \[ h^{1,1}({\widehat{V}}^*) = h^{3,3}({\widehat{V}}^*) = 433,\;\; h^{2,2}({\widehat{V}}^*) = 1820.\] On the other hand, it is easy to check that \[ h^{3,1}(V) = h^{1,3}(V) = 433, h^{2,2}(V) = 1816.\] We observe that $h^{2,2}({\widehat{V}}^*) - h^{2,2}(V) = 4 \neq 0$. The duality (\ref{d-rel}) fails, because $V$ has $4$ isolated singular points: $ z_0^4 + z_1^4 =0$, $z_2 = z_3=z_4 =z_5=0$. A small neigbourhood of each such a point is analytically isomorphic to ${\Bbb C}^4/\pm id$. Unfortunately, it is impossible to resolve these singularities of $V$ if one wants to keep the canonical class trivial.} \label{exam2} \end{exam} The last example shows that we need some new Hodge numbers, so called {\em string-theoretic Hodge numbers} (or simply {\em stringy Hodge numbers}) $h^{p,q}_{\rm st}(X)$ for singular Calabi-Yau varieties $X$, so that for each mirror pair $(V, V^*)$ of singular Calabi-Yau varieties we would have \begin{equation} h^{p,q}_{\rm st}(V) = h^{d-p,q}_{\rm st}(V^*), \;\; 0 \leq p, q \leq d. \label{d-rel2} \end{equation} (In particular, we would have $h^{2,2}_{\rm st}(V) = h^{2,2}(V) + 4 =1820$ in \ref{exam2}.) \bigskip Let us fix our terminology. In this paper we want to consider the following most general definition of Calabi-Yau varieties with singularities: \begin{dfn} {\rm A normal projective algebraic variety $X$ is called a {\bf Calabi-Yau variety} if $X$ has at worst Gorenstein canonical singularities and the canonical line bundle on $X$ is trivial. } \label{d-cy} \end{dfn} \begin{rem} {\rm One usually requires that the cohomologies of the structure sheaf ${\cal O}_X$ of a Calabi-Yau variety $X$ satisfy some additional vanishing conditions \[ h^i(X, {\cal O}_X) = 0, \;\;\;\; 0 < i < d= dim\,X. \] However, in the connection with the mirror symmetry, we don't want to restrict ourselves to these additional assumptions. } \end{rem} So far (see \cite{BD,BB}), our approach to the topological mirror symmetry test was restricted to the following steps: First, one chooses a class of ``allowed'' Gorenstein canonical singularities on Calabi-Yau varieties. Second, one defines the notion of {\em stringy Hodge numbers} $h^{p,q}_{\rm st}$ for Calabi-Yau with the allowed singularities. Third, one proves (\ref{d-rel2}) for already known examples mirror pairs $(V,V^*)$ of Calabi-Yau varieties with the allowed singularities. For instance, a definition of stringy Hodge numbers for projective algebraic varieties with Gorenstein quotient singularities or with Gorenstein toroidal singularities has been introduced in \cite{BD}, and all steps of the above program has been succesfully accomplished for Calabi-Yau complete intersections in toric varieties in \cite{BB}. One could naturally ask whether there exists a way to define stringy Hodge numbers for projective algebraic varieties with {\em arbitrary} Gorenstein canonical singularities. One of purposes in this paper is to show that in general the answer is ``no''. Nevertheless, there always exists a way to formulate the topological mirror symmetry test for Calabi-Yau varieties with arbitrary Gorenstein canonical singularities. Our main idea consists of an association with every projective variety $X$ with at worst Gorenstein singularities a rational function $E_{\rm st}(X; u,v) \in {\Bbb Z}[[u,v]] \cap {\Bbb Q}(u,v)$, which we call {\em stringy $E$-function}. If $E_{\rm st}(X; u,v)$ is a polynomial, then the coefficients of this polynomial allow us to define the stringy Hodge numbers $h^{p,q}_{\rm st} (X)$ by the formula \[ E_{\rm st}(X; u,v) = \sum_{p,q} (-1)^{p+q} h^{p,q}_{\rm st}(X)u^p v^q. \] If $E_{\rm st}(X; u,v)$ is not a polynomial, then we can't define stringy Hodge numbers of $X$ and say that they {\em do not exist}. However, even in this bad situation we could think about $E_{\rm st}(X; u,v)$ as if it were a generating function for Hodge numbers of a smooth projective manifold, i.e., as \[ \sum_{p,q} (-1)^{p+q} h^{p,q}u^p v^q. \] In particular, we shall see that the function $E_{\rm st}(X; u,v)$ always satisfies the following Poincar\'e duality: \[ E_{\rm st}(X; u,v)= (uv)^d E_{\rm st}(X; u^{-1},v^{-1}). \] For this reason, one can always formulate the topological mirror symmetry test for a mirror pair $(V,V^*)$ of $d$-dimensional Calabi-Yau varieties with arbitrary singularities as the equality \[ E_{\rm st}(V; u,v)= (-u)^{d}E_{\rm st}(V^*; u^{-1},v), \] whenever the stringy Hodge numbers of $V$ and $V^*$ exist or not. This is exactly the formula which has been proved for Calabi-Yau complete intersections in toric varieties \cite{BB}. The paper is organized as follows. In section 3 we introduce stringy $E$-functions and discuss their properties. In section 4 we compute these functions for toric varieties. In section 5 we give further examples of $E$-functions and formulate some natural open questions. In Appendix we explain the proof of the main technical statement \ref{key} which rely upon a non-Archimedian integration over the space of arcs. This part of the paper is strongly influenced by Kontsevich's idea of {\em motivic integration} \cite{K} and a recent paper of Denef and Loeser \cite{DL1}. \bigskip \bigskip \noindent {\bf Acknowledgements:} It is my pleasure to thank Professors Yujiro Kawamata, Maxim Kontsevich, Shigefumi Mori, Miles Reid, Yuji Shimizu, Joseph Steenbrink and Duco van Straten for useful discussions concerning this topic and the Taniguchi Foundation for its financial support. \newpage \section{Basic definitions and notations} Let $X$ be a normal irreducible algebraic variety of dimension $d$ over ${\Bbb C}$, $Z_{d-1}(X)$ the group of Weil divisors on $X$, ${\rm Div }(X) \subset Z_{d-1}(X) $ the subgroup of Cartier divisors on $X$, $Z_{d-1}(X) \otimes {\Bbb Q}$ the group of Weil divisors on $X$ with coefficients in ${\Bbb Q}$, $K_X \in Z_{d-1}(X)$ a canonical divisor of $X$. We denote by $b_i(X)$ the rank of the $i$-th cohomology group with compact supports $H^i_c(X, {\Bbb Q})$. By the {\bf usual Euler number} of $X$ we always mean $$ e(X) = \sum_{i \geq 0} (-1)^i b_i(X). $$ The groups $H^i_c(X, {\Bbb C})$, $( 0 \leq i \leq 2d)$ carry a natural mixed Hodge structure \cite{P-D}. We denote by $h^{p,q}\left(H^i_c(X, {\Bbb C})\right)$ the dimension of the $(p,q)$-type Hodge component in $H^i_c(X, {\Bbb C})$ and define the {\bf $E$-polynomial} by the formula \[ E(X; u,v) := \sum_{p,q} e^{p,q}(X) u^p v^q, \] where \[ e^{p,q}(X): = \sum_{i \geq 0} (-1)^i h^{p,q}\left(H^i_c(X, {\Bbb C})\right). \] In particular, one has $e(X) = E(X;1,1)$. \begin{rem} {\rm For our purpose, it will be very important that $E$-polynomials have properties which are very similar to ones of the usual Euler characteristic: (i) if $X = X_1 \cup \cdots \cup X_k$ is a disjoint union of Zariski locally closed strata $X_1, \ldots, X_k$, then \[ E(X; u,v) = \sum_{i=1}^k E(X_i; u,v); \] (ii) if $X = X_1 \times X_2$ is a product of two algebraic varieties $X_1$ and $X_2$, then \[ E(X; u,v) = E(X_1; u,v) \cdot E(X_2; u,v); \] (iii) if $X$ admits a Zariski locally trivial fibering over $Z$ such that each fiber of $f\,: \, X \to Z$ is isomorphic to affine space ${{\Bbb C}}^n$, then \[ E(X; u,v) = E({{\Bbb C}}^n; u,v) \cdot E(Z; u,v)= (uv)^n E(Z; u,v). \] } \label{e-poly} \end{rem} Recall the following fundamental definition from the Minimal Model Program \cite{KMM}: \begin{dfn} {\rm A variety $X$ is said to have at worst {\bf log-terminal} (resp. {\bf canonical }) singularities, if the following conditions are satisfied: (i) $K_X$ is an element of ${\rm Div}(X) \otimes {\Bbb Q}$ (i.e. $X$ is ${\Bbb Q}$-Gorenstein). (ii) for a resolution of singularities $\rho \,: \, Y \rightarrow X$ such that the exceptional locus of $\rho$ is a divisor $D$ whose irreducible components $D_1, \ldots, D_r$ are smooth divisors with only normal crossings, we have \[ K_{Y} = \rho^* K_X + \sum_{i=1}^r a_i D_i \] with $a_i > -1 $ (resp. $a_i \geq 0)$ for all $i$, where $D_i$ runs over all irreducible components of $D$. } \label{log-t} \end{dfn} \section{Stringy Hodge numbers} \begin{dfn} {\rm Let $X$ be a normal irreducible algebraic variety with at worst log-terminal singularities, $\rho\, : \, Y \rightarrow X$ a resolution of singularities as in \ref{log-t}, $D_1, \ldots, D_r$ irreducible components of the exceptional locus, and $I: = \{1, \ldots, r\}$. For any subset $J \subset I$ we set \[ D_J := \left\{ \begin{array}{ll} \bigcap_{ j \in J} D_j & \mbox{\rm if $J \neq \emptyset$} \\ Y & \mbox{\rm if $J = \emptyset$} \end{array} \right. \;\;\;\; \;\;\;\; \,\mbox{\rm and} \;\;\;\; \;\;\;\; D_J^{\circ} := D_J \setminus \bigcup_{ j \in\, I \setminus J} D_j. \] We define an algebraic function $E_{\rm st}(X; u,v)$ in two variables $u$ and $v$ as follows: \[ E_{\rm st}(X; u,v) := \sum_{J \subset I} E(D_J^{\circ}; u,v) \prod_{j \in J} \frac{uv-1}{(uv)^{a_j +1} -1} \] (it is assumed $\prod_{j \in J}$ to be $1$, if $J = \emptyset$). We call $E_{\rm st}(X; u,v)$ {\bf stringy $E$-function of} $X$. } \label{def-main} \end{dfn} \begin{rem} {\rm Since all $a_i \in {\Bbb Q}$, but not necessary $a_i \in {\Bbb Z}$, $E_{\rm st}(X; u,v)$ {\bf is not a rational} function in $u,v$ in general. On the other hand, if all singularities of $X$ are Gorenstein, then the numbers $a_1, \ldots, a_r$ are nonnegative integers, hence, $E_{\rm st}(X; u,v)$ is a {\bf rational function}. Moreover, it is easy to see that $E_{\rm st}(X; u,v)$ is an element of ${\Bbb Z}[[u,v]] \cap {\Bbb Q}(u,v)$. } \end{rem} \begin{dfn} {\rm In the above situation, we call the rational number \[ e_{\rm st}(X) : = \lim_{u,v \to 1} E_{\rm st} (X; u,v) = \sum_{J \subset I} e(D_J^{\circ}) \prod_{j \in J} \frac{1}{a_j +1} \] the {\bf stringy Euler number} of $X$.} \end{dfn} For $E_{\rm st}(X; u,v)$ and $e_{\rm st}(X)$ to be well-defined we need the following main technical statement: \begin{theo} The function $E_{\rm st}(X; u,v)$ doesn't depend on the choice of a resolution $\rho\,: \, Y \rightarrow X$, if all irreducible components of the exceptional divisor $D$ are normal crossing divisors. Moreover, it is enough to demand the normal crossing condition only for those irreducible components $D_i$ of $D$ for which $a_i \neq 0$ $($see the formula in {\rm \ref{log-t}}$)$. \label{key} \end{theo} \begin{rem} {\rm A complete proof of this statement will be postponed to Appendix. Its main idea, due to Kontsevich, is based on an interpretation of the formula for $E_{\rm st}(X; u,v)$ as a some sort of a ``motivic non-Archimedian integral'' over the space of arcs $J_{\infty}(Y)$ which imitates properties of $p$-adic integrals over ${\Bbb Z}_p$-points of algebraic schemes over ${\Bbb Z}_p$ (see \cite{D1}). Some details of this idea can be found in papers of Denef and Loeser \cite{DL1,DL2} (see also \cite{D2,DL}). } \end{rem} \begin{coro} If $X$ is smooth, then $ E_{\rm st}(X; u,v) = E(X; u,v)$. \label{smooth} \end{coro} Our next statement shows that although $E_{\rm st} (X; u,v)$ is not necessary a polynomial, or even a rational function, it satisfies the Poincar\'e duality as well as usual $E$-polynomials of smooth projective varieties. \begin{theo} {\sc (Poincar\'e duality)} Let $X$ be a projective ${\bf Q}$-Gorenstein algebraic variety of dimension $d$ with at worst log-terminal singularities. Then $E_{\rm st} (X; u,v)$ has the following properties: {\rm (i) } \[ E_{\rm st} (X; u,v) = (uv)^d E_{\rm st} (X; u^{-1},v^{-1}); \] {\rm (ii)} \[ E_{\rm st} (X; 0, 0) = 1. \] \label{p-d} \end{theo} \noindent {\em Proof.} (i) Let $\rho\, : \, Y \rightarrow X$ be a resolution with the normal crossing exceptional divisors $D_1, \ldots, D_r$. Using the stratification \[ D_J = \bigcup_{J' \subset J} D_{J'}^{\circ}, \] we obtain \[ E (D_J; u,v) = \sum_{J' \subset J} E ( D_{J'}^{\circ}; u,v). \] Applying the last equality to each closed stratum $D_{J'}$, we come to the following formula \[ E (D_J^{\circ}; u,v) = \sum_{J' \subset J} (-1)^{|J| -|J'|} E(D_{J'}; u,v). \] Now we can rewrite the formula for $E(X; u,v)$ as follows \[ E_{\rm st} (X; u,v) = \sum_{J \subset I} \left(\sum_{J' \subset J} (-1)^{|J| -|J'|} E(D_{J'}; u,v) \right)\prod_{j \in J} \frac{uv-1}{(uv)^{a_j +1} -1} = \] \[ = \sum_{J \subset I} E(D_J; u,v) \prod_{j \in J} \left( \frac{uv -1}{(uv)^{a_j +1} -1} -1 \right) =\sum_{J \subset I} E(D_J; u,v) \prod_{j \in J} \left( \frac{uv -(uv)^{a_j + 1}}{(uv)^{a_j +1} -1} \right) . \] It remains to observe that each term of the last sum satisfies the required duality, because \[ (uv)^{|J|} \prod_{j \in J} \left( \frac{(uv)^{-1} -(uv)^{-a_j - 1}}{(uv)^{-a_j -1} -1} \right) = \prod_{j \in J} \left( \frac{uv -(uv)^{a_j + 1}}{(uv)^{a_j +1} -1} \right) \] and \[ (uv)^{d - |J|)}E(D_J; u^{-1},v^{-1}) = E(D_J; u,v) \] (the last equation follows from the Poincar{\'e} duality for smooth projective subvarieties $D_J \subset Y$ of dimension $d-|J|$). (ii) By substitution $u=v=0$ in the expression for $E_{\rm st}(X; u,v)$ obtained in the proof of (i), we get \[ E_{\rm st} (X; u,v) = E(Y; 0,0). \] It remains to use the equality $E(Y; 0,0) = 1$, which follows from the Poincar\'e duality for $Y$. \hfill $\Box$ \begin{dfn} {\rm Let $X$ be a projective algebraic variety with at worst Gorenstein canonical singularities. Assume that $E_{\rm st}(X; u,v) $ is a polynomial: $E_{\rm st}(X; u,v) = \sum_{p,q} a_{p,q} u^p v^q. $ Then we define the {\bf stringy Hodge numbers} of $X$ to be \[ h^{p,q}_{\rm st} (X): = (-1)^{p+q} a_{p,q}. \] } \end{dfn} \begin{rem} {\rm It follows immediately from \ref{p-d} that if $E_{\rm st}(X; u,v)$ is a polynomial, then the degree of $E_{\rm st}(X; u,v)$ is equal to $2d$. Moreover, in this case one has $h_{\rm st}^{0,0}(X) = h_{\rm st}^{d,d}(X) =1$ and $h^{p,q}_{\rm st}(X)$ are integers satisfying $h_{\rm st}^{p,q}(X) = h_{\rm st}^{q,p}(X)$ $\forall p,q$.} \end{rem} \begin{conj} Let $X$ be a projective algebraic variety with at worst Gorenstein canonical singularities. Assume that $E_{\rm st}(X; u,v) $ is a polynomial. Then all stringy Hodge numbers $h^{p,q}_{\rm st} (X)$ are nonnegative. \end{conj} Now we consider a geometric approach to a computation of stringy Hodge numbers. \begin{dfn} {\rm Let $X$ be a projective algebraic variety with at worst canonical Gorenstein singularities. A birational projective morphism $\rho\;: \; Y \rightarrow X$ is called a {\bf crepant desingularization of} $X$ if $Y$ is smooth and $\rho^*K_X = K_Y$. } \end{dfn} \begin{theo} Assume that a ${{\Bbb Q}}$-Gorenstein algebraic variety with at worst log-terminal singularities $X$ admits a projective birational morphism $\rho\, : \, Y \rightarrow X$ such that $\rho^* K_X = K_{Y}$. Then $ E_{\rm st} (X; u,v) = E_{\rm st} (Y; u,v)$. In particular, if $\rho$ is a crepant desingularization $($i.e. if $Y$ is smooth$)$, then $E_{\rm st} (X; u,v) = E(Y; u,v)$ and therefore $h^{p,q}_{\rm st}(X) = h^{p,q}(Y) \;\; \forall p, q.$ \label{d-crep} \end{theo} \noindent {\em Proof.} Let $\alpha\, : \, Z \rightarrow Y$ be a resolution of singularities of $Y$ such that the exceptional locus of the birational morphism $\alpha$ is a divisor $D$ with normal crossing irreducible components $D_1, \ldots, D_r$. We have \[ K_Z = \alpha^* K_Y + \sum_{i =1}^r a_i D_i. \] The composition $\alpha \circ \rho\, : \, Z \rightarrow X$ is a resolution of singularities of $X$ and we have \[ K_Z = \alpha^* (\rho^*K_X) + \sum_{i =1}^r a_i D_i, \] since $\rho^* K_X = K_{Y}$. Denote by $D'$ the exceptional locus of $\alpha \circ \rho$. Then $Supp\, D' \supset Supp\, D$, i.e., if one writes \[ K_Z = \alpha^* (\rho^*K_X) + \sum_{i =1}^{s} a_i' D_i', \] where $D_1', \ldots, D_s'$ are irreducible components of $D'$, then $a_j' =0$ for all $j \in \{1, \ldots, s \}$ such that $D_j' \subset Supp\, D' \setminus Supp\, D$. By \ref{key}, we obtain the same formulas for the $E$-functions of $Y$ and $X$ in terms of the desingularizations $\rho \circ \alpha$ and $\alpha$, i.e., $E_{\rm st} (X; u,v) = E_{\rm st} (Y; u,v)$. The last statement of this theorem follows immediatelly from \ref{smooth}. \hfill $\Box$ \begin{rem} {\rm Let $X$ be an $n$-dimensional Calabi-Yau variety in the sense of \ref{d-cy}. In \cite{B1} we have shown that \[ \frac{d^2}{d u^2} E_{\rm st}(X; u,1)|_{u =1} = \frac{3n^2 - 5n}{12} e_{\rm st}(X). \] This identity of the above type for smooth varieties $X$ has appeared in the papers of T. Eguchi et. al \cite{EHX,EMX} in the connection with the Virasoro algebra. If $X$ is smooth, then the above identity follows from the Hirzebruch-Riemann-Roch formula (see \cite{LB1,LW}). By \ref{d-crep}, this immediately implies the identity for any $X$ which admits a crepant desingularization. If $X$ is a $K3$-surface, then the identity is equivalent to the equality $e(X) =24$. We remark that for smooth Calabi-Yau $4$-folds $X$ with $h^{1,0}(X) = h^{2,0}(X) = h^{3,0}(X) =0$ the identity is equivalent to the equality \[ e(X) = 6( 8 - h^{1,1}(X) + h^{2,1}(X) - h^{3,1}(X)) \] (see also \cite{W2}). } \end{rem} \section{Stringy $E$-function of toric varieties} Let us consider the case when $X$ is a normal $d$-dimensional ${\Bbb Q}$-Gorenstein toric variety associated with a rational polyhedral fan $\Sigma \subset N_{{\Bbb R}} = N \otimes {{\Bbb R}} $, where ${N}$ is a free abelian group of rank $d$. Denote by $X_{\sigma}$ the torus orbit in $X$ corresponding to a cone $\sigma$ ($codim_X X_{\sigma} = dim\, \sigma$). Then the property of $X$ to be ${\Bbb Q}$-Gorenstein is equivalent to existence of a continious function $\varphi_K\, : \, N_{{\Bbb R}} \rightarrow {{\Bbb R}}_{\geq 0}$ satisfying the conditions (i) $\varphi_K (e) = 1$ , if $e$ is a primitive integral generator of a $1$-dimensional cone $\sigma \in \Sigma$ (ii) $\varphi_K$ is linear on each cone $\sigma \in \Sigma$. \bigskip The following statement is well-known in toric geometry (see e.g. \cite{KMM}, Prop. 5-2-2): \begin{prop} Let $\rho\: : \; X' \to X$ be a toric desingularization of $X$, which is defined by a subdivision $\Sigma'$ of the fan $\Sigma$. Then the irreducible components $D_1, \ldots, D_r$ of the exceptional divisor $D$ of the birational morphism $\rho$ have only normal crossings and they one-to-one correspond to primitive integral generators $e_1', \ldots, e_r'$ of those $1$-dimensional cones $\sigma' \in \Sigma'$ which do not belong to $\Sigma$. Moreover, in the formula \[ K_{X'} = \rho^* K_X + \sum_{i=1}^r a_i D_i, \] one has $a_i = \varphi_K(e_i') -1$ $\forall i \in \{1, \ldots, r\}$. \end{prop} \begin{coro} Any normal ${\Bbb Q}$-Gorenstein toric variety has at worst log-terminal singularities. \end{coro} Let $\sigma^{\circ}$ be the relative interior of $\sigma$ (we put $\sigma^{\circ} = 0$, if $\sigma = 0$). We give the following explicit formula for the function $E_{\rm st}(X; u, v)$: \begin{theo} \[ E_{\rm st}(X; u,v) = (uv -1)^d \sum_{ \sigma \in \Sigma} \sum_{n \in \sigma^{\circ} \cap N} (uv)^{-\varphi_K(n)}. \] \label{st-tor} \end{theo} \noindent {\em Proof.} Let $\rho\: : \; X' \to X$ be a toric desingularization of $X$ defined by a regular simplicial subdivision $\Sigma'$ of the fan $\Sigma$, $D_1, \ldots, D_r$ the irreducible components of the exceptional divisor $D$ of $\rho$ corresponding to primitive integral generators $e_1', \ldots, e_r' \in N$, and $I = \{1, \ldots, r\}$. First of all we remark that the condition $\bigcap_{j \in J} D_j \neq \emptyset$ for some nonempty $J \subset I$ is equivalent to the fact that \[ \sigma_J := \{ \sum_{j \in J} \lambda_j e_j' \; | \; \lambda_j \in {\Bbb R}_{\geq 0} \} \] is a cone of $\Sigma'$. Let $\Sigma'(J)$ be the star of the cone $\sigma_J \subset \Sigma'$, i.e., $\Sigma'(J)$ consists of the cones $\sigma' \in \Sigma'$ such that $\sigma' \supset \sigma_J$. We denote by $\Sigma_{\circ}'(J)$ the subfan of $\Sigma'(J)$ consisting of those cones $\sigma' \in \Sigma'(J)$ which do not contain any $e_i'$ where $i \not\in J$. Then the fan $\Sigma'(J)$ defines the toric subvariety $D_J \subset X'$ and the fan $\Sigma_{\circ}'(J)$ defines the open subset $D_J^{\circ} \subset D_J$. The canonical stratification by torus orbits \[ X' = \bigcup_{ \sigma' \in \Sigma'} X_{\sigma'}' \] induces the following stratifications \[ D_J^{\circ} = \bigcup_{ \sigma' \in \Sigma_{\circ}'(J)} X_{\sigma'}', \;\;\;\;\; \emptyset \neq J \subset I. \] So we have \begin{equation} E(D^{\circ}_J; u,v) = \sum_{ \sigma' \in \Sigma_{\circ}'(J)} (uv -1)^{d - dim\, \sigma'} \;\;\;\;\; \emptyset \neq J \subset I. \label{n-emp} \end{equation} Let $\Sigma'(\emptyset)$ be the subfan of $\Sigma'$ consisting of those cones $\sigma' \in \Sigma'$ which do not contain any element of $\{ e_1', \ldots, e_r' \}$. Then $\Sigma'(\emptyset)$ defines the canonical stratification of \[ X' \setminus D = \sum_{\sigma' \in \Sigma'(\emptyset)} X_{\sigma'}'. \] In particular, one has \begin{equation} E(X' \setminus D; u,v) = \sum_{ \sigma' \in \Sigma'(\emptyset)} (uv -1)^{d - dim\, \sigma'}. \label{emp} \end{equation} The smoothness of $X'$ implies that $\sigma_J \cap N$ is a free semigroup with the basis $\{ e_j' \}_{j \in J}$. For this reason we can express the power series \[ \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)} \] as a product of $|J|$ geometric series: \[ \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)} = \prod_{j \in J} \frac{ (uv)^{-\varphi_K(e_j')} } { 1 - (uv)^{-\varphi_K(e_j')}} = \prod_{j \in J} \frac{ (uv)^{-a_j -1} } { 1 - (uv)^{-a_j -1}} = \prod_{j \in J} \frac{1}{(uv)^{a_j +1} -1}. \] Hence, \begin{equation} \prod_{j \in J} \frac{ uv -1 }{ (uv)^{a_j + 1} -1} = (uv -1)^{|J|} \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)}. \label{factor} \end{equation} Combining (\ref{n-emp}), (\ref{emp}) and (\ref{factor}), we come to the formula \begin{equation} E_{\rm st}(X; u,v) = E(X \setminus D; u,v) + \sum_{\emptyset \neq J \subset I} E(D_J^{\circ}; u,v)(uv -1)^{|J|} \left( \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)} \right) = \label{formu} \end{equation} \[ = \sum_{ \sigma' \in \Sigma'(\emptyset)} (uv -1)^{d - dim\, \sigma'} + \sum_{\emptyset \neq J \subset I} \left( \sum_{ \sigma' \in \Sigma_{\circ}'(J)} (uv -1)^{d + |J|- dim\, \sigma'} \right) \left( \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)} \right). \] Note that \[ \Sigma' = \Sigma(\emptyset) \cup \bigcup_{\emptyset \neq J \subset I} \Sigma_{\circ}'(J). \] If $ \sigma' = \{ \lambda_{1} e_{i_1}' + \cdots + \lambda_{k} e_{i_k}' \; | \; \lambda_1, \ldots, \lambda_k \in {\Bbb R}_{\geq 0} \}$ is a $k$-dimensional cone of $\Sigma'(\emptyset)$, then the value of $\varphi_K$ is $1$ on all elements of $\{ e_{i_1}', \ldots, e_{i_k}\}$ and therefore \[ (uv-1)^d \sum_{n \in (\sigma')^{\circ} \cap N} (uv)^{-\varphi_K(n)} = (uv-1)^d \left(\frac{ (uv)^{-1}}{ 1- (uv)^{-1}}\right)^k= (uv -1)^{d-k}. \] On the other hand, if $\sigma' = \{ \lambda_{1} e_{i_1}' + \cdots + \lambda_{k} e_{i_k}' \; | \; \lambda_1, \ldots, \lambda_k \in {\Bbb R}_{\geq 0} \}$ is a $k$-dimensional cone of $\Sigma'(J)$ $(\emptyset \neq J \subset I)$, then \[ (uv-1)^d \sum_{n \in (\sigma')^{\circ} \cap N} (uv)^{-\varphi_K(n)} = (uv-1)^d \left(\frac{ (uv)^{-1}}{ 1- (uv)^{-1}}\right)^{k-|J|} \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)} = \] \[= (uv -1)^{d + |J|-k} \sum_{ n \in \sigma^{\circ}_J \cap N} (uv)^{- \varphi_K(n)}, \] since the value of $\varphi_K$ is $1$ on all elements of $\{ e_{i_1}', \ldots, e_{i_k}' \} \setminus \{ e_j' \}_{j \in J}$. It remains to apply (\ref{formu}) and use the fact that $\Sigma'$ is a subdivision of $\Sigma$, i.e., \[ \bigcup_{\sigma \in \Sigma} \sigma^{\circ} \cap N = \bigcup_{\sigma' \in \Sigma'} (\sigma')^{\circ} \cap N. \] \hfill $\Box$ \begin{prop} If all singularities of a toric variety $X$ are Gorenstein, then $E_{\rm st}(X; u,v)$ is a polynomial. \end{prop} \noindent {\em Proof.} According to \cite{R}, $X$ is Gorenstein if and only if $\varphi_K(n) \in {\Bbb Z}$ for all $n \in N$. By \ref{st-tor}, it is sufficient to prove that \[ (uv -1)^d \sum_{n \in \sigma^{\circ} \cap N} (uv)^{-\varphi_K(n)} \] is a polynomial in $uv$ for any $\sigma \in \Sigma$. The latter follows from the fact that \[ (1-t)^{dim\, \sigma} \sum_{n \in \sigma^{\circ} \cap N} t^{\varphi_K(n)} = T(\sigma,t) \] is a polynomial in $t$ of degree $dim\, \sigma$ \cite{DKh}. \hfill $\Box$ Recall the following definition introduced by M. Reid \cite{R}(4.2): \begin{dfn} {\rm Let $X$ be an arbitary $d$-dimensional normal ${\Bbb Q}$-Gorenstein toric variety defined by a fan $\Sigma$. Denote by $\Sigma^{(d)}$ the set of all $d$-dimensional cones in $\Sigma$. If $\sigma \in \Sigma^{(d)}$ is an arbitrary cone, then define {\bf shed of $\sigma$} to be the pyramid \[ {\rm shed}\, \sigma = \sigma \cap \{ y \in N_{{\Bbb R}}\, : \, \varphi_K(y) \leq 1 \}. \] Furthermore, define {\bf shed of $\Sigma$} to be \[ {\rm shed}\, \Sigma = \bigcup_{\sigma \in \Sigma^{(d)}} {\rm shed}\, \sigma. \] } \end{dfn} \begin{dfn} {\rm Let $\sigma \in \Sigma^{(d)}$ be an arbitrary cone. Define $vol(\sigma)$ to be the volume of ${\rm shed}\, \sigma$ with respect to the lattice $N \subset N_{{\Bbb R}}$ multiplied by $d!$. We set \[ vol(\Sigma) := \sum_{\sigma \in \Sigma^{(d)}} vol(\sigma). \] } \end{dfn} \begin{rem} {\rm It is easy to see that $vol(\sigma)$ is a positive integer. Moreover, $vol(\sigma)=1$ if and only if $\sigma$ is generated by a ${\Bbb Z}$-basis of $N$. } \end{rem} \begin{dfn} {\rm Let $X,X'$ and $X''$ be normal $d$-dimensional ${\Bbb Q}$-Gorenstein toric varieties. Assume that we are given two equivariant projective birational morphisms $g\,:\, X \to X''$ and $h\, :\, X' \to X''$ such that $K_X^{-1}$ is $g$-ample , $K_{X'}$ is $h$-ample, and both $g$ and $h$ are isomorphisms in codimension $1$. Then the birational rational map $f= h^{-1} \circ g \, : \, X \dasharrow X'$ is called a {\bf toric flip}. } \end{dfn} The next statement playing important role in termination of toric flips is due to M. Reid (see p. 417 in \cite{R}): \begin{prop} Let $f \, : \, X \dasharrow X'$ be a toric flip of arbitrary ${\Bbb Q}$-Gorenstein toric varieties $X$ and $X'$ associated with fans $\Sigma$ and $\Sigma'$. Then \[ vol(\Sigma) > vol(\Sigma'). \] \label{v-m} \end{prop} Now we observe the following: \begin{prop} Let $X$ be an arbitrary normal ${\Bbb Q}$-Gorenstein toric variety defined by a fan $\Sigma$. Then the stringy Euler number $e_{\rm st}(X)$ equals $vol(\Sigma)$. In particular, one always has $e_{\rm st}(X) \in {\Bbb Z}$. \end{prop} \noindent {\em Proof.} By \ref{d-crep}, it is enough to prove the statement only for the case of a simplicial fan $\Sigma$ (one can always subdivide an arbitrary fan $\Sigma$ into a simplicial fan $\Sigma'$ such that ${\rm shed}\, \Sigma = {\rm shed}\, \Sigma'$). Let $\sigma \in \Sigma$ be a $d$-dimensional simplicial cone with primitive lattice generators $e_1, \ldots, e_d$. Then $vol(\sigma)$ is equal to the index of the subgroup in $N$ generated by $e_1, \ldots, e_d$. By a direct computation, one obtains that \[ \lim_{u,v \to 1} (uv -1)^d \sum_{n \in \sigma^{\circ} \cap N} (uv)^{-\varphi_K(n)} = vol(\sigma). \] Now if $\sigma \in \Sigma$ be a $k$-dimensional simplicial cone with primitive lattice generators $e_1, \ldots, e_k$ $(k <n)$. Then, as in the previous case $k =d$, we compute that \[ \lim_{u,v \to 1} (uv -1)^k \sum_{n \in \sigma^{\circ} \cap N} (uv)^{-\varphi_K(n)}= vol(\sigma) \] is a positive integer. Therefore, \[ \lim_{u,v \to 1} (uv -1)^d \sum_{n \in \sigma^{\circ} \cap N} (uv)^{-\varphi_K(n)} = vol(\sigma) \lim_{u,v \to 1} (uv -1)^{d-k} = 0. \] It remains to apply \ref{st-tor}. \hfill $\Box$ \begin{coro} Let $f \, : \, X \dasharrow X'$ be a toric flip of arbitrary ${\Bbb Q}$-Gorenstein toric varieties $X$ and $X'$. Then \[ e_{\rm st}(X) > e_{\rm st}(X'). \] \label{e-m} \end{coro} \section{Further examples and open questions} \begin{exam} {\rm Let $X_0$ be a $(d-1)$-dimensional smooth projective Fano variety with a very ample line bundle $L$ such that $L^{\otimes k} \cong K_{X_0}^{-l}$ for some positive integers $k,l$. We define $X$ to be the cone over $X_0$ in a projective embedding defined by $L$, i.e., $X$ is obtained by the contraction to a singular point $p \in X$ of the section $D$ in the ${\Bbb P}^1$-bundle $Y := {\Bbb P}({\cal O}_{X_0} \oplus L)$ corresponding to the embedding ${\cal O}_{X_0} \hookrightarrow {\cal O}_{X_0} \oplus L$. Let us compute the function $E_{\rm st}(X; u,v)$ by considering the contraction morphism $\rho\, : \, Y \rightarrow X$ as a desingularization of $X$. By a direct computation, one obtains \[ K_Y = \rho^*K_X + \left( \frac{k}{l} -1 \right) D, \] i.e., $X$ has at worst log-terminal singularities. Since the stratum $X \setminus p = Y \setminus D$ is isomorphic to the total space of the line bundle $L$ over $X_0$, we have \[ E(X \setminus p ; u,v) = (uv) E(X_0; u,v). \] Using the isomorphism $X_0 \cong D$, we obtain \[ E_{\rm st}(X; u,v) = E(X \setminus p ; u,v) + \frac{uv -1}{(uv)^{k/l} -1} E(X_0; u,v) = \] \[ = E(X_0; u,v)\left( uv + \frac{uv -1}{(uv)^{k/l} -1} \right) = \frac{(uv)^{k/l\; +1} -1}{(uv)^{k/l} -1} E(X_0; u,v). \] Therefore, one has \[ e_{\rm st}(X) = \frac{k+l}{k} e(X_0). \] } \label{fano} \end{exam} \medskip \begin{exam} {\rm It is not true in general that $E_{\rm st}(X; u,v)$ is a polynomial even if $X$ has at worst Gorenstein canonical singularities: Let us take $X_0$ in Example \ref{fano} to be a smooth quadic of dimension $d-1 \geq 2$. Then $k = d-1$, $l =1$ and \[ E(X_0; u,v) = \left\{ \begin{array}{ll} {\displaystyle \frac{(1 + (uv)^{\frac{d-1}{2}})((uv)^{\frac{d+1}{2}} -1)}{uv -1}} \; & \; \mbox{\rm if $d-1$ is even} \\ {\displaystyle \frac{(uv)^{d} -1}{uv -1}}\; &\; \mbox{\rm if $d-1$ is odd} \end{array} \right. \] Hence, for the quadric cone $X$ over $X_0$ we have \[ E_{\rm st} (X; u,v) = \left\{ \begin{array}{ll} {\displaystyle \left( \frac{(uv)^{d} -1}{(uv)^{d-1} -1)} \right) \left( \frac{(1 + (uv)^{\frac{d-1}{2}})((uv)^{\frac{d+1}{2}} -1)}{uv -1} \right) } \; & \; \mbox{\rm if $d-1$ is even,} \\ {\displaystyle \left( \frac{(uv)^{d} -1}{(uv)^{d-1} -1)} \right) \left( \frac{(uv)^{d} -1}{uv -1} \right) } \; &\; \mbox{\rm if $d-1$ is odd}. \end{array} \right. \] \[ e_{\rm st} (X) = \left\{ \begin{array}{ll} {\displaystyle \frac{d(d+1)}{d-1} } \; & \; \mbox{\rm if $d-1$ is even,} \\ {\displaystyle \frac{d^2}{d-1}} \; &\; \mbox{\rm if $d-1$ is odd}. \end{array} \right. \] The function $E_{\rm st} (X; u,v)$ is not a polynomial and $e_{\rm st}(X) \not\in {\Bbb Z}$ for $d \geq 4$. In particular, stringy Hodge numbers of $X$ do not exist if $d \geq 4$. If $d =3$, then $E_{\rm st}(X;u,v)$ equals $(uv +1)( (uv)^2 + uv +1)$ and we obtain $h^{1,1}(X) = h^{2,2}(X) =2$, $e_{\rm st}(X) =6$. } \end{exam} \begin{exam} {\rm Let $X \subset {\Bbb C}^4$ be a $3$-dimensional hypersurface defined by the equation $x^2 + y^2 + z^2 + t^3 = 0$, $\rho_0\, : \, V_0 \to {\Bbb C}^4$ the blow up of the point $q=(0,0,0,0) \in {\Bbb C}^4$, and $X_0 := \rho_0^{-1}(X)$. Then $X_0$ is smooth and the exceptional locus of the birational morphism $\rho_0\, : \, X_0 \to X$ consists of an irreducible divisor $D_0 \subset X_0$ which is isomorphic to a singular quadric $Q \subset {\Bbb P}^3$ defined by the equation $z_0^2 - z_1z_2 =0$. Denote by $p_0 \in D_0$ the unique singular point in $D_0$. Let $\rho_1\, : \, Y \to X_0$ be the blow up of $p_0$ on $X_0$, $D_1$ the birational transform of $D_0$, and $D_2 = \rho_1^{-1}(p_0)$. Then the composition $\rho = \rho_1 \circ \rho_0\; :\; Y \to X$ is a resolution of $X$ with normal crossing divisors $D_1, D_2$. The unique singularity $q \in X$ is terminal. On the other hand, one has \[ K_Y = \rho^*K_X + 1 \cdot D_1 + 2 \cdot D_2, \] \[ E(D_{\emptyset}; u,v) = (uv)^3 -1, \; E(D_1^{\circ};u,v) = (uv +1)uv, \; E(D_2^{\circ};u,v) = (uv)^2, \; E(D_{\{1,2\}}; u,v) = uv +1. \] Hence, the stringy $E$-function of the Gorenstein variety $X$ is the following \[ E_{\rm st}(X; u,v)= (uv)^3 -1 + (uv +1)uv \frac{uv-1}{(uv)^2 -1} + (uv)^2 \frac{uv-1}{ (uv)^3 -1 } + (uv +1) \frac{(uv-1)^2}{((uv)^2 -1)((uv)^3 -1 )} = \] \[ = (uv)^2\frac{(uv)^3 +(uv)^2 + 2uv +1}{(uv)^2 + uv +1}, \] i.e., $E_{\rm st}(X; u,v)$ is not a polynomial and $e_{\rm st}(X) = 5/3 \not\in {\Bbb Z}$. } \end{exam} \begin{rem} {\rm It is known that canonical Gorenstein singularities of surfaces are exactly $ADE$-rational double points which always admit crepant desingularizations. By \ref{d-crep}, $E_{\rm st}(X; u,v)$ is a polynomial and $e_{\rm st}(X) \in {\Bbb Z}$ for arbitrary algebraic surface $X$ with at worst canonical singularities. One can prove that $e_{\rm st}(X) \in {\Bbb Z}$ also for arbitrary algebraic surface $X$ with at worst log-terminal singularities. } \end{rem} \begin{ques} Let $X$ be a geometric quotient of ${{\Bbb C}}^n$ modulo an action of a semisimple subgroup $G \subset SL(n, {\Bbb C})$. Is it true that $E(X; u,v)$ is a polynomial? \end{ques} \begin{rem} {\rm The answer is known to be positive if $G$ is commutative ($X$ is a Gorenstein toric variety), or if $G$ is finite (see \cite{BD}). } \end{rem} \begin{conj} Let $f\; : \; X \dasharrow Y$ be a flip of two projective $3$-dimensional varieties with log-terminal singularities. Is it true that $e_{\rm st} (X) > e_{\rm st}(Y)$? \end{conj} \begin{rem} {\rm As we observed in \ref{e-m}, the statement holds true if $X$ and $Y$ are toric varieties and $f$ is a toric flip.} \end{rem} \begin{conj} Let $X$ be a $d$-dimensional algebraic variety with at worst Gorenstein canonical singularities. Then the denominator of the rational number $e_{\rm st}(X)$ is bounded by a constant $C(d)$ depending only on $d$. \end{conj} \begin{rem} {\rm In Example \ref{fano}, it follows from the boundedness of the Fano index $k \leq dim\, X_0 + 1 =d$ that $e_{\rm st}(X) \in \frac{1}{d!}{\Bbb Z}$. We expect that if $dim\, X = 3$, then $e_{\rm st}(X) \in \frac{1}{n} {\Bbb Z}$, where $n \in \{1,2,3,4,6\}$. Some evidences for that can be found in \cite{JK}. } \end{rem} \section{Appendix: non-Archimedian integrals} Let $X$ be a smooth $n$-dimensional complex manifold, $p \in X$ a point, and $\Delta_0:= \{z\, : \, |z| < \delta \} \subset {\Bbb C}$ a disc with center $0$ of any small radius $\delta$. Two germs of holomorphic mappings $y_1, y_2\, : \, \Delta_0 \to X$ such that $y_1(0)=y_2(0)=p$ are called {\bf $l$-equivalent} if their derivatives in $0$ coincide up to order $l$. The set of $l$-equivalent germs is denoted by $J_l(X,p)$ and called the {\bf jet space of order $l$ at $p$} (see \cite{G-G}, Part A). It is well-known that the union $$J_l(X) = \bigcup_{p \in X} J_l(X,p)$$ is a complex manifold of dimension $(l+1)n$, which is a holomorphic affine bundle over $X$ (but not a vector bundle for $l \geq 2$!). The complex manifold $J_l(X)$ is called the {\bf jet space of order $l$ of $X$}. There are canonical mappings $j_l\, : \, J_{l+1}(X) \to J_l(X)$ $(l \geq 0)$ whose fibers are isomorphic to affine linear spaces ${\Bbb C}^n$. We denote by $J_{\infty}(X)$ the projective limit of $J_l(X)$ and by $\pi_l$ the canonical projection $J_{\infty}(X) \to J_l(X)$. The space $J_{\infty}(X)$ is known as the {\bf space of arcs of $X$}. We denote by $J_{\infty}(X,p)$ the projective limit of $J_l(X,p)$ and call it {\bf set of arcs at $p$}. Let $R$ be the formal power series ring ${\Bbb C}[[t]]$, i.e., the inverse limit of finite dimensional ${\Bbb C}$-algebras $R_l: = {\Bbb C}[t]/(t^{l+1})$. If $X$ is $n$-dimensional smooth quasi-projective algebraic variety over ${\Bbb C}$, then the set of points in $J_{\infty}(X)$ (resp. $J_l(X)$) coincides with the set of $R$-valued (resp. $R_l$-valued) points of $X$. From now on we shall consider only the spaces $J_{\infty}(X)$, where $X$ is a smooth quasi-projective algebraic variety. In this case, $J_l(X)$ is a smooth quasi-projective algebraic variety for all $l \geq 0$. Our further terminology is influenced by the theory of Gaussian measures in infinite dimensional linear topological spaces (see the book of Gelfand and Vilenkin \cite{G-V}, Ch. IV). \begin{dfn} {\rm A set $C \subset J_{\infty}(X)$ is called {\bf cylinder set} if there exists a positive integer $l$ such that $C = \pi^{-1}_l(B_l(C))$ for some constructible subset $B_l(C) \subset J_l(X)$ (i.e., for a finite union of Zariski locally closed subsets). Such a constructible subset $B_l(C)$ will be called the $l$-{\bf base of} $C$. By definition, the empty set $\empty \subset J_{\infty}(X)$ is a cylinder set and its $l$-base in $J_l(X)$ is assumed to be empty for all $l \geq 0$.} \end{dfn} \begin{rem} {\rm Let $C \subset J_{\infty}(X)$ be a cylinder set with an $l$-base $B_l(X)$. (i) It is clear that $B_{l+1}(C):= j^{-1}_l(B_l(C)) \subset J_{l+1}(X)$ is the $(l+1)$-base of $C$ and $B_{l+1}(X)$ is a Zariski locally trivial affine bundle over $B_l(C)$, whose fibers are isomorphic to the affine space ${\Bbb C}^n$. (ii) Using (i), it is a standard exercise to show that finite unions, intersection and complements of cylinder sets are again cylinder sets. \label{cyl} } \end{rem} Next statement follows straightforward from the definition of jet spaces of order $l$: \begin{prop} Let $p \in X$ be a point of a smooth complex quasi-projective algebraic $n$-fold $X$ and $z_1, \ldots, z_n$ are local holomorphic coordinates at $p$. Then the local coordinate functions $z_1, \ldots, z_n$ define canonical isomorphisms of quasi-projective algebraic varieties \[ J_{l}(X,p) \cong J_{l}({{\Bbb C}}^n, 0) \;\; \forall l \geq 0. \] In particular, one obtains a canonical isomorphism of algebraic pro-varieties \[ J_{\infty}(X,p) \cong J_{\infty}({{\Bbb C}}^n, 0) \] which induces a bijection between cylinder sets in $J_{\infty}(X,p)$ and cylinder sets in $J_{\infty}({{\Bbb C}}^n, 0)$, where $J_{\infty}({{\Bbb C}}^n, 0)$ can be identified with the set of all $n$-tuples of formal power series $(p_1(t), \ldots, p_n(t)) \in R^n$ having the property $(p_1(0), \ldots, p_n(0)) = (0, \ldots, 0)$. \label{point} \end{prop} \begin{coro} For any point $c = (c_1, \ldots, c_n) \in {\Bbb C}^n$, there exists a canonical isomorphism of algebraic pro-varieties \[ J_{\infty}({{\Bbb C}}^n, c) \cong J_{\infty}({\Bbb C}^n) \] which is defined by the mapping \[ \varphi_c \; :\; (x_1(t), \ldots, x_n(t)) \mapsto ((t-c_1)^{-1}x_1(t), \ldots, (t-c_n)^{-1} x_n(t)). \] The mapping $\varphi_c$ induces a bijection between cylinder sets in $J_{\infty}({\Bbb C}^n,c)$ and cylinder sets in $J_{\infty}({{\Bbb C}}^n)$. \label{point1} \end{coro} Recall the following property of constructible sets (cf. \cite{G-D}, Cor. 7.2.6): \begin{prop} Let $K_1 \supset K_2 \supset \cdots $ be an infinite decreasing sequence of constructible subsets of a complex algebraic variety $V$. Assume that $K_i$ is nonempty for all $i \geq 1$. Then \[ \bigcap_{i =1}^{\infty} K_i \neq \emptyset. \] \label{const-b} \end{prop} \noindent {\em Proof.} Denote by $\overline{K}_i$ the Zariski closure of $K_i$. By notherian property, there exists a positive integer $m$ such that $\overline{K}_i = \overline{K}_{i+1}$ for all $i \geq m$. Let $Z$ be an irreducible component of $\overline{K}_m$. Then for any $i \geq m$ there exists a nonempty Zariski open subset $U_i \subset Z$ such that $U_i$ is contained in $K_i$. By theorem of Baire, $\bigcup_{i =1}^{\infty} U_i \neq \emptyset$. Therefore, $\bigcup_{i =1}^{\infty} K_i \neq \emptyset$. \hfill $\Box$ The following property of cylinder sets will be important: \begin{theo} Assume that a cylinder set $C \subset J_{\infty}(X)$ is contained in a countable union $\bigcup_{i =1}^{\infty} C_i$ of cylinder sets $C_i$. Then there exists a positive integer $m$ such that $C \subset \bigcup_{i =1}^{m} C_i$. \label{cover} \end{theo} \noindent {\em Proof.} Without loss of generality, we can assume that all $C_i$ are contained in $C$ (otherwise we could consider cylinder sets $C \cap C_i$ instead of $C_i$). Define new cylinder sets \[ Z_k := C \setminus \left( C_1 \cup \cdots \cup C_k \right), \; \;( k \geq 1). \] Thus, we obtain a desreasing sequence \[ Z_1 \supset Z_2 \supset Z_3 \supset \cdots \] of cylinder sets whose intersection is empty. Assume that none of sets $Z_1, Z_2, \ldots$ is empty. Then there exits an increasing sequence of positive integers $l_1 < l_2 < l_3 \cdots $ such that $Z_k = \pi^{-1}_{l_k} (B_{l_k}(Z_k))$ for some nonempty constructible sets $B_{l_k}(Z_k) \subset J_{l_k}(X)$ for all $k \geq 1$. Then $\pi_l(Z_k) \neq \emptyset$ for all $l \geq 0$, $k \geq 1$. By theorem of Chevalley, $\pi_l(Z_k) \subset J_l(X)$ is constructible for all $l \geq 0$, $k \geq 1$. On the other hand, using \ref{const-b}, we obtain \[ B_l:= \bigcap_{k \geq 1} \pi_l(Z_k) \neq \emptyset, \;\; \forall l \geq 0. \] Since $B_0 \neq \emptyset$, we can choose a closed point $p \in B_0$ and set \[ Z_k(p) := Z_k \cap J_{\infty}(X, p). \] Then the cylinder sets $Z_k(p)$ are nonempty for all $k \geq 1$ and form a desreasing sequence \[ Z_1(p) \supset Z_2(p) \supset Z_3(p) \supset \cdots \] whose intersection is empty. Now we show that the last property leads to a contradiction. By \ref{point}, we can restrict ourselves to the case $X = {\Bbb C}^n$ and $p = p_0 \in {\Bbb C}^n$. Using the isomorphism \[ \varphi_{p_0}\; : \; J_{\infty}({{\Bbb C}}^n, p_0) \cong J_{\infty}({\Bbb C}^n)\;\; \mbox{\rm (see \ref{point1})}, \] and the induced bijection between cylinder sets in $J_{\infty}({\Bbb C}^n,p_0)$ and $J_{\infty}({{\Bbb C}}^n)$, we can repeat the same arguments for $J_{\infty}({{\Bbb C}}^n)$ and choose a next point $p_1 \in J_1({\Bbb C}^n, p_0) \cong J_0({\Bbb C}^n)$ such that $p_1 \in B_1$ and $j_0(p_1) = p_0$. By induction, we obtain an infinite sequence of points $p_l \in B_l$ such that $j_l(p_{l+1}) = p_l$. It remains to show that the projective limit of $\{ p_l\}_{l \geq 0}$ is a point $q \in J_{\infty}({{\Bbb C}}^n, p_0)$ which is contained in all cylinder sets $Z_k$ $(k \geq 1)$. Indeed, we have $p_{l_l} \in B_{l_k} \subset \pi_{l_k}(Z_k) = B_{l_k}(Z_k)$. Since $Z_k$ is a cylinder set with the $l_k$-base $B_{l_k}(Z_k)$ and $\pi_{l_k}(q) = p_{l_k}$, we obtain $q \in Z_k$. Therefore, $q \in \bigcap_{k \geq 1} Z_k \neq \emptyset$. Contradiction. \hfill $\Box$ \begin{dfn} {\rm Let ${\Bbb Z}[\tau^{\pm 1}]$ be the Laurent polynomial ring in $\tau$ with coefficients in ${\Bbb Z}$ and ${A}$ the group algebra of $({\Bbb Q}, +)$ with coefficients in ${\Bbb Z}[\tau^{\pm 1}]$. We denote by $\theta^s \in {A}$ the image of $s \in {\Bbb Q}$ under the natural homomorphism $({\Bbb Q}, +) \to ({A}^*, \cdot)$, where ${A}^*$ is the multiplicative group of invertible elements in ${A}$ ($\theta \in {A}$ is considered as a transcendental element over ${\Bbb Z}[\tau^{\pm 1}]$). We also write $${A}:= {\Bbb Z}[ \tau^{\pm 1}][\theta^{{\Bbb Q}}] $$ and identify the ring ${A}$ with the direct limit of the subrings ${A}_N := {\Bbb Z}[ \tau^{\pm 1}][\theta^{\frac{1}{N}{\Bbb Z}}] \subset {A}$, where $N$ runs over the set of all natural numbers. } \end{dfn} \begin{dfn} {\rm We consider a topology on ${A}$ defined by the {\bf non-Archimedian norm} \[ \| \cdot \| \; : \; {A} \to {\Bbb R}_{\geq 0} \] which is uniquely characterised by the properties: (i) $\| ab \| = \| a\| \cdot \| b\|$, $\forall a,b \in {\cal A}$; (ii) $\| a + b \| = \max \{ \| a\|, \| b \| \}$, $\forall a,b \in {\cal A}$ if $\| a\| \neq \| b \|$; (ii) $\| a \| = 1$, $\forall a \in {\Bbb Z}[ \tau^{\pm 1}] \setminus \{0 \}$; (iii) $\| \theta^s \| = e^{-s}$ if $s \in {\Bbb Q}$. \noindent The {\bf completion} of ${A}$ (resp. of ${A}_N$) with respect to the norm $\| \cdot \|$ will be denoted by $\widehat{A}$ (resp. by $\widehat{A}_N$). We set \[ \widehat{A}_{\infty} := \bigcup_{N \in {\Bbb N}} \widehat{A}_N \subset \widehat{A}. \]} \end{dfn} \begin{rem} {\rm The noetherian ring $\widehat{A}_N$ consists of Laurent power series of $\theta^{1/N}$ with coefficients in ${\Bbb Z}[ \tau^{\pm 1}]$. The ring $\widehat{A}$ consists consists of formal infinite sums \[ \sum_{i=1}^{\infty} a_i \theta^{s_i},\;\;a_i \in {\Bbb Z}[ \tau^{\pm 1}], \] where $s_1 < s_2 < \cdots $ is a sequence of rational numbers having the property \[ \lim_{i \to \infty} s_i = \infty. \]} \end{rem} \begin{dfn} {\rm Let $W \subset V$ is a constructible subset in a complex quasi-projective algebraic variety $V$. Assume that \[ W = W_1 \cup \cdots \cup W_k \] is a union of pairwise nonintersecting Zariski locally closed subsets $W_1, \ldots, W_k$. Then we define {\bf $E$-polynomial of} $W$ as follows: \[ E(W; u,v) := \sum_{i=1}^k E(W_i; u,v). \]} \label{const-e} \end{dfn} \begin{rem} {\rm Using \ref{e-poly}(i), it is easy to check that the above definition does not depend on the choice of the decomposion of $W$ into a finite union of pairwise nonintersecting Zariski locally closed subsets.} \end{rem} Now we define a {\bf non-Archimedian cylinder set measure} on $J_{\infty}(X)$. \begin{dfn} {\rm $C \subset J_{\infty}(X)$ be a cylinder set. We define the {\bf non-Archimedian volume} $Vol_X(C) \in {A}_1$ of $C$ by the following formula: \[ Vol_X(C) := E(B_l(C); \tau \theta^{-1}, \tau^{-1} \theta^{-1}) \theta^{(l+1)2n} \in {A}_1, \] where $C = \pi^{-1}_l(B_l(C))$ and $E(B_l(C); u,v)$ is the $E$-polynomial of the $l$-base $B_l(C) \subset J_l(X)$. If $C= \emptyset$, we set $Vol_X(C) :=0$. } \end{dfn} \begin{prop} The definition of $Vol_X(C)$ does not depend on the choice of an $l$-base $B_l(C)$. \end{prop} \noindent {\em Proof.} By \ref{e-poly}(iii) and \ref{cyl}(i), $$E(B_{l+1}(C); \tau \theta^{-1}, \tau^{-1} \theta^{-1}) = E({{\Bbb C}}^n; \tau \theta^{-1}, \tau^{-1} \theta^{-1}) \cdot E(B_l(C); \tau \theta^{-1}, \tau^{-1} \theta^{-1}) = $$ $$ = \frac{1}{(\theta^2)^n} E(B_l(C); \tau \theta^{-1}, \tau^{-1} \theta^{-1}).$$ This implies the required independence on $l$ by induction arguments. \hfill $\Box$ \begin{prop} The non-Archimedian cylinder set measure has the following properties: {\rm (i)} If $Z_1$ and $Z_2$ are two cylinder sets such that $Z_1 \subset Z_2$, then \[ \| Vol_X(Z_1) \| \leq \| Vol_X(Z_2) \|. \] {\rm (ii)} If $Z_1, \ldots, Z_k$ are cylinder sets, then \[ \| Vol_X(Z_1 \cup \cdots \cup Z_k) \| = \max_{i =1}^k \| Vol_X(Z_i) \|. \] \label{prop12} \end{prop} \noindent {\em Proof.} (i) It follows from \ref{const-e} that for any constructible set $W$ one has \[ \| E(W; \tau \theta^{-1}, \tau^{-1} \theta^{-1}) \| = e^{2dim\,W}. \] Therefore \[ \| Vol_X(Z_i) \| = \| E(B_l(Z_i); \tau \theta^{-1}, \tau^{-1} \theta^{-1}) \| e^{-2n(l+1)} = e^{2dim\,B_l(Z_i) - 2n(l+1)}, \;\; i =1,2. \] The inclusion $Z_1 \subset Z_2$ implies that there exists a nonnegative integer $l$ such that $B_l(Z_1) \subset B_l(Z_2)$, i.e., $dim\, B_l(Z_1) \leq dim\, B_l(Z_2)$. This implies the statement. (ii) Using our arguments in (i), it remains to remark that \[ dim (W_1 \cup \cdots \cup W_k) = \max_{i =1}^k \, dim\, W_i. \] \hfill $\Box$ \begin{rem} {\rm It follows immediately from \ref{e-poly}(i) that \[ Vol_X(C)= Vol_X(C_1) + \cdots + Vol_X(C_k) \] if $C$ is a finite disjoint union of cylinder sets $C_1, \ldots, C_k$. This show that $Vol_X(\cdot)$ is a finitely additive measure. The next example shows that an extension of $Vol_X(\cdot)$ to a countable additive measure with values in $\widehat{A}_1$ needs additional accuracy.} \label{union} \end{rem} \begin{exam} {\rm Let $C \subset R= {\Bbb C}[[t]]$ be the set consisting of all power series $\sum_{i \geq 0} a_i t^i$ such that $a_i \neq 0$ for all $i \geq 0$. For any $k \in {\Bbb Z}_{\geq 0}$, we define $C_k \subset R$ to be the set consisting of all power series $\sum_{i \geq 0} a_i t^i$ such that $a_i \neq 0$ for all $0 \leq i \leq k$. We identify $R$ with $J_{\infty}({{\Bbb C}})$. Then every $C_k \subset J_{\infty}({{\Bbb C}})$ is a cylinder set and $Vol_{{{\Bbb C}}}(C_k)= (\theta^2 -1)^{k+1}$. Moreover, we have \[ C_0 \supset C_1 \supset C_2 \supset \cdots\; , \;\; \mbox{\rm and} \;\; C = \bigcap_{k\geq 0} C_k. \] However, the sequence \[ Vol_{{{\Bbb C}}}(C_0), \; Vol_{{{\Bbb C}}}(C_1), \; Vol_{{{\Bbb C}}}(C_2), \; \ldots \] does not converge in $\widehat{A}_1$.} \end{exam} \begin{dfn} {\rm We say that a subset $C \subset J_{\infty}(X)$ is {\bf measurable} if for any positive real number $\varepsilon$ there exists a sequence of cylinder sets $C_0(\varepsilon), C_1(\varepsilon), C_2(\varepsilon), \cdots $ such that $$ \left( C \Delta C_0(\varepsilon) \right) \subset \bigcup_{i \geq 1} C_i(\varepsilon) $$ and $\| Vol_X(C_i({\varepsilon}))\| < \varepsilon$ for all $i \geq 1$. If $C$ is measurable, then the element \[ Vol_X(C) := \lim_{\varepsilon \to 0} C_0(\varepsilon) \in \widehat{A}_1 \] will be called the {\bf non-Archimedian volume} of $C$. } \end{dfn} \begin{theo} In the above definition, the limit $\lim_{\varepsilon \to 0} C_0(\varepsilon)$ exists and does not depend on the choice of the sequences $C_0(\varepsilon), C_1(\varepsilon), C_2(\varepsilon), \cdots $. \end{theo} \noindent {\em Proof.} Let $C$ be a measurable set. For two positive real numbers $\varepsilon$, $\varepsilon'$ we choose two sequences \[ C_0(\varepsilon), C_1(\varepsilon), C_2(\varepsilon), \cdots \;\;\; \mbox{\rm and} \;\;\; C_0'(\varepsilon'), C_1'(\varepsilon'), C_2'(\varepsilon'), \cdots, \] such that \[ \| Vol_X(C_i({\varepsilon}))\| < \varepsilon, \;\;\; \| Vol_X(C_i'({\varepsilon'}))\| < \varepsilon' \;\;\;(\forall i \geq 1),\] and \[ \left( C \Delta C_0(\varepsilon) \right) \subset \bigcup_{i \geq 1} C_i(\varepsilon), \;\;\; \left( C \Delta C_0'(\varepsilon') \right) \subset \bigcup_{i \geq 1} C_i'(\varepsilon'). \] Then we have \[ \left( C_0(\varepsilon) \Delta C_0'(\varepsilon') \right) \subset \left( \bigcup_{i \geq 1} C_i({\varepsilon}) \right) \cup \left( \bigcup_{i \geq 1} C_i'({\varepsilon'}) \right). \] Since $\left( C_0(\varepsilon) \Delta C_0'(\varepsilon') \right)$ is a cylinder set, it follows from \ref{cover} that there exist two positive integers $L(\varepsilon)$ and $L(\varepsilon')$ such that \[ \left( C_0(\varepsilon) \Delta C_0'(\varepsilon') \right) \subset \left( \bigcup_{i = 1}^{L(\varepsilon)} C_i({\varepsilon}) \right) \cup \left( \bigcup_{i = 1}^{L(\varepsilon')} C_i'({\varepsilon'}) \right). \] By \ref{prop12}, we obtain \[ \| Vol_X \left( C_0(\varepsilon) \Delta C_0'(\varepsilon') \right) \| \leq \max \{ \varepsilon, \varepsilon' \}. \] Using the inclusions \[ C_0(\varepsilon) \setminus \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right) \subset \left( C_0(\varepsilon) \Delta C_0'(\varepsilon') \right), \;\; C_0'(\varepsilon') \setminus \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right) \subset \left( C_0(\varepsilon) \Delta C_0'(\varepsilon')\right) , \] we obtain the inequalities \[ \| Vol_X \left( C_0(\varepsilon) \setminus \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right) \right) \|, \; \;\; \| Vol_X \left( C_0'(\varepsilon') \setminus \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right) \right) \| \leq \max \{ \varepsilon, \varepsilon' \}. \] Using \ref{union} and disjoint union decompositions $C_0(\varepsilon)= \left( C_0(\varepsilon) \setminus \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right) \right) \cup \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right)$ and $C_0'(\varepsilon')= \left( C_0'(\varepsilon') \setminus \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right) \right) \cup \left( C_0(\varepsilon) \cap C_0'(\varepsilon') \right)$, we conclude that \[ \| Vol_X C_0(\varepsilon) - Vol_X C_0'(\varepsilon') \| \leq \max \{ \varepsilon, \varepsilon' \}. \] Now standard arguments show that both limits \[ \lim_{\varepsilon \to 0} C_0(\varepsilon), \;\; \lim_{\varepsilon' \to 0} C_0(\varepsilon') \] exist and coincide. \hfill $\Box$ The proof of the following statement is a standard exercise: \begin{prop} Measurable sets possess the following properties: {\rm (i)} Finite unions, finite intersections of measurable sets are measurable. {\rm (ii)} If $C$ is a disjoint union of nonintersecting measurable sets $C_1, \ldots, C_m$, then \[ Vol_X(C) = Vol_X(C_1) + \cdots + Vol_X(C_m). \] {\rm (iii)} If $C$ is measurable, then the complement $\overline{C}:= J_{\infty}(X) \setminus C$ is measurable. {\rm (iv)} If $C_1, C_2, \ldots, C_m, \ldots$ is an infinite sequence of nonintersecting measurable sets having the property \[ \lim_{i \to \infty} \| Vol_X(C_i) \| =0, \] then \[ C = \bigcup_{i =1}^{\infty} C_i \] is measurable and \[ Vol_X(C) = \sum_{i =1}^{\infty} Vol_X(C_i). \] \label{bool} \end{prop} \begin{dfn} {\rm We shall say that a subset $C \subset J_{\infty}(X)$ has {\bf measure zero} if for any positive real number $\varepsilon$ there exists a sequence of cylinder sets $C_1(\varepsilon), C_2(\varepsilon), \cdots $ such that $C \subset \bigcup_{i \geq 1} C_i(\varepsilon)$ and $\| Vol_X(C_i({\varepsilon}))\| < \varepsilon$ for all $i \geq 1$. } \end{dfn} \begin{dfn} {\rm Let $Z \subset X$ be a Zariski closed algebraic subvariety. For any point $x \in Z$, we denote by ${\cal O}_{X,x}$ the ring of germs of holomorphic functions at $x$. Let $I_{Z,x} \subset {\cal O}_{X,x}$ be the ideal of germs of holomorphic functions vanishing on $Z$. We set \[ J_{l}(Z,x) := \{ y \in J_{l}(X,x)\, : \, g(y) =0 \;\; \forall \; g \in I_{Z,x} \}, \;\; l \geq 1, \] \[ J_{\infty}(Z,x) := \{ y \in J_{\infty}(X,x)\, : \, g(y) =0 \;\; \forall \; g \in I_{Z,x} \} \] and \[ J_{\infty}(Z) := \bigcup_{x \in Z} J_{\infty}(Z,x), \;\; J_{\infty}(X,Z) := J_{\infty}(X) \setminus J_{\infty}(Z). \] The space $J_{\infty}(Z) \subset J_{\infty}(X)$ will be called {\bf space of arcs with values in $Z$}. If $W \subset X$ is a Zariski locally closed subset, i.e., $W = Z \cap U$ for some closed $Z$ and open $U$, then we set \[ J_{\infty}(X,W) := J_{\infty}(Z) \cap J_{\infty}(X, X \setminus U). \]} \end{dfn} \begin{prop} Let $W$ be an arbitrary Zariski locally closed subset in a smooth irredicible quasi-projective manifold $X$. Then $J_{\infty}(X,W) \subset J_{\infty}(X)$ is measurable. Moreover, one has \[ Vol_X(J_{\infty}(X,W)) = \left\{ \begin{array}{ll} 0 \; & \; \mbox{\rm if $dim\, W < dim\, X$} \\ Vol_X(J_{\infty}(X))\; &\; \mbox{\rm if $dim\, W = dim\, X$.} \end{array} \right. \] \label{subv} \end{prop} \noindent {\em Proof.} Using \ref{bool}, one immediatelly obtains that it suffices to prove the statement only in the case when $X$ is affine and $W$ is smooth irreducible subvariety of codimension $1$. Then $B_l(Z):= \pi_l(J_{\infty}(Z)) \subset J_l(X)$ is a complete intersection of $(l+1)$ divisors, i.e., $dim\, B_l(Z) = (n-1)(l+1)$. Let $C_l$ be the cylinder sets with the $l$-base $B_l(Z)$. Then $C_l$ contains $J_{\infty}(W)$ and $\| Vol_X(C_l) \| = e^{-2(l+1)}$. Taking $l$ arbitrary large, we obtain that $\| Vol_X(J_{\infty}(W)) \|= 0$. \hfill $\Box$ \begin{dfn} {\rm A function $F\, : \, J_{\infty}(X) \to {\Bbb Q} \cup \{\infty\}$ is called {\bf measurable}, if $F^{-1}(s)$ is measurable for all $s \in {\Bbb Q} \cup \{\infty\}$.} \end{dfn} \begin{dfn} {\rm A measurable function $F\, : \, J_{\infty}(X) \to {\Bbb Q} \cup \{\infty\}$ is called {\bf exponentially integrable} if $F^{-1}(\infty)$ has measure zero and the series \[ \sum_{s \in {\Bbb Q}} \| Vol_X(F^{-1}(s)) \| e^{-2s} \] is convergent. If $F$ is exponentially integrable, then the sum \[ \int_{ J_{\infty}(X)} e^{-F} := \sum_{s \in {\Bbb Q}} Vol_X(F^{-1}(s)) \theta^{2s} \in \widehat{A} \] will be called the {\bf exponential integral of ${F}$ over $ J_{\infty}(X) $}. } \end{dfn} \begin{dfn} {\rm Let $D \subset Div(X)$ be a subvariety of codimension $1$, $x \in D$ a point, and $g \in {\cal O}_{X,x}$ the local equation for $D$ at $x$. For any $y \in J_{\infty}(X,D)$, we denote by $\langle D, y \rangle_x$ the order of $g(y)$ at $t =0$. The number $\langle D, y \rangle_x$ will be called the {\bf intersection number} of $D$ and $y$ at $x \in X$. We define the function \[ F_D\; : \; J_{\infty}(X) \to {\Bbb Z}_{\geq 0}\cup \{\infty\} \] as follows: \[ F_D(y) = \left\{ \begin{array}{ll} 0 \; & \; \mbox{\rm if $\pi_0(y) = x \not\in D$} \\ \langle D, y \rangle_x \; &\; \mbox{\rm if $\pi_0(y) \in D$, but $y \not\in J_{\infty}(D)$} \\ \infty \; &\; \mbox{\rm if $y \in J_{\infty}(D)$} \end{array} \right. \] If $D = \sum_{i =1}^m a_iD_i \in Div(X) \otimes {\Bbb Q}$ is a ${\Bbb Q}$-divisor (i.e., $a_i \in {\Bbb Q}$ and $D_i$ is irreducible subvariety $\forall i \in \{1, \ldots,m \}$), we set \[ F_D := \sum_{i =1}^m a_i F_{D_i}, \] where the symbol $\infty$ is assumed to have the properties $\infty \pm \infty = \infty$, $s \cdot \infty = \infty$ ($\forall s \in {\Bbb Q}$). } \end{dfn} \begin{prop} Let $D \subset Div(X) \otimes {\Bbb Q}$ be an arbitrary ${\Bbb Q}$-Cartier divisor. Then the function \[ F_D \;: \; J_{\infty}(X) \to {\Bbb Q} \cup \{\infty\} \] is mesurable. Moreover, one has \[ \int_{ J_{\infty}(X)} e^{- F_{D_1 + D_2}} = \int_{ J_{\infty}(X)} e^{-F_{D_1} - F_{D_2}}\;\; \;\; \forall \; D_1, D_2 \in Div(X) \otimes {\Bbb Q} \] provided both integrals exist. \label{func} \end{prop} \noindent {\em Proof.} If $D \subset Div(X)$ be a subvariety of codimension $1$, then there exists a finite open covering by affine subsets $X = \bigcup_{i =1}^k U_i$ such that $D \cap U_i \subset U_i$ is a principal divisor $(f_i)$. Since each subset $J_{\infty}(U_i) \subset J_{\infty}(X)$ is a cylinder set, it suffices to show that the $F_D^{-1}(l) \cap J_{\infty}(U_i)$ is a cylinder set with some $(l+1)$-base for all $l \in {\Bbb Z}_{\geq 0}$ (cf. \ref{cyl}(ii)). The latter follows directly from a local definition of $\langle D, \cdot \rangle$. The rest of the proof follows from the equality: \[ F^{-1}(s) = \sum_{s_1 + s_2 = s} F_1^{-1}(s_1) \cap F_2^{-1}(s_2), \] if $F = F_1 + F_2$. \hfill $\Box$ \begin{theo} Let $\rho\, : \, X' \to X$ be a projective morphism of smooth complex varieties, $W = \sum_{i =1}^r e_i W_i$ the Cartier divisor defined by the equality \[ K_{X'} = \rho^* K_X + \sum_{i =1}^r e_i W_i.\] Then a function $F\, : \, J_{\infty}(X) \to {\Bbb Q} \cup \{\infty \}$ is exponentially integrable if an only if $F \circ \rho + F_{W}$ is exponentially integrable. Moreover, if the latter holds, then \[ \int_{J_{\infty}(X)} e^{-F} = \int_{J_{\infty}(X')} e^{- F \circ \rho- F_{W}}. \] \label{morph} \end{theo} \noindent {\em Proof.} First we remark that if $C \subset J_{\infty}(X)$ is a cylinder set, then ${\rho}^{-1}(C) \subset J_{\infty}(X')$ is again a cylinder set. Moreover, since $\rho\, : \, X' \to X$ is projective and birational, the induced mapping of the spaces of arcs establishes a bijection \[ J_{\infty}(X') \setminus J_{\infty}(W_1 \cup \cdots \cup W_r) \; \leftrightarrow \; J_{\infty}(X) \setminus J_{\infty}(\rho(W_1 \cup \cdots \cup W_r)). \] For any integer $k \in {\Bbb Z} \cup \{\infty\}$, we denote by $U_k(X',W)$ the cylinder set consisting of all arcs $y' \in J_{\infty}(X')$ such that $F_{W}(y') = k$. Then for any cylinder $C \subset J_{\infty}(X)$ we have a disjoint union decomposition: \[ {\rho}^{-1}(C)= \bigcup_{k \in {\Bbb Z} \cup \{\infty\}} {\rho}^{-1}(C) \cap U_k(X',W). \] In particular, if we choose $s \in {\Bbb Q} \cup \{\infty\}$ and set \[ C(k, s) : = {\rho}^{-1}(F^{-1}(s)) \cap U_k(X',W), \] then we obtain \[ (F \circ \rho)^{-1}(s) = \bigcup_{k \in {\Bbb Z} \cup \{\infty\}} C(k, s)\;\; \mbox{\rm and} \;\; F^{-1}(s) = \bigcup_{k \in {\Bbb Z} \cup \{\infty\}} \rho(C(k, s)). \] The key observation is the fact that $k= F_W(y')$ is the order of vanishing of the Jacobian of $\rho$ at any arc $y' \in U_k(X',W)$. Using this fact, we obtain \[ Vol_X( \rho(C(k, s))) = \theta^{2k} Vol_{X'}(C(k, s)). \] Therefore, \[ \sum_{s \in {\Bbb Q} \cup \{ \infty \}} \| Vol_X(F^{-1}(s)) \| e^{-2s} = \sum_{s \in {\Bbb Q} \cup \{ \infty \}} \sum_{k \in {\Bbb Z} \cup \{\infty\} } \| Vol_X( \rho(C(k, s))) \| e^{-2s} = \] \[ = \sum_{s \in {\Bbb Q} \cup \{ \infty \} } \sum_{k \in {\Bbb Z} \cup \{\infty\} } \| Vol_{X'}( C(k, s) ) \| e^{-2s +2k} = \sum_{s \in {\Bbb Q} \cup \{ \infty \}} \| Vol_{X'}((\rho \circ F + F_W)^{-1}(s)) \| e^{-2s}. \] This implies that the sum $$\sum_{s \in {\Bbb Q} \cup \{ \infty \}} \| Vol_X(F^{-1}(s)) \| e^{-2s}$$ converges if and only if the sum $$ \sum_{s \in {\Bbb Q} \cup \{ \infty \}} \| Vol_{X'}((\rho \circ F + F_W)^{-1}(s)) \| e^{-2s}$$ converges. In the latter case we obtain the equality \[ \sum_{s \in {\Bbb Q} } Vol_X(F^{-1}(s)) \theta^{-2s} = \sum_{s \in {\Bbb Q} } Vol_{X'}((\rho \circ F + F_W)^{-1}(s)) \theta^{-2s}. \] \hfill $\Box$ \begin{theo} Let $D:= a_1D_1 + \cdots + a_r D_r \in Div(X) \otimes {\Bbb Q}$ be a ${\Bbb Q}$-divisor. Assume that the subset of all irreducible components $D_i$ with $a_i \neq 0$ form a system of normal crossing divisors. Then $F_D$ is exponentially integrable on $J_{\infty}(X)$ if and only $a_i > -1$ for all $i \in \{1, \ldots, r \}$. If the latter holds, then \[ \int_{J_{\infty}(X)} e^{-F_D} = \sum_{J \subset I} E(D^{\circ}_J; \tau \theta^{-1},\tau^{-1} \theta^{-1} ) (\theta^{-2} -1)^{|J|}\prod_{j \in J} \frac{1}{1 - \theta^{2(1 + a_j)}} \] \label{int-for} \end{theo} \noindent {\em Proof.} Denote by $J_{\infty}(D) \subset J_{\infty}(X)$ the subset of all arcs contained in the union $D_1 \cup \cdots \cup D_r$. Since $Vol_X(J_{\infty}(D)) =0$, it is sufficient to investigate the exponential integral of $F_D$ only over $J_{\infty}(X,D):= J_{\infty}(X) \setminus J_{\infty}(D)$. For any $(m_1, \ldots, m_r) \in {\Bbb Z}^r_{\geq 0}$, we denote \[ U_{m_1, \ldots, m_r}(X,D):= \{ y \in J_{\infty}(X,D)\; : \; \langle D_1, y \rangle = m_1, \ldots, \langle D_1, y \rangle = m_r \}. \] If $J \subset I:= \{1, \ldots, r\}$, then we set \[ U(X, D_J^{\circ} ) := \{ y \in J_{\infty}(X,D)\: : \; \pi_0(y) \in D_J^{\circ} \} \] and \[ M_J:= \{ (m_1, \ldots, m_r) \in {\Bbb Z}^r_{\geq 0}\; : \; m_j > 0 \Leftrightarrow j \in J \}. \] Thus, we obtain the following two stratifications of $J_{\infty}(X,D)$: \[ J_{\infty}(X,D): = \bigcup_{J \subset I} U(X, D_J^{\circ} ) \] and \[ J_{\infty}(X,D): = \bigcup_{(m_1, \ldots, m_r) \in {\Bbb Z}^r_{\geq 0} } U_{m_1, \ldots, m_r}(X,D), \] where $$ U(X, D_J^{\circ}) = \bigcup_{(m_1, \ldots, m_r) \in M_J } U_{m_1, \ldots, m_r}(X,D). $$ We remark that the value of $F_D$ on the cylinder set $U_{m_1, \ldots, m_r}(X,D)$ equals $\sum_{j \in J} {2(m_j+1)a_j}$. On the other hand, we have \[ Vol_X(U_{m_1, \ldots, m_r}(X,D)) = E(D^{\circ}_J;\tau \theta^{-1},\tau^{-1} \theta^{-1}) \prod_{l =1}^r (\theta^{-2} -1) \theta^{2(m_l+1)} = \] \[ = E(D^{\circ}_J;\tau \theta^{-1},\tau^{-1} \theta^{-1}) (\theta^{-2} -1)^{|J|} \prod_{j \in J} \theta^{2(m_j+1)}, \;\;\mbox{\rm if $(m_1, \ldots, m_r) \in M_J$}. \] The function $F_D$ is exponentially integrable if and only if the series \[ \sum_{J \subset I} \;\; \sum_{(m_1, \ldots, m_r) \in M_J } \| Vol_X(U_{m_1, \ldots, m_r}(X,D))\| \prod_{l=1}^r e^{-2 (m_j+1)a_j} = \] \[ = \sum_{J \subset I} \;\; \sum_{(m_1, \ldots, m_r) \in M_J } e^{n-|J| +r} \prod_{l=1}^r e^{-2 (m_j+1)(a_j+1)} \] is convergent. The latter holds if an only if $a_l+1 > 0$ for all $ l \in \{ 1,\ldots, r \}$. Using the stratifications $U(X, D_J^{\circ})$ by $U_{m_1, \ldots, m_r}(X,D)$, we can rewrite the integral as follows \[ \int_{J_{\infty}(X)} e^{-F_D} = \sum_{J \subset I} E(D^{\circ}_J; \tau \theta^{-1},\tau^{-1} \theta^{-1} ) (\theta^{-2} -1)^{|J|} \sum_{m \in M_J} \prod_{j \in J} \theta^{2(m_j+1)(1 + a_j)}. \] \hfill $\Box$ \bigskip \bigskip \noindent {\bf Proof of Theorem \ref{key}}: Let $\rho_1\, : \, Y_1 \to X$ and $\rho_2\, : \, Y_2 \to X$ be two resolutions of singularities such that \[ K_{Y_1} = \rho_1^* K_X + D_1, \;\; K_{Y_2} = \rho_2^* K_X + D_2 \] where \[ D_1 = \sum_{i=1}^{r_1} a_i' D_i' \; \;\; \mbox{\rm and} \;\; D_2 = \sum_{i=1}^{r_2} a_i'' D_i'', \;\;\;(a_i', a_i'' > -1). \] Choosing a resolution of singularities $\rho_0\, : \, Y_0\to X$ which dominates both resolutions $\rho_1$ and $\rho_2$, we obtain two morphisms $\alpha_1\, : \, Y_0\to Y_1$ and $\alpha_2\, : \, Y_0 \to Y_2$ such that $\rho_0 = \rho_1 \circ \alpha_1 = \rho_2 \circ \alpha_2$. We set $F = F_{ K_{Y_0} - \rho_0^* K_X}$. Since \[ K_{Y_0} - \rho_0^* K_X = ( K_{Y_0} - \alpha_i^* K_{Y_i}) + \alpha_i^*D_i, \;\; (i =1,2), \] we obtain \[ \int_{J_{\infty}(Y_1)} e^{-F_{D_1}} = \int_{J_{\infty}(Y_0)} e^{-F} = \int_{J_{\infty}(Y_2)} e^{-F_{D_2} }\;\;\; \mbox{\rm (see \ref{morph})}. \] On the other hand, by \ref{int-for}, we have \[ \int_{J_{\infty}(Y_i)} e^{-F_{D_i}} = E_{\rm st}(X; \tau \theta^{-1}, \tau^{-1} \theta^{-1}), \;\; i \in \{1,2\}.\] Making the substitutions \[ u = \tau \theta^{-1}, v= \tau^{-1} \theta^{-1}, \] we obtain that the definition of the stringy $E$-function $E_{\rm st}(X; u,v)$ doesn't depend on the choice of resolutions $\rho_1$ and $\rho_2$. \hfill $\Box$ Now we can deduce a Hodge-theoretic version of the results in \cite{B}: \begin{coro} If two smooth $n$-dimensional projective varieties $X_1$ and $X_2$ having trivial canonical classes are birationally isomorphic, then the Hodge numbers of $X_1$ and $X_2$ are the same. \end{coro} \noindent {\em Proof. } Let $\alpha\, : \, X_1 \dasharrow X_2$ be birational map. By Hironaka's theorem we can resolve the indeterminancy locus of $\alpha$ and construct some projective birational morphisms \[ \beta_1 \; : \; X \to X_1 \;\; \mbox{\rm and } \;\; \beta_2 \; : \; X \to X_2 \] such that $\alpha = \beta_2 \circ \beta_1^{-1}$. Since the canonical classes of both $X_1$ and $X_2$ are trivial, it follows from \ref{morph} that \[ \int_{J_{\infty}(X_1)} e^{0} = \int_{J_{\infty}(X)} e^{- F_{W}} = \int_{J_{\infty}(X_2)} e^{0}, \] where $W$ is the canonical class of $X$. On the other hand, \[ \int_{J_{\infty}(X_i)} e^{0} = Vol_{X_i}(J_{\infty}(X_i)) = E(X_i; \tau \theta^{-1}, \tau^{-1} \theta^{-1})\theta^{2n} \;\; ( i =1,2). \] This implies the equality for the Hodge numbers of $X_1$ and $X_2$. \hfill $\Box$

9711

alg-geom/9711016

en

https://arxiv.org/abs/alg-geom/9711016

alg-geom/9711016

Eko Hironaka

Eriko Hironaka

Boundary Manifolds of Line Arrangements

Latex, 22 pages, 15 figures

In this paper we describe the complement of real line arrangements in the complex plane in terms of the boundary three-manifold of the line arrangement. We show that the boundary manifold of any line arrangement is a graph manifold with Seifert fibered vertex manifolds, and depends only on the incidence graph of the arrangement. When the line arrangement is defined over the real numbers, we show that the homotopy type of the complement is determined by the incidence graph together with orderings on the edges emanating from each vertex.

alg-geom

\section{Introduction} Let ${\cal L}$ be a finite union of lines in the complex plane ${\Bbb C}^2$. Two line arrangements ${\cal L}_1$ and ${\cal L}_2$ are said to be {\it topologically equivalent} if there is a homeomorphism of pairs $$ ({\Bbb C}^2,{\cal L}_1) \rightarrow ({\Bbb C}^2,{\cal L}_2). $$ The {\it incidence graph} $\Gamma_{\cal L}$ associated to ${\cal L}$ is the bipartite graph with line-vertices corresponding to the set of lines ${\cal A}$ in ${\cal L}$, point-vertices corresponding to the set of points of intersection ${\cal P}$ on ${\cal L}$ and edges $e(p,L)$ and $e(L,p)$ whenever $p \in L$. A morphism between incidence graphs is a morphism of graphs preserving the vertex labelings, so that line-vertices go to line-vertices and point-vertices go to point-vertices. Two line arrangements ${\cal L}_1$ and ${\cal L}_2$ are said to be {\it combinatorially equivalent} if there is an isomorphism of incidence matrices $\Gamma_{{\cal L}_1} \rightarrow \Gamma_{{\cal L}_2}$. Our motivation is to understand, given a line arrangement ${\cal L}$ in the complex plane ${\Bbb C}^2$, to what extent the topology of the pair $({\Bbb C}^2,{\cal L})$ is determined by the combinatorics of ${\cal L}$. It is known that the cohomology of the complement $E_{\cal L} = {\Bbb C}^2 \setminus {\cal L}$ only depends on the incidence graph $\Gamma_{\cal L}$ \cite{O-T:Arr}, \cite{G-M:Strat}, while the homotopy type and fundamental group of $E_{\cal L}$ depend on more information \cite{Ryb:Fund}. In this paper, we describe the homotopy type of the complement $E_{\cal L}$, when ${\cal L}$ is defined over the real numbers, in terms of the boundary 3-manifold $M_{\cal L}$ of a regular neighborhood of ${\cal L}$ in ${\Bbb C}^2$. We do this by describing $M_{\cal L}$ and $E_{\cal L}$ in terms of a graph of manifolds over the incidence graph $\Gamma_{\cal L}$. Let $\Gamma = \{{\cal V},{\cal E}\}$ be a directed graph, with vertices ${\cal V}$ and edges ${\cal E}$. Assume that for each oriented edge $e \in {\cal E}$ its opposite $\overline e$ is also contained in ${\cal E}$. Denote by $i(e)$ the initial point of $e$ and $t(e)$ the terminal point of $e$. Thus, $i(e) = t(\overline{e})$ and $t(e) = i(\overline{e})$. A {\it graph manifold} $\{M_v, M_e, \phi_e\}$ over a directed graph $\Gamma$ is a collection of connected vertex manifolds $M_v$, for $v \in {\cal V}$, connected edge manifolds $M_e$, with $M_e = M_{\overline e}$, for $e \in {\cal E}$, and inclusions $\phi_e: M_e \rightarrow M_{t(e)}$ which are isomorphisms onto a boundary component of $M_{t(e)}$, and which induce endomorphisms on fundmantal groups (cf. \cite{Wal:Klasse}). A {\it morphism} of graph manifolds over a given graph $\Gamma$ is a collection of continuous maps between corresponding vertex and edge manifolds which commute with the maps $\phi_e$. The manifold associated to a graph manifold is the space obtained by gluing the $M_v$ together along their boundary components according to the maps $\phi_e$. \begin{theorem} If ${\cal L}$ is a complex line arrangement, and $M_{\cal L}$ is the boundary 3-manifold of ${\cal L}$, then $M_{\cal L}$ has the structure of a graph manifold over $\Gamma_{\cal L}$ whose vertex manifolds are Seifert fibered 3-manifolds. Furthermore, if ${\cal L}_1$ and ${\cal L}_2$ are two line arrangements and $$ \alpha : \Gamma_{{\cal L}_1} \rightarrow \Gamma_{{\cal L}_2}, $$ is an isomorphism of incidence graphs, then there is a compatible isomorphism $$ \beta : M_{{\cal L}_1} \rightarrow M_{{\cal L}_2} $$ of graph manifolds. \end{theorem} Any decomposition of a 3-manifold by incompressible surfaces gives rise to a presentation of the manifold as a graph manifold. Let $H_d$ be a thickened $d$-component oriented Hopf link in $S^3$, where each pair of component links having linking number $1$. Then $S^3 \setminus H_d$ is a 3-manifold with boundary 2-tori and each boundary torus has a natural framing $(\mu,\lambda)$. Theorem 1 is a consequence of the following more explicit description of $M_{\cal L}$. The omitted case is treated separately in the beginning of Section 2. \begin{theorem} Let ${\cal L}$ be any line arrangement in ${\Bbb C}^2$ so that each line $L \in {\cal A}$ contains at least one point in ${\cal P}$. Then the boundary 3-manifold $M_{\cal L}$ has a torus decomposition into pieces $$ M_v = S^3 \setminus H_d, $$ where $v$ ranges over vertices in $\Gamma_{\cal L}$. If $v$ is a point-vertex, there is a one-to-one correspondence between the $d$ boundary components of $M_v$ and the edges emanating from $v$ in $\Gamma_{\cal L}$. If $v$ is a line-vertex, there is a one-to-one correspondence between $d-1$ of the $d$ boundary components of $M_v$ and the edges emanating from $v$ in $\Gamma_{\cal L}$. Given an edge $e = e(p,L)$, the corresponding boundary components $T_{L,p} \subset M_{v_p}$ and $T_{p,L} \subset M_{v_L}$ are identified by $\phi_e$ and $\phi_{\overline{e}}$ according to the relation: \begin{eqnarray*} \mu_{p,L} &=& \lambda_{L,p}\\ \lambda_{p,L} &=& \mu_{L,p} + \lambda_{L,p}, \end{eqnarray*} where $(\mu_{L,p},\lambda_{L,p})$ is the induced framing of $T_{L,p}$ and $(\mu_{p,L},\lambda_{p,L})$ is the induced framing of $T_{p,L}$. \end{theorem} Theorem 2, proved in Section 2, implies that the fundamental group of $M_{\cal L}$ is torsion free and residually finite \cite{Serre:Trees}, \cite{Hemp:Res} (see Corollary 2.4). Given two topological space $X \subset Y$, let $Y/X$ be the space $Y$ with $X$ collapsed to a point. For real line arrangements, the homotopy type of the complement can be described as follows. \begin{theorem} Let ${\cal L}$ be a real line arrangement. Then there is a continuous map $$ f : \Gamma_{\cal L} \rightarrow M_{\cal L} $$ such that $f(v) \in M_v$ for all vertices $v$ of $\Gamma_{\cal L}$, and the complement $E_{\cal L} = {\Bbb C}^2 \setminus {\cal L}$ is homotopy equivalent to $M_{\cal L}/f(\Gamma_{\cal L})$. \end{theorem} The map $f$ of Theorem 3 depends on the {\it ordered incidence graph} $\widetilde{\Gamma_{\cal L}}$ of ${\cal L}$, the incidence graph $\Gamma_{\cal L}$ together with an ordering of the edges emanating from each vertex. (An ordered graph is similar to a {\it fat graph}, which has a cyclic ordering on the edges at each vertex \cite{Pen:Per}.) A morphism of ordered incidence graphs is a morphism of incidence graphs which preserves orderings of edges. Let ${\cal L}$ be a line arrangement defined over the real numbers. We construct a model for the homotopy type of $E_{\cal L}$ which only depends on the ordered graph $\widetilde {\Gamma_{\cal L}}$ associated to ${\cal L}$ and is simple to describe (cf. \cite{Falk:Hom}, \cite{Lib:Hom}, \cite{O-T:Arr}, \cite{Ran:Fund} and \cite{Sal:Top}.) \begin{theorem} Let ${\cal L}$ be a real line arrangement. Choose basepoints $b_v \in M_v$ for each vertex $v$ of $\Gamma_{\cal L}$. For each edge $e$ in $\Gamma_{\cal L}$, let $\tau_e$ be a path on $M_{i(e)} \cup M_{t(e)}$ from $b_{i(e)}$ to $b_{t(e)}$ which intersects $M_e$ in one point. Identify $\Gamma_{\cal L}$ with its associated singular $1$-complex. Then there is a continuous mapping $$ f: \Gamma_{\cal L} \rightarrow M_{\cal L} $$ satisfying the following conditions: \begin{description} \item{(i)} $f(v)= b_v$ for all vertices $v \in {\cal V}$; \item{(ii)} if $p_1,\dots,p_s$ are the ordered points on $L \cap {\cal P}$, $e = e(L,p)$, and $p = p_j$, then $$ f(e) = \left \{\begin{array}{ll} \tau_{e}&\qquad\mbox{if $j=1,2$}\\ g_{L,p_2}\dots g_{L,p_{j-1}}\tau_{e} &\qquad\mbox{if $j>2$}, \end{array} \right . $$ where $L_1,\dots,L_r$ are the ordered lines through $p$, and $$ g_{L_j,p}= \mu_{L_1,p} \dots \mu_{L_j,p} \mu_{L_{j-1},p}^{-1} \dots \mu_{L_1,p}^{-1}; $$ \item{(iii)} $f(e) = f(\overline{e})^{-1}$ for all edges $e$; and \item{(iv)} $E_{\cal L}$ is homotopy equivalent to the space $M_{\cal L}/f(\Gamma_{\cal L})$. \end{description} \end{theorem} Section 3 contains proofs of Theorem 3 and Theorem 4. In Section 4 we show why the construction fails for general complex line arrangements and algebraic plane curves. The author thanks P. Orlik and M. Falk for helpful discussions during the writing of this paper. \section{Boundary manifolds} Let ${\cal L}$ be a line arrangement in the complex plane ${\Bbb C}^2$, let ${\cal A}$ be the set of lines in ${\cal L}$ and let ${\cal P}$ be the points of intersection on ${\cal L}$. In this section we describe the boundary manifold of ${\cal L}$ in terms of the incidence graph $\Gamma_{\cal L}$. \heading{Case of disconnected incidence graph.} Consider the case when ${\cal L}$ consists of $k$ non-intersecting lines. This is the only case where $\Gamma_{\cal L}$ is not a connected graph and is a finite union of vertices, one for each line in ${\cal A}$. The boundary 3-manifold $M_{\cal L}$ is then a disjoint union of $k$ solid tori. The complement $E_{\cal L}$ is equal to the product $$ {\Bbb C} \setminus \{k\ \mbox{points}\} \times {\Bbb C}. $$ Thus, $M_{\cal L}/f(\Gamma_{\cal L})$ is homotopy equivalent to $E_{\cal L}$ for any map $f : \Gamma_{\cal L} \rightarrow M_{\cal L}$ which sends each $v_L$ to a point on the line $L$. Throughout this paper, unless otherwise stated we will assume that ${\cal P}$ is non-empty and therefore $\Gamma_{\cal L}$ is connected. \heading{Incidence graph.} The {\it (point/line) incidence graph} $\Gamma_{\cal L}$ of ${\cal L}$ is a bipartite graph with {\it point-vertices} $$ v_p, \qquad p \in {\cal P} $$ and {\it line-vertices} $$ v_L, \qquad L \in {\cal A}. $$ The edges of $\Gamma_{\cal L}$ are of the form $$ e(p,L)\ \mathrm{or}\ e(L,p), \qquad p\in {\cal P}, L \in {\cal A},\ \mathrm{and}\ p \in L. $$ The graph $\Gamma_{\cal L}$ is a directed graph. The {\it initial point} of $e = e(p,L)$ is defined to be $i(e) = v_p$ and the {\it terminal point} is defined to be $t(e) = v_L$. Similarly, if $e = e(L,p)$, then $i(e) = v_L$ and $t(e) = v_p$. We say that $e(L,p)$ and $e(p,L)$ are {\it conjugates} of each other and write $e(L,p) = \overline{e(p,L)}$. We will also think of $\Gamma_{\cal L}$ as a singular 1-complex whose zero cells map to the vertices and whose one cells map to the edges so that $e$ and $\overline{e}$ are identified but with opposite orientations. The endpoints of each edge $e$ are attached to $i(e)$ and $t(e)$ in the obvious way. \heading{Graph manifold.} Let $\Gamma = \{{\cal V},{\cal E}\}$ be a directed graph such that for each edge $e \in {\cal E}$, there is a conjugate edge $\overline{e} \in {\cal E}$ so that $i(e) = t(\overline{e})$ and $t(e) = i(\overline{e})$. A {\it graph manifold} ${\cal M}$ over $\Gamma$ is a collection $(\{M_v\},\{M_e\},\{\phi_e\})$ where \begin{description} \item{(i)} $M_v$ is a connected manifold with boundary for all $v \in {\cal V}$; \item{(ii)} $M_e$ is a connected compact manifold without boundary for all $e \in {\cal E}$; \item{(iii)} $M_e = M_{\overline{e}}$ for all $e \in {\cal E}$; \item{(iv)} $$ \phi_e : M_e \rightarrow M_{t(e)} $$ is an embedding of $M_e$ onto a boundary component of $M_{t(e)}$; \item{(v)} the map $\phi_e$ induces an endomorphism $$ {\phi_e}_* : \pi_1(M_e) \rightarrow \pi_1(M_{t(e)}) $$ on fundamental groups. \end{description} The underlying space $M$ associated to a graph manifold ${\cal M}$ is defined to be the space obtained by gluing together the vertex manifolds $M_v$ along their boundary components so that $\phi_e(M_e)$ is identified with $\phi_{\overline{e}}(M_e)$ by $\phi_e(q) = \phi_{\overline{e}}(q)$ for all $q \in M_e$. Any connect sum of manifolds glued along incompressible boundary components can be thought of as a graph of manifolds. \heading{Graphs of groups.} As with graph complexes, one can talk about the graph of groups associated to a graph manifold. A graph of groups ${\cal G}$ over a directed graph $\Gamma$ is a collection of groups $G_v$, for each vertex $v \in {\cal V}$, groups $G_e = G_{\overline{e}}$, for each edge $e \in {\cal E}$, and group endomorphisms $$ \psi_e : G_e \rightarrow G_{t(e)}. $$ The collection of fundamental groups of the vertex and edge manifolds of a graph of manifolds is a graph of groups. The underlying group of a graph of groups is obtained from the vertex and edges groups by a combination of amalgamated products and HNN extensions. If ${\cal G}$ is the graph of groups associated to a graph of manifolds ${\cal M}$ then the underlying group of ${\cal G}$ is the fundamental group of the underlying manifold $M$ of ${\cal M}$. \heading{Regular neighborhood.} Consider ${\Bbb C}^2$ as a metric space with the usual distance function $$ d((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}. $$ Let $\epsilon > 0$ be such that $d(p,L) > 2 \epsilon$ for all pairs $(p,L) \in {\cal P} \times {\cal A}$ such that $p \notin L$. For each $p \in {\cal P}$, let $N_p$ be the ball of radius $\epsilon$ around $p$ in ${\Bbb C}^2$ and let $S_p$ be the boundary of $N_p$. Then $N_p$ is a Milnor ball around $p$ and the pair $(N_p, N_p \cap {\cal L})$ is a cone over $(S_p, S_p \cap {\cal L})$ (see \cite{Milnor:Sing}). Note that $L \cap N_p \neq \emptyset$ if and only if $p \in L$. Let $$ \delta_p= \min\{d(L_1\cap S_p, L_2\cap S_p)\} $$ where the minimum is taken over $L_1,L_2 \in {\cal A}$ and $p \in L_1 \cap L_2$. Take $\delta > 0$ such that $\delta_p > 2\delta$ for all $p \in {\cal P}$. For each line $L \in {\cal A}$, let $$ N_L = \{q \in {\Bbb C}^2\ : \ d(q,L) \leq \delta\}. $$ By choosing $\delta$ smaller if necessary, we can assume that for each edge $e(p,L)$ in $\Gamma_{\cal L}$, $N_L$ meets $S_p$ transversally. Let $$ N_{\cal L} = \bigcup_{p \in {\cal P}} N_p \cup \bigcup_{L \in {\cal A}} N_L. $$ We call $N_{\cal L}$ a {\it regular neighborhood} of ${\cal L}$ in ${\Bbb C}^2$. \heading{Boundary manifold.} Consider the projective compactification ${\Bbb P}^2$ of ${\Bbb C}^2$ and let $L_\infty$ be the line at infinity, so that ${\Bbb P}^2 = {\Bbb C}^2 \cup L_\infty$. Let $Q$ be a point on $L_\infty$ not on the projective closure $\overline{{\cal L}}$ of ${\cal L}$. Let $\overline L$ be the projective closure of $L$ for all $L \in {\cal L}$. For each $L \in {\cal A}$, let $B_L$ be the closure of $N_L$ in ${\Bbb P}^2$. Then $B_L$ is a closed tubular neighborhood of $\overline L$ in ${\Bbb P}^2$. Let $B_\infty$ be a closed tubular neighorhood of $L_\infty$ not containing any element of ${\cal P}$. Let $S_L, S_\infty$ be the boundaries of $B_L$ and $B_\infty$, respectively. The boundary $M_{\cal L}$ of $N_{\cal L}$ has a deformation retraction onto $M_{\cal L} \setminus B_\infty$. Hereafter, for simplicity, we will replace $M_{\cal L}$ by $M_{\cal L} \setminus B_\infty$. For each $p \in {\cal P}$, note that $$ \bigcup_{L \in {\cal A}} N_L $$ intersects $S_p$ transversally in disjoint 2-tori. Set $$ M_p = S_p \setminus \bigcup_{L \in {\cal A}} N_L. $$ For each $L \in {\cal A}$, let $$ M_L = S_L \setminus (B_\infty \cup \bigcup_{p \in {\cal P}} N_p). $$ Then $M_{\cal L}$ is a connect sum of the $M_p$ and $M_L$. By our construction we have the following. \begin{lemma} The submanifolds $M_p$ and $M_L$ in $M_{\cal L}$ intersect in a boundary component if and only if $e(p,L)$ is an edge in $\Gamma_{\cal L}$, and this intersection is a $2$-dimensional torus. \end{lemma} Recall that a Hopf fibration $h : S^3 \rightarrow S^2$ is an oriented circle bundle such that the fibers have intersection number 1. Given any $d$ points ${\cal Q}$ in $S^2$, the preimage $h^{-1}({\cal Q})$ is called a $d$-component Hopf link (see Figure \ref{fig:Hopf}). The pair $(S^3, h^{-1}({\cal Q}))$ does not depend on which $d$ points are chosen. Let $H_d$ be a thickening of $h^{-1}({\cal Q})$. \makefig{3-component Hopf link.}{fig:Hopf} {\psfig{figure=figs/Hopf,height=1in}} \begin{lemma} For each $p \in {\cal P}$ there is a natural identification of the pair $(S_p,M_p)$ with $(S^3,S^3 \setminus H_d)$, where $d$ is the degree of $v_p$. For each $L \in {\cal A}$, there is a natural identification of the pair $(S_L,M_L)$ with the pair $(S^3,S^3 \setminus H_{d+1})$, where $d$ is the degree of $v_L$. \end{lemma} \heading{Proof.} We begin with $M_p$. Identify $S_p$ with the 3-sphere $S^3$ and $L_\infty$ with $S^2$. Then the pencil of lines through $p$ determines a projection $\rho: S_p \rightarrow L_\infty$ giving a commutative diagram $$ \cd { &S_p &\mapright{=} &S^3\cr &\mapdown{\rho} &&\mapdown{h}\cr &L_\infty &\mapright{=} &S^2 } $$ where $h$ is the standard Hopf fibration. \makefig{Neighborhood of a point $p \in {\cal P}$.}{fig:pnbd} {\psfig{figure=figs/pnbd,height=1.5in}} Let $L_1,\dots,L_d \subset {\cal L}$ be the lines passing through $p$. Under the identification of $S_p$ with $S^3$, $M_p$ is identified with the complement in $S^3$ of $$ h^{-1}(\bigcup_{i=1}^d D_i), $$ where each $D_i$ is a small disk around the point in $S^2$ corresponding to $\overline{L_i} \cap L_\infty$. We thus have a natural identification of the pair $(S_p,M_p)$ with $(S^3,S^3 \setminus H_d)$. Describing $M_L$ is for the most part the dual picture to the above. Let $S_L$ be the boundary of a tubular neighborhood of $S_{\overline L}$. Let $B_Q$ be a ball neighborhood of $Q$ in ${\Bbb P}^2$ and let $S_Q$ be its boundary 3-sphere. Then $S_L$ and $S_Q$ are canonically identified as 3-spheres fibering over a 2-sphere. The pencil of lines through $Q$ defines a commutative diagram $$ \cd { &S_Q &\mapright{=} &S^3 \cr &\mapdown{\rho} && \mapdown{h}\cr &\overline L &\mapright{=} &S^2 } $$ where again $h$ is the standard Hopf fibration. Under the identification of $S_Q$ with $S_L$, $B_\infty$ equals $h^{-1}(D_\infty)$ where $D_\infty$ is a small disk neighborhood of $\overline L \cap L_\infty$. \makefig{Neighborhood of a line $L \in {\cal A}$.}{fig:lnbd} {\psfig{figure=figs/lnbd,height=1.5in}} Let $p_1,\dots,p_d \in {\cal P}$ be the points in ${\cal P}$ which lie on $L$. Then the identification of $S_L$ with $S_Q$ gives an identification of $M_L$ with $$ S_Q \setminus h^{-1}(D_\infty \cup \bigcup_{i=1}^d D_i), $$ where each $D_i$ is a small disk neighborhood of $p_i$ in $\overline L$. This identifies $(S_L,M_L)$ with $(S^3, S^3 \setminus H_{d+1})$. \qed Lemma 2.1 and Lemma 2.2 imply the following. \begin{proposition} For any line arrangement ${\cal L}$, $M_{\cal L}$ is a Haken 3-manifold with a torus decomposition into Seifert fibered manifolds. This torus decomposition gives $M_{\cal L}$ the structure of a graph manifold over $\Gamma_{\cal L}$. \end{proposition} \heading{Proof.} We only need to check when the torus boundary components of $S^3 \setminus H_d$ are incompressible. This is true as long as $d > 1$, which holds unless ${\cal L}$ is a union of parallel lines.\qed \begin{corollary} The fundamental group $\pi_1(M_{\cal L})$ is torsion free and residually finite. \end{corollary} \heading{Proof.} The fundamental group of $\pi_1(M_{\cal L})$ is a graph of groups whose vertex groups are $\pi_1(S^3 \setminus H_d) = {\Bbb Z} \times F_d$, where $F_d$ is a free group on $d-1$ generators, and are torsion free. It follows that $\pi_1(M_{\cal L})$ is also torsion free (see, for example, Section 1.3, Corollary 2 and Section 1.4, Proposition 5 of \cite{Serre:Trees}.) Since $M_{\cal L}$ is a 3-manifold with a torus decomposition along incompressible tori into Seifert fibered pieces, by \cite{Hemp:Res}, Theorem 1.1, the fundamental is residually finite. \qed \heading{The gluing maps.} Let $T$ be a 2-dimensional torus. A {\it framing} $(\mu,\lambda)$ on $T$ is a choice of generators for its fundamental group $\pi_1(T)$. As stated in Proposition 2.3, the edge manifolds of $M_{\cal L}$ considered as a graph manifold are all 2-dimensional tori, so they are isomorphic to $T$. To prove Theorem 2, we need to describe the gluing maps $$ \phi_e : T \rightarrow M_p $$ and $$ \phi_{\overline{e}} : T \rightarrow M_L $$ for each edge $e = e(p,L)$ on the graph $\Gamma_{\cal L}$. By Lemma 2.2, the vertex manifolds $M_p$ and $M_L$ are naturally identified with $E_d = S^3 \setminus H_d$ for some integer $d \ge 2$. Furthermore, the complex structure on $M_p$ and $M_L$ determines an orientation on the core loops of $H_d$. The oriented pair $(S^3,H_d)$ determines a framing on the boundary components of $E_d$ and hence on the boundary components of $M_p$ and $M_L$. Let $T$ be a boundary component of $E_d$ and let $\ell$ be its core curve in $S^3$. The framing $(\mu,\lambda)$ on $T$ is given as follows. Choose a basepoint $x$ on $T$. The loop $\mu$ is a positively oriented meridian loop around the core curve based at $x$. If one considers $T$ as an $S^1$-bundle over $\ell$, then $\mu$ is a loop going once in the positive direction around the fiber of the bundle. The loop $\lambda$ is a loop based at $x$ ``parallel" to the core curve. That is, it is a positively oriented loop whose linking number with the core curve is zero. These definitions uniquely determine $\mu$ and $\lambda$ up to homotopy. Let $p \in {\cal P}$, $L \in {\cal A}$ be such that $p \in L$. We will write $T_{L,p}$ for the boundary component on $M_p$ corresponding to $L$, and $T_{p,L}$ for the boundary component on $M_L$ corresponding to $p$. Let $(\mu_{L,p}, \lambda_{L,p})$ be the framing on $T_{L,p}$ and $(\mu_{p,L},\lambda_{p,L})$ be the framing on $T_{p,L}$. To describe $\phi_e$ up to homotopy, it suffices to describe what it does to the framings. \begin{lemma} Let $e = e(L,p)$ be an edge of $\Gamma_{\cal L}$. If we identify $T$ with $T_{L,p}$ by the map \begin{eqnarray*} \phi_e : T &\rightarrow& T_{L,p} \subset M_p\\ \phi_e(\mu) &=& \mu_{L,p}\\ \phi_e(\lambda) &=& \lambda_{L,p}, \end{eqnarray*} then \begin{eqnarray*} \phi_{\overline{e}} : T &\rightarrow& T_{p,L} \subset M_L\\ \phi_{\overline{e}}(\mu) &=& \mu_{p,L} + \lambda_{p,L}\\ \phi_{\overline{e}}(\lambda) &=& \mu_{p,L} \end{eqnarray*} gives the gluing map for the graph manifold $M_{\cal L}$. \end{lemma} \makefig{Intersection of $S_L$ and $S_p$.}{fig:gluing1} {\psfig{figure=figs/gluing1,height=1.5in}} \heading{Proof.} We will study how $S_L$ and $S_p$ meet by changing coordinates so that $L$ is given by the equation $y=0$, the point $p$ is the origin, $p = (0,0)$. Then $S_L$ and $S_p$ are given by $$ S_L = \overline{\{(x,y) \in {\Bbb C}^2\ : \ |y| = 1\}}, $$ and \begin{eqnarray*} S_p &=& \{(x,y) \in {\Bbb C}^2\ : \ |x| \leq 1, |y| = 1\} \cup \{(x,y) \in {\Bbb C}^2\ : \ |x| = 1, |y| \leq 1\}\\ &=& {\Bbb T}_1 \cup {\Bbb T}_2 \end{eqnarray*} as shown in Figure \ref{fig:gluing1}. Then $S_L \cap S_p = T$, where $T$ is the 2-torus $$ T = \{(x,y) \in {\Bbb C}^2\ : \ |x|=1, |y|=1\} $$ and is the common boundary of ${\Bbb T}_1$ and ${\Bbb T}_2$. As before let $T_{L,p}$ be the boundary component of $S_p$ corresponding to $L$ and $T_{p,L}$ be the boundary component of $S_L$ corresponding to $p$. Since the components of the Hopf link are all unknots, the framing on the link boundary components is given as follows. Each link component has a solid torus neighborhood $N$ in $S^3$. The complement of $N$ is also a solid torus $N^c$. The framing on the boundary of $N$ is given by $(\mu,\lambda)$ where $\mu$ contracts in $N$ and $\lambda$ contracts in $N^c$. Thus, $T_{L,p}$ has the framing \begin{eqnarray*} \mu_{L,p} &=& (1, \exp(2\pi i t))\\ \lambda_{L,p} &=& (\exp(2\pi i t),1) \end{eqnarray*} and $T_{p,L}$ has the framing \begin{eqnarray*} \mu_{p,L} &=& (\exp(2\pi i t), 1)\\ \lambda_{p,L} &=& (\exp(2 \pi i t), \exp (2 \pi i t)). \end{eqnarray*} This implies $\mu_{p,L} = \lambda_{L,p}$ and $\lambda_{p,L} = \mu_{L,p} + \lambda_{L,p}$, as illustrated in Figure \ref{fig:gluing2}. \qed \makefig{Gluing map.}{fig:gluing2} {\psfig{figure=figs/gluing2,height=1.75in}} \heading{Proof of Theorem 2.} We have shown that $M_{\cal L}$ is a graph manifold whose vertex manifolds are $S^3 \setminus H_d$, where $d$ is the degree of the vertex if the vertex is a point-vertex and $d-1$ is the degree of the vertex if the vertex is a line-vertex. The identifications of the edge manifolds with the boundary components of the vertex manifolds can be described in terms of the natural framings on the boundary components of $S^3 \setminus H_d$ by the edge maps described in Lemma 2.5. This completes the proof of Theorem 2.\qed \section{Real line arrangements} In this section, we concentrate on real line arrangements, and will assume that ${\cal L}$ is defined by equations of the form $$ y=a_i x + b_i, \qquad i=1,\dots,k, $$ where $a_i,b_i \in {\Bbb R}$. We will show how to reconstruct the homotopy type of the complement of ${\cal L}$ in terms of the ordered graph of ${\cal L}$. \heading{Ordered graphs.} \makefig{Ordered graph associated to the Ceva arrangement.}{fig:graph} {\psfig{figure=figs/graph,height=2in}} The ordered graph associated to ${\cal L}$ is the incidence graph $\Gamma_{\cal L}$ together with some extra structure. For real line arrangements, we will order the edges emanating from each vertex of the incidence graph $\Gamma_{\cal L}$ as follows. If $v_p$ is the point-vertex associated to the point $p \in {\cal P}$, then we order the edges $e(p,L_1),\dots,e(p,L_r)$ emanating from $p$ by the slopes of $L_1,\dots,L_r$ in decreasing order. If $v_L$ is the line-vertex associated to the line $L \in {\cal A}$, we order the edges $e(L,p_1),\dots,e(L,p_s)$ emanating from $v_L$ by the $x$-coordinates of $p_1,\dots,p_s$ in decreasing order. Since we assume that none of the lines are parallel to the $y$-axis, this is well-defined. The incidence graph $\Gamma_{\cal L}$ of ${\cal L}$ endowed with these orderings on the edges emanating from vertices is called the {\it ordered graph} associated to ${\cal L}$. For example, Figure \ref{fig:graph} gives the ordered graph of the Ceva arrangement. The global ordering of the points in ${\cal P}$ by their $x$-coordinate and the lines in ${\cal A}$ by their slope determines the ordered graph. The orderings near $v_{L_1}$ and $v_{p_5}$ are given are given in Figure \ref{fig:graph}. \heading{The skeleton of a line arrangement.} Let $M_{\cal L}$ be the boundary manifold of ${\cal L}$ and let $$ \alpha: M_{\cal L} \rightarrow {\cal L} $$ be the natural projection map, well-defined up to homotopy. Let $\Sigma_{\cal L}$ be the real part of ${\cal L}$ with all infinite ends removed. We will call $\Sigma_{\cal L}$ the skeleton of ${\cal L}$. Figure \ref{fig:skeleton} gives the skeleton of the Ceva arrangement. \makefig{Skeleton associated to the Ceva arrangement.}{fig:skeleton} {\psfig{figure=figs/skeleton,height=1in}} We start by describing the homotopy type of $E_{\cal L}$ in terms of $M_{\cal L}$ and $\Sigma_{\cal L}$. \begin{lemma} There is an embedding $$ g : \Sigma_{\cal L} \rightarrow M_{\cal L} $$ so that \begin{description} \item{(i)} $\alpha\circ g$ is the identity map; \item{(ii)} the image of $g$ contracts in $E_{\cal L}$; and \item{(iii)} $E_{\cal L} = M_{\cal L}/g(\Sigma_{\cal L})$. \end{description} \end{lemma} \heading{Proof.} The embedding $g$ is given as follows. If $(x,y) \in N_p$, for some $p=(x_p,y_p) \in {\cal P}$, then $$ g(x,y) = (x,y + i \max\{{\sqrt{\epsilon^2-(y-y_p)^2},\delta}\}), $$ otherwise $$ g(x,y) = (x,y+i\delta). $$ The image $g(\Sigma_{\cal L})$ contracts in $E_{\cal L}$, since in $E_{\cal L}$ it is isotopic to $\Sigma_{\cal L} + (0,i\epsilon) \subset {\Bbb R}^2 + (0,i\epsilon)$. No point $(x,y+i\epsilon) \in {\Bbb R}^2 + (0,i\epsilon)$ satisfies an equation of the form $$ y=ax +b $$ where $a$ and $b$ are real numbers. Thus, ${\Bbb R}^2 + (0,i\epsilon)$ is contained in $E_{\cal L}$. Since ${\Bbb R}^2 + (0,i\epsilon)$ is contractable, we see that $g(\Sigma_{\cal L})$ is also contractable. Now let us consider the projection of $\rho : {\Bbb C}^2 \rightarrow {\Bbb C}$ onto the first coordinate. Let $\rho_{\cal L}$ be the restriction of $\rho$ to $E_{\cal L}$. Then $\rho_{\cal L}$ is a topological fibration over the complement of ${\cal Q} = \rho({\cal P})$. \makefig{Contraction of the $x$-coordinate plane to $V_Q$.}{fig:contraction} {\psfig{figure=figs/contraction,width=4in}} For each $q \in {\cal Q}$ let $$ D_q = \{x \in {\Bbb C}\ : \ |x - q|\leq 1\} $$ and let $S_q$ be the boundary of $D_q$. Assume (by expanding coordinates if necessary) that the $S_q$ do not intersect one another. Let $W_Q$ be the union of the $S_q$ and real line segments joining the $S_q$, and let $V_Q$ be the union of $W_Q$ and the $D_Q$ as in Figure \ref{fig:contraction}. The deformation retraction of ${\Bbb C}$ onto $V_Q$ extends to a deformation retraction of $E_{\cal L}$ onto $\rho_{\cal L}^{-1}(V_Q)$. The fibers of $\rho_{\cal L}$ over $V_{{\cal Q}}$ split up into 3 types: those over the line segments in $W_{{\cal Q}}$; those over $S_q$; and those over the $q$. These retract to the outlined and shaded regions shown in Figure \ref{fig:fibers} which we will write as $F_x$, for $x \in I$, $F_s$, for $s \in S$ and $F_q$, respectively. \makefig{Fibers over $V_{{\cal Q}}$.}{fig:fibers} {\bigskip\psfig{figure=figs/fibers,height=2.75in}} We reconstruct the homotopy type of $E_{\cal L}$ from this picture as follows. First notice that the space $\rho^{-1}|_{M_{\cal L}}(D_q)$ has a deformation retraction to $$ F_{D_q} = D_q \times F_q \cup \bigcup_{s \in S_q}F_s. $$ Under this retraction $g(\Sigma_{\cal L})$ maps to the right most points of the circles in $F_s$, $F_x$ and $F_q$ union the line segments depicted on $F_{s_1}$ and $F_{s_2}$. The union of the right-most points of the circles on $F_s$ as $s$ ranges in $S_q$ bounds a $2$-disk $$ \{p + (0,iy): |y| \leq \epsilon\} = D_q \times \{p + (0,i\epsilon)\} $$ in the deformation of $\rho^{-1}|_{\cal L}(D_q)$. Thus, if we include all right-most points of circles in the fibers over $F_x,F_s,$ and $F_q$, union the line segments on $F_{s_1}$ and $F_{s_2}$, we obtain a set $G$ which retracts onto $g(\Sigma_{\cal L})$ in $E_{\cal L}$. \qed \heading{Proof of Theorem 4.} The existence of a map $f : \Gamma_{\cal L} \rightarrow M_{\cal L}$ with property $(iii)$ of Theorem 4 follows from Lemma 3.1, since $\Gamma_{\cal L}$ can be continuously deformed to $\Sigma_{\cal L}$. One can see this by noting that the figures in Figure \ref{fig:compare} are homotopy equivalent. \makefig{Incidence graph and skeleton.}{fig:compare} {\psfig{figure=figs/compare,height=1in}} We will explicitly describe the homotopy type of a map $f : \Gamma_{\cal L} \rightarrow M_{\cal L}$ satisfying the conditions of Theorem 4 in terms of the ordered graph. We begin by defining a map $\sigma : \Gamma_{\cal L} \rightarrow \Sigma_{\cal L}$. For the point-vertices, we simply send $v_p$ to the point $p$ on $\Sigma_{\cal L}$. For the line vertices, there is no canonical choice. Let $L \in {\cal A}$. If $\mathrm{degree}(v_L) = 1$, then let $\sigma(v_L) = p$ where $p = {\cal P} \cap L$. Otherwise, let $p_1,p_2,\dots,p_r$ be the points in ${\cal P} \cap L$. Choose a point on the line segment on $\Sigma_{\cal L}$ strictly between $p_1$ and $p_2$ and map $v_L$ to this point. For the edges of $\Gamma_{\cal L}$, once we have determined where the vertices go, the edges map to the unique straight line segment on $\Sigma_{\cal L}$ connecting the endpoints. This defines a continuous map $\sigma: \Gamma_{\cal L} \rightarrow \Sigma_{\cal L}$. The composition $f = g \circ \sigma$ gives a map satisfying properties (i),(iii) and (iv) of Theorem 4. For each vertex $v$ on $\Gamma$, let $z_v = f(v)$. (We will write $z_p = z_{v_p}$ and $z_L = z_{v_L}$.) To describe the homotopy type of the map $f$, it suffices to describe the homotopy type of $f(e)$ for each edge $e$ in $\Gamma$. The path $\sigma(e)$ breaks up into segments each passing over one point $p \in {\cal P}$. Let $I_{L,p}$ be a straight line segment on $\Sigma_{\cal L} \cap L$ such that $I_{L,p} \cap {\cal P} = p$. We will describe the homotopy type of $g(I_{L,p})$. \makefig{Path $\gamma_L$ lifting to $L_0$.}{fig:gamma} {\psfig{figure=figs/gamma,width=4in}} First fix $L$. Consider the path $\gamma_L$ on the complex plane shown in Figure \ref{fig:gamma}. Here ${\cal Q} = \{q_1,\dots,q_s\} = \rho({\cal P} \cap L)$. The projection map $\rho$ is one-to-one when restricted to $L$. Since $\rho^{-1}(\gamma_L) \cap L$ is contained in $L_0$, $b_L = \rho^{-1}(\gamma_L) \cap L$ defines a contractable subset of $L_0$. Let $h : \gamma_L \rightarrow M_L$ be defined by $$ h(x,y) = \rho|_L^{-1}(x,y) + (0,\delta i), $$ for each $(x,y) \in \gamma_L$. Then $h(\gamma_L)$ defines a contractible subset of $M_L$. We can assume that the image contains the basepoint $z_L$. \makefig{Generators for $M_p$.}{fig:generators} {\psfig{figure=figs/generators,height=0.75in}} Now fix $p \in L \cap {\cal P}$. Let $q = \rho(p)$ and let $J_p$ be the arc segment of $\gamma_L$ near $q$. For each fiber $F_s$ over points $s \in J_p$, $h(J_p) \cap F$ is the rightmost point in the circle (see Figure \ref{fig:homotopy}) on $F$ corresponding to $L \cap F$. As noted in the proof of Lemma 3.1, the path $g(I_p)$ is isotopic to the path whose intersection with each fiber $F_s$ over an interior point $s$ of $J_p$ is the right-most point of the large circle and whose intersection with the fibers $F_{s_1}$ and $F_{s_2}$ is a path from the right-most point of the large circle to the right-most point of the inner circle associated to $L$ (see Figure {fig:fibers}). The large disk in the fiber $F_s$ over $s \in J_p$ rotates in the counter-clockwise direction by 180 degrees as $s$ moves from right to left on $J_p$, as illustrated in Figure \ref{fig:homotopy}. \makefig{Homotopy type of $f(e(p,L))$.}{fig:homotopy} {\psfig{figure=figs/homotopy,height=2.5in}} Note that the basepoint $z_p$ for $M_p$ is the right-most point on the large circle on $F_{s_1}$ and that $b_L \cap F_s$ is the right-most point on the inner circle corresponding to $L$ for all $s \in J_p$. Let $\tau_{L,p}$ be the straight line segment from $b_L \cap F_{s_1}$ to $z_p$ (see Figure \ref{fig:homotopy}.) Let $\mu_{L_1,p},\dots,\mu_{L_r,p}$ be the generators for $\pi_1(M_p)$ as in Figure \ref{fig:generators}. Then $g(I_p)h(J_p)^{-1}$ is homotopy equivalent to the path $abc$ on $F_{s_1}$ pictured in Figure \ref{fig:paths}. In this example $L_1,L_2,L_3$ are the lines passing through $p$ in the order of their slopes and $L = L_2$, then $h(J_p)$ is homotopy equivalent to $g(I_p)h(J_p)^{-1} = abc = \tau_{L,p}\mu_{L_3,p}^{-1}\mu_{L_2,p}\mu_{L_3,p} \tau_{L,p}^{-1}$. \makefig{Homotopy type of $g(I_p)h(J_p)^{-1}$}{fig:paths} {\psfig{figure=figs/paths,height=1.25in}} In the general case, we have the following lemma. \begin{lemma} Let $p \in {\cal P}$ and let $I_p$ be a line segment on $\Sigma_{\cal L} \cap L$ such that $I_p \cap {\cal P} = \{p\}$ and $p$ lies in the interior of $I_p$. Let $L_1,\dots,L_r$ be an ordering by slope of the lines in ${\cal A}$ passing through $p$. Let $j$ be such that $L_j = L$. Then the homotopy class of the difference $g(I_p)h(J_p)^{-1}$ on $M_{\cal L}$ is given by the element $$ g_{L,p}=\tau_{L,p}\mu_{L_r,p}^{-1}\mu_{L_{r-1},p}^{-1}\dots \mu_{L_j,p} \mu_{L_{j+1},p} \dots \mu_{L_r,p}\tau_{L,p}^{-1}. $$ \end{lemma} We now describe the homotopy type of the map $$ f : \Gamma_{\cal L} \rightarrow M_{\cal L} $$ in terms of the ordered graph $\Gamma_{\cal L}$. Fix $p \in {\cal P}$ and let $e(p,L_1),\dots,e(p,L_r)$ be the ordered edges emanating from $v_p$. For each line $L \in {\cal A}$, let $e(L,p_1),\dots,e(L,p_s)$ be the ordered edges emanating from $v_L$. For any edge $e(L,p)$ in $\Gamma_{\cal L}$, replace $\tau_{L,p}$ by the path from $z_L \in M_L$ to $z_p \in M_p$ which goes along $h(\gamma_L)$ and the original $\tau_{L,p}$. This does not change the homotopy type of $\tau_{L,p}$. Note that the choices of $\tau_{L,p}$, $z_p$, $z_L$ a continuous map $$ \ell : \Gamma_{\cal L} \rightarrow M_{\cal L} $$ where $\ell(v) = z_v$, for vertices $v$ and $\ell(e(L,p)) = \tau_{L,p}$ for edges $e(L,p)$. We will call this map $\ell$ the {\it lifting} of $\Gamma_{\cal L}$ in $M_{\cal L}$ defined by the collection $\tau_{L,p}$. Let $g_{L,p} \in \pi_1(M_p,z_p)$ be defined as in Lemma 3.2, except that its basepoint is at $z_L$. We have shown the following. \begin{lemma} Let $$ f : \Gamma_{\cal L} \rightarrow M_{\cal L} $$ be the continuous map on $\Gamma_{\cal L}$ considered as a $1$-complex defined by the following data: \begin{description} \item{(i)} for the point-vertex $v_p$, $f(v_p) = z_p$; \item{(ii)} for the line-vertex $v_L$, $f(v_L) = z_L$; and \item{(iii)} for the edge $e(L,p)$, if $e(L,p_1),\dots,e(L,p_s)$ are the ordered edges emanating from $v_L$, and $p = p_j$, then $$ f(e(L,p)) = \left \{ \begin{array}{ll} \tau_{L,p}&\qquad \hbox{if $j = 1,2$;}\\ g_{L,p_2}\dots g_{L,p_{j-1}} \tau_{L,p} &\qquad \hbox{if $j>2$} \end{array} \right . $$ \end{description} and $f(e(p,L)) = f(e(L,p))^{-1}$. Then $E_{\cal L}$ is homotopy equivalent to $M_{\cal L} / f(\Gamma_{\cal L})$. \end{lemma} For each pair $(p_0,L_0) \in {\cal P} \times {\cal A}$ with $p_0 \in L_0$, the element $g_{L_0,p_0}$ depends on the ordering of the edges $e(p_0,L)$ emanating from $v_{p_0}$ and $f(e(L_0,p_0))$ depends on the ordering the edges $e(L_0,p)$ emanating from $v_{L_0}$. This completes the proof of Theorem 4.\qed \heading{Fundamental group.} Choose a vertex $v$ of $\Gamma_{\cal L}$. We present the fundamental group $G_{\cal L} = \pi_1(M_{\cal L},f(v))$ in the notation of \cite{Serre:Trees} (p. 41). For each edge $e = e(L,p)$ in $\Gamma_{\cal L}$, let $\tau_e = \tau_{L,p}$ and let $\tau_{\overline e} = \tau_{e}^{-1}$. The corresponding lifting of $\Gamma_{\cal L}$ into $M_{\cal L}$ determines a presentation of $\pi_1(M_{\cal L})$ as follows. Any element of $G_{\cal L}$ can be written as a word \begin{eqnarray} \gamma_1\tau_{e_1}\dots\gamma_k\tau_{e_k}\gamma_{k+1} \end{eqnarray} satisfying \begin{description} \item{(i)} $t(e_i) = i(e_{i+1})$, for $i=1,\dots,k$; \item{(ii)} $t(e_{k}) = i(e_1) = v$; \item{(iii)} each $\gamma_i$ is an element of $\pi_1(M_{t(e_i)})$. \end{description} The relations on $G_{\cal L}$ are generated by elements of the form $$ \tau_e\gamma\tau_{\overline{e}} (\gamma')^{-1}, $$ where $\gamma = \phi_e(\kappa)$ and $\gamma' = \phi_{\overline{e}}(\kappa)$ for some $\kappa \in G_e$. A word of the form (1) is said to have length $k$. The length thus depends on the word and not its equivalence class in $G_{\cal L}$. A word of the form (1) is {\it reduced} if whenever $e_i = \overline{e_{i+1}}$, for some $i=1,\dots,{k-1}$, then $\gamma_{i+1} \notin \phi_{e_{i}}(\pi_1(M_{e_i}))$. The following theorem is useful for determining properties of groups which are graphs of groups. \begin{proposition}(\cite{Serre:Trees}, Theorem 11.) A reduced word of the form (1) is trivial if and only if it has length 1 and $\gamma_1 = 1$. \end{proposition} For example, we have the following result. \begin{corollary} The map $f : \Gamma_{\cal L} \rightarrow M_{\cal L}$ induces an endomorphism on fundamental groups. \end{corollary} \heading{Proof.} Let $G_{\cal L}$ be the image of $$ f_* : \pi_1(\Gamma_{\cal L},v) \rightarrow \pi_1(M_{\cal L},f(v)). $$ Then $G_{\cal L}$ consists of elements of the form \begin{eqnarray} f(e_1) \dots f(e_k) = f_* (e_1\dots e_k) \end{eqnarray} where $e_1\dots e_k$ is a closed loop on $\Gamma_{\cal L}$ and $i(e_1) = t(e_k) = v$. Suppose an element of the form (2) is trivial in $G_{\cal L}$ and $k > 1$. We will show that we can write the same element in the form (2) with smaller $k$. By Theorem 5, $k \ge 2$ and there is at least one $i$ so that $e_i = \overline {e_{i+1}}$. In this case $e_i e_{i+1} = 1$, so the element can be represented with smaller $k$. \qed We will now present the fundamental groups of $M_{\cal L}$ and of $E_{\cal L}$ purely from the ordered graph $\Gamma_{\cal L}$. Take any ordered bipartite graph $\widetilde{\Gamma}$ with ``line" and ``point" vertices, such that each directed edge $e(p,L)$ (or $e(L,p)$) joins a point-vertex $v_p$ to a line-vertex $v_L$ (or vice versa.) Choose a vertex $v_0$ on $\widetilde{\Gamma}$. For $v$, a vertex on $\widetilde{\Gamma}$, let $e_{v,1},\dots,e_{v,d}$ be the ordering of edges emanating from $v$. Set $$ G_v = \langle\ \lambda_v , \mu_{1,v},\dots,\mu_{d,v} : \lambda_v\mu_{j,v} = \mu_{j,v}\lambda_v, \quad j = 1,\dots,d \ \rangle . $$ (Note that in the previous notation $\lambda_{j,v} = \lambda_v\mu_{j,v}$, for each $j = 1,\dots,\mathrm{degree}(v)$.) Let ${\cal T}_\Gamma$ be the collection of closed loops on $\Gamma$ based at $v_0$. Then, for any $\tau \in {\cal T}_\Gamma$, $\tau = e_1\dots e_r$, where $t(e_j) = i(e_{j+1})$, for $j=1,\dots,r-1$ and $i(e_1) = t(e_r) = v_0$. The following is a corollary of Theorem 2. \begin{corollary} Let ${\cal L}$ be a complex line arrangement. Then $\pi_1(M_{\cal L})$ can be presented as follows: each element of $\pi_1(M_{\cal L})$ can be written as $$ \gamma_0 e_1\gamma_1 e_2 \dots e_s \gamma_s, $$ where $e_1\dots e_s \in {\cal T}_{\Gamma_{\cal L}}$ and $\gamma_j \in G_{t(e_j)}$, for $j=1,\dots,s$, and the relations are generated by \begin{eqnarray*} \alpha e(p,L) \mu_{k,v_L} e(L,p) \beta &=& \alpha \lambda_{v_p} \mu_{j,v_p}^{-1}\beta, \\ \alpha e(p,L) \lambda_{v_L} \mu_{k,v_L}^{-1} e(L,p) \beta &=& \alpha \lambda_{v_p} \beta, \end{eqnarray*} if $e(p,L)$ is the $j$th edge emanating from $v_p$, and $e(L,p)$ is the $k$th edge emanating from $v_L$. \end{corollary} Note that the indexing of the edges at each vertex does not effect the isomorphism class of the group presented. By Theorem 4, $\pi_1(E_{\cal L})$ is a quotient of $\pi_1(M_{\cal L})$ and we have the following presentation of $\pi_1(E_{\cal L})$. \begin{corollary} Let ${\cal L}$ be a real line arrangement. Then the fundamental group of the complement $\pi_1(E_{\cal L})$ is $\pi_1(M_{\cal L})$ with the additional relations $$ f(e_1)\dots f(e_s) = 1 $$ where $f$ is as in Theorem 4 and $e_1 \dots e_s \in {\cal T}_\Gamma$. \end{corollary} \section{Complex line arrangements} Let ${\cal L}$ be an arbitrary complex line arrangement consisting of lines ${\cal A}$ and points of intersection ${\cal P}$. The proofs of Theorem 3 and Theorem 4 given in Section 3 rely on the existence of a continuous map $$ g : \Sigma_{\cal L} \rightarrow M_{\cal L} $$ from the skeleton $\Sigma_{\cal L}$ of ${\cal L}$ to the boundary 3-manifold $M_{\cal L}$ whose image contracts in $E_{\cal L}$. We will show that Theorem 3 and Theorem 4 cannot be applied to arbitrary complex line arrangements. To do this we extend the definition of skeleton for complex line arrangements, and show that the map $g$ and hence the map $$ f : \Gamma_{\cal L} \rightarrow M_{\cal L} $$ with the required properties does not exist in general. \heading{Knotted 1-complexes.} Let $\Sigma \subset {\Bbb R}^3$ be a compact 1-complex. We will call $\Sigma$ a {\it graph knot}. Let $N_\Sigma$ be a thickening of $\Sigma$ in ${\Bbb R}^3$ and let $M_\Sigma$ be its boundary. Then $N_\Sigma$ is a handlebody and $M_\Sigma$ is an oriented surface. Let $\alpha' : N_\Sigma \rightarrow \Sigma$ be the contraction map and let $\alpha: M_\Sigma \rightarrow \Sigma$ be the restriction of $\alpha'$ to $M_\Sigma$. We say $\Sigma$ is {\it unknotted} if there is an embedding $$ g : \Sigma \rightarrow M_\Sigma $$ such that $\alpha\circ g$ is the identity on $\Sigma$, and the image of $g$ contracts in ${\Bbb R}^3$. We say that $\Sigma$ is {\it knotted} otherwise. Dehn's Lemma (\cite{Rolfsen:Knots}, p. 101) implies the following. \begin{lemma} A graph knot $\Sigma \subset {\Bbb R}^3$ is knotted if there is an embedding $$ \sqcup S^1 \rightarrow \Sigma $$ whose image is a nontrivial knot or link in ${\Bbb R}^3$. \end{lemma} \heading{Skeleta for complex arrangements.} Let ${\cal L}$ be an arbitrary complex line arrangement and let $$ \rho : {\Bbb C}^2 \rightarrow {\Bbb C} $$ be projection onto the $x$-coordinate. Assume that $\rho|_L$ is one-to-one for all lines $L \in {\cal A}$. Let ${\cal Q} = \rho({\cal P})$. Given an ordering of the points $q_1\dots,q_s \in {\cal Q}$, let $I = I_1 \cup \dots \cup I_{s-1}$ be a union of embedded line segments in ${\Bbb C}$ such that each $I_i$ has endpoints $q_i$ and $q_{i-1}$ and no pair $I_i$ and $I_j$ intersect except possibly at their endpoints. We will call $\Sigma_{\cal L}(I) = \rho^{-1}(I) \cap {\cal L}$ the {\it skeleton} of ${\cal L}$ associated to $I$. (The 1-complex $\Sigma_{\cal L}(I)$ considered as a subset of ${\Bbb R}^2 \times I$ is also known as a {\it braided wiring diagram} and has been used to describe the fundamental group and homotopy type of the complement of arbitrary complex line arrangements and algebraic plane curves \cite{Arv:Fund}, \cite{C-S:Braided}.) The skeleton $\Sigma_{\cal L}$ defined for real line arrangements comes from ordering the $q_i$ in the order in which they lie on the real line ${\Bbb R}$ and taking $I$ to be the line segment on ${\Bbb R}$ connecting the smallest to the largest. We will call this $\Sigma_{\cal L}$ the {\it standard skeleton} for the real line arrangement ${\cal L}$. As with real line arrangements, we have the following. \begin{lemma} For any complex line arrangement ${\cal L}$ and any ordering of ${\cal Q}$, the 1-complex $\Sigma_{\cal L}(I)$ is homotopy equivalent to the graph $\Gamma_{\cal L}$. \end{lemma} We can think of $\Sigma_{\cal L}(I)$ as a subset of ${\Bbb R}^3 = {\Bbb R}^2 \times {\Bbb R}$ by embedding $I \in {\Bbb R}$. This makes $\Sigma_{\cal L}(I)$ a graph knot. When ${\cal L}$ is a real line arrangement and $\Sigma_{\cal L}$ is the standard skeleton $\Sigma_{\cal L}$ is unknotted. Conversely, if ${\cal L}$ is a complex line arrangement such that for all orderings the associated skeleton $\Sigma_{\cal L}(I)$ is knotted, then ${\cal L}$ is not topologically equivalent to a real line arrangement. Lemma 3.1 then generalizes as follows. \begin{theorem} Let ${\cal L}$ be a complex line arrangement and let $\alpha : M_{\cal L} \rightarrow {\cal L}$ be the natural projection map of the boundary manifold $M_{\cal L}$ onto ${\cal L}$. Then for some choice of $I$, $\Sigma_{\cal L}(I)$ is unknotted if and only if there is an embedding $$ g : \Sigma_{\cal L}(I) \rightarrow M_{\cal L}, $$ such that \begin{description} \item{(i)} $\alpha \circ g$ is the identity map; and \item{(ii)} the complement $E_{\cal L}$ is homotopy equivalent to $$ M_{\cal L} /g(\Sigma_{\cal L}(I)). $$ \end{description} \end{theorem} \heading{Proof.} The proof is the same as for Lemma 3.1. \qed It follows that if $\Sigma_{\cal L}(I)$ is knotted for all choices of $I$, then Theorem 4 cannot be extended to ${\cal L}$. For general algebraic plane curves, one can also describe the boundary 3-manifold $M_{\cal C}$ as a graph of manifolds over a suitably defined incidence graph $\Gamma_{\cal C}$, but Theorem 5 provides an obstruction for mimicing Theorem 4 in this setting.

9711

alg-geom/9711023

en

https://arxiv.org/abs/alg-geom/9711023

alg-geom/9711023

David A. Bayer

Dave Bayer, Bernd Sturmfels

Cellular Resolutions of Monomial Modules

Plain TeX, with epsf.tex, 17 pages, 2 illustrations, author-supplied PostScript file available at http://www.math.columbia.edu/~bayer/papers/cellular.ps

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals.

alg-geom

\section{} Cellular resolutions \noindent Fix a subset $\setdef{\enma{\mathbf{a}}_j}{j\in I} \subset \enma{\mathbb{Z}}^n\!$, for an index set $I$ which need not be finite, and let $M $ be the monomial module generated by the monomials $m_j = \enma{\mathbf{x}}^{\enma{\mathbf{a}}_j}$, $ j \in I$. Let $X$ be a {\it regular cell complex\/} having $I$ as its set of vertices, and equipped with a choice of an {\it incidence function} $\varepsilon(F,F')$ on pairs of faces. We recall from [BH, Section 6.2] that $\,\varepsilon\,$ takes values in \set{0,1,-1}, that $\varepsilon(F,F') = 0$ unless $F'$ is a facet of $F$, that $\varepsilon(\{j\},\emptyset) =1$ for all vertices $j\in I$, and that for any codimension 2 face $F'$ of $F$, $$\varepsilon(F,F_1) \varepsilon(F_1,F') \,\, + \,\, \varepsilon(F,F_2) \varepsilon(F_2,F') \quad = \quad 0$$ where $F_1$, $F_2$ are the two facets of $F$ containing $F'$. The prototype of a regular cell complex is the set of faces of a convex polytope. The incidence function $\varepsilon$ defines a differential $\partial$ which can be used to compute the homology of $X$. Define the {\it augmented oriented chain complex} $\widetilde{C}(X;k) = \mathop{\hbox{$\bigoplus$}}_{F\in X}\; kF$, with differential $$ \partial F \quad = \quad \sum_{F'\in X} \, \varepsilon(F,F') \, F'.$$ The {\it reduced cellular homology group} $\widetilde{H}_i(X;k)$ is the $i$th homology of $\widetilde{C}(X;k)$, where faces of $X$ are indexed by their dimension. The {\it oriented chain complex} $C(X;k) = \mathop{\hbox{$\bigoplus$}}_{F\in X,\, F\ne\emptyset}\; kF$ is obtained from $\widetilde{C}(X;k)$ by dropping the contribution of the empty face. It computes the ordinary homology groups $H_i(X;k)$ of $X$. The cell complex $X$ inherits a $\enma{\mathbb{Z}}^n$-grading from the generators of $M$ as follows. Let $F$ be a nonempty face of $X$. We identify $F$ with its set of vertices, a finite subset of $I$. Set $m_F := \mathop{\rm lcm}\nolimits \setdef{m_j}{j\in F}$. The exponent vector of the monomial $m_F$ is the {\it join} $ \, \enma{\mathbf{a}}_F := \bigvee\setdef{\enma{\mathbf{a}}_j}{j\in F}$ in $\enma{\mathbb{Z}}^n $. We call $\enma{\mathbf{a}}_F $ the {\it degree} of the face $F$. Homogenizing the differential $\partial$ of $C(X;k)$ yields a $\enma{\mathbb{Z}}^n$-graded chain complex of $S$-modules. Let $SF$ be the free $S$-module with one generator $F$ in degree $\enma{\mathbf{a}}_F$. The {\it cellular complex} $\, \enma{\mathbf{F}}_X \,$ is the $\enma{\mathbb{Z}}^n$-graded $S$-module $\mathop{\hbox{$\bigoplus$}}_{F\in X,\, F\ne\emptyset} SF$ with differential $$ \partial \, F \quad = \quad \sum_{F'\in X,\, F'\ne\emptyset} \; \varepsilon(F,F') \; {m_F \over m_{F'}} \; F'.$$ The homological degree of each face $F$ of $X$ is its dimension. For each degree $\enma{\mathbf{b}}\in\enma{\mathbb{Z}}^n$, let $X_{\preceq\enma{\mathbf{b}}}$ be the subcomplex of $X$ on the vertices of degree $\preceq\enma{\mathbf{b}}$, and let $X_{\prec\enma{\mathbf{b}}}$ be the subcomplex of $X_{\preceq\enma{\mathbf{b}}}$ obtained by deleting the faces of degree \enma{\mathbf{b}}. For example, if there is a unique vertex $j$ of degree $\preceq\enma{\mathbf{b}}$, and $\enma{\mathbf{a}}_j=\enma{\mathbf{b}}$, then $X_{\preceq\enma{\mathbf{b}}} = \set{\set{j},\emptyset}$ and $X_{\prec\enma{\mathbf{b}}}=\set{\emptyset}$. A full subcomplex on no vertices is the acyclic complex \set{}, so if there are no vertices of degree $\preceq\enma{\mathbf{b}}$, then $X_{\preceq\enma{\mathbf{b}}} = X_{\prec\enma{\mathbf{b}}} = \set{}$. The following proposition generalizes [BPS,~Lemma~2.1] to cell complexes: \proposition{propD} The complex $\enma{\mathbf{F}}_X$ is a free resolution of $M$ if and only if $X_{\preceq\enma{\mathbf{b}}}$ is acyclic over $k$ for all degrees $\enma{\mathbf{b}}$. In this case we call $\enma{\mathbf{F}}_X$ a {\it cellular resolution} of $M$. \stdbreak\noindent{\bf Proof. } The complex $\enma{\mathbf{F}}_X$ is $\enma{\mathbb{Z}}^n$-graded. The degree \enma{\mathbf{b}}\ part of $\enma{\mathbf{F}}_X$ is precisely the oriented chain complex $C(X_{\preceq\enma{\mathbf{b}}};k)$. Hence $\enma{\mathbf{F}}_X$ is a free resolution of $M$ if and only if $H_0(X_{\preceq\enma{\mathbf{b}}};k) \cong k$ for $\enma{\mathbf{x}}^\enma{\mathbf{b}} \in M$, and otherwise $H_i(X_{\preceq\enma{\mathbf{b}}};k) = 0$ for all $i$ and all $\enma{\mathbf{b}}$. This condition is equivalent to $\widetilde{H}_i(X_{\preceq\enma{\mathbf{b}}};k) = 0$ for all $i$ (since $\enma{\mathbf{x}}^\enma{\mathbf{b}} \in M \,$ if and only if $ \emptyset \in X_{\preceq\enma{\mathbf{b}}}$) and thus to $X_{\preceq\enma{\mathbf{b}}}$ being acyclic. \Box \remark{remU} Fix $\enma{\mathbf{b}}\in\enma{\mathbb{Z}}^n$. The set of generators of $M$ of degree $\preceq\enma{\mathbf{b}}$ is finite. It generates a monomial module $M_{\preceq \enma{\mathbf{b}}}$ isomorphic to an ideal (up to a degree shift). \corollary{makefinite} The cellular complex $\enma{\mathbf{F}}_X$ is a resolution of $M$ if and only if the cellular complex $\enma{\mathbf{F}}_{X_{\preceq\enma{\mathbf{b}}}}$ is a resolution of the monomial ideal $M_{\preceq\enma{\mathbf{b}}}$ for all $\enma{\mathbf{b}} \in \enma{\mathbb{Z}}^n$. \stdbreak\noindent{\bf Proof. } This follows from \ref{propD} and the identity $\,(X_{\preceq\enma{\mathbf{b}}})_{\preceq\enma{\mathbf{c}}} = X_{\preceq \, \enma{\mathbf{b}} \wedge\enma{\mathbf{c}}}$. \Box \remark{distinct} A cellular resolution $\enma{\mathbf{F}}_X$ is a minimal resolution if and only if any two comparable faces $F' \subset F$ of the complex $X$ have distinct degrees $\,\enma{\mathbf{a}}_F \not= \enma{\mathbf{a}}_{F'}$. \vskip .2cm The simplest example of a cellular resolution is the {\it Taylor resolution\/} for monomial ideals [Tay]. The Taylor resolution is easily generalized to arbitrary monomial modules $M$ as follows. Let $\setdef{m_j}{j \in I}$ be the minimal generating set of $M$. Define $\Delta$ to be the simplicial complex consisting of all finite subsets of $I$, equipped with the standard incidence function $\varepsilon(F,F') = (-1)^j$ if $F\setminus F'$ consists of the $j$th element of $F$. The Taylor complex of $M$ is the cellular complex $\enma{\mathbf{F}}_\Delta$. \proposition{propT} The Taylor complex $\enma{\mathbf{F}}_\Delta$ is a resolution of $M$. \stdbreak\noindent{\bf Proof. } By \ref{propD} we need to show that each subcomplex $\Delta_{\preceq\enma{\mathbf{b}}}$ of $\Delta$ is acyclic. $\Delta_{\preceq\enma{\mathbf{b}}}$ is the full simplex on the set of vertices \setdef{j\in I}{\enma{\mathbf{a}}_j \preceq \enma{\mathbf{b}}}. This set is finite by \ref{remU}. Hence $\Delta_{\preceq\enma{\mathbf{b}}}$ is a finite simplex, which is acyclic. \Box The Taylor resolution $\enma{\mathbf{F}}_\Delta$ is typically far from minimal. If $M$ is infinitely generated, then $\Delta$ has faces of arbitrary dimension and $\enma{\mathbf{F}}_\Delta$ has infinite length. Following [BPS, \S 2] we note that every simplicial complex $X\subset\Delta$ defines a submodule $ \enma{\mathbf{F}}_X \subset \enma{\mathbf{F}}_\Delta$ which is closed under the differential $\partial$. We call $\enma{\mathbf{F}}_X$ the {\it restricted Taylor complex\/} supported on $X$. $\enma{\mathbf{F}}_X$ is a resolution of $M$ if and only if the hypothesis of \ref{propD} holds, with cellular homology specializing to simplicial homology. \example{exA} Consider the monomial ideal $M=\idealfour{a^2b}{ac}{b^2\!}{bc^2}$ in $S=k[a,b,c]$. \nextfigtoks Figure~\numtoks\ shows a truncated ``staircase diagram'' of unit cubes representing the monomials in $S \backslash M$, and shows two simplicial complexes $X$ and $Y$ on the generators of $M$. Both are two triangles sharing an edge. Each vertex, edge or triangle is labeled by its degree. The notation {\bf 210}, for example, represents the degree $(2,1,0)$ of $a^2b$. \draw{70}{diagonals}{ \setext{.0685}{.475}{\sepad{3pt}{3pt}{$ac$}} \nwtext{.107}{.1}{\nwpad{1pt}{1pt}{$a^2b$}} \nwtext{.248}{.352}{\wpad{2pt}{$b^2$}} \swtext{.185}{.81}{\swpad{3pt}{3pt}{$bc^2$}} % \setext{.353}{.94}{\sepad{2pt}{2pt}{\stack{4pt}{$ac$}{\sevenbf 101}}} \etext{.358}{.518}{\epad{3pt}{\sevenbf 211}} \netext{.353}{.105}{\nepad{2pt}{2pt}{\stack{4pt}{\sevenbf 210}{$a^2b$}}} \stext{.48}{.919}{\spad{3pt}{\sevenbf 112}} \ctext{.48}{.518}{\sevenbf 121} \ntext{.48}{.122}{\npad{3pt}{\sevenbf 220}} \swtext{.605}{.94}{\swpad{2pt}{2pt}{\stack{4pt}{$bc\hexp 2$}{\sevenbf 012}}} \wtext{.602}{.518}{\wpad{3pt}{\sevenbf 022}} \nwtext{.605}{.105}{\nwpad{2pt}{2pt}{\stack{4pt}{\sevenbf 020}{$b\hexp 2$}}} % \ctext{.438}{.38}{\sevenbf 221} \ctext{.522}{.659}{\sevenbf 122} % \setext{.742}{.94}{\sepad{2pt}{2pt}{\stack{4pt}{$ac$}{\sevenbf 101}}} \etext{.747}{.518}{\epad{3pt}{\sevenbf 211}} \netext{.742}{.105}{\nepad{2pt}{2pt}{\stack{4pt}{\sevenbf 210}{$a^2b$}}} \stext{.869}{.919}{\spad{3pt}{\sevenbf 112}} \ctext{.869}{.518}{\sevenbf 212} \ntext{.869}{.122}{\npad{3pt}{\sevenbf 220}} \swtext{.994}{.94}{\swpad{2pt}{2pt}{\stack{4pt}{$bc\hexp 2$}{\sevenbf 012}}} \wtext{.991}{.518}{\wpad{3pt}{\sevenbf 022}} \nwtext{.994}{.105}{\nwpad{2pt}{2pt}{\stack{4pt}{\sevenbf 020}{$b\hexp 2$}}} % \ctext{.827}{.659}{\sevenbf 212} \ctext{.911}{.38}{\sevenbf 222} % \ntext{.13}{0}{\npad{10pt}{$M$}} \ntext{.48}{0}{\npad{10pt}{$X$}} \ntext{.869}{0}{\npad{10pt}{$Y$}} } \noindent By \ref{propD}, the complex $X$ supports the minimal free resolution $\enma{\mathbf{F}}_X =$ $$ 0 \rightarrow S^2 \rightarrowmat{2pt}{4pt}{\!-b&0\cr c&0\cr 0&-b\cr \!-a&c\cr 0&-a&\cr} S^5 \rightarrowmat{2pt}{4pt}{c&b&0&0&0\cr \!-ab&0&bc&b\sexp2&0\cr 0&0& \,-a&0&b\cr 0&-a\sexp2&0&-ac&\,-c\sexp2&\,\cr} S^4 \rightarrowmat{3pt}{6pt}{a^2b & ac & bc^2 & b^2 \cr} M \rightarrow 0. $$ The complex $Y$ fails the criterion of \ref{propD}, and hence $\enma{\mathbf{F}}_Y$ is not exact: if $\enma{\mathbf{b}}=(1,2,1)$ then $Y_{\preceq\enma{\mathbf{b}}}$ consists of the two vertices $ac$ and $b^2$, and is not acyclic. \Box We next present four examples which are not restricted Taylor complexes. \example{exBH} Let $M$ be a {\it Gorenstein ideal of height $3$} generated by $m$ monomials. It is shown in [BH1, \S 6] that the minimal free resolution of $M$ is the cellular resolution $\, \enma{\mathbf{F}}_X \,:\, 0 \rightarrow S \rightarrow S^m \rightarrow S^m \rightarrow S \rightarrow 0 \,$ supported on a {\it convex $m$-gon}. \example{excogen} % A monomial ideal $M$ is {\it co-generic} if its no variable occurs to the same power in two distinct irreducible components $\,\langle x_{i_1}^{r_1}, x_{i_2}^{r_2}, \ldots, x_{i_s}^{r_s} \rangle \,$ of $M$. It is shown in [Stu2] that the minimal resolution of a co-generic monomial ideal is a cellular resolution $\enma{\mathbf{F}}_X$ where $X$ is the complex of bounded faces of a {\it simple polyhedron}. \example{exD} Let $u_1,\ldots,u_n$ be distinct integers and $M$ the module generated by the $\, n\, !\, $ monomials $\, x_{\pi(1)}^{u_1} x_{\pi(2)}^{u_2} \cdots x_{\pi(n)}^{u_n} \,$ where $\pi$ runs over all permutations of $\,\{1,2,\ldots,n \} $. Let $X$ be the complex of all faces of the {\it permutohedron} [Zie, Example 0.10], which is the convex hull of the $\,n \,! \,$ vectors $\bigl(\pi(1),,\ldots,\pi(n)\bigr)$ in $\enma{\mathbb{R}}^n$. It is known [BLSWZ, Exercise~2.9] that the $i$-faces $F$ of $X$ are indexed by chains $$ \emptyset \;=\; A_0 \subset A_1 \subset \;\;\ldots\;\; \subset A_{n-i-1} \subset A_{n-i} \;=\; \{u_1,u_2,\ldots,u_n\} . $$ We assign the following monomial degree to the $i$-face $F$ indexed by this chain: $$ \enma{\mathbf{x}}^{\enma{\mathbf{a}}_F} \quad = \quad \prod_{j=1}^{n-i} \; \prod_{r \in A_j \backslash A_{j-1}} \!\! x_r^{\max\{ A_j \backslash A_{j-1}\}}. $$ It can be checked (using our results in \S 2) that the conditions in \ref{propD} and \ref{distinct} are satisfied. Hence $\enma{\mathbf{F}}_X$ is the minimal free resolution of $M$. \Box \example{exR} \def\enma{\RR\bbPP^2}{\enma{\enma{\mathbb{R}}\enma{\mathbb{P}}^2}} Let $S=k[a,b,c,d,e,f]$. Following [BH, page 228] we consider the Stanley-Reisner ideal of the minimal triangulation of the {\it real projective plane} \enma{\RR\bbPP^2}, $$M \;=\; \ideal{abc,\; abf,\; ace,\; ade,\; adf,\; bcd,\; bde,\; bef,\; cdf,\; cef}.$$ The dual in \enma{\RR\bbPP^2}\ of this triangulation is a cell complex $X$ consisting of six pentagons. The ten vertices of $X$ are labeled by the generators of $M$. We illustrate $ X \simeq $ \enma{\RR\bbPP^2} \ as the disk shown on the left in \nextfigtoks Figure~\numtoks; antipodal points on the boundary are to be identified. The small pictures on the right will be discussed in Example 2.14. \draw{70}{projplane}{ \ntext{.161}{.475}{\npad{2pt}{$abc$}} \stext{.0915}{.406}{\swpad{4pt}{3pt}{$ace$}} \stext{.233}{.406}{\sepad{4pt}{3pt}{$abf$}} \nwtext{.169}{.728}{\npad{1pt}{$bcd$}} % \stext{.12}{1}{\swpad{2pt}{2pt}{$bef$}} \stext{.202}{1}{\swpad{2pt}{2pt}{$cef$}} \swtext{.28}{.866}{\swpad{2pt}{1pt}{$cdf$}} \wtext{.322}{.626}{\wpad{2pt}{\mimic{$a$}{$adf$}}} \wtext{.322}{.373}{\wpad{2pt}{\mimic{$a$}{$ade$}}} \nwtext{.278}{.132}{\nwpad{2pt}{1pt}{$bde$}} \ntext{.202}{.006}{\nwpad{1pt}{2pt}{$bef$}} \ntext{.12}{.006}{\nwpad{1pt}{2pt}{$cef$}} \netext{.043}{.14}{\nepad{2pt}{1pt}{$cdf$}} \etext{.0015}{.373}{\epad{2pt}{\mimic{$a$}{$adf$}}} \etext{.0015}{.626}{\epad{2pt}{\mimic{$a$}{$ade$}}} \setext{.04}{.85}{\sepad{2pt}{1pt}{$bde$}} % \ctext{.161}{.88}{$\widehat{a}$} \ctext{.061}{.31}{$\widehat{b}$} \ctext{.261}{.31}{\mimic{$\widehat{b}$}{$\widehat{c}$}} \ctext{.161}{.25}{$\widehat{d}$} \ctext{.241}{.63}{\mimic{$\widehat{f}$}{$\widehat{e}$}} \ctext{.081}{.63}{$\widehat{f}$} % \ntext{.51}{.264}{\npad{10pt}{\stack{6pt}{$a\,=\,0$}{6 cycles}}} \ntext{.714}{.264}{\npad{10pt}{\stack{6pt}{$a\,=\,1$}{6 cycles}}} \ntext{.918}{.264}{\nwpad{10pt}{2pt}{\stack{6pt}{$b+c+d\,=\,1$}{10 cycles}}} } \noindent If $\mathop{\rm char }\nolimits k \ne 2$ then $X$ is acyclic over $k$ and the cellular complex $\enma{\mathbf{F}}_X$ coincides with the minimal free resolution $\, 0 \rightarrow S^6 \rightarrow S^{15} \rightarrow S^{10} \rightarrow M $. If $\mathop{\rm char }\nolimits k = 2$ then $X$ is not acyclic over $k$, and the cellular complex $\enma{\mathbf{F}}_X$ is not a resolution of $M$. \Box Returning to the general theory, we next present a formula for the {\it Betti number} $\beta_{i,\enma{\mathbf{b}}} = \dim\mathop{\rm Tor}\nolimits_i(M,k)_\enma{\mathbf{b}} \,$ which is the number of minimal $i$th syzygies in degree $\enma{\mathbf{b}}$. The degree $\enma{\mathbf{b}} \in \enma{\mathbb{Z}}^n$ is called a {\it Betti degree} of $M$ if $\beta_{i,\enma{\mathbf{b}}}\not= 0$ for some $i$. \theorem{propA} If $\, \enma{\mathbf{F}}_X$ is a cellular resolution of a monomial module $M$ then $$\beta_{i,\enma{\mathbf{b}}} \quad = \quad \dim H_i(X_{\preceq\enma{\mathbf{b}}}, X_{\prec\enma{\mathbf{b}}};k) \quad = \quad \dim \widetilde{H}_{i-1} (X_{\prec\enma{\mathbf{b}}};k),$$ where $H_*$ denotes relative homology and $\widetilde{H}_*$ denotes reduced homology. \stdbreak\noindent{\bf Proof. } We compute $\mathop{\rm Tor}\nolimits_i(M,k)_\enma{\mathbf{b}}$ as the $i$th homology of the complex of vector spaces $(\enma{\mathbf{F}}_X\otimes_S k)_\enma{\mathbf{b}}$. This complex equals the chain complex $\widetilde{C}(X_{\preceq\enma{\mathbf{b}}},X_{\prec\enma{\mathbf{b}}};k)$ which computes the relative homology with coefficients in $k$ of the pair $(X_{\preceq\enma{\mathbf{b}}},X_{\prec\enma{\mathbf{b}}})$. Thus $$\mathop{\rm Tor}\nolimits_i(M,k)_\enma{\mathbf{b}} \quad = \quad H_i(X_{\preceq\enma{\mathbf{b}}},X_{\prec\enma{\mathbf{b}}};k).$$ Since $X_{\preceq\enma{\mathbf{b}}}$ is acyclic, the long exact sequence of homology groups looks like $$ 0 \,=\, \widetilde{H}_i(X_{\preceq\enma{\mathbf{b}}}; k) \,\rightarrow \, H_i(X_{\preceq\enma{\mathbf{b}}},X_{\prec\enma{\mathbf{b}}}; k) \, \rightarrow \, \widetilde{H}_{i-1}(X_{\prec\enma{\mathbf{b}}}; k) \, \rightarrow \, \widetilde{H}_{i-1}(X_{\preceq\enma{\mathbf{b}}}; k) \,=\, 0 .$$ We conclude that the two vector spaces in the middle are isomorphic. \Box A subset $Q\subset\enma{\mathbb{Z}}^n$ is an {\it order ideal} if $\enma{\mathbf{b}}\in Q$ and $\enma{\mathbf{c}} \in \enma{\mathbb{N}}^n$ implies $\enma{\mathbf{b}}-\enma{\mathbf{c}}\in Q$. For a $\enma{\mathbb{Z}}^n$-graded cell complex $X$ and an order ideal $Q$ we define the {\it order ideal complex\/} $\,X_Q \, = \, \bigsetdef{F\in X}{\enma{\mathbf{a}}_F\in Q}$. Note that $\,X_{\prec \enma{\mathbf{b}}}$ and $\,X_{\preceq \enma{\mathbf{b}}}$ are special cases of this. \corollary{corK} If $\enma{\mathbf{F}}_X$ is a cellular resolution of $M$ and $Q\subset\enma{\mathbb{Z}}^n$ an order ideal which contains the Betti degrees of $M$, then $\enma{\mathbf{F}}_{X_Q}$ is also a cellular resolution of $M$. \stdbreak\noindent{\bf Proof. } By \ref{makefinite} and the identity $\,(X_Q)_{\preceq \enma{\mathbf{b}}} = (X_{\preceq \enma{\mathbf{b}}})_Q$, it suffices to prove this for the case where $M$ is a monomial ideal and $X$ is finite. We proceed by induction on the number of faces in $X \backslash X_Q$. If $X_Q = X$ there is nothing to prove. Otherwise pick $\enma{\mathbf{c}} \in \enma{\mathbb{Z}}^n \backslash Q$ such that $X_{\preceq \enma{\mathbf{c}}} = X$ and $X_{\prec \enma{\mathbf{c}}} \not= X$. Since $\enma{\mathbf{c}}$ is not a Betti degree, \ref{propA} implies that the complex $X_{\prec \enma{\mathbf{c}}}$ is acyclic. For any $\enma{\mathbf{b}} \in \enma{\mathbb{Z}}^n $, the complex $\,(X_{\prec \enma{\mathbf{c}}})_{\preceq \enma{\mathbf{b}}}\,$ equals either $X_{\prec \enma{\mathbf{c}}}$ or $X_{\preceq\, \enma{\mathbf{b}} \wedge \enma{\mathbf{c}}}$ and is hence acyclic. At this point we replace $X$ by the proper subcomplex $X_{\prec \enma{\mathbf{c}}}$, and we are done by induction. \Box By \ref{propT} and \ref{propA}, the Betti numbers $\beta_{i,\enma{\mathbf{b}}}$ of $M$ are given by the reduced homology of $\Delta_{\prec\enma{\mathbf{b}}}$. Let us compare that formula for $\beta_{i,\enma{\mathbf{b}}}$ with the following formula which is due independently to Hochster [Ho] and Rosenknop [Ros]. \corollary{propH} The Betti numbers of $M$ satisfy $ \,\beta_{i,\enma{\mathbf{b}}} \, = \,\dim \widetilde{H}_i(K_\enma{\mathbf{b}};k)$ where $K_\enma{\mathbf{b}}$ is the simplicial complex $\{\, \sigma \subseteq\set{1,\ldots,n}\,|\, M \! \mtext{has a generator of degree} \! \!\preceq\enma{\mathbf{b}}- \sigma\}$. Here each face $\sigma$ of $K_\enma{\mathbf{b}}$ is identified with its characteristic vector in $\{0,1\}^n$. \stdbreak\noindent{\bf Proof. } For $i \in \{1,\ldots,n\}$ consider the subcomplex of $\Delta_{\prec \enma{\mathbf{b}}}$ consisting of all faces $F$ with degree $\enma{\mathbf{a}}_F \preceq \enma{\mathbf{b}} - \{i\}$. This subcomplex is a full simplex. Clearly, these $n$ simplices cover $\Delta_{\prec \enma{\mathbf{b}}}$. The nerve of this cover by contractible subsets is the simplicial complex $K_\enma{\mathbf{b}}$. Therefore, $K_\enma{\mathbf{b}}$ has the same reduced homology as $\Delta_{\prec \enma{\mathbf{b}}}$. \Box \section{} The hull resolution \noindent Let $M$ be a monomial module in $T = k[x_1^{\pm 1}\!\!,\, \dots ,\, x_n^{\pm 1}]$. In this section we apply convexity methods to construct a canonical cellular resolution of $M$. For $\enma{\mathbf{a}} \in \enma{\mathbb{Z}}^n$ and $t \in \enma{\mathbb{R}}$ we abbreviate $t^\enma{\mathbf{a}} = (t^{a_1}, \dots, t^{a_n})$. Fix any real number $t $ larger than $ (n+1) \,!\, = 2 \cdot 3 \cdot \cdots \cdot (n+1)$. We define $P_t$ be the convex hull of the point set $$\setdef{ t^\enma{\mathbf{a}} }{\enma{\mathbf{a}} \mtext{is the exponent of a monomial} \enma{\mathbf{x}}^\enma{\mathbf{a}}\in M} \quad \subset \quad \enma{\mathbb{R}}^n.$$ The set $P_t$ is a closed, unbounded $n$-dimensional convex polyhedron. \proposition{propO} The vertices of the polyhedron $P_t$ are precisely the points $ t^\enma{\mathbf{a}} = (t^{a_1},\ldots,t^{a_n}) $ for which the monomial $ \enma{\mathbf{x}}^\enma{\mathbf{a}} = x_1^{a_1}\! \cdots x_n^{a_n} $ is a minimal generator of $M$. \stdbreak\noindent{\bf Proof. } Suppose $\enma{\mathbf{x}}^\enma{\mathbf{a}} \in M$ is not a minimal generator of $M$. Then $M$ contains both $\enma{\mathbf{x}}^{\enma{\mathbf{a}}+\enma{\mathbf{e}}_i} = \enma{\mathbf{x}}^\enma{\mathbf{a}} x_i$ and $\enma{\mathbf{x}}^{\enma{\mathbf{a}}-\enma{\mathbf{e}}_i} = \enma{\mathbf{x}}^\enma{\mathbf{a}} /x_i$ for some $i$. The line segment $\mathop{\rm conv}\nolimits \{t^{\enma{\mathbf{a}}-\enma{\mathbf{e}}_i},t^{\enma{\mathbf{a}}+\enma{\mathbf{e}}_i} \}$ lies in $P_t$ and contains $t^\enma{\mathbf{a}}$ in its relative interior. Therefore $t^\enma{\mathbf{a}}$ is not a vertex of $P_t$. Next, suppose $\enma{\mathbf{x}}^\enma{\mathbf{a}} \in M$ is a minimal generator of $M$. Let $\enma{\mathbf{v}} = t^{-\enma{\mathbf{a}}}$, so $\enma{\mathbf{v}}\cdot t^{\enma{\mathbf{a}}} = n$. For any other exponent $\enma{\mathbf{b}}$ of a monomial in $M$, we have $b_i \ge a_{i}+1$ for some $i$, so $$ \enma{\mathbf{v}} \cdot t^\enma{\mathbf{b}} \quad = \quad \sum_{j=1}^n t^{b_j - a_j} \quad \geq \quad t^{b_i-a_{i}} \quad \ge \quad t \quad > \quad (n+1) \,! \quad > \quad n .$$ Thus, the inner normal vector $\enma{\mathbf{v}}$ supports $t^{\enma{\mathbf{a}}}$ as a vertex of $P_t$. \Box \corollary{} $\, P_t \,\,\, = \,\,\, \enma{\mathbb{R}}_+^n \,\, + \,\,\mathop{\rm conv}\nolimits\, \setdef{ t^\enma{\mathbf{a}} }{\enma{\mathbf{x}}^\enma{\mathbf{a}} \mtext{is a minimal generator} \enma{\mathbf{x}}^\enma{\mathbf{a}} \mtext{of} M}$. Our first goal is to establish the following combinatorial result. \theorem{indepthm} The face poset of the polyhedron $P_t$ is independent of $t$ for $\,t > (n+1) \,!$. The same holds for the subposet of all bounded faces of $P_t$. \stdbreak\noindent{\bf Proof. } The face poset of $P_t$ can be computed as follows. Let $C_t \subset \enma{\mathbb{R}}^{n+1}$ be the cone spanned by the vectors $\,(t^\enma{\mathbf{a}} ,1 ) \,$ for all minimal generators $\enma{\mathbf{x}}^\enma{\mathbf{a}}$ of $M$ together with the unit vectors $\,(\enma{\mathbf{e}}_i, 0)\,$ for $i=1,\ldots,n$. The faces of $P_t$ are in order-preserving bijection with the faces of $C_t$ which do not lie in the hyperplane ``at infinity'' $\,x_{n+1} = 0$. A face of $P_t$ is bounded if and only if the corresponding face of $C_t$ contains none of the vectors $(\enma{\mathbf{e}}_i,0)$. It suffices to prove that the face poset of $C_t$ is independent of $t$. For any $(n+1)$-tuple of generators of $C_t$ consider the sign of their determinant $$ \mathop{\rm sign}\nolimits \,\, \mathop{\rm det}\nolimits \bmatrix{2pt}{ \enma{\mathbf{e}}\ssub{i_0} & \;\;\cdots & \enma{\mathbf{e}}\ssub{i_r} & \;\;\; t^{\enma{\mathbf{a}}\ssub{j_1}} & \;\;\cdots & t^{\enma{\mathbf{a}}\ssub{j_{n-r}}} & \;\quad\cr 0 & \cdots & 0 & 1 & \cdots & 1 \cr} \quad \in \quad \{-1,0,+1 \}. \eqno (1) $$ The list of these signs forms the (possibly infinite) {\it oriented matroid} associated with the cone $C_t$. It is known (see e.g.~[BLWSZ]) that the face poset of $C_t$ is determined by its oriented matroid. It therefore suffices to show that the sign of the determinant in (1) is independent of $t$ for $t > (n+1) \, !$. This follows from the next lemma. \Box \lemma{factoriallem} Let $a_{ij}$ be integers for $\,1 \leq i,j \leq r$. Then the Laurent polynomial $ f(t) = \mathop{\rm det}\nolimits \bigl( (t^{a_{ij}})_{1 \leq i,j \leq r} ) \,$ either vanishes identically or has no real roots for $t > r \,!$. \stdbreak\noindent{\bf Proof. } Suppose that $f$ is not zero and write $\,f(t) = c_\alpha t^\alpha + \sum_\beta c_\beta t^\beta$, where the first term has the highest degree in $t$. For $t > r!$ we have the chain of inequalities $$ | \sum_\beta c_\beta \cdot t^\beta | \, \leq \, \sum_\beta |c_\beta | \cdot t^\beta \, \leq \, (\sum_\beta |c_\beta | ) \cdot t^{\alpha-1} \, < \, r \, ! \cdot t^{\alpha-1} \, < \, t^\alpha \, \le \, |c_\alpha \cdot t^\alpha| .$$ Therefore $f(t)$ is nonzero, and $\, \mathop{\rm sign}\nolimits\bigl(f(t)\bigr) = \mathop{\rm sign}\nolimits(c_\alpha)$. \Box In the proof of \ref{indepthm} we are using \ref{factoriallem} for $r=n+1$. Lev Borisov and Sorin Popescu constructed examples of matrices which show that the exponential lower bound for $t$ is necessary in \ref{factoriallem}, and also in \ref{indepthm}. We are now ready to define the hull resolution and state our main result. The {\it hull complex} of a monomial module $M$, denoted $\mathop{\rm hull}\nolimits(M)$, is the complex of bounded faces of the polyhedron $P_t$ for large $t$. \ref{indepthm} ensures that $\mathop{\rm hull}\nolimits(M)$ is well-defined and depends only on $M$. The vertices of $\mathop{\rm hull}\nolimits(M)$ are labeled by the generators of $M$, by \ref{propO}, and hence the complex $\,\mathop{\rm hull}\nolimits(M) \,$ is $\enma{\mathbb{Z}}^n$-graded. Let $\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M)}$ be the complex of free $S$-modules derived from $\mathop{\rm hull}\nolimits(M)$ as in Section~1. \theorem{thmH} The cellular complex $\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M)}$ is a free resolution of $M$. \stdbreak\noindent{\bf Proof. } Let $X = (\mathop{\rm hull}\nolimits(M))_{\preceq\enma{\mathbf{b}}}$ for some degree \enma{\mathbf{b}}; by \ref{propD} we need to show that $X$ is acyclic. This is immediate if $X$ is empty or a single vertex. Otherwise choose $\,t > (n+1)\,! \,$ and let $\enma{\mathbf{v}}=t^{-\enma{\mathbf{b}}}$. If $t^\enma{\mathbf{a}}$ is a vertex of $X$ then $\enma{\mathbf{a}} \prec\enma{\mathbf{b}}$, so $$\enma{\mathbf{v}} \cdot t^{\enma{\mathbf{a}}} \quad = \quad t^{-\enma{\mathbf{b}}} \cdot t^{\enma{\mathbf{a}}} \quad < \quad t^{-\enma{\mathbf{b}}} \cdot t^\enma{\mathbf{b}} \quad = \quad n,$$ while for any other $\enma{\mathbf{x}}^\enma{\mathbf{c}} \in M$ we have $c_{i}\ge b_i+1$ for some $i$, so $$\enma{\mathbf{v}} \cdot t^{\enma{\mathbf{a}}} \quad = \quad t^{-\enma{\mathbf{b}}} \cdot t^{\enma{\mathbf{c}}} \quad \ge \quad t^{c_i-b_i} \quad \ge \quad t \quad > \quad n.$$ Thus, the hyperplane $H$ defined by $\enma{\mathbf{v}}\cdot\enma{\mathbf{x}} = n$ separates the vertices of $X$ from the remaining vertices of $P_t$. Make a projective transformation which moves $H$ to infinity. This expresses $X$ as the complex of bounded faces of a convex polyhedron, a complex which is known to be contractible, e.g.~[BLSWZ, Exercise 4.27 (a)]. \Box We call $\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M)}$ the {\it hull resolution} of $M$. Let us see that the hull resolution generalizes the {\it Scarf complex} introduced in [BPS]. This is the simplical complex $$\Delta_M \quad = \quad \setdef{F \subseteq I}{m_F \ne m_G \mtext{for all} G \subseteq I \mtext{other than} F}.$$ The Scarf complex $\Delta_M$ defines a subcomplex $\enma{\mathbf{F}}_{\Delta_M}$ of the Taylor resolution $\enma{\mathbf{F}}_\Delta$. \proposition{scarfinhull} For any monomial module $M$, the Scarf complex ${\Delta_M}$ is a subcomplex of the hull complex $\mathop{\rm hull}\nolimits(M)$. \stdbreak\noindent{\bf Proof. } Let $F = \{ \enma{\mathbf{x}}^{\enma{\mathbf{a}}_1},\! \ldots,\enma{\mathbf{x}}^{\enma{\mathbf{a}}_p}\}$ be a face of $\Delta_M$ with $\,m_F = \mathop{\rm lcm}\nolimits(F) = \enma{\mathbf{x}}^\enma{\mathbf{u}}$. Consider any injective map $\sigma : \{1,\ldots,p\} \rightarrow \{1,\ldots,n\}$ such that $a_{i,\sigma(i)} = u_i$ for all $i$. Compute the inverse of the $p \times p$-matrix $(t^{a_{i,\sigma(j)}})$, and let $\enma{\mathbf{v}}^\sigma(t)'$ be the sum of the column vectors of that inverse matrix. By augmenting the $p$-vector $\enma{\mathbf{v}}^\sigma(t)'$ with additional zero coordinates, we obtain an $n$-vector $\enma{\mathbf{v}}^\sigma(t)$ with the following properties: \item{(i)} $\, \enma{\mathbf{v}}^\sigma(t) \cdot t^{\enma{\mathbf{a}}_1} = \enma{\mathbf{v}}^\sigma(t) \cdot t^{\enma{\mathbf{a}}_2} = \cdots = \enma{\mathbf{v}}^\sigma(t) \cdot t^{\enma{\mathbf{a}}_p} \,=\, 1$; \item{(ii)} $\, v^\sigma_j(t) \, = \, 0 \,$, for all $j \not\in \mathop{\rm image}\nolimits(\sigma)$; \item{(iii)} $\, v^\sigma_j(t) \, = \, t^{-u_j} \,+ \,$ {\sl lower order terms in $t$}, for all $j \in \mathop{\rm image}\nolimits(\sigma)$. By taking a convex combination of the vectors $\enma{\mathbf{v}}^\sigma(t)$ for all possible injective maps $\sigma$ as above, we obtain a vector $\enma{\mathbf{v}}(t)$ with the following properties: \item{(iv)} $\, \enma{\mathbf{v}}(t) \cdot t^{\enma{\mathbf{a}}_1} = \enma{\mathbf{v}}(t) \cdot t^{\enma{\mathbf{a}}_2} = \cdots = \enma{\mathbf{v}}(t) \cdot t^{\enma{\mathbf{a}}_p} \,=\, 1$; \item{(v)} $\, v_j(t) \, = \, c_j \cdot t^{-u_j} \,+ \, $ {\sl lower order terms in $t\,$} with $c_j > 0 $, for all $j \in \{1,\ldots,n\}$. For any $\enma{\mathbf{x}}^\enma{\mathbf{b}} \in M$ which is not in $F$ there exists an index $\ell$ such that $b_\ell \geq u_\ell + 1$. This implies $\, \enma{\mathbf{v}}(t) \cdot t^\enma{\mathbf{b}} \geq c_\ell \cdot t^{b_\ell-u_\ell} \,+ \, $ {\sl lower order terms in} $t$, and therefore $\, \enma{\mathbf{v}}(t) \cdot t^\enma{\mathbf{b}} > 1\,$ for $t \gg 0$. We conclude that $F$ defines a face of $P_t$ with inner normal vector $\enma{\mathbf{v}}(t)$. \Box A binomial first syzygy of $M$ is called {\it generic} if it has full support, i.e., if no variable $x_i$ appears with the same exponent in the corresponding pair of monomial generators. We call $M$ {\it generic} if it has a basis of generic binomial first syzygies. This is a translation-invariant generalization of the definition of genericity in [BPS]. \lemma{genlem} If $M$ is generic, then for any pair of generators $m_i$, $m_j$ either the corresponding binomial first syzygy is generic, or there exists a third generator $m$ which strictly divides the least common multiple of $m_i$ and $m_j$ in all coordinates. \stdbreak\noindent{\bf Proof. } Suppose that the syzygy formed by $m_i$ and $m_j$ is not generic, and induct on the length of a chain of generic syzygies needed to express it. If the chain has length two, then the middle monomial $m$ divides $\,\mathop{\rm lcm}\nolimits(m_i,m_j)$. Moreover, because the two syzygies involving $m$ are generic, this division is strict in each variable. If the chain is longer, then divide it into two steps. Either each step represents a generic syzygy, and we use the above argument, or by induction there exists an $m_j$ strictly dividing the degree of one of these syzygies in all coordinates, and we are again done. \Box \lemma{notunder} Let $M$ be a monomial module and $F$ a face of $\mathop{\rm hull}\nolimits(M)$. For every monomial $m \in M$ there exists a variable $x_j$ such that $\mathop{\rm deg}\nolimits_{x_j}(m) \geq \mathop{\rm deg}\nolimits_{x_j}(m_F)$. \stdbreak\noindent{\bf Proof. } Suppose that $m = \enma{\mathbf{x}}^\enma{\mathbf{u}}$ strictly divides $m_F$ in each coordinate. Let $t^{\enma{\mathbf{a}}_1},\ldots,t^{\enma{\mathbf{a}}_p}$ be the vertices of $F$ and consider their barycenter $\,\enma{\mathbf{v}}(t) = {1 \over p} \cdot (t^{\enma{\mathbf{a}}_1}+ \cdots + t^{\enma{\mathbf{a}}_p})\,\in \, F$. The $j$th coordinate of $\enma{\mathbf{v}}(t)$ is a polynomial in $t$ of degree equal to $\,\mathop{\rm deg}\nolimits_{x_j}(m_F)$. The $j$th coordinate of $t^\enma{\mathbf{u}}$ is a monomial of strictly lower degree. Hence $\,t^{\bf u} < \enma{\mathbf{v}}(t)\,$ coordinatewise for $t \gg 0$. Let $\enma{\mathbf{w}}$ be a nonzero linear functional which is nonnegative on $\enma{\mathbb{R}}_+^n $ and whose minimum over $P_t$ is attained at the face $F$. Then $\enma{\mathbf{w}} \cdot \enma{\mathbf{v}}(t) = \enma{\mathbf{w}} \cdot \enma{\mathbf{a}}_1 = \cdots = \enma{\mathbf{w}} \cdot \enma{\mathbf{a}}_p $, but our discussion implies $\enma{\mathbf{w}} \cdot t^\enma{\mathbf{u}} < \enma{\mathbf{w}} \cdot \enma{\mathbf{v}}(t) $, a contradiction. \Box \theorem{genmin} If $M$ is a generic monomial module then $\mathop{\rm hull}\nolimits(M)$ coincides with the Scarf complex $\Delta_M$ of $M$, and the hull resolution $\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M)} = \enma{\mathbf{F}}_{\Delta_M}$ is minimal. \stdbreak\noindent{\bf Proof. } Let $F$ be any face of $\mathop{\rm hull}\nolimits(M)$ and $\enma{\mathbf{x}}^{\enma{\mathbf{a}}_1}, \ldots,\enma{\mathbf{x}}^{\enma{\mathbf{a}}_p}$ the generators of $M$ corresponding to the vertices of $F$. Suppose that $F$ is not a face of $\Delta_M$. Then either \item{(i)} $\, \mathop{\rm lcm}\nolimits(\enma{\mathbf{x}}^{\enma{\mathbf{a}}_1}, \ldots,\enma{\mathbf{x}}^{\enma{\mathbf{a}}_{i-1}}, \enma{\mathbf{x}}^{\enma{\mathbf{a}}_{i+1}},\ldots,\enma{\mathbf{x}}^{\enma{\mathbf{a}}_p}) = m_F \,$ for some $i \in \{1,\ldots,p\}$, or \item{(ii)} there exists another generator $\enma{\mathbf{x}}^\enma{\mathbf{u}}$ of $ M$ which divides $m_F$ and such that $t^\enma{\mathbf{u}} \not\in F$. Consider first case (i). By \ref{notunder} applied to $m = {\bf x}^{\enma{\mathbf{a}}_i} $ there exists $x_j$ such that $\mathop{\rm deg}\nolimits_{x_j}(\enma{\mathbf{x}}^{\enma{\mathbf{a}}_i}) = \mathop{\rm deg}\nolimits_{x_j}(m_F)$, and hence $\mathop{\rm deg}\nolimits_{x_j}(\enma{\mathbf{x}}^{\enma{\mathbf{a}}_i}) = \mathop{\rm deg}\nolimits_{x_j}(\enma{\mathbf{x}}^{\enma{\mathbf{a}}_k})$ for some $k \not= i$. The first syzygy between $\enma{\mathbf{x}}^{\enma{\mathbf{a}}_i}$ and $\enma{\mathbf{x}}^{\enma{\mathbf{a}}_k}$ is not generic, and, by \ref{genlem}, there exists a generator $m$ of $M$ which strictly divides $\mathop{\rm lcm}\nolimits(\enma{\mathbf{x}}^{\enma{\mathbf{a}}_i},\enma{\mathbf{x}}^{\enma{\mathbf{a}}_k})$ in all coordinates. Since $\mathop{\rm lcm}\nolimits(\enma{\mathbf{x}}^{\enma{\mathbf{a}}_i},\enma{\mathbf{x}}^{\enma{\mathbf{a}}_k})$ divides $m_F$, we get a contradiction to \ref{notunder}. Consider now case (ii). For any variable $x_j$ there exists $i \in \{1,\ldots,p\}$ such that $\mathop{\rm deg}\nolimits_{x_j}(\enma{\mathbf{x}}^{ \enma{\mathbf{a}}_i}) = \mathop{\rm deg}\nolimits_{x_j}(m_F) \geq \mathop{\rm deg}\nolimits_{x_j}(\enma{\mathbf{x}}^\enma{\mathbf{u}})$. If the inequality $\geq$ is an equality $=$, then the first syzygy between $\enma{\mathbf{x}}^\enma{\mathbf{u}}$ and $\enma{\mathbf{x}}^{\enma{\mathbf{a}}_i}$ is not generic, and \ref{genlem} gives a new monomial generator $m$ which strictly divides $m_F$ in all coordinates, a contradiction to \ref{notunder}. Therefore $\geq$ is a strict inequality $>$ for all variables $x_j$. This means that $\enma{\mathbf{x}}^\enma{\mathbf{u}}$ strictly divides $m_F$ in all coordinates, again a contradiction to \ref{notunder}. Hence both (i) and (ii) lead to a contradiction, and we conclude that every face of $\mathop{\rm hull}\nolimits(M)$ is a face of $\Delta_M$. This implies $\mathop{\rm hull}\nolimits(M) = \Delta_M$ by \ref{scarfinhull}. The resolution $\enma{\mathbf{F}}_{\Delta_M}$ is minimal because no two faces in $\Delta_M$ have the same degree. \Box In this paper we are mainly interested in nongeneric monomial modules for which the hull complex is typically not simplicial. Nevertheless the possible combinatorial types of facets seem to be rather limited. Experimental evidence suggests: \conjecture{} Every face of $\mathop{\rm hull}\nolimits(M)$ is affinely isomorphic to a subpolytope of the $(n-1)$-dimensional permutohedron and hence has at most $\, n \, ! \,$ vertices. By \ref{exD} it is easy to see that any subpolytope of the $(n-1)$-dimensional permutohedron can be realized as the hull complex of suitable monomial ideal. The following example, found in discussions with Lev Borisov, shows that the hull complex of a monomial module need not be locally finite: \example{lev} Let $n=3$ and $M$ the monomial module generated by $x_1^{-1} x_2$ and $\setdef{x_2^i x_3^{-i}}{i\in\enma{\mathbb{Z}}}$. Then every triangle of the form $\setthree{x_1^{-1} x_2}{ x_2^i x_3^{-i}}{x_2^{i+1} x_3^{-i-1}}$ is a facet of $\mathop{\rm hull}\nolimits(M)$. In particular, the vertex $x_1^{-1} x_2$ of $\mathop{\rm hull}\nolimits(M)$ has infinite valence. \Box For a generic monomial module $M$ we have the following important identity $$ \, \mathop{\rm hull}\nolimits(M_{\preceq \enma{\mathbf{b}}}) \quad = \quad \mathop{\rm hull}\nolimits(M)_{\preceq \enma{\mathbf{b}}}.$$ See equation (5.1) in [BPS]. This identity can fail if $M$ is not generic: \example{} Consider the monomial ideal $M=\idealfour{a^2b}{ac}{b^2\!}{bc^2}$ studied in \ref{exA} and let $\enma{\mathbf{b}}=(2,1,2)$. Then $\mathop{\rm hull}\nolimits(M_{\preceq \enma{\mathbf{b}}})$ is a triangle, while $\mathop{\rm hull}\nolimits(M)_{\preceq \enma{\mathbf{b}}}$ consists of two edges. The vertex $b^2$ of $\mathop{\rm hull}\nolimits(M)$ ``eclipses'' the facet of $\mathop{\rm hull}\nolimits(M_{\preceq \enma{\mathbf{b}}})$ \Box The hull complex $\mathop{\rm hull}\nolimits(M)$ is particularly easy to compute if $M$ is a squarefree monomial ideal. In this case we have $P_t = P_1$ for all $t$. Moreover, if all square-free generators of $M$ have the same total degree, then the faces of their convex hull are precisely the bounded faces of $P_t$. \ref{thmH} implies the following corrollary. \corollary{squarefree} Let $\enma{\mathbf{a}}_1,\ldots,\enma{\mathbf{a}}_p$ be $0$-$1$-vectors having the same coordinate sum. Then their boundary complex, consisting of all faces of the convex polytope $\,P = \mathop{\rm conv}\nolimits \{\enma{\mathbf{a}}_1,\ldots,\enma{\mathbf{a}}_p\}$, defines a cellular resolution of the ideal $\,M = \langle \enma{\mathbf{x}}^{\enma{\mathbf{a}}_1},\ldots, \enma{\mathbf{x}}^{\enma{\mathbf{a}}_p} \rangle$. \example{exRb} \ref{squarefree} applies to the Stanley-Reisner ideal of the real projective plane in \ref{exR}. Here $P$ is a 5-dimensional polytope with 22 facets, corresponding to the 22 cycles on the $2$-complex $X$ of length $\le 6$. Representatives of these three cycle types, and supporting hyperplanes of the corresponding facets of $P$, are shown on the right in \ref{rpfig}. This example illustrates how the hull resolution encodes combinatorial information without making arbitrary choices. \Box \section{} Lattice ideals \noindent Let $L\subset\enma{\mathbb{Z}}^n$ be a lattice. In this section we study (cellular) resolutions of the lattice module $M_L$ and of the lattice ideal $I_L$. Let $S[L]$ be the group algebra of $L$ over $S$. We realize $S[L]$ as the subalgebra of $\,k[x_1,\ldots,x_n, z_1^{\pm 1}\!\!,\, \dots ,\, z_n^{\pm 1}] \,$ spanned by all monomials $\enma{\mathbf{x}}^\enma{\mathbf{a}} \enma{\mathbf{z}}^\enma{\mathbf{b}}$ where $\enma{\mathbf{a}} \in \enma{\mathbb{N}}^n$ and $\enma{\mathbf{b}} \in L$. Note that $\,S \,= \,S[L]/ \langle \,\enma{\mathbf{z}}^\enma{\mathbf{a}} - 1 \,\,| \,\, \enma{\mathbf{a}} \in L\, \rangle $. \lemma{lem1} The lattice module $M_L$ is an $S[L]$-module, and $\,M_L \otimes_{S[L]} S \,= \, S/I_L$. \stdbreak\noindent{\bf Proof. } The $k$-linear map $\,\phi : S[L] \rightarrow M_L , \, \enma{\mathbf{x}}^\enma{\mathbf{a}} \enma{\mathbf{z}}^\enma{\mathbf{b}} \mapsto \enma{\mathbf{x}}^{\enma{\mathbf{a}} + \enma{\mathbf{b}}} \,$ defines the structure of an $S[L]$-module on $M_L$. Its kernel $\,\mathop{\rm ker}\nolimits(\phi) \,$ is the ideal in $S[L]$ generated by all binomials $\,\enma{\mathbf{x}}^\enma{\mathbf{u}} - \enma{\mathbf{x}}^\enma{\mathbf{v}} \enma{\mathbf{z}}^{\enma{\mathbf{u}}-\enma{\mathbf{v}}}$ where $\enma{\mathbf{u}},\enma{\mathbf{v}} \in \enma{\mathbb{N}}^n$ and $\enma{\mathbf{u}} - \enma{\mathbf{v}} \in L$. Clearly, we obtain $I_L$ from $\mathop{\rm ker}\nolimits(\phi)$ by setting all $\enma{\mathbf{z}}$-variables to $1$, and hence $\,(S[L]/\mathop{\rm ker}\nolimits(\phi) )\otimes_{S[L]} S \,= \, S/I_L$. \Box We define a $\enma{\mathbb{Z}}^n$-grading on $S[L]$ via $\,\mathop{\rm deg}\nolimits(\enma{\mathbf{x}}^\enma{\mathbf{a}} \enma{\mathbf{z}}^\enma{\mathbf{b}}) = \enma{\mathbf{a}} + \enma{\mathbf{b}}$. Let ${\cal A}$ be the category of $\enma{\mathbb{Z}}^n$-graded $S[L]$-modules, where the morphisms are $\enma{\mathbb{Z}}^n$-graded $S[L]$-module homomorphisms of degree $\enma{\mathbf{0}}$. The polynomial ring $S = k[x_1,\ldots,x_n]$ is graded by the quotient group $\enma{\mathbb{Z}}^n/L$ via $\,\mathop{\rm deg}\nolimits(\enma{\mathbf{x}}^\enma{\mathbf{a}}) = \enma{\mathbf{a}} + L$. Let ${\cal B}$ be the category of $\enma{\mathbb{Z}}^n/L$-graded $S$-modules, where the morphisms are $\enma{\mathbb{Z}}^n/L$-graded $S$-module homomorphisms of degree $\enma{\mathbf{0}}$. Clearly, $M_L$ is an object in ${\cal A}$, and $\,M_L \otimes_{S[L]} S \,= \, S/I_L \,$ is an object in ${\cal B}$. \theorem{thmU} The categories ${\cal A}$ and ${\cal B}$ are equivalent. \stdbreak\noindent{\bf Proof. } Define a functor $\pi:{\cal A} \rightarrow {\cal B}$ by the rule $\,\pi(M) := M \otimes_{S[L]} S$. This functor weakens the $\enma{\mathbb{Z}}^n$-grading of objects in ${\cal A}$ to a $\enma{\mathbb{Z}}^n/L$-grading. The properties of $\pi$ cannot be deduced from the tensor product alone, which is poorly behaved when applied to arbitrary $S[L]$-modules; e.g., $S$ is not a flat $S[L]$-module. Further, the categories ${\cal A}$ and ${\cal B}$ are {\sl not isomorphic}; we are only claiming that they are {\sl equivalent}. We apply condition iii) of [Mac, \S IV.4, Theorem 1]: It is enough to prove that $\pi$ is full and faithful, and that each object $N\in{\cal B}$ is isomorphic to $\pi(M)$ for some object $M\in{\cal A}$. To prove that $\pi$ is full and faithful, we show that for any two modules $M$, $M'\in {\cal A}$ it induces an identification $\mathop{\rm Hom}\nolimits_{\cal A}(M,M') = \mathop{\rm Hom}\nolimits_{\cal B}(\pi(M),\pi(M'))$. Because each module $M\in{\cal A}$ is $\enma{\mathbb{Z}}^n$-graded, the lattice $L\subset S[L]$ acts on $M$ as a group of automorphisms, i.e. the multiplication maps $\enma{\mathbf{z}}^\enma{\mathbf{b}}: M_\enma{\mathbf{a}} \rightarrow M_{\enma{\mathbf{a}}+\enma{\mathbf{b}}}$ are isomorphisms of $k$-vector spaces for each $\enma{\mathbf{b}} \in L$, compatible with multiplication by each $x_i$. For each $\alpha\in\enma{\mathbb{Z}}^n/L$, the functor $\pi$ identifies the spaces $M_\enma{\mathbf{a}}$ for $\enma{\mathbf{a}}\in\alpha$ as the single space $\pi(M)_\alpha$. A morphism $f:M\rightarrow M'$ in ${\cal A}$ is a collection of $k$-linear maps $f_\enma{\mathbf{a}}: M_\enma{\mathbf{a}}\rightarrow M'_\enma{\mathbf{a}}$, compatible with the action by $L$ and with multiplication by each $x_i$. A morphism $g:\pi(M)\rightarrow \pi(M')$ in ${\cal B}$ is a collection of $k$-linear maps $g_\alpha:\pi(M)_\alpha\rightarrow \pi(M')_\alpha$, compatible with multiplication by each $x_i$. For each $\alpha\in\enma{\mathbb{Z}}^n/L$, the functor $\pi$ identifies the maps $f_\enma{\mathbf{a}}$ for $\enma{\mathbf{a}}\in\alpha$ as the single map $\pi(f)_\alpha$. It is clear from this discussion that $\pi$ takes distinct morphisms to distinct morphisms. Given a morphism $g\in\mathop{\rm Hom}\nolimits_{\cal B}(\pi(M),\pi(M'))$, define a morphism $f\in\mathop{\rm Hom}\nolimits_{\cal A}(M,M')$ by the rule $f_\enma{\mathbf{a}}=g_\alpha$ for $\enma{\mathbf{a}}\in\alpha$. We have $\pi(f)=g$, establishing the desired identification of Hom-sets. Hence $\pi$ is full and faithful. Finally, let $N = \mathop{\hbox{$\bigoplus$}}_{ \alpha \in \enma{\mathbb{Z}}^n/L} N_\alpha$ be any object in ${\cal B}$. We define an object $M = \oplus_{\enma{\mathbf{a}} \in \enma{\mathbb{Z}}^n} M_\enma{\mathbf{a}}$ in ${\cal A}$ by setting $M_\enma{\mathbf{a}} := N_\alpha$ for each $\enma{\mathbf{a}} \in \alpha$, by lifting each multiplication map $x_i:N_\alpha \rightarrow N_{\alpha+\enma{\mathbf{e}}_i}$ to maps $x_i:M_\enma{\mathbf{a}} \rightarrow M_{\enma{\mathbf{a}}+\enma{\mathbf{e}}_i}$ for $\enma{\mathbf{a}} \in \alpha$, and by letting $\enma{\mathbf{z}}^\enma{\mathbf{b}}$ act on $M$ as the identity map from $M_\enma{\mathbf{a}} $ to $M_{\enma{\mathbf{a}}+\enma{\mathbf{b}}}$ for $\enma{\mathbf{b}} \in L$. The module $M$ satisfies $\pi(M) = N$, showing that $\pi$ is an equivalence of categories. \Box \ref{thmU} allows us to resolve the lattice module $M_L\in {\cal A}$ in order to resolve the quotient ring $\pi(M_L) = S/I_L\in {\cal B}$, and conversely. \corollary{corU} A $\enma{\mathbb{Z}}^n$-graded complex of free $S[L]$-modules, $$ C : \qquad \;\cdots \; \rightarrowbox{8pt}{$f_2$}\; S[L]^{\beta_1} \rightarrowbox{8pt}{$f_1$}\; S[L]^{\beta_0} \;\rightarrowbox{8pt}{$f_0$}\; S[L] \;\rightarrow \; M_L \;\rightarrow \; 0, $$ is a (minimal) free resolution of $M_L$ if and only if its image $$ \pi(C)\,: \, \;\cdots \; \rightarrowbox{8pt}{$\pi(f_2)$}\; S^{\beta_1} \rightarrowbox{8pt}{$\pi(f_1)$}\; S^{\beta_0} \;\rightarrowbox{8pt}{$\pi(f_0)$}\; S \;\rightarrow \; S/I_L \;\rightarrow \; 0 ,$$ is a (minimal) $\enma{\mathbb{Z}}^n/L$-graded resolution of $S/I_L$ by free $S$-modules. \stdbreak\noindent{\bf Proof. } This follows immediately from \ref{lem1} and \ref{thmU}. \Box Since $S[L]$ is a free $S$-module, every resolution $C$ as in the previous corollary gives rise to a resolution of $M_L$ as a $\enma{\mathbb{Z}}^n$-graded $S$-module. We demonstrate in an example how resolutions of $M_L$ over $S$ are derived from resolutions of $S/I_L$ over $S$. \example{exU} Let $S=k[x_1,x_2,x_3]$ and $L=\mathop{\rm ker}\nolimits \bmatrix{2pt}{1 & 1 & 1 \cr} \subset \enma{\mathbb{Z}}^3$. Then $\enma{\mathbb{Z}}^3 /L \simeq \enma{\mathbb{Z}}$, $I_L = \ideal{x_1-x_2,x_2-x_3}$, and $M_L$ is the module generated by all monomials of the form $\ x_1^i x_2^j x_3^{-i-j}$. The ring $S/I_L$ is resolved by the Koszul complex $$ 0 \longrightarrow S(-2) \rightarrowmat{4pt}{4pt}{\!\!x_2 - x_3 \cr x_2 - x_1 \cr} S(-1)^2 \rightarrowmat{4pt}{6pt}{x_1 - x_2 & x_2 - x_3 \cr} S \longrightarrow S/I_L . $$ This is a $\enma{\mathbb{Z}}^3/L$-graded complex of free $S$-modules. An inverse image under $\pi$ equals $$ \eqalign{ 0 \longrightarrow S[L]\bigl(-(1,1,0)\bigr) & \rightarrowmat{4pt}{4pt}{\!\!x_2 - x_3 z_2 z_3^{-1}\cr x_2- x_1 z_2 z_1^{-1} \cr} S[L]\bigl(-(1,0,0)\bigr) \oplus S[L]\bigl(-(0,1,0)\bigr) \cr & \qquad \qquad \rightarrowmat{4pt}{6pt}{x_1 - x_2 z_1 z_2^{-1} & x_2 - x_3 z_2 z_3^{-1} \cr} S[L] \longrightarrow M_L .\cr} $$ Writing each term as a direct sum of free $S$-modules, for instance, $\, S[L]\bigl(-(1,1,0)\bigr) = \oplus_{i+j+k=2} S \bigl(-(i,j,k) \bigr)$, we get a $\enma{\mathbb{Z}}^3$-graded minimal free resolution of $M_L$ over $S$: $$ 0 \,\, \rightarrow \mathop{\hbox{$\bigoplus$}}_{i+j+k=2}\! S \bigl(-(i,j,k) \bigr) \,\, \rightarrow \mathop{\hbox{$\bigoplus$}}_{i+j+k=1}\! S \bigl(-(i,j,k) \bigr)^2 \,\, \rightarrow \mathop{\hbox{$\bigoplus$}}_{i+j+k=0}\! S \bigl(-(i,j,k) \bigr) \rightarrow M_L. \Box $$ \vskip .2cm Our goal is to define and study cellular resolutions of the lattice ideal $I_L$. Let $X$ be a $\enma{\mathbb{Z}}^n$-graded cell complex whose vertices are the generators of $M_L$. Each cell $F \in X$ is identified with its set of vertices, regarded as a subset of $L$. The cell complex $X$ is called {\it equivariant} \ if $\,F \in X$ and $\enma{\mathbf{b}} \in L$ implies that $F + \enma{\mathbf{b}} \,\in \, X$, and if the incidence function satisfies $\,\varepsilon(F,F') = \varepsilon(F+\enma{\mathbf{b}},F'+\enma{\mathbf{b}})$ for all $\enma{\mathbf{b}} \in L$. \lemma{ } If $X$ is an equivariant $\enma{\mathbb{Z}}^n$-graded cell complex on $M_L$ then the cellular complex $\enma{\mathbf{F}}_X$ has the structure of a $\enma{\mathbb{Z}}^n$-graded complex of free $S[L]$-modules. \stdbreak\noindent{\bf Proof. } The group $L$ acts on the faces of $X$. Let $X/L$ denote the set of orbits. For each orbit ${\cal F} \in X/L $ we select a distinguished representative $F \in {\cal F}$, and we write $\mathop{\rm Rep}\nolimits(X/L)$ for the set of representatives. The following map is an isomorphism of $\enma{\mathbb{Z}}^n$-graded $S$-modules, which defines the structure of a free $S[L]$-module on $\enma{\mathbf{F}}_X$: $$ \mathop{\hbox{$\bigoplus$}}_{F \in \mathop{\rm Rep}\nolimits(X/L)} \!\!\!\! S[L] \cdot e_F \quad \simeq \quad \mathop{\hbox{$\bigoplus$}}_{F \in X }S \cdot e_F \, \, \,\,\, = \,\,\, \enma{\mathbf{F}}_X , \quad \, \enma{\mathbf{z}}^\enma{\mathbf{b}} \cdot e_F \,\, \mapsto \,\, e_{F + \enma{\mathbf{b}}}\ .$$ The differential $\partial $ on $\enma{\mathbf{F}}_X$ is compatible with the $S[L]$-action on $\enma{\mathbf{F}}_X$ because the incidence function is $L$-invariant. For each $F \in \mathop{\rm Rep}\nolimits(X/L)$ and $\enma{\mathbf{b}} \in L$ we have $$ \eqalign{ \partial(\enma{\mathbf{z}}^\enma{\mathbf{b}} \cdot e_F ) \quad &= \quad \partial ( e_{F+\enma{\mathbf{b}}}) \quad = \quad \sum_{F'\in X,\, F'\ne\emptyset} \; \varepsilon(F \! + \! \enma{\mathbf{b}},F' \! + \!\enma{\mathbf{b}}) \; {m_{F \!+\! \enma{\mathbf{b}}} \over m_{F'\! +\! \enma{\mathbf{b}}}} \; e_{F'+\enma{\mathbf{b}}} \cr & = \,\, \sum_{F'\in X,\, F'\ne\emptyset} \; \! \! \varepsilon(F,F') \; {m_{F} \over m_{F'}} \; \enma{\mathbf{z}}^\enma{\mathbf{b}} \cdot e_{F'} \quad = \quad \enma{\mathbf{z}}^\enma{\mathbf{b}} \cdot \partial( e_F ) . \cr} $$ Clearly, the differential $\partial$ is homogeneous of degree $0$, which proves the claim. \Box \corollary{} If $X$ is an equivariant $\enma{\mathbb{Z}}^n$-graded cell complex on $M_L$ then the cellular complex $\enma{\mathbf{F}}_X$ is exact over $S$ if and only if it is exact over $S[L]$. \stdbreak\noindent{\bf Proof. } The $\enma{\mathbb{Z}}^n$-graded components of $\enma{\mathbf{F}}_X$ are complexes of $k$-vector spaces which are independent of our interpretation of $\enma{\mathbf{F}}_X$ as an $S$-module or $S[L]$-module. \Box If $X$ is an equivariant $\enma{\mathbb{Z}}^n$-graded cell complex on $M_L$ such that $\enma{\mathbf{F}}_X$ is exact, then we call $\enma{\mathbf{F}}_X$ an {\it equivariant cellular resolution} of $M_L$. \corollary{cor17} If $\enma{\mathbf{F}}_X$ is an equivariant cellular (minimal) resolution of $M_L$ then $\, \pi(\enma{\mathbf{F}}_X)\,$ is a (minimal) resolution of $S/I_L$ by $\enma{\mathbb{Z}}^n/L$-graded free $S$-modules. \Box We call $\,\pi(\enma{\mathbf{F}}_X)\,$ a {\it cellular resolution} of the lattice ideal $I_L$. Let $Q$ be an order ideal in the quotient poset $\enma{\mathbb{N}}^n/L$. Then $Q + L$ is an order ideal in $\enma{\mathbb{N}}^n +L$, and the restriction $\,\enma{\mathbf{F}}_{X_{Q+L}}\,$ is a complex of $\enma{\mathbb{Z}}^n$-graded free $S[L]$-modules. We set $\,\pi(\enma{\mathbf{F}}_X)_Q \,:= \,\pi(\enma{\mathbf{F}}_{X_{Q+L}})$. This is a complex of $\enma{\mathbb{Z}}^n/L$-graded free $S$-modules. \ref{corK} implies \proposition{restr} If $\pi(\enma{\mathbf{F}}_X)$ is a cellular resolution of $I_L$ and $Q$ is an order ideal in $\enma{\mathbb{N}}^n/L$ which contains all Betti degrees then $\,\pi(\enma{\mathbf{F}}_{X})_Q\,$ is a cellular resolution of $I_L$. In what follows we shall study two particular cellular resolutions of $I_L$. \theorem{nicethm} The Taylor complex $\Delta$ on $M_L$ and the hull complex $\mathop{\rm hull}\nolimits(M_L)$ are equivariant. They define cellular resolutions $\,\pi(\enma{\mathbf{F}}_\Delta)\,$ and $\,\pi(\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M_L)}) \,$ of $I_L$. \stdbreak\noindent{\bf Proof. } The Taylor complex $\Delta$ consists of all finite subsets of generators of $M_L$. It has an obvious $L$-action. The hull complex also has an $L$-action: if $\,F = \mathop{\rm conv}\nolimits \bigl( \set{ t^{\enma{\mathbf{a}}_1},\ldots,t^{\enma{\mathbf{a}}_s} } \bigr) \,$ is a face of $\mathop{\rm hull}\nolimits(M_L)$ then $\,\enma{\mathbf{z}}^b \cdot F = \mathop{\rm conv}\nolimits \bigl( \set{ t^{\enma{\mathbf{a}}_1+\enma{\mathbf{b}}},\ldots,t^{\enma{\mathbf{a}}_s+\enma{\mathbf{b}}} } \bigr) \,$ is also a face of $\mathop{\rm hull}\nolimits(M_L)$ for all $\enma{\mathbf{b}} \in L$. In both cases the incidence function $\varepsilon$ is defined uniquely by the ordering of the elements in $L$. To ensure that $\varepsilon$ is $L$-invariant, we fix an ordering which is $L$-invariant; for instance, order the elements of $L$ by the value of an $\enma{\mathbb{R}}$-linear functional whose coordinates are $\enma{\mathbb{Q}}$-linearly independent. Both $\pi(\enma{\mathbf{F}}_\Delta)$ and $\pi(\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M_L)})$ are cellular resolutions of $I_L$ by \ref{cor17}. \Box The {\it Taylor resolution} $\pi(\enma{\mathbf{F}}_\Delta)$ of $I_L$ has the following explicit description. For $\alpha \in \enma{\mathbb{N}}^n/L$ let $\mathop{\rm fiber}\nolimits(\alpha)$ denote the (finite) set of all monomials $\enma{\mathbf{x}}^\enma{\mathbf{b}}$ with $\enma{\mathbf{b}} \in \alpha$. Thus $S_\alpha = k \cdot \mathop{\rm fiber}\nolimits(\alpha)$. Let $E_i(\alpha)$ be the collection of all $i$-element subsets $I$ of $\mathop{\rm fiber}\nolimits(\alpha)$ whose greatest common divisor $\mathop{\rm gcd}\nolimits(I)$ equals $1$. For $I \in E_i(\alpha)$ set $\mathop{\rm deg}\nolimits(I) := \alpha$. \proposition{explicit} The Taylor resolution $\pi(\enma{\mathbf{F}}_\Delta)$ of a lattice ideal $I_L$ is isomorphic to the $\enma{\mathbb{Z}}^n/L$-graded free $S$-module $\,\, \mathop{\hbox{$\bigoplus$}}_{\alpha \in \enma{\mathbb{N}}^n/L} S \cdot E_i(\alpha)\,$ with the differential $$ \partial(I)\quad = \quad \sum_{m \in I} \mathop{\rm sign}\nolimits(m,I) \cdot \mathop{\rm gcd}\nolimits(I \backslash \set{m})\cdot [I \backslash \set{m}]. \eqno (3.1) $$ In this formula, $\,[I \backslash \set{m}] $ denotes the element of $ \,E_{i-1}\bigl( \alpha - \mathop{\rm deg}\nolimits(\mathop{\rm gcd}\nolimits(I \backslash \set{m}))\bigr)$ which is obtained from $I \backslash \set{m}$ by removing the common factor $\,\mathop{\rm gcd}\nolimits(I \backslash \set{m})$. \stdbreak\noindent{\bf Proof. } For $\enma{\mathbf{b}} \in \enma{\mathbb{Z}}^n$ let $F_i(\enma{\mathbf{b}})$ denote the collection of $i$-element subsets of generators of $M_L$ whose least common multiple equals $\enma{\mathbf{b}}$. For $J \in F_i(\enma{\mathbf{b}})$ we have $\mathop{\rm lcm}\nolimits(J) = \enma{\mathbf{x}}^\enma{\mathbf{b}}$. The Taylor resolution $\enma{\mathbf{F}}_\Delta$ of $M_L$ equals $\, \mathop{\hbox{$\bigoplus$}}_{\enma{\mathbf{b}} \in \enma{\mathbb{N}}^n + L} S \cdot F_i(\enma{\mathbf{b}})\,$ with differential $$ \partial(J) \quad = \quad \sum_{m \in J} \mathop{\rm sign}\nolimits(m,J) \cdot {\mathop{\rm lcm}\nolimits(J) \over \mathop{\rm lcm}\nolimits( J \backslash \set{m})} \cdot J \backslash \set{m}. \eqno (3.2) $$ There is a natural bijection between $F_i(\enma{\mathbf{b}})$ and $E_i(\enma{\mathbf{b}}+L)$, namely, $\,J \, \mapsto \, \set{ \enma{\mathbf{x}}^\enma{\mathbf{b}} / \enma{\mathbf{x}}^\enma{\mathbf{c}} \,\,|\,\,\enma{\mathbf{x}}^\enma{\mathbf{c}} \in J} \, = \, I $. Under this bijection we have $\,{\enma{\mathbf{x}}^\enma{\mathbf{b}} \over \mathop{\rm lcm}\nolimits( J \backslash \set{m})} = \mathop{\rm gcd}\nolimits(I \backslash \set{m})$. The functor $\pi$ identifies each $F_i(\enma{\mathbf{b}})$ with $E_i(\enma{\mathbf{b}}+L)$ and it takes (3.2) to (3.1). \Box \corollary{} Let $Q$ be an order ideal in $\enma{\mathbb{N}}^n/L$ which contains all Betti degrees. Then $\,\pi(\enma{\mathbf{F}}_\Delta)_Q = \mathop{\hbox{$\bigoplus$}}_{\alpha \in Q} S E_i(\alpha)\,$ with differential (3.1) is a cellular resolution of $I_L$. \stdbreak\noindent{\bf Proof. } This follows from \ref{restr}, \ref{nicethm} and \ref{explicit}. \Box \example{} {\sl (Generic lattice ideals) } The lattice module $M_L$ is generic (in the sense of \S 2) if and only if the ideal $I_L$ is generated by binomials with full support. Suppose that this holds. It was shown in [PS] that the Betti degrees of $I_L$ form an order ideal $Q$ in $\enma{\mathbb{N}}^n/L$. \ref{genmin} and \ref{restr} imply that the resolution $\, \pi(\enma{\mathbf{F}}_\Delta)_Q $ is minimal and coincides with the hull resolution $\pi(\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M_L)})$. \Box \vskip .2cm The remainder of this section is devoted to the hull resolution of $I_L$. We next show that the hull complex $\mathop{\rm hull}\nolimits(M_L)$ is locally finite. This fact is nontrivial, in view of \ref{lev}. It will imply that the hull resolution has finite rank over $S$. Write each vector $\enma{\mathbf{a}} \in L \subset \enma{\mathbb{Z}}^n$ as difference $\enma{\mathbf{a}} = \enma{\mathbf{a}}^+ - \enma{\mathbf{a}}^-$ of two nonnegative vectors with disjoint support. A nonzero vector $\enma{\mathbf{a}} \in L$ is called {\it primitive} if there is no vector $\enma{\mathbf{b}} \in L \backslash \set{ \enma{\mathbf{a}}, \enma{\mathbf{0}} }$ such that $\enma{\mathbf{b}}^+ \leq \enma{\mathbf{a}}^+$ and $\enma{\mathbf{b}}^- \leq \enma{\mathbf{a}}^-$. The set of primitive vectors is known to be finite [St, Theorem 4.7]. The set of binomials $\,\enma{\mathbf{x}}^{\enma{\mathbf{a}}^+} - \enma{\mathbf{x}}^{\enma{\mathbf{a}}^-}\,$ were $\enma{\mathbf{a}}$ runs over all primitive vectors in $L$ is called the {\it Graver basis} of the ideal $I_L$. The Graver basis contains the universal Gr\"obner basis of $I_L$ [St, Lemma 4.6]. \lemma{graver} If $\set{\enma{\mathbf{0}},\enma{\mathbf{a}}}$ is an edge of $\mathop{\rm hull}\nolimits(M_L)$ then $\enma{\mathbf{a}}$ is a primitive vector in $L$. \stdbreak\noindent{\bf Proof. } Suppose that $\enma{\mathbf{a}}= (a_1,\ldots,a_n) $ is a vector in $L$ which is not primitive, and choose $\,\enma{\mathbf{b}} = (b_1,\ldots,b_n) \in L \backslash \set{ \enma{\mathbf{a}}, \enma{\mathbf{0}}}$ such that $\enma{\mathbf{b}}^+ \leq \enma{\mathbf{a}}^+$ and $\enma{\mathbf{b}}^- \leq \enma{\mathbf{a}}^-$. This implies $\,t^{b_i} + t^{a_i-b_i} \le 1 + t^{a_i} \,$ for $t \gg 0$ and $i \in \set{1,\ldots,n}$. In other words, the vector $\,t^{\enma{\mathbf{b}}} + t^{\enma{\mathbf{a}}-\enma{\mathbf{b}}}\,$ is componentwise smaller or equal to the vector $\, t^\enma{\mathbf{0}} + t^\enma{\mathbf{a}} $. We conclude that the midpoint of the segment $ \, \mathop{\rm conv}\nolimits \set{ t^\enma{\mathbf{0}} , t^\enma{\mathbf{a}}}\, $ lies in $\,\mathop{\rm conv}\nolimits \set{t^{\enma{\mathbf{b}}} , t^{\enma{\mathbf{a}}-\enma{\mathbf{b}}}} + \enma{\mathbb{R}}_+^n$, and hence $\,\mathop{\rm conv}\nolimits \set{ t^\enma{\mathbf{0}} , t^\enma{\mathbf{a}}}$ is not an edge of the polyhedron $\,P_t \, = \, \mathop{\rm conv}\nolimits \set{\, t^\enma{\mathbf{c}} \,: \, \enma{\mathbf{c}} \in L} \,+ \, \enma{\mathbb{R}}_+^n$. \Box \theorem{ } The hull resolution $\pi(\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M_L)})$ is finite as an $S$-module. \stdbreak\noindent{\bf Proof. } By \ref{graver} the vertex $\enma{\mathbf{0}}$ of $\mathop{\rm hull}\nolimits(M_L)$ lies in only finitely many edges. It follows that $\enma{\mathbf{0}}$ lies in only finitely many faces of $\mathop{\rm hull}\nolimits(M_L)$. The lattice $L$ acts transitively on the vertices of $\mathop{\rm hull}\nolimits(M_L)$, and hence every face of $\mathop{\rm hull}\nolimits(M_L)$ is $L$-equivalent to a face containing $\enma{\mathbf{0}}$. The faces containing $\enma{\mathbf{0}}$ generate $\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M_L)}$ as an $S[L]$-module, and hence they generate $\pi(\enma{\mathbf{F}}_{\mathop{\rm hull}\nolimits(M_L)})$ as an $S$-module. \Box A minimal free resolution of a lattice ideal $I_L$ generally does not respect symmetries, but the hull resolution does. The following example illustrates this point. \example{exE} {\sl (The hypersimplicial complex as a hull resolution)} \hfill \break The lattice $\,L \,=\, \mathop{\rm ker}\nolimits_\enma{\mathbb{Z}} \pmatrix{ 1 \! & \!1 & \cdots & 1 }\,$ in $\enma{\mathbb{Z}}^n$ defines the toric ideal $$ I_L \quad = \quad \langle \, x_i - x_j \,\,: \,\, 1 \leq i < j \leq n \, \rangle .$$ The minimal free resolution of $I_L$ is the Koszul complex on $n-1$ of the generators $x_i-x_j$. Such a minimal resolution does not respect the action of the symmetric group $S_n$ on $I_L$. The hull resolution is the Eagon-Northcott complex of the matrix {\smallmath $\,\bmatrix{2pt}{1\;\, & 1\;\, & \cdots & 1\;\, \cr x_1 & x_2 & \cdots & x_n \cr}$}. This resolution is not minimal but it retains the $S_n$-symmetry of $I_L$. It coincides with the {\sl hypersimplicial complex} studied by Gel'fand and MacPherson in [GM, \S 2.1.3]. The basis vectors of the hypersimplicial complex are denoted $\Delta_{\ell}^I$ where $I$ is a subset of $\set{1,2,\ldots,n}$ with $|I| \geq 2$ and $\ell$ is an integer with $ 1 \leq \ell \leq |I| - 1$. We have $\,\Delta_1^{\set{i,j}} \mapsto x_i - x_j\,$ and the higher differentials act as $$ \Delta_\ell^I \quad \mapsto \quad \sum_{i \in I} \mathop{\rm sign}\nolimits(i,I) \cdot x_i \cdot \Delta^{I \setminus \! \set{i}}_{\ell-1} \, - \, \sum_{i \in I} \mathop{\rm sign}\nolimits(i,I) \cdot \Delta^{I \setminus \! \set{i}}_{\ell}, $$ where the first sum is zero if $\ell=1$ and the second sum is zero if $\ell = \abs{I}-1$. \Box \remark{curious} Our study suggests a {\sl curious duality} of toric varieties, under which the coordinate ring of the primal variety is resolved by a discrete subgroup of the dual variety. More precisely, the hull resolution of $I_L$ is gotten by taking the convex hull in $\enma{\mathbb{R}}^n$ of the points $t^\enma{\mathbf{a}}$ for $\enma{\mathbf{a}} \in L$. The Zariski closure of these points (as $t$ varies) is itself an affine toric variety, namely, it is the variety defined by the lattice ideal $\,I_{L^\perp}$ where $L^\perp$ is the lattice dual to $L$ under the standard inner product on $\enma{\mathbb{Z}}^n$. For instance, in \ref{exE} the primal toric variety is the line $\,(t , t , \ldots , t)$ and the dual toric variety is the hypersurface $x_1 x_2 \cdots x_n = 1$. That hypersurface forms a group under coordinatewise multiplication, and we are taking the convex hull of a discrete subgroup to resolve the coordinate ring of the line $\,(t , t , \ldots , t)$. \Box \example{exV} {\sl (The rational normal quartic curve in $P^4$)} \vskip .1cm \noindent Let $L = \mathop{\rm ker}\nolimits_\enma{\mathbb{Z}} {\smallmath\bmatrix{2pt}{ 0 & 1 & 2 & 3 & 4 \cr 4 & 3 & 2 & 1 & 0 \cr}}$. The minimal free resolution of the lattice ideal $I_L$ looks like $ 0 \rightarrow S^3 \rightarrow S^8 \rightarrow S^6 \rightarrow I_L $. The primal toric variety in the sense of \ref{curious} is a curve in $P^4$ and the dual toric variety is the embedding of the $3$-torus into affine $5$-space given by the equations $ \, x_2 x_3^2 x_4^3 x_5^4 = x_1^4 x_2^3 x_3^2 x_4 = 1$. Here the hull complex $\mathop{\rm hull}\nolimits(M_L)$ is simplicial, and the hull resolution of $I_L$ has the format $ 0 \rightarrow S^4 \rightarrow S^{16} \rightarrow S^{20} \rightarrow S^9 \rightarrow I_L $. The nine classes of edges in $\mathop{\rm hull}\nolimits(M_L)$ are the seven quadratic binomials in $I_L$ and the two cubic binomials $\,x_3 x_4^2 - x_1 x_5^2 , \, x_2^2 x_3 - x_1^2 x_5 $. \vskip 1.2cm \noindent {\bf Acknowledgements. } We thank Lev Borisov, David Eisenbud, Irena Peeva, Sorin Popescu, and Herb Scarf for helpful conversations. Dave Bayer and Bernd Sturmfels are partially supported by the National Science Foundation. Bernd Sturmfels is also supported by the David and Lucille Packard Foundation and a 1997/98 visiting position at the Research Institute for Mathematical Sciences of Kyoto University. \bigskip \bigskip \references \itemitem{[AB]} R.~Adin and D.~Blanc, Resolutions of associative and Lie algebras, preprint, 1997. \itemitem{[BHS]} I.~Barany, R.~Howe, H.~Scarf: The complex of maximal lattice free simplices, {\sl Mathematical Programming} {\bf 66} (1994) Ser.~A, 273--281. \itemitem{[BPS]} D.~Bayer, I.~Peeva and B.~Sturmfels, Monomial resolutions, preprint, 1996. \itemitem{[BLSWZ]} A.~Bj\"orner, M.~Las~Vergnas, B.~Sturmfels, N.~White and G.~Ziegler, {\sl Oriented Matroids}, Cambridge University Press, 1993. \itemitem{[BH]} W.~Bruns and J.~Herzog, {\sl Cohen-Macaulay Rings}, Cambridge University Press, 1993. \itemitem{[BH1]} W.~Bruns and J.~Herzog, On multigraded resolutions, {\sl Math.~Proc.~Cambridge Philos.~Soc.} {\bf 118} (1995) 245--257. \itemitem{[GM]} I.~M.~Gel'fand and R.~D.~MacPherson: Geometry in Grassmannians and a generalization of the dilogarithm, {\sl Advances in Math.} {\bf 44} (1982), 279--312. \itemitem{[Ho]} M. Hochster, Cohen-Macaulay rings, combinatorics and simplicial complexes, in {\sl Ring Theory II}, eds.~B.R.~McDonald and R.~Morris, Lecture Notes in Pure and Appl.~Math.~{\bf 26}, Dekker, New York, (1977), 171--223. \itemitem{[KS]} M.~Kapranov and M.~Saito, Hidden Stasheff polytopes in algebraic K-theory and the space of Morse functions, preprint, 1997, paper \# 192 in {\tt http://www.math.uiuc.edu/K-theory/}. \itemitem{[Mac]} S. MacLane, {\sl Categories for the Working Mathematician}, Graduate Texts in Mathematics, No.~5, Springer-Verlag, New York, 1971. \itemitem{[PS]} I.~Peeva and B.~Sturmfels, Generic lattice ideals, to appear in {\sl Journal of the American Math.~Soc.} \itemitem{[Ros]} I.~Z.~Rosenknop, Polynomial ideals that are generated by monomials (Russian), {\sl Moskov. Oblast. Ped. Inst. Uw cen Zap.} {\bf 282} (1970), 151-159. \itemitem{[Stu]} B.~Sturmfels, {\sl Gr\"obner Bases and Convex Polytopes}, AMS University Lecture Series, Vol. 8, Providence RI, 1995. \itemitem{[Stu2]} B.~Sturmfels, The co-Scarf resolution, to appear in {\sl Commutative Algebra and Algebraic Geometry}, Proceedings Hanoi 1996, eds.~D.~Eisenbud and N.V.~Trung, Springer Verlag. \itemitem{[Tay]} D.~Taylor, {\sl Ideals Generated by Monomials in an $R$-Sequence}, Ph.~D.~thesis, University of Chicago, 1966. \itemitem{[Zie]} G.~Ziegler, {\sl Lectures on Polytopes}, Springer, New York, 1995. \vskip 1.8cm \noindent Dave Bayer, Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA, {\tt [email protected]}. \vskip .4cm \noindent Bernd Sturmfels, Department of Mathematics, University of California, Berkeley, CA 94720, USA, {\tt [email protected]}. \bye

9711

alg-geom/9711024

en

https://arxiv.org/abs/alg-geom/9711024

alg-geom/9711024

V. Shokurov

V.V. Shokurov

Complements on surfaces

105 pages. Minor typos corrected

J. of Math. Sci., Vol. 102 (2000), No. 2, 3876--3932

The main result of the paper is a boundedness for $n$-complements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated.

alg-geom

\section{Introduction} \label{int} An introduction to complements can be found in \cite[\S 5]{Sh2}. See also the end of this section. \refstepcounter{subsec} \label{contex} \begin{example}~Let $S=\mathbb P_E(F_2)$ be a ruled surface over a non-singular curve $E$ of genus $1$, which corresponds to a non-splitting vector bundle $F_2/E$ of rank 2 \cite[Theorem 5 (i)]{At}. We denote the ruling by $f\colon S\to E$. It has a single section $\mathbb P_E(F_1)$ in its linear and even numerical equivalence class (by \cite[2.9.1]{Sh1}, cf. arguments below). We identify the section with $E$. Note that $E|_E\sim \det F_2\sim 0$, where $\sim$ denotes {\it linear\/} equivalence (cf. \cite[Proposition 2.9]{H}). On the other hand $(K_S+E)|_E\sim K_E\sim 0$ according to the Adjunction Formula. Thus $(K_S+2E)|_E\sim 0$ \cite[Lemma 2.10]{H}. But $K_S+2E\equiv 0/E$, where $\equiv$ denotes {\it numerical\/} equivalence. So, $K_S+2E=f^*((K_S+2E)|_E)\sim 0$ \cite[2.9.1]{Sh1}. Equivalently, we may choose $-K_S=2E$ as an anti-canonical divisor. In particular, $-K_S$ is nef because $E^2=0$. The latter implies also that the cone $\NEc{S}$ is two-dimensional and has two extremal rays: \begin{itemize} \item the first one $R_1$ is generated by a fibre $F$ of the ruling $f$, \item the second one $R_2$ is generated by the section $E$. \end{itemize} We contend that $E$ is the only curve in $R_2$. Indeed if there is a curve $C\not= E$ in $R_2$, then $C\equiv m E$ where $m=(C.F)$ is the degree of $C$ as a multi-section of $f$. We may suppose that $m$ is minimal. Then as above, $C\sim m E$ which induces another fibering with projection $g\colon S\to \mathbb P^1$. All fibres of $g$ are non-singular curves of genus $1$. However, some of them may be multiple. According to our assumptions, any multiple fibre $G_i$ has a multiplicity $m_i=m\ge 2$, and we have at least two of them. So, if $G_1=E$ then another multiple fibre $G_2\equiv E$ gives another section in $R_2$. This is impossible. Thus in general a nef anti-canonical divisor $-K$ may not be semi-ample, and we may have no complements when $-K$ is just nef. The same may occur in the log case. \end{example} \refstepcounter{subsec} \label{posch} \begin{example} Now let $E$ be a fibre of an elliptic fibering $f:S\to Z$ such that \begin{itemize} \item $E$ is a non-singular elliptic curve, \item $m$ is the multiplicity of $f$ in $E$, i.~e., $m$ is the minimal positive integer with $m E\sim 0$ locally$/Z$. \end{itemize} Then, for a canonical divisor $K$, $K\sim a E$ locally$/Z$ with a unique $0\le a\le m-1$. Moreover in the characteristic $0$, $a=m-1$, or $K+E\sim 0$, which means that $K+bE$, with some real $0\le b\le 1$, has a $1$-complement locally$/Z$. However if the characteristic $p$ is positive, one can construct such a fibering with $m=p^k$ and any $0\le a\le m-1$. For instance, see \cite[Example 8.2]{KU} for $m=p^2$ and $0\le a\le p-1$. So, $K+bE$, for $b$ close to $1$, has no $n$-complements with bounded $n$ because such a complement would have the form $K+E\sim (a+1)E$ and $n(K+E)\sim n(a+1)E\not\sim 0$ when $n,a\ll p$ and $m\ge p$. So, $K+E$ can be an $n$-complement having rather high index $n|m$. However, it is high in the Archimedean sense, but $n$ is small $p$-adically when $m=p^k$. \end{example} So, everywhere below we suppose that the characteristic is $0$. \refstepcounter{subsec} \theoremstyle{definition} \newtheorem*{conjcs}{\thesection.\thesubsec. Conjecture on complements} \begin{conjcs} \label{conjcs} We consider log pairs $(X,B)$ with boundaries $B$ such that $K+B$ is log canonical. Then complements in any given dimension $d$ are {\it bounded}. This means that there exists a finite set $N_d$ of natural numbers such that any contraction $f:X\to Z$, satisfying certain conditions which we discuss below, and of $\dim X\le d$, has locally$/Z$ an $n$-complement for some {\it index\/} $n\in N_d$. Of course, we need to assume that \begin{description} \item{(EC)} there exists at least some $n$-complement. \end{description} In particular, if $B=0$ then the linear system $|-nK|\not=\emptyset$ (cf. Corollaries~\ref{int}.\ref{nonvan1}-\ref{nonvan2} and (NV) in Remark~\ref{indcomp}.\ref{rel}) and its generic element has good singularities in terms of complements (see explanations before Corollary~\ref{int}.\ref{nonvan1}). Maybe it is enough the existence of complements (EC). More realistic and important for applications are the following conditions \begin{description} \item{(SM)} the multiplicities $b_i$ of $B$ are {\it standard}, i.~e., $b_i=(m-1)/m$ for a natural number $m$, or $b_i=1$ (as for $m=\infty$); and \item{(WLF)} $(X/Z,B)$ is a {\it weak log Fano\/} which means that $-(K+B)$ is nef and big$/Z$. \end{description} Note that these conditions imply (EC) according to the proof of \cite[Proposition 5.5]{Sh2}. By Example~\ref{int}.\ref{contex} a condition that $-(K+B)$ is just nef, is not sufficient even for (EC). \end{conjcs} \refstepcounter{subsec} \theoremstyle{plain} \newtheorem*{mainth}{\thesection.\thesubsec. Main Theorem} \begin{mainth} The complements in dimension $2$ are bounded under the condition (WLF) and \begin{description} \item{{\rm (M)}} the multiplicities $b_i$ of $B$ are {\rm standard\/}, i.~e., $b_i=(m-1)/m$ for a natural number $m$, or $b_i\ge 6/7$. \end{description} More precisely, for almost all contractions and all contractions of relative dimension $0$ and $1$, we can take a complementary index in $RN_2=\{1,2,3,4,6\}$. \end{mainth} Here {``}almost all" means up to a bounded family of contractions in terms of moduli. Really, this concerns moduli spaces in the global case or that of log Del Pezzos. Moduli itself may not be the usual one according to Remark~\ref{int}.\ref{mod}. By the {\it global\/} case we mean that $Z=\mathop{\rm pt.}$. The other cases are {\it local\/}$/Z$. \refstepcounter{subsec} \begin{definition} \label{dexc} (Cf. \cite[Theorem 5.6]{Sh2}.) We say that a complement $K+B^+$ is {\it non-exceptional\/} if it is not Kawamata log terminal whenever $Z=\mathop{\rm pt.}$, and it is not purely log terminal on a log terminal resolution whenever $Z\not=\mathop{\rm pt.}$ (cf. \cite[Theorem~5.6]{Sh2}). The former will be called {\it global\/} and the latter {\it local\/}. Respectively, the corresponding log pair $(X/Z,B)$ is called {\it non-exceptional\/} if it has a non-exceptional complement. For instance, surface Du Val singularities of types $\mathbb A_*$ and $\mathbb D_*$ are non-exceptional even though they have trivial complements that are canonical (cf. Example~\ref{int}.\ref{tcomp}). They have respectively other $1$- and $2$-complements that are non-exceptional. On the other hand, the pair $(X/Z,B)$ is {\it exceptional\/} if each of its complements $K+B^+$ is {\it exceptional\/}. The latter means that $K+B^+$ is Kawamata log terminal whenever $Z=\mathop{\rm pt.}$, and it is {\it exceptionally\/} log terminal whenever $Z\not=\mathop{\rm pt.}$. The exceptional log terminal property means the purely log terminal property on a log terminal resolution. The Du Val singularities of exceptional types $\mathbb E_6,\mathbb E_7$ and $\mathbb E_8$ are really exceptional from this point of view as well (cf.~\cite[Example 5.2.3]{Sh2}). If the pair $(X/Z,B)$ is exceptional and $Z\not=\mathop{\rm pt.}$, then for any complement $K+B^+$, any possible divisor (at most one as one can prove) with log discrepancy $0$ for $K+B^+$ has the center over the given point in $Z$. Otherwise we can find another complement $B^{+'}>B^+$ that is non-exceptional (see the Proof of Theorem~\ref{lcomp}.\ref{lmainth}: General case in Section~\ref{lcomp}). For instance, in the Main Theorem the $n$-complements with $n$ not in $RN_2$ are over $Z=\mathop{\rm pt.}$ and exceptional as we see later in Theorems~\ref{lcomp}.\ref{lmainth} and \ref{gcomp}.\ref{gmainth}. So, the theorem states that the global exceptions are bounded. Some of them will be discussed in Section~\ref{ecomp}. In higher dimension, it is conjectured that such complements and the corresponding pairs $(X/\mathop{\rm pt.},B)$ under conditions (WLF) and (SM) are bounded. Of course, this is not true if we drop (WLF) (see Example~\ref{int}.\ref{tcomp}). However, it may hold formally and even in the local case under certain conditions as suggested in Remark~\ref{int}.\ref{exc}. $N_2$ in the Conjecture on complements differs from $RN_2$ in the Main Theorem in exceptional cases, which are still not classified completely. Nonetheless we define corresponding {\it exceptional\/} indexes as $EN_2=N_2\setminus RN_2$. We say that $RN_2$ is {\it regular\/} part of $N_2$ in the Conjecture for dimension $2$. Note that $RN_2=N_1=RN_1\cup EN_1$ where $RN_1=\{1,2\}$ and $ER_1=\{3,4,6\}$. \end{definition} \refstepcounter{subsec} \begin{conjecture} \label{econj} In general we may define $EN_d=N_2\setminus N_{d-1}$ and conjecture that $RN_d=N_{d-1}$, or, equivalently, that $EN_d$ really corresponds to exceptions. Evidence supporting this is related to our method in the proof of the Main Theorem and Borisov-Alekseev's conjecture~\cite{Al}. Some extra conditions on multiplicities are needed: (SM) or an appropriate version of (M) (cf. (M)' in Remark~\ref{gcomp}.\ref{mrem}.2, (M)'' in \ref{csing}.\ref{singth}.1, and the statement~\ref{indcomp}.\ref{indth}.1). \end{conjecture} Basic examples of complements may be found in~\cite[5.2]{Sh2}. \refstepcounter{subsec} \begin{example-p} \label{tcomp} {\it Trivial complements.} Let $X/Z$ be a contraction having on $X$ only log canonical singularities and with $K\equiv 0/Z$, e.~g., Abelian variety, K3 surface, Calabi-Yau 3-fold or a fibering of them. Then it is known that locally$/Z$, $K$ is semi-ample: $K\sim_{\mathbb Q} 0$ or $n K\sim 0$ for some natural number $n$ (at least in the log terminal case, cf.~\cite[Remark~6-1-15(2)]{KMM}). Such minimal $n$ is known as the {\it index\/} of $X/Z$, to be more precise, over a point $P\in Z$. Indices are {\it global\/} when $Z=P=\mathop{\rm pt.}$ and {\it local\/} otherwise. So, if we consider thought $X/Z$ as a log pair with $B=0$, then (EC) is fulfilled for the above $n$. In the global case the Conjecture on complements suggests that we can find such $n$ in $N_d$ for a given dimension $d=\dim X$. Moreover, in that case, we may replace $B=0$ by $B$ under (SM) according to the Monotonicity~\ref{indcomp}.\ref{monl1} below, and assuming that $X$ may be semi-normal. In the local case we need an additional assumption on a presence of a log canonical singularity$/P$, i.~e., there exists an exceptional or non-exceptional divisor $E$ with the log discrepancy $0$ for $K+B$ and with $\centr{X}{E}/P$. Otherwise a complement may be {\it non-trivial\/}: $B^+>B$. This really holds in dimensions $1$ and $2$ by Corollary~\ref{int}.\ref{ind2} below. In dimension $3$ it is know that any global index $n$ divides the Beauville number $$b_3=2^5\cdot 3^3\cdot 5^2\cdot 7\cdot 11 \cdot 13\cdot 17\cdot 19$$ when $X/\mathop{\rm pt.}$ has at most terminal singularities and $B=0$. Note that the similar $2$-dimensional number is $b_2=12$. Its proper divisors give $N_1=RN_2$. So, we conjecture that $N_2$ consists of (the proper) divisors of the Beauville number. Perhaps we have something like this in higher dimensions. Of course, if in a given dimension $d$ indices are bounded, we have a universal index $I_d$ as their least common multiple. The case of the Conjecture on complements under consideration in dimension $d$ is equivalent to an existence of such $I_d$. As always in mathematics, it is known only that $I_1=12$ and that $I_2$ exists (see Corollary~\ref{int}.\ref{ind2}). We suggest that $I_2\approx b_3$. In particular, this means that $b_3$ corresponds to non-exceptional cases. Note also that in the global case, according to our definition, contractions and their complements are non-exceptional whenever they are Kawamata log terminal, for instance, Abelian varieties. These varieties have unbounded moduli in any dimension $\ge 2$, but their index is $1$. So, from the formal viewpoint of Remark~\ref{int}.\ref{exc} below, we treat them as a regular case. Note that we are still discussing the case that is opposite to the assumption (WLF) in the Main Theorem. Indeed, if we consider a birational contraction $X/Z$ then the indices are not bounded. Take, for instance, $X=Z$ as a neighborhood of a quotient singularity. Nonetheless, the above should hold when $X$ is really log canonical in $P$ as we suppose. In the surface case such singularities with $B=0$ are known as {\it elliptic\/} and they have in a certain sense a Kodaira dimension $0$. Their universal index is $12$. The Conjecture on complements implies an existence of such a universal number in any dimension. An inductive explanation may be presented in the log canonical case (cf. Remark~\ref{int}.\ref{exc}). Using the LMMP it is possible to reduce to the case when $B$ has a reduced component $E$. Then by the Adjunction formula the index of $K+B$ near $E$ coincides with the same for restriction $K_E+B_E=(K+B)|_E$. So, it is not surprising that indices for dimension $d-1=\dim E$ should divide indices in dimension $d$. If the above restriction is epimorphic for Cartier multiples of $K_E+B_E$ then both indices coincide over a neighborhood of $P\in Z$. This holds, for example, when $X/Z$ is birational. We see in Section~\ref{lcomp} that this works in fibre cases as well. Finally, note that in the fibre case the complements may be non-trivial when the above condition on a log canonicity is not fulfilled. Let $X$ be smooth and $B=0$. Then the index of $K$ in a neighborhood of a fibre $F\subset X$ over $P$ may be arbitrarily high. The point is that contraction $X/Z$ itself is not smooth in those cases and fibre $F$ is multiple. For instance, $K$ and $F$ have multiplicity $m$ near fibre $F$ of type $mI_0$ in an elliptic fibering. However, there exists $1$-complement $K+F$ in a neighborhood of $F$. This explains in particular why complements are fruitful even in a well-known case as elliptic fiberings. More details may be found in Section~\ref{lcomp}. In the positive characteristic $p$, it does not work well according to Example~\ref{int}.\ref{posch}. Maybe it works modulo $p$ factors. Each {\it trivial\/} complement, i.~e., a complement with $B^+=B$, defines a cover $\widetilde X\to X$ of degree $d$, which is unramified in divisors with respect to $B$: a covering is $B$-{\it unramified\/} if it preserves the boundary $B$, i.~e., inequality $\mult{D_i}{B}=b_i\ge (m_j-1)/m_j$, where $m_j$ is any ramification multiplicity$/D_i$ \cite[2.1.1]{Sh2}, holds for each prime $D_i$ in $X$. The conjecture in this case states a boundedness of such degrees. In terms of a local or global algebraic fundamental group $\pi (X/Z)$, they correspond to normal subgroups of finite index and with a cyclic quotient. A general conjecture here states that $\pi (X/Z)$ is quasi-Abelian, and even finite in exceptional cases (see Remark~\ref{int}.\ref{exc}). The structure of the fundamental group is interesting even for non-singular $X$. \end{example-p} In the previous example we know that the condition (EC) is satisfied. So, according to the general philosophy and to conjectures there, the following results are not surprising. \refstepcounter{subsec} \begin{corollary} \label{mcorol} In the Main Theorem we may replace the condition (WLF) by (EC). \end{corollary} This will be proven in Sections~\ref{lcomp} and \ref{gcomp}. More advanced results can be found in Remark~\ref{gcomp}.\ref{mrem}. Here we consider only a {\it trivial\/} case. \begin{proof}[Proof: Numerically trivial case and under (SM)] So, we suppose that $K+B\equiv 0/Z$ and it has a trivial complement $B^+=B$ (cf. Monotonicity~\ref{indcomp}.\ref{monl1}). Moreover we consider the global case when $Z=\mathop{\rm pt.}$ (for the local case see Section~\ref{lcomp}). We need to bound the index of $K+B$. If $B=0$ and $X$ has only canonical singularities it is well-known that the index of $K$ divides $I_1=12$. On the other hand, pairs $(X,B)$ with $B\not=0$ and surfaces $X$ with non-canonical singularities are bounded whenever they are $\varepsilon$-log canonical for any fixed $\varepsilon >0$, for instance, for $\varepsilon\ge 1/7$ \cite[Theorem~7.7]{Al}. So, their indices are bounded as well. Note, that by (SM) the possible multiplicities $b_i$ satisfy the d.c.c. (descending chain condition). Thereby we need to bound the indices in the non-$\varepsilon$-log canonical case for an appropriate $\varepsilon$. We take $\varepsilon=1/7$. After a log terminal resolution we may suppose that $B$ has a multiplicity $b=b_i>6/7$ in a prime divisor $D=D_i$. The resolution has the same indices (cf. \cite[Lemma~5.4]{Sh2}). Moreover, for the regular complements or for almost all $(S,B)$, all $b_i=1$, when $b_i\ge 6/7$, and the index is in $RN_2$. So, there exists a real $c>0$ such that all $b_i=1$ whenever $B_i\ge c$, and we have then a regular complement. To reduce this to the Main Theorem we apply the LMMP. (Cf. Tsunoda's \cite[Theorem~2.1]{T}, which states that $K+B$ has the index $\le 66$, when $B$ is reduced and $K+B\equiv 0$.) We have an extremal contraction $f:X\to Z$ because $K+B-b D$ is negative on a covering family of curves. If $f$ is birational we replace $X$ by $Z$. Then $D$ is not contracted since $(K+B-b D.D)=-b D^2>0$. If $Z$ is a curve, then $f$ is a ruling, and, according to a 1-dimensional result as our corollary, applied to the generic fibre , $b_i=1$ and $K+B$ has a regular index near the generic fibre of $f$. Note that $D$ is a multi-section of $f$ of a multiplicity at most $2$. If $D$ is a $1$-section we reduce the problem to a $1$-dimensional case after an adjunction $K_D+B_D=(K+B)|_D$. In that case we use the same arguments as in Example~\ref{int}.\ref{contex} (cf. Lemma~\ref{indcomp}.\ref{indcomp1}). We can take a complement with an even index when $K+B$ has index $2$ near the generic fibre. Otherwise $D$ is an irreducible double section of $f$. In that case the index is in $RN_2$ near each fibre. The same holds for entire $X$, except for a case when $B=D$, $K_D+B_D=(K+B)_D\sim 0$ and $2(K+B)\sim 0$ (see Lemma~\ref{indcomp}.\ref{indcomp2}). Finally, if $Z=\mathop{\rm pt.}$, we may apply the Main Theorem for $B-\delta D$ for some $\delta>0$. The main difficulties are here. \end{proof} \refstepcounter{subsec} \begin{corollary} \label{ind2} Under (SM), $I_2$ exists. Moreover, for the trivial complements, each $b_i=1$ whenever $b_i\ge I_2/(I_2+1)$, and , for the almost all trivial complements, each $b_i=1$ whenever $b_i\ge 6/7$. In the global case when $(X,B)$ is not Kawamata log terminal, and in the local case we can replace inexplicit $I_2$ by $I_1=12$, (SM) by (M). Then each $b_i=1$ whenever $b_i\ge 6/7$. If we have an infinite number of exceptional divisors with log discrepancies $0$ over $P$, then we can replace $I_1=12$ even by $2$. \end{corollary} \begin{proof}[Proof: Global case] We suppose that $Z=\mathop{\rm pt.}$. We consider other cases in Section~\ref{lcomp}. Then the results follow from the above arguments and by \cite[Theorem~7.7]{Sh2}. The non-Kawamata log terminal case corresponds to regular complements by the Main Theorem (cf. the Inductive Theorem and Theorem~\ref{gcomp}.\ref{gmainth} below). For global and local cases, with an infinite number of exceptional divisors with log discrepancies $0$, see \ref{indcomp}.\ref{indth}.2. \end{proof} \refstepcounter{subsec} \label{ind3} \begin{corollary} Under (SM), let $P\in (X,B)$ be a 3-fold log canonical singularity as in Example~\ref{int}.\ref{tcomp} having the trivial complement. Then its index divides $I_2$. Moreover, each $b_i=1$ whenever $b_i\ge I_2/(I_2+1)$. If $(X,B)$ has at least two exceptional divisors$/P$ with the log discrepancy $0$, then we can replace inexplicit $I_2$ by $I_1=12$. The same holds with one exceptional or non-exceptional divisor with the center passing through $P$ but $\not=P$. If $(X,B)$ on some resolution has a triple point in an exceptional locus of divisors$/P$ with the log discrepancy $0$, then we can replace $I_1=12$ even by $2$. \end{corollary} \begin{proof} According to the very definition of the trivial complements, we have an exceptional divisor $E/P$ with the log discrepancy $0$. By the LMMP (cf. \cite[Theorem~3.1]{Sh3}), we can make a crepant resolution $g:Y\to X$ just of this $E$ and $E$ is in the reduced part of boundary $B^Y$. By the adjunction $K_E+B_E=(K_Y+B^Y)|_E\equiv 0$ and $B_E$ has only standard coefficients. So, it has a trivial complement. Then by Corollary~\ref{int}.\ref{ind2} and by the local inverse adjunction $K_Y+B^Y$ has the same Cartier index and it divides $I_2$ whenever the restriction $K_E+B_E=(K_Y+B^Y)|_E$ is Kawamata log terminal. Here we may use a covering trick as well. Then, by the Kawamata-Viehweg vanishing, restrictions $m(K_E+B_E)=m(K_Y+B^Y)|_E$ are epimorphic for $m$ dividing the index of $K_E+B_E$. So, $K+B$ has the same index and it divides $I_2$. Other special cases follow from special cases in Corollary~\ref{int}.\ref{ind2} (cf. \cite[Corollary~5.10]{Sh2}). Then the index will divide $I_1=12$. Here a difficulty is related to an application of the above surjectivity to a semi-normal surface $E$. However, its combinatoric is quite simple. \end{proof} We may simulate examples on well-known varieties as in the next example. \refstepcounter{subsec} \label{simul} \begin{example} Let $X=\mathbb P^d$ and $B=\sum b_i L_i$ where prime divisors $L_i$ are hyperplanes in a generic position and there are at most $l$ of them. (The latter holds when $\min\{b_i\not=0\}>0$.) Then for any boundary coefficients $0\le b_i\le 1$, the log pair $(X,B)$ is log terminal. If we have a reduced component $L_i$ in $B$ we can restrict our problem on $L_i$. So, really new complements are related to the case when all $b_i<1$ that we assume below. Note that if all $b_i$ are rational (or $\sum b_i<d+1$), the condition (EC) follows from an inequality $\sum b_i\le d+1$. An existence of an $n$-complement is equivalent to another inequality $$\deg\lfloor (n+1)B\rfloor= \sum \lfloor (n+1)b_i\rfloor/n\le d+1.$$ So, the Conjecture on complements states that we may choose such $n$ in a finite set $N_d$. The space $$ T_d^l=\{(b_1,\dots,b_l)\mid b_i\in [0,1] \text{ and } \sum b_i\le d+1\} $$ is compact and the vectors $v=(b_1,\dots,b_l)$ without $n$-complements form a union of convex rational polyhedra: $b_i\ge c_i$. In the intersection they give a vector $v$ without any $n$-complements. We can also assume that it is maximal $\sum b_i=d+1$, and $b_i>0$. But this is impossible, because there exists an infinite set of approximations with natural numbers $n$ that $(n+1)b_i=N_i+\delta_i, i=1,2,$ where $|\delta_i-b_i|<\varepsilon\ll 1$, and $N_i$ is integer. Indeed, then each $\lfloor (n+1)b_i\rfloor=N_i$, and $$ \sum \lfloor (n+1)b_i\rfloor/n=\sum N_i/n= \sum b_i+\sum (b_i-\delta_i)/n\le d+1+l\varepsilon /n. $$ Hence $\sum \lfloor (n+1)b_i\rfloor/n\le d+1$, when $l\varepsilon<1$, because $d$ is integer. We can find the required approximation from another one $nb_i=N_i+\varepsilon_i$, where $\varepsilon_i=\delta_i-b_i$, $|\varepsilon_i|<\varepsilon\ll 1$, and $N_i$ is integer. It has required solutions $n$, according to the Kronecker Theorem \cite[Theorem~IV in Section 5 of Chapter III]{Ca}, with $L_i=b_i x$ and $\alpha_i=0$. Indeed, for every integer numbers $u_i$, $\sum u_i\alpha_i=0$ is integer. Explicitly, it is known up to $d=2$. For $d=1$ \cite[Example 5.2.1]{Sh2}. For $d=2$, under (SM) $n\le 66$ due to Prokhorov~\cite[Example~6.1]{P2} (cf. \cite[ibid]{T}). We may generalize this and suppose that $d$ is any positive rational (and, maybe, even real), the hyperplanes are in a non-generic position, and with hypersurfaces rather than hyperplanes of fixed degrees. Moreover, we can consider complements with $n$ divided by a given natural number $m$ (cf. Lemma~\ref{indcomp}.\ref{compl3}). (Even for $l=\infty$, as for $d=1$. However, the boundedness is unknown for $l=\infty$ and $d>1$.) There exists a corresponding local case when we consider a contraction $X/Z$ with a fibre $F=\mathbb P^d$ and $B=\sum b_i L_i$ where prime divisors $L_i$ intersect $F$ in hyperplanes in a generic position. Of course, more interesting cases are related to a non-generic position and with hypersurfaces rather than hyperplanes. They also give non-trivial examples of trivial complements. For instance, if $C\subset \mathbb P^2$ is a plane curve of degree $6$ with one simple triple point singularity, then $K+C/2$ is log terminal and has a trivial $2$-complement. The corresponding double cover $\widetilde X\to \mathbb P^2$ produces a K3 surface $\widetilde X$ with a single Du Val singularity of type $\mathbb D_4$ over the triple singularity of $C$. For a generic $C$, after a resolution of $\widetilde X$ we get a non-singular K3 surface $Y$ with the Picard lattice of rank $5$ and an involution on $Y$ which is identical on the lattice. Note that generic curve $C$ is a generic trigonal curve of genus $7$. But this is a different tale. \end{example} \refstepcounter{subsec} \begin{example-p} \label{gquot} Other interesting complements correspond to {\it Galois quotients\/}. Let $G:X$ an (effective) action of a finite group $G$ on a log pair $(X/Z,B)$, with a boundary $B$ under (SM), and $-(K+B)$ is nef. For instance, $X=\mathbb P^n$ or a Fano variety, Abelian variety, Calabi-Yau 3-fold, or an identical contraction of a non-singular point. Then on the quotient $f:X\to Y=X/G$ we have a unique boundary $B_Y$ such that $K+B=f^*(K_Y+B_Y)$. Hence, $(Y/Z,B_Y)$ will be a log pair of the same type as $(X/Z,B)$. The (minimal) complement index $n$ in this case is an invariant of the action of the group. In the global $1$-dimensional case $X=\mathbb P^1$ there exists an action of a finite group $G\subset PGL(1)$ with $n\in N_1$. All exceptional cases have the quotient description. But in dimension $d\ge 1$ it may be that some complement indices do not correspond to quotients of $PGL(d)$ and not all exceptional cases have the quotient description for $X=\mathbb P^d$ or some other non-singular Del Pezzos. Even in the $1$ dimensional case $Y=\mathbb P^1$ with $B=(n-1)P/n +(m-1)Q/m$ and $n>m\ge 2$ does not correspond to a Galois quotient. In higher dimensions we expect more asymmetry as in a modern cosmology. But among symmetric minority we may meet real treasures. Quotients $X/G$, for finite groups of automorphisms of K3 surfaces $X$, reflect geometry of pairs $(X,G)$. According to Nikulin, such pairs $(X,G)$ are bounded when $G$ is non-trivial (cf. \cite[Theorem~18.5]{BPV}), in particular, with a non-symplectic action. On the other hand each Abelian surface $X$ has an involution. So, pairs $(X,\mathbb Z_2)$ are not bounded in that case. However, we anticipate that the pairs $(X,G)$ with $K\equiv 0$ and non-singular $X$ are bounded when $(X,G)$ is quite non-trivial (non-regular). For instance, $X$ has irregularity $0$ and $G\not\cong \mathbb Z_m$ with proper divisors $m|12$. We may say that such pairs $(X,G)$ are {\it exceptional\/} for dimension $2$. Indeed, if we have such a group $G$ then it has a fixed point, according to the classification of algebraic surfaces. If the fixed point is not isolated it produces a non-trivial boundary on $X/G$. Otherwise it gives a non Du Val singularity on $X/G$ when the action is non-symplectic. Then $X/G$ with a boundary belongs to a bounded family by Corollary~\ref{int}.\ref{mcorol} and Alekseev~\cite[ibid]{Al}. Perhaps this holds sometimes for symplectic actions as well as for $K3$ surfaces. \end{example-p} \refstepcounter{subsec} \begin{remark} \label{mod} Moduli spaces mentioned after the Main Theorem may have real parameters corresponding to boundary coefficients. If we would like to have usual algebraic moduli we can forget boundaries or impose a condition as (SM). A little bit more generally, we may suppose that $1$ is the only accumulation point for possible boundary multiplicities. \end{remark} \refstepcounter{subsec} \begin{remark} \label{exc} In addition to Conjecture~\ref{int}.\ref{econj} we suggest that regular $RN_d=N_{d-1}$ is enough for global non-exceptional and any local complements in dimension $d$. We verify it in this paper for $RN_2=N_1$. An explanation is related to~\cite[Lemma 5.3]{Sh2}. If we really have log canonical singularities we can induce the problem from a lower dimension (cf.~\cite[Proof of Theorem 5.6]{Sh2} and the Induction Theorem~\ref{indcomp}.\ref{indth} below). It means that in this case we need only indices from $N_{d-1}$ for dimension $d$. The same holds, if we can increase $B$ to $B'$ preserving all requirements on $K+B$ for $K+B'$ and $K+B'$ having a log canonical singularity. A good choice is to simulate construction of complements. So, we take $B'=B^+=B+H/n$ where $H\in |-nK-\lfloor (n+1)B\rfloor|$ for some $n$. If $K+B^+$ is log canonical it gives an $n$-complement. This complement is bounded when $n\in N_d$ or $N_{d-1}$. Otherwise we have a log singular case in which $H$ is called a {\it singular\/} element and $K+B^+$ is a {\it singular\/} complement. Keel and McKernan referred to such $H$ as a tiger but it looks more like a must. In this case we may consider a weighted combination $B'=a B+(1-a)B^+$ and for an appropriate $0<a<1$, log divisor $K+B'$ will be log canonical but not Kawamata log terminal. Therefore we have no reduction to dimension $d-1$ when we have no singular elements or such complements. These cases correspond to exceptions. In particular, $|-nK-\lfloor (n+1)B\rfloor|=\emptyset$ for each $n\in RN_d$ in the exceptional case. However, $|-nK-\lfloor (n+1)B\rfloor|\not=\emptyset$ for some $n\in RN_d$ in the regular cases. (Cf. Corollaries~\ref{int}.\ref{nonvan1}-\ref{nonvan3} below for $d=2$.) In the local case singular elements are easy to construct adding a pull back of a hyperplane section of the base $Z$. In the global case singular elements defines an ideal sheaf of a type of Nadel's multipliers ideal sheaf. If we have no singular elements and $B=0$ it implies an existence of a K\"ahler-Einstein metric with a good convergence in singularities (cf.~\cite{Na}). It is then known that $X$ is stable in the sense of Bogomolov (cf. \cite[Proposition~1.6]{Si}) and exceptions with $B=0$ should be bounded. An algebraic counterpart of this idea is the Borisov-Alekseev's conjecture. So, we may expect the same for exceptions with $B\not=0$ at least under (SM). This will be proven for dimension $2$ in Section~\ref{gcomp}. We discuss some exceptions in Section~\ref{ecomp}. We think the same or something close holds in any dimension (which will be discussed elsewhere), as well as for exceptional and non-exceptional cases in the {\it formal sense\/}. This means that they have an appropriate index: some $n\in RN_d$ in the regular case, and only some $n\in ER_d$ in the exceptional case. In the local cases we replace $d$ by $d-1$. For instance, any Abelian variety is exceptional according to Definition~\ref{int}.\ref{dexc}, but they are non-exceptional from the formal point of view. Their moduli are unbounded as we anticipate in the non-exceptional case. On the other hand, an elliptic fibre of type III or any other exceptional type is formally exceptional. In a certain sense their moduli are bounded. However the log terminal singularities of this type, i.~e., with the same graph of a minimal resolution, will be unbounded, but again bounded if we suppose the $\varepsilon$-log terminal property. For instance, the Du Val or surface canonical singularities of the exceptional types are bounded up to a certain degree (cf. Corollaries~\ref{csing}.\ref{exbound}-\ref{exboundf}). This is a local version of the Borisov-Alekseev's conjecture. \end{remark} \refstepcounter{subsec} \begin{remark} Complements and their indices allow us to classify contractions. In particular, we divide them into exceptional and regular. This implies the same in some special situations. For instance, any finite (and even reductive) group representation, or more generally, action on a Fano or on an algebraic variety with a numerically semi-negative canonical divisor, can be treated as exceptional or regular, and they have such an invariant as the index in accordance with their quotients in Example~\ref{int}.\ref{gquot}. In particular, this works for subgroups of $PGL$'s. In the case of $G\subset PGL(1)$ all the exceptional subgroups correspond to exceptional boundary structures on $\mathbb P^1\cong \mathbb P^1/G$. Crystallographic groups give other possible examples. We can apply the same ideas to a classification of surface quotients, log terminal or canonical, or elliptic singularities, or their higher dimensional analogs. The same holds for elliptic fiberings or other fiberings with complements. Quite possibly, the most important applications of complements are still to come; perhaps they will be related to classifications of contractions of 3-folds $X$ semi-negative with respect to $K$. They may help to choose an appropriate model for Del Pezzo, elliptic fiberings and for conic bundles. A strength of these methods is that they apply in the most general situation when we have log canonical singularities and contractions are extremal in an algebraic sense, or even just contractions. This also shows a weakness, because the distance to very special applications, when we have such restrictions as terminal singularities and/or extremal properties contractions, may be quite long and difficult. In addition, exceptional cases are still not classified completely and explicitly. A primitive sample we give in \ref{ecomp}.\ref{emainth}.3. \end{remark} Another application to a log uniruledness is given in \cite{KM}. (Cf. Remark~\ref{clasf}.\ref{rull}.) Now we explain the statement of the Main Theorem and outline a plan of its proof. We also derive some corollaries. Let $S$ be a normal algebraic surface, and $C+B$ be a boundary (or subboundary) in it. We assume that $C=\lfloor C+B\rfloor$ is the reduced part of the boundary, and $B=\{C+B\}$ is its fractional part. Let $f:S\to Z$ be a contraction. Fix a point $P\in Z$. We need to find $n\in RN_2$, such that $K+C+B$ has an $n$-complement locally over a neighborhood of $P\in Z$, or prove that other possibilities are bounded. More precisely, the latter ones are only global: $Z=P$, and $(S,C+B)$ are bounded. In particular, underlying $S$ are bounded. On the other hand, for $(S/Z,C+B)$ having an $n$-complement, there exists an element $D\in |-n K-n C-\lfloor(n+1)B\rfloor|$ such that, for $B^+=\lfloor (n+1)B\rfloor/n+D/n$, $K+C+B^+$ has only log canonical singularities (cf. \cite[Definition~5.1]{Sh2}). In the local case the liner system means a local one$/P$. As in Corollary~\ref{int}.\ref{mcorol} we can prove more. \refstepcounter{subsec} \label{nonvan1} \begin{corollary} Let $(S/Z,C+B)$ be a {\rm quasi\/} log Del Pezzo, { \rm i.~e., the pair with nef$/Z$ $-(K+C+B)$ and under (EC)\/}, and (M) holds for $B$. Then for almost all of them there exists an index $n\in RN_2$ such that $|-n K-n C-\lfloor (n+1)B\rfloor|\not=\emptyset$ or, equivalently, we have a non-vanishing $R^0f_*\mathcal O(-n K-n C-\lfloor (n+1)B\rfloor)\not=0$ in $P$. More precisely, the same hold for $n\in RN_1=\{1,2\}$, whenever $K+C+\lfloor (n+1)B\rfloor/n$ is not exceptionally log terminal, {\rm i.~e., there exists an infinite set of exceptional divisors$/P$ with the log discrepancy $0$.} Moreover, we need the condition (M) only in the global case with only Kawamata log terminal singularities, in particular, with $C=0$, and the exceptions for $RN_2$ belong to it. In this case we can choose a required $n$ in $N_2$. \end{corollary} The last two statements will be proven in Section~\ref{gcomp} and a more precise picture is given in Example~\ref{csing}.\ref{sabclas}. Note also that $n$ in the local case in the corollary depends on $P$. We prove that $N_2$ is finite but still we have no explicit description of it. By the Monotonicity~\ref{indcomp}.\ref{monl1} $\lfloor(n+1)B\rfloor\ge n B$ under the condition (SM), or even (M) if $n\in RN_2$. \refstepcounter{subsec} \label{nonvan2} \begin{corollary} Suppose again conditions (EC) and (M). Then for almost all of log pairs $(S/Z,B+C)$ there exists an index $n\in RN_2$ such that $|-n (K+C+B)|\not=\emptyset$ or, equivalently, we have a non-vanishing $R^0f_*\mathcal O(-n(K+C+B))\not=0$ in $P$. More precisely, the same hold for $n\in RN_1=\{1,2\}$, whenever $K+C+B$ is not exceptionally log terminal. The exceptions for $RN_2$ belong only to the global case with Kawamata log terminal singularities, in particular, with $C=0$. In this case we can choose $n$ in $N_2$ but in (M) instead of $b_i\ge 6/7$ we should require that $b_i\ge m/(m+1)$ with maximal $m\in N_2$. In particular, if the boundary $C+B$ is reduced, i.~e., $B=0$, then $|-n (K+C)|\not=\emptyset$ or $R^0f_*\mathcal O(-n(K+C))\not=0$ in $P$. \end{corollary} The last non-vanishing is non-trivial even when $S$ is a Del Pezzo surface with quotient singularities, because it states that $|-nK|\not=\emptyset$ or $h^0(S,-nK)\not=0$ for bounded $n$. Such Del Pezzo surfaces form an unbounded family and moreover the indices of $K$ for them are unbounded, as are their ranks of the Picard group. \refstepcounter{subsec} \label{nonvan3} \begin{corollary} Again under (EC) and (M): A log pairs $(S/Z,B+C)$ is exceptional when $Z=\mathop{\rm pt.}$, $(S,B+C)$ is Kawamata log terminal, in particular, $C=0$, and, for each $n\in RN_2$, $|-n K-\lfloor (n+1)B\rfloor|=\emptyset$ or $|-n (K+B)|=\emptyset$, equivalently, we have a vanishing $h^0(S,-n K-\lfloor (n+1)B\rfloor)=0$ or $h^0(S,-n(K+B))=0$. In particular, if the boundary $B$ is reduced, i.~e., $B=0$, then $|-n K|=\emptyset$ or $h^0(S,-nK)=0$. \end{corollary} Quite soon, in the proof of the global case in the Inductive Theorem, we see that an inverse holds at least in the weak Del Pezzo case and in the formal sense. It leads to the following question. \refstepcounter{subsec} \begin{question} Under (WLF) and (SM), does any $(X/Z,B)$ with a formal regular complement have a real non-exceptional regular complement in the sense of Definition~\ref{int}.\ref{dexc}? In general it is not true completely ($\reg{(X,B^+)}=1$), for instance, locally for non log terminal singularities with $2$-complements of type $\mathbb E2_1^0$ (see Section~\ref{clasf}); the singularity is exceptionally log terminal. It holds for log terminal singularities of types $\mathbb A_m$ and $\mathbb D_m$. But it is unknown even for surfaces $X=S/\mathop{\rm pt.}$. \end{question} The inverse does not hold if we drop (WLF), for instance, for Abelian varieties, as it was mentioned in Example~\ref{int}.\ref{tcomp}. However, it looks possible for $\mathbb P^1\times E$ where $E$ is an Abelian, in particular, an elliptic curve. In this case $-K$ has a positive numerical dimension. Moreover, each $1$-complement in such a case has log singularities. Alas, these complements are not quite non-exceptional as well. We explain it in Section~\ref{csing} in terms of $\reg{(S,B^+)}$, which specifies the question for $1$ and $2$-complements with $\reg{(S,B^+)}=1$. The above corollaries show that it is easier to construct a required $D\in |-n K-n C-\lfloor(n+1)B\rfloor|$ with small $n$ when $K+C+B$ has more log singularities. An expansion of this fact is related to the Inductive Theorem in Section~\ref{indcomp}, an analog of arguments in \cite[Theorem 5.6]{Sh2}. As in the last theorem, under conditions of the Inductive Theorem we extend or lift $D$ and a complement from its $1$-dimensional restriction or projection. So, we say that we have an {\it inductive\/} case or complement, and the latter has regular indices and is non-exceptional in the global case. Then we may try to change $(X/Z,B)$ in such a way that a type of complement will be preserved and new $(X/Z,B)$ will satisfy the Inductive Theorem. For instance, we can increase $B$ which will be used for local complements in Section~\ref{lcomp}. Here, in the global case, arises a problem with standard multiplicities. Fortunately, all the other cases are global and Kawamata log terminal where we use a reduction to the Inductive Theorem or to a Picard number $1$ case which, in addition, is $1/7$-log terminal in points. In the latter case we apply Alekseev's \cite{Al}. This we discuss in Section~\ref{gcomp}. \section{Inductive complements} \label{indcomp} \refstepcounter{subsec} \label{noncex} \begin{example}~Let $S=\mathbb P_E(F_3)$ be a ruled surface over a non-singular curve $E$ of genus $1$, which corresponds to a non-splitting vector bundle $F_3/E$ of rank 2 \cite[p.~141]{BPV} with the odd determinant. We denote the ruling by $f\colon S\to E$. It has a single section $\mathbb P_E(F_1)$ in its linear class by a Riemann-Roch and a vanishing below. We identify the section with $E$. Note that $E|_E\sim \det F_2\sim O$ for a single point $O\in E$ (cf. \cite[Proposition 2.9 in Ch.V]{H}). So, we have a natural structure of an elliptic curve on $E$ with $O$ as a zero. On the other hand $(K_S+E)|_E\sim K_E\sim 0$ according to the Adjunction Formula. Thus $(K_S+2E-f^*O)|_E\sim 0$ \cite[Lemma 2.10 in Ch. V]{H}. But $K_S+2E\equiv 0/E$. So, as in Example~\ref{int}.\ref{contex}, $K+2E-f^*O\sim 0$ and we may chose $-K_S=2E-f^*O$ as an anti-canonical divisor. In particular, $K_S^2=0$, and $-K_S$ is nef because $F_3$ is not splitting. The latter also implies that the cone $\NEc{S}$ is two-dimensional and has two extremal rays: \begin{itemize} \item the first one $R_1$ is generated by a fibre $F$ of the ruling $f$, \item the second one $R_2$ is generated by $-K_S$. \end{itemize} Since $-K_S=2E-f^*O$, ray $R_2$ has no sections. We contend that {\it $R_2$ has three unramified double sections $C_i$, $1\le i\le 3$. Each of them is $C_i\sim 2E- f^*(O+\theta_i)=-K_S-f^*\theta_i$, where $\theta_i$ is a non-trivial element of the second order in $\Pic{E}$. Hence $K_S$ has $2$-complements $C_i$: $2(K_S+C_i)\sim 0$ but $K_S+C_i\not\sim 0$, but none $1$-complement. The sections are non-singular curves of genus $1$. Moreover, $R_2$ is contractible with $\cont{R_2}:S\to \mathbb P^1$ having $4$-sections also non-singular and of genus $1$ as generic fibres . The curves $C_i$ are the only multiple (really, double) fibres of $\cont{R_2}$.\/} Since $-K_S+m E$ is ample for integer $m>0$, we have vanishings $h^i(S,m E)=h^i(K_S-K_S+m E)=0$ for $i>0$ according to Kodaira. Hence by the Riemann-Roch, $$ h^0(S,2E)=2E(2E-K_S)/2=3. $$ Similarly, $h^0(S,E)=1$. On the other hand a restriction of $|2E|$ on $E$ is epimorphic and free, because $h^1(S,E)=0$ and the restriction has degree $2$ on $E$. Thus $|2E|$ is free and defines a finite morphism $g:S\to\mathbb P^2$ of degree $4$. According to the restrictions, $E$ goes to a line $L=g(E)$, and a generic fibre $F=f^*P$ of $f$ goes to a conic $Q=g(F)$, which is tangent to $L$ in $g(P)$. Otherwise, $|2E-F|\not=\emptyset$, and we have a single double section $C\equiv -K_S$ passing through $Q\sim 2O-P$. This defines a contraction onto $E$ which is impossible by Kodaira's formula \cite[Theorem~12.1 in Ch. V]{BPV}, since $-K_S$ is nef. Thus $Q=g(F)$ is a conic and $g$ embeds $F$ onto $Q\subset\mathbb P^2$. Then, locally over $g(P)$, by the projection formula, $g^*(L|_Q)=(g^*L)|_F=2E|_F=2P$. Thus $Q$ is tangent to $L$ in $g(P)$. Note, that the same image gives a fibre $F'=f^*P'$ for $P'\sim 2O-P$, because $F_3$ is invariant for an automorphism (even involution) $i:S\to S$ induced by the involution $P\mapsto P'$ on $E$. The latter holds by a uniqueness of $F_3$ with the determinant $O$. Therefore we have a $1$-dimensional linear system $|4E-f^*O|$, a generic element of which is a non-singular $4$-section of genus $1$. This gives $g=\cont{R_2}$. Since all non-multiple fibres of $g$ are isomorphic, then $g_*K_{S/E}\equiv\chi(\mathcal O_S)=0$. Hence by Kodaira's formula we have degenerations. (Equivalently, the second symmetric product of $F_3$ is not spanned on $E$; cf. \cite[p.~3]{Md}.) Double fibres are the only possible degenerations, because their components are in $R_2$ but not sections of $f$. They give double sections $C_i$ of $f$ in $R_2$. On the other hand $2K_S\sim -4E+2f^*O\sim-g^*P$ for generic $P\in \mathbb P^1$. Thus $K_S\sim_{\mathbb Q} -g^*P/2$. In particular, $K_S$ has $2$-complement $g^*P/2$. Again due to Kodaira we have $3$ double fibres: $C_i$, $1\le 1\le 3$. Note that $2 C_i\sim g^*P\sim -2K_S\sim 4E-2f^*O$. Thus we get all properties of $C_i$, except for $\theta_i\not= 0$. This follows from a monodromy argument since we have exactly $3$ such $\theta_i \in \Pic{E}$ with $2\theta_i=0$, and $3$ double sections $C_i$ of $f$ in $R_2$. (The corresponding double cover $C_i/E$ is given by $\theta_i$.) \end{example} \refstepcounter{subsec} \label{elcomp} \begin{corollary} Let $S/E$ be an extremal ruling over a non-singular curve of genus $1$. It can be given as a projectivization $S=\mathbb P_E(F)$ for a vector bundle $F/E$ of rank $2$. Then $S/E$ or $F/E$ has \begin{itemize} \item a splitting type if and only if $K_S$ has a $1$-complement; \item an exception in Example~\ref{indcomp}.\ref{noncex} if and only if $K_S$ has $2$-complement; and \item an exception in Example~\ref{int}.\ref{contex} if and only if $K_S$ does not have complements at all. \end{itemize} In addition, the cone $\NE{S}$ is closed and generated by two curves or extremal rays: \begin{itemize} \item the first ray $R_1$ is generated by a fibre $F$ of the ruling $S/E$, \item the second one $R_2$ is given by one of curves $G$ with $G^2\le 0$ given by a splitting in a splitting case: $K_S+G+G'\sim 0$; other cases were discussed in Examples~\ref{int}.\ref{contex} and \ref{indcomp}.\ref{noncex}. \end{itemize} \end{corollary} \begin{proof} We need only to consider the splitting case when $F=V\oplus V'$ is a direct sum of two line bundles. Then we have two disjoint sections $G=\mathbb P_E(V)$ and $G'=\mathbb P_E(V')$. We may suppose that $G^2\le 0$. By the adjunction formula, $K_S+G+G'\equiv 0$, moreover, $\sim 0$ by arguments in Example~\ref{int}.\ref{contex}. Thus $K$ has a $1$-complement, and $G$ generates $R_2$. The latter holds for some $C$ with $C^2\le 0$. We assume that curve $C$ is an irreducible multi-section of $f$. If $C^2<0$, then $C=G$. Otherwise $C$ is disjoint from $G$ and $G'$, and $(K_S.C)=(K_S+G+G'.C)=0=(K_S+G+G'.G)$. Thus $C=G$. If $C^2=G^2=0$, $R_2$ is generated by $G$ as well. \end{proof} \refstepcounter{subsec} \label{indth} \theoremstyle{plain} \newtheorem*{indth}{\thesection.\thesubsec. Inductive Theorem} \begin{indth} Let $(S/Z,C+B)$ be a surface log contraction such that \begin{description} \item{{\rm (NK)}} $K+C+B$ is not Kawamata log terminal, for instance, $C\not=0$, and \item{{\rm (NEF)}} $-(K+C+B)$ is nef. \end{description} Then it has locally$/Z$ a regular complement, {\rm i.~e., $K+C+B$ has $1-,2-,3-,4-$ or $6$-complement\/}, under (WLF) of Conjecture~\ref{int}.\ref{conjcs}, or assuming (M) under any one of the following conditions: \begin{description} \item{{\rm (RPC)}} $\NEc{S/Z}$ is rationally polyhedral with contractible faces$/Z$, or \item{{\rm (EEC)}} there exists an {\rm effective\/} complement, {\rm i.~e., a boundary $B'\ge B$ such that $K+C+B'$ is log canonical and $\equiv 0/Z$,} or \item{{\rm (EC)+(SM)}}, or \item{{\rm (ASA)}} {\rm anti\/} log canonical divisor $-(K+C+B)$ {\rm semi-ample}$/Z$, or \item{{\rm (NTC)}} there exists a {\rm numerically trivial contraction\/} $\nu : X\to Y/Z$, {\rm i.~e., $\nu$ contracts the curves $F\subset S/Z$ with $(K+C+B.F)=0$.} \end{description} \end{indth} \begin{statement1} We can drop (M) in the theorem, but then it states just a boundedness of $n$-complements. More precisely, $n\in\{1,2,3,4, 5,6, 7,8, 9,10, 11,12, 13,14$, $15,16, 17,18, 19,20, 21,22, 23,24,25, 26, 27,28, 29,30, 31,35, 36,40, 41$, $42, 43$, $56, 57\}$ \end{statement1} \begin{statement2} If there exists an infinite number of exceptional divisors with the log discrepancy $0$ then we have a $1$ or $2$-complement. If we drop (M), we have in addition $6$-complements. \end{statement2} There exist (formally) non-regular complements in the Inductive Theorem when (M) is not assumed. Similarly, we have examples with only $6$-complements in \ref{indcomp}.\ref{indth}.2. \refstepcounter{subsec} \label{nonrcomp} \begin{example} Let $f:S\to \mathbb P^1$ be an extremal ruling $\mathbb F_n$ with a negative section $C$. Take a divisor $B=V+H$ with a vertical part $f^*(E)$, where $E\ge 0$, and a horizontal part $H=\sum d_i D_i$ with $d_i\in \mathbb Z/m\cap (0,1)$ for some natural number $m$. The latter can be given, for instance, as generic sections of $f$. Suppose that the different $B_C=(B)_C$ has only the standard multiplicities, $V=f^*B_C$, $\deg (K_C+B_C)=0$, and $H$ disjoint from $C$. Then $K+C+B$ as $K_C+B_C$ have $n$-complements only for $n$ such that $n$ is divided by the index of $K_C+B_C$ (cf. Monotonicity Lemmas~\ref{indcomp}.\ref{monl1} and~\ref{indcomp}.\ref{monl3} below). Then we do not have $n$ complements for $m=(n+1)$ when $K+C+B\equiv 0$ (cf. Monotonicity~\ref{indcomp}.\ref{monl2}.1 below). Therefore we have a non-regular $n(n+1)$-complement when $K_C+B_C$ has only one regular $n$-complement, and $n(n+1)\ge 7$. Moreover, we can have no regular complements in this case. For instance, this holds, when $n=6$. In general non-regular complements in the Inductive Theorem have a similar nature, as we see in its proof, with the following modifications. The ruling may not be extremal, $S$ may be singular, and $B_C$ may have non-standard multiplicities. To find complements in such cases, we use Lemmas~\ref{indcomp}.\ref{indcomp1}, \ref{indcomp}.\ref{compl2}-\ref{compl3}. If we take $V$ with the horizontal multiplicities $1/3$, and such $H$ that $(C,B_C)$ has just $2$-complements, then $K+C+B$ will have $6$-complements as the minimal. \end{example} The following result clarifies relations of conditions in the Inductive Theorem. \refstepcounter{subsec} \label{relcond} \begin{proposition} Assuming that $K+C+B$ is log canonical and nef$/Z$, $$ {\rm (WLF)}\Longrightarrow {\rm (RPC)}\Longrightarrow {\rm (NTC)}\Longleftrightarrow {\rm (ASA)}\Longleftrightarrow {\rm (EEC)}\Longleftarrow {\rm (EC)}+{\rm (SM)} $$ with the following exception. For ${\rm (WLF)}\Longrightarrow{\rm (RPC)}$: \begin{description} \item{\rm (EX1)} contraction $f:S\to Z$ is birational, and up to a log terminal resolution, $C=C+B$ is a curve with nodal singularities of arithmetic genus $1$, $/P$, $K+C\equiv 0/Z$, and $S$ has only canonical singularities and outside $C$. \end{description} For ${\rm (EEC)}\Longrightarrow{\rm (NTC)}$: \begin{description} \item{\rm (EX2)} $Z=\mathop{\rm pt.}$, $K+C+B$ has a numerical dimension $1$, $B'$ and $E$ are unique, $(K+C+B')$ has a log {\rm non-torsion\/} singularity of {\rm genus\/} $1$ and of {\rm numerical dimension\/} $1$, {\rm i.~e., on a crepant log resolution $C+B'=C$ form a curve with only nodal singularities, with the connected components of genus $1$, and $E|_{C}\equiv 0$, but not $\sim_{\mathbb Q}$.} \end{description} Nonetheless, ${\rm (WLF)}\Longrightarrow{\rm (NTC)}$ always, and ${\rm (WLF)}\Longrightarrow{\rm (RPC)}$ always in the analytic category or in the category of algebraic spaces. In (EX2) there exists a $1$-complement. \end{proposition} \refstepcounter{subsec} \begin{remcor} \label{rel} In particular, for surfaces, (WLF) always implies (ASA). In other words, if $-(K+C+B)$ is log canonical, nef and big$/Z$, then it is semi-ample. It is well-known when $K+B+C$ is Kawamata log terminal \cite[Remark~3-1-2]{KMM} (cf. with arguments in the proof below). For log canonical singularities, it is an open question in dimension $3$ and higher. In dimension $2$ we can replace the last two conditions by a non-vanishing \begin{description} \item{(NV)} $-(K+C+B)\sim_{\mathbb R} E/Z$ where $E$ is effective, \end{description} and $E^2>0$. (For a definition of $\sim_{A}$ see \cite[Defenition~2.5]{Sh3}.) Note also that (EEC) implies (NV) but not vise-versa (cf. Example~\ref{int}.\ref{contex}). So, in general nef $-(K+C+B)/Z$ is not semi-ample if it is not big and not $\equiv 0/Z$. In the latter case $E=0$ and according to the semi-ampleness conjecture for $K+C+D/Z$ we anticipate semi-ampleness \cite[Conjecture~2.6]{Sh3} (cf. \cite[Remark~6-1-15(2)]{KMM} and Remark~\ref{int}.\ref{tcomp}). This is the main difficulty in a construction of complements: (EC). In dimension $2$, at least (NV) {\it holds when $-(K+C+B)/Z$ is nef\/}. However as in Example~\ref{int}.\ref{contex} $K+C+B+E$ may not be log canonical. Does such a non-vanishing hold in higher dimensions? In any case it implies a log generalization of a Campana-Peternell's problem in dimension $2$ \cite[11.4]{CP}, \cite{Md} and cf.~\cite{Gr}: $-(K+C+B)$ is ample$/Z$ if and only if $-(K+C+B)$ is positive on all curves $F\subset Y/Z$. Indeed, then $E^2>0$ and $-(K+C+B)$ is ample due to Nakai-Moishezon~\cite[Corollary 5.4]{BPV}, or we can use the implication ${\rm (WLF)}\Longrightarrow{\rm (ASA)}$. Therefore we have a complement in a weak form (EEC). Again by Example~\ref{int}.\ref{contex} we cannot replace the above positivity be a weaker version: nef and $(K+C+B.F)=0$ only for a finite set of curves. As we will see in a proof of (NV), the nef property of $-(K+C+B)$ implies (EEC) in most cases and in a linear form. For instance, the only exception is $S$ of Example~\ref{int}.\ref{contex} when $S$ is not rational. Do exceptions exist when $S$ is rational? It appears that $-(K+C+B)$ satisfies (ASA) in most of cases as well. The exceptions again are unknown to the author. For $3$-folds, similar questions are more difficult, for example, the Campana-Peternell's problem. In dimensional $2$, the most difficult cases are related to non-rational or non-rationally connected surfaces, or more precisely, to extremal fiberings over curves $E$ of genus $1$. They are projectivizations of rank $2$ vector bundles$/E$. So, similar cases are of prime interest for $3$-folds: projectivizations $X$ of rank $3$ vector bundles$/E$, of rank $2$ over Abelian or K3 surfaces. Is their cone $\NE{X}$ always closed rationally polyhedral and generated by curves as for Fanos? What are the complements for $K$? (Cf. Corollary~\ref{indcomp}.\ref{elcomp}.) According to the same example, we really need at least a contractibility of extremal faces in (RPC) for (EEC) or (EC), moreover for regular complements in the theorem. As in (EX1), we sometimes have just the rational polyhedral property but not contractibility of any face. If $-(K+C+B)$ is only nef, the cone may not be locally polyhedral near $(K+C+B)^\perp$ \cite[Example 4.6.4]{CKM}. The same example gives an exception of (EX2). In dimension $\ge 3$, (ASA) is better than (NTC): (ASA) implies (NTC) easily, but the converse is harder as we see below. In addition, in a proof of the Inductive Theorem below, we will see that a semi-ampleness of $-(K+B)$ on $S$ is enough. Fortunately, it is good for an induction in higher dimensions. This will be developed elsewhere. It appears that the same should hold for (NTC); then it would be best in this circumstance. (NTC) also implies the Campana-Peternell's problem (but not a solution of the latter). \end{remcor} \begin{proof} First we check that (WLF) implies (NTC), and, except for (EX1), (RPC). For the former, it is enough to prove (ASA). For the latter, it is enough to prove that $\NEc{S/Z}$ is rationally polyhedral and, except for (EX1), with the contractible faces near a contracted face $\NEc{S/Z}\cap (K+C+B)^\perp$. Indeed, the cone satisfies both properties locally outside the face by the LMMP. A contraction in any face of $\NEc{S/Z}\cap (K+C+B)^\perp$ preserves this outside property. To prove (ASA) in the $\mathbb Q$-factorial case we may use the modern technique: to restrict a Cartier multiple of $-(K+C+B)$ on the log singularities of $K+C+B$. Then a multiple of $-(K+C+B)$ will be free in such singularities, because any nef Cartier divisor is free on a point, on a rational curve, and on a curve of arithmetic genus $1$ with at most nodal singularities assuming that in the last case the divisor is canonical when it is numerically trivial. We meet only this case after the restriction. Then we may apply traditional arguments to eliminate the base points outside the log singularities. For surfaces, however we may use more direct arguments, which work in the positive characteristics. Since $-(K+C+B)$ is big we have a finite set of curves $F/P$ with $(K+C+B.F)=0$. We check that each of them generates an extremal ray and (RPC) or (EX1) hold near them. After a log crepant blow-up \cite[Theorem~3.1]{Sh3}, we assume that $S$ has only rational singularities where as $K+C+B$ is log terminal. In particular, $C$ has only nodal singularities. In most cases we can easily check (RPC) and derive (ASA) from this as shown below. For instance, if $-(K+C+B)$ is ample it follows from the LMMP. Otherwise we change $C+B$ such that it will hold except for three cases: (EX1), or \begin{description} \item{\rm (i)} $C$ is a chain of rational curves$/P$, $(K+C+B.C+B')=0$ in $C+B'$ is contractible to a rational elliptic singularity, where $C+B'/P$ is the connected component of $C$ in $B$; or \item{\rm (ii)} $Z=\mathop{\rm pt.}$ and there exists a ruling $g:S\to C$ with a section $C$, with $C^2<0$, which is a non-singular curve of genus $1$, and $S$ has only canonical singularities, and outside $C$ and $B$, $B$ has no components in fibres and does not intersect $C$. \end{description} In (EX1) we obviously have the rational polyhedral property but contractions of some faces may not exist in general. In the last two cases we check (RPC) directly near $(K+C+B)^\perp$. According to \cite[Lemma~6.17]{Sh3}, we suppose that $-(K+C+B)=D=\sum d_i D_i$ is an effective divisor with irreducible curves $D_i$. Moreover, we assume that $\Supp D$ contains an ample divisor $H$. So, if $K+C+B+\delta D$ is log canonical for some $\delta>0$, then $K+C+B+\delta D-\gamma H$ gives a required boundary $C+B\colon =C+B+\delta D-\gamma H$. For instance, it works when $K+C+B$ is Kawamata log terminal. So, $C\not=0$ in the exceptional cases which are considered below. The curve $C$ is connected by \cite[Lemma~5.7 and the proof of Theorem~6.9]{Sh2} (see also \cite[Theorem~17.4]{KC}). More precisely, $K+C+B+\delta D$ is not log canonical whenever $\Supp D$ has a component $C_i$ which is in the reduced part $C$ as well (cf. \cite[1.3.3]{Sh2}). Moreover, any component $C_i$ of $C$ is $/P$, in $D$, $(K+C+B.C_i)=0$ and $C_i^2<0$. If $C_i$ is not$/P$ or $C_i/P$ with $(K+C+B.C_i)<0$, we can decrease the boundary multiplicity in $C_i$. By an induction and connectedness of $C$ this gives a reduction to the Kawamata log terminal case. If $C_i^2\ge 0$ we can do the same. Since $H$ and $D$ intersect $C_i$, $C_i\subseteq\Supp{D}$. In most cases, $C$ is a chain of non-singular rational curves, because $C$ is connected with nodal singularities and $$\deg K_C\le (C.K+C)\le (C.K+C+B)=-(C.D)\le 0.$$ The only exception arises when $\deg K_C=0$, $C$ has the arithmetic genus 1, $B=0$, and $S$ is non-singular in a neighborhood of $C$. To check (RPC) locally, we need to check (in the exceptional cases) that $D=-(K+C+B)$, that each nef $\mathbb R$-divisor $D$ near $-(K+C+B)$ is semi-ample, and that only curves $F/Z$ with $(K+C+B.F)=0$ are contracted by $D$. Indeed, $D$ is big as $-(K+C+B)$. So, we may assume that $D$ is effective as above and $D\ge H$. So, $D$ is numerically trivial only on a finite set of curves $F$ and $F^2<0$. We may also assume that $D$ is quite close to $-(K+C+B)$, namely, $(K+C+B.F)=0$. In other words we need to check that any set of curves $F$ with $(K+C+B.F)=0$ is contractible in an algebraic category. If the singularities after the contraction are rational we can pull down $D$ to a $\mathbb R$-Cartier divisor. So, the pull down of $D$ is ample according to Nakai-Moishezon \cite[Corollary~5.4]{BPV}. Hence $D$ is semi-ample. Of course, it applies directly when $D$ is a $\mathbb Q$-divisor. Otherwise we may present it as weighted combination of such divisors (cf. \cite[Step~2 in the Proof of Theorem~2.7]{Sh3}), because a small perturbation of coefficients of $D$ preserves it positivity on all curves. (Essential ones are components of $\Supp D$.) The cone is rational finite polyhedral near $K+C+B$ because the set of curves $F$ is finite. If the curves $F$ with exceptional curves on a minimal resolution form a tree of rational curves, we could then contract any set of curves $F$ by Grauert's and Artin's criteria \cite[Theorems~2.1 and 3.2]{BPV}. In addition, the singularities after contraction are rational log canonical. It works in case (i) when $C$ is a chain of non-singular rational curves $C_i/P$. Indeed, we can do contractions inductively, because $F^2<0$. First, for $F$ not in $C$. Since the boundary multiplicity in $F$ is $<1$, $F$ is a non-singular rational curve. And by a classification of the log terminal singularities, $F$ with the curves of a resolution form a tree of rational curves. So, $F$ is contractible (by the LMMP as well). Moreover, this preserves the log terminality and rationality of singularities. By \cite[Lemma~5.7]{Sh2} $C$ will form a chain of rational curves again. Finally, $C$ will be the whole curve where $K+C+B\equiv 0/Z$. Since all singularities are log terminal, we have the required resolution of $C$. Then we can contract $C$ inductively too. This effectively gives case (i) because $B=0$ near $C$. Otherwise we can decrease $B$ and replace $C+B$ by a Kawamata log terminal boundary. Now we suppose that $C$ is a curve$/P$ with only nodal singularities of arithmetic genus $1$. If $K+C+B\equiv 0/P$, then we have the exceptional case (EX1), because $-(K+C+B)$ is big$/P$. Since $S$ is non-singular in a neighborhood of $C$, it easy to check that other curves$/P$ on a minimal resolution form a chain of rational curves intersecting simply $C$. Moreover, they are $(-1)$ or $(-2)$-curves. So, we can contract any set of such curves with any proper subset of $C$ due to Artin. Here the only problem is to contract the whole $C$. Alas, this is not always true in the algebraic category, e.~g., after a monoidal transform in a generic point of $C$. The cone in this case is rational polyhedral, but contractions may not be defined for some faces including the components of $C$. Nonetheless $-(K+C+B)$ is semi-ample because $K+C+B$ is semi-ample in this case. Suppose now that $K+C+B\not\equiv 0/Z$. Then $C$ is non-singular and it is case (ii). Indeed we have an extremal contraction $S\to Y/Z$ negative with respect to $K+B$. It is not to a point or onto a curve because $(K+C+B.C)=(K+C.C)=0$, and respectively $C$ cannot be a section, if $C$ is singular. So, the contraction is birational. By a classification of such contractions it contracts a curve which does not intersect $C$. After that we have again $K+C+B\not\equiv 0/Z$. Induction on the Picard number$/Z$ gives a contradiction. If $C$ is non-singular the only possible case is when, after a finite number of birational contractions, we have an extremal ruling which induces a ruling $g$ as in (ii) with section $C$. In particular then $Z=\mathop{\rm pt.}$. $S$ has only canonical singularities and outside $C$. There are also no components of $\Supp B$ in fibres of $g$ and intersecting $C$. In other words, $K+C+B$ is canonical in points, and there are no components of $B$ in fibres of $g$. Indeed, as we know there are no components of $B$ in fibres of $g$ which intersect $C$. After extremal contractions$/C$ it holds for any component in fibres of $g$. So, after a blow-up in a non-canonical point we get a contradiction. This implies that it is really case (ii). Now we verify (RPC) near $(K+C+B)^\perp$. As above in case (i) we need to check that any set of curves $F$ with $(K+C+B.F)=0$ is contractible. Again, as in case (i), any such $F$ besides $C$ is rational. Hence such $F$ belongs to a fibre of $g$. So, we have a finite number of them. As above this implies the polyhedral property for $\NEc{S}$. In addition, a required contraction corresponds to a face in $\NEc{S}\cap (K+C+B)^{\bot}$, and the latter is generated by curves in fibres of $g$ and, perhaps, $C$. We only want to check the existence of the contraction in this face. If $C$ does not belong to such a face, it follows from the relative statement$/C$. Otherwise after making contractions of the curves of the face in fibres , we suppose that $F$ is generated by $C$. Any birational contractions, with disjoint contracted loci, commute. So, it is enough to establish a contraction of $C$, after contractions of the curves in fibres of $g$, which does not intersect $C$. Equivalently, we can assume that the fibres of $g$ are irreducible. Then $S$ is non-singular, $g$ is extremal, and $\rho (S)=2$, where $\rho(S)$ denotes the Picard number. Since $C^2<0$, we contract $C$ by $h\colon S\to S'$ to a point at least in the category of normal algebraic spaces. However, we can pull down $K+C+B$ to a $\mathbb R$-Cartier divisor $-D'=K_{S'}+h(C+B)$ because $(K+C+B)|_{C}$ is semi-ample over a neighborhood of $h(C)$. This follows from an existence of $1$-complement locally (cf. \cite[Corollary~5.10]{Sh2}) and even globally as we see later in the big case of the Inductive Theorem. Since $D'$ is nef and big, and $\rho(S')=1$, then $D'$ is ample and $-(K+C+B)=h^*D'$ is semi-ample. This completes the proof of ${\rm (WLF)}\Longrightarrow{\rm (RPC)}$. (RPC) implies (NTC) by the definition. (ASA) implies (NTC), because any semi-ample divisor $D$ defines a contraction which contracts the curves $F$ with $(D.F)=0$. A converse and other arguments in this proof are related to the semi-ampleness of log canonical divisors~\cite[Theorem~11.1.3]{KC} (it assumes $\mathbb Q$-boundaries that can be improved up to $\mathbb R$-boundaries as in \cite[Theorem~2.7]{Sh3}). So, let $\nu :X\to Y/Z$ be a numerical contraction. Then, by the semi-ampleness$/Z$, $K+C+B=\nu^*D$ for a $\mathbb R$-divisor on $Y$ which is numerically positive on each curve of $Y/Z$. Then it is easy to see that $D$ is ample$/Z$ and hence semi-ample$/Z$ except for the case when $\nu$ is birational and $Z=\mathop{\rm pt.}$. However, this case is also known because for complete surfaces $-(K+C+B)$ is ample when $(K+C+B.F)<0$ on each curve $F\subset S$ which follows from (NV) as explained in Remark-Corollary~\ref{indcomp}.\ref{rel}. In the corollary we need only to verify (NV) whenever $-(K+C+B)$ is nef$/Z$. We do it now. After a crepant resolution we assume that $K+C+B$ is log terminal. When $-(K+C+B)$ is numerically big, then (NV) follows from \cite[Lemma~6.17]{Sh2}. On the other hand if the numerical dimension of $-(K+C+B)$ is $0/Z$ then by \cite[Theorem~2.7]{Sh3} $-(K+C+B)\sim_{\mathbb R} 0/Z$. In other cases, $Z=\mathop{\rm pt.}$ and $D=-(K+C+B)$ has the numerical dimension $1$, i.~e., $D$ is nef, $D\not\equiv 0$ and $D^2=0$. We assume that (NTC) and (ASA) do not hold. Otherwise $-(K+C+B)$ is semi-ample as we already know and (NV) holds. Now we reduce (NV) to the case when $D$ is a $\mathbb Q$-divisor, or, equivalently, $B$ is a $\mathbb Q$-divisor (cf. proof in the big case of the Inductive Theorem). If $C+B$ has a big prime component $F$, then $-(K+C+B-\varepsilon F)$ satisfy (WLF) and $-(K+C+B)$ is not assumed (NTC). On the other hand, each component $F$ of $B$ with $F^2<0$ can be contracted by the LMMP or due to Artin. If this contraction is crepant, we preserve the numerical dimension, the log terminally and (NV). Otherwise $(K+C+B.F)<0$ and we can decrease the multiplicity of $B$ in $F$ which, as in the above big case, gives (NV) by (WLF). So, we assume that each $F$ in $B$ with $F^2<0$ is contracted, i.~e., for each prime component $F$ in $B$, $F^2=0$. By the same reason such components $F$ are disjoint. Hence, if we slightly decrease each irrational $b_i$ to a rational value, we get a $\mathbb Q$-boundary $B'\le B$ and a divisor $-(K+C+B')$ which is nef and with the same numerical dimension. Assuming (NV), but not (NTC), we have a unique effective $D'\sim_{\mathbb Q} -(K+C+B')$. By the uniqueness $D$ has positive multiplicities in components $F$ for any such change. Moreover they are bigger or equal to the change. So, $D\sim_{\mathbb Q} -(K+C+B)$ and (NV) holds. We assume now that $D$ is a $\mathbb Q$-divisor. After a minimal crepant resolution we suppose also that $S$ is non-singular. Then, for positive Cartier multiples $m D$, we have vanishing $h^2(S,m D)=h^0(S,K-m D)=0$, at lease for $m\gg 0$, because $D$ has positive numerical dimension. Therefore by the Riemann-Roch: \begin{align*} h^0(S,m D)\ge& h^1(S,m D)+m D(m D -K)/2+ \chi(\mathcal O_S)\\ =&-m D K/2+\chi(\mathcal O_S)= m D(C+B)+\chi(\mathcal O_S)\ge \chi(\mathcal O_S). \end{align*} In particular, $h^0(S,m D)\ge 1$ when $\chi(\mathcal O_S)\ge 1$. For instance, when $S$ is rational. Assuming now that $S$ is non-rational, we check then that, after crepant blow-downs, $S$ is an extremal or minimal ruling over a non-singular curve of genus $1$. Indeed, we have a ruling $g:S\to E$ over a non-singular curve of genus $1$ or higher, since $D=K+C+B\not=0$, and $D\not=0/E$. If $g$ is not extremal we have a divisorial contraction in a fibre of $g$, which is positive with respect to $D$. Hence after a contraction we have (WLF). Since $S$ is not rationally connected it is only possible when $C\not=0$ by \cite[Theorem~9]{Sh6}. If $C\not=0$, a classification of log canonical singularities and \cite[Corollary~7]{Sh6} implies that we have a connectedness by rational and elliptic curves in $C$. Hence $E$ has genus $1$, and $C$ is a section of $g$. If $S$ then, is not extremal we have a crepant blow-down of a $(-1)$-curve in a fibre of $g$. In addition, $\chi(\mathcal O_S)=0$ and, for each $F$ in $C+B$, $(D.F)=0$ by the above Riemann-Roch. In particular, $C+B$ does not have components in fibres . The cone $\NE{S}=\NEc{S}$ is a closed angle with two sides: \begin{itemize} \item $R_1$ is generated by a fibre of the ruling $g$, \item the second one, $R_2$, is generated by a multi-section $F$ with $F^2\le 0$. \end{itemize} Note that $(D.R_1)>0$ and $D=-(K+C+B)$ is nef. Hence, if $F^2<0$, we can take $F=C$ as a section. In this case, we find an effective divisor $D=B'-B$ increasing a component of $B\not=0$, or taking another disjoint section $F$ otherwise. Then $K+C+B+D=K+C+B'\sim_{\mathbb R} 0$ by the semi-ampleness \cite[Theorem~2.7]{Sh3}, because $K+C+B'$ is log canonical \cite[Theorem~6.9]{Sh2}. Finally, if $F^2=0$, then $D=-(K+C+B)$, and any component of $C+B$ generates $R_2$ as well. Moreover, each curve in $R_2$ is non-singular and of genus $1$. They are disjoint in $R_2$. There exist at most two of them when (NTC) is not assumed. As above $D=B'-B$, for some $B'>B$, such that $K+C+B'\equiv 0$. Using \cite{At}, it is possible to check that $K+C+B'\sim 0$ for an appropriate choice or $K+C+B'$ is log canonical, and log canonical, except for the case in Example~\ref{int}.\ref{contex}. In the latter case, we have a single curve $E$ in $R_2$, $C+B=a E$ with $a\in [0,1]$, and $K\sim -2E$. Hence $D=-(K+C+B)\sim (2-a)E>0$. This completes the proof of (NV). Now suppose (EEC): there exists an effective divisor $E=B'-B=(K+C+B')-(K+C+B)\equiv -(K+C+B)/Z$ which is nef. If $E$ is numerically big$/Z$, then $E$ is semi-ample$/Z$. This implies (NTC). In other cases the numerical dimension of $E$ is $1$ or $0/Z$. In the latter case $K+C+B\equiv -E=0/Z$ is semi-ample. This implies (ASA) and (NTC). Hence we assume that $E$ has the numerical dimension $1/Z$ and $Z=\mathop{\rm pt.}$. Then we can reduce this situation to the case when $E$ is an isolated reduced component in $C+B'$. Otherwise $E$ is contractible by the LMMP. Indeed, if $K+C+B'$ is purely log terminal near a connected component $F$ of $E$, we may increase $C+B'$ in $F$: for small $\varepsilon>0$, a log canonical divisor $K+C+B'+\varepsilon F\equiv \varepsilon F$ is log canonical and semi-ample. This implies (NTC). After a crepant blow-up, we can suppose that $K+C+B'$ is log terminal, and a log singularity is on each connected component $F$ of $E$. Then essentially by Artin we can contract the log terminal components in $F$. So, $E$ is reduced in $C+B'$, and each connected component $F$ of $E$ has the numerical dimension $1$. We check that $F$ is semi-ample when $F$ is not isolated in $C+B'$. This implies (NTC). Let $D$ be another component of $B'$ intersecting $F$ and with a positive multiplicity $b$ in $B$. Then we can subtract a nef and big positive linear combination $\delta D+\varepsilon F$ from $K+C+B'$ which gives the nef and big anti log divisor $-(K+B'')=-(K+C+B')+\delta D +\varepsilon F$. That is (WLF), which implies (RPC) and (NTC). Hence we need to consider now the case when $C+B'$ has as a reduced and isolated component $F=\Supp{E}$. We also assume that $K+C+B'$ is log terminal. Then $F$ has only nodal singularities. Since $E$ is nef and of numerical dimension $1$, $E|_F\equiv 0$. If $E|_F\not\sim_{\mathbb R} 0$, we have (EX2), but not (NTC). Nonetheless in these cases $K+C+B'\sim 0$ or, equivalently, $K+C+B$ has $1$-complement. In particular, $C+B'$ is reduced. Indeed, if $E|_F\not\sim_{\mathbb R} 0$, then $F$ is a curve of arithmetic genus $1$. To find the index of $K+C+B'$ we may replace $S$ by its terminal resolution, when $S$ is non-singular. If $S$ is rational we reduce the problem to a minimal case, where $F=C=C+B'\sim -K$. Hence $K+C+B'$ has index $1$. Otherwise $S$ is not rationally connected. Hence $F$ is a non-singular curve with at most two components of genus $1$, $S$ has a ruling $g:S\to G$ over a non-singular curve $G$ of genus $1$. If $g$ has a section $G$ in $F$, then $C$ has another section $G'$, whereas $C=C+B'=G+G'$ and $K+C+B'=K+C\sim 0$. Indeed, after contractions of $(-1)$-curves, which do not intersect a component of $F$, where $E|_F\equiv 0$, but $\not\sim_{\mathbb R} 0$, we preserve the last property on an extremal $g$. It has section $G$ in $F$ such that $E|_G\not\sim_{\mathbb R} 0$ or, equivalently, $G|_G\not\sim_{\mathbb Q} 0$. According to [At], this is a splitting case, i.~e., $G$ generates extremal ray $R_2$, and it has only two curves $G$ and another section $G'$. The latter holds because $R_2$ is not contractible. So, we can assume that $F=C+B'$ is a double section of $g$. This is possible only when the ruling $g$ is minimal. Otherwise after a contraction of a $(-1)$-curve in a fibre of $g$ intersecting $F$, we get big $F\equiv -K$. It is impossible by (ASA) for $-K$, since $K$ has no log singularities and hence $S$ is rationally connected. So, $S/G$ is extremal with another extremal ray $R_2$ generated by $F$, and $F^2=0$. According to a classification of minimal rulings over $G$, this is a non-splitting case, because (NTC) is not assumed. By Examples~\ref{int}.\ref{contex} and \ref{indcomp}.\ref{noncex} this is impossible for the other rulings: either we have no complements, or we have (NTC) by Corollary~\ref{indcomp}.\ref{elcomp}. Note that (NTC) does not hold in (EX2), for example, due to uniqueness of $B'$ and $E$. In these cases $K+C+B'\sim 0$, but not only $\equiv 0$ by Corollary~\ref{indcomp}.\ref{elcomp}. In addition, $B'$ and $E$ are unique in (EX2), because otherwise a weighted linear combination of different complements $B'$ gives a complement $C+B'$ which has no log singularity in a component of $F$. In other cases $E|_F\sim_{\mathbb R} 0$. Let $G$ be a connected component of $E$ in a reduced part of $C+B'$. We prove then that a multiple of $G$ is movable at least algebraically. This implies (NTC), because $K+B+C'+\varepsilon G'$ is semi-ample for small $\varepsilon>0$ and a divisor $G'$ disjoint from $\Supp{(C+B')}$, including $E$, which is algebraically equivalent to a multiple of $G$. This implies $\equiv$ as well. Note, that the LMMP implies, after log terminal contractions for $K+C+B'$, that $C+B'$ is reduced, or (NTC) holds. Indeed, if $B'$ has a prime component $D$ with multiplicity $0<b<1$, then $D$ is in $B$ and disjoint from $F$. After log terminal contractions of such divisors with $D^2<0$, we assume that $D^2\ge 0$ for others. Since $D^2>0$ implies (WLF) and (RPC) for $K+C+B'-\varepsilon D$, then $D^2>0$ implies (NTC) for $E$. The same holds for $D^2=0$ by the semi-ampleness of $K+C+B'+\varepsilon D$. Hence $C+B'$ is reduced after contractions. Now we can use a covering trick \cite[Example~2.4.1]{Sh2}, because $K+C+B'\sim_{\mathbb Q} 0$. If $S$ is rational, the latter holds for any numerically trivial divisor. If $S$ is non-rational, then as in (EX2) above we check that $K+C+B'\sim 0$ or $\sim_{\mathbb Q} 0$ of index $2$ by Corollary~\ref{indcomp}.\ref{elcomp}. So, after an algebraic covering, which is ramified only in $C+B'$ we suppose that $K+C+B'$ has index $1$. A new reduced boundary is an inverse image of $C+B'$. The same holds for $G$. After a crepant log resolution we assume that $S$ is non-singular and $G$ is a Cartier divisor. The curve $G$ has only nodal singularities and genus $1$. Contracting the $(-1)$-curves in $G$, we assume that each non-singular rational component of $G$ is a $(-m)$-curve with $m\ge 2$. Then all such curves are $(-2)$-curves and we can take reduced $G$. The latter is obvious in other cases, too. Indeed, after a normalization, we suppose that $D$ is a $\mathbb Q$-divisor with multiplicities $\le 1$ and with one component $D$ with multiplicity $1$. Then $K+C+B'-G$ is log terminal near $G$, $\equiv 0$ on $G$ and has multiplicity $0$ in $D$. Since each such $D$ is not a $(-1)$-curve, log divisor $K+C+B'-G$ is trivial near $G$ and $G$ is a reduced Cartier divisor. According to our assumptions and reductions, $G|G\sim_{\mathbb Q} 0$. We prove that a multiple $mG$ is linearly movable. First, we suppose that $S$ is rational. Let $m G|G\sim 0$ for an integer $m>0$. Then by a restriction sequence on $G$, we have a non-vanishing $h^1(S,(m-1)G)\not=0$ whenever $G$ is linearly fixed and the restriction is not epimorphic. Hence by the Riemann-Roch, \begin{align*} h^0(S,(m-1)G)&\ge h^1(S,(m-1)G)+(m-1)G((m-1)G- K)/2+\chi(\mathcal O_S)\\ &\ge 1+0+1=2 \end{align*} and $(m-1)G$ is linearly movable. In other cases, $S$ is not rationally connected. Moreover, $F$ is a non-singular curve of genus $1$ or a pair of them, and there exists a ruling $g: S\to G'$ to a non-singular curve $G'$ of genus $1$. If $G$ is a section of $g$, then we can suppose that $S$ is extremal$/G'$ after contractions of $(-1)$-curves, disjoint from $G$, in fibres $/G'$. By \cite{At}, (EEC) and Example~\ref{int}.\ref{contex}, it is a splitting case, i.~e., $g$ has a section $G'$ in $C+B'$, and $G'$ is disjoint from $G$. Then as in Example~~\ref{int}.\ref{contex} we verify that $G-G'\sim_{\mathbb Q} 0$, because $g$ is extremal and $(G-G')|_G=G|_G\sim_{\mathbb Q} 0$. This means that a multiple of $G$ is linearly movable. In other cases $C+B'=F=G$ is a double section of $g$. As above $S/G'$ is minimal since $S$ is not rationally connected. Then again $G$ is linearly movable when $S$ has a splitting type. If $G$ is a double section, then $C+B'=G$ and we have (NTC) by Corollary~\ref{indcomp}.\ref{elcomp}. Really, we have no such a case after the covering trick. As in \cite[Proposition~5.5]{Sh2}, (ASA) implies (EEC). Finally, (EC)+(SM) implies (EEC) by a monotonicity result below. \end{proof} \refstepcounter{subsec} \theoremstyle{plain} \label{monl1} \newtheorem*{monlemma}{\thesection.\thesubsec. Monotonicity Lemma} \begin{monlemma} Let $r=(n-1)/n$ for a natural number $n\not=0$. Then, for any natural number $m\not=0$, $$\lfloor(m+1)r\rfloor /m \ge r.$$ Moreover, for $n\ge m+1$, $$\lfloor(m+1)r\rfloor /m = 1.$$ \end{monlemma} \begin{proof} Indeed, since $m r$ has denominator $n$ and $r=1-1/n$, then $$\lfloor(m+1)r\rfloor= \lfloor m r+r\rfloor \ge m r.$$ This implies the inequality. Since $\lfloor(m+1)r\rfloor /m$ has denominator $m$, and it is always $\le 1$, we get the equation. \end{proof} \begin{proof}[Proof of the Inductive Theorem: Big case] Suppose that $-(K+C+B)$ is big (as in \cite[Theorem~5.6]{Sh2}). (Cf. \cite[Theorem~19.6]{KC}.) The proof is based on the Kawamata-Viehweg vanishing and the log singularities connectedness~\cite[Lemma~5.7 and Theorem~6.9]{Sh2}. Kawamata states that the vanishing works even for $\mathbb R$-divisors. In our situation it is easy to replace $B$ by a new $\mathbb Q$-divisor $\le B$ with the same regular complements. We can make a small decrease of each irrational $b_i$ in prime $D_i$ with $(K+B+C.D_i)<0$. Finally, after this procedure $B$ will be rational. Equivalently, each $b_i$ in $D_i$ with $(K+B+C.D_i)=0$ is rational. Since $-(K+C+B)$ is big, each such $D_i$ is contractible and corresponding $b_i$ is rational. \end{proof} A higher dimensional version can be done similarly (cf. Proof of Theorem~\ref{csing}.\ref{singth}). Other cases in the Inductive Theorem are more subtle and need more preparation. (A higher dimensional version of that will be presented elsewhere.) But first we derive some corollaries. \refstepcounter{subsec} \begin{corollary} If $(S,B)$ is a weak log Del Pezzo with $(K+B)^2\ge 4$, then it has a regular complement. \end{corollary} \begin{proof} We need to find $B'>0$ such that $(S,B+B')$ is a weak log Del Pezzo, but $K+B+B'$ is not Kawamata log terminal. By the Riemann-Roch formula and arguments in the proof of \cite[Lemma~1.3]{Sh1}, it follows from the inequality: \begin{eqnarray*} (N(-K-B).N(-K-B)-K)/2 &=N(N+1)(K+B)^2/2+(-K-B.B)/2\\ &\ge 2N(N+1)> 2N(2N+1)/2. \end{eqnarray*} \end{proof} It implies non-vanishings as in Corollaries~\ref{int}.\ref{nonvan1}-\ref{nonvan3}, in particular, the following non-vanishing. \refstepcounter{subsec} \begin{corollary} If $(S,0)$ is a weak Del Pezzo with log canonical singularities and $K^2\ge 4$, then $|-12K|\not=\emptyset$ or $h^0(X,-12K)\not=0$. \end{corollary} \refstepcounter{subsec} \begin{corollary} A weak log Del Pezzo $(S,B)$ is exceptional only when $(K+B)^2< 4$. \end{corollary} \refstepcounter{subsec} \begin{definition} Let $D$ be a divisor of a complete algebraic variety $X$. We say that $D$ has a {\it type of numerical dimension\/} $m$ if $m$ is the maximum of the numerical dimension for effective $\mathbb R$-divisors $D'$ with $\Supp{D'}\subseteq\Supp{D}$. Similarly, we define a {\it type of linear dimension\/} where we replace the numerical dimension of $D'$ by the Iitaka dimension of $D'$. Note that both $0\le m\le \dim X$. A {\it big\/} type is a type with $m=\dim X$. \end{definition} Here are the basic properties. \begin{itemize} \item The linear type $m$ is the Iitaka dimension of $D$ when $D$ is an effective $\mathbb R$-divisor. \item Such $D$ is movable if and only if $D$ has the linear type $m\ge 1$. \item The linear type is a birational invariant for log {\it isomorphisms}, i.~e., we add exceptional divisors for extractions and contract only divisors in $D$. In particular, this holds for the extractions and flips. \item For arbitrary extraction, the numerical type is not decreasing. \item For arbitrary log transform, the linear type is not decreasing. \end{itemize} For surfaces we can prove more. \refstepcounter{subsec} \label{types} \begin{p-definition} Let $X=S$ be a complete surface, and $D$ be a divisor. Then the numerical type of $D$ is not higher than that of the linear type. Moreover, if $(S,C+B)$ is a log surface such that \begin{itemize} \item $K+C+B$ log terminal, \item $K+C+B\equiv 0$ , and \item $\Supp D$ is {\it divisorially\/} disjoint from $LCS{(S,C+B)}$, i.e, they have no divisorial components in common. \end{itemize} Then the numerical type of $D$ is the same as the linear one. In addition to the big type we have a {\it fibre\/} type for a type of dimension $1$, and that of an {\it exceptional} type for a type of dimension $0$. More precisely, $D$ is a fibre {\it geometrically\/}, i.~e., there exists a fibre contraction $g:S\to Y$ with an algebraic fibre $D'=g^*P$, for $P$ in a non-singular curve $Y$, having $\Supp{D'}\subseteq\Supp{D}$, if and only if $D$ has a fibre type. In addition, $D$ is supported in fibres of $g$ and $g$ is defined uniquely by $D$. Respectively, $D$ is exceptional {\it geometrically\/}, i.~e., there exists a birational contraction of $D$ to a $0$-dimensional locus, if and only if $D$ has an exceptional type. The condition $K+C+B\equiv 0$ can be replaced by (ASA). \end{p-definition} \begin{proof} We use the semi-ampleness \cite[Theorem~2.7]{Sh3} for $K+C+B+\varepsilon D' \sim_{\mathbb R}D'$ with small $\varepsilon>0$. Note that $K+C+B\sim_{\mathbb R} 0$. If $D$ has a fibre type we have a fibering $g:S\to Y$ with a required fibre $D'=g^*P$. If $D$ has a horizontal irreducible component $D''$, then $D'+\varepsilon D''$, will have a big type for small $\varepsilon >0$. {\it Horizontal\/} means not in fibres, {\it vertical\/} in fibres. If $D$ is of an exceptional type, we can contract $D$ due to Artin or by the LMMP (cf. with the proof of Proposition~\ref{indcomp}.\ref{relcond}). Finally, if $K+C+B$ satisfies (ASA), we have a numerical complement $K+C+B'\equiv 0$ with the required properties. \end{proof} \begin{proof}[Proof of the Inductive Theorem: Strategy] In this section we assume that $Z=\mathop{\rm pt.}$. The local cases, when $\dim Z\ge 1$, we discuss in Section~\ref{lcomp} where we will obtain better results. Using a log terminal blow-up \cite[Example~1.6 and Lemma~5.4]{Sh2}, we reduce the problem to the case when $K+C+B$ is log terminal. By our assumption it has a non-trivial reduced component $C\not=0$. We would like to induce complements from lower dimensions (in most cases, from $C$). First, we prove the theorem in \begin{description} \item{ Case~I:} $C$ is not a chain of rational curves. \end{description} After that we assume that $C$ is a chain of rational curves. By Proposition~\ref{indcomp}.\ref{relcond}, we can assume (NTC) or, equivalently, (ASA). So, we have a numerical contraction $\nu : S\to Y$ for $-(K+C+B)$, where $Y$ is a non-singular curve or a point, the latter for a {\it while\/}. In the exceptional cases we have regular complements. Let $\kappa^*$ be the numerical dimension of $-(K+C+B)$, equivalently, $\dim Y$. We construct complements in different cases according to configuration of $C+B$ with respect to $\nu$. For $\kappa^*=1$, we distinguish two cases: \begin{description} \item{Case~II:} $\nu$ has a multi-section in $C$. \item{Case~III:} $C$ is in a fibre of $\nu$. \end{description} For $\kappa^*=0$, most of cases are inserted in above ones. A pair $(S,K+C+B)$ with $\kappa^*=0$ is considered as Case~II, when $D=\Supp{B}$ has a big type. As we see in a proof below, $\nu$ will naturally arise when we need it, and such $\nu$ will be a contraction to a curve. Such a pair is considered as Case~II, when $D$ has a fibre type and $C+D$ is of a big type. Equivalently, $C$ has a multi-section for $\nu=g$ with $g$ of Proposition~\ref{indcomp}.\ref{types}. Moreover, then $C$ has a double section of $g$. In this case we will have a complement by Lemma~\ref{indcomp}.\ref{indcomp2} . Such a pair is considered as Case~III, when $D$ has a fibre type, but now $C+B$ and $D$ sit only in fibres of $\nu=g$. One component of $D$ gives a (geometric) fibre of $\nu$. Contraction $\nu=g :S\to Y$ plays here the same role as $\nu$ in Case~III with $\kappa^*=1$. Finally, \begin{description} \item{Case IV:} $D$ has an exceptional type. \end{description} Then we contract the boundary $B$ to points. Case~I could be distributed in the other cases. We prefer to simplify the geometry of $C$, which simplifies slightly the proofs of Cases II-IV. We try to reduce each case to the Big one for $K+C+B$, or for $K+C+B'$ with $B'=\lfloor (n+1)B\rfloor/n$, when a complement index $n$ is suggested. There are two obstacles here. First, we cannot change $B$ or $B'$ in such a way. For instance, in Case~III with curves of genus $1$ in fibres. This leads to a separation into the cases. Second, we cannot preserve complements of some indices. For instance, decreasing $B$ as, Lemma~\ref{indcomp}.\ref{monl2} shows. Here is a situation as in Example~\ref{indcomp}.\ref{nonrcomp}, and it is a main difficulty in Case~II, when we try to induce a (regular) complement from $C$. To resolve this difficulty, we use Lemmas \ref{indcomp}.\ref{indcomp1} and \ref{indcomp}.\ref{indcomp2} below. In the cases when we cannot induce regular complements, we find others by Lemmas~\ref{indcomp}.\ref{compl1}-\ref{compl3}. This occurs only when (M) is not assumed. In Case~III, the most difficult situation, when $\nu$ has curves of genus $1$ in fibres, is quite concrete. Then we use the Kodaira's classification of degenerate fibres. We also use {\it indirectly\/} a Kodaira's formula \cite[p. 161]{BPV} for canonical divisor $K$. A possible alternative approach to Cases II-III is to use directly an analog of Kodaira's formula: \begin{equation} \label{adj} K+C+B\sim_{\mathbb R} \nu^*(K_Y+\nu_*((K+C+B)_{S/Y})+B_Y) \end{equation} for a certain boundary $B_Y$, which can be found locally$/Y$. We assume that $K+C+B\equiv 0/Y$. So, this is a fibering of genus $1$ log curves. In arbitrary dimension $n=\dim S$, formula (\ref{adj}), with a boundary $D$ instead of $\nu_*((K+C+B)_{S/Y})+B_Y$, was proposed, but not proved, in the first draft of the paper. Formula (\ref{adj}) also plays an important role in a proof of an adjunction in codimension $n$. In this context, a similar formula was proved, in our surface case, by Kawamata \cite{K2}, where $B_Y$ corresponds to a divisorial part and $f_*((K+C+B)_{S/Y})$ to a moduli part. However, we have three difficulties in its application. First, a relative log canonical divisor $(K+C+B)_{S/Y}$ is nowhere defined, and its properties are nowhere be found. Second, the divisorial part is given but not very explicitly. Third, we have $\sim_{\mathbb R}$ or $\sim_{\mathbb Q}$ for $\mathbb Q$-boundaries. So, we need to control indices for a complement on $(Y,B_Y)$ and for $\sim_{\mathbb Q}$ in (\ref{adj}) for an induced complement. Finally, in Case~IV we decrease $C$ (cf. proof of Corollary~\ref{int}.\ref{mcorol}) and use a covering trick. In this case we have trivial regular complements. A modification of a log model may change {\it types\/}, i.e., possible indices, of complements. Nonetheless for blow-down we can always induce a complement of the same index from above by \cite[Lemma~5.4]{Sh2}. An inverse does not hold in general. For instance, there are no complements after many blow-ups in generic points. Lemmas~\ref{indcomp}.\ref{mlem1} and \ref{gcomp}.\ref{mlem3} give certain sufficient conditions when we can induce complements from below. \end{proof} \refstepcounter{subsec} \begin{lemma} \label{mlem1} (Cf.~\cite[Lemma 5.4]{Sh2}.) In the notation of \cite[Definition~5.1]{Sh2}, let $f\colon X\to Y$ be a birational contraction such that $K_X+S+\lfloor (n+1)D\rfloor/n$ is numerically non-negative on a {\rm sufficiently general\/} curve$/Y$ in each exceptional divisor of $f$. Then $$K_Y+f(S+D)\ n-complementary\ \ \Longrightarrow \ K_X+S+D\ \ n-complementary.$$ \end{lemma} By a {\it sufficiently general} curve in a variety $Z$ we mean a curve which belongs to a covering family of curves in $Z$ (cf. \cite[Conjecture]{Sh4}). Note that for such curve $C$ and any effective $\mathbb R$-Cartier divisor $D$, $(D.C)\ge 0$. \refstepcounter{subsec} \begin{example-c} \label{mlem2} If $S+D=S$ is in a neighborhood of exceptional locus for $f$, and $K_X+S+D$ is nef$/Y$, then we can pull back the complements, i.e., for any integer $n>0$, $$K_Y+f(S+D)\ \ n-complementary\ \ \Longrightarrow \ K_X+S+D\ \ n-complementary.$$ \end{example-c} \refstepcounter{subsec} \label{negat} \theoremstyle{plain} \newtheorem*{negatl}{\thesection.\thesubsec. Negativity of a proper modification} \begin{negatl} (Cf.~\cite[Negativity~1.1]{Sh2}.) Let $f:X\to S$ be a proper modification morphism, and $D$ be a $\mathbb R$-Cartier divisor. Suppose that \begin{description} \item{\rm (i)} $f$ contracts all components $E_i$ of $D$ with negative multiplicities, {\rm i.e., such components are exceptional for $f$\/}; \item{\rm (ii)} $D$ is numerically non-positive on a {\rm sufficiently general\/} curve$/S$ in each exceptional divisor $E_i$ of (i). \end{description} Then $D$ is effective. \end{negatl} \begin{proof} First, it is a local problem$/S$. Second, according to Hironaka, we may assume that $f$ is projective$/S$, and $X$ is non-singular, in particular, $\mathbb Q$-factorial. The pull back of $D$ will satisfy the assumptions. The proper inverse image of a sufficiently general curve will be again sufficiently general. Third, there exists an effective Cartier divisor $H$ in $S$ such that \begin{description} \item{\rm (iii)} the support of $f^*H$ contains the components of $D$ with negative multiplicities; and \item{\rm (iv)} $f^{-1}H$ is positive on the sufficiently general curves in (ii) (it is meaningful since $f^{-1}H$ is Cartier). \end{description} For example, we may take, as $H$, a general hyperplane through the direct image of an effective and relatively very ample divisor$/S$, and through the images of the components of $D$ with negative multiplicities. According to (iii-iv), Cartier divisor $E=f^*H-f^{-1}H$ is effective with positive multiplicities for the components of $D$ having negative multiplicities, and \begin{description} \item{\rm (v)} negative on the sufficiently general curves in (ii). \end{description} Thus there exists a positive real $r$ such that $D+rE\ge 0$ and has an exceptional component $E_i$ with 0 multiplicity unless $D\ge 0$. However then $(D+rE.C)\ge 0$ for a general curve $C$ in $E_i$, which contradicts (ii) and (v). \end{proof} \begin{proof}[Proof of Lemma~\ref{indcomp}.\ref{mlem1}] We take a crepant pull back: $$K_X+D^{+X}=f^*(K_Y+D^+).$$ It satisfies \cite[5.1.2-3]{Sh2} as $K_Y+D^+$, and we need to check \cite[5.1.1]{Sh2} only for the exceptional divisors. For them it follows from our assumption and the Negativity~\ref{indcomp}.\ref{negat}. \end{proof} \refstepcounter{subsec} \label{monl2} \begin{monlemma} Let $r$ be a real and $n$ be a natural number. Then $$\lfloor(n+1)(r-\varepsilon)\rfloor /n = \lfloor(n+1)r\rfloor /n, $$ for any small $\varepsilon>0$, if and only if $r\not\in\mathbb Z/(n+1)$. \end{monlemma} \begin{statement1} Note, that for $r=k/(n+1)>0$, we have $\lfloor(n+1)r\rfloor /n=k/n>r$. So, $r>0$ is not in $Z/(n+1)$ if $\lfloor(n+1)r\rfloor /n=k/n\le r$. \end{statement1} \begin{proof}[Proof of the Inductive Theorem: Case~I] By (ASA) and \cite[Proposition~5.5]{Sh2}, we can assume that $K+C+B\equiv 0$. Then by \cite[Theorem~6.9]{Sh2}, or by the LMMP and \cite[Lemma~5.7]{Sh2}, $C$ has a single connected component, except for the case when $C$ consists of two non-singular disjoint irreducible components $C_1$ and $C_2$ such that $S$ has a ruling $g:S\to C_1\cong C_2/Y$ with sections $C_1$ and $C_2$ (cf. Theorem~\ref{clasf}.\ref{clasfth} below). By \cite[Lemma~5.7]{Sh2}, an existence of the ruling follows from the LMMP applied to $K+C_2+B$. A terminal model cannot be a log Del Pezzo surface, because $C_i$ will always be disjoint and non-exceptional during the LMMP (cf. Theorem~\ref{clasf}.\ref{dipole}). In this case we can construct complements to $K+C+B$ with given $C=C_1+C_2$. (Cf. Lemma~\ref{indcomp}.\ref{indcomp1} and~\ref{indcomp}.\ref{indcomp2} below.) Making a blow-up of the generic point of $C_1$ and contracting then the complement component of the fibre we reduce to the case when $C_2$ is big. Then, for small rational $\varepsilon>0$, $-(K+C+B-\varepsilon C_2)=-(K+C+B)+\varepsilon C_2$ is nef and big which gives a required complement by the Big case above. More precisely, we have the same complement as $K_{C_1}+(B)_{C_1}$ on $C_1$, where $(B)_{C_1}$ denotes different \cite[Adjunction~3.1]{Sh2}. The blow-ups preserve complements by \cite[Lemma~5.4]{Sh2}. For contractions we may use Example~\ref{indcomp}.\ref{mlem2}. In this case it is easily verified directly as well. So, we suppose that $C$ is connected. We also assume that each component of $C$ is a non-singular rational curve. Otherwise $C$ is a (non-singular, if we want) irreducible curve of genus $1$ with $B=0$ and non-singular $S$ in a neighborhood of $C$ [Sh2, Properties 3.2 and Proposition 3.9]. In this case we suppose that $S$ is non-singular everywhere after a minimal resolution. We chose $B$ which is crepant for the resolution. Then the LMMP and a classification of contractions in a 2-dimensional minimal model program gives a ruling $f\colon S\to C'$ with a surjection $C\to C'$, or $S=\mathbb P^2$ with a cubic $C+B=C$. Since a blow-up in any point of $\mathbb P^2$ gives a ruling, then $\mathbb P^2$ is the only possible case, when $S=\mathbb P^2$, and $C=C+B\sim -K_{\mathbb P^2}$. Hence $K+C+B=K+C\sim 0$ on $S=\mathbb P^2$, and we have a $1$-complement. Similarly, a $1$ or $2$-complement holds in the case with the ruling $f$ and double covering $C\to C'$ (cf. Lemma~\ref{indcomp}.\ref{indcomp2}). We want to check only that $B=0$ and $2(K+C+B)=2(K+C)\sim 0$ but not only $\equiv 0$. After contractions of exceptional curves of the first kind intersecting $C$ in fibres of $f$ we may suppose that $f$ is extremal, i.e., with irreducible fibres. This preserves all types of complements by Example~\ref{indcomp}.\ref{mlem2}. Note also that the same reduction holds for the ruling $f\colon S\to C'=C$ having a section $C$. Since $K+C+B\equiv 0$ and covering $C\to C'$ is double, we have no boundary components in fibres of $f$ and $B=0$. So, if $C$ is rational, then $f$ is a rational ruling $\mathbb F_n$, and $K+C+B=K+C\sim 0$. Otherwise $C'$ has genus $1$, and $(K+C)$ is $1$- or $2$-complementary by Corollary~\ref{indcomp}.\ref{elcomp}. If $S$ is a surface of Example~\ref{indcomp}.\ref{noncex}, then $C=C_i$ and we have a $2$-complement. The surface $S$ of Example~\ref{int}.\ref{contex} is impossible by (ASA). If $f$ is a splitting case: $-K_S\sim G+G'$. Therefore $C\equiv G+G'$, as well as $C\sim 2G$ and $2G'$, because $C\cap G=C\cap G'=G\cap G'=\emptyset$. Thus $2C\sim 2(G+G')\sim -2K_S$ which gives a $2$-complement (but not $1$-complement). Next we consider a case when $C=C'$ is a non-singular curve of genus $1$ and a section of $f$. Boundary $B=\sum b_i D_i\not=0$ has only horizontal components $D_i$. The curves $D_i$ are non-rational. Hence $D_i^2\ge 0$ by the LMMP. On the other hand, we have a trivial $n$-complement for $(K+C+B)|_C=K_{C'}\sim 0$ for any natural number $n$. Thus we have an $n$-complement when we have $D_i^2>0$ with multiplicity $b_i\not\in\mathbb Z/(n+1)$. Indeed, we can then construct an $n$-complement as in the Big case, for $(K+C+B-\varepsilon D_i)$ with small $\varepsilon>0$. For $(K+B+C)$, we have the same complement by the Monotonicity Lemma~\ref{indcomp}.\ref{monl2}. Since $\mathbb Z/2\cap\mathbb Z/3=\emptyset$ in the unit interval $(0,1)$, then $K+C+B$ is $1$- or $2$-complementary, whenever some $D_i^2>0$. Otherwise all $D_i^2=0$. By Corollary~\ref{indcomp}.\ref{elcomp}, the curves $D_i$ and $C'$ are in $R_2$, and are all disjoint non-singular curves (of genus $1$). Hence it is enough to find an $n$-complement in a generic fibre$/Z$, which we have for $n=1$ or $2$ by \cite[Example~5.2.1]{Sh2}. Finally, we suppose that $C$ is a (connected) wheel of rational curve. Then by the arguments of the case when $C$ is a non-singular curve of genus $1$, we see that $S$ is a rational ruling $S\to C'$ with a double covering $C\to C'$, or $S=\mathbb P^2$ with a cubic $C+B=C$. In both cases we have a $1$-complement. \end{proof} \refstepcounter{subsec} \label{monl3} \begin{monlemma} Let $r<1$ be a rational number with a positive integer denominator $n$, and $m$ be a positive integer, such that $n|m$. Then $$\lfloor(m+1)r\rfloor /m \le r.$$ Moreover, this gives $=$ if and only if $r\ge 0$. \end{monlemma} \begin{proof} Let $r=k/n$. Then $$\lfloor(m+1)r \rfloor /m= \lfloor(m+1)k/n\rfloor/m= (k m/n+\lfloor k/n\rfloor)/m= r+\lfloor r \rfloor/m\le r,$$ and $=r$ if and only if $r\ge 0$. \end{proof} \refstepcounter{subsec} \begin{corollary} Let $m$ be a natural number, and $D$ be a subboundary of index $m$ in codimension 1 and without reduced part, i.e., $m D$ is integral with the multiplicities $<m$. Then $$m D\ge \lfloor (m+1)D\rfloor,$$ and $=$ if and only if $D$ is a boundary. \end{corollary} \refstepcounter{subsec} \label{ncomp} \begin{lemma} Let $C=\lfloor C+B\rfloor$ be the reduced component in a boundary $C+B$ on a surface $S$, and $C'\subseteq C$ be a complete curve such that \begin{description} \item{\rm (i)} $K+C+B$ is (formally) log terminal in a neighborhood of $C'$; \item{\rm (ii)} $(K+C+B)|_{C'}$ has an $n$-complement; and \item{\rm (iii)} $-(K+C+B)$ is nef on $C'$. \end{description} Then $(C_i.K+C+D)\le 0$ on each component $C_i\subseteq C'$ with $D=\lfloor (n+1)B\rfloor/n$; and $K+C+D$ is log canonical in a neighborhood of $C'$. \end{lemma} In (i), {\it formally\/} means locally in an analytic or etale topology. This can be defined formally as well. \begin{proof} First, we may suppose that $C'$ is connected. Second, by [Sh2, the proof of Theorem 5.6] (cf. Proof of the Big case in the Inductive Theorem and that of the Local case in Section~\ref{lcomp}), the lemma holds when $C'$ is contractible because then we have an $n$-complement in a neighborhood of $C'$. Third, it is enough for an analytic contraction, because under our assumptions in most cases, it will be algebraic due to Artin or by the LMMP. It works and we get a rational singularity after the contraction when $C'$ is not isolated in $C+B$. Otherwise $B=0$ and $C+B=C+D=C'$. Then the lemma follows from (i) and (iii). Finally, the contraction exists when a certain numerical condition on the intersection form on $C'$ is satisfied. This will be negative after making sufficiently many monoidal transforms in generic points of $C'$. Such a crepant pull back of $K+C+D$ preserves the assumptions (i)-(iii) and the statements. \end{proof} \refstepcounter{subsec} \label{indcomp1} \begin{lemma} Let $(S,C+B)$ be a complete log surface with a ruling $f\colon S\to Z$ such that \begin{description} \item{\rm (i)} there exists a section $C_1\hookrightarrow S$ of $f$ which is in the reduced part $C$; \item{\rm (ii)} $(K+C+B)|_{C}$ has an $n$-complement for some natural $n>0$; \item{\rm (iii)} $C+D=C+\lfloor (n+1)B)\rfloor/n$ gives an $n$-complement near the generic fibre of $f$, {\rm i.e., $K+C+D$ is numerically trivial on it}; \item{\rm (iv)} $-(K+C+B)$ is nef; and \item{\rm (v)} $K+C+B$ is (formally) log terminal in a neighborhood of $C$, if we do not assume that $K+C+B$ is log canonical everywhere but just $C+B\ge 0$ outside $C$. \end{description} Then $K+C+B$ has an $n$-complement. \end{lemma} \begin{proof} The above numerical property (iv) and \cite[Theorem 6.9]{Sh2} imply that $K+C+B$ is log canonical everywhere (cf.~\cite[the proof of Theorem 5.6]{Sh2}). Making a crepant log blow-up we may assume that $K+C+B$ is log terminal everywhere, essentially by \cite[Lemma 5.4]{Sh2}. Since $f$ is a ruling, then $\NEc{S/Z}$ is rational polyhedral and generated by curves in fibres of $f$ (cf.~(EX1) in \ref{indcomp}.\ref{relcond}). Note that any contraction of a curve $E\not\subseteq C$ in fibres of $f$ will preserve (i)-(v): (ii) by \cite[Lemma~5.3]{Sh2} because the boundary coefficients of $(K+C+B)|_{C}$ are not increasing. This implies (v) as well. We consider simultaneously the boundary $C+D$ as in (iii). After contractions of curve $E\not\subseteq C$ in fibres of $f$ with $(E.K+C+D)\ge 0$, we may suppose that $-(K+C+D)$ is nef. Indeed, it is true for the fibres and on section $C_1$. Since $K+C+D\equiv 0/Z$, applying the LMMP to $f$ we may suppose that $f$ is extremal. Then $\NEc{S}$ is generated by a fibre and a section. Note that $(C_i.K+C+D)\le 0$ for the curves $C_i\subseteq C$ by Lemma~\ref{indcomp}.\ref{ncomp}. It is enough to construct an $n$-complement after such contractions by Lemma~\ref{indcomp}.\ref{mlem1}. The boundary coefficients of $D$ belong to $\mathbb Z/n$. In addition, by (iii) \begin{description} \item{\rm (vi)} $K+C+D$ is numerically trivial$/Z$. \end{description} Therefore again by Lemma~\ref{indcomp}.\ref{mlem1}, we may assume that the fibres of $f$ are irreducible or in $C$. Since $C_1$ is a section, we increase $C+D$ in fibres to $B^+$ in such a way that $(K+C+B^+)|_{C_1}$ is given by an $n$-complement in (ii). We contend that $K+B^+$ gives an $n$-complement of $K+C+B$, too. First, note that \cite[5.1.1]{Sh2} holds by the construction. Second, as above, $K+B^+\equiv 0$ because it is true for the fibres and section $C_1$. Third, $K+B^+$ is log canonical in a neighborhood of $C_1$ by the Inverse Adjunction \cite[Corollary~9.5]{Sh2}. So, as above, $K+B^+$ is log canonical everywhere, i.e., \cite[5.1.2]{Sh2}. Finally, we need to check that $n(K+B^+)\sim 0$. In particular it means that $n B^+$ is integral. Since the log terminal singularities are rational as well as any contractions of curves in fibres of $f$, we can replace $(S/Z,C+B^+)$ by any other crepant birational model. For instance, we can suppose that $S$ is non-singular, and all fibres of $f$ are irreducible. Then $S$ is a non-singular minimal ruling$/Z$ with section $C_1$. In that case $n(K+B^+)$ is integral and $\sim 0$ by the Contraction Theorem because these hold for $n(K+B^+)|_{C_1}$. The latter will be preserved after any above crepant modification. \end{proof} \refstepcounter{subsec} \label{indcomp2} \begin{lemma} Lemma~\ref{indcomp}.\ref{indcomp1} holds even if we drop (iii), but at the same time, change (i) in it by \begin{itemize} \item there exists a curve $C_1$ in $C$ with a covering $C_1\to Z$ of degree $d\ge 2$, \end{itemize} except for the case $C+B=C_i$ in Example~\ref{indcomp}.\ref{noncex}, when $n$ is odd. Moreover, then $d=2$ always. In the exceptional case we have a $2n$-complement for any natural number $n$. \end{lemma} \refstepcounter{subsec} \label{invinv} \begin{lemma} Let $f\colon X\to Y$ be a conic bundle contraction with a double section $C$. If a divisor $D\equiv 0$ over generic points of codimension 2 in $Y$, and, for any component $C_i$ of $C$, $C_i\not\subseteq D$, then the different $D_{C^{\nu}}$ is invariant under the involution $I$ given by the double covering $C^{\nu}\to Y$ on the normalization $C^{\nu}$. \end{lemma} \begin{statement1} The same holds for $C$ assuming that $K+C$ is log canonical in codimension 2 (cf. {\rm \cite[Theorem~12.3.4]{KC}\/}). \end{statement1} \begin{proof}[Proof-Commentary] First, taking hyperplane sections, we reduce the lemma to the case of a surface ruling $X\to Y$ with a double curve $C^{\nu}$ over $Y$. Second, we can drop $D$, because it is pull backed from $Y$. Finally, according to the numerical definition of the different \cite{Sh2}, and because $K+C\equiv 0/Y$ we may replace $X$ by any crepant model $(X,D)$. In particular, we may suppose that $X$ is non-singular with an extremal ruling $f$. According to M. Noether, we may assume that $C$ is non-singular as well. Then $D\equiv 0/Y$, because it is supported in fibres of $f$. So, we may drop $D$ again. In this case the different is $0$ and invariant. The same works for~\ref{indcomp}.\ref{invinv}.1. We need the log canonical condition on $K+C+D$ only to define $(K+C+D)|_{C}$. \end{proof} \refstepcounter{subsec} \label{invcomp} \begin{lemma} Let $C_1$ be a component of a semi-normal curve $C$ with a finite Galois covering $f\colon C_1\to C'$ of a {\rm main type\/} $\mathbb A$, and let $B$ be a Weil $\mathbb R$-boundary supporting in the normal part of $C$ and Galois invariant on $C_1$. Then $K+B$ has an $n$-complement which is Galois invariant on $C_1$, if and only if it has an $n$-complement. \end{lemma} \begin{proof}[Proof-Commentary] The type $\mathbb A$ means that we have branchings at most over two points $Q_1$ and $Q_2\in C'$ in each irreducible component of $C'$. According to \cite[Example~5.2.2]{Sh2}, $K+B$ has an $n$-complement if and only if $$K+\lfloor B\rfloor +\lfloor (n+1)\{B\}\rfloor/n$$ is non-positive on all components of $C$. This is a numerical condition which can be preserved, if we first replace $C$ by $C_1$, and even by any irreducible component with the Galois covering $f:C=C_1\to C'$ given by the stabilizer of this component. Intersections with other components we include into the boundary with multiplicity $1$. We also assume that $f$ is not an isomorphism. If $C$ is singular, $B^+=0$ is invariant. If $C$ is non-singular, by the Monotonicity Lemma~\ref{indcomp}.\ref{monl3}, we suppose that $n B$ is integral. If $K+B\equiv 0$ then we have a required complement by the above criterion, and it is invariant by our conditions. Otherwise $\deg (K+B)<0$, and $C$ is rational. Then under our conditions on the branchings we have them only over two points $Q_1$ and $Q_2\in C'$. In addition, we have unique points $P_1/Q_1$ and $P_2/Q_2$ respectively with maximal ramification indices $\deg f-1$. Indeed, $$ -2=\deg K_C= (\deg f)(K_{C'}+(\frac{r_1-1}{r_1})Q_1+ (\frac{r_2-1}{r_2})Q_2)=-(\deg f)(\frac1{r_1}+ \frac1{r_2}), $$ where $r_i|\deg f$ is the ramification multiplicity in $P_i/Q_i$. So, we can maximally extend $B$ in $P_1$ and $P_2$ preserving the following properties \begin{itemize} \item $B$ is Galois invariant; \item $n B$ is integral; and \item $\deg (K+B)\le 0$. \end{itemize} Then $\deg (K+B)=0$, because $\deg K=-2$. Again by the numerical condition we are done. \end{proof} \refstepcounter{subsec} \begin{example} For other types of Galois action we may lose Lemma~\ref{indcomp}.\ref{invcomp}. Let us consider, for example, a type $\mathbb D$. In this case we have a Galois covering $f \colon C\to C'$ such that \begin{itemize} \item the curves $C$ and $C'$ are isomorphic to $\mathbb P^1$; \item $f$ is branching over three points $Q_1,Q_2$ and $Q_3$; \item $f$ has $2$ branching points $P_{1,1}$ and $P_{1,2}/Q_1$ with multiplicities $d$ where $\deg f=2d\ge 4$; and \item $f$ has $d$ simple branching points $P_{i,1},...,P_{i,d}/Q_i$ with $i=2$ and $3$. \end{itemize} So, if $d=2m+1$ is odd and $n=d$, then for $$B=\frac m d (P_{1,1}+P_{1,2})+ \frac 1 d (\sum P_{2,i}),$$ $\deg (K+B)=-2+2m/d+d/d=-1/d$, whereas $B$ is invariant. However, any $d$-complement will be $B+(1/d)P$ which will not be invariant for any choice of point $P$. \end{example} \begin{proof}[Proof of Lemma~\ref{indcomp}.\ref{indcomp2}] (iv) in~\ref{indcomp}.\ref{indcomp1} implies that $d=2$. So, $D=0$ near the generic fibre of $f$, and (iii) in~\ref{indcomp}.\ref{indcomp1} is satisfied. Hence by Lemma~\ref{indcomp}.\ref{indcomp1} we may assume that $C_1$ is irreducible. By (v) in~\ref{indcomp}.\ref{indcomp1} and \cite[Lemma~3.6]{Sh2}, $C_1=C_1^{\nu}$ is {\it non-singular\/}, except for the case when $C_1$ is a Cartesian leaf, i.~e., an irreducible curve of arithmetic genus $1$ with a single nodal singularity. In such a case $C+B=C+D=C_1$ and $K+C+B\equiv 0$ by the LMMP. This gives an $n$-complement for any $n$ because $S$ is rational. The double covering $C_1\to Z$ is given by an involution $I\colon C_1\to C_1$. By (iii) in \ref{indcomp}.\ref{indcomp1}, $K+C+B\equiv 0/Z$. Thus any contraction in fibres$/Z$ is crepant. They preserve the different $(C-C_1+B)_{C_1}$. Thus we may suppose that $S/Z$ is extremal. Then $D=C-C_1+B=(K+C+B)-(K+C_1)\equiv -(K+C_1)\equiv 0/Z$ and, according to Lemma~\ref{indcomp}.\ref{invinv}, $(C-C_1+B)_{C_1}$ is invariant under $I$. So, it has an invariant $n$-complement by Lemma~\ref{indcomp}.\ref{invcomp}, when $C_1$ is rational. Otherwise $C_1=C+B$ is a non-singular curve of genus $1$ and we have again an invariant $n$-complement and for any $n$: $0$. Then we may construct $B^+$ as in the proof of Lemma~\ref{indcomp}.\ref{indcomp1}, and, after a reduction to extremal $f$, check that $B^+$ gives an $n$-complement, except for the case when $S$ is non-rational and, by Corollary~\ref{indcomp}.\ref{elcomp}, $n$ is odd. In the latter case, by the same corollary we are in the situation of Example~\ref{indcomp}.\ref{noncex}. Indeed, in a splitting case, $n(K+C+D)=n(K+C+B)= n(K+C_1)=n(K_S+G+G')\sim 0$ for any $n$. On the other hand, in the exceptional case, $2n(K+C+D)=2n(K+C+B)= 2n(K+C_1)=2n(K_S+C_i)\sim 0$ for any $n$. \end{proof} \refstepcounter{subsec} \label{invadj} \begin{lemma} Suppose that in a surface neighborhood $S$ of a point $P$, we have a boundary $B=C+\sum b_iD_i$ with distinct prime divisors $D_i$ such that \begin{description} \item{\rm (i)} $C$ is reduced and irreducible curve through $P$; \item{\rm (ii)} each $b_i=(m-1)/m$ for some integer $m>0$; and \item{\rm (iii)} $K+B$ is log terminal. \end{description} Then $$(K+B)|_{C}=\frac{n-1}n P$$ with some integer $n>0$, and $m\mid n$. \end{lemma} \begin{statement1} Formally at most one component $D_i$, with $b_i>0$, is passing through $P$. If we replace (iii) by \begin{description} \item{\rm (iv)} $K+B$ is log canonical, but not formally log terminal in $P$, \end{description} then $$(K+B)|_{C}=P\ ,$$ and formally at most two components $D_i$, with $b_i>0$, are passing through $P$. Moreover, both have multiplicities $b_i=1/2$, whenever we have two of them. \end{statement1} \begin{proof} By \cite[Theorem~5.6]{Sh2} and (i)-(iii), $K+B$ has a $1$-complement $K+C+\sum D_i$ in a neighborhood of $P$. Thus $K+C+\sum D_i$ is log canonical there. So, by \cite[Corollary~3.10]{Sh2} in a neighborhood of $P$, a single divisor $D_i$ passes through $P$, and $D_i|_{C}=1/l$ where $l$ is the index of $P$. Therefore $$(K+B)|_{C}=(\frac{l-1}l+\frac{m-1}{ml})P= \frac{n-1}n P$$ with $n=l m$ (cf. \cite[the proof of Lemma~4.2]{Sh2}). \ref{indcomp}.\ref{invadj}.1 follows from the above calculations and \cite[Corollary~9.5]{Sh2}. \end{proof} \refstepcounter{subsec} \label{adjunct} \begin{corollary} If a pair $(X,B)$ is log canonical and $B$ satisfies (M) or (SM), then for any reduced divisor $C$ in the reduced part $\lfloor B\rfloor$, the different $(B-C)_C$ satisfies respectively (M) or (SM). \end{corollary} \begin{proof} Hyperplane sections reduce the proof to a surface case $X=S$. It is enough to consider locally a log terminal case. For (M), note that $(l7-1)/l 7\ge 6/7$ for any natural number $l=n/7$. \end{proof} \refstepcounter{subsec} \label{compl1} \begin{lemma} Let $b_i\in (0,1)$ and $n$ be a natural number. Then pairs (i)-(ii) and (iii)-(iv) of conditions below are equivalent: \begin{description} \item{\rm (i)} $\sum b_i\le 1$, \item{\rm (ii)} $\sum \lfloor (n+1)b_i\rfloor/n>1$, \item{\rm (iii)} $b_i\in \mathbb Z/(n+1)$, and \item{\rm(iv)} $\sum b_i=1$. \end{description} \end{lemma} \begin{proof} The inequality (ii) holds if and only if, for natural numbers $k_i$, $b_i\ge k_i/(n+1)$ and $\sum k_i\ge n+1$. Hence by (i) we have (iii)-(iv): $1\ge \sum b_i\ge \sum k_i/(n+1)\ge 1$. The converse follows from the same computation. \end{proof} \refstepcounter{subsec} \label{compl2} \begin{corollary} In the notation of \cite[Example~5.2.2]{Sh2}, let $X=C$ be a chain of rational curves with boundary $B$. Then $(X,B)$ is $1$-complementary, but not $2$-complementary only when \begin{itemize} \item all $b_i'\in \mathbb Z/3$ and $\deg B'=1$, or \item all $b_i''\in \mathbb Z/3$ and $\deg B''=1$. \end{itemize} In either of these cases we have $4$- and $6$-complements. \end{corollary} \begin{proof} As in \cite[Example~5.2.2]{Sh2}. \end{proof} \refstepcounter{subsec} \label{compl3} \begin{lemma} In the notation of \cite[Example~5.2.1]{Sh2}, let $X=C$ be an irreducible rational curve with boundary $B$, and $n\in N_1$ be the minimal complementary index for $(X,B)$. Then $(X,B)$ is $(n+1)m$-complementary for a bounded $m$. More precisely, if $\deg B<2$: \begin{description} \item{\rm for $n=1$:} $m\in\{1,2,3,4,5,6,7,8,9,11\}$; \item{\rm for $n=2$:} $m\in\{1,2,3,4,5,6,7,8,10\}$; \item{\rm for $n=3$:} $m\in\{1,3,4,5,6\}$; \item{\rm for $n=4$:} $m\in\{2,3,4,5,6,8\}$; and \item{\rm for $n=6$:} $m\in\{3,4,5,6,8\}$. \end{description} \end{lemma} \begin{proof}[Proof-Remark] As in \cite[Example~5.2.2]{Sh2}. But it is better to use a computer. There exists a program by Anton~Shokurov for $\deg B<2$, which can be easily modified for $\deg B=2$. On the other hand we can decrease $B$. Then by Lemma~\ref{indcomp}.\ref{monl2}, we have the same complementary indices $(n+1)m$ as in the case with $\deg B<2$, except for the case when $\deg B=2$ and $B$ has index $(n+1)m+1$. Then we have an $(n+1)m+1$-complement. From a theoretical point of view, under the assumption that the number of the elements in $\Supp{B}$ is bounded, we can prove this result for any $n$, if we check that, for any $B$, there exists an $(n+1)m$-complement. Of course, here is the difficult case for $\deg B=2$ and $B$ having irrational multiplicities, which has been done in Example~\ref{int}.\ref{simul}. Finally, we anticipate that the lemma holds for arbitrary $n$ and without the assumption that $n$ is the minimal complementary index. For the letter, if $\deg B<2$, a computer check shows that \begin{description} \item{\rm for $n=2$:} $m\in\{1,2,3,\dots,15,16,18\}$; \item{\rm for $n=3$:} $m\in\{1,2,3,\dots,24,25,27\}.$ \end{description} \end{proof} \begin{proof}[Addition in a proof of~\ref{indcomp}.\ref{indth}.1] To obtain the indices in~\ref{indcomp}.\ref{indth}.1, we unify $(n+1)m$ and $(n+1)m+1$ for $n$ and $m$ in Lemma~\ref{indcomp}.\ref{compl3}. \end{proof} \begin{proof}[Addition in a proof of~\ref{indcomp}.\ref{indth}.2] To obtain the indices in~\ref{indcomp}.\ref{indth}.2, we add a $6$-complement of Lemma~\ref{indcomp}.\ref{compl2}, which follows from the proof below. \end{proof} \begin{proof}[Proof of the Inductive Theorem: Case~II] Here, we assume that $C$ has a multi-section $C_1$ of $\nu$. If $ C_1$ is not a section we have a regular complement from $C_1$ by Lemma~\ref{indcomp}.\ref{indcomp2} and \cite[Examples~5.1.1-2]{Sh2}. In the exceptional cases, we take $n=1$ or $2$, which gives regular complements again. Hence we may assume that $C$ has a single section $C_1$. Note that $C$ is then connected by \cite[Theorem 6.9]{Sh2}, and by our assumptions $C$ is a chain of rational curves. We also assume that $B$ has a big type when $\kappa^*=0$. Note also that $B\not=0$: $B$ has horizontal components for any contraction on a curve, when $\kappa^*=0$. Otherwise $C$ is a double section. Let $B_C=(B)_C$ be the different for adjunction $(K+C+B)|_{C}=K_{C}+B_C$. Then according to our construction the numerical dimension of $-(K+C+B)$ is equal to that of $-(K_{C}+B_C)$, i.~e. equal to $\kappa^*$, at least on $C_1$. Suppose that $K_C+B_C$ is $n$-complementary and $\kappa^*=1$. Let $b_i$ be a multiplicity of $B$ in a horizontal component $D_i$ such that $b_i\not\in \mathbb Z/(n+1)$. Then $-(K+C+B)+\varepsilon D_i= -(K+C+B-\varepsilon D_i)$ is big and with the same $n$-complements by the Monotonicity Lemma~\ref{indcomp}.\ref{monl2}. Therefore $K+C+B$ is $n$-complementary unless all horizontal $b_i\in\mathbb Z/(n+1)\cap (0,1)$. In this case we find a complement for another index $m\in (n+1)\mathbb N$ by Lemmas~\ref{indcomp}.\ref{monl3} and \ref{indcomp}.\ref{indcomp1}. We have an $m$-complement for such $m$ by Corollary~\ref{indcomp}.\ref{compl2} and Lemma~\ref{indcomp}.\ref{compl3}. If $C$ is not irreducible we have just regular complements. Example~\ref{indcomp}.\ref{nonrcomp} shows that we need non-regular complements as well. In addition, for multiplicities under (M) or (SM), we have only regular complements. Indeed, then $b_i=1/2$, because $K+C+B\equiv 0/Z$. By \cite[Example~5.2.2]{Sh2}, we have for $K+C+B$ a (formally) regular complement, except for the case with $n=1$, when $K_C+B_C$ is $1$-complementary, but not $n$-complementary for all other regular indices $n$. Note that $B_C$ also satisfies (M) and (SM) by Corollary~\ref{indcomp}.\ref{adjunct}. Thus it is possible only when $C$ is irreducible (otherwise we have a $2$-complement) and $B_C$ is reduced. Additionally, by \cite[Example~5.2.1]{Sh2} and in its notation, $b_1\ge b_2\ge 1/2$ and all other $b_i=0$. Hence we have a $2$-complement, which concludes the case under (M) and (SM). The same holds for an appropriate $\nu$ when $\kappa^*=0$. Indeed, we have (RPC) by Proposition~\ref{indcomp}.\ref{relcond}, because $(K+B+C)-\varepsilon D'$ satisfies (WLF) for some $D'$ with $\Supp{D'}=\Supp{B}$ and small $\varepsilon>0$. Then we use arguments in the proof of Lemma~\ref{indcomp}.\ref{indcomp1}. We mean that we contract exceptional curves $E$ with $(K+C+B'.E)>0$, where $B'=\lfloor (n+1)B\rfloor/n$. This preserves the situation and $n$-complements by the same reasons. (RPC) is also preserved. For a terminal model, we have either a fibre contraction such that $(K+C+B'.E)>0$ for its generic fibre, or $-(K+C+B')$ is nef on the model. In the former case, the contraction induces a contraction $\nu$ as for $\kappa^*=1$. Indeed, $C$ is a section of $\nu$. Otherwise $C$ is in a fibre of $\nu$. After the above contractions, $(K+C+B'.C)>0$, which contradicts Lemma~\ref{indcomp}.\ref{ncomp}. In the other cases we assume that $-(K+C+B')$ is nef after such contractions. However, $K+C+B'$ may not be log terminal, but just log canonical and only near $C$. So, if we would like to use Lemma~\ref{indcomp}.\ref{indcomp1} and the Big case, we need the following preparation. We contract all connected components of the exceptional type in $B'$. After that, by Proposition~\ref{indcomp}.\ref{types}, we have a semi-ample divisor $D'$ with $\Supp D'=\Supp B'$. We contend that one can apply Lemmas~\ref{indcomp}.\ref{indcomp1}-\ref{indcomp2} and the Big case in that situation with $C$ instead of $\lfloor C+B'\rfloor$. In the lemmas we suppose that $C$ has a multi-section for a given contraction $f:S\to Z$ and $K+C+B'\equiv 0/Z$. To verify the lemmas and the Big case, we replace $K+C+B'$ by $K+C+B'-\varepsilon D'$. The condition (ii) in \ref{indcomp}.\ref{indcomp1} follows from that for $K+C+B$ and $K+C+B'$ by \cite[Lemma~5.3]{Sh2}. Since $K+C+B'\equiv 0/Z$, we have (iii) in \ref{indcomp}.\ref{indcomp1} by Lemma~\ref{indcomp}.\ref{monl3}. Other conditions (iv) and (v) follow from the construction. For instance, (v) holds because otherwise we have a log canonical, but not log terminal, point $P\not\in \Supp{B'}$. Then $P$ is not log terminal for $K+C$ and $K+C+B$. But that is impossible by the LMMP for $K+C+B$ versus $K+C+B'$. Finally, $K+C+B'$ and $K+C+B'-\varepsilon D'$ have the same $n$-complements for small $\varepsilon>0$ by the Monotonicity~\ref{indcomp}.\ref{monl2}. We continue the case with nef $-(K+C+B')$. If $-(K+C+B')$ is big, we have an $n$-complement as in the Big case, as well for $K+C+B'$ and $K+C+B$ by our construction. If $-(K+C+B')$ has the numerical dimension $1$, we have a numerical contraction $f=\nu :S\to Z$ and $K+C+B'$ is $n$-complementary as in the above case with $\kappa^*=1$, whenever $f$ has a multi-section in $C$. The same holds and by the same reasons (cf. a proof in Case~III below) if $B'$ has a horizontal component. If $B'$ is in fibres of $f$, then $B$ (an image of $B$) is also in fibres of $f$, because $K+C+B\equiv K+C+B'\equiv K+C\equiv 0/Z$. But $B$ has a big type. Thus $B'$ has a horizontal component. Finally, $K+C+B'\equiv 0$. By the Monotonicity~\ref{indcomp}.\ref{monl2} we will have an $n$-complement, when $B'$ is of a big type. As above, if $B'$ has a fibre type and in fibres, this is only possible when $C$ is a multi-section for $g$ given by $B'$. Then $K+C+B'$ and $K+C+B$ are $n$-complementary by Lemmas~\ref{indcomp}.\ref{indcomp1} and~\ref{indcomp}.\ref{indcomp2}. Since $B$ has a big type, the case, when $B'=0$ and $K+C=K+C+B'\equiv 0$, is impossible. \end{proof} \begin{proof}[Proof of the Inductive Theorem: Case~III] If $\kappa^*=1$ and we have a horizontal element $D_i$ in $B$, we can reduce this case to the Big one for $K+C+B-\varepsilon D_i$ whenever $D_i$ is chosen properly. As in the proof of Lemma~\ref{indcomp}.\ref{ncomp}, we can find an $n$-complement, with $n\in RN_2$, near $C$ (cf. Proof of the Local case in the Inductive Theorem in Section~\ref{lcomp}). In particular, by~\ref{indcomp}.\ref{monl2}.1, we have horizontal $D_i$ with the multiplicity $b_i$ of $B$ such that $b_i\not\in \mathbb Z/(n+1)$ (cf. Lemma~\ref{indcomp}.\ref{compl1}). Then we can use the above reduction by Lemma~\ref{indcomp}.\ref{monl2}. If $\kappa^*=0$ we have $\nu$ given by $B$ and $B$ have only vertical components. Thus below we assume that $B$ has only vertical components with respect to $\nu:S\to Y$, and $\nu$ is a fibering with curve fibres of genus $1$. $C$ is also vertical and, according to Kodaira, a modification of its fibre has type $\rm I_b^*$, II, II$^*$, III, III$^*$, or IV, IV$^*$. Near such fibres we have respectively $n=2$-, $6$-, $4$- or $3$-complements (cf. Classification~\ref{lcomp}.\ref{class}). In most cases it can be extended to an $n$-complement on $S$. In other cases we have $n(n+1)$-complements. Under (M) or (SM), in the latter cases $n=1$, and we have regular $n(n+1)=2$. We can make it as in the proofs of Lemma~\ref{indcomp}.\ref{indcomp1} and Case~II. We consider contractions of curves $E\not\subseteq C$ with $(K+C+B'.E)\ge 0$ for $B'=\lfloor (n+1)B\rfloor/n$. In particular, $B'$ has multiplicities in $\mathbb Z/n$. Since $K+C+B=K+C+B'=K\equiv 0/Y$, we can make $\nu$ extremal outside a fibre $C$, i.~e., all other fibres are irreducible. Then $K+C+B'$ is positive on fibres of some fibering $f:S\to Z$, or $-(K+C+B')$ is nef. Indeed, we can contract components of $C$ preserving the numerical properties of $K+C+B'$ because $K+C+B'\equiv 0/Y$ on each component of $C$. A terminal model will be extremal and its cone has two extremal rays: \begin{itemize} \item the first one $R_1$ is generated by a fibre $F$ of $\nu$, \item the second one $R_2$ is generated by a multi-section section $E$. \end{itemize} If $(K+C+B'.E)> 0$, then $E$ induces a required fibering $f$. We need to check that if $E$ is contracted to a point, then we have a required fibre contraction on $S$. The former contraction induces a birational contraction $f:S\to Z$, for a birational inverse image of $E$. After that $K+C+B'$ will be nef and big. Moreover, it will also be positive on each curve $E\not\subseteq C$. Again we can find extremal contractions of such $E$ in $S$ subtracting $C$. Finally, they give a required fibering, because a terminal model cannot have the Picard number $1$ by Lemma~\ref{indcomp}.\ref{ncomp}. In the case of such a fibre contraction (ruling), we return to the original $S$. By Lemma~\ref{indcomp}.\ref{ncomp}, $C$ will have a section of induced $f:S\to Z$. So, the horizontal multiplicities of $B$ are in $\mathbb Z/(n+1)$ by Lemma~\ref{indcomp}.\ref{compl1}. Then we use Lemma~\ref{indcomp}.\ref{indcomp1}. After contractions in a fibre of $C$ for $\nu$, we have a fibre $C$ with the same $B_C$, because $K+C+B\equiv 0/Y$. On the other hand, by Corollary~\ref{indcomp}.\ref{adjunct}, $B_C$ satisfies (M) and (SM) whereas $\deg (K_C+B_C)=0$. Thus by Monotonicities~\ref{indcomp}.\ref{monl1} and \ref{indcomp}.\ref{monl3}, $K_C+B_C$ is $n$-complementary if and only if $B_C$ has index $n$. Therefore $K_C+B_C$ is $m n$-complementary for any natural number $m$. Then, for $n:=(n+1)n$, we have (ii)-(iii) in Lemma~\ref{indcomp}.\ref{indcomp1} on $S$. Thus $K+C+B$ is $n(n+1)$-complementary. Under (M) or (SM), $n=1$. In other cases $-(K+C+B')$ is nef after the above contractions. We also assume that all fibres of $\nu$, except for fibre $C$, are irreducible. By \cite[Theorem~6.9]{Sh2} after a complement (cf. with the proof of Case~I), $K+C+B'$ is log terminal, except for the case when $C$ and $C'=\lfloor B'\rfloor$ are irreducible curves, and $K+C+B'\equiv 0$. Then $K+C+B'$ and $K+C+B$ are $n$-complementary as in the proof of Case~I. So, we suppose that $K+C+B'$ is log terminal. If $K+C+B'\equiv 0$, then $B'$ has a fibre type, and we have an $n$-complement as in Case~II. Indeed, we can use Lemma~\ref{indcomp}.\ref{indcomp1} when there exists a ruling $f:S\to Y$. Otherwise, by the LMMP with $K+C+B-\varepsilon D$ for an algebraic fibre $D=\nu^*(\nu(C))$, we have a birational contraction of a curve $E\not\subseteq C$ with $(K+C+B'.E)=0$. Since $E$ is a multi-section of $\nu$ and $B'$ has a fibre type before the contraction, it will have a big type afterwards. Therefore, decreasing $B'$ after contraction, we have an $n$-complement as in the big case by Lemmas~\ref{indcomp}.\ref{monl2} and \ref{indcomp}.\ref{mlem1}. Finally, if $(K+C+B'.E)<0$ we increase $B'$ adding vertical components, but not in $C$, in such a way that the new $B'$ will again have multiplicities in $\mathbb Z/n$, and $(K+C+B'.E)=0$. Then $K+C+B'\equiv 0$, which is the above case. To verify $K+C+B'\equiv 0$, note that $\nu$ is also a numerical contraction for $K+C+B'$ with old $B'$, and new $B':=B+\nu^* D$ for an effective divisor $D$ on $Y$. Before increasing $B'$, we classify and choose an appropriate model for fibre $C$. We suppose that $C$ is minimal: all $(-1)$-curves of $C$ are contracted whenever $S$ is non-singular near $C$. It is possible only when $C$ is reducible and the $(-1)$-curve is not an edge in the chain of $C$. Then fibre $C$ has one of the following types (see \cite{BPV} and the Classification~\ref{lcomp}.\ref{class}): \begin{description} \item{($\rm I_b^*$)} A minimal resolution of fibre $C$ has a graph $\widetilde D_{4+b}$, where $b+1\ge 1$ is the number of irreducible components of $C$. All curves of the resolved fibre are $(-2)$-curves. $B_C=(1/2)(P_1+P_2+P_3+P_4)$. \item {(II)} $B_C=(1/2)P_1+(2/3)P_2+(5/6)P_3$. \item {(III)} $B_C=(1/2)P_1+(3/4)(P_2+P_3)$. \item {(IV)} $B_C=(2/3)P_1+(2/3)P_2+(2/3)P_3$. \end{description} Curve $C$ is irreducible exactly in the cases $(\rm I_0^*)$ and (II)-(IV). Moreover, in cases II-IV, $C$ is $(-1)$- or $(-2)$-curve, which splits our classification into Kodaira's cases II-IV and II$^*$-IV$^*$, respectively (cf. Classification~\ref{lcomp}.\ref{class}). Respectively, for I$_b^*$ and (II)-(IV), $n=2$ and $6,4,3$. Type I$_b$ disappeared by our conditions on $C$. When $C$ is a $(-1)$-curve, we transform fibre $C$ of type (II)-(IV) into a standard one $F_0$ of type II-IV in Kodaira's classification \cite[p. 158]{BPV}. Then $S$ is non-singular near $F_0$, $F_0$ has multiplicity $1$ for $\nu$, and, near $F_0$, modified $C=(5/6)F_0, (3/4)F_0,$ and $(2/3)F_0$ respectively. Now the log singularity $C$ is hidden in a point on $F_0$, and new $C$ is not reduced. By Lemma~\ref{indcomp}.\ref{mlem1}, we preserve the $n$-complements. During the crepant modification all boundary multiplicities have denominator $n$. If $C$ is a $(-2)$-curve, then for types (II)-(IV) all curves of the resolved fibre are $(-2)$-curves too. Fibre $F_0=nC$ has, respectively, multiplicity $6,4$ and $3$. In case $(\rm I_b^*)$, $F_0=2 C$. This is an algebraic fibre. Now we choose $E$. Since $K+C+B'$ is not nef, there exists an extremal contraction $f:S\to Z$ negative with respect to $K+C+B'$. It is not to a point. First, suppose that $f$ gives a ruling with a generic fibre $E$, i.~e., $E$ is a $0$-curve or $(K.E)=-2$. Then $(C+B'.E)\le 2-1/n$, because $E$ only crosses $C+B'$ in non-singular points, and $(K+C+B'.E)\le -(1/n)$, when $<0$. If $E$ is a section of $\nu$, we can add to $K+C+B'$ a few copies of the generic fibre $F$ of $\nu$ with multiplicity $1/n$. Otherwise $E$ is a double section of $\nu$, except for cases II$^*$-IV$^*$, which we consider later. This follows from inequality $(C+B'.E)\le 2-1/n$, because $\nu$ has multiplicity $\le 2$ in $F_0$ and even $1$ in the cases II-IV. On the other hand, $\mult{F_0}{C}\ge (n-1)/n\ge 2/3$ in the cases II-IV. So, $E$ is a double section, and, if $B'\not= 0$, then $(B'.E)\le 1/n$: in case $\rm I_b^*$, $(B'.E)=(C+B'.E)-(C.E)\le 1-1/n=1/2$, and, in the cases II-IV, $(B'.E)=(C+B'.E)-(((n-1)/n)F_0.E)\le (2-1/n)-2(n-1)/n=1/n$. Therefore such $B'=(1/n)F_1$ and $(F_1.E)=1$, where $F_1\not= F_0$ is an irreducible fibre of $\nu$. Then we increase $B'$ to $(2/n)F_1$. If $B'=0$, we increase $B'$ by $(1/n)F$ in a generic fibre $F/Y$. If $f$ is birational, $f$ contracts a $(-1)$-curve $E$ ($(-1)$-curve on a minimal resolution). Again in the cases I$_b^*$ and II-IV, $0>(C+K+B'.E)\ge -(1/n)$. Moreover, $E$ is a section of $\nu$, and we can increase $B'$ adding $(1/n)F$, when $K+C+B'$ has index $n$ near $E$. In case $\rm I_b^*$, $E$ passes through $P_i$. In the other cases $K+C+B'$ does not have index $n$ somewhere near section $E$. It gives a singularity of $S$ on $E$ outside $C$. On the above log model, $E$ has another singularity $P_3$ of $S$. (In the cases II-IV, section $E$ crosses $F_0$ only in a (single) non-singular point of $F_0$.) Hence, by a classification of surface log contractions, there exists just the single singularity outside $C$, and we increase $B'$ only in a corresponding fibre $F_1$. After that we need to check only that $K+C+B'$ will have index $n$ near $F_1$. After a crepant modification we suppose that $S$ is minimal$/Y$ and non-singular near $F_1$. On the other hand, $(K+C+B'.E)=0$ and $K+C+B'$ has index $n$ outside $F_1$. Since $E$ is a section, $K+C+B$ has index $n$ near $E$ or in a point $F_1\cap E$. Using a classification of degenerations due to Kodaira, we obtain that $K+C+B'$ has index $n$ everywhere in fibre $F_1$ and in $S$. Finally, we consider the cases with II$^*$-IV$^*$. When $f$ is a ruling, there exists a $(-1)$-curve $E$ in a singular fibre of $f$. In this case we choose $E$ through $P_3$. If $f$ is birational again we have such a curve $E$. In both cases $C$ passes through a singular point $P_j\in C$ of $S$. This time: $0>(C+K+B'.E)\ge -(n-1)/n$. More precisely, if $E$ is passing through point $P_j$ of type $\mathbb A_i$, then $(C+K+B'.E)\ge -i/(i+1)$. Since $E$ is not of a big type, then on a minimal resolution it can intersect only $(-m)$-curves of the resolution with $m\ge i+1$ and $=$ only when $f$ is a ruling. But then $(C+K+B'.E)\ge -i/(i+1)+(i-1)/(i+1)=-1/(i+1)$, and $\ge -i/(i+1)(i+2)$ when $E$ is contractible. In particular, if $E$ is a section, then $i+1=n$ and $-(n-1)/n(n+1)>-1/n$. So, $E$ has at most two singularities of $S$ with the m.l.d. $<1$ (the m.l.d. is the minimal log discrepancy), and no intersections with $B'$ in other points in such a case. So, we can complete $B'$ as above when $E$ is a section. Note also that in the ruling case, $E$ is a section, except for the case when $E$ has only one singularity $P_3$ of type $\mathbb A_3$, and $E$ intersects the middle curve on a minimal resolution of this point. Then $n=4$, $E$ is double section, $B'$ intersects $E$ simply in a single branching point $Q\in E/Y$, and $(K+C+B'.E)=1/2$ in $P_3$ and $1/4$ in $Q$. Such intersection means the intersection of $E$ on a minimal crepant resolution with the boundary. So, we complete $B'$ in the corresponding fibre. Then $(K+C+B'.E)=1/2$ in $Q$ too for new $B'$. We need to verify that, for such $B'$, $K+B'$ will have index $n$. This can be done on a Kodaira model $C'$. If $E$ passes through a singular point on Kodaira's model $C'$ after contractions to the central curves as in our models $(I_b^*)$ and (II)-(IV), then for the fibre $C'$ of type I$_b^*$, $E$ passes through a simple Du Val singularity, $B'=(1/2)C'$, and $K+(1/2)C'$ has index $4$. Another possible type, assuming that $C'$ is not reduced in $B'$, is only of type IV$^*$ with $B'=(3/4)C'$ and $E$. Again $K+(1/2)C'$ has index $4$. If $E$ passes transversally through a non-singular point of $C'$, then the multiplicity of $C'$ is $2$, $B'=(1/2)C'$ and $K+(1/2)C'$ has index $4$. If $E$ does not transversally pass $C'$, the multiplicity of $C'$ is $1$, $(C'.E)=2$ near $C'$, $B'=(1/4)C'$ and $K+(1/4)C'$ has index $4$. In the other cases, $E$ is contractible, and $E$ is an $l$-section with $2\le l\le 4$. According to the classification of such contractions, $E$ has a simple single intersection with exceptional curve over $P_j$, and it is an edge curve. Moreover, $E$ has at most two singularities. In the latter case $B'\not= 0$ only in two corresponding fibres $C$ and $C'$, because $-i/(i+1)(i+2)\ge 1/6$ and $(K+C+B'.E)<0$. In addition, $(K+C+B'.E)=1/(i+1)$ near $P_j$, and $l=n/(i+1)$. We increase $B'$ to the numerically trivial case. Near $C'$, $(K+C+B'.E)=i/(i+1)$ for new $B'$. For such $B'$, we verify, that $K+B'$ has index $n$ near $C'$. If $i=1$, $n=4$ and $l=2$, it was proved above in the ruling case. If $i=1$, $n=6$ and $l=3$, we can proceed similarly. Since $E$ is $3$-section, the type I$_b^*$ is only possible when $B'=(1/3)C'$ with $E$ passing a singular point. Then $K+B'$ has index $6$. For type I$_b$, $B'=(1/6)C'$ when $C'$ has the multiplicity $1$ and $B'=(1/2)$ when $C'$ has the multiplicity $3$. In both cases, $K+B'$ has index $6$. The same holds for types II-IV whereas the multiplicity of $C'$ is $1$. In the case III$^*$, $B=(2/3)C'$ and $K+(2/3)C'$ has index $6$. In the case IV$^*$, $B=(1/2)C'$ and $K+(1/3)C'$ has index $6$. In the other case with two singularities, $i=2$, $n=6$ and $l=2$. For type I$_b$, $B'=(2/3)C'$ when the multiplicity of $C'$ is $2$, and $B'=(1/3)C'$ when the multiplicity is $1$. The same hold for types II-IV whereas the multiplicity of $C'$ is $1$. For type I$_b^*$, $B'=(1/3)C'$ whereas $E$ does not pass through singularities. Types II$^*$-IV$^*$ are impossible in this situation. In all these cases $K+C+B'$ has index $6$. Finally, $P_j$ is the only singularity of $S$ on $E$. Then it is enough to construct $B'$ near $E$. So, it is possible to do this when $E$ has the only (maximally branching) point with simple intersection with a fibre somewhere over $Y$. On the other hand, if $B'\not =0$ is in a fibre $C'$ with a non-branching point, then, near $C'$, $B'\ge (1/n)C'$ and $(B'.E)\ge l/n\ge 2/n$. So, $n\ge 4$, and, for $n=4$, $(K+C+B'.E)=1/4$ in $P_j$, and $l=2$ is impossible. So, $n=6$, $(K+C+B'.E)=1/3$ in $P_j$, and we can increase $C'$ up to $(1/3)C'$, when $B'$ does not have other components. Otherwise we have a third fibre $C''$ with a simple only intersection and a $6$-complement. In all other cases we have two branchings of $E/Y$ in one fibre $C'$, $l=4$, and these branchings are in a fibre $C'$ with $B'\ge (1/n)C'$. (Cf. Lemma~\ref{indcomp}.\ref{invcomp}.) Then $n=6$, $P_j=P_2$, $(K+C+B'.E)=2/3$ in $P_2$, and this case is impossible. \end{proof} \begin{proof}[Proof of the Inductive Theorem: Case~IV] Here we suppose that $K+C+B\equiv 0$ and $B$ has an exceptional type. Then $B_C$ satisfies (SM), by Lemma~\ref{indcomp}.\ref{adjunct} after a contraction of $B$. On the other hand, $\deg (K_C+B_C)=(K+C+B.C)=0$. Thus we have an $n$-complement on $C$ and near $C$ (exactly) for such $n$ that $B_C$ has index $n$. Let us take such $n$. Therefore $K+C+B=K+C+B'\equiv 0$ and has index $n$ near $C$, where $B'=\lfloor (n+1)B\rfloor/n$. This follows from \cite[Proposition~3.9]{Sh2}, when $B$ is contracted. To establish that $K+C+B$ gives an $n$-complement, we need to check that $K+C+B$ has index $n$ everywhere in $S$. After a contraction of $B$ we assume that $C+B=C$. Using the LMMP for $K$, we reduce the situation to the case when there exists an extremal fibre contraction $f:S\to Z$. If $F$ is to a curve $Z$, we use Lemma~\ref{indcomp}.\ref{indcomp2}. Perhaps we change $n$ by $2n$ when $n=1$ or $3$. Otherwise $Z=\mathop{\rm pt.}$, $S$ has the Picard number $1$, and $C$ is ample. Then we use a covering trick with a cyclic $n$-covering $g:T\to S$. On $T$, $K_T+D=g^*(K+C)$ has index $1$ near ample $D=g^{-1}C$. By \cite[Proposition~3.9]{Sh2}, $D$ is a non-singular curve (of genus $1$) and $T$ is non-singular near $D$. Hence $T$ is rational and $K_T+D\sim 0$ (cf. proof of Case~I). Therefore, $K+C+B$ has index $n$. \end{proof} \begin{proof}[Proof of \ref{indcomp}.\ref{indth}.2: Global case] From the proof of the Inductive Theorem. \end{proof} \section{Local complements} \label{lcomp} In the {\it local case\/}, {\rm when $Z\ge 1$\/}, we can drop most of the assumptions in the Main and Inductive Theorems, and in other results. \refstepcounter{subsec} \label{lmainth} \begin{theorem} Let $(S/Z,C+B)$ be a surface log contraction such that \begin{itemize} \item $\dim Z\ge 1$, and \item $-(K+C+B)$ is nef. \end{itemize} Then it has locally$/Z$ a regular complement, {\rm i.~e., $K+C+B$ has $1-,2-,3-,4-$ or $6$-complement\/}. \end{theorem} \begin{proof}[Proof of the Theorem~\ref{lcomp}.\ref{lmainth}: Special case] Here (NK) of the Inductive Theorem is assumed in the following strict form: \begin{description} \item{(MLC)} ({\it maximal log canonical\/}) $K+C+B$ is not Kawamata log terminal {\it in\/} a fibre$/P$, near which we would like to find a complement, i.~e., $C$ has a component in the fibre, or an exceptional divisor, with the log discrepancy $0$ for $K+C+B$, has the center in the fibre. \end{description} But as in Theorem~\ref{lcomp}.\ref{lmainth}, we drop other assumptions in the Inductive Theorem except for (NEF). After a log terminal blow-up, we assume that $K+C+B$ is log terminal, and by our assumption $C\not=0$. Moreover, $C$ has a component in a fibre of $f$. According to the Big case, we suppose that $Z$ is a non-singular curve, and $K+C+B\equiv 0/Z$. Hence $f$ is a fibering of log curves of genus $1$. Note that $C$ is connected near the fibre by the LMMP and \cite[Lemma~5.7]{Sh2}. Let $B_C=(B)_C$ be the different for adjunction $(K+C+B)|_{C}=K_{C}+B_C$. Then $K_C+B_C\equiv 0/P$. Take a regular $n$ such that $K_C+B_C$ is $n$-complementary near the fibre. We contend that $K+C+B$ is $n$-complementary near the fibre too. Divisor $-(K+C+B')$ is nef on a generic fibre of $f$ where $B'=\lfloor (n+1)B\rfloor$. Indeed, by the LMMP and Proposition~\ref{indcomp}.\ref{ncomp}, the same holds for the fibre after a contraction of non-reduced components of $B$ in the fibre$/P$. We decrease $C+B$ in a horizontal component whenever one exists. Then, by Lemma~\ref{indcomp}.\ref{monl2} and \ref{indcomp}.\ref{monl2}.1, for an appropriate choice of the horizontal component, we have the same $n$-complements and use the Big case. So, we can assume that $C+B$ has no horizontal components. Thus $f$ is a fibering of curve of genus 1. Then, as in Case~IV in the proof of the Inductive Theorem, $B$ has an exceptional type. As there we can verify that $B_C$ satisfies (SM), $K_C+B_C\equiv 0$, and $K_C+B_C$ has index $n$. Moreover, $K+C+B$ has (local) index $n$ near the fibre$/P$, and $B'=B$. Therefore, to check that $K+C+B$ has a (trivial) $n$-complement near the fibre, we need to check only that $K+C+B$ has a {\it global} index $n$, i.~e., \begin{equation} \label{indn} n(K+C+B)\sim 0/P. \end{equation} The log terminal contractions, i.~e., contractions of the components in $B$, preserve the (formal) log terminal property of $K+C+B$ and (\ref{indn}) according to \cite[2.9.1]{Sh1} \cite[3-2-5]{KMM}. Thus we assume after contractions that $C+B=C$ is the fibre$/P$ and $K+C$ is (formally) log terminal. Such a model of the fibre is its {\it weak log canonical model\/}. It is not unique. For instance, we can blow up a nodal point of $C$. However, it is unique if we impose the following {\it minimal\/} property: \begin{itemize} \item all $(-1)$-curves of $C$ are contracted whenever $S$ is non-singular near $C$. \end{itemize} Such a model will be called {\it log minimal}. Its uniqueness follows from the MMP and a classification below (see Classification~\ref{lcomp}.\ref{class}). Note also that it can be non log terminal, but it is always formally log terminal. We check (\ref{indn}) for the minimal $n$ such that $K_C+B_C$ has index $n$, i.~e., for the index of $K_C+B_C$. This essentially follows from Kodaira's classification of elliptic fibres \cite[Section 7 of Ch. V]{BPV} and his formula for a canonical divisor of an elliptic fibering \cite[Theorem~12.1 in Ch. V]{BPV} \cite{Shf}. See also the Classification~\ref{lcomp}.\ref{class} below. The latter gives a (non-standard) classification of the degenerations for a fibering with the generic curve of genus 1. \end{proof} First, we add types $_m\rm I_b,b\ge 0,$ to types of log models in the proof of Case~III in the Inductive Theorem. In the following cases, $S$ is non-singular near $C$. \begin{description} \item{($_m\rm I_0$)} $C$ is a non-singular curve of genus $1$, and $f^*P=m C$. \item{($_m\rm I_1$)} $C$ is an irreducible rational curve of genus $1$ with one node, and $f^*P=m C$. \item{($_m\rm I_b$)} $C$ is a wheel of $b\ge 2$ irreducible non-singular rational curves $C_i$, and $f^*P=m C$. Each $C_i$ is a $(-2)$-curve. \end{description} \refstepcounter{subsec} \label{class} \theoremstyle{plain} \newtheorem*{class}{\thesection.\thesubsec. Classification of degenerations in genus $1$ (Kodaira)} \begin{class} Any degeneration of non-singular curves of genus $1$ has up to a birational transform a log minimal model of one of the following types: $_m\rm I_b$, $_m\rm I_b^*$, {\rm II, II$^*$, III, III$^*$}, or {\rm IV, IV$^*$}. Each of these models has a unique birational transform into a Kodaira model with the same label. In addition, for the log model of type $_m\rm I_b$, $_m\rm I_b^*$, {\rm II, II$^*$, III, III$^*$}, or {\rm IV, IV$^*$}, $K+C$ has, respectively, the index $1,2,6,4$ or $3$. \end{class} Note that $K+C$ is log terminal, except for the type $_m\rm I_1$, when $C$ is a Cartesian leaf. \begin{proof} Adding a multiplicity of the fibre we can suppose (MLC) (cf. below Proof of Theorem~\ref{lcomp}.\ref{lmainth}: General case). Then, according to the proof the Special case in Theorem~\ref{lcomp}.\ref{lmainth}, we have a log minimal model $C/P$. According to a classification of formally log terminal singularities, $C$ is a connected curve with only nodal singularities. On the other hand $(C,B_C)$ has a log genus $1$. Thus $C$ has an arithmetic genus $\le 1$. If the genus is $0$, then $C$ is a chain of rational curves, and the possible types were given in the proof of Case~III in the Inductive Theorem. If $C$ is not irreducible then, by (SM) and since $K_C+B_C\equiv 0$, $B_C$ is the same as in the type $\rm I^*_b$. So, each $P_i$ is a simple double singularity of $S$, and a minimal resolution gives a graph $\widetilde D_{4+b}$. This fibre has the type $\rm I_b^*$ due to Kodaira. In that case, $n=2$, $f^*P=2 C$, and $K\sim 0/P$. Thus $2(K+C)\sim 0/P$. The same holds when $C$ is irreducible, and $B_C$ is the same as in the type $\rm I^*_b$. In the other cases with the genus $0$, $C$ is an irreducible non-singular rational curve. Then $\deg B_C=2$ and under (SM) we have only $B_C$ as in types $\rm I^*_b$, (II), (III), or (IV) as above. We need to consider only the types (II)-(IV). In all of them we have three singularities $P_i$. If $C$ is $(-1)$-curve, each of them is simple, i.~e., has a resolution with one irreducible curve. Otherwise $C$ is $(-2)$-curve and the singularities are Du Val. This gives respectively Kodaira's types II-IV and II$^*$-IV$^*$. For instance, the curves, in a minimal resolution of points $P_i$ in type~IV, are $(-3)$-curves. Now we can easily transform fibres of the types~II, III and IV into the same due Kodaira, because $C$ is a $(-1)$-curve. In type~III, $C$ will be transformed into three $(-2)$-curves with a simple intersection in a single point. The latter is a blow-down of the old $C$. (Cf. \cite[p. 158]{BPV}. After the transform, $C=((n-1)/n)F_0$, where $F_0$ is the modified fibre (cf. Proof of the Inductive Theorem: Case~III). For types I$_b^*$ and II$^*$-IV$^*$, the transform is a minimal resolution. Then, for Kodaira's types~II-IV, and II$^*$-IV$^*$, $K\sim 0/P$ and $F_0\sim 0/P$. (More generally, $D\sim 0/P$ for any integral $D$ such that $D\equiv 0/P$, and for the types $\rm I_b^*$, II-IV and II$^*$-IV$^*$. This follows from the fact that $F_0$ does not have non-trivial unramified coverings.) Hence, on Kodaira's model, $n(K+C)=n(K+((n-1)/n)F_0)=nK+(n-1)F_0\sim 0/P$. An alternative approach will be discussed at the end of the proof. In the other cases $B_C=0$, and the genus is $1$. Since $K+C$ is formally log terminal, $S$ is non-singular near $C$ \cite[3.9.2]{Sh2}, $C$ is a curve with only nodal singularities and of arithmetic genus $1$. In particular, $n=1$. If $C$ is irreducible, then for some natural number $m$, $f^*P=m C$, and $K\sim (m-1)C/P$ by Kodaira's formula \cite[ibid]{BPV} (cf. formula (\ref{adj}), with $B_Y=P$, in Section~\ref{indcomp}). So, $K+C\sim 0/P$. Similarly, we can do the next case, when $C$ is reducible. Since $K_C+B_C\equiv 0$, the irreducible components $C_i$ of $C$ form a wheel as in $_m\rm I_b$. By the minimal property, $K/P$ is nef. On the other hand, $K\equiv 0$ in a generic fibre. Hence $K\equiv 0/P$, each $C_i$ is a $(-2)$-curve, and $f^*P=m C$. So, in this case, a log minimal model $C/P$ coincides with a Kodaira model of the type $_m\rm I_b$, $K\sim (m-1)C/P$, and $K+C\sim 0/P$. Finally, we may also use a covering trick \cite[Section~2]{Sh2} to reduce a proof of (\ref{indn}) to a case with $n=1$ or to the type $_m\rm I_b$. The latter is a {\it crucial\/} fact: $K+C\sim 0/P$ for the type $_m\rm I_b$. It can be induced from dimension $1$. \end{proof} \begin{proof}[Proof of Corollary~\ref{int}.\ref{ind2}: Local case] This is the Special case because any local trivial complement satisfies (MLC). On the other hand, any regular $n$ divides $I_1=12$. \end{proof} \begin{proof}[Proof of \ref{indcomp}.\ref{indth}.2: Local case] From \cite[Example~5.2.2]{Sh2}, because the complements are induced from the $1$-dimensional case. \end{proof} \begin{proof}[Proof of the Theorem~\ref{lcomp}.\ref{lmainth}: General case] According to the Big case in the Inductive Theorem, we suppose that $Z$ is a non-singular curve, and $K+C+B\equiv 0/Z$. By \cite[Lemma~5.3]{Sh2} we can increase $B$. We do it in such a way that $K+C+B+p f^*P$ is {\it maximally log canonical\/} for some real $p\ge 0$: \begin{itemize} \item $K+C+B+p f^*P$ is log canonical, but \item $K+C+B+p' f^*P$ is not so for any $p'>p$. \end{itemize} Such a $p$ exists, and (MLC) is equivalent to these conditions. \end{proof} \begin{proof}[Proof of Corollary~\ref{int}.\ref{mcorol}, Main and Inductive Theorems: Local case] Follows from Theorem~\ref{lcomp}.\ref{lmainth}. \end{proof} \section{Global complements} \label{gcomp} \refstepcounter{subsec} \begin{theorem} \label{gmainth} Let $(S,C+B)$ be a complete algebraic log surface such that \begin{description} \item (M) of the Main Theorem holds, and \item{{\rm (NEF)}} $-(K+C+B)$ is nef. \end{description} Then its complements are bounded under any one of the following conditions: \begin{description} \item{{\rm (WLF)}} of Conjecture~\ref{int}.\ref{conjcs}; \item{{\rm (RPC)}} of the Inductive Theorem; \item{{\rm (EEC)}} of the Inductive Theorem; \item{{\rm (EC)+(SM)}} of Conjecture~\ref{int}.\ref{conjcs}; \item{{\rm (ASA)}} of the Inductive Theorem; or \item{{\rm (NTC)}} of the Inductive Theorem. \end{description} More precisely, for almost all such $(S,C+B)$, we can take a regular index in $RN_2$. The non-regular complements are exceptional in the sense of Definition~\ref{int}.\ref{dexc}. \end{theorem} \refstepcounter{subsec} \label{unb} \begin{lemma} There exists $c>0$ such that all $b_i\le 1-c$, for any log surface $(S,B)$, under the assumptions of Theorem~\ref{gcomp}.\ref{gmainth}, and such that $\rho(S)=1$, $S$ is $1/7$-log terminal, and $(S,B)$ does not have the regular complements. \end{lemma} \begin{proof} If $B=0$, any $c>0$ fits. Otherwise, $S$ is a log Del Pezzo. Such an $S$ is bounded by \cite[Theorem 6.9]{Al}. By (M) the same holds for $(S,\Supp{B})$. So, we may assume that $S$ is fixed, as are the irreducible components of $\Supp B=\cup D_i$. Consider a domain $$\mathcal D= \{D=\sum d_i D_i\mid K+D \text{ is log canonical and } -(K+B) \text{ is nef}\}.$$ It is a {\it closed\/} polyhedron by \cite[Property~1.3.2]{Sh2} and by a polyhedral property of $\NEc{S}$. Take $c=1-d$ where $d=\max \{d_i\}$ for $D\in\mathcal D\cap\{D\ge B\}$. \end{proof} \begin{proof}[Proof: Strategy] We are looking for the exceptions. Thus we assume that $(S,B)$ does not have $1-,2-,3-,4-$ and $6$-complements. We prove that $(S,C+B)$ belongs to a bounded family. Equivalently, $(S,\Supp{(C+B)})$ is bounded. Moreover, we verify that complements are bounded and exceptional as well. We will suppose (ASA) or (NTC) by Proposition~\ref{indcomp}.\ref{relcond}. For the exceptions in the proposition, we have regular complements. According to~\cite[Theorem~2.3]{Sh3} and \cite[Lemma~5.4]{Sh2}, we can suppose that $(S,C+B)$ is log terminal. In particular, $S$ has only rational singularities. So, $S$ is projective. Moreover, then $(S,C+B)$ is Kawamata log terminal by the Inductive Theorem. In particular, $C=0$. The change preserves (ASA). In addition, we suppose that $K+B$ is $1/7$-log terminal in the closed points of $S$. Otherwise we make a crepant blow-up of the exceptional curves $E$ with a log discrepancy $\le 1/7$. This preserves all our assumptions. We have a finite set of such $E$ by \cite[Corollary~1.7]{Sh3}. In other words, now $K+B$ is $1/7$-log terminal in the closed points. If $K+B$ is $1/7$-log terminal everywhere, or, equivalently, if $B$ does not have an irreducible component $D_i$ with $\mult{D_i}{B}\ge 6/7$ and satisfies (SM), then $S=(S,B)$ is bounded by (M) and \cite[Theorem~6.9]{Al}, except for the case, when $B=0$ and $S$ has only canonical singularities. In the former case we have a bounded complement. If $(S,B)$ satisfies (WLF), we construct a complement as in \cite[Proposition~5.5]{Sh2}. Similarly, we proceed in the other cases by (ASA). Since $(S,B)$ is bounded in a strict sense, i.~e., in an algebraic moduli sense, a freeness of $-(K+B)$ is bounded. In the case when $B=0$ and $S$ has only canonical singularity, $(S,0)$ has a regular complement according to (ASA) and to a classification of surfaces. In such a case, we can even suppose that $S$ is non-singular. Now we can assume that $B$ has an irreducible component $D_i$ with $\mult{D_i}{B}\ge 6/7$. Then we reduce all required boundednesses to a case with the minimal Picard number $\rho=\rho(S)=1$. We find a birational contraction $g: S\to S_{\rm min}$ such that $S_{\rm min}$ has all the above properties and $\rho(S_{\rm min})=1$. Moreover, $g$ does not contract the irreducible components $D_i$ with $\mult{D_i}{B}\ge 6/7$, and \begin{description} \item{\rm (BPR)} there exists a boundary $B'\ge B$, with $\Supp{B'}$ in divisors $D_i$ having $\mult{D_i}{B}\ge 6/7$, such that $g$ contracts only curves $E$ with log discrepancies $\le 1$ for $K_{\rm min}+B'_{\rm min}$, and $-(K_{\rm min}+B'_{\rm min})$ is nef, where $K_{\rm min}=K_{S_{\rm min}}$ and $B'_{\rm min}=g(B')$. \end{description} In particular, $B_{\rm min}=g(B)\not=0$. This reduction will be called a {\it minimization\/}. It uses the Inductive Theorem and the Main Lemma below. By the LMMP $-(K_{\rm min}+B_{\rm min})$ is nef. Hence $B_{\rm min}$ and $-K_{\rm min}$ are ample, because $\rho(S_{\rm min})=1$ and $B_{\rm min}\not= 0$. So, $S_{\rm min}$ is a log Del Pezzo surface. Since $K_{\rm min}+B_{\rm min}$ is $1/7$-log terminal in the closed points, then, by \cite[Monotonicity~1.3.3]{Sh2}, $S_{\rm min}$ does the same. Therefore, due to Alekseev, we have a bounded family of such Del Pezzo surfaces \cite[Theorem 6.9]{Al}. For $\Supp{B_{\rm min}}$ we have only a bounded family of possibilities, because all $b_i\ge 1/2$ and $\rho(S_{\rm min})=1$. The condition (BPR) above guarantees a {\it boundedness for the partial resolution\/} $g$. First, by the Inductive Theorem $K_{\rm min}+B'_{\min}$ is Kawamata log terminal and $B'_{\rm min}$ is reduced, because $K_{\rm min}+B'_{\rm min}$ does not have the regular complements as $K_{\rm min}+B_{\rm min}$. Hence the multiplicities of $B'_{\rm min}$ and $B$ are {\it universally\/} bounded by Lemma~\ref{gcomp}.\ref{unb} and (M): all $b_i\le 1-c$ for some $c>0$. Thus $(S,\Supp{B})$ is bounded, because it resolves only exceptional (for $S_{\rm min}$) divisors $E$ with log discrepancies $\le 1$ for $K_{\rm min}+B'_{\rm min}$ by (BPR) (cf. \cite[Second Main Theorem and Corollary~6.22]{Sh3}). This will be done more explicitly in Theorem~\ref{ecomp}.\ref{emainth} below. For another approach see the following remark. \end{proof} \refstepcounter{subsec} \begin{remark} In the strategy above, $(S,\Supp{B})$ is bounded by \cite[Theorem~6.9]{Al} and Lemma~\ref{gcomp}.\ref{unb}. Indeed, $K+B$ is $\varepsilon$-log terminal for any $c>\varepsilon>0$. However, we prefer a more effective and explicit property (BPR) (cf. Proof of Theorem~\ref{ecomp}.\ref{emainth} in Section~\ref{ecomp}). \end{remark} In the same style as Lemma~\ref{indcomp}.\ref{mlem1}, we can prove its following improvement. \refstepcounter{subsec} \label{mlem3} \theoremstyle{plain} \newtheorem*{mainl}{\thesection.\thesubsec. Main Lemma} \begin{mainl} In the notation of \cite[Definition~5.1]{Sh2}, let $f\colon X\to Y$ be a birational contraction such that \begin{description} \item{\rm (i)} $K_X+S+D$ is numerically non-negative on a {\rm sufficiently general\/} curve$/Y$ in each exceptional divisor of $f$; and \item{\rm (ii)} for each multiplicity $d_i=\mult{D_i}{D}$ of a prime divisor $D_i$ in $D$, $\lfloor (n+1)d_i\rfloor/n\ge d_i$, whenever a non-exceptional on $Y$ divisor $D_i$ intersects an exceptional divisor of $f$. \end{description} Then $$K_Y+f(S+D)\ n-complementary\ \ \Longrightarrow \ K_X+S+D\ \ n-complementary.$$ In addition, we can assume that $D$ is just a subboundary. \end{mainl} \refstepcounter{subsec} \label{mlem4} \begin{example-c} By the Monotonicity Lemma \ref{indcomp}.\ref{monl1}, (ii) in \ref{gcomp}.\ref{mlem3} holds whenever all coefficients are standard, i.~e., they satisfy (SM). Respectively, (i) in \ref{gcomp}.\ref{mlem3} holds when $K+S+D$ is nef$/Y$. Then by the Main Lemma we can pull back the complements, i.e., for any integer $n>0$, $$K_Y+f(S+D)\ \ n-complementary\ \ \Longrightarrow \ K_X+S+D\ \ n-complementary.$$ \end{example-c} \begin{proof}[Proof of the Main Lemma] We take a crepant pull back: $$K_X+D^{+X}=f^*(K_Y+D^+).$$ It satisfies \cite[5.1.2-3]{Sh2} as $K_Y+D^+$, and we need to check \cite[5.1.1]{Sh2} only for the exceptional divisors. For them it follows from our assumption and the Negativity~\ref{indcomp}.\ref{negat}. Indeed, on the exceptional prime divisors $D_i$, $D^{+X}\ge D$ and has multiplicities in $\mathbb Z/n$. Hence $D^{+X}\ge S+\lfloor (n+1)D\rfloor/n$ according to the Monotonicity~\ref{indcomp}.\ref{monl3} above and \cite[Lemma~5.3]{Sh2}. Indeed, for any multiplicity $d_i^+<1$ in $D^{+X}$, we have $d_i^+\ge\lfloor (n+1)d_i^+\rfloor/n\ge \lfloor (n+1)d_i\rfloor/n$. \end{proof} \begin{proof}[Proof of Theorem~\ref{gcomp}.\ref{gmainth}: Minimization] Let $D$ denote a boundary with the coefficients $$d_i=\cases 1&\text{ if }b_i\ge 6/7,\\ b_i &\text{ otherwise.}\endcases$$ Hence by the Monotonicity Lemma~\ref{indcomp}.\ref{monl1}, for any $n\in RN_2$, we have \begin{itemize} \item $\lfloor(n+1)B\rfloor/n= \lfloor D\rfloor+\lfloor(n+1)\{D\}\rfloor/n\ge D\ge B$. \end{itemize} Therefore, $K+D$ is log canonical. Indeed, locally by \cite[Corollary~5.9]{Sh2}, there exists an $n$-complement $(S,B^+)$ with $n\in RN_2$. In addition, $B^+\ge \lfloor(n+1)B\rfloor/n\ge D$. Hence by \cite[Monotonicity~1.3.3]{Sh2}, $K+D$ is log canonical. Since $B$ has a multiplicity $b_i\ge 6/7$, then $D$ has a non-trivial reduced part and $K+D$ is not Kawamata log terminal. By the Inductive Theorem, $-(K+D)$ does not satisfy (ASA), because $K+D$ as $K+B$ does not have the regular complements. Moreover, we contend that when $\rho=\rho(S)>1$, then for any $\mathbb R$-divisor $F$, such that $D\ge F\ge B$ and $-(K+F)$ is semi-ample, there exist an exceptional curve $E$ and a divisor $B'$ such that \begin{itemize} \item $(K+D.E)>0$ and $\mult{E}{B}\le 5/6$, \item $D\ge B'\ge F$, $(K+B'.E)=0$, and $-(K+B')$ is semi-ample. \end{itemize} Then we contract $E$ to a point: $h:S\to Z$, and replace $(S,B)$ by $(Z,h(B))$. On the first $S$ we take $F=B$. Then we take $F=h(B')$. The contraction preserves the properties. In particular, $(Z,h(B))$ does not have the regular complements by the Main Lemma. We contract only curves with $(K+B.E)\le 0$, and, by the Local case and (M), with $b_i\le 5/6$. Indeed, near $E$ we have a regular complement $(S,B^+)$, $B^+\ge D$, $\mult{E}{B^+}=\mult{E}{D}=1$, and $(K+D.E)\le (K+B^+.E)=0$. Hence, $K+B$ will always be $1/7$-log terminal, and we do not contract the curves with $b_i\ge 6/7$. Contracted $E$, or any other exceptional divisor of $S$ with a log discrepancy $\le 1$ for $K+B'$, will have the same log discrepancy for $K_Z+h(B')$. By \cite[1.3.3]{Sh2}, these discrepancies do not increase for $K+F'$ with any $F'\ge h(B')$. Thus all contracted $E$ will have log discrepancies $\le 1$ for $K+B'$. Finally, an induction on $\rho$ gives required $S_{\rm min}=S$ with $\rho=1$. We find $E$ case by case with respect to the numerical dimension $\kappa^*$ of $-(K+B)$. First, (WLF) when $\kappa^*=2$. Then we have (RPC). In particular, $-(K+B)$ is not nef, because it is not semi-ample. Then there exists an exceptional curve $E$ with $(E.K+D)>0$. Since $\rho>1$, otherwise we have a fibre extremal contraction $S\to Z$, which is positive with respect to $K+D$. The latter is impossible by (M), because $-(K+B)$ is nef. Therefore, we need to find $B'$ and $E$ with above properties. Take a closed polyhedron $$ \mathcal D=\{B'\mid D\ge B'\ge F,\text{ and} -(K+B')\text{ is nef}\}. $$ It is polyhedral by (RPC). Take a maximal $B'$ in $\mathcal D$. Then $-(K+B')$ satisfies (ASA). It is Kawamata log terminal by the Inductive Theorem, because $B'\ge B$. So, we cannot increase $B'$ only because $(K+B'.E)=0$ for some extremal curve and is positive when we increase $B'$. By (M) it is possible only for birational contractions. Properties (WLF) and (RPC) will be preserved. Second, $\kappa^*=1$, and we have a numerical contraction $\nu :S\to Y$ for $K+B$. By (M) the horizontal multiplicities satisfy (SM), and $D=B$ in the horizontal components. Thus $-(K+D)\equiv 0/Y$. Otherwise we have a vertical exceptional curve $E$ with $(K+D.E)>0$. As above we contract $E$. This time, we can take $B'=B$, because $K+B\equiv 0/Y$. After such contractions, $-(K+D)\equiv 0/Y$ and is not nef, because it is not semi-ample. Note that $Y$ is rational, because $K+B$ is negative on the horizontal curves. As above we have no extremal fibre contractions, positive with respect to $K+D$. Thus we have an exceptional (horizontal) curve $E$ with $(E.K+D)>0$. After that contraction we will have (WLF) and do as above. Third, $\kappa^*=0$ or $K+B\equiv 0$. In this case we take $B'=B$ and need only to contract some $E$ with $(K+D.E)>0$. If $B$ has a big type, we again have (WLF) and (RPC). If $B'$ has a fibre type, then, by Proposition~\ref{indcomp}.\ref{types}, we have a fibration $S\to Y$ of genus $1$ curves, whereas $B'$ and $D$ have only vertical components. As above, after contractions, we suppose that $K+D\equiv 0/Y$. Since $B\not=0$ and forms a fibre, we have a horizontal extremal curve $E$ with the required properties. After its contraction, we have (WLF). Finally, $B$ has an exceptional type. Then decreasing $B$ in the non-standard multiplicities, we can find $E$, which is outside $\lfloor D\rfloor$, but intersects $\lfloor D\rfloor$. Thus $(K+D.E)>0$. If after a contraction of such $E$, we change a type of $B$, we return to a corresponding type: big or fibre. \end{proof} \begin{proof}[Proof of Theorem~\ref{gcomp}.\ref{gmainth}: Bounded complements] Here, we check that complements are bounded. Since $(S,\Supp{B})$ is bounded, it is enough to establish that complements, for all $$ B'\in \mathcal D=\{ \Supp{B'}=\Supp{B}, -(K+B')\text{ is nef and log canonical}\}, $$ are bounded. Note that each $K+B'$ is semi-ample by Proposition~\ref{indcomp}.\ref{types}, because $K+B$ is semi-ample, Kawamata log terminal, and (NV) of Remark~\ref{indcomp}.\ref{rel} holds for $K+B'$. Thus, for each $\mathbb Q$-boundary $B'$ we have an $n$-complement such that $n B'$ is integral. Therefore we have $n$-complements near $B'$ by the Monotonicity~\ref{indcomp}.\ref{monl2}. Hence we have bounded complements, according to Example~\ref{int}.\ref{simul}. Indeed, we can restrict our problem on any ample non-singular curve; as we see later in Section~\ref{ecomp}, the cases with non-standard coefficients are reduced to a case with $\rho(S)=1$. A more explicit approach will be given in Theorem~\ref{ecomp}.\ref{emainth}. \end{proof} \begin{proof}[Proof of Theorem~\ref{gcomp}.\ref{gmainth}: Exceptional complements] As in the strategy we assume (ASA), a log terminal property for $K+B$ and an absence of the regular complements. Then we have a (non-regular) complement $(S,B^+)$. Here we check that $K+B^+$ is Kawamata log terminal for any (such) complement. After a crepant blow-up we suppose that $K+B^+$ has a reduced component, and derive a contradiction. Let $D$ denote a boundary with the coefficients $$d_i=\cases 1&\text{ if }b_i^+=1,\\ 6/7&\text{ if }1>b_i^+\ge 6/7,\\ b_i &\text{ otherwise.}\endcases$$ Then $B^+\ge D$ and $D$ satisfies (SM). By the Monotonicity~\ref{indcomp}.\ref{monl1}, in the non-reduced components of $D$ and $B^+$, $\lfloor(n+1)B\rfloor/n \le \lfloor(n+1)D\rfloor/n$, for any $n\in RN_2$. Hence, by \cite[Lemmas~5.3-4]{Sh2}, $(S,D)$ does not have the regular complements. Hence $-(K+D)$ does not satisfy (ASA) by the Inductive Theorem. We contend then that $\rho>1$ and we have an exceptional curve $E$ with $(K+D.E)>0$, and automatically $\mult{E}{D}<1$. Indeed, if $\rho=1$, then $K+B^+\equiv 0$, log canonical, and $B^+\not=0$, $K$ are ample. Hence $-(K+D)$ is nef, because $B^+\ge D$. This is impossible by the Inductive theorem. Therefore, $\rho>1$. If we have an exceptional curve $E$ with $(K+D.E)>0$, we contract this curve. Again we will have no regular complements by Example~\ref{gcomp}.\ref{mlem4}. Such a contraction will be to a rational singularity, because $E$ will not be in $\lfloor D\rfloor$. We prove case by case that such an $E$ exists, except for a case when $K+B\equiv 0$. Indeed, if we have (WLF) or $\kappa^*=2$, then we have (RPC), and the latter can be preserved after a crepant log resolution above. Take a weighted linear combination of $B$ and $B^+$. In addition, $-(K+C)$ is not nef when (ASA) fails. Thus we have an extremal contraction $S\to Z$ which is positive with respect to $K+D$, and $\dim Z\ge 1$. If $Z$ is a curve, $K+B^+\equiv 0/Z$, but $K+D$ is numerically positive$/Z$, which is impossible for $B^+\ge D$ as in the above case. Hence we have $E$. An induction on $\rho$ and contractions of such $E$ give a contradiction in this case. Suppose now that $\kappa^*=1$ and we have a numerical contraction $\nu :S\to Y$ for $K+B$. By (M) the horizontal multiplicities satisfy (SM), and $B^+=D=B$ in the horizontal components. Thus $-(K+D)$ is nef on the horizontal curves. Moreover, it is nef. Otherwise we have a vertical exceptional curve $E$ with $(K+D.E)>0$. As above we contract $E$. Finally, $-(K+D)$ is nef, and $-(K+D)\equiv 0$ by the Inductive Theorem. It is possible only when $B^+=D$. But then we have (ASA) which does not hold in our case too. So, we get $\kappa^*=0$ or $K+B\equiv 0$. Now let $D$ denote a support of the non-standard multiplicities in $B$. If $D$ has a big type, then we get (WLF) for $K+B-\varepsilon D'$ for some effective $\mathbb R$-divisor $D'$ with $\Supp{D'}\le D$ by Proposition~\ref{indcomp}.\ref{types}. Again we do not have regular complements by the Monotonicity~\ref{indcomp}.\ref{monl2}: the non-standard multiplicities $>6/7$ under (M). We do the same when $D$ has a fibre type. Finally, $D$ has an exceptional type and we contract $D$ to points. (Cf. Proof of Corollary~\ref{int}.\ref{mcorol}: Numerically trivial case in Section~\ref{int}.) So, $K+B\equiv 0$, satisfies (SM), but does not have the regular complements by Lemma~\ref{gcomp}.\ref{mlem3} and the Monotonicity~\ref{indcomp}.\ref{monl1}. Then we have only the trivial complements. In that case $B^+=B$, which contradicts a Kawamata log terminal property of $K+B$. This gives a contradiction to our assumption on an existence of a non-regular and non-exceptional complement $(S,B^+)$. \end{proof} \begin{proof}[Proof of the Main Theorem: Global case] Follows from Theorem~\ref{gcomp}.\ref{gmainth}. \end{proof} Now we slightly improve Proposition~\ref{indcomp}.\ref{relcond}. \refstepcounter{subsec} \label{drel} \begin{proposition} Assuming that $K+C+B$ is log canonical and nef$/Z$, $$ {\rm (EC)}\Longrightarrow {\rm (NTC)}\Longleftrightarrow {\rm (ASA)} $$ with the exception (EX2) of Proposition~\ref{indcomp}.\ref{relcond}. Nonetheless, in (EX2) there exists a $1$-complement. Moreover, we can replace (EC) by its weaker form: \begin{description} \item{\rm (EC)'} there exists a boundary $B'$ such that $K+B'$ is log canonical and $\equiv 0/Z$, \end{description} {\rm i.~e., (EC) for $S$\/}. \end{proposition} \begin{proof} Let $(S,B')$ be as in (EC)'. If we replace $C+B$ by a weighted linear combination of $C+B$ and $B'$ we can suppose that $K+C+B$ and $K+B'$ have the same log singularities: \begin{itemize} \item the exceptional and non-exceptional divisors with the log discrepancy $0$, and \item the exceptional and non-exceptional divisors with the log discrepancies $<1$. \end{itemize} Note that (EX2) will mean that $B'$ is unique and $B'\ge B+C$, i.~e., (EEC) holds. After a log terminal resolution we suppose that $K+B'$ is log terminal. By the above properties, a support $D$ of curves, where $C+B>B'$, is disjoint divisorially from $\LCS{(S,B')}$. If $D$ has an exceptional type, we can contract it, when $K+C+B\equiv 0$ on $D$. Then $B'\ge B$ and we have (EEC), which implies the proposition by Proposition~\ref{indcomp}.\ref{relcond}. If $K+C+B$ is negative somewhere on $D$, then (WLF) and (RPC) hold for $K+C+B-\varepsilon D'$ with some $\varepsilon>0$, and $D'$ having $\Supp{D'}\le D$. On the other hand, if $D$ has a big type, (WLF) and (RPC) hold for $K+C+B-\varepsilon D'$ with some $\varepsilon>0$, and nef and big $D'$ having $\Supp{D'}\le D$. Here we may have one exception (EX1), when $K+C+B$ satisfies (NTC). In addition, the proposition holds when $K+C+B\equiv 0$ by its semi-ampleness. Finally, $D$ has a fibre type and $-(K+C+B)$ has a numerical dimension $1$ and $Z=\mathop{\rm pt.}$ (a global case). If a fibering given by $D$ does not agree with $K+C+B$, i.e., $(K+C+B.F)<0$ on a generic fibre. Then $K+C+B-\varepsilon F'$ satisfies (WLF) for a divisor $F'$ with $\Supp{F'}\le D$, which defines the fibering. Otherwise, the fibering gives a numerical contraction for $K+C+B$. \end{proof} \begin{proof}[Proof of Corollary~\ref{int}.\ref{mcorol}: Global case] Follows from Theorem~\ref{gcomp}.\ref{gmainth} and Proposition~\ref{gcomp}.\ref{drel}. \end{proof} \begin{proof}[Proof of Corollary~\ref{int}.\ref{nonvan1}] Again in the global case, it follows from Theorem~\ref{gcomp}.\ref{gmainth} and Proposition~\ref{gcomp}.\ref{drel}. In the local case, use Theorem~\ref{lcomp}.\ref{lmainth}. \end{proof} \refstepcounter{subsec} \label{mrem} \begin{remcor} We can improve most of the above results as well. \ref{gcomp}.\ref{mrem}.1. In the Main and Inductive Theorems we can replace (WLF) by (EC)' of Proposition~\ref{gcomp}.\ref{drel}. We anticipate that the Main Theorem and Corollary~\ref{int}.\ref{mcorol} hold without (M) as does the Inductive Theorem. Of course, then exceptional complements may be unbounded (cf. Example~\ref{indcomp}.\ref{nonrcomp}). \ref{gcomp}.\ref{mrem}.2. By the Monotonicity~\ref{indcomp}.\ref{monl1}, in Corollaries~\ref{int}.\ref{ind2}-\ref{ind3}, we can replace (SM) by \begin{description} \item{\rm (M)'} the multiplicities $b_i$ of $B$ are {\rm standard\/}, i.~e., $b_i=(m-1)/m$ for a natural number $m$, or $b_i\ge I/(I+1)$, where $I$ is maximal among the indexes under (SM): $I|I_2$. \end{description} (Cf. Classification~\ref{csing}.\ref{singth}.1 below.) \end{remcor} \section{Exceptional complements} \label{ecomp} In this section we start a classification of the exceptional complements. By the Main and Inductive Theorems, they arise only in the global case $(S,B)$ when $K+B$ is Kawamata log terminal. By Remark~\ref{gcomp}.\ref{mrem}.1, we can assume just (EC)' and (M) as additional conditions. In a classification we describe such $(S,B)$, which will also be called {\it exceptional\/}, and their {\it minimal\/} complements. Here we do this completely in a few cases. An importance of a complete classification of the exceptions will be illustrated in Section~\ref{csing}. We will continue the classification elsewhere. Since the exceptional complements are bounded, the following invariant \begin{align*} \delta (S,B)=\#\{E\mid E &\text{ is an exceptional or non-exceptional divisor}\\ &\text{ with the log discrepancy } a(E)\le 1/7\text{ for }K+B\} \end{align*} is also bounded. It is independent on the crepant modifications. \refstepcounter{subsec} \label{emainth} \begin{theorem} $\delta\le 2$. \end{theorem} \begin{statement1} If $\delta =0$, then $(K,B)$ is $1/7$-log terminal, and $B$ has only multiplicities in $\{0,1/2,2/3,3/4,4/5,5/6\}$. However, the m.l.d. of $K+B$ is only $>1/7$. \end{statement1} A minimum of such m.l.d.'s exists, but it is yet unknown explicitly. In the other cases, $\delta\ge 1$, the m.l.d. of $K+B$ is $\ge 1/7$, and after a crepant resolution we assume that $K+B$ is $1/7$-log terminal in the closed points. To classify the original $(S,B)$ we need to find crepant birational contractions of the $1/7$-log terminal pairs $(S,B)$. To classify the latter pairs we consider their minimizations $g:S\to S_{\rm min}$ as in the strategy of the proof of Theorem~\ref{gcomp}.\ref{gmainth}. In this section some results on $(S_{\rm min},B_{\rm min})$ and their classification are given. According to the strategy, Theorem~\ref{ecomp}.\ref{emainth} is enough to prove for $(S_{\rm min},B_{\rm min})$. So, we assume in this section \begin{itemize} \item $\rho (S)=1$, \item $K+B$ is $1/7$-log terminal in the closed points, \item $B$ has a multiplicity $b_i\ge 6/7$, \item $-(K+B)$ is nef, but \item $K+D$ is ample for $D=\lfloor(n+1)B\rfloor/n$ with any $n\in RN_2$. \end{itemize} To find all such $1/7$-log terminal pairs $(S,B)$ with $\rho(S)>1$ we need to find $K+B'\equiv 0$ with $B'\ge B$ and $\rho(S)=1$. The former pairs are crepant partial resolutions of $(S,B')$. See the strategy and (BPR) in Section~\ref{gcomp}. Let $C=\lfloor D\rfloor$ denote a support of the curves $C_i$ with $\mult{C_i}{B}\ge 6/7$, and $D$ be the same as in the Minimization of Section~\ref{gcomp}. Let $F$ be the rest of $B$ or, equivalently, be the fractional part of $D$: $F=\{D\}=\sum b_i D_i$ for $D_i$ with $b_i=\mult{D_i}{B}\le 5/6$. By the Inductive Theorem, $K+D$ is ample for such $D$. \begin{statement2} For $\delta=1$, a curve $C$ is irreducible, has only nodal singularities and at most one node. The arithmetic genus of $C$ is $\le 1$. Divisor $F$ does not pass the node. \end{statement2} Abe found a classification in the {\it elliptic\/} case when $C$ has arithmetic genus $1$ \cite{Ab}. \begin{statement3} For $\delta=2$, $C=C_1+C_2$, where $C_1$ and $C_2$ are irreducible curves with only normal crossings in non-singular points of $S$. Divisor $F$ does not pass $C_1\cap C_2$ and $b_1+b_2<13/7$, where $b_i=\mult{C_i}{B}$. Constant $c$ below is as in Lemma~\ref{gcomp}.\ref{unb} For $C$, we have only the following three configurations: \begin{description} \item {$(\text{I}_2)$} $C=C_1+C_2$ and $C_i$'s form a wheel; and \item {$(\text{A}_2)$} $C=C_1+C_2$, and $C_i$'s form a chain. \end{description} Moreover, the curves $C_i$ are non-singular rational $m_1\ge m_2\ge 0$-curves, except for Case~{\rm ($\rm A_2^6$)} below. In the case $(\text{A}_2)$, the only possible cases are: \begin{description} \item{\rm ($\rm A_2^1$)} $S=\mathbb P^2$ whereas $C_1$ and $C_2$ are straight lines, $F=\sum d_i D_i$ and $1<\sum d_i\deg D_i\le 3-b_1-b_2$, assuming that $K+B$ is log terminal. \item{\rm ($\rm A_2^2$)} $S$ is a quadratic cone, whereas $C_1$ is its section and $C_2$ is its generator, $2b_1+b_2\le 8/3$. $F=(2/3)D_1$, where $D_1$ is another section not passing the vertex. $c=1/21$. \item{\rm ($\rm A_2^3$)} $S$ is a normal rational cubic cone whereas $C_1$ is its section and $C_2$ is its generator, $3b_1+b_2\le 7/2$. $F=(1/2)D_1$ where $D_1$ is also a section. Both sections $C_1$ and $D_1$ do not pass the vertex and $\# C_1\cap D_1\ge 2$. $c=1/14$. \item{\rm ($\rm A_2^4$)} $S$ has $B=(6/7)(C_1+C_2)+(1/2)D_1$, $m_1=1$ and $m_2=0$, whereas $S$ has only two singularities $P_1\in C_1$ and $P_2\in C_2$, and $D_1$ is a non-singular rational $1$-curve with a single simple intersection with $C_2$, a single simple intersection with $C_1$ and with another single intersection with $C_1$ in $P_1$. Singularity $P_i$ is Du Val of type $\mathbb A_i$. \item{\rm ($\rm A_2^5$)} $S$ has $B=(6/7)(C_1+C_2)+(1/2)D_1$, $m_1=1$ and $m_2=0$, whereas $S$ has only two singularities $P_1\in C_1$ and $P_2\in C_2$, and $D_1$ is a non-singular rational $1$-curve with a single simple intersection with $C_2$, a single simple intersection with $C_1$ and with another single intersection with $C_1$ in $P_1$. Singularity $P_1$ is simple with $(-3)$-curve in a minimal resolution, singularity $P_2$ is Du Val of type $\mathbb A_3$. $c=1/7$, and $(S,B)$ is $14$-complementary and the complement is trivial. \item{\rm ($\rm A_2^6$)} $S$ has $B=(6/7)(C_1+C_2)$, whereas $S$ has only two singularities $P_1,P_2\in C_2$. Curve $C_1$ has arithmetic genus $1$ and has only nodal singularities; at most $1$. Curve $C_2$ is a rational non-singular $(-1)$-curve. Singularities $P_i$ are Du Val of type $\mathbb A_i$. $c=1/7$, and $(S,B)$ is $7$-complementary and the complement is trivial. In the case $(\text{I}_2)$ the only possible cases are: \item{\rm ($\rm I_2^1$)} $S$ is a quadratic cone whereas $C_1$ and $C_2$ are its two distinct sections, $b_1+b_2\le 7/4$. $F=(1/2)L$, where $L$ is a generator of cone $S$. $c=3/28$. \item{\rm ($\rm I_2^2$)} $S$ has $B=(6/7)(C_1+C_2)$, $m_1=1,m_2=2$, whereas $S$ has only two singularities $P_i\in C_i$. Singularities $P_i$ are Du Val of type $\mathbb A_i$. $c=1/7$, and $(S,B)$ is $7$-complementary and the complement is trivial. \end{description} \end{statement3} \refstepcounter{subsec} \label{logter} \begin{proposition} Under the assumptions of this section, $K+D$ is formally log terminal, except for the case when $P$ is non-singular and near $P$, $D=C+(1/2)C'$ with non-singular irreducible curves $C$ and $C'$ having a simple tangency; $\mult{C}{B}< 13/14$. \end{proposition} Note that the latter log singularity appears only on cones: Cases (A$_2^{1-3}$) when $D_1$ is tangent to $C_1$. \begin{proof} By the proof of the Minimization in Section~\ref{gcomp}, $K+D$ is log canonical, and ($1/7$-)log terminal outside $C$. So, we need to check a log terminal property formally (locally in an analytic topology) in the points $P\in C$. First, suppose that $C\not= D$ in a neighborhood (even Zariski) of $P$. Then $K+C$ is purely log terminal and $C$ is non-singular by \cite[Lemma~3.6]{Sh2}. But $S$ may have a singularity of index $m$ in $P$ \cite[Proposition~3.9]{Sh2}. If we have formally two distinct prime divisors (two branches) $D_1$ and $D_2$ through $P$ in $\Supp (D-C)$, then, by~\ref{indcomp}.\ref{invadj}.1, $K+D$ will be log canonical in $C$ only when $b_1=b_2=1/2$ and, in a neighborhood of $P$, $K+D=K+C+(1/2)(D_1+D_2)$. (We would like to remind the reader that all non-reduced $b_i=(n-1)/n$ with $n=1,2,3,4,5$ or $6$.) So, if $P$ is singular, then by a classification of surface log canonical singularities, the curves $E$ of a minimal resolution form a chain, whereas a birational transform of $C$ intersects simply one end of the chain, and that of $D_i$ intersects simply another end. The intersection points are outside the intersections of the curves $E$. Thus the log discrepancy $a=(E,K+B)$, in any $E$ for $K+B$, is the same as $a(E,K+b C+D_1)$ for $K+b C+D_1$, where $b=\mult{C}{B}\ge 6/7$ near $P$. Therefore, by Corollary~\ref{indcomp}.\ref{adjunct}, $a=a(E,K+B)=1-\mult{P}{(b C)_{D_1}} \le 1-\mult{P}{((6/7)C)_{D_1}}\le 1-6/7=1/7$ for an exceptional divisor $E$. This is impossible by our assumptions. Hence $P$ is non-singular. Then the monoidal transform in $P$ gives $E$ with the same property. Hence we may have formally at most one irreducible component (or a single branch) $C'$ of $\Supp (D-C)$ through $P$. By the form of it and a classification of log canonical singularities \cite{K1}, $K+D$ will be log terminal in $P$, except for the case when $C'$ has multiplicity $1/2$ in $D$, and $B$ intersects only an end curve in a minimal resolution of $P$, or $D=C$ near $P$. Again, as in the above case, we have a contradiction with assumptions, except for the case when $P$ is non-singular and $C'$ has a simple tangency with $C$ in $P$. Such singularity will be $1/7$-log terminal for $K+B=b C+(1/2)C'$, only when $b<13/14$. Finally, we suppose that $C=D$ near $P$, then by \cite[Theorem~9.6~(6)]{K1} $P$ has type $\mathbb D_m$ with $m\ge 3$ when $C$ is formally irreducible in $P$. Then it has an exceptional divisor $E$ on a minimal resolution with $a(E,K+B)\le 1-b\le 1/7$, which is again impossible. Indeed, $a(E,K+C)=0$, $a(E,K)\le 1$ and $a(E,K+B)=a(E,K+b C)=a(E,K+C)+(1-b)\mult{E}{C}= (1-b)a(E,K)\le 1/7$. Otherwise by \cite[Theorem~9.6~(7)]{K1}, $C$ has two branches in $P$ and again, by the $1/7$ log terminal property in $P$, $P$ will be non-singular. Hence $K+B=K+C$ is formally log terminal here. \end{proof} \refstepcounter{subsec} \label{node} \begin{corollary} $C$ has only nodal singularities, and only in non-singular points of $S$. $C$ is connected. Moreover, each irreducible component of $C$ intersects all other such components. \end{corollary} \begin{proof} The first statement follows from Proposition~\ref{ecomp}.\ref{logter}. Since $\rho=1$, each curve on $S$ is ample, which proves the rest. \end{proof} Let $g$ be the arithmetic genus of $C$. \refstepcounter{subsec} \label{genus} \begin{proposition} $g\le 1$. \end{proposition} In its proof and in a proof of Theorem~\ref{ecomp}.\ref{emainth} below, we use the following construction. We reconstruct $S$ into a non-singular minimal rational model $S'$. Make a minimal resolution $S^{\rm min}\to S$ of $S$, and then contract $(-1)$-curves: $S^{\rm min}\to S'$, where $S'$ is minimal. Then $S'=\mathbb P^2$ or $S'=\mathbb F_m$, because $S$ and $S'$ are rational. By Corollary~\ref{ecomp}.\ref{node}, the resolution $S^{\rm min}\to S$ preserves $C$ up to an isomorphism. A birational transform of a curve $C_i$ or another one on $S'$ are denoted again by $C_i$ or as the other one respectively. According to the LMMP, $(S',B')$ is log canonical and $-(K_{S'}+B')$ is nef, because the same holds for $K+B$ and its crepant blow-up $K^{\rm min}+B^{\rm min}$, where $K^{\rm min}=K^{S^{\rm min}}$ and $B'$ is the image of $B^{\rm min}$. An image of $B^{\rm min}$ is not less than a birational image of $B$. So, $-(K_{S'}+B)$ is nef for $S'=\mathbb P^2$. \begin{statement1} Moreover, on a minimal rational model $S'$, $g(C)\le 1$ and that $g(C')=0$ for each (proper) $C'\subset C$, except for Case~($\rm A_2^6$) in \ref{ecomp}.\ref{emainth}.3. \end{statement1} \begin{proof} Suppose that $g\ge 2$. $S^{\rm min}\to S'$ cannot contract all $C$, because it has to be a tree of non-singular rational curves. However, we may increase $g$ after contraction of some components of $C$ and other curves on $S^{\rm min}$. If $S'=\mathbb P^2$, then $-(K_{S'}+B)$ and $-(K_{S'}+(6/7)C)$ are nef, and $\deg C\le 3$, which is only possible for $g\le 1$. Therefore $S'=\mathbb F_m$ with $m\ge 2$. Indeed, original $S'\not\cong \mathbb F_0$, because $\rho=1$. So, if final $S'=\mathbb F_0$, we had before a contraction of $(-1)$-curve. Then we can reconstruct $S$ into $S'=\mathbb F_1$, and so into $\mathbb P^2$. By Corollary~\ref{ecomp}.\ref{node}, we have at most one fibre $F$ of $\mathbb F_m$ in $C$. If a unique {\it negative\/} section $\Sigma$ is not in $C$, then $\sigma=\mult{\Sigma}{B'}\le 2 -2\times (6/7)=2/7<1/3$, because $C$ is not a section of $\mathbb F_m$ over the generic point of $\Sigma$. (Otherwise $C$ will be rational with only double singularities, and $g=0$.) By the nef property of $-(K+B)$, we have inequality $0\ge (K_{S'}+B'.\Sigma)\ge (K_{S'}+(2/7)\Sigma.\Sigma)$. This implies that $\Sigma$ is $(-2)$-curve, and $m=2$. If we had before a contraction of a $(-1)$-curve we can reconstruct $S'$ to the above $S=\mathbb P^2$. Hence $(S'=S^{\rm min},B'=B^{\rm min})$ is the minimal resolution of $S$, and $C\cap \Sigma=\emptyset$ (cf. Lemma~\ref{ecomp}.\ref{ineq} below). Since $\rho(S)=1$, $S$ is a quadratic cone (or a quadric of rank $3$ in $\mathbb P^3$) with a double section $C=D$ ($\sim -K$) not passing through its vertex. So, by the Adjunction $g=1$. Finally, $\Sigma$ is a component of $C$, then we have another component $\Sigma '$ in $C$, which is also a section$/\Sigma$. If $C=\Sigma+\Sigma'$ has $g\ge 2$, then $(\Sigma.\Sigma')\ge 3$. This is impossible, because $0\ge (K_{S'}+B'.\Sigma)\ge (K_{S'}+\Sigma+(6/7)\Sigma'.\Sigma)= \deg (K_{\Sigma}+(6/7)(\Sigma'|_{\Sigma})) \ge -2+3(6/7)=4/7$. One last case $C=\Sigma+\Sigma'+F$, whereas $C$ has $g\ge 2$, only when $(\Sigma'+F.\Sigma)\ge 3$, because $\Sigma\cap\Sigma'\cap F=\emptyset$ by the log canonical property: $3\times (6/7)>2$. Then we may act as above replacing $\Sigma'+F$ by $\Sigma'$. Now we prove \ref{ecomp}.\ref{genus}.1. If $C$ has a component, say $C_1$, of the arithmetic genus $g\ge 1$. Then according to the above, $g=1$, $C=C_1+C_2$, where $C_2$ is non-singular rational ($m_2$)-curve, and $C_1$ intersects $C_2$ in one point. Moreover, if $m_2\ge 0$, then $C_2$ is not exceptional on $S'$ and $C$ in $S'$ has genus $\ge 2$, which is impossible as we know. On the other hand, $(K_X.C_2)<0$ since $\rho (S)=1$. Therefore $m_2=-1$. However, this is only possible for Case~(A$_2^6$) by the following lemma. \end{proof} \refstepcounter{subsec} \label{move} \begin{lemma} Each non-singular irreducible rational (proper) component $C_i\subset C$ is movable on a minimal resolution of $S$, i.e., $C_i$ is an $m$-curve with $m\ge 0$, except for Case~{\rm ($\rm A_2^6$)} in \ref{ecomp}.\ref{emainth}.3. \end{lemma} \refstepcounter{subsec} \label{ineq} \begin{lemma} Let $P$ be a log singularity $(S,B)$ such that \begin{description} \item{\rm (i)} $B\ge b C>0$, where $C$ is an irreducible curve through $P$; \item{\rm (ii)} $B$ is a boundary, and \item{\rm (iii)} $P$ is a singularity of $S$. \end{description} Let $E$ be a curve on a minimal resolution of $P$ intersecting the proper inverse image of $C$, and let $d=1-a(E,K+B)$ be the multiplicity of the boundary on the resolution for the crepant pull back. Then $d\ge ((m-1)/m)b$ where $m=-E^2$. Moreover, $d\ge (1/2)b$ always, and $=$ holds only when $P$ is simple Du Val, $B=b C$ near $P$, and $K+C$ is log terminal in $P$. Otherwise $d\ge (2/3)b$. In addition, $d\ge ((m-1)/m)b$ whenever $P$ is log terminal for $K+C$ of index $m$. In this case we may also include in $C$ components of $B$ with standard multiplicities {\rm (cf. Lemma \ref{indcomp}.\ref{invadj})}. \end{lemma} \begin{proof} Let $C$ be the inverse image of $C$ in $S^{\rm min}$. Since the resolution is minimal all multiplicities of $B^{\rm min}$ are non-negative. So, we may consider a contraction of only $E$, and suppose that $B=b C$. Then we may find $d$ from the following equation $(K+b C+d E.E)=0$. If $E$ is singular or non-rational, $d\ge 1\ge (1/2)b$ and even $\ge (2/3)b$ because $b\le 1$ by (ii). Otherwise $E$ is $(-m)$-curve with $m\ge 2$. Hence $d=(m-2)/m+(C.E)(1/m)b\ge ((m-2)/m+ (1/m))b= ((m-1)/m)b\ge (1/2)b$, because $0\le b\le 1$. Moreover $=$ only for $m=2$, $B=b C$ near $P$, and $(C.E)=1$ in $P$ when $b>0$. The next calculation shows that $P$ is a simple Du Val singularity when $b>0$ and $d= (1/2)b$. If $m\ge 3$, then $d\ge (2/3)b$. The same holds if we replace $E$ by a pair of intersected $(-2)$-curves. Finally, $d(b)=1-a(E,K+B)$ is a linear function of $b$. So, it is enough to check the last inequality for $b=0$ and $1$. For $b=0$, $a\le 1$ and $d\ge 0$ by (iii). This gives the required inequality. For $b=1$, $a(E,K+B)\le a(E,K+C)=1/m$ by \cite[3.9.1]{Sh2}. Hence $d\ge (m-1)/m\ge ((m-1)/m)b$. We may also include in $C$ components with standard coefficients by Lemma~\ref{indcomp}.\ref{invadj}. \end{proof} \begin{proof}[Proof of Lemma~\ref{ecomp}.\ref{move}] Since $-K$ is ample on $S$ we should only eliminate the case when $C_i$ is a $(-1)$-curve. Since $C^2_i>0$ on $S$ and by Proposition~\ref{ecomp}.\ref{logter}, $C_i$ has at least two singularities $P_1$ and $P_2$. They are distinct from the intersection points $P=(C\setminus C_i)\cap C_i$. Such an intersection point $P$ exists because $C_i\not= C$ and by Corollary~\ref{ecomp}.\ref{node}. So, we may calculate $(C_i.K+B)$ on a minimal resolution $S^{\rm min}/S$. We denote again by $C'$ and $C_i$ respectively a proper inverse image of $C'=C\setminus C_i$ and $C_i$. Over $P_1$ and $P_2$ we have, respectively, single (non-singular rational) curves $E_1$ and $E_2$ intersecting $C_i$ in $S^{\rm min}$. Let $b$, $d$, $b_1\le b_2$ be the multiplicities of $B^{\rm min}$ in $C'$ (in any component through $P$), $C_i,E_1,E_2$ respectively. Then by our assumptions $b,d\ge 6/7$. On the other hand $b_1\ge (1/2)d$ because $P_1$ is singular, and $b_1=d/2$ is attained only for a simple Du Val singularity in $P_1$. Since $C^2_i>0$, then $P_2$ is not such a singularity and $b_2\ge (2/3)d$ by Lemma~\ref{ecomp}.\ref{ineq}. Otherwise we have a third singularity $P_3$ of $S$ in $C_i$, and $b_3=\mult{E_3}{B^{\rm min}}\ge (1/2)d$, where $E_3$ intersects $C_i$ in $S^{\rm min}$. For three points and more: $(K+B.C_i)\ge -1+b-d+b_1+b_2+b_3\ge -1+b+d/2\ge -1+6/7+3/7=2/7>0$. Therefore we have only two singularities and $0\ge (K+B.C_i)= (K^{\rm min}+B^{\rm min}.C_i)\ge (K^{\rm min}+b C'+d C_i+b_1 E_1+b_2 E_2.C_i)\ge -1+b-d+b_1+b_2\ge -1+b-d+(1/2)d+(2/3)d=-1+b+(1/6)d\ge -1+6/7+1/7=0$. Hence we have the equality, whereas $C_i=C_2$ and $C'=C_1$, $B=(6/7)(C_1+C_2)$, $P_1$ is a simple Du Val singularity, and $P_2$ is a Du Val singularity of type $\mathbb A_2$. Otherwise it is a simple singularity with $(-3)$-exceptional curve, because $b_2=(2/3)d$ and $C_2^2>0$. However, this is impossible for $m_2=-1$. So, we can resolve singularities $P_1$ and $P_2$ by $(-2)$-curves $E_1$ and $E_2,E_3$ respectively, where $E_2$ intersects $C_2$ on the resolution. If we contract successively $C_2,E_2$ and $E_3$, we transform $E_1$ into a ($1$)-curve which is tangent to the transform of $C_1$ with order $3$. Hence $S'=\mathbb P^2$ and $C_1$ is a cubic in it. Finally, $K+B\equiv 0$ and we have a trivial $7$-complement according to the computation. \end{proof} \begin{proof}[Proof of Theorem~\ref{ecomp}.\ref{emainth}] $C$ has at most $\delta=3$ irreducible components $C_i$ by Proposition~\ref{ecomp}.\ref{genus}. Otherwise $g\ge 2$ and $2$ is attained when $C$ has four non-singular rational components with one intersection point for each pair of components by Corollary~\ref{ecomp}.\ref{node}. Thus we prove~\ref{ecomp}.\ref{emainth}.1-2, and can assume that $C$ has at least $\delta\ge 2$ components $C_i,1\le i\le \delta$. The $1/7$-log terminal property of $K+B$ implies that $F$ does not pass $C_1\cap C_2$ and $b_1+b_2<13/7$. By Proposition~\ref{ecomp}.\ref{logter}, $F$ does not pass the nodes of $C$: $2\times (6/7)+1/2>2$. In particular, $F$ does not pass $C_i\cap C_j$ for $i\not= j$. Except for Case~(A$_2^6$), each component $C_i$ is non-singular and rational. By~\ref{ecomp}.\ref{genus}.1 this holds on $S'$. Then, for $S$, it is implied by Lemma~\ref{ecomp}.\ref{move}. So, excluding Case~(A$_2^6$) in what follows, we have only the following three configurations of non-singular rational curves $C_i$: \begin{description} \item {$(\text{I}_3)$} $C=C_1+C_2+C_3$ and $C_i$'s form a wheel; \item {$(\text{I}_2)$} $C=C_1+C_2$ and $C_i$'s form a wheel; and \item {$(\text{A}_2)$} $C=C_1+C_2$, and $C_i$'s form a chain. \end{description} This follows from Corollary~\ref{ecomp}.\ref{node} and Proposition~\ref{ecomp}.\ref{genus}. To eliminate some of these cases we prove that each $C_i$, with two nodes $P_1$ and $P_2$ in $C$, is an $m_i$-curve with $m_i\ge 1$. We know or can suppose that $C_i$ is non-singular rational, or an $m_i$-curve. Moreover, by Lemma~\ref{ecomp}.\ref{move}, $m_i\ge 0$. We suppose that $m_i=0$, and derive a contradiction. Since $C^2_i>0$ on $S$, then $C_i$ has at least one singularity $P$ of $S$, $P\not=P_1$ and $P_2$. As above we may calculate $(K+B.C_i)$ on a minimal resolution $S^{\rm min}/S$. We denote by $C'$, $C''$ branches of $C\setminus C_i$ in $P_1$ and $P_2$ respectively. We identify them with their proper inverse images on $S^{\rm min}$. Over $P$ in $S^{\rm min}$, we have, respectively, one (non-singular rational) curve $E$ intersecting $C_i$. Let $b'$, $b''$, $d$, $b$ be the multiplicities of $B^{\rm min}$ in $C',C'',C_i,E$. Then by our assumptions $b',b'',d\ge 6/7$. On the other hand, $b\ge (1/2)d$ by Lemma~\ref{ecomp}.\ref{ineq}. Therefore we get a contradiction: $0\ge (K+B.C_i)= (K^{\rm min}+B^{\rm min}.C_i)\ge (K^{\rm min}+b' C'+b'' C''+d C_i+b E.C_i)\ge -2+b'+b''+b\ge -2+b'+b''+(1/2)d\ge -2+6/7+6/7+3/7=1/7$. Now we are ready to verify that $g(C)=0$ and we have Case~($\rm A_2$), except for two Cases ($\rm I_2^1$) and (I$_2^2$) in \ref{ecomp}.\ref{emainth}.3 with configuration (I$_2$). We want to eliminate Case~($\text{I}_3$) and the other cases in ($\text{I}_2$). According to that which was proved above and to the construction, $S'$ again has the same curves $C_i$ as components of $C$: $m_i'$-curves with $m_i'\ge m_i\ge 1$. First, we consider $S'=\mathbb P^2$. Since $4\times (6/7)>3$, they are all $1$-curves in Case~($\text{I}_3$). Hence there are no contractions of $(-1)$-curves onto $C\subset S'$ for $S^{\rm min}\to S'$. In particular, we preserve curve $E$ over any singularity $P\in C$ of $S$, or a curve $E$ with a standard multiplicity $0<\mult{E}{B}<1$. Either has multiplicity $b\ge 3/7$. So, we get $0\ge \deg(K_{S'}+b_1 C_1+b_2 C_2+b_3 C_3+b E)\ge -3+b_1+b_2+b_3+b\ge -3+3\times (6/7)+3/7=0$. Hence $b=3/7$ and at most one of curves $C_i$, say $C_1$, has a singularity. This is impossible when $S\not=S'$ because then $S$ is a rational cone by the condition $\rho (S)=1$, and because then $S^{\rm min}=\mathbb F_1$, when $S'=\mathbb P^2$. So, $S=S'=\mathbb P^2$ and $F=0$. But then $(S,B)$ is $1$-complementary with $B^+=C$, which contradicts our assumptions. We can do the same in Case~($\text{I}_2$), when the ($1$)-curve $C_i$ does not have singularities of $S$, because then $S=S'=\mathbb P^2$ with $F=0$ and $1$-complement $B^+=C$. The case when $C_1$ and $C_2$ are both ($1$)-curves on $S'$ is only possible when $g(C)=0$. So, we have a ($1$)-curve, say $C_1$, on $S$ and on $S'$ with a single simple Du Val singularity $P_1$ of $S$. It is only possible in Case~(I$_2^2$). More precisely, $m_1=m_1'=1,m_2=2,m_2'=4, B'=(6/7)(C_1+C_2)+(3/7)E_1$ and the line $E_1$ is tangent to the conic $C_2$ in a point $P\not\in C_1\cap C_2$ on $S'$. The inverse transform $S'-\to S$ can be done as follows. Surface $S^{\rm min}$ is obtained by successive monoidal transforms: first, in $P$ which gives the ($-1$)-curve $E_2$, then in $E_1\cap E_2$, which gives the ($-1$)-curve $E_3$, then in $E_1\cap E_3$, which gives the ($-1$)-curve $E_4$. Curves $E_1,E_2$ and $E_3$ are ($-2$)-curves on $S^{\rm min}$ and $B^{\rm min}=(6/7)(C_1+C_2)+(3/7)E_1+(2/7)E_2+(4/7)E_3$. To obtain $S$ we contract $E_1$ to $P_1$ and $E_2,E_3$ to $P_2$. Now we suppose that $S'=\mathbb F_{m}$ with $m\ge 2$ but never $=\mathbb P^2$. Since $C_i$ are $m_i$-curves with $m_i\ge 1$, they are sections of $\mathbb F_{m}/\Sigma$, $C_i\not=\Sigma$, and only in Case~($\text{I}_2^1$). Indeed, as in the proof of Proposition~\ref{ecomp}.\ref{genus} $\sigma\le 2/7$, and $\Sigma$ is an exceptional $(-2)$-curve in $S$. Hence $S^{\rm min}=S'=\mathbb F_2$ and $S$ is a quadric of rank 3, having just one singularity. Moreover, since $\sigma\le 2/7$, this is only possible in Case~($\rm I_2^1$) with two conic sections (not through the singularity) $C_1$ and $C_2$. However $\deg B\le 4$ with respect to $C_1\sim C_2$. Therefore, $F=(1/2)L$, where $L$ is a generator of the quadric, $c=3/28$, and $b_1+b_2\le 7/4$. Since $\mathbb F_0\not=S^{\rm min}$, we reduce $S'=\mathbb F_0$ to one of the above cases. In particular, we have proved that $\delta\le 2$. In our assumptions $\delta=2$. We know also that $C_1$ and $C_2$ are $m_1$ and $m_2$-curves, and say $m_1\ge m_2\ge 0$. Excluding Case~($\rm I_2$) in what follows, we suppose ($\rm A_2$): $\# C_1\cap C_2=1$. First, we consider cases with $S'=\mathbb F_m$, but never $=\mathbb P^2$, in particular, $m\ge 2$. As above $S'=\mathbb F_m$ is possible only when $C_1$ is a section and $C_2$ is a fibre of $\mathbb F_m/\Sigma$. If both $C_1$ and $C_2$ are sections of $S'=\mathbb F_m$ and $m\ge 2$, then $m=2,S^{\rm min}=S'=\mathbb F_2$, and $\Sigma\not=C_1$ and $C_2$. So, $(C_1.C_2)\ge 2$ which is impossible under (A$_2$). By the same reason, $C_1$ is a section but not a multi-section. Since $C_2\subset S'$ is a $0$-curve, $m_2=m_2'=0$, and there were no contractions on $C_2$. So, by Proposition~\ref{ecomp}.\ref{logter}, \begin{description} \item $K+D$ is log terminal near $C_2$ on $S$. \end{description} Otherwise $F=\{D\}\ge (1/2)D_1$ where curve $D_1$ is tangent to $C_2$. Since $(K+D.C_2)>0$, we have on $C_2$ a singular point of $S$ or one more (non-tangency) intersection point with $F$. This gives one more curve $D_2$ on $S'$ with $d_2=\mult{D_2}{B'}\ge (1/2)b_2\ge 3/7$ by Lemma~\ref{ecomp}.\ref{ineq}. But this is impossible: $0\ge (K+B.C_2)\ge -2+b_1+2(1/2)+d_2\ge -2+6/7+1+3/7=2/7$. Since $K+D$ is log terminal near $C_2$ on $S$ and $(K+D.C_2)>0$, we have at least in total two singularities of $S$ on $C_2$ or intersection points $P_1$ and $P_2$ with $F$. Moreover, one of them is not \begin{description} \item{\rm ($\Atd{1}{}$)} a simple Du Val singularity of $S$, near the singularity $F=0$; nor \item{\rm ($\Atd{1}{*}$)} a simple (in a non-singular point of $S$) intersection point with a component $D_i$ of $F$ with $\mult{D_i}{F}=1/2$, near the point $F=(1/2)D_i$. \end{description} Otherwise we have in total three singularities or intersection points with $F$, which is impossible, because then $0\ge (K_{S'}+B'.C_2)\ge -2+6/7+3\times (3/7)=1/7>0$ by Lemma~\ref{ecomp}.\ref{ineq}. We assume that $P_2$ is not ($\Atd{1}{}$) nor ($\Atd{1}{*}$). So, if $P_2$ is a non-singular point of $S$, then $F$ has a component $D_2$ passing through $P_2$ with $\mult{D_2}{F}=(i_2-1)/i_2$ and $i_2\ge 3$. Moreover, near $P_2$, $D_2$ has a simple intersection with $C_2$ and $F=((i_2-1)/i_2)D_2$. In addition, $P_1$ has type ($\Atd{1}{}$) or ($\Atd{1}{*}$). Otherwise some $S'=\mathbb P^2$. Indeed, $0\ge (K+B.C_2)\ge -2+6/7+2\times (4/7)=0$. Then $K+B,K^{\rm min}+B^{\rm min}$ and $K_{S'}+B'\equiv 0$, $b_1=b_2=6/7$, and the modifications are crepant. By Lemma~\ref{ecomp}.\ref{ineq}, $K+D$ has the index $3$ in each $P_i$. Moreover, $B'=(6/7)(C_1+C_2)+(4/7)(E_1+E_2)$ for divisors $E_1, E_2$ on $S^{\rm min}$ and $S'$ over $P_1$ and $P_2$. Hence each $P_i$ is a singularity and $\rho(S^{\rm min})\ge 3$. So, we may suppose that $S'=\mathbb F_m$ and $m\ge 3$. Then $\Sigma=E_i$ for some $E_i$. This gives a contradiction $4/7=\sigma\ge 1/3+(1/3)b_2=1/3+2/7$. Assuming that \begin{description} \item $P_1$ has type ($\Atd{1}{}$) and this is the only singularity of $S$ on $C_2$, \end{description} we verify then that $(S,B)$ has type (A$_2^2$). Indeed, a fractional component $F$ of $D$, with multiplicity $(l-1)/l, l\ge 3$, intersects $C_2$. Since $\rho(S)=1$, $S$ is a quadric cone, $S^{\rm min}=S'=\mathbb F_2$, and $0\ge (K+B.C_2)\ge -2+b_1+(1/2)b_2+(l-1)/l\ge -2+6/7+3/7+(l-1)/l$. Hence $(l-1)/l\le 5/7$, and $l=3$ which gives Case~(A$_2^2$). In particular, $F=(2/3)D_1$ for a section $D_1$ not passing $P_1$. The next case when \begin{description} \item $P_1$ has type ($\Atd{1}{}$) and $P_2$ is a singularity of $S$, \end{description} is reduced to $\mathbb P^2$. Indeed, $\rho(S^{\rm min})\ge 3$, and we can suppose that $m\ge 3$. Then $\Sigma\not=E_1$, where $E_1$ is the exceptional curve$/P_1$ on $S^{\rm min}$ or $S'$, because $E_1$ will be a section of $S'=\mathbb F_m$ with $E_1^2\ge -2$ and even $\ge m$ on $S'$. On the other hand, $0\ge (K+B.C_2)\ge -2+b_1+\sigma+(1/2)b_2\ge -2+6/7+\sigma+3/7$ and $\sigma\le 5/7$. But, since $(K+B'.\Sigma)\le 0$, $5/7\ge \sigma\ge (m-2)/m+b_2/m\ge (m-2)/m+(1/m)(6/7)$, which gives $m\le 4$. As above after a modification we assume that $m\le 3$. So, $m=3$. Curves $C_1$ and $E_1$ do not intersect $\Sigma$ simultaneously. Otherwise we have a contradiction: $5/7\ge \sigma\ge 1/3+(b_1+b_2+d_1)/3\ge 1/3+5/7$, where $d_1=\mult{E_1}{B'}=(1/2)b_2$. Therefore $(E_1.C_1)\ge 3$ and the intersection points $E_1\cap C_1$ are outside of $\Sigma$. We need to make at least $3$ blow-ups in $E_1$ to disjoint $E_1$ and $C_1$ on $S^{\rm min}$. Hence we can get $S'=\mathbb P^2$. In the next case \begin{description} \item $P_1$ has type ($\Atd{1}{*}$). \end{description} Then $P_2$ is a singularity of $S$, because $C_2^2>0$ and $m_2=m_2'=0$. We suppose that $P_2$ has index $l\ge 3$ for $K+D$. Then we get type (A$_2^3$). We have no other singularities on $C_2$, equivalently, $B=b_2 C_2$ and $S$ is non-singular near each other point. So, $\Sigma$ is a curve in a resolution of $P_2$ intersecting $C_2$. It is an ($-m$)-curve on $S'=\mathbb F_m$ with $m\ge 2$. Singularity $P_1$ gives a fractional component: $F\ge (1/2)D_1$, where $D_1$ is a section of $S'=\mathbb F_m$. As above, $0\ge(K+B.C_2)\ge -2+b_1+\sigma+1/2\ge -2+6/7+\sigma+1/2$, and $\sigma\le 9/14$. By Lemma~\ref{ecomp}.\ref{ineq}, $9/14\ge \sigma\ge ((l-1)/l)b_2\ge ((l-1)/l)6/7$. So, $l\le 4$. Moreover, for $l=4$ we have equations: $b_2=6/7$ and $\sigma=9/14$. As in the last part of the proof of Lemma~\ref{ecomp}.\ref{ineq}, this is possible only when $F=0$ near $P_2$ and $P_2$ is a Du Val singularity of type $\mathbb A_3$. But then $m=2$ and we can reconstruct $S'$ into $\mathbb P^2$. Therefore $l=3$. Moreover, according to the same reasons this is not a Du Val singularity of type $\mathbb A_2$. Thus $P_2$ is a simple singularity which can be resolved by ($-3$)-curve. So, $m=l=3$ and $S$ is a cubic cone. This is Case~(A$_2^3$). Moreover, $F=(1/2)D_1$, and both sections $C_1$ and $D_1$ do not pass the vertex. Since $K+B$ is log terminal $\# C_1\cap D_1\ge 2$. Since $K+B$ has a non-positive degree, $3b_1+b_2\le 7/2$ and $c=1/14$. Finally, we consider the case when some $S'=\mathbb P^2$. We suppose that $C_1$ and $C_2$ are respectively $m_1$- and $m_2$-curves on $S$ with $m_1\ge m_2\ge 0$, and $m_1'$- and $m_2'$-curves on $S'=\mathbb P^2$ with $m_1', m_2'\ge 1$, and $m_1'\ge m_2'$ whenever $m_1=m_2$. Note that $K+D$ is log terminal in this case. By Proposition~\ref{ecomp}.\ref{logter}, if $K+D$ is not log terminal, then $F\ge (1/2)D_1$ where curve $D_1$ is tangent to $C$, as it is for $C$ and $D_1$ in $S'=\mathbb P^2$. If $D_1$ is a line in $\mathbb P^2$, then $3\times (6/7)+1/2>3$ and $K+B'$ is ample. Therefore $m_1'=m_2'=1$ and $D_1$ is a conic in $\mathbb P^2$. It was checked above that $D_1$ on $S$ is not tangent to the $0$-curves $C_i$. In other words, $D_1$ is tangent to $C_i$ with $m_i=m_i'=1$. In addition, we have on $C_i$ a singular point of $S$ or one more (non-tangency) intersection point with $F$. This gives one more curve $D_2$ with $d_2=\mult{D_2}{B'}\ge (1/2)b_i\ge 3/7$ by Lemma~\ref{ecomp}.\ref{ineq}. But this is impossible: $0\ge \deg(K_{S'}+B')\ge \deg(K_{S'}+b_1 C_1+b_2 C_2+(1/2)D_1+d_2 D_2)\ge -3+b_1+b_2+2(1/2)+d_2\ge -3+2\times (6/7)+1+3/7=1/7$. If $m_1=m_2=0$, we have contraction $S^{\rm min}\to S'=\mathbb F_0= \mathbb P^1\times\mathbb P^1$ given by the linear system $|C_1+C_2|$ on $S^{\rm min}$. Such cases we consider later. In other cases $m_1\ge 1$. We verify that the latter is possible only for types (A$_2^1$), (A$_2^4$) or (A$_2^5$). So, first we check that $m_1=1$. Otherwise, $m_1'=4\ge m_1\ge 2$ and $C_1$ is a conic on $S'=\mathbb P^2$; $m_2'=1$ and $C_2$ is a line on $S'=\mathbb P^2$, because $4\times (6/7)>3$. Since $K+B'$ is log terminal, $C_1$ and $C_2$ have two intersection points $R_1$ and $R_2$. Moreover, $m_2=0$, because $C_1+C_2$ has configuration (A$_2$) on $S$ by our assumptions. As we know, in total $C_2$ has two singularities of $S$ or intersections with $F$. On the other hand, we have contractions of curves onto $C_2$ for $S^{\rm min}\to S'$ only over one of points $R_i$. Thus there exists point $P_1\not= R_1$ and $R_2$ on $C_2$ which is singular on $S$ or belongs to $F$. Moreover, \begin{description} \item $P_1$ on $S$ has type ($\Atd{1}{}$) or ($\Atd{1}{*})$. \end{description} Otherwise, by Lemma~\ref{ecomp}.\ref{ineq}, $P_1$ gives on $S'=\mathbb P^2$ a curve $D_1\not= C_1$ and $C_2$ with $d_1=\mult{E_1}{B'}\ge (2/3)b_2\ge 4/7$. Then $K+B'$ is ample, because $3\times (6/7)+4/7>3$. It is impossible. By the same reasons, $d_1=3/7$, $P_1$ has type ($\Atd{1}{}$), $D_1$ is ($-2$)-curve on $S^{\rm min}$, $b_1=b_2=6/7$, $F=0$, $K+B,K^{\rm min}+B^{\rm min}$ and $K_{S'}+B'\equiv 0$, and the modifications are crepant. If $\# D_1\cap C_1=2$, then we should make at least two blow-ups in this intersection to disjoint $D_1$ and $C_1$ on $S^{\rm min}$. According to (A$_2$) for $C_1+C_2$ on $S$ and $S^{\rm min}$, we need to make at least one blow-up in $R_1$ or $R_2$. So, $m_1\le 1$. This contradicts our assumptions. Thus $D_1$ is tangent with $C_1$. But again to disjoint $D_1$ and $C_1$ we need two blow-ups. That leads to the same contradiction. As we see later, the latter case is possible for $m_1=1$ in type (I$_2^2$), or for $\rho=2$ and $m_1=m_2=0$ as will be discussed below. So, $m_1=1$, and we have contraction $S^{\rm min}\to S'=\mathbb P^2$ given by the linear system $|C_1|$ on $S^{\rm min}$; $m_1'=m_2'=1$. So, $C_1$ and $C_2$ are lines on minimal $S'=\mathbb P^2$. If $m_2=1$, we have no contractions onto $C$ for $S^{\rm min}\to S'=\mathbb P^2$ and no singularities of $S$ on $C$. Then $S=\mathbb P^2$, and this is the first exception (A$_2^1$). Indeed, if, say, $C_1$ has a singularity, it gives rise to a curve $E_1\subset S^{\rm min}$ and $S'=\mathbb P^2$, which intersects $C_2$ on both $S'$ and $S^{\rm min}$. It is impossible. Therefore, $m_2=0$. Moreover, by the same arguments $C_2$ has at most one singularity $P_2$ of $S$, and $C_1$ has a singularity $P_1$ of $S$ (except for (A$_2^1$)). Really, $S$ has a singularity $P_2\in C_2$, because $m_2=0$; $P_2$ is a single singularity of $S$ on $C_2$. It corresponds to point $P_2\in C_2\subset S'=\mathbb P^2$ into which we contract and only once a $(-1)$-curve. So, $F\ge (1/2)D_1$ for some curve $D_1\not=C_1$ and $C_2$ on $S$. Moreover, $D_1$ is a curve on $S'=\mathbb P^2$. On the other hand, $P_1$ gives another curve $E_1$ on $S'=\mathbb P^2$ with $d_1=\mult{E_1}{B'}\ge ((m-1)/m)b_1$, where $m\ge 2$ is the index of $K+D$ in $P_1$. Moreover, by the construction $E_1$ is a line on $S'=\mathbb P^2$. Curve $D_1$ is also a line on $S'=\mathbb P^2$. Otherwise, $0\ge \deg(K_{S'}+B')\ge \deg(K_{S'}+b_1 C_1+b_2 C_2+(1/2)D_1+d_1 E_1)\ge -3+b_1+b_2+2(1/2)+d_1\ge -3+2\times (6/7)+1+3/7=1/7$. By the same reasons, other components of $F$ are contracted on $S'=\mathbb P^2$. In other words, $\Supp{B'}=C_1+C_2+D_1+E_1$. Equivalently, $D_1$ is the only component of $F$ which is passing a non-singular point of $S$ on $C_1$. Moreover, lines $C_1,C_2,D_1$ and $E_1$ are in general position in $S'=\mathbb P^2$, because the intersection point $P_3$ of $E_1$ and $D_1$ does not belong to $C$. Curve $D_1$ crosses $C_1$ in a single non-singular point of $S$. We contend that $D_1$ passes $P_1$ on $S$ too. Indeed, we can increase $B$ to $B''$ in $D_1$ in such a way that $K+B''\equiv 0$. Then $K^{\rm min}+(B'')^{\rm min}$ and $K_{S'}+B'''\equiv 0$ and the modifications are crepant, where $B'''=b_1 C_1+b_2 C_2+d''D_1+d_1'E_1$ with $d_1'=\mult{E_1}{(B'')^{\rm min}}$ and with $d''=\mult{D_1}{B''}$, and $(B'')^{\rm min}$ corresponds to the crepant resolution $S^{\rm min}\to S$. By the above arguments or the Inductive Theorem, $d_1'$ and $d''<1$. However $\delta=d_1'+d''-1\ge 0$, if we assume that $D_1$ and $E_1$ are disjoint on $S^{\rm min}$. Moreover, we have contractions of curves for $S^{\rm min}\to S'$ onto $D_1$ only over the intersection point $P_3$, because their multiplicities in $(B'')^{\rm min}$ are non-negative. Curves on $S^{\rm min}/P_3$ form a chain with $D_1$ and $E_1$. So, $D_1$ is a rational $n$-curve with $n\le 0$ on $S$ with a single singularity $P_3$ of type $\mathbb A_l$. Moreover, $n=0$, because $D_1^2>0$ on $S$. Also $l\ge 2$, because $S$ is not a cone: it has too many singularities. In addition, by Lemma~\ref{ecomp}.\ref{ineq} and our description of the modification near $D_1$, $\delta\ge d_2=\mult{E_2}{(B'')^{\rm min}}\ge (2/3)d''\ge (2/3)(1/2)\ge 1/3$, where $E_3/P_3$ in $S^{\rm min}$ intersects $D_1$. This gives a contradiction: $0=\deg(K_{S'}+B''')= \deg(K_{S'}+b_1 C_1+b_2 C_2+d''D_1+d_1'E_1)\ge -3+b_1+b_2+d''+d_1'\ge -3+2\times (6/7)+1+1/3=1/21$. Note another contradiction, that the $(-1)$-curve$/P_3$, which is non-exceptional on $S$, does not intersect $C_2$ on $S$. Therefore, $D_1$ passes $P_1$, and, by the log terminal property of $K+D$ and since $D_1$ intersects $E_1$ on $S^{\rm min}$, $P_1$ is a simple singularity with a single ($-l$)-curve $E_1$ on $S^{\rm min}/P_1$. In addition, $\Supp F=D_1$. Arguing as above, we can verify that $P_2$ is a Du Val singularity of type $\mathbb A_l$, or $S^{\rm min} \to S'$ contracts only ($-2$)-curves and one ($-1$)-curves -- successive blow-ups of $P_2\in S'=\mathbb P^2$. In addition $F=0$ near $P_2$. Hence $P_2$ has index $l+1$ for $K+D$, and $l\ge 2$, because $S$ is not a cone (except for (A$_2^1$)). This gives $d_2=\mult{E_2}{B'}=(l/(l+1))b_2\ge (l/(l+1))(6/7)$ by Lemma~\ref{ecomp}.\ref{ineq} in divisor $E_2/P_2$ on $S^{\rm min}$ intersecting $C_2$. So, in this case $0\ge (K+B.C_2)\ge -2+b_1+\mult{D_1}{F}+d_2\ge -2+6/7+1/2+(l/(l+1))(6/7)$. Hence $l/(l+1)\le 3/4$ and $l\le 3$. This gives types (A$_2^4$) and (A$_2^5$) for $l=2$ and $l=3$ respectively. The same inequality with $l=2$ gives $\mult{D_1}{F}\le 4/7$ and so $F=(1/2)D_1$. If $l=3$, $K+(6/7)(C_1+C_2)+(1/2)D_1\equiv 0$. Finally, we prove that the cases with $m_1=m_2=0$ are impossible. First, we verify that each $C_i$ has at least two singularities of $S$. Otherwise $F\ge (1/2)D_1$ where curve $D_1\not= C_1$ and $C_2$ intersects one of theses curves, say $C_1$, in a non-singular point and only in this point. Curve $C_1$ has a singular point $Q_1$ of $S$, and $C_2$ does so for $P_1$, because $C^2_i>0$. Let $F_1/Q_1$ and $E_1/P_1$ be respectively curves on $S^{\rm min}$ which intersect $C_1$ and $C_2$. They give different generators $F_1$ and $E_1$ of $S'=\mathbb P^1\times\mathbb P^1$ with multiplicities $f_1=\mult{F_1}{B'}\ge (1/2)b_1$ and $e_1=\mult{E_1}{B'}\ge (1/2)b_1$. Moreover, if $\Supp{F}$ passes $Q_1$, equivalently, a component $D_i$ of $F$ passes $Q_1$, then $f_1\ge (1/2)b_1+(1/2)\mult{D_i}{B}\ge 3/7+1/4$, but $0\ge (K+B.C_1)\ge -2+b_2+f_1+\mult{D_i}{B}\ge -2+6/7+3/7+1/4+1/2=1/28$. By the same reasons, $D_1$ intersects $C_1$ only in one point. Thus $\Supp{F}=D_1$ does not pass $Q_1$. If $D_1$ does not intersect $C_2$ in a non-singular point of $S$, then $D_1$ is also a generator $D_1\sim F_1$. However, $D_1$ intersects transversally in one point $P$ another generator $E_1$ on $S'=\mathbb P^1\times \mathbb P^1$, and $\Supp{B'}=D_1+E_1$ near $P$. On $S^{\rm min}$, $D_1$ and $E_1$ cannot be disjoint by a chain of rational curves$/P$, because we can assume as above that $K+B\equiv 0$ and the modifications are crepant. Then a ($-1$)-curve$/P$ will be a curve on $S$ non-intersecting $C_1$. It is impossible. Hence $D_1$ passes each singular point of $S$ on $C_2$ and we have two of them, or $D_1$ intersects $C_2$ in a non-singular point of $S$. The former case is impossible: $0\ge (K+B.C_2)\ge -2+b_2+2\times((1/4)+(1/2)b_2) \ge-2+6/7+1/2+6/7=3/4$ as above with $f_1$. So, $D_1$ intersects $C$ on $S$ in two non-singular points: one on each $C_i$. Then $D_1\sim F_1+E_1$ or has bi-degree $(1,1)$ on $S'=\mathbb P^1\times\mathbb P^1$. Moreover, $B'=e_1 E_1+f_1F_1+(\mult{D_1}{F} )D_1$, and $D_1$ passes the intersection point $P$ of $E_1$ and $F_1$, because $D_1$ does not pass singular points of $C$. Assuming as above that $K+B\equiv 0$ and the modifications are crepant, we can check that $S^{\rm min}\to S'$ contracts only curves over $P$. Since $\rho(S)=1$, all these curves but one are contracted on $S$, to singularities $P_1$, $Q_1$ and maybe one on $D_1$. However, this is only possible when we contract $E_1$ and $F_1$ after the first monoidal transform in $P$ which does not produce singularities of $S$ at all. Otherwise as above, a ($-1$)-curve$/P$ will be a curve on $S$ non-intersecting some $C_i$. So, $F$ intersects $C$ only in singular points of $S$. Hence we have at least four singularities of $S$: $Q_1,Q_2\in C_1$ and $P_1,P_2\in C_2$. They give respectively different generators $F_1\sim F_2$ and $E_1\sim E_2$ with multiplicities $f_i=\mult{F_i}{B'}\ge (1/2)b_1$ and $e_i=\mult{E_i}{B'}\ge (1/2)b_2$. If for two of these multiplicities, say $e_1$ and $f_1$, $e_1+f_1<1$, then we cannot disjoint $E_1$ and $F_1$ on $S^{\rm min}$ under assumption $K+B\equiv 0$. So, we may suppose that $e_1$ and $e_2\ge 1/2$. Then by Lemma~\ref{ecomp}.\ref{ineq}, $e_1$ and $e_2\ge (2/3)b_1\ge 4/7$. This gives an equation in the inequality $0\ge (K+B.C_2)\ge -2+b_1+e_1+e_2\ge -1+6/7+2\times (4/7)=0$. Moreover, $e_1=e_2=4/7, B=(6/7)(C_1+C_2)$, $F=0$, $K+B\equiv 0$ and the modifications are crepant, Indeed, since $1/4+(1/2)(6/7)=19/28>4/7$, $C_2$ has exactly two singularities $P_1$ and $P_2$ of the index $3$ for $K+D$ or $K+C_2$. Thus they are Du Val of type $\mathbb A_2$. On the other hand, $S$ has two singularities $Q_1$ and $Q_2\in C_1$. We can assume that $Q_2$ is not simple Du Val. If $Q_1$ is also not simple Du Val, then again both are Du Val of type $\mathbb A_2$. So, $B'=(6/7)(C_1+C_2)+(4/7)(E_1+E_2+F_1+F_2)$, where $F_i$ is curves with $d=4/7$ over $Q_i$. All curves $C_1\sim E_1\sim E_2$ and $C_2\sim F_1\sim F_2$ are generators of corresponding rulings of $S'=\mathbb P^1\times\mathbb P^1$. On $S^{\rm min}$ there exists a curve $E'$ with $\mult{E'}{B^{\rm min}}=2/7$. For instance, the second curve in a minimal resolution of $P_1$. Such curves could only be over the intersection of $P=E_i\cap F_i$, which is impossible because they have only one curve $E''/P$ with $a(E'',K_{S'}+B')\le 1$. It is a blow-up of $P$ with $a(E'',K_{S'}+B')=6/7$. Therefore $Q_1$ is simple Du Val, $f_1=(1/2)b_1=3/7,f_2=2-b_2-f_1=2-6/7-3/7=5/7$, and $B'=(6/7)(C_1+C_2)+(4/7)(E_1+E_2)+(3/7)F_1+(5/7)F_2$. However, this is possible only for a surface $S$ with $\rho(S)=2$. Indeed, to disjoint $F_2$ and $E_1$ we should make two successive monoidal transformations: first in $F_2\cap E_1$ which gives $E_3$, and then in $F_2\cap E_3$ which gives $E_4$. Both curves $E_1$ and $E_3$ are ($-2$)-curves on $S^{\rm min}$: over points $F_1\cap E_i$, the contraction $S^{\rm min}\to S'$ is just one monoidal transform. On the other hand, $F_2$ will be a ($-4$)-curve and $E_4$ is a contractible ($-1$)-curve passing $Q_2$ and $P_1$. Hence $\rho (S)\ge 2$. One can check that $\rho (S)=2$, and after a contraction of $E_4$ on $S$ we get Case~(I$_2^2$) in \ref{ecomp}.\ref{emainth}.3. Now we can find minimal complements in~\ref{ecomp}.\ref{emainth}.3 as in \cite[Example 5.2.1]{Sh2}, however we did it only in a few trivial cases. Indeed, by Proposition~\ref{ecomp}.\ref{logter}, we should care only about the numerical property, not singularities. \end{proof} \section{Classification of surface complements} \label{clasf} Take $r\in N_2$. A classification of $r$-complements or of complements of index $r$ means a classification of surface log canonical pairs $(S/Z,B)$ with $r(K+B)\sim 0/Z$. We assume that $S/Z$ is a contraction: in the local case, according to the Main Theorem, such that there exists a log canonical center$/P$ (cf. Example~\ref{int}.\ref{tcomp} and Section~\ref{lcomp}). This classification implies a classification of log surfaces $(S/Z,B')$ with such complements. For instance, the minimal complementary index $r$ is an important invariant of $(S/Z,B')$. Moreover, for exceptional $r\in EN_2=\{7,\dots\}$ $K+B$ is to be assumed Kawamata log terminal and $S$ is complete with $Z=\mathop{\rm pt.}$. Such cases are bounded. They have been partially described in Section~\ref{ecomp}. Here we focus our attention on basic invariants for regular complements with $r\in RN_2=\{1,2,3,4,6\}$. For indices $1,2,3,4$ and $6$, types of complements are denoted respectively by $\mathbb A_m^n,\mathbb D_m^n, \mathbb E3_m^n,\mathbb E4_m^n$ and $\mathbb E6_m^n$, where \begin{itemize} \item $n$ is the number of reduced (and formally) irreducible components in $B$ (over a neighborhood of a given $P\in Z$ in the local case), and \item $m$ is the number of reduced exceptional divisors of $B^{\rm min}$ on a crepant minimal formally log terminal resolution (in most cases, this is a minimal resolution cf. Example~\ref{clasf}.\ref{notation} below) $(S^{\rm min},B^{\rm min})\to (S,B)$ (over a neighborhood of $P\in Z$), equivalently, the number of exceptional divisors on the minimal resolution with the log discrepancy $0$. \end{itemize} We also assume that, for the types $\mathbb A_m^n$ and $\mathbb D_m^n$, $\Supp{B^{\rm min}}$ is a connected and singular curve, otherwise we denote them respectively $\mathbb E1_m^n$ and $\mathbb E2_m^n$. Equivalently, exactly in types $\mathbb E1-2_m^n$ among types $\mathbb A_m^n$ and $\mathbb D_m^n$, the number of exceptional divisors with the log discrepancy $0$ is finite. The same numerical invariants can be defined in any dimension up to the LMMP. But as we see in the next section, a simplicial space associated to reduced components is a more important invariant. \refstepcounter{subsec} \label{clasfth} \begin{theorem} If $m=n=0$, then $Z=\mathop{\rm pt.}$, and $K+B$ is Kawamata log terminal of a regular index $r\in RN_2$. It is possible only for types $\mathbb Er_0^0$. The complements of this type are bounded when $B\not=0$ or $S$ has non-log terminal point {\rm \cite{Al}}. Otherwise $B=0$ and $S$ a complete surface with canonical singularities and $r K\sim 0$ (their classification is well known up to a minimal resolution and can be found in any textbook on algebraic surfaces, e.~g., {\rm \cite{BPV}} and {\rm \cite{Shf}}). In the other types we suppose that $m+n\ge 1$. Then they have a non-empty locus of the log canonical singularities $\LCS{(S,B)}$. It is the image of $\LCS{(S^{\rm min},B^{\rm min})}= \Supp{\lfloor B^{\rm min}\rfloor}$ and it has at most two components: Two only for exceptional types $\mathbb Er_n^m$ with $n+m=2$ and in the global case. If $\LCS{(S,B)}$ is connected, it is a point if and only if $n=0$. Otherwise it is a connected curve $C=\Supp{\lfloor B\rfloor}$ with at most nodal singularities and of the arithmetic genus $g\le 1$ for any sub-curve $C'\subseteq C/P$. Moreover, if $g=1$ then $C'=C$, this case is only possible for types $\mathbb E1_1^0$ and $\mathbb A_m^n$ with $n\ge 1$, whereas $C=C'$ is respectively a non-singular curve of genus $1$ or a Cartesian leaf when $n=1$ and $C$ is a wheel of $n$ rational curves when $n\ge 2$. In all other cases $C$ is a chain of $n$ rational curves. The singularities of $(S,B)$ outside of $\LCS{(S,B)}$ are log terminal of index $r$. In particular they are only canonical when $r=1$; moreover, only of type $\mathbb A_i$ when $m+n\ge 1$. If $B=0$ then $n=0$ and $\LCS{(S,B)}$ is the set of elliptic singularities of $S$. For types $\mathbb A_m^n$ and $\mathbb D_m^n$, any natural numbers $m,n$ are possible. If $n=0$, then it is a global case with $B=0$, $K\sim 0$ and $S$ has a single elliptic singularity. In the exceptional types $\mathbb Er_m^n$, $n+m\le 2$ and any $m,n$ under this condition are possible; $n+m=2$ is only possible in the global case. The number of the connected components of $\LCS{(S,B)}$ is $m+n$. Equivalently, each such component is irreducible. Moreover, it is a point or a non-singular curve, respectively, of genus $1$ for types $\mathbb E1_m^n$ and of genus $0$ for the other types, when $C/P$. \end{theorem} \begin{proof} The most difficult part is related to connectedness \cite[Theorem~6.9]{Sh2}. Other statements follow an adjunction, except for the statement on types of canonical singularities for $1$-complements $(S/Z,B)$. Essentially it was proved in Section~\ref{indcomp}. So, let $(S/Z,B)$ be a $1$-complement. After a formal log resolution we can assume that $\LCS{(S,B)}=B=C$ is a reduced curve and $S$ non-singular near $C$. We can also assume that $C$ is minimal, i.~e., does not contain $(-1)$-curves. We verify that the singularities of $S$ have type $\mathbb A_i$ using an induction on extremal contractions. By the LMMP we have an extremal contraction $g:S\to T/Z$ with respect to $K$ if there exists a curve$/P$ not in $C$. If this is a contraction of a curve $C'$ to a point, then it is to a non-singular point and $S$ has only singularities of type $\mathbb A_i$ near $C'$, because $S$ has only canonical singularities. If this is a contraction of a fibre type, then it is a ruling which can have singularities only when $C$ has an irreducible component $C'$ as a double section. Then the only possible singularities are simple double. If $T=\mathop{\rm pt.}$, then components of $C$ are ample and $S=\mathbb P^2$. Note that if we have no contractions we have no singularities, because the latter ones are apart of $C$. \end{proof} Of course, our notation is similar to the classical one (however, with some twists). \refstepcounter{subsec} \label{notation} \begin{example} For instance, a singularity $P\in (S/S,H)$ of type $\mathbb A_m$ with the generic hyperplane through $P$ has type $\mathbb A_m^2$ in our notation. But type $\mathbb D_m$ corresponds to $\mathbb D_{m-2}^1$ with some reduced and irreducible $H$. We have more differences for elliptic fiberings $(S/S,E)$. For instance, the Kodaira type $_m\rm I_b$ is our $\mathbb A_0^b$. Let $(S/S,L_1+(1/2)(L_2+L_3))$ be a singularity as in the plane $S=\mathbb P^2$ in the intersection of three lines $L_i$. Then it has type $\mathbb D_1^1$, because its minimal log resolution is a monoidal transform in this point. \end{example} \refstepcounter{subsec} \label{toric} \begin{example} Each toric variety $X$ has a natural $1$-complement structure $(X,D)$ where $D$ is $D=\sum D_i$ with the orbit closures $D_i$. So, the number of elements in this sum $n$ is the number of edges in the fan. A toric surface $S$ with $n$ ages has type $\mathbb A_m^n$. In addition, $n=\rho(S)+2$. This characterizes toric surfaces. \end{example} \refstepcounter{subsec} \label{rhob} \begin{theorem} Let $(S/Z,B)$ be with log canonical $K+B$ and nef$/Z$ divisor $-(K+B)$. Then $\rho(S/Z)\ge \sum b_i-2$, where $\rho(S/Z)$ is the rank of Weil group modulo algebraic equivalence$/Z$, or just the Picard number when the singularities of $S$ are rational. Moreover, $=$ holds if and only if $K+B\equiv 0$ and $S/Z$ is formally toric with $C=\lfloor B\rfloor\subseteq D$. In addition, in the case $=$ and reduced $B=C$, $(S/Z,C)$ is formally toric with $C=D$ (see Example~\ref{ecomp}.\ref{toric}). \end{theorem} {\it Formally toric\/}$/Z$ means formally equivalent to a toric contraction, or locally$/Z$ in analytic topology, when the base field is $\mathbb C$. \begin{proof} First, we can assume that $\LCS{(S,C)}\not=\emptyset$. In the local case we can do this adding pull back divisors as in the proof of the General Case in Theorem~\ref{lcomp}.\ref{lmainth}. In the global case, after contractions, we can assume that $\rho(S)=1$. If $\LCS{(S,C)}=\emptyset$, the inequality will be improved after contractions. If $B$ has at most one component $C_i$ with $b_i=\mult{B_i}{B}>0$, then $\rho(S)=1>b_i-2$. Otherwise we have at least two curves $C_i$ and $C_j$ with $b_i$ and $b_j>0$. We can also assume that $K+B\equiv 0$. If $\LCS{(S,B)}=\emptyset$ we can change $b_i$ and $b_j$ in such a way that $K+B\equiv 0$ and $b_i+b_j$ is not decreasing. Indeed $K+B\equiv 0$ gives a linear equation on $b_i$ and $b_j$. Then we get $\LCS{(S,B)}\not=\emptyset$, or we get $b_i$ or $b_j=0$. An induction on the number of curves in $\Supp{B}$ gives the log singularity or the inequality. Second, we can replace $(S,C)$ by its log terminal minimal resolution $(S^{\rm lt},C^{\rm lt})$ over $C=\LCS{(S,B)}\not=0$. We preserve all the statements. The contraction will be toric because it contracts curves of $D$. If every curve $C'/P$ is in $C$, we have the local case and by the adjunction and \cite[Corollary~3.10]{Sh2}, we reduce our inequality to a $1$-dimensional case on $C''/P$. In addition, for $=$, $S/P$ is non-singular and toric which is possible to check case by case. Here we use the monotonicity $(m-1)/m+\sum k_i d_i/m\ge d_i$ when $k_i\ge 1$, and even $>d_i$, when $m\ge 2, k_i\ge 1$ and $1>d_i$ (cf. [ibid]). Third, we could assume that $K+B\equiv 0$ on $C/P$ after birational contractions. This improves the inequality. If we return to a Kawamata log terminal case, we can find a complement $K+B'\equiv 0/Z$ with $B'\ge B$. This again improves the inequality. By \cite[Theorem~6.9]{Sh2}, we assume that $C/P$ is connected. Otherwise $Z=\mathop{\rm pt.}$, $C=C_1+C_2$, and after contractions we can assume that each fractional component, i.~e., each component of $B-C$, intersects some $C_i$ (cf. the arguments for the connected case). Then we reduce the problem to a $1$-dimensional case on $C_i$. Therefore we assume that there exists a curve $C'/P$ not in $C$. Then we have an extremal contraction $g:S\to T$ which is numerically non-negative for a divisor $H$ with $\Supp H=C$ and $g$ is numerically negative for $K+B-\varepsilon H$ with some $\varepsilon>0$. If $C$ has an exceptional type we take such an $H$ that is negative on $C$. Otherwise we assume that $H$ is nef on $C$, and even ample in the big case. So, such a $g$ preserves birationally $C$, whenever birational, or $H$ is ample and we consider this case as a contraction to $Z=\mathop{\rm pt.}$ below. After birational contractions, $f$ has a fibre type. If it is to a point, then $Z=\mathop{\rm pt.}$ and $C$ has at most two components which are intersected by other components of $B$. We can choose them and reduce the problem to a $1$-dimensional case as above. If $g$ is a ruling we can do similarly when $C$ has an ample component. Otherwise $C$ is in a fibre of $g$. As in the first step, we can assume that we have at most one other fibre component. This implies the inequality. Otherwise we obtain a case when $C$ is not connected. Note, that we preserve inequality only when we contract a curve $C_i/P$ with $(K+B.C)=0$ and $b_i=1$. Such a transform preserves the formally toric property. So, $D$ contains $C$ always. \end{proof} We hope that in general $\rho(X/Z)\ge - \dim X+\sum b_i$, where $\rho$ is the Weil-Picard number, i.~e., the rank of the Weil divisors modulo algebraic equivalence. Moreover, $=$ holds exactly for formally toric varieties and $\lfloor B\rfloor\subseteq D$. For instance, this implies that locally $\sum b_i\le \dim X$ when the singularity is $\mathbb Q$-factorial and $\rho=0$. If the singularity is not $\mathbb Q$-factorial we have a stronger inequality $\sum b_i\le \dim X-1$ for $B=\sum b_i D_i$ with $\mathbb Q$-Cartier $D_i$ (cf. \cite[Theorem~18.22]{KC}). \refstepcounter{subsec} \begin{corollary} Let $(S/Z,C)$ be as in Theorem~\ref{clasf}.\ref{rhob}. The following statements are equivalent. \begin{itemize} \item $(S/Z,C)$ is a surface $1$-complement of type $\mathbb A_m^n$ with $n=\rho(S/Z)+2$; \item $\rho(S/Z)= n-2$, where $n$ is the number of (formally) irreducible components in $C$ ; and \item $(S/Z,C)$ is formally toric. \end{itemize} \end{corollary} For instance, if $\rho(S)=1$ and $Z=\mathop{\rm pt.}$, then a $1$-complement $(S,C)$ of type $\mathbb A_m^n$ is toric with $D=C$, if and only if $n=3$. In other cases $n\le 2$. Most of the above results work over non-algebraically closed fields of the characteristic $0$. \refstepcounter{subsec} \label{arith} \begin{example} If $C$ is a non-singular curve of genus $0$, it always has a $1$-complement. But it has type $\mathbb A_0^2$ only when it has a $k$-point. Otherwise it has type $\mathbb A_0^1$ and its Fano index is $1$. \end{example} A complement with connected $\LCS{(S,B)}$ can be called a {\it monopoly\/}. Other complements are {\it dipoles\/}. \refstepcounter{subsec} \label{dipole} \begin{theorem} Any exceptional complement $(S/\mathop{\rm pt.},B)$ of type $\mathbb Er_0^2$ has a ruling $g:S\to Z$ with a normal curve $Z$ and with two sections in $\LCS{(S,B)}$; the genus of $Z$ is $1$ for $(r=1)$-complements and $0$ in other cases. \end{theorem} \begin{proof} We obtain the ruling after birational contractions which are with respect to a curve $C_i$ in $\LCS{(S,B)}$, $C_i^2\le 0$. Cf. Proof of the Inductive Theorem: Case I. In addition, if $g(Z)\ge 1$ then $B=0$ and we have a $1$-complement $(S,B)$. \end{proof} \refstepcounter{subsec} \label{dipolec} \begin{corollary} Let $(S,B)\to (S',B')$ be a normalization of a connected semi-normal log pair $(S',B')$ with $B'$ under {\rm (M)\/}. Then $(S',B')$ has an $r$-complement, if $(S,B)$ has a complement of type $\mathbb Er_m^n$. \end{corollary} \begin{proof} The divisors $B^+$ in the normal part of $S$ belong to fibres of the contraction $g$ on components of the normalization of $S$ after a log terminal resolution. The latter has the same $r$ for each component of $S$ because they are induced from curves of non-normal singularities on $S'$. \end{proof} For the other dipoles, $\LCS{(S,B)}$ is a pair of points or a point and a curve. \refstepcounter{subsec} \label{rull} \begin{remark} The ruling induces a pencil $\{C_t\}$ of rational curves through the points. Similarly, in other cases we can find \begin{description} \item{(PEN)} {\it a log proper pencil $\{C_t\}$ of log genus $1$ curves\/}, i.~e., $C_t$ does not intersect $\LCS{(X,B)}$, when its normalization has genus $1$, and, for the corresponding map $g_t:(\mathbb P^1,0+\infty)\to (X,B)$ onto $C_t$, $g_t(Q)\not\in \LCS{(X,B)}$ when $C_t$ has genus $0$, $t$ is generic, and $Q\not=\infty$ and $\not= 0$. \end{description} This implies easy cases in the Keel-McKernan Theorem on the log rational covering family \cite[Theorem~1.1]{KM}. The difficult cases are exceptional and bounded. Perhaps it can be generalized in a weighted form for fractional boundaries or m.l.d's. \end{remark} \section{Classification of 3-fold log canonical singularities} \label{csing} \refstepcounter{subsec} \label{singth} \begin{theorem} Let $(X/Z,B)$ be a birational contraction $f:X\to Z$ of a log 3-fold $X$ \begin{itemize} \item with boundary $B$ under {\rm (SM)\/} and \item nef $-(K+B)$. \end{itemize} Then it has an $n$-complement $(X/Z,B^+)$ over a neighborhood of any point $P\in Z$ such that \begin{itemize} \item $n\in N_2$ and \item $K+B^+$ is not Kawamata log terminal over $P$. \end{itemize} \end{theorem} \begin{statement1} We can replace {\rm (SM)\/} by \begin{description} \item{{\rm (M)''}} the multiplicities $b_i$ of $B$ are {\rm standard\/}, i.~e., $b_i=(m-1)/m$ for a natural number $m$, or $b_i\ge l/(l+1)$ where $l=\max\{r\in N_2\}$. \end{description} \end{statement1} \refstepcounter{subsec} \label{singl} \begin{lemma} Let $(S/Z,B)$ be a surface log pair and $(S^{\rm min},B^{\rm min})$ be its crepant minimal resolution. Suppose that $B$ satisfies {\rm (SM)}. Then $$K+B \ n-complementary\ \ \Longrightarrow \ K^{\rm min}+B^{\rm min}\ \ n-complementary.$$ Moreover, we could replace $S^{\rm min}$ by any resolution $S'\to S$ with subboundary $B'=B^{S'}$. For any $n\in N_2$, we can replace {\rm (SM)\/} by {\rm (M)''\/}. \end{lemma} \begin{proof} Follows from the Main Lemma~\ref{gcomp}.\ref{mlem3} or can be done it the same style. The last statement follows from the Monotonicity~\ref{indcomp}.\ref{monl1}. \end{proof} \begin{proof}[Sketch Proof of Theorem~\ref{csing}.\ref{singth}] First, we can assume that $K+B$ is strictly log terminal$/Z$ and $B$ has reduced part $S=\lfloor B\rfloor\not=\emptyset/P$. For this we add a multiple of an effective divisor $D=f^*(H)$ for a hyperplane section $H$ of $Z$ through $P\in Z$. However, $B+d D$ may contradict (SM) in $D$. We drop $D$ after a log terminal resolution of $(S,B+d D)$. In its turn this can spoil the nef condition for $-(K+B)$. This will be preserved when $D$ or even a divisor $D'\ge D$ below is nef$/P$. If not we can do this after modifications in extremal rays on which $D'=B^+-B\ge D$ is negative, where $B^+$ is a complement for $B+d D$, i.~e., $K+B^+\equiv 0/P$. We drop then $D'$. To do the log flops with respect to $D'$ we use the LSEPD trick \cite[10.5]{Sh2}. Finally, if $K+B$ is not strictly log terminal, it holds after a log terminal resolution of $(S,B)$. Second, $(S/Z,B_S)$ is a semi-normal and connected surface over a neighborhood of $P$ where $B_S$ is non-singular in codimension $1$ and again under (SM) or, for \ref{csing}.\ref{singth}.1, under (M)''. This follows from the LMMP or \cite[Theorem 17.4]{KC} and Corollary~\ref{indcomp}.\ref{adjunct}. Adjunctions $(K+B)|_{S_i}$ on each component are log terminal \cite[3.2.3]{Sh2}. We have a complement $(S/Z,B_S^+)$ on $S/Z$ and hence on each $S_i$. This gives (EC)'. Third, we can glue complements from the irreducible components of $S$. If one of them has no regular complements, then $S$ is normal and there is nothing to verify. In the other cases we have $r$-complements with $r\in RN_2$. Moreover if we have a complement of type $\mathbb Er_m^n$ on some component $S_i$, then $m=0$ by the log terminality of the adjunction $(K+B)|_{S_i}$, and $S$ is a wheel or a chain of its irreducible components. Then we can glue complements by Corollary~\ref{clasf}.\ref{dipolec}. Finally, we have the complements of type $\mathbb D_m^n$ with $r=2$. They are induced from a $1$-dimensional non-irreducible case when we always have a $2$-complement (cf. \cite[Example~5.2.2]{Sh2}). (However if we have non-standard coefficients $b_i\le 2/3$ we need to use higher complements (cf. Lemma~\ref{indcomp}.\ref{compl3}).) Finally, we can act as in the proof of \cite[Theorem~5.6]{Sh2} (cf. \cite[Theorem 19.6]{KC}). So, an $r$-complement of $(X/Z,B)$ is induced from the $r$-complement of $(S/Z, B_S)$. We can lift the $r$-complement on any log resolution by Lemma~\ref{csing}.\ref{singl}. \end{proof} In particular, we divide $3$-fold birational contractions into two types: exceptional and regular. By Mori's results all small contractions in the terminal case are formally regular. This time the exceptions are not bounded. For instance, if $(X,S)$ is a simple compound Du Val singularity: $$ x^2+y^2+z^2+w^d=0 $$ with quadratic cone surface $S$ given by $x^2+y^2+z^2=0$. It is (not formally) an exceptional complement. Such singularities are not isomorphic for different $d$ even formally. However, they have many common finite invariants: the m.l.d., the index of complement, the index of $K$, etc.. They are bounded up to isomorphism of a certain degree or order. \refstepcounter{subsec} \label{exbound} \begin{corollary} Under the assumptions of Theorem~\ref{csing}.\ref{singth}, for any $\varepsilon>0$, the exceptional contractions $(X/Z,B)$ and their complements $(X/Z,B^+)$ are bounded with respect to the m.l.d. and discrepancies, when $K+B$ is $\varepsilon$-log terminal: over a neighborhood of $P\in Z$, the set of the m.l.d.'s $a(\eta,B,X)$ for points $\eta$ and the discrepancies $a(E,B^+,X)$, $a(E,B,X)\le \delta$ for any $\delta>0$ is finite. It holds also under (M)'', if the contraction is not divisorial. Otherwise we should assume that the set of $b_i\le 1-\varepsilon$ is finite. \end{corollary} \begin{proof} According to our assumptions, $b_i$ belongs to a finite set. Indeed, if $b_i\le (1-\varepsilon)$. So, it is enough to verify the finiteness for discrepancies in exceptional divisors $E$ of $X$. Indeed, the latter implies that we can consider the m.l.d.'s $>3$. Such does not exist. Note that $a=a(E,B^+,X)=a(E,B^+,Y)$ form a finite set because $K+B^+$ has a finite set of indices $n\in N_2$. The index $N$ of $S$ is bounded$/P$ as well because it is bounded locally on $Y$. So, we assume that $N$ is the universal index. To verify the local case, after a $\mathbb Q$-factorialization, we can suppose that $X$ is $\mathbb Q$-factorial. We assume this below always. Then the boundedness follows from the boundedness of quotient singularities on $S$ by the exceptional property. In particular, for any point $Q\in S$, the local fundamental groups of $S\setminus Q$ are bounded. Along curves in $S$ the index is bounded ($\le 6$) by \cite[Proposition~3.9]{Sh2}. After a covering branching over such curves, $S$ does not pass through codimension $2$ singularities of $Y$. Then we can argue as in the proof of \cite[Corollary~3.7]{Sh2} (cf. \cite[Lemma~1.1.5]{P1}). The index in such singularities will be bounded by orders of cyclic quotients of the fundamental groups. This also implies that any divisor near $S$ has a bounded index. Now we consider discrepancies $a(E,B,X)$. Especially, for $E=S/P$. If it is not exceptional, the finiteness follows from (SM) and the $\varepsilon$-log terminal property. The same holds for the other non-exceptional $E$ on $X$. For exceptional $E=S$, and for any other exceptional $E$, to compute discrepancies, we choose an appropriate strictly log terminal model $g:Y\to X$, on which $S$ is the only exceptional divisor$/X$. Then $B^Y=g^{-1}B+(1-a(S,B,X))S= B_Y^+-a(S,B,B)S-D$ where $D=g^{-1}(B^+-B)$ is effective. Moreover, the multiplicities of $D$ and $B^+-B$ form a finite set. Take rather generic curve $C\subset S/X/P$. Then \begin{eqnarray*} 0=(K_Y+B^Y.C)&=(K_Y+B_Y^+.C)-a(S,B,X)(S.C)-(D.C)\\ &=-a(S,B,X)(S.C)-(D.C) \end{eqnarray*} implies $$ a=a(S,B,X)=-(D.C)/(S.C). $$ Since $a>\varepsilon>0$, and $(D.C)=(D|_S.C)$ belongs to a finite set, $(S.C)$ is bounded from below. On the other hand $(S.C)<0$. So, $(S.C)$ and $a(S,B,X)$ belong to a finite set. This implies the statement for discrepancies $a(E,B,X)=a(E,B^Y,Y)=a(E,B^+,Y)+ a(S,B,X)\mult{E}{S}+\mult{E}{D}$ with centers near $S$ or intersecting $S$. For the other centers, it is enough to verify that the index of $D$ is also bounded there. By \cite[Theorem~3.2]{Sh3}, we need to verify that the set of exceptional divisors with discrepancies $a(E,B_Y,Y)<1+1/l$ is bounded (cf. the proof \cite[Proposition~4.4]{Sh3}). Since $D$ is effective and $K_Y+B^+$ has index $\le l$, then $a(E,B_Y^+,Y)\le 1$. We need to verify that such $E$, with centers not intersecting $S$, are bounded. We will see that this bound has the form $\le A(1/\varepsilon)$. Take a terminal resolution $W/Y$ of the above exceptional divisors for $K_Y+B_Y^+$. It does not change the intersection $(S.C)$ for any curve $C\subset S/Z/P$. (This time $C$ may be not$/X$.) As above we have inequality: $$ 0\ge (K+B.g(C))=(K_Y+B^Y.C)=-a(S,B,X)(S.C)-(D.C), $$ or $(S.C)\ge -(D.C)/a(S,B,X)>-(D.C)/\varepsilon$. So, $(S,C)$ belongs to a finite set even if we assume that $C$ is ample on $C$ and on its other such models of $S$. Then we apply the LMMP to $K_W+B_W^+-S$. More precisely, we make flops for $K_W+B_W^+$ with respect to $-S$, or in extremal rays $R$ with $(S.R)>0$. So, we decrease $(K_W+B_W^+-S.C)$ and increase $(S.C)$ and strictly when the support $|R|$ has divisorial intersection with $S$. So, the number of exceptional divisors on $W/X$ is bounded, because we contract all of them during such a LMMP. \end{proof} We have proved more. \refstepcounter{subsec} \label{exboundf} \begin{corollary} For the exceptional and $\varepsilon$-log terminal $(X/Z,B)$, the fibres $f^{-1}P$ are bounded. In particular, if $X/Z$ is small the number of curves$/P$ is bounded. \end{corollary} \refstepcounter{subsec} \begin{corollary} For each $\varepsilon>0$, there exists a finite set $M(\varepsilon)$ in $(+\infty,\varepsilon]$ such that $(X/Z,B)$ is not exceptional, whenever the m.l.d. of $(X,B)/P\ge \varepsilon$ and is not in $M(\varepsilon)$. \end{corollary} \begin{proof} $M(\varepsilon)$ is the set of the m.l.d.'s for the exceptional complements which are $>\varepsilon$. \end{proof} \refstepcounter{subsec} \begin{corollary} There exists a natural number $n$ such that any small contraction $X/Z$ of a $3$-fold $X$ with terminal singularities has a regular complement, whenever it has a singularity $Q/P$ in which $K$ has the index $\ge n$. A similar bound exists for the number of curves$/P$ (cf. Corollary~\ref{csing}.\ref{exboundf}). \end{corollary} \begin{proof} Take $n=1/A$, where $A=\min M(1)$. Note that any terminal {\it singularity\/} of index $n$ has the m.l.d.$=1/n$. \end{proof} This result is much weaker than Mori's on the good element in $|-K|$, when $X$ is formally $\mathbb Q$-factorial$/Z$, and has at most $1$ curve$/P$. Indeed, as was remarked to the author by Prokhorov, then there exists a good element $D\in |-K|$ according to Mori and Kollar. (It is unknown whether it holds when the number of curves$/P$ is $>1$ or when $X/Z$ is not $\mathbb Q$-factorial.) So, $(X/Z,D)$ and $(X/Z,0)$ have a regular complement by \cite[Theorem~5.12]{Sh2}. In general, the corollary shows that exceptional cases are the most difficult in combinatorics. On the other hand we anticipate a few exceptions (maybe, none) among them in the terminal case. \refstepcounter{subsec} \begin{example} Let $(X/Z,B)$ be a divisorial contraction with a surface $B=E/P$, and let $(K+B)_E=K_E+B_E$ be of type (A$_2^6$) or (I$_2^2$) in~\ref{ecomp}.\ref{emainth}.3. Then $(X/Z,E)$ has a trivial $7$-complement. In this case, the singularity $P$ in $Z$ is (maximal) log canonical of the index $7$, but it is not log terminal. \end{example} \refstepcounter{subsec} \begin{example} Let $(X/Z,B)$ be a divisorial contraction with a surface $B=E=\mathbb P^2/P$, and let $(K+B)_E=K_E+B_E$ be of type (A$_2^1$) in~\ref{ecomp}.\ref{emainth}.3. Moreover suppose that $B_E=b_1 L_1+b_2 L_2 +(2/3)L_3+(1/2)L_4$, where $L_i$ are straight lines in a general position and $b_1,b_2=6/7$. Note that in such a situation coefficients $b_i$ are always standard \cite[Proposition 3.9]{Sh2} and we have a finite choice of them. Then $(X/Z,E)$ is not regular, it has a $42$-complement. Moreover, in this case, $P$ is a purely log terminal singularity, but {\it not terminal\/} or {\it canonical\/}, except for the case when $X$ has only Du Val singularities along curves $L_i$ and $K\equiv 0/Z$. In particular, $P$ has a crepant desingularization. Here we do not discuss an existence of such singularities. Indeed, $E$ has index $42$. So, for a straight line $L$ in $E$, $(E.L)<0$ and $42(E.L)$ is an integer. By the adjunction, $(K+E.L)=(K_E+B_E.L)=-5/42$ and $(E.L)\ge -5/42$, when $P$ is canonical. Moreover, $(E.L)> -5/42$ in such cases, except for the above exception with $K\equiv 0/Z$. For $(E.L)=-5/42$, $X/Z$ is a crepant blow-up with log canonical singularities along $L_i$. Hence in the other cases, $(E.C)=-m/42$ with some integer $m=1,2,3$ or $4$. Then $(K.L)=-(5-m)/42$ and discrepancy $d=d(E,0,Z)=(5-m)/m$. If $P$ is terminal of the index $N\ge 2$, so it should have at least all $N-1$ discrepancies $i/N, 1\le i< N$ \cite[Theorem~3.2]{Sh3}. For instance, if $m=1$ and $X$ has only Du Val singularities, then $d=4$ and $N\ge 21$. On the other hand, making blow-ups over lines $L_i$, it is possible to construct a (minimal log) resolution $Y/X$ with at most $1+1+2+2\times 6=16$ exceptional divisors. Then all other divisors have discrepancies $>1$. (Moreover the divisors in the resolution give only a discrepancy $1/7<1$.) So, it is not a terminal singularity. Similarly we can exclude other cases. Therefore $P$ is canonical or worse. Of course, this approach uses a classification of terminal singularities. But it can be replaced by the following arguments. Let $m=4$, then $d=1/4$ and, for any exceptional divisor $E'/P$, the discrepancy $d(E',0,Z)=d(E',K-(1/4)E,X)= d(E',K,X)+(1/4)\mult{E'}{E}$. For instance, $L_4$ is a simple Du Val singularity and, for $E'/L_4$ on its minimal resolution, $d(E',0,Z)=d(E',K,X)+(1/4)\mult{E'}{E}= 0+(1/4)(1/2)=1/8$. But, if $L_3$ is a simple singularity, i.e., a divisor $E'/L_3$ is unique on a minimal resolution, then $d(E',0,Z)=d(E',K,X)+(1/4)\mult{E'}{E}= -1/3+(1/4)(1/3)=-1/4$. In this case $P$ is not canonical. Otherwise, $L_3$ is a Du Val singularity of type $\mathbb A_2$. The same works for other singularities $L_i$, whenever $d< 1$, that they are Du Val too. Otherwise they have a discrepancy $<0$. It can be checked by induction on the number of divisors on a minimal resolution. But then on a minimal resolution $g:Y\to X$, \begin{eqnarray*} -4/42=(E.L)&=(g^*E.g^{-1}L)= (g^{-1}E.g^{-1}L)+(\mult{L_i}{E}.L)\\ &=I+(1/2)+(2/3)+2\times(6/7)= I+3-5/42, \end{eqnarray*} where $I=(g^{-1}E.g^{-1}L)$ is an integer. This is impossible. Similarly, we can make other cases. \end{example} \refstepcounter{subsec} \begin{p-definition} Let $(X,B)$ be a log terminal pair. Put $D=\sum D_i=\lfloor B\rfloor$. Then we can define a simplicial space $\R{(X,B)}$: \begin{itemize} \item its $l$-simplex is an irreducible component $\Delta_l$ in intersection of $l+1$ distinct irreducible components $D_{i_0},\dots D_{i_l}$: $$ \Delta_l\subseteq D_{i_0}\cap\dots\cap D_{i_l}; $$ \item $\Delta_{l'}$ is a face of $\Delta_l$ if $\Delta_{l'}\supseteq \Delta_l$; and \item the intersection of two simplices $\Delta_l$ and $\Delta_{l'}$ consists of a finite set of simplices $\Delta_{l''}$ such that $$ \Delta_{l''}\supseteq \Delta_l\cup\Delta_{l'}. $$ \end{itemize} The simplices $\Delta_l$ give a triangulation of $\R{(X,B)}$ or a simplicial complex if and only if we have real global normal crossings in the generic points: all the intersections $D_{i_0}\cap\dots\cap D_{i_l}$ are irreducible. The latter can be obtained for an appropriate log terminal resolution $(S'/S,B')$. If $(X,B)$ has a log terminal resolution $(Y/X,B_Y)$, then the {\it topological type of $\R{(Y,B_Y)}$ is independent on such a resolution\/}. So, we denote it by $\R{(X,B)}$. The topology of $\R{(X,B)}$ reflects a complexity of log singularities for $(X,B)$ and in particular of $\LCS(X,B)=\lfloor B\rfloor$ when $(X,B)$ is log terminal. Put $\reg{(X,B)}=\dim \R{(X,B)}$. When $X/Z$ is a contraction, we {\it assume\/} that components $\Delta_l$ are irreducible formally or in the analytic topology on $Z$, i.~e., we consider irreducible branches over a neighborhood of $P\in Z$. \end{p-definition} \begin{proof} According to Hironaka, it is enough to verify that a monoidal transform in $\Delta_l\subset X$ gives a barycentric triangulation of $\Delta_l$ in $\R{(X,B)}$. \end{proof} \refstepcounter{subsec} \label{sabclas} \begin{example} If $(S,B)$ is a surface singularity, then $\R{(S,B)}$ is a graph of $\LCS{(S',B')}$ for a log terminal resolution $(S',B')\to (S,B)$. Moreover, $\R{(S,B)}$ is homeomorphic to a circle $S^1$, to a segment $[0,1]$, to a point, or to an empty set, when $(S,B)$ is log canonical. Additionally, the case $S^1$ is only possible when $B=0$ near the singularity and it is elliptic with a wheel of rational curves for a minimal resolution. Now let $(S/Z,B)$ an $r$-complement. Then \begin{itemize} \item $\reg{(S,B)}=1$ if $(S/Z,B)$ has type $\mathbb A_m^n$ or $\mathbb D_m^n$, \item $\reg{(S,B)}=0$ when $(S/Z,B)$ has type $\mathbb Er_m^n$ with $(2\ge )m+n\ge 1$, and \item $\reg{(S,B)}=-\infty$ when $(S/Z,B)$ has type $\mathbb Er_m^n$ with $m+n= 0$, i.~e., when $K+B$ is Kawamata log terminal. \end{itemize} If $(S,B)=(\mathbb P^2, L)$ where $L=\sum L_i$ with $n$ lines $L_i$ in a generic position, then $\R{(\mathbb P^2,L)}$ is a complete graph with $n$ points. So, it is a manifold with boundary only when $n\le 3$ or $-(K+L)$ is nef. \end{example} We have something similar in dimension $3$. \refstepcounter{subsec} \label{topsm} \begin{p-definition} Let $(X/Z,B)$ be an $n$-fold contraction. Then the space $\R{(X,B)}$ has the following property: \begin{description} \item {\rm (DIM)} $\R{(X,B)}$ is a compact topological space of real dimension $\reg{(X,B)}\le n-1=\dim X-1$. \end{description} Suppose now that \begin{itemize} \item $-(K+B)$ is nef$/Z$, and \item $(X,B)$ is log canonical. \end{itemize} Then locally$/Z$ \begin{description} \item {\rm (CN)} $\R{(X,B)}$ is connected, whenever $-(K+B)$ is big$/Z$, or, for $n\le 3$, consists of two points; moreover, the latter is possible only if $\dim Z\le \dim X-1= 2$ and there exists a (birationally unique) conic bundle structure on $X/Z$ with $2$ reduced and disjoint sections $D_1$ and $D_2$ in $B'$ for a log terminal resolution $(X',B')\to (X,B)$; $\R{(X,B)}=\{D_1,D_2\}$; \item {\rm (MB)} $\R{(X,B)}$ is a manifold with boundary; in addition, it is a manifold when $(X,B)$ is a $1$-complement and $B$ is over a given point $P\in Z$. \end{description} In particular, we associate to each birational contraction $(X/Z,B)$ or a singularity, when $X\to Z$ is an isomorphism, a connected manifold $\R{(X,B)}$ locally over a point $P\in Z$, which will be called a {\it type\/} of $(X/Z,B)$. The {\it regularity\/} $\reg{(X,B)}$ characterizes its topological difficulty. If $(X/Z,B)$ is an $n$-complement, that is not Kawamata log terminal over $P$ as in Theorem~\ref{csing}.\ref{singth}, then $\R{(X,B)}\not=\emptyset$ and $\reg{(X,B)}\ge 0$. If $(X/Z,B)$ is an arbitrary log canonical singularity, we associate with it a topological manifold of the maximal dimension (and maximal for inclusions) for some complements $\R{(X,B^+)}$ and its {\it complete\/} regularity is $\reg{(X,B^+)}$. \end{p-definition} \begin{proof} (DIM) holds in any dimension by the very definition. For $n=3$-folds, (CN) follows from the LMMP and proofs of \cite[Connectedness Lemma~5.7 and Theorem~6.9]{Sh2}. The connectedness, when $-(K+B)$ is big$/Z$, was proved in \cite[Theorem~17.4]{KC}. (MB) is a local question on $\R{(X,B)}$ near each point $\Delta_0=D_{i_0}$. But this is a global question on $Y=D_{i_0}$ that $\R{(Y,B_Y)}$ satisfies (MB). More precisely, a neighborhood of $\Delta_0$ is a cone over $\R{(Y,B_Y)}$. But we may assume (MB) for $\R{(Y,B_Y)}$ by the adjunction and an induction on $\dim X$. \end{proof} \refstepcounter{subsec} \begin{corollary} Under the assumptions of Theorem~\ref{csing}.\ref{singth}, let $(X/Z,B^+)$ be an $n$-complement with the minimal index $n$, then \begin{itemize} \item $\reg{(X,B^+)}=2$ and $\R{(X,B^+)}$ is a real compact surface with boundary only when $n=1$ or $2$; $\R{(X,B^+)}$ is closed only for $n=1$; \item $\reg{(X,B^+)}=1$ and $\R{(X,B^+)}$ is a real curve with boundary only when $n=1,2,3,4$ or $6$; and \item $\reg{(X,B^+)}=0$ and $\R{(X,B^+)}$ is a point only when $n\in N_2$; such complements and contractions are exceptional. \end{itemize} \end{corollary} \begin{proof} Follows from the proof of Theorem~\ref{csing}.\ref{singth}. \end{proof} A topology of log singularities can be quite difficult when $\reg{(X,B^+)}=2$. By a real surface below we mean a connected compact manifold with boundary of dimension $2$. It is {\it closed\/} when the boundary is empty. \refstepcounter{subsec} \begin{example} For any closed real surface $S$, there exists a $3$-fold $1$-complement $(X,B)$ such that $\R{(X,B)}$ is homeomorphic to $S$. First, we take a triangulation $\{\Delta_i\}$ of $S$. Second, we immerse its dual into $\mathbb P^3$ in such a way that each point $\Delta_0$ is presented by a plane $L_i$. The plane is in a generic position. Third, we make a blow-up in such intersections $L_i\cap L_j$ which does not correspond to a segment $\Delta_1=L_i\cap L_j$. Then we get an algebraic surface $B=\sum L_i$ such that $K+B$ is log terminal and $\R{(X,B)}$ is the triangulation. So, $\R{(X,B)}$ is homeomorphic to $S$. Moreover, $K+B$ is numerically trivial on the $1$-dimensional skeleton or on each curve $C_{i,j}=\Delta_1=L_i\cap L_j$, because we have exactly two triple-points on each $\Delta_1$ or each $\Delta_1$ belongs exactly to two simplices of the triangulation. Fourth, we contract all $C_{i,j}$ and something else from $B$ and $X$ to a point which gives the required singularity. Indeed, after the LMMP for $K+B$, we can use a semi-ampleness of $K+B$, when it has a general type. Note that the birational contractions or flips do not touch $C_{i,j}$. This can be verified on each $L_i$ by the adjunction. By the same reason we have no surface contraction on $L_i$ or flips intersecting $B$, but not in $B$. However, terminal singularities and flips are possible outside of $B$ or inside of $B$, which preserves our assumptions on $B$. To secure the big property for $K+B$ we can add similarly $B'$ on which $K+B$ is big. Then $K+B$ has the log Kodaira dimension $3$ when $B'$ has more than two connected components, otherwise $(K+B+B')|_{B'}$ will not be big on some of the components. The same holds for $2$-complements with arbitrary real non-closed surface $S$ with boundary. We need not to contract birationally some $L_i$ and replace them by $(1/2)D$ for generic $D\in |2L_i|$. This can be done by the above combinatorics, when $(K+B+B')_{L_i}$ is big. \end{example} \refstepcounter{subsec} \label{ball} \begin{corollary} Let $P\in (X,B)$ be a log canonical singularity such that \begin{itemize} \item $X$ is $\mathbb Q$-factorial, and even formally or locally in the analytic topology of $X$, when there exists a non-normal curve in $P$ as a center of log canonical discrepancy $0$ for $K+B$; \item $\{B\}\not=0$ in $P$, i.~e., $B$ has a fractional component through $P$; and \item $K+\lfloor B\rfloor$ is purely log terminal near $P$. \end{itemize} Then $\R{(X,B)}$ has type $\mathbb B^r$, where $\mathbb B^r$ is a ball of dimension $r=\reg{(X,B)}$. Moreover, we can drop the conditions when $r\le 0$, or we can drop the first condition when \begin{itemize} \item $B$ has a fractional $\mathbb R$-Cartier component $F$, i.~e., $0<F\le\{B\}$. \end{itemize} \end{corollary} \begin{proof} According to our assumptions, $K+\lfloor B\rfloor$ is purely log terminal in $P$. Then $X$ is the only log minimal model of $(X,\lfloor B\rfloor)$ over $X$ \cite[1.5.7]{Sh2}. Now we take a log terminal resolution $(Y/X,B_Y)$ and consider formally the LMMP$/X$ for $\lfloor B_Y\rfloor$. According to the above, the final model will be $(X,\lfloor B\rfloor)$ with $\reg{(X,\lfloor B\rfloor)}\le 1$ and it has a trivial homotopy type. On the other hand $\R{(X,B)}=\R{(Y,B_Y)}$. So, it is enough to check that contractions and flips preserve the homotopy type. If the centers of the flip or contraction are not in $\LCS{(X,B)}=\lfloor B_Y\rfloor$, then we even have a homeomorphism. If we have a divisorial contraction of a divisor $D_i$ in $\lfloor B_Y\rfloor$, it induces a fibre contraction $D_i\to Z/X$. If $Z$ is a curve$/P$, then according to our conditions this is a curve on another component $D_j$ in $\lfloor B_Y\rfloor$, because any contraction of $Y/X$ is divisorial. Note that each exceptional divisor of $Y/X$ belongs to $\lfloor B_Y\rfloor$, and $\R{(Y,B_Y)}$ is a gluing of a cone with vertex $\Delta_0=D_i$ over $\R{(D_i,B_{D_i})}$ and in the latter. It is homotopy to $\mathbb B^{r-1}$, because $-(K_Y+\lfloor B_Y\rfloor)$ is nef$/X$ on all double curves $\Delta_1=D_i\cap D_j$. The surgery drops this cone. We have a similar picture when $Z=\mathop{\rm pt.}$ or $Z$ is not over $P$. In the latter case $Z$ has at most one curves$/P$ formally. (If non-formally, algebraically, then at most two double curves$/P$, which in addition are connected, by our assumptions.) Finally, let $Y-\to Y^+/X$ be a flip in a curve $C_{i,j}=\Delta_1=D_i\cap D_j/P$. Then according to our assumptions we have a flip on a third surface $D_k$ in $B_Y$ with $(D_k.C_{i,j})=1$ on $Y$. The surgery deletes the segment $\Delta_1=C_{i,j}$ and the interior of the triangle $\Delta_2=D_i\cap D_j \cap D_k$. Formally, it looks like a blow-up in $\Delta_1$ which is the barycentric triangulation in $\Delta_1$ and then we {``}contract" the resolution divisor on a curve on $D_k$. If $r=1$, we can also assume that at the start $\{B\}\equiv -(K_Y+\lfloor B_Y\rfloor)/Z$ is nef $Z$ \cite[Theorem~5.2]{Sh3}. Indeed, we may consider the LMMP for $B_Y+\varepsilon \{B\}$. Since $-(K_Y+\lfloor B_Y\rfloor)$ is big$/Z$, we have a birational contraction $Y\to Z/X$; it contracts all surfaces to a curve $C$ by Theorem~\ref{clasf}.\ref{dipole}. Then any fractional component of $B$ is positive on $C$, which contradicts our assumption, because $Z/X$ is small. If $B$ has the fractional component $F$, we can apply an induction on the number of irreducible curves $C_i/P$ on a formal $\mathbb Q$-factorialization $X'/X$. Indeed, if $X'/X$ blows up $1$ such curve $C_i$, it is irreducible rational, and it has at most to points in which $K+B$ is not log terminal. So we glue at most two balls $\mathbb B^r$ in $\mathbb B^{r-1}$. Note that $F$ passes through $C_i$ on $X'$. Finally, we have no such curves when $X$ is formally $\mathbb Q$-factorial in $P$. This drops the first condition. \end{proof} In general, for $r=2$ we can verify that the formal Weil-Picard number in $P$ is at least $q=h^1(\R{(X,B)},\mathbb R) =2-\chi(\R{(X,B)})$ (the topological genus), where $\chi$ is the topological Euler characteristic. We may consider $q$ as the {\it genus\/} of the singularity $P$. \refstepcounter{subsec} \label{balls} \begin{corollary} Under the assumptions of Theorem~\ref{csing}.\ref{singth}, let $(X/Z,B^+)$ be an $n$-complement with the minimal index $n$, and $P$ be $\mathbb Q$-factorial, formally, when there exists a non-normal curve in $P$ as a center of log canonical discrepancy $0$ for $K+\lfloor B^+\rfloor$, and purely log terminal $K+B$, then $\R{(X,B)}$ has type $\mathbb B^r$ with $r=\reg{(X,B)}$, or $S^2$. The latter is only possible for $n=1$. \end{corollary} If $P$ is an isolated singularity, we may drop the formal condition for an appropriate complement with $n\ge 2$. For $n=1$, we can take a $2$-complement as in the proof below. For $n\ge 3$, we can drop even the $\mathbb Q$-factorial property. \begin{proof} We need to consider only the case with $n=1$. Then we have a reduced component $D$ through $P$ in $B^+$. If we replace $D$ by a generic $(1/2)D'$, where $D'\in |2D|$ is rather generic, we obtain a $2$-complement $B'$, with $\R{(X,B')}$ homeomorphic to $\mathbb B^2$. Then $\R{(X,B^+)}$ could be obtained from this by gluing a cone over $\R{(D,B_D)}$. The letter is $[0,1]$ or $S^1$. That gives respectively $\mathbb B^2$ or $S^2$. \end{proof} In the investigation of the m.l.d.'s for $3$-folds we can assume that the point $P$ in $(X,B)$ is \begin{description} \item{(T2)} $\mathbb Q$-factorial (even formally) and terminal in codimension $1$ and $2$. \end{description} Otherwise after a crepant resolution we can reduce the problem to (T2) or to dimension $2$ or $1$. \refstepcounter{subsec} \begin{corollary} The conjecture on discrepancies {\rm \cite[Cojecture 4.2]{Sh3}} holds for $3$-fold log singularities $(X,B)$ with $b_i \in \Gamma$ under {\rm (M)''} and {\rm (T2)}, when $\reg{(X,B^+)}\le 1$. Moreover, in such a case the only clusters of $A(\Gamma,3)$ are \begin{description} \item{\rm (0)} $0$, when $\reg{(X,B^+)}=0$, {\rm i.~e., in the exceptional cases\/}; \item{\rm (1)} $0$ and $$A(\Gamma',2)$$ when $\reg{(X,B^+)}=1$, where $\Gamma'=\{0,1/2,2/3,3/4,4/5, 5/6\}$, and where we consider only $2$-dimensional singularities $(S,B^+)$ with the boundary multiplicities in $\Gamma'$ and $\reg{(S,B^+)}=0$, {\rm i.~e., the exceptional case.} \end{description} The cluster points are rational, our $A(\Gamma,3)$ is closed when $1\in \Gamma$, and the only cluster of the clusters is $0$. \end{corollary} \begin{proof}[Sketch Proof] Note that $\Gamma$ satisfies (M)'', but may not be the d.c.c.. In particular, $\Gamma$ may not be standard. If $\reg{(X,B)}=0$, then by Corollary~\ref{csing}.\ref{exbound}, for any $\varepsilon>0$, we have a finite subset $$\{a\in A(\Gamma,3)\mid a\ge \varepsilon\}$$ of corresponding $A(\Gamma,3)$. Of course, the corollary was proved for the standard $\Gamma$, but in our case, according to Corollary~\ref{indcomp}.\ref{adjunct} and because $\reg{(X,B^+)}=0$ all $b_i\le (l-1)/l$, where $l=\max\{r\in N_2\}$. So, such $b_i$ is standard due to (M)''. If $\reg{(X,B)}=1$, we can use the arguments of \cite{Sh5} and Theorem~\ref{clasf}.\ref{dipole}. The latter almost reduces our case to the $2$-dimensional situation. Almost means except for the edge components in the chain of the reduced part $S$ of $B^+$ as in the proof of Theorem~\ref{csing}.\ref{singth} (cf. the proof of Corollary~\ref{csing}.\ref{ball}). The clusters can be realized as m.l.d.'s for $(X,B)$ with a reduced part $S=\lfloor B\rfloor$ and the the fractional multiplicities in $\Gamma'$. \end{proof} So, to complete the conjecture on discrepancies for $3$-folds someone needs to consider the singularities with $2$-complements of type $\mathbb B^2$ by Corollary~\ref{csing}.\ref{balls}. This case is related to $1$- or $2$-complements. The former are closed to toric complements, where the conjecture was verified by Borisov \cite{B}. We also see that $\reg{(X,B^+)}$ conjecturally may have interpretations in terms of clusters: the first cluster of $A(\Gamma,n)$ with $r=\reg{(X,B^+)}$ are $A(\Gamma,n-1)$ with $\reg{(X,B^+)}=r-1$, in particular, they are rational when $\Gamma$ is standard. \refstepcounter{subsec} \begin{remark} We anticipate that most of the results in this section hold in any dimension and for any regularity $r=\reg{(X,B)}$ or $\reg{(X,B^+)}$. \end{remark}

9711

alg-geom/9711018

en

https://arxiv.org/abs/alg-geom/9711018

alg-geom/9711018

Elizabeth Gasparim

Elizabeth Gasparim

Chern Classes of Bundles over Rational Surfaces

A mistake in the original was corrected

Consider the blow up $\pi: \widetilde{X} \to X$ of a rational surface $X$ at a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and define $V =(\pi_*\widetilde{V})^{\vee \vee}.$ Friedman and Morgan gave the following bounds for the second Chern classes $j \leq c_2(\widetilde{V}) - c_2(V) \leq j^2.$ We show that these bounds are sharp.

alg-geom

\section{Introduction.} In this paper our basic setting will be the following. $X$ will be a rational surface, $\pi : \widetilde {X} \rightarrow X$ the blow-up of $X$ at point $x \in X$ and $\ell$ the exceptional divisor. $\widetilde {V}$ will be a rank two bundle over the surface $\widetilde X$ satisfying $det\, \widetilde {V} \simeq {\cal O}_ {\widetilde X}$ and $\widetilde{V}|_{\ell} \simeq {\cal O}(j) \oplus {\cal O}(-j), \, j \geq 0.$ Let $V = \pi_* {\widetilde{V}}^{\vee \vee}.$ Friedman and Morgan [2] gave the following estimate for the second Chern classes $j \leq c_2(\widetilde{V}) - c_2(V) \leq j^2.$ We show that these bounds are sharp by giving an algebraic procedure to calculate $c_2(\widetilde{V}) - c_2(V)$ directly from the transition matrix defining $\widetilde{V}$ in a neighborhood of $\ell.$ Since $\widetilde{V}|_{\widetilde{X} - {\ell}} = \pi^* V|_{X - p} $ it is natural to localize our study of $\widetilde{V}$ to a neighborhood of the exceptional divisor. If $x$ is the blown-up point, we choose an open set $U \ni x$ that is biholomorphic to ${\bf C}^2.$ Then $\pi^* (U)$ gives a neighborhood $N_{\ell} \sim \widetilde{{\bf C}^2}.$ We remark that since $\widetilde{X}$ is rational we can only guaranty the existence of a neighborhood of $x$ that is biholomorphic to ${\bf C}^2$ minus a finite number of points. However, all our calculations use holomorphic functions, which always extend over these points since we are in dimension 2. Therefore me may from the start assume without loss of generality that $N(\ell) = \widetilde{{\bf C}^2}.$ We use the following results: \begin{theorem}:([5], Thm. 2.1) Let $E$ be a holomorphic bundle on $ \widetilde{\bf C}^2 $ with $E_{\ell} \simeq {\cal O}(j) \oplus {\cal O}(-j). $ Then $E$ has a transition matrix of the form $\left(\matrix {z^j & p \cr 0 & z^{-j} \cr }\right)$ from $U$ to $V,$ where $p = \sum_{i = 1}^{2j-2} \sum_{l = i-j+1}^{j-1}p_{il}z^lu^i.$ \end{theorem} \begin{corollary}([6], Cor. 4.1) \label{topzer}Every holomorphic rank two vector bundle over $\widetilde{X}$ with vanishing first Chern class is topologically determined by a triple $(V,j,p)$ where $V$ is a rank two holomorphic bundle on $X$ with vanishing first Chern class, $j$ is a non-negative integer, and $p$ is a polynomial. \end{corollary} It follows that for each polynomial $p$ ( in 3 variables $z,$ $z^{-1},$ and $u$ ) there is a canonical construction that assings a bundle $\widetilde{V}$ over $\widetilde{X}$ whose topological type is determined by $p.$ We remark that N. Buchdahl [1] has shown that these bounds are sharp in a more general setting by entirely different methods. Following Friedman and Morgan [2, p. 302], we define the sheaf $Q$ by the exact sequence \begin{equation}\label{seq} 0 \rightarrow \pi_* \widetilde {V} \rightarrow V \rightarrow Q \rightarrow 0. \label{eq- seq} \end{equation} where $Q$ is a sheaf supported only at $x.$ From the exact sequence (\ref{seq}) it follows immediately that $ c_2\pi_*(\widetilde{V}) - c_2(V) = l(Q),$ where $l$ stands for length. An application of Grothendieck-Riemann-Roch (cf [2]) gives that $c_2( \widetilde{V}) - c_2(V) = l(Q) + l(R^1 \pi_* \widetilde{V}).$ Since $Q$ and $R^1\pi_* \widetilde {V}$ are supported at $x$ we can compute their lengths by looking at $\widetilde {V}|_{N_{\ell}}$ and at $\pi$ as the blow-up map $\pi: \widetilde{\bf C}^2 \rightarrow {\bf C}^2.$ The lengths of $Q$ and $R^1\pi_* \widetilde{V}$ can be explicitly computed using the Theorem on Formal Functions of Grothendieck (see[3]). \section{ Calculation of Chern classes.} \subsection{The upper bound occurs for $p =0.$} \begin{guess}\label{j^2}: If the bundle $\widetilde{V}$ splits on a neighborhood of the exceptional divisor, then $c_2(\widetilde{V}) - c_2(V) = j^2.$ \end{guess} \noindent {\bf Proof}: We show that for $\widetilde{V}|_{N(\ell)} = {\cal O}(j) \oplus {\cal O}(-j),$ we have $l(R^1\pi_* \widetilde{V}) = j(j+1)/2$ and $l(Q) = j(j-1)/2,$ which we state as lemmas. It follows that $c_2(\widetilde{V}) - c_2(V) = l(Q) + l(R^1\pi_* \widetilde{V}) = j(j+1)/2 + j(j-1)/2 = j^2.$\hfill\vrule height 3mm width 3mm We now prove the lemmas we just used. \begin{lemma}\label{l(Q)} If the bundle $\widetilde{V}$ splits on a neighborhood of the exceptional divisor, then $l(Q) = j(j+1)/2.$ \end{lemma} \noindent{\bf Proof}: Since the length of $Q$ equals the dimension of $Q_x^{\wedge}$ as a $k(x)$-vector space, we need to study the map $ (\pi_* \widetilde {V}^{\wedge}_x) \rightarrow V^{\wedge}_x$ and compute the dimension of the cokernel as a $k(x)$-vector space. But as $V^{\wedge}_x = (\pi_* \widetilde{V}^{\wedge}_x)^{\vee \vee},$ we need to compute the ${\cal O}^{\wedge}_x$-module structure on $M = (\pi_* \widetilde{V_x})^{\wedge}$ and study the natural map $M \hookrightarrow M^{\vee \vee}$ of ${\cal O}^{\wedge}_x$-modules. By the Formal Functions Theorem $$M \simeq \lim_{\longleftarrow}H^0(\ell_n, \widetilde{V}|{\ell_n})$$ as ${\cal O}^{\wedge}_x ( \simeq {\bf C}[[x,y]]$)- modules, where $\ell_n \simeq N_{\ell} \times {\cal O}_x/m^{n+1}_x$ is the n-th infinitesimal neighborhood of $\ell.$ To do this we will have to isolate the action of $\displaystyle{{\bf C}[[x,y]]} \over{ (x,y) ^{n+1}}$ on $H^0(\ell_n, \widetilde{V}|{\ell_n}).$ We first write the blow-up of ${\bf C}^2$ with two charts $U \sim V \sim {\bf C}^2 $ with $(z,u) \mapsto (z^{-1}, zu)$ in $U \cap V.$ Then the blow-up map $\pi : \widetilde{{\bf C}^2} \rightarrow {\bf C}^2$ is given on the $U$ chart by $ (x,y) = \pi(z,u) = (u,zu).$ We give the natural action of $x$ and $y$ on this space; that is, $x$ acts by multiplication by $u$ and $y$ acts by multiplication by $zu.$ This yields $M = {\bf C}[[x,y]]<\alpha,\beta_0,\beta_1,...,\beta_j>,$ where $$\alpha = \left(\matrix{ 1 \cr 0}\right),\,\, \beta_0 = \left(\matrix{ 0 \cr 1}\right),\,\, \beta_1 = \left(\matrix{ 0 \cr z}\right),\,\, \cdots,\,\, \beta_j = \left(\matrix{ 0 \cr z^j}\right).$$ With relations: $$\left\{\matrix {x\beta_1 - y \beta_0 \cr x \beta_2 - y \beta_1 \cr \vdots \cr x\beta_j - y \beta_{j-1}}\right., \,\, \, \left\{\matrix{x^2\beta_2 - y^2 \beta_0 \cr \vdots \cr x^2\beta_j - y^2 \beta_{j-2}}\right., \,\, \,\cdots \,\,\,, \,\,\, \left\{\matrix{ x^j \beta_j - y^j \beta_0.}\right. $$ All together there are $j(j+1)/2$ relations. Now writing the generators of $M^{\vee}$ and $M^{\vee \vee}$ we see that $coker(M \hookrightarrow M^{\vee \vee})$ is a $j(j+1)/2$ dimensional vector space over ${\bf C} \simeq k(x),$ hence $l(Q) = j(j+1)/2.$\hfill\vrule height 3mm width 3mm \begin{lemma}\label{oplus} If we have a split extension on a neighborhood of the exceptional divisor, that is, when $\widetilde{V}|_{N(\ell)} \simeq {\cal O}(j) \oplus {\cal O}(-j),$ then $l(R^1\pi_* \widetilde{V}) = j(j-1)/2.$ \end{lemma} \noindent {\bf Proof}: Here again we follow the same method using the Theorem on Formal Functions and compute that $$h^1(\ell_n, \widetilde{V}|_{\ell_n}) = \left\{\begin{array}{ll} (j-1)\,\, & for \,\, n = 0 \\ (j-1)+(j-2)+ \cdots + (j-n)\,\, & for \,\, 1\leq n \leq j-1 \\ j(j-1)/2 \,\, & for \,\, n \geq j-1. \end{array} \right.$$ Also $ Ker \left(H^1(\ell_n, \widetilde{V}|_{\ell_n}) \rightarrow H^1(\ell_{n-1},\widetilde{V}|_{\ell_{n-1}})\right)$ has dimension $(j-n)$ for $2 \leq n \leq j-1.$ Computing the inverse limit (cf. Lang [4]) this implies that $$ l(R^1\pi_* \widetilde{V}) = dim\,\, \inftwo{\lim}{\longleftarrow}{n}\,\, H^1(\ell_n, \widetilde{V}|_{\ell_n}) = 1+2+ \cdots + (j-1) = j(j-1)/2.$$ This can also be verified by invoking the short exact sequence on p.388 of Hartshorne [3].\hfill\vrule height 3mm width 3mm \vspace{5 mm} \noindent {\bf Remark}: We can also proof Proposition \ref{j^2} in a simpler way, by explicitly constructing a generic section of $\widetilde{V}$ and counting its zeros. \subsection{The lower bound occurs for $p =u.$} \begin{theorem} If $\widetilde{V}$ is a bundle corresponding to the triple $(V,j,u)$ (according to Cor. 1.2), then $c_2(\widetilde{V})- c_2(V) = j.$ \end{theorem} \noindent {\bf Proof}: We show that $l(R^1\pi_* \widetilde{V}) = j-1$ and $l(Q) = 1,$ which we state as lemmas. It follows that $c_2(\widetilde{V}) - c_2(V) = (j-1) +1 = j.$\hfill\vrule height 3mm width 3mm \begin{lemma} If $\widetilde{V} $ is given by $(V,j,u)$ then $l(Q) = 1.$ \end{lemma} \noindent {\bf Proof}: The bundle $\widetilde{V}$ is given over $N(\ell)$ (according to Theorem 2.1) by the transition matrix $\left(\matrix{ z^j & u \cr 0 & z^{-j}}\right).$ Here calculations are similar to the ones in the proof of Lemma 2.2. We set $M = (\pi_* \widetilde{V_x})^{\wedge}$ and study the natural map $\rho: M \hookrightarrow M^{\vee \vee}$ of ${\cal O}^{\wedge}_x$-modules. The value of $l(Q)$ is the dimension of cokernel of $\rho.$ The ${\cal O}_x$ - module structure of $M$ is completely determined by the structure of the sections of the bundle $V'.$ Writing sections of $V'$ in the form $\left( \matrix{ a \cr b}\right) = \left( \matrix{ \sum a_{ik}z^ku^i \cr \sum b_{ik}z^k u^i }\right)$ it is simple to see which restrictions are imposed in the coefficients $a_{ik}$ and $b_{ik}.$ Some terms of the general form for the sections are $$\left( \matrix{ a \cr b}\right) = b_{00}\left( \matrix{ 0 \cr 1}\right) + b_{01}\left( \matrix{ 0 \cr z}\right) +b_{0j}\left( \matrix{ -u \cr z^j }\right) + \cdots + a_{j0}\left( \matrix{ u^j \cr 0 }\right)+ \cdots .$$ In fact, one verifies that these terms are enough to generate all the sections. It then follows that $M = <\beta_{00}, \beta_{01},\beta_{0j},\alpha{jo}>/R$ where $\beta_{00} = \left( \matrix{ 0 \cr 1}\right), \beta_{01} = \left( \matrix{ 0 \cr z}\right), \beta_{0j} = \left( \matrix{ -u \cr z^j }\right), \alpha_{j0} = \left( \matrix{ u^j \cr 0 }\right)$ and $R$ is the set of relations $$\left\{ \begin{array} {l} x \, \beta_{01} - y \, \beta_{00} = 0 \cr \alpha_{j0} + x^{j-1} \beta_{0j} - y^{j-1}\beta_{01} = 0\cr \end{array}\right..$$ Using the second relation, one eliminates $\alpha_{j0}$ from the set of generators and gets a simpler presentation $M \simeq <\beta_{00}, \beta_{01},\beta_{0j}>/R'$ where $R'$ now has the single relation $x \, \beta_{01} - y \, \beta_{00} = 0.$ It is now a matter of simple algebra to find that $M^{\vee} = < a, b > $ is free on two generators, where $a = \left\{\begin{array} {l} \beta_{00} \mapsto x \cr \beta_{01} \mapsto y \cr \beta_{0j} \mapsto 0 \cr \end{array}\right.$ and $b = \left\{\begin{array} {l} \beta_{00} \mapsto 0 \cr \beta_{01} \mapsto 0 \cr \beta_{0j} \mapsto 1 \cr \end{array}\right..$ Then naturally $M^{\vee\vee} = < a^* , b^*>$ is generated by the dual basis, namely $a^* = \left\{\begin{array} {l} a \mapsto 1 \cr b \mapsto 0 \cr \end{array}\right.$ and $b^* = \left\{\begin{array} {l} a \mapsto 0 \cr b \mapsto 1 \cr \end{array}\right..$ The map $\rho$ is given by evaluation and we have $im \,\rho = < x\, a^*, y \, a^*, b^*>$ and therefore the cokernel is $coker \, \rho = < \overline{a^*}>$ and $l(Q) = dim \, coker \, \rho = 1.$ \hfill\vrule height 3mm width 3mm \begin{lemma} If $\widetilde{V} $ is given by $(V,j,u)$ then $l(R^1\pi_*\widetilde{V}) = j-1.$ \end{lemma} \noindent {\bf Proof}: We claim that $H^1(\ell_n, \widetilde{V}|_{\ell_n}) $ is generated by the 1-cocycles $\left(\matrix{z^k \cr 0}\right)$ for $ -j \leq k \leq -1$ and that the maps $H^1(\ell_n, \widetilde{V}|_{\ell_n}) \rightarrow H^1(\ell_{n-1}, \widetilde{V}|_{\ell_{n-1}}) $ are the identity. Hence $l(R^1\pi_*\widetilde{V}) = dim\,\, \inftwo{\lim}{\longleftarrow}{n}\,\, H^1(\ell_n, \widetilde{V}|_{\ell_n}) = j-1.$ In fact, if $T$ is the transition matrix for $\widetilde{V}|_{N(\ell)}$ then the equality $$B = \sum_{i = 0}^\infty\sum_{k = -\infty}^\infty\left(\matrix{ 0 \cr b_{ik} z^ku^i}\right) = \sum_{i = 0}^\infty\sum_{k =0}^\infty \left(\matrix{ 0 \cr b_{ik}z^ku^i}\right) + T^{-1} \sum_{i = 0}^\infty\sum_{k = -\infty}^{-1} \left(\matrix{ b_{ik}z^ku^{i+1} \cr b_{ik}z^{k-j}u^i } \right)$$ shows that $B$ is a coboundary, since the first term on the r.h.s. is holomorphic in $U$ and the last term of the r.h.s. is holomorphic in $V.$ As a consequence every 1-cocycle has a representative of the form $\alpha = \sum_{i = 0}^\infty\sum_{k = -\infty}^\infty \left(\matrix{ a_{ik}z^ku^i \cr 0 }\right).$ Analogously $ A = \sum_{i = 0}^\infty\sum_{k =0}^\infty \left(\matrix{ a_{ik} z^ku^i \cr 0 }\right) + T^{-1} \sum_{i = 0}^\infty\sum_{k = -\infty}^{-1} \left(\matrix{ a_{ik}z^ku^i \cr 0 } \right)$ is a coboundary. Therefore the only terms that give nonzero cohomology classes in $\alpha$ are the ones with indices $k$ for $-j \leq k \leq -1$ and the claim follows. \hfill\vrule height 3mm width 3mm Acknowledgements: I would like to thank N. Buchdahl and E. Ballico for pointing out a mistake in the original. Special thanks go to R. Hartshorne for useful comments.

9711

alg-geom/9711012

en

https://arxiv.org/abs/alg-geom/9711012

alg-geom/9711012

Goettsche Lothar

Lothar Goettsche

A conjectural generating function for numbers of curves on surfaces

amslatex 13 pages

10.1007/s002200050434

I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta\le 6$ and Kleiman-Piene for the case $\delta\le 8$. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso-Harris for the Severi degrees in $\P_2$. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.

alg-geom

P(n){P(n)} \def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}} \defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}} \def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}} \def\Inc#1{Z_{#1}} \def\inc#1{\zeta_{#1}} \def\boh#1{pt_{#1} \def\Boh#1{Pt_{#1}} \def\mah#1{al_{#1} \def\Mah#1{Al_{#1}} \def\pr#1{W_{#1}} \def\alpha{\alpha} \def\Gamma{\Gamma} \def\bar \Gamma{\bar \Gamma} \def\gamma{\gamma} \def\bar \eta{\bar \eta} \def{\cal Z}{{\cal Z}} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\cal W}{{\cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def{\text{\rm {Sym}}}{{\text{\rm {Sym}}}} \def\stil#1{\tilde S^#1} \def\pi{\pi} \defg{g} \def\phi{\varphi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \begin{document} \title[A conjectural generating function for numbers of curves] {A conjectural generating function for numbers of curves on surfaces} \keywords{Severi degrees, Gromov-Witten invariants, nodal curves, modular forms} \author{Lothar G\"ottsche} \address{International Center for Theoretical Physics, Strada Costiera 11, P.O. Box 586, 34100 Trieste, Italy } \email{gottsche@@ictp.trieste.it} \begin{abstract} I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher \cite{V2} for the case $\delta\le 6$ and Kleiman-Piene \cite{K-P} for the case $\delta\le 8$. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso-Harris for the Severi degrees in ${\Bbb P}_2$. We verify the conjecture for genus $2$ curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points. \end{abstract} \maketitle \section{Introduction} Let $L$ be a line bundle on a projective algebraic surface $S$. In the case $\delta\le 6$ Vainsencher \cite{V2} proved formulas for the numbers $t^S_\delta(L)$ of $\delta$-nodal curves in a general $\delta$-dimensional sub-linear system of $|L|$. By a refining Vainsechers approach Kleiman-Piene \cite{K-P} extended the results to $\delta\le 8$. The formulas hold under the assumption that $L$ is a sufficiently high power of a very ample line bundle. In this paper we want to give a conjectural generating function for the numbers $t^S_\delta(L)$. We will have only partial success: We are able to express the $t^S_\delta(L)$ in terms of five universal generating functions in one variable $q$. Three of these are Fourier developments of (quasi-)modular forms, the other two we have not been able to identify: the formulas of \cite{C-H} for the Severi degrees on ${\Bbb P}_2$ yield an algorithm for computing their coefficients and I computed them up to degree 28. As the functions are universal, one would hope that there exists a nice closed expression for them. If the canonical divisor of the surface $S$ is numerically trivial, only the quasimodular forms appear in the generating function. Thus we obtain (conjecturally) the numbers of curves of arbitrary genus on a K3 surface and on an abelian surface as the Fourier coefficients of quasimodular forms. The formulas generalize the calculation of \cite{Y-Z} for the numbers of rational curves on K3 surfaces. The fact that for K3 surfaces and abelian surfaces the numbers can be expressed solely in terms of quasimodular forms might be related to physical dualities. The numbers $t^S_\delta(L)$ will on a general surface (including also ${\Bbb P}_2$) count all curves with the prescribed numbers of nodes, including the reducible ones. It seems however that on an abelian surface one is actually counting irreducible curves, and in the case of K3 surfaces one can restrict attention to the case that $|L|$ contains only irreducible reduced curves. Both in the case of abelian surfaces and K3 surfaces I do then expect the generating function to count the curves of given genus in sub-linear systems of $|L|$, even if $L$ is only assumed to be very ample and not a high multiple of an ample line bundle. The curves can then have worse singularities than nodes and a curve $C$ should be counted with a multiplicity determined by the singularities of $C$, as in \cite{B}, \cite{F-G-vS}. In particular nodal (or more generally immersed) curves should count with multiplicity $1$. In the case of K3 surfaces and primitive line bundle $L$ the numbers of these curves have in the meantime been computed in \cite{Br-Le}. The coefficients of the unknown power series are by the recursion of \cite{C-H} the solutions to a highly overdetermined system of linear equations, the same is true for a similar recursion obtained by Vakil \cite{Va} for rational ruled surfaces and the results of \cite{Ch} on ${\Bbb P}_2$ and ${\Bbb P}_1\times {\Bbb P}_1$. This gives an additional check of the conjecture. Finally we compute the numbers of genus $2$ curves on an abelian surface. I thank B. Fantechi for many useful discussions without which this paper could not have been written and P. Aluffi for pointing out \cite{V2} to me. I thank Don Zagier for useful comments and discussions which improved the formulation of conjecture \ref{mainconj}, R. Vakil for giving me a preliminary version of \cite{Va} and S. Kleiman, R. Piene for useful comments. This paper was started during my stay at the Mittag Leffler Institute. \section{Statement of the conjecture} Let $S$ be a projective algebraic surface and $L$ a line bundle on $S$. In this paper by a curve on $S$ we mean an effective reduced divisor on $S$. A nodal curve on $S$ is a reduced (not necessarily irreducible) divisor on $S$, which has only nodes as singularities. We denote by $K_S$ the canonical bundle and by $c_2(S)$ the degree of the second Chern class. For two line bundles $L$ and $M$ let $LM$ denote the degree of $c_1(L)\cdot c_1(M)\in H^4(S,{\Bbb Z})$. In \cite{V2} (for $\delta\le 6$) and \cite{K-P} (for $\delta\le 8$) formulas for the number $a^S_{\delta}(L)$ of $\delta$-nodal curves in a general $\delta$-dimensional linear sub-system $V$ of $|L|$ were proved. Here general means that $V$ lies in an open subset of the Grassmannian of $\delta$-dimensional subspaces of $|L|$. The number is expressed as a polynomial in $c_2(S)$, $K_S^2$, $LK_S$ and $L^2$ of degree $\delta$. The formulas are valid if $L$ is a sufficiently high multiple of an ample line bundle. In other words, for such $L$, the locally closed subset $W^S_\delta(L)$ (with the reduced structure) of elements in $|L|$ defining $\delta$-nodal curves has codimension $\delta$, and its degree is given by a polynomial as above. It is clear and was noted before \cite{K-P} that there should be a formula for all $\delta$: \begin{conj} \label{aconj} For all $\delta\in {\Bbb Z}$ there exist universal polynomials $T_\delta(x,y,z,t)$ of degree $\delta$ ($T_\delta=0 $ if $\delta<0$) with the following property. Given $\delta$ and a pair of a surface $S$ and a very ample line bundle $L_0$ there exists an $m_0>0$ such that for all $m\ge m_0$ and for all very ample line bundles $M$ the line bundle $L:=L_0^{\otimes m}\otimes M$ satisfies $$ a^S_{\delta}(L)=T_\delta(L^2,LK_S,K_S^2,c_2(S) ). $$ \end{conj} \begin{rem} Note that the statement is slightly stronger than that of \cite{V2} in the case $\delta\le 6$. We expect $L$ does not have to be a high power of a very ample line bundle but that it suffices that $L$ is sufficiently ample. In fact the results of the final two sections below suggest that there might exist a universal constant $C >0$ (independent of $S$ and $L$), such that, if $L$ is $C\delta$-very ample (see section 5), then the conjecture holds for up to $\delta$-nodes. \end{rem} Assuming conjecture \ref{aconj} we will in future just write $$ T^S_\delta(x,y):=T_\delta(x,y,K_S^2,c_2(S) ),\quad t^S_\delta(L):=T^S_\delta(L^2,LK_S)), \quad T(S,L):=\sum_{\delta\ge 0} t^S_\delta(L) x^\delta.$$ The aim of this note is to give a conjectural formula for the generating function $T(S,L)$, and to give some evidence for it. We start by noting that conjecture 2.1 imposes rather strong restrictions on the structure of $T(S,L)$. The point is that the conjecture applies to all surfaces, including those with several connected components. In this case we will write $|L|$ for ${\Bbb P}(H^0(L))$. By definition $W^S_\delta(L)$ includes only $f\in |L|$ which do not vanish identically on a component of $S$. \begin{prop}\label{constr} Assume conjecture \ref{aconj}. Then there exist universal power series $A_1$, $A_2$, $A_3$, $A_4\in {\Bbb Q}[[x]]$ such that $$T(S,L)=\exp(L^2 A_1+LK_S A_2+ K_S^2 A_3 +c_2(S) A_4).$$ \end{prop} \begin{pf} Fix $\delta_0\in {\Bbb Z}_{>0}$. Assume that $S=S_1\sqcup S_2$ and that $L_1:=L|_{S_1}$ and $L_2:=L|_{S_2}$ are both sufficiently ample so that the $W^{S_i}_\delta(L_i)$ have codimension $\delta$ and degree $t^{S_i}_\delta(L_i)$ in $|L_i|$ for $i=1,2$ and $\delta<\delta_0$. Fix $\delta<\delta_0$. The application $(f+g)\mapsto (f,g)$ defines a surjective morphism $p:U\to |L_1|\times |L_2|$, defined on the open subset $U\subset |L|$ where neither $f$ not $g$ vanish identically. The fibres of $p$ are lines in $|L|$. Obviously $$ W^S_\delta(L)=p^{-1}\Big(\coprod_{\delta_1+\delta_2=\delta} W^{S_1}_{\delta_1}(L_1)\times W^{S_2}_{\delta_2}(L_2)\Big). $$ In particular $W^S_\delta(L)$ has codimension $\delta$ in $|L|$ and modulo the ideal generated by $x^{\delta_0}$ we have $T(S,L)\equiv T(S_1,L_1) T(S_2,L_2).$ Now choose $n\in {\Bbb Z}_{>0}$ such that conjecture \ref{aconj} holds for $Z_1:=({\Bbb P}_2,{\cal O}(n))$, $Z_2:=({\Bbb P}_2,{\cal O}(2n))$, $Z_3:=({\Bbb P}_2,{\cal O}(3n))$ and $Z_3:=({\Bbb P}_1\times {\Bbb P}_1,{\cal O}(n,n))$ for all $\delta\le \delta_0$. Let $S=S(a_1,a_2,a_3,a_4)$ be the disjoint union of $a_i$ copies of each of the $Z_i$ with $a_i\in Z_{\ge 0}$. Then by the above argument $T(S,L)=\prod_i T(Z_i)^{a_i}$ which can be written in the form $\exp(L^2 A_1+LK_S A_2+ K_S^2 A_3 +c_2(S) A_4),$ for universal power series $A_1$, $A_2$, $A_3$, $A_4$. On the other hand we note that the $4$-tuple $(L^2,LK_S,K_S^2,c_2(S))$ takes on the $Z_i$ the linearly independent values $(n^2,-3n,9,3)$, $(4n^2,-6n,9,3)$, $(9n^2,-9n,9,3)$, $(2n^2,-4n,8,4)$. Thus the image of the $S(a_1,a_2,a_3,a_4)$ is Zariski-dense in ${\Bbb Q}^4$, and the result follows. \end{pf} We recall some facts about quasimodular forms from \cite{K-Z}. We denote by ${\cal H}:=\bigl\{\tau\in {\Bbb C}\bigm|\Im(\tau)>0\bigr\}$ the complex upper half plane, and for $\tau \in {\cal H}$ we write $q:=e^{2\pi i\tau}$. A modular form of weight $k$ for $SL(2,{\Bbb Z})$ is a holomorphic function $f$ on ${\cal H}$ satisfying $$ f\Big(\frac{a\tau + b}{c\tau + d}\Big)=(c\tau+d)^k f(\tau), \quad \tau\in {\cal H},\ \quad\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in SL(2,{\Bbb Z}) $$ and having a Fourier series $f(\tau)=\sum_{n=0}^\infty a_n q^n.$ Writing $\sigma_k(n):=\sum_{d|n}d^{k}$, the Eisenstein series $$ G_k(\tau)=-\frac{B_k}{2k}+\sum_{n>0}\sigma_{k-1}(n)q^n,\quad k\ge 2, \quad B_k=\text{$k$th Bernoulli number} $$ are for even $k>2$ modular forms of weight $k/2$, while $G_2(\tau)$ is only a quasimodular form. Another important modular form is the discriminant $$ \Delta(\tau)=q\prod_{k>0} (1-q^k)^{24}=\eta(\tau)^{24} $$ where $\eta(\tau)$ is the Dedekind $\eta$ function. For the precise definition of quasimodular forms see \cite{K-Z}. They are essentially the holomorphic parts of almost holomorphic modular forms. The ring of modular forms for $SL(2,{\Bbb Z})$ is just ${\Bbb Q}[G_4,G_6]$, while the ring of quasimodular forms is ${\Bbb Q}[G_2,G_4,G_6]$. We denote by $D$ the differential operator $D:=\frac{1}{2\pi i}\frac{d}{ d\tau}=q\frac{d}{d\,q}$. Unlike the ring of modular forms the ring of quasimodular forms is closed under differentiation, i.e. for a quasimodular form $f$ of weight $k$ the derivative $Df$ is a quasimodular form of weight $k+2$. In fact every quasimodular form has a unique representation as a sum of derivatives of modular forms and of $G_2$ (see \cite{K-Z}). \begin{conj}\label{mainconj} There exist universal power series $B_1$, $B_2$ in $q$ such that $$ \sum_{\delta\in {\Bbb Z}}t^S_\delta(L) (DG_2(\tau))^{\delta}= \frac{(DG_2(\tau)/q)^{\chi(L)}\,B_1(q)^{K_S^2}\, B_2(q)^{LK_S}} {(\Delta(\tau)\,D^2G_2(\tau)/q^2)^{\chi({\cal O}_S)/2}}. $$ \end{conj} \begin{rem} \noindent (1) Using the fact that $DG2(\tau)/q$, $B_1(q)$, $B_2(q)$, $\Delta(\tau)D^2G_2(\tau)/q^2$ are power series in $q$ starting with $1$, and by the standard formulas $\chi({\cal O}_S)=(K_S^2+c_2(S))/12$, $\chi(L)=(L^2-LK_S)/2+\chi({\cal O}_S)$ one sees that conjecture \ref{mainconj} expresses the $t^S_\delta(L)$ as polynomials of degree $\delta$ in $L^2,\, K_SL,\, K_S^2,\, c_2(S)$. \noindent (2) I have checked that conjecture \ref{mainconj} reproduces the formulas of Vainsencher and Kleiman-Piene for $\delta\le 8$. This determines $B_1(q)$ and $B_2(q)$ up to degree $q^8$. In remark \ref{Srem} below we use the formulas of \cite{C-H} for the Severi degrees in ${\Bbb P}_2$ to determine the coefficients of $B_1(q)$ and $B_2(q)$ up to degree 28 (they are given here up to degree 20). \begin{align*} B_1&(q)\, \equiv\, 1-q-5\,{q}^{2}+39\,{q}^{3}-345\,{q}^{4}+2961\,{q}^{5}-24866\,{q}^{6}+ 207759\,{q}^{7}-1737670\,{q}^{8}\\ &+14584625\,{q}^{9}-122937305\,{q}^{10} +1040906771\,{q}^{11}-8852158628\,{q}^{12}+75598131215\,{q}^{13}\\ &- 648168748072\,{q}^{14}+5577807139921\,{q}^{15}-48163964723088\,{q}^{16 }+417210529188188\,{q}^{17}\\ &-3624610235789053\,{q}^{18}+ 31575290280786530\,{q}^{19}-275758194822813754\,{q}^{20}+O ({q}^{ 21} )\\ B_2&(q)\, \equiv \, 1+5\,q+2\,{q}^{2}+35\,{q}^{3}-140\,{q}^{4}+986\,{q}^{5}-6643\,{q}^{6}+ 48248\,{q}^{7}-362700\,{q}^{8}\\ &+2802510\,{q}^{9}-22098991\,{q}^{10}+ 177116726\,{q}^{11}-1438544962\,{q}^{12}+11814206036\,{q}^{13}\\ &- 97940651274\,{q}^{14}+818498739637\,{q}^{15}-6888195294592\,{q}^{16}+ 58324130994782\,{q}^{17}\\ &-496519067059432\,{q}^{18}+4247266246317414\,{ q}^{19}-36488059346439524\,{q}^{20}+O(q^{21}). \end{align*} \end{rem} \begin{rem} We give a reformulation of the conjecture. We define, for all $l,m,r\in {\Bbb Z}$ \begin{align*} n^S_r(l,m)&:=T^S_{l+\chi({\cal O}_S)-1-r}(2l+m,m)\\ m^S_g(l,m)&:=n_{g-m-2+\chi({\cal O}_S)}(l,m) \end{align*} If $L$ is sufficiently ample with respect to $\delta=\chi(L)-1-r$ and $S$ (and thus in particular $\chi(L)=H^0(S,L)$ and $r \ge 0$), then $n^S_{r}((L^2-LK_S)/2,LK_S)$ counts the $\delta$-nodal curves in a general $r$-codimensional sub-linear system of $|L|$. Then $$ \sum_{l\in {\Bbb Z}} n_r^S(l,m)q^{l} =B_1(q)^{K_S^2}B_2(q)^{m}\big(DG_2(\tau)\big)^r \frac{D^2 G_2(\tau)}{(\Delta(\tau)D^2G_2(\tau))^{\chi({\cal O}_S)/2}}. $$ An irreducible $\delta$-nodal curve $C$ on $S$ has geometric genus $g(C)=(L^2+LK_S)/2+1-\delta$, we take the same definition also if $C$ is reducible. For $L$ sufficiently ample $m^S_g((L^2-LK_S)/2,LK_S)$ counts the nodal curves $C$ with $g(C)=g$ in a general $g-LK_S +\chi({\cal O}_S)-2$-codimensional sub-linear system of $|L|$. \end{rem} \begin{pf} If $f(q)$ and $g(q)$ are power series in $q$ and $g(q)$ starts with $q$, then we can develop $f(q)$ as a power series in $g(q)$ and $$ \text{Coeff}_{g(q)^k} f(q)=\text{Res}_{g(q)=0}\frac{f(q)dg(q)}{g(q)^{k+1}} =\text{Coeff}_{q^{0}}\frac{f(q)Dg(q)}{g(q)^{k+1}}.$$ We apply this with $g(q)=DG_2(\tau)$. \end{pf} \section{Counting curves on K3 surfaces and abelian surfaces} Let now $S$ be a surface with numerically trivial canonical divisor. We denote $n^S_r(l):=n^S_r(l,0)$, i.e. for $L$ sufficiently ample $n^S_r(L^2/2)$ is the number of $\chi(L)-r-1$-nodal curves in an $r$-codimensional sub-linear system of $L$. $n^S_{r}(l)$ can be expressed in terms of quasimodular forms. For $S$ a K3 surface, $A$ an abelian surface and $F$ an Enriques or bielliptic surface we get \begin{align} \label{fK3} \sum_{l\in{\Bbb Z}} n^S_{r}(l)q^l &= \big(DG_2(\tau)\big)^r/\Delta(\tau)\\ \label{fA} \sum_{l\in {\Bbb Z}} n^A_{r}(l)q^l&= \big(DG_2(\tau)\big)^r D^2G_2(\tau),\\ \sum_{l\in {\Bbb Z},r\ge 0} n^A_{r}(l)q^l\frac{z^r}{r!} &=\frac{1}{z}D\big(\exp\big(D G_2(\tau) z\big)\big)\nonumber \\ \sum_{l\in {\Bbb Z},r\ge 0} n^F_{r}(l)q^l &=\big(DG_2(\tau)\big)^r \big( (D^2G_2(\tau))/\Delta(\tau)\big)^{1/2} \end{align} Note that $m_g^S(l)=n_g^S(l), \ m_g^F(l)=n_{g-1}^F(l), \ m_g^A(l)=n_{g-2}^A(l).$ \begin{rem}\label{exprem} In the case of an abelian surface or a K3 surface we expect that the numbers $n^A_{r}(l)$ and $n^S_{r}(l)$ have a more interesting geometric significance. \noindent(1) Let $(S,L)$ be a polarized K3 surface with $Pic(S)={\Bbb Z} L$. Then the linear system $|L|$ contains only irreducible curves. The number $n^S_r(L^2/2)$ is a count of curves $C\in |L|$ of geometric genus $r$ passing through $r$ general points on $S$. In this case the numbers of rational curves have been calculated in \cite{Y-Z} and \cite{B} and (\ref{fK3}) is a generalization to arbitrary genus. The rational curves that are counted are not necessarily nodal. In the count a rational curve $C$ is assigned the Euler number $e(\bar JC)$ of its compactified Jacobian (which is $1$ if $C$ is immersed) as multiplicity. Let $\overline M_{g,n}(S,\beta)$ be the moduli space of $n$-pointed genus $g$ stable maps of homology class $\beta$ (see \cite{K-M}). It comes equipped with an evaluation map $\mu$ to $S^n$. In \cite{F-G-vS} it is shown that for a rational curve $C$ on a K3 surface $e(\bar JC)$ is just the multiplicity of $\overline M_{0,0}(S,[L])$ at the point defined by the normalization of $C$. Here $[L]$ denotes the homology class Poincar\'e dual to $c_1(L)$. In other words $n^S_{0}(L^2/2)$ is just the length of the $0$-dimensional scheme $\overline M_{0,0}(S,[L])$. I expect that for curves of arbitrary genus the corresponding result should hold: If $S$ and $L$ are general and $x$ is a general point in $S^r$ then the fiber $\mu^{-1}(x)\subset \overline M_{r,r}(S,[L])$ should be a finite scheme and $n^S_{r}(L^2/2)$ should just be its length. More generally $n_r^S(L^2/2)$ should be a generalized Gromov-Witten invariant as defined and studied in \cite{Br-Le} in the symplectic setting and in \cite{Be-F2} in the algebraic geometric setting. In the meantime this invariant has been computed in \cite{Br-Le} for curves of arbitrary genus on K3 surfaces confirming the conjecture. \noindent(2) Let $A$ be an abelian surface with a very ample line bundle $L$. We claim that in general all the curves counted in $n^A_{r}(L^2/2)$ will be irreducible and reduced: The set of $\delta$-nodal curves in $|L|$ has expected dimension $\chi(L)-\delta-1=L^2/2-\delta-1$. On the other hand the set of reducible $\delta$-nodal curves $C_1+\ldots +C_n\in |L|$ with $C_i\in |L_i|$ has expected dimension $$ \sum_{i=1}^n(L_i^2/2 -1 ) -\delta+\sum_{1\le i\ne j\le n}L_i L_j= L^2/2-\delta-n. $$ Therefore in general $n^A_r(L^2/2)$ should count the irreducible curves $C\in |L|$ of geometric genus $r+2$ passing through $r$ general points. Again we expect that this result holds in a modified form also if $L$ is not required to be sufficiently ample, and if not all the curves in $|L|$ are immersed. The moduli space $\overline M_{g,n}(A,[L])$ is naturally fibered over $Pic^L(A)$. The fibers are the spaces $\overline M_{g,n}(A,|M|)$ of stable maps $\varphi: W\to A$ with $\varphi_*(W)$ a divisor in $|M|$, where $c_1(M)=c_1(L)$. Again for $A$ and $L$ general and $x$ a general point in $A^r$ the number $n^A_{r}(L^2/2)$ should be the length of the fiber $\mu^{-1}(x)\subset \overline M_{r+2,r}(A,|L|)$. \end{rem} We want to show conjecture \ref{mainconj} and the expectations of remark \ref{exprem} for abelian surfaces in a special case. \begin{thm} Let $A$ be an abelian surface with an ample line bundle $L$ such that $c_1(L)$ is a polarization of type $(1,n)$. Assume that $A$ does not contain elliptic curves. Write again $\sigma_1(n)=\sum_{d|n} d$. Then the number of genus $2$ curves in $|L|$ is $n^2\sigma_1(n)$. Moreover all these curves are irreducible and immersed, and the moduli space $\overline M_{2,0}(A,|L|)$ consists of $n^2\sigma_1(n)$ points corresponding to the their normalizations. \end{thm} \begin{pf} We will denote by $[C]$ the homology class of a curve $C$ and by $[L]$ the Poincar\'e dual of $c_1(L)$. For a divisor $D$ we write $c_1(D)$ for $c_1({\cal O}(D))$. Let $\varphi:C\to A$ be a morphism from a connected nodal curve of arithmetic genus $2$ to $A$, with $\varphi_*([C])=[L]$. Put $D:=\varphi(C)$. As $A$ does not contain curves of genus $0$ or $1$, $C$ must be irreducible and smooth, and $\varphi$ must be generically injective. In particular $[D]=[L]$. Let $J(C)$ be the Jacobian of $C$. We freely use standard results about Jacobians of curves, see (\cite{L-B} chap. 11) for reference. For each $c\in C$ the Abel-Jacobi map $\alpha_c:C\to J(C)$ is an embedding with $\alpha_c(c)=0$. We write $C_c:=\alpha_c(C)$ and $\theta_C:=c_1(\alpha_c(C))$. By the Torelli theorem the isomorphism class of $C$ is determined uniquely by the pair $(J(C),\theta_C)$. For all $a\in A$ we denote by $t_a$ the translation by $a$. By the universal property of the Jacobian there is a unique isogeny $\widetilde \varphi:J(C)\to A$ such that $\varphi=t_{\varphi(c)}\circ \widetilde\varphi \circ \alpha_c$ for all $c\in C$. As $\widetilde \varphi$ is \'etale $\varphi$ is an immersion and $\varphi:C\to D$ is the normalization map. We also see that $\widetilde\varphi^*(c_1(L))=n\theta_C$. On the other hand, let $(B,\gamma)$ be a principally polarized abelian surface and $\psi:B\to A$ an isogeny with $\psi^*(c_1(L))=n\gamma$. By the criterion of Matsusaka-Ran and the assumption that $A$ does not contain elliptic curves we obtain that $(B,\gamma)=(J(C),\theta_C)$ for $C$ a smooth curve of genus $2$ and $\psi=\widetilde \varphi$ for a morphism $\varphi:C\to A$ with $\varphi_*([C])=[L]$. $\widetilde \varphi$ depends only on $\varphi$ up to composition with a translation in $A$ and $\varphi=t_{\varphi(c)}\circ\widetilde\varphi\circ\alpha_c$ is determined by $\widetilde\varphi$ up to translation in $A$. By the universal property of $J(C)$ applied to the embedding $\alpha_c$, an automorphism $\psi$ of $C$ induces an automorphism $\widehat \psi$ of $J(C)$. If $\epsilon$ is an automorphism of $J(C)$ with $\epsilon^*(\theta_C)=\theta_C$, then it is $\widehat\psi$ for some automorphism $\psi$ of $C$. ($H^0(J(C),{\cal O}(C_c))=1$, and $a\mapsto t_a^*({\cal O}(C_c))$ defines an isomorphism $J(C)\to Pic^{\theta_C}(J(C))$.) Therefore we see that the set $M_1$ of morphisms $\varphi:C\to A$ from curves of genus $2$ with $\varphi_*([C])=[L]$ modulo composition with automorphisms of $C$ and with translations of $A$ can be identified with the set $M_2$ of morphisms $\psi:B\to A$ from a principally polarized abelian surface $(B,\gamma)$, such that $\psi^*(c_1(L))=n\gamma$ modulo composition with automorphisms $\eta:B\to B$ with $\eta^*(\gamma)=\gamma$. We write $A={\Bbb C}^2/\Gamma$ and $B={\Bbb C}^2/\Lambda$. Then $c_1(L)$ is given by an alternating form $a:\Gamma\times\Gamma\to {\Bbb Z}$ such that there is a basis $x_1,x_2,y_1,y_2$ of $\Gamma$ with $a(x_1,y_1)=1$, $a(x_2,y_2)=n$, $a(x_1,x_2)=a(y_1,y_2)=0$. A homomorphism $\psi:B\to A$ is given by a linear map $\widehat \psi:{\Bbb C}^2\to {\Bbb C}^2$ with $\psi(\Lambda)\subset \Gamma$. We see that $M_2$ can be identified with the set $M_3$ of sublattices $\Lambda\subset \Gamma$ of index $n$ with $a(\Lambda,\Lambda)\subset n{\Bbb Z}$. We claim that $M_3$ has $\sigma_1(n)$ elements. First we want to see that this claim implies the theorem. Let $Pic^L(A)$ be the group of line bundles on $A$ with first Chern class $c_1(L)$. By \cite{L-B} proposition 4.9 the morphism $\varphi_L:A\to Pic^L(A); a\mapsto t^*_aL$ is \'etale of degree $n^2$. By the claim this means that for each $L_1\in Pic^L(A)$ the linear system $|L_1|$ contains precisely $n^2\sigma_1(n)$ curves of genus $2$. Finally we show the claim. Via the basis $x_1,y_1,x_2,y_2$ (in that order) we identify $\Gamma$ with ${\Bbb Z}^2\times {\Bbb Z}^2$. We see that $\Lambda$ must be of the form $\Lambda'\times {\Bbb Z}^2$, where $\Lambda'$ is a sublattice of index $n$ in ${\Bbb Z}^2$, satisfying $b(\Lambda',\Lambda')\subset n{\Bbb Z}$, for the alternating form $b$ defined by $a(x_1,y_1)=1$. Let $\Lambda'$ be a sublattice of ${\Bbb Z}^2$ of index $n$. We claim that $b(\Lambda',\Lambda')\subset n{\Bbb Z}$. Then the result follows by the well-known fact that the number of sublattices of index $n$ in a rank two lattice is $\sigma_1(n)$. Let $L_1:=p_2(\Lambda')$ for the second projection ${\Bbb Z}^2\to {\Bbb Z}$ and put $L_2:=ker(p_2|_{\Lambda'})$. Then $L_1=d_1{\Bbb Z}$ and $L_2=d_2{\Bbb Z}$ for $d_1,d_2\in {\Bbb Z}$ with $d_1d_2=n$. Choose $x=(k,d_1)\in \Lambda'\cap p_2^{-1}(d_1)$. Then $\Lambda'$ is generated by $L_2$ and $x$, and in particular $b(\Lambda',\Lambda')\subset n{\Bbb Z}$ \end{pf} \section{Severi degrees on ${\Bbb P}_2$ and rational ruled surfaces} The Severi degree $N^{d,\delta}$ is the number of plane curves of degree $d$ with $\delta$ nodes passing through $(d^2+3d)/2-\delta$ general points. In \cite{R2} a recursive procedure for determining the $N^{d,\delta}$ is shown, and in \cite{C-H} a different recursion formula is proven. In the number $N^{d,\delta}$ also reducible curves are included, they however occur only if $d\le \delta+1$, furthermore the numbers of irreducible curves can be determined from them (\cite{C-H} see also \cite{Ge}). For simplicity I will write $t_\delta(d):=t^{{\Bbb P}_2}_{\delta}({\cal O}(d))$. If $d$ is sufficiently large with respect to $\delta$, then $N^{d,\delta}$ should be equal to $t_{\delta}(d)$: \begin{conj}\label{Sconj} If $\delta\le 2d-2$, then $N^{d,\delta}=t_{\delta}(d)$. \end{conj} \begin{rem}\label{Srem} (1) Conjecture \ref{Sconj} and \cite{C-H} provide an effective method of determining the coefficients of the two unknown power series $B_1(q)$ and $B_2(q)$. Using a suitable program I computed the $N^{d,\delta}$ via the recursive formula from \cite{C-H} for $d\le 16$ and $\delta\le 30$. We write $x=DG_2(\tau)$. By conjectures \ref{mainconj} and \ref{Sconj} one has for all $d>0$ moduli the ideal generated by $x^{2d-1}$ the identity \begin{equation}\label{expeqn} \sum_{\delta\in {\Bbb Z}}N^{d,\delta}x^\delta\equiv \exp(d^2C_1(x)+dC_2(x)+C_3(x)) \end{equation} Here $C_1(x)$ is known by conjecture \ref{mainconj} and the first $k$ coefficients of $C_2(x) $ and $C_3(x)$ determine the first $k$ coefficients of $B_1(q)$ and $B_2(q)$. Taking logarithms on both sides gives, for any two degrees $d_1<d_2$ and $\delta\le 2d_1-2$, a system of two linear equations for the coefficients of $x^\delta$ in $C_2(x)$ and $C_3(x)$. Note that this also gives a test of the conjecture. It is a nontrivial fact that the generating function has the special shape (\ref{expeqn}). In particular each pair $d_1<d_2$ with $\delta\le 2d_1-2$ already determines the coefficients of $x^\delta$. \noindent(2) The conjecture implies in particular that for $\delta\le 2d-2$ the numbers $N^{d,\delta}$ are given by a polynomial $P_\delta(d)$ of degree $2\delta$ in $d$. This was already conjectured in \cite{D-I}. Denote by $p_\mu(\delta)$ the coefficient of $d^{2\delta-\mu}$ in $P_\delta$. In \cite{D-I} a conjectural formula for the leading coefficients $p_\mu(\delta)$ for $\mu\le 6$ is given. Kleiman and Piene have determined $p_7(\delta)$ and $p_8(\delta)$ \cite{K-P}. In \cite{Ch} the $N^{d,\delta}$ for $\delta\le 4$ are computed using the recursive method of \cite{R2}, and as an application $p_0( \delta)$ and $p_1(\delta)$ are determined. Using conjectures \ref{mainconj} and \ref{Sconj} there is an algorithm to determine the $p_\mu(\delta)$. Again we use the formula (\ref{expeqn}), and collect terms. From knowing the coefficients of $C_2(x)$ and $C_3(x)$ up to degree 28 we get the $p_\mu(\delta)$ for $\mu\le 28$. For $\mu\le 8$ they coincide with those from \cite{D-I},\cite{Ch} and \cite{K-P}. Let $[\ ]$ denote the integer part. For $\mu\le 28$ we observe that $p_\mu(\delta)$ is of the form $$p_\mu(\delta)=\frac{3^{\delta-[\mu/2]}}{(\delta-[\mu/2])! }Q_\mu(\delta),$$ where $Q_\mu(\delta)$ is a polynomial of degree $[\mu/2]$ in $\delta$ with integer coefficients, which have only products of powers of $2$ and $3$ as common factors. In particular \begin{align*} Q_8(\delta)&=-2^4(282855\, \delta^4-931146\, \delta^3 +417490\, \delta^2+425202\, \delta+1141616),\\ Q_9(\delta)&=-2^33^2(128676\, \delta^4+268644\, \delta^3-1011772\, \delta^2- 1488377\, \delta-1724779),\\ Q_{10}(\delta)&=2^4 3^2(4345998\, \delta^5-15710500\, \delta^4-3710865\, \delta^3+7300210\, \delta^2\\ &\qquad +57779307\, \delta+98802690). \end{align*} Note that for $d>\delta+1$ all the curves are irreducible, so that in this case we get a conjectural formula for the stable Gromov-Witten invariants of ${\Bbb P}_2$, i.e. the numbers of irreducible curves $C$ of degree $d$ with $g(C)\ge\binom{d-2}{2}$. \end{rem} \begin{rem}\label{sigmarem} Let $\Sigma_e$ be a rational ruled surface, $E$ the curve with $E^2=-e$ and $F$ a fiber of the ruling. For simplicity denote $t^e_\delta(n,m):=t^{\Sigma_e}_{\delta}({\cal O}(nF+mE))$. \cite{Ch} determined the $N^{2,m}_{0,\delta}$ for $\delta\le 9$ as polynomials of degree $\delta$ in $m$ and several numbers $N^{3,m}_{0,\delta}$ In \cite{Va} a recursion formula very similar to that of \cite{C-H} is proved for the generalized Severi degrees $N^{n,m}_{e,\delta}$, i.e. the number of $\delta$-nodal curves in $|nF+mE|$ which do not contain $E$ as a component. Using a suitable program I computed the $N^{n,m}_{e,\delta}$ for $e\le 4 $, $\delta\le 10$, $n\le 11$ and $m\le 8$. These results are compatible with conjecture \ref{mainconj}, if one conjectures that for $(n,m)\ne (1,0)$ one has $N^{n,m}_{e,\delta}=t^{e}_{\delta}(n,m)$ if and only if $\delta\le min(2m,n-em)$ or $\delta\le min(2m,2n)$ in case $e=0$. \end{rem} \begin{rem} A slightly sharpened version of conjecture \ref{Sconj} can be reformulated as saying that $N^{d,\delta}=t_{\delta}(d)$ if and only if $H^0({\Bbb P}_2,{\cal O}(d))>\delta$ and the locus of nonreduced curves in $|{\cal O}(d)|$ has codimension bigger then $\delta$. In a similar way the conjecture of remark \ref{sigmarem} can be reformulated as saying that $N^{n,m}_{e,\delta}=t^{e}_{\delta}(n,m)$ if and only if $H^0(\Sigma_e,{\cal O}(nF+mE))>\delta$ and the locus of curves in $|{\cal O}(nF+mE)|$ which are nonreduced or contain $E$ as a component has codimension bigger then $\delta$. One would expect that nonreduced curves contribute to the count of nodal curves, and the recursion formula of \cite{Va} only counts curves not containing $E$. Therefore it seems that, at least in the case of ${\Bbb P}_2$ and of rational ruled surfaces, $t^{S}_{\delta}(L)$ is the actual number of $\delta$-nodal curves in a general $\delta$-codimensional linear system, unless this cannot be expected for obvious geometrical reasons. \end{rem} \section{Connection with Hilbert schemes of points} Let again $S$ be an algebraic surface and let $L$ be a line bundle on $S$. Let $S^{[n]}$ be the Hilbert scheme of finite subschemes of length $n$ on $S$, and let $Z_n(S)\subset S\times S^{[n]}$ denote the universal family with projections $p_n:Z_n(S)\to S$, $q_n:Z_n(S)\to S^{[n]}$. Then $L_n:=(q_n)_*(p_n)^*(L)$ is a locally free sheaf of rank $n$ on $S^{[n]}$. \begin{defn} Let $S^{\delta}_2\subset S^{[3\delta]}$ be the closure (with the reduced induced structure) of the locally closed subset $S^{\delta}_{2,0}$ which parametrizes subschemes of the form $\coprod_{i=1}^{\delta} {\hbox{\rom{Spec}\,}}({\cal O}_{S,x_i}/{m}^2_{S,x_i})$, where $x_1,\ldots x_{\delta}$ are distinct points in $S$. It is easy to see that $S^{\delta}_2$ is birational to $S^{[\delta]}$. We put $ d_n(L):=\int_{S^\delta_2}c_{2\delta}(L_{3\delta}). $ \end{defn} Following \cite{B-S} we call $L$ $k$-very ample if for all subschemes $Z\subset S$ of length $k+1$ the natural map $H^0(S,L)\to H^0(L\otimes{\cal O}_Z)$ is surjective. If $L$ and $M$ are very ample, then $L^{\otimes k}\otimes M^{\otimes l}$ is $(k+l)$-very ample. \begin{prop} \label{port} Assume $L$ is $(3\delta-1)$-very ample, then a general $\delta$-dimensional linear subsystem $V\subset |L|$ contains only finitely many curves $C_1,\ldots,C_s$ with $\ge \delta$ singularities. There exist positive integers $n_1,\ldots,n_s$ such that $\sum_i n_i=d_\delta(L)$. If furthermore $L$ is $(5\delta-1)$-very ample ($5$-very ample if $\delta=1$), then the $C_i$ have precisely $\delta$ nodes as singularities. \end{prop} \begin{pf} Assume first that $L$ is $(3\delta-1)$-very ample. We apply the Thom-Porteous formula to the restrictions of the evaluation map $H^0(S,L)\otimes{\cal O}_{S^{[3\delta]}}\to L_{3\delta}$ to $S^{\delta}_2$ and to $S^{\delta}_{2}\setminus S^{\delta}_{2,0}$. As $L$ is $(3\delta-1)$-very ample the evaluation map is surjective. Then (\cite{Fu} ex. 14.3.2) applied to $S^{\delta}_2$ gives that for a general $\delta$-dimensional sublinear system $V\subset |L|$ the class $d_n(L)$ is represented by the class of the finite scheme $W$ of $Z\in S^{\delta}_2$ with $Z\subset D$ for $D\in V$. The scheme structure of $W$ might be nonreduced. The application of (\cite{Fu} ex. 14.3.2) to $S^{\delta}_{2}\setminus S^{\delta}_{2,0}$ and a dimension count give that $W$ lies entirely in $S^{\delta}_{2,0}$. Now assume that $L$ is $(5\delta-1)$-very ample. Let $V\subset |L|$ again be general $\delta$-dimensional subsystem of $|L|$. The Porteous formula applied to the restriction of $L_{3\delta+3}$ to $S^{\delta+1}_2$ and a dimension count shows that there will be no curves in $V$ with more than $\delta$ singularities. Let $S^{\delta}_{3,0}\subset S^{[5\delta]}$ be the locus of schemes of the form $Z_1\sqcup Z_2\ldots \sqcup Z_{\delta}$, where each $Z_i$ is of the form ${\hbox{\rom{Spec}\,}}({\cal O}_{S,x_i}/({m}^3+xy))$ with $x,y$ local parameters at $x_i$ and let $S^{\delta}_{3}$ be the closure. If a curve $C$ with precisely $\delta$ singularities does not contain a subscheme corresponding to a point in $S^{\delta}_{3}\setminus S^{\delta}_{3,0}$, then it has $\delta$ nodes as only singularities. It is easy to see that $S^{\delta}_{3,0}$ is smooth of dimension $4\delta$. Applying the Porteous formula to the restriction of $L_{5n}$ to $S^{\delta}_{3}\setminus S^{\delta}_{3,0}$ and a dimension count we see that all the curves in $V$ with $\delta$ singularities have precisely $\delta$ nodes. \end{pf} \begin{conj}\label{dconj} $d_\delta(L)= T_\delta(L^2 , LK_S, K^2_S, c_2(S)).$ \end{conj} Conjecture \ref{dconj} gives the hope of proving conjecture \ref{mainconj} via the study of the cohomology of Hilbert schemes of points. \begin{rem} Note that one can generalize the above to singular points of arbitrary order: Let $\mu=(m_1,\ldots,m_{l(\mu)})$ where $m_i\in{\Bbb Z}_{\ge 2}$. Let $N(\mu):=\sum_{i=1}^s \binom{m_i+1}{2}$ and let $S_\mu$ be the closure in $S^{[N(\mu)]}$ of the subset of schemes of the form $\coprod_{i=1}^{l(\mu)} {\hbox{\rom{Spec}\,}}({\cal O}_{S,x_i}/{m}_{S,x_i}^{m_i})$. Denote $ d_\mu(L):=\int_{S_\mu} c_{2l(\mu)}(L_{N(\mu)}). $ We call a curve $D\in|L|$ of type $\mu$ if there are distinct points $x_1,\ldots,x_{l(\mu)}$ in $S$, such that the ideal ${\cal I}_{D,x_i}$ is contained in ${m}_{S,x_i}^{m_i}$. A straightforward generalization of the proof of proposition \ref{port} shows that, for $V$ a general $(N(\mu)-2l(\mu))$-dimensional linear subsystem of an $N(\mu)$-very ample line bundle $|L|$, $d_\mu(L)$ counts the finite number of curves of type $\mu$ in $V$ with positive multiplicities. Again I expect $d_\mu(L)$ to be a polynomial of degree $l(\mu)$ in $L^2$, $LK_S$, $K_S^2$ and $c_2(S)$. In a similar way one can also deal with cusps instead of nodes. \end{rem}

9711

alg-geom/9711022

en

https://arxiv.org/abs/alg-geom/9711022

alg-geom/9711022

Francisco Jose Plaza Martin

J. M. Mu\~noz Porras, F. J. Plaza Mart\'in

Equations of the moduli of pointed curves in the infinite Grassmannian

Final version to appear in Journal of Differential Geometry. Some mistakes involving addition formulas for points of arbitrary connected components of $\gr(V)$ have been removed. 35 pages. LaTeX

The main result of this paper is the explicit computation of the equations defining the moduli space of triples $(C,p,z)$ (where $C$ is an integral and complete algebraic curve, $p$ a smooth rational point and $z$ a formal trivialization around $p$) in the infinite Grassmannian of $k((t))$. This is achieved by introducing infinite Grassmannians, tau and Baker-Ahkiezer functions algebraically and by proving an Addition Formula for tau functions.

alg-geom

\section{Introduction} The Krichever morphism gives an immersion of the moduli space, $\M_\infty$, of triples $(C,p,\phi)$ (where $C$ is an algebraic curve, $p$ a smooth point of $C$ and $\phi$ a certain isomorphism) in the infinite Grassmannian (see \cite{SS,SW}). The aim of this paper is to give an explicit system of equations defining the subscheme $\M_\infty$ of $\grv$, this problem is solved in \S6. We have adopted an algebraic point of view (\cite{AMP}) and most of the results of this paper are valid over arbitrary base fields. Consequently, we have included several paragraphs addressing certain aspects of the theory of soliton equations and infinite Grassmannians for arbitrary base fields (these facts are well known when the base field is $\mathbb C$, \cite{SW,DJKM,KSU}). This allows us to give the foundations for a theory of soliton equations valid over arbitrary base fields, extending the previous results of G. Anderson (\cite{An}) for the case of $p$-adic fields. We hope that the techniques developed in this paper will clarify the ``arithmetic properties'' of the theory of KP equations. When the base field is $\mathbb C$, our equations for $\M_\infty$ (as a subscheme of the infinite Grassmannian, $\grv$) are equivalent to a system of infinite partial differential equations {\ref{thm:pde-mod}} which are different from the equations of the KP hierarchy. To clarify the relation between the KP hierarchy and our equations {\ref{thm:pde-mod}}, let us consider the chain of closed immersions: $$\M_\infty\overset{\text{\peqq Krichever}}\hookrightarrow \grv \overset{\text{\peqq Pl\"ucker}}\hookrightarrow {\mathbb P}^\infty$$ where ${\mathbb P}^\infty$ is a suitable infinite dimensional projective space. It is well known (\cite{Pl2,SS}) that the image of $\grv$ in ${\mathbb P}^\infty$ is defined by the Pl\"ucker equations, which are equivalent to the KP hierarchy ($\operatorname{char}(k)=0$). Then, the image of $\M_\infty$ in ${\mathbb P}^\infty$ will be defined by the following system of differential equations: $$\left\{\aligned &\text{ the KP equations (given in Theorem {\ref{thm:KP-eq}}),} \\ &\text{ the p.d.e.'s given in Theorem {\ref{thm:pde-mod}},} \\ &\text{ the p.d.e.'s given in Corollary {\ref{cor:pde-tau}}.3.} \endaligned\right.$$ In particular, we deduce a characterization ({\ref{cor:pde-tau}}), in terms of partial differential equations, of the infinite formal series $\tau(t)\in {\mathbb C}\{\{t_1,t_2,\dots\}\}$, which are the $\tau$-functions of a triple $(C,p,\phi)\in\M_\infty$. Let us note that the results of Shiota (\cite{Sh}) give a necessary and suficient condition for a theta function of a principally polarized abelian variety to be the theta function of a Jacobian, but do not solve the problem of characterizing, in terms of differential equations, the formal $\tau$-functions defined by Jacobian theta functions; this problem is solved in Corollary {\ref{cor:pde-tau}}. The paper is organized as follows. A survey on infinite Grassmannians is given in \S2. In \S3 the algebraic analogue of the group of maps $S^1\to{\mathbb C}^*$ of Segal-Wilson is constructed and interpreted as the Jacobian of the formal curve. The action of that group on the Grassmannian is used in \S4 in order to introduce tau and Baker-Ahkiezer functions. At the end of this section the Addition Formulae for tau functions are stated and proved. Equations defining the infinite Grassmannian of $k((z))$ in a suitable infinite dimensional projective space are computed in \S5. Over an arbitrary field it is defined by the well known Bilinear Residue Identity (equation \ref{eq:residue}) whose proof is an easy consequence of Theorem \ref{4:thm:BA}. When the base field is ${\mathbb C}$ and using Schur polynomials, this identity turns out to be equivalent to the KP Hierarchy. The last section, \S6, contains the main result of this paper. A relative generalization of the Krichever map allows us to obtain a closed immersion of the moduli functor of pointed curves, whose rational points are the triples $(C,p,\phi)$, into the infinite Grassmannian of $k((z))$. We prove then its representability and give a characterization that permits to compute its equations. We are very grateful to the referee for his comments, which help us to improve the paper and to remove some initial mistakes. \section{Infinite Grassmannians and Determinant Bundles} \subsection{Infinite Grassmannians}\label{subsect:grass} In order to define the ``infinite'' Grassmannian of a vector space $V$ (over a field $k$) one should consider some extra structure on it. This structure consists of a family $\B$ of subspaces of $V$ such that the following conditions hold: \begin{enumerate} \item $A,B\in\B\quad \implies\quad A+B,A\cap B\in\B$, \item $A,B\in\B\quad \implies\quad \dim(A+B)/A\cap B <\infty$, \item $\cap_{A\in\B} A=(0)$, \item the canonical homomorphism $V\to\limpl{A\in\B}V/A$ is an isomorphism, \item the canonical homomorphism $\limil{A\in\B}A\to V/B$ is a surjection (for $B\in\B$). \end{enumerate} Let us interpret these conditions. First, $V$ is endowed with a linear topology where a family of neigbourhoods of $(0)$ is precisely $\B$. Then, the last three claims mean that the topology is separated; $V$ is complete and every finite dimensional subspace of $V/B$ is a neigbourhood of $(0)$. \begin{exam}\label{2:exam} In the study of the moduli space of pointed curves the fundamental example is $V=k((z))$ and $\B$ consists of the set of subspaces $A\subseteq V$ containing $z^n\cdot k[[z]]$ as a subspace of finite codimension (for an integer $n$). \end{exam} \begin{exam} Other examples of pairs $(V,\B)$ satisfying the above requeriments are ($V$ an arbitrary $k$-vector space): \begin{itemize} \item $(V, \B:=\{(0)\})$; \item $V$ and $\B$ the set of all finite dimensional subspaces of $V$. \end{itemize} \end{exam} In the sequel, we shall fix a pair $(V,\B)$ satisfying the above conditions and a subspace $V_+\in\B$. Following \cite{AMP} (see also \cite{KSU}), there exists a Grassmannian scheme $\gr(V,\B)$, whose rational points are the set: $$\left\{\begin{gathered} \text{ subspaces $L\subseteq V$, such that $L\cap V_+$ }\\ \text{ and ${V}/{L+V_+}$ are of finite dimension } \end{gathered}\right\}$$ (which we shall call discrete subspaces of $V$) that coincides with the usual infinite Grassmannian defined by Pressley-Segal in \cite{PS}, and Segal-Wilson in \cite{SW}. In order to construct this scheme, we shall give its functor of points $\fu{\gr}(V,\B)$ and prove that it is representable in the category of $k$-schemes. To this end, we need some notation: given a morphism $T\to S$ of $k$-schemes a sub-$\o_S$-module $B\subseteq V_S:=V\otimes_k\o_S$, we denote: \begin{itemize} \item $\w B_T:=\lim (B/(A_S\cap B) \underset k\otimes\o_T)$. \item $\w{(V/B)}_T:=\lim ((V_S/(A_S+B))\underset k\otimes\o_T)$. \end{itemize} \begin{defn}\label{defn:discrete} Given a $k$-scheme $S$, a discrete submodule of $\w V_S$ is a sheaf of quasi-coherent $\o_S$-submodules $\L \subset \w V_S$ such that: $\L_T\subset \w V_T$ for every morphism $T\to S$; and, for each $s\in S$ there exists an open neighborhood $U$ of $s$ and a commensurable $k$-vector subspace $B\in\B$ such that $\L_U\cap \w B_U$ is free of finite type and $\w V_U/\L_U+ \w B_U=0$. \end{defn} \begin{defn} The Grassmannian functor of a pair $(V,\B)$, $\fu{\gr}(V,\B)$, is the contravariant functor over the category of $k$-schemes defined by: $$\fu{\gr}(V,\B)(S)=\left\{ \text{discrete sub-$\o_S$-modules of $\w V_S$}\right\}$$ \end{defn} \begin{rem} Note that if $V$ is a finite dimensional $k$-vector space and $\B=\{(0)\}$, then $\fu{\gr}(V,\{(0)\})$ is the usual Grassmannian functor defined by Grothendieck \cite{EGA}~I.9.7. \end{rem} \begin{thm} The functor $\fu{\gr}(V,\B)$ is representable by a reduced and separated $k$-scheme $\gr(V,\B)$. The discrete sub\-module corresponding to the identity: $$Id\in \fu{\gr} (V,\B)\left(\gr(V,\B)\right)$$ will be called the {\bf universal submodule} and will be denoted by: $$\L_V\subset \w V_{\gr(V,\B) }$$ \end{thm} \begin{pf} The proof is modeled on the Grothendieck construction of finite Grassmannians \cite{EGA}. Given a vector subspace $A\in \B $, define the functor $\fu{F_A}$ over the category of $k$-schemes by: $$\fu{F_A}(S)=\{\text{ sub-$\o_S$-modules $\L\subset \w V_S$ such that $\L\oplus \w A_S=\w V_S$ }\}$$ and show that it is representable by an affine and integral $k$-scheme, $F_A$. The properties on $\B$ imply that $\{\fu{F_A}, A\in \B \}$ is a covering of $\fu{\gr}(V,\B)$ by open subfunctors (see \cite{AMP}) and the result follows. \end{pf} \begin{rem} In this subsection infinite-dimensional Grassmannian sche\-mes have been constructed in an abstract way. Choosing particular vector spaces $(V,\B)$ we obtain different classes of Grassmannians. Two examples are relevant: \begin{itemize} \item $V=k((z))$, $V_+=k[[z]]$. In this case, $\gr(k((z)),k[[z]])$ is the algebraic version of the Grassmannian constructed by Pressley-Segal, and Segal-Wilson (\cite{PS,SW}). This Grassmannian is particularly well adapted for studying problems related to the moduli space of pointed curves (over arbitrary fields), to the moduli space of vector bundles and to the KP-hierarchy. \item Let $(X,\o_X)$ be a smooth, proper and irreducible curve over the field $k$ and let $V$ be the adeles ring over the curve and $V_+=\underset p \prod\w{\o_p}$ (recall the first example). In this case $\gr(V,\B)$ is an adelic Grassmannian which will be useful for studying arithmetic problems over the curve $X$ or problems related to the classification of vector bundles over a curve (non abelian theta functions...). \newline Instead of adelic Grassmannians, we could define Grassmannians associated with a fixed divisor on $X$ in an analogous way. \newline These adelic Grassmannians will also be of interest in the study of conformal field theories over Riemann surfaces in the sense of Witten (\cite{W}). \end{itemize} \end{rem} \subsection{Determinant Bundles}\label{subsect:det} In this subsection we recall from \cite{AMP} the construction the determinant bundle over the Grassmannian, in the sense of Knudsen and Mumford \cite{KM}. This allow us to define determinants algebraically and over arbitrary fields (for example for $k={\mathbb Q}$ or $k={\mathbb F}_q$). We shall denote the Grassmannian $\gr(V,\B)$ simply by $\grv$. \begin{defn} For each $A\in \B $ and each $L\in\grv(S)$ we define a complex, $\c_A(L)$, of $\o_S$-modules by: $$\c_A(L) \equiv \ldots\to 0\to L\oplus \hat A_S @>\delta>> \hat V_S\to 0\to\ldots$$ $\delta$ being the addition homomorphism. \end{defn} It is not difficult to prove that $\c_A(L)$ is a perfect complex of $\o_S$-modules, and therefore its determinant and its index are well defined. Recall that the index of a point $L\in\grv(S)$ is the locally constant function $i_L\colon S\to\Z$ defined by: $$i_L(s)=\text{ Euler-Poincar\'e characteristic of } \c_{V_+}(L)\otimes k(s)$$ $k(s)$ being the residual field of the point $s\in S$ (see \cite{KM} for details). By easy calculation, we have that if $V$ is a finite-dimensional $k$-vector space, $\B=\{(0)\}$ and $L\in\grv(S)$, then $i_L=\operatorname{codim}_k(L)$. And for the general case, the index of any rational point $L\in\grv(\spk)$ is exactly: $$\dim_k(L\cap V_+)-\dim_k V/(L+ V_+)$$ Let $\gr^n(V)$ be the subset over which the index takes values equal to $n\in\Z$. Then, the decomposition of $\grv$ in connected components is: $$\grv=\underset{n\in\Z}\coprod\gr^n(V)$$ Given a point $L\in\grv(S)$ and $A\in \B $, we denote by $\det\c_A(L)$ the determinant sheaf of the perfect complex $\c_A(L)$ in the sense of \cite{KM}. Note that this determinant does not depend on $A$ (up to isomorphisms), since for $A,B\in\B$ there exists a canonical isomorphism: \beq \detd\c_A\iso\detd\c_B\otimes\wedge^{max}( A/{A\cap B})\otimes \wedge^{max}(B/{A\cap B})^* \label{2:eqn:det-iso}\end{equation} Hence we define the determinant bundle over $\gr^0(V)$, $\det_V$, as the invertible sheaf: $$\det\c_{V_+}(\L_V)$$ ($\L_V$ being the universal submodule over $\gr^0(V)$). Observe that the choice of other subspace of $\B$, $A$, instead of $V_+$ might shift the labelling of the connected components by a constant and modify the determinant bundle by an isomorphism. Now, if $L\in\gr^0(V)$ is a rational point, and $L\cap V_+$ and ${V}/{L+ V_+}$ are therefore $k$-vector spaces of the same dimension, then one has an isomorphism: $$\det_V(L)\simeq \wedge^{max}(L\cap V_+)\otimes\wedge^{max}({V}/{(L+V_+)})^\ast$$ That is, our determinant coincides, over the rational points, with the determinant bundles of Pressley-Segal and Segal-Wilson (\cite{PS,SW}). Furthermore, this construction gives the usual determinant bundle when $V$ is finite-dimensional. \subsection{Sections of the Determinant bundle} It is well known that the determinant bundle has no global sections. We shall therefore explicitly construct global sections of the dual of the determinant bundle over the connected component $\gr^0(V)$ of index zero. We use the following notations: $\wedge^\punto E$ is the exterior algebra of a $k$-vector space $E$, $\wedge^r E$ its component of degree $r$, and $\wedge E$ is the component of higher degree when $E$ is finite-dimensional. Given a perfect complex $\c$ over $k$-scheme $X$, we shall write $\detd\c$ to denote the dual of the invertible sheaf $\det\c$. To explain how global sections of the invertible sheaf $\detd\c$ can be constructed, recall that if $f:E\to F$ is a homomorphism between finite-dimensional $k$-vector spaces of equal dimension, then it induces a homomorphism: $$\wedge(f):k\to \wedge F\otimes (\wedge E)^*$$ Thus, considering $E @>{f}>> F$ as a perfect complex, $\c$, over $\spk$, we have defined a {\bf canonical section} $\wedge(f)\in H^0(\spk,\detd\c)$. Let us now consider a perfect complex $\c\equiv(E@>{f}>>F)$ of sheaves of $\o_X$-modules over a $k$-scheme $X$, with Euler-Poincar\'e characteristic ${\mathcal X}(\c)=0$. Using the above argument, we construct a canonical section $det(f\vert_U)\in H^0(U,\detd\c)$ for every open subscheme of $X$, $U$, over which $\c$ is quasi-isomorphic to a complex of finitely-generated free modules. Since the construction is canonical, the functions: $$g_{UV}:= det(f\vert_U)\vert_{U\cap V}\cdot det(f\vert_V)\vert_{U\cap V}^{-1}$$ satisfy the cocycle condition. Then, these local sections glue $\{det(f\vert_U)\}$ and give a canonical global section: $$\det(f)\in H^0(X,\detd\c)$$ (see \cite{AMP} for more details). If the complex $\c$ is acyclic, one has an isomorphism: $$\aligned \o_X &\iso\det^\ast\c\\ 1 &\mapsto det(f) \endaligned$$ Let us consider the perfect complex $\c_A\equiv(\L\oplus A @>{\delta_A}>> V)$ over $\grv$ defined in {\ref{subsect:det}} ($\L$ being the universal discrete submodule over $\grv$) for a given $A\in\B$. Since $\c_A\vert_{F_A}$ is acyclic, one then has an isomorphism: $$\aligned \o_X\vert_{F_A}& @>\sim>> \detd\c_A\vert_{F_A}\\ 1 &\mapsto s_A=det(\delta_A)\vert_{F_A} \endaligned$$ By the above argument it is easy to prove that the section $s_A\in H^0(F_A,\detd\c_A)$ can be extended in a canonical way to a global section of $\detd\c_A$ over $\gr^0(V)$, which will be called the {\bf canonical section $\omega_A$ of $\detd\c_A$}. (We restrict ourselves to cases where $F_A\subseteq \gr^0(V)$, or, what amounts to the same, $\dim_k( A/{A\cap V_+})-\dim_k({V_+}/{A\cap V_+})=0$). This result allows us to compute many global sections of $\detd_V=\det\c_{V_+}$ over $\gr^0(V)$: given $A\in \B $ such that $F_A\subseteq\gr^0(V)$, the isomorphism $\detd\c_A\iso\detd_V$ is not canonical (recall the formula \ref{2:eqn:det-iso}). Therefore, the determination of an isomorphism $\detd\c_A\iso\detd_V$ depends on the choice of bases for the vector spaces $ A/{A\cap V_+}$ and ${V_+}/{A\cap V_+}$. For a detailed discussion of the construction of sections see \cite{AMP,Pl2}). \subsection{Computations for the infinite Grassmannian: $V=k((z))$}\label{subsect:comp-inf-grass} Let $(V,\B)$ be as in Example \ref{2:exam} and take $V_+=k[[z]]$. Let $\mathcal S$ be the set of Young's diagrams (also called Maya or Ferrers diagrams) of virtual cardinal zero; or equivalently, the sequences $\{s_0,s_1,\ldots\}$ of integer numbers satisfying the following conditions: \begin{enumerate} \item the sequence is strictly increasing, \item there exists $s\in\Z$ such that $\{s,s+1,s+2,\ldots\}\subseteq\{s_0,s_1,\ldots\}$, \item $\#(\{s_0,s_1,\ldots\}-\{0,1,\ldots\})= \#(\{0,1,\ldots\}-\{s_0,s_1,\ldots\})$ \end{enumerate} For notation's sake, we define $e_i:=z^i$. For each $S\in{\mathcal S}$, let $A_S$ be the vector subspace of $V$ generated by $\{e_{s_i}, i\ge 0\}$. By the third condition one has: $$\dim_k({A_S}/{A_S\cap V_+})= \dim_k({V_+}/{A_S\cap V_+})$$ and hence $A_S\in \B $ and $F_{A_S}\subseteq\gr^0(V)$. Further, $\{F_{A_S},S\in{\mathcal S}\}$ is a covering of $\gr^0(V)$. Define linear forms $\{e^*_i\}$ of $V^*$ by the following condition: $$e^*_i(e_j)\,:=\,\delta_{ij}$$ For each finite set of increasing integers, $J=\{j_1,\dots,j_r\}$, let us define $e_J:=e_{j_1}\wedge\ldots\wedge e_{j_r}$ and $e^*_J:=e^*_{j_1}\wedge\ldots\wedge e^*_{j_r}$. Given $S\in{\mathcal S}$, choose $J,K\subseteq\Z$ such that $\{e_j\}_{j\in J}$ is a basis of ${V_+}/{A_S\cap V_+}$ and $\{e^*_k\}_{k\in K}$ of $({A_S}/{A_S\cap V_+})^*$ . We have seen that tensor by $e_J\otimes e^*_K$ defines an isomorphism: $$ H^0(\gr^0(V),\detd\c_{A_S}) @>{\,\otimes(e_J\otimes e^*_K)\,}>> H^0(\gr^0(V),\detd_V)$$ \begin{defn}\label{defn:global-section} For each $S\in{\mathcal S}$, $\Omega_S$ is the global section of $\detd_V$ defined by: $$\Omega_S=\omega_{A_S}\otimes e_J\otimes e^*_K$$ We shall denote by $\Omega_+$ the canonical section of $\detd_V$. \end{defn} Let $\Omega({\mathcal S})$ be the $k$-vector subspace of $ H^0(\gr^0(V),\detd_V)$ generated by the global sections $\{\Omega_S,S\in{\mathcal S}\}$. We define the Pl\"ucker morphism: $$\aligned{\frak p}_V: \gr^0(V) &\to \P\Omega(S)^*:=\proj \operatorname{Sym}\Omega(S) \\ L &\mapsto \{\Omega_S(L)\}\endaligned$$ as the morphism of schemes associated to the surjective sheaf homomorphism: $$\Omega(S)_{\gr^0(V)}\to\detd_V$$ by the universal property of $\P$. \begin{rem}\label{rem:grass=plucker} Once the Pl\"ucker morphism is introduced it can be proved that the Pl\"ucker equations are in fact the defining equations for $\gr^0(V)$ when $\operatorname{char}(k)=0$ (\cite{Pl2}). This is the property that Sato used in \cite{SS} to define his Universal Grassmann Manifold (UGM); that is, a point of the UGM is a point of an infinite dimensional projective space (with countable many coordinates) satisfying all the Pl\"ucker relations. \end{rem} \begin{rem}\label{rem:indexn} For studing the index $n$ connected component of the Grassmannian, $\gr^n(V)$ the same constructions are applied since the homothety of $k((z))$ defined by $z^{-n}$ induces isomorphisms: $$\begin{gathered} \gr^n(V)\,\iso\,\gr^0(V) \\ H^0(\gr^0(V),\detd_V)\,\iso \, H^0(\gr^n(V),\detd_n) \end{gathered}$$ where $\detd_n$ is defined by $\detd\c_{z^n\cdot V_+}$. Let us distinguish the global section given by the image of $\Omega_+$ and denote it by $\Omega_+^{n}$. \end{rem} \subsection{The Grassmannian of the dual space}\label{subsect:grass-dual} Let $(V,\B)$ be as usual. Consider $V$ as a linear topological space. Now, a submodule $L\subseteq \w V_S$ ($S$ a $k$-scheme) carries a linear topology: that given by $\{L\cap \w A_S\}_{A\in\B}$ as neigbourhoods of $(0)$. We introduce the following notation: $$\begin{aligned} L^* &\,:=\, \hom_{\o_S}(L,\o_S) \\ L^c &\,:=\, \{f\in L^* \text{ continuous }\} \\ L^\circ &\,:=\, \{f\in (\w V_S)^* \,\vert\, f\vert_L\equiv 0 \} \\ L^\diamond &\,:=\, \{f\in (\w V_S)^c \,\vert\, f\vert_L\equiv 0 \} \end{aligned}$$ where $\o_S$ has the discrete topology. Observe that given two subspaces $A,B\in\B$ such that $B\subseteq A$ the following claims hold: \begin{itemize} \item there is a canonical isomorphism $A^\diamond / B^\diamond \iso (A/B)^*$. (This implies that $(A^\diamond + B^\diamond)/A^\diamond \cap B^\diamond$ is finite dimensional). \item $(A+B)^\diamond= A^\diamond \cap B^\diamond$. \item $(A\cap B)^\diamond= A^\diamond + B^\diamond$. \end{itemize} Consider the following family of subspaces of $V^c$: $$\B^\diamond\,:=\,\{A^\diamond\text{ where }A\in\B\}$$ In order to make explicit the meaning of the expression ``Grassmannian of the dual space'', $(V^c,\B^\diamond)$, we need the following: \begin{lem}[\cite{Pl2}]\hfill \begin{enumerate} \item $V^c=\limil{A\in\B}A^\diamond$; \item $V=\limil{A\in\B} A$; \item $\cap_{A\in\B} A^\diamond =(0)$; \item $V^c=\limpl{A\in\B} V^c/A^\diamond$. \end{enumerate} \end{lem} The Lemma and these considerations imply the following: \begin{thm} The family $\B^\diamond$ satisfies the conditions of \ref{subsect:grass}, and therefore the infinite Grassmannian of the pair $(V^c,\B^\diamond)$ exists. \end{thm} If a subspace $V_+\in\B$ is chosen, then we shall consider the subspace $V_+^\diamond$ for the pair $(V^c,\B^\diamond)$. Now, we shall construct a canonical isomorphism between the Grassmannian of $V$ and that of $V^c$. The expression of this isomorphism for the rational points will be that given by incidence: $$\aligned I:\gr(V,\B) &\longrightarrow \gr(V^c,\B^\diamond) \\ L\quad &\longmapsto\quad L^\diamond \endaligned $$ Let $\L$ be the universal sheaf of $\grv$. Consider the following sub-$\o_{\grv}$-module of $\w V^c_{\grv}$ defined as: $$\L^\diamond\,=\, \{ \omega\in \w V^c_{\grv} \text{ such that }\omega\vert_{\L}\equiv 0 \}$$ Let us check that $\L^\diamond$ is in fact a $\grv$-valued point of $\gr(V^c)$, and that it does induce the desired morphism. By the definition of the Grassmannian, this can be done locally. Recall that $\{F_A\}_{A\in \B }$ is a covering of $\grv$. Let $\L^\diamond\vert_{F_A}$ be the restriction of $\L^\diamond$ (as sub-$\o_{\grv}$-module of $\w V^c_{\grv}$) to the open subscheme $F_A\hookrightarrow \grv$. Since $\w V_{F_A}\simeq \L\vert_{F_A}\oplus\w A_{F_A}$ canonically, one has: $$\L^\diamond\vert_{F_A} \,\simeq\, (\w A_{F_A})^c$$ in a canonical way. Recalling the Definition {\ref{defn:discrete}}, the conclusion follows. To finish, we compute $I^*\det_{V^c}$. Observe that there exists a canonical morphism of complexes of $\o_{\grv}$-modules (written vertically): $$\CD \L_V^\diamond\oplus V_+^\diamond @>>> V^* \\ @VVV @VVV \\ V^c @>>> \L_V^*\oplus V_+^* \endCD$$ (where $\L_V$ is the universal submodule of $\grv$) and one easily checks that this is in fact a quasi-isomorphism. Since the inverse image of the universal submodule of $\gr(V^c)$ is $\L_V^\diamond$, one has the following formulae: $$I^*\det_{V^*}\simeq \detd_V\qquad\qquad I^*(i)=-i$$ ($i$ is the index function). \section{``Formal Geometry'' of Local Curves} \subsection{Formal Groups} We are first concerned with the algebraic analogue of the group $\Gamma$ (\cite{SW}~\S~2.3) of continuous maps $S^1\to{\mathbb C}^*$ acting as multiplication operators over the Grassmannian. The main difference between our definition of the group $\Gamma$ and the definitions offered in the literature (\cite{SW,PS}) is that in the algebro-geometric setting the elements $\sum_{-\infty}^{+\infty}g_k\,z^k$ with infinite positive and negative coefficients do not make sense as multiplication operators over $k((z))$. In this sense, our approach is close to that of \cite{KSU}. The main idea for defining the algebraic analogue of the group $\Gamma$ is to construct a (formal) scheme whose set of rational points is precisely the multiplicative group $k((z))^*$ (see \cite{AMP}). \begin{defn} The contravariant functor, $\kz$, over the category of $k$-schemes with values in the category of commutative groups is defined by: $$S\rightsquigarrow \kz(S)\,:=\, H^0(S,\o_S)((z))^*$$ Where for a $k$-algebra $A$, $A((z))^*$ is the group of invertible elements of the ring $A((z)):=A[[z]][z^{-1}]$ of Laurent series with coefficients in $A$. \end{defn} Note that for each $k$-scheme $S$ and $f\in\kz(S)$, the function: $$\aligned S&\to \Z \\ s &\mapsto v_s(f):= \text{order of $f_s\in k(s)((z))$} \endaligned$$ is locally constant. From this fact we deduce that given an irreducible affine $k$-scheme, $S=\sp(A)$, we have that: $$\aligned \kz(S)=&\coprod_{n\in \Z}\left\{f\in A((z))^* \,\vert\, v(f)=n\right\} \\ =&\coprod_{n\in \Z}\left\{\gathered \text{series }\,a_{n-r}\,z^{n-r}+\dots+a_n\,z^n+\dots\text{ such that}\\ a_{n-r},\dots,a_{n-1}\text{ are nilpotent and }a_n\in A^* \endgathered\right\} \endaligned$$ \begin{thm} The subfunctor $\kz_{red}$ of $\kz$ defined by: $$S\rightsquigarrow \kz_{red}(S)\,:=\, \coprod_{n\in \Z}\left\{z^n+\sum_{i> n} a_i\,z^i \quad a_i\in H^0(S,\o_S)\right\}$$ is representable by a group $k$-scheme whose connected component of the origin will be denoted by $\Gamma_+$. \end{thm} \begin{pf} It suffices to observe that the functor: $$S\rightsquigarrow \left\{z^n+\sum_{i> n} a_i\,z^i \quad a_i\in H^0(S,\o_S)\right\}$$ is representable by the scheme $\sp(\limil{l} k[x_1,\dots,x_l])=\limpl{l}\A^l_k$, where the group law is given by the multiplication of series; that is: $$\aligned k[x_1,\dots]&\to k[x_1,\dots]\otimes_k k[x_1,\dots] \\ x_i &\mapsto x_i\otimes 1+\sum_{j+k=i}x_j\otimes x_k+1\otimes x_i \endaligned$$ \end{pf} \begin{thm} Let $\kz_{nil}$ be the subfunctor of $\kz$ defined by: $$S\rightsquigarrow \kz_{nil}(S)\,:=\, \coprod_{n>0}\left\{\gathered \text{finite series }\,a_n\,z^{-n}+\dots+a_1\,z^{-1}+1\\ \text{such that $a_i\in H^0(S,\o_S)$ are nilpotent} \endgathered\right\}$$ There exists a formal group $k$-scheme $\Gamma_-$ such that: $$\hom_{\text{for-sch}}(S,\Gamma_-)=\kz_{nil}(S)$$ for every $k$-scheme $S$. \end{thm} \begin{pf} Note that $\Gamma_-$ is the direct limit in the category of formal schemes (\cite{EGA} {\bf I}.10.6.3) of the schemes representing the subfunctors: {\small $$S\rightsquigarrow \Gamma^n_-(S)=\left\{\gathered \,a_n\,z^{-n}+\dots+a_1\,z^{-1}+1\text{ such that }a_i\in H^0(S,\o_S)\\ \text{ and the ${\text{n}}^{\text{th}}$ power of the ideal $(a_1,\dots,a_n)$ is zero} \endgathered\right\}$$} And its associated ring (that is, as ringed space) is: $$k\{\{x_1,\dots\}\}=\underset n\limp k[[x_1,\dots,x_n]]$$ the morphisms of the projective system being: $$\aligned k[[x_1,\dots,x_{n+1}]]&\to k[[x_1,\dots,x_n]] \\ x_i&\mapsto x_i\qquad\text{for }i=1,\dots,n-1 \\ x_{n+1}&\mapsto 0 \endaligned$$ It is now easy to show that $\Gamma_-=\sf(k\{\{x_1,\dots\}\})$ (with group law given by multiplication of series) satisfies the desired condition. We shall call the ring $k\{\{x_1,\dots\}\}$ the ring of ``infinite'' formal series in infinite variables (which is different from the ring of formal series in infinite variables. For instance, $x_1+x_2+\ldots\in k\{\{x_1,\dots\}\}$). \end{pf} \begin{rem} Our group scheme $\Gamma$ is the algebraic analogue of the $\Gamma$ group of Segal-Wilson \cite{SW}. Note that the indices ``-'' and ``+'' do not coincide with the Segal-Wilson notations. Replacing $k((z))$ by $k((z^{-1}))$, we obtain their notation. \end{rem} Let us define the exponential maps for the groups $\Gamma_-$ and $\Gamma_+$. Let $\A_n$ be the $n$ dimensional affine space over $\spk$ with the additive group law, and ${\hat\A}_n$ the formal group obtained as the completion of $\A_n$ at the origin. We define ${\hat\A}_\infty$ as the formal group scheme $\limil{n}{\hat\A}_n$. Obviously it holds that: $${\hat\A}_\infty=\sf k\{\{y_1,\dots\}\}$$ with the additive group law. \begin{defn} If the characteristic of $k$ is zero, the exponential map for $\Gamma_-$ is the following isomorphism of formal group schemes: $$\aligned {\hat\A}_\infty & @>{\exp}>> \Gamma_- \\ \{a_i\}_{i>0} &\mapsto \exp(\sum_{i>0}a_i\,z^{-i}) \endaligned$$ This is the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}& @>{\qquad\exp^*\qquad}>> k\{\{y_1,\dots\}\}\\ x_i &\mapsto \text{ coefficient of $z^{-i}$ in the series } \exp(\sum_{j>0}y_j\,z^{-j})\endaligned$$ \end{defn} \begin{defn}\label{defn:exp-gamma-minus} If the characteristic of $k$ is positive, the exponential map for $\Gamma_-$ is the following isomorphism of formal schemes: $$\aligned {\hat\A}_\infty &\to \Gamma_- \\ \{a_i\}_{i>0}&\mapsto \prod_{i>0}(1-a_i\,z^{-i})\endaligned$$ which is the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}&@>\exp^*>> k\{\{y_1,\dots\}\}\\ x_i &\mapsto \text{ coefficient of $z^{-i}$ in the series } \prod_{i>0}(1-a_i\,z^{-i})\endaligned$$ \end{defn} Note that this latter exponential map is not a isomorphism of groups. Considering over ${\hat\A}_\infty$ the law group induced by the isomorphism, $\exp$, of formal schemes, we obtain the Witt formal group law. The exponential map for $\Gamma_+$ is defined in a analogous way; one only has to replace $z^{-i}$ by $z^i$ in the above expressions. (See \cite{B} for the connection of these definitions and the Cartier-Dieudonn\'e theory). The following property gives the structure of $\kz$. \begin{thm} The natural morphism of functors of groups over the category of $k$-schemes: $$\fu{\Gamma_-}\times\fu{{\mathbb G}_m}\times\fu{\Gamma_+}\to\kz$$ is injective and for $char(k)=0$ gives an isomorphism with $\kz_0$ (the connected component of the origin in the functor of groups $\kz$). The functor on groups $\kz_0$ is therefore ``representable'' by the (formal) $k$-scheme: $$\Gamma=\Gamma_-\times{\mathbb G}_m\times\Gamma_+$$ \end{thm} \subsection{``Formal Geometry''} It should be noted that the formal group scheme $\Gamma_-$ has properties formally analogous to the Jacobians of algebraic curves: one can define formal Abel maps and prove formal analogues of the Albanese property of the Jacobians of smooth curves. Let $\hat C=\sf(k[[t]])$ be a formal curve. The Abel morphism of degree $1$ is defined as the morphism of formal schemes: $$\phi: \hat C\to \Gamma_-$$ given by the $\hat C$-valued point of $\Gamma_-$: $\phi(t)=(1-\frac{t}z)^{-1}=1+\sum_{i>0}^{}\frac{t^i}{z^i}$; that is, the morphism induced by the ring homomorphism: $$\aligned k\{\{x_1,\dots\}\}&\to k[[t]]\\ x_i\,&\mapsto t^i\endaligned$$ Note that the Abel morphism is the algebro-geometric version of the function $q_\xi(z)$ used by Segal-Wilson (\cite{SW} page~32) to study the Baker-Akhiezer function. Let us explain further why we call $\phi$ the ``Abel morphism'' of degree 1. If $char(k)=0$, composing $\phi$ with the inverse of the exponential map, affords: $$\bar\phi:\hat C @>{\phi}>> \Gamma_- @>{\exp^{-1}}>> {\hat\A}_\infty$$ and since $(1-\frac{t}z)^{-1}=\exp(\sum_{i>0}^{}\frac{t^i}{i\,z^i})$ (see \cite{SW} page~33), $\bar\phi$ is the morphism defined by the ring homomorphism: $$\aligned k\{\{y_1,\dots\}\} &\to k[[t]]\\ y_i&\mapsto \frac{t^i}i \endaligned$$ or in terms of the functor of points: $$\aligned\hat C& @>{\bar\phi}>> {\hat\A}_\infty\\ t&\mapsto \{t,\frac{t^2}2,\frac{t^3}3,\dots\} \endaligned$$ Observe that given the basis $\omega_i=t^i\,dt$ of the differentials $\Omega_{\hat C}=k[[t]]dt$, $\bar\phi$ can be interpreted as the morphism defined by the ``abelian integrals'' over the formal curve: $$\bar\phi(t)=\left( \int_0^t\omega_0,\int_0^t\omega_1,\dots,\int_0^t\omega_i,\dots \right)$$ which coincides precisely with the local equations of the Abel morphism for smooth algebraic curves over the field of complex numbers. \begin{rem}\label{rem:uni-ele} The above introduced notation is also motivated by the following two facts: \begin{itemize} \item $(\Gamma_-,\phi)$ satisfies the Albanese property for $\hat C$; that is, every morphism $\psi:\hat C\to X$ in a commutative group scheme (which sends the unique rational point of $\hat C$ to the $0\in X$) factors through the Abel morphism and a homomorphism of groups $\Gamma_-\to X$. This property follows from the following fact: the direct limit $\limi_{n}S^n\w C$ exists (as a formal scheme) and it is naturally isomorphic to $\Gamma_-$). \item Observe that for each element: $$u\in\Gamma_-(S)\subseteq\fu{k((z))^*}(S)=H^0(S,\o_S)((z))^*$$ we can define a fractionary ideal of the formal curve $\hat C_S$ by: $I_u=u\cdot\o_S((z))$. We can therefore interpret the formal group $\Gamma_-$ as a kind of Picard scheme over the formal curve $\hat C$ (see \cite{C,B2}). The universal element of $\Gamma_-$ is the invertible element of $\kz(\Gamma_-)$ given by: $$v(x,z)=1+\underset {i \geq 1}\sum x_i\,z^{-i}\in k((z))\hat \otimes k\{\{x_{ 1},x_{ 2},\dots\}\}$$ This universal element will be the formal analogue of the universal invertible sheaf for the formal curve $\hat C$. \end{itemize} \end{rem} \section{$\tau$-functions and Baker-Akhiezer functions} The first part of this section is devoted to defining the $\tau$-function and the Baker-Akhiezer function algebraically over an arbitrary base field $k$. We then prove an analogue for $\tau$ of the Addition Formulae for theta functions that will allow a characterization of the Baker-Akhiezer function, which is quite close to proposition 5.1 of \cite{SW}. Following on with the analogy between the groups $\Gamma$ and $\Gamma_-$ and the Jacobian of the smooth algebraic curves, we shall perform the well known constructions for the Jacobians of the algebraic curves for the formal curve $\hat C$ and the group $\Gamma$: Poincar\'e bundle over the dual Jacobian and the universal line bundle over the Jacobian. In the formal case, these constructions are essentially equivalent to defining the $\tau$-functions and the Baker-Akhiezer functions. Let us denote by $\gr^0(V)$ index zero connected component of the Grassmannian of $V=k((z))$ and by $\Gamma$ the group $\Gamma_-\times {\mathbb G}_m \times \Gamma_+$. Let $$\Gamma \times \gr^0(V) \overset \mu \longrightarrow \gr^0(V) $$ be the action of $\Gamma$ over the Grassmannian by homotheties. We define the Poincar\'e bundle over $\Gamma\times \gr^0(V)$ as the invertible sheaf: $${\mathcal P}=\mu^*\det_V^*$$ $p_2:\Gamma\times\gr^0(V) \longrightarrow \gr^0(V)$ being the natural projection. For each point $U\in \gr^0(V)$, let us define the Poincar\'e bundle over $\Gamma\times \Gamma$ associated with $U$ by: $${\mathcal P}_U=(1\times \mu_U)^*{\mathcal P}=m^*({\mu_U}^*\det_V^*)$$ where $\mu_U:\Gamma\to \Gamma(U) \subset \gr^0(V)$ is the action of $\Gamma$ on the orbit of $U$ and $ m :\Gamma\times\Gamma \to \Gamma $ is the group law. The sheaf of $\tau$-functions of a point $U\in \gr^0(V)$ (not necessarily a geometric point), $\tilde {\L_\tau}(U)$, is the invertible sheaf over $\Gamma\times \{U\}$ defined by: $$\tilde{\L_\tau}(U)={\mathcal P}\vert_{\Gamma\times\{U\}}$$ The restriction homomorphism induces the following homomorphism between global sections: \beq H^0\left(\Gamma\times \gr^0(V), \mu^*\det_V^*\right)\to H^0\left(\Gamma\times \{U\},\tilde {\L_\tau}(U) \right) \label{eq:restriction} \end{equation} \begin{defn}\label{defn:tau0} The $\tau$-function of the point $U\in\gr^0(V)$ over $\Gamma$ is defined as the image $\tilde {\tau }_U$ of the section $\mu^*\Omega_+$ by the homomorphism {\ref{eq:restriction}} ($\Omega_+$ being the global section defined in {\ref{defn:global-section}}). \end{defn} Obviously $\tilde {\tau }_U$ is not a function over $\Gamma\times \{U\}$ since the invertible sheaf $\tilde {\L_\tau}(U)$ is not trivial, but this definition is essentially the $\tau$-function defined by M. and Y. Sato and Segal-Wilson (\cite{SS,SW}). Their $\tau$-function is obtained by restricting the invertible sheaf $\tilde {\L_\tau}(U)$ to the subgroup $\Gamma_-\subset \Gamma$. From \cite{AMP} we know that the invertible sheaf over $\Gamma_-$: $${\L_\tau}(U)= \tilde {\L_\tau}(U)\vert_{\Gamma_-\times \{U\}}$$ is trivial, and that: $$\sigma_0(g)=g\cdot\delta_U$$ (where $g\in \Gamma_-)$ and $\delta_U$ be a non-zero element in the fibre of ${\L_\tau}(U)$ over the point $(1,U)$ of $\Gamma\times\{U\}$) is a global section of ${\L_\tau}(U)$ without zeroes; and, therefore, it gives a trivialization. With respect to this, the global section of ${\L_\tau}(U)$ defined by $\tilde{\tau}_U$ is identified with the function $\tau_U\in \o(\Gamma_-)=k\{\{x_1,\dots\}\}$ given by Segal-Wilson \cite{SW}: $${\tau}_U(g)=\frac{\tilde{\tau}_U}{\sigma_0}= \frac{\mu^*\Omega_+}{\sigma_0}=\frac{\Omega_+(gU)}{g\cdot\delta_U}$$ Observe that the $\tau$-function ${\tau}_U$ is not a series of infinite variables but rather an element of the ring $k\{\{x_1, \dots\}\}$. \begin{rem} In the literature (\cite{SS,AD,KNTY}) one also finds another definition of the $\tau$-function: for a geometric point $\tilde U\in\det_V$ in the fibre of $U\in\gr^0(V)$, the $\bar\tau$-function of $\tilde U$ is defined as the element $\bar \tau(U)\in H^0\left (\detd_V\right)^*\otimes k(U)$ ($k(U)$ being the residual field of $U$). The deep relationship between both definitions emerges through the so called boso\-ni\-zation isomorphism. To introduce this isomorphism certain preliminaries are necessary. Since the subgroup $\Gamma_+$ of $\Gamma$ acts freely on $\gr^0(V)$, the orbits of the rational points of $\gr^0(V)$ under the action of $\Gamma_+$ are isomorphic to $\Gamma_+$ (as schemes). Let $X$ be the orbit of $V_-=z^{-1}\cdot k[z^{-1}]\subset V$ under $\Gamma_+$. The restrictions of $\det_V$ and $\detd_V$ to $X$ are trivial invertible sheaves. Bearing in mind that the points of $X$ are $k$-vector subspaces of $V$ whose intersection with $V_+$ is zero, one has that the section $\Omega_+$ of $\det_V^*$ defines a canonical trivialization of $\detd_V$ over $X$. Now, the {\bf bosonization isomorphism} is the canonical isomorphism: $$\Omega(S)\longrightarrow \o(\Gamma_+)$$ ($\Omega(S)$ is defined in {\ref{defn:global-section}}) induced by the restriction homomorphism: $$B: H^0(\gr^0(V),\detd_V)\rightarrow H^0(X,\detd_V\vert_X)$$ and the isomorphism: $$ H^0(X,\detd_V\vert_X)\iso\o(\Gamma_+)$$ (associated to the trivialization defined by $\Omega_+$). Finally, note that there exists an isomorphism of $k$-vector spaces: $$\o(\Gamma_+)^*=k[x_1,\dots]^*\to \o(\Gamma_-)=k\{\{x_1,x_2,\dots\}\}$$ identifying the Schur polynomial $F_S$ with the linear form: $$F_{S'}\mapsto F_S(F_{S'})=\delta_{S,S'}$$ (for details see the first chapter of \cite{Mc}). Now, the composition of the homomorphism $B^*$ (the dual of $B$) and the above isomorphism gives: $$\tilde B^*:\o(\Gamma_+)^*=k\{\{x_1,x_2,\dots\}\} \longrightarrow H^0\left (\detd_V\right)^*$$ The relationship between $\tau_U$ and $\bar \tau (\tilde U)$ is: $\tilde B^*(\tau_U)=\bar \tau (\tilde U)$ (up to a non-zero constant). The transformations of vertex operators under this isomorphism can now be explicitely computed when the characteristic is zero (\cite{DJKM}). In our approach, vertex operators are to be understood as the formal Abel morphisms which will be studied below. \end{rem} Once we have defined the $\tau$-function algebraically, we can define the Baker-Akhiezer functions using formula~5.14. of \cite{SW}; this procedure is used by several authors. However, we prefer to continue with the analogy with the classical theory of curves and Jacobians and define the Baker-Akhiezer functions as a formal analogue of the universal invertible sheaf of the Jacobian. Let us consider the composition of morphism: $$\tilde \beta:\hat C\times \Gamma\times \gr^0(V) \overset {\phi\times Id}\longrightarrow \Gamma \times\Gamma\times\gr^0(V)\overset {m\times Id} \longrightarrow\Gamma\times \gr^0(V)$$ $\phi:\hat C=\sf k[[z]] \to \Gamma$ being the Abel morphism (taking values in $\Gamma_-\subset \Gamma$) and $m:\Gamma\times \Gamma\to\Gamma$ the group law. \begin{defn} The sheaf of Baker-Akhiezer functions is the invertible sheaf over $\hat C\times \Gamma\times \gr^0(V)$ defined by: $$\tilde\L_{\tiny BA}=(\phi\times Id)^*(m\times Id)^*{\mathcal P}$$ Let us define the sheaf of Baker-Akhiezer functions at a point $U\in \gr^0(V)$ as the invertible sheaf: $$\tilde\L_{\tiny BA}(U)=\tilde\L_{\tiny BA}\vert_{\hat C\times \Gamma\times \{U\}}=\tilde {\beta_U}^*\tilde {\L_\tau}(U)$$ where $\tilde {\beta_U}^*$ is the following homomorphism between global sections: $$ H^0(\Gamma\times \{U\},\tilde {\L_\tau}(U)) \overset{\tilde{\beta_U}^*}\longrightarrow H^0(\hat C\times\Gamma\times \{U\},\tilde\L_{\tiny BA}(U))$$ \end{defn} By the definitions, $\tilde\L_{\tiny BA}(U)\vert_{\hat C\times\Gamma_-\times \{U\}}$ is a trivial invertible sheaf over $\hat C \times \Gamma_-$. \begin{defn} The Baker-Akhiezer function of a point $U\in \gr^0(V)$ is $\psi_U=v^{-1}\cdot \beta^*_U(\tau_U)$ where $\beta^*$ is the restriction homomorphism: $$H^0\left( \Gamma\times \{U\},\tilde {\L_\tau}(U)\right)\overset{\beta^*}\longrightarrow H^0\left(\hat C\times \Gamma\times \{U\},\tilde\L_{\tiny BA}(U)\right)$$ induced by $\tilde{\beta_U}^*$. \end{defn} Note that from the definition one has the following expression for the Baker-Akhiezer function: \beq \psi_U(z,g)=v(g,z)^{-1}\cdot \frac{\tau_U\left(g\cdot \phi_1(z)\right)}{\tau_U(g)} \label{eq:BA}\end{equation} and that the Baker-Ahkiezer function of $V_-=z^{-1}\,k[z^{-1}]$ is the universal invertible element $v^{-1}$ (see remark {\ref{rem:uni-ele}}). When the characteristic of the base field $k$ is zero, we can identify $\Gamma_-$ with the additive group scheme ${\hat\A}_{\infty}$ through the exponential. Therefore, the latter expression is the classical expression for the Baker-Akhiezer functions (\cite{SW}~5.16): $$\psi_U(z,t)=\exp(-\sum t_i\,z^{-i})\cdot \left(\frac{\tau_U(t+[z])}{\tau_U(t)}\right)$$ where $[z]=(z,\frac{1}{2}z^2,\frac{1}{3} z^3,\dots)$ and $t=(t_1,t_2,\dots)$ and, through the exponential map, $v(t,z)=\exp(\sum t_i\,z^{-i})$. For the general case, we obtain explicit expressions for $\psi_U$ as a function over $\hat C\times {\hat\A}_{\infty}$ but considering in ${\hat\A}_{\infty}$ the group law, $*$, induced by the exponential {\ref{defn:exp-gamma-minus}}: $$\psi_U(z,t)=v(t,z)^{-1}\cdot \frac{\tau_U\left(t* \phi(z)\right)}{\tau_U(t) }$$ \begin{rem} Note that our definitions of $\tau$-functions and Baker-Akhiezer functions are valid over arbitrary base schemes. One then has the notion of $\tau$-functions and Baker-Akhiezer functions for families of elements of $\gr^0(V)$ and, if we consider the Grassmannian of $\Z((z))$ one then has $\tau$-functions and Baker-Akhiezer functions of the rational points of $\gr\left(\Z((z))\right)$ and the geometric properties studied in this paper have a translation into arithmetic properties of the elements of $\gr\left(\Z((z))\right)$. The results stated by Anderson in \cite{An} are a particular case of a much more general setting valid not only for $p$-adic fields but also for arbitrary global numbers field. \end{rem} The fundamental property of the $\tau$-function is the analogue of the Addition Formulae. Let $\phi_N$ be the Abel morphism of degree $N$ ($N>0$); that is, the morphism $\hat C^N\to\Gamma_-$ given by $\prod_{i=1}^N(1-\frac{x_i}z)^{-1}$, where $\hat C^N=\sf(A)$ ($A=k[[x_1,\dots,x_N]]$). Let $\Gamma_-=\sf k\{\{t_1,\dots\}\}$ and observe that, in this setting, $t_i$ should be interpreted as the coefficient of $z^{-i}$ in $\prod(1-\frac{x_j}z)^{-1}$. For simplicity's sake, $U_A$ will denote the point $U\hat\otimes A\in\gr^0(V)(A)$ for a rational point $U\in\grv$, and $V$ (resp. $V_+$) will denote $\w V_A$ (resp. $(\w{V_+})_A$). Now, for our $\tau$-function we shall give formal analogues of the addition formula of \cite{SS} and of corollary~2.19 of \cite{F} for theta functions and of Lemma~4.2 of \cite{Kon} for $\tau$-funtions. Let us begin with an explicit computation for $\tau$. \begin{lem}\label{lemma:add-for} Let $U$ be a rational point of $\gr^0(V)$. Assume that $V/V_++z^N\cdot U=0$ and $V_+\cap z^N\cdot U$ is $N$-dimensional, and let $\{f_1,\dots,f_N\}$ be a basis of the latter. One then has that: $$\phi_N^*\tau_U = \frac{1}{\prod_{i< j}(x_i-x_j)}\cdot det \pmatrix f_1(x_1) & \dots & f_1(x_N) \\ \vdots & &\vdots \\ f_N(x_1) & \dots & f_N(x_N) \endpmatrix$$ as functions on $\hat C^N$ (up to elements of $k^*$). \end{lem} \begin{pf} By the very definition of the $\tau$-function and by the properties of $Det$, it follows that $\phi_N^*\tau_U$ equals the determinant of the inverse image of the complex: $$\L\to\left({ V}/{ V_+}\right)_{\gr^0(V)}$$ by the morphism $\hat C^N\to\Gamma_-\to\gr^0(V)$, which is precisely: $${\mathcal C}^\punto\equiv g\cdot U_A\to{A((z))}/{A[[z]]}=V/V_+$$ Let us define the following homomorphism of $A$-modules: $$\aligned \alpha_N:A[[z]] & \to A^N \\ f(z)&\mapsto\left(f(x_1),\dots,f(x_N)\right)\endaligned$$ whose kernel is the ideal generated by $\prod_{i=1}^N (z-x_i)$. One thus has the following exact sequence of complexes of $A$-modules: {\small $$\minCDarrowwidth17pt\CD 0 @>>> \prod(z-x_i)\cdot { V_+} @>>> { V_+} @>>> {{ V_+}}/{\prod(z-x_i)\cdot { V_+}} @>>> 0 \\ @. @V{\beta}VV @V{(\pi,\alpha_N)}VV @V{\bar\alpha_N}VV \\ 0 @>>> V/{z^N\cdot U_A} @>>> \left(V/{z^N\cdot U_A}\right)\oplus A^N @>>> A^N @>>> 0 \endCD$$ } The complex on the middle (right hand side resp.) will be denoted by ${\mathcal C}^\punto_1$ (${\mathcal C}^\punto_2$ resp.). Further, note that the complex on the left hand side is quasi-isomorphic to ${\mathcal C}^\punto$. \noindent Observe that: \begin{itemize} \item $\det({\mathcal C}^\punto_2)$ is isomorphic to the ideal of the diagonals (as an $A$-module), and the section $det(\bar\alpha_N)$ is exactly $\prod_{i<j}(x_i-x_j)$, \item $det(\beta)=det((\pi,\alpha_N))\cdot det(\bar\alpha_N)^{-1}$. \end{itemize} Moreover, we also have another exact sequence: {\small $$\minCDarrowwidth17pt\CD 0 @>>> { V_+}\cap z^N\cdot U_A @>>> { V_+} @>>> {{ V_+}}/{{ V_+}\cap z^N\cdot U_A} @>>>0 \\ @. @V{\alpha^U_N}VV @V{(\alpha_N,\pi)}VV @V{\simeq}VV \\ 0 @>>> A^N @>>> A^N \oplus{ V}/{z^N\cdot U_A} @>>> { V}/{z^N\cdot U_A} @>>> 0 \endCD$$ } Let ${\mathcal C}^\punto_3$ denote the complex on the right hand side. The hypothesis implies that the non-trivial homomorphism of the complex ${\mathcal C}^\punto_3$ is an isomorphism. They also imply that $\det({\mathcal C}^\punto_3)\in k^*$. From both these exact sequences one has the following relation: $$det(\beta)=\frac{det(\alpha^U_N)}{\prod_{i<j}(x_i-x_j)}\qquad \text{(up to an element of $k^*$)}$$ In order to compute $det(\alpha^U_N)$, let us choose $\{f_1,\dots,f_N\}$ a basis of ${ V_+}\cap z^N\cdot U_A$ (as an $A$-module). The matrix associated to $\alpha^U_N$ is now: $$\pmatrix f_1(x_1) & \dots & f_1(x_N) \\ \vdots & &\vdots \\ f_N(x_1) & \dots & f_N(x_N) \endpmatrix$$ and the formula follows. \end{pf} This proof implies that (in the same conditions) $\phi_N^*\tau_{hU}=\prod_i h(x_i)\cdot\phi_N^*\tau_U$, where $h\in\Gamma_+$, and also the following theorem. \begin{thm}[Addition Formulae \cite{SS}] Let $U\in\gr^0(V)$ be a rational point; then, for all $n\leq N$ (natural numbers) the functions: $$\left\{ \prod_{j<k}(x_{i_k}-x_{i_j}) \cdot \phi_{i_1\dots i_n}^*\tau_U \right\}_{0<i_1<\dots<i_n\leq N}$$ satisfy the Pl\"ucker equations. Here, $\phi_{i_1\dots i_n}$ denotes the morphism $\hat C^N\to\Gamma_-$ given by $\prod_j(1-\frac{x_{i_j}}z)^{-1}$. \end{thm} We finish this section with a characterization of the Baker-Akhiezer function. The importance of this result will be clear in the next section. Further, this characterization will show the close relationship between the Baker-Akhiezer of a point $U\in\gr^0(V)$ and a basis of $U$ as $k$-vector space. (Compare with Proposition 5.1 of \cite{SW} and Proposition 4.8 of \cite{KNTY}). \begin{thm}\label{4:thm:BA} Let $U\in\gr^0(V)$ be a rational point. Then: $$\psi_U(z,t)= z\cdot\sum_{i\geq 1} \psi_U^{(i)}(z) p_i(t)$$ where $\psi_U^{(i)}(z)\in U$ and $p_i(t)\in k\{\{t_1,\dots\}\}$ is independent of $U$. Furthermore, let $U$ be in $F_{A_S}$ for a sequence $S$ and $g$ be $\prod_{j=1}^n(1-\frac{x_j}z)^{-1}\in\Gamma_-$. Define $s_i$ by $\Z-S=\{s_1,s_2,\dots\}$ (as in {\ref{subsect:comp-inf-grass}}). Then $\psi_U^{(i)}(z)$ has a pole in $z=0$ of order $s_i$, and $p_i(x)$ is a homogeneous polynomial in the $x$'s of degree $i-1$. \end{thm} \begin{pf} Since $\Gamma_-=\sf k\{\{t_1,\dots\}\}$ is the direct limit of the symmetric products of $\hat C$ (recall the relation between the $x$'s and $t$'s), let us compute $\psi_U\vert_{\hat C^N}$ using {\ref{lemma:add-for}} for $N>>0$. Choose $N$ such that $z^{-N-i}k[[z]]\cap U$ is $N+i$-dimensional for $i=0,1$, and let $\{f_1(z),\dots,f_N(z)\}$ be a basis of $z^{-N}k[[z]]\cap U$, and $\{f_1(z),\dots,f_{N+1}(z)\}$ of $z^{-N-1}k[[z]]\cap U$. With no loss of generality, we can assume that $f_i$ has a pole at $z=0$ of order $s_i$. Recall that for $g=\prod_{j=1}^n(1-\frac{x_j}z)^{-1}$ the Baker-Akhiezer function is: \beq \psi_U(z,g)=g(z)^{-1}\cdot \frac{\tau_U(g\cdot\phi(z))}{\tau_U(g)} \label{eq:eq1} \end{equation} Let $\phi_{1,N}$ be the morphism $\hat C\times\hat C^N\to\Gamma_-$ given by $g\cdot(1-\frac{\bar z}z)^{-1}$, and $\phi_N$ the morphism $\hat C^N\to\Gamma_-$ given by $g$. Here, the ring of $\hat C^N$ is $k[[x_1,\dots,x_N]]$ and the ring of $\hat C\times\hat C^N$ is $k[[\bar z]]\hat\otimes k[[x_1,\dots,x_N]]$. By lemma {\ref{lemma:add-for}} one sees that: $$\frac{\phi^*_{1,N}\tau_U}{\phi^*_N\tau_U} =(-1)^N\cdot\prod_{i=1}^N(\bar z-x_i)^{-1}\cdot \left(\bar f_{N+1}(\bar z)\prod_{i}x_i+ \sum_{i=1}^N\bar f_i(\bar z)p_i(x_1,\dots,x_N)\right)$$ where $\bar f_i(z)=z^{N+1}f_i(z)$ and $p_i$ is a symmetric polynomial. Now, from the above expression for the Baker-Akheizer function, one has: $$\psi_U(z,t)\vert_{\hat C^N}= z\cdot\sum_{i=1}^{N+1}f_i(z)p_i(x_1,\dots,x_N)$$ and therefore $\psi_U(z,t)= z\cdot\sum_{i>0}f_i(z)p_i(t)$, where $f_i(z)\in U$ and $p_i(t)\in k\{\{t_1,\dots\}\}$. Further, since $\{\psi_U(z,t)\vert_{\hat C^N}\}_{N\in{\mathbb N}}$ is an element of an inverse limit and $p_{N+1}(x)\vert_{\hat C^N}$ is known, one can compute $p_i$ explicitly. From the choice of $f_i$ and the properties of $p_i$ one concludes. \end{pf} Now, all the above definitions and results on tau and BA functions can be generalized to the case of $U\in\gr^n(V)$, a point in an arbitrary connected component. \begin{defn}[$\tau$ and BA functions for arbitrary points] Let $U$ be a point of $\gr^n(V)$ ($n\in\Z$). Define its $\tau$-function by Definition \ref{defn:tau0} but replacing $\Omega_+$ by $\Omega_+^n$ (see Remark \ref{rem:indexn}). Define its Baker-Akhiezer function by formula \ref{eq:BA}. \end{defn} Observe that from the definition of $\tau$-function, we have that: $$\tau_U(g)\,=\,\frac{\Omega_+^n(gU)}{g\cdot \delta_U}\,=\, \frac{\Omega_+(g\cdot z^{-n}U)}{g\cdot \delta_U}$$ Furthermore, Lemma \ref{lemma:add-for} can be generalized in the following way: let $U$ be a rational point of $\gr^n(V)$. Assume that $V/V_++z^{N-n}\cdot U=0$ and $V_+\cap z^{N-n}\cdot U$ is $N$-dimensional, and let $\{f_1,\dots,f_N\}$ be a basis of the latter. One then has that: $$\phi_N^*\tau_U = \frac{1}{\prod_{i< j}(x_i-x_j)}\cdot det \pmatrix f_1(x_1) & \dots & f_1(x_N) \\ \vdots & &\vdots \\ f_N(x_1) & \dots & f_N(x_N) \endpmatrix$$ as functions on $\hat C^N$ (up to elements of $k^*$). Similarly, the very definition of Baker-Ahkiezer function implies that: $$\psi_U(z,t)\,=\,\psi_{z^{-n}U}(z,t)$$ for a point $U\in\gr^n(V)$. Moreover, the proof of Theorem \ref{4:thm:BA} shows that in this case: \beq \psi_U(z,t)\,=\,z^{1-n}\cdot\sum_{i\geq 1} \psi_U^{(i)}(z) p_i(t) \label{eq:BA=sum}\end{equation} where $\psi_U^{(i)}(z)\in U$ and $p_i(t)\in k\{\{t_1,\dots\}\}$ is independent of $U$. \section{Bilinear Identity and KP Hierarchy} This section has two well defined parts. In the first part the famous Residue Bilinear Identity is deduced, while in the second one the equivalence with the KP hierarchy is shown, which was already known (see \cite{DJKM}). Nevertheless, the importance lies not in the result but in the proofs, since the methods used will also allow us to prove the fundamental theorems of our paper. The essential ingredient comes from the relationship between $\gr(k((z)))$ and $\gr(k((z))^c)$ outlined in {\ref{subsect:grass-dual}}. However, in our case there exists a metric on $V$; namely, that induced by the residue pairing: $$(f,g) \,=\, \res_{z=0} (f\cdot g) dz $$ Let us denote by $\bar z^i$ the element of $V^c$ such that $\bar z^i(z^j)=\delta_{ij}$. One can formally write $V^c=k((\bar z^{-1}))$ (as $k$-vector spaces), and it may be seen that the homomorphism induced by the residue: $$\aligned V&\to V^c \\ z^i &\mapsto \bar z^{-i-1} \endaligned$$ is in fact an isomorphism, and sends $V_+$ to $V_+^\diamond$. It therefore induces an isomorphism $\gr(V^c,\B^\diamond)\iso\gr(V,\B)$. The composition of the latter isomorphism and the isomorphism $I$ constructed in {\ref{subsect:grass-dual}} gives a non-trivial automorphism of the Grassmannian: $$\aligned R:\grv &\longrightarrow \grv \\ L &\longmapsto L^{\perp} \endaligned$$ Trivial calculation shows that $R^*\det_V\simeq \det_V$, and that the index of a point $L\in\grv$ is exactly the opposite of the index of $R(L)=L^\perp$. Therefore, this induces an involution: $$R^*: H^0(\gr^n(V),\detd_{n})\to H^0(\gr^{-n}(V),\detd_{-n})$$ It is now straightforward to see that $R^*\Omega_+^{-n}=\Omega_+^{n}$. The last remarkable fact about $R$ is that for a given point $U\in\grv$ the morphism: $$\mu_U:\Gamma\to \grv$$ ($\mu_U$ being that induced by the action of $\Gamma$ on $\grv$) is equivariant with respect to ``passing to the inverse'' in $\Gamma$ and $R$ in $\grv$. In other words: $$(g\cdot U)^\perp= g^{-1}\cdot U^\perp$$ From the latter two observations, one has trivially $\Omega_+^n(g\cdot U^\perp)=\Omega_+^n((g^{-1}\cdot U)^\perp)=\Omega_+^{-n}(g^{-1}\cdot U)$, and hence: $$\tau_{U^\perp}(g)=\tau_U(g^{-1})$$ This motivates us to give the following: \begin{defn} The adjoint Baker-Akheizer function of a point $U\in\grv$ is: $$\psi^*_U(z,g)=\psi_{U^\perp}(z,g^{-1})$$ (Recall that: $\psi_{U^\perp}(z,g)= g^{-1}\frac{\tau_U(g^{-1}*\phi(z)^{-1})}{\tau_U(g^{-1})}$). \end{defn} Note that in characteristic zero, the definition gives: $$\psi^*_U(z,t)=\exp(\sum_{i>0}t_i\,z^{-i})\frac{\tau_U(t-[z])}{\tau_U(t)}$$ By the very definition of the Baker-Akhiezer function and formula \ref{eq:BA=sum}, it follows that: $$\left(\psi_{U^\circ}(\bar z^{-1},t')\right) \left(\psi_U(z,t)\right) \,=\,0$$ or, in other words: $$\res_{z=0} \psi_U(z,t)\psi_{U^\perp}(z,t')\frac{dz}{z^2}\,=\,0$$ Using the adjoint Baker-Akhiezer function of $U$ as $\psi_{U^\perp}$, then the latter equation can be rewritten as: \beq \res_{z=0} \psi_U(z,t)\psi^*_{U}(z,t')\frac{dz}{z^2}\,=\,0 \label{eq:residue} \end{equation} \begin{thm}\label{thm:res-equa} Let $U$ and $U'$ be two rational points of the Grassmannian and let us assume they have the same index. Then, it holds that: $$\res_{z=0} \psi_U(z,t)\psi^*_{U'}(z,t')\frac{dz}{z^2}\,=\,0\quad\iff\quad\ U=U'$$ \end{thm} \begin{pf} The converse is already shown. For the direct one, observe that the identity and formula \ref{eq:BA=sum} imply that: ${U'}^\perp\subseteq U^\perp$. Recalling that both have the same index and that the metric is non-degenerate, the result follows. \end{pf} When the base field is the field of complex numbers, $\mathbb C$, it is well known that the above results (valid over arbitrary fields) turn out to be equivalent to the KP hierarchy (see \cite{DJKM,F2}). The goal now is to show how the KP hierarchy is obtained from the Residue Bilinear Identity (when $k={\mathbb C}$). In this procedure we shall use Schur polynomials and their properties in a very fundamental way. However, the advantage of this is that we can deal in a similar way with the residue condition that characterizes the moduli space of pointed curves in the Grassmannian, and this will allow us to compute its equations explicitly. Let us begin with some preliminaries on symmetric polynomials following the first chapter of \cite{Mc}. Let $k$ be a ring of characteristic zero, and let $\mathcal S$ denote the subring of $k\{\{x_1,\dots\}\}$ consisting of symmetric polynomials. Given a decreasing sequence $\lambda$ of natural numbers $\lambda_1\geq\dots\geq\lambda_n$ define the associated Schur polynomial as: $$\chi_\lambda(x)=\frac{det(x_j^{\lambda_i+n-i})}{det(x_j^{n-i})}$$ It is also known that there exists a non-degenerated metric in $\mathcal S$ for which the Schur polynomials are an orthonormal basis. Denote by $\chi^*_\lambda$ the linear form ${\mathcal S}\to k$ defined by: $$\chi^*_\lambda(\chi_\mu(x))=\delta_{\lambda,\mu}$$ Therefore, the identity morphism $Id:{\mathcal S}\to {\mathcal S}$ can be expressed as $\sum_\lambda\chi_\lambda(x) \chi^*_\lambda$. That is, for each element $p(x)\in{\mathcal S}$ one has that: $$p(x)=\sum_\lambda \chi_\lambda(x)\chi^*_\lambda(p)$$ \begin{rem}\label{rem:Taylor} \begin{enumerate} \item Note that if $k$ is a field, then one can express $\chi^*_\lambda$ in terms of differential operators. Explicitly, this is: $$\chi^*_\lambda=\chi_\lambda(\tilde\partial_x)\vert_{x=0}$$ where $\tilde\partial_x= (\frac{\partial}{\partial x_1},\frac12\frac{\partial}{\partial x_2},\dots)$. The operator $\sum_\lambda \chi_\lambda(x)\chi_\lambda(\tilde\partial_x)\vert_{x=0}$ will be called Taylor expansion operator. Analogously, for more than one set of variables, for instance $x=(x_1,x_2,\dots)$ and $y=(y_1,y_2,\dots)$, the identity can be written as: $$Id=\sum_{\lambda,\mu}\chi_\lambda(x)\chi_\mu(y) \chi_\lambda(\tilde\partial_x)\chi_\mu(\tilde\partial_y)\vert_{x=y=0}$$ \item Assume we have two set of variables, as above. Observe that (applying the Taylor expansion operator to $f$): $$\left(\sum_\lambda\chi_\lambda(y)\chi^*_\lambda(\tilde\partial_x)\right)f(x) \vert_{x=0}=f(y)$$ That is, this operator replaces $x_i$ by $y_i$. \item Similarly, by replacing $x_i$ by $\frac{z^i}i$ and denoting by $p_j$ the polynomials defined by $\exp(\sum_{i>0} x_i z^i)=\sum_{j\geq 0} p_j(x)z^j$, one has: $$\text{coefficient of $z^j$ in $f(z)$ }=\, p_j(\tilde\partial_x)\vert_{x=0}f(x)$$ (note that $p_j(\tilde\partial_x)\vert_{x=0}(p_i(x))=\delta_{ij}$). \item Now, observe that: $$P(\partial_y)\vert_{y=0}f(x+y)=P(\partial_x)f(x)$$ where $P$ is a polynomial, $f$ a function, and $x+y$ denotes $(x_1+y_1,x_2+y_2,\dots)$. It therefore follows that: $$\text{coefficient of $z^j$ in }f(x+[z])\,=\, p_j(\tilde\partial_y)\vert_{y=0}f(x+y) \,=\, p_j(\tilde\partial_x)f(x)$$ (where $[z]$ is $(z,\frac{z^2}2,\dots)$), and hence: $$f(x+[z])=\sum_j z^jp_j(\tilde\partial_x)f(x)$$ That is, the operator $\sum_j z^jp_j(\tilde\partial_x)$ (also written as $\exp(\sum z^i\tilde\partial_{x_i})$) replaces $x_i$ by $x_i+\frac{z^i}i$. Finally, this operator relates the $\tau$-function and the Baker-Akhiezer function by: $$\psi_U(z,t)=\exp(-\sum t_iz^{-i})\frac {\exp\left(\sum z^i\tilde\partial_t\right)\tau_U(t)}{\tau_U(t)}$$ \end{enumerate} \end{rem} We can now state the main result of this subsection in the form given in \cite{F2}. \begin{thm}[KP Equations]\label{thm:KP-eq} The condition: $$\operatorname{Res}_{z=0}\psi_U(z,t)\cdot\psi^*_U(z,t')\frac{dz}{z^2}=0$$ for a rational point $U\in\grv$ is equivalent to the infinite set of equations (indexed by a pair of Young diagrams $\lambda_1,\lambda_2$): $$\left(\sum p_{\beta_1}(\tilde\partial_t)D_{\lambda_1,\alpha_1}(-\tilde\partial_t) \cdot p_{\beta_2}(-\tilde\partial_{t'})D_{\lambda_2,\alpha_2}(\tilde\partial_{t'}) \right)\vert_{t=t'=0}\tau_U(t)\cdot \tau_U(t')\,=\,0$$ where the sum is taken over the 4-tuples $\{\alpha_1,\beta_1,\alpha_2,\beta_2\}$ of integers such that $-\alpha_1+\beta_1-\alpha_2+\beta_2=1$, and $D_{\lambda,\alpha}=\sum_\mu \chi_\mu$ where $\mu$ is a Young diagram such that $\lambda-\mu$ is a horizontal $\alpha$-strip. \end{thm} \begin{pf} First, observe that $\Gamma_-$ (and equivalently $\Gamma_+$) acts on $\grv$ by homotheties preserving the determinant sheaf. Then, by straightforward computation one shows that $\tau_U(t)=\sum_{\lambda}\Omega_\lambda(U) \chi_\lambda(t)$ (the sum taken over the set of Young diagrams). Now recall two basic facts about functions $f(x)\in k\{\{x_1,x_2,\dots\}\}$: the coefficient of $z^\beta$ in $f([z])$ is precisely $p_\beta(\tilde\partial_y)f(y)\vert_{y=0}$ ($[z]$ being $(z,\frac12z^2,\dots)$); and $\chi_\lambda(\tilde\partial_y)\vert_{y=0}f(x+y)= \chi_\lambda(\tilde\partial_x)f(x)$. From them it follows that: $$\text{coefficient of $z^\beta$ in }f(x+[z])\,=\, p_\beta(\tilde\partial_x)f(x)$$ Let us now begin, properly speaking, with the proof of the theorem. The residue condition is trivially equivalent to: $$\text{coefficient of $z$ in } \exp(-\sum t_iz^{-i})\tau_U(t+z)\exp(\sum t'_iz^{-i})\tau_U(t'-z)\,=\,0$$ This coefficient is given by the sum: $$\sum p_{\alpha_1}(-t)p_{\beta_1}(\tilde\partial_t)\tau_U(t)\cdot p_{\alpha_2}(t')p_{\beta_2}(-\tilde\partial_{t'})\tau_U(t')$$ over the 4-tuples $\{\alpha_1,\beta_1,\alpha_2,\beta_2\}$ of integers such that $-\alpha_1+\beta_1-\alpha_2+\beta_2=1$. But now a necessary and sufficient condition for an element $f(t,t')\in k\{\{t,t'\}\}$ to be zero is that $\chi_{\lambda_1}(\tilde\partial_t)\chi_{\lambda_2}(\tilde\partial_{t'}) \vert_{t=t'=0}$ applied to $f$ must be zero for all pairs of Young diagrams $\lambda_1,\lambda_2$. Simple computation together with formula {\bf I}.5.16 of \cite{Mc} gives the following infinite set of quadratic differential equations on $\tau_U$: $$\left(\sum p_{\beta_1}(\tilde\partial_t)D_{\lambda_1,\alpha_1}(-\tilde\partial_t) \cdot p_{\beta_2}(-\tilde\partial_{t'})D_{\lambda_2,\alpha_2}(\tilde\partial_{t'}) \right)\vert_{t=t'=0}\tau_U(t)\cdot \tau_U(t')\,=\,0$$ where $D_{\lambda,\alpha}=\sum_\mu\chi^*_\mu$ (the sum taken over the set of Young diagrams $\mu$ such that $\lambda-\mu$ is a horizontal $\alpha$-strip). \end{pf} \begin{rem} Recalling Remark {\ref{rem:grass=plucker}} one obtains four different ways to characterize the set of rational points of the infinite Grassmannian $\gr({\mathbb C}((z)))$ into the infinite dimensional projective space $\P\Omega(S)^*$; namely, the Pl\"ucker equations (\cite{Pl2}), the Bilinear Residue Identity (see Proposition 4.15 of \cite{KNTY}, the KP hierarchy (\cite{DJKM}) and the Hirota's Bilinear equations (\cite{DJKM}). In the last two characterizations the differential operators introduced in Remark \ref{rem:Taylor} are needed. \end{rem} \section{Characterization and Equations of the moduli space of pointed curves in the Grassmannian} In this section, and since our goal is to compute equations for the moduli of pointed curves, we study a slightly modified Krichever construction; namely, the application: $$\aligned\{(C,p,\phi)\} &\to \grv \\ (C,p,\phi) &\mapsto \phi\left(H^0(C-p,\o_C)\right) \endaligned$$ Here and in the sequel $\grv$ will denote the infinite Grassmannian of the data $(V=k((z)),\B,V_+=k[[z]])$ as in Example \ref{2:exam}. Let us introduce some more notation. Given a $k$-scheme $S$ define: $$\begin{aligned} \o_S[[z]] \,&=\, \limpl{n}\o_S[z]/z^n \\ \o_S((z)) \,&=\, \limil{m}z^{-m}\o_S[[z]] \end{aligned}$$ Given a flat curve $\pi:C\to S$ and a section $\sigma: S\to C$ defining a Cartier divisor $D$ denote: $$\widehat\o_{C,D} \,=\, \limpl{n}\o_C/\o_C(-n)$$ where $\o_C(-1)$ is the ideal sheaf of $D$. Observe that $\widehat\o_{C,D}$ is supported along $\sigma(S)$ and it is then a sheaf of $\o_S$-algebras. We also define the following sheaf of $\o_S$-algebras: $$\widehat \Sigma_{C,D}\,=\,\limil{m}\widehat\o_{C,D}(m)$$ \begin{defn} Let $S$ be a $k$-scheme. Define the functor $\tilde\M^g_{\infty}$ over the category of $k$-schemes by: $$S\rightsquigarrow \tilde\M^g_{\infty}(S)= \{\text{ families $(C,D,\phi)$ over $S$ }\}$$ where these families satisfy: \begin{enumerate} \item $\pi:C\to S$ is a proper flat morphism, whose geometric fibres are integral curves of arithmetic genus $g$, \item $\sigma:S\to C$ is a section of $\pi$, such that when considered as a Cartier Divisor $D$ over $C$ it is smooth, of relative degree 1, and flat over $S$. (We understand that $D\subset C$ is smooth over $S$, iff for every closed point $x\in D$ there exists an open neighborhood $U$ of $x$ in $C$ such that the morphism $U\to S$ is smooth). \item $\phi$ is an isomorphism of $\o_S$-algebras: $$\widehat \Sigma_{C,D}\,\iso\, \o_S((z))$$ \end{enumerate} \end{defn} On the set $\tilde\M^g_{\infty}(S)$ one can define an equivalence relation, $\sim$: $(C,D,\phi)$ and $(C',D',\phi')$ are said to be equivalent, if there exists an isomorphism $C\to C'$ (over $S$) such that the first family goes to the second under the induced morphisms. \begin{defn} The moduli functor of pointed curves of genus $g$, $\M^g_{\infty}$, is the functor over the category of $k$-schemes defined by the sheafication of the functor: $$S\rightsquigarrow {\tilde\M^g_{\infty}(S)}/{\sim}$$ (the superindex $g$ will be left out to denote the union over all $g\geq 0$). \end{defn} \begin{prop} The sheaf $\limil{m}\pi_*\o_{C,D}(m)$ is an $S$-valued point of $\gr^{1-g}(\widehat \Sigma_{C,D})$ for all $(C,D,\phi)\in\M^g_{\infty}$. \end{prop} \begin{pf} Consider the following exact sequence: $$0\to \o_{C,D}(-n)\to \o_{C,D}(m)\to \o_{C,D}(m)/\o_{C,D}(-n)\to 0$$ for $n,m\geq 0$. Take now $\pi_*$ and recall that: $$\pi_*\left(\o_{C,D}(m)/\o_{C,D}(-n)\right)\,\iso\, \widehat \o_{C,D}(m)/\widehat\o_{C,D}(-n)$$ since it is concentrated on $\sigma(S)$. One then has the following long exact sequence: $$0\to\pi_*\o_{C,D}(-n)\to\pi_* \o_{C,D}(m)\to \widehat\o_{C,D}(m)/\widehat\o_{C,D}(-n)\to R^1\pi_*\o_{C,D}(-n)$$ and hence: $$0\to\pi_*\o_{C,D}(-n)\to\limil{m}\pi_* \o_{C,D}(m)\to \widehat\Sigma_{C,D}/\widehat\o_{C,D}(-n)\to R^1\pi_*\o_{C,D}(-n)$$ Recalling now from \cite{Al} a Grauert type theorem that holds in this case: there exists $\{U_i\}$ a covering of $S$ and $\{m_i\}$ integers such that $\pi_*\o_C(m_i)_{U_i}$ is locally free of finite type, and $R^1\pi_*\o_C(m_i)_{U_i}=0$; the result follows. \end{pf} Observe now that $\phi$ induces an isomorphism of infnite Grassmannians: $$\gr^{1-g}(\widehat \Sigma_{C,D})\,\iso\,\gr^{1-g}(V)$$ in a natural way since there exists an integer $r$ such that: $$z^{n+r}\o_S[[z]]\,\subseteq\,\phi(\widehat\o_{C,D}(n))\,\subseteq \,z^{n-r}\o_S[[z]]\qquad\forall n$$ Summing up: $$\phi\left(\limil{m}\pi_*\o_{C,D}(m)\right)$$ is a $S$-valued point of $\gr^{1-g}(V)$ for all $(C,D,\phi)\in\M^g_{\infty}$. In other words, we have defined a morphism of functors: $$\aligned K:\M_{\infty}&\longrightarrow \grv\\ (C,D,\phi)&\longmapsto \phi\left(\limil{n}\pi_*\o_C(n)\right) \endaligned $$ which will be called the {\bf Krichever morphism}.(Note that $K$ considered for the $\sp({\mathbb C})$-valued points is the usual Krichever map, see \cite{K,PS,SW}). Now let us state the following characterization of the image of $K$, which is well known in the complex case: \begin{thm}\label{thm:krich-char} A point $U\in\grv(S)$ lies in the image of the Krichever morphism, if and only if $\o_S\subset U$ and $U\cdot U\subseteq U$ (where $\cdot$ is the product of $V$). \end{thm} \begin{pf} Assume we have such a point $U$ of $\grv(S)$. Since $\M_{\infty}$ and $\grv$ are sheaves, we can assume that $S$ is affine, $S=\sp(B)$. Now, $U\subset B((z))$ is a sub-$B$-algebra and has a natural filtration $U_n$ by the degree of $z^{-1}$. Let $\mathcal U$ denote the associated graded algebra. It is not difficult to prove that $C=\proj_B({\mathcal U})$ is a curve over $B$. Let $I$ be the ideal sheaf generated by the elements $a\in{\mathcal U}$ such that the homogeneous localization ${\mathcal U}_{(a)}^0$ is isomorphic to $\mathcal U$. Since $I$ is locally principal, it defines a section $\sigma:S\to C$. Finally, from the inclusion $U\subset \o_S((z))$ one easily deduces an isomorphism $\phi:\widehat \Sigma_{C,D}\iso\o_S((z))$. An easy calculation shows that this construction and the Krichever morphism are the inverse of each other. \end{pf} \begin{thm} $\M_\infty$ is representable by a closed subscheme of $\grv$. \end{thm} \begin{pf} By the preceding characterization of $\M_\infty$, it will suffice to recall the following result of \cite{Al}: Let $A$ be a $k$-algebra, and $V$ a sheaf of $A$-modules over the category of $A$-algebras. Let $M$, $M'$ be two quasi-coherent subsheaves of $V$, such that $V/M'$ is isomorphic to a inverse limit of finite type free $A$-modules. There then exists an ideal $I\subseteq A$ such that every morphism $f:A\to B$ with the property $M_B\subseteq M'_B$ (as subsheaves of $V_B$) factorizes through $A/I$. (The subindex $B$ denotes the canonically induced sheaf of $B$-modules over the category of $B$-algebras). Assuming this result, and since $\hat V_S/L$ is isomorphic to a inverse limit of finite type $A$-modules for $L\in\grv(S)$, one deduces that the conditions of {\ref{thm:krich-char}} are closed. Let us now prove the claim. By the hypothesis one has that $(V/M')_B\iso V_B/M'_B$ for all morphisms $A\to B$. Let $f$ be a morphism $A\to B$, then the canonical inclusion $j:(M+M')/M'\to V/M'$ gives: $$j_B^i:((M+M')/M')_B @>>> (V/M')_B\iso \limpl{i} L_i\to L_i$$ where $L_i$ are finite type free $B$-modules. One has that $M_B\subseteq M'_B$ if and only if $j_B^i$ is identically zero for all $i$. Recall that given a sub-$A$-module $\bar M$ of a finite type free $A$-module $L=A^n$, the inclusion morphism $\bar j:\bar M\to L$ assigns to each $\bar m\in\bar M$ a set of coordinates $(\bar j_1(\bar m),\dots,\bar j_n(\bar m))$, and hence $\bar j_B$ is identically zero if and only if $f:A\to B$ factorizes through the ideal generated by $\{j_i(\bar m)\,\vert\, \bar m\in \bar M, i=1,\dots, n\}$. This concludes the proof. \end{pf} Our aim, now, is to give explicit characterizations of the set of rational points $U$ satisfying the conditions of theorem {\ref{thm:krich-char}}. \begin{thm} Let $S$ be a Young diagram of virtual cardinal $n$, and let $U$ be a rational point of $F_{A_S}\subset\gr^n(V)$. The following three conditions are equivalent: \begin{enumerate} \item $k\subset U$ and $U\cdot U\subseteq U$, \item $U\cdot U=U$, \item $0\notin S$ and $\res_{z=0}\psi_U(z,t)\psi_U(z,t')\psi^*_U(z,t'')z^{n-3}dz=0$. \end{enumerate} \end{thm} \begin{pf} $1\implies2$ is trivial. For $2\implies1$, one has only to check that $k\subset U$. But recall that the element $u$ of $U-\{0\}\subset k((z))-\{0\}$ of highest order is unique (up to a non-zero scalar) and should therefore satisfy $u=\lambda\cdot u\cdot u$ and hence $u=\lambda^{-1}\in k-\{0\}$. $1\implies3$ First, note that $k\subset U$ implies $0\notin S$. It is now clear by {\ref{eq:residue}} that the first condition implies the third. $3\implies1$ If the residue condition is verified, it then follows that $U^\perp\subseteq (U\cdot U)^\perp$ and therefore $U\cdot U\subseteq U$, as desired. Now, since $0\notin S$ an element $u\in U$ of highest order must have non-negative order, and since $u\cdot u\in U$, one concludes that $u=\lambda\in k-\{0\}$. \end{pf} \begin{prop} Let $S$ be a Young diagram. A necessary and sufficient condition for the existence of a rational point $U\in F_{A_S}$ such that $U\cdot U=U$, is that $0\notin S$ and $\Z-S$ should be closed under addition. \end{prop} \begin{pf} Obvious. \end{pf} This condition will be called the Weierstrass gap property (WGP). Let us denote by $\gr_W(V)$ the open subscheme of $\grv$ consisting of the union of the open subsets $F_{A_S}$ such that $S$ satisfies WGP. Then one has: \begin{cor}\label{cor:equationofm} The subset $\{U\in\gr^n(V)\,\vert\,k\subset U\text{ and } U\cdot U\subseteq U\}$ is given by one of the following equivalent conditions: \begin{enumerate} \item $$\gr_W(V)\cap\left\{ U\in\grv\,\vert\, \res_{z=0}\psi_U(z,t)\psi_U(z,t')\psi^*_U(z,t'')z^{n-3}dz=0 \,\right\}$$ \item $$\left\{ \begin{aligned} \res_{z=0}&\psi^*_U(z,t)\frac{dz}{z^{n+1}}\,=\,0 \\ \res_{z=0}&\psi_U(z,t)\psi_U(z,t')\psi^*_U(z,t'')z^{n-3}dz\,=\,0 \end{aligned}\right.$$ \end{enumerate} \end{cor} \begin{pf} The first one is obvious. For the second one we only need to show that the condition $k\subset U$ is equivalent to $\res_{z=0}\psi^*_U(z,t)\frac{dz}{z^{n+1}}=0$. However, from the proof of Theorem {\ref{thm:res-equa}} and formula \ref{eq:BA=sum} is easily deduced that $\res_{z=0}f(z)\cdot\psi^*_U(z,t)\frac{dz}{z^{n+1}}=0$ if and only if $f(z)\in U$. \end{pf} \begin{rem} Assume that $U\in F_{A_S}\subset\gr^n(V)(\spk)$ ($S$ the sequence associated to a Young diagram) is a point that lies on the image of the Krichever morphism; that is, there exists $(C,p,\phi)\in\M_\infty^g$ such that $K(C,p,\phi)=U$. Note that by the very construction one has an isomorphism $H^0(C-p,\o_C)\iso U$ and hence: \begin{itemize} \item $n=1-g$, \item the arithmetic genus of $C$ equals $\#({\mathbb Z}_{<0}\cap S)$, \item the set of Weierstrass gaps of $C$ at $p$ is precisely ${\mathbb Z}_{<0}\cap S$. \end{itemize} \end{rem} \begin{thm}\label{thm:pde-mod} The condition: $$\res_{z=0}\psi_U(z,t)\psi_U(z,t')\psi^*_U(z,t'') \frac{dz}{z^{2+g}}\,=\,0$$ for a rational point $U\in\gr^{1-g}(V)$ is equivalent to the infinite set of equations: $$P(\lambda_1,\lambda_2,\lambda_3)\vert \Sb t=0 \\ t'=0 \\ t''=0 \endSb \left(\tau_U(t)\cdot \tau_U(t')\cdot\tau_U(t'')\right)\,=\,0$$ for every three Young diagrams $\lambda_1,\lambda_2,\lambda_3$, where $P(\lambda_1,\lambda_2,\lambda_3)$ is the differential operator defined by: $$\sum p_{\beta_1}(\tilde\partial_t)D_{\lambda_1,\alpha_1}(-\tilde\partial_t)\cdot p_{\beta_2}(\tilde\partial_{t'})D_{\lambda_2,\alpha_2}(-\tilde\partial_{t'}) \cdot p_{\beta_3}(-\tilde\partial_{t''})D_{\lambda_3,\alpha_3}(\tilde\partial_{t''}) $$ where the sum is taken over the 6-tuples $\{\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3\}$ of integers such that $-\alpha_1+\beta_1-\alpha_2+\beta_2-\alpha_3+\beta_3=1+g$. \end{thm} \begin{rem} The meaning of the Residue Identity $Res_{z=0}\psi\psi\psi^*z^{-(g+2)}dz=0$ (where $\psi$ is the Baker function of $(C,p,z)$) is the following: for all sections $s_1,s_2\in U=H^0(C-p,\o_C)$, $\omega\in U^\perp=H^0(C-p,\Omega_C)$ the differential $s_1\cdot s_2\cdot \omega$ has residue zero at $p$. Or, what amounts to the same: let $D_i$ be the divisor of poles of $s_i$ ($i=1,2$) and $D^*$ that of $\omega$, then $D_1+D_2+D^*\,=\, K+m\cdot p$ for some non negative integer $m$ and some canonical divisor $K$. \end{rem} These differential equations are the equations of the moduli of curves (the image of the functor $K$) in the infinite Grassmannian. A very important fact is that a theta function of a Jacobian variety satisfies these differential equations, which are not obtained from the KP equations. Moreover: \begin{cor}\label{cor:pde-tau} A formal series $\tau(t)\in k\{\{t_1,t_2,\dots\}\}$ is the $\tau$-func\-tion associated with a rational point of $\M_\infty^g\subset \gr^{1-g}(V)$ (and may therefore be written in terms of the theta function of a Jacobian variety) if and only if it satisfies the following set of equations: \begin{enumerate} \item the KP equations (given in theorem {\ref{thm:KP-eq}}), \item the p.d.e.'s given in theorem {\ref{thm:pde-mod}}, \item the p.d.e.'s: $$\left(\sum_{-\alpha+\beta=1-g} p_{\beta}(-\tilde\partial_t)D_{\lambda,\alpha}(\tilde\partial_t) \right)\vert_{t=0}\tau_U(t)=0\quad \text{for all Young diagrams }\lambda$$ \end{enumerate} \end{cor} \begin{pf} Note that the third condition is $\res_{z=0}\psi^*_U(z,t)\frac{dz}{z^{2-g}}=0$ but given in terms of partial differential equations. \end{pf} \begin{rem} These technics have been used in \cite{Pl} for the study of the moduli space of Prym varieties and to generalize the characterizations of Jacobians given by Mulase (\cite{Mul}) and Shiota (\cite{Sh}) \end{rem} \begin{rem} An open problem now is to re--state Corollary \ref{cor:equationofm} as a characterization for a pseudodifferential operator to come from algebro-geometric data. \end{rem} \vskip2truecm

9711

alg-geom/9711001

en

https://arxiv.org/abs/alg-geom/9711001

alg-geom/9711001

Sandra DiRocco

M. C. Beltrametti, S. Di Rocco and A. J. Sommese

On higher order embeddings of Fano threefolds by the anticanonical linear system

22 pages (http://www.math.kth.se/~sandra/Welcome, http://www.nd.edu/~sommese/)

The map given by the anticanonical bundle of a Fano manifold is investigated with respect to a number of natural notions of higher order embeddings of projective manifolds. This is of importance in the understanding of higher order embeddings of the special varieties of adjunction theory, which are usually fibered by special Fano manifolds. An analysis is carried out of the higher order embeddings of the special varieties of adjunction theory that arise in the study of the first and second reductions. Special attention is given to determining the order of the anticanonical embeddings of the three dimensional Fano manifolds which have been classified by Iskovskih, Mori, and Mukai and also of the Fano complete intersections in $\pn N$.

alg-geom

\section*{Introduction} Let $X$ be an $n$-dimensional connected complex projective manifold. There are three natural notions (see (\ref{kthemb})) of the ``order'' of an embedding given by a line bundle $L$. From strongest to weakest they are $k$-jet ampleness \cite{Plenum}, $k$-very ampleness \cite{BSCo}, and $k$-spannedness \cite{Duke}. For $k=1$ (respectively $k=0$) all three notions are equivalent to very ampleness (respectively spannedness at all points of $X$ by global sections). We consider the most natural notion to be $k$-very ample, which by definition means that given any $0$-dimensional subscheme $\sZ$ of $X$ of length $k+1$, the map $ H^0(X,L)\to H^0(\sZ,L_{\sZ})$, is onto. There has been considerable work on deciding the order of the embeddings relative to these notions for the line bundles that come up on the standard classes of varieties. In this article we investigate what the ``order'' of the embedding by $|-K_X|$ is. In \S \ref{Background} we recall some known results that we will need in the sequel. We also prove some foundational results on $k$-very ampleness that are not in the literature, e.g., Lemma (\ref{Tensor}) and the useful lower bound for the degree and the number of sections of $L$, Proposition (\ref{h0Est}). In \S \ref{FCIsection} we give the $k$-jet ampleness, $k$-very ampleness, and $k$-spannedness of $-K_X$ for Fano manifolds $X\subset \pn N$ which are complete intersections of hypersurfaces in $\pn N$. In \S \ref{kAd} we work out the order of the embeddings of the special varieties that come up in the study of the first and second reductions of adjunction theory. Fano manifolds of special types come up naturally as the fibers of degenerate adjunction morphisms. In \S \ref{Mukai} we continue the investigation of the Fano manifolds that came up in \S \ref{kAd}, those with $-K_X\cong (n-2)L$ and $n\ge 4$. In \S \ref{FanoS}, by using Fujita's classification of Del Pezzo threefolds, we completely settle the order of the embeddings of $3$-dimensional Fano manifolds with very ample anticanonical bundle $-K_X$. For further discussion and a guide to most of the published papers on $k$-very ampleness, we refer to the book \cite{Book} by the first and third author. We also call attention to \cite{Sandra1} of the second author, which do a thorough investigation for surfaces of the questions analogous to those we ask for higher dimensions. The third author would like to thank the University of Notre Dame and the Alexander von Humboldt Stiftung for their support. We would like to thank the referee for helpful comments. \section{Background material}\label{Background} \setcounter{theorem}{0} Throughout this paper we deal with complex projective manifolds $V$. We denote by $\sO_V$ the structure sheaf of $V$ and by $K_V$ the canonical bundle. For any coherent sheaf $\sF$ on $V$, $h^i(\sF)$ denotes the complex dimension of $H^i(V,\sF)$. Let $L$ be a line bundle on $V$. $L$ is said to be {\em numerically effective} ({\em nef}, for short) if $L\cdot C\geq 0$ for all effective curves $C$ on $V$. $L$ is said to be {\em big} if $\grk(L)=\dim V$, where $\grk(L)$ denotes the Kodaira dimension of $L$. If $L$ is nef then this is equivalent to be $c_1(L)^n>0$, where $c_1(L)$ is the first Chern class of $L$ and $n=\dim V$. \begin{prgrph*}{Notation}\label{Notation} In this paper, we use the standard notation from algebraic geometry. Let us only fix the following. \begin{enumerate} \item[] $\approx$ (respectively $\sim$) denotes linear (respectively numerical) equivalence of line bundles; \item[] $|L|$, the complete linear system associated with a line bundle $L$ on a variety $V$; \item[] $\Gamma(L)=H^0(L)$ denotes the space of the global sections of $L$. We say that $L$ is spanned if it is spanned at all points of $V$ by $\Gamma(L)$; \item[] $\grk(V):=\grk(K_V)$ denotes the Kodaira dimension, for $V$ smooth; \item[] $b_2(V)=\sum_{p+q=2}h^{p,q}$ denotes the second Betti number of $V$, for $V$ smooth, where $h^{p,q}:=h^q(\Omega_V^p)$ denotes the Hodge $(p,q)$ number of $V$. \end{enumerate} Line bundles and divisors are used with little (or no) distinction. Hence we freely use the additive notation. \end{prgrph*} \begin{prgrph*}{Reductions}\label{Red} (See e.g., \cite[Chapters 7, 12]{Book}) Let $(\widehat{X},\widehat{L})$ be a smooth projective variety of dimension $n\geq 2$ polarized with a very ample line bundle $\widehat{L}$. A smooth polarized variety $(X,L)$ is called a {\em reduction} of $(\widehat{X},\widehat{L})$ if there is a morphism $r:\widehat{X}\to X$ expressing $\widehat{X}$ as the blowing up of $X$ at a finite set of points, $B$, such that $L:=(r_*\widehat{L})^{**}$ is an ample line bundle and $\widehat{L}\approx r^*L-[r^{-1}(B)]$ (or, equivalently, $K_{\widehat{X}}+(n-1)\widehat{L}\approx r^*(K_X+(n-1)L)$). Note that there is a one to one correspondence between smooth divisors of $|L|$ which contain the set $B$ and smooth divisors of $|\widehat{L}|$. Except for an explicit list of well understood pairs $(\widehat{X},\widehat{L})$ (see in particular \cite[\S\S 7.2, 7.3]{Book}) we can assume: \begin{enumerate} \item[{\rm a)}] $K_{\widehat{X}}+(n-1)\widehat{L}$ is spanned and big, and $K_X+(n-1)L$ is very ample. Note that in this case this reduction, $(X,L)$, is unique up to isomorphism. We will refer to it as {\em the first reduction} of $(\widehat{X},\widehat{L})$. \item[{\rm b)}] $K_X+(n-2)L$ is nef and big, for $n\geq 3$. \end{enumerate} Then from the Kawamata-Shokurov basepoint free theorem (see e.g., \cite[(1.5.2)]{Book}) we know that $|m(K_X+(n-2)L|$, for $m\gg 0$, gives rise to a morphism $\vphi:X\to Z$, with connected fibers and normal image. Thus there is an ample line bundle $\sK$ on $Z$ such that $K_X+(n-2)L\approx \vphi^*\sK$. Let $\sD:=(\vphi_*L)^{**}$. The pair $(Z,\sD)$, together with the morphism $\vphi:X\to Z$ is called the {\em second reduction} of $(\widehat{X},\widehat{L})$. The morphism $\vphi$ is very well behaved (see e.g., \cite[\S\S 7.5, 7.6 and \S\S 12.1, 12.2]{Book}). In particular $Z$ has terminal, $2$-factorial isolated singularities and $\sK\approx K_Z+(n-2)\sD$. Moreover $\sD$ is a $2$-Cartier divisor such that $2L\approx \vphi^*(2\sD)-\Delta$, for some effective Cartier divisor $\Delta$ on $X$ which is $\vphi$-exceptional (see \cite[(7.5.6), (7.5.8)]{Book}). \end{prgrph*} \begin{prgrph*}{Nefvalue} (See e.g., \cite[\S 1.5]{Book}) Let $V$ be a smooth projective variety and let $L$ be an ample line bundle on $V$. Assume that $K_V$ is not nef. Then from the Kawamata rationality theorem (see e.g., \cite[(1.5.2)]{Book}) we know that there exists a rational number $\tau$ such that $K_V+\tau L$ is nef and not ample. Such a number, $\tau$, is called the {\em nefvalue} of $(V,L)$. Since $K_V+\tau L$ is nef, it follows from the Kawamata-Shokurov base point free theorem (see e.g., \cite[(1.5.1)]{Book}) that $|m(vK_V+uL)|$ is basepoint free for all $m\gg 0$, where $\tau=u/v$. Therefore, for such $m$, $|m(K_V+\tau L)|$ defines a morphism $f:V\to {\Bbb P}_{\comp}$. Let $f=s\circ \Phi$ be the Remmert-Stein factorization of $f$, where $\Phi:V\to Y$ is a morphism with connected fibers onto a normal projective variety $Y$ and $s:Y\to {\Bbb P}_{\comp}$ is a finite-to-one morphism. For $m$ large enough such a morphism, $\Phi$, only depends on $(V,L)$ (see \cite[\S 1.5]{Book}). We call $\Phi:V\to Y$ the {\em nefvalue morphism} of $(V,L)$. \end{prgrph*} \begin{prgrph*}{Special varieties}\label{Special} (See e.g., \cite[\S 3.3]{Book}) Let $V$ be a smooth variety of dimension $n$ and let $L$ be an ample line bundle on $V$. We say that $V$ is a {\em Fano manifold} if $-K_V$ is ample. We say that $V$ is a {\em Fano manifold of index } $i$ if $i$ is the largest positive integer such that $K_V\approx -iH$ for some ample line bundle $H$ on $V$. Note that $i\leq n+1$ (see e.g., \cite[(3.3.2)]{Book}) and $n-i+1$ is referred to as the {\em co-index} of $V$. We say that a Fano manifold, $(V,L)$, is a {\em Del Pezzo manifold} (respectively a {\em Mukai manifold}) if $K_V\approx -(n-1)L$ (respectively $K_V\approx -(n-2)L$). We also say that $(V,L)$ is a {\em scroll} (respectively a {\em quadric fibration}; respectively a {\em Del Pezzo fibration}; respectively a {\em Mukai fibration}) over a normal variety $Y$ of dimension $m$ if there exists a surjective morphism with connected fibers $p:V\to Y$ such that $K_V+(n-m+1)L\approx p^*\sL$ (respectively $K_V+(n-m)L\approx p^*\sL$; respectively $K_V+(n-m-1)L\approx p^*\sL$; respectively $K_V+(n-m-2)L\approx p^*\sL$) for some ample line bundle $\sL$ on $Y$. \end{prgrph*} \begin{prgrph*}{$k$-th order embeddings}\label{kthemb} Let $V$ be a smooth algebraic variety. We denote the Hilbert scheme of $0$-dimensional subschemes $(\sZ,\sO_{\sZ})$ of $V$ with ${\rm length}(\sO_\sZ)=r$ by $V^{[r]}$. Since we are working in characteristic zero, we have ${\rm length}(\sO_\sZ)=h^0(\sO_\sZ)$. We say that a line bundle $L$ on $V$ is $k$-{\em very ample} if the restriction map $\Gamma(L)\to\Gamma(\sO_\sZ(L))$ is onto for any $\sZ\in V^{[k+1]}$. Note that $L$ is $0$-very ample if and only if $L$ is spanned by global sections, and $L$ is $1$-very ample if and only if $L$ is very ample. Note also that for smooth surfaces with $k\leq 2$, $L$ being $k$-very ample is equivalent to $L$ being $k$-{\em spanned} in the sense of \cite{Duke}, i.e., $\Gamma(L)$ surjects on $\Gamma (\sO_\sZ(L))$ for any {\em curvilinear} $0$-cycle $\sZ \in V^{[k+1]}$, i.e., any $0$-dimensional subscheme, $\sZ\subset V$, such that ${\rm length}(\sO_\sZ)=k+1$ and $\sZ\subset C$ for some smooth curve $C$ on $V$ (\cite[(0.4), (3.1)]{Duke}). Let $x_1,\ldots,x_r$ be $r$ distinct points on $V$. Let ${\frak m}_i$ be the maximal ideal sheaves of the points $x_i\in V$, $i=1,\ldots,r$. Note that the stalk of ${\frak m}_i$ at $x_i$ is nothing but the maximal ideal, ${\frak m}_i\sO_{V,x_i}$, of the local ring $\sO_{V,x_i}$, $i=1,\ldots,r$. Consider the $0$-cycle $\sZ= x_1+\cdots+x_r$. We say that $L$ is $k$-{\em jet ample at} $\sZ$ if, for every $r$-tuple $(k_1,\ldots,k_r)$ of positive integers such that $\sum_{i=1}^rk_i=k+1$, the restriction map $$\Gamma(L)\to\Gamma(L\otimes (\sO_V/\otimes_{i=1}^r{\frak m}_i^{k_i})) \left(\cong \oplus_{i=1}^r\Gamma(L\otimes (\sO_V/{\frak m}_i^{k_i})\right)$$ is onto. Here ${\frak m}_i^{k_i}$ denotes the $k_i$-th tensor power of ${\frak m}_i$. We say that $L$ is $k$-{\em jet ample} if, for any $r\geq 1$ and any $0$-cycle $\sZ= x_1+\cdots+x_r$, where $x_1,\ldots,x_r$ are $r$ distinct points on $V$, the line bundle $L$ is $k$-jet ample at $\sZ$. Note that $L$ is $0$-jet ample if and only if $L$ is spanned by its global sections and $L$ is $1$-jet ample if and only if $L$ is very ample. Note also that if $L$ is $k$-jet ample, then $L$ is $k$-very ample (see \cite[(2.2)]{Plenum} and compare also with (\ref{SpecialProp})). We will use over and over through the paper the fact \cite[(1.3)]{BSMZ}, that if $L$ is a $k$-very ample line bundle on $V$, then $L\cdot C\geq k$ for each irreducible curve $C$ on $V$. We refer to \cite{BSAq}, \cite{BSCo} and \cite{BSMZ}, and \cite{Plenum} for more on $k$-spannedness, $k$-very ampleness and $k$-jet ampleness respectively. \end{prgrph*} \begin{definition}\label{image} Let $p:X\to Y$ be a holomorphic map between complex projective schemes. Let $\sZ$ be a $0$-dimensional subscheme of $X$ defined by the ideal sheaf $\sJ_\sZ$. Then the {\em image} $p(\sZ)$ of $\sZ$ is the $0$-dimensional subscheme of $Y$ whose defining ideal is $\sI=\{g\in\sO_Y\;|\;g\circ p\in \sJ_\sZ\}$. \end{definition} We need the following general fact. \begin{lemma}\label{GenFa} Let $p:X\to Y$ be a morphism of quasiprojective varieties $X$, $Y$. Let $\sZ$ be a $0$-dimensional subscheme of $X$ of length $k$. Then $p(\sZ)$ has length $\leq k$. \end{lemma} \proof Let $x_1,\ldots,x_t$ be $t$ distinct points such that ${\rm Supp}(\sZ)=\{x_1,\ldots,x_t\}$. Let ${\rm length}(\sO_\sZ)=k$ and ${\rm length}(\sO_{\sZ,x_i})=k_i$, where $\sO_{\sZ,x_i}$ denotes the stalk of $\sO_\sZ$ at $x_i$, $i=1,\ldots,t$. Then $k=\sum_{i=1}^tk_i$. Set $\sZ':=p(\sZ)$ and let $\sJ_\sZ$, $\sJ_{\sZ'}$ be the ideal sheaves of $\sZ$, $\sZ'$ respectively. Arguing by contradiction, assume that ${\rm length}(\sO_{\sZ'})> k$. Then there are $k+1$ linearly independent functions, $g_0=1,g_1,\ldots,g_k$, in $\sO_{\sZ'}$. Let $y_1,\ldots,y_s$, $s\leq t$, be the images of the points $x_i$, $i=1,\ldots,t$. For each $j=1,\ldots,s$, consider the vector subspace of $\comp^{k+1}$ defined by $$V_j:=\{(\lambda_0,\ldots,\lambda_k)\in \comp^{k+1}\;|\;\sum_{i=0}^k\lambda_i g_i\in {\frak m}_j\},$$ where ${\frak m}_j$ denote the ideal sheaf of $y_j$. Since $$\dim V_j\geq k+1-k_j,\;j=1,\ldots,s,\ \ {\rm and}\ \ \sum_{j=1}^sk_j\leq k<k+1$$ we conclude that $$\dim\cap_{j=1}^s V_j\geq \sum_{j=1}^s\dim V_j-(s-1)(k+1)\ge k+1-\sum_{j=1}^sk_j\ge 1.$$ Thus, there exist $\lambda_0,\ldots,\lambda_k\in \comp$ such that $\sum_{i=0}^k\lambda_i g_i=0$ at $y_j$ for each $j=1,\ldots,s$. Thus $$p^*\left(\sum_{i=0}^k\lambda_i g_i\right)=\sum_{i=0}^k\lambda_i p^*g_i=0$$ at $x_i$ for each $i=1,\ldots,t$. It follows that $\sum_{i=0}^k\lambda_i p^*g_i\in\sJ_\sZ$ and hence $\sum_{i=0}^k\lambda_i g_i\in\sJ_{\sZ'}$, or $\sum_{i=0}^k\lambda_i g_i=0$ in $\sO_{\sZ'}$. This contradicts the assumption that $1,g_1,\ldots,g_k$ are linearly independent.\qed If $X_1$, $X_2$ are projective schemes and $\sF_1$, $\sF_2$ are sheaves on $X_1$, $X_2$ respectively, we will denote $$\sF_1\bx\sF_2:=p_1^*\sF_1\otimes p_2^*\sF_2,$$ where $p_1$, $p_2$ are the projections on the two factors. The following is the $k$-very ample version of Lemma (3.2) of \cite{Embedding}. \begin{lemma}\label{Tensor} Let $X_1$, $X_2$ be complex projective schemes and $L_1$, $L_2$ line bundles on $X_1$, $X_2$ respectively. For $i=1,2$ assume that $L_i$ is $k_i$-very ample and let $k:=\min\{k_1,k_2\}$. Then $L_1\bx L_2$ is $k$-very ample on $X_1\bx X_2$. Furthermore if $L_1\bx L_2$ is $k'$-very ample on $X_1\bx X_2$ then $L_1$, $L_2$ are $k'$-very ample. \end{lemma} \proof Let $(\sZ,\sO_\sZ)$ be a $0$-dimensional subscheme of length $k+1$ on $X_1\times X_2$. Let $p_i:X_1\times X_2\to X_i$, $i=1,2$, the projections on the two factors. Let $\sZ_i:={p_i}(\sZ)$ be the $0$-dimensional subschemes of $X_i$ obtained as images of $\sZ$, as in (\ref{image}), and let $\sI_{{\sZ}_i}$ be the ideal sheaves defining $\sZ_i$, $i=1,2$. Let $J_i:=p_i^*{\sI_{{\sZ}_i}}$, $i=1,2$. There exist generating function germs $f\in J_i$ of type $f=g\circ p_i$, $g\in \sI_{{\sZ}_i}$, $i=1,2$. Therefore for each point $z\in \sZ_{\mathop{\rm red}\nolimits}$, $f(z)=g(p_i(z))=0$, $i=1,2$. This means that $z$ belongs to the subscheme of $X_1\times X_2$ defined by the ideal sheaf of the image of $J_1\bx J_2$ in $\sO_{X_1\times X_2}$. Hence we have an inclusion of ideal sheaves $(J_1,J_2)\subset \sI_{\sZ}$. But $(J_1,J_2)$ defines the subscheme $\sJ$ whose structural sheaf is $$\sO_{\sJ}=p_1^*\sO_{\sZ_1}\otimes p_2^*\sO_{\sZ_2}=\sO_{\sZ_1}\bx\sO_{\sZ_2}.$$ Thus we have a surjection \begin{equation}\label{surj} \sO_{\sZ_1}\bx\sO_{\sZ_2}\to \sO_\sZ\to 0. \end{equation} On the other hand, by the Kunneth formula, we have \begin{equation}\label{K1} H^0(L_1\bx L_2) = H^0(p_1^*L_1\otimes p_2^* L_2)\\ =H^0(L_1)\otimes H^0(L_2), \end{equation} as well as, \begin{eqnarray*}\label{K2} H^0(L_1\bx L_2\otimes\sO_{\sZ_1}\bx\sO_{\sZ_2})&=& H^0(\sO_{\sZ_1}(L_1)\bx\sO_{\sZ_2}(L_2))\\ &=&H^0(\sO_{\sZ_1}(L_1))\otimes H^0(\sO_{\sZ_2}(L_2)). \end{eqnarray*} Therefore, from (\ref{surj}) and (\ref{K1}), we get a surjection \begin{equation}\label{s1} H^0(\sO_{\sZ_1}(L_1))\otimes H^0(\sO_{\sZ_2}(L_2))\to H^0(\sO_\sZ(L_1\bx L_2)). \end{equation} By Lemma (\ref{GenFa}) we have that $\sZ_1$ is of length $\leq k+1\leq k_1+1$. Therefore, since $L_1$ is $k_1$-very ample, the restriction map $H^0(L_1)\to H^0(\sO_{\sZ_1}(L_1))$ is onto. Similarly we have that $H^0(L_2)\to H^0(\sO_{\sZ_2}(L_2))$ is onto. Thus by using (\ref{K1}) and (\ref{s1}) we get a surjection $$H^0(L_1\bx L_2)\to H^0(\sO_\sZ(L_1\bx L_2))\to 0.$$ This shows that $L_1\bx L_2$ is $k$-very ample. To show the last part of the statement note that ${L_1\bx L_2}_{|X_1\times x_2}\cong L_1$ for each $x_2\in X_2$. Then $L_1$ is $k'$-very ample if $L_1\bx L_2$ is $k'$-very ample. Similarly for $L_2$. \qed The following result is a useful partial generalization of \cite[Lemma 1.1] {BSMZ} (compare also with \cite[(3.1)]{Plenum}). \begin{lemma}\label{kBlowupLemma}Let $L$ denote a $k$-very ample line bundle on an $n$-dimensional projective manifold $X$. Assume that $k\ge 2$ and let $\pi : \hatX\to X$ denote the blowup of $X$ at a finite set $\{x_1,\ldots,x_{k-1}\}\subset X$ of distinct points. Let $E_i:= \pi^{-1}(x_i)$ for $1\le i\le k-1$. Then $\pi^*L-(E_1+\cdots+E_{k-1})$ is very ample. \end{lemma} \proof Assume that $h^0(L)=N+1$, and we use $|L|$ to embed $X$ into $\pn N$. Consider a linear subspace $\pn t\subset\pn N$, where $t\leq k-1$. Assume that $\pn t$ met $X$ in a positive dimensional set. Then we can assume without loss of generality that it is a curve. If not we can choose a $\pn {t-1}$ contained in the $\pn t$ which will meet $X$ in a set of dimension at most one less. Clearly $t>1$ since otherwise $\pn t$ would be a line contained in $X$, contradicting the $k$-very ampleness assumption with $k\geq 2$. Choose a $\pn {t-1}$ in the $\pn t$ meeting $X$ in a finite set. This is possible since $\pn t$ meets $X$ in some points and a curve, say $C$. By using the very ampleness assumption we conclude that the $\pn {t-1}$ meets the curve $C$ in at most $t$ points. Choose a hyperplane $H\subset\pn N$ meeting the $\pn t$ in the $\pn {t-1}$. Then $H$ meets the curve $C$ in at most $t\leq k-1$ points. Since $H$ restricts to $L$ on $X$, this contradicts the fact that $L\cdot C\geq k$. Therefore we conclude that any $\pn t\subset \pn N$ with $t\le k-1$ meets $X$ scheme theoretically in a $0$-dimensional subscheme, say $\sX$. Furthermore ${\rm length}(\sO_{\sX})\leq t+1$. Indeed otherwise $X\cap \pn t$ would contain a $0$-cycle of length $t+2\leq k+1$ which spans a $\pn {t+1}$ since $L$ is $k$-very ample. Thus by taking the projective space $P:=\pn {k-2}$ generated by $\{x_1,\ldots,x_{k-1}\}$, we see that this $\pn {k-2}$ meets $X$ in precisely $\{x_1,\ldots,x_{k-1}\}$. Thus the blowup $\grs : Z\to \pn N$ of $\pn N$ at this $P$ has $\hatX$ as the proper transform of $X$ and the induced map from $\hatX$ to $X$ is $\pi$. Since $\grs^*\sO_{\pn N}(1)-\grs^{-1}(P)$ is spanned by global sections and $\hatL:=\pi^*L-(E_1+\cdots+E_{k-1})$ is the pullback to $\hatX$ of $\grs^*\pnsheaf N 1-\grs^{-1}(P)$ we see that $\hatL$ is spanned by global sections. Thus we conclude that global sections of $\hatL$ give a map $\phi :\hatX\to \pn{N-k+1}$. Applying the fact that the $\pn{k-1}$ generated by the image in $\pn N$ of any $0$-dimensional subscheme $\sZ\subset X$ of length $k$ meets $X$ scheme theoretically in $\sZ$, it follows that the rational map from $\hatX$ to $\pn {N-k+1}$ induced by the projection of $\pn N$ from $P$ is one-to-one on $\hatX$. To see that $\hatL$ is very ample we consider the points $x\in \hatX\setminus\cup_{i=1}^{k-1} E_i$ and the points $x\in \cup_{i=1}^{k-1}E_i$ separately. First assume that $x\in\hatX\setminus\cup_{i=1}^{k-1} E_i$. Choose a tangent vector $\tau_x$ at $x$. Let $u_1,\ldots,u_n$ be a choice of local coordinates defined in a neighborhood of $x$, all zero at $x$, and with $\tau_x$ tangent to the $u_n$ axis. Let ${\frak a}_x$ denote the ideal sheaf which equals $\sO_X$ away from $x$, and at $x$ is defined by $(u_1,\ldots,u_{n-1},u_n^2)$. We can choose a zero dimensional subscheme $\sZ$ of $X$ of length $k+1$ that is defined by ${\frak a}_x\otimes {\frak m}_{x_1}\otimes\cdots\otimes {\frak m}_{x_{k-1}}$. Since $H^0(L)\to H^0(L\otimes \sO_{\sZ})$ is onto by the definition of $k$-very ampleness, we conclude that the map $\phi$ given by global sections of $\hatL$ has nonzero differential evaluated on the tangent vector $\tau_x$. It follows that the global sections of $\hatL$ embed away from $\cup_{i=1}^{k-1}E_i$. To finish consider a point $x\in \cup_{i=1}^{k-1}E_i$. Without loss of generality we can assume, by relabeling if necessary, that $x\in E_1$. We thus have $x_1=\pi(x)$. Choose a tangent vector $\tau_x$ at $x$. Note that since the line bundle $\hatL$ is spanned and since $\hatL_{E_1}\cong\pnsheaf {n-1}1$, the restriction $\phi_{E_1}$ is an embedding. Thus we can assume that $\tau_x$ is not tangent to $E_1$. Let $\tau=d\pi_x(\tau_x)\in \sT_{X,x_1}$, where $d\pi_x:\sT_{\hatX,x}\to \sT_{X,x_1}$ is the differential map and $\sT_{\hatX,x}$, $\sT_{X,x_1}$ are the tangent bundles to $\hatX$, $X$ respectively. Let $u_1,\ldots,u_n$ be a choice of local coordinates defined in a neighborhood of $x_1$, all zero at $x_1$, with $\tau$ tangent to the $u_n$ axis, and such that the proper transform of the $u_n$ axis is tangent to $\tau_x$. Let ${\frak b}_x$ denote the ideal sheaf which equals $\sO_X$ away from $x$, and at $x$ is defined by $(u_1,\ldots,u_{n-1},u_n^3)$. We can choose a zero dimensional subscheme $\sZ$ of $X$ of length $k+1$ that is defined by ${\frak b}_x$ if $k=2$ and by ${\frak b}_x\otimes {\frak m}_{x_2}\otimes\cdots\otimes {\frak m}_{x_{k-1}}$ if $k\ge 3$. Since $H^0(L)\to H^0(L\otimes \sO_\sZ)$ is onto by the definition of $k$-very ampleness, we conclude that global sections of $\hatL$ give a map with nonzero differential evaluated on the tangent vector $\tau_x$. It follows that the global sections of $\hatL$ embed $\hatX$. \qed \begin{rem*}There is a nice interpretation of this result in terms of $X^{[k]}$, the Hilbert scheme of $0$-dimensional subschemes of $X$ of length $k$. There is a natural line bundle $\sL$ on $X^{[k]}$ induced by $L$ on $X$. The fact that $L$ is $k$-very ample is equivalent to $\sL$ being very ample \cite{CG}. Given a set $\{x_1,\ldots,x_{k-1}\}\subset X$ of $k-1$ distinct points of $X$, the subscheme of all $0$-dimensional subschemes of $X$ of length $k$ containing $\{x_1,\ldots,x_{k-1}\}$ is naturally isomorphic to $X$ blown up at $\{x_1,\ldots,x_{k-1}\}\subset X$. Under this identification the very ample line bundle $\sL$ restricts to the line bundle $\pi^*L-(E_1+\cdots+E_{k-1})$ of the above lemma. \end{rem*} We have the following estimate for the degree and the number of sections of $L$. \begin{proposition}\label{h0Est}Let $L$ denote a $k$-very ample line bundle on an $n$-dimensional projective manifold $X$. Assume that $k\ge 2$. Then $L^n\ge 2^n+k-2$ and $h^0(L)\ge 2n +k-1$. Moreover $h^0(L)= 2n +k-1$ implies that $L^n = 2^n+k-2$. \end{proposition} \proof Let $\pi : \hatX\to X$ denote the blowup of $X$ at a finite set $\{x_1,\ldots,x_{k-1}\}\subset X$ of distinct points. Let $E_i:= \pi^{-1}(x_i)$ for $1\le i\le k-1$. By Lemma (\ref{kBlowupLemma}), $\pi^*L-(E_1+\cdots+E_{k-1})$ is very ample, and thus $\pi^*L-(E_1+\cdots+E_{k-1})-E_1$ is spanned. This implies $(\pi^*L-(2E_1+E_2\cdots+E_{k-1}))^n\ge 0$ and thus that $L^n \geq 2^n+k-2$ since $E_i^n=(-1)^n$, $i=1,\ldots,k-1$. If $h^0(L)\le 2n +k-1$, then since $h^0(\hatL)=h^0(L)-k+1$, we have $h^0(\hatL)\le 2n$. The argument in \cite[Theorem (4.4.1), i)]{BSS2} shows that $h^0(\hatL)= 2n$, so that $h^0(L)= 2n+k-1$, and ${\hatL}^n=2^n-1$. Since ${\hatL}^n=L^n-k+1$, we are done. \qed \section{Fano complete intersections}\label{FCIsection}\setcounter{theorem}{0} Our first result gives the $k$-jet ampleness, $k$-very ampleness, and $k$-spannedness of $-K_X$ for Fano manifolds $X\subset \pn N$ which are scheme theoretically complete intersections of hypersurfaces in $\pn N$. We say that a curve $\ell$ on $X$ is a line if $\sO_X(1)\cdot \ell=1$. \begin{theorem}\label{CompIntFano}Let $X$ be a positive dimensional connected projective submanifold of $\pn N$, which is a complete intersection of hypersurfaces of $\pn N$ of degree $d_i$, $i=1,\ldots, r:=N-\dim X$. If the anticanonical bundle, $-K_X$ is ample and if $X$ is not a degree $2$ curve, then $X$ contains a line. In particular $-K_X$ is $(N+1-\sum_{i=1}^rd_i)$-jet ample, but not $(N+2-\sum_{i=1}^rd_i)$-spanned. \end{theorem} \proof Since the curve case of this result is trivial, assume that $\dim X\ge 2$. Since $X$ is a complete intersection, $K_X =\pnsheaf N{-N-1+\sum_{i=1}^rd_i}$, where $d_1,\ldots,d_r$ are the degrees of the hypersurfaces which intersect transversely in $X$. Since $-K_X$ is very ample, we conclude that $\sum_{i=1}^rd_i\le N$, and $-K_X$ is $(N+1-\sum_{i=1}^rd_i)$-jet ample (see \cite[Corollary (2.1)]{Plenum}). If we show that $X$ contains a line $\ell$ it will follow that $-K_X\cdot \ell=N+1-\sum_{i=1}^rd_i$, and thus $-K_X$ is not $(N+2-\sum_{i=1}^rd_i)$-spanned. Thus we must only show that $X$ contains a line. Let $G$ denote the Grassmannian $\grass 2 {N+1}$ of $2$-dimensional complex vector subspaces of $\comp^{N+1}$. Let $\sF$ denote the tautological rank $2$ quotient bundle of $G\times \comp^{N+1}$. Note that $G\times \comp^{N+1}$ is naturally identified with $G\times H^0(\pnsheaf N 1)$. Under this identification $\proj \sF\subset G\times \pn N$ is identified with the universal family of linear $\pn 1$'s contained in $\pn N$. The image in $s'\in H^0(\sF)$ of a section $s$ of $\pnsheaf N 1$ vanishes at points of $G$ corresponding to lines in $s^{-1}(0)$. Further a section $s$ of $\pnsheaf N d$ maps naturally to a section $s'$ of $\sF^{(d)}$, the $d$-th symmetric tensor product $\sF$. Here $s'$ vanishes at points of $G$ corresponding to the lines contained in $s^{-1}(0)$. Thus if $X$ is defined by sections $s_1,\ldots,s_r$ of $\pnsheaf N {d_1},\ldots,\pnsheaf N {d_r}$, the lines on $X$ correspond to the common zeroes of the images, $s'_1,\ldots,s'_r$, in $\sF^{(d_1)},\ldots,\sF^{(d_r)}$. Thus if we show that $$c_{{\mathop{\rm rank}\nolimits}\sF^{(d_1)}}(\sF^{(d_1)})\wedge\cdots\wedge c_{{\mathop{\rm rank}\nolimits}\sF^{(d_r)}}(\sF^{(d_r)})$$ is a nontrivial cohomology class, then it follows that $s'_1,\ldots,s'_r$ must have common zeroes and $X$ must contain lines. Note that ${\mathop{\rm rank}\nolimits}\sF^{(d_i)}=d_i+1$. For odd $d_i$ we have \begin{equation}\label{OddchernClasses} c_{d_i+1}(\sF^{(d_i)})= (d_i+1)^2c_2(\sF)\prod_{t=1}^{\frac{d_i-1}{2}} \left(t(d_i-t)c_1^2(\sF)+(d_i-2t)^2c_2(\sF)\right) \end{equation} and for even $d_i$ we have \begin{equation}\label{EvenchernClasses} c_{d_i+1}(\sF^{(d_i)})=(d_i+1)^2c_2(\sF) \frac{d_i}{2}c_1(\sF)\prod_{t=1}^{\frac{d_i}{2}-1} \left(t(d_i-t)c_1^2(\sF)+(d_i-2t)^2c_2(\sF)\right). \end{equation} Since $\sF$ is spanned both $c_1(\sF)$ and $c_2(\sF)$ are semipositive classes (see \cite[Example (12.1.7)]{Fulton}). Therefore, since all the monomial terms in the above formulae (\ref{OddchernClasses}) and (\ref{EvenchernClasses}) have positive coefficients, we have that $c_{d_1+1}(\sF^{(d_1)})\wedge\cdots\wedge c_{d_r+1}(\sF^{(d_r)})$ is not zero if $$\left(c_2(\sF)\wedge c_1^{d_1-1}(\sF)\right)\wedge\cdots \wedge\left(c_2(\sF)\wedge c_1^{d_r-1}(\sF)\right)$$ is not zero. Since the zero set of $r$ general sections of $\sF$ vanishes on a subgrassmannian $G':=\grass 2 {N+1-r}\subset G$ corresponding to an inclusion $\comp^{N+1-r}\subset \comp^{N+1}$ of vector spaces, this is the same as showing that $$c_1(\sF_{G'})^{\sum_{i=1}^r(d_i-1)}$$ is a nonzero cohomology class. Since $\det\sF$ is the very ample line bundle on $G$ which gives the Pl\"ucker embedding, we see that this is nonzero if $-r+\sum_{i=1}^rd_i\le \dim G'=2(N-1-r)$, i.e., if $\sum_{i=1}^rd_i\le 2N-2-r$. If this is false, then since $\sum_{i=1}^rd_i\le N$, we conclude that $N>2 N-2-r$ which implies that $r+1\ge N$. Since $r=N-\dim X$ this implies that $X$ is a curve. \qed See \cite{KS} for some related results about hypersurfaces in $\pn 1$-bundles over Grassmannians. \begin{rem*}We follow the notation used in Theorem \ref{CompIntFano} and its proof. If $N> \sum_{i=1}^rd_i$ then the argument of Barth and Van de Ven \cite{BVdV} applies to show that there is a family of lines covering $X$ with an $(N-\sum_{i=1}^rd_i-1)$-dimensional space of lines through a general point. \end{rem*} \section{Adjunction structure in case of $k$-very ampleness}\label{kAd} \setcounter{theorem}{0} Let $\widehat{X}$ be a smooth connected $n$-fold, $n\geq 3$, and let $\widehat{L}$ be a $k$-very ample line bundle on $\widehat{X}$, $k\geq 2$. In this section we describe the first and the second reduction of $(\widehat{X},\widehat{L})$. We have the following general fact. \begin{lemma}\label{nefv}Let $\widehat{X}$ be a smooth connected $n$-fold, $n\geq 3$, and let $\widehat{L}$ be a $k$-very ample line bundle on $\widehat{X}$, $k\geq 2$. Let $\tau$ be the nefvalue of $(\widehat{X},\widehat{L})$. Then $\tau\leq \frac{n+1}{k}$. \end{lemma} \proof Let $\Phi:\widehat{X}\to W$ be the nefvalue morphism of $(\widehat{X},\widehat{L})$. Let $C$ be an extremal curve contracted by $\Phi$. Then $(K_\hatX+\tau \hatL)\cdot C=0$ yields $k\tau \leq \tau \hatL\cdot C=-K_\hatX\cdot C\leq n+1$, or $\tau\leq\frac{n+1}{k}$.\qed We can now prove the following structure result. \begin{theorem}\label{General}Let $\widehat{X}$ be a smooth connected $n$-fold, $n\geq 3$, and let $\widehat{L}$ be a $k$-very ample line bundle on $\widehat{X}$, $k\geq 2$. Then either $(\widehat{X},\widehat{L})\cong (\pn 3,\pnsheaf 3 2)$ with $k=2$, or the first reduction $(X,L)$ of $(\widehat{X},\widehat{L})$ exists and $\widehat{X}\cong X$, $L\cong \hatL$. Furthermore either: \begin{enumerate} \item[{\rm i)}] $(\widehat{X},\widehat{L})\cong (\pn 3,\sO_{\pn 3}(2))$, $k=2$; \item[{\rm ii)}] $(\widehat{X},\widehat{L})\cong (\pn 3,\sO_{\pn 3}(3))$, $k=3$; \item[{\rm iii)}] $(\widehat{X},\widehat{L})\cong (\pn 4,\sO_{\pn 4}(2))$, $k=2$; \item[{\rm iv)}] $(\widehat{X},\widehat{L})\cong (Q,\sO_Q(2))$, $Q$ hyperquadric in $\pn 4$, $k=2$; \item[{\rm v)}] there exists a morphism $\psi:\hatX\to C$ onto a smooth curve $C$ such that $2K_\hatX+3\hatL\approx \psi^*H$ for some ample line bundle $H$ on $C$ and $(F,{\hatL}_F)\cong(\pn 2,\sO_{\pn 2}(2))$ for any fiber $F$ of $\psi$, $k=2$; \item[{\rm vi)}] $(\widehat{X},\widehat{L})$ is a Mukai variety, i.e., $K_\hatX\approx -(n-2)\hatL$ and either $n=4,5$ and $k=2$ or $n=3$ and $k\leq 4$; \item[{\rm vii)}] $(\widehat{X},\widehat{L})$ is a Del Pezzo fibration over a curve such that $(F,{\widehat{L}}_F)\cong (\pn 3,\sO_{\pn 3}(2))$ for each general fiber $F$, $n=4$, $k=2$, \end{enumerate} or there exists the second reduction $(Z,\sD)$, $\vphi:X\to Z$ of $(\widehat{X},\widehat{L})$. In this case the following hold. \begin{enumerate} \item[{\rm 1)}] If $n\geq 4$, then $X\cong Z$; \item[{\rm 2)}] If $n=3$, then either $X\cong Z$ or $k=2$ and $\vphi$ only contracts divisors $D\cong \pn 2$ such that $L_D\cong \sO_{\pn 2}(2)$; furthermore $\sO_D(D)\cong \sO_D(-1)$ and $Z$ is smooth. \end{enumerate}\end{theorem} \proof We use general results from adjunction theory for which we refer to \cite{Book}. From \cite[(9.2.2)]{Book} we know that $K_{\widehat{X}}+(n-1)\widehat{L}$ is spanned unless either $(\widehat{X},\widehat{L})\cong (\pn n,\sO_{\pn n}(1))$, or $\widehat{X}\subset \pn {n+1}$ is a quadric hypersurface and $L\approx\sO_{\pn {n+1}}(1)_{|\widehat{X}}$, or $(\widehat{X},\widehat{L})$ is a scroll over a curve. Since $L\cdot C\geq 2$ for each curve $C$ on $\widehat{X}$, all the above cases are excluded. Thus we can conclude that $K_{\widehat{X}}+(n-1)\widehat{L}$ is spanned. Then from \cite[(7.3.2)]{Book} we know that $K_{\widehat{X}}+(n-1) \widehat{L}$ is big unless either $(\widehat{X},\widehat{L})$ is a Del Pezzo variety, i.e., $K_{\widehat{X}}\approx -(n-1)\widehat{L}$, or $(\widehat{X},\widehat{L})$ is a quadric fibration over a smooth curve, or $(\widehat{X},\widehat{L})$ is a scroll over a normal surface. Then, as above, the quadric fibration and the scroll cases are excluded, so that $(\widehat{X},\widehat{L})$ is a Del Pezzo variety. In this case $\tau=n-1$, so that Lemma (\ref{nefv}) gives $2\leq k\leq \frac{n+1}{n-1}$. Hence $n=3$. By looking over the list of Del Pezzo $3$-folds (see \cite[(8.11)]{Fujita}) we conclude that $(\widehat{X},\widehat{L})\cong (\pn 3,\sO_{\pn 3}(2))$ in this case. Thus we can assume that the first reduction, $(X,L)$, of $(\widehat{X},\widehat{L})$ exists and in fact $\widehat{X}\cong X$, since otherwise we can find a line $\ell$ on $\widehat{X}$ such that $\widehat{L}\cdot \ell=1$. From \cite[(7.3.4), (7.3.5), (7.5.3)]{Book}) we know that on $\widehat{X}\cong X$ the line bundle $K_X+(n-2)L$ is nef and big unless either \begin{enumerate} \item[{\rm a)}] $(X,L)\cong (\pn 3,\sO_{\pn 3}(3))$; \item[{\rm b)}] $(X,L)\cong (\pn 4,\sO_{\pn 4}(2))$; \item[{\rm c)}] $(X,L)\cong (Q,\sO_Q(2))$, $Q$ hyperquadric in $\pn 4$; \item[{\rm d)}] there exists a morphism $\psi:X\to C$ onto a smooth curve $C$ such that $2K_X+3L\approx \psi^*H$ for some ample line bundle $H$ on $C$ and $(F,L_F)\cong(\pn 2,\sO_{\pn 2}(2))$ for any fiber $F$ of $\psi$; \item[{\rm e)}] $K_X\approx -(n-2)L$, i.e., $(X,L)$ is a Mukai variety; \item[{\rm f)}] $(X,L)$ is a Del Pezzo fibration over a smooth curve under the morphism, $\Phi_L$, given by $|m(K_X+(n-2)L)|$ for $m\gg 0$; \item[{\rm g)}] $(X,L)$ is a quadric fibration over a normal surface under $\Phi_L$; or \item[{\rm h)}] $(X,L)$ is a scroll over a normal threefold under $\Phi_L$.\end{enumerate} Cases a), b), c), d), e) lead to cases ii), iii), iv), v), vi) respectively. Case f) gives case vii). To see this, let $F$ be a general fiber of $\Phi_L$ and let $L_F$ be the restriction of $L$ to $F$. Let $\tau_F$ be the nefvalue of $(F,L_F)$. Then $K_F+(n-2)L_F$ is trivial, and hence $\tau_F=n-2=\dim F-1$. Therefore the same argument as above, by using again \cite[(8.11)]{Fujita}, gives $\dim F=3$ and $(F,L_F)\cong (\pn 3,\sO_{\pn 3}(2))$. Note that in case e), one has $\tau=n-2$, so that Lemma (\ref{nefv}) yields $2\leq k\leq \frac{n+1}{n-2}$. Thus either $n=4,5$ and $k=2$, or $n=3$ and $k\leq 4$. In cases g), h) we can find a line $\ell$ on $X$ such that $L\cdot \ell=1$, so that they are excluded. Thus we can assume that the second reduction, $(Z,\sD)$, $\vphi:X\to Z$, of $(\widehat{X}, \widehat{L})$ exists. Use the structure results of the second reduction (see \cite[(7.5.3), (12.2.1)]{Book}). If $n\geq 4$ we see that we can always find a line $\ell$ on $X$ such that $L\cdot\ell=1$. Then $X\cong Z$. If $n=3$, either $X\cong Z$ or $\vphi$ contracts divisors $D\cong \pn 2$ such that $L_D\cong \sO_{\pn 2}(2)$. Then $\sO_D(D)\cong \sO_D(-1)$ and $Z$ is smooth in this case.\qed The following is an immediate consequence of Theorem (\ref{General}). \begin{corollary} \label{ample}Let $\widehat{X}$ be a smooth connected $n$-fold, $n\geq 3$, and let $\widehat{L}$ be a $k$-very ample line bundle on $\widehat{X}$, $k\geq 2$. Then $K_{\widehat{X}}+(n-2)\widehat{L}$ is ample if $n\geq 4$ unless $k=2$ and either $(\widehat{X},\widehat{L})\cong (\pn 4,\sO_{\pn 4}(2))$, or $n=4,5$ and $(\widehat{X},\widehat{L})$ is a Mukai variety, or $n=4$ and $(\widehat{X},\widehat{L})$ is a Del Pezzo fibration over a curve such that $(F, {\widehat{L}}_F)\cong(\pn 3,\sO_{\pn 3}(2))$ for each fiber $F$.\end{corollary} The results of this section justify the study of Mukai varieties of dimension $n=3,4,5$, polarized by a $k$-very ample line bundle, $k\geq 2$. \section{Mukai varieties of dimension $n\geq 4$}\label{Mukai} \setcounter{theorem}{0} Let $(X,L)$ be a Mukai variety of dimension $n\geq 3$, i.e., $K_X\approx -(n-2)L$, polarized by a $k$-very ample line bundle $L$, $k\geq 2$ (see \cite{Mu1}, \cite{Mu2} for classification results of Mukai varieties). Since the nefvalue, $\tau$, of such pairs $(X,L)$ is $\tau=n-2$, we immediately see from Lemma (\ref{nefv}) that either $n=4,5$, $k=2$, or $n=3$, $k\leq 4$ (compare with the proof of (\ref{General})). We have the following result. \begin{theorem}\label{MuThm} Let $(X,L)$ be a Mukai variety of dimension $n\geq 4$ polarized by a $k$-very ample line bundle $L$, $k\geq 2$. Then either \begin{enumerate} \em\item\em $n=4$, $k=2$, $(X,L)\cong (Q,\sO_Q(2))$, $Q$ hyperquadric in $\pn 5$, or \em\item\em $n=5$, $k=2$, $(X,L)\cong (\pn 5, \sO_{\pn 5}(2))$.\end{enumerate}\end{theorem} \proof By the above we know that $k=2$ and $n=4,5$. Let $V$ be the $3$-fold section obtained as transversal intersection of $n-3$ general members of $|L|$. Let $L_V$ be the restriction of $L$ to $V$. Note that $K_V\approx -L_V$, so that $V$ is a Fano $3$-fold, and $L_V$ is $k$-very ample. We denote by $r$ the index of $V$. Then $L_V\approx -K_V=rH$ for some ample line bundle $H$ on $V$. Note that we have $r\geq 2$ since otherwise Shokurov's theorem \cite{Sho} (see also \cite{R}) applies to say that either $V=\pn 1\times \pn 2$ or there exists a line $\ell$ on $V$ with respect to $H$. In the latter case $(H\cdot\ell)_V=(L\cdot\ell)_X=1$, which contradicts the assumption $k\geq 2$. The following argument rules out the former case $V=\pn 1\times \pn 2$. If $V=\pn 1\times \pn 2$, as corollary of the extension theorem \cite[Prop. III]{SoAmple} we conclude that $X$ is a linear $\pn 3$-bundle over $\pn 1$, $p:X\to \pn 1$, with the restriction $p_V$ giving the map $V=\pn 1\times\pn 2\to \pn 1$. By taking the direct image of $$0\to \sO_X\to L\to L_V\to 0$$ we get the exact sequence $$0\to \sO_{\pn 1}\to \sE\to \sE/\sO_{\pn 1}\to 0,$$ where $\sE:=p_*L$. Since ${\Bbb P}(\sE/\sO_{\pn 1})\cong \pn 1\times \pn 2$ we conclude that $\sE/\sO_{\pn 1}=\sO_{\pn 1}(1)\oplus\sO_{\pn 1}(1)\oplus \sO_{\pn 1}(1)$. Thus $\deg(\sE/\sO_{\pn 1})=3=\deg(\sE)< {\rm rank}(\sE)=4$. Therefore $L$ cannot be ample. By Lefschetz theorem we have $H\approx H'_V$ for some line bundle $H'$ on $X$, as well as $L\approx rH'$. Hence in particular $H'$ is ample. We have $K_X+r(n-2)H'\approx \sO_X$, so that $r(n-2)\leq n+1$ by a well known result due to Maeda (see e.g., \cite[(7.2.1)]{Book}). If $r=4,3$ we find numerical contradictions since we are assuming $n\geq 4$. Thus $r=2$ and $n\leq 5$. If $n=5$ we have $K_X\approx -6H'$ and if $n=4$ we have $K_X\approx -4H'$. By the Kobayashi-Ochiai theorem (see e.g., \cite[(3.6.10)]{Book}) we get in the former case $(X,H')\cong (\pn 5,\sO_{\pn 5}(1))$, or $(X,L)\cong (\pn 5,\sO_{\pn 5}(2))$ as in case $1$) of the statement, and in the latter case $(X,H')\cong (Q,\sO_Q(1))$, $Q$ hyperquadric in $\pn 5$, or $(X,L)\cong (Q,\sO_Q(2))$ as in case $2$) of the statement. \qed \begin{rem*}\label{j1} Note that in both cases 1), 2) of Theorem (\ref{MuThm}) the line bundle $L$ is in fact $2$-jet ample (see \cite[Corollary (2.1)]{Plenum}). \end{rem*} \section{The Fano $3$-fold case}\label{FanoS} \setcounter{theorem}{0} In this section we classify the $3$-dimensional Mukai varieties $(X,L)$ polarized by a $k$-very ample line bundle $L$, $k\geq 2$, i.e., we classify all Fano $3$-folds $X$ such that the anticanonical divisor $-K_X$ is $k$-very ample, $k\geq 2$. \begin{prgrph*}{A special case}\label{SpecialEx} Let us start by studying a particular case. This case has a special interest also because it gives a simple explicit example of a line bundle which is $2$-very ample but not $2$-jet ample. This example is case 4) in the Iskovskih-Shokurov's list \cite[Table 21]{IS} of Fano $3$-folds of first species, i.e., $b_2(X)=1$. \begin{proposition}\label{SpecialProp} Let $X$ be a smooth double cover of $\pn 3$, $p:X\to \pn 3$, branched along a quartic. Then $L:=-K_X$ is $2$-very ample but not $2$-jet ample. \end{proposition} \proof We have $L:=-K_X=p^*\sO_{\pn 3}(2)$. First we show that $L$ is not $2$-jet ample. Let $R$ be the ramification divisor of $p$. Let $\sZ$ be a length $3$ zero dimensional subscheme of $X$ such that ${\rm Supp}(\sZ)=\{x\}$ with $x\in R$. We can assume that $R$ is defined at $x$ by a local coordinate, $s$, i.e., $R=\{s=0\}$ at $x$. Consider local coordinates $(s,v,w)$ on $X$ at $x$. Let $y\in \pn 3$ be a point, belonging to the branch locus of $p$, such that $y=p(x)$. We can consider local coordinates $(t,v,w)$ on $\pn 3$ at $y$, where $p^*t=s^2$. We have \begin{equation}\label{Decomp} H^0(L)\cong H^0(p_*p^*\sO_{\pn 3}(2))\cong H^0(\sO_{\pn 3}(2))\oplus H^0(\sO_{\pn 3}). \end{equation} Therefore we can find a base, $\sB$, of $H^0(L)$ given by the pullback of sections of $\sO_{\pn 3}(2)$ and one more section, $\grs\in H^0(\sO_{\pn 3})$, which in local coordinates around $x$ is of the form $\lambda s$ with $\lambda$ a holomorphic function that doesn't vanish at $x$. That is, recalling that $s^2=p^*t$, $v=p^*v$, $w=p^*w$, $$\sB=<1,s^2,v,w,s^4,v^2,w^2,s^2v,s^2w,vw,\grs>.$$ On the other hand, $H^0(L/{\frak m}_x^3)(=H^0(\sO_\sZ(L)))$ contains the elements $sv$, $sw$ which are not images of elements of the base $\sB$. This shows that the restriction map $H^0(L)\to H^0(\sO_\sZ(L))$ is not surjective. Thus $L$ is not $2$-jet ample. We prove now that $L$ is $2$-very ample. Consider a $0$-dimensional subscheme $\sZ$ of $X$ of length $3$. Recalling (\ref{Decomp}), the fact that $\sO_{\pn 3}(2)$ is $2$-very ample and Lemma (\ref{GenFa}), we see that the restriction map $H^0(L)\to H^0(\sO_\sZ(L))$ is always surjective except possibly in the case when ${\rm Supp}(\sZ)$ is a single point, $x$, belonging to the ramification divisor of the cyclic covering $p$. Thus, let us assume ${\rm Supp}(\sZ)=\{x\}$ and consider the ideals $\sJ_i:=(\sJ_\sZ,{\frak m}_x^i)$, where $\sJ_\sZ$, ${\frak m}_x$ denote the ideal sheaves of $\sZ$, $x$ respectively, the sheaves $\sO_i:=\sO_X/\sJ_i$, the maps $p_i:\sO_\sZ\to \sO_i$, $i=1,2,3$, and the cofiltration $\sO_3\to\sO_2\to\sO_1\to 0.$ Note that the following hold true. \begin{itemize} \item If $H^0(\sO_\sZ)$ is generated by only terms of degree $\leq 1$ in the local coordinates $s$, $v$, $w$ on $X$ at the point $x$ then (as observed before in the proof that $L$ is not $2$-jet ample) the image of $H^0(L)$ can generate $H^0(\sO_\sZ(L))$, so the restriction map $H^0(L)\to H^0(\sO_\sZ(L))$ is surjective in this case and we are done; \item ${\rm length}(\sO_1)=1$; \item ${\rm length}(\sO_i)\neq{\rm length}(\sO_{i+1})$, $i=1,2$. Indeed, otherwise, $(\sJ_\sZ,{\frak m}_x^i)=(\sJ_\sZ,{\frak m}_x^{i+1})$ so that ${\frak m}_x^i\subset \sJ_{\sZ}$ and therefore $H^0(\sO_\sZ)$ is generated by only constant terms or linear terms. By the above we are done in this case.\end{itemize} Thus we are reduced to consider the case when ${\rm length}(\sO_2)=2$, ${\rm length}(\sO_3)=3$. We claim that $H^0(\sO_\sZ)$ contains at least one quadratic term in $s$, $v$, $w$. Indeed, if not, ${\frak m}_x^2\subset \sJ_\sZ$ and hence we would have $(\sJ_\sZ, {\frak m}_x^2)=(\sJ_\sZ,{\frak m}_x^3)$, which gives the contradiction ${\rm length}(\sO_2)={\rm length}(\sO_3)$. Furthermore, since ${\rm length}(\sO_X/(\sJ_\sZ,{\frak m}_x^2))=2$ and ${\rm length}(\sO_X/{\frak m}_x^2)=4$, we conclude that $\sJ_\sZ$ contains two independent linear terms, say $f$, $g$, not belonging to ${\frak m}_x^2$. Write $$f=as+bv+cw,\;\;g=ds+ev+hw,$$ where the coefficients $a$, $b$, $c$, $d$, $e$, $h$ belong to $\sO_{X,x}$. Let $\sB$ be the base of $H^0(L)$ constructed in the first part of the proof, where we showed that $L$ is not $2$-jet ample. Following that argument we see that $L$ is $2$-very ample as soon as we show that the elements $sv$, $sw$ can be written in $\sO_\sZ$ in terms of elements of $\sB$. We go on by a case by case analysis. First, assume $a\neq 0$, i.e., $a$ invertible in $\sO_{X,x}$ and write $$asw=w(as+bv+cw)-bvw-cw^2.$$ Since $as+bv+cw=f=0$ in $\sO_\sZ$, up to dividing by $a$, we can express $sw$ in terms of $vw, w^2\in \sB$ in $\sO_\sZ$. Similarly, writing $$asv=v(as+bv+cw)-bv^2-cvw,$$ we conclude that $sv$ can be expressed in terms of $v^2, vw\in\sB$ in $\sO_\sZ$. If $d\neq 0$ we get the same conclusion. Thus it remains to consider the case when $a=d=0$. In this case $f=bv+cw$, $g=ev+hw$ in $\sO_\sZ$. Then, solving with respect to $v$, $w$, and noting that $\left(\begin{array}{ll} b &c\\ e &h\end{array}\right)$ is a rank two matrix since $f$, $g$ are independent, we can express $v$, $w$ as linear functions of $f$, $g$ in $\sO_\sZ$. Since $f, g\in \sJ_\sZ$, we conclude that $v, w\in \sJ_\sZ$ and hence $sv$, $sw$ belong to $\sJ_\sZ$. Therefore $sv=sw=0$ in $\sO_\sZ$. \qed \end{prgrph*} The following general result is a consequence of Fujita's classification \cite{Fupapers}, \cite[(8.11)]{Fujita} of Del Pezzo $3$-folds (see also \cite{MoMu}, \cite{IS} and \cite{Murre} for a complete classification of Fano $3$-folds). Note that in each case of the theorem below the line bundle $L$ is in fact $k$-very ample (see \cite[Corollary (2.1)]{Plenum}, Lemma (\ref{Tensor}) and Proposition (\ref{SpecialProp}). \begin{theorem}\label{Fano} Let $X$ be a Fano threefold. Assume that $L:=-K_X$ is $k$-very ample, $k\ge 2$. Then either: \begin{enumerate} \em\item\em $X$ is a divisor on $\pn 2\times\pn 2$ of bidegree $(1,1)$, $L=\sO_X(2,2)$, $k=2$; \em\item\em $X=\pn 1\times\pn 2$, $L=\sO_{\pn 1}(2)\bx\sO_{\pn 2}(3)$, $k=2$; \em\item\em $X=V_7$, the blowing up of $\pn 3$ at a point, $L=2(q^*\sO_{\pn 3}(2)-E)$, $q:V_7\to \pn 3$, $E$ the exceptional divisor, $k=2$; \em\item\em $X=\pn 1\times\pn 1\times\pn 1$, $L=\sO_{\pn 1}(2)\bx\sO_{\pn 1}(2)\bx\sO_{\pn 1}(2)$, $k=2$; \em\item\em $X=\pn 3$, $L=\sO_{\pn 3}(4)$, $k=4$; \em\item\em $X$ is a hyperquadric in $\pn 4$, $L=\sO_X(3)$, $k=3$; \em\item\em $X$ is a cubic hypersurface in $\pn 4$, $L=\sO_X(2)$, $k=2$; \em\item\em $X$ is the complete intersection of two quadrics in $\pn 5$, $L=\sO_X(2)$, $k=2$; \em\item\em $X$ is a double cover of $\pn 3$, $p:X\to \pn 3$, branched along a quartic; $L=p^*\sO_{\pn 3}(2)$ is $2$-very ample but not $2$-jet ample; \em\item\em $X$ is the section of the Grassmannian $\grass 2 5$ {\rm (}of lines in $\pn 4${\rm )} by a linear subspace of codimension $3$, $L=\sO_X(2)$, $k=2$. \end{enumerate} \end{theorem} \proof Let $r$ be the index of $X$. Then $L:=-K_X=rH$ for some ample line bundle $H$ on $X$. Note that we have $r\geq 2$ since otherwise Shokurov's theorem \cite{Sho} applies to say that either $X=\pn 1\times\pn 2$ or there exists a line $\ell$ with respect to $H$. In the former case we are in case $2$) of the statement. In the latter case $H\cdot\ell=L\cdot\ell=1$, which contradicts the assumption $k\geq 2$. If $r=4,3$, by using the Kobayashi-Ochiai theorem (see e.g., \cite[(3.6.1)]{Book}) we find cases $5$), $6$) of the statement respectively. Thus we can assume $r=2$. In this case $(X,H)$ is a Del Pezzo $3$-fold described as in \cite[(8.11)]{Fujita}. A direct check shows that the cases listed in \cite[(8.11)]{Fujita} lead to cases $1$), $3$), $4$), $7$), $8$), $9$), $10$) of the statement. Recall that case $9$) is discussed in Proposition (\ref{SpecialProp}). Notice that the case of $X={\Bbb P}(\sT)$, for the tangent bundle $\sT$ of $\pn 2$, as in \cite[(8.11), $6$)]{Fujita} gives our case $1$) (see Remark (\ref{PT}) below). Note also that case $1$) of \cite[(8.11)]{Fujita}, when $(X,H)$ is a weighted hypersurface of degree $6$ in the weighted projective space ${\Bbb P}(3,2,1,\ldots,1)$ with $H^3=1$, is ruled out since $L=2H$ is not even very ample. To see this notice that there exist a smooth surface $S$ in $|H|$ and a smooth curve $C$ in $|H_S|$ (see \cite[(6.1.3), (6.14)]{Fujita}). On $S$ we have $K_S\approx -H_S$, so that $K_S^2=1$. Therefore $C$ is a smooth elliptic curve with $H\cdot C=H^3=1$, i.e., $L\cdot C=2$.\qed \begin{rem*}\label{PT} It is a standard fact that ${\Bbb P}(\sT)$, for the tangent bundle $\sT:=\sT_{\pn n}$ of $\pn n$, is embedded in $\pn n\times \pn n$ as a divisor of bidegree $(1,1)$ (see also \cite{Sato}). To see this, note that $\sT(-1)$ is spanned with $n+1$ sections. Thus letting $\xi$ denote the tautological line bundle of $\proj{\sT(-1)}$, the map $f:{\Bbb P}(\sT)\to \pn n$ associated to $|\xi|$ is an embedding on fibers of the bundle projection $p:{\Bbb P}(\sT)\to \pn n$. The product map $(f,p):{\Bbb P}(\sT)\times {\Bbb P}(\sT)\to \pn n\times \pn n$ is thus an embedding with image a divisor $D$ such that $\sO_{\pn n\times \pn n}(D)_{|F}\cong\pnsheaf {n-1}1$ on the fibers $F$ of $p$. The fibers $(F',\xi_{F'})$ of $f$ are $\cong \pnpair {n-1}1$. To see this for $F'$, a generic fiber of $f$, note that $c_1(p^*\pnsheaf n 1)^{n-1}\cdot F'=c_1(p^*\pnsheaf n 1)^{n-1}\cdot c_1(\xi)^n$. Using the defining equation for the Chern classes of $\sT(-1)$, we see that this equals $p^*\left(c_1(\pnsheaf n 1)^{n-1}\cdot c_1(\sT(-1))\right)\cdot c_1(\xi)^{n-1}= c_1(\pnsheaf n 1)^{n-1}\cdot c_1(\sT(-1))=1.$ Since $D$, $f(D)$, and the generic fiber of $f$ are all connected, it follows that all fibers of $f$ are connected. From this it follows that all fibers of $f$ are isomorphic if the automorphism group of $D$ acts transitively on $D$. This can be seen by observing that given two nonzero tangent vectors of $\pn n$ there is an automorphism of $\pn n$ which takes one tangent vector to the other. \end{rem*} \begin{rem*}\label{kspanj} Note that the line bundles $L$ in (\ref{MuThm}) and (\ref{Fano}) are of course $k$-spanned and also $k$-jet ample, with the only exception of Case $9$) in (\ref{Fano}) (see (\ref{kthemb}), (\ref{SpecialProp}) and \cite[Corollary (2.1)]{Plenum}). Thus (\ref{MuThm}) and (\ref{Fano}) also give the classification of Mukai varieties $(X,L)$ of dimension $n\geq 3$ polarized by either a $k$-spanned or a $k$-jet ample line bundle $L$, with the only exception, for $k$-jet ampleness, of Case $9$) in (\ref{Fano}). \end{rem*}

9711

alg-geom/9711004

en

https://arxiv.org/abs/alg-geom/9711004

alg-geom/9711004

Anvar Mavlyutov

Anvar R. Mavlyutov

Approximation by smooth curves near the tangent cone

Extended version of the paper with an application to studying schemes of algebras included

Journal of Algebra, Volume 266, Issue 1, 2003, pages 154-161

We show that through a point of an affine variety there always exists a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This result has an application to the study of affine schemes.

alg-geom

\section{Plane curves approximating varieties near the tangent cone.}\label{s:pc} In this section, we generalize the notion of multiplicity of intersection used in \cite[Chapter~3,~\S4, Definition~1]{clo}. Proposition~3 in \cite[Chapter~9,~\S6]{clo}, has a criterion which says that a line lies in the tangent space of the variety iff the line meets the variety with multiplicity at least 2. Here, we show that through a singular point of an affine variety we can always draw a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This basic result does not seem to appear anywhere in the literature. Let $K$ be an arbitrary field. An algebraic curve $C\subset{\Bbb A}^{n}$ can be parameterized at its smooth point $p$ by Taylor's series $$G(t)=p+t\cdot v_1+t^2\cdot v_2+\cdots,$$ where $t$ is a local parameter. Using this parameterization, we can define multiplicity of intersection of an affine variety with the curve $C$ at the point $p$. \begin{defn} Let $m$ be a nonnegative integer and $X\subset{\Bbb A}^{n}$ an affine variety, given by $\text{\bf I}(X)\subset K[x_1,\ldots,x_n]$. Suppose that we have a curve $C$, smooth at a point $p$ and with the parameterization as above. Then $C$ {\bf meets} $X$ {\bf with multiplicity} $m$ at $p$ if $t=0$ is a zero of multiplicity at least $m$ of the series $f\circ G(t)$ for all $f\in\text{\bf I}(X)$, and for some $f$, of multiplicity exactly $m$. We denote this multiplicity of intersection by $I(p,C,X)$. \end{defn} \begin{rem} This definition does not depend on the choice of the parameterization. To check this one can use the following criterion: $t=0$ is a zero of multiplicity $m$ of $h(t)\in K[[t]]$ if and only if $h(0)=h'(0)=\cdots=h^{(m-1)}(0)=0$ but $h^{(m)}(0)\ne 0$. \end{rem} Also, we denote by $T_{p}X$ ($TC_{p}X$) the tangent space (cone) of a variety $X$ at a point $p$. The next result contains a necessary condition for a line to be in the tangent cone. Unfortunately, it is not a sufficient one. \begin{thm}\label{t:m} Let $X\subset{\Bbb A}^{n}$ be an affine variety and $p\in X$ a singular point. Then for each $v\in TC_{p}X$ there exists a smooth curve $C\subset{\Bbb A}^{n}$ through $p$ such that $v\in T_{p}C$ and $I(p,C,X)\geq3$. \end{thm} \begin{pf} Let $v\in TC_{p}X$. We can choose affine coordinates $x_{1},\ldots,x_{n}$, so that $p=(0,\ldots,0)$. Let $C$ be a curve through $p$ parameterized by $G(t)=t\cdot v+t^{2}\cdot\gamma$, where $v\in TC_{p}X$, $\gamma=(\gamma_1,\ldots,\gamma_{n})$. It suffices to show that there is $\gamma$, which is zero or not a scalar multiple of $v$, such that $t=0$ is a zero of multiplicity $\geq3$ of $f\circ G(t)$ for all $f\in\text{\bf I}(X)$. In this case, $C$ will be a smooth curve with the local parameter $t$ at $p$. For each $f\in\text{\bf I}(X)$, denote by $l_{f}(x)=l_{1}x_{1}+\cdots+l_{n}x_{n}$ and $q_{f}(x)$ the linear and quadratic parts of $f$. According to this notation we define $$W=\{(l_{1},\ldots,l_{n},q_{f}(v)):\,f\in\text{\bf I}(X)\}\subset k^{n+1},$$ which is a subspace since $\text{\bf I}(X)$ is. For all $f\in\text{\bf I}(X)$, we have $$f(t\cdot v+t^{2}\cdot\gamma)=l_{f}(t\cdot v)+l_{f}(t^{2}\cdot\gamma)+ q_{f}(t\cdot v)+\text{ terms of degree }\geq3\text{ in }t.$$ Since $v\in T_{p}X$, we get $l_{f}(v)=0$ for any $f\in\text{\bf I}(X)$. Hence, $$f(t\cdot v+t^{2}\cdot\gamma)\equiv t^{2}(l_{1}\gamma_{1}+\cdots+l_{n}\gamma_{n}+q_{f}(v))$$ modulo terms of degree $\geq3$ in $t$. So we need to resolve equations $a_{1}\gamma_{1}+\cdots+a_{n}\gamma_{n}+a_{n+1}=0$ in variables $\gamma_{1},\ldots,\gamma_{n}$ for all possible $(a_{1},\ldots,a_{n+1})\in W$. Consider the scalar product $k^{n+1}\times k^{n+1}\rightarrow k$, which sends $a\times b$ to $a\cdot b:= \sum^{n+1}_{i=1}a_{i}b_{i}$. Using this scalar product, we will find the vector $(\gamma_{1},\ldots,\gamma_{n},1)\in W^{\perp}:= \{b\in k^{n+1}:\,b\cdot a=0\text{ for all } a\in W\}$. If $W\subseteq k^{n}\times\{0\}$, then we can put $\gamma=(0,\ldots,0)$ to obtain the required curve. Otherwise, consider the projection $\pi:\,k^{n+1}\rightarrow k^{n}\times\{0\}$, which assigns $(a_{1},\ldots,a_{n},0)$ to $(a_{1},\ldots,a_{n+1})$. And, let $\tilde{\pi}:\,W\rightarrow k^{n}\times\{0\}$ be induced by $\pi$. The projection $\tilde{\pi}$ is injective, because $v\in TC_{p}X$ and $l_{f}=0$ imply $q_{f}(v)=0$. If we denote $\widetilde{W}=\tilde{\pi}(W)$, then $\dim W^{\perp}=\dim\widetilde{W}^{\perp}$, by injectivity of $\tilde{\pi}$. Now, if $W^{\perp}$ is of the form $W_{1}\times\{0\}$, then by construction $\widetilde{W}^{\perp}=W_{1}\times k$, contradicting with the equality of the dimensions. Therefore, there exists a vector $(\gamma_{1},\ldots,\gamma_{n},1)\in W^{\perp}$, i.e., $l_{f}(\gamma)+q_{f}(v)=0$ for all $f\in I(X)$. We claim that this $\gamma$ is linearly independent of $v$. Indeed, since $W$ is not included into $k^{n}\times\{0\}$, we get $q_{f}(v)\ne0$ for some $f\in\text{\bf I}(X)$, which implies $l_{f}(\gamma)\ne0$. On the other hand, since $v\in T_{p}X$, we get $l_{f}(v)=0$. Thus, we have found the desired $\gamma$. \end{pf} \begin{rem} This result cannot be improved in the following sense. For the curve $X$, given by the equation $x^{2}=y^{3}$, in the affine plane, and $p=(0,0)$ there is no smooth curve $C$ such that $I(p,C,X)\geq4$. \end{rem} \begin{rem} In the case of analytic varieties the theorem is also valid. \end{rem} \section{An application to the affine schemes of algebras.}\label{s:ap} This section shows that at least theoretically one may still be able to answer the problem of Shafarevich discussed in the introduction by solving quadratic equations arising from the second order obstructions to deformations of algebras. In particular, we find that the vectors, contributing to the discrepancy between the tangent spaces to the scheme of associative multiplications and the scheme of the degree 3 nilpotent multiplications, can be easily ``killed'' by the obstructions in many cases. Then Theorem~\ref{t:m} implies that the tangent cones of the reduced schemes are the same, which means that the irreducible component of one variety is the component of the other one. Let us recall the notation from \cite{sh,am}. The affine scheme $C_n$ of all multiplications on a fixed $n$-dimensional vector space $V$ over a field $K$ (${\rm char} K\ne2$), which represent associative commutative algebras, is given by the equations of commutativity and associativity: $$c_{ij}^k=c_{ji}^k,\qquad \sum_{s=1}^n c_{ij}^s c_{sk}^l=\sum_{s=1}^n c_{is}^l c_{jk}^s$$ in the structure constants of multiplication $e_ie_j=\sum_{k=1}^n c_{ij}^k e_k$ of the basis $\{e_1,\ldots,e_n\}$. Similar equations determine the affine scheme $A_n$ of commutative nilpotent degree 3 multiplications. The reduced schemes associated to the above schemes are denoted $C_n^{\rm red}$ and $A_n^{\rm red}$, respectively. From \cite{sh}, the irreducible components of $A_n^{\rm red}$ have a very simple description $$A_{n,r}=\{N\in A_n^{\rm red} |\, \dim N^2\le r\le \dim {\rm Ann}_N N\},$$ $1\le r\le(n-1)(n-r+1)/2$, where $N$ denotes the algebra represented by the corresponding multiplication, $N^2$ is the square of the algebra and $\rm Ann$ is the annihilator. Let $d:=n-r$, then, for $r=1,2$ and $r>(d^2-1)/3$, Shafarevich in \cite{sh} showed that $A_{n,r}$ is not a component of $C_n^{\rm red}$, by constructing a line, contained in $C_n^{\rm red}$ but not in $A_n^{\rm red}$, through a point of $A_{n,r}$. For other $r$, one has to compare the tangent spaces to the non-reduced schemes $A_n$ and $C_n$. The smooth set of $A_n^{\rm red}$ is the union of $$U_{n,r}=\{N\in A_n^{\rm red} |\, \dim N^2= r= \dim {\rm Ann}_N N\}.$$ If $W\subset V$ is a subspace of dimension $r$, then the space $S_{n,r}=L(S^2(V/W),W)$ of all linear maps from the symmetric product of $V/W$ to $W$ is naturally included as an affine subspace into $A_{n,r}$. The group $G={\rm GL}(V)$ acts on $C_n$ and $G S_{n,r}=A_{n,r}$. Then, the tangent space to $A_{n,r}$ at a point $N\in S_{n,r}$ is $$T_N A_{n,r}=L(S^2(N/N^2),N^2)+T_N G N,$$ where the tangent space $T_N G N$ to the orbit is the space of maps from $L(S^2N,N)$ given by the coboundaries (in a Hochschild complex, see \cite{am}) $x\circ y= x\varphi(y)-\varphi(xy)+y\varphi(x)$ for some linear map $\varphi:N@>>>N$. The tangent space $T_N C_n$ always includes $T_N A_{n,r}+F$, where the space $F$ consists of $x\circ y=xf(y)+yf(x)$ for some $f\in L(N/N^2,K)\subset L(N,K)$. To explain this difference Shafarevich embedded the scheme $C_n$ into the scheme $\widetilde{C}_n$ of commutative associative multiplications on a $(n+1)$-dimensional space which represent algebras with a unit $e$: $$C_n\hookrightarrow\widetilde{C}_n, \qquad N\mapsto N\oplus Ke.$$ In this situation, the subgroup $\widetilde G$ of $GL(V\oplus Ke)$ which fixes the unit acts on $\widetilde{C}_n$. It turns out that $T_N GN+F$ is the tangent space to $\widetilde{G} S_{n,r}$, whence the equality \begin{equation}\label{e:tan} T_N \widetilde{C}_n=T_N A_{n,r}+F \end{equation} was enough to conclude that $A_{n,r}$ is the component of $C_n^{\rm red}$. The equality was shown in \cite{sh} for $3\le r\le (d+1)(d+2)/6$, and this also holds $r=(5d^2-8d)/16$ and $d$ divisible by 4 by \cite[Section~2.1]{am}. The original Shafarevich's method is very special to this situation and is impractical to use it in general to explain the nilpotents in the structure sheaf of a scheme. We expect that the equality (\ref{e:tan}) holds for all $3\le r<(d^2-1)/3$, which would leave only one case $r=(d^2-1)/3$ unsettled. However, in this last case the tangent space to $C_n$ at a generic point in $A_{n,r}$ is too big: \begin{equation}\label{e:for} T_N C_n= L(S^2(N/N^2),N^2)+T_N G N+L(S^2N_1,N_1), \end{equation} where $N=N_1\oplus N^2$ is a fixed decomposition. To show this one have to use Lemma~1 in \cite{am}: a generic algebra $N\in A_{n,r}$, for $r\le(d^2-1)/3$, can be given by $d$ generators and a $(d(d+1)/2-r)$-dimensional space of homogeneous degree 2 relations among the generators, so that the nilpotence of the third degree follows from this relations. But, for $r=(d^2-1)/3$, this condition implies that there are no nontrivial relations among the homogeneous degree 2 relations, i.e., all of the relations $\sum_i n_iz_i=0$ in $S^3N_1$, $n_i\in N_1$, $z_i\in Z:=ker(\mu:S^2N_1@>>>N^2)$ ($\mu$ is the multiplication on $N$), are induced by a trivial in $N_1\otimes Z$ element $\sum_i n_i\otimes z_i$. Indeed, $\dim S^3N_1=d(d+1)(d+2)/6$, while the number of equations that we get from the homogeneous degree 2 relations is equal to $$\dim N_1\otimes Z=d(d(d+1)/2-(d^2-1)/3)=d(d+1)(d+2)/6.$$ So, if there is a nontrivial relation among the homogeneous degree 2 relations, there would be not enough equations to deduce $N^3=0$. Since the only restriction in \cite[Lemma~1]{am} for a vector from $L(S^2N_1,N_1)\subset L(S^2N,N)$ to be tangent to the scheme $C_n$ was arising from the nontrivial relations, we conclude (\ref{e:for}). Anan'in suggested in \cite{am} to use the second order obstructions to deformations of algebras to eliminate the excessive space $L(S^2N_1,N_1)$. To be precise, suppose that $N\in A_{n,r}$ is a smooth point of $C_n^{\rm red}$ and $\circ\in L(S^2N_1,N_1)$. Then, if $\circ$ is tangent to $C_n^{\rm red}$, the sum $\circ+L(S^2(N/N^2),N^2)$ also lies in the tangent space $T_N C_n^{\rm red}$. A deformation of a commutative algebra on a vector space $V$ with multiplication $x\cdot y$ can be thought as a smooth curve $$x\cdot y+(x\circ y)t+(x\star y)t^2+\cdots$$ in the affine space $L(S^2V,V)$, where $t$ is a local parameter. It is a well know fact in algebraic geometry that through a smooth point $p$ of a variety $X$ we can find a smooth curve inside $X$, whose tangent vector at $p$ is a given one in $T_p X$. Therefore, the smooth curves in $C_n^{\rm red}$ with tangent vectors $\circ+L(S^2(N/N^2),N^2)$ give a lot of equations (the second order obstructions) arising from the associativity of multiplication: $$(x\tilde\circ y)\tilde\circ z-x\tilde\circ(y\tilde\circ z)= x(y\star z)-(xy)\star z+x\star(yz)-(x\star y) z$$ for some $\star\in L(S^2N,N)$, where $\tilde\circ\in\circ+L(S^2(N/N^2),N^2)$. A nice thing about these quadratic equations on $\circ$ is that one can linearize them: $$(x\circ y)* z+(x*y)\circ z-x\circ(y* z)-x*(y\circ z)= x(y\star z)-(xy)\star z+x\star(yz)-(x\star y) z$$ for all $*\in L(S^2(N/N^2),N^2)$. These equations have been used to prove Theorem~1 in \cite{am} which implies that $A_{n,r}$ is the component of $C_n^{\rm red}$ for $d(d+1)/9\le r\le[d/3](d-3)$ (almost all cases that are not covered by \cite{sh}) if a certain algebra $N\in A_{n,r}$ is a smooth point of $C_n^{\rm red}$. Unfortunately, it seems impossible to prove that $N$ is the smooth point unless we know the tangent space at $N$. Now, we can apply Section~\ref{s:pc}. Let $N\in A_{n,r}$ with multiplication $x\cdot y$ and $\circ\in TC_N C_n^{\rm red}$. By Theorem~\ref{t:m}, there exists a smooth plane curve $C$ $$x\cdot y+(x\circ y)t+(x\star y)t^2$$ through $N$ in the affine space $L(S^2V,V)$ of commutative multiplications on the vector space $V$, such that $\circ$ is the tangent vector to the curve at $N$ and the multiplicity of intersection $I(p,C,C_n^{\rm red})\ge3$. Hence, the associativity equations in the structure constants imply \begin{equation}\label{e:ob} (x\circ y)\circ z-x\circ(y\circ z)= x(y\star z)-(xy)\star z+x\star(yz)-(x\star y) z \end{equation} for some $\star\in L(S^2N,N)$. Note that we do not assume the smoothness condition at $N$. We want to deduce the effective part of the obstructions to $\circ$. As in \cite[\S1]{am}, decompose $\circ$ and $\star$ into the sum of linear maps $f_{ij}^k$ and $g_{ij}^k$ in $L(N_i\otimes N_j,N_k)$, respectively, where $1\le i,j,k\le2$ and $N=N_1\oplus N_2$, $N_2:=N^2$. For a sufficiently generic $N\in A_{n,r}$, besides commutativity, $f_{ij}^k$ satisfy $$f_{12}^1=f_{22}^1=f_{22}^2=0,$$ \begin{equation}\label{e:co} xf_{11}^1(y,z)+f_{12}^2(x,yz)=f_{12}^2(z,xy)+f_{11}^1(x,y)z, \end{equation} by \cite[\S1]{am}. Moreover, $f_{12}^2$ satisfying (\ref{e:co}) is unique for a given $f_{11}^1$, while its existence condition is described in \cite[Lemma~1]{am}. The missing $f_{11}^2\in L(S^2(N/N^2),N^2)$ is always tangent to $C_n$. From (\ref{e:ob}) we get \begin{equation}\label{e:ob1} f_{11}^1(f_{11}^1(x,y),z)-f_{11}^1(x,f_{11}^1(y,z))= -g_{12}^1(z,xy)+g_{12}^1(x,yz), \end{equation} \begin{equation}\label{e:ob2} f_{12}^2(x,f_{12}^2(y,zt))-f_{12}^2(y,f_{12}^2(x,zt))= xg_{12}^1(y,zt)-yg_{12}^1(x,zt) \end{equation} and $$f_{12}^2(f_{11}^1(x,y),zt)-f_{12}^2(x,f_{12}^2(y,zt))=xg_{12}^1(y,zt)- g_{22}^2(xy,zt),$$ where $x,y,z,t\in N_1$. The last equation determines $g_{22}^2$, and, one can check that commutativity of $g_{22}^2$ (surprisingly) follows from (\ref{e:co}), (\ref{e:ob1}) and (\ref{e:ob2}). So, the only restrictions for $\circ$ to lie in the tangent cone to $C_n$ are (\ref{e:co}), (\ref{e:ob1}) and (\ref{e:ob2}). The same argument as in \cite[\S1]{am} shows that $g_{12}^1$, satisfying (\ref{e:ob1}), is unique for a given $f_{11}^1$, and the existence condition is similar to \cite[Lemma~1]{am}. We have the following result: \begin{thm}\label{t:m1} The variety $A_{n,r}$ is an irreducible component of $C_n^{\rm red}$ when $r\le(d^2-1)/3$ if and only if, for some sufficiently generic $N\in A_{n,r}$, a solution $\circ\in L(S^2N,N)$ to (\ref{e:co}), (\ref{e:ob1}) and (\ref{e:ob2}) has its part $f^1_{11}=0$ on the kernel of the multiplication $\mu:S^2N_1@>>>N^2$ of the algebra $N$. \end{thm} \begin{pf} If $A_{n,r}$ is an irreducible component of $C_n^{\rm red}$, then, for some sufficiently generic $N\in A_{n,r}$, the tangent cone to $C_n^{\rm red}$ at $N$ coincides with $$T_N A_{n,r}=L(S^2(N/N^2),N^2)+T_N G N.$$ But we know that the solution to (\ref{e:co}), (\ref{e:ob1}) and (\ref{e:ob2}) gives rise to a vector $\circ$ from the tangent cone. Hence, $f^1_{11}=0$ on the kernel of the multiplication $\mu:S^2N_1@>>>N^2$, because the vectors of $T_N G N$ are of the form $x\circ y= x\varphi(y)-\varphi(xy)+y\varphi(x)$ for some linear map $\varphi:N@>>>N$, while the vectors from $L(S^2(N/N^2),N^2)$ clearly satisfy the property. Conversely, suppose that a vector $\circ\in L(S^2N,N)$ belongs to the tangent cone to the scheme $C_n$ at $N$. So, by the discussion above, it satisfies (\ref{e:co}), (\ref{e:ob1}) and (\ref{e:ob2}). Hence, $\circ$ has $f^1_{11}=0$ on the kernel of the multiplication of some sufficiently generic $N\in A_{n,r}$. Then $f_{11}^1$ with $f_{12}^2(x,yz)=-xf_{11}^1(y,z)$ and $g_{12}^1(x,yz)=-f_{11}^1(x,f_{11}^1(y,z))$ are unique solutions to (\ref{e:co}), (\ref{e:ob1}) and (\ref{e:ob2}). This shows that $\circ$ is actually from $T_N A_{n,r}$. \end{pf} We will finish this section showing the result of \cite[Theorem~2]{am} with application of the above theorem and without the use of Shafarevich's embedding $C_n\hookrightarrow\widetilde{C}_n$. \begin{cor} $A_{n,r}$ is a component of $C_n^{\rm red}$ for the values $r=(5d^2-8d)/16$, $d$ is divisible by 4. \end{cor} \begin{pf} In \cite[Section~2.1]{am}, it was already shown that $T_N{C}_n=T_N A_{n,r}+F$ for a sufficiently generic algebra $N$. This implies that the part $f_{11}^1$ of a vector from $T_N{C}_n$ is determined by $f(x)y+f(y)x$, for some $f\in L(N/N^2,K)$, on the kernel of the multiplication $\mu$ on $N$. The algebra $N$ had the property that the square of all $d$ generators is zero and for each generator $u$ there was a distinct generator $v$ such that their product $uv$ also vanishes in $N$. Taking $x=y=u$ and $z=v$ in (\ref{e:ob1}), we have $$f_{11}^1(2f(u)u,v)-f_{11}^1(u,f(u)v+f(v)u)=0.$$ Since $uv$ and $uu$ are in the kernel of multiplication $\mu$, we further get $$2f(u)(f(u)v+f(v)u)-f(u)(f(u)v+f(v)u)-f(v)(2f(u)u)=0,$$ whence $f(u)^2v-f(u)f(v)u=0$. Therefore, $f(u)=0$ for all generators, and $f_{11}^1=0$ on the kernel of $\mu$ as it was required in Theorem~\ref{t:m1}. \end{pf}

9711

alg-geom/9711009

en

https://arxiv.org/abs/alg-geom/9711009

alg-geom/9711009

Michael Finkelberg

Boris Feigin, Michael Finkelberg, Alexander Kuznetsov, Ivan Mirkovi\'c (Landau Institute of Theoretical Physics, Independent University of Moscow and University of Massachusetts at Amherst)

Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

version published in AMS Translations

Amer. Math. Soc. Transl. Ser. 2, vol. 194 (1999), 113--148

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

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis numbered by the irreducible components of certain algebraic cycles, isomorphic to the intersections of semiinfinite orbits in the affine Grassmannian of $G$. We compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$. They carry a natural pure Tate Hodge structure, and their generating functions are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we prove that Kontsevich's space of stable maps provides a natural resolution of $Q$.

alg-geom

\section{Introduction} \subsection{} This paper is a sequel to ~\cite{fm}. We will make a free use of notations, conventions and results of {\em loc. cit.} One of the main results of the present work is a computation of local $\IC$ stalks of the Schubert strata closures in the spaces $\CZ^\alpha$. We prove that the generating functions of these stalks are given by the {\em generic} (or {\em periodic) Kazhdan-Lusztig polynomials}, see the Theorem ~\ref{main}. We understand that this result was known to G.Lusztig for a long time, cf. ~\cite{l2} ~\S11. His proof was never published though, and as far as we understand, it differs from ours: for example we never managed to find a direct geometric proof of the key property ~\cite{l1} ~11.1.(iv) of Lusztig's $R$-polynomials. Our proof uses the standard convolution technique. The only nonstandard feature is the check of pointwise purity of the $\IC$ sheaves involved (Theorem ~\ref{finkel}). Usually one proceeds by finding global transversal slices. We were not able to find the good slices, and instead reduced the proof to the purity properties of $\IC$ sheaves on the affine Grassmannian. \subsection{} The central result of this work is the geometric construction of the universal enveloping algebra $U(\fn_+^L)$ of the nilpotent subalgebra of the Langlands dual Lie algebra $\fg^L$ (Theorem ~\ref{!}). This construction occupies the section 2. The geometric incarnation $\CA$ of $U(\fn_+^L)$ naturally acts on the global intersection cohomology $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$ of all Quasimaps' spaces, and this action extends to the geometrically defined $\fg^L$-action (section 4). In case $\bG=SL_n$ such action was constructed in ~\cite{fk}, and the present paper grew out of attempts to generalize the results of {\em loc. cit.} to the case of arbitrary simple $\bG$. For $\fg=\frak{sl}_n$ the geometric construction of $U(\fn_+^L)$ given in {\em loc. cit.} used the Laumon resolution $\CQ_L^\alpha$ of the Quasimaps' spaces $\CQ^\alpha$, and provided $U(\fn_+^L)$ with the geometrically defined Poincar\'e-Birkhoff-Witt basis. The present construction also provides $U(\fn_+^L)$ with a special basis numbered by the irreducible components of the semiinfinite orbits' intersections in the affine Grassmannian. We want to stress that the two bases are {\em different}: the latter one looks more like a canonical basis of ~\cite{l}. Another advantage of the new construction of $U(\fn_+^L)$ is its local nature. It allows one to define a $\fg^L$-action on the global cohomology of Quasimaps' spaces with coefficients in sheaves more general than just $\IC(\CQ^\alpha)$. Namely, given a perverse sheaf $\CF\in\CP(\oCG_\eta,\bI)$ on the affine Grassmannian we defined in ~\cite{fm} its {\em convolution} $\bc^\alpha_\CQ(\CF)$ --- a perverse sheaf on $\CQ^{\eta+\alpha}$. In section 7 we construct the $\fg^L$-action on $\oplus_{\alpha\in Y}H^\bullet(\CQ^{\eta+\alpha},\bc^\alpha_\CQ(\CF))$. In case $\CF$ is $\bG[[z]]$-equivariant we conjecture that the resulting $\fg^L$-module is {\em tilting}. As in ~\cite{fk}, this conjecture is motivated by an analogy with the semiinfinite cohomology of quantum groups. Recall that for a $\bG[[z]]$-equivariant sheaf $\CF$ on the affine Grassmannian the action of $\fg^L$ on its global cohomology was constructed in ~\cite{mv}. The relation between the various $\fg^L$-actions on global cohomology will be discussed in a separate paper. \subsection{} Let us list the other points of interest in this paper. In section 3 we compute the stalks of $\IC(\CZ^\alpha)$ (Theorem ~\ref{simple} and Corollary ~\ref{berezin}). Their generating functions are expressed in terms of Lusztig's $q$-analogue $\CK^\alpha(t)$ of Kostant partition function. In case $\bG=SL_n$ this result was proved in ~\cite{ku} using the Laumon resolution of the Quasimaps' space $\CQ^\alpha$. The proof in general case uses the {\em Beilinson-Drinfeld} incarnation of $\CZ^\alpha$ (see ~\cite{fm} ~\S6). Formally, Theorem ~\ref{simple} is just a particular case of the Theorem ~\ref{main} computing $\IC$-stalks of the general Schubert strata closures in $\CZ^\alpha$. But the argument goes the other way around: we deduce ~\ref{main} as a rather formal corollary of ~\ref{simple}. It is well known that the generating function of $\IC$ stalks of $\bG[[z]]$-orbits' closures in the affine Grassmannian is also given by $\CK^\alpha(t)$ in the stable range (see ~\cite{lus}). This coincidence is explained in section 3: though the local singularities of $\CZ^\alpha$ and $\oCG_\eta$ {\em are different}, their ``skeleta'' (intersections of semiinfinite orbits in $\oCG_\eta$, and the {\em central fiber} in $\CZ^\alpha$) {\em are the same}. In section 4 we prove that $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$ carries a natural pure Tate Hodge structure and compute its generating function (Theorems ~\ref{ostrik}, ~\ref{feigin}). In section 5 we prove that the adjacency order on the set of Schubert strata in $\CZ$ is equivalent to Lusztig's order (see ~\cite{l1}) on the set of {\em alcoves}. The proof uses a map $\pi:\ \CQ^\alpha_K\lra\CQ^\alpha$ from Kontsevich's space of {\em stable maps} to Drinfeld's Quasimaps' space. This map is constructed in the Appendix. The construction is just an application of Givental's ``Main Lemma'' (see ~\cite{g}). His proof of the ``Main Lemma'' has not satisfied all of its readers, so we include the complete proof into the Appendix. In section 7 we collect various conjectures on the structure of $\fg^L$-modules of geometric origin. For a mysterious reason these conjectures involve tilting modules --- either over $\fg^L$ itself, or over the related quantum group. The conjecture ~\ref{roman} was recently proved in ~\cite{fkm}. The conjecture ~\ref{denis} may be viewed as a description of an ``automorphic sheaf'' on the moduli space of $\bG$-torsors corresponding to the trivial $\bG^L$-local system on $\BP^1$. Finally, in ~\ref{romka} we propose a direct geometric construction of the $\fn^L_\pm$-action on $H^\bullet(\CG,\CF)$ for a $\bG[[z]]$-equivariant perverse sheaf $\CF$ on the affine Grassmannian $\CG$. Surprisingly enough, no direct construction of $\fn^L_\pm$-action has been found so far (cf. ~\cite{mv} for the direct construction of the action of the dual Cartan ${\fh}^L\subset\fg^L$). \subsection{} We are deeply grateful to V.Lunts and L.Positselsky whose explanations helped us at the moments of despair. We are very much obliged to R.Bezrukavnikov who found a serious gap in the original version of geometric construction of $U(\fn^L_+)$. It is a pleasure to thank S.Arkhipov, D.Gaitsgory, and V.Ostrik for the inspiring discussions around the tilting conjectures in section 7. We are very much obliged to D.Gaitsgory for the careful reading of this paper, and for suggesting a lot of drastic simplifications of the arguments and proofs of the conjectures therein. They will all appear in his forthcoming publication. \section{Local Ext-algebra $\CA$} \subsection{} Throughout this paper $C$ will denote a genus zero curve $\BP^1$ with the marked points $0,\infty$. The complement $C-\infty$ is the affine line $\BA^1$. The {\em twisting} map $\sigma_{\beta,\gamma}:\ \CQ^\beta\times C^\gamma \lra\CQ^{\beta+\gamma}$ defined in ~\cite{fm} ~3.4.1 restricts to the map $\varsigma_{\beta,\gamma}:\ \CZ^\beta\times\BA^\gamma\lra\CZ^{\beta+\gamma}$. We denote its image by $\del_\gamma\CZ^{\beta+\gamma}$. The map $\varsigma_{\beta,\gamma}:\ \CZ^\beta\times\BA^\gamma\lra \del_\gamma\CZ^{\beta+\gamma}$ is finite. Moreover, in case $\beta=0$, the space $\CZ^\beta$ is just a point, and the map $\varsigma_{0,\gamma}:\ \BA^\gamma\lra\del_\gamma\CZ^\gamma$ is an isomorphism. We will identify $\del_\gamma\CZ^\gamma$ with $\BA^\gamma$ via this map. {\bf Definition.} $\CA_\alpha:= \Ext^\bullet_{\CZ^\alpha}(\IC(\del_\alpha\CZ^\alpha), \IC(\CZ^\alpha));\ \CA:=\oplus_{\alpha\in\BN[I]}\CA_\alpha$. Here the $\Ext^\bullet_{\CZ^\alpha}$ is taken in the constructible derived category on $\CZ^\alpha$. {\em A priori} $\CA_\alpha$ is a graded vector space. In this section we will show that it is concentrated in the degree $|\alpha|$, and we will define a structure of cocommutative $\BN[I]$-graded bialgebra on $\CA$. \subsection{} \label{Weil} To unburden the notations we will denote the $\IC$ sheaf $\IC(\CZ^\alpha)$ by $\IC^\alpha$ (see ~\cite{fm} 10.7.1). We denote the closed embedding of $\BA^\alpha=\del_\alpha\CZ^\alpha$ into $\CZ^\alpha$ by $\fs_\alpha$. Then $\CA_\alpha =\Ext^\bullet_{\CZ^\alpha}(\IC(\del_\alpha\CZ^\alpha),\IC^\alpha)= \Ext^\bullet_{\del_\alpha\CZ^\alpha} (\IC(\del_\alpha\CZ^\alpha),\fs_\alpha^!\IC^\alpha)= H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)[-|\alpha|]$ since $\IC(\del_\alpha\CZ^\alpha)=\ul\BC[|\alpha|]$. {\bf Theorem.} a) $H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ is concentrated in degree 0; b) $\dim H^0(\BA^\alpha,\fs_\alpha^!\IC^\alpha)=\CK(\alpha)$ where $\CK(\alpha)$ stands for the Kostant partition function. The proof of the Theorem will occupy the subsections ~\ref{raz}--\ref{dva}. \subsection{} \label{raz} We denote the closed embedding of the origin 0 into $\BA^\alpha$ by $\iota_\alpha$. Since $\fs_\alpha^!\IC^\alpha$ is constructible with respect to the diagonal stratification of $\BA^\alpha$, we have the canonical isomorphism $H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)= \iota_\alpha^*\fs_\alpha^!\IC^\alpha$. Recall the projection $\pi_\alpha:\ \CZ^\alpha\lra\BA^\alpha$ (see ~\cite{fm} ~7.3). We denote the {\em central fiber} $\pi_\alpha^{-1}(0)$ by $\CF^\alpha$, and we denote its closed embedding into $\CZ^\alpha$ by $\iota_\alpha$. The Cartan group $\bH$ acts on $\CZ^\alpha$ contracting it to the fixed point set $(\CZ^\alpha)^\bH=\del_\alpha\CZ^\alpha$. The projection $\pi_\alpha$ is $\bH$-equivariant. Hence the canonical morphism $\fs_\alpha^!\IC^\alpha\lra\pi_{\alpha!}\IC^\alpha$ of sheaves on $\BA^\alpha$ is an isomorphism. By the proper base change we have the canonical isomorphism $\iota_\alpha^*\pi_{\alpha!}\IC^\alpha=\pi_{\alpha!}\iota_\alpha^*\IC^\alpha= H^\bullet_c(\CF^\alpha,\iota_\alpha^*\IC^\alpha)$. Combining all the above isomorphisms, we conclude that $H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)= H^\bullet_c(\CF^\alpha,\iota_\alpha^*\IC^\alpha)$. \subsection{} \label{leq0} In this subsection we prove that $H^\bullet_c(\CF^\alpha,\iota_\alpha^*\IC^\alpha)$ is concentrated in nonpositive degrees. To this end we study the intersection of the {\em fine stratification} of $\CZ^\alpha$ (see ~\cite{fm} ~8.4.1) with the central fiber $\CF^\alpha$. Recall the description of the central fiber given in {\em loc. cit.} ~6.4.1, ~6.4.2 in terms of semiinfinite orbits in the affine Grassmannian: $\CF^\alpha=\ol{T}_{-\alpha}\cap S_0$. \subsubsection{Lemma} The intersection of $\CF^\alpha$ with a fine stratum $\oZ^\gamma\times(\BC^*)^{\beta-\gamma}_\Gamma,\ \gamma\leq\beta\leq\alpha, \Gamma\in\fP(\beta-\gamma)$, is nonempty iff $\gamma=\beta$. In this case $\CF^\alpha\cap\oZ^\beta\simeq T_{-\beta}\cap S_0$. {\em Proof.} Evident. $\Box$ \subsubsection{Corollary} \label{strat F} Let $\CS$ be a stratum of the fine stratification of $\CZ^\alpha$. Then either $\CS\cap\CF^\alpha=\emptyset$, or $\dim(\CS\cap\CF^\alpha)=\frac{1}{2}\dim\CS$. \subsubsection{} According to the above Lemma, we have $\CF^\alpha=\sqcup_{\beta\leq\alpha}(\oZ^\beta\cap\CF^\alpha)$ --- a partition into locally closed subsets. The restriction of $\IC^\alpha$ to $\oZ^\beta\cap\CF^\alpha$ is concentrated in the degrees $\leq-\dim\oZ^\beta= -2|\beta|$. Moreover, by the definition of $\IC$ sheaf, if $\beta<\alpha$, the above inequality is strict. Since $\dim(\oZ^\beta\cap\CF^\alpha)=|\beta|$ we conclude that $H^\bullet_c(\oZ^\beta\cap\CF^\alpha,\IC^\alpha|_{\oZ^\beta\cap\CF^\alpha})$ is concentrated in nonpositive degrees, and if $\beta<\alpha$ it is concentrated in strictly negative degrees. Applying the Cousin spectral sequence associated with the partition $\CF^\alpha=\sqcup_{\beta\leq\alpha}(\oZ^\beta\cap\CF^\alpha)$ we see that $H^\bullet_c(\CF^\alpha,\iota_\alpha^*\IC^\alpha)$ is concentrated in nonpositive degrees. \subsubsection{} The above spectral sequence shows also that $H^0_c(\CF^\alpha,\iota_\alpha^*\IC^\alpha)=H^0_c(\oZ^\alpha\cap\CF^\alpha, \IC^\alpha|_{\oZ^\alpha\cap\CF^\alpha})=H^0_c(\oZ^\alpha\cap\CF^\alpha, \ul\BC[2|\alpha|])=H^{2|\alpha|}_c(\oZ^\alpha\cap\CF^\alpha),\BC)= H^{2|\alpha|}_c(T_{-\alpha}\cap S_0)$. The latter term has a canonical basis of irreducible components of $T_{-\alpha}\cap S_0$. According to ~\cite{mv}, the number of irreducible components equals $\CK(\alpha)$. This completes the proof of ~\ref{Weil} b). \subsection{} \label{dva} It remains to show that $H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ is concentrated in nonnegative degrees, or dually, that $H^\bullet_c(\BA^\alpha,\fs_\alpha^*\IC^\alpha)$ is concentrated in nonpositive degrees. To this end we study the intersection of the fine stratification of $\CZ^\alpha$ with $\BA^\alpha=\del_\alpha\CZ^\alpha$. \subsubsection{Lemma} The intersection of $\del_\alpha\CZ^\alpha$ with a fine stratum $\oZ^\gamma\times(\BC^*)^{\beta-\gamma}_\Gamma,\ \gamma\leq\beta\leq\alpha, \Gamma\in\fP(\beta-\gamma)$, is nonempty iff $\gamma=0$. In this case $\del_\alpha\CZ^\alpha$ contains the stratum $\oZ^0\times(\BC^*)^\beta_\Gamma \simeq(\BC^*)^\beta_\Gamma$. {\em Proof.} Evident. $\Box$ \subsubsection{} If $\Gamma=\{\{\gamma_1,\ldots,\gamma_k\}\}$ is a partition of $\beta$ then $\dim(\BC^*)^\beta_\Gamma=k$, and due to the factorization property ~\cite{fm} ~7.3, the stalk of $\IC^\alpha$ at (any point in) the stratum $(\BC^*)^\beta_\Gamma$ is isomorphic to $\IC^{\alpha-\beta}_\Gamma\simeq \IC^0_{\{\{\alpha-\beta\}\}}\otimes\bigotimes_{r=1}^k\IC^0_{\{\{\gamma_r\}\}}$ (notations of {\em loc. cit.} ~ 10.7.1, ~10.7.2). {\em Lemma.} The restriction of $\IC^\alpha$ to the stratum $(\BC^*)^\beta_\Gamma$ is concentrated in the degrees $\leq-2k$ (here $k$ is the number of elements in the partition $\Gamma$). {\em Proof.} In view of the above product formula, it suffices to check that for any $r=1,\ldots,k$ the simple stalk $\IC^0_{\{\{\gamma_r\}\}}$ is concentrated in degree $\leq-2$. This is the stalk of $\IC^{\gamma_r}$ at the one-dimensional (diagonal) stratum $(\BC^*)^{\gamma_r}_{\{\{\gamma_r\}\}}\subset\CZ^{\gamma_r}$. By the definition of $\IC$ sheaf, its restriction to any nonopen $l$-dimensional stratum is concentrated in degrees $<-l$. This completes the proof of the Lemma. $\Box$ \subsubsection{} We consider the partition of $\del_\alpha\CZ^\alpha$ into locally closed subsets: $\del_\alpha\CZ^\alpha=\sqcup(\BC^*)^\beta_\Gamma$. According to the above Lemma, the restriction of $\IC^\alpha$ to a $k$-dimensional stratum is concentrated in degrees $\leq-2k$. Applying the Cousin spectral sequence associated with this partition we conclude, exactly as in subsection ~\ref{leq0}, that $H^\bullet_c(\BA^\alpha,\fs_\alpha^*\IC^\alpha)$ is concentrated in nonpositive degrees. Dually, $H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ is concentrated in nonnegative degrees. This completes the proof of the Theorem ~\ref{Weil}. $\Box$ \subsubsection{Corollary} $\CA_\alpha= \Ext^\bullet_{\CZ^\alpha}(\IC(\del_\alpha\CZ^\alpha),\IC^\alpha)$ is concentrated in the degree $|\alpha|$. {\em Proof.} See the beginning of ~\ref{Weil}. $\Box$ \subsection{} \label{rest} In the rest of this section we construct the multiplication map $\CA_\beta\otimes\CA_\alpha\lra\CA_{\beta+\alpha}$. We start with the following Lemma. \subsubsection{Lemma} \label{normal} $\IC(\del_\alpha\CZ^{\alpha+\beta})= (\varsigma_{\beta,\alpha})_*\IC(\CZ^\beta\times\BA^\alpha)= (\varsigma_{\beta,\alpha})_*(\IC^\beta\boxtimes\ul\BC[|\alpha|])$. {\em Proof.} The map $\varsigma_{\beta,\alpha}$ is finite and generically one-to-one. $\Box$ \subsubsection{Remark} In fact, $\varsigma_{\beta,\alpha}$ is the normalization map, but we do not need nor prove this fact. \subsubsection{} \label{ef} We denote by $\IC_\alpha^\beta$ the sheaf $\IC(\del_\alpha\CZ^{\alpha+\beta})$. In particular, $\del_0\CZ^\beta=\CZ^\beta$, and $\IC_0^\beta=\IC^\beta$. Consider the restriction of the map $\varsigma_{\beta+\alpha,\gamma}:\ \CZ^{\beta+\alpha}\times\BA^\gamma\lra\del_\gamma\CZ^{\gamma+\beta+\alpha}$ to $\del_\alpha\CZ^{\beta+\alpha}\times\BA^\gamma$. Evidently, $\varsigma_{\beta+\alpha,\gamma}(\del_\alpha\CZ^{\beta+\alpha}\times\BA^\gamma) =\del_{\alpha+\gamma}\CZ^{\gamma+\beta+\alpha}$. The map $\varsigma_{\beta+\alpha,\gamma}$ restricted to $\del_\alpha\CZ^{\beta+\alpha}\times\BA^\gamma$ is finite, so the semisimple perverse sheaf $(\varsigma_{\beta+\alpha,\gamma})_* \IC(\del_\alpha\CZ^{\beta+\alpha}\times\BA^\gamma)= (\varsigma_{\beta+\alpha,\gamma})_*(\IC_\alpha^\beta\boxtimes\ul\BC[|\gamma|])$ contains the direct summand $\IC_{\alpha+\gamma}^\beta$ with multiplicity one. Let $$\IC_{\alpha+\gamma}^\beta\stackrel{e}{\lra} (\varsigma_{\beta+\alpha,\gamma})_*(\IC_\alpha^\beta\boxtimes\ul\BC[|\gamma|]) \stackrel{f}{\lra}\IC_{\alpha+\gamma}^\beta$$ denote the corresponding inclusion and projection. \subsection{} We include the spaces $\CA_\alpha$ into the wider framework. For $\alpha,\beta,\gamma\in\BN[I]$ we define $$\CA_\beta^{\alpha,\gamma}:=\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}} (\IC^\gamma_{\alpha+\beta},\IC^{\beta+\gamma}_\alpha)$$ In particular, $\CA_\beta=\Ext^\bullet_{\CZ^\beta}(\IC^0_\beta,\IC^\beta_0)= \CA_\beta^{0,0}$. Now for $\delta\in\BN[I]$ we construct the {\em stabilization map} $\tau^{\alpha,\gamma}_{\beta,\delta}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha,\gamma+\delta}$, and the {\em costabilization map} $\xi^{\alpha,\gamma}_{\beta,\delta}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha+\delta,\gamma}$. \subsubsection{} \label{bezruk} We choose an open subset $U'\stackrel{j'}{\hookrightarrow} \CZ^{\alpha+\beta+\gamma+\delta}$ (resp. $U\stackrel{j}{\hookrightarrow}\CZ^{\alpha+\beta+\gamma}$) such that $U'\cap\del_{\alpha+\beta}\CZ^{\alpha+\beta+\gamma+\delta}=W':= \oZ^{\gamma+\delta}\times\BA^{\alpha+\beta}$ (resp. $U\cap\del_{\alpha+\beta}\CZ^{\alpha+\beta+\gamma}=W:= \oZ^\gamma\times\BA^{\alpha+\beta}$) (the open subset formed by all the quasimaps of defect {\em exactly} $\alpha+\beta$). We have $\Ext^\bullet_{U'}(\IC^{\gamma+\delta}_{\alpha+\beta}, \IC^{\beta+\gamma+\delta}_\alpha)= \Ext^\bullet_{W'}(\IC^{\gamma+\delta}_{\alpha+\beta}, \varsigma_{\gamma+\delta,\alpha+\beta}^!\IC^{\beta+\gamma+\delta}_\alpha)$, and $\Ext^\bullet_U(\IC^\gamma_{\alpha+\beta}, \IC^{\beta+\gamma}_\alpha)= \Ext^\bullet_W(\IC^\gamma_{\alpha+\beta}, \varsigma_{\gamma,\alpha+\beta}^!\IC^{\beta+\gamma}_\alpha)$, where $\varsigma_{\gamma,\alpha+\beta}$ stands for the finite map $\CZ^\gamma\times\BA^{\alpha+\beta}\to \del_{\alpha+\beta}\CZ^{\alpha+\beta+\gamma}\hookrightarrow \CZ^{\alpha+\beta+\gamma}$. Now by the Lemma ~\ref{normal} and the factorization property, there exists a unique complex of sheaves $\CM$ on $\BA^{\alpha+\beta}$ such that $(\varsigma_{\gamma+\delta,\alpha+\beta}^! \IC^{\beta+\gamma+\delta}_\alpha)|_{W'} =\ul\BC[2|\gamma+\delta|]\boxtimes\CM$, and $(\varsigma_{\gamma,\alpha+\beta}^!\IC^{\beta+\gamma}_\alpha)|_W= \ul\BC[2|\gamma|]\boxtimes\CM$. As $\IC^\gamma_{\alpha+\beta}|_W= \ul\BC[2|\gamma|]\boxtimes\ul\BC[|\alpha+\beta|]$, we get the restriction map $j^*:\ \Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}} (\IC^\gamma_{\alpha+\beta},\IC^{\beta+\gamma}_\alpha)\to \Ext^\bullet_{\oZ^\gamma\times\BA^{\alpha+\beta}} (\IC(\oZ^\gamma)\boxtimes\IC(\BA^{\alpha+\beta}),\IC(\oZ^\gamma)\boxtimes\CM)$. We can project the right hand side to the summand $\Id\otimes\Ext^\bullet_{\BA^{\alpha+\beta}}(\IC(\BA^{\alpha+\beta}),\CM)$. The resulting map from $\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}} (\IC^\gamma_{\alpha+\beta},\IC^{\beta+\gamma}_\alpha)$ to $\Ext^\bullet_{\BA^{\alpha+\beta}} (\IC(\BA^{\alpha+\beta}),\CM)$ will be also denoted by $j^*$. Multiplying with $\Id\in\Ext^0_{\CZ^{\gamma+\delta}}(\IC^{\gamma+\delta}, \IC^{\gamma+\delta})$, we get an embedding from $\Ext^\bullet_{\BA^{\alpha+\beta}}(\IC(\BA^{\alpha+\beta}),\CM)$ to $\Ext^\bullet_{\CZ^{\gamma+\delta}\times\BA^{\alpha+\beta}} (\IC^{\gamma+\delta}\boxtimes\IC(\BA^{\alpha+\beta}), \IC^{\gamma+\delta}\boxtimes\CM)$. The key observation is that there exists a unique morphism $c:\ \IC^{\gamma+\delta}\boxtimes\CM\to \varsigma_{\gamma+\delta,\alpha+\beta}^! \IC^{\beta+\gamma+\delta}_\alpha$ extending the isomorphism $(\varsigma_{\gamma+\delta,\alpha+\beta}^! \IC^{\beta+\gamma+\delta}_\alpha)|_{W'} =\ul\BC[2|\gamma+\delta|]\boxtimes\CM$ from $W'$ to $\CZ^{\gamma+\delta}\times\BA^{\alpha+\beta}$. In effect, let $\ic^{\beta+\gamma+\delta}_\alpha$ be the mixed Hodge counterpart of $\IC^{\beta+\gamma+\delta}_\alpha$. It is a pure Hodge module of weight $w=2|\beta+\gamma+\delta|+|\alpha|$. Then $(\varsigma_{\gamma+\delta,\alpha+\beta}^! \ic^{\beta+\gamma+\delta}_\alpha)|_{W'} =\ul\BQ[2|\gamma+\delta|]\boxtimes\bM$ for some Hodge module $\bM$ on $\BA^{\alpha+\beta}$. It follows from the Proposition ~\ref{stunt} below that $\bM$ is a pure (hence semisimple) Hodge complex of weight $|\alpha+2\beta|$. Thus $\ic^{\gamma+\delta}\boxtimes\bM$ is a subquotient of the direct sum of cohomology of the mixed Hodge complex $\varsigma_{\gamma+\delta,\alpha+\beta}^!\ic^{\beta+\gamma+\delta}_\alpha$. The latter complex has weight $\geq w$, while $\ic^{\gamma+\delta}\boxtimes\bM$ is pure of weight $w$. The desired morphism $c$ is constructed by induction in the cohomology degree using the Proposition 5.1.15 of ~\cite{bbd}. Finally, we define $$\tau^{\alpha,\gamma}_{\beta,\delta}:\ \CA_\beta^{\alpha,\gamma}=\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}} (\IC^\gamma_{\alpha+\beta},\IC^{\beta+\gamma}_\alpha)\lra \Ext^\bullet_{\CZ^{\alpha+\beta+\gamma+\delta}} (\IC^{\gamma+\delta}_{\alpha+\beta},\IC^{\beta+\gamma+\delta}_\alpha)= \CA_\beta^{\alpha,\gamma+\delta}$$ as the following composition: $\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}} (\IC^\gamma_{\alpha+\beta},\IC^{\beta+\gamma}_\alpha) \stackrel{j^*}{\lra} \Ext^\bullet_{\BA^{\alpha+\beta}} (\IC(\BA^{\alpha+\beta}),\CM) \stackrel{\Id\otimes ?}{\lra} \Ext^\bullet_{\CZ^{\gamma+\delta}\times\BA^{\alpha+\beta}} (\IC^{\gamma+\delta}\boxtimes\IC(\BA^{\alpha+\beta}), \IC^{\gamma+\delta}\boxtimes\CM) \stackrel{c}{\lra} \Ext^\bullet_{\CZ^{\gamma+\delta}\times\BA^{\alpha+\beta}} (\IC^{\gamma+\delta}\boxtimes\IC(\BA^{\alpha+\beta}), \varsigma_{\gamma+\delta,\alpha+\beta}^! \IC^{\beta+\gamma+\delta}_\alpha)= \Ext^\bullet_{\CZ^{\alpha+\beta+\gamma+\delta}} (\IC^{\gamma+\delta}_{\alpha+\beta},\IC^{\beta+\gamma+\delta}_\alpha)$. Quite obviously, the result does not depend on a choice of $U$ and $U'$. \subsubsection{} To define the costabilization map $\xi^{\alpha,\gamma}_{\beta,\delta}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha+\delta,\gamma}$ we note that $\CA_\beta^{\alpha,\gamma}:=\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}} (\IC^\gamma_{\alpha+\beta},\IC^{\beta+\gamma}_\alpha)= \Ext^\bullet_{\CZ^{\alpha+\beta+\gamma}\times\BA^\delta} (\IC^\gamma_{\alpha+\beta}\boxtimes\ul\BC[|\delta|], \IC^{\beta+\gamma}_\alpha\boxtimes\ul\BC[|\delta|])$ maps by $(\varsigma_{\alpha+\beta+\gamma,\delta})_*$ to $\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma+\delta}} ((\varsigma_{\alpha+\beta+\gamma,\delta})_* (\IC^\gamma_{\alpha+\beta}\boxtimes\ul\BC[|\delta|]), (\varsigma_{\alpha+\beta+\gamma,\delta})_* (\IC^{\beta+\gamma}_\alpha\boxtimes\ul\BC[|\delta|]))$. According to ~\ref{ef}, the first argument of the latter Ext contains the canonical direct summand $e\IC^\gamma_{\alpha+\delta+\beta}$, while the second argument of the latter Ext canonically projects by $f$ to $\IC^{\beta+\gamma}_{\alpha+\delta}$. Thus, the latter Ext canonically projects to $\Ext^\bullet_{\CZ^{\alpha+\beta+\gamma+\delta}} (\IC^\gamma_{\alpha+\delta+\beta}, \IC^{\beta+\gamma}_{\alpha+\delta})=\CA_\beta^{\alpha+\delta,\gamma}$. Composing this projection with the above map $(\varsigma_{\alpha+\beta+\gamma,\delta})_*$ we obtain the desired map $\xi^{\alpha,\gamma}_{\beta,\delta}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha+\delta,\gamma}$. \subsection{Proposition} \label{abs non} For $\alpha,\beta,\gamma,\delta,\epsilon\in\BN[I]$ we have a) $\tau_{\beta,\epsilon}^{\alpha,\gamma+\delta}\circ \tau_{\beta,\delta}^{\alpha,\gamma}= \tau_{\beta,\delta+\epsilon}^{\alpha,\gamma}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha,\gamma+\delta+\epsilon}$; b) $\xi^{\alpha+\delta,\gamma}_{\beta,\epsilon}\circ \xi^{\alpha,\gamma}_{\beta,\delta}= \xi^{\alpha,\gamma}_{\beta,\delta+\epsilon}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha+\delta+\epsilon,\gamma}$; c) $\xi^{\alpha,\gamma+\delta}_{\beta,\epsilon}\circ \tau^{\alpha,\gamma}_{\beta,\delta}= \tau^{\alpha+\epsilon,\gamma}_{\beta,\delta}\circ \xi^{\alpha,\gamma}_{\beta,\epsilon}:\ \CA_\beta^{\alpha,\gamma}\lra\CA_\beta^{\alpha+\epsilon,\gamma+\delta}$. {\em Proof.} Routine check. $\Box$ \subsection{Definition} \label{multip} Let $a\in\CA_\alpha=\CA_\alpha^{0,0},\ b\in\CA_\beta=\CA_\beta^{0,0}$. We define the product $a\cdot b$ as the following composition: $$a\cdot b:=\tau_{\alpha,\beta}^{0,0}(a)\circ\xi_{\beta,\alpha}^{0,0}(b) \in\CA_{\alpha+\beta}^{0,0}=\CA_{\alpha+\beta}$$ The associativity of this multiplication follows immediately from the Proposition ~\ref{abs non}. \subsection{} \label{stab} To reward the patient reader who has fought his way through the above notation, we repeat here the definition of $\tau_{\alpha,\beta}^{\gamma,\delta}$ in the particular case of $\gamma=\delta=0$. This is the only case needed in the definition of multiplication (the general case is needed to prove the associativity), and the definition simplifies somewhat in this case. So we are going to define the map $$\tau_{\alpha,\beta}^{0,0}:\ \CA_\alpha =\Ext^\bullet_{\CZ^\alpha}(\IC(\del_\alpha\CZ^\alpha),\IC^\alpha)\lra \Ext^\bullet_{\CZ^{\alpha+\beta}}(\IC(\del_\alpha\CZ^{\alpha+\beta}), \IC^{\alpha+\beta})$$ First of all, tensoring with $\Id\in\Ext^0_{\CZ^\beta}(\IC^\beta,\IC^\beta)$, we get the map from $\Ext^\bullet_{\CZ^\alpha}(\IC(\del_\alpha\CZ^\alpha),\IC^\alpha)= \Ext^\bullet_{\BA^\alpha}(\IC(\BA^\alpha),\fs_\alpha^!\IC^\alpha)$ to $\Ext^\bullet_{\CZ^\beta\times\BA^\alpha}(\IC^\beta\boxtimes\IC(\BA^\alpha), \IC^\beta\boxtimes\fs_\alpha^!\IC^\alpha)$. Now the same argument as in ~\ref{bezruk} shows that there is a unique morphism $c$ from $\IC^\beta\boxtimes\fs_\alpha^!\IC^\alpha$ to $\varsigma_{\beta,\alpha}^!\IC^{\alpha+\beta}$ extending the factorization isomorphism from the open part $\oZ^\beta\times\BA^\alpha$ of $\CZ^\beta\times\BA^\alpha$. Thus $c$ induces the map from $\Ext^\bullet_{\CZ^\beta\times\BA^\alpha}(\IC^\beta\boxtimes\IC(\BA^\alpha), \IC^\beta\boxtimes\fs_\alpha^!\IC^\alpha)$ to $\Ext^\bullet_{\CZ^\beta\times\BA^\alpha}(\IC^\beta\boxtimes\IC(\BA^\alpha), \varsigma_{\beta,\alpha}^!\IC^{\alpha+\beta})$. The latter space equals $\Ext^\bullet_{\CZ^{\alpha+\beta}} ((\varsigma_{\beta,\alpha})_*(\IC^\beta\boxtimes\IC(\BA^\alpha)), \IC^{\alpha+\beta})= \Ext^\bullet_{\CZ^{\alpha+\beta}}(\IC(\del_\alpha\CZ^{\alpha+\beta}), \IC^{\alpha+\beta})$. Finally, we define $\tau_{\alpha,\beta}^{0,0}$ as the composition of above maps. \subsection{} We close this section with a definition of comultiplication $\Delta:\ \CA_{\alpha}\lra\oplus_{\beta+\gamma=\alpha} \CA_\beta\otimes\CA_\gamma$. \subsubsection{} We choose two disjoint open balls $\Omega,\Upsilon\subset\BC= \BA^1$. Then $\Omega^\beta\times\Upsilon^\gamma\subset\BA^{\beta+\gamma}= \BA^\alpha$. According to the factorization property of $\CZ^\alpha$, we have $(\fs_\alpha^!\IC^\alpha)|_{\Omega^\beta\times\Upsilon^\gamma}= \fs_\beta^!\IC^\beta|_{\Omega^\beta}\boxtimes \fs_\gamma^!\IC^\gamma|_{\Upsilon^\gamma}$. We define the comultiplication component $\Delta_{\beta,\gamma}:\ \CA_{\alpha}\lra \CA_\beta\otimes\CA_\gamma$ as the following composition: $\CA_\alpha=H^0(\BA^\alpha,\fs_\alpha^!\IC^\alpha)\lra H^0(\Omega^\beta,\fs_\beta^!\IC^\beta)\otimes H^0(\Upsilon^\gamma,\fs_\gamma^!\IC^\gamma)= H^0(\BA^\beta,\fs_\beta^!\IC^\beta)\otimes H^0(\BA^\gamma,\fs_\gamma^!\IC^\gamma)=\CA_\beta\otimes\CA_\gamma$. Quite evidently, the result does not depend on a choice of $\Omega$ and $\Upsilon$. This comultiplication is manifestly cocommutative and coassociative. It also commutes with the multiplication defined in ~\ref{multip}, so it defines the structure of cocommutative bialgebra on $\CA$. \subsubsection{} \label{crit} Consider the main diagonal stratum $\BA^1\stackrel{u_\alpha}{\hookrightarrow}\BA^\alpha$. We will denote by $U^\alpha\stackrel{j}{\hookrightarrow}\BA^\alpha$ the open inclusion of the complement $U^\alpha=\BA^\alpha-\BA^1$. We say that a cohomology class $a\in H^0(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ {\em is supported on the main diagonal} if $0=j^*a\in H^0(U^\alpha,\fs_\alpha^!\IC^\alpha)$. The following criterion will prove very useful in section ~4. {\em Lemma.} $a\in\CA_\alpha$ is supported on the main diagonal iff $a$ is primitive, i.e. $\Delta(a)=1\otimes a+a\otimes1$. {\em Proof.} a) If $a\in\CA_\alpha$ is supported on the main diagonal then, obviously, $\Delta_{\beta,\gamma}(a)=0$ when $\beta>0<\gamma$. b) The converse follows by the Mayer-Vietoris type argument since $U^\alpha$ may be covered by the open subsets of the type $\Omega^\beta\times\Upsilon^\gamma,\ \beta>0<\gamma. \quad \Box$ \section{Simple $\IC$-stalks} \subsection{} \label{simple} Let $\check\CR{}^+\subset Y$ denote the set of positive coroots. For $\alpha\in\BN[I]$ the following $q$-analogue of the Kostant partition function $\CK(\alpha)$ was introduced in ~\cite{lus}: $$\CK^\alpha(t):=\sum_\kappa t^{-|\kappa|}$$ Here the sum is taken over the set of partitions of $\alpha$ into a sum of positive coroots $\ctheta\in\check\CR{}^+$, and for a partition $\kappa$ the number of elements in $\kappa$ is denoted by $|\kappa|$. Finally, $t$ is a formal variable of degree 2. We have $\CK(\alpha)=\CK^\alpha(1)$. In this section we prove the following Theorem: {\bf Theorem.} $\CK^\alpha(t)$ is the generating function of the simple stalk $\IC^0_{\{\{\alpha\}\}}$ (see ~\cite{fm} ~10.7.1), that is (notations of ~\ref{raz}), a) for odd $k$ we have $H^k\iota_\alpha^*\fs_\alpha^*\IC^\alpha=0$; b) $\dim H^{2k}\iota_\alpha^*\fs_\alpha^*\IC^\alpha$ equals the coefficient $\CK^\alpha_k$ of $t^k$ in $\CK^\alpha(t)$. \subsubsection{Corollary} \label{berezin} The generating function of the stalk $\IC^0_\Gamma,\ \Gamma=\{\{\gamma_1,\ldots,\gamma_k\}\}$ (see ~\cite{fm} ~10.7.1) equals $\prod_{r=1}^k\CK^{\gamma_r}(t)$. The Corollary follows immediately from the above Theorem and the factorization property. In its turn, the Corollary (along with the Proposition ~10.7.3 of ~\cite{fm}) implies the Parity vanishing conjecture ~10.7.4 of {\em loc. cit.} \subsection{} We start with the proof of ~\ref{simple} ~a). Recall that the Cartan group $\bH$ acts on $\CZ^\alpha$ contracting it to the fixed point set $(\CZ^\alpha)^\bH=\del_\alpha\CZ^\alpha=\BA^\alpha$. Moreover, if $\BC^*$ is the group of automorphisms of $C=\BP^1$ preserving $0,\infty$, then $\bH\times\BC^*$ acts on $\CZ^\alpha$ contracting it to the fixed point set $(\CZ^\alpha)^{\bH\times\BC^*}=\fs_\alpha\circ\iota_\alpha(0)$. Applying the Corollary ~14.3 of ~\cite{bl} we see that the equivariant cohomology $H^\bullet_{\bH\times\BC^*}(\CZ^\alpha,\IC^\alpha)$ is isomorphic to $H^\bullet_{\bH\times\BC^*}(\cdot)\otimes H^\bullet(\CZ^\alpha,\IC^\alpha)$, and $H^\bullet_\bH(\CZ^\alpha,\IC^\alpha)\cong H^\bullet_\bH(\cdot)\otimes H^\bullet(\CZ^\alpha,\IC^\alpha)= H^\bullet_\bH(\cdot)\otimes \iota_\alpha^*\fs_\alpha^*\IC^\alpha$. On the other hand, according to the Theorem ~B.2 of ~\cite{em}, the natural map $$H^\bullet_\bH(\cdot)\otimes H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)= H^\bullet_\bH(\BA^\alpha,\fs_\alpha^!\IC^\alpha)\lra H^\bullet_\bH(\CZ^\alpha,\IC^\alpha)$$ is an isomorphism after certain localization in $H^\bullet_\bH(\cdot)$. Now $H^\bullet_\bH(\cdot)$ is concentrated in even degrees, and the above localization is taken with respect to a homogeneous multiplicative subset. By the Theorem ~\ref{Weil}, $H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ is concentrated in degree 0, hence the (localized) $H^\bullet_\bH(\cdot)\otimes H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)= H^\bullet_\bH(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ is concentrated in even degrees, hence the (localized) $H^\bullet_\bH(\CZ^\alpha,\IC^\alpha)\cong H^\bullet_\bH(\cdot)\otimes H^\bullet(\CZ^\alpha,\IC^\alpha)= H^\bullet_\bH(\cdot)\otimes \iota_\alpha^*\fs_\alpha^*\IC^\alpha$ is concentrated in even degrees, hence $\iota_\alpha^*\fs_\alpha^*\IC^\alpha$ is concentrated in even degrees. This completes the proof of the Theorem ~\ref{simple} ~a). \subsubsection{Corollary of the proof} \label{cor} $\dim(\iota_\alpha^*\fs_\alpha^*\IC^\alpha)= \chi(\iota_\alpha^*\fs_\alpha^*\IC^\alpha)=\CK(\alpha)$. \subsection{} To prove ~\ref{simple} ~b) we need to introduce and recall some notation. \subsubsection{} As we have seen in ~\ref{strat F}, the fine stratification of $\CZ^\alpha$ induces a partition of the central fiber $\CF^\alpha$ into the locally closed subsets: $\CF^\alpha=\sqcup_{\beta\leq\alpha} (\oZ^\beta\cap\CF^\alpha)=:\sqcup_{\beta\leq\alpha}\oF^\beta$. Moreover, $\oF^\beta\simeq T_{-\beta}\cap S_0$. The closed embedding of the smallest point stratum $\oF^0$ into $\CF^\alpha$ will be denoted by $\fs_\alpha$. Recall that the closed embedding $\CF^\alpha\hookrightarrow\CZ^\alpha$ is denoted by $\iota_\alpha$. Certainly, we have $\iota_\alpha^*\fs_\alpha^*\IC^\alpha=\fs_\alpha^*\iota_\alpha^*\IC^\alpha$. \subsubsection{} \label{Hodge} We will use the machinery of {\em mixed Hodge modules}. The sheaf $\IC^\alpha$ has its mixed Hodge counterpart $\ic^\alpha$ --- the irreducible pure Hodge module on $\CZ^\alpha$ of weight $2|\alpha|$. The following Theorem is a strong version of ~\ref{simple}. {\bf Theorem.} a) for odd $k$ we have $H^k\fs_\alpha^*\iota_\alpha^*\ic^\alpha=0$; b) $H^{2k}\fs_\alpha^*\iota_\alpha^*\ic^\alpha$ is the direct sum of $\CK^\alpha_k$ copies of the Tate Hodge structure $\BQ(k+|\alpha|)$. \subsubsection{} \label{Grass} Recall that for a dominant $\eta\in Y^+$ we have a $\bG(\CO)$-orbit $\CG_\eta$ and its closure $\oCG_\eta$ in the affine Grassmannian $\CG$. Recall that $w_0$ is the longest element in the Weyl group $\CW_f$ of $\bG$. For a sufficiently dominant $\eta\in Y^+$, the difference $\eta-w_0(\alpha)=: \vartheta$ is also dominant, and $\CG_\eta\subset\oCG_\vartheta$. Let $\ic(\oCG_\vartheta)_\eta$ denote the stalk of the Hodge module $\ic(\oCG_\vartheta)$ at any point in $\CG_\eta$. G.Lusztig has proved the following theorem in ~\cite{lus} (in fact, he used the language of Weil sheaves instead of Hodge modules): {\bf Theorem.} Let $\eta\in Y^+$ be sufficiently dominant. Then a) for odd $k$ we have $H^k(\ic(\oCG_\vartheta)_\eta[-2|\eta|])=0$; b) $H^{2k}(\ic(\oCG_\vartheta)_\eta[-2|\eta|])$ is the direct sum of $\CK^\alpha_k$ copies of the Tate Hodge structure $\BQ(k+|\alpha|)$. \subsubsection{} \label{hard} Comparing ~\ref{Grass} and ~\ref{Hodge} we see that the Theorems ~\ref{Hodge} and ~\ref{simple} follow from the following {\bf Theorem.} $\ic(\oCG_\vartheta)_\eta[-2|\eta|]\cong\fs_\alpha^*\iota_\alpha^*\ic^\alpha$. The rest of this section is devoted to the proof of this Theorem. \subsection{} We consider the intersection of semiinfinite orbits $\ol{T}_{w_0\eta-\alpha} \cap S_{w_0\eta}$ in the affine Grassmannian $\CG$. It lies in the closure $\oCG_\vartheta$, and the translation by the Cartan loop $w_0\eta(z)\in\bG((z))$ identifies it with $\ol{T}_{-\alpha}\cap S_0\simeq\CF^\alpha$. Moreover, the partition of $\ol{T}_{w_0\eta-\alpha}\cap S_{w_0\eta}$ into intersections with $\bG(\CO)$-orbits $\CG_{\eta-w_0\beta},\ 0\leq\beta\leq\alpha$, corresponds under this identification to the partition $\CF^\alpha=\sqcup_{0\leq\beta\leq\alpha}\oF^\beta$. We denote the locally closed embedding $\CF^\alpha\simeq \ol{T}_{w_0\eta-\alpha}\cap S_{w_0\eta}\hookrightarrow\oCG_\vartheta$ by $\bi_\alpha$. Thus, $\ic(\oCG_\vartheta)_\eta=\fs_\alpha^*\bi_\alpha^*\ic(\oCG_\vartheta)$. \subsubsection{Conjecture} $\bi_\alpha^*\ic(\oCG_\vartheta)[-2|\eta|]\cong \iota_\alpha^*\ic^\alpha$. This conjecture would imply the Theorem ~\ref{hard}. Unfortunately, we cannot prove this conjecture at the moment. Instead of it, we will prove its version in the $K$-group of mixed Hodge modules. \subsection{Proposition} \label{stunt} a) $\fs_\alpha^*\iota_\alpha^*\ic^\alpha$ is a pure Hodge complex of weight $2|\alpha|$; b) $\fs_\alpha^!\iota_\alpha^*\ic^\alpha$ is a pure Hodge structure of weight $2|\alpha|$. {\em Proof.} a) The natural map $H^\bullet(\CZ^\alpha,\ic^\alpha)\lra \fs_\alpha^*\iota_\alpha^*\ic^\alpha$ is an isomorphism since $\ic^\alpha$ is smooth along the fine stratification. Now $\ic^\alpha$ is pure of weight $2|\alpha|$, and $H^\bullet(?)$ increases weights, while both $\iota_\alpha^*(?)$ and $\fs_\alpha^*(?)$ decrease weights. b) We already know that $\fs_\alpha^!\iota_\alpha^*\ic^\alpha= H^\bullet(\BA^\alpha,\fs_\alpha^!\ic^\alpha)$, and both $H^\bullet(?)$ and $\fs_\alpha^!(?)$ increase weights. On the other hand, $\fs_\alpha^!\iota_\alpha^*\ic^\alpha= \pi_{\alpha!}\iota_\alpha^*\ic^\alpha$, and both $\pi_{\alpha!}(?)$ and $\iota_\alpha^*(?)$ decrease weights. $\Box$ \subsection{} The above Proposition implies that to prove the Theorems ~\ref{hard} and ~\ref{Hodge} it suffices to identify the classes of $[\fs_\alpha^*\iota_\alpha^*\ic^\alpha]$ and $[\ic(\oCG_\vartheta)_\eta]$ in the $K$-group of mixed Hodge structures $K(MHM(\cdot))$. We will proceed by induction in $\alpha$. The first step $\alpha=0$ being trivial, we may suppose that our Theorems are established for all $\beta<\alpha$. Let us denote the complement $\CF^\alpha-\oF^0$ by $U$, and let us denote its open embedding into $\CF^\alpha$ by $j$. By factorization, $\iota_\alpha^*\ic^\alpha|_{\oF^\beta}$ is the constant Hodge module with the stalk $\fs_{\alpha-\beta}^*\iota_{\alpha-\beta}^*\ic^{\alpha-\beta}[2|\beta|]$. Thus, by induction, we know that the classes of $[j^*\iota_\alpha^*\ic^\alpha]$ and $[j^*\bi_\alpha^*\ic(\oCG_\vartheta)]$ in the $K$-group of mixed Hodge modules $K(MHM(U))$ coincide. Let us consider the following exact triangles of mixed Hodge complexes: \begin{equation} \label{1} \fs_\alpha^!\iota_\alpha^*\ic^\alpha\lra \fs_\alpha^*\iota_\alpha^*\ic^\alpha\lra \fs_\alpha^*j_*j^*\iota_\alpha^*\ic^\alpha\lra \fs_\alpha^!\iota_\alpha^*\ic^\alpha[1]\lra\ldots \end{equation} \begin{equation} \label{2} \fs_\alpha^!\bi_\alpha^*\ic(\oCG_\vartheta)\lra \fs_\alpha^*\bi_\alpha^*\ic(\oCG_\vartheta)\lra \fs_\alpha^*j_*j^*\bi_\alpha^*\ic(\oCG_\vartheta)\lra \fs_\alpha^!\bi_\alpha^*\ic(\oCG_\vartheta)[1]\lra\ldots \end{equation} \subsubsection{} We already know that $\fs_\alpha^*\iota_\alpha^*\ic^\alpha$ lives in the even negative degrees, and $\fs_\alpha^!\iota_\alpha^*\ic^\alpha$ lives in degree 0. Hence the differential $\fs_\alpha^!\iota_\alpha^*\ic^\alpha\lra \fs_\alpha^*\iota_\alpha^*\ic^\alpha$ vanishes (since the category of mixed Hodge structures has the cohomological dimension 1), and $\fs_\alpha^*j_*j^*\iota_\alpha^*\ic^\alpha= \fs_\alpha^!\iota_\alpha^*\ic^\alpha[1]\oplus \fs_\alpha^*\iota_\alpha^*\ic^\alpha$. \subsubsection{} By ~\cite{mv} we know that $\fs_\alpha^!\bi_\alpha^*\ic(\oCG_\vartheta)$ lives in degree 0 (and has dimension $\CK(\alpha)$). Now the same argument as above proves that $\fs_\alpha^*j_*j^*\bi_\alpha^*\ic(\oCG_\vartheta)= \fs_\alpha^!\bi_\alpha^*\ic(\oCG_\vartheta)[1]\oplus \fs_\alpha^*\bi_\alpha^*\ic(\oCG_\vartheta)$. Moreover, by ~\cite{mv} we know that $\fs_\alpha^!\bi_\alpha^*\ic(\oCG_\vartheta)$ is a direct summand of $H^\bullet(\oCG_\vartheta,\ic(\oCG_\vartheta))$, and hence it is a pure Tate Hodge module of weight $2|\alpha|$. We deduce that $\fs_\alpha^*j_*j^*\bi_\alpha^*\ic(\oCG_\vartheta)$ is a Tate Hodge module. \subsubsection{} By the induction assumption, the classes of $[\fs_\alpha^*j_*j^*\bi_\alpha^*\ic(\oCG_\vartheta)]$ and $[\fs_\alpha^*j_*j^*\iota_\alpha^*\ic^\alpha]$ in $K(MHM(\cdot))$ coincide. So they both lie in the subgroup $K(TMHM(\cdot))$ of Tate Hodge modules. We have $[\fs_\alpha^*j_*j^*\iota_\alpha^*\ic^\alpha]= [\fs_\alpha^*\iota_\alpha^*\ic^\alpha]- [\fs_\alpha^!\iota_\alpha^*\ic^\alpha]$, and no cancellations occur in the RHS. In effect, all graded parts of $\fs_\alpha^*\iota_\alpha^*\ic^\alpha$ have weights $<2|\alpha|$ (since they live in negative degrees), while $\fs_\alpha^!\iota_\alpha^*\ic^\alpha$ is pure of weight $2|\alpha|$ (see ~\ref{stunt}). We deduce that both $[\fs_\alpha^*\iota_\alpha^*\ic^\alpha]$ and $[\fs_\alpha^!\iota_\alpha^*\ic^\alpha]$ lie in $K(TMHM(\cdot))$. Thus $\fs_\alpha^!\iota_\alpha^*\ic^\alpha$ is a sum of $\CK(\alpha)$ copies of $\BQ(|\alpha|)$, hence $\fs_\alpha^!\iota_\alpha^*\ic^\alpha\cong \fs_\alpha^!\bi_\alpha^*\ic(\oCG_\vartheta)$. It follows that $[\fs_\alpha^*\iota_\alpha^*\ic^\alpha]= [\fs_\alpha^*\bi_\alpha^*\ic(\oCG_\vartheta)]$. This completes the proof of the Theorem ~\ref{hard} along with the Theorems ~\ref{Hodge} and ~\ref{simple}. $\Box$ \subsection{Remark} It is easy to describe the semisimple complex $\fs_\alpha^!\IC^\alpha$ on $\BA^\alpha$. For a Kostant partition $\kappa\in\fK(\alpha)$ let $\BA^\alpha_\kappa\subset\BA^\alpha$ denote the closure of the corresponding diagonal stratum. It is known that the normalization $\tilde\BA{}^\alpha_\kappa\stackrel{N^\kappa}{\lra}\BA^\alpha_\kappa$ is isomorphic to an affine space. We have $\fs_\alpha^!\IC^\alpha\simeq \bigoplus_{\kappa\in\fK(\alpha)}N^\kappa_*\ul\BC$. \section{$\fg^L$-action on the global $\IC$-cohomology} \subsection{} \label{ostrik} We compute the global Intersection Cohomology of $\CQ^\alpha$ following ~\S2 of ~\cite{fk}. First we introduce the open subset $\ddQ^\alpha\subset\CQ^\alpha$ formed by all the quasimaps $\phi$ such that the defect of $\phi$ does not meet $\infty\in C$. Twisting by the multiples of $\infty$ we obtain the partition into locally closed subsets $\CQ^\alpha=\sqcup_{\beta\leq\alpha} \ddQ^\beta$. The evaluation at $\infty$ defines the projection $p_\alpha:\ \ddQ^\alpha\lra\bX$ which is a locally trivial fibration with a fiber isomorphic to $\CZ^\alpha$. By the Poincar\'e duality and Theorem ~\ref{simple} the generating function of $H^\bullet_c(\CZ^\beta,\IC^\beta)$ is given by $\CK^\beta(t^{-1})$. By the parity vanishing, the spectral sequence of fibration $p_\beta:\ \ddQ^\beta\lra\bX$ degenerates and provides us with the isomorphism of graded vector spaces $H^\bullet_c(\ddQ^\beta,\IC(\CQ^\beta)) \cong H^\bullet(\bX,\ul\BC[\dim\bX])\otimes H^\bullet_c(\CZ^\beta,\IC^\beta)$. By the factorization property, $\IC(\CQ^\alpha)|_{\ddQ^\beta}\simeq \IC(\CQ^\beta)\otimes\IC^0_{\{\{\alpha-\beta\}\}}$. Finally, the parity vanishing implies the degeneration of the Cousin spectral sequence associated with the partition $\CQ^\alpha=\sqcup_{\beta\leq\alpha}\ddQ^\beta$. Combining all the above equalities we arrive at the formula for the generating function of $H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$. To write it down in a neat form we will consider the generating function $P_\bG(t)$ of $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$. To record the information on $\alpha$ we will consider this generating function as a formal cocharacter of $\bH$ with coefficients in the Laurent polynomials in $t$. Formal cocharacters will be written multiplicatively, so that the cocharacter corresponding to $\alpha$ will be denoted by $e^\alpha$. Finally, for the reasons which will become clear later (see Proposition ~\ref{h}), we will make the following rescaling. We will attach to $H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$ the cocharacter $e^{\alpha+2\crho}$. With all this in mind, the Poincar\'e polynomial $P_\bG(t)$ of $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$ is calculated as follows: {\bf Theorem.} $$P_\bG(t)=\frac{e^{2\check\rho}t^{-\frac{1}{2}\dim\bX} \sum_{w\in\CW_f}t^{l(w)}} {\prod_{\ctheta\in{\check\CR}{}^+}(1-te^\ctheta)(1-t^{-1}e^\ctheta)}$$ where $l(w)$ stands for the length of $w\in\CW_f$, and $2\check\rho$ stands for the sum of positive coroots $\ctheta\in{\check\CR}{}^+$. $\Box$ \subsubsection{} \label{feigin} The above argument along with the Theorem ~\ref{Hodge} establishes also the following Theorem: {\bf Theorem.} a) For odd $k$ we have $H^{k-2|\alpha|-\dim\bX}(\CQ^\alpha,\ic(\CQ^\alpha))=0$; b) $H^{2k-2|\alpha|-\dim\bX}(\CQ^\alpha,\ic(\CQ^\alpha))$ is the direct sum of a few copies of the Tate Hodge module $\BQ(k). \quad \Box$ \subsection{} \label{prim} In the rest of this section we equip $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$ with the structure of the Langlands dual Lie algebra $\fg^L$-module with the character $\frac{|\CW_f|e^{2\check\rho}} {\prod_{\ctheta\in{\check\CR}{}^+}(1-e^\ctheta)^2}$. We start with the following Proposition describing the subspace Prim$(\CA)$ of the primitive elements of bialgebra $\CA$. {\bf Proposition.} a) If $\alpha\in\BN[I]$ is not a positive coroot, that is $\alpha\not\in\cR^+$, then $\CA_\alpha$ does not contain primitive elements; b) dim(Prim$(\CA)\cap\CA_\alpha)\leq1$ for any $\alpha\in\BN[I]$. {\em Proof.} We will use the Criterion ~\ref{crit}. Consider the long exact sequence of cohomology (notations of ~\ref{crit}): $$\ldots\lra H^\bullet(\BA^1,u_\alpha^!\fs_\alpha^!\IC^\alpha)\lra H^\bullet(\BA^\alpha,\fs_\alpha^!\IC^\alpha)\lra H^\bullet(U^\alpha,\fs_\alpha^!\IC^\alpha)\lra\ldots$$ Applying Poincar\'e duality to the Theorem ~\ref{simple} we see that $u_\alpha^!\fs_\alpha^!\IC^\alpha$ is the direct sum of constant sheaves living in nonnegative degrees, and $H^0(u_\alpha^!\fs_\alpha^!\IC^\alpha)$ is nonzero iff $\alpha\in\cR^+$. In this case $H^0(u_\alpha^!\fs_\alpha^!\IC^\alpha)=\ul\BC$. We deduce that $H^0(\BA^1,u_\alpha^!\fs_\alpha^!\IC^\alpha)$ is nonzero iff $\alpha\in\cR^+$, and in this case $\dim H^0(\BA^1,u_\alpha^!\fs_\alpha^!\IC^\alpha)=1$. Now an element $a\in H^0(\BA^\alpha,\fs_\alpha^!\IC^\alpha)$ is supported on the main diagonal iff it comes from $H^0(\BA^1,u_\alpha^!\fs_\alpha^!\IC^\alpha)$. This completes the proof of the Proposition. $\Box$ \subsubsection{Remark} Later on we will see that in fact the converse of ~\ref{prim} ~a) is also true (see ~\ref{!}). \subsection{Definition} \label{a} a) $\ba$ is the Lie algebra formed by the primitive elements Prim$(\CA)$; b) $\fa\subset\ba$ is the Lie subalgebra generated by $\oplus_{i\in I}\CA_i \subset$ Prim$(\CA)$. \subsection{} \label{Serre} We are going to prove that $\fa=\ba\cong\fn^L_+$ --- the nilpotent subalgebra of the Langlands dual Lie algebra $\fg^L$. We choose generators $e_i\in\CA_i-0$, and start with the following Proposition: {\bf Proposition.} The generators $e_i$ satisfy the Serre relations of $\fn^L_+$. {\em Proof.} The corresponding commutator has the weight lying out of $\cR^+$, and hence vanishes by ~\ref{prim} ~a). $\Box$ \subsubsection{Remark} \label{choice} In fact, there is a preferred choice of generators $e_i$. Note that $\CZ^i\cong\BA^2$, and $\del_i\CZ^i\cong\BA^1$. So $\IC^i$ contains a subsheaf $\ul\BZ[2]$, and $H^0(\BA^1,\fs_i^!\IC^i)\supset H^0(\BA^1,\fs_i^!\ul\BZ[2])=\BZ$. The element $1\in\BZ$ in the RHS corresponds to the canonical element $e_i$ in the LHS. We will use this choice of $e_i$ in what follows. \subsection{} \label{smirnoff} We construct the action of $\CA$ on $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$ closely following ~\ref{rest}--\ref{stab}. \subsubsection{} First we construct the {\em stabilization map} $\ft_{\alpha,\beta}:\ \CA_\alpha\lra\Ext^{|\alpha|}_{\CQ^{\alpha+\beta}} (\IC(\del_\alpha\CQ^{\alpha+\beta}),\IC(\CQ^{\alpha+\beta}))$ where $\del_\alpha\CQ^{\alpha+\beta}$ stands for the image of the twisting map $\sigma_{\beta,\alpha}:\ \CQ^\beta\times C^\alpha\lra\CQ^{\alpha+\beta}$ (see ~\cite{fm} ~3.4.1). To this end we note that the sheaf $\fs_\alpha^!\IC^\alpha$ on $\BA^\alpha$ extends smoothly to the sheaf $\fF_\alpha$ on $C^\alpha$, and $H^0(\BA^\alpha,\fs_\alpha^!\IC^\alpha)=H^0(C^\alpha,\fF_\alpha)= \Ext^{|\alpha|}_{C^\alpha}(\IC(C^\alpha),\fF_\alpha)$. Furthermore, tensoring with $\Id\in\Ext^0(\IC(\CQ^\beta),\IC(\CQ^\beta))$, we get the map from $\CA_\alpha=\Ext^{|\alpha|}_{C^\alpha} (\IC(C^\alpha),\fF_\alpha)$ to $\Ext^{|\alpha|}_{\CQ^\beta\times C^\alpha} (\IC(\CQ^\beta)\boxtimes\IC(C^\alpha),\IC(\CQ^\beta)\boxtimes \fF_\alpha)$. Now the same argument as in ~\ref{bezruk} shows that there is a unique morphism $c$ from $\IC(\CQ^\beta)\boxtimes\fF_\alpha$ to $\sigma_{\beta,\alpha}^!\IC(\CQ^{\alpha+\beta})$ extending the factorization isomorphism from an open part $\oQ^\beta\times\BA^\alpha\subset \CQ^\beta\times C^\alpha$. Thus $c$ induces the map from $\Ext^{|\alpha|}_{\CQ^\beta\times C^\alpha} (\IC(\CQ^\beta)\boxtimes\IC(C^\alpha),\IC(\CQ^\beta)\boxtimes\fF_\alpha)$ to $\Ext^{|\alpha|}_{\CQ^\beta\times C^\alpha} (\IC(\CQ^\beta)\boxtimes\IC(C^\alpha), \sigma_{\beta,\alpha}^!\IC(\CQ^{\alpha+\beta}))$. The latter space equals $\Ext^{|\alpha|}_{\CQ^{\alpha+\beta}} ((\sigma_{\beta,\alpha})_*(\IC(\CQ^\beta)\boxtimes\IC(C^\alpha)), \IC(\CQ^{\alpha+\beta}))=\Ext^{|\alpha|}_{\CQ^{\alpha+\beta}} (\IC(\del_\alpha\CQ^{\alpha+\beta}),\IC(\CQ^{\alpha+\beta}))$. Finally, we define $\ft_{\alpha,\beta}$ as the composition of above maps. \subsubsection{} \label{bezr} Second, we construct the {\em costabilization map} $$\fx_{\beta,\alpha}:\ H^\bullet(\CQ^\beta,\IC(\CQ^\beta))\lra H^{\bullet-|\alpha|}(\del_\alpha\CQ^{\alpha+\beta}, \IC(\del_\alpha\CQ^{\alpha+\beta}))$$ To this end we note that exactly as in ~\ref{normal}, we have $\IC(\del_\alpha\CQ^{\alpha+\beta})=(\sigma_{\beta,\alpha})_* \IC(\CQ^\beta\times C^\alpha)=(\sigma_{\beta,\alpha})_* (\IC(\CQ^\beta)\boxtimes\ul\BC[|\alpha|])$. Let $[C^\alpha]\in H^{-|\alpha|}(C^\alpha,\ul\BC[|\alpha|])$ denote the fundamental class of $C^\alpha$. Now, for $h\in H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ we define $\fx_{\beta,\alpha}(h)$ as $h\otimes[C^\alpha]\in H^{\bullet-|\alpha|}(\CQ^\beta\times C^\alpha,\IC(\CQ^\beta)\boxtimes \ul\BC[|\alpha|])=H^{\bullet-|\alpha|}(\del_\alpha\CQ^{\alpha+\beta}, (\sigma_{\beta,\alpha})_*(\IC(\CQ^\beta)\boxtimes\ul\BC[|\alpha|]))= H^{\bullet-|\alpha|}(\del_\alpha\CQ^{\alpha+\beta}, \IC(\del_\alpha\CQ^{\alpha+\beta}))$. \subsection{Definition} Let $a\in\CA_\alpha,\ h\in H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$. We define the action $a(h)\in H^\bullet(\CQ^{\alpha+\beta},\IC(\CQ^{\alpha+\beta}))$ as the action of $\ft_{\alpha,\beta}(a)$ on the global cohomology applied to $\fx_{\beta,\alpha}(h)$. Let us stress that the action of $\CA_\alpha$ {\em preserves cohomological degrees.} \subsubsection{} For $a\in\CA_\alpha,\ b\in\CA_\beta,\ h\in H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ we have $a(b(h))=a\cdot b(h)$. The proof is entirely similar to the proof of associativity of the multiplication in $\CA$. \subsection{} \label{F} For $\beta\in\BN[I],\ i\in I$ both $H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ and $H^\bullet(\CQ^{\beta+i},\IC(\CQ^{\beta+i}))$ are Poincar\'e selfdual. We define the map $$f_i:\ H^\bullet(\CQ^{\beta+i},\IC(\CQ^{\beta+i}))\lra H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$$ as the dual of the map $$e_i:\ H^\bullet(\CQ^\beta,\IC(\CQ^\beta))\lra H^\bullet(\CQ^{\beta+i},\IC(\CQ^{\beta+i}))$$ (see ~\ref{choice}). It follows from ~\ref{Serre} that the maps $f_i,\ i\in I$, satisfy the Serre relations of $\fn_-^L$. \subsubsection{} \label{f} Let us sketch a more explicit construction of the operators $f_i$. Let $\sigma$ denote the closed embedding $\del_i\CQ^{\beta+i}\hookrightarrow \CQ^{\beta+i}$, and let $j$ denote the embedding of the complementary open subset $U$. We have an exact sequence of perverse sheaves $$0\to\sigma^*j_{!*}\IC(U)[-1]\to j_!\IC(U)\to\IC(\CQ^{\beta+i})\to0$$ defining an element $\ff'_i$ in $\Ext^1(\IC(\CQ^{\beta+i}),\sigma^*j_{!*}\IC(U)[-1]$. The weight considerations as in ~\ref{bezruk} show that $\sigma^*j_{!*}\IC(U)[-1]$ canonically surjects to $\IC(\del_i\CQ^{\beta+i})$, and thus $\ff'_i$ gives rise to $\ff_i\in\Ext^1(\IC(\CQ^{\beta+i}), \IC(\del_i\CQ^{\beta+i})$. The action of $\ff_i$ on the global cohomology defines the same named map $\ff_i:\ H^\bullet(\CQ^{\beta+i},\IC(\CQ^{\beta+i}))\lra H^{\bullet+1}(\del_i\CQ^{\beta+i},\IC(\del_i\CQ^{\beta+i}))$. Now recall that $H^\bullet(\del_i\CQ^{\beta+i},\IC(\del_i\CQ^{\beta+i}))= H^\bullet(\CQ^\beta,\IC(\CQ^\beta))\otimes H^\bullet(C,\ul\BC[1])$. So the projection of $H^\bullet(C,\ul\BC[1])=\BC[1]\oplus\BC[-1]$ to $\BC[-1]$ defines the projection $p:\ H^{\bullet+1}(\del_i\CQ^{\beta+i},\IC(\del_i\CQ^{\beta+i}))\lra H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$. Finally, for $h\in H^\bullet(\CQ^{\beta+i},\IC(\CQ^{\beta+i}))$ we have $f_i(h)=p\ff_i(h)\in H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$. \subsubsection{} \label{e} We leave to the reader the absolutely similar elementary construction of the operators $e_i$. We only mention that the corresponding local element $\fe_i\in \Ext^1_{\CQ^{\beta+i}}(\IC(\del_i\CQ^{\beta+i}),\IC(\CQ^{\beta+i}))$ comes from the star extension of the constant sheaf on an open subset of $\CQ^{\beta+i}$. \subsection{Proposition} \label{fe} If $i,j\in I,\ i\not=j$, then $e_if_j=f_je_i:\ H^\bullet(\CQ^\beta,\IC(\CQ^\beta))\lra H^\bullet(\CQ^{\beta+i-j},\IC(\CQ^{\beta+i-j}))$. {\em Proof.} Using the elementary constructions ~\ref{e}, ~\ref{f} of $e_i$ and $f_j$ we reduce the Proposition to the local calculation in a smooth open subset $U$ of $\CQ^{\beta+i}$. This local calculation is nothing else than the following fact. Let $D=D_1\cup D_2$ be a divisor with normal crossings in $U$, consisting of two smooth irreducible components. If we shriek extend the constant sheaf on $U-D$ through $D_1$ and then star extend it through $D_2$ we get the same result if we first star extend the constant sheaf on $U-D$ through $D_2$, and then shriek extend it through $D_1. \quad \Box$ \subsection{} \label{h} For $i\in I$ we define the endomorphism $h_i$ of $H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ as the scalar multiplication by $\langle\beta+2\crho,i'\rangle$ where $i'\in X$ is the simple root. {\bf Proposition.} $e_if_i-f_ie_i=h_i:\ H^\bullet(\CQ^\beta,\IC(\CQ^\beta))\lra H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$. {\em Proof.} Let $U\subset\CQ^{\beta+i}$ be an open subset such that $W:=U\cap\del_i\CQ^{\beta+i}$ consists of quasimaps of defect {\em exactly} $i$. Then $W=\oQ^\beta\times C$ is a smooth divisor in $U$. Let $T_WU$ be the normal bundle to $W$ in $U$. This is a line bundle on $W$. For $\phi\in\oQ^\beta$ we can restrict $T_WU$ to $C=\phi\times C\subset \oQ^\beta\times C$, and the degree of the restriction does not depend on a choice of $\phi$. We denote this degree by $d_{\beta,i}$. Using the elementary construction of $e_i,f_i$ we see that the commutator $[e_i,f_i]$ acts as the scalar multiplication by $d_{\beta,i}$. The equality $d_{\beta,i}=\langle\beta+2\crho,i'\rangle$ is proved in ~\cite{fk} ~4.4. Let us make a few comments on this proof. As stated, it works in the case $\bG=SL_n$, moreover, it uses the Laumon resolution $\CQ^\alpha_L\lra\CQ^\alpha$. In fact, the proof uses not the whole space $\CQ^\alpha_L$ but the open subspace $\CU_\alpha^L\subset\CQ^\alpha_L$ (see {\em loc. cit.} ~4.2.1). This open subset projects isomorphically into $\CQ^\alpha$. More precisely, $\CU_\alpha^L\subset\CQ^\alpha$ consists of all quasimaps with defect {\em at most a simple coroot}. Now one can see that the calculations in $\CU_\alpha^L$ carried out in ~\S4 of {\em loc. cit.} do not use any specifics of $SL_n$ and carry over to the case of arbitrary $\bG$. This completes the proof of the Proposition. $\Box$ \subsection{} \label{!} Combining the results of ~\ref{Serre}, ~\ref{F}, ~\ref{fe}, ~\ref{h} we see that $e_i,f_i,h_i,\ i\in I$, generate the action of the Langlands dual Lie algebra $\fg^L$ on $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha))$. Since $\fg^L$ is a {\em simple} Lie algebra, we obtain an {\em embedding} $\fg^L\hookrightarrow$End $(\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha)))$. The image of $\fn_+^L\subset\fg^L$ under this embedding coincides with the image of the Lie algebra $\fa\subset\CA$ (see ~\ref{a}) in End$(\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^\alpha,\IC(\CQ^\alpha)))$. We deduce that $\fa\cong\fn_+^L$. Hence we obtain the embedding of the enveloping algebras $U(\fn_+^L)\cong U(\fa)\hookrightarrow\CA$. Comparing the graded dimensions we see that this embedding is in fact an isomorphism. In particular, Prim$(\CA)=\fa\cong\fn_+^L$. We have proved the following Theorem. {\bf Theorem.} The bialgebra $\CA$ is isomorphic to the universal enveloping algebra $U(\fn_+^L)$. \section{Closures of Schubert strata and combinatorics of alcoves} \subsection{} \label{fuck} We recall some combinatorics from ~\cite{l1} and ~\cite{l4}. Let $E$ be an $\BR$-vector space $Y\otimes_\BZ\BR$. We equip it with a scalar product $(|)$ extending it by linearity from the basis $I:\ (i|j)\df\ d\frac{i\cdot j}{d_id_j}$. Here $i\cdot j$ is a part of Cartan datum (see ~\cite{l}, 1.1), $d_i\df\ \frac{i\cdot i}{2}$, and $d\df\ \operatorname{max}_{i\in I}d_i$. The Weyl group $\CW_f$ acts on $E$ by orthogonal reflections; it is generated by the reflections $s_i(y)\df\ y-\langle y,i'\rangle i$ (notations of {\em loc. cit.}, 2.2; we have extended the pairing $\langle,\rangle:\ Y\times X\lra\BZ$ by linearity to $\langle,\rangle:\ E\times X\lra\BR$.) We shall regard $\CW_f$ as acting on $E$ {\em on the right}. The set $I\CW_f$ is the set $\check\CR$ of {\em coroots}, and the set $I\CW_f\cap\BN[I]$ is the set $\check\CR^+$ of {\em positive coroots}. Recall that $\crho=\frac{1}{2}\sum_{\ctheta\in\check\CR^+}\ctheta\in E$. We consider the following collection $$\fF\df\ \{H_{\theta,n},\ \theta\in\CR,\ n\in\BZ\}$$ of affine hyperplanes in $E$: $$H_{\theta,n}=\{y\in E | \langle y,\theta\rangle=n\}$$ Each $H\in\fF$ defines an orthogonal reflection $y\mapsto y\sigma_H$ in $E$ with fixed point set $H$. Let $\Omega$ be the group of affine motions generated by the $\sigma_H\ (H\in\fF)$. We shall regard $\Omega$ as acting on the right on $E$. The connected components of $E-\cup_{H\in\fF}$ are called {\em alcoves}. The group $\Omega$ acts simply transitively on the set $\fA$ of alcoves. We shall denote by $A^-_0$ the following alcove: $$A^-_0\df\ \{y\in E | \langle y,i'\rangle<0\ \forall i\in I;\ \langle y,\theta_0\rangle>-1\}$$ where $\theta_0\in\CR^+$ is the highest root. Then $\Omega$ is generated by the reflections in the walls of $A^-_0$. The subgroup of $\Omega$ generated by the reflections in the walls $H_{i',0}$ is just the Weyl group $\CW_f$. Note that in ~\cite{l4}, \S\S1,8 the group $\Omega$ is called the {\em affine Weyl group} and denoted by $W$ (not to be confused with the {\em dual affine Weyl group} $W^\sharp$). We will follow the notations of ~\cite{l1} instead. In particular, we will call $\CW$ the group defined as in {\em loc. cit.}, 1.1. It acts on $\fA$ simply transitively {\em on the left}, commuting with the action of $\Omega$. As a Coxeter group it is canonically isomorphic to the affine Weyl group $\Omega$. In the notations of {\em loc. cit.} the subgroup $\CW_f\subset\Omega$ is nothing else then $\Omega_0$, and $w_0=\omega_0$. Recall that the set $S$ of Coxeter generators of $\CW$ can be represented as the faces (walls) of $A^-_0$. The generator corresponding to the wall $H_{i',0},\ i\in I,$ will be denoted $s_i$, and the generator corresponding to the wall $H_{\theta_0,-1}$ will be denoted by $s_0$. It is easy to see that the intersection of $\Omega$ with the group of translations of $E$ is (the group of translations by vectors in) $Y$. Thus we obtain a normal subgroup $Y\subset\Omega$. It is known that $\Omega$ is a semidirect product of $\CW_f$ and $Y$ (see ~\cite{l4}, 1.5). In particular, each element $\omega\in\Omega$ can be uniquely written in the form $\omega=w\chi,\ w\in \CW_f,\ \chi\in Y$. Combining this observation with the action of $\Omega$ on $\fA$ we obtain a bijection $$\CW_f\times Y\iso\fA:\ (w,\chi)\mapsto A^-_0w\chi$$ We will use this bijection freely, so we will write $(w,\chi)$ for an alcove often. We will denote by $\beta_0\in\check\CR^+$ the coroot dual to the highest root $\theta_0$, and by $s_{\beta_0}\in \CW_f$ the reflection in $\CW_f$ taking $y\in E$ to $y-\langle y,\theta_0\rangle\beta_0$. \subsubsection{Lemma} \label{s0} For any $i\in I$ we have $s_i(w,\chi)=(s_iw,\chi)$, and $s_0(w,\chi)=(s_{\beta_0}w,\chi-\beta_0w)$. {\em Proof.} Clear. $\Box$ \subsubsection{} \label{dim} Recall that for a pair of alcoves $A,B\in\fA$ the {\em distance} $d(A,B)$ was defined in ~\cite{l1}, 1.4. For $\chi=\sum_{i\in I}a_ii\in Y$ we define $|\chi|\in\BZ$ as follows: $$|\chi|\df\ \sum_{i\in I}a_i$$ For $w\in\CW_f$ let $l(w)$ denote the usual length function: $$l(w)=\sharp(\check\CR^+w\cap-\check\CR^+)=\dim\bX_w$$ {\em Lemma.} Let $B=(w,\chi)$. Then $d(A^-_0,B)=2|\chi|+l(w)$. {\em Proof.} For $A=(w_1,\chi_1),B=(w_2,\chi_2)$ let us define $d'(A,B)\df\ 2|\chi_2|-2|\chi_1|+l(w_2)-l(w_1)$. We want to prove $d'(A,B)=d(A,B)$. To this end it suffices to check the properties ~\cite{l1}(1.4.1),(1.4.2) for $d'$ instead of $d$. This in turn follows easily from the Lemma ~\ref{s0}. $\Box$ \subsubsection{} \label{leq} Recall a few properties of the partial order $\leq$ on $\fA$ introduced in ~\cite{l1}, 1.5. \begin{equation} \label{01} (w_1,\chi)\leq(w_2,\chi)\Leftrightarrow w_1\leq w_2, \end{equation} where $w_1\leq w_2$ stands for the usual Bruhat order on $\CW_f$. \begin{equation} \label{02} (w_1,\chi_1)\leq(w_2,\chi_2)\Leftrightarrow (w_1,\chi_1+\chi)\leq(w_2,\chi_2+\chi)\ \forall\chi\in Y, \end{equation} \begin{equation} \label{03} (w,0)\leq(ws_{\check\theta},\check{\theta})\ \forall \check{\theta}\in\check\CR^+, \end{equation} where $s_{\check\theta}\in\CW_f$ is the reflection in $\CW_f$ taking $y\in E$ to $y-\langle y,\theta\rangle\check{\theta},\ \theta$ being a root dual to the coroot $\check{\theta}$. \begin{equation} \label{04} s_0(w,\chi)\leq(w,\chi)\Leftrightarrow\beta_0w>0. \end{equation} The proofs are easy if not contained explicitly in ~\cite{l1}. For the converse we have the following {\em Lemma.} Let $\preceq$ be the minimal partial order on $\fA$ enjoying the properties ~(\ref{01}), ~(\ref{02}), ~(\ref{03}). Then $A\preceq B\Leftrightarrow A\leq B$. {\em Proof.} Easy. $\Box$ \subsection{} \label{closure} Recall the notations of ~\cite{fm} ~8.4, ~9.1. The Schubert stratification of $\CZ^\alpha_\chi$ reads as follows: $$\CZ^\alpha_\chi=\bigsqcup_{w\in\CW_f}^{0\leq\beta\leq\alpha} \dZ^\beta_{w,\chi-\alpha+\beta}$$ or, after a change of variable $\eta=\chi-\alpha+\beta$, $$\CZ^\alpha_\chi=\bigsqcup_{w\in\CW_f}^{\chi-\alpha\leq\eta\leq\chi} \dZ^{\eta-\chi+\alpha}_{w,\eta}$$ We want to describe the closure of a stratum $\dZ^{\eta-\chi+\alpha}_{w,\eta}$. {\bf Theorem.} Fix a pair of alcoves $A=(w,\eta),B=(y,\xi)$. For any $\chi\geq\eta,\xi$ and $\alpha\in\BN[I]$ sufficiently dominant (such that $\eta-\chi+\alpha\geq10\crho\leq\xi-\chi+\alpha$) we have a) $\dim\dZ^{\xi-\chi+\alpha}_{y,\xi}-\dim\dZ^{\eta-\chi+\alpha}_{w,\eta}= d(A,B)$; b) $\dZ^{\eta-\chi+\alpha}_{w,\eta}$ lies in the closure of $\dZ^{\xi-\chi+\alpha}_{y,\xi}$ iff $A\leq B$. {\em Proof.} a) follows immediately comparing the Lemmas ~\ref{dim} and ~\cite{fm} ~8.5.2. The proof of b) occupies the rest of this section. \subsection{} Let $\th\in\CR^+$ be a positive root. Let $\fg_\th$ denote the corresponding ${\frak{sl}}_2$ Lie subalgebra in $\fg$, and let $\bG_\th$ denote the corresponding $SL_2$-subgroup in $\bG$. Take any $w\in\CW_f$ and let $y=s_\th w$. We will view $w,y$ as $\bH$-fixed points in $\bX$. The $\bG_\theta$-orbit $\bG_\th w=\bG_\th y$ is a smooth rational curve of degree $\ctheta$ in $\bX$ (here $\check\th$ stands for the dual coroot of the root $\th$). We will view it as a closed subset of $\bX$ rather than a parametrized curve. Let us denote this curve by $L_{w,y}$. It is fixed by the $\bH$ action on $\bX$ (as a subset, not pointwise). \subsubsection{Lemma} Let $L\subset\bX$ be an irreducible rational curve fixed by the Cartan action. Then $L=L_{w,y}$ for some $w,y=s_\th w\in\CW_f$. {\em Proof.} Since $L$ is a fixed curve it must contain a fixed point $w\in\CW_f$. Let $f:\hat L\to L\subset\bX$ be the normalization of $L$. The $\bH$-action on $L$ extends to the $\bH$-action on $\hat L$. We choose an $\bH$-fixed point in $f^{-1}(w)$ and preserve the name $w$ for this point. Let $t$ be a formal coordinate at $w\in\hat L$. The map $f$ gives rise to the homomorphism $f^*:\widehat\CO_{\bX,w}\to\BC[[t]]=\widehat\CO_{\hat L,w}$ of $X$-graded rings from the completion of the local ring $\CO_{\bX,w}$ to the ring of formal power series in $t$. We have $\widehat\CO_{\bX,w}=\BC[[T^*_w\bX]]$ --- the ring of formal power series on the tangent space of $\bX$ at the point $w$, and the grading is induced by the grading of the cotangent space $$ T^*_w\bX=\bigoplus\limits_{\vth\in w\CR^+}\BC x_\vth, $$ where $x_\vth\in\fg^*$ is an $\bH$-eigenvector with the eigenvalue $\vartheta$. Since the roots in $w\CR^+$ are pairwise linearly independent, it follows that there exists a root $\th\in w\CR^+$ and a positive integer $n$ such that the map $f^*$ is given by $$ f^*(x_\vth)=\begin{cases} t^n, &\text{ if }\vth=\th\\ 0, &\text{ otherwise.} \end{cases} $$ Therefore the formal jet of the map $f:\hat L\to\bX$ coincides with the formal jet of the map $\vphi_{w,y}^n:\BP^1\to\bX$ given by the composition of the $n$-fold covering $\BP^1\to L_{w,y}$ ramified over the points $w$ and $y$, and of the embedding $L_{w,y}\to\bX$. Now irreducibility of $L$ implies $L=L_{w,y}. \quad \Box$ \subsubsection{Remark} The fixed curves $L_{w,y}$ and $L_{w',y'}$ intersect nontrivially iff $w=w'$ or $w=y'$ or $y=w'$ or $y=y'$. In effect, a point of intersection of two fixed curves has to be a fixed point. \subsection{Definition} Let $f:\CC\to\bX$ be a stable map (see ~\cite{ko}) from a genus 0 curve $\CC$ into the flag variety $\bX$. Let $w,y\in\CW_f$. If $f(\CC)\cap\bX_w\ne\emptyset$ and $f(\CC)\cap\overline\bX_y\ne\emptyset$ we will say that the pair $(w,y)$ is $(f,\CC)$-connected. If $\deg\CC=\ga$ we will say also that $(w,y)$ is $\ga$-connected. \subsubsection{Lemma} \label{conn} A pair $(w,y)$ is $\ga$-connected iff there exists a collection $(\check\th_1,\dots,\check\th_k)$ of positive coroots $\check\th_r\in\check\CR^+$ such that $\check\th_1+\ldots+\check\th_k\le\ga$ and $s_{\th_k}\dots s_{\th_1}w\le y$. {\em Proof.} Suppose the pair $(w,y)$ is $(f,\CC)$-connected, where $f:\CC\to\bX$ is a stable map of degree $\gamma$. Then we have $f(\CC)\cap\bX_w\ne\emptyset$. Acting on $f$ by an element of the Borel subgroup we can assume that $w\in f(\CC)$. Acting by the Cartan subgroup $\bH$ we can degenerate $(f,\CC)$ into an $\bH$-fixed stable map $f':\CC'\to\bX$. Since $w\in f(\CC)$ and $w$ is an $\bH$-fixed point, we will have $w\in f'(\CC')$. Similarly, since $f(\CC)\cap\overline\bX_y\ne\emptyset$ and $\overline\bX_y$ is a closed $\bH$-invariant subspace, we will have $f'(\CC')\cap\overline\bX_y\ne\emptyset$, hence $f'(\CC')\ni y'$ for some $y'\in\CW_f$ such that $y'\le y$. The image of an $\bH$-fixed stable map is a connected union of $\bH$-fixed curves, hence there exists a sequence $L_1,\dots,L_k$ of $\bH$-fixed curves such that $w\in L_1$, $y'\in L_k$ and $L_r\cap L_{r+1}\ne\emptyset$. We can assume that $L_r=L_{w_r,w_{r+1}}$ ($r=1,\dots,k$), where $w_1=w$, $w_{r+1}=s_{\th_r}w_r$ and $w_{k+1}=y'$. Since $$ \ga=\deg f=\deg f'\ge\deg L_1+\ldots+\deg L_k=\check\th_1+\ldots+\check\th_k $$ the Lemma follows. $\Box$ \subsubsection{Corollary} \label{codim1} If $w\not\le y$ and $(w,y)$ is $\ga$-connected then $$ l(w)\le l(y)+2|\ga|-1 $$ and equality holds only if $\ga=\check\th$, $l(s_\th)=2|\ctheta|-1$ and $s_\th w=y$ for some $\ctheta\in\cR^+$. {\em Proof.} By the Lemma \ref{conn} we have \begin{multline*} l(y)\ge l(s_{\th_k}\dots s_{\th_1}w)\ge l(w)-l(s_{\th_1})-\ldots-l(s_{\th_k})\ge\\ \ge l(w)-(2|\ctheta_1|-1)-\ldots-(2|\ctheta_k|-1)\ge l(w)-2|\ga|+k\ge l(w)-2|\ga|+1. \end{multline*} The Corollary follows. $\Box$ \subsection{Proposition} \label{criter} A Schubert stratum $\dZ^{\eta-\chi+\al}_{w,\eta}$ lies in the closure of $\dZ^{\xi-\chi+\al}_{y,\xi}$ iff $\eta\le\xi$ and the pair $(w,y)$ is $(\xi-\eta)$-connected. {\em Proof.} We start with the ``only if'' part. Both strata lie in the closed subspace $\CZ^{\xi-\chi+\al}_\xi\subset\CZ^\al_\chi$, hence we can consider them as subspaces in $\CZ^{\xi-\chi+\al}_\xi$. Let $\CQ^{\xi-\chi+\al}_K= \overline M_{0,0}(\BP^1\times\bX,(1,\xi-\chi+\al))$ be the Kontsevich space of stable maps from the genus zero curves without marked points to $\BP^1\times\bX$ of bidegree $(1,\xi-\chi+\alpha)$ (see ~\cite{ko}). The map $\pi:\ \CQ^{\xi-\chi+\al}_K\lra\CQ^{\xi-\chi+\al}$ is constructed in the Appendix. For a stable map $f:\ \CC\lra\BP^1\times\bX$ in $\CQ^{\xi-\chi+\al}_K$ we denote by $f':\ \CC\lra\BP^1$ (resp. $f'':\ \CC\lra\bX$) its composition with the first (resp. second) projection. Let $\CK^{\xi-\chi+\al}_\xi\subset\CQ^{\xi-\chi+\al}_K$ be the preimage of the subspace $\CZ^{\xi-\chi+\al}_\xi\subset\CQ^{\xi-\chi+\al}$ under the map $\pi:\CQ_K^{\xi-\chi+\al}\to\CQ^{\xi-\chi+\al}$. It is a locally closed subspace consisting of all stable maps $f:\CC\to\BP^1\times\bX$ such that ${f'}^{-1}(\infty)$ is a point (i.e. $\CC$ has no vertical component over $\infty\in\BP^1$), and $f''({f'}^{-1}(\infty))=w_0\in\bX$. Consider the preimage of the Schubert stratification of $\CZ^{\xi-\chi+\al}_\xi$: $$ \CK^{\xi-\chi+\al}_\xi=\bigsqcup_{w\in\CW_f}^{\chi-\al\le\eta\le\chi} \dK^{\eta-\chi+\al}_{w,\eta}. $$ Here $\dK^{\eta-\chi+\al}_{w,\eta}$ is just the subspace of all stable maps $f:\CC\to\BP^1\times\bX$ such that $$ \deg\CC_1=\xi-\eta\quad\text{and}\quad f''(P)\in\bX_w, $$ where $\CC_1={f'}^{-1}(0)$ is the vertical component of $\CC$ over the point $0\in\BP^1$, and $\CC_0$ is the main component of $\CC$ (the one which projects isomorphically onto $\BP^1$ under $f'$); the point $P$ is the intersection of these components: $P=\CC_0\cap\CC_1$; and as always, $f''({f'}^{-1}(\infty))=w_0$. It follows that $\dZ^{\eta-\chi+\al}_{w,\eta}$ lies in the closure of $\dZ^{\xi-\chi+\al}_{y,\xi}$ iff the map $$ \pi:\dK^{\eta-\chi+\al}_{w,\eta}\bigcap \overline{\dK^{\xi-\chi+\al}_{y,\xi}} \to\dZ^{\eta-\chi+\al}_{w,\eta} $$ is surjective. In particular, the above intersection must be non-empty. Consider the subspace $\CK^{\xi-\chi+\al}_{y,\xi}\subset\CK^{\xi-\chi+\alpha}_\xi$ formed by the stable maps $f:\CC\to\BP^1\times\bX$ such that $f(\CC)\cap(\{0\}\times\overline\bX_y)\ne\emptyset$. In other words, $\CK^{\xi-\chi+\al}_{y,\xi}\subset\CK^{\xi-\chi+\alpha}_\xi$ is formed by the stable maps such that $f''(\CC_1)\cap\overline\bX_y\ne\emptyset$. The subspace $\CK^{\xi-\chi+\al}_{y,\xi}$ is obviously closed and irreducible. It contains $\dK^{\xi-\chi+\al}_{y,\xi}$ as an open subspace, hence $$ \CK^{\xi-\chi+\al}_{y,\xi}=\overline{\dK^{\xi-\chi+\al}_{y,\xi}}. $$ The intersection $\dK^{\eta-\chi+\al}_{w,\eta}\bigcap \overline{\dK^{\xi-\chi+\al}_{y,\xi}}$ consists of all stable maps such that $$ \deg\CC_1=\xi-\eta,\qquad f''(P)\in\bX_w\quad\text{and}\quad f''(\CC_1)\cap\overline\bX_y\ne\emptyset. $$ Therefore the intersection is non-empty only if the pair $(w,y)$ is $(\xi-\eta)$-connected (recall that $P\in\CC_1$). This completes the proof of the ``only if'' part. On the other hand, if $(w,y)$ is $(\xi-\eta)$-connected, then for any stable map $(f,\CC)$ we can replace the component $\CC_1$ by a stable curve of degree $(\xi-\eta)$, connecting the point $f''(P)\in\bX_w$ with $\overline\bX_y$. Such replacement will not affect $\pi(f,\CC)$, but the new curve will lie in the intersection $\dK^{\eta-\chi+\al}_{w,\eta}\bigcap\overline{\dK^{\xi-\chi+\al}_{y,\xi}}$, hence the above intersection maps surjectively onto $\dZ^{\eta-\chi+\al}_{w,\eta}$ and the Proposition follows. $\Box$ \subsection{} Now we can prove the Theorem ~\ref{closure}. We define the {\em adjacency order} $\preceq$ on $\fA$ as follows. We say that $A=(w,\eta)\preceq(y,\xi)=B$ if for any $\chi\geq\eta,\xi$ and sufficiently dominant $\alpha\in\BN[I]$ the stratum $\dZ^{\eta-\chi+\alpha}_{w,\eta}$ lies in the closure of $\dZ^{\xi-\chi+\alpha}_{y,\xi}$. First we check that $\preceq$ satisfies the relations ~(\ref{01}--\ref{03}) of ~\ref{leq}. To this end let us rephrase these properties via criterion \ref{criter}. The property ~(\ref{01}) means that $(w_1,w_2)$ is $0$-connected iff $w_1\le w_2$. The property ~(\ref{02}) expresses the fact that the adjacency of strata depends on difference of degrees only, which is evident from the criterion \ref{criter}. The property ~(\ref{03}) means that the pair $(w,s_\th w)$ is $\check\th$-connected. This is indeed so since the curve $L_{w,s_\th w}$ has degree $\check\th$ and connects $w$ with $s_\th w$. Now the adjacency order $\preceq$ is clearly generated by the adjacencies in codimension 1. So it remains to check that the only adjacencies in codimension 1 are the ones of type ~(\ref{03}) or type ~(\ref{01}) with $l(w_1)=l(w_2)-1$. Assume that $\dZ^{\eta-\chi+\al}_{w,\eta}$ lies in the closure of $\dZ^{\xi-\chi+\al}_{y,\xi}$ and has codimension 1. Since $$ \dim\dZ^{\eta-\chi+\al}_{w,\eta}=2|\eta-\chi+\al|+l(w)-\dim\bX,\quad \dim\dZ^{\xi-\chi+\al}_{y,\xi}=2|\xi-\chi+\al|+l(y)-\dim\bX $$ it follows that $l(w)=l(y)+2|\xi-\eta|-1$. On the other hand, by the criterion \ref{criter} the pair $(w,y)$ is $(\xi-\eta)$-connected. If $\xi=\eta$ then $l(w)=l(y)-1$ and we are in the situation of type ~(\ref{01}). If $\xi>\eta$ then $w\not\le y$ and by the Corollary \ref{codim1} it follows that $\xi-\eta=\check\th$ and $y=s_\th w$ for some $\th\in\CR^+$, i.e. we are in the situation of type ~(\ref{03}). The Theorem follows. $\Box$ \subsection{} \label{who} Recall that $\dQ^\gamma\subset\CQ^\gamma$ is the open subset formed by all the quasimaps without defect at $0\in C$ (i.e. defined at $0\in C$, see ~\cite{fm} ~8.1). For $w\in\CW_f$ we define the locally closed subset $\dQ^\gamma_w\subset\dQ^\gamma$ formed by all the quasimaps $\phi$ such that $\phi(0)\in\bX_w\subset\bX$ (cf. {\em loc. cit.} ~8.4). Its closure will be denoted by $\CQ^\gamma_w$; it coincides with the closure of the fine Schubert stratum $\oQ^\gamma_w$ defined in {\em loc. cit.} ~8.4.1. {\bf Corollary of the Proof.} Let $\gamma\leq\beta\in\BN[I],\ w,y\in\CW_f$. Then $\CQ^\beta_y\supset\CQ^\gamma_w$ iff $(y,\beta)=:B\geq A:=(w,\gamma)$. Also, $\dim\CQ^\beta_y-\dim\CQ^\gamma_w=d(A,B). \quad \Box$ \subsection{} \label{t'ma} We will denote the closure of a fine Schubert stratum $\oZ^\beta_y$ by $\CZ^\beta_y$ (note that it was denoted by $\oCZ^\beta_y$ in ~\cite{fm}. We hope this will cause no confusion.) Similarly, the closure of $\oZ^\beta_{y,\xi}$ (see {\em loc. cit.} ~9.4) will be denoted by $\CZ^\beta_{y,\xi}$. We promote ``alcovic'' notations. Suppose we are in the situation of the above theorem, i.e. $A=(w,\eta)\leq(y,\xi)=B;\ \chi\geq\eta,\xi$, and $\alpha\in\BN[I]$ is sufficiently dominant. We introduce new variables $\gamma=\eta-\chi+\alpha$ and $\beta=\xi-\chi+\alpha$ (note that both $\gamma$ and $\beta$ necessarily lie in $\BN[I]$). Then the above Theorem claims that $\dZ^\gamma_{w,\eta}$ lies in the closure $\CZ^\beta_{y,\xi}$ of $\dZ^\beta_{y,\xi}$. We will denote this closure by $\CZ^\beta_B$; thus we have the closed embedding $\CZ^\gamma_A\hra\CZ^\beta_B$. With our new notations, it is only natural to rename the snop $\CL(w,\eta)$ (see {\em loc. cit.} ~9.4) into $\CL(A)$, and $\CM(w,\eta)$ into $\CM(A)$, and $\CalD\CM(w,\eta)$ into $\CalD\CM(A)$. \subsubsection{} \label{mrak} Recall (see {\em loc. cit.} ~9.3) that we call $\alpha\in\BN[I]$ sufficiently dominant, and write $\alpha\gg0$, iff $\alpha\geq10\crho$. For such $\alpha$ we defined the open subvariety $\ddZ^\alpha\subset\CZ^\alpha$. For $\beta\leq\alpha$ the closed subvariety $\CZ^\beta_y\cap\ddZ^\alpha\subset \ddZ^\alpha$ is nonempty iff $\beta\gg0$. In this case it will be denoted by the same symbol $\CZ^\beta_y$; we hope this will cause no confusion. Again, for $0\ll\gamma\leq\beta$ we have $\CZ^\beta_y\supset\CZ^\gamma_w$ iff $(y,\beta)=:B\geq A:=(w,\gamma)$. Also, $\dim\CZ^\beta_y-\dim\CZ^\gamma_w=d(A,B).$ \section{Mixed snops and convolution} \subsection{} In this section we compute the stalks of irreducible snops. First we prove that they carry a natural Tate Hodge structure. We start with preliminary results about affine Grassmannian. Recall the stratification of the affine Grassmannian $\CG$ by the Iwahori orbits $\CG_{w,\eta}$ described e.g. in ~\cite{fm} ~10.4. The orbit closure $\oCG_{w,\eta}$ is partitioned into its intersections with semiinfinite orbits: $\oCG_{w,\eta}= \sqcup_{\alpha\in Y}(\oCG_{w,\eta}\cap T_\alpha)$. The irreducible perverse sheaf $\IC(\oCG_{w,\eta})$ has the mixed Hodge module counterpart $\ic(\oCG_{w,\eta})$. {\bf Proposition.} a) $H^\bullet_c(\oCG_{w,\eta}\cap T_\alpha, \ic(\oCG_{w,\eta}))$ is a pure Hodge complex of weight $\dim\oCG_{w,\eta}$. It is a direct sum of Tate Hodge structures; b) $H^\bullet_c(\oCG_{w,\eta}\cap\ol{T}_\alpha, \ic(\oCG_{w,\eta}))$ is a pure Hodge complex of weight $\dim\oCG_{w,\eta}$. It is a direct sum of Tate Hodge structures. {\em Proof.} a) Let us denote the locally closed embedding of $\oCG_{w,\eta}\cap T_\alpha$ (resp. $\oCG_{w,\eta}\cap S_\alpha$) into $\oCG_{w,\eta}$ by $i_T$ (resp. $i_S$). One can construct a natural isomorphism $H^\bullet(\oCG_{w,\eta}\cap S_\alpha,i_S^!\ic(\oCG_{w,\eta}))\iso H^\bullet_c(\oCG_{w,\eta}\cap T_\alpha,i_T^*\ic(\oCG_{w,\eta}))$ (see ~\cite{mv}, Opposite parabolic restrictions). On the other hand, $\ic(\oCG_{w,\eta})$ is a pure Hodge module of weight $\dim\oCG_{w,\eta}$, while $H^\bullet_c(?)$ and $i_T^*$ decrease weights, and $H^\bullet(?)$ and $i_S^!$ increase weights. It remains to prove that $H^\bullet_c(\oCG_{w,\eta}\cap T_\alpha, \ic(\oCG_{w,\eta}))$ is a direct sum of Tate Hodge structures. To this end consider the Cousin spectral sequence associated with the partition into locally closed subsets $\oCG_{w,\eta}= \sqcup_{\alpha\in Y}(\oCG_{w,\eta}\cap T_\alpha)$. We have $E_2^{p,q}\Longrightarrow H^{p+q}(\oCG_{w,\eta},\ic(\oCG_{w,\eta}))$, and $E_2^{p,q}=\oplus H^{p+q}_c(\oCG_{w,\eta}\cap T_\alpha,i_T^*\ic(\oCG_{w,\eta}))$; the sum is taken over $\alpha$ such that the codimension of $\oCG_{w,\eta}\cap T_\alpha$ in $\oCG_{w,\eta}$ equals $q$. For the weight reasons the differentials in this spectral sequence vanish, and it collapses at the second term. Hence $H^\bullet(\oCG_{w,\eta},\ic(\oCG_{w,\eta}))\cong\oplus_{\alpha\in Y} H^\bullet_c(\oCG_{w,\eta}\cap T_\alpha,i_T^*\ic(\oCG_{w,\eta}))$. On the other hand, the LHS is well known to be a direct sum of Tate Hodge structures. So a) is proved. Now b) follows by the application of Cousin spectral sequence associated with the partition into the locally closed subsets $\oCG_{w,\eta}\cap\ol{T}_\alpha=\sqcup_{\beta\geq\alpha} (\oCG_{w,\eta}\cap T_\beta). \quad \Box$ \subsubsection{Remark} \label{zhopa} Let $g\in\bG$ lie in the normalizer $N(\bH)$ of the Cartan subgroup $\bH$, and let $T_\alpha^g$ denote $g(T_\alpha)$: the action of $g$ on the semiinfinite orbit in $\CG$. The same argument as above proves that $H^\bullet_c(\oCG_{w,\eta}\cap T_\alpha^g, \ic(\oCG_{w,\eta}))$ is a pure Hodge complex of weight $\dim\oCG_{w,\eta}$; it is a direct sum of Tate Hodge structures. \subsection{} \label{purity} According to the Theorem ~\ref{Hodge}, the simple stalk $\fs_\alpha^*\iota_\alpha^*\ic^\alpha=\ic^0_{\{\{\alpha\}\}}$ is pure of weight $2|\alpha|$, and it is a direct sum of Tate Hodge structures. By factorization, the Hodge module $\ic^\alpha$ on $\CZ^\alpha$ is pointwise pure, and all its stalks are direct sums of Tate Hodge structures. Hence the Hodge module $\ic(\CQ^\alpha)$ on $\CQ^\alpha$ is pointwise pure, and all its stalks are direct sums of Tate Hodge structures. In what follows, such Hodge modules will be called {\em pointwise pure Tate Hodge modules}. Recall the proalgebraic variety $\fQ^\alpha$ introduced in ~\cite{fm} ~10.6. The theory of mixed Hodge modules on such varieties is developed in ~\cite{kt1}. In particular, we will be interested in the irreducible Hodge module $\ic(\fQ^\alpha)$, cf. ~\cite{fm} ~10.7.3. The same argument as in {\em loc. cit.} ~12.7 proves that $\ic(\fQ^\alpha)$ is a pointwise pure Tate Hodge module. For the future reference, let us collect the above facts into a Theorem. {\bf Theorem}. The Hodge modules $\ic^\alpha,\ic(\CQ^\alpha),\ic(\fQ^\alpha)$ are the pointwise pure Tate Hodge modules. \subsection{Theorem} \label{finkel} $\ic(\CQ^\alpha_w)$ (see ~\ref{who}) is a pointwise pure Tate Hodge module. {\em Proof.} The usual argument with factorization reduces the proof to the study of the stalk of $\ic(\CQ^\alpha_w)$ at the fine Schubert stratum $\oQ^0_y\times(\BP^1-0)^\beta_\Gamma\times0^{\alpha-\beta} \subset\CQ^\alpha,\ \beta\leq\alpha,\ y\in\CW_f$ (see {\em loc. cit.} ~8.4.1). Moreover, by the same factorization argument, this stalk will not change if we add $\gamma\in\BN[I]$ to both $\alpha$ and $\beta$. We will choose $\gamma$ sufficiently dominant to ensure that $\alpha+\gamma\in Y^+$. So we may and will suppose that $\alpha$ is dominant. Furthermore, we can choose any point in the stratum $\oQ^0_y\times(\BP^1-0)^\beta_\Gamma\times0^{\alpha-\beta}$, and we will choose a point $\phi=(\fL_\lambda)_{\lambda\in X^+}\in \oQ^\beta_y$ where $\fL_\lambda=\CV_\lambda^{\bN_+^y}(\langle\beta-\alpha, \lambda\rangle\cdot0-\langle D,\lambda\rangle)$, and $\bN_+^y$ is the nilpotent subgroup conjugated to $\bN_+$ by any representative $\dot{y}$ of $y$ in the normalizer of $\bH$ (see {\em loc. cit.} ~3.3); $D\in(\BP^1-0)^\beta_\Gamma$. Recall the setup of the Theorem ~12.12 of {\em loc. cit.} We consider the Hodge module $\CF=\ic(\oCG_{w,\alpha})$ on $\oCG_\alpha$ (recall that we assumed $\alpha\in Y^+$), and the convolution $\bc^0_\CQ(\CF)=\bq_*(\ic(\fQ^0)\otimes\bp^*\CF) [-\dim\ul\fM^\alpha]$ which is a mixed Hodge module on $\CQ^\alpha$ smooth along the fine Schubert stratification. By the Proposition ~12.3b) of {\em loc. cit.} $\bc^0_\CQ(\ic(\oCG_{w,\alpha}))= \bq_*\ic(\CG\CQ^0_{w,\alpha})$. By the Decomposition Theorem, $\bq_*\ic(\CG\CQ^0_{w,\alpha})$ is a direct sum of irreducible Hodge modules on $\CQ^\alpha$. By ~10.4.2 of {\em loc. cit.} $\bq(\CG\CQ^0_{w,\alpha})=\CQ^\alpha_w$, and hence $\bq_*\ic(\CG\CQ^0_{w,\alpha})$ contains a direct summand $\ic(\CQ^\alpha_w)$. Thus it suffices to prove that $\bc^0_\CQ(\CF)$ is a pointwise pure Tate Hodge module. More precisely, we are interested in the stalk of $\bc^0_\CQ(\CF)$ at the point $\phi\in\oQ^\beta_y$. As in the proof of ~12.9.1 of {\em loc. cit.}, we have $\bq^{-1}(\phi)=\oCG_{w,\alpha}\cap\ol{T}{}_{\beta-\alpha}^y= \sqcup_{\gamma\geq0}(\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y)$ where $T_{\beta-\alpha+\gamma}^y$ is $\dot{y}(T_{\beta-\alpha+\gamma})$: the action of $\dot{y}\in\bG$ on the semiinfinite orbit in $\CG$. According to the Lemma ~13.1 of {\em loc. cit.}, we have $\ic(\CG\CQ^0_{w,\alpha})|_{\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y}= \ic(\oCG_{w,\alpha})|_{\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y}\otimes \ic^\gamma_\Gamma$. By ~\ref{zhopa}, $H^\bullet_c(\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y, \ic(\CG\CQ^0_{w,\alpha}))$ is a pure direct sum of Tate Hodge structures, and by ~\ref{purity}, $\ic^\gamma_\Gamma$ is also a pure direct sum of Tate Hodge structures; thus $H^\bullet_c(\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y, \ic(\CG\CQ^0_{w,\alpha})\otimes\ic^\gamma_\Gamma)$ is also a pure direct sum of Tate Hodge structures. Hence the Cousin spectral sequence associated to the partition of $\bq^{-1}(\phi)=\oCG_{w,\alpha}\cap\ol{T}{}_{\beta-\alpha}^y$ into the locally closed subsets $\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y$ collapses at the second term for the weight reasons. We deduce that $\bc^0_\CQ(\CF)_{(\phi)}= H^\bullet(\bq^{-1}(\phi),\ic(\CG\CQ^0_{w,\alpha}))\cong \oplus_{\gamma\geq0} H^\bullet_c(\oCG_{w,\alpha}\cap T_{\beta-\alpha+\gamma}^y, \ic(\CG\CQ^0_{w,\alpha})\otimes\ic^\gamma_\Gamma)$ is also a pure direct sum of Tate Hodge structures. This completes the proof of the Theorem. $\Box$ \subsection{} \label{this} The subsections ~\ref{this}--\ref{long} are quite parallel to \S\S10--12 of ~\cite{fm}. \subsubsection{} Let $\sG$ be the usual affine flag manifold $\bG((z))/\bI$. It is the ind-scheme representing the functor of isomorphism classes of triples $(\CT,\tau,\ft)$ where $\CT$ is a $\bG$-torsor on $\BP^1$, and $\tau$ is its section (trivialization) defined off 0, while $\ft$ is its $\bB$-reduction at 0. It is equipped with a natural action of proalgebraic Iwahori group $\bI$, and the orbits of this action are naturally numbered by the affine Weyl group $\CW:\ \sG=\sqcup_{x\in\CW}\sG_x$. An orbit $\sG_y$ lies in the closure $\osG_x$ iff $y\leq x$ in the usual Bruhat order on $\CW$. The orbit closure $\osG_x$ has a natural structure of projective variety. The natural projection $\pr:\ \sG\lra\CG$ is a fibration with the fiber $\bX$. All these facts are very well known, see e.g. ~\cite{kt}. \subsubsection{} Let $\sM$ be the affine flag scheme representing the functor of isomorphism classes of $\bG$-torsors on $\BP^1$ equipped with trivialization in the formal neighbourhood of $\infty$ and with $\bB$-reduction at 0 (see ~\cite{ka} and ~\cite{kt1}). It is equipped with a natural action of the opposite Iwahori group $\bI_-\subset\bG[[z^{-1}]]$, and the orbits of this action are naturally numbered by the affine Weyl group $\CW:\ \sM=\sqcup_{x\in\CW}\sM_x$. An orbit $\sM_y$ lies in the closure $\usM_x$ iff $y\geq x$ in the usual Bruhat order on $\CW$. For any $x\in\CW$ the union of orbits $\sM^x:=\sqcup_{\CW\ni y\leq x}\sM_y$ forms an open subscheme of $\sM$. This subscheme is a projective limit of schemes of finite type, all the maps in projective system being fibrations with affine fibers. Moreover, $\sM^x$ is equipped with a free action of a prounipotent group $\bG^x$ (a congruence subgroup in $\bG[[z^{-1}]]$) such that the quotient $\usM^x$ is a smooth scheme of finite type. The natural projection $\pr:\ \sM\lra\fM$ (see ~\cite{fm} ~10.6) forgetting a $\bB$-reduction at 0 is a fibration with the fiber $\bX$. Restricting a trivialization of a $\bG$-torsor from $\BP^1-0$ to the formal neighbourhood of $\infty$ we obtain the closed embedding $\bi:\ \sG\lra\sM$. The intersection of $\sM_x$ and $\sG_y$ is nonempty iff $y\leq x$, and then it is transversal. Thus, $\osG_x\subset\sM^x$. According to ~\cite{kt}, the composition $\osG_x\hookrightarrow\sM^x\lra\usM^x$ is a closed embedding. \subsubsection{} Following ~10.6.3 of ~\cite{fm} we define for {\em arbitrary} $\alpha\in Y$ the scheme $\osQ^\alpha$ (resp. $\sQ^\alpha$) representing the functor of isomorphism classes of triples $(\CT,\ft,(\fL_\lambda)_{\lambda\in X^+})$ where $\CT$ is a $\bG$-torsor trivialized in the formal neighbourhood of $\infty\in\BP^1$, and $\ft$ is its $\bB$-reduction at 0, while $\fL_\lambda\subset\CV_\lambda^\CT,\ \lambda\in X^+$, is a collection of line subbundles (resp. invertible subsheaves) of degree $\langle-\alpha,\lambda\rangle$ satisfying the Pl\"ucker conditions (cf. {\em loc. cit.}). The evident projection $\osQ^\alpha\lra\sM$ (resp. $\sQ^\alpha\lra\sM$) will be denoted by $\obp$ (resp. $\bp$). The open embedding $\osQ^\alpha\hra\sQ^\alpha$ will be denoted by $\bj$. Clearly, $\bp$ is projective, and $\obp=\bp\circ\bj$. The free action of prounipotent group $\bG^x$ on $\sM^x$ lifts to the free action of $\bG^x$ on the open subscheme $\bp^{-1}(\sM^x)\subset\sQ^\alpha$. The quotient is a scheme of finite type $\usQ^{\alpha,x}$ equipped with the projective morphism $\bp$ to $\usM^x$. There exists a $\bI_-$-invariant stratification $\fS$ of $\sQ^\alpha$ such that $\bp$ is stratified with respect to $\fS$ and the stratification $\sM=\sqcup_{x\in\CW}\sM_x$. One can define perverse sheaves and mixed Hodge modules on $\sQ^\alpha$ smooth along $\fS$ following the lines of ~\cite{kt1}. In particular, we have the irreducible Hodge module $\ic(\sQ^\alpha)$. \subsubsection{} \label{following} Following ~8.4.1 of ~\cite{fm} we introduce the {\em fine stratification} of $\sQ^\alpha$ according to defects of invertible subsheaves: $$\sQ^\alpha=\bigsqcup^{\alpha\geq\beta\geq\gamma}_{\Gamma\in\fP(\beta-\gamma)} \osQ^\gamma\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta}$$ and the {\em fine Schubert stratification} of $\sQ^\alpha$: $$\sQ^\alpha= \bigsqcup^{\alpha\geq\beta\geq\gamma}_{w\in\CW_f,\ \Gamma\in\fP(\beta-\gamma)} \osQ^\gamma_w\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta}$$ where $\osQ^\gamma_w\subset\osQ^\gamma$ is defined as follows. A collection of line subbundles $\fL_\lambda\subset\CV^\CT_\lambda$ defines a collection of lines $L_\lambda\subset(\CV^\CT_\lambda)_{(0)}$ in the fiber over $0\in C$. This collection of lines defines a $\bB$-reduction of $\CT$ at $0\in C$, and if the relative position of this reduction and $\ft$ is $w\in\CW_f$, we say that the triple $(\CT,\ft,(\fL_\lambda)_{\lambda\in X^+})$ lies in $\osQ^\gamma_w$. The closure of $\osQ^\gamma_w$ is denoted by $\sQ^\gamma_w\subset \sQ^\gamma$. Also, for an alcove $A=(w,\gamma)$ we will often write $\osQ_A$ for $\osQ^\gamma_w$, and $\sQ_A$ for $\sQ^\gamma_w$. An open subset of $\sQ^\gamma_w$ formed by the triples $(\CT,\ft,(\fL_\lambda)_{\lambda\in X^+})$ such that the collection $(\fL_\lambda)_{\lambda\in X^+}$ has no defect at $0\in C$ will be denoted by $\dsQ^\gamma_w$. The {\em refined stratification} of $\sQ^\alpha$ is a refinement of the fine Schubert stratification: it additionally subdivides $\osQ^\gamma_w$ into the strata $\osQ^{\gamma,x}_w:=\bp^{-1}(\sM_x)$. The map $\bp:\ \sQ^\alpha\lra\sM$ is clearly stratified with respect to the refined stratification of $\sQ^\alpha$ and the stratification $\sM=\sqcup_{x\in\CW}\sM_x$. \subsubsection{} \label{parallel} For $\beta\geq\alpha$ we have a closed embedding $\sQ^\alpha\hookrightarrow \sQ^\beta$, sending a triple $(\CT,\tau,(\fL_\lambda)_{\lambda\in X^+})$ to $(\CT,\tau, (fL_\lambda(\langle\beta-\alpha,\lambda\rangle\cdot0))_{\lambda\in X^+})$. These embeddings form an inductive system, and we will denote its union by $\sQ$. A {\em perverse sheaf} or a {\em mixed Hodge module on $\sQ$ supported on $\sQ^\alpha$} is just the same as a perverse sheaf or a mixed Hodge module on $\sQ^\alpha$. Let $\aleph(\sQ)$ be the additive category formed by the (finite) direct sums of irreducible Hodge modules $\ic(\sQ^\gamma_w)$ and their Tate twists on $\sQ$. Let $\mho(\sQ)$ be the additive category formed by the direct sums of Goresky-MacPherson sheaves $\IC(\sQ^\gamma_w)$ and their shifts in the derived category. The subcategories of sheaves supported on $\sQ^\alpha\subset \sQ$ will be denoted by $\aleph(\sQ^\alpha)$ and $\mho(\sQ^\alpha)$. \subsubsection{Remark} \label{brav} The category of perverse sheaves on $\sQ$ of finite length, with all the irreducible constituents of the form $\IC(\sQ^\gamma_w)$, is equivalent to the category $\PS$ defined in ~\cite{fm}. This is the good working definition of $\PS$ promised in ~\cite{fm} instead of the ugly provisional definition given there. \subsubsection{} \label{similar} Similarly, let $\aleph(\sG)$ be the additive category formed by the (finite) direct sums of irreducible Hodge modules $\ic(\osG_x)$ and their Tate twists on $\sG$. Let $\mho(\sG)$ be the additive category formed by the direct sums of Goresky-MacPherson sheaves $\IC(\osG_x)$ and their shifts in the derived category. The subcategories of sheaves supported on $\osG_x\subset\sG$ will be denoted by $\aleph(\osG_x)$ and $\mho(\osG_x)$. For $\alpha\in\BN[I]$ let $\aleph(\CQ^\alpha)$ be the additive category formed by the (finite) direct sums of irreducible Hodge modules $\ic(\CQ^\gamma_w),\ \gamma\leq\alpha$, and their Tate twists on $\CQ^\alpha$. Let $\mho(\CQ^\alpha)$ be the additive category formed by the direct sums of Goresky-MacPherson sheaves $\IC(\CQ^\gamma_w),\ \gamma\leq\alpha$, and their shifts in the derived category. For $\alpha\in\BN[I]$ sufficiently dominant let $\aleph(\ddZ^\alpha)$ be the additive category formed by the (finite) direct sums of irreducible Hodge modules $\ic(\CZ^\gamma_w),\ 0\ll\gamma\leq\alpha$, and their Tate twists on $\ddZ^\alpha$ (see ~\ref{mrak}). Let $\mho(\ddZ^\alpha)$ be the additive category formed by the direct sums of Goresky-MacPherson sheaves $\IC(\CZ^\gamma_w),\ 0\ll\gamma\leq\alpha$, and their shifts in the derived category. Finally, let $\aleph(\PS)$ be the additive category formed by the collections $\fH^\alpha_\chi\in\aleph(\ddZ^\alpha_\chi)$ together with factorization isomorphisms as in ~\cite{fm} ~9.3. Let $\mho(\PS)$ be the additive category formed by the direct sums of irreducible snops $\CL(A)\in\PS$ (see ~\ref{t'ma}) and their shifts in the derived category. \subsection{} \label{that} For $\fF,\fG\in\mho(\sG)$ their convolution $\fF\star\fG\in\mho(\sG)$ was studied e.g. in ~\cite{kt}. We will define the convolution $\mho(\sG)\times\mho(\sQ)\lra\mho(\sQ)$. Its construction occupies the subsections ~\ref{that}--\ref{long}. \subsubsection{} \label{degree 0} Let $x\in\CW,\ w\in\CW_f,\ \alpha\in Y$. We define the {\em convolution diagram} $\sG\sQ^\alpha_{w,x}$ as the cartesian product of $\osG_x$ and $\sQ^\alpha_w$ over $\sM$. Thus we have the following cartesian diagram (cf. ~\cite{fm} ~12.2): $$ \begin{CD} \sG\sQ^\alpha_{w,x} @>\bi>> \sQ^\alpha_w \\ @V{\bp}VV @V{\bp}VV \\ \osG_x @>{\bi}>> \sM \end{CD} $$ The same argument as in {\em loc. cit.} ~12.2 proves that $\ic(\sQ^\alpha_w)\otimes\bp^*\bi_*\ic(\osG_x)[-\dim\usM^x]\cong \ic(\sG\sQ^\alpha_{w,x})$, and moreover, for any $\fF\in\aleph(\osG_x)$ the complex $\ic(\sQ^\alpha_w)\otimes\bp^*\bi_*\fF[-\dim\usM^x]$ is in fact a semisimple Hodge module (living in cohomological degree 0). Furthermore, the Verdier duality $\CalD$ takes $\ic(\sQ^\alpha_w)\otimes\bp^*\bi_*\fF[-\dim\usM^x]$ to $\ic(\sQ^\alpha_w)\otimes\bp^*\bi_*\CalD\fF[-\dim\usM^x]$ (see {\em loc. cit.} ~12.1.b). \subsubsection{} \label{convolution} It is well known that $\CW_f\backslash\CW/\CW_f=Y^+$. Let us denote the double coset of $x\in\CW$ by $\eta\in Y^+$. Then the image $\pr(\osG_x)$ in the affine Grassmannian $\CG$ lies in $\oCG_\eta$. Suppose $\eta+\alpha\in\BN[I]$. Comparing the above definition of $\sG\sQ^\alpha_{w,x}$ with {\em loc. cit.} ~12.2 we obtain the natural projection $\pr:\ \sG\sQ^\alpha_{w,x}\lra\CG\CQ^\alpha_\eta$. Recall the map $\bq:\ \CG\CQ^\alpha_\eta\lra\CQ^{\eta+\alpha}$ defined in {\em loc. cit.} ~11.2. For $\fF\in\mho(\oCG_x)$ we define the {\em convolution} $$\fF\star\IC(\sQ^\alpha_w):=\bq_*\pr_* (\ic(\sQ^\alpha_w)\otimes\bp^*\bi_*\fF[-\dim\usM^x])$$ By the decomposition theorem $\fF\star\IC(\sQ^\alpha_w)\in\mho(\CQ^{\eta+\alpha})$. By additivity, the convolution extends to the functor $$\star:\ \mho(\osG_x)\times\mho(\sQ^\alpha)\lra\mho(\CQ^{\eta+\alpha})$$ By the last sentence of ~\ref{degree 0} this functor commutes with Verdier duality: $$\CalD\fF\star\CalD\fH\iso\CalD(\fF\star\fH)$$ \subsection{} \label{label} Let $\alpha\in Y,\ w\in\CW_f$. Suppose a fine Schubert stratum $\osQ^\gamma_w\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta}$ lies in $\sQ^\alpha_y$ (see ~\ref{following}). The stalk of $\ic(\sQ^\alpha_y)$ at (any point $\phi$ in) the stratum $\osQ^\gamma_w\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta}$ will be denoted by $\ic(\sQ^\alpha_y)_{\beta,\gamma,\Gamma,w}$. Let $\xi\in Y^+$ be dominant enough so that $\xi+\gamma\in\BN[I]$ (hence $\xi+\alpha\in\BN[I]$). Then we may consider the stalk $\ic(\CQ^{\xi+\alpha}_y)_{\xi+\beta,\xi+\gamma,\Gamma,w}$ of $\ic(\CQ^{\xi+\alpha}_y)$ at (any point in) the stratum $\oQ^{\xi+\gamma}_w\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta} \subset\CQ^{\xi+\alpha}$. {\bf Theorem.} $\ic(\sQ^\alpha_y)_{\beta,\gamma,\Gamma,w}$ is isomorphic, up to a shift, to $\ic(\CQ^{\xi+\alpha}_y)_{\xi+\beta,\xi+\gamma,\Gamma,w}$. {\em Proof.} The argument is parallel to the one in ~\cite{fm} ~12.6, ~12.7. Let us consider another copy $\CG^1$ of affine Grassmannian: the ind-scheme representing the functor of isomorphism classes of pairs $(\CT,\tau')$ where $\CT$ is a $\bG$-torsor on $C$, and $\tau'$ is its section (trivialization) {\em defined off $1\in C$}. Restricting a trivialization to the formal neighbourhood of $\infty\in C$ we obtain the closed embedding $\bi^1:\ \CG^1\hookrightarrow\fM$. Furthermore, since $\tau'$ identifies the fiber of $\CT$ over $0\in C$ with $\bG$, the set of $\bB$-reductions of $\CT$ at 0 is canonically isomorphic to $\bX$. Choosing the reduction $\ft=e=\bB\in\bX$ we obtain the same named closed embedding $\bi^1:\ \CG^1\hookrightarrow\sM$. We consider the cartesian product $\CG^1\sQ^\alpha_{y,\eta}$ of $\oCG_\eta$ and $\sQ^\alpha_y$ over $\sM$. Thus we have the following cartesian diagram (cf. ~\ref{degree 0}): $$ \begin{CD} \CG^1\sQ^\alpha_{y,\eta} @>\bi^1>> \sQ^\alpha_y \\ @V{\bp}VV @V{\bp}VV \\ \oCG^1_\eta @>{\bi^1}>> \sM \end{CD} $$ The same argument as in ~\ref{degree 0} proves that up to a shift $\ic(\sQ^\alpha_y)\otimes\bp^*\bi^1_*\ic(\oCG^1_\eta)\cong \ic(\CG^1\sQ^\alpha_{y,\eta})$. Recall the convolution diagram $\CG\CQ^\alpha_\eta$ (see ~\cite{fm} ~11.1, ~12.2). The point $0\in C$ played a special role in the definition of this diagram. Let us replace 0 by 1 in the definition of $\CG\CQ^\alpha_\eta$. Let us call the result $\CG^1\CQ^\alpha_\eta$. One defines the map $\bq^1:\ \CG^1\CQ^\alpha_\eta\lra\CQ^{\eta+\alpha}$ as in {\em loc. cit.} ~11.2. Then the Proposition ~12.6 of {\em loc. cit.} states that the restriction of $\bq^1$ to $\dQ^{1,\eta+\alpha}$ is an isomorphism, where $\dQ^{1,\eta+\alpha}\subset\CQ^{\eta+\alpha}$ is an open subset formed by all the quasimaps defined at $1\in C$ (i.e. without defect at 1). A moment of reflection shows that there is no difference between the definitions of $\CG^1\sQ^\alpha_{w_0,\eta}$ and $\CG^1\CQ^\alpha_\eta$. Identifying them and imbedding $\CG^1\sQ^\alpha_{y,\eta}$ into $\CG^1\sQ^\alpha_{w_0,\eta}$ we obtain the map $\bq^1:\ \CG^1\sQ^\alpha_{y,\eta}\lra\CQ^{\eta+\alpha}$ which is a closed embedding over $\dQ^{1,\eta+\alpha}$. Clearly, the image of this embedding coincides with $\CQ^{1,\eta+\alpha}_y:=\CQ^{\eta+\alpha}_y \cap\dQ^{1,\eta+\alpha}$. Evidently, $\CQ^{1,\eta+\alpha}_y$ is open in $\CQ^{\eta+\alpha}_y$. Now let us consider the stalk of $\ic(\sQ^\alpha_y)$ at a point $\phi=(\CT,\ft,(\fL_\lambda)_{\lambda\in X^+})$ in the stratum $\osQ^\gamma_w\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta}$. Suppose that the isomorphism class of $\bG$-torsor $\CT$ equals $\eta\in Y^+$, i.e. $\CT\in\fM_\eta$. The stalk in question does not depend on a choice of $\CT\in\fM_\eta$ and the defect $D\in(C-0)^{\beta-\gamma}_\Gamma$. In particular, we may (and will) suppose that $\CT\in\bi^1(\CG^1_\eta)$, and $D\in(C-0-1)^{\beta-\gamma}$. We have seen that up to a shift $\ic(\sQ^\alpha_y)_\phi\otimes\bp^*\bi^1_*\ic(\oCG_\eta)_\CT\cong \ic(\CG^1\sQ^\alpha_{y,\eta})_\phi$. But the stalk of $\ic(\oCG^1_\eta)$ at any point $\CT\in\CG^1_\eta$ is isomorphic, up to a shift, to the trivial Tate Hodge structure $\BQ(0)$. We deduce that up to a shift $\ic(\sQ^\alpha_y)_\phi\cong\ic(\CG^1\sQ^\alpha_{y,\eta})_\phi$. On the other hand, the latter stalk is isomorphic to the stalk of $\ic(\CQ^{1,\eta+\alpha}_y)$ at the point $\bq^1(\phi)$. This point lies in $\oQ^{\eta+\gamma}_w\times(C-0)^{\beta-\gamma}_\Gamma\times0^{\alpha-\beta}$. Thus we have proved that up to a shift $\ic(\sQ^\alpha_y)_{\beta,\gamma,\Gamma,w}$ is isomorphic to $\ic(\CQ^{\eta+\alpha}_y)_{\eta+\beta,\eta+\gamma,\Gamma,w}$. The standard use of factorization shows that the latter stalk is independent of the shift $\eta\mapsto\xi$. This completes the proof of the Theorem. $\Box$ \subsubsection{Corollary} \label{pure Tate} The Hodge module $\ic(\sQ^\alpha_y)$ is pointwise pure Tate. $\Box$ \subsubsection{Corollary} Let $\beta,\gamma\in Y,\ w,y\in\CW_f$. Then $\sQ^\beta_y\supset\sQ^\gamma_w$ iff $(y,\beta)=:B\geq A:=(w,\gamma)$. {\em Proof.} $\sQ^\beta_y\supset\sQ^\gamma_w$ iff the stalk of $\ic(\sQ^\beta_y)$ at the generic point of $\sQ^\gamma_w$ does not vanish. Also, $\CQ^\beta_y\supset\CQ^\gamma_w$ iff the stalk of $\ic(\CQ^\beta_y)$ at the generic point of $\CQ^\gamma_w$ does not vanish. Now recall that the order on alcoves is translation invariant, and apply ~\ref{who}. $\Box$ \subsection{} \label{short} Before we finish the construction of convolution $\star:\ \mho(\sG)\times\mho(\sQ)\lra\mho(\sQ)$ we have to define the numerous functors between the categories $\aleph$ and $\mho$ of ~\ref{parallel} and ~\ref{similar}. First of all, we choose an additive equivalence $\fE^\PS_\sQ:\ \aleph(\sQ)\lra\aleph(\PS)$ sending $\ic(\sQ^\eta_w),\ \eta\in Y,\ w\in\CW_f$, to the collection $\ic(\CZ^\alpha_{w,\eta})\in\aleph(\ddZ^\alpha_\eta)$ (see ~\ref{mrak} and ~\cite{fm} ~9.3). Let $\fE_\PS^\sQ:\ \aleph(\PS)\lra\aleph(\sQ)$ be an inverse equivalence. We preserve the name $\fE^\PS_\sQ$ for an additive equivalence $\mho(\sQ)\lra\mho(\PS)$ sending $\IC(\sQ^\eta_w),\ \eta\in Y,\ w\in\CW_f$, to $\CL(w,\eta)\in\mho(\PS)$ and commuting with shifts. Let $\fE_\PS^\sQ:\ \mho(\PS)\lra\mho(\sQ)$ be an inverse equivalence. For $\alpha\in\BN[I]$ let $\aleph_{\gg0}(\CQ^\alpha)\subset\aleph(\CQ^\alpha)$ be the direct summand subcategory formed by the direct sums of irreducible Hodge modules $\ic(\CQ^\gamma_w),\ 0\ll\gamma\leq\alpha$, and their Tate twists. Let $\mho_{\gg0}(\CQ^\alpha)\subset\mho(\CQ^\alpha)$ be the direct summand subcategory formed by the direct sums of Goresky-MacPherson sheaves $\IC(\CQ^\gamma_w),\ 0\ll\gamma\leq\alpha$, and their shifts in derived category. Let $\fE_{\CQ^\alpha}^{\gg0}:\ \aleph(\CQ^\alpha)\lra\aleph_{\gg0}(\CQ^\alpha),\ \mho(\CQ^\alpha)\lra\mho_{\gg0}(\CQ^\alpha)$ denote the projections to the direct summands, and let $\fE^{\CQ^\alpha}_{\gg0}:\ \aleph_{\gg0}(\CQ^\alpha)\lra\aleph(\CQ^\alpha),\ \mho_{\gg0}(\CQ^\alpha)\lra\mho(\CQ^\alpha)$ denote the embeddings of direct summands. Let $\ss_\alpha$ denote the locally closed embedding $\ddZ^\alpha\hookrightarrow\CQ^\alpha$. One can easily see that for $\gamma\gg0$ we have $\ss_\alpha^*\ic(\CQ^\gamma_w)[-\dim\bX]=\ic(\CZ^\gamma_w)$. Hence the functor $\ss_\alpha^*[-\dim\bX]$ defines an equivalence $\fE^{\ddZ^\alpha}_{\CQ^\alpha}:\ \aleph_{\gg0}(\CQ^\alpha)\lra\aleph(\ddZ^\alpha),\ \mho_{\gg0}(\CQ^\alpha)\lra\mho(\ddZ^\alpha)$. Let $\fE_{\ddZ^\alpha}^{\CQ^\alpha}:\ \aleph(\ddZ^\alpha)\lra\aleph_{\gg0}(\CQ^\alpha),\ \mho(\ddZ^\alpha)\lra\mho_{\gg0}(\CQ^\alpha)$ be an inverse equivalence. Finally, for $\xi\in Y$ let $\xi_*:\ \aleph(\sQ)\lra\aleph(\sQ)$ (resp. $\mho(\sQ)\lra\mho(\sQ)$) be an additive equivalence taking $\ic(\sQ^\alpha_w)$ to $\ic(\sQ^{\xi+\alpha}_w)$ (resp. $\IC(\sQ^\alpha_w)$ to $\IC(\sQ^{\xi+\alpha}_w)$ and commuting with shifts). \subsection{} \label{long} Finally, we are in a position to define the convolution $\star:\ \mho(\sG)\times\mho(\sQ)\lra\mho(\sQ)$. To this end recall the convolution $\mho(\osG_x)\times\mho(\sQ^\alpha)\lra\mho(\CQ^{\eta+\alpha})$ defined in ~\ref{that}. To stress the dependence on $x\in\CW$ and $\alpha\in Y$ we will call this convolution $\star^\alpha_x$ (recall that $\eta\in Y^+= \CW_f\backslash\CW/\CW_f$ is just the double coset of $x$). Let $\fF\in\mho(\osG_x),\ \fH\in\mho(\sQ^\alpha)$. The collection $\fE^{\ddZ^\beta}_{\CQ^\beta}\circ\fE^{\gg0}_{\CQ^\beta} (\fF\star^{\beta-\eta}_x((\beta-\alpha-\eta)_*\fH)) \in\mho(\ddZ^\beta_{\alpha+\eta}),\ \beta\in\BN[I]$, defines an object in $\mho(\PS)$. Applying $\fE_\PS^\sQ$ to this object we obtain the desired convolution $\fF\star^\alpha_x\fH\in\mho(\sQ)$. Note that if $y\leq x,\ \gamma\leq\alpha$, and $\fF$ is actually supported on $\osG_y$, while $\fH$ is actually supported on $\sQ^\gamma$, then we have the canonical isomorphism $\fF\star^\gamma_y\fH\equiv\fF\star^\alpha_x\fH$. Hence the convolutions $\star^\alpha_x:\ \mho(\osG_x)\times\mho(\sQ^\alpha)\lra \mho(\sQ)$ extends to the convolution $\star:\ \mho(\sG)\times\mho(\sQ)\lra\mho(\sQ)$. As we have seen in ~\ref{that}, this convolution commutes with Verdier duality. Moreover, for $\fF,\fG\in\mho(\sG),\ \fH\in\mho(\sQ)$ one can easily construct a canonical isomorphism $\fF\star(\fG\star\fH)\cong(\fF\star\fG)\star\fH$. Finally, composing this convolution with the mutually inverse equivalences $\fE^\PS_\sQ:\ \mho(\sQ)\rightleftharpoons\mho(\PS)\ :\fE_\PS^\sQ$ we obtain the same named convolution functor $\star:\ \mho(\sG)\times\mho(\PS)\lra\mho(\PS)$. It is also associative with respect to convolution in $\mho(\sG)$, and commutes with Verdier duality. \subsection{} \label{main} Let $K(\sQ)$ (resp. $K(\PS)$) denote the $K$-group of the category $\mho(\sQ)$ (resp. $\mho(\PS)$): it is generated by the isomorphism classes of objects in $\mho(\sQ)$ with relations $[K_1\oplus K_2]=[K_1]+[K_2]$. Both $K(\sQ)$ and $K(\PS)$ are equipped with the structure of $\BZ[v,v^{-1}]$-modules: $-v$ acts as the shift $[-1]$. $K(\PS)$ is a free $\bzw$-module with a basis $[\CL(A)],\ A\in\fA$ (see ~\ref{t'ma}). Similarly, $K(\sQ)$ is a free $\bzw$-module with a basis $[\CL'(A)],\ A\in\fA$ where for $A=(w,\eta)$ we denote by $\CL'(A)$ the Goresky-MacPherson sheaf $\IC(\sQ^\eta_w)=\IC(\sQ_A)$ (see ~\ref{following}). We denote by $\hK(\PS)\supset K(\PS)$ (resp. $\hK(\sQ)\supset K(\sQ)$) the completed module formed by the possibly infinite sums $\sum_{A\in\fA}a_A[\CL(A)]$ (resp. $\sum_{A\in\fA}a_A[\CL'(A)]$) $a_A\in\bzw$ such that $\exists B\in\fA$ with the property $(a_A\ne0\ \Rightarrow\ A\leq B)$. We introduce another (topological) basis $\{[\CM'(A)],\ A\in\fA\}$ of $\hK(\sQ)$. It is characterized by the following property. The stalk at a fine Schubert stratum $\osQ_B\subset\sQ_B$ defines the functional $f_B:\ \hK(\sQ)\lra\bzw$. We require $f_A([\CM'(A)])=f_A([\CL'(A)])$ and $f_B([\CM'(A)])=0$ for $B\ne A$. In a similar way one defines a (topological) basis $\{[\CM(A)],\ A\in\fA\}$ of $\hK(\PS)$. The main goal of this section is a computation of the transition matrix between the bases $[\CL(A)]$ and $[\CM(A)]$. We will make a free use of the notations and results of the excellent exposition ~\cite{s}. In particular, the polynomials $\ol{q}_{B,A}$ (generic Kazhdan-Lusztig polynomials) are introduced in {\em loc. cit.}, Theorems ~6.1, ~6.4. We will prove the following {\bf Theorem.} a) $[\CL(A)]=\sum_{B\leq A}(-1)^{d(A,B)}\ol{q}_{B,A}[\CM(B)]$; b) $[\CL'(A)]=\sum_{B\leq A}(-1)^{d(A,B)}\ol{q}_{B,A}[\CM'(B)]$. The proof of the Theorem occupies the subsections ~\ref{start}--\ref{finish}. \subsection{} \label{start} For the time being let us define $\fq_{B,A}\in\bzw$ (resp. $\fq'_{B,A}\in\bzw$) by the requirement $[\CL(A)]=\sum_{B\leq A}\fq_{B,A}[\CM(B)]$ (resp. $[\CL'(A)]=\sum_{B\leq A}\fq'_{B,A}[\CM'(B)]$). Thus we have to prove $\fq_{B,A}=\fq'_{B,A}=(-1)^{d(A,B)}\ol{q}_{B,A}$. By the Corollary ~\ref{pure Tate} the Hodge module $\ic(\sQ_A)$ is pointwise pure Tate. This allows us to reconstruct the stalk $\ic(\sQ_A)_B$ of $\ic(\sQ_A)$ at $\osQ_B\subset\sQ_A$ from the polynomials $\fq'_{B,A}$. Namely, suppose that the stalk of $\ic(\sQ_A)$ at $\osQ_A$ is the Hodge structure $\BQ(0)$ living in cohomological degree $\delta_A$ (this is the definition of $\delta_A$). Then for an odd integer $k$ we have $H^{\delta_A+k}\ic(\sQ_A)_B=0$. For arbitrary integer $k$ the cohomology $H^{\delta_A+2k}\ic(\sQ_A)_B$ is a sum of a few copies of $\BQ(k)$. Finally, $\dim H^{\delta_A+k}\ic(\sQ_A)_B$ equals the coefficient of $(-v)^{-d(B,A)+k}$ in $\fq'_{B,A}$ (for arbitrary $k\in\BZ$). The same discussion applies to the stalks $\ic(\CZ_A)_B$ of $\ic(\CZ_A)$. But the stalks $\ic(\CZ_A)_B$ and $\ic(\sQ_A)_B$ are isomorphic. In effect, if $A=(w,\alpha),\ B=(y,\beta)$, and $\xi\in Y^+$ is big enough, then by the Theorem ~\ref{label} $\ic(\sQ_A)_B$ is isomorphic to the stalk of $\ic(\CQ^{\xi+\alpha}_w)$ at $\oQ^{\xi+\beta}_y$ (up to a shift; but we may disregard shifts: as we have just seen the cohomological degrees contain exactly the same information as the Hodge structures by the pointwise Tate purity). And the latter stalk is isomorphic to $\ic(\CZ_A)_B$ since $\ic(\CZ^{\xi+\alpha}_w)=\ss_{\xi+\alpha}^*\ic(\CQ^{\xi+\alpha}_w)[-\dim\bX]$ where $\ss_{\xi+\alpha}$ stands for the locally closed embedding of $\ddZ^{\xi+\alpha}$ into $\CQ^{\xi+\alpha}$. This implies that the functor $\fE_\PS^\sQ:\ \mho(\PS)\lra\mho(\sQ)$ inducing an isomorphism of $\hK(\PS)$ and $\hK(\sQ)$ sending $[\CL(A)]$ to $[\CL'(A)]$ {\em also sends $[\CM(A)]$ to $[\CM'(A)]$}. In other words, $\fE_\PS^\sQ$ preserves stalks. In particular, $\fq_{B,A}=\fq'_{B,A}$ for all $B\leq A$. From now on we will identify $\hK(\PS)$ and $\hK(\sQ)$, we will write $[\CL(A)]$ for $[\CL'(A)]$, and $[\CM(A)]$ for $[\CM'(A)]$. \subsubsection{Remark} As we have seen, the stalk of $\ic(\CQ^\alpha_w)$ at a fine Schubert stratum $\oQ^\beta_y$ can be reconstructed in terms of polynomials $\fq_{B,A}$ for $A=(w,\alpha),B=(y,\beta)$. Since we know the simple stalks $\ic^0_{\{\{\gamma\}\}}$, the factorization property allows us to reconstruct all the other stalks of $\ic(\CQ^\alpha_w)$. \subsection{} \label{kato} We know the simple stalks $\ic^0_{\{\{\gamma\}\}}$ by the Theorems ~\ref{simple}, ~\ref{Hodge}. They coincide with the stalks of $\ic(\sQ^\alpha)=\ic(\sQ^\alpha_{w_0})$. Thus we deduce the following formula: $$[\CL(w_0,\alpha)]=\sum_{w\in\CW_f}^{\beta\in\BN[I]} (-v)^{l(w)-l(w_0)}\CK^\beta(v^2)[\CM(w,\alpha-\beta)]$$ Comparing with Kato's Theorem (see ~\cite{kat} or ~\cite{s} ~6.3) we see that for any $B\in\fA$ and $A=(w_0,\alpha)$ we have $\fq_{B,A}=(-1)^{d(A,B)}\ol{q}_{B,A}$. \subsection{} \label{last} The $K$-group $K(\sG):=K(\mho(\sG))$ is also a $\bzw$-module in a natural way ($-v$ acts as a shift $[-1]$). The convolution $\star:\ \mho(\sG)\times \mho(\sG)\lra\mho(\sG)$ makes $K(\sG)$ into a $\bzw$-algebra. This is the affine Hecke algebra $\CH$ as described e.g. in ~\cite{s} ~\S2. The basis $\{H_x,\ x\in\CW\}$ corresponds to perverse shriek extensions of constant sheaves on orbits $\sG_x$. The involution $H\mapsto\ol{H}$ of {\em loc. cit.} is induced by the Verdier duality $\CalD:\ \mho(\sG)\lra\mho(\sG)$. Let $s_i\in\CW,\ i\in I\sqcup0$ be a simple reflection (see ~\ref{fuck}). Then $\osG_{s_i}\subset\sG$ is a projective line, to be denoted by $\osG_i$. The class $[\IC(\osG_i)]\in\CH$ equals $H_{s_i}-v^{-1}$ (we identify $H_e$ with $1\in\CH$). Following W.Soergel, we will denote $[\IC(\osG_i)]$ by $\tC_i=\tC_{s_i}$. The convolution $\star:\ \mho(\sG)\times\mho(\sQ)\lra\mho(\sQ)$ gives rise to the structure of $\CH$-module on $\hK(\sQ)$. {\bf Proposition.} Let $i\in I\sqcup0,\ A\in\fA$. If $s_iA>A$ then $\tC_i[\CM(A)]=[\CM(s_iA)]-v^{-1}[\CM(A)]$, and if $s_iA<A$ then $\tC_i[\CM(A)]=[\CM(s_iA)]-v[\CM(A)]$. {\em Proof.} As everything is invariant with respect to translations, we may (and will) assume that $A=(w,\alpha)$ for $\alpha\in\BN[I]$. Consider the convolution diagram $\sG\sQ^\alpha_{w,s_i}\stackrel{\bi} {\hookrightarrow}\sQ^\alpha_w$. We will denote by $\sG\dsQ^\alpha_{w,s_i}$ the intersection of $\sG\sQ^\alpha_{w,s_i}$ with $\dsQ^\alpha_w\subset \sQ^\alpha_w$ (see ~\ref{following}). To compute $\tC_i[\CM(A)]$ we have to compute the stalks of $(\bq\circ\pr)_!\IC(\sG\dsQ^\alpha_{w,s_i})$. Recall that $\bq\circ\pr$ maps $\sG\sQ^\alpha_{w,s_i}$ to $\CQ^{\eta+\alpha}$ where $\eta\in Y^+=\CW_f\backslash\CW/\CW_f$ is the double coset of $s_i$. Note that the double coset of $s_0$ equals $\beta_0$ (see ~\ref{fuck}) --- the coroot dual to the highest root. And the double coset of $s_i,\ i\in I$, equals $0\in Y^+$. So in any case we have the map $\bq\circ\pr:\ \sG\dsQ^\alpha_{w,s_i}\lra\CQ^{\beta_0+\alpha}$. According to ~\ref{label} the sheaf $\IC(\sG\dsQ^\alpha_{w,s_i})$ is constant along the fibers of $\bq\circ\pr$. So the Proposition follows from the following Claim. \subsubsection{Claim} \label{claim} Let $\phi\in\dQ^\beta_y\subset\CQ^{\beta_0+\alpha}$, and let $B=(y,\beta)\in\fA$. Then a) if $B\ne A,s_iA$, then $(\bq\circ\pr)^{-1}(\phi)=\emptyset$; b) if $B$ equals $A$ or $s_iA$, and $s_iA>A$, then $(\bq\circ\pr)^{-1}(\phi)$ consists of exactly one point; c) if $B$ equals $A$ or $s_iA$, and $s_iA<A$, then $(\bq\circ\pr)^{-1}(\phi)$ is isomorphic to the affine line $\BA^1$. \subsubsection{Remark} If $i\in I$ then the double coset of $s_i$ equals 0, and the above map $\bq$ is just an isomorphism. In this case the convolution boils down to the usual convolution on $\bX$. The order relation between $s_iA$ and $A$ also reduces to the usual Bruhat order on $\CW_f$ (see ~\ref{s0} and ~\ref{leq}). So ~\ref{claim} follows in this case from the standard facts about the geometry of $\bX$. \subsubsection{} In the general case note first that the (isomorphism type of the) fiber $(\bq\circ\pr)^{-1}(\phi)$ is independent of $\phi\in\dQ^\beta_y$. The proof is absolutely similar to the one in ~\cite{fm} ~12.6.b). It makes use of the {\em local convolution diagram} (see {\em loc. cit.} ~11.5). We spare the reader the bulk of notation needed to carry this notion over to our situation. So we may (and will) choose $\phi$ lying in the fine Schubert stratum $\oQ^0_y\times(C-0)^\beta_\Gamma\times0^{\alpha+\beta_0-\beta} \subset\dQ^\beta_y$. Moreover, we will assume $\phi=y\times D\times 0^{\alpha+\beta_0-\beta}$ where $D\in(C-0)^\beta_\Gamma$, and $y\in\bX_y=\oQ^0_y$ is the $\bH$-fixed point. We may (and will) view the fiber $(\bq\circ\pr)^{-1}(\phi)$ as a subset of $\osG_i$. As such, it is isomorphic to the intersection of $\osG_i$ with the orbit $T^y_{y,\beta-\alpha-\beta_0}$ of $\bN_-^y((z))$ in $\sG$ passing through the $\bH$-fixed point $(y,\beta-\alpha-\beta_0)\in\Omega$ (we use the identification $\CW_f\times Y\iso\Omega$ of ~\ref{fuck} along with the fact that the set of $\bH$-fixed points of $\sG$ is naturally identified with $\Omega$), cf. the proof of ~12.6.b) in {\em loc. cit.} Here as always $\bN_-^y\subset\bG$ denotes the subgroup $\dot{y}\bN_-\dot{y}{}^{-1}$ for a representative $\dot{y}\in N(\bH)$ of $y$. Now the Claim follows by the standard Bruhat-Tits theory. This completes the proof of the Proposition. $\Box$ \subsection{} \label{finish} Comparing the above Proposition with ~\cite{l1} or ~\cite{s} ~4.1 we see that the $\CH$-module $\hK(\sQ)$ is isomorphic to Lusztig's completed {\em periodic Hecke module} $\hP$ (see ~\cite{s} ~\S6), and this isomorphism $\varphi$ takes $[\CM(A)]\in\hK(\sQ)$ to a basic element $A\in\hP$. Moreover, comparing ~\ref{kato} with the Theorem 6.4.2 (due to Kato) of {\em loc. cit.} we see that for $A=(w_0,\alpha)$ we have $\varphi([\CL(A)])=\utP_A$ (notations of {\em loc. cit.} ~\S6). The Verdier duality $\CalD:\ \mho(\sQ)\lra\mho(\sQ)$ gives rise to an involution $N\mapsto\CalD N$ of $\hK(\sQ)$. As Verdier duality commutes with the convolution, this involution is compatible with the involution $H\mapsto\ol{H}$ of $\CH$, that is, $\CalD(H(N))=\ol{H}(\CalD N)$. Certainly, $\CalD[\CL(A)]=[\CL(A)]$ for any $A\in\fA$. Recall the Lusztig's involution $P\mapsto\ol{P}$ of the periodic module $\hP$ (see e.g. {\em loc. cit.} ~4.3). It is also compatible with the involution on the Hecke algebra, i.e. $\ol{H(P)}=\ol{H}(\ol{P})$. It also preserves the elements $\utP_A,\ A\in\fA$ (see {\em loc. cit.} ~6.4). Since the elements $\utP_A,\ A=(w_0,\alpha)\in\fA$ clearly generate the $\CH$-module $\hP$, we conclude that $\varphi(\CalD N)=\ol{\varphi(N)}$ for any $N\in\hK(\sQ)$. In other words, we may identify $\CH$-modules $\hP$ and $\hK(\sQ)$ with their involutions and standard bases. Now the Theorem ~6.4.2 of {\em loc. cit.} carries over to $\hK(\sQ)$ and claims that for any $A\in\fA$ the element $\varphi^{-1}(\utP_A)=\sum_B(-1)^{d(A,B)}\ol{q}_{B,A}[\CM(B)]$ is the only $\CalD$-invariant element of $\hK(\sQ)$ lying in $[\CM(A)]+\sum_{B\leq A}v^{-1}\BZ[v^{-1}][\CM(B)]$. But $[\CL(A)]$ is also $\CalD$-invariant and it also lies in $[\CM(A)]+\sum_{B\leq A}v^{-1}\BZ[v^{-1}][\CM(B)]$ by the definition of Goresky-MacPherson extension. This completes the proof of the Theorem ~\ref{main}. $\Box$ \section{Tilting Conjectures} \subsection{} Recall the notations of ~\cite{fm} ~12.12, ~12.13. So let $\eta\in Y^+,\ \beta\in Y,\ \eta+\beta\in\BN[I]$. For a sheaf $\CF\in\CP(\oCG_\eta,\bI)$ we consider the complex $\bc^\beta_\CQ(\CF)= \bq_*(\IC(\fQ^\beta)\otimes\bp^*\CF)[-\dim\ufM^\eta]$ on $\CQ^{\eta+\beta}$. If $\eta+\beta\not\in\BN[I]$ we set $\CQ^{\eta+\beta}=\emptyset$, and $\bc^\beta_\CQ(\CF)=0$. We will define a natural action of $\fg^L$ on $\oplus_{\beta\in Y}H^\bullet(\CQ^{\eta+\beta},\bc^\beta_\CQ(\CF))= \oplus_{\beta\in Y}H^\bullet(\CG\CQ^\beta_\eta,\IC(\fQ^\beta)\otimes \bp^*\CF)$ closely following ~\ref{smirnoff}--\ref{h}. \subsubsection{} For $\beta\in Y,\ \alpha\in\BN[I]$ we have the usual {\em twisting map} $\sigma_{\beta,\alpha}:\ \CG\CQ^\beta_\eta\times C^\alpha \lra\CG\CQ^{\alpha+\beta}_\eta$ (resp. $\fQ^\beta\times C^\alpha\lra \fQ^{\alpha+\beta}$). We define $\del_\alpha\CG\CQ^{\alpha+\beta}_\eta$ (resp. $\del_\alpha\fQ^{\alpha+\beta}$) as the image of this map. We will construct the {\em stabilization map} $\fv_{\alpha,\beta}:\ \CA_\alpha\lra \Ext_{\CG\CQ^{\alpha+\beta}_\eta}^{|\alpha|} (\IC(\del_\alpha\fQ^{\alpha+\beta})\otimes\bp^*\CF, \IC(\fQ^{\alpha+\beta})\otimes\bp^*\CF)$. We will follow the construction of ~\ref{bezr}. First, tensoring with $\Id\in\Ext^0_{\CG\CQ^\beta_\eta}(\IC(\fQ^\beta)\otimes\bp^*\CF, \IC(\fQ^\beta)\otimes\bp^*\CF)$, we obtain the map from $\CA_\alpha=\Ext^{|\alpha|}_{C^\alpha}(\IC(C^\alpha),\fF_\alpha)$ to $\Ext^{|\alpha|}_{\CG\CQ^\beta_\eta\times C^\alpha} (\IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\IC(C^\alpha), \IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\fF_\alpha)$ (notations of ~\ref{bezr}). As in ~\ref{bezr}, there is a canonical map $c$ from $\IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\fF_\alpha$ to $\sigma_{\beta,\alpha}^! (\IC(\fQ^{\alpha+\beta})\otimes\bp^*\CF)$ extending the factorization isomorphism from an open part $\oGQ^\beta_\eta\times\BA^\alpha\subset \CG\CQ^\beta_\eta\times C^\alpha$. Thus $c$ induces the map from $\Ext^{|\alpha|}_{\CG\CQ^\beta_\eta\times C^\alpha} (\IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\IC(C^\alpha), \IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\fF_\alpha)$ to $\Ext^{|\alpha|}_{\CG\CQ^\beta_\eta\times C^\alpha} (\IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\IC(C^\alpha), \sigma_{\beta,\alpha}^! (\IC(\fQ^{\alpha+\beta})\otimes\bp^*\CF))$. The latter space equals $\Ext^{|\alpha|}_{\CG\CQ^{\alpha+\beta}_\eta} ((\sigma_{\beta,\alpha})_* (\IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes\IC(C^\alpha)), \IC(\fQ^{\alpha+\beta})\otimes\bp^*\CF)= \Ext_{\CG\CQ^{\alpha+\beta}_\eta}^{|\alpha|} (\IC(\del_\alpha\fQ^{\alpha+\beta})\otimes\bp^*\CF, \IC(\fQ^{\alpha+\beta})\otimes\bp^*\CF)$. Finally, we define $\fv_{\alpha,\beta}$ as the composition of above maps. \subsubsection{} We construct the {\em costabilization map} $$\fy_{\beta,\alpha}:\ H^\bullet(\CG\CQ_\eta^\beta, \IC(\fQ^\beta)\otimes\bp^*\CF)\lra H^{\bullet-|\alpha|}(\del_\alpha\CG\CQ_\eta^{\alpha+\beta}, \IC(\del_\alpha\fQ^{\alpha+\beta})\otimes\bp^*\CF)$$ To this end we note that exactly as in ~\ref{normal}, we have $\IC(\del_\alpha\fQ^{\alpha+\beta})=(\sigma_{\beta,\alpha})_* \IC(\fQ^\beta\times C^\alpha)=(\sigma_{\beta,\alpha})_* (\IC(\fQ^\beta)\boxtimes\ul\BC[|\alpha|])$. Let $[C^\alpha]\in H^{-|\alpha|}(C^\alpha,\ul\BC[|\alpha|])$ denote the fundamental class of $C^\alpha$. Now, for $h\in H^\bullet(\CG\CQ_\eta^\beta, \IC(\fQ^\beta)\otimes\bp^*\CF)$ we define $\fy_{\beta,\alpha}(h)$ as $h\otimes[C^\alpha]\in H^{\bullet-|\alpha|}(\CG\CQ_\eta^\beta\times C^\alpha, \IC(\fQ^\beta)\otimes\bp^*\CF\boxtimes \ul\BC[|\alpha|])=H^{\bullet-|\alpha|}(\del_\alpha\CG\CQ_\eta^{\alpha+\beta}, (\sigma_{\beta,\alpha})_*(\IC(\fQ^\beta)\otimes\bp^*\CF \boxtimes\ul\BC[|\alpha|]))= H^{\bullet-|\alpha|}(\del_\alpha\CG\CQ_\eta^{\alpha+\beta}, \IC(\del_\alpha\fQ^{\alpha+\beta})\otimes\bp^*\CF)$. \subsection{Definition} \label{Roma} Let $a\in\CA_\alpha,\ h\in H^\bullet(\CG\CQ_\eta^\beta, \IC(\fQ^\beta)\otimes\bp^*\CF)$. We define the action $a(h)\in H^\bullet(\CG\CQ_\eta^{\alpha+\beta}, \IC(\fQ^{\alpha+\beta})\otimes\bp^*\CF)$ as the action of $\fv_{\alpha,\beta}(a)$ on the global cohomology applied to $\fy_{\beta,\alpha}(h)$. Let us stress that the action of $\CA_\alpha$ {\em preserves cohomological degrees.} \subsubsection{} For $a\in\CA_\alpha,\ b\in\CA_\beta,\ h\in H^\bullet(\CG\CQ_\eta^\beta,\IC(\fQ^\beta)\otimes\bp^*\CF)$ we have $a(b(h))=a\cdot b(h)$. The proof is entirely similar to the proof of associativity of the multiplication in $\CA$. \subsection{} \label{FF} For $\beta\in Y$ the graded space $H^\bullet(\CG\CQ_\eta^\beta,\IC(\fQ^\beta)\otimes\bp^*\CF)$ is up to a shift Poincar\'e dual to $H^\bullet(\CG\CQ_\eta^\beta,\IC(\fQ^\beta)\otimes\bp^*\CF)$ (cf. ~\cite{fm} ~12.1b). We define the map $$f_i:\ H^\bullet(\CG\CQ_\eta^{\beta+i}, \IC(\fQ^{\beta+i})\otimes\bp^*\CF)\lra H^\bullet(\CG\CQ_\eta^\beta,\IC(\fQ^\beta)\otimes\bp^*\CF)$$ as the dual of the map $$e_i:\ H^\bullet(\CG\CQ_\eta^\beta,\IC(\fQ^\beta)\otimes\bp^*\CF)\lra H^\bullet(\CG\CQ_\eta^{\beta+i},\IC(\fQ^{\beta+i})\otimes\bp^*\CF)$$ (see ~\ref{choice}). It follows from ~\ref{Serre} that the maps $f_i,\ i\in I$, satisfy the Serre relations of $\fn_-^L$. \subsubsection{} \label{ff} We safely leave to the reader an elementary construction of the operators $f_i,e_i$ absolutely similar to the one in ~\ref{f},~\ref{e}. \subsection{} \label{hh} For $i\in I$ we define the endomorphism $h_i$ of $H^\bullet(\CG\CQ_\eta^\beta,\IC(\fQ^\beta)\otimes\bp^*\CF)$ as the scalar multiplication by $\langle\beta+2\crho,i'\rangle$ where $i'\in X$ is the simple root. Exactly as in ~\ref{fe},~\ref{h}, one proves that the operators $e_i,f_i,h_i,\ i\in I$, generate the action of the Langlands dual Lie algebra $\fg^L$ on $\oplus_{\beta\in Y}H^\bullet(\CG\CQ^\beta_\eta,\IC(\fQ^\beta)\otimes\bp^*\CF)= \oplus_{\beta\in Y}H^\bullet(\CQ^{\eta+\beta},\bc^\beta_\CQ(\CF))$. In the particular case $\eta=0,\ \CF$ is the irreducible skyscraper sheaf, we have $\bc^\beta_\CQ(\CF)=\IC(\CQ^\beta)$, and we return to the $\fg^L$-action on $\oplus_{\beta\in\BN[I]}H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ defined in ~\ref{!}. We will formulate a few conjectures concerning this action, following ~\cite{fk} ~\S6. \subsection{} \label{roman} Let $\CN^L\subset\fg^L$ be the nilpotent cone. The Lie algebra $\fg^L$ acts on the cohomology $H^{\dim\bX}_{\fn_-^L}(\CN^L,\CO)$ of the structure sheaf of $\CN^L$ with supports in $\fn_-^L$. The character of this module is well known to be $\frac{|\CW_f|e^{2\check\rho}} {\prod_{\ctheta\in{\check\CR}{}^+}(1-e^\ctheta)^2}$. Thus it coincides with the character of $\fg^L$-module $\oplus_{\beta\in\BN[I]}H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$. {\bf Conjecture.} $\fg^L$-modules $\oplus_{\beta\in\BN[I]}H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ and $H^{\dim\bX}_{\fn_-^L}(\CN^L,\CO)$ are isomorphic. As explained in ~\cite{fk} ~\S6 the module $H^{\dim\bX}_{\fn_-^L}(\CN^L,\CO)$ is {\em tilting}, and to prove the Conjecture it suffices to check that $\oplus_{\beta\in\BN[I]}H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ is tilting as well. To this end it is enough to check that the action of the algebra $\CA=U(\fn_+^L)$ on $\oplus_{\beta\in\BN[I]}H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ is free. \subsection{} \label{roma} For $\eta\in Y^+$ the irreducible $\fg^L$-module with the highest weight $\eta$ is denoted by $W_\eta$. The Theorem ~13.2 of ~\cite{fm} implies that the character of $\fg^L$-module $\oplus_{\beta\in Y}H^\bullet(\CQ^\beta,\bc^\beta_\CQ(\IC(\oCG_\eta)))$ equals char($W_\eta)\times\frac{|\CW_f|e^{2\check\rho}} {\prod_{\ctheta\in{\check\CR}{}^+}(1-e^\ctheta)^2}$. {\bf Conjecture.} There is an isomorphism of $\fg^L$-modules $\oplus_{\beta\in Y}H^\bullet(\CQ^\beta,\bc^\beta_\CQ(\IC(\oCG_\eta)))$ and $W_\eta\otimes H^{\dim\bX}_{\fn_-^L}(\CN^L,\CO)$. This Conjecture is again equivalent to the statement that the $\fg^L$-module $\oplus_{\beta\in Y}H^\bullet(\CQ^\beta,\bc^\beta_\CQ(\IC(\oCG_\eta)))$ is tilting. To check this it suffices to prove that the action of $\CA=U(\fn_+^L)$ on $\oplus_{\beta\in Y}H^\bullet(\CQ^\beta,\bc^\beta_\CQ(\IC(\oCG_\eta)))$ is free. \subsection{} \label{romka} Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. According to ~\cite{mv}, $H^\bullet(\CG,\CF)$ is equipped with a canonical $\fg^L$-action. While the action of ${\fh}^L\subset\fg^L$ is defined pretty explicitly in geometric terms in ~\cite{mv}, the actions of $\fn^L_-,\fn^L_+\subset\fg^L$ were not constructed as explicitly so far. Motivated by the Conjecture ~\ref{roma} above we propose the following conjectural constructions of the $\fn^L_\pm$-actions on $H^\bullet(\CG,\CF)$. \subsubsection{} Let $\CQ^{\eta+\beta}=\sqcup_{0\leq\gamma\leq\eta+\beta}\dQ^\gamma$ be the partition of $\CQ^{\eta+\beta}$ according to the defect at $0\in C$. Let $\CQ^{\eta+\beta}=\cup_{0\leq\gamma\leq\eta+\beta}\CQ^\gamma$ be the corresponding filtration by closed subspaces. In particular, we have $\bX=\dQ^0=\CQ^0\subset\CQ^{\eta+\beta}$. The perverse sheaf $\bc_\CQ^\beta(\CF)$ is isomorphic to a direct sum of a few copies of the sheaves $\IC(\CQ^\gamma)$. More precisely, the multiplicity of $\IC(\CQ^\gamma)$ in $\bc_\CQ^\beta(\CF)$ equals $H^{2|\beta-\gamma|}_c(T_{\gamma-\beta},\CF)$ (see the Theorem ~13.2 of ~\cite{fm}). In particular, $H^{2|\beta|}_c(T_{-\beta},\CF)\otimes\IC(\CQ^0)= H^{2|\beta|}_c(T_{-\beta},\CF)\otimes\ul\BC_\bX[n]$ is canonically a direct summand of $\bc_\CQ^\beta(\CF)$. Hence $H^{2|\beta|}_c(T_{-\beta},\CF)= H^{2|\beta|}_c(T_{-\beta},\CF)\otimes H^{-n}(\bX,\ul\BC[n])$ is canonically a direct summand of $H^\bullet(\CQ^{\eta+\beta},\bc_\CQ^\beta(\CF))= H^\bullet(\CG\CQ^\beta_\eta,\IC(\fQ^\beta)\otimes\bp^*\CF)$. We conjecture that $\oplus_{\beta\in Y} H^{2|\beta|}_c(T_{-\beta},\CF)=\oplus_{\beta\in Y} H^{2|\beta|}_c(T_{-\beta},\CF)\otimes H^{-n}(\bX,\ul\BC[n])$ is an $\CA^{opp}=U(\fn^L_-)$-submodule of $\oplus_{\beta\in Y} H^\bullet(\CG\CQ^\beta_\eta,\IC(\fQ^\beta)\otimes\bp^*\CF)$ (with respect to the action defined in ~\ref{Roma}), and the resulting action of $U(\fn^L_-)$ on $\oplus_{\beta\in Y}H^{2|\beta|}_c(T_{-\beta},\CF)=H^\bullet(\CG,\CF)$ coincides with the action of ~\cite{mv}. \subsubsection{} Similarly, $H^{2|\beta|}_c(T_{-\beta},\CF)= H^{2|\beta|}_c(T_{-\beta},\CF)\otimes H^{n}(\bX,\ul\BC[n])$ is canonically a direct summand of $H^\bullet(\CQ^{\eta+\beta},\bc_\CQ^\beta(\CF))= H^\bullet(\CG\CQ^\beta_\eta,\IC(\fQ^\beta)\otimes\bp^*\CF)$. We conjecture that the kernel of the natural projection $\oplus_{\beta\in Y} H^\bullet(\CG\CQ^\beta_\eta,\IC(\fQ^\beta)\otimes\bp^*\CF)\lra \oplus_{\beta\in Y} H^{2|\beta|}_c(T_{-\beta},\CF)=\oplus_{\beta\in Y} H^{2|\beta|}_c(T_{-\beta},\CF)\otimes H^{n}(\bX,\ul\BC[n])$ is invariant under the action of $\CA=U(\fn^L_+)$ defined in ~\ref{FF}, and the induced action of $U(\fn^L_+)$ on $\oplus_{\beta\in Y}H^{2|\beta|}_c(T_{-\beta},\CF)=H^\bullet(\CG,\CF)$ coincides with the action of ~\cite{mv}. \subsection{} \label{denis} Recall the moduli scheme $\fM$ of $\bG$-torsors on $\BP^1$ equipped with the formal trivialization at $\infty\in\BP^1$. Its stratification according to the isomorphism classes of $\bG$-torsors $\fM=\sqcup_{\eta\in Y^+}\fM_\eta$ was described in ~\cite{fm} ~10.6. Let $\ol\fM_\eta$ denote the closure of the stratum $\fM_\eta$, and let $\IC(\ol\fM_\eta)$ denote the corresponding $\IC$-sheaf. According to the Decomposition Theorem, for $\alpha\in Y$, the direct image $\bp_*\IC(\fQ^\xi)$ is a semisimple complex on $\fM$, isomorphic to a direct sum of various $\IC(\ol\fM_\eta)$ with shifts and multiplicities. We have $\bigoplus_{\xi\in Y}\bp_*\IC(\fQ^\xi)= \bigoplus_{\eta\in Y^+}K^\bullet_\eta\otimes\IC(\ol\fM_\eta)$ for (infinite dimensional in general) graded multiplicities spaces $K_\eta$. One can introduce the stalkwise action of $\fg^L$ on $\bigoplus_{\xi\in Y}\bp_*\IC(\fQ^\xi)$ entirely similar to ~\ref{hh}. In other words, one obtains the action of $\fg^L$ on the multiplicities spaces $K^\bullet_\eta$ for any $\eta\in Y^+$. For instance, $K^\bullet_0=\oplus_{\beta\in\BN[I]}H^\bullet(\CQ^\beta,\IC(\CQ^\beta))$ by definition. The action of $\fg^L$ on $K^\bullet_0$ was described in ~\ref{roman}. We will extend the conjecture ~\ref{roman} to the case of arbitrary $\eta\in Y^+$. Recall the Lusztig's quantum groups $\fu\subset\fU$, and the representations' category $\fC$, introduced in ~\cite{fm} ~1.3. The highest weights of the regular block $\fC^0$ of the category of $\fU$-modules are contained in the $\CW$-orbit $\CW\cdot0\subset X$ (the strong linkage principle). Let $m_\eta\in\CW$ be the shortest representative of the double coset $\CW_f\eta\CW_f$ in $\CW$. Let $T(m_\eta\cdot0)\in\fC^0$ be the indecomposable {\em tilting} $\fU$-module with the highest weight $m_\eta\cdot0$ (see ~\cite{s}). For example, $T(m_0\cdot0)$ is the trivial module. Finally, recall the notion of semiinfinite cohomology $H^{\frac{\infty}{2}+\bullet}(\fu,?)$ introduced in ~\cite{a}. It is known that for $T\in\fC^0$ the semiinfinite cohomology $H^{\frac{\infty}{2}+\bullet}(\fu,T)$ carries a natural structure of $\fn_-^L$-integrable $\fg^L$-module. {\bf Conjecture.} For $\eta\in Y^+$ there is an isomorphism of graded $\fg^L$-modules $K^\bullet_\eta\simeq H^{\frac{\infty}{2}+\bullet}(\fu,T(m_\eta\cdot0))$. Under this isomorphism, the weight $\xi\in Y$ part of the RHS corresponds to the multiplicity $_{(\xi)}K^\bullet_\eta\subset K^\bullet_\eta$ of $\IC(\ol\fM_\eta)$ in $\bp_*\IC(\fQ^\xi)$. {\bf Corollary of the Conjecture.} Suppose $\eta-2\check{\rho}\in Y^+$. Then $K^\bullet_\eta$ is concentrated in degree 0, and is isomorphic to the irreducible $\bG^L$-module $W_{\eta-2\check\rho}$. {\em Remark.} S.Arkhipov has recently proved that $H^{\frac{\infty}{2}+\bullet}(\fu,T(0))\simeq H^{\dim\bX}_{\fn^L_-}(\CN^L,\CO)$. Together with the results of ~\cite{fkm} this establishes the $\eta=0$ case of the above conjecture. \section{Appendix. Kontsevich resolution of Quasimaps' space} \subsection{} For $\alpha=\sum_{i\in I}a_ii\in\BN[I]$ let $\CQ^\alpha_K= \ol{M}_{0,0}(\BP^1\times\bX),(1,\alpha))$ be the Kontsevich space of stable maps from the genus zero curves without marked points to $\BP^1\times\bX$ of bidegree $(1,\alpha)$ (see ~\cite{ko}). In this Appendix we construct a regular birational map $\pi:\ \CQ^\alpha_K\lra\CQ^\alpha$. To this end recall that $\bX$ is a closed subvariety in $\prod_{i\in I}\BP(V_{\omega_i})$ given by Pl\"ucker relations. Drinfeld's space $\CQ^\alpha$ is a closed subvariety in $\prod_{i\in I}\BP H^0(C,V_{\omega_i}\otimes\CO(a_i))$ also given by Pl\"ucker relations. So to construct the desired map it suffices to solve the following problem. For a vector space $V$ one has 2 compactifications of the space $M^a$ of algebraic maps $f:\ C\lra\BP(V)$ of degree $a\in H_2(\BP(V),\BZ)$. Kontsevich compactification $M^a_K=\overline{M}_{0,0}(C\times\BP(V),(1,a))$ is the space of all stable maps from the genus 0 curves into the product $C\times\BP(V)$ of bidegree $(1,a)$; the embedding of $M^a$ into $M^a_K$ takes $f$ to its graph $\Gamma_f$. Drinfeld compactification $M^a_D$ is the space of invertible subsheaves of degree $-a$ in $V\otimes\CO_C$. Twisting by $\CO(a)$ we identify $M^a_D$ with $\BP H^0(C,V\otimes\CO(a))$. The embedding of $M^a$ into $M^a_D$ takes $f$ to the line subbundle $\CL_f:=p_*q^*\CO_{\BP(V)}(-1)\subset V\otimes\CO_C$ where $C\stackrel{p}{\longleftarrow}\Gamma_f\stackrel{q}{\lra}\BP(V)$ are the natural projections. We want to prove that the identification $M^a_K\supset M^a=M^a\subset M^a_D$ extends to the regular map $\pi:\ M^a_K\lra M^a_D$. This problem was addressed by A.Givental in ~\cite{g}, ~``Main Lemma''. Unfortunately, the validity of his proof is controversial. In the rest of the Appendix we prove that Id: $M^a\iso M^a$ extends to the regular map $\pi:\ M^a_K\lra M^a_D$. \subsection{} \label{kuznec} One of the most important properties of the Drinfeld space $M^a_D$ is the following. {\em Lemma.} Let $S$ be a scheme and $\CL\subset V\otimes\CO_{C\times S}$ be an invertible subsheaf in the trivial vector bundle $V\otimes\CO_{C\times S}$ on $C\times S$ of degree $-a$ over $S$ (i.e.\ for any $s\in S$ the restriction $\CL_s$ of $\CL$ to the fiber $C\times s\subset C\times S$ is equal to $\CO(-a)$). If the open subset $U=\{(x,s)\in C\times S\ |\ \CL_{(x,s)}\to V\otimes\CO_{x,s} \text{ is embedding}\}$ contains generic points of all fibers of $C\times S$ over $S$ then the map $f:S\to M^a_D$ sending each point $s\in S$ to the subsheaf $\CL_s\subset V\otimes\CO_{C\times s}\cong C\otimes\CO_C$ is regular. {\em Proof.} Let $p:C\times S\to S$ be the natural projection and $L=p_*(\CL\otimes\CO_C(a))$. It is evident that for any $s\in S$ we have $$ \begin{array}{rcrcl} H^0(p^{-1}(s),\CL\otimes\CO_C(a))&=&H^0(C,\CO_C)&=&\BC,\\ H^{>0}(p^{-1}(s),\CL\otimes\CO_C(a))&=&H^{>0}(C,\CO_C)&=&0 \end{array} $$ and the map $$ H^0(p^{-1}(s),\CL\otimes\CO_C(a))\to H^0(p^{-1}(s),V\otimes\CO_C(a)) $$ is injection, hence $L$ is a line subbundle in $p_*(V\otimes\CO_C(a))=H^0(C,V\otimes\CO_C(a))\otimes\CO_S$. The map $f$ is just the regular map associated with the subbundle $L$. $\Box$ \subsection{} Recall the statement of Givental's ``Main Lemma''. {\bf ``Main Lemma''} a) The map Id: $M^a\iso M^a$ extends to the regular map $\pi:M^a_K\to M^a_D$; b) Let $(\vphi:\CC\to C\times \BP(V))\in M^a_K$ be a stable map and let $\vphi':\CC\to C$, $\vphi'':\CC\to \BP(V)$ denote the induced maps. Let $\CC_0$ denote the irreducible component of $\CC$ such that $\vphi':\CC_0\to C$ is dominant, and let $\CC_1,\dots,\CC_m$ be the connected components of $\CC\setminus\CC_0$. Finally, let $x_1=\vphi'(\CC_1),\dots,x_m=\vphi'(\CC_m)$. Then $\pi(\CC)$ is the subsheaf with normalisation equal to $\vphi'_*(\vphi''_{|\CC_0})^*\CO_{\BP(V)}(-1)$ and with defect at points $x_1$, \dots $x_m$ of degree $\vphi''_*[\CC_1]$, \dots, $\vphi''_*[\CC_m]$ respectively. {\em Proof.} a) Consider the space $\tmk=\overline M_{0,1}(C\times \BP(V);(1,a))$ of stable maps with one marked point. There are natural maps ${\sf f}:\tmk\to M^a_K$ forgeting the marked point, and $\ev:\tmk\to C\times \BP(V)$ evaluating at the marked point. Let $p:\tmk\to C\times M^a_K$ denote the product of the map $p_1\circ\ev:\tmk @>\ev>> C\times \BP(V) @>p_1>> C$ and of the map ${\sf f}$. Let $q:\tmk\to \BP(V)$ denote the map $p_2\circ\ev:\tmk @>\ev>> C\times \BP(V) @>p_2>> \BP(V)$. The map $p$ is an isomorphism over the open subspace $U\subset C\times M^a_K$ of pairs $(x,\CC)$ such that $x\in\vphi'(\CC_0-\CC^{\operatorname{sing}})$. Consider the sheaf $\CB=p_*q^*\CO_{\BP(V)}(-1)\subset p_*q^*(V\otimes\CO_{\BP(V)})= V\otimes\CO_{C\times M^a_K}$. Its reflexive hull $\CB^{**}$ is a reflexive rank 1 sheaf on the smooth orbifold $C\times M^a_K$, hence $\CB^{**}$ is an invertible sheaf. Note that since $\CB$ is invertible over $U$, we have $\CB^{**}_{|U}=\CB_{|U}$. Since $U$ contains $C\times M^a$, the restriction of $\CB^{**}$ to the generic fiber of $C\times M^a_K$ over $M^a_K$ is $\CO(-a)$, hence the restriction of $\CB^{**}$ to any fiber is $\CO(-a)$ as well. Over the set $U$, the embedding $\CB^{**}_{|U}\to V\otimes\CO_U$ is an embedding of vector bundles (because over $U$ the map $p$ is an isomorphism). On the other hand, $U$ contains generic points of all fibers of $C\times M^a_K$ over $M^a_K$, therefore we can apply the Lemma ~\ref{kuznec} and the first part of the ``Main Lemma'' follows. b) Let $s=(\vphi:\CC\to C\times \BP(V))\in M^a_K$ be a stable map, and let $L\subset V\otimes\CO_C$ denote a subsheaf $\pi(s)\in M^a_D$; let $L'=\vphi'_*({\vphi''_{|\CC_0}}^*)\CO_{\BP(V)}(-1)$. The Lemma ~\ref{kuznec} implies $L=\CB^{**}_s$. We want to prove that the normalization $\widetilde L$ of $L$ equals $L'$. Denote $C-\{x_1,\dots,x_m\}$ by $C^0$. Then $U\bigcap(C\times s)=C^0\times s$, hence $ L_{|C^0\times s}={\CB_s}_{|C^0\times s}= (\vphi'_*{\vphi''}^*\CO_{\BP(V)}(-1))_{|C^0\times s}= (\vphi'_*(\vphi''_{|\CC_0})^*\CO_{\BP(V)}(-1))_{|C^0\times s}= L'_{|C^0\times s}, $ i.e. $L$ and $L'$ coincide on the open subset of $C\times s$. Since $L'$ is a line subbundle in $V\otimes\CO_{C\times s}$ this means that $L$ is a subsheaf in $L'$ and the quotient $L'/L$ is concentrated at the points $x_1,\dots,x_m$. Therefore $\widetilde L=L'$ and the defect of $L$ is concentrated at the points $x_1,\dots,x_m$. Note that in the case $m=1$ the ``Main Lemma'' follows. In the general case we proceed as follows. Let $c_k=\vphi''_*[\CC_k]$, $(k=1,\dots,m)$. Let ${\bf M}_K^k=\{(\psi:{\sf C}\to C\times \BP(V))\}\subset M^a_K$ denote the subspace of all stable maps such that a curve ${\sf C}$ has two irreducible components ${\sf C}={\sf C}_0\cup{\sf C}_1$, $\psi_*[{\sf C}_0]=(1,a-c_k)$, $\psi_*[{\sf C}_1]=(0,c_k)$ and $\psi'({\sf C}_1)=x_k$. Then we have $s\in\overline{\bf M}_K^1\bigcap\overline{\bf M}_K^2\bigcap\dots \bigcap\overline{\bf M}_K^m.$ Let ${\bf M}_D^k\subset M^a_D$ denote the subspace of all invertible subsheaves with defect of degree $c_k$ concentrated at the point $x_k$. According to the case $m=1$ we have $\pi({\bf M}_K^k)\subset{\bf M}_D^k$, hence $\pi(s)\in\overline{\bf M}_D^1\bigcap\overline{\bf M}_D^2\bigcap\dots \bigcap\overline{\bf M}_D^m.$ Since the degree of defect at a given point increases under specialization, the degrees of the defect of $\pi(s)$ at the points $x_k$ are greater or equal than $c_k$, hence they are equal to $c_k$, and the ``Main Lemma'' follows. $\Box$

9711

alg-geom/9711020

en

https://arxiv.org/abs/alg-geom/9711020

alg-geom/9711020

Kalle Karu

Kalle Karu

Semistable reduction in characteristic 0 for families of surfaces and three-folds

AMS-LaTeX, 7 pages

Let F:X->B be a morphism of varieties in characteristic zero. The problem of semistable reduction of F was stated as a problem in the combinatorics of polyhedral complexes by Abramovich and Karu (alg-geom/9707012). In this paper we solve the combinatorial problem when the relative dimension of F is not bigger than 3.

alg-geom

\section{Introduction} In \cite{ak} the semistable reduction of a morphism $F:X\rightarrow} \newcommand{\dar}{\downarrow B$ was stated as a problem in the combinatorics of polyhedral complexes. In this paper we solve it in the case when the relative dimension of $F$ is no bigger than three. First we recall the setup of the problem from \cite{ak}. The ground field $k$ will be algebraically closed of characteristic zero. \begin{dfn} A flat morphism $F:X\rightarrow} \newcommand{\dar}{\downarrow B$ of nonsingular projective varieties is semistable if in local analytic coordinates $x_1,\ldots,x_n$ at $x\in X$ and $t_1,\ldots,t_m$ at $b\in B$ the morphism $F$ is given by \[ t_i = \prod_{j=l_{i-1}+1}^{l_i} x_j \] where $0=l_0<l_1<\ldots<l_m\leq n$. \end{dfn} The conjecture of semistable reduction states that \begin{conj}\label{conj-ssr} Let $F: X\rightarrow} \newcommand{\dar}{\downarrow B$ be a surjective morphism with geometrically integral generic fiber. There exist an alteration (proper surjective generically finite morphism) $B'\rightarrow} \newcommand{\dar}{\downarrow B$ and a modification (proper biratonal morphism) $X'\rightarrow} \newcommand{\dar}{\downarrow X\times_B B'$ such that $X'\rightarrow} \newcommand{\dar}{\downarrow B'$ is semistable. \end{conj} Conjecture~\ref{conj-ssr} was proved in \cite{te} (main theorem of Chapter~2) in case when $B$ is a curve. A weak version of the conjecture was proved in \cite{ak} for arbitrary $X$ and $B$. In both cases the proof proceeds by reducing $F$ to a morphism of toroidal embeddings, stating the problem in terms of the associated polyhedral complexes, and solving the combinatorial problem. \subsection{Polyhedral complexes} We consider (rational, conical) polyhedral complexes $\Delta=(|\Delta|,\{\sigma\},\{N_\sigma\})$ consisting of a collection of lattices $N_\sigma\cong{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}^n$ and rational full cones $\sigma\subset N_\sigma\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ with a vertex. The cones $\sigma$ are glued together to form the space $|\Delta|$ so that the usual axioms of polyhedral complexes hold: \begin{enumerate} \item If $\sigma\in\Delta$ is a cone, then every face $\sigma'$ of $\sigma$ is also in $\Delta$, and $N_{\sigma'}=N_\sigma|_{{\operatorname{Span}}(\sigma')}$. \item The intersection of two cones $\sigma_1\cap\sigma_2$ is a face of both of them, $N_{\sigma_1\cap\sigma_2} = N_{\sigma_1}|_{{\operatorname{Span}}(\sigma_1\cap\sigma_2)} = N_{\sigma_2}|_{{\operatorname{Span}}(\sigma_1\cap\sigma_2)}$. \end{enumerate} A morphism $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ of polyhedral complexes $\Delta_X=(|\Delta_X|,\{\sigma\},\{N_\sigma\})$ and $\Delta_B=(|\Delta_B|,\{\tau\},\{N_\tau\})$ is a compatible collection of linear maps $f_\sigma: (\sigma,N_\sigma)\rightarrow} \newcommand{\dar}{\downarrow(\tau,N_\tau)$; i.e. if $\sigma'$ is a face of $\sigma$ then $f_{\sigma'}$ is the restriction of $f_\sigma$. We will only consider morphisms $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ such that $f_\sigma^{-1}(0)\cap\sigma=\{0\}$ for all $\sigma\in\Delta_X$. \begin{dfn} A surjective morphism $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ such that $f^{-1}(0)=\{0\}$ is semistable if \begin{enumerate} \item $\Delta_X$ and $\Delta_B$ are nonsingular. \item For any cone $\sigma\in\Delta_X$, we have $f(\sigma)\in\Delta_B$ and $f(N_\sigma)=N_{f(\sigma)}.$ \end{enumerate} We say that $f$ is weakly semistable if it satisfies the two properties except that $\Delta_X$ may be singular. \end{dfn} The following two operations are allowed on $\Delta_X$ and $\Delta_B$: \begin{enumerate} \item Projective subdivisions $\Delta_X'$ of $\Delta_X$ and $\Delta_B'$ of $\Delta_B$ such that $f$ induces a morphism $f':\Delta_X'\rightarrow} \newcommand{\dar}{\downarrow\Delta_B'$; \item Lattice alterations: let $\Delta_X'=(|\Delta_X|,\{\sigma\},\{N_\sigma'\}), \Delta_B'=(|\Delta_B|,\{\tau\},\{N_\tau'\})$, for some compatible collection of sublattices $N_\tau'\subset N_\tau$, $N_\sigma'=f^{-1}(N'_\tau)\cap N_\sigma$, and let $f':\Delta_X'\rightarrow} \newcommand{\dar}{\downarrow\Delta_B'$ be the morphism induced by $f$. \end{enumerate} \begin{conj}\label{main-conj} Given a surjective morphism $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$, such that $f^{-1}(0)=\{0\}$, there exists a projective subdivision $f':\Delta_X'\rightarrow} \newcommand{\dar}{\downarrow\Delta_B'$ followed by a lattice alteration $f'':\Delta_X''\rightarrow} \newcommand{\dar}{\downarrow\Delta_B''$ so that $f''$ is semistable. \[ \begin{array}{lclcl} \Delta_{X''} & \rightarrow} \newcommand{\dar}{\downarrow & \Delta_{X'} & \rightarrow} \newcommand{\dar}{\downarrow & \Delta_{X} \\ \downarrow f'' & & \downarrow f' & & \downarrow f \\ \Delta_{B''} & \rightarrow} \newcommand{\dar}{\downarrow & \Delta_{B'} & \rightarrow} \newcommand{\dar}{\downarrow & \Delta_{B} \end{array} \] \end{conj} The importance of Conjecture~\ref{main-conj} lies in the fact that it implies Conjecture~\ref{conj-ssr} (Proposition~8.5 in \cite{ak}). In the case when $\dim(\Delta_B)=1$, Conjecture~\ref{main-conj} was proved in \cite{te} (main theorem of Chapter~3). In \cite{ak} (Theorem~0.3) the conjecture was proved with semistable replaced by weakly semistable. The main result of this paper is \begin{th}\label{main-thm} Conjecture~\ref{main-conj} is true if $f$ has relative dimension $\leq 3$. Hence, Conjecture~\ref{conj-ssr} is true if $F$ has relative dimension $\leq 3$. \end{th} The relative dimension of a linear map $f:\sigma\rightarrow} \newcommand{\dar}{\downarrow\tau$ of cones $\sigma, \tau$ is $\dim(\sigma)-\dim(f(\sigma))$. The relative dimension of $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ is by definition the maximum of the relative dimensions of $f_\sigma:\sigma\rightarrow} \newcommand{\dar}{\downarrow\tau$ over all $\sigma\in\Delta_X$. If $F:X\rightarrow} \newcommand{\dar}{\downarrow B$ is a morphism of toroidal embeddings of relative dimension $d$, then the associated morphism of polyhedral complexes $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ has relative dimension $\leq d$ because in local models the relative dimension of $F$ is no bigger than the rank of the kernel of $f: N_\sigma\rightarrow} \newcommand{\dar}{\downarrow N_\tau$. Thus, the second statement of the theorem follows from the first. \subsection{Notation} We will use notations from \cite{te} and \cite{fu}. For a cone $\sigma\in N\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ we write $\sigma={\langle}v_1,\ldots,v_n{\rangle}$ if $v_1,\ldots,v_n$ lie on the 1-dimensional edges of $\sigma$ and generate it. If $v_i$ are the first lattice points along the edges we call them primitive points of $\sigma$. For a simplicial cone $\sigma$ with primitive points $v_1,\ldots,v_n$, the multiplicity of $\sigma$ is \[ m(\sigma,N_\sigma) = [N_\sigma:{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}} v_1\oplus\ldots\oplus{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}} v_n]. \] A polyhedral complex $\Delta$ is nonsingular if and only if $m(\sigma,N_\sigma)=1$ for all $\sigma\in\Delta$. To compute the multiplicity of $\sigma$ we can count the representatives $w\in N_\sigma$ of classes of $N_\sigma/{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}} v_1\oplus\ldots\oplus{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}} v_n$ in the form \[ w=\sum_{i}\alpha_i v_i, \qquad 0\leq\alpha_i<1.\] Such points $w$ were called Waterman points of $\sigma$ in \cite{te}. Also notice that since the multiplicity of a face of $\sigma$ is no bigger than the multiplicity of $\sigma$, to compute the multiplicity of $\Delta$ it suffices to consider maximal cones only. If $\Delta_X$ and $\Delta_B$ are simplicial, we we say that $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ is simplicial if $f(\sigma)\in\Delta_B$ for all $\sigma\in\Delta_X$. Assume that $f$ is simplicial. Let $u_1,\ldots,u_n$ be the primitive points of $\Delta_B$, and $m_1,\ldots,m_n$ positive integers. By taking the $(m_1,\ldots,m_n)$ sublattice at $u_1,\dots,u_n$ we mean the lattice alteration $N_\tau'={\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}[m_{i_1} u_{i_1},\ldots, m_{i_l}u_{i_l}]$ where $\tau\in\Delta_B$ has primitive points $u_{i_1},\dots,u_{i_l}$. For cones $\sigma_1,\sigma_2\in\Delta$ we write $\sigma_1\leq\sigma_2$ if $\sigma_1$ is a face of $\sigma_2$. A subdivision $\Delta'$ of $\Delta$ is called projective if there exists a homogeneous piecewise linear function $\psi:|\Delta|\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ taking rational values on the lattice points (a good function for short) such that the maximal cones of $\Delta'$ are exactly the maximal pieces in which $\psi$ is linear. \subsection{Acknowledgment} The suggestion to write up the proof of semistable reduction for low relative dimensions came from Dan Abramovich. \section{Joins} For cones $\sigma_1,\sigma_2\in{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}^N$ lying in complementary planes: $\mbox{Span}(\sigma_1)\cap\mbox{Span}(\sigma_2)=\{0\}$, the join of $\sigma_1$ and $\sigma_2$ is $\sigma_1*\sigma_2=\sigma_1+\sigma_2$. Let $\sigma$ be a simplicial cone $\sigma=\sigma_1*\ldots*\sigma_n$. If $\sigma_i'$ is a subdivision of $\sigma_i$ for all $i=1,\ldots,n$, we define the join \[ \sigma' = \sigma_1' * \ldots * \sigma_n' \] as the set of cones $\rho = \rho_1+\ldots+\rho_n$, where $\rho_i\in\sigma_i'$. Let $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ be a simplicial map of simplicial complexes. For $u_i$ a primitive point of $\Delta_B$, $i=1,\ldots,n$, let $\Delta_{X,i}=f^{-1}({\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+ u_i)$ be the simplicial subcomplex of $\Delta_X$. If $\Delta_{X,i}'$ is a subdivision of $\Delta_{X,i}$ for $i=1,\ldots,n$, we can define the join \[ \Delta_X' = \Delta'_{X,1} *\ldots*\Delta'_{X,n} \] by taking joins inside all cones $\sigma\in\Delta_X$. This is well defined by the assumption that $f^{-1}(0)=\{0\}$. \begin{lem} If $\Delta_{X,i}'$ are projective subdivisions of $\Delta_{X,i}$ then the join $\Delta_{X}'$ is a projective subdivision of $\Delta_{X}$. \end{lem} {\bf Proof.} Let $\psi_i$ be good functions for $|\Delta_{X,i}'|$. Extend $\psi_i$ linearly to the entire $|\Delta_X'|$ by setting $\psi_i(|\Delta_{X,j}'|)=0$ for $j\neq i$. Clearly, $\psi = \sum_i \psi_i$ is a good function defining the subdivision $\Delta_{X}'$. \qed\\ Consider $f|_{\Delta_{X,i}}: \Delta_{X,i}\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+ u_i$. By the main theorem of Chapter~2 in \cite{te} there exist a subdivision $\Delta_{X,i}'$ of $\Delta_{X,i}$ and an $m_i\in{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$ such that after taking the $m_i$-sublattice at $u_i$ we have $f'|_{\Delta_{X,i}'}$ semistable. Now let $\Delta_X'$ be the join of $\Delta_{X,i}'$, and take the $(m_1,\ldots,m_n)$-sublattice at $(u_1,\ldots,u_n)$. Then $f':\Delta_X'\rightarrow} \newcommand{\dar}{\downarrow\Delta_B'$ is a simplicial map and $f'|_{\Delta_{X,i}'}$ is semistable. We can also see that the multiplicity of $\Delta_X'$ is not bigger than the multiplicity of $\Delta_X$. Let $\sigma\in\Delta_X$ have primitive points $v_i$ and let $\sigma'\subset\sigma$ be a maximal cone in the subdivision with primitive points $v_i'$. The multiplicity of $\sigma'$ is the number of Waterman points $w'\in N_\sigma'$ \[ w'=\sum_{i} \alpha_{i} v_i', \qquad 0\leq\alpha_{i}<1.\] We show that the set of Waterman points of $\sigma'$ can be mapped injectively into the set of Waterman points of $\sigma$, hence the multiplicity of $\sigma'$ is not bigger than the multiplicity of $\sigma$. Write \[ w'=\sum_{i}(\beta_i+b_i) v_i, \qquad 0\leq\beta_{i}<1, \qquad b_i\in{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}_+.\] Then $w=\sum_{i}\beta_i v_i \in N_\sigma$ is a Waterman point of $\sigma$. If different $w_1', w_2'$ give the same $w$, then $w_1'-w_2' \in N_\sigma'\cap{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}\{v_i\} = {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}\{v_i'\}$, hence $w_1'-w_2'=0$. \section{Modified barycentric subdivisions} Let $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ be a simplicial morphism of simplicial complexes. Consider the barycentric subdivision $BS(\Delta_B)$ of $\Delta_B$. The 1-dimensional cones of $BS(\Delta_B)$ are ${\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+\hat{\tau}$ where $\hat{\tau}=\sum u_i$ is the barycenter of a cone $\tau\in\Delta_B$ with primitive points $u_1,\ldots,u_m$. A cone $\tau'\in BS(\Delta_B)$ is spanned by $\hat{\tau}_1,\ldots,\hat{\tau}_k$, where $\tau_1\leq\tau_2\leq\ldots\leq\tau_k$ is a chain of cones in $\Delta_B$. In general, $f$ does not induce a morphism $BS(\Delta_X)\rightarrow} \newcommand{\dar}{\downarrow BS(\Delta_B)$. For that we need to modify the barycenters $\hat{\sigma}$ of cones $\sigma\in\Delta_X$. \begin{dfn} The data of {\bf modified barycenters} consists of \begin{enumerate} \item A subset of cones $\tilde{\Delta}_X\subset\Delta_X$. \item For each $\sigma\in\tilde{\Delta}_X$ a point $b_\sigma\in \mbox{int}(\sigma)\cap N_\sigma$ such that $f(b_\sigma)\in{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+\hat{\tau}$ for some $\tau\in\Delta_B$. \end{enumerate} \end{dfn} Recall that for any total order $\prec$ on the set of cones in $\Delta_X$ refining the partial order $\leq$, the barycentric subdivision $BS(\Delta_X)$ can be realized as a sequence of star subdivisions at the barycenters $\hat{\sigma}$ of $\sigma\in\Delta_X$ in the descending order $\prec$. \begin{dfn} Given modified barycenters $(\tilde{\Delta}_X,\{b_\sigma\})$ and a total order $\prec$ on $\Delta_X$ refining the partial order $\leq$, the {\bf modified barycentric subdivision} $MBS_{\tilde{\Delta}_X,\{b_\sigma\},\prec}(\Delta_X)$ is the sequence of star subdivisions at $b_\sigma$ for $\sigma\in\tilde{\Delta}_X$ in the descending order $\prec$. \end{dfn} To simplify notations, we will write $MBS(\Delta_X)$ instead of $MBS_{\tilde{\Delta}_X,\{b_\sigma\},\prec}(\Delta_X)$. By definition, $MBS(\Delta_X)$ is a projective simplicial subdivision of $\Delta_X$. As in the case of the ordinary barycentric subdivision, the cones of $MBS(\Delta_X)$ can be characterized by chains of cones in $\Delta_X$. We may assume that the 1-dimensional cones of $\Delta_X$ are all in $\tilde{\Delta}_X$. For a cone $\sigma\in\Delta_X$ let $\tilde{\sigma}$ be the maximal face of $\sigma$ (w.r.t. $\prec$) in $\tilde{\Delta}_X$. Given a chain of cones $\sigma_1\leq\ldots\leq\sigma_k$ in $\Delta_X$, the cone spanned by $b_{\tilde{\sigma}_1}, \ldots, b_{\tilde{\sigma}_k}$ is a subcone of $\sigma_k$. Let $C(\Delta_X)$ be the set of all such cones corresponding to chains $\sigma_1\leq\ldots\leq\sigma_k$ in $\Delta_X$. \begin{prp} $C(\Delta_X)=MBS(\Delta_X)$. \end{prp} {\bf Proof.} Let $BS(\Delta_X)$ be the ordinary barycentric subdivision of $\Delta_X$. Both $C(\Delta_X)$ and $MBS(\Delta_X)$ are obtained from $BS(\Delta_X)$ by moving the barycenters $\hat{\sigma}$ (and everything attached to them) to the new position $b_{\tilde{\sigma}}$ for all $\sigma\in\Delta_X$ in the descending order $\prec$. \qed\\ \begin{cor}\label{cor-simpl} If $f(\tilde{\sigma})=f(\sigma)$ for all $\sigma\in\Delta_X$ then $f$ induces a simplicial map $f':MBS(\Delta_X)\rightarrow} \newcommand{\dar}{\downarrow BS(\Delta_B)$. \end{cor} {\bf Proof.} Let $\sigma'\in MBS(\Delta_X)$ correspond to a chain $\sigma_1\leq\ldots\leq\sigma_k$. Then we have a chain of cones $f(\sigma_1)\leq \ldots\leq f(\sigma_k)$ in $\Delta_B$. The assumption that $f(\tilde{\sigma}_i)=f(\sigma_i)$ implies that $f(b_{\tilde{\sigma}_i})\in{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+ \widehat{f}(\sigma_i)$, hence the cone ${\langle}b_{\tilde{\sigma}_1},\ldots,b_{\tilde{\sigma}_k}{\rangle}$ maps onto the cone ${\langle}\widehat{f}(\sigma_1), \ldots, \widehat{f}(\sigma_k){\rangle}\in BS(\Delta_B)$. \qed\\ The hypothesis of the corollary is satisfied, for example, if for any $\sigma\in\Delta_X$ with $f(\sigma)=\tau\in\Delta_B$ and for any face $\sigma_1\leq\sigma$ such that $\sigma_1\in\tilde{\Delta}_X$, $f(\sigma_1)\neq\tau$, there exists $\sigma_2\in\tilde{\Delta}_X$ such that $\sigma_1\leq\sigma_2\leq\sigma$ and $f(\sigma_2)=\tau$: \[ \begin{array}{ccccc} \sigma_1 & \leq & \sigma_2 & \leq & \sigma \\ \downarrow & & \downarrow & & \downarrow \\ \tau_1 & \leq & \tau & = & \tau \\ \end{array} \] Indeed, $\tilde{\sigma}\neq\sigma_1$ because $\sigma_1\prec\sigma_2$. \subsection{Example} \label{example} Assume that $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ is a simplicial map of simplicial complexes taking primitive points of $\Delta_X$ to primitive points of $\Delta_B$ (e.g. $\Delta_X$ is simplicial and $f$ is weakly semistable). Then for a cone $\sigma\in\Delta_X$ such that $f:\sigma\stackrel{\simeq}{\rightarrow} \newcommand{\dar}{\downarrow}\tau$, we have $f(\hat{\sigma})=\hat{\tau}$. Let $\tilde{\Delta}_X=\bar{\Delta}_X = \{\sigma\in\Delta_X: f|_\sigma \mbox{is injective}\}$, $b_\sigma=\hat{\sigma}$. In this case $\tilde{\sigma}$ is the maximal face of $\sigma$ (w.r.t. $\prec$) such that $f|_\sigma$ is injective. Clearly, the hypothesis of the lemma is satisfied, and we have a simplicial map $f':MBS(\Delta_X)\rightarrow} \newcommand{\dar}{\downarrow BS(\Delta_B)$. Next we compute the multiplicity of $MBS(\Delta_X)$. Let $\sigma\in\Delta_X$ have primitive points $v_1,\ldots,v_n$, and let $\sigma'\subset\sigma$ be a maximal cone in the subdivision, corresponding to the chain \[ \langle v_1\rangle \leq \langle v_1,v_2\rangle\leq\ldots\leq \langle v_1,\ldots,v_n\rangle. \] Since $\tilde{\rho}\subset\rho$ for any $\rho$, the primitive points of $\sigma'$ can be written as \[ \begin{array}{lllllll} v_1' &=& a_{11} v_1 & & & & \\ v_2' &=& a_{21} v_1 & + & a_{22} v_2 & & \\ & \cdots & & & & & \\ v_n' &=& a_{n 1} v_1 & + & \ldots & + & a_{n n} v_n \end{array} \] for some $0\leq a_{i j}$. The multiplicity of $\sigma'$ is $a_{1 1}\cdot a_{2 2} \cdots a_{n n}$ times the multiplicity of $\sigma$. In case when $b_\rho$ are barycenters $\hat{\rho}$, all $a_{i j}\leq 1$, hence the multiplicity of $\sigma'$ is not bigger than the multiplicity of $\sigma$. \section{Reducing the multiplicity of $\Delta_X$} Let $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ be weakly semistable and $\Delta_X$ simplicial (i.e. $\Delta_B$ is nonsingular, $\Delta_X$ is simplicial, and $f$ is a simplicial map taking primitive points of $\Delta_X$ to primitive points of $\Delta_B$). Notice that if $\bar{\Delta}_X$ is as in Example~\ref{example}, then $\bar{\Delta}_X$ is nonsingular, and $f(\hat{\sigma})=\widehat{f}(\sigma)$ for any $\sigma\in\bar{\Delta}_X$. A singular simplicial cone $\sigma\in\Delta_X$ with primitive points $v_1,\ldots,v_n$ contains a Waterman point $w\in N_\sigma$, \[ w=\sum_{i} \alpha_{i} v_i, \qquad 0\leq\alpha_{i}<1, \qquad \sum_i\alpha_i>0.\] The star subdivision of $\sigma$ at $w$ has multiplicity strictly less than the multiplicity of $\sigma$. We will show in this section that if every singular cone of $\Delta_X$ contains a Waterman point $w$ mapping to a barycenter of $\Delta_B$, then there exists a modified barycentric subdivision $MBS(\Delta_X)$ having multiplicity strictly less than the multiplicity of $\Delta_X$, such that $f$ induces a simplicial map $f':MBS(\Delta_X)\rightarrow} \newcommand{\dar}{\downarrow BS(\Delta_B)$. For every singular cone $\sigma\in\Delta_X$ choose a point $w_\sigma$ as follows. By assumption, there exists a Waterman point $w\in\sigma$ mapping to a barycenter of $\Delta_B$: $f(w)=\hat{\tau}$. Write $f(\sigma)=\tau*\tau_0$ and choose a face $\sigma_0\leq\sigma$ such that $f:\sigma_0\stackrel{\simeq}{\rightarrow} \newcommand{\dar}{\downarrow}\tau_0$. Set $w_\sigma=w+\hat{\sigma}_0$; then \[ f(w_\sigma) = f(w)+f(\hat{\sigma}_0)=\hat{\tau}+\hat{\tau}_0 = \widehat{f}(\sigma)\] Having chosen the set $\{w_\sigma\}$, we may remove some of the points $w_\sigma$ if necessary so that every simplex $\rho\in\Delta_X$ contains at most one $w_\sigma$ in its interior. With $\bar{\Delta}_X$ as in Example~\ref{example}, let $\tilde{\Delta}_X =\bar{\Delta}_X \cup \{\rho\in\Delta_X| w_\sigma\in\mbox{int $(\rho)$ for some singular $\sigma$}\}$, $b_\rho = \hat{\rho}$ if $\rho\in\bar{\Delta}_X$, and $b_\rho = w_\sigma$ if $w_\sigma\in\mbox{int}(\rho)$. By construction, $(\tilde{\Delta}_X,\{b_\rho\})$ satisfies the hypothesis of Corollary~\ref{cor-simpl}, hence $f$ induces a simplicial map $f':MBS(\Delta_X)\rightarrow} \newcommand{\dar}{\downarrow BS(\Delta_B)$. Before we compute the multiplicity of $MBS(\Delta_X)$, we choose a particular total order $\prec$ on $\Delta_X$. Extend $\leq$ on $\Delta_X$ to a partial order $\prec_0$ by declaring that $\sigma_1\prec_0\sigma_2$ for all (nonsingular) $\sigma_1\in\bar{\Delta}_X$ and singular $\sigma_2\in\Delta_X$. Let $\prec$ be an extension of $\prec_0$ to a total order on $\Delta_X$. With such $\prec$, if $\sigma\in\Delta_X$ is singular, then $b_{\tilde{\sigma}}$ is one of the points $w_\rho$. As in Example~\ref{example}, the multiplicity of $MBS(\Delta_X)$ is not bigger than the multiplicity of $\Delta_X$. If $\sigma\in\Delta_X$ is singular we show by induction on the dimension of $\sigma$ that the multiplicity of $MBS(\sigma)$ is strictly less than the multiplicity of $\sigma$. Let $v_1,\ldots,v_N$ be the primitive points of $\sigma$, and consider the cone $\sigma'=\langle b_{\tilde{\sigma}},v_1,\ldots,v_{N-1}\rangle$ in the star subdivision of $\sigma$ at $b_{\tilde{\sigma}}=\sum_i a_i v_i$. To show that every maximal cone of $MBS(\sigma)$ contained in $\sigma'$ has multiplicity less than the multiplicity of $\sigma$, we have three cases: \begin{enumerate} \item If $a_N$ = 0, then $\sigma'$ is degenerate. \item If $0<a_N<1$, then the multiplicity of ${\langle}b_{\tilde{\sigma}},v_1,\ldots,v_{N-1}{\rangle}$ is less than the multiplicity of $\sigma$, and since all $b_\rho=\sum_i c_i v_i$ have coefficients $0\leq c_i \leq 1$, further subdivisions at $b_\rho$ do not increase the multiplicity of ${\langle}b_{\tilde{\sigma}},v_1,\ldots,v_{N-1}{\rangle}$. \item If $a_N=1$, then $b_{\tilde{\sigma}}=w+\hat{\rho}$ for some $\rho\leq\sigma$ and $w\in{\langle}v_1,\ldots,v_{N-1}{\rangle}$ a Waterman point. Hence ${\langle}v_1,\ldots,v_{N-1}{\rangle}$ is singular and, by induction, every maximal cone in $MBS({\langle}v_1,\ldots,v_{N-1}{\rangle})$ has multiplicity less than the multiplicity of ${\langle}v_1,\ldots,v_{N-1}{\rangle}$. Then also every maximal cone in ${\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+ b_{\tilde{\sigma}}*MBS({\langle}v_1,\ldots,v_{N-1}{\rangle})$ has multiplicity less than the multiplicity of ${\langle}b_{\tilde{\sigma}},v_1,\ldots,v_{N-1}{\rangle}$. \end{enumerate} \section{Families of surfaces and 3-folds.} {\bf Proof of Theorem~\ref{main-thm}.} It is not difficult to subdivide $\Delta_X$ and $\Delta_B$ so that $\Delta_X$ is simplicial, $\Delta_B$ is nonsingular, and $f:\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$ is a simplicial map (e.g. Proposition~4.4 and the remark following it in \cite{ak}). Applying the join construction we can make $f|_{\Delta_{X,i}}$ semistable without increasing the multiplicity of $\Delta_X$. We will show below that every singular simplex of $\Delta_X$ contains a Waterman point mapping to a barycenter of $\Delta_B$. By the previous section, there exist a modified barycentric subdivision and a simplicial map $f':MBS(\Delta_X)\rightarrow} \newcommand{\dar}{\downarrow BS(\Delta_B)$, with multiplicity of $MBS(\Delta_X)$ strictly less than the multiplicity of $\Delta_X$. Since $f'$ is simplicial and $BS(\Delta_B)$ nonsingular, the proof is completed by induction. Restrict $f$ to a singular simplex $f:\sigma\rightarrow} \newcommand{\dar}{\downarrow\tau$, where $\sigma$ has primitive points $v_{i j}, i=1,\ldots,n, j=1,\ldots,J_i$, $\tau$ has primitive points $u_1\ldots,u_n$, and $f(v_{i j})=u_i$. Since $\sigma$ is singular, it contains a Waterman point \[ w=\sum_{i,j} \alpha_{i j} v_{i j}, \qquad 0\leq\alpha_{i j}<1, \] where not all $\alpha_{i j}=0$. Restricting to a face of $\sigma$ if necessary we may assume that $w$ lies in the interior of $\sigma$, hence $0<\alpha_{i j}$. Since $f(w)\in N_\tau$, it follows that $\sum_j \alpha_{ij}\in{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$ for all $i$. In particular, if $J_{i_0}=1$ for some $i_0$ then $\alpha_{i_0 1}=0$, and $w$ lies in a face of $\sigma$. So we may assume that $J_i>1$ for all $i$. Since the relative dimension of $f$ is $\sum_i (J_i-1)$, we have to consider all possible decompositions $\sum_i (J_i-1) \leq 3$, where $J_i>1$ for all $i$. The cases when the relative dimension of $f$ is 0 or 1 are trivial and left to the reader. If the relative dimension of $f$ is 2, then either $J_1=3$, or $J_1=J_2=2$. In the first case, we have that ${\langle}v_{11},v_{12},v_{13}{\rangle}$ is singular, contradicting the semistability of $f|_{\Delta_{X,1}}$. In the second case, $\alpha_{11}+\alpha_{12}, \alpha_{21}+\alpha_{22} \in {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$ and $0< \alpha_{i j} < 1$ imply that $\alpha_{11}+\alpha_{12}= \alpha_{21}+\alpha_{22}=1$. Hence $f(w)=u_1+u_2$ is a barycenter. In relative dimension 3, either $J_1=4$, or $J_1=3,J_2=2$, or $J_1=J_2=J_3=2$. In the first case, we get a contradiction with the semistability of $f|_{\Delta_{X,1}}$; the third case gives $\alpha_{11}+\alpha_{12}= \alpha_{21}+\alpha_{22}=\alpha_{31}+\alpha_{32}=1$ as for relative dimension $2$. In the second case either $\alpha_{11}+\alpha_{12}+\alpha_{13} = \alpha_{21}+\alpha_{22}=1$ and $w$ maps to a barycenter, or $\alpha_{11}+\alpha_{12}+\alpha_{13} = 2, \alpha_{21}+\alpha_{22}=1$ and $(\sum v_{i j})-w$ maps to a barycenter. \qed\\

9711

alg-geom/9711034

en

https://arxiv.org/abs/alg-geom/9711034

alg-geom/9711034

Vladimir Masek

Vladimir Masek

Minimal discrepancies of hypersurface singularities

12 pages, AMS-LaTeX 1.2

We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov's conjecture is true for log-terminal threefolds.

alg-geom

\subsection*{Contents} \begin{enumerate} \item[0.] Introduction \item[1.] Generalities about discrepancy coefficients \item[2.] Minimal discrepancies and Shokurov's conjecture \item[3.] Proof of the main result (Theorem 1) \item[4.] Minimal discrepancies of log-terminal threefold singularities \end{enumerate} \section{Introduction} Let $Y$ be a normal, \ensuremath{\mathbb{Q}\,}-Gorenstein projective variety, and let $f:X \to Y$ be a resolution of singularities. The discrepancy divisor $\Delta=K_X-f^*K_Y$ plays a key role in the geometry of $Y$. For example, the singularities allowed on a minimal (resp. on a canonical) model of $Y$ are defined in terms of $\Delta$. Also, effective results for global generation of linear systems on singular threefolds (cf. \cite{elm}) involve certain coefficients of $\Delta$. There are many difficult conjectures, and several important results (at least in dimension $\leq 3$), regarding the discrepancy coefficients of $Y$ (i.e., the coefficients of $\Delta$). In this paper we study a special case of the following problem: \vspace{3pt} \noindent {\bf Shokurov's conjecture.} (cf. \cite{sho}, \cite{kollar}) \emph{ If $\dim(Y) = n$, and $y \in Y$ is a singular point, then $\md_y(Y) \leq n-2$}. The \emph{minimal discrepancy} of $Y$ at $y$, $\md_y(Y)$, is defined as $$ \md_y(Y) = \inf \{ \ord_F(\Delta) \mid f(F) = \{ y \} \ \ \}. $$ \vspace{3pt} The main result of this paper is an elementary computation of $\md_y(Y)$ for a large class of hypersurface singularities: \begin{Theorem} Assume that the germ $(Y,y)$ is analytically equivalent to a hypersurface singularity $(Y',0) \subset ({\mathbb{A}}_{\,\ensuremath{\mathbb{C}\,}}^{n+1},0)$, given by $$ Y'=\{ (y_1, \dots, y_{n+1}) \mid G(y_1, \dots, y_{n+1}) = 0 \} ;\quad G(0, \dots , 0) = 0. $$ For an $n$-tuple $(a_1, \dots, a_n)$ of positive integers, write $G(t^{a_1}u_1, \dots, t^{a_n}u_n, t) = \\ t^A \phi(u_1, \dots, u_n) + t^{A+1} \psi(u_1, \dots, u_n, t)$ with $\phi(u_1, \dots, u_n) \neq 0$. Note that $\phi$ is always a polynomial of degree at most $A$, even if $G$ is a power series. Assume that $\phi$ has at least one irreducible factor with exponent 1 in its factorization. Then $\md_y(Y) \leq d$, where $d=(a_1+\cdots+a_n)-A$. \end{Theorem} This criterion applies, for example, to hypersurface singularities of rank at least $2$ (if the singularity $(Y,0)$ is defined by $G(y_1,\dots,y_{n+1}) =0$, then we define its rank as the rank of the quadratic part of $G$). It applies also to terminal (and, more generally, $cDV$) singularities in dimension 3. Shokurov's conjecture for terminal threefolds was proved by D. Markushevich in \cite{mark}, using the language of toric geometry, Newton diagrams, admissible weights, etc., and using the fact that the singularities are isolated. Our proof shows that the result is completely elementary, and works for non-isolated singularities as well. \vspace{5pt} The paper is organized as follows: In \S1 we discuss discrepancy coefficients in general. Everything in this section is well-known to the experts; we wrote it mainly to fix our notations and terminology. We discuss in some detail the invariance of certain definitions under analytic equivalence of germs; we couldn't find a satisfactory reference in the literature. (N. Mohan Kumar pointed out to us that the matter is not completely trivial.) In \S2 we introduce minimal discrepancies and prove some easy reductions of Shokurov's conjecture. In \S3 we prove Theorem 1, and in \S4 we carry out the computations for log-terminal threefold singularities. \vspace{5pt} We would like to express our gratitude to L.~Ein, P.~Ionescu, R.~Lazarsfeld, K.~Matsuki, and N.~Mohan Kumar; our many conversations were very useful. \section{Generalities about discrepancy coefficients} In this section we recall several definitions and results regarding discrepancy coefficients, cf. \cite{reid}, \cite{ckm}, \cite{kmm}. \vspace{5pt} {\bf (1.1)} Let $f:X \to Y$ be a birational morphism of normal projective varieties of dimension $n$ over \ensuremath{\mathbb{C}\,}. A prime Weil divisor $F \subset X$ is \emph{$f$-exceptional} if $\dim f(F) \leq n-2$. The closed subset $f(F) \subset Y$ is called the \emph{center} of $F$ on $Y$. More generally, a \ensuremath{\mathbb{Q}\,}-Weil divisor $D=\sum a_j F_j$ is $f$-exceptional if all the irreducible components $F_j$ are $f$-exceptional. Let $\Exc(f) = \cup \{ F_j \mid \textup{$F_j \subset X$ a prime $f$-exceptional divisor} \}$; thus $D$ is $f$-exceptional if and only if $\Supp(D) \subset \Exc(f)$. \vspace{5pt} {\bf (1.2)} Choose a canonical divisor $K_Y$ on $Y$. Assume that $Y$ is \ensuremath{\mathbb{Q}\,}-Gorenstein, with global index $r$; i.e., $m K_Y$ is Cartier for some integer $m \geq 1$, and $r$ is the smallest such integer. Then we can define a \ensuremath{\mathbb{Q}\,}-divisor $f^*K_Y$ on $X$ by $f^*K_Y = \tfrac{1}{r} f^*(r K_Y)$. On the other hand, there is a unique canonical divisor $K_X$ on $X$ such that the \ensuremath{\mathbb{Q}\,}-divisor $\Delta = K_X - f^*K_Y$ is $f$-exceptional. ($K_X$ is obtained as follows: let $\omega$ be a rational differential $n$-form on $Y_{\textup{reg}}$, the smooth locus of $Y$; then $f^*\omega$ extends uniquely to a rational form on $X$, which we still denote by $f^*\omega$. If $\omega$ is chosen such that $K_Y = \divisor_Y(\omega)$, then $K_X = \divisor_X(f^*\omega)$.) The divisor $\Delta = K_X - f^*K_Y$ is called the \emph{discrepancy divisor} of $f$. Note that $K_Y$ varies in a linear equivalence class on $Y$, and correspondingly $K_X$ varies in its own linear equivalence class on $X$; however, $\Delta$ is uniquely determined by $f$: indeed, if $\omega' = \phi \, \omega$ on $Y_{\textup{reg}}$, for some rational function $\phi \in \ensuremath{\mathbb{C}\,}(Y_{\textup{reg}}) = \ensuremath{\mathbb{C}\,}(Y)$, then $f^*\omega' = (f^*\phi)(f^*\omega)$, and $\divisor_X(f^*\phi) = f^* \divisor_Y(\phi)$. \vspace{5pt} {\bf (1.3)} Write $\Delta = \sum a_j F_j$; the rational numbers $a_j$ are called \emph{discrepancy coefficients}. Now consider another birational morphism $f' : X' \to Y$ (with $X'$ a normal projective variety of dimension $n$). ${f'}^{-1} \circ f$ is a birational map $g: X \, \cdots \! \to X'$. Let $F_j \subset X$ be an $f$-exceptional divisor which intersects the regular locus $\Reg(g)$ of $g$, and assume that $g$ is an isomorphism at the generic point of $F_j$; i.e., $\overline{g(F_j \cap \Reg(g))}$ is a \emph{divisor} $F_j'$ on $X'$. Then $F_j'$ is an $f'$-exceptional divisor; in fact, $F_j$ and $F_j'$ have the same center on $Y$, $f(F_j)=f'(F_j')$. Moreover, if $a_j'$ is the coefficient of $F_j'$ in $\Delta' = K_{X'} - {f'}^* K_Y$, then $a_j' = a_j$. This is seen by resolving the indeterminacies of the map $g$ to a morphism $\tilde{g}:\tilde{X} \to X'$ ($\tilde{X}$ being obtained after a finite sequence of blowing-ups from $X$, cf. \cite[p.144, Consequence (1) of Corollary 1]{hironaka}) and calculating in two ways the discrepancy coefficient of $\tilde{F}_j$ = proper transform of $F_j$ on $\tilde{X}$. Thus, in fact, the discrepancy coefficient (and the center on $Y$) depends only on the discrete valuation of the rational function field $\ensuremath{\mathbb{C}\,}(Y)$ determined by $F_j$ (or $F_j'$). \vspace{5pt} {\bf (1.4.)} Conversely, let $v$ be any discrete valuation of $\ensuremath{\mathbb{C}\,}(Y)$. Then $v$ is associated to a certain divisor $F^0 \subset X^0$ for some birational morphism $f^0 : X^0 \to Y$; in fact, by Hironaka's embedded resolution of singularities, if we start with \emph{any} birational morphism $f:X \to Y$ as before, we can find a suitable $f^0$ with $X^0$ smooth, $\Exc(f^0)$ a divisor with normal crossings, and $X^0$ obtained from $X$ by a finite sequence of blowing-ups along smooth centers. Then $f^0(F^0) \subset Y$ depends only on $v$; this closed subset is called the \emph{center of $v$ on $Y$}. $v$ is \emph{$Y$-exceptional} if this center has dimension at most $n-2$, and in this case $v$ has a well-defined discrepancy coefficient with respect to $Y$. \vspace{5pt} {\bf (1.5)} Let $f:X \to Y$ be as before, and let $F_j \subset X$ be $f$-exceptional. The computation of the discrepancy coefficient $a_j$ is local on $X$; i.e., we may replace $Y$ with an open neighborhood of the generic point of $f(F_j)$, and $X$ with an open neighborhood of the generic point of $F_j$. From this point of view, the projectivity requirement is irrelevant. In particular, we may consider discrepancy coefficients for \emph{germs} $(Y,y)$ of algebraic varieties; one such coefficient is associated to each $Y$-exceptional discrete valuation of $\ensuremath{\mathbb{C}\,}(Y)$ whose center on $Y$ contains $y$. Moreover, the requirement that $X$ be normal is also irrelevant in some situations; for example, if $F_j$ is a Cartier divisor on $X$ (or at least on some open subset $U \subset X$ with $F_j \cap U \neq \emptyset$), then the generic point of $F_j$ has a \emph{nonsingular} open neighborhood in $X$, and we may replace $X$ with this neighborhood if we are interested only in the discrepancy coefficient of $F_j$. \vspace{5pt} {\bf (1.6)} {\bf Definition.} A projective variety $Y$ as before (i.e. normal, \ensuremath{\mathbb{Q}\,}-Gorenstein, $n$-dimensional) has \emph{only terminal (canonical, log-terminal, log-canonical) singularities} if all discrepancy coefficients of $Y$ are $>0$ (resp. $\geq 0$, $>-1$, $\geq -1$). Similarly, $Y$ is terminal (canonical, etc.) at a point $y$, or the germ $(Y,y)$ is terminal (etc.), if all discrepancy coefficients of discrete valuations with center containing $y$ are $>0$ (resp. $\geq 0$, etc.) \begin{Proposition} \label{p2} \textup{(cf. \cite[Proposition 6.5]{ckm})} Let $f:X \to Y$ be a proper birational morphism, with $X$ smooth and $\Exc(f)$ with only normal crossings. Let $\Delta = K_X - f^*K_Y = \sum a_j F_j$, and let $\alpha = \min \{ a_j \}$. If $-1 \leq \alpha \leq 1$, then \emph{all} the discrepancy coefficients of $Y$ are $\geq \alpha$ (even for those discrete valuations of $\ensuremath{\mathbb{C}\,}(Y)$ which are $Y$-exceptional but do not correspond to divisors on $X$). \end{Proposition} \vspace{3pt} In particular, to check whether $Y$ (or a germ $(Y,y)$) is terminal (etc.), it suffices to examine the discrepancy coefficients of a single resolution of singularities $f$ as above. \vspace{3pt} We reproduce the proof here for the reader's convenience (cf. \cite{ckm}); the same computation will be used again in (1.7) and in (2.1). \begin{proof} As explained in (1.4), it suffices to consider a single blowing-up of $X$ along a smooth center $Z \subset X$. Let $h:X' \to X$ be this blowing-up, $F_j' = h^{-1} F_j$ (proper transform), and $F'$ = the exceptional divisor of $h$. Let $r = \codim_X(Z) \geq 2$. Since $\Exc(f) = \cup \, F_j$ has only normal crossings, $Z$ is contained in at most $r$ of the divisors $F_j$; say $Z \subset F_1, \ldots, F_s$, $s \leq r$. Let $f' = f \circ h : X' \to Y$, and $\Delta' = K_{X'} - {f'}^*K_Y$; then \begin{multline*} \Delta' = K_{X'} - h^* f^* K_Y = K_{X'} - h^*(K_X-\Delta) = \\ = K_{X'} - h^* K_X + h^*(\sum a_j F_j) = (r-1) F' + \sum a_j F_j' + (\sum_{j=1}^s a_j) F'; \end{multline*} the discrepancy coefficient of $F'$ is therefore $a' = (r-1) + (\sum_1^s a_j)$. If $\alpha \leq 0$, we have $\sum_1^s a_j \geq s \alpha \geq r \alpha$ (because $s \leq r$ and $\alpha \leq 0$), and therefore $a' \geq (r-1) + r \alpha \geq \alpha$ (because $r > 1$ and $\alpha \geq -1$). If $\alpha > 0$, then we get $a' \geq r-1 \geq 1 \geq \alpha$. \end{proof} \emph{Remarks.} \ \emph{1.} The condition $\alpha \leq 1$ can always be achieved, as follows: let $f:X \to Y$ be a resolution of singularities, as in the statement of the proposition; choose a smooth subvariety $T \subset X$ of codimension $2$, such that $T \not\subseteq \Exc(f)$; and replace $f$ with $f \circ g$, where $g:\tilde{X} \to X$ is the blowing-up of $X$ along $T$. The computation used in the proof of the proposition shows that the exceptional divisor of $g$ has discrepancy coefficient $1$ relative to $Y$. \vspace{3pt} \emph{2.} If $\alpha < -1$, then the infimum of all discrepancy coefficients relative to $Y$ is $-\infty$; see \cite[Claim 6.3]{ckm}. (We prove a more precise statement in \S2, Lemma \ref{notlc}.) In general, the infimum of all discrepancy coefficients is called the \emph{(total) discrepancy} of $Y$, notation: $\discrep(Y)$. Thus $\discrep(Y) = -\infty$ if $Y$ is not log-canonical; if $Y$ \emph{is} log-canonical, then $-1 \leq \discrep(Y) \leq 1$, and $\discrep(Y)$ can be calculated by examining a single resolution of singularities $f:X \to Y$ as in the proposition. \vspace{3pt} \emph{3.} If $0 \leq \alpha \leq 1$, the proof shows that every $Y$-exceptional discrete valuation of $\ensuremath{\mathbb{C}\,}(Y)$, other than those associated to the exceptional divisors of $f$, has discrepancy coefficient $\geq 1$. \vspace{5pt} {\bf (1.7)} Let $(Y,y)$ be an algebraic germ, as before, and let $(Y^{an},y)$ be the corresponding analytic germ; note that $Y$ normal and irreducible $\implies$ $Y^{an}$ normal and irreducible. Also, $Y$ \ensuremath{\mathbb{Q}\,}-Gorenstein $\implies$ $Y^{an}$ \ensuremath{\mathbb{Q}\,}-Gorenstein. The theory of discrepancy divisors, discrepancy coefficients, terminal singularities, etc., can be developed in parallel in the category of germs of Moishezon analytic spaces; the results discussed so far are identical in the two categories. An interesting question arises when we try to compare the discrepancy coefficients for $(Y,y)$ and $(Y^{an},y)$. For example, is it true that $(Y,y)$ is terminal if and only if $(Y^{an},y)$ is terminal? (If this is true, then ``terminal'' depends only on the analytic equivalence class of an algebraic germ.) In general, the field of meromorphic functions of $Y^{an}$, $\mathcal{M}(Y^{an})$, has many discrete valuations which vanish identically on $\ensuremath{\mathbb{C}\,}(Y)$; therefore the question is non-trivial. The answer is given by the following observation: \begin{Proposition} \label{dcan} Let $f:X \to (Y,y)$ be a proper birational morphism with $X$ smooth and $\Exc(f)$ with normal crossings. Let $\{F_j\}_{j \in J}$ be the $f$-exceptional divisors on $X$, and let $\Delta = \sum a_j F_j$. Then the set of \emph{all} discrepancy coefficients of $(Y,y)$ is completely determined by the following combinatorial data: \begin{itemize} \item[(1)] The finite set $J$; \item[(2)] The rational numbers $a_j$ (one for each $j \in J$); and \item[(3)] For each subset $I \subset J$, the logical value of $\displaystyle{\operatornamewithlimits{\bigcap}_{j \in I} F_j} \neq \emptyset$ \textsc{(true \textnormal{or }false)}. \end{itemize} \end{Proposition} This observation (and its proof below) is valid in the algebraic as well as in the analytic case. In particular, the set of all ``algebraic'' and the set of all ``analytic'' discrepancy coefficients of $(Y,y)$ coincide. (We may start with the same algebraic resolution $f:X \to (Y,y)$ in the analytic category, as $f^{an}:X^{an} \to (Y^{an}, y)$; then the initial combinatorial data for $f^{an}$ is the same as for $f$.) \begin{proof} Let $v$ be a $Y$-exceptional discrete valuation of $\ensuremath{\mathbb{C}\,}(Y)$ with center containing~${y}$. By \cite[Main Theorem II]{hironaka}, there exists a finite succession of blowing-ups $f_i : X_{i+1} \to X_i$ along $Z_i \subset X_i$, where $0 \leq i < N$ and $X_0 = X$, with the following properties: \begin{itemize} \item[(i)] $v$ corresponds to a divisor on $X_N$; \item[(ii)] $Z_i$ is smooth and irreducible; and \item[(iii)] If $E_0=\Exc(f)$, and $E_{i+1}=f_i^{-1}(E_i)_{\text{red}} \cup f_i^{-1}(Z_i)_{\text{red}}$, $0\leq i<N$, then $E_i$ has only normal crossings with $Z_i$. \end{itemize} (Recall what this means, from \cite[Definition 2]{hironaka}: at each point $x \in Z_i$ there is a regular system of parameters of $\mathcal{O}_{X_i,x}$, say $(z_1, \dots, z_n)$, such that each component of $E_i$ which passes through $x$ has ideal in $\mathcal{O}_{X_i,x}$ generated by one of the $z_j$, and the ideal of $Z_i$ in $\mathcal{O}_{X_i,x}$ is generated by some of the $z_j$.) Let $f_1 : X_1 \to X$ be the blowing-up along a smooth irreducible subvariety $Z \subset X$, of codimension $r \geq 2$, such that $\Exc(f) = \cup_{j \in J} F_j$ has only normal crossings with $Z$. Say $Z \subset F_j$ if and only if $j \in \{j_1, \dots, j_s \}$; $s \leq r$. Considering $g = f \circ f_1 : X_1 \to Y$, we get a new element $j'$ added to $J$, $J_1 = J \cup \{j'\}$, where $j'$ corresponds to the exceptional divisor $F'$ of $f_1$. The corresponding number is $a_{j'}=(r-1) + (a_{j_1}+ \cdots + a_{j_s})$. Since the $F_j$ have only normal crossings with $Z$, the ``intersection data'' for $J_1$ is completely determined by the data for $J$, plus the following combinatorial data for $Z$: \begin{itemize} \item[(4)] For each $I \subset J$, the non-negative integer $d_I = \dim \left( Z \cap \left[ \displaystyle{\operatornamewithlimits{\bigcap}_{j \in I} F_j} \right] \right) $. \end{itemize} (Note that this collection of data contains, in particular, the codimension $r$ of $Z$, in the form $d_{\emptyset} = r$, and also the information about which $F_j$'s contain $Z$, in the form $Z \subset F_j \Leftrightarrow d_{\{j\}} = r$.) Finally, which such functions $\{d_I\}_{I \subset J}$ are possible is completely determined by the ``intersection data'' for $J$. Since every discrepancy coefficient of $Y$ is obtained after a finite number of such elementary operations on the combinatorial data (corresponding to a succession of blowing-ups along smooth centers), the result follows by induction. \end{proof} \section{Minimal discrepancies and Shokurov's conjecture} {\bf (2.1)} {\bf Definition.} Let $(Y,y)$ be an algebraic or analytic germ (as always, we assume it is normal, \ensuremath{\mathbb{Q}\,}-Gorenstein, $n$-dimensional). The \emph{minimal discrepancy of $Y$ at $y$}, $\md_y(Y)$, is the infimum of all discrepancy coefficients of discrete valuations of $\ensuremath{\mathbb{C}\,}(Y)$, resp. $\mathcal{M}(Y^{an})$, whose center on $Y$ is $y$. \begin{Lemma} \label{notlc} If $(Y,y)$ is not log-canonical at $y$, then $\md_y(Y) = -\infty$. \end{Lemma} \begin{proof} Let $f \colon X \to (Y,y)$ be a resolution of singularities with $\Exc(f)$ having only normal crossings. Let $F_j \subset X$ be an $f$-exceptional divisor with $y \in f(F_j)$ and discrepancy coefficient $a_j < -1$. Since $f^{-1}(y)$ is a union of $f$-exceptional divisors, and $F_j$ meets $f^{-1}(y)$, there is at least one exceptional divisor $F_i$ with $f(F_i) = \{ y \} $ and $F_i \cap F_j \neq \emptyset$. We may assume that $F_i$ and $F_j$ are distinct (if $F_j \subset f^{-1}(y)$ and it is the only component of the fiber, we may blow up $X$ at a point of $F_j$; then take the exceptional divisor of this blowing-up in place of $F_i$, and the proper transform of $F_j$ in place of $F_j$). Set $Z = F_i \cap F_j$; then $Z$ is a smooth subvariety of codimension $2$ in $X$, and is not contained in any other exceptional divisor. Let $a_i$ be the discrepancy coefficient of $F_i$. Let $g: \tilde{X} \to X$ be the blowing-up of $X$ along $Z$. Let $F'$ be its exceptional divisor, with discrepancy coefficient $a'$ relative to $Y$. Then $a' = 1 + a_j + a_i$ (see the proof of Proposition \ref{p2} in \S1). Moreover, $F'$ has center $\{ y \}$ on $Y$, and intersects the proper transform $F_j'$ of $F_j$ on $\tilde{X}$ (which has discrepancy coefficient $a_j' = a_j$ relative to $Y$). Note that $a_j < 1 \implies a' < a_i$. In fact, since all the discrepancy coefficients of $(Y,y)$ are integer multiples of $\frac{1}{r}$ (if $r$ is the index of $K_Y$ at $y$), we see that $a' \leq a_i - \frac{1}{r}$. Therefore the proof may be completed by induction. \end{proof} \vspace{5pt} {\bf (2.2)} Recall the statement of Shokurov's conjecture from the Introduction. The lemma we have just proved shows that the conjecture is true for non-log-canonical singularities. Shokurov's conjecture is trivially true for curves (there are no singular normal points in dimension 1). It is also true in dimension 2: if $(Y,y)$ is a normal singularity and $f:X \to (Y,y)$ is the \emph{minimal} desingularization, then \emph{all} the coefficients of $\Delta=K_X-f^*K_Y$ are $\leq 0$ (see, for example, \cite[Lemma 1.4]{elm}). \vspace{5pt} {\bf (2.3)} The following lemma shows that the conjecture can be reduced to the case of singularities of index one: \begin{Lemma} \label{indexone} Let $\varphi : Y' \to Y$ be a finite morphism of normal, \ensuremath{\mathbb{Q}\,}-Gorenstein varieties. Assume that $\varphi$ is \'{e}tale in codimension one. Let $y'$ be a point of $Y'$, and $y = \varphi(y')$. Then $\md_y(Y) \leq \md_{y'}(Y')$. \end{Lemma} In particular, if $(Y,y)$ has index $r$, then there exists a $\varphi:Y'\to Y$ as in the lemma, with $Y'$ having index one (the ``index-one cover'', cf. \cite[Definition 6.8]{ckm}). Thus it would suffice to prove Shokurov's conjecture for singularities of index one. \begin{proof} (cf. \cite[proof of Lemma 2.2]{elm}) Let $f' : X' \to Y'$ be a resolution of singularities of $Y'$ such that $\md_{y'}(Y')=\ord_{F'}(\Delta')\geq-1$, where $\Delta'=K_{X'}-{f'}^*K_{Y'}$ and $F'$ is a divisor on $X'$ with $f'(F')=\{y'\}$. (If $\md_{y'}(Y') = -\infty$, let $\alpha < -1$ be a rational number, and choose $f', F'$ such that $f'(F') = \{y'\}$ and $\ord_{F'}(\Delta') < \alpha$.) Let $f:X \to Y$ be a resolution of singularities of $Y$. By blowing up $X$, then $X'$, if necessary, we may assume that $\psi = f^{-1} \circ \varphi \circ f' : X' \to X$ is a morphism and that $\psi(F')$ is a divisor $F \subset X$. Let $\Delta = K_X - f^*K_Y$ and $a=\ord_F(\Delta)$. Let $t$ be the ramification index of $\psi$ along $F'$. Then: \begin{align*} K_{X'} &= \psi^*K_X + (t-1)F' + \textup{ other terms } \\ &= \psi^*(f^*K_Y + aF + \textup{ other terms }) + (t-1)F' + \textup{ other terms } \\ &= \psi^*f^*K_Y + (ta+t-1)F' + \textup{ other terms } \\ &= {f'}^*\varphi^*K_Y + (ta+t-1)F' + \textup{ other terms } \\ &= {f'}^*K_{Y'} + (ta+t-1)F' + \textup{ other terms } \end{align*} (note that $\varphi^*K_Y = K_{Y'}$, because $\varphi$ is \'{e}tale in codimension one). Therefore $\ord_{F'}(\Delta') = ta+t-1$. If $\ord_{F'}(\Delta') \geq -1$, we get $\ord_F(\Delta) = a \leq \ord_{F'}(\Delta') = \md_{y'}(Y')$; indeed, $t \geq 1$, and therefore $ta \leq ta + (t-1)(1+\ord_{F'}(\Delta')) = t\ord_{F'}(\Delta')$. If $\ord_{F'}(\Delta') < \alpha < -1$, then $\ord_F(\Delta) = a \leq \tfrac{1}{t} \ord_{F'}(\Delta') < \tfrac{1}{t}\alpha$, with $1 \leq t \leq \deg(\varphi)$ and $\alpha$ an arbitrarily negative rational number. Since $f(F) = f\psi(F') = \varphi f'(F') = \varphi(\{y'\}) = \{y\}$, the lemma is proved. \end{proof} \vspace{5pt} {\bf (2.4)} Finally, we show that $\md_y(Y)$ is an analytic invariant. In fact, we show that the set of all discrepancy coefficients for divisors with center $\{y\}$ on $Y$ is the same in the algebraic and in the analytic category: \begin{Proposition} \label{mdan} Let $f:X \to (Y,y)$ be a resolution of singularities, as in Proposition \ref{dcan}. Then the set of all discrepancy coefficients for divisors with center $\{y\}$ on $Y$ is completely determined by the combinatorial data (1), (2), (3) in Proposition~\ref{dcan}, plus: \begin{itemize} \item[(3+)] For each $j \in J$, the logical value of ``$f(F_j)=\{y\}$'' \textsc{(true \textnormal{or} false)}. \end{itemize} \end{Proposition} \begin{proof} Let $f_1:X_1 \to X$ be the blowing-up of a smooth subvariety $Z\subset X$, as in the proof of Proposition \ref{dcan}. Put $g=f \circ f_1$, and let $F'$ be the exceptional divisor of $f_1$. Then $[g(F')=\{y\}] \Leftrightarrow [f(F_j)=\{y\}$ for at least one of the $F_j$'s containing $Z$]. Indeed, if $Z \subset F_j$ and $f(F_j)=\{y\}$, then $g(F') = f(Z) \subset \{y\}$, so that in fact $g(F')=\{y\}$. Conversely, $g(F') = \{y\} \implies Z \subset f^{-1}(y)$. As $Z$ is irreducible and $f^{-1}(y)$ is a union of divisors $F_j$ with $f(F_j)=\{y\}$, $Z$ must be contained in at least one such $F_j$. Therefore the ``extended'' combinatorial data for $g$ (including the information in (3+)) can be obtained from the ``extended'' combinatorial data for $f$. The conclusion follows by induction. \end{proof} \section{Proof of the main result (Theorem 1)} {\bf (3.1)} Recall the statement of Theorem 1 from the Introduction. By (2.4), we may assume that $Y$ is the hypersurface $G=0$ in $\ensuremath{\mathbb{A}\,}^{n+1}$, with $y=0$. For convenience, denote $\ensuremath{\mathbb{A}\,}^{n+1}$ by $V$; thus $Y \subset V$. Let $U = \ensuremath{\mathbb{A}\,}^{n+1}$; write the coordinates in $V$ as $(y_1, \dots, y_{n+1})$, and the coordinates in $U$ as $(u_1, \dots, u_n, t)$. Let $f:U \to V$ be the birational morphism defined by $y_{n+1}=t; y_i = t^{a_i} u_i, i=1, \dots, n$. Let $E \subset U$ be the hyperplane $(t=0)$; then $\Exc(f)=E$. \vspace{5pt} {\bf (3.2)} Let $\bar{Y} \subset U$ be the proper transform of $Y$ by $f$, $\bar{f} : \bar{Y} \to Y$ the restriction of $f$ to $\bar{Y}$, and $\bar{E} = E |_{\bar{Y}}$ (as a Cartier divisor). By hypothesis, $f^*Y = \bar{Y} + AE$, and $\bar{E}$ has equation $\phi(u_1, \dots, u_n) = 0$ in $E \cong \ensuremath{\mathbb{A}\,}^n$. Since $\phi$ has at least one irreducible factor with exponent 1, $\bar{E}$ has at least one irreducible component with multiplicity one: $\bar{E}=F_1+ \cdots $. As explained in (1.5), since $\bar{Y}$ is smooth in a neighborhood of the generic point of $F_1$, and $F_1$ is the exceptional divisor which will produce the desired discrepancy coefficient, we need not worry about the normality of $\bar{Y}$. \vspace{5pt} {\bf (3.3)} Take $\omega = dy_1 \wedge \cdots \wedge dy_{n+1}$ on $V$; then $f^*\omega = t^{a_1+\cdots+a_n}du_1 \wedge \cdots \wedge du_n \wedge dt$ on $U$, so that $K_U - f^*K_V = (a_1 + \cdots a_n)E$. The adjunction formula gives $K_Y = K_V + Y |_Y$ and $K_{\bar{Y}} = K_U + \bar{Y} |_{\bar{Y}}$. Therefore we have: \begin{align*} K_{\bar{Y}} - \bar{f}^* K_Y &= (K_U+\bar{Y}) |_{\bar{Y}} - \bar{f}^*(K_V+Y|_Y) \\ &= (K_U - f^*K_V + \bar{Y} - f^*Y) |_{\bar{Y}} \\ &= ((a_1 + \cdots + a_n) E - AE ) |_{\bar{Y}} \\ &= d \bar{E} = d F_1 + \cdots . \end{align*} (Recall that $d = (a_1 + \cdots + a_n) - A$.) Thus the discrepancy coefficient of $F_1\subset\bar{Y}$ with respect to $Y$ is equal to $d$. Since $\bar{f}(F_1)=\{y\}$, Theorem 1 is proved. \qed \vspace{5pt} {\bf (3.4) Example.} Let $(Y,y)$ be a singular germ of multiplicity 2. That is, we may assume that $Y$ is a hypersurface in $\ensuremath{\mathbb{A}\,}^{n+1}$ given by an equation $G=0$, with $y=0$, $G$ and all its first-order partial derivatives at 0 equal to zero, and some second-order partial derivative of $G$ at 0 not equal to zero. If $(Y,0)$ has rank at least 2, then $\md_y(Y) \leq n-2$ (as predicted by Shokurov's conjecture). Indeed, consider the usual blowing-up of $V$ at 0; that is, take $a_1=\cdots=a_n=1$. The hypothesis means that $A=2$, and --- after a linear change of parameters, if necessary --- $\phi(u_1,\dots,u_n) = u_1^2 + \cdots + u_r^2$, where $r \geq 2$ is the rank of the singularity. Thus $d=(a_1+\cdots+a_n)-A=n-2$ in this case, and $\phi$ is irreducible (if $r \geq 3$), resp. a product of two distinct irreducible (linear) factors, if $r = 2$. \section{Minimal discrepancies of log-terminal threefold singularities} Let $(Y,y)$ be a three-dimensional log-terminal singularity. In this section we will show that $\md_y(Y) \leq 1$ (so that Shokurov's conjecture is true in this case). \vspace{5pt} {\bf (4.1)} As shown in (2.3), we may assume that $(Y,y)$ has index one. Then $(Y,y)$ is canonical (the index-one cover of a log-terminal singularity is again log-terminal, by Propositon \ref{p2}, and therefore canonical). In this case, M.~Reid \cite[Theorem 2.2]{reid} proved that either $(Y,y)$ is a $cDV$ point (see below), or there exists a proper birational morphism $f:Y'\to Y$ with $f^* K_Y=K_{Y'}$ and $f^{-1}(y)$ containing at least one prime divisor of $Y'$. Of course, in the latter case we have $\md_y(Y) = 0$. There only remains to consider the case when $(Y,y)$ is a compound Du Val ($cDV$) point; that is, $(Y,y)$ is analytically equivalent to a hypersurface singularity at the origin $0 \in \ensuremath{\mathbb{A}\,}^4$, with equation $G=0$, $$ G(y_1, y_2, y_3, t) = f(y_1,y_2,y_3) + t g(y_1,y_2,y_3,t), $$ where $f(y_1,y_2,y_3)=0$ defines a Du Val singularity (rational double point) of a surface at $0 \in \ensuremath{\mathbb{A}\,}^3$. To simplify notation, we write $\mathbf{y}$ for $y_1,y_2,y_3$ and $\mathbf{u}$ for $u_1,u_2,u_3$. By (2.4), we may assume that $(Y,y)$ \emph{is} the hypersurface $(G=0)\subset \ensuremath{\mathbb{A}\,}^4$, with $y=0$. By Theorem 1, it suffices to find $a_1,a_2,a_3 \geq 1$ such that $$ G(t^{a_1}u_1,t^{a_2}u_2,t^{a_3}u_3,t)=t^A \phi(\mathbf{u})+t^{A+1}\psi(\mathbf{u},t) $$ with $\phi(\mathbf{u}) \neq 0$, $(a_1+a_2+a_3)-A = 1$, and $\phi$ having at least one irreducible factor with exponent one in its prime decomposition. \vspace{5pt} {\bf (4.2)} We will do a case-by-case analysis, according to the type of singularity; $(Y,0)$ is of type $cA_n$, $cD_n$, or $cE_n$, if the surface singularity $f(\mathbf{y})=0 \subset \ensuremath{\mathbb{A}\,}^3$ is of type $A_n$, $D_n$, or $E_n$. In each case, $f(\mathbf{y})$ is completely known. $g(\mathbf{y},t)$, on the other hand, is not. Of course, we have $g(0,0,0,0)=0$, or else $(Y,0)$ would be a smooth point. We will not make any other assumptions about $g$. Write $g = g_1 + g_2 + \cdots$, where $g_i$ is a homogeneous form of degree $i$, and $o_k(g) = g_k + g_{k+1} + \cdots \,\, (k \geq 1)$. A note on terminology: we distinguish between \emph{form} and \emph{polynomial}; for instance, a quadratic polynomial is the sum of a quadratic form, a linear form, and a constant term. We say that a polynomial (or a form) \emph{contains} a certain monomial if the coefficient of the monomial in that polynomial is non-zero. We say that a monomial \emph{contains} $y_1$ if that monomial is divisible by $y_1$. \vspace{5pt} {\bf (4.3) Case} $\mathbf{cA_n}$: \ \ $f(\mathbf{y}) = y_1^2 + y_2^2 + y_3^{n+1} \quad (n \geq 1)$. Then $(Y,0)$ is a singularity of multiplicity 2 and rank at least 2. This case is therefore covered by the Example discussed in (3.4). \vspace{5pt} {\bf (4.4) Case} $\mathbf{cD_n}$: \ \ $f(\mathbf{y}) = y_1^2 + y_2 y_3^2 + y_3^{n-1} \quad (n \geq 4)$. If $g_1(\mathbf{y},t) \neq 0$, then the quadratic part of $G = f + t g$ is $y_1^2 + t g_1(\mathbf{y},t)$. If this quadratic part has rank at least 2, then the conclusion follows from (3.4). If it has rank 1, i.e. if $y_1^2+tg_1(\mathbf{y},t)$ is the square of a linear form, then a linear change of variable, $y_1' = y_1 + \alpha y_2 + \beta y_3 + \gamma t$, transforms the equation $G=0$ into a similar one with $g_1(\mathbf{y},t)=0$. So we need to consider only the case $g_1=0$. Note that a similar argument applies to singularities of type $cE_n$. Assume that $g_1=0$. Then put $a_1=2, a_2=a_3=1$; that is, put $y_1=t^2 u_1, y_2 = t u_2, y_3 = t u_3$. We have: $$ G(t^2u_1,tu_2,tu_3,t) = t^3\phi(\mathbf{u}) + t^4\psi(\mathbf{u},t), $$ where $\phi(\mathbf{u}) = u_2 u_3^2 + \delta_{n,4} u_3^3 + [\text{terms of degree $\leq 2$ in the $u_j$}]$; $\delta_{n,4}=1$ if $n=4$, otherwise $\delta_{n,4} = 0$. (The terms of lower degree come from $t g_2(t^2u_1,tu_2,tu_3,t)$, with $g_2$ --- the quadratic component of $g(\mathbf{y},t)$. Note that not \emph{all} the terms in $tg_2$ contribute to $\phi(\mathbf{u})$: as $y_1=t^2u_1$, the monomials in $g_2(\mathbf{y},t)$ which contain $y_1$ give rise to monomials containing $t$ to the fourth or higher power.) The proof in this case is complete, for $(a_1+a_2+a_3)-A = (2+1+1)-3 = 1$, and $\phi$ has at least one irreducible factor with exponent one (otherwise $\phi$ would have to be the cube of a linear polynomial in $\mathbf{u}$; that linear polynomial would have to contain $u_2$, because $\phi$ contains $u_2 u_3^2$, and then $\phi$, being the cube of that linear polynomial, would contain $u_2^3$, which is not the case). \vspace{5pt} {\bf (4.5) Case} $\mathbf{cE_6}$: \ \ $f(\mathbf{y}) = y_1^2 + y_2^3 + y_3^4$. As in (4.4), we may assume that the linear part $g_1(\mathbf{y},t)$ of $g(\mathbf{y},t)$ is equal to zero. In $g_2$ (the quadratic part of $g$), separate the monomials which contain $y_1$ from those that don't: $g_2(\mathbf{y},t) = y_1 L(\mathbf{y},t) + Q(y_2,y_3,t)$, where $L$ is a linear form and $Q$ is a quadratic form. Put $y_1 = t^2 u_1, y_2 = t u_2, y_3 = t u_3$; then $$ G(t^2u_1, tu_2, tu_3, t) = t^3\phi(\mathbf{u}) + t^4\psi(\mathbf{u},t), $$ with $\phi(\mathbf{u}) = u_2^3 + Q(u_2,u_3,1)$. If $\phi$ is not the cube of a linear polynomial, then we complete the proof just as in (4.4). However, in this case it might be that $\phi$ \emph{is} a perfect cube. If this is so, then $y_2^3 + t Q(y_2,y_3,t)$ is the cube of a linear form in $y_2,y_3,t$. A linear change of variable, $y_2' = y_2 + \alpha y_3 + \beta t$, reduces the proof to the case $Q=0$. This argument stands also in the cases $cE_7$ and $cE_8$, discussed below. There only remains to consider the case $g_2(\mathbf{y},t)=y_1L(\mathbf{y},t)$, where $L$ is a linear form (possibly zero). In this case put $a_1=a_2=2, a_3=1$, i.e. $y_1=t^2u_1, y_2=t^2u_2, y_3=tu_3$. Then: \begin{gather*} G(\mathbf{y},t) = y_1^2 + y_2^3 + y_3^4 + t[y_1L(\mathbf{y},t) + o_3(g)], \quad \text{and} \\ G(t^2u_1,t^2u_2,tu_3,t) = t^4 \phi(\mathbf{u}) + t^5 \psi(\mathbf{u},t), \end{gather*} where $\phi(\mathbf{u})=u_1^2+u_3^4+[\text{terms of degree $\leq 3$ in the $u_j$}]$, and the expression in brackets does not contain $u_1^2$ (so that $u_1^2$ doesn't cancel out from $\phi$). Note that $\deg(\phi)=4$, and $\phi$ cannot be the square of a quadratic polynomial in the $u_j$ (if it were, then $\phi$ would contain the mixed product $u_1 u_3^2$, because it contains $u_1^2$ and $u_3^4$ but no $u_1^4$; the monomial $u_1u_3^2$ could only arise from a monomial $t(y_1y_3^2 t^k)$ of $tg(\mathbf{y},t)$, $k \geq 0$; but $t(t^2u_1)(tu_3)^2t^k = t^{5+k}u_1u_3^2$, so $u_1u_3^2$ cannot be a monomial of $\phi$). Therefore $\phi$ has an irreducible factor with exponent one, and the conclusion follows --- note that $(a_1+a_2+a_3)-A = (2+2+1)-4 = 1$. \vspace{5pt} {\bf (4.6) Case} $\mathbf{cE_7}$: \ \ $f(\mathbf{y}) = y_1^2 + y_2^3 + y_2 y_3^3$. We may again assume that $g_1=0$, as in (4.4), and that $g_2=y_1L(\mathbf{y},t)$ with $L$ a linear form (possibly zero), as in (4.5). Write $L(\mathbf{y},t) = L_1(y_1,y_2)+L_2(y_3,t)$, and $g_3(\mathbf{y},t) = C_1(\mathbf{y},t) + C_2(y_3,t)$, where $C_1$ and $C_2$ are cubic forms such that every monomial of $C_1$ contains $y_1$ or $y_2$. Put $a_1=a_2=2, a_3=1$; then \begin{gather*} \begin{split} G(\mathbf{y},t)=y_1^2+y_2^3+y_2 y_3^3+t [ y_1 L_1(y_1,y_2)&+ y_1 L_2(y_3,t) \\ &+ C_1(\mathbf{y},t)+C_2(y_3,t)+o_4(g)],\quad\text{and} \end{split} \\ G(t^2u_1,t^2u_2,tu_3,t) = t^4 \phi(\mathbf{u}) + t^5 \psi(\mathbf{u},t), \qquad\qquad \qquad\qquad \end{gather*} where $\phi(\mathbf{u}) = u_1^2 + u_1 L_2(u_3,1) + C_2(u_3,1)$. Note that $(a_1+a_2+a_3)-A = (2+2+1)-4 = 1$. If $\phi$ has degree 3 (i.e. if $C_2(y_3,t)$ contains $y_3^3$), then $\phi$ has an irreducible factor with exponent one, because $\phi$ cannot be a perfect cube (it contains $u_1^2$ but no $u_1^3$); in this case the proof is complete. Otherwise $\phi$ has degree 2 (it contains $u_1^2$). Then either $\phi$ has an irreducible factor with exponent one (and then the proof is complete), or else $\phi$ is the square of a linear polynomial. In the latter case, $y_1^2 + t y_1 L_2(y_3,t) + t C_2 (y_3,t)$ is a perfect square. The (non-linear) change of variable $y_1' = y_1 + \tfrac{1}{2} t L_2(y_3,t)$ transforms the equation $G=0$ into a similar one with $L_2=C_2=0$ (the verification is straightforward). Therefore we may assume that $G$ has the form: $$ G(\mathbf{y},t) = y_1^2 + y_2^3 + y_2 y_3^3 + t [ y_1 L_1(y_1,y_2) + C_1(\mathbf{y},t) + o_4(g)], $$ where $L_1$ is a linear form, and $C_1$ is a cubic form such that every monomial of $C_1$ contains $y_1$ or $y_2$. (The same argument carries over unchanged to the last case, $cE_8$.) Now put $a_1=3,a_2=2,a_3=1$; that is, $y_1=t^3u_1, y_2=t^2u_2, y_3=tu_3$. We have: $$ G(t^3u_1, t^2u_2, tu_3,t) = t^5 \phi(\mathbf{u}) + t^6 \psi(\mathbf{u},t), $$ where $\phi(\mathbf{u}) = u_2 u_3^3 + u_2 p(u_3) + q(u_3)$; $u_2 p(u_3)$ corresponds to the monomials of $C_1(\mathbf{y},t)$ of the form $y_2 y_3^k t^{2-k}, k=0,1,2$ (all other monomials of $C_1$ produce at least a sixth power of $t$; recall that all monomials of the cubic form $C_1$ contain $y_1$ or $y_2$), and $q(u_3)$ corresponds to the monomials of $g_4$ of the form $y_3^k t^{4-k}, k=0, \dots, 4$. Note that $\phi$ has degree exactly one as a polynomial in $u_2$, and therefore $\phi$ cannot be the square of another polynomial in the $u_j$. Since $\phi$ has (total) degree 4, it must have an irreducible factor with exponent one. As $(a_1+a_2+a_3)-A = (3+2+1)-5 = 1$, the proof is complete in this case. \vspace{5pt} {\bf (4.7) Case} $\mathbf{cE_8}$: \ \ $f(\mathbf{y}) = y_1^2 + y_2^3 + y_3^5$. As in the previous case, we may assume that $$ G(\mathbf{y},t) = y_1^2 + y_2^3 + y_3^5 + t [ y_1 L_1(y_1,y_2) + C_1(\mathbf{y},t) + o_4(g)], $$ where $L_1$ is a linear form and $C_1$ is a cubic form such that every monomial of $C_1$ contains $y_1$ or $y_2$. If we take $a_1=3, a_2=2, a_3=1$, i.e. $y_1=t^3 u_1, y_2=t^2 u_2, y_3=tu_3$, we get $$ G(t^3u_1, t^2u_2, tu_3, t) = t^5 \phi(\mathbf{u}) + t^6 \psi(\mathbf{u},t) $$ with $\phi(\mathbf{u}) = u_3^5 + u_2 p(u_3) + q(u_3)$, where $p(u_3)$ and $q(u_3)$ are exactly as in the previous case. $(a_1+a_2+a_3)-A = (3+2+1) - 5 = 1$; so the proof is complete if $\phi$ has an irreducible factor with exponent one. Since $\deg(\phi) = 5$, if $\phi$ does not have an irreducible factor with exponent one then $\phi = M^2 N^3$ for two linear polynomials $M=M(u_2,u_3)$ and $N = N(u_2,u_3)$ (possibly equal). In particular, if this is the case then $\phi$ cannot have degree exactly one as a polynomial in $u_2$, and therefore $p(u_3)=0$; i.e., $C_1(\mathbf{y}, t)$ contains no monomials of the form $y_2 y_3^k t^{2-k}\,(k=0,1,2)$. As every monomial of $C_1$ contains $y_1$ or $y_2$, this means that every monomial of $C_1$ actually contains $y_1$ or $y_2^2$. Now $\phi(\mathbf{u}) = u_3^5 + q(u_3) \text{ ($q$ of degree at most four) } = M^2(u_3) N^3(u_3)$. If we write $g_4(\mathbf{y},t) = F_1(\mathbf{y},t) + F_2(y_3,t)$, with $F_1,F_2$ forms of degree 4 such that every monomial of $F_1$ contains $y_1$ or $y_2$, then $q(u_3)=F_2(u_3,1)$. $u_3^5+q(u_3) = M^2(u_3)N^3(u_3)$ means that $y_3^5+F_2(y_3,t) = \tilde{M}^2(y_3,t)\tilde{N}^3(y_3,t)$, with $\tilde{M}, \tilde{N}$ linear forms in $y_3,t$ ($M(u_3)=\tilde{M}(u_3,1)$, etc.) A linear change of variable $y_3'=y_3 + \alpha t$ reduces $G$ to the form $$ G(\mathbf{y},t) = y_1^2 + y_2^3 + y_3^3 ( y_3 + at)^2 + t[ y_1 L(y_1,y_2) + C_1(\mathbf{y},t) + F_1(\mathbf{y},t) + o_5(g)], $$ where $a \in \ensuremath{\mathbb{C}\,}$ (possibly $a=0$), every monomial of the cubic form $C_1$ contains $y_1$ or $y_2^2$, and every monomial of the quartic form $F_1$ contains $y_1$ or $y_2$. Put $a_1=3, a_2=a_3=2$; that is, $y_1=t^3 u_1, y_2=t^2u_2, y_3=t^2 u_3$. Then $$ G(t^3u_1, t^2u_2, t^2u_3,t) = t^6 \phi(\mathbf{u}) + t^7 \psi(\mathbf{u},t) $$ with $\phi(\mathbf{u}) = u_1^2 + u_2^3 + [\text{terms of degree at most 2 in the $u_j$}]$, and $u_1^2$ is not among the terms inside the brackets. Therefore $\phi$ has an irreducible factor with exponent one; as $(a_1+a_2+a_3)-A = (3+2+2) - 6 = 1$, the proof is complete in all cases. \qed \vspace{5pt} {\bf (4.8)} \emph{Remarks.} \ \emph{1.} If $(Y,y)$ is a terminal threefold singularity of index one, then $\md_y(Y) = 1$ ($\md_y(Y) \geq 1$ because all the discrepancy coefficients of $Y$ at $y$ are positive integers). On the other hand, Kawamata \cite{kaw} proved that the minimal discrepancy of a terminal threefold singularity of index $r \geq 2$ is $\frac{1}{r}$. \vspace{3pt} \emph{2.} Our computations in \S4 seem related to those in \cite{mark}, except for the fact that Markushevich uses the toric language. At first glance, it looks like his proof uses the assumption that the singularity is isolated; however, this is needed only to reduce the equation $G=0$ to various standard forms, and -- as our elementary computations in (4.3) -- (4.7) show --- this can be done without assuming the singularity is isolated. This allows us to conclude that Shokurov's conjecture is true for canonical (and log-terminal) threefold singularities, rather than just for terminal singularities.

9306

alg-geom/9306004

en

https://arxiv.org/abs/alg-geom/9306004

alg-geom/9306004

O. Debarre, K. Hulek, J. Spandaw

Very ample linear systems on abelian varieties

29 pages, typeset with AMS-LaTeX 1.1

Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$ is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the following: (1) If $d \ge g+2$, then $\phi_L\colon X \to {\bf P}^{d-1}$ defines a birational morphism onto its image. (2) If $d > 2^g$, then $L$ is very ample. We show the latter by checking it on a suitable rank-$(g-1)$-degeneration.

alg-geom

\section{Introduction} \pagenumbering{arabic} Let $M$ be an ample line bundle on a abelian variety $X$. (Throughout this paper, we shall work over the complex numbers ${\Bbb C}$. Hence by abelian variety we always mean complex abelian variety.) It is known that $M^n$ is very ample for $n\ge3$ (Lefschetz theorem) and that $M^2$ is very ample if and only if $|M|$ has no fixed divisor (Ohbuchi theorem). Very little is known for line bundles on $X$ which are not non-trivial powers of ample line bundles. In this article we consider exclusively the case of line bundles $L$ of type $(1,\ldots,1,d)$, which are exactly the pullbacks of principal polarizations by cyclic isogenies of degree $d$. When $X$ is an abelian surface, such an $L$ is base-point-free if and only if $d\ge3$ and $(X,L)$ is not a product \cite[lemma~10.1.2]{BL}; it is very ample if and only if $d\ge5$ and $X$ contains no elliptic curve $E$ such that $L.E\le 2$ \cite[corollary~10.4.2]{BL}. In higher dimensions, we show that for $(X,L)$ {\em generic} of type $(1,\ldots,1,d)$ and dimension $g$, the linear system $|L|$ is base-point-free if and only if $d\ge g+1$ (see proposition~\ref{75}). Moreover, the morphism $\phi_L: X \to {\Bbb P}^{d-1}$ that it defines is birational onto its image if and only if $d\ge g+2$ (see proposition~\ref{76}). These results are part of a general conjectural picture: for $d>g$, the morphism $\phi_L$ should be an embedding outside of a set of dimension $2g+1-d$. In particular, $L$ should be very ample if and only if $d\ge 2g+2$. (For $g>2$, Barth and Van de Ven have shown that $L$ cannot be very ample for $d\le 2g+1$ \cite{B,VV}.) We show that for $(X,L)$ generic of type $(1,\ldots,1,d)$ and dimension $g$, the line bundle $L$ is very ample for $d>2^g$, by checking it on a rank-$(g-1)$ degeneration (see corollary~\ref{171}). The same result was proved in \cite{BLR} by a different method for $g=3$ and $d\ge13$. For $g=3$, this leaves only the case $d=8$ open (the polarization is never very ample in that case for the degenerations we consider). As shown by Koll\'ar in \cite{K}, our result in dimension 3 implies the following version of a conjecture of Griffiths and Harris: for $d$ odd, $d\ge 9$, the degree of any curve on a very general hypersurface of degree $6d$ in ${\Bbb P}^4$ is divisible by $d$. This paper was completed during a visit of the first author at the University of Hannover. The authors are grateful to the DFG for financial support which made this visit possible. \section{Linear systems on abelian varieties} In this section we focus on the behaviour of morphisms $\phi_L: A\to {\Bbb P}(H^0(A,L)^{\ast})$, where $A$ is a {\em generic} abelian variety and $L$ an ample line bundle on $A$, such that $h^0(A,L)=d$ or, equivalently, $L^g = g! d$. \begin{thm}\label{157} Let $A$ be an abelian variety of dimension $g$ and let $\phi: A\to {\Bbb P}^{d-1}$ be a finite morphism. Then \begin{assertions} \item the ramification locus of $\phi$ has dimension at least $2g-d$ \item if $F$ is a closed subset of $A$ such that the restriction of $\phi$ to $A- F$ is an embedding, then $\dim(F)\ge 2g+1-d$, except if $(g,d)=(1,3)$ or $(2,5)$. \end{assertions} \end{thm} \begin{pf*} The ramification of $\phi$ is the locus where $\operatorname{d}\!\phi:TA \to \phi^{\ast} T {\Bbb P}^{d-1}$ has rank less than $g$. Since $\phi$ is finite, $\phi^{\ast} T{\Bbb P}^{d-1}$ is ample, hence so is $\phi^{\ast} T{\Bbb P}^{d-1} \otimes T^{\ast} A$. By \cite[theorem~1.1]{FL}, this locus is non-empty if $g\ge d-1-(g-1)=d-g$ and has dimension at least $g-(d-g)$. This proves (i). To prove (ii), we follow ideas of Van de Ven (\cite{VV}). Assume that $\dim(F) \le 2g-d$. The intersection of $\phi(A)$ with $s=2g-d+1$ generic hyperplanes is a smooth irreducible $(g-s)$-dimensional scheme $S$ contained in $\phi(A-F)$. Note that $S$ sits in ${\Bbb P}^{2(g-s)}$, so that we can do a Chern class computation as in \cite{VV}. Set $l=c_1({\cal O}_S(1))$. Then $c(TS)=(1+l)^{-s}$ and the exact sequence \[ 0 \to TS \to T{\Bbb P}^{2(g-s)}|_S \to N \to 0 \] gives \( c(N)=(1+l)^d \). It follows (\cite[prop.~3]{VV}) that \( (\deg S)^2 = c_{g-s}(N) =\binom{d}{g-s} l^{g-s} \). Since $l^{g-s}=\deg S =\deg A$, we finally get \( \deg A=\binom {d}{g+1} \). Since the degree of any ample line bundle on an abelian variety of dimension $g$ is divisible by $g!$, the conclusion is that $\binom{d}{g+1}$ is divisible by $g!$. We may assume that $g<d\le 2g+1$. We will let the reader check that this can only happen for $g=1$ or 2. (For the case $d=2g+1$, see \cite{VV}.) Cases where $d=g+1$ are trivially excluded and so is $(g,d)=(2,4)$. This finishes the proof of (ii). \end{pf*} One might expect that there is equality in (i) and~(ii) for a generic abelian variety, but the answer probably depends on the {\em type} of the polarization $\phi^{\ast} {\cal O}_{{\Bbb P}}(1)$. From now on, we will restrict ourselves to polarizations of type $(1,\ldots,1,d)$. We shall need the following facts. Let $L$ be a line bundle of type $(1,\ldots,1,d)$ on an abelian variety $A$. We define the groups ${\cal G}(L)$ and $K(L)$ as in \cite[chapter 6]{BL}. Then, $K(L)$ is isomorphic to $({\Z/ d\Z})^2$, and there is a central extension: \[ 1 \to {\C^{\ast}} \to {\cal G}(L) \buildrel{\pi}\over{\to} K(L) \to 0. \] The group ${\cal G}(L)$ operates on $H^0(A,L)$ \cite[p.~295]{M2}. If $\tilde\epsilon$ is an element of ${\cal G}(L)$ of order $d$, it generates a maximal level subgroup of ${\cal G}(L)$ (in the sense of \cite[p.~291]{M2}, hence there exists a non-zero section $s$ of $L$, unique up to multiplication by a non-zero scalar, such that $\tilde\epsilon\cdot s=s$ \cite[prop.~3]{M2}. In particular, for $0\le\lambda< d$, there exists a non-zero section $s_{\lambda}$ such that $\tilde\epsilon\cdot s_{\lambda}=e^{2i\pi\lambda/d} s_{\lambda}$. Given $A$, $L$ and $\tilde\epsilon$, the ordered set $\{ s_0,\ldots ,s_{d-1}\}$ is well-defined up to multiplication of its elements by non-zero scalars; it is a basis for $H^0(A,L)$, which we will call a {\em canonical} basis. \begin{pro}\label{75} Let $(A,L)$ be a generic polarized abelian variety of dimension $g$ and type $(1,\ldots,1,d)$. Then $|L|$ is base-point-free if and only if $d>g$. \end{pro} \begin{pf*} If $d\le g$, then $d$ elements in $|L|$ always intersect, since $L$ is ample. Now let $d>g$. It is enough to exhibit one example. The construction is the same as in \cite{BLR}. Let $E_1,\ldots,E_g$ be elliptic curves, let $\epsilon_j$ be a point of order $d$ on $E_j$, for $j=1,\ldots,g$ and let $\pi: E_1\times\cdots\times E_g\to A$ be the quotient by the subgroup generated by $\epsilon_j-\epsilon_k$, for $1\le j<k\le g$. Let $0_j$ be the origin of $E_j$, let $L_j={\cal O}_{E_j}(d 0_j)$ and let $M$ be the polarization $\bigotimes_{j=1}^g {\rm pr}_j^*L_j$ on $E_1\times\cdots\times E_g$. Pick a lift $\tilde\epsilon_j$ of $\epsilon_j$ of order $d$ in ${\cal G}(L_j)$. Then the $\tilde\epsilon_j\tilde\epsilon_1^{-1}$, for $1< j\le g$, generate a level subgroup of ${\cal G}(M)$ hence, by \cite[prop.~1]{M2}, there exists a polarization $L$ on $A$ of type $(1,\ldots,1,d)$ such that $\pi^*L=M$. Moreover, if $\{ s_{j,\lambda}\} _{0\le\lambda\le d-1}$ is a canonical basis for $(E_j,L_j,\tilde\epsilon_j)$, then $\{ s_{1,\lambda}s_{2,\lambda}\cdots s_{g,\lambda}\} _{0\le\lambda\le d-1}$ is a basis for $\pi^*H^0(A,L)$.... If all these sections vanish at some point $(e_1,\ldots,e_g)$ and $d>g$, then at least two different $s_{j \alpha}$, $s_{j \beta}$ must vanish at $e_j$ for some $j$. But this cannot happen since $\operatorname{div}(s_{j\beta})$ and $\operatorname{div}(s_{j\alpha})$ are distinct and are both translates of the divisor $\sum_{l=1}^d (l\epsilon_j)$. Hence $L$ is base-point-free. \end{pf*} \begin{rems} \normalshape \begin{assertions} \item A similar argument shows that for $0<d\le g$, the generic base locus has dimension exactly $g-d$. \item By an argument similar to the one used in \cite{M3}, the following can be shown. Let ${\cal A}$ be a moduli space of polarized abelian varieties of dimension~$g$ and degree~$g+1$. Then the locus of polarized abelian varieties for which the corresponding linear system has a base point is either ${\cal A}$ or a divisor. The proposition implies that it is a divisor when the type is $(1,\ldots,1,g+1)$. It may happen that this locus be everything (e.g. for type $(1,2,2)$ (cf. also \cite{NR})). \end{assertions} \end{rems} In view of theorem~\ref{157}, it is tempting to make the following \begin{con} Let $(A,L)$ be a generic polarized abelian variety of dimension $g>2$ and type $(1,\ldots,1,d)$ with $d>g$ and let $\phi: A\to {{\Bbb P}}^{d-1}$ be the morphism associated with $|L|$ (cf. proposition~\ref{75}). Then the ramification of $\phi$ has dimension $2g-d$ and there exists a closed subset $F$ of $A$ of dimension~$2g+1-d$ such that the restriction of $\phi$ to $A-F$ is an embedding. \end{con} It is of course understood that a set of negative dimension is empty. In particular, the conjecture implies that for $d\ge g+2$, the morphism $\phi$ should be birational onto its image. This is proven below (proposition~\ref{76}). For $d\ge 2g+2$, the line bundle $L$ should be very ample. In \S5, we will prove that this is the case for $d>2^g$ (see theorem~\ref{71}). The following proposition shows that, to prove the conjecture for given $d$ and~$g$, it is enough to exhibity {\em one} polarized abelian variety, or a suitable degeneration, for which it holds. \begin{pro}\label{77} Consider a commutative diagram \[ \begin{CD} X @>f>> Y \\ @VgVV @VVhV \\ W @= W, \end{CD} \] where $X$, $Y$, $W$ are analytic (resp. algebraic) varieties and $f$, $g$ and $h$ are proper. Let $p$ be a point in $W$, let $X_p$ be the fibre of $f$ over $p$ and assume that there is a closed (resp. Zariski closed) subset $F_p$ of $X_p$ such that the restriction of $f$ to $X_p - F_p$ is unramified (an embedding). Then, for all points $w$ in an open (resp. Zariski open) neighbourhood of $p$ in $W$, there exists a closed (resp. Zariski closed) subset $F_w$ of $X_w$ with $\dim(F_w)\le \dim(F_p)$ such that the restriction of $f$ to $X_w -F_w$ is unramified (an embedding). \end{pro} \begin{pf*} Let $G$ be the support of $\Omega_{X/Y}$. Since $\Omega_{X/Y} \otimes {\cal O}_{X_p}\cong \Omega_{X_p/Y_p}$ and $f|_{X_p-F_p}$ is unramified, $G\cap X_p$ is contained in $F_p$. By semicontinuity of the dimension of the fibres of a morphism \cite[prop.~3.4, p.~134]{Fi}, this proves the first part of the proposition, since $f$ is unramified outside of $G$. Assume now that $f|_{X_p-F_p}$ is an embedding. Let $Z$ be the union of the components of $X\times_Y X$ other than the diagonal $\Delta_X$, whose image in $W$ contains $p$. Then, by definition of $\Omega_{X/Y}$, the set $Z\cap\Delta_X$ is contained in $G\times G$; moreover, since $f|_{X_p-F_p}$ is injective, $Z_p-\Delta_{X_p}$ is contained in $F_p' \times F_p'$, where $F_p'=f^{-1}(f(F_p))$. Therefore, $Z_p$ is contained in $F_p'\times F_p'$. Since $F_p'-F_p$ embeds in $f(F_p)$, one has $\dim(F_p')=\dim(F_p)$. This finishes the proof: one can take for $F_w$ the first projection of $Z_w$ in $X_w$. \end{pf*} \begin{pro}\label{76} Let $(A,L)$ be a generic polarized abelian variety of dimension~$g$ and type $(1,\ldots,1,d)$ with $d\ge g+2$. Then the morphism $\phi: A \to {{\Bbb P}}^{d-1}$ associated with $|L|$ is birational onto its image. \end{pro} \begin{pf*} Let ${\cal A}_{g,d,\theta}$ be the moduli space of abelian varieties $A$ of dimension $g$, with a polarization $L$ of type $(1,\ldots,1,d)$ and a point $\tilde\epsilon$ of order $d$ in ${\cal G}(L)$. Let $P=(A,L,\tilde\epsilon)$ be a point in ${\cal A}_{g,d,\theta}$, and let $\{ s_0,\ldots ,s_{d-1}\}$ of be a corresponding canonical basis for $H^0(A,L)$. We first claim that for $P$ in a dense open set $U$ of ${\cal A}_{g,d,\theta}$, no $g+1$ distinct sections in the canonical basis have a common zero. Since ${\cal A}_{g,d,\theta}$ is irreducible \cite[chapter 8, \S3]{BL}, it is enough to find one point $P$ for which this property holds. This follows directly from the proof of proposition 1: keeping the same notation, the point $\tilde\epsilon_1$ of ${\cal G}(L_1)$ corresponds to a point $\tilde\epsilon$ of ${\cal G}(L)$ of order $d$ \cite[prop.~2]{M2}, and the basis for $H^0(A,L)$ given in the proof is canonical for $(A,L,\tilde\epsilon)$; no $g+1$ elements of this basis vanish simultaneously. Now let $(A,L,\tilde\epsilon)$ be an element of $U$ with $A$ {\em simple} (i.e. such that $A$ contains no non-trivial abelian subvarieties). The choice of a canonical basis defines a morphism \[ \phi :A\to {\Bbb P} ^{d-1}. \] Assume that $\phi$ is not birational over its image. Then there exists a component $D$ of $A\times_{{\Bbb P} ^{d-1}}A$, distinct from the diagonal, such that the first projection $D\to A$ is surjective. In particular, $\dim (D)\ge g$. Let $m$ be the morphism \begin{align*} m: D &\to A\\ (x,y) &\mapsto x-y. \end{align*} Let $a$ be a generic point in $m(D)$, let $F$ be an irreducible component of $m^{-1}(a)$ and let $G=\operatorname{pr}_2(F)$. Then \[ F=\{ (x+a,x) ;\, x\in G\}. \] This implies that $\phi(x) =\phi(x+a)$ for all $x\in G$, hence $L|_G \cong \tau_{-a}^{\ast} L|_G$. (For any $x\in A$, $\tau_x: A\to A$ denotes translation over $x$.) Therefore, $a$ lies in the kernel of \begin{align*} A & \to \operatorname {Pic}^0 (\Gamma)\\ x &\mapsto (\tau_x^{\ast} L \otimes L^{-1})|_\Gamma. \end{align*} Since $A$ is simple, it follows that either $a$ is torsion, in which case $m$ is constant with image $a$, or else $m^{-1}(a)$ is finite, in which case $m$ is surjective. In the first case, all elements of $|L|$ are invariant by translation by $a$, which implies $a=0$ and contradicts our choice of $D$. In the second case, there exists $x\in A$ such that $\phi (x)=\phi(x+\epsilon)$, where $\epsilon$ is the image of $\tilde\epsilon$ in $K(L)$. Since \[ \phi (x+\epsilon)=(s_0(x),\omega s_1(x),\ldots ,\omega^{d-1}s_{d-1}(x)), \] where $\omega= e^{2i\pi/d}$, it follows that all of the $d$ sections in the canonical basis but one vanish at $x$. When $d\ge g+2$, this contradicts the fact that $P\in U$. It follows that $\phi$ is birational onto its image. \end{pf*} \section{Degeneration of abelian varieties}\label{70} \subsection{Abelian varieties}\label{50} We consider the Siegel space of degree $g$, i.e. \[ {\cal H}_g = \{ \tau \in M(g \times g, {\Bbb C});\, \tau = \tr \tau, \operatorname{Im} \tau >0\}. \] Fix an integer $d\ge1$. Then every point \[ \tau=\T\in{\cal H}_g \] defines a $(1,\ldots,1,d)$-polarized abelian variety, namely \[ A_{\tau} = {\Bbb C}^g / L_{\tau}, \] where $ L_{\tau}$ is the lattice spanned by the rows of the period matrix \[ \Omega_{\tau}=\begin{pmatrix} \tau\\ D \end{pmatrix},\qquad D=\operatorname{diag}(1,\ldots,1,d). \] We are interested in certain degenerations of these abelian varieties, namely those which arise if $\tau_{11},\ldots, \tau_{g-1,g-1}$ go to $i \infty$. We shall first treat the principally polarized case, i.e. $d=1$. The general case can then be derived easily from this. We will employ the following notation \begin{align*} {\bold{e}}: {\Bbb C} &\to {\C^{\ast}}\\ z &\mapsto e^{2\pi i z}. \end{align*} Let \[ t_{ij}={\bold{e}}(\tau_{ij}). \] Let $z_1,\ldots,z_g$ be the standard coordinates on ${\Bbb C}^g$ and let \[ w_i={\bold{e}}(z_i). \] The abelian variety $A_{\tau}$ can be written as a quotient \[ A_{\tau}={\Bbb Z}^g\backslash ({\C^{\ast}})^g, \] where $l=(l_1,\ldots,l_g)\in{\Bbb Z}^g$ operates on $({\C^{\ast}})^g$ by \begin{align*}\label{3} l(w_1,\ldots,w_g) &=(w_1',\ldots,w_g')\\ w_i' &=\prod_{j=1}^g t_{ij}^{l_j}w_i. \end{align*} We are interested in what happens as $t_{11},\ldots,t_{g-1,g-1}$ go to zero. \subsection{Toroidal embedding} Recall that $\gamma\in \operatorname{Sp}(2g,{\Bbb Z})$ operates on ${\cal H}_g$ by \[ \gamma = \begin{pmatrix} A & B\\ C & D \end{pmatrix}: \tau \mapsto (A \tau + B)(C \tau + D)^{-1}. \] (Here $A$, $B$, $C$ and $D$ are $(g \times g)$-matrices.) The quotient \( \operatorname{Sp}(2g,{\Bbb Z})\backslash {\cal H}_g \) is the moduli space of principally polarized abelian varieties of dimension~$g$. The symplectic group $\operatorname{Sp}(2g,{\Bbb Z})$ contains the lattice subgroup \[ {\cal P} = \left\{ \begin{pmatrix} {\bold{1}} & N\\ 0 & {\bold{1}} \end{pmatrix};\, N \in \operatorname{Sym}(g, {\Bbb Z}) \right\}, \] which acts on ${\cal H}_g$ by $\tau \mapsto \tau + N$. We consider the partial quotient \[ {\cal P} \backslash {\cal H}_g \subset \operatorname{Sym}(g,{\Bbb Z})\backslash \operatorname{Sym}(g,{\Bbb C}). \] Using the coordinates $t_{ij}$, one can make the identification \[ \operatorname{Sym}(g,{\Bbb Z})\backslash \operatorname{Sym}(g,{\Bbb C}) \cong ({\C^{\ast}})^{{\frac 1 2} g(g+1)}. \] We use the standard coordinates $T_{ij}$ on $\operatorname{Sym}(g,{\Bbb C}) \cong {\Bbb C}^{{\frac 1 2} g(g+1)}$ and consider the embedding \[ \phi_0: ({\C^{\ast}})^{{\frac 1 2} g(g+1)} \to {\Bbb C}^{{\frac 1 2} g(g+1)} \] given by \begin{equation}\label{1} T_{ij}=\begin{cases} \prod_{k=1}^g t_{ki}& \text{if $i=j$}\\ t_{ij}^{-1} & \text{if $i\neq j$.} \end{cases} \end{equation} The image of $({\C^{\ast}})^{{\frac 1 2} g(g+1)}$ under $\phi_0$ is the standard torus $({\C^{\ast}})^{{\frac 1 2} g(g+1)}$ in ${\Bbb C}^{{\frac 1 2} g(g+1)}$ and the inverse of $\phi_0$ is given by \begin{equation} t_{ij}=\begin{cases} \prod_{k=1}^g T_{ki}& \text{if $i=j$}\\ T_{ij}^{-1} & \text{if $i\neq j$.} \end{cases} \end{equation} The reason why we are interested in the map $\phi_0$ is that it is closely related to the so-called principal cone (see \cite[p.~93]{N}), which plays an essential role in the reduction theory of quadratic forms. The embedding $\phi_0$ is also a central building block for toroidal compactifications of ${\cal A}_g$. In fact we have \begin{lem} The embedding given by $\phi_0$ is the toroidal embedding corresponding to the principal cone . \end{lem} \begin{pf*} Let $n_{ij}\in\operatorname{Sym}(g,{\Bbb Z})$ be the matrix defined by \[ (n_{ij})_{kl}=\begin{cases} 1 & \text{if $\{i,j\}=\{k,l\}$}\\ 0 & \text{otherwise.} \end{cases} \] The set $\{n_{ij}\}_{1\le i\le j\le g}$ is a basis of $\operatorname{Sym}(g,{\Bbb Z})$. We call it the standard basis. We define another basis $\{n'_{ij}\}_{1\le i\le j\le g}$ by \[ n'_{ij}=n_{ii}+n_{jj}-n_{ij}. \] This basis defines the principal cone. In the notation of \cite[p.~5]{O}, we have \[ T_{ij}={\bold{e}}(m'_{ij})\quad\text{and}\quad t_{ij}={\bold{e}}(m_{ij}), \] where $\{ m_{ij}\}$ is the dual of the standard basis and $\{ m'_{ij}\}$ is the dual of the basis which defines the principal cone. The relation between the two bases for $\operatorname{Sym}(g,{\Bbb Z})$ gives a relation between the two dual bases and this yields the required relation between $T_{ij}$ and $t_{ij}$. \end{pf*} \subsection{Mumford's construction} Let \[ X_0 = \phi_0({\cal P} \backslash {\cal H}_g) \subset {\Bbb C}^{{\frac 1 2} g(g+1)}. \] Recall from section \ref{50} that $A_{\tau}$ is the quotient of $({\C^{\ast}})^g$ by the rank-$g$ lattice generated by \begin{equation} \begin{aligned}\label{51} r_i&=(t_{1i},\ldots,t_{gi})\\ &=(T_{1i}^{-1},\ldots,\prod_{k=1}^g T_{ki},\ldots,T_{gi}^{-1}) \qquad\text{($i=1,\ldots,g$).} \end{aligned} \end{equation} This shows that there is a family of abelian varieties ${\cal A}_0 \to X_0$ such that for each point $[\tau]\in X_0$ the fibre ${\cal A}_{[\tau]}$ is isomorphic to $A_{\tau}$. We now want to add \lq \lq boundary points\rq \rq, i.e. we consider the set $X$, which is defined to be the interior of the closure of $X_0$ in ${\Bbb C}^{{\frac 1 2} g(g+1)}$. Mumford's construction enables us to extend the family ${\cal A}_0 \to X_0$ to a family ${\cal A} \to X$ by adding degenerate abelian varieties over the boundary points. For this purpose we consider \[ A={\Bbb C}[T_{ij}],\quad I= (T_{ij})\subset A. \] Let $K$ be the quotient field of $A$ and consider the torus \[ \tilde{G}=\operatorname {Spec} A[w_1,\ldots,w_g,w_1^{-1},\ldots,w_g^{-1}] = ({\C^{\ast}})^g \times \operatorname {Spec} A. \] By $\tilde{G}(K)$ we denote the $K$-valued points of $\tilde{G}$. The character group ${\Bbb X}=\operatorname {Hom}(({\C^{\ast}})^g,{\C^{\ast}})$ is spanned by $w_1,\ldots, w_g$. Finally, we consider the lattice ${\Bbb Y}\subset \tilde{G}(K)$ which is spanned by the $r_1,\ldots,r_g$ from~(\ref{51}). Following the terminology of \cite{M}, we shall call ${\Bbb Y}$ the period lattice. Let \begin{align*} \Phi:{\Bbb Y} &\to {\Bbb X}\\ r_i &\mapsto w_i. \end{align*} \begin{lem}\label{52} The homomorphism $\Phi$ is a polarization in the sense of Mumford. \end{lem} \begin{pf*} Let $y=\sum y_i r_i$ and $z=\sum z_i r_i\in{\Bbb Y}$. Then the character $X^{\Phi(y)} =\prod w_i^{y_i}$ given by $\Phi(y)$ satisfies \begin{equation}\label{2} \begin{align*} X^{\Phi(y)}(z)&=\prod_{i=1}^g \left( \Big( \prod_{j\neq i} T_{ij}^{-z_j} \Big) \Big( \prod_{k=1}^g T_{ki}\Big)^{z_i} \right)^{y_i}\\ &= \prod_{i=1}^g T_{ii}^{y_i z_i}\cdot \prod_{i<j} T_{ij}^{(y_i-y_j)(z_i-z_j)}. \end{align*} \end{equation} Hence $X^{\Phi(y)}(z)=X^{\Phi(z)}(y)$ and $X^{\Phi(y)}(y)\in I$ unless $y=0$. It follows that $\Phi$ is a polarization in the sense of Mumford. \end{pf*} \begin{rem} \normalshape In fact $\Phi:{\Bbb Y} \to {\Bbb X}$ is an isomorphism, hence it is a principal polarization in the sense of \cite[chapter II]{HKW}. \end{rem} Before we can explain Mumford's construction, we still have to choose a star $\Sigma\subset{\Bbb X}$. For $\alpha$, $\beta\in{\Bbb R}^g$, we say that $\alpha\ge\beta$ if $\alpha_i\ge \beta_i$ for $i=1,\ldots,g$. If furthermore $\alpha\neq \beta$, then we say that $\alpha>\beta$. Now set \[ \Sigma'=\{ \alpha\in\{0,\pm 1\}^g;\, \alpha\ge0\text{ or }\alpha\le0\}. \] We identify this set with \[ \Sigma =\{X^{\alpha}=\prod w_i^{\alpha_i};\, \alpha\in\Sigma'\}\subset{\Bbb X}. \] This is a star in the sense of \cite{M}. For technical reasons we note \begin{lem}\label{53} For all $y\in {\Bbb Y}$ and for all $\alpha \in \Sigma$, we have $X^{\Phi(y)+\alpha}(y)\in A$. \end{lem} \begin{pf*} The calculations in the proof of lemma \ref{52} show that \[ X^{\Phi(y)+\alpha_i}(y)=\prod_{i=1}^g T_{ii}^{y_i(y_i+\alpha_i)}\cdot \prod_{i<j} T_{ij}^{(y_i-y_j)(y_i-y_j+\alpha_i-\alpha_j)}. \] The claim now follows since $z(z+\beta)\ge0$ if $z\in{\Bbb Z}$ and $\beta\in\{0,\pm 1\}$ and $\alpha_i$, $\alpha_i-\alpha_j\in \{0,\pm 1\}$ for all $\alpha\in\Sigma$. \end{pf*} We are now ready to explain Mumford's construction. As in \cite{M, HKW} we consider the graded ring \[ {\cal R} = \sum_{k=0}^{\infty} \{ K[\ldots,X^{\alpha},\ldots]_{\alpha \in {\Bbb Y}} / (X^{\alpha + \beta} - X^{\alpha} X^{\beta}, X^0 -1) \} \theta^k, \] where $\theta$ is an indeterminate of degree 1 and all other elements have degree 0. Let $R_{\Phi,\Sigma}$ be the subring of ${\cal R}$ given by \[ R_{\Phi,\Sigma} = A[\ldots, X^{\Phi(y)+\alpha}(y)X^{2\Phi(y)+\alpha}, \ldots]_{\alpha\in\Sigma, y\in{\Bbb Y}} \] By lemma \ref{53} \[ R_{\Phi,\Sigma} \subset A[\ldots,X^{\alpha}\theta,\ldots]_{\alpha\in{\Bbb Y}}. \] Let \[ \tilde{P}=\operatorname {Proj} R_{\Phi,\Sigma}. \] This is a scheme over $\operatorname {Spec} A$. The group ${\Bbb Y}$ acts on $\tilde{P}$ and the desired extension ${\cal A}\to X$ of ${\cal A}_0 \to X_0$ is given by \[ {\cal A} = {\Bbb Y} \backslash (\tilde{P}|_X). \] The scheme $\tilde{P}$ is covered by the affine open sets \[ U_{\alpha,y}=\operatorname {Spec} R_{\alpha,y}, \] where \[ R_{\alpha,y}=A[\ldots,\frac{X^{\Phi(z)+\beta}(z)}{X^{\Phi(y)+\alpha}(y)} X^{2\Phi(z-y)+\beta-\alpha},\ldots]_{\beta\in\Sigma, z\in{\Bbb Y}}, \] as $\alpha$ runs through $\Sigma$ and $y$ runs through ${\Bbb Y}$. The action of $y\in{\Bbb Y}$ on $\tilde{P}$ identifies $U_{\alpha,0}$ with $U_{\alpha,y}$, so it suffices to calculate \begin{gather*} U_{\alpha}=U_{\alpha,0}=\operatorname {Spec} R_{\alpha,0}\\ R_{\alpha}=R_{\alpha,0}=A[\ldots,M_{\beta,z},\ldots]_{\beta\in\Sigma, z\in{\Bbb Y}}, \end{gather*} where \begin{align*}\label{54} M_{\beta,z}&=X^{\Phi(z)+\beta}(z)X^{2\Phi(z)+\beta-\alpha}\\ &=\prod_{i=1}^g T_{ii}^{z_i(z_i+\beta_i)}\cdot \prod_{i<j} T_{ij}^{(z_i-z_j)(z_i-z_j+\beta_i-\beta_j)}\cdot \prod_{i=1}^g w_i^{2z_i+\beta_i-\alpha_i}. \end{align*} We are not interested in {\em all} degenerations of abelian varieties arising from this construction, but only in those which correspond to $\tau_{11},\ldots, \tau_{g-1,g-1} \to i \infty$. Hence we can fix the entries $\tau_{ij}$ for $i\neq j$ in the matrix $\tau$. This corresponds to fixing the coordinates $T_{ij}=t_{ij}^{-1}$ for $i\neq j$ and hence defines an affine subspace \[ L=L(\tau_{ij}) \subset {\Bbb C}^{{\frac 1 2} g(g+1)}. \] We can use $T_{11},\ldots,T_{gg}$ as coordinates on $L$. For the sake of simplicity, we introduce the notation \begin{align*} \tau_i&=\tau_{ii}\\ T_i&=T_{ii},\qquad i=1,\ldots,g, \end{align*} and consider the ring \[ A'={\Bbb C}[T_1,\ldots,T_g]. \] By abuse of notation we shall denote the restriction of $\tilde{P}$ (resp. $U_{\alpha}$) to $L$ also by $\tilde{P}$ (resp. $U_{\alpha}$). Then \[ U_{\alpha}=\operatorname {Spec} R_{\alpha}, \] where now \[ R_{\alpha}=A'[\ldots,M_{\beta,z},\ldots]_{\beta\in\Sigma, z\in{\Bbb Y}}, \] and \[ M_{\beta,z}=\prod_{i=1}^g T_{i}^{z_i(z_i+\beta_i)}\cdot \prod_{i=1}^g w_i^{2z_i+\beta_i-\alpha_i}. \] \begin{pro}\label{55} For $\alpha\in\Sigma$, we have $R_{\alpha}=A'[X_1,\ldots,X_g,Y_1,\ldots,Y_g]$, where \[ X_i=\begin{cases} T_i^{\alpha_i} w_i &\text{if $\alpha\ge0$}\\ w_i & \text{if $\alpha\le 0$} \end{cases} \qquad Y_i=\begin{cases} w_i^{-1} &\text{if $\alpha\ge0$}\\ T_i^{-\alpha_i} w_i^{-1} & \text{if $\alpha\le 0$.} \end{cases} \] \end{pro} \begin{pf*} This follows easily from the observation that $z(z+\beta)-(2z+\beta)\ge -1$ if $z\in{\Bbb Z}$ and $\beta\in\{0,\pm 1\}$. \end{pf*} \begin{cor}\label{79} The scheme $\tilde{P}$ is smooth. \end{cor} \begin{pf*} It is enough to show that all $U_{\alpha}$ are smooth. If $\alpha=0$, then $U_{\alpha}= {\Bbb C}^g \times ({\C^{\ast}})^g$. Now let $\alpha\neq 0$. We treat the case $\alpha=(1,\ldots,1)$, the other cases being similar. Then \[ U_{(1,\ldots,1)}=\operatorname {Spec} ({\Bbb C}[T_1,\ldots,T_g,Z_1,\ldots,Z_{2g}] / (Z_i Z_{i+g} -T_i)). \] Projecting onto $\operatorname {Spec} {\Bbb C}[T_1,\ldots,T_g,Z_1,\ldots,Z_{2g}]$ shows that $U_{(1,\ldots,1)}\cong {\Bbb C}^{2g}$. \end{pf*} Now we consider the set \[ V=X \cap L. \] One easily checks that $V$ contains the lines where all $T_i$ but one are zero. Let $\tilde{P}_V$ be the restriction of $\tilde{P}$ to $V$. \begin{pro} \begin{assertions} \item The group of periods ${\Bbb Y}$ acts freely and properly discontinuously on $\tilde{P}_V$. \item The quotient ${\cal A}_V = {\Bbb Y} \backslash \tilde{P}_V$ is smooth. Moreover, the family ${\cal A}_V \to V$ is flat. It extends the family ${\cal A}_0|_{V\cap X_0}$. In particular, the general fibre is a smooth abelian $g$-fold. \end{assertions} \end{pro} \begin{pf*} \begin{assertions} \item This can be done as in \cite[theorem 3.14 (i)]{HKW}. \item Smoothness follows from (i) and corollary \ref{79}. The family is flat since ${\cal A}_V$ is smooth of dimension $2g$ and every fibre has dimension $g$. It extends ${\cal A}_0|_{V\cap X_0}$ by construction. \end{assertions} \end{pf*} \subsection{Description of the degenerate abelian varieties} We now want to describe the fibre of ${\cal A}_V$ over a point $p=(0,\ldots,0,T_g)$ with $T_g\neq0$. We shall denote this fibre by $A_p$ and we shall denote the fibres of $\tilde{P}$, $U_{\alpha}$ and $U_{\alpha,y}$ over this point by $\tilde{P}_p$ , $(U_{\alpha})_p$ and $(U_{\alpha,y})_p$ respectively. Recall that \[ (U_{\alpha})_p = \operatorname {Spec} (R_{\alpha})_p, \] where \[ (R_{\alpha})_p = {\Bbb C} [X_1,\ldots,X_g, Y_1,\ldots,Y_g] = {\Bbb C} [X_1,\ldots,X_{g-1}, Y_1,\ldots,Y_{g-1}, w_g, w_g^{-1}] \] with $X_i$ and $Y_i$ as in proposition \ref{55}. We first consider $(U_{\alpha})_p$. Clearly $(U_{0})_p = ({\C^{\ast}})^g$. In general, $(U_{\alpha})_p$ consists of $2^h$ irreducible components, where \[ h = \# \{ i\, ;\, 1 \le i \le g-1, \alpha_i\neq0 \}. \] It is singular for $h\ge1$. Its regular part is the disjoint union of $2^h$ tori. These tori can be described as follows. Consider \[ w_i = T_i^{-\beta_i},\qquad i=1,\ldots,g, \] where $\beta_i\in \{0, -\alpha_i\}$ for $i=1,\ldots,g-1$. Outside the hyperplanes $\{T_i=0\}$, this defines a section of the torus bundle $U_0$. Note that $U_0 = U_{\alpha}$ on $L- \cup_i \{T_i=0\}$. This section can be extended to a section of $U_{\alpha}$ over $p$, where it meets exactly one of the $2^h$ tori whose union is the smooth part of $(U_{\alpha})_p$. This shows also that $(U_{\alpha})_p \subset (U_{\beta})_p$ if and only if $\beta\ge\alpha\ge0$ or $\beta\le\alpha\le0$. The subgroup $\langle r_g \rangle$ of ${\Bbb Y}$ generated by $r_g$ acts on $U_0$ and hence also on $(U_p)_0$. It also acts on $\operatorname {Spec}{\Bbb C}[w_g, w_g^{-1}] = {\C^{\ast}}$ by \[ r_g(w_g) = {\bold{e}}(\tau_g) w_g. \] The inclusion of rings \[ {\Bbb C}[w_g, w_g^{-1}] \subset (R_{\alpha})_p \] defines a map \[ (U_0)_p \to {\C^{\ast}}, \] which is equivariant with respect to the action of $\langle r_g \rangle$. In this way we get a semi-abelian variety of rank $g-1$, i.e. an extension \[ 0 \to ({\C^{\ast}})^{g-1} \to {\Bbb Z} \backslash (U_0)_p \to E_{\tau_g,1} \to 0, \] where $E_{\tau_g,1}$ is the elliptic curve \[ E_{\tau_g,1} = {\Bbb C} / ({\Bbb Z} + {\Bbb Z} \tau_g). \] The closure of $(U_0)_p$ in $\tilde{P}_p$ has the structure of a $({\Bbb P}^1)^{g-1}$-bundle over ${\C^{\ast}}$. Taking the quotient by $\langle r_g \rangle$ gives rise to a $({\Bbb P}^1)^{g-1}$-bundle over the elliptic curve $E_{\tau_g,1}$. We now return our attention to $\tilde{P}_p$. Recall that \[ \tilde{P}_p = \cup_{\alpha\in\Sigma,y\in{\Bbb Y}} (U_{\alpha,y})_p. \] It follows from our observations above that the regular part of $\tilde{P}_p$ is the union of countably many tori. These tori can be labelled in a natural way by elements $(l_1,\ldots,l_{g-1})\in{\Bbb Z}^{g-1}$: the section given outside the union of the hyperplanes $\{T_i=0\}$ by \[ w_i=T_i^{-l_i} \] can be extended over the point $p$ where it meets exactly one of the tori contained in $\tilde{P}_p$. We shall label this torus by $(l_1,\ldots,l_{g-1})$. The element $r_i$ of ${\Bbb Y}$ ($i=1,\ldots,g-1$) then maps the torus $(l_1,\ldots,l_i,\ldots,l_{g-1})$ to the torus $(l_1,\ldots,l_i-1,\ldots,l_{g-1})$, whereas $r_g$ maps each of these tori to itself. We can now summarize our discussion in \begin{pro}\label{56} Let $A_p = {\Bbb Y} \backslash \tilde{P}_p$ be the fibre of ${\cal A}_ V$ over the point $(0,\ldots,0,T_g)$ of~$V$ with $T_g\neq0$. Then the following holds \begin{assertions} \item The regular part $A_p^{\text{reg}}$ of $A_p$ is a semi-abelian variety of rank~$g-1$. More precisely, there exists an extension \[ 0 \to ({\C^{\ast}})^{g-1} \to A_p^{\text{reg}} \to E_{\tau_g,1} \to 0. \] \item The normalization of $A_p$ is a $({\Bbb P}^1)^{g-1}$-bundle over the elliptic curve $E_{\tau_g,1}$. The identifications given by the normalization map are induced by the following identifications on ${({\Bbb P}^1)}^{g-1}\times{\C^{\ast}}$ \begin{align*} r_i: & ( w_1,\ldots, w_{i-1},\infty, w_i,\ldots, w_g) \mapsto\\ &\qquad (t_{i1} w_1,\ldots,t_{i,i-1} w_{i-1},0,t_{i,i+1} w_{i+1}, \ldots,t_{ig} w_g), \end{align*} where $i$ runs through $\{1,\ldots,g-1\}$. \end{assertions} \end{pro} \begin{rems}\label{201} \normalshape \begin{assertions} \item Let \[ a_i = [\tau_{ig}] \in E_{\tau_g,1}\qquad i=1,\ldots,g-1. \] Then the action of $r_i$ lies over the translation $x\mapsto x+a_i$ on the elliptic curve $E_{\tau_g,1}$. \item The singularities of $A_p$ can be read off from proposition~\ref{55}, but can also be understood in terms of the identifications described in proposition~\ref{56}~(ii). For every $h\in \{0,\ldots,g-1\}$, there is a locally closed $(g-h)$-dimensional subset of $A_p$, where $2^h$ smooth branches meet and where the Zariski tangent space has dimension $g+h$. The \lq \lq worst\rq \rq\ singularities of $A_p$ occur along an elliptic curve isomorphic to $E_{\tau_g,1}$, where $2^{g-1}$ smooth branches meet. \end{assertions} \end{rems} \subsection{The $(1,\ldots,1,d)$-polarized case.} We now turn to the case of general $d\ge 1$, i.e. we consider the period matrix \[ \Omega_{\tau}=\begin{pmatrix} \tau\\ D \end{pmatrix},\qquad D=\operatorname{diag}(1,\ldots,1,d). \] Dividing out by the last $g$ rows of this period matrix gives a torus $({\C^{\ast}})^g$ with coordinates \[ w_i= \begin{cases} {\bold{e}}(z_i) & i=1,\ldots,g-1\\ {\bold{e}}(z_g/d) & i=g. \end{cases} \] Let \[ t_{ij}= \begin{cases} {\bold{e}}(\tau_{ij}) &\text{if $i,j=1,\ldots,g-1$}\\ {\bold{e}}(\tau_{ig}/d) &\text{if $i=g$ or $j=g$}. \end{cases} \] Then the first $g$ rows of $\Omega_{\tau}$ act on $({\C^{\ast}})^g$ by multiplication by \[ \begin{cases} (t_{i1},\ldots,t_{ig}) &\text{for the $i$-th row, where $i=1,\ldots,g-1$}\\ (t_{1g}^d,\ldots,t_{gg}^d) &\text{for the $g$-th row.} \end{cases} \] Changing the polarization from a principal one to a polarization of type $(1,\ldots,1,d)$ corresponds to changing the group of periods ${\Bbb Y}$ to the subgroup \[ {\Bbb Y}' = \langle r_1,\ldots, r_{g-1}, r_g^d \rangle. \] We shall, therefore, consider the family \[ {\cal A}_ V = {\Bbb Y}' \backslash \tilde{P}_ V. \] Now the general element is a smooth abelian variety with a polarization of type $(1, \ldots, 1,d)$. Proposition~\ref{56} remains unchanged with the one exception that the base curve $E_{\tau_g,1}$ has to be replaced with the elliptic curve \[ E_{\tau_g,d} = {\Bbb C} / ({\Bbb Z} d + {\Bbb Z} \tau_g). \] \section{Degeneration of the polarization}\label{80} Here we shall always consider the case of polarizations of type $(1,\ldots,1,d)$. What we have done in the previous section was to extend the family ${\cal A}_0|_{V\cap X_0}$ to a family ${\cal A}_V$ over $V$. We would now like to construct a relative polarization on ${\cal A}_0|_{V\cap X_0}$ which extends to the family ${\cal A}_V$. Although this can be done, we shall actually do slightly less: since we are only interested in degenerations belonging to points $(0,\ldots,0,T_g)$ with $T_g\neq0$, we shall restrict ourselves to small neighbourhoods of such points. For each $m\in{\Bbb R}^g$, consider the theta-function \begin{align*} \theta_{m,0}: {\cal H}_g\times{\Bbb C}^g &\to {\Bbb C}\\ (\tau,z) &\mapsto \sum_{q\in{\Bbb Z}^g}{\bold{e}}({\frac 1 2} (q+m) \tau \tr (q+m)+(q+m)\tr z), \end{align*} Let \[ s(\tau)=(-{\frac 1 2} \tau_1,\ldots,-{\frac 1 2} \tau_{g-1},0) \] and \[ r=(0,\ldots,0,{\frac{1}{d}}). \] For $k\in{\Bbb Z}$, we can then consider the functions \begin{align*} \theta_k: {\cal H}_g\times {\Bbb C}^g &\to {\Bbb C}\\ (\tau,z) &\mapsto \theta_{kr,0}(\tau,z+s(\tau)). \end{align*} Note that this depends only on the class of $k$ in ${\Z/ d\Z}$. These functions all have the same automorphy factor and hence are sections of a line bundle ${\cal L}_\tau$ on $A_\tau$. In fact, ${\cal L}_\tau$ represents the $(1,\ldots,1,d)$-polarization on $A_\tau$ and the $\theta_k$ define a basis of the space of sections of this line bundle \cite[p.~75]{I}. Let \[ \tau'=\begin{pmatrix \] and set \[ \tau''=\tr (\tau_{1g},\ldots,\tau_{g-1,g}). \] Finally we consider the analogue of the functions $\theta_k$ in one variable, i.e. the functions \begin{align*} \vartheta_k: {\cal H}_1\times{\Bbb C} &\to {\Bbb C}\\ (\tau,z) &\mapsto \sum_{q\in{\Bbb Z}}{\bold{e}}({\frac 1 2} (q+k/d)^2 \tau +(q+k/d)z), \end{align*} which also depend only on $k\in{\Z/ d\Z}$. \begin{pro}\label{57} With the notation of \S\ref{70}, the functions $\theta_k$ can be written in the form \[ \theta_k(\tau,z)=\sum_{q\in{\Bbb Z}^{g-1}} c_q(\tau') \vartheta_k(\tau_g,z_g+q \tau'') \prod_{i=1}^{g-1} t_i^{{\frac 1 2} q_i(q_i-1)}w_i^{q_i}, \] with \[ c_q(\tau')=\prod_{0<i<j<g} t_{ij}^{q_i q_j}. \] \end{pro} \begin{pf*} This follows from a straightforward computation. \end{pf*} \begin{rem} \normalshape The shift $z\mapsto z+s(\tau)$ was introduced in order to obtain integer exponents of the variables $t_i$ in the above description. \end{rem} {}From now on, we fix $\tau'$ and $\tau''$. Let $p=(0,\ldots,0,T_g)$ be an element of $V$ with $T_g\neq0$. For small neighbourhoods $W$ of $p$ in $V$, it follows from proposition~\ref{57} that we may consider the $\theta_k$ as holomorphic functions on $W\times ({\C^{\ast}})^{g-1}\times{\Bbb C}$ (with coordinates $(T_1,\ldots,T_g;w_1,\ldots,w_{g-1}, z_g)$). Let $W_0=W\cap X_0$ and let ${\cal A}_{W_0}$, resp. ${\cal A}_W$ be the restriction of the family ${\cal A}$ to $W_0$, resp. $W$. Since the automorphy factors of the functions $\theta_k$ do not depend on $k$, there exists a line bundle ${\cal L}_0$ on ${\cal A}_{W_0}$ such that the functions $\theta_k$ are sections of this line bundle. The line bundle ${\cal L}_0$ defines a relative polarization on ${\cal A}_{W_0}$. Our aim is to extend the line bundle ${\cal L}_0$ and its sections $\theta_k$ to ${\cal A}_W$. \begin{pro} The line bundle ${\cal L}_0$ on ${\cal A}_{W_0}$ can be extended to a line bundle ${\cal L}$ on ${\cal A}_W$. Moreover, the sections of ${\cal L}_0$ defined by the functions $\theta_k$ can be extended to sections of ${\cal L}$. \end{pro} \begin{pf*} Using proposition~\ref{57}, this can be done in the same way as in \cite[prop.~II.5.13]{HKW} or as in \cite[prop.~4.1.3]{HW}. Therefore, we shall not give all technical details, but only an outline of the proof. We first consider the open part ${\cal U}$ of $\tilde{P}_W$ given by the union of the open sets $(U_{0,y})_W$, where $y\in{\Bbb Y}$. The codimension of the complement of ${\cal U}$ in $\tilde{P}$ is~2. Each open set $(U_{0,y})_W$ is a trivial torus of rank~$g$ over~$W$. For $y=0$, we can use coordinates $(T_1,\ldots,T_g;w_1,\ldots,w_{g-1},w_g)$ to identify $(U_0)_W$ with $W\times ({\C^{\ast}})^g$. Since the function $\vartheta_k(\tau_g, z_g+q\tau'')$ can be expressed in terms of the coordinate $w_g={\bold{e}}(z_g)$, we can view the functions $\theta_k$ as functions on $(U_0)_W$. Similarly, using the action of ${\Bbb Y}$, we can choose coordinates for $(U_{0,y})_W$ for every $y\in{\Bbb Y}$, such that this open set is also identified with $W\times ({\C^{\ast}})^g$. In this way we can consider the $\theta_k$ also as functions on $(U_{0,y})_W$. We can think of ${\cal U}$ as a complex manifold which is obtained by glueing the open sets $(U_{0,y})_W$. For every $y\in{\Bbb Y}$, we consider the $\theta_k$ as sections of the trivial line bundle on $(U_{0,y})_W$. Using the automorphy factors of the functions $\theta_k$, we can glue these trivial line bundles and obtain a line bundle ${\cal L}_{\cal U}$ on ${\cal U}$. This can be done in such a way that the action of ${\Bbb Y}$ on ${\cal U}$ lifts to an action of ${\Bbb Y}$ on ${\cal L}_{\cal U}$. Hence this line bundle descends to a line bundle on ${\Bbb Y}\backslash {\cal U}$. By construction, the functions $\theta_k$ define sections of this line bundle. We have now extended the line bundle ${\cal L}_0$ to an open set of ${\cal A}_W$, whose complement has codimension~2. To extend the bundle to the whole of ${\cal A}_W$, one can either use the Remmert-Stein extension theorem (cf. \cite{HKW, HW}) or one can perform a similar construction using all sets $(U_{\alpha,y})_W$ ($\alpha\in\Sigma$, $y\in{\Bbb Y}$). \end{pf*} For future reference we also note \begin{pro}\label{72} Let $S=\{0,1\}^{g-1}$. Then \begin{equation*}\label{10} \lim_{t_1,\ldots,t_{g-1}\to 0} \theta_k(\tau,z)= \sum_{q\in S} c_q(\tau') \vartheta_k(\tau_g,z_g+q\tau'') w^q, \end{equation*} where \[ w^q=\prod_{i=1}^{g-1} w_i^{q_i}. \] \end{pro} \begin{pf*} This follows from proposition~\ref{57}. \end{pf*} \begin{rem}\label{200} \normalshape We denote the restriction of the line bundle ${\cal L}$ to $A_p$ by ${\cal L}_p$. The functions from proposition~\ref{72} give $d$ sections of ${\cal L}_p$. \end{rem} \section{Very ampleness in the case $d>2^g$} Let $d\ge1$. Given a symmetric matrix \[ \tau'=\begin{pmatrix, \] an element $\tr\tau''=(\tau_{1g},\ldots,\tau_{g-1,g})$ of ${\Bbb C}^{g-1}$ and an element $\tau_g$ of ${\cal H}_1$, we have constructed in \S\ref{70} a degenerate abelian variety $A_p$ of dimension~$g$, whose normalization is a ${({\Bbb P}^1)}^{g-1}$-bundle over the elliptic curve $E=E_{\tau_g,d}={\Bbb C}/({\Bbb Z} d +{\Bbb Z} \tau_g)$. Moreover, there is a commutative diagram \[ \begin{matrix} E & \subset & A_p & \stackrel{\phi}{\longrightarrow} & {{\Bbb P}}^{d-1}\\ \big\uparrow && \big\uparrow\vcenter{\rlap{$\scriptstyle{\rho}$}} && \big\uparrow\\ {\Bbb C}\times\{0,\ldots,0\} &\subset & {\Bbb C}\times {({\Bbb P}^1)}^{g-1} & \stackrel{\Phi}{\longrightarrow} & {\Bbb C}^d, \end{matrix} \] where the map $\Phi$ is defined (with the notation of \S\ref{80}) by \begin{gather*} \Phi = (\phi_0,\ldots,\phi_{d-1})\\ \phi_k(z_g;w_1,\ldots,w_{g-1})= \sum_{q\in S} c_q(\tau')\vartheta_k(\tau_g,z_g+q\tau'')w^q \end{gather*} (cf. proposition~\ref{72}). We want to study the rational map $\phi$. Recall that when $\phi$ is a morphism, $\phi^{\ast}{\cal O}_{{\Bbb P}^{d-1}}(1)$ is the line bundle ${\cal L}_p$ on $A_p$ defined in remark~\ref{200} and note that $\phi(E)$ is a normal elliptic curve of degree $d$ in ${\Bbb P}^{d-1}$. For any $z\in{\Bbb C}$, we let $[z]$ be the image of $z$ in $E$ and we set $a_i=[\tau_{ig}]$ for $i=1,\ldots,g-1$, and $a=\tr (a_1,\ldots,a_{g-1})$. Recall that $S=\{0,1\}^{g-1}$. {}From now on, we assume that $\tau''$ is generic. More precisely, it suffices that \begin{equation}\label{20} \text{{\em the points $a_1,\ldots,a_{g-1}$ of $E$ are independent over ${\Bbb Z}$}}. \end{equation} Then, for any $x\in E$, the subset \[ I(x)=\{x+qa;\, q\in S\} \] of $E$ has $2^{g-1}$ elements. The following lemma is a consequence of the Riemann-Roch theorem. \begin{lem}\label{13} Any set of at most $d-1$ points on $\phi(E)$ is linearly independent. \end{lem} \begin{pro}\label{18} Let $\tau''$ be generic. If $d>2^{g-1}$, then $\phi_0,\ldots,\phi_{d-1}$ have no common zeroes. In particular, $\phi$ is a morphism and ${\cal L}_p$ is base-point-free. \end{pro} \begin{pf*} For any $Z=(z_g;w_1,\ldots,w_{g-1})$ in ${\Bbb C}\times{({\Bbb P}^1)}^{g-1}$, the vector $\Phi(Z)$ is a linear combination of the vectors $(\vartheta_k(\tau_g, z_g +q\tau''))_{k\in{\Z/ d\Z}}$, whose coefficients are not all zero. The proposition then follows from lemma~\ref{13}. \end{pf*} {}From now on, we assume that $d>2^{g-1}$. Let $z_g\in{\Bbb C}$; the morphism $\rho$ induces an isomorphism between $\{z_g\}\times{({\Bbb P}^1)}^{g-1}$ and a closed subscheme of $A_p$\,, which depends only on $x=[z_g]$. We will denote it by~$F_x$. Note that $I(x)\subset F_x$. Since $d>2^{g-1}$, the points of $\phi(I(x))$ are linearly independent. It follows from proposition~\ref{72}, that the restriction of $\phi$ to $F_x$ is a Segre embedding. \begin{pro} Let $\tau''$ be generic. If $d>2^g$, then $\phi$ is injective. \end{pro} \begin{pf*} From proposition~\ref{56}, we see that the restriction of $\rho$ to ${\Bbb C}\times({\Bbb P}^1 -\{\infty\})^{g-1}$ induces a bijection \[ B_p\to A_p, \] where $B_p$ is an open subset of the normalization of $A_p$, fibred over $E$ with fibres isomorphic to $({\Bbb P}^1 -\{\infty\})^{g-1}$. For $x\in E$, we let $F_x^0$ be the image in $A_p$ of the fibre of $x$. It is a subset of $F_x$. Let $x$, $y$ be two points of $E$. Since $d>2^g$, the points of $\phi(I(x)\cup I(y))$ are linearly independent. It follows that if $\phi(F_x^0)$ and $\phi(F_y^0)$ meet, then $x\in I(y)$ and $y\in I(x)$. Condition~(\ref{20}) then implies $x=y$. Hence, for $x\neq y$, the sets $\phi(F_x^0)$ and $\phi(F_y^0)$ do not meet. Since $\phi|_{F_x}$ is an embedding, the lemma is proved. \end{pf*} \begin{thm}\label{71} Let $\tau''$ be generic. For $d>2^g$, the morphism $\phi$ is an embedding. In particular, ${\cal L}_p$ is very ample. \end{thm} \begin{pf*} It remains to prove that the differential is injective on the Zariski tangent spaces. After reordering the coordinates, we may assume that we are at a point \[ P=\rho(z_g;0,\ldots,0,v_{h+1},\ldots,v_{g-1}), \] where $z_g\in{\Bbb C}$, $v_{h+1},\ldots,v_{g-1}\in{\Bbb P}^1-\{0,\infty\}$. The Zariski tangent space of $A_p$ at $P$ has dimension $g+h$ (see remark~\ref{201}). Moreover, $A_p$ has $2^h$ smooth branches at $P$, which are indexed by subsets $K$ of $\{1,\ldots,h\}$. The branch corresponding to $K$ is \[ \rho(\{z_g-\tau_K''\}\times({\Bbb P}^1)^{g-1}), \] where $\tau_K''=\sum_{i\in K} \tau_{ig}$ and in this branch, the point $P_K$ above $P$ is \[ \rho(z_g - \tau_K''; v_1',\ldots,v_{g-1}'), \] where \[ v_l' = \begin{cases} \infty & \text{if $l\in K$}\\ 0 & \text{if $l\in K'=\{1,\ldots,h\}- K$}\\ \prod_{i\in K} t_{il}^{-1} v_l & \text{for $h<l<g$} \end{cases} \] (see proposition~\ref{56}~(ii)). We change the coordinates around $P_K$ by setting \[ w_i'' = \begin{cases} (w_i')^{-1} &\text{if $i\in K$}\\ w_i' &\text{otherwise.} \end{cases} \] In these coordinates, $\Phi$ is given by \[ \Phi(z_g;w_1'',\ldots,w_{g-1}'')= \Big( \sum_{q\in S} c_q \vartheta_k(\tau_g,z_g +q\tau'') (w'')^{{\bold{1}} -q_K} (w'')^{q_{K'} +q_L} \Big)_{k\in{\Z/ d\Z}}, \] where \begin{align*} {\bold{1}} &= (1,\ldots,1)\in S\\ L &= \{ h+1,\ldots,g-1\} \end{align*} and, for $M\subset \{1,\ldots,g-1\}$ and $q\in S$, we define $q_M\in S$ by \[ (q_M)_i = \begin{cases} q_i &\text{if $i\in M$}\\ 0 &\text{otherwise.} \end{cases} \] We need to calculate the corresponding Jacobian matrix at $P_K$ (whose new coordinates are $(z_g-\tau_K''; 0,\ldots,0,v_{h+1}',\ldots,v_{g-1}')$). \noindent {\em Derivative with respect to $x$.\/} In the sum, we need only consider indices $q$ such that \[ q_i = \begin{cases} 1 &\qquad\text{if $i\in K$}\\ 0 &\qquad\text{if $i\in K'$.} \end{cases} \] After setting $r=q-{\bold{1}}_K$, we get \[ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_{{\bold{1}}_K +r} \vartheta_k' (\tau_g,z_g+r\tau'') (v')^{r_L} \Big)_{k\in{\Z/ d\Z}}. \] Since \[ c_{{\bold{1}}_K +r}= \Big( \prod\begin{Sb} i<j\\ i,j\in K\end{Sb} t_{ij}\Big) \Big( \prod\begin{Sb}i\in K\\ j\in L\end{Sb}t_{ij}^{r_{j}}\Big) c_r \] and \[ (v')^{r_L}=\prod\begin{Sb}i\in K\\ l\in L\end{Sb}t_{il}^{-r_l}v_l^{r_l}, \] we get a non-zero multiple of the vector \[ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_{r} \vartheta_k' (\tau_g,z_g+r\tau'') v^{r} \Big)_{k\in{\Z/ d\Z}}. \] \noindent {\em Derivative with respect to $w_{\beta}''$, $\beta\in K$.\/} In the sum, we need only consider indices $q$ such that \[ q_i = \begin{cases} 1 &\qquad\text{if $i\in K -\{\beta\}$}\\ 0 &\qquad\text{if $i\in K'\cup \{\beta\}$.} \end{cases} \] A similar calculation yields (after setting $r=q-{\bold{1}}_{K-\{\beta\}}$) a non-zero multiple of the vector \[ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_r \vartheta_k(\tau_g, z_g-\tau_{\beta g}+r \tau'')v^r \big( \prod_{j\in L} t_{\beta j}^{-r_j} \big) \Big)_{k\in{\Z/ d\Z}}. \] \noindent {\em Derivative with respect to $w_{\beta}''$, $\beta\in K'$.\/} We get a non-zero multiple of \[ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_r \vartheta_k(\tau_g, z_g+\tau_{\beta g}+r \tau'')v^r \big( \prod_{j\in L} t_{\beta j}^{r_j} \big) \Big)_{k\in{\Z/ d\Z}}. \] \noindent {\em Derivative with respect to $w_{\beta}''$, $\beta\in L$.\/} We get a non-zero multiple of \[ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\\ r_{\beta}=1\end{Sb} c_r \vartheta_k(\tau_g, z_g+r \tau'')v^r \Big)_{k\in{\Z/ d\Z}}. \] Altogether, letting $K$ vary among all subsets of $\{1,\ldots,h\}$, we see that the closure of $\operatorname{d}\!\phi (T_P A_p)$ in ${\Bbb P}^{d-1}$ is spanned by the following $(g+h+1)$ points \begin{gather*} \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_{r} \vartheta_k (\tau_g,z_g+r\tau'') v^r \Big)_{k\in{\Z/ d\Z}}\\ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_{r} \vartheta_k' (\tau_g,z_g+r\tau'') v^r \Big)_{k\in{\Z/ d\Z}}\\ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\end{Sb} c_r \vartheta_k(\tau_g, z_g+\epsilon \tau_{\beta g}+r \tau'')v^r \big( \prod_{j\in L} t_{\beta j}^{\epsilon q_j} \big) \Big)_{k\in{\Z/ d\Z}} \end{gather*} for all $(\beta,\epsilon)\in \{1,\ldots,h\}\times\{-1,1\}$, and \[ \Big( \sum\begin{Sb} r\in S\\ r_1 = \cdots =r_h =0\\ r_{\beta}=1\end{Sb} c_r \vartheta_k(\tau_g, z_g+r \tau'')v^r \Big)_{k\in{\Z/ d\Z}} \] for all $\beta\in\{h+1,\ldots,g-1\}$. Since $d>2^g\ge2^{g-h}(h+1)$, it follows from lemma~\ref{13} that the $2^{g-h}(h+1)$ vectors \begin{gather*} \big( \vartheta_k(\tau_g, z_g + r\tau'') \big)_{k\in{\Z/ d\Z}}\\ \big( \vartheta_k'(\tau_g, z_g + r\tau'') \big)_{k\in{\Z/ d\Z}}\\ \big( \vartheta_k(\tau_g, z_g + \epsilon \tau_{\beta g} + r\tau'') \big)_{k\in{\Z/ d\Z}} \end{gather*} for $r\in S$ with $r_1=\cdots=r_h=0$, $\epsilon=\pm 1$ and $\beta\in \{1,\ldots,h\}$, are linearly independent in ${\Bbb C}^d$. This implies that the $(g+h+1)$ vectors above are linearly independent in ${\Bbb C}^d$ and proves that the image of the differential of $\phi$ at $P$ has dimension $g+h$. The differential of $\phi$ is therefore everywhere injective, which proves the theorem. \end{pf*} \begin{cor}~\label{171} Let $(A,L)$ be a generic polarized abelian variety of dimension~$g$ and type $(1,\ldots,1,d)$. For $d>2^g$, the line bundle $L$ is very ample. \end{cor} \begin{rem} \normalshape Similar calculations show that for given $\tau_g$ and $\tau''$ satisfying condition~(4), and for a generic choice of the matrix $\tau'$, the morphism $\phi$ is unramified for $d\ge 2^g-g(g-3)/2$ and is an embedding for $d>2^g -g(g-3)/2$. For $g\ge4$, this improves slightly on the bound in theorem~\ref{71}. However, $\phi$ is never an embedding for $g=3$ and $d=8$: for a generic choice of $\tau'$, it is unramified and identifies (transversally) a finite number of pairs of smooth points of $A_p$. \end{rem}

9306

alg-geom/9306011

en

https://arxiv.org/abs/alg-geom/9306011

alg-geom/9306011

David Cox

Victor V. Batyrev (Essen) and David A. Cox (Amherst College)

On the Hodge Structure of Projective Hypersurfaces in Toric Varieties

43 pages, LaTeX Version 2.09

This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$.

alg-geom

\section{The definition of a simplicial toric variety ${\bf P}_{\Sigma}$} Let $M$ be a free abelian group of rank $d$, $N = {\rm Hom}(M, {\bf Z})$ the dual group. We denote by $M_{\bf R}$ (resp. by $N_{\bf R}$) the ${\bf R}$-scalar extension of $M$ (resp. of $N$). \begin{opr} {\rm A convex subset $\sigma \subset N_{\bf R}$ is called a {\em rational $k$-dimensional simplicial cone} $(k \geq 1)$ if there exist $k$ linearly independent elements $e_1, \ldots, e_k \in N$ such that \[ \sigma = \{ \mu_1 e_1 + \cdots + \mu_k e_k \mid l_i \in {\bf R}, l_i \geq 0 \}. \] We call $e_1, \ldots, e_k \in N$ {\em integral generators of} $\sigma$ if for every $e_i$ $(1 \leq i \leq k)$ and any non-negative rational number $\mu$, $\mu \cdot e_i \in N$ only when $\mu \in {\bf Z}$. The origin $0 \in N_{\bf R}$ is the {\em rational $0$-dimensional simplicial cone}, and the set of integral generators of this cone is empty.} \end{opr} \begin{opr} {\rm A rational simplicial cone $\sigma'$ is called {\em a face} of a rational simplicial cone $\sigma$ (we write $\sigma' \prec \sigma)$ if the set of integral generators of $\sigma'$ is a subset of the set of integral generators of $\sigma$. } \label{face} \end{opr} \begin{opr} {\rm A finite set $\Sigma = \{ \sigma_1, \ldots , \sigma_s \}$ of rational simplicial cones in $N_{\bf R}$ is called {\em a rational simplicial complete d-dimensional fan} if the following conditions are satisfied: \begin{description} \item[{\rm (i)}] if $\sigma \in \Sigma$ and $\sigma' \prec \sigma$, then $\sigma' \in \Sigma$; \item[{\rm (ii)}] if $\sigma$, $\sigma'$ are in $\Sigma$, then $\sigma \cap \sigma' \prec \sigma$ and $\sigma \cap \sigma' \prec \sigma'$; \item[{\rm (iii)}] $N_{\bf R} = \sigma_1 \cup \cdots \cup \sigma_s$. \end{description} The set of all $k$-dimensional cones in $\Sigma$ will be denoted by $\Sigma^{(k)}$.} \end{opr} \begin{exam} {\rm Let $w = \{ w_1, \ldots, w_{d+1}\}$ be the set of positive integers satisfying the condition ${\rm gcd}(w_i) =1$. Choose $d+1$ vectors $ e_1, \ldots, e_{d+1}$ in a $d$-dimensional real space $V$ such $V$ is spanned by $e_1, \ldots, e_{d+1}$ and there exists the linear relation \[ w_1 e_1 + \cdots + w_{d+1} e_{d+1} =0. \] Define $N$ to be the lattice in $V$ consisting of all integral linear combinations of $e_1, \ldots, e_{d+1}$. Obviously, $N_{\bf R} = V$. Let $\Sigma(w)$ be the set of all possible simplicial cones in $V$ generated by proper subsets of $\{ e_1, \ldots, e_{d+1} \}$. Then $\Sigma(w)$ is an example of a rational simplicial complete $d$-dimensional fan. } \label{weig} \end{exam} We want to show how every rational simplicial complete $d$-dimensional fan $\Sigma$ defines a a compact $d$-dimensional complex algebraic variety ${\bf P}_{\Sigma}$ having only quotient singularities. For instance, if $\Sigma$ is a fan $\Sigma(w)$ from Example \ref{weig}, then the corresponding variety ${\bf P}_{\Sigma}$ will be the $d$-dimensional weighted projective space ${\bf P}( w_1, \ldots, w_{d+1})$. The standard defintion of ${\bf P}( w_1, \ldots, w_{d+1})$ describes it as a quotient of ${\bf C}^{d+1} \setminus \{0\}$ by the diagonal action of the multiplicative group ${\bf C}^*$: \[ (z_1, \ldots, z_{d+1} ) \rightarrow (t^{w_1}z_1, \ldots, t^{w_{d+1}}z_{d+1}),\;\; t \in {\bf C}^*. \] In Definition \ref{def.tor}, we will show that the toric variety ${\bf P}_{\Sigma}$ can be constructed in a similar manner. We first need some definitions. \begin{opr} {\rm (\cite{cox}) Let $S(\Sigma) = {\bf C}\lbrack z_1, \ldots, z_n \rbrack$ be the polynomial ring over ${\bf C}$ with variables $z_1, \ldots, z_n$, where $\Sigma^{(1)} = \{\rho_1,\dots,\rho_n\}$ are the $1$-dimensional cones of $\Sigma$. Then, for $\sigma \in \Sigma$, let $\widehat{z}_\sigma = \prod_{\rho_i \not\subset \sigma} z_i$, and let $B(\Sigma) = \langle \widehat{z}_\sigma : \sigma \in \Sigma\rangle \subset S(\Sigma)$ be the ideal generated by the $\widehat{z}_\sigma$'s. } \label{def.B} \end{opr} \begin{opr} \label{def.Z} {\rm Let ${\bf A}^n = {\rm Spec}\, S(\Sigma)$ be the $n$-dimensional affine space over ${\bf C}$ with coordinates $z_1,\dots,z_n$. The ideal $B(\Sigma) \subset S(\Sigma)$ gives the variety \[ Z(\Sigma) = {\bf V}(B(\Sigma)) \subset {\bf A}^n,\] and we get the Zariski open set \[ U(\Sigma) = {\bf A}^n \backslash Z(\Sigma).\]} \end{opr} \begin{opr} {\rm Consider the injective homomorphism $\alpha : M \rightarrow {\bf Z}^n$ defined by \[ \alpha(m) = (\langle m,e_1\rangle, \dots, \langle m,e_n\rangle ). \] The cokernel of this map is ${\bf Z}^n/\alpha(M) \simeq Cl(\Sigma)$, and we define ${\bf D}(\Sigma) : = {\rm Spec}\,{\bf C} \lbrack Cl(\Sigma) \rbrack$ to be $(n-d)$-dimensional commutative affine algebraic $D$-group whose group of characters is isomorphic to $Cl(\Sigma)$ (see \cite{ham}). } \label{diag} \end{opr} \begin{rem} {\rm Notice that the finitely generated abelian group $Cl(\Sigma)$ defines ${\bf D}(\Sigma)$ not only as an abstract commutative affine algebraic $D$-group, but also as a canonically embedded into $({\bf C}^*)^n = {\rm Spec}\,{\bf C} \lbrack {\bf Z}^n \rbrack$ subgroup associated with the surjective homomorphism ${\bf Z}^n \rightarrow {\bf Z}^n / \alpha(M) \simeq Cl(\Sigma)$. So we obtain the canonical diagonal action of ${\bf D}(\Sigma)$ on the affine space ${\bf A}^n$. Obviously, $U(\Sigma)$ is invariant under this action.} \end{rem} Now we are ready to formulate the key theorem. \begin{theo} Let $\Sigma$ be a rational simplicial complete $d$-dimensional fan. Then the canonical action of ${\bf D}(\Sigma)$ on $U(\Sigma)$ has the geometric quotient \[ {\bf P}_{\Sigma} = U(\Sigma)/{\bf D}(\Sigma) \] which is a com\-pact com\-plex $d$-di\-men\-sional al\-geb\-raic va\-riety having only abel\-ian quotient singularities (and hence ${\bf P}_\Sigma$ is a $V$-manifold). \label{key.theo} \end{theo} \noindent {\bf Proof. } The existence of the quotient $U(\Sigma)/{\bf D}(\Sigma)$ was proved in the analytic case by Audin \cite{aud} and in the algebraic case by Cox \cite{cox}. It remains to show that ${\bf P}_\Sigma$ has only abelian quotient singularities. First observe that $Z(\Sigma)$ is defined by the vanishing of $\widehat{z}_\sigma$ for the $d$-dimensional cones $\sigma \in \Sigma^{(d)}$. For such a $\sigma$, set $U_\sigma = \{z \in {\bf A}^n: \widehat{z}_\sigma \ne 0\}$. Thus $U(\Sigma) = \bigcup_{\sigma \in \Sigma^{(d)}} U_\sigma$, and it suffices to show that $U_\sigma/{\bf D}(\Sigma)$ has abelian quotient singularities. If the $1$-dimensional cones of $\Sigma$ are $\Sigma^{(1)} = \{\rho_1, \ldots, \rho_n \}$, put $G(\Sigma) = \{ e_1, \ldots , e_n \}$ where $e_i$ is the integral generator of $\rho_i$. Now renumber $G(\Sigma)$ so that the generators of $\sigma$ are $e_1,\dots,e_d$. Then $U_\sigma = {\bf C}^d\times({\bf C}^*)^{n-d}$. To see how ${\bf D}(\Sigma)$ acts on this set, consider the commutative diagram \begin{equation} \begin{array}{ccccccccc} &&&& 0 && 0 &&\\ &&&& \downarrow && \downarrow &&\\ &&&& {\bf Z}^{n-d} & = & {\bf Z}^{n-d} &&\\ &&&& \downarrow && \downarrow &&\\ 0 & \to & M & \to & {\bf Z}^n & \to & Cl(\Sigma) & \to & 0\\ && \parallel && \downarrow && \downarrow &&\\ 0 & \to & M & \to & {\bf Z}^d & \to & Cl(\sigma) & \to & 0\\ &&&& \downarrow && \downarrow &&\\ &&&& 0 && 0 && \end{array} \label{diagram} \end{equation} where the map $M \to {\bf Z}^d$ is $m \mapsto (\langle m,e_1\rangle, \dots, \langle m,e_d\rangle)$. Since $e_1,\dots,e_d$ are linearly independent, it follows that $Cl(\sigma)$ is a finite group. Then the last column of the diagram gives the exact sequence \begin{equation} 1 \to {\bf D}(\sigma) \to {\bf D}(\Sigma) \to ({\bf C}^*)^{n-d} \to 1. \label{eqD} \end{equation} Since $U_\sigma = {\bf C}^d\times({\bf C}^*)^{n-d}$ and ${\bf D}(\sigma)$ acts naturally on ${\bf C}^d$, it follows that the map ${\bf C}^d \to U_\sigma$ defined by $(t_1,\dots,t_d) \mapsto (t_1,\dots,t_d,1,\dots,1)$ is equivariant. It is also easy to see that the induced map ${\bf C}^d/{\bf D}(\sigma) \to U_\sigma/{\bf D}(\Sigma)$ is an isomorphism. Since ${\bf D}(\sigma)$ is a finite abelian group, the theorem is proved. \hfill $\Box$ \begin{rem} {\rm Audin \cite{aud} and Cox \cite{cox} have shown that ${\bf P}_{\Sigma}$ is isomorphic to the complete toric variety associated with the simplicial fan $\Sigma$ in the usual sense of the theory \cite{dan1,oda} (see \cite{cox} for other people who have discovered this result). Also, \cite{cox} shows that ${\bf A}_\sigma = U_\sigma/{\bf D}(\Sigma) \subset {\bf P}_\Sigma$ is the affine toric open associated to the cone $\sigma \in \Sigma$. We can thus use Theorem \ref{key.theo} as the {\em definition} of toric variety. } \end{rem} \begin{opr} {\rm The complete $d$-dimensional algebraic variety \[ {\bf P}_{\Sigma} = U(\Sigma)/{\bf D}(\Sigma) \] is called the {\em toric variety associated with the complete simplicial fan} $\Sigma$.} \label{def.tor} \end{opr} This construction of a toric variety makes it easy to see the torus action. \begin{opr} {\rm Denote by ${\bf T}(\Sigma)$ the quotient of $({\bf C}^*)^n$ by the subgroup ${\bf D}(\Sigma)$. When $\Sigma$ is fixed, we denote ${\bf T}(\Sigma)$ simply by ${\bf T}$. } \end{opr} \begin{rem} {\rm The group ${\bf T}(\Sigma)$ is isomorphic to the torus $N\otimes{\bf C}^* = ({\bf C}^*)^d$. Since the group $({\bf C}^*)^n$ is an open subset of $U(\Sigma)$, and it acts canonically on $U(\Sigma)$, ${\bf T}(\Sigma)$ is an open subset of ${\bf P}_{\Sigma}$ having the induced action by regular automorphisms of ${\bf P}_{\Sigma}$.} \end{rem} \section{The structure of $Z(\Sigma) = {\bf A}^n \setminus U(\Sigma)$} We will next discuss the closed subvariety $Z(\Sigma) = {\bf A}^n \backslash U(\Sigma)$. It has an interesting combinatorial interpretation. \begin{opr} {\rm (\cite{bat.class}) Let the $1$-dimensional cones of $\Sigma$ be $\Sigma^{(1)} = \{\rho_1, \ldots, \rho_n \}$, and put $G(\Sigma) = \{ e_1, \ldots , e_n \}$ where $e_i$ is the integral generator of $\rho_i$. We call a subset ${\cal P} =\{ e_{i_1}, \ldots , e_{i_p} \} \subset G(\Sigma)$ a {\em primitive collection} if $\{e_{i_1}, \ldots , e_{i_p}\}$ is not the set of generators of a $p$-dimensional simplicial cone in $\Sigma$, while any proper subset of ${\cal P}$ generates a cone in $\Sigma$.} \end{opr} \begin{exam} {\rm Let $\Sigma$ be a fan $\Sigma(w)$ from Example \ref{weig}. Then there exists only one primitive collection in $G(\Sigma(w))$ which coincides with $G(\Sigma(w))$ itself.} \label{primweig} \end{exam} \begin{opr} {\rm Let ${\cal P} =\{ e_{i_1}, \ldots , e_{i_p} \}$ be a primitive collection in $G(\Sigma)$. Define ${\bf A}({\cal P})$ to be the $(n-p)$-dimensional affine subspace in ${\bf A}^n$ having the equations \[ z_{i_1} = \cdots = z_{i_p} = 0. \]} \end{opr} \begin{rem} {\rm Since every primitive collection ${\cal P}$ has at least two elements, the codimension of ${\bf A}({\cal P})$ is at least $2$.} \end{rem} \begin{lem} Let $Z(\Sigma) \subset {\bf A}^n$ be the variety defined in Definition \ref{def.Z}. Then the decomposition of $Z(\Sigma)$ into its irreducible components is given by \[ Z(\Sigma) = \bigcup_{\cal P} {\bf A}({\cal P}), \] where ${\cal P}$ runs over all primitive collections in $G(\Sigma)$. \label{zdecomp} \end{lem} \noindent {\bf Proof. } First, it follows from the definition of $Z(\Sigma) = {\bf V}(\widehat{z}_\sigma : \sigma \in \Sigma)$ that \begin{equation} \label{eqz} Z(\Sigma) = \bigcup_{\cal Q} {\bf A}({\cal Q}), \end{equation} where ${\cal Q}$ runs over all subsets ${\cal Q} \subset G(\Sigma)$ which are not the set of generators of any cone in $\Sigma$. Then note that the set of all such ${\cal Q}$'s are partially ordered by inclusion, and ${\cal Q} \subset {\cal Q}'$ implies ${\bf A}({\cal Q}') \subset {\bf A}({\cal Q})$. Hence, in the above union, it suffices to use the minimal ${\cal Q}$'s, which are precisely the primitive collections. It follows that these give the irreducible components of $Z(\Sigma)$. \hfill $\Box$ \begin{rem} {\rm In \cite{bat.class}, the first author conjectured that for smooth complete toric varieties, the number of primitive collections could be bounded in terms of the Picard number $\rho$ ($= n-d$). Since primitive collections correspond to irreducible components of $Z(\Sigma)$ by the above lemma, we can reformulate this conjecture as follows.} \end{rem} \begin{conj} For any $d$-dimensional smooth complete toric variety with Picard number $\rho$ defined by a complete regular fan $\Sigma$, there exists a constant $N(\rho)$ depending only on $\rho$ such that $Z(\Sigma) \subset {\bf A}^n$ has at most $N(\rho)$ irreducible components. \end{conj} We next study the codimension of $Z(\Sigma)$. When ${\bf P}_\Sigma$ is a weighted projective space, it follows from Example \ref{primweig} that $Z(\Sigma) = \{0\}$, which is as small as possible. It turns out that in most other cases, $Z(\Sigma)$ is considerably larger. The precise result is as follows. \begin{prop} \label{prop.codim} Let ${\bf P}_\Sigma$ be a complete simplicial toric variety of dimension $d$, and let $Z(\Sigma) \subset {\bf A}^n$ be as above. Then either \begin{enumerate} \item $2 \le {\rm codim}\,Z(\Sigma) \le [{1\over2}d]+1$, or \item $n = d+1$ and $Z(\Sigma) = \{0\}$. \end{enumerate} \end{prop} \noindent {\bf Proof. } To illustrate the range of techniques that can be brought to bear on this subject, we will give two proofs of this result. For the first proof, we make the additional assumption that ${\bf P}_\Sigma$ is projective. A projective embedding of ${\bf P}_\Sigma$ is given by a strictly convex support function on $\Sigma$, which determines a convex polytope $\Delta \subset M_{\bf R}$ (see \cite{oda}). Now consider the dual polytope $\Delta^* \subset N_{\bf R}$. Combinatorially, $\Delta^*$ is closely related to the fan $\Sigma$---in fact, $\Sigma$ is the cone over $\Delta^*$. Let $k = [{1\over2}d]+1$ and assume that ${\rm codim}\, Z(\Sigma) > k$. Now pick any subset ${\cal Q} \subset G(\Sigma)$ consisting of $k$ elements (this corresponds to picking $k$ vertices of $\Delta^*$). Then ${\rm codim}\, Z(\Sigma) > k$ implies that ${\bf A}({\cal Q}) \not\subset Z(\Sigma)$. From equation (\ref{eqz}), it follows that ${\cal Q}$ must be the set of generators for some face $\sigma \in \Sigma$. In terms of the polytope $\Delta^*$, this shows that every set of $k$ vertices are the vertices of some face of $\Delta^*$. Since $k > [{1\over2}d]$, standard results about convex polytopes (see Chapter 7 of \cite{grunbaum}) imply that $\Delta^*$ is a simplex, which proves that the number of 1-dimensional cones is $n = d+1$. Our second proof, which applies to arbitrary complete simplicial toric varieties, uses the Stanley-Reisner ring of a certain monomial ideal associated with $\Sigma$. \begin{opr} {\rm Let $\Sigma$ be a complete simplicial fan of dimension $d$. The {\em Stanley-Reisner ideal} $I(\Sigma)$ is the ideal in $S = S(\Sigma)$ generated by all monomials $z_{i_1} \cdots z_{i_p}$ such that $\{ e_{i_1}, \ldots, e_{i_p} \}$ is a primitive collection in $G(\Sigma)$. The quotient $R(\Sigma) = S(\Sigma)/I(\Sigma)$ is called the {\em Stanley-Reisner ring} of the fan $\Sigma$. } \end{opr} As explained in \cite{reisner,stanley}, the ring $R(\Sigma)$ comes from the simplicial complex given by all subsets of $G(\Sigma)$ which correspond to cones of $\Sigma$ (this is because $\Sigma$ is a simplicial fan). But since $\Sigma$ is also complete, the simplicial complex is a triangulation of the $d-1$ sphere. This has some very strong consequences about the ring $R(\Sigma)$. For instance, by \S1 of \cite{reisner}, the irreducible components of the affine variety ${\rm Spec}\, R(\Sigma)$ correspond to maximal faces of the simplicial complex. It follows that ${\rm Spec}\, R(\Sigma)$ is a union of $d$-dimensional linear subspaces in ${\bf A}^n$, one for each $d$-dimensional cone of $\Sigma$. \begin{rem} {\rm One should not confuse $Z(\Sigma)$ with ${\rm Spec}\, R(\Sigma)$. For a weighted projective space as in Example \ref{weig}, $Z(\Sigma) = \{ 0 \}$, but $I(\Sigma)$ is a principal ideal generated by the monomial $z_1 \cdots z_{d+1}$, i.e., ${\rm Spec}\, R(\Sigma)$ is the union of $d+1$ linear subspaces of codimension $1$ in ${\bf A}^{n} = {\bf A}^{d+1}$. } \end{rem} The ring $R(\Sigma)$ has many nice properties. First, it has the natural grading by elements of ${\bf Z}_{\geq 0}$ induced from the ${\bf Z}_{\geq 0}$-grading of the polynomials ring $S$. Second, since the simplicial complex is a triangulation of the $d-1$ sphere, it follows from Chapter II, \S5 of \cite{stanley} that $R(\Sigma)$ is a graded Gorenstein ring of dimension $d$. Then, by \cite{BE}, the minimal free resolution \[ 0 \rightarrow P_{n-d} \stackrel{d_{n-d}}\rightarrow P_{n - d -1} \stackrel{d_{n-d-1}}\rightarrow \cdots \stackrel{d_{2}}\rightarrow P_1 \stackrel{d_{0}}\rightarrow P_0 \rightarrow R(\Sigma) \rightarrow 0 \] of $R(\Sigma)$ as a module over $S$ satisfies the duality property \begin{equation} P_i \cong {\rm Hom}_S (P_{n-d-i}, P_{n-d}) \label{duality} \end{equation} where each $d_i$ is a graded homomorphism of degree $0$ between the graded $S$-modules $P_i$ and $P_{i-1}$. Further, the module $P_0$ is isomorphic to $S$, and since we know the generators of $I(\Sigma)$, we get an isomorphism \begin{equation} P_1 \cong \bigoplus_{{\cal P} \subset G(\Sigma)} S(-|{\cal P}| ) \label{p1} \end{equation} where ${\cal P}$ runs over all primitive collections in $G(\Sigma)$, and $|{\cal P}|$ is the cardinality of ${\cal P}$. Finally, the duality (\ref{duality}) shows that $P_{n-d}$ is a free $S$-module of rank 1. We claim that $P_{n-d} \cong S(-n)$. For each $i$ between $0$ and $n-d$, let $h(i)$ be the minimal integer $h$ such that $P_i$ has a nonzero element of degree $h$. The minimality of the free resolution implies that $0 = h(0) < h(1) < \cdots < h(n-d-1) < h(n-d)$. Since $P_{n-d}$ has rank 1, we have $P_{n-d} = S(-h(n-d))$. Hence it suffices to show $h(n-d) = n$. The Hilbert-Poincare series of $S(-j)$ is $H(S(-j),t) = t^j/(1-t)^n$, so that the free resolution of $R(\Sigma)$ implies that \[H(R(\Sigma),t) = \sum_{i=0}^{n-d} (-1)^i H(P_i,t) = {\hbox{polynomial of degree $h(n-d)$} \over (1-t)^n}.\] However, Theorem 1.4 of Chapter II of \cite{stanley} shows that \[ H(R(\Sigma),t) = {\hbox{polynomial of degree $d$} \over (1-t)^d}.\] Comparing these two expressions, we conclude $h(n-d) = n$. Once we know $P_{n-d} = S(-n)$, the isomorphism (\ref{p1}) and the duality (\ref{duality}) give an isomorphism \begin{equation} P_{n-d-1} \cong \bigoplus_{{\cal P} \subset G(\Sigma)} S(-n+|{\cal P}|). \label{pn-d-1} \end{equation} Assume now that $n > d+ 1$. Then $h(n-d-1) > \cdots > h(1)$ implies \begin{equation} h(n -d -1) \geq h(1) + n - d - 2. \label{ineq} \end{equation} {}From the isomorphisms (\ref{p1}) and (\ref{pn-d-1}), we see that there exist primitive collections ${\cal P}_1$ and ${\cal P}_2$ such that \[ h(n-d-1) = n - |{\cal P}_1|\quad\hbox{and}\quad h(1) = |{\cal P}_2|. \] Then the inequality (\ref{ineq}) implies that \[ |{\cal P}_1|+ |{\cal P}_2| \leq d + 2, \] and thus \[ \min(|{\cal P}_1|,|{\cal P}_2|) \le [\textstyle{1\over2}d] + 1.\] On the other hand, Lemma \ref{zdecomp} implies \[ \min_{{\cal P} \subset G(\Sigma)} |{\cal P}| = \min_{{\cal P} \subset G(\Sigma)} {\rm codim}\, {\bf A}({\cal P}) = {\rm codim}\, Z(\Sigma). \] This proves ${\rm codim}\, Z(\Sigma) \le [{1\over2}d] + 1$ when $n > d+1$. On the other hand, the equality $n = d+1$ is possible only if the minimal projective resoluton of $R(\Sigma)$ consists of $P_0 \cong S$ and $P_1 \cong S(-n)$, which means that $I(\Sigma)$ is the principal ideal generated by the monomial $z_1 \cdots z_n$ of degree $n$. This implies that $\{z_1,\dots,z_n\}$ is the unique primitive collection, and we obtain $Z(\Sigma) = \{0 \}$. This completes the proof of the proposition. \hfill$\Box$ \medskip The condition $n = d+1$ is closely related to ${\bf P}_\Sigma$ being a weighted projective space. \begin{lem} Let $\Sigma$ be a complete simplicial fan with $n = d+1$ 1-dimensional cones. Then there is a weighted projective space ${\bf P}(w_1,\dots,w_{d+1})$ and a finite surjective morphism \[ {\bf P}(w_1,\dots,w_{d+1}) \to {\bf P}_\Sigma. \] Furthermore, if $G(\Sigma) = \{e_1,\dots,e_{d+1}\}$, then the following are equivalent: \begin{enumerate} \item ${\bf P}_\Sigma$ is a weighted projective space. \item ${\bf D}(\Sigma) \cong {\bf C}^*$ (or equivalently, $Cl(\Sigma) \cong {\bf Z}$). \item $e_1,\dots,e_{d+1}$ generate $N$ as a ${\bf Z}$-module. \end{enumerate} \label{lem.nd} \end{lem} \noindent {\bf Proof. } First note that (1) $\Rightarrow$ (2) is immediate. Further, if (3) holds, then Example \ref{weig} shows that ${\bf P}_\Sigma$ is a weighted projective space. It remains to show (2) $\Rightarrow$ (3). But if $Cl(\Sigma) \cong {\bf Z}$, then taking the dual of the exact sequence \[ 0 \to M \stackrel{\alpha}\rightarrow {\bf Z}^{d+1} \to {\bf Z} \to 0\] from Definition \ref{diag} gives an exact sequence \[ 0 \to {\bf Z} \to {\bf Z}^{d+1} \to N \to 0,\] and it follows that $e_1,\dots,e_{d+1}$ generate $N$. Finally, given an arbitrary fan $\Sigma$ with $n = d+1$, let $N' \subset N$ be the sublattice generated by $e_1,\dots,e_{d+1}$. Then the fan $\Sigma$ induces a fan $\Sigma'$ for the lattice $N'$, and the natural inclusion $(N',\Sigma') \to (N,\Sigma)$ induces a finite surjection ${\bf P}_{\Sigma'} \to {\bf P}_\Sigma$ by Corollary 1.16 of \cite{oda}. By the above, ${\bf P}_{\Sigma'}$ is a weighted projective space, and the proposition is proved. \hfill $\Box$ \medskip Lemma \ref{lem.nd} implies that a smooth complete toric variety with $n=d+1$ is ${\bf P}^d$ (this fact is well-known). Thus we get the following corollary of Proposition \ref{prop.codim}. \begin{coro} Let ${\bf P}_\Sigma$ be a smooth complete toric variety. Then either \begin{enumerate} \item $2 \le {\rm codim}\, Z(\Sigma) \le [{1\over2}d]+1$, or \item ${\bf P}_\Sigma = {\bf P}^d$. \end{enumerate} \end{coro} We end this section with some comments about combinatorially equivalent fans. \begin{opr} {\rm Two rational simplicial complete $d$-dimensional fans $\Sigma$ and $\Sigma'$ are called {\em combinatorially equivalent} if there exists a bijective mapping $\Sigma \rightarrow \Sigma'$ respecting the face-relation ``$\prec$" (see Definition \ref{face}). } \end{opr} \begin{rem} {\rm It is easy to see that the closed subset $Z(\Sigma) \subset {\bf A}^n$ depends only on the combinatorial structure of $\Sigma$, i.e., for two combinatorially equivalent fans $\Sigma$ and $\Sigma'$, we can assume that we are in the same affine space ${\bf A}^n$, and then we have $Z(\Sigma) = Z(\Sigma')$ and hence $U(\Sigma) = U(\Sigma')$. } \end{rem} It follows that all toric varieties coming from combinatorially equivalent fans are quotients of the same open set $U \subset {\bf A}^n$, though the group actions will differ. For example, all $d$-dimensional weighted projective spaces come from combinatorially equivalent fans and are quotients of $U = {\bf A}^{d+1} - \{0\}$. \section{Quasi-smooth hypersurfaces} Throughout this section, let ${\bf P} = {\bf P}_\Sigma$ be a fixed $d$-dimensional complete simplicial toric variety. The action of ${\bf D} = {\bf D}(\Sigma)$ on ${\bf A}^n$ induces an action on $S = S(\Sigma) = {\bf C}[z_1,\dots,z_n]$. The decomposition of this representation gives a grading on $S$ by the character group $Cl(\Sigma)$ of ${\bf D}$. A polynomial $f$ in the graded piece of $S$ corresponding to $\beta \in Cl(\Sigma)$ is said to be ${\bf D}$-homogeneous of degree $\beta$. Such a polynomial $f$ has a zero set ${\bf V}(f) \subset {\bf A}^n$, and ${\bf V}(f)\cap U(\Sigma)$ is stable under ${\bf D}$ and hence descends to a hypersurface $X \subset {\bf P}$ (this is because ${\bf P}$ is a geometric quotient---see \cite{cox} for more details). We call ${\bf V}(f) \subset {\bf A}^n$ the {\em affine quasi-cone} of $X$. \begin{opr} {\rm If a hypersurface $X \subset {\bf P}$ is defined by a ${\bf D}$-homogeneous polynomial $f$, then we say that $X$ is {\em quasi-smooth} if the affine quasi-cone ${\bf V}(f)$ is smooth outside $Z = Z(\Sigma) \subset {\bf A}^n$. } \label{quasi} \end{opr} To see what this definition says about the hypersurface $X$, we need the concept of a $V$-submanifold of a $V$-manifold. As usual, a $d$-dimensional variety $W$ is a $V$-manifold if for every point $p \in W'$, there is an analytic isomorphism of germs $({\bf C}^d/G,0) \cong (W,p)$ where $G \subset GL(d,{\bf C})$ is a finite small subgroup. (Recall that being {\em small} means there are no elements with 1 as an eigenvalue of multiplicity $d-1$, i.e., no complex rotations other than the identity). In this case, we say that $({\bf C}^d/G,0)$ is a {\em local model} of $W$ at $p$. \begin{opr} {\rm If a $d$-dimensional variety $W$ is a $V$-manifold, then a subvariety $W' \subset W$ is a {\em $V$-submanifold} if for every point $p \in W'$, there is a local model $({\bf C}^d/G,0) \cong (W,p)$ such that $G \subset GL(d,{\bf C})$ is a finite small subgroup and the inverse image of $W'$ in ${\bf C}^d$ is smooth at $0$. } \end{opr} \begin{rem} {\rm In \cite{prill}, Prill proved that $G$ is determined up to conjugacy by the germ $(W,p)$. Hence, if $W' \subset W$ is a $V$-submanifold for one local model at $p$, then it is a $V$-submanifold for all local models at $p$. It is also easy to see that a $V$-submanifold of a $V$-manifold $W$ is again a $V$-manifold. However, the converse is false: a subvariety of $W$ which is a $V$-manifold need not be a $V$-submanifold. This is because the singularities of a $V$-submanifold are intimately related to the singularities of the ambient space.} \end{rem} We will omit the proof of the following easy lemma. \begin{lem} If $G \subset GL(d,{\bf C})$ is a small and finite, then a subvariety $W' \subset {\bf C}^d/G$ is a $V$-submanifold if and only if the inverse image of $W'$ in ${\bf C}^d$ is smooth. \label{local} \end{lem} We can now describe when a hypersurface of the toric variety ${\bf P}$ is a $V$-submanifold. \begin{prop} If a hypersurface $X \subset {\bf P}$ is defined by a ${\bf D}$-homogeneous polynomial $f$, then $X$ is quasi-smooth if and only if $X$ is a $V$-submanifold of ${\bf P}$. \label{prop.quasi} \end{prop} \noindent {\bf Proof. } We will use the notation of the proof of Theorem \ref{key.theo}. Thus, let $\sigma$ be a $d$-dimensional cone of $\Sigma$ and assume that $\sigma$ is generated by $e_1,\dots,e_d$. As we saw in the proof of Theorem~\ref{key.theo}, $\sigma$ gives an affine open set ${\bf A}_\sigma = {\bf C}^d/{\bf D}(\sigma) \cong U_\sigma/{\bf D}$ of ${\bf P}$. We claim that ${\bf D}(\sigma) \subset ({\bf C}^*)^d$ is a small subgroup. Suppose that $g = (\lambda,1,\dots,1) \in {\bf D}(\sigma)$. We can regard $g$ as a homomorphism $g : Cl(\sigma) \to {\bf C}^*$, and diagram (\ref{diagram}) shows that $g = 1$ on the image of $(\langle m,e_1\rangle,\dots,\langle m,e_d\rangle)$ whenever $m \in M$. This means that $\lambda^{\langle m,e_1\rangle} = 1$ for all $m \in M$. Since $e_1$ is not the multiple of any element of $N$ and $M = {\rm Hom}(N,{\bf Z})$, it follows that $\lambda = 1$ and hence $g$ is the identity. Since ${\bf A}_\sigma = {\bf C}^d/{\bf D}(\sigma)$, Lemma \ref{local} implies that $X\cap {\bf A}_\sigma$ is a $V$-submanifold if any only if the inverse image of $X\cap{\bf A}_\sigma$ in ${\bf C}^d$ is smooth. Since the map ${\bf C}^d/{\bf D}(\sigma) \cong U_\sigma/{\bf D}$ is induced by $(t_1,\dots,t_d) \mapsto (t_1,\dots,t_d,1,\dots,1)$, it follows that the inverse image of $X\cap{\bf A}_\sigma$ is ${\bf C}^d$ is defined by $g = 0$, where \[ g(t_1,\dots,t_d) = f(t_1,\dots,t_d,1,\dots,1).\] Thus $X\cap{\bf A}_\sigma$ is a $V$-submanifold if and only if the subvariety $g = 0$ is smooth in ${\bf C}^d$. We need to relate this to the smoothness of ${\bf V}(f)\cap U_\sigma$. Recall the decomposition $U_\sigma = {\bf C}^d\times({\bf C}^*)^{n-d}$. Using the surjection ${\bf D} \to ({\bf C}^*)^{n-d}$ from equation (\ref{eqD}), we see that ${\bf V}(f)\cap U_\sigma$ is smooth if and only if it is smooth at all points of the form $(t_1,\dots,t_d,1,\dots,1)$. Now comes the key observation. \begin{lem} ${\bf V}(f)\cap U_\sigma$ is smooth at $t = (t_1,\dots,t_d,1,\dots,1)$ if and only if one of the partials $f_{z_i}(t)$ is nonzero for some $1 \le i \le d$. \label{key.lem} \end{lem} \begin{rem} {\rm This lemma and the previous paragraphs show that ${\bf V}(f)\cap U_\sigma$ is smooth if and only if $X\cap{\bf A}_\sigma$ is a $V$-submanifold. Since the quasi-cone of $X$ is smooth outside $Z$ if and only if ${\bf V}(f)\cap U_\sigma$ is smooth for all $d$-dimensional $\sigma$, Proposition \ref{prop.quasi} follows from Lemma \ref{key.lem}.} \end{rem} \noindent {\bf Proof. } Suppose $f_{z_i}(t) = 0$ for $1 \le i \le d$. Now take any $j > d$. Since $e_1,\dots,e_d$ are a basis of $N_{\bf Q}$, we can write $e_j = \sum_{i=0}^d \phi_i e_i$. By Lemma \ref{euler.lem} below, there is a constant $\phi(\beta)$ such that \[ \phi(\beta) f = z_j{\partial f \over \partial z_j} - \sum_{i=1}^d \phi_i z_i {\partial f \over \partial z_i}.\] Evaluating this at $t \in {\bf V}(f)$ and using $f_{z_i}(t) = 0$ for $1 \le i \le d$, we see that $f_{z_j}(t) = 0$ for all $1 \le j \le n$. Hence, in order to be smooth at $t$, at least one of the first $d$ partials must be nonvanishing. \hfill$\Box$ \medskip To complete the proof of Proposition \ref{prop.quasi}, we need to prove the following lemma. \begin{lem} \label{euler.lem} Suppose that we have complex numbers $\phi_1,\dots,\phi_n$ with the property that $\sum_{i=0}^n \phi_i e_i = 0$ in $N_{\bf C}$. Then, for any class $\beta \in Cl(\Sigma)$, there is a constant $\phi(\beta)$ with the property that for any ${\bf D}$-homogeneous polynomial $f \in S$ of degree $\beta$, we have \[ \phi(\beta) f = \sum_{i=1}^n \phi_i z_i {\partial f \over \partial z_i}.\] \end{lem} \noindent {\bf Proof. } From $\phi_1,\dots,\phi_n$, we get a map $\tilde\phi : {\bf Z}^n \to {\bf C}$ defined by $(a_1,\dots,a_n) \mapsto \sum_{i=1}^n \phi_i a_i$. Furthermore, under the map $\alpha: M \to {\bf Z}^n$, note that $m \in M$ maps to $\sum_{i=1}^n \phi_i \langle m,e_i\rangle = \big\langle m,\sum_{i=1}^n \phi_i e_i\big\rangle = 0$. Thus $\tilde\phi$ induces a map $\phi : {\bf Z}^n/\alpha(M) \cong Cl(\Sigma) \to {\bf C}$. Note that every monomial $f = \prod_{i=1}^n z_i^{a_i}$ of $S$ is ${\bf D}$-homogeneous. If we let $\beta = \deg f = \sum_{i=1}^n a_i \deg z_i$, then the desired formula for $\phi(\beta) f$ follows immediately. By linearity, the formula then holds for all ${\bf D}$-homogeneous poynomials of degree $\beta$. \hfill$\Box$ \medskip In light of this lemma, we make the following definition. \begin{opr} {\rm The identity $\phi(\beta) f = \sum_{i=1}^n \phi_i z_i f_{z_i}$ from Lemma \ref{euler.lem} is called the {\em Euler formula} determined by $\phi_1,\dots,\phi_n$.} \label{euler.def} \end{opr} \begin{rem} {\rm It follows easily from the proof of Lemma \ref{euler.lem} that the set of all Euler formulas form the vector space ${\rm Hom}(Cl(\Sigma), {\bf C})$. For example, if ${\bf P}$ is a weighted projective space, then ${\rm Hom}(Cl(\Sigma), {\bf C}) \cong {\bf C}$, where a basis is given by the usual Euler formula for homogeneous polynomials. Note also that ${\rm Hom}(Cl(\Sigma),{\bf C}) = {\rm Lie}({\bf D})$, so that an Euler formula can be regarded as a vector field $\sum_{i=1}^n \phi_i z_i{\partial\over\partial z_i}$ which is tangent to the orbits of ${\bf D}$.} \end{rem} We will end this section with a discussion of how quasi-smooth hypersurfaces of ${\bf P}$ relate to Danilov's theory of ``toroidal pairs'' \cite{dan3}. \begin{opr} {\rm Let $W'$ be a subvariety of an algebraic variety $W$. Then the pair $(W,W')$ is {\em simplicially toroidal} if for every point $p \in W$, there is a simplicial cone $\sigma$ such that the germ of $(W,W')$ at $p$ is analytically isomorphic to the germ of $({\bf A}_\sigma,D)$ at the origin, where ${\bf A}_\sigma$ is the toric variety of $\sigma$ and $D$ is an irreducible torus-invariant subvariety of ${\bf A}_\sigma$. } \label{def.dan} \end{opr} For hypersurfaces of ${\bf P}$, this concept is equivalent to being quasi-smooth. \begin{prop} A hypersurface $X \subset {\bf P}$ is quasi-smooth if and only if the pair $({\bf P},X)$ is simplicially toroidal. \label{toroid.qsmooth} \end{prop} \noindent {\bf Proof. } First suppose we have a pair $({\bf A}_\sigma,D)$ as in Definition \ref{def.dan}. We know that ${\bf A}_\sigma \cong {\bf C}^d/{\bf D}(\sigma)$, and since the inverse image of $D$ is a coordinate subspace, we see that $D$ is a $V$-submanifold of ${\bf A}_\sigma$. Thus $({\bf P},X)$ simplicially toroidal implies that $X$ is a $V$-submanifold and hence quasi-smooth by Proposition \ref{prop.quasi}. Conversely, if $X$ is quasi-smooth, then it is a $V$-submanifold of ${\bf P}$. Thus, given $p \in X$, there is a local model $({\bf C}^d/G,0) \cong ({\bf P},p)$ such that $G \subset ({\bf C}^*)^d$ is small and finite, and the inverse image $Y \subset {\bf C}^d$ of $X \subset {\bf P}$ is smooth at the origin. Then $Y$ is defined by some equation $h = 0$, and since $Y$ is $G$-invariant, there is a character $\lambda : G \to {\bf C}^*$ such that $h(g\cdot t) = \lambda(g)h(t)$ for all $g \in G$. Since $Y$ is smooth at the origin, we can also assume that the partial derivative $h_{z_1}(0)$ is nonvanishing. Then the map $\phi : {\bf C}^d \to {\bf C}^d$ defined by $(t_1,\dots,t_d) \mapsto (h(t_1,\dots,t_d),t_2,\dots,t_d)$ is a local analytic isomorphism which carries $Y$ to a coordinate hyperplane. Note that $\phi$ is equivariant provided that $g = (g_1,\dots,g_d) \in G$ acts on the target space via \[ g\cdot (t_1,\dots,t_d) = (\lambda(g)t_1,g_2 t_2,\dots,g_d t_d).\] This gives a map $\psi : G \to ({\bf C}^*)^d$. We thus have a local model $({\bf C}^d/\psi(G),0)$ where the inverse image of $X$ is a coordinate hyperplane. Since the torus $({\bf C}^*)^d/\psi(G)$ acts on the affine variety ${\bf C}^d/\psi(G)$, it follows from Theorem 1.5 of \cite{oda} that ${\bf C}^d/\psi(G)$ is an affine toric variety. Thus ${\bf C}^d/\psi(G) \cong {\bf A}_\sigma$ for some cone $\sigma$, and note that $\sigma$ must be simplicial. Finally, the image in ${\bf C}^d/\psi(G)$ of a coordinate hyperplane is an irreducible torus-invariant subvariety. This proves that $({\bf P},X)$ has the appropriate simplicial toroidal local model, and the proposition is proved. \hfill $\Box$ \begin{rem} {\rm Let ${\bf A}_{\sigma} = {\rm Spec}\, {\bf C} \lbrack \check{\sigma} \cap M \rbrack$ be an affine $d$-dimensional toric variety correponding to a $d$-dimensional rational simplicial cone $\sigma \in \Sigma$, where $\check{\sigma}$ is the dual cone in $M_{\bf R}$. There are two important cases where one can explicitly describe local toroidal models for a quasi-smooth hypersurface $X$ in ${\bf A}_{\sigma}$: {\sc Case I.} $X$ has transversal intersections with all orbits ${\bf T}_\tau \subset {\bf A}_{\sigma}$ of the action of the torus ${\bf T}$ on ${\bf A}_{\sigma}$. Then at the point of intersection of $X$ with a $1$-dimensional stratum ${\bf T}_{\tau_0}$ corresponding to a $(d-1)$-dimensional face $\tau_0 \prec \sigma$, the local toroidal model of $X$ is the $(d-1)$-dimensional affine toric variety ${\bf A}_{\tau_0}$. {\sc Case II.} $X$ contains the single closed ${\bf T}_\sigma$-orbit $p_{\sigma} \in {\bf A}_{\sigma}$ and is ``tangent" to a closure of a $(d-1)$-dimensional ${\bf T}_\sigma$-orbit corresponding to a $1$-dimensional cone $\rho \prec \sigma$. In this case, the local toroidal model of $X$ at $p_{\sigma}$ is the $(d-1)$-dimensional affine toric variety ${\bf A}_{\sigma/\rho}$, where $\sigma/\rho$ is the $(d-1)$-dimensional projection of the cone $\sigma \subset N_{\bf R}$ to the quotient $N_{\bf R} / {\bf R}\rho$. } \end{rem} \section{${\bf T}$-linearized sheaves on ${\bf P}$} In this section we will assume that ${\bf P}$ is a complete toric variety. Recall from \cite{mam} the notion of a linearization of a sheaf on an algebraic variety having a regular action of an algebraic group. We will apply this to sheaves on ${\bf P}$ with its action by the torus ${\bf T}$. \begin{opr} {\rm Let $\mu_{\bf t} : {\bf P} \rightarrow {\bf P}$ be the automorphism of ${\bf P}$ defined by $t \in {\bf T}$. Then a {\em {\bf T}-linearization of a sheaf ${\cal E}$} is a family of isomorphisms \[ \phi_t\; : \; \mu_{t}^*\, {\cal E} \cong {\cal E} \] satisfying the co-cycle condition \[ \phi_{{t_1}\cdot {t}_2} = \phi_{{t}_2} \circ \mu^*_{{t}_2} \phi_{{t}_1},\;{\rm for\; all} \;{t}_1, { t}_2 \in {\bf T}. \]} \end{opr} \begin{rem} {\rm If ${\cal E}$ is a ${\bf T}$-linearized sheaf on ${\bf P}$, then for any ${\bf T}$-invariant open subset ${U} \subset {\bf P}$, the group ${\bf T}$ has the natural linear representation in the space of global sections $H^0({ U}, {\cal E})$. Thus $H^0(U, {\cal E})$ splits into a direct sum of subspaces $H^0(U, {\cal E})_{m}$ corresponding to characters $ m \in M$ of ${\bf T}$. } \end{rem} \begin{opr} \label{def.polytope} {\rm (\cite{kempf}) Let ${\cal E}$ be a ${\bf T}$-linearized sheaf on ${\bf P}$. Then the polytope \[ \Delta({\cal E}) = {\rm Conv}\, \{ m \in M : H^0({\bf P}, {\cal E})_{m} \neq 0 \} \] is called the {\em support polytope for} ${\cal E}$.} \end{opr} \begin{opr} {\rm Let ${\cal L}$ be a ${\bf T}$-linearized invertible sheaf on ${\bf P}$, $\sigma \in \Sigma^{(d)}$ a $d$-dimensional cone in $\Sigma$. We denote by $m_{\sigma}({\cal L})$ the unique element of $M$ with the property that \[ \{m \in M : H^0(X_{\sigma}, {\cal L})_m \ne 0\} = m_\sigma({\cal L}) + \check\sigma\cap M,\] where ${\bf A}_\sigma \subset {\bf P}$ is the affine toric variety determined by $\sigma$. } \end{opr} \begin{rem} {\rm The existence of $m_\sigma({\cal L})$ follows from \S6.2 of \cite{dan1}, and it is unique because $\sigma$ is $d$-dimensional. Note also that the mapping $h : N_{\bf R} \to {\bf R}$ defined by $h(n) = \langle m_\sigma({\cal L}),n\rangle$ (for $n \in \sigma$) is the support function for the invertible sheaf ${\cal L}$. We should also mention that every invertible sheaf ${\cal L}$ on ${\bf P}$ has a ${\bf T}$-linearization and that any two linearizations of ${\cal L}$ differ by a homomorphism ${\bf T} \to {\bf C}^*$, i.e., by an element of $M$. } \end{rem} \begin{prop} \label{delta} {\rm (\cite{dan1})} Let ${\cal L}$ be a ${\bf T}$-linearized invertible sheaf on ${\bf P}$. Then the corresponding polyhedron $\Delta = \Delta({\cal L})$ is equal to the intersection \[ \bigcap_{\sigma \in \Sigma^{(d)}} ( m_{\sigma}({\cal L}) + \check\sigma). \] \end{prop} \begin{opr} {\rm Let $\tau$ be a $k$-dimensional cone in $\Sigma$, and let ${\rm St}\,(\tau)$ the set of all cones $\sigma \in \Sigma$ such that $\tau \prec \sigma$. Consider the $(d-k)$-dimensional fan $\Sigma(\tau)$ consisting of projections of cones $\sigma \in {\rm St}\,(\tau)$ into $N_{\bf R} / {\bf R}\tau$. The corresponding $(d-k)$-dimensional toric subvariety in ${\bf P}$ will be denoted by ${\bf P}_{\tau}$.} \label{def.orbit} \end{opr} \begin{rem} {\rm The toric subvariety ${\bf P}_{\tau}$ is the closure of a $(d-k)$-dimensional orbit ${\bf T}_{\tau}$ of ${\bf T}$. Any ${\bf T}_{\tau}$-linearized sheaf ${\cal E}$ on ${\bf P}_{\tau}$ can be considered as a ${\bf T}$-linearized sheaf on ${\bf P}$. } \end{rem} \begin{opr} {\rm Let ${\cal L}$ be a ${\bf T}$-linearized ample invertible sheaf on ${\bf P}$. For any $k$-dimensional cone in $\tau \in \Sigma$, we denote by $\Delta_{\tau}$ the face of $\Delta$ of codimension $k$ defined as \[ \bigcap_{\sigma \in \Sigma^{(d)},\, \tau \prec \sigma} ( m_{\sigma}({\cal L}) + \check\sigma \cap \tau^{\perp}). \]} \end{opr} The next statement follows immediately from the ampleness criterion for invertible sheaves \cite{dan1}. \begin{prop} One has the one-to-one correspondence between $(d-k)$-dimensional faces $\Delta_{\tau}$ of the polytope $\Delta = \Delta({\cal L})$ and $k$-dimensional cones $\tau \in \Sigma$ reversing the face-relation. Moreover, the $(d-k)$-dimensional polytope $\Delta_{\tau}$ is the support polytope for the ${\bf T}$-linearized sheaf ${\cal O}_{{\bf P}_{\tau}} \otimes {\cal L}$. \label{ample} \end{prop} We next study the relation between $H^0({\bf P},{\cal L})$ and the coordinate ring $S = {\bf C}[z_1,\dots,z_n]$. \begin{lem} If ${\cal L}$ is a ${\bf T}$-linearized invertible sheaf on ${\bf P}$, and let $\beta \in Cl(\Sigma)$ be the class of ${\cal L}$. Then there is a natural isomorphism \[ H^0({\bf P},{\cal L}) \cong S_\beta,\] where $S_\beta$ is the graded piece of $S$ corresponding to $\beta$. This isomorphism is determined uniquely up to a nonzero constant in ${\bf C}$. \label{lem.iso} \end{lem} \noindent {\bf Proof. } Since ${\bf P}$ is complete, there is a one-to-one correspondance between ${\bf T}$-linearized invertible sheaves and ${\bf T}$-invariant Cartier divisors (see \S2.2 of \cite{oda}). Thus there is a ${\bf T}$-invariant Cartier divisor such that ${\cal L} \cong {\cal O}_{\bf P}(D)$ as ${\bf T}$-linearized sheaves. Note that this isomorphism is unique up to a nonzero constant. However, in \cite{cox}, it is shown that $D$ determines an isomorphism $H^0({\bf P},{\cal O}_{\bf P}(D)) \cong S_\beta$, and the lemma follows immediately.\hfill$\Box$ \begin{rem} {\rm If in addition ${\bf P}$ is simplicial, then the polynomial $f \in S_\beta$ corresponding to a global section of ${\cal L}$ determines a hypersurface $X \subset {\bf P}$ as in \S3, and one can check that $X$ is exactly the zero section of the global section.} \end{rem} \begin{opr} \label{def.nondeg} {\rm Let $f$ be a global section of an ample invertible sheaf ${\cal L}$ on ${\bf P}$. Then the hypersurface $X = \{ p \in {\bf P} : f(p) = 0 \}$ is called {\em nondegenerate} if for any $\tau \in \Sigma$, the affine hypersurface $X \cap {\bf T}_{\tau}$ is a smooth subvariety of codimension $1$ in ${\bf T}_{\tau}$. } \end{opr} \begin{rem} {\rm The nondegeneracy of a global section $f$ is equivalent to $f$ being $\Delta$-regular, as defined in \cite{bat.var} (where $\Delta = \Delta({\cal L})$). For a proof of this, see \cite{khov}.} \end{rem} \begin{prop} Let $f$ be a generic global section of an ample ${\bf T}$-linearized invertible sheaf ${\cal L}$ on ${\bf P}$. Then $X = \{p \in {\bf P} : f(p) = 0 \}$ is a nondegenerate hypersurface. Moreover, every nondegenerate hypersurface $X \subset {\bf P}$ is quasi-smooth. \label{prop.generic} \end{prop} \noindent {\bf Proof. } As observed in \cite{dan2}, the first part of the statement follows from Bertini's theorem, and it was proved in \cite{dan1} that $X$ is simplicially toroidal. Thus, it remains to apply Proposition \ref{toroid.qsmooth}. \hfill $\Box$ \begin{rem} {\rm One should remark that a quasi-smooth hypersurface in a toric variety ${\bf P}$ need not be nondegenerate.} \end{rem} We will conclude this section by studying the relation between ${\bf T}$-linearized sheaves on ${\bf P}$ and graded $S$-modules. It is known (see \cite{cox}) that every $Cl(\Sigma)$-graded $S$-module $F$ determines a quasi-coherent sheaf $\widetilde{F}$ on ${\bf P}$. What extra structure on $F$ is needed in order to induce a ${\bf T}$-linearization on $\widetilde{F}$? To state the answer, note that $S$ has a natural grading by ${\bf Z}^n$ which is compatible with its grading by $Cl(\Sigma)$ via the map ${\bf Z}^n \to Cl(\Sigma)$ from Definition \ref{diag}. Then any ${\bf Z}^n$-graded module can be regarded as a $Cl(\Sigma)$-graded module. \begin{prop} If $F$ is a ${\bf Z}^n$-graded $S$-module, then the sheaf $\widetilde{F}$ on ${\bf P}$ has a natural ${\bf T}$-linearization. Furthermore, if ${\bf P}$ is simplicial and $N$ is generated by $e_1,\dots,e_n$, then every ${\bf T}$-linearized quasi-coherernt sheaf on ${\bf P}$ arises in this way. \end{prop} \noindent {\bf Proof. } Given $\sigma \in \Sigma$, let $S_\sigma$ be the localization of $S$ at $\widehat{z}_\sigma = \prod_{e_i \notin \sigma} z_i$, and let $(S_\sigma)_0$ be the elements of degree $0$ with respect to $Cl(\Sigma)$. Then the module $(F\otimes S_\sigma)_0$ determines a sheaf on the affine piece ${\bf A}_\sigma = {\rm Spec}((S_\sigma)_0)$ of ${\bf P}$, and, as explained in \S3 of \cite{cox}, these sheaves patch to give $\widetilde{F}$. Since $S_\sigma$ and $F\otimes S_\sigma$ have natural ${\bf Z}^n$ gradings, the groups $(S_\sigma)_0$ and $(F\otimes S_\sigma)_0$ have natural gradings by $M$ (which is the kernel of ${\bf Z}^n \to Cl(\Sigma)$). The $M$ grading on $(S_\sigma)_0$ determines the action of ${\bf T}$ on ${\bf A}_\sigma$, and the $M$ grading on $(F\otimes S_\sigma)_0$ then determines a ${\bf T}$-linearization. These clearly patch to give a ${\bf T}$-linearization of $\widetilde{F}$. To prove the final part of the proposition, note that $Cl(\Sigma)$ is torsion free since $e_1,\dots,e_n$ generate $N$. Thus the map ${\bf Z}^n \to Cl(\Sigma)$ has a left inverse $\phi : Cl(\Sigma) \to {\bf Z}^n$. Now, given $\alpha \in Cl(\Sigma)$, consider the $S$-module $S(\alpha)$, which has the $Cl(\Sigma)$ grading $S(\alpha)_\beta = S_{\alpha+\beta}$. We can give $S(\alpha)$ a grading by ${\bf Z}^n$ where $S(\alpha)_u = S_{u+\phi(\alpha)}$ for $u \in {\bf Z}^n$ (and we are using the usual ${\bf Z}^n$ grading of $S$). By the above, this grading on $S(\alpha)$ gives a ${\bf T}$-linearized sheaf ${\cal O}_{\bf P}(\alpha)$. If ${\cal F}$ is a quasi-coherent sheaf on ${\bf P}$, then \S3 of \cite{cox} implies that \[ F = \bigoplus_{\alpha \in Cl(\Sigma)} H^0({\bf P}, {\cal F} \otimes {\cal O}_{\bf P}(\alpha))\] is a $Cl(\Sigma)$-graded $S$-module whose associated sheaf is ${\cal F}$. The module structure on $F$ comes from the natural isomorphism $H^0({\bf P},{\cal O}_{\bf P}(\alpha)) \cong S_\alpha$. Now assume that ${\cal F}$ has a ${\bf T}$-linearization. Then each ${\cal F} \otimes {\cal O}_{\bf P}(\alpha)$ has a natural ${\bf T}$-linearization, so that $H^0({\bf P},{\cal F} \otimes {\cal O}_{\bf P}(\alpha))$ has a grading by $M$. Then define a ${\bf Z}^n$ grading on $F$ by setting \[ F_u = H^0({\bf P},{\cal F} \otimes {\cal O}_{\bf P}(\alpha))_{u-\phi(\alpha)},\] where $u$ maps to $\alpha$ under the map ${\bf Z}^n \to Cl(\Sigma)$. Since $\phi$ is a homomorphism, it is easy to check that $F$ becomes a ${\bf Z}^n$-graded $S$-module and that this grading induces the given ${\bf T}$-linearization on ${\cal F}$.\hfill$\Box$ \begin{rem} {\rm In Definition \ref{def.module}, we describe graded $S$ modules which give the sheaves $\Omega_{\bf P}^p$. The definition shows that these modules have a canonical ${\bf Z}^n$ grading, which gives the usual ${\bf T}$-linearization on $\Omega_{\bf P}^p$. Note that we use the ${\bf Z}^n$ grading in a crucial way in the proof of Proposition \ref{prop.dminus1}. } \end{rem} \section{Local properties of differential forms} Let $\Omega^p_{\bf P} $ denote the sheaf of $p$-differential forms of Zariski on ${\bf P}$. This means $\Omega^p_{\bf P} = j_*\Omega^p_W$, where $W$ is the smooth part of ${\bf P}$, $\Omega^p_W$ is the usual sheaf of $p$-forms on $W$, and $j : W \to {\bf P}$ is the natural inclusion. The sheaf $\Omega^p_{\bf P} $ has a canonical ${\bf T}$-linearization which induces $M$-graded decompositions of sections of $\Omega^p_{\bf P} $ over the ${\bf T}$-invariant the affine open subsets ${\bf A}_\sigma \subset {\bf P}$. \begin{opr} {\rm Let $m$ be an element of $\check\sigma\cap M$, where $\sigma \in \Sigma$. We denote by $\Gamma(m)$ the ${\bf C}$-subspace in $M_{\bf C}$ generated by elements of the minimal face of $\check\sigma$ containing $m$. } \end{opr} \begin{prop} \label{prop.A} {\rm (\cite{dan1}, \S 4) } Let ${\bf A}_{\sigma} \subset {\bf P}$ be the affine open corresponding to a $d$-dimensional cone $\sigma \in \Sigma$. Then the sections over ${\bf A}_\sigma$ of the {\bf T}-linearized sheaf $\Omega_{\bf P}^p$ decompose into a direct sum of $M$-homogeneous components as follows: \[ \Omega_{{\bf A}_{\sigma}}^p : = H^0({\bf A}_\sigma, \Omega_{{\bf P}}^p) = \bigoplus_{m \in \check\sigma \cap M} \Lambda^p \Gamma(m). \] \end{prop} Besides $\Omega^p_{\bf P}$, we also have the sheaves $\Omega_{\bf P}^p({\rm log}\, D)$ of differential $p$-forms with logarithmic poles along $D = {\bf P} \setminus {\bf T}$ (see \S 15 of \cite{dan1}). These sheaves have the weight filtration \[ {\cal W} : 0 \subset W_0 \Omega_{\bf P}^p({\rm log}\, D) \subset W_1 \Omega_{\bf P}^p({\rm log}\, D) \subset \cdots \subset W_p \Omega_{\bf P}^p({\rm log}\, D) = \Omega_{\bf P}^p({\rm log}\, D) \] defined by \[ W_k \Omega_{\bf P}^p({\rm log}\, D) = \Omega_{\bf P}^{p-k} \wedge \Omega_{\bf P}^k({\rm log}\, D). \] Note in particular that $W_0 \Omega_{\bf P}^p({\rm log}\, D) \cong \Omega_{\bf P}^p$ and $W_p \Omega_{\bf P}^p({\rm log}\, D) \cong {\cal O}_{\bf P} \otimes \Lambda^p M$. We have the following local description of the weight filtration. \begin{prop} {\rm (\cite{dan1}, \S 15.6)} If $\sigma$ is a $d$-dimensional cone in $\Sigma$, then the sections over ${\bf A}_\sigma$ of the {\bf T}-linearized sheaf $W_k \Omega_{\bf P}^p({\rm log}\, D)$ decompose into a direct sum of $M$-homogeneous components as follows: \[ W_k \Omega_{{\bf A}_\sigma}^p({\rm log}\, D) := H^0({\bf A}_\sigma, W_k \Omega_{\bf P}^p({\rm log}\, D)) = \bigoplus_{m \in \check\sigma \cap M} \Lambda^{p-k}\Gamma(m) \wedge \Lambda^{k} M_{\bf C}. \] \label{log.local} \end{prop} The successive quotients of the weight filtration are described using the Poincar\'e residue map. Recall from Definition \ref{def.orbit} that ${\bf P}_\tau$ is the Zariski closure of the ${\bf T}$-orbit of ${\bf P}$ corresponding to $\tau \in \Sigma$. \begin{theo} {\rm (\cite{dan1}, \S 15.7)} For any integer $k$ $(0 \leq k \leq p)$, there is a short exact sequence \[ 0 \rightarrow W_{k-1} \Omega_{\bf P}^p({\rm log}\, D) \rightarrow W_{k} \Omega_{\bf P}^p({\rm log}\, D) \stackrel{\rm Res}{\rightarrow} \bigoplus_{\dim\tau = k} \Omega_{{\bf P}_{\tau}}^p \rightarrow 0 \] where ``$\dim\tau = k$'' means the sum is over all $k$-dimensional cones in $\tau \in \Sigma$, and ${\rm Res}$ is the Poincar\'e residue map. \label{poin} \end{theo} This short exact sequence has a natural ${\bf T}$-action and splits into $M$-homogeneous components. Let's examine what happens over an affine toric chart ${\bf A}_{\sigma}$, where $\sigma$ is a $d$-dimensional cone in $\Sigma$. Assume that the generators of $\sigma$ are $\{e_1,\dots,e_d\}$. Then, for any subset $\{ i_1, \ldots, i_k \} \subset \{ 1, \dots, d \}$, we denote by ${\bf A}_{i_1 \dots i_k}$ the closed affine subvariety in ${\bf A}_{\sigma}$ corresponding to the cone $\tau_{i_1\dots i_k} = {\bf R}_{\geq 0} e_{i_1} + \cdots + {\bf R}_{\geq 0} e_{i_k}$. By \S 15.7 of \cite{dan1}, we get the following local description of the residue map: \begin{prop} \label{loc.poin} Given $m \in \check\sigma\cap M$, let $\omega_m$ be an element of $\Lambda^{p-k}\Gamma(m)$, and let $\omega'$ be an element of $\Lambda^{k} M_{\bf C}$. Then the image of the $m$-homogeneous element $\omega_m \wedge \omega' \in W_k \Omega_{{\bf A}_\sigma}^p({\rm log}\, D)$ under the residue map \[ {\rm Res} : W_k \Omega_{{\bf A}_\sigma}^p({\rm log}\, D) \rightarrow \bigoplus_{1 \leq i_1 < \cdots < i_k \leq d} \Omega_{{\bf A}_{i_1 \dots i_k}}^{p-k} \] is given by \[ {\rm Res}(\omega_m \wedge \omega' )_{i_1 \dots i_k} = \omega'(e_{i_1}, \ldots , e_{i_k}) \cdot \omega_m \in \Lambda^{p-k}\Gamma(m). \] \end{prop} \section{Globalization of the Poincar\'e residue map} Let ${\cal L}$ be an ample {\bf T}-linearized invertible sheaf on the complete toric variety ${\bf P}$, and let $\Delta = \Delta({\cal L})$ be as in Definition \ref{def.polytope}. Tensoring by $\cal L$ the short exact sequence in Theorem \ref{poin}, we obtain the exact sequence \[ 0 \rightarrow W_{k-1} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L} \rightarrow W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L} \stackrel{\rm Res}{\rightarrow} \bigoplus_{\dim \tau = k} \Omega_{{\bf P}_{\tau}}^{p-k} \otimes {\cal L} \rightarrow 0. \] The goal of this section is to give an explicit description of the map of spaces of global sections \begin{equation} \gamma : H^0 ({\bf P}, W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}) \rightarrow \bigoplus_{\dim \tau = k} H^0 ( {\bf P}, \Omega_{{\bf P}_{\tau}}^{p-k} \otimes {\cal L}) \label{global.poin} \end{equation} induced by the Poincar\'e residue map. \begin{opr} {\rm Let ${\cal L}$ be a ${\bf T}$-linearized ample invertible sheaf, which determines the convex polytope in $\Delta = \Delta({\cal L}) \subset M_{\bf R}$. For any $m \in M$, we denote by $\Gamma_{\Delta}(m)$ the ${\bf C}$-subspace in $M_{\bf C}$ generated by all vectors $s - s'$, where $s, s' \in \Delta_m$, and $\Delta_m$ is the minimal face of $\Delta$ containing $m$. } \label{glob.diff} \end{opr} First we notice the following properties: \begin{prop} \label{glob1} The space of global sections of ${\bf T}$-linearized sheaf $\Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}$ decomposes into a direct sum of $M$-homogeneous components as follows: \[ H^0 ({\bf P}, W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}) = \bigoplus_{m \in \Delta \cap M} \Lambda^{p-k}\Gamma_{\Delta}(m) \wedge \Lambda^{k} M_{\bf C} \] \end{prop} \noindent {\bf Proof. } The statement follows from Proposition \ref{log.local} and Proposition \ref{delta}. \hfill $\Box$ \begin{prop} \label{glob2} Let $\tau$ be a $k$-dimensional cone in $\Sigma$, $\Delta_{\tau}$ the corresponding face of $\Delta$ of codimension $k$ $($see Proposition \ref{ample}$)$. Then the space of global sections of ${\bf T}$-linearized sheaf $\Omega_{{\bf P}_{\tau}}^p \otimes {\cal L}$ decomposes into a direct sum of $M$-homogeneous components as follows: \[ H^0 ({\bf P}, \Omega_{{\bf P}_{\tau}}^{p-k} \otimes {\cal L}) = \bigoplus_{m \in \Delta_{\tau} \cap M} \Lambda^{p-k}\Gamma_{\Delta_{\tau}}(m) \] \end{prop} \noindent {\bf Proof. } The statement follows from Proposition \ref{prop.A} and Proposition \ref{delta}. \hfill $\Box$ \medskip The linear mapping $\gamma$ of (\ref{global.poin}) is the direct sum $\bigoplus_{\dim \tau = k} \gamma_{\tau}$, where \[ \gamma_{\tau} \;:\; H^0 ({\bf P}, W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}) \rightarrow H^0 ( {\bf P}, \Omega_{{\bf P}_{\tau}}^{p-k} \otimes {\cal L}). \] By Proposition \ref{loc.poin}, we can then describe $\gamma_\tau$ as follows. \begin{prop} \label{morph} Let $\omega_m$ be an element of $\Lambda^{p-k}\Gamma_{\Delta}(m)$ and $\omega'$ be an element of $\Lambda^{k} M_{\bf C}$. Thus $\omega_m \wedge \omega'$ is an $m$-homogeneous element in $H^0 ({\bf P}, W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L})$. Choose a $k$-dimensional cone $\tau$ with generators $e_{i_1}, \ldots, e_{i_k}$. Then \begin{enumerate} \item $\gamma_{\tau}(\omega_m \wedge \omega') = 0$ if $m \not\in \Delta_{\tau}$; \item $\gamma_{\tau}(\omega_m \wedge \omega') = \omega'(e_{i_1}, \ldots , e_{i_k}) \cdot \omega_m$ if $m \in \Delta_{\tau}$. \end{enumerate} \end{prop} \section{A generalized theorem of Bott-Steenbrink-Danilov} In Theorem 7.5.2 of \cite{dan1}, Danilov formulated without proof the following vanishing theorem generalizing for complete toric varieties the well-known theorem of Bott and Steenbrink: \begin{theo} Let $\cal L$ be an ample invertible sheaf on $\bf P$. Then for any $p \geq 0$ and $ i > 0$, one has \[ H^i({\bf P}, \Omega^p_{{\bf P}} \otimes {\cal L}) =0. \] \end{theo} We prove now for simplicial toric varieties a more general vanishing theorem: \begin{theo} \label{theo.bott} Let $\cal L$ be an ample invertible sheaf on a complete simplicial toric variety $\bf P$. Then for any $p \geq 0$, $k \geq 0$ and $ i > 0$, one has \[ H^i({\bf P}, W_k\Omega^p_{{\bf P}}({\rm log}\, D) \otimes {\cal L}) = 0. \] \end{theo} \noindent {\bf Proof. } We prove this theorem using induction on $p-k$. For $p-k =0$, the sheaf $W_p\Omega^p_{{\bf P}}({\rm log}\, D) \otimes {\cal L} = \Omega_{\bf P}^p({\rm log}\, D)\otimes{\cal L} = \Lambda^p M_{\bf C}\otimes {\cal L}$ is the direct sum of ${ d \choose p} $ copies of $\cal L$. Thus, the vanishing property for $W_p\Omega^p_{{\bf P}}({\rm log}\, D) \otimes {\cal L}$ is implied by the following general vanishing property for the ample invertible sheaf $\cal L$. \begin{prop} {\rm (\cite{dan1}, \S 7.3)} Let ${\cal L}$ be an ample invertible sheaf on a complete toric variety ${\bf P}$. Then \[ H^i({\bf P}, {\cal L}) = 0\;\; {\rm for}\;\;i > 0. \] \end{prop} On the other hand, for any $k$ ($0 \leq k \leq p$) we can apply the induction assumption to $W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}$ and $\Omega_{{\bf P}_{\tau}}^p \otimes {\cal L}$ appearing in the short exact sequence \[ 0 \rightarrow W_{k-1} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L} \rightarrow W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L} \stackrel{\rm Res}{\rightarrow} \bigoplus_{\dim\tau = k } \Omega_{{\bf P}_{\tau}}^{p-k} \otimes {\cal L} \rightarrow 0. \] The required vanishing properties of $W_{k-1} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}$ follows now from the following lemma. \begin{lem} The mapping \[ \gamma : H^0 ({\bf P}, W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L}) \rightarrow \bigoplus_{\dim\tau = k} H^0 ( {\bf P}, \Omega_{{\bf P}_{\tau}}^{p-k} \otimes {\cal L}) \] is surjective. \end{lem} \noindent {\bf Proof. } Since ${\cal L}$ is ample, there exists a one-to-one correspondence between $i$-di\-men\-sional cones of $\Sigma$ and $(d-i)$-dimensional faces of the convex polytope $\Delta = \Delta({\cal L})$ (see Proposition \ref{ample}). Choose a $k$-dimensional cone $\tau_0 \in \Sigma$. We know from Proposition \ref{glob2} that the $m$-homogeneous component of $H^0 ( {\bf P}_{\tau_0}, \Omega_{{\bf P}_{\tau_0}}^{p-k} \otimes {\cal L})$ is non-zero only if $m \in \Delta_{\tau_0}$, and when the latter holds, Definition \ref{glob.diff} shows that the $m$-component is determined by the minimal face $\Delta_{\tau_1}$ of $\Delta_{\tau_0}$ containing $m$. This means that $\tau_1 \in \Sigma$ is a cone of dimension $c \geq k$ containing $\tau_0$. Fix such a lattice point $m$ and the corresponding $\tau_0 \subset \tau_1$. We can assume that $\tau_1$ is a face of a $d$-dimensional cone $\sigma \in \Sigma$ with generators $e_1, \ldots, e_d \in N$ such that $e_1, \ldots, e_c$ are generators of $\tau_1$ and $e_1, \ldots, e_k$ are generators of $\tau_0$. To describe $\Gamma_{\Delta_{\tau_0}}(m)$, suppose that $\Delta$ is defined by inequalities $\langle m',e_i\rangle \ge -a_i$ for $1 \le i \le n$. Then $\Delta_{\tau_1} \subset \Delta$ is the face obtained by requiring $\langle m',e_i\rangle = -a_i$ for $1 \le i \le c$, which implies that the subspace $\Gamma_{\Delta_{\tau_0}}(m) \subset M_{\bf C}$ is defined $\langle m',e_i\rangle = 0$ for $1 \le i \le c$. Now let $h_1, \ldots, h_d \in M_{\bf Q}$ form the dual basis to the basis $e_1, \ldots, e_d$ of $N_{\bf Q}$. It follows immediately that $h_{c+1},\dots,h_d$ form a basis of $\Gamma_{\Delta_{\tau_0}}(m)$. Thus, by Proposition \ref{glob2}, the $m$-homogeneous component of $H^0 ( {\bf P}, \Omega_{{\bf P}_{\tau_0}}^{p-k} \otimes {\cal L})$ has a basis consisting of $(p-k)$-vectors \[ \omega_{j_1 \ldots j_{p-k}} = h_{j_1} \wedge \cdots \wedge h_{j_{p-k}} \] where $\{ j_1, \ldots, j_{p-k} \}$ is a subset of $\{ c+1, \ldots, d\}$. For any such a $(p-k)$-vector $\omega_{j_1 \ldots j_{p-k}}$, the $p$-vector \[ h_{1} \wedge \cdots \wedge h_{{k}} \wedge \omega_{j_1 \ldots j_{p-k}} \] defines an element in the $m$-homogeneous component of the space $H^0 ({\bf P}, W_{k} \Omega_{\bf P}^p({\rm log}\, D) \otimes {\cal L})$ by Proposition \ref{glob1}. Furthermore, Proposition \ref{morph} shows that \[ \gamma_{\tau_0}(h_{1} \wedge \cdots \wedge h_{{k}} \wedge \omega_{j_1 \ldots j_{p-k}}) = h_1\wedge\cdots \wedge h_k (e_1\wedge \cdots \wedge e_k)\cdot\omega_{j_1 \ldots j_{p-k}} = \omega_{j_1 \ldots j_{p-k}}.\] Thus, to prove the lemma, it suffices to show that for $k$-dimensional cones $\tau \in \Sigma$, we have \[ \gamma_{\tau}(h_{1} \wedge \cdots \wedge h_{k} \wedge \omega_{j_1 \ldots j_{p-k}}) = 0 \; \; {\rm for}\; \tau \neq \tau_0. \] However, by Proposition \ref{morph}, $\gamma_{\tau}(h_{1} \wedge \cdots \wedge h_{{k}} \wedge \omega_{j_1 \ldots j_{p-k}}) = 0$ for $k$-dimensional cones $\tau \in \Sigma$ such that $m \notin \Delta_{\tau}$. Since $\Delta_{\tau_1}$ is the minimal face containing $m$, the condition $m \notin \Delta_\tau$ holds whenever $\tau$ is not a face of $\tau_1$. It remains to see what happens when $\tau$ is a $k$-dimensional face of $\tau_1$. In this case, the generators of $\tau$ are $e_{i_1}, \ldots , e_{i_k}$, where $\{ i_1, \ldots, i_k \} \subset \{ 1, \ldots, c\}$. Since the $e_i$ are dual to the $h_i$, the value $h_{1} \wedge \cdots \wedge h_{k}(e_{i_1}, \ldots , e_{i_k})$ is nonzero only if $\tau = \tau_0$. From Proposition \ref{morph}, it follows that $\gamma_{\tau}(h_{1} \wedge \cdots \wedge h_{{k}} \wedge \omega_{j_1 \ldots j_{p-k}}) = 0$ when $\tau \ne \tau_0$. \hfill $\Box$ \section{Differential forms and graded $S$-modules} As usual, $\Omega_{\bf P}^p$ denotes the sheaf of Zariski differential $p$-forms on a complete simplicial toric variety ${\bf P} = {\bf P}_\Sigma$. We will study these sheaves using certain graded modules over the polynomial ring $S = {\bf C}[z_1,\dots,z_n]$, which is graded by $Cl(\Sigma)$. Given a Cartier divisor $X \subset {\bf P}$, our goal is to describe $H^0({\bf P},\Omega^p_{\bf P}(X))$ in terms of $S$. Recall that the fan $\Sigma$ lies in $N_{\bf R} \cong {\bf R}^d$ and that $M = {\rm Hom}(N,{\bf Z})$. Also recall that $e_1,\dots,e_n$ are the generators of the 1-dimensional cones of $\Sigma$. \begin{opr} {\rm Given $p$ between $0$ and $d$, define $\widehat{\Omega}^p_S$ by the exact sequence of graded $S$-modules: \[ 0 \to \widehat{\Omega}_S^p \to S \otimes \Lambda^p M \stackrel{\gamma}\rightarrow \bigoplus_{i=1}^n (S/z_i S)\otimes\Lambda^{p-1} (M\cap e_i^\perp), \] where the $i^{\rm th}$ component of $\gamma$ is $\gamma_i(g\otimes \omega) = g \bmod z_i \otimes \langle e_i,\omega\rangle$. A careful description of the interior product $\langle e_i,w\rangle \in \Lambda^{p-1}(M\cap e_i^\perp)$ may be found in \S3.2 of \cite{oda}.} \label{def.module} \end{opr} By \cite{cox}, we know that every finitely generated graded $S$-module $F$ gives a coherent sheaf $\widetilde F$ on ${\bf P}$. \begin{lem} The sheaf $\widetilde{\widehat{\Omega}}{}^p_S$ on ${\bf P}$ associated to the graded $S$-module $\widehat{\Omega}^p_S$ is naturally isomorphic to $\Omega^p_{\bf P}$. \label{lem.tilde} \end{lem} \noindent {\bf Proof. } From \S3 of \cite{cox}, we get $\widetilde{S} = {\cal O}_{\bf P}$. The primitive element $e_i$ generates the cone $\rho_i \in \Sigma$, which by Definition \ref{def.orbit} gives the ${\bf T}$-invariant divisor $D_i = {\bf P}_{\rho_i} \subset {\bf P}$. By the example before Corollary 3.8 of \cite{cox}, the ideal $z_i S$ gives the ideal sheaf of $D_i$. Since $F \mapsto \widetilde{F}$ is exact (see Proposition 3.1 of \cite{cox}), it follows that $S/z_i S$ gives the sheaf ${\cal O}_{D_i}$. Then the exactness of $F \mapsto \widetilde{F}$, applied to the sequence defining $\Omega_S^p$, gives the exact sequence of sheaves: \[ 0 \to \widetilde{\widehat{\Omega}}{}^p_S \to {\cal O}_{\bf P}\otimes \Lambda^p M \stackrel{\gamma}\to \bigoplus_{i=1}^n {\cal O}_{D_i}\otimes \Lambda^{p-1} (M\cap e_i^\perp)\ .\] By Theorem 3.6 of \cite{oda}, we can identify $\widetilde{\widehat{\Omega}}{}^p_S$ with $\Omega^p_{\bf P}$. \hfill$\Box$ \medskip We next discuss the twists of the $\Omega^p_{\bf P}$. First recall that if $\beta \in Cl(\Sigma)$ and $F$ is a graded $S$-module, then $F(\beta)$ is the graded $S$-module defined by $F(\beta)_\gamma = F_{\beta+\gamma}$ for $\gamma \in Cl(\Sigma)$. \begin{opr} {\rm Given $\beta \in Cl(\Sigma)$, the sheaf on ${\bf P}$ associated to the graded $S$-module $\widehat{\Omega}_S^p(\beta)$ is denoted $\Omega_{\bf P}^p(\beta)$.} \end{opr} \begin{rem} {\rm If ${\cal O}_{\bf P}(\beta)$ is the sheaf associated to $S(\beta)$, then there is a natural isomorphism $\Omega^p_{\bf P}(\beta) \cong \Omega^p_{\bf P}\otimes {\cal O}_{\bf P}(\beta)$ whenever $\beta$ is the class of a Cartier divisor. However, when $\beta$ is not Cartier, the sheaves $\Omega^p_{\bf P}(\beta)$ and $\Omega^p_{\bf P}\otimes {\cal O}_{\bf P}(\beta)$ may be nonisomorphic.} \end{rem} \begin{prop} \label{prop.sections} For any divisor class $\beta \in Cl(\Sigma)$, there is a natural isomorphism \[ H^0({\bf P},\Omega^p_{\bf P}(\beta)) \cong (\widehat{\Omega}^p_S)_\beta.\] \end{prop} \noindent {\bf Proof. } The sequence in Definition \ref{def.module} remains exact after shifting by $\beta$. Then, taking the associated sheaves on ${\bf P}$, we get an exact sequence \[ 0 \to \Omega^p_{\bf P}(\beta) \to {\cal O}_{\bf P}(\beta)\otimes \Lambda^p M \stackrel{\gamma}\to \bigoplus_{i=1}^n {\cal O}_{D_i}(\beta) \otimes \Lambda^{p-1} (M\cap e_i^\perp)\ .\] Since taking global sections is left exact, we get the exact sequence \[0 \to H^0({\bf P},\Omega^p_S(\beta)) \to H^0({\bf P},{\cal O}_{\bf P}(\beta)) \otimes \Lambda^p M \stackrel{\gamma}\to \bigoplus_{i=1}^n H^0({\bf P},{\cal O}_{D_i}(\beta)) \otimes \Lambda^{p-1} (M\cap e_i^\perp)\ .\] Using the natural isomorphism $H^0({\bf P},{\cal O}_{\bf P}(\beta)) \cong S_\beta$ from Proposition 1.1 of \cite{cox}, we get \begin{equation} 0 \to H^0({\bf P},\Omega^p_S(\beta)) \to S_\beta \otimes \Lambda^p M \stackrel{\gamma}\to \bigoplus_{i=1}^n H^0({\bf P},{\cal O}_{D_i}(\beta)) \otimes \Lambda^{p-1} (M\cap e_i^\perp). \label{eq.first} \end{equation} However, since $z_i$ vanishes on $D_i$, the map $\gamma$ factors \[S_\beta \otimes \Lambda^p M \to \bigoplus_{i=1}^n (S/z_i S)_\beta \otimes \Lambda^{p-1} (M\cap e_i^\perp) \to \bigoplus_{i=1}^n H^0({\bf P},{\cal O}_{D_i}(\beta)) \otimes \Lambda^{p-1} (M\cap e_i^\perp).\] Assume for the moment that $(S/z_i S)_\beta \to H^0({\bf P},{\cal O}_{D_i}(\beta))$ is injective. Then (\ref{eq.first}) gives the exact sequence \[ 0 \to H^0({\bf P},\Omega^p_S(\beta)) \to S_\beta \otimes \Lambda^p M \stackrel{\gamma}\to \bigoplus_{i=1}^n (S/z_i S)_\beta \otimes \Lambda^{p-1} (M\cap e_i^\perp),\] and the isomorphism $H^0({\bf P},\Omega^p_S(\beta)) \cong (\widehat{\Omega}^p_S)_\beta$ follows immediately from Definition \ref{def.module}. It remains to show that the natural map $(S/z_i S)_\beta \to H^0({\bf P},{\cal O}_{D_i}(\beta))$ is injective. Let $\sigma$ be a cone of $\Sigma$ containing $e_i$, and let ${\bf A}_\sigma \subset {\bf P}$ be the corresponding affine toric variety. Then it suffices to show that the $(S/z_i S)_\beta \to H^0({\bf A}_\sigma,{\cal O}_{D_i}(\beta))$ is injective. However, as explained in \S3 of \cite{cox}, we have \[ H^0({\bf A}_\sigma,{\cal O}_{D_i}(\beta)) \cong ((S/z_i S)(\beta) \otimes S_\sigma)_0 = ((S/z_iS)\otimes S_\sigma)_\beta,\] where $S_\sigma$ is the localization of $S$ at $\widehat{z}_\sigma = \prod_{e_j \notin \sigma} z_j$. Thus we need to show that \[ (S/z_i S)_\beta \to ((S/z_i S)\otimes S_\sigma)_\beta\] is injective. This follows easily since $z_i$ doesn't divide $\widehat{z}_\sigma$, and the proposition is proved. \hfill $\Box$ \medskip We next want to relate $\widehat{\Omega}^p_S$ to the usual module $\Omega^p_S$ of $p$-forms in $dz_1,\dots,dz_n$. First note that $\Omega^\cdot_S$ can be given the structure of a graded $S$-algebra by declaring that $dz_i$ and $z_i$ have the same degree in $Cl(\Sigma)$. This means that if $z_i \in S_{\beta_i}$, then we have an isomorphism of graded $S$-modules \[ \Omega^p_S \cong \bigoplus_{1 \le i_1 < \cdots < i_p \le n} S(-\beta_{i_1} - \cdots - \beta_{i_p}).\] \begin{lem} There is a natural inclusion $\widehat{\Omega}^p_S \subset \Omega^p_S$ of graded $S$-modules. \end{lem} \noindent {\bf Proof. } Consider the diagram \begin{equation} \begin{array}{ccccccc} 0 & \to & \widehat{\Omega}^p_S & \to & S\otimes\Lambda^p M & \stackrel{\gamma}\to & \bigoplus_{i=1}^n (S/z_i S)\otimes\Lambda^{p-1} M \\ & & & & \downarrow & & \downarrow \\ 0 & \to & \Omega^p_S & \to & S\otimes\Lambda^p {\bf Z}^n & \stackrel{\delta}\to & \bigoplus_{i=1}^n (S/z_i S)\otimes\Lambda^{p-1} {\bf Z}^n \label{dia.omega} \end{array} \end{equation} The first row is exact by the definition of $\widehat{\Omega}^p_S$ and the inclusion $M\cap e_i^\perp \subset M$. To understand the second row, let $h_1,\dots,h_n$ be the standard basis of ${\bf Z}^n$. Then the map $\Omega^p_S \to S\otimes\Lambda^p{\bf Z}^n$ is defined by \[ dz_{i_1}\wedge\cdots\wedge dz_{i_p} \mapsto z_{i_1}\cdots z_{i_d}\otimes h_{i_1}\wedge\cdots\wedge h_{i_d},\] and for $g\otimes\omega \in S\otimes\Lambda^p{\bf Z}^n$, the $i^{\rm th}$ component of $\delta(g\otimes\omega)$ is $\delta_i(g\otimes\omega) = g \bmod z_i \otimes \langle h_i^*,\omega\rangle$, where $h_1^*,\dots,h_n^*$ is the dual basis to $h_1,\dots,h_n$. It is easy to check that the sequence on the bottom is exact. The vertical maps in (\ref{dia.omega}) are induced by the map $\alpha : M \to {\bf Z}^n$ from Definition \ref{diag}, and since the dual $\alpha^* : {\bf Z}^n \to N$ maps $h_i^*$ to $e_i$, it follows that diagram (\ref{dia.omega}) commutes. This gives the desired inclusion. \hfill $\Box$ \medskip The final step is to relate $\widehat{\Omega}^p_S$ to rational $p$-forms on ${\bf P}$ with poles on a hypersurface $X \subset {\bf P}$. We will assume that $X$ is defined by $f = 0$ for some $f \in S_\beta$. Thus $\beta \in Cl(\Sigma)$ is the class of $X$, and we will also assume that $X$ is a Cartier divisor. Then $\Omega^p_{\bf P}(X) = \Omega^p_{\bf P} \otimes {\cal O}_{\bf P}(X)$. Recall that local sections of $\Omega^p_{\bf P}(X)$ are rational $p$-forms which become holomorphic when multiplied by the local equation of $X$. \begin{prop} \label{prop.pform} If $X$ is a Cartier divisor on ${\bf P}$ defined by $f = 0$ for $f \in S_\beta$, then \[ H^0({\bf P},\Omega^p_{\bf P}(X)) = \Bigl\{ {\omega \over f} : \omega \in (\widehat{\Omega}^p_S)_\beta \Bigr\}.\] \end{prop} \noindent {\bf Proof. } We first observe that multiplication by $1/f$ gives an isomorphism ${\cal O}_{\bf P}(\beta) \cong {\cal O}_{\bf P}(X)$. To see this, we will work locally on an affine open ${\bf A}_\sigma \subset {\bf P}$, where $\sigma \in \Sigma$. Recall that ${\bf A}_\sigma = {\rm Spec}\,(S_\sigma)_0$, where $S_\sigma$ is the localization of $S$ at $\widehat{z}_\sigma = \prod_{e_i \notin \sigma} z_i$. Since $X$ is Cartier, Lemma 3.4 of \cite{cox} shows that there is a monomial $z^D \in S_\beta$ which is invertible in $S_\sigma$. It follows that $f/z^D = 0$ is the local equation of $X$ on ${\bf A}_\sigma$. Hence \[ H^0({\bf A}_\sigma,{\cal O}_{\bf P}(X)) = {1 \over f/z^D} \cdot H^0({\bf A}_\sigma,{\cal O}_{\bf P}) = {1 \over f/z^D} \cdot (S_\sigma)_0 = {1 \over f} \cdot (S_\sigma)_\beta ,\] so that $(1/f)S$ is the graded $S$-module that gives ${\cal O}_{\bf P}(X)$. Thus $1/f : S(\beta) \to (1/f)S$ is a graded isomorphism which induces ${\cal O}_{\bf P}(\beta) \cong {\cal O}_{\bf P}(X)$. This proves that we have an isomorphism $\Omega^p_{\bf P}(\beta) \cong \Omega^p_{\bf P}(X)$ which is given by multiplication by $1/f$ when we represent each sheaf by a graded $S$-module. Then the desired result follows immediately from Proposition \ref{prop.sections}. \hfill $\Box$ \begin{rem} {\rm There is a direct way of seeing that $\omega/f$ descends to $p$-form on ${\bf P}$. In diagram (\ref{dia.omega}), we can think of $S\otimes\Lambda^p{\bf Z}^n$ as $\Omega^p_S({\rm log}\,\tilde{D})$, where $\tilde{D}$ is the union of the coordinate hyperplanes. With this interpretation, the standard basis of ${\bf Z}^n$ is $dz_i/z_i$, and the map $\Omega^p_S \to S\otimes \Lambda^p{\bf Z}^n$ is given by $dz_i \mapsto z_i\otimes dz_i/z_i$. Now fix a basis $m_1,\dots,m_d$ of $M$ and let \[ t_j = \prod_{i=1}^n z_i^{\langle m_j,e_i\rangle}\] for $1 \le j \le d$. The $t_j$ are invariant under the group ${\bf D} = {\bf D}(\Sigma)$ and hence descend to rational functions on ${\bf P}$ (in fact, they are coordinates for the torus ${\bf T} \subset {\bf P}$). Note that $dt_j/t_j = \sum_{i=1}^n\langle m_j,e_i\rangle dz_i/z_i = \alpha(m_j)$, where $\alpha : M \to {\bf Z}^n$ is the map from Definition \ref{diag}. Then, in $S\otimes \Lambda^p{\bf Z}^n$, we can write \begin{eqnarray*} \omega & = & \sum_{j_1 < \cdots < j_d} g_{{j_1}\cdots{j_d}} \otimes \alpha(m_{j_1})\wedge\cdots\wedge \alpha(m_{j_d})\\ & = & \sum_{j_1 < \cdots < j_d} g_{{j_1}\cdots{j_d}} \otimes dt_{j_1}/t_{j_1}\wedge\cdots\wedge dt_{j_d}/t_{j_d} \end{eqnarray*} where $g_{{j_1}\cdots{j_d}} \in S_\beta$. Thus $\omega/f$ can be written \[{\omega \over f} = \sum_{j_1 < \cdots < j_d} {g_{{j_1}\cdots{j_d}} \over f} \otimes dt_{j_1}/t_{j_1}\wedge\cdots\wedge dt_{j_d}/t_{j_d}.\] Since $g_{{j_1}\cdots{j_d}}$ and $f$ have the same degree, $\omega/f$ descends to a rational $p$-form on ${\bf P}$.} \label{rem.dz} \end{rem} \section{Differential forms of degree $d$ and $d-1$} In this section, we will find module generators for $\widehat{\Omega}^d_S$ and $\widehat{\Omega}^{d-1}_S$, where $d$ is the dimension of the complete simplicial toric variety ${\bf P}$. This will enable us to give an explicit description of $H^0({\bf P},\Omega^p_{\bf P}(X))$ for $p = d$ and $d-1$. \begin{opr} {\rm Fix an integer basis $m_1,\dots,m_d$ for the lattice $M$. Then, given a subset $I = \{i_i,\dots,i_d\} \subset \{1,\dots,n\}$ consisting of $d$ elements, define \[ \det(e_I) = \det(\langle m_j,e_{i_k}\rangle_{\scriptscriptstyle{1 \le j,k \le d}}). \] We also define $dz_I = dz_{i_1}\wedge\cdots\wedge dz_{i_d}$ and $\widehat{z}_I = \textstyle{\prod_{i \notin I}} z_i$.} \label{def.eI} \end{opr} \begin{rem} {\rm Although $\det(e_I)$ and $dz_I$ depend on how the elements of $I$ are ordered, their product $\det(e_I) dz_I$ does not.} \end{rem} \begin{opr} \label{def.omegao} {\rm We define the $d$-form $\Omega_0 \in \Omega^d_S$ by the formula \[\Omega_0 = \sum_{|I| = d} \det(e_I) \widehat{z}_I dz_I,\] where the sum is over all $d$ element subsets $I \subset \{1,\dots,n\}$.} \end{opr} \begin{exam} {\rm Suppose that ${\bf P}$ is the weighted projective space ${\bf P}(w_1,\dots,w_{d+1})$. Then one can check that up to a nonzero constant (which depends on the basis of $M$ chosen in Definition \ref{def.eI}), we have \[ \Omega_0 = \sum_{i=1}^{d+1} (-1)^i w_i z_i dz_1\wedge\cdots \wedge \widehat{dz_i} \wedge\cdots \wedge dz_{d+1}.\] This form appears in 2.1.3 of \cite{dolgach}, where it is denoted $\Delta(dT_0\wedge\cdots\wedge dT_r)$. Also, when ${\bf P} = {\bf P}^d$, we have $w_i = 1$ for all $i$, and we recover the form $\Omega$ in Corollary 2.11 of \cite{griff1}.} \end{exam} \begin{prop} $\widehat{\Omega}^d_S \subset \Omega^d_S$ is a free $S$-module of rank 1 generated by $\Omega_0 \in \Omega^d_S$. \label{prop.dmodule} \end{prop} \noindent {\bf Proof. } Using the basis $m_1,\dots,m_d$ of $M$, we see that $S\otimes\Lambda^d M$ is free of rank 1 with generator $m_1\wedge\cdots\wedge m_d$. Furthermore, since $\langle e_i,m_1\wedge\cdots\wedge m_d\rangle$ is nonzero in the rank 1 {\bf Z}-module $\Lambda^{d-1}(M\cap e_i^\perp)$, the definition of $\widehat{\Omega}^d_S$ shows that for $A \in S$, $A\, m_1\wedge\cdots\wedge m_d$ lies in $\widehat{\Omega}^d_S$ if and only if $A$ is divisible by $z_1\cdots z_n$. Thus $\widehat{\Omega}^d_S \subset S\otimes\Lambda^d M$ is free of rank 1 with $z_1\dots z_n\, m_1\wedge\cdots\wedge m_d$ as generator. As in Remark \ref{rem.dz}, we will denote the standard basis of ${\bf Z}^n$ by $dz_1/z_1,\dots,dz_n/z_n$. Then, using the map $\alpha : M \to {\bf Z}^n$, we have $\alpha(m) = \sum_{i=1}^n \langle m,e_i\rangle dz_i/z_i$. Thus, inside $S\otimes\Lambda^d{\bf Z}^n$, the generator $z_1\cdots z_n\,m_1\wedge\cdots\wedge m_d$ of $\widehat{\Omega}^d_S$ is \begin{eqnarray*} z_1\cdots z_n \textstyle{\left(\sum_{i=1}^n \langle m_1,e_i\rangle {dz_i\over z_i}\right) \wedge \cdots \wedge \left(\sum_{i=1}^n \langle m_d,e_i\rangle {dz_i\over z_i}\right)} & = & z_1\cdots z_n \sum_{|I| = d} \det(e_I) (\textstyle{\prod_{i \in I} z_i^{-1}}) dz_I\\ & = & \sum_{|I| = d} \det(e_I) \widehat{z}_I dz_I\ =\ \Omega_0, \end{eqnarray*} and the proposition is proved. \hfill$\Box$ \begin{rem} \label{rem.omegad} {\rm If $z_i \in S_{\beta_i}$ for $1 \le i \le n$, then we set $\beta_0 = \sum_{i=1}^n \beta_i.$ Since $\Omega_0$ has degree $\beta_0$, the above proposition gives an isomorphism of graded $S$-modules \[ \widehat{\Omega}^d_S \cong S(-\beta_0).\] Furthermore, since $\beta_0 \in Cl(\Sigma)$ is the class of ${\bf P} \backslash {\bf T} = \sum_{i=1}^n D_i$, we get the well-known isomorphism of sheaves \[ \Omega^d_{\bf P} \cong {\cal O}_{\bf P}\bigl(-\textstyle{\sum_{i=1}^n} D_i\bigr).\]} \end{rem} If we combine Propositions \ref{prop.dmodule} and \ref{prop.pform}, we get the following way of representing $d$-forms on $P$ with poles on $X$. \begin{theo} \label{theo.dform} Let $X \subset {\bf P}$ be a Cartier divisor defined by $f = 0$, where $f \in S_\beta$. Then \[ H^0({\bf P},\Omega^d_{\bf P}(X)) = \Bigl\{ {A\Omega_0 \over f} : A \in S_{\beta-\beta_0}\Bigr\},\] where $\beta_0 = \sum_{i=1}^n \beta_i$ and $z_i \in S_{\beta_i}$. \end{theo} We next describe generators for $\widehat{\Omega}^{d-1}_S$. \begin{opr} \label{def.omegai} {\rm Given $i$ between $1$ and $n$, we define the $(d-1)$-form $\Omega_i \in \Omega^{d-1}_S$ by the formula \[ \Omega_i = \sum_{\scriptstyle{ |J| = d-1 \atop i \notin J}} \det(e_{J\cup\{i\}}) \widehat{z}_{J\cup\{i\}} dz_J, \] where $\det(e_{J\cup\{i\}})$ is computed by ordering the elements of $J\cup\{i\}$ so that $i$ is {\em first\/} (this ensures that $\det(e_{J\cup\{i\}}) dz_J$ is well-defined).} \end{opr} \begin{exam} {\rm When ${\bf P} = {\bf P}^1\times{\bf P}^1$, we have $e_1 = (1,0)$, $e_2 = -e_1$, $e_3 = (1,0)$ and $e_4 = -e_3$, and one can check that \label{exam.p1p1} \begin{eqnarray*} \Omega_1 & = & z_2(z_4dz_3 - z_3dz_4) \\ \Omega_2 & = & -z_1(z_4dz_3 - z_3dz_4) \\ \Omega_3 & = & z_3(z_1dz_2 - z_2dz_1) \\ \Omega_4 & = & -z_4(z_1dz_2 - z_2dz_1) \end{eqnarray*} In this case, one can show that $\widehat{\Omega}^1_S$ is the submodule of $\Omega^1_S$ generated by $z_1dz_2-z_2dz_1$ and $z_4dz_3 - z_3dz_4$ (this will also follow from Proposition \ref{prop.dminus1} below).} \end{exam} As the above example indicates, the $\Omega_i$ may be multiples of other forms. This happens whenever there is a $j$ such that $e_j = -e_i$. In this situation, note that $\det(e_{J\cup\{i\}}) = 0$ when $j \in J$ since the matrix will have one column which is the negative of another. But when $j \notin J$, then $z_j$ divides $\widehat{z}_{J\cup\{i\}}$. It follows that $e_j = -e_i$ implies \[ \Omega_i = z_j \sum_{\scriptstyle{|J| = d-1 \atop i,j \notin J}} \det(e_{J\cup\{i\}}) \widehat{z}_{J\cup\{i,j\}} dz_J. \] Hence we get the following definition. \begin{opr} {\rm Let \begin{eqnarray*} {\cal I}_0 & = & \bigl\{ i : -e_i \notin \{e_1,\dots,e_n\}\bigr\} \\ {\cal I}_1 & = & \bigl\{ (i,j) : i < j,\ e_i = -e_j\bigr\} \end{eqnarray*} Furthermore, for $(i,j) \in {\cal I}_1$, we define the $(d-1)$-form \[ \Omega_{ij} = \sum_{\scriptstyle{|J| = d-1 \atop i,j \notin J}} \det(e_{J\cup\{i\}}) \widehat{z}_{J\cup\{i,j\}} dz_J. \]} \end{opr} \begin{rem} {\rm For $(i,j) \in {\cal I}_1$, we have $\Omega_i = z_j\Omega_{ij}$, and since $\det(e_{J\cup\{i\}}) = -\det(e_{J\cup\{j\}})$ (remember $e_j = -e_i$), we get $\Omega_j = - z_i\Omega_{ij}$ as in Example \ref{exam.p1p1}.} \end{rem} \begin{prop} With the above notation, $\widehat{\Omega}^{d-1}_S$ is the submodule of $\Omega^{d-1}_S$ generated by the $\Omega_i$ for $i \in {\cal I}_0$ and the $\Omega_{ij}$ for $(i,j) \in {\cal I}_1$. \label{prop.dminus1} \end{prop} \noindent {\bf Proof. } We begin with the exact sequence \begin{equation} 0 \to \widehat{\Omega}^{d-1}_S \to S\otimes\Lambda^{d-1} \stackrel{\gamma}\to \bigoplus_{i=1}^n (S/z_i S)\otimes\Lambda^{d-2}M \label{seq.dminus1} \end{equation} which follows from the definition of $\widehat{\Omega}^{d-1}_S$ and the inclusion $M\cap e_i^\perp \subset M$. If $m_1,\dots,m_d$ is an integer basis of $M$, then we get the bases \[ \begin{array}{rl} \hbox{basis of $\Lambda^{d-1}M$}: & \omega_j = (-1)^{j+1}m_1\wedge \cdots \wedge \widehat{m_j} \wedge \cdots \wedge m_d,\quad 0 \le j \le d \\ \hbox{basis of $\Lambda^{d-2}M$}: & \omega_{jk} = (-1)^{j+k}m_1\wedge \cdots \wedge \widehat{m_j} \wedge \cdots \wedge \widehat{m_k} \wedge \cdots \wedge m_d,\quad 0 \le j < k \le d \end{array}\] We will also set $\omega_{jk} = -\omega_{kj}$ when $j > k$. The signs in the definitions of $\omega_j$ and $\omega_{jk}$ were chosen to make interior products easy to calculate. In fact, one can check that if $m_1^*,\dots,m_d^*$ is the dual basis to $m_1,\dots,m_d$, then \[ \langle m_k^*,\omega_j\rangle = \cases{0 & $k = j$ \cr \omega_{kj} & $k \ne j$.\cr}\] Since $e_i = \sum_{k=1}^d \langle m_k,e_i\rangle m_k^*$, it follows that \begin{equation} \langle e_i,\omega_j\rangle = \sum_{k \ne j} \langle m_k, e_i\rangle \omega_{kj}. \label{eq.interior} \end{equation} For later purposes, we also note that as in the proof of Proposition \ref{prop.dmodule}, the form $\omega_j$ can be written inside $S\otimes\Lambda^{d-1}{\bf Z}^n$ as \begin{equation} \sum_{|J| = d-1} (-1)^{j+1} \det(e^j_J)(\textstyle{\prod_{l \in J}} z_l^{-1}) dz_J \label{eq.omegaj} \end{equation} where $\det(e_J^j)$ is the determinant of the $(d-1)\times(d-1)$ submatrix of $(\langle m_k,e_l\rangle_{\scriptscriptstyle{1 \le k \le d, l \in J}})$ obtained by deleting the $j^{\rm th}$ row. The next observation is that $S$ has a ${\bf Z}^n$ grading coming from the action of $({\bf C}^*)^n$ on ${\bf A}^n$. The graded pieces are 1-dimensional, each spanned by a single monomial. This induces a grading on $S\otimes\Lambda^{d-1}M$ and $S/z_i S\otimes\Lambda^{d-2}M$, and the map $\gamma$ of (\ref{seq.dminus1}) is a graded homomorphism. Thus the entire exact sequence has a ${\bf Z}^n$ grading. It following that in studying $\widehat{\Omega}^{d-1}_S \subset S\otimes\Lambda^{d-1}M$, we can restrict our attention to ``homogeneous'' elements of $S\otimes\Lambda^{d-1}M$, i.e., elements of the form $\sum_{j=1}^d c_j z^D\otimes \omega_j$, where $\omega_j$ is as above, $c_j \in {\bf C}$, and $z^D$ is a monomial. Take such an element $\sum_{j=1}^d c_jz^D\otimes\omega_j \in S\otimes\Lambda^{d-1} M$. Then, in the exact sequence (\ref{seq.dminus1}), equation (\ref{eq.interior}) shows that the $i^{\rm th}$ component of $\gamma(\sum_{j=1}^d c_jz^D\otimes\omega_j)$ is given by \[ \sum_{j=1}^d\sum_{k\ne j} (c_jz^D \langle m_k,e_i\rangle \bmod z_i)\otimes \omega_{kj} = \sum_{j<k} (\langle c_jm_k-c_km_j,e_i\rangle z^D)\bmod z_i)\otimes \omega_{kj}.\] Thus it follows that $\sum_{j=1}^d c_j z^D\otimes\omega_j$ lies in $\widehat{\Omega}^{d-1}_S$ if and only if \begin{equation} \hbox{$z_i$ divides $\langle c_j m_k-c_km_j,e_i\rangle z^D$ for all $i$, $j$ and $k$.} \label{criterion} \end{equation} For $1 \le i \le n$, the above criterion shows that $\sum_{j=1}^d \langle m_j,e_i\rangle \widehat{z}_i\, \omega_j$ lies in $\widehat{\Omega}^{d-1}_S$, where $\widehat{z}_i = \prod_{l \ne i}z_l$. Furthermore, using equation (\ref{eq.omegaj}), we see that in $S\otimes\Lambda^{d-1}{\bf Z}^n$, this form equals \[ \sum_{j=1}^d \langle m_j,e_i\rangle \widehat{z}_i \Bigl(\textstyle{\sum_{|J| = d-1}} (-1)^{j+1} \det(e^j_J)(\textstyle{\prod_{l \in J}} z_l^{-1})\Bigr) dz_J, \] which can be written as \[ \sum_{|J| = d-1} \Bigl(\textstyle{\sum_{j=1}^d} (-1)^{j+1}\langle m_j,e_i\rangle \det(e_J^j)\Bigr) \widehat{z}_{J\cup\{i\}} dz_J = \sum_{|J| = d-1} \det(e_{J\cup\{i\}}) \widehat{z}_{J\cup\{i\}} dz_J = \Omega_i \] since $\langle m_j,e_i\rangle$ for $1 \le j \le d$ gives the first column of the matrix in $\det(e_{J\cup\{i\}})$. Similarly, if $(i,j) \in {\cal I}_1$, one can show that $\sum_{j=1}^d \langle m_j,e_i\rangle \widehat{z}_{ij}\, \omega_j$ lies in $\widehat{\Omega}^{d-1}_S$ and equals $\Omega_{ij}$. Thus $\Omega_i$, $1 \le i \le n$, and $\Omega_{ij}$, $(i,j) \in {\cal I}_1$ lie in $\widehat{\Omega}^{d-1}_S$. To prove that these forms generate $\widehat{\Omega}^{d-1}_S$, suppose $\omega = \sum_{j=1}^d c_j z^D\otimes \omega_j$ satisfies equation (\ref{criterion}). We can assume that at least one $c_j \ne 0$. There are three cases to consider: {\sc Case I.} Suppose that $z_i$ divides $z^D$ for all $i$. It suffices to consider $\omega = z_1\dots z_n\, \omega_j$, and we can assume that $e_1,\dots, e_d$ are linearly independent. Then for $1 \le i \le d$, the $z_i\Omega_i = \sum_{j=1}^n \langle m_j,e_i\rangle z_1\cdots z_n\, \omega_j$ have the same span as the $z_1\dots z_n\, \omega_j$ since $(\langle m_j,e_i\rangle_{\scriptscriptstyle{1 \le k,i \le d}})$ is invertible. Thus $z_1\cdots z_n\, \omega_j$ lies in the submodule generated by the $\Omega_i$. {\sc Case II.} Suppose that $z_i$ doesn't divide $z^D$ for some $i \in {\cal I}_0$ . Then (\ref{criterion}) implies that $\langle c_jm_k - c_km_j, e_i\rangle = 0$ for all $j,k$. Thus $e_i$ is orthogonal to the codimension 1 sublattice spanned by $c_jm_k-c_km_j$. Since $e_i$ is primitive, we see that $e_i$ is determined uniquely up to $\pm$. In particular, if there were some $j \ne i$ where $z_j$ also didn't divide $z^D$, then we would have $e_j = -e_i$, which is impossible since $i \in {\cal I}_0$. This shows that $\widehat{z}_i$ divides $z^D$. Furthermore, $0 = \langle c_jm_k-c_km_j,e_i\rangle = c_j\langle m_k,e_i\rangle - c_k\langle m_j,e_i\rangle$ shows that for some constant $\lambda$, we have $c_j = \lambda\langle m_j,e_i\rangle$ for all $j$. Thus $\omega$ is a multiple of $\Omega_i$. {\sc Case III.} Suppose that $z_i$ doesn't divide $z^D$ for some $i \notin {\cal I}_0$. We can assume $(i,j) \in {\cal I}_1$, and the argument of Case II shows that $\widehat{z}_{ij}$ divides $z^D$. Then, as in Case II, we see that $\omega$ is a multiple of $\Omega_{ij}$. Since $\Omega_i = z_j\Omega_{ij}$ and $\Omega_j = -z_i\Omega_{ij}$ for $(i,j) \in {\cal I}_1$, we only need the the $\Omega_i$ for $i \in {\cal I}_0$ along with the $\Omega_{ij}$ to generate $\widehat{\Omega}^{d-1}_S$. This proves the proposition. \hfill$\Box$ \medskip We get the following description of $(d-1)$-forms on ${\bf P}$ with poles on $X$. \begin{theo} \label{theo.dminus1} Let $X \subset {\bf P}$ be a Cartier divisor defined by $f = 0$, where $f \in S_\beta$. Then \[ H^0({\bf P},\Omega^{d-1}_{\bf P}(X)) = \Bigl\{ {\sum_{i \in {\cal I}_0} A_i\Omega_i + \sum_{(i,j) \in {\cal I}_1} A_{ij} \Omega_{ij}\over f} : A_i \in S_{\beta-\beta_0+\beta_i},\ A_{ij} \in S_{\beta-\beta_0+\beta_i + \beta_j} \Bigr\},\] where $\beta_0 = \sum_{i=1}^n \beta_i$ and $z_i \in S_{\beta_i}$. However, if $S_\beta \subset B(\Sigma) = \langle \widehat{z}_\sigma : \sigma \in \Sigma\rangle$ (see Definition \ref{def.B}), then \[ H^0({\bf P},\Omega^{d-1}_{\bf P}(X)) = \Bigl\{ {\sum_{i=1}^n A_i\Omega_i \over f} : A_i \in S_{\beta-\beta_0+\beta_i} \Bigr\}.\] \end{theo} \noindent {\bf Proof. } Since $\Omega_i$ has degree $\beta_0 - \beta_i$ and $\Omega_{ij}$ has degree $\beta_0 - \beta_i - \beta_j$, the first part of the theorem follows immediately from Propositions \ref{prop.pform} and \ref{prop.dminus1}. For the second part, suppose that $S_\beta \subset B(\Sigma)$, and consider a form $A\, \Omega_{ij}$ where $A \in S_{\beta -\beta_0 + \beta_i + \beta_j}$. If $z^D$ is a monomial that appears in $A$, then $z^D\,\widehat{z}_{ij} \in S_\beta \subset B(\Sigma)$. This implies that $z^D\,\widehat{z}_{ij}$ is divisible by $\widehat{z}_\sigma$ for some $\sigma \in \Sigma$. But $\sigma$ can't contain both $e_i$ and $e_j$ since $e_i = -e_j$. Thus $z_i$ or $z_j$ divides $\widehat{z}_\sigma$, so that $z_i$ or $z_j$ divides $z^D\,\widehat{z}_{ij}$. It follows that $z^D$ is divisible by $z_i$ or $z_j$, and thus $z^D\,\Omega_{ij}$ is a multiple of either $z_i\,\Omega_{ij} = -\Omega_j$ or $z_j\,\Omega_{ij} = \Omega_j$. Hence $A\,\Omega_{ij}$ is in the submodule generated by the $\Omega_i$, and the theorem is proved. \hfill $\Box$ \begin{rem} {\rm The reader can check that when ${\bf P} = {\bf P}^d$, the description of $H^0({\bf P},\Omega^{d-1}_{\bf P}(X))$ given above generalizes equation (4.4) from \cite{griff1}.} \end{rem} To exploit the second part of Theorem \ref{theo.dminus1}, we need to know when $S_\beta \subset B(\Sigma)$. \begin{lem} \label{lem.finB} If $\beta \in Cl(\Sigma)$ is the class of an ample divisor on ${\bf P}$, then $S_\beta \subset B(\Sigma)$. \end{lem} \noindent {\bf Proof. } Given $z^D = \prod_{i=1}^n z_i^{a_i} \in S_\beta$, our hypothesis implies that $D = \sum_{i=1}^n a_i D_i$ is an ample divisor on ${\bf P}$. Thus ${\cal L} = {\cal O}_{\bf P}(D)$ is an ample ${\bf T}$-linearized invertible sheaf. Let $\Delta = \Delta({\cal L}) \subset M_{\bf R}$ be its associated convex polytope. Since $\Delta$ is defined by the inequalities $\langle m,e_i\rangle \ge -a_i$, we have $0 \in \Delta$ since $a_i \ge 0$. Then let $\Delta_0$ be the minimal face of $\Delta$ containing $0$. We know that for some $\sigma \in \Sigma$, the face $\Delta_\sigma$ of $\Delta$ corresponding to $\sigma$ is $\Delta_0$. We claim that $\widehat{z}_\sigma$ divides our monomial $z^D$. To see why this is true, suppose we have some $z_i$ which doesn't divide $z^D$. This means $a_i = 0$. If $\rho_i$ is the 1-dimensional cone of $\Sigma$ generated by $e_i$, then the corresponding facet of $\Delta$ is $\Delta_{\rho_i} = \Delta\cap\{m : \langle m,e_i\rangle \ge 0\}$ since $a_i = 0$. Thus $0 \in \Delta_{\rho_i}$, which implies $\Delta_\sigma \subset \Delta_{\rho_i}$ by the minimality of $\Delta_\sigma$. It follows that $\rho_i \subset \sigma$. We have thus proved that $a_i = 0$ implies $e_i \in \sigma$, and it follows immediately that $\widehat{z}_\sigma = \prod_{e_i \notin \sigma} z_i$ divides $z^D$. Thus $z^D \in B(\Sigma)$, and the lemma is proved. \hfill $\Box$ \medskip If we combine this lemma with Theorem \ref{theo.dminus1}, we get the following useful corollary. \begin{coro} \label{coro.dminus1} Let $X \subset {\bf P}$ be an ample Cartier divisor defined by $f = 0$, where $f \in S_\beta$. Then \[ H^0({\bf P},\Omega^{d-1}_{\bf P}(X)) = \Bigl\{ {\sum_{i=1}^n A_i\Omega_i \over f} : A_i \in S_{\beta-\beta_0+\beta_i} \Bigr\}.\] where $\beta_0 = \sum_{i=1}^n \beta_i$ and $z_i \in S_{\beta_i}$. \end{coro} \section{Cohomology of the complement of an ample $\;\;\;\;\;$ divisor} In this section, ${\bf P}$ will be a $d$-dimensional complete simplicial toric variety and $X \subset {\bf P}$ will be the zero locus of a global section of a ${\bf T}$-linearized ample invertible sheaf ${\cal L}$. If $\beta \in Cl(\Sigma)$ is the class of ${\cal L}$, then Lemma \ref{lem.iso} shows that $X$ is defined by an equation $f = 0$ for some $f \in S_\beta$. We will also assume that $X$ is quasi-smooth (by Proposition \ref{prop.generic}, this is true for generic $f \in S_\beta$). Our goal is to compute the cohomology of ${\bf P} \backslash X$ in terms of $f \in S$. We will also study the cohomology of $X$. Our results will generalize classical results of Griffiths, Dolgachev and Steenbrink (see \cite{griff1,dolgach,steen}). Since $X$ is a $V$-submanifold of the $V$-manifold ${\bf P}$, we can compute $H^\cdot({\bf P} \backslash X)$ using the complex $\Omega_{\bf P}^\cdot({\log X})$ (we always use cohomology with coefficients in ${\bf C}$). Furthermore, the Hodge filtration $F^\cdot$ on $H^{p+q}({\bf P}\backslash X)$ comes from the spectral sequence $H^q({\bf P},\Omega_{\bf P}^p({\log X})) \Rightarrow H^{p+q}({\bf P}\backslash X)$ which degenerates at $E_1$ (see \S15 of \cite{dan1}). Thus we obtain isomorphisms \[ Gr_F^p\, H^d({\bf P}\backslash X) \cong H^{d-p}({\bf P},\Omega_{\bf P}^p(\log X))\] for $p = 0,\dots,d$. Morever, $\Omega_{\bf P}^p(\log X)$ has the following resolution. \begin{prop} If $X$ is a quasi-smooth hypersurface of a complete simplicial toric variety ${\bf P}$, then there is a canonical exact sequence \[ 0 \rightarrow \Omega^p_{\bf P} ({\rm log}\; X) \rightarrow \Omega^p_{\bf P} (X) \stackrel{d}{\rightarrow} \Omega^{p+1}_{\bf P} (2X)/ \Omega^{p+1}_{\bf P} (X) \stackrel{d}{\rightarrow} \ldots \] \[ \ldots \stackrel{d}{\rightarrow} \Omega^{d-1}_{\bf P} ((d -p)X)/ \Omega^{d-1}_{\bf P} ((d-p-1)X) \stackrel{d}{\rightarrow} \Omega^{d}_{\bf P} ((d -p +1)X)/ \Omega^d_{\bf P} ((d-p)X) \rightarrow 0. \] \end{prop} \noindent {\bf Proof. } The proof of this is similar to the proof of Theorem 6.2 in \cite{bat.var}. \hfill$\Box$ \medskip If we combine the above exact sequence with the Bott-Steenbrink-Danilov vanishing theorem (see Theorem \ref{theo.bott}), then we obtain the following corollary. \begin{coro} \label{coro.log} There are natural isomorphisms \[ Gr_F^p\, H^d({\bf P}\backslash X) \cong H^{d-p}({\bf P}, \Omega_{\bf P}^p ({\rm log}\, X)) \cong \frac{H^0({\bf P}, \Omega^{d}_{\bf P} ((d -p +1)X)} {H^0({\bf P}, \Omega^d_{\bf P} ((d -p)X) + dH^0({\bf P}, \Omega^{d-1}_{\bf P} ((d -p)X)}. \] \end{coro} Before we can state our next result, we need some definitions. \begin{opr} {\rm Let $f \in S_\beta$ be a nonzero polynomial. Then the {\em Jacobian ideal $J(f) \subset S$} is the ideal of $S = {\bf C}[z_1,\dots,z_n]$ generated by the partial derivatives $\partial f/\partial z_1,\dots,\partial f/\partial z_n$. Also, the {\em Jacobian ring $R(f)$} is the quotient ring $S/J(f)$.} \end{opr} \begin{rem} {\rm Since $f \in S_\beta$, we have $\partial f/\partial z_i \in S_{\beta-\beta_i}$, where $z_i \in S_{\beta_i}$. Thus $J(f)$ is a graded ideal of $S$, so that $R(f)$ has a natural grading by the class group $Cl(\Sigma)$.} \end{rem} \begin{lem} \label{lem.fJ} If $f \in S_\beta$, where $\beta \ne 0$, then $f \in J(f)$. \end{lem} \noindent {\bf Proof. } We will prove this using the Euler formulas from Definition \ref{euler.def}. Let $\prod_{i=1}^n z_i^{b_i}$ be a monomial appearing in $f$. Then, in $Cl(\Sigma)$, we have $\beta = [\sum_{i=1}^n b_i D_i]$, where $D_i$ is the divisor in ${\bf P}$ corresponding to $e_i$. If we can find $\phi_1,\dots,\phi_n \in {\bf C}$ such that $\sum_{i=1}^n \phi_i e_i = 0$, then Lemma \ref{euler.lem} tells us that \[ ({\textstyle{\sum_{i=1}^n \phi_ib_i}})\,f = \sum_{i=1}^n \phi_i z_i {\partial f \over \partial z_i}.\] Thus we need to find a relation $\sum_{i=1}^n \phi_i e_i = 0$ such that $\sum_{i=1}^n \phi_i b_i \ne 0$. To do this, pick $j$ such that $b_j > 0$. By completeness, $-e_j$ must lie in some cone $\sigma \in \Sigma$. Since $e_j$ can't be a generator of $\sigma$, we get $-e_j = \sum_{i \ne j} \phi_i e_i$, where $\phi_i \ge 0$, so that setting $\phi_j = 1$ gives $\sum_{i=1}^n \phi_i e_i = 0$. Since $b_j > 0$ and $b_i \ge 0$ for $i \ne j$, we have $\sum_{i=1}^n \phi_i b_i > 0$. \hfill $\Box$ \medskip We can now state the first main result of this section. \begin{theo} \label{theo.main1} Let ${\bf P}$ be a $d$-dimensional complete simplicial toric variety, and let $X \subset {\bf P}$ be a quasi-smooth ample hypersurface defined by $f \in S_\beta$. If $R(f)$ is the Jacobian ring of $f$, then there is a canonical isomorphism \[ Gr_F^p H^d({\bf P}\backslash X) \cong R(f)_{(d-p+1)\beta-\beta_0},\] where $z_i \in S_{\beta_i}$ and $\beta_0 = \sum_{i=1}^n \beta_i$. \end{theo} \noindent {\bf Proof. } The arguments are similar to those used in the classical case (see, for example, \cite{pet.steen}). By Theorem \ref{theo.dform}, we have \[ H^0({\bf P},\Omega^d_{\bf P}((d-p+1)X)) = \Bigl\{ {A\Omega_0 \over f^{d-p+1}} : A \in S_{(d-p+1)\beta-\beta_0}\Bigr\},\] so that the map $\phi(A\Omega_0/f^{d-p+1}) = A$ defines a bijection \[\phi: H^0({\bf P},\Omega^d_{\bf P}((d-p+1)X)) \cong S_{(d-p+1)\beta -\beta_0}.\] By Corollary \ref{coro.log}, it suffices to show that the subspace \[ H^0({\bf P}, \Omega^d_{\bf P} ((d -p)X) + dH^0({\bf P}, \Omega^{d-1}_{\bf P} ((d -p)X) \subset H^0({\bf P},\Omega^d_{\bf P}((d-p+1)X)) \] maps via $\phi$ to $J(f)_{(d-p+1)\beta-\beta_0} \subset S_{(d-p+1)\beta-\beta_0}$. If $p = d$, then the desired result follows immediately since $H^0({\bf P},\Omega_{\bf P}^d)$, $H^0({\bf P},\Omega_{\bf P}^{d-1})$ and $J(f)_{\beta-\beta_0}$ all vanish (the last because $\partial f/\partial z_i \in S_{\beta-\beta_i}$). Now assume $p < d$. Since $A\Omega_0/f^{d-p} = fA\Omega_0/f^{d-p+1}$, we see that $\phi(H^0({\bf P},\Omega_{\bf P}^d((d-p)X))) = fS_{(d-p)\beta-\beta_0}$. It remains to see what happens to $dH^0({\bf P},\Omega^{d-1}_{\bf P}((d-p)X))$. By Corollary \ref{coro.dminus1}, we know that \[ H^0({\bf P},\Omega^{d-1}_{\bf P}((d-p)X)) = \Bigl\{ {\sum_{i=1}^n A_i\Omega_i \over f^{d-p}} : A_i \in S_{(d-p)\beta-\beta_0+\beta_i} \Bigr\}.\] Since $d-p \ne 0$ and $f \in J(f)$ by Lemma \ref{lem.fJ}, the theorem now follows easily from the following lemma. \begin{lem} If $A \in S_{(d-p)\beta-\beta_0+\beta_i}$, then \[ d\Bigl({A \Omega_i \over f^{d-p}}\Bigr) = {(f \partial A/\partial z_i - (d-p) A \partial f/\partial z_i)\Omega_0 \over f^{d-p+1}}.\] \end{lem} \noindent {\bf Proof. } First note that \begin{equation} \label{eq.mess} d\Bigl({A\Omega_i \over f^{d-p}}\Bigr) = {1\over f^{d-p+1}}(f dA \wedge \Omega_i + f A d\Omega_i - (d-p)df\wedge \Omega_i). \end{equation} This equals $B\Omega_0/f^{d-p+1}$ for some $B \in S_{(d-p+1)\beta-\beta_0}$, where \[\Omega_0 = \sum_{|I| = d} \det(e_I) \widehat{z}_I dz_I\] (see Definition \ref{def.omegao}). Pick $I_0 \subset \{1,\dots,n\}$ such that $|I_0| = d$, $i \in I_0$ and $\det(e_{I_0}) \ne 0$. To find $B$, it suffices to determine the coefficient of $dz_{I_0}$ in (\ref{eq.mess}). From Definition \ref{def.omegai}, we have \[ \Omega_i = \sum_{\scriptstyle{ |J| = d-1 \atop i \notin J}} \det(e_{J\cup\{i\}}) \widehat{z}_{J\cup\{i\}} dz_J. \] Since neither $z_i$ nor $dz_i$ appear in $\Omega_i$, $dz_{I_0}$ doesn't appear in $fAd\Omega_i$. Furthermore, if we set $J_0 = I_0 \backslash \{i\}$, then the coefficient of $dz_{I_0} = dz_i\wedge dz_{J_0}$ in $fdA\wedge \Omega_i - (d-p)df \wedge \Omega_i$ is \[f \textstyle{\partial A \over\partial z_i} dz_i \wedge \det(e_{J_0\cup\{i\}}) \widehat{z}_{J_0\cup\{i\}} dz_{J_0} - (d-p)A\textstyle{\partial f \over \partial z_i}dz_i \wedge \det(e_{J_0\cup\{i\}}) \widehat{z}_{J_0\cup\{i\}} dz_{J_0}\] \[ = (f \partial A/\partial z_i - (d-p)A\partial f/\partial z_i)\det(e_{I_0}) \widehat{z}_{I_0} dz_{I_0}.\] This shows that $B = f \partial A/\partial z_i - (d-p)A\partial f/\partial z_i$ and completes the proof of the lemma. \hfill$\Box$ \medskip We next study the cohomology of the hypersurface $X$. Our first result is a Lefschetz theorem. \begin{prop} \label{prop.lef} Let $X$ be a quasi-smooth hypersurface of a $d$-dimensional complete simplicial toric variety ${\bf P}$, and suppose that $X$ is defined by $f \in S_\beta$. If $f \in B(\Sigma)$ (see Definition \ref{def.B}), then the natural map $i^* : H^i({\bf P}) \to H^i(X)$ is an isomorphism for $i < d-1$ and an injection for $i = d-1$. In particular, this holds if $X$ is an ample hypersurface. \end{prop} \noindent {\bf Proof. } In the affine space ${\bf A}^n$, $f \in B(\Sigma)$ implies that $Z(\Sigma) = {\bf V}(B(\Sigma)) \subset {\bf V}(f)$. Thus ${\bf A}^n \backslash {\bf V}(f) \subset {\bf A}^n - Z(\Sigma) = U(\Sigma)$ is affine. Since ${\bf P} = U(\Sigma)/{\bf D}(\Sigma)$, it follows that ${\bf P} \backslash X = ({\bf A}^n \backslash {\bf V}(f))/{\bf D}(\Sigma)$ is affine. This implies $H^i({\bf P}\backslash X) = 0$ for $i > d$ by Corollary 13.6 of \cite{dan1}. Now consider the Gysin sequence \[ \dots \to H^{i-2}(X) \stackrel{i_!}\to H^i({\bf P}) \to H^i({\bf P}\backslash X) \to H^{i-1}(X) \stackrel{i_!}\to H^{i+1}({\bf P}) \to \dots\] (see the proof of Theorem 3.7 of \cite{dan.hov}). Since the Gysin map $i_!$ is dual to $i^*$ under Poincar\'e duality, we see that $i^*$ has the desired property. Finally, if $X$ is ample, then Lemma \ref{lem.finB} implies that $f \in B(\Sigma)$. \hfill$\Box$ \medskip This result shows that the ``interesting'' part of the cohomology of $X$ occurs in dimension $d-1$ and consists of those classes which don't come from ${\bf P}$. Hence we get the following definition. \begin{opr} {\rm The {\em primitive cohomology group} $PH^{d-1}(X)$ is defined by the exact sequence \[ 0 \to H^{d-1}({\bf P}) \to H^{d-1}(X) \to PH^{d-1}(X) \to 0.\]} \end{opr} \begin{rem} {\rm Since $H^{d-1}({\bf P})$ and $H^{d-1}(X)$ have pure Hodge structures, $PH^{d-1}(X)$ is also pure. Its Hodge components are denoted $PH^{p,d-1-p}(X)$.} \end{rem} \begin{prop} \label{prop.prim} When $X \subset {\bf P}$ is ample, there is an exact sequence \[ 0 \to H^{d-2}({\bf P}) \stackrel{\cup[X]}\to H^d({\bf P}) \to H^d({\bf P}\backslash X) \to PH^{d-1}(X) \to 0\] where $[X] \in H^2({\bf P})$ is the cohomology class of $X$. \end{prop} \noindent {\bf Proof. } By (3.3) of \cite{pet.steen}, we have a commutative diagram \begin{equation} \label{eq.gysin} \begin{array}{ccc} H^i({\bf P}) && \\ \ \downarrow\,\scriptstyle{i^*} & \searrow\!\!{}^{\cup [X]} & \\ H^i(X) & \stackrel{i_!}\longrightarrow\hfill & \!\!H^{i+2}({\bf P}) \end{array} \end{equation} When $i = d-1$, $\cup [X]$ is an isomorphism by Hard Lefschetz, and it follows that $PH^{d-1}$(X) is isomorphic to the kernel of $i_! : H^{d-1}(X) \to H^{d-1}({\bf P})$. Then the Gysin sequence gives us an exact sequence \[ H^{d-2}(X) \stackrel{i_!}\to H^d({\bf P}) \to H^d({\bf P}\backslash X) \to PH^{d-1}(X) \to 0.\] Since $\cup [X] : H^{d-2}({\bf P}) \to H^d({\bf P})$ is injective by Hard Lefschetz and $i^* : H^{d-2}({\bf P}) \to H^{d-2}(X)$ is an isomorphism by Proposition \ref{prop.lef}, the desired exact sequence now follows easily using (\ref{eq.gysin}) with $i = d-2$.\hfill$\Box$ \medskip The maps in Proposition \ref{prop.prim} are all morphisms of mixed Hodge structures (with appropriate shifts). Since $H^i({\bf P})$ vanishes for $i$ odd and has only $(p,p)$ classes for $i$ even (see Theorem 3.11 of \cite{oda}), we get the following corollary of Proposition \ref{prop.prim}. \begin{coro} When $p \ne d/2$, there is a natural isomorphism \[Gr_F^p H^d({\bf P}\backslash X) \cong PH^{p-1,d-p}(X),\] and when $p = d/2$, there is an exact sequence \[ 0 \to H^{d-2}({\bf P}) \stackrel{\cup[X]}\to H^d({\bf P}) \to Gr_F^{d/2}H^d({\bf P}\backslash X) \to PH^{d/2-1,d/2}(X) \to 0.\] \end{coro} If we combine this with Theorem \ref{theo.main1} and replace $p$ with $p+1$, then we get the following theorem. \begin{theo} \label{theo.main} Let ${\bf P}$ be a $d$-dimensional complete simplicial toric variety, and let $X \subset {\bf P}$ be a quasi-smooth ample hypersurface defined by $f \in S_\beta$. If $R(f)$ is the Jacobian ring of $f$, then for $p \ne d/2-1$, there is a canonical isomorphism \[R(f)_{(d-p)\beta-\beta_0} \cong PH^{p,d-1-p}(X)\] where $z_i \in S_{\beta_i}$ and $\beta_0 = \sum_{i=1}^n \beta_i$. For $p = d/2-1$, we have an exact sequence \[ 0 \to H^{d-2}({\bf P}) \stackrel{\cup[X]}\to H^d({\bf P}) \to R(f)_{(d/2+1)\beta-\beta_0} \to PH^{d/2-1,d/2}(X) \to 0.\] \end{theo} \begin{rem} \label{rem.main} {\rm Notice that when $d$ is odd, we always have $R(f)_{(d-p)\beta-\beta_0} \cong PH^{p,d-1-p}(X)$. The same conclusion holds whenever ${\bf P}$ is a toric variety with the property that $\cup [X] : H^{d-2}({\bf P}) \to H^d({\bf P})$ is an isomorphism. The latter holds when ${\bf P}$ is a weighted projective space, which explains why the classical case is so nice. Notice also that in all cases, we always have a surjection \[R(f)_{(d-p)\beta-\beta_0} \to PH^{p,d-1-p}(X) \to 0.\]} \end{rem} \begin{rem} {\rm One consequence of our results is that there is a natural map \[ H^d({\bf P}) \to R(f)_{(d/2+1)\beta-\beta_0}.\] It would be interesting to have an explicit description of this map.} \end{rem} \section{Cohomology of affine hypersurfaces} In a recent paper \cite{bat.var}, the first author obtained some results on the cohomology of the affine hypersurface $Y = X\cap{\bf T} \subset {\bf T}$, where ${\bf T}$ is the torus contained in a projective toric variety ${\bf P}$ and $X \subset {\bf P}$ is an ample hypersurface. We will show how the results of \cite{bat.var} can be expressed in terms of various graded ideals of the ring $S = {\bf C}[z_1,\dots,z_n]$. We begin with a definition. \begin{opr} {\rm If $X \subset {\bf P}$ is a hypersurface, then we let $Y = X\cap {\bf T} \subset {\bf T}$ be its intersection with the torus ${\bf T} \subset {\bf P}$. The primitive cohomology group $PH^{d-1}(Y)$ is then defined to be the cokernel of the map $H^{d-1}({\bf T}) \to H^{d-1}(Y)$, where as usual we use cohomology with coefficients in ${\bf C}$.} \end{opr} \begin{rem} {\rm In \cite{dan.hov}, it is shown that $H^{d-1}({\bf T}) \to H^{d-1}(Y)$ is injective. Note also that $PH^{d-1}(Y)$ has a natural mixed Hodge structure. The Hodge filtration on $PH^{d-1}(Y)$ will be denoted $F^\cdot$.} \end{rem} One of the main results of \cite{bat.var} is a description of the Hodge filtration on $PH^{d-1}(Y)$. To state this in terms of the ring $S$, we need the following ideal of $S$. \begin{opr} {\rm Given $f \in S_\beta$, let $J_0(f) \subset S$ denote the ideal generated by $z_i \partial f/\partial z_i$ for $1 \le i \le n$. We then let $R_0(f)$ denote the quotient ring $S/J_0(f)$.} \end{opr} \begin{rem} {\rm Since $f \in S_\beta$, we see that $J_0(f)$ is a graded ideal of $S$, and hence $R_0(f)$ has a natural grading by the class group $Cl(\Sigma)$.} \end{rem} \begin{theo} \label{theo.aff} {\rm (\cite{bat.var})} If $X \subset {\bf P}$ is a nondegenerate (see Definition \ref{def.nondeg}) ample divisor defined by $f \in S_\beta$ and $Y = X\cap {\bf T}$ is the corresponding affine hypersurface in the torus ${\bf T}$, then there is a natural isomorphism \[ Gr^p_F PH^{d-1}(Y) \cong R_0(f)_{(d-p)\beta}.\] \end{theo} \noindent {\bf Proof. } We first show how certain constructions in \cite{bat.var} can be formulated in terms of the ring $S$. Since $X \subset {\bf P}$ is ample, we get the $d$-dimensional convex polyhedron $\Delta \subset M_{\bf R}$. Recall that $\Delta$ is defined by the inequalities $\langle m,e_i\rangle \ge -b_i$, where $b_i \ge 0$ and $X$ is linearly equivalent to $D = \sum_{i=1}^n b_i D_i$. As in \cite{bat.var}, we define the ring $S_\Delta$ to be the subring of ${\bf C}[t_0,t_1^{\pm1},\dots,t_d^{\pm1}]$ spanned over ${\bf C}$ by all Laurent monomials of the form $t_0^k t^m =t_0^k t_1^{m_1}\cdots t_d^{m_d}$ where $k \ge 0$ and $m \in k\Delta$. This ring is graded by setting $\deg(t_0^kt^m) = k$. We should think of $t_1,\dots,t_d$ as coordinates on the torus ${\bf T}$ and $t_0$ as an auxillary variable to keep track of the grading. A first observation is that there is a natural isomorphism of graded rings \begin{equation} \rho : S_\Delta \cong \bigoplus_{k=0}^\infty S_{k\beta} \subset S \label{eq.translate} \end{equation} which is defined by \[ \rho(t_0^k t^m) = \prod_{i=1}^n z_i^{kb_i + \langle m,e_i\rangle}.\] Since the integer points of $k\Delta$ naturally give a basis of $H^0({\bf P},{\cal O}_{\bf P}(kD))$, Proposition 1.1 of \cite{cox} shows that we get the desired ring isomorphism. In particular, $f \in S_\beta$ corresponds to an element $\sum_{m \in \Delta\cap M} a_m t_0 t^m \in (S_\Delta)_1$. Setting $t_0 = 1$, we get the Laurent polynomial $g = \sum_{m \in \Delta\cap M} a_m t^m$ formed using the lattice points of $\Delta$. Since $t_1,\dots,t_d$ are coordinates on the torus ${\bf T}$, one can show that $Y$ is naturally isomorphic to the subvariety of ${\bf T}$ defined by $g = 0$. Following \cite{bat.var}, we use the Laurent polynomial $g$ to define $F = t_0 g -1$, and then we set $F_i = t_i \partial F/\partial t_i$ for $i = 0,1,\dots,d$. Note that $F_i \in (S_\Delta)_1$ for all $i$. Finally, we define $J_{g,\Delta}$ to be the ideal of $S_\Delta$ generated by the $F_i$. Then Corollary 6.10 of \cite{bat.var} gives an isomorphism \begin{equation} Gr_F^p PH^{d-1}(Y) \cong (S_\Delta/J_{g,\Delta})_{d-p}. \end{equation} To prove the theorem, it suffices to show that for all $k \ge 0$, the isomorphism $\rho$ of (\ref{eq.translate}) maps the graded piece $(J_{g,\Delta})_k \subset (S_\Delta)_k$ to $J_0(f)_{k\beta} \subset S_{k\beta}$. First observe that $F_0 = t_0 g$ maps to $f$ under the map $\rho$. To write this explicitly, we introduce the following notation: given $m \in \Delta\cap M$, let $z^{D(m)} = \prod_{i=1}^n z_i^{b_i + \langle m,e_i\rangle}$. Thus $f = \sum_{m \in \Delta\cap M} a_m z^{D(m)}$. Next note that $t_1,\dots,t_d$ are a basis of the character group of ${\bf T}$ and hence determine a basis of $M$. If $h_1,\dots,h_d \in N$ is the dual basis, then one computes that for $i > 0$, the polynomial $F_i = t_0t_i \partial g/\partial t_i$ maps to \[ \tilde f_i = \sum_{m\in \Delta\cap M} a_m\langle m,h_i\rangle z^{D(m)}.\] In contrast, note that \[ z_i \partial f/\partial z_i = \sum_{m\in\Delta\cap M} a_m(b_i + \langle m,e_i\rangle) z^{D(m)} = b_i f + \sum_{m\in\Delta\cap M} a_m \langle m,e_i\rangle z^{D(m)}.\] It suffices to show that each $z_i\partial f/\partial z_i$ is a ${\bf C}$-linear combination of $f,\tilde f_1,\dots,\tilde f_d$ and conversely. To prove this, first note that $e_i$ can be expressed in terms of the $h_1,\dots,h_d$, and hence $z_i\partial f/\partial z_i$ is a ${\bf C}$-linear combination of $f,\tilde f_1,\dots,\tilde f_d$. Going the other way, we can label $e_1,\dots,e_n$ so that $e_1,\dots,e_d$ are linearly independent. Then $h_i$ can be expressed in terms of $e_1,\dots,e_d$, and it follows easily that $f,\tilde f_1,\dots,\tilde f_d$ are ${\bf C}$-linear combinations of $f,z_1\partial f/\partial z_1,\dots,z_n\partial f/\partial z_n$. To complete the proof, observe that $f \in J_0(f)$ follows easily from the proof of Lemma \ref{lem.fJ}.\hfill $\Box$ \medskip We next study the weight filtration on $PH^{d-1}(Y)$. Since $Y$ is quasi-smooth, we have $W_{d-2}H^{d-1}(Y) = 0$, which implies that $W_{d-2}PH^{d-1}(Y) = 0$. It follows that $W_{d-1}PH^{d-1}(Y)$ has a pure Hodge structure. We can identify this Hodge structure as follows. \begin{prop} \label{prop.weight} If $X \subset {\bf P}$ is a nondegenerate ample hypersurface, then there is a natural isomorphism of Hodge structures \[ PH^{d-1}(X) \cong W_{d-1}PH^{d-1}(Y).\] \end{prop} \noindent {\bf Proof. } Let $D = {\bf P} \setminus {\bf T}$, and recall from Theorem \ref{poin} that the complex $\Omega_{\bf P}^\cdot(\log D)$ has a weight filtration $W_\cdot$ with the property that \[ Gr^W_k \Omega^\cdot_{\bf P}(\log D) \cong \bigoplus_{\dim \tau = k} \Omega_{{\bf P}_\tau}^\cdot.\] The spectral sequence of this filtered complex gives the weight filtration on $H^\cdot({\bf T})$, and since the spectral sequence degenerates at $E_2$, we get an exact sequence \[ \bigoplus_{i=1}^n H^{d-3}(D_i) \to H^{d-1}({\bf P}) \to Gr_{d-1}^W H^{d-1}({\bf T}) \to 0,\] where $D = \sum_{i=1}^n D_i$. Notice also that $Gr_{d-1}^W H^{d-1}({\bf T}) = W_{d-1}H^{d-1}({\bf T})$ since ${\bf T}$ is smooth. Since $X \subset {\bf P}$ is nondegenerate, the same argument applies to $D\cap X = X \setminus Y$, so that we also have an exact sequence \[ \bigoplus_{i=1}^n H^{d-3}(D_i\cap X) \to H^{d-1}(X) \to Gr_{d-1}^W H^{d-1}(Y) \to 0,\] and as noticed earlier, $Gr_{d-1}^W H^{d-1}(Y) = W_{d-1} H^{d-1}(Y)$. To see how this applies to primitive cohomology, consider the commutative diagram: \[ \begin{array}{ccccccc} && 0 && 0 && \\ && \uparrow && \uparrow && \\ && PH^{d-1}(X) & \stackrel{\alpha}\to & W_{d-1}PH^{d-1}(Y) && \\ && \uparrow && \uparrow && \\ \bigoplus_{i=1}^n H^{d-3}(D_i\cap X) & \to & H^{d-1}(X) & \to & W_{d-1}H^{d-1}(Y) & \to & 0\\ \uparrow && \uparrow && \uparrow && \\ \bigoplus_{i=1}^n H^{d-3}(D_i) & \to & H^{d-1}({\bf P}) & \to & W_{d-1}H^{d-1}({\bf T}) & \to & 0 \end{array} \] The columns are exact by the definition of primitive cohomology and the strictness of the weight filtration, and we've already seen that the bottom two rows are exact. It follows easily that the map $\alpha : PH^{d-1}(X) \to W_{d-1}PH^{d-1}(Y)$ exists and is surjective. Notice also that $\alpha$ is a morphism of Hodge structures. However, each $D_i$ is a $(d-1)$-dimensional complete simplicial toric variety, and $D_i\cap X \subset D_i$ is quasi-smooth since $X$ is nondegenerate. Thus Proposition \ref{prop.lef} implies that \[ \bigoplus_{i=1}^n H^{d-3}(D_i) \to \bigoplus_{i=1}^n H^{d-3}(D_i\cap X)\] is an isomorphism. An easy diagram chase then shows that $\alpha$ is injective, and the proposition is proved.\hfill$\Box$ \medskip In order to interpret this proposition in terms of the polynomial ring $S$, we will need the following ideal. \begin{opr} {\rm Given the ideal $J_0(f) = \langle z_1\partial f\partial z_1,\dots,z_n \partial f/\partial z_n\rangle \subset S$, we get the ideal quotient $J_1(f) = J_0(f)\colon z_1\cdots z_n$. We put $R_1(f) = S/J_1(f)$. } \end{opr} \begin{theo} {\rm (\cite{bat.var})} If $X \subset {\bf P}$ is a nondegenerate ample divisor defined by $f \in S_\beta$, then there is a natural isomorphism \[ PH^{p,d-1-p}(X) \cong R_1(f)_{(d-p)\beta-\beta_0},\] where $z_i \in S_{\beta_i}$ and $\beta_0 = \sum_{i=1}^n \beta_i$. \label{theo.r1} \end{theo} \noindent {\bf Proof. } As in Definition 2.8 of \cite{bat.var}, consider the ideal $I^{(1)}_\Delta \subset S_\Delta$ spanned by all monomials $t_0^k t^m$ such that $m$ is an interior point of $k\Delta$. Then let $H = \oplus H_i$ denote the image of the homogeneous ideal $I^{(1)}_\Delta$ in the graded Artinian ring $S_\Delta/J_{g,\Delta}$, where $J_{g,\Delta}$ is as in the proof of Theorem \ref{theo.aff}. Proposition 9.2 of \cite{bat.var} tells us that under the isomorphism $(S_\Delta/J_{f,\Delta})_{d-p} \cong Gr_F^p PH^{d-1}(Y)$, the subspace $H_{d-p}$ maps to $W_{d-1} Gr_F^p PH^{d-1}(Y)$. If we combine this with Proposition \ref{prop.weight}, then we get an isomorphism \[ H_{d-p} \cong PH^{p,d-1-p}(X). \] Under the isomorphism $\rho$ of (\ref{eq.translate}), a monomial $t_0^k t^m$ maps to $z^{D(m)} = \prod_{i=1}^n z_i^{b_i + \langle m,e_i\rangle}$. Since $m$ is in the interior of $k\Delta$ if and only if $\langle m,e_i\rangle > -b_i$ for all $i$, we see that $t_0^kt^m \in (I^{(1)}_\Delta)_k$ exactly when $z^{D(m)}$ is divisible by $z_1\cdots z_n$. Thus $\rho((I^{(1)}_\Delta)_k) = \langle z_1\cdots z_n\rangle_{k\beta}$. Since we know by the proof of Theorem \ref{theo.aff} that $\rho$ maps $(J_{g,\Delta})_k$ to $J_0(f)_{k\beta}$, it follows easily that $H_k \subset (S_\Delta/J_{g,\Delta})_k$ is isomorphic to the image of $\langle z_1\cdots z_n\rangle_{k\beta}$ in $(S/J_0(f))_{k\beta}$. This last subspace is isomorphic to $(S/J_1(f))_{k\beta-\beta_0} = R_1(f)_{k\beta-\beta_0}$, and the theorem is proved. \hfill$\Box$ \begin{rem} {\rm It is interesting to compare this result to Theorem \ref{theo.main}, which gives a natural surjection \begin{equation} R(f)_{(d-p)\beta-\beta_0} \to PH^{p,d-1-p}(X) \to 0 \label{eq.main} \end{equation} when $X$ is quasi-smooth. The theorem just proved shows that, under the stronger hypothesis that $X$ is nondegenerate, there is an isomorphism \[ PH^{p,d-1-p}(X) \cong R_1(f)_{(d-p)\beta-\beta_0}.\] One can show that the composition of these maps in induced by the obvious inclusion of ideals \[ J(f) = \langle \partial f/\partial z_i\rangle \subset \langle z_i \partial f/\partial z_i \rangle\colon z_1\cdots z_n = J_1(f).\] Since the map of (\ref{eq.main}) is an isomorphism for $p \ne d/2-1$, it follows that the ideals $J(f)$ and $J_1(f)$ agree in degrees $(d-p)\beta-\beta_0$ for $p \ne d/2-1$, though for $p = d/2-1$, we get an exact sequence \[ 0 \to H^{d-2}({\bf P}) \stackrel{\cup [X]}\to H^d({\bf P}) \to (J_1(f)/J(f))_{(d/2+1)\beta-\beta_0} \to 0.\] When ${\bf P}$ is a weighted projective space, the ideals $J(f)$ and $J_1(f)$ are equal in all degrees. For in this case, $f$ being quasi-smooth means that the $\partial f/\partial z_i$ form a regular sequence, while $f$ being nondegenerate means that the $z_i \partial f/\partial z_i$ form a regular sequence. Then standard results from commutative algebra (see part ($\gamma$) of (1.2) of \cite{scheja}) imply that $\langle \partial f/\partial z_i\rangle = \langle z_i\partial f/\partial z_i\rangle\colon z_1\cdots z_n$, which gives the desired equality. In the general case, the precise relation between $J(f)$ and $J_1(f)$ is not well understood.} \end{rem} \section{A generalized Euler short exact sequence and $\;\;\;\;\;$ applications} We begin with a generalization of the classical Euler short exact sequence. \begin{theo} Let $D_1, \ldots, D_n$ the irreducible components of $D = {\bf P} \setminus {\bf T}$, and let $d$ be the dimension of ${\bf P}$. Then there exists the following short exact sequence \begin{equation} 0 \rightarrow \Omega^1_{\bf P} \rightarrow \bigoplus_{i =1}^n {\cal O}_{\bf P}(-D_i) \rightarrow {\cal O}^{n-d}_{\bf P} \rightarrow 0. \label{euler.seq} \end{equation} \end{theo} \begin{rem} {\rm When ${\bf P}$ is projective space, the above short exact sequence coincides with the well-known Euler short exact sequence. So we call (\ref{euler.seq}) the {\em generalized Euler exact sequence}.} \end{rem} \noindent {\bf Proof. } There are the following two short exact sequences: \begin{equation} 0 \rightarrow \Omega^1_{\bf P} \rightarrow \Omega^1_{\bf P}(\log D) \stackrel{r}{\rightarrow} \bigoplus_{i =1}^n {\cal O}_{D_i} \rightarrow 0 \label{log.seq} \end{equation} and \begin{equation} 0 \rightarrow \bigoplus_{i =1}^n {\cal O}_{\bf P}(-D_i) \rightarrow {\cal O}_{\bf P}^{n} \stackrel{p}{\rightarrow} \bigoplus_{i =1}^n {\cal O}_{D_i} \rightarrow 0, \label{ideal.seq} \end{equation} where $r$ is the Poincar\'e residue map. The short exact sequence \[ 0 \rightarrow M \rightarrow {\bf Z}^n \rightarrow Cl(\Sigma) \rightarrow 0 \] shows that $Cl(\Sigma)$ has rank $n-d$. Since $\Omega^1_{\bf P}(\log D) \cong \Lambda^1 M \otimes_{\bf Z} {\cal O}_{\bf P}$, we can tensor this sequence with ${\cal O}_{\bf P}$ to obtain \[ 0 \rightarrow \Omega^1_{\bf P}(\log {\bf D}) \stackrel{s}{\rightarrow} {\cal O}_{\bf P}^n {\rightarrow} {\cal O}_{\bf P}^{n-d} \rightarrow 0. \] By global properties of the Poincar\'e residue map (Section 6), one has $r = p \circ s$. So $s$ induces an injective map $\imath : \Omega^1_{\bf P} \hookrightarrow \bigoplus_{i =1}^n {\cal O}_{\bf P}(-D_i)$, and then the short exact sequences (\ref{log.seq}) and (\ref{ideal.seq}) fit into the following commutative diagram: \begin{center} \begin{tabular}{ccccccccc} & $ $ & $ 0 $ & $ $ & $ 0 $ & $ $ & $ $ & $ $ & \\ & $ $ & $ \downarrow $ & $ $ & $ \downarrow $ & $ $ & $ $ & $ $ & \\ $ 0 $ & $ \rightarrow $ & $ \Omega^1_{\bf P} $ & $ \stackrel{\imath}\rightarrow $ & $ \bigoplus_{i=1}^n {\cal O}_{\bf P}(-D_i) $ & $ \rightarrow $ & $ {\cal Q} $ & $ \rightarrow $ & $ 0 $ \\ & $ $ & $ \downarrow $ & $ $ & $ \downarrow $ & $ $ & $ \downarrow $ & $ $ & \\ $ 0 $ & $ \rightarrow $ & $ \Omega^1_{\bf P}(\log {\bf D}) $ & $ \stackrel{s}{\rightarrow} $ & $ {\cal O}_{\bf P}^n $ & $ \rightarrow $ & $ {\cal O}_{\bf P}^{n-d} $ & $ \rightarrow $ & $ 0$ \\ & $ $ & $\; \; \downarrow {\scriptstyle r} $ & $ $ & $ \;\; \downarrow {\scriptstyle p} $ & $ $ & $ \downarrow $ & $ $ & \\ $ 0 $ & $ \rightarrow $ & $ \bigoplus_{i=1}^n {\cal O}_{D_i} $ & $ = $ & $ \bigoplus_{i=1}^n {\cal O}_{D_i} $ & $ \rightarrow $ & $ 0 $ & $ $ & \\ & $ $ & $ \downarrow $ & $ $ & $ \downarrow $ & $ $ & $ $ & $ $ & \\ & $ $ & $ 0 $ & $ $ & $ 0 $ & $ $ & $ $ & $ $ & \\ \end{tabular} \end{center} \noindent Since the first two columns are exact, the snake lemma implies that the sheaf ${\cal Q}$ is isomorphic to ${\cal O}_{\bf P}^{n-d}$. \hfill $\Box$ \medskip As a first application of the Euler exact sequence, we will show how to find generators of $H^0({\bf P},\Omega^{d-1}_{\bf P}(X))$, where $X \subset {\bf P}$ is an ample hypersurface defined by $f \in S_\beta$. If we apply the functor $Hom(*, \Omega^d_{\bf P}(X))$ to (\ref{euler.seq}), then we obtain the exact sequence \[ 0 \rightarrow (\Omega^d_{\bf P}(X))^{n-d} \rightarrow \bigoplus_{i =1}^n Hom ({\cal O}_{\bf P}(-D_i),\Omega^d_{\bf P}(X)) \rightarrow \Omega^{d-1}_{\bf P}(X) \rightarrow 0, \] since $Hom(\Omega^k_{\bf P}, \Omega^d_{\bf P}) = \Omega^{d-k}_{\bf P}$ (see \cite{dan1}). Then we get the short exact sequence of global sections \[ 0 \rightarrow H^0({\bf P}, (\Omega^d_{\bf P}(X))^{n-d}) \rightarrow \bigoplus_{i =1}^n H^0( {\bf P}, Hom ({\cal O}_{\bf P}(-D_i),\Omega^d_{\bf P}(X))) \rightarrow H^0({\bf P}, \Omega^{d-1}_{\bf P}(X)) \rightarrow 0 \] since $H^1({\bf P},\Omega^d_{\bf P}(X)) = 0$ by the Bott-Steenbrink-Danilov vanishing theorem. However, by Remark \ref{rem.omegad}, we know that $\Omega_{\bf P}^d \cong {\cal O}_{\bf P}(-D)$, and it follows that we have an isomorphism of ${\bf T}$-linearized sheaves \[ Hom ({\cal O}(-D_i),\Omega^d_{\bf P}(X)) \cong {\cal O}_{\bf P}(X - D +D_i).\] Then Lemma \ref{lem.iso} implies that we have an isomorphism \[ H^0( {\bf P}, Hom ({\cal O}(-D_i),\Omega^d_{\bf P}(X))) \cong S_{\beta - \beta_0 + \beta_i}, \] where as usual $\beta_i$ is the class of $D_i$ and $\beta = \sum_{i=1}^n \beta_i$ is the class of $D$. We have thus proved the following result. \begin{prop} When $X$ is an ample hypersurface of ${\bf P}$ defined by $f \in S_\beta$, then there exists a surjective homomorphism \[ \bigoplus_{i =1}^n S_{\beta - \beta_0 + \beta_i} \rightarrow H^0({\bf P}, \Omega^{d-1}_{\bf P}(X)). \] \end{prop} \begin{rem} {\rm The reader should compare this proposition with Theorem \ref{theo.dminus1} and Corollary \ref{coro.dminus1}.} \end{rem} We conclude this section with a discussion of the tangent sheaf of a toric variety. \begin{opr} {\rm Let ${\cal T}_{\bf P}$ be the sheaf $Hom(\Omega^1_{\bf P},{\cal O}_{\bf P})$. We call ${\cal T}_{\bf P}$ the {\em tangent sheaf} of ${\bf P}$. } \end{opr} \begin{rem} {\rm If ${\bf P}$ is smooth, then ${\cal T}_{\bf P}$ coincides with the usual tangent sheaf $\Theta_{\bf P}$ of ${\bf P}$.} \end{rem} Applying $Hom(*,{\cal O}_{\bf P})$ to (\ref{euler.seq}), we obtain the short exact sequence \[ 0 \rightarrow {\cal O}_{\bf P}^{n-d} \rightarrow \bigoplus_{i =1}^n {\cal O}(D_i) \rightarrow {\cal T}_{\bf P} \rightarrow 0. \] Since $H^1({\bf P}, {\cal O}_{\bf P}) = 0$, we get \[ {\rm dim}\, H^0({\bf P}, {\cal T}_{\bf P}) = \sum_{i =1}^n ({\rm dim}\, S_{\beta_i} - 1) + d. \] It was proved by second author in \cite{cox} that \[ {\rm dim\,Aut}({\bf P}) = \sum_{i =1}^n ({\rm dim}\, S_{\beta_i} - 1) + d. \] Thus the global sections of ${\cal T}_{\bf P}$ can be identified with the Lie algebra of ${\rm Aut}({\bf P})$. \section{Moduli of ample hypersurfaces} This section will study how the Jacobian ring is related to the moduli of the hypersurfaces $X \subset {\bf P}$ coming from sections of an ample invertible sheaf ${\cal L}$ on ${\bf P}$. As usual, we assume that ${\bf P}$ is a complete simplicial toric variety. We first study the automorphisms of ${\bf P}$ which preserve a given divisor class. Let ${\rm Aut}({\bf P})$ denote the automorphism group of ${\bf P}$. \begin{opr} {\rm Given $\beta \in Cl(\Sigma)$, let ${\rm Aut}_\beta({\bf P})$ denote the subgroup of ${\rm Aut}({\bf P})$ consisting of those automorphisms which preserve $\beta$. } \end{opr} \begin{rem} {\rm If ${\rm Aut}{}^0({\bf P})$ is the connected component of the identity of ${\rm Aut}({\bf P})$, then the results of \S3 of \cite{cox} imply that ${\rm Aut}{}^0({\bf P})$ is a subgroup of finite index in ${\rm Aut}_\beta({\bf P})$.} \end{rem} When we describe ${\bf P}$ as the quotient $U(\Sigma)/{\bf D}(\Sigma)$, note that ${\rm Aut}({\bf P})$ doesn't act on $U(\Sigma)$. However, in \cite{cox}, it is shown that there is an exact sequence \[ 1 \to {\bf D}(\Sigma) \to \widetilde{\rm Aut}({\bf P}) \to {\rm Aut}({\bf P}) \to 1\] where $\widetilde{\rm Aut}({\bf P})$ is the group of automorphisms of ${\bf A}^n$ which preserve $U(\Sigma)$ and normalize ${\bf D}(\Sigma)$. An element $\phi \in \widetilde{\rm Aut}({\bf P})$ induces an automorphism $\phi : S \to S$ which for all $\gamma \in Cl(\Sigma)$ satisfies $\phi(S_\gamma) = S_{\phi(\gamma)}$. \begin{opr} {\rm Given $\beta \in Cl(\Sigma)$, let $\widetilde{\rm Aut}{}_\beta({\bf P})$ denote the subgroup of $\widetilde{\rm Aut}({\bf P})$ consisting of those automorphisms which preserve $\beta$.} \end{opr} The group $\widetilde{\rm Aut}_\beta({\bf P})$ has the following obvious properties. \begin{lem} There is a canonical exact sequence \[ 1 \to {\bf D}(\Sigma) \to \widetilde{\rm Aut}_\beta({\bf P}) \to {\rm Aut}_\beta({\bf P}) \to 1.\] Furthermore, there is a natural action of $\widetilde{\rm Aut}_\beta({\bf P})$ on $S_\beta$. \end{lem} \begin{rem} {\rm Let $\widetilde{\rm Aut}{}^0({\bf P})$ be the connected component of the identity of $\widetilde{\rm Aut}({\bf P})$. In \cite{cox}, it is shown that $\widetilde{\rm Aut}{}^0({\bf P})$ is naturally isomorphic to the group ${\rm Aut}_g(S)$ of $Cl(\Sigma)$-graded automorphisms of $S$. Then $\widetilde{\rm Aut}{}^0({\bf P}) \subset \widetilde{\rm Aut}_\beta({\bf P})$, and the action of $\widetilde{\rm Aut}_\beta({\bf P})$ on $S_\beta$ is compatible with the action of ${\rm Aut}_g(S)$. } \end{rem} If $\beta \in Cl(\Sigma)$ is an ample class, then we know that $X = \{p \in {\bf P}: f(p) = 0\} \subset {\bf P}$ is quasi-smooth for generic $f \in S_\beta$ (see Proposition \ref{prop.generic}). Then \[ \{f \in S_\beta : f\ \hbox{is quasi-smooth}\}/\widetilde{\rm Aut}_\beta({\bf P})\] should be the coarse moduli space of quasi-smooth hypersurfaces in ${\bf P}$ in the divisor class of $\beta$. The problem is that $\widetilde{\rm Aut}_\beta({\bf P})$ need not be a reductive group, so that the quotient may not exist. However, it is well-known (see, for example, \S2 of \cite{cox.tu.don}) that there is a nonempty invariant open set \[ U \subset \{f \in S_\beta : f\ \hbox{is quasi-smooth}\}\] such that the geometric quotient \[ U/\widetilde{\rm Aut}_\beta({\bf P})\] exists. \begin{opr} {\rm We call the quotient $U/\widetilde{\rm Aut}_\beta({\bf P})$ a {\em generic coarse moduli space for hypersurfaces of ${\bf P}$ with divisor class $\beta$}.} \end{opr} We can relate the Jacobian ring $R(f)$ to the generic coarse moduli space as follows. \begin{prop} If $\beta$ is ample and $f \in S_\beta$ is generic, then $R(f)_\beta$ is naturally isomorphic to the tangent space of the generic coarse moduli space of quasi-smooth hypersurfaces of ${\bf P}$ with divisor class $\beta$. \end{prop} \noindent {\bf Proof. } First note that $\widetilde{\rm Aut}{}^0({\bf P}) \subset \widetilde{\rm Aut}_\beta({\bf P})$ has finite index. Thus, by shrinking $U$ if necessary, we may assume that \[ U/\widetilde{\rm Aut}{}^0({\bf P}) \to U/\widetilde{\rm Aut}_\beta({\bf P})\] is \'etale. Hence it suffices to identify $R(f)_\beta$ with the tangent space to $U/\widetilde{\rm Aut}{}^0({\bf P})$. Shrinking $U$ further, we may assume that the map \[ U \to U/\widetilde{\rm Aut}{}^0({\bf P})\] is smooth (see \S2 of \cite{cox.tu.don}). Then the tangent space to a point of the generic moduli space is naturally isomorphic to the quotient of $S_\beta$ (= tangent space of $U$) modulo the tangent space to the orbit of ${\rm Aut}_g(S) = \widetilde{\rm Aut}{}^0({\bf P})$ acting on $f \in S_\beta$. Hence, to prove the proposition, we need tos show that $J(f)_\beta$ is the tangent space to the orbit of $f$. But the tangent space to the orbit is given by the action of the Lie algebra of ${\rm Aut}_g(S)$ on $f$. Since the Lie algebra consists of all derivations of $S$ which preserve the grading, its elements can be written in the form $\sum_{i=1}^n A_i \partial/\partial z_i$, where $A_i$ and $z_i$ lie in the same graded piece $S_{\beta_i}$ of $S$ for all $i$. Thus the action of the Lie algebra on $f$ gives the subspace $\{ \sum_{i=1}^n A_i \partial f/\partial z_i : A_i \in S_{\beta_i}\} = J(f)_\beta$, and the proposition is proved.\hfill$\Box$

9306

alg-geom/9306002

en

https://arxiv.org/abs/alg-geom/9306002

alg-geom/9306002

Alexis Kouvidakis

Picard groups of Hilbert schemes of curves

13 pages, LaTeX

We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the classes of the generators in terms of some ``natural'' divisor classes.

alg-geom

\section{Introduction} Given positive integers $d,g,r$, let ${\frak H}_{d,g,r} $ denote the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$ and genus $g$ in ${\Bbb P}^r$. In general, the geometric structure of ${\frak H}_{d,g,r} $ is difficult to describe and in most of the cases is unknown but for $d \geq 2g+1$ and $r \leq d-g$, it turns out that ${\frak H}_{d,g,r} $ is smooth and irreducible, see \cite{Ha}, \cite{CS}. The natural forgetful map ${\frak H}_{d,g,r} \longrightarrow {\frak M}_g $, where ${\frak M}_g $ is the moduli space of smooth curves of genus $g \geq 4$, is onto. The purpose of this paper is to describe the Picard group of ${\frak H}_{d,g,r} $ over the integers when $d \geq 2g+1$ and $r \leq d-g$. From now on, we are going to exclude from the Hilbert scheme those points which represent curves with automorphisms. By doing so, the description of the Picard group is not effected, since the locus of such points is of big codimension when $g \geq 4$. Let ${\frak M}^0_g $ denote the moduli space of smooth, irreducible curves of genus $g \geq 4$ without automorphisms. Over ${\frak M}^0_g $ we have the universal family $\pi : {\frak C}_g \longrightarrow {\frak M}^0_g$. To that we can associate the family $\psi : {\frak J} ^d \longrightarrow {\frak M}^0_g $, the universal Picard variety of degree $d$, whose fiber over $[C] \in {\frak M}^0_g $ is $J^d(C)$. Over ${\frak J} ^d$ one can construct a projective fibration $ \phi : {\frak P}_d \longrightarrow {\frak J} ^d $ whose fiber over a point $[L] \in J^d (C)$ is ${\Bbb P} ({\Bbb C}^{r+1} \otimes H^0(C,L))$. Note that for $r=0$, this is just the universal symmetric product ${\frak C}_g^{(d)} \longrightarrow {\frak J} ^d $ of degree $d$. The existence of such a bundle is based on the existence of the bundle ${\frak C}_g^{(d)} \longrightarrow {\frak J} ^d $ and the existence of a local section (in the analytic topology) of the map $\pi : \, {\frak C}_g \longrightarrow {\frak M}^0_g $. The variety ${\frak H}_{d,g,r} $ can be included in ${\frak P}_d$ as follows. An element $h$ in ${\frak H}_{d,g,r} $ corresponds to a smooth, irreducible, non-degenerate curve $C$ of degree $d$ and genus $g$ in ${\Bbb P} ^r$. Let $H_i, \; i=1, \ldots , r+1$, denote the hyperplane section $X_i=0$. Then, define $L\stackrel{\rm def}{=}{\cal O}(1)|_C \in J^d(C)$ and $s_i \stackrel{\rm def}{=} H_i|_C \in H^0(C,L)$. We then correspond to $h$ the point $[C,L, <s_1, \ldots ,s_{r+1}>] \in {\frak P}_d$ and the map is obviously one to one. By doing so, one can factor the canonical map ${\frak H}_{d,g,r} \longrightarrow {\frak M}^0_g $ as $$ {\frak H}_{d,g,r} \stackrel{i}{ \hookrightarrow} {\frak P}_d \stackrel{\phi }{\longrightarrow } {\frak J} ^d \stackrel{\psi }{\longrightarrow } {\frak M}^0_g . $$ The complement of ${\frak H}_{d,g,r} $ in ${\frak P}_d$ corresponds to those tuples $[C,L, <s_1, \ldots ,s_{r+1}>] $ for which the space of sections $<s_1, \ldots ,s_{r+1}>$ has either a base point or does not separate points and tangent directions on the curve or the dimension of the span$<s_1, \ldots ,s_{r+1}> $ is $\leq r$. In the first case the map of $C$ to ${\Bbb P} ^r$ defined by the above data is of degree $<d$, in the second the map is not an embedding and in the third the map is degenerate. Over ${\frak H}_{d,g,r} $ we have the universal curve $q: \, {\frak C}_{\frak H} \longrightarrow {\frak H}_{d,g,r} $. By construction ${\frak C}_{\frak H} \subset {\frak H}_{d,g,r} \times {\Bbb P} ^r$. On ${\frak C}_{\frak H} $, we have the tautological bundle ${\cal F}$ which is the pull back by the projection map $\eta : {\frak C}_{\frak H} \longrightarrow {\Bbb P}^r$ of ${\cal O}(1)$ on ${\Bbb P} ^r$ . If $\pi : \, {\frak C}_g \longrightarrow {\frak M}^0_g $ is the universal curve over ${\frak M}^0_g $ and $\psi $ the map as above, then we denote by ${\frak C}_{g,d}$ the fiber product ${\frak C}_{g,d} \stackrel{\rm def}{=}{\frak C}_g \times _{{\frak M}^0_g } {\frak J} ^d $. Let $\nu $ denote the projection map ${\frak C}_{g,d}\longrightarrow {\frak J} ^d $. The basic diagram we are going to use is the following: \begin{equation} \begin{diagram}[{\frak C}_{g,d}aaa] \node{} \node{{\frak C}_{\frak H}} \arrow{e,t}{{ \alpha }} \arrow{s,r}{ {q} } \arrow{sw,l}{{\eta }} \node{{\frak C}_{g,d}} \arrow{e,t}{{\beta }} \arrow{s,r}{{\nu }} \node{{\frak C}_g } \arrow{s,r}{{\pi } }\\ \node{{\Bbb P} ^r}\node{{\frak H}_{d,g,r} } \arrow{e,t}{{\phi } } \node{{\frak J} ^d} \arrow{e,t}{{\psi }} \node{{\frak M}^0_g } \end{diagram} \label{diag1} \end{equation} \vskip.2in Let $\omega _q $ denote the relative dualizing sheaf of the map $q: \, {\frak C}_{\frak H} \longrightarrow {\frak H}_{d,g,r} $. Then we have on ${\frak H}_{d,g,r} $ the following three natural divisor classes: \begin{tabbing} nn \= nnnnn \kill \> $A = q_{*}({\cal F}^2)$, \\ \> $B = q_{*} ({\cal F} \omega _q)$,\\ \> $C =q_{*} (\omega _q ^2)$. \end{tabbing} In this paper we describe the Picard group of ${\frak H}_{d,g,r} $ and we express the classes of its generators in terms of the classes $A, B, C$ given above. This is the content of Theorem \ref{main}. \section{Intersection calculations.} \noindent We are going to use the following results \begin{enumerate} \item The Picard group of ${\frak M}^0_g $ is freely generated over the integers by the determinant $\lambda $ of the Hodge bundle. In other words, $\lambda =det \pi _{*} \omega_{\pi }$, where $\pi :{\frak C}_g \longrightarrow {\frak M}^0_g $ is the universal curve, see \cite{AC2}. \item The Picard group of the universal Jacobian $\psi :{\frak J} ^d\longrightarrow {\frak M}^0_g $ is freely generated over the integers by the pull back $\psi ^*\lambda$ and a line bundle ${\cal L }_d$. The later is uniquely defined up to the pull back of line bundles from ${\frak M}^0_g $ and has the following characteristic property: its restriction to a fiber $J^d(C)$ has class $ k_d \theta $, where $k_d=\frac{2g-2}{g.c.d.(2g-2,d-g+1)}$ and $\theta $ denotes the class of the theta divisor, see \cite{K1}. \item On ${\frak J} ^d \times _{{\frak M}^0_g } {\frak C}_g $ there is a line bundle ${\cal P }_d $ with the property ${\cal P }_d|_{[L] \times C} \cong L^{\otimes s_d}$, where $s_d=gcd(2g-2,d+g-1)$ is minimum with this property, see Application in Section 5 of \cite{K2}. \end{enumerate} There are various ways to construct the bundles ${\cal L}_d$ and ${\cal P}_d$ in $2.$ and $3.$ above. In the following we give a construction which is the most convenient for our purposes. We start with some notation. On the $d$-th symmetric product of a curve we denote by $\Delta $ the diagonal or its class in the Chow ring. We denote by $x$ the class of the image of a coordinate plane from the $d$-th ordinary product under the canonical map. Let $u: C^{(d)} \longrightarrow J^d(C)$ be the Abel-Jacobi map and let $\omega _u$ the relative dualizing sheaf. Given a line bundle $M$ on $C$, then we associate to that a line bundle $L_M$ on the symmetric product $C^{(d)}$ as follows: Take a meromorphic section of M written in the form $D_1-D_2$, where $D_1,\, D_2$ are effective divisors which are sums of distinct points. Define on $C^{(d)}$ the divisors $X_{D_i}, \; i=1,\, 2$ with support $\{D \in C_d, \; \mbox{s.t.}\; D \cap D_i \neq \emptyset \}$. Then we define $L_M$ to be the line bundle ${\cal O}(X_{D_1}) \otimes {\cal O}(X_{D_2})^{-1}$. It is easy to see that this is independent from the choice of $D_1$ and $D_2$. For $M=K$, we denote by $L_K$ the line bundle which corresponds to the canonical divisor $K$. \begin{rem} {\rm For $ d \geq 2g+1 $, the Abel-Jacobi map $u: C^{(d)} \longrightarrow J^d(C)$ is a fibration of projective spaces of dimension $r=d-g$. In that case, if $A \in J^d(C)$, then $L_M|_{u^{-1}(A) \simeq {\Bbb P}^{\, r}} \simeq {\cal O}_{{\Bbb P}^{\, r}} ({\rm deg}M)$.} \label{rem21} \end{rem} We describe now the analogue of $L_K$ for families of curves. Consider the diagram \begin{equation} \begin{diagram}[aa] \node{{\frak C}_g ^{\times d}} \arrow{s,l}{\pi _i} \arrow[2]{e,t}{c} \arrow{se,t}{\chi} \node[2]{{\frak C}_g ^{(d)}} \arrow{sw,t}{\chi ^{'}} \\ \node{{\frak C}_g} \arrow{e,t}{\pi } \node{{\frak M}^0_g } \label{diag2} \end{diagram} \end{equation} \noindent where the maps $ c, \chi , \chi ^{'} , \pi $ are the canonical ones and $\pi _i $ is the $i$-th projection. For $l$ big enough, let $\sigma (l)$ be a generic section of $ H^0({\frak C}_g , \omega _{\pi } \otimes \pi ^{*}\lambda ^{\otimes l}) \cong H^0({\frak M}^0_g , \pi _{*} \omega _{\pi } \otimes \lambda ^{\otimes l}) $. Since $\pi _* \omega _{\pi }$ is a locally free sheaf of rank $g$, we may assume that the section $\sigma (l)$ does not vanish identically on the fibers of $\pi $ over a Zariski open ${\cal U}$ of ${\frak M}^0_g $ with complement of codimension $g$. Then, as above, we can construct over ${\cal U}$ a divisor whose restriction to the fiber $C^{(d)}$ is the divisor $X_{\sigma (l)|_{C}}$ (following the above notation). The corresponding line bundle extends uniquely to a line bundle on ${\frak C}_g ^{(d)}$. We denote that by ${\cal L}_{\sigma (l)}$. On ${\frak C}_g ^{\times d}$ we define ${\cal L}_K = \otimes _{i=1}^d \pi ^{*}_i \omega _{\pi }$. Then we have: \begin{lem} Following the above notation, we have \[ c^{*}{\cal L}_{\sigma (l)} \cong {\cal L}_K \otimes \chi ^{*}(\lambda ^{\otimes dl}). \] \label{pull back} \end{lem} {\sc {Proof:}} By construction we have that $c^{*}{\cal L}_{\sigma (l)} \cong \otimes _{i=1}^d \pi _i ^{*} {\cal O}(\sigma(l)) \cong \otimes _{i=1}^d \pi _i ^{*} ( \omega _{\pi } \otimes \pi ^{*}\lambda ^{\otimes l}) \cong {\cal L}{_K} \otimes \chi ^{*}(\lambda ^{\otimes dl}) $. \begin{flushright} $\Box$ \end{flushright} \begin{defi} {\rm We define on ${\frak C}_g^{(d)} $ the line bundle ${\cal L}_{\omega } \stackrel{\rm def}{=} {\cal L}_{\sigma (l)} \otimes \chi ^{'*} (\lambda ^{-dl})$. } \end{defi} \begin{rem} {\rm The characteristic property of $ {\cal L}_{\omega } $ is that $c^{*} {\cal L}_{\omega } \cong {\cal L}_K$. This is an application of the above Lemma \ref{pull back}. Note also that the line bundle ${\cal L}_{\omega } $ does not depend on the choice of the number $l$ and that the restriction of ${\cal L}_{\omega }$ to a fiber $C^{(d)}$ is isomorphic to $L_K$.} \label{rem22} \end{rem} Before we continue we need two more things: The first is that the universal symmetric product carries a universal bundle ${\cal D}$. Indeed, if $\delta : {\frak C}^{(d-1)} \times {\frak C}_g \longrightarrow {\frak C}_g^{(d)} \times {\frak C}_g $ is the map sending the pair $(D,p)$ to $(D+p,p)$, then the line bundle ${\cal D}$ corresponds to the divisor which is the image of the above map. The second is the MacDonnald's formula which expresses the class of the pull back by the Abel-Jacobi map of the theta divisor on the Jacobian in terms of the classes $x$ and $\Delta $. That is, $u^{*} \theta = (d+g-1)x - \frac{\Delta }{2}$. \\ \\ \noindent {\bf {Construction of (normalized) ${\cal L}_d$:}} Consider the Abel-Jacobi map $u :\, {\frak C}_g ^{(d)} \longrightarrow {\frak J} ^d $. The bundle $\frac{d+g-1}{s_d} {\cal L}_{\omega } -k_d \frac{\Delta }{2}$ is trivial on the fibers of $u$: Indeed, by Remark \ref{rem21}, we have ${\cal L}_{\omega }|_{u^{-1}(L)} \cong {\cal O}(2g-2)$ and, by MacDonnald's formula, we have $\frac{\Delta}{2}|_{u^{-1}(L)} \cong {\cal O}(d+g-1)$. We thus get by the see saw principle, see \cite{Mu1}, that the above bundle descents to a bundle on ${\frak J} ^d$ and this is exactly the bundle ${\cal L}_d$. That ${\cal L}_d$ has class $k_d \theta$, it is again an application of the MacDonnald's formula.\\ \\ \noindent {\bf {Construction of ${\cal P}_d$: }} Consider the diagram \begin{equation} \begin{diagram}[aaaaaaaaa] \node{{\frak C}_g ^{(d)} \times_{{\frak M}^0_g} {\frak C}_g } \arrow{e,t}{\tilde {u}} \arrow{s,r}{ p_1 } \node{{\frak J} ^d \times_{{\frak M}^0_g } {\frak C}_g } \arrow{s,r} {{\nu }} \\ \node{{\frak C}_g ^{(d)} } \arrow{e,t}{u } \node{{\frak J} ^d} \end{diagram} \label{diag3} \end{equation} On ${\frak C}_g ^{(d)} \times_{{\frak M}^0_g } {\frak C}_g $ consider the line bundle $p_1^{*}(m{\cal L}_{\omega } -n \frac{\Delta }{2} )+ s_d {\cal D}$, where $n,\, m$ are integers satisfying $n(d+g-1)-m(2g-2)=s_d$. One can see again that this is trivial on the fibers of $\tilde {u}$ (note that ${\cal D}|_{u^{-1}(L) \times \{ p \} }\cong {\cal O}(1)$) and that its restriction on $\{ D \} \times C$ is isomorphic to $D^{\otimes s_d}$. The bundle ${\cal P}_d $ is the descent of that one on ${\frak J} ^d \times_{{\frak M}^0_g } {\frak C}_g $. Note that different choices of $n$, give rise to line bundles ${\cal P}_d$ which differ by the pull back of a line bundle from ${\frak J} ^d$. In the following we can fix one such $n$, for example choose $n$ to be the smallest positive integer such that the number $\frac{n(d+g-1) - s_d}{2(g-1)}$ is an integer, or equivalently, such that the number $\frac{nd-s_d}{g-1} + n$ is an even integer. We continue with some lemmas about intersections. In the following we denote a line bundle and its first Chern class in the Chow group by the same symbol. \begin{lem} If $p_1 : \, {\frak C}_g ^{(d)} \times_{{\frak M}^0_g} {\frak C}_g \longrightarrow {\frak C}_g ^{(d)}$ denotes the first projection, then $$ p_{1*}{\cal D}^2 = -{\cal L}_{\omega } + \Delta . $$ \label{D} \end{lem} \vskip-.2in {\sc {Proof:}} Consider the diagram \begin{equation} \begin{diagram}[aaaaaaaa] \node{{\frak C}_g ^{\times d}} \arrow{e,t}{\delta _{i,d+1}} \arrow{se,t}{id} \node{{\frak C}_g ^{\times d} \times_{{\frak M}^0_g } {\frak C}_g} \arrow{e,t}{\tilde {c}} \arrow{s,r}{ q_1 } \node{{\frak C}_g ^{(d)} \times_{{\frak M}^0_g } {\frak C}_g } \arrow{s,r}{p_1} \\ \node{{\frak C}_g } \node{{\frak C}_g ^{\times d}} \arrow{w,t}{\pi _i} \arrow{e,t}{c } \node{{\frak C}_g ^{(d)}} \end{diagram} \label{diag4} \end{equation} \noindent The map $\delta_{i,d+1}$ is the $i,d+1$-diagonal embedding, and $c$ the canonical map. The maps $q_1$ and $\pi _i$ are the projections. The map $c$ is flat and its pull back defines an injection in the intersection rings. By the commutativity and Remark \ref{rem22}, it is enough to show that $q_{1*} \tilde{c}^{*}({\cal D} ^2)= -{\cal L}_K +2\Delta$, where we denote also by $\Delta $ the sum of the big diagonals in ${\frak C}_g ^{\times d}$. We have \[ \tilde{c}^{*}({\cal D} ^2)=\tilde{c}^{*}({\cal D})^2 = ( \sum _{i=1}^{d} \Delta _{i,d+1})^2 = \sum_{1 \leq i,j \leq d} \Delta _{i,d+1} \, \Delta _{j, d+1}. \] There are two cases to consider \begin{eqnarray*} \mbox{if} \;\; & i\neq j & \;\; \mbox{then} \;\;\; q_{1*}( \Delta _{i,d+1} \Delta_{j, d+1})=\Delta _{i,j}\\ \mbox{if} \;\; & i=j & \;\; \mbox{then} \;\;\; q_{1*} \Delta _{i,d+1}^2 = \delta _{i,d+1}^{*} \Delta _{i,d+1} = \pi _i^{*} \omega _{\pi _i}^{-1}. \end{eqnarray*} Therefore \begin{eqnarray*} q_{1*} \tilde{c}^{*}({\cal D} ^2) & = & 2 \sum _{1\leq i<j \leq d}\Delta _{i,j} + \sum_{i=1}^d \omega _{\pi _i}^{-1} \\ & = & 2\Delta - {\cal L}_K. \end{eqnarray*} \begin{flushright} $\Box$ \end{flushright} \begin{lem} Following the notation of diagram \ref{diag3}, we have the following (where again we keep the same notation for a line bundle and its first Chern class): \begin{enumerate} \item $\nu _{*}( {\cal P }_d ^2) = s_d^2 \, \frac{nd-s_d}{g-1}\,{\cal L }_d $, \item $\nu _{*} ({\cal P }_d \omega_{\nu }) = s_d n \, {\cal L }_d$, \item $\nu _{*}(\omega _{\nu }^2) = 12 \, \psi ^*\lambda$, \end{enumerate} where $n$ is the integer defined in the construction of ${\cal P}_d$, see Section 2. \label{push forward} \end{lem} {\sc {Proof:}} Again, since $u$ is flat and its pull back defines an injection in the chow rings, it is enough to prove that $p_{1*} \tilde {u} ^{*} ({\cal P}_d^2) = s_d^2 \, \frac{nd-s_d}{g-1}\, u^*{\cal L}_d$. By the construction of ${\cal P}_d$, we have \[ \tilde {u} ^{*} ({\cal P}_d^2) = \left[ s_d\, {\cal D}+ p_1^{*}(m {\cal L}_{\omega } - n \frac{\Delta }{2}) \right] ^2= s_d^2 \, {\cal D}^2+ 2 s_d \, {\cal D} \, p_1^{*} (m {\cal L}_{\omega } - n \frac{\Delta }{2}) + p_1^{*}(m {\cal L}_{\omega } - n \frac{\Delta }{2})^2. \] Thus, \begin{eqnarray*} p_{1*} \tilde{u}^{*}{\cal P}_d^2 & = & s_d^2 \, p_{1*}{\cal D}^2 + 2 s_d\, p_{1*} {\cal D} (m {\cal L}_{\omega } - n \frac{\Delta }{2}) = s_d^2 \, (-{\cal L}_{\omega }+\Delta ) + 2 d s_d \, (m {\cal L}_{\omega } - n \frac{\Delta }{2}) \\ & = & (2 d m s_d -s_d ^2) {\cal L}_{\omega } + (2 s_d^2 - 2 d n s_d ) \frac{\Delta }{2}. \end{eqnarray*} The first coefficient can be written as \begin{eqnarray*} 2 d m s_d -s_d ^2 & = & \frac{d s_d }{g-1} (n(d+g-1)-s_d)-s_d^2 = \frac{d s_d }{g-1} \, n (d+g-1)- \frac{d s_d ^2 }{g-1}- s_d ^2 \\ & = & \frac{d s_d }{g-1} \, n (d+g-1)- \frac {s_d ^2}{g-1}(d+g-1) = \frac{s_d }{g-1}(d+g-1)(dn-s_d). \end{eqnarray*} The second coefficient can be written as \[ 2 s_d^2 - 2 d n s_d = -2s_d (dn -s_d). \] We thus have \begin{eqnarray*} p_{1*} \tilde {u}^* {\cal P}_d^2 & = & \frac{s_d}{g-1} (d+g-1)(dn-s_d) \, {\cal L}_{\omega }-2s_d(dn-s_d)\frac{\Delta }{2}\\ & = & \frac{s_d^2}{g-1}(dn-s_d) \frac{d+g-1}{s_d} \, {\cal L}_{\omega } - s_d ^2 \frac{dn-s_d}{g-1} k_d \, \frac{\Delta }{2} \\ & = & s_d ^2 \frac{dn-s_d }{g-1} \, (\frac{d+g-1}{s_d } \, {\cal L}_{\omega } -k_d \frac{\Delta }{2})= s_d ^2 \frac{dn-s_d }{g-1} \, u^{*} {\cal L}_d. \end{eqnarray*} This proves the first part of the lemma. For the second part, consider the diagram \vskip-.3in \begin{equation} \begin{diagram}[aaa] \node{ } \node[2] { } \node[2]{{\frak C}_g } \\ \node { } \node{{\frak C}_g ^{(d-1)} \times_{{\frak M}^0_g } {\frak C}_g } \arrow[2]{e,t}{\delta } \arrow{sw,t}{\pi _2} \arrow{se,t} {\sigma } \node[2]{{\frak C}_g ^{(d)} \times_{{\frak M}^0_g } {\frak C}_g } \arrow{ne,t}{p_2} \arrow {sw,t}{p_1} \arrow[2]{e,t}{\tilde {u}} \node[2]{{\frak J} ^d \times _{{\frak M}^0_g } {\frak C}_g} \arrow{sw,t}{\nu } \\ \node{ {\frak C}_g } \node[2]{{\frak C}_g ^{(d)} }\arrow[2]{e,t}{u} \node[2]{{\frak J} ^d} \end{diagram} \label{diag5} \end{equation} \noindent The map $\sigma $ in the diagram is the addition map. We have \[ \tilde {u}^{*} ({\cal P}_d \omega _{\nu }) = (p_1^{*}(m{\cal L}_{\omega } - n \frac{\Delta }{2}) + s_d {\cal D}) \omega _{p_1 } = p_1^{*}(m{\cal L}_{\omega } - n \frac{\Delta }{2}) p_2^{*} \omega _{\pi } + s_d \, {\cal D} p_2^{*} \omega _{\pi }. \] By applying $p_{1*}$ and using the definition of ${\cal D}$, we get \[ p_{1*} \tilde{u}^{*} ({\cal P}_d \omega _{\nu })= (2g-2) (m{\cal L}_{\omega } - n \frac{\Delta }{2}) + s_d \, \sigma _{*} \delta ^{*} p_2^{*} \omega _{\pi }. \] Since $p_2 \circ \delta = \pi _2$, we have that $ \sigma _{*} \delta ^{*} p_2^{*} \omega _{\pi }= \sigma _* \pi _{2}^* \omega _{\pi }$. By the definition of ${\cal L}_{\omega}$ and by some diagram chasing it is easy to see that $ \sigma _{*}\pi _2 ^{*} \omega _{\pi }= {\cal L}_{\omega }$. We thus get \begin{eqnarray*} p_{1*} \tilde{u} ^{*} ({\cal P}_d \omega _{\nu }) & = & ((2g-2)m +s_d ){\cal L}_{\omega } - n (2g-2)\frac{\Delta }{2} \\ & = & n(d+g-1) {\cal L}_{\omega } - n (2g-2)\frac{\Delta }{2} = n s_d \, u^{*}{\cal L}_d. \end{eqnarray*} This proves the second part. The third part is an immediate consequence of the fact $\pi _{*} (\omega _{\pi }^2) = 12 \lambda $, see \cite{Mu2}. \begin{flushright} $\Box$ \end{flushright} \section{The Hilbert scheme} Let $C$ be a smooth curve of genus $g \geq 4$ and $d>2g-2 $, $r \leq d-g $ given integers. We give now estimates for the dimension of the complement of ${\frak H}_{d,g,r} $ in ${\frak P}_d$. Let $L$ denote a line bundle of degree $d$ on $C$. The complement of ${\frak H}_{d,g,r} $ in the fiber ${\frak P}_d^L = {\Bbb P} ({\Bbb C}^{r+1} \otimes H^0(C,L))$ of ${\frak P}_d \longrightarrow {\frak J} ^d$ over $L$, consists of two (maybe overlapping) loci ${\frak U}^L_{\rm deg}$ and ${\frak U}^L_{\rm nemb}$. The first contains those points which define degenerate maps and the second those for which the corresponding map is not an embedding of degree $d$. We have the following lemmas: \begin{lem} Following the above notation, we have that ${\rm codim}_{{\frak P}_d^L} {\frak U}^L_{\rm deg } = d-g-r+1 $. In particular, if $3 \leq r < d-g $, then ${\rm codim}_{{\frak P}_d^L} {\frak U}^L_{\rm deg } \geq 2$ and if $r= d-g $, then ${\rm codim}_{{\frak P}_d^L} {\frak U}^L_{\rm deg } =1$. In the later case, ${\frak U}^L_{\rm deg }$ is an irreducible divisor of degree $d-g+1$ in the projective space ${\frak P}_d^L$. \label{complement1} \end{lem} {\sc {Proof:}} The above locus ${\frak U}^L_{\rm deg} $ corresponds to those tuples $<s_1, \ldots , s_{r+1}> \in {\Bbb C}^{r+1} \otimes H^0(C,L)$ which span a space of dimension $\leq r$. Since dim$H^0(C,L)=d-g+1$, this proves the Lemma. \begin{flushright} $\Box$ \end{flushright} \begin{rem} {\rm For $r=d-g$, the assumption that $d \geq 2g+1$, implies that $ {\frak U}^L_{\rm nemb} \subset {\frak U}^L_{\rm deg} $ with codimension $\geq 1$. } \label{cased-g} \end{rem} \begin{lem} For $ 4 \leq r < d-g $, we have that $ {\rm codim}_{{\frak P}_d^L} {\frak U}^L_{\rm nemb} \geq 2 $. For $3=r < d-g$, the locus $ {\frak U}^L_{\rm nemb} $ is an irreducible divisor of degree $2(d-1)(d-2)-4g$ in the projective space ${\frak P}_d^L $. \label{grass} \end{lem} {\sc {Proof:}} The space ${\frak P}_d^L \setminus {\frak U}^L_{\rm deg}$ maps in a natural way to the Grassmanian ${\bf Gr}(r+1,H^0(C,L))$ which parametrizes linear series $g^r_d$ of $L$ on $C$. Thus, for $r < d-g$, there is a rational map $\alpha : {\frak P}_d^L \longrightarrow {\bf Gr}(r+1,H^0(C,L))$ which is not defined in a codimension $\geq 2$ locus, see Lemma \ref{complement1}. The fiber of $\alpha $ is isomorphic to ${\bf PGL}(r+1)$. The locus ${\frak U}^L_{\rm nemb} \setminus {\frak U}^L_{\rm deg} $ is the pull back of the correspondent locus in ${\bf Gr}(r+1,H^0(C,L))$. It is enough to prove the Lemma on the ``level'' of ${\bf Gr}(r+1,H^0(C,L))$. Consider the curve $C$ embedded in ${\Bbb P} (H^0(C,L)^{\vee})$ by the complete linear system of $L$. Maps defined by the $g^r_d$'s, correspond to projections from $(d-g-r-1)$-dimensional projective planes in the above space. Maps which are not embeddings of degree $d$, correspond to projections from those planes which intersect the secant variety of $C$. The later is a locus in the dual of the Grassmanian ${\bf Gr}(r+1,H^0(C,L))$ of codimension equal to $ r-2 $ (the dimension of the secant variety is 3). Thus, if $r\geq 4$, then the codimension of $ {\frak U}^L_{\rm nemb} $ is $\geq 2$. We turn now to the case $r=3$. Observe first that the pull back by $\alpha $ of the generator ${\cal O}_{\bf Gr}(1)$ of the Picard group of the Grassmanian to ${\frak P}_d^L$ is isomorphic to ${\cal O}_{{\frak P}_d^L}(r+1)$. By the above discussion, one can see that ${\frak U}^L_{\rm nemb}$ is an irreducible divisor; the formula for its degree is a consequence of the previous observation and of the fact that the degree of the secant variety is $\frac{(d-1)(d-2)}{2}-g$. \begin{flushright} $\Box$ \end{flushright} \bigskip We have the following lemmas. The proof of the first is an immediate consequence of Lemmas \ref{complement1} and \ref{grass} above. \begin{lem} For $d \geq 2g+1$ and $3<r<d-g$, we have an isomorphism of Picard groups $\mbox{Pic} \, {\frak H}_{d,g,r} \cong \mbox{Pic} \, {\frak P}_d$. \label{equal picard} \end{lem} \begin{lem} The bundle ${\frak P}_d \longrightarrow {\frak J} ^d$ admits a line bundle whose restriction to a fiber is isomorphic to ${\cal O}(s_d)$. The number $s_d $ is minimum with this property. \label{number} \end{lem} {\sc {Proof:}} Lets consider first the same question for the bundle $u:\, {\frak C}_g^{(d)} \longrightarrow {\frak J} ^d $. Let ${\cal L}$ be a line bundle on ${\frak C}_g^{(d)} $ whose restriction on the fiber ${\Bbb P} H^0(C,L)$ is isomorphic to ${\cal O}(t)$. By \cite{K1}, pg. 844 , the class of its restriction is of the form $(2g-2)m x - n \frac{\Delta }{2}$, where $m, \, n$ are integers. Since the restrictions of $x$ and $\frac{\Delta}{2} $ on a fiber of the Abel-Jacobi map have classes $c_1{\cal O}(1)$ and $c_1{\cal O}(d+g-1)$ respectively, we conclude that $s_d|t$. One can see that the same is true for the bundle ${\frak P}_d$. The proof is similar to that of Lemma 4, in \cite{K2}. On the other hand, one can construct a line bundle ${\cal R}$ on ${\frak H}_{d,g,r} $ whose restriction on the fibers is isomorphic to ${\cal O}(s_d)$ as follows. We turn back to diagram \ref{diag1}. The pull back by the map $\eta $ of ${\cal O}(1)$ to ${\frak C}_{\frak H} $, is the tautological bundle ${\cal F}$. By the see-saw principle we have that there exists a bundle ${\cal R}$ on ${\frak H}_{d,g,r} $ such that \[ q^{*} {\cal R} \cong {\cal F}^{\otimes s_d} \otimes {\alpha }^{*} {\cal P}_d ^{-1}. \] It is easy to see that the restriction of ${\cal R}$ to the fibers of the map $\phi $ is ${\cal O}(s_d)$. \begin{flushright} $\Box$ \end{flushright} \bigskip The above constructed line bundle ${\cal R}$ is the generator of the relative Picard group of ${\frak H}_{d,g,r} $ over $ {\frak J} ^d$. To summarize, we have that the Picard group of ${\frak H}_{d,g,r} $, when $d \geq 2g+1 $ and $3<r<d-g $, is freely generated over the integers by the line bundles $\psi ^{*} \phi ^{*} \lambda, \; \phi ^{*} {\cal L}_d$ and ${\cal R}$. \section{The classes of the generators} In this section we express the classes of the above generators of Pic${\frak H}_{d,g,r} $, in terms of the naturally defined classes $A, \; B, \; C $ of Section 1. For the notation, see diagram 1. \begin{lem} If $A = q_{*}({\cal F}^2), \;\; B = q_{*} ({\cal F} \omega _q)$ and $ C =q_{*}( \omega _q^2)$, we have \begin{enumerate} \item $ A = \frac {nd-s_d}{g-1} \, \phi ^{*} {\cal L}_d + 2 \frac{d}{s_d } {\cal R} $, \item $ B= n \, \phi ^{*} {\cal L}_d + \frac{2g-2}{s_d}\, {\cal R} $, \item $ C = 12 \, \phi ^{*} \psi ^{*} \lambda $, \end{enumerate} where $n$ is the integer defined in the construction of ${\cal P}_d$ in Section 2. \label{classes} \end{lem} \begin{rem} {\rm Note that the coefficient matrix has determinant of absolute value 24.} \end{rem} \noindent {\sc {Proof:}} For the first one: \begin{eqnarray*} s_d^2 \, {\cal F}^2 & = & (q^{*}{\cal R} + \alpha ^{*} {\cal P}_d)^2 \\ & = & q^{*}{\cal R}^2 + 2 \, q^{*}{\cal R} \, \alpha ^{*} {\cal P}_d + \alpha ^{*} {\cal P}_d^2. \end{eqnarray*} Therefore by applying $q_{*}$, we get \begin{eqnarray*} s_d^2 \, q_{*} {\cal F}^2 & = & 2 \, q_{*} \alpha ^{*} {\cal P}_d \, {\cal R} + q_{*} \alpha ^{*} {\cal P}_d^2 \\ & = & 2 d s_d \, {\cal R} + \phi ^{*} \nu _{*} {\cal P}_d^2 \\ & \stackrel {\mbox{Lem \ref {push forward}}}{=} & 2 d s_d \, {\cal R} + s_d^2 \, \frac{nd-s_d}{g-1} \phi ^{*} {\cal L}_d. \end{eqnarray*} and this proves the first part. For the second part: By the definition of ${\cal R}$ and by multiplying by $\omega _q$ we have \[ {\cal F} \omega _q = \frac{1}{s_d} \, q^{*}{\cal R} \, \omega _q + \frac{1}{s_d} \, \alpha ^{*} {\cal P}_d \, \omega _q, \] and so, \begin{eqnarray*} q_{*}({\cal F} \omega _q) & = & \frac{1}{s_d} \, {\cal R} \, q_{*} \omega _q + \frac{1}{s_d} \, q_{*} \alpha ^{*}( {\cal P}_d \omega _{\nu }) \\ & = & \frac{2g-2}{s_d} \, {\cal R} + \frac{1}{s_d} \, \phi ^{*} \nu _{*} ({\cal P}_d \omega _{\nu }) \\ & \stackrel {\mbox{Lem \ref {push forward}}}{=} & \frac{2g-2}{s_d}\, {\cal R} + n \phi ^{*}\, {\cal L}_d. \end{eqnarray*} The third part is an immediate consequence of Lemma \ref{push forward}. \begin{flushright} $\Box$ \end{flushright} \bigskip The above Lemma \ref{classes} gives the expression of the generators in terms of the classes $A,\, B, \, C $. We thus have \begin{theor} For $d \geq 2g+1 $ and $3 < r < d-g$, the Picard group of ${\frak H}_{d,g,r} $ is freely generated over $\Bbb Z$ by the three line bundles $\psi ^{*} \phi ^{*} \lambda, \; \phi ^{*} {\cal L}_d$ and ${\cal R}$. Their classes can be expressed as \begin{enumerate} \item $ {\cal R}= \frac {1}{2} n \, A- \frac {1}{2}\frac{nd - s_d}{g-1}\, B , $ \item $ \phi ^{*} {\cal L}_d = - \frac {g-1}{s_d} \, A + \frac{d}{s_d} \, B , $ \item $ \phi ^{*} \psi ^{*} \lambda = \frac{1}{12} \, C , $ \end{enumerate} where $n$ is the integer defined in the construction of ${\cal P}_d$, see Section 2. \label{main} \end{theor} \begin{rem} {\rm Note that in the above theorem the numbers $\frac{g-1}{s_d}$ and $ \frac{d}{s_d} $ are either integers or half integers. The number $ \frac{nd - s_d}{g-1}$ is an integer by its definition. } \end{rem} \begin{rem} {\rm The cases $r=3 < d-g $ and $r=d-g$ have to be treated separately: For $r=3 < d-g $ the Picard group of ${\frak H}_{d,g,3}$ fits in an exact sequence \[ 0 \longrightarrow {\Bbb Z} \stackrel {\mu }{ \longrightarrow} {\Bbb Z}^{\oplus 3} \stackrel {p }{ \longrightarrow} \mbox{Pic} \, {\frak H}_{d,3,g} \longrightarrow 0 , \] where the map $\mu $ is the multiplication by $2(d-1)(d-2) -4g $ in the first factor. The case is similar for $r=d-g$. Here $\mu $ is the multiplication by $d-g+1$. We leave to the reader to find in this case the analogue of Theorem \ref{main}.} \end{rem} \begin{rem} {\rm One can pursue the above discussion further and calculate the Picard groups over the integers of the Severi varieties and the Hurwitz schemes in the case when the degree $d$ is big with respect to the genus $g$. For the Severi varieties, the results of Diaz-Harris, see \cite{DH1}, \cite{DH2}, imply that the union of the Severi variety $V_{d,g}$ with the three irreducible, independent divisors $CU, \; TN, \; TR$ of ${\frak P}_d$ which are defined in the above mentioned papers, has complement of codimension $\geq 2$ in ${\frak P}_d$. Since Pic${\frak P}_d$ is isomorphic to ${\Bbb Z}^{\oplus 3}$, this implies that Pic$V_{d,g}$ is torsion. Furthermore, by using the expressions of $CU, \; TN, \; TR$ in terms of $A,B,C$ given in \cite{DH1} and the above Lemma \ref{classes}, one can calculate that torsion group. A similar result should be obtained for the Hurwitz scheme by using the analogues results of Mockizuki, see \cite{Mo}. In the case of the Severi varieties, the calculations of the author lead to rather messy formulas. } \end{rem}

9306

alg-geom/9306001

en

https://arxiv.org/abs/alg-geom/9306001

alg-geom/9306001

Alexis Kouvidakis and Tony Pantev

The automorphism group of the moduli space of semi stable vector bundles

48 pages, LaTeX

Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve $C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms of $ {\cal S}{\cal U}(r, L_0) $. The analogue of this result is carried out for the space ${\cal U}(r,d) $ of semi stable vector bundles of rark $r$ and degree $d$. As an application of the technics we use, we give in the appendix at the end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal S}{\cal U}(r, L_0) $ for any rank $r$ and degree $d$.

alg-geom

\section{Introduction} To every smooth curve $C$ one can associate a natural variety $\urd$ - the moduli space of semi stable vector bundles of rank $r$ and degree $d$ on $C$. This canonical object has a rich geometrical structure which reflects in a beautiful way the geometry of the curve $C$ and its deformations. In the abelian case $r=1$ this relationship is classical and goes back to the Riemann inversion problem, the theory of Jacobi theta-functions and the Torelli theorem. Understanding the subtleties of the non-abelian case has resulted over the years in a multitude of new ideas and unexpected connections with other branches of mathematics. One of the remarkable similarities between the Jacobian varieties and the higher rank moduli spaces is the existence of theta line bundles on them, see \cite{dn}. These are ample determinantal line bundles, naturally arising from the interpretation of $\urd$ as a moduli space and provide valuable information about the projective geometry of $\urd$. In this paper we describe the groups of automorphisms and of polarized automorphisms of $\urd$ for $r > 1$. To understand the problem better, consider first the case $r = 1$. It is well known that for any integer $d$, the group of automorphisms of the Jacobian $J^{d}(C) = {\cal U}(1,d)$ is isomorphic to the semi-direct product \[ J^{0}(C) \rtimes {\rm Aut}^{\rm group}(J^{0}(C)) \cong {\rm Aut}(J^{d}(C)), \] where ${\rm Aut}^{\rm group}(J^{0}(C))$ is the group of group automorphisms of $J^{0}$. The above isomorphism depends on the choice of a point $L_{0} \in J^{d}(C)$ and is given explicitly by $(\xi,\phi) \rightarrow T_{\xi}\circ T_{L_{0}}\circ \phi \circ T_{L_{0}}^{-1}$. Moreover, the subgroup of automorphisms preserving the class of the theta bundle is just \[ {\rm Aut}^{\theta}(J^{d}(C)) \cong \left\{ \begin{array}{ll} J^{0}(C) \rtimes ((\pm {\rm id})\times {\rm Aut}(C)) & C {\rm -not \; hyperelliptic,} \\ J^{0}(C) \rtimes {\rm Aut}(C) & C{\rm -hyperelliptic}. \end{array} \right. \] Some natural automorphisms of $\urd$ can be obtained by analogy with this case. For example, the Jacobian $J^{0}(C)$ acts by translations on the moduli space and, as it turns out, it actually coincides with the identity component of the automorphism group. Furthermore, we have a natural morphism \begin{equation} \det : \urd \longrightarrow J^{d}(C) \label{eq01} \end{equation} sending a bundle $E$ to its determinant $\det (E)$. The fibers of this morphism are moduli spaces on its own right, e.g. the fiber $\det^{-1}(L_{0})$ over a point $L_{0}$ is just the moduli space $\su$ of semistable bundles of rank $r$ and fixed determinant $L_{0}$. One can use the fibration (\ref{eq01}) to construct some natural automorphisms of $\urd$, that is, certain automorphisms of the fiber $\su$ can be glued with certain automorphisms of the base $J^{d}(C)$ to produce global automorphisms of the moduli space. After choosing a point $L_{0} \in J^{d}(C)$ we can trivialize the bundle (\ref{eq01}) by pulling it back to an \'{e}tale cover of $J^{d}(C)$: \begin{equation} {\divide\dgARROWLENGTH by 9 \begin{diagram} \node{\su\times J^{0}(C)} \arrow[3]{e,t}{(E,L) \rightarrow E\otimes L} \arrow{s,l}{p_{2}} \node[3]{\urd} \arrow{s,r}{\det} \\ \node{J^{0}(C)} \arrow[3]{e,b}{L \rightarrow L^{\otimes r}\otimes L_{0}} \node[3]{J^{d}(C)} \end{diagram}} \label{eq02} \end{equation} The top and the bottom rows of the diagram ({\ref{eq02}) are Galois covers with Galois group the group of $r$-torsion points $J^{0}(C)[r]$. Consider the subgroup ${\rm Aut}_{[r]}(J^{0}(C)) \subset {\rm Aut}(J^{0}(C))$ consisting of group automorphisms of $J^{0}(C)$ which act trivially on the torsion points $J^{0}(C)[r]$. Then given a translation, $T_{\mu}$, by a torsion point $\mu \in J^{0}(C)[r]$ and $\varphi \in {\rm Aut}_{[r]}(J^{0}(C))$, the automorphism $T_{\mu} \times \varphi$ of $\su\times J^{0}(C)$ commutes with the action of the Galois group and hence descends to an automorphism of $\urd$. In addition, every symmetry of the curve $C$ will induce an automorphism of $\urd$. In the special case $r | 2d$, by choosing $\nu \in J^{\frac{2d}{r}}(C)$ with property $\nu^{\otimes r} = L_{0}^{\otimes 2}$, we can construct an additional automorphism $\delta$ of order two: \[ \begin{array}{cccc} \delta : & \urd & \longrightarrow & \urd \\ & E & \longrightarrow & E^{\vee}\otimes \nu \end{array} \] To combine these, consider the subgroups ${\frak U}{\frak G}^{\circ}_{[r]} \subseteq {\frak U}{\frak G}_{[r]} \subseteq {\rm Aut}(J^{0}(C))\times {\rm Aut}(C)$ defined by \begin{defi} \label{def01} \[ {\frak U}{\frak G}^{\circ}_{[r]} = \{ (\phi,\sigma) \; | \; \phi^{-1}\circ \sigma^{*} \in {\rm Aut}_{[r]}(J^{0}(C))\},\] and \[ {\frak U}{\frak G}_{[r]} = \{ (\phi,\sigma) \; | \; \phi^{-1}\circ (\pm {\bf 1})\circ \sigma^{*} \in {\rm Aut}_{[r]}(J^{0}(C)) \}. \] \end{defi} The group ${\frak U}{\frak G}^{\circ}_{[r]}$ is of index two in ${\frak U}{\frak G}_{[r]}$ and we have a natural homomorphism: \begin{equation} \begin{array}{ccl} J^{0}(C) \rtimes {\frak U}{\frak G}^{\circ}_{[r]} & \longrightarrow & {\rm Aut}(\urd ) \\ (\xi,\phi,\sigma) & \longrightarrow & (E \mapsto \sigma ^* E \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ^{-1} )) \end{array} \label{eq03} \end{equation} where $\eta \in J^{0}(C)$ is an arbitrary line bundle with the property $\eta ^{\otimes r} = \det E \otimes L_0 ^{-1}$. If in addition $r \, | \, 2d$, then the homomorphism (\ref{eq03}) can be extended to \begin{equation} \label{eq04} \begin{array}{ccl} J^{0}(C) \rtimes {\frak U}{\frak G}_{[r]} & \longrightarrow & {\rm Aut}(\urd ) \\ (\xi,\phi,\sigma) & \longrightarrow & E \mapsto \left\{ \begin{array}{ll} \sigma ^* E \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ^{-1} ) & {\rm if} \; \phi^{-1}\circ \sigma^{*} \in {\rm Aut}_{[r]}(J^{0}(C)) \\ \sigma ^* E^{\vee} \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta) & {\rm if} \; \phi^{-1}\circ ({\bf - 1}) \circ \sigma^{*} \in {\rm Aut}_{[r]}(J^{0}(C)) \\ \end{array} \right. \end{array} \end{equation} The commutative diagram (\ref{eq02}) suggests that the subgroups (\ref{eq03}) (or (\ref{eq04}) in the case $r \, | \, 2d$) will not differ too much from the full automorphism group, if the moduli space $\su $ doesn't have excess automorphisms. The variety $\su $ has a lot of advantages. It is a Fano variety of \linebreak Picard number one and by a theorem of Narasimhan and Ramanan, \cite{nr}, $H^{0}(\su ,T\su) = 0$ unless $g=2$, $r=2$ and $d$ even. Therefore, in general, ${\rm Aut}(\su )$ is finite and contains the group $J^{0}(C)[r]$. In the case $r = 2$, odd degree and $g = 2$ the group ${\rm Aut}({\cal S}{\cal U}^{s}(2,L_0) )$ was described by Newstead \cite{ne} as a consequence of his proof of the Torelli theorem for the variety ${\cal S}{\cal U}^{s}(2,L_0)$. To generalize his result we adopt a different viewpoint - we use Hitchin's abelianization to prove our main theorem \begin{theor} Let $C$ be a curve without automorphisms. Then the automorphism group of the moduli space $\su$ can be described as follows \begin{enumerate} \item If $r \nmid 2d$, then the natural map \[ \begin{array}{ccl} J^{0}(C)[r] & \longrightarrow & {\rm Aut}(\su ) \\ \mu & \longrightarrow & (E \mapsto E\otimes \mu) \end{array} \] is an isomorphism, and \item If $r \, | \, 2d$, then the natural map \[ \begin{array}{ccl} J^{0}(C)[r] \rtimes {\Bbb Z}/2{\Bbb Z} & \longrightarrow & {\rm Aut}(\su ) \\ (\mu, \varepsilon) & \longrightarrow & (E \mapsto \delta^{\varepsilon}(E)\otimes \mu) \end{array} \] is isomorphism for $r \geq 3$ and has kernel ${\Bbb Z}/2{\Bbb Z}$ for $r = 2$. \end{enumerate} \label{theor1} \end{theor} \noindent The strategy of the proof is to lift the automorphism $\Phi$ to a birational automorphism of the moduli space of Higgs bundles and then study the induced automorphism $\tilde{\Phi} _{\tilde {d}} $ of the family $\unprymd$ of smooth spectral Pryms. To determine the map $\tilde{\Phi} _{\tilde {d}} $ explicitly we study the ring of relative correspondences over the universal spectral curve and show that the family of groups $\unprym$ does not have non-trivial global group automorphisms. Furthermore, we show that all the sections of $\unprymd$ come from pull-back of line bundles on the base curve $C$ and conclude that $\tilde{\Phi} _{\tilde {d}} $ must be a multiplication by $\pm 1$ along the fibers of $\unprymd$ followed by a translation by a pull-back bundle. To finish the proof, we use the rational map from a spectral Prym to the moduli space $\su$ to recover from $\tilde{\Phi} _{\tilde {d}} $ the original automorphism $\Phi$. For curves with automorphisms the statement of the main theorem has to be modified appropriately. To avoid some technical complications which occur in the case $g = 2$, we assume that $g \geq 3$ throughout the paper. Consider the subgroups ${\frak S}{\frak G}^{\circ}_{[r]} \subseteq {\frak S}{\frak G}_{[r]} \subseteq J^{0}(C)\times {\rm Aut}(C)$ defined as \begin{defi}\label{def02} \[ {\frak S}{\frak G}^{\circ}_{[r]} = \{ (\xi, \sigma) \; | \; \xi^{r} = L_{0}\otimes\sigma^{*}(L_{0}^{-1}) \}, \] and \[ {\frak S}{\frak G}_{[r]} = \{ (\xi, \sigma) \; | \; \xi^{r} = L_{0}\otimes\sigma^{*}(L_{0}^{\pm 1}) \}. \] \end{defi} Again ${\frak S}{\frak G}^{\circ}_{[r]}$ is a normal subgroup of index two in ${\frak S}{\frak G}_{[r]}$ and we have \begin{theor} Let $C$ be any smooth curve of genus $g \geq 3$. We have \begin{enumerate} \item If $r \nmid 2d$, then the natural map \[ \begin{array}{ccl} {\frak S}{\frak G}^{\circ}_{[r]} & \longrightarrow & {\rm Aut}(\su ) \\ (\xi , \sigma) & \longrightarrow & (E \mapsto \sigma^{*}E\otimes \xi) \end{array} \] is surjective, and \item If $r \, | \, 2d$, then the natural map \[ \begin{array}{ccl} {\frak S}{\frak G}_{[r]}& \longrightarrow & {\rm Aut}(\su ) \\ (\xi, \sigma) & \longrightarrow & E \mapsto \left\{ \begin{array}{ll} \sigma^{*}E\otimes \xi & {\rm if} \; (\xi, \sigma) \in {\frak S}{\frak G}^{\circ}_{[r]} \\ \sigma^{*}E^{\vee}\otimes \xi & {\rm if} \; (\xi, \sigma) \in {\frak S}{\frak G}_{[r]} \setminus {\frak S}{\frak G}^{\circ}_{[r]} \end{array} \right. \end{array} \] is surjective. \end{enumerate} \label{theor2} \end{theor} \begin{rem} {\rm The maps in the above Theorem \ref{theor2} are actually isomorphisms. Our proof of the injectivity involves some standard arguments from Prym theory similar to those in \cite{nr2}; the proof is not included here because of its length. } \end{rem} \noindent The first part of Theorem \ref{theor2} gives a positive answer to a question posed by Tyurin in the survey paper \cite{ty}. The main theorem, together with the observation that every automorphism of $\urd $ must lift to an automorphism of $\su \times J^{0}(C)$, see Lemma \ref{lemurd3}, gives \begin{theor} Let $C$ be a smooth curve of genus $g \geq 3$. Then the automorphisms of the full moduli space $\urd $ can be described as follows \begin{enumerate} \item If $r \nmid 2d$, then the map {\rm (}\ref{eq03}{\rm )} \[ \begin{array}{ccl} J^{0}(C) \rtimes {\frak U}{\frak G}^{\circ}_{[r]} & \hookrightarrow & {\rm Aut}(\urd ) \\ (\xi,\phi,\sigma) & \rightarrow & (E \mapsto \sigma ^* E \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ^{-1} )) \end{array} \] is surjective, and \item If $r | 2d$, then the map {\rm (}\ref{eq04}{\rm )} \[\begin{array}{ccl} J^{0}(C) \rtimes {\frak U}{\frak G}_{[r]} & \hookrightarrow & {\rm Aut}(\urd ) \\ (\xi,\phi,\sigma) & \rightarrow & E \mapsto \left\{ \begin{array}{ll} \sigma ^* E \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ^{-1} ) & {\rm if} \; \phi^{-1}\circ \sigma^{*} \in {\rm Aut}_{[r]}(J^{0}(C))\\ \sigma ^* E^{\vee} \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta) & {\rm if} \; \phi\circ \sigma^{*} \in {\rm Aut}_{[r]}(J^{0}(C)) \\ \end{array} \right. \end{array} \] is surjective. \end{enumerate} \label{theor3} \end{theor} The description of the polarized automorphisms of $\urd$ follows easily from Theorem \ref{theor3}: \begin{theor} For any curve $C$ of genus $g \geq 3$ the automorphisms of the moduli space $\urd$ which preserve the class of the theta bundle are those belonging to the image of the subgroup \linebreak $J^{0}(C) \rtimes ((\pm {\rm id})\times {\rm Aut}(C)) \subseteq J^{0}(C) \rtimes {\frak U}{\frak G}_{[r]}$ under the map {\rm (}\ref{eq04}{\rm ).} \label{theor4} \end{theor} As an application of the technics we use in this work, we prove in the Appendix at the end of the paper the Torelli Theorem for the moduli space of vector bundles: \begin{theor} Let $C_1$, $C_2$ be smooth curves of genus $g \geq 3$ and $L_1$, $L_2$ line bundles of degree $d$ on $C_1$, $C_2$ respectively. If ${\cal S}{\cal U}_{C_1}(r, L_1) \simeq {\cal S}{\cal U}_{C_2}(r, L_2)$, then $C_1 \simeq C_2$. \label{Torelli} \end{theor} The paper is organized as follows. In the first section we gather all the facts about the moduli space of Higgs bundles, the linear system of spectral curves and the Hitchin map which we are going to use later on. We have included the proofs of some statements which seem to be well known to the experts in the field but which can not be found in the standard sources, as well as, the proofs of some facts about the discriminant locus in the Hitchin base which help us simplify our main argument. In the second section we construct the induced automorphism $\tilde{\Phi} _{\tilde {d}} $ and study its first properties. The third section contains a discussion of the relative Picard of the universal spectral curve and its sections over a Zariski open set in the Hitchin base. In section four we analyze the ring of relative correspondences on the fibers of the universal spectral curve whose description is the key ingredient in the proof of the main theorem. In the fifth section we give the proof of the main theorem and discuss the case of curves with symmetries. In the sixth section we derive the description of ${\rm Aut}(\urd )$. \bigskip \noindent {\bf Acknowledgments:} We are very grateful to Ron Donagi for his constant support and for many valuable discussions and suggestions. We would like to thank Eyal Markman for explaining to us the proof of Lemma \ref{lem14} and George Pappas for helpful conversations. \bigskip \noindent {\bf Notation and Conventions} \medskip \noindent $\su \, :$ The moduli space of semi stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$. \\ ${\cal S} {\cal U} ^s(r, L_0) \, :$ The moduli space of stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$. \\ ${\cal X}(r, L_0) \, : $ The total space of the cotangent bundle $T^* \sustable$. \\ ${\cal M}(r, L_0) \, : $ The moduli space of semi stable Higgs pairs. \\ $H : {\cal M}(r, L_0) \longrightarrow W \, : $ The Hitchin map. \\ $ C \, : $ A smooth curve of genus $g \geq 3$. \\ $ \omega _C \, : $ The canonical bundle on $C$. \\ $ S^{\circ} \, : $ The total space of the canonical bundle $\omega _C$. \\ $ S \, : $ The space ${\Bbb P}({\cal O} \oplus \omega _C)$. \\ $ \alpha : S \longrightarrow C \, : $ The canonical map. \\ $ Y \in H^0(S, {\cal O}_S(1) ) \, : $ The infinity section of $S$. \\ $ X \in H^0(S, {\cal O}_S(1) \oplus \alpha ^* \omega _C) \, : $ The zero section of $S$. \\ $ x= X/Y \, : $ The tautological section of $S^{\circ} $. \\ $ Y_{\infty } \, : $ The divisor at infinity ${\rm div}(Y) \subset S $. \\ $ X_0 \, : $ The zero divisor ${\rm div}(X) \subset S^{\circ} $. \\ $ \overline{W} = H^0(C, {\cal O}) \oplus H^0(C, \omega _C^{\otimes 2}) \oplus \cdots \oplus H^0(C, \omega _C^{\otimes r}) $. \\ $ W_k= H^0(C, \omega _C^{\otimes k}) $. \\ $ W \simeq W_2 \oplus \cdots \oplus W_r$, embedded in $\overline{W}$ as $\{ 1 \} \oplus W_2 \oplus \cdots \oplus W_r$. \\ $ W^{\infty } \simeq W_2 \oplus \cdots \oplus W_r$, embedded in $\overline{W}$ as $\{ 0 \} \oplus W_2 \oplus \cdots \oplus W_r$. \\ $ {\cal D} \, : $ The discriminant divisor. \\ $ W^{{\rm reg}} = W \setminus {\cal D}$. \\ $ \spec \, : $ The spectral curve of genus $\tilde{g} =r^2(g-1)+1$ associated to $s \in W$. \\ $ \pi _s : \spec \longrightarrow C \, : $ The projection to the base curve $C$. \\ $\pi : \unspec \longrightarrow C \, : $ The universal spectral curve and its projection to $C$. \\ $ \beta : \unspec \longrightarrow S^{\circ} \, : $ The map to $S^{\circ} $. \\ $ {\rm Prym}( \tilde {C}_s, C) \, : $ The prymian of the map $\pi _s : \spec \longrightarrow C$. \\ $ \unprym \, : $ The universal prymian associated to the map $\pi : \unspec \longrightarrow C $. \\ $ {\rm Prym}_{\tilde{d}}(\tilde{C}_s , C) \, : $ The ``prymian'' of degree $\tilde{d}=d+r(r-1)(g-1)$, i.e. the set $\{ \tilde{L} \in J^{\tilde{d}} (\spec ) \; | \; \det \pi _* \tilde{L} = L_0 \}$. \\ $ \unprymd \, : $ The universal spectral ``prymian'' of degree $\tilde{d}$. \section{Spectral curves} \subsection{Linear systems of spectral curves} Let $C$ be a smooth curve of genus $g$. Let $S^{\circ}$ be the total space of the line bundle $\omega_{C} \rightarrow C$ and let $S = {\Bbb P}(\omega_{C}\oplus {\cal O}_{C})$ be the projective extension of $\omega_{C}$. Denote by $\alpha : S \rightarrow C$ the natural projection and let ${\cal O}_{S}(1)$ be the relative hyperplane bundle on $S$ corresponding to the vector bundle $\omega_{C}\oplus {\cal O}_{C}$. \begin{defi} An $r$-sheeeted spectral curve is an element $\widetilde{C}$ of the linear system $| \alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)|$ having the property $\widetilde{C} \subset S^{\circ}$ \label{def11} \end{defi} The adjective spectral refers to another interpretation of the curve $\widetilde{C}$ which we recall next. For any vector bundle $E$ of rank $r$ over $C$ and any $\omega_{C}$-twisted endomorphism $\theta \in H^{0}(C,{\rm End}E\otimes\omega_{C})$ of $E$ there is a suitable notion of a characteristic polynomial. Define the $i$-th characteristic coefficient $s_{i} \in H^{0}(C,\omega_{C}^{\otimes i})$ of $\theta$ as $s_{i} = (-1)^{i+1}{\rm tr}(\wedge^{i}\theta)$. The characteristic polynomial $P(x)$ of $\theta$ can be written formally as $P(x) = x^{r} + s_{1}x^{r-1} + \ldots + s_{r}$. Geometrically the polynomial $P(x)$ corresponds to a subcheme $\widetilde{C}_{s} \subset S^{\circ}$ which via the projection $\alpha$ is an $r$-sheeted branch cover of $C$. Indeed, let $X \in H^{0}(C,\alpha^{*}\omega_{C}\otimes{\cal O}_{S}(1))$ be the zero section of the ruled surface $S$ and let $Y \in H^{0}(C,{\cal O}_{S}(1))$ be the infinity section of $S$. By setting $x := \frac{X}{Y}$, i.e. $x$ is the tautological section of $\alpha^{*}\omega_{C} \rightarrow S^{\circ}$, we can reinterpret $Y^{\otimes r}P(x) = X^{r} + s_{1}X^{r-1}Y + \ldots s_{r}Y^{r}$ as a section of $\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)$ whose divisor is contained in $S^{\circ}$, that is - a spectral curve, for more details see \cite{bnr}, \cite{h1}. \begin{rem} \label{rem11} {\rm We can view the space $\widetilde{W} := {\displaystyle \oplus^{r}_{i=1}}H^{0}(C,\omega_{C}^{\otimes i})$ as the space of characteristic polynomials of $\omega_{C}$-twisted endomorphisms of rank $r$ vector bundles. The vector space $\widetilde{W}$ can be identified explicitly as the locus of spectral curves in $| \alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)|$ as follows. The push forward map $\alpha_{*}$ induces an isomorphism \[ \alpha_{*} : H^{0}(S,\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)) \longrightarrow H^{0}(C,\alpha_{*}(\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r))).\] On the other hand, \[ \begin{array}{lcl} \alpha_{*}(\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)) & = & \omega_{C}^{\otimes r} \otimes \alpha_{*}({\cal O}_{S}(r)) = \omega_{C}^{\otimes r} \otimes ({\rm Sym}^{r}({\cal O}_{C} \oplus \omega_{C}^{-1}) = \\ & = & {\cal O}_{C} \oplus \omega_{C}\oplus \ldots \oplus \omega_{C}^{r}. \end{array} \] Therefore we get an isomorphism \[ \alpha_{*} : H^{0}(S,\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)) \longrightarrow \oplus_{i=0}^{r} H^{0}(C,\omega_{C}^{\otimes i}) ,\] whose inverse is \[ \begin{array}{ccc} \oplus_{i=0}^{r} H^{0}(C,\omega_{C}^{\otimes i}) & \longrightarrow & H^{0}(S,\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)) \\ (s_{0}, s_{1}, \ldots , s_{r}) & \longrightarrow & s_{0}X^{r} + s_{1}X^{r-1}Y + \ldots + s_{r}Y^{r}. \end{array} \] Here we have identified the spaces $H^{0}(C,\omega_{C}^{\otimes i})$ and $\alpha^{*}H^{0}(C,\omega_{C}^{\otimes i}) \subset H^{0}(C, \alpha^{*}\omega_{C}^{\otimes i})$ via $\alpha$. For any $s = (s_{0}, s_{1}, \ldots , s_{r})$ denote by $\widetilde{C}_{s}$ the divisor ${\rm Div}( s_{0}X^{r} + s_{1}X^{r-1}Y + \ldots + s_{r}Y^{r})$ on $S$. Let ${\rm Div}(X) = X_{0}$ and ${\rm Div}(Y) = Y_{\infty}$. By definition the curve $\widetilde{C}_{s}$ is spectral if $\widetilde{C}_{s} \subset S^{\circ}$, or equivalently, if $\widetilde{C}_{s} \cap Y_{\infty} = \varnothing$. Since $X_{0} \cap Y_{\infty} = \varnothing$ it follows that $\widetilde{C}_{s} \cap Y_{\infty} \neq \varnothing$ if and only if $Y_{\infty} \subset \widetilde{C}_{s}$, i.e. if and only if $s_{0} = 0$. Thus the locus of all spectral curves in the projective space $| \alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)|$ is the affine open set $\widetilde{W} = \{ \widetilde{C}_{s} \, | \, s_{0} \neq 0 \}$. }\end{rem} \bigskip For the purposes of this paper we will be mainly concerned with the slightly smaller space \[ W := H^{0}(C,\omega_{C}^{\otimes 2})\oplus H^{0}(C,\omega_{C}^{\otimes 3})\oplus \ldots \oplus H^{0}(C,\omega_{C}^{\otimes r}),\] consisting of the characteristic polynomials of traceless twisted endomorphisms of rank $r$ vector bundles. Let $h : \unspec \rightarrow W$ be the family of spectral curves parametrized by $W$. To construct the variety $\unspec $ consider the line bundle $p_{S}^{*}(\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)) \rightarrow W\times S$. There is a natural tautological section ${\frak c} \in H^{0}(W\times S, p_{S}^{*}(\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)))$ given by \[ {\frak c}(((s_{2},\ldots ,s_{r});p)) = X^{r}(p) + s_{2}(p)X^{r-2}(p)Y^{2}(p) + \ldots + s_{r}(p)Y^{r}(p),\] and $\unspec = {\rm Div}({\frak c})$ is just the divisor of this section. The universal family $\unspec$ is proper over $W$ and admits a natural compactification to a projective variety $\overline{\cal C}$ - the universal family for the linear system ${\Bbb P}(H^{0}(C,{\cal O}_{C})\oplus W)$. Set $\overline{W} := H^{0}(C,{\cal O}_{C})\oplus W$. Next we are going to study the linear system ${\Bbb P}(\overline{W})$ and the total spaces of the universal families $\unspec$ and $\overline{\cal C}$. \begin{lem} \label{lem11} The linear system ${\Bbb P}(\overline{W})$ is base point free. Furthermore, if $r \geq 3$ the morphism $f_{{\Bbb P}(\overline{W})} : S \rightarrow {\Bbb P}(\overline{W})^{\vee}$ is an inclusion when restricted to $S^{\circ}$ and contracts the infinity section $Y_{\infty}$. \end{lem} {\bf Proof.} Let $p \in S$. If $p \in Y_{\infty}$, then the curve ${\rm Div}(X^{r}) \in {\Bbb P}(\overline{W})$ does not pass through $p$. If $p \notin Y_{\infty}$, then choose a section $s_{r} \in H^{0}(C, \omega_{C}^{\otimes r})$ not vanishing at $\alpha(p)$. The curve ${\rm Div}(s_{r}Y^{r}) \in {\Bbb P}(\overline{W})$ does not pass trough $p$. Consider the morphism \[f_{|\overline{W}|} : S \longrightarrow {\Bbb P}(\overline{W})^{\vee}.\] Since a curve $\widetilde{C} \in | \alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)|$ is either spectral or contains $Y_{\infty}$, see Remark \ref{rem11}, it follows that $f_{{\Bbb P}(\overline{W})}$ contracts $Y_{\infty}$. If $p\neq q \in S$ are points in the same fiber $F$ of $\alpha$, then they are separated by ${\Bbb P}(\overline{W})$ because by pulling ${\Bbb P}(\overline{W})$ back on $F$ we get the linear system \[ {\Bbb P}({\rm Span}(x_{0}^{r},x_{0}^{r-2}x_{1}^{2}, \ldots ,x_{1}^{r})) \subset |{\cal O}_{F}(r)|, \] which for $r \geq 3$ separates the points on $F \cong {\Bbb P}^{1}$. If $p\neq q \in S$ are points in two different fibers of $\alpha$, then by choosing a section $s_{r} \in H^{0}(C, \omega_{C}^{\otimes r})$ satisfying $s_{r}(\alpha(p)) = 0$ and $s_{r}(\alpha(q)) \neq 0$, we get a curve ${\rm Div}(s_{r}Y^{r})$ passing trough $p$ and not passing through $q$. The above considerations and the local triviality of $S^{\circ} \rightarrow C$ show that ${\Bbb P}(\overline{W})$ separates also the tangent directions because for every point $p \in S^{\circ}$ the morphism $f_{{\Bbb P}(\overline{W})}$ maps the fiber to hrough $p$ and a suitable translate of $X_{0}$ into two transversal curves in ${\Bbb P}(\overline{W})^{\vee}$. \begin{flushright} $\Box$ \end{flushright} \bigskip Here are some immediate corollaries from the above lemma which are going to be usefull later on. \begin{cor} \label{cor11} The generic spectral curve $\spec$ is smooth. \end{cor} {\bf Proof.} Follows from the fact that ${\Bbb P}(\overline{W})$ does not have base points and from Bertini's theorem. \begin{flushright} $\Box$ \end{flushright} \begin{cor} \label{cor12} The total spaces of the universal families $\unspec$ and $\overline{\cal C}$ are smooth. \end{cor} {\bf Proof.} Since $\unspec \subset \overline{\cal C}$ is a Zariski open set it is enough to show that $\overline{\cal C}$ is smooth. Let $\pi : \overline{U} \rightarrow {\Bbb P}(\overline{W})^{\vee}$ be the universal bundle of linear hypersurfaces in ${\Bbb P}(\overline{W})$. The total space $\overline{\cal C}$ fits in the fiber product diagram \[ \begin{diagram} \node{\overline{\cal C}} \arrow{e} \arrow{s} \node{\overline{U}} \arrow{s,r}{\pi} \\ \node{S} \arrow{e,b}{f_{{\Bbb P}(\overline{W})}} \node{{\Bbb P}(\overline{W})^{\vee}} \end{diagram} \] But both $S$ and $\overline{U}$ are smooth varieties and moreover the projection $\pi : \overline{U} \rightarrow {\Bbb P}(\overline{W})^{\vee}$ is a smooth map. Therefore the fiber product $\overline{\cal C} = S_{f_{|\overline{W}|}} \!\! \times_{\pi}{\cal P}$ is also smooth. \begin{flushright} $\Box$ \end{flushright} \bigskip Let $\overline{W}_{r-1,r}$ (resp. $W_{r-1,r}^{\infty}$) be the linear subsystem of $\overline{W}$ consisting of points of the form $(s_0,0,\ldots , s_{r-1}, s_{r})$ (resp. $(0,0,\ldots , s_{r-1}, s_{r})$). According to Remark \ref{rem11}, $\overline{W}_{r-1,r}$ (resp. $W_{r-1,r}^{\infty}$) can be embedded naturally in the vector space $H^{0}(S, \alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r))$ and thus, $|(\overline{W}_{r-1,r}|$ (resp. $|(W^{\infty}_{r-1,r}|$) can be viewed as a linear subsystem of $|\alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)|$. The arguments in the proof of Lemma \ref{lem11} yield the following corollary. \begin{cor} \label{cor13} The linear system $\overline{W}_{r-1,r}$ (resp. $W_{r-1,r}^{\infty}$) is base point free. Furthermore if $r \geq 3$, then the morphism $f_{|\overline{W}_{r-1,r}|} : S \rightarrow {\Bbb P}(\overline{W}_{r-1,r})^{\vee}$ (resp. $f_{|W^{\infty} _{r-1,r}|} : S \rightarrow {\Bbb P}(W^{\infty}_{r-1,r})^{\vee}$) is an inclusion when restricted to $S^{\circ}$ and contracts the infinity section $Y_{\infty}$. \end{cor} \bigskip In the next section we are going to prove the irreducibility of various discriminant loci. The following fact is an essential ingredient in the proof of Corollary \ref{cor16} and is also of independent interest. \begin{prop} \label{prop10} Let $\widetilde{C}$ be a spectral curve of degree $r$ and let $\Sigma \subset \widetilde{C}$ be an irreducible component of $\widetilde{C}$. Then $\Sigma$ is a spectral curve of degree $l \leq r$. \end{prop} {\bf Proof.} Since $\Sigma \subset \widetilde{C} \subset S^{\circ}$, we only need to show that \[ {\cal O}_{S}(\Sigma) = {\cal O}_{S}(l)\otimes\alpha^{*}\omega_{C}^{\otimes l}, \] for some $l \leq r$. But ${\rm Pic}(S) = {\Bbb Z}\cdot {\cal O}_{S}(1) \oplus \alpha^{*}{\rm Pic}(C)$, and hence \[ {\cal O}_{S}(\Sigma) = {\cal O}_{S}(l)\otimes\alpha^{*}L, \] for some line bundle $L$ on $C$. Since $\Sigma \subset S^{\circ}$ we have $\Sigma\cdot Y_{\infty} = 0$ which yields \[ 0 = {\cal O}_{S}(\Sigma)\cdot{\cal O}_{S}(1) = {\cal O}_{S}(l)\cdot {\cal O}_{S}(1) + \alpha^{*}L\cdot {\cal O}_{S}(1) = - l(2g-2) + \deg L,\] i.e. $\deg L = l(2g - 2)$. On the other hand the line bundle ${\cal O}_{S}(l)\otimes\alpha^{*}L$ has a section $\Sigma$ which does not vanish on the infinity divisor $Y_{\infty}$. But the sections of ${\cal O}_{S}(l)\otimes\alpha^{*}L$ vanishing on $Y_{\infty}$ are $H^{0}(S,{\cal O}_{S}(l)\otimes\alpha^{*}L\otimes {\cal O}_{S}(-Y_{\infty})) = H^{0}(S,{\cal O}_{S}(l-1)\otimes\alpha^{*}L)$. By pushing forward ${\cal O}_{S}(l-1)\otimes\alpha^{*}L$ on $C$ we get \[ \begin{array}{lcl} H^{0}(S,{\cal O}_{S}(l-1)\otimes\alpha^{*}L) & = & H^{0}(C,\alpha_{*}({\cal O}_{S}(l-1))\otimes L) = H^{0}(C,{\rm Sym}^{l-1}({\cal O}_{C}\oplus \omega_{C}^{-1})\otimes L) = \\ & = & H^{0}(C,L\oplus (L\otimes\omega_{C}^{-1})\oplus \ldots \oplus (L\otimes \omega_{C}^{-(l-1)})) = \oplus_{i=0}^{l-1} H^{0}(C, L\otimes \omega_{C}^{-i}). \end{array} \] Similarly $H^{0}(S,{\cal O}_{S}(l)\otimes\alpha^{*}L) = \oplus_{i=0}^{l} H^{0}(C, L\otimes \omega_{C}^{-i})$ and hence \[ H^{0}(S,{\cal O}_{S}(l)\otimes\alpha^{*}L)/H^{0}(S,{\cal O}_{S}(l-1)\otimes\alpha^{*}L) \cong H^{0}(C, L\otimes \omega_{C}^{-l}).\] The section of the divisor $\Sigma$ gives a non-zero element in $H^{0}(S,{\cal O}_{S}(l)\otimes\alpha^{*}L)/H^{0}(S,{\cal O}_{S}(l-1)\otimes\alpha^{*}L)$ and therefore $H^{0}(C, L\otimes \omega_{C}^{-l}) \neq 0$. Since $\deg (L\otimes \omega_{C}^{-l}) = 0$ we get that $L\otimes \omega_{C}^{-l} \cong {\cal O}_{C}$. \begin{flushright} $\Box$ \end{flushright} \begin{rem} \label{rem10} {\rm The meaning of the above proposition becomes transparent if we use the interpretation of the spectral curves as characteristic polynomials of $\omega_{C}$-twisted endomorphisms. Let $F \rightarrow \widetilde{C}$ be a line bundle. Then $F$ can be viewed as a sheaf on $S$ supported on $\widetilde{C} \subset S^{\circ}$. Consider the rank $r$ vector bundle $E = \alpha_{*}F$ on $C$. The push-forward of the homomorphism \[ F \stackrel{\otimes x}{\longrightarrow} F\otimes\alpha^{*}\omega_{C}\] is a $\omega_{C}$-twisted endomorphism of $E$: \[ \theta : E \longrightarrow E\otimes\omega_{C}.\] In this way the curve $\widetilde{C}$ then can be described as the zero scheme of the section \[ \det (\alpha^{*}\theta - x\cdot {\rm id}_{\alpha^{*}E}) \in H^{0}(S, \alpha^{*}\omega_{C}^{\otimes r}\otimes {\cal O}_{S}(r)).\] Let now $\Sigma \subset \widetilde{C}$ be a component of $\widetilde{C}$. Consider the sheaf $F\otimes {\cal O}_{\Sigma}$ on $S$ supported on $\Sigma$. Let $E' := \alpha_{*}(F\otimes {\cal O}_{\Sigma})$ and let \[ \theta' : E' \longrightarrow E'\otimes\omega_{C}\] be the push forward of the homomorphism \[ F\otimes {\cal O}_{\Sigma} \stackrel{\otimes x}{\longrightarrow} (F\otimes {\cal O}_{\Sigma})\otimes\alpha^{*}\omega_{C}.\] The commutative diagram of (torsion) sheaves on $S^{\circ}$ \[ \begin{diagram} \node{F} \arrow{e,t}{\otimes x} \node{F\otimes\alpha^{*}\omega_{C}} \\ \node{F\otimes {\cal O}_{\Sigma}} \arrow{n} \arrow{e,t}{\otimes x} \node{(F\otimes {\cal O}_{\Sigma})\otimes\alpha^{*}\omega_{C}} \arrow{n} \end{diagram} \] then pushes down to a commutative diagram of vector bundles on $C$: \[ \begin{diagram} \node{E} \arrow{e,t}{\theta} \node{E\otimes\omega_{C}} \\ \node{E'} \arrow{e,t}{\theta'} \arrow{n} \node{E'\otimes\omega_{C}} \arrow{n} \end{diagram} \] In particular $E'$ is $\theta$ invariant and $\theta' = \theta_{|E'}$. Then $\Sigma$ is the zero scheme of the section \[det (\alpha^{*}\theta' - x\cdot {\rm id}_{\alpha^{*}E'}) \in H^{0}(S, \alpha^{*}\omega_{C}^{\otimes l}\otimes {\cal O}_{S}(l)),\] where $l = {\rm rk}E'$, i.e. $\Sigma$ is a spectral curve.} \end{rem} \begin{cor} \label{cor14} The locus $R \subset W$ consisting of reducible spectral curves has codimension \linebreak ${\rm codim}(R,W) \geq g-1$ \end{cor} {\bf Proof.} Set $W_{i} := H^{0}(C,\omega_{C}^{\otimes i})$. Let $R_{r}$ be the image of $R$ under the natural projection $W \rightarrow W_{r}$. It suffices to show that ${\rm codim}(R_{r},W_{r}) \geq g-1$ or equivalently, since $R_{r}$ is invariant under dilations, that ${\rm codim}({\Bbb P}(R_{r}),{\Bbb P}(W_{r})) \geq g-1$. For every $i = 1, \ldots, r-1$ consider the variety $S_{i} \subset {\Bbb P}(W_{r})\times {\Bbb P}(W_{i})$ defined by \[ S_{i} = \{ (D,G) \; | \; D \geq G\}. \] Since a component of a spectral curve is a spectral curve we have \[ {\Bbb P}(R_{r}) \subset p_{{\Bbb P}(W_{r})}(S_{1}) \cup \ldots \cup p_{{\Bbb P}(W_{r})}(S_{r-1}),\] and therefore ${\rm codim}({\Bbb P}(R_{r}),{\Bbb P}(W_{r})) \geq \min_{1\leq i \leq r-1}\{ {\rm codim}(p_{{\Bbb P}(W_{r})}(S_{i}),{\Bbb P}(W_{r}))\}$. Furthermore \linebreak $\dim (p_{{\Bbb P}(W_{r})}(S_{i})) = \dim S_{i}$ because the map $p_{{\Bbb P}(W_{r})} : S_{i} \longrightarrow {\Bbb P}(W_{r})$ is finite on its image. To compute $\dim S_{i}$ look at the second projection \begin{equation} \label{eq11} p_{{\Bbb P}(W_{i})} : S_{i} \longrightarrow {\Bbb P}(W_{i}). \end{equation} The map (\ref{eq11}) is obviously onto and for any $G \in {\Bbb P}(W_{i})$ we get by Riemann-Roch \[ \dim p_{{\Bbb P}(W_{i})}^{-1}(G) = h^{0}(C,\omega_{C}^{\otimes r}(-G)) - 1 = h^{0}(C,\omega_{C}^{\otimes (r-i)}) - 1 \leq (2(r-i) - 1)(g-1). \] Consequently by the fiber-dimesion theorem we get $\dim S_{i} \leq (2r - 2)(g-1) - 1$ and hence \[ {\rm codim}(p_{{\Bbb P}(W_{r})}(S_{i}),{\Bbb P}(W_{r})) = g-1, \] for any $i = 1, \ldots , r-1$. \begin{flushright} $\Box$ \end{flushright} \subsection{Discriminant loci} \label{ss11} As we saw in Corollary \ref{cor11} the generic spectral curve is smooth. Therefore the space $W$ of spectral curves contains a natural divisor parametrizing the singular spectral curves: \[ {\cal D} := \{ s \in W \, | \, \spec{\rm - is \; not \; smooth} \}, \] which due to Corollary \ref{cor12} is just the discriminant locus for the map $h : \unspec \rightarrow W$. The divisor ${\cal D}$ is irreducible. To see this consider the full family $\widetilde{W}$ of spectral curves and its discriminant divisor $\widetilde{\cal D}$. We have a natural inclusion $W \hookrightarrow \widetilde{W}$ and clearly ${\cal D} = W \cap \widetilde{\cal D}$. The first step will be to show that $\widetilde{\cal D}$ is irreducible and the second to deduce the irreducibility of ${\cal D}$ from that. To achieve this we need to introduce some auxilliary objects. Let ${\Bbb A}{\rm ff} $ be the additive group of the vector space $H^{0}(C,\omega_{C})$. There is a natural affine action of ${\Bbb A}{\rm ff}$ on the bundle $\omega_{C}$ which can be considered as an action on its total space \[ \begin{array}{lccl} \tau : & {\Bbb A}{\rm ff} & \longrightarrow & {\rm Aut}(S^{\circ}) \\ & \gamma & \longrightarrow & (p \stackrel{\tau_{\gamma}}{\mapsto} p + \gamma(\alpha(p))). \end{array} \] The action $\tau$ on $S^{\circ}$ has a natural lift to an action on the bundle $\alpha^{*}\omega_{C} \rightarrow S^{\circ}$ which gives an affine action of ${\Bbb A}{\rm ff}$ on the space of its global sections: $\gamma \rightarrow (\mu \mapsto \mu + \alpha^{*}\gamma)$. Consequently we obtain a polynomial action on the space of spectral curves \[ \begin{array}{lccl} \rho : & {\Bbb A}{\rm ff} & \longrightarrow & {\rm Aut}(\widetilde{W}) \\ & \gamma & \longrightarrow & ( x^{r} + s_{1}x^{r-1} + \ldots + s_{r} \stackrel{\rho_{\gamma}}{\mapsto} (x+\gamma)^{r} + s_{1}(x+\gamma)^{r-1} + \ldots + s_{r}). \end{array} \] For every $\gamma \in {\Bbb A}{\rm ff}$ we have a commutative diagram \begin{equation} \label{eq12} \begin{diagram} \node{\unspec} \arrow{e,t}{\tau_{\gamma}} \arrow{s,r}{h} \node{\unspec} \arrow{s,l}{h} \\ \node{\widetilde{W}} \arrow{e,t}{\rho_{\gamma}} \node{\widetilde{W}} \end{diagram} \end{equation} where $\tau_{\gamma}$ is the automorphism of the universal spectral curve induced by $\tau_{\gamma} \in {\rm Aut}(S^{\circ})$. \begin{prop} \label{prop12} The discriminant divisor $\widetilde{\cal D} \subset \widetilde{W}$ is irrreducible. \end{prop} {\bf Proof.} Consider the incidence correspondence $\widetilde{\Gamma}\subset \widetilde{W}\times S^{\circ}$ defined by \[ \widetilde{\Gamma} := \{ (s,p) \in \widetilde{W}\times S^{\circ} \; | \; {\rm Ord}_{p}\spec \geq 2 \}. \] Since $\widetilde{\cal D} = p_{\widetilde{W}}(\widetilde{\Gamma})$ it suffices to show that $\widetilde{\Gamma}$ is irreducible. The commutativity of the diagram (\ref{eq12}) implies that $\widetilde{\Gamma}$ is $\rho$-invariant. Furthermore, if $s' = \rho_{\gamma}(s)$, then $s_{1}' = s_{1} + r\gamma$ and hence $\rho$ is a free action. Therefore we can form the quotient variety $\widetilde{\Gamma}/{\Bbb A}{\rm ff}$. Since the fibration \[ \widetilde{\Gamma} \longrightarrow \widetilde{\Gamma}/{\Bbb A}{\rm ff} \] is a locally trivial affine bundle, the irreducibility of $\widetilde{\Gamma}$ is equivalent to the irreducibility of $\widetilde{\Gamma}/{\Bbb A}{\rm ff}$. The fibers of the projection \[ p_{C} := \alpha\circ p_{S^{\circ}} : \widetilde{\Gamma} \longrightarrow C \] are $\rho$-invariant and by taking the quotient we obtain a morphism \[ p_{C} : \widetilde{\Gamma}/{\Bbb A}{\rm ff} \longrightarrow C.\] Let $\Xi \subset \widetilde{W}\times C$ be the incidence correspondence \[ \Xi = \{ (s,a) \in \widetilde{W}\times C \; | \; \spec \; {\rm is \; singular \; at \; the \; point} \; \alpha^{-1}(a) \cap X_{0} \}. \] More explicitly by using the Jacobian criterion for smoothnes one gets, see \cite{bnr} Remark 3.5, \[ \Xi = \{ (s,a) \in \widetilde{W}\times C \; | \; {\rm Div}(s_{r}) \geq 2a, \; {\rm Div}(s_{r-1}) \geq a \}.\] The fact that $\omega_{C}^{\otimes r}$ is very ample for any $r\geq 2$, implies that the fibration $\Xi \rightarrow C$ is a vector bundle of rank $\dim\widetilde{W} - 3$. On the other hand we have a natural morphism of fibrations \[ \begin{diagram} \node{\Xi} \arrow[2]{e,t}{t} \arrow{se} \node[2]{\widetilde{\Gamma}/{\Bbb A}{\rm ff}} \arrow{sw,r}{p_{C}} \\ \node[2]{C} \end{diagram} \] where $t((s,a)) = {\rm Orb}_{{\Bbb A}{\rm ff}}((s,\alpha^{-1}(a)\cap X_{0}))$. It is easy to see that $t$ is onto. Indeed, if $(s,p) \in \widetilde{\Gamma}$, then ${\rm Ord}_{p}(\spec) \geq 2$. Choose $\gamma \in H^{0}(C,\omega_{C})$ with the property $\gamma(\alpha(p)) = p$. Then \[(\rho_{-\gamma},\tau_{-\gamma})\cdot (s,p) = (\rho_{-\gamma}(s), \alpha^{-1}(\alpha(p))\cap X_{0})), \] and hence \[ {\rm Orb}_{{\Bbb A}{\rm ff}}((s,p)) = {\rm Orb}_{{\Bbb A}{\rm ff}}((\rho_{-\gamma}(s), \alpha^{-1}(\alpha(p))\cap X_{0}))). \] Consequently $t$ is onto and $\widetilde{\Gamma}/{\Bbb A}{\rm ff}$ is irreducible. \begin{flushright} $\Box$ \end{flushright} \begin{rem} \label{rem12} {\rm The variety $\widetilde{\Gamma}/{\Bbb A}{\rm ff}$ (and therefore $\widetilde{\Gamma}$) is actually smooth. Indeed, let $\Xi_{a}$ be the fiber of the bundle $\Xi$ over the point $a \in C$. Let $(s,a), (s',a) \in \Xi_{a}$ be such that $t((s,a)) = t((s',a))$. Then \[ {\rm Orb}_{{\Bbb A}{\rm ff}} ((s,\alpha^{-1}(a)\cap X_{0})) = {\rm Orb}_{{\Bbb A}{\rm ff}} ((s',\alpha^{-1}(a)\cap X_{0})),\] and therefore $s' = \rho_{\gamma}(s)$ for some $\gamma \in H^{0}(C,\omega_{C})$ satisfying $\gamma(a) = 0$. Let $\Xi_{0} \subset \Xi$ be the subbundle \[ \Xi_{0} = \{ (s,a) \in \Xi \; | \; {\rm Div}(s_{1}) \geq a \}.\] The morphism $t$ descends to an isomorphism of fibrations \[ \begin{diagram} \node{\Xi /\Xi_{0}} \arrow[2]{e,t}{t} \arrow{se} \node[2]{\widetilde{\Gamma}/{\Bbb A}{\rm ff}} \arrow{sw,r}{p_{C}} \\ \node[2]{C} \end{diagram} \] and thus $\widetilde{\Gamma}/{\Bbb A}{\rm ff} \cong \Xi /\Xi_{0}$ is smooth. Furthermore, observe that $p_{\widetilde{W}} : \widetilde{\Gamma} \rightarrow \widetilde{\cal D}$ is birational because for the generic $s \in \widetilde{\cal D}$ the curve $\spec$ has a unique ordinary double point. Therefore $\widetilde{\Gamma}$ can be viewed as a natural desingularization of $\widetilde{\cal D}$. } \end{rem} \begin{cor} \label{cor15} The discriminant divisor ${\cal D} \subset W$ is irreducible. \end{cor} {\bf Proof.} Consider the incidence correspondence \[ \Gamma = \{ (s,p) \; | \; {\rm Ord}_{p}(\spec ) \geq 2 \}. \] Let $q_{|\Gamma} :\Gamma \rightarrow \widetilde{\Gamma}/{\Bbb A}{\rm ff}$ be the restriction to $\Gamma$ of the natural quotient morphism $q : \widetilde{\Gamma} \rightarrow \widetilde{\Gamma}/{\Bbb A}{\rm ff}$. If $(s,p) \in \widetilde{\Gamma}$, then $(\rho_{-\frac{1}{r}\cdot s_{1}}(s), \tau_{-\frac{1}{r}\cdot s_{1}}(s)) \in \Gamma$. Combined with the fact that $\rho$ acts freely, this yields \[ \Gamma \cap {\rm Orb}_{{\Bbb A}{\rm ff}} ((s,p)) = \{ (\rho_{-\frac{1}{r}\cdot s_{1}}(s), \tau_{-\frac{1}{r}\cdot s_{1}}(s)) \}. \] Consequently $q_{|\Gamma}$ is an isomorhism and in particular $\Gamma$ and ${\cal D}$ are irreducible. \begin{flushright} $\Box$ \end{flushright} There are two other discriminant loci whose irreducibility will be usefull. Define $W_{r} := H^{0}(C,\omega_{C}^{\otimes r})$ and $W_{r-1,r} := H^{0}(C,\omega_{C}^{\otimes (r-1)})\oplus H^{0}(C,\omega_{C}^{\otimes r})$. Let ${\cal D}_{r} = {\cal D}\cap W_{r}$ and ${\cal D}_{r-1,r} ={\cal D}\cap W_{r-1,r}$ be the discriminant divisors for the families of spectral curves $W_{r}$ and $W_{r-1,r}$ respectively. \begin{prop} \label{prop13} The discriminant divisor ${\cal D}_{r} \subset W_{r}$ is irreducible. \end{prop} {\bf Proof.} Consider the incidence correspondence $\Gamma_{r} \subset W_{r}\times C$ defined by \[ \Gamma_{r} = \{ (s_{r},a) \; |\; {\rm Div}(s_{r}) \geq 2a \}.\] Using the Jacobian criterion it is easy to check that a curve $\spec$, $s \in W_{r}$ is singular if and only if $s_{r}$ has a double zero. Therefore ${\cal D}_{r} = p_{W_{r}}(\Gamma_{r})$ and it suffices to show that $\Gamma_{r}$ is irreducible. The divisor $\Gamma_{r}$ is equivariant with respect to the natural ${\Bbb C}^{\times}$ action on $W_{r}$ (extended trivially to $W_{r}\times C$). Thus the problem reduces to showing that ${\Bbb P}(\Gamma_{r}) \subset {\Bbb P}(W_{r})\times C$ is irreducible. Consider the projection $p_{C} : {\Bbb P}(\Gamma_{r}) \rightarrow C$. By Riemann-Roch $p_{C}$ is onto and ${\rm codim}(p_{C}^{-1}(a),{\Bbb P}(\Gamma_{r})) \linebreak = 2$ for every $a \in C$. Moreover, $p_{C}^{-1}(a) \subset {\Bbb P}(W_{r})$ is a linear subspace and hence all the fibers of $p_{C}$ are equidimensional and irreducible. Therefore ${\Bbb P}(\Gamma_{r})$ is irreducible. \begin{flushright} $\Box$ \end{flushright} \begin{rem} \label{rem13} {\rm The divisor ${\Bbb P}({\cal D}_{r})$ is a divisor in the projective space of the complete linear system ${\Bbb P}(H^{0}(C,\omega_{C}^{\otimes r}))$. This allows us to give more geometric description of ${\Bbb P}({\cal D}_{r})$: \[ {\Bbb P}({\cal D}_{r}) = {\Bbb P}(\{ H \subset {\Bbb P}(W_{r}^{\vee}){\rm - hyperplane}\; | \; H \supset {\Bbb P}(T_{a}f_{|W_{r}|}(C)) \; {\rm for \; some \;} a \}), \] that is, ${\Bbb P}({\cal D}_{r}) = f_{|W_{r}|}(C)^{\vee}$ is the dual hypersurface of the $r$-canonical model of the curve $C$.} \end{rem} \begin{prop} \label{prop14} The divisor ${\cal D}_{r-1,r} \subset W_{r-1,r}$ is irreducible. \end{prop} {\bf Proof.} Consider the incidence variety \[ \Gamma_{r-1,r} = \{ (a,b,p) \in W_{r-1,r}\times S^{\circ} \; | \; {\rm Ord}_{p}(x^{r} + ax + b) \geq 2 \}. \] Again $p_{W_{r-1,r}}(\Gamma_{r-1,r}) = {\cal D}_{r-1,r}$ so it suffices to show that $\Gamma_{r-1,r}$ is irreducible. Consider the projection \[ p_{S^{\circ}} : \Gamma_{r-1,r} \longrightarrow S^{\circ}. \] Let $(U,z)$ be a coordinate chart on $C$. Then we can choose natural coordinates $(w,z) : \alpha^{-1}(U) \widetilde{\rightarrow} {\Bbb C}\times U$ on $\alpha^{-1}(U)$, so that the tautological section $x$ is $x = w\alpha^{*}(dz)$. Let $(z_{0},w_{0}) \in \alpha^{-1}(U)$. Then for any $(a,b) \in W_{r-1,r}$ one has in a neighborhood of $(z_{0},w_{0})$: \[ \begin{array}{lclcl} a & = & (a_{0} + a_{1}(z-z_{0}) + \ldots)dz^{\otimes (r-1)} & = & f(z)dz^{\otimes (r-1)} \\ b & = & (b_{0} + b_{1}(z-z_{0}) + \ldots)dz^{\otimes r} & = & g(z)dz^{\otimes r}. \end{array} \] Therefore in the local coordinates $(z,w)$ the equation of the curve $x^{r} + ax + b$ becomes \begin{equation} \label{eq13} w^{r} + f(z)w + g(z) = 0. \end{equation} The conditions for (\ref{eq13}) to have singularity at $(z_{0},w_{0})$ are \[ \begin{array}{rcl} b_{0} & = & 0 \\ (rw^{r-1} + f(z))_{|(z_{0},w_{0})} & = & 0 \\ (f'(z)w + g'(z))_{|(z_{0},w_{0})} & = & 0 \end{array} \] or equivalently \begin{equation} \label{eq14} \begin{array}{rcl} b_{0} & = & 0 \\ rw^{r-1}_{0} +a_{0} & = & 0 \\ a_{1}w_{0} + b_{1} & = & 0. \end{array} \end{equation} The equations (\ref{eq14}) determine a codimension 3 affine subspace of $W_{r-1,r}$ and hence ${\Gamma_{r-1,r}}_{|\alpha^{-1}(U)} \longrightarrow \alpha^{-1}(U)$ is a trivial affine bundle. Therefore $\Gamma_{r-1,r}$ is an affine bundle over $S^{\circ}$ and hence is irreducible. \begin{flushright} $\Box$ \end{flushright} \begin{cor} \label{cor16} Let $h : \unspec \rightarrow W$ be the universal spectral curve and let $h_{r-1,r} : \unspec_{|W_{r-1,r}} \rightarrow W_{r-1,r}$ and $h_{r} : \unspec_{|W_{r}} \rightarrow W_{r}$ be the subfamilies parametrized by $W_{r-1,r}$ and $W_{r}$ respectively. Then the divisors $h^{-1}({\cal D})$, $h_{r-1,r}^{-1}({\cal D}_{r-1,r})$ and $h^{-1}_{r}({\cal D}_{r})$ are irreducible. \end{cor} {\bf Proof.} Recall the following standard lemma, see \cite{sh}, \begin{lem} \label{lem12} Let $f : X \rightarrow Y$ be a proper map between algebraic varieties of pure dimension. If $Y$ is irreducible, the general fiber of $f$ is irreducible and all the fibers are equidimensional then $X$ is irreducible. \end{lem} As we saw above each of the discriminant loci ${\cal D}_{r-1,r}$, ${\cal D}_{r}$ and ${\cal D}$ is irreducible. According to Lemma \ref{lem12} it suffices to show that the generic fiber of each of the maps $h$, $h_{r-1}$ and $h_{r-1,r}$ is irreducible. But, by Corollary \ref{cor14}, there exists a point $s \in {\cal D}_{r} \subset {\cal D}_{r-1,r} \subset {\cal D}$ such that $\spec$ is irreducible which finishes the proof. \begin{flushright} $\Box$ \end{flushright} \subsection{The Hitchin map} \label{ss13} For a line bundle $L_{0} \in {\rm Pic}^{d}(C)$ denote by $\su$ the moduli space of semistable vector bundles of rank $r$ and determinant $L_{0}$. It is well known that $\su$ is a normal projective variety whose smooth locus coincides with the locus $\sustable$ of stable vector bundles, see \cite{se}. Since $C$ is a curve, the deformations of any vector bundle $E \rightarrow C$ are unobstructed and the Kodaira-Spencer map \begin{equation} \label{eq15} T_{[E]}\sustable \longrightarrow H^{1}(C,{\rm End}^{o}E), \end{equation} is an isomorphism; here ${\rm End}^{o}E$ is the bundle of traceless endomorphisms of $E$. Using the isomorphism (\ref{eq15}) and Serre's duality, one gets a canonical identification \[ T_{[E]}^{*}\sustable \cong H^{0}(C,{\rm End}^{o}E\otimes\omega_{C}), \] of the fiber of the cotangent bundle to $\sustable$ at the point $[E]$ and the space of $\omega_{C}$-twisted traceless endomorphisms. Therefore the collection of characteristic coefficients defined in Section \ref{ss11} gives rise to a morphism \begin{equation} \label{eq16} \begin{array}{lccl} H \; : \; & T^{*}\sustable & \longrightarrow & W \\ & (E,\theta) & \longrightarrow & (s_{2}(\theta), s_{3}(\theta),\ldots , s_{r}(\theta)). \end{array} \end{equation} Hitchin \cite{h1} was the first one to study the morphism (\ref{eq16}) in various situations. He discovered many of its remarkable properties and in particular he showed that $H$ endows the (holomorphic) symplectic manifold $\cotangent := T^{*}\sustable$ with a structure of an algebraically completely integrable hamiltonian system. More specifically, he showed that there exists a partial compactification \begin{equation} \label{eq17} \begin{diagram} \node{\cotangent} \arrow[2]{e} \arrow{se,r}{H} \node[2]{\higgs} \arrow{sw,r}{H} \\ \node[2]{W} \end{diagram} \end{equation} with general fiber an abelian variety and such that the fibers of (\ref{eq16}) embed as Zariski open sets in the fibers of $H : \higgs \rightarrow W$. The variety $\higgs$ is again a moduli space which parametrizes the equivalence classes of semistable Higgs pairs - that is, pairs $(E,\theta)$ of a bundle and an $\omega_{C}$-twisted endomorphism such that for every $\theta$-invariant subbundle $U \subset E$ the usual inequality $\mu(U) \leq \mu(E)$ for the slopes of $U$ and $E$ holds. The characteristic coefficients map (\ref{eq16}) is a well defined morphism on the whole variety $\higgs$ and is called the Hitchin map of $\higgs$. There is a natural ${\Bbb C}^{\times}$-action on the moduli space $\higgs$ \[ (E,\theta) \longrightarrow (E,t\theta), \] which induces via the Hitchin map a weighted ${\Bbb C}^{\times}$-action on the vector space $W$. A number $t \in {\Bbb C}^{\times}$ acts on $W$ by multiplying the piece $H^{0}(C,\omega_{C}^{\otimes i})$ with $t^{i}$. The compactification diagram (\ref{eq17}) and its generalizations have been studied extensively in the last years, see \cite{bnr}, \cite{h1},\cite{h2}, \cite{ni}, \cite{si}. For our purposes, the most convenient is the approach in \cite{bnr} which we proceed to describe. \begin{prop}[Beauville-Narasimhan-Ramanan]\label{propbnr} Assume that $\pi_{s} : \spec \rightarrow C$ is an integral spectral curve. Then the push-forward map $\pi_{s*}$ induces a canonical bijection between \begin{itemize} \item Isomorphism classes of rank one torsion-free sheaves $F$ on $\spec$ of Euler characteristic $d - r(g-1)$ satisfying $\det (\pi_{s*}F) = L_{0}$. \item Isomorphism classes of Higgs pairs $(E,\theta)$ satisfying $\det E = L_{0}$, ${\rm tr}\theta = 0$, $H(\theta) = s$. \end{itemize} \end{prop} \begin{rem} \label{rem14} {\rm Let $s \in W$ be such that $\spec$ is integral. Then every Higgs pair satisfying $H(\theta) = s$ is stable. Indeed, if we assume that there exists a proper $\theta$-invariant subbundle $U \subset E$, then the spectral curve of $(U,\theta_{|U})$ will be contained in $\spec$ which contadicts the integrality of $\spec$. Consequently $E$ does not have $\theta$-invariant subbundles and in particular $(E,\theta)$ is stable. Under the bijection in Proposition \ref{propbnr} this translates into the well known fact that the moduli functor of rank one torsion free sheaves on an integral curve possesing only planar singularities is representable by an irreducible fine moduli space - the compactified Jacobian of the curve, see \cite{aik}. Therefore the push-forward map $\pi_{s*}$ induces an isomorphism \[ \pi_{s*} : {\rm Prym}_{\tilde{d}}(\spec, C) \longrightarrow H^{-1}(s) , \] where ${\rm Prym}_{\tilde{d}}(\spec, C) = \{ F \in \overline{J}_{d - r(g-1)}(\spec) \; | \; \det (\pi_{s*}F) = L_{0} \}$, and $\overline{J}_{d - r(g-1)}(\spec)$ is the generalized Jacobian parametrizing rank one torsion free sheaves of Euler characteristic $d - r(g-1)$ on $\spec$.} \end{rem} \bigskip \noindent \begin{rem} \label{rem15} {\rm If the curve $\spec$ is smooth, then the Prymian ${\rm Prym}_{\tilde{d}}(\spec, C)$ is in a natural way torsor over an abelian variety. Indeed, it is easy to see that \[ \det (\pi_{s*}F) = {\rm Nm}_{\pi_{s*}}(F)\otimes \det (\pi_{s*}{\cal O}_{\spec}),\] for any line bundle $F$ on $\spec$. Since $\pi_{s*}{\cal O}_{\spec} = {\cal O}_{C}\oplus \omega_{C}^{-1} \oplus \ldots \oplus \omega_{C}^{-(r-1)}$, see \cite{bnr}, we get \[ {\rm Prym}_{\tilde{d}}(\spec, C) = \{ F \in J^{\tilde{d}}(\spec) \; | \; {\rm Nm}_{\pi_{s*}}(F) = L_{0}\otimes \omega_{C}^{\otimes \frac{r(r-1)}{2}} \},\] with $\tilde{d} = d + r(r-1)(g-1)$.} \end{rem} \bigskip \noindent \begin{rem} \label{rem16} {\rm We will need more explicit geometric description for the fiber $H^{-1}(s)$ of the Hitchin map over a generic point $s \in {\cal D}$ of the discriminant. Observe first that there exists a point $s \in {\cal D}$ for which $\spec$ is irreducible and has a unique ordinary double point as singularity. Indeed, let $s_{r} \in H^{0}(C,\omega_{C}^{\otimes r})$ be a section having a unique double zero $a \in C$. It follows then by the Jacobian criterion for smoothness that the curve \[ \spec : X^{r} + s_{r}Y^{r} = 0, \] has a unique node lying over $a$. Consequently, by upper semicontinuity there is a non-empty Zariski open set ${\cal D}^{o} \subset {\cal D}$ satisfying \[ {\cal D}^{o} = \{s \in {\cal D} \; | \; \spec \; {\rm is \; irreducible \; and \; has \; a \; unique \; ordinary \; double \; point} \}. \] Let now $s \in {\cal D}^{o}$ and say $p \in \spec$ is its ordinary double point. Set $a = \pi_{s}(p)$. To study the geometric properties of ${\rm Prym}_{\tilde{d}}(\spec, C)$, we recall the construction of the compactified Jacobian $\overline{J}_{d - r(g-1)}(\spec)$ in this simple case. Let \[ \begin{diagram} \node{\spec^{\nu}} \arrow{e,t}{\nu} \arrow{se,r}{\pi_{s}^{\nu}} \node{\spec} \arrow{s,r}{\pi_{s}} \\ \node[2]{C} \end{diagram} \] be the normalization of the curve $\spec$. Let $\nu^{-1}(p) = \{ p^{+}, p^{-} \} \subset \spec^{\nu}$. For any rank one torsion free sheaf $F \rightarrow \spec$ we have the exact sequence \begin{equation} \label{eq18} 0 \longrightarrow F \longrightarrow \nu_{*}\nu^{*}F \longrightarrow {\Bbb C}_{p} \longrightarrow 0. \end{equation} Therefore $\chi(F) + 1 = \chi(\nu_{*}\nu^{*}F) = \chi(\nu^{*}F)$ and hence the pull-back induces a morphism \begin{equation} \label{eq19} \nu^{*} : \overline{J}_{d - r(g-1)}(\spec) \longrightarrow J_{d + 1 - r(g-1)}(\spec^{\nu}), \end{equation} where $J_{d + 1 - r(g-1)}(\spec^{\nu})$ is the component of the Picard group of $\spec^{\nu}$ consisting of line bundles of Euler characteristic $d + 1 - r(g -1)$. One can show that (\ref{eq19}) is a bundle with structure group ${\Bbb C}^{\times}$ and with fibers isomorphic to a ${\Bbb P}^{1}$ glued at two points. To identify the bundle (\ref{eq19}) explicitly, consider a Poincare bundle \[ \begin{diagram}[J] \node{{\cal P}} \arrow{s} \\ \node{J_{d + 1 - r(g-1)}(\spec^{\nu})\times\spec^{\nu}} \end{diagram} \] and set ${\cal P}_{+} = {\cal P}_{|J_{d + 1 - r(g-1)}(\spec^{\nu})\times \{p^{+}\} }$ and ${\cal P}_{-} = {\cal P}_{|J_{d + 1 - r(g-1)}(\spec^{\nu})\times \{p^{-}\} }$. The ${\Bbb P}^1$-bundle \[ {\Bbb P}({\cal P}_{+}\oplus {\cal P}_{-}) \longrightarrow J_{d + 1 - r(g-1)}(\spec^{\nu}) \] does not depend on the choice of the Poincare bundle ${\cal P}$ and is furnished with two sections $X_{+}$ and $X_{-}$ corresponding to ${\cal P}_{+}$ and ${\cal P}_{-}$ respectively. Furthermore, there is a bundle isomorphism \[ \begin{diagram} \node{\overline{J}_{d - r(g-1)}(\spec)} \arrow[2]{e,t}{\cong} \arrow{se,r}{\nu^{*}} \node[2]{{\Bbb P}({\cal P}_{+}\oplus {\cal P}_{-})/X_{+}\sim X_{-}} \arrow{sw} \\ \node[2]{J_{d + 1 - r(g-1)}(\spec^{\nu})} \end{diagram} \] To describe the subvariety ${\rm Prym}_{\tilde{d}}(\spec, C) \subset \overline{J}_{d +1 - r(g-1)}(\spec)$ in these terms, we need the following lemma \begin{lem} \label{lem13} If $F_{1}, F_{2} \in \overline{J}_{d + 1- r(g-1)}(\spec)$ are such that \[ \det (\pi_{s*}F_{1}) = \det (\pi_{s*}F_{2}), \] then \[ \det (\pi_{s*}^{\nu} \nu^{*}F_{1}) = \det (\pi_{s*}^{\nu} \nu^{*}F_{2}). \] \end{lem} {\bf Proof.} For any $F \in \overline{J}_{d + 1- r(g-1)}(\spec)$ we have the short exact sequence (\ref{eq18}). After taking direct images, we get \[ {\divide\dgARROWLENGTH by 4 \begin{diagram} \node{0} \arrow[2]{e} \node[2]{\pi_{s*}F} \arrow[2]{e} \node[2]{\pi_{s*}\nu_{*}\nu^{*}F} \arrow[2]{e} \arrow{s,=,-} \node[2]{\pi_{s*}{\Bbb C}_{p}} \arrow[2]{e} \arrow{s,=,-} \node[2]{R^{1}\pi_{s*}F} \arrow{s,=,-} \\ \node[5]{\pi_{s*}^{\nu} \nu^{*}F} \node[2]{{\Bbb C}_{a}} \node[2]{0} \end{diagram}} \] i.e. we have a short exact sequence of sheaves on $C$: \[ 0 \longrightarrow \pi_{s*}F \longrightarrow \pi_{s*}^{\nu} \nu^{*}F \longrightarrow {\Bbb C}_{a} \longrightarrow 0.\] Therefore the bundle $\pi_{s*}F$ is a Hecke transform of the bundle $\pi_{s*}^{\nu} \nu^{*}F$ with center at an $(r-1)$-dimensional subspace of the fiber $(\pi_{s*}^{\nu} \nu^{*}F)_{a}$. Thus \[ \det(\pi_{s*}F) = \det(\pi_{s*}^{\nu} \nu^{*}F)\otimes {\cal O}_{C}(-x), \] which yields the statement of the lemma. \begin{flushright} $\Box$ \end{flushright} According to the above lemma, if a point $F \in \overline{J}_{d + 1- r(g-1)}(\spec)$ belongs to the subvariety ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$, then every point in the fiber $(\nu^{*})^{-1}(\nu^{*}(F))$ also belongs to this subvariety. Define ${\rm Prym}(\spec^{\nu},C) := \{ F \in J_{d + 1 - r(g-1)}(\spec^{\nu}) \; | \; \det (\pi_{s*}^{\nu}F) = L_{0}(-a) \}$. Then ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$ fits in the fiber square \[ \begin{diagram} \node{{\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)} \arrow{e} \arrow{s,l}{\nu^{*}} \node{\overline{J}_{d + 1- r(g-1)}(\spec)} \arrow{s,r}{\nu^{*}} \\ \node{{\rm Prym}(\spec^{\nu},C)} \arrow{e} \node{J_{d + 1 - r(g-1)}(\spec^{\nu})} \end{diagram} \] In particular, ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$ is a bundle over the abelian variety ${\rm Prym}(\spec^{\nu},C)$ whose total space is singular along a divisor and has a smooth normalization which is a ${\Bbb P}^{1}$-bundle over ${\rm Prym}(\spec^{\nu},C)$.} \end{rem} \bigskip The total space $\cotangent$ of the cotangent bundle of $\sustable$ is a Zariski open set in the moduli space of Higgs bundles $\higgs$. It has the advantage that we have morphisms both to $\sustable$ and $W$: \[ \begin{diagram} \node[2]{\cotangent} \arrow{sw,l}{\Pi} \arrow{se,l}{H} \\ \node{\sustable} \node[2]{W} \end{diagram} \] whose fibers are generically transversal. The natural projection $\Pi : \cotangent \rightarrow \sustable$ is onto by definition and the Hitchin map $H$ is dominant by an argument of Beauville-Narasimhan-Ramanan, see \cite{bnr}. In particular, this implies that for the generic element $s \in W$ the push-forward map $\pi_{s*} : H^{-1}(s)\cap \cotangent \rightarrow \sustable$ is dominant. In specific geometric situations it is important to know what is the codimension of the locus of those $s \in W$ for which $\pi_{s*}$ is not dominant. A general result to this extend can be deduced easily from the following Lemma \ref{lem14} communicated to us by E. Markman. First we introduce some notation. \begin{defi} A vector bundle $E \in \urd$ is called very stable if it does not have non-zero nilpotent $\omega_{C}$-twisted endomorphism. \end{defi} By a result of Drinfeld and Laumon, see \cite{la}, \cite{bnr}, very stable vector bundles always exist. Let $U \subset \sustable$ be the non-empty Zariski open set consisting of very stable bundles. For an $E \in \cotangent$ denote by ${\cal X}_{E} := T^{*}_{[E]}\sustable = H^{0}(C,{\rm End}^{o}E\otimes \omega_{C}) \subset \cotangent$ the fiber of the cotangent bundle at $[E]$. \begin{lem} \label{lem14} For any $E \in U$ the Hitchin map \begin{equation} \label{eq111} H_{E} = H_{|{\cal X}_{E}} : {\cal X}_{E} \longrightarrow W \end{equation} is surjective. \end{lem} {\bf Proof.} Since $E$ is very stable we have $H_{E}^{-1}(0) = 0$ and hence the dimension of the generic fiber of $H_{E}$ is zero. On the other hand, $\dim {\cal X}_{E} = \dim W$ and therefore $H_{E} : {\cal X}_{E} \rightarrow W$ is dominant. Denote by $W^{o} \subset W$ the Zariski open subset \[ W^{o} = \{ s \in W \; | \; \dim H_{E}^{-1}(s) = 0 \}. \] Clearly the subvariety ${\cal X}_{E}$ is preserved by the ${\Bbb C}^{\times}$ action on $\higgs$. Furthermore the ${\Bbb C}^{\times}$-equivariance of $H$ implies that $W^{o}$ is a ${\Bbb C}^{\times}$-invariant subset of $W$. Since $H_{E}^{-1}(0) = \{ 0 \}$, we have that \begin{equation}\label{eq110} H_{E} : {\cal X}_{E}\setminus \{ 0 \} \longrightarrow W\setminus \{ 0 \} \end{equation} is a ${\Bbb C}^{\times}$-equivariant morphism and thus descends to a morphism \[ \overline{H}_{E} : {\Bbb P}({\cal X}_{E}) \longrightarrow {\Bbb P}_{{\rm weight}} , \] where ${\Bbb P}_{{\rm weight}} = ( W\setminus \{ 0 \})/{\Bbb C}^{\times}$ is the corresponding weighted projective space. The morphism $\overline{H}_{E}$ is dominant because its range contains the Zariski open set $( W^{o}\setminus \{ 0 \})/{\Bbb C}^{\times} \subset {\Bbb P}_{{\rm weight}}$. But ${\Bbb P}({\cal X}_{E})$ is a projective variety and therefore $\overline{H}_{E}({\Bbb P}({\cal X}_{E}))$ is projective and thus $\overline{H}_{E}$ is onto. We obtain a commutative diagram of ${\Bbb C}^{\times}$-bundles over ${\Bbb P}({\cal X}_{E})$ and ${\Bbb P}_{{\rm weight}}$ respectively \[ \begin{diagram} \node{{\cal X}_{E}\setminus \{ 0 \}} \arrow{e,t}{H_{E}} \arrow{s} \node{W\setminus \{ 0 \}} \arrow{s} \\ \node{{\Bbb P}({\cal X}_{E})} \arrow{e,t}{\overline{H}_{E}} \node{{\Bbb P}_{{\rm weight}}} \end{diagram} \] Since the ${\Bbb C}^{\times}$ on the fibers of ${\cal X}_{E}\setminus \{ 0 \} \rightarrow {\Bbb P}({\cal X}_{E})$ and $W\setminus \{ 0 \} \rightarrow {\Bbb P}_{{\rm weight}}$ is simply transitive, we obtain that the map (\ref{eq110}) (and consequently the map (\ref{eq111})) is surjective. \begin{flushright} $\Box$ \end{flushright} \begin{cor} \label{cor17} For any $s \in W$ the rational map \[ \begin{diagram}[\pi_{s*} = \Pi_{|H^{-1}(s)} : \; \; H^{-1}(s)] \node{\pi_{s*} = \Pi_{|H^{-1}(s)} : \; \; H^{-1}(s)} \arrow{e,..} \node{\su} \end{diagram} \] is dominant. \end{cor} {\bf Proof.} By Lemma \ref{lem14} the range $\Pi(H^{-1}(s))$ contains the nonempty Zariski open set $U$. \begin{flushright} $\Box$ \end{flushright} \section{The abelianization of an automorphism} \label{s2} \subsection{The map $\varphi_{W}$} \label{ss21} Start with an automorphism $\Phi$ of $\su$. We would like to abelianize it, i.e. to lift $\Phi$ to the moduli space of Higgs bundles and induce from it an automorphism of the family of spectral Pryms. The hope is that in this way we will get a simpler picture because of the fact that the automorphisms of the abelian varieties are pretty well understood. As we will see in the subsequent sections, this hope turns out to be unsubstantiated in that simple form since the spectral Pryms are not generic abelian varieties and can have non-trivial group automorphisms. This is the reason why we have to work with the whole family of Pryms rather then the individual Prym varieties and to make essential use of its ``rigidity'' as a group scheme over $W$. To construct the abelianized version of $\Phi$, observe first that the codifferential of $\Phi$ provides a natural lift of $\Phi_{|\sustable}$ to the total space of the cotangent bundle \[ d\Phi^{*} : \cotangent \longrightarrow \cotangent.\] The automorphism $d\Phi^{*}$ commutes with the projection $\Pi : \cotangent \rightarrow \sustable$ by deffinition. The behaviour of $d\Phi^{*}$ with respect to the Hitchin map is described by the following proposition. \begin{prop} \label{prop21} There exists an automorphism $\varphi_{W} \in {\rm Aut}(W)$ making the following diagram commutative \begin{equation} \label{eq21} \begin{diagram} \node{\cotangent} \arrow{e,t}{d\Phi^{*}} \arrow{s,l}{H} \node{\cotangent} \arrow{s,r}{H} \\ \node{W} \arrow{e,b}{\varphi_{W}} \node{W} \end{diagram} \end{equation} \end{prop} {\bf Proof.} Let $s \in W\setminus {\cal D}$. Then the spectral curve $\spec$ is smooth and (see Remark \ref{rem14}) $H^{-1}(s) = {\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$. Consider the fiber \[ H^{-1}_{\cal X}(s) := H^{-1}(s)\cap \cotangent = \{ F \in {\rm Prym}_{\tilde{d}}(\tilde{C}_s , C) \; | \; \alpha_{*}F {\rm - stable} \}. \] The argument in Remark 5.2 of \cite{bnr} gives ${\rm codim}(H^{-1}_{\cal X}(s), H^{-1}(s)) \geq 2$. Therefore since ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C) = H^{-1}(s)$ is smooth, the regular map $H\circ d\Phi^{*} : H^{-1}_{\cal X}(s) \rightarrow W$ extends by Hartogs theorem to a morphism $\overline{H\circ d\Phi^{*}} : H^{-1}(s) \rightarrow W$. But ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$ is projective and $W$ is an affine space; hence \[ \overline{H\circ d\Phi^{*}}({\rm Prym}_{\tilde{d}}(\tilde{C}_s , C) ) = {\rm point \; in \;} W. \] Denote this point by $\varphi_{W}(s)$. In this way we obtain a rational map $\varphi_{W}$ on $W$, with possible singularities along ${\cal D}$ which makes the diagram (\ref{eq21}) commutative. This implies that $\varphi_{W}$ is ${\Bbb C}^{\times}$-equivariant since $d\Phi^{*}$ is linear on the fibers of the cotangent bundle and $H$ is ${\Bbb C}^{\times}$-equivariant. Thus $\varphi_{W}$ descends to a rational map \[ \begin{diagram}[\overline{\varphi}_{W} : \; \; {\Bbb P}_{{\rm weight}}] \node{\overline{\varphi}_{W} : \; \; {\Bbb P}_{{\rm weight}}} \arrow{e,..} \node{{\Bbb P}_{{\rm weight}}} \end{diagram}. \] But ${\Bbb P}_{{\rm weight}}$ is a normal projective variety and therefore $\overline{\varphi}_{W}$ is not defined in a locus of codimension $\geq 2$. Finally the ${\Bbb C}^{\times}$-equivariance of $\varphi_{W}$ together with the fact that ${\Bbb C}^{\times}$ acts transitively on the fibers of $W\setminus \{ 0 \} \rightarrow {\Bbb P}_{{\rm weight}}$ yield that $\varphi_{W}$ itself is not defined on a locus of codimension $\geq 2$. Since $\varphi_{W}$ is a map from a vector space to a vector space, it extends by Hartogs to the whole $W$. \begin{flushright} $\Box$ \end{flushright} Since we want to reduce the study of the automorphism $\Phi$ to the study of a suitable fiberwise automorphism of the family of spectral Pryms, the best situation for us would have been if we could say that $\varphi_{W} = {\rm id}$. Unfortunately, at this stage the only information we can extract from the above proposition is that $\varphi_{W}$ commutes with the ${\Bbb C}^{\times}$ action on $W$. In the next lemma we use this to determine some further properties of $\varphi_{W}$. \begin{lem} \label{lem21} The map $\varphi_{W}$ preserves the subspace $W_{r-1,r} \subset W$ and moreover $\phibase := {\varphi_{W}}_{| W_{r-1,r} }$ is of the form $\phibase = (\phi_{r-1},\phi_{r})$ where \[ \phi_{r-1} : H^{0}(C,\omega_{C}^{\otimes (r-1)}) \longrightarrow H^{0}(C,\omega_{C}^{\otimes (r-1)}) \] and \[ \phi_{r} : H^{0}(C,\omega_{C}^{\otimes r}) \longrightarrow H^{0}(C,\omega_{C}^{\otimes r}) \] are linear maps. \end{lem} {\bf Proof.} Consider the algebra of regular functions $S := {\Bbb C}[W]$ on $W$. Decomposing $S$ into isotypical components with respect to the ${\Bbb C}^{\times}$ action, we get a new grading on it: \[ S = \otimes_{n \geq 0} S_{n}, \] where $S_{n}$ consists of all functions on $W$ on which a number $t \in {\Bbb C}^{\times}$ acts by a multiplication by $t^{n}$. More explicitly, if we choose coordinates $x_{i\scriptsize{1}}, \ldots , x_{in_{i}}$ in $W_{i} = H^{0}(C,\omega_{C}^{\otimes i})$, then the elements in $S_{n}$ are just polynomials in $x_{ij}$'s with admissible multidegrees $\{ a_{ij}\}$ satisfying \[ \sum_{i,j} i\cdot a_{ij} = n .\] Any automorphism of $W$ commuting with the ${\Bbb C}^{\times}$ action will preserve this grading and in particular will preserve the subspaces of the form $W_{i}\oplus W_{i+1}\oplus \ldots \oplus W_{r}$ for any $i$. In paticular, since $\varphi_{W}$ commutes with the ${\Bbb C}^{\times}$ action we have \[ \phibase : W_{r-1,r} \longrightarrow W_{r-1,r}.\] Moreover, since ${\rm g.c.d.}(r-1,r) = 1$, it follows that $\phibase$ has the required form. \begin{flushright} $\Box$ \end{flushright} \subsection{Extension to the family of Pryms} \label{ss22} As we have seen in Section \ref{ss13} the fiber of the Hitchin map $H : \higgs \longrightarrow W$ over every point $s \in W^{\rm reg} := W\setminus {\cal D}$ is isomorphic to an abelian variety. Moreover, one can show, see \cite{ni}, that ${\cal D}$ is exactly the locus of critical values of $H$. Define $\unprymd$ to be the preimage of $W^{\rm reg}$ under the Hitchin map. Then $\unprymd$ is a smooth variety, see \cite{ni}, and we have a smooth fibration $H : \unprymd \rightarrow W^{\rm reg}$. Our next goal is to extend the automorhism $d\Phi^{*}$ to an automorphism of the family $\unprymd$. The first result in this direction is the following proposition \begin{prop} \label{prop22} The map $\varphi_{W}$ preserves the discriminant divisor ${\cal D} \subset W$. \end{prop} {\bf Proof.} Let $s \in W$. Corollary \ref{cor17} guarantees that ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C) \cap \cotangent \neq \varnothing$ and hence, due to Proposition \ref{prop21}, $d\Phi^{*}$ induces a birational automorphism between $H^{-1}(s)$ and $H^{-1}(\varphi_{W}(s))$. Let $s \in {\cal D}^{o}$. If we assume that $\varphi_{W}(s) \not\in {\cal D}$, then $\widetilde{C}_{\varphi_{W}(s)}$ is smooth and ${\rm Prym}(\widetilde{C}_{\varphi_{W}(s)},C)$ is isomorphic to an abelian variety. Therefore ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$ is birational to an abelian variety. On the other hand ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C)$ is birationally isomorphic to a ${\Bbb P}^{1}$ bundle over an abelian variety, see Remark \ref{rem16}, which is a contradiction because such a bundle has Kodaira dimension $- \infty$. Consequently $\varphi_{W}({\cal D}^{o}) \subset {\cal D}$. But $\varphi_{W}$ is an automorphism and hence is a closed map which yields $\varphi_{W}({\cal D}) = {\cal D}$ since the discriminant is irreducible, see Corollary \ref{cor15}. \begin{flushright} $\Box$ \end{flushright} \begin{cor} \label{cor21} Let $\phi_{r}$ be the map defined in Lemma \ref{lem21}. Then there exists a number $\lambda \in {\Bbb C}^{\times}$ and an automorphism $\sigma$ of the curve $C$ such that \[ \phi_{r} = m_{\lambda}\circ \sigma^{*} , \] where $m_{\lambda}$ is the dilation by $\lambda$. \end{cor} {\bf Proof.} By Lemma \ref{lem21} the map $\phi_{r}$ is a linear map. On the other hand, Proposition \ref{prop22} implies in particular that the map $\phi_{r}$ preserves the discriminant divisor ${\cal D}_{r}$. Therefore the induced map $\overline{\phi}_{r}$ on the projective space ${\Bbb P}(W)$ preserves the divisor ${\Bbb P}({\cal D}_{r})$. But, by Remark \ref{rem14}, the divisor ${\Bbb P}({\cal D}_{r})$ is the dual variety of the $r$-canonical model of $C$ and therefore the dual map $\overline{\phi}_{r}^{\vee}$ preserves the curve $f_{{\Bbb P}(W_{r})}(C) \subset {\Bbb P}(W_{r})^{\vee}$. Let $\sigma$ be the induced automorphism of $C$. Then, since $f_{{\Bbb P}(W_{r})}(C)$ spans ${\Bbb P}(W_{r})^{\vee}$ and $\overline{\phi}_{r}^{\vee}$ is linear, it follows that $\overline{\phi}_{r} = \sigma^{*}$. \begin{flushright} $\Box$ \end{flushright} \begin{rem} \label{rem21} {\rm If the curve $C$ does not have automorphisms, then the above corollary implies that $\phi_{r}$ is just a dilation.} \end{rem} The proposition above yields the commutative diagram \[ \begin{diagram} \node{\unprymd} \arrow{e,t,..}{d\Phi^{*}} \arrow{s,l}{H} \node{\unprymd} \arrow{s,r}{H} \\ \node{W^{\rm reg}} \arrow{e,b}{\varphi_{W}} \node{W^{\rm reg}} \end{diagram} \] Furthermore, as we saw in the proof of Proposition \ref{prop22} the map $d\Phi^{*}$ gives a birational isomorphism between $H^{-1}(s)$ and $H^{-1}(\varphi_{W}(s))$ for any $s$. In particular, in the case $s \in W^{\rm reg}$ we get a birational automorphism between abelian varieties and hence $d\Phi^{*}_{|H^{-1}(s)}$ extends to a biregular isomorphism between $H^{-1}(s)$ and $H^{-1}(\varphi_{W}(s))$. Therefore $d\Phi^{*}$ extends as a continuous automorphism $\tilde{\Phi} _{\tilde {d}} $ of the whole variety $\unprymd$ and since $\unprymd$ is smooth, we get: \begin{prop} \label{prop23} There exists a biregular automorphism $\tilde{\Phi} _{\tilde {d}} $ extending $d\Phi^{*}$ which fits in the commutative diagram \[ \begin{diagram} \node{\unprymd} \arrow{e,t}{\tilde{\Phi} _{\tilde {d}} } \arrow{s,l}{H} \node{\unprymd} \arrow{s,r}{H} \\ \node{W^{\rm reg}} \arrow{e,b}{\varphi_{W}} \node{W^{\rm reg}} \end{diagram} \] \end{prop} \section{The fibration $\unspec \longrightarrow \wbase$ } \label{section3 } \subsection{The Picard group of $\unspec $ } \label{picunspec} We start with some notation: For a vector space $V$, the points of the Grassmanian $Gr(k,V)$ correspond to codimension $k$ linear subspaces of $V$. Also, for a linear space of sections $W$ on $S$ and a given point $p$ on $S$, $W_p$ denotes the linear subspace of sections of $W$ vanishing at $p$. We now make the assumption that $r \geq 3$. We are going first to prove our main Theorem \ref{theor1} in the case $r \geq 3$ and next, in Section \ref{rank2}, we outline the modifications needed for the proof of the rank $2$ case. For simplicity denote the space $W_{r-1,r} $ by $B$ and the spaces $\overline{W}_{r-1,r}$ and $\wbase = W_{r-1,r} \setminus {\cal D}$ by $\overline{B}$ and $B^{reg } $ respectively. In the following we are going to use repeatedly Proposition 6.5 in \cite{ha}, which compares the Picard group of a variety $X$ with that of a Zariski open subset $U$ of $X$. \\ \\ {\bf Notation.} From now on we will denote by $\unspec $ the part of the universal spectral curve sitting over $B^{\rm reg}$, unless otherwise stated. \\ \\ We have the diagram: \vskip.1in \[ \begin{diagram} \node{ \unspec } \arrow{se,l}{\pi } \arrow{s,l}{\beta} \arrow{e,t}{h} \node{B^{\rm reg} } \\ \node{S^{\circ } } \arrow{e,t}{\alpha } \node{C} \end{diagram} \] \vskip.1in \begin{prop} The Picard group of the variety $\unspec $ is isomorphic to the pull back of the Picard group of the base curve $C$ by the map $\pi $, i.e. ${\rm Pic} \, \unspec \simeq \pi ^* {\rm Pic} \, C $. \label{proppicunspec} \end{prop} \noindent {\bf Proof.} The variety $\unspec $ can be constructed as follows. Consider the diagram: \vskip.1in \begin{equation} \begin{diagram}[aaaaaaaa] \node{f^{*}_{| \overline B |} U^{\rm reg}|_{S^{\circ}} \simeq \unspec } \arrow{e,t}{ } \arrow{s,l} {\beta } \node{ U^{\rm reg} \subset U } \arrow{e} \arrow{s,r} { } \node{Gr(1,\overline{B}) \times \overline {B}} \arrow{sw,l}{p_1} \arrow{s,r}{p_2} \\ \node{\;\;\;\;\;\;\; S^{\circ} \subset S } \arrow{e,b}{ f_{| \overline {B} |}} \node{Gr(1,\overline{B}) } \node{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \overline{B} \supseteq B \supseteq {\cal D} } \end{diagram} \label{diagunspec} \end{equation} \vskip.1in \noindent The map $ f_{| \overline {B} |}$ is the map associated to the base point free linear system $| \overline {B} |$ i.e. it sends a point $p$ to the point $[\overline {B}_p] $ representing the linear subspace $\overline {B}_p$. The bundle $U$ is the universal bundle over the Grassmanian. The fiber of the bundle $U \cap p_2^{-1}(B) $ over a point $[H]$ in the Grassmanian is isomorphic to $H \cap B$. The variety $U \cap p_2^{-1}(B) $ is an affine bundle over $ f_{| \overline {B} |}(S^{\circ} )$ with fiber over the point $f_{| \overline {B} |}(p)$ isomorphic to $B _p$. Define $U^{\rm reg}\stackrel{\rm df}{=} U \cap p_2^{-1}(B \setminus {\cal D} ) =U \cap p_2^{-1} (B ^{\rm reg})$. Then $ \unspec \simeq f^{*}_{| \overline B |} U^{\rm reg}|_{S^{\circ}} $. Note that by Corollary \ref{cor16}, the divisor $ f_{| \overline {B} |}^* ( p_2 ^{-1} (\Delta ) \cap U)$ is irreducible. Also it is linearly equivalent to zero since ${\rm Pic} \, B $ is trivial. \noindent To complete the proof of the proposition, we have: \[ \begin{array}{rlll} {\rm Pic} \, \unspec & \simeq & {\rm Pic} \, f^{*}_{|\overline{B }|} U^{\rm reg}|_{S^{\circ} } & \\ & \simeq & {\rm Pic} \, f^{*}_{|\overline {B }|} ( U \cap p_2^{-1}(B \setminus {\cal D} ) ) |_{S^{\circ}} & \\ & \simeq & {\rm Pic} \, f^{*}_{|\overline {B }|} ( U \cap p_2^{-1}(B ) ) |_{S^{\circ} } & \mbox{since}\;\; U \cap p_2^{-1} {\cal D} \;\; \mbox {is an irreducible divisor lin. equiv. to} \; \; 0, \\ & \simeq & \beta ^* {\rm Pic} \, S^{\circ} & \mbox {since} \;\; f^{*}_{|\overline {B }|} ( U \cap p_2^{-1}(B ) ) |_{S^{\circ} }\;\; \mbox{is an affine bundle over} \;\; S^{\circ} , \\ & \simeq & \pi ^* {\rm Pic} \, C & \mbox{since} \;\; S^{\circ} \;\; \mbox{is an affine bundle over} \;\; C . \end{array} \] \begin{flushright} $\Box$ \end{flushright} \subsection{The Picard group of $\unspec _p $} \label{picunspecp} For a fixed point $p$ in $S^{\circ }$ we denote by $\unspec _p$ the subvariety of $\unspec $ consisting of those curves whose image in $S^{\circ}$ passes through $p$. $\unspec _p$ sits over the subspace $B^{\rm reg} _p $ of $B^{\rm reg} $, consisting of those sections vanishing at $p$. Let $h: \unspec _p \longrightarrow B_p^{\rm reg} $ denote again the restriction of the map $h$ on $\unspec _p$. We are going to calculate the Picard group of $ \unspec _p $. Observe that the above fibration $h$ has a section corresponding to the point $p$. Take the restriction of the map $\beta $ to $\unspec _p$ i.e. $\beta: \unspec _p \longrightarrow S^{\circ}$. Let $H_{p}$ be the preimage of $p$ via the map $\beta $. By a construction similar to that for the variety $\unspec $ in the above Section \ref{picunspec}, one can see that the fibration $\beta: \unspec _p \setminus H_p \longrightarrow S^{\circ} \setminus \{ p \} $ is the complement of an irreducible divisor linearly equivalent to zero in an affine fibration. We thus have ${\rm Pic} \, (\unspec _p\setminus H_p) \simeq \beta ^* {\rm Pic} \, ( S^{\circ} \setminus \{ p \}) \simeq \beta ^* {\rm Pic} \, S^{\circ} \simeq \pi ^* {\rm Pic} \, C$. Since $H_p$ is an irreducible divisor in $\unspec _p$, we conclude that ${\rm Pic} \, \unspec _ p $ is generated by $\pi ^* {\rm Pic} \, C$ and $H_p$. We proceed by showing that we actually have: \begin{prop} $ {\rm Pic} \, \unspec _p \simeq \pi ^* {\rm Pic} \, C \oplus {\Bbb Z}[H_p] $. \label{proppicunspecp} \end{prop} \noindent {\bf Proof.} Pick a generic pencil ${\Bbb P} ^1 $ in $\overline {B}_p$ such that all the fibers of the restriction of the compactified universal spectral curve over the pencil are irreducible, except the one over the infinity point $b$ which has exactly two irreducible components, see Corollary \ref{cor14}. We may also assume that none of the singular points of the fibers lies on the $N= 2r^2(g-1)$ base points of the system. The restriction of the compactified universal spectral curve over ${\Bbb P} ^1 $, is a smooth surface $X$ which is the blow up of the surface $S$ over the $N$ base points $p_1=p, \ldots ,p_N$ of the pencil. We have the following picture: \vspace{4in} \noindent In the above picture, $b$ is the point at infinity, $b_1, \ldots ,b_k$ correspond to the singular fibers and $E_1, \ldots ,E_N$ are the exceptional divisors. The curve over $b$ consist of two components $\tilde {Y}_{\infty }$ and $\tilde{C}_b^{'}$, one of which, $\tilde {Y}_{\infty }$, is the proper transform of the divisor at infinity on $S$. The intersection of $X$ with the variety $\unspec _p $ is $X^{\circ } = X \setminus h^{-1}(b,b_1, \ldots ,b_k)$. We define $E_i^{\circ} = E_i \cap X^{\circ}$. Note that $E_1^{\circ} = H_p \cap X$. To prove the claim, it is enough to show that on $X^{\circ }$ the line bundle $E^{\circ} $ is independent from $\pi ^* {\rm Pic} \, C$. We have \begin{equation} {\rm Pic} \, X \simeq \beta ^* {\rm Pic} \, S \oplus {\Bbb Z}[E_i]_{i=1, \ldots , N} \simeq \pi ^* {\rm Pic} \, C \oplus {\Bbb Z}[\tilde {Y}_{\infty }] \oplus_{i=1} ^N (\oplus {\Bbb Z}[E_i]). \label{eqpicunspecp1} \end{equation} Now $\beta ^* (rY_{\infty} + r \alpha ^* \omega _C ) = h ^{-1}(\mbox{point}) + \sum _{i=1}^NE_i $, where the equality stands for the linear equivalence of divisors. Therefore, $r \tilde {Y}_{\infty } + r \pi ^* \omega _C = h ^{-1}(\mbox{point}) + \sum _{i=1}^NE_i = \tilde {C} ^{'}_b + \tilde {Y}_{\infty }+ \sum _{i=1}^NE_i$. In other words, \begin{equation} \tilde {C} ^{'}_b= (r-1) \tilde{Y}_{\infty} + r \pi ^* \omega _C - \sum _{i=1}^NE_i . \label{eqpicunspecp2} \end{equation} All the fibers of $h$ define linear equivalent divisors on $X$ and the restriction on $X \setminus h ^{-1} (b)$ of the line bundles corresponding to the divisors $\tilde {C} ^{'}_b$ and $\tilde {Y}_{\infty }$ are trivial. We thus get \[ \begin{array}{rlll} {\rm Pic} \, X ^{\circ } & \simeq & {\rm Pic} \, X \setminus h ^{-1}(b) \simeq {\rm Pic} \, X \left/ (\tilde {Y}_{\infty } =0, \tilde {C} ^{'}_b =0 ) \right. & \\ & \simeq & \pi ^* {\rm Pic} \, C \oplus_{i=1} ^N (\oplus {\Bbb Z}[E^{\circ }_i]) \left/ (r\pi ^* \omega _C = \sum _{i=1}^N E^{\circ} _i ) \right. & \mbox{by} \;\; (\ref{eqpicunspecp1}) \;\; \mbox{and} \;\; (\ref{eqpicunspecp2}). \end{array} \] Assume now that $\pi ^* L + m H_p = 0 $ on $\unspec _p$. Then $\pi ^* L + m E^{\circ }_1 = 0$ on $X^{\circ}$. By the description of the ${\rm Pic} \, X ^{\circ } $ we conclude that $\pi ^* L =0$ and $m=0$ and this completes the proof of the proposition. \begin{flushright} $\Box$ \end{flushright} \subsection{ The sections of $ H: \unjacd \longrightarrow \wbase$ } \label{sections} \begin{prop} The only sections of the map $ H: \unjacd \longrightarrow \wbase = B^{\rm reg}$ are those coming from a pull back of a fixed line bundle on $C$. In other words, if $\sigma : B^{\rm reg} \longrightarrow \unjacd $ is a section of $H$, then $\sigma (s) = [\pi _s ^* M] $ where $M$ is a fixed line bundle on $C$. In particular, if $r$ does not divide $\tilde {d}$, then the map $H$ has no sections. \label{propsections} \end{prop} \noindent We start with two lemmas: \begin{lem} On $\unjacd \times _{B^{\rm reg}} \unspec $ there exists a line bundle ${\cal P}^{\tilde d} $ such that ${\cal P}^{\tilde d} |_{[L] \times \spec } \simeq L^{\otimes n}$ for some integer $n$. \label{lempoincare} \end{lem} {\bf Proof.} See \cite{mr}. \begin{flushright} $\Box$ \end{flushright} \begin{rem} { \rm One can actually prove that the minimum such positive integer $n$ is equal to ${\rm g.c.d.(r,d)}$. } \label{remsections1} \end{rem} \begin{lem} If $\sigma $ is a section of the map $H$ whose image lies in the locus $\pi ^* {\rm Pic} \, C$ i.e. $\sigma (s)=[\pi ^* M_s] $, where $M_s$ is a line bundle on $C$, then $M_s=M$ for all $s \in B^{\rm reg} $. \label{lemsectionimage} \end{lem} \noindent {\bf Proof.} Consider the map $\gamma: \unspec \longrightarrow \unjacd \times _{B^{\rm reg} } \unspec $ which sends a point $q$ sitting over $s \in B^{\rm reg} $ to $( \sigma (s), q )$. Then $ [\gamma ^* {\cal P}^{\tilde d} |_{[L]} ] = n \sigma (s)$. By the description of ${\rm Pic} \, \unspec $, see Proposition \ref{proppicunspec}, we get that $\gamma ^* {\cal P}^{\tilde d} \simeq \pi ^* M_1$ for a fixed $M_1$ in ${\rm Pic} \, C$. Hence, $n [\pi ^* M_s] = n \sigma (s) = [\pi ^* M_1]$ for all $s\in B^{\rm reg} $. Since the map $\pi ^* $ is one to one, see Remark 3.10 in \cite{bnr}, we conclude that the map $B^{\rm reg} \longrightarrow {\rm Pic} \, C$ that sends $s$ to $[M_s]$ has finite image. Since $B^{\rm reg} $ is connected, the map is constant. \begin{flushright} $\Box$ \end{flushright} \bigskip \noindent {\bf Proof of Proposition \ref{propsections}.} Say that $\sigma $ is not coming from a pull back. By the above Lemma \ref{lemsectionimage}, we may assume that there exists an $s_0 \in B^{\rm reg} $ such that $\sigma (s_0) $ is not of the form $\pi ^* A$ for some $A$ in ${\rm Pic} \, C$. Take a point $p$ on $S^{\circ} $ that lies on $\tilde {C}_{s_0} \subseteq S^{\circ} $. The family of curves $\unspec _p $ has a section and therefore on $J^{\tilde {d}} (\unspec _p) \times _{B^{\rm reg} _p} \unspec _p $ there exists a Poincare bundle ${\cal P}^{\tilde d} _p $, see \cite{mr}. We have \[ [\gamma ^* {\cal P}^{\tilde d} |_{\spec }] =n \sigma (s) \;\; \mbox{and}\;\; [\gamma ^* {\cal P}^{\tilde d} _p|_{\spec }] = \sigma (s) \;\; \mbox{for all} \;\; s \in B^{\rm reg} _p. \] Therefore, $n \gamma ^* {\cal P}^{\tilde d} _p|_{\spec } \simeq \gamma ^* {\cal P}^{\tilde d} |_{\spec }$ for all $s\in B^{\rm reg} _p$. Since ${\rm Pic} \, B^{\rm reg} _p $ is trivial, we conclude by the see-saw principle, see \cite{mu}, that \begin{equation} n \gamma ^* {\cal P}^{\tilde d} _p \simeq \gamma ^* {\cal P}^{\tilde d}|_{\unspec _p } \;\; \mbox{on}\;\; \unspec _p. \label{eqpicunspecp3} \end{equation} On the other hand, by the description of the Picard groups of $\unspec _p $ and $\unspec $, see Propositions \ref{proppicunspec} and \ref{proppicunspecp}, we have \begin{equation} \gamma ^* {\cal P}^{\tilde d} _p \simeq \pi ^* M + m H_p \;\; \mbox{and} \;\; \gamma ^* {\cal P}^{\tilde d} \simeq \pi ^* L. \label{eqpicunspecp4} \end{equation} Hence, by (\ref{eqpicunspecp3}) and (\ref{eqpicunspecp4}), we have on $\unspec _p $ that $n \pi ^* M + n m H_p \simeq \pi ^* L $. Proposition \ref{proppicunspecp} implies that $m=0$ i.e. $\gamma ^* {\cal P}^{\tilde d} _p \simeq \pi ^* M $. But then, $\sigma (s_0) = [\gamma ^* {\cal P}^{\tilde d} _p |_{\tilde {C}_{s_0}}] = [ \pi ^* M]$ which contradicts the assumption on $\sigma (s_0) $. \begin{flushright} $\Box$ \end{flushright} \section{The ring of correspondences} \label{section4} \subsection{The map $ \beta ^{'} : \fibprod \longrightarrow S^{\circ} \times S^{\circ} $ } \label{soso} Throughout the Sections \ref{section4} and \ref{section5}, we will denote by $\phi $ the map $\phi _{r-1,r} : B \longrightarrow B$. According to Lemma \ref{lem21}, we have that $\phi _{r-1,r}$ has the form $\phi _{r-1,r}=(\phi _{r-1}, \phi _r)$ where $\phi _i : H^0(C, \omega _C^i) \longrightarrow H^0(C, \omega _C^i)$ are linear automorphisms for $i=r-1, r$. Furthermore, by Proposition \ref{prop22}, we have that $\phi ({\cal D}) = {\cal D}$. Therefore, the restriction of the map $\phi $ on $B^{\rm reg} $, which we will denote again by $\phi $, induces an automorphism $\phi : B^{\rm reg} \longrightarrow B^{\rm reg} $. In this section we study the fiber product $\fibprod $ defined by the diagram \vskip.1in \[ {\divide\dgARROWLENGTH by 8 \begin{diagram} \node{S^{\circ} \times S^{\circ}} \node[3]{\fibprod }\arrow[3]{w,t}{\beta ^{'}} \arrow[2]{e,t}{ } \arrow[2]{s,l}{ } \node[2]{\unspec } \arrow{s,r} {h }\\ \node{ } \node[3]{ } \node[2]{ B^{\rm reg} } \arrow{s,r}{\phi} \\ \node{ } \node[3]{\unspec} \arrow[2]{e,t}{h} \node[2]{B^{\rm reg}} \end{diagram}} \] \vskip.1in \noindent We define $\overline {\phi } : \overline{B} \longrightarrow \overline{B}$ to be the map which extends $\phi$ to $\overline{B}$ as $ \overline {\phi }= (1,\phi _{r-1}, \phi _r) $. To investigate the map $\beta ^{'}$ , we define in the product $S \times S$ the following loci $A$ and $\Gamma $. \begin{defi} {\rm \[ \begin{array}{rll} A & \stackrel{\rm df }{=} & S^{\circ} \times S^{\circ} \setminus {\rm Im}\beta^{'} \\ \Gamma & \stackrel{\rm df }{=} & \{ (p,q) \in S \times S \;\; \mbox{such that} \;\; \overline{B}_p =\overline {\phi } (\overline{B}_q) \} \end{array} \] } \end{defi} \noindent On $S \times S \setminus (A \cup \Gamma) $ we can define a map $f$ to the Grassmanian $Gr(2,\overline{B})$ of codimension $2$ linear subspaces of $\overline{B}$, by sending the point $(p,q)$ to the class of the plane $\overline{B}_p \cap \overline {\phi }(\overline{B}_q) $. We have the following diagram: \vskip.1in \begin{equation} \begin{diagram}[aaaaaaaaaaaaaaaaaaaaa] \node{\;\;\;\;\;\; \fibprod |_{S^{\circ} \times S^{\circ} \setminus \Gamma} \simeq f^{*} U^{\rm reg}|_{S^{\circ} \times S^{\circ} \setminus \Gamma } } \arrow{e,t}{ } \arrow{s,l} {\beta ^{'} } \node{ U^{\rm reg} \subset U } \arrow{e} \arrow{s,r} { } \node{Gr(2,\overline{B}) \times \overline {B}} \arrow{sw,l}{p_1} \arrow{s,r}{p_2} \\ \node{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; S^{\circ} \times S^{\circ} \setminus (A \cup \Gamma) \subseteq S \times S \setminus (A \cup \Gamma) } \arrow{e,t}{ f} \node{Gr(2,\overline{B}) } \node{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \overline{B} \supseteq B \supseteq {\cal D} } \end{diagram} \label{diagfibprod2} \end{equation} \vskip.1in \noindent The notation is in complete analogy with that of the Diagram (\ref{diagunspec}). As in the case of $\unspec $, the variety $ \fibprod |_{S^{\circ} \times S^{\circ} \setminus (A \cup \Gamma)} $ is an affine bundle over the image of $S^{\circ} \times S^{\circ} \setminus (A \cup \Gamma) $ under the map $f$. \begin{lem} ${\rm dim}A \leq 2$ \label{lemdimA} \end{lem} \noindent {\bf Proof.} Let $B^{\infty }$ denote the space of sections $W_{r-1,r}^{\infty}$, i.e. the linear subspace of $\overline{W}$ consisting of points of the form $(0,0, \ldots , 0, s_{r-1}, s_r)$. We have \[ \begin{array}{ll} A & = \{ (p,q) \in S^{\circ} \times S^{\circ} \;\; \mbox{such that} \;\; U|_{f(p,q)} \cap p_2^{-1} B =\varnothing \} \\ & = \{ (p,q) \in S^{\circ} \times S^{\circ} \;\; \mbox{such that} \;\; B_p \cap \phi (B _q) = \varnothing \} \\ & = \{ (p,q) \in S^{\circ} \times S^{\circ} \;\; \mbox{such that} \;\; B^{\infty}_p = \overline {\phi} (B^{\infty} _q) \} \end{array} \] \noindent By Corollary \ref{cor13}, the linear system defined by $B^{\infty } $ separates points on $S^{\circ} $. Therefore, given a point $p \in S^{\circ} $ there exists at most one $q \in S^{\circ} $ with $(p,q) \in A$ and this proves the lemma. \begin{flushright} $\Box$ \end{flushright} \bigskip We give now a better description of the locus $\Gamma $. Consider the diagram \vskip.1in \[ \begin{diagram} \node{S } \arrow{e,t}{^{f_{| \overline{B}| }}} \arrow{se,r}{\overline{\phi} ^{\, *} \, f_{| \overline{B}| } } \node{{\Bbb P}( \overline{B}^{\, \vee}) } \arrow{s,r} {\overline {\phi} ^{\, *} }\\ \node { }\node { {\Bbb P}( \overline{B}^{\, \vee}) }\end{diagram} \] \vskip.1in \noindent In the above diagram the map $\overline{\phi} ^{\, *} $ is the induced automorphism of ${\Bbb P}( \overline{B}^{\, \vee})$ by the linear automorphism $\overline{\phi} $. We denote by $\pi _1 $ and $\pi _2 $ the two projections of $\Gamma $ to the surface $S$. It is easy to see that \[ \begin{array}{rll} & \pi _1 (\Gamma) & = \{ p \in S \;\; \mbox{such that} \;\; f_{| \overline{B}| } (p) \in \overline{\phi} ^{\, *} \, f_{| \overline{B}| } (S) \} \\ & & = f_{| \overline{B}| }^{-1} \left( f_{| \overline{B}| }(S) \cap \overline{\phi} ^{\, *} \, f_{| \overline{B}| } (S) \right) \\ \mbox{and similar} \;\;\;\;\;\;\;\;\;\; \\ & \pi _2(\Gamma ) & = (\overline{\phi} ^{\, *} \, f_{| \overline{B}| })^{-1} \left( f_{| \overline{B}| }(S) \cap \overline{\phi} ^{\, *} \, f_{| \overline{B}| } (S) \right) \end{array} \] We describe the fibers of the map $\pi _1$. If $p \in Y_{\infty }$, then it is easy to see that $\pi _1 ^{-1} (p)$ consists of the whole divisor $ Y_{\infty} $. If $p \in S^{\circ} $, then $\pi _1 ^{-1} (p)$ consists of at most one point since the system $\overline{B}$ separates points on $S^{\circ} $. By the above discussion we conclude that \begin{lem} ${\rm dim} \left( \Gamma \cap (S^{\circ} \times S^{\circ} ) \right) \leq 2$. \label{lemdimB} \end{lem} \subsection{The Picard group of $\fibprod $ } \label{picfibprod} We are going to use the following diagram \vskip.1in \[ \begin{diagram} \node{\fibprod } \arrow{se,l}{\pi ^{'}} \arrow{s,l}{\beta {'}} \\ \node{S^{\circ} \times S^{\circ} } \arrow{e,t} {\alpha ^{'} } \node{C \times C} \end{diagram} \] \vskip.1in \noindent to calculate the Picard group of $\fibprod $. \begin{lem} ${\rm Pic} \, ( \fibprod |_{S^{\circ} \times S^{\circ} \setminus \Gamma} ) \simeq \pi ^{' *} {\rm Pic} \, (C \times C )$. \label{lempicfibprod1} \end{lem} \noindent {\bf Proof.} We have $\fibprod |_{S^{\circ} \times S^{\circ} \setminus \Gamma} \simeq f^{*} U^{\rm reg}|_{S^{\circ} \times S^{\circ} \setminus \Gamma } \simeq f^{*} U^{\rm reg}|_{S^{\circ} \times S^{\circ} \setminus (\Gamma \cup A )} $ since the fiber over $A$ is empty. Therefore, \noindent $ {\rm Pic} \, ( \fibprod |_{S^{\circ} \times S^{\circ} \setminus \Gamma} ) \simeq $ \nopagebreak \vskip-.1in \[ \begin{array}{lll} \;\;\; \simeq & {\rm Pic} \, f^{*} U^{\rm reg}|_{S^{\circ} \times S^{\circ} \setminus (\Gamma \cup A )} & \\ \;\;\; \simeq & {\rm Pic} \, f^* (U \cap p_2^{-1}(B))|_{S^{\circ} \times S^{\circ} \setminus (\Gamma \cup A)} & \mbox{since the preimage of} \;\, {\cal D} \;\, \mbox{is irred. divisor lin. equiv. to}\;\; 0, \\ \;\;\; \simeq & \beta ^{' *} {\rm Pic} \, ( S^{\circ} \times S^{\circ} \setminus (\Gamma \cup A)) & \mbox{since it is an affine bundle over} \;\; S^{\circ} \times S^{\circ} \setminus (\Gamma \cup A), \\ \;\;\; \simeq & \beta ^{' *} {\rm Pic} \, ( S^{\circ} \times S^{\circ} ) & \mbox {since} \;\; {\rm codim}(\Gamma \cup A) \geq 2 , \\ \;\;\; \simeq & {\rm Pic} \, (C \times C ) & \mbox{since} \;\; S^{\circ} \times S^{\circ} \;\; \mbox{is a rank two affine bundle over} \;\; C \times C. \end{array} \] \begin{flushright} $\Box$ \end{flushright} \bigskip The variety $\fibprod $ splits as $$\fibprod = \fibprod |_{S^{\circ} \times S^{\circ} \setminus \Gamma} \, \amalg \, \fibprod |_{\Gamma}. $$ \noindent To calculate its Picard group, we have to consider the following two cases.\\ \\ {\bf Case A:} dim$\, \Gamma \cap (S^{\circ} \times S^{\circ} ) \leq 1$: Then one can easily see that dim$\, \fibprod |_{\Gamma} \leq {\rm dim}(\fibprod ) -2$. We thus have: \begin{lem} In case A, the ${\rm Pic} \, ( \fibprod ) \simeq {\rm Pic} \, ( \fibprod|_{S^{\circ} \times S^{\circ} \setminus \Gamma} ) \simeq \pi ^{' *} {\rm Pic} \,(C \times C) $. \label{lemcasea} \end{lem} \noindent {\bf Case B:} dim$\, \Gamma \cap (S^{\circ} \times S^{\circ} ) =2 $: Following the notation we used in the description of $\Gamma $ in the above Section \ref{soso}, we have $\pi _1(\Gamma ) = S$. This implies that $\overline {\phi} ^{\, *} $ induces an automorphism of $f_{|\overline{B}|} (S)$. The following summarizes the basic properties of the induced automorphism $\overline {\phi} ^{\, *} $. \begin{enumerate} \item The point $p_{\infty}=f_{|\overline{B}|}(Y_{\infty})$ remains fixed. \item $\overline {\phi} ^{\, *}$ sends a fiber of the map $\alpha $ to a fiber: otherwise we get a ${\Bbb P} ^1 $ cover of the curve $C$, which contradicts the assumption that genus $g(C) >0$. \item Since $C$ has no automorphisms, $\overline {\phi} ^{\, *}$ sends a fiber of the map $\alpha $ to the same fiber. \end{enumerate} Therefore $\overline {\phi} ^{\, *} $ induces an automorphism $\chi : S^{\circ} \longrightarrow S^{\circ} $ which preserves the fibers of the map $\alpha: S^{\circ} \longrightarrow C$. \begin{lem} Let $s\in B$ and let $\spec $ denote also the image of the corresponding spectral curve on $S^{\circ} $. Then $\chi (\spec ) = \tilde{C}_{\phi ^{-1}(s)}$. \label{lemsurfautom} \end{lem} \noindent {\bf Proof.} Let $H_s$ denote the hyperplane in $ {\Bbb P}( \overline{B}^{\, \vee})$ corresponding to the section $s$. Then $$ \spec \simeq f_{|\overline{B}|}(\spec ) = H_s \cap f_{|\overline{B}|}(S^{\circ} ) $$ and $$ \tilde{C}_{\phi ^{-1}(s)} \simeq f_{|\overline{B}|}(\tilde{C}_{\phi ^{-1}(s)}) = H_{\phi ^{-1}(s)} \cap f_{|\overline{B}|}(S^{\circ} ). $$ It is $\overline{\phi }^{\, *} (H_s) = H_{\phi ^{-1}(s)}$ and so, \[ \begin{array}{llll} f_{|\overline{B}| }(\tilde{C}_{\phi ^{-1}(s)}) & = & \overline{\phi }^{\, *} (H_s) \cap f_{|\overline{B}| }( S^{\circ} ) & \\ & = & \overline{\phi }^{\, *} (H_s) \cap \overline{\phi }^{\, *} f_{|\overline{B}| } ( S^{\circ} ) & \mbox{since} \;\; \overline{\phi }^{\, *} f_{|\overline{B}| } ( S^{\circ} ) = f_{|\overline{B}| } ( S^{\circ} ) \;\; \mbox{in} \;\; {\Bbb P}(\overline{B}^*), \\ & = & \overline{\phi }^{\, *} (H_s \cap f_{|\overline{B}| } ( S^{\circ} ) ) & \mbox{since} \;\; \overline{\phi }^{\, *} \;\; \mbox{is an automorphism,} \\ & = & \overline{\phi }^{\, *} f_{|\overline{B}| } (\spec ). & \end{array} \] \begin{flushright} $\Box$ \end{flushright} \bigskip By the above Lemma \ref{lemsurfautom}, the automorphism $\chi $ on $S^{\circ} $ induces an automorphism $\psi $ of $\unspec $ over $B$ which makes the following diagram commutative \vskip.1in \[ \begin{diagram} \node{\unspec } \arrow{s,l}{h} \arrow{e,t}{\psi } \node{\unspec } \arrow{s,r} {h} \\ \node{B } \arrow{e,t}{^{\phi ^{-1}}} \node{B } \end{diagram} \] \vskip.1in \noindent Since $S^{\circ} $ is the total space of the line bundle $\omega _C $ on $C$, the map $\chi $ acts by a dilation and a translation by a section of $\omega _C$ on the fibers i.e. it has the form $\chi = m_{\lambda } \circ T_s$ where $\lambda \in {\Bbb C}^*$ and $s \in H^0(C, \omega _C) $. We actually claim that $s=0$ and that $\phi ^{-1} $ has the form $\phi ^{-1}(s_{r-1}, s_r)= (\lambda ^{-(r-1)} s_{r-1}, \lambda ^{-r} s_r)$. Indeed, \begin{equation} \chi \circ h^{-1}(s_{r-1}, s_r)= h^{-1}( \phi ^{-1}(s_{r-1}, s_r)). \label{eqpicfibprod1} \end{equation} To prove the first claim, we apply (\ref{eqpicfibprod1}) to $(s_{r-1}, s_r)=(0,0)$. Then, $\chi \circ h^{-1}(0,0)= h^{-1}(0,0)$ since $\phi ^{-1}$ is a linear map. But $h^{-1}(0,0)$ is the curve $x^r=0$, and so, $\chi \circ h^{-1}(0,0)$ is the curve $(\lambda x +s)^r=0$, see beginning of Section \ref{ss11}. This implies that $s=0$. For the second claim: $h^{-1}(s_{r-1}, s_r)$ is the curve $x^r +s_{r-1}x +s_r=0$ and so, $\chi \circ h^{-1}(s_{r-1}, s_r)$ is the curve $\lambda ^r x^r +\lambda s_{r-1}x +s_r=0$ i.e. the curve $x^r +\lambda ^{-(r-1)} s_{r-1}x + \lambda^{-r} s_r=0$ i.e. the curve $h^{-1}(\lambda ^{-(r-1)} s_{r-1}, \lambda^{-r} s_r)$ which completes the proof. \begin{cor} $\phi (s_{r-1}, s_r)=(\lambda ^{r-1} s_{r-1}, \lambda^{r} s_r). $ \label{corformphi} \end{cor} We now proceed with our discussion about the Picard group of $\fibprod $ in the case of dim$\Gamma \cap (S^{\circ} \times S^{\circ} ) =2 $. We have that $\fibprod |_{\Gamma}= \Delta _{\lambda}= \{ (p, \psi ^{-1}(p)), p \in \unspec \}$. Note that if $\lambda=1$, then $\Delta _{\lambda} $ is exactly the diagonal in the fiber product $\tilde {\cal C \;} _{ h} \!\!\times _{h} \tilde {\cal C}$. Combining the latter with Lemma \ref{lempicfibprod1} we get: \begin{prop} In case B, the ${\rm Pic} \, ( \fibprod ) $ is generated by $\pi ^{' *} {\rm Pic} \, (C \times C)$ and $ \Delta _{\lambda}$. \label{proppicfibprodb} \end{prop} \begin{rem} { \rm It is easy to see that ${\rm Pic} \, (\fibprod ) \simeq \pi ^{' *} {\rm Pic} \, (C \times C) \oplus {\Bbb Z}[ \Delta _{\lambda}] $. } \label{rempicfibprod1} \end{rem} \section{Proof of the main Theorem} \label{section5} \subsection{The form of $\tilde{\Phi }$} \label{formphi} We start with a definition. Let $C$ and $C_1$ are two smooth curves. Given a line bundle ${\cal L}$ on the product $C \times C_1 $, then ${\cal L}$ induces a map $$ \psi _{\cal L}: {\rm Pic} \, C \longrightarrow {\rm Pic} \, C_1 $$ \noindent defined by $ \psi _{\cal L} ({\cal O}(p)) = \alpha ^*({\cal L}|_{C_1^p})$, where $C_1^p$ is the fiber in the product over the point $p \in C$ and $\alpha : C_1 \longrightarrow C_1^p $ the natural isomorphism. The extension of the definition to a point $[L] \in {\rm Pic} \, C$ is given by taking a meromorphic section of the bundle $L$. In the same way, whenever we have a line bundle on a fiber product of two families of curves we get a map between their relative Picard groups Consider the diagram \[ \begin{diagram} \node{} \node{\fibprod} \arrow{se,l}{\pi _1} \arrow{s,l}{\pi ^{'}} \arrow{sw,l}{\pi _2} \arrow{e,t}{h^{'}} \node{B^{\rm reg}} \\ \node{C} \node{C \times C} \node {C} \end{diagram} \] \vskip.1in \noindent Let ${\cal L}$ be a line bundle on $C \times C$. Then $\pi ^{' *} {\cal L}$ is a line bundle on the fiber product $\fibprod $. Following the above notation, it is easy to see that \begin{lem} $\psi _{\pi ^{'*} {\cal L}} = \pi _2 ^* \, \psi _{\cal L} \, {\rm Nm}_1 $, where ${\rm Nm}_1$ is the norm map of $\pi _1$. \label{lemblabla1} \end{lem} Note that on the level of fibers we have \[ \begin{array}{cccc} \psi _ {\pi ^{'*} {\cal L}} \,: & {\rm Pic} \, \spec & \longrightarrow & {\rm Pic} \, \tilde{C}_{\phi (s) }\\ & {\cal O}(p) & \longmapsto & \pi ^*_ {\phi (s) } \, \psi _{\cal L} \, {\rm Nm}_s( {\cal O}(p)) \end{array} \] \noindent where the subscripts refer to the restriction on the corresponding fiber over a point of $B^{\rm reg} $. \begin{rem} {\rm We recall the following fact about maps of abelian torsors and induced maps of abelian schemes. Let $p: {\cal G} \longrightarrow Z $ be an abelian scheme and $\pi : {\cal T} \longrightarrow Z$ a ${\cal G}$-torsor. Given two elements $t_1$ and $t_2$ in the same fiber of ${\cal T}$ over $z \in Z$, we denote by $t _1 - t _2 $ the unique element $g \in {\cal G}$ over $z \in Z$, with the property: $T_g t_2 =t_1$. Let now $\phi _{\cal T} : {\cal T} \longrightarrow {\cal T}$ be an automorphism of the abelian torsor ${\cal T}$ i.e. an automorphism that sends a fiber of the map $\pi $ to another fiber and preserves the action of ${\cal G}$. To that, one can associate a {\em group} automorphism $\phi _{\cal G}$ of the abelian scheme ${\cal G}$ as follows. Given $g \in {\cal G}_z$ i.e. an element of ${\cal G}$ sitting over $z \in Z$, choose an element $t_z \in {\cal T}_z$ and define \[ \phi _{\cal G} (g_z)= \phi _{\cal T} (T_{g_z} t_z) - \phi _{\cal T}( t_z) . \] It is easy to check that this is independent from the choice of $t_z$ in the fiber ${\cal T}_z$ and that it defines a group automorphism. Note that given $t_1$ and $t_2$ two elements in the same fiber ${\cal T}_z $, then \[ \phi _{\cal T} (t_1) - \phi _{\cal T}(t_2) = \phi _{\cal G} (t_1 -t_2) . \] \noindent } \label{remtorsor} \end{rem} Following the notation of Proposition \ref{prop23}, let $\tilde{\Phi} : \unprym \longrightarrow \unprym $ be the group automorphism associated to the automorphism $\tilde{\Phi} _{\tilde {d}} : \unprymd \longrightarrow \unprymd $ of $\unprym$-torsors. We determine now the form of the map $\tilde{\Phi} $. Consider the map \[ \begin{array}{cccc} \mu \,: & \fibprod & \longrightarrow & \unprym _{\; H} \!\! \times _{h} \unspec \\ & (p_s, q_{\phi (s)}) & \longmapsto & \left( \tilde{\Phi} (rp_s - \pi^*_s {\rm Nm}_s (p_s)), q_{\phi (s)} \right) \end{array} \] where the notation for a point on a curve, stands also for the line bundle which the point defines. According to Lemma \ref{lempoincare}, on the product $\unprym _{\; H} \!\! \times _{h} \unspec$ we have a line bundle ${\cal P} $ with the property ${\cal P} |_{ [L] \times \tilde{C}_{\phi (s)}} \simeq n \, L $ for some integer $n$. Hence, \begin{equation} \mu ^* {\cal P} |_{ [p_s ] \times \tilde{C}_{\phi (s)}} \simeq n \tilde{\Phi} (rp_s - \pi^*_s {\rm Nm}_s (p_s)). \label{eqformphi1} \end{equation} By using (\ref{eqformphi1}) and the knowledge of the Picard group of $\fibprod$ we will derive the form of the map $\tilde{\Phi} $. \\ \\ {\bf Case A:} dim$\, \Gamma \cap (S^{\circ} \times S^{\circ} ) \leq 2$. Then, by Lemma \ref{lemcasea}, we have that ${\rm Pic} \, (\fibprod) \simeq \pi ^{'*} {\rm Pic} \, (C \times C)$. We thus get \[ \mu ^* {\cal P} \simeq \ \pi ^{'*} {\cal L} \;\; \mbox{and so,} \;\; \mu ^* {\cal P}|_{[p_s ] \times \tilde{C}_{\phi (s)} } \simeq \ \pi ^{'*} {\cal L} |_{[p_s ] \times \tilde{C}_{\phi (s)} } , \] i.e. \begin{equation} \tilde{\Phi} (nrp_s -n \pi^*_s {\rm Nm}_s (p_s) ) \simeq \pi ^*_ {\phi } \psi _{\cal L} {\rm Nm}_s(p_s) . \label{eqformphi2} \end{equation} \noindent Take the map $\tilde{\Phi} _s : {\rm Prym}( \tilde {C}_s, C) \longrightarrow {\rm Prym}(\tilde{C}_{\phi (s)} , C)$ Now given a line bundle $\tilde{L}_s \in {\rm Prym}( \tilde {C}_s, C) $, choose a line bundle $\tilde{M}_s$ in ${\rm Prym}( \tilde {C}_s, C) $ such that $\tilde{L}_s = nr \tilde{M}_s$. By choosing a meromorphic section, we can write $\tilde{M}_s = {\cal O}( \sum _i (p_i^1 -p_i^2) )$ where ${\rm Nm}(\sum _i (p_i^1 -p_i^2))=0$. We have \[ \begin{array}{rlll} \tilde{\Phi} _s({\cal L}_s) & = & \tilde{\Phi} _s (rn {\cal M}_s) = \tilde{\Phi} _s \left( \sum _i rn (p_i^1 -p_i^2) \right)= & \\ & = & \sum _i \left( \tilde{\Phi} _s (rn p_i^1- n\pi _s ^* {\rm Nm}_s (p_i^1)) - \tilde{\Phi} _s (rn p_i^2- n\pi _s ^* {\rm Nm}_s (p_i^2)) \right) \\ & & + \sum _i \tilde{\Phi} _s ( n\pi _s ^* {\rm Nm}_s (p_i^1- p_i^2)) \\ & = & \sum _i \pi ^*_{\phi (s)} \psi _{\cal L} {\rm Nm}_s (p_i^1) - \sum _i \pi ^*_{\phi (s)} \psi _{\cal L} {\rm Nm}_s (p_i^2) & \mbox{by} \;\; (\ref{eqformphi2}), \\ & = & \pi ^*_{\phi (s)} \psi _{\cal L} {\rm Nm}_s (\sum_i (p_i^1 -p_i^2)) & \\ & = & 0 & \mbox{since} \;\; {\rm Nm}(\sum _i (p_i^1 -p_i^2))=0. \end{array} \] \noindent The later contradicts the fact that $\tilde{\Phi} $ is an isomorphism, which means that case A cannot occur. Therefore dim$\Gamma \cap (S^{\circ} \times S^{\circ} ) = 2$, i.e. we are in case B which we examine bellow:\\ \\ {\bf Case B:} dim$\Gamma \cap (S^{\circ} \times S^{\circ} ) = 2$. By the discussion in Section \ref{picfibprod}, the ${\rm Pic} \, (\fibprod) $ is generated by $\pi ^{'*} {\rm Pic} \, (C \times C)$ and the divisor $\Delta _{\lambda}$. Hence, $$ \mu ^* {\cal P} \simeq \pi ^{'*} {\cal L}+ n \Delta _{\lambda}. $$ Working as before and using the definition of $\Delta _{\lambda}$, we conclude that $\tilde{\Phi} ({\cal L}) = n \psi ^*({\cal L}) $. Since $\tilde{\Phi} $ and $\psi ^*$ are isomorphisms, we get that $n=\pm 1$. To summarize, \begin{prop} $\tilde{\Phi} = \pm \psi ^*$, where $\psi $ is the map defined in Section \ref{picfibprod}. \label{propplusminus} \end{prop} \subsection{The conclusion of the proof} \label{conclusionproof} We now conclude the proof of the main Theorem \ref{theor1} in the case of $r \geq 3$ by examining the two cases $\tilde{\Phi} = \pm \psi ^*$. We start with some notation. We denote by $\psi _{\tilde {d}}^* : \unprymd \longrightarrow \unprymd $ the pull back of the map $\psi : \unspec \longrightarrow \unspec $. Note that $\psi ^* = \psi _0^*$. We have the diagram: \vskip.1in \[ \begin{diagram} \node{\unprymd } \arrow{s,r}{H} \arrow{e,t} { ^{ \psi ^*_{ \tilde{d} }} } \node{\unprymd } \arrow{s,r} {H} \\ \node{B^{\rm reg} } \arrow{e,t}{\phi } \node{B^{\rm reg} } \end{diagram} \] \vskip.1in \begin{lem} Let $x$ denote an element in $\unprymd$. We have that \begin{enumerate} \item If $\tilde{\Phi} = \psi ^* $, then $\psi _{\tilde{d}}^{*-1} \tilde{\Phi} _{\tilde {d}} (x)-x $ is independent from $x$ on the fibers of the map $H$. \item If $\tilde{\Phi} = -\psi ^* $, then $\psi _{\tilde{d}}^{*-1}\tilde{\Phi} _{\tilde {d}} (x)+x $ is independent from $x$ on the fibers of the map $H$. \end{enumerate} \label{lempsi} \end{lem} \noindent {\bf Proof.} For the first: Let $y$ be apoint in the fiber of $H$ through $x$. Since $\psi^{*-1} \tilde{\Phi} =1$, it is enough to show that $\psi _{\tilde{d}}^{*-1} \tilde{\Phi} _{\tilde {d}} (x) - \psi _{\tilde{d}}^{*-1}\tilde{\Phi} _{\tilde {d}} (y) =\psi ^{*-1} \tilde{\Phi} (x-y)$. Since $\psi ^ {*-1} \tilde{\Phi}$ is the group homomorphism associated to the map $\psi _{\tilde{d}}^{*-1} \tilde{\Phi} _{\tilde {d}} $ of abelian torsors, the later is true by Remark \ref{remtorsor} For the second: It is enough to show that $\psi _{\tilde{d}}^{*-1} \tilde{\Phi} _{\tilde {d}} (x) - \psi _{\tilde{d}}^{*-1}\tilde{\Phi} _{\tilde {d}} (y) =y-x $. But since $\tilde{\Phi} = -\psi ^* $ we have $ \psi ^{*-1} \tilde{\Phi} (x-y)=y-x $ and this case follows as well. \begin{flushright} $\Box$ \end{flushright} \begin{prop} Let ${\bf -1}$ denote the inversion along the fibers of $\unprym $. Then \begin{enumerate} \item If $\tilde{\Phi} = \psi ^* $, then $\tilde{\Phi} _{\tilde {d}} = T_{\pi ^* \mu} \circ \psi _{\tilde {d}}^* $, where $\mu$ is an $r$-torsion line bundle on the base curve $C$. \item If $\tilde{\Phi} = -\psi ^* $, then $ r|2d$ and $\tilde{\Phi} _{\tilde {d}} = T_{\pi ^* \nu _1} \circ(\psi _{\tilde {d}}^* ) \circ ({ \bf -1})$, where $ \nu $ is a line bundle on the base curve $C$ which satisfies $ \nu _1 ^{\otimes r}=L_0^{\otimes 2} \otimes \omega _C ^{\otimes r(r-1)} $. \end{enumerate} \label{propaaa} \end{prop} \noindent {\bf Proof.} For the first: According to Lemma \ref{lempsi}, on each fiber $H^{-1}(s) = {\rm Prym}_{\tilde{d}}(\spec, C) $ of the map $H$, the maps $\tilde{\Phi} _{\tilde {d}} $ and $\psi _{\tilde {d}}^* $ differ by a translation by a unique element $a(s) \in {\rm Prym}( \tilde {C}_s, C) $. This defines a section of the map $H : \unjac \longrightarrow B^{\rm reg} $ and therefore by Proposition \ref{propsections}, it must have the form $\pi ^* M$ for some fixed line bundle $\mu $ on $C$. Since the image of the section is in the $\unprym $ we get that $\mu $ is an $r$-torsion line bundle. For the second: According to Lemma \ref{lempsi}, we can construct a section of the map $H : J^{2 \tilde{d}} (\unspec ) \longrightarrow B^{\rm reg} $ by assigning to the point $s \in B^{\rm reg}$ the line bundle $\psi _{\tilde{d}}^{*-1}\tilde{\Phi} _{\tilde {d}} (x)+x $ for some point $x \in {\rm Prym}_{\tilde{d}}(\spec, C) $. By Proposition \ref{propsections}, we get that $r|2 \tilde{d}$. Since $\tilde{d}= d +r(r-1)(g-1)$ this is equivalent to $r|2d$. The same proposition implies that the above section must have the form $\pi ^* \nu _1$ for some fixed line bundle $\nu _1$ on $C$. To complete the proof of the second part of the proposition, observe that the map $\psi ^*$ commutes with the translations by an element of the form $\pi ^*\nu _1$ and that ${\rm Nm} \, \pi ^* \nu _1 = L_0 ^{\otimes 2} \otimes \omega _C ^{\otimes r(r-1)}$. \begin{flushright} $\Box$ \end{flushright} \bigskip We are ready now to complete the proof of the Theorem \ref {theor1} in the case $r \geq 3$. Pick an element element $s \in B^{\rm reg} $. Then, by Corollary \ref{cor17}, the prymian ${\rm Prym}_{\tilde{d}}(\tilde{C}_s , C) $ maps dominantly to $\su $. Let ${\cal V}$ be its image. It is enough to prove the theorem for the restriction of the map $\Phi $ on ${\cal V}$. We have the following commutative diagram: \vskip.1in \[ \begin{diagram} \node{{\rm Prym }_{\tilde {d}}(\spec ,C)} \arrow{s,l}{\pi _{s *}} \arrow{e,t} {\tilde{\Phi} _{\tilde{d} _{ } }} \node{ {\rm Prym }_{\tilde {d}}(\tilde{C}_{\phi (s)} ,C) } \arrow{s,r} {\pi _{\phi (s) *} } \\ \node{{\cal V} } \arrow{e,t}{\Phi } \node{{\cal V} } \end{diagram} \] \vskip.1in \noindent where the maps $\pi _{s *}$ and $\pi _{\phi (s) *}$ are rational maps. Assume first that $\tilde{\Phi} = \psi ^* $. Let $E$ a vector bundle in ${\cal V}$. Then, there exists a line bundle $\tilde {L}_s$ in ${\rm Prym }_{\tilde {d}}(\spec ,C)$ such that $\pi _{s *} (\tilde{L}_s) = E$. By Proposition \ref{propaaa}, the above diagram and the fact that the map $\psi $ commutes with the projections $\pi _{s *}$ and $\pi _{\phi (s) *}$ to the base curve $C$, we conclude that $$ \Phi (E) = \Phi \pi _{s *} (\tilde{L}_s) = \pi _{\phi (s) *} \tilde{\Phi} _{\tilde {d}} (\tilde{L}_s) = \pi _{\phi (s) *} (\psi _{\tilde {d}}^* (\tilde{L}_s) \otimes \pi _{\phi (s) *} \mu ) =\pi _{\phi (s) *} \psi _{\tilde {d}}^* (\tilde{L}_s) \otimes \mu = E \otimes \mu . $$ Assume next that $\tilde{\Phi} =-\psi ^* $. Then, following the above notation, we have $$ \Phi (E) = \Phi \pi _{s*} (\tilde{L}_s) = \pi _{\phi (s) *} \tilde{\Phi} _{\tilde {d}} (\tilde{L}_s) = \pi _{\phi (s) *} (\psi _{\tilde {d}}^* (\tilde{L}_s^{-1} ) \otimes \pi _{\phi (s) *} \nu _1) = \pi _{\phi (s) *} \psi _{\tilde {d}}^* (\tilde{L}_s^{-1}) \otimes \nu _1 = \pi _{s *} (\tilde{L}_s^{-1}) \otimes \nu _1 . $$ We claim that $ \pi _{s *} (\tilde{L}_s^{-1}) = E^{\vee } \otimes \omega _C^{-(r-1)}$. Indeed, consider the map $\pi _s : \spec \longrightarrow C$. By relative duality we have \[ {\cal R}^0 \pi _{s *} (\tilde{L}_s^{-1}) \simeq {\cal R}^0 \pi _{s *} (\omega _{\pi _s} \otimes \tilde{L}_s )^{\vee} . \] By the adjunction formula we have, see e.g. \cite{h1}, that $\omega_{\pi _s } \simeq \pi _s ^* \omega _C ^{r-1}$. We thus get \[ \pi _{s *} (\tilde{L}_s^{-1}) \simeq E^{\vee} \otimes \omega_C^{-(r-1)} . \] By choosing $\nu = \nu _1 \otimes \omega_C ^{-(r-1)}$, we get that $ \phi (E) = E^{\vee } \otimes \nu $ where $\nu ^{\otimes r} =L_0 ^{\otimes 2}$. Thus, we have shown the surjectivity of the maps (1.) and (2.) in the statement of Theorem \ref{theor1}. To show the injectivity of the map (1.), pick up a point $\mu \neq 0 \in J^{0}[r]$ and consider the set of fixed points $\su^{\langle T_{\mu} \rangle}$. A vector bundle $E$ is fixed under the action of $T_{\mu}$, if we have an isomorphism \[ \kappa : E \longrightarrow E\otimes\mu .\] If $p \, | \, r$ is the order of the torsion point $\mu$, then the $\mu$-twisted Higgs bundle $(E,\kappa)$ gives a $p$-sheeted unramified spectral cover $\pi_{\mu} : C_{\mu} \rightarrow C$ and a semistable vector bundle $F_{\kappa}$ of rank $r/p$ on it with the property $\pi_{\mu *}(F_{\kappa}) = E$, see \cite{nr2} for details. Therefore the locus $\su^{\langle T_{\mu} \rangle}$ can be identified with the image of the moduli space ${\cal S}{\cal U}_{C_{\mu}}(r/p)$ under the pushforward map $\pi_{\mu *}$ and hence is a proper subvariety. We will sketch the proof for the injectivity of the map (2.) in the case $r \geq 3$ and when $L_{0} = {\cal O}$. The modifiations of the argument for general $L_{0}$ are minor and are left to the reader. If $L_{0} = {\cal O}$, then it suffices to check that the map $E \rightarrow E^{\vee}$ is not the identity on ${\cal S}{\cal U}(r,{\cal O})$. But if $E$ is a stable vector bundle satisfying $E \cong E^{\vee}$, then the bundle $E^{\otimes 2}$ has a unique (up to scaling) non-zero section $t$. But $H^{0}(C,E^{\otimes 2}) = H^{0}(C,{\rm Sym}^{2}E)\oplus H^{0}(C,\wedge^{2}E)$ and therefore either $H^{0}(C,E^{\otimes 2}) = 0$ or $H^{0}(C,\wedge^{2}E) = 0$. This implies that the isomorphism $t$ between $E$ and $E^{\vee}$ is either symmetric or skew-symmetric. Thus, $E$ is either orthogonal or symplectic and hence it lies either in the moduli space of stable $SO(r)$-bundles or in the moduli space of stable $Sp(r)$-bundles. But for $r \geq 3$, those are proper subvarieties of ${\cal S}{\cal U}(r, {\cal O})$ and this yields that the map (2.) is injective for $r \geq 3$. \begin{rem} {\rm For $r=2$, the moduli space ${\cal S}{\cal U}(2, {\cal O})$ coincides with the moduli space of $Sp(2)$-bundles; if $E \in {\cal S}{\cal U}(2, {\cal O})$, then $E \simeq E ^{\vee }$}. \label{remrank2} \end{rem} \subsection{The rank $2$ case} \label{rank2} The proof in the rank $2$ case is a modification of the proof of the rank $\geq 3$ case. The main difference is that the linear system defined by $B= W_2=H^0(C, \omega _C^2)$ does not separate the points on the surface $S^{\circ} $: A section $s \in B$ corresponds to the curve $x^2 + s=0$ on $S^{\circ} $. If $m_{-1}$ is the dilation by $-1$ on $S^{\circ} $, then a curve in the linear system $|B|$ that passes through a point $p$, passes also through the point $m_{-1}(p)$. Therefore the map $f_{|B|}$ sends $S^{\circ} $ in a $2:1$ way to the projective space. We now show briefly the adjustments for the rank $2$ case of the argument we used in the rank $\geq 3$ case. At first one can see e.g. by Corollary \ref{cor21}, that the map $\phi = \phi _2$ is a multiplication by a $\lambda \in {\Bbb C}^*$. Following a similar argument with that of Section \ref{picunspec}, we can prove that the Picard group of $\unspec \longrightarrow B^{\rm reg} $ is again $$ {\rm Pic} \, \unspec = \pi ^* {\rm Pic} \, C . $$ We also have \begin{equation} {\rm Pic} \, \unspec _p = {\rm Pic} \, \pi ^* C \oplus {\Bbb Z} [H_p] \label{eqrank21} \end{equation} but in this case the technicalities of the proof are slightly different: Let $p^{'} = m_{-1}(p)$. On the fiber of $S^{\circ} $ over $c= \alpha(p) =\alpha (p^{'})$, only the points $p $ and $p^{'}$ belong in the image of the map $\beta $. We write $\unspec _p = \unspec _p\setminus (H_p \cup H_{p^{'}}) \amalg (H_p \cup H_{p^{'}})$. Then ${\rm Pic} \, \unspec _p$ is generated by ${\rm Pic} \,(C\setminus \{ c \})$ and $H_p, H_{p^{'}}$. Observe that $\pi ^*(c) = H_p+ H_{p^{'}}$. We thus have that ${\rm Pic} \, \unspec _p $ is generated by ${\rm Pic} \, \pi ^* C $ and $H_p$. To prove that this is a direct sum, we again take a pencil as in the proof of Proposition \ref {proppicunspecp}, but now the curve at infinity embedded on $S$, consists of the $Y_{\infty }$ and a bunch of $N_1$ fibers over the points $c_1, \ldots , c_{N_1}$ on C, which pass through the $N= 2N_1$ base points of the pencil. Therefore the fiber over the infinity point $b$ on $X$ consists of the infinity divisor $\tilde{Y}_{\infty}$ and a bunch of $N_1$ divisors $A_1, \ldots , A_{N_1}$ which satisfy $\pi ^* (c_i) = A_i + E_i + E^{'}_i$. Using that we get $$ {\rm Pic} \, X^{\circ } = \pi ^* {\rm Pic} \, C \oplus _{i=1}^{N_1} ( \oplus {\Bbb Z}[E_i]) \oplus _{i=1}^{N_1} (\oplus {\Bbb Z}[E^{'}_i] ) \left/ (\pi ^*(c_i)=E_i +E^{'}_i)_{i=1, \ldots N_1} \right. . $$ Working as in the rank $\geq 3$ case, this proves relation (\ref{eqrank21}). To prove the analogue of Proposition \ref{propsections}, we proceed in the same way as in the rank $\geq 3$ case. For the Picard group of $\fibprod $: Let again $\psi :\unspec \longrightarrow \unspec $ be the analogue of the map introduced in Section \ref{picfibprod}. Then the locus $\Gamma $ consists of two components namely $\Gamma _1 = \{ (p, \psi (p)) \;\; \mbox{for} \;\; p \in \unspec \}$ and $\Gamma _2 = \{ (p, \psi (m_{-1}(p))) \;\; \mbox{for} \;\; p \in \unspec \}$. By observing that $\pi ^{'*} (\mbox{Diagonal in} \;\; C \times C ) = \Gamma _1 + \Gamma _2 $ we conclude that ${\rm Pic} \, (\fibprod )$ is generated by ${\rm Pic} \, (C \times C)$ and $\Gamma _1$. The rest of the argument, combined with Remark \ref{remrank2}, proceeds as in the rank $r \geq 3 $ case. \subsection{Curves with automorphisms} \label{automorphisms} For the rank $r \geq 3$ case, the only modification that has to be done, is in the argument of Section \ref{picfibprod}, about the induced automorphism of the embedded surface $f_{|\overline{B} |}$. For the rank $2$ case, the only modification is in the use of Corollary \ref{cor21}. We leave to the reader to fill up the details for the proof of Theorem \ref{theor2}. \section{The automorphisms of $\urd $} \label{urd} \begin{prop} Let $\Phi $ be an automorphism of $\urd $. Then $\Phi $ factors through the determinant map i.e. we have the following commutative diagram \begin{equation} \begin{diagram}[aaaaaaa] \node{\urd } \arrow{e,t}{\Phi } \arrow{s,l} {\rm det} \node{ \urd } \arrow{s,r} {\rm det }\\ \node{J^d(C) } \arrow{e,t}{ \phi _d} \node{ J^d(C) } \end{diagram} \label{diagurd1} \end{equation} \label{propurd1} \end{prop} \noindent {\bf Proof.} We would like to show that the map ${\rm det} \circ \phi : {\rm det}^{-1}(L_0) \longrightarrow J^d(C)$ is constant for all $L_0 \in J^d(C)$. By definition, ${\rm det}^{-1}(L_0) = \su $ and ${\rm dim} \, \su = (r^2-1)(g-1) > g, \;\; g \geq 2 $. Therefore the map ${\rm det} \circ \phi : \su \longrightarrow J^d(C)$ has positive dimentional fibers and so, no pull back of a line bundle from $J^d(C)$ can be ample. According to \cite{dn}, ${\rm Pic} \, \su = {\Bbb Z}[\Theta ]$. Since $\su $ is a projective variety, $\Theta $ is an ample divisor. Let ${\cal L}$ be a line bundle on $J^d(C)$. We thus have $({\rm det} \circ \phi) ^*{\cal L} \simeq n \Theta$ for some integer $n$. We claim that $n=0$: If not, then say first that $n >0$. Then $ n \Theta$ is ample and so $({\rm det} \circ \phi) ^*{\cal L} $ is ample, which is a contradiction. Next, if $n<0$, then the same argument applied to the line bundle ${\cal L}^{-1}$ leads to a contradiction. Therefore $({\rm det} \circ \phi) ^*{\cal L} ={\cal O}$ for all ${\cal L} \in {\rm Pic} \, J^d(C)$. Since $\su $ is irreducible, this implies that the map is constant. Indeed, if the image has positive dimension, then there exists a positive divisor on that - the hyperplane section of its embedding in the projective space. But then the pull back of that divisor on $\su $ by the map ${\rm det} \circ \phi $ defines a non - trivial line bundle, which is a contradiction. \begin{flushright} $\Box$ \end{flushright} \bigskip As in the case of the automorphisms of $\su $, an automorphism $\Phi $ of $\urd $ induces an automorphism $\tilde{\Phi} _{\tilde {d}} $ of the Jacobian fibration $H: \unjacd \longrightarrow W^{\rm reg}$ which sends a fiber of the Hitchin map $H$ to a fiber of $H$. Let $N_1 : \unjacd \longrightarrow J^d(C)$ be the map defined by $N_1 = {\rm det} \circ \pi _*$. Note that ${\rm Nm} = T_{\frac{r(r-1)}{2} \omega _C } \circ N_1$. \begin{lem} Let $s $ be a point in $W^{\rm reg}$. Then the following diagram commutes \[ \begin{diagram} \node{J^{\tilde {d}}(\spec ) } \arrow{e,t}{\tilde{\Phi} _{\tilde{d} _{ }} } \arrow{s,l} {\rm N_1} \node{ J^{\tilde {d}}( \tilde{C}_{\phi _W (s)}) } \arrow{s,r} {\rm N_1 }\\ \node{J^d(C) } \arrow{e,t}{ \phi _d} \node{ J^d(C) } \end{diagram} \] \label{lemurd1} \end{lem} \noindent {\bf Proof.} For the proof we are going to use the following commutative diagram \begin{equation} \begin{diagram} \node{{\cal X}(r,d) } \arrow{e,t}{d\Phi ^* } \arrow{s,l} {\pi _*} \node{{\cal X}(r,d) } \arrow{s,r} {\pi _*}\\ \node{\urd } \arrow{e,t}{\Phi } \arrow{s,l} {{\rm det}} \node{\urd} \arrow{s,r} {{\rm det}}\\ \node{J^d(C) } \arrow{e,t}{ \phi _d} \node{ J^d(C) } \end{diagram} \label{diagurd3} \end{equation} \noindent It is enough to show that $N_1 \tilde{\Phi} _{\tilde {d}} = \phi _d N_1 $ on a Zariski open $U$ in $ J^{\tilde {d}}(\spec ) $. Choose $U$ to be the intersection of the cotangent bundle ${\cal X}(r,d) $ to $\urd $ with the Jacobian $J^{\tilde {d}}(\spec )$. According to Corollary \ref{cor17}, this is a non empty Zariski open in $\urd$. The proof of the Lemma is now a consequence of the commutativity of the above diagram. \begin{flushright} $\Box$ \end{flushright} \bigskip Let $\tilde{\Phi} $ and $\phi $ be the group maps associated to $\tilde{\Phi} _{\tilde {d}} $ and $\phi _d$, see Remark \ref{remtorsor}. By using the above Lemma \ref{lemurd1}, it is easy to see that \begin{cor} The following diagram is commutative \begin{equation} \begin{diagram} \node{J^0 (\spec ) } \arrow{e,t}{\tilde{\Phi} } \arrow{s,l} {\rm Nm} \node{ J^0( \tilde{C}_{\phi _W (s)} ) } \arrow{s,r} {\rm Nm }\\ \node{J^0(C) } \arrow{e,t}{ \phi } \node{ J^0(C) } \end{diagram} \label{diagurd4} \end{equation} where ${\rm Nm}$ is the norm map. \label{corurd1} \end{cor} \begin{lem} Following the notation of Corollary \ref{corurd1} , the diagram bellow is commutative \[ \begin{diagram} \node{J^0 (\spec ) } \arrow{e,t}{\tilde{\Phi} } \node{ J^0( \tilde{C}_{\phi _W (s)} )} \\ \node{J^0(C) }\arrow{n,l} {\pi ^*}\arrow{e,t}{ \phi } \node{ J^0(C) } \arrow{n,r}{\pi ^*} \end{diagram} \] \label{lemurd2} \end{lem} \noindent {\bf Proof.} Note first that \begin{equation} \tilde{\Phi} \pi ^* (L) = \pi ^* (M) \;\; \mbox{for some fixed line bundle} \;\; M . \label{equrd1} \end{equation} Indeed, $\tilde{\Phi} $ is a global automorphism on $\unjac $ and so, $\tilde{\Phi} \, \pi ^* (L)$ defines a section of the map $H: \unjac \longrightarrow W^{\rm reg}$. Hence, by Proposition \ref{propsections}, it must have the form $ \pi ^* (M) $ for some fixed line bundle $M$ on $C$. By applying the norm map to (\ref{equrd1}), we get ${\rm Nm} \tilde{\Phi} \pi ^* (L) ={\rm Nm} \, \pi ^* (M) $. Corollary \ref{corurd1} implies that $r \phi (L) =r M$. Let $n_r$ denote the multiplication by $r$. By composing both sides of (\ref{equrd1}) by $n_r$ and by using the last relation we get that $n_r \tilde{\Phi} \pi ^* (L) = n_r \pi ^*\phi (L) $. Since the map $n_r $ is onto, this completes the proof. \begin{flushright} $\Box$ \end{flushright} \begin{lem} For any $E \in \urd $ and $M \in J^0(C)$ we have $ \Phi (E \otimes M)= \Phi (E) \otimes \phi (M)$. \label{lemurd3} \end{lem} \noindent {\bf Proof.} It suffices to prove the relation for all $E$ in a Zariski open $V$ of $\urd $. Take $\spec $ a smooth spectral curve and let $V$ be the image of $ {\cal X}(r,d) \cap J^{\tilde{d}}(\spec )$ on $\urd$. According to Corollary \ref{cor17}, this is a Zariski open of $\urd $. Then $E=\pi _* \tilde {L} $. We have \[ \begin{array}{rlll} \Phi (E \otimes M) & = & \Phi (\pi _* \tilde {L} \otimes M) \\ & = &\Phi (\pi _* (\tilde {L} \otimes \pi ^* M)) & \mbox{by the projection formula}, \\ & = & \pi _* \tilde{\Phi} _{\tilde {d}} (\tilde{L} \otimes \pi ^* M) & \mbox{by diagram (\ref {diagurd3})}, \\ & = & \pi _* (\tilde{\Phi} _{\tilde {d}} (\tilde{L}) \otimes \tilde{\Phi} (\pi ^* M) ) & \mbox {by the definition of the group map} \;\; \tilde{\Phi} , \\ & = & \pi _* (\tilde{\Phi} _{\tilde {d}} (\tilde{L}) \otimes \pi ^* \phi ( M) ) & \mbox{by Lemma \ref{lemurd2}}, \\ & = & \pi _* \tilde{\Phi} _{\tilde {d}} (\tilde{L}) \otimes \phi (M) & \mbox{by the projection formula},\\ & = & \Phi \pi _* (\tilde {L}) \otimes \phi (M) \\ & = & \Phi (E) \otimes \phi (M). \end{array} \] \begin{flushright} $\Box$ \end{flushright} \bigskip \noindent {\bf Proof of Theorem \ref{theor3}.} Let $\Phi :\urd \longrightarrow \urd$ be a given automorphism and let $\phi _d$ the induced automorphism on $J^d(C)$ as in Lemma \ref{lemurd1}. Given a vector bundle $E \in \urd $, we can find a vector bundle $E_{L_0} \in \su $ and a line bundle $\eta \in J^0(C)$ such that $E= E_{L_0} \otimes \eta $. Note that $ \eta ^{\otimes r} = det E \otimes L_0 ^{-1}$. The above decomposition is unique up to a choice of an $r$-torsion point. Let $\xi _1$ be a line bundle such that $ \xi _1 ^{ \otimes r}= \phi _d(L_0) \otimes L_0 ^{-1} $. Then, $ T_{ \xi _1 ^{ -1} } \circ \Phi $ induces an automorphism of $\su $. By the main Theorem \ref{theor2}, we have two cases.\\ \\ {\bf Case 1:} $ T_{ \xi _1 ^{ -1} } \circ \Phi (E_{L_0}) = \sigma ^* E_{L_0} \otimes \mu $, where $\sigma $ is an automorphism of the curve $C$ and $\mu $ a line bundle which satisfies $ \mu ^{ \otimes r} = L_0 \otimes \sigma ^* L_0 ^{-1}$. We choose now $ \xi = \xi _1 \otimes\mu $ and so, $\xi ^{\otimes r} =\phi _d(L_0) \otimes \sigma ^* L_0 ^{-1} $. We have \[ \begin{array}{rlll} \Phi (E) & = & \Phi (E_{L_0} \otimes \eta ) \\ & = & \Phi ( E_{L_0}) \otimes \phi (\eta ) & \mbox{by Lemma \ref{lemurd3}}, \\ & = & T_{\xi _1} (T_{ \xi _1^{-1} } \Phi (E_{L_0})) \otimes \phi (\eta ) & \\ & = & \sigma ^* E_{L_0} \otimes \xi \otimes \phi (\eta ) & \\ & = & \sigma ^* (E \otimes \eta ^{-1}) \otimes \xi \otimes \phi (\eta ) & \mbox{by the definition of} \;\; E_{L_0} , \\ & = & \sigma ^* E \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ^{-1} ) . \end{array} \] Therefore, \[ \Phi (E) = \sigma ^* E \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ^{-1}) \;\; \mbox {where} \;\; \eta ^{\otimes r} = {\rm det} E \otimes L_0 ^{-1}\;\; \mbox{and} \;\; \xi ^{\otimes r} = \phi _d(L_0) \otimes \sigma ^* L_0 ^{-1} . \] Note that, due to Lemma \ref{lemurd3}, the map $\phi ^{-1} \circ \sigma ^* $ is the identity on the set of $r$-torsion points in $J^0(C)$.\\ \\ {\bf Case 2:} $ T_{ \xi _1^{ -1} } \circ \Phi (E_{L_0}) = \sigma ^* E_{L_0}^{\vee} \otimes \nu $, where $\sigma $ is an automorphism of the curve $C$ and $\nu $ is a line bundle which satisfies $ \nu ^{\otimes r}= L_0 \otimes \sigma ^* L_0 $. In this case we choose $\xi = \xi _1 \otimes \nu $ and so, $ \xi ^{\otimes r} = \phi _d (L_0) \otimes \sigma ^* L_0 $. We have \[ \begin{array}{rlll} \Phi (E) & = & \Phi ( E_{L_0}) \otimes \phi (\eta ) & \mbox{by Lemma \ref{lemurd3}}, \\ & = & T_{\xi _1 } (T_{ \xi _1^{ -1} } \Phi (E_{L_0})) \otimes \phi (\eta ) & \\ & = & \sigma ^* E_{L_0}^ {\vee} \otimes \xi \otimes \phi (\eta ) & \\ & = & \sigma ^* (E^{\vee } \otimes \eta ) \otimes \xi \otimes \phi (\eta ) & \mbox{by the definition of} \;\; E_{L_0} ,\\ & = & \sigma ^* E^{\vee} \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ) . \end{array} \] Therefore, \[ \Phi (E) = \sigma ^* E ^{\vee} \otimes \xi \otimes \phi (\eta ) \otimes \sigma ^*(\eta ) \;\; \mbox {where} \;\; \eta ^{\otimes r} = {\rm det} E \otimes L_0^{-1}\;\; \mbox{and} \;\; \xi ^{\otimes r} = \phi _d(L_0) \otimes \sigma ^* L_0 . \] Again, due to Lemma \ref{lemurd3}, the map $\phi \circ \sigma ^* $ has to be the identity map on the set of $r$-torsion points in $J^0(C)$.

9207

alg-geom/9207001

en

https://arxiv.org/abs/alg-geom/9207001

alg-geom/9207001

Richard Hain

Richard Hain (Duke Univeristy)

Completions of mapping class groups and the cycle $C - C^-$

31 pages, LaTeX

In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the archemidean height of the cycle C - C- in the Jacobian of the curve C. We also develop some of the basic theory of relative completions.

alg-geom

\section{Introduction}\label{intro} The classical Malcev (or unipotent completion) of an abstract group $\pi$ is a prounipotent group ${\cal P}$ defined over ${\Bbb Q}$ together with a homomorphism $\phi : \pi \to {\cal P}$. It is characterized by the property that if $\psi : \pi \to {\cal U}$ is a homomorphism of $\pi$ into a prounipotent group, then there is a unique homomorphism $\Psi: {\cal P} \to {\cal U}$ of prounipotent groups such that $\psi = \Psi \phi$. $$ \matrix{ \pi &\stackrel{\phi}{\to} &{\cal P} \cr & \mapse{\psi} &\mapdown{\Psi} \cr && {\cal U} \cr } $$ When $H_1(\pi;{\Bbb Q}) = 0$, the unipotent completion is trivial, and it is therefore a useless tool for studying mapping class groups. Deligne has suggested a notion of {\it relative Malcev completion}: Suppose $\Gamma$ is an abstract group and that $\rho : \Gamma \to S$ is a homomorphism of $\Gamma$ into a semisimple linear algebraic group defined over ${\Bbb Q}$. Suppose that $\rho$ has Zariski dense image. The {\it completion of $\Gamma$ relative to $\rho$} is a proalgebraic group ${\cal G}$ over ${\Bbb Q}$, which is an extension of $S$ by a prounipotent group ${\cal U}$, and a homomorphism $\tilde{\rho} : \Gamma \to {\cal G}$ which lifts $\rho$. When $S$ is the trivial group, it reduces to the unipotent completion. The relative completion is characterized by a universal mapping property which generalizes the one in the unipotent case (see (\ref{ump})). Denote the mapping class group associated to a surface of genus $g$ with $r$ boundary components and $n$ ordered marked distinct points by $\Gamma_{g,r}^n$. The mapping class group $\Gamma_{g,r}^n$ has a natural representation $\rho : \Gamma_{g,r}^n \to Sp_g({\Bbb Z})$ obtained from the action of $\Gamma_{g,r}^n$ on the first homology group of the underlying compact Riemann surface. Its kernel is, by definition, the Torelli group $T_{g,r}^n$. One can therefore form the completion of $\Gamma_{g,r}^n$ relative to $\rho$. It is a proalgebraic group ${\cal G}_{g,r}^n$ which is an extension $$ 1 \to {\cal U}_{g,r}^n \to {\cal G}_{g,r}^n \to Sp_g \to 1 $$ of $Sp_g$ by a prounipotent group. The homomorphism $$ \tilde{\rho}: \Gamma_{g,r}^n \to {\cal G}_{g,r}^n $$ induces a homomorphism $T_{g,r}^n \to {\cal U}_{g,r}^n$. This, in turn, induces a homomorphism ${\cal T}_{g,r}^n \to {\cal U}_{g,r}^n$ from the unipotent completion of the Torelli group into ${\cal U}_{g,r}^n$. Our main result is: \medskip \noindent{\bf Theorem.}{\sl When $g\ge 3$, the natural homomorphism ${\cal T}_{g,r}^n \to {\cal U}_{g,r}^n$ is surjective with nontrivial kernel which is contained in the center of ${\cal T}_{g,r}^n$ and which is isomorphic to ${\Bbb Q}$ whenever $g \ge 8$.} \medskip We prove this by relating the central extension above to the line bundle over the moduli space of genus three curves associated to the archimedean height of the algebraic cycle $C - C^-$ in the jacobian of a curve $C$ of genus 3.\footnote{When the genus $g$ of $C$ is $\ge 3$, one can relate this central extension to the height pairing between the cycles $C^{(a)} - {C^{(a)}}^-$ and $C^{(b)} - {C^{(b)}}^-$ in ${\rm Jac\,} C$, where $a+b = g-1$ and $C^{(r)}$ denotes the $r$th symmetric power of $C$. We chose not to do this in order to keep the Hodge theory straightforward.} This theorem is related to, and complements, the work of Morita \cite{morita}. The constant 8 in the theorem can surely be improved, possibly to 3.\footnote{The optimal constant is the smallest integer $d$ such that $H^2(Sp_g({\Bbb Z}),A)$ vanishes for all rational representations $A$ of $Sp_g({\Bbb Q})$ whenever $g\ge d$.} One reason for introducing relative completions of fundamental groups of varieties, and of mapping class groups in particular, is that their coordinate rings are, under suitable conditions, direct limits of variations of mixed Hodge structure over the variety. This result and some of its applications to the action of the mapping class group of a surface $S$ on the lower central series of $\pi_1(S,\ast)$ will be presented elsewhere. Part of a general theory of relative completions is worked out in Section \ref{basic}. Many of the results of that section were worked out independently and contemporaneously by Eduard Looijenga. I would like to thank him for his correspondence. I would also like to thank P.~Deligne for his correspondence, and the Mathematics Department of the University of Utrecht for its hospitality and support during a visit in May, 1992 when this paper was written. \medskip \noindent{\bf Conventions:} The group $Sp_g(R)$ will denote the group of automorphisms of a free $R$ module of dimension $2g$ which preserve a unimodular skew symmetric bilinear form. In short, elements of $Sp_g(R)$ are $2g \times 2g$ matrices. \section{Relative completion} \label{rel_malcev} Fix a field $F$ of characteristic zero. Suppose that $\pi$ is a abstract group and that $\rho : \pi \to S$ is a representation of $\pi$ into a linear algebraic group $S$ defined over $F$. Assume that the image of $\rho$ is Zariski dense. In this section we define the {\it completion of $\pi$ relative to $\rho$}. When $S$ is the trivial group, this reduces to the Malcev completion (a.k.a.\ unipotent completion) which is defined, for example, in \cite{quillen}, \cite{chen:malcev} and \cite{sullivan}. The idea of relative completion is due to Deligne \cite{deligne:variations} and is a refinement of the idea of the ``algebraic hull'' of a group introduced by Hochschild and Mostow \cite[p.~1140]{hoch-mostow}. To construct the relative completion of $\pi$ with respect to $\rho$, consider all commutative diagrams of the form $$ \matrix{ 1 & \to & U & \to & E & \to & S & \to &1 \cr & & & & \tilde{\rho}\uparrow & \nearrow\rho &&&\cr &&&& \pi &&&&\cr} $$ where $E$ is a linear algebraic group over $F$, $U$ a unipotent subgroup of $E$, and where $\tilde{\rho}$ is a lift of $\rho$ to $E$ whose image is Zariski dense. All morphisms in the top row are algebraic group homomorphisms. One can define morphisms of such diagrams in the obvious way. \begin{proposition} \label{invsys} The set of such diagrams forms an inverse system. \end{proposition} \noindent{\bf Proof.~} If $$ \matrix{ 1 & \to & U_\alpha & \to & E_\alpha & \to & S & \to &1 \cr &&&&&&&&\cr 1 & \to & U_\beta & \to & E_\beta & \to & S & \to &1 \cr} $$ are two extensions of $S$ by a unipotent algebraic group, then one can form the fibered product $$ \matrix{ E & \to& E_\alpha \cr \downarrow &&\downarrow \cr E_\beta & \to & S .\cr} $$ The natural homomorphism $E \to S$ is surjective with kernel the unipotent group $U_\alpha \times U_\beta$. Now suppose that $\rho_\alpha : \pi \to E_\alpha$ and $\rho_\beta : \pi \to E_\beta$ are lifts of $\rho : \pi \to S$ to $E_\alpha$ and $E_\beta$, respectively, both with Zariski dense image. They induce a homomorphism $\rho_{\alpha\beta} : \pi \to E$ which lifts both $\rho_\alpha$ and $\rho_\beta$. Denote the Zariski closure of the image of $\rho_{\alpha\beta}$ in $E$ by $E_{\alpha\beta}$. Then $E_{\alpha\beta}$ is a linear algebraic group and the kernel of the natural homomorphism $E_{\alpha\beta} \to S$ is unipotent as it is a subgroup of $U_\alpha \times U_\beta$. The natural map $E_{\alpha\beta}\to S$ is surjective as the image of $\rho$ is Zariski dense in $S$. \quad $\square$ \begin{definition} The {\it completion ${\cal P}_F$ of $\pi$ (over $F$) relative to $\rho : \pi \to S$} is defined to be the proalgebraic group $$ {\cal P}_F = \lim_\leftarrow E, $$ where the inverse limit is taken over all commutative diagrams $$ \matrix{ 1 & \to & U & \to & E & \to & S & \to &1 \cr & & & & \mapupalt{\tilde{\rho}} & \mapne{\rho} &&&\cr &&&& \pi &&&&\cr} $$ whose top row is an extension of $S$ by a unipotent group in the category of linear algebraic groups over $F$, and where $\tilde{\rho}$ has dense image. The homomorphisms $\tilde{\rho} : \pi \to E$ induce a canonical homomorphism $\pi \to {\cal P}_F$. \end{definition} Often we will simply say that $\pi \to {\cal P}_F$ is the {\it $S$-% completion\/} of $\pi$. The coordinate ring of ${\cal P}_F$ is the direct limit of the coordinate rings of the groups $E$. It will be denoted ${\cal O}({\cal P}_F)$. It is a commutative Hopf algebra with antipode. There is a natural surjection ${\cal P}_F \to S$ whose kernel is a prounipotent group. When $S$ is the trivial group, we obtain the classical Malcev completion. The $S$-Malcev completion is characterized by the following easily verified universal mapping property. \begin{proposition} \label{ump} Suppose that ${\cal E}$ is a linear proalgebraic group defined over $F$, and that ${\cal E} \to S$ is a homomorphism of proalgebraic groups with prounipotent kernel. If $\varphi : \pi \to {\cal E}$ is a group homomorphism, then there is a unique homomorphism $\tau : {\cal P}_F \to {\cal E}$ of pro-algebraic groups over $F$ such that the diagram $$ \matrix{ &&{\cal P}_F&& \cr &\hat{\rho}\nearrow& &\searrow& \cr \pi & & \mapdown{\tau} && S \cr &\varphi \searrow && \nearrow& \cr && {\cal E}&& \cr} $$ commutes. \quad $\square$ \end{proposition} Suppose that $G$ is a (pro)algebraic group over the field $F$. Suppose that $k$ is a field extension of $F$. We shall denote $G$, viewed as an algebraic group over $k$ by extension of scalars, by $G(k)$. The following assertion follows directly from the universal mapping property. \begin{corollary} If $k$ is a field extension of the field $F$, then there is a natural homomorphism ${\cal P}_{k} \to {\cal P}_F(k)$. \quad $\square$ \end{corollary} We will show in the next two sections that, with some extra hypotheses, this homomorphism is always an isomorphism. \section{A construction of the Malcev completion} \label{classical} There is an explicit algebraic construction of the Malcev completion which is due to Quillen \cite{quillen}. We will use it to show that the Malcev completion of a group over $F$ is isomorphic to the $F$ points of its Malcev completion over ${\Bbb Q}$. Denote the group algebra of a group $\pi$ over a commutative ring $R$ by $R\pi$. The {\it augmentation\/} is the homomorphism $\epsilon : R\pi \to R$ defined by taking each $\gamma\in \pi$ to 1. The kernel of the augmentation is called the {\it augmentation ideal\/} and will be denoted by $J_R$, or simply $J$ when there is no chance of confusion. With the coproduct $\Delta : R\pi \to R\pi \otimes R\pi$ defined by $\Delta (\gamma) = \gamma \otimes \gamma$, for all $\gamma \in \pi$, $R\pi$ has the structure of a cocommutative Hopf algebra. The powers of the augmentation ideal define a topology on $R\pi$ that is called the {\it $J$-adic topology}. The {\it $J$-adic completion\/} of the group ring is the $R$ module $$ R\pi~\widehat{\!}{\;} := \lim_\leftarrow R\pi/J^l. $$ The completion of $J$ will be denoted by $\widehat{J}$. Since the coproduct is continuous, it induces a coproduct $$ \Delta : R\pi~\widehat{\!}{\;} \to R\pi \widehat\otimes R\pi, $$ where $\widehat\otimes$ denotes the completed tensor product. This gives $R\pi~\widehat{\!}{\;}$ the structure of a complete Hopf algebra. The proof of the following proposition is straightforward. \begin{proposition} \label{first-quot} If $\pi$ is a group and $R$ a ring, then the function $ \pi \to J_R/J_R^2 $ defined by taking $\gamma \in \pi$ to the coset of $\gamma-1$, induces an $R$-module isomorphism $$ H_1(\pi,R) \approx J_R / J_R^2.\quad \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3 $$ \end{proposition} Now let $R$ be a field $F$ of characteristic zero. The logarithm and exponential maps are mutually inverse homeomorphisms $$ \log : 1 + \widehat{J}_F \to \widehat{J}_F \hbox{ and } \exp : \widehat{J}_F \to 1+ \widehat{J}_F. $$ The set of {\it primitive elements\/} ${\goth p}$ of $F\pi~\widehat{\!}{\;}$ is defined by $$ {\goth p} = \{ X \in \widehat{J}_F : \Delta(X) = X \otimes 1 + 1 \otimes X \}. $$ With the bracket $[X,Y] = XY - YX$, it has the structure of a Lie algebra. The topology of $F\pi~\widehat{\!}{\;}$ induces a topology on ${\goth p}$, giving it the structure of a complete topological Lie algebra. The set of {\it group-like\/} elements ${\cal P}$ of $F\pi~\widehat{\!}{\;}$ is defined by $$ {\cal P} = \{ X \in F\pi\hat {\ } : \Delta(X) = X \otimes X \hbox{ and } \epsilon(X) = 1 \}. $$ It is a subgroup of the group of units of $F\pi~\widehat{\!}{\;}$. The logarithm and exponential maps restrict to mutually inverse homeomorphisms $$ \log : {\cal P} \to {\goth p} \hbox{ and } \exp : {\goth p} \to {\cal P}. $$ The filtration of $F\pi~\widehat{\!}{\;}$ by the powers of $J$ induces filtrations of ${\cal P}$ and ${\goth p}$: Set $$ {\goth p}^l = {\goth p} \cap \widehat{J}^l \hbox{ and } {\cal P}^l = {\cal P} \cap (1 + \widehat{J}^l). $$ These satisfy $$ {\goth p} = {\goth p}^1 \supseteq {\goth p}^2 \supseteq {\goth p}^3 \cdots $$ and $$ {\cal P} = {\cal P}^1 \supseteq {\cal P}^2 \supseteq {\cal P}^3 \cdots . $$ These filtrations define topologies on ${\goth p}$ and ${\cal P}$. Both are separated and complete. For each $l$ set $$ {\goth p}_l = {\goth p} / {\goth p}^{l+1} \hbox{ and } {\cal P}_l = {\cal P} / {\cal P}^{l+1}. $$ It follows easily from (\ref{first-quot}) that if $H_1(\pi,{\Bbb Q})$ is finite dimensional (e.g. $\pi$ is finitely generated), then each ${\cal P}_l$ is a linear algebraic group. Since the logarithm and exponential maps induce isomorphisms between $1+\widehat{J}/\widehat{J}^{l+1}$ and $\widehat{J}/\widehat{J}^{l+1}$, and since $$ {\goth p}_l \subseteq \widehat{J}/\widehat{J}^{l+1} \hbox{ and } {\cal P}_l \subseteq 1 + \widehat{J}/\widehat{J}^{l+1}, $$ it follows that the logarithm and exponential maps induce polynomial bijections between ${\goth p}_l $ and ${\cal P}_l$. Consequently, when $H_1(\pi,{\Bbb Q})$ is finite dimensional, each of the groups ${\cal P}_l$ is a unipotent algebraic group over $F$ with Lie algebra ${\goth p}_l$. It follows that if $H_1(\pi,F)$ is finite dimensional, then ${\cal P}$ is a prounipotent group over $F$. The composition of the canonical inclusion of $\pi$ into $F\pi$ followed by the completion map $F\pi \to F\pi~\widehat{\!}{\;}$ yields a canonical map $ \pi \to F\pi~\widehat{\!}{\;}$. Since the image of this map is contained in ${\cal P}$, there is a canonical homomorphism $\pi \to {\cal P}$. The composition of the natural homomorphism $\pi \to {\cal P}$ with the quotient map ${\cal P} \to {\cal P}_l$ yields a canonical homomorphism $\pi \to {\cal P}_l$. \begin{proposition} \label{dense} If $H_1(\pi,F)$ is finite dimensional, then each of the homomorphisms $\pi \to {\cal P}_l$ has Zariski dense image. \end{proposition} \noindent{\bf Proof.~} Denote the Zariski closure of the image of $\pi$ in ${\cal P}_l$ by ${\cal Z}_l$. Each ${\cal Z}_l$ is an algebraic subgroup of ${\cal P}_l$. Since the composite $$ H_1(\pi;F) \to H_1({\cal Z}_l) \to H_1({\cal P}_l) $$ is an isomorphism, it follows that the second map is surjective. Since ${\cal P}_l$ is unipotent, this implies that the inclusion ${\cal Z}_l \hookrightarrow {\cal P}_l$ is surjective. \quad $\square$ \begin{theorem} \label{description} If $H_1(\pi,F)$ is finite dimensional, then the natural map $\pi \to {\cal P}$ is the Malcev completion of $\pi$ over $F$. \end{theorem} \noindent{\bf Proof.~} By the universal mapping property (\ref{ump}), there is a canonical homomorphism from the Malcev completion ${\cal U}_F$ of $\pi$ to ${\cal P}$. It follows from (\ref{dense}) that this homomorphism is surjective. We now establish injectivity. Suppose that $U$ is a unipotent group defined over $F$. This means that $U$ can be represented as a subgroup of the group of upper triangular unipotent matrices in $GL_n(F)$ for some $n$. A representation $\rho : \pi \to U$ induces a ring homomorphism $$ \tilde{\rho} : F\pi \to gl_n(F). $$ Since the representation is unipotent, it follows that $\tilde{\rho}(J)$ is contained in the set of nilpotent upper triangular matrices. This implies that the kernel of $\tilde{\rho}$ contains $J^n$. Consequently, $\rho$ induces a homomorphism $$ \bar{\rho} : F\pi /J^n \to gl_n(F). $$ Since the image of $J$ is contained in the nilpotent upper triangular matrices, the image of the subgroup ${\cal P}_{n-1}$ of $1+J/J^n$ is contained in the group of unipotent upper triangular matrices. Because the image of $\pi$ is Zariski dense in ${\cal P}_{n-1}$, it follows that the image of ${\cal P}_{n-1}$ is contained in $U$. That is, there is a homomorphism ${\cal P}_{n-1} \to U$ of linear algebraic groups over $F$ such that the diagram $$ \matrix{ \pi &\to &{\cal P}_{n-1} \cr & \mapse{\rho} &\downarrow \cr && U \cr } $$ commutes. It follows that ${\cal U}_F \to {\cal P}$ is injective. \quad $\square$ \begin{corollary} \label{forms} If ${\cal P}_F$ is the Malcev completion of the group $\pi$ over $F$, then the natural homomorphism $$ {\cal P}_F \to {\cal P}_{\Bbb Q}(F) $$ is an isomorphism. \end{corollary} \noindent{\bf Proof.~} This follows as ${\cal P}_F$ is the set of group-like elements of $F\pi~\widehat{\!}{\;}$, while ${\cal P}_{{\Bbb Q}}(F)$ is the set of group-like elements of ${\Bbb Q} \pi ~\widehat{\!}{\;} \otimes F\approx F\pi~\widehat{\!}{\;}$. \quad $\square$ \section{Basic theory of relative completion} \label{basic} In this section we establish several basic properties of relative completion. We begin by recalling some basic facts about group extensions. Once again, $F$ will denote a fixed field of characteristic zero. All algebraic groups will be linear. Suppose that $L$ is an abstract group and that $A$ is an $L$ module. The group of congruence classes of extensions $$ 0 \to A \to G \to L \to 1, $$ where the natural action of $L$ on $A$ is the given action, is naturally isomorphic to $H^2(L;A)$. The identity is the semidirect product $L \semi A$ (\cite[Theorem 4.1, p.~112]{maclane}). If $H^1(L;A)$ vanishes, then any 2 splittings $s_0,s_1 : L \to L \semi A$ are conjugate via an element of $A$ (\cite[Prop.\ 2.1,p.~106]{maclane}). That is, there exists $a \in A$ such that $s_1 = a s_0 a^{-1}$. If $S$ is a connected semisimple algebraic group over $F$, and if $A$ is a rational representation of $S$, then every extension $$ 0 \to A \to G \to S \to 1 $$ in the category of algebraic groups over $F$ splits. Moreover, any 2 splittings $s_0,s_1 : S \to G$ are conjugate by an element of $A$ \cite[p.~185]{humphreys}. These results extend to the case when the kernel is unipotent. For this we need the following construction. \begin{para} {\bf Construction.} \label{construction} Suppose that \begin{equation}\label{G} 1 \to U \to G \to L \to 1 \end{equation} is an extension of an abstract group $L$ by a group $U$. Suppose that $Z$ is a central subgroup of $U$ and that the extension \begin{equation}\label{G/Z} 1 \to U/Z \to G/Z \to L \to 1 \end{equation} is split. Let $s: L \to G/Z$ be a splitting. Pulling back the extension $$ 1 \to Z \to G \to G/Z \to 1 $$ along $s$, one obtains an extension \begin{equation}\label{E_s} 1 \to Z \to E_s \to L \to 1. \end{equation} This determines, and is determined by, a class $\zeta_s$ in $H^2(L;Z)$ which depends only on $s$ up to inner automorphisms by elements of $G$. The extension (\ref{E_s}) splits if and only if $\zeta_s = 0$. \end{para} \begin{proposition}\label{split} If $\zeta_s = 0$, then the extension (\ref{G}) splits. Moreover, if any 2 splittings of (\ref{G/Z}) are conjugate, and if $H^1(L;Z) = 0$, then any 2 splittings of (\ref{G}) are conjugate. \end{proposition} \noindent{\bf Proof.~} If $\zeta_s$ vanishes, there is a splitting $\sigma: L \to E_s$. By composing $\sigma$ with the inclusion of $E_s \hookrightarrow G$, one obtains a splitting of the extension (\ref{G}). Suppose now that $H^1(L;Z)$ vanishes, and that any 2 splittings of $L \to G/Z$ are conjugate. If $s_0,s_1 : L \to G$ are 2 splittings of (\ref{G}), then their reductions $\overline{s}_0, \overline{s}_1 : L \to G/Z$ mod $Z$ are conjugate. We may therefore assume that $s_0$ and $s_1$ agree mod $Z$. The images of $s_0$ and $s_1$ are then contained in the subgroup $E_s$ of $G$ determined by the section $\overline{s}_0 = \overline{s}_1 : L \to G/Z$. Since $H^1(L;Z) = 0$, there is an element $z$ of $Z$ which conjugates $s_0$ into $s_1$. \quad $\square$ \medskip A similar argument, combined with the facts about extensions of algebraic groups at the beginning of this section, can be used to prove the following result. \begin{proposition} Suppose that $S$ is a connected semisimple algebraic group over $F$. If $$ 1 \to U \to G \to S \to 1 $$ is an extension in the category of algebraic groups over $F$, and if $U$ is unipotent, then the extension splits, and any two splittings $S \to G$ are conjugate. \quad $\square$ \end{proposition} By choosing compatible splittings and then taking inverse limits, we obtain the analogous result for proalgebraic groups. \begin{proposition} Suppose that $S$ is a connected semisimple algebraic group over $F$. If $$ 1 \to {\cal U} \to {\cal G} \to S \to 1 $$ is an extension in the category of proalgebraic groups over $F$, and if ${\cal U}$ is prounipotent, then the extension splits, and any two splittings $S \to {\cal G}$ are conjugate. \quad $\square$ \end{proposition} Suppose that $\Gamma$ is a abstract group, $S$ an algebraic group over $F$, and that $\rho : \Gamma \to S$ a representation with Zariski dense image. Denote the image of $\rho$ by $L$ and its kernel by $T$. Thus we have an extension $$ 1 \to T \to \Gamma \to L \to 1. $$ Let $\Gamma \to {\cal G}_F$ be the completion over $F$ of $\Gamma$ relative to $\rho$ and ${\cal U}_F$ its prounipotent radical. There is a commutative diagram $$ \matrix{ 1 & \to & T & \to & \Gamma & \to & L & \to 1 \cr && \downarrow && \downarrow && \downarrow && \cr 1 & \to & {\cal U}_F & \to & {\cal G}_F & \to & S & \to 1 \cr} $$ Denote the unipotent (i.e., the Malcev) completion of $T$ over $F$ by ${\cal T}_F$. The universal mapping property of ${\cal T}_F$ gives a homomorphism $\Phi : {\cal T}_F \to {\cal U}_F$ of prounipotent groups whose composition with the natural map $T\to {\cal T}_F$ is the canonical map $T \to {\cal U}_F$. Denote the kernel of $\Phi$ by ${\cal K}_F$. \begin{proposition} Suppose that $H_1(T;F)$ is a finite dimensional. If the action of $L$ on $H_1(T;F)$ extends to a rational action of $S$, then the kernel ${\cal K}_F$ of $\Phi$ is central in ${\cal T}_F$. \end{proposition} \noindent{\bf Proof.~} First, $\Gamma$ acts on the completion $FT~\widehat{\!}{\;}$ of the group algebra of $T$ by conjugation. This action preserves the filtration by the powers of $\widehat{J}$, so it acts on the associated graded algebra $$ Gr^\bullet_J FT~\widehat{\!}{\;} = \bigoplus_{m=0}^\infty \widehat{J}^m/\widehat{J}^{m+1}. $$ If $H_1(T;F)$ is finite dimensional, each truncation $FT/J^l$ of $FT~\widehat{\!}{\;}$ is finite dimensional. This implies that each of the groups ${\rm Aut\,} FT/J^l$ is an algebraic group. Since ${\rm Aut\,} Gr^\bullet_J FT~\widehat{\!}{\;}$ is generated by $J/J^2 = H_1(T;F)$, it follows that when $H_1(T;F)$ is finite dimensional, ${\rm Aut\,} FT~\widehat{\!}{\;}$, the group of augmentation preserving algebra automorphisms of $FT~\widehat{\!}{\;}$, is a proalgebraic group which is an extension of a subgroup of ${\rm Aut\,} H_1(T;F)$ by a prounipotent group. If the action of $\Gamma$ on $H_1(T;F)$ factors through a rational representation $S \to {\rm Aut\,} H_1(T;F)$ of $S$, then we can form a proalgebraic group extension $$ 1 \to J^{-1} {\rm Aut\,} FT~\widehat{\!}{\;} \to {\cal E} \to S \to 1 $$ of $S$ by the prounipotent radical of ${\rm Aut\,} FT~\widehat{\!}{\;}$ by pulling back the extension $$ 1 \to J^{-1} {\rm Aut\,} FT~\widehat{\!}{\;} \to {\rm Aut\,} FT~\widehat{\!}{\;} \to {\rm Aut\,} H_1(T;F) $$ along $S\to {\rm Aut\,} H_1(T;F)$. The representation $\Gamma \to {\rm Aut\,} FT~\widehat{\!}{\;}$ lifts to a representation $\Gamma \to {\cal E}$ whose composition with the projection ${\cal E}\to S$ is $\rho : \Gamma \to S$. This induces a homomorphism ${\cal G}_F \to {\cal E}$. Since the composite $$ {\cal T}_F \to {\cal G}_F \to {\cal E} \to {\rm Aut\,} FT~\widehat{\!}{\;} $$ is the action of ${\cal T}_F \subseteq FT~\widehat{\!}{\;}$ on $FT~\widehat{\!}{\;}$ by inner automorphisms, it follows that the kernel of this map is the center $Z({\cal T}_F)$ of ${\cal T}_F$. It follows that the kernel of ${\cal T}_F \to {\cal U}_F$ is a subgroup of $Z({\cal T})$. \quad $\square$ \medskip The following result and its proof were communicated to me by P.~Deligne \cite{deligne:letter}. \begin{proposition} Suppose that $S$ is semisimple and that the natural action of $L$ on $H_1(T;F)$ extends to a rational representation of $S$. If $H^1(L;A) = 0$ for all rational representations $A$ of $S$, then $\Phi$ is surjective. \end{proposition} \noindent{\bf Proof.~} The homomorphism ${\cal T}_F \to {\cal U}_F$ is surjective if and only if the induced map $H_1(T;F) \to H_1({\cal U}_F)$ is surjective. Let $A$ be the cokernel of this map. This is a rational representation of $S$ as both $H_1(T;F)$ and $H_1({\cal U}_F)$ are. By pushing out the extension $$ 1 \to {\cal U}_F\to {\cal G}_F \to S \to 1 $$ along the map ${\cal U}_F\to H_1(U) \to A$, we obtain an extension $$ 1 \to A \to G \to S \to 1 $$ of algebraic groups and a homomorphism $\Gamma \to G$ which lifts $\rho$ and has Zariski dense image in $G$. Let $$ 1 \to A \to G^L \to L \to 1 $$ be the restriction of this extension to $L$. Since $A$ is the cokernel of the map ${\cal T}_F \to H_1(U)$, the image of $\Gamma$ in $G$ is $\Gamma /T = L$. So $\rho$ induces a homomorphism $\tilde{\rho} : L \to G$ which has Zariski dense image. The image of $\tilde{\rho}$ lies in $G^L$ and induces a splitting $\sigma : L \to G^L$ of the projection $G^L \to L$. Since $G$ is an algebraic group, there is a splitting $s : S \to G$ of the projection in the category of algebraic groups. This restricts to a splitting $s' : L \to G$. Since $H^1(L;A)$ vanishes, there exists $a\in A$ which conjugates $s'$ into $\sigma$. Thus the image of $\sigma$ is contained in the algebraic subgroup $as(S)a^{-1}$ of $G$. Since the image of $\sigma$ in $G$ is Zariski dense, it follows that $A$ must be trivial. \quad $\square$ \medskip A direct consequence of this result is a criterion for the map $\rho : \Gamma \to S$ itself to be the $\rho$ completion. This criterion is satisfied by arithmetic groups in semisimple groups where each factor has ${\Bbb Q}$ rank $\ge 2$ \cite{ragunathan}. \begin{corollary} \label{comp_L} Suppose that $\rho : \Gamma \to S$ is a homomorphism of an abstract group into a semisimple algebraic group. If $H^1(L;A)$ vanishes for all rational representations of $S$, and if $H^1(T;F)$ is zero (e.g., $\rho$ injective), then the relative completion of $\Gamma$ with respect to $\rho$ is $\rho : \Gamma \to S$. \quad $\square$ \end{corollary} Next we consider the problem of imbedding an extension of $L$ by a unipotent group $U$ in an extension of $S$ by $U$. \begin{proposition}\label{lift} Suppose that $$ 1 \to U \to G \to L \to 1 $$ is a split extension of abstract groups, where $U$ is a unipotent group over $F$ and where the action of $L$ on $H^1(U)$ extends to a rational representation of $S$. If $H^1(L;A)$ vanishes for all rational representations of $S$, then there is an extension $$ 1 \to U \to \tilde{G} \to S \to 1 $$ of algebraic groups such that the original extension is the restriction of this one to $L$. \end{proposition} \noindent{\bf Proof.~} Since the first extension splits, we can write it as a semi direct product $$ G = L \semi U. $$ Denote the Lie algebra of $U$ by ${\goth u}$. The group of Lie algebra automorphisms of ${\goth u}$ is an algebraic group over $F$. It can be expressed as an extension $$ 1 \to J^{-1} {\rm Aut\,} {\goth u} \to {\rm Aut\,} {\goth u} \to {\rm Aut\,} H_1({\goth u}) $$ where the kernel consists of those automorphisms which act trivially on $H^1({\goth u})$, and therefore on the graded quotients of the lower central series of ${\goth u}$. It is a unipotent group. We can pull this extension back along the representation $S \to {\rm Aut\,} H_1({\goth u})$ to obtain an extension $$ 1 \to J^{-1} {\rm Aut\,} {\goth u} \to \tilde{A} \to S \to 1 $$ The representation $L \to {\rm Aut\,} {\goth u}$ lifts to a homomorphism $L \to \tilde{A}$. This induces a homomorphism of the $S$-% completion of $L$ into $\tilde A$. By (\ref{comp_L}), the completion of $L$ is simply the inclusion $L \hookrightarrow S$. That is, the representation $L \to \tilde A$ extends to an algebraic group homomorphism $S \to \tilde A$. This implies that the representation of $L$ on ${\goth u}$ extends to a rational representation $S \to {\rm Aut\,} {\goth u}$. We can therefore form the semi direct product $S \semi U$, which is an algebraic group. The homomorphism $L \semi U \to S \semi U$ exists because of the compatibility of the actions of $L$ and $S$ on $U$. \quad $\square$ \medskip Combining this with (\ref{split}), we obtain: \begin{corollary}\label{split-lift} Suppose that ${\cal U}$ is a prounipotent group over $F$ with $H_1({\cal U})$ finite dimensional, and suppose that ${\cal Z}$ is a central subgroup of ${\cal U}$. Suppose that \begin{equation}\label{original} 1 \to {\cal U} \to G \to L \to 1 \end{equation} is an extension of abstract groups where the action of $L$ on $H_1({\cal U})$ extends to a rational representation of $S$. Suppose that $$ 1 \to {\cal U}/{\cal Z} \to {\cal G} \to S \to 1 $$ is an extension of proalgebraic groups which gives the extension $$ 1 \to {\cal U}/{\cal Z} \to G/{\cal Z} \to L \to 1 $$ when restricted to $L$. If the class in $H^2(L;Z)$ given by (\ref{construction}) vanishes, then there exists an extension $$ 1 \to {\cal U} \to \tilde{{\cal G}} \to S \to 1 $$ of proalgebraic groups whose restriction to $L$ is the extension (\ref{original}). \quad $\square$ \end{corollary} By pushing out the extension $$ 1 \to T \to \Gamma \to L \to 1 $$ along the homomorphism $T \to {\cal T}_F$, we obtain a ``fattening'' $\hat{\Gamma}$ of $\Gamma$. Let ${\cal G}^L$ be the inverse image of $L$ in ${\cal G}_F$. Using the universal mapping property of pushouts, one can show easily that the natural homomorphism $\Gamma \to {\cal G}_F$ induces homomorphism $\hat{\Gamma} \to {\cal G}^L$. These groups fit into a commutative diagram of extensions: $$ \matrix{ 1 & \to & T & \to & \Gamma & \to & L & \to & 1 \cr && \downarrow && \downarrow && \| && \cr 1 & \to & {\cal T}_F & \to & \hat{\Gamma} & \to & L & \to & 1 \cr && \downarrow && \downarrow && \| && \cr 1 & \to & {\cal U}_F & \to & {\cal G}^L & \to & L & \to & 1 \cr && \| && \downarrow && \downarrow && \cr 1 & \to & {\cal U}_F & \to & {\cal G}_F & \to & S & \to & 1 \cr} $$ Next we introduce conditions we need to impose on our extension for the remainder of the section. \begin{para} \label{conditions} We will now assume that the extension $\Gamma$ of $L$ by $T$ satisfies the following conditions. First, $H_1(T;F)$ is finite dimensional and the action of $L$ on it extends to a rational representation of $S$. Next, we assume that $H^1(L;A)$ vanishes for all rational representations of $S$. Finally, we add the new condition that $H^2(L;A)$ vanishes for all {\it nontrivial} irreducible rational representations of $S$. These conditions are satisfied by arithmetic groups in semisimple groups where each factor has real rank at least 8 \cite{borel:triv,borel:twisted}. \end{para} The following result is an immediate consequence of (\ref{split-lift}). \begin{proposition}\label{central} If the conditions (\ref{conditions}) are satisfied, then ${\cal K}_F = \ker \Phi$ is contained in the center of the thickening $\hat{\Gamma}$ of $\Gamma$. \quad $\square$ \end{proposition} Applying the construction (\ref{construction}) to the thickening $\hat{\Gamma}$ of $\Gamma$ and a splitting of ${\cal G}^L \to L$, we obtain an extension \begin{equation}\label{obstn} 0 \to {\cal K}_F \to G \to L \to 1 \end{equation} which is unique up to isomorphism. It follows from (\ref{central}) that this is a central extension. Since $H_1(L;F)$ vanishes, there is a universal central extension with kernel an $F$ vector space. It is the extension $$ 0 \to H_2(L;F) \to \tilde{L} \to L \to 1 $$ with cocycle the identity map $$ \left\{H_2(L;F) \stackrel{id}{\to} H_2(L;F)\right\}\in {\rm Hom}(H_2(L;F),H_2(L;F)) \approx H^2(L;H_2(L;F)). $$ The central extension (\ref{obstn}) is classified by a linear map $\psi_F : H_2(L;F) \to {\cal K}_F$. Because all splittings $s: L \to {\cal G}^L$ are conjugate (\ref{split}), the class of this extension is independent of the choice of the splitting. Since ${\cal K}_F$ is an abelian unipotent group, ${\cal K}_F(k) = {\cal K}_F\otimes k$ for all fields $k$ which contain $F$. The homomorphism $\psi_F$ satisfies the following naturality property: \begin{proposition} If $k$ is an extension field of $F$, then the diagram $$ \matrix{ H_2(L;k) &\to & {\cal K}_k \cr || && \downarrow \cr H_2(L;F)\otimes k & \to & {\cal K}_F(k) \cr } $$ commutes. \quad $\square$ \end{proposition} The next result bounds the size of ${\cal K}_F$. \begin{proposition}\label{surj} If the conditions (\ref{conditions}) hold, then the natural map $\psi_F : H_2(L;F) \to {\cal K}_F$ is surjective. \end{proposition} \noindent{\bf Proof.~} As above, we shall denote by $G$ the central extension of $L$ by ${\cal K}_F$. Let $A$ be the cokernel of $\psi_F$ and $E$ the cokernel of $\psi_F : H_2(L;F) \to G$. Then $E$ is a central extension of $L$ by $A$. Because the composite $H_2(L;F) \to {\cal K}_F \to A$ is trivial, this extension is split. From (\ref{split}) it follows that the extension $$ 1 \to {\cal T}_F/{\rm im\,} \psi_F \to \hat{\Gamma}/{\rm im\,} \psi_F \to L \to 1 $$ is split. By (\ref{lift}), this implies that there is a proalgebraic group ${\cal E}$ which is a semidirect product of $S$ by ${\cal T}_F/{\rm im\,} \psi_F$ into which $\hat{\Gamma}/{\rm im\,}\psi_F$ injects. The map of $\Gamma$ to ${\cal E}$ induces a map ${\cal G}_F \to {\cal E}$. Since the kernel of the map ${\cal T}_F \to {\cal E}$ is ${\rm im\,} \psi_F$, it follows that ${\cal K}_F$ is contained in ${\rm im\,} \psi_F$. \quad $\square$ \begin{corollary}\label{indep} If the conditions (\ref{conditions}) hold, then the natural map ${\cal G}_k \to {\cal G}_F(k)$ associated to a field extension $k:F$ is an isomorphism. \end{corollary} \noindent{\bf Proof.~} By (\ref{forms}), the natural map ${\cal T}_F \to {\cal T}_F(k)$ is an isomorphism. Since ${\cal T}_K \to {\cal U}_K$ is surjective for all fields $K$, the natural map ${\cal U}_k \to {\cal U}_F(k)$ is also surjective. Consequently, the natural map $K_k \to K_F(k)$ is injective. Since ${\cal K}_F$ is abelian unipotent, ${\cal K}_F(k) = {\cal K}_F \otimes k$. But it follows from (\ref{surj}) that ${\cal K}_k \to {\cal K}_F(k)$ is surjective, and therefore an isomorphism. \quad $\square$ \section{The Johnson homomorphism} \label{johnson_homo} This section is a brief review of the construction of Johnson's homomorphism \cite{johnson:survey,johnson:2}. There are two equivalent ways, both due to Johnson, to define a homomorphism $$ T_g \to \Lambda^3 H_1(C;{\Bbb Z}) $$ where $C$ is a compact Riemann surface of genus $g$. Choose a base point $x$ of $C$. The first construction uses the action of $T_g^1$ on $\pi_1(C,x)$. Denote the lower central series of $\pi_1(C,x)$ by $$ \pi_1(C,x) = \pi^1 \supseteq \pi^2 \supseteq \pi^3 \supset \cdots $$ The first graded quotient $\pi^1/\pi^2$ is $H_1(C;{\Bbb Z})$. The second is naturally isomorphic to $\Lambda^2 H_1(C;{\Bbb Z})/\langle q\rangle$, where $q : \Lambda^2 H^1(C;{\Bbb Z}) \to H^2(C;{\Bbb Z}) \approx{\Bbb Z}$ is the cup product. If $\gamma_j$, $j=1,\ldots,2g$, are generators of $\pi_1(C,x)$, then the residue class of the commutator $\gamma_j\gamma_k\gamma_j^{-1}\gamma_k^{-1}$ modulo $\pi^3$ is the element $c_j\wedge c_k$ of $\Lambda^2 H_1(C;{\Bbb Z})/\langle q \rangle$, where $c_j$ denotes the homology class of $\gamma_j$. The form $q$ is just the equivalence class of the standard relation in $\pi_1(C,x)$. If $\phi : (C,x) \to (C,x)$ is a diffeomorphism which represents an element of $T_g^1$, then $\phi$ acts trivially on the homology of $C$. It therefore acts as the identity on each graded quotient of the lower central series of $\pi_1(C,x)$. Define a function $\pi \to \pi$ by taking $\gamma$ to $\phi(\gamma)\gamma^{-1}$. Since $\phi$ acts trivially on $H_1(C)$, it follows that this map takes $\pi^l$ into $\pi^{l+1}$. In particular, it induces a well defined function $$ \tilde{\tau}(\phi) : H_1(C;{\Bbb Z}) \to \Lambda^2 H_1(C;{\Bbb Z})/\langle q \rangle $$ between the first two graded quotients of $\pi$, which is easily seen to be linear. Using Poincar\'e duality, $\tilde{\tau}(\phi)$ can be regarded as an element of $$ H_1(C;{\Bbb Z}) \otimes \left(\Lambda^2 H_1(C;{\Bbb Z})/\langle q \rangle\right). $$ The map $\phi \mapsto \tilde{\tau}(\phi)$ induces a group homomorphism $$ T_g^1 \to H_1(C;{\Bbb Z}) \otimes \left(\Lambda^2 H_1(C;{\Bbb Z})/\langle q \rangle\right). $$ and therefore a homomorphism $$ \hat{\tau} : H_1(T_g^1) \to H_1(C;{\Bbb Z}) \otimes \left(\Lambda^2 H_1(C;{\Bbb Z})/\langle q \rangle\right). $$ There is a natural inclusion $$ \Lambda^3 H_1(C;{\Bbb Z}) \to H_1(C;{\Bbb Z}) \otimes \left(\Lambda^2 H_1(C;{\Bbb Z})/\langle q \rangle\right). $$ defined by $$ x\wedge y\wedge z \mapsto x\otimes (y\wedge z) + y \otimes (z \wedge x) + z \otimes (x \wedge y). $$ Johnson has proved that the image of $\hat{\tau}$ is contained in the image of this map, so that $\hat{\tau}$ induces a homomorphism $$ \tau_g^1 : H_1(T_g^1) \to \Lambda^3 H^1(C;{\Bbb Z}). $$ It is not difficult to check that this homomorphism is $Sp_g({\Bbb Z})$ equivariant. This story can be extended to $T_g$ as follows. There is a natural extension $$ 1 \to \pi_1(C,x) \to T_g^1 \to T_g \to 1. $$ Applying $H_1$, we obtain the diagram $$ \matrix{ H_1(C;{\Bbb Z}) & \to & H_1(T_g^1) &\to & H_1(T_g) &\to & 0\cr &&\mapdown{\tau_g^1} && &&\cr &&\Lambda^3 H_1(C;{\Bbb Z})&&&&\cr} $$ whose top row is exact. Identify the lower group with $H_3({\rm Jac\,} C;{\Bbb Z})$. Since ${\rm Jac\,} C$ is a group with torsion free homology, the group multiplication induces a product on its homology which is called the {\it Pontrjagin product}. The composite of $\tau_g^1$ with the map from $H_1(C)$ takes a class in $H_1({\rm Jac\,} C)$ to its Pontrjagin product with $q =[C] \in H_2({\rm Jac\,} C;{\Bbb Z})$. It follows that $\tau_g^1$ induces a map $$ \tau_g : H_1(T_g) \to \Lambda^3 H_1(C;{\Bbb Z})/\left(q\wedge H_1(C;{\Bbb Z})\right). $$ The following fundamental theorem is due to Dennis Johnson. \begin{theorem}\label{johnson:3} When $g\ge 3$, the homomorphisms $\tau_g^1$ and $\tau_g$ are isomorphisms modulo 2 torsion. \end{theorem} The second way to construct a map $T_g^1 \to \Lambda^3 H_1(C;{\Bbb Z})$ is as follows. Suppose that $\phi$ represents an element of $T_g^1$. Let $M_\phi \to S^1$ be the bundle over the circle constructed by identifying the point $(z,1)$ of $C\times [0,1]$ with the point $(\phi(z),0)$. Since $\phi$ fixes the basepoint $x$, the map $t \to (x,t)$ induces a section of $M_\phi \to S^1$. Similarly, one can construct the bundle of jacobians; this is trivial as $\phi$ acts trivially on $H_1(C)$. One can imbed $M_\phi$ in this bundle of jacobians using this section of basepoints. Let $p$ be the projection of the bundle of jacobians onto one of its fibers. Then one has the 3 cycle $p_\ast M_\phi$ in ${\rm Jac\,} C$. \begin{proposition}\label{tau_2} {\rm \bf \cite{johnson:survey}} When $g\ge 3$, the homology class $$ p_\ast [M_\phi] \in H_3({\rm Jac\,} C;{\Bbb Z}) \approx \Lambda^3 H_1(C;{\Bbb Z}) $$ is $\tau_g^1(\phi)$. \quad $\square$ \end{proposition} This can be proved, for example, by checking that both maps agree on what Johnson calls ``bounding pair" maps. These maps generate the Torelli group when $g\ge 3$ \cite{johnson:1}. \section{The cycle $C - C^-$} \label{cycle} In this section we relate the algebraic cycle $C - C^-$ to Johnson's homomorphism. Let $C$ be a compact Riemann surface. Denote its jacobian by ${\rm Jac\,} C$. This is defined to be ${\rm Pic}^0 C$, the group of divisors of degree zero on $C$ modulo principal divisors. Each divisor $D$ of degree zero may be written as the boundary of a topological 1-chain: $D = \partial \gamma$. Taking $D$ to the functional $\omega \mapsto \int_\gamma \omega$ on the space $\Omega(C)$ of holomorphic 1-forms yields a well defined map \begin{equation}\label{AJ} {\rm Pic}^0 C \to \Omega(C)^\ast/H_1(C;{\Bbb Z}). \end{equation} This is an isomorphism by Abel's Theorem. For each $x\in C$, we have an Abel-Jacobi map $$ \nu_x : C \to {\rm Jac\,} C $$ which is defined by $\nu_x(y) = y - x$. This map is an imbedding if the genus $g$ of $C$ is $\ge 1$. Denote its image by $C_x$. This is an algebraic 1-cycle in ${\rm Jac\,} C$. Denote the involution $D\mapsto -D$ of ${\rm Jac\,} C$ by $i$, and the image of $C_x$ under this involution by $C_x^-$. Since $i$ induces $-id$ on $H^1({\rm Jac\,} C;{\Bbb Z})$, and since $i^\ast$ is a ring homomorphism, we see that $i^\ast$ acts as $(-1)^k$ on $H_k({\rm Jac\,} C)$. It follows that $C_x$ and $C_x^-$ are homologically equivalent. Set $Z_x = C_x - C_x^-$. This is a homologically trivial 1-cycle. Griffiths has a construction which associates to a homologically trivial analytic cycle in a compact K\"ahler manifold a point in a complex torus. His construction is a generalization the construction of the map (\ref{AJ}). We review this construction briefly. Suppose that $X$ is a compact K\"ahler manifold, and that $Z$ is an analytic $k$-cycle in $X$ which is homologous to 0. Write $Z = \partial \Gamma$, where $\Gamma$ is a topological $2k+1$ chain in $X$. Define $$ F^p H^m(X) = \bigoplus_{s \ge p} H^{s,m-s}(X). $$ Each class in $F^pH^m(X)$ can be represented by a closed form where each term of a local expression in terms of local holomorphic coordinates $(z_1,\ldots, z_n)$ has at least $p$ $dz_j$s. Integrating such representatives of classes in $F^{k+1}H^{2k+1}(X)$ over $\Gamma$ gives a well defined functional $$ \int_\Gamma : F^{k+1} H^{2k+1}(X) \to {\Bbb C}. $$ The choice of $\Gamma$ is unique up to a topological $2k+1$ cycle. So $Z$ determines a point of the complex torus $$ J_k(X) := F^{k+1} H^{2k+1}(X)^\ast/H_{2k+1}(X;{\Bbb Z}). $$ This group is called the {\it $k$th intermediate jacobian of $X$}. In our case, the cycle $Z_x$ determines a point $\zeta_x(C)$ in the intermediate jacobian $$ J_1({\rm Jac\,} C) = F^2H^3({\rm Jac\,} C)^\ast/H_3({\rm Jac\,} C; {\Bbb Z}). $$ The homology class of $C_x$ is easily seen to be independent of $x$. Taking the Pontrjagin product with $[C]$ defines an injective map $$ H_1({\rm Jac\,} C;{\Bbb Z}) \hookrightarrow H_3({\rm Jac\,} C;{\Bbb Z}). $$ The dual $H^3({\rm Jac\,} C) \to H^1(C)$ is a morphism of Hodge structures of type $(-1,-1)$ --- that is, $H^{s,t}$ gets mapped into $H^{s-1,t-1}$. It follows that this map induces an imbedding $$ \Phi : {\rm Jac\,} C \hookrightarrow J_1({\rm Jac\,}(C)) $$ of complex tori. We shall denote the cokernel of this map by $JQ_1({\rm Jac\,} C)$. It is trivial when $g < 3$. The following result is not difficult; a proof may be found in \cite{pulte}. \begin{proposition} If $x,y \in C$, then $$ \zeta_x(C) - \zeta_y(C) = 2 \Phi(x-y). $$ In particular, the image of $\zeta_x(C)$ in $JQ_1({\rm Jac\,} C)$ is independent of $x$. \end{proposition} Denote the common image of the $\zeta_x(C)$ in $JQ_1({\rm Jac\,} C)$ by $\zeta(C)$. We now suppose that the genus $g$ of $C$ is $\ge 3$. Fix a level $l$ so that the moduli space ${\cal M}_g^n(l)$ of curves of genus $g$ and $n$ marked points with a level $l$ structure is smooth. (Any $l\ge 3$ will do for the time being.) Denote the space of principally polarized abelian varieties of dimension $g$ with a level $l$ structure by ${\cal A}_g(l)$. One can easily construct bundles ${\cal J}_m \to {\cal A}_g(l)$ of complex tori whose fiber over the abelian variety $A$ is $J_m(A)$. The pullback of ${\cal J}_0$ along the period map ${\cal M}_g(l) \to {\cal A}_g(l)$ is the bundle of jacobians associated to the universal curve. The imbedding ${\cal J}_0 \hookrightarrow {\cal J}_1$ defined over ${\cal M}_g(l)$ extends to all of ${\cal A}_g(l)$ as the homology class $[C] \in H_2({\rm Jac\,} C;{\Bbb Z})$ extends to a class $q \in H_2(A;{\Bbb Z})$ for every abelian variety $A$. Denote the bundle of quotient tori by ${\cal Q} \to {\cal A}_g(l)$. Denote the level $l$ congruence subgroup of $Sp_g({\Bbb Z})$ by $L(l)$. Since ${\cal A}_g(l)$ is an Eilenberg-Mac Lane space $K(L(l),1)$, and since the fiber of ${\cal J}_1 \to {\cal A}_g(l)$ is a $K(H_3({\rm Jac\,} C ;{\Bbb Z}),1)$, it follows that ${\cal J}_1$ is also an Eilenberg-Mac Lane space whose fundamental group is an extension of $L(l)$ by $H_3({\rm Jac\,} C;{\Bbb Z})$. Since this bundle has a section, viz., the zero section, this extension splits. The action of $L(l)$ on $H_3({\rm Jac\,} C;{\Bbb Z})$ is the restriction of the third exterior power of the fundamental representation of $Sp_g$. There is a similar story with ${\cal J}_1$ replaced by ${\cal Q}$. We shall denote the quotient $\Lambda^3 H_1(C;{\Bbb Z})/\left([C]\cdot H_1(C;{\Bbb Z})\right)$ by $Q\Lambda^3 H_1(C)$. This is the fundamental group of $JQ_1({\rm Jac\,} C)$. \begin{proposition} The spaces ${\cal J}_1$ and ${\cal Q}$ are Eilenberg-Mac Lane spaces with fundamental groups $$ \pi_1({\cal J}_1,0_C) \approx L(l) \semi \Lambda^3 H_1(C;{\Bbb Z}) $$ and $$ \pi_1({\cal Q},0_C) \approx L(l) \semi Q\Lambda^3 H_1(C), $$ respectively. Here $0_C$ denotes the identity element in the fiber over ${\rm Jac\,} C$. \end{proposition} The normal functions $(C,x) \mapsto \zeta_x(C)$ and $C \mapsto \zeta(C)$ give lifts of the period map: $$ \matrix{ && {\cal J}_1 \cr & \mapne{\zeta_g^1} & \downarrow \cr {\cal M}_g^1(l) & \to & {\cal A}_g(l) \cr } \qquad \matrix{ && {\cal Q} \cr & \mapne{\zeta_g} & \downarrow \cr {\cal M}_g(l) & \to & {\cal A}_g(l) \cr } $$ These induce maps of fundamental groups $$ {\zeta_g^1}_\ast : \Gamma_g^1(l) \to L(l)\semi \Lambda_3 H_1(C) $$ and $$ {\zeta_g}_\ast : \Gamma_g(l) \to L(l) \semi Q\Lambda^3 H_1(C) $$ Since these maps commute with the canonical projections to $L(l)$, these induce $L(l)$ equivariant maps $$ \zeta_g^1 : H_1(T_g^1) \to \Lambda^3 H_1(C;Z), \quad \zeta_g^1 : H_1(T_g) \to Q\Lambda^3 H_1(C;Z) $$ The following result follows easily from (\ref{tau_2}). The factor of 2 arises as both $C$ and $C^-$ each contribute a copy of the Johnson homomorphism. \begin{proposition}\label{double} The map $\zeta_g^n$ is twice Johnson's map $\tau_g^n$ for $n=0,1$. \quad $\square$ \end{proposition} It is natural to try to give a ``motivic description'' of the Johnson homomorphism rather than of twice it. Looijenga (unpublished) has done this by constructing a normal function which compares the cycle $C_x$ to a fixed topological (but not algebraic) cycle in ${\rm Jac\,} C$ which is homologous to $C_x$. At the cost of being more abstract, we give another description which does not make use of Looijenga's topological cycle. We will only consider the pointed case, the unpointed case being similar. For each pointed curve $(C,x)$, the cycle $C_x$ determines a point $c_x$ in the Deligne cohomology group $H^{2g-2}_{\cal D}({\rm Jac\,}(C),{\Bbb Z}(g-1))$. This group is an extension of the Hodge classes $H^{g-1,g-1}_{\Bbb Z}({\rm Jac\,} C)$ by the intermediate jacobian $J_1({\rm Jac\,} C)$: $$ 0 \to J_1({\rm Jac\,} C) \to H^{2g-2}_{\cal D}({\rm Jac\,}(C),{\Bbb Z}(g-1)) \to H^{g-1,g-1}_{\Bbb Z}({\rm Jac\,} C) \to 0 $$ One can consider the bundle over ${\cal A}_g(l)$ whose fiber over the abelian variety $A$ is $H^{2g-2}_{\cal D}(A,{\Bbb Z}(g-1))$. The subbundle whose fiber over $A$ is the $J_1(A)$ coset of the class of the polarization $q \in H^{g-1,g-1}_{\Bbb Z}(A)$ is a principal ${\cal J}_1$ torsor over ${\cal A}_g(l)$. Denote it by ${\cal Z} \to {\cal A}_g(l)$. The cycle gives a lift $$ \matrix{ && {\cal Z} \cr & \mapne{c} & \downarrow \cr {\cal M}_g^1(l) & \to & {\cal A}_g(l) \cr} $$ of the period map. The total space ${\cal Z}$ is an Eilenberg Mac Lane space whose fundamental group is an extension of $L(l)$ by $\Lambda^3 H_1(C;Z)$. The map induced by $c$ on fundamental groups induces the Johnson homomorphism $\tau_g^1$. This follows directly from (\ref{tau_2}). \section{Completion of mapping class groups} Denote the completion of the mapping class group $\Gamma_{g,r}^n$ with respect to the canonical representation $\Gamma_{g,r}^n \to Sp_g$ by ${\cal G}_{g,r}^n$ and its prounipotent radical by ${\cal U}_{g,r}^n$. Denote the Malcev completion of the Torelli group $T_{g,r}^n$ by ${\cal T}_{g,r}^n$ and the kernel of the natural homomorphism ${\cal T}_{g,r}^n \to {\cal U}_{g,r}^n$ by ${\cal K}_{g,r}^n$. These groups are all defined over ${\Bbb Q}$ by (\ref{indep}). For all $g \ge 3$, and all arithmetic subgroups $L$ of $Sp_g({\Bbb Z})$, $H^1(L;A)$ vanishes for all rational representations $A$ of $Sp_g$. By the results of Borel \cite{borel:triv,borel:twisted}, the hypotheses (\ref{conditions}) are satisfied by all arithmetic subgroups $L$ of $Sp_g({\Bbb Z})$ and $H^2(L;{\Bbb Q})$ is one dimensional when $g\ge 8$. This, combined with (\ref{surj}) yields the following result. \begin{proposition} For all $g \ge 3$ and all $n, r \ge 0$, the natural map ${\cal T}_{g,r}^n \to {\cal U}_{g,r}^n$ is surjective. When $g\ge 8$, the kernel ${\cal K}_{g,r}^n$ is either trivial or isomorphic to ${\Bbb Q}$. \quad $\square$ \end{proposition} Our main result is: \begin{theorem}\label{main} For all $g \ge 3$ and all $r,n\ge 0$, the kernel ${\cal K}_{g,r}^n$ is non-trivial, so that ${\cal K}_{g,r}^n \approx {\Bbb Q}$ when $g\ge 8$. \end{theorem} Let $\lambda_1,\ldots,\lambda_g$ be a fundamental set of weights of $Sp_g$. For a dominant integral weight $\lambda$, denote the irreducible representation with highest weight $\lambda$ by $V(\lambda)$. In this section we will reduce the proof of Theorem \ref{main} to the proof of the the case $g=3$ and $r=n=0$. The following assertion follows easily by induction on $n$ and $r$ from Johnson's result. \begin{proposition} If $g \ge 3$, then for all $n,r\ge 0$, there is an $Sp_g$ equivariant isomorphism $$ H_1(T_{g,r}^n;{\Bbb Q}) \approx V(\lambda_3) \oplus V(\lambda_1)^{n+r}. \quad\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3 $$ \end{proposition} The important fact for us is that the multiplicity of $\lambda_3$ in $H_1(T_{g,r}^n;{\Bbb Q})$ is always 1. Denote the second graded quotient of the lower central series of ${\cal T}_{g,r}^n$ by ${\cal V}_{g,r}^n$. The commutator induces a linear surjection $$ \Lambda^2 H_1(T_{g,r}^n;{\Bbb Q}) \to {\cal V}_{g,r}^n $$ which is $Sp_g$ equivariant. By Schur's lemma, there is a unique copy of the trivial representation in $\Lambda^2 V(\lambda_3)$. Let $\beta_{g,r}^n : {\Bbb Q} \to {\cal V}$ be the composite $$ {\Bbb Q} \hookrightarrow \Lambda^2 V(\lambda_3) \to \Lambda^2 H_1(T_{g,r}^n;{\Bbb Q}) \to {\cal V}_{g,r}^n, $$ where the first map is the inclusion of the trivial representation. Consider the following assertions: \medskip \noindent{$\boldmath A_{g,r}^n$:} \qquad The map $\beta_{g,r}^n$ is injective. \medskip \begin{proposition}\label{red_1} If $h\ge g\ge 3$, $s\ge r\ge 0$ and $m\ge n\ge $, then $A_{g,r}^n$ implies $A_{h,s}^m$. Furthermore $A_{g,1}$ implies $A_g$. In particular, $A_3$ implies $A_{g,r}^n$ for all $g \ge 3$ and $n,r \ge 0$. \end{proposition} \noindent{\bf Proof.~} It is easy to see that $\beta_{g,r}^n$ is the composite of $\beta_{g,s}^m$ with the canonical quotient map ${\cal V}_{g,s}^m \to {\cal V}_{g,r}^n$ whenever $m\ge n$ and $s\ge r$. So $A_{g,r}^n$ implies $A_{g,s}^m$. Moreover, when $r\ge 1$, the composition of $\beta_{g,r}^n$ with the map ${\cal V}_{g,r}^n \to {\cal V}_{g+1,r}^n$ induced by any one of the natural maps $\Gamma_{g,r}^n \to \Gamma_{g+1,r}^n$ is $\beta_{g+1,r}^n$. So, in this case, $A_{g,r}^n$ implies $A_{g+1,r}^n$. To see that $A_{g,1}$ implies $A_g$, consider the group extension $$ 1 \to \pi_1(T^\ast_1 C,\vec{v}) \to T_{g,1} \to T_g \to 1 $$ where $T^\ast_1 C$ denotes the unit cotangent bundle of $C$. It is not difficult to show that ${\cal V}_{g,1}$ is the direct sum of ${\cal V}_g$ and the second graded quotient of the lower central series of $\pi_1(T^\ast_1 C,\vec{v})$, from which the assertion follows. \quad $\square$ \medskip The assertions $A_{g,r}^n$ can be proved using Harer's computations of $H^2(\Gamma_{g,r}^n;{\Bbb Q})$ \cite{harer:h2,harer:h3}. However, we will prove $A_3$ in the course establishing the rest of Theorem \ref{main} directly, without appeal to Harer's computation. Denote the fattening (see discussion following (\ref{split})) of the mapping class group $\Gamma_{g,r}^n$ by $\hat{\Gamma}_{g,r}^n$. This is an extension $$ 1 \to {\cal T}_{g,r}^n \to \hat{\Gamma}_{g,r}^n \to Sp_g({\Bbb Z}) \to 1. $$ Dividing out by the commutator subgroup of ${\cal T}_{g,r}^n$, we obtain an extension \begin{equation}\label{aux} 0 \to H_1(T_{g,r}^n;{\Bbb Q}) \to {\cal E}_{g,r}^n \to Sp_g({\Bbb Z}) \to 1. \end{equation} This extension is split. This can be seen using a straightforward generalization of (\ref{double}). Now suppose $A_{g,r}^n$ holds. Then there is a quotient $G_{g,r}^n$ of ${\cal T}_{g,r}^n$ which is an extension of $H_1(T_{g,r}^n;{\Bbb Q})$ by ${\Bbb Q}$. The cocycle of the extension being a non-zero multiple of the polarization $$ \theta \in \Lambda^2 V(\lambda_3) \subseteq H^2(T_{g,r}^n;{\Bbb Q}). $$ Dividing $\hat{\Gamma}_{g,r}^n$ by the kernel of ${\cal T}_{g,r}^n \to G_{g,r}^n$, we obtain an extension \begin{equation}\label{ext} 1 \to G_{g,r}^n \to E_{g,r}^n \to Sp_g({\Bbb Z}) \to 1. \end{equation} Since the extension (\ref{aux}) splits, we can apply the construction (\ref{construction}) to obtain an extension $$ 0 \to {\Bbb Q} \to H_{g,r}^n \to Sp_g({\Bbb Z}) \to 1 $$ or equivalently, a class $e_{g,r}^n \in H^2(Sp_g({\Bbb Z});{\Bbb Q})$. By (\ref{split}), the non-triviality of this class is the obstruction to splitting the extension (\ref{ext}), which, by (\ref{lift}) is the obstruction to imbedding it in an algebraic group extension of $Sp_g$ by $G_{g,r}^n$. This proves the following statement. \begin{proposition} If $A_{g,r}^n$ holds and if $e_{g,r}^n$ is non-zero, then ${\cal K}_{g,r}^n$ is non-trivial. \end{proposition} To reduce the proof of Theorem (\ref{main}) to the genus 3 case, we need to relate the classes $e_{g,r}^n$. \begin{proposition}\label{red_2} For fixed $g \ge 3$, the classes $e_{g,r}^n$ are all equal. The image of $e_{g+1,1}$ under the natural map $H^2(Sp_{g+1}({\Bbb Z});{\Bbb Q}) \to H^2(Sp_g({\Bbb Z});{\Bbb Q})$ is $e_{g,1}$. \end{proposition} \noindent{\bf Proof.~} Both statements follow from the naturality of the construction. \quad $\square$ \medskip Combining (\ref{red_1}) and (\ref{red_2}), we have: \begin{proposition}\label{main_3} If $A_3$ holds and $e_3$ is non trivial, then Theorem \ref{main} is true. \end{proposition} \section{Proof of Theorem \protect\ref{main}} \label{proof} We prove Theorem \ref{main} by proving (\ref{main_3}). In this section we assume that the reader is familiar with mixed Hodge theory. We will use the notation and conventions of \cite[\S\S 2--3]{hain:heights}. The moduli space ${\cal A}_g$ can be thought of as the moduli space of principally polarized Hodge structures of weight $-1$, level 1 and dimension $2g$; the abelian variety $A \in {\cal A}_g$ corresponds to the Hodge structure $H_1(A)$ and its natural polarization. We can construct bundles over ${\cal A}_g$ by considering moduli spaces of various mixed Hodge structures derived from such a Hodge structure of weight $-1$. To guarantee that we have a smooth moduli space, we fix a level $l$ so that ${\cal A}_g(l)$ is smooth. For a Hodge structure $H\in{\cal A}_g$, where $g \ge 3$, with principal polarization $q$, we define $QH$ to be the Hodge structure $$ \left[\Lambda^3 H/\left( q\wedge H\right)\right] \otimes {\Bbb Z}(-1) $$ which is of weight $-1$. Denote the dual Hodge structure $$ {\rm Hom}(QH,{\Bbb Z}(1)) \approx \ \ker\left\{ \_\wedge q : \Lambda^{2g-3} H \to \Lambda ^{2g-1} H\right\}\otimes {\Bbb Z}(2-g) $$ by $PH$. The set of all mixed Hodge structures with weight graded quotients ${\Bbb Z}$ and $QH$ is naturally isomorphic to the complex torus $$ J(QH) := QH_{\Bbb C} / \left(F^0 QH_{\Bbb C} + QH_{\Bbb Z} \right). $$ If $H = H_1({\rm Jac\,} C)$, then $J(QH)$ is the torus $JQ_1({\rm Jac\,} C)$ defined in \S\ref{cycle}. As we let $H$ vary over ${\cal A}_g(l)$, we obtain the bundle ${\cal Q} \to {\cal A}_g(l)$ of intermediate jacobians constructed in \S\ref{cycle}. The set of all mixed Hodge structures with weight graded quotients $QH$ and ${\Bbb Z}(1)$ is naturally isomorphic to the complex torus $$ J(PH) := PH_{\Bbb C} / \left(F^0 PH_{\Bbb C} + PH_{\Bbb Z} \right). $$ This torus is the dual of $J(QH)$. Performing this construction for each $H$ in ${\cal A}_g(l)$, we obtain a bundle ${\cal P} \to {\cal A}_g(l)$ of complex tori. We now construct a line bundle over the fibered product $$ {\cal P}\times_{{\cal A}_g(l)} {\cal Q} \to {\cal A}_g(l). $$ It is the {\it biextension line bundle}. For a Hodge structure $H\in {\cal A}_g$, let $B(H)$ be the set of mixed Hodge structure with weight graded quotients canonically isomorphic to ${\Bbb Z}$, $QH$, and ${\Bbb Z}(1)$. Set $$ G_{\Bbb Z} = \pmatrix{ 1 & QH_{\Bbb Z} & {\Bbb Z}(1) \cr 0 & 1 & PH_{\Bbb Z} \cr 0 & 0 & 1 \cr } \quad G = \pmatrix{ 1 & QH_{\Bbb C} & {\Bbb C} \cr 0 & 1 & PH_{\Bbb C} \cr 0 & 0 & 1 \cr } $$ $$ F^0 G_{\Bbb Z} = \pmatrix{ 1 & F^0 QH & 0 \cr 0 & 1 & F^0 PH \cr 0 & 0 & 1 \cr }. $$ There is a natural isomorphism $$ B(H) = G_{\Bbb Z}\backslash G / F^0 G. $$ The natural projection $$ B(H) \to J(QH) \times J(PH), $$ which takes $V \in B(H)$ to $(V/W_{-2},W_{-1}V)$, is a principal ${\Bbb C}^\ast$ bundle. Doing this construction over ${\cal A}_g(l)$, we obtain a ${\Bbb C}^\ast$ bundle $$ {\cal B} \to {\cal Q}\times_{{\cal A}_g(l)} {\cal P} $$ Denote the corresponding line bundle by ${\cal L} \to {\cal Q}\times_{{\cal A}_g(l)} {\cal P}$. Since the fiber of ${\cal B} \to {\cal A}_g(l)$ over $H$ is the nilmanifold $B(H)$, which is an Eilenberg-Mac Lane space, it follows that ${\cal B}$ is also an Eilenberg-Mac Lane space whose fundamental group of ${\cal B}$ is an extension \begin{equation}\label{extn} 1 \to G_{\Bbb Z} \to \pi_1({\cal B},\ast) \to L(l) \to 1. \end{equation} Taking $H\in {\cal A}_g(l)$ to the split biextension ${\Bbb Z} \oplus H \oplus {\Bbb Z}(1)$ defines a section of the bundle ${\cal B} \to {\cal A}_g$. It follows that the extension (\ref{extn}) is split. \begin{proposition} $\pi_1({\cal B},\ast) \approx L(l) \semi G_{\Bbb Z}\quad $. \quad $\square$ \end{proposition} We now restrict ourselves to genus 3 and consider the problem of lifting the period map ${\cal M}_3(l) \to {\cal A}_3(l)$ to ${\cal B}$. The idea is that such a lifting will be the period map of a variation of mixed Hodge structure. To this end we define certain algebraic cycles which are just more canonical versions of the cycle $C-C^-$ considered in \S\ref{cycle}. Suppose that $C$ is a curve of genus 3 and that $\alpha$ is a theta characteristic of $C$ --- i.e., $\alpha$ is a square root of the canonical divisor $\kappa_C$. Denote the algebraic cycle corresponding to the canonical inclusion $C \hookrightarrow {\rm Pic}^1C$ by $C$. Denote the involution $x \mapsto \alpha -x$ of ${\rm Pic}^1 C$ by $i^\alpha$. For $D \in {\rm Pic}^0C$, denote the translation map $x \mapsto x + D$ by $$ \tau^D : {\rm Pic}^1 C \to {\rm Pic}^1 C. $$ For $D \in {\rm Pic}^0 C$, define $C_D = \tau^D_\ast C$ and $Z_{\alpha,D} = C_D - i^\alpha_\ast C_D$. Each $Z_{\alpha,D}$ is homologous to zero. Set $Z_\alpha = Z_{\alpha,0}$. Let $\Theta_\alpha$ be the {\it theta divisor} $$ \left\{ x+y - \alpha : x,y \in C \right\} \subseteq {\rm Pic}^0 C $$ and $\Delta$ be the {\it difference divisor} $$ \left\{x - y : x, y \in C \right\} \subseteq {\rm Pic}^0 C. $$ The following fact is easily verified. \begin{proposition}\label{disjoint} The cycles $Z_\alpha$ and $Z_{\alpha,D}$ have disjoint supports if and only if $D \not\in \Theta_\alpha \cup \Delta$. \quad $\square$ \end{proposition} Now choose a point $\delta \in {\rm Pic}^0 C$ of order 2 such that $\beta := \alpha + \delta$ is an {\it even} theta characteristic. (i.e., $h^0(C,\beta) = 0$ or 2.) \begin{proposition}\label{hyperelliptic} The cycles $Z_\alpha$ and $Z_{\alpha,\delta}$ have disjoint supports except when $C$ is hyperelliptic and either $\delta$ is the difference between two distinct Weierstrass points or $\alpha + \delta$ is the hyperelliptic series. \end{proposition} \noindent{\bf Proof.~} By (\ref{disjoint}), $Z_\alpha$ and $Z_{\alpha,\delta}$ intersect if and only if $\delta \in \Delta$ or $\delta \in \vartheta_\alpha$. In the first case, there exist $x,y \in C$ such that $x-y = \delta\neq 0$. So $2x - 2y=0$, which implies that $C$ is hyperelliptic and that $x$ and $y$ are distinct Weierstrass points. In the second case, there are $x,y \in C$ such that $$ x+y = \alpha + \delta, $$ which implies that $\alpha + \delta$ is an effective theta characteristic. Since $\alpha + \delta$ is even by assumption, $h^0(C,\alpha + \delta) = 2$, which implies that $C$ is hyperelliptic and that $\alpha + \delta$ is the hyperelliptic series. \quad $\square$ \medskip Next we use these cycles to construct various variations of mixed Hodge structure whose period maps give lifts of the period map ${\cal M}_3(l) \to {\cal A}_3(l)$ to ${\cal Q}\times_{{\cal A}_3(l)} {\cal P}$, and to ${\cal B}$ generically. For each $D \in {\rm Pic}^0 C$, one has the extension of mixed Hodge structure $$ 0 \to H_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \to H_3({\rm Pic}^1 C, Z_{\alpha,D};{\Bbb Z}(-1)) \to {\Bbb Z} \to 0 $$ where the generator 1 of ${\Bbb Z}$ is the image of any relative class $[\Gamma]$ with $\partial \Gamma = Z_{\alpha,D}$. Pushing this extension out along the projection $$ H_3({\rm Pic}^1 C) \to QH_3({\rm Pic}^1 C) $$ one obtains an extension \begin{equation}\label{Q} 0 \to QH_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \to E \to {\Bbb Z} \to 0. \end{equation} This extension determines the same point in $J(QH_3({\rm Pic}^1 C))\approx Q_1({\rm Jac\,} C)$ as the cycle $C_x - C_x^-$, and is independent of the choice of $D$. Dually, one can consider the extension \begin{equation}\label{extension} 0 \to {\Bbb Z}(1) \to H_3({\rm Pic}^1 C - Z_{\alpha,D}; {\Bbb Z}(-1)) \to H_3({\rm Pic}^1 C; {\Bbb Z}(-1)) \to 0 \end{equation} which comes from the Gysin sequence. Here the canonical generator of ${\Bbb Z}(1)$ is the boundary of any 4 ball which is transverse to $Z_{\alpha,D}$ and has intersection number 1 with it. \begin{proposition}\label{map} If $D$ is a point of order 2 in ${\rm Jac\,} C$, then there is a morphism of Hodge structures $$ H_1(C,{\Bbb Z}) \to H_3({\rm Pic}^1 C - Z_{\alpha,D}; {\Bbb Z}(-1)) $$ whose composition with the natural map $$ H_3({\rm Pic}^1 C - Z_{\alpha,D}; {\Bbb Z}(-1)) \to H_3({\rm Pic}^1 C ; {\Bbb Z}(-1)) $$ is the map $\underline{\phantom{x}}\times [C]$ given by Pontrjagin product with $[C]$. \end{proposition} \noindent{\bf Proof.~} The exact sequence of Hodge structures $$ 0\to H_1(C;{\Bbb Z}) \stackrel{\times[C]}{\longrightarrow} H_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \to QH_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \to 0 $$ induces an exact sequence of Ext groups $$ 0\to {\rm Ext}^1(QH_3({\rm Pic}^1 C,{\Bbb Z}(-1)),{\Bbb Z}(1)) \to \phantom{xxxxxxxxxxxxxxxxxxxxx} $$ $$ \phantom{xxxxxxxxxxxxx} {\rm Ext}^1(H_3({\rm Pic}^1 C,{\Bbb Z}(-1)),{\Bbb Z}(1)) \to {\rm Ext}^1(H_1(C;{\Bbb Z}),{\Bbb Z}(1)) \to 0. $$ This sequence may be identified naturally with the sequence $$ 0 \to JPH_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \to JH_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \stackrel{\phi}{\to} {\rm Jac\,} C \to 0, $$ where $\phi$ is the map induced by the morphism of Hodge structures $H_3({\rm Jac\,} C;{\Bbb Z}) \to H_1(C;{\Bbb Z})$ defined by $$ x\times y \times z \mapsto (x\cdot y)z + (y\cdot z) x + (z\cdot x) y. $$ Here $x,y,z$ are elements of $H_1(C;{\Bbb Z})$, $\times$ denotes the Pontrjagin product, and $(\phantom{x}\cdot\phantom{y})$ denotes the intersection pairing. To prove the assertion, it suffices to show that the image in ${\rm Jac\,} C$ of the class of the extension (\ref{extension}) vanishes. Lefschetz duality gives an isomorphism of (\ref{extension}) with the the extension $$ 0 \to H_3({\rm Pic}^1 C;{\Bbb Z}(-1)) \to H_3({\rm Pic}^1 C,Z_{\alpha,D};{\Bbb Z}(-1)) \to {\Bbb Z} \to 0 $$ It follows directly from \cite[(6.7)]{pulte} that the image of this extension under the map $\phi$ is $K_C - 2 (\alpha + D) = 0$. \quad $\square$ \medskip Taking $D=0$ and dividing out by this copy of $H_1(C;{\Bbb Z})$, we obtain an extension \begin{equation}\label{P} 0 \to {\Bbb Z}(1) \to F \to QH_3({\rm Pic}^1 C; {\Bbb Z}(-1)) \to 0. \end{equation} Let ${\cal M}_3(l,\alpha,\delta)$ be the moduli space of genus 3 curves with a level $l$ structure, a distinguished even theta characteristic $\alpha$, and a distinguished point $\delta \in {\rm Pic}^0 C$ of order 2 such that $\alpha + \delta$ is also an even theta characteristic. Denote the universal jacobian over ${\cal M}_3(l,\alpha,\delta)$ by ${\cal J} \to {\cal M}_3(l,\alpha,\delta)$. The period maps for the extensions (\ref{Q}), (\ref{P}), respectively, define maps ${\cal J} \to {\cal Q}$ and ${\cal J} \to {\cal P}$. These induce a map $\phi$ into their fibered product over ${\cal A}_3(l)$ such that the diagram $$ \matrix{ {\cal J} & \mapright{\phi} & {\cal Q} \times_{{\cal A}_3(l)} {\cal P} \cr \downarrow & & \downarrow \cr {\cal M}_3(l,\alpha,\delta) & \to & {\cal A}_3(l) \cr } $$ commutes. Pulling back the ${\Bbb C}^\ast$ bundle ${\cal B} \to {\cal Q}\times_{{\cal A}_3(l)}{\cal P}$ along $\phi$ gives a ${\Bbb C}^\ast$ bundle ${\cal L}^\ast \to {\cal J}$. Denote the corresponding line bundle by ${\cal L}$. Denote the relative difference divisor in ${\cal J}$ by ${\cal D}$ and the relative theta divisor associated to $\alpha$ by $\vartheta_\alpha$ \begin{lemma}\label{total_chern} The Chern class of this line bundle is the divisor ${\cal J}$ is $2{\cal D} - 4\vartheta_\alpha$. \end{lemma} \noindent{\bf Proof.~} We construct a meromorphic section of ${\cal L}$. A point of ${\cal J}$ is a curve $C$ and a point $D$ of ${\rm Jac\,} C$. According to (\ref{disjoint}), the cycles $Z_\alpha$ and $Z_{\alpha,D}$ have disjoint supports when $D \not\in \Delta\cup \Theta_\alpha$. In this case we can consider the mixed Hodge structure $H_3({\rm Pic}^1 C - Z_{\alpha}, Z_{\alpha,D};{\Bbb Z}(-1))$.\footnote{This is everywhere a local system. One has to replace it with another group when $C$ is hyperelliptic and either $\alpha$ or $\alpha + \delta$ is the hyperelliptic series. For details, see the footnote on page 887 of \cite{hain:heights}.} Dividing this biextension out by the image of the composite $$ H_1(C,{\Bbb Z}) \to H_3({\rm Pic}^1 C - Z_{\alpha}; {\Bbb Z}(-1)) \to H_3({\rm Pic}^1 C - Z_\alpha, Z_{\alpha,D}; {\Bbb Z}(-1)) $$ of the map of (\ref{extension}) with the natural inclusion produces a biextension $b_{C,\alpha}$ with weight graded quotients canonically isomorphic to $$ {\Bbb Z}, \quad QH_3({\rm Jac\,} C;{\Bbb Z}(-1)),\quad {\Bbb Z}(1); $$ the generator 1 of ${\Bbb Z}$ corresponding to any $\Gamma$ with $\partial \Gamma = Z_{\alpha,D}$, and the canonical generator $2\pi i$ of ${\Bbb Z}(1)$ being the class of the boundary of any small 4 ball which is transverse to $Z_{\alpha}$ and having intersection number 1 with it. This defines a lift $$ \tilde{\phi} : {\cal J} - ({\cal D}\cup \vartheta_\alpha) \to {\cal B} $$ of the map $\phi : {\cal J} \to {\cal Q} \times_{{\cal A}_3(l)}{\cal P}$. It therefore defines a nowhere vanishing holomorphic section $s$ of ${\cal L} \to {\cal J}$ on the complement of ${\cal D} \cup \vartheta_\alpha$. It follows from \cite[(3.4.3)]{hain:heights} that $s$ extends to a meromorphic section of ${\cal L}$ on all of ${\cal J}$. Consequently, the Chern class of this bundle is supported on the divisor ${\cal D} \cup \vartheta_\alpha$. Since the divisors ${\cal D}$ and $\vartheta_\alpha$ are irreducible, the Chern class can be computed by restricting to a general enough fiber. With the aid of (\ref{map}), and the formula \cite[(3.2.11)]{hain:heights}, one can easily show that the height of the biextension $b_{C,\alpha}$ equals that of $$ H_3({\rm Pic}^1 C - Z_\alpha, {\Bbb Z}_{\alpha,D};{\Bbb Z}(-1)). $$ It follows from the main theorem of \cite{hain:heights} that the divisor of $s$ restricted to ${\rm Jac\,} C$ is $2D - 4\Theta_\alpha$ for all $C$. The result follows. \quad $\square$ \medskip The point $\delta$ of order 2 is a section of the bundle ${\cal J} \to {\cal M}_3(l,\alpha,\delta)$. A lift $\zeta : {\cal M}_3(l,\alpha,\delta) \to {\cal Q}\times_{{\cal A}_3(l)}{\cal P}$ of the period map can be defined by composing $\delta$ with $\phi$. The pullback of the line bundle ${\cal L}$ along $\delta$ equals the pullback of the biextension line bundle to ${\cal M}_3$. It follows that the Chern class of this line bundle is $2 \delta^\ast ({\cal D} - 2 \vartheta_\alpha).$ \begin{proposition}\label{chern_class} If $\alpha$ and $\alpha +\delta$ are even theta characteristics, then the pushforward of the divisor $\delta^\ast ({\cal D} - 2 \vartheta_\alpha)$ in ${\cal M}_3(l,\alpha,\delta)$ to ${\cal M}_3(l)$ is $28.35$ times the hyperelliptic locus. Consequently, the line bundle $$ \delta^\ast{\cal L}\in {\rm Pic}\, {\cal M}_3(l,\alpha,\delta)\otimes{\Bbb Q} $$ is non-trivial. \end{proposition} In the proof of this result, we will need the following fact. \begin{lemma}\label{transverse} The section $\delta: {\cal M}_3(l,\alpha,\delta) \to {\cal J}$ is transverse to the divisors ${\cal D}$ and $\vartheta_\alpha$. \end{lemma} \noindent{\bf Proof.~} We first prove that $\delta$ intersects $\vartheta_\alpha$ transversally. We view ${\cal M}_3(l,\alpha,\delta)$ as a subvariety of ${\cal J}$ via the section $\delta$. Let $(C,\alpha,\delta)$ be a point in $\vartheta_\alpha$. Then, by (\ref{hyperelliptic}), $C$ is hyperelliptic, and $\alpha + \delta$ is the hyperelliptic series. So $h^0(\alpha) = 0$ and $h^0(\alpha+\delta) = 2$. Let $Z_0$ be the period matrix of $C$ with respect to some symplectic basis of $H_1(C;{\Bbb Z})$, and $\theta_\alpha(u,Z)$ the theta function which defines $\vartheta_\alpha$. The point $\delta$ of order 2 may be viewed as a function $\delta(Z)$. Set $\delta_0 = \delta(Z_0)$. Since $\delta_0 \in \Theta_\alpha$, $\theta_\alpha(\delta_0,Z_0)=0$. We have to show that there exist $a,b$ such that $$ {\partial \over \partial Z_{ab}} \theta_\alpha(\delta(Z),Z)|_{Z_0} \neq 0. $$ By Riemann's Theorem \cite[p.\ 348]{griffiths-harris}, the multiplicity of $\delta$ on $\Theta_\alpha$ is $h^0(\alpha +\delta) = 2$. That is, \begin{equation}\label{vanishing} {\partial \theta_\alpha \over \partial u_a} (\delta_0,Z_0) = 0 \end{equation} for all indices $a$, but there exist indices $a,b$ such that $$ {\partial^2 \theta_\alpha \over \partial u_a\partial u_b} (\delta_0,Z_0) \neq 0. $$ Substituting (\ref{vanishing}) into the chain rule, we have $$ {\partial \theta_\alpha \over \partial Z_{ab}}(\delta(Z),Z)|_{Z_0} = \sum_{j=1}^g {\partial \theta_\alpha\over \partial u_j} (\delta_0,Z_0) {\partial u_j\over \partial Z_{ab}}(\delta_0,Z_0) + {\partial \theta_\alpha \over \partial Z_{ab}}(\delta_0,Z_0) = {\partial \theta_\alpha \over \partial Z_{ab}}(\delta_0,Z_0). $$ Plugging this into the heat equation, we obtain the desired result: $$ {\partial \theta_\alpha \over \partial Z_{ab}}(\delta(Z),Z)|_{Z_0}= {\partial \theta_\alpha \over \partial Z_{ab}}(\delta_0,Z_0) = 2\pi i (1 + \delta_{ab}) {\partial^2 \theta_\alpha \over \partial u_a \partial u_b} (\delta_0,Z_0) \neq 0. $$ To prove that $\delta$ is transverse to ${\cal D}$, we use an argument suggested to us by Nick Katz. Consider a family of curves $C \to {\rm Spec\,} {\Bbb C}[\epsilon]/(\epsilon^2)$ over the dual numbers. There is a relative difference divisor $\Delta\to {\rm Spec\,} {\Bbb C}[\epsilon]/(\epsilon^2)$ contained in the Picard scheme ${\rm Pic}^0 C \to {\rm Spec\,} {\Bbb C}[\epsilon]/(\epsilon^2)$. Suppose that we have a point of order 2 $$ \delta : {\rm Spec\,} {\Bbb C}[\epsilon]/(\epsilon^2) \to {\rm Pic}^0 C, $$ defined over the dual numbers which lies in $\Delta$. To prove transversality, it suffices to show that $C$ is hyperelliptic over the dual numbers. But this is immediate as $\delta$ gives a 2:1 map $C \to {\Bbb P}^1$ defined over the dual numbers. \quad $\square$ \medskip \noindent{\bf Proof of (\ref{chern_class}).} Denote the hyperelliptic series of a hyperelliptic curve by $H$. It follows from (\ref{total_chern}) and (\ref{transverse}) that $$ \delta^\ast {\cal D} = {\cal H}_\Delta\quad\hbox{and}\quad \delta^\ast \vartheta_\alpha = {\cal H}_0 $$ where $$ {\cal H}_\Delta = \left\{ (C,\alpha,\delta) : C \hbox{ is hyperelliptic }, \delta \in \Delta\hbox{ and } h^0(C,\alpha + \delta) \hbox{ is even}\right\} $$ and $$ {\cal H}_0 = \left\{ (C,\alpha,\delta) : C \hbox{ is hyperelliptic }, \alpha \neq H, \alpha + \delta = H \right\}. $$ If $C$ is hyperelliptic and if $\delta = x - y \in \Delta$, then $x$ and $y$ are distinct Weierstrass points, and $$ H + \delta = 2y + x -y = x + y $$ which is an odd theta characteristic. It follows that $$ {\cal H}_\Delta = \left\{ (C,\alpha,\delta) : C \hbox{ is hyperelliptic }, \alpha \neq H, \delta \in \Delta\hbox{ and } h^0(C,\alpha + \delta) \hbox{ even}\right\}. $$ Apart from the hyperelliptic series, every even theta characteristic on a hyperelliptic curve $C$ is of the form $$ -H + p_1 + p_2 + p_3 + p_4 = - H + q_1 + q_2 + q_3 + q_4 $$ where $p_1,\ldots,p_4, q_1,\ldots,q_4$ are the Weierstrass points. If $\alpha = -H + p_1 + p_2 + p_3 + p_4 $ and $\alpha + x-y$ is an even theta characteristic, then it is not difficult to show that $x-y = q_i - p_j$ for some $i,j$. So, for each even theta characteristic $\alpha \neq H$, there are 16 points $\delta\in \Delta$ such that $\alpha + \delta$ is also an even theta characteristic. Since there are 35 even theta characteristics $\alpha \neq H$, it follows that ${\cal H}_\Delta$ has degree 16.35 over the hyperelliptic locus ${\cal H}$ of ${\cal M}_3(l)$. Since there is only one point $\delta$ of order 2 such that $\alpha + \delta = H$, ${\cal H}_0$ has degree 35 over ${\cal H}$. Putting this together we see that the pushforward of $\delta^\ast{\cal L}$ to ${\cal M}_3$ is $$ \pi_\ast (2{\cal H}_\Delta - 4{\cal H}_0) = (2.16.35 - 4.35){\cal H} = 28.35 {\cal H}. \quad \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3 $$ We are now ready to prove (\ref{main_3}). Denote the subgroup of $\Gamma_3$ which corresponds to ${\cal M}_3(l,\alpha,\delta)$ by $\Gamma_3(l,\alpha,\delta)$, and its intersection with $T_3$ by $T_3(\alpha,\delta)$. \medskip \noindent{\bf Proof of (\ref{main_3}).} Let $N$ be the line bundle over ${\cal A}_3(l)$ which is the determinant of $R^1f_\ast {\cal O}$, where $f: {\cal J} \to {\cal A}_3(l)$ is the universal abelian variety. The restriction of this to ${\cal M}_3(l)$ is ${\cal H}/9$ \cite[p.~134]{harris_j}. We shall also denote its pullback to ${\cal Q}\times_{{\cal A}_3(l)}{\cal P}$ by $N$. It follows from (\ref{chern_class}) that the line bundle ${\cal L} \otimes N^{\otimes(-9.28.35)}$ pulls back to the trivial line bundle over ${\cal M}_3(l,\alpha,\delta)$. There is therefore a lift of the period map $$ {\cal M}_3(l,\alpha,\delta) \to \left({\cal L} \otimes N^{\otimes(-9.28.35)}\right)^\ast $$ to the ${\Bbb C}^\ast$ bundle associated to ${\cal L} \otimes N^{\otimes(-9.28.35)}$. This induces a group homomorphism $$ \Gamma_3(l,\alpha,\delta) \to \pi_1(\left({\cal L} \otimes N^{\otimes(-9.28.35)}\right)^\ast,\ast). $$ This last group is an extension $$ 1 \to G_{\Bbb Z} \to \pi_1(\left({\cal L} \otimes N^{\otimes(-9.28.35)}\right)^\ast,\ast) \to L(l) \to 1. $$ It follows from (\ref{double}) and the fact that $T_3(\alpha,\delta)$ has finite index in $T_3$ that the image of the map $$ H_1(T_3(\alpha,\delta);{\Bbb Q}) \to H_1(G_{\Bbb Z};{\Bbb Q}) = QH_3({\rm Jac\,} C;{\Bbb Q}) \oplus PH_3({\rm Jac\,} C;{\Bbb Q}) \approx V(\lambda_3)^2 $$ is the diagonal copy of $V(\lambda_3)$. The restriction of the extension $$ 1 \to {\Bbb Q} \to G_{\Bbb Q}\to V(\lambda^3)^2 \to 1 $$ to the the diagonal is the extension given by the polarization of $V(\lambda_3)$. It follows that the homomorphism $$ {\Bbb Q} \hookrightarrow \Lambda^2 H_1(T_3;{\Bbb Q}) \to {\Bbb Q} $$ given by evaluating the bracket on the polarization is an isomorphism, as claimed. Second, the element of $H^2(L(l);{\Bbb Q})$ which corresponds to the extension $$ 0 \to {\Bbb Q} \to H \to L(l) \to 1 $$ constructed from the Torelli group is just the Chern class of the pullback of the line bundle ${\cal L} \otimes N^{\otimes(-9.28.35)}$ pulled back to ${\cal A}_3(l)$ along the zero section of ${\cal Q} \times_{{\cal A}_3(l)}{\cal P} \to {\cal A}_3(l)$. This is just $-9.28.35c_1(N)$, which is nonzero in $H^2({\cal A}_3(l);{\Bbb Q})$. Consequently, the extension is non-trivial as claimed. \quad $\square$ \medskip \bibliographystyle{plain}

9705

alg-geom/9705011

en

https://arxiv.org/abs/alg-geom/9705011

alg-geom/9705011

Alex Degtyarev

Alexander Degtyarev (Steklov Math. Inst., St.Petersburg, and Bilkent University)

On the Pontrjagin-Viro form

AMSTeX with amsppt and epsf.tex, 24 pages with 4 figures

Topology, ergodic theory, real algebraic geometry, 71--94, Amer. Math. Soc. Transl. Ser. 2, v202, Amer. Math. Soc., Providence, RI, 2001

A new invariant, the Pontrjagin-Viro form, of algebraic surfaces is introduced and studied. It is related to various Rokhlin-Guillou-Marin forms and generalizes Mikhalkin's complex separation. The form is calculated for all real Enriques surfaces for which it is well defined.

alg-geom

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9705

alg-geom/9705014

en

https://arxiv.org/abs/alg-geom/9705014

alg-geom/9705014

Paul Bressler

P. Bressler, R. Nest, B. Tsygan

Riemann-Roch theorems via deformation quantization

We give a proof of a conjecture of P. Schapira and J.-P. Schneiders on the characteristic classes of D-modules.

alg-geom

\section{Introduction} In this note we outline the proof of a formula expressing the Euler class of a perfect complex of modules over a symplectic deformation quantization of a complex manifold in terms of the Chern character of the associated symbol complex, the $\hat A$-class of the manifold and a characteristic class of the deformation quantization. As a consequence we obtain the conjecture of P.Schapira and J.-P.Schneiders (\cite{SS}, Conjecture 8.5, see Conjecture \ref{conj:ch}) and the corollaries thereof. In order to formulate the conjecture one needs a construction of a local Chern character (cf. \cite{SS}, p.93). Such a construction is provided in Section \ref{section:char} and appears to be new. Specifically, we define the Euler class and the Chern character of a perfect complex of sheaves of modules over a sheaf of algebras by ``sheafifying'' the Dennis trace map and the Goodwillie-Jones map defined by R.McCarthy in the generality of exact categories in \cite{McC}. The consrtuction, as described in Section \ref{section:char}, applies to sheaves of algebras on a topological space, but, in fact, can be carried out for a sheaf of algebras on a site (using, for example, the work of J.F.Jardine). We reduce the Riemann-Roch theorem fot the Euler class to the Riemann-Roch theorem for periodic cyclic cochains of deformed sheaves of algebras of functions on a symplectic manifold. We reduce the latter to the local Riemann-Roch theorem for periodic cyclic cochains of the Weyl algebra. The latter result is contained in \cite{NT1}, \cite{NT2} and draws on the ideas in \cite{FT2}. For the sake of brevity we restrict our attention to the ``absolute'' case. Analogous constructions can be carried out for families of symplectic manifolds without any additional difficulties. Two techniques whose discussion is not included here are \begin{itemize} \item Chern--Weil construction of Lie algebra cocycles with coefficients ``constant up to homotopy'' (used to reduce the Riemann-Roch theorem to a local statement); \item noncommutative differential calculus for periodic cyclic cochains needed in the proof of the local Riemann-Roch theorem. \end{itemize} Admittedly, our presentation of the preliminaries is exteremely sketchy A more detailed exposition will be given in a future paper. Here are the contents of the paper in brief. In Section \ref{section:ell} we recall some of the basic definitions and facts about elliptic pairs leading up to Conjecture \ref{conj:ch}. In Section \ref{section:quant} we review basic properties of quantized rings of functions on symplectic manifolds and relate them to algebras of differential and microdifferential operators. This relationship allows us to place the traditional Riemann-Roch theorem into the context of deformation quantization. Section \ref{section:char} contains the background material on Hochschild and cyclic homology, and their relationship to algebraic K-theory necessary to define the Euler class and the Chern character of a perfect complex of sheaves of modules over a sheaf of algebras. In Section \ref{section:cx-man} we summarize the resuts on the Hochschild and cyclic homology of algebras of (micro)differential operators as well as quantized function algebras and compare characteristic classes and trace density maps. Section \ref{section:RR} is devoted to the statement of our main technical result (Theorem \ref{thm:main}) which may be considered as a local Riemann-Roch type theorem for periodic cyclic chains of the quantized ring of functions. We apply Theorem \ref{thm:main} to characteristic classes of perfect complexes of modules over the quantized rings of functions as well as the algebras of differential and microdifferential operators to obtain Conjecture \ref{conj:ch} and analogous statements as corollaries. Section \ref{section:formalRR} leads up to the statement of Theorem \ref{thm:formalRR} which is the analog of Theorem \ref{thm:main} in the particular case when the symplectic manifold in question is the formal neighborhood of the origin in a symplectic vector space. In Section \ref{section:GF} we review the basic facts concerning Fedosov connections and Gel'fand-Fuchs cohomology necessary to, loosely speaking, establish a map from Theorem \ref{thm:formalRR} to Theorem \ref{thm:main}. As a consequence of the existence of this (Gel'fand-Fuchs) map the former theorem implies the latter. The authors would like to thank J.-L.Brylinski and one another for inspiring discussions. \section{The index theory for elliptic pairs}\label{section:ell} The application of our results to the index theory of elliptic pairs is of particular importance. Here we recall the results of \cite{SS} restricting ourselves to the ``absolute'' case for the sake of simplicity. \subsection{} Let $X$ be a complex manifold. An elliptic pair $({\cal M}^\bullet,F^\bullet)$ on $X$ consists of an object ${\cal M}^\bullet$ of $\operatorname{D}^b_{good}({\cal D}^{op}_X)$ (the bounded derived category of complexes of right ${\cal D}_X$-modules with {\em good} \footnote{A ${\cal D}_X$-module is called {\em good} if it admits a good filtration in a neighborhood of every compact subset of $X$.} cohomology and an object $F^\bullet$ of $\operatorname{D}^b_{{\Bbb R}-c}(X)$ (the bounded derived category of ${\Bbb R}$-constructible complexes of sheaves of vector spaces over ${\Bbb C}$ on X) which satisfy \[ \operatorname{char}({\cal M}^\bullet)\cap\operatorname{SS}(F^\bullet)\subseteq T^*_XX\ . \] If \[ \operatorname{Supp}({\cal M}^\bullet,F^\bullet)\overset{def}{=}\operatorname{Supp}{\cal M}^\bullet\cap\operatorname{Supp} F^\bullet \] (where the support of a complex of sheaves is understood to be the cohomological support) is compact one has \[ \dim H^\bullet(X;F^\bullet\otimes{\cal M}^\bullet\otimes^{{\Bbb L}}_{{\cal D}_X}{\cal O}_X) <\infty \ . \] Thus, the Euler characteristic \[ \chi(X;F^\bullet\otimes{\cal M}^\bullet\otimes^{{\Bbb L}}_{{\cal D}_X}{\cal O}_X) \overset{def}{=}\sum_{i} (-1)^i \dim H^i(X;F^\bullet\otimes{\cal M}^\bullet\otimes^{{\Bbb L}}_{{\cal D}_X}{\cal O}_X) \] is defined. As particular cases one obtains \begin{itemize} \item the Euler characteristic $\chi(X;{\cal L}^\bullet)$ of $X$ with coefficients in a compactly supported perfect complex ${\cal L}^\bullet$ of ${\cal O}_X$-modules (taking $F^\bullet = {\Bbb C}_X$, ${\cal M}^\bullet = {\cal L}^\bullet\otimes_{{\cal O}_X}{\cal D}_X$); \item the index of an elliptic complex ${\cal L}^\bullet$ on a compact real analytic manifold $X_0$ (taking $X$ to be a complexification of $X_0$ such that ${\cal L}^\bullet$ extends to a complex $\tilde{{\cal L}^\bullet}$ of ${\cal O}_X$-modules and differential operators, $F^\bullet = {\Bbb C}_{X_0}$, ${\cal M}^\bullet = \operatorname{Diff}({\cal O}_X,\tilde{{\cal L}^\bullet})$). \end{itemize} The index theorem for elliptic pairs (\cite{SS}, Theorem 5.1) says that, for an elliptic pair $({\cal M}^\bullet,F^\bullet)$ with compact support on $X$ of dimension $\dim_{{\Bbb C}}X = d$ \[ \chi(X;F^\bullet\otimes{\cal M}^\bullet\otimes^{{\Bbb L}}_{{\cal D}_X}{\cal O}_X) = \int_{T^*X}\mu\operatorname{eu}({\cal M}^\bullet)\smile\mu\operatorname{eu}(F^\bullet) \] where $\mu\operatorname{eu}({\cal M}^\bullet)\in H^{2d}_{\operatorname{char}({\cal M}^\bullet)}(T^*X;{\Bbb C})$ and $\mu\operatorname{eu}(F^\bullet)\in H^{2d}_{\operatorname{SS}(F^\bullet)}(T^*X;{\Bbb C})$ are defined in \cite{SS}. The class $\mu\operatorname{eu}(F^\bullet)$ is the characteristic cycle of the constructible complex $F^\bullet$ as defined by Kashiwara (see \cite{KS} for more details). For example, if $Y\subset X$ is a closed real analytic submanifold, one has $\mu\operatorname{eu}({\Bbb C}_Y) = [T^*_YX]$. With regard to $\mu\operatorname{eu}({\cal M}^\bullet)$ P.Schapira and J.-P.Schneiders conjectured that it is related to a certain characteristic class of the symbol $\sigma({\cal M}^\bullet)$ of ${\cal M}^\bullet$ (\cite{SS}, Conjecture 8.5, see Conjecture \ref{conj:ch}). \subsection{The Riemann-Roch type formula} Suppose that ${\cal M}^\bullet$ is an object of $\operatorname{D}^b_{good}({\cal D}^{op}_X)$. Let $\pi : T^*X\to X$ denote the projection. If ${\cal M}^\bullet$ admits a gobal {\em good} filtration (and this is the case when $X$ is compact) then the symbol complex of ${\cal M}$ is defined by \[ \sigma({\cal M}^\bullet) = \pi^{-1}gr{\cal M}^\bullet\otimes_{\pi^{-1}gr{\cal D}_X} {\cal O}_{T^*X}\ . \] The assumption that the filtration is good amounts to the fact that $\sigma({\cal M}^\bullet)$ has ${\cal O}_{T^*X}$-coherent cohomology, and the characteristic variety is defined by \[ \operatorname{char}({\cal M}^\bullet)\overset{def}{=}\operatorname{Supp}\sigma({\cal M}^\bullet)\ . \] For $\Lambda$ a closed subvariety of $T^*X$ let $K^0_\Lambda(T^*X)$ denote the Grothendieck group of perfect complexes of ${\cal O}_{T^*X}$-modules supported on $\Lambda$ (i.e. acyclic on the complement of $\Lambda$ in $T^*X$). For $\Lambda$ containing $\operatorname{char}({\cal M})$ let $\sigma_\Lambda({\cal M}^\bullet)$ denote the class of $\sigma({\cal M}^\bullet)$ in $K^0_\Lambda(T^*X)$. \begin{remark} As is easy to show, both the characteristic variety and the class of the symbol in the Grothendieck group are independent of the choice of the good filtration, thus the existence of a good filtration locally is sufficient to define $\sigma_\Lambda({\cal M}^\bullet)$. \end{remark} One can define the Chern character (see \eqref{map:ch}) \[ ch_\Lambda : K^0_\Lambda(T^*X)\to\bigoplus_i H^{2i}_\Lambda(T^*X;{\Bbb C})\ . \] For $\alpha$ an element of a graded object let $[\alpha]^p$ denote the homogeneous component of $\alpha$ of degree $p$. In \cite{SS}, P.Schapira and J.-P.Schneiders make the following conjecture. \begin{conj}\label{conj:ch} For ${\cal M}^\bullet$ in $\operatorname{D}^b_{good}({\cal D}^{op}_X)$, $\Lambda$ a conic subvariety of $T^*X$ containing $\operatorname{char}({\cal M}^\bullet)$ \[ \mu\operatorname{eu}({\cal M}^\bullet) = \left[ ch_\Lambda(\sigma({\cal M}^\bullet)\smile \pi^*Td(TX)\right]^{2d} \] \end{conj} We will obtain the above conjecture as a corollary of a Riemann-Roch type formula in the context of deformation quantization of symplectic manifolds. \section{Deformation quantization} \label{section:quant} \subsection{Review of deformation quantization} A {\em deformation quantization} of a manifold $M$ is a formal one parameter deformation of the structure sheaf ${\cal O}_M$, i.e. a sheaf of algebras ${\Bbb A}^\hbar_M$ flat over ${\Bbb C} [[\hbar]]$ together with an isomoprhism of algebras ${\Bbb A}^\hbar_M\otimes_{{\Bbb C} [[\hbar]]}{\Bbb C}\to{\cal O}_M$. The formula \[ \lbrace f, g\rbrace = \frac{1}{\hbar}[\tilde f,\tilde g ] + \hbar\cdot {\Bbb A}^\hbar_M\ , \] where $f$ and $g$ are two local sections of ${\cal O}_M$ and $\tilde f$, $\tilde g$ are their respective lifts ${\Bbb A}^\hbar_M$, defines a Poisson structure on $M$ called the Poisson structure associated to the deformation quantization ${\Bbb A}^\hbar_M$. \begin{remark} The definition of ``deformation quantization'' as given above is essentially the one given by \cite{BFFLS} in the case of $C^\infty$ manifolds. It is not at all clear whether the scope of generality of the above definition above is sufficiently broad ... \end{remark} The deformation quantization ${\Bbb A}^\hbar_M$ is called {\em symplectic} if the associated Poisson structure is nondegenerate. In this case $M$ is symplectic. In what follows we will only consider symplectic deformation quantizations, so assume that ${\Bbb A}^\hbar_M$ is symplectic from now on. It is known (and not difficult to show) that all symplectic deformation quantizations of $M$ of dimension $\dim_{{\Bbb C}}M = 2d$ are locally isomorphic to the standard deformation quantization of ${\Bbb C}^{2d}$. That is, for any point $x\in M$ and small neighborhoods $U$ of $x$ and $U'$ of the origin in ${\Bbb C}^{2d}$ there is an isomorphism \begin{equation}\label{loc-iso-h} {\Bbb A}^\hbar_{{\Bbb C}^{2d}}(U')\overset{def}{=} {\cal O}_{{\Bbb C}^{2d}}(U')[[\hbar ]]\overset{\sim}{\to}{\Bbb A}^\hbar_M(U) \end{equation} of algebras over ${\Bbb C} [[\hbar ]]$, continuous in the $\hbar$-adic topology, where the product on ${\Bbb A}^\hbar_{{\Bbb C}^{2d}}(U')$ is given, in coordinates $x_1,\ldots,x_d,\xi_1,\ldots,\xi_d$ on ${\Bbb C}^{2d}$ by the standard Weyl product \[ (f\ast g)(\underline x,\underline\xi) = \\ exp\left(\frac{\sqrt{-1}\hbar}{2}\sum_{i=1}^d\left( \frac{\partial\ }{\partial\xi_i}\frac{\partial\ }{\partial y_i}- \frac{\partial~}{\partial\eta_i}\frac{\partial~}{\partial x_i}\right)\right) f(\underline x,\underline\xi)g(\underline y,\underline\eta) \vert_{\overset{\underline x=\underline y}{\underline\xi =\underline\eta}} \] where $\underline x = (x_1,\ldots,x_d),\ \underline\xi = (\xi_1,\ldots,\xi_d), \ \underline y = (y_1,\ldots,y_d),\ \underline\eta = (\eta_1,\ldots,\eta_d)$. Note that the reduction of \eqref{loc-iso-h} modulo $\hbar$ is an isomorphism of Poisson algebras \[ {\cal O}_{{\Bbb C}^{2d}}(U')\overset{\sim}{\to}{\cal O}_M(U) \] with the Poisson brackets associated to the standard symplectic structure on ${\Bbb C}^{2d}$ and to the deformation quantization respectively. In particular (the images of) $x_1,\ldots,x_d,\xi_1,\ldots,\xi_d$ form a Darboux coordinate system on $U\subset M$. To a symplectic deformation quantization ${\Bbb A}^\hbar_M$ one associates a characteristic class $\theta\in H^2(M;\frac{1}{\sqrt{-1}\hbar}{\Bbb C}[[\hbar]])$ with the property that the coefficient of $\frac{1}{\sqrt{-1}\hbar}$ is the class of the symplectic form associated to the deformation quantization. \subsection{Microlocalization}\label{subsec:microloc} We start with a coherent ${\cal D}_X$-module ${\cal M}$ equipped with a good filtration $F_\bullet{\cal M}$ and consider the graded module \[ {\cal R}{\cal M}\overset{def}{=} \bigoplus_q F_q{\cal M}\cdot\hbar^q \] over the graded ring \[ {\cal R}{\cal D}_X \overset{def}{=} \bigoplus_q F_q{\cal D}_X\cdot\hbar^q \subset {\cal D}_X [\hbar ] \] where $F_\bullet{\cal D}_X$ is the filtration by the order of the differential operator. There exists a deformation quantization ${\Bbb A}^\hbar_{T^*X}$ of $T^*X$, i.e. a formal deformation of the structure sheaf ${\cal O}_{T^*X}$ and faithfully flat maps \[ \pi^{-1}{\cal R}{\cal D}_X @>>> {\cal R}{\cal E}_X @>>> {\Bbb A}^\hbar_{T^*X} \] of algebras over ${\Bbb C} [\hbar ]$ where ${\cal E}_X$ is the sheaf (on $T^*X$) of microdifferential operators. The characteristic class $\theta$ of the deformation ${\Bbb A}^\hbar_{T^*X}$ is equal to $\displaystyle\frac12\pi^*c_1(TX)$ (note that the sympledctic form is exact in this case). Consider the ``microlocalization'' \[ \mu{\cal M}\overset{def}{=}\pi^{-1}{\cal R}{\cal M}\otimes_{\pi^{-1}{\cal R}{\cal D}_X}{\Bbb A}^\hbar_{T^*X} \] of the filtered ${\cal D}_X$-module ${\cal M}$. Note that $\mu{\cal M}$ is $\hbar$-torsion free. Then, clearly, there is an isomorphism \[ \sigma({\cal M})\overset{\sim}{=}\mu{\cal M}\otimes_{{\Bbb A}^\hbar_{T^*X}}{\cal O}_{T^*X}\overset{\sim}{=} \mu{\cal M}/\mu{\cal M}\cdot\hbar\ . \] Defining the symbol of an $\hbar$-torsion free ${\Bbb A}^\hbar_{T^*X}$-module ${\cal N}$ by \[ \sigma({\cal N}) = {\cal N}\otimes_{{\Bbb A}^\hbar_{T^*X}}{\cal O}_{T^*X} \] we have \[ \sigma({\cal M})\overset{\sim}{=}\sigma(\mu{\cal M})\ . \] \section{Characteristic classes of perfect complexes}\label{section:char} In this section we construct the Euler class (with values in Hochschild homology) and the Chern character (with values in negative cyclic homology) for a perfect complex of sheaves of modules over a sheaf of algebras over a topological space. \subsection{Review of Hochschild and cyclic homology} Let $k$ denote a commutative algebra over a field of characteristic zero and let $A$ be a flat $k$-algebra with $1_A\cdot k$ contained in the center, not necessarily commutative. Let $C_p(A)\overset{def}{=}A^{\otimes_k p+1}$ and let \begin{eqnarray*} b : C_p(A) & @>>> & C_{p-1}(A) \\ a_0\otimes\cdots\otimes a_p & \mapsto & (-1)^p a_pa_0\otimes\cdots\otimes a_{p-1} + \\ & & \sum_{i=0}^{p-1}(-1)^ia_0\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_p\ . \end{eqnarray*} Then $b^2 = 0$ and the complex $(C_\bullet, b)$, called {\em the standard Hochschild complex of $A$} represents $A\otimes^{{\Bbb L}}_{A\otimes_kA^{op}}A$ in the derived category of $k$-modules. The map \begin{eqnarray*} B : C_p(A) & @>>> & C_{p+1}(A) \\ a_0\otimes\cdots\otimes a_p & \mapsto & \sum_{i=0}^p (-1)^{pi} 1\otimes a_i\otimes\cdots\otimes a_p\otimes a_0\otimes\cdots\otimes a_{i-1} \end{eqnarray*} satisfies $B^2 = 0$ and $[B,b] = 0$ and therefore defines a map of complexes \[ B: C_\bullet(A) @>>> C_\bullet(A)[-1]\ . \] For $i,j,p\in{\Bbb Z}$ let \begin{eqnarray*} CC^-_p(A) & = & \prod_{\overset{i\geq 0}{i+j = p \mod 2}} C_{i+j}(A) \\ CC^{per}_p(A) & = & \prod_{i+j = p \mod 2} C_{i+j}(A)\ . \end{eqnarray*} The complex $(CC^-_\bullet(A),B+b)$ (respectively $(CC^{per}_\bullet(A),B+b)$) is called the {\em negative} (respectively {\em periodic}) {\em cyclic complex of $A$}. There are inclusions of complexes \[ CC^-_\bullet(A)[-2]\hookrightarrow CC^-_\bullet(A)\hookrightarrow CC^{per}_\bullet(A) \] and the short exact sequence \[ 0 @>>> CC^-_\bullet(A)[-2] @>>> CC^-_\bullet(A) @>>> C_\bullet(A) @>>> 0\ . \] Suppose that $X$ is a topological space and ${\cal A}$ is a flat sheaf of $k$-algebras on $X$ such that there is a global section $1\in\Gamma(X;{\cal A})$ which restricts to $1_{{\cal A}_x}$ and $1_{{\cal A}_x}\cdot k$ is contained in the center of ${\cal A}_x$ for every point $x\in X$. Let $C_\bullet({\cal A})$ (respectively $CC^-_\bullet({\cal A}),\ CC^{per}_\bullet({\cal A})$) denote the complex of sheaves of $k$-modules associated to the presheaf with value $C_\bullet({\cal A}(U))$ (respectively $CC^-_\bullet({\cal A}(U)),\ CC^{per}_\bullet({\cal A}(U))$) on an open subset $U$ of $X$. Then $C_\bullet({\cal A})$ represents ${\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A}$ in the derived category of sheaves of $k$-modules on $X$. \subsection{Perfect complexes} We briefly recall the notion of perfection as introduced in \cite{Ill}. A module over ${\cal A}$ is said to be {\em free of finite type} if it is isomorphic to ${\cal A}^{\oplus n}$ for some $n\in{\Bbb Z}$. A complex ${\cal P}^\bullet$ of (sheaves of) ${\cal A}$-modules is called {\em strictly perfect} if \begin{enumerate} \item ${\cal P}^p = 0$ for almost all $p\in{\Bbb Z}$; \item for any $p\in{\Bbb Z}$ and every point $x\in X$ there exists a neighborhood $U$ of $x$ such that ${\cal P}^p\vert_U$ is a direct summand of a free ${\cal A}\vert_U$-module of finite type. \end{enumerate} A complex ${\cal F}^\bullet$ of (sheaves of) ${\cal A}$-modules is called {\em perfect} if for any point $x\in X$ there exists an open neighborhood $U$ of $x$, a strictily perfect complex ${\cal P}^\bullet$ of ${\cal A}\vert_U$-modules and a quasiisomorphism ${\cal P}^\bullet\to{\cal F}^\bullet\vert_U$. \subsection{The Euler class in Hochschild homology} For a perfect complex ${\cal F}^\bullet$ of ${\cal A}$ modules we define the {\em Lefschetz map} as the morphism in the derived category of sheaves of $k$-modules given by the composition \[ \operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal F}^\bullet) @<{\overset{\sim}{=}}<< (\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal A})\otimes_k{\cal F}^\bullet) \otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A} @>{ev\otimes\operatorname{id}}>> {\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A} \] and will denote it by ${\cal L}_{{\cal A}}({\cal F}^\bullet)$ or simply by ${\cal L}_{{\cal A}}$. We define the {\em Euler map} of ${\cal F}^\bullet$ as the morphism in the derived category given by the composition \[ k @>{1\mapsto\operatorname{id}}>> \operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal F}^\bullet) @>{{\cal L}_{{\cal A}}}>> {\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A} \] and will denote it by $\operatorname{Eu}_{{\cal A}}({\cal F}^\bullet)$. Suppose that $Z$ is a closed subset of $X$ such that $Z\supseteq\operatorname{Supp}{\cal F}^\bullet\overset{def}{=}\bigcup_p\operatorname{Supp} H^p{\cal F}^\bullet$. Then, clearly, the canonical morphism $\operatorname{\bold R}\Gamma_Z(\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal F}^\bullet))\to \operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal F}^\bullet)$ is an isomorphism. Thus, after applying the functor $\operatorname{\bold R}\Gamma(X;\operatorname{\bold R}\Gamma_Z(\bullet))$ and passing to cohomology, the morphism $\operatorname{Eu}_{{\cal A}}({\cal F}^\bullet)$ determines a cohomology class in $H^0_Z(X;{\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A})$ which we will refer to as the {\em Euler class} and will denote by $\operatorname{eu}_{{\cal A}}^Z({\cal F}^\bullet)$. For $Z$ a closed subset of $X$ let $K^0_Z({\cal A})$ denote the Grothendieck group of perfect complexes of ${\cal A}$-modules supported on $Z$; set $K^0({\cal A}) \overset{def}{=}K^0_X({\cal A})$. The Euler class defined above determines a homomorphism of groups \[ \operatorname{eu}_{{\cal A}}^Z : K^0_Z({\cal A}) @>>> H^0_Z(X;{\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A})\ . \] \subsection{Review of K-theory} Recall that, to a category ${\cal C}$ with cofibrations and weak equivalences one associates the simplicial category $S_\bullet{\cal C}$ (the $S$-construction) whose definition can be found in \cite{W}. The $K$-theory spectrum $K({\cal C})$ of ${\cal C}$ is defined by $K({\cal C}) = \Omega\vert S_\bullet{\cal C}\vert$ which describes the zeroth $\Omega^\infty$-space whose deloopings are given by iterating the $S$-construction. The groups $K_i({\cal C})$ are defined by \[ K_i({\cal C}) = \pi_iK({\cal C})\ . \] To a ring $A$ one associates the spectrum $K^{naive}(A)$ (respectively $K(A)$) defined as the $K$-theory spectrum of the category of strictly perfect (respectively perfect) complexes of $A$-modules. The natural inclusion of the former category into the latter induces the natural morphism of spectra \[ K^{naive}(A) @>>> K(A)\ . \] The inclusion of the category ${\cal P}_A$ of finitely generated projective $A$-modules into the category of strictly perfect complexes induces the natural weak equivalence \[ K({\cal P}_A) @>>> K^{naive}(A)\ . \] \subsection{Hochschild and cyclic homology revisited} We summarize some of the results of \cite{McC} concerning the Hochschild and cyclic homology of exact categories. To a $k$-linear additive category ${\cal C}$ ($k$ a ring) one associates {\em the cyclic nerve} $CN_\bullet{\cal C}$ which is a cyclic $k$-module. In the case when ${\cal C}$ is a $k$-linear additive category with one object, i.e. a $k$-algebra $A$, one has $CN_\bullet{\cal C}\overset{\sim}{=} C_\bullet(A)$. In what follows we will not make notational distinctions between complexes and corresponding simplicial Abelian groups. To a category ${\cal C}$ with cofibrations and weak equivalences one associates the Hochschild complex $C_\bullet({\cal C})$ by \[ C_\bullet({\cal C}) = \Omega\vert CN_\bullet S_\bullet{\cal C}\vert = \operatorname{Tot}(CN_\bullet S_\bullet{\cal C})[-1]\ . \] Using the cyclic structure on $C_\bullet({\cal C})$ one defines the cyclic (respectively the negative cyclic, respectively the periodic cyclic) complex of ${\cal C}$ which we denote by $CC_\bullet({\cal C})$ (respectively $CC^-_\bullet({\cal C})$, respectively $CC^{per}_\bullet({\cal C})$). There is natural commutative diagram \[ \begin{CD} CC^-_\bullet({\cal C}) @>>> CC^{per}_\bullet({\cal C}) @>>> CC_\bullet({\cal C})[2] \\ @VVV @VVV @VV{\operatorname{id}}V \\ C_\bullet({\cal C}) @>>> CC_\bullet({\cal C}) @>>> CC_\bullet({\cal C})[2] \end{CD} \] with rows exact triangles. Let ${\cal P}_A$ denote the category of finitely generated projective $A$-modules. There is a natural quasiisomorphism of cyclic $k$-modules \[ C_\bullet(A) @>>> C_\bullet({\cal P}_A) \] which induces quasiisomorphisms of respective cyclic (negative cyclic, periodic cyclic) complexes. The natural inclusion \[ S_\bullet{\cal C} @>>> CN_\bullet S_\bullet{\cal C} \] induces the natural morphism of spectra \[ \operatorname{Eu}_{{\cal C}} : K({\cal C}) @>>> C_\bullet({\cal C}) \] which is the Dennis trace map in the case ${\cal C} = {\cal P}_A$. The morphism $\operatorname{Eu}$ has a natural lifting \[ ch_{{\cal C}} : K({\cal C}) @>>> CC^-_\bullet({\cal C}) \] which is the Chern character (the Goodwillie-Jones map) in the case ${\cal C} = {\cal P}_A$. One can show that the (exact) inclusion of ${\cal P}_A$ into the category of strictly perfect complexes of $A$-modules induces quasiisomorphisms of respective Hochschild, cyclic, negative cyclic, and periodic cyclic complexes. In particular one has the Euler class and the Chern character \[ \operatorname{Eu}_A : K^{naive}(A) @>>> C_\bullet(A),\ \ \ ch_A : K^{naive}(A) @>>> CC^-_\bullet(A)\ . \] \subsection{Presheaves of spectra} The following is a summary of the definitions and constructions of Section 3 of \cite{M}. We refer the reader to \cite{M} (and references therein, particularly \cite{T}) for further details. To a presheaf ${\cal S}$ of spectra on $X$ (and, more generally, a presheaf with values in a category with filtered colimits, products and coproducts) one can associate the functorial cosimplicial Godement resolution in the usual way. The stalk ${\cal S}_x$ of ${\cal S}$ at $x\in X$ is defined by \[ {\cal S}_x = \operatorname{colim} {\cal S}(U) \] where the colimit is taken over all open neighborhoods $U$ of $x$. For $U$ an open subset of $X$ let \[ T({\cal S})(U) = \prod_{x\in U}{\cal S}_x\ . \] Then $T({\cal S})$ is a presheaf on $X$, $T$ is a functor from presheaves to presheaves and in fact a monad. Thus, $T$ gives rise to the functorial (in ${\cal S}$) cosimplicial object $T^\bullet({\cal S})$ in presheaves on $X$ called the {\em Godement resolution of} ${\cal S}$. For ${\cal S}$ a presheaf of spectra on $X$ and an open subset $U$ of $X$ let \[ {\Bbb H}(U;{\cal S}) = \underset{\Delta}{\operatorname{holim}}\ T^\bullet({\cal S})(U)\ . \] The assignment $U\mapsto{\Bbb H}(U;{\cal S})$ determines a presheaf of spectra on $X$. Note that there is a natural morphism of presheaves of spectra \[ {\cal S} @>>> {\Bbb H}(\bullet ;{\cal S})\ . \] For $Z\subset X$ a closed subset let $\Gamma_Z{\cal S}$ denote the presheaf of spectra whose value on an open set $U\subset X$ is defined to be the homotopy fiber of the restriction map ${\cal S}(U)\to {\cal S}(U\setminus Z)$. In particular there is a canonical morphism $\Gamma_Z{\cal S}\to{\cal S}$. Set \[ {\Bbb H}_Z(U;{\cal S})\overset{def}{=} \underset{\Delta}{\operatorname{holim}}\ \Gamma_ZT^\bullet({\cal S})(U)\ . \] \subsection{The Euler class and the Chern character revisited} Let $K_Z({\cal A})$ (respectively $K^{naive}_Z({\cal A})$) denote the $K$-theory spectrum of the category of perfect complexes (respectively strictly perfect complexes) of sheaves of ${\cal A}$-modules on $X$ supported on $Z$ (i.e. acyclic on the complement of $Z$ in $X$). The inclusion of the category of strictly perfect complexes into the category of all perfect complexes induces the natural morphism of spectra $K^{naive}_Z({\cal A})\to K_Z({\cal A})$. The assignment $U\mapsto K^{naive}({\cal A}(U))$ (respectively $U\mapsto K^{naive}({\cal A}\vert_U)$, $U\mapsto K({\cal A}\vert_U)$) determines a presheaf of spectra on $X$. The functor $M\mapsto M\otimes_{{\cal A}(U)}{\cal A}\vert_U$ and the inclusion of strictly perfect complexes into perfect complexes induce morphism of presheaves of spectra \begin{equation}\label{maps:KKK} K^{naive}({\cal A}(\bullet)) @>>> K^{naive}({\cal A}\vert_\bullet) @>>> K({\cal A}\vert_\bullet)\ . \end{equation} \begin{lemma} The morphisms \eqref{maps:KKK} induce equivalences on stalks. \end{lemma} \begin{cor} The morphism \eqref{maps:KKK} induce equivalences \begin{equation}\label{maps:HHH} {\Bbb H}_Z(X;K^{naive}({\cal A}(\bullet))) @>>> {\Bbb H}_Z(X;K^{naive}({\cal A}\vert_\bullet)) @>>> {\Bbb H}_Z(X;K({\cal A}\vert_\bullet))\ . \end{equation} \end{cor} Combing the equivalences \eqref{maps:HHH} with the the Euler class (Dennis trace map) and the canonical morphism $K_Z({\cal A})\to{\Bbb H}_Z(X;K({\cal A}\vert_\bullet))$ we obtain the morphism of spectra \[ \operatorname{Eu} :K_Z({\cal A}) @>>> {\Bbb H}_Z(X;C_\bullet({\cal A})) = \operatorname{\bold R}\Gamma_Z(X;C_\bullet({\cal A}))\ . \] Using the Chern character instead of the Euler class we obtain the morphism \begin{equation}\label{map:ch} ch :K_Z({\cal A}) @>>> {\Bbb H}_Z(X;CC^-_\bullet({\cal A})) = \operatorname{\bold R}\Gamma_Z(X;CC^-_\bullet({\cal A}))\ . \end{equation} All in all, we have constructed Euler class and Chern character for a perfect complex. \section{Characteristic classes on complex manifolds} \label{section:cx-man} \subsection{Notations and conventions} In what follows we will be considering, for a complex manifold $X$, the sheaves of algebras ${\cal O}_X$, ${\cal D}_X$, ${\cal E}_{T^*X}$, and ${\Bbb A}^\hbar_{T^*X}$. All of these are sheaves of topological vector spaces. In what follows all tensor products are understood to be projective tensor products. In particular, let \begin{eqnarray*} {\cal O}_X^{\frak{e}} & = & {\cal O}_X\widehat\otimes{\cal O}_X \\ {\cal D}_X^{\frak{e}} & = & {\cal D}_X\widehat\otimes{\cal D}_X^{op} \\ {\cal E}_X^{\frak{e}} & = & {\cal E}_X\widehat\otimes{\cal E}_X \\ ({\Bbb A}^\hbar_M)^{\frak{e}} & = & {\Bbb A}^\hbar_M \widehat\otimes_{{\Bbb C}[[\hbar]]}({\Bbb A}^\hbar_M)^{op}\ . \end{eqnarray*} If ${\cal A}$ is one of ${\cal O}_X$, ${\cal D}_X$, ${\cal E}_X$, ${\Bbb A}^\hbar_M$ then ${\cal A}$ has a natural structure of an ${\cal A}^{\frak{e}}$-module and there is a natural map ${\cal A}\otimes{\cal A}^{op}\to{\cal A}^{\frak{e}}$ which induces a map of complexes \[ {\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A} @>>> {\cal A}\otimes^{{\Bbb L}}_{{\cal A}^{\frak{e}}}{\cal A}\ . \] In what follows we will only consider the composition \[ \operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal F}^\bullet) @>{{\cal L}_{{\cal A}}}>> {\cal A}\otimes^{{\Bbb L}}_{{\cal A}\otimes_k{\cal A}^{op}}{\cal A} @>>> {\cal A}\otimes^{{\Bbb L}}_{{\cal A}^{\frak{e}}}{\cal A}\ , \] refer to it as the Lefschetz map and denote it by ${\cal L}_{{\cal A}}$. Similarly, the Euler map $\operatorname{Eu}_{{\cal A}}({\cal F}^\bullet)$ will denote the composition \[ k @>{1\mapsto\operatorname{id}}>> \operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal A}}({\cal F}^\bullet,{\cal F}^\bullet) @>{{\cal L}_{{\cal A}}}>> {\cal A}\otimes^{{\Bbb L}}_{{\cal A}^{\frak{e}}}{\cal A}\ . \] \subsection{Some examples of Hochschild and cyclic homology} Here we recall some facts about the Hochschild and cyclic homology of ${\cal O}_X$, ${\cal D}_X$, ${\cal E}_X$ and ${\Bbb A}^\hbar_{T^*X}$ for a complex manifold $X$ of dimension $\dim_{{\Bbb C}}X = d$ from \cite{B}. By the theorem of Hochschild-Kostant-Rosenberg there is an isomorphism in the derived category of ${\cal O}_X$-modules \[ \mu_{{\cal O}_X} : {\cal O}_X\otimes^{{\Bbb L}}_{{\cal O}_X^{\frak{e}}}{\cal O}_X @>>> \bigoplus_p\Omega^p_X [p] \] given, in terms of the standard Hochschild complex representing ${\cal O}_X\otimes^{{\Bbb L}}_{{\cal O}_X\otimes{\cal O}_X}{\cal O}_X$, by the formula \[ a_0\otimes\cdots\otimes a_p \mapsto \frac{1}{p!} a_0 da_1\wedge\cdots\wedge da_p \ . \] The same formula gives the quaiisomorphism \begin{equation}\label{map:HKR} \tilde\mu_{{\cal O}} : CC^{per}_\bullet({\cal O}_X) @>>> \prod_{p\in{\Bbb Z}}\Omega^\bullet_X [2p] \end{equation} of the periodic cyclic complex of ${\cal O}_X$ and the de Rham complex of $X$, since it is easily verified that the diagram \[ \begin{CD} C_\bullet({\cal O}_X) @>{B}>> C_\bullet({\cal O}_X)[-1] \\ @V{\mu}VV @V{\mu}VV \\ \bigoplus_p\Omega^p_X [p] @>{d}>> \bigoplus_p\Omega^p_X [p-1] \end{CD} \] (where $d$ denotes exterior differentiation) is commutative. In view of the fact that the canonical map ${\Bbb C}_X\to\Omega^\bullet_X$ is a quasiisomorphism we will view the map $\tilde\mu_{{\cal O}}$ as the isomorphism (in the derived category) \[ \tilde\mu_{{\cal O}} : CC^{per}_\bullet({\cal O}_X) @>>> \prod_{p\in{\Bbb Z}}{\Bbb C}_X [2p]\ . \] The inverse to $\tilde\mu_{{\cal O}}$ is provided by the map of periodic cyclic complexes induced by the inclusion ${\Bbb C}_X\hookrightarrow{\cal O}_X$. According to J.-L.Brylinski (\cite{Bry}) \[ H^p({\cal D}_X\otimes^{{\Bbb L}}_{{\cal D}_X^{\frak{e}}}{\cal D}_X)\overset{\sim}{=} \left\lbrace\begin{array}{lll} 0 & \text{if} & p\neq -2d \\ {\Bbb C}_X & \text{if} & p = -2d \end{array}\right. \] Consider a point $x\in X$, an open neighborhood $U$ of $x$ in $X$ and a local coordinate system $x_1,\ldots,x_d$ centered at $x$. The class of the Hochschild cycle $Alt(1\otimes x_1\otimes\cdots x_d\otimes \displaystyle\frac{\partial}{\partial x_1}\otimes\cdots\otimes \displaystyle\frac{\partial}{\partial x_d})$ represents a global section $\Phi^{{\cal D}}_U$ of $H^{-2d}({\cal D}_U\otimes^{{\Bbb L}}_{{\cal D}_U^{\frak{e}}}{\cal D}_U)$. Let \[ \mu_{{\cal D}} : {\cal D}_X\otimes^{{\Bbb L}}_{{\cal D}_X^{\frak{e}}}{\cal D}_X \to {\Bbb C}_X [2d] \] denote the isomorphism in the derived category of sheaves which corresponds to the global section of $H^{-2d}({\cal D}_X\otimes^{{\Bbb L}}_{{\cal D}_X^{\frak{e}}}{\cal D}_X)$ determined by the condition that it restricts to $\Phi^{{\cal D}}_U$ for every sufficiently small open set $U\subset X$. Let \[ \mu^\hbar_{{\cal D}} : {\cal D}_X\otimes^{{\Bbb L}}_{{\cal D}_X^{\frak{e}}}{\cal D}_X \to {\Bbb C}_X [\hbar^{-1},\hbar]] [2d] \] denote the composition $\displaystyle\frac{1}{\hbar^d}\mu_{{\cal D}}$. Similar result hold for the sheaf ${\cal E}_X$ of microdifferential operators. Specifically, \[ H^p({\cal E}_X\otimes^{{\Bbb L}}_{{\cal E}_X^{\frak{e}}}{\cal E}_X)\overset{\sim}{=} \left\lbrace\begin{array}{lll} 0 & \text{if} & p\neq -2d \\ {\Bbb C}_{T^*X} & \text{if} & p = -2d \end{array}\right. \] Let $\xi_i$ denote the symbol of $\displaystyle\frac{\partial}{\partial x_i}$ where $x_1,\ldots,x_d$ are local coordinates on $X$ as before. The class of the Hochschild cycle $Alt(1\otimes x_1\otimes\cdots\otimes x_d\otimes\xi_i\otimes\cdots\otimes\xi_d)$ determines a global section $\Phi^{{\cal E}}_U$ of $H^{-2d}({\cal E}_U\otimes^{{\Bbb L}}_{{\cal E}_U^{\frak{e}}}{\cal E}_U)$. Let \[ \mu_{{\cal E}} : {\cal E}_X\otimes^{{\Bbb L}}_{{\cal E}_X^{\frak{e}}}{\cal E}_X \to {\Bbb C}_{T*X} [2d] \] denote the isomorphism in the derived category of sheaves which corresponds to the global section of $H^{-2d}({\cal E}_X\otimes^{{\Bbb L}}_{{\cal E}_X^{\frak{e}}}{\cal E}_X)$ determined by the condition that it restricts to $\Phi^{{\cal E}}_U$ for every sufficiently small open set $U\subset X$. Let \[ \mu^\hbar_{{\cal E}} : {\cal E}_X\otimes^{{\Bbb L}}_{{\cal E}_X^{\frak{e}}}{\cal E}_X \to {\Bbb C}_{T^*X} [\hbar^{-1},\hbar]] [2d] \] denote the composition $\displaystyle\frac{1}{\hbar^d}\mu_{{\cal E}}$. Suppose that ${\Bbb A}^\hbar_M$ is a symplectic deformation quantization of a complex manifold $M$ of dimension $\dim_{{\Bbb C}} M = 2d$. The sheaf of algebras ${\Bbb A}^\hbar_M[\hbar^{-1}]$ exhibits properties similar to those of ${\cal E}_X$. We will show that \[ H^p({\Bbb A}^\hbar_M \otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_M)^{\frak{e}}} {\Bbb A}^\hbar_M)[\hbar^{-1}]\overset{\sim}{=} \left\lbrace\begin{array}{lll} 0 & \text{if} & p\neq -2d \\ {\Bbb C}_M[\hbar^{-1},\hbar ]] & \text{if} & p = -2d \end{array}\right. \] Consider a ``local trivialization'' of ${\Bbb A}^\hbar_M$ as in \eqref{loc-iso-h}. It induces an isomorphism \begin{equation}\label{loc-iso-hoch} H^{-2d}({\Bbb A}^\hbar_{U'}\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_{U'})^{\frak{e}}} {\Bbb A}^\hbar_{U'})[\hbar^{-1}] @>{\overset{\sim}{=}}>> H^{-2d}({\Bbb A}^\hbar_{U}\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_{U})^{\frak{e}}} {\Bbb A}^\hbar_{U})[\hbar^{-1}]\ . \end{equation} The expression $Alt(1\otimes x_1\otimes\cdots\otimes x_d\otimes \displaystyle\frac{\xi_1}{\hbar}\otimes\cdots\otimes\displaystyle \frac{\xi_d}{\hbar})$ represents a (non-trivial) global section of $H^{2d}({\Bbb A}^\hbar_{U'}\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_{U'})^{\frak{e}}} {\Bbb A}^\hbar_{U'})[\hbar^{-1}]$ whose image $\Phi^{{\Bbb A}}_U$ under the isomorphism \eqref{loc-iso-hoch} is independent of the local trivialization \eqref{loc-iso-h}. Let \[ \mu_{{\Bbb A}}^\hbar : {\Bbb A}^\hbar_M\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_M)^{\frak{e}}} {\Bbb A}^\hbar_M[\hbar^{-1}] @>>> {\Bbb C}_M[\hbar^{-1},\hbar ]][2d] \] denote the isomorphism in the derived category of sheaves which corresponds to the global section of $H^{2d}({\Bbb A}^\hbar_M\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_M)^{\frak{e}}} {\Bbb A}^\hbar_M)[\hbar^{-1}]$ determined by the condition that it restricts to $\Phi^{{\Bbb A}}_U$ for every sufficiently small open set $U\subset M$ as above. We now turn to the deformation quantization ${\Bbb A}^\hbar_{T^*X}$ as in \ref{subsec:microloc} The compatibility of all of the maps defined above is expressed by the commutativity of the following diagram: \[ \begin{CD} \pi^{-1}{\cal D}_X @>>> {\cal E}_X @>>> {\Bbb A}^\hbar_{T^*X}[\hbar^{-1}] \\ @VVV @VVV @VVV \\ \pi^{-1}\left({\cal D}_X\otimes^{{\Bbb L}}_{{\cal D}_X^{\frak{e}}}{\cal D}_X\right) @>>> {\cal E}_X\otimes^{{\Bbb L}}_{{\cal E}_X^{\frak{e}}}{\cal E}_X @>>> {\Bbb A}^\hbar_{U}\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_{U})^{\frak{e}}} {\Bbb A}^\hbar_{U}[\hbar^{-1}] \\ @V{\mu^\hbar_{{\cal D}}}VV @V{\mu^\hbar_{{\cal E}}}VV @V{\mu^\hbar_{{\Bbb A}}}VV \\ \pi^{-1}{\Bbb C}_X[\hbar^{-1},\hbar]][2d] @>>> {\Bbb C}_{T^*X}[\hbar^{-1},\hbar]][2d] @>>> {\Bbb C}_{T^*X}[\hbar^{-1},\hbar]][2d] \end{CD} \] Although we will restrict ourselves to the discussion of the cyclic homology of ${\Bbb A}^\hbar_M[\hbar^{-1}]$, analogs of the statements below hold for the algebras ${\cal D}_X$ and ${\cal E}_X$. Since there are no nontirvial morphisms ${\Bbb C}_M[\hbar^{-1},\hbar]]\to{\Bbb C}_M[\hbar^{-1},\hbar]][-1]$ in the derived category it follows that the map $B : C_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}]\to C_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}][-1]$ represents the trivial morphism (in the derived category) and, consequently, there are isomorphisms (in the derived category) \begin{equation*} CC^-_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}]\overset{\sim}{\to} \prod_{p = 0}^{\infty} C^\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}][-2p]\overset{\sim}{\to} \prod_{p = 0}^{\infty} {\Bbb C}_M[\hbar^{-1},\hbar]][2d - 2p] \end{equation*} and \begin{equation}\label{map:TR} CC^{per}_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}]\overset{\sim}{\to} \prod_{p = -\infty}^{\infty} C^\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}][-2p]\overset{\sim}{\to} \prod_{p = -\infty}^{\infty} {\Bbb C}_M[\hbar^{-1},\hbar]][2d - 2p] \end{equation} which are induced by $\mu^\hbar_{{\Bbb A}}$ on each factor. We will denote the latter composition by $\tilde\mu^\hbar_{{\Bbb A}}$. It is not difficult to show that the inverse to $\tilde\mu^\hbar_{{\Bbb A}}$ is provided by the map of periodic cyclic complexes induced by the inclusion ${\Bbb C}_M[[\hbar]]\hookrightarrow{\Bbb A}^\hbar_M$. \subsection{Euler classes of ${\cal D}_X$-, ${\cal E}_X$- and ${\Bbb A}^\hbar_{T^*X}$-modules} Consider a perfect complex ${\cal M}^\bullet$ of ${\cal D}_X$-modules and a closed subvariety $\Lambda$ of $T^*X$ containing $\operatorname{char}({\cal M}^\bullet)$. It is well known that $\operatorname{char}({\cal M}^\bullet) = \operatorname{Supp}(\pi^{-1}{\cal M}^\bullet \otimes_{\pi^{-1}{\cal D}_X}{\cal E}_X)$ and that the microlocal Euler class $\mu\operatorname{eu}({\cal M}^\bullet)\in H^{2d}_\Lambda(T^*X;{\Bbb C})$ depends only on the microlocalization $\pi^{-1}{\cal M}^\bullet\otimes_{\pi^{-1}{\cal D}_X}{\cal E}_X$. In fact, it is not difficult to establish the equality \[ \mu\operatorname{eu}({\cal M}^\bullet) = \mu_{{\cal E}}\left(\operatorname{eu}^\Lambda_{{\cal E}} (\pi^{-1}{\cal M}^\bullet\otimes_{\pi^{-1}{\cal D}_X}{\cal E}_X)\right)\ . \] Consider a perfect complex ${\cal N}^\bullet$ of ${\cal E}_X$-modules and a closed subset $\Lambda$ of $T^*X$ containing $\operatorname{Supp}({\cal N}^\bullet)$. The commutativity of the diagram \[ \begin{CD} \operatorname{\bold R}\underline{\operatorname{Hom}}_{{\cal E}_X}({\cal N}^\bullet,{\cal N}^\bullet) & @>>> & \operatorname{\bold R}\underline{\operatorname{Hom}}_{{\Bbb A}^\hbar_{T^*X}[\hbar^{-1}]}({\cal N}^\bullet\otimes_{{\cal E}_X} {\Bbb A}^\hbar_{T^*X}[\hbar^{-1}],{\cal N}^\bullet\otimes_{{\cal E}_X} {\Bbb A}^\hbar_{T^*X}[\hbar^{-1}]) \\ @V{{\cal L}_{{\cal E}}}VV & & @VV{{\cal L}_{{\Bbb A}^\hbar_{T^*X}[\hbar^{-1}]}}V \\ {\cal E}\otimes^{{\Bbb L}}_{{\cal E}_X^{\frak{e}}}{\cal E}_X & @>>> & {\Bbb A}^\hbar_{T^*X}\otimes^{{\Bbb L}}_{({\Bbb A}^\hbar_{T^*X})^{\frak{e}}} {\Bbb A}^\hbar_{T^*X}[\hbar^{-1}] \end{CD} \] implies the identity $\mu^\hbar_{{\cal E}}\circ\operatorname{Eu}_{{\cal E}}({\cal N}^\bullet) = \mu^\hbar_{{\Bbb A}}\circ\operatorname{Eu}_{{\Bbb A}^\hbar_{T^*X}[\hbar^{-1}]} ({\cal N}^\bullet\otimes_{{\cal E}_X}{\Bbb A}^\hbar_{T^*X}[\hbar^{-1}])$, and, consequently, the equality \[ \mu^\hbar_{{\cal E}}\left(\operatorname{eu}^\Lambda_{{\cal E}}({\cal N}^\bullet)\right) = \mu^\hbar_{{\Bbb A}}\left(\operatorname{eu}^\Lambda_{{\Bbb A}} ({\cal N}^\bullet\otimes_{{\cal E}_X}{\Bbb A}^\hbar_{T^*X}[\hbar^{-1}])\right)\ . \] Thus, calculation of microlocal Euler classes reduces to calculation of Euler classes for ${\Bbb A}^\hbar_M[\hbar^{-1}]$-modules. \section{Riemann-Roch type theorems} \label{section:RR} \subsection{The Riemann-Roch theorem for periodic cyclic cocycles} The following theorem constitutes the central result of this note. \begin{thm}\label{thm:main} The diagram (in the derived category of Abelian sheaves on $M$) \[ \begin{CD} CC^{per}_\bullet({\Bbb A}^\hbar_M) @>{\sigma}>> CC^{per}_\bullet({\cal O}_M) \\ @V{\iota}VV @VV {\tilde\mu_{{\cal O}}\smile \widehat A(TM)\smile e^\theta}V \\ CC^{per}_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}] @>{\tilde\mu^\hbar_{{\Bbb A}}}>> \displaystyle\prod_{p=-\infty}^\infty {\Bbb C}_M[\hbar^{-1},\hbar]][-2p] \end{CD} \] is commutative. \end{thm} The proof of Theorem \ref{thm:main} is postponed until the later sections. In Section \ref{section:GF} we introduce the methods of Gel'fand-Fuchs cohomology and reduce (see Corollary \ref{cor:reduction}) Theorem \ref{thm:main} to an analogous statement (Theorem \ref{thm:formalRR}) in the case when $M$ is a formal neighborhood of the origin in a symplectic vector space over ${\Bbb C}$ which is formulated in Section \ref{section:formalRR}. The rest of this section is devoted to corollaries of Theorem \ref{thm:main}. \subsection{Riemann-Roch for ${\Bbb A}^\hbar$-modules} The commutativity of the diagram \[ \begin{CD} K^0_\Lambda({\Bbb A}^\hbar_M[\hbar^{-1}]) @<{\iota}<< K^0_\Lambda({\Bbb A}^\hbar_M) @>{\sigma}>> K^0_\Lambda({\cal O}_M) \\ @V{ch^\Lambda_{{\Bbb A}^\hbar[\hbar^{-1}]}}VV @V{ch^\Lambda_{{\Bbb A}^\hbar}}VV @VV{ch^\Lambda_{{\cal O}}}V \\ H^0_\Lambda(M;CC^{per}_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}]) @<{\iota}<< H^0_\Lambda(M;CC^{per}_\bullet({\Bbb A}^\hbar_M)) @>{\sigma}>> H^0_\Lambda(M;CC^{per}_\bullet({\cal O}_M)) \end{CD} \] yields the following. \begin{cor}\label{cor:rrAch} Suppose that $M$ is a complex manifold, ${\Bbb A}^\hbar_M$ is a symplectic deformation quantization of $M$, ${\cal M}^\bullet$ is a perfect complex of ${\Bbb A}^\hbar_M$-modules and $\Lambda$ is a closed subvariety of $M$ containing $\operatorname{Supp}({\cal M}^\bullet)$. Then \begin{equation}\label{eq:rrAch} \mu^\hbar_{{\Bbb A}}\left(ch^\Lambda_{{\Bbb A}^\hbar_M[\hbar^{-1}]} ({\cal M}^\bullet[\hbar^{-1}])\right) = \tilde\mu_{{\cal O}}(ch^\Lambda_{{\cal O}}(\sigma({\cal M}^\bullet))) \smile\widehat A(TM)\smile e^\theta \end{equation} in $H_\Lambda^\bullet(M;{\Bbb C}[\hbar^{-1},\hbar ]])$, where $\theta$ is the characteristic class of the deformation quantization ${\Bbb A}^\hbar_M$. \end{cor} Note that the class $\widehat A(E)$ is defined for any symplectic vector bundle $E$, for example by choosing a reduction of the (symplectic) structure group of $E$ to the unitary group. Recall that, for an element $\alpha$ of a graded object, $[\alpha]^p$ denotes the homogeneous component of $\alpha$ of degree $p$. \begin{cor} Under the assumptions of Corollary \ref{cor:rrAch} \begin{equation}\label{eq:rrAeu} \mu^\hbar_{{\Bbb A}}\left(\operatorname{eu}^\Lambda_{{\Bbb A}^\hbar_M[\hbar^{-1}]} ({\cal M}^\bullet[\hbar^{-1}])\right) = \left[\tilde\mu_{{\cal O}}(ch^\Lambda_{{\cal O}}(\sigma({\cal M}^\bullet))) \smile\widehat A(TM)\smile e^\theta) \right]^{\dim_{{\Bbb C}}M} \end{equation} in $H_\Lambda^\bullet(M;{\Bbb C}[\hbar^{-1},\hbar ]])$. \end{cor} \subsection{Riemann-Roch for ${\cal D}$- and ${\cal E}$-modules} If $M = T^*X$ for a complex manifold $X$, and ${\Bbb A}^\hbar_{T^*X}$ is the deformation quantization with the characteristic class $\theta = \frac12\pi^*c_1(X)$, then \[ \widehat A(TM)\smile e^\theta = \pi^*Td(TX) \] and the right hand side of \eqref{eq:rrAeu} is, clearly, independent of $\hbar$; if ${\cal M}^\bullet[\hbar^{-1}]$ is obtained by an extension of scalars from a complex of ${\cal E}_X$-modules, then, clearly, so is the left hand side. Thus, we obtain Conjecture \ref{conj:ch} of P.Schapira and J.-P.Schneiders. \begin{cor} Suppose that $X$ is a complex manifold, $({\cal M}^\bullet,F_\bullet)$ is a perfect complex of ${\cal D}_X$-modules with a good filtration and $\Lambda$ is a closed subvariety of $T^*X$ containing $\operatorname{char}{\cal M}^\bullet$. Then \[ \mu\operatorname{eu}_\Lambda({\cal M}^\bullet) = \left[ ch^\Lambda_{{\cal O}_{T^*X}}(\sigma({\cal M}^\bullet)) \smile\pi^*Td(TX)\right]^{2\dim_{{\Bbb C}}X} \] in $H_\Lambda^{2\dim_{{\Bbb C}}X}(T^*X;{\Bbb C})$. \end{cor} \section{The Riemann-Roch formula in the formal setting} \label{section:formalRR} The Weyl algebra of a symplectic vector space $(V,\omega)$ over ${\Bbb C}$ may be considered as a symplectic deformation quantization of the completion of $V$ at the origin. In this section we introduce the notations and the facts necessary to state the analogue of Theorem \ref{thm:main} in this setting. In what follows $(V,\omega)$ is viewed as a symplectic manifold. \subsection{The Weyl algebra} Here we briefly recall the definition and the basic properties of the Weyl algebra $W = W(V)$ of a (finite dimensional) symplectic vector space $(V,\omega)$ over ${\Bbb C}$. Let $V^\ast = \operatorname{Hom}_{{\Bbb C}}(V,{\Bbb C})$. Let $I=I(V)$ denote the kernel of the map (of ${\Bbb C}$-algebras) \[ \operatorname{Sym}^\bullet (V^\ast)\otimes{\Bbb C}[\hbar] @>>> {\Bbb C}\ . \] Let \[ \widehat\operatorname{Sym}^\bullet(V^\ast)[[\hbar]] = \varprojlim \frac{\operatorname{Sym}^\bullet (V^\ast)\otimes{\Bbb C}[\hbar]}{I^n} \] and let $\widehat I = \widehat I(V)$ denote the kernel of the map $\widehat\operatorname{Sym}^\bullet(V^\ast)[[\hbar]]\to{\Bbb C}$. The Moyal-Weyl product on $\widehat\operatorname{Sym}^\bullet(V^\ast)[[\hbar]]$ is defined by the formula \begin{equation}\label{formula:MW} f\ast g = \sum_{n=0}^\infty\frac{1}{n!}\left(\frac{\sqrt{-1}\hbar}{2}\right)^n \omega(d^nf,d^ng)\ , \end{equation} where \[ d^n : \widehat\operatorname{Sym}^\bullet(V^\ast) @>>> \widehat\operatorname{Sym}^\bullet(V^\ast)\otimes \operatorname{Sym}^n(V) \] assigns to a jet of a function the symmetric tensor composed of its $n$-th order partial deriviatives, and $\omega$ is extended naturally to a bilinear form on $\operatorname{Sym}^n(V)$. The Moyal-Weyl product endows $\widehat\operatorname{Sym}^\bullet(V^\ast)[[\hbar]]$ with a structure of an associative algebra with unit over ${\Bbb C}[[\hbar]]$ which contains $\widehat I$ as a twosided ideal. Moreover, the Moyal-Weyl product is continuous in the $\widehat I$-adic topology. Let $W=W(V)$ denote the topological algebra over ${\Bbb C}[[\hbar]]$ whose underlying ${\Bbb C}[[\hbar]]$-module is $\widehat\operatorname{Sym}^\bullet(V^\ast)[[\hbar]]$ and the multiplication is given by the Moyal-Weyl product. Let $F_pW = \widehat I^{-p}$. Then $(W,F_\bullet)$ is a filtered ring. Note also that the center of $W$ is equal to ${\Bbb C}[[\hbar]]$ and $\left[W,W\right] = \hbar\cdot W$. Clearly, the association $(V,\omega)\mapsto W(V)$ is functorial. In particular, the group $Sp(V)$ acts naturally on $W(V)$ by continuous algebra automorphisms. \subsection{Derivations of the Weyl algebra} Let $\frak{g} = \frak{g}(V)$ denote the Lie algebra of continuous, ${\Bbb C}[[\hbar]]$-linear derivations of $W$. Then there is a central extension of Lie algebras \[ 0 @>>> \frac{1}{\sqrt{-1}\hbar}{\Bbb C}[[\hbar]] @>>> \frac{1}{\sqrt{-1}\hbar}W @>>> \frak{g} @>>> 0\ , \] where the Lie algebra structure on $\frac{1}{\sqrt{-1}\hbar}W$ is given by the commutator (note that $\left[\frac{1}{\sqrt{-1}\hbar}W,\frac{1}{\sqrt{-1}\hbar}W\right] \subseteq\frac{1}{\sqrt{-1}\hbar}W$) and the second map is defined by $\frac{1}{\sqrt{-1}\hbar}f\mapsto \frac{1}{\sqrt{-1}\hbar}\left[f,\bullet \right]$. Let \[ F_p\frak{g} = \left\{D\in\frak{g}\ \vert\ D(F_iW)\subseteq F_{i+p}W \ \text{for all $i$}\right\} \] Then $(\frak{g},F_\bullet)$ is a filtered Lie algebra and the action of $\frak{g}$ on $W$ respects the filtrations, i.e. $\left[F_p\frak{g},F_q\frak{g}\right]\subseteq F_{p+q}\frak{g}$ and $F_p\frak{g}F_qW\subseteq F_{p+q}W$. The following properties of the filtered Lie algebra $(\frak{g},F_\bullet)$ are easily verified: \begin{enumerate} \item $Gr^F_p\frak{g} = 0$ for $p>1$ (in particular $\frak{g}=F_1\frak{g}$), hence $Gr^F_1\frak{g}$ is Abelian; \item the composition $\frac{1}{\sqrt{-1}\hbar}V\hookrightarrow \frac{1}{\sqrt{-1}\hbar}W\to\frak{g}\to Gr^F_1\frak{g}$ is an isomorphism; \item the composition $\frak{sp}(V)\to\operatorname{Sym}^2(V^\ast)\to \frac{1}{\sqrt{-1}\hbar}\widehat I\to F_0\frak{g}\to Gr^F_0\frak{g}$ is an isomophism; \item under the above isomorphisms the action of $Gr^F_0\frak{g}$ on $Gr^F_1\frak{g}$ is identified with the natural action of $\frak{sp}(V)$ on $V$ (in particular there is an isomorphism $\frak{g}/F_{-1}\frak{g}\overset{\sim}{=} V\ltimes\frak{sp}(V)$); \item the Lie algebra $F_{-1}\frak{g}$ is pro-nilpotent. \end{enumerate} In what follows $\frak{h}$ will denote the Lie subalgebra of $\frak{g}$ which is the image of the embedding $\frak{sp}(V)\hookrightarrow\frak{g}$. \subsection{The Weyl algebra as a deformation quantization} The (commutative) algebra $W/\hbar\cdot W$ is naturally isomorphic to the completion $\widehat{\cal O} =\widehat{\cal O}_V$ of the ring ${\cal O}_V$ of regular functions on $V$ at the origin, i.e. with respect to the powers of the maximal ideal $\frak{m}$ of functions which vanish at $0\in V$. The natural surjective map \[ \sigma : W(V) @>>> \widehat{\cal O}_V \] is strictly compatible with the $\widehat I$-adic filtration on $W$ and the $\frak{m}$-adic filtration on $\widehat{\cal O}_V$. The Lie algebra $\frak{g}$ acts by derivations on $\widehat{\cal O}$ by the formula \[ D(f) = \sigma(D(\tilde f))\ , \] where $D\in \frak{g}$ and $\tilde f\in W$ is such that $\sigma(\tilde f)=f$. Thus, $\sigma$ is a map of $\frak{g}$-modules. \subsection{The Hochschild homology of the Weyl algebra} We recall the calculation of the Hochschild homology of the Weyl algebra (\cite{FT1}, \cite{Bry}). The Hochschild homology of $W$ may be computed using the Koszul resolution of $W$ as a $W^{\frak{e}}\overset{def}{=} W\widehat\otimes_{{\Bbb C}[[\hbar]]}W^{op}$ module. The Koszul complex $(K^\bullet,\partial)$ is defined by \[ K^{-q} = W\otimes{\bigwedge}^q V^\ast\otimes W \] with the differential acting by \begin{eqnarray*} \partial(f\otimes v_1\wedge\ldots\wedge v_q\otimes g) & = & \sum_i (-1)^i fv_i\otimes v_1\wedge\ldots\wedge\widehat v_i\wedge\ldots\wedge v_q \\ & + & \sum_i (-1)^i f\otimes v_1\wedge\ldots\wedge\widehat v_i\wedge\ldots \wedge v_q\otimes v_ig\ . \end{eqnarray*} Here we consider $V^\ast$ embedded in $W$ and $\bigwedge^qV^\ast$ embedded in $(V^\ast)^{\otimes q}$. The map $K^\bullet\to W$ of complexes of $W^{\frak{e}}$-modules with the only nontrivial component (in degree zero) given by multiplication is easily seen to be a quasiisomorphism. The map \[ K^\bullet(W)\overset{def}{=}K^\bullet\otimes_{W^{\frak{e}}}W @>>> C_\bullet(W) \] defined by \[ f\otimes v_1\wedge\ldots\wedge\ldots\wedge v_q\otimes g\otimes h \mapsto fhg\otimes Alt(v_1\otimes\cdots\otimes v_q) \] is easily seen to be a quasiisomorphism. Hence it induces a quasiisomorphism \[ K^\bullet(W)\otimes_{{\Bbb C}[[\hbar]]}{\Bbb C}[\hbar^{-1},\hbar]] @>>> C_\bullet(W)\otimes_{{\Bbb C}[[\hbar]]}{\Bbb C}[\hbar^{-1},\hbar]]\ . \] Below we will use $(\bullet)[\hbar^{-1}]$ to denote $(\bullet)\otimes_{{\Bbb C}[[\hbar]]}{\Bbb C}[\hbar^{-1},\hbar]]$. Under the natural isomorphism \[ K^{-q}(W) @>>> W\otimes{\bigwedge}^qV^\ast \] defined by \[ f\otimes v_1\wedge\ldots\wedge v_q\otimes g\otimes h\mapsto fhg\otimes v_1\wedge\ldots\wedge v_q \] the induced differential acts on the latter by \[ \partial(f\otimes v_1\wedge\ldots\wedge v_q) = \sum_i [f,v_i]\otimes v_1\wedge\ldots\wedge\widehat v_i\wedge\ldots\wedge v_q\ . \] Suppose that $\dim V = 2d$. Let $\widehat\Omega^\bullet =\widehat\Omega^\bullet_V$ denote the de Rham complex of $V$ with formal coefficients (i.e. $\widehat\Omega^q_V =\Omega^q_V\otimes_{{\cal O}_V} \widehat{\cal O}_V$). The map \[ W\otimes{\bigwedge}^q V^\ast @>>> \widehat\Omega^{2d-q}[[\hbar]] \] given, by \[ f\otimes v_1\wedge\ldots\wedge v_q \mapsto f\cdot\iota_{v_1}\cdots\iota_{v_q}(\omega^{\wedge d}) \] (where $f\in\widehat{\cal O}$) is easily seen to determine an isomorphism of complexes \[ (K^\bullet(W),\partial) @>>> (\widehat\Omega^\bullet[[\hbar]], \hbar\cdot d)[2d]\ . \] The formal Poincare Lemma implies that \[ H^q(K^\bullet\otimes_{W^{\frak{e}}}W[\hbar^{-1}]) \overset{\sim}{=} HH_q(W;W)[\hbar^{-1}]\overset{\sim}{=}\left\lbrace\begin{array}{ll} 0 & \text{if $q\neq 2d$} \\ {\Bbb C}[\hbar^{-1},\hbar]] & \text{if $q=2d$} \end{array}\right. \] The canonical generator of $HH_{2d}(W;W)[\hbar^{-1}]$ is represented by the cycle $1\otimes\frac{1}{d!}\omega^{\wedge d}\in W\otimes {\bigwedge}^{2d}V^\ast$. If $x_1,\ldots ,x_d,\xi_1,\ldots ,\xi_d$ is (dual to) a symplectic basis of $V$ (so that $\omega = \sum_i x_i\wedge\xi_i$), then the canonical generator is represented by the cycle $Alt(1\otimes x_1\otimes\cdots\otimes x_d\otimes \xi_1\otimes\cdots\otimes\xi_d)\in C_{2d}(W)$. We will denote this cycle (and its class) by $\Phi = \Phi_V$. Note also that $\Phi$ corresponds under the above (quasi)isomorphisms to the cocycle $1\in\widehat\Omega^0$. Observe that the Lie algebra $\frak{g}$ acts on all of the complexes introduced above (by Lie deriviative) and all maps defined above are, in fact, $\frak{g}$-equivariant. The cycle $\Phi$ is not invariant under the action of $\frak{g}$. It is, however, invariant under the action of the subalgebra $\frak{h}$. \subsection{Characteristic classes in Lie algebra cohomology} We will presently construct the classes in relative Lie algebra cohomology of the pair $(\frak{g},\frak{h})$ which enter the Riemann-Roch formula in the present setting. \subsubsection{The trace density} Since $\frak{h}$ acts semi-simply on $C_\bullet(W)[\hbar^{-1}]$ and $\widehat\Omega^\bullet[\hbar^{-1},\hbar]]$, the quasiisomorphism \[ \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d] @>>> C_\bullet(W)[\hbar^{-1}] \] constructed above admits an $\frak{h}$-equivariant splitting \[ \mu^\hbar_{(0)} : C_\bullet(W)[\hbar^{-1}] @>>> \widehat\Omega^\bullet_V[\hbar^{-1},\hbar]][2d] \] which is a quasiisomorphism. We will consider the map $\mu^\hbar_{(0)}$ as a relative Lie algebra cochain \[ \mu^\hbar_{(0)}\in C^0(\frak{g},\frak{h};\operatorname{Hom}^0(C_\bullet(W)[\hbar^{-1}], \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d]))\ . \] \begin{lemma} $\mu^\hbar_{(0)}$ extends to a cocycle $\mu^\hbar = \sum_p\mu^\hbar_{(p)}$ with \[ \mu^\hbar_{(p)}\in C^p(\frak{g},\frak{h};\operatorname{Hom}^{-p}(C_\bullet(W)[\hbar^{-1}], \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d]))\ . \] Moreover, any two such extensions are cohomologous. \end{lemma} Since the complex $C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet_{{\Bbb C}[[\hbar]]} (C_\bullet(W)[\hbar^{-1}],\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d]))$ represents (in the derived category) the object $\operatorname{\bold R}\operatorname{Hom}^\bullet_{(\frak{g},\frak{h})}(C_\bullet(W)[\hbar^{-1}], \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d])$, $\mu^\hbar$ represents a well defined isomorphism \[ \mu^\hbar : C_\bullet(W)[\hbar^{-1}] @>>> \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d] \] in the derived category of $(\frak{g},\frak{h})$-modules. The image of $\mu^\hbar$ under the functor of forgetting the module structure is $\mu^\hbar_{(0)}$. Cup product with $\mu^\hbar$ induces the quasiisomorphism of complexes \[ \mu^\hbar : C^\bullet(\frak{g},\frak{h};C_\bullet(W)[\hbar^{-1}]) @>>> C^\bullet(\frak{g},\frak{h};\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d]) \] unique up to homotopy. \begin{lemma} $\mu^\hbar_{(0)}$ extends to an $\frak{h}$-equivariant quasiisomorphism of complexes \[ \tilde\mu^\hbar_{(0)} : CC^{per}_\bullet(W)[\hbar^{-1}] @>>> \prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2p]\ . \] \end{lemma} We will consider the map $\tilde\mu^\hbar_{(0)}$ as a relative Lie algebra cochain \[ \tilde\mu^\hbar_{(0)}\in C^0(\frak{g},\frak{h};\operatorname{Hom}^0(CC^{per}_\bullet(W)[\hbar^{-1}], \prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2p]))\ . \] \begin{lemma}\label{lemma:formalTR} $\tilde\mu^\hbar_{(0)}$ extends to a cocycle $\tilde\mu^\hbar = \sum_p\tilde\mu^\hbar_{(p)}$ with \[ \mu^\hbar_{(p)}\in C^p(\frak{g},\frak{h};\operatorname{Hom}^{-p}(CC^{per}_\bullet(W)[\hbar^{-1}], \prod_{q\in{\Bbb Z}}\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2q]))\ . \] Moreover, any two such extensions are cohomologous. \end{lemma} Since the complex $C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(W)[\hbar^{-1}],\prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2p]))$ represents the object $\operatorname{\bold R}\operatorname{Hom}^\bullet_{(\frak{g},\frak{h})}(CC^{per}_\bullet(W)[\hbar^{-1}], \prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d])$, $\tilde\mu^\hbar$ represents a well defined isomorphism \[ \tilde\mu^\hbar : CC^{per}_\bullet(W)[\hbar^{-1}] @>>> \prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet[\hbar^{-1},\hbar]][2d] \] in the derived category of $(\frak{g},\frak{h})$-modules. The image of $\tilde\mu^\hbar$ under the functor of forgetting the module structure is $\tilde\mu^\hbar_{(0)}$. Cup product with $\tilde\mu^\hbar$ induces the quasiisomorphism of complexes \[ \tilde\mu^\hbar : C^\bullet(\frak{g},\frak{h};CC^{per}_\bullet(W)[\hbar^{-1}]) @>>> C^\bullet(\frak{g},\frak{h};\prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2p]) \] unique up to homotopy. The natural inclusion \[ \iota : CC^{per}_\bullet(W) @>>> CC^{per}_\bullet(W)[\hbar^{-1}] \] is a morphism of complexes of $(\frak{g},\frak{h})$-modules, therefore determines a cocycle \[ \iota\in C^0(\frak{g},\frak{h};\operatorname{Hom}^0 (CC^{per}_\bullet(W), CC^{per}_\bullet(W)[\hbar^{-1}]))\ . \] The cup product of $\iota$ and $\tilde\mu^\hbar$ is a cocycle \[ \tilde\mu^\hbar\smile\iota\in C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(W),\prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2p]) \] of (total) degree zero which prepresents the morphism \[ \tilde\mu^\hbar\circ\iota : CC^{per}_\bullet(W) @>>> \prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet[\hbar^{-1},\hbar]][2p] \] in the derived category of $(\frak{g},\frak{h})$-modules. \subsubsection{The symbol and the Hochschild-Kostant-Rosenberg map} \label{sssection:HKR} The maps \[ \sigma : CC^{per}_\bullet(W) @>>> CC^{per}_\bullet(\widehat{\cal O}) \] (induced by $\sigma : W\to\widehat{\cal O}$) and \[ \tilde\mu : CC^{per}_\bullet(\widehat{\cal O}) @>>> \prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet[2p] @>>> \prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet[2p][\hbar^{-1},\hbar]] \] (defined by $f_0\otimes\cdots\otimes f_p\mapsto\frac{1}{p!} f_0df_1\wedge\ldots\wedge df_p$) are morphisms of complexes of $(\frak{g},\frak{h})$-modules, therefore determine cocycles \[ \sigma\in C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(W), CC^{per}_\bullet(\widehat{\cal O}))) \] and \[ \tilde\mu\in C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(\widehat{\cal O}),\prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet [\hbar^{-1},\hbar]][2p]))\ . \] \subsubsection{The characteristic class of the deformation} \label{sssection:charcl} The central extension of Lie algebras \[ 0 @>>> \frac{1}{\sqrt{-1}\hbar}{\Bbb C}[[\hbar]] @>>> \frac{1}{\sqrt{-1}\hbar}W @>>> \frak{g} @>>> 0 \] restricts to a trivial extension of $\frak{h}$, therefore is classified by a class $\theta\in H^2(\frak{g},\frak{h};\frac{1}{\sqrt{-1}\hbar}{\Bbb C}[[\hbar]])$ represented by the cocycle \[ \theta : X\wedge Y\mapsto \widetilde{[X,Y]} - [\widetilde{X},\widetilde{Y}] \] where $\widetilde{(\ )}$ is a choice of a ${\Bbb C}[[\hbar]]$-linear splitting of the extension. \subsubsection{The $\widehat A$-class}\label{sssection:formalA} Let $\nabla : \frak{g}\to\frak{h}$ denote an $\frak{h}$-equivariant splitting of the inclusion $\frak{h}\hookrightarrow\frak{g}$ and let \[ R(X,Y) = [\nabla (X),\nabla (Y)]-\nabla ([X,Y]) \] for $X,\ Y\in\frak{g}$. Then, $R\in C^2(\frak{g},\frak{h};\frak{h})$ is a cocycle (where $\frak{h}$ is considered as a trivial $(\frak{g},\frak{h})$-module). The $k$-fold cup product of $R$ with itself is a cocycle $R^{\smile k}\in C^{2k}(\frak{g},\frak{h};\frak{h}^{\otimes k})$. The splitting $\nabla$ determines the Chern-Weil map \[ CW : \widehat\operatorname{Sym}^\bullet(\frak{h}^\ast)^{\frak{h}} @>>> C^{2\bullet}(\frak{g},\frak{h};{\Bbb C}) \] of complexes (with $\widehat\operatorname{Sym}^\bullet(\frak{h}^\ast)^{\frak{h}}$ endowed with the trivial differential) which is defined as the composite \[ \operatorname{Sym}^k(\frak{h}^\ast)^{\frak{h}} @>>> C^0(\frak{h};\operatorname{Sym}^k(\frak{h}^\ast)) @>>> C^0(\frak{h};\operatorname{Sym}^k(\frak{h}^\ast))\otimes C^{2k}(\frak{g},\frak{h}; \frak{h}^{\otimes k}) @>>> C^{2k}(\frak{g},\frak{h};{\Bbb C}) \] where the first map is the natural inclusion (of the cocycles), the second map is $P\mapsto P\otimes R^{\smile k}$, and the last map is induced by the natural pairing on the coefficients. Let $\widehat A$ denote the image under the Chern-Weil map of \[ \frak{h}\ni X\mapsto\det\left(\frac{ad(\frac{X}{2})}{exp(ad(\frac{X}{2})) - exp(ad(-\frac{X}{2}))}\right)\ . \] \subsection{Riemann-Roch formula in Lie algebra cohomology} The following theorem is the analog of the Theorem \ref{thm:main} in the setting of this section. \begin{thm}\label{thm:formalRR} The cocycle $\tilde\mu^\hbar\smile\iota - \tilde\mu\smile\widehat A\smile e^\theta\smile\sigma$ is cohomolgous to zero in $C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(W),\prod_{p\in{\Bbb Z}} \widehat\Omega^\bullet [\hbar^{-1},\hbar]][2p]))$. \end{thm} \begin{cor} The diagram \[ \begin{CD} CC^{per}_\bullet(W) @>{\sigma}>> CC^{per}_\bullet(\widehat{\cal O}) \\ @V{\iota}VV @VV{\tilde\mu\smile\widehat A\smile e^\theta}V \\ CC^{per}_\bullet(W)[\hbar^{-1}] @>{\tilde\mu^\hbar}>> \prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet_V[\hbar^{-1},\hbar]][2p] \end{CD} \] in the derived category of $(\frak{g},\frak{h})$-modules is commutative. \end{cor} \begin{cor} The diagram \[ \begin{CD} C^\bullet(\frak{g},\frak{h};CC^{per}_\bullet(W)) @>{\sigma}>> C^\bullet(\frak{g},\frak{h};CC^{per}_\bullet(\widehat{\cal O})) \\ @V{\iota}VV @VV{\tilde\mu\smile\widehat A\smile e^\theta}V \\ C^\bullet(\frak{g},\frak{h};CC^{per}_\bullet(W)[\hbar^{-1}]) @>{\tilde\mu^\hbar}>> C^\bullet(\frak{g},\frak{h}; \prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet [\hbar^{-1},\hbar]][2p]) \end{CD} \] is homotopy commutative. \end{cor} A proof of Theorem \ref{thm:formalRR} may be found in \cite{NT1}, \cite{NT2}. In Section \ref{section:GF} we will show that Theorem \ref{thm:main} reduces to Theorem \ref{thm:formalRR}. \section{Gel'fand-Fuchs cohomology}\label{section:GF} In this section we introduce the machinery of Fedosov connections and Gel'fand-Fuchs cohomology and reduce Theorem \ref{thm:main} to the analogous statement in the particular case when $M$ is the formal neighborhood of the origin in a symplectic vector space over ${\Bbb C}$. Suppose given a complex manifold $M$ and a symplectic deformation quantization ${\Bbb A}^\hbar_M$ of $M$. Let $\omega\in H^0(M;\Omega^2_M)$ denote the associated symplectic form. \subsection{The sheaf of Weyl algebras} The sheaf of Weyl algebras ${\Bbb W}_M$ on $M$ is the sheaf of topological algebras over the sheaf of topological algebras ${\cal O}_M[[\hbar]]$ (equipped with the $\hbar$-adic topology) defined as follows. Let $\Theta_M$ denote the sheaf of holomorphic vector fields on $M$. Let ${\cal I}$ denote the kernel of the augmentation map \[ \operatorname{Sym}_{{\cal O}_M}^\bullet(\Theta_M)\otimes_{{\cal O}_M}{\cal O}_M[\hbar] @>>> {\cal O}_M\ . \] The completion $\widehat\operatorname{Sym}_{{\cal O}_M}^\bullet(\Theta_M)[[\hbar]]$ of $\operatorname{Sym}_{{\cal O}_M}^\bullet(\Theta_M)\otimes_{{\cal O}_M}{\cal O}_M[\hbar]$ in the ${\cal I}$-adic topology is a topological ${\cal O}_M[[\hbar]]$-module. The Weyl multiplication on $\widehat\operatorname{Sym}_{{\cal O}_M}^\bullet(\Theta_M)[[\hbar]]$ is defined by \eqref{formula:MW}. Then ${\Bbb W}_M$ is the sheaf of ${\cal O}_M[[\hbar]]$-algebras whose underlying sheaf of ${\cal O}_M[[\hbar]]$-modules is $\widehat\operatorname{Sym}_{{\cal O}_M}^\bullet(\Theta_M)[[\hbar]]$ and the multiplication is given by the Moyal-Weyl product. Note that the center of ${\Bbb W}_M$ is ${\cal O}_M[[\hbar]]$, and $\left[{\Bbb W}_M,{\Bbb W}_M\right] = \hbar\cdot{\Bbb W}_M$. Let $\widehat{\cal I}$ denote the kernel of the canonical map ${\Bbb W}_M\to{\cal O}_M$. The Weyl multiplication is continuous in the $\widehat{\cal I}$-adic topology. Let $F_p{\Bbb W}_M = \widehat{\cal I}^{-p}$. Then $({\Bbb W}_M,F_\bullet)$ is a filtered ring, i.e. $F_p{\Bbb W}_M\cdot F_q{\Bbb W}_M\subseteq F_{p+q}{\Bbb W}_M$. Note that the quotients $F_p{\Bbb W}_M/F_q{\Bbb W}_M$ are locally free ${\cal O}_M$-modules of finite rank. \subsection{Review of the Fedosov construction} We refer the reader to \cite{F} and \cite{NT3} for a detailed exposition of the construction of deformation quantizations via Fedosov connections. Let ${\cal A}^{p,q}_M$ denote the sheaf of complex valued $C^\infty$-forms of type $(p,q)$ on $M$, ${\cal A}^r_M = \oplus_{p+q=s}{\cal A}^{p,q}_M$. Let $F_\bullet{\cal A}^\bullet_M$ denote the Hodge filtration and $d$ the de Rham differential. Then $(({\cal A}^\bullet_M,d),F_\bullet)$ is a filtered differential graded algebra (i.e. $F_r{\cal A}^\bullet_MF_s{\cal A}^\bullet_M\subseteq F_{r+s}{\cal A}^\bullet_M$ and $d(F_r{\cal A}^\bullet_M)\subseteq F_r{\cal A}^\bullet_M$). Let ${\cal A}^p_M({\Bbb W}_M)={\cal A}^p_M\otimes_{{\cal O}_M}{\Bbb W}_M$. Then \[ {\cal A}^\bullet_M({\Bbb W}_M)\overset{def}{=}\bigoplus_p {\cal A}^p_M({\Bbb W}_M)[-p] \] has a natural structure of a sheaf of graded algebras. Let \[ F_r{\cal A}^\bullet_M({\Bbb W}_M) = \sum_{p+q=r}F_p{\cal A}^\bullet_M \otimes_{{\cal O}_M}F_p{\Bbb W}_M\ . \] Then $({\cal A}^\bullet_M({\Bbb W}_M),F_\bullet)$ is a filtered graded algebra over $({\cal A}^\bullet_M,F_\bullet)$ (i.e.\linebreak $F_r{\cal A}^p_M({\Bbb W}_M)F_s{\cal A}^q_M({\Bbb W}_M)\subseteq F_{r+s}{\cal A}^{p+q}_M({\Bbb W}_M)$ and $F_r{\cal A}^p_MF_s{\cal A}^q_M({\Bbb W}_M)\subseteq F_{r+s}{\cal A}^{p+q}_M({\Bbb W}_M)$). Let $F_p{\Bbb A}^\hbar_M = \hbar^{-p}\cdot{\Bbb A}^\hbar_M$. One can show that there exists a map \[ \nabla : {\cal A}^\bullet_M({\Bbb W}_M) @>>> {\cal A}^\bullet_M({\Bbb W}_M)[1] \] which has the following properties: \begin{enumerate} \item $\nabla\left(F_p{\cal A}^\bullet_M({\Bbb W}_M)\right)\subseteq F_p{\cal A}^\bullet_M({\Bbb W}_M)$; the induced maps \[ Gr^F_p\nabla : Gr^F_p{\cal A}^\bullet_M({\Bbb W}_M) @>>> Gr^F_p{\cal A}^\bullet_M({\Bbb W}_M)[1] \] are ${\cal O}_M[[\hbar]]$-linear differential operators of order one; \item $\nabla^2 = 0$; \item $({\cal A}^\bullet_M({\Bbb W}_M),\nabla)$ is a sheaf of differential graded algebras over the $C^\infty$ de Rham complex $({\cal A}^\bullet_M,d)$ (in particular $H^\bullet({\cal A}^\bullet_M({\Bbb W}_M),\nabla)$ is a sheaf of graded algebras over the constant sheaf ${\Bbb C}_M[[\hbar]]$); \item there is a filtered quasiisomorphism \[ ({\Bbb A}^\hbar_M,F_\bullet) @>>> (({\cal A}^\bullet_M({\Bbb W}_M),\nabla),F_\bullet) \] of differential graded algebras over ${\Bbb C}_M[[\hbar]]$ (in particular $H^p(F_q{\cal A}^\bullet_M({\Bbb W}_M),\nabla)=0$ for $p\neq 0$); \end{enumerate} therefore $\nabla$ is determined by its component \[ \nabla : {\cal A}^0_M({\Bbb W}_M) @>>> {\cal A}^1_M({\Bbb W}_M) \] which has all the properties of a flat connection on ${\Bbb W}_M$. Let $H = Sp(\dim M)$ and let $P @>{\pi}>> M$ denote the $H$-principal bundle of symplectic frames in $TM$, identify $TM$ with the vector bundle assiciated to the standard representation of $H$. Recall that, for an $\frak{h}$-module $V$, the subcomplex $\left[\pi_*{\cal A}_P^\bullet\otimes V\right]^{basic}\subset\pi_*{\cal A}_P^\bullet$ is defined by the pull-back diagram \[ \begin{CD} \left[\pi_*{\cal A}_P^\bullet\otimes V\right]^{basic} @>>> C^\bullet(\pi_*{\cal T}_P,\frak{h};V) \\ @VVV @VVV \\ \pi_*{\cal A}_P^\bullet\otimes V @>>> C^\bullet(\pi_*{\cal T}_P;V) \end{CD} \] where ${\cal T}_P$ denotes the sheaf of Lie algebras of $C^\infty$ vector fields on $P$. Let $W$ denote the Weyl algebra of the standard representation of $H$. Then ${\Bbb W}_M$ is identified with the sheaf of sections of the associated bundle $P\times_HW$ and pull-back by $\pi$ \[ {\cal A}_M^\bullet({\Bbb W}_M) @>>> \pi_*{\cal A}_P^\bullet\widehat\otimes W \] identifies ${\Bbb W}_M$-valued forms on $M$ with the subcomplex of basic $W$-valued forms on $P$. The flat connection $\nabla$ gives rise to a basic $\frak{g}(=Der(W))$-valued 1-form $A\in H^0(P;{\cal A}_P^1\widehat\otimes\frak{g})$ which satisfies the Maurer-Cartan equation $dA+\frac12\lbrack A,A\rbrack = 0$ (so that $(d + A)^2=0$). Then, pull back by $\pi$ induces the isomorphism of filtered complexes \[ ({\cal A}_M^\bullet({\Bbb W}_M),\nabla) @>>> \left(\left\lbrack\pi_*{\cal A}_P^\bullet\widehat\otimes W\right\rbrack^{basic}, d+A\right)\ . \] Given $A$ as above and a (filtered) topological $\frak{g}$-module $L$ such that the action of $\frak{h}\subset\frak{g}$ integrates to an action of $H$. Set \[ ({\cal A}_M^\bullet(L),\nabla)\overset{def}{=} \left(\left\lbrack\pi_*{\cal A}_P^\bullet\widehat\otimes L\right\rbrack^{basic}, d+A\right)\ . \] Note that the association $L\mapsto ({\cal A}_M^\bullet(L),\nabla)$ is functorial in $L$. In particular it extends to complexes of $\frak{g}$-modules. Taking $L={\Bbb C}$, the trivial $\frak{g}$-module, we recover $({\cal A}_M^\bullet,d)$. For any complex $(L^\bullet,d_L)$ of $\frak{g}$-modules as above the complex $({\cal A}_M^\bullet(L^\bullet),\nabla + d_L)$ has a natural structure of a differential graded module over $({\cal A}_M^\bullet,d)$. The Gel'fand-Fuchs map \[ GF : C^\bullet(\frak{g},\frak{h};L) @>>> {\cal A}_M^\bullet(L) \] (the sourse understood to be the constant sheaf) is defined by the formula \[ GF(c)(X_1,\ldots,X_p) = c(A(X_1),\ldots,A(X_p))\ , \] where $c\in C^p(\frak{g},\frak{h};L)$ and $X_1,\ldots,X_p$ are locally defined vector fields. It is easy to verify that $GF$ takes values in basic forms and is a map of complexes. Note also, that $GF$ is natural in $L$. In particular the definition above has an obvious extension to complexes of $\frak{g}$-modules. We now proceed to apply the above constructions to particular examples of complexes $L^\bullet$ of $\frak{g}$-modules. In all examples below the $\frak{g}$-modules which appear have the following additional property which is easy to verify, namely, \[ \text{$H^p({\cal A}_M^\bullet(L),\nabla) = 0$ for $p\neq 0$}\ . \] If $(L^\bullet,d_L)$ is a complex of $\frak{g}$-modules with the above property, then the inclusion \begin{equation}\label{map:incl-ker} (\ker(\nabla),d_L\vert_{\ker(\nabla)})\hookrightarrow ({\cal A}_M^\bullet(L^\bullet),\nabla + d_L) \end{equation} is a quasiisomorphism. In such a case the map \eqref{map:incl-ker} induces an isomoprhism \[ \operatorname{\bold R}\Gamma(M;\ker(\nabla)) @>>> \Gamma(M;{\cal A}_M^\bullet(L^\bullet)) \] in the derived category of complexes since the sheaves ${\cal A}_M^\bullet(L^\bullet)$ are soft and the Gel'fand-Fuchs map induces the (natural in $L^\bullet$) morphism \begin{equation}\label{map:GF} GF : C^\bullet(\frak{g},\frak{h};L) @>>> \operatorname{\bold R}\Gamma(M;\ker(\nabla)) \end{equation} in the derived category. For $L = W$ (respectively $\frak{g}$, $CC^{per}_\bullet(W)$, $\widehat{\cal O}$, ${\Bbb C}$, $\widehat\Omega^\bullet$, $CC^{per}_\bullet(\widehat{\cal O})$), $\ker(\nabla) = {\Bbb A}^\hbar_M$ (respectively $Der({\Bbb A}^\hbar_M)$, $CC^{per}_\bullet({\Bbb A}^\hbar_M)$, ${\cal O}_M$, ${\Bbb C}_M$, $\Omega^\bullet_M$, $CC^{per}_\bullet({\cal O}_M)$). We leave it to the reader to identify $\ker(\nabla)$ for other (complexes of) $\frak{g}$-modules which appear in Section \ref{section:formalRR} by analogy with the above examples. The relationship between the Lie algebra cocycles defined in Section \ref{section:formalRR} and morphism defined in Section \ref{section:cx-man} established by the Gel'fand-Fuchs map \eqref{map:GF} is as follows. \subsubsection{The trace density} The image of $\tilde\mu^\hbar$ (defined in Lemma \ref{lemma:formalTR}) under (the map on cohomology in degree zero induced by) \begin{multline*} GF : C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(W)[\hbar^{-1}],\prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet [\hbar^{-1},\hbar]][2d-2p]) @>>> \\ \operatorname{\bold R}\Gamma(M;\underline{\operatorname{Hom}}^\bullet(CC^{per}_\bullet({\Bbb A}^\hbar_M)[\hbar^{-1}], \prod_{p\in{\Bbb Z}}\Omega^\bullet_M[\hbar^{-1},\hbar]][2d-2p])) \end{multline*} is the morphism $\tilde\mu^\hbar_{{\Bbb A}}$ defined in \eqref{map:TR}. Similarly, the image of $\tilde\mu^\hbar\smile\iota$ under $GF$ is the morphism $\tilde\mu^\hbar_{{\Bbb A}}\circ\iota$. \subsubsection{The symbol and the Hochschild-Kostant-Rosenberg map} The image of $\sigma$ (defined in \ref{sssection:HKR}) under (the map on cohomology in degree zero induced by) \[ GF : C^\bullet(\frak{g},\frak{h}; \operatorname{Hom}^\bullet(CC^{per}_\bullet(W), CC^{per}_\bullet(\widehat{\cal O})) @>>> \operatorname{\bold R}\Gamma(M;\underline{\operatorname{Hom}}^\bullet(CC^{per}_\bullet({\Bbb A}^\hbar_M), CC^{per}\bullet({\cal O}_M)) \] is the morphism \[ \sigma : CC^{per}_\bullet({\Bbb A}^\hbar_M) @>>> CC^{per}_\bullet({\cal O}_M)\ . \] The image of $\tilde\mu$ (defined in \ref{sssection:HKR}) under (the map on cohomology in degree zero induced by) \begin{multline*} GF : C^\bullet(\frak{g},\frak{h};\operatorname{Hom}^\bullet (CC^{per}_\bullet(\widehat{\cal O}),\prod_{p\in{\Bbb Z}}\widehat\Omega^\bullet[-2p])) @>>> \\ \operatorname{\bold R}\Gamma(M;\underline{\operatorname{Hom}}^\bullet(CC^{per}_\bullet({\cal O}_M), \prod_{p\in{\Bbb Z}}\Omega^\bullet_M[-2p])) \end{multline*} is the morphism $\tilde\mu_{{\cal O}}$ defined in \eqref{map:HKR}. \subsubsection{The characteristic class of the deformation} The image of the cocycle $\theta\in C^2(\frak{g},\frak{h}; \frac{1}{\sqrt{-1}\hbar}{\Bbb C}[[\hbar]])$ (defined in \ref{sssection:charcl}) under the map \[ GF: C^\bullet(\frak{g},\frak{h};\frac{1}{\sqrt{-1}\hbar}{\Bbb C}[[\hbar]]) @>>> \frac{1}{\sqrt{-1}\hbar}{\cal A}^\bullet_M[[\hbar]] \] is the characteristic class $\theta$ of the deformation quantization ${\Bbb A}^\hbar_M$ defined in \cite{F} and \cite{D}. \subsubsection{The $\widehat A$-class} The composition \[ \widehat\operatorname{Sym}^\bullet(\frak{h})^{\frak{h}} @>{CW}>> C^\bullet(\frak{g},\frak{h};{\Bbb C}) @>{GF}>> {\cal A}_M^\bullet \] is easily seen to be the usual Chern-Weil homomorphism. In particular we have \[ GF(\widehat A) = \widehat A(TM)\ , \] where $\widehat A$ is defined in \ref{sssection:formalA}. Combining the above facts we obtain the following proposition. \begin{prop} The image of the cocycle $\tilde\mu^\hbar\smile\iota - \tilde\mu\smile\widehat A\smile e^\theta\smile\sigma$ (see Theorem \ref{thm:formalRR}) is the morphism $\iota\circ\tilde\mu^\hbar_{{\Bbb A}}-(\tilde\mu_{{\cal O}}\widehat A\smile e^\theta)\circ\sigma$ (see Theorem \ref{thm:main}). \end{prop} \begin{cor}\label{cor:reduction} Theorem \ref{thm:formalRR} implies Theorem \ref{thm:main}. \end{cor}

9705

alg-geom/9705024

en

https://arxiv.org/abs/alg-geom/9705024

alg-geom/9705024

Ionut Ciocan-Fontanine

Aaron Bertram, Ionut Ciocan-Fontanine, and William Fulton

Quantum multiplication of Schur polynomials

LaTeX, 20 pages with 11 figures

We give formulas for the products of classes of Schubert varieties in the quantum cohomology rings of Grassmannians, in terms of the combinatorics of partitions and tableaux.

alg-geom

\section{ Introduction} The Giambelli formula writes the class of a Schubert variety in a Grassmannian as a Schur polynomial, and rules of Pieri and Littlewood-Richardson give explicit formulas for multiplying these classes. The quantum cohomology ring of a Grassmannian is known, together with formulas for the classes of Schubert varieties \cite{B}. Our aim here is to present and discuss formulas for their quantum multiplication. Fix positive integers $k$ and $l$, and set $n=k+l$. Let $X:=Gr(l,n)$ be the Grassmannian of $l$-planes in ${\bf C} ^n$. Let $\sigma_1,\dots ,\sigma_k$ be indeterminates, with $\sigma_i$ of degree $i$. Define polynomials $Y_r=Y_r(\sigma_1,\dots ,\sigma_k)$ by the formulas \begin{equation} {Y_r={\rm det}(\sigma_{1+j-i})_{1\leq i,j\leq r}.} \end{equation} In the classical cohomology ring $H^*(X)$, if $\sigma_i$ maps to the $i^{\rm th}$ Chern class of the universal quotient bundle, then $(-1)^rY_r$ maps to the $r^{\rm th}$ Chern class of the universal subbundle. This leads to one of the standard presentations of the classical cohomology ring of the Grassmannian: \begin{equation} H^*(X)={\bf Z} [\sigma_1,\dots ,\sigma_k]/(Y_{l+1},\dots ,Y_n). \end{equation} The (small) quantum cohomology ring $QH^*(X)$ is an algebra over ${\bf Z} [q]$, with $q$ a variable of degree $n$. It has the presentation (\cite{ST}, \cite{B}, \cite{FP}) \begin{equation} QH^*(X)={\bf Z}[q,\sigma_1,\dots ,\sigma_k]/(Y_{l+1},\dots ,Y_{n-1},Y_n+(-1)^kq). \end{equation} We identify a partition $\lambda$ with its Young diagram; we write $\lambda\subset l\times k$ for a partition $\lambda =(\lambda_1\geq\dots\geq\lambda_l\geq 0)$ with $\lambda_1\leq k$, i.e., for a partition whose Young diagram fits inside an $l\times k$ rectangle. \vskip 10pt \epsfxsize 9cm \centerline{\epsfbox{rectangle.eps}} \vskip 20pt There is a Schubert variety $\Omega_{\lambda}\subset X$ for each $\lambda\subset l\times k$; $\Omega_{\lambda}$ consists of $l$-dimensional subspaces of ${\bf C}^n$ such that ${\rm dim}( L\cap{\bf C}^{k+i-\lambda_i}) \geq i$, for $1\leq i\leq l$. The classical Giambelli formula says that the class of $\Omega_{\lambda}$ in $H^*(X)$ is $\sigma_{\lambda}$, where \begin{equation} \sigma_{\lambda}={\rm det}(\sigma_{\lambda_i+j-i})_{1\leq i,j\leq l}. \end{equation} These classes $\sigma_{\lambda}$ give a ${\bf Z}$-basis for $H^*(X)$. Their product in this ring can therefore be written \begin{equation} \sigma_{\lambda}\cdot\sigma_{\mu}=\sum N_{\lambda\mu}^{\nu}\sigma_{\nu}. \end{equation} Here $\lambda,\mu$ and $\nu$ are partitions inside the $l\times k$ rectangle, with sizes $ | \nu | = | \lambda | + | \mu | $. That the coefficients $N_{\lambda\mu}^{\nu}$ are nonnegative follows from the fact that the Schubert varieties can be translated by the action of $GL_n({\bf C})$ so that they meet properly. In fact, Grassmannians are one class of varieties for which explicit closed expressions are known for multiplying Schubert classes, which also show the coefficients to be nonnegative. The formula for multiplication is known as the {\em Littlewood-Richardson rule}, and states that $N_{\lambda\mu}^{\nu}$ is the number of tableaux on the skew shape $\nu/\lambda$ of content $\mu$, whose word is a reverse lattice word. Here $\nu/\lambda$ is the complement of the Young diagram of $\lambda$ in the Young diagram of $\nu$. A {\em tableau} of {\em content} $\mu=(\mu_1,\dots ,\mu_l)$ is a numbering of the boxes of $\nu/\lambda$ with $\mu_1$ $1$'s, $\mu_2$ $2$'s, $\dots$ , up to $\mu_l$ $l$'s, which are weakly increasing across rows, and strictly increasing down columns (these are often called {\em semistandard tableaux}). The {\em word} of a tableau is the list of its entries, read from left to right in rows, from bottom to top. A word is {\em reverse lattice} if, from any point in it to the end, there are at least as many $1$'s as there are $2$'s, at least as many $2$'s as $3$'s, and so on. An example with $\lambda =(2,1)$, $\nu =(3,3,3,1)$, $\mu =(3,2,2)$ is shown in the figure below. \vskip 20pt \epsfxsize 55mm \centerline{\epsfbox{lattice.eps}} \vskip 20pt The word $3123121$ of the first tableau is a reverse lattice word, while the word $2133121$ of the second is not. The rule follows from the corresponding identity for Schur functions \begin{equation} s_{\lambda}\cdot s_{\mu}=\sum N_{\lambda\mu}^{\nu}s_{\nu}, \end{equation} where $s_{\lambda}={\rm det}(s_{\lambda_i+j-i})$, with $s_1,s_2,\dots $ indeterminates. In this case there is no need to restrict to partitions inside any rectangle; the map $$s_i\longmapsto \sigma_i\in H^*(X)$$ takes $s_{\lambda}$ to $0$ if $\lambda$ is not contained in the $l\times k$ rectangle (see \cite{M}, \cite{F2}). The Pieri formula is a special case, \begin{equation} s_p\cdot s_{\lambda}=\sum s_{\nu}, \end{equation} the sum over all $\nu$ obtained from $\lambda$ by adding $p$ boxes, with no two in the same column. Repeated, it gives a formula \begin{equation} s_{\mu _1}\cdot\dots\cdot s_{\mu _r}\cdot s_{\lambda}=\sum K_{\lambda\mu}^{\nu}s_{\nu}. \end{equation} Here $\mu _1,\dots ,\mu_r$ are nonnegative integers, the sum is over all $\nu$ with $ | \nu | = | \lambda | +\sum_{i=1}^r\mu_i$, and $K_{\lambda\mu}^{\nu}$ is the {\em Kostka number}: the number of tableaux on $\nu/\lambda$ with content $\mu$. The usual Kostka number $K_{\nu\mu}$ is $K_{\lambda\mu}^{\nu}$, where $\lambda=\emptyset$ and $\mu$ is a partition. It is a consequence of (8) that $K_{\lambda\mu}^{\nu}$ is independent of the order of terms in $(\mu _1,\dots ,\mu_r)$. We turn now to the quantum analogues of these formulas. We identify $QH^*(X)$ with the ring presented in (3), and we regard $\sigma_1,\dots ,\sigma_k$ as elements of this ring. For any partition $\lambda =(\lambda_1,\dots ,\lambda_r)$, we let $$\sigma_{\lambda}={\rm det}(\sigma_{\lambda _i +j-i})_{1\leq i,j\leq r},$$ an element of $QH^*(X)$. The classes $\sigma_{\lambda}$, for $\lambda\subset l\times k$ form a ${\bf Z}[q]$ basis for $QH^*(X)$. A Schubert variety $\Omega_{\lambda}$ determines a class in $QH^*(X)$. It is a result of \cite{B} that this class is still given by $\sigma_{\lambda}$; unlike other flag varieties (see \cite{C-F}, \cite{FGP}), there are no correction terms involving the variable $q$. It follows that in $QH^*(X)$ there are formulas \begin{equation} \sigma_{\lambda}\cdot\sigma_{\mu}=\sum q^mN_{\lambda\mu}^{\nu}(l,k)\sigma_{\nu}, \end{equation} the sum over $m\geq 0$ and $\nu\subset l\times k$, but with $ | \nu | = | \lambda | + | \mu | -mn$. The coefficients $N_{\lambda\mu}^{\nu}(l,k)$ are again nonnegative, this time for \lq\lq quantum geometric" reasons: $N_{\lambda\mu}^{\nu}(l,k)$ is the number (properly counted) of rational curves of degree $m$ that meet general Schubert varieties $\Omega_{\lambda}$, $\Omega_{\mu}$, and $\Omega_{\nu^{\vee}}$, where $\nu^{\vee}=(k-\nu_l,\dots ,k-\nu_1)$ (cf. \cite{B}, \cite{FP}). The coefficients $N_{\lambda\mu}^{\nu}(l,k)$ are uniquely determined from the presentation (3) of the ring $QH^*(X)$. Similarly, for $0\leq\mu _1,\dots ,\mu_r\leq k$ and $\lambda\subset l\times k$, \begin{equation} \sigma_{\mu _1}\cdot\dots\cdot \sigma_{\mu _r}\cdot \sigma_{\lambda}=\sum q^mK_{\lambda\mu}^{\nu}(l,k)\sigma_{\nu}, \end{equation} the sum over $m\geq 0$, $\nu\subset l\times k$, with $ | \nu | = | \lambda | -mn+\sum_{i=1}^r\mu_i$. Our goal in this paper is to give explicit formulas for these {\em quantum Littlewood-Richardson numbers} $N_{\lambda\mu}^{\nu}(l,k)$ and {\em quantum Kostka numbers} $K_{\lambda\mu}^{\nu}(l,k)$. Note that they depend on our given numbers $l$ and $k$, as well as on $\lambda$, $\mu$ and $\nu$. Our result is most satisfying for the quantum Kostka numbers: we show that they are the number of tableaux satisfying a certain condition. Our algorithm for the quantum Littlewood-Richardson numbers is efficient, but, since it involves signs, it does not show them to be nonnegative. If $\lambda_1>k$, it follows from the definition that $\sigma_{\lambda}=0$. If $\lambda_{l+1}>0$, however, $\sigma_{\lambda}$ need not vanish. Our basic algorithm gives a formula for all such polynomials $\sigma_{\lambda}$. This can be expressed in terms of removing rim $n$-hooks from $\lambda$. There is a {\em rim hook} of ${\lambda}$ corresponding to each of its boxes. It is a {\em rim $n$-hook} if it contains exactly $n$ boxes (which is the same number of boxes as the hook of the corresponding box). The figure below shows a rim hook with $n=10$. \vskip 20pt \epsfxsize 35mm \centerline{\epsfbox{hook.eps}} \vskip 20pt Although quantum cohomology is not functorial, isomorphic varieties do have isomorphic quantum cohomology rings. The natural isomorphism of $QH^*(Gr(l,n))$ with $QH^*(Gr(k,n))$ leads to interesting dual versions of our results, which are discussed in Section 4. The content of this paper is algebraic and combinatorial; quantum cohomology is used only for motivation, and for the nonnegativity of the quantum Littlewood-Richardson numbers. \vskip 10pt This work was carried out in the stimulating atmosphere of the Mittag-Leffler Institute. We thank F. Sottile and C. Greene for help with the literature on rim hooks. The first author was partially supported by the National Science Foundation. The second author was supported by a Mittag-Leffler Institute postdoctoral fellowship. The third author was supported by a Tage Erlander Guest Professorship, and the National Science Foundation. \section{The rim hook algorithm} For positive integers $k,l$, set $n=k+l$, and set $$\Lambda(l,k)={\bf Z} [q,\sigma_1,\ldots,\sigma_k]/(Y_{l+1}(\sigma),\ldots,Y_{n-1}(\sigma),Y_n(\sigma)+(-1)^kq)\;\; ,$$ where $Y_p(\sigma)={\rm det}(\sigma_{1+j-i})_{1\leq i,j\leq p}$. For any partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $\sigma_{\lambda}$ be the image of ${\rm det}(\sigma_{\lambda_i+j-i})_{1\leq i,j\leq r}$ in $\Lambda(l,k)$. Let $\lambda$ be a partition with $\lambda_1\leq k$, and let $\widetilde{\lambda}_i$ be the number of boxes in the $i^{\rm th}$ column of $\lambda$; i.e., the conjugate partition $\widetilde{\lambda}$ is $(\widetilde{\lambda}_1,\dots ,\widetilde{\lambda}_k)$. One can start at the bottom of any column and move upward and to the right until one has counted off $n$ boxes along the rim. This will be a rim $n$-hook, unless this process ends in a box directly to the left of the last box in some column. We call this an {\em illegal $n$-rim}. \vskip 20pt \epsfxsize 25mm \centerline{\epsfbox{illegal.eps}} \vskip 20pt If this process starts in column $r$ and ends in column $s$, the column lengths of the shape that remains are $$(\widetilde{\lambda}_1,\dots ,\widetilde{\lambda}_{r-1},\widetilde{\lambda}_{r+1}-1, \widetilde{\lambda}_{r+2}-1,\dots ,\widetilde{\lambda}_{s}-1,\widetilde{\lambda}_{r}-r+s-n, \widetilde{\lambda}_{s+1},\dots ,\widetilde{\lambda}_k).$$ The $n$-rim is illegal exactly when $\widetilde{\lambda}_{r}-r+s-n=\widetilde{\lambda}_{s+1}-1$, or \begin{equation} \widetilde{\lambda}_{r}-r-n=\widetilde{\lambda}_{s+1}-(s+1). \end{equation} Otherwise $s$ is the unique integer $\geq r$ with \begin{equation} \widetilde{\lambda}_{s}-s>\widetilde{\lambda}_{r}-r-n>\widetilde{\lambda}_{s+1}-(s+1). \end{equation} The partition $\mu$ obtained after removing such an $n$-rim hook from $\lambda$ is characterized by \begin{equation} \widetilde{\mu}=(\widetilde{\lambda}_1,\dots ,\widetilde{\lambda}_{r-1},\widetilde{\lambda}_{r+1}-1, \dots ,\widetilde{\lambda}_{s}-1,\widetilde{\lambda}_{r}-r+s-n, \widetilde{\lambda}_{s+1},\dots ,\widetilde{\lambda}_k). \end{equation} The {\em width} of the rim $n$-hook is the number $s-r+1$ of columns it occupies. It is also possible that there is no $n$-rim starting in column $r$, which happens if $\widetilde{\lambda}_r+\lambda_1-r<n$; so if $\widetilde{\lambda}_1+\lambda_1\leq n$, then $\lambda$ contains no $n$-rim. \begin{mlemma}{\nonumber} $({\rm A})$ If $\lambda$ contains an illegal $n$-rim, or if $\lambda_{l+1}>0$ and $\lambda$ contains no $n$-rim, then $\sigma_{\lambda}=0$. $({\rm B})$ If $\mu$ is the result of removing a rim $n$-hook from $\lambda$, then $$\sigma_{\lambda}=(-1)^{k-w}q\sigma_{\mu},$$ where $w$ is the width of the rim $n$-hook removed. \end{mlemma} {\it Proof:} Let $y_r=\sigma_{(1^r)}$ for all integers $r\geq 1$, with $y_0=1$ and $y_i=0$ for $i<0$. We know by (1) and (3) that $y_i=0$ for $l<i<n$, and $y_n=(-1)^{k-1}q$. By expanding the determinant expression for $y_r$ along the top row, we have $$y_r=\sigma_1y_{r-1}-\sigma_2y_{r-2}+\dots +(-1)^{k-1}\sigma_ky_{r-k}.$$ From this it follows by induction that \begin{equation} y_{mn+j}=(-1)^{m(k-1)}q^my_j \qquad {\rm for} \quad 0\leq j\leq n-1,\ m\geq 0. \end{equation} For any sequence $m=(m_1,\dots ,m_k)$ of integers, set \begin{equation} \tau_m={\rm det}(y_{m_i+j-i})_{1\leq i,j\leq k}. \end{equation} If $\lambda$ is a partition with $\lambda_1\leq k$, it is a basic identity for symmetric polynomials (cf. \cite{M}, $(2.9')$) that \begin{equation} \sigma_{\lambda}=\tau_{\widetilde{\lambda}} \end{equation} If $1\leq r<s\leq k$ and the $r^{\rm th}$ row of the matrix $(y_{m_i+j-i})_{1\leq i,j\leq k}$ is successively interchanged with the $(r+1)^{\rm st}$ row, then the $(r+2)^{\rm nd}$ row, and so on to the $s^{\rm th}$ row, one finds \begin{equation} \tau_m=(-1)^{s-r}\tau_{m'}\ , \end{equation} where $m'=(m_1,\dots ,m_{r-1},m_{r+1}-1,\dots ,m_s-1,m_r-r+s,m_{s+1},\dots ,m_k)$. Suppose the $n$-rim starting in column $r$ of $\lambda$ ends in column $s$. Set $m=\widetilde{\lambda}$. If this rim is illegal, it follows from (11) that $(y_{m_i'+j-i})$ has two identical rows, so $\sigma_{\lambda}=\tau_m=0$. This proves the first part of $({\rm A})$. If $\widetilde{\lambda}_r\geq l$ for some $r$ between $1$ and $k$, applying (14) to the entries of row $s$ of $(y_{m_i'+j-i})$ yields \begin{equation}\tau_{m'}=(-1)^{k-1}q\tau_{m''}\ , \end{equation} where $m''=(m'_1,\dots ,m'_{s-1},m'_s-n,m'_{s+1},\dots ,m'_k)$. If the $n$-rim starting in column $r$ of $\lambda$ is a rim hook, and it ends in column $s$, then by (17) and (18) $$\tau_m=(-1)^{k-1+s-r}q\tau_{m'''}\ ,$$ where $m'''=(m_1,\dots ,m_{r-1},m_{r+1}-1,\dots ,m_s-1,m_r-r+s-n,m_{s+1},\dots ,m_k)$. Part $({\rm B})$ of the lemma follows from this, using (13) and (16). On the other hand, if $\lambda_{l+1}>0$ and $\lambda$ contains no $n$-rim, we have $m_1>l$, $m_1+\lambda_1\leq n$, with $m=\widetilde{\lambda}$. Then $\sigma_{\lambda}={\rm det}(y_{m_i+j-i})_{1\leq i,j\leq \lambda_1}$, which vanishes since its top row is zero. This completes the proof of $({\rm A})$.\hspace*{\fill}\hbox{$\Box$} \vskip 12pt Given any partition $\lambda$ with $\lambda_1\leq k$, one can succesively apply the lemma, removing rim $n$-hooks until one arrives at a partition $\mu$ which has no rim $n$-hooks. If $\mu\subset l\times k$, and $R_1,\dots ,R_m$ are the rim hooks removed, then \begin{equation} \sigma_{\lambda}=\epsilon(\lambda/\mu)q^m\sigma_{\mu}, \end{equation} where $\epsilon(\lambda/\mu)=(-1)^{\sum(k-{\rm width}(R_i))}$. If $\mu$ is not contained in the $l\times k$ rectangle, then \begin{equation} \sigma_{\lambda}=0. \end{equation} One sees from (19) that if $\mu\subset l\times k$, then $\mu$ is independent of choice of the $m$ rim hooks removed, as is $\sum{\rm width}(R_i)$ mod($2$). This is a general fact: $\mu$ is the {\em $n$-core} of $\lambda$, which is uniquely determined by $\lambda$ and $n$ (\cite{JK}, \S 2.7, cf. \cite{M}, \S 1). Consider the specialization of (6) by $s_{\lambda}\longmapsto \sigma_{\lambda}$, using (19) and (20). This gives \begin{equation} \sigma_{\lambda}\cdot\sigma_{\mu}=\sum_{m,\nu}q^mN_{\lambda\mu}^{\nu}(l,k)\sigma_{\nu},\;\;\; N_{\lambda\mu}^{\nu}(l,k)=\sum_{\varrho}\epsilon(\varrho/\nu)N_{\lambda\mu}^{\varrho}. \end{equation} Here $\lambda,\mu\subset l\times k$, the first sum is over $m\geq 0$ and $\nu\subset l\times k$ with $$|\lambda|+|\mu|=|\nu|+mn,$$ the second sum is over $\varrho$ obtained from $\nu$ by adding $m$ rim $n$-hooks, and $N_{\lambda\mu}^{\varrho}$ are the {\em classical} Littlewood-Richardson coefficients. This gives a formula for the quantum Littlewood-Richardson numbers in terms of the classical numbers. From the identification of $\Lambda(l,k)$ with $QH^*(X)$ we deduce: \begin{corollary} For $\lambda,\mu,\nu\subset l\times k$, with $|\lambda|+|\mu|=|\nu|+mn$ for some $m\geq 0$, the number of rational curves of degree $m$ that meet general translates of the Schubert varieties $\Omega_{\lambda}$, $\Omega_{\mu}$, and $\Omega_{\nu^{\vee}}$ is equal to $$\sum\epsilon(\varrho/\nu)N_{\lambda\mu}^{\varrho},$$ where the sum is over all $\varrho$ with $\varrho_1\leq k$ that can be obtained from $\nu$ by adding $m$ rim $n$-hooks. In particular, $\sum\epsilon(\varrho/\nu)N_{\lambda\mu}^{\varrho}\geq 0$.\hspace*{\fill}\hbox{$\Box$} \end{corollary} \begin{numexample} {\rm $k=5$, $l=5$, $\lambda=(5,4,4,2,2)$, $\mu=(3,2,1)$, $\nu=(2,1)$:} \vskip 20pt \epsfxsize 13cm \centerline{\epsfbox{qlrex.eps}} \vskip 20pt \centerline{$N_{\lambda\mu}^{\nu}(5,5)=2-1=1$.} \end{numexample} \begin{numexample} {\rm This example exhibits the dependence of $N_{\lambda\mu}^{\nu}(l,k)$ on both $l$ and $k$, as well as an instance where a quantum Littlewood-Richardson number (which is a nontrivial sum of signed classical ones) vanishes. Let $\lambda=(3,3,2,1)$, $\mu=(4,3,2,1)$, $\nu=(4,2,2,1)$. By removing a rim $10$-hook from each of the partitions $\varrho_1=(6,5,3,3,2)$, $\varrho_2=(5,5,3,3,2,1)$, and $\varrho_3=(4,4,3,3,2,1,1,1)$, one obtains the shape $\nu$, and these are the only partitions with this property that appear in the product $s_{\lambda}\cdot s_{\mu}$ . The classical Littlewood-Richardson coefficients are $$N_{\lambda\mu}^{\varrho_1}=6,\qquad N_{\lambda\mu}^{\varrho_2}=8,\qquad N_{\lambda\mu}^{\varrho_3}=2.$$ Our algorithm gives $$ N_{\lambda\mu}^{\nu}(4,6)=N_{\lambda\mu}^{\varrho_1}-N_{\lambda\mu}^{\varrho_2}+N_{\lambda\mu}^{\varrho_3}=6-8+2=0, $$ $$ N_{\lambda\mu}^{\nu}(5,5)=N_{\lambda\mu}^{\varrho_2}-N_{\lambda\mu}^{\varrho_3}= 8-2=6, $$ and $$ N_{\lambda\mu}^{\nu}(6,4)=N_{\lambda\mu}^{\varrho_3}=2. $$ } \end{numexample} \vskip 10pt The quantum Pieri formula, for multiplying any $\sigma_{\lambda}$ by a class $\sigma_p$, was given in \cite{B}: for $\lambda\subset l\times k$, $1\leq p\leq k$, \begin{equation} \sigma_p\cdot\sigma_{\lambda}=\sum\sigma_{\mu}+\sum q\sigma_{\nu}. \end{equation} Here the first (classical) sum is over $\mu$ with $$k\geq \mu_1\geq\lambda_1\geq\mu_2\geq\lambda_2\geq\dots\geq\mu_l\geq\lambda_l\geq 0,$$ and $|\mu|=|\lambda|+p $; the second sum is over $\nu$ with $$\lambda_1-1\geq\nu_1\geq\lambda_2-1\geq\nu_2\geq\dots\geq\lambda_l-1\geq\nu_l\geq 0,$$ and $|\nu|=|\lambda|+p-n$. These $\nu$ are the partitions obtained from $\lambda$ by removing $n-p$ boxes from its border rim, taking at least one box from each of the $l$ rows. The proof in \cite{B} combined algebra with some quantum geometry. This Pieri formula follows easily from the Main Lemma. Indeed, one has the formula for multiplying Schur functions $$s_p\cdot s_{\lambda}=\sum s_{\mu},$$ the sum over $\mu$ obtained from $\lambda$ by adding $p$ boxes, no two in the same column. Terms with $\mu_1>k$, or with $\mu_1<k$ and $\mu_{l+1}>0$ vanish by $({\rm A})$ of the Main Lemma. Those with $\mu_{l+1}=0$ give the first sum in (22). Those with $\mu_{l+1}>0$ and $\mu_1=k$ have a rim $n$-hook that occupies all $k$ columns; by $({\rm B})$ of the Main Lemma, these give the second sum in (22). \begin{numexample} {\rm $k=4$, $l=4$, $\lambda=(2,2,1,1)$, $p=3$:} \vskip 20pt \epsfxsize 40mm \centerline{\epsfbox{pieri.eps}} \vskip 20pt \centerline{$\sigma_3\cdot\sigma_{(2,2,1,1)}=\sigma_{(4,2,2,1)}+q\sigma_1$.} \end{numexample} \bigskip \section{Quantum Kostka numbers} \bigskip For a partition $\nu\subset l\times k$, and a nonnegative integer $m$, we denote by $\nu[m]$ the partition obtained from $\nu$ by adding $m$ rim $n$-hooks to $\nu$, each starting in the first column and ending in the $k^{\rm th}$ column. Note that the height of each added rim is $l+1$, i.e., its lowest box in the first column is $l$ rows below its highest box in the $k^{\rm th}$ column. Let $\lambda\subset\varrho$ be partitions, with $\varrho_1\leq k$. We call a (semistandard) tableau $T$ on $\varrho/\lambda$ {\em proper} if, for each entry $i$ of $T$ in the first column and row $l+p$, $p\geq 1$, the box in the $k^{\rm th}$ column and row $p$ is either in $\lambda$, or it is in $\varrho/\lambda$ and it has an entry of $T$, at most equal to $i$. \vskip 20pt \epsfxsize 9cm \centerline{\epsfbox{examples.eps}} \vskip 20pt For any sequence $\mu=(\mu_1,\dots ,\mu_r)$ of nonnegative integers, each at most $k$, and partitions $\lambda,\nu\subset l\times k$, with $|\lambda|+\sum\mu_i=|\nu|+mn$ for some $m\geq 0$, define the {\em quantum Kostka number} $K_{\lambda\mu}^{\nu}(l,k)$ to be the number of proper tableaux on $\nu[m]/\lambda$ with content $\mu$. When $m=0$, this is the classical Kostka number $K_{\lambda\mu}^{\nu}$ (the coefficient of $s_{\nu}$ in $s_{\mu_r}\cdot\ldots\cdot s_{\mu_1}\cdot s_{\lambda}$). \begin{proposition} For $\lambda\subset l\times k$ and $0\leq \mu_1,\dots ,\mu_r\leq k$, $$\sigma_{\mu_r}\cdot\dots\cdot\sigma_{\mu_1}\cdot\sigma_{\lambda}=\sum K_{\lambda\mu}^{\nu}(l,k)q^m\sigma_{\nu},$$ the sum over $\nu\subset l\times k$ and $m\geq 0$, with $|\nu|+mn=|\lambda|+\sum\mu_i$. \end{proposition} Note in particular that $K_{\lambda\mu}^{\nu}(l,k)$ is independent of the ordering of the terms in $\mu=(\mu_1,\dots ,\mu_r)$. The proof will follow from three lemmas. Fix $\lambda$ and $\mu=(\mu_1,\dots ,\mu_r)$ as in the proposition. For any tableau $T$ of content $\mu$ on a skew shape $\varrho/\lambda $, and $1\leq i\leq r$, let $T_i$ be the subtableau of $T$ obtained by discarding all entries strictly larger than $i$; $T_i$ is a tableau of content $(\mu_1,\dots ,\mu_i)$ on some shape $\varrho(i)/\lambda$. Set $\varrho(0)=\lambda$. \begin{lemma} A tableau $T$ of content $\mu$ on $\varrho/\lambda$ is proper if and only if $$\widetilde{\varrho(i)}_1-\widetilde{\varrho(i)}_k\leq l,\ \ \ {\it for}\ \ 1\leq i\leq r.$$ \end{lemma} {\it Proof:} The condition of the definition is satisfied for an entry $i$ in the first column of $T$ if and only if the $k^{\rm th}$ column of $\varrho(i)$ is no more than $l$ boxes shorter than its first column.\hspace*{\fill}\hbox{$\Box$} \vskip 12pt For any $T$ that is not proper, let $U(T)=T_i$, where $i$ is minimal not satisfying the condition of the lemma. We call $U$ the {\em improper kernel} of $T$, and $T$ a {\em descendent} of $U$ if $U(T)=U$. Note that each $\varrho(i)$ has at most one more box in any column than $\varrho(i-1)$. It follows that if $U=U(T)$ has shape $\alpha/\lambda$, then (since $\alpha=\varrho(i)$ and $i$ is minimal) \begin{equation} \widetilde{\alpha}_1-\widetilde{\alpha}_k=l+1. \end{equation} For any tableau $T$ on a shape $\varrho/\lambda$, write $\sigma_T$ for the element $\sigma_{\varrho}$ in $\Lambda(l,k)$. \begin{lemma} Let $U$ be a tableau on $\alpha/\lambda$ with content $(\mu_1\dots ,\mu_i)$ for some $1\leq i\leq r$. Suppose $\widetilde{\alpha}_1-\widetilde{\alpha}_k=l+1$. Then $$\sum\sigma_T=0,$$ the sum over all $T$ of content $\mu$ with $T_i=U$. \end{lemma} {\it Proof:} The sum $\sum\sigma_T$ in the statement is equal to $$\sigma_{\mu_r}\cdot\sigma_{\mu_{r-1}}\cdot\dots\cdot\sigma_{\mu_{i+1}}\cdot\sigma_{\alpha}$$ in $\Lambda(l,k)$. It therefore suffices to show that $\sigma_{\alpha}=0$. From the assumption $\widetilde{\alpha}_1-\widetilde{\alpha}_k=l+1$ it follows that the $n$-rim starting at the bottom of the first column will end in the $(k-1)^{\rm st}$ column, just to the left of the last box in the $k^{\rm th}$ column. \vskip 20pt \epsfxsize 8cm \centerline{\epsfbox{nproper.eps}} \vskip 20pt \noindent This is an illegal $n$-rim, so $\sigma_{\alpha}=0$ by $({\rm A})$ of the Main Lemma.\hspace*{\fill}\hbox{$\Box$} \begin{lemma} Let $\varrho$ be a partition with $\varrho_1\leq k$ and $\widetilde{\varrho}_1-\widetilde{\varrho}_k\leq l$. Then there is a unique partition $\nu\subset l\times k$ and $m\geq 0$ with $\varrho=\nu[m]$. Moreover, $\sigma_{\varrho}=q^m\sigma_{\nu}$ in $\Lambda(l,k)$. \end{lemma} {\it Proof:} If $\widetilde{\varrho}_1\leq l$, then $\nu=\varrho$ and $m=0$. Otherwise, the condition $\widetilde{\varrho}_1-\widetilde{\varrho}_k\leq l$ implies that $\varrho$ contains a rim $n$-hook of width $k$ and height $l+1$. Removing this rim hook leaves a shape $\beta$ with $\widetilde{\beta}_1-\widetilde{\beta}_k\leq l$, and $\sigma_{\varrho}=q\sigma_{\beta}$ by $({\rm B})$ of the Main Lemma. The proof concludes by induction on the number of boxes.\hspace*{\fill}\hbox{$\Box$} \vskip 12pt To prove the proposition, from the classical expansion of Schur functions (cf. \cite{F2}, \S 2, or \cite{M}, \S I.5), we have $$\sigma_{\mu_r}\cdot\dots\cdot\sigma_{\mu_1}\cdot\sigma_{\lambda}=\sum \sigma_{T},$$ the sum over tableaux $T$ of content $\mu$ on some shape $\varrho(T)/\lambda$. The shape $\varrho=\varrho(T)$ for any proper $T$ satisfies the condition of Lemma 3 (by Lemma 1), so the sum of the $\sigma_T$ for $T$ proper gives the right side of the equation in the proposition. The $T$ that are not proper are divided into equivalence classes indexed by their improper kernels $U=U(T)$. Each sum $\sum\sigma_T$ over $T$ descended from $U$ vanishes by Lemma 2, which concludes the proof of the proposition.\hspace*{\fill}\hbox{$\Box$} \vskip 12pt This proposition gives another algorithm for computing the quantum product $\sigma_{\lambda}\cdot\sigma_{\mu}$, for $\lambda$ and $\mu$ partitions in $l\times k$: Expand $$\sigma_{\mu}={\rm det}(\sigma_{\mu_i+j-i})=\sum_{\tau\in S_l}{\rm sign}(\tau)\prod_{i=1}^l\sigma_{\mu_i+\tau(i)-i}$$ and use the proposition to expand each $\prod\sigma_{\mu_i+\tau(i)-i}\cdot\sigma_{\lambda}$. This writes $\sigma_{\lambda}\cdot\sigma_{\mu}$ as an alternating sum of sums of $\sigma_T$'s as $T$ runs over proper tableaux of content $(\mu_1+\tau(1)-1,\dots ,\mu_l+\tau(l)-l)$ on shapes $\nu[m]/\lambda$. Note that among the sum with $\tau=id$ are all the tableaux whose word is a reverse lattice word. It is known from the classical Littlewood-Richardson rule that the other tableaux cancel in such a sum, but it is unclear how proper and nonproper tableaux behave under such a cancellation. \bigskip \section{Duality} \bigskip Although the functoriality of ordinary cohomology does not extend to quantum cohomology, isomorphic varieties do have isomorphic quantum cohomology rings. The canonical isomorphism of $Gr(l,n)$ with $Gr(k,n)$ gives an isomorphism of $QH^*(Gr(l,n))$ with $QH^*(Gr(k,n))$, which takes the class of a Schubert variety $\Omega_{\lambda}$ to the class of a Schubert variety $\Omega_{\widetilde{\lambda}}$, for $\lambda\subset l\times k$. Our basic algorithm from Section 2 is not at all invariant under the involution $\lambda\mapsto\widetilde{\lambda}$, however, so this isomorphism leads to some interesting combinatorial identities. We begin with an algebraic proof of this isomorphism. To avoid cofusion, set $$\Lambda(k,l)={\bf Z}[q,\tau_1,\ldots,\tau_l ]/(Y_{k+1}(\tau),\ldots Y_{n-1}(\tau), Y_n(\tau)+(-1)^lq).$$ \begin{numproposition} The map $\sigma_i\mapsto Y_i(\tau)$, $1\leq i\leq k$, determines an isomorphism $\Lambda(l,k)\stackrel{\sim}{\rightarrow}\Lambda(k,l)$, with inverse taking $\tau_i$ to $Y_i(\sigma)$, $1\leq i\leq l$. For any $\lambda\subset l\times k$, this isomorphism takes $\sigma_{\lambda}={\rm det}(\sigma_{\lambda_i+j-i})_{1\leq i,j\leq l}$ to $\tau_{\widetilde{\lambda}}={\rm det}(\tau_{\widetilde{\lambda}_i+j-i})_{1\leq i,j\leq k}$. \end{numproposition} {\it Proof:} Consider the ring $$\Lambda={\bf Z} [q,\sigma_1,\ldots,\sigma_k,\tau_1,\ldots,\tau_l ]/I\; ,$$ where $I$ is generated by the elements $$\sigma_1-\tau_1,\;\sigma_2-\sigma_1\tau_1+\tau_2,\;\ldots,\; \sigma_k\tau_{l-1}-\sigma_{k-1}\tau_l,\;\sigma_k\tau_l-q.$$ These relations identify $\tau_i$ with $Y_i(\sigma)$ for $1\leq i\leq l$, and then prescribe that $Y_i(\sigma)=0$ for $l<i<n$, and that $Y_n(\sigma)=(-1)^{k-1}q$. This identifies $\Lambda$ with $\Lambda(l,k)$. By symmetry, it also identifies $\Lambda$ with $\Lambda(k,l)$, and the composite $\Lambda(l,k)\cong\Lambda\cong\Lambda(k,l)$ is the isomorphism of the proposition. The fact that $\sigma_{\lambda}$ and $\tau_{\widetilde{\lambda}}$ correspond under this isomorphism follows from the general identity \cite{M}, $(2.9')$, and the fact that all $\sigma_i$ (resp. $\tau_i$) occuring in the matrix for $\sigma_{\lambda}$ (resp. $\tau_{\widetilde{\lambda}}$) have $i<n$.\hspace*{\fill}\hbox{$\Box$} \vskip 12pt The following dual of the quantum Pieri formula follows from this proposition. We include a direct proof, for contrast with the proof of the Pieri formula in Section 2. \begin{numproposition} Let $p\leq l$ be a positive integer and let $\lambda\subset l\times k$ be a partition. The following holds in $\Lambda(l,k)$: \begin{equation}\label{pieri2} \sigma_{\lambda}\cdot\sigma_{(1^p)}=\sum\sigma_{\mu}+q\sum\sigma_{\nu}, \end{equation} the first sum over $\mu\subset l\times k$, obtained by adding $p$ boxes to $\lambda$, with no two in the same row, and the second sum over $\nu$ obtained from $\lambda$ by removing $n-p$ boxes in its border rim, with at least one from each of the $k$ columns. Note that there are no such $\nu$ if $\lambda_1<k$. Equivalently, the partitions $\nu$ occuring in the second sum can be characterized by $ | \nu | = | \lambda | +p-n$ and $$ \widetilde{\lambda}_1-1\geq\widetilde{\nu}_1\geq\widetilde{\lambda}_2-1\geq\widetilde{\nu}_2 \geq\dots\geq\widetilde{\lambda}_k-1\geq\widetilde{\nu}_k\geq 0.$$ \end{numproposition} {\it Proof:} The classical Pieri formula for Schur polynomials states that \begin{equation}\label{star} s_{\lambda}\cdot s_{(1^p)}=\sum s_{\pi}, \end{equation} sum over all $\pi$ obtained by adding $p$ boxes to $\lambda$, with no two in the same row. We need to show that the terms with $\pi_{l+1}>0$ give the second sum in (\ref{pieri2}). Let ${\cal P}$ denote the set of partitions $\pi$ that occur in (\ref{star}), with $\pi_1\leq k$, $\pi_{l+1}>0$, and the $n$-core of $\pi$ contained in the $l\times k$ rectangle. Only partitions in ${\cal P}$ can contribute to the second sum in (\ref{pieri2}), by $({\rm A})$ of the Main Lemma. The formula (\ref{pieri2}) will follow therefore from the following two claims: \vskip 10pt \noindent{\bf Claim 1.} If $\nu$ is as in the proposition, then there exists a unique partition $\pi\in{\cal P}$ such that $\nu$ is obtained from $\pi$ by removing an $n$-rim. Moreover, $\epsilon(\pi/\nu)=1$. \vskip 12pt \noindent {\bf Claim 2.} Let $\pi\in{\cal P}$ be any partition not arising in 1. Then there exists exactly one other partition $\pi '\in{\cal P}$ with the same $n$-core $\varrho$ as $\pi$. Moreover, $\epsilon(\pi/\varrho)= -\epsilon(\pi '/\varrho)$. \vskip 10pt {\it Proof of Claim $1$:} The partition $\pi$ is obtained by adding $p$ boxes to $\lambda$ as follows: \noindent For each $i\in\{ 2,3,\dots ,k\}$, add $\widetilde{\nu}_{i-1}-\widetilde{\lambda}_i+1$ boxes to the $i^{\rm th}$ column of $\lambda$. The condition $\widetilde{\lambda}_1-1\geq\widetilde{\nu}_1\geq\widetilde{\lambda}_2-1\geq\widetilde{\nu}_2 \geq\dots\geq\widetilde{\lambda}_k-1\geq\widetilde{\nu}_k\geq 0$ ensures that no two of these new boxes are in the same row; the total number of boxes added according to this recipe is $$\sum_{i=2}^k\widetilde{\nu}_{i-1}-\widetilde{\lambda}_i+1= | \nu | -\widetilde{\nu}_k- | \lambda | +\widetilde{\lambda}_1+k-1=p-(l+1-\widetilde{\lambda}_1+\widetilde{\nu}_k).$$ Now add $l+1-\widetilde{\lambda}_1+\widetilde{\nu}_k$ boxes to the first column of $\lambda$ to get a partition $\pi$. Then $\pi$ contains the rim $n$-hook consisting of these $p$ added boxes together with the $n-p$ boxes in $\lambda/\nu$; its width is $k$, so $\epsilon(\pi/\nu)=1$. This is the only way a $\pi$ in ${\cal P}$ can arise from $\nu$ by adding a rim $n$-hook. \vskip 12pt {\it Proof of Claim $2$:} Let $\pi$ be a partition in ${\cal P}$. The $n$-rim of $\pi$ starting at the bottom of the first column is a rim $n$-hook whose removal gives the $n$-core $\varrho$ of $\pi$. This rim hook consists of some boxes in the rim of $\lambda$ and some of the $p$ added boxes. Let $t$ be the maximum index such that this rim hook contains at least one box from the $t^{\rm th}$ column of $\lambda$. If $t=k$, then $\pi$ is one of the partitions considered in 1. We therefore assume that $t\neq k$. The rim hook must either $(a)$ end with a box from $\lambda$ in the $t^{\rm th}$ column of $\pi$, or $(b)$ end with an added box in the $(t+1)^{\rm st}$ column. In case $(a)$, $\pi '$ is obtained from $\pi$ by removing $\widetilde{\varrho}_t-\widetilde{\pi}_{t+1}+1$ boxes from the first column, and adding them to the $(t+1)^{\rm st}$ column of $\pi$; note that $\pi '$ satisfies $(b)$, has the same $n$-core $\varrho$, and $\epsilon(\pi/\varrho)= -\epsilon(\pi '/\varrho)$. Conversely, if $\pi$ satisfies $(b)$, the only $\pi '$ with the same $n$-core as $\pi$ is obtained from $\pi$ by removing $\widetilde{\pi}_{t+1}-\widetilde{\varrho}_{t+1}$ boxes from the $(t+1)^{\rm st}$ column and adding them to the first column. \hspace*{\fill}\hbox{$\Box$} \vskip 12pt We turn next to a dual form of the proposition in Section 3; this is {\em not} obtained by applying the duality isomorphism of Proposition 4.1. Call a filling $T$ of the boxes of a skew shape $\varrho / \lambda$ a {\em conjugate tableau} if its entries are strictly increasing across rows and weakly increasing down columns. Call a conjugate tableau {\em proper} if for each entry $i$ of $T$ in the first column and row $l+p$, $p \geq 1$, the box in the $k^{\rm th}$ column and row $p$ is either in $\lambda$, or it has an entry of $T$ {\em strictly} smaller than $i$. For a sequence $\mu =(\mu_1,\ldots ,\mu_r)$ of nonnegative integers, and partitions $\lambda ,\nu \subset l \times k$, with $|\lambda|+\sum\mu_i=|\nu|+mn$, $m\geq 0$, define the {\em conjugate quantum Kostka number} $\widetilde{K}_{\lambda\mu}^{\nu}(l,k)$ to be the number of proper conjugate tableaux on $\nu[m] / \lambda$ with content $\mu$. \begin{numproposition} For $\lambda \subset l \times k$ and $0\leq \mu_1\leq \ldots \leq \mu_r\leq l$, $$ \sigma_{(1^{\mu_r})}\cdot\ldots\cdot\sigma_{(1^{\mu_1})}\cdot\sigma_{\lambda}= \sum\widetilde{K}_{\lambda\mu}^{\nu}(l,k) q^m \sigma_{\nu}, $$ the sum over $\nu\subset l \times k$ and $m\geq 0$ with $|\nu|+mn=|\lambda|+\sum\mu_i$. \end{numproposition} {\it Proof:} As in Section 3, for any conjugate tableau $T$ on shape $\varrho/\lambda$ with content $\mu$, let $T_i$ be the part of $T$ of content $(\mu_1,\ldots,\mu_i)$, of shape $\varrho(i)/\lambda$. This time $T$ is proper if and only if, for $1\leq i\leq r$, $\widetilde{\varrho(i)}_1-\widetilde{\varrho(i)}_k\leq l$, with the inequality strict if $T$ has an entry $i$ in the $k^{\rm th}$ column. If $T$ is not proper, let $U(T)=T_i$ for $i$ minimal not satisfying the condition of the lemma; call $T$ a descendent of $U(T)$. We consider conjugate tableaux $U$ of content $(\mu_1,\ldots,\mu_i)$ which are not proper, but such that $U_{i-1}$ (of content $(\mu_1,\ldots,\mu_{i-1})$) is proper and fixed. Consider the $n$-rim of the shape $\varrho(U)$ that starts in the first column. If this rim is illegal, then $\sigma_{\varrho(U)}=0$. It follows as in Lemma 2 of \S 3 that $\sum\sigma_T=0$, the sum over all $T$ descended from $U$. For each $U$ whose $n$-rim is a rim hook we will construct another conjugate tableau $U'$ of content $(\mu_1,\ldots,\mu_i)$, with $U'_{i-1}=U_{i-1}$, whose $n$-rim is also a rim hook. The results of removing the rim $n$-hooks from $U'$ and $U$ will be the same, but the rim hooks will end in adjacent columns, so $\sigma_{\varrho(U')}+\sigma_{\varrho(U)}=0$. And we will have $(U')'=U$. It will follow that $\sum\sigma_T=0$, the sum over all $T$ descended from $U$ or $U'$. The proof of the proposition concludes by an application of Lemma 3 of \S 3. There are two cases, $(a)$ and $(b)$, which are interchanged by the involution $U\longleftrightarrow U'$. Let $V=U_{i-1}$, with shape $\alpha/\lambda$ (with $\alpha=\lambda$ if $i=1$). Case $(a)$ occurs when the $n$-rim in $\varrho(U)$ ends in a box of $\alpha$, and $(b)$ occurs when it ends in a box of $\varrho(U)/\alpha$. If $U$ is as in case $(a)$, the $n$-rim ends in a column $t$ with $t<k$, since $U$ is not proper; $U'$ is obtained from $U$ by taking $\widetilde{\varrho(U)}_t-\widetilde{\varrho(U)}_{t+1}$ $i$'s from the end of the first column and putting them at the end of column $t+1$. (That this many $i$'s can be removed from the first column follows from the fact that $V$ is proper.) Conversely, if $U$ is in case $(b)$, and the $n$-rim ends in column $t+1$, the $i$'s in the $n$-rim in column $t+1$ are removed and placed in the first column.\hspace*{\fill}\hbox{$\Box$} \vskip 12pt It follows from Proposition 4.1 that for $\lambda,\mu,\nu\subset l\times k$, with $|\lambda|+|\mu|=|\nu|+mn,\; m\geq 0$, \begin{equation}\label{qlrdual} N_{\lambda\mu}^{\nu}(l,k)= N_{\widetilde{\lambda}\widetilde{\mu}}^{\widetilde{\nu}}(k,l). \end{equation} Similarly, from Proposition 4.3, \begin{equation}\label{qkdual} K_{\lambda\mu}^{\nu}(l,k)= \widetilde{K}_{\widetilde{\lambda}\mu}^{\widetilde{\nu}}(k,l), \end{equation} for $\lambda,\nu\subset l\times k$, and $\mu =(\mu_1,\ldots ,\mu_r),\; 0\leq\mu_i\leq k$. The numbers on the right are the conjugate quantum Kostka numbers, calculated by regarding $\widetilde{\lambda}$ and $\widetilde{\nu}$ in the $k\times l$ rectangle. Equivalently, if $\nu[\widetilde{m}]$ is the partition obtained from $\nu$ by adding $m$ rim $n$-hooks, each occupying rows $1$ through $l$, then $\widetilde{K}_{\widetilde{\lambda}\mu}^{\widetilde{\nu}}(k,l)$ is the number of tableaux $T$ of content $\mu$ on $\nu[\widetilde{m}]/\lambda$ such that for each entry $i$ of $T$ in the first row and column $k+p,\; p\geq 1$, the box in the $l^{\rm th}$ row and column $p$ is either in $\lambda$, or it has an entry of $T$ strictly smaller than $i$. We do not know a correspondance between the sets of tableaux that would explain equation $(\ref{qkdual})$. \begin{example} {\rm $k=5$, $l=4$, $\lambda=(5,3,3,1)$, $\mu=(5,2,2)$, $\nu=(2,1)$:} \vskip 10pt \epsfxsize 9cm \centerline{\epsfbox{qkex1.eps}} \vskip 10pt \centerline{$K_{\lambda\mu}^{\nu}(4,5)=2$.} \vskip 10pt {\rm Note that there are in fact no nonproper tableaux of content $\mu$ on the shape $\nu[m]/\lambda$.} \vskip 10pt \epsfxsize 8cm \centerline{\epsfbox{qkex2.eps}} \vskip 10pt \centerline{$\widetilde{K}_{\widetilde{\lambda}\mu}^{\widetilde{\nu}}(5,4)=2$.} \vskip 10pt {\rm As opposed to the previous case, there are many nonproper conjugate tableaux on $\widetilde{\nu}[m]/\lambda$. One such tableau is shown below.} \vskip 10pt \epsfxsize 2cm \centerline{\epsfbox{nprop2.eps}} \vskip 10pt \end{example} Since $$K_{\lambda\mu}^{\nu}(l,k)=\sum\epsilon(\varrho/\nu)K_{\lambda\mu}^{\varrho},$$ where the sum is over all $\varrho$ that can be obtained from $\nu$ by adding $m$ rim $n$-hooks, and $K_{\lambda\mu}^{\varrho}$ is the classical Kostka number, equation $(\ref{qkdual})$ gives a nontrivial relation among classical Kostka numbers. \bigskip \vskip .2truein

9705

alg-geom/9705026

en

https://arxiv.org/abs/alg-geom/9705026

alg-geom/9705026

Shihoko Ishii

Shihoko Ishii

Minimal model theorem for toric divisors

AMS-Latex, text 14 pages, figures 2 pages. The figures are not submitted because of a technical reason. A person who wants the figure pages is asked to contact to the Author. She will send a hard copy of the figures by postal mail

Minimal model conjecture for a proper variety $X$ is that if $\kappa(X)\geq 0$, then $X$ has a minimal model with the abundance and if $\kappa =-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$ with $-K_Y$ relatively ample. In this paper, we prove this conjecture for a $\D$-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of ambient toric variety. Furthermore, for such a divisor $X$ with $\kappa(X)\geq 0$ we construct a projective minimal model with the abundance in a different way; by means of "puffing up" of the polytope, which gives an algorithm of a construction of a minimal model.

alg-geom

\section{\bf Introduction} Let $k$ be an algebraically closed field of arbitrary characteristic. Varieties in this paper are all defined over $k$. Let $X$ be a proper algebraic variety. A proper algebraic variety $Y$ is called a minimal model of $X$, if (1) $Y$ is birationally equivalent to $X$, (2) $Y$ has at worst terminal singularities and (3) the canonical divisor $K_Y$ is nef. A minimal model $Y$ is said to have the abundance if the linear system $|mK_Y|$ is basepoint free for sufficiently large $m$. The minimal model conjecture states: an arbitrary proper variety with $\kappa\geq 0$ has a minimal model with the abundance and an arbitrary proper variety with $\kappa=-\infty$ has a birationally equivalent model $Y$ with at worst terminal singularities and a fibration $Y\to Z$ to a lower dimensional variety with $-K_Y$ relatively ample. The conjecture holds true for 2-dimensional case which is known as a classical result. For 3-dimensional case the conjecture for $k={\Bbb C}$ is proved by Mori \cite{M} and Kawamata \cite{kawamata}, while it is not yet proved for higher dimensional case. As a special case of higher dimension, Batyrev \cite{Batyrev} proved, among other results, the existence of a minimal model for a ${\Delta}$-regular anti-canonical divisor of a Gorenstein Fano toric variety ${T_N(\D)}$. In this paper we prove the minimal model conjecture for every ${\Delta}$-regular divisor $X$ on a toric variety of arbitrary dimension by means of successive contractions of extremal rays and flips. Furthermore for such a divisor with $\kappa \geq 0$, we construct a projective minimal model with the abundance in a different way; by means of "puffing up" of the polytope corresponding to the adjoint divisor. By this method one can concretely construct a projective minimal model. As a corollary, for a field $k$ of characteristic 0, the minimal model conjecture holds for a general member of a basepoint free linear system on a proper toric variety over $k$. The half of this work was done during the author's stay at the Johns Hopkins University on April 1996. She expresses her gratitude to Professors Shokurov and Kawamata who made her stay possible. She is also grateful to the Johns Hopkins University for their hospitality. She would like to thank Professor Reid who gave useful suggestions and Professor Batyrev who called her attention to this problem and pointed out an error of the first draft of this paper. \section{\bf The minimal model theorem for toric divisors} \begin{defn} (\cite{Batyrev}) A divisor $X$ of a toric variety ${T_N(\D)}$ defined by a fan ${\Delta}$ is called ${\Delta}$-regular, if for every $\tau \in {\Delta}$ the intersection $X\cap orb( \tau)$ is either a smooth divisor of $orb( \tau)$ or empty. \end{defn} \begin{defn} Let $V$ and $V'$ are toric varieties defined by fans ${\Delta}$ and ${\Delta}'$ respectively and $f: V' - \to V$ a toric birational map: i.e. ${\Delta}'$ is obtained by successive subdivisions and converse of subdivisions from ${\Delta}$. Let $T$ be the maximal orbit in $V$. If an irreducible divisor $X$ on $V$ satisfies $X\cap T\neq \phi$, the divisor $X'=\overline{f^{-1}(X\cap T)}$ on $V'$ is called the proper transform of $X$ on $V'$. \end{defn} \begin{defn} Let $X$ a divisor on a normal variety $V$ such that $K_V+X$ is a ${\Bbb Q}$-Cartier divisor and $f:V'\to V$ a birational morphism. Let $X'$ be the proper transform of $X$. If $$K_{V'}+X'=f^*(K_V+X)+\sum_i a_iE_i,$$ where $E_i$'s are the exceptional divisors of $f$, then $a_i$ is called the discrepancy of $K_V+X$ at $E_i$ \end{defn} \begin{defn} Let $V$ be a toric variety defined by a simplicial fan ${\Delta}$ and $X$ an irreducible divisor on $V$. The divisor $K_V+X$ is called terminal, if the following hold: (1) there exists a morphism $f:V'=T_N({\Delta}')\to V$ corresponding to a non-singular subdivision ${\Delta}'$ of ${\Delta}$ (${\Delta}'\neq {\Delta}$) such that the proper transform $X'$ of $X$ on $V'$ is ${\Delta}'$-regular, in particular $X\cap T\neq \phi$ for the maximal orbit $T$ in $V$, and (2) for every such morphism as in (1) the discrepancy of $K_V+X$ at every exceptional divisor on $V'$ is positive. \end{defn} \begin{lem} \label{terminal} If $V={T_N(\D)} $ is non-singular and an irreducuble divisor $X$ on $V$ is ${\Delta}$-regular, then $K_V+X$ is terminal \end{lem} \begin{pf} For every non-singular subdivision ${\Delta}'$ of ${\Delta}$, where ${\Delta}'\neq {\Delta}$, the proper transform $X'$ of $X$ by the corresponding morphism $f:V'=T_N({\Delta}')\to V$ is ${\Delta}'$-regular by 3.2.1 of \cite{Batyrev}. Since $X'=f^*X$ and $K_{V'}=f^*K_V+\sum_ia_iE_i$, where $a_i>0$ for every exceptional divisor $E_i$ on $V'$, it follows that the discrepancy of $K_{V}+X$ at each $E_i$ is positive. \end{pf} \begin{prop} Let $V$ be a toric variety defined by a simplicial fan ${\Delta}$ and $X$ an irreducible divisor on $V$. Then the divisor $K_V+X$ is terminal if and only if the following hold: (i) there exists a morphism $f:V'=T_N({\Delta}')\to V$ corresponding to a non-singular subdivision ${\Delta}'$ of ${\Delta}$ (${\Delta}'\neq {\Delta}$) such that the proper transform $X'$ of $X$ on $V'$ is ${\Delta}'$-regular. (ii) for one such morphism as in (i) the discrepancy of $K_V+X$ at every exceptional divisor on $V'$ is positive. \end{prop} \begin{pf} Let $f:V'=T_N({\Delta}')\to V$ be the morphism satisfying the condition (i) and (ii) and $g:V''\to V$ be another morphism satisfying (i). Take a nonsingular toric variety $\tilde V$ which dominates both $V'$ and $V''$. Then by \ref{terminal}, $K_{V'}+X'$ is terminal. Therefore the discrepancy of $K_V+X$ at every exceptional divisor on $\tilde V$ is positive which yields the positivity of it at every exceptional divisor on $V''$. \end{pf} \begin{lem} \label{terminal sing} Let $V$ be a toric variety defined by a simplicial fan ${\Delta}$ and $X$ an irreducible divisor on $V$. If the divisor $K_V+X$ is terminal, then $V$ has at worst terminal singularities. \end{lem} \begin{pf} This follows from the fact that a discrepancy of $K_V$ is greater than or equal to that of $K_V+X$. \end{pf} Here we summerize the results of Reid (\cite{Reid}) which are used in this section. \begin{prop} \label{reid} (\cite{Reid}) Let $V$ be the toric variety defined by a proper simplicial fan ${\Delta}$. (i) $NE(V)=\sum_{i=1}^r{\Bbb R}_{\geq 0}[\ell_i]$, where $\ell_i$'s are 1-dimensional strata on $V$. Here each ${\Bbb R}_{\geq 0}[\ell_i]$ is called an extremal ray. (ii) For every extremal ray $R$ there exist a toric morphism $\varphi_R:V\to V'$ which is an elementary contraction in the sense of Mori theory: $\varphi_R{\cal O}_V={\cal O}_{V'}$ and $\varphi _RC=pt$ if and ony if $[C]\in R$. Let $A\subset V$ and $B\subset V'$ be the loci on which $\varphi_R$ is not an isomorphism, then $\varphi_R|_A:A \to B$ is a flat morphism and all of whose fibers are weighted projective spaces of the common dimension. (iii) If $\varphi_R:V\to V'=T_N({\Delta}')$ is birational and not isomorphic in codimension one, then the exceptional set of $\varphi_R$ is an irreducible divisor and ${\Delta}'$ is proper simplicial. Here this $\varphi_R$ is called a divisorial contraction. (iv) If $\varphi_R:V\to V'=T_N({\Delta}')$ is isomorphic in codimension one, then there exists the following commutative diagram: $$\matrix & & \tilde V & & \\ &\psi \swarrow & &\searrow \psi_1 & \\ V & & & &V_1=T_N({\Delta}_1)\\ &\varphi_R \searrow& &\swarrow \varphi_1& \\ & & V' & & \endmatrix$$ % such that ${\Delta}_1$ is proper simplicial, ${\Delta}_1(1)={\Delta}(1)$, all morphisms are elementary contractions of extremal rays, $\psi$ and $\psi_1$ are birational morphisms with the exceptional divisor $D$, $\varphi_R$ and $\varphi_1$ are birational morphisms with the exceptional sets $\psi(D)$ and $\psi_1(D)$ respectively, and identifying $N_1(V)$ and $N_1(V_1)$, $-R$ is an extremal ray in $NE(V_1)$ and $\varphi_1=\varphi_{-R}$. Here the birational map $\varphi_1^{-1}\circ \varphi_R:V- \to V_1$ is called a flip. \end{prop} \begin{lem} \label{elementary} Let $V$ be a toric variety defined by a proper simplicial fan ${\Delta}$ and $X$ an irreducible divisor such that $K_V+X$ is terminal. Let $R$ be an extremal ray such that $(K_V+X)R<0$. Then the following hold: (i) if $\varphi_R:V\to V'=T_N({\Delta}')$ is a divisorial contraction, then $K_{V'}+X'$ is terminal, where $X'$ is the proper transform of $X$ on $V'$; (ii) let $\varphi_R:V\to V'$ be isomorphic in codimension one; for the diagram $$\matrix & & \tilde V & & \\ &\psi \swarrow & &\searrow \psi_1 & \\ V & & & &V_1=T_N({\Delta}_1)\\ &\varphi_R \searrow& &\swarrow \varphi_1& \\ & & V' & & \endmatrix$$ of (iv), \ref{reid}, let $X_1$ be the proper transform of $X$ on $V_1$, $D$ the exceptional divisor of $\psi$ and $\psi_1$, $\alpha$ the discrepancy of $K_V+X$ at $D$ and $\alpha'$ the discrepancy of $K_{V_1}+X_1$ at $D$; then $\alpha <\alpha'$ and $K_{V_1}+X_1$ is terminal. \end{lem} \begin{pf} For the proof of (i), first one should remark that $V'$ is ${\Bbb Q}$-factorial, because ${\Delta}'$ is simplicial. Let $E$ be the exceptional divisor for $\varphi_R$. \begin{claim} $ER<0$. \end{claim} For the proof of the claim, take an irreducible divisor $H$ on $V'$ such that $H\supset \varphi_R(E)$. Then $\varphi^*H=[H]+aE$ with $a>0$, where $[H]$ is the proper transform of $H$ on $V$. Since $ (\varphi_R^*H) R=0$ and $[H]R>0$, it follows that $aER<0$ which completes the proof of the claim. Denote $K_V+X$ by $\varphi_R^*(K_{V'}+X')+bE$, then $b>0$. In fact, by $(K_V+X)R<0$, $\varphi_R^*(K_{V'}+X')R=0$ and $ER<0$, it follows that $b>0$. Let $\overline {\Delta}$ be a non-singular subdivision of ${\Delta}$ such that the proper transform $\overline X$ of $X$ on $\overline V=T_N(\overline {\Delta})$ is $\overline {\Delta}$-regular. Since $K_V+X$ is terminal, the discrepancy of $K_V+X$ at every exceptional divisor for $\overline V \to V$ is positive. By this, and $b>0$, it follows that the discrepancy of $K_{V'}+X'$ at every exceptional divisor for $\overline V \to V'$ is positive. For the proof of (ii), take a curve $\ell$ on $\tilde V$ such that $\psi_1(\ell)=pt$ and $\psi(\ell)\neq pt$. This is possible, because if a curve contracted by both $\psi$ and $\psi_1$ exists, then the extremal rays corresponding to $\psi$ and $\psi_1$ coincide which implies $V\simeq V_1$ and $\varphi_R=\varphi_1$ a contradiction to $\varphi_1=\varphi_{-R}$ in (iv) of \ref{reid}. For this $\ell$, one can prove that $D\ell <0$ in the same way as in the claim above. Now as $\psi_*(\ell)$ is contracted to a point by $\varphi_R$, $[\psi_*(\ell)]\in R$, therefore $\psi^*(K_V+X)\ell=(K_V+X)\psi_*(\ell)<0$. By intersecting $\ell$ with $K_{\tilde V}+\tilde X= \psi^*(K_V+X)+\alpha D$, we obtain $$(K_{\tilde V}+\tilde X)\ell< \alpha D\ell.$$ Here the left hand side is $\psi_1^*(K_{V_1}+X_1)\ell + \alpha'D\ell$, and $\psi_1^*(K_{V_1}+X_1)\ell=0$ because of the definition of $\ell$. This proves that $\alpha<\alpha'$. To prove the last statement, take a non-singular subdivision $\tilde {\tilde {\Delta}}$ of $\tilde {\Delta}$ such that the proper transform $\tilde {\tilde X}$ of $\tilde X$ is $\tilde {\tilde {\Delta}}$-regular. Let $\lambda:\tilde {\tilde V}=T_N(\tilde {\tilde {\Delta}})\to \tilde V$ be the corresponding morphism. Then $K_{\tilde {\tilde V}}+\tilde {\tilde X}= \lambda^*\psi^*(K_V+X)+\sum_i\beta_iE_i$, where $\beta_i>0$ for every exceptional divisor $E_i$, because $K_V+X$ is terminal. Now by substituting $\psi^*(K_V+X)=\psi_1^*(K_{V_1}+X_1)+ (\alpha'-\alpha)D$ into the equality above, the discrepancy of $K_{V_1}+X_1$ at every exceptional divisor on $\tilde {\tilde V}$ turns out to be positive. \end{pf} \begin{thm} \label{theorem} Let $V$ be a toric variety defined by a proper simplicial fan ${\Delta}$ and $X$ an irreducible divisor on $V$ such that $K_V+X$ is terminal. Then there exists a sequence of birational toric maps: $$V=V_1 \stackrel{\varphi_1}{-\to} V_2 \stackrel{\varphi_2}{-\to} \cdots \stackrel{\varphi_{r-1}}{-\to}V_r$$ where (i) each $\varphi_i$ is either a divisorial contraction or a flip, in particular $V_i$ is defined by a proper simplicial fan; (ii) for the proper transform $X_i$ of $X$ on $V_i$ $(i=1,\ldots , r)$, $K_{V_i}+X_i$ is terminal; (iii) either that $K_{V_r}+X_r$ is nef or that there exists an extremal ray $R$ on $V_r$ such that $(K_{V_r}+X_r)R<0$ and the elementary contraction $\varphi_R: V_r\to Z$ is a fibration to a lower dimensional variety $Z$. \end{thm} \begin{pf} If $K_V+X$ is nef, then the statement is obvious. If $K_V+X$ is not nef, then there is an extremal ray $R$ such that $(K_V+X)R<0$. Take the elementary contraction $\varphi_R:V\to V'$. If $\dim V'< \dim V$, then the statement holds. So assume that $\varphi _R$ is birational. If $\varphi_R$ is divisorial, then define $\varphi_1:=\varphi_R:V\to V'=:V_2$. If $\varphi_R$ is not divisorial, then let $\varphi_1:V-\to V_2$ be the flip. Then in both cases, $K_{V_2}+X_2$ is terminal by Lemma \ref{elementary}. Now if $K_{V_2}+X_2$ is nef, then the proof is completed. If it is not nef, make the same procedure as above. By the successive procedure, one obtains a sequence of divisorial contractions and flips: $$V=V_1 \stackrel{\varphi_1}{-\to} V_2 \stackrel{\varphi_2}{-\to} \cdots \stackrel{\varphi_{r-1}}{-\to}V_r \cdots .$$ It is sufficient to prove that the sequence terminates at finite stage. Let us assume that there exists such a sequence of infinite length. Since the divisorial contraction makes the Picard number strictly less, the number of divisorial contractions in the sequence is finite. So we may assume that there is $m_0\in {\Bbb N}$ such that $\varphi_m$'s are all flips for $m\geq m_0$. By (iv) of \ref{reid} the set of one dimensional cones of the fan defining $V_m$ ($m\geq m_0$) are common. As the number of such fans is finite, there are numbers $m<m'$ such that $\varphi_{m'-1}\circ \cdots \circ \varphi_m: V_m -\to V_{m'}$ is identity. For each flip $\varphi_j$ $(j=m,\ldots , m'-1)$, take the dominating variety $V_j'$ as in (iv) of \ref{reid}: $$\matrix & & V'_j & & \\ &\psi_j \swarrow & &\searrow \psi'_j & \\ V_j & & & &V_{j+1} \endmatrix.$$ Let $D_j$ be the exceptional divisor of $\psi_j$ and $\psi_{j+1}$. Then take a proper toric variety $\tilde V=T_N(\tilde {\Delta})$ which dominates all $V'_j$, $j=m,\ldots , m'-1$ and on which the proper transform $\tilde X$ of $X_j$'s is $\tilde {\Delta}$-regular. This is possible, because $K_{V_j}+X_j$'s are terminal. Here one should note that the set of exceptional divisors on $\tilde V$ for all morphisms $\tilde V\to V_j$ $(j=m,\ldots ,m'-1)$ are common. For every $j=m,\ldots , m'-1$, the discrepancy $\alpha$ of $K_{V_j}+X_j$ at $D_j$ is less than the discrepancy $\alpha'$ of $K_{V_{j+1}}+X_{j+1}$ at $D_j$ by \ref{elementary}. By this fact, for every exceptional divisor $E$ on $\tilde V$, the discrepancy $\alpha_E$ of $K_{V_j}+X_j$ at $E$ and the discrepancy $\alpha'_E$ of $K_{V_{j+1}}+X_{j+1}$ at $E$ satisfy $\alpha_E\leq \alpha'_E$ and for at least one exceptional divisor $E$, $\alpha_E<\alpha'_E$. Therefore comparing $K_{V_m}+X_m$ and $K_{V_{m'}}+X_{m'}$, there exists an exceptional divisor on $\tilde V$ at which the discrepancy of $K_{V_m}+X_m$ is less than that of $K_{V_{m'}}+X_{m'}$, which is the contradiction to that $V_m\to V_{m'}$ is the identity. \end{pf} To apply the theorem above to the minimal model problem for a toric divisor, one needs the following lemma. \begin{lem} \label{basic lemma} (Lemma 2.7, \cite{weight}) Let $Y\subset Z$ be an irreducible Weil divisor on a variety $Z$. Assume that $Z$ admits at worst ${\Bbb Q}$-factorial log-terminal singularities. Let $\varphi:\tilde Y \to Y$ be a resolution of singularities on $Y$. Assume $K_{\tilde Y}=\varphi^*((K_Z+Y)|_{Y})+\sum_im_i E_i$ with $m_i>-1$ for all $i$, where $E_i$'s are the exceptional divisors of $\varphi$. Then $Y$ is normal, and $Y$ has at worst log-terminal singularities. In particular, if $m_i> 0$ for all $i$, then $Y$ has at worst terminal singularities. \end{lem} \begin{cor} \label{MMT} Let $V$ be a toric variety defined by a proper fan ${\Delta}$ and $X$ a ${\Delta}$-regular divisor on $V$. If $\kappa(X)\geq 0$, then $X$ has a minimal model with the abundance. If $\kappa(X)=-\infty$, then $X$ is birationally equivalent to a proper variety $Y$ with at worst terminal singularities and a fibration $\varphi:Y\to Z$ to a lower dimensional variety $Z$ with $-K_{Y}$ relatively ample. \end{cor} \begin{pf} Let $V_1$ be the toric variety defined by a non-singular subdivision ${\Delta}_1$ of ${\Delta}$ and $X_1$ be the proper transform of $X$ on $V_1$. Then $X_1$ is ${\Delta}_1$-regular and therefore $K_{V_1}+X_1$ is terminal by \ref{terminal}. Then one obtains a sequence: $$V_1 \stackrel{\varphi_1}{-\to} V_2 \stackrel{\varphi_2}{-\to} \cdots \stackrel{\varphi_{r-1}}{-\to}V_r$$ as in Theorem \ref{theorem}. One can prove that for each $j=1, \ldots , r$, $X_j$ has at worst terminal singularities. In fact, take a morphism $\varphi:\tilde V \to V_j$ corresponding to a non-singular subdivison $\tilde {\Delta}$ of the fan ${\Delta}_j$ of $V_j$ such that the proper transform $\tilde X$ of $X$ is $\tilde {\Delta}$-regular. Then, as $K_{V_j}+X_j$ is terminal, it follows that $$(K_{\tilde V}+\tilde X)|_{\tilde X}= \varphi^*((K_{V_j}+X_j)|_{X_j})+\sum_ia_iE_i|_{\tilde X} \ \ \ \ (a_i>0\ \text{for\ all}\ i).$$ Here the left hand side is the canonical divisor $K_{\tilde X}$ of a non-singular variety $\tilde X$. Therefore by Lemma \ref{basic lemma}, one sees that $X_j$ has at worst terminal singularities. By (iii) of \ref{theorem} there are two cases for $V_r$. \begin{case} $K_{V_r}+X_r$ is nef. Then the linear system $|m(K_{V_r}+X_r)|$ is basepoint free for some $m\in {\Bbb N}$. This is proved by a slight modification of the proof of Toric Nakai Criterion (2.18, \cite{Oda88}). Therefore $|mK_{X_r}|$ is basepoint free, which implies that $X_r$ is a minimal model with the abundance. In this case, $\kappa(X)=\kappa(X_r)\geq 0$. \end{case} \begin{case} There exists an extremal ray $R$ on $V_r$ such that $(K_{V_r}+X_r)R<0$ and the elementary contraction $\varphi_R: V_r\to Z$ is a fibration to a lower dimensional variety $Z$. Under this situation, first consider the case: {\sl Subcase.} $\dim X_r> \dim \varphi_R(X_r)$. Let $ F$ be a fiber of $\varphi_R$. Then by (ii) of \ref{reid}, $F$ is a weighted projective space and $(K_{V_r}+X_r)C<0$ for every curve $C$ in $F$, which implies that $-(K_{V_r}+X_r)$ is relatively ample over $Z$. Hence $-K_{X_r}$ is relatively ample over $\varphi_R(X_r)$. This yields that $\kappa(X)=\kappa(X_r)=-\infty$, and $\varphi_R|_{X_r}:X_r\to \varphi_R(X_r)$ is a desired fibration. {\sl Subcase.} $\dim X_r=\dim \varphi_R(X_r)$. In this case $\dim Z=\dim V_r-1$ and every fiber $\ell$ of $\varphi_R: V_r \to Z$ is ${\Bbb P}^1$ by (ii) of \ref{reid}. Therefore $K_{V_r}\ell=-2$. On the other hand, because $\varphi|_{X_r}$ is generically finite, $X_r\ell>0$. Here, since $V_r$ has at worst terminal singularities by \ref{terminal sing}, the singular locus has codimension greater than 2 and therefore the divisor $X_r$ is a Cartier divisor along a general fiber $\ell$, which yields that $X_r\ell$ is an integer. By $(K_{V_r}+X_r)\ell<0$, it follows $X_r\ell=1$. It implies that $\varphi_R|_{X_r}:X_r\to Z$ is a birational morphism, therefore $X_r$ is rational. So $X$ and $X_r$ are birationally equivalent to ${\Bbb P}^n$ which has ample anti-canonical divisor and of course $\kappa(X)=-\infty$. \end{case} \end{pf} \begin{cor} Let the ground field $k$ be of characteristic zero. Let $V$ be a proper toric variety, $|L|$ a linear system without a basepoint and $X$ a general member of $|L|$. Then the statements of Corollary \ref{MMT} hold for $X$. \end{cor} \begin{pf} By the Bertini's Theorem, $X$ is ${\Delta}$-regular. \end{pf} \begin{cor} \label{kappa} Let $V$ be a toric variety defined by a proper fan ${\Delta}$ and $X$ a ${\Delta}$-regular divisor on $V$. Assume $\kappa(X)\geq 0$. Then there exists a non-singular subdivision $\tilde {\Delta}$ of ${\Delta}$ such that $\tilde V=T_N(\tilde {\Delta})$ and the proper transform $\tilde X$ of $X$ on $\tilde V$ satisfy the following: $$\kappa(\tilde V, K_{\tilde V}+\tilde X)\geq 0.$$ \end{cor} \begin{pf} Use the notation of the proof of \ref{MMT}. Take a nonsingular subdivision $\tilde {\Delta}$ of both ${\Delta}$ and ${\Delta}_r$ which is the fan of $V_r$. Then the proper transform $\tilde X$ of $X$ on $\tilde V=T_N(\tilde {\Delta})$ is $\tilde {\Delta}$-regular. Since $K_{V_r}+X_r$ is terminal and $|m(K_{V_r}+X_r)|$ is basepoint free for some $m\in {\Bbb N}$, $$0\neq \Gamma(V_r, m(K_{V_r}+X_r))\subset \Gamma(\tilde V, m(K_{\tilde V}+\tilde X)).$$ \end{pf} \section{\bf Divisors and Polytopes} \begin{say} Here we summerize the basic notion of an invariant divisor of a toric variety and the corresponding polytope which will be used in the next section. In this paper, a polytope in an ${\Bbb R}$-vector space means the intersection of finite number of half-spaces $\{{\bold m} | f_i({\bold m})\geq a_i\}$ for linear functions $f_i$. \end{say} \begin{say} Let $M$ be the free abelian group ${{\Bbb Z}}^n$ $(n\geq 3)$ and $N$ be the dual $Hom_{{\Bbb Z}}(M, {{\Bbb Z}})$. We denote $M\otimes _{{\Bbb Z}}{{\Bbb R}}$ and $N\otimes_{{\Bbb Z}}{{\Bbb R}}$ by ${M_{{\Bbb R}}}$ and ${N_{\bR}}$, respectively. Define $M_{{\Bbb Q}}$ and $N_{{\Bbb Q}}$ in the same way. Then one has the canonical pairing $(\ \ ,\ \ ):N\times M \to {\Bbb Z}$, which can be canonically extended to $ (\ \ ,\ \ ):{N_{\bR}}\times M_{{\Bbb R}} \to {\Bbb R}$. For a fan ${{\Delta}}$ in ${{N_{\bR}}}$, we construct the toric variety $T_N({{\Delta}})$. The fan ${\Delta}$ is always assumed to be proper, i.e. the support $|{\Delta}|={N_{\bR}}$. Denote by ${\Delta}(k)$ the set of $k$-dimensional cones in ${\Delta}$. Denote by ${{\Delta}}[1]$ the set of primitive vectors ${{\bold q}}=(q_1,\ldots , q_r)\in {N}$ whose rays ${{\Bbb R}}_{\geq 0}{{\bold q}}$ belong to ${{\Delta}(1)}$. For ${{\bold q}}\in {{\Delta}}[1]$, denote by $D_{{\bold q}}$ the corresponding divisor which is denoted by $\overline{orb\ {{\Bbb R}}_{\geq 0}{{\bold q}}}$ in \cite{Oda75}. Denote by $U_{\sigma}$ the invariant affine open subset which contains ${orb\ {\sigma}}$ as the unique closed orbit. \end{say} \begin{defn} For ${\bold p} \in {N_{\bR}}$ and a subset $K\subset {M_{\bR}}$, define $${\bold p}(K):=\inf_{{\bold m}\in K}({\bold p}, {\bold m})$$ \end{defn} \begin{defn} \label{support function} Let ${\Delta}$ be a proper fan in ${N_{\bR}}$. A continuous function $h:{N_{\bR}} \to {\Bbb R}$ is called a ${\Delta}$-support function, if (1) $h|_{\sigma}$ is ${\Bbb R}$-linear for every cone $\sigma \in {\Delta}$ and (2) $h$ is ${\Bbb Q}$-valued on $N_{{\Bbb Q}}$. A ${\Delta}$-support function $h$ is called integral if (2') $h$ is ${\Bbb Z}$-valued on $N$. \end{defn} \vskip.5truecm \begin{prop} \label{correspondence} For a ${\Delta}$-support function $h$, define $D_h=-\sum_{{\bold p}\in {\Delta}[1]}h({\bold p})D_{{\bold p}}$. Then the correspondence $h \mapsto D_h$ gives a bijective map: \{${\Delta}$-support functions\} $\simeq$ \{invariant ${\Bbb Q}$-Cartier divisors on ${T_N(\D)}$\}. Here $D_h$ is a Cartier divisor, if and only if $h$ is integral. \end{prop} \vskip.5truecm \begin{defn} For a ${\Delta}$-support function $h$, define $${\Box_h}:=\{m\in M_{{\Bbb R}}|({\bold p},{\bold m})\geq h({\bold p}), \forall {\bold p}\in {N_{\bR}}\},$$ and call it the polytope associated with $h$ or with $D_h$. Actually it is a polytope by \ref{boundary} and compact since the fan ${\Delta}$ is proper. \end{defn} \vskip.5truecm \begin{prop} \label{basepoint free} (see \cite{Oda88}) For an integral ${\Delta}$-support function $h$, the following are equivalent: (i) the linear system $|D_h|$ is basepoint free; (ii) $h$ is upper convex; i.e. for arbitrary ${\bold n}, {\bold n}'\in {N_{\bR}}$, $h({\bold n})+h({\bold n}')\leq h({\bold n}+{\bold n}')$; (iii) ${\Box_h}=$ the convex hull of $\{h_{\sigma}|\sigma \in {\Delta}(n)\} $, where $h_{\sigma}$ is a point of $M$ which gives the linear function $h|_{\sigma}$ for $\sigma \in {\Delta}(n)$. \end{prop} \vskip.5truecm \begin{prop} \label{ample} (see \cite{Oda88}) For a ${\Delta}$-support function $h$, the following are equivalent: (i) the ${\Bbb Q}$-Cartier divisor $D_h$ is ample; (ii) $h$ is strictly upper convex; i.e. $h$ is upper convex and $h({\bold n})+h({\bold n}')< h({\bold n}+{\bold n}')$, if there is no cone $\sigma$ such that ${\bold n}, {\bold n}'\in \sigma$; (iii) ${\Box_h}$ is of dimension $n$ and the correspondence $\sigma \mapsto h_{\sigma}$ gives the bijective map ${\Delta}(n) \simeq $ \{The vertices of ${\Box_h}$\}, where $h_{\sigma}$ is a point of $M_{{\Bbb Q}}$ which gives the linear function $h|_{\sigma}$ for $\sigma \in {\Delta}(n)$. \end{prop} \vskip.5truecm Now we show simple lemmas which are used in the next section. \begin{lem} \label{inf} Let $h$ be a ${\Delta}$-support function. If $h_{\sigma}\in {\Box_h}$ for every $\sigma\in {\Delta}(n)$, then $h({\bold p})={\bold p}({\Box_h})$ for every ${\bold p}\in {N_{\bR}}$, and the polytope ${\Box_h}$ is the convex hull of the set $\{h_{\sigma}\}$. \end{lem} \begin{pf} By the definition of ${\Box_h}$, $h({\bold p})\leq({\bold p},{\bold m})$ for all ${\bold m}\in {\Box_h}$. Therefore $h({\bold p})\leq {\bold p}({\Box_h})$. Let $\sigma$ be the cone in ${\Delta}(n)$ such that ${\bold p}\in \sigma$, then $h({\bold p})=({\bold p}, h_{\sigma})\geq {\bold p} ({\Box_h})$, since $h_{\sigma}\in {\Box_h}$. For the second assertion, assume a vertex ${\bold m}\in {\Box_h}$ does not belong to the convex hull of $\{h_{\sigma}\}$. Then there exists ${\bold p}\in {N_{\bR}}$ such that $({\bold p}, {\bold m})<({\bold p},h_{\sigma})$ for every $\sigma \in {\Delta}(n)$, where the left hand side is greater than or equal to $h({\bold p})$ by the definition of ${\Box_h}$. This is a contradiction, because for $\sigma\in {\Delta}(n) $ such that ${\bold p} \in \sigma$, $h({\bold p})=({\bold p}, h_{\sigma})$. \end{pf} \vskip.5truecm \begin{lem} \label{boundary} Denote an invariant divisor $D_h=\sum_{{\bold p}\in {\Delta}[1]}m_{{\bold p}}D_{{\bold p}}$. Then $${\Box_h}=\bigcap_{{\bold p}\in {\Delta}[1]}\{{\bold m}\in {M_{\bR}}| ({\bold p},{\bold m})\geq -m_{{\bold p}}\}.$$ \end{lem} \begin{pf} By \ref{correspondence}, $m_{{\bold p}}=-h({\bold p})$, then the inclusion ${\Box_h}\subset \bigcap_{{\bold p}\in {\Delta}[1]}\{{\bold m}\in {M_{\bR}}| ({\bold p},{\bold m})\geq -m_{{\bold p}}\} $ is obvious. Take an element ${\bold m}$ from the right hand side. For an arbitrary ${\bold p} \in {N_{\bR}}$, take $\sigma \in {\Delta}(n)$ such that ${\bold p} \in \sigma$. Let $\sigma$ be spanned by ${\bold p}_1,{\bold p}_2,\ldots ,{\bold p}_s$ $({\bold p}_i\in {\Delta}[1])$, then ${\bold p}=\sum a_i{\bold p}_i$ with $a_i\geq 0$. One obtains that $({\bold p},{\bold m})= \sum a_i({\bold p}_i,{\bold m})\geq \sum a_ih({\bold p}_i) =\sum a_i({\bold p}_i,h_{\sigma})=h({\bold p})$, which shows that ${\bold m}$ belongs to ${\Box_h}$. \end{pf} \vskip.5truecm \begin{defn} \label{contribution} Let $\Box$ be a polytope in ${M_{\bR}}$ defined by $\bigcap_{i=1}^rH_i$, where $H_i=\{{\bold m}\in {M_{\bR}}| ({\bold p}_i,{\bold m})\geq a_i\}$. We say that $H_i$ contributes to $\Box$, if $\Box \cap \{{\bold m}\in {M_{\bR}}| ({\bold p}_i,{\bold m})= a_i\}\neq \phi$. And we say that $H_i$ contributes properly to $\Box$, if $\bigcap_{j\neq i}H_j\neq \Box$. \end{defn} \begin{defn} \label{dual fan} Let $\Box$ be an $n$-dimensional compact polytope in ${M_{\bR}}$. Define the dual fan $\Gamma_{\Box}$ of $\Box$ as follows: $\Gamma_{\Box}=\{\gamma^*\}$, where $\gamma$ is a face of $\Box$ and $\gamma^*:=\{{\bold n}\in{N_{\bR}}|$ the function ${\bold n}|_{\Box}$ attains the minimal value at all points of $\gamma$\}. Then $\Gamma_{\Box}$ turns out to be a proper fan. \end{defn} \vskip.5truecm \begin{say} \label{projective} If ${\Delta}$ is the dual fan of the polytope ${\Box_h}$ corresponding to a ${\Delta}$-support function $h$, then by \ref{ample} $D_h$ is ample, therefore the variety ${T_N(\D)}$ turns out to be a projective variety. \end{say} \section{\bf The construction of a minimal model} \begin{say} In this section we concretely construct a projective minimal model with the abundance for a ${\Delta}$-regular toric divisor $X$ with $\kappa(X)\geq 0$ by means of a polytope of the adjoint divisor. Let $V$ be a toric variety defined by a proper fan ${\Delta}$ and $X$ a ${\Delta}$-regular divisor with $\kappa(X)\geq 0$. To construct a minimal model of $X$ we may assume that $V$ is non-singular and $\kappa(V, K_V+X)\geq 0$, by Corollary \ref{kappa}. \end{say} \begin{say} \label{construction} {\bf The construction} Let $h$ be a ${\Delta}$-support function such that $ K_{{T_N(\D)}}+X \sim D_h$. Then, by $\kappa ({T_N(\D)} , K_{{T_N(\D)}}+X) \geq 0$, it follows that ${\Box_h} \neq \phi$. Let ${\Delta}[1]=\{{\bold p}_1,\ldots ,{\bold p}_s\}$ and $H_i=\{{\bold m} \in {M_{\bR}}|({\bold p}_i,{\bold m}) \geq h({\bold p}_i)\}$. Then by \ref{boundary} ${\Box_h}=\bigcap_{i=1}^sH_i$. Assume that $H_1,\ldots ,H_r$ $(r\leq s)$ are all that contribute to ${\Box_h}$. For $\epsilon_i>0$ $i=1,\ldots , r$, define $H_{i, \epsilon _i} :=\{{\bold m}\in {M_{\bR}}| ({\bold p}_i,{\bold m})\geq h({\bold p}_i)-\epsilon_i\}$, $\partial H_{i, \epsilon _i} :=\{{\bold m}\in {M_{\bR}}| ({\bold p}_i,{\bold m})= h({\bold p}_i)-\epsilon_i\}$ and $\Box({\epsilon}):=\bigcap_{i=1}^rH_{i,\epsilon_i}$, where $\epsilon = (\epsilon_1,\ldots , \epsilon_r)$. Here one should note that the polytope ${\Box_h}$ may not be of the maximal dimension. By "puffing up" this, one get a polytope $\Box({\epsilon})$ of the maximal dimension. The subset $Z=\{\epsilon \in {\Bbb R}_{>0}^r| \bigcup \partial H_{i,\epsilon_i}$ is not of normal crossings\} is Zariski closed and the complement ${\Bbb R}_{>0}^r\setminus Z$ is divided into finite number of chambers. Take a chamber $W$ such that: (\ref{construction}.1) $0\in \overline{W}$; (\ref{construction}.2) every $H_{i,\epsilon_i}$ $ (i=1,\ldots, r)$ contibutes properly to $\Box({\epsilon})$ for $\epsilon\in W$. Then the dual fan $\Sigma$ of $\Box({\epsilon})$ is common for every $\epsilon \in W$ and it is simplicial, because $\bigcup\partial H_{i,\epsilon_i}$ is of normal crossings. Let ${X(\Sigma)}$ be the proper transform of $X$ in ${T_N(\Sigma)}$. we claim that ${X(\Sigma)}$ is a minimal model of $X$ with the abundance. One can see that ${T_N(\Sigma)}$ is projective, because an invariant ${\Bbb Q}$-Cartier divisor $\sum_{{\bold p}_i\in\Sigma[1]}(h({\bold p}_i)-\epsilon_i)D_{{\bold p}_i}$ with all $\epsilon_i$ rational and $\epsilon \in W$ is ample since $\Sigma$ is the dual fan of the corresponding polytope to this divisor (\ref{projective}). Hence the projectivity of ${X(\Sigma)}$ follows automatically. \end{say} \vskip.5truecm \begin{say} Now we are going to prove that ${X(\Sigma)}$ satisfies desired conditions for a minimal model. First note that $\Sigma[1]=\{{\bold p}_1,\ldots , {\bold p}_r\}$, by \ref{construction}.2. Next note that every ${\Bbb Q}$-Weil divisor on ${T_N(\Sigma)}$ is a ${\Bbb Q}$-Cartier divisor, because $\Sigma$ is simplicial and therefore ${T_N(\Sigma)}$ has quotient singularities. \end{say} \begin{claim} \label{h and k} The divisor $K_{{T_N(\Sigma)}}+{X(\Sigma)}$ is linearly equivalent to an invariant divisor $-\sum_{i=1}^rh({\bold p}_i)D_{{\bold p}_i}$. Let $k$ be the $\Sigma$-support function corresponding to this divisor, then $h({\bold p}_i)=k({\bold p}_i)$ for $i=1,\ldots , r$ and ${\Box_h}={\Box_k}$. \end{claim} \begin{pf} The first assertion follows from that the divisor $K_{{T_N(\Sigma)}}+{X(\Sigma)}$ is the proper transform of $K_{{T_N(\D)}}+X \sim -\sum_{i=1}^sh({\bold p}_i)D_{{\bold p}_i}$. The second assertion is obvious and the last assertion follows from \ref{boundary} and the fact that $H_1,\ldots , H_r$ are all that contribute to ${\Box_h}$. \end{pf} \begin{claim} \label{k in box} For all $\sigma \in \Sigma(n)$, it follows that $k_{\sigma}\in {\Box_k}$. \end{claim} \begin{pf} Let $\{{\epsilon^{(m)}}\}_m$ be series of rational points in $W$ which converge to $0$. Let ${k^{(m)}}$ be the $\Sigma$-support function corresponding to a ${\Bbb Q}$-Cartier divisor $\sum_{i=1}^r(-h({\bold p}_i)+{\epsilon^{(m)}} _i)D_{{\bold p}_i}$. Then by \ref{boundary} it follows that $\Box_{{k^{(m)}}}=\Box({\epsilon^{(m)}})$, and therefore by \ref{projective} the divisor is ample. Replacing $\{{\epsilon^{(m)}}\}_m$ by suitable subsequence, one can assume there exists $\lim_{m\to \infty}{k^{(m)}}_{\sigma}$ for every $\sigma\in \Sigma (n)$. Indeed, replacing by suitable subsequence, one may assume that ${\epsilon^{(m)}}_i\geq {\epsilon^{(m+1)}}_{i}$ for every $i$, then $\Box_{{\epsilon^{(m)}}}\supset \Box_{\epsilon^{m+1}}\supset\cdots$; therefore for every $\sigma\in \Sigma(n)$ and $m$ it follows that ${k^{(m)}}_{\sigma}\in \Box_{\epsilon^{(1)}}$ which is compact; so $\{{k^{(m)}}_{\sigma}\}$ have an accumulating point. Let $k'_{\sigma}:=\lim_{m\to \infty}{k^{(m)}}_{\sigma}$, then $k'_{\sigma}\in {\Box_k}$, because the ampleness of $D_{{k^{(m)}}}$ yields ${k^{(m)}}_{\sigma}\in \Box({\epsilon^{(m)}})$. The collection $\{k'_{\sigma}\}_{\sigma\in\Sigma(n)}$ defines a function $k'$ on ${N_{\bR}}$. In fact, for every $m$, ${k^{(m)}}_{\sigma}={k^{(m)}}_{\tau}$ as a function on $\sigma\cap \tau$, which yields that $k'_{\sigma}=k'_{\tau}$ as a function on $\sigma\cap \tau$. Now one obtains that $k'=k$. This is proved as follows: for every ${\bold p}_i\in \Sigma[1]$ take $\sigma\in \Sigma(n)$ such that ${\bold p}_i \in \sigma$; $k'({\bold p}_i)=({\bold p}_i,k'_{\sigma})=\lim_{m\to \infty}({\bold p}_i,{k^{(m)}}_{\sigma}) = \lim_{m\to \infty}(h({\bold p}_i)-{\epsilon^{(m)}}_i)=h({\bold p}_i)=k({\bold p}_i)$, since ${k^{(m)}}_{\sigma}$ is on the hyperplane $({\bold p}_i,{\bold m})=h({\bold p}_i)-{\epsilon^{(m)}}_i$. Hence it follows that $k=k'$ and therefore $k_\sigma =k'_\sigma $ for every $\sigma\in \Sigma (n)$, which shows that $k_{\sigma}\in {\Box_k}$. \end{pf} Now by \ref{inf} and \ref{basepoint free} the linear system $|mD_k|=|m(K_{{T_N(\Sigma)}}+{X(\Sigma)})|$ has no basepoint for such $m$ that $mD_k$ is a Cartier divisor. \begin{say} Let ${\tilde \Sigma}$ be a non-singular subdivision of $\Sigma$ and ${\Delta}$. Let $$\matrix & \psi \nearrow & {T_N(\D)}\\ T_N({\tilde \Sigma})& & \\ & \varphi \searrow & {T_N(\Sigma)} \endmatrix$$ be the corresponding morphisms and $X({\tilde \Sigma})$ the proper transform of $X$ in $T_N({\tilde \Sigma})$. Since $X({\tilde \Sigma})$ is ${\tilde \Sigma}$-regular by \cite{Batyrev}, it is non-singular and $\varphi |_{X({\tilde \Sigma})}$ is birational. \end{say} \begin{claim} \label{positive} It follows that $$K_{T_N({\tilde \Sigma})}+X({\tilde \Sigma})=\varphi^*(K_{{T_N(\Sigma)}}+X(\Sigma))+ \sum_{{\bold p}\in {\tilde \Sigma}[1]\setminus \Sigma[1]}m_{{\bold p}}D_{{\bold p}},$$ where $m_{{\bold p}}>0 $ for ${\bold p}$ such that $D_{{\bold p}}\cap X({\tilde \Sigma})\neq\phi$. \end{claim} \begin{pf} Denote $$K_{T_N({\tilde \Sigma})}+X({\tilde \Sigma})=\psi^*(K_{{T_N(\D)}}+X)+\sum_{{\bold p}\in{\tilde \Sigma}[1] \setminus{\Delta}[1]}\alpha_{{\bold p}}D_{{\bold p}},$$ then $\alpha_{\bold p}>0$ for ${\bold p}$ such that $D_{{\bold p}}\cap X({\tilde \Sigma})\neq\phi$, since $X$ is non-singular. Putting $\alpha_{\bold p}=0$ for ${\bold p}\in{\Delta}[1]$, one obtains that $K_{T_N({\tilde \Sigma})}+X({\tilde \Sigma})\sim \sum_{{\bold p} \in {\tilde \Sigma}[1]} (-h({\bold p})+\alpha_{{\bold p}})D_{{\bold p}}$, as $K_{{T_N(\D)}}+X\sim D_h$. On the other hand, $$K_{T_N({\tilde \Sigma})}+X({\tilde \Sigma})=\varphi^*(K_{T_N(\Sigma)}+X(\Sigma))+ \sum_{{\bold p}\in {\tilde \Sigma} [1]\setminus \Sigma [1]}m_{\bold p} D_{\bold p}.$$ Putting $m_{\bold p}=0$ for ${\bold p}\in\Sigma[1]$, one obtains that $K_{T_N({\tilde \Sigma})}+X({\tilde \Sigma})\sim \sum_{{\bold p} \in {\tilde \Sigma}[1]} (-k({\bold p})+m_{{\bold p}})D_{{\bold p}}$, as $K_{{T_N(\Sigma)}}+X(\Sigma)\sim D_k$. Therefore $\sum_{{\bold p} \in {\tilde \Sigma}[1]} (-h({\bold p})+\alpha_{{\bold p}})D_{{\bold p}}\sim \sum_{{\bold p} \in {\tilde \Sigma}[1]} (-k({\bold p})+m_{{\bold p}})D_{{\bold p}}$. As $h({\bold p})=k({\bold p})$ and $\alpha_{\bold p}=m_{\bold p}=0$ for ${\bold p}\in \Sigma[1]$, one obtains that $$\sum_{{\bold p}\in{\tilde \Sigma}[1]\setminus\Sigma[1]}((-h({\bold p})+\alpha_{\bold p})- (-k({\bold p})+m_{\bold p}))D_{\bold p} \sim 0.$$ Here $D_{\bold p}$ $({\bold p}\in{\tilde \Sigma}[1]\setminus\Sigma[1])$ are all exceptional for $\varphi$. Then the divisor above is not only linearly equivalent to 0 but also equal to 0. Therefore $(-h({\bold p})+\alpha_{\bold p})- (-k({\bold p})+m_{\bold p})=0$ for every ${\bold p}\in {\tilde \Sigma}[1]\setminus\Sigma[1]$, where $k({\bold p})={\bold p} ({\Box_h})$ by $k_\sigma \in {\Box_k}={\Box_h}$ and \ref{inf}. Now consider the divisor $D_{\bold p}$ such that $D_{\bold p}\cap X({\tilde \Sigma})\neq 0$. For ${\bold p}\in {\tilde \Sigma}[1] \setminus {\Delta}[1]$, $m_{\bold p}={\bold p}({\Box_h})-h({\bold p})+\alpha_{\bold p}\geq \alpha_{\bold p} > 0$. For ${\bold p}\in {\Delta}[1]\setminus\Sigma[1]$, it follows that $m_{\bold p}={\bold p}({\Box_h})-h({\bold p})>0$, because $\{{\bold m}|({\bold p},{\bold m})\geq h({\bold p})\}$ does not contribute to ${\Box_h}$ by the definition of $\Sigma$ (c.f \ref{construction}). This completes the proof. \end{pf} \vskip.5truecm \begin{say} Since ${T_N(\Sigma)}$ has at worst quotient singularities, one can apply Lemma\ref{basic lemma} to our situation and obtain that ${X(\Sigma)}$ is normal and has at worst terminal singularities. And the linear system of $mK_{X(\Sigma)}=m(K_{{T_N(\Sigma)}}+{X(\Sigma)})|_{{X(\Sigma)}}$ $(m\gg 0)$ has no basepoint, because $|m(K_{{T_N(\Sigma)}}+{X(\Sigma)})|$ is basepoint free as is noted after the proof of \ref{k in box}. This completes the proof of that ${X(\Sigma)}$ is a projective minimal model with the abundance. \end{say} \begin{say} Pursuing elementary contractions and flips is like groping for a minimal model in the dark. The reason why the discussion of this section goes well without contractions nor flips is because in toric geometry every exceptional divisor is visible as a vector in the space $N$. Then one can prepare so that every discrepancy of adjoint divisor is positive (cf. \ref{positive}), which makes the singularities terminal. In the discussion, one puffed up the polytope of the adjoint divisor and took its dual fan $\Sigma$. This implies that in ${T_N(\Sigma)}$ the adjoint divisor is the limit of a sequence of ample divisors (cf. \ref{k in box}), which makes the adjoint divisor nef; or equivalently semi-ample. \end{say} \vskip 1truecm \section{\bf Examples} In this section the base field $k$ is always assumed to be of characteristic zero. Let $M$ be ${\Bbb Z}^3$ and $N$ be its dual. \begin{exmp} \label{exmp1} Let ${\bold p}_i$ $(i=1,\ldots , 6)$ and ${\bold q}_j$ $(j=1, \ldots , 8)$ be points in $N$ as follows: ${\bold p}_1=(1,0,0),$ ${\bold p}_2=(-1,0,0),$ ${\bold p}_3=(0,1,0),$ ${\bold p}_4=(0,-1,0),$ ${\bold p}_5=(0,0,1),$ ${\bold p}_6=(0,0,-1),$ ${\bold q}_1=(1,1,1),$ ${\bold q}_2=(-1,-1,-1),$ ${\bold q}_3=(1,1,-1),$ ${\bold q}_4=(-1,-1,1),$ ${\bold q}_5=(1,-1,1),$ ${\bold q}_6=(-1,1,-1),$ ${\bold q}_7=(-1,1,1),$ ${\bold q}_8=(1,-1,-1).$ Let them generate one-dimensional cones ${\Bbb R}_{\geq 0}{\bold p}_i$, ${\Bbb R}_{\geq 0}{\bold q}_j$ and construct a fan ${\Delta}$ with these cones as in Figure 1. Here note that Figure 1 is the picture of the fan which is cut by a hypersphere with the center the origin and unfolded onto the plane. This fan is the dual fan of the polytope of Figure 2 and it is easy to check that it is non-singular. Let $X$ be a general member of a base-point-free linear system $|\sum_{i=1}^6D_{{\bold p}_i} +2\sum_{j=1}^8D_{{\bold q}_j}|$. Let $h$ be the ${\Delta}$-support function such that $K_{{T_N(\D)}}+X\sim D_h$. Then the polytope ${\Box_h}$ is one point $\bigcap_{i=1}^6\{{\bold m}| ({\bold p}_i,{\bold m})\geq 0\}$and the half spaces contributing to this polytope are $\{{\bold m}| ({\bold p}_i,{\bold m})\geq 0\}$, $i=1,\ldots ,6$, because $K_{{T_N(\D)}}+X\sim \sum D_{{\bold q}_j}$. Therefore, for a sufficiently small general $\epsilon$, the polytope $\Box(\epsilon)=\bigcap_i\{{\bold m}\in {M_{\bR}}|({\bold p}_i,{\bold m})\geq -\epsilon_i\}$ is a hexahedron whose picture is as Figure 3. The dual fan $\Sigma$ of $\Box(\epsilon)$ (Figure 4) gives a minimal model ${X(\Sigma)}$ of $X$. Since $K_{{T_N(\Sigma)}}+{X(\Sigma)} \sim 0$, it follows that $\kappa (X)= 0$. \end{exmp} \begin{exmp} Let ${\bold p}_i$ and ${\bold q}_j$ be as in \ref{exmp1} and ${\Delta}$ be the fan with the cones generated by these vectors as Figure 5. This fan is the dual fan of the polytope of Figure 6 and it is easy to check that it is non-singular. Let $X$ be a general member of a base-point-free linear system $|2D_{{\bold p}_1}+2D_{{\bold p}_2}+\sum_{i=3}^6D_{{\bold p}_i}+3\sum_{j=1}^8D_{{\bold q}_j}|$. Let $h$ be the ${\Delta}$-support function such that $K_{{T_N(\D)}}+X\sim D_h$. Then the polytope ${\Box_h}$ is a segment $\bigcap_{i=1}^2\{{\bold m}|({\bold p}_i,{\bold m})\geq -1\}\cap \bigcap_{i=3}^6 \{{\bold m} | ({\bold p}_i,{\bold m})\geq 0\}$ and the half spaces contributing to this polytope are $\{{\bold m}|({\bold p}_i,{\bold m})\geq -1\}$ $(i=1,2)$ and $\{{\bold m} | ({\bold p}_i,{\bold m})\geq 0\}$ $(i=3,\ldots ,6)$, because $K_{{T_N(\D)}}+X\sim D_{{\bold p}_1}+D_{{\bold p}_2}+ 2\sum D_{{\bold q}_j}$. Therefore, for a sufficiently small general $\epsilon$, the polytope $\Box(\epsilon)= (\bigcap_{i=1}^2\{{\bold m}\in {M_{\bR}}|({\bold p}_i,{\bold m})\geq -1-\epsilon_i\}) \cap (\bigcap_{i=3}^6\{{\bold m}\in {M_{\bR}}|({\bold p}_i,{\bold m})\geq -\epsilon_i\})$ is a hexahedron whose picture is as Figure 7. The dual fan $\Sigma$ of $\Box(\epsilon)$ (Figure 4) gives a minimal model ${X(\Sigma)}$ of $X$. Since ${\Box_h}$ is of one dimension, $\dim \Gamma({T_N(\Sigma)} , m(K_{{T_N(\Sigma)}}+{X(\Sigma)} ))$ grows in order 1, and therefore $\dim \Phi _{|m(K_{{T_N(\Sigma)}}+{X(\Sigma)} )|}({T_N(\Sigma)})=1$. This shows that $\dim \Phi_{|mK_{{X(\Sigma)}}|}({X(\Sigma)})\leq 1$. As the dual fan of the polytope of ${X(\Sigma)} \sim 2D_{{\bold p}_1}+2D_{{\bold p}_2}+ \sum_{i=3}^6D_{{\bold p}_i}$ is $\Sigma$, ${X(\Sigma)}$ is ample by \ref{projective}. Hence ${X(\Sigma)}$ intersects all fibers of $\Phi _{|m(K_{{T_N(\Sigma)}}+{X(\Sigma)} )|}$, which shows that $\kappa (X)=1$. \end{exmp} \makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother

9705

alg-geom/9705012

en

https://arxiv.org/abs/alg-geom/9705012

alg-geom/9705012

Mella Massimiliano

Massimiliano Mella

Existence of good divisors on Mukai varieties

LaTex, 10 pages, nofig

A Mukai variety is a Fano n-fold of index n-2. In this paper we study the fundamental divisor of a Mukai variety with at worst log terminal singularities. The main result is a complete classification of log terminal Mukai varieties which have not good divisors, examples of "bad" varieties are given. In such a way we also give a shorter proof of Mukai Conjecture, solved in our previous paper alg-geom/9611024.

alg-geom

\section*{Introduction} A normal projective variety $X$ is called {\sf Fano} if a multiple of the \hbox{anticanonical} Weil divisor, $-K_X$, is an ample Cartier divisor. The importance of Fano \hbox{varieties} is twofold, from one side they give, has predicted by Fano \cite{Fa}, \hbox{examples} of non rational varieties having plurigenera and irregularity all zero (cfr \cite{Is}); on the other hand they should be the building block of uniruled variety. Indeed recently, Minimal Model Theory predicted that every uniruled variety is birational to a fiber space whose general fiber is a Fano variety with terminal singularities (cfr \cite{KMM}). The index of a Fano variety $X$ is the number $$i(X):=sup\{t\in {\bf Q}: -K_X\equiv tH,\mbox{\rm for some ample Cartier divisor $H \}$}.$$ It is known that $0<i(X)\leq dimX+1$ and if $i(X)\geq dim X$ then $X$ is either an hyperquadric or a projective space by the Kobayashi--Ochiai criterion. Smooth Fano n-folds of index $i(X)=n-1$, {\sf del Pezzo n-folds}, have been classified by Fujita \cite{Fu} and terminal Fano n-folds of index $i(X)>n-2$ have been independently classified by Campana--Flenner \cite{CF} and Sano \cite{Sa}. If $X$ has log terminal singularities, then $Pic(X)$ is torsion free and therefore, the $H$ satisfying $-K_X\equiv i(X)H$ is uniquely determined and is called the {\sf fundamental divisor} of $X$. Mukai announced, \cite{Mu}, the classification of smooth Fano n-folds $X$ of index $i(X)=n-2$, under the assumption that the linear system $|H|$ contains a smooth divisor. It is usually said that a Fano variety $X$ has {\sf good divisors} if the generic element of the fundamental divisor of $X$ has at worst the same singularities as $X$. Our main Theorem is the following \vspace{.5cm}\noindent {\bf Theorem 1} {\it Let $X$ be a Mukai variety with at worst log terminal singularities. Then $X$ has good divisors except in the following cases: \begin{itemize} \item[-] $X$ is a singular terminal Gorenstein 3-fold which is a complete intersection of a quadric and a sestic in ${\bf P}(1,1,1,1,2,3)$ \item[-] $X$ is a terminal not Gorenstein 3-fold and the canonical cover of $X$ is a complete intersection of a quadric and a quartic in ${\bf P}(1,1,1,1,1,2)$. \end{itemize} In both exceptional cases the generic element of the fundamental divisor has canonical singularities.} \vspace{.5cm} In particular this proves Mukai hypothesis and therefore the result of Mukai \cite{Mu} provide a complete classification of smooth Fano n-folds of index $i(X)=n-2$, {\sf Mukai manifolds}, see also \cite{CLM} for a different approach. The ancestors of the theorem, and indeed the lighthouses that directed its proof, are Shokurov's proof for smooth Fano 3-folds, \cite{Sh} and Reid's extension to canonical Gorenstein 3-folds using the Kawamata's base point free technique \cite{Re}. This technique was then applied by Wilson in the case of smooth Fano \hbox{4-folds} of index 2, \cite{Wi}, afterwards Alexeev, \cite{Al} did it for log terminal Fano \hbox{n-folds} of index $i(X)>n-2$ and recently Prokhorov used it to prove Mukai Conjecture in dimension 4 and 5, \cite{Pr1} \cite{Pr2} \cite{Pr3}. Theorem 1 was proved for smooth Mukai variety in a first version of this paper,\cite{Me}, using Helmke's inductive procedure,\cite{He}. Our main tools will be Kawamata's base point free technique, Kawamata's notion of centers of log canonical singularities, \cite{Ka1} and his subadjunction formula for codimension $\leq 2$ minimal centers \cite{Ka2}. These tools allows to replace difficult non vanishing arguments by a simple Riemann--Roch calculation. While working on this subject I had several discussions with M. Andreatta, I would like to express him my deep gratitude. I would also like to thank A. Corti and Y. Prokhorov for valuable comments and Y. Kawamata for signaling a gap in the first version of this paper. This research was partially supported by the Istituto Nazionale di Alta Matematica Francesco Severi (senior grant 96/97) and the Consiglio Nazionale delle Ricerche. \section{Preliminary results} We use the standard notation from algebraic geometry. In particular it is compatible with that of \cite{KMM} to which we refer constantly, everything is defined over {\bf C}. In the following $\equiv$ (respectively $\sim$) will indicate numerical (respectively linear) equivalence of divisors. Let $\mu:Y\rightarrow X$ a birational morphism of normal varieties. If $D$ is a {\bf Q}-divisor ({\bf Q}-Cartier) then is well defined the strict transform $\mu_*^{-1}D$ (the pull back $\mu^*D$). For a pair $(X,D)$ of a variety $X$ and a {\bf Q}-divisor $D$, a log resolution is a proper birational morphism $\mu:Y\rightarrow X$ from a smooth $Y$ such that the union of the support of $\mu_*^{-1}D$ and of the exceptional locus is a normal crossing divisor. \begin{Definition} Let $X$ be a normal variety and $D=\sum_id_iD_i$ an effective {\bf Q}-divisor such that $K_X+D$ is {\bf Q}-Cartier. If $\mu:Y\rightarrow X$ is a log resolution of the pair $(X,D)$, then we can write $$K_Y+\mu_*^{-1}D=\mu^*(K_X+D)+F$$ with $F=\sum_je_jE_j$ for the exceptional divisors $E_j$. We call $e_j\in {\bf Q}$ the discrepancy coefficient for $E_j$, and regard $-d_i$ as the discrepancy coefficient for $D_i$. The pair $(X,D)$ is said to have {\sf log canonical} (LC) (respectively {\sf Kawamata log terminal} (KLT), {\sf log terminal} (LT)) singularities if $d_i\leq 1$ (resp. $d_i< 1$, $d_i=0$) and $e_j\geq -1$ (resp. $e_j>-1$, $e_j>-1$) for any $i,j$ of a log resolution $\mu:Y\rightarrow X$. The {\sf log canonical threshold} of a pair $(X,D)$ is $lct(X,D):= sup\{t\in{\bf Q}$: $(X,tD)$ is LC$\}$. \label{lc} \end{Definition} \begin{Definition} A {\sf log-Fano variety} is a pair $(X,\Delta)$ with KLT singularities and such that for some positive integer $m$, $-m(K_X+\Delta)$ is an ample Cartier divisor. The index of a log-Fano variety $i(X,\Delta):=sup \{t\in {\bf Q}: -(K_X+\Delta)\equiv tH$ for some ample Cartier divisor $H \}$ and the $H$ satisfying $-(K_X+\Delta)\equiv i(X,\Delta)H$ is called fundamental divisor. In case $\Delta=0$ we have log terminal Fano variety. \end{Definition} \begin{Proposition}[\cite{Al}] Let $(X,\Delta)$ be a log-Fano n-fold of index r, $H$ the fundamental divisor in $X$. If $r\geq n-2$ then $dim |H|\geq n-1$. \label{al} \end{Proposition} Let us recall the notion and properties of minimal centers of log canonical singularities as introduced in \cite{Ka1} \begin{Definition}[\cite{Ka1}] Let $X$ be a normal variety and $D=\sum d_iD_i$ an effective {\bf Q}-divisor such that $K_X+D$ is {\bf Q}-Cartier. A subvariety $W$ of $X$ is said to be a {\sf center of log canonical singularities} for the pair $(X,D)$, if there is a birational morphism from a normal variety $\mu:Y\rightarrow X$ and a prime divisor $E$ on $Y$ with the discrepancy coefficient $e\leq -1$ such that $\mu(E)=W$. The set of all centers of log canonical singularities is denoted by $CLC(X,D)$. \end{Definition} \begin{Theorem}[\cite{Ka1},\cite{Ka2}] Let $X$ be a normal variety and $D$ an effective {\bf Q}-Cartier divisor such that $K_X+D$ is {\bf Q}-Cartier. Assume that $X$ is LT and $(X,D)$ is LC. \begin{itemize} \item[i)] If $W_1,W_2\in CLC(X,D)$ and $W$ is an irreducible component of $W_1\cap W_2$, then $W\in CLC(X,D)$. In particular, if $(X,D)$ is not KLT then there exist minimal elements in $CLC(X,D)$. \item[ii)] If $W\in CLC(X,D)$ is a minimal center then $W$ is normal \item[iv)] (subadjunction formula) If $W$ is a minimal center for $CLC(X,D)$ and $cod W\leq 2$ then there exists an effective {\bf Q}-divisors $\Delta$ on $W$ such that $(K_X+D)_{|W}\equiv K_W+\Delta$ and $(W,\Delta)$ is KLT. \end{itemize} \label{clc} \end{Theorem} \section{Proof of Theorem 1} Let me sketch the idea of the proof, to make it more transparent. By Bertini Theorem if $X$ has not good divisors then the generic element of $|H|$ has a center of "bad" singularities contained in $Bsl|H|$. We will derive a contradiction producing a section of $|H|$ not vanishing identically on $Z$. The following lemma will be frequently used to this purpose. \begin{Lemma} Let $X$ be a log terminal Fano n-fold, with $n\geq 3$ and $H$ an ample Cartier divisor with $-K_X\equiv (n-2)H$ and $G$ a {\bf Q}-Cartier divisor. Assume that $(X,G)$ is LC, $Z\in CLC(X,G)$ is a minimal center and $G\equiv \gamma H$, with $\gamma< cod Z-1$. Then there is a section of $H$ not vanishing identically on $Z$. \label{KV} \end{Lemma} \par \noindent{\bf Proof. }\nopagebreak First let us perturb $G$ to construct a {\bf Q}-divisor $G_1\equiv \gamma_1 H$ such that $\gamma_1< codZ-1$ and $Z$ is the only element in $CLC(X,G_1)$. Let $M\in |mH|$, for $m\gg 0$, a general member among Cartier divisors containing $Z$. Let \hbox{$G^{\prime}= G+\epsilon M$,} and $\gamma^{\prime} =lct(X,G^{\prime})$. Define $G_1:=1/\gamma^{\prime} G^{\prime}$ then $G_1\equiv \gamma_1 H$ and for $\epsilon\ll 1/m$ we have $\gamma_1<cod Z$. Furthermore $(X,G_1)$ is LC and $Z$ is the only element element of $CLC(X,G_1)$. Since we have the strict inequality $\gamma_1<cod Z-1$, we may furthermore assume, by Kodaira Lemma, that there exists a log resolution $\mu:Y\rightarrow X$ of $(X,G_1)$ such that $$K_Y+E-A+F=\mu^*(K_X+G_1)+P,$$ where $\mu(E)=Z$, $A$ is an integral \hbox{$\mu$-exceptional} divisor, $\lfloor F\rfloor=0$ and $P$ is an ample {\bf Q}-divisor. Let $$ N(t):= -K_Y-E-F+A+\mu^*(tH),$$ then $N(t)\equiv \mu^*(t+(n-2)-\gamma_1)H+P$ and $N(t)$ is ample for $t+(n-2)-\gamma_1> 0$, hence by hypothesis this is true whenever $t\geq -n+1+cod Z$. Thus K--V vanishing yields \begin{equation} H^i(Y,\mu^*(tH)-E+A)=0\hspace{.7cm} H^i(E,(\mu^*(tH)+A)_{|E_0})=0 \label{van} \end{equation} for $i>0$ and $t\geq -n+1+cod Z$. In particular there is the following surjection $$ H^0(Y,\mu^*H+A)\rightarrow H^0(E,\mu^*H+A)\rightarrow 0. $$ Since $A$ is effective and $\mu$-exceptional, then $H^0(Y,\mu^*H+A)\simeq H^0(X,H)$, thus any section in $H^0(Y,\mu^*H+A)$ not vanishing on $E$, pushes forward to give a section of $H$ not vanishing on $Z$. To conclude the proof it is, therefore, enough to prove that $h^0(E,\mu^*H+A)>0$. Let $p(t)=\chi(E,\mu^*tH+A)$, then by equation (\ref{van}), $p(0)\geq 0$ and $p(t)=0$ for $0>t\geq -n+1+cod Z=-dim Z+1$. Since $deg p(t)= dim Z$ and $p(t)>0$ for $t\gg 0$ then $h^0(E,\mu^*H+A)=p(1)>0$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par We will first prove that log terminal Mukai varieties always have a log terminal fundamental divisor. \begin{Theorem} Let $X$ be a Mukai variety with log terminal singularities and $K_X\equiv -(n-2)H$. Then the general element in $|H|$ has log terminal singularities. \label{muklt} \end{Theorem} \par \noindent{\bf Proof. }\nopagebreak By Proposition \ref{al} $dim |H|\geq 2$. Let $S\in |H|$ a generic section and assume that $S$ has worse than LT singularities. Let $\gamma=lct(X,S)$, then by our assumption $\gamma\leq 1$, see for instance \cite[1.4]{Al}. Let $Z\in CLC(X,\gamma S)$ a minimal center. By Bertini Theorem $Z\subset Bsl|H|$ therefore by Lemma \ref{KV} $cod Z\leq 2$. \claim{ $\gamma< cod Z$.} \par \noindent{\bf Proof. }\nopagebreak(of the Claim) $\gamma\leq 1\leq cod Z$ and the equality could hold only if $Z$ were a fixed component of $|H|$ of multiplicity 1. Let $S=Z+B$, then by connectedness $W:=Z\cap B\neq \emptyset$ and $W$ is properly contained in $Z$. $S$ is a Cartier divisor singular along the codimension 2 subscheme $W$ therefore $W\in CLC(X,S)$ and $Z$ cannot be minimal in $CLC(X,S)$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par Let us perturb $S$, as in Lemma \ref{KV}, to construct a {\bf Q}-divisor $S_1\equiv \gamma_1 H$ such that $\gamma_1< codZ$ and $Z$ is the only element in $CLC(X,\gamma S)$. Let $\nu:Y\rightarrow X$ a log resolution of $(X,S_1)$, with $$K_Y-A+E+F=\nu^*(K_X+S_1),$$ where $A$ is $\nu$-exceptional, $\nu(E)=Z$ and $\lfloor F\rfloor=0$. Let $$N:=\nu^*H+A-F-E-K_Y\equiv \nu^*(n-2)H,$$ then $N$ is nef and big and by K--V vanishing we have, \begin{equation} H^1(Y,\nu^*H-E+A)=0 \label{van1} \end{equation} \claim{ $h^0(Z,H_{|Z})>0$} Let us first prove that the claim gives us a contradiction. Since $A$ does not contain $E$ and is effective then $H^0(Z,H_{|Z})\hookrightarrow H^0(E,(\mu^*H+A)_{|E})$ therefore by the vanishing (\ref{van1}) and the claim there exists a section in $H^0(Y,\mu^*H+A)$ not vanishing on $E$. Thus there exists a section of $H$ not vanishing on $Z$, giving a contradiction and proving the theorem. \par \noindent{\bf Proof. }\nopagebreak(of the claim) $cod Z\leq 2$ thus we can apply subadjunction formula of Theorem \ref{clc}. There exists an effective {\bf Q}-divisor $\Delta$ on $Z$ such that $$-(n-2-\gamma_1)H_{|Z}\equiv(K_X+S_1)_{|Z}\equiv K_Z+\Delta$$ and $(Z,\Delta)$ is KLT. That is to say $(Z,\Delta)$ is a log Fano variety with $i(Z)\geq dimZ-2$. and the claim follows directly from Proposition \ref{al}. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par The next step is to prove that canonical Mukai varieties have canonical fundamental divisor. \begin{Theorem} Let $X$ be a Mukai variety with canonical singularities. Then the general element in $|H|$ has canonical singularities. \label{mukcan} \end{Theorem} \par \noindent{\bf Proof. }\nopagebreak By Theorem \ref{muklt} the general element $S\in |H|$ has LT singularities. Let $\mu:Y\rightarrow X$ a log resolution of $(X,S)$, with $\mu^*S=\overline{S}+ \sum r_i E_i$, where $\overline{S}$ is base point free, and $K_Y=\mu^*K_X+\sum a_i E_i$. Let us assume that $S$ has not canonical singularities, then, maybe after reordering the indexes, we have $a_0<r_0$. Since $S$ is generic then $\mu(E_i)\subset Bsl|H|$, for all $i$ with $r_i>0$. Let $D=S+S^1$, with $S^1\in |H|$ a generic section. First observe that $\mu$ is a log resolution of $(X,D)$. Then $ (X,D)$ is not LC, infact $a_0+1<r_0+r^1_0$, where $r^1_0\geq 1$ is the multiplicity of $S_1$ at the center of the valuation associated to $E_0$. Let $\gamma=lct(X,D)<1$ and $W$ a minimal center of $CLC(X,\gamma D)$. $X$ has canonical singularities therefore whenever $cod \mu(E_j)\leq 2$ then $a_j\in {\bf N}$. Thus, since $(X,D)$ is not LC and $S$ is LT, we have $cod W\geq 3$. We can therefore derive a contradiction by Lemma \ref{KV}. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \Remark{ If $X$ is a terminal Mukai variety of dimension$\geq 4$ then by the above proposition we immediately get that the generic element $S$ of the fundamental divisor $|H|$ is terminal. In fact outside $Bsl|H|$ $S$ is terminal by Bertini Theorem and along the base locus the discrepancy must be positive, since the generic section of $H_{|S}$ is canonical.} What remains to be done is to study terminal 3-folds with $-K_X\equiv H$, let us start with some examples \vspace{.2cm}\par \noindent{\bf Example } Let us consider $X_{2,6}\subset {\bf P}(1,1,1,1,2,3)$ given by the following equations \begin{eqnarray*} F_2(x_0,\ldots,x_3)=0\\ x_5^2+F_3(x_0,\ldots,x_4)x_5+F_6(x_0,\ldots,x_4)=0 \end{eqnarray*} Assume that $F_6$ contains the monomial $x_4^3$ then $$X\cap \{x_0=x_1=x_2=x_3=0\}=\{[(0:0:0:0:-1:1)]\}=\{p\},$$ thus $X$ is on the smooth locus of ${\bf P}(1,1,1,1,2,3)$ and $S$ and $Q$ are Cartier along $X$. In particular adjunction formula holds $X$ is Gorenstein and $Bsl|-K_X|=\{p\}$. Since $Q$ is singular at $p$ then $X$ is singular at $p$ and therefore all elements in $|-K_X|$ are singular. \vspace{.2cm}\par \noindent{\bf Example } Let us now consider $Y_{2,4}\subset {\bf P}(1,1,1,1,1,2)$ given by the following equations \begin{eqnarray*} x_5+F_2(x_0,\ldots,x_4)=0\\ F_4(x_0,\ldots,x_5)=0 \end{eqnarray*} Since the first equation is linear then $X$ is isomorphic to a quartic in ${\bf P}^4$. Let us choose this quartic with two simple nodes at $(0,0,0,\pm1,1)$ and consider the involution $\sigma$ on ${\bf P}(1,1,1,1,1,2)$ given by $$(x_0:x_1:x_2:x_3:x_4:x_5)\mapsto(x_0:x_1:x_2:-x_3:-x_4:-x_5).$$ Let $\pi:Y\rightarrow X=Y/ \sigma$ the quotient, then $X$ is a 3-fold with a $cA_1$ point, \hbox{$p=\pi([(0:0:0:\pm1:1:0)])$,} and 8 points of singular index 2, the fixed points of the involution. Furthermore $-K_X\equiv H$ and $Bsl|H|=\{p\}$, therefore all elements in $H$ are singular. To prove the theorem we have to show that the above are the unique possibilities for a terminal Mukai variety which has not good divisors. \begin{Theorem} Let $X$ be a terminal Mukai 3-fold, assume that all the divisor in the linear system $|H|$ are singular, then $X$ is one of the following: \begin{itemize} \item[-]\cite{Mo} if $X$ is Gorenstein then $X$ is a complete intersection of a quadric and a sestic in ${\bf P}(1,1,1,1,2,3)$ \item[-] if $X$ is not Gorenstein then the canonical cover of $X$ is a complete intersection of a quadric and a quartic in ${\bf P}(1,1,1,1,1,2)$. \end{itemize} \end{Theorem} \par \noindent{\bf Proof. }\nopagebreak Let $S\in |H|$ a generic element, then $S$ has at worse canonical singularities, by Theorem \ref{mukcan}. By K--V vanishing we get that $S$ is either a K3 surface or an Enriques surface. Assume now that $Bsl|H|\neq\emptyset$ and $S$ is singular, then by \cite{SD} and \cite{Cos} we know that $S$ has an $A_1$ singularity and $Bsl|H|=Bsl|H_{|S}|=\{x\}$, see also \cite{Shi} and \cite{Prok}. Let $f:Y\rightarrow X$ the blow up of the point $x$, with exceptional divisor $E$. Then $f^*S=\tilde{S}+E$ and $\tilde{S}\cdot E^2=-2$ furthermore $$0=f^*S\cdot E^2=\tilde{S}\cdot E^2+E^3,$$ thus $E^3=2$. By \cite{SD} and \cite{Cos}, $\tilde{S}_{|\tilde{S}}$ is an elliptic pencil, therefore $\tilde{S}^3=0$. This gives $$0=\tilde{S}^3=(f^*S-E)^3=\varphi^*H^3-2,$$ thus $H^3=2$, in case $X$ is not Gorenstein see also \cite{Prok}. Assume that $X$ is not Gorenstein then the generic element $E\in |H|$ is a canonical Enriques surface with an $A_1$ singularity. Let $\pi:Y\rightarrow X$ the cyclic cover associated to ${\cal O}_X(K_X+H)$, \cite{YPG}. Let $S=\pi^{-1}E$ the pull back of a generic section $E\in |H|$ and $C=S_{|S}$. By connectedness $S$ is a canonical K3 surface and $S^3=4$. By Riemann--Roch theorem $h^0(C,S_{|C})=3$ and $h^0(C,nS_{|C})=2(2n-1)$, for $n>1$. Let $\{x_2,x_3,x_4\}$ be a basis of $H^0(C,S_{|C})$. Let $\psi:S\stackrel{\scriptstyle - - >}{ } S^{\prime}$ the map defined by the sections of $H^0(S,C)$. If $\psi$ is not birational or is not a morphism, then, by \cite[Th 5.2, Sect. 2.7]{SD} (see also \cite[Cor 2.2]{Shi}), $H^0(C,S_{|C})^{\otimes 2}\neq H^0(C,2S_{|C})$, in other words there is a section $x_5\in H^0(C,2S_{|C})$ which is not in $H^0(C,S_{|C})^{\otimes 2}$, and there is a quadratic relation of the kind $F_2(x_2,x_3,x_4,x_5)=0$. Going further we get that nothing new happens for $H^0(C,3S_{|C})$, while we get a relation in $H^0(C,4S_{|C})$, of type \hbox{$F_4(x_2,x_3,x_4,x_5)=0$.} This is enough to describe $Y$ as a complete intersection of a quadric and a quartic in ${\bf P}(1,1,1,1,1,2)$. To conclude observe that we can explicitly write down an involution on $Y$ with only a finite number of fixed points as $$(x_0:x_1:x_2:x_3:x_4:x_5)\mapsto (x_0:x_1:x_2:-x_3:-x_4:-x_5).$$ If the map $\psi$ is a birational morphism then, by \cite[Th 6.1]{SD}, $Y$ is a quartic in ${\bf P}^4$, embedded by $H^0(Y,\pi^*H)$. Note that the involution $\pi$ must be the restriction of a projective transformation and cannot, therefore, fix only finitely many points. If $X$ is Gorenstein then the generic element $S\in |H|$ is a canonical K3 surface. Let $C=S_{|S}$ then with a similar calculation we get that \hbox{$h^0(C,H_{|C})=2$} and $h^0(C,nH_{|C})=(2n-1)$, for $n>1$. This time $H_{|S}$ is not base point free by hypothesis, thus there is a new section $x_4$ in $H^0(C,2H_{|C})$ which gives rise to a quadratic relation, and a new section $x_5$ in $H^0(C,3H_{|C})$, which gives rise to a relation in $H^0(C,6H_{|C})$. Therefore we have the description given in the proposition. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par What remains to be done is to prove Mukai hypothesis. \begin{Theorem} Let $X$ be a smooth Mukai variety. Then $X$ has good divisors. \end{Theorem} \par \noindent{\bf Proof. }\nopagebreak If $X$ is smooth and the generic element in $|H|$ is not smooth then by the surjection $$H^0(X,H)\rightarrow H^0(H,H_{|H})\rightarrow 0,$$ and previous Theorems we know that $Bsl|H|=\{x\}$. Let $H_i\in |H|$, for $i=1,\ldots, n-1$ generic elements and $D=H_1+\cdots+H_{(n-1)}$, then the minimal center of $CLC(X,D)$ is $x$ and $(X,D)$ is not LC at $x$, since $2(n-1)>n$. We can therefore derive a contradiction by Lemma \ref{KV}. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \small

9705

alg-geom/9705021

en

https://arxiv.org/abs/alg-geom/9705021

alg-geom/9705021

Stavros Garoufalidis

Stavros Garoufalidis, James Pommersheim

Values of zeta functions at negative integers, Dedekind sums and toric geometry

LaTeX2e, 24 pages with 3 figures

This is an expanded version. We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of the above mentioned relations and explicit computations of the various zeta values and Dedekind sums involved.

alg-geom

\section{Introduction} \lbl{sec.intro} In the present paper, we study relations among special values of zeta functions of real quadratic fields, properties of generalized Dedekind sums and Todd classes of toric varieties. The main theme of the paper is the use of toric geometry to explain in a conceptual way properties of the values of zeta functions and Dedekind sums, as well to provide explicit computations. Both toric varieties and zeta functions associate numerical invariants to cones in lattices; however, with different motivations and applications. Though we will focus on the case of two-dimensional cones in the present paper, we introduce notation and definitions that are valid for cones of arbitrary dimension. The reasons for this added generality is clarity, as well as preparation for the results of a subsequent publication. \subsection{Zeta functions} \lbl{sub.history} We begin by reviewing the first source of numerical invariants of cones: the study of zeta functions. Given a number field $K$, its zeta function is defined (for $Re(s)$ sufficiently large) by: \begin{equation*} \zeta(K,s)=\sum_{\alpha} \frac{1}{Q(\alpha)^s} \end{equation*} where the summation is over all nonzero ideals, and $Q$ is the norm. The above function admits a meromorphic continuation in $\mathbb C$, with a simple pole at $s=1$, and regular everywhere else. Lichtenbaum \cite{Li} conjectured a specific behavior of the zeta function $\zeta(K,s)$ at nonpositive integers related to the global arithmetic of the number field. In the special case of a totally real number field $K$, Lichtenbaum conjectured that the values of the zeta function at negative integers are rational numbers which involve the rank of the algebraic (or \'etale) $K$-theory of $K$. It turns out (see \cite{Sh1} and \cite[Section 2]{Za1}) that for a totally real field $K$ the zeta function can be decomposed as a sum $\zeta(K,s)= \sum_I \zeta(I,s)$ where the sum is over the (finite set of) narrow ideal classes. For each narrow ideal class $I$, there is a finite set $\{ \sigma^{\circ}_{i,I} \}_i$ of open cones in a rank $[K:\mathbb Q]$ lattice $M_I$ in $K$ such that $\zeta(I^{-1},s)=\sum_i \zeta_{Q,\sigma^{\circ}_{i,I}}(s)$ where $Q$ is a multiple of the norm on $K$ and where for an open cone $\tau^{\circ}$ we have $$ \zeta_{Q,\tau^{\circ}}(s)=\sum_{a \in \tau^\circ \cap M_I}\frac{1}{Q(a)^s}. $$ The problem of calculating the zeta values $\zeta_{Q,\tau^{\circ}}(-n)$ for $n \geq 0$ for all triples $(M,Q,\tau^{\circ})$ that arise from totally real fields has attracted a lot of attention by several authors. Klingen \cite{Kl} and Siegel \cite{Si1,Si2} (using analytic methods) proved that the values of the zeta functions of totally real fields at nonpositive integers are rational numbers and provided an algorithm for calculating them. Meanwhile, Shintani \cite{Sh1} (using algebraic and combinatorial methods) gave an independent calculation of the zeta values $\zeta_{Q,\tau^{\circ}}(-n)$ for an arbitrary $Q$ which is a product of linear forms. Meyer \cite{Me} and Zagier \cite{Za3} gave another calculation of the zeta values $\zeta_{Q,\tau^{\circ}}(-n)$ for rank two lattices. Related results have also been obtained by P. Cassou-Nogu\`{e}s, \cite{C-N1, C-N2, C-N3}. More recently, Hayes \cite{Hs}, Sczech \cite{Sc1, Sc2} and Stevens \cite{St} have constructed $\text{PGL}_m(\mathbb Q)$ cocycles which, among other things, provide a calculation of the zeta values (at nonpositive integers) of totally real number fields in terms of generalized Dedekind sums. We now specialize to the case of real quadratic fields. Given a narrow ideal class $I$, there is a cone $\sigma_I$ (i.e., the convex hull of two rays from the origin) in a rank two lattice $M \subseteq K$, such that: $\zeta(I^{-1},s)= \zeta_{Q,\sigma_I}(s).$ Here $Q$ is a multiple of the norm form and $\zeta_{Q,\sigma}$ is given by the following definition: \begin{definition} \lbl{def.zeta} Let $M$ be an two-dimensional lattice, and let $V$ denote the real vector space $ M_{\mathbb R}=M\otimes \mathbb R$. Let $Q: M_{\mathbb R} \to \mathbb R$ be a {\em nonzero quadratic homogenous function} i.e., a function satisfying $Q(a v)=a^2 Q(v)$ for $a \in \mathbb R , v \in M_{\mathbb R}$ and such that $Q$ is not identically $0$. We set \begin{equation} \zeta_{Q,\tau}(s)= \sum_{ a \in \tau \cap M} \frac{wt(\tau, a)}{Q(a)^s} \end{equation} where $\tau$ is a rational two-dimensional cone in $M$ and $wt(\tau, \cdot ): M \to \mathbb Q$ is the weight function defined by: \begin{equation} \lbl{eq.we} wt(\tau, a)= \begin{cases} 1 & \text{if $a$ lies in the interior of $\tau$} \\ 1/2 & \text{if $a$ lies in the boundary of $\tau$, and $a \neq 0$} \\ 0 & \text{otherwise.} \end{cases} \end{equation} \end{definition} \noindent We will assume throughout that $Q$ is positive on all of $\tau$; that is, for all $a\in\tau$, $a\neq 0$, we have $Q(a)>0$. The above zeta function, defined for $Re(s)$ sufficiently large, \cite{Za2} can be analytically continued to a meromorphic function on $\mathbb C$, regular everywhere except at $1$. In \cite{Za2}, Zagier showed that after possibly multiplying $M$ by a totally positive number and $Q$ by a nonzero rational number (which only multiplies values of the zeta function by a nonzero constant), every triple $(M,Q,\tau)$ which arise from real quadratic field can be constructed by means of a finite sequence $b=(b_0, \dots,b_{r-1})$ of integers greater than $1$ and not all equal to $2$, which will be denoted by $(M_b,Q_b,\tau_b)$. In addition, there are canonical vectors $A_0, \dots, A_r$ in $M_b$ (that depend on $b$) that subdivide the cone $\tau_b= \langle A_0, A_r \rangle$ (i.e., the cone whose extreme rays are $A_0$ and $A_r$) in $M_b$ in $r$ nonsingular cones $\langle A_i, A_{i+1} \rangle$. For a detailed discussion, see Section \ref{sub.real}. We will now give an explicit formula for the zeta values $ \zeta_{Q_{b}, \tau_b}(-n)$ (for $ n \geq 0$), which in turn enables one to calculate the values of the zeta function of a real quadratic field at nonpositive integers. While both the motivation and the proof involve concepts from the theory of toric geometry, the formula can be stated and understood without a knowledge of toric varieties. We will state the formula here, and in the next subsection, we will discuss concepts from toric geometry which are necessary for the proof and which lead to a conceptual understanding of the present formula. Let $\lambda_m$ be defined by the power series: \begin{equation} \lbl{eq.lm} \frac{h}{1-e^{-h}} = \sum_{m=0}^\infty \lambda_m h^m \end{equation} thus we have: $\lambda_m = (-1)^m B_m/m!$ where $B_m$ is the $m^{th}$ Bernoulli number. (See also Definition \ref{def.dedsum} below.) Note that if $m>1$ is odd, then $\lambda_m=0$. For $d\geq 2$ even, define homogeneous polynomials $P_d(X,Y), R_d(X,Y)$ of degree $d-2$ by: \begin{eqnarray*} P_d(X,Y)& = & \epsilon_{d}\sum_{i+j=d,i,j>0} \lambda_i \lambda_j X^{i-1} Y^{j-1},\\ R_d(X,Y)& = & \frac{X^{d-1}+Y^{d-1}}{X+Y}=X^{d-2}-X^{d-3}Y+\dots+Y^{d-2}, \end{eqnarray*} \noindent where $\epsilon_{2}=-1$ and $\epsilon_{d}=1$ for all even $d>2$. We then have: \begin{theorem} \lbl{cor.zv} The values $\zeta_{Q_{b}, \tau_b}(-n)$ for $n \geq 0$ are polynomials in $b_i$ with rational coefficients (symmetric under cyclic permutation of the $b_i$) given explicitly as follows: \begin{eqnarray} \lbl{eq.zva} \zeta_{Q_{b}, \tau_b}(-n) & = & (-1)^n n! \left\{ \sum_{i=1}^r P_{2n+2}\left(\pd x , \pd y \right) \diamond e^{- Q_b( x A_{i-1} + y A_i)} \right. \\ & & \left. + \lambda_{2n+2} \sum_{i=1}^r b_i R_{2n+2}\left(\pd x , \pd y \right) \diamond e^{- Q_b( x A_{i-1} + y A_{i+1})} \right\} \end{eqnarray} The diamond symbol indicates the evaluation of a partial derivative at the origin. \end{theorem} \noindent In particular, we obtain the formula of Zagier \cite[Equation 3.3]{Za4}: \begin{eqnarray} \lbl{eq.z0} \zeta_{Q_{b}, \tau_b}(0) & = & \frac{1}{12} \sum_{i=0}^{r-1} (b_i-3). \end{eqnarray} \subsection{Toric geometry} \lbl{sub.toric} The second source of numerical invariants of cones comes from the theory of {\em toric varieties}. For a general reference on toric varieties, see \cite{Dan} or \cite{Fu}. Founded in the 1970s, the subject of toric varieties provides a strong link between algebraic geometry and the theory of convex bodies in a lattice. To each lattice polytope (the convex hull of a finite set of lattice points) is associated an algebraic variety with a natural torus action. This correspondence enables one to translate important properties and theorems about lattice polytopes into the language of algebraic geometry, and vice-versa. One important example of this is the very classical problem of counting the number of lattice points in a polytope. The early pioneers in the subject of toric varieties found that this problem, translated into algebraic geometry, becomes the problem of finding the Todd class of a toric variety. Much progress has been made over the past ten years on the Todd class problem, and this has led to a greater understanding of the lattice point counting question. One approach to computing the Todd class of a toric variety is the fundamental work of R. Morelli \cite{Mo}. He settled a question of Danilov by proving a local formula expressing the Todd class of a toric variety as a cycle. Another approach, introduced in by the first author in \cite{P1, P2}, is to express the Todd class as a polynomial in the torus-invariant cycles. Dedekind sums appear as coefficients in these polynomials, and this leads to lattice point formulas in terms of Dedekind sums, as well as new reciprocity laws for Dedekind sums \cite{P1}. Cappell and Shaneson \cite{CS} subsequently announced an extension of the program of \cite{P1} in which they proposed formulas for the Todd class of a toric variety in all dimensions. In \cite{P3} it was shown that the polynomials of \cite{P1, P2} can be expressed nicely as the truncation of a certain power series whose coefficients were shown to be polynomial-time computable using an idea of Barvinok \cite{Ba}. A beautiful power series expression for the equivariant Todd class of a toric variety was given by Brion and Vergne in \cite{BV2}. Guillemin \cite{Gu} also proved similar Todd class formulas from a symplectic geometry point of view. Furthermore, in \cite{BV}, Brion and Vergne use the Todd power series of \cite{BV2} to give a formula for summing any polynomial function over a polytope. The power series studied in \cite{P3} and \cite{BV,BV2} are, in fact, identical (see Section \ref{sec.thm.1}) and play a central role in the present paper. A detailed discussion of the properties of these power series, which we call the {\em Todd power series of a cone}, is contained in Section \ref{sec.thm.1}. In this section, we state two theorems about the Todd power series of a two-dimensional cone that we will need in our study of the zeta function. Given independent rays $\rho_1, \dots , \rho_n$ from the origin in an $n$-dimensional lattice $N$, the convex hull of the rays in the vector space $N_{\mathbb R} =N\otimes\mathbb R$ forms an $n$-dimensional cone $\sigma= \langle \rho_1, \dots , \rho_n \rangle $. Cones of this type (that is, ones generated by linearly independent rays) are called {\em simplicial}. Let $\mathcal C^n(N)$ denote the set of $n$-dimensional simplicial cones of $N$ with ordered rays. There is then a canonical function \begin{equation*} \mathfrak t : \mathcal C^n (N) \to \mathbb Q\[x_1, \dots, x_n \], \end{equation*} invariant under lattice automorphisms, which associates to each cone $\sigma$ a power series $\mathfrak t_{\sigma}$ with rational coefficients, called the Todd power series of $\sigma$. Several ways of characterizing this function are given in Section \ref{sec.thm.1}. These include an $N$-additivity property (See Proposition \ref{prop.j1}), an exponential sum over the cone (Proposition \ref{prop.j2}) and an explicit cyclotomic sum formula (Proposition \ref{prop.j3}.) In the case of a two-dimensional cone $\sigma$, the coefficient of $xy$ of $\mathfrak t_\sigma(x,y)$ was identified as a {\em Dedekind sum} \cite{P1}. Furthermore, in \cite{P1} it was shown that {\em reciprocity formulas} for Dedekind sums follow from an $N$-additivity formula for $\mathfrak t$. Zagier's higher-dimensional Dedekind sums \cite{Za4} were later shown to appear as coefficients \cite{BV2}. It is natural to conjecture that, in the two-dimensional case, {\em all} the coefficients of $\mathfrak t$ are generalized Dedekind sums and that reciprocity properties of generalized Dedekind sums will be related to the $N$-additivity formula of $\mathfrak t$. Indeed, this is the case: see Theorem \ref{thm.f=s}. We now present an explicit link between Todd power series and zeta functions. The following theorem expresses the values of zeta function at negative integers in terms of the Todd power series of a two-dimensional cone. Note that the idea of considering the Todd power series as a differential operator, and applying this to an integral over a shifted cone is not new: this was introduced in \cite{KP} and developed further in \cite{BV}. First we introduce some notation which is standard in the theory of toric varieties. If $\tau$ is a cone in a lattice $M$, the dual cone $\check\tau$ is a cone in the dual lattice $N=\text{Hom}(M,\mathbb Z)$, defined by $$ \check\tau=\{v\in N | \langle v, u \rangle \ge 0\ \text{for all}\ u\in\tau\}. $$ The dual of an $n$-dimensional simplicial cone $\tau=\langle\rho_1,\dots, \rho_n\rangle$ is generated by the rays $u_i$ a normal to the $n-1$-dimensional faces of $\tau$; hence $\check\tau$ is also a simplicial cone. Given $h=(h_1,\dots,h_n)\in\mathbb R^n$, we define the {\em shifted cone} $\tau(h)$ to be the following cone in $M_{\mathbb R}$: $$\tau(h)= \{ m \in M_{\mathbb R}| \langle u_i, m \rangle \geq -h_i \text{ for all } i=1, \dots, n \}. $$ Here (and throughout) we have identified each ray $u_i$ with the primitive lattice point on that ray, that is, the nonzero lattice point on $u_i$ closest to the origin. Given an $n$-dimensional simplicial cone $\tau=\langle \rho_1,\dots,\rho_n\rangle$ in an $n$-dimensional lattice $M$, the {\em multiplicity} of $\tau$, denoted by $\text{mult}(\tau)$, is defined to be the index in $M$ of the sublattice $\mathbb Z\rho_1+\dots +\mathbb Z\rho_n$ (again identifying the rays $\rho_i$ with their primitive lattice points.) Thus the multiplicity of $\tau$ is simply the volume of the parallelepiped formed by the vectors from the origin to the primitive lattice points on the rays of $\tau$. \begin{theorem} \lbl{thm.1} Let $\tau$ be a two-dimensional cone of multiplicity $q$ in a two-dimensional lattice $M$, and let $\sigma=\check\tau$ be the dual cone in $N=Hom(M,\mathbb Z)$. Then for all $n \geq 0$, we have: $$\zeta_{Q,\tau}(-n)= (-1)^n n! \left\{ (\mathfrak t_{\sigma})_{(2n+2)} \left(\frac{\partial}{\partial h_1},\frac{\partial}{\partial h_2} \right) -\delta_{n,0} \frac{q}{2} \frac{\partial^2}{\partial h_1 \partial h_2} \right\} \diamond \int_{\tau(h)} exp(-Q(u))du, $$ where all derivatives above are evaluated at $h_1=h_2=0$ and $\delta_{n,0}=1$ (resp. $0$) if $n=0$ (resp. $ n \neq 0$). \end{theorem} \noindent In this equation, $(\mathfrak t_{\sigma})_d$ denotes the degree $d$ part of the Todd power series thought of as an (infinite order) constant coefficients differential operator acting on the function $h \to \int_{\tau(h)} exp(-Q (u))du$. The coefficients of $\mathfrak t_{\sigma}$ for a two-dimensional cone may be expressed explicitly in terms of continued fractions. We now state this formula. Let $\sigma$ be a two-dimensional cone in a lattice $N$. Then there exist a (unique) pair of relatively prime integers $p,q$ with $0< p \le q$ such that $\sigma$ is lattice equivalent (equivalent under the automorphism group of the lattice) to the cone $\sigma_{(p,q)} \overset{\text{def}}{=} \langle (1,0),(p,q) \rangle$ in $\mathbb Z^2$. Such a cone will be called a cone of type $(p,q)$. Let $ b_i, h_i, k_i, X_i$ are defined in terms of the negative-regular continued fraction expansions: \begin{equation} \lbl{eq.bis} \frac{q}{p} = [b_1,\dots,b_{r-1}], \qquad \frac{ h_i}{ k_i} \overset{\text{def}}{=} [b_1,\dots,b_{i-1}], \qquad X_i \overset{\text{def}}{=} - h_i x+(q k_i -p h_i)y. \end{equation} (Throughout, we use such bracketed lists to denote finite negative-regular continued fractions.) We then have the following continued fraction expression for the degree $d$ part $(\mathfrak t_{\sigma})_d$ of the Todd power series $\mathfrak t_{\sigma}$. \begin{theorem} \lbl{thm.toddd} For $\sigma$ of type $(p,q)$ as above, and for $d\geq 2$ an even integer, we have: \begin{eqnarray*} (\mathfrak t_{\sigma})_{d}(x,y)& = & qxy \sum_{i=1}^r P_d( X_{i-1}, X_i) +\lambda_d qxy\sum_{i=1}^{r-1} b_i R_d( X_{i-1}, X_{i+1}) \\ & & -\lambda_d(x X_1^{d-1}+y X_{r-1}^{d-1})+\frac12 \delta_{d,2}qxy. \end{eqnarray*} If $d\geq 1$ is odd, then $ (\mathfrak t_{\sigma})_{d}(x,y) =\frac12\lambda_{d-1}q^{d-2}xy(x^{d-2}+y^{d-2}). $ \end{theorem} \begin{remark} We have stated the formula above in the form we will need it for our study of zeta functions. However, from the toric geometry point of view, it is more natural to use the continued fraction expansion of $\displaystyle\frac{q}{q-p}$ instead. This formula will be given in Section \ref{sub.proof12}. \end{remark} \begin{remark} The formula for the Todd operator in Theorem \ref{thm.toddd} is reminiscent of formulas in quantum cohomology, even though we know of no conceptual explanation for this fact. \end{remark} \subsection{Dedekind sums} \lbl{sub.dedekind} Finally, we calculate the coefficients of the Todd power series of two-dimensional cones in terms of a particular generalization $s_{i,j}$ (defined below) of the classical Dedekind sums. For an excellent review of the properties of the classical Dedekind sum, see \cite{R-G}. Several generalizations of the classical Dedekind sums were studied by T. Apostol, Carlitz, C. Meyer, D. Solomon, and more recently and generally by U. Halbritter \cite{AV, Ha, Me, So}. These papers investigate the sums ( and several other generalizations of them, which we will not consider here) given in the following definition: \begin{definition} \lbl{def.dedsum} For relatively prime integers $p,q$ (with $ q \neq 0$), and for nonnegative integers $i,j$, we define the following {\em generalized Dedekind sum}: \begin{equation} \lbl{eq.dsum} s_{i,j}(p,q)= \frac{1}{i! j!} \sum_{a=1}^{q}\widehat{\text{Ber}}_i(\langle \frac{a}{q} \rangle) \widehat{\text{Ber}}_j(\langle -\frac{ap}{q} \rangle) \end{equation} where for a real number $x$, we denote by $\langle x \rangle $ the (unique) number such that $\langle x \rangle \in (x + \mathbb Z) \cap (0,1]$. Here $\text{Ber}_m$ denotes the $m^{th}$ {\em Bernoulli polynomial}, defined by the power series $ \sum_{m=0}^{\infty} \text{Ber}_m (x) \frac{t^m}{m!} =\frac{ t e^{tx}}{1 - e^t}$, and $\widehat{\text{Ber}}_m$ denotes the restriction of the $m^{th}$ Bernoulli polynomial $\text{Ber}_m$ to $(0,1]$, with the boundary condition $\widehat{\text{Ber}}_m(1) \overset{\text{def}}{=} \frac{1}{2} ( \text{Ber}_m(1) + \text{Ber}_m(0)) = B_m + \delta_{m,1}/2$, where $B_m$ is the $m^{th}$ Bernoulli number, defined by $B_m=\text{Ber}_m(0)$. \footnote{There seem to be two conventions for denoting Bernoulli numbers; one, which we follow here, is used often in number theory texts. Due to the facts that $B_{2n+1}=0$ for $n\geq 1$, and that the sign of $B_{2n}$ alternates, the other convention, which is often used in intersection theory, defines the $n^{th}$ Bernoulli number to be $(-1)^{n+1} B_{2n}$.} \end{definition} By definition, the sum $s_{1,1}(-p,q)$ coincides with the classical Dedekind sum $s(p,q)$ (cf. \cite{R-G}.) An important property of generalized Dedekind sums is a reciprocity formula which leads to an evaluation in terms of negative-regular continued fractions. This reciprocity formula was most conveniently written by D. Solomon in terms of an additivity formula of a generating power series \cite[Theorem 3.3]{So}. On the other hand, the Todd operator also satisfies an additivity property. We denote by $f_{i,j}(p,q) $ (for nonnegative integers $i,j$) the coefficient of $x^i y^j$ of the power series $\mathfrak t_{\sigma_{(p,q)}}(x,y)$ ({\em abbreviated} by $\mathfrak t_{(p,q)}(x,y)$). We then have: \begin{theorem} \lbl{thm.f=s} Let $p,q\in\mathbb Z$ be relatively prime with $q \neq 0$. If $i,j>1$, then we have: \begin{equation} \lbl{eq.spp} f_{i,j}(p,q)=q^{i+j-1} (-1)^i s_{i,j}(p,q). \end{equation} If $i=1$ or $j=1$, the above equation is true when the correction term $$ q^{i+j-1}(-1)^{i+j}\frac{B_iB_j}{i!j!} $$ is added to the right hand side. \end{theorem} \begin{corollary} Fixing $r$, for all nonnegative integers $i,j$ and with the notation of \eqref{eq.bis}, $s_{i,j}(p,q)$ are polynomials in $b_i, 1/q$. \end{corollary} \begin{remark} For $i=j=1$, the above theorem was obtained in \cite{P1}, and was a motivation for the results of the present paper. \end{remark} \subsection{Is toric geometry needed?} \lbl{sub.needed} Some natural questions arise, at this point: \begin{itemize} \item Is toric geometry needed? \item How do the statement and proof of Theorem \ref{cor.zv} differ from the statement and proof of Zagier's formula \cite{Za2}? \end{itemize} With respect to the first question, Theorem \ref{cor.zv} (an evaluation of zeta functions associated to real quadratic fields) is stated without reference to toric geometry, and a close examination reveals that its proof is based on an analytic Lemma \ref{prop.1} and on the two dimensional analogue of the Euler-MacLaurin formula given by Proposition \ref{prop.11}. In addition, two dimensional cones can be canonically subdivided into nonsingular cones, so in a sense the two dimensional analogue of the Euler-MacLaurin formula can be obtained by the classical (one dimensional) one, as is used by Zagier \cite{Za2}. Furthermore, number theory offers, for every $m \geq 2$, a canonical {\em Eisenstein cocycle} of $\text{PGL}_m(\mathbb Q)$ \cite{Sc1, Sc2} that expresses, among other things, the generalized Dedekind sums $s_{i,j}$ in terms of negative-regular continued fraction expansions like the ones of Theorems \ref{thm.toddd} and \ref{thm.f=s}, and the values (at nonpositive integers) of zeta functions of totally real fields (of degree $m$) in terms of generalized Dedekind sums. See also \cite{So, St}. On the other hand, toric geometry constructs for every simplicial cone $\sigma$ in an $m$-dimension- al lattice $N$, a canonical Todd power series $\mathfrak t_\sigma$ satisfying an $N$-additivity property (see Proposition \ref{prop.j1} below). The coefficients of the Todd power series are generalized Dedekind sums, and the power series itself is intimately related to the Euler-MacLaurin summation formula. As a result, for $m=2$, we provide a toric geometry explanation of Theorems \ref{cor.zv} and \ref{thm.f=s}. With respect to the second question, Zagier \cite{Za2} obtained similar formulas for the values of the zeta function of a quadratic number field at negative integers. Zagier used additivity in the lattice $M$, whereas we use additivity in the dual lattice $N$. $N$-additivity implies that these zeta values are a priori polynomials in $b_i$ and $1/q$; moreover these polynomials are directly defined in terms of the Todd operator and the quadratic form $Q$. Thus the present paper is an example of the rather subtle difference between $N$-additivity and $M$-additivity. This difference can be seen by contrasting the additivity property of the zeta function $\zeta_{Q,\tau}$ with the additivity property satisfied by Todd operator. The modified Todd operator $\mathfrak s_\sigma$ (introduced in Section \ref{sub.general}) is $N$-additive under subdivisions of the cone $\sigma$, i.e., it satisfies equation \eqref{eq.j1} below. On the other hand, the function $\zeta_{Q, \cdot}(s): \mathcal C^2(M) \to \mathbb C$ is $M$-additive, i.e., it satisfies: \begin{equation} \lbl{eq.Mad} \zeta_{Q,\tau}(s) = \zeta_{Q, \tau_1}(s) + \zeta_{Q, \tau_2}(s) \end{equation} whenever $\tau$ is subdivided into cones $\tau_1$ and $\tau_2$. This $M$-additivity of $\zeta_{Q, \cdot}(s)$ is due to the particular choice of the weight function $wt$, and is purely set-theoretic. In contrast, the $N$-additivity of $\mathfrak s_{\sigma}$ is somewhat deeper, and has the advantage that only cones of maximal dimension are involved. \subsection{Plan of the proof} \lbl{sub.plan} In Section \ref{sec.thm.1}, we review well known properties of Todd power series and prove Theorems \ref{thm.1} and \ref{thm.toddd}. In Section \ref{sec.calc}, we review the relation between the zeta functions that we consider and the zeta functions of real quadratic fields. We give a detailed construction of zeta functions, and prove Theorem \ref{cor.zv}. Finally, in Section \ref{sec.dedsum} we review properties of generalized Dedekind sums and prove Theorem \ref{thm.f=s}. \subsection{Acknowledgments} We wish to thank M. Rosen for encouraging conversations during the academic year 1995-96. We also thank B. Sturmfels and M. Brion for their guidance. We especially wish to thank W. Fulton for enlightening and encouraging conversations since our early years of graduate studies. \section{The Todd power series of a cone} \lbl{sec.thm.1} In this section we study properties of the Todd power series $\mathfrak t_{\sigma}$ associated to a simplicial cone $\sigma$. Section \ref{sub.general} provides an introduction and statements of several previously discovered formulas for the Todd power series. In Section \ref{sub.2dim} we make these formulas explicit for two-dimensional cones. Section \ref{sub.proof12} contains a proof of the explicit continued fraction formula for the Todd power series of a two-dimensional cone. Finally, in Section \ref{sub.thm.1} we prove Theorem \ref{thm.1} which links Todd power series with the problem of evaluating zeta functions an nonpositive integers. \subsection{General properties of the Todd power series} \lbl{sub.general} Todd power series were studied in connection with the Todd class of a simplicial toric variety in \cite{P3}. Independently, they were introduced in \cite{BV2} in the study of the equivariant Todd class of a simplicial toric variety. In addition, these power series appear in Brion and Vergne's formula for counting lattice points in a simple polytope \cite{BV}, which is an extension of Khovanskii and Pukhlikov's formula \cite{KP}. In these remarkable formulas, the power series in question are considered as differential operators which are applied to the volume of a deformed polytope. The result yields the number of lattice points in the polytope, or more generally, the sum of any polynomial function over the lattice points in the polytope. Below (Proposition \ref{prop.11}), we give a version of this formula expressing the sum of certain functions over the lattice points contained in a simplicial cone. The Todd power series considered in the works cited above are also closely related to the fundamental work of R. Morelli on the Todd class of a toric variety. \cite{Mo}. A precise connection is given in \cite[Section 1.8]{P3}. We now state some of the properties of the Todd power series of a simplicial cone. Our purpose is twofold: we will need these properties in our application to zeta functions, and we wish to unite the approaches of the works cited above. Here, we follow the notation of \cite{P3}. Let $\sigma=\langle \rho_1, \dots \rho_n\rangle$ be an $n$-dimensional simplicial cone in an $n$-dimensional lattice $N$. The Todd power series $\mathfrak t_{\sigma}$ of $\sigma$ is a power series with rational coefficients in variables $x_1, \dots, x_n$ be corresponding to the rays of $\sigma$. These power series, when evaluated at certain divisor classes, yield the Todd class of any simplicial toric variety \cite[Theorem 1]{P3}. To state the properties of $\mathfrak t_{\sigma}$, it will be useful to consider the following variant $\mathfrak s$ of the power series $\mathfrak t$ defined in \cite{P3} by: $$ \mathfrak s_{\sigma}(x_1,\dots,x_n)=\frac1{\text{mult}(\sigma) x_1\cdots x_n} \mathfrak t_{\sigma}(x_1,\dots,x_n), $$ which is a Laurent series in $x_1, \dots, x_n$. The Todd power series $\mathfrak t_{\sigma}$ and $\mathfrak s_{\sigma}$ are characterized by the following proposition \cite[Theorem 2]{P3}, which states that $\mathfrak s$ is {\it additive} under subdivisions (after suitable coordinate changes), and gives the value of $\mathfrak s$ on nonsingular cones. An $n$-dimensional cone is called nonsingular if it is generated by rays forming a basis of the lattice. It is well-known that any cone may be subdivided into nonsingular cones, and that such a subdivision determines a resolution of singularities of the corresponding toric variety (cf. \cite[Section 2.6]{Fu}.) \begin{proposition} \lbl{prop.j1} If $\Gamma$ is a simplicial subdivision of $\sigma$ then \begin{equation} \lbl{eq.j1} \mathfrak s_{\sigma}(X)=\sum_{\gamma\in\Gamma_{(n)}} \mathfrak s_{\gamma}(\gamma^{-1}\sigma X). \end{equation} where the sum is taken over the $n$-dimensional cones of the subdivision $\Gamma$. Here $X$ denotes the column vector $(x_1,\dots,x_n)^t$ and we have identified each cone ($\sigma$ and $\gamma$) with the $n$-by-$n$ matrix whose columns are the coordinates of the rays of that cone. For nonsingular cones, $\sigma$, we have the following expression for $\mathfrak s_{\sigma}$: $$ \mathfrak s_{\sigma}(x_1,\dots,x_n)=\prod_{i=1}^n\frac{1}{1-e^{-x_i}}. $$ \end{proposition} \begin{remark} It follows immediately that for any cone $\sigma$, $\mathfrak s_{\sigma}$ is a rational function of the $e^{x_i}$. \end{remark} The Laurent series $\mathfrak s_{\sigma}$ in $x_1, \dots , x_n$ may also be expressed as an exponential sum over the lattice points in the cone, in the spirit of the important earlier work of M. Brion \cite{Br}. \begin{proposition} \lbl{prop.j2} Let $\check\sigma$ denote the dual cone in the lattice $M=\text{Hom}(N,\mathbb Z )$. We then have $$ \mathfrak s_{\sigma}(x_1,\dots,x_n)=\sum_{m\in\check\sigma\cap M} e^{-(\langle m,\rho_1 \rangle x_1 + \cdots + \langle m,\rho_n \rangle x_n )}, $$ The equality is one of rational functions in the $e^{x_i}$. \end{proposition} \begin{proof} As noted in \cite{Br}, since $\langle m,\rho_i \rangle \geq 0$ for $m\in\check\sigma\cap M$, the right hand side has a meaning in the completion of $\mathbb C[y_1,\dots,y_n]$ with respect to the ideal $(y_1,\dots,y_n)$, where $y_i$ stands for $e^{-x_i}$. While it is not obvious, the right hand side is an rational function of the $e^{x_i}$. See \cite[p.654]{Br}. The left hand side is a rational function of the $e^{x_i}$ by Proposition \ref{prop.j1}. By \cite[p.655]{Br} and the second formula of Proposition \ref{prop.j1}, these two rational functions are equal on nonsingular cones. As any cone can be subdivided into nonsingular cones, it suffices to verify that the right hand side satisfies the additivity formula of Proposition \ref{prop.j1}. But this follows again from Brion's work. See the proposition of \cite[p.657]{Br}, for example. Intuitively, this additivity can be seen from the fact that a sum of exponentials over a cone containing a straight line vanishes formally. \end{proof} The $\mathfrak s_{\sigma}$ also have an explicit expression in terms of cyclotomic sums, due to Brion and Vergne. Following \cite{BV2}, we introduce the following notation. Let $u_1,\dots, u_n$ denote the primitive generators of the dual cone $\check\sigma$. Thus we have $\langle u_i,\rho_j \rangle =0$ if $i\ne j$, and $\langle u_i,\rho_i \rangle \in \mathbb Z $, but does not necessarily equal $1$. Let $N_{\sigma}$ be the subgroup of $N$ generated by the $\rho_i$, $i=1,\dots,n$, and let $G_{\sigma}= N/N_{\sigma}$. Then $G_{\sigma}$ is an abelian group of order $\text{mult}( \sigma)$, which we denote by $q$. Define characters $a_i$ of $G_{\sigma}$ by $$ a_i(g)=e^{2\pi i\frac{\langle u_i, g \rangle}{\langle u_i, \rho_i \rangle}}. $$ \begin{proposition} \lbl{prop.j3} The Laurent series $\mathfrak s_{\sigma}$ coincides with Brion and Vergne's formula expressing their Todd differential operator. Namely, we have $$ \mathfrak s_{\sigma}(x_1,\dots,x_n)=\frac 1q \sum_{g\in G_{\sigma}}\prod_{i=1}^n \frac 1{1-a_i(g)e^{-x_i}}. $$ \end{proposition} \begin{proof} By inspection, the right hand side is a rational function in the variables $y_i=e^{-x_i}$ which takes the value $1$ when $y_i=0$ for all $i$. Such rational functions embed into the completion of the ring $\mathbb C[y_1,\dots,y_n]$ with respect to the ideal $(y_1, \dots , y_n)$. To prove the proposition, it is enough to show that the right hand side and the right hand side of Proposition \ref{prop.j2} define the same element of this completion. To do so, we expand the right hand side, getting: $$ \frac 1q \sum_{k_1,\dots,k_n\ge 0} \sum_{g\in G_{\sigma}} \prod_{i=1}^n \biggl[a_i(g)e^{-x_i}\biggr]^{k_i},$$ which becomes $$ \frac 1q\sum_{k_1,\dots,k_n\ge 0} e^{-(k_1 x_1 +\cdots + k_n x_n)} \sum_{g\in G_{\sigma}} a_1^{k_1}\cdots a_n^{k_n}(g). $$ This last sum over $G_{\sigma}$ is either $q$ or $0$, depending on whether $a_1^{k_1}\cdots a_n^{k_n}$ is the trivial character of $G_{\sigma}$ or not. Comparing the above with the right hand side of Proposition 2, we see that it suffices to show that $a_1^{k_1}\cdots a_n^{k_n}$ is the trivial character of $G_{\sigma}$ if and only if there exists $m\in \check\sigma$ such that $\langle m,\rho_i \rangle = k_i$ for all $i$. This straightforward lattice calculation is omitted. \end{proof} \begin{remark} The sum in the proposition above appears in the important work of Ricardo Diaz and Sinai Robins \cite{DR}. They use such sums to give an explicit formula for the number of lattice points in a simple polytope. Interestingly, their techniques, which come from Fourier analysis, are seemingly unrelated to the toric geometry discussed above. \end{remark} \subsection{Properties of Todd power series of two-dimensional cones} \lbl{sub.2dim} In this section, we state properties of the power series $\mathfrak t_{\sigma}$ for a two-dimensional cone $\sigma$. It is not hard to see that if $\sigma$ is any two-dimensional cone, then there are relatively prime integers $p,q$ such that $\sigma$ is lattice-equivalent to the cone $$ \sigma_{(p,q)} \overset{\text{def}}{=} \langle (1,0), (p,q) \rangle \subset \mathbb Z^2. $$ Here $q$ is determined up to sign and $p$ is determined modulo $q$. Thus we may arrange to have $q>0$ and $0\leq p<q$. This gives a complete classification of two-dimensional cones up to lattice isomorphism. In discussing Todd power series we will abbreviate $\mathfrak t_{\sigma_{(p,q)}}$ by $\mathfrak t_{(p,q)}$. The explicit cyclotomic formula of Proposition \ref{prop.j3} may be written as: \begin{proposition} \lbl{prop.2d.cycl} The Todd power series of a two-dimensional cone is given by $$ \mathfrak t_{(p,q)}(x,y)= \sum_{\omega^q=1} \frac{xy}{(1- \omega^{-p} e^{-x})(1 - \omega e^{-y})} $$ \end{proposition} \begin{proof} With the coordinates above and the notation of Proposition \ref{prop.j3}, we have $u_1=(q,-p)$, $u_2=(0,1)$, and $G_{\sigma}$ consists of the lattice points $(0,k), k=0,\dots, q-1.$ The desired equation now follows directly from Proposition \ref{prop.j3}. \end{proof} In the two-dimensional case, the additivity formula of Proposition \ref{prop.j1} can be expressed as an explicit {\em reciprocity} law. This, and a {\em periodicity} relation for $\mathfrak s$ are contained in the following theorem. \begin{proposition} \lbl{prop.2d.reci} Let $p$ and $q$ be relatively prime positive integers. Then \begin{eqnarray} \lbl{eq.reci} \mathfrak s_{(p,q)}(x - \frac{p}{q} y, \frac{1}{q} y)+ \mathfrak s_{(q,p)}(y- \frac{q}{p}x , \frac{1}{p} x) & = & \mathfrak s_{(0,1)}(x,y) \\ \lbl{eq.peri} \mathfrak s_{(p+q,q)}(x,y) & = & \mathfrak s_{(p,q)}(x,y) \end{eqnarray} \end{proposition} \begin{proof} The quadrant $\langle (1,0),(0,1) \rangle$ may be subdivided into cones $\gamma_1=\langle (1,0), (p,q) \rangle$ and $\gamma_2=\langle (0,1), (p,q) \rangle$. The cone $\gamma_1$ is of type $(p,q)$, and $\gamma_2$ is of type $(q,p)$. Applying the additivity formula (Proposition \ref{prop.j1}) to this subdivision yields the first equation above. The second equation follows from the fact that the cones $\langle (1,0), (p,q) \rangle$ and $\langle (1,0), (p+q,q) \rangle$ are lattice isomorphic. \end{proof} Let $\mathfrak s^{ev}$ (resp. $\mathfrak s^{odd}$) denote the part of $\mathfrak s$ of even (resp. odd) total degree. \begin{corollary} \lbl{cor.oneonly} Given a two-dimensional lattice $N$, the function $\mathfrak s: \mathcal C^2(N) \to \mathbb Q\(x,y\)$ (where $\mathbb Q\(x,y\)$ is the function field of the power series ring $\mathbb Q\[x,y\]$) is uniquely determined by properties \eqref{eq.reci} and \eqref{eq.peri} and its {\em initial condition} $\mathfrak s_{(0,1)}(x,y)= 1/((1 - e^{-x})(1 - e^{-y}))$. Since equations \eqref{eq.reci} and \eqref{eq.peri} are homogenous with respect to the degrees of $x$ and $y$, it follows that $\mathfrak s^{ev}$ (resp. $\mathfrak s^{odd}$) satisfies \eqref{eq.reci} and \eqref{eq.peri} with initial condition $\mathfrak s_{(0,1)}^{ev}$ (resp. $\mathfrak s_{(0,1)}^{odd}$). \end{corollary} \begin{remark} \lbl{rem.oneo} Corollary \ref{cor.oneonly} has a converse. >From equations \eqref{eq.reci} and \eqref{eq.peri} it follows that $\mathfrak s_{(0,1)}$ satisfies the following relations: $$ \mathfrak s_{(0,1)}(x,y) = \mathfrak s_{(0,1)}(x-y,y) + \mathfrak s_{(0,1)}(y-x, x) \text{ and } \mathfrak s_{(0,1)}(x,y) = \mathfrak s_{(0,1)}(y,x) $$ Conversely, one can show that given any element $g(x,y) \in \mathbb Q\(x,y\)$ satisfying: $$ g(x,y) = g(x-y,y) + g(y-x, x) \text{ and } g(x,y) = g(y,x) $$ there is a unique function $\mathfrak g :\mathcal C^2(N) \to \mathbb Q\(x,y\)$ so that $\mathfrak g_{(0,1)} = g$. \end{remark} \subsection{Continued fraction expansion for the Todd power series} \lbl{sub.proof12} In this section, we prove Theorem \ref{thm.toddd}, which expresses the coefficients in the Todd power series of a two-dimensional cone in terms of continued fractions. Before doing so, we first formulate an equivalent version which is more natural from the point of view of toric varieties. The continued fraction expansion in this second version of the formula corresponds directly to a desingularization of the cone. Given relatively prime integers $p,q$ with $p>0$ and $0\leq p<q$, let $ a_i, \gamma_i, \delta_i, L_i$ be defined in terms of the negative continued fraction expansions: \begin{equation*} \frac{q}{q-p} = [a_1,\dots, a_{s-1}], \qquad \frac{ \gamma_i}{ \delta_i} \overset{\text{def}}{=} [a_1,\dots,a_{i-1}], \qquad L_i \overset{\text{def}}{=} \gamma_i x+(q \delta_i+(p-q) \gamma_i)y. \end{equation*} We then have the following continued fraction expression for the degree $d$ part $(\mathfrak t_{\varpi})_d$ of the Todd power series of a two-dimensional cone $\varpi$. \begin{theorem} \lbl{thm.todd} For $\varpi$ a cone of type $(p,q)$ as above, and for $d\geq 2$ an even integer, we have: \begin{equation*} (\mathfrak t_{\varpi})_{d}(x,y)=-qxy \sum_{i=1}^s P_d(L_{i-1}, L_i) -\lambda_d qxy\sum_{i=1}^{s-1} a_i R_d( L_{i-1}, L_{i+1}) +\lambda_d(x L_1^{d-1}-y L_{s-1}^{d-1}) \end{equation*} If $d\geq 1$ is odd, then $ (\mathfrak t_{\varpi})_{d}(x,y) =\frac12\lambda_{d-1}q^{d-1}xy(x^{d-2}+y^{d-2}). $ \end{theorem} \begin{proof} It will be convenient to choose coordinates so that $\varpi= \langle (0,-1), (q,q-p) \rangle$ (which is easily seen to be lattice equivalent to the cone $\langle (1,0),(p,q) \rangle$). We now subdivide $\varpi$ into nonsingular cones. It is well-known that for two-dimensional cones this can be done in a canonical way, and that the resulting subdivision has an explicit expression in terms of continued fractions \cite[Section 2.6]{Fu}. In our coordinate system, the rays of this nonsingular subdivision of $\varpi$ are given by \begin{eqnarray*} \beta_0&=&(0,-1)\\ \beta_1&=&(1,0)\\ \beta_2&=&(a_1,1)\\ &\dots&\\ \beta_s &=&(q,q-p). \end{eqnarray*} Thus we have $ \beta_{i+1} + \beta_{i-1} = a_i \beta_i$, which implies that $ \beta_i=( \gamma_i, \delta_i)$. The cone $\varpi$ is subdivided into cones $\varpi_i= \langle \beta_{i-1}, \beta_i \rangle$, $i=1,\dots s$. The $N$-additivity formula of Proposition \ref{prop.2d.reci} implies that: $$ \mathfrak s_{\varpi}(x,y)=\sum_{i=1}^s \mathfrak s_{\varpi_i} (\varpi_i^{-1}\varpi (x,y)^t), $$ where we again we have identified the $2$-dimensional cone $\varpi$ with the 2-by-2 matrix whose columns are the primitive generators of $\varpi$. One easily sees that this becomes $$ \mathfrak s_{\varpi}(x,y)=\sum_{i=1}^s \mathfrak s_{\varpi_i} (-L_{i-1}, L_i) $$ Rewriting the equation in terms of the $\mathfrak t_{\varpi}$ yields $$ \mathfrak t_{\varpi}(x,y)=-qxy\sum_{i=1}^s \frac{\mathfrak t_{\varpi_i} (-L_{i-1}, L_i)}{L_{i-1}L_i} $$ Since every $\varpi_i$ is nonsingular, we have $$ \mathfrak t_{\varpi_i}(X,Y)=g(X)g(Y) $$ where $g(z)=z/(1-e^{-z}).$ Now consider the degree $d$ part of the above. We will assume $d>2$ is even and leave the other (easier) cases to the reader. The degree $d$ part of $g(L_{i-1})g(L_{i})$ may be written as: $$ -L_{i-1}L_iP_d(-L_{i-1},L_i) + \lambda_d (L_{i-1}^d+L_i^d). $$ Summing the first term above yields the first term in the equation of the theorem. So it suffices to examine the remaining term: $$ -\lambda_dqxy\sum_{i-1}^s \frac{L_{i-1}^d+L_i^d}{L_{i-1}L_i}. $$ Using the relation $L_{i-1}+L_{i+1}=a_iL_i$, the sum above may be rewritten as $$ \sum_{i=1}^{s-1} a_i \frac{L_{i-1}^{d-1}+L_{i+1}^{d-1}}{L_{i-1}+L_{i+1}} +\biggl(\frac{L_1^{d-1}}{L_0} + \frac{L_{s-1}^{d-1}}{L_s} \biggr). $$ Keeping in mind that $L_0=-qy$ and $L_s=qx$, Theorem \ref{thm.todd} follows easily. \end{proof} \begin{corollary} \lbl{cor.use1} For a two-dimensional cone $\sigma= \langle \rho_1, \rho_2 \rangle$ of multiplicity $q$ we have: \begin{equation} \lbl{eq.evenf} \mathfrak t_{\sigma}(h_1, h_2) - \frac{1}{2}\mathfrak t_{\langle \rho_1 \rangle}( q h_1)h_2 - \frac{1}{2}\mathfrak t_{\langle \rho_2 \rangle}( q h_2) h_1 = \mathfrak t_{\sigma}^{ev}(h_1, h_2) - \frac{q h_1 h_2}{2} \end{equation} where $\mathfrak t_{\sigma}^{ev}$ is the even total degree part of the power series $\mathfrak t_{\sigma}$. \end{corollary} \begin{proof} It follows immediately from (and in fact is equivalent to) the formula for the odd part of the Todd power series $\mathfrak t_{\sigma}$ given by Theorem \ref{thm.todd}. \end{proof} We now prove Theorem \ref{thm.toddd}. Let $\sigma$ be a two-dimensional cone of type $(p,q)$ in a lattice $N$, and let $ a_i, h_i, k_i$ and $ X_i$ be as in Theorem \ref{thm.toddd}. The dual cone $\check{\sigma}$ in the dual lattice $M$ is easily seen to have type $(-p,q)$ and so we may choose coordinates in $M$ so that $\check{\sigma}= \langle (0,-1), (q,p) \rangle$ in $\mathbb Z^2$. Furthermore, the negative-regular continued fraction expansion of $q/p$ corresponds naturally to the desingularization of the dual cone $\check{\sigma}$. Explicitly, the desingularization of $\check\sigma$ is given by the subdivision \begin{eqnarray*} \rho_0&=(0,-1)\\ \rho_1&=(1,0)\\ \rho_2&=(b_1,1)\\ &\dots\\ \rho_r &=(q,p). \end{eqnarray*} One has $ \rho_{i+1} + \rho_{i-1} =b_i\rho_i$ and thus $ \rho_i=( h_i, k_i)$. Applying Theorem \ref{thm.todd} to $\check{\sigma}$ expresses $\mathfrak t_{(-p,q)}$ in terms of the $b_i$. However $\mathfrak t_{(-p,q)}$ is related to $\mathfrak t_{(p,q)}$ via the relation $$\mathfrak t_{(p,q)}(x,y)= \mathfrak t_{(-p,q)}(-x,y) + \frac{qxy}{1-e^{-qy}}.$$ In this way, we obtain an expression for $\mathfrak t_{(p,q)}$ in terms of the $b_i$, which concludes the proof of Theorem \ref{thm.toddd}. \subsection{Zeta function values in terms of Todd power series} \lbl{sub.thm.1} In this section, we prove Theorem \ref{thm.1} which expresses values of the zeta function of a two-dimensional cone in terms of the Todd power series. Three ingredients are involved in the proof of this theorem: an asymptotic series formula (Proposition \ref{prop.1}), a polytope summation formula (Proposition \ref{prop.111}) and a cone summation formula (Proposition \ref{prop.11}). We begin with the first ingredient: \begin{lemma}\cite[Proposition 2]{Za2} \lbl{prop.1} Let $\phi(s)= \sum_{\lambda > 0} a_\lambda \lambda^{-s}$ be a Dirichlet series where $\{ \lambda \}$ is a sequence of positive real numbers converging to infinity. Let $E(t)= \sum_{\lambda > 0} a_\lambda e^{- \lambda t}$ be the corresponding exponential series. Assume that $E(t)$ has the following asymptotic expansion as $t \to 0$: \begin{equation} E(t) \sim \sum_{n=-1}^{\infty} c_{n} t^n \end{equation} Then it follows that \begin{itemize} \item $\phi(s)$ can be extended to a meromorphic function on $\mathbb C$. \item $\phi(s)$ has a simple pole at $s=1$, and no other poles. \item The values of $\phi$ at nonpositive integers are given by: $\phi(-n)= (-1)^{n} n! c_{n}$. \end{itemize} \end{lemma} We now present our second ingredient, a polytope summation formula. This proposition is a variant of the lattice point formula of \cite{BV2}. A polytope of dimension $n$ is called {\em simple} if each of its vertices lies on exactly $n$ facets ($n-1$-dimensional faces) of the polytope. \begin{proposition} \lbl{prop.111} Let $N$ be an $n$-dimensional lattice, $P$ a simple lattice polytope in $M$ and $\Sigma$ its associated fan in $N$. For every analytic function $\phi: M_\mathbb R \to \mathbb R$, we have the following asymptotic expansion as $t \to 0$: \begin{equation} \sum_{a \in P \cap M} \phi(t a) \sim \mathfrak t_{\Sigma}\left(\frac{\partial}{\partial h}\right) \diamond \int_{P(h)} \phi(t u) du \end{equation} where $\mathfrak t_\Sigma$ is the Todd power series of \cite[Definition 10]{BV2}. \end{proposition} \begin{proof} First of all, the meaning of the right hand side is as follows: we consider the degree $k$ Taylor expansion $\phi=\phi_k + R_k$ of $\phi$, where $\phi_k$ is a polynomial in $M_\mathbb R$ of degree $k$ and $R_k$ is the remainder satisfying $\lim_{a\to 0} |a|^k R_k(a)=0$. It follows that $\int_{P(h)} P_k(tu) du$ is a polynomial in $t$ and $h$ of degree $k$ (with respect to $t$) and that $ \int_{P(h)} R_k(tu) du=o(t^k)$ at $t=0$ (with the notation that $f(t)=o(t^k)$ if and only if $\lim_{t\to 0} f(t) t^{-k}=0$). Thus, $$ \mathfrak t_\Sigma\left(\frac{\partial}{\partial h}\right) \diamond \int_{P(h)} \phi(t u) du = \mathfrak t_\Sigma\left(\frac{\partial}{\partial h}\right) \diamond \int_{P(h)} \phi_k(t u) du + o(t^k). $$ On the other hand, \begin{eqnarray*} \sum_{a \in P \cap M} \phi(t a) & = & \sum_{a \in P \cap M} \phi_k(t a) + \sum_{a \in P \cap M} R_k(t a) \\ & = & \sum_{a \in P \cap M} \phi_k(t a) + o(t^k). \end{eqnarray*} Brion-Vergne \cite[theorem 11]{BV2} prove that for every polynomial function (such as $\phi_k$) on $M_\mathbb R$ we have: $$ \sum_{a \in P \cap M} \phi_k(t a) = \mathfrak t_{\Sigma}\left(\frac{\partial}{\partial h}\right) \diamond \int_{P(h)} \phi_k(t u) du, $$ which concludes the proof. \end{proof} We call a function $\phi: \mathbb R^n \to \mathbb R$ {\em rapidly decreasing} if it is analytic and for every constant coefficients differential operator $D$, and every subset $I$ of $[n]=\{1,\dots, n\}$, the restriction $D(\phi)|_I$ obtained by setting $x_i=0$ for $i \not\in I$ is in $L^1(\mathbb R_+^{I})$. Examples of rapidly decreasing functions can be obtained by setting $\phi=P \exp(Q)$ where $P$ is a polynomial on $\mathbb R^n$ and $Q: \mathbb R^n \to \mathbb R$ is totally positive, i.e., its restriction to $\mathbb R_{+}^{I}$ takes positive values for every subset $I$ of $[n]$. \begin{proposition} \lbl{prop.11} Let $N$ be an $n$-dimensional lattice. For every $\sigma \in \mathcal C^n(N)$, and every rapidly decreasing analytic function $\phi: M_\mathbb R \to \mathbb R$, we have the following asymptotic expansion as $t\to 0$: \begin{equation} \lbl{eq.p3} \sum_{a \in \check{\sigma} \cap M} \phi(t a) \sim \mathfrak t_{\sigma}\left(\frac{\partial}{\partial h}\right) \diamond \int_{\check{\sigma}(h)} \phi(t u) du \end{equation} \end{proposition} \begin{proof} First of all, the right hand side of the above equation has the following meaning: consider the decomposition $\mathfrak t_\sigma=\sum_k \mathfrak t_{\sigma,k}$ of the power series $\mathfrak t_\sigma$, where $\mathfrak t_{\sigma,k}$ is a homogenous polynomial of degree $k$. A change of variables $v=tu$ implies that $$ \mathfrak t_{\sigma,k}\left(\frac{\partial}{\partial h}\right) \diamond \int_{\check{\sigma}(h)} \phi(t u) du = \mathfrak t_{\sigma,k}\left(\frac{\partial}{\partial h}\right) \diamond \int_{\check{\sigma}(th)} \phi(v) dv/t^n = t^{k-n} \mathfrak t_{\sigma,k}\left(\frac{\partial}{\partial h}\right) \diamond \int_{\check{\sigma}(h)} \phi(v) dv $$ is a multiple of $ t^{k-n}$, thus the right hand side is defined to be the Laurent power series in $t$ given by $$ \sum_{k=0}^\infty t^{k-n} \mathfrak t_{\sigma,k}\left(\frac{\partial}{\partial h}\right) \diamond \int_{\check{\sigma}(h)} \phi(v) dv. $$ \noindent For the proof of the proposition, truncate in some way the cone $\check{\sigma}$ in $M$ to obtain a simple convex polytope $P$, with associated fan $\Sigma$. Since $\cup_{r > 0} r P=\check{\sigma}$, using the convergence properties of $\phi$, we obtain as $r \to \infty$: \begin{eqnarray*} \sum_{a \in \check{\sigma} \cap M} \phi(t a) & = & \lim_r \sum_{a \in r P \cap M} \phi(t a) \\ & \sim & \lim_{r} \mathfrak t_{\Sigma}\left(\frac{\partial}{\partial h}\right) \diamond \int_{(rP)(h)} \phi(t u) du \\ & = & \lim_{r} \mathfrak t_{\Sigma}\left(\frac{\partial}{\partial h}\right) \diamond \int_{(rtP)(th)} \phi(v) dv/t^n \\ & = & \lim_{r} \sum_{k} \mathfrak t_{\Sigma,k}t^{k-n}\left(\frac{\partial}{\partial h}\right) \diamond \int_{(rtP)(h)} \phi(v) dv \\ & = & \sum_{k} \lim_r \mathfrak t_{\Sigma,k}t^{k-n}\left(\frac{\partial}{\partial h}\right) \diamond \int_{(rtP)(h)} \phi(v) dv \\ & = & \sum_{k} \mathfrak t_{\sigma,k}t^{k-n}\left(\frac{\partial}{\partial h}\right) \diamond \int_{P(h)} \phi(v) dv, \end{eqnarray*} which concludes the proof. \end{proof} In case of a two-dimensional lattice $N$ and a rapidly decreasing function $\phi: M_\mathbb R \to \mathbb R$, using the weight function $wt$ of equation \eqref{eq.we} and inclusion-exclusion, we obtain the following \begin{corollary} \lbl{cor.weights} For a two-dimensional cone $\sigma = \langle \rho_1, \rho_2 \rangle$ of multiplicity $q$ in $N$, we have the asymptotic expansion as $t \to 0$: \begin{equation} \lbl{eq.p3.weight} \sum_{a \in \check{\sigma} \cap M} wt(\check{\sigma}, a) \phi(ta) \sim \left\{ \mathfrak t_{\sigma}^{ev}\left(\frac{\partial}{\partial h_1},\frac{\partial}{\partial h_2}\right) -\frac{q}{2} \frac{\partial^2}{\partial h_1 \partial h_2} \right\} \diamond \int_{\check{\sigma}(h)} \phi(tu)du, \end{equation} where the right hand side lies in the formal power series ring $t^{-2} \mathbb R \[t^2 \]$. \end{corollary} \begin{proof}[Proof of Theorem \ref{thm.1}] Recall that $\tau$ is a cone in $M$, $\sigma$ is its dual in $N$ and $Q$ is homogenous quadratic, totally positive on $\tau$; thus $e^{-Q}$ is rapidly decreasing on $M_\mathbb R$. Corollary \ref{cor.weights} implies that the generating function \begin{equation} \lbl{eq.gf} Z_{Q,\tau}(t)= \sum_{ a \in \tau \cap M} wt(\tau, a)e^{-t Q(a)} =\sum_{ a \in \tau \cap M} wt(\tau, a)e^{-Q(t^{1/2}a)}, \end{equation} satisfies the hypothesis of Lemma \ref{prop.1}, which in turn yields Theorem \ref{thm.1}. \end{proof} \begin{remark} \lbl{rem.notice} Notice that the above proof of Theorem \ref{thm.1} used crucially the fact that $B_1 = -\frac{1}{2}$ and the definition of the weight function $wt$. If we had weighted the sum defining the zeta function in any other way, the resulting variation of Theorem \ref{thm.1} would not hold. \end{remark} \section{Zeta functions of number fields} \lbl{sec.calc} \subsection{Parametrizing triples $(M_n,Q_b,\tau_b)$} \lbl{sub.real} In this section, we review in detail the construction of the triple $(M_b,Q_b,\tau_b)$ mentioned in Section \ref{sub.history} that is associated to a sequence $b=(b_0, \dots, b_{r-1})$ of integers greater than $1$ and not all equal to $2$. Our notation and construction is borrowed from \cite{Za2}. Given a sequence $b$ as above, we extend it to a sequence of integers parametrized by the integers by defining $b_k = b_{k \bmod r}$. Furthermore, for an integer $k$, we define $$w_k= \[ b_k, \dots, b_{k+ r-1} \] = b_k - \cfrac{1}{b_{k+1} - \cfrac{1}{b_{k+2} - \cdots }} $$ where $\[b_k, \dots, b_{k+ r-1} \]$ denotes the infinite continued fraction with period $r$. Note that $w_k = w_{k+r}$ for all integers $k$, and that, by definition, $$w_0= b_0 - \cfrac{1}{b_1 - \cfrac{1}{ \cdots b_{r-1} -\cfrac{1}{w_0}}} $$ from which it follows that $w_0$ satisfies a quadratic equation $ A_b w^2 + B_b w + C_b = 0$ where $A_b,B_b,C_b$ are (for a fixed $r$) polynomials in $b_i$ with integer coefficients.\footnote{not to be confused with $A_0,\dots, A_{r}$ introduced below, despite the rather poor choice of notation.} Let $D_b= B_b^2 - 4 A_b C_b$ be the discriminant. Since $b_i > 1$, it follows that $D_b > 0$, and that the roots of the quadratic equation are $w_0=( -B_b + \sqrt{D_b})/(2 A_b)$ and $w_0 '=( -B_b - \sqrt{D_b})/(2 A_b)$. We thus have that $w_0 > 1 > w_0 '$. We define the complex numbers $A_0=1$, $A_{k-1}= A_k w_k$ for $k \in \mathbb Z$. Since $w_k = b_k - \frac{1}{w_{k+1}}$ it follows that \begin{equation} \lbl{eq.Ar} A_{k-1} + A_{k+1} = b_k A_k \end{equation} for all integers $k$. Let us define $M_b= \mathbb Z w_0 + \mathbb Z $ to be the rank two lattice with the oriented basis $\{w_0, 1 \}$. Note that $M_b$ is a rank two lattice in the real quadratic field $K_b = \mathbb Q(\sqrt{D_b})$. Due to the fact that $A_{-1} = w_0$ and the recursion relation \eqref{eq.Ar}, it follows that $M_b= \mathbb Z A_k + \mathbb Z A_{k+1}$ for all integers $k$, and thus $\epsilon M_b = M_b$ where $\epsilon = w_0 \dots w_{r-1}$. Thus $U_b= \{ \epsilon^m | m \in \mathbb Z \} $ acts on $M_b$ by multiplication, and the zeta function of (the inverse of) the narrow ideal class of $M_b$ satisfies \begin{equation*} \zeta(M_b, s)= N(M_b)^s \sum_{\substack{a \in M_b/U_b \\ a \gg 0 }} \frac{1}{Q_b(a)^s}, \end{equation*} where in the above summation we exclude $0$, and $N(M_b)$ is a constant, defined in \cite{Za2}, which is not so crucial for our purposes. We are therefore interested to describe the quotient $M_b/U_b$. The action of $U_b$ on $M_b$ implies that $\epsilon A_k = A_{k-r}$. The matrix of multiplication by $\epsilon$ on $M_b$, in terms of the basis $\{ \beta_1, \beta_2 \} \overset{\text{def}}{=} \{ A_{-1}, A_0 \}$ can be calculated as follows. Recall first that in terms of the above basis of $M_b$, we have that: \begin{equation} \lbl{eq.aaa} A_k = - p_k A_{-1} + q_k A_0 \text{ where } \frac{p_k}{q_k}= [ b_0, \dots, b_{k-1} ] \end{equation} which implies that $$ \epsilon^{-1} \tByo {\beta_1} {\beta_2} = \tByo {A_{r-1}} {A_r} = \tByt {-p_{r-1}} {- p_r} {q_{r-1}} {q_r} \tByo {\beta_1} {\beta_2} , $$ from which it follows that the matrix of $\epsilon$ is given by \begin{equation} \lbl{eq.ea} \tByt {q_r} {p_r} {-q_{r-1}} {-p_{r-1}} = \tByt { b_0 q - p} {q} {- b_0 p' + (p p' -1)/q} {- p'} = \tByt {b_0} {-1} {1} {0} \dots \tByt {b_{r-1}} {-1} {1} {0} \end{equation} where \begin{equation} \lbl{eq.pq} \frac{q}{p} \overset{\text{def}}{=} [b_1, \dots, b_{r-1}], \end{equation} (with the understanding that $q=1, p=0$ if $r=1$), $ p'=\text{numerator} [b_1, \dots, b_{r-2}] $, which is a multiplicative inverse of $p$ mod $q$ (with the understanding that $p'=0$ (resp. $1$) if $r=1$ (resp. $2$)). \begin{lemma} \lbl{claim.b} The orbit space $M_b/U_b$ has as a fundamental domain a semiopen two-dimensional simplicial cone whose closure is the cone $\langle A_0, A_r \rangle$ in $K_b$. Furthermore, the (closed) cone $\tau_b=\langle A_0, A_r \rangle$ in $M_b$ is of type $(-p,q)$ and the dual cone $\sigma_b$ in $N_b$ is of type $(p,q)$ where $ 0 \leq p < q$ as in equation \eqref{eq.pq} above. \end{lemma} \begin{proof} >From Figure \ref{Aray}, it follows that a fundamental domain of $M_b/U_b$ is a semiopen two-dimensional cone whose closure is the cone $\tau_b =\langle A_r, A_0 \rangle$ in $M_b$. The cone $\langle A_r, A_0 \rangle$ is canonically subdivided into nonsingular cones $\langle A_{i+1}, A_i \rangle$ (for $i=0, \dots, r-1$), and using equation \eqref{eq.Ar}, it follows that $\tau_b$ is of type $(c_1, q)$ where $0 \leq c_1 < q$ and $ \frac{q}{q - c_1} = [b_1, \dots, b_{r-1}]$. Therefore the dual cone $\sigma_b$ in $N$ is of type $(p,q)$ where $p= q-c_1$. Thus $(p,q)$ satisfies equation \eqref{eq.pq}. \end{proof} \begin{figure}[htpb] $$ \eepic{Aray}{0.03} $$ \caption{A fundamental domain of $M_b/U_b$.}\lbl{Aray} \end{figure} Since $M_b \subseteq \mathbb Q(\sqrt{D_b})$ is a rank two lattice of a real quadratic field, we can define a homogenous function $Q_b: M_{\mathbb R} \to \mathbb R$ by $$Q_b(x w_0 + y)= C_b x^2 - B_b xy + A_b y^2 $$ This agrees with the (normalized) norm function of the real quadratic field $\mathbb Q(\sqrt{D_b})$ considered by Zagier \cite[p 138]{Za2}. We close this section with the following: \begin{proposition} \lbl{prop.qij} Let $l,m\in\mathbb Z$ with $l\ne m$. Then the coefficients of $x^2, xy$ and $y^2$ of the quadratic form $Q_b(x A_l + y A_m)$ are given by polynomials in $b_i$ with integer coefficients. \end{proposition} \begin{proof} We first claim that in terms of the basis $\{ A_r, A_0 \}$ we have: \begin{equation} \lbl{eq.change} Q_b(x A_r + y A_0)= q Q_{\Theta}(x,y) \end{equation} where $Q_{\Theta}(x,y)= x^2 + \Theta xy + y^2$ and $\Theta= b_0 q - p - p'$. The above claim follows from the fact that $A_b, B_b, C_b$ can be calculated (using their definition) in terms of the $b_i$ as follows: \begin{equation*} A_b = q, \text{ \ \ \ } B_b = - b_0 q + p - p', \text{ and } C_b = q_{r-1}= b_0 p' +(1-p p')/q, \end{equation*} together with a change of variables formula from $\{ A_{-1}, A_0 \}$ to $\{ A_{r}, A_0 \}$ given by equation \eqref{eq.aaa}. We then claim that if $l,m$ are distinct integers, then \begin{equation} \lbl{eq.qlm} Q_b(x A_l + y A_m)= \frac{1}{q} Q_\Theta(H_l x + H_m y, K_l x + K_m y) \end{equation} where $\displaystyle{ \qquad \frac{ h_i}{ k_i} = [ b_1, \dots, b_{i-1}], \qquad H_i x + K_i y \overset{\text{def}}{=} X_i \overset{\text{def}}{=} - h_i x + (q k_i -p h_i)y. }$ This follows from equation \eqref{eq.change} using the change of variables given in equation \eqref{eq.aaa}. We now claim that the coefficients of $x^2, xy $ and $y^2$ in equation \eqref{eq.qlm} are polynomials in the $b_i$ with integer coefficients. Indeed, observe that $ h_i , k_i$ are polynomials in $b_i$ with integer coefficients. Since $H_i=- h_i, K_i = q k_i -p h_i = -p h_i \bmod q$, it follows that: \begin{eqnarray*} Q_\Theta(H_l x + H_m y, K_l x + K_m y) & = & ( h_l x + h_m y)^2 + \Theta ( h_l x + h_m y)(p h_l x + p h_m y) \\ & & + (p h_l x + p h_m y)^2 \bmod q \\ & = & ( h_l x + h_m y)^2(1 + \Theta p + p^2) \bmod q. \end{eqnarray*} Since $\Theta = q b_0 - p - p'$, and $p p' \equiv 1 \bmod q$, it follows that $1 + \Theta p + p^2 = 0 \bmod q$, which finishes the proof of the proposition. \end{proof} \begin{remark} \lbl{rem.compare} The notation of Theorem \ref{thm.toddd} (and its proof, in Section \ref{sub.proof12}) corresponds to the notation of this section. Indeed, the cone $\sigma_b$ in $N$ is of type $(p,q)$ where $p,q$ are given in terms of equation \eqref{eq.pq}. The rays $ \rho_i$ (for $i=1, \dots r-1$) that desingularize the cone $\tau_b$ in $M$ are given by $ \rho_i = A_i$. In addition, the $ h_i, k_i, X_i$ that appear in Theorem \ref{thm.toddd} match the notation Proposition \ref{prop.qij}. \end{remark} \subsection{Proof of Theorem \ref{cor.zv}} \lbl{sub.explicit} \begin{proof} The main idea is to use Theorem \ref{thm.1} which calculates the zeta values in terms of the Todd operator of $\sigma_b$, and Theorem \ref{thm.toddd} which expresses the Todd operator in terms of the $b_i$. The expression that we obtain for the zeta values differ from the one of equation \eqref{eq.zva} by an error term, which vanishes identically, as one can show by an explicit calculation. Now, for the details, we follow the notation of Section \ref{sub.real}. We begin by calculating the integral $\int_{\tau_b(h_1, h_2)} e^{-Q(u) } du$. Using the parametrization $\mathbb R^2 \to M_{\mathbb R}$ given by: $(w_1,w_2) \to w_1A_r+w_2A_0$, it follows that the preimage of $\tau_b(x, y)$ in $\mathbb R^2$ is given by $\{ (w_1, w_2) | w_1 \geq -x/q, w_2 \geq -y/q \}$. Thus we have: \begin{eqnarray*} \int_{\tau_b(x, y)} e^{-Q_b(u) } du & = & q \int_{w_1= -x/q}^{\infty} \int_{w_2= -y/q}^{\infty} e^{-Q_b(q w_1, -p w_1 + w_2)} d w_2 d w_1 \end{eqnarray*} Differentiating, we get \begin{eqnarray*} \left(\frac{\partial}{\partial x}\right) \left(\frac{\partial}{\partial y}\right) \diamond \int_{\tau_b(x, y)} e^{-Q_b(u) } du & = \frac{1}{q} e^{-Q_b(- x A_r/q - y A_0/q)} = \frac{1}{q} e^{-1/q Q_{\Theta}(x, y)} \end{eqnarray*} Using the following elementary identity: $$ \left(\alpha \pd {\bar x} + \beta \pd {\bar y} \right)^i \left(\gamma \pd {\bar x} + \delta \pd {\bar y} \right)^j \Big|_{\substack{ \bar x = \alpha a + \gamma b \\ \bar y = \beta a + \delta b }} f( \bar x, \bar y)= \left(\pd x \right)^i \left(\pd y \right)^j \Big|_{\substack {x=a \\ y=b}} f(\alpha x + \gamma y, \beta x + \delta y) $$ (and temporarily abbreviating $\pd x$ by $x$) we obtain that \begin{eqnarray*} \left(q xy X_l^a X_m^b \right) \diamond \int_{\tau_b(x, y)} e^{-Q_b(u) } du & = & X_l^a X_m^b \diamond e^{-1/q Q_{\Theta}(x, y)} \\ & = & \left(\pd x\right)^a\left(\pd y\right)^b \diamond e^{-1/q Q_\Theta(H_l x + H_m y, K_l x +K_m y)} \\ & = & \left(\pd x\right)^a\left(\pd y\right)^b \diamond e^{-Q_b(x A_l + y A_m)}, \quad\text{ as well as} \end{eqnarray*} \begin{multline*} (x X_1^{2n+1} + y X_{r-1}^{2n+1}) \diamond \int_{\tau_b(x, y)} e^{-Q_b(u) } du = \\ \frac{1}{q^{n+1}} \int_{0}^{\infty} \left( \left(\pd x + p \pd y \right)^{2n+1} + \left(\pd x + p' \pd y\right)^{2n+1} \right) \big|_{x=0} e^{-Q_{\Theta}(x,y)} dy \end{multline*} The above, together with Theorem \ref{thm.toddd}, implies that: \begin{eqnarray*} \zeta_{Q_{b}, \tau_b}(-n) & = & (-1)^n n! \left\{ \sum_{i=1}^r P_{2n+2} \left(\pd x , \pd y \right) \diamond e^{- Q_b( x A_{i-1} + y A_i)} \right. \\ & & \left. + \lambda_{2n+2} \sum_{i=1}^{r} b_i R_{2n+2}\left(\pd x , \pd y \right) \diamond e^{- Q_b( x A_{i-1} + y A_{i+1})} + E_{2n+2}(b) \right\} \end{eqnarray*} where \begin{eqnarray*} E_{2n+2}(b) & = & \lambda_{2n+2} \left\{ -b_0 R_{2n+2}\left(\pd x , \pd y \right) e^{-1/q(L x^2 + M xy + N y^2)} \right. \\ & & \left. -\frac{1}{q^{n+1}} \int_{0}^{\infty} \left( \left(\pd x + p \pd y\right)^{2n+1} + \left(\pd x + p' \pd y\right)^{2n+1} \right) \big|_{x=0} e^{-Q_{\Theta}(x,y)} dx \right\} \end{eqnarray*} and $$ L = q p' b_0 + 1 - p p' , \qquad M = q b_0 \Theta_b +2(pp' -1), \qquad N = q p b_0 + 1 - p p'. $$ \newline Fixing $r$, the length of the sequence $(b_0, \dots, b_{r-1})$, Lemma \ref{lem.zag} below can be used to express $E_{2n+2}(b)$ as a polynomial in $1/q, p, p', b_0$. An explicit but lengthy calculation implies that $E_{2n+2}(b)=0$ for any $b$. We have thus succeeded in expressing $\zeta_{Q_{b}, \sigma_b}(-n)$ as a polynomial in $b_i$ with rational coefficients, symmetric under cyclic permutation, (since $Q_b(x A_{l} + y A_m)$ is symmetric under cyclic permutation). This concludes the proof of Theorem \ref{cor.zv}. Equation \eqref{eq.z0} follows easily using $\lambda_1= 1/2, \lambda_2=1/12$. \end{proof} \begin{lemma}\cite{Za2} \lbl{lem.zag} For every $i,j \geq 0$ with $i+j$ even, we have: \begin{equation*} \left(\frac{\partial}{\partial x}\right)^i \left(\frac{\partial}{\partial y}\right)^j \diamond e^{-(a x^2 + b xy + c y^2)}= i! j! (-1)^{(i+j)/2} a^{i/2} c^{j/2} \sum_{i_2=0; i_2 \equiv i \bmod 2}^{min \{i,j \}} \frac{ (a^{-1/2} b c^{-1/2})^{i_2}}{((i-i_2)/2)! i_2 ! ((j -i_2)/2)!} \end{equation*} Furthermore, for every $n \geq 1$ we have: \begin{equation*} \int_{y=0}^{\infty} \left(\frac{\partial}{\partial x}\right)^{2n-1} \diamond e^{-(a x^2 + b xy + c y^2)} d y = - \frac{(2n-1)!}{2 c^{n}} \sum_{r=0}^{n-1} (-1)^r \frac{(n-1-r)!}{r! (2n-1-2r)!} a^r b^{2n-1-2r} c^r \end{equation*} \end{lemma} \section{Dedekind sums in terms of Todd power series} \lbl{sec.dedsum} In this section we give a proof of Theorem \ref{thm.f=s}. To do this we will use Proposition \ref{prop.j2}, which can be used to express the coefficients $f_{i,j}$ as a sum of rational numbers, which turn out to equal products of certain values of the Benoulli polynomials. (Note, in contrast, that Proposition \ref{prop.j3} expresses this same number in terms of roots of unity instead of rational numbers.) Let $p$ and $q$ be as in the statement of Theorem \ref{thm.f=s}, and let $\sigma$ be the cone $\langle (1,0),(p,q) \rangle$ in $\mathbb Z^2$. We let $\rho_1=(1,0)$, and $\rho_2=(p,q)$ denote the generators of this cone. The left hand side of Theorem \ref{thm.f=s}, $f_{i,j}(p,q)$, equals the coefficient of $x^iy^j$ in the power series $\mathfrak t_{\sigma}(x,y)$. We now compute this power series using Proposition \ref{prop.j2}. This proposition contains an expansion for the power series $\mathfrak s_{\sigma}$, which is equivalent to the following expansion of $\mathfrak t_{\sigma}$: $$ \mathfrak t_{\sigma}(x,y)=qxy\sum_{m\in\check\sigma\cap M} e^{-(\langle m,\rho_1 \rangle x + \langle m,\rho_2 \rangle y )}. $$ Every point of $\check\sigma\cap M$ can be written uniquely as a nonnegative integral combination of the generators $u_1=(q,-p)$ and $u_2=(0,1)$ of $\check\sigma$, plus a lattice point in the semiopen parallelepiped $$ P=\{cu_1+du_2 | c,d\in [0,1) \}. $$ Using $\langle u_1,\rho_1 \rangle =\langle u_2,\rho_2 \rangle = q$, it follows that $$ \mathfrak t_{\sigma}(x,y)=qxy \frac1{1-e^{-qx}}\frac1{1-e^{-qy}} \sum_{m\in P\cap M} e^{-(\langle m,\rho_1 \rangle x + \langle m,\rho_2 \rangle y )}. $$ One finds also that $$P=\biggl\{(k,\{\frac{pk}{q}\}-\frac{pk}{q}) : k=0,\dots,q-1 \biggr\}, $$ where $\{x\}\in [0,1)$ denotes the fractional part of $x$. (Note this is slightly different from $\langle x \rangle\in (0,1]$, which appears in the definition of $s_{i,j}$: by definition, $\langle 0 \rangle=1$, while $\{0\}=0$.) One then finds that $$ \mathfrak t_{\sigma}(x,y)=qxy \frac1{1-e^{-qx}}\frac1{1-e^{-qy}} \sum_{k=0}^{q-1} e^{-q( \frac{k}{q}x +\{\frac{kp}{q}\} y )}. $$ We may then compute $f_{i,j}(p,q)$ as the coefficient of $x^iy^j$ in the above expression. It is convenient to replace $x$ and $y$ with $-x$ and $-y$, which introduces a factor of $(-1)^{i+j}$. We obtain $$ f_{i,j}(p,q)=(-1)^{i+j}q^{i+j-1} \sum_{k=0}^{q-1} \text{coeff}\biggl(x^i; \frac{xe^{\frac{k}{q}x}}{1-e^x}\biggr) \text{coeff}\biggl(y^i; \frac{ye^{\{\frac{kp}{q}\}y}}{1-e^y}\biggr). $$ It is then clear that we can write the above sum in terms of values of the Bernoulli polynomials, as follows: $$ f_{i,j}(p,q)=(-1)^{i+j}q^{i+j-1} \sum_{k=0}^{q-1} \text{Ber}_i\biggl(\frac{k}{q}\biggr) \text{Ber}_j\biggl(\{\frac{kp}{q}\}\biggr). $$ Using the identity $$ \text{Ber}_j(\lambda)=(-1)^j \text{Ber}_j(1-\lambda), $$ we may rewrite our expression as $$ f_{i,j}(p,q)=(-1)^{i}q^{i+j-1} \sum_{k=0}^{q-1} \text{Ber}_i\biggl(\frac{k}{q}\biggr) \text{Ber}_j\biggl(\langle -\frac{kp}{q} \rangle\biggr). $$ The sum on the right hand side is easily seen to equal the sum defining $s_{i,j}(p,q)$, except for a possible discrepancy in the $k=0$ term. It follows easily from the definitions that the $k=0$ terms actually match unless $i=j=1$, or $i=1$ and $j$ is even, or $j=1$ and $i$ is even. In these cases, we need the correction terms which appear in the statement of the Theorem. \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9705

alg-geom/9705013

en

https://arxiv.org/abs/alg-geom/9705013

alg-geom/9705013

Frank Sottile

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

Identities of Structure Constants for Schubert polynomials and Orders on S_n

11 pages, Latex2e, 7 figures, uses epsf.sty. Summary of alg-geom/9703001 for an audience of combinatorialists. To appear in conference abstracts of FPSAC'97 (Formal Power Series and Algebraic Combinatorics, Vienna, July, 1997)

We use geometry to prove a number of new identities among the Littlewood-Richardson coefficients for Schubert polynomials (Schubert classes in a flag manifold). For many of these identities, there is a companion result about the Bruhat order which should imply the identity, were it known how to express these coefficients in terms of the Bruhat order. This analysis leads the determination of many of these constants. We conclude with an outline of geometric proofs for these identities.

alg-geom

\section*{Introduction} Extending work of Demazure~\cite{Demazure} and of Bernstein, Gelfand, and Gelfand~\cite{BGG}, in 1982 Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert} defined remarkable polynomial representatives for Schubert classes in the cohomology of a flag manifold, which they called Schubert polynomials. For each permutation $w$ in ${\mathcal S}_\infty$, the group of permutations of ${\mathbb N} :=\{1,2,\ldots\}$ which fix all but finitely many numbers, there is a Schubert polynomial ${\mathfrak S}_w\in{\mathbb Z}[x_1,x_2,\ldots]$. The collection of all Schubert polynomials forms an additive basis for this polynomial ring. Thus the identity \begin{equation}\label{eq:structure} {\mathfrak S}_u \cdot {\mathfrak S}_v \quad =\quad \sum_w c^w_{u\, v} {\mathfrak S}_w \end{equation} defines integral {\em structure constants} $c^w_{u\, v}$ for the ring of polynomials with respect to its Schubert basis. Littlewood-Richardson coefficients are a special case of the $c^w_{u\, v}$ as any Schur symmetric polynomial is a Schubert polynomial. The $c^w_{u\, v}$ are non-negative integers: Evaluating a Schubert polynomial at certain Chern classes gives a Schubert class in the cohomology of the flag manifold. Hence, $c^w_{u\, v}$ enumerates the flags in a suitable triple intersection of Schubert varieties. It is an open problem to give a combinatorial interpretation or a bijective formula for these constants. All known formulas express $c^w_{u\, v}$ in terms of chains in the Bruhat order. For instance, the Littlewood-Richardson rule~\cite{Littlewood_Richardson}, a special case, may be expressed in this form. Other formulas for these constants, particularly Monk's formula~\cite{Monk}, Pieri-type formulas~\cite{Lascoux_Schutzenberger_polynomes_schubert,% sottile_pieri_schubert,Winkel_multiplication,Kirillov_Maeno}, and the other formulas of~\cite{sottile_pieri_schubert}, are all of this form. For quantum Schubert polynomials~\cite{Fomin_Gelfand_Postnikov,Ciocan_partial}, the Pieri-type formulas~\cite{Ciocan_partial,Postnikov} are also of this form. We present a number of new identities among the $c^w_{u\,v}$ which are consistent with the expectation that they can be expressed in terms of chains in the Bruhat order. In addition, we give a formula (Theorem~\ref{thm:skew_shape}) for many of the $c^w_{u\,v}$ when ${\mathfrak S}_v$ is a symmetric polynomial. These identities impose stringent conditions on any combinatorial interpretation for the coefficients. They also point to some potentially beautiful combinatorics once such an interpretation is known. These results are expanded on and proven in a manuscript, ``{S}chubert polynomials, the {B}ruhat order, and the geometry of flag manifolds''~\cite{bergeron_sottile_symmetry}. For a background on Schubert polynomials, we recommend the original papers~\cite{Lascoux_Schutzenberger_polynomes_schubert,% Lascoux_Schutzenberger_structure_de_Hopf,% Lascoux_Schutzenberger_symmetry,% Lascoux_Schutzenberger_interpolation,% Lascoux_Schutzenberger_schub_LR_rule,% Lascoux_Schutzenberger_operators}, the survey~\cite{Lascoux_historique}, or the book~\cite{Macdonald_schubert}. For their relation to geometry, we recommend the book~\cite{Fulton_tableaux}. \section{Chains in the Bruhat order}\label{sec:suborders} Let $(a,b)$ denote the transposition interchanging $a<b$. The {\em Bruhat order} $\leq $ on the symmetric group ${\mathcal S}_n$ is defined by its covers: $$ u\ \lessdot\ u(a,b) \quad \Longleftrightarrow\quad \ell(u) + 1 \ =\ \ell(u(a,b)). $$ It also appears as the index of summation in Monk's formula~\cite{Monk}: $$ {\mathfrak S}_u \cdot {\mathfrak S}_{(k,\,k{+}1)} \quad =\quad {\mathfrak S}_u \cdot (x_1+\cdots+x_k) \quad =\quad \sum {\mathfrak S}_{u(a, b)}, $$ the sum over all $a\leq k<b$ where $\ell(u(a,b))=\ell(u)+1$. This suggests the following notion, which appeared in~\cite{Lascoux_Schutzenberger_symmetry}. A {\em coloured chain} is a (saturated) chain in the Bruhat order together with an element of $\{a,a+1,\ldots,b-1\}$ for each cover $u\lessdot u(a,b)$ in that chain. Let $I$ be any subset of ${\mathbb N}$. An {\em $I$-chain} is a coloured chain whose colours are chosen from the set $I$. If $u\leq w$ in the Bruhat order, let $f^w_u(I)$ count the $I$-chains from $u$ to $w$. Iterating Monk's formula, we obtain: $$ \left(\sum_{i\in I}\:{\mathfrak S}_{(i,i{+}1)}\right)^m \quad =\quad \sum_v\, f^v_e(I)\; {\mathfrak S}_v. $$ Multiplying this expression by ${\mathfrak S}_u$, expanding the products using (\ref{eq:structure}) and Monk's formula, and equating coefficients of ${\mathfrak S}_w$, we obtain: \begin{thm}\label{thm:chains} Let $u,w\in{\mathcal S}_\infty$ and $I\subset{\mathbb N}$. Then $$ f^w_u(I)\quad =\quad \sum_v\, c^w_{u\,v}\, f^v_e(I). $$ \end{thm} This number, $f^v_e(I)$, is non-zero precisely when $v$ is minimal in its coset $vW_{\overline{I}}$, where $W_{\overline{I}}$ is the parabolic subgroup~\cite{Bourbaki_Groupes_IV} of ${\mathcal S}_\infty$ generated by the transpositions $\{(i,i{+}1)\:|\: i\not\in I\}$. We say that $u$ is comparable to $w$ in the {\em $I$-Bruhat order} if there is an $I$-chain from $u$ to $w$. In \S\S\ref{sec:k-bruhat} and~\ref{sec:Schur_identities}, we consider this order when $I=\{k\}$. Any eventual combinatorial interpretation of the constants $c^w_{u\,v}$ should give a bijective proof of Theorem~\ref{thm:chains}. We expect there will be a combinatorial interpretation of the following form: Let $u,v,w\in{\mathcal S}_\infty$, and $I\subset{\mathbb N}$ be such that $v$ is minimal in $vW_{\overline{I}}$. (There always is such an $I$.) Then, for any $I$-chain $\gamma$ from $e$ to $v$, $$ c^w_{u\, v}\quad =\quad\# \left\{ \mbox{\begin{minipage}[c]{2.15in} $I$-chains from $u$ to $w$ satisfying \,{\em some} condition \,imposed by \,$\gamma$ \end{minipage}} \right\}. $$ \section{The $k$-Bruhat order}\label{sec:k-bruhat} The $k$-Bruhat order, $\leq_k$, is the $\{k\}$-Bruhat order of \S\ref{sec:suborders}. It has another description: \begin{thm}\label{thm:k-length} Let $u,w\in {\mathcal S}_\infty$ and $k\in{\mathbb N}$. Then $u\leq _k w$ if and only if \begin{enumerate} \item[I.] $a\leq k < b$ implies $u(a)\leq w(a)$ and $u(b)\geq w(b)$. \item[II.] If $\/a<b$, $u(a)<u(b)$, and $w(a)>w(b)$, then $a\leq k< b$. \end{enumerate} \end{thm} Considering covers shows conditions I and II are necessary. Sufficiency follows from a greedy algorithm: \begin{alg}[Produces a chain in the $k$-Bruhat order]\label{alg:chain} \mbox{ } \noindent{\tt input: }Permutations $u,w\in {\mathcal S}_\infty$ satisfying conditions I and II of Theorem~\ref{thm:k-length}. \noindent{\tt output: }A chain in the $k$-Bruhat order from $w$ to $u$. Output $w$. While $u\neq w$, do \begin{enumerate} \item[1] Choose $a\leq k$ with $u(a)$ minimal subject to $u(a)< w(a)$. \item[2] Choose $b>k$ with $u(b)$ maximal subject to $w(b)<w(a)\leq u(b)$. \item[3] (Then $w(a,b)\lessdot_k w$.) Set $w:=w(a,b)$, output $w$. \end{enumerate} At every iteration of\/ {\rm 1}, $u,w$ satisfy conditions I and II of Theorem~\ref{thm:k-length}. Moreover, this algorithm terminates in $\ell(w)-\ell(u)$ iterations and the sequence of permutations produced is a chain in the $k$-Bruhat order from $w$ to $u$. \end{alg} Observe that Algorithm~\ref{alg:chain} may be stated in terms of the permutation $\zeta:=w u^{-1}$: \medskip \noindent{\tt input: }{\it A permutation $\zeta\in {\mathcal S}_\infty$.} \noindent{\tt output: }{\it Permutations $\zeta,\zeta_1,\ldots,\zeta_m=e$ such that if $u\leq_k \zeta u$, then $$ u\ \lessdot_k\ \zeta_{m-1}u\ \lessdot_k\ \cdots\ \lessdot_k\ \zeta_1u\ \lessdot_k\ \zeta u $$ is a saturated chain in the $k$-Bruhat order. Output $\zeta$. While $\zeta\neq e$, do \begin{enumerate} \item[1] Choose $\alpha$ minimal subject to $\alpha< \zeta(\alpha)$. \item[2] Choose $\beta$ maximal subject to $\zeta(\beta)<\zeta(\alpha)\leq\beta$. \item[3] $\zeta:=\zeta(\alpha,\beta)$, output $\zeta$. \end{enumerate}} To see this is equivalent to Algorithm~\ref{alg:chain}, set $\alpha=u(a)$ and $\beta=u(b)$ so that $w(a)=\zeta(\alpha)$ and $w(b)=\zeta(\beta)$. Thus $w(a, b) = \zeta u(a, b) = \zeta(\alpha,\beta) u$. This observation is generalised considerably in Theorem~\ref{thm:B} ({\em i}) below. \section{Identities when ${\mathfrak S}_v$ is a Schur polynomial}\label{sec:Schur_identities} The Schur symmetric polynomial $S_\lambda(x_1,\ldots,x_k)$ is the Schubert polynomial ${\mathfrak S}_{v(\lambda,k)}$, where $v(\lambda,k)$ is a {\em Grassmannian permutation}, a permutation with unique descent at $k$. Here $\lambda_{k+1-j} = v(j)-j$ and $v(\lambda,k)$ is minimal in its coset $v(\lambda,k) W_{\overline{\{k\}}}$. Consider the constants $c^w_{u\:v(\lambda,k)}$ which are defined by the identity: $$ {\mathfrak S}_u \cdot S_\lambda(x_1,\ldots,x_k) \quad =\quad \sum_{w} c^w_{u\:v(\lambda,k)}{\mathfrak S}_w. $$ By Theorem~\ref{thm:chains}, the $c^w_{u\:v(\lambda,k)}$ are related to the enumeration of chains in the $k$-Bruhat order. These $c^w_{u\:v(\lambda,k)}$ share many properties with Littlewood-Richardson coefficients: If $\lambda, \mu$, and $\nu$ are partitions with at most $k$ parts, then the Littlewood-Richardson coefficients $c^\nu_{\mu\,\lambda}$ are defined by the identity $$ S_\mu(x_1,\ldots,x_k)\cdot S_\lambda(x_1,\ldots,x_k) \quad = \quad \sum_\nu c^\nu_{\mu\,\lambda} S_\nu(x_1,\ldots,x_k). $$ The $ c^\nu_{\mu\,\lambda}$ depend only upon $\lambda$ and the skew partition $\nu/\mu$. That is, if $\kappa$ and $\rho$ are partitions with at most $l$ parts and $\kappa/\rho = \nu/\mu$, then for any partition $\lambda$, $$ c^\nu_{\mu\,\lambda} \quad=\quad c^\kappa_{\rho\,\lambda}. $$ Moreover, $c^\kappa_{\rho\,\lambda}$ is the coefficient of $S_\kappa(x_1,\ldots,x_l)$ when $S_\rho(x_1,\ldots,x_l)\cdot S_\lambda(x_1,\ldots,x_l)$ is expressed as a sum of Schur polynomials. The order type of the interval in Young's lattice from $\mu$ to $\nu$ is determined by $\nu/\mu$. If $u\leq_k w$, let $[u,w]_k$ be the interval between $u$ and $w$ in the $k$-Bruhat order. Permutations $\zeta$ and $\eta$ are {\em shape equivalent} if there exist sets of integers $P=\{p_1<\cdots<p_n\}$ and $Q=\{q_1<\cdots<q_n\}$, where $\zeta$ (respectively $\eta$) acts as the identity on ${\mathbb N}- P$ (respectively ${\mathbb N}- Q$), and $$ \zeta(p_i)\ =\ p_j \quad \Longleftrightarrow\quad \eta(q_i)\ =\ q_j. $$ \begin{thm}[Skew Coefficients]\label{thm:B} Suppose $u\leq_kw$ and $x\leq_l z$ where $wu^{-1}$ is shape equivalent to $zx^{-1}$. Then \begin{enumerate} \item[({\em i})] $[u,w]_k\simeq[x,z]_l$. When $wu^{-1}=zx^{-1}$, this isomorphism is given by $v\mapsto vu^{-1}x$. \item[({\em ii})] For any partition $\lambda$ with length at most the minimum of $l$ and $k$, $$ c^w_{u\,v(\lambda,k)} = c^z_{x\,v(\lambda,l)}. $$ \end{enumerate} \end{thm} By Theorem~\ref{thm:B} ({\em ii}), we may define the {\em skew coefficient} $c^\zeta_\lambda$ for $\zeta\in {\mathcal S}_\infty$ and $\lambda$ a partition by $c^\zeta_\lambda := c^{\zeta u}_{u\,v(\lambda,k)}$ for any $u \in {\mathcal S}_\infty$ with $u\leq_k \zeta u$. (There always is a $u$ and $k$ with $u\leq_k \zeta u$.) Moreover, $c^\zeta_\lambda$ depends only upon $\lambda$ and the shape equivalence class of $\zeta$. Theorem~\ref{thm:B} ({\em i}) is proven using combinatorial arguments. The key lemma is that if $u\lessdot_k(\alpha,\beta)u\leq_k w$ and $y\leq_k z$ with $wu^{-1}=zy^{-1}$, then $y\lessdot_k(\alpha,\beta)y\leq_k z$. The identity of Theorem~\ref{thm:B} ({\em ii}) is proven using geometric arguments ({\em cf.}~\S5): It follows from an equality of homology classes, which we show by explicitly computing the intersection of two Schubert varieties in a flag manifold and the image of that intersection under a projection to a Grassmannian. By Theorem~\ref{thm:B} ({\em i}), we may define a partial order $\preceq$ on ${\mathcal S}_\infty$ as follows: Set $\eta\preceq \zeta$ if there exists $u\in {\mathcal S}_\infty$ and $k$ such that $u\leq_k\eta u\leq_k\zeta u$. This partial order has a rank function defined by $|\zeta|:=\ell(\zeta u)-\ell(u)$, whenever $u\leq_k \zeta u$. Both the definition of $\preceq$ and Theorem~\ref{thm:B} ({\em i}) are illustrated by the following example: Let $\zeta = (24)(153)$ and $\eta=(35)(174)$. Then $\zeta$ nd $\eta$ are shape equivalent. Also $21345 \leq_2 45123 = \zeta\cdot 21345$ and $3215764\leq_3 5273461= \eta\cdot 3215764$. We illustrate the intervals $[21342,\,45123]_2$, $[3215764,\,5273461]_3$, and $[e,\zeta]_\preceq$ below $$\epsfxsize=5.in \epsfbox{fig1.eps}$$ The order and rank function may be defined independent of $u$ and $k$: Define $\mathrm{up}_\zeta:= \{\alpha\;|\; \zeta(\alpha)>\alpha\}$ and $\mathrm{down}_\zeta:=\{\alpha\;|\; \zeta(\alpha)<\alpha\}$. Then $\eta\preceq\zeta$ if and only if \begin{enumerate} \item $\alpha<\eta(\alpha)\Rightarrow \eta(\alpha)\leq \zeta(\alpha)$, \item $\alpha>\eta(\alpha)\Rightarrow \eta(\alpha)\geq \zeta(\alpha)$, and \item If $\alpha,\beta\in\mathrm{up}_\zeta$ or $\alpha,\beta\in\mathrm{down}_\zeta$ with $\alpha<\beta$ and $\zeta(\alpha)<\zeta(\beta)$, then $\eta(\alpha)<\eta(\beta)$. \end{enumerate} Similarly, $|\zeta|$ equals the difference of $\#\{(\alpha,\beta)\in \zeta(\mbox{up}_\zeta)\times\zeta(\mbox{down}_\zeta)\,|\, \alpha>\beta\}$ and $$ \begin{array}{c} \#\{(a,b)\in \mbox{up}_\zeta\times\mbox{down}_\zeta\,|\, a>b\} +\ \#\{(a,b)\in \mbox{up}_\zeta\times\mbox{up}_\zeta\,|\, a>b \mbox{ and } \zeta(a)<\zeta(b)\} \\ +\ \#\{(a,b)\in \mbox{down}\zeta\times\mbox{down}\zeta\,|\, a>b \mbox{ and } \zeta(a)<\zeta(b)\}. \end{array} $$ This new order is preserved by many group-theoretic operations: For $\zeta\in {\mathcal S}_n$, let $\overline{\zeta}:=w_0\zeta w_0$, conjugation by the longest element of ${\mathcal S}_n$. For $P: p_1<p_2<\cdots \subset{\mathbb N}$ and $\zeta\in{\mathcal S}_\infty$, define the homomorphism $\phi_P:{\mathcal S}_\infty \rightarrow {\mathcal S}_\infty$, by requiring that $\phi_P(\zeta)\in {\mathcal S}_\infty$ act as the identity on ${\mathbb N}-P$ and $\phi_P(\zeta)(p_i)=p_{\zeta(i)}$. Note that $\zeta$ and $\phi_P(\zeta)$ are shape equivalent. \begin{thm}\label{thm:new_order} Suppose $\zeta,\eta,\xi\in{\mathcal S}_\infty$. \begin{enumerate} \item[({\em i})] The restriction of $\preceq$ to Grassmannian permutations $v(\lambda,k)$ gives Young's lattice of partitions with at most $k$ parts. \item[({\em ii})] For $\eta\preceq\zeta$, the map $\xi\mapsto \xi\eta^{-1}$ induces an isomorphism $[\eta,\zeta]_\preceq \stackrel{\sim}{\longrightarrow} [e,\zeta\eta^{-1}]_\preceq$. \item[({\em iii})] For any infinite set $P\subset {\mathbb N}$, $\phi_P:{\mathcal S}_\infty \rightarrow {\mathcal S}_\infty$ is an injection of graded posets. \item[({\em iv})] The map $\eta\mapsto \eta\zeta^{-1}$ is an order reversing bijection between $[e,\zeta]_\preceq$ and $[e,\zeta^{-1}]_\preceq$. \item[({\em v})] The homomorphism $\zeta\mapsto \overline{\zeta}$ on ${\mathcal S}_n$ induces an automorphism of $({\mathcal S}_n,\preceq)$. \end{enumerate} \end{thm} These properties are easy consequences of the definitions. This order is studied further in~\cite{bergeron_sottile_order}. Figure~\ref{fig:new_order.S_4} shows $\preceq$ on ${\mathcal S}_4$. \begin{figure}[htb]\label{fig:new_order.S_4} $$\epsfxsize=4in \epsfbox{fig2.eps}$$ \caption{ $\preceq$ on ${\mathcal S}_4$} \end{figure} Some of these structure constants $c^w_{u\,v(\lambda,k)}$ may be expressed in terms of chains in the Bruhat order. If $u\lessdot_k u(a,b)$ is a cover in the $k$-Bruhat order, label that edge of the Hasse diagram with the integer $u(b)$. The {\em word} of a chain in the $k$-Bruhat order is the sequence of labels of edges in the chain. \begin{thm}\label{thm:skew_shape} Suppose $u\leq_k w$ and $wu^{-1}$ is shape equivalent to $v(\mu,l)\cdot v(\nu,l)^{-1}$, for some $l$ and partitions $\mu\subset\nu$. Then, for any partition $\lambda$ and standard Young tableau $T$ of shape $\lambda$, $$ c^w_{u\,v(\lambda,k)}\ =\ \#\left\{\begin{array}{cc}\mbox{Chains in $k$-Bruhat order from $u$ to $w$ whose word} \\\mbox{ has recording tableau $T$ under Schensted insertion} \end{array}\right\}. $$ \end{thm} Since a chain in the $k$-Bruhat order is a $\{k\}$-chain in the sense of \S\ref{sec:suborders}, Theorem~\ref{thm:skew_shape} gives a combinatorial proof of Theorem~\ref{thm:chains} when $wu^{-1}$ is shape equivalent to a skew partition. The key step is when $w$ and $u$ are Grassmannian permutations. In that case, symmetry of the Schensted algorithm reduces the theorem to showing the `diagonal word' of a tableau is Knuth-equivalent to its reading word. By the {\em diagonal word}, we mean the entries of a tableau read first by their diagonal, and then in increasing order in each diagonal. For instance, this tableau has diagonal word $7\,58\,379\,148\,26\,26\,5\,8$. $$ \epsfxsize=.9in \epsfbox{fig3.eps} $$ There are many permutations $u,w$ which do not satisfy the hypotheses of Theorem~\ref{thm:skew_shape}, but for which the conclusion holds. For example, any $u,w$ for which $wu^{-1}=(143652)$ satisfy the conclusion, but $(143652)$ is not shape equivalent to any skew partition. However, some hypotheses are necessary. Let $u=312645$ and $w=561234$. Here is the labeled Hasse diagram of $[u,w]_2$: $$ \epsfxsize=1.9in \epsfbox{fig4.eps} $$ There are six chains in this interval from which we obtain these recording tableaux: $$ \epsfxsize=4.3in \epsfbox{fig5.eps}\ \raisebox{6pt}{.} $$ This list omits the tableau \begin{picture}(17,17)(0,2) \put(0,0){\line(0,1){17}} \put(0,0){\line(1,0){17}}\put(8.5,0){\line(0,1){17}} \put(0,8.5){\line(1,0){17}}\put(17,0){\line(0,1){17}} \put(0,17){\line(1,0){17}} \put(1.5,1.5){\scriptsize 1} \put(10,1.5){\scriptsize 2} \put(1.5,10){\scriptsize 3} \put(10,10){\scriptsize 4} \end{picture}, and the third and fourth tableaux are identical. \smallskip We calculate $c^w_{u,v(\lambda,2)}$ using Theorem~5 of~\cite{sottile_pieri_schubert}, or Theorem 4 (1) of~\cite{sottile_geometry_schubert}: $$ c^w_{u\; v(\begin{picture}(12,3) \put(0,0){\line(0,1){3}} \put(0,0){\line(1,0){12}} \put(3,0){\line(0,1){3}} \put(0,3){\line(1,0){12}} \put(6,0){\line(0,1){3}} \put(9,0){\line(0,1){3}} \put(12,0){\line(0,1){3}} \end{picture}\,,2)}\ =\ c^w_{u\; v(\,\begin{picture}(9,6) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){9}} \put(3,0){\line(0,1){6}} \put(0,3){\line(1,0){9}} \put(6,0){\line(0,1){3}} \put(0,6){\line(1,0){3}} \put(9,0){\line(0,1){3}} \end{picture}\,,2)}\ =\ c^w_{u\; v(\,\begin{picture}(6,6)(0,0) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){6}} \put(3,0){\line(0,1){6}} \put(0,3){\line(1,0){6}} \put(6,0){\line(0,1){6}} \put(0,6){\line(1,0){6}} \end{picture}\,,2)}\ =\ 1. $$ If a skew Young diagram $\kappa$ is the disjoint union of two incomparable diagrams $\rho$ and $\theta$, then $$ c^\kappa_\lambda\ =\ \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}c^\rho_\mu c^\theta_\nu. $$ Similarly, we say that a product $\zeta\cdot\eta$ is {\em disjoint} if the two permutations $\zeta$ and $\eta$ have disjoint supports and $|\zeta\cdot\eta| = |\zeta|+ |\eta|$. We have: \begin{thm}[Disjointness] Suppose $\zeta\cdot\eta$ is disjoint. Then \begin{enumerate} \item The map $(\alpha,\beta)\mapsto \alpha\cdot\beta$ induces an isomorphism $[e,\zeta]_\preceq\times[e,\eta]_\preceq \stackrel{\sim}{\longrightarrow}[e,\zeta\cdot\eta]_\preceq$. \item For all $\lambda$, \ ${\displaystyle c^{\zeta\cdot\eta}_\lambda\ =\ \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}c^\zeta_\mu c^\eta_\nu}$. \end{enumerate} \end{thm} The next identity has no analogy with the classical Littlewood-Richardson coefficients. Let $(1\,2\,\ldots\, n)$ be the permutation which cyclicly permutes $[n]$. \begin{thm}[Cyclic Shift]\label{thm:D} Suppose $\zeta\in S_n$ and $\eta = \zeta^{(1\,2\,\ldots\,n)}$. Then, for any partition $\lambda$, $c^\zeta_\lambda = c^\eta_\lambda$. \end{thm} This is proven using geometric arguments similar to those which establish Theorem~\ref{thm:B} ({\em ii}). Combined with Theorem~\ref{thm:chains}, we obtain: \begin{cor}\label{cor:equal_chains} If $u\leq_k w$ and $x\leq_k z$ with $w u^{-1},z x^{-1}\in{\mathcal S}_n$ and $(wu^{-1})^{(1\,2\,\ldots\,n)} = zx^{-1}$, then the two intervals $[u,w]_k$ and $[x,z]_k$ each have the same number of chains. \end{cor} It would be interesting to give a bijective proof of Corollary~\ref{cor:equal_chains}. The two intervals $[u,w]_k$ and $[x,z]_k$ of Corollary~\ref{cor:equal_chains} are typically non-isomorphic: In ${\mathcal S}_4$, let $u=1234$, $x=2134$, and $v=1324$. If $\zeta=(1243)$, $\eta=(1423)= \zeta^{(1234)}$, and $\xi=(1342)=\eta^{(1234)}$, then $$ u \ \leq_2\ \zeta u,\quad x\ \leq_2\ \eta x,\quad\mbox{and} \quad v\ \leq_2 \ \xi v. $$ We illustrate the intervals $[u,\zeta u]_2$, $[x,\eta x]_2$, and $[v,\xi v]_2$. $$\epsfxsize=4in \epsfbox{fig6.eps}$$ \section{Substitutions} The identities of \S\ref{sec:Schur_identities} require a more general study of the behaviour of Schubert polynomials under certain specializations of the variables. This leads to a number of new formulas and identities. For $w\in{\mathcal S}_{n+1}$ and $1\leq p\leq n+1$, let $w/_p\in {\mathcal S}_n$ be defined by deleting the $p$th row and $w(p)$th column from the permutation matrix of $w$. If $y\in{\mathcal S}_n$ and $1\leq q\leq n+1$, then $\varepsilon_{p,q}(y)\in {\mathcal S}_{n+1}$ is the permutation such that $\varepsilon_{p,q}(y)/_p = y$ and $\varepsilon_{p,q}(y)(p)=q$. The index of summation in a particular case of the Pieri-type formula~\cite{Lascoux_Schutzenberger_polynomes_schubert,% sottile_pieri_schubert,Winkel_multiplication}, $$ {\mathfrak S}_v \cdot (x_1\cdots x_{p-1}) \quad =\quad \sum_{v \stackrel{c_p}{\llra} w} {\mathfrak S}_w, $$ defines the relation $v\stackrel{c_p}{\llra} w$. More concretely, $v\stackrel{c_p}{\llra} w$ if and only if there is a chain in the $(p-1)$-Bruhat order: $$ v\ \lessdot_{p-1}\ (\alpha_1,\beta_1) v\ \lessdot_{p-1}\ \cdots\ \lessdot_{p-1}\ (\alpha_{p-1},\beta_{p-1})\cdots (\alpha_1,\beta_1) v \ =\ w $$ with $\beta_1>\beta_2>\cdots>\beta_{p-1}$. Define $\Psi_p:{\mathbb Z}[x_1,x_2,\ldots] \rightarrow {\mathbb Z}[x_1,x_2,\ldots]$ by $$ \Psi_p(x_j)\ =\ \left\{\begin{array}{ll} x_j&\mbox{ if } j<p\\ 0& \mbox{ if } j=p\\ x_{j-1}&\mbox{ if } j>p\end{array}\right.. $$ \begin{thm}\label{thm:A} Let $u,w\in {\mathcal S}_\infty$. Suppose $w(p)=u(p)$ and $\ell(w)-\ell(u)=\ell(w/_p)-\ell(u/_p)$, for some positive integer $p$. Then \begin{enumerate} \item[({\em i})] $\varepsilon_{p,u(p)} : [u/_p,w/_p] \stackrel{\sim}{\longrightarrow} [u,w]$. \item[({\em ii})] For any $v\in {\mathcal S}_\infty$, $$ c^w_{u\, v}\quad =\quad \sum_{\stackrel{\mbox{\scriptsize $y\in{\mathcal S}_\infty$}}% {v \stackrel{c_p}{\llra} \varepsilon_{p,1}(y)}} c^{w/_p}_{u/_p\: y}. $$ \item[({\em iii})] For any $v\in {\mathcal S}_\infty$, $$ \Psi_p({\mathfrak S}_v) = \sum_{\stackrel{\mbox{\scriptsize $y\in{\mathcal S}_\infty$}}% {v \stackrel{c_p}{\llra} \varepsilon_{p,1}(y)}} {\mathfrak S}_y. $$ \end{enumerate} \end{thm} The first statement is proven using combinatorial arguments, while the second and third are proven by computing certain maps on cohomology. Since $c^w_{u\,v}=c^w_{v\,u}=c^{w_0 u}_{v\:w_0w}$, Theorem~\ref{thm:A} ({\em ii}) gives a recursion for $c^w_{u\, v}$ when one of $wu^{-1}, wv^{-1}$, or $w_0uv^{-1}$ has a fixed point and the condition on lengths is satisfied. Theorem~\ref{thm:A} ({\em iii}) is both a generalization and a strengthening of the transition equations of~\cite{Lascoux_Schutzenberger_schub_LR_rule}. We give an example: Let $v=413652$. Consider the part of the labeled Hasse diagram in the 2-Bruhat order above $v$ with decreasing edge labels. Then the leaves are those $w$ with $v\stackrel{c_3}{\longrightarrow}w$. $$ \epsfxsize=2.2in \epsfbox{fig7.eps} $$ Of these, only the two underlined permutations are of the form $\varepsilon_{3,1}(y)$: $$ 631452\ =\ \varepsilon_{3,1}(52341)\quad\mbox{and}\quad 531642\ =\ \varepsilon_{3,1}(42531). $$ Thus Theorem~\ref{thm:A} ({\em iii}) asserts that $\Psi_3({\mathfrak S}_{413652}) = {\mathfrak S}_{52341} + {\mathfrak S}_{42531}.$ Indeed, \begin{eqnarray*} {\mathfrak S}_{413652} &=& x_1^4x_2x_4x_5 + x_1^3x_2^2x_4x_5 + x_1^3x_2x_4^2x_5 +\\ &\ & x_1^4x_2x_3x_4 + x_1^4x_2x_3x_5 + x_1^4x_3x_4x_5 + x_1^3x_2^2x_3x_4 + x_1^3x_2^2x_3x_5 + x_1^3x_2x_3^2x_4 +\\ &\ & x_1^3x_2x_3^2x_5 + x_1^3x_2x_3x_4^2 + x_1^3x_3^2x_4x_5 + x_1^3x_3x_4^2x_5 + 2\cdot x_1^3x_2x_3x_4x_5, \end{eqnarray*} so $$ \Psi_3({\mathfrak S}_{413652}) = x_1^4x_2x_3x_4 + x_1^3x_2^2x_3x_4 + x_1^3x_2x_3^2x_4. $$ Since $$ {\mathfrak S}_{52341}\ =\ x_1^4x_2x_3x_4\quad\mbox{ and}\quad {\mathfrak S}_{42531}\ =\ x_1^3x_2^2x_3x_4 + x_1^3x_2x_3^2x_4, $ we see that $$ \Psi_3({\mathfrak S}_{413652}) = {\mathfrak S}_{52341} + {\mathfrak S}_{42531}. $$ We also compute the effect of other substitutions of the variables in terms of the Schubert basis: For $P\subset {\mathbb N}$, set $P^c:= {\mathbb N}-P$ and list the elements of $P$ and $P^c$ in order: $$ P\ :\ p_1<p_2<\cdots\qquad \qquad P^c\ :\ p^c_1<p^c_2<\cdots $$ Define $\Psi_P:{\mathbb Z}[x_1,x_2,\ldots]\rightarrow {\mathbb Z}[y_1,y_2,\ldots, z_1,z_2,\ldots]$ by: $$ \Psi_P(x_{p_j}) \ =\ y_j\qquad\mbox{and}\qquad \Psi_P(x_{p^c_j}) \ =\ z_j. $$ \begin{thm}\label{thm:substitution} Let $P\subset {\mathbb N}$ be as above. Then there exists an (explicitly described) infinite set $\Pi_P\subset {\mathcal S}_\infty$ such that for any $w\in {\mathcal S}_n$ and $\pi\in I_P - {\mathcal S}_{3n}$, $$ \Psi_P({\mathfrak S}_w)\ =\ \sum_{u,\, v} c^{(u\times v)\cdot \pi}_{\pi\; w}\; {\mathfrak S}_u(y)\;{\mathfrak S}_v(z). $$ \end{thm} This generalizes~\cite[1.5]{Lascoux_Schutzenberger_structure_de_Hopf} (See also~\cite[4.19]{Macdonald_schubert}), where it is shown that the coefficients are nonnegative when $P=[n]$. Theorem~\ref{thm:substitution} gives infinitely many identities of the form $c^{(u\times v)\cdot \pi}_{\pi\; w} = c^{(u\times v)\cdot \sigma}_{\sigma\; w}$ for $\pi,\sigma\in \Pi_P$. Moreover, for these $u,v$, and $\pi$, we have $[\pi,\; (u\times v)\cdot \pi]\simeq [e,u]\times[e,\,v]$, which is suggestive of a chain-theoretic basis for these identities. \section{Outline of geometric proofs} Many of these results are proven with arguments from geometry. Our main technique is as follows: If $u,w\in {\mathcal S}_n$, then $$ c^w_{u\,v(\lambda,k)}\ =\ \#\left( X_{w_0w}\bigcap X'_u \bigcap X''_{v(\lambda,k)}\right), $$ where $X_{w_0w}$, $X'_u$, and $X''_{v(\lambda,k)}$ are Schubert varieties in general position in the manifold ${\mathbb F}\ell_n$ of complete flags in ${\mathbb C}^n$. We reduce this to a computation in $\mbox{\em Grass}(k,n)$, the Grassmann manifold of $k$-planes in ${\mathbb C}^n$. Let $\pi_k:{\mathbb F}\ell_n \rightarrow\mbox{\em Grass}(k,n)$ be the projection that sends a complete flag to its $k$-dimensional subspace. Since $X''_{v(\lambda,k)}=\pi_k^{-1}(\Omega''_\lambda)$, where $\Omega''_\lambda$ is a Schubert subvariety of $\mbox{\em Grass}(k,n)$, we have $$ c^w_{u\,v(\lambda,k)}\ =\ \# \pi_k\left( X_{w_0w}\bigcap X'_u\right) \bigcap \Omega''_\lambda. $$ Thus it suffices to study $\pi_k\left(X_{w_0w}\bigcap X'_u\right)\subset \mbox{\em Grass}(k,n)$, equivalently, its fundamental cycle in homology, as $$ \left[\pi_k\left(X_{w_0w}\bigcap X'_u\right)\right] \ =\ \sum_\lambda c^w_{u\,v(\lambda,k)}S_{\lambda^c}, $$ where $S_{\lambda^c}$ is the homology class dual to the fundamental cycle of $\Omega''_\lambda$. To prove Theorems 3.1 ({\em ii}) and Theorem 3.5, we first use Theorem 4.1 ({\em ii}) to reduce to the case of $k=l$ and $wu^{-1}=zx^{-1}$. Then we explicitly compute a dense subset of $X_{w_0w}\bigcap X'_u$ and its image, $Y_{w,u}$, in $\mbox{\em Grass}(k,n)$. This analysis shows that, up to the action of the general linear group, $Y_{w,u}$ depends only upon $wu^{-1}$, up to conjugation by $(12\ldots n)$, whenever $wu^{-1}\in {\mathcal S}_n$. For Theorem 4.2, we study maps $$ \Psi_P\ :\ {\mathbb F}\ell_n\times {\mathbb F}\ell_m \ \longrightarrow {\mathbb F}\ell_{n+m} $$ where $\Psi_P$ `shuffles' pairs of flags together to obtain a longer flag, according to a set $P: 1\leq p_1<\cdots<p_n\leq m$. We show that $\Psi_P(X_u\times X_v)$ is an intersection of two Schubert varieties, which enables the computation of the map $(\Psi_P)_*$ on homology. Then Poincar\'e duality determines the map $(\Psi_P)^*$ on the Schubert basis of cohomology. By construction, $(\Psi_P)^*$ acts by the substitution $\Psi_P$ of \S 4. In the case $m=1$, these computations become more precise, and we obtain Theorem 4.1 ({\em i}) and ({\em ii}). Finally, for Theorem 3.4, suppose $\zeta\cdot\eta$ is disjoint and $u\leq_{k+l}(\zeta\cdot\eta)u$ with $u\in {\mathcal S}_{n+m}$. Then, set $P=u^{-1}{\rm supp}(\zeta)$ and consider the commutative diagram: $$ \setlength{\unitlength}{2.2pt} \begin{picture}(107,34) \put(0,3){$\mbox{\it Grass}_k{\mathbb C}^n\times\mbox{\it Grass}_l{\mathbb C}^m$} \put(75,3){$\mbox{\it Grass}_{k+l}{\mathbb C}^{n+m}$} \put(13.8,26){${\mathbb F}\ell_n\times{\mathbb F}\ell_m$} \put(82,26){${\mathbb F}\ell_{n+m}$} \put(58,7){$\varphi_{k,l}$} \put(60,30){$\psi_P$} \put(6.5,16){$\pi_k\times\pi_l$} \put(91,16){$\pi_{k+l}$} \put(53,4.5){\vector(1,0){20}} \put(40,27.5){\vector(1,0){40}} \put(25.5,23){\vector(0,-1){14}} \put(89,23){\vector(0,-1){14}} \end{picture} $$ Here, $\varphi_{k,l}$ maps a pair $(H,K)\in \mbox{\it Grass}_k{\mathbb C}^n\times\mbox{\it Grass}_l{\mathbb C}^m$ to the sum $H\oplus K$ in $\mbox{\it Grass}_{k+l}{\mathbb C}^{n+m}$. We show there exists $x\in{\mathcal S}_n$, $y\in {\mathcal S}_m$ and $\zeta',\eta'$ shape-equivalent to $\zeta$ and $\eta$ such that $x \leq_k \zeta' x$, $y \leq_l \eta' y$, and $$ \Psi_P\left(X_{w_0\zeta'x}\bigcap X'_x\right)\times \left(X_{w_0\eta'y}\bigcap X'_y\right)\ =\ X_{w_0(\zeta\cdot\eta)u}\bigcap X'_u. $$ Thus to compute $\pi_{k+l}(X_{w_0(\zeta\cdot\eta)u}\bigcap X'_u)$, or rather its homology class, it suffices to compute the map $(\varphi_{k,l})_*$ on homology, which is $$ (\varphi_{k,l})_* S_\lambda\ =\ \sum_{\mu,\nu}c^\lambda_{\mu\,\nu} S_\mu\otimes S_\nu. $$ For more details on these proofs and other aspects of this note, see~\cite{bergeron_sottile_symmetry}.

9705

alg-geom/9705007

en

https://arxiv.org/abs/alg-geom/9705007

alg-geom/9705007

Kota Yoshioka

Kota Yoshioka

Some notes on the moduli of stable sheaves on elliptic surfaces

AMS-LaTex, 18 pages

Under some assumptions, we compute the Picard group of moduli of stable sheaves on Abelian surfaces. Considering relative moduli spaces, it is sufficient to consider the moduli of stable sheaves on the product of elliptic curves. By using elementary transformatios (which was used by Friedman to treat rank 2 moduli spaces on elliptic surfaces), we treat this case. We also show that some moduli spaces on P^2 are rational.

alg-geom

\section{Introduction} Let $X$ be a smooth projective surface over $\Bbb C$ and $H$ an ample divisor on $X$. Let $M_H(r,c_1,\Delta)$ be the moduli of stable sheaves $E$ of rank $r$ on $X$ with $c_1(E)=c_1 \in \operatorname{NS}(X)$ and $\Delta(E)=\Delta$, where $\Delta(E):=c_2(E)-\{(\operatorname{rk}(E)-1)/2\operatorname{rk}(E)\} (c_1(E)^2)$. In this note, we shall consider the moduli spaces on elliptic surfaces. Let $\pi:X \to C$ be an elliptic surface such that every singular fibre is irreducible and $f$ a fibre of $\pi$. We assume that $X$ is regular, $(c_1,f)$ is odd and $H$ is sufficiently close to $f$. Then Friedman [F] showed that $M_H(2,c_1,\Delta)$ is birationally equivalent to $S^n (J^d X)$, where $n=\dim M_H(2,c_1,\Delta)/2$, $2d+1=(c_1,f)$ and $J^d X$ is an elliptic surface over $C$ whose generic fibre is the set of line bundles of degree $d$. In this note, we shall generalize it to the case where $r$ and $(c_1,f)$ are relatively prime. As an application, we shall show that $M_H(r,kH,\Delta)$ is a rational variety for the case where $(X,H) =(\Bbb P^2, \cal O_{\Bbb P^2}(1))$ and $(r,3k)=1$. We also consider moduli spaces on Abelian surfaces. In particular, we shall compute a generator of $H^2(M_H(r,c_1,\Delta),\Bbb Z)$. For general surfaces, Li [Li1, Li2] considered the structure of $H^i(M_H(2,c_1,\Delta),\Bbb Q)$, $i \leq 2$ and $\operatorname{Pic}(M_H(2,c_1,\Delta))\otimes \Bbb Q$ for $\Delta \gg 0$. For the integral cohomologies, Mukai [Mu3, Mu5] and O'Grady [O] investigated the structure of $H^2(M_H(r,c_1,\Delta),\Bbb Z)$ and the Picard group, if $X$ is a K3 surface. By the same method as in [Y2], we get a generator of $H^2(M_H(r,c_1,\Delta),\Bbb Z)$, if $X$ is a ruled surface. Our results for Abelian surfaces are similar to these results. In section 1, we shall consider the birational structure of $M_H(r,c_1,\Delta)$. Our method is the same as that in Friedman [F] and Maruyama [M2]. That is, we shall use elementary transformations. For simplicity, we assume that $X$ is regular. Let $E$ be an element of $M_H(r,c_1,\Delta)$. Since $H$ is sufficiently close to the fibre, $E_{|\pi^{-1}(\eta)}$ is a stable vector bundle on $\pi^{-1}(\eta)$. Then there is a stable vector bundle $E_1$ such that $E_{1|l}$ is semi-stable in the sense of Simpson [S] for all fibres $l$, and $E$ is obtained from $E_1$ by successive elementary transformations along coherent sheaves of pure dimension 1 on fibres. Let $E_2$ be a stable vector bundle such that $E_{2|\pi^{-1}(\eta)} \cong E_{1|\pi^{-1}(\eta)}$, $E_{2|l}$ is semi-stable in the sense of Simpson and $\det E_{2|l} \cong \det E_{1|l}$ for all fibres $l$. By using the irreducibility of $l$, we shall show that $E_2 \cong E_1 \otimes \pi^* L$, where $L \in \operatorname{Pic}(C)$. Then we can easily show that $S^n (J^d X)$ is birationally equivalent to an irreducible component of $M_H(r,c_1,\Delta)$, where $n=\dim M_H(r,c_1,\Delta)/2$ and $d$ is an integer. By the dimension counting of non-locally free part (cf. [Y1, Thm. 0.4]), we see that every irreducible component contains vector bundles (the non-locally free part is of codimension $r-1$). Let $E$ be a vector bundle of $M_H(r,c_1,\Delta)$. We note that $\operatorname{Ext}^2(E,E(-l))_0 \cong \operatorname{Hom}(E,E(K_X+l))^{\vee}_0=0$ for all fibres $l$, where $\operatorname{Ext}^i(E,E(D))_0$ is the trace free part of $\operatorname{Ext}^i(E,E(D))$. Then $\operatorname{Ext}^1(E,E)_0 \to \operatorname{Ext}^1(E_{|l},E_{|l})_0$ is surjective. Considering the deformation space of $E_{|l}$, we shall show that $S^n (J^d X)$ is birationally equivalent to $M_H(r,c_1,\Delta)$. In section 2, we shall treat the moduli spaces on $\Bbb P^2$. Let $V \subset H^0(\Bbb P^2,K_{\Bbb P^2}^{\vee})$ be a linear pencil which contains an elliptic curve $C$. Since $(K_{\Bbb P^2},H)<0$, we can deform $E \in M_H(r,c_1,\Delta)$ to a sheaf $E' \in M_H(r,c_1,\Delta)$ such that $E'_{|C}$ is semi-stable. If $(c_1,H)$ and $r$ are relatively prime, then $E'_{|C}$ is a stable vector bundle. Let $\Bbb P^2 \to \Bbb P^1$ be the rational map defined by $V$ and $Y \to \Bbb P^2$ the blow-ups of $\Bbb P^2$ which defines the morphism $Y \to \Bbb P^1$. Then $M_H(r,c_1,\Delta)$ is birationally equivalent to a component of a moduli space $M_{H'}(r,c_1,\Delta)$, where $H'$ is an ample divisor on $Y$ which is sufficiently close to the fibre in $\operatorname{NS}(Y)$. Since $M_{H'}(r,c_1,\Delta)$ is birationally equivalent to a symmetric product of $Y$, we get that $M_H(r,c_1,\Delta)$ is rational. We also prove that the moduli of simple torsion free sheaves on Del Pezzo surfaces are irreducible. In section 3, we shall consider the moduli spaces on an Abelian surface. We assume that $c_1 \mod r\operatorname{NS}(X)$ is a primitive element of $\operatorname{NS}(X)/r\operatorname{NS}(X)$. Mukai [Mu1] gave a complete description of $M_H(r,c_1,\Delta)$ in the case where $\dim M_H(r,c_1,\Delta)=2$. Hence we assume that $\dim M_H(r,c_1,\Delta) \geq 4$. By using a quasi-universal family [Mu3], we shall construct a generator of $H^i(M_H(r,c_1,\Delta),\Bbb Z)$ for $i=1,2$, where $H$ is a general polarization (Theorem \ref{thm:H2}). Our method is the same as in G\"{o}ttsche and Huybrechts [G-H], that is, we shall deform $X$ to a product of elliptic curves. Then $M_H(r,c_1,0)$ is isomorphic to $X$ and $M_H(r,c_1,\Delta)$ is birationally equivalent to $X \times Hilb_X^{r\Delta}$. Since both spaces have trivial canonical bundles, there are closed subsets $Z_1 \subset M_H(r,c_1,\Delta)$ and $Z_2 \subset X \times Hilb_X^{r\Delta}$ such that $\operatorname{codim}(Z_1) \geq 2$, $\operatorname{codim}(Z_2) \geq 2$ and $M_H(r,c_1,\Delta) \setminus Z_1 \cong (X \times Hilb_X^{r\Delta}) \setminus Z_2$. Hence we get an isomorphism $H^i(M_H(r,c_1,\Delta),\Bbb Z) \cong H^i(X \times Hilb_X^{r\Delta}, \Bbb Z)$, $i=1,2$. Constructing a family of stable sheaves parametrized by $X \times Hilb_X^{r\Delta} \setminus Z_2$ directly, we shall construct a generator of $H^i(M_H(r,c_1,\Delta),\Bbb Z)$, $i=1,2$. By using deformation of $X$ and the result in [Y4], we shall also show that the Betti numbers of $M_H(2,c_1,\Delta)$ are the same as those of $M_H(1,0,2\Delta)$ (Theorem \ref{thm:B}). We next show that the morphism $M_H(r,c_1,\Delta) \to \operatorname{Pic}^0(X) \times X$ defined in [Y2, Sect. 5] is an Albanese map, if $\dim M_H(r,c_1,\Delta) \geq 4$. Combining all together, we also describe the Picard group of $M_H(r,c_1,\Delta)$ (Theorem \ref{thm:Pic}). I would like to thank Professors A. Ishii and M. Maruyama for valuable discussions. \vspace{ 1pc} Notation.\newline Let $X$ be a smooth projective surface over $\Bbb C$ and $H$ an ample divisor on $X$. For a scheme $S$, we denote the projection $S \times X \to S$ by $p_S$. We denote the N\'{e}ron-Severi group of $X$ by $\operatorname{NS}(X)$. For an $x \in \operatorname{NS}(X) \otimes \Bbb Q$, we set $P(x):=(x,x-K_X)/2+\chi(\cal O_X)$. For a torsion free sheaf $E$ on $X$, we set $$ \Delta(E):=c_2(E)-\frac{\operatorname{rk}(E)-1}{2\operatorname{rk} (E)}(c_1(E)^2). $$ We denote the trace free part of $\operatorname{Ext}^i(E,E(D))$ by $\operatorname{Ext}^i(E,E(D))_0$. In this note, we only use the notion of (semi-)stability in the sense of Mumford. Let $M_H(r,c_1,\Delta)$ be the moduli of stable sheaves $E$ of rank $r$ on $X$ with $c_1(E)=c_1 \in \operatorname{NS}(X)$ and $\Delta(E)=\Delta$. We denote the open subscheme of $M_H(r,c_1,\Delta)$ consisting of stable vector bundles by $M_H(r,c_1,\Delta)_0$. \section{Moduli spaces on elliptic surfaces} \subsection{Preliminaries} Let $\pi:X \to C$ be an elliptic surface such that every fibre is irreducible. We denote the algebraically equivalence class of a fibre by $f$. Let $\eta$ be the generic point of the base curve $C$. Let $J^d X \to C$ be the elliptic surface over $C$ such that the generic fibre is the set of line bundles of degree $d$ on $X_{|\pi^{-1}(\eta)}$. For a coherent sheaf $E$ on a fibre $l$, we set \begin{align*} \operatorname{rk}(E) &:=\operatorname{length}_{\cal O_{\eta_l}}(E \otimes \cal O_{\eta_l}),\\ \deg(E) &:=\chi(E), \end{align*} where $\eta_l$ is the generic point of $l$. A coherent sheaf $E$ of pure dimension 1 on a fibre $l$ is semi-stable if $$ \frac{\chi(F)}{\operatorname{rk}(F)} \leq \frac{\chi(E)}{\operatorname{rk}(E)} $$ for all subsheaves $F \ne 0$ of $E$. \begin{lem}\label{lem:k} Let $L$ be a relatively ample divisor on $X$. Let $D$ be a divisor on $X$ such that $(D,f)\ne 0$ and $(D,L+kf)=0$ for some positive number $k$. Then, \begin{equation}\label{eq:2} (D^2) \leq \frac{-1}{(L,f)^2}((L^2)+2k(L,f)). \end{equation} \end{lem} \begin{pf} We set $D=aL+bf+D'$, where $a, b \in \Bbb Q$ and $(D',L)=(D',f)=0$. By the Hodge index theorem, $({D'}^2)\leq 0$. Hence $(D^2)=((aL+bf)^2)+({D'}^2) \leq ((aL+bf)^2)=a^2(L^2)+2ab(L,f)$. Thus we may assume that $D=aL+bf$. $(D,L+kf)=0$ implies that $b(L,f)=-a(L,L+kf)$. Hence $((aL+bf)^2)=-a^2((L^2)+2k(L,f))$. Since $(L,f) \ne 0$, we get that $|a| \geq 1/|(L,f)|$. Hence \eqref{eq:2} holds. \end{pf} \begin{lem} Let $r$ be a positive integer and $c_1$ an algebraically equivalence class on $X$ such that $(c_1,f)$ and $r$ are relatively prime. Let $L$ be an ample divisor on $X$. Then \begin{equation*} M_{L+nf}(r,c_1,\Delta) =\left\{E\left|\begin{split} &\text{$E$ is torsion free of rank $r$ with $(c_1(E),\Delta(E))$}\\ &\text{$=(c_1,\Delta)$ and $E_{|\pi^{-1}(\eta)}$ is stable} \end{split} \right. \right\} \end{equation*} for $n>(r^3(L,f)^2\Delta-2(L^2))/4(L,f)^2$. We denote this space by $M(r,c_1,\Delta)$. \end{lem} \begin{pf} The proof is similar to that in [Y3, Prop. 6.2] (in [Y3], we used slightly different definition of $\Delta$). \end{pf} Since $\operatorname{Ext}^2(E,E)_0 \cong \operatorname{Hom}(E,E)_0^{\vee}=0$, $E \in M(r,c_1,\Delta)$, $M(r,c_1,\Delta)$ is smooth of dimension $2r\Delta- (r^2-1)\chi(\cal O_X)+\dim \operatorname{Pic}^0(X)$. For a stable sheaf $E \in M(r,c_1,\Delta)$, $\chi(E_{|f})=(c_1,f)$ and $\chi(E \otimes k_x)=r$ are relatively prime, where $E$ is locally free at $x \in X$ and $k_x$ is the structure sheaf of $x$. Hence there is a universal family (cf. [M1, Thm. 6.11]). If we fix the rank $r$ and the equivalence class $c_1 \mod \pi^*H^1(C,\Bbb Z)$, then we may denote $M(r,c_1,\Delta)$ by $M(\Delta)$. In fact, $c_1 \mod r\pi^* H^2(C,\Bbb Z)$ is determined by $r\Delta$ and the isomorphic class of $M_H(r,c_1,\Delta)$ is determined by $r$, $c_1 \mod r\pi^* H^2(C,\Bbb Z)$ and $\Delta$. \begin{lem}\label{lem:D} Let $E$ be a vector bundle of rank $r$ on $X$ such that $(c_1(E),f)=d$, and let $F$ be a coherent sheaf of pure dimension 1 on a fibre $l$ with $\operatorname{rk}(F)=r_1$ and $\deg(F)=d_1$. Let $E \to F$ be a surjective homomorphism and $E'$ the kernel. Then \begin{equation} \Delta(E')=\Delta(E)+\frac{rd_1-r_1d}{r}. \end{equation} \end{lem} \begin{pf} For a coherent sheaf $G$ on $X$, $\chi(G)=\operatorname{rk} (G) P(c_1(G)/\operatorname{rk} G)-\Delta (G)$. Since $\chi (E)=\chi (E')+\chi(F)$, \begin{align*} \Delta(E')-\Delta(E) & = d_1-r(P(c_1(E)/\operatorname{rk} E)-P(c_1(E')/\operatorname{rk} E'))\\ & = d_1-\frac{r_1 d}{r}. \end{align*} \end{pf} The following is a special case of Maruyama [M2, 3.8]. \begin{cor}\label{cor:1} Let $E$ be a vector bundle on $X$ such that $E_{|\pi^{-1}(\eta)}$ is a semi-stable vector bundle. Then there is a vector bundle $E'$ on $X$ such that $E'_{|l}$ is semi-stable for every fibre $l$ and $E$ is obtained from $E'$ by successive elementary transformations along coherent sheaves of pure dimension 1 on fibres. \end{cor} \begin{pf} We note that $\Delta(E) \geq 0$. We shall prove our claim by induction on $\Delta(E)$. We assume that there is a fibre $l$ such that $E_{|l}$ is not semi-stable. Then there is a surjective homomorphism $E_{|l} \to F$ such that $F$ is of pure dimension 1 and $\chi(E_{|l})/\operatorname{rk} (E_{|l})>\chi(F)/\operatorname{rk} F$. We shall consider the following elementary transformation along $F$: \begin{equation*} 0 \to E_1 \to E \to F \to 0. \end{equation*} Since $\operatorname{depth}_{\cal O_x}F=1$, $x \in C$ and $X$ is smooth, we see that $\operatorname{proj-dim}_{\cal O_x} F=\dim X-\operatorname{depth}_{\cal O_x}F=1$. Hence $E_1$ is also locally free. By Lemma \ref{lem:D}, we get that $\Delta(E_1)<\Delta(E)$. Hence we obtain our corollary. \end{pf} \subsection{General element of $M(\Delta)$} Let $E$ be a general element of $M(\Delta)$. We shall consider the Harder-Narasimhan filtration of the restriction $E_{|l}$ of $E$ to fibres $l$. In particular, we shall show that $E_{|l}$ is semi-stable for all singular fibres $l$. \begin{lem}\label{lem:def} Let $C$ be a projective curve and $\cal O_C(1)$ an ample divisor on $C$. Let $L$ be a line bundle on $C$. Let $Q$ be the subscheme of $Quot_{\cal O_C(-n)^{\oplus N}/C}$ parametrizing quotients $\cal O_C(-n)^{\oplus N} \to E$ such that $(\mathrm{i})$ $E$ is a locally free sheaf of rank r with $\det E = L$ and $(\mathrm{ii})$ $H^1(C,E(n))=0$. Then $Q$ is smooth and irreducible. \end{lem} \begin{pf} Let $\lambda:\cal O_C(-n)^{\oplus N} \to E$ be a quotient which belongs to $Q$. Then we see that $\operatorname{Ext}^1(\ker \lambda,E)=0$. Since $\operatorname{Hom}(\ker \lambda,E) \to \operatorname{Ext}^1(E,E) \overset{\operatorname{tr}}\to H^1(C,\cal O_C)$ is surjective, $Q$ is smooth. For $l \geq n$, there is an exact sequence $ 0 \to \cal O_C^{\oplus (r-1)} \to E(l) \to L(rl) \to 0$. We set $\Bbb P:=\Bbb P(\operatorname{Ext}^1(L(rl),\cal O_C^{\oplus (r-1)})^{\vee})$. We shall consider the universal extension: $$ 0 \to \cal O_{\Bbb P \times C}^{\oplus (r-1)} \to \cal E \to L(rl) \otimes \cal O_{\Bbb P}(-1) \to 0. $$ Let $\Bbb P'$ be the open subscheme of $\Bbb P$ of points $y$ such that $H^1(C,\cal E_y)=0$. Then $p_{\Bbb P'*}(\cal E)$ is a locally free sheaf on $\Bbb P'$. Let $\phi:\Bbb A \to \Bbb P'$ be the vector bundle associated to the locally free sheaf $\cal Hom(\cal O_{\Bbb P'}^{\oplus N},p_{\Bbb P'*}(\cal E))$. Then there is a homomorphism $\Lambda:\cal O_{\Bbb A \times C}^{\oplus N} \to (\phi \times 1)^* \cal E$. Let $\Bbb A'$ be the open subscheme of $\Bbb A$ such that $\Lambda$ is surjective. Then there is a surjective morphism $\Bbb A' \to Q$, and hence $Q$ is irreducible. \end{pf} \begin{prop} Let $M(\Delta)^0$ be the open subscheme of $M(\Delta)$ of elements $E$ such that $E_{|l}$ is semi-stable for every singular fibre $l$. Then $M(\Delta)^0$ is a dense subscheme of $M(\Delta)$. \end{prop} \begin{pf} Let $E$ be an element of $M(\Delta)$. Since $E_{|\pi^{-1}(\eta)}$ is stable, we see that $\operatorname{Ext}^2(E,E(-l))_0 \cong \operatorname{Hom}(E,E(l+K_X))_0^{\vee}=0$. Hence we get that $\operatorname{Ext}^1(E,E)_0 \to \operatorname{Ext}^1(E_{|l},E_{|l})_0$ is surjective. Let $m$ be the multiplicity of $l$ and set $l=ml'$. By Corollary \ref{cor:1}, there is a vector bundle $E_1$ on $X$ such that $E_{1|l}$ is semi-stable and $\det(E_{1|l})=\det(E_{|l})\otimes \cal O_l(kl')$. Since $(r,m)=1$, replacing $E_1$ by $E_1 \otimes \cal O_X(jl')$, we may assume that $\det(E_{1|l})=\det(E_{|l})$. By using Lemma \ref{lem:def}, we see that $E_{|l}$ deforms to a semi-stable vector bundle. Hence we see that $E$ deforms to a sheaf $E'$ such that $E'_{|l}$ is semi-stable. Thus $M(\Delta)^0$ is an open dense subscheme of $M(\Delta)$. \end{pf} \begin{lem} Let $l$ be a smooth fibre. Let $h:=\{(r_1,d_1),(r_2,d_2),\dots ,(r_s,d_s)\}$ be a sequence of pairs of integers such that $r_i>0$, $1 \leq i \leq s$ and $d_1/r_1>d_2/r_2>\dots>d_s/r_s$. Let $D_h$ be the subset of $M(r,c_1,c_2)$ of elements $E$ such that the Harder-Narasimhan filtration of $E_{|l}$ : $0 \subset F_1 \subset F_2 \subset \dots \subset F_s=E_{|l}$ satisfies that $\operatorname{rk}(F_i/F_{i-1})=r_i$ and $\deg(F_i/F_{i-1})=d_i$, $1 \leq i \leq s$. Then $\operatorname{codim}(D_h)\geq \sum_{i<j}r_jd_i-r_id_j$. In particular, if $\operatorname{codim}(D_h)=1$, then $s=2$ and $r_2d_1-r_1d_2=1$. \end{lem} \begin{pf} Let $\Def(E_{|l})$ be the local deformation space of $E_{|l}$ of fixed determinant and $\Def(E_{|l})_h$ the subset of $\Def(E_{|l})$ of elements $G$ such that the Harder-Narasimhan filtration of $G$ : $0 \subset F_1 \subset F_2 \subset \dots \subset F_s=G$ satisfies that $\operatorname{rk}(F_i/F_{i-1})=r_i$ and $\deg(F_i/F_{i-1})=d_i$, $1 \leq i \leq s$. We assume that $\Def(E_{|l})_h$ is not empty. We note that $\operatorname{Ext}^1(E,E)_0 \to \operatorname{Ext}^1(E_{|l},E_{|l})_0$ is surjective. It is known that $\operatorname{codim}(\Def(E_{|l})_h)=\sum_{i<j}r_jd_i-r_id_j$ (cf. [A-B, Thm. 7.14]). Hence we get our lemma. \end{pf} Let $(r_1,d_1)$ be the pair of integers such that $0<r_1<r$ and $rd_1-r_1d=1$. Let $M(\Delta)^1$ be the open subscheme of $M(\Delta)^0$ of elements $E$ such that $E_{|l}$ is stable, or the Harder-Narasimhan filtration of $E_{|l}$ is $0 \subset F \subset E_{|l}$ for all fibres $l$, where $F$ is a stable vector bundle of rank $r_1$ on $l$ with $\deg(F)=d_1$. Then $M(\Delta)^1$ is an open dense subscheme of $M(\Delta)^0$. \subsection{Vector bundles on elliptic curves} The following is due to Atiyah [A]. \begin{lem}\label{lem:bdle} Let $C$ be a smooth elliptic curve. Let $r$ be a positive integer and $d$ an integer such that $(r,d)=1$. Then,\newline $(1)$ There is a stable vector bundle of rank $r$ and degree $d$. \newline $(2)$ Let $(r_1,d_1)$ be the pair of integers such that $r_1d-rd_1=1$ and $0<r_1<r$. Let $E_1$ be a stable vector bundle of rank $r_1$ and degree $d_1$. Then every stable vector bundle $E$ of rank $r$ and degree $d$ is defined by an exact sequence \begin{equation} 0 \to E_1 \to E \to E_2 \to 0, \end{equation} where $E_2$ is a stable vector bundle of rank $r_2:=r-r_1$ and degree $d_2:=d-d_1$.\newline $(3)$ Let $0 \subset F_1 \subset F_2 \subset \dots \subset F_s=E$ be the Harder-Narasimhan filtration of a vector bundle $E$. Then $E \cong \oplus_{i=1}^s E_i$, where $E_i:=F_i/F_{i-1}$. \end{lem} \begin{pf} (1) We shall prove our claim by induction on $r$. If $r=1$, then our claim obviously holds. Let $(r_1,d_1)$ be the pair of integers such that $r_1d-rd_1=1$ and $0<r_1<r$. We set $r_2:=r-r_1$ and $d_2:=d-d_1$. By induction hypothesis, there are stable vector bundles $E_i$ of rank $r_i$ and degree $d_i$, $i=1,2$. Since $d_1/r_1<d_2/r_2$, $\operatorname{Hom}(E_2,E_1)=0$. By using the Riemann-Roch theorem, we get that $\operatorname{Ext}^1(E_2,E_1) \cong \Bbb C$. Let $0 \to E_1 \to E \to E_2 \to 0$ be a non-trivial extension. We shall show that $E$ is stable. If $E$ is not stable, then there is a semi-stable subsheaf $G$ of $E$ such that $\deg G/\operatorname{rk} G>d/r$. Since $G$ and $E_2$ are semi-stable and $G \to E\to E_2$ is not zero, $\deg G/\operatorname{rk} G \leq d_2/r_2$. We assume that $\deg G/\operatorname{rk} G<d_2/r_2$. Then we see that $1/rr_2=d_2/r_2-d/r>d_2/r_2-\deg G/\operatorname{rk} G \geq 1/r_2\operatorname{rk} G$, which is a contradiction. Hence $\deg G/\operatorname{rk} G=d_2/r_2$. Then we get that $\operatorname{rk} G=r_2$ and $\deg G=d_2$. Hence $G \cong E_2$, which is a contradiction. (2) Let $E$ be a stable vector bundle of rank $r$ and degree $d$. Then $\operatorname{Ext}^1(F_1,E) \cong \operatorname{Hom}(E,F_1)^{\vee}=0$. By the Riemann-Roch theorem, there is a non-zero homomorphism $\varphi:E_1 \to E$. We shall show that $\varphi$ is injective and $\operatorname{coker} \varphi$ is stable. Since $E_1$ and $E$ are stable, $d_1/r_1 \leq \deg \varphi(E_1)/\operatorname{rk} \varphi(E_1)<d/r$. In the same way as in the proof of (1), we see that $\operatorname{rk} \varphi(E_1)=r_1$ and $\deg \varphi(E_1)=d_1$. Hence we get that $E_1 \cong \varphi(E_1)$. We set $E_2:=\operatorname{coker} \varphi$. We assume that there is a quotient $G$ of $E_2$ such that $G$ is semi-stable and $d_2/r_2>\deg G/\operatorname{rk} G$. Since $G$ is a quotient of $E$, we get that $d/r<\deg G/\operatorname{rk} G$. Hence we get that $d/r<\deg G/\operatorname{rk} G<d_2/r_2$. Then $1/rr_2=d_2/r_2-d/r>d_2/r_2-\deg G/\operatorname{rk} G \geq 1/r_2\operatorname{rk} G$, which is a contradiction. Hence $E_2$ is a stable vector bundle. (3) Since $\deg E_i/\operatorname{rk} E_i>\deg E_j/\operatorname{rk} E_j$, $i<j$, the Serre duality implies that $\operatorname{Ext}^1(E_j,E_i)=0$, $i<j$. By the induction on $s$, we see that $E \cong \oplus_i E_i$. \end{pf} \begin{lem}\label{lem:elm} Let $(r,d)$ (resp. $(r_1,d_1)$, $(r_2,d_2)$) be the pair in Lemma \ref{lem:bdle}. Let $E$ be a vector bundle of rank $r$ on $C$ with degree $d$ and $E_2$ a stable vector bundle of rank $r_2$ on $C$ with degree $d_2$.\newline (1) If $E$ is stable, then $\operatorname{Hom}(E,E_2) \cong \Bbb C$ and a non-zero homomorphism is surjective.\newline (2) Let $F_1$ (resp. $F_2$) be a stable vector bundle of rank $r_1$ and degree $d_1$ (resp. rank $r_2$ and degree $d_2$). We assume that $E \cong F_1 \oplus F_2$ and there is a surjective homomorphism $\varphi:E \to E_2$ such that $\ker \varphi$ is also stable. Then $E_2 \cong F_2$ and $\operatorname{Hom}(E,E_2) \cong \Bbb C^{\oplus 2}$. \end{lem} \begin{pf} (1) Since $E$ is stable, $\operatorname{Ext}^1(E,E_2) \cong \operatorname{Hom}(E_2,E)^{\vee}=0$. By the Riemann-Roch theorem, we see that $\dim \operatorname{Hom}(E,E_2)=1$. In the same way as in the proof of Lemma \ref{lem:bdle}, we see that a non-zero homomorphism $E \to E_2$ is surjective. (2) If $E_2 \not\cong F_2$, then $\ker \varphi \cong \ker(\varphi_{|F_1})\oplus F_2$. Since $\varphi_{|F_1}:F_1 \to E_2$ is surjective, $\ker(\varphi_{|F_1}) \ne 0$. Hence $E_2 \cong F_2$. By the Riemann-Roch theorem, $\operatorname{Hom}(F_1,E_2) \cong \Bbb C$. Therefore $\operatorname{Hom}(E,E_2) \cong \Bbb C^{\oplus 2}$. \end{pf} Let $C_0$ be the open subscheme of $C$ such that $\pi:X_0:=X \times_C C_0 \to C_0$ is smooth. We assume that $\pi$ has a section $\sigma$. We denote the relative moduli space of stable vector bundles of rank $r$ on fibres with degree $d$ by $\cal M_{X_0/C_0}(r,d) \to C_0$. We assume that $(r,d)=1$. We shall construct a family of stable vector bundles $\cal E_{r,d}$ on $X_0 \times_{C_0}X_0$ and show that $\cal M_{X_0/C_0}(r,d) \cong X_0$ as a $C_0$-scheme, by using induction on $r$. If $r=1$, then $\cal E_{1,d}:=\cal O_{X_0 \times _{C_0}X_0}((d+1)\sigma-\Delta)$ is a universal family, where $\Delta$ is the diagonal of $X_0 \times _{C_0}X_0$. Let $(r_1,d_1)$ be the pair of integers such that $r_1d-rd_1=1$ and $0<r_1<r$. We set $r_2=r-r_1$ and $d_2=d-d_1$. Let $E$ be a vector bundle on $X_0$ such that $E_{|l}$ is a stable vector bundle of rank $r_2$ and $\det E_{|l} \cong \cal O_l(d_2\sigma)$ for every fibre $l$. By using Lemma \ref{lem:bdle}, we see that $\cal L:=\operatorname{Ext}^1_{p_{X_0}}(E,\cal E_{r_1,d_1})$ is a line bundle on $X_0$. Then there is the universal extension \begin{equation} 0 \to \cal E_{r_1,d_1} \to \cal E_{r,d} \to E \otimes p_{X_0}^*(\cal L) \to 0, \end{equation} which parametrizes stable vector bundles of rank $r$ on fibres with degree $d$. Hence there is a morphism $X_0 \to \cal M_{X_0/C_0}(r,d)$. By our construction, this morphism is injective. By ZMT, it is an isomorphism. \begin{lem}\label{lem:hom} Let $E$ and $E'$ be semi-stable vector bundles on a multiple fibre $l=ml'$ such that $\operatorname{rk} E=\operatorname{rk} E'$, $\det E \cong \det E'$, and $\chi(E)=\chi(E')=d$. Then, \begin{equation} \operatorname{Hom}(E,E')= \begin{cases} \Bbb C,& \text{ if $E \cong E'$},\\ 0,& \text{ otherwise}. \end{cases} \end{equation} \end{lem} \begin{pf} We set $L:=\cal O_X(-l')_{|l'}$. We note that $\operatorname{rk}(E \otimes L^{\otimes k})=r$ and $\chi(E \otimes L^{\otimes k})=d/m$ for $0 \leq k \leq m-1$. Since $(r,d)=1$ and $E$ is semi-stable, $E \otimes L^{\otimes k}$ is a stable sheaf on $ml'$. Thus $0 \subset E(-(m-1)l') \subset E(-(m-2)l') \subset \dots \subset E(-l') \subset E$ is a Jordan-H\"{o}lder filtration of $E$. Since the order of $L \in Pic^0(l')$ is $m$ and $(m,r)=1$, $\det E \cong \det E'$ and the stabilities of $E_{|l'}$ and $E_{|l'}$ imply that $\operatorname{Hom}(E_{|l},E'\otimes L^{\otimes k})=0$ for $1 \leq k \leq m-1$. Let $\varphi:E \to E'$ be a non-zero homomorphism. We shall show that $\varphi$ is an isomorphism. Since $\operatorname{Hom}(E_{|l},E'\otimes L^{\otimes k})=0$ for $1 \leq k \leq m-1$, we see that $\varphi_{|l'}:E_{|l'} \to E'_{|l'}$ is not zero, which implies that $E_{|l'} \cong E'_{|l'}$. By Nakayama's lemma, $\varphi$ is an isomorphism. Then it is easy to see that $\operatorname{Hom}(E,E') \cong \Bbb C$. \end{pf} \begin{lem}\label{lem:5-10} Let $E,E'$ be vector bundles of rank $r$ on $X$ such that $E_{|l}$ and $E'_{|l}$ are semi-stable for all fibres $l$ and $\det E \cong \det E'$. Then there is a line bundle $L$ on $C$ such that $E \cong E' \otimes \pi^*(L)$. \end{lem} \begin{pf} We note that $E_{|\pi^{-1}(\eta)} \cong E'_{|\pi^{-1}(\eta)}$. By the upper semi-continuity of $h^0(l,E^{'\vee}\otimes E_{|l})$, there is a non-zero homomorphism $E'_{|l} \to E_{|l}$ for every fibre $l$. Since $E_{|l}$ and $E'_{|l}$ are semi-stable, Lemma \ref{lem:hom} implies that $E_{|l} \cong E'_{|l}$ and $H^0(l,E^{'\vee}\otimes E_{|l}) \cong \Bbb C$. By the base change theorem, we get that $L:=\pi_*(E^{'\vee} \otimes E)$ is a line on $C$ and $\pi^*(L) \otimes E' \to E$ is an isomorphism. \end{pf} \begin{cor} $M(\Delta)$ is not empty if and only if $\Delta\geq \Delta_0:=\frac{(r^2-1)}{2r} \chi(\cal O_X)$. \end{cor} \begin{pf} We set $\Delta':=\min\{\Delta|M(\Delta) \ne \emptyset \}$. Lemma \ref{lem:5-10} implies that $\dim \operatorname{Pic}^0(X)=\dim M(\Delta')=2r\Delta'-(r^2-1)\chi(\cal O_X) +\dim \operatorname{Pic}^0(X)$. Hence we get our claim \end{pf} \begin{rem} Let $E$ be an element of $M(\Delta_0)$. By Lemma \ref{lem:5-10}, there is a surjective morphism $\operatorname{Pic}^0(X) \to M(\Delta_0)$ sending $L \in \operatorname{Pic}^0(X)$ to $E \otimes L$. Hence we get that $M(\Delta_0)=\operatorname{Pic}^0(X)/\Phi(E)$, where $\Phi(E):=\{L \in \operatorname{Pic}^0(X)|E \otimes L \cong E \}$. In particular, if $\operatorname{Pic}^0(X)=\operatorname{Pic}^0(C)$, then $M(\Delta_0)=\operatorname{Pic}^0(X)$. \end{rem} \vspace{1pc} \subsection{Construction of a family} We assume that $\pi:X \to C$ has a section and show that $M(\Delta)$ is birational to $M(\Delta_0) \times S^nX$, where $n:=r(\Delta-\Delta_0)$. Let $\cal E$ be a universal family on $M(\Delta_0) \times X$. Let $(r_1,d_1)$ be the pair of integers such that $r_1d-rd_1=-1$ and $0 < r_1<r$, and let $\cal E_{r_1,d_1}$ be the vector bundle on $X_0 \times_{C_0} X_0$. Let $j:X_0 \times_{C_0} X_0 \to X_0 \times X$ be the immersion. We denote the projection $M(\Delta_0) \times X_0 \to M(\Delta_0)$ by $q_1$ and $M(\Delta_0) \times X_0 \to X_0$ by $q_2$. By Lemma \ref{lem:elm}, $\cal L:=\operatorname{Hom}_{p_{M(\Delta_0) \times X_0}} ((q_1 \times 1_X)^*\cal E,(q_2 \times 1_X)^* j_*\cal E_{r_1,d_1})$ is a line bundle on $M(\Delta_0) \times X_0$, and there is a surjective homomorphism: $(q_1 \times 1_X)^*\cal E \to (q_2 \times 1_X)^*j_*\cal E_{r_1,d_1} \otimes p_{M(\Delta_0) \times X_0}^*(\cal L)^{\vee}$. Let $p_i:X_0^n:=X_0 \times X_0 \times \dots \times X_0 \to X_0$ be the $i$-th projection, $1 \leq i \leq n$. Then there is a homomorphism \begin{equation} \Lambda:\widetilde{\cal E} \to \oplus_{i=1}^n (q_2 \circ (1_{M(\Delta_0)} \times p_i) \times 1_X)^*j_* \cal E_{r_1,d_1}\otimes \cal L_i, \end{equation} where $\widetilde{\cal E}$ is the pull-back of $\cal E$ to $M(\Delta_0) \times X_0^n \times X$ and $\cal L_i=(1_{M(\Delta_0)} \times p_i \times 1_X)^* p_{M(\Delta_0) \times X_0}^*(\cal L)^{\vee}.$ We set $\Gamma:=\{(x_1,x_1,\dots,x_n) \in X_0^n| \pi(x_i)=\pi(x_j) \text{ for some $i \ne j$}\}$. Then $\Lambda_1:=\Lambda_{|M(\Delta_0) \times (X_0^n \setminus \Gamma) \times X}$ is a surjective homomorphism. We set $\cal F:=\ker \Lambda_1$. By Lemma \ref{lem:D}, $\cal F$ is a family of stable vector bundles on $X$. Hence there is a morphism $M(\Delta_0) \times (X_0^n \setminus \Gamma) \to M(\Delta)$. By our construction, this morphism is $\frak S_n$-invariant, and hence we get a morphism $\nu:M(\Delta_0) \times (X_0^n /\frak S_n) \to M(\Delta)$. By our construction, it is injective. Since $\dim S^n X=2n=\dim M(\Delta)- \dim M(\Delta_0)$, ZMT implies that $M(\Delta_0) \times (X_0^n /\frak S_n) \to M(\Delta)$ is an immersion. We set $M(\Delta)^2:=\nu(M(\Delta_0) \times (X_0^n /\frak S_n))$. We shall show that $M(\Delta)^2$ is dense. For this purpose, we shall estimate the dimension of $M(\Delta)^1 \setminus M(\Delta)^2$. \begin{lem} $\dim(M(\Delta)^1 \setminus M(\Delta)^2) = 2n-1+\dim M(\Delta_0)$. \end{lem} \begin{pf} For a $E \in M(\Delta)^1$ and a smooth fibre $l$, we assume that $E_{|l}$ is not stable. By the definition of $M(\Delta)^1$, we see that $E_{|l} \cong E_1 \oplus E_2$, where $E_1$ (resp. $E_2$) is a stable vector bundle of rank $r_1$ and degree $d_1$ (resp. rank $r_2$ and degree $d_2$). We set $E':=\ker(E \to E_1)$. Then there is an exact sequence \begin{equation}\label{eq:1} 0 \to E_1 \to E'_{|l} \to E_2 \to 0. \end{equation} Then $E$ is obtained by the inverse transform from $E'$ : \begin{equation} 0 \to E \to E'(l) \to E_2 \to 0. \end{equation} By \eqref{eq:1}, $E'_{|l}$ is stable or $E'_{|l} \cong E_1 \oplus E_2$. By Lemma \ref{lem:D}, $\Delta(E')=\Delta(E)-1/r$. Conversely, for $E' \in M(\Delta-1/r)^1$, we shall consider a surjective homomorphism $\psi:E' \to F_2$ such that the kernel of $E'_{|l} \to F_2$ is stable, where $F_2$ is a stable vector bundle of rank $r_2$ on a smooth fibre $l$ with degree $d_2$. If $\ker \psi \otimes \cal O_X(l)$ belongs to $M(\Delta)^1 \setminus M(\Delta)^2$, then (i) $E'_{|l}$ is stable and $E'$ belongs to $M(\Delta-1/r)^1 \setminus M(\Delta-1/r)^2$, or (ii) $E'_{|l}$is not stable and $F_2$ is a direct summand of $E'_{|l}$. Since $\#\{l|\text{$E'_{|l}$ is not stable}\} \leq n-1$, by using Lemma \ref{lem:elm}, we see that \begin{align*} \dim(M(\Delta)^1 \setminus M(\Delta)^2)& = \max\{\dim (M(\Delta-1/r)^1 \setminus M(\Delta-1/r)^2)+2, \dim M(\Delta-1/r)^1+1\}\\ & =2n-1+\dim M(\Delta_0). \end{align*} \end{pf} \begin{thm}\label{thm:1} $M(\Delta)$ is irreducible and birational to $M(\Delta_0) \times S^n (J^{d_1}X)$, where $n:=r(\Delta-\Delta_0)$. \end{thm} \begin{pf} If $\pi:X \to C$ has a section, we have proved our theorem. For general cases, we shall consider a Galois covering $\gamma:C' \to C$ such that $\pi':X \times _C C' \to C'$ has a section $\sigma'$. Let $C_1$ be an open subscheme of $C_0$ such that $\gamma^{-1}(C_1) \to C_1$ is etale. We set $X_1':=\pi^{-1}(C_1) \times_{C} C'$. Let $\cal E_{r_1,d_1}'$ be the vector bundle on $X_1' \times_{\gamma^{-1}(C_1)} X_1'$ and $j':X_1' \times_{\gamma^{-1}(C_1)} X_1' \cong X_1' \times_{C_1} X_1 \hookrightarrow X_1' \times X_1$ the inclusion. Let $X_1' \to J^{d_1}X$ be the morphism induced by $\cal E_{r_1,d_1}'$. For a $g \in \operatorname{Gal}(C'/C)$, let $\tilde{g}:X_1' \to X_1'$ be the automorphism of $X_1'$ sending $(x,y) \in \pi^{-1}(C_1) \times_C C'$ to $(x+(d_1-1)(\sigma'(g(y))-\sigma'(y)),g(y))$. Then it defines an action of $\operatorname{Gal}(C'/C)$ to $X_1'$. By the construction of $\cal E_{r_1,d_1}'$, we see that $\det(\cal E_{r_1,d_1}')_{|\tilde{g}((x,y))} \cong \det(\cal E_{r_1,d_1}')_{|(x,y)}$. Hence $(\cal E_{r_1,d_1}')_{|\tilde{g}((x,y))} \cong (\cal E_{r_1,d_1}')_{|(x,y)}$. Thus the morphism $X_1' \to J^{d_1}X$ is $\operatorname{Gal}(C'/C)$-invariant. Then we get that $X_1'/\operatorname{Gal}(C'/C) \to J^{d_1}X$ is an immersion. Replacing $j_* \cal E_{r_1,d_1}$ by $j'_* \cal E_{r_1,d_1}'$, we can construct a family of stable vector bundles $\cal F$ parametrized by $M(\Delta_0) \times ((X_1')^n \setminus \Gamma')$, where $\Gamma'$ is the pull-back of $\Gamma$ to $(X_1')^n$. Hence we get a morphism $M(\Delta_0) \times ((X_1')^n \setminus \Gamma') \to M(\Delta)$. By the construction, $\operatorname{Gal}(C'/C) \times \frak S_n$ acts on $((X_1')^n \setminus \Gamma')$, and this morphism is $\operatorname{Gal}(C'/C)\times \frak S_n$-invariant. Hence we get a morphism $M(\Delta_0) \times ((J^{d_1}X_1)^n \setminus \Gamma)/\frak S_n \to M(\Delta)$. Then we see that $M(\Delta)$ is birationally equivalent to $M(\Delta_0) \times S^n (J^{d_1}X)$. \end{pf} \section{Moduli spaces on Del Pezzo surfaces} \subsection{} We shall apply Theorem \ref{thm:1} to moduli spaces on Del Pezzo surfaces. \begin{thm} We assume that $X=\Bbb P^2$ and set $H:=\cal O_{\Bbb P^2}(1)$. Then $M_H(r,kH,\Delta)$ is a rational variety if $(r,3k)=1$. \end{thm} \begin{pf} Let $V \subset H^0(\Bbb P^2,\cal O_{\Bbb P^2}(3))$ be a pencil such that every member $D \in V$ is irreducible and $\#\{P|P \in \cap_{D \in V} D \}=9$. Let $\phi:Y \to \Bbb P^2$ be the blow-ups of $\Bbb P^2$ at base points of $V$. Then there is an elliptic fibration $\pi:Y \to \Bbb P^1$ such that every fibre is isomorphic to a member $D$ of $V$. We set \begin{equation} N:=\{E \in M_H(r,kH,\Delta)_0| \text{$\phi^*E_{|\pi^{-1}(\eta)}$ is stable } \}, \end{equation} where $\eta$ is the generic point of $\Bbb P^1$. Let $E$ be a stable vector bundle of rank $r$ on $\Bbb P^2$ with $c_1(E)=kH$. Then $\operatorname{Ext}^2(E,E(-3))_0 \cong \operatorname{Hom}(E,E)_0^{\vee}=0$. Let $D \in V$ be a smooth elliptic curve. Then we get the surjective homomorphism $\operatorname{Ext}^1(E,E)_0 \to \operatorname{Ext}^1(E_{|D},E_{|D})_0$. Hence $\Def(E) \to \Def(E_{|D})$ is submersive. Since $(r,\deg(E_{|D}))=(r,3k)=1$, we can deform $E$ to a stable sheaf $F$ such that $F_{|D}$ is a stable vector bundle on $D$. By the openness of stability, $F_{|\pi^{-1}(\eta)}$ is a stable vector bundle. Hence $N$ is an open dense subscheme of $M_H(r,kH,\Delta)$ and there is an open immersion $\phi^*:N \to M(r,k\phi^* H,\Delta)$. By Theorem \ref{thm:1}, $N$ is bitarional to $S^nY$, where $n=r\Delta-(r^2-1)/2$. Since $S^nY$ is a rational variety, we get our theorem. \end{pf} \begin{defn} $Spl(r,c_1,\Delta)$ is the moduli space of simple torsion free sheaves $E$ of rank $r$ with $c_1(E)=c_1$ and $\Delta(E)=\Delta$. \end{defn} We shall next consider the irreducibility of $Spl(r,c_1,\Delta)$ for Del Pezzo surfaces. \begin{prop}\label{prop:irr} Let $\pi:X \to \Bbb P^1$ be a rational elliptic surface with a section $\sigma$. For a $c_1 \in \operatorname{NS}(X)$ such that $(c_1,f)$ and r are relatively prime, we shall consider the moduli space $M(\Delta)=M(r,c_1,\Delta)$. Then $M(\Delta)$ is irreducible and rational. \end{prop} \begin{pf} We note that $\sigma$ is a $(-1)$-curve. Let $\phi:X \to Y$ be the contraction of $\sigma$. Since the characteristic of $\Bbb C$ is 0, $\pi_* \cal O_X$ is locally free of rank 1, and hence $\pi_* K_X^{\vee}(\sigma)\cong \pi_* K_X^{\vee}$. Then we get that $H^0(Y,K_Y^{\vee}) \cong H^0(X,K_X^{\vee}(\sigma))\cong H^0(X,K_X^{\vee}) \cong \Bbb C^{\oplus 2}$. By the Riemann-Roch theorem, $H^1(Y,K_Y^{\vee})=0$. Let $\delta:\cal Y \to S$ be a smooth family of 8-points blow-ups of $\Bbb P^2$ such that $H^1(\cal Y_s,K_{\cal Y_s}^{\vee})=0$ for all $s \in S$ and $\cal Y_{s_0}=Y$ for some $s_0 \in S$. Let $\xi$ be the generic point of $S$. By the base change theorem, $\delta_*(K_{\cal Y/S}^{\vee})$ is a locally free sheaf of rank 2 and $\delta_*(K_{\cal Y/S}^{\vee})\otimes k(s) \to H^0(K_{Y_s}^{\vee}), s \in S$ is an isomorphism. We set $\cal O_{\cal Z}:=\operatorname{coker}(\delta^* \delta_*(K_{\cal Y/S}^{\vee}) \to K_{\cal Y/S}^{\vee}) \otimes K_{\cal Y/S}$. Then $\cal O_{\cal Z} \otimes k(s)$ defines a reduced one point of $Y_s$. Thus $\cal Z$ defines a section of $\delta$. Let $\phi_S:\cal X \to \cal Y$ be the blow-up of $\cal Y$ along $\cal Z$ and set $\epsilon:=\delta \circ \phi$. Then there is a morphism $\pi_S:\cal X \to \Bbb P:= \Bbb P(\epsilon_*(K_{\cal Y/S}^{\vee}))$, which defines a family of elliptic fibrations. Choosing a sufficiently general family, we may assume that $\pi_{S|\xi}:\cal X_{\xi} \to \Bbb P^1_{k(\xi)}$ is an elliptic surface such that every fibre is irreducible. Let $\cal O_{\cal X}(1)$ be a relative ample line bundle on $\cal X$ which is sufficiently close to the pull-back of an ample line bundle on $\Bbb P$. For a line bundle $\cal L$ on $\cal X$ such that $c_1(\cal L_{s_0})= c_1$, we shall consider the relative moduli space $\cal M(r,\cal L,\Delta) \to S$ of stable sheaves $E$ of rank $r$ on $\cal X_s, s \in S$ such that $c_1(E)=\cal L_s$ and $\Delta(E)=\Delta$. By Maruyama [M1, Cor. 5.9.1, Prop. 6.7], $\cal M(r,\cal L,\Delta)$ is smooth and projective over $S$. By Theorem \ref{thm:1}, the generic fibre is irreducible, and hence every fibre is irreducible. Thus $M(\Delta)$ is irreducible. Since $M(\Delta)$ contain an irreducible component which is birational to $S^n X$ for some $n$ ( see the proof of Theorem \ref{thm:1}), $M(\Delta)$ is a rational variety. \end{pf} \begin{lem}\label{lem:spl} Let $\phi:\widetilde{X} \to X$ be a one point blow-up of a surface $X$ and $E$ a simple torsion free sheaf of rank $r$ on $X$ which is locally free at the center of the blow-up. Let $C_1$ be the exeptional divisor of $\phi$ and $\phi^*E \to \cal O_{C_1}^{\oplus k}$, $0<k<r$ a surjective homomorphism. We set $E':=\ker(\phi^* E \to \cal O_{C_1}^{\oplus k})$. Then $E'$ is also a simple torsion free sheaf. \end{lem} \begin{pf} We note that $\operatorname{Ext}^1(\cal O_{C_1}^{\oplus k},E) \cong H^1(C_1,E^{\vee} \otimes \cal O_{C_1}(K_{\widetilde{X}})^{\oplus k}) \cong H^1(C_1,\cal O_{C_1}(-1)^{\oplus rk})=0$. By the exact sequence $0 \to E' \to E \to \cal O_{C_1}^{\oplus k} \to 0$, we see that $\operatorname{Hom}(E,E) \cong \operatorname{Hom}(E',E)$. Since $\operatorname{Hom}(E',E') \to \operatorname{Hom}(E',E)$ is injective, we get that $\operatorname{Hom}(E',E')=\Bbb C$. \end{pf} \begin{cor}\label{cor:spl} Let $E$ be a simple torsion free sheaf of rank $r$ on $X$ with $c_1(E)=c_1$ and $\Delta(E)=(\Delta)$ which is locally free at the center of a blow-up $\phi:\widetilde{X} \to X$, and $E'$ the kernel of a surjective homomorphism $\phi^* E \to \cal O_{C_1}^{\oplus k}$, $0 \leq k <r$. We set $\Delta(E')=\Delta'$. Then, if $Spl(r,\phi^* c_1-kC_1,\Delta')$ is irreducible, $Spl(r,c_1,\Delta)$ is also irreducible. \end{cor} \begin{pf} Let $Spl(r,\phi^* c_1,\Delta)^0$ be the open subscheme of $Spl(r,\phi^* c_1,\Delta)$ of elements $E$ such that $E_{|C_1} \cong \cal O_{C_1}^{\oplus r}$. Then $\phi^*:Spl(r,c_1,\Delta)' \to Spl(r,\phi^* c_1,\Delta)^0$ is an isomorphism, where $Spl(r,c_1,\Delta)'$ is the open dense subspace of $Spl(r,c_1,\Delta)$ consisting of $E$ such that $E$ is locally free at the center of the blow-up. For an $E \in Spl(r,\phi^* c_1,\Delta)^0$, the quotients $\phi^* E \to \cal O_{C_1}^{\oplus k}$ is parametrized by the Grassmannian variety $G(H^0(C_1,E_{|C_1}),k)$. Let $Spl(r,\phi^* c_1-kC_1,\Delta')^0$ be the open subscheme of $Spl(r,\phi^* c_1-kC_1,\Delta')$ of elements $E'$ such that $E'_{C_1} \cong \cal O_{C_1}(1)^{\oplus k} \oplus \cal O_{C_1}^{\oplus (r-k)}$. By using Lemma \ref{lem:spl}, we can show that there is an open subscheme $U$ of $Spl(r,\phi^* c_1-kC_1,\Delta')^0$ and a surjective morphism $U \to Spl(r,\phi^* c_1,\Delta)^0$ such that every fibre is a Grassmannian variety. Hence, the irreducibility of $Spl(r,\phi^* c_1-kC_1,\Delta')$ implies that of $Spl(r,c_1,\Delta)$. \end{pf} \begin{prop} Let $X$ be a Pel Pezzo surface and $c_1$ an element of $\operatorname{NS}(X)$. Then $Spl(r,c_1,\Delta)$ is irreducible. \end{prop} \begin{pf} This follows from Corollary \ref{cor:spl} and Proposition \ref{prop:irr}. \end{pf} \section{Moduli spaces on Abelian surfaces} \subsection{} For a manifold $V$ and $\alpha \in H^*(V,\Bbb Z)$, $[\alpha]_i \in H^i(V,\Bbb Z)$ denotes the $i$-th component of $\alpha$. Let $K(V)$ be the Grothendieck group of $V$. Let $p:X \to \Spec(\Bbb C)$ be an Abelian surface over $\Bbb C$. We set \begin{equation} \begin{cases} H^{ev}(X, \Bbb Z):= H^0(X,\Bbb Z) \oplus H^2(X,\Bbb Z) \oplus H^4(X,\Bbb Z)\\ H^{odd}(X,\Bbb Z):= H^1(X,\Bbb Z) \oplus H^3(X,\Bbb Z). \end{cases} \end{equation} Let $E_0$ be an element of $M_H(r,c_1,\Delta)$. We set \begin{equation} H(r,c_1,\Delta):= \{\alpha \in H^{ev}(X,\Bbb Z)|[p_*((\operatorname{ch} E_0)\alpha)]_0=0 \}. \end{equation} Let $\cal F$ be a quasi-universal family of similitude $\rho$ on $M_H(r,c_1,\Delta) \times X$ [Mu3, Thm. A.5]. Then Mukai [Mu3, Mu5] and Drezet [D, D-N] defines a homomorpism \begin{equation} \kappa_2:H(r,c_1,\Delta) \to H^2(M_H(r,c_1,\Delta),\Bbb Z) \end{equation} such that \begin{equation} \kappa_2(\alpha)=\frac{1}{\rho}[p_{M_H(r,c_1,\Delta)*}(\operatorname{ch}(\cal F)\alpha)]_2. \end{equation} \begin{rem} In the notation of Mukai [Mu5, Sect. 5], $\kappa_2(\alpha)=-\theta_v(\alpha^{\vee})$ and $H(r,c_1,\Delta)=v^{\bot}$, where $v:=(r,c_1,(c_1^2)/2r-\Delta) \in H^{ev}(X, \Bbb Z)$ is the Chern character of $E_0$. and $\vee:H^{ev}(X,\Bbb Z) \to H^{ev}(X,\Bbb Z)$ is the automorphism sending $\alpha=\alpha_0+\alpha_2+\alpha_4,\; \alpha_i \in H^{2i}(X,\Bbb Z)$ to $\alpha^{\vee}=\alpha_0-\alpha_2+\alpha_4$. Since we used Drezet's notation in [Y2,Y3], we shall use Drezet's homomorphism in this note. \end{rem} We also consider the homomorphism: \begin{equation} \kappa_1:H^{odd}(X,\Bbb Z) \to H^1(M_H(r,c_1,\Delta),\Bbb Z) \end{equation} such that \begin{equation} \kappa_1(\alpha)=\frac{1}{\rho}[p_{M_H(r,c_1,\Delta)*}(\operatorname{ch}(\cal F)\alpha)]_1. \end{equation} We note that $\kappa_1$ and $\kappa_2$ do not depend on the choice of $\cal F$. In this section, we shall prove the following theorem. \begin{thm}\label{thm:H2} Let $c_1$ be an element of $\operatorname{NS}(X)$ such that $c_1 \mod r H^2(X,\Bbb Z)$ is a primitive element of $H^2(X,\Bbb Z/r \Bbb Z)$ and $H$ a general ample divisor. We assume that $\dim M_H(r,c_1,\Delta)=2r\Delta+2 \geq 6$. Let $\frak a:M_H(r,c_1,\Delta) \to \operatorname{Alb}(M_H(r,c_1,\Delta))$ be an Albanese map. Then the following holds.\newline $(1)$ $\kappa_1$ is an isomorphism and $\kappa_2$ is injective. \newline $(2)$ \begin{equation}\label{eq:H2} \begin{split} H^2(M_H(r,c_1,\Delta),\Bbb Z) &= \kappa_2(H(r,c_1,\Delta)) \oplus \frak a^* H^2(\operatorname{Alb}(M_H(r,c_1,\Delta),\Bbb Z)\\ &=\kappa_2(H(r,c_1,\Delta))\oplus \bigwedge^2 \kappa_1(H^{odd}(X,\Bbb Z)). \end{split} \end{equation} $(3)$ \begin{equation}\label{eq:NS} \operatorname{NS}(M_H(r,c_1,\Delta)) = \kappa_2(H(r,c_1,\Delta)_{alg}) \oplus \frak a^* \operatorname{NS}(\operatorname{Alb}(M_H(r,c_1,\Delta)), \end{equation} where $H(r,c_1,\Delta)_{alg}: =(H^0(X,\Bbb Z) \oplus \operatorname{NS}(X) \oplus H^4(X,\Bbb Z))\cap H(r,c_1,\Delta)$. \end{thm} \subsection{} We first assume that $X$ is a product of elliptic curves. Let $C_1$ and $C_2$ be elliptic curves and set $X=C_1 \times C_2$. We set $C_k^i:=C_k$ and $X^i:=C_1^i \times C_2^i$ for $i=0,1,\dots,n,a$, and $k=1,2$. Let $\Delta_k^{i,j}$ be the diagonal of $C_k^i \times C_k^j=C_k \times C_k$. Let $p_k^i$ be a point of $C_k^i$. We also denote $c_1(\cal O(p_k^i))$ by $p_k^i$. For simplicity, we denote the pull-backs of $p_k^i$ and $\Delta_k^{i,j}$ to $X^0 \times Y_0 \times X^a$ by $p_k^i$ and $\Delta_k^{i,j}$ respectively. Let $\Delta_X^{i,j,k}$ be the pull-back of the diagonal of $X^i \times X^j \times X^k$ to $X^1 \times X^2 \times \dots \times X^n$ and $\Delta_X^{i,j}$ that of $X^i \times X^j$ to $X^1 \times X^2 \times \dots \times X^n$. We set $Z:=\cup_{i<j<k}\Delta_X^{i,j,k}$. Let $\phi:Y \to (X^1 \times X^2 \times \dots \times X^n) \setminus Z$ be the blow-up of $(X^1 \times X^2 \times \dots \times X^n) \setminus Z$ at the subscheme $\cup_{i<j}\Delta_X^{i,j} \setminus Z$, We set $E^{i,j}:=\phi^{-1}(\Delta_X^{i,j} \setminus Z)$. For $\alpha \in H^*(X,\Bbb Z)$ and the projection $\varpi_i:X^0 \times Y_0 \times X^a \to X^i=X$, $i=0,1,\dots,n,a$, we denote the pull-back of $\alpha$ to $X^0 \times Y_0 \times X^a$ by $\alpha^i$. Then $H^2(\operatorname{Hilb}_X^n,\Bbb Z) \cong H^2(Y,\Bbb Z)^{\frak S_n}$ and $H^2(Y,\Bbb Z)^{\frak S_n}$ is generated by $\sum_{i=1}^n e^i$, $\sum_{i<j}(f^i \cdot g^j-g^i \cdot f^j)$ and $\sum_{i<j}E^{i,j}$ where $e \in H^2(X,\Bbb Z)$ and $f,g \in H^1(X, \Bbb Z)$. Let $\frak a:X^0 \times \operatorname{Hilb}_X^n \to X^0 \times X$ be the Albanese map such that $\frak a((x,I_Z))=(x,\sum_{i=1}^n x_i)$ for reduced subscheme $Z=\cup_i\{x_i \}$. \begin{lem}\label{lem:Ch} $(1)$ Let $F$ be a vector bundle on $C_2^0 \times C_2^a$ such that $F_{|\{t\} \times C_2^a}$, $t \in C_2^0$ is a stable vector bundle of rank $r$ on $C_2^a$ with $\det F_{|\{t\} \times C_2^a} \cong \cal O(\Delta_2^{0,a}+(d-1)p_2^a)_{|\{t\} \times C_2^a}$. Then, \begin{equation} \begin{cases} c_1(F)=\Delta_2^{0,a}+(d-1)p_2^a+(r_1-1+kr)p_2^0,\; k \in \Bbb Z\\ \operatorname{ch}_2(F)=\frac{1}{2r}(c_1(F)^2). \end{cases} \end{equation} If $k=0$, then $\operatorname{ch}_2(F)=d_1 p_2^0 \cdot p_2^a$.\newline $(2)$ Let $F_i$ ($1 \leq i \leq n$) be a vector bundle on $C_2^i \times C_2^a$ such that $F_{i|\{t\} \times C_2^a}$, $t \in C_2^i$ is a stable vector bundle of rank $r_2$ on $C_2^a$ with $\det F_{i|\{t\} \times C_2^a} \cong \cal O(\Delta_2^{i,a}+(d_2-1)p_2^a)_{|\{t\} \times C_2^a}$. Then, \begin{equation} \begin{cases} c_1(F_i)=\Delta_2^{i,a}+(d_2-1)p_2^a+(r_1-1+kr)p_2^i,\; k \in \Bbb Z\\ \operatorname{ch}_2(F_i)=\frac{1}{2r_2}(c_1(F_i)^2). \end{cases} \end{equation} If $k=0$, then $\operatorname{ch}_2(F_i)=d_1 p_2^i \cdot p_2^a$. \end{lem} \begin{pf} We shall only prove (1). We set $c_1(F)=\Delta_2^{0,a}+(d-1)p_2^a+(r_1-1+x)p_2^0$, $x \in \Bbb Z$. Since $F_{|\{t\} \times C_2^a}$, $t \in C_2^0$ is a stable vector bundle, $\Delta(F)=c_2(F)-(c_1(F)^2)(r-1)/2r=0$. Hence we get that $\operatorname{ch}_2(F)=-(c_2(F)-(c_1(F)^2)/2)=(c_1(F)^2)/2r$. We note that $c_2(F)=(d(r_1+x)-1)(r-1)/r$ is an integer. Hence $d(r_1+x)-1=rd_1+rx$ is a multiple of $r$. Since $(r,d)=1$, $x$ is a multiple of $r$. We also see that $(c_1(F)^2)/2r=d_1 p_2^0 \cdot p_2^a$ for the case $x=0$. \end{pf} Let $F$ and $F_i$ be vector bundles in Lemma \ref{lem:Ch} and assume that $k=0$. We also denote the pull-backs of $F$ and $F_i$ to $C_2^0 \times C_2^i \times C_2^a$ by $F$ and $F_i$ respectively. Let $q_{C_2^0 \times C_2^i}: C_2^0 \times C_2^i \times C_2^a \to C_2^0 \times C_2^i$ be the projection. We set $\cal L:=\operatorname{Hom}_{q_{C_2^0 \times C_2^a}}(F,F_i)$. Then $c_1(\cal L)=-\Delta_2^{0,i}$. \begin{pf} By using the Grothendieck-Riemann-Roch theorem and the above lemma, we see that \begin{align*} c_1(\cal L)&=[q_{C_2^0 \times C_2^i*}(\operatorname{ch}(F^{\vee})\operatorname{ch}( F_i))]_2\\ &=[q_{C_2^0 \times C_2^i*}(r-c_1(F)+\frac{1}{2r}(c_1(F)^2)) (r_2-c_1(F_i)+\frac{1}{2r_2}(c_1(F_i)^2))]_2\\ &=[q_{C_2^0 \times C_2^i*}(rr_2+(rc_1(F_i)-r_2c_1(F))+ \frac{1}{2rr_2}((rc_1(F_i)-r_2c_1(F))^2))]_2\\ &=\frac{1}{2rr_2}[q_{C_2^0 \times C_2^i*}((rc_1(F_i)-r_2c_1(F))^2)]_2\\ &=-\Delta_2^{0,i}. \end{align*} \end{pf} Let $Y_0$ be the complement of the closed subset $W:=\cup_{i<j<k}(\widetilde{\Delta}_1^{i,j} \cap \widetilde{\Delta}_1^{j,k}) \cup \cup_{i<j} (\widetilde{\Delta}_1^{i,j} \cap E^{i,j})$ of $Y$, where $\Delta_1^{i,j}=\widetilde{\Delta}_1^{i,j} \cup E^{i,j}$. Since $\operatorname{codim} W=2$, $H^2(X^0 \times Y_0,\Bbb Z) \cong H^2(X^0 \times Y,\Bbb Z)$. We shall construct a family of stable sheaves on $X$ parametrized by $X^0 \times Y_0$. For simplicity, we denote the pull-backs of $F$ and $F_i$ to $X^0 \times Y_0 \times X^a$ by $F$ and $F_i$ respectively. Then there is a homomorphism: \begin{equation} \Lambda:F \otimes \cal O(\Delta_1^{0,a}-p_1^a) \to \oplus_{i=1}^n (F_{i|\Delta_1^{i,a}} \otimes L^i), \end{equation} where $L^i$ is a line bundle on $X^0 \times Y_0 \times X^a$ such that $c_1(L^i)=\Delta_1^{0,i}-p_1^i+\Delta_2^{0,i}$. Let $\cal E$ be the kernel of this homomorphism and $\cal Q$ the cokernel. Then $\cal Q \cong \oplus_{i<j}((F_i/G_j)_{|\Delta_1^{i,a} \cap \widetilde{\Delta}_1^{i,j}} \otimes L^i \oplus(F_i \otimes L^i_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}}))$, where $G_i:=\ker(F \to F_i)$. We first assume that $r_1 \leq r_2$. Then $G_{j|\Delta_1^{i,a}} \to F_{i|\Delta_1^{i,a}}$ is injective and $(F_i/G_j)_{|\Delta_1^{i,a}}$ is flat over $X^0 \times Y_0$. Hence we see that \begin{equation} \operatorname{Tor}^{\cal O_{X^0 \times Y_0}}_2((F_i/G_j)_{|\Delta_1^{i,a} \cap \widetilde{\Delta}_1^{i,j}},k(x))=0,\; x \in X^0 \times Y_0. \end{equation} Since $F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}}$ is also flat over $X^0 \times Y_0$, we get that \begin{equation} \operatorname{Tor}^{\cal O_{X^0 \times Y_0}}_2 (F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}},k(x))=0,\; x \in X^0 \times Y_0. \end{equation} Hence we see that $\operatorname{Tor}^{\cal O_{X^0 \times Y_0}}_1(\operatorname{im}(\Lambda),k(x))=0$, which implies that $\cal E$ is flat over $X^0 \times Y_0$ and $\cal E \otimes k(x)$ is torsion free. Then $\cal E$ defines a family of stable sheaves on $X$ parametrized by $ X^0 \times Y_0.$ It defines a morphism $X^0 \times Y_0 \to M(r,c_1,\Delta)$, which is $\frak S_n$-invariant. Hence we get a morphism $\nu:X^0 \times (Y_0/\frak S_n) \to M(r,c_1,\Delta)$. Let $\overline{\kappa_2}:H(r,c_1,\Delta) \to H^2(X^0 \times Y_0,\Bbb Z)/\frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z)$ be the homomorphism sending $\alpha \in H(r,c_1,\Delta)$ to $[p_{X^0 \times Y_0*} (\operatorname{ch}(\cal E) \alpha)]_2 \mod \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z)$. Since $\kappa_2$ does not depend on the choice of quasi-universal families, we shall compute the image of $\overline{\kappa_2}$. \begin{align*} \operatorname{ch}(\cal E) &= \operatorname{ch}(F \otimes \cal O(\Delta_1^{0,a}-p_1^a))- \sum_{i=1}^n \operatorname{ch}(F_i \otimes L^i_{|\Delta_1^{i,a}})+ \sum_{i<j}\operatorname{ch}(F_i\otimes L^i_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}}) \\ &\phantom{ (r+c_1(F)+d_1p_2^0\cdot p_2^a)(1+\Delta_1^{0,a})-} +\sum_{i<j}\operatorname{ch}(F_i/G_j \otimes L^i_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}})\\ &= (r+c_1(F)+d_1p_2^0\cdot p_2^a)(1+\Delta_1^{0,a}-p_1^a-p_1^0 \cdot p_1^a) -\sum_{i=1}^n \Delta_1^{i,a} (r_2+c_1(F_i)+d_1p_2^i\cdot p_2^a)(\operatorname{ch} L^i)\\ &\phantom{ (r+c_1(F)}+ \sum_{i<j}\operatorname{ch}(F_i \otimes L^i_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}})+\sum_{i<j}\operatorname{ch}(F_i/G_j \otimes L^i_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}}). \end{align*} Since $[p_{X^0 \times Y_0*} (\operatorname{ch}(F \otimes \cal O(\Delta_1^{0,a}-p_1^a)) \alpha^a)]_2 \equiv 0,\; \sum_{i=1}^n\Delta_1^{0,i}-p_1^i \equiv 0 \mod \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z)$, we get that \begin{align*} \overline{\kappa_2}(\alpha)= & -\sum_{i=1}^n [p_{X^0 \times Y_0*} (\Delta_1^{i,a} (r_2+c_1(F_i)+d_1p_2^i\cdot p_2^a)(1+\Delta_2^{0,i})\alpha^a)]_2\\ &\phantom{-\sum_{i=1}^n [p_{X^0 \times Y_0*}} +\sum_{i<j}[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}})\alpha^a)]_2\\ &\phantom{-\sum_{i=1}^n [p_{X^0 \times Y_0*}(\Delta_1^{i,a}} +\sum_{i<j}[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i/G_j \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}})\alpha^a)]_2. \end{align*} Let $\alpha=x_1+x_2p_1+x_3p_2+x_4p_1 \cdot p_2 +D$ be an element of $H(r,c_1,\Delta)$, $D \in H^1(C_1, \Bbb Z) \otimes H^1(C_2, \Bbb Z)$. Then we see that $0=[p_*((\operatorname{ch} E_0) \alpha)]_0= [p_*((r+dp_2-r_2np_1-d_2n p_1 \cdot p_2)\alpha)]_0 =-d_2n x_1-r_2 n x_3+d x_2+rx_4$. Thus $\alpha$ satisfies \begin{equation}\label{eq:perp} d x_2+r x_4=d_2n x_1+r_2 n x_3. \end{equation} By a simple calculation, we get that \begin{equation} \left\{ \begin{aligned} &[p_{X^0 \times Y_0*}(\operatorname{ch} (F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}})]_2 = d_2 \Delta_2^{0,i}+d_1 p_2^i\\ &[p_{X^0 \times Y_0*}(\operatorname{ch} (F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}})p_2^a )]_2 =r_2 \Delta_2^{0,i}+r_1 p_2^i\\ &[p_{X^0 \times Y_0*}(\operatorname{ch} (F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}})p_1^a )]_2 =d_2 p_1^i\\ &[p_{X^0 \times Y_0*}(\operatorname{ch} (F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}})D^a )]_2 =D^i\\ &[p_{X^0 \times Y_0*}(\operatorname{ch} (F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}})(p_1^a \cdot p_2^a))]_2 =r_2 p_1^i, \end{aligned} \right. \end{equation} \begin{equation} \left\{ \begin{aligned} &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i/G_j \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}}))]_2= (2d_2-d)\widetilde{\Delta}_1^{i,j} \\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i/G_j \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}})p_2^a)]_2 =(2r_2-r) \widetilde{\Delta}_1^{i,j}\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i/G_j \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}})p_1^a)]_2=0\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i/G_j \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}})D^a)]_2=0\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i/G_j \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{\widetilde{\Delta}_1^{i,j}})(p_1^a \cdot p_2^a))]_2=0, \end{aligned} \right. \end{equation} and \begin{equation} \left\{ \begin{aligned} &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}}))]_2=d_2 E^{i,j}\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}})p_2^a)]_2=r_2 E^{i,j}\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}})p_1^a)]_2=0\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}})D^a)]_2=0\\ &[p_{X^0 \times Y_0*}(\operatorname{ch}(F_i \otimes \cal O(\Delta_2^{0,i})_{|\Delta_1^{i,a}} \otimes \cal O_{E^{i,j}})(p_1^a \cdot p_2^a))]_2=0, \end{aligned} \right. \end{equation} where $D \in H^1(C_1, \Bbb Z) \otimes H^1(C_2, \Bbb Z)$. Hence we get that \begin{equation} \begin{align*} \overline{\kappa_2}(\alpha)=& -\sum_{i=1}^n(d_2 x_1+r_2 x_3)\Delta_2^{0,i}- \sum_{i=1}^n(d_2 x_2+r_2 x_4)p_1^i -\sum_{i=1}^n(d_1 x_1+r_1 x_3)p_2^i-\sum_{i=1}^n D^i\\ &+\sum_{i<j}((2d_2-d)x_1+(2r_2-r) x_3)\widetilde{\Delta}_1^{i,j} +\sum_{i<j}(d_2 x_1+r_2 x_3)E. \end{align*} \end{equation} We note that \begin{equation} \left\{ \begin{aligned} &\sum_{i=1}^n \Delta_2^{0,i} \equiv \sum_{i=1}^n p_2^i \mod \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z)\\ &\sum_{i<j}\Delta_1^{i,j} \equiv 2(n-1)\sum_{i=1}^n p_1^i \mod \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z)\\ & \widetilde{\Delta}_1^{i,j}=\Delta_1^{i,j}-E^{i,j}. \end{aligned} \right. \end{equation} Therefore we get that \begin{equation} \overline{\kappa_2}(\alpha) =y_1(\sum_{i=1}^n p_2^i) +y_2 (\sum_{i=1}^n p_1^i) +y_3(\sum_{i<j} E^{i,j})-\sum_{i=1}^n D^i, \end{equation} where \begin{equation} \begin{cases} y_1=-(dx_1+r x_3)\\ y_2=-\{(d_2x_2+r_2 x_4)-2(n-1)((2d_2-d)x_1+(2r_2-r)x_3)\}\\ y_3=(d_1x_1+r_1x_3)\\ y_4=dx_2+rx_4-n(d_2x_1+r_2x_3). \end{cases} \end{equation} Since $dr_1-rd_1=d_2r-dr_2=1$, the homomorphism $\psi:\Bbb Z^{\oplus 4} \to \Bbb Z^{\oplus 4}$ sending $(x_1,x_2,x_3,x_4)$ to $(y_1,y_2,y_3,y_4)$ is an isomorphism. The condition \eqref{eq:perp} implies that $y_4=0$. Therefore, \begin{equation} \overline{\kappa_2}: H(r,c_1,\Delta) \to H^2(X^0 \times Y_0,\Bbb Z)^{\frak S_n}/ \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z) \end{equation} is an isomorphism. Since $H^2(X^0 \times Y_0,\Bbb Z)^{\frak S_n} \cong H^2(X^0 \times \operatorname{Hilb}_X^n,\Bbb Z)$, we get that \begin{equation} H(r,c_1,\Delta) \to H^2(X^0 \times \operatorname{Hilb}_X^n,\Bbb Z)/ \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z) \end{equation} is an isomorphism. We next treat the case $r_1>r_2$. Since $G_j \to F_i$ is surjective, we get that \begin{equation} \begin{align*} \overline{\kappa_2}(\alpha)=& -\sum_{i=1}^n(d_2 x_1+r_2 x_3)\Delta_2^{0,i}- \sum_{i=1}^n(d_2 x_2+r_2 x_4)p_1^i -\sum_{i=1}^n(d_1 x_1+r_1 x_3)p_2^i\\ &-\sum_{i=1}^n D^i+\sum_{i<j}(d_2 x_1+r_2 x_3)E. \end{align*} \end{equation} In the same way as in the case $r_1 \leq r_2$, we see that \begin{equation} H(r,c_1,\Delta) \to H^2(X^0 \times \operatorname{Hilb}_X^n,\Bbb Z)/ \frak a^* H^2(\operatorname{Alb}(X^0 \times \operatorname{Hilb}_X^n),\Bbb Z) \end{equation} is an isomorphism. Therefore $\kappa_2$ is injective and $H^2(M_H(r,c_1,\Delta))$ is generated by $\operatorname{im}(\kappa_2)$ and $\operatorname{im}(\frak a)$. By using similar computations, we see that $\kappa_1$ is an isomorphism. Hence Theorem \ref{thm:H2} (1), (2) hold for this case. \subsection{} We next treat general cases. Replacing $c_1$ by $c_1+r c_1(H)$, we may assume that $c_1$ belongs to the ample cone. \begin{prop}\label{prop:1} Let $(X,L)$ be a pair consisting of abelian surface $X$ and an ample divisor $L$ of type $(d_1,d_2)$, where $d_1$ and $d_2$ are positive integers of $d_1|d_2$ and $(r, d_1)=1$. Then Theorem \ref{thm:H2} $(1)$, $(2)$ hold for $M_H(r,c_1(L),\Delta)$, where $H$ is a general polarization. \end{prop} \begin{pf} Let $(X,L)$ be a pair consisting of abelian surface $X$ and an ample divisor $L$ of type $(d_1,d_2)$, where $d_1$ and $d_2$ are positive integers of $d_1|d_2$ and $(r, d_1)=1$. We shall choose an ample line bundle $H$ on $X$ which is not lie on walls. Let $T$ be a connected smooth curve and $(\cal X, \cal L)$ a pair of a smooth family of abelian surface $p_T:\cal X \to T$ and a relatively ample line bundle $\cal L$ of type $(d_1,d_2)$. For points $t_0, t_1 \in T$, we assume that $(\cal X_{t_0},\cal L_{t_0}) =(X,L)$ and $\cal X_{t_1}$ is an abelian surface of $\operatorname{NS}(\cal X_{t_1}) \cong \Bbb Z$. Let $g:Pic_{\cal X/T} \to T$ be the relative Picard scheme. We denote the connected component of $Pic_{\cal X/T}$ containing the section of $g$ which corresponds to the family $\cal L$ by $Pic^{\xi}_{\cal X/T}$. Since $Pic^0_{\cal X/T} \cong Pic^{\xi}_{\cal X/T}$, $Pic^{\xi}_{\cal X/T} \to T$ is a smooth morphism. Let $h:\overline{\cal M}_{\cal X/T}(r,\xi,\Delta) \to T$ be the moduli scheme parametrizing $S$-equivalence classes of $\cal L_t$-semi-stable sheaves $E$ on $\cal X_t$ with $(\operatorname{rk}(E),c_1(E),\Delta(E))=(r,c_1(\cal L_t),\Delta)$ [Ma1]. Let $D$ be the closed subset of $\overline{\cal M}_{\cal X/T}(r,\xi,\Delta)$ consisting of properly $\cal L_t$-semi-stable sheaves on $\cal X_t$. Since $h$ is a proper morphism, $h(D)$ is a closed subset of $T$. Since $h(D)$ does not contain $t_1$ and $T$ is an irreducible curve, $h(D)$ is a finite point set. Replacing $T$ by the open subscheme $T \setminus (h(D)\setminus \{t_0\})$, we may assume that $\cal L_t$-semi-stable sheaves are $\cal L_t$-stable for $t \ne t_0$. Let $s:\cal Spl_{\cal X/T}(r,\xi,\Delta) \to T$ be the moduli of simple sheaves $E$ on $\cal X_t, t \in T$ with $(\operatorname{rk}(E),c_1(E),\Delta(E))=(r,c_1(\cal L_t),\Delta)$ [A-K, Thm. 7.4]. Let $U_1$ be the closed subset of $s^{-1}(T \setminus \{t_0 \})$ consisting of simple sheaves on $\cal X_t$, $t \in T \setminus \{t_0 \}$ which are not stable with respect to $\cal L_t$ and $\overline{U_1}$ the closure of $U_1$ in $\cal Spl_{\cal X/T}(r,\xi,\Delta)$. Let $U_2$ be the closed subset of $s^{-1}(t_0)$ consisting of simple sheaves which are not semi-stable with respect to $H$. Then we can show that $\overline{U_1} \cap s^{-1}(t_0)$ is a subset of $U_2$ (see the second paragraph of the proof of Lemma \ref{lem:1}). We set $\cal M:=\cal Spl_{\cal X/T}(r,\xi,\Delta) \setminus (\overline{U_1} \cup U_2)$. Then $\cal M$ is an open subspace of $\cal Spl_{\cal X/T}(r,\xi,\Delta)$ which is of finite type and contains all $H$-stable sheaves on $\cal X_{t_0}$. By using valuative criterion of separatedness and properness, we get that $s:\cal M \to T$ is a proper morphism. In fact, since $\cal M \times_T (T \setminus \{t_0\}) \to T \setminus \{t_0\}$ is proper, it is sufficient to check these properties near the fibre $\cal X_{t_0}$. The separatedness follows from base change theorem and stability with respect to $H$ (cf. [A-K, Lem. 7.8]), and the properness follows from the following lemma (Lemma \ref{lem:1}) and the projectivity of $\cal M_{t_0}$. Since $Pic^{\xi}_{\cal X/T} \to T$ is a smooth morphism, [Mu2, Thm. 1.17] implies that $s:\cal M \to T$ is a smooth morphism. Let $\frak a_T:\cal M \to \operatorname{Alb}_{\cal M/T}$ be the family of Albanese map over $T$. Let $\cal F_T$ be a quasi-universal family of similitude $\rho$ on $\cal M \times_T \cal X$ and we shall consider the homomorphism \begin{equation} \begin{cases} \kappa_{1,t}:H^{odd}(\cal X_t,\Bbb Z) \to H^1(\cal M_t,\Bbb Z)\\ \kappa_{2,t}:H(r,c_1(\cal L_t),\Delta) \to H^2(\cal M_t,\Bbb Z) \end{cases} \end{equation} such that $\kappa_{i,t}(\alpha_{i,t})= \frac{1}{\rho}[p_{\cal M_t*}((\operatorname{ch} \cal F_t)\alpha_t)]_i $, where $\alpha_{1,t} \in H^{odd}(\cal X_t,\Bbb Z)$, $\alpha_{2,t} \in H(r,c_1(\cal L_t),\Delta)$. We assume that $\cal X_{t_0}$ is a product of elliptic curves. Since $p_T$ and $s$ are smooth, Theorem \ref{thm:H2} (1),(2) for the pair $(\cal X_{t_0},\cal L_{t_0})$ imply that Theorem \ref{thm:H2} (1),(2) also hold for all pairs $(\cal X_t, \cal L_t)$, $t \in T$. By the connectedness of the moduli of $(d_1,d_2)$-polarized abelian surfaces (cf. [L-B, 8]), \eqref{eq:H2} holds for all pairs $(X,L)$ of $(d_1,d_2)$-polarized abelian surfaces. \end{pf} The following is due to Langton [L]. \begin{lem}\label{lem:1} Let $R$ be a discrete valuation ring, $K$ the quotient field of $R$, and $k$ the residue field of $R$. Let $\Spec(R) \to T$ be a dominant morphism such that $\Spec(k) \to T$ defines the point $t_0$. For a stable sheaf $E_K$ on $X_K$, there is a $R$-flat coherent sheaf $E$ on $X_R$ such that $E \otimes_R K=E_K$ and $E \otimes_R k$ is a $H$-stable sheaf. \end{lem} \begin{pf} Let $E^0$ be an $R$-flat coherent sheaf on $X_R$ such that $E^0 \otimes_R K=E_K$ and $E^0_k:=E^0 \otimes_R k$ is torsion free. If $E_k^0$ is $H$-stable, then we put $E=E^0$. We assuime that $E^0_k$ is not $H$-stable. Let $F ^0_k (\subset E^0_k)$ be the first filter of the Harder-Narasimhan filtration of $E^0_k$ with respect to $H$. We set $E^1:=\ker(E^0 \to E^0_k/F^0_k)$. Then $E^1$ is an $R$-flat coherent sheaf on $X_R$ with $E^1_K=E_K$. If $E^1_k$ is not $H$-stable, then we shall consider the first filter $F_k^1$ of the Harder-Narasimhan filtration of $E^1_k$ and set $E^2:=\ker(E^1 \to E^1_k/F^1_k)$. Continuing this procedure successively, we obtain a decreasing sequence of $R$-flat coherent sheaves on $X_R$: $E^0 \supset E^1 \supset E^2 \supset \cdots$. We assume that this sequence is infinite. Then in the same way as in [L, Lem. 2], we see that there is an integer $i$ such that $E^i \otimes_R \widehat{R}$ has a subsheaf $F$ of rank $r'$ with $F \otimes_R k=F_k^i$, where $\widehat{R}$ is the completion of $R$. We set $\widehat{K}:=K \otimes_R \widehat{R}$ and $D:=\det(E^i\otimes_R \widehat{R})^{\otimes r'}\otimes \det(F)^{\otimes (-r)}$. Let $P(x)$ be the Hilbert polynomial of $D$ with respect to $\cal L_{\widehat{R}}$. Let $V$ be a locally free sheaf on $\cal X$ such that there is a surjective homomorphism $V \otimes_{\cal O_T} \widehat{R} \to D$, and we shall consider the quot scheme $\cal Q:=\operatorname{Quot}_{V/\cal X/T}^{P(x)}$. Then $D$ defines a morphism $\tau:\Spec(\widehat{R}) \to \cal Q$ such that $D=(\tau \times_T 1_{\cal X})^* \cal D$, where $\cal D$ is the universal quotient. Let $\cal Q_0$ be the connected component of $\cal Q$ which contains the image of $\Spec(\widehat{R})$. Since $\Spec(\widehat{R}) \to T$ is dominant, $\frak q:\cal Q_0 \to T$ is dominant, and hence surjective. Since $E^i_{\widehat{K}} \cong E_K \otimes_K \widehat{K}$ is a stable sheaf on $X_{\widehat{K}}$, $(\cal D_{q_1},\cal L_{q_1})=(\cal D_{\widehat{K}},\cal L_{\widehat{K}})>0$, where $q_1$ is a point of $\frak q^{-1}(t_1)$. Since $\operatorname{NS}(\cal X_{t_1}) \cong \Bbb Z$, we get that $c_1(\cal D_{q_1})=l c_1(\cal L_{q_1})$, $l>0$. Hence we obtain that $(\cal D_{\tau(t_0)}^2)>0$ and $(\cal D_{\tau(t_0)},\cal L_{\tau(t_0)})>0$. By the Riemann-Roch theorem and the Serre duality, we see that $\cal D_{\tau(t_0)}$ is an effective divisor. Therefore $(\cal D_{\tau(t_0)},H)>0$, which is a contradiction. Hence there is an integer $n$ such that $E^n \otimes_R k$ is $H$-stable. \end{pf} {\it Proof of Theorem} \ref{thm:H2} (3). Let $\kappa_2':H(r,c_1,\Delta)\otimes \Bbb C \to H^2(M(r,c_1,\Delta),\Bbb C)$ be the homomorphism induced by $\kappa_2$. We note that $H^{2,0}(X)$ and $H^{0,2}(X)$ are subsets of $H(r,c_1,\Delta)\otimes \Bbb C$. Since $\operatorname{ch}_i(\cal F)$ is of type $(i,i)$, we see that \begin{equation} \begin{cases} \kappa_2'(H^{2,0}(X)) \subset H^{2,0}(M_H(r,c_1,\Delta))\\ \kappa_2'(\oplus_{p=0}^2 H^{p,p}(X)) \subset H^{1,1}(M_H(r,c_1,\Delta))\\ \kappa_2'(H^{0,2}(X)) \subset H^{0,2}(M_H(r,c_1,\Delta)). \end{cases} \end{equation} Since $H(r,c_1,\Delta)\otimes \Bbb C=H^{2,0}(X)\oplus (\oplus_{p=0}^2 H^{p,p}(X)) \cap H(r,c_1,\Delta)\otimes \Bbb C \oplus H^{0,2}(X)$ and $\frak a^*$ preserves the type, we obtain that $$ H^{1,1}(M_H(r,c_1,\Delta))=\kappa_2'((\oplus_{p=0}^2 H^{p,p}(X))\cap H(r,c_1,\Delta)\otimes \Bbb C)\oplus \frak a^* (H^{1,1}(\operatorname{Alb}(M_H(r,c_1,\Delta)). $$ Hence we get Theorem \ref{thm:H2} (3). \qed Combining [Y4, Thm. 2.1] with the proof of Proposition \ref{prop:1}, we get the following theorem. \begin{thm}\label{thm:B} Let $X$ be an abelian surface defined over $\Bbb C$ and $c_1 \in \operatorname{NS}(X)$ a primitive element. Then $$ P(M_H(2,c_1,\Delta),z)=P(M_H(1,0,2\Delta),z) $$ for a general polarization $H$, where $P(\quad \; ,z)$ is the Poincar\'{e} polynomial. \end{thm} \subsection{} We shall next consider the Albanese variety of $M_H(r,c_1,\Delta)$. Let $\cal P$ be the Poincar\'{e} line bundle on $\widehat{X} \times X$, where $\widehat{X}$ is the dual of $X$. For an element $E_0 \in M_H(r,c_1,\Delta)$, let $\alpha_{E_0}:M_H(r,c_1,\Delta) \to X$ be the morphism sending $E \in M_H(r,c_1,\Delta)$ to $\det p_{\widehat{X}!}((E-E_0)\otimes(\cal P-\cal O_{\widehat{X} \times X})) \in \operatorname{Pic}^0(\widehat{X})=X$, and $\det_{E_0}:M_H(r,c_1,\Delta) \to \widehat{X}$ the morphism sending $E$ to $\det E \otimes \det E_0^{\vee} \in \widehat{X}$ (cf. [Y3, Sect. 5]). We shall show that $\frak a_{E_0}:=\det_{E_0} \times \alpha_{E_0} $ is the Albanese map of $M_H(r,c_1,\Delta)$. Let $B$ be an effective divisor on $X$. Then we see that \begin{align*} &\det p_{\widehat{X}!}((E-E_0)\otimes \cal O_B \otimes (\cal P-\cal O_{\widehat{X} \times X}))\\ =& \det p_{\widehat{X}!}((\det E_{|B}-\det E_{0|B})\otimes (\cal P-\cal O_{\widehat{X} \times X}))\\ =& \zeta (\det\nolimits_{E_0}(E)), \end{align*} where $\zeta:\widehat{X} \to X$ is the morphism sending $L \in \widehat{X}$ to $\otimes_i \cal P_{\widehat{X} \times \{x_i\}} \in \operatorname{Pic}^0(\widehat{X})=X$, $L \cdot B = \sum_i x_i$. Therefore if $\frak a_{E_0}$ is the Albanese map for $M_H(r,c_1,\Delta)$, then $\frak a_{E_0(B)}$ is the Albanese map for $M_H(r,c_1+rc_1(\cal O_X(B)), \Delta)$. Hence we may assume that $c_1$ belongs to the ample cone. In the notation of Proposition \ref{prop:1}, we assume that there is a section $\sigma:T \to \cal M$ of $s$. Then we can also construct a morphism $\frak a_{\sigma}:\cal M \to \operatorname{Pic}^0_{\cal X/T} \times_T\cal X$. In fact, it is sufficient to construct the morphism on small neighbourhoods $U$ (in the sense of classical topology) of each points. By using a universal family on $U \times_T \cal X$, we get the morphism. Since $s: \cal M \to T$ and $\operatorname{Pic}^0_{\cal X/T}\times_T \cal X \to T$ are smooth over $T$, it is sufficient to prove that \begin{equation} \frak a_{E_0}^*:H^1(\widehat{X} \times X,\Bbb Z) \to H^1(M_H(r,c_1,\Delta),\Bbb Z) \end{equation} is an isomorphism, if $X$ is a product of elliptic curves. In order to prove this assertion, we shall show that \begin{equation}\label{eq:E0} \frak a_{E_0}^*:\operatorname{Pic}^0(\widehat{X} \times X) \to \operatorname{Pic}^0(M(r,c_1,\Delta)) \end{equation} is an isomorphism. Let $\cal E$ be a universal family on $M(r,c_1,\Delta)$. For simplicity, we set $M:=M(r,c_1,\Delta)$. Let $\widehat{X} \times X \to \operatorname{Pic}^0(X \times \widehat{X})$ be the isomorphism sending $(\hat{x},x) \in \widehat{X} \times X$ to $\cal P_{|\{\hat{x}\} \times X} \otimes \cal P_{|\widehat{X} \times \{x\}}$. We set $\cal R:=\det p_{\widehat{X} \times M!}((\cal E-E_0 \otimes \cal O_M) \otimes (\cal P-\cal O_{\widehat{X} \times X}))$. By the construction of $\alpha_{E_0}$, we get that $\cal R \cong (1_{\widehat{X}} \times \alpha_{E_0})^* \cal P \otimes L$, where $L$ is the pull-back of a line bundle on $M$. Since $\cal R_{|\{0\} \times M}\cong \cal O_M$, we get that $L\cong \cal O_{\widehat{X} \times M}$. Hence we see that \begin{equation}\label{eq:6-12-1} \begin{split} \alpha^*_{E_0}(\cal P_{|\{\hat{x}\}\times X}) &= \det p_{M!}((\cal E-E_0 \otimes \cal O_M) \otimes(\cal P_{|\{\hat{x}\}\times X}-\cal O_X))\\ &=\det p_{M!}(\cal E\otimes(\cal P_{|\{\hat{x}\}\times X}-\cal O_X)). \end{split} \end{equation} In the same way, we see that \begin{equation}\label{eq:6-12-2} \begin{split} \det\nolimits_{E_0}^*(\cal P_{|\widehat{X} \times \{x\}})&= (\det \cal E \otimes \det E_0^{\vee} \otimes \det \cal E_{|M \times \{0\}}^{\vee})_{|M \times \{x\}}\\ &=\det p_{M!}(\cal E \otimes (k_x-k_0)), \end{split} \end{equation} where $0 \in X$ is the zero of the group low. In order to prove \eqref{eq:E0}, we shall consider the pull-backs of $\alpha^*_{E_0}(\cal P_{|\{\hat{x}\}\times X})$ and $\det_{E_0}^*(\cal P_{|\widehat{X} \times \{x\}})$ to $X^0 \times Y_0$. We denote the zero of $C_1$ and $C_2$ by $0_1$ and $0_2$ respectively. For a point $q_k$ of $C_k$, $k=1,2$, we set $l_k:=q_k-0_k$. We also denote the pull-back of $l_k$ to $X=C_1 \times C_2$ by $l_k$. In the same way as in , we denote $\varpi_i^{!} (G)$, $i=0,1,\dots,n,a$ by $G^i$, $G \in K(X)$. We also denote $\cal O_X(D)^i$ by $\cal O_{X^0 \times Y_0}(D^i)$. By simple calculations, we see that \begin{equation}\label{eq:6-12-3} \left\{ \begin{aligned} & \det p_{X^0 \times Y_0!}(\cal E\otimes(\cal O_X(l_1)-\cal O_X)^a) =\cal O_{X^0 \times Y_0}(dl^0_1-d_2\sum_{i=1}^n l_1^i)\\ & \det p_{X^0 \times Y_0!}(\cal E\otimes(\cal O_X(l_2)-\cal O_X)^a) = \cal O_{X^0 \times Y_0}(\sum_{i=1}^n l_2^i)\\ & \det p_{X^0 \times Y_0!}(\cal E\otimes(k_{(q_1,0_2)}-k_{(0_1,0_2)})^a) = \cal O_{X^0 \times Y_0}(rl_1^0-r_2\sum_{i=1}^n l_1^i)\\ & \det p_{X^0 \times Y_0!}(\cal E\otimes(k_{(0_1,q_2)}-k_{(0_1,0_2)})^a) =\cal O_{X^0 \times Y_0}(l_2^0). \end{aligned} \right. \end{equation} Since $d_2 r-d r_2=1$ and $\operatorname{Pic}^0(X^0 \times \operatorname{Hilb}_X^n) \cong \operatorname{Pic}^0(X^0 \times Y_0)^{\frak S_n}$, \eqref{eq:6-12-1}, \eqref{eq:6-12-2} and \eqref{eq:6-12-3} implies that \eqref{eq:E0} holds. We set \begin{equation} K(r,c_1,\Delta):=\{\alpha \in K(X)| \chi(\alpha \otimes E_0)=0, E_0 \in M_H(r,c_1,\Delta)\}. \end{equation} Let $\{U_i \}$ be an open covering of $M_H(r,c_1,\Delta)$ such that there are universal family $\cal F_i$ on each $U_i \times X$ and $\cal F_{i|(U_i \cap U_j) \times X} \cong \cal F_{j|(U_i \cap U_j) \times X}$. Since the action of $\cal O_{U_i}^{\times}$ to $\det p_{U_i!}(\cal F_i \otimes \alpha)$ is trivial, we get a line bundle $\tilde{\kappa}(\alpha)$ on $M_H(r,c_1,\Delta)$. Thus we obtain a homomorphism \begin{equation} \tilde{\kappa}:K(r,c_1,\Delta) \to \operatorname{Pic}(M_H(r,c_1,\Delta)). \end{equation} We note that there is a commutative diagram: \begin{equation} \begin{CD} K(r,c_1,\Delta) @>{\tilde{\kappa}}>> \operatorname{Pic}(M_H(r,c_1,\Delta))\\ @V{\operatorname{ch}}VV @VV{c_1}V\\ H(r,c_1,\Delta) @>{\kappa_2}>> H^2(M_H(r,c_1,\Delta),\Bbb Z) \end{CD} \end{equation} Let $K^2$ be the subgroup of $K(r,c_1,\Delta)$ generated by $k_P-k_0$, $P \in X$ and $N$ the kernel of the Albanese map $K^2 \to X$. Since $\ker (\operatorname{ch})$ is generated by $\cal O_X(D)-\cal O_X$, $\cal O_X(D) \in \operatorname{Pic}^0(X)$ and $k_P-k_0$, $P \in X$, $\eqref{eq:6-12-1}$ and $\eqref{eq:6-12-2}$ implies that $\tilde{\kappa}$ induces an isomorphism $\ker(\operatorname{ch})/N \to \operatorname{Pic}^0(M_H(r,c_1,\Delta))$. By using Theorem \ref{thm:H2} (3), we get the following theorem, which is similar to [Y2, Thm. 0.1]. \begin{thm}\label{thm:Pic} Under the same assumption as in Theorem \ref{thm:H2}, the following holds.\newline $(1)$ $\frak a_{E_0}:M_H(r,c_1,\Delta) \to \widehat{X} \times X$ is an Albanese map.\newline $(2)$ $\tilde{\kappa}:K(r,c_1,\Delta)/N \to \operatorname{Pic}(M_H(r,c_1,\Delta))$ is injective.\newline $(3)$ $\operatorname{Pic}(M_H(r,c_1,\Delta))/\frak a_{E_0}^*(Pic(\widehat{X} \times X))$ is generated by $\tilde{\kappa}(K(r,c_1,\Delta))$.\newline $(4)$ $\frak a_{E_0}^*(\operatorname{Pic}(\widehat{X} \times X)) \cap \tilde{\kappa}(K(r,c_1,\Delta)) \cong X \times \widehat{X}$. \end{thm} \section{appendix} In this appendix, we shall show the following. \begin{prop}\label{prop:1-24} Let $L$ be an ample line bundle on $X$. We assume that $\chi(L)=(c_1(L)^2)/2$ and $r$ are relatively prime. Then $M_H(r,c_1(L),\Delta) \cong M_H(r,L,\Delta) \times \widehat{X}$, where $M_H(r,L,\Delta)$ is the moduli space of determinant $L$. In particular, $P(M_H(2,L,\Delta),z)=P(\operatorname{Hilb}_X^{2\Delta},z)$ for a general polarization $H$. \end{prop} \begin{pf} For a stable sheaf $E \in M_H(r,c_1(L),\Delta)$, $\lambda(E)$ denotes the point of $\widehat{X}$ which correspond to the line bundle $\det(E) \otimes L^{-1}$. Let $\phi_L:X \to \widehat{X}$ be the morphism sending $x \in X$ to $T^*_x L \otimes L^{-1}$, and $\varphi: \widehat{X} \to X$ the morphism such that $\phi_L \circ \varphi=n^2_{\widehat{X}}$, where $T_x: X \to X$ is the translation defined by $x$ and $n^2=\chi(L)^2=\deg \phi_L$. Since $(r,n^2)=1$, there are integers $k$ and $k'$ such that $rk+n^2 k'=1$. We denote the Poincar\'{e} line bundle on $X \times \widehat{X}$ by $\cal P$. Let $A:M_H(r,c_1(L),\Delta) \to M_H(r,L,\Delta)\times \widehat{X}$ be the morphism sending $F \in M_H(r,c_1(L),\Delta)$ to $(T^*_{-k'\varphi \circ \lambda(F)} (F \otimes \cal P_{-k\lambda(F)}), \lambda(F))$ and $B:M_H(r,L,\Delta)\times \widehat{X} \to M_H(r,c_1(L),\Delta)$ the morphism sending $(E,x) \in M_H(r,L,\Delta)\times \widehat{X}$ to $T^*_{k'\varphi(X)}E \otimes \cal P_{kx}$. For an element $(E,x)$ of $M_H(r,L,\Delta)\times \widehat{X}$, $\det(T^*_{k'\varphi(x)}E \otimes \cal P_{kx}) \cong T_{k'\varphi(x)}^*L \otimes \cal P_{rkx} \cong L \otimes \cal P_{k'\phi_L \circ \varphi (x)} \otimes \cal P_{rkx} \cong L \otimes \cal P_{(n^2k'+rk)x}=L \otimes \cal P_{x}$. Hence $\lambda \circ B((E,x))=x$. Then it is easy to see that $A \circ B$ and $B \circ A$ are identity morphisms. Hence $A:M_H(r,c_1(L),\Delta) \to M_H(r,L,\Delta) \times \widehat{X}$ is an isomorphism. \end{pf} Let $\Bbb D(X)$ and $\Bbb D(\widehat{X})$ be the derived categories of $X$ and $\widehat{X}$ respectively. Let $\cal S:\Bbb D(X) \to \Bbb D(\widehat{X})$ be the Fourier-Mukai transform [Mu4]. Then the morphism $\alpha:=\alpha_{E_0}$ defined in {\bf 3.4} satisfies that $\alpha(E)=\det \cal S(E) \otimes (\det\cal S(E_0))^{-1}$. Thus $\alpha_{E_0}$ is also defined by Fourier-Mukai transform. By using [M4], we shall treat the case $2r\Delta=2$ (at least, Mukai treated the case where $X$ is a principally polarized Abelian surface). \begin{prop} Let $L$ be an ample divisor. If $2r\Delta=2$, then for a general polarization $H$, $\alpha:M_H(r,L,\Delta) \to X$ is an isomorphism. \end{prop} \begin{pf} Since $rc_2-(r-1)(L^2)/2=1$ and $\chi(L)=(L^2)/2$, $r$ and $\chi(L)$ are relatively prime. We shall choose an element $E$ of $M_H(r,L,\Delta)$ and let $\xi:X \times \widehat{X} \to M(r,c_1(L),\Delta)$ be the morphism sending $(x,y) \in X \times \widehat{X}$ to $T_x^*E \otimes \cal P_y$. Then $\lambda \circ \xi(x,y)=\phi_L(x)+ry$. Let $f:X \to X \times \widehat{X}$ be the morphism such that $f(x)=(rx,-\phi_L(x))$. Since $\# \ker \phi_L=\chi(L)^2$ and $r$ are relatively prime, $f$ is injective. Let $g:\widehat{X} \to X \times \widehat{X}$ be the morphism such that $g(y)=(k'\varphi(y),ky)$. Then $f \times g:X \times \widehat{X} \to X \times \widehat{X}$ is an isomorphism. In fact, if $(rx+k'\varphi(y), -\phi_L(x)+ky)=(0,0)$, then $\phi_L(rx+k'\varphi(y))=r\phi_L(x)+n^2k'y=0$. Hence $y=(n^2k'+rk)y=0$. Since $f$ is injective, $x=0$, which implies that $f \times g$ is injective. Therefore $f \times g$ is an isomorphism. Then we get a morphism $\xi \circ f: X \to M(r,L,\Delta)$. Replacing $E$ by $E \otimes L^{\otimes m}$, we may assume that there is an exact sequence $0 \to \cal O_X^{\oplus (r-1)} \to E \to I_Z \otimes L \to 0$, where $I_Z$ is the ideal sheaf of a codimension 2 subscheme $Z$ of $X$. By our assumption on Chern classes, $1/r=\Delta(E)=\deg Z-(r-1)/r\chi(L)$. For simplicity, we denote $\det \cal S (?)$ by $\delta(?)$. Then we see that $\delta(T^*_x E \otimes \cal P_y) =\delta(I_{T_{-x}(Z)} \otimes T^*_xL \otimes \cal P_y) =\delta(I_{Z-(\deg Z)x} \otimes L \otimes \cal P_{\phi_L(x)+y}) =\delta(L \otimes \cal P_{\phi_L(x)+y})\otimes \cal P_{-Z+(\deg Z)x} =\det T^*_{\phi_L(x)+y}(\cal S(L)) \otimes \cal P_{-Z+(\deg Z)x} =\delta (L) \otimes \cal P_{\phi_{\delta(L)}(\phi_L(x)+y)+(\deg Z)x-Z}$. Hence $\alpha \circ \xi \circ f(x) =\alpha \circ \xi \circ f(0)+(r-1)\phi_{\delta(L)}\circ \phi_L(x)+r(\deg Z)x$. By the proof of [Mu4, Prop.1.23], $\phi_{\delta(L)}(\phi_L(x))=-\chi(L)x$. Since $r\deg Z=1+(r-1)\chi(L)$, we get that $\alpha\circ \xi \circ f(x)=\alpha \circ \xi \circ f(0)+x$. Thus $\alpha\circ \xi \circ f(x)$ is an isomorphism. Therefore we get that $\alpha:M_H(r,L,\Delta) \to X$ is an isomorphism. \end{pf}

9705

alg-geom/9705016

en

https://arxiv.org/abs/alg-geom/9705016

alg-geom/9705016

Rahul Pandharipande

Rahul Pandharipande

A geometric construction of Getzler's relation

Latex2e 14 pages

A geometric construction of Getzler's cohomological relation in the moduli space of 4 pointed elliptic curves is given by a push-forward of a natural rational equivalence in a space of admissible covers. In particular, Getzler's relation is shown to be a rational equivalence. The recursion for the elliptic Gromov-Witten invariants of P^2 predicted by Eguchi, Hori, and Xiong from the Virasoro conjecture is proven via Getzler's equation and the WDVV-equations.

alg-geom

\section{\bf{Introduction}} Let $\overline{M}_{1,4}$ be the moduli space of Deligne-Mumford stable 4-pointed elliptic curves. E. Getzler has determined the natural $\mathbb{S}_4$-module structure on the vector space $H^*(\overline{M}_{1,4}, \mathbb{Q})$ using modular operads and Deligne's mixed Hodge theory [G1]. The dimension of the $\mathbb{S}_4$-invariant space $H^4(\overline{M}_{1,4}, \mathbb{Q})^{\mathbb{S}_4}$ is 7. $\overline{M}_{1,4}$ has a natural stratification by dual graph type. Define an $\mathbb{S}_4$-invariant stratum of $\overline{M}_{1,4}$ to be an $\mathbb{S}_4$-orbit of closed strata of $\overline{M}_{1,4}$. The number of invariant dimension 2 strata in $\overline{M}_{1,4}$ is 9. The classes of these invariant strata in cohomology must therefore satisfy at least 2 linear relations. The first relation is evident. Let $\bigtriangleup_0$ be the boundary stratum with generic element corresponding to a 4-pointed nodal rational curve. There is a natural map $\overline{M}_{0,6} \rightarrow \bigtriangleup_0$ obtained by identifying the markings 5 and 6. Pushing forward the basic divisor linear equivalence on $\overline{M}_{0,6}$ to $\bigtriangleup_0$ yields a relation among the dimension 2 boundary strata of $\overline{M}_{1,4}$ contained in $\bigtriangleup_0$. In [G2], Getzler computes the $9 \times 9$ intersection pairing of the invariant strata in $\overline{M}_{1,4}$. This intersection matrix is found to have rank 7. The invariant strata therefore span $H^4(\overline{M}_{1,4}, \mathbb{Q})^{\mathbb{S}_4}$. The null space of the intersection matrix is computed to find a new relation among these strata in cohomology. A direct construction of Getzler's new relation via a rational equivalence in the Chow group $A_2(\overline{M}_{1,4}, \mathbb{Q})$ is presented here. The idea is to equate cycles corresponding to different degenerations of elliptic curves in the Chow group of a space of admissible covers. The push-forward of this equivalence to $\overline{M}_{1,4}$ yields Getzler's equation. The strategy of the construction is the following. First, select a point: $$\zeta=[\mathbf P^1, p_1,p_2,p_3,p_4]\in M_{0,4}.$$ Consider the space of degree 2 maps $\pi: E \rightarrow \mathbf P^1$ from 4-pointed nonsingular elliptic curves to $\mathbf P^1$ satisfying: \begin{enumerate} \item[(i)] $\pi$ is not ramified at the markings of E, \item[(ii)] The $i^{th}$ marking of $E$ lies over $p_i$. \end{enumerate} This space has dimension 4 and may be compactified (after ordering the ramification points) by the space of pointed admissible double covers $\overline{H}_\zeta$. These spaces are discussed in Section \ref{admi}. There are two natural dominant maps: $$\lambda:\overline{H}_\zeta \rightarrow \overline{M}_{1,4},$$ $$\rho: \overline{H}_\zeta \rightarrow Sym^4 (\mathbf P^1).$$ The map $\lambda$ is the map to moduli while $\rho([\pi])$ is defined by the degree 4 ramification divisor of $\pi$. There are two natural surfaces in $Sym^4(\mathbf P^1)$: the loci where the generic divisor shapes are 2+2 and 1+3 respectively. These surfaces are linearly equivalent (up to a scalar) in $A_2(Sym^4(\mathbf P^1), \mathbb{Q})$ since $A_2(Sym^4 (\mathbf P^1),\mathbb{Q})=\mathbb{Q}$. The intersection pull-back via $\rho$ of these surfaces yields a rational equivalence of codimension 2 cycles in $\overline{H}_\zeta$. This is a {\em boundary} linear equivalence because a double cover by an nonsingular elliptic curve is branched over 4 distinct points. The $\lambda$ push-forward of this relation yields Getzler's relation in $A_2(\overline{M}_{1,4}, \mathbb{Q})$. C. Faber has determined the Chow ring of $\overline{M}_3$ in [Fa]. There are 12 dimension 2 boundary strata. In [Fa], the rank $A_2(\overline{M}_3, \mathbb{Q})=7$ is computed and the strata are shown to span. Four geometric relations are constructed among the dimension 2 boundary strata. Faber deduces a fifth relation via the ring axioms and further knowledge of $A_*(\overline{M}_3, \mathbb{Q})$. There is a map $\tau: \overline{M}_{1,4} \rightarrow \overline{M}_3$ obtained by identifying the marked point 1 with 2 and 3 with 4. The $\tau$ push-forward of Getzler's relation is easily seen to yield a relation among the dimension 2 strata of $\overline{M}_3$. This provided a geometric construction of Faber's fifth relation. Getzler's new relation yields a differential equation for elliptic Gromov-Witten invariants analogous to the WDVV-equations in genus 0 (see [G2]). Eguchi, Hori, and Xiong and, independently, S. Katz have proposed the full (gravitational) potential function is annihilated by a certain representation of the Virasoro algebra. The Virasoro conjecture (together with the topological recursion relations in genus 1) predicts a remarkably simple recursion for the elliptic Gromov-Witten invariants of $\mathbf P^2$ [EHX]. In the last section, this elliptic recursion is proven via Getzler's equation and the WDVV-equations. The author wishes to thank D. Abramovich, P. Belorousski, C. Faber, E. Getzler, T. Graber, S. Katz, and A. Kresch for conversations about $\overline{M}_{1,4}$ and related issues. This research was completed at the Mittag-Leffler Institute. The author was partially supported by an NSF post-doctoral fellowship. \section{\bf Moduli of maps} \label{admi} An {\em admissible cover of degree $2$ with $4$ marked points and $4$ branch points} consists of a morphism $\pi: E\rightarrow D$ of pointed curves $$[E, \ P_1, \ldots,P_4], \ \ [D,\ p_1, \ldots, p_4, q_a, \ldots ,q_d]$$ satisfying the following conditions: \begin{enumerate} \item[(i)] $E$ is a connected, reduced, nodal curve of arithmetic genus 1. \item[(ii)] The markings $P_i$ lie in the nonsingular locus $E_{ns}$. \item[(iii)] $\pi(P_i)=p_i$. \item[(iv)] $[D, p_1, \ldots, p_4, q_a, \ldots, q_d]$ is an $8$-pointed stable curve of genus $0$. \item[(v)] $\pi^{-1}(D_{ns})= E_{ns}$. \item[(vi)] $\pi^{-1}(D_{sing})=E_{sing}$. \item[(vii)] $\pi|_{E_{ns}}$ is \'etale except over the points $q_j$ where $\pi$ is simply ramified. \item[(viii)] If $x\in E_{sing}$, then: \begin{enumerate} \item[(a)] $x \in E_1 \cap E_2 $ where $E_1, E_2$ are distinct components of $E$. \item[(b)] $\pi(E_1), \pi(E_2)$ are distinct components of $D$. \item[(c)] The ramification numbers at $x$ of the two morphisms: $$\pi: E_1 \rightarrow \pi(E_1), \ \ \ \pi:E_2 \rightarrow \pi(E_2)$$ are equal. \end{enumerate} \end{enumerate} Let $\overline{H}$ be the space of 4-pointed admissible degree 2 covers of $\mathbf P^1$ branched at 4 points. $\overline{H}$ is an irreducible variety parametrizing admissible covers. There are natural maps \begin{equation} \label{mmm} \lambda:\overline{H} \rightarrow \overline{M}_{1,4}, \ \ \pi:\overline{H} \rightarrow \overline{M}_{0,8} \end{equation} obtained from the domain and range of the admissible cover respectively. The projection $\pi$ is a finite map to $\overline{M}_{0,8}$. For each marking $i \in\{1,2,3,4\}$, there is a natural $\mathbb{Z}/2\mathbb{Z}$-action given by switching the sheet of the $i^{th}$-marking of $E$. These actions induce a product action in which the diagonal $\bigtriangleup$ acts trivially. Define the group $\mathbf{G}$ by: $$\mathbf{G}= (\mathbb{Z}/2\mathbb{Z})^4/\bigtriangleup.$$ The action of $\mathbf{G}$ on $\overline{H}$ is generically free and commutes with the projection $\pi$. Therefore, the quotient $\overline{H}/\mathbf{G}$ naturally maps to $\overline{M}_{0,8}$. In fact, since $\overline{H}/\mathbf{G} \rightarrow \overline{M}_{0,8}$ is finite and birational, it is an isomorphism. Spaces of admissible covers were defined in [HM]. The methods there may be used to construct spaces of pointed admissible covers. A quick route to $\overline{H}$ via Kontsevich's space of stable maps will be presented here. Let $\mu:U\rightarrow \overline{M}_{0,8}$ be the 8-pointed universal curve. Let $\overline{M}_{1,4}(\mu,2)$ be the relative space of stable maps of double covers of the universal curve. $\overline{M}_{1,4}(\mu,2)$ is a projective variety (see [K], [KM], [FP], [BM]). Let $H$ denote the quasi-projective subvariety of $\overline{M}_{1,4}(\mu,2)$ corresponding to 4-pointed maps $\mu: E \rightarrow D$ satisfying: \begin{enumerate} \item[(i)] $E$ is a nonsingular 4-pointed elliptic curve, $D$ is a nonsingular 8-pointed $\mathbf P^1$. \item[(ii)] The markings of $E$ lie over the first four markings of $D$. \item[(iii)] The four ramification points of $\mu$ lie over the last four markings of $D$. \end{enumerate} $H$ is an \'etale $8$-sheeted cover of $M_{0,8}$ with a free $\mathbf{G}$-action. Let $\overline{H}$ be the closure of $H$ in $\overline{M}_{1,4}(\mu,2)$. The points of the closure are seen to parameterize pointed admissible covers. The basic maps (\ref{mmm}) are obtained from the restriction of maps defined on $\overline{M}_{1,4}(\mu,2)$. The $\mathbf{G}$-action is deduced from the universal properties of $\overline{M}_{1,4}(\mu,2)$. $\overline{H}$ and $\overline{M}_{0,8}$ are 5 dimensional spaces. Let $\zeta = [\mathbf P^1, p_1, p_2, p_3, p_4]\in M_{0,4}$. Let $X_\zeta$ be the fiber over $\zeta$ of the natural map $\overline{M}_{0,8} \rightarrow \overline{M}_{0,4}$ determined by the first four markings. $X_{\zeta}$ is a nonsingular 4 dimensional variety. Another way to view $X_\zeta$ is the following. Let $\mathbf P^1[8]$ be the Fulton-MacPherson configuration space [FM]. Let $\gamma_1, \ldots, \gamma_4$ be the first four projections $\gamma_i : \mathbf P^1[8] \rightarrow \mathbf P^1$. $X_\zeta$ is naturally identified with the intersection: $$\gamma_1^{-1}(p_1) \cap \gamma_2^{-1}(p_2) \cap \gamma_3^{-1}(p_3) \cap \gamma_4^{-1}(p_4) \subset \mathbf P^1[8].$$ In particular, the four markings corresponding to the ramification points yield four natural projections $\gamma_a, \ldots, \gamma_d$ from $X_\zeta$ to $\mathbf P^1$. $X_\zeta$ is a compactification of the space of $8$ points on $\mathbf P^1$ where the first four have a specified cross ratio. The boundary of $X_\zeta$ is a divisor with normal crossings. Moreover, $X_\zeta$ has a stratification by graph type. Let $\overline{H}_\zeta= \pi^{-1}(X_\zeta)$. $\overline{H}_\zeta$ is a natural compactification of the space of 4-pointed nonsingular elliptic double covers of $\mathbf P^1$ satisfying: \begin{enumerate} \item[(i)] The ramification points on $E$ are ordered and distinct from the markings. \item[(ii)] The markings lie over a fixed 4-tuple $\zeta \in M_{0,4}$. \end{enumerate} $\mathbf{G}$ acts on $\overline{H}_\zeta$ with quotient $\overline{H}_\zeta/\mathbf{G}=X_\zeta$. The restriction $\lambda: \overline{H}_\zeta \rightarrow \overline{M}_{1,4}$ is a generically finite dominant map of degree $48$. \section{\bf Two Surfaces} \label{2sur} Let $S_{2,2}$ and $S_{1,3}$ be the irreducible surfaces in $Sym^4 (\mathbf P^1) \stackrel{\sim}{=} \mathbf P^4$ corresponding to divisors of shape $2+2$ and $1+3$ respectively. The degrees of these surfaces are $4$ and $6$ respectively. Therefore, $$\frac{[S_{2,2}]}{4} = \frac{[S_{1,3}]}{6}$$ in $A_2(Sym^4(\mathbf P^1))$. Chow groups here will always be taken with $\mathbb{Q}$-coefficients. Index four $\mathbf P^1$'s by the letters $\{a,b,c,d\}$. Define: $$\mathbf P^1_4= \mathbf P^1_a \times \cdots \times \mathbf P^1_b.$$ The following convenient notation will be used for the diagonal subvarieties of $\mathbf P^1_4$. Let $(ab)$ denote diagonal where the coordinates of $a$ and $b$ coincide. Let $(ab,cd)= (ab)\cap (cd)$. Similarly, let $(abc)$ denote the diagonal where the coordinates of $a$, $b$, and $c$ coincide. Let $\mathbf P^1_4 \rightarrow Sym^4(\mathbf P^1)$ be the $\mathbb{S}_4$-quotient map. The inverse image of $S_{2,2}$ in $\mathbf P^1_4$ is $(ab,cd)\cup(ac,bd)\cup(ad,bc)$. Similarly, the inverse image of $S_{1,3}$ is $(abc)\cup(acd)\cup(abd)\cup(bcd)$. \begin {lm} \label{pbck} Let $\mathbf{A}$ be a finite group. Let $Z$ be an irreducible algebraic variety with an $\mathbf{A}$-action and a quotient morphism $\alpha: Z \rightarrow Z/\mathbf{A}$. There exists a pull-back $\alpha^*: A_*(Z/\mathbf{A}) \rightarrow A_*(Z)$ defined by: $$\alpha^*([V])= {|\text{\em Stab}(V)|}\cdot {[\alpha^{-1}(V)_{red}]}$$ where $V$ is an irreducible subvariety of $Z/\mathbf{A}$, the scheme $\alpha^{-1}(V)_{red}$ is the reduced preimage of $V$, and $|\text{\em Stab}(V)|$ is the size of the generic stabilizer of points over $V$. \end{lm} By Lemma \ref{pbck}, there is an equality in $A_2(\mathbf P_4^1)$: \begin{equation} \label{prodd} [ab,cd] + [ac,bd] + [ad,bc] = [abc] + [acd] + [abd] + [bcd]. \end{equation} This equality can of course be checked directly. Lemma \ref{pbck} will be used in Section \ref{tacal}. See [V] for a proof of Lemma \ref{pbck}. \section{Refined Intersection} \label{reff} The notation of Sections \ref{admi} and \ref{2sur} is followed here. The four projections $\gamma_a,\ldots, \gamma_d$ from $X_\zeta$ to $\mathbf P^1$ yield a map: $$\Gamma: X_\zeta \rightarrow \mathbf P^1_4.$$ The pull-back of the relation (\ref{prodd}) by the map $\Gamma^*: A^*(\mathbf P^1_4) \rightarrow A^*(X_\zeta)$ is now analyzed. First, the left side (\ref{prodd}) is considered. The scheme theoretic inverse $\Gamma^{-1}(ab)$ is a reduced union of boundary divisors of $X_\zeta$. Therefore, since the boundary in $X_\zeta$ has normal crossings, $$\Gamma^{-1}(ab,cd)= \Gamma^{-1}(ab) \cap \Gamma^{-1}(cd)$$ is a reduced union of 2 and 3 dimensional strata $X_\zeta$. In fact, in the notation of Appendices B and C , $$\Gamma^{-1}(ab,cd)= S_1 \cup S_2 \cup S_2 \cup S_3 \cup T_3 \cup T_4.$$ $S_2$ occurs with 2 different round marking assignments. While the strata $S_i$ are of expected dimension 2, $T_3$ and $T_4$ are of excess dimension. Excess intersection theory yields a dimension 2 cycle supported on $T_3$ and $T_4$. There is a bundle sequence on the divisor $T=T_3 \cup T_4$ which is exact away from the 1 dimensional intersections of $T$ with the 2 dimensional components of $\Gamma^{-1}(ab,cd)$: $$0\rightarrow {\mathcal{O}}_T(T) \rightarrow \Gamma^*(N_{ab,cd}) \rightarrow L \rightarrow 0.$$ Here, $N_{ab,cd}$ is the normal bundle of $(ab,cd)$ in $\mathbf P^1_4$, the first map is obtained from the differential of $\Gamma$, and $L$ is the excess bundle. The first Chern class of $L$ is the excess cycle on $T$ (see [Fu]). A standard calculation yields the excess cycle: $$2 R_1+ \frac{1}{2} R_2+ 2 R_3 + 2 R_4 + S_4 +S_7 +S_8.$$ The intersection pull-back is thus determined by: \begin{equation*} \Gamma^*[ab,cd] = 2 R_1+ \frac{1}{2} R_2+ 2 R_3 + 2 R_4 +S_1+ 2 S_2 + S_3+ S_4 +S_7 +S_8. \end{equation*} The above equalities are in $A_2(X_\zeta)/K$ (see Appendix B). The scheme theoretic inverse $\Gamma^{-1}(abc)$ is a union of boundary divisors of $X_\zeta$: $$\Gamma^{-1}(abc)= T_1 \cup T_2 \cup T_3 \cup T_4.$$ The excess bundle $L$ is again determined by the natural exact sequence on $T=\Gamma^{-1}(abc)$: $$0\rightarrow {\mathcal{O}}_T(T) \rightarrow \Gamma^*(N_{abc}) \rightarrow L \rightarrow 0.$$ A calculation yields the excess class in $A_2(X_\zeta)/K$: $$ \Gamma^*[abc] = \frac{6}{10}R_1 + \frac{1}{2} R_2 + \frac{13}{10}R_3 +\frac{1}{2} R_4 + \frac{6}{10} R_5 + \frac{1}{10} R_6 +\frac{1}{6} R_7 $$ $$+ S_4 + S_5 + S_6 +S_7 +S_8 +S_9. $$ The $\Gamma$ pull-back of the left side of (\ref{prodd}) in $A_2(X_\zeta)/K$ is simply $3\Gamma^*[ab,cd]$. Similarly, the pull-back of the right side is $4\Gamma^*[abc]$. Hence, the above calculation yield an equality in $A_2(X_\zeta)/K$: \begin{equation} \label{mainn} \frac{36}{10}R_1 - \frac{1}{2} R_2 + \frac{8}{10}R_3 + 4 R_4 - \frac{24}{10} R_5 - \frac{4}{10} R_6 -\frac{4}{6} R_7 = \end{equation} $$-3S_1-6S_2-3S_3+S_4+4S_5+4S_6+S_7+S_8+4S_9.$$ \section{Push-Forwards} \label{tacal} The application of $\lambda_* \pi^*: A_2(X_\zeta) \rightarrow A_2(\overline{M}_{1,4})$ to relation (\ref{mainn}) yields a boundary relation in $A_2(\overline{M}_{1,4})$. The calculation of $\pi^*$ is obtained by Lemma 1. It is checked the generic stabilizers of $\mathbf{G}$ over the strata are trivial except in the following cases: \begin{eqnarray*} |\text{Stab}(R_4)| & = & 2 \\ |\text{Stab}(R_5)| & = & 2 \\ |\text{Stab}(R_6)| & = & 2 \\ |\text{Stab}(R_7)| & = & 4 \\ |\text{Stab}(S_6)| & = & 2 \\ |\text{Stab}(S_9)| & = & 2. \end{eqnarray*} The elements $\lambda_* \pi^*(R_i)$ are tabulated below. See Appendix A for the notation in $A_2(\overline{M}_{1,4})$. These elements lie in the linear span of the 4 invariant strata of $A_2(\overline{M}_{1,4})$ with nonsingular elliptic components. \begin{tabbing} $\lambda_* \pi^*(R_*)+$ \= +=+ \= $++ \bigtriangleup_{*.*}$ \= + \= $++ \bigtriangleup_{*,*}$ \= + \= $\bigtriangleup_{*,*}$ \= + \= + \= $\bigtriangleup_{*,*}$ \kill $\lambda_* \pi^*(R_1)$ \> = \> \> \> \> 2 \> $\bigtriangleup_{2,4}$ \> + \> 6 \>$\bigtriangleup_{3,4}$\\ $\lambda_* \pi^*(R_2)$ \> = \> $96 \bigtriangleup_{2,2}$ \>+ \> $32 \bigtriangleup_{2,3}$ \> \> \> \> \>\\ $\lambda_* \pi^*(R_3)$ \> = \> \> \> \> 12 \> $\bigtriangleup_{2,4}$ \> \> \> \\ $\lambda_* \pi^*(R_4)$ \> = \> \> \> $16 \bigtriangleup_{2,3}$ \> \> \> \> \>\\ $\lambda_* \pi^*(R_5)$ \> = \> \> \> \> \> \> \> 12\> $\bigtriangleup_{3,4}$\\ $\lambda_* \pi^*(R_6)$ \> = \> \> \> \> 32 \> $\bigtriangleup_{2,4}$ \> + \> 12 \> $\bigtriangleup_{3,4}$\\ $\lambda_* \pi^*(R_7)$ \> = \> \> \> $48 \bigtriangleup_{2,3}$ \> \> \> \> \> \end{tabbing} The elements $\lambda_* \pi^*(S_i)$ lie in the linear span of the 5 invariant strata contained in the boundary divisor $\bigtriangleup_0 \subset \overline{M}_{1,4}$ corresponding to a 4-pointed nodal rational curve. \begin{tabbing} $\lambda_* \pi^*(S_*)+$ \= +=+ \= $++* \bigtriangleup_{*.*}$ \= $++* \bigtriangleup_{*,*}$ \= $++* \bigtriangleup_{*,*}$ \= $++* \bigtriangleup_{*}$ \= $++* \bigtriangleup_{*}$ \kill $\lambda_* \pi^*(S_1)$ \> = \> $\frac{2}{3} \bigtriangleup_{0,2}$ \> \> \> $+2 \bigtriangleup_{a}$ \> $+\frac{16}{3} \bigtriangleup_{b}$ \\ $\lambda_* \pi^*(S_4)$ \> = \> $4 \bigtriangleup_{0,2}$ \> \> \> \> \\ $\lambda_* \pi^*(S_5)$ \> = \> \> \> $2\bigtriangleup_{0,4}$ \> \> \\ $\lambda_* \pi^*(S_6)$ \> = \> \> $2 \bigtriangleup_{0,3}$ \> \> \> \\ $\lambda_* \pi^*(S_7)$ \> = \> \> $2 \bigtriangleup_{0,3}$ \> \> \> \\ \end{tabbing} \noindent The push-forwards $\lambda_*\pi^*(S_2)$, $\lambda_*\pi^*(S_3)$, $\lambda_*\pi^*(S_8)$, and $\lambda_*\pi^*(S_9)$ all vanish. In the above formulas, the ordinary coarse moduli fundamental classes in $\overline{M}_{1,4}$ are used on the right. The orbifold classes differ by a factor of two for $\bigtriangleup_{2,4}$, $\bigtriangleup_{3,4}$, and $\bigtriangleup_{0,4}$. These push forwards are easy to compute. A representative example will be given. Consider the graph $R_4$: \begin{center} \input{r4.pictex} \end{center} \noindent with a fixed labelling of the marked points and the round markings. The three components of the graph have been labeled $U$, $V$, and $W$. The labeled graph now corresponds to an irreducible stratum $R$ of $X_\zeta$. An admissible cover lying above this stratum consists of the following data: an elliptic double cover of $V$ ramified at $a$, $b$, $c$, and the intersection $V \cap U$, a rational double cover of $U$ ramified at $d$ and the intersection $V\cap W$, and an disjoint \'etale double cover of $W$. The double cover of $W$ is a union of two $\mathbf P^1$'s. The markings $\{1,2,3\}$ can be distributed in 4 ways (up to isomorphism) on the 2 components over $W$: $$(12,3),(13,2), (23,1), (123).$$ This yields four components of $\pi^{-1} (R)$. The component $(12,3)$ pushes forward by $\lambda$ to: \begin{center} \input{ex1.pictex} \end{center} \noindent The $\lambda$ push-forward of the component $(123)$ lies in the divisor stratum of $\overline{M}_{1,4}$: \begin{center} \input{ex3.pictex} \end{center} \noindent and is easily seen to equal: \begin{center} \input{ex2.pictex} \end{center} \noindent Adding all the permutations and multiplying by the automorphism factor 2 associated to $R_4$ yields $\lambda_* \pi^*(R_4)= 16 \ \bigtriangleup_{2,3}$. \section{Getzler's relation} The calculations of $\lambda_* \pi^*$ in Section \ref{tacal} and the linear equivalence (\ref{mainn}) yield the relation: \begin{equation} \label{anss} 48 \bigtriangleup_{2,2}-16 \bigtriangleup_{2,3}-4 \bigtriangleup_{2,4}+12\bigtriangleup_{3,4} \end{equation} $$+2 \bigtriangleup_{0,2} +10 \bigtriangleup_{0,3} +8 \bigtriangleup_{0,4} -6 \bigtriangleup_a -16 \bigtriangleup_b=0$$ in $A_2(\overline{M}_{1,4})$. The basic linear equivalence in $A_2(\overline{M}_{1,4})$ obtained from $\bigtriangleup_0$ is (see [G2]): \begin{equation} \label{ratt} \bigtriangleup_{0,2}+ 3\bigtriangleup_{0,3} + 3 \bigtriangleup_{0,4} -3 \bigtriangleup_a -4 \bigtriangleup_b=0 \end{equation} Equation (\ref{anss}) minus twice equation (\ref{ratt}) yields: $$4 \cdot (12 \bigtriangleup_{2,2}-4 \bigtriangleup_{2,3}- \bigtriangleup_{2,4}+3 \bigtriangleup_{3,4} + \bigtriangleup_{0,3}+ \frac{1}{2} \bigtriangleup_{0,4} - 2 \bigtriangleup_b)=0$$ which is Getzler's relation [G2]. \begin{tm} Getzler's relation in $H^4(\overline{M}_{1,4}, \mathbb{Q})$ is obtained from a rational equivalence in $A_2(\overline{M}_{1,4}, \mathbb{Q})$. \end{tm} \section{Elliptic invariants of $\mathbf P^2$} For $d\geq 1$, let $N^{(0)}_d$ and $N_d^{(1)}$ be the rational and elliptic Gromov-Witten invariants of $\mathbf P^2$. Let $\Gamma$ and $E$ be the (quantum) rational and elliptic potentials: \begin{eqnarray} \Gamma(y_1,y_2) & = & \sum_{d\geq 1} N^{(0)}_d e^{dy_1} \frac{y_2^{3d-1}}{(3d-1)!}, \\ E(y_1,y_2) & = & -\frac{y_1}{8} + \sum_{d\geq 3} N_d^{(1)} e^{dy_1} \frac{y_2^{3d}} {(3d)!}. \end{eqnarray} The variables $y_0, y_1, y_2$ correspond to the fundamental, line, and point classes of $\mathbf P^2$ respectively. The elliptic sum starts in degree 3 since $N_1^{(1)}$ and $N_2^{(1)}$ vanish. Let $\tilde{E}= E+ \frac{1}{8} y_1$. The composition axiom for Gromov-Witten invariants ([KM], [RT], [BM]) and Getzler's relation immediately yield partial differential equations satisfied by the elliptic potential. Getzler uses these to obtain a complete recursion determining the numbers $N_d^{(1)}$ [G2]. The numbers $N_d^{(1)}$ have been obtained via more classical techniques in algebraic geometry in [CH], [R]. The relation predicted by Eguchi, Hori, and Xiong from the Virasoro conjecture is: \begin{equation} \label{egu} \frac{N_d^{(1)}}{(3d-1)!} = \frac{1}{12} \binom {d}{3} \frac{ N_d^{(0)} } {(3d-1)!} + \sum_{d_1+d_2=d, d_i>0} \frac{3d_1^2 d_2-2d_1d_2}{9} \frac{N^{(0)}_{d_1}}{(3d_1-1)!} \frac{N_{d_2}^{(1)}}{(3d_2)!}. \end{equation} While the equation suggests divisor geometry, no proof via such an approach is known to the author. The relation is established here via Getzler's equation and the WDVV-equations. The first step is to rewrite the recursion (\ref{egu}) in differential form: $$\tilde{E}_2 = \frac{\Gamma_{111}-3\Gamma_{11}+2\Gamma_{1}}{72} + \frac{\Gamma_{11}\tilde{E}_1}{3} -\frac{2\Gamma_1 \tilde{E}_1}{9}.$$ The subscripts denote partial differentiation by $y_1$ and $y_2$ respectively. Next, the equation $$ \tilde{E}_1= \frac{y_2}{3}\tilde{E}_2 $$ is used to obtain: $$ \tilde{E}_2 \ (1-\frac{1}{9}y_2 \Gamma_{11}+\frac{2}{27}y_2 \Gamma_1) =\frac{\Gamma_{111}-3\Gamma_{11}+2\Gamma_{1}}{72}.$$ Finally, we see: \begin{equation} \label{www} E= -\frac{y_1}{8}+ \int \frac{{\Gamma_{111} -3\Gamma_{11}+2\Gamma_{1}}}{72} {(1-\frac{1}{9} y_2 \Gamma_{11}+ \frac{2}{27} y_2 \Gamma_1)^{-1}} \ dy_2. \end{equation} Formula (\ref{www}) is equivalent to (\ref{egu}). In order to prove (\ref{egu}), we consider a differential equation obtained from Getzler's relation. Let $$\pi: \overline{M}_{1,4+3d-6}(\mathbf P^2,d) \rightarrow \overline{M}_{1,4}$$ be the natural projection. Let $$e_i: \overline{M}_{1,4 +3d-6} \rightarrow \mathbf P^2$$ be the natural evaluation maps. The codimension 2 Gromov-Witten class $$\pi_* \big( \ [\overline{M}_{1,4+3d-6}(\mathbf P^2,d)]^{Vir} \cap \prod_{i=1}^{3d-2} e_i^*([\text{point}]) \ \big)$$ intersected with Getzler's relation yields the following differential equation: \begin{equation} \label{fff} 36E_{11} \Gamma_{122}^2 -48 E_{12}\Gamma_{112}\Gamma_{122} -48E_{22}\Gamma_{222}-12 E_1 \Gamma_{1122} \Gamma_{122} \end{equation} $$ +24 E_1 \Gamma_{112}\Gamma_{1222} +24 E_2 \Gamma_{2222} +2 \Gamma_{1222}\Gamma_{1112}$$ $$+ \frac{1}{2} \Gamma_{12222}\Gamma_{111}+ \frac{3}{2} \Gamma_{22222} -3 \Gamma_{1122}^2 =0$$ It is easily seen by examining the coefficient recursions that (\ref{fff}) uniquely determines the potential $E$ in form (7) from $\Gamma$. Therefore, to establish (\ref{egu}), it suffices to prove $(\ref{www})$ determines a solution of (\ref{fff}). Replace the terms $E_1$, $E_{11}$, $E_{12}$ in (\ref{fff}) by: \begin{eqnarray*} E_1 & = & \frac{y_2}{3} E_2 - \frac{1}{8}, \\ E_{11} & = & \frac{y_2^2}{9} E_{22} + \frac{y_2}{9} E_2, \\ E_{12} & = & \frac{y_2}{3} E_{22} + \frac{1}{3} E_2. \end{eqnarray*} Equation (\ref{www}) then may be used to replace the terms $E_2$ and $E_{22}$ by derivatives of $\Gamma$. A partial differential equation for $\Gamma$ is then obtained. Equation (\ref{www}) defines a solution of (\ref{fff}) if and only if $\Gamma$ satisfies this differential equation. The WDVV-equation for $\Gamma$ is: \begin{equation} \label{wd} \Gamma_{222} -\Gamma^2_{112} + \Gamma_{111}\Gamma_{122}=0. \end{equation} $\Gamma$ also satisfies: \begin{equation} \Gamma_1= \frac{y_2}{3} \Gamma_2 + \frac{1}{3} \Gamma. \label{ll} \end{equation} A simple check in symbolic algebra shows the required differential equation for $\Gamma$ is implied by the WDVV-equation and its first two $y_2$-derivatives together with (\ref{ll}). In fact, the required differential equation for $\Gamma$ modulo equation (\ref{ll}) is seen to be of the form: \begin{equation} \label{fform} A \Psi + B \Psi_2+ C \Psi^2_{2} + D \Psi_{22} =0. \end{equation} $A$ , $B$, $C$, and $D$ are polynomials in $y_2$, $\Gamma$, and the derivatives of $\Gamma$. $\Psi$ is the left side of the WDVV-relation (\ref{wd}), and the subscripts denote partial differentiation by $y_2$. \begin{eqnarray*} A & = & -2^7 \cdot (36y_2\Gamma_{22} + 24y_2^2\Gamma_{222}- 4\Gamma_{1}\Gamma_{11} -9 \Gamma^2_{11}+\frac{4}{3}\Gamma_{1}^2 \\ & & \ \ \ \ \ \ \ \ \ +12 \Gamma_1\Gamma_{111} -18 \Gamma_{11}\Gamma_{111} +27 \Gamma_{111}^2) \\ B & = & -2^6 \cdot (-60 \Gamma +375 \Gamma_1 -\frac{1521}{2} \Gamma_{11} +\frac{783}{2} \Gamma_{111} \\ & & \ \ \ \ \ \ \ \ \ +2y_2 \Gamma_1^2-6 y_2\Gamma_1\Gamma_{11}+8 y_2 \Gamma_1 \Gamma_{111} \\ & & \ \ \ \ \ \ \ \ \ -\frac{11}{2}y_2\Gamma_{11}^2-9y_2\Gamma_{11}\Gamma_{111} +\frac{9}{2}y_2\Gamma_{111}^2) \\ C & = & -2^6 \cdot y_2^4 \\ D & = & -2^5 \cdot(y_2 \Gamma_{111}+9) \cdot (3y_2\Gamma_{11}-2y_2\Gamma_1-27) \end{eqnarray*} This concludes the proof of recursion (\ref{egu}). \begin{tm} The Eguchi-Hori-Xiong formula holds for the elliptic Gromov-Witten invariants of $\mathbf P^2$. \end{tm} \section*{Appendix A \ \ Strata of $\overline{M}_{1,4}$ in dimension 2} Getzler's notation for the complete set of $\mathbb{S}_4$-invariant dimension 2 strata of $\overline{M}_{1,4}$ is followed: \begin{center} \input{g1.pictex} \end{center} \vspace{+5pt} \begin{center} \input{g2.pictex} \end{center} \vspace{+5pt} \begin{center} \input{g3.pictex} \end{center} \noindent The short bars indicate the marked points $\{1,2,3,4\}$. Nonsingular genus 1 components (labeled by 1) occur in the first row. Each graph denotes the sum of strata in the corresponding $\mathbb{S}_4$-orbit. For example, $\bigtriangleup_{2,2}$ and $\bigtriangleup_{2,3}$ are sums of 3 and 12 strata respectively. \section*{Appendix B \ \ Strata of $X_\zeta$ in dimension 2} A graph $G$ below {\em together} with an assignment $\mu$ of the round markings to the set $\{ a,b,c,d \}$ corresponds to an $\mathbb{S}_4$-invariant dimension 3 stratum of $\overline{M}_{0,8}$. The 8 point marking set is $\{ 1,2,3,4,a,b,c,d \}$ as in Section \ref{admi}, and the $\mathbb{S}_4$-action is on the first four markings. By intersecting these strata of $\overline{M}_{0,8}$ with $X_\zeta$, a sum of dimension 2 strata of $X_\zeta$ is associated to $(G, \mu)$. For example, consider $R_3$. After an assignment of the round markings to the set $\{ a,b,c,d\}$, there are 4 strata of $\overline{M}_{0,8}$ in the $\mathbb{S}_4$-orbit. These yield a sum of four strata of $X_\zeta$. Similarly, $S_3$ together with an assignment of the round markings yields a sum 12 strata of $X_\zeta$. The assignment $\mu$ of the round markings will be suppressed since the calculation of the map $$\lambda_* \pi^*: A_2(X_\zeta) \rightarrow A_2(\overline{M}_{1,4})$$ does not depend upon $\mu$. Let $K=\text{Ker}(\lambda_* \pi^*)$. Let $\mu$ and $\mu'$ be two round marking assignments for a fixed graph $G$. The two cycles corresponding to $(G, \mu)$ and $(G, \mu')$ are equal in $A_2(X_\zeta)/K$. \begin{center} \input{r1.pictex} \end{center} \vspace{+5pt} \begin{center} \input{r2.pictex} \end{center} \vspace{+5pt} \begin{center} \input{s1.pictex} \end{center} \vspace{+5pt} \begin{center} \input{s2.pictex} \end{center} \vspace{+5pt} \begin{center} \input{s3.pictex} \end{center} \noindent This is not a complete list. Only the strata which are required for the computations in Section \ref{reff} are given. \section*{Appendix C \ \ Strata of $X_\zeta$ in dimension 3} As in Appendix B, a graph below together with an assignment of the round markings to the set $\{a,b,c,d\}$ corresponds to a sum of dimension 3 strata of $X_\zeta$. \begin{center} \input{t.pictex} \end{center}

9705

alg-geom/9705022

en

https://arxiv.org/abs/alg-geom/9705022

alg-geom/9705022

Sandra Di Rocco

Sandra Di Rocco, Kristian Ranestad

On surfaces in P^6 with no trisecant lines

21 pages, DVI file available at http://www.math.kth.se/~sandra/Welcome Mittag-Leffler Report No 13, 1996/97 Latex

We construct examples of smooth surfaces S in P^6 with no trisecant lines. This list includes examples of surfaces not cut out by quadrics. We prove that unless S has a finite number of disjoint $(-1)$-lines, and each one meets some other line L on the surface with L^2\leq -2, and S is not an inner projection from P^7 the surface belongs to the list of examples.

alg-geom

\section*{ {\vskip-1cm \Large Introduction}} The study of varieties embedded in $\pn{N}$ with no trisecant lines is a very classical problem in algebraic geometry. The simplest case, i.e. the case of space curves goes back to Castelnuovo.\\ For surfaces the problem has been studied in codimension $2$ and $3$. In \cite{au} Aure classifies smooth surfaces in $\pn{4}$ with no trisecant through the general point in the space. In \cite{bau} Bauer classifies smooth surfaces in $\pn{5}$ with no trisecant lines through the general point on the surface.\\ Here we treat the case of smooth surfaces in $\pn{6}$ with no trisecant lines at all. Of course this includes all surfaces which are cut out by quadrics, but there are some more examples.\\ We proceed as follows: \begin{itemize} \item In the first two sections we construct examples of surfaces in $\pn{6}$ with no trisecant lines. Standard examples are used to construct new examples, via linkage. In each case we indicate whether the ideal is generated by quadrics or not. The cases of surfaces containing lines and surfaces with no lines are treated separately. \item In section \ref{nlist} we give a complete list of surfaces with no proper trisecant lines and no lines on it. \item In section \ref{list2} we give a classification of surfaces with lines on the surface but no proper trisecant lines. These include scrolls, conic bundles, surfaces with an isolated $(-1)$-line and inner projections from $\pn{7}$, i.e. projections of smooth surfaces in $\pn{7}$ from a smooth point on the surface. \end{itemize} The list of cases produced in sections \ref{nlist} and \ref{list2} cover all the examples constructed in sections \ref{noline} and \ref{wline}. Of course this does not cover all the possibilities. In fact, surfaces with a finite number of disjoint $(-1)$-lines need not be {\it inner projections} from $\pn{7}$. If in addition every $(-1)$-line meets some other line $L$ of selfintersection $L^2\leq -2$ on the surface our methods do not apply. However, checking the cases with less than one hundred $(-1)$-lines give no new examples, so we conjecture that the list is in fact complete.\\ The main results of this work are summarized in the following:\\ \newpage {\rm M{\sc ain theorem.}} {\it Let $S$ be a smooth surface embedded in $\pn{6}$ with no trisecant lines. Unless $S$ has a finite number of disjoint $(-1)$-lines, and each one meets some other line $L$ on the surface with $L^2\leq -2$, and $S$ is not an inner projection from $\pn{7}$ the surface belongs to the following list:} \begin{center} \begin{tabular}{|c|c|c|c|} \hline surface &degree&linear system& example \\ \hline\hline ${\bf P}^2$&1,4&${\cal O}_{{\bf P}^2}(1)$, ${\cal O}_{{\bf P}^2}(2)$& \ref{nl1} \\\hline Rational scrolls&2,3,4,5&linearly normal& \\\hline Elliptic scrolls&7&linearly normal& \ref{ell} \\\hline Rational&4,5,6& anticanonical (Del Pezzo)& \\\hline $Bl_7({\bf P}^2)$&8&$6l-\sum 2E_i$&\ref{nl2}\\\hline Rational&6& conic bundle& \ref{con1}\\ \hline Rational&7& conic bundle& \ref{con2}\\ \hline Rational&8& conic bundle& \ref{con3}\\ \hline $Bl_{8}({\bf P}^2)$&8&$4l-\sum_1^8 E_i$& \ref{p48}\\ \hline $K3$&8&Complete intersection (1,2,2,2)&\ref{nl3}\\\hline $Bl_{9}({\bf P}^1\times {\bf P}^1 )$&9&$3(1,1)-\sum_1^9 E_i$&\ref{p49}\\ \hline $Bl_{11}({\bf P}^2)$&10&$6l-\sum_1^5 2E_i-\sum_1^6 E_i$&\ref{p410}\\ \hline $K3$&10&nontrigonal of genus 6&\ref{nl3},\ref{nl7} \\ \hline $Bl_1(K3)$&11& nontetragonal of genus 7 $p^{\ast}(\ol{\cH})-E$ & \ref{prk3}\\ \hline $Bl_{11}({\bf P}^2)$&12&$9l-\sum_1^5 3E_i-\sum_1^6 2E_i$&\ref{nl4} \\ \hline $Bl_1(K3)$&12&$p^{\ast}(\ol{\cH})-2E$&\ref{nl5}\\ \hline Elliptic&12& minimal with $p_g=2$&\ref{elliptic}\\ \hline Abelian&14&$(1,7)$-polarization&\ref{nl8}\\ \hline general type&16& complete intersection (2,2,2,2)&\ref{nl9}\\ \hline \end{tabular} \end{center} The authors would like to thank the Mittag-Leffler Institute for its support and its warm environment, which made this collaboration possible and most enjoyable. \section*{\vskip-1cm \Large Notation} The groundfield is the field of complex numbers ${\bf C}$. We use standard notation in algebraic geometry, as in \cite{HAr}. $S$ is always assumed to be a non singular projective surface.\\ By abuse of notation $\cH_S$ will denote the hyperplane section and the line bundle giving the embedding, with no distinction.\\ When $S$ is the blow up of $\ol{S}$ in $n$ points $S$ will be denoted by $Bl_n(\ol{S})$. \section{ \Large Construction of surfaces with no lines}\label{noline} \subsection{ \large Surfaces defined by quadrics}\label{ex1} If $S$ is a surface whose ideal $I_S$ is generated by quadrics then clearly it cannot have trisecant lines. The first examples in our list are then: \\ \begin{para}\label{nl1}The Veronese surface in $\pn{5}$, i.e. $S=\pn{2}$ embedded in $\pn{5}$ by the linear system $|\cO_{\pn{2}}(2)|$.\end{para} \begin{para}\label{nl2} Del Pezzo surfaces of degree $8$ \end{para} Let $S$ be the blow up of $\pn{2}$ in $7$ points embedded in $\pn{6}$ by the linear system $|-2K_S|$. The line bundle $\cH_S=-2K_S$ is $2$-very ample and thus embeds $S$ without trisecant lines. See \cite{sa} for the definition and the proof of the $2$-very ampleness. One can construct this surface in $\pn{6}$ as the intersection of the cone over a Veronese surfaces in $\pn{5}$ with a quadric hypersurface. Thus the surface is cut out by quadrics, and it has no lines as soon as the quadric does not contain the vertex of the cone. These surfaces are cut out by 7 quadrics in $\pn{6}$. \begin{para}\label{nl3} General minimal nontrigonal {\it $K3$-surface of degree $8$ or $10$}.\end{para} Consider a nontrigonal $K3$ surface of degree $8$ in $\pn{5}$. It is the complete intersection of $3$ quadrics, and the general one has Picard group generated by the hyperplane section so it has no lines.\par Similarly a nontrigonal $K3$ surface of degree $10$ in $\pn{6}$ is a linear section of the Pl\"ucker embedding of the Grassmannian $Gr(2,5)$ intersected with a quadric hypersurface. Again the surface is cut out by quadrics, in fact 6 quadrics, and the general one has Picard group generated by the hyperplane section, so it has no lines. \begin{para}\label{nl7}Minimal tetragonal $K3$-surfaces of degree $10$\end{para} Let $V$ be the cubic cone over $\pn{1}\times\pn{2}$, and consider two general quadrics $\Qs{1},\Qs{2}$, containing a quadric surface $S_2$ in $\pn{3}$. The complete intersection $V\cap \Qs{1}\cap\Qs{2}=S\cup S_2$ produces a smooth surface $S$ of degree $10$ in $\pn{6}$. A trisecant line of $S$ would be a line in $S_2$. But the intersection curve $S\cap S_2$ is an elliptic quartic curve, of type $(2,2)$ on $S_2$, so there is no trisecant. Again using adjunction one can see that $p_g(S)=1$ and $K_S\cdot \cH_S=0$ and that the ruling $\cF$ of $V$ restricted to $S$ form a quartic elliptic curve on $S$. It follows that $S$ is a $K3$ surface of degree $10$ in $\pn{6}$, with a pencil of elliptic quartic curves on it. Furthermore it is cut out by 6 quadrics.\\ One can also construct $S$ by linkage from the rational surface $Bl_7(\pn{2})$ embedded in $\pn{5}$ by the line bundle $4l-2E-\sum_1^6E_i$. These surfaces are cut out by 4 quadrics in $\pn{5}$ ( see \ref{con1}), and in a complete intersection $(2,2,2,2)$ in $\pn{6}$ they are linked to $K3$-surfaces of the above type. \begin{para}\label{elliptic} Two families of elliptic surfaces of degree $12$ \end{para} Let $V$ be a rational normal $4$-fold scroll of degree $3$ in $\pn{6}$ and consider $S=V\cap{\cal Q}_1\cap{\cal Q}_2$, where ${\cal Q}_1,{\cal Q}_2$ are two general quadrics, which do not have any common point in the singular locus of $V$. Note that this is possible only if the cubic 4-fold has vertex a point or a line. This gives us two separate cases. In both cases the intersection of $V$ with the two quadrics is a smooth surface. Now $K_{\ol V}=-4\cH+\cF$, where $\ol V$ is the $\pn{3}$-bundle over $\pn{1}$ which is mapped to $V$ by $\cH$, and $\cF$ is a member of the ruling. Then $K_S=\cF|_S$ gives a fibration of elliptic quartic curves without multiple fibers onto $\pn{1}$.\\ $S$ does not have proper trisecant lines since it is cut out by quadrics. Moreover $S$ cannot contain lines. In fact if there were a line $L\subset S$, then $L\subset V$ and $L\subset {\cal Q}_i$. The biggest component of the Fano variety of lines in $V$ is the 5-dimensional component of lines in the pencil of $\pn{3}$s. Since a line imposes three conditions on a quadric, we see that in order for $L$ to be on two quadrics we need at least a $6$-dimensional family of lines on $V$. Thus we have constructed two families of elliptic surfaces of degree 12, one on the cubic 4-fold cone with vertex a point, and one on the cubic 4-fold cone with vertex a line. In the first case any two canonical curves span $\pn{6}$, while in the other case any two canonical curves span a $\pn{5}$. These surfaces are cut out by 5 quadrics in $\pn{6}$. \begin{para}\label{nl9} Complete intersections of $4$ quadric hypersurfaces in $\pn{6}$.\end{para} This is a degree $16$ surface of general type. Since the Fano variety of lines in a quadric has codimension $3$ in the Grassmannian of lines in $\pn{6}$ and this Grassmannian has dimension $10$, there are no lines in a general complete intersection of $4$ quadrics. A similar argument is made more precise in \ref{nl5} below. \subsection{\large Non minimal $K3$-surfaces of degree 12}\label{nl5} Consider $4$ general quadrics $\cQ_1,...,\cQ_4\sub\pn{6}$ containing the Veronese surface $S_4\sub\pn{5}$ and let $S$ be the residual surface of degree 12 in $\pn{6}$. Let $V$ be the complete intersection of a general subset of three of the quadrics $\cQ_1,...,\cQ_4$. By adjunction we see that $C=S\cap S_4=\cH_V|_{S_4}-K_{S_4}= 5l$, where $l$ is the generator of $Pic(S_4)$. Then $K_S=(\cH_V-S_4)_S$ and $K_S\cdot \cH_S=((\cH_V+S_4)_S)\cdot \cH_S=12-10=2$. The exact sequence \begin{equation}\label{E1} 0\to\cO_V(\cH_V-S_4)\to\cO_V(\cH)\to\cO_{S_4}(\cH)\to 0 \end{equation} and the fact that $h^0(\cO_{S_4}(\cH))=6$ gives $h^0(\cO_V(\cH_V-S_4))=1$ and $h^1(\cO_V((\cH_V-S_4))=h^2(\cO_V(((\cH_V-S_4))=0$. Plugging those values in the long exact sequence of \begin{equation}\label{E2} 0\to\cO_V(-\cH)\to\cO_V(\cH_V-S_4)\to\cO_S(K_S)\to 0 \end{equation} we get $p_g(S)=1$, $q=0$ and thus $\chi(S)=2$. Thus the canonical curve is a $(-1)$- conic section or two disjoint $(-1)$-lines. It is the residual to the intersection $C$ of $S$ with $S_4$ in a hyperplane section of $S$, so each component intersect the curve $C$ in at least two points. If the canonical curve is two disjoint lines these would be secants to $S_4$. But a secant line to $S_4$ is a secant line to a unique conic section on $S_4$. Any quadric which contains a secant line must therefore contain the plane of this conic section. Therefore $S$ contains no secant line to $S_4$ as soon as $S$ is irreducible. We conclude that the canonical curve is a $(-1)$-conic section. The surface $S$ is the blow up of a minimal $K3$-surface and $K_S^2=-1$. We now show that $S$ contains no lines. First assume that the line does not intersect $S_4$. Let $G$ be the Grassmannian of $4$-dimensional subspaces of quadrics in $\pn{6}$ which contain $S_4$. Consider the incidence variety $$I=\{(L,Q_4)\in Gr(2,7)\times G|L\sub\cap_{\cQ\in Q_4} \cQ, L\cap S_4=\emptyset\}$$ and let $p:I\to Gr(2,7)$ and $q: I\to G$ the two projections. Since the lines in $\pn{6}$ form a $10$-dimensional family and $L\sub \Qs{i}$ is a codimension $3$ condition it is clear that codim$(q(p^{-1}(L)))=12$ and that codim$(q(p^{-1}(Gr(2,7)))\geq 12-10=2$. This means that we can choose $Q_4$ general enough so that there is no line $L\sub\cap_{\cQ\in Q_4} \cQ$ which is disjoint from $S_4$.\\ Similarly consider $I=\{(L,Q_4)\in Gr(2,7)\times G|L\sub\cap_{\cQ\in Q_4} \cQ,\, L\cap S_4\not=\emptyset\}$ and $p,q$ as before where now $p:I\to \ol{G}= \{L\in Gr(2,7)|L\cap S_4\not=\emptyset\}$. The intersection of a line in $\pn{6}$ with a surface imposes $3$ conditions so that dim$\ol{G}=7$. Since a line through a point on $S_4$ imposes 2 conditions on quadrics through $S_4$ we have codim$(q(p^{-1}(l)))=8$ in this case, it follows that codim$(q(p^{-1}(\ol{G})))\geq 8-7=1$ so we can assume that there are no lines on $S$ intersecting $S_4$ in one point. We are left to examine the case when $L$ meets $S_4$ in at least 2 points, i.e. when $L$ is a secant line to the Veronese surface $S_4$. But as above this is impossible as long as $S$ is irreducible.\\ Thus $S$ has no lines on it and it is the blow up of a $K3$ surface in one point and $\cH_S=p^*(\ol{\cH}-2E)$, where $p:S\to\ol{S}$ is the blow up map and $\ol{\cH}$ is a line bundle on $\ol{S}$ of degree $16$. Again $S$ has no trisecant line because such a line would necessarily be a line in the Veronese surface. Notice that the conic sections on the Veronese surface each meet $S$ in 5 points. In fact the Veronese surface is the union of the 5-secant conic sections to $S$ and is therefore contained in any quadric which contains $S$. It is straightforward to check that the surfaces $S$ are cut out by $4$ quadrics and $3$ cubics in $\pn{6}$. \subsection{\large A family of rational surfaces of degree 12}\label{nl4} Let $S=Bl_{11}(\pn{2})$ be polarized by the line bundle $H=9l-\sum_1^5 3E_i-\sum_6^{11}2E_j$. Assume that the $11$ points blown up are in general position. More precisely we require that the following linear systems are empty for all possible sets of distinct indices: \begin{itemize} \item $|E_i-E_j|$ \item $|l-E_i-E_j-E_k|$ \item $|2l-\sum_{k=1}^6 E_{i_k}|$ \item $|3l-\sum_{k=1}^{10}E_{i_k}|$ and $|3l-2E_{i_0}-\sum_{k=1}^7E_{i_k}|$ \item $|4l-2\sum_{k=1}^2E_{i_k}-\sum_{k=3}^{11}E_{i_k}|$ and $|4l-2\sum_{k=1}^3E_{i_k}-\sum_{k=4}^{10}E_{i_k}|$ \item $|5l-2\sum_{k=1}^5E_{i_k}-\sum_{k=6}^{11}E_{i_k}|$ \item $|6l-2\sum_{k=1}^{8}E_{i_k}-\sum_{k=7}^{11}E_{i_k}|$, $|6l-3E_{i_0}-2\sum_{k=1}^{5}E_{i_k}-\sum_{k=6}^{10}E_{i_k}|$,\\ $|6l-2\sum_{k=1}^{9}E_{i_k}-E_{i_{10}}|$ \end{itemize} \begin{lemma} $H$ is a very ample line bundle on $S$. \end{lemma} \noindent{\em Proof.}\, $H$ is shown to be very ample in \cite{li}. We report here a different short proof which relies on \begin{lemma} {\rm (Alexander)\cite[Lemma 0.15]{kr}.} If $H$ has a decomposition $$H=C+D,$$ where $C$ and $D$ are curves on $S$, such that ${\rm dim}|C|\geq 1$, and if the restriction maps $H^0(\cO_S(H))\to H^0(\cO_D(H))$ and $H^0(\cO_S(H))\to H^0(\cO_C(H))$ are surjective, and $|H|$ restricts to a very ample linear system on $D$ and on every $C$ in $|C|$, then $|H|$ is very ample on $S$. \end{lemma} \ Consider the reducible hyperplane section: $$H=D_1+D_2=(3l-\sum_1^8E_i)+(6l-2\sum_1^5E_i-\sum_6^8E_j-2\sum_9^{11}E_k)$$ Then $D_1$ is embedded as a degree 6 elliptic curve and $D_2$ as a sextic curve of genus $2$. Moreover a general element of $|D_2|$ is irreducible by the choice of points in general position and all the elements in the pencil $|D_1|$ are irreducible. It follows that $H_{D_1}$ and $H_{D_2}$ are very ample. The fact that both the maps $H^0(S,H)\to H^0(D_1,H_{D_1})$ and $H^0(S,H)\to H^0(D_2,H_{D_2})$ are surjective concludes the argument. \hspace*{\fill}Q.E.D.\vskip12pt plus 1pt \begin{lemma} There are no lines on $S$.\end{lemma} \noindent{\em Proof.}\, Assume $L=al-\sum_1^{5}a_i E_i-\sum_6^{11}b_j E_j$ is a line on $S$. Then looking at the intersection of $L$ with the twisted cubics $E_i$, $i=1,...,5$, and the intersection of $L$ with the conics $E_j$, $j=6,...,11$ we derive the bounds $0\leq a_i\leq 2$ and $0\leq b_i\leq 2$. This implies that $1=H\cdot L=9a-\sum_1^5 3a_i-\sum_6^{11}2b_i\geq 9a-30-24$, i.e. $a\leq 4$. The only numerical possibilities are: \begin{itemize} \item $L=E_i-E_j$; \item $L=3l-2E_{i_1}-\sum_{k=2}^5E_{i_k}-\sum_{k=6}^9E_{i_k}\quad \{i_1,\dots ,i_5\}=\{1,\dots ,5\}$,\\ $6\leq i_6<\dots <i_9\leq 11$; \item $L=2l-\sum_1^5E_i-E_j$, $6\leq j\leq 11$; \item $L=l-E_i-E_j-E_k\quad 1\leq i<j\leq 5<k\leq 11$. \end{itemize} But those are empty linear systems by the general position hypothesis. \hspace*{\fill}Q.E.D.\vskip12pt plus 1pt \begin{proposition}$S$ is a rational surface in $\pn{6}$ with no trisecant lines. \end{proposition} \noindent{\em Proof.}\, Consider the reducible hyperplane section $H=\Fd+\Fu$ where \begin{itemize} \item $\Fd=-K_S+E_i+E_j$ $i,j=6\dots,11$, i.e. an elliptic quartic curve. \item $\Fu=H-\Fd=6l-2\sum_1^5E_k-\sum_6^{11}E_k-E_i-E_j$, i.e. a curve of degree $8$ of genus $3$. \end{itemize} Fix a curve $\Fd$. Any trisecant line $L$, would together with $\Fd$ span a hyperplane, so there is some reducible hyperplane section $H=\Fd+\Fu$ for which $L$ is a trisecant. Since neither $\Fu$ nor $\Fd$ have trisecants, $L$ must in fact intersect both of these curves, so $\Fd$ and $L$ spans at most a $\pn{4}$. This means that we can always find a curve $C\in |\Fu|$ passing through the points in $L\cap\Fd$, and such that $L$ is a trisecant to $C\cup \Fd$. If $C$ is irreducible this is impossible since $C$ has no trisecant lines\\ Assume it is reducible and write $C=A+B$, were $A$ and $B$ are irreducible with $deg(A)\leq deg(B)$. Then the following cases could occur: \begin{itemize} \item[(a)] $A$ is a plane conic; \item[(b)] $A$ is a plane cubic or a twisted cubic; \item[(c)] $A$ is a quartic curve; \end{itemize} Let $A=\al l-\sum_1^5\as{i}E_i-\sum_6^{11}\be{j}E_j$.\\ Assume $\al=0$. If $A=E_k$ for $k\in\{1,...,5\}$ then $B=\Fd -E_k$, which is impossible by the general position hypothesis. If $A=E_k$ for $k\in\{6,...,11\}\setminus\{i,j\}$ then $B=\Fu -E_k$, this possibility will be analyzed more closely below.\\ Assume now $\al>0$, i.e. $A\neq E_i$, then by intersection properties and the assumption that $A$ and $B$ are effective divisors, $0\leq\as{i}\leq 2$, $0\leq\be{j}\leq 1$ and $1\leq\al\leq 6$. \\ \noindent (a) Going over the possibilities for $\al,\as{i}$ and $\be{j}$ gives no result by the general position hypothesis.\\ \noindent (b) Examining the possible choices for $\al,\as{i}$ and $\be{j}$ we get: \begin{itemize} \item $A=l-E_m-E_n, 1\leq m<n\leq 5$ residual to $B=5l-2\sum_{k=1}^5E_k+E_m+E_n-\sum_{k=6}^{11} E_k-E_i-E_j$ \item $A=2l-\sum_1^5E_k$ residual to $B=4l-\sum_1^{11}E_k-E_i-E_j$ \item $3l-\sum_{k=1}^{11}E_k+E_m, 1\leq m\leq 5$ and $3l-\sum_1^5E_k-E_m-E_i-E_j$ \item $4l-2\sum_{k=1}^{11}E_k-E_m-E_n, 1\leq m<n\leq 5$ \item $5l-2\sum_1^5E_i-\sum_6^{11}E_j$ \end{itemize} In the first two cases the residual curve $B$ does not exist by the general position hypothesis. Likewise the curve $A$ does not exist in the last cases.\\ \noindent(c) Similar computations leads to \begin{itemize} \item $A=l-E_m-E_n$, $1\leq m\leq 5$ and $6\leq n\leq 11$, whose residual curve does not exist. \item $A=2l-\sum_1^5E_k+E_m-E_n$, $1\leq m\leq 5<n\leq 11$, whose residual curve does not exist. \item $A=F_{k,l}$, $B=F_{m,n}$ with $\{i,j,k,l,m,n\}=\{6,\dots,11\}$ \end{itemize} We are left with the cases $C=E_k+(6l-2\sum_1^{5}E_m-\sum_6^{11}E_n-E_i-E_j-E_k)$ or $C=F_{k,l}+F_{m,n}$.\\ Notice that in both cases the projective spaces spanned by the two components, $<A>,<B>$, intersect in a line $L=<A>\cap<B>$. Moreover neither $A$ nor $B$ admits trisecant lines and $A\cap B=2$. It follows that $A\cap <B>=A\cap B=B\cap <A>$. Any trisecant line $L$ to $C$, must meet $A$ ( or respectively $B$) in two points and thus it is contained in $<A>$, which implies $L\cap B\sub A\cap B$. But this means that $L$ is a trisecant line for $A$, which is impossible. \hspace*{\fill}Q.E.D.\vskip12pt plus 1pt These surfaces are cut out by 3 quadrics and 4 cubics in $\pn{6}$. \subsection{ \large Abelian surfaces}\label{nl8} Recently Fukuma \cite{ab1}, Bauer and Szemberg \cite{ab2} have proved that the general $(1,7)$-polarized abelian surface in $\pn{6}$ does not have any trisecant lines. The argument uses a generalization of Reider's criterium to higher order embeddings. These surfaces are not contained in any quadrics. \section{ \Large Construction of surfaces with lines}\label{wline} In this section we will construct examples of surfaces with no proper trisecant lines but with lines on it. In all the cases below the surfaces are cut out by quadrics, so naturally there are no proper trisecant lines. In fact, we do not know of any surface with lines on it, but with no proper trisecant which is not cut out by quadrics. Natural examples of surfaces with at least a one dimensional family of lines are given by surfaces of minimal degree, i.e. the rational normal scrolls of degree $N-1$ in $\pn{N},3\leq N\leq 6$. Similarly the Del Pezzo surfaces of degree 4, 5 and 6 form natural families of surfaces cut out by quadrics. They have a finite number of lines on them. We construct other rational and nonrational surfaces. \subsection{\large Rational surfaces} \begin{para}\label{con1}Conic bundles of degree $6$\end{para} Consider a cubic 3-fold scroll $V\sub\pn{5}$ and let $S=V\cap \cQ$ be the surface of degree $6$ given by the intersection with a general quadric hypersurface. $S$ is smooth as soon as $\cQ$ avoids the vertex of $V$. Therefore we have two cases, when $V$ is smooth and when $V$ is the cone over a smooth cubic surface scroll in $\pn{4}$.\\ Let $\ol{V}$ be the $\pn{2}$-scroll which is mapped to $V$ by $\cH$, and let $\cF$ be a member of the ruling. By abuse of notation we denote the pullback of $S$ to $\ol{V}$ by $S$, it is isomorphic anyway. Then by adjunction $$K_S=(-3\cH+\cF+2\cH)|_S=-\cH|_S+F|_S$$ and thus $(K_S)^2=2$ and $K_S\cdot\cH_S=-4$. Furthermore $p_g(S)=q(S)=0$, so $S$ is rational. The ruling of the scroll define a conic bundle structure on $S$, and there are 6 singular fibers, i.e. 12 $(-1)$-lines in the fibers since $(K_S)^2=2$. These surfaces are cut out by 4 quadrics in $\pn{5}$. \begin{para}\label{con2}Conic bundles of degree $7$\end{para} Consider a rational normal 3-fold scroll $V\sub\pn{6}$ of degree 4, and let $\cQ$ be a general quadric hypersurface containing a member of the ruling $\cF$. Then the complete intersection $V\cap\cQ=\cF\cup S$, where $S$ is a surface of degree $7$ in $\pn{6}$. As soon as $V$ is smooth, $S$ is also smooth. In fact in this case $S$ would be singular if $V$ was singular.\ By adjunction $$K_S=(-3\cH+\cF+2\cH)|_S=-\cH|_S+F|_S,$$ where $\cH$ is a hyperplane section. Thus $(K_S)^2=3$ and $K_S\cdot\cH_S=-5$. Like in the previous case the ruling of $V$ define a conic bundle structure on $S$ with 5 singular fibers, i.e. 10 $(-1)$-lines altogether in the fibers. If $S$ had a trisecant line, $L$, then $L$ would be a line in $\cF$ intersecting the curve $C=S\cap\cF$ in three points. In this case $C$ is conic section so there is no trisecant. In fact one can also show that $S$ is cut out by 8 quadrics. \begin{para}\label{con3}Conic bundles of degree $8$\end{para} Let $V$ be a rational normal $3$-fold scroll of degree $4$ in $\pn{6}$, and let $\cQ$ be a general quadric hypersurface not containing any singular point on $V$. Then $V$ is smooth or is a cone with vertex a point, and $S=V\cap\cQ$ is a smooth surface of degree $8$ in $\pn{6}$. We get two cases like in \ref{con1}. Proceeding with notation like in that case we get $K_S=2\cF_S-\cH_S$ and $K_S^2=0$. Thus we get conic bundles with sectional genus $3$ with $8$ singular fibers, i.e. 16 $(-1)$-lines in fibers. These surfaces are cut out by $7$ quadrics. \begin{para}\label{p48}A family of surfaces of degree $8$\end{para} Let $V$ the a cone over $\pn{1}\times\pn{2}$ as in the previous example and let $\Qs{1},\Qs{2}$ be two quadrics containing a $F=\pn{3}$ of the ruling, in particular they pass through the vertex on the cone. Then $V\cap\Qs{1}\cap\Qs{2}=S\cup F$, where $S$ is a smooth surface of degree $8$ in $\pn{6}$. In fact let $E$ be the exceptional divisor in the blow up $\ol{V}$ of $V$ at the vertex, and let $\ol{S}$ be the strict transform of $S$. With notation as above $\ol{S}=(2\cH-\cF-E)\cap (2\cH-\cF-E)$ on $\ol{V}$. The exceptional divisor $E$ is isomorphic to $\pn{1}\times\pn{2}$ and $E|_E=-\cH$ via this isomorphism. Furthermore the self intersection of $E|_{\ol{S}}$ is \[\begin{array}{ll}[(2\cH_V-\cF-E)^2]\cdot E\cdot E&=[4\cH^2-4\cH F+2\cF E-a\cH E+E^2]\cdot E\\ &=2H^2\cdot\cF-E^3=2-3=-1 \end{array} \] By adjunction $K_S=-\cF_S$ and thus $|-K_S|$ is a pencil of elliptic curves with one base point at the vertex of $V$. It follows that $S$ is rational of degree $8$, sectional genus $3$ and $K_S^2=1$. The adjunction $|\cH+K_S|$ maps the surface birationally to $\pn{2}$, so the the surface is $Bl_{8}({\bf P}^2)$ with $\cH=4l-\sum_1^8 E_i$. These surfaces are cut out by 6 quadrics in $\pn{6}$. \begin{para}\label{p49}Two families of surfaces of degree $9$\end{para} Let $\ol{V}$ be a $\pn{3}$-bundle of degree $3$ over $\pn{1}$ with ruling $\cF$ and let $V$ be its image rational normal $4$-fold of degree $3$ in $\pn{6}$ under the map defined by $\cH)$ as in \ref{elliptic}. Let $\Qs{1},\Qs{2}$ be general quadrics with no common point in the vertex of $V$ which contain a smooth cubic surface $S_3=V\cap\pn{4}$, for some general $\pn{4}\sub\pn{6}$. Then $V\cap\Qs{1}\cap\Qs{2}=S_3\cup S$, and $S$ is smooth as soon as the vertex of $V$ is a line or a point. The curve of intersection $C=S_3\cap S$ is then a curve of degree $6$ represented in $S_3$ by the divisor $2\cH_{S_3}$. Thus $S$ cannot have trisecants since $C$ has no trisecant lines.\\ Since $S_3$ meets each ruling of $V$ in a line, $S$ which is linked to $S_3$ in 2 quadrics meets each ruling of $V$ in a twisted cubic curve. Therefore $S$ is rational. Furthermore, $K_{\ol{V}}=-4\cH+\cF$, so by adjunction $K_S=(-4\cH+\cF+4\cH)|_S-S_3|_S=-C+\cF_S$. Thus $K_S\cdot\cH_S=(\cF_S-C)\cdot \cH_S=-3$, and $S$ has sectional genus $4$. Notice that, by adjunction, $C\cdot \cF_S=2$. Let $D=\cH-C$, then $D$ has degree 3 and moves in a pencil, so it must be a twisted cubic curve, with $D^2=0$. The genus formula implies that $C\cdot D=3$, and so $C^2=3$. Therefore $K_S^2=(\cF_S-C)^2=-1$. >From the two types of cubic scrolls $V$ we get two types of surfaces $S$. Both are rational surfaces of degree $9$, sectional genus $4$ and $K_S^2=-1$. In both case $S$ has two pencils of twisted cubic curves, but in one case any two of the curves in a pencil span $\pn{6}$, while in the other case any two of them span a $\pn{5}$. The adjunction $|\cH+K_S|$ maps the surface birationally to a smooth quadric surface in $\pn{3}$, so the surface is $Bl_{9}({\bf P}^1\times {\bf P}^1)$ with $\cH=3(1,1)-\sum_1^9 E_i$. The two families correspond to the cases when the 9 points on the quadric is the complete intersection of two rational quartic curves (of type $(3,1)$ and $(1,3)$ respectively) or not. These surfaces are cut out by 6 quadrics in $\pn{6}$. \begin{para}\label{p410}A family of surfaces of degree $10$\end{para} Consider the Del Pezzo surface $S_6$ of degree $6$ in $\pn{6}$, and $4$ general quadrics containing it, $\Qs{1},\Qs{2},\Qs{3},\Qs{4}$. Then $\Qs{1}\cap\Qs{2}\cap\Qs{3}\cap\Qs{4}=S\cup S_6$, where $S$ is a smooth surface of degree $10$ in $\pn{6}$. The exact sequences (\ref{E1}), (\ref{E2}) of \ref{nl5} applied in this case show that $S$ is rational. The adjunction formula gives $K_S\cdot \cH_S=(\cH-S_6)_S\cdot\cH_S=-2$, so the sectional genus is $5$. The intersection curve $C=S\cap{S_6}=2\cH_{S_6}$ on $S_6$, which means that $S$ has no trisecant lines. These surfaces are cut out by 5 quadrics in $\pn{6}$. The adjoints of the surfaces in \ref{nl4} are of this type. \subsection{\large Non rational surfaces} There are also non rational surfaces without proper trisecant lines. The first examples are \begin{para}\label{ell} Elliptic scrolls\end{para} The elliptic normal scrolls of degree 7 for which the minimal self intersection of a section is $1$, are cut out by $7$ quadrics (cf. \cite{hks}).\\ Finally there are \begin{para}\label{prk3}A family of non minimal $K3$-surfaces.\end{para} Consider an inner projection of a general nontrigonal and nontetragonal $K3$-surface $\ol{S}$ of degree $12$ in $P^7$ cf. \cite{Mu}. The surface $S$ is the projection from a point $p\in \ol{S}$, $\pi_p:\ol{S} \to S$. Then $S$ is a $K3$-surface of degree $11$ in $\pn{6}$ with one line, i.e. the exceptional line over $p$. Any trisecant of $S$ will come from a trisecant of $\ol{S}$ or from a four secant $\pn{2}$ to $\ol{S}$ through $p$. But a normally embedded $K3$-surface with a trisecant is trigonal and with a $4$-secant plane is tetragonal which is avoided by assumption. So $S$ has no trisecant. These surfaces are cut out by $5$ quadrics in $\pn{6}$. \section{ \Large Complete list of surfaces with no lines}\label{nlist} Throughout this section we will assume that $S$ is a surface embedded in $\pn{6}$ by the line bundle $\cH$, with no lines on it and no trisecant lines. $C$ will denote the general smooth hyperplane section of $S$.\\ With this hypothesis the invariants of the surfaces will give zero in the two formulas of Le Barz in \cite{leb}. Let \begin{itemize} \item $n=$degree$(S)$; \item $k=K_S^2$; \item $c=c_2(S)$; \item $e=K_S\cdot\cH$; \end{itemize} Then the formula of Le Barz for the number of trisecant lines meeting a fixed $\pn{4}\sub\pn{6}$ is: \begin{equation}\label{A} D_3=2 n^3 - 42 n^2 + 196 n-k(3n-28)+c(3n-20) - e(18n - 132)\quad (=0) \end{equation} and the formula for the number of lines in $\pn{6}$ which are tangential trisecants, i.e. tangent lines that meet the surface in a scheme of length at least 3, is: \begin{equation}\label{B} T_3=6 n^2 - 84 n+k(n-28) -c(n-20)+ e(4n-84)\quad (=0) \end{equation} We set these equal to 0. Next we use the Castelnuovo bound for an irreducible curve of degree $n$ in $\pn{N}$ \cite{ACGH}: $$p(N)=[\frac{n-2}{N-1}](n-N-([\frac{n-2}{N-1}]-1)\frac{N-1}{2})$$ where $[x]$ means the greatest integer $\leq x$, and refined versions of it given by the following two theorems of Harris and Ciliberto: \begin{theorem}\label{HA}\cite{ha}, \cite[3.4]{ci} Let $p_1=\frac{n^2}{10}-\frac{n}{2}$ and let $C$ be a reduced, irreducible curve in $\pn{5}$ of degree $n$ and genus $g$. Then \begin{itemize} \item[(a)] If $g(C)> p_1$ then $C$ lies on a surface of minimal degree in $\pn{5}$; \item[(b)] if $g=p_1$ and $n\geq 13$ then $C$ lies on a surface of degree $\leq r$. \end{itemize} \end{theorem} \begin{theorem}\cite[Th. 3.7]{ci}\label{p3} Let $p_3(n,r)=\frac{n^2}{2(r-1)+3}+O(n)$ and $p_2(n,r)=\frac{n^2}{2(r-1)+4}+O(n)$, where $0\leq O(n)\leq 1$.\\ Let $C$ be a reduced, irreducible, non degenerate curve in $\pn{6}$, $r\geq 6$, of degree $n$ and genus $p$. Then \begin{itemize} \item if $n>(r+1)$ and $p>p_3(n,r)$ then $C$ lies on a surface of degree $\leq r$; \item if $2r+3\leq n\leq 5r+2$ and $p>p_2(n,r)$ then $C$ lies on a surface of degree $\leq r$. \end{itemize} \end{theorem} \subsection{ \large The cases with $n\leq 11$} Notice that surfaces in $\pn{3}$ and in $\pn{4}$ have trisecant lines or contain lines. The only surfaces in $\pn{5}$ which do not contain lines or have trisecants are the Veronese surfaces and the general complete intersections $(2,2,2)$, i.e. the general nontrigonal $K3$-surfaces of degree 8.\\ Then assuming $N\geq 5$ and imposing the following conditions : \begin{itemize} \item (\ref{A}) and (\ref{B}); \item $g\leq p(5)$ ($S$ spans $\pn{6}$); \item $c+k=12\cdot$integer; \item $k\leq 3c$ (Miyaoka) and $kn\leq e^2$ ( Hodge index Theorem) \end{itemize} numerical computations give a list of possible invariants for $n\leq 11$: \begin{center} \begin{tabular}{|c||c|c|c|c|} \hline &$n $ & $e$& $k$& $c$\\ \hline\hline (1)&$4$ &$ -6$ &$9$& $3$\\ \hline (2)&$ 8$& $ -4$& $2$& $10$\\ \hline (3)&$ 8$&$0$&$ 0$&$ 24$ \\\hline (4)&$10$ &$0$&$ 0$&$ 24$ \\\hline \end{tabular} \end{center} The examples \ref{nl1}, \ref{nl2}, \ref{nl3}, \ref{nl7} in section 1 have these invariants, and it is easy to see that these are the only ones. In the first two cases any smooth surface in the family would have no lines, but for $K3$-surfaces it is easy to construct degenerations to smooth surfaces with one or several lines. These lines would be $(-2)$-lines on the $K3$-surface. Therefore the four cases are: \begin{itemize} \item $(S,\cH)=(\pn{2},\Os{\pn{2}}(2)$ is a Veronese surface in $\pn{5}$; \item $(S,\cH)=(Bl_7(\pn{2}),-2K_S)$ is a Del Pezzo surface in $\pn{6}$; \item $S$ is a general nontrigonal $K3$ surface of degree $8$ in $\pn{5}$; \item $S$ is a general tetragonal or nontetragonal $K3$ surface of degree $10$ in $\pn{6}$. \end{itemize} \subsection{ \large When $n\geq 12$} Eliminating $c$ from (\ref{A}) and (\ref{B}) gives $$k=\frac{[n^4-32n^3+332n^2-1120n]-e(3n^2-80+480)}{8n}$$ By \ref{HA} if $n\geq 11$ then we get for the general hyperplane section $C$ of $S$ that $g(C)\leq p_1$ or $C$ lies on a surface of minimal degree in $\pn{5}$. In the last case $C$ would lie on a rational normal scroll, or on the Veronese surface.\\ If $C\sub S_4$, where $S_4$ is a scroll of degree $4$ in $\pn{5}$ with hyperplane section $\cH$ and ruling $\cF$, then $C=2\cH+b\cF$ on $S_4$ and thus $S$ is a conic bundle with $K_S\cdot\cH=n-12$. But $(K_S+\cH)^2$ gives $K_S^2=24-3n$, which implies the existence of $3n-16$ singular fibers, i.e. $(-1)$-lines in $S$.\\ Assume now $S_4$ is a Veronese surface, then $S\sub V$, where $V$ is the cone over $S_4$. Let $\ol{V}$ be the blowup of $V$ in the vertex, then $\ol{V}$ is a $\pn{1}$-bundle over $\pn{2}$. In this case $S\in|2\cH+b\cF|$ on $\ol{V}$ where $\cF$ is the pullback of a line from $\pn{2}$. Consider $[(\cH-2\cF)^2]\cdot S=-2b$, where $(\cH-2\cF)$ is the contracted divisor. Since $S$ is smooth $b=0$ and $S$ is a Del Pezzo surface of degree $n=8<11$ in $\pn{6}$ (cf. \ref{nl2}).\\ We can therefore assume that $C$ is not contained in a surface of minimal degree in $\pn{5}$.\\ The Hodge index theorem, i.e. $k\leq \frac{e^2}{n}$ implies $$n^4-32n^3+332n^2-1120n\leq 8e^2+e(3n^2-80+480)$$ and the bound $e<\frac{n^2}{5}-2n$ yields $ n\leq 27$ and the following list:\\ \begin{center} \begin{tabular}{|c||c|c|c|c|} \hline &$n $ & $e$& $k$& $c$\\ \hline\hline (a)& $ 12$&$-2$&$ -3$&$ 3$ \\\hline (b)& $ 12$&$0$&$ -2$&$ 14$\\\hline (c)& $ 12$&$ 2$&$-1$&$ 25$\\\hline (d)& $ 12$&$4$&$ 0$&$36$ \\ \hline (e)& $ 14$&$ 0$&$ 0$&$ 0 $ \\\hline (f)& $ 16$&$16$&$16$&$80 $ \\ \hline (g)& $20$&$ 40$&$70$&$ 206$ \\ \hline \end{tabular} \end{center} Let us examine the various cases:\\ \noindent \underline{case (a)}\\ Since $\chi({\cal O}_S)=0$, $K_S^2=-3$ and $\cH_S\cdot K_S=-2$, the surface $S$ is the blow up of an elliptic ruled surface in $3$ points.\\ Since $S$ has no lines the exceptional curves have degree at least 2. Now $(K_S+\cH_S)^2$=$h^0(K_S+\cH_S)=5$, so by Reider's criterium $S$ is embedded in $\pn{4}$ via $|K_S+\cH_S|$. But there are no nonminimal elliptic ruled surfaces of degree 5 in $\pn{4}$, so this case does not occur.\\ \noindent\underline{case (b)}\\ This is a rational surface with $K_S^2=-2$. The adjoint bundle embeds $S$ as a surface of degree $10$ and genus $5$ in $\pn{6}$. Assuming that $S$ has no lines $|2K_S+\cH_S|$ will blow down $(-1)$- conics and embedd the blown down surface as a Del Pezzo surface of degree 4 in $\pn{4}$. Therefore $K_S+\cH_S=6l-\sum_1^5 2E_i-\sum_6^{11} E_i$ and $\cH_S=9l-\sum_1^5 3E_i-\sum_6^{11}2 E_i$. This is example \ref{nl4} in section \ref{noline}.\\ \underline{case (c)}\\ This is the blow up of a $K3$ surface in one point, $\pi:S\to\ol{S}$ and $\cH_S=\pi^{\ast}(\cH_{\ol{S}})-2E$ where $\cH_{\ol{S}}^2=16$.\\ Using the exact sequence: $$0\to {\cal I}_S\otimes {\cal O}_{\pn{6}}(2)\to {\cal O}_{\pn{6}}(2)\to {\cal O}_S(2)\to 0$$ and the fact that $h^1(S,{\cal O}_S(2))=0$ and thus $h^0(S,{\cal O}_S(2))=24$ we see that $h^0({\cal I}_S\otimes {\cal O}_{\pn{6}}(2))\geq 28-24=4$ and therefore $S$ lies on at least $4$ quadric hypersurfaces.\\ If ${\cal Q}_1\cap ... \cap {\cal Q}_4=S\cup S_4$ where $S_4$ is the residual degree $4$ surface then $S_4$ must be a Veronese surface and $S$ is as in example \ref{nl5}.\\ Assume now that $\cap \cQ_i$ is not a complete intersection, i.e. $\cap \cQ_i=V$, a threefold in $\pn{6}$. Consider a general $\pn{4}\sub\pn{6}$, then $C=V\cap \pn{4}$ is a curve in $W=\cap\cQ_i\cap \pn{4}$. This only occurs when $C$ lies on a cubic scroll in $\pn{4}$. It is easy to see that this happens only if $V$ lies on a cubic scroll in $\pn{6}$. In this case $S$ lies in a cubic $4$-fold scroll in $\pn{6}$. Let $\cF$ be the general fiber and $\cF_S=\cF\cap S$. $\cF_S$ is a pencil of smooth curves on $S$ and $\cF_S^2=0$ or$\cF_S^2=1$. Moreover since $\cF_S\sub\pn{3}$ we know that $\cF$ has no trisecant lines only if the degree$(\cF_S)\leq 4$. Since $S$ is a non-minimal $K3$-surface it cannot have a pencil of rational curves, so $\cF_S$ is an elliptic quartic curve. Since $S$ has degree 12 it must be the complete intersection of the cubic scroll and two quadrics, but this is not $K3$ cf. example \ref{nl5}, so this is not possible.\\ \underline{case (d)}\\ Since $K_S^2=0$ and $\chi(S)=3$ this is an elliptic surface of degree $12$ and $K_S\cdot\cH_S=4$. Let $K_S=mF+\sum(m_i-1)F)$, where $F$ is the general fiber of the elliptic fibration $S\to B$ and $m_iF$ the multiple fibers. From $K_S\cdot \cH_S=4$ and $m=\chi({\cal O}_S)+2g(B)-2\geq 1$ we see that the only possibility is $m=1$ and thus $g(B)=0$, $K_S=F$ and no multiple fibers. So $|K_S|$ gives a fibration over $\pn{1}$. The canonical curves are elliptic quartic curves, they each span a $\pn{3}$, and these $\pn{3}$s generate a cubic scroll. The surface have degree 12 and the canonical pencil have no basepoints so $S$ must be complete intersection of the scroll and two quadric hypersurfaces. This is example \ref{elliptic} in section \ref{noline}. \\ \underline{case e)}\\ Since $\chi(S)=K^2_S=c_2(S)=0$ the surface $S$ must be minimal abelian or bielliptic. Following Serrano's analysis cf. \cite{Ser} of ample divisors on bielliptic surfaces, one sees that any minimal bielliptic surface of degree 14 in $\pn{6}$ has an elliptic pencil of plane cubic curves, i.e. it has a 3-dimensional family of trisecants. The general abelian surface have no trisecant however, cf \cite{ab1, ab2}. This is example \ref{nl8}.\\ \underline{case f)}\\ The surface must be the complete intersection of $4$ quadric hypersurfaces in $\pn{6}$, example \ref{nl9} in section \ref{noline}.\\ \underline{case g)}\\ In this case $g(\cH)=p_1$ and thus by (b) in \ref{HA} if the general hyperplane section $C$ is not contained in a minimal surface, it lies on a surface $S_5$ of degree $5$ in ${\bf P}^5$ i.e an anticanonically embedded Del Pezzo surface or the cone over an elliptic quintic curve in $\pn{4}$.\\ In either case the sectional genus of $C$ implies that $C$ is the intersection of $S_5$ with a quartic hypersurface, and each line on $S_5$ will be a 4-secant line to $C$. This excludes case (g).\\ The previous results summarizes as follows: \begin{theorem} Let $S$ be a smooth surface embedded in $\pn{6}$ with no lines. Then $S$ has no trisecants if and only if it belongs to one of the cases listed in the table below: \begin{center} \begin{tabular}{|c|c|c|c|} \hline surface &degree&linear system& example \\ \hline\hline ${\bf P}^2$&4&${\cal O}_{\bf P}^2(2)$& \ref{nl1} \\\hline $Bl_7({\bf P}^2)$&8&$6l-\sum 2E_i$&\ref{nl2}\\\hline $K3$&8&complete intersection (1,2,2,2)&\ref{nl3}\\\hline $K3$&10&nontrigonal of genus 6&\ref{nl3},\ref{nl7} \\ \hline $Bl_{11}({\bf P}^2)$&12&$9l-\sum_1^53E_i-\sum_1^6 2E_i$&\ref{nl4} \\ \hline $Bl_1(K3)$&12&$p^*(\ol{\cH})-2E$&\ref{nl5}\\ \hline Elliptic&12& minimal with $p_g=2$&\ref{elliptic}\\ \hline Abelian&14& (1,7) polarization&\ref{nl8}\\ \hline general type&16& complete intersection (2,2,2,2)&\ref{nl9}\\ \hline \end{tabular} \end{center} \end{theorem} \section{\Large List of surfaces with lines}\label{list2} For surfaces with lines and no proper trisecants the formula (\ref{B}) for the number of tangential trisecants is not necessarily 0. If $S$ has finitely many lines every $(-1)$-line contributes with multiplicity 1 to the formula. Therefore we cannot use this formula as it stands in this case. We will use several alternative approaches instead, they will recover all our examples, but will not completely treat all cases, as is made precise in theorem 0.0.1. One approach we will use is a projection to $\pn{4}$ to get some numerical relations replacing (\ref{B}). \begin{lemma}\label{lemma} Let $S$ be a smooth surface in $\pn{6}$ with no proper trisecant lines. Assume that $L\subset S$ is a line on the surface, and let $\pi_L:S\to \pn{4}$ be the projection of $S$ from $L$, i.e. the morphism defined by $|\cH_S-L|$. Then $\pi_L$ is the composition of the contraction of any line on $S$ which meet $L$ and an embedding. In particular, if $S$ has finitely many lines, $\pi_L(S)$ is smooth unless there are some line $L_1$ on $S$ meeting $L$ with $L_1^2\leq -2$. \end{lemma} \noindent{\em Proof.}\, Let $P$ be a plane in $\pn{6}$ which contains $L$. Let $Z_P$ be residual to $L$ in $S\cap P$. If $Z_P$ is finite, its degree is at most 1, otherwise $S$ would have a trisecant in $P$. If $Z_P$ contains a curve, this curve would have to be a line which would coincide with $Z_P$, since again $S$ has no trisecants. \hspace*{\fill}Q.E.D.\vskip12pt plus 1pt With this, let us first examine the cases of surfaces with at least a one dimensional family of lines. \begin{proposition} Let $S$ be a scroll in $\pn{6}$. Then $S$ has no proper trisecant lines if and only if $S$ is \begin{itemize} \item a rational normal scroll; \item an elliptic normal scroll, with minimal self intersection $E_0^2=e=1$, cf \ref{ell} . \end{itemize} \end{proposition} \noindent{\em Proof.}\, If $S$ is a rational normal scroll or elliptic normal scroll with $e=1$, then it is cut out by quadrics and therefore has no trisecant lines.\\ Assume now $S$ is a scroll with no trisecant lines and let $L$ be a line on it. Consider the projection of $S$ from $L$: $$\pi_L:S\to P^4$$ If there is no line on $S$ meeting $L$ then $\pi_L$ is an embedding by \ref{lemma}. Since the only smooth scrolls in $\pn{4}$ are the rational cubic scrolls and the elliptic quintic scrolls, $S$ must be as in the statement.\\ If $\pi_L$ is not a finite map, i.e. there is a line section $L_0$, then $S$ must be rational and normal. Assume in fact $S$ not normal, i.e. $S=\pn{}(\Os{}(1)\oplus\Os{}(b))$ with $b\geq 5$, and consider the projection: $$\pi_{L_0}:S\to \pn{4}$$ The image curve $\pi_{L_0}(S)$ is a rational non normal curve in $\pn{4}$ and therefore it has a trisecant line, $L_t$. Then the linear span $\pn{3}=<L_t,L_0>$ contains three rulings of $S$, therefore also a pencil of trisecant lines to $S$.\hspace*{\fill}Q.E.D.\vskip12pt plus 1pt If $S$ is not a scroll, it has only a finite number of lines. Notice that only lines with self intersection $(-1)$ contributes to the formula (\ref{B}) in \cite{leb}. Therefore we will assume that $S$ has at least one $(-1)$-line, $L$, on it and examine separately the following overlapping cases: \begin{itemize} \item $L$ is isolated, i.e. it does not intersect other lines of negative self-intersection. \item $L$ intersects some $(-1)$-lines. \item $L$ can be contracted so that it is the exceptional line of an inner projection from $\pn{7}$. \end{itemize} \subsection{ \large Surfaces with at least one isolated line} Assume now that $S$ has a finite number of $(-1)$-lines and at least one $L$ is isolated, i.e. it does not intersect any other line. Then the projection: $$\pi_L:S\to \pn{4}$$ is an embedding by \ref{lemma}. Let $\pi_L^*(\cH)+L=\cH_S$, and notice that $K_{\pi_L(S)}=K_S$. Using the same invariants $n,e,k,c$ for $S$ we have that \begin{itemize} \item degree$(\pi_L(S))=n-3$; \item $K_{\pi_L(S)}\cdot\cH=e+1$; \item $K_{\pi_L(S)}^2=k$; \item $c_2(\pi_L(S))=c$; \end{itemize} and thus the double point formula in $\pn{4}$ gives \begin{equation}\label{D} (n-3)(n-13)-5e-k+c+29=0 \end{equation} Proceeding as in the previous section, i.e. using the assumptions: \begin{itemize} \item (\ref{A}) and (\ref{D}); \item $g\leq p(5)$; \item $c+k=12\cdot$integer; \item $k\leq 3c$ and $kn\leq e^2$; \end{itemize} numerical computations give the following set of invariants: \begin{center} \begin{tabular}{|c||c|c|c|c|} \hline &$n $ & $e$& $k$& $c$\\ \hline\hline (a)& $ 8$&$-8$&$ 5$&$ -5$ \\\hline (b)& $ 8$&$-4$&$ 1$&$ 11$\\\hline (c)& $ 9$&$ -3$&$ -1$&$ 13 $ \\\hline (d)& $ 10$&$ -2$&$ -2$&$ 14 $ \\\hline (e)& $ 11$&$ 1$&$ -1$&$ 25 $ \\\hline \end{tabular} \end{center} \noindent \underline{case (a)}\\ $\chi(\Os{S})=0$ and $K_S^2=5$, this is impossible.\\ \noindent\underline{cases (b), (c), (d)}\\ Since $e=\cH_S\cdot K_S\leq -2$ and $\chi(\Os{S})=1$ the surfaces $S$ are rational. The adjunction morphism defined by $|\cH_S+K_S|$ is birational and maps $S$ onto $\pn{2}$, a quadric in $\pn{3}$ and a Del Pezzo surface of degree 4 in $\pn{4}$ respectively in the three cases. Thus we recover the surfaces constructed in \ref{p48}, \ref{p49} and \ref{p410}.\\ \noindent\underline{case(e)}\\ Since $\cH_S\cdot K_S=1$ and $\chi(\Os{S})=2$ the surface $S$ must be a $K3$-surface blown up in one point. Thus we recover the inner projection of the general $K3$-surface of degree $12$ in $\pn{7}$ described in \ref{prk3}. The line $L$ is the only exceptional line on the surface.\\ We have then proved: \begin{proposition}\label{proj4} Let $S$ be a surface in $\pn{6}$ with no trisecant lines and with at least one isolated $(-1)$-line. then $S$ is one of the following: \begin{itemize} \item A rational surface of degree $8$ and genus $3$, as in \ref{p48}; \item A rational surface of degree $9$ and genus $4$, as in \ref{p49}; \item A rational surface of degree $10$ and genus $5$, as in \ref{p410}; \item A non minimal $K3$-surface of degree $11$ and genus $ 7$, as in \ref{prk3}. \end{itemize} \end{proposition} \subsection{ \large Conic bundles}\label{conic} Assume now that $S$ has at least two $(-1)$-lines $L_1$ and $L_2$ which meet.\\ Then $(L_1+L_2)^2=0$ and $L_1+L_2$ or some multiple of it moves in an algebraic pencil of conic sections. Thus $S$ is a conic bundle.\\ If there is a line $L$ of selfintersection $L^2\leq -2$ intersecting $L_1$, then this line intersects all the members of the pencil and therefore is mapped onto the base curve. This would imply then that $S$ is a rational conic bundle in $\pn{6}$. Moreover the general hyperplane section $C\in|\cH_S|$ is a hyperelliptic curve which, since we are in the smooth case, must be nonspecial. This implies $h^1(S,\cH_S)=0$ and thus $7=1+\frac{n-e}{2}$ by Riemann Roch. Since $(K_S+\cH_S)^2=n+2e+k=0$ we derive: \begin{itemize} \item $e=n-12$; \item $k=24-3n$; \item $c=3n-12$. \end{itemize} Substituting those values in the first Le Barz equation, (\ref{A}), we get an equation of degree $3$ in the variable $n$ and therefore three possible cases, which are easily seen to be the ones constructed in \ref{con1}, \ref{con2} and \ref{con3}.\\ Let us now assume that there is no line $L$ of selfintersection $L^2\leq -2$ intersecting $L_1$. Then the projection: $$\pi_{L_1}:S\to \pn{4}$$ will be composed by the contraction of $L_2$ and possibly other $-1$-lines intersecting $L_1$, and an embedding by \ref{lemma}. The conic bundle structure is of course preserved. Therefore the image surface in $\pn{4}$ is rational of degree 4 or 5, or it is an elliptic conic bundle of degree $8$ cf. \cite{BR, ES}. It is clear that the two first come from the surfaces of type \ref{con2} and \ref{con3}. The elliptic conic bundle has two plane quartic curves on it in $\pn{4}$ which are bisections on the surface. On $S$ their preimage would have degree $5$ or $6$ depending on the intersection with $L_1$. But this is a curve of genus $3$ so the degree upstairs must be $6$ if $S$ is smooth. A curve of degree $6$ and genus $3$ has trisecants, so this excludes this case. \subsection{ \large Inner projections from $\pn{7}$} In this section we will investigate the remaining cases, i.e. surfaces with $r$ $(-1)$-lines on it, which do not intersect. In other words $S$ has $r$ exceptional curves of the first kind $E_1,...,E_r$, which can therefore be contracted.\\ Let us assume that we can contract at least one of the $E_i$'s down to $x\in\ol{S}$ which is not a base point for $\ol{\cH}$, where $\cH=p^*( \ol{\cH})-E$ and $p$ is the projection map $p :S\to \ol{S}$. Then we can assume that $\ol{\cH}$ embeds $\ol{S}$ in $\pn{7}$ with no trisecant lines and with $(r-1)$ $(-1)$-lines.\\ Notice that the existence of $r$ $(-1)$-lines on $S$ gives a contribution of $4r$ in the number of tangential trisecants, which in terms of the formula as stated in (\ref{B}) means: $$ T_3=4r$$ Moreover the existence of $(r-1)$ $(-1)$-lines on $\ol{S}$ gives a contribution of $-(r-1)$ in the formula for trisecant lines of surfaces in $\pn{7}$. Using the same invariants $n,e,k,c$ for $S$, as in \ref{nlist}, we have: \begin{itemize} \item degree($\ol{S}$)=$n+1$; \item $K_{\ol{S}}\cdot\ol{\cH}=e-1$; \item $K_{\ol{S}}^2=k+1$; \item $c_2(\ol{S})=c-1$; \end{itemize} Plugging those invariants in the formula of Le Barz we get: \begin{equation}\label{C} S_3:=n^3-27n^2+176n+108+c(3n-37)-k(3n-53)-e(15n-177)=6r-6 \end{equation} We can then assume: \[ \left\{\begin{array}{ll} D_3&=2 n^3 - 42 n^2 + 196 n-K(3n-28)+c(3n-20) - e(18n - 132)=0\\ T_3&=6 n^2 - 84 n+k(n-28) -c(n-20)+ e(4n-84)=4r\\ S_3&=n^3-27n^2+176n+108+c(3n-37)-k(3n-53)-e(15n-177) \\&=-6r+6 \end{array} \right. \] Thus \[ \begin{array}{ll} 2S_3&=-2(6r-6)\\ &=D_3-12n^2+156n+108-k(3n-78)+c(3n-54)-e(12n-222)\\ &=D_3-3T_3+6n^2-96n+216-6k+6c-30e\\ &=-12r+6n^2-96n+216-6k+6c-30e \end{array} \] and hence \begin{equation} \label{ss}12=6n^2-96n+216-6k+6c-30e\end{equation} Let $\ol{C}$ be a general hyperplane section of $\ol{S}$, of degree $n+1$ and genus $g$. If $n+1\geq 15$, by \ref{p3} either $g\leq p_2$ or $\ol{C}$ lies on a surface of degree $5$ or $6$ in $\pn{6}$. If $\ol{C}\sub S_5$, where $S_5$ is a surface of degree $5$ in $\pn{6}$, then $S_5$ is a rational normal scroll and we can write $\ol{C}=2\cH+b\cF$, since we are assuming $\ol{C}$ has no trisecant lines. Like in section \ref{nlist}, $S_5$ can be viewed as a conic bundle and thus this case has already been studied in \ref{conic}.\\ If $\ol{C}\sub S_6$, where $S_6$ is a surface of degree $6$ in $\pn{6}$, then $S_6$ is a Del Pezzo surface embedded via the anticanonical bundle $-K_{S_6}$, i.e. $S_6=Bl_3(\pn{2})$, or it is the cone over an elliptic curve of degree 6 in $\pn{5}$. On the cone the curve $\ol{C}$ would intersect any ruling at most twice. Since it is smooth it passes through the vertex at most once, so the degree $n+1\leq 13$. In case of a Del Pezzo surface, write $\ol{C}=al-a_1E_1-a_2E_2-a_3E_3$, then the fact that $\ol{C}$ has no trisecants yields: $a_i\leq 2$, $i=1,2,3$ and $a-a_i-a_j\leq 2$, which implies $a\leq 6$. Since $n\geq 14$ the only possibility would be $a=6$ and $a_i<2$ for some $i$, but this is impossible since we need $a-a_i-a_j\leq 2$. \\ Thus we can assume $g\leq p_3$ (cf. \ref{p3}) when $n\geq 13$. Then easy computations involving (\ref{ss}), (\ref{A}), $kn\leq e^2$ (Hodge Index theorem) yield $n\leq 15$.\\ This bound and \begin{itemize} \item(\ref{A}), (\ref{B}), (\ref{C}); \item $k\cdot n\leq e^2$ and $k\leq 3c$; \item $n+e+2\leq p(5)$ (Castelnuovo bound); \item $n+2e+k=(K_S+\cH_S)^2>0$ (not a conic bundle case). \end{itemize} give the following numerical invariants: \begin{center} \begin{tabular}{|c||c|c|c|c|c|} \hline &$n $ & $e$& $k$& $c$&r\\ \hline\hline (a)& $ 8$&$-4$&$ 1$&$ 11$&$8$ \\\hline (b)& $ 9$&$-3$&$ -1$&$ 13$&$9$\\\hline (c)& $ 10$&$-2$&$ -2$&$14$&$6$ \\ \hline (d)& $ 11$&$ 1$&$ -1$&$ 25 $ &$1$\\\hline \end{tabular} \end{center} One can immediately see that we recovered the examples \ref{p48}, \ref{p48}, \ref{p410} and \ref{prk3} respectively. The results of this section summarize as follows: \begin{proposition} Let $S$ be a surface embedded in $\pn{6}$ with no trisecant lines and having $r$ $(-1)$-lines on it. Assume that $S$ is the inner projection of a smooth surface $\ol{S}\sub\pn{7}$, then $S$ is as in Proposition \ref{proj4} i.e. \begin{itemize} \item a rational surface of degree $8$ and genus $3$ with eight $(-1)$-lines, as in \ref{p48}; \item a rational surface of degree $9$ and genus $4$ with nine $(-1)$-lines, as in \ref{p49}; \item a rational surface of degree $10$ and genus $5$ with six $(-1)$-lines, as in \ref{p410}; \item A non minimal $K3$-surface of degree $11$ and genus $7$ with only one $(-1)$-line, as in \ref{prk3}. \end{itemize} \end{proposition} \section*{\vskip-1cm \Large Conclusion} We may summarize sections \ref{nlist}, \ref{list2} in the following \begin{proposition} Let $S$ be a surface in $\pn{6}$ with no trisecant lines. Unless $S$ is not an inner projection from $\pn{7}$ and every $(-1)$-line on $S$, if there are any, meet some other line $L$ on the surface with $L^2\leq -2$, the surface belongs to the list of examples in sections \ref{noline} and \ref{wline}. \end{proposition} We have made computations as in section \ref{nlist} fixing the number $r$ of $(-1)$-lines on the surface. Checking up to $r=100$ give no contribution to the list we have produced.\\ This numerical observation and the fact that the examples constructed in section \ref{wline} cover all the cases listed in section \ref{list2} lead us to make the following \begin{con} Let $S$ be a surface in $\pn{6}$ with no trisecant line, then the surface belongs to the list of examples in sections \ref{noline} and \ref{wline}. \end{con} \small

9705

alg-geom/9705001

en

https://arxiv.org/abs/alg-geom/9705001

alg-geom/9705001

Antony Maciocia

Antony Maciocia

Generalized Fourier-Mukai Transforms

13 pages, AMSLaTeX 1.2

J. reine angew. Math., 480 (1996) 197-211

The paper sets out a generalized framework for Fourier-Mukai transforms and illustrates their use via vector bundle transforms. A Fourier-Mukai transform is, roughly, an isomorphism of derived categories of (sheaves) on smooth varieties X and Y. We show that these can only exist if the first Chern class of the varieties vanishes and, in the case of vector bundle transforms, will exist if and only if there is a bi-universal bundle on XxY which is "strongly simple" in a suitable sense. Some applications are given to abelian varieties extending the work of Mukai.

alg-geom

\section*{Introduction}{} A Fourier transform could be loosely be described as `pullback a function to $\mathbb{R}^n\cross\mathbb{R}^n$ multiply by $\exp(2\pi i\langle x,y\rangle)$ and take the direct image (integrate) with respect to the second variable'. If we regard this as a transformation from $L^2$ integrable functions to themselves then this satisfies certain useful properties such as having an inverse, the Parseval theorem, the convolution theorem, etc. The Fourier-Mukai transform was introduced in \cite{Muk1} and is formally analogous to the Fourier transform but acts on the derived category of (bounded) complexes of sheaves on an abelian variety ${\mathbb T}$ and maps this to the same category but for the dual abelian ${\hat{\mathbb T}}$ variety. It too satisfies useful properties such as the Fourier Inversion Theorem (FIT), the Parseval Theorem, the Convolution Theorem, etc (see \cite{Muk2}). More precisely, the role of $\exp(2\pi i\langle x,y\rangle)$ is played by the Poincar\'e line bundle over ${\mathbb T}\cross{\hat{\mathbb T}}$ and the direct image needs to be derived. In \cite{Muk3}, Mukai showed that a similar transform gave rise to a FIT on the level of $K$-Theory using the universal bundle $\mathbb{E}$ over $S\cross{\mathcal M}(S)$ instead of the Poincar\'e bundle, where $S$ is a K3 surface and ${\mathcal M}(S)\cong S$ is a two dimensional moduli space of simple sheaves on $S$. This was shown to give rise to a FIT in \cite{BBH} and which satisfies the Parseval Theorem. The question arises: when do such transformations give rise to Inversion Theorems in more general contexts? We shall go some way to answering this question in this paper by classifying such transformations and giving conditions that they give rise to Inversion Theorems. We shall restrict our attention in the examples to the case of holomorphic varieties although it is not too hard to extend our results to the case of varieties over more general fields. The theorems are at their strongest in dimensions 1 and 2 but restricted forms apply to higher dimensions as well. Our first aim is to give a rough classification of such transforms. In section \ref{s:applics} we study two applications of the theory of Fourier-Mukai transforms to the case of abelian varieties. A new transform is constructed which is based on a certain component of the moduli space of simple bundles on abelian varieties. This is then used to deduce quite quickly that each Hilbert scheme of points arises as a component of the moduli space of simple torsion-free sheaves. This is expressed in \thmref{t:hilbismod}. We can also use these the general theory to prove that if the torus acts freely and effectively on any moduli component then the Euler characteristic of the sheaves parametrised by that component must be $\pm1$ (see \thmref{t:eulercl}). \begin{notation} We shall let $X$ and $Y$ be two smooth varieties and we shall consider pairs of functors $\mathbf{R}\Phi:D(X)\to D(Y)$ and $\mathbf{R}\hat\Phi:D(Y)\to D(X)$, where $D(X)$ denotes the derived category of bounded complexes of coherent sheaves on $X$. In this paper, for the sake of clarity, we give names to the various possible types of such pairs. We use the terms invertible correspondence, adjunction, Verdier and Fourier-Mukai type. These are not mutually exclusive conditions. They are each treated in the first four sections. We use the notation $T_E$ to denote the functor $F\mapsto E\mathbin{\buildrel{\mathbf L}\over{\tensor}} F:D(X)\to D(X)$, where $E\in D(X)$. \end{notation} \section{Invertible correspondences and resolutions of the diagonal} Let $X\lLa{x}Z\lRa{y}Y$ be flat maps of smooth quasi-projective varieties. Let $D(S)$ denote the derived category of bounded complexes of coherent sheaves on $S$. Fix two objects $P$ and $Q$ in $D(Z)$ and define two functors $\mathbf{R}\Phi_P:D(X)\to D(Y)$ and $\mathbf{R}\hat\Phi_Q:D(Y)\to D(X)$ by $$\displaylines{\mathbf{R}\Phi_P(-)=\mathbf{R} y_*(x^*-\mathbin{\buildrel{\mathbf L}\over{\tensor}} P)\cr \mathbf{R}\hat\Phi_Q(-)=\mathbf{R} x_*(y^*-\mathbin{\buildrel{\mathbf L}\over{\tensor}} Q).\cr}$$ Let $Z\lLa{p_y} Z_y\lRa{p'_y} Z$ (respectively $Z_x$) be the pullback of $y$ (respectively $x$) along itself. Let $q_x:Z_y\to X\cross X$ and $q_y:Z_x\to Y\cross Y$ be the maps defined by $(x\raise2pt\hbox{$\scriptscriptstyle\circ$} p_y,x\raise2pt\hbox{$\scriptscriptstyle\circ$} p'_y)$ and $(y\raise2pt\hbox{$\scriptscriptstyle\circ$} p_x,y\raise2pt\hbox{$\scriptscriptstyle\circ$} p'_x)$. \begin{thm} $\mathbf{R}\Phi_P$ and $\mathbf{R}\hat\Phi_Q$ give an equivalence of categories (after a shift of complexes) if and only if the following two conditions hold:\\ (i) $\mathbf{R} q_{x*}(p_y^* P\mathbin{\buildrel{\mathbf L}\over{\tensor}} p^{\prime*}_yQ)\cong {\mathscr O}_\Delta[r]$\\ (ii) $\mathbf{R} q_{y*}(p^{\prime*}_x P\mathbin{\buildrel{\mathbf L}\over{\tensor}} p_x^* Q)\cong {\mathscr O}_{\Delta'}[r]$,\\ where ${\mathscr O}_\Delta$ is the structure sheaf of the diagonal $\Delta\subset X\cross X$, ${\mathscr O}_\Delta'$ is the structure sheaf of diagonal $\Delta'$ in $Y\cross Y$, and $r$ is an integer. All isomorphisms are quasi-isomorphisms of complexes. \label{th:koszul} \end{thm} In other words we must have that the LHS's of (i) and (ii) are resolutions of the both diagonals if we are to have an inversion theorem for such transforms. The proof of the sufficiency of the conditions is, of course, well known and fairly trivial, but the proof of necessity does require some care and so we reproduce it here. \begin{proof}\footnote{I am grateful to the referee for pointing out a simplification in the original version of this proof} Without loss of generality we set $r=0$. Condition (i) will be equivalent to the fact that $\mathbf{R}\Phi_P$ has a left inverse and (ii) will be the corresponding statement for $\mathbf{R}\hat\Phi_Q$. Hence, it suffices to consider only (i). Now, $\mathbf{R}\hat\Phi_Q\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Phi_P(E)=\mathbf{R} x_*\bigl(y^*\mathbf{R} y_*(x^*E\mathbin{\buildrel{\mathbf L}\over{\tensor}} P)\mathbin{\buildrel{\mathbf L}\over{\tensor}} Q\bigr)$. Using $Z_y$ and the base-change formula we can write this as $$\mathbf{R} x_*\bigl(\mathbf{R} p'_{y*}(p^*_yx^* E\mathbin{\buildrel{\mathbf L}\over{\tensor}} p^*_yP)\mathbin{\buildrel{\mathbf L}\over{\tensor}} Q\bigr).$$ Using the projection formula (and the hypotheses on $X$, $Y$ and $Z$ we have $$\mathbf{R}(x\raise2pt\hbox{$\scriptscriptstyle\circ$} p'_y)_*\bigl((x\raise2pt\hbox{$\scriptscriptstyle\circ$} p_y)^* E\mathbin{\buildrel{\mathbf L}\over{\tensor}} p^*_y P\mathbin{\buildrel{\mathbf L}\over{\tensor}} p^{\prime*}_y Q\bigr).$$ Let $p_1$ and $p_2$ denote the projections $X\cross X\to X$. Then $x\raise2pt\hbox{$\scriptscriptstyle\circ$} p_y=p_1\raise2pt\hbox{$\scriptscriptstyle\circ$} q_x$ and $x\raise2pt\hbox{$\scriptscriptstyle\circ$} p'_y=p_2\raise2pt\hbox{$\scriptscriptstyle\circ$} q_x$. Substituting these and using the projection formula again we have \begin{equation} \mathbf{R} p_{2*}\bigl(p_1^*E\mathbin{\buildrel{\mathbf L}\over{\tensor}}\mathbf{R} q_{x*}(p^*_y P\mathbin{\buildrel{\mathbf L}\over{\tensor}} p^{\prime *}_y Q)\bigr).\label{eq:psiophi} \end{equation} Suppose condition (i) holds then $$\mathbf{R}\hat\Phi_Q\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Phi_P=\mathbf{R} p_{2*}(p^*_2 E\mathbin{\buildrel{\mathbf L}\over{\tensor}}{\mathscr O}_{\Delta})=E.$$ This proves the sufficiency of (i). To see that it is necessary observe that \ref{eq:psiophi} still holds. Assume that $\mathbf{R}\hat\Phi_Q\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Phi_P\mathbin{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}}\operatorname{Id}$ and put $E={\mathscr O}_\alpha$, the structure sheaf of a point $\alpha\in X$. Let $A_{\scriptscriptstyle\bullet}$ be a (bounded) complex of locally-free ${\mathscr O}_{X\cross X}$-modules representing $\Gamma=\mathbf{R} q_{x*}(p^*_y P\mathbin{\buildrel{\mathbf L}\over{\tensor}} p^{\prime *}_y Q)$. By assumption, we have the quasi-isomorphism \begin{equation} \mathbf{R} p_{2*}(\Gamma\mathbin{\buildrel{\mathbf L}\over{\tensor}} p_1^*{\mathscr O}_\alpha) \simeq{\mathscr O}_\alpha.\label{eq:alpha} \end{equation} Then $$R^*p_{2*}(\Gamma\mathbin{\buildrel{\mathbf L}\over{\tensor}} p_1^*{\mathscr O}_\alpha)=H^*\bigl(p_{2*}(A_{\scriptscriptstyle\bullet}\tensor p_1^*{\mathscr O}_\alpha)\bigr).$$ But this is concentrated in position 0 and so we see that $H_i(A_{\scriptscriptstyle\bullet}\tensor p_1^*{\mathscr O}_\alpha)=0$ for all $i\neq0$ and so $\Gamma$ is flat over $p_1$. Now the fibres of $\Gamma$ are given by $\Gamma_{(\alpha,\beta)}= \mathbf{R} p_{2*}(\Gamma\tensor p_1^*{\mathscr O}_\alpha\tensor p_2^*{\mathscr O}_\beta)$. We can rewrite this as $\mathbf{R} p_{2*}(\Gamma\tensor p_1^*{\mathscr O}_\alpha)\tensor{\mathscr O}_\beta ={\mathscr O}_\alpha\tensor{\mathscr O}_\beta$. This is zero if $\alpha\neq\beta$. Hence, $\Gamma$ is supported on $\Delta_X$ and the case $\alpha=\beta$ implies that it has rank 1 everywhere along this diagonal. If we now substitute $E={\mathscr O}_X$ then a standard hypercohomology argument shows that it must also be trivial. \end{proof} \begin{dfn} We shall call functors $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ a {\it invertible correspondence transforms} if they satisfy the conditions of \thmref{th:koszul}. These can also be thought of as transformations satisfying the `Fourier Inversion Theorem' or FIT. \end{dfn} The invertible correspondence transforms should be compared to the notion of tilting transforms for categories of modules over associative rings. Details of this can be found in \cite{Ric}. As an intermediate example one can consider the Beilinson spectral sequence which could be viewed as a composition of derived functors from the derived category of coherent sheaves on ${\mathbb C} P^n$ to the derived category of finitely generated modules over a suitable algebra (the path algebra of a certain quiver) see \cite{Bei1} and \cite{Bei2}. This also works for more general varieties, see King \cite{Ki}. We shall see in section 3 that when we consider invertible correspondences for varieties then their existence imposes quite strong condition on the varieties. \section{Transforms of Adjunction Type} In this section we shall look briefly at the categorical aspects of equivalences of such transforms. The following is well known: \begin{thm} {\rm(See \cite{McL})}. Let $F:\mathcal{A}\to\mathcal{B}$ and $G:\mathcal{B}\to\mathcal{A}$ be two functors of categories. If $F$ and $G$ give rise to an equivalence of categories (i.e. $F\raise2pt\hbox{$\scriptscriptstyle\circ$} G$ and $G\raise2pt\hbox{$\scriptscriptstyle\circ$} F$ are naturally equivalent to their respective identity functors) then $F\adj G\adj F$. Conversely, if $F\adj G\adj F$ and $F$ is fully faithful such that it surjects on objects up to isomorphism (we shall say {\em quasi-surjects}) then $F$ and $G$ determine an equivalence of categories. \end{thm} \begin{dfn} We say that a pair of functors $F$ and $G$ are a {\it transform of adjunction type} if $F\adj G\adj F$. \end{dfn} The theorem says that invertible correspondence transforms are adjunction transforms. When we have an adjunction type transform then we often already know that the functors are inverses on one side. For example, suppose that we already know that the adjunction gives rise to $GF\mathrel{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}}\operatorname{Id}$. Recall that an adjunction gives rise to natural transformations $\eta:\operatorname{Id}\mathbin{\mathop{\longrightarrow}\limits^{\scriptscriptstyle\bullet}} GF$ and $\epsilon:FG\mathbin{\mathop{\longrightarrow}\limits^{\scriptscriptstyle\bullet}}\operatorname{Id}$ called the unit and counit of the adjunction $F\adj G$ similarly $G\adj F$ gives rise to $\eta':\operatorname{Id}\mathbin{\mathop{\longrightarrow}\limits^{\scriptscriptstyle\bullet}} FG$ and $\epsilon':GF\mathbin{\mathop{\longrightarrow}\limits^{\scriptscriptstyle\bullet}}\operatorname{Id}$. Then it is well known that $G$ is faithful if and only if $\epsilon_a$ is epi for all $a$ and $G$ is full if and only if $\epsilon_a$ is split monic (i.e. has a left inverse). The same holds for $F$ if we replace $\epsilon$ by $\epsilon'$. If we already know that $\epsilon'$ is a natural isomorphism then $F$ must be full and faithful and that $G$ is full. But $GFa\cong a$ for each $a$ and so $G$ must quasi-surject on objects. It now follows that $F$ and $G$ are an equivalence of categories if and only if $G$ is faithful. We summarise this in the following. \begin{prop}\label{p:parseval} Suppose that $F$ and $G$ form a transform of adjunction type such that $GF\mathbin{\mathop{\longrightarrow}\limits^{\scriptscriptstyle\bullet}}\operatorname{Id}$ is an isomorphism. Then $F$ satisfies the {\em Parseval Theorem}: $$\operatorname{Hom}(a,b)\cong\operatorname{Hom}(Fa,Fb)$$ Furthermore, $F$ and $G$ form an equivalence of categories if and only if $G$ satisfies the Parseval Theorem.\label{p:semiadj} \end{prop} \section{Transforms of Verdier Type} We now look at more specific transforms. We limit ourselves to smooth projective varieties $X$ and $Y$ and let $Z=X\cross Y$. Let $n=\dim X$ and $m=\dim Y$. Note that we have $$\mathbf{R}\hat\Phi(F)=\mathbf{R} x_*\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,y^*F),$$ where $Q=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,{\mathscr O})$ and $P$ is a sheaf. We shall see that this choice of $Q$ is forced on us up to the pullback of (a power of) the canonical bundle and a shift. Recall that we have Grothendieck-Verdier duality: $$\operatorname{Hom}_{D(X)}(\mathbf{R} x_*F,G)\cong\operatorname{Hom}_{D(X\cross Y)}(F,x^*G\mathbin{\buildrel{\mathbf L}\over{\tensor}} y^*\omega_Y[n]).$$ $$\operatorname{Hom}_{D(Y)}(\mathbf{R} y_*F,G)\cong\operatorname{Hom}_{D(X\cross Y)}(F,y^*G\mathbin{\buildrel{\mathbf L}\over{\tensor}} x^*\omega_X[m]).$$ (see Hartshorne \refb{Hartres}{III.11}). Applying this and the classical adjunction $f^*\adj\mathbf{R} f_*$ to $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ we obtain \begin{prop} $$\mathbf{R}\Phi\adj T_{\omega_X}\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat\Phi[n] \quad\text{and}\quad \mathbf{R}\hat\Phi\adj T_{\omega_Y}\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Phi[m],$$ where $Q=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,{\mathscr O}_{X\cross Y})$. \end{prop} \begin{proof} This is just a computation using the adjunctions above: \begin{align*} \operatorname{Hom}_{D(Y)}(\mathbf{R} y_*(x^*F\mathbin{\buildrel{\mathbf L}\over{\tensor}} P),G)&\cong \operatorname{Hom}_{D(X\cross Y)}(x^*F\mathbin{\buildrel{\mathbf L}\over{\tensor}} P,y^*G\tensor x^*\omega_X[n])\\ &=\operatorname{Hom}_{D(X)}(F,\mathbf{R} x_*\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,y^*G)\tensor\omega_X[n]) \end{align*} Similarly for the other adjunction. \end{proof} \begin{remark} We could use $Q=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,x^*L\tensor y^*M)$ for line bundles $L$ on $X$ and $M$ on $Y$. This does not affect the adjunctions except we have to twist by one or other of these line bundles or their duals.\label{r:libQ} \end{remark} \begin{prop}\label{p:veradj} Let $\mathbf{R}\Delta_X=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(-,\omega_X)[n]$ and $\mathbf{R}\Delta_Y=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(-,\omega_Y)[m]$. Then $$\mathbf{R}\Delta_Y\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Phi_P\mathbin{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}} T_{\omega_Y}\raise2pt\hbox{$\scriptscriptstyle\circ$} \mathbf{R}\Phi_{Q}\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Delta_X[m],$$ $$\mathbf{R}\Delta_X\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat\Phi_P\mathbin{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}} T_{\omega_X} \raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Hat\Phi_{Q}\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Delta_Y[n],$$ where $Q=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,{\mathscr O})$ \label{p:dual} \end{prop} \begin{proof} We use local Verdier duality: \begin{align*} \mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(\mathbf{R} y_*(x^*F\mathbin{\buildrel{\mathbf L}\over{\tensor}} P),\omega_Y)&\cong \mathbf{R} y_*\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(x^*F\tensor x^*\omega^*_X,\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,y^*\omega_Y[n+m]))\\ &\cong\mathbf{R} y_*(x^*\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(F,\omega_X)\mathbin{\buildrel{\mathbf L}\over{\tensor}}\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,y^*\omega_Y)[n+m])\\ &\cong\mathbf{R} y_*(x^*\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(F,\omega_X[n])\mathbin{\buildrel{\mathbf L}\over{\tensor}}\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,{\mathscr O}))[m]\tensor \omega_Y \end{align*} Similarly for $\mathbf{R}\hat\Phi$.\end{proof} \begin{thm}\label{c:koko} Let $X$ and $Y$ be two smooth projective varieties of dimensions $n$ and $m$ respectively. Let $P$ and $Q$ be two complexes of coherent sheaves on $X\cross Y$. We assume that $X\neq Y$ and $P$ is a sheaf of rank at least 1. Define two functors $D(X)\to D(Y)$ and $D(Y)\to D(X)$ by $\mathbf{R}\Phi(E)=\mathbf{R} y_*(x^*E\mathbin{\buildrel{\mathbf L}\over{\tensor}} P)$ and $\mathbf{R}\hat\Phi(F)=\mathbf{R} x_*(y^*F\mathbin{\buildrel{\mathbf L}\over{\tensor}} Q)$ respectively, where $x$ and $y$ are the projections from $X\cross Y$ to $X$ and $Y$ respectively. If $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ form an equivalence of categories then $\omega_X^k={\mathscr O}_X$, $\omega_Y^k={\mathscr O}_Y$ for some integer $k$, $n=m$ and $Q=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,x^*\omega_X)[n]=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,y^*\omega_Y)[n]$. In particular, we must have $c_1(X)=0$. \end{thm} \begin{proof} This follows from \propref{p:veradj} and the fact that adjoints are unique up to natural isomorphism. We may assume that $Q$ takes the form $\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,x^*\omega_x)[n]$ because the adjoint of $\mathbf{R}\Phi_P$ is $\mathbf{R}\Hat\Phi_{\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,x^*\omega_X)[n]}$ by duality. The adjoints give $$\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}\bigl(\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,{\mathscr O}),{\mathscr O}\bigr)\tensor x^*\omega^*_X\tensor y^*\omega_Y[m-n]=P$$ which can only happen if $n=m$ and, since $P$ has support on the whole of the product, $\omega_X^k={\mathscr O}_X$ and $\omega_Y^k={\mathscr O}_Y$ for some integer $k$ as required. In fact, what we need is that $P\tensor x^*\omega_X\tensor y^*\omega^*_Y\cong P$. \end{proof} \begin{remark} In other words, an invertible correspondence transform given by, for example, a torsion-free sheaf $P$ for smooth projective varieties can only exist if the dimensions of the varieties are the same and their canonical bundles are trivial or torsion (with order dividing the rank of $P$). Moreover, the transform complex $Q$ must be the (derived) dual of $P$ (up to a twist and shift). Relative versions of the results of this section are also available where we we have $Z=X\cross_S Y$ for some scheme $S$. In fact, we can clearly strengthen the conclusion to say that the canonical bundles must act trivially on the the restrictions of $P$ to $X\cross\{y\}$ and $\{x\}\cross Y$. This will happen, for example, for relative transforms where the fibres have trivial canonical bundle. \end{remark} \begin{dfn} We say that a pair of transforms $\mathbf{R}\Phi_P$ and $\mathbf{R}\hat\Phi_Q$ is {\em of Verdier type} if $\omega_X={\mathscr O}_X$, $\omega_Y={\mathscr O}_Y$, $\dim X=\dim Y$ and $Q=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P,{\mathscr O}_{X\cross Y})$ (up to a shift). So we have the chain of implications $$\hbox{invertible correspondence}\implies\hbox{Verdier} \implies\hbox{adjunction}$$. \end{dfn} \section{Fourier-Mukai Transforms} \begin{dfn} Let $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ be transforms of Verdier type as in the last section but with the additional constraint that $P$ is a locally-free sheaf over $X\cross Y$. We also assume that they give rise to an equivalence of categories. Then we say that they are {\it transforms of Fourier-Mukai type}. In other words, the transforms are both of invertible correspondence type and Verdier type with some constraint on the choice of $P$. \end{dfn} Let $\operatorname{Spl}(S)$ denote the moduli space of simple sheaves on $S$. Recall that Mukai \cite{Muk3} proves that this space is smooth when $S$ is an abelian surface or a K3 surface. \begin{prop}\label{c:nisr} Suppose that $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ form a pair of functors which are of Fourier-Mukai type. Then $n=r$, where $n=\dim X$ and $r$ is given in \thmref{th:koszul}. Furthermore, if $X$ and $Y$ admit smooth moduli of simple torsion-free sheaves then $X$ is a component of $\operatorname{Spl}(Y)$ and $Y$ is a component of $\operatorname{Spl}(X)$. \end{prop} \begin{proof} In the proof we only assume that $P$ is torsion-free and flat over both projections. For this we use Proposition 2.26 of \cite{Muk4}. This states that if $f:Z\to Y$ is a proper map of noetherian schemes and $F$ is a coherent sheaf on $Z$ flat over $Y$ and $S\subset Y$ is a locally complete intersection then if $H^i(f^{-1}(y),F_y)=0$ for all $i<\operatorname{codim} S$ and $y\not\in S$ then $R^if_*F=0$ for the same set of $i$. We apply this to $S=\Delta\subset X\cross X$. Then $\operatorname{codim} S=n$ and so if $r<n$ we have that the cohomology sheaves $R^iq_{x*}(\Gamma)$ of $\mathbf{R} q_{x*}(\Gamma)$ vanish for for $i>r$, where $$\Gamma=\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(p^*_xP,p^*_yP).$$ Hence $(R^rq_{x*}(\Gamma))_{(a,a')}\cong H^r(Y;\Gamma_{(a,a')})=0$ for $(a,a')\in X\cross X\setminus\Delta$. Note that $\Gamma_{(a,a')}\cong\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(P_a,P_{a'})$. This implies that $R^rq_{x*}(\Gamma)=0$ and so $R^iq_{x*}(\Gamma)=0$ for all $i$; a contradiction. Hence $n=r$. Let $P_a$ denote the restriction of $P$ to $\{a\}\cross Y$. We know that $h^n(Y;\operatorname{\mathcal{H}\mathnormal{om}}(P_a,P_a))=\dim\operatorname{Ext}^n(P_a^{**},P_a)=1$ and so $\dim\operatorname{Ext}^n(P_a,P_a)=1$ because $P_a\to P_a^{**}$ induces a surjection on Ext groups. Then Serre duality implies that $P_a$ is simple. Similarly $P_b$ is simple for all $b\in Y$. On the other hand, $\operatorname{Ext}^n(P_a,P_{a'})=0$ for $a\neq a'$ and so $P_a\not\cong P_{a'}$ and hence $X\subset\operatorname{Spl}(Y)$. Suppose that $E\in\operatorname{Spl}(Y)$ is not in $X$ but has the same Chern character as $P_a$. If $\operatorname{Ext}^i(P_a,E)=0$ for all $i$ and all $a\in X$ then $\mathbf{R}\hat\Phi(E)=0$. But $\mathbf{R}\hat\Phi$ is an equivalence of categories and so this is impossible. For a fixed $a$ the set $Z_a=\{E\in\operatorname{Spl}(Y):\operatorname{Ext}^i(P_a,E)=0,\;\forall i\}$ is Zariski open in $\operatorname{Spl}(Y)$ and non-empty as it contains $X\setminus\{a\}$. But $(U_a\cap Z_a)\setminus X$ is empty for any open neighbourhood $U_a$ of $P_a$ in $\operatorname{Spl}(Y)$. This shows that a component of $\operatorname{Spl}(Y)$ which contains $X$ is smooth at each point of $X$ and has the same dimension as $X$. \end{proof} \begin{remark} In the case when $n=1$ or $n=2$ we can proceed more directly and explicitly. Assume for simplicity that $P$ is locally-free and consider first the case $n=1$. Note that $X$ must be an elliptic curve by \ref{c:koko}. The fact that $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ are invertible correspondences mean that $$\mathbf{R} q_{x*}\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(p^{\prime*}_y P^*,p_y^* P)=\mathbf{R} q_{x*}\Lambda\cong{\mathscr O}_\Delta[-r]$$ for some $r=0$ or 1. Serre duality implies that $R^1 q_{x*}\Lambda\neq0$ and so $r=1$. Then $\chi(P_b,P_c)=0$ for $c\neq b$ and hence for $b=c$ as well. We also have $\dim\operatorname{Ext}^1(P_b,P_b)=1$ and so $P_b$ are simple sheaves with a 1 dimensional moduli which contains $Y$. Hence, $\dim Y=1$ as required. When $n=2$ we argue similarly. Then again $\chi(P^*_a\tensor P_{a'})=0$ for all $a,a'\in X$. Note that \ref{c:koko} shows that the canonical bundles are trivial and so Serre duality implies that $r\neq0$; otherwise $H^0(P_b^*\tensor P_b)\neq0$ so $H^2(P_b^*\tensor P_b)\neq0$, and hence $R^2q_{x*}\Lambda\neq0$. But if $r=1$ then the support of $\{b\in Y:H^1(X,P^*_b\tensor P_c)\neq0\}$ is 0-dimensional Mukai's result above implies that $R^1q_{x*}\Lambda=0$, a contradiction. Hence we must have $r=2$. It also follows that $P_b$ are all simple. Then $Y$ is contained in the moduli space of such $P_b$'s and hence has dimension at most 2 because $\chi(P_b,P_b)=0$. It cannot have dimension 1. \end{remark} \section{Bi-universal sheaves}{} In this section we ask the following question. If $X$ is a smooth complex projective variety with trivial canonical bundle and $Y={\mathcal M}(X)$ is a smooth moduli of simple sheaves on $X$ (also assumed to have trivial canonical bundle) then when do we obtain a Fourier-Mukai transform from $X$ to $Y$? We answer this question by considering various universal sheaves on the product $X\cross Y$. \begin{dfn} Following Mukai, we say that a sheaf $\mathbb{E}$ is {\it semi-universal} on $X\cross Y$ if for all $y\in Y$, $\mathbb{E}_b\cong E^{\oplus\sigma}$ for some $\sigma\in{\mathbb Z}$ and $\mathbb{E}$ is a universal deformation. If $\sigma=1$ we say that $\mathbb{E}$ is a {\it universal sheaf} (as usual). If $X$ is isomorphic to a moduli of simple sheaves on $Y$ such that $\mathbb{E}_a\cong E^{\oplus\sigma'}$ and $\mathbb{E}_b\cong E^{\oplus\sigma}$ we say that $\mathbb{E}$ {\it is bi-semi-universal}. If either $\sigma$ or $\sigma'$ is 1 then we call $\mathbb{E}$ {\it sesqui-universal} and if $\sigma'=\sigma=1$ then we call $\mathbb{E}$ {\it bi-universal}. In all these cases we say that $\mathbb{E}$ is {\em strongly} universal (etc.) if whenever $b\neq b'$ then $\operatorname{Ext}^i(\mathbb{E}_b,\mathbb{E}_{b'})=0$ for all $i$ and similarly for $a\neq a'$. \end{dfn} Mukai has shown that if $Y$ is a (representable) component of the moduli of simple sheaves on $X$ then a semi-universal sheaf always exists (see \refb{Muk3}{Thm~A.5}). \begin{remark} Of course, in dimension 2, if the component of the moduli space consists of, say, stable bundles then the strong condition will always hold. \end{remark} In the following we let $P=\mathbb{E}$. We also assume that $\dim Y=\dim X=n$. \begin{prop} If $\mathbb{E}$ is strongly semi-universal then $\mathbf{R}\Phi\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat\Phi\mathbin{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}}\operatorname{Id}^{\oplus\sigma^2}[n]$. In particular, $\mathbf{R}\hat\Phi$ is faithful. If $\mathbb{E}$ is strongly bi-semi-universal then $\sigma=\sigma'$. \end{prop} \begin{proof} The conditions on $\mathbb{E}$ ensures that $R^iq_{x*}\Gamma=0$ for $i<n$ and for $i=n$ is supported on the diagonal. That it is trivial and given by such a direct sum follows from the fact that $\mathbf{R}\hat\Phi({\mathscr O}_b)\cong(\mathbb{E}_b^{\oplus\sigma})^*[n]$ and $\mathbf{R}\Phi(\mathbb{E}_b)={\mathscr O}_b^{\oplus\sigma}$. Then $\mathbf{R}\Phi\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat\Phi$ is faithful and so $\mathbf{R}\hat\Phi$ is. If $\mathbb{E}$ is bi-universal then we also have $\mathbf{R}\Phi(\mathbb{E}^*_b)\cong{\mathscr O}_b^{\oplus\sigma}$. So $\mathbf{R}\Phi\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat\Phi({\mathscr O}_b)=\mathbf{R}\Phi(\mathbb{E}_b^{\oplus\sigma})^*= {\mathscr O}_b^{\oplus\sigma^2}$ On the other hand, $\mathbf{R}\Phi\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat\Phi=\operatorname{Id}^{\oplus\sigma'{}^2}$ and so $\sigma^2=\sigma'{}^2$. \end{proof} \begin{cor} If $\mathbb{E}$ is strongly sesqui-universal then it must be strongly bi-universal. \end{cor} \begin{cor} From \ref{p:semiadj} we see that if $\mathbb{E}$ is strongly universal then the following are equivalent \begin{enumerate} \item $\mathbb{E}$ gives rise to a Fourier-Mukai transform. \item $\mathbb{E}$ is sesqui-universal. \item $\mathbf{R}\Phi$ satisfies the Parseval Theorem. \end{enumerate} \end{cor} \section{Examples}{} \begin{example} We shall now look at some examples of Fourier-Mukai transforms. The first is the Mukai transform itself. For this we set $P=\P$, the Poincar\'e bundle on ${\mathbb T}\cross{\hat{\mathbb T}}$, where $X={\mathbb T}$ is an abelian variety and $Y={\hat{\mathbb T}}\cong\Pic^0{\mathbb T}$ is its dual abelian variety. It was shown in \cite{Muk2} that $\mathbf{R}{\mathcal F}=\mathbf{R}\Phi_\P$ and $\mathbf{R}\hat{\mathcal F}=\mathbf{R}\hat\Phi_\P$ are of Fourier-Mukai type. In this particular case we also obtain a convolution theorem (see \refb{Muk2}{}) as well. Note also that $X\neq Y$. A relative version can also be found in \cite{Muk4}. The Grothendieck-Riemann-Roch Theorem can be used to compute the Chern characters of the transforms: $$\operatorname{ch}(\mathbf{R}{\mathcal F}(E))_i=(-1)^i\operatorname{ch}(E)_{n-i},$$ where we identify $H^i({\mathbb T})$ with $H^{n-i}({\hat{\mathbb T}})$ via Poincar\'e duality. It is conventional in this case to use $\mathbf{R}\hat\Phi_{\P}$ instead of $\mathbf{R}\hat\Phi_{\P^*}$. \end{example} \begin{example} The second example first appeared in \cite{Muk4} and was shown to be a Fourier-Mukai transform by Bartocci, Bruzzo and Hern\'andez Ruip\'erez \cite{BBH}. In this case $X$ is a K3 surface satisfying suitable conditions for the existence of $Y$, a 2-dimensional space of stable bundles on $X$ which is isomorphic to $X$. Then $P$ is a bi-universal bundle on $X\cross Y$. The Grothendieck-Riemann-Roch Theorem also tells us the Chern characters. \end{example} \begin{example} Consider a smooth projective variety $X$ with trivial canonical bundle and $h^{0,p}=0$ for $p\neq0,n$ (for example, a K3 surface). Let $P={\mathscr I}_\Delta$ be the ideal sheaf of the diagonal in $X\cross X$. Then $P$ is strongly bi-universal. This follows because $P_a\cong{\mathscr I}_a$ and, if $a\neq a'$ then $\operatorname{Ext}^i({\mathscr I}_a,{\mathscr I}_{a'})=0$ for all $i$ as can be easily seen using the long exact sequences induced by the structure sequences of $a$ and $a'$. Note that $Q$ cannot be represented by a single sheaf. The Chern character of the resulting Fourier-Mukai transform of $E\in D(X)$ is $$\bigl(\chi(E)-\operatorname{ch}_0(E),-\operatorname{ch}_{1}(E),\ldots,-\operatorname{ch}_{i}(E),\ldots,-\operatorname{ch}_{n-1}(E),-\operatorname{ch}_n(E)\bigr), $$ as can be easily seen from the Grothendieck-Riemann-Roch formula or directly from the structure sequence of $\Delta$. \end{example} \begin{example}\label{e:Mrl1} Consider an abelian surface ${\mathbb T}$ polarised by $\ell$ such that $\ell^2=2r$, with dual polarisation $\hat\ell$ of ${\hat{\mathbb T}}$. Let ${\mathcal M}={\mathcal M}(r,\ell,1)$ be the moduli space of stable bundles with the given Chern characters. Such module spaces have been also considered by Mukai (\cite{Muk1}). We shall prove in \propref{p:7.1} below that this is projective and non-empty. It is easy to see that if $E\in{\mathcal M}$ then $R{\mathcal F}(E)= \hat L^*\tensor\P_x$ for some (assumed symmetric) $\hat L\in\hat\ell$ and $\P_x\in\Pic{\hat{\mathbb T}}\cong{\mathbb T}$. This implies that ${\mathcal M}\cong{\mathbb T}$ under $E\mapsto x$. A result of Mukai (\refb{Muk3}{Appendix 2}) implies that a universal sheaf $\mathbb{E}$ exists over ${\mathbb T}\cross{\mathcal M}$. In fact it is possible to write this down explicitly. Let $\pi_{ij}$ be the projection maps to the $i$th and $j$th factors of ${\mathbb T}\cross{\hat{\mathbb T}}\cross{\mathbb T}$ and $\pi_i$, the projections to the $i$th factor. Then it is easy to check that $\mathbb{E}=\mathbf{R}\pi_{13*}(\pi_2^*\hat L^*\tensor\pi_{23}^*\P\tensor\pi_{12}^*\P^*)$. This also shows that $\mathbb{E}$ is bi-universal. Then $\mathbf{R}\Phi_{\mathbb{E}}$ is of Fourier-Mukai type. Using Grothendieck-Riemann-Roch (or \lemref{l:phiisf}) we find \begin{align*} \operatorname{ch}(\mathbf{R}\Phi(E))_0&=\operatorname{ch}_0(E)+\operatorname{ch}_1(E)\cdot\ell+r\operatorname{ch}_2(E),\\ \operatorname{ch}(\mathbf{R}\Phi(E))_1&=\operatorname{ch}_1(E)+\operatorname{ch}_2(E)\ell,\\ \operatorname{ch}(\mathbf{R}\Phi(E))_2&=\operatorname{ch}_2(E). \end{align*} We shall study this example in more detail below. \end{example} \section{Two Applications}\label{s:applics} We shall now look at two applications of the general theory of Fourier-Mukai transforms. The first is a special case of Example \ref{e:Mrl1} and the second is a generalisation of that example. First, we let $X={\mathbb T}$ be an abelian surface with a polarisation $\ell$. Let ${\mathcal M}={\mathcal M}(r,\ell,1)$ denote the moduli space of (Gieseker) stable bundles of the given Chern character. Choose symmetric representative line bundles $L\in\ell$ and $\hat L\in\hat\ell$ in $\ell$ and the dual polarisation respectively. \begin{prop}\label{p:7.1} The moduli space ${\mathcal M}$ is isomorphic to ${\mathbb T}$. \end{prop} \begin{proof} Consider the collection $\{\mathbf{R}\hat{\mathcal F}(\hat L\tensor\P_x)\mid x\in{\mathbb T}\}$. Note that $R^i\hat{\mathcal F}(\hat L)=0$ unless $i=0$ and so this set consists of vector bundles of Chern character $(r,\ell,1)$. Moreover, these are all $\mu$-stable since the projective bundle $B={\mathbb P} R^0\hat{\mathcal F}(\hat L)$ has fibres consisting of the linear systems $|\hat L\tensor\P_x|$ which are just translates of $|\hat L|$. So $B$ admits a flat connection given by translation of this linear system. This connection then induces a projectively anti-self-dual connection on each of the elements of the collection. This implies that the bundles are all $\mu$-stable. Since the collection is non-empty and the Mukai transform is an isomorphism of schemes we see that ${\mathcal M}\cong{\mathbb T}$. \end{proof} We have seen in \ref{e:Mrl1} that a strongly bi-universal sheaf $\mathbb{E}$ exists over ${\mathbb T}\cross{\mathcal M}$. We shall prove the following \begin{thm}\label{t:modideal} Let $({\mathbb T},\ell)$ be a polarised torus with $\ell^2=2r$. There is a component of the moduli space of simple sheaves over ${\mathbb T}$ with Chern characters $(rn-1,n\ell,n)$ is canonically isomorphic to $\operatorname{Hilb}^n{\mathbb T}\cross{\hat{\mathbb T}}$. \end{thm} We shall see that this isomorphism is given by $\mathbf{R}\Phi$. We introduce the following terminology, again following Mukai. \begin{dfn} We say that a sheaf $E$ on $X$ satisfies $\Phi$-WIT$_i$ if for all $j\neq i$, $R^j\Phi(E)=0$. In other words, $\mathbf{R}\Phi(E)$ is again a sheaf. We just write WIT$_i$ for ${\mathcal F}$-WIT$_i$ and $\Phi$-WIT if we don't want to specify $i$. \end{dfn} \begin{lemma}\label{l:PhiofP} The flat line bundles $\P_{\Hat x}$ satisfy $\Phi$-WIT$_0$ and $R^0\Phi(\P_{\Hat x})=\P_{\Hat x}$. \end{lemma} \begin{proof} This follows from Lemma \ref{l:phiisf} below since $\P_{\Hat x}$ satisfies WIT$_2$ with transform ${\mathscr O}_{-{\Hat x}}$. \end{proof} \begin{lemma} The ideal sheaf ${\mathscr I}_S$ of a 0-dimensional subscheme $S$ of ${\mathbb T}$ satisfies $\Phi$-WIT$_1$ and the transform can be written as $A/{\mathscr O}$, where $A$ admits a filtration whose factors are elements of ${\mathcal M}(r,\ell,1)$. More generally, $\mathbf{R}\Phi({\mathscr I}_S\tensor\P_{\Hat x})=A/\P_{\Hat x}$. \end{lemma} Note that ${\mathscr I}_S$ never satisfies WIT. We could say that ${\mathscr I}_S$ is a ``half-WIT'' sheaf. \begin{proof} Observe first that the structure sheaf ${\mathscr O}_S$ of $S$ satisfies $\Phi$-WIT$_0$ and its transform is a sheaf $A$. Since ${\mathscr O}_S$ is built up from a series of extensions of structure sheaves of single points we see that $A$ admits a filtration whose factors are $\mathbf{R}\Phi({\mathscr O}_x)=E_x$ for $x\in S$. If we apply $\mathbf{R}\Phi$ to the structure sequence of $S$ twisted by $\P_{\Hat x}$ then we obtain the long exact sequence $$0\longrightarrow R^0\Phi({\mathscr I}_S\tensor\P_{\Hat x})\longrightarrow\P_{\Hat x}\lRa{f} A\longrightarrow R^1\Phi({\mathscr I}_S\tensor\P_{\Hat x})\lra0$$ using \lemref{l:PhiofP}. Since $f=\mathbf{R}\Phi(\P_{\Hat x}\to{\mathscr O}_S)$ it must be non-zero as $\P_{\Hat x}$ and ${\mathscr O}_S$ both satisfy $\Phi$-WIT. Since $A$ is locally-free and the rank of $\P_{\Hat x}$ is one we see that $f$ must inject. This proves the lemma. \end{proof} Observe that $\operatorname{ch}(R^1\Phi({\mathscr I}_S))=(r|S|-1,|S|\ell,|S|)$. To prove the theorem it suffices to show that $R^1\Phi({\mathscr I}_S\tensor\P_{\Hat x})$ is simple. But this follows immediately from the Parseval Theorem (\ref{p:parseval}) since ${\mathscr I}_S$ is simple. Then the map ${\mathscr I}_S\tensor\P_{\Hat x}\mapsto\mathbf{R}\Phi({\mathscr I}_S\tensor\P_{\Hat x})[1]$ gives an injection $\operatorname{Hilb}^n{\mathbb T}\cross{\hat{\mathbb T}}\to\operatorname{Spl}(rn-1,n\ell,n)$. Since the moduli of simple sheaves on ${\mathbb T}$ with this Chern character is smooth of dimension $2n-2$ then the image must be single reducible component. Since $\mathbf{R}\Phi$ is an equivalence of derived categories, it also preserves the holomorphic deformation structure of the spaces. In particular, the tangent spaces are canonically isomorphic and so the map is a diffeomorphism and preserves the complex structures. One can also see this algebraically by observing that the Fourier-Mukai transforms are actually a natural isomorphism of moduli functors and so give an isomorphism of coarse moduli schemes. In fact, the transform also preserves the natural symplectic structures which are simply given as a composition of derived morphisms. This completes the proof of the theorem.$\qquad\square$ \medskip For our second application, we consider an abelian variety ${\mathbb T}$ of any dimension $n$ and consider a moduli space ${\mathcal M}$ of stable bundles on ${\mathbb T}$ of dimension $n$ which is isomorphic to ${\mathbb T}$. Suppose further that this isomorphism is given by the translation action of ${\mathbb T}$ on ${\mathcal M}$ by pullback. Then Mukai has shown that there is a (strongly) semi-universal sheaf on ${\mathbb T}\cross{\mathcal M}$. We let $E_0$ be the base point determined by $0\in{\mathbb T}\cong{\mathcal M}$. \begin{prop} Define a complex $\mathbb{E}$ in $D({\mathbb T}\cross{\mathbb T})$ given by $$\mathbb{E}=\mathbf{R}\pi_{13*}(\pi_2^*E_0^*\tensor\pi_{23}^*\P^*\tensor\pi_{12}^*\P),$$ where $\pi_{ij}$ denotes the projection maps to the $i$th and $j$th factors of ${\mathbb T}\cross{\hat{\mathbb T}}\cross{\mathbb T}$, $\pi_i$ is the projection to the $i$th factor and $\P$ is the Poincar\'e bundle on ${\mathbb T}\cross{\hat{\mathbb T}}$. If we assume that $E_0$ satisfies WIT then $\mathbb{E}$ is a bi-universal sheaf. \end{prop} This proposition follows more or less immediately from the following lemma. \begin{lemma}\label{l:phiisf} Let $\mathbb{E}$ be given as in the proposition (we do not assume that $E_0$ satisfies WIT) then $$\mathbf{R}\Phi\mathbin{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}}(-1)^*\mathbf{R}\hat{\mathcal F}\raise2pt\hbox{$\scriptscriptstyle\circ$} T_{\mathbf{R}{\mathcal F}(E_0)}\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}{\mathcal F}$$ and $$\mathbf{R}\hat\Phi\mathbin{\mathop{\cong}\limits^{\scriptscriptstyle\bullet}}(-1)^*\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\hat{\mathcal F}\raise2pt\hbox{$\scriptscriptstyle\circ$} T_{\mathbf{R}{\mathcal F}(E^*_0)}\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}{\mathcal F}[n]$$ \end{lemma} \begin{proof} These are just hideous computations of derived functors. Let $\hat E=\mathbf{R}{\mathcal F}(E_0)$. Consider the commuting diagram \begin{equation*} \begin{CD} {\mathbb T}\cross{\hat{\mathbb T}} @<\pi_{12}<< {\mathbb T}\cross{\hat{\mathbb T}}\cross{\mathbb T} @>\pi_{23}>> {\hat{\mathbb T}}\cross{\mathbb T}\\ @VpVV @VV\pi_{13}V @VV p V \\ {\mathbb T} @<x<< {\mathbb T}\cross{\mathbb T} @>y>> {\mathbb T} \end{CD} \end{equation*} Then \begin{equation}\label{eq:aa} \mathbf{R} y_*(x^*F\mathbin{\buildrel{\mathbf L}\over{\tensor}}\mathbb{E})= \mathbf{R} y_*(x^*F\mathbin{\buildrel{\mathbf L}\over{\tensor}}\mathbf{R}\pi_{13*} (\pi_2^*\hat E\tensor\pi^*_{23}\P^*\tensor\pi_{12}^*\P) \end{equation} Note that $x^*F=x^*\mathbf{R} p_*p^*F=\mathbf{R}\pi_{13*}\pi^*_{12}p^*F=\mathbf{R}\pi_{13*}\pi_1^*F$ so $$\hbox{(\ref{eq:aa})}=\mathbf{R} y_*\mathbf{R}\pi_{13*}(\pi_1^*F\tensor\pi_2^*\hat E \tensor\pi_{23}^*\P^*\tensor\pi_{12}^*\P).$$ Which we can write as $$\mathbf{R} p_*\bigl(\mathbf{R}\pi_{23*}(\pi_1^*F\tensor\pi_{12}^*\P)\tensor\P^*\tensor q^*\hat E\bigr)$$ since $\pi_2=q\raise2pt\hbox{$\scriptscriptstyle\circ$}\pi_{23}$. But $\mathbf{R}\pi_{23*}(\pi_1^* F\tensor\pi_{12}^*\P)= \mathbf{R}\pi_{23*}\pi_{12}^*(p^*F\tensor\P)=q^*\mathbf{R} q_*(p^*F\tensor\P) =q^*\mathbf{R}{\mathcal F}(F)$. Then $$\hbox{(\ref{eq:aa})}=\mathbf{R} p_*\bigl(q^*(\mathbf{R}{\mathcal F}(F)\tensor\hat E)\tensor\P^*\bigr)$$ as required. The second equation follows similarly if we observe that $$\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(\mathbb{E},{\mathscr O})=\mathbf{R}\pi_{13*}(\pi_2^*\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(\hat E,{\mathscr O})\tensor\pi_{23}^*\P \tensor\pi_{12}^*\P^*)$$ and $\pi_{23}^*\P\tensor\pi_{12}^*\P^*=\pi_{23}^*\P^*\tensor\pi_{12}^*\P$. But $\mathbf{R}\operatorname{\mathcal{H}\mathnormal{om}}(\hat E,{\mathscr O})=(-1)^*\mathbf{R}{\mathcal F}(E^*_0)[n]$. \end{proof} \begin{cor} $$\mathbf{R}\Phi(-)=(-1)^*E_0\mathbin{\mathop{*}\limits^{\mathbf{R}}}(-)[-n]\qquad\hbox{and}\qquad \mathbf{R}\hat\Phi(-)=(-1)^*E_0^*\mathbin{\mathop{*}\limits^{\mathbf{R}}}(-),$$ where $\mathbin{\mathop{*}\limits^{\mathbf{R}}}$ denotes the (derived) convolution product: $\mathbf{R} m_*(p_1^*(-)\tensor p_2^*(-))$ and $m,p_1,p_2:{\mathbb T}\cross{\mathbb T}\to{\mathbb T}$ are the multiplication map and the projections onto the factors respectively. In particular, $E^*_0\mathbin{\mathop{*}\limits^{\mathbf{R}}} E_0\cong{\mathscr O}$. \end{cor} \begin{proof} This follows immediately from the lemma and the convolution theorem for the Mukai transform (see \refb{Muk2}{3.7}). \end{proof} \begin{proof} (of proposition) Observe that the fibres of $\mathbb{E}$ over $\{a\}\cross\{b\}$ are given by $H^{n-i}(\mathbf{R}{\mathcal F}(E_0)\tensor\P_{b}\tensor\P_{-a})$. This shows that $\mathbb{E}|_{{\mathbb T}\cross\{b\}}\cong E_b$ and so $\mathbb{E}^{\oplus\sigma}$ is isomorphic to the semi-universal bundle on ${\mathbb T}\cross{\mathcal M}$ via ${\mathbb T}\mathbin{\mathop{\rightarrow}\limits^\sim}{\mathcal M}$. This implies that $\mathbb{E}$ is universal. But since $\mathbb{E}_{\{a\}\cross{\mathbb T}}\cong E_{-a}$ we see that it is also bi-universal. \end{proof} \begin{thm}\label{t:eulercl} If ${\mathcal M}$ is a moduli space of stable bundles over an abelian variety ${\mathbb T}$ of dimension $n$ satisfying WIT on which ${\mathbb T}$ acts freely and effectively by pullback, then for each $E\in{\mathcal M}$ the transform sheaf $\mathbf{R}{\mathcal F}(E)$ is a line bundle. In particular, $|\chi(E)|=1$. \end{thm} \begin{proof} The proposition shows that $\mathbf{R}\Phi$ and $\mathbf{R}\hat\Phi$ are of Fourier-Mukai type. If we substitute the expressions of \lemref{l:phiisf} into $\mathbf{R}\hat\Phi\raise2pt\hbox{$\scriptscriptstyle\circ$}\mathbf{R}\Phi=\operatorname{Id}$ then we obtain $$\mathbf{R}{\mathcal F}(E^*_0)\tensor\mathbf{R}{\mathcal F}(E_0)={\mathscr O}.$$ This implies that $\mathbf{R}{\mathcal F}(E_0)$ is an invertible sheaf. \end{proof} This allows us to generalise \thmref{t:modideal} to arbitrary polarised abelian varieties and so we can state the following theorem. \begin{thm}\label{t:hilbismod} Let ${\mathbb T}$ be an abelian variety with a polarisation $\ell$. Then there is a non-empty component ${\mathcal M}$ of the moduli space of simple sheaves on ${\mathbb T}$ with Chern character $$\Bigl(\frac{m}{n!}(\hat\ell^n)^*-(-1)^n,-\frac{m}{(n-1)!}(\hat\ell^{n-1})^*, \ldots,(-1)^{n-1}m\hat\ell^*,(-1)^nm1^*\Bigl),$$ where $\alpha\mapsto\alpha^*$ denotes $H^i({\hat{\mathbb T}},{\mathbb C})\cong H^{2n-i}({\mathbb T},{\mathbb C})$, which is isomorphic to $\operatorname{Hilb}^m{\mathbb T}\cross{\hat{\mathbb T}}$. In particular, each Hilbert scheme of points on an abelian variety arises as a moduli space of simple sheaves on that variety. \end{thm} \begin{proof} The proof is essentially the same as that of \thmref{t:modideal}. The equivalent statement to \propref{p:7.1} holds because $R^0{\mathcal F}(L)$ has a natural Hermitian-Einstein connection via the flat connection on ${\mathbb P}(R^0{\mathcal F}(L))$. Then the moduli space of Gieseker stable vector bundles of Chern character $((\ell^n)^*/n!,(\ell^{n-1})^*/(n-1)!,\ldots,1^*)$ gives rise to a Fourier-Mukai transform $R\Phi$ as before. An ideal sheaf ${\mathscr I}_S$ of a zero-dimensional subscheme is $\Phi$-WIT and the transform is a simple sheaf by the Parseval Theorem. \end{proof} Analogous results also hold in the case of the K3 surface (see \cite{BM}). This theorem strengthens the results of Mukai which give a series of birational isomorphisms between Hilbert schemes of points and components of the moduli of simple sheaves on abelian varieties (see \cite{Muk5} Theorem 2.7, Theorem 2.17 and Theorem 2.20). \section{Discussion} The Fourier-Mukai transforms are very useful tools in the study of moduli spaces of simple or stable sheaves as well as to the study of more direct questions about the geometry of the underlying varieties. It is therefore an important programme to identify them for any given variety with trivial canonical bundle. As the application demonstrates it is possible to identify many moduli spaces with Hilbert schemes of points. This lends weight to the conjecture that all the components of the moduli spaces of simple sheaves on projective varieties with trivial canonical bundle are isomorphic to (deformations of) punctual Hilbert schemes. The theory of Fourier-Mukai transforms presents a number of immediate conjectures: \begin{conjecture} If $\mathbf{R}\Phi$ is a Fourier-Mukai functor from $X$ to $Y$ then $X$ is a holomorphic deformation of $Y$. \end{conjecture} The Mukai transform shows that $X$ need not be naturally isomorphic to $Y$ when $X$ is an abelian variety with no principal polarisations. But the geometry of $X$ and $Y$ are identical in this case. This is essentially the question of the extent to which $D(X)$ determines $X$. For a K3 surface, no examples are currently known for which $Y$ is not identical to $X$. For abelian varieties, it is possible to find transforms when $X$ and $Y$ are simply isogenous. A more specific question which arises based on the theorems above is \begin{problem} Given a Chern character, find a Fourier-Mukai transform for which each simple sheaf in a component of the moduli space of simple sheaves with this Chern character satisfies $\Phi$-WIT. This would be particularly useful for abelian varieties where the sheaves might be half-WITs. \end{problem} It would also be important to know when strongly bi-universal sheaves exist in more general contexts because the resulting transforms are still of Verdier type and may even have one sided inverses. \begin{conjecture} Given a smooth projective variety $X$ with $K_X={\mathscr O}_X$ and $Y$ a projective component of $\operatorname{Spl}(X)$ of dimension $\dim X$ with a universal sheaf $\mathbb{E}$ on $X\cross Y$. Then $\mathbb{E}$ is bi-universal. \end{conjecture} \noindent All currently known examples satisfy this conjecture, for example, this is always true for an abelian surface. Another interesting problem is to construct such Fourier-Mukai transforms for Calabi-Yau three-folds. One could then study the analytic versions of the transforms (in analogy with the Nahm transform for instantons on complex tori) and use these to solve the Hermitian-Yang-Mills equations on such 3-folds. An obvious question then arises about whether one can find a Fourier-Mukai transform for which $Y$ is a mirror of $X$.

9705

alg-geom/9705003

en

https://arxiv.org/abs/alg-geom/9705003

alg-geom/9705003

Michael Finkelberg

Michael Finkelberg, Alexander Kuznetsov, Ivan Mirkovi\'c (Independent University of Moscow and University of Massachusetts at Amherst)

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

8 pages, AmsLatex 1.1

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps' space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to Q^D_\alpha$.

alg-geom

\section{Introduction} \subsection{} Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space is noncompact. Some remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$ (Quasiflags) of $\CQ_\alpha$ were constructed by Drinfeld and Laumon respectively. In ~\cite{k} it was proved that the natural map $\pi:\ \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps' space $\CQ^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$ is irreducible. The proof is based on the {\em factorization property} of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi:\ \CQ^L_\alpha\to\CQ^D_\alpha$. We are grateful to P.Schapira for the illuminating correspondence. This note is a sequel to ~\cite{k} and ~\cite{fk}. In fact, the local geometry of $\CQ^D_\alpha$ was the subject of ~\cite{k}; the global geometry of $\CQ^D_\alpha$ was the subject of ~\cite{fk}, while the microlocal geometry of $\CQ^D_\alpha$ is the subject of the present work. We will freely refer the reader to ~\cite{k} and ~\cite{fk}. \section{Reductions of the main theorem} \subsection{Notations} \subsubsection{} \label{not} We choose a basis $\{v_1,\ldots,v_n\}$ in $V$. This choice defines a Cartan subgroup $H\subset G=SL(V)=SL_n$ of matrices diagonal with respect to this basis, and a Borel subgroup $B\subset G$ of matrices upper triangular with respect to this basis. We have $\CB=G/B$. Let $I=\{1,\ldots,n-1\}$ be the set of simple coroots of $G=SL_n$. Let $R^+$ denote the set of positive coroots, and let $2\rho=\sum_{\theta\in R^+}\theta$. For $\alpha=\sum a_ii\in\BN[I]$ we set $|\alpha|:=\sum a_i$. Let $X$ be the lattice of weights of $G,H$. Let $X^+\subset X$ be the set of dominant (with respect to $B$) weights. For $\lambda\in X^+$ let $V_\lambda$ denote the irreducible representation of $G$ with the highest weight $\lambda$. Recall the notations of ~\cite{k} concerning Kostant's partition function. For $\gamma\in\BN[I]$ a {\em Kostant partition} of $\gamma$ is a decomposition of $\gamma$ into a sum of positive coroots with multiplicities. The set of Kostant partitions of $\gamma$ is denoted by $\fK(\gamma)$. There is a natural bijection between the set of pairs $1\leq q\leq p\leq n-1$ and $R^+$, namely, $(p,q)$ corresponds to $i_q+i_{q+1}+\ldots+i_p$. Thus a Kostant partition $\kappa$ is given by a collection of nonnegative integers $(\kappa_{p,q}), 1\leq q\leq p\leq n-1$. Following {\em loc. cit.} (9) we define a collection $\mu(\kappa)$ as follows: $\mu_{p,q}=\sum_{r\leq q\leq p\leq s}\kappa_{s,r}$. Recall that for $\gamma\in\BN[I]$ we denote by $\Gamma(\gamma)$ the set of all partitions of $\gamma$, i.e. multisubsets (subsets with multiplicities) $\Gamma=\lbr \gamma_1,\ldots,\gamma_k\rbr $ of $\BN[I]$ with $\sum_{r=1}^k\gamma_r=\gamma,\ \gamma_r>0$ (see e.g. ~\cite{k}, 1.3). The configuration space of colored effective divisors of multidegree $\gamma$ (the set of colors is $I$) is denoted by $C^\gamma$. The diagonal stratification $C^\gamma=\sqcup_{\Gamma\in\Gamma(\gamma)} C^\gamma_\Gamma$ was introduced e.g. in {\em loc. cit.} Recall that for $\Gamma=\lbr \gamma_1,\ldots,\gamma_k\rbr $ we have $\dim C^\gamma_\Gamma=k$. \subsubsection{} For the definition of Laumon's Quasiflags' space $\CQ^L_\alpha$ the reader may consult ~\cite{la} 4.2, or ~\cite{k} 1.4. It is the space of complete flags of locally free subsheaves $$0\subset E_1\subset\dots\subset E_{n-1}\subset V\otimes\CO_C=:\CV$$ such that rank$(E_k)=k$, and $\deg(E_k)=-a_k$. It is known to be a smooth projective variety of dimension $2|\alpha|+\dim\CB$. \subsubsection{} For the definition of Drinfeld's Quasimaps' space $\CQ^D_\alpha$ the reader may consult ~\cite{k} 1.2. It is the space of collections of invertible subsheaves $\CL_\lambda\subset V_\lambda\otimes\CO_C$ for each dominant weight $\lambda\in X^+$ satisfying Pl\"ucker relations, and such that $\deg\CL_\lambda=-\langle\lambda,\alpha\rangle$. It is known to be a (singular, in general) projective variety of dimension $2|\alpha|+\dim\CB$. The open subspace $\CQ_\alpha\subset\CQ^D_\alpha$ of genuine maps is formed by the collections of line subbundles (as opposed to invertible subsheaves) $\CL_\lambda\subset V_\lambda\otimes\CO_C$. In fact, it is an open stratum of the stratification by the {\em type of degeneration} of $\CQ^D_\alpha$ introduced in ~\cite{k} 1.3: $$\CQ^D_\alpha=\bigsqcup_{\beta\leq\alpha}^{\Gamma\in\Gamma(\alpha-\beta)} \fD^{\beta,\Gamma}_\alpha$$ We have $\fD_{\alpha,\emptyset}=\CQ_\alpha$, and $\fD^{\beta,\Gamma}_\alpha= \CQ_\beta\times C^{\alpha-\beta}_\Gamma$ (see {\em loc. cit.} 1.3.5). The space $\CQ^D_\alpha$ is naturally embedded into the product of projective spaces $$\BP_\alpha=\prod_{1\leq p\leq n-1} \BP(\Hom(\CO_C(-\langle\omega_p,\alpha\rangle), V_{\omega_p}\otimes\CO_C))$$ and is closed in it (see {\em loc. cit.} 1.2.5). Here $\omega_p$ stands for the fundamental weight dual to the coroot $i_p$. The fundamental representation $V_{\omega_p}$ equals $\Lambda^pV$. \subsection{} \label{main} We will study the characteristic cycle of the Goresky-MacPherson perverse sheaf (or the corresponding regular holonomic $D$-module) $IC_\alpha$ on $\CQ^D_\alpha$. As $\CQ^D_\alpha$ is embedded into the smooth space $\BP_\alpha$, we will view this characteristic cycle $SS(IC_\alpha)$ as a Lagrangian cycle in the cotangent bundle $T^*\BP_\alpha$. {\em A priori} we have the following equality: $$SS(IC_\alpha)=\overline{T^*_{\CQ_\alpha}\BP_\alpha}+ \sum_{\beta<\alpha}^{\Gamma\in\Gamma(\alpha-\beta)}m^{\beta,\Gamma}_\alpha \overline{T^*_{\fD^{\beta,\Gamma}_\alpha}\BP_\alpha},$$ closures of conormal bundles with multiplicities. {\bf Theorem.} $SS(IC_\alpha)=\overline{T^*_{\CQ_\alpha}\BP_\alpha}$ is irreducible. In the following subsections we will reduce the Theorem to a statement about geometry of Laumon's resolution. \subsection{} We fix a coordinate $z$ on $C$ identifying it with the standard $\BP^1$. We denote by $\CQ^\infty_\alpha\subset\CQ^D_\alpha$ the open subspace formed by quasimaps which are genuine maps in a neighbourhood of the point $\infty\in C$. In other words, $(\CL_\lambda\subset V_\lambda\otimes\CO_C)_{\lambda\in X^+} \in\CQ^\infty_\alpha$ iff for each $\lambda$ the invertible subsheaf $\CL_\lambda\subset V_\lambda\otimes\CO_C$ is a line subbundle in some neighbourhood of $\infty\in C$. Evidently, $\CQ^\infty_\alpha$ intersects all the strata $\fD_{\beta,\Gamma}$. Thus it suffices to prove the irreducibility of the singular support of Goresky-MacPherson sheaf of $\CQ^\infty_\alpha$. There is a well-defined map of evaluation at $\infty\in C$: $$\Upsilon_\alpha:\ \CQ^\infty_\alpha\lra\CB$$ It is compatible with the stratification of $\CQ^\infty_\alpha$ and realizes $\CQ^\infty_\alpha$ as a (stratified) fibre bundle over $\CB$. In effect, $G$ acts naturally both on $\CQ^\infty_\alpha$ (preserving stratification) and on $\CB$; the map $\Upsilon_\alpha$ is equivariant, and $\CB$ is homogeneous. We denote the fiber $\Upsilon_\alpha^{-1}(B)$ over the point $B\in\CB$ by $\CZ_\alpha$. It inherits the stratification $$\CZ_\alpha= \bigsqcup_{\beta\leq\alpha}^{\Gamma\in\Gamma(\alpha-\beta)} \CZ\fD^{\beta,\Gamma}_\alpha$$ from $\CQ^\infty_\alpha$ and $\CQ^D_\alpha$. It is just the transversal intersection of the fiber $\Upsilon_\alpha^{-1}(B)$ with the stratification of $\CQ^\infty_\alpha$. As in ~\cite{k} 1.3.5 we have $\CZ\fD^{\beta,\Gamma}_\alpha\iso\CZ_\beta\times (C-\infty)^{\alpha-\beta}_\Gamma$. Hence it suffices to prove the irreducibility of the singular support $SS(IC(\CZ_\alpha))$ of Goresky-MacPherson sheaf $IC(\CZ_\alpha)$ of $\CZ_\alpha$. \subsection{Factorization} The Theorem 6.3 of ~\cite{fm} admits the following immediate Corollary. Let $(\phi_\beta,\gamma_1x_1,\ldots,\gamma_kx_k)=\phi_\alpha\in \CZ_\beta\times(C-\infty)^{\alpha-\beta}_\Gamma=\CZ\fD^{\beta,\Gamma}_\alpha \subset\CZ_\alpha$. Consider also the points $(\phi_r,\gamma_rx_r)= \phi_{\gamma_r}\in\CZ_0\times(C-\infty)^{\gamma_r}_{\{\{\gamma_r\}\}}= \CZ\fD^{0,\{\{\gamma_r\}\}}_{\gamma_r}\subset\CZ_{\gamma_r},\ 1\leq r\leq k$. {\bf Proposition.} There is an analytic open neighbourhood $U_\alpha$ (resp. $U_\beta$, resp. $U_{\gamma_r},\ 1\leq r\leq k$) of $\phi_\alpha$ (resp. $\phi_\beta$, resp. $\phi_{\gamma_r},\ 1\leq r\leq k$) in $\CZ_\alpha$ (resp. $\CZ_\beta$, resp. $\CZ_{\gamma_r},\ 1\leq r\leq k$) such that $$U_\alpha\iso U_\beta\times\prod_{1\leq r\leq k}U_{\gamma_r}$$ $\Box$ Recall the nonnegative integers $m^{\beta,\Gamma}_\alpha$ introduced in ~\ref{main}. The Proposition implies the following Corollary. {\bf Corollary.} $m^{\beta,\Gamma}_\alpha=\prod_{1\leq r\leq k} m^{0,\{\{\gamma_r\}\}}_{\gamma_r}$. $\Box$ Thus to prove that all the multiplicities $m^{\beta,\Gamma}_\alpha$ vanish, it suffices to check the vanishing of $m^{0,\{\{\gamma\}\}}_\gamma$ for arbitrary $\gamma>0$. \subsection{} It remains to prove that the conormal bundle $T^*_{\fD^{0,\{\{\gamma\}\}}_\gamma}\BP_\alpha$ to the closed stratum of $\CQ_\gamma$ enters the singular support $SS(IC_\alpha)$ with multiplicity 0. To this end we choose a point $(B,\gamma0)=\phi\in\CB\times C= \CQ_0\times C^\gamma_{\{\{\gamma\}\}}=\fD_\gamma^{0,\{\{\gamma\}\}} \subset\CQ_\gamma\subset\BP_\gamma$. We also choose a sufficiently generic meromorphic function $f$ on $\BP_\gamma$ regular around $\phi$ and vanishing on $\fD_\gamma^{0,\{\{\gamma\}\}}$. According to the Proposition 8.6.4 of ~\cite{ks}, the multiplicity in question is 0 iff $\Phi_f(IC_\gamma)_\phi=0$, i.e. the stalk of vanishing cycles sheaf at the point $\phi$ vanishes. To compute the stalk of vanishing cycles sheaf we use the following argument, borrowed from ~\cite{bfl} ~\S1. As $\pi:\ \CQ^L_\gamma\lra\CQ^D_\gamma$ is a small resolution of singularities, up to a shift, $IC_\alpha=\pi_*\uQ$. By the proper base change, $\Phi_f\pi_*\uQ=\pi_*\Phi_{f\circ\pi}\uQ$. So it suffices to check that $\Phi_{f\circ\pi}\uQ|_{\pi^{-1}(\phi)}=0$. Let us denote the differential of the function $f$ at the point $\phi$ by $\xi$ so that $(\phi,\xi)\in T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$. Then the support of $\Phi_{f\circ\pi}\uQ|_{\pi^{-1}(\phi)}$ is {\em a priori} contained in the {\em microlocal fiber} over $(\phi,\xi)$ which we define presently. \subsubsection{Definition} Let $\varpi:\ A\to B$ be a map of smooth varieties. For $a\in A$ let $d_a^*\varpi:\ T^*_{\varpi(a)}B\lra T^*_aA$ denote the codifferential, and let $(b,\eta)$ be a point in $T^*B$. Then the {\em microlocal fiber} of $\varpi$ over $(b,\eta)$ is defined to be the set of points $a\in \varpi^{-1}(b)$ such that $d^*_a\varpi(\eta)=0$. \subsubsection{} \label{prop} Thus we have reduced the Theorem ~\ref{main} to the following Proposition. {\bf Proposition.} For a sufficiently generic $\xi$ such that $(\phi,\xi) \in T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$, the microlocal fiber of Laumon's resolution $\pi$ over $(\phi,\xi)$ is empty. Equivalently, the cone $\cup_{E_\bullet\in\pi^{-1}(\phi)}\Ker (d^*_{E_\bullet}\pi)$ is a proper subvariety of the fiber of $T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$ at $\phi$. \subsection{Piecification of a simple fiber} The fiber $\pi^{-1}(\phi)$ was called the {\em simple fiber} in ~\cite{k} ~\S2. It was proved in {\em loc. cit.} ~2.3.3 that $\pi^{-1}(\phi)$ is a disjoint union of (pseudo)affine spaces $\fS(\mu(\kappa))$ where $\kappa$ runs through the set $\fK(\gamma)$ of Kostant partitions of $\gamma$ (for the notation $\mu(\kappa)$ see ~\ref{not} or ~\cite{k} ~(9)). Another way to parametrize these pseudoaffine pieces was introduced in ~\cite{fk} ~2.11. Let us recall it here. We define nonnegative integers $c_p, 1\leq p\leq n-1$, so that $\gamma=\sum_{p=1}^{n-1}c_pi_p$. \subsubsection{Definition} $\CalD(\gamma)$ is the set of collections of nonnegative integers $(d_{p,q})_{1\leq q\leq p\leq n-1}$ such that a) For any $1\leq q\leq p\leq r\leq n-1$ we have $d_{r,q}\leq d_{p,q}$; b) For any $1\leq p\leq n-1$ we have $\sum_{q=1}^pd_{p,q}=c_p$. \subsubsection{Lemma} The correspondence $\kappa=(\kappa_{p,q})_{1\leq q\leq p \leq n-1}\mapsto(d_{p,q}:=\sum_{r=p}^{n-1}\kappa_{r,q})_{1\leq q\leq p \leq n-1}$ defines a bijection between $\fK(\gamma)$ and $\CalD(\gamma)$. $\Box$ \subsubsection{} Using the above Lemma we can rewrite the parametrization of the pseudoaffine pieces of the simple fiber as follows: $$\pi^{-1}(\phi)=\bigsqcup_{\fd\in\CalD(\gamma)}\fS(\fd)$$ In these terms the dimension formula of ~\cite{k} ~2.3.3 reads as follows: for $\fd=(d_{p,q})_{1\leq q\leq p\leq n-1}$ we have $\dim\fS(\fd)= \sum_{1\leq q<p\leq n-1}d_{p,q}$. Note also that $\sum_{1\leq q\leq p\leq n-1}d_{p,q}= \sum_{1\leq p\leq n-1}c_p=|\gamma|$. \subsection{Proposition} \label{red} For arbitrary $\fd=(d_{p,q})_{1\leq q\leq p\leq n-1}\in\CalD(\gamma)$ and arbitrary quasiflag $E_\bullet\in\fS(\fd)\subset\pi^{-1}(\phi)$ we have $\dim\Ker (d_{E_\bullet}\pi)<\sum_{1\leq p\leq n-1}d_{p,q} +\sum_{1\leq q\leq p\leq n-1}d_{p,q}-1$. This Proposition implies the Proposition ~\ref{prop} straightforwardly. In effect, $\codim\Ker (d_{E_\bullet}\pi)=\dim\CQ^L_\gamma-\dim\Ker (d_{E_\bullet}\pi)>2|\gamma|+\dim\CB-\sum_{1\leq p\leq n-1}d_{p,p} -\sum_{1\leq q\leq p\leq n-1}d_{p,q}+1= \dim\CB+1+\sum_{1\leq q<p\leq n-1}d_{p,q}$. Hence the subspace $\Ker (d^*_{E_\bullet}\pi)\subset T^*_\phi\BP_\gamma$ has codimension greater than $\dim\CB+1+\sum_{1\leq q<p\leq n-1}d_{p,q}$. Recall that $\dim\fD_\gamma^{0,\{\{\gamma\}\}}=\dim\CB+1$. Hence the codimension of $\Ker (d^*_{E_\bullet}\pi)\cap T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$ in the fiber of $T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$ at $\phi$ is greater than $\sum_{1\leq q<p\leq n-1}d_{p,q}=\dim\fS(\fd)$. Hence the cone $\cup_{E_\bullet\in\fS(\fd)}\Ker (d^*_{E_\bullet}\pi)$ is a proper subvariety of the fiber of $T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$ at $\phi$. The union of these proper subvarieties over $\fd\in\CalD(\gamma)$ is again a proper subvariety of the fiber of $T^*_{\fD_\gamma^{0,\{\{\gamma\}\}}}\BP_\gamma$ at $\phi$ which concludes the proof of the Proposition ~\ref{prop}. \subsection{Fixed points} It remains to prove the Proposition ~\ref{red}. To this end recall that the Cartan group $H$ acts on $V$ and hence on $\CQ^L_\alpha$. The group $\BC^*$ of dilations of $C={\Bbb P}^1$ preserving $0$ and $\infty$ also acts on $\CQ^L_\alpha$ commuting with the action of $H$. Hence we obtain the action of a torus $\BT:=H\times\BC^*$ on $\CQ^L_\alpha$. It preserves the simple fiber $\pi^{-1}(\phi)$ and its pseudoaffine pieces $\fS(\fd),\ \fd\in\CalD(\gamma)$, for evident reasons. It was proved in ~\cite{fk} ~2.12 that each piece $\fS(\fd),\ \fd=(d_{p,q})_{1\leq q\leq p\leq n-1}$ contains exactly one $\BT$-fixed point $\delta(\fd)=(E_1,\ldots,E_{n-1})$. Here $$ {\arraycolsep=1pt \begin{array}{llrlcrlcccrlc} E_1 & = E_{1,1} \\ E_2 & = E_{2,1} &\opl& E_{2,2} \\ \ \vdots && \vdots &&& \vdots \\ E_{n-1} & = E_{n-1,1} &\opl& E_{n-1,2} &\opl& \dots &\opl& E_{n-1,n-1} \\ \end{array} } $$ and $E_{p,q}=\CO(-d_{p,q})\subset\CO v_q\subset\CV=V\otimes\CO_C$ with quotient sheaf $\dfrac\CO{\CO(-d_{p,q})}$ concentrated at $0\in C$. \subsubsection{} \label{key} Now the $\BT$-action contracts $\fS(\fd)$ to $\delta(\fd)$. Since the map $\pi$ is $\BT$-equivariant, and the dimension of $\Ker (d_{E_\bullet}\pi)$ is lower semicontinuous, the Proposition ~\ref{red} follows from the next one. {\bf Key Proposition.} For arbitrary $\fd=(d_{p,q})_{1\leq q\leq p\leq n-1}\in\CalD(\gamma)$ ($\gamma\ne0$) we have $\dim\Ker (d_{\delta(\fd)}\pi)<\sum_{1\leq p\leq n-1}d_{p,p} +\sum_{1\leq q\leq p\leq n-1}d_{p,q}-1$. The proof will be given in the next section. \subsubsection{Remark} In general, the pieces $\fS(\fd)$ of the simple fiber are not equisingular, i.e. $\dim\Ker (d_{E_\bullet}\pi)$ is not constant along a piece. The simplest example occurs for $G=SL_3,\ \gamma=2i_1+2i_2$. Then the simple fiber is a singular 2-dimensional quadric. Its singular point is the fixed point of the 1-dimensional piece $\fS(\fd)$ where $d_{1,1}=2,d_{2,1}=d_{2,2}=1$. At this point we have $\dim\Ker (d_{\delta(\fd)}\pi)=3$ while at the other points in this piece we have $\dim\Ker (d_{E_\bullet}\pi)=2$. \section{The proof of the Key Proposition} \subsection{Tangent spaces} Let $\Omega$ be the following quiver: $\Omega=1\lra 2\lra\ldots\lra n-1$. Thus the set of vertices coincides with $I$. A quasiflag $(E_1\hra E_2\hra \ldots\hra E_{n-1}\subset\CV)\in\CQ^L_\gamma$ may be viewed as a representation of $\Omega$ in the category of coherent sheaves on $C$. If we denote the quotient sheaf $\CV/E_p$ by $Q_p,\ 1\leq p\leq n-1$, we have another representation of $\Omega$ in coherent sheaves on $C$, namely, $$Q_\bullet:=(Q_1\twoheadrightarrow Q_2\twoheadrightarrow \ldots\twoheadrightarrow Q_{n-1})$$ \subsubsection{Exercise} $T_{E_\bullet}\CQ^L_\gamma=\Hom_\Omega(E_\bullet, Q_\bullet)$ where $\Hom_\Omega(?,?)$ stands for the morphisms in the category of representations of $\Omega$ in coherent sheaves on $C$. \subsubsection{} Consider a point $\CL_\bullet= (\CL_1,\ldots,\CL_{n-1})\in\BP_\gamma$. Here $\CL_p\subset V_{\omega_p}\otimes\CO_C$ is an invertible subsheaf, the image of morphism $\CO_C(-\langle\omega_p,\gamma\rangle)\hra V_{\omega_p} \otimes\CO_C$. {\em Exercise.} $T_{\CL_\bullet}\BP_\gamma=\prod_{p=1}^{n-1}\Hom(\CL_p, V_{\omega_p}\otimes\CO_C/\CL_p)$. \subsubsection{} Recall that for $E_\bullet\in\CQ^L_\gamma$ we have $\pi(E_\bullet)=\CL_\bullet\in\BP_\gamma$ where $\CL_p=\Lambda^pE_p$ for $1\leq p\leq n-1$. {\em Exercise.} For $h_\bullet=(h_1,\ldots,h_{n-1})\in T_{E_\bullet}\CQ^L_\gamma$ we have $d_{E_\bullet}\pi(h_\bullet)= (\Lambda^1h_1,\Lambda^2h_2,\ldots,\Lambda^{n-1}h_{n-1})\in T_{\CL_\bullet}\BP_\gamma$. \subsection{} From now on we fix $\gamma>0,\ \fd\in\CalD(\gamma),\ \delta(\fd)=:E_\bullet$. To unburden the notations we will denote the tangent space $T_{E_\bullet}\CQ^L_\gamma$ by $T$. Since $\QL\gamma$ is a smooth $(2|\gamma|+\dim\CB)$-dimensional variety it suffices to find a subspace $N\subset T$ of dimension $$ 2|\gamma|+\dim\CB-\sum_{1\le p\le n-1}d_{p,p}-\sum_{1\le q\le p\le n-1}d_{p,q}+1= \sum_{1\le q<p\le n-1}(d_{p,q}+1)+1 $$ such that $d_{E_\bullet}\pi|_{N}$ is injective. \subsection{} Let $N_0=\opll_{n-1\ge p>q\ge 1}\Hom(\CO(-d_{p,q}),\CO)$. We have $\dim N_0=\sum_{n-1\ge p>q\ge 1}(d_{p,q}+1)$. Recall that we have canonically $T=\Hom_\Om(E_\bullet,Q_\bullet)$, where $$ Q_p=\CV/E_p=\left(\opll_{q=1}^p\left(\dfrac\CO{\CO(-d_{p,q})}\right)v_q\right) \opl\left(\opll_{q=p+1}^n\CO v_q\right). $$ \subsection{} Let us define a map $\nu_0:N_0\to T$ assigning to an element $(f_{p,q})\in N_0$ a morphism $\nu_0(f_{p,q}):= F\in\Hom_\bullet(E_\bullet,Q_\bullet)$ of graded coherent sheaves, where $F|_{E_{p,q}}=\opll_{r=p+1}^n F_{p,q}^r$, and $$ F_{p,q}^r:E_{p,q}\to\CO v_r\subset Q_p\quad \text{is defined as the composition}\quad E_{p,q} \subset E_{r,q}=\CO(-d_{r,q}) @>f_{r,q}>> \CO v_r $$ \subsection{}{\bf Lemma.} The map $F:E_\bullet\to Q_\bullet$ is a morphism of representations of the quiver $\Om$. \begin{pf} We need to check the commutativity of the following diagram $$ \begin{CD} E_p @>>> E_{p'} \\ @VFVV @VFVV \\ Q_p @>>> Q_{p'} \end{CD} $$ Since $E_p$ and $Q_{p'}$ are canonically decomposed into the direct sum it suffices to note that for any $q\le p\le p'<r$ the following diagram $$ \begin{CD} E_{p,q} @>>> E_{p',q} @>>> E_{r,q} \\ @VF_{p,q}^rVV @VF_{p',q}^rVV @Vf_{r,q}VV \\ \CO v_r @= \CO v_r @= \CO v_r \end{CD} $$ commutes and for any $q\le p<r\le p'$ the following diagram $$ \begin{CD} E_{p,q} @>>> E_{r,q} @>>> E_{p',q} \\ @VF_{p,q}^rVV @Vf_{r,q}VV @V0VV \\ \CO v_r @= \CO v_r @>>> \left(\dfrac\CO{\CO(-d_{r,q})}\right)v_r \end{CD} $$ commutes as well. \end{pf} \subsection{} Let $N_1={\Bbb C}$. Let $p_0=\min\{1\le p\le n-1\ |\ d_{p,p}>0\}$ and pick a non-zero element $\ff\in\Hom(\CO(-d_{p,p}),\frac\CO{\CO(-d_{p,p})})$. Define the map $\nu_1:N_1\to T$ by assigning to $1\in N_1$ the element $\fF\in\Hom_\Om(E_\bullet,Q_\bullet)$ defined on $E_{p,p}$ as the composition $$ E_{p,p}=\CO(-d_{p,p}) @>\ff>> \frac\CO{\CO(-d_{p,p})}v_p\subset Q_p $$ and with all other components equal to zero. \subsection{} Let $\CM(r,d;\CV)$ denote the space of rank $r$ and degree $d$ subsheaves in $\CV$. \subsubsection{} \label{lem} Let $E\subset\CV$ be a rank $k$ and degree $d$ subsheaf in the vector bundle $\CV$. Let $\CV/E=\CT\opl \CF$ be a decomposition of the quotient sheaf into the sum of the torsion $\CT$ and a locally free sheaf $\CF$. Consider the map $\det:\CM(r,d;\CV)\to\CM(1,d;\Lambda^k\CV)$ sending $E$ to $\Lambda^kE$. Then the restriction of its differential $d_E\det:T_E\CM(r,d;\CV)=\Hom(E,\CV/E)\to \Hom(\Lambda^kE,\Lambda^kV/\Lambda^kE)=T_{\Lambda^kE}\CM(1,d;\Lambda^k\CV)$ to the subspace $\Hom(E,\CF)\subset\Hom(E,\CV/E)$ factors as $\Hom(E,\CF)\cong\Hom(\Lambda^kE,\Lambda^{k-1}E\ot \CF)\subset \Hom(\Lambda^kE,\Lambda^kV/\Lambda^kE)$. Therefore it is injective. \subsubsection{}\label{lem1} Let $E=\CO^{\opl(r-1)}\opl\CO(-d)$ be a subsheaf in $\CV=\CO^{\opl n}$. Then the restriction of differential $d_E\det$ to the subspace $\Hom(E,\CT)\subset\Hom(E,\CV/E)$ is injective. This immediately follows from the following fact. Let $\TE=\CO^{\opl r}\subset\CV$ be the normalization of $E$ in $\CV$, that is, the maximal vector subbundle $\TE\subset\CV$ such that $\TE/E$ is torsion. Then $\CT=\TE/E\cong\Lambda^k\TE/\Lambda^kE\subset\Lambda^k\CV/\Lambda^kE$. \subsubsection{}\label{rem} Clearly, the subsheaves $\Lambda^{k-1}E\ot \CF\subset\Lambda^k\CV/\Lambda^kE$ and $\CT\subset\Lambda^k\CV/\Lambda^kE$ do not intersect. \subsubsection{} It follows from \ref{lem}, \ref{lem1} and \ref{rem} that the composition $d_{E_\bullet}\pi\circ(\nu_0\opl\nu_1):N_0\opl N_1\to T_{\pi(E_\bullet)}\CP$ is injective, hence $N:=(\nu_0\opl\nu_1)(N_0\opl N_1)\subset T_{E_\bullet}\QD\gamma$ enjoys the desired property. Namely, $d_{E_\bullet}\pi|_N$ is injective, and $\dim N=\sum_{1\leq q<p\leq n-1}(d_{p,q}+1)+1$. This completes the proof of the Key Proposition ~\ref{key} along with the Main Theorem ~\ref{main}. $\Box$

9705

alg-geom/9705004

en

https://arxiv.org/abs/alg-geom/9705004

alg-geom/9705004

Misha Verbitsky

Misha Verbitsky

Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface

Arguments improved, errors corrected, rigor added. Sections 8 and 9 were totally rewritten, Tex-type: LaTeX 2e

Geom. Funct. Anal. 8 (1998), no. 4, 732--782

Let X be a hyperkaehler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a K3 surface M, the Hilbert scheme classifying zero-dimensional subschemes of M admits a hyperkaehler structure. We show that for M generic, there are no trianalytic subvarieties of the Hilbert scheme. This implies that a generic deformation of the Hilbert scheme of K3 has no complex subvarieties.

alg-geom

\section{Introduction} \hfill For basic results and definitions of hyperk\"ahler geometry, see \cite{_Besse:Einst_Manifo_}. \hfill This Introduction is independent from the rest of this paper. \subsection{An overview} An {\bf almost hypercomplex} manifold $M$ is a manifold equipped with an action of the quaternion algebra ${\Bbb H}$ on its tangent bundle. The manifold $M$ is called {\bf hypercomplex} if for all algebra embedding ${\Bbb C} \stackrel{\iota}\hookrightarrow \Bbb H$, the corresponding almost complex structure $I_\iota$ is integrable. A manifold $M$ is called {\bf hyperk\"ahler} if, on top of that, $M$ is equipped with a Riemannian metric which is K\"ahler with respect to the complex structures $I_\iota$, for all embeddings ${\Bbb C} \stackrel{\iota}\hookrightarrow \Bbb H$. The complex structures $I_\iota$ are called {\bf induced complex structures}; the corresponding K\"ahler manifold is denoted by $(M, I_\iota)$. For a more formal definition of a hyperk\"ahler manifold, see \ref{_hyperkahler_manifold_Definition_}. The notion of a hyperk\"ahler manifold was introduced by E. Calabi, \cite{_Calabi_}. Clearly, the real dimension of $M$ is divisible by 4. For $\dim_{\Bbb R} M= 4$, there are only two classes of compact hyperk\"ahler manifolds: compact tori and K3 surfaces. Let $M$ be a complex surface and $M^{(n)}$ be its $n$-th symmetric power, $M^{(n)} = M^n/S_n$. The variety $M^{(n)}$ admits a natural desingularization $M^{[n]}$, called {\bf Hilbert scheme of points}, or {\bf Hilbert scheme} for short. For its construction and additional results, see the lectures of H. Nakajima, \cite{_Nakajima_}. Most importantly, $M^{[n]}$ admits a hyperk\"ahler metrics whenever the surface $M$ is compact and hyperk\"ahler (\cite{_Beauville_}). This way, Beauville constructed two series of examples of hyperk\"ahler manifolds, associated with a torus \footnote{There is a natural torus action on its Hilbert scheme; to obtain the Beauville's hyperk\"ahler manifold, we must take the quotient by this action.} and a K3 surface. It was conjectured that all hyperk\"ahler manifolds $X$ with $H^1(X) =0$, $H^{2,0}(X)={\Bbb C}$ are deformationally equivalent to one of these examples. We study the complex and hyperk\"ahler geometry of $M^{[n]}$ for $M$ a ``sufficiently generic'' K3 surface, in order to construct counterexamples to this conjecture. \hfill Let $M$ be a hyperk\"ahler manifold. A {\bf trianalytic subvariety} of $M$ is a closed subset which is complex analytic with respect to any of the induced complex structures. It was proven in \cite{_Verbitsky:Symplectic_II_} that for all induced complex structures $I$, except maybe a countable number, all complex subvarieties of $(M,I)$ are trianalytic (see also \ref{_generic_are_dense_Proposition_}). This reduces the study of complex subvarieties of ``sufficiently generic'' deformations\footnote{By {\bf deformations of $M$} we understand complex manifolds which are deformationally equivalent to $M$.} of $M$ to the study of trianalytic subvarieties. Trianalytic subvarieties of hyperk\"ahler manifolds were studied at length in \cite{_Verbitsky:Symplectic_II_} and \cite{_Verbitsky:DesinguII_}. Of the results obtained in this study, the most important ones are Desingularization Theorem (\ref{_desingu_Theorem_}) and the cohomological criterion of trianalyticity (\ref{_G_M_invariant_implies_trianalytic_Theorem_}). The aim of the present paper is to obtain the following theorem. \hfill \theorem \label{_no_triana_subva_Introdu_Theorem_} Let $M$ be a complex K3 surface without automorphisms, $\c H$ a hyperk\"ahler structure on $M$, and $I$ an induced complex structure on $M$ which is Mumford-Tate generic in the class of induced complex structures.% \footnote{For the definition of Mumford-Tate generic, see \ref{_generic_manifolds_Definition_}.} Let $M^{[n]}$ be a Hilbert scheme of points on the complex surface $(M, I)$. Pick any hyperk\"ahler structure on $M^{[n]}$, compatible with the complex structure. Then, $M^{[n]}$ has no proper trianalytic subvarieties. {\bf Proof:} This is \ref{_no_triana_subva_of_Hilb_Theorem_}. \blacksquare \hfill In the forthcoming paper, we construct a 21-dimensional family of compact hyperk\"ahler manifolds $M_x$, with \begin{equation}\label{_non-decompo_Equation_} H^{1}(M) =0, \ \ H^{2,0}(M)={\Bbb C} \end{equation} which {\it have} proper trianalytic subvarieties. This leads to assertion that these manifolds are not deformations of $M^{[n]}$. As another application of \ref{_no_triana_subva_Introdu_Theorem_}, we obtain that a generic complex deformation of $M^{[n]}$ has no proper closed complex subvarieties (\ref{_no_comple_subva_of_gen_Hilb_Corollary_}). A version of this result is true for a compact torus. For a generic complex structure $I$ on a complex torus $T$, the complex manifold $(T,I)$ has no proper subvarieties. This is easy to see from the fact that the group $H^{p,p}(T,I) \cap H^{2p}(T, {\Bbb Z})$ is empty. For a Hilbert scheme of K3, this Hodge-theoretic argument does not work. In fact, there are integer cycles in $H^{2p,2p}(M^{[n]}, I)$ for all complex structures on $M^{[n]}$. As \ref{_no_comple_subva_of_gen_Hilb_Corollary_} implies, these cycles cannot be represented by subvarieties. This gives a counterexample to the Hodge conjecture. Of course, for a generic complex structure $I$, the manifold $(M^{[n]}, I)$ is not algebraic. There are many other counterexamples to the Hodge conjecture for non-algebraic manifolds. \hfill In our approach to the study of trianalytic subvarieties of the Hilbert scheme, we introduce the concept of the {\bf universal subvariety} of the Hilbert scheme (\ref{_Unive_subva_Definition_}). For a complex surface $M$, the local automorphisms $\gamma:\; U {\:\longrightarrow\:} U$ of $M\supset U$ act on the corresponding open subsets $U^{[n]}\subset M^{[n]}$ of the Hilbert scheme. A universal subvariety of $M^{[n]}$ is a subvariety which is preserved by all automorphisms obtained this way (see \ref{_Unive_subva_Definition_} for a more precise statement). We show that a trianalytic subvariety of a Hilbert scheme of a K3 surface $M$ is universal, assuming that $M$ is Mumford-Tate generic with respect to some hyperk\"ahler structure and has no complex automorphisms (\ref{_triana_subva_universal_Theorem_}). \subsection{Contents} The paper is organized as follows. \begin{itemize} \item Section \ref{_basics_hyperka_Section_} is taken with preliminary conventions and basic theorems. We define hyperk\"ahler manidols and formulate Yau's theorem on the existence of hyperk\"ahler structures on compact holomorphically symplectic manifolds of K\"ahler type. Furthermore, we define trianalytic subvarieties of hyperk\"ahler manifolds and recall the basic properties of trianalytic subvarieties, following \cite{_Verbitsky:Symplectic_II_} and \cite{_Verbitsky:DesinguII_}. There are no new results in Section \ref{_basics_hyperka_Section_}, and nothing unknown to the reader acquainted with the literature. \item In Section \ref{_appendix_subva_produ_Section_}, we apply the desingularization theorem to the subvarieties of $M^n$, where $M$ is a generic K3 surface. We classify trianalytic subvarieties of $M^n$ and describe them explicitly. This section is independent from the rest of this paper. \item We study the Hilbert scheme $M^{[n]}$ of a smooth holomorphic symplectic complex surface $M$ in Section \ref{_Hilbert_sche_Section_}. We give its definition and explain the construction of the holomorphic symplectic structure on $M^{[n]}$. By Yau's proof of Calabi conjecture (\ref{_symplectic_=>_hyperkahler_Proposition_}), this implies that $M^{[n]}$ admits a hyperk\"ahler structure. Using perverse sheaves, we write down the cohomology of $M^{[n]}$ in terms of diagonals in the symmetric power $M^{(n)}$. This is done using the fact that the standard projection $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ is a semi-small resolution of the symmetric power $M^{(n)}$. These results are well known (\cite{_Nakajima_}). Further on, we apply the same type arguments to the trianalytic subvarieties $X\subset M^{[n]}$. Using the holomorphic symplectic geometry, we show that the map $\tilde X\stackrel {\pi\circ n}{\:\longrightarrow\:} \pi(X)$ is a semi-small resolution, where $\tilde X\stackrel {n}{\:\longrightarrow\:} X$ is the hyperk\"ahler desingularization of $X$ (\ref{_desingu_Theorem_}). This gives an expression for the cohomology of $\tilde X$. We don't use this result anywhere in this paper. \item Section \ref{_Hilbert_sche_Section_} is heavily based on perverse sheaves (\cite{_Asterisque_100_}), and does not use results of hyperk\"ahler geometry, except Desingularization Theorem (\ref{_desingu_Theorem_}). \item The following four sections (Sections \ref{_unive_subva_Section_}--\ref{_triana_unive_subva_Section_}) are dedicated to the study of universal properties of the Hilbert scheme. \begin{itemize} \item In Section \ref{_unive_subva_Section_}, we give a definition of a universal subvariety of a Hilbert scheme. A relative dimension of a universal subvariety is a dimension of the generic fiber of the projection $\pi:\; X {\:\longrightarrow\:} \pi(X)$, where $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ is the standard projection of the Hilbert scheme to the symmetric power of $M$. We classify and describe explicitly the universal subvarieties of relative dimension $0$. Results of Section \ref{_unive_subva_Section_} are in no way related to the hyperk\"ahler geometry. \item In Section \ref{_subva_gene_fini_Section_}, we study the Hilbert scheme of a K3 surface $M$, assuming that $M$ is Mumford-Tate generic with respect to some hyperk\"ahler structure. We consider subvarieties $X\subset M^{[n]}$, such that $X$ is projected generically finite to $\pi(X)\subset M$, and $\pi(X)$ is a diagonal in $M^{(n)}$. We use the theory of Yang-Mills connections and Uhlenbeck--Yau theorem, in order to show that such subvarieties are universal, in the sense of Section \ref{_unive_subva_Section_}. \item Section \ref{_specia_subva_Section_} is completely parallel to Section \ref{_unive_subva_Section_}. We define {\bf special subvarieties} of the Hilbert scheme, which are similar to the universal subvarieties, with some conditions relaxed. Whereas universal subvarieties are subvarieties which are fixed by all local automorphisms of $M^{[n]}$ coming from $M$, special subvarieties are the subvarieties fixed by all the local automorphisms coming from $M$ which preserve a finite subset $S\subset M$. As in Section \ref{_unive_subva_Section_}, we classify and describe explicitely the special subvarieties of relative dimension zero. Using results of Section \ref{_subva_gene_fini_Section_}, we study the subvarieties $X\subset M^{[n]}$, such that $X$ is projected generically finite to $\pi(X)$, where $M$ is a generic K3 surface. We show that all such subvarieties are special of relative dimension zero. \item In Section \ref{_triana_unive_subva_Section_}, we study the deformations of subvarieties of the Hilbert scheme of a K3 surface. The deformations of special subvarieties of relative dimension zero are easy to study using the explicit description given in Section \ref{_specia_subva_Section_}. The deformations of trianalytic subvarieties were studied at length in \cite{_Verbitsky:Deforma_}. In conjunction, these results lead to the assertion that all trianalytic subvarieties of $M^{[n]}$ are universal, in the sense of Section \ref{_unive_subva_Section_}. \end{itemize} \item In Section \ref{_last_Section_}, we study the second cohomology of universal subvarieties $\c X_\alpha\stackrel \phi\hookrightarrow M^{[n]}$ of the Hilbert scheme $M^{[n]}$, in assumption that $\c X_\alpha$ is trianalytic. First of all, we show that $\c X_\alpha$ is birationally equivalent to a hyperka\"hler manifold which is a product of Hilbert schemes of $M$. Using Mukai's theorem, which states that second cohomology of hyperka\"hler manifolds is a birational invariant, we obtain a structure theorem for $H^2(\c X_\alpha)$. Assuming that $\c X_\alpha$ is not a product of hyperka\"hler manifolds, we show that the pullback map $\phi^*:\; H^2(M^{[n]}) {\:\longrightarrow\:} H^2(\c X_\alpha)$ is an isomorphism, and compute this map explicitly. The second cohomology of a hyperka\"hler manifold $X$ is equipped with a canonical non-degenerate quadratic form $(\cdot,\cdot)_{\c B}$, defined up to a constant multiplier. This form is invariant under the natural $SU(2)$-action on $H^2(X)$. We compute the pullback of the form $(\cdot,\cdot)_{\c B}$ under the map $\phi^*:\; H^2(M^{[n]}) {\:\longrightarrow\:} H^2(\c X_\alpha)$, and show that it cannot be $SU(2)$-invariant. Thus, $\phi^*$ is not compatible with the $SU(2)$-action on the second cohomology. This implies that $\phi$ cannot be compatible with the hyperka\"hler structures on $\c X_\alpha$, $M^{[n]}$. Therefore, $M^{[n]}$ contains no trianalytic subvarieties. \end{itemize} \section{Hyperk\"ahler manifolds} \label{_basics_hyperka_Section_} \subsection{Hyperk\"ahler manifolds} This subsection contains a compression of the basic and best known results and definitions from hyperk\"ahler geometry, found, for instance, in \cite{_Besse:Einst_Manifo_} or in \cite{_Beauville_}. \hfill \definition \label{_hyperkahler_manifold_Definition_} (\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ endowed with three complex structures $I$, $J$ and $K$, such that the following holds. \begin{description} \item[(i)] the metric on $M$ is K\"ahler with respect to these complex structures and \item[(ii)] $I$, $J$ and $K$, considered as endomorphisms of a real tangent bundle, satisfy the relation $I\circ J=-J\circ I = K$. \end{description} \hfill The notion of a hyperk\"ahler manifold was introduced by E. Calabi (\cite{_Calabi_}). \hfill Clearly, a hyperk\"ahler manifold has a natural action of the quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$. Therefore its complex dimension is even. For each quaternion $L\in \Bbb H$, $L^2=-1$, the corresponding automorphism of $TM$ is an almost complex structure. It is easy to check that this almost complex structure is integrable (\cite{_Besse:Einst_Manifo_}). \hfill \definition \label{_indu_comple_str_Definition_} Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying $L^2=-1$. The corresponding complex structure on $M$ is called {\bf an induced complex structure}. The $M$ considered as a K\"ahler manifold is denoted by $(M, L)$. In this case, the hyperk\"ahler structure is called {\bf combatible with the complex structure $L$}. Let $M$ be a compact complex variety. We say that $M$ is {\bf of hyperk\"ahler type} if $M$ admits a hyperk\"ahler structure compatible with the complex structure. \hfill \hfill \definition \label{_holomorphi_symple_Definition_} Let $M$ be a complex manifold and $\Theta$ a closed holomorphic 2-form over $M$ such that $\Theta^n=\Theta\wedge\Theta\wedge...$, is a nowhere degenerate section of a canonical class of $M$ ($2n=dim_{\Bbb C}(M)$). Then $M$ is called {\bf holomorphically symplectic}. \hfill Let $M$ be a hyperk\"ahler manifold; denote the Riemannian form on $M$ by $<\cdot,\cdot>$. Let the form $\omega_I := <I(\cdot),\cdot>$ be the usual K\"ahler form which is closed and parallel (with respect to the Levi-Civitta connection). Analogously defined forms $\omega_J$ and $\omega_K$ are also closed and parallel. A simple linear algebraic consideration (\cite{_Besse:Einst_Manifo_}) shows that the form $\Theta:=\omega_J+\sqrt{-1}\omega_K$ is of type $(2,0)$ and, being closed, this form is also holomorphic. Also, the form $\Theta$ is nowhere degenerate, as another linear algebraic argument shows. It is called {\bf the canonical holomorphic symplectic form of a manifold M}. Thus, for each hyperk\"ahler manifold $M$, and an induced complex structure $L$, the underlying complex manifold $(M,L)$ is holomorphically symplectic. The converse assertion is also true: \hfill \theorem \label{_symplectic_=>_hyperkahler_Proposition_} (\cite{_Beauville_}, \cite{_Besse:Einst_Manifo_}) Let $M$ be a compact holomorphically symplectic K\"ahler manifold with the holomorphic symplectic form $\Theta$, a K\"ahler class $[\omega]\in H^{1,1}(M)$ and a complex structure $I$. Let $n=\dim_{\Bbb C} M$. Assume that $\int_M \omega^n = \int_M (Re \Theta)^n$. Then there is a unique hyperk\"ahler structure $(I,J,K,(\cdot,\cdot))$ over $M$ such that the cohomology class of the symplectic form $\omega_I=(\cdot,I\cdot)$ is equal to $[\omega]$ and the canonical symplectic form $\omega_J+\sqrt{-1}\:\omega_K$ is equal to $\Theta$. \hfill \ref{_symplectic_=>_hyperkahler_Proposition_} immediately follows from the conjecture of Calabi, pro\-ven by Yau (\cite{_Yau:Calabi-Yau_}). \blacksquare \hfill Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$ with the group of unitary quaternions. This gives a canonical action of $SU(2)$ on the tangent bundle, and all its tensor powers. In particular, we obtain a natural action of $SU(2)$ on the bundle of differential forms. \hfill \lemma \label{_SU(2)_commu_Laplace_Lemma_} The action of $SU(2)$ on differential forms commutes with the Laplacian. {\bf Proof:} This is Proposition 1.1 of \cite{_Verbitsky:Symplectic_I_}. \blacksquare Thus, for compact $M$, we may speak of the natural action of $SU(2)$ in cohomology. \hfill Further in this article, we use the following statement. \hfill \lemma \label{_SU(2)_inva_type_p,p_Lemma_} Let $\omega$ be a differential form over a hyperk\"ahler manifold $M$. The form $\omega$ is $SU(2)$-invariant if and only if it is of Hodge type $(p,p)$ with respect to all induced complex structures on $M$. {\bf Proof:} This is \cite{_Verbitsky:Hyperholo_bundles_}, Proposition 1.2. \blacksquare \subsection{Trianalytic subvarieties in compact hyperk\"ahler manifolds.} In this subsection, we give a definition and a few basic properties of trianalytic subvarieties of hyperk\"ahler manifolds. We follow \cite{_Verbitsky:Symplectic_II_}. \hfill Let $M$ be a compact hyperk\"ahler manifold, $\dim_{\Bbb R} M =2m$. \hfill \definition\label{_trianalytic_Definition_} Let $N\subset M$ be a closed subset of $M$. Then $N$ is called {\bf trianalytic} if $N$ is a complex analytic subset of $(M,L)$ for any induced complex structure $L$. \hfill \hfill Let $I$ be an induced complex structure on $M$, and $N\subset(M,I)$ be a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$. Consider the homology class represented by $N$. Let $[N]\in H^{2m-2n}(M)$ denote the Poincare dual cohomology class. Recall that the hyperk\"ahler structure induces the action of the group $SU(2)$ on the space $H^{2m-2n}(M)$. \hfill \theorem\label{_G_M_invariant_implies_trianalytic_Theorem_} Assume that $[N]\in H^{2m-2n}(M)$ is invariant with respect to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic. {\bf Proof:} This is Theorem 4.1 of \cite{_Verbitsky:Symplectic_II_}. \blacksquare \remark \label{_triana_dim_div_4_Remark_} Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4. \hfill \definition \label{_generic_manifolds_Definition_} Let $M$ be a complex manifold admitting a hyperk\"ahler structure $\c H$. We say that $M$ is {\bf of general type} or {\bf generic} with respect to $\c H$ if all elements of the group \[ \bigoplus\limits_p H^{p,p}(M)\cap H^{2p}(M,{\Bbb Z})\subset H^*(M)\] are $SU(2)$-invariant. We say that $M$ is {\bf Mumford--Tate generic} if for all $n\in {\Bbb Z}^{>0}$, all the cohomology classes \[ \alpha \in \bigoplus\limits_p H^{p,p}(M^n)\cap H^{2p}(M^n,{\Bbb Z})\subset H^*(M^n) \] are $SU(2)$-invariant. In other words, $M$ is Mumford--Tate generic if for all $n\in {\Bbb Z}^{>0}$, the $n$-th power $M^n$ is generic. Clearly, Mumford--Tate generic implies generic. \hfill \proposition \label{_generic_are_dense_Proposition_} Let $M$ be a compact manifold, $\c H$ a hyperk\"ahler structure on $M$ and $S$ be the set of induced complex structures over $M$. Denote by $S_0\subset S$ the set of $L\in S$ such that $(M,L)$ is Mumford-Tate generic with respect to $\c H$. Then $S_0$ is dense in $S$. Moreover, the complement $S\backslash S_0$ is countable. {\bf Proof:} This is Proposition 2.2 from \cite{_Verbitsky:Symplectic_II_} \blacksquare \hfill \ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following immediate corollary: \corollary \label{_hyperkae_embeddings_Corollary_} Let $M$ be a compact holomorphically symplectic manifold. Assume that $M$ is of general type with respect to a hyperk\"ahler structure $\c H$. Let $S\subset M$ be closed complex analytic subvariety. Then $S$ is trianalytic with respect to $\c H$. \blacksquare \hfill In \cite{_Verbitsky:hypercomple_}, \cite{_Verbitsky:Desingu_}, \cite{_Verbitsky:DesinguII_}, we gave a number of equivalent definitions of a singular hyperk\"ahler and hypercomplex variety. We refer the reader to \cite{_Verbitsky:DesinguII_} for the precise definition; for our present purposes it suffices to say that all trianalytic subvarieties are hyperk\"ahler varieties. The following Desingularization Theorem is very useful in the study of trianalytic subvarieties. \hfill \theorem \label{_desingu_Theorem_} (\cite{_Verbitsky:DesinguII_}) Let $M$ be a hyperk\"ahler or a hypercomplex variety, $I$ an induced complex structure. Let \[ \widetilde{(M, I)}\stackrel n{\:\longrightarrow\:} (M,I)\] be the normalization of $(M,I)$. Then $\widetilde{(M, I)}$ is smooth and has a natural hyperk\"ahler structure $\c H$, such that the associated map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} (M,I)$ agrees with $\c H$. Moreover, the hyperk\"ahler manifold $\tilde M:= \widetilde{(M, I)}$ is independent from the choice of induced complex structure $I$. \blacksquare \subsection{Simple hyperk\"ahler manifolds} \definition \label{_simple_hyperkahler_mfolds_Definition_} (\cite{_Beauville_}) A connected simply connected compact hy\-per\-k\"ah\-ler manifold $M$ is called {\bf simple} if $M$ cannot be represented as a product of two hyperk\"ahler manifolds: \[ M\neq M_1\times M_2,\ \text{where} \ dim\; M_1>0 \ dim\; M_2>0 \] Bogomolov proved that every compact hyperk\"ahler manifold has a finite covering which is a product of a compact torus and several simple hyperk\"ahler manifolds. Bogomolov's theorem implies the following result (\cite{_Beauville_}): \hfill \theorem\label{_simple_mani_crite_Theorem_} Let $M$ be a compact hyperk\"ahler manifold. Then the following conditions are equivalent. \begin{description} \item[(i)] $M$ is simple \item[(ii)] $M$ satisfies $H^1(M, {\Bbb R}) =0$, $H^{2,0}(M) ={\Bbb C}$, where $H^{2,0}(M)$ is the space of $(2,0)$-classes taken with respect to any of induced complex structures. \end{description} \blacksquare \subsection{Hyperholomorphic bundles} \label{_hyperholo_Subsection_} This subsection contains several versions of a definition of hyperholomorphic connection in a complex vector bundle over a hyperk\"ahler manifold. We follow \cite{_Verbitsky:Hyperholo_bundles_}. \hfill Let $B$ be a holomorphic vector bundle over a complex manifold $M$, $\nabla$ a connection in $B$ and $\Theta\in\Lambda^2\otimes End(B)$ be its curvature. This connection is called {\bf compatible with a holomorphic structure} if $\nabla_X(\zeta)=0$ for any holomorphic section $\zeta$ and any antiholomorphic tangent vector $X$. If there exist a holomorphic structure compatible with the given Hermitian connection then this connection is called {\bf integrable}. \hfill One can define a {\bf Hodge decomposition} in the space of differential forms with coefficients in any complex bundle, in particular, $End(B)$. \hfill \theorem \label{_Newle_Nie_for_bu_Theorem_} Let $\nabla$ be a Hermitian connection in a complex vector bundle $B$ over a complex manifold. Then $\nabla$ is integrable if and only if $\Theta\in\Lambda^{1,1}(M, \operatorname{End}(B))$, where $\Lambda^{1,1}(M, \operatorname{End}(B))$ denotes the forms of Hodge type (1,1). Also, the holomorphic structure compatible with $\nabla$ is unique. {\bf Proof:} This is Proposition 4.17 of \cite{_Kobayashi_}, Chapter I. $\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill \definition \label{_hyperho_conne_Definition_} Let $B$ be a Hermitian vector bundle with a connection $\nabla$ over a hyperk\"ahler manifold $M$. Then $\nabla$ is called {\bf hyperholomorphic} if $\nabla$ is integrable with respect to each of the complex structures induced by the hyperk\"ahler structure. As follows from \ref{_Newle_Nie_for_bu_Theorem_}, $\nabla$ is hyperholomorphic if and only if its curvature $\Theta$ is of Hodge type (1,1) with respect to any of complex structures induced by a hyperk\"ahler structure. As follows from \ref{_SU(2)_inva_type_p,p_Lemma_}, $\nabla$ is hyperholomorphic if and only if $\Theta$ is a $SU(2)$-invariant differential form. \hfill \example \label{_tangent_hyperholo_Example_} (Examples of hyperholomorphic bundles) \begin{description} \item[(i)] Let $M$ be a hyperk\"ahler manifold, $TM\otimes_{\Bbb R} {\Bbb C}$ a complexification of its tangent bundle equipped with Levi--Civita connection $\nabla$. Then $\nabla$ is integrable with respect to each induced complex structure, and hence, Yang--Mills. \item[(ii)] For $B$ a hyperholomorphic bundle, all its tensor powers are also hyperholomorphic. \item[(iii)] Thus, the bundles of differential forms on a hyperk\"ahler manifold are also hyperholomorphic. \end{description} \subsection{Stable bundles and Yang--Mills connections.} This subsection is a compendium of the most basic results and definitions from the Yang--Mills theory over K\"ahler manifolds, concluding in the fundamental theorem of Uhlenbeck--Yau \cite{_Uhle_Yau_}. \hfill \definition\label{_degree,slope_destabilising_Definition_} Let $F$ be a coherent sheaf over an $n$-dimensional compact K\"ahler manifold $M$. We define $\deg(F)$ as \[ \deg(F)=\int_M\frac{ c_1(F)\wedge\omega^{n-1}}{vol(M)} \] and $\text{slope}(F)$ as \[ \text{slope}(F)=\frac{1}{\text{rank}(F)}\cdot \deg(F). \] The number $\text{slope}(F)$ depends only on a cohomology class of $c_1(F)$. Let $F$ be a coherent sheaf on $M$ and $F'\subset F$ its proper subsheaf. Then $F'$ is called {\bf destabilizing subsheaf} if $\text{slope}(F') \geq \text{slope}(F)$ A holomorphic vector bundle $B$ is called {\bf stable} \footnote{In the sense of Mumford-Takemoto} if it has no destabilizing subsheaves. \hfill Later on, we usually consider the bundles $B$ with $deg(B)=0$. \hfill Let $M$ be a K\"ahler manifold with a K\"ahler form $\omega$. For differential forms with coefficients in any vector bundle there is a Hodge operator $L: \eta{\:\longrightarrow\:}\omega\wedge\eta$. There is also a fiberwise-adjoint Hodge operator $\Lambda$ (see \cite{_Griffi_Harri_}). \hfill \definition \label{Yang-Mills_Definition_} Let $B$ be a holomorphic bundle over a K\"ahler manifold $M$ with a holomorphic Hermitian connection $\nabla$ and a curvature $\Theta\in\Lambda^{1,1}\otimes End(B)$. The Hermitian metric on $B$ and the connection $\nabla$ defined by this metric are called {\bf Yang-Mills} if \[ \Lambda(\Theta)=constant\cdot \operatorname{Id}\restrict{B}, \] where $\Lambda$ is a Hodge operator and $\operatorname{Id}\restrict{B}$ is the identity endomorphism which is a section of $End(B)$. Further on, we consider only these Yang--Mills connections for which this constant is zero. \hfill A holomorphic bundle is called {\bf indecomposable} if it cannot be decomposed onto a direct sum of two or more holomorphic bundles. \hfill The following fundamental theorem provides examples of Yang-\--Mills \linebreak bundles. \theorem \label{_UY_Theorem_} (Uhlenbeck-Yau) Let B be an indecomposable holomorphic bundle over a compact K\"ahler manifold. Then $B$ admits a Hermitian Yang-Mills connection if and only if it is stable, and this connection is unique. {\bf Proof:} \cite{_Uhle_Yau_}. \blacksquare \hfill \proposition \label{_hyperholo_Yang--Mills_Proposition_} Let $M$ be a hyperk\"ahler manifold, $L$ an induced complex structure and $B$ be a complex vector bundle over $(M,L)$. Then every hyperholomorphic connection $\nabla$ in $B$ is Yang-Mills and satisfies $\Lambda(\Theta)=0$ where $\Theta$ is a curvature of $\nabla$. \hfill {\bf Proof:} We use the definition of a hyperholomorphic connection as one with $SU(2)$-invariant curvature. Then \ref{_hyperholo_Yang--Mills_Proposition_} follows from the \hfill \lemma \label{_Lambda_of_inva_forms_zero_Lemma_} Let $\Theta\in \Lambda^2(M)$ be a $SU(2)$-invariant differential 2-form on $M$. Then $\Lambda_L(\Theta)=0$ for each induced complex structure $L$.\footnote{By $\Lambda_L$ we understand the Hodge operator $\Lambda$ associated with the K\"ahler complex structure $L$.} {\bf Proof:} This is Lemma 2.1 of \cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare \hfill Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure. For any stable holomorphic bundle on $(M, I)$ there exists a unique Hermitian Yang-Mills connection which, for some bundles, turns out to be hyperholomorphic. It is possible to tell when this happens (though in the present paper we never use this knowledge). \hfill \theorem Let $B$ be a stable holomorphic bundle over $(M,I)$, where $M$ is a hyperk\"ahler manifold and $I$ is an induced complex structure over $M$. Then $B$ admits a compatible hyperholomorphic connection if and only if the first two Chern classes $c_1(B)$ and $c_2(B)$ are $SU(2)$-invariant.\footnote{We use \ref{_SU(2)_commu_Laplace_Lemma_} to speak of action of $SU(2)$ in cohomology of $M$.} {\bf Proof:} This is Theorem 2.5 of \cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare \section{Trianalytic subvarieties of powers of K3 surfaces} \label{_appendix_subva_produ_Section_} \subsection{Trianalytic subvarieties of a product of a K3 surface with itself} Let $M$ be any manifold, $M^n = M\times ... \times M$ its $n$-th product with itself. We define the ``natural'' subvarieties of $M$, recursively, as follows. \begin{equation}\label{_recu_subva_in_M^n_Equation_} \begin{minipage}[m]{0.8\linewidth} \begin{description} \item[(i)] Natural subvarieties of $M$ are $M$ and points. \item[(ii)] Let $Z\subset M^n$ by a natural subvariety. The following subvarieties of $M^{n+1}$ are natural. \begin{description} \item[a.] $Z_M := Z\times M \subset M^n \times M = M^{n+1}$ \item[b.] $Z_t:= Z\times \{t\} \subset M^n \times M= M^{n+1}$, depending on a point $t\in M$. \item[c.] $Z_i := \bigg\{ (m_1, ... m_{n+1}) \in Z\times M \;\;|\;\; m_i=m_{n+1} \bigg\}$ depending on a number $i\in \{ 1, ..., n\}$ \end{description} \end{description} \end{minipage} \end{equation} The main result of this section is the following theorem. \hfill \theorem \label{_subva_product_Therem_} Let $M$ be a hyperk\"ahler K3-surface which has no hyperk\"ahler automorphisms, and $X\subset M^{n}$ an irreducible trianalytic subvariety. Then $X$ is ``natural'', in the sense of \eqref{_recu_subva_in_M^n_Equation_}. \hfill {\bf Proof:} Let \[ \check\Pi_{n+1}:\; M^{n+1}{\:\longrightarrow\:} M^n\] be the natural projection $m_1, ..., m_{n+1} {\:\longrightarrow\:} m_1, ..., m_n$. Clearly, $\check\Pi_{n+1}(X)$ is irreducible and trianalytic. Using induction, we may assume that \begin{equation}\label{_indu_assu_for_subva_in_product_Equation_} \begin{minipage}[m]{0.8\linewidth} a trianalyiic subvariety $X\subset M^k$ is natural, in the sense of \eqref{_recu_subva_in_M^n_Equation_}, for $k\leq n$. \end{minipage} \end{equation} Clearly, by \eqref{_indu_assu_for_subva_in_product_Equation_} $\check\Pi_{n+1}(X)$ is of the type \eqref{_recu_subva_in_M^n_Equation_}. All varieties $Z$ of type \eqref{_recu_subva_in_M^n_Equation_} are isomorphic to $M^k$, for $k=\dim_{\Bbb H} Z$. Thus, $X$ is realized as a subvariety in \[ \check\Pi_{n+1}(X)\times M = M^{\dim_{\Bbb H}\check\Pi_{n+1}(X)+1}. \] Unless $\dim_{\Bbb H} \check\Pi_{n+1}(X) =n$, \eqref{_indu_assu_for_subva_in_product_Equation_} implies that $X$ is a ``natural'' subvariety. Thus, to prove \eqref{_subva_product_Therem_}, we may assume that $\check\Pi_{n+1}(X) = M^n$. For a point $t\in M$, let $X_t:= \bigg\{ (m_1, ... m_{n+1}) \in X \;\; |\;\; m_{n+1}=t \bigg\}$. The subvariety $X_t\subset M^n$ is not necessarily irreducible, because the components of $X_t$ may ``flow together'' as $t$ changes, so that $X = \bigcup_{t\in M} X_t$ is still irreducible, while the $X_t$'s are not. However, all components of $X_t$ must be deformationally equivalent in the class of ``natural'' subvarieties of $M^n$, in order for $X$ to be irreducible. Since $X$ is irreducible and $X= \cup_t X_t$, then either $X_t = X$ for one value of $t$ (and in this case $X= M^{n}\times \{t\}$), or $X_t \neq\emptyset$ for $t$ in a positive-dimensional trianalytic subset of $M$. Since $\dim_{\Bbb H} M =1$, this subset coinsides with $M$. Using \eqref{_indu_assu_for_subva_in_product_Equation_}, we obtain that all irreducible components of $X_t$ are of the type \eqref{_recu_subva_in_M^n_Equation_}. All ``natural'' subvarieties of $M^n$ of complex codimension $2=\dim_{\Bbb C} M$ are given by either $m_i=m_j$ for some distinct fixed indices $i, j$, or by $m_i=t$ for a fixed index $i$ and a fixed point $t\in M$. We proceed on case-by-case basis. \begin{description} \item[(i)] For some $t$, $X_t$ contains a component $X_t^{i,j}$ given by $m_i=m_j$ for distinct fixed indices $i, j$. Since $X_t^{i,j}$ is rigid in the class of natural subvarieties, and $X$ is irreducible, this implies that $X_t$ contains $M^n_{i,j}\times\{t\}$ for all $t$, where $M^n_{i,j}\subset M^n$ is a subvariety given by $m_i=m_j$. Since $\dim X= M^n_{i,j} \times M$, $X$ irreducible and $M^n_{i,j} \times M\subset X$, this implies that $X=M^n_{i,j} \times M$. This proves \ref{_subva_product_Therem_} (case (i)). \item[(ii)] For some $t$, $X_t$ contains a component $X^i_t(m)$, given by $m_i=m$, for a fixed index $i$ and a fixed point $m\in M$. Deforming $X^i_t(m)$ in the class of natural subvarieties, we obtain again $X^i_t(m')$, with different $m'$. Taking a union of all $X^i_t(m)\subset X_t$, for some fixed $i$ and varying $t$ and $m$, we obtain a closed subvariety of $X$ of the same dimension as $X$. Since $X$ is irreducible, {\it all} components of $X_t$ are given by $m_i=m$, for a fixed index $i$ and a fixed point $m\in M$. Consider a trianalytic subvariety $Z\subset M^2$, \[ Z:= \{ (m, t) \in M^2 \;\;|\;\; X^i_t(m)\subset X_t \} \] To prove \ref{_subva_product_Therem_} (case (ii)), it suffices to show that $Z$ is natural, in the sense of \eqref{_recu_subva_in_M^n_Equation_}. We reduced \ref{_subva_product_Therem_} to the case of trianalytic subvarieties in $M^2$. \end{description} The following lemma finishes the proof of \ref{_subva_product_Therem_}. \hfill \lemma \label{_subva_M^2_Lemma_} Let $M$ be a hyperk\"ahler K3-surface which has no hyperk\"ahler automorphisms, and $X\subset M^2$ a closed irreducible trianalytic subvariety of $M^2$. Then $X$ is a ``natural'' subvariety of $M^2$, in the sense of \eqref{_recu_subva_in_M^n_Equation_}. \hfill {\bf Proof:} Let $\pi_1, \pi_2:\; M^2 {\:\longrightarrow\:} M$ be the natural projections. Assume that neither $\pi_1(X)$ nor $\pi_2(X)$ is a point, and $X\subsetneqq M^2$ (otherwise $X$ obviously satisfies \eqref{_recu_subva_in_M^n_Equation_}). Let $\tilde X\stackrel n {\:\longrightarrow\:} X$ be the desingularization of $X$, given by \ref{_desingu_Theorem_}. Consider the maps $p_i:\; \tilde X {\:\longrightarrow\:} M$, $i= 1,2$, given by $p_i:= n\circ \pi_i$. Since $\dim X = \dim M$, and $p_i$ is non-trivial, these maps have non-degenerate Jacobians in general point. Fix an induced complex structure $I$ on $M$, and consider $X$, $\tilde X$, $M^2$ as complex varieties and $p_i$ as holomorphic maps. Let $\Theta_M$ be the holomorphic symplectic form of $M$. Then $p_i^* \Theta_M$ gives a section of the canonical class of $\tilde X$. Since $\tilde X$ is compact and hyperk\"ahler, any non-zero section of the canonical class is nowhere degenerate. Thus, $p_i^* \Theta_M$ is nowhere degenerate, and the Jacobian of $p_i$ nowhere vanishes. Therefore, $p_i$ is a covering. Since $X$ is irreducible, $\tilde X$ is connected, and since $M$ is simply connected, $p_i$ is isomorphism. Since $M$ has no hyperk\"ahler automorphisms, except identity, $X$ is a graph of an identity map. This proves \ref{_subva_M^2_Lemma_} and \ref{_subva_product_Therem_}. \blacksquare \hfill \corollary \label{_complex_subva_M^n_Corollary_} Let $M$ be a complex K3 surface with no complex automorphisms. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is Mumford-Tate generic with respect to $\c H$. Let $X$ be an irreducible complex subvariety of $M^n$. Then $X$ is a ``natural'' subvariety of $M^n$, in the sense of \eqref{_recu_subva_in_M^n_Equation_}. {\bf Proof:} By \ref{_hyperkae_embeddings_Corollary_}, $X$ is trianalytic. Now \ref{_complex_subva_M^n_Corollary_} is implied by \ref{_subva_product_Therem_}. \blacksquare \subsection{Subvarieties of symmetric powers of varieties} \label{_subva_of_symme_special_Subsection_} In this section, we fix the notation regarding the ``natural'' subvarieties of the symmetric powers of complex varieties. \hfill Let $M$ be a complex variety and $M^{(n)}$ its symmetric power, $M^{(n)}=M^n/S_n$. The space $M^{(n)}$ has a natural stratification by {\bf diagonals} $\Delta_{(\alpha)}$, which are numbered by Young diagrams \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n.\] This stratification is constructed as follows. Let $\sigma:\; M^n {\:\longrightarrow\:} M^{(n)}$ be the natural finite map (a quotient by the symmetric group). Then \begin{equation} \label{_Del-a_alpha_definition_Equation_} \begin{split} \Delta_{(\alpha)} := & \sigma \left (\left\{ (x_1, x_2, ... , x_n)\in M^n \ \ \bigg | \;\; \right.\right.\\ & x_1 = x_2 = ... = x_{n_1}, \ \ x_{n_1+1} = x_{n_1+2} = ... = x_{n_1+n_2}, \ \\ & \left.\left. \vphantom{\ \ \bigg | \;\;} ..., \ \ x_{\sum_{i=1}^{k-1} n_i+1} = x_{\sum_{i=1}^{k-1} n_i+2} = ... = x_n \right\} \right) \end{split} \end{equation} where $\sigma:\; M^n {\:\longrightarrow\:} M^{(n)}$ is the natural quotient map. \hfill Consider a Young diagram $\alpha$, \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n. \] As in \eqref{_Del-a_alpha_definition_Equation_}, $\alpha$ corresponds to a diagonal $\Delta_{(\alpha)}$, which is a closed subvariety of $M^{(n)}$. Fix a subset $\c A \subset \{1, ..., k\}$, and let $\phi:\; \c A {\:\longrightarrow\:} M$ be an arbitrary map. Then $\Delta_{(\alpha)}(\c A, \phi)\subset \Delta_{(\alpha)}$ is given by \begin{equation} \label{_Delta(A,phi)_alpha_definition_Equation_} \begin{split} \Delta_{(\alpha)}(\c A, \phi) := & \sigma \left (\left\{ (x_1, x_2, ... , x_n)\in M^n \ \ \bigg | \;\; \right.\right.\\ & x_1 = x_2 = ... = x_{n_1}, \ \ x_{n_1+1} = x_{n_1+2} = ... = x_{n_1+n_2}, \ \\ & ..., \ \ x_{\sum_{i=1}^{k-1} n_i+1} = x_{\sum_{i=1}^{k-1} n_i+2} = ... = x_n,\\ & \text{\ \ and\ \ } \forall i \in \c A, \ x_i = \phi(i) \left.\left. \vphantom{\ \ \bigg | \;\;} \right\} \right), \end{split} \end{equation} where $\sigma:\; M^n {\:\longrightarrow\:} M^{(n)}$ is the standard projection. For ``sufficiently generic'' K3 surfaces, all complex subvarieties in $M^n$ are given by \ref{_complex_subva_M^n_Corollary_}. From \ref{_complex_subva_M^n_Corollary_}, it is easy to deduce the following result. \hfill \proposition\label{_subva_in_M^(n)_Proposition_} Let $M$ be a complex K3 surface with no complex automorphisms. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is Mumford-Tate generic with respect to $\c H$. Let $X$ be an irreducible complex subvariety of $M^n$. Then $X= \Delta_{(\alpha)}(\c A, \phi)$ for appropriate $\alpha, \c A, \phi$. \blacksquare \section{Hilbert scheme of points} \label{_Hilbert_sche_Section_} For the definitions and results related to the Hilbert scheme of points on a surface, see the excellent lecture notes of H. Nakajima \cite{_Nakajima_}. \subsection{Symplectic structure of the Hilbert scheme} \label{_symple_on_Hi_Subsection_} \definition Let $M$ be a complex surface. The $n$-th {\bf Hilbert scheme of points}, also called {\bf Hilbert scheme} of $M$ (denoted by $M^{[n]})$ is the scheme classifying the zero-dimensional subschemes of $M$ of length $n$. \hfill There is a natural projection $\pi:\; M^{[n]} {\:\longrightarrow\:} M^{(n)}$ associating to a subscheme $S\subset M$ the set of points $x_i\in Sup(S)$ of support of $S$, taken with the multiplicities equal to the length of $S$ in $x_i$. It is well-known that a Hilbert scheme of a smooth surface is a smooth manifold, and the fibers of $\pi$ are irreducible and reduced (see, e. g. \cite{_Nakajima_}). \hfill For our purposes, the most important property of the Hilbert scheme is the existence of the non-degenerate holomorphic symplectic structure, for $M$ holomorphically symplectic. \hfill Let $X$ be an irreducible complex analytic space, which is reduced in generic point, and $\Omega^1 X$ the sheaf of Ka\"hler differentials on $X$. We denote the exterior square $\Lambda^2_{{\cal O}_X}\Omega^1 X$ by $\Omega^2 X$. The sections of $\Lambda^2_{{\cal O}_X}\Omega^1 X$ are called {\bf 2-forms on $X$}. We say that two-forms $\omega_1, \omega_2$ are {\bf equal up to a torsion} if $\omega_1=\omega_2$ in the generic point of $X$. \hfill \proposition \label{_Hilbert_symple_Proposition_} (Beauville) Let $M$ be a smooth complex surface equipped with a nowhere degenerate holomorphic 2-form. Then \begin{description} \item[(i)] the Hilbert scheme $M^{[n]}$ is equipped with a natural, nowhere degenerate holomorphic 2-form $\Theta_{M^{[n]}}$. \item[(ii)] Consider the Cartesian square \begin{equation}\label{_M^[n]_Cartesian_Equation_} \begin{CD} \tilde M^{[n]} @>{\tilde \pi}>> M^{n} \\ @VV{\tilde \sigma}V @VV{\sigma}V \\ M^{[n]} @>{\pi}>> M^{(n)} \end{CD} \end{equation} Let $\Theta_{M^{n}}$ be the natural symplectic form on $M^n$. Then the pullback $\tilde\sigma^* \Theta_{M^{[n]}}$ is equal to the pullback $\tilde \pi^* \Theta_{M^{n}}$, outside of the subvariety $D\subset \tilde M^{[n]}$ of codimension 2. \item[(iii)] The complex analytic space $\tilde M^{[n]}$ is irreducible, and $\tilde\sigma^* \Theta_{M^{[n]}}$ is equal up to a torsion to $\tilde \pi^* \Theta_{M^{n}}$. \end{description} \hfill {\bf Proof:} In \cite{_Beauville_}, A. Beauville proved the conditions (i) and (ii). Clearly, (ii) implies that $\tilde\sigma^* \Theta_{M^{[n]}}$ is equal up to a torsion to $\tilde \pi^* \Theta_{M^{n}}$, assuming that $\tilde M^{[n]}$ is irreducible. It remains to show that $\tilde M^{[n]}$ is irreducible. \hfill The following argument can be easily generalized to a more general type of Cartesian squares. We only use that the arrow $\pi$ is rational, $\sigma$ is finite and generically etale, and the varieties $M^n$, $M^{[n]}$ are irreducible. \hfill Let $U$ be the general open stratum of $\tilde M^{[n]}$, \[ U := \tilde M^{[n]}\backslash \bigcup_\alpha \tilde \sigma^{-1}\Delta_{[\alpha]} \] The map $\tilde \pi:\; U {\:\longrightarrow\:} M^n$ is an open embedding. Therefore, the variety $U$ is irreducible. To prove that $\tilde M^{[n]}$ is irreducible, we need to show that for all points $x\in \tilde M^{[n]}$, there exists a sequence $\{x_i\}\subset U$ which converges to $x$. Since $M^{[n]}$ is irreducible, there exists a sequence $\{ \underline{x_i}\}\in \tilde\sigma(U)$ converging to $\tilde \sigma(x)$. Consider the sequence $\{\pi(\underline{x_i})\} \subset M^{(n)}$. The general stratum $\tilde \pi(U)$ of $M^n$ is identified with $U$, since $\tilde \pi\restrict U$ is an isomorphism. Lifting $\{\pi(\underline{x_i})\}$ to $M^n$, we obtain a sequence $\{x_i\}\subset \tilde \pi(U)=U$. Taking a subsequence of $\{x_i\}$, we can assure that it converges to a point in a finite set $\tilde\sigma^{-1}(\tilde\sigma(x))$. By an appropriate choice of the lifting, we obtain a sequence converging to any point in $\tilde\sigma^{-1}(\tilde\sigma(x))$, in particular, $x$. This proves that $\tilde M^{[n]}$ is irreducible. \blacksquare \hfill \remark {}From \ref{_Hilbert_symple_Proposition_} and Calabi-Yau theorem (\ref{_symplectic_=>_hyperkahler_Proposition_}), it follows immediately that $M^{[n]}$ admits a hyperk\"ahler structure, if $M$ is a K3 surface or a compact torus. However, this hyperk\"ahler structure is not in any way related to the hyperk\"ahler structures on $M$. \hfill \remark In the preliminary version of \cite{_Nakajima_}, it was stated without proof that $\tilde\sigma^* \Theta_{M^{[n]}} =\tilde \pi^* \Theta_{M^{n}}$. This statement seems to be subtle, and I was unable to find the proof. However, a weaker version of this equality can be proven. \hfill \proposition \label{_simplec_on_subva_Proposition_} Let $A_{M^{[n]}}:= \tilde\sigma^* \Theta_{M^{[n]}}$, $A_{M^n}:= \tilde \pi^* \Theta_{M^{n}}$, be the forms defined in \ref{_Hilbert_symple_Proposition_}. Then for all closed complex subvarieties $X\stackrel i \hookrightarrow \tilde M^{[n]}$, the 2-forms $i^* A_{M^{[n]}}$, $i^*A_{M^n}\in \Omega^2 X$ are equal outside of singularities of $X$. \hfill {\bf Proof:}\footnote{The proof is based on ideas of D. Kaledin.} The forms $A_{M^{[n]}}$, $A_{M^n}$ are equal up to torsion. Therefore, their difference lies in the torsion subsheaf of $\Omega^2 \tilde M^{[n]}$. To prove that the 2-forms $i^* A_{M^{[n]}}$, $i^*A_{M^n}\in \Omega^2 X$ are equal outside of singularities of $X$ it suffices to show the following: for all torsion sections $\omega \in \Omega^2 \tilde M^{[n]}$, the pullback $i^*\omega$ lies in the torsion of $\Omega^2 X$ Let $\Delta_{[\alpha]}$ be a stratum of $M^{[n]}$, defined by a Young diagram $\alpha$ as in Subsection \ref{_Young_Hilb_Subsection_}, and $X\stackrel i \hookrightarrow \tilde M^{[n]}$ an irreducible component of $\tilde \sigma^{-1}(X)$, considered as a complex subvariety of $\tilde M^{[n]}$. As a first step in proving \ref{_simplec_on_subva_Proposition_}, we show that for all torsion sections $\omega \in \Omega^2 \tilde M^{[n]}$, the pullback $i^*\omega$ lies in the torsion of $\Omega^2 X$, for this particular choice of $X$. Consider a generic point $x\in\Delta_{[\alpha]}$, and let $V\subset M^{[n]}$ be a neighbourhood of $x$ in $M^{[n]}$, $U\subset \Delta_{[\alpha]}$ be a neighbourhood of $x\in\Delta_{[\alpha]}$. For an appropriate choice of $V$, $U$, these varieties are equipped with a locally trivial fibration $V \stackrel p{\:\longrightarrow\:} U$, inverse to a natural embedding $U\hookrightarrow V$. Assume also that $U$ consists entirely of generic points of $\Delta_{[\alpha]}$. Let $\tilde x$ be a point of $\tilde \sigma^{-1}(x) \cap X$, and $\tilde V$ be a component of $\tilde \sigma^{-1}(V)$ which contains $\tilde x$, and $\tilde U:= \tilde \sigma^{-1}(U) \cap \tilde V$. Since $U$ consists of generic points of $\Delta_{[\alpha]}$, and $\tilde \sigma$ is finite, the map $\tilde \sigma:\; \tilde U {\:\longrightarrow\:} U$ is etale. Therefore, $\tilde V$ is equipped with a locally trivial fibration $\tilde V \stackrel {\tilde p}{\:\longrightarrow\:} \tilde U$, inverse to a natural embedding $\tilde U\hookrightarrow\tilde V$. Using this fibration, we decompose the sheaf of differentials on $\tilde V$ as follows: \[ \Omega^1 \tilde V = {\tilde p}^*\Omega^1 \tilde U \oplus \Omega^1_{\tilde p} \tilde V \] where $\Omega^1_{\tilde p} \tilde V$ is the sheaf of relative differentials of $\tilde V$ along $\tilde p$. Clearly, for all sections $\omega \in \Omega^1_{\tilde p} \tilde V$, the pullback of $\omega$ under $\tilde U \hookrightarrow\tilde V$ is zero. On the other hand, $\tilde U$ is smooth, and therefore ${\tilde p}^*\Omega^1 \tilde U$ has no torsion. Thus, the torsion-component of $\Omega^1 \tilde V$ is contained in $\Omega^1_{\tilde p} \tilde V$ and vanishes on $\tilde U$. A similar argument implies that the torsion-component of $\Omega^2 \tilde V$ is contained in \[ \Omega^2_{\tilde p} \tilde V\oplus {\tilde p}^*\Omega^1 \tilde U \otimes \Omega^1_{\tilde p} \tilde V \subset \Omega^2 \tilde V \] and also vanishes on $\tilde U$. Therefore, all torsion components on $\Omega^2 \tilde M^{[n]}$ vanish on $X$, where $X$ is an irreducible component of the preimage of the stratum of $M^{[n]}$. Consider a stratification of $\tilde M^{[n]}$ by such $X$'s. For any subvariety $Y\subset \tilde M^{[n]}$, let $\tilde \Delta_{[\alpha]}$ be the smallest stratum of $\tilde M^{[n]}$ which contains $Y$. Then, the set $Y_g$ of generic points of $Y$ is contained in the set $\tilde U_{[\alpha]}$ of the generic points of $\tilde \Delta_{[\alpha]}$. But, as we have seen, the restrictions of the forms $A_{M^{[n]}}$, $A_{M^n}$ to $\tilde U_{[\alpha]}$ coinside. Therefore, restrictions of $A_{M^{[n]}}$, $A_{M^n}$ to $Y_g \subset \tilde U_{[\alpha]}$ are equal. This proves \ref{_simplec_on_subva_Proposition_}. \blacksquare \hfill In the situation similar to the above, we will say that the forms $A_{M^{[n]}}$, $A_{M^n}$ are {\bf equal on subvarieties}. \hfill From the fact that $A_{M^{[n]}}$, $A_{M^n}$ are equal on subvarieties we immediately obtain the following. \hfill \claim \label{_triana_finite_in_gene_Claim_} Let $X\subset M^{[n]}$ be a complex subvariety of the Hilbert scheme. Assume that the holomorphic symplectic form is non-degenerate in the generic point of $X$ (this happens, for instance, when $X$ is trianalytic). Then the restriction $\pi\restrict X$ of $\pi:\;M^{[n]}{\:\longrightarrow\:} M^{(n)}$ to $X$ is finite in generic point of $X$ {\bf Proof:} This result is easily implied by \ref{_simplec_on_subva_Proposition_}. For details of the proof, the reader is referred to \ref{_holo_symple_semismall_Proposition_}, \ref{_triana_speci_semismall_Corollary_}. \blacksquare \hfill The rest of this section is not used directly anywhere in this paper. A reader who does not like perverse sheaves is invited to skip the rest and proceed to Section \ref{_unive_subva_Section_}. \subsection{Cohomology of the Hilbert scheme} For perverse sheaves, we freely use terminology and results of \cite{_Asterisque_100_} . For the computation of cohomology of the Hilbert scheme via perverse sheaves, see \cite{_Nakajima_}. \hfill \definition Let $X$ be an irreducible complex variety and $F$ a perverse sheaf on $M$. The $F$ is called {\bf a Goresky-MacPherson sheaf}, or {\bf a sheaf of GM-type} if it has no proper subquotient perverse sheaves with support in $Z\subsetneq X$. For an arbitrary perverse sheaf $F$, consider the Goresky-MacPherson subquotient $F_{GM}$ of $F$, such that for a nonempty Zariski open set $U\subset X$, $F\restrict U = F_{GM}\restrict U$. Such subquotient is obviously unique; we call it the Goresky-MacPherson extension of $F$. The Intersection Cohomology sheaf $IC(X)$ is the Goresky-MacPherson extension of the constant sheaf ${\Bbb C}_X$. \hfill \definition Let $X$ be a complex variety. The $X$ is called {\bf homology rational} if the constant sheaf ${\Bbb C}_X$ on $X$, considered as a complex of sheaves, is isomorphic to the Intersection Cohomology perverse sheaf. The variety $X$ is called {\bf weakly homology rational} if the Intersection Cohomology sheaf (considered as a complex of sheaves) is constructible (i. e., the cohomology of this complex of sheaves are zero in all but one degree). \hfill \remark Clearly, for a homology rational variety, the Intersection Cohomology coinsides with the standard cohomology. \hfill \claim \label{_quotient_by_finite_homo_ratio_Claim_} Let $f:\; X {\:\longrightarrow\:} Y$ be a finite dominant morphism of complex varieties. Assume that $X$ is smooth. Then $Y$ is weakly homology rational. Moreover, if $Y$ is also normal, then $Y$ is homology rational. {\bf Proof:} Well known. \blacksquare \hfill \definition Let $\pi:\; X {\:\longrightarrow\:} Y$ be a morphism of complex varieties. Consider a stratification $\{C_i\}$ of $Y$, defined in such a way that the restriction of $f$ to the open strata $\pi^{-1}(C_i)$ is a locally trivial fibration. The map $f$ is called {\bf a semismall resolution of $Y$} if $X$ is smooth, and for all $i$, the dimension of the fibers of $\pi:\; \pi^{-1}(C_i) {\:\longrightarrow\:} C_i$ is at most half the codimension of $C_i \subset Y$. \hfill \proposition \label{_cohomo_semismall_gene_Proposition_} Let $\pi:\; X {\:\longrightarrow\:} Y$ be a semismall resolution, associated with the stratification $Y = \coprod C_i$. Let $Y_i$ be the closed strata of the corresponding stratification of $Y$, $Y_i = \bar{C_i}$, and $X_i$ the corresponding closed strata of $X$, $Y_i = \pi^{-1}(X_i)$. Consider the Goresky-MacPherson sheaf $V_i$ associated with the sheaf $R^{l_i}\pi_* {\Bbb C}_{X_i}$ where $l_i=\frac{\operatorname{codim}_{\Bbb C} Y_i}{2}$ (for $\operatorname{codim}_{\Bbb C} Y_i$ odd, we put $V_i=0$) and ${\Bbb C}_{X_i}$ a constant sheaf on $X_i$. Assume that $X$ is a K\"ahler manifold. Then $R^\bullet \pi_* {\Bbb C}_X$ is a direct sum of the perverse sheaves $V_i$ shifted by $l_i$. {\bf Proof:} In algebraic situation, this is proven using the weight arguments and $l$-adic cohomology (\cite{_Asterisque_100_}). To adapt this argument in K\"ahler situation, one uses the mixed Hodge modules of M. Saito, \cite{_Saito_}. \blacksquare \hfill \theorem \label{_Hilbert_sche_semismal_cohomo_Theorem_} \cite{_Gott_Sorg_} Let $M$ be a complex surface, $M^{[n]}$ its Hilbert scheme, $M^{(n)}$ the symmetric power and \[ \pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}\] the standard projection map. Consider the stratification of $M^{(n)}$ by the diagonals $\Delta_{(\alpha)}$, parametrized by the Young diagrams $\alpha$ (see \eqref{_Del-a_alpha_definition_Equation_}). Then $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ is a semismall resolution associated with this stratification. Moreover, the sheaves $V_i$ of \ref{_cohomo_semismall_gene_Proposition_} are constant sheaves ${\Bbb C}_{\Delta_{(\alpha)}}$. \footnote{The varieties $\Delta_{(\alpha)}$ are normal and obtained as quotients of smooth manifolds by group action. Thus, all $\Delta_{(\alpha)}$ are homology rational by \ref{_quotient_by_finite_homo_ratio_Claim_}. Thus, the sheaves ${\Bbb C}_{\Delta_{(\alpha)}}$ are GM-type.} {\bf Proof:} The map $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ is a semismall resulution, which is easy to check by counting dimensions (see \ref{_holo_symple_semismall_Proposition_} for a conceptual proof). Now, the first assertion of \ref{_Hilbert_sche_semismal_cohomo_Theorem_} is a straightforward application of \ref{_cohomo_semismall_gene_Proposition_}. The second assertion is much more subtle; see \cite{_Nakajima_} for details and further reference. \blacksquare \hfill \corollary \label{_cohomo_of_Hilbert_explicit_Corollary_} The $i$-th cohomology of $M^{[n]}$ are isomorphic to \begin{equation} \label{_decompo_of_cohomo_Equation_} \bigoplus_\alpha {\text{\Large \it H}} ^{\text{\large i} + \frac{\operatorname{codim}\Delta_{(\alpha)}}{2}}\left(\Delta_{(\alpha)}\right) \end{equation} {\bf Proof:} By \ref{_Hilbert_sche_semismal_cohomo_Theorem_}, \[ R^\bullet\pi_* {\Bbb C}_{M^{[n]}} = \oplus {\Bbb C}_{\Delta_{(\alpha)}} \left[ \frac{\operatorname{codim}\Delta_{(\alpha)}}{2}\right ], \] where $[\cdots]$ denotes the shift by that number. \blacksquare \subsection[Holomorphically symplectic manifolds and semismall resolutions]{Holomorphically symplectic manifolds \\and semismall resolutions} \hfill \definition Let $\pi:\; X {\:\longrightarrow\:} Y$ be a morphism of complex varieties, and $\c Y$, $\c X$ be stratification of $Y$, $X$. Denote by $Y_i$ the strata of $\c Y$. We say that $\c Y$ and $\c X$ are compatible, if the preimages $\pi^{-1}(Y_i)$ coinside with the strata $X_i$ of $\c X$, all the strata of $\c Y$ are non-singular, and the maps $\pi\restrict{X_i}:\; X_i {\:\longrightarrow\:} Y_i$ are locally trivial fibrations. \hfill \proposition \label{_holo_symple_semismall_Proposition_} Let $\pi:\; X {\:\longrightarrow\:} Y$ be a generically finite, dominant morphism of complex varieties. Assume that $X$ is smooth and equipped with a holomorphically symplectic form $\Theta_X$. Moreover, assume that there exists a Cartesian square \[ \begin{CD} \tilde X @>{\tilde \pi}>> \tilde Y\\ @VV{\tilde \sigma}V @VV{\sigma}V \\ X @>{\pi}>> Y \end{CD} \] with finite dominant morphisms as vertical arrows and birational morphisms as horisontal arrows. \footnote{By a {\bf birational morphism} we understand a morphism $\phi:\; X_1 {\:\longrightarrow\:} X_2$ of complex varieties such that the inverse of $\phi$ is rational.} Assume that $\tilde Y$ is a holomorphically symplectic manifold, and the pullbacks of the holomorphic symplectic forms $\Theta_X$, $\Theta_{\tilde Y}$ via ${\tilde \pi}$ and $\tilde \sigma$ are equal on subvarieties, in the sense of Subsection \ref{_symple_on_Hi_Subsection_}. Assume, finally, that there exist compatible stratifications $\{X_i\}$, $\{\tilde X_i\}$, $\{\tilde Y_i\}$ such that $\Theta_{\tilde Y}\restrict{\tilde Y_i}$ is non-degenerate in the generic points of $\tilde Y_i$. Then $\pi:\; X {\:\longrightarrow\:} Y$ is a semismall resolution. \hfill {\bf Proof:} Let $r(X_i)$ be the rank of the radical of $\Theta_X\restrict{X_i}$ in the generic point of the stratum $X_i$. Similarly, let $r(\tilde X_i)$ the rank of the radical of $\tilde \sigma^*\Theta_X\restrict{\tilde X_i}$ in the generic point of the stratum $\tilde X_i$. Since $\tilde \sigma$ is finite dominant, we have $r(\tilde X_i)= r(X_i)$. By definition, $\tilde Y_i = \tilde\pi(\tilde X_i)$ is (generically) a non-degenerate symplectic subvariety of $\tilde Y$. Since the forms $\Theta_{\tilde X}$ and $\tilde \pi^* \Theta_{\tilde Y}$ are equal on subvarieties, and $\Theta_{\tilde Y}\restrict{\tilde Y_i}$ is non-degenerate in the generic points of $\tilde Y_i$, we have \begin{equation}\label{_r_X_i_=_dim_fib_Equation_} r(\tilde X_i) = \dim_{\Bbb C} \left(\tilde\pi^{-1}(y)\right), \end{equation} for $y\in \tilde Y_i$ a generic point. Let $w(X_i)$ be the number $\dim(X_i) - r(X_i)$. The following linear-algebraic claim immediately implies that \begin{equation}\label{_codim_Y_i_>=_2r_X_i_Equation_} \dim_{\Bbb C} X - w(X_i) \geq 2 r(X_i) \end{equation} \blacksquare \hfill \claim Let $W$ be a symplectic vector space, $\Theta$ the symplectic form, $V\subset W$ a subspace, $r(V)$ the rank of the radical $\Theta\restrict V$ and $w(V):= \dim V - r(V)$. Then $\dim W - w(V) \geq 2 r(V)$. {\bf Proof:} Clear. \blacksquare \hfill Comparing \eqref{_r_X_i_=_dim_fib_Equation_} and \eqref{_codim_Y_i_>=_2r_X_i_Equation_}, we obtain that \[ \operatorname{codim}_{\Bbb C} Y_i \geq 2 \dim_{\Bbb C} \left(\tilde\pi^{-1}(y)\right), \] for $y\in \tilde Y_i$ a generic point. Finally, since $\sigma:\; \tilde Y {\:\longrightarrow\:} Y$ is finite dominant, we have $\dim_{\Bbb C} \left(\tilde\pi^{-1}(y)\right) = \dim_{\Bbb C} \left(\pi^{-1}(\sigma(y))\right)$. Thus, $\operatorname{codim}_{\Bbb C} Y_i \geq 2 \dim_{\Bbb C} \left(\pi^{-1}(y)\right)$ for $y\in Y_i$ a generic point. This finishes the proof of \ref{_holo_symple_semismall_Proposition_}. \blacksquare \hfill \corollary \label{_triana_speci_semismall_Corollary_} Let $M$ be a complex K3 surface or a compact complex 2-dimensional torus, $M^{[n]}$ its Hilbert scheme and $M^{(n)}$ the symmetric power of $M$, with $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ being the standard map. Consider an arbitrary hyperk\"ahler structure $\c H$ on $M^{[n]}$ compatible with the complex structure. Let $Z\subset M^{[n]}$ be a subvariety which is trianalytic with respect to $\c H$, and $n:\; X {\:\longrightarrow\:} Z$ be the desingularization of $Z$. Assume that $M$ is Mumford-Tate generic with respect to some hyperk\"ahler structure. Then $\pi \circ n :\; X {\:\longrightarrow\:} Y$ is a semismall resolution of $Y:=\pi(Z)$. \hfill {\bf Proof:} Assume that $Z$ is irreducible. Since the desingularization $X$ is hyperk\"ahler, this manifold is holomorphically symplectic, and the holomorphic symplectic form $\Theta_{X}$ on $X$ is obtained as a pullback of the holomorphic symplectic form $\Theta_{M^{[n]}}$ on $M^{[n]}$. To simplify notations, we denote $\pi \circ n$ by $\pi$. Let \begin{equation}\label{_commu_squa_with_subva_Equation_} \begin{CD} \tilde X @>{\tilde \pi}>> \tilde Y\\ @VV{\tilde \sigma}V @VV{\sigma}V \\ X @>{\pi}>> Y \end{CD} \end{equation} be the Cartesian square, with $\tilde Y$ obtained as an irreducible component of the preimage $\sigma^{-1}(Y)\subset M^n$. We intend to show that the square \eqref{_commu_squa_with_subva_Equation_} satisfies the assumptions of \ref{_holo_symple_semismall_Proposition_}. For each morphism of complex varieties, there exists a stratification, compatible with this morphism. Take a set of compatible stratifications $\{X_i\}$, $\{\tilde Y_i\}$, $\{\tilde X_i\}$. By \ref{_hyperkae_embeddings_Corollary_}, any stratification of $\tilde Y$ consists of trianalytic subvarieties because all closed complex subvarieties of $M^n$ are trianalytic. Applying \ref{_holo_symple_semismall_Proposition_} to the map $\pi:\; X {\:\longrightarrow\:} Y$ and the Cartesian square \eqref{_commu_squa_with_subva_Equation_}, we immediately obtain \ref{_triana_speci_semismall_Corollary_}. \blacksquare \section{Universal subvarieties of the Hilbert scheme} \label{_unive_subva_Section_} The Sections \ref{_unive_subva_Section_}--\ref{_triana_unive_subva_Section_} are independent from the rest of this paper. The only result of Sections \ref{_unive_subva_Section_}--\ref{_triana_unive_subva_Section_} that we use is \ref{_one-to-one_triana_Corollary_}. Let $M$ be a smooth complex surface, $M^{[n]}$ its Hilbert scheme. An automorphism $\gamma$ of $M$ gives an automorphism $\gamma^{[n]}$ of $M^{[n]}$; similarly, an infinitesimal automorphism of $M$ (that is, a holomorphic vector field) gives an infinitesimal automorphism of $M^{[n]}$. \hfill \definition \label{_Unive_subva_Definition_} Let $M$ be a surface, $M^{[n]}$ its Hilbert scheme and $X\subset M^{[n]}$ a closed complex subvariety. Then $X$ is called {\bf universal} if for all open $U\subset M$, and all global or infinitesimal automorphisms $\gamma\in \Gamma(T(M))$, the subvariety $X_U := X \cap U^{[n]}$ is preserved by $\gamma^{[n]}$. \hfill The universal subvarieties are described more explicitly in the following subsection \subsection{Young diagrams and universal subvarieties of the Hilbert scheme} \label{_Young_Hilb_Subsection_} Let $M$ be a smooth surface, $M^{(n)}$ its symmetric power, $M^{[n]}$ its Hilbert scheme and $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ the natural map. For a Young diagram \[ \alpha = (n_1\geq n_2\geq ... \geq n_k), \ \ \sum n_i =n, \] we defined a subvariety $\Delta_{(\alpha)}\subset M^{(n)}$ \eqref{_Del-a_alpha_definition_Equation_}. Let $\Delta_{[\alpha]}:= \pi^{-1}(\Delta_{(\alpha)})$ the the corresponding subvariety in $M^{[n]}$. Let $a$ be the general point of $\Delta_{(\alpha)}$, i. e. the one satisfying \begin{equation}\label{_gene_of_Delta_alpha_Equation_} \begin{split} a = &\sigma(a_1, ..., a_n),\ \ \text{where} \ \ a_1 = a_2 = ... = a_{n_1} \\& a_{n_1+1} = a_{n_1+2} = ... = a_{n_1+n_2} \ ..., \\ & a_{\sum_{i=1}^{k-1} n_i+1} = a_{\sum_{i=1}^{k-1} n_i+2} = ... = a_n \\ & \text{and the points $a_1$, $a_{n_1+1}$, ..., $a_{\left(\sum_{i=1}^{k-1} n_i\right)+1}$ are pairwise unequal} \end{split} \end{equation} Let $F_\alpha(a):= \pi^{-1}(a) \subset \Delta_{[\alpha]}$ be the general fiber of the projection $\pi:\; \Delta_{[\alpha]} {\:\longrightarrow\:} \Delta_{(\alpha)}$. By definition, $F_\alpha(a)$ parametrizes $0$-dimensional subschemes $S\subset M$ with $Sup(S)= \{a_i\}$ and prescribed multiplicites \[ \operatorname{length}_{a_i}S=n_i. \] Let $\hat {\cal O}_{a_i}$ be the adic completion of ${\cal O}_M$ at $a_i$, and $G_{a_i} := \operatorname{Aut}(\hat {\cal O}_{a_i})$. Clearly, the group $G_a:= \prod_i G_{a_i}$ acts naturally on $F_\alpha(a)$. We are interested in $G_a$-invariant subvarieties of $F_\alpha(a)$. \hfill \lemma \label{_inva_subva_F_alpha_identifi_Lemma_} Let $\alpha$ be a Young diagram, $\Delta_{(\alpha)}$ the corresponding subvariety of $M^{(n)}$ and $a, b$ the points of $\Delta_{(\alpha)}$ satisfying \eqref{_gene_of_Delta_alpha_Equation_}. Let $F_\alpha(a)$, $F_\alpha(b)$ be the corresponding fibers of $\pi:\; \Delta_{[\alpha]} {\:\longrightarrow\:} \Delta_{(\alpha)}$. Consider the groups $G_a$, $G_b$ acting on $F_\alpha(a)$, $F_\alpha(b)$ as above. Then \begin{description} \item[(i)] there exist a canonical bijective correspondence $\theta$ between $G_a$-invariant subvarieties in $F_\alpha(a)$ and $G_b$-invariant subvarieties in $F_\alpha(b)$. \item[(ii)] For any complex automorphism $\gamma\; M {\:\longrightarrow\:} M$ such that $\gamma(a)=b$, the corresponding map $\gamma:\; F_\alpha(a){\:\longrightarrow\:} F_\alpha(b)$ maps $G_a$-invariant subvarieties of $F_\alpha(a)$ to $G_b$-invariant subvarieties of $F_\alpha(b)$ and induces the correspondence $\theta$. \end{description} {\bf Proof:} Let $(a_1, ... a_n)$, $(b_1, ... b_n)$ be the points of $M^n$ satisfying \eqref{_gene_of_Delta_alpha_Equation_}, such that $a = \sigma(b_1, ... b_n)$, $b=\sigma(b_1, ... b_n)$. Let $U$ be an open subset of $M$ containing $a_i$, $b_i$, $i = 1, ... , n $. Let $\gamma:\; U {\:\longrightarrow\:} U$ be a complex automorphism of $U$ such that $\gamma(a_i) = b_i$. Since $a$, $b$ satisfy \eqref{_gene_of_Delta_alpha_Equation_}, for an appropriate choice of $U$, such $\gamma$ always exists. Clearly, $\gamma$ identifies $F_\alpha(a)$ and $F_\alpha(b)$. This identification is {\it not} unique, since it depends on the choice of $\gamma$, but every two such identifications differ by a twist by $G_a$, $G_b$. This proves \ref{_inva_subva_F_alpha_identifi_Lemma_}. \blacksquare \hfill By \ref{_inva_subva_F_alpha_identifi_Lemma_}, the set of $G_a$-invariant subvarieties of $F_\alpha(a)$ is independent from $a$. Denote this set by $\c Z_\alpha$. For each $Y\in \c Z_\alpha$, and a generic point $a\in \Delta_{(\alpha)}$, denote the corresponding subvariety of $F_\alpha(a)$ by $Y(a)$. Let $\c Z_\alpha(Y)$ be the union of $Y(a)$ for all $a\in \Delta_{(\alpha)}$ satisfying \eqref{_gene_of_Delta_alpha_Equation_}. \hfill \theorem\label{_inva_subva_from_Young_Theorem_} Let $\alpha$ be a Young diagram, $\Delta_{(\alpha)}\subset M^{(n)}$ the corresponding diagonal in $M^{(n)}$ and $a\in \Delta_{(\alpha)}$ a general point (that is, one satisfying \eqref{_gene_of_Delta_alpha_Equation_}). Let $Y\in \c Z_\alpha$ be a $G_a$-invariant subvariety of $F_\alpha(a)= \pi^{-1}(a)\subset M^{[\alpha]}$, and $\c Z_\alpha(Y)$ the corresponding subvariety of $M^{[\alpha]}$. Then $\c Z_\alpha(Y)$ is a universal subvariety of $M^{[\alpha]}$, in the sense of \ref{_Unive_subva_Definition_}. Moreover, all irreducible universal subvarieties of $M^{[\alpha]}$ can be obtained this way. \hfill {\bf Proof:} The statement of \ref{_inva_subva_from_Young_Theorem_} is local by $M$. Thus, to prove that $\c Z_\alpha(Y)$ is preserved by infinitesimal automorphisms, it suffices to show that $\c Z_\alpha(Y)$ is preserved by all global automorphisms of $M$. Let $\gamma:\; M {\:\longrightarrow\:} M$ be an automorphism. Denote by $\Delta_{(\alpha)}^\circ\subset \Delta_{(\alpha)}$ the set of all $a$ satisfying \eqref{_gene_of_Delta_alpha_Equation_}. Clearly, $\gamma$ preserves \[ \Delta_{[\alpha]}^\circ:= \pi^{-1}\left(\Delta_{(\alpha)}^\circ\right). \] To show that $\gamma$ preserves $\c Z_\alpha(Y)$, it suffices to prove that, for all $a, b \in \Delta_{(\alpha)}^\circ$, $\gamma(a)=b$, the automorphism $\gamma$ maps $F_\alpha(a)$ to $F_\alpha(b)$. This is \ref{_inva_subva_F_alpha_identifi_Lemma_} (ii). We obtained that $\c Z_\alpha(Y)$ is universal. \hfill Let $X$ be an irreducible universal subvariety in $M^{[n]}$. Then $\pi(X)\subset M^{(n)}$ is preserved by the automorphisms of $M^{(n)}$ coming from $M$. For $M$ Stein, the only subvarieties of $M^{(n)}$ preserved by infinitesimal automorphisms are unions of diagonals. Since $X$ is irreducible, so is $\pi(X)$. We obtain that $\pi(X)$ is a diagonal $\Delta_{(\alpha)}$ corresponding to some Young diagram $\alpha$. It remains to prove that $X\cap F_\alpha(a)$ is $G_a$-invariant, for all $a\in \Delta_{(\alpha)}^\circ$. This is clear, because infinitesimal automorphisms of $M$ fixing $\{a_i\}$ generate the group $G_a =\prod_i \operatorname{Aut}(\hat {\cal O}_{a_i})$, and since $X$ is invariant under such automorphisms, $X\cap F_\alpha(a)$ is $G_a$-invariant. \ref{_inva_subva_from_Young_Theorem_} is proven. \blacksquare \subsection{Universal subvarieties of relative dimension 0} \definition Let $M$ be a smooth complex surface, $M^{[n]}$ its Hilbert scheme, $\alpha$ a Young diagram corresponding to a diagonal $\Delta_{(\alpha)} \subset M^{(n)}$. Let $a\in \Delta_{(\alpha)}$ be a general point, and $F_\alpha(a):= \pi^{-1}(a)$ the corresponding fiber of $\pi:\; M^{[n]} {\:\longrightarrow\:} M^{(n)}$. Consider a $G_\alpha(a)$-invariant subvariety $Y$ of $F_\alpha(a)$. Let $Z\subset M^{[n]}$ be a corresponding universal subvariety, $Z = Z_\alpha(Y)$ (\ref{_inva_subva_from_Young_Theorem_}). Then the {\bf relative dimension} of $Z$ is the dimension of $Y$. \hfill In this subsection, we classify the universal subvarieties of relative dimension 0. \hfill Let $\alpha = (n_1\geq n_2\geq ... \geq n_k)$ be a Young diagram, $\sum n_i =n$. Clearly, \begin{equation}\label{_F_alpha_via_F_0_Equation_} F_\alpha(a) \cong F_0(n_1) \times F_0(n_2) \times ... , \end{equation} where $F_0(i)$ is the classifying space of $0$-dimensional subschemes of length $i$ in ${\Bbb C}^2$ with support in $0\in {\Bbb C}^2$. Let $G_0 = \operatorname{Aut}({\Bbb C}[[x,y]])$ be the group of automorphisms of the ring of formal series, acting on $F_0(i)$. By \eqref{_F_alpha_via_F_0_Equation_}, the $k$-th power of $G_0$ acts on $F_\alpha(a)$. This gives an isomorphism $G_0^k \cong G_\alpha(a)$. Assume that $n_i = \frac{m_i\cdot (m_i+1)}{2}$, for some positive integer $m_i$. Consider a $G_0$-invariant point $s_i\in F_0(n_i)$, given by \begin{equation} \label{_inva_point_quotie_by_power_Equation_} s_{m_i} = {\Bbb C}[[x,y]] / {\mathfrak m}^{m_i}, \end{equation} where $\mathfrak m\subset {\Bbb C}[[x,y]]$ is the maximal ideal generated by $x$ and $y$. Let $\{s_1\} \times\{s_2\} \times ... \times\{s_k\}$ be the $G_0^k$-invariant point of $\prod F_0(n_i)$. Using the isomorphism \eqref{_F_alpha_via_F_0_Equation_}, we obtain a $G_\alpha(a)$-invariant point $a$ of $F_\alpha(a)$. Denote by $\c X_\alpha$ the corresponding universal subvariety of $M^{[n]}$. It has relative dimension 0. The aim of this subsection is to show that all universal subvarieties of relative dimension $0$ are obtained this way. \hfill \proposition \label{_unive_subva_rela_dime_0_Proposition_} Let $X\subset M^{[n]}$ be a universal subvariety of relative dimension $0$. Then where exists a Young diagram \[ \alpha = (n_1\geq n_2\geq ... \geq n_k), \ \ \sum n_i =n, \] and positive integers $m_1, ..., m_k$, such that $n_i = \frac{m_i\cdot (m_i+1)}{2}$, and $X = \c X_\alpha$. \hfill {\bf Proof:} Let $a$ be a general point of $\Delta_\alpha$, and $s\in F_\alpha(a)$ a point of the zero-dimensional variety $F_\alpha(a)\cap X$. Consider the varieties $F_0(i)$ defined above, and the action of $G_0 = \operatorname{Aut}({\Bbb C}[[x,y]])$ on $F_0(i)$. Let $x_i\in F_0(n_i)$ be the points of $F_0(k)$, such that under an isomorphism \eqref{_F_alpha_via_F_0_Equation_}, $s$ corresponds to $\{x_1\} \times \{x_2\} \times ... \times \{x_k\}$. Then the points $x_i$ are $G_0$-invariant. To finish the proof of \ref{_unive_subva_rela_dime_0_Proposition_}, it remains to prove the following lemma. \hfill \lemma \label{_G_0_inva_poins_in_F_0_Lemma_} Let $s\in F_0(i)$ be a $G_0$-invariant point. Then $i= \frac{j\cdot(j+1)}{2}$ and $s$ is given by \eqref{_inva_point_quotie_by_power_Equation_}. {\bf Proof:} The group $GL(2, {\Bbb C})$ acts on ${\Bbb C}[[x,y]]$ by automorphisms. Clearly, this $GL(2, {\Bbb C})$-action is factorized through the natural action of \[ G_0 = \operatorname{Aut}({\Bbb C}[[x,y]]). \] We show (\ref{_GL(2)_inva_ideals_in_series_Sublemma_} below) that all $GL(2, {\Bbb C})$-invariant ideals in ${\Bbb C}[[x,y]]$ are powers of the maximal ideal. Since $x = {\Bbb C}[[x,y]]/ I$ for some $G_0$-, and hence, $GL(2, {\Bbb C})$-invariant ideal of ${\Bbb C}[[x,y]]$, this will finish the proof of \ref{_G_0_inva_poins_in_F_0_Lemma_}. We reduced \ref{_unive_subva_rela_dime_0_Proposition_} to the following result. \hfill \sublemma \label{_GL(2)_inva_ideals_in_series_Sublemma_} Consider the natural action of $GL(2, {\Bbb C})$ on \[ A= {\Bbb C}[[x,y]].\] Let $I$ be a proper $GL(2, {\Bbb C})$-invariant ideal in $A$. Then $I$ is a power of the maximal ideal. {\bf Proof:} Consider the $GL(2)$-invariant filtration \[ A_0 \subset A_0 \oplus A_1 \subset A_0\oplus A_1 \oplus A_2 \subset ... \subset A \] where $A_i\subset A$ consists of homogeneous polynomials of degree $i$. Let $l$ be the minimal number for which $I\cap A_l \neq 0$. Since $I$ and $A_l$ are $GL(2)$-invariant, the intersection $I \cap A_l$ is also $GL(2)$-invariant. The space $A_l$ is an irreducible representation of $GL(2)$, and thus, $I \supset A_l$. Therefore, $I = A_l \cdot A$, and $I$ is $l$-th power of the maximal ideal. This finishes the proof of \ref{_GL(2)_inva_ideals_in_series_Sublemma_}, \ref{_G_0_inva_poins_in_F_0_Lemma_}, and \ref{_unive_subva_rela_dime_0_Proposition_}. \blacksquare \section{Subvarieties of $M^{[n]}$ which are generically finite over $M^{(n)}$, for $M$ a generic K3 surface} \label{_subva_gene_fini_Section_} Throughout this section, $M$ is a smooth complex surface, $M^{[n]}$ the Hilbert scheme of $M$, $M^{(n)}$ the $n$-th symmetric power of $M$ and $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ the natural map. Let $f:\; X {\:\longrightarrow\:} Y$ be a morphism of complex varieties. We say that $f$ is {\bf generically finite} if there exist an open dense subset $X_0\subset X$ such that the map $f\restrict{X_0}:\; X_0 {\:\longrightarrow\:} f(X_0)$ is finite. The morphism $f$ is called {\bf generically one-to-one} if there exist an open dense subset $X_0\subset X$ such that the map $f\restrict{X_0}:\; X_0 {\:\longrightarrow\:} f(X_0)$ is an isomorphism. \hfill The main result of this section is the following theorem. \hfill \theorem\label{_gene_fini_universa_Theorem_} Let $M$ be a complex K3 surface. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is Mumford-Tate generic with respect to $\c H$ (\ref{_generic_manifolds_Definition_}). Let $X\subset M^{[n]}$ be an irreducible complex analytic subvariety such that the restriction of $\pi$ to $X$ is generically finite. Assume that there exists a Young diagram $\alpha$ such that the subvariety $\pi(X)\subset M^{(n)}$ coinsides with $\Delta_{(\alpha)}$. Then $X$ is a universal subvariety (\ref{_Unive_subva_Definition_}) of $M^{[n]}$. \hfill \remark The relative dimension of the universal subvariety $X\subset M^{[n]}$ is zero, because $\pi\restrict X$ is generically finite. Thus, \ref{_unive_subva_rela_dime_0_Proposition_} can be applied to this situation. We obtain that, under assumptions of \ref{_gene_fini_universa_Theorem_}, $\pi\restrict X :\; X {\:\longrightarrow\:} \Delta_{(\alpha)}$ is generically one-to-one. \hfill The proof of \ref{_gene_fini_universa_Theorem_} takes the rest of this section. \subsection{Fibrations arising from the Hilbert scheme} We work in assumptions of \ref{_gene_fini_universa_Theorem_}. Let $\Delta^\circ_{(\alpha)}\subset \Delta_{(\alpha)}$ be the set of general points of $\Delta_{(\alpha)}$, defined by \eqref{_gene_of_Delta_alpha_Equation_}. Consider the fibration $\pi:\; \Delta^\circ_{[\alpha]}{\:\longrightarrow\:} \Delta^\circ_{(\alpha)}$, where $\Delta^\circ_{[\alpha]}= \pi^{-1}(\Delta^\circ_{(\alpha)})$. Let $M^{(l)}_\circ$ be the $M^{(l)}$ with all diagonals deleted: \[ M^{(l)}_\circ = \bigg\{ (x_1 , ... x_l) \in M^{(l)}\ \ |\ \ x_i\neq x_j \text{\ \ for all \ \ } i\neq j \bigg\} \] We write $\alpha = (n_1 \geq n_2 \geq ... \geq n_k )$ as \begin{equation}\label{_n'_i_defi_Equation_} \begin{split} \alpha = &\left(n_1= n_2 = ... =n_{n_1'} > n_{n'_1+1} = ... = n_{n'_1 +n'_2} > \vphantom{ n_{\sum_{i=1}^{k'-1} n_i' -1} } \right. ... \\& \left. ... > n_{\sum_{i=1}^{k'-1} n_i' +1} = ... = n_{\sum_{i=1}^{k'-1} n_i' -1} = n_{\sum_{i=1}^{k'} n_i'}\right), \end{split} \end{equation} where $\sum_{i=1}^{k'} n_i' =k$. \hfill \claim\label{_Delta_without_diags_product_Claim_} The manifold $\Delta^\circ_{(\alpha)}$ is naturally isomorphic to $\prod_i M^{(n_i')}_\circ$. {\bf Proof:} Clear. \blacksquare \hfill Let $D^\circ_{(\alpha)}$ be the universal covering of $\Delta^\circ_{(\alpha)}$. {}From \ref{_Delta_without_diags_product_Claim_} it is clear that \begin{equation}\label{_D_alpha_is_M^K'_Equation_} D^\circ_{(\alpha)} = \prod_i M^{n_i'}_\circ\subset M^{k'}, \end{equation} where $M^{n_i'}_\circ$ is $M^{n_i'}$ without diagonals. We define $D^\circ_{[\alpha]}$ as a fibered product, in such a way that the square \begin{equation}\label{_D_alpha_Cartesian_defi_Equation_} \begin{CD} D^\circ_{[\alpha]} @>>> \Delta^\circ_{[\alpha]}\\ @VVpV @VV\pi V \\ D^\circ_{(\alpha)} @>>> \Delta^\circ_{(\alpha)} \end{CD} \end{equation} is Cartesian. The map $D^\circ_{[\alpha]}\stackrel p {\:\longrightarrow\:} D^\circ_{(\alpha)}$ is a locally trivial fibration. We determine the fibers of $p$ in terms of the isomorphism \eqref{_D_alpha_is_M^K'_Equation_} as follows. \hfill Consider the vector bunlde $J^i(M)$ over $M$, with the fibers $J^i(M)\restrict x = {\cal O}_M/ {\mathfrak m}_x^i$, where ${\mathfrak m}_x$ is the maximal ideal on ${\cal O}_M$ corresponding to $x$. Clearly, the bundle $J^i(M)$ has a natural ring structure. Let $G^i(M)$ be the fibration over $M$ with the fibers $G^i(M)\restrict x$ classifying the ideals $I\subset J^i(M)$ of codimension $i$. Consider again the equation \eqref{_n'_i_defi_Equation_}. Let $N:\; \{ 1 , .. k'\} {\:\longrightarrow\:} {\Bbb Z}^+$ be the map \[ l {\:\longrightarrow\:} n_{\sum_{i=1}^{l-1}}, \] i. e., $1$ is mapped to $n_1$, $2$ to the biggest value of $n_i$ not equal to $n_1$, $3$ to the third biggest, etc. For a locally trivial fibrations $Y_1$, $Y_2$ over $X_1$, $X_2$, we denote the external product by $Y_1 \newboxtimes Y_2$. This is a fibration over $X_1\times X_2$, with the fibers which are products of fibers of $Y_1$, $Y_2$. The iterations of $\newboxtimes$ (for three or more fibrations) are defined in the same spirit. \hfill \claim Under the isomorphism \eqref{_D_alpha_is_M^K'_Equation_}, the locally trivial fibration $p:\; D^\circ_{[\alpha]}{\:\longrightarrow\:} D^\circ_{(\alpha)}$ is isomorphic to the fibration \[ \newboxtimes\limits_{i=1}^{k'} G^{N(i)}(M) \restrict {D^\circ_{(\alpha)}} \] over $D^\circ_{(\alpha)} \subset M^{k'}$. {\bf Proof:} Clear. \blacksquare \hfill Let $D_{[\alpha]}{\:\longrightarrow\:} D_{(\alpha)}$ be the fibration $\newboxtimes\limits_{i=1}^{k'} G^{N(i)}(M) {\:\longrightarrow\:} M^{k'}$. We consider $D^\circ_{[\alpha]}$, $D^\circ_{(\alpha)}$ as open subsets in $D_{[\alpha]}$, $D_{(\alpha)}$. \hfill Let $X\subset M^{[n]}$ be a closed subvariety, $\pi(X) = \Delta_{(\alpha)}$, and $\pi:\; X {\:\longrightarrow\:} \Delta_{(\alpha)}$ generically finite. Consider $X\cap \Delta^\circ_{[\alpha]}$ as a closed subvariety of $\Delta^\circ_{[\alpha]}$. Let $\tilde X$ be an irreducible component of $n^{-1} (X) \subset D^\circ_{[\alpha]}$, where $n:\; D^\circ_{[\alpha]}{\:\longrightarrow\:} \Delta^\circ_{[\alpha]}$ is the horisontal arrow of \eqref{_D_alpha_Cartesian_defi_Equation_}. Clearly, the closure of $\tilde X$ in $D_{[\alpha]}$ is a closed complex subvariety of $D_{[\alpha]}$. We denote this subvariety by $Z$. By construction, $Z$ is irreducible (it is an image of an irreducible variety) and generically finite over $D_{(\alpha)} = M^{k'}$. \hfill Consider the fibration $G^m(M) {\:\longrightarrow\:} M$ constructed above. Assume that $m = \frac{l\cdot (l+1)}{2}$ for a positive integer $l$. Then the fibration $G^m(M) {\:\longrightarrow\:} M$ has a canonical section $s:\; M {\:\longrightarrow\:} G^m(M)$, defined by $x {\:\longrightarrow\:} {\cal O}_M /{\mathfrak m}_x^l$, where ${\mathfrak m}_x\subset {\cal O}_M$ is the maximal ideal corresponding to $x$. \hfill \proposition\label{_subvarie_of_G^i_Proposition_} Let $M$ be a complex K3 surface. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is generic with respect to $\c H$. Let $Y\subset G^m(M)$ be a closed irreducible subvariety of the total space of the fibration $G^m(M) \stackrel p {\:\longrightarrow\:} M$. Assume that $Y$ is generically finite over $M$. Then $m = \frac{l\cdot (l+1)}{2}$ for some positive integer $l$, and $Y$ is the image of the natural map $s:\; M {\:\longrightarrow\:} G^m(M)$ constructed above. \hfill \ref{_subvarie_of_G^i_Proposition_} is proven in Subsection \ref{_fibra_ove_K3_Subsection_}. Presently, we are going to explain how \ref{_subvarie_of_G^i_Proposition_} implies \ref{_gene_fini_universa_Theorem_}. \hfill Consider the map $p:\; D_{[\alpha]} {\:\longrightarrow\:} M^{k'}$, and the closed subvariety $Z\subset D_{[\alpha]}$ constructed from $X$ as above. Let $(m_1, ... , m_{k'-1}) \in M^{k'-1}$ be a point such that the map \[ p:\; Z \cap p^{-1}(\{(m_1, ... , m_{k'-1})\} \times M) {\:\longrightarrow\:} \{(m_1, ... , m_{k'-1})\} \times M) \] is generically finite. The set $S$ of such $(m_1, ... , m_{k'-1})$ is open and dense in $M^{k'-1}$. Let $\Psi_i:\; D_{[\alpha]} {\:\longrightarrow\:} G^{N(i)}(M)$ be the natural projection to the $i$-th component of the product $D_{[\alpha]} = \newboxtimes\limits_{i=1}^{k'} G^{N(i)}(M)$. By \ref{_subvarie_of_G^i_Proposition_}, $N(k')= \frac{l\cdot(l-1)}{2}$ and the subvariety \[ \Psi(Z \cap p^{-1}(\{(m_1, ... , m_{k'-1})\} \times M))\] coinsides with image of the map \[ s_{k'}:\; M {\:\longrightarrow\:} G^{N(k')}(M).\] Since $Z$ is irreducible, \[ \Psi_{k'}(Z \cap p^{-1}(\{(m_1, ... , m_{k'-1})\} \times M)) \subset G^{N(k')}(M) \] is independent from the choice of $(m_1, ... , m_{k'-1})\in S$. Therefore, $\Psi_{k'}(Z) = \operatorname{im}(s_{k'})$. A similar argument shows that $\Psi_{i}(Z) = \operatorname{im}(s_{i})$, for all $i= 1, ... , k'$. Thus, $Z$ is an image of the section of the map $p:\; D_{[\alpha]} {\:\longrightarrow\:} M^{k'}$ given by $\newboxtimes\limits_{i=1}^{k'} s_i$. This implies \ref{_gene_fini_universa_Theorem_}. We reduced \ref{_gene_fini_universa_Theorem_} to \ref{_subvarie_of_G^i_Proposition_}. \subsection{Projectivization of stable bundles} Let $M$ be a compact K\"ahler manifold. We understand stability of holomorphic vector bundles over $M$ in the sense of Mumford--Takemoto (\ref{_degree,slope_destabilising_Definition_}). A polystable bundle is a direct sum of stable bundles of the same slope. Let $V$ be a polystable bundle, and ${\Bbb P} V$ its projectivization. Consider the unique Yang-Mills connection on $V$ (\ref{Yang-Mills_Definition_}). This gives a natural connection $\nabla_V$ on the fibration ${\Bbb P} V{\:\longrightarrow\:} M$. \hfill \proposition\label{_subvarie_of_PV_Proposition_} Let $M$ be a compact complex simply connected manifold of hyperk\"ahler type. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is generic with respect to $\c H$. Consider $M$ as a K\"ahler manifold, with the K\"ahler metric induced from $\c H$. Let $V$ be a polystable bunlde over $M$, and ${\Bbb P} V\stackrel \pi{\:\longrightarrow\:} M$ its projectivization. Consider a closed irreducible subvariety $X\subset {\Bbb P} V$ such that $\pi(X) = M$. Then $X$ is preserved by the connection $\nabla_V$ in ${\Bbb P} V$. \hfill {\bf Proof:} Let $x\in M$ be a point of $M$ such that in a neighbourhood $U\subset M$ of $x$, the projection $\pi:\; X {\:\longrightarrow\:} M$ is a locally trivial fibration. Assume that $U$ is open and dense in $M$. Let $X_x$ be the fiber of $\pi:\; X {\:\longrightarrow\:} M$ in $x$, and $V_x:= V\restrict x$ the fiber of $V$. Consider the Hilbert scheme $H$ classifying the subvarieties $Y\subset {\Bbb P} V_x$ with the same Hilbert polynomial as $Y$. Then $H$ can be naturally embedded to the projectivization of a linear space $W_x$, where $W_x$ is a certain tensor power of $V_x$, depending on the Hilbert polynomial of $X_x$. Consider the corresponding bundle $W$, which is related to $V$ in the same way as $W_x$ to $V_x$. Then, $W$ is a tensor power of $V$, and hence, $W$ is equipped with a unique Yang-Mills connection. Consider the corresponding connection $\nabla_W$ on the projectivization ${\Bbb P}W$. Let $X_0$ denote $\pi^{-1}(U) \cap X$. The locally trivial fibration $\pi\restrict{X_0}:\; X_0{\:\longrightarrow\:} U$ gives a section $s$ of ${\Bbb P} W\restrict{X_0}$. To prove that $X$ is preserved by the connection $\nabla_V$ in ${\Bbb P} V$, it suffices to show that $X_0$ is preserved by $\nabla_V$, or that $\operatorname{im} s$ is preserved by $\nabla_W$. This is implied by the following lemma, which finishes the proof of \ref{_subvarie_of_PV_Proposition_}. \hfill \lemma\label{_secti_preserved_by_nabla_Lemma_} In assumptions of \ref{_subvarie_of_PV_Proposition_}. let $U\subset M$ be a dense open set, such that $\pi\restrict{X_0}:\; X_0{\:\longrightarrow\:} U$ is an isomorphism, where \[ X_0 = \pi^{-1}(U) \cap X\subset {\Bbb P}V. \] Then $\nabla_V$ preserves $X$. \hfill {\bf Proof:} Since $M$ is generic with respect to $\c H$, all its complex subvarieties have complex codimension at least 2. Thus, we may assume that the complement $M \backslash U$ is a complex subvariety of codimension at least 2. Consider the restriction $V\restrict U$. Then $X_0$ gives a one-dimensional subbundle $L$ of $V\restrict U$. Let $V'= i_* L \subset i_* V\restrict U$ be the direct image of $L$ under the embedding $U\stackrel i \hookrightarrow M$. Since $M \backslash U$ is a complex subvariety of codimension at least 2, the natural map $V {\:\longrightarrow\:} i_* V\restrict U$ is an isomorphism. Therefore, $V'$ is a coherent subsheaf in $V$. To prove \ref{_secti_preserved_by_nabla_Lemma_} it suffices to show that $V'$ is preserved by the connection. Since $M$ is generic with respect to $\c H$, all integer $(1,1)$-classes of cohomology have degree 0 (\ref{_Lambda_of_inva_forms_zero_Lemma_}). Therefore, $\operatorname{slope}(V')=\operatorname{slope}(V)=0$ and $V'$ is a destabilising subsheaf of $V$. Since $V$ is polystable, this implies that $V'$ is a direct summand of $V$, and the Yang-Mills connection in $V$ preserves the decomposition $V = V' \oplus {V'}^{\bot}$, where ${V'}^{\bot}$ is the orthogonal complement of $V'$ with respect to any Yang-Mills metric on $V$. \ref{_secti_preserved_by_nabla_Lemma_} is proven. This finishes the proof of \ref{_subvarie_of_PV_Proposition_}. \blacksquare \hfill \corollary In assumptions of \ref{_subvarie_of_PV_Proposition_}, let $X$ be generically finite over $M$. Assume that $M$ is simply connected. Then $\pi:\; X{\:\longrightarrow\:} M$ is an isomorphism. {\bf Proof:} Since $X$ is preserved by the connection, the map $\pi:\; X{\:\longrightarrow\:} M$ is a finite covering. Since $M$ is simply connected, and $X$ is irreducible, $\pi:\; X{\:\longrightarrow\:} M$ is one-to-one. \blacksquare \subsection{Fibrations over K3 surfaces} \label{_fibra_ove_K3_Subsection_} The purpose of this subsection is to prove \ref{_subvarie_of_G^i_Proposition_}. Consider the fibration $G^m(M)\stackrel p {\:\longrightarrow\:} M$ over the K3 surface $M$. Recall that $G^m(M)$ was defined as a fibration with fibers classifying the codimension-$m$ ideals in $J^m(M)$, where $J^m(M)$ is the bundle of rings $J^m(M)\restrict x = {\cal O}_M/{\mathfrak m_x}^m$. There is a decreasing filtration \begin{equation}\label{_filtra_on_J_Equation_} J^i(M) \supset {\mathfrak W}(M) \supset {\mathfrak W}^2(M) \supset ..., \end{equation} with \[ {\mathfrak W}^i(M)\restrict x = {\mathfrak m_x}^i \cdot {\cal O}_M/{\mathfrak m_x}^m \] Consider the bundle $V = {\mathfrak W}^{l-1}(M)/{\mathfrak W}^{l}(M)$. \hfill \lemma\label{_V_stable_Lemma_} Let $M$ be a complex K3 surface which is generic with respect to some hyperka\"hler structure. Let $V$ the holomorphic vector bundle defined above, $V= {\mathfrak W}^{l-1}(M)/{\mathfrak W}^{l}(M)$. Then $V$ is isomorphic to a symmetric power of the cotangent bundle of $M$. Moreover, $V$ is (Mumford-Takemoto) stable for all K\"ahler structures on $M$, and has no proper subbundles. {\bf Proof:} The first assertion is clear. Let us prove stability of $V$. {}From Yau's proof of Calabi conjecture, it follows that for all K\"ahler classes on $M$, $M$ is equipped with the hyperk\"ahler metric in the same K\"ahler class. The Levi-Civita connection on the cotangent bundle $\Lambda^1(M)$ of a hyperk\"ahler manifold $M$ is hyperholomorphic (\ref{_hyperho_conne_Definition_}), and hence Yang-Mills (\ref{Yang-Mills_Definition_}). Therefore, $\Lambda^1(M)$ is stable, and $V$ polystable (a tensor power of a Yang-Mills bundle is again Yang-Mills). The holonomy group of $\Lambda^1(M)$ is obviously isomorphic to $SU(2)$. Therefore, the holonomy group of $V= S^l(\Lambda^1(M))$ is also $SU(2)$. The symmetric power of the tautological representation of $SU(2)$ is obviously irreducible. Therefore, the holonomy representation of $V$ is irreducible, and $V$ cannot be represented as a direct sum of vector bundles. To show that $V$ has no proper subbundles, we notice that $H^{1,1}(M) \cap H^2(M, {\Bbb Z})=0$ because $M$ is generic with respect to $\c H$. Therefore, all coherent sheaves on $M$ have first Chern class zero. We obtain that a proper subbundle of $V$ is destabilizing, which contradicts stability of $V$. \blacksquare \hfill Let $J^m_{gr}(M)$ be the graded sheaf of rings associated with the filtration \eqref{_filtra_on_J_Equation_}. Conside the fibration $G_{gr}^m(M)$ with the points classifying codimension-$m$ ideals in the fibers of $J^m_{gr}(M)$. There is a natural map $G^m(M)\stackrel \phi{\:\longrightarrow\:} G_{gr}^m(M)$ associating to an ideal its associated graded quotient. Composing $\phi$ with the map $s:\; M {\:\longrightarrow\:} G^{\frac{l(l-1)}{2}}(M)$, we obtain the section $s_{gr}:\; M {\:\longrightarrow\:} G_{gr}^{\frac{l(l-1)}{2}}(M)$ of the natural projection $p_{gr}:\; G_{gr}^{\frac{l(l-1)}{2}}(M){\:\longrightarrow\:} M$. The following \ref{_subvarie_of_G^i_gr_Proposition_} obviously implies \ref{_subvarie_of_G^i_Proposition_}. \hfill \proposition\label{_subvarie_of_G^i_gr_Proposition_} Let $M$ be a complex K3 surface. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is generic with respect to $\c H$. Let $Y\subset G^m_{gr}(M)$ be a closed irreducible subvariety of the total space of the fibration $G^m_{gr}(M) \stackrel p {\:\longrightarrow\:} M$. Assume that $Y$ is generically finite over $M$. Then $m = \frac{l\cdot (l+1)}{2}$ for some positive integer $l$, and $Y$ is the image of the natural map $s_{gr}:\; M {\:\longrightarrow\:} G^m_{gr}(M)$ constructed above. \hfill {\bf Proof:} By \ref{_V_stable_Lemma_}, the bundle $J^m_{gr}(M)$ is polystable. As usually, applying the Uhle\-n\-beck-\--Yau theorem (\ref{_UY_Theorem_}), we endow the fibration $G^m_{gr}(M) \stackrel p {\:\longrightarrow\:} M$ with a natural connection $\nabla$. {}From \ref{_subvarie_of_PV_Proposition_} it is easy to deduce that the image of $\pi:\; X {\:\longrightarrow\:} G^m_{gr}(M)$ is preserved by the connection $\nabla$. Since $Y$ is generically finite over $M$, the natural projection $Y\stackrel p {\:\longrightarrow\:} M$ is a finite covering. Since $M$ is simply connected, this map is an isomorphism. For $x\in M$, let $t_x \in X$ be the ideal of $J_{gr}^m(M)$ such that $p(t_x) = x$. Denote by $l$ the maximal number such that $t_x \not\supset {\mathfrak W}^{l-1}(M)$ for some $x$. Consider the space \begin{multline*} A_x:= {\mathfrak W}^{l-1}(M)\cap t_x \bigg/{\mathfrak W}^{l}(M)\ \ \ \text{\LARGE $\subset$} \ \ \ {\mathfrak W}^{l-1}(M) \bigg/ {\mathfrak W}^{l}(M).\\ \end{multline*} Let $w= \dim A_x$. Since $t_x$ is preserved by the connection $\nabla$, the number $w$ does not depend on $x$. This gives a $w$-dimensional subbundle $A$ in $V= {\mathfrak W}^{l}(M)\subset {\mathfrak W}^{l-1}(M)$. By \ref{_V_stable_Lemma_}, $A$ is either $V$ or empty. Since $t_x \not\supset {\mathfrak W}^{l-1}(M)$, $A=0$. Since $t_x$ is an ideal, this implies that $t_x \subset {\mathfrak W}^{l}(M)$. By definition of $l$, it is the maximal number for which $t_x \not\supset {\mathfrak W}^{l-1}(M)$, and thus, $t_x \supset {\mathfrak W}^{l}(M)$. Therefore, $t_x=l$. This proves \ref{_subvarie_of_G^i_gr_Proposition_}. We finished the proof of \ref{_subvarie_of_G^i_Proposition_} and \ref{_gene_fini_universa_Theorem_}. \blacksquare \section{Special subvarieties of the Hilbert scheme} \label{_specia_subva_Section_} \subsection{Special subvarieties} \label{_special_subva_Subsection_} \definition\label{_specia_subva_Definition_} (See also \ref{_Unive_subva_Definition_}). Let $M$ be a complex surface, $\c A$ a finite set, $\phi:\; \c A {\:\longrightarrow\:} M$ an arbitrary map. For $i\in \c A$, consider the local ring ${\cal O}_{\phi(i)}$ of germs of holomorphic functions in $\phi(i)$. For $U\subset M$, consider the set $A_U$ of all automorphisms (global or infinitesimal) of $U$ which fix the image $\operatorname{im}\phi\subset M$ and act trivially on ${\cal O}_{\phi(i)}$. For $\gamma\in A_U$, we denote by $\gamma^{[n]}$ the corresponding automorphism of the Hilbert scheme $U^{[n]}$. A closed subvariety $X\subset M^{[n]}$ is called {\bf special} if for all $U\subset M$, all $\gamma\in A_U$, $X\cap U^{[n]} $ is fixed by $\gamma^{[n]}$. \hfill We are going to characterize special subvarieties more explicitly, in the spirit of \ref{_inva_subva_from_Young_Theorem_}. \hfill Let $M$ be a complex surface, $\alpha$ a Young diagram. \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n, \] $\c A\subset \{1, ... , k\}$, and $\phi:\; \c A {\:\longrightarrow\:} M$ an arbitrary map. Consider a subvariety $\Delta_{(\alpha)}(\c A, \phi)$ of $M^{(n)}$ defined as in \eqref{_Delta(A,phi)_alpha_definition_Equation_}. A generic point $a \in \Delta_{(\alpha)}(\c A, \phi)$ is the one satisfying \begin{equation}\label{_gene_of_Delta_alpha(A,phi)_Equation_} \begin{split} a = &\sigma(x_1, ..., x_n),\ \ \text{where} \ \ x_1 = x_2 = ... = x_{n_1} \\& x_{n_1+1} = x_{n_1+2} = ... = x_{n_1+n_2} \ ..., \\ & x_{\sum_{i=1}^{k-1} n_i+1} = x_{\sum_{i=1}^{k-1} n_i+2} = ... = x_n \\[4mm] & \begin{minipage}[t]{0.7\linewidth} and \begin{description} \item[(i)] $x_i = \phi(i)$ for all $i\in \c A$ \item[(ii)] the points $x_1$, $x_{n_1+1}$, ..., $x_{\left(\sum_{i=1}^{j-1} n_i\right)+1}, ..., $ are pairwise unequal for all $j\notin \c A$ \item[(iii)] the points $x_1$, $x_{n_1+1}$, ..., $x_{\left(\sum_{i=1}^{j-1} n_i\right)+1}, ..., $ don't belong to the set $\phi(\c A)$, for all $j\notin \c A$ \end{description} \end{minipage} \end{split} \end{equation} \hfill We split the Young diagram \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n, \] onto two diagrams, $ \alpha_{\c A} = (n_{a_1} \geq n_{a_2} ...\geq n_{a_l}), $ with $a_i$ running through $\c A$, and $ \check\alpha_{\c A} = (n_{b_1} \geq n_{b_2} ...\geq n_{b_{k-l}}), $ where $b_i$ runs through $\{1, ... , k\} \backslash \c A$. Consider the Hilbert scheme $M^{[n_a]}$, where $n_a = \sum n_{a_i}$. The map $\phi:\; {\c A} {\:\longrightarrow\:} M$ gives a point $\Phi\in \Delta_{(\alpha_{\c A})}\subset M^{(n_a)}$ (see Subsection \ref{_speci_of_K3_Subsection_} for details). Let $F_\phi:= \pi^{-1}(\Phi)$ be the fiber of the standard projection $\pi:\; M^{[n_a]}{\:\longrightarrow\:} M^{(n_a)}$. For $y\in \Delta_{(\check\alpha_{\c A})}$ a generic point of $\Delta_{(\check\alpha_{\c A})}$, let $F_{\check\alpha_{\c A}}(y)$ be the fiber of $\pi:\; M^{[n_b]}{\:\longrightarrow\:} M^{(n_b)}$ over $y$. Clearly, for $z\in \Delta_{(\alpha)}(\c A, \phi)$ a generic point, the fiber of $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ over $z$ is isomorphic to $F_\phi\times F_{\check\alpha_{\c A}}(y)$. This isomorphism is not canonical, but is defined up to a twist by the action of the group $G_y$ (see \ref{_inva_subva_F_alpha_identifi_Lemma_} for details). \hfill Fix a $G_y$-invariant subvariety $E\subset F_\phi\times F_{\check\alpha_{\c A}}(y)$. For a generic point $z\in \Delta_{(\alpha)}(\c A, \phi)$, consider a subvariety $E_z\subset \pi^{-1}(z)\subset M^{[n]}$ corresponding to $E$ under the isomorphism \begin{equation} \label{_fiber_gene_special_subva_Equation_} \pi^{-1}(z) \cong F_\phi\times F_{\check\alpha_{\c A}}(y). \end{equation} Let $\Delta_{[\alpha]}(\c A, \phi, E)$ be the closure of the union of $E_z$ for all $z\in\Delta_{(\alpha)}(\c A, \phi)$ satisfying \eqref{_gene_of_Delta_alpha(A,phi)_Equation_}. Clearly, $\Delta_{[\alpha]}(\c A, \phi, E)$ is a closed subvariety in $M^{[n]}$. \hfill \theorem\label{_special_subva_explici_Theorem_} Let $M$ be a complex surface, $\c A$ a finite set, $\phi:\; \c A {\:\longrightarrow\:} M$ an arbitrary map, and $X\subset M^{[n]}$ a special subvariety, associated with $\phi$. Then \begin{description} \item[(i)] there exist a Young diagram $\alpha$ \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n, \] an injection $\c A \hookrightarrow \{1, ... , k\}$, and a $G_y$-invariant subvariety \[ E\subset F_\phi\times F_{\check\alpha_{\c A}}(y),\] such that $X=\Delta_{[\alpha]}(\c A, \phi, E)$, where $F_\phi\times F_{\check\alpha_{\c A}}(y)$ and $\Delta_{[\alpha]}(\c A, \phi, E)$ are varieties constructed above. \item[(ii)] Conversely, $\Delta_{(\alpha)}(\c A, \phi, E)$ is a special subvariety of $M^{[n]}$ for all $\c A$, $\pi$, $E$. \end{description} \hfill {\bf Proof:} We use the notation of \ref{_specia_subva_Definition_}. For sufficiently small $U$, the automorphisms from $A_{U\backslash \operatorname{im} \phi}$ act $n$-transitively on $U\backslash \operatorname{im} \phi$. This implies that $\pi(X) = \Delta_{(\alpha)}(\c A, \phi)$, for an appropriate Young diagram \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n, \] and an embedding $\c A \hookrightarrow \{1, 2, ..., k\}$. Let $x$ be a generic point of $\Delta_{(\alpha)}(\c A, \phi)$. Consider the isomorphism $\pi^{-1}(x) \cong F_\phi\times F_{\check\alpha_{\c A}}(y)$ of \eqref{_fiber_gene_special_subva_Equation_}, and the action of $G_y$ on $F_\phi\times F_{\check\alpha_{\c A}}(y)$. Clearly, $A_U$ acts on $\pi^{-1}(x)$ as $G_y$. Therefore, the intersection $E_x:= X \cap \pi^{-1}(x)$ is $G_y$-invariant. We intend to show that $X=\Delta_{[\alpha]}(\c A, \phi, E_x)$ For $x, y\in M^{(n)}$ generic points of $\Delta_{(\alpha)}(\c A, \phi)$, there exists $U\supset \{x,y\}$ and an automorphism $\gamma:\; U {\:\longrightarrow\:} U$ such that $\gamma^{(n)}(x) =y$, for $\gamma^{(n)}:\; U^{(n)} {\:\longrightarrow\:} U^{(n)}$ the induced by $\gamma$ automorphism of $U^{(n)}$. Since $X$ is a special subvariety, $\gamma^{[n]}$ maps $E_x$ to $E_y:= X \cap \pi^{-1}(y)$. By definition, $\Delta_{[\alpha]}(\c A, \phi, E_x)$ is a closure of the union of all $\gamma^{[n]}(E_x)$, for all $U\subset M$ and $\gamma \in A_U$. On the other hand, $X$ is a closure of the union of all $E_y$, where $y$ runs through all generic points of $\Delta_{(\alpha)}(\c A, \phi)$. Thus, $X$ and $\Delta_{[\alpha]}(\c A, \phi, E_x)$ coinside. This proves \ref{_special_subva_explici_Theorem_} (i). \ref{_special_subva_explici_Theorem_} (ii) is clear. \blacksquare \subsection{Special subvarieties of the Hilbert scheme of K3} \label{_speci_of_K3_Subsection_} \theorem \label{_all_subva_are_special_Theorem_} Let $M$ be a complex K3 surface admitting a hyperk\"ahler structure $\c H$ such that $M$ is generic with respect to $\c H$, $M^{[n]}$ its Hilbert scheme and $M^{(n)}$ its symmetric power. Let $X\subset M^{[n]}$ be a closed irreducible subvariety such that $X$ is generically finite over $\pi(X) \subset M^{(n)}$. Assume that $M$ has no holomorphic automorphisms. Then $X$ is a special subvariety of $M^{[n]}$, in the sense of \ref{_specia_subva_Definition_}. \hfill {\bf Proof:} {}From \ref{_subva_in_M^(n)_Proposition_} it follows that $\pi(X) = \Delta_{(\alpha)}(\c A, \phi)$ for appropriate $\c A$, $\alpha$ and $\phi$. As previously, we split the Young diagram $\alpha$ onto $ \alpha_{\c A} = (n_{a_1} \geq n_{a_2} ...\geq n_{a_l}), $ with $a_i$ running through $\c A$, and $ \check\alpha_{\c A} = (n_{b_1} \geq n_{b_2} ...\geq n_{b_{k-l}}), $ where $b_i$ runs through $\{1, ... , k\} \backslash \c A$. Let $n_a:= \sum n_{a_i}$, $n_b:= \sum n_{b_i}$. Consider the natural map \begin{equation}\label{_M_na_prod_M_nb_Equation_} M^{(n_a)} \times M^{(n_b)} \stackrel s {\:\longrightarrow\:} M^{(n)}, \end{equation} defined in such a way as that to map $\Delta_{(\alpha_{\c A})}\times \Delta_{(\check\alpha_{\c A})}$ to $\Delta_{(\alpha)}$. This map is obviously finite. Let $x=(x_1, ... , x_{n_a})\in \Delta_{(\alpha_{\c A})}$ $y=(y_1, ... , y_{n_a})\in \Delta_{(\check\alpha_{\c A})}$ be the points satisfying $x_i \neq y_j$ $\forall i, j$. Then the fiber of $\pi:\; M^{[n]} {\:\longrightarrow\:} M^{(n)}$ in $s(x,y)$ is naturally isomorphic to the product $\pi^{-1}(x) \times \pi^{-1}(y)$, where the first $\pi$ is the standard projection $\pi:\; M^{[n_a]} {\:\longrightarrow\:} M^{(n_a)}$ and the second one is the standard projection $\pi:\; M^{[n_b]} {\:\longrightarrow\:} M^{(n_b)}$. Denote thus obtained map \begin{equation}\label{_M_na_prod_M_nb_fibers_Equation_} \pi^{-1}(x) \times \pi^{-1}(y) \tilde{\:\longrightarrow\:} \pi^{-1}(s(x,y)) \end{equation} by $\theta$. Together, the maps \eqref{_M_na_prod_M_nb_Equation_}, \eqref{_M_na_prod_M_nb_fibers_Equation_} give a correspondence \[ \c D \subset \bigg( \Delta_{[\alpha_{\c A}]}\times \Delta_{[\check\alpha_{\c A}]}\bigg) \times\Delta_{[\alpha]} \] which is generically one-to-one over the first component and generically finite over the second one. Denote the corresponding projections from $\c D$ by $\pi_1$, $\pi_2$. Consider $X$ (the subvariety of $M^{[n]}$ given as data of \ref{_all_subva_are_special_Theorem_}) as a subvariety of $\Delta_{[\alpha]}$. Let $\c D_X:=\pi_1(\pi_2^{-1}(X))$. and $\Phi\in \Delta_{(\alpha_{\c A})}$ be the point given by $\phi$, \[ \Phi= \left(\underbrace{\phi(a_1), ..., \phi(a_1)}_ {n_{a_1} \text{\ times}} , \ \underbrace{\phi(a_1), ..., \phi(a_2)}_ {n_{a_2} \text{\ times}}, ... \right). \] Let $p_1$, $p_2$ be the projections of $\Delta_{[\alpha_{\c A}]}\times \Delta_{[\check\alpha_{\c A}]}$ to its components. Since $\pi(X) =\Delta_{(\alpha)}(\c A, \phi)$, and $X$ is generically finite over $\pi(X)$, the subvariety \[ D_X\subset \Delta_{[\alpha_{\c A}]}\times \Delta_{[\check\alpha_{\c A}]} \] is generically finite over $\{\Phi\}\times\Delta_{(\check\alpha_{\c A})}$. Therefore, $p_2(\c D_2)\subset \Delta_{[\check\alpha_{\c A}]}$ is generically finite over $\{\Phi\}\times\Delta_{(\check\alpha_{\c A})}$. Applying \ref{_gene_fini_universa_Theorem_}, we obtain that $p_2(\c D_X)$ is a universal subvariety of $\Delta_{[\check\alpha_{\c A}]}$ The varieties $\Delta_{[\alpha_{\c A}]}$, $\Delta_{[\check\alpha_{\c A}]}$ are equipped with the local action of the automorphisms $A_U$ (see \ref{_specia_subva_Definition_}). Since $p_2(\c D_X)$ is universal, \[ \c D_X\subset\{\Phi\}\times \Delta_{[\check\alpha_{\c A}]} \subset \Delta_{[\alpha_{\c A}]}\times \Delta_{[\check\alpha_{\c A}]} \] is fixed by the $A_U$-action. Therefore, $p_2(\c D_2)\subset \Delta_{[\check\alpha_{\c A}]}$ is also fixed by $A_U$. By construction, $\pi_2(\c D_X)=X$, and thus, $X$ is fixed by $A_U$, i. e., special. \blacksquare \subsection{Special subvarieties of relative dimension 0} \definition Let $M$ be a complex surface, $M^{[n]}$ its Hilbert scheme and $X\subset M^{[n]}$ an irreducible special subvariety. The {\bf relative dimension of $X$} is the dimension of the generic fiber of the projection $\pi\restrict X :\; X {\:\longrightarrow\:} \pi(X)$, where $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ is the standard morphism. \hfill Let $\Delta_{(\alpha)}(\c A, \phi)\subset M^{(n)}$ be the subvariety defined as in Subsection \ref{_subva_of_symme_special_Subsection_}. Split $\alpha$ onto $\alpha_{\c A}$ and $\check \alpha_{\c A}$, as in Subsection \ref{_special_subva_Subsection_}: $ \alpha_{\c A} = (n_{a_1} \geq n_{a_2} ...\geq n_{a_r}), $ with $a_i$ running through $\c A$, and $ \check\alpha_{\c A} = (n_{b_1} \geq n_{b_2} ...\geq n_{b_{k-r}}), $ where $b_i$ runs through $\{1, ... , k\} \backslash \c A$. Let $\Phi\in \Delta_{(\alpha_{\c A})}$ be the point defined in Subsection \ref{_speci_of_K3_Subsection_}. Consider the variety $\pi^{-1}(\Phi)\subset M^{[n_a]}$, where $n_a = \sum n_{a_i}$. \hfill \noindent \proposition\label{_speci_unive_dime_zero_Proposition_} \begin{description} \item[(i)] There exists a special subvariety $X\subset M^{[n]}$ such that $\pi(X)= \Delta_{(\alpha)}(\c A, \phi)$ if and only if all the numbers $n_{b_i}$ are of form $\frac{l\cdot (l+1)}{2}$, for integer $l$'s. \item[(ii)] Let ${\mathfrak S}(\alpha, \c A, \phi)$ be the set of all such subvarieties. Assume that all the numbers $n_{b_i}$ are of form $\frac{l\cdot (l+1)}{2}$, for integer $l$'s. Then ${\mathfrak S}(\alpha, \c A, \phi)$ is in bijective correspondence with the set of points of $\pi^{-1}(\Phi)\subset M^{[n_a]}$. \end{description} {\bf Proof:} Using notation of the proof of \ref{_all_subva_are_special_Theorem_}, we consider the subvariety $p_2(\c D_X)\subset \Delta_{(\check\alpha_{\c A})}$. We have shown that this subvariety is universal of relative dimension 0. Therefore, ${\mathfrak S}(\alpha, \c A, \phi)$ is nonepmpty if and only if all $n_{b_i}$ are of the form $\frac{l\cdot (l+1)}{2}$, for integer $l$'s. Assume that ${\mathfrak S}(\alpha, \c A, \phi)$ is nonempty. Consider the unique universal subvariety $S\subset \Delta_{[\check\alpha_{\c A}]}$ of relative dimension 0, constructed in \ref{_unive_subva_rela_dime_0_Proposition_}. As in \ref{_inva_subva_from_Young_Theorem_}, a universal subvariety of $\Delta_{[\check\alpha_{\c A}]}$ corresponds to a $G_y$-invariant subvariety of the general fiber of the projection $\Delta_{[\check\alpha_{\c A}]} {\:\longrightarrow\:} \Delta_{(\check\alpha_{\c A})}$. Since $S$ is of relative dimension 0, the corresponding $G_y$-invariant subvariety is a point. Denote this point by $s$. Choose a point $f\in \pi^{-1}(\Phi)$. Using the notation of \ref{_special_subva_explici_Theorem_}, and an isomorphism \eqref{_M_na_prod_M_nb_fibers_Equation_}, we construct a special subvariety $X\subset M^{[n]}$, $X = \Delta_{[\alpha]}(\c A, \phi, \{s\}\times \{f\})$. {}From \ref{_special_subva_explici_Theorem_} it follows that all special subvarieties of relative dimension 0 are obtained this way. Since $s$ is defined canonically, the only freedom of choice we have after $\alpha, \c A, \phi$ are fixed is the choice of $f\in \pi^{-1}(\Phi)$. This finishes the proof of \ref{_speci_unive_dime_zero_Proposition_}. \blacksquare \hfill \proposition \label{_symple_subva_special_Proposition_} Let $M$ be a complex K3 surface with no complex automorphisms, $M^{[n]}$ its Hilbert scheme and $\Omega$ be the canonical holomorphic symplectic form on $M^{[n]}$. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is Mumford-Tate generic with respect to $\c H$. Let $X$ be an irreducible complex subvariety of $M^n$, such that the restriction $\Omega\restrict X$ is non-degenerate somewhere in $X$.\footnote{Clearly, for $X$ trianalytic, $X_{ns}$ the non-singular part of $X$, $\Omega\restrict{X_{ns}}$ is nowhere degenerate.} Then $X\subset M^{[n]}$ is a special subvariety of relative dimension 0. {\bf Proof:} By \ref{_all_subva_are_special_Theorem_}, to prove that $X$ is special it suffices to show that $X$ is generically finite over $\pi(X)$. This follows \ref{_triana_finite_in_gene_Claim_}. \blacksquare \hfill \corollary\label{_one-to-one_triana_Corollary_} Let $M$ be a complex K3 surface which is Mumford-Tate generic with respect to some hyperka\"hler structure, $M^{[n]}$ its Hilbert scheme and $M^{(n)}$ its symmetric power. Assume that $M$ has no holomorphic automorphisms. Consider an arbitrary hyperka\"hler structure on $M^{[n]}$ whcih is compatible with the complex structure. Let $X\subset M^{[n]}$ be a trianalytic subvariety of $M^{[n]}$. Then $X$ is generically one-to-one over $\pi(X)\subset M^{(n)}$ {\bf Proof:} By \ref{_symple_subva_special_Proposition_}, $X$ is a special subvariety of relative dimension 0. Now \ref{_one-to-one_triana_Corollary_} follows from an explicit description of special subvarieties of relative dimension 0, given in the proof of \ref{_speci_unive_dime_zero_Proposition_}. \blacksquare \section[Trianalytic and universal subvarieties of the Hilbert scheme of a general K3 surface] {Trianalytic and universal subvarieties of the \\Hilbert scheme of a general K3 surface} \label{_triana_unive_subva_Section_} The aim of this section is to show that all trianalytic subvarieties of the Hilbert scheme of a generic K3 surface are universal (\ref{_triana_subva_universal_Theorem_}). \subsection{Deformations of trianalytic subvarieties} We need the following general results on the structure of deformations of trianalytic subvarieties, proven in \cite{_Verbitsky:Deforma_}. \footnote{The Desingularization Theorem (\ref{_desingu_Theorem_}) significantly simplifies some of the proofs of \cite{_Verbitsky:Deforma_}. This simplification is straightforward.} \hfill Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure and $X\subset M$ a trianalytic subvariety. Consider $(X, I)$ as a closed complex subvariety of $(M,I)$. Let $\operatorname{Def}_I(X)$ be a Douady space of $(X,I) \subset (M,I)$, that is, a space of deformations of $(X,I)$ inside of $(M,I)$. By \ref{_G_M_invariant_implies_trianalytic_Theorem_}, for $X'\subset (M,I)$ a complex deformation of $X$, the subvariety $X'\subset M$ is trianalytic. In particular, $X'$ is equipped with a natural singular hyperk\"ahler structure (\cite{_Verbitsky:Deforma_}), i. e., with a metric and a compatible quaternionic structure. \hfill \noindent \theorem\label{_deforma_triana_Theorem_} \begin{description} \item[(i)] The $\operatorname{Def}_I(X)$ is a singular hyperk\"ahler variety, which is independent from the choice of an induced complex structure $I$. \item[(ii)] Consider the universal family $\pi:\; \c X {\:\longrightarrow\:} \operatorname{Def}_I(X)$ of subvarieties of $(M,I)$, parametrized by $\operatorname{Def}_I(X)$. Then the fibers of $\pi$ are isomorphic as hyperk\"ahler varieties. \item[(iii)] Applying the desingularization functor to $\pi:\; \c X {\:\longrightarrow\:} \operatorname{Def}_I(X)$, we obtain a projection $\pi:\; \tilde X\times Y {\:\longrightarrow\:} Y$, where $Y$ is a desingularization of $\operatorname{Def}_I(X)$ and $\tilde X$ is a desingularization of $X$. \item[(iv)] The variety $\tilde X\times Y$ is equipped with a natural hyperk\"ahler immersion to $M$. \end{description} {\bf Proof:} \ref{_deforma_triana_Theorem_} (i) and (ii) is proven in \cite{_Verbitsky:Deforma_}, and \ref{_deforma_triana_Theorem_} (iii) is a trivial consequence of \ref{_deforma_triana_Theorem_} (ii) and the functorial property of the hyperk\"ahler desingularization. To prove \ref{_deforma_triana_Theorem_} (iv), we notice that $\c X$ is equipped with a natural morphism $f:\; \c X {\:\longrightarrow\:} M$, which is compatible with the hyperk\"ahler structure. Let $n:\;\tilde X\times Y{\:\longrightarrow\:} \c X$ be the desingularization map. Clearly, the composition $\tilde X\times Y\stackrel n {\:\longrightarrow\:} \c X \stackrel f{\:\longrightarrow\:} M$ is compatible with the hyperk\"ahler structure. A morphism compatible with a hyperk\"ahler structure is necessarily an isometry, and an isometry is always an immersion. \blacksquare \subsection{Deformations of trianalytic special subvarieties} Let $M$ be a K3 surface, without automorphisms, which is Mumford-Tate generic with respect to some hyperk\"ahler structure. In this Subsection, we study the deformations of the special subvarieties of $M^{[n]}$. \hfill By \ref{_unive_subva_rela_dime_0_Proposition_}, universal subvarieties of $M^{[n]}$ are rigid. For the special subvarieties, a description of its deformations is obtained as an easy consequence of \ref{_speci_unive_dime_zero_Proposition_}. \hfill \claim Let $X\subset M^{[n]}$ be a special subvariety of relative dimension 0, \[ X = \Delta_{[\alpha]}(\c A, \phi, \{s\}\times \{\psi\})\] associated with $\Delta_{(\alpha)}(\c A, \phi)$ and $\psi\in F_{\phi}$ as in \ref{_speci_unive_dime_zero_Proposition_}. Then the deformations of $X$ are locally parametrized by varying $\phi:\; \c A {\:\longrightarrow\:} M$ and $\psi\in F_{\phi}$. \blacksquare \hfill Let $a_1$, ... , $a_r$ enumerate $\c A \subset \{ 1, ... , k\}$. Unless all $n_{a_i}=1$, the dimension of $F_\phi$ is non-zero. Thus, the union $\c X$ of all deformation of $X= \Delta_{[\alpha]}(\c A, \phi, \{s\}\times \{\psi\})$ is not generically finite over $\pi(\c X)=\Delta_{[\alpha]}(\c A, \phi)$. Together with \ref{_deforma_triana_Theorem_} and \ref{_symple_subva_special_Proposition_}, this suggests the following proposition. \hfill \proposition \label{_n_i_=1_for_i_in_A_Proposition_} Let $M$ be a complex K3 surface with no automorphisms which is Mumford-Tate generic with respect to some hyperk\"ahler structure. Consider the Hilbert scheme $M^{[n]}$ as a complex manifold. Let $\c H$ be an arbitrary hyperk\"ahler structure on $M^{[n]}$ agreeing with this complex structure. Consider an irreducible trianalytic subvariety $X\subset M^{[n]}$. By \ref{_symple_subva_special_Proposition_}, $X$ is a special subvariety of $M^{[n]}$, $X= \Delta_{[\alpha]}(\c A, \phi, \{\psi\}\times \{s\})$. Then $n_{i}=1$ for all $i\in \c A$. \hfill {\bf Proof:} Consider $X$ as a complex subvariety in the complex variety $M^{[n]}$. The corresponding Douady space is described by \ref{_deforma_triana_Theorem_}. Consider the diagonal $\Delta_{(\alpha)}\subset M^{(n)}$. For a general point $a\in \Delta_{(\alpha)}\subset M^{(n)}$, the fiber $\pi^{-1}(a)$ is naturally decomposed as in \eqref{_M_na_prod_M_nb_fibers_Equation_}: \[ \pi^{-1}(x) \times \pi^{-1}(y) \tilde{\:\longrightarrow\:} \pi^{-1}(r(x,y)), \] for $a = r(x,y)$, where $r:\; M^{(n_a)}\times M^{(n_b)}{\:\longrightarrow\:} M^{(n)}$ is a morphism of \eqref{_M_na_prod_M_nb_Equation_}. Consider the subvariety $\pi^{-1}(y)\times \{s\}\subset \pi^{-1}(a)$. This subvariety is clearly $G_a$-invariant, and applying \ref{_inva_subva_from_Young_Theorem_}, we obtain a universal subvariety of $M^{[n]}$. Denote this universal subvariety by $\Delta_{[\alpha]}(\c A)$. Let $\c X$ be the union of all complex deformations of $X\subset M^{[n]}$. From \ref{_deforma_triana_Theorem_} it is clear that $\c X$ is trianalytic; from \ref{_speci_unive_dime_zero_Proposition_} it follows that $\c X = \Delta_{[\alpha]}(\c A)$. By \ref{_deforma_triana_Theorem_}, $\c X$ is trianalytic in $M^{[n]}$. {}From \ref{_symple_subva_special_Proposition_} it follows that all trianalytic subvarieties $\c X\subset M^{[n]}$ are generically finite over $\pi(\c X)\subset M^{(n)}$. Consider the Young diagram $\alpha_{\c A}= (n_{a_1}\geq n_{a_2} \geq ...)$. The generic fiber of thus obtained generically finite map \begin{equation} \label{_Delta_alpha(A)_to_Delta_Equation_} \pi:\; \Delta_{[\alpha)}(\c A){\:\longrightarrow\:} \Delta_{(\alpha)} \end{equation} is isomorphic to $\pi^{-1}(y)$, where $y$ is a generic point of $\Delta_{(\alpha_{\c A})}$, and $\pi$ a projection $\pi:\; \Delta_{[(\alpha_{\c A})]} {\:\longrightarrow\:} \Delta_{(\alpha_{\c A})}$. The dimension of this fiber is equal to $\sum (n_{a_i}-1)$. By construction, $a_i$ enumerates $\c A\subset \{1, ... , k \}$. Thus, $n_{i}=0$ for all $i\in \c A$. This proves \ref{_n_i_=1_for_i_in_A_Proposition_}. \blacksquare \subsection{Special subvarieties and holomorphic symplectic form} The aim of the Subsection is the following statement. \hfill \proposition \label{_holo_symple_degene_on_speci_Proposition_} Let $M$ be a complex surface equipped with a holomorphically symplectic form, $M^{[n]}$ its $n$-th Hilbert scheme, and \[ X=\Delta_{[\alpha]}(\c A, \phi, \{\psi\}\times \{s\}) \] the special subvariety of relative dimension 0. Consider the holomorphic symplectic form $\Omega$ on $M^{[n]}$. Assume that for all $i\in \c A$, $n_i=1$. Assume, furthermore, that the normalizarion $\tilde X$ of $X$ is smooth, and the pullback of $\Omega$ to $\tilde X$ is a nowhere degenerate holomorphic symplectic form on $\tilde X$. Then $\c A$ is empty. \footnote{To say that $\c A$ is empty is the same as to say that $X$ is a universal subvariety of $M$.} \hfill {\bf Proof:} Assume that $\c A$ is nonempty. By \ref{_G_M_invariant_implies_trianalytic_Theorem_}, all complex deformations of $X$ are trianalytic. Clearly, $\Delta_{[\alpha]}(\c A, \phi', \{\psi\}\times \{s\})$ is a deformation of $X$ for every $\phi':\; \c A {\:\longrightarrow\:} M$. Therefore, we may assume that $\phi:\; \c A {\:\longrightarrow\:} M$ is an embedding. Let $x\in \Delta_{(\alpha)}$ be an arbitrary point. We represent $x$ as in \eqref{_Del-a_alpha_definition_Equation_}. The {\bf $i$-th component} of $x$ is $x_{\left(\sum_{i=1}^{i} n_1\right)+1}$, in notation of \eqref{_Del-a_alpha_definition_Equation_}. The components are defined up to a permutation $i {\:\longrightarrow\:} j$, for $i$, $j$ satisfying $n_i=n_j$. Fix $p, q\in \{1, ... , k\}$, $p\in \c A$, $q\notin \c A$. Let $\Pi_{pq}:\; X {\:\longrightarrow\:} \{\phi(q)\} \times M$ be the map associating to $x\in X$ the $n_p$-th and $n_q$-th components of $\pi(x) \in \Delta_{(\alpha)} \subset M^{(n)}$. Clearly, the map $\Pi_{pq}$ is correctly defined. Let $\check \Pi_{pq}:\; X {\:\longrightarrow\:} M^{(n-n_p-n_q)}$ be the map associating to $x\in X$ the rest of components $x_{\left(\sum_{i=1}^{i} n_1\right)+1}$, $i \neq p, q$ of $x$. Let $2^M$ be the set of subsets of $M$. Consider the map $c:\; M^{(i)}{\:\longrightarrow\:} 2^M$ associating to $x\in M^{(i)}$ the corresponding subset of $M$. For $t= (\phi(q), t_0)\in \{\phi(q)\} \times M$, denote by $c(t)$ the subset $\{ \phi(q), t_0\} \subset M$. Let $X_0\subset X$ be the set of all $x\in X$ such that $c(\Pi_{pq}(x))$ does not intersect eith $c(\check\Pi_{pq}(x))$. A subvariety $C\subset X$ is called non-degenerately symplectic if the holomorphic symplectic form on $X$ is nowhere degenerate on $C$. For any $t\in\check \Pi_{pq}(X)\subset M^{(n-n_p-n_q)}$, the intersection $X_t :={\check \Pi_{pq}}^{-1}(t) \cap X_0$ is smooth. The holomorphically symplectic form in a tangent space to a zero-dimensional sheaf $S\in M^{[n]}$, $Sup(S) = A \coprod B$ can be computed separately for the part with support in $A$ and the part with support in $B$. Thus, $X_t$ must be non-degenerately symplectic. On the other hand, $X_t={\check \Pi_{pq}}^{-1}(t) \cap X_0$ is easy to describe explicitly. Let $M_0= M\backslash c(t)$, where $c(t)$ is again $t$ considered as a subset on $M$. Then $X_t$ is canonically isomorphic to a blow-up of $M_0$ in $\{\phi(q)\}$. This blow-up is obviously not non-degerenerately symplectic. We obtained a contradiction. This concludes the proof of \ref{_holo_symple_degene_on_speci_Proposition_}. \blacksquare \subsection{Applications for trianalytic subvarieties} \ref{_holo_symple_degene_on_speci_Proposition_} implies the following theorem, which is the main result of this section. \hfill \theorem\label{_triana_subva_universal_Theorem_} Let $M$ be a complex K3 surface with no automorphisms which is Mum\-ford-\-Tate generic with respect to some hyperk\"ahler structure. Consider the Hilbert scheme $M^{[n]}$ as a complex manifold. Let $\c H$ be an arbitrary hyperk\"ahler structure on $M^{[n]}$ agreeing with this complex structure. Consider a trianalytic subvariety $X\subset M^{[n]}$. Then $X$ is a universal subvariety of $M^{[n]}$ of relative dimension 0. {\bf Proof:} By \ref{_n_i_=1_for_i_in_A_Proposition_}, $X$ is a special subvariety of $M^{[n]}$, $X=\Delta_{[\alpha]}(\c A, \phi, \{\psi\}\times \{s\})$. The $X$ is non-degenerately symplectic because it si trianalytic. Applying \ref{_holo_symple_degene_on_speci_Proposition_}, we obtain that $\c A$ is empty, and $X$ is universal in $M^{[n]}$. \blacksquare \hfill \corollary In assumptions of \ref{_triana_subva_universal_Theorem_}, $\operatorname{codim}_{\Bbb C} X \geq 4$, unless $X= M^{[n]}$. {\bf Proof:} \ref{_special_subva_explici_Theorem_} classifies universal subvarieties of relative dimension 0. All such subvarieties correspond to diagonals $\Delta_{(\alpha)}\subset M^{(n)}$, with \[ \alpha = (n_1 \geq n_2 ...\geq n_k), \ \ \sum n_i = n, \] with all $n_i$ of form $\frac{l(l+1)}{2}$, with integer $l$'s. Thus, for $X\neq M^{[n]}$ we have $n_1\geq 3$. On the other hand, $\operatorname{codim}_{\Bbb C} \Delta_{(\alpha)} = 2 \sum (n_i-1)$, so $\operatorname{codim}_{\Bbb C} \Delta_{(\alpha)}\geq 4$. Finally, since $X$ is of relative dimension $0$, $\dim X = \dim \Delta_{(\alpha)}$, so $\operatorname{codim}_{\Bbb C} X =\operatorname{codim}_{\Bbb C} \Delta_{(\alpha)}\geq 4$. \blacksquare \section{Universal subvarieties of the Hilbert scheme and algebraic properties of its cohomology} \label{_last_Section_} \subsection{Birational types of algebraic subvarieties of relative dimension 0} Let $M^{[n]}$ be a Hilbert scheme, $\alpha$ a Young diagram satisfying assumptions of \ref{_unive_subva_rela_dime_0_Proposition_} and $\c X_\alpha\subset M^{[n]}$ the corresponding universal subvariety of relative dimension 0. Consider the natural map $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ mapping the Hilbert scheme to the symmetric product of $n$ copies of $M$. Clearly, $\pi(\c X_\alpha)= \Delta_{(\alpha)}$, where $\Delta_{(\alpha)}$ is the stratum of $M^{(n)}$ corresponding to the Young diagram $\alpha$ as in Subsection \ref{_subva_of_symme_special_Subsection_}. Moreover, from the definition of $\c X_\alpha$ it is evident that $\pi:\; \c X_\alpha{\:\longrightarrow\:} \Delta_{(\alpha)}$ is a birational isomorphism. \hfill Let \[ \alpha = \left(\vphantom{\sum} n_1 = , ... , n_{i_1}> n_{i_1+1}=, ... , =n_{i_1+i_2} > ,..., > n_{1+\sum_{j=1}^{l-1} i_j}= , ..., n_{\sum_{j=1}^{l} i_j}\right) \] be a diagram satisfying assumptions of \ref{_unive_subva_rela_dime_0_Proposition_}. Then $\Delta_{(\alpha)}$ is birationally isomorphic to the product $\prod_{j=1}^l M^{(i_j)}$. Thus, $\c X_\alpha$ is birational to a hyperka\"hler manifold $\prod_{j=1}^l M^{[i_j]}$. \hfill \theorem\label{_hype_bira_Theorem_} (Mukai) Let $f:\; X_1 {\:\longrightarrow\:} X_2$ be a birational isomorphism of compact complex manifolds of hyperka\"hler type. Then the second cohomology of $X_1$ is naturally isomorphic to the second cohomology of $X_2$, and this isomorphism is compatible with the Hodge structure and the Bogomolov-Beauville\footnote{For the definition and properties of Bogomolov-Beauville form, see Subsection \ref{_B_B_Hilbe_Subsection_}.} form on $H^2(X_1)$, $H^2(X_2)$. {\bf Proof:} Well known; see, e. g. \cite{_Mukai:Sugaku_}, \cite{_Huybrechts_}. \blacksquare \hfill We obtain that $H^2(\c X_\alpha)$ is isomorphic to $\oplus_{j=1}^l H^2(M^{[i_j]})$. Therefore, $\dim H^{2,0}(\c X_\alpha)>1$ unless $l=1$. On the other hand, by Bogomolov's decomposition theorem (\ref{_simple_mani_crite_Theorem_}), a hyperka\"hler manifold with $H^1(X, {\Bbb R})=0$, $\dim H^{2,0}(X)>1$ is canonically isomorphic to a product of two hyperka\"hler manifolds of positive dimension. We obtain the following result. \hfill \proposition \label{_triana_bira_M^(l)_Proposition_} Let $M$ be a complex K3 surface with no complex automorphisms, admitting a hyperk\"ahler structure $\c H$ such that $M$ is Mumford-Tate generic with respect to $\c H$. Let $\c X_\alpha$ be a trianalytic subvariety of $M^{[n]}$, which is by \ref{_triana_subva_universal_Theorem_} universal and corresponds to a Young diagram $\alpha = (n_1\geq n_2\geq ...\geq n_l)$. Assume that $\c X_\alpha$ is not isomorphic to a product of two hyperka\"hler manifolds of positive dimension. Then $n_1 = n_2 =, ..., =n_l$, and $\c X_\alpha$ is birationally equivalent to $M^{[l]}$. \blacksquare \subsection {The Bogomolov-Beauville form on the Hilbert scheme} \label{_B_B_Hilbe_Subsection_} Let $X$ be a simple\footnote{See \ref{_simple_mani_crite_Theorem_} for the definition of ``simple''.} hyperka\"hler manifold. It is well known that $H^2(X)$ is equipped with a natural non-degenerate symmetric pairing \[ (\cdot, \cdot)_{\c B}:\; H^2(X) \times H^2(X) {\:\longrightarrow\:} {\Bbb C} \] which is compatible with the Hodge structure and with the $SU(2)$-action. This pairing is defined up to a constant multiplier, and it is a topological invariant of $X$. For a formal definition and basic properties of this form, see \cite{_Beauville_} (Remarques, p. 775), and also \cite{_Verbitsky:cohomo_}, \cite{_coho_announce_}. \hfill For a Hilbert scheme $M^{[n]}$ of points on a K3 surface, the form $(\cdot, \cdot)_{\c B}$ can be computed explicitly as follows. \hfill Consider the map $\pi:\; M^{[n]}{\:\longrightarrow\:} M^{(n)}$ from the Hilbert scheme to the symmetric power of $M$. Clearly, the space $H^2(M^{(n)})$ is naturally isomorphic to $H^2(M)$. Let $\Delta_{n}\subset M^{[n]}$ be the singular locus of the map $\pi$, and $[\Delta_{n}]\in H^2(M^{[n]})$ its fundamental class. The following proposition gives a full description of the Bogomolov-Beauville form on $H^2(M^{[n]})$ in terms of the Poincare form on $H^2(M)$. \hfill \proposition\label{_B-B_on_H^2(M^[n])_Proposition_} Let $M$ be a K3 surface, and $M^{[n]}$ its Hilbert scheme of points. Consider the pullback map $\pi^*:\; H^2(M) = H^2(M^{(n)}) {\:\longrightarrow\:} H^2(M^{[n]})$. Then \begin{description} \item[(i)] The map $\pi^*:\; H^2(M) {\:\longrightarrow\:} H^2(M^{[n]})$ is an embedding. We have a direct sum decomposition \begin{equation} \label{_H^2_M^[n]_decompo_Equation_} H^2(M^{[n]}) = \pi^*(H^2(M)) \oplus {\Bbb C} \cdot [\Delta_{n}], \end{equation} where $[\Delta_{n}]\in H^2(M^{[n]})$ is the cohomology class defined above. \item[(ii)] The decomposition \eqref{_H^2_M^[n]_decompo_Equation_} is orthogonal with respect to the Bogomolov-Beauville form $(\cdot, \cdot)_{\c B}$. The restriction of $(\cdot, \cdot)_{\c B}$ to \[ H^2(M) = \pi^*(H^2(M))\subset H^2(M^{[n]})\] is equal to the Poincare form times constant. Since the form $(\cdot, \cdot)_{\c B}$ is defined up to a constant multiplier, we may assume that, after a rescaling, $(\cdot, \cdot)_{\c B}\restrict {H^2(M)}$ is equal to the Poincare form. \item[(iii)] After a rescaling required by (ii), we have \[ ( [\Delta_{n}], [\Delta_{n}])_{\c B} = -2 (n-1). \] \end{description} {\bf Proof:} Well known; see, for instance, \cite{_Huybrechts:hyperkahle_} 2.2. \blacksquare \hfill Further on, we always normalize the Bo\-go\-mo\-lov-\-Beau\-ville form on \linebreak $H^2(M^{[n]})$ as in \ref{_B-B_on_H^2(M^[n])_Proposition_} (ii). \subsection{Frobenius algebras associated with vector spaces} Let $X$ be a compact hyperk\"ahler manifold. The algebraic structure of $H^*(X)$ is studied using the general theory of Lefschetz-Frobenius algebras, introduced in \cite{_Lunts-Loo_}. For details of definitions and computations, the reader is referred to \cite{_Verbitsky:cohomo_}, \cite{_coho_announce_}. \hfill \definition Let $A= \bigoplus\limits^{2d}_{i=0} A_i$ be a graded commutative associative algebra over a field of characteristic zero. Assume that $A_{2d}$ is 1-dimensional, and the natural linear form $\epsilon:\; A {\:\longrightarrow\:} A_{2d}$ projecting $A$ to a $A_{2d}$ gives a non-degenerate scalar product $a, b {\:\longrightarrow\:} \epsilon(ab)$. Then $A$ is called {\bf a graded commutative Frobenius algebra}, or Frobenius algebra for short. \hfill \proposition \label{_S^*_algebra_Fro_Proposition_} Let $V$ be a vector space equipped with a non-\-de\-ge\-ne\-rate scalar product, and $n$ a positive integer number. Then there exist a unique up to an isomorphism Frobenius algerba \[ A(V, n) = A_0 \oplus A_2 \oplus .. \oplus A_{4n} \] such that \begin{description} \item[(i)] \begin{equation*}\begin{split} A_{2i} = S^i(V), \ \ \ &\text{for $i\leq n$} \\ A_{2i} = S^{2n-i}(V), \ \ \ &\text{for $i\geq n$} \end{split} \end{equation*} and \item[(ii)] For an operator $g\in SO(V)$, consider the corresponding endomorphism of $S^*(V)$. This way, $g$ might be considered as a linear operator on $A$. Then $g$ is an algebra automorphism. \end{description} {\bf Proof:} \ref{_S^*_algebra_Fro_Proposition_} is elementary. For a complete proof of existence and uniqueness of $A(V, n)$, see \cite{_Verbitsky:cohomo_}. \blacksquare \hfill The importance of the algebra $A(V, n)$ is explained by the following theorem. \hfill \theorem\label{_stru_of_H^*_Theorem_} \cite{_Verbitsky:cohomo_} Let $X$ be a compact connected simple hyperk\"ahler manifold. Consider the space $V= H^2(X)$, equipped with the natural scalar product of Bogomolov-Beauville (Subsection \ref{_B_B_Hilbe_Subsection_}). Let $A$ be a subalgebra of $H^*(X)$ generated by $H^2(M)$. Then $A$ is naturally isomorphic to $A(V,n)$. \blacksquare \subsection{A universal embedding from a K3 to its Hilbert scheme} Consider a universal embedding $M \stackrel \phi \hookrightarrow M^{[n]}$, $n = \frac{k (k-1)}{2}$, $n >1$, mapping a point $x\in M$ to subscheme given by the ideal $({\mathfrak m}_x)^k$, where ${\mathfrak m}_x$ is the maximal ideal of $x$. Pick a hyperka\"hler structure $\c H$ on $M^{[n]}$. The aim of this subsection is to prove the following result. \hfill \proposition \label{_ima_of_M_not_triana_Proposition_} The image of $\phi$ is not trianalytic in $M^{[n]}$. \hfill The proof of \ref{_ima_of_M_not_triana_Proposition_} takes the rest of this section. Together with \ref{_unive_subva_rela_dime_0_Proposition_} and \ref{_gene_fini_universa_Theorem_}, this result immediately implies the following corollary. \hfill \corollary Let $M$ be a complex K3 surface without automorphisms. Assume that $M$ admits a hyperk\"ahler structure $\c H$ such that $M$ is Mumford-Tate generic with respect to $\c H$ (\ref{_generic_manifolds_Definition_}). Pick a hyperka\"hler structure $\c H'$ on its Hilbert scheme $M^{[n]}$ ($n>1$). Let $X\subset M^{[n]}$ be a trianalytic subvariety of $M^{[n]}$. Then $\dim_{\Bbb H} X>1$. \blacksquare \hfill {\bf Proof of \ref{_ima_of_M_not_triana_Proposition_}:} Consider the map \[ \phi^*:\; H^4(M^{[n]}) {\:\longrightarrow\:} H^4(M) = {\Bbb C}. \] To prove that $\operatorname{im} \phi$ is not trianalytic in $M^{[n]}$, it suffices to show that $\phi^*$ is not $SU(2)$-invariant. Let $H^4_r(M^{[n]})$ be the subspace of $H^4(M^{[n]})$ generated by $H^2(M^{[n]})$, and $f:\; H^4_r(M^{[n]}){\:\longrightarrow\:} {\Bbb C}$ the restriction of $\phi^*$ to $H^4_r(M^{[n]})\subset H^4(M^{[n]})$. Since the subspace $H^4_r(M^{[n]})\subset H^4(M^{[n]})$ is $SU(2)$-invariant, the map $f$ should be $SU(2)$-invariant if $\phi^*:\; H^4(M^{[n]}) {\:\longrightarrow\:} H^4(M) = {\Bbb C}$ is $SU(2)$-invariant. \hfill By \ref{_stru_of_H^*_Theorem_}, the space $H^4_r(M^{[n]})$ is naturally isomorphic to \[ S^2H^2(M^{[n]}).\] Thus, $f$ can be considered as a map $f:\; S^2H^2(M^{[n]}){\:\longrightarrow\:} {\Bbb C}$. Consider the Bo\-go\-mo\-lov-\-Beau\-ville form as another such map $B:\; S^2 H^2 (M^{[n]}){\:\longrightarrow\:} {\Bbb C}$. From \ref{_B-B_on_H^2(M^[n])_Proposition_}, it is clear that $f = B + 2 (n-1) d^2$, where $d:\; H^2(M^{[n]}){\:\longrightarrow\:} C$ is the projection of $H^2(M^{[n]})$ to the component ${\Bbb C} = {\Bbb C} \cdot [\Delta_n]$ of the decomposition \eqref{_H^2_M^[n]_decompo_Equation_}. Since $B$ is $SU(2)$-invariant, the map $f$ is $SU(2)$-invariant if and only if the map $d^2:\; S^2H^2(M^{[n]}){\:\longrightarrow\:} {\Bbb C}$ is $SU(2)$-invariant. Therefore, the following claim is sufficient to prove \ref{_ima_of_M_not_triana_Proposition_}. \hfill \claim \label{_d^2_not_SU(2)_inv_Claim_} In the above notations, the vector $d^2\in S^2H^2(M^{[n]})^*$ is not $SU(2)$-invariant. \hfill {\bf Proof:} Let $V$ be the $SU(2)$-subspace of $S^2H^2(M^{[n]})^*$ generated by $d^2$. Acting on $d^2$ by various $g\in \mathfrak{su}(2)$, we can obtain any element of type $d \cdot g(d)$ (this follows from Leibnitz rule). Therefore, $V = SU(2) \cdot d^2$ contains $d\otimes V_0$, where $V_0 \subset H^2(M^{[n]})^*$ is the $SU(2)$-subspace of $H^2(M^{[n]})^*$ generated by $d$. Acting on $d \cdot g(d)$ by various $h\in SU(2)$, we obtain any element of type $h(d) \cdot h(gd)$. This implies that $V = SU(2) \cdot d\otimes V_0$ contains $S^2(V_0)$. We obtained that $V = S^2 V_0$. Clearly, $d^2$ is $SU(2)$-invariant if and only if $V$ is 1-dimensional. Thus, $d^2$ is $SU(2)$-invariant if and only if $V_0$ is 1-dimensional, that is, if $d$ is $SU(2)$-invariant. On the other hand, the map $d$ is an orthogonal projection to ${\Bbb C} \cdot [\Delta_n]\subset H^2(M^{[n]})$. Thus, $d$ is $SU(2)$-invariant if and only if $[\Delta_n]\in H^2(M^{[n]})$ is $SU(2)$-invariant. The class $[\Delta_n]\in H^2(M^{[n]})$ is a fundamental class of a subvariety $\Delta_n \subset M^{[n]}$. By \ref{_G_M_invariant_implies_trianalytic_Theorem_}, $[\Delta_n]$ is $SU(2)$-invariant if and only if $\Delta_n \subset M^{[n]}$ is trianalytic. The trianalytic subvarieties are hyperka\"hler, outside of singularities. Since $\Delta_n$ is a divisor, it has odd complex dimension and cannot be hyperka\"hler. Thus, the class $[\Delta_n]$ is not $SU(2)$-invariant, and the map $d^2:\; S^2H^2(M^{[n]}){\:\longrightarrow\:} {\Bbb C}$ is not $SU(2)$-invariant. This proves \ref{_d^2_not_SU(2)_inv_Claim_}. \ref{_ima_of_M_not_triana_Proposition_} is proven. \blacksquare \subsection{Universal subvarieties of the Hilbert scheme and Bo\-go\-mo\-lov-\-Beau\-ville form} In this subsection, we show that the subvarieties $\c X_\alpha\subset M^{[n]}$, obtained as in \ref{_triana_bira_M^(l)_Proposition_}, are not trianalytic. We prove this using the explicit calculation of the Bogomolov-Beauville form on $M^{[i]}$ (\ref{_B-B_on_H^2(M^[n])_Proposition_}) and the following result. \hfill \claim \label{_pullback_B-B_SU(2)_inv_Proposition_} Let $\phi:\; X \hookrightarrow Y$ be a morphism of compact hyperka\"hler manifolds. Consider the corresponding pullback map \[ \phi^*:\; H^2(Y) {\:\longrightarrow\:} H^2(X).\] Let \[ \Psi:\; S^2H^2(Y) {\:\longrightarrow\:} S^2H^2(X) \] be the symmetric square of $\phi^*$, and $B_Y\in S^2H^2(Y)$ the vector corresponding to the Bogomolov-Beauville pairing. Then $\Psi(B_Y)$ is $SU(2)$-invariant, with respect to the natural action of $SU(2)$ on $S^2 H^2(X)$. \hfill {\bf Proof:} It is well known that, for every morphism of hyperka\"hler varieties, the pullback map is compatible with the $SU(2)$-action in the cohomology. To see this, one may notice that the $SU(2)$-action is obtained from the Hodge-type grading associated with induced complex structures, and the pullback is compatible with the Hodge structure. Now, $B_Y$ is $SU(2)$-invariant, and therefore, $\Psi(B_Y)$ is also $SU(2)$-invariant. \blacksquare \hfill Let $M$ be a K3 surface, $M^{[n]}$ its Hilbert scheme and $\c X_\alpha$ be a universal subvariety of $M^{[n]}$ of relative dimension 0, obtained from the Young diagram $(n_1 = , ..., = n_l)$ as in \ref{_triana_bira_M^(l)_Proposition_}. Assume that $\c X_\alpha$ is trianalytic with respect to some hyperka\"hler structure on $M^{[n]}$. The manifold $\c X_\alpha$ is birational to $M^{[l]}$ (\ref{_triana_bira_M^(l)_Proposition_}). By \ref{_hype_bira_Theorem_}, there is a natural isomorphism $H^2(\c X_\alpha)\cong H^2(M^{[l]})$, and this isomorphism is compatible with the Bogomolov-Beauville form. Consider the map $\phi^*:\; H^2(M^{[n]}) {\:\longrightarrow\:} H^2(\c X_\alpha) = H^2(M^{[l]})$. Recall that $H^2(M)$ is considered as a subspace of $H^2(M^{[i]})$, for all $i$ (\ref{_B-B_on_H^2(M^[n])_Proposition_} (i)) Clearly, $\phi^*$ acts as identity on the subspaces $H^2(M) \subset H^2(M^{[n]})$, $H^2(M) \subset H^2(M^{[l]})$. Consider the pullback $\phi^*([\Delta_n])\in H^2(\c X_\alpha) = H^2(M^{[l]})$ of $[\Delta_n]\in H^2(M^{[n]})$. \hfill \lemma\label{_pullba_of_Delta_Lemma_} In the above notations, $\phi^*([\Delta_n]) = \frac{n}{l} [\Delta_l]$. \hfill {\bf Proof:} Let $t\in H^2(M) \subset H^2(M^{[l]})$ be the component of $\phi^*([\Delta_n])$ corresponding to the decomposition \eqref{_H^2_M^[n]_decompo_Equation_}. Since the decomposition \eqref{_H^2_M^[n]_decompo_Equation_} is integer, $t$ is an integer cohomology class. Let $\c M$ be a universal K3 surface, considered as a fibration over the moduli space $D$ of marked K3 surfaces. The construction of the Hilbert scheme can be applied to the fibers of $\c M$. We obtain a universal Hilbert scheme $\c M^{[n]}$, which is a fibration over $D$. Since $\c X_\alpha$ is a universal subvariety of $M^{[n]}$, there is a corresponding fibration over $D$ as well. Consider the class $t\in H^2(M)$ as a function $t(I)$ of the complex structure $I$ on $M$. Since the cohomology class $t(I)$ is integer, and $D$ is connected, $t(I)$ it is independent from $I\in D$. On the other hand, $t(I)$ has type $(1,1)$ with respect to $I$. There are no non-zero cohomology classes $\eta\in H^2(M)$ which have type $(1,1)$ with respect to all complex structures on $M$. Thus, $t=0$. We obtain that $\phi^*([\Delta_n]) = r [\Delta_l]$, where $r$ is some integer number. It remains to check that $r= \frac{n}{l}$. Recall that $\frac{n}{l}$ is an integer number which is equal to $\frac{k(k-1)}{2}$, for some $k\in {\Bbb Z}$, $k>1$. The points of the Hilbert scheme $M^{[i]}$ correspond to ideals $I\in{\cal O}_{M}$, $\dim {\cal O}_{M}/I =i$. Consider a rational map $\xi:\; M^{[l]} {\:\longrightarrow\:} M^{[n]}$ mapping an ideal $I\subset {\cal O}_{M}$ to $I^k$. Let $S\subset M^{[l]}$ be the union of all strata $\Delta_{[\alpha]}$ of codimension more than 1. It is easy to check that $\xi$ is well defined outside of $S$: for all ideals $I\in M^{[l]}\backslash S$, the ideal $I^k$ satisfies $\dim {\cal O}_{M}/I^k =n$. Consider the pullback map on the cohomology associated with the morphism $\psi:\; M^{[l]}\backslash S {\:\longrightarrow\:} M^{[n]}$. Clearly, $H^2(M^{[l]}\backslash S)= H^2(M^{[l]})$. The map $\phi^*:\; H^2(M^{[n]}) {\:\longrightarrow\:} H^2(\c X_\alpha) = H^2(M^{[l]})$ is equal to \[ \xi^*:\; H^2(M^{[n]}) {\:\longrightarrow\:} H^2(M^{[l]}\backslash S)= H^2(M^{[l]}) \] Let $p\in \pi_2(M^{[l]}\backslash S)$ be an element of the second homotopy group corresponding to $[\Delta_l]$ under Gurevich isomorphism. It remains to show that $\xi(p)$ is equal to $\frac{n}{l}$ times the element of the second homotopy group $\pi_2(M^{[n]})$ corresponding to $[\Delta_n]$. The following observation is needed to understand the geometry of $\Delta_i$. \begin{equation}\label{_Arti_leng_2_Equation_} \begin{minipage}[m]{0.8\linewidth} Closed Artinian subschemes $\xi\subset M$ of length 2 with support in $x\in M$ are in one to one correspondence with the vectors of projectivization of $T_x M$. \end{minipage} \end{equation} Therefore, generic points of $\Delta_i$ correspond to the triples $(\c X, x, \lambda)$, where $\c X$ is a non-ordered set of $(i-1)$ distinct points of $M$, $x\in M\backslash \c X$ a point of $M$ and $\lambda\in {\Bbb P} T_x M$ a line in $T_x M$. Fix $\c X$, $x$ and an isomorphism ${\Bbb P} T_x M\cong {\Bbb C} P^1$. We pick a map $p_x:\; S^2 {\:\longrightarrow\:} M^{[l]}$ in such a way that $p_x(\theta)= (\c X, x, \theta)$. Clearly, the corresponding element of $\pi_2(M^{[l]})$ is mapped to $[\Delta_l]$ by Gurevich's isomorphism. To simplify notations, we assume that $l=2$. It is easy to do the case of general $l$ in the same spirit as we do $l=2$. Let $U$ be a neighbourhood of $x$ in $M$. Taking $U$ sufficiently small, we may assume that $U$ is equipped with coordinates. Let $R_\lambda$ be a parallel translation of $U$ along these coordinates in the direction of $\lambda$, for $\lambda \in {\Bbb C}^2$. The map $R_\lambda$ is defined in a smaller neighbourhood $U'$ of $x$: $R_\lambda:\; U_1 {\:\longrightarrow\:} U$. The coordinates give a natural identification ${\Bbb C} P^1 \cong {\Bbb P} T_y M$, for all $y\in U$. Let $\lambda\in T_x M\backslash 0$ be a vector corresponding to $\theta\in {\Bbb P} T_x M$. Clearly, then, \[ p_x(\theta) = \lim\limits_{t\mapsto 0, t\in {\Bbb R}\backslash 0} \bigg\{x, R_{t\lambda}(x) \bigg\}, \] where the pair $\{x, R_{t\lambda}(x) \}$ is considered as a point in $M^{[2]}$. Let $y_i \in U_1^{[\frac{n}{2}]}\backslash \Delta_{\frac{n}{2}}$ be a sequence of points converging to $\phi'(x)$, where $\phi':\; M \mapsto M^{[\frac{n}{2}]}$ maps a maximal ideal of $x\in M$ to its $k$-th power. Denote the support of $y_i$ by $S_{y_i}$. Clearly, \[ \phi(p_x(\theta)) = \lim\limits_{t\mapsto 0, t\in {\Bbb R}\backslash 0} \left(\vphantom{\prod} \lim \limits_{i\mapsto\infty}\{y_i, R_{t\lambda}(y_i) \}\right). \] Taking limits in different order, we obtain \[ \phi(p_x(\theta))= \lim \limits_{i\mapsto\infty}\left(\vphantom{\prod} \lim\limits_{t\mapsto 0, t\in {\Bbb R}\backslash 0} \{y_i, R_{t\lambda}(y_i) \} = \lim \limits_{i\mapsto\infty} y_i(\theta)\right), \] where $y_i(\theta)\in \Delta_n$ is a point corresponding to a closed subscheme in $M^{[n]}$ with support $S_{y_i}$, of length 2 at every point of its support. For $y\in S_{y_i}$, consider the restriction $y_i(\theta)\restrict y$ of $y_i(\theta)$ to $y$, which is a closed Artinian subscheme of length 2 with support in $y$. Clearly, $y_i(\theta)\restrict y$ corresponds to $\theta\in {\Bbb P}T_y M$ as in \eqref{_Arti_leng_2_Equation_}. Varying $\theta$, we obtain a map $p_i:\; {\Bbb C} P^1 {\:\longrightarrow\:} \Delta_n \backslash S$, $\theta {\:\longrightarrow\:} y_i(\theta)$. By construction, this map is homotopic to $\phi(p_x)$. On the other hand, it is clear that $p_{y_i}$ is homotopic to $Card(S_{y_i})=\frac{n}{l}$ times a homotopy class represented by a map $p_x:\; {\Bbb C} P^1 {\:\longrightarrow\:} \Delta_n \backslash S$. This proves \ref{_pullba_of_Delta_Lemma_}. \blacksquare \hfill The Beauville-Bogomolov form identifies the second cohomology with its dual. Thus, this form can be considered as a tensor in the symmetric square of the second cohomology. To show that the embedding $\c X_\alpha \hookrightarrow M^{[n]}$ is not trianalytic, we compute the pullback $\phi^* B_{M^{[n]}}$ of the Beauville-Bogomolov tensor $B_{M^{[n]}}\in S^2 H^2(M^{[n]})$. Consider the decomposition \eqref{_H^2_M^[n]_decompo_Equation_} \begin{equation}\label{_H^2_M^[n]_decompo_again_Equation_} H^2(M^{[i]}) = H^2(M) \oplus {\Bbb C} \cdot \Delta_i. \end{equation} Let $P\in S^2H^2(M)$ be the tensor corresponding to the Poincare pairing. \ref{_B-B_on_H^2(M^[n])_Proposition_} computes the form $(\cdot,\cdot)_{\c B}\in S^2H^2(M)^*$ in terms of Poincare form and the decomposition \eqref{_H^2_M^[n]_decompo_again_Equation_}: $(\cdot,\cdot)_{\c B}= P - 2(n-1)d^2$. Therefore, the dual tensor can be written as $B_{M^{[i]}} = P - \frac{1}{2(n-1)}[\Delta_i]^2$. We have shown that $\phi^*$ acts as an identity on the first summand of \eqref{_H^2_M^[n]_decompo_again_Equation_}, and maps $[\Delta_n]$ to $\frac{n}{l}[\Delta_l]$. Therefore, \begin{equation} \label{_B_Ml_in_terms_of_phi_B_Mn_Equation_} \phi^* B_{M^{[n]}} = P - \frac{1}{2(n-1)}\frac{n}{l}[\Delta_l]^2 = B_{M^{[l]}} + \left(\frac{1}{2(l-1)} -\frac{1}{2(n-1)}\frac{n}{l}\right) [\Delta_l]^2 \end{equation} By \ref{_d^2_not_SU(2)_inv_Claim_}, $[\Delta_l]^2$ is not $SU(2)$-invariant. Since $B_{M^{[l]}}$ {\it is} $SU(2)$-invariant, $\phi^* B_{M^{[n]}}$ is $SU(2)$-invariant if and only if the coefficient of $[\Delta_l]^2$ in \eqref{_B_Ml_in_terms_of_phi_B_Mn_Equation_} vanishes: \begin{equation} \label{_coeffi_pullback_formula_Equation_} (l-1)^{-1} -(n-1)^{-1}\frac{n}{l} =0 \end{equation} Clearly, this happens only if $(n- \frac{n}{l})= n-1$, i. e. when $\frac{n}{l}=1$. By definition, $\frac{n}{l} = \frac{k(k+1)}{2}\geq 3$. Therefore, $\phi^* B_{M^{[n]}}$ is not $SU(2)$-invariant, and $\c X_\alpha$ is not trianalytic in $M^{[n]}$.\footnote{ We do not need the whole strength of \ref{_pullba_of_Delta_Lemma_} to show that the tensor $\phi^* B_{M^{[n]}}$ is not $SU(2)$-invariant. Let $t$ be the coefficient of \ref{_pullba_of_Delta_Lemma_}, $\phi^*([\Delta_n]) = t [\Delta_l]$. To show that $\phi^* B_{M^{[n]}}$ is not $SU(2)$-invariant, we need only to check that $n-1 \neq t (l-1)$, where $n= l\frac{k(k+1)}{2}$. Since $l\frac{k(k+1)}{2}-1$ is not divisible by $l-1$ for most $l$, $k$, the inequality $n-1 \neq t (l-1)$ holds automatically for most $l$, $n$.} Comparing this with \ref{_pullback_B-B_SU(2)_inv_Proposition_}, and \ref{_triana_bira_M^(l)_Proposition_}, we obtain the following result. \hfill \theorem \label{_no_triana_subva_of_Hilb_Theorem_} Let $M$ be a complex K3 surface without automorphisms. Assume that $M$ is Mumford-Tate generic with respect to some hyperka\"hler structure. Consider the Hilbert scheme $M^{[n]}$ of points on $M$. Pick a hyperk\"ahler structure on $M^{[n]}$ which is compatible with the complex structure. Then $M^{[n]}$ has no proper trianalytic subvarieties. \blacksquare \hfill \remark It is easy to see that a generic K3 surface has no complex automorphisms. \hfill \corollary \label{_no_comple_subva_of_gen_Hilb_Corollary_} Let $M$ be a complex K3 surface. Consider its Hilbert scheme $M^{[n]}$. Let $\c M$ be the generic deformation of $M^{[n]}$. Then $\c M$ has no complex subvarieties. {\bf Proof:} Follows immediately from \ref{_no_triana_subva_of_Hilb_Theorem_} and \ref{_hyperkae_embeddings_Corollary_}. \blacksquare \hfill {\bf Acknowledegments:} I am grateful to R. Bezrukavnikov and A. Beilinson for lecturing me on perverse sheaves and semismall resolutions, H. Nakajima for sending me the manuscript of \cite{_Nakajima_}, V. Lunts for fascinating discourse on the Mumford-Tate group, and to P. Deligne, M. Grinberg, D. Kaledin, D. Kazhdan and T. Pantev for valuable discussions. My gratitude to P. Deligne and D. Kaledin, who found important errors in the earlier versions of this paper.

9705

alg-geom/9705027

en

https://arxiv.org/abs/alg-geom/9705027

alg-geom/9705027

Kota Yoshioka

Kota Yoshioka

An application of exceptional bundles to the moduli of stable sheaves on a K3 surface

12 pages, AMS-Latex

Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v. If v is primitive, than it is expected that M(v) is deformation equivalent to some Hilbert scheme and weight two hogde structure can be described by H^*(X,Z). These are known by Mukai, O'Grady and Huybrechts if rank is 1 or 2, or the first Chern class is primitive. Under some conditions on the dimension of M(v), we shall show that these assertion are true. For the proof, we shall use Huybrechts's results on symplectic manifolds.

alg-geom

\section{Introduction} Let $X$ be a smooth projective surface defined over ${\Bbb C}$ and $L$ an ample divisor on $X$. For a coherent sheaf $E$ on $X$, let $v(E):=\operatorname{ch}(E)\sqrt{\operatorname{Td}(X)} \in H^*(X,{\Bbb Q})$ be the Mukai vector of $E$, where $\operatorname{Td}(X)$ is the Todd class of $X$. We denote the moduli of stable sheaves of Mukai vector $v$ by $M_L(v)$, where the stability is in the sense of Simpson [S]. For a regular surface $X$, exceptional vector bundles which were introduced by Drezet and Le-Potier [D-L] are very useful tool for analysing $M_L(v)$. For example, if $X={\Bbb P}^2$, then Maruyama [Ma1] showed the rationality of some moduli spaces and Ellingsrud and Str\o mme [E-S] computed relations of a generator of $H^*(M_L(v),{\Bbb Z})$. Drezet also obtained many interesting results [D1], [D2]. If $X$ is a K3 surface, G\"{o}ttsche and Huybrechts [G-H] computed Hodge numbers of rank 2 moduli spaces. Moreover Huybrechts [H1] showed that $M_L(v)$ is deformation equivalent to a rank 1 moduli space ( Hilbert scheme of points). Motivated by their results, we shall treat other rank cases. In particular, we shall prove the following asymptotic result. \begin{thm} Let $v=r+\xi+a \omega,\; \xi \in H^2(X,{\Bbb Z})$ be a primitive Mukai vector of $r>0$ and $\langle v^2 \rangle/2 >r^2$. Then $M_L(v)$ is deformation equivalent to $\operatorname{Hilb}_X^{\langle v^2 \rangle/2+1}$ and $$ \theta_v:v^{\perp} \to H^2(M(v),{\Bbb Z}) $$ is an isometry which preserves hodge structures, where $L$ is a general ample divisor. \end{thm} The second assertion is known by Mukai [Mu3] and O'Grady [O] if $r \leq 2$ or $\xi$ is primitive. For the proof of the second assertion, Mukai lattice and Mukai's reflection defined by an exceptional bundle give a clear picture. During preparation of this paper, the author noticed that Huybrechts [H2] proved birational irreducible symplectic manifolds are deformation equivalent. Then primitive first Chern class cases (Theorem \ref{thm:1}) follow from O'Grady's description of $M_L(v)$ (see [H2, Cor. 4.8] and [O]). Since our method is most successful for these cases and is needed to treat other cases (Theorem \ref{thm:2}), we shall first treat these cases. \section{Preliminaries} \subsection{Notation} Let $M$ be a complex manifold. For a cohomology class $x \in H^*(M,{\Bbb Z})$, $[x]_i \in H^{2i}(X,{\Bbb Z})$ denotes the $2i$-th component of $x$. Let $p:X \to \operatorname{Spec}({\Bbb C})$ be a K3 surface over ${\Bbb C}$. We shall recall the Mukai lattice [Mu2]. For $x,y \in H^*(X,{\Bbb Z})$, Mukai defined a symmetric bilinear form $$ \langle x, y \rangle:=-p_*(x^{\vee} y) , $$ where $\vee:H^*(X,{\Bbb Z}) \to H^*(X,{\Bbb Z})$ be the homomorphism sending $x \in H^*(X,{\Bbb Z}) \to x-2[x]_1 \in H^*(X,{\Bbb Z})$. For a coherent sheaf $E$ on $X$, let $v(E):=\operatorname{ch}(E)(1+\omega) \in H^*(X,{\Bbb Z})$ be the Mukai vector of $E$, where $\omega$ is the fundamental class of $X$. Then the Riemann-Roch theorem implies that $\chi(E,F)=-\langle v(E),v(F) \rangle$ for coherent sheaves $E$, $F$ on $X$. Let $N$ be a line bundle on $X$. Since $\langle x \operatorname{ch}(N),y \operatorname{ch}(N) \rangle= \langle x,y \rangle$, the homomorphism $T_N:H^*(X,{\Bbb Z}) \to H^*(X,{\Bbb Z})$ sending $x$ to $x \operatorname{ch}(N)$ is an isometry. Let $L$ be an ample divisor on $X$. For $v \in H^*(X,{\Bbb Z})$, let $M_L(v)$ be the moduli of stable sheaves of Mukai vector $v$, where the stability is in the sense of Simpson [S]. By Mukai [Mu1], $M_L(v)$ is smooth of dimension $ \langle v^2 \rangle+2$. We denote the projection $S \times X \to S$ by $p_S$. We set $$ v^{\perp}:=\{x \in H^*(X,{\Bbb Z})| \langle v, x \rangle=0 \}. $$ Then Mukai constructed a natural homomorphism $$ \theta_v:v^{\perp} \longrightarrow H^2(M_L(v),{\Bbb Z})_f $$ by $$ \theta_v(x):=\frac{1}{\rho} [p_{M_L(v)*}(\operatorname{ch}({\cal E})(1+\omega)x^{\vee})]_1, $$ where $H^2(M_L(v),{\Bbb Z})_f$ is the torsion free quotient of $H^2(M_L(v),{\Bbb Z})$ and ${\cal E}$ a quasi-universal family of similitude $\rho$. We note that an isometry $T_N, \;N \in \operatorname{Pic}(X)$ satisfies that \begin{equation}\label{eq:N} \theta_{T_N(v)}(T_N(x))=\theta_v(x). \end{equation} \section{Fundamental lemma} We shall prove the following lemma whose proof is quite similar to [Y, Lem. 1.8]. \begin{lem}\label{lem:key} Let $X$ be a smooth projective surface of $\operatorname{NS}(X) \cong {\Bbb Z} H$. For a coherent sheaf $F$ of $c_1(F)=dH$, we set $\deg(F)=d$. Let $r$ and $d$ be relatively prime positive integers and let $r_1$ and $d_1$ be the integers which satisfy $r_1d-rd_1=1$ and $0<r_1 \leq r$. We set $r_2:=r-r_1$ and $d_2:=d-d_1$. $(1)$ Let $E_1$ be a stable vector bundle of rank $r_1$ and $\deg(E_1)=d_1$ and $E_2$ a stable sheaf of rank $r_2$ and $\deg(E_2)=d_2$. Then every non-trivial extension $$ 0 \to E_1 \to E \to E_2 \to 0 $$ defines a stable sheaf. $(2)$ Let $E_1$ be a stable vector bundle of rank $r_1$ and $\deg(E_1)=d_1$ and $E$ a stable sheaf of rank $r$ and $\deg(E)=d$. Let $V$ be a subvector space of $\operatorname{Hom}(E_1,E)$. Then $V \otimes E_1 \to E$ is injective or surjective in codimension 1. Moreover if $V \otimes E_1 \to E$ is injective, then the cokernel is stable. \end{lem} \begin{pf} (1) We first treat the case $r_1<r$. If $E$ is not stable, then there is a semi-stable subsheaf $G$ of $E$ such that $\deg (G)/\operatorname{rk} G>d/r$. Since $G$ and $E_2$ are semi-stable and $\phi:G \to E\to E_2$ is not zero, $\deg (G)/\operatorname{rk} G \leq d_2/r_2$. We assume that $\deg (G)/\operatorname{rk} G<d_2/r_2$. Then we see that $1/rr_2=d_2/r_2-d/r>d_2/r_2-\deg (G)/\operatorname{rk} G \geq 1/r_2\operatorname{rk} G$, which is a contradiction. Hence $\deg (G)/\operatorname{rk} G=d_2/r_2$. Since $G$ is semi-stable and $E_2$ is stable, $\ker \phi$ is semi-stable of $\deg (\ker \phi)/\operatorname{rk}(\ker \phi)=d_2/r_2$ and $\phi$ is surjective in codimension 1. Hence $G$ is isomorphic to $E_2$ in codimension 1. Let $e \in \operatorname{Ext}^1(E_2,E_1)$ be the extension class. By the homomorphism $\operatorname{Ext}^1(E_2,E_1) \to \operatorname{Ext}^1(G,E_1)$, $e$ goes to $0$. Since $E_1$ is a vector bundle, $\operatorname{Ext}^1(E_2/G,E_1)=0$. Hence $\operatorname{Ext}^1(E_2,E_1) \to \operatorname{Ext}^1(G,E_1)$ is injective. Thus we get that $e=0$, which is a contradiction. If $r_1=r$, then $r=r_1=1$ and $d=d_2=1$. In this case, it is sufficient to prove that $E$ is torsion free. Let $G$ be the torsion subsheaf. Since $E_1$ is locally free, $G \to E_2$ is injective. Since $E_2$ is stable, $\Supp E_2/G$ is of codimension 2. In the same way, we see that the extension class is trivial, which is a contradiction. (2) We shall prove our claim by induction on $\dim V$. We assume that $\dim V=1$. Let $\varphi:E_1 \to E$ be a non-zero homomorphism. We first treat the case $r_1<r$. Since $E_1$ and $E$ are stable, $d_1/r_1 \leq \deg (\varphi(E_1))/\operatorname{rk} \varphi(E_1)<d/r$. In the same way as in the proof of (1), we see that $\operatorname{rk} \varphi(E_1)=r_1$ and $\deg (\varphi(E_1))=d_1$. Hence we get that $E_1 \cong \varphi(E_1)$. We set $E_2:=\operatorname{coker} \varphi$. We assume that there is a quotient $G$ of $E_2$ such that $G$ is semi-stable and $d_2/r_2>\deg (G)/\operatorname{rk} G$. Since $G$ is a quotient of $E$, we get that $d/r<\deg (G)/\operatorname{rk} G$. Hence we get that $d/r<\deg (G)/\operatorname{rk} G<d_2/r_2$. Then $1/rr_2=d_2/r_2-d/r>d_2/r_2-\deg (G)/\operatorname{rk} G \geq 1/r_2\operatorname{rk} G$, which is a contradiction. Hence the support of the torsion submodule of $E_2$ is 0 dimensional and the torsion free quotient is stable. Since $E_1$ is locally free, $E_2$ is torsion free and hence stable. If $r_1=r \;( \;= 1\;)$, then $\varphi$ is injective. We shall show that $\operatorname{coker} \varphi$ is of pure dimension $1$. Let $T$ be a subsheaf of dimension 0 and $\widetilde{T}$ the pull-back of $T$ to $E$. Then $E_1 \to \widetilde{T}$ is an isomorphism in codimension 1. Since $E_1^{\vee \vee} \cong \widetilde{T}^{\vee \vee}$ and $E_1$ is locally free, $E_1=\widetilde{T}$. Hence we get that $T=0$. We set $C:=\Supp E_2$. Since $H$ is a generator of the N\'{e}ron-Severi group, $C$ is reduced and irreducible. Hence $E_2$ is a torsion free sheaf of rank 1 on $C$, which shows that $E_2$ is a stable sheaf. We shall treat general cases. Let $\varphi_1,\varphi_2,\dots,\varphi_n$ be a basis of $V$ and $V'$ the subspace generated by $\varphi_2,\dots,\varphi_n$. We note that $r_1d_2-r_2d_1=1$. We assume that $r_1<r_2$. By induction hypothesis, $V' \otimes E_1 \to \operatorname{coker}(\varphi_1)$ is injective or surjective in codimension 1. If $r_1 > r_2>0$, in the same way, we see that $\varphi_2:E_1 \to \operatorname{coker}(\varphi_1)$ is surjective. If $r_1=r_2$ or $r_2=1$, then $r_1=d_2=1$. Since every degree 1 curve is reduced and irreducible, $V' \otimes E_1 \to \operatorname{coker}(\varphi_1)$ is injective or surjective in codimension 1. Hence we get our claim. \end{pf} \begin{cor}\label{cor:uni} Under the same assumption of Lemma \ref{lem:key} $(1)$, the universal extension $$ 0 \to E_1\otimes \operatorname{Ext}^1(E_2,E_1)^{\vee} \to E \to E_2 \to 0 $$ defines a stable sheaf. \end{cor} \section{Moduli of stable sheaves on K3 surfaces} \subsection{Stable sheaves of pure dimension 1} Let $H$ be an ample divisor on $X$. We set $v:=H+a \omega \in H^*(X,{\Bbb Z})$. For a pure dimensional sheaf $E$ of $v(E)=v$ and an ample divisor $L$, $\chi(E \otimes L^{\otimes n})=(H,L)n+b$. Hence $E$ is semi-stable with respect to $L$, if $$ \frac{\chi(F)}{(c_1(F),L)} \leq \frac{\chi(E)}{(c_1(E),L)} $$ for any proper subsheaf $F$ of $E$, and $E$ is stable if the inequality is strict. We can generalize the concept of the chamber. For a sheaf $F$ of pure dimension 1, We set $D:=\chi(F)c_1(E)-\chi(E)c_1(F)$ and $W_D:=\{x \in \operatorname{Amp}(X)|(x,D)=0\}$. Then $(D^2)=\chi(F)^2(c_1(E)^2)-2\chi(E)\chi(F)(c_1(E),c_1(F)) +\chi(E)^2(c_1(F)^2)$. If $W_D$ is not empty, then the Hodge index theorem implies that $(D^2) \leq 0$. Hence the choice of $\chi(F)$ is finite, which shows that the number of non-empty walls $W_D$ is finite. We shall call a chamber a connected component of $\operatorname{Amp}(X) \setminus \cup_D W_D$. If $c_1(E)$ is primitive, then $M_L(v)$ is compact for a general $L$. As is well known ([G-H, Prop. 2.2], cf. [Y, Prop. 3.3]), the choice of polarizations is not so important. Hence we denote $M_L(v)$ by $M(v)$. \subsection{Correspondence} Let $r$ and $d$ be relatively prime positive integers and let $r_1$ and $d_1$ be the integers which satisfy $r_1d-rd_1=1$ and $0<r_1 \leq r$. We shall consider a K3 surface $X$ such that $\operatorname{Pic}(X)= {\Bbb Z} H$, where $H$ is an ample divisor. We assume that there are Mukai vectors $v_1,v \in H^*(X,{\Bbb Z})$ such that \begin{equation} \begin{cases} v_1=r_1+d_1H+a_1 \omega,\\ v=r+dH+a \omega,\\ \langle v_1^2 \rangle =-2, \end{cases} \end{equation} where $a_1,a \in {\Bbb Z}$. Since $\langle v_1^2 \rangle=-2$, there is a unique stable vector bundle $E_1$ of $v(E_1)=v_1$, that is $M(v_1)=\{E_1 \}$. We note that $E_1$ satisfies that \begin{equation} \begin{cases} \operatorname{Hom}(E_1,E_1)={\Bbb C},\\ \operatorname{Ext}^1(E_1,E_1)=0,\\ \operatorname{Ext}^2(E_1,E_1)={\Bbb C}. \end{cases} \end{equation} \begin{defn} For $i \geq 1$, $$ M(v)_i:=\{E \in M(v)| \dim \operatorname{Hom}(E_1,E)=-\langle v_1,v \rangle-1+i \} $$ is a locally closed subscheme of $M(v)$ with reduced structure. \end{defn} \begin{defn} For $w:=v-mv_1$ with $[w]_0 \geq 0$, we set $$ N(mv_1,v,w):=\{E_1^{\oplus m} \subset E| E \in M(v) \}. $$ Let $\pi_v:N(mv_1,v,w) \to M(v)$ is the projection and $N(mv_1,v,w)_i=\pi_v^{-1}(M(v)_i)$. \end{defn} We shall construct $N(mv_1,v,w)$ as a scheme over $M(v)$. Let $Q(v)$ be the open subscheme of a canonical quot scheme $\operatorname{Quot}_{{\cal O}_X^{\oplus N}/X}$ such that $Q(v)/PGL(N) \cong M(v)$ and ${\cal O}_{Q(v) \times X}^{\oplus N} \to {\cal E}_v$ be the universal quotient on $Q(v) \times X$. Let $W_2 \to W_1 \to E_1 \to 0$ be a locally free resolution of $E_1$ such that $\operatorname{Ext}_{p_{Q(v)}}^i(W_j \boxtimes {\cal O}_{Q(v)},{\cal E}_v)=0$ for $i>0$ and $j=1,2$. We set ${\cal W}_j:= \operatorname{Hom}_{p_{Q(v)}}(W_j \boxtimes {\cal O}_{Q(v)},{\cal E}_v)$, $j=1,2$. We shall consider the Grassmannian bundle $\xi:{\Bbb G}=Gr({\cal W}_1,m) \to Q(v)$ parametrizing $m$-dimensional subspaces of $({\cal W}_1)_t, t \in Q(v)$. Let ${\cal U} \to \xi^*{\cal W}_1$ be the universal subbundle. For the composition $\Psi:{\cal U} \to \xi^*{\cal W}_1 \to \xi^*{\cal W}_2$, we set $D:=\{t \in {\Bbb G}| \Psi_t=0\}$. Then there is a universal family of homomorphisms on $D \times X$: $$ E_1 \boxtimes {\cal U} \to \xi^*{\cal E}_v. $$ By Lemma \ref{lem:key}, $m$-dimensional subspace $U$ of $\operatorname{Hom}(E_1,E)$ defines an injective homomorphism $E_1 \otimes U \to E$. Since ${\cal W}_1,{\cal W}_2, {\cal U}$ and $\Psi$ are $GL(N)$-equivariant, We get $N(mv_1,v,w) \to M(v)$ as a scheme $D/PGL(N) \to M(v)$. Since $F:=\operatorname{coker}(E_1 \otimes U \to E)$ is a stable sheaf, we also obtain a morphism $\pi_w:N(mv_1,v,w) \to M(w)$. We shall consider the fiber of $\pi_w$. For $F \in M(w)_{i+m}$, we get that $\dim \operatorname{Ext}^1(F,E_1)=i-1+m$. Let $U^{\vee}$ be a $m$-dimensional subspace of $\operatorname{Ext}^1(F,E_1)$. Then the element $e \in U \otimes \operatorname{Ext}^1(F,E_1)$ which corresponds to the inclusion $U^{\vee} \subset \operatorname{Ext}^1(F,E_1)$ defines an extension $$ 0 \to U \otimes E_1 \to E \to F \to 0. $$ By the choice of the extension class and Corollary \ref{cor:uni}, $E$ is a stable sheaf. Thus the fiber of $F$ is the Grassmannian $Gr(i-1+m,m)$. \begin{lem}\label{lem:cod} For $i \geq 1+\langle v,v_1 \rangle$, $$ \operatorname{codim} M(v)_i=(i-1)(i-1-\langle v,v_1 \rangle). $$ In particular, if $\langle v,v_1 \rangle \leq 0$, then $M(v)_1$ is an open and dense subscheme of $M(v)$. \end{lem} \begin{pf} Let $E$ be an element of $M(v)_i$. Then $\dim \operatorname{Ext}^1(E_1,E)=-\chi(E_1,E)+\dim \operatorname{Hom}(E_1,E) =i-1$. We shall consider the universal extension $$ 0 \to E_1^{\oplus (i-1)} \to G \to E \to 0. $$ By Corollary \ref{cor:uni}, $G$ is a stable sheaf of $\operatorname{rk}(G)=(i-1)r_1+r$ and $\deg(G)=(i-1)d_1+d$. Hence we get that $\operatorname{Ext}^2(E_1,G)=0$. Simple calculations show that \begin{align*} \dim \operatorname{Hom} (E_1,G)&=i-1+\dim \operatorname{Hom}(E_1,E)\\ &=2i-2-\langle v_1,v \rangle,\\ \dim \operatorname{Ext}^1(E_1,G)&=0. \end{align*} Hence $G$ belongs to $M(v(G))_1$. We set $k:=2i-2-\langle v_1,v \rangle$. Then the choice of $i-1$-dimensional subspace of $V:=\operatorname{Hom} (E_1,G)$ is parametrized by the Grassmannian $Gr(k,i-1)$. Since $v(G)=v+(i-1)v_1$, we see that \begin{align*} \dim M(v(G))&=\langle v(G)^2 \rangle+2\\ &=-2(i-1)^2+2(i-1)\langle v_1,v \rangle +\langle v^2 \rangle+2. \end{align*} Therefore we see that \begin{align*} \dim M(v)_i &=\dim M(v(G))+\dim Gr(k,i-1)\\ &=-(i-1)(i-1-\langle v_1,v \rangle)+\dim M(v), \end{align*} which implies our claim. \end{pf} \begin{rem}\label{rem:gr} We can also see that $M(v)_i$ is an \'{e}tale locally trivial $Gr(k,i-1)$-bundle over $M(v(G))_1$. Moreover we can show that $\pi_v:N(mv_1,v,w)_i \to M(v)_i$ is an \'{e}tale locally trivial $Gr(l,m)$-bundle and $\pi_w:N(mv_1,v,w)_i \to M(w)_{i+m}$ is an \'{e}tale locally trivial $Gr(i-1+m,m)$-bundle, where $l:=i-1-\langle v_1,v \rangle$. \end{rem} \subsection{A special case} We set $k(s):=s r_1^2+rr_1-r^2$, $s \in {\Bbb Z}$. For a positive integer $k(s)$, we shall consider a K3 surface $X$ such that $\operatorname{Pic}(X)={\Bbb Z} H$ of $(H^2)=2k(s)$. We set \begin{equation} \begin{cases} v_1=r_1+d_1H+(d_1^2r+d_1^2sr_1-r_1d^2+2d)\omega\\ v=r+dH+((2dd_1r_1-rd_1^2)s+d^2(r_1-r))\omega. \end{cases} \end{equation} Then a simple calculation shows that \begin{equation} \begin{cases} \langle v_1^2 \rangle=-2\\ \langle v^2 \rangle=2s\\ \langle v,v_1 \rangle=-1. \end{cases} \end{equation} \begin{lem}\label{lem:1} We assume that $r_1 \geq r/2$. Under the following assumptions, $k(s)>0$. \begin{enumerate} \item $r=2$ and $s \geq 3$, \item $r_1 \geq (2/3) r$ and $s \geq 1$. \end{enumerate} \end{lem} Let us consider the reflection $R_{v_1}$ of $H^*(X,{\Bbb Z})$ defined by $v_1$: $$ x \mapsto x+\langle x,v_1 \rangle v_1. $$ We set $w:=R_{v_1}(v)=v-v_1$. By Lemma \ref{lem:cod}, $N(v_1,v,w)$ gives a birational correspondence between $M(v)$ and $M(w)$. \begin{lem}\label{lem:smoo} $N(v_1,v,w)$ is smooth and irreducible. \end{lem} \begin{pf} In the same way as in [G-H], it is sufficient to prove that $\operatorname{Hom}(E,E_2)={\Bbb C}$. We assume that $\dim \operatorname{Hom}(E,E_2)>1$. Let $\phi:E \to E_2$ be a non-zero homomorphism. By Lemma \ref{lem:key}, $\phi$ is surjective in codimension 1 and we also see that $\ker \phi$ is stable. Since $[v(\ker \phi)]_i=[v_1]_i$, $i=0,1$, the stability of $\ker \phi$ implies that $[v(\ker \phi)]_2 \leq [v_1]_2$. Since $[v(\phi(E))]_2 \leq [w]_2$ and $[v(\phi(E))]_2+[v(\ker \phi)]_2 =[v]_2$, we must have $v(\ker \phi)=v_1$ and $v(\phi(E)) =w$. Hence $\phi$ is surjective and $\ker \phi \cong E_1$. Let $$ 0 \to E_1 \otimes \operatorname{Ext}^1(E,E_1)^{\vee} \to G \to E \to 0 $$ be the universal extension. Then $\ker \phi$ is determined by a $m$-dimensional subspace of $ \operatorname{Hom}(E_1,G)$ containing $\operatorname{Ext}^1(E,E_1)^{\vee}$, where $m=\dim \operatorname{Ext}^1(E,E_1)+1$. We shall consider the composition $\phi':G \to E \to E_2$. Then it defines a universal extension (up to the action of $\operatorname{Aut}(\operatorname{Ext}^1(E_2,E_1)^{\vee})$) $$ 0 \to E_1 \otimes \operatorname{Ext}^1(E_2,E_1)^{\vee} \to G \to E_2 \to 0. $$ Since $G$ is stable, $\phi'$ is determined by a subspace $\operatorname{Ext}^1(E_2,E_1)^{\vee}$ of $\operatorname{Hom}(E_1,G)$ up to multiplication by constants. Hence $\dim \operatorname{Hom}(E,E_2)=1$. \end{pf} Let us consider the relation between $\theta_v$ and $\theta_{w}$. By our assumption on $v$, there is a universal family ${\cal E}$ on $M(v) \times X$ ([M1]). Then $N(v_1,v,w)$ is constructed as a projective bundle ${\Bbb P}(V)$, where $V:=\operatorname{Ext}^2_{p_{M(v)}}({\cal E},{\cal O}_{M(v)}\boxtimes(E_1 \otimes K_X))$, and there is a universal homomorphism $\Psi:{\cal O}_{N(v_1,v,w)}\boxtimes E_1 \to \pi_v^*{\cal E} \otimes {\cal O}_{N(v_1,v,w)}(1)$. We denote the cokernel by ${\cal G}$. Then ${\cal G}$ is flat over $N(v_1,v,w)$. Since $\operatorname{codim}(M(v)_i) \geq 2$ for $i \geq 2$, we see that $\det p_{N(v_1,v,w)!} ({\cal G} \otimes q^*(E_1^{\vee})) \cong {\cal O}_{N(v_1,v,w)}$. Therefore we see that \begin{align*} [p_{N(v_1,v,w)*}(\operatorname{ch}({\cal E}\otimes {\cal O}_{N(v_1,v,w)}(1)) (1+\omega)x^{\vee})]_1 & =[p_{N(v_1,v,w)*}(\operatorname{ch}({\cal G})(1+\omega)x^{\vee})]_1\\ & =[p_{N(v_1,v,w)*}(\operatorname{ch}({\cal G})(1+\omega) (x+\langle v_1,x \rangle v_1)^{\vee})]_1. \end{align*} Hence we get that $\theta_v(x)=\theta_{R_{v_1}(v)}(R_{v_1}(x))$ for $x \in v^{\perp}$. \begin{prop}\label{prop:bilin} The following diagram is commutative. \begin{equation*} \begin{CD} v^{\perp} @>{R_{v_1}}>> w^{\perp}\\ @V{\theta_v}VV @VV{\theta_{w}}V\\ H^2(M(v),{\Bbb Z})_f @>>{\pi_{w*}\pi_v^*}> H^2(M(w),{\Bbb Z})_f \end{CD} \end{equation*} \end{prop} Let us consider the structure of $N(v_1,v,w)_i$ more closely. For simplicity we assume that $i=2$. We set $u:=v+v_1$. Let $Q(u)$ be the open subscheme of a quot scheme $\operatorname{Quot}_{{\cal O}_X^{\oplus N}/X}$ such that $Q(u)/PGL(N) \cong M(u)$ and let ${\cal O}_{Q(u) \times X}^{\oplus N} \to {\cal E}_u$ be the universal quotient on $Q(u) \times X$. We set $V:=\operatorname{Hom}_{p_{Q(u)_1}}(E_1 \boxtimes {\cal O}_{Q(u)_1},{\cal E}_u)$, where $Q(u)_1$ is the pull-back of $M(u)_1$ to $Q(u)$. Then $V$ is a locally free sheaf of rank $3$. We shall consider projective bundles $\xi_1:{\Bbb P}_1:={\Bbb P}(V^{\vee}) \to Q(u)_1$and $\xi_2:{\Bbb P}_2:={\Bbb P}(V)\to Q(u)_1$. Let \begin{align*} &0 \to {\cal Q}_1 \to \xi_1^*V^{\vee} \to {\cal O}_{{\Bbb P}_1}(1) \to 0\\ &0 \to {\cal Q}_2 \to \xi_2^*V \to {\cal O}_{{\Bbb P}_2}(1) \to 0 \end{align*} be the universal bundles on ${\Bbb P}_1$ and ${\Bbb P}_2$ respectively. Let ${\cal Z} \subset {\Bbb P}_1 \times {\Bbb P}_2$ be the incidence correspondence. Let $\eta_i:{\cal Z} \to {\Bbb P}_i$ $i=1,2$ be the projections. For simplicity, we denote $\eta_1^* {\cal O}_{{\Bbb P}_1}(m) \otimes \eta_2^* {\cal O}_{{\Bbb P}_2}(n)$ by ${\cal O}_{{\cal Z}}(m,n)$. There is a universal family of filtrations \begin{equation} 0 \subset {\cal O}_{{\cal Z}}(-1,0) \subset \eta^*_2 {\cal Q}_2 \subset \eta_2^* \xi_2^* V. \end{equation} Then there are homomorphisms \begin{align*} \alpha:& E_1 \boxtimes {\cal O}_{{\Bbb P}_1} \to E_1 \boxtimes \xi_1^*V \otimes {\cal O}_{{\Bbb P}_1}(1) \to (\xi_1 \times id_X)^*{\cal E}_u \otimes {\cal O}_{{\Bbb P}_1 \times X}(1) \\ \beta:& E_1 \boxtimes {\cal Q}_2 \to E_1 \boxtimes \xi_2^*V \to (\xi_2 \times id_X)^*{\cal E}_u. \end{align*} By the construction of the homomorphisms, $\alpha$ and $\beta$ are injective and ${\cal E}_v:=\operatorname{coker} \alpha$ and ${\cal E}_w:=\operatorname{coker} \beta$ are flat family of stable sheaves of Mukai vectors $v$ and $w$ respectively. Then we get the following exact and commutative diagram. \begin{equation} \begin{CD} @. @. 0 @. 0 @.\\ @.@. @VVV @VVV\\ 0 @>>> E_1 \boxtimes {\cal O}_{{\cal Z}}(-1,0) @>>> E_1\boxtimes \eta^*_2 {\cal Q}_2 @>>> E_1 \boxtimes {\cal O}_{{\cal Z}}(1,-1) @>>> 0 \\ @. @| @VVV @VV{\psi}V @. \\ 0 @>>> E_1 \boxtimes {\cal O}_{{\cal Z}}(-1,0) @>>> (\xi_1 \eta_1 \times id_X)^*{\cal E}_u @>>> (\eta_1 \times id_X)^*{\cal E}_v @>>> 0\\ @.@. @VVV @VVV @.\\ @.@. (\eta_2 \times id_X)^*{\cal E}_w @= (\eta_2 \times id_X)^*{\cal E}_w @.\\ @.@.@VVV @VVV@.\\ @. @. 0 @. 0 @. \end{CD} \end{equation} The homomorphism $\psi$ defines a morphism ${\cal Z} \to N(v_1,v,w)_2$. Obviously this morphism is $PGL(N)$ invariant, and hence we get a morphism $f:{\cal Z}/PGL(N) \to N(v_1,v,w)$. Conversely, let ${\cal F}_v$ be a family of stable sheaves which belong to $M(v)_2$ and let $\psi:E_1 \boxtimes {\cal O}_S \to {\cal F}_v$ be a family of homomorphisms which belong to $N(v_1,v,w)_2$ and are parametrized by a scheme $S$ over $M(v)_2$. Then $L:=\operatorname{Ext}^1_{p_S}({\cal F}_v,E_1 \boxtimes {\cal O}_S)$ is a line bundle. Hence we get an extension $$ 0 \to E_1 \boxtimes L^{\vee} \to {\cal F}_u \to {\cal F}_v \to 0. $$ We set ${\cal F}_w:=\operatorname{coker} \psi$ and ${\cal G}:=\ker({\cal F}_u \to {\cal F}_w)$. Then ${\cal F}_w$ is a family of stable sheaves which belong to $M(w)$. We get the following exact and commutative diagram. \begin{equation} \begin{CD} @. @. 0 @. 0 @.\\ @.@. @VVV @VVV\\ 0 @>>> E_1 \boxtimes L^{\vee} @>>> {\cal G} @>>> E_1 \boxtimes {\cal O}_{S} @>>> 0 \\ @. @| @VVV @VV{\psi}V @. \\ 0 @>>> E_1 \boxtimes L^{\vee} @>>> {\cal F}_u @>>> {\cal F}_v @>>> 0\\ @.@. @VVV @VVV @.\\ @.@. {\cal F}_w @= {\cal F}_w @.\\ @.@.@VVV @VVV@.\\ @. @. 0 @. 0 @. \end{CD} \end{equation} We set $Q:=\operatorname{Hom}_{p_S}(E_1 \boxtimes {\cal O}_S,{\cal G})$. Then it is easy to see that $E_1 \boxtimes Q \to {\cal G}$ is an isomorphism. $E_1 \boxtimes L^{\vee} \to {\cal F}_3$ defines a morphism $S \to N(v_1,u,v)_1$. The filtration of vector bundles $$ 0 \subset L^{\vee} \subset Q \subset \operatorname{Hom}_{p_S}(E_1 \boxtimes {\cal O}_S,{\cal F}_u) $$ defines a lifting $S \to {\cal Z}/PGL(N)$. In particular, we obtain a morphism $g:N(v_1,v,w)_2 \to {\cal Z}/PGL(N)$. Then $g$ is the inverse of $f$. We also see that ${\Bbb P}_1/PGL(N)=N(v_1,u,v)_1 \to M(v)_2$ and ${\Bbb P}_2/PGL(N)=N(2v_1,u,w)_1 \to M(w)_3$ are isomorphisms. Therefore $\pi_{w}\pi^{-1}_v:M(v)\; - - > M(w)$ is an elementary transformation in codimension 2. By using [H1, Cor. 5.5] or [H2, Cor. 4.7], we obtain the following theorem. \begin{thm}\label{thm:corres} $M(v)$ is deformation equivalent to $M(w)$. \end{thm} \begin{rem} In the notation of Lemma \ref{lem:cod} and Remark \ref{rem:gr}, we shall consider \'{e}tale locally trivial $Gr(2i-1,i-1) \times Gr(2i-1,i)$-bundle $M(v)_i \times_{M(v(G))_1} M(w)_{i+1} \to M(v(G))_1$. Let ${\cal Z}$ be the incidence correspondence. Then we also see that ${\cal Z}$ is isomorphic to $N(v_1,v,w)_i$. \end{rem} \subsection{Cohomologies of $M(v)$} \begin{lem}\label{lem:trans} Assume that $\rho(X) \geq 2$. Let $v=l(r+\xi)+a \omega,\; \xi \in H^2(X,{\Bbb Z})$ be a Mukai vector such that $r+ \xi$ is primitive. Then there is a line bundle $L$ such that $(1)$ $\xi':=[T_L(v)]_1/l=\xi+rL$ is primitive, $(2)$ $\xi'$ is ample, and $(3)$ $({\xi'}^2) \geq 4$. \end{lem} \begin{pf} Let $L'$ be a primitive ample divisor on $X$ such that $L'$ and $\xi$ are linearly independent. Let $n$ be an integer such that $r$ and $n$ are relatively prime. Since $L'$ is primitive, and $[T_{nL'}(v)]_1/l=\xi+rnL'$, $[T_{nL'}(v)]_1/l$ is primitive. Hence for a sufficiently large integer $n$, $L:=nL'$ satisfies our claims. \end{pf} \begin{prop}\label{prop:deform} Let $X_1$ and $X_2$ be K3 surfaces, and let $v_1:=l(r+\xi_1)+a_1 \omega \in H^*(X_1,{\Bbb Z})$ and $v_2:=l(r+\xi_2)+a_2 \omega \in H^*(X_2,{\Bbb Z})$ be Mukai vectors such that $(1)$ $r+\xi_1$ and $r+\xi_2$ are primitive, $(2)$ $\langle v_1^2 \rangle=\langle v_2^2 \rangle=2s$, and $(3)$ $a_1 \equiv a_2 \mod l$. Then $M(v_1)$ and $M(v_2)$ are deformation equivalent. \end{prop} \begin{pf} We may assume that $\xi_1$ and $\xi_2$ are ample. We assume that $\rho(X_i)=1$ for some $i$. Let $T_i$ be a connected smooth curve and $({\cal X}_i, {\cal L}_i)$ a pair of a smooth family of K3 surfaces $p_{T_i}:{\cal X}_i \to T_i$ and a relatively ample line bundle ${\cal L}_i$. For points $t_0, t_1 \in T_i$, we assume that $(({\cal X}_i)_{t_0},({\cal L}_i)_{t_0}) =(X_i,\xi_i)$ and $({\cal X}_i)_{t_1}$ is a K3 surface of $\rho(({\cal X}_i)_{t_1}) \geq 2$. We can construct an algebraic space ${\cal M}_{{\cal X}_i/T_i}(v_i) \to T_i$ which is smooth and proper over $T_i$, and is a family of moduli of stable sheaves on geometric fibers of Mukai vector $v_i$ with respect to general polarizations on fibers ([G-H],[O],[Y]). Replacing $X_i$ by $({\cal X}_i)_{t_1}$, we may assume that $\rho(X_i) \geq 2$ for $i=1,2$. By Lemma \ref{lem:trans}, we may assume that $(1)$ $\xi_i$ is primitive, $(2)$ $\xi_i$ is ample, and $(3)$ $(\xi_i^2) \geq 4$. Let $\pi:Z \to {\Bbb P}^1$ be an elliptic K3 surface of $\rho(Z)=2$ and let $f$ be a fiber of $\pi$ and $C$ a section of $\pi$. Then by using deformations of $(X_i,\xi_i)$ and $M(v_i)$, we see that $M(v_i)$ is deformation equivalent to $M(v_i')$, where $v_i':=l(r+(C+k_i f))+a_i \omega$, $i=1,2$. Since $a_1 \equiv a_2 \mod l$, we set $n:=(a_1-a_2)/l$. Then it is easy to see that $T_{n f}(v_2')=v_1'$. \end{pf} \begin{rem}\label{rem:theta} By the proof of [Mu2, Thm. A.5], there is a quasi-universal family ${\cal F}$ on ${\cal M}_{{\cal X}_i/T_i}(v_i) \times_{T_i} {\cal X}_i$. Since ${\cal L}$ is relatively ample, we can consider a locally free resolution of ${\cal F}$. In particular, we can construct a family of homomorphisms $(\theta_{v_i})_t:(v_i)^{\perp}_t \to H^2({\cal M}_{({\cal X}_i)_t/k(t)}(v_i),{\Bbb Z})$, where $(v_i)^{\perp}_t \subset H^*(({\cal X}_i)_t,{\Bbb Z})$. This fact will be used in the proof of Corollary \ref{cor:period} (cf. [Y, Prop. 3.3]). \end{rem} We get another proof of [H2, Cor. 4.8]. \begin{thm}\label{thm:1} Let $v=r+ \xi+a \omega,\; \xi \in H^2(X,{\Bbb Z})$ be a Mukai vector such that $r>0$ and $r+\xi$ is primitive. Then $M(v)$ and $\operatorname{Hilb}_X^{\langle v^2 \rangle/2+1}$ are deformation equivalent. If $r=0$, $\xi$ is ample, and $(\xi^2) \geq 4$, then the same results hold. \end{thm} \begin{pf} If $\langle v^2 \rangle =0$, then Mukai [Mu1] showed that $M(v)$ is a K3 surface. Hence we assume that $\langle v^2 \rangle =2s>0$. We first assume that $r \ne 2$. Let $v$ be a Mukai vector in Theorem \ref{thm:corres}. We assume that $d=r-1$. Since $r_1=r-1$, $M(v)$ is deformation equivalent to $\operatorname{Hilb}_X^{s+1}$. By Proposition \ref{prop:deform}, $M(v)$ is deformation equivalent to $\operatorname{Hilb}_X^{s+1}$ for every $v$. We shall treat the rank 2 case. Let $s$ be a positive integer. By Theorem \ref{thm:corres} and Lemma \ref{lem:1}, there is a Mukai vector $v$ and $w$ such that $\langle v^2 \rangle=\langle w^2 \rangle=2s$, $[v]_0=2$, $[w]_0=7$, and $M(v)$ is deformation equivalent to $M(w)$. By using Proposition \ref{prop:deform}, we see that $M(v)$ is deformation equivalent to $\operatorname{Hilb}_X^{\langle v^2 \rangle/2+1}$ foe every $v$. \end{pf} We also get another proof of [O]. \begin{cor}\label{cor:period} Under the same assumption of Theorem \ref{thm:1} and $\langle v^2\rangle \geq 2$, $M(v)$ is an irreducible symplectic manifold and $$ \theta_v : v^{\perp} \longrightarrow H^2(M(v),{\Bbb Z}) $$ is an isometry which preserves hodge structures. \end{cor} \begin{pf} The rank one case easily follows from [B] (cf. [Mu3],[O]). By using \eqref{eq:N}, Proposition \ref{prop:bilin}, Remark \ref{rem:theta}, and the proofs of Proposition \ref{prop:deform} and Theorem \ref{thm:1}, we get our corollary. \end{pf} \section{Non-primitive first Chern class cases} In this section, we shall consider a more general case and get some partial results. Let $r$ and $d$ be relatively prime non-negative integers. Then there are integers $r_1$ and $d_1$ such that $dr_1-d_1r=1$ Let $l$ be a positive integer such that $l$ and $r_1$ are relatively prime. We shall choose $r_1$ of $0<r_1<lr$. Then there are unique pair of integers $p,q$ of $pr_1-ql=-1$ and $0 \leq p < l$. We set $k(s):=r_1(qr+r_1s)-r^2, s \in {\Bbb Z}$. Let $X$ be a K3 surface such that $\operatorname{Pic}(X)={\Bbb Z} H$ and $(H^2)=2k(s)$. We set \begin{equation}\label{eq:4-1} \begin{cases} v=lr+ld H+\{l((1+dr_1)d_1s+d^2qr_1-rd^2)-p \}\omega,\\ v_1=r_1+d_1 H+\{r_1(-d^2+d_1^2s)+d_1^2rq+2d \}\omega. \end{cases} \end{equation} Then we see that \begin{equation} \begin{cases} \langle v_1^2 \rangle=-2,\\ \langle v^2 \rangle =2l(ls+rp),\\ \langle v_1,v \rangle=-1. \end{cases} \end{equation} We set $w:=v-v_1$ and we shall consider the relation between $M(v)$ and $M(w)$. \begin{lem}\label{lem:a1} Let $E_1$ be a stable vector bundle of $\operatorname{rk}(E_1)=r_1$ and $\deg(E_1)=d_1$. $(1)$ Let $E$ be a $\mu$-stable sheaf of $\operatorname{rk}(E)=lr$ and $\deg(E)=ld$. Then every non-zero homomorphism $\phi:E_1 \to E$ is injective and $\operatorname{coker} \phi$ is a stable sheaf. $(2)$ Let $E$ be a $\mu$-stable sheaf of $\operatorname{rk}(E)=lr$ and $\deg(E)=ld$. For a non-trivial extension $$ 0 \to E_1 \to E' \to E \to 0, $$ $E'$ is a stable sheaf. $(3)$ Let $E_2$ be a stable sheaf of $\operatorname{rk}(E_2)=lr-r_1$ and $\deg(E_2)=ld-d_1$. Then every non-trivial extension $$ 0 \to E_1 \to E \to E_2 \to 0, $$ defines a $\mu$-semi-stable sheaf. $(4)$ Let $E'$ be a stable sheaf of $\operatorname{rk}(E')=lr+r_1$ and $\deg(E')=ld+d_1$. Then every non-zero homomorphism $\phi:E_1 \to E'$ is injective and $\operatorname{coker} \phi$ is $\mu$-semi-stable. \end{lem} \begin{pf} (1) Since $d_1/r_1=\deg(E_1)/\operatorname{rk}(E_1) \leq \deg(\phi(E_1))/\operatorname{rk}(\phi(E_1)) < \deg(E)/\operatorname{rk}(E)=d/r$, we get that $1/rr_1=d/r-d_1/r_1 \geq d/r-\deg(\phi(E_1))/\operatorname{rk}(\phi(E_1)) \geq 1/r \operatorname{rk}(\phi(E_1))$. Hence we obtain that $\operatorname{rk}(\phi(E_1))=r_1$, which implies that $\phi$ is injective. We assume that $\operatorname{coker} \phi$ is not stable. Then there is a semi-stable quotient sheaf $G$ of $\operatorname{coker} \phi$ such that $\deg(G)/\operatorname{rk}(G) <\deg(\operatorname{coker} \phi)/\operatorname{rk}(\operatorname{coker} \phi) =(ld-d_1)/(lr-r_1)$. Since $G$ is a quotient of $E$, we get that $\deg(G)/\operatorname{rk}(G)>d/r$. Hence we obtain that $1/(lr-r_1)r= (ld-d_1)/(lr-r_1)-d/r>\deg(G)/\operatorname{rk}(G)-d/r \geq 1/r \operatorname{rk}(G)$, which is a contradiction. Therefore $G$ is a stable sheaf. (2) We assume that $E'$ is not stable. Let $G$ be a destabilizing semi-stable subsheaf of $E'$. We assume that $\phi:G \to E$ is not surjective in codimension 1. Then the $\mu$-stability of $E$ implies that $d/r > \deg(\phi(G))/\operatorname{rk}(\phi(G)) \geq \deg(G)/\operatorname{rk}(G)>\deg(E')/\operatorname{rk}(E')=(ld+d_1)/(lr+r_1)$. Hence we see that $1/r(lr+r_1)>d/r-\deg(G)/\operatorname{rk}(G) \geq 1/r \operatorname{rk}(G)$, which is a contradiction. Thus $\phi$ is surjective in codimension 1. If $\deg(\phi(G))/\operatorname{rk}(\phi(G)) > \deg(G)/\operatorname{rk}(G)$, then we also get a contradiction, and hence $\ker \phi$ is $\mu$-semi-stable of $\deg(\ker \phi)/\operatorname{rk}(\ker \phi)=\deg(G)/\operatorname{rk}(G)$. Thus $\deg(\ker \phi)/\operatorname{rk}(\ker \phi)>d_1/r_1$, which contradicts the stability of $E_1$. Therefore $G \to E$ is an isomorphism in codimension 1. In the same way as in the proof of Lemma \ref{lem:key}, we see that the extension is split, which is a contradiction. (3) We assume that $E$ is not $\mu$-semi-stable. Let $G$ be a semi-stable subsheaf of $E$ such that $d/r<\deg(G)/\operatorname{rk}(G)$. It is sufficient to show that $\phi:G \to E_2$ is an isomorphism in codimension 1. We note that $d/r<\deg(G)/\operatorname{rk}(G) \leq \deg(\phi(G))/\operatorname{rk}(\phi(G)) \leq (ld-d_1)/(lr-r_1)$. Then we get that $1/r(lr-r_1)=(ld-d_1)/(lr-r_1)-d/r \geq \deg(\phi(G))/\operatorname{rk}(\phi(G))-d/r \geq 1/\operatorname{rk}(\phi(G))r$. Thus we obtain that $\operatorname{rk}(\phi(G))=(lr-r_1)$ and $\deg(\phi(G))/\operatorname{rk}(\phi(G)) = (ld-d_1)/(lr-r_1)$. Therefore $\phi$ is surjective in codimension 1. We set $G':=\ker \phi$. We assume that $G' \ne 0$. Since $\operatorname{rk}(G)=\operatorname{rk}(G')+lr-r_1$, $\deg(G)=\deg(G')+ld-d_1$ and $d/r-\deg(G)/\operatorname{rk}(G)<0$, we get that $r \deg(G') \geq d \operatorname{rk}(G')$. Hence $\deg(G')/\operatorname{rk}(G') > \deg(E_1)/\operatorname{rk}(E_1)$, which is a contradiction. Therefore $G \to E_2$ is an isomorphism in codimension 1. The proof of (4) is similar to that of (1). \end{pf} Let $M(v)^{\mu}$ be the open subscheme of $M(v)$ consisting of $\mu$-stable sheaves. In the same way as in section 3, we shall define $N(v_1,v,w)$ and an open subscheme $N(v_1,v,w)^{\mu}:=\{E_1 \subset E| E \in M(v)^{\mu} \}$. \begin{lem}\label{lem:smooth2} $N(v_1,v,w)^{\mu}$ is smooth and irreducible. \end{lem} \begin{pf} For $E_1 \subset E \in N(v_1,v,w)^{\mu}$, we set $E_2:=E/E_1$. It is sufficient to show that $\operatorname{Hom}(E,E_2) \cong {\Bbb C}$. Let $\phi:E \to E_2$ be a non-zero homomorphism. Then we see that $\ker \phi \cong E_1$ and $\phi$ is surjective. We assume that $\dim \operatorname{Hom}(E,E_2)>1$. We set ${\Bbb P}:={\Bbb P}(\operatorname{Hom}(E,E_2)^{\vee})$. Since $E_1$ is simple, we get an exact sequence $$ 0 \to E_1 \boxtimes {\cal O}_{{\Bbb P}}(n) \to E \to E_2 \boxtimes {\cal O}_{{\Bbb P}}(1) \to 0, $$ where $n \in {\Bbb Z}$. We note that \begin{align*} \det p_{{\Bbb P}!}(E_1^{\vee} \otimes E_1 \boxtimes {\cal O}_{{\Bbb P}}(n)) &\cong {\cal O}_{{\Bbb P}}(2n),\\ \det p_{{\Bbb P}!}(E_1^{\vee} \otimes E_2 \boxtimes {\cal O}_{{\Bbb P}}(1))&\cong {\cal O}_{{\Bbb P}}(-1). \end{align*} Hence we see that ${\cal O}_{{\Bbb P}} \cong {\cal O}_{{\Bbb P}}(2n) \otimes {\cal O}_{{\Bbb P}}(-1)$, which is a contradiction. \end{pf} By using Lemma \ref{lem:a1} and similar correspondence $N(v_1,v+v_1,v)$, we see that $\operatorname{codim}_{M(v)^{\mu}}(M(v)^{\mu}_i) \geq 2$ for $i \geq 2$. Indeed, we get that $\dim M(v)_i^{\mu}=\dim N(v_1,v+v_1,v)_{i-1}-(i-2) =\dim M(v+v_1)_{i-1}+2 \leq \dim M(v)-2$. In particular $N(v_1,v,w)$ defines a birational correspondence. By virtue of Lemma \ref{lem:smooth2}, we also see that $\operatorname{codim}_{M(v)^{\mu}}(M(v)_i^{\mu}) \geq 3$ for $i \geq 3$. By using [H2, Cor. 4.7], we obtain the following. \begin{prop}\label{prop:corres2} We assume that $M(v)^{\mu} \ne \emptyset$. Then $M(v)$ is deformation equivalent to $M(w)$, in particular $M(v)$ is an irreducible symplectic manifold. Moreover if $\operatorname{codim}_{M(v)}(M(v) \setminus M(v)^{\mu}) \geq 2$, then the following diagram is commutative. \begin{equation*} \begin{CD} v^{\perp} @>{R_{v_1}}>> w^{\perp}\\ @V{\theta_v}VV @VV{\theta_{w}}V\\ H^2(M(v),{\Bbb Z}) @>>{\pi_{w*}\pi_v^*}> H^2(M(w),{\Bbb Z}) \end{CD} \end{equation*} \end{prop} We shall give an estimate of $\operatorname{codim}_{M(v)}(M(v) \setminus M(v)^{\mu})$, which depends on $l$ and $\langle v^2 \rangle$. \begin{lem}\label{lem:mu} We assume that $\langle v^2 \rangle/2l \geq l$. Then $$ \dim(M(v))-\dim(M(v) \setminus M(v)^{\mu}) \geq \frac{\langle v^2 \rangle}{2l}-l+1. $$ In particular $\operatorname{codim}_{M(v)}(M(v) \setminus M(v)^{\mu}) \geq 2$ for $\langle v^2 \rangle/2l >l$. \end{lem} \begin{pf} Let $E$ be a $\mu$-semi-stable sheaf of $v(E)=v$ and let $0 \subset F_1 \subset F_2 \subset \cdots \subset F_t=E$ be a Jprdan-H\"{o}lder filtration of $E$ with respect to $\mu$-stability. We set $E_i:=F_i/F_{i-1}$. Then the moduli number of this filtration is bounded by \begin{align*} & \sum_{i \leq j} (\dim \operatorname{Ext}^1(E_j,E_i)-\dim \operatorname{Hom}(E_j,E_i))+1\\ \leq & -\sum_{i \leq j}\chi(E_j,E_i)+\sum_{i \leq j}\dim \operatorname{Ext}^2(E_j,E_i)+1\\ \leq & -\chi(E,E)+\sum_{i>j}\chi(E_j,E_i)+ \sum_{i<j}\dim \operatorname{Ext}^2(E_j,E_i)+t+1. \end{align*} We set $v(E):=lr+ldH+a \omega$ and $v(E_i):=l_i r+l_i dH+a_i \omega$. Since $\langle v(E_i),v(E_j) \rangle=l_i l_j d^2 H^2-r(l_i a_j+l_j a_i)$, we see that \begin{align*} \sum_{i>j}\chi(E_j,E_i) &= -\sum_{i>j} \langle v(E_j),v(E_i) \rangle\\ &=-\sum_{i>j}\left\{\frac{l_i(l_j d^2 H^2-2r a_j)}{2} +\frac{l_j(l_i d^2 H^2-2r a_i)}{2} \right \}\\ &=-\sum_{i>j}\left\{\frac{l_i\langle v(E_j)^2 \rangle}{2l_j}+ \frac{l_j\langle v(E_i)^2 \rangle}{2l_i} \right \}\\ &=-\sum_{i}\frac{(l-l_i)\langle v(E_i)^2 \rangle}{2l_i}. \end{align*} We set $\max_i \{l_i\}=(l-k)$. Let $i_0$ be an integer such that $ \langle v(E_{i_0})^2 \rangle \geq 0$. Since $\sum_i l_i=l$, we obtain that $t \leq k+1$. Since $l-l_i-k \geq 0$ and $\langle v(E_i)^2 \rangle \geq -2$, we get that \begin{align*} \sum_{i>j} \langle v(E_j),v(E_i) \rangle &=k\sum_i \frac{\langle v(E_i)^2 \rangle}{2l_i}+ \sum_i \frac{(l-l_i-k)\langle v(E_i)^2 \rangle}{2l_i}\\ & \geq k\frac{\langle v(E)^2 \rangle}{2l}- \sum_{i \ne i_0} (l-l_i-k)\\ & \geq k\frac{\langle v(E)^2 \rangle}{2l}-(l-1-k)k. \end{align*} If $r>1$ or $l_i>1$ for some $i$, then for a general filtration, there are $E_i$ and $E_j$ such that $\operatorname{Ext}^2(E_j,E_i)=0$. Therefore we get that $\sum_{i<j} \dim \operatorname{Ext}^2(E_j,E_i) \leq (k+1)k/2-1$ for a general filtration. Then the moduli number of these filtrations is bounded by \begin{align*} \langle v^2 \rangle- k\frac{\langle v^2 \rangle}{2l} +(l-1-k)k+\frac{(k+1)k}{2}-1+t+1 &\leq (\langle v^2 \rangle+2)- \left(k\frac{\langle v^2 \rangle}{2l} -lk+\frac{k(k-1)}{2}+1 \right)\\ & \leq (\langle v^2 \rangle+2)- \left(\frac{\langle v^2 \rangle}{2l} -l+1 \right). \end{align*} If $l_i=1$ for all $i$ and $r=1$, then $\sum_{i<j}\dim \operatorname{Ext}^2(E_j,E_i) \leq k(k+1)/2=(l-1)l/2$, and hence we obtain that the moduli number is bounded by $$ (\langle v^2 \rangle+2)- \left(\frac{\langle v^2 \rangle}{2l} -l+\frac{(l-1)(l-2)}{2} \right). $$ Therefore we obtain that $$ \dim(M(v))-\dim(M(v) \setminus M(v)^{\mu}) \geq \frac{\langle v^2 \rangle}{2l}-l+1 $$ unless $r=1$ and $l=2$. If $r=1$ and $l=2$, then the moduli number of filtrations $$ 0 \subset T_Z \subset E $$ of $c_2(I_Z)=c_2(E)$ is $(\langle v^2 \rangle+2)-(\langle v^2 \rangle/2l-l)$. In this case, we can easily show that $E^{\vee \vee} \cong {\cal O}_X^{\oplus 2}$, and hence the moduli number of these $E$ is bounded by $3 c_2(E)-3 \leq (\langle v^2 \rangle+2)-(\langle v^2 \rangle/2l-l+1)$. Therefore we obtain our lemma. \end{pf} \begin{thm}\label{thm:2} Let $v=lr+l \xi+a \omega, \;\xi \in H^2(X,\Bbb Z)$ be a primitive Mukai vector such that $r>0$ and $r+\xi$ is primitive. If $\langle v^2 \rangle/2 > \max\{rl(rl-1),l^2\}$, then $M(v)$ is deformation equivalent to $\operatorname{Hilb}_X^{\langle v^2 \rangle/2+1}$ and $$ \theta_v : v^{\perp} \longrightarrow H^2(M(v),{\Bbb Z}) $$ is an isometry which preserves hodge structures. \end{thm} \begin{pf} By Proposition \ref{prop:deform} and Lemma \ref{lem:mu}, it is sufficient to find a Mukai vector $v'=l(r+\xi')+a' \omega$ which satisfies (1) $r+\xi'$ is primitive, (2) $\langle {v'}^2 \rangle= \langle {v}^2 \rangle$, (3) $a' \equiv a \mod l$, and (4) $M(v')$ satisfies our claims. By Proposition \ref{prop:corres2}, it is sufficient to find a Mukai vector $v'$ of the form \eqref{eq:4-1}. Hence $a \equiv -p \mod l$. We have to choose integers $r_1,d_1$ and $d$. We note that $pr_1 \equiv -1 \mod l$, and $r_1$ and $r$ are relatively prime. We can easily choose such an integer $r_1$ of $0<r_1<lr$. Then we can choose $d$ and $d_1$ of $d r_1-d_1r=1$. Since $\langle v^2 \rangle/2l > r(rl-1)$, we see that $k(s)=(r_1 \langle v^2 \rangle/2l+r)r_1/l-r^2>0$. Hence we get a Mukai vector $v'$ of the form \eqref{eq:4-1} \end{pf} Tohru Nakashima taught the author the following remark. \begin{rem}\label{rem:exist} If $r=1$ and $\xi=0$, then $rl(rl-1)<l^2$. Hence the assertions of Theorem \ref{thm:2} hold for $\langle v^2 \rangle/2l > l$. It is known that $M(v)^{\mu}\ne \emptyset$ if and only if $\langle v^2 \rangle/2l \geq l$, and $\langle v^2 \rangle/2l \ne l$ if $v$ is primitive. \end{rem} \vspace{1pc} {\it Acknowledgement.} The author would like to thank Masaki Maruyama for giving him a preprint of Ellingsrud and Str\o mme ([E-S]), which inspired him very much, Tohru Nakashima for teaching him Remark \ref{rem:exist}, and Hiraku Nakajima for valuable discussions.

9705

alg-geom/9705025

en

https://arxiv.org/abs/alg-geom/9705025

alg-geom/9705025

Daniel Huybrechts

D. Huybrechts

Compact Hyperkaehler Manifolds: Basic Results

Latex 2e, 44 pages

The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, we discuss the projectivity of such manifolds and their ample resp. Kaehler cones. It is proved that the period map surjects any non-empty connected component of the moduli space of marked manifolds onto the corresponding period domain. We also establish an anlogue of the transitivity of the Weyl-action on the set of chambers of a K3 surface. As a converse of a higher-dimensional version of the `Main Lemma' of Burns and Rapoport we prove that two birational irreducible symplectic manifolds are deformation equivalent. This compares nicely with a result of Batyrev and Kontsevich.

alg-geom

\section{Preliminaries}\label{prel} {\bf \refstepcounter{theorem}\label{IrredSympl} \thetheorem} --- A complex manifold $X$ is called {\it irreducible symplectic} if {\it i)} $X$ is compact and K\"ahler, {\it ii)} $X$ is simply connected, and {\it iii)} $H^0(X,\Omega^2_X)$ is spanned by an everywhere non-degenerate two-form $\sigma$. Any holomorphic two-form $\sigma$ induces a homomorphism ${\cal T}_X\to \Omega_X$, which we also denote by $\sigma$. The two-form is everywhere non-degenerate if and only if $\sigma:{\cal T}_X\to\Omega_X$ is bijective. Note that {\it iii)} implies $h^{2,0}(X)=h^{0,2}(X)=1$ and $K_X\cong {\cal O}_X$. In particular, $c_1(X)=0$. Any irreducible symplectic manifold $X$ has even complex dimension which we will fix to be $2n$. Irreducible symplectic manifolds occupy a distinguished place in the list of higher dimensional K\"ahler manifolds. Together with Calabi-Yau manifolds they are the only irreducible simply connected K\"ahler manifolds with $c_1(X)=0$ (cf.\ \cite{Beauville1}). \bigskip {\bf \refstepcounter{theorem}\label{HypK} \thetheorem} --- A compact connected $4n$-dimensional Riemannian manifold $(M,g)$ is called {\it hyperk\"ahler} ({\it irreducible hyperk\"ahler}) if its holonomy is contained in (equals) ${\rm Sp}(n)$. If $(M,g)$ is hyperk\"ahler, then the quaternions ${\mathbb H}$ act as parallel endomorphisms on the tangent bundle of $M$. This is a consequence of the holonomy principle: Every tensor at a point in $M$ that is invariant under the holonomy action can be extended to a parallel tensor over $M$. In particular, any $\lambda\in{\mathbb H}$ with $\lambda^2=-1$ gives rise to an almost complex structure on $M$. As it turns out, these almost complex structures are all integrable \cite{Salamon2}. After having fixed a standard basis $I$, $J$, and $K:=IJ$ of ${\mathbb H}$ any $\lambda\in{\mathbb H}$ with $\lambda^2=-1$ can be written as $\lambda=aI+bJ+cK$ with $a^2+b^2+c^2=1$. Note that the metric $g$ is K\"ahler with respect to every such $\lambda\in S^2$. The corresponding K\"ahler form is denoted by $\omega_\lambda:=g(\lambda\,.\,,\,.\,)$. Thus, a hyperk\"ahler metric $g$ on a manifold $M$ defines a family of complex K\"ahler manifolds $(M,\lambda,\omega_\lambda)$, where $\lambda\in S^2\cong\IP^1$. \bigskip {\bf \refstepcounter{theorem}\label{Comparison} \thetheorem} --- We briefly sketch the relation between irreducible symplectic and irreducible hyperk\"ahler manifolds. For details we refer to \cite{Beauville1}. If $X$ is irreducible symplectic and $\alpha\in H^2(X,{\mathbb R})$ is a K\"ahler class on $X$, then there exists a unique Ricci-flat K\"ahler metric $g$ with K\"ahler class $\alpha$. This follows from Yau's solution of the Calabi-conjecture. Then $g$ is an irreducible hyperk\"ahler metric on the underlying real manifold $M$. Moreover, for one of the complex structures, say $I$, one has $X=(M,I)$. Conversely, if $(M,g)$ is hyperk\"ahler and $I,J,K$ are complex structures as above, then $\sigma:=\omega_J+i\omega_K$ is a holomorphic everywhere non-degenerate two-form on $X=(M,I)$. If $M$ is compact and $g$ is irreducible hyperk\"ahler, then $M$ is simply connected and $H^0((M,I),\Omega_{(M,I)}^2)=\sigma\cdot\IC$, i.e.\ $X$ is irreducible symplectic. Thus, irreducible symplectic manifolds with a fixed K\"ahler class and compact irreducible hyperk\"ahler manifolds are the holomorphic respectively metric incarnation of the same object. We will use the two names accordingly. \bigskip {\bf \refstepcounter{theorem}\label{Notat} \thetheorem} --- Let $X$ be an irreducible symplectic manifold. For later use we introduce the following notations: The {\it K\"ahler cone} ${\cal K}_X\subset H^{1,1}(X)_{\mathbb R}:=H^{1,1}(X)\cap H^2(X,{\mathbb R})$ is the open convex cone of all K\"ahler classes on $X$. If $\alpha\in H^{1,1}(X)_{\mathbb R}$, then one defines $$P(X):=(\sigma\cdot\IC\oplus\bar\sigma\cdot\IC)\cap H^2(X,{\mathbb R}) \phantom{XXX}{\rm and}\phantom{XXX}F(\alpha):=P(X)\oplus\alpha\cdot{\mathbb R},$$ where $\sigma$ is a holomorphic two-form spanning $H^0(X,\Omega_X^2)$. If $\alpha\in{\cal K}_X$ and the induced hyperk\"ahler structure is $(M,g,I,J,K)$, where $X=(M,I)$, then $\sigma=\omega_J+i\omega_K$ (up to scalar factors) and, therefore, $P(X)=[\omega_J]\cdot{\mathbb R}\oplus[\omega_K]\cdot{\mathbb R}$ and $F(\alpha)=[\omega_I]\cdot{\mathbb R}\oplus[\omega_J]\cdot{\mathbb R}\oplus[\omega_K]\cdot{\mathbb R}$. Hence, $F(\alpha)\subset H^2(X,{\mathbb R})=H^2(M,{\mathbb R})$ is independent of the complex structure and depends only on the metric $g$ on $M$. Sometimes, $F(\alpha)$ is called the {\it HK 3-space} associated with the hyperk\"ahler metric $g$. \bigskip {\bf \refstepcounter{theorem}\label{Cohomology} \thetheorem} --- Let $X$ be an irreducible symplectic manifold. The following identities are immediate consequences of the definition: $$\begin{array}{rcl} H^1(X,{\cal O}_X)&=&0,~~ H^2(X,{\cal O}_X)\cong\IC\\ H^0(X,{\cal T}_X)&\cong&H^0(X,\Omega_X)=0\\ H^1(X,{\cal T}_X)&\cong& H^1(X,\Omega_X).\\ \end{array}$$ In the following (\ref{hypcoh},\ref{irredcoh},\ref{pairing}) we state some results concerning the cohomological structure of irreducible symplectic manifolds. Most of them are due to Fujiki. For details and other results in this direction we refer to Enoki's survey article \cite{Enoki}, to the original paper of Fujiki \cite{Fujiki2} and to the more recent articles by Looijenga and Lunts \cite{LL} and Verbitsky \cite{Verbitsky}. \bigskip {\bf \refstepcounter{theorem}\label{hypcoh} \thetheorem} --- Let $(M,g)$ be a compact irreducible hyperk\"ahler manifold of real dimension $4n$. Let $F$ denote the associated HK $3$-space spanned by the three K\"ahler forms $[\omega_I],[\omega_J],[\omega_K]$. The Lefschetz operator $L_\lambda: =L_{\omega_\lambda}$ on the K\"ahler manifold $(M,\lambda,\omega_\lambda)$ for $\lambda\in S^2$ acts on the cohomology $H^*(M,{\mathbb R})$ and allows one to define the primitive cohomology $H^k((M,\lambda),{\mathbb R})_{\rm pr}:= \ker(L_\lambda^{2n-k+1}:H^k(M,{\mathbb R})\to H^{4n-k+2}(M,{\mathbb R}))$ ($k\leq2n$). Recall, the {\it Lefschetz decomposition} on the compact K\"ahler manifold $(M,\lambda,\omega_\lambda)$ is the direct sum decomposition $$H^k(M,{\mathbb R})=H^k((M,\lambda),{\mathbb R})=\bigoplus_{(k-\ell)\geq(k-2n)^+}L_\lambda^{k-\ell}(H^{2\ell-k}((M,\lambda),{\mathbb R})_{\rm pr})$$ and the {\it Hard Lefschetz Theorem} asserts that for $k\leq 2n$ $$L_\lambda^{2n-k}:H^k(M,{\mathbb R})\cong H^{4n-k}(M,{\mathbb R}).$$ The latter implies $$L_\lambda^{k-\ell}:H^{2\ell-k}((M,\lambda),{\mathbb R})_{\rm pr}\cong L_\lambda^{k-\ell}(H^{2\ell-k}((M,\lambda),{\mathbb R})_{\rm pr}).$$ Combining the statements for all $\lambda\in S^2$ one obtains the following results \cite{Fujiki2}: Let $N^*\subset H^*(M,{\mathbb R})$ denote the subalgebra generated by $F$ and let $H^*(M,{\mathbb R})_F:=\bigoplus H^k(M,{\mathbb R})_F=\bigoplus \{\alpha\in H^k((M,\lambda),{\mathbb R})_{\rm pr}| ~{\rm for~all~}\lambda\in S^2\}$. Then $$H^k(M,{\mathbb R})=\bigoplus_{2(k-l)\geq(k-2n)^+} N^{k-\ell}H^\ell(M,{\mathbb R})_F$$ and for $k\leq n$ $$N^{k-\ell}\otimes_{\mathbb R} H^\ell(M,{\mathbb R})_F\cong N^{k-\ell}H^\ell(M,{\mathbb R})_F.$$ The first statement in particular says $$H^2(M,{\mathbb R})=F\oplus H^2(M,{\mathbb R})_F.$$ Note that for any $\lambda\in S^2$ the space $H^2(M,{\mathbb R})_F$ is contained in $H^{1,1}((M,\lambda),{\mathbb R})$. \bigskip {\bf \refstepcounter{theorem}\label{irredcoh} \thetheorem} --- Let $X$ be an irreducible symplectic manifold and let $0\ne\sigma\in H^0(X,\Omega_X^2)$ be fixed. By the holonomy principle one easily obtains (cf.\ \cite{Beauville1}): $$H^0(X,\Omega_X^p)\cong\left\{\begin{array}{lcl} 0&&p\equiv1(2)\\ \Lambda^{p/2}\sigma\cdot\IC&&p\equiv0(2).\\ \end{array}\right.$$ Fujiki also proved holomorphic versions of the Lefschetz decomposition and the Hard Lefschetz Theorem. They allow one to compute other cohomology groups of the form $H^q(X,\Omega_X^p)$ as follows: Let $L_\sigma:H^q(X,\Omega^p_X)\to H^q(X,\Omega_X^{p+2})$ and $L_{\bar\sigma}:H^q(X,\Omega_X^p)\to H^{q+2}(X,\Omega_X^p)$ be the map given by the cup-product with $\sigma$ and $\bar\sigma$, respectively, and let $H^q(X,\Omega^p_X)_\sigma:= \ker(L_\sigma^{n-p+1})$ and $H^q(X,\Omega_X^p)_{\bar\sigma}:=\ker(L_{\bar\sigma}^{n-q+1})$. Then $$H^q(X,\Omega_X^p) =\bigoplus_{(p-\ell)\geq(p-n)^+} L_\sigma^{p-\ell}H^q(X,\Omega_X^{2\ell-p})_\sigma = \bigoplus_{(q-\ell)\geq(q-n)^+}L_{\bar\sigma}^{q-\ell}H^{2\ell-q}(X,\Omega_X^p)_{\bar\sigma}$$ and $$\begin{array}{crcll} L_\sigma^{n-p}:&H^q(X,\Omega_X^p)&\cong&H^q(X,\Omega_X^{2n-p})~~~~~&{\rm for}~p\leq n\\ L_{\bar\sigma}^{n-q}:&H^q(X,\Omega_X^p)&\cong&H^{2n-q}(X,\Omega_X^p)~~~~~&{\rm for}~q\leq n.\\ \end{array}$$ The following special cases will be used frequently $$\begin{array}{crcl} L_\sigma^{n-1}:&H^q(X,\Omega_X)&\cong&H^q(X,\Omega_X^{2n-1})~~~~~~{\rm for~all}~q\\ L_{\bar\sigma}^{n-1}:&H^1(X,\Omega_X^p)&\cong&H^{2n-1}(X,\Omega_X^p)~~~~~~{\rm for~all}~p.\\ \end{array}$$ The first isomorphism can be deduced using the isomorphism $\sigma:{\cal T}_X\to\Omega_X$ and the perfect pairing $\Omega_X\otimes\Omega_X^{2n-1}\to K_X\cong{\cal O}_X$: The induced map $\Omega_X\cong{\cal T}_X\cong(\Omega_X)\makebox[0mm]{}^{{\scriptstyle\vee}}\cong\Omega_X^{2n-1}$ is just $L_\sigma^{n-1}$. The complex conjugate of the first isomorphism gives the second. Combining both we get $$L^{n-1}_{\sigma\bar\sigma}:=L_\sigma^{n-1}\circ L_{\bar\sigma}^{n-1}:H^1(X,\Omega_X)\cong H^{2n-1}(X,\Omega_X^{2n-1}).$$ In Sect.\ \ref{projectivity} we will also need a version of this isomorphism on the level of smooth forms. \bigskip {\bf \refstepcounter{theorem}\label{pairing} \thetheorem} --- The following natural pairings are perfect: $$\begin{array}{ccccl} H^1(X,{\cal T}_X)&\otimes &H^{2n-1}(X,\Omega_X^{2n-1})&\to&H^{2n}(X,\Omega_X^{2n-2})\cong({\bar\sigma}^n\sigma^{n-1})\cdot\IC\\ H^1(X,{\cal T}_X)&\otimes& H^1(X,\Omega_X)&\to&H^2(X,{\cal O}_X)\cong\bar\sigma\cdot\IC\\ \end{array}$$ Using Serre duality it suffices to show that the induced homomorphisms $H^1(X,{\cal T}_X)\to{\rm Hom}(H^{2n-1}(X,\Omega_X^{2n-1}), H^{2n}(X,\Omega_X^{2n-2}))\cong{\rm Hom}(H^0(X,\Omega_X^2),H^1(X,\Omega_X))$ and $H^1(X,{\cal T}_X)\to {\rm Hom}(H^1(X,\Omega_X),H^2(X,{\cal O}_X))\cong {\rm Hom}(H^{2n-2}(X,\Omega^{2n}_X),H^{2n-1}(X,\Omega_X^{2n-1}))$ are bijective. Using the general form of the holomorphic Hard Lefschetz Theorem above one finds that these maps are also given by the cup-product. Then the perfectness of the first pairing follows from the fact that $\sigma:{\cal T}_X\to\Omega_X$ is bijective and the proof of the second uses in addition that the composition of $L^n_\sigma:H^1(X,{\cal T}_X)\to H^1(X,\Omega_X^{2n-1})$ followed by $L^{n-1}_{\bar\sigma}:H^1(X,\Omega^{2n-1}_X)\to H^{2n-1}(X,\Omega_X^{2n-1})$ is bijective. Note that in particular for any $0\ne\alpha\in H^1(X,\Omega_X)$ (or $\in H^{2n-1}(X,\Omega_X^{2n-1})$) the natural map $\alpha:H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ (respectively $H^1(X,{\cal T}_X)\to H^{2n}(X,\Omega_X^{2n-2})$) is surjective. We will also use the following fact: Let $\beta\in H^{2n-1}(X,\Omega_X^{2n-1})$, then the cup-product with $\beta$ sends $v\in H^1(X,{\cal T}_X)$ to $\beta\cdot v\in H^{2n}(X,\Omega_X^{2n-2})$. Hence $(\beta\cdot v)\sigma$ can be integrated over $X$. On the other hand, $\beta$ can be regarded as a linear form on $H^1(X,\Omega_X)$, and thus can be evaluated on the image $\alpha$ of $v$ under the isomorphism $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$. Clearly, the two expressions $\int_X(\beta\cdot v)\sigma$ and $\beta(\alpha)=\int_\beta\alpha$ agree. A similar statement holds for $\beta\in H^1(X,\Omega_X)$. Here we have $\int (\beta\cdot v)\sigma^n{\bar\sigma}^{n-1}=\int(\beta\alpha)(\sigma\bar\sigma)^{n-1}=c q_X(\beta,\alpha)$, where $c$ is a positive number (for the definition of $q_X$ see \ref{quadraticform}). \bigskip {\bf \refstepcounter{theorem}\label{quadraticform} \thetheorem} --- Due to work of Beauville \cite{Beauville1} there exists a natural quadratic form on the second cohomology of an irreducible symplectic manifold generalizing the intersection pairing on a K3 surface. There are several approaches towards this quadratic form \cite{Beauville1,Fujiki2,Enoki,Bo3,Verbitsky} most of them are intimately interwoven with the deformation theory of such manifolds. Let us state the main facts. Let X be an irreducible symplectic manifold and let $\sigma\in H^0(X,\Omega^2_X)$ such that $\int(\sigma\bar\sigma)^n=1$. Define a quadratic form on $H^2(X,{\mathbb R})$ by $$f(\alpha)=\frac{n}{2}\int(\sigma\bar\sigma)^{n-1}\alpha^2+(1-n)\left(\int\sigma^{n-1}{\bar\sigma}^n\alpha\right)\cdot\left(\int\sigma^n{\bar\sigma}^{n-1}\alpha\right).$$ Writing $\alpha$ according to the Hodge decomposition as $\alpha=\lambda\sigma +\beta+\bar\lambda\bar\sigma$, where $\beta$ is a $(1,1)$-form, then $$f(\alpha)=\frac{n}{2}\int(\sigma\bar\sigma)^{n-1}\beta^2+\lambda\bar\lambda.$$ Clearly, with respect to this quadratic form $H^{1,1}(X)$ is orthogonal to $H^{2,0}(X)\oplus H^{0,2}(X)$. Moreover, $f(\sigma)=0$ and $f(\sigma+\bar\sigma)=1>0$. If $\alpha$ is a K\"ahler class on $X$ and $F=F(\alpha)$, then the decomposition $H^2(X,{\mathbb R})=F\oplus H^2(X,{\mathbb R})_F$ is orthogonal with respect to $f$. The restriction of $f$ to $F$ only depends on the underlying manifold $M$ and the hyperk\"ahler metric but not on the complex structure. One also has the following extremely useful formula \beeq{extrform} v(\alpha)^2f(\beta)=f(\alpha)\left((2n-1)v(\alpha)\int\alpha^{2n-2}\beta^2-(2n-2)\left(\int\alpha^{2n-1}\beta\right)^2\right) \end{eqnarray} for any two classes $\alpha$ and $\beta$, where $v(\alpha)=\int\alpha^{2n}$. Applying this formula to a K\"ahler class $\alpha$ and $\beta=\sigma+\bar\sigma$ yields $v(\alpha)=2f(\alpha)(2n-1)\int\alpha^{2n-2}(\sigma\bar\sigma)$. By the Hodge-Riemann bilinear relations the integral on the right hand side is positive. Hence: For any K\"ahler class $\alpha$ the quadratic form $f$ restricted to $F(\alpha)$ is positive definite. For $0\ne\beta\in H^2(X,{\mathbb R})_F$ the above formula shows $v(\alpha)f(\beta)=(2n-1)f(\alpha)\int\alpha^{2n-2}\beta^2$. Thus $f$ restricted to $H^2(X,{\mathbb R})_F$ is a positive multiple of the standard Hodge-Riemann bilinear form and, therefore, negative definite. Yet another consequence of formula (\ref{extrform}) is the following: Since $v(\sigma+\bar\sigma)>0$ and $f(\sigma+\bar\sigma)=1$, for any $\beta\in H^2(X,\IQ)$ close to $\sigma+\bar\sigma$ one has $f(\beta)>0$ and $v(\beta)\in\IQ$. Hence $$f(\alpha)/f(\beta)=\frac{1}{v(\beta)^2}\left((2n-1) v(\beta)\int\beta^{2n-2}\alpha^2-(2n-2)\left(\int\beta^{2n-1}\alpha\right)^2\right)\in\IQ$$ for all $\alpha\in H^2(X,\IQ)$. The upshot is: There exists a positive constant $c\in{\mathbb R}$ such that $q_X:=c\cdot f$ is a {\it primitive integral quadratic form} on $H^2(X,{\mathbb Z})$ of index $(3,b_2(X)-3)$. For any K\"ahler class $\alpha$ the decomposition $F(\alpha)\oplus H^2(X,{\mathbb R})_{F(\alpha)}$ is orthogonal with respect to $q_X$. Also note $q_X(\sigma)=0$ and $q_X(\sigma+\bar\sigma)>0$. By means of the integral quadratic form $q_X$ one can establish a close link between rational classes of dimension one and those of codimension one. Namely, if $c$ is the positive constant such that $c\cdot f=q_X$, then $cL_{\sigma\bar\sigma}^{n-1}$ defines an isomorphism $H^{1,1}(X)_\IQ\cong H^{2n-1,2n-1}(X)_\IQ$. (Note that this is quite similar to the Hard Lefschetz Theorem with respect to a Hodge class.) Indeed, if $\alpha\in H^{1,1}(X)$ and $\beta:=c L_{\sigma\bar\sigma}^{n-1}(\alpha)$, then for any $\gamma\in H^2(X,\IQ)$ one has $$\int\beta\gamma=\int\beta\gamma^{1,1}=c\int L_{\sigma\bar\sigma}^{n-1}(\alpha)\gamma^{1,1}=\frac{2}{n}q_X(\alpha,\gamma^{1,1})=\frac{2}{n}q_X(\alpha,\gamma).$$ Now, if $\alpha$ is rational then $q_X(\alpha,\gamma)\in\IQ$ and hence $\int\beta\gamma\in\IQ$ for all $\gamma\in H^2(X,\IQ)$. Thus $\beta\in H^{4n-2}(X,\IQ)$. Conversely, if $\beta$ is rational, then $\int\beta\gamma\in\IQ$ and therefore $q_X(\alpha,\gamma)\in\IQ$. Since $q_X$ is non-degenerate and integral, this shows $\alpha\in H^2(X,\IQ)$. \bigskip {\bf \refstepcounter{theorem}\label{posconedef} \thetheorem} --- The {\it positive cone} ${\cal C}_X$ is by definition the component of $\{\alpha\in H^{1,1}(X)_{\mathbb R}|q_X(\alpha)>0\}$ that contains the K\"ahler cone ${\cal K}_X$ (cf.\ \ref{Notat}). Note that one has a Hodge Index Theorem with respect to $q_X$: A $(1,1)$-class $\beta$ which is orthogonal to a K\"ahler class (with respect to $q_X$) satisfies $q_X(\beta)<0$ or is zero. In particular, if $\alpha\in {\cal K}_X$ then $q_X(\alpha,\,.\,)$ is positive on ${\cal C}_X$. Moreover, if $\alpha\in{\cal K}_X$ then $\int\alpha^{2n-1}\beta>0$ for $\beta\in{\cal C}_X$. Indeed, if not then one could find $\beta\in{\cal C}_X$ with $\int\alpha^{2n-1}\beta=0$, i.e.\ $\beta\in H^2(X,{\mathbb R})_{F(\alpha)}$, which would imply $q_X(\beta)<0$. This is absurd. \bigskip {\bf \refstepcounter{theorem}\label{Todd} \thetheorem} --- For the following we refer to Fujiki's paper \cite{Fujiki2} but we also wish to draw the reader's attention to the more recent preprint of Looijenga and Lunts \cite{LL}. For any integral class $\alpha\in H^{2j}(X,{\mathbb Z})$ one has the form of degree $2n-j$ that sends $\beta\in H^2(X,{\mathbb Z})$ to $\int\alpha\beta^{2n-j}\in{\mathbb Z}$. E.g.\ for $j=0$ and $\alpha=1$ this is $v(\beta)=\int\beta^{2n}$. Fujiki shows that for any $\alpha\in H^{4j}(X,{\mathbb Z})$ contained in the subalgebra generated by the Chern classes of $X$ there exists a constant $c\in \IQ$ such that \beeq{extrform2} \int\alpha\beta^{2(n-j)}=cq_X(\beta)^{n-j}~~{\rm for~any}~ \beta\in H^2(X,\IQ). \end{eqnarray} In particular, for $j=0$ and $\alpha=1$ this yields $v(\beta)=\int\beta^{2n}=cq_X(\beta)^n$ and in this case one has $c>0$. In fact, the result can be slightly generalized to all classes $\alpha$ which are of type $(2j,2j)$ on all small deformations of $X$. As an application of (\ref{extrform2}) one has that the Hirzebruch-Riemann-Roch formula on an irreducible symplectic manifold takes the following form: If $L$ is a line bundle on $X$ then $$\chi(L)=\sum \frac{a_i}{(2i)!}q_X(c_1(L))^i,$$ where the $a_i$'s are constants only depending on $X$. Indeed, $\chi(L)=\sum\int ch_{i}(L)td_{2n-i}(X)$ and $(2i)! ch_{2i}(L)=c_1(L)^{2i}$. Since $td_i(X)$ is a polynomial in the Chern classes, one has $td_i(X)=0$ for $i\equiv 1(2)$ (use $c_i(X)=0$ for $i$ odd) and $c_1(L)^{2i}td_{2n-2i}=a_iq_X(c_1(L))^i$. This will be crucial in the proof of Theorem \ref{birat}. Another application of the relation between $q_X(\alpha)$ and $v(\alpha)$ is the fact that if $X$ is irreducible symplectic, $\alpha$ a K\"ahler class, and $Y\subset X$ an effective divisor, then $q_X(\alpha,[Y])>0$. One possible proof goes as follows: If $[Y]\in{\cal C}_X$ then certainly $q_X(\alpha,[Y])>0$ by the Hodge Index Theorem. Since $\int_Y\alpha^{2n-1}>0$ and hence $\int\alpha^{2n-1}(-[Y])<0$, the case $-[Y]\in{\cal C}_X$ can be excluded (cf.\ \ref{posconedef}). Thus it remains to verify the claim for $q_X([Y])\leq0$. Of course, $q_X(\alpha,[Y])>0$ if and only if $q_X(\alpha+[Y])> q_X(\alpha)+q_X([Y])$. Thus it suffices to show $q_X(\alpha+[Y])> q_X(\alpha)$. Replacing $\alpha$ by $k\beta$ with $k\gg0$ and $\beta$ fix, this is equivalent to $\int (k\beta+[Y])^{2n}=c q_X(k\beta+[Y])^{n}> c q_X(k\beta)^n=\int(k\beta)^{2n}$. Since $\int_Y\beta^{2n-1}>0$, this follows immediately. \bigskip {\bf \refstepcounter{theorem}\label{deformationallg} \thetheorem} --- A {\it deformation} of a compact manifold $X$ is a smooth proper holomorphic map ${\cal X}\to S$, where $S$ is an analytic space and the fibre over a distinguished point $0\in S$ is isomorphic to $X$. We will say that a certain property holds for the {\it generic} fibre, if for an open (in the analytic topology) dense set $U\subset S$ and all $t\in U$ the fibre ${\cal X}_t$ has this property. The property holds for the {\it general} fibre if such a set $U$ exists that is the complement of the union of countably many nowhere dense closed (in the analytic topology) subsets. One knows that for any compact K\"ahler manifold $X$ there exists a {\it semi-universal} deformation ${\cal X}\to {\it Def}(X)$, where ${\it Def}(X)$ is a germ of an analytic space and the fibre ${\cal X}_0$ over $0\in {\it Def}(X)$ is isomorphic to $X$. The {\it Zariski tangent space} of ${\it Def}(X)$ is naturally isomorphic to $H^1(X,{\cal T}_X)$. If $H^0(X,{\cal T}_X)=0$, i.e.\ if $X$ does not allow infinitesimal automorphisms, then ${\cal X}\to {\it Def}(X)$ is universal, i.e.\ for any deformation ${\cal X}_S\to S$ of $X$ there exists a uniquely determined holomorphic map $S\to {\it Def}(X)$ such that ${\cal X}_S\cong{\cal X}\times_{{\it Def}(X)}S$. By a result of Tian \cite{Tian} and Todorov \cite{Todorov} the base space ${\it Def}(X)$ is smooth if $K_X\cong {\cal O}_X$ (for an algebraic proof of this result we refer to \cite{Kawamata} and \cite{R}, see also \cite{Bo2} for irreducible symplectic manifolds). In this case one says that $X$ deforms {\it unobstructed}. Hence, if $X$ is an irreducible symplectic manifold then there exists a universal deformation ${\cal X}\to {\it Def}(X)$ of $X$ with ${\it Def}(X)$ smooth of dimension $h^1(X,{\cal T}_X)=h^1(X,\Omega_X)=h^{1,1}(X)$. Note that any small deformation of an irreducible symplectic manifold is irreducible symplectic (\cite{Beauville1}, see also \ref{exbydef}). Also note that the universal deformation ${\cal X}\to{\it Def}(X)$ is in fact universal for any fibre ${\cal X}_t$ with $t$ close to $0$. \bigskip {\bf \refstepcounter{theorem}\label{deformationhk} \thetheorem} --- Let $\alpha$ be a K\"ahler class on an irreducible symplectic manifold $X$ and let $(M,g)$ be the underlying hyperk\"ahler manifold. As briefly explained above, there exists a sphere $S^2\cong\IP^1$ of complex structures on $M$ induced by the hyperk\"ahler metric $g$. This gives rise to a compact manifold ${\cal X}(\alpha)$ and a smooth holomorphic map $${\cal X}(\alpha)\to \IP^1$$ such that for any $\lambda\in S^2\cong\IP^1$ the fibre over $\lambda$ is isomorphic to $(M,\lambda)$. In particular, $X$ occurs as the fibre over $I$. The space ${\cal X}(\alpha)$ together with the projection is called the {\it twistor space} of the hyperk\"ahler manifold $(M,\lambda)$. For more details and the precise relation between the metric $g$ and the twistor space see \cite{HKR}. The twistor space ${\cal X}(\alpha)\to \IP^1$ induces a non-trivial (local) map $\IP^1\to {\it Def}(X)$ and hence a one-dimensional subspace of $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$. Fujiki \cite{Fujiki2} shows that this subspace is spanned by $\alpha$. In other words, the Kodaira-Spencer class of the twistor space of $X$ with respect to the K\"ahler class $\alpha$ is identified with $\alpha$ under the isomorphism $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$ induced by the holomorphic two-form $\sigma$ on $X$. Note that one can also prove the unobstructedness of $X$ along this line: Any class in ${\cal K}_X\subset H^1(X,\Omega_X)$ can be realized as a Kodaira-Spencer class of a smooth curve and ${\cal K}_X$ is open in the real part $H^{1,1}(X)_{\mathbb R}$ of $H^{1,1}(X)$ (cf.\ \cite{Fujiki2}). \bigskip {\bf \refstepcounter{theorem}\label{deformationlb} \thetheorem} --- At several points in this article we will have to deal with joint deformations of an irreducible symplectic manifold $X$ together with a line bundle $L$ on $X$. Again, there exists a universal deformation $({\cal X},{\cal L})\to {\it Def}(X,L)$, i.e.\ a deformation ${\cal X}\to {\it Def}(X,L)$ of ${\cal X}_0=X$ and a line bundle ${\cal L}$ on ${\cal X}$ with ${\cal L}_0:={\cal L}|_{{\cal X}_0}\cong L$ such that any other deformation $({\cal X}_S,{\cal L}_S)\to S$ of this sort is isomorphic to the pull-back of $({\cal X},{\cal L})$ via a uniquely determined map $S\to {\it Def}(X,L)$. The Zariski tangent space of ${\it Def}(X,L)$ is canonically isomorphic to $H^1(X,{\cal D}(L))$, where ${\cal D}(L)$ is the sheaf of differential operators on $L$ of order $\leq1$. Using the symbol sequence and $H^1(X,{\cal O}_X)=0$ one shows $H^1(X,{\cal D}(L))=\ker(c_1(L):H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X))$. By \ref{pairing} the contraction $c_1(L):H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)=\bar\sigma\cdot\IC$ is surjective if $0\ne c_1(L)\in H^1(X,\Omega_X)$. Moreover, in this case the space ${\it Def}(X,L)$ is a smooth hypersurface of ${\it Def}(X)$. Another consequence of \ref{pairing} is that for two line bundles $L$ and $M$ with $c_1(L)$ and $c_1(M)$ linearly independent the two hypersurfaces ${\it Def}(X,L)$ and ${\it Def}(X,M)$ intersect transversally. \bigskip {\bf \refstepcounter{theorem}\label{periodmapdef} \thetheorem} --- Let $\Gamma$ be a lattice of index $(3,b-3)$. By $q_\Gamma$ we denote its quadratic form. A {\it marked} irreducible symplectic manifold is a tuple $(X,\varphi)$ consisting of an irreducible symplectic manifold $X$ and an isomorphism $\varphi:H^2(X,{\mathbb Z})\cong \Gamma$ compatible with $q_X$ and $q_\Gamma$. The {\it period} of $(X,\varphi)$ is by definition the one-dimensional subspace $\varphi(H^{2,0}(X))\subset \Gamma_\IC$ considered as a point in the projective space $\IP(\Gamma_\IC)$. If ${\cal X}\to {\it Def}(X)$ is the universal deformation of ${\cal X}_0=X$, then a marking $\varphi$ of $X$ naturally defines markings $\varphi_t$ of all the fibres ${\cal X}_t$. Thus we can define the {\it period map} $${\cal P}:{\it Def}(X)\to \IP(\Gamma_\IC)$$ as the map that takes $t$ to the period of $({\cal X}_t,\varphi_t)$. Note ${\cal P}$ is holomorphic. Its tangent map is given by the contraction $$H^1(X,{\cal T}_X)\to {\rm Hom}(H^{2,0}(X), H^{1,1}(X))\subset {\rm Hom}(H^{2,0}(X), H^{2}(X,\IC)/H^{2,0}(X)).$$ By \ref{quadraticform} the holomorphic two-form $\sigma$ on $X$ satisfies $q_X(\sigma)=0$ and $q_X(\sigma+\bar\sigma)>0$. Hence the image of ${\cal P}$ is contained in the {\it period domain} $Q\subset \IP(\Gamma_\IC)$ defined as $\{x\in\IP(\Gamma_\IC)|q_\Gamma(x)=0,~~ q_\Gamma(x+\bar x)>0\}$, which is an open (in the analytic topology) subset of the non-singular quadric defined by $q_\Gamma$. Beauville proved in \cite{Beauville1} the Local Torelli Theorem: {\it For any marked irreducible symplectic manifold $(X,\varphi)$ the period map ${\cal P}:{\it Def}(X)\to Q$ is a local isomorphism.} \bigskip {\bf \refstepcounter{theorem}\label{hypersurfaces} \thetheorem} --- Let $(X,\varphi)$ be a marked irreducible symplectic manifold. For any $\alpha \in H^2(X,{\mathbb R})$ we define $$S_\alpha:=\{t\in {\it Def}(X)|q_\Gamma(\varphi(\alpha),{\cal P}(t))=0\},$$ i.e.\ $S_\alpha$ is the pull-back of the hyperplane defined by $q_\Gamma(\varphi(\alpha),\,.\,)$. By the properties of $q_X$, the set $S_\alpha$ is easily identified as the set of points $t\in {\it Def}(X)$ such that $\alpha$ is of type $(1,1)$ on ${\cal X}_t$. Analogously, if $\alpha\in H^{4n-2}(X,{\mathbb R})$, which can be considered as a linear form on $H^2(X,{\mathbb R})$, then $$S_\alpha:=\{t\in {\it Def}(X)|\alpha(\varphi^{-1}({\cal P}(t)))=0\}.$$ Using the holomorphic Lefschetz theorem \ref{irredcoh} one finds that $S_\alpha$ in this situation is the set of $t\in {\it Def}(X)$ such that $\alpha$ is of type $(2n-1,2n-1)$ on the fibre ${\cal X}_t$. Using the perfectness of the natural pairings \ref{pairing} one proves the following results \cite{Beauville1,Fujiki2}: \begin{itemize} \item[--] For $0\ne\alpha\in H^2(X,{\mathbb R})$ or $0\ne\alpha\in H^{4n-2}(X,{\mathbb R})$ the set $S_\alpha$ is a smooth possibly empty hypersurface of ${\it Def}(X)$. \item[--] If $0\in S_\alpha$, i.e.\ $\alpha\in H^{1,1}(X)_{\mathbb R}$ respectively $\alpha\in H^{2n-1,2n-1}(X)_{\mathbb R}$, the tangent space $T_0S_\alpha$ of $S_\alpha$ is the kernel of the surjection $\alpha:H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ respectively $\alpha:H^1(X,{\cal T}_X)\to H^{2n}(X,\Omega_X^{2n-2})$. \item[--] If $\alpha,\alpha'\in H^{1,1}(X)$ are linearly independent, then $S_\alpha$ and $S_{\alpha'}$ intersect transversally in $0\in {\it Def}(X)$. The analogous statement holds true for linearly independent classes $\alpha,\alpha'\in H^{2n-1,2n-1}(X)$. \item[--] If $\alpha\in H^2(X,{\mathbb Z})$, then there exists a line bundle ${\cal L}$ on ${\cal X}|_{S_\alpha}$ such that $c_1({\cal L}_t)=\alpha$ for all $t\in S_\alpha$. Moreover, if $\alpha=c_1(L)$, then $S_\alpha={\it Def}(X,L)$. \end{itemize} \bigskip {\bf \refstepcounter{theorem}\label{twistor2} \thetheorem} --- In \ref{deformationhk} we discussed the twistor space ${\cal X}(\alpha)\to \IP^1$ of an irreducible symplectic manifold $X$ endowed with a K\"ahler class $\alpha$. The base of the twistor space can be identified via the period map as follows: Recall, $F(\alpha)_\IC=\sigma\cdot\IC\oplus\bar\sigma\cdot\IC\oplus\alpha\cdot\IC$. If $\varphi$ is a marking of $X$, then the period map ${\cal P}:\IP^1\to \IP(\Gamma_\IC)$ defines an isomorphism of $\IP^1$ with $Q\cap\IP(\varphi(F(\alpha)_\IC))$. Thus, $\IP^1$, as the base space of the twistor space, is naturally identified with the quadric $T(\alpha)\subset \IP(F(\alpha)_\IC)$ defined by $q_X(\beta)=0$. In the sequel, we will denote the twistor space by ${\cal X}(\alpha)\to T(\alpha)$. This can be slightly generalized as follows. If $\alpha$ is just any class of type $(1,1)$ on $X$ we define $T(\alpha):={\it Def}(X)\cap{\cal P}^{-1}(\IP(\varphi(F(\alpha)_\IC)))$ and ${\cal X}(\alpha)$ as the restriction of the universal family to $T(\alpha)$. Note that for a K\"ahler class $\alpha$ the space $T(\alpha)$ means the complete base $\IP^1$ of the twistor space, but in general it is only defined as a closed subset of ${\it Def}(X)$. Fujiki \cite{Fujiki1} proved the following very useful result: {\it If $\alpha$ is a K\"ahler class and ${\cal X}(\alpha)\to T(\alpha)$ is the associated twistor space, then the general fibre contains neither effective divisors nor curves.} Indeed, the tangent space of the twistor space is spanned by the image $v$ of $\alpha$ under the isomorphism $H^1(X,\Omega_X)\cong H^1(X,{\cal T}_X)$. On the other hand, if $D\subset X$ is an effective divisor and $C\subset X$ is a curve, then the tangent space of the hypersurface $S_{[D]}$ (respectively $S_{[C]}$) is the kernel of the map $H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ (respectively $H^1(X,{\cal T}_X)\to H^{2n}(X,\Omega_X^{2n-2})$). Since up to scalar factors $\int_X ([D]\cdot v) \sigma^{n}{\bar\sigma}^{n-2}=q_X(\alpha,[D])>0$ and $\int_X ([C]\cdot v)\sigma=\int_C\alpha\ne0$ (use \ref{Todd} resp.\ \ref{pairing} and that $\alpha$ is K\"ahler), the hypersurfaces $S_{[D]}$ and $S_{[C]}$ meet $T(\alpha)$ locally only in $0$. Or equivalently, neither $D$ nor $C$ deforms in the twistor space. Since there are only countably many such classes $[D]$ and $[C]$, this proves the claim. \bigskip {\bf \refstepcounter{theorem}\label{moduli} \thetheorem} --- Last but not least we fix the notation for the moduli space of marked irreducible symplectic manifolds. Let $\Gamma$ be a lattice of index $(3,b-3)$, where $b\geq3$. Then $ {\mathfrak M}_\Gamma=\{(X,\varphi)\}/\sim$, where $(X,\varphi)$ is a marked irreducible symplectic manifold (we usually also fix the dimension $2n$) and $(X,\varphi)\sim(X',\varphi')$ if and only if there exists an isomorphism $f:X\cong X'$ such that $f^*=\pm(\varphi^{-1}\circ\varphi')$. The Local Torelli Theorem (\cite{Beauville1}, \ref{periodmapdef}) allows one to patch the local charts ${\it Def}(X)$. Thus ${\mathfrak M}_\Gamma$ carries the structure of a non-separated (i.e.\ non-Hausdorff) complex manifold. The period map can be considered as a holomorphic map ${\cal P}:{\mathfrak M}_\Gamma\to Q\subset\IP(\Gamma_\IC)$. \section{Examples}\label{examples} For the reader's convenience we collect the known examples of irreducible symplectic manifolds. In all cases the verification is reduced to the conditions of Definition \ref{IrredSympl} and not to an explicit construction of an irreducible hyperk\"ahler metric. Explicit examples of irreducible hyperk\"ahler metrics on compact manifolds would be highly desirable. \bigskip {\bf \refstepcounter{theorem}\label{K3} \thetheorem. K3 surfaces.} {\it A complex manifold of dimension two is irreducible symplectic if and only if it is a K3 surface.} By definition a K3 surface is a compact connected surface with trivial canonical bundle and vanishing first Betti number. One can show that any K3 surface is deformation equivalent to a smooth quartic hypersurface in $\IP^3$ and, therefore, simply connected. That a K3 surface is K\"ahler is due to Siu \cite{Siu} (see also \cite{Periodes}). Examples of Guan \cite{Guan} show that in higher dimensions not every simply connected holomorph-symplectic manifold is K\"ahler. \bigskip {\bf \refstepcounter{theorem}\label{HilbK3} \thetheorem. Hilbert schemes of K3 surfaces.} {\it If $S$ is a K3 surface, then ${\rm Hilb}^n(S)$ is irreducible symplectic (cf.\ \cite{Beauville1})} By the Hilbert scheme ${\rm Hilb}^n(S)$ we mean the Douady space of zero-dimensional subspaces $(Z,{\cal O}_Z)$ of $S$ of length $\dim_\IC{\cal O}_Z=n$. Strictly speaking, ${\rm Hilb}^n(S)$ is a scheme only if $S$ is algebraic. In general, it is just a complex space. Using that $S$ is smooth, compact, connected, and of dimension two, one shows that ${\rm Hilb}^n(S)$ is a smooth compact connected manifold of dimension $2n$. By results of Varouchas \cite{Varouchas} the Hilbert scheme is K\"ahler if the underlying surface is K\"ahler which is the case for K3 surfaces. Beauville then concluded that for any K3 surface $S$ the Hilbert scheme ${\rm Hilb}^n(S)$ is irreducible symplectic by showing that ${\rm Hilb}^n(S)$ admits a unique (up to scalars) everywhere non-degenerate holomorphic two-form and that it is simply connected. For $n=2$ this result was also obtained by Fujiki. It is interesting to note that for $n>1$ one has $b_2({\rm Hilb}^n(S))=23$. Moreover, the second cohomology $H^2({\rm Hilb}^n(S),{\mathbb Z})$ endowed with the natural quadratic form $q_X$ (cf.\ \ref{quadraticform}) is isomorphic to the lattice $H^2(S,{\mathbb Z})\oplus (-2(n-1)\cdot{\mathbb Z})$. \bigskip {\bf \refstepcounter{theorem}\label{Kummer} \thetheorem. Generalized Kummer varieties.} {\it If $A$ is a two-dimensional torus, then ${\rm K}^{n+1}(A)$ is irreducible symplectic (cf.\ \cite{Beauville1}).} The generalized Kummer variety ${\rm K}^{n+1}(A)$ is by definition the fibre over $0\in A$ of the natural morphism ${\rm Hilb}^{n+1}(A)\to S^{n+1}(A)\rpfeil{5}{\Sigma}A$, where $\Sigma$ is the summation and $0\in A$ is the zero-point of the torus. ${\rm Hilb}^{n+1}(A)$ itself also admits an everywhere non-degenerate two-form, but neither is this two-form unique nor is ${\rm Hilb}^{n+1}(A)$ simply connected. That both conditions are satisfied for ${\rm K}^{n+1}(A)$ was shown by Beauville. That ${\rm K}^{n+1}(A)$ is K\"ahler follows again from the results in \cite{Varouchas}. The second Betti number of ${\rm K}^{n+1}(A)$ is $7$ (cf.\ \cite{Beauville1}). I usually refer to the examples provided by the Hilbert schemes of K3 surfaces and by the generalized Kummer varieties as the two standard series of examples of irreducible symplectic manifolds. Note that by means of these examples we have in any real dimension $4n$ at least two different compact real manifolds admitting irreducible hyperk\"ahler metrics. That they are not diffeomorphic (in fact, not even homeomorphic) follows from a comparison of their second Betti numbers. \bigskip {\bf \refstepcounter{theorem}\label{exbydef} \thetheorem. New examples by deformation.} {\it Any small deformation of an irreducible symplectic manifold is irreducible symplectic (cf.\ \cite{Beauville1}).} The stability results in \cite{KSIII} say that any small deformation of a compact K\"ahler manifold is again K\"ahler. Since the Hodge number $h^{2,0}$ is constant in families of compact K\"ahler manifolds, any small deformation of an irreducible symplectic manifold admits a unique non-trivial non-degenerate two-form which is everywhere non-degenerate. In fact, using the splitting theorem one can show that any K\"ahler deformation of an irreducible symplectic manifold is again irreducible symplectic. For the details see \cite{Beauville1}. It seems to be an open problem if any deformation of an irreducible symplectic manifold is K\"ahler and, therefore, irreducible symplectic. Deforming the Hilbert scheme ${\rm Hilb}^n(S)$ of a K3 surface or the generalized Kummer variety ${\rm K}^{n+1}(A)$ provides new examples of irreducible symplectic manifolds. Indeed, for $n>1$ one has $\dim {\it Def}({\rm Hilb}^n(S))=h^{1,1}({\rm Hilb}^n(S))=h^{1,1}(S)+1=\dim {\it Def}(S)+1(=21)$ and $\dim {\it Def}({\rm K}^{n+1}(A))=h^{1,1}({\rm K}^{n+1}(A))=h^{1,1}(A)+1=\dim {\it Def}(A)+1(=5)$. Thus one can think of the deformations of ${\rm Hilb}^n(S)$ that remain Hilbert schemes as a hypersurface in the full deformation space of ${\rm Hilb}^n(S)$. An analogous result holds true for Kummer varieties. \bigskip {\bf \refstepcounter{theorem}\label{exbybir} \thetheorem. New examples by birational transformation.} {\it If a compact K\"ahler manifold admits an everywhere non-degenerate two-form and is birational to an irreducible symplectic manifold, then it is irreducible symplectic as well.} This follows easily from the observation that $\pi_1(X)=\pi_1(Y)$ and $h^{2,0}(X)=h^{2,0}(Y)$ for two birational compact manifolds $X$ and $Y$. Can one drop the assumption that the manifold is K\"ahler? Or equivalently, is any compact manifold that is birational to an irreducible symplectic manifold and that admits an everywhere non-degenerate two-form, automatically K\"ahler? Note also the following \begin{lemma}\label{hodgeunderbir}--- If $X$ and $X'$ are birational irreducible symplectic manifolds, then there exists a natural isomorphism $H^2(X,{\mathbb Z})\cong H^2(X',{\mathbb Z})$ compatible with the Hodge structures and the quadratic forms $q_X$ and $q_{X'}$ \cite{Enoki, Mu2,OG}. \end{lemma} {\em Proof}. We provide three slightly different descriptions of this isomorphism. First, let us fix maximal open subsets $U\subset X$ and $U'\subset X'$ such that $U\cong U'$ and ${\rm codim}(X\setminus U),{\rm codim} (X'\setminus U')\geq2$ (see the remarks after \ref{assump}). Then one defines the isomorphism as the composition $H^2(X,{\mathbb Z})\cong H^2(U,{\mathbb Z})\cong H^2(U',{\mathbb Z})\cong H^2(X',{\mathbb Z})$. Since these isomorphisms commute with $H^0(X,\Omega_X^2)\cong H^0(U,\Omega_U^2)\cong H^0(U',\Omega_{U'}^2) \cong H^0(X',\Omega_{X'}^2)$, the isomorphism $H^2(X,{\mathbb Z})\cong H^2(X',{\mathbb Z})$ is compatible with the Hodge structures. The second description is in terms of the closure of the graph $Z\subset X\times X'$ of the birational map $X- - \to X'$. Then $[Z]_*:H^2(X,{\mathbb Z})\to H^2(X',{\mathbb Z})$, defined by $\alpha\mapsto p'_*([Z].p^*\alpha)$, equals the above isomorphism. We also write $[Z]_*$ for the map $\beta\mapsto p_*([Z].{p'}^*\beta)\in H^2(X,{\mathbb Z})$. Using the first description, one finds $[Z]_*\circ [Z]_*={\rm id}$. At one point in the discussion later on we will need $q{_X'}([Z]_*\alpha,\beta)=q_X(\alpha,[Z]_*\beta)$, which follows easily from the compatibility with the quadratic form, proved below, and $[Z]_*\circ [Z]_*={\rm id}$. Yet another description goes as follows: Let $\tilde Z\to Z$ be a desingularization. Then $H^2(\tilde Z,{\mathbb Z})\cong H^2(X,{\mathbb Z})\oplus\bigoplus_i[E_i]\cdot{\mathbb Z}$ and $H^2(\tilde Z,{\mathbb Z})\cong H^2(X',{\mathbb Z})\oplus\bigoplus_i[E_i]\cdot{\mathbb Z}$ , where the $E_i$'s are the exceptional divisors of $\tilde Z\to X$ (or, equivalently, of $\tilde Z\to X'$). The isomorphism $H^2(X,{\mathbb Z})\cong H^2(X',{\mathbb Z})$ is then given as the composition of the natural inclusion followed by the projection. In either of the three descriptions one sees that the isomorphism maps the class of an effective divisor in $X$ to the class of an effective divisor in $X'$. To see the compatibility with the quadratic forms $q_X$ and $q_{X'}$, let $\sigma$ and $\sigma'$ be two-forms on $X$ and $X'$, respectively, with $\int_X(\sigma\bar\sigma)^{n}=1=\int_{X'}(\sigma'{\bar\sigma}')^n$. Their pull-backs to $\tilde Z$ coincide. Using the description of $q_X$ and $q_{X'}$ given in \ref{quadraticform}, one finds that both can be defined on $\tilde Z$ and that via $[Z]_*$ they coincide if $\sigma^{n-1}|_{E_i}=0$. The latter equality now follows from the observation that the fibers (which are of positive dimension) of the morphisms $E_i\to X$ and $E_i\to X'$ are different and, hence, the rank of $\sigma$ (or $\sigma'$) on $E_i$ drops at least by two. \hspace*{\fill}$\Box$ \bigskip There exist non-trivial birational transformations: Mukai introduced the notion of {\it elementary transformations} \cite{Mu1}: Let $X$ be an irreducible symplectic manifold and assume that a smooth $\IP^m$-bundle $P:=\IP(F)\rpfeil{5}{\phi}Y$ can be embedded into $X$ as a submanifold of codimension $m$. Then ${\cal N}_{P/X}\cong\Omega_\phi$ and, hence, the exceptional divisor of the blow-up $\tilde X\to X$ of $X$ along $P$ is isomorphic to $\IP(\Omega_\phi)$. Regarding $\IP(\Omega_\phi)$ as the incidence variety in $\IP(F)\times_Y\IP(F\makebox[0mm]{}^{{\scriptstyle\vee}})$ yields another projection $\IP(\Omega_\phi)\to\IP(F\makebox[0mm]{}^{{\scriptstyle\vee}})$. Using the blow-down criterion of Fujiki and Nakano one extends this projection to a blow-down $\tilde X\to X' (:=elm_P(X))$. It is not hard to see that $X'$ also admits a unique everywhere non-degenerate two-form and that $X'$ is simply connected. If $X'$ is K\"ahler (is this always true?) then it is also irreducible symplectic. \begin{example}\label{bevdeb}--- First, let us recall an example of Beauville \cite{Beauville2}: If $S\subset \IP^3$ is a smooth quartic hypersurface, then the generic line $\ell\subset \IP^3$ meets $S$ in four distinct points. Thus one defines a rational map ${\rm Hilb}^2(S)- - \to {\rm Hilb}^2(S)$ by sending $[Z]\in {\rm Hilb}^2(S)$ to $(\ell_Z\cap S)\setminus\{Z\}$, where $\ell_Z$ is the uniquely defined line through $Z$. If $S$ does not contain any line then this map extends to an automorphism of ${\rm Hilb}^2(S)$. On the other hand, if $S$ contains $k$ lines $\ell_1,\ldots,\ell_k$, then ${\rm Hilb}^2(S)- - \to {\rm Hilb}^2(S)$ is the elementary transformation along the naturally embedded planes $\IP^2\cong {\rm Hilb}^2(\ell_i)\subset {\rm Hilb}^2(S)$. But, not any elementary transformation of an irreducible symplectic manifold $X$ is again isomorphic to $X$! One should rather think of Beauville's example as an exception. However, it is usually not easy to show that $X'=elm_P(X)$ is not isomorphic to $X$. One explicit example was given by Debarre in \cite{Debarre}: Let $S$ be a K3 surface such that ${\rm Pic}(S)={\cal O}(C)\cdot{\mathbb Z}$, where $C$ is a $(-2)$-curve. The Hilbert scheme $X:={\rm Hilb}^n(S)$ contains the projective space ${\rm Hilb}^n(C)\cong S^n(C)\cong\IP^n$ as above. Let $X'$ be the elementary transformation of $X$ along this projective space. If $X$ and $X'$ were isomorphic, then this would yield a birational automorphism of $X$. Debarre then shows that, due to the special shape of the surface $S$, the induced map on the second cohomology would be trivial and that this implies that the birational automorphism can be extended to an isomorphism, which is absurd. Due to \ref{hodgeunderbir} the two manifolds $X$ and $X'$ have isomorphic periods, but in order to obtain an honest counterexample to the Global Torelli Theorem one has in addition to ensure that $X'$ is K\"ahler. This is not clear in general, but if $S$ is close to an algebraic K3 surface $S_0$ as in the example above, then $X'$ is close to $elm_{\IP^n}({\rm Hilb}(S_0))\cong {\rm Hilb}^n(S_0)$. But any small deformation of a K\"ahler manifold is K\"ahler; hence there are examples for which $X'$ is K\"ahler. Note that in Debarre's example the manifolds are not projective. It would be interesting to find also a projective counterexample to the Global Torelli Theorem. Good candidates are the moduli space of stable sheaves on a K3 surface and the Hilbert scheme of the surface, which in some cases are birational, but most likely not isomorphic (cf.\ \ref{modK3}). \end{example} The following examples are related to one of the standard series by deformation or birational transformation as described above. In particular, the proof that they are irreducible symplectic is reduced to the proof of this fact for either a Hilbert scheme or a Kummer variety. \bigskip {\bf \refstepcounter{theorem}\label{fano} \thetheorem. Fano varieties of cubics \cite{BD}.} Let $Y\subset \IP^5$ be a smooth cubic hypersurface and let $X:=F(Y)$ be the Fano variety of lines on $Y$. Then $X$ is smooth and projective of dimension $4$. All cubics are deformation equivalent and, hence, so are the corresponding Fano varieties. Beauville and Donagi showed that for a special cubic $Y$ the Fano variety $X=F(Y)$ is isomorphic to the Hilbert scheme ${\rm Hilb}^2(S)$ of a special K3 surface of degree $14$ in $\IP^8$. Hence, for an arbitrary cubic $Y$ the Fano variety $X=F(Y)$ is a deformation of ${\rm Hilb}^2(S)$ and, therefore, irreducible symplectic. The map $Y\mapsto F(Y)$ identifies ${\it Def}(Y)$ with a hypersurface of ${\it Def}(X)$ parametrizing algebraic deformations of ${\rm Hilb}^2(S)$. It would be interesting to understand the space of algebraic deformations of ${\rm Hilb}^n(S)$ for other K3 surfaces. \bigskip {\bf \refstepcounter{theorem}\label{reljac} \thetheorem. Relative Jacobians \cite{Markushevich}.} Markushevich constructed an explicit example of a projective irreducible symplectic manifold which is completely integrable: Let $\pi:S\to \IP^2$ be a generic double cover ramified along a sextic. Then $S$ is a K3 surface. The dual space ${\IP^2}\makebox[0mm]{}^{{\scriptstyle\vee}}$ can be regarded as the base space of the family of hyperelliptic curves (of genus $2$) of the form $\pi^{-1}(\ell)$, where $\ell\subset\IP^2$ is a line. Then the compactified relative Jacobian $X\to{\IP^2}\makebox[0mm]{}^{{\scriptstyle\vee}}$ of this family of curves is smooth projective and admits an everywhere non-degenerate two-form. Since the map that sends $[Z]\in {\rm Hilb}^2(S)$ to the divisor $(Z)$ on the curve $\pi^{-1}(\ell_{\pi(Z)})$, where $\ell_{\pi(Z)}$ is the line through $\pi(Z)$, defines a birational map ${\rm Hilb}^2(S)- -\to X$, the manifold $X$ is irreducible symplectic. In fact, $X$ can be seen as an example of a moduli space of simple sheaves (or rather stable sheaves with pure support as treated below). Similar examples were considered by Beauville and Mukai. But, as it is not surprising, in all cases the resulting spaces are birational to some Hilbert scheme of the underlying K3 surfaces. \bigskip {\bf \refstepcounter{theorem}\label{modK3} \thetheorem. Moduli spaces of sheaves on K3 surfaces.} Let $S$ be a K3 surface. Consider $M_H(v)$ -- the moduli space of sheaves with primitive Mukai vector $v$ which are stable with respect to a generic polarization $H$ (see \cite{OG} or \cite{HL} for the notation). This space is a projective variety and by the general smoothness criterion it is non-singular. Note that also ${\rm Hilb}^n(S)$ can be considered as such a moduli space, namely the moduli space of stable rank one sheaves. Mukai \cite{Mu1} constructed on $M_H(v)$ an everywhere non-degenerate two-form. Later it was shown that $M_H(v)$ is indeed irreducible symplectic (in \cite{GottscheHuybrechts} for the rank two case and in \cite{OG} for arbitrary rank). The proof in both cases goes roughly as follows: Any smooth deformation of $(S,H)$ induces a deformation of $M_H(v)$ which is smooth as long as $H$ stays $v$-generic. Using this one reduces the proof to the case of a very special K3 surface (e.g.\ an elliptic surface in \cite{OG}), where one can show that the moduli space is birational to the Hilbert scheme of the same dimension. Hence we have: {\it The moduli spaces $M_H(v)$ are irreducible symplectic manifolds whenever $v$ is primitive and $H$ is $v$-generic.} \section{Projectivity}\label{projectivity} Let $X$ be an irreducible symplectic manifold. The goal of this section is to show that the second cohomology $H^2(X,{\mathbb Z})$ endowed with the natural weight-two Hodge structure and the quadratic form $q_X$ (cf.\ \ref{quadraticform}) determines whether $X$ is a projective variety. The following list collects known projectivity criteria that either motivate the main result (Theorem \ref{proj}) or are essential for its proof. \bigskip \refstepcounter{theorem}\label{Moish}{\bf \thetheorem} --- Let $X$ be a compact complex manifold. Then $X$ is projective if and only if $X$ is K\"ahler and Moishezon (cf.\ \cite{Moishezon}, \cite{Peternell}). \refstepcounter{theorem}\label{Kod}{\bf \thetheorem} --- Let $X$ be a compact complex manifold. Then $X$ is projective if and only if $X$ admits a K\"ahler form $\omega$ such that its cohomology class $[\omega]\in H^2(X,\IC)$ is integral (cf.\ \cite{Kodaira2}, \cite{GriffithsHarris}). \refstepcounter{theorem}\label{Chow}{\bf \thetheorem} --- Let $X$ be a compact complex surface. Then $X$ is projective if and only if $X$ is Moishezon (cf.\ \cite{Kodaira1}, \cite{BPV}). \refstepcounter{theorem}\label{possurf}{\bf \thetheorem} --- Let $X$ be a compact complex surface. Then $X$ is projective if and only if there exists a line bundle $L$ on $X$ such that $\int c_1^2(L)>0$ (cf.\ \cite{BPV}). \bigskip It is easy to see that a surface $X$ that admits a line bundle $L$ with $\int c_1^2(L)>0$ is Moishezon. Indeed, the Hirzebruch-Riemann-Roch formula shows that either $h^0(X,L^m)\sim m^2$ or $h^0(X,L^{-m}\otimes K_X)\sim m^2$. For manifolds of dimension $>2$ the existence of a big line bundle $L$, i.e.\ a line bundle with $\int c_1(L)^{\dim(X)}>0$, is not sufficient to conclude that $X$ is Moishezon, for the Hirzebruch-Riemann-Roch formula only gives $\sum h^{2i}(X,L^m)\sim m^{\dim(X)}$. Let us begin with a result due to Fujiki \cite{Fujiki1} (see also \cite{Campana}), which in particular shows that projective irreducible symplectic manifolds are dense in the moduli space. Some of the techniques used in later chapters are based on the proof of this result. Thus, we decided to reproduce it here. \begin{theorem}\label{fujiki}{\rm \cite{Fujiki1}}--- Let $X$ be an irreducible symplectic manifold, let $0\in S\subset {\it Def}(X)$ be a smooth analytic subset of positive dimension and let ${\cal X}\to S$ denote the restriction of the Kuranishi family to $S$. Then any open neighbourhood $U\subset S$ of $0\in S$ contains a point $t\ne0$ such that the fibre ${\cal X}_t$ over $t$ is projective. \end{theorem} {\em Proof}. We may assume that $S$ is one-dimensional. Then, its tangent space at zero $T_0S$ is a line in $T_0{\it Def}(X)=H^1(X,{\cal T}_X)$ spanned by, say, $0\ne v\in H^1(X,{\cal T}_X)$. By \ref{pairing} the induced map $\tilde v:H^1(X,\Omega_X)\to H^2(X,{\cal O}_X)\cong\IC$ is surjective. Since the K\"ahler cone ${\cal K}_X$ is an open subset of $H^{1,1}(X)_{\mathbb R}= H^{1,1}(X)\cap H^2(X,{\mathbb R})$, which spans $H^1(X,\Omega_X)$ as a complex vector space, there exists a K\"ahler form $\omega$ on $X$ such that $\tilde v([\omega])\ne0$. Consider the corresponding hypersurface $S_{[\omega]}\subset {\it Def}(X)$ of deformations of $X$ where $[\omega]$ stays of type $(1,1)$ (cf.\ \ref{hypersurfaces}). The tangent space of $S_{[\omega]}$ at $0$ is the kernel of the map $H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)\cong\IC$ induced by the product with $[\omega]\in H^1(X,\Omega_X)$. Since $\tilde v([\omega])\ne0$, the tangent spaces $T_0S_{[\omega]}$ and $T_0S$ have zero intersection. Hence, $S_{[\omega]}$ and $S$ meet transversally in $0\in {\it Def}(X)$. Shrinking $S$ we can also assume that $S_{[\omega]}\cap S=\{0\}$. Next, pick classes $\alpha_i\in H^2(X,\IQ)$ converging to $[\omega]$ and consider the associated hypersurfaces $S_{\alpha_i}\subset {\it Def}(X)$. Then the hypersurfaces $S_{\alpha_i}$ converge to $S_{[\omega]}$ and therefore $S_{\alpha_i}\cap S\ne\emptyset$ for $i\gg0$. Moreover, if we choose $\alpha_i$ such that they are not of type $(1,1)$ on $X$, then $0\not\in S_{\alpha_i}\cap S$. Hence, there exist points $t_i\in (S_{\alpha_i}\cap S)\setminus\{0\}$ converging to $0$. Using the isomorphism $H^2({\cal X}_t,{\mathbb Z})\cong H^2(X,{\mathbb Z})$, the classes $\alpha_i$ are considered as rational classes of type $(1,1)$ on ${\cal X}_{t_i}$. Intuitively, the classes $\alpha_i$ on ${\cal X}_{t_i}$ converge to the K\"ahler class $[\omega]$ on $X={\cal X}_0$ and thus should be K\"ahler for $i\gg0$. This can be made rigorous as follows: Fix a diffeomorphism ${\cal X}\cong X\times S$ compatible with the projections to $S$. By a result of Kodaira and Spencer (Thm.\ 3.1 in \cite{KSI}, Thm.\ 15 in \cite{KSIII}) there exists a real two-form $\tilde \omega$ on $X\times S$ such that the restriction $\tilde \omega_t$ of $\tilde\omega$ to ${\cal X}_t=X\times\{t\}$ is a K\"ahler form for all $t\in S$ and $\tilde \omega_0=\omega$. One also finds two-forms $\tilde\omega_i$ on $X\times S$ such that $(\tilde\omega_i)_t$ is harmonic with respect to $\tilde\omega_t$ and $[(\tilde\omega_i)_t]=\alpha_i$ (Sect.\ 2 in \cite{KSI}). In particular, $(\tilde\omega_i)_{t_i}$ is a harmonic $(1,1)$-form on ${\cal X}_{t_i}$ representing $\alpha_i$. Since $(\tilde\omega_i)_{t_i}$ converges to $\omega$, the harmonic $(1,1)$-form $(\tilde\omega_i)_{t_i}$ is a K\"ahler form on ${\cal X}_{t_i}$ for $i\gg0$. Hence, $\alpha_i$ is a Hodge class on ${\cal X}_{t_i}$ for $i\gg0$. In particular, ${\cal X}_{t_i}$ is projective for $i\gg0$.\hspace*{\fill}$\Box$ \bigskip For the proof of the main result of this section we need to recall some facts about positive forms and currents. Let $X$ be an irreducible symplectic manifold and let $\omega$ be a fixed K\"ahler form on $X$. By ${\cal A}^{p,p}(X)$ (resp. ${\cal A}^{p,p}(X)_{\mathbb R}$) we denote the space of smooth (real) $(p,p)$-forms on $X$. A form $\varphi\in{\cal A}^{p,p}(X)_{\mathbb R}$ is called {\it positive} if locally it can be written in the form $i\alpha_1\wedge\bar\alpha_1\wedge\ldots\wedge i\alpha_p\wedge\bar\alpha_p$, where the $\alpha_i$'s are smooth $(1,0)$-forms. For $p=2n-1$ we introduce the convex cone $C_{pos}\subset{\cal A}^{2n-1,2n-1}(X)_{\mathbb R}$ that is spanned by the positive forms. \begin{remark} --- Of course, $\omega$ itself is a positive $(1,1)$-form. Conversely, if $\varphi$ is a $d$-closed positive $(1,1)$-form, then $\varphi+\varepsilon\omega$ is a K\"ahler form for any $\varepsilon>0$. \end{remark} For the next lemma recall that $\sigma^{n-1}$ defines an isomorphism of complex bundles ${T^*_X}^{1,0}\cong {T^*_X}^{2n-1,0}$ (use the argument in \ref{irredcoh}). Analogously, $\bar\sigma^{n-1}$ defines an isomorphism ${T^*_X}^{0,1}\cong {T^*_X}^{0,2n-1}$. Passing to form valued sections of these bundles one obtains isomorphisms ($p\geq0$): $$\sigma^{n-1}:{\cal A}^{1,p}(X)\cong{\cal A}^{2n-1,p}(X)~~~~~{\rm and}~~~~~ \bar\sigma^{n-1}:{\cal A}^{p,1}(X)\cong{\cal A}^{p,2n-1}(X)$$ and, since $\sigma\bar\sigma$ is real, also $(\sigma\bar\sigma)^{n-1}:{\cal A}^{1,1}(X)_{\mathbb R}\cong {\cal A}^{2n-1,2n-1}(X)_{\mathbb R}.$ \begin{lemma}\label{posi} --- Let $\psi\in {\cal A}^{1,1}(X)_{\mathbb R}$. If $(\sigma\bar\sigma)^{n-1}\psi\in{\cal A}^{2n-1,2n-1}(X)_{\mathbb R}$ is positive, then also $\psi$ is positive. \end{lemma} {\em Proof}. If $(\sigma\bar\sigma)^{n-1}\psi$ is positive it can locally be written as $$\begin{array}{rcl} (\sigma\bar\sigma)^{n-1}\psi&=&i\alpha_1\wedge\bar\alpha_1\wedge\ldots\wedge i\alpha_{2n-1}\wedge\bar\alpha_{2n-1}\\ &=&i^{2n-1}(-1)^{(2n-1)(n-1)}\alpha_1\wedge\ldots\wedge\alpha_{2n-1}\wedge\bar\alpha_1\wedge\ldots\wedge\bar\alpha_{2n-1}.\\ \end{array}$$ First, observe that $i^{2n-1}(-1)^{(2n-1)(n-1)}=i$. Second, there exists a $(1,0)$-form $\beta$ such that $\sigma^{n-1}\wedge\beta=\alpha_1\wedge\ldots \wedge\alpha_{2n-1}$. Hence, $(\sigma\bar\sigma)^{n-1}\psi=i(\sigma\bar \sigma)^{n-1}(\beta\wedge\bar\beta)$. Therefore, $\psi=i\beta\wedge\bar\beta$, i.e.\ $\psi$ is positive.\hspace*{\fill}$\Box$ \bigskip The choice of the K\"ahler form $\omega$ induces a Hodge decomposition $${\cal A}^{p,p}(X)={\cal H}^{p,p}(X)\oplus({\rm im} (d)\oplus {\rm im} (d^*))\cap{\cal A}^{p,p}(X).$$ The space ${\cal H}^{p,p}(X)$ of harmonic $(p,p)$-forms is naturally isomorphic to $H^{p,p}(X)$. Since $\sigma\bar\sigma=\omega_J^2+\omega_K^2$ is harmonic, where $(I,J,K)$ is the hyperk\"ahler structure associated with $(\omega,I)$ (cf.\ \ref{Comparison}), the isomorphism in \ref{irredcoh} can be understood as the isomorphism given by $$\begin{array}{ccc} {\cal H}^{1,1}(X)_{\mathbb R}&\cong&{\cal H}^{2n-1,2n-1}(X)_{\mathbb R}\\ \alpha&\mapsto&(\sigma\bar\sigma)^{n-1}\alpha\\ \end{array}$$ By definition a $(1,1)$-{\it current} $T$ is a continuous linear map ${\cal A}^{2n-1,2n-1}(X)\to\IC$. The topology on ${\cal A}^{2n-1,2n-1}(X)$ is the one induced by the Hodge metric. $T$ is {\it positive} if $T(\varphi)\geq0$ for all $\varphi\in C_{pos}$. The current $T$ is {\it closed} if $T(d\gamma)=0$ for all $\gamma$ or, equivalently, if $T$ factorizes over ${\cal A}^{2n-1,2n-1}(X)/({\rm im} (d)\cap{\cal A}^{2n-1,2n-1}(X))$. Any closed $(1,1)$-current $T$ gives rise to a cohomology class $[T]\in H^{1,1}(X)$ which is given by restricting $T$ to ${\cal H}^{2n-1,2n-1}(X)$ and identifying the dual space of ${\cal H}^{2n-1,2n-1}(X)$ with ${\cal H}^{1,1}(X)$. A current $T$ is {\it real} if it is the natural extension of a continuous linear map ${\cal A}^{2n-1,2n-1}(X)_{\mathbb R}\to{\mathbb R}$. \begin{proposition} --- Let $X$ be an irreducible symplectic manifold and let $\alpha\in H^{1,1}(X)_{\mathbb R}$ such that $q_X(\alpha,\,.\,)$ is positive on the K\"ahler cone ${\cal K}_X$. Then there exists a closed positive real $(1,1)$-current $T$ such that $[T]=\alpha$. \end{proposition} {\em Proof}. The proof is inspired by an argument of Peternell in \cite{Peternell}. Let $A:={\cal A}^{2n-1,2n-1}(X)_{\mathbb R}/({\rm im} (d)\cap{\cal A}^{2n-1,2n-1}(X)_{\mathbb R})$, let $C$ be the image of $C_{pos}$ under the projection ${\cal A}^{2n-1,2n-1}(X)_{\mathbb R}\to A$, and let $B:={\cal H}^{2n-1,2n-1}(X)_{\mathbb R}$. Using Hodge decomposition one has $A={\cal H}^{2n-1,2n-1}(X)_{\mathbb R}\oplus ({\rm im} (d^*)\cap{\cal A}^{2n-1,2n-1}(X)_{\mathbb R})$. Thus $B$ can be considered as a (finite dimensional) subspace of $A$. The class $\alpha$ defines a continuous linear map $f_0:B\to {\mathbb R}$. A closed positive real $(1,1)$-current $T$ with $[T]=\alpha$ is a continuous linear extension $f:A\to {\mathbb R}$ of $f_0$ with $f|_C\geq0$. The existence of such an extension is due to the following general fact: If $A$ is a topological vector space, $B\subset A$ a subspace, and $C\subset A$ a convex cone such that $C\cap(-C)=0$ and $B\cap \stackrel{{\scriptscriptstyle o}}{C} \ne \emptyset$, then any continuous linear function $f_0:B\to {\mathbb R}$ with $f_0|_{B\cap C}\geq0$ can be extended to a continuous linear function $f:A\to {\mathbb R}$ with $f|_C\geq0$. (\cite{Bourbaki}, Ch.II, Sect.\ 3). Let us now verify the assumptions in our situation: Clearly, $C$ is a convex cone as it is the image of $C_{pos}$. If $\varphi,-\varphi'\in C_{pos}$ with $\varphi=\varphi'+d\gamma$ then the positivity of $\omega$ implies $0\leq\int\varphi\omega=\int\varphi'\omega\leq0$ and hence $\int\varphi\omega=\int\varphi'\omega=0$. Since $\omega$ is K\"ahler, we get $\varphi=\varphi'=0$. Therefore, $C\cap(-C)=0$. The intersection $B\cap \stackrel{\scriptscriptstyle o}{C}$ is not empty as it contains the class $\omega^{2n-1}$ which is an inner point of $C_{pos}$ (cf.\ \cite{Harvey}). Last but not least, one checks $f_0|_{B\cap C}\geq0$. Indeed, if $\varphi\in {\cal H}^{2n-1,2n-1}(X)_{\mathbb R}$ is positive, then there exists $\psi\in{\cal H}^{1,1}(X)_{\mathbb R}$ with $(\sigma\bar\sigma)^{n-1}\psi=\varphi$ and $\psi $ is positive by Lemma \ref{posi}. Hence, $\psi+\varepsilon\omega$ is K\"ahler for all $\varepsilon>0$. The assumption on $\alpha$ now implies $q_X(\alpha,[\psi+\varepsilon\omega])>0$ for all $\varepsilon>0$ and hence $q_X(\alpha,[\psi])\geq0$. Therefore, $\int\varphi\alpha=\int(\sigma\bar\sigma)^{n-1}\psi\alpha= cq_X([\psi],\alpha)\geq0$ for some positive constant $c$. Hence, $f_0(\varphi)\geq0$.\hspace*{\fill}$\Box$ \begin{remark} --- {\it i)} Demailly \cite{Demailly} introduced the notion of pseudo-effective classes: A class $\alpha\in H^{1,1}(X)$ is called pseudo-effective if it can be represented by a closed positive $(1,1)$-current. They span a closed cone $H^{1,1}_{psef}(X)\subset H^{1,1}(X)$. The proposition says that the cone dual to the K\"ahler cone is contained in $H^{1,1}_{psef}(X)$. {\it ii)} By \ref{posconedef} any class $\alpha\in {\cal C}_X$ is positive on ${\cal K}_X$. Hence ${\cal C}_X\subset H^{1,1}_{psef}(X)$. In fact, as ${\cal C}_X$ is an open cone, any class $\alpha\in{\cal C}_X$ can be represented by a closed positive real $(1,1)$-current $T$ such that $T-\varepsilon\omega$ is still positive for some $\varepsilon>0$. \end{remark} As an immediate consequence of \cite{Demailly} we obtain \begin{corollary}\label{demcor} --- If $X$ is a projective irreducible symplectic manifold and $L$ is a line bundle such that $c_1(L)\in{\cal C}_X$, then $L$ has maximal Kodaira-dimension, i.e.\ $h^0(L^k)$ grows like $c\cdot k^{2n}$ for some $c>0$. In particular, ${\cal C}_X\cap H^2(X,{\mathbb Z})$ is contained in the effective cone. \hspace*{\fill}$\Box$ \end{corollary} The main result of this section is the following \begin{theorem}\label{proj}--- Let $X$ be an irreducible symplectic manifold. Then $X$ is projective if and only if there exists a line bundle $L$ on $X$ with $q_X(c_1(L))>0$. \end{theorem} {\em Proof}. If $X$ is projective then $c_1(L)$ of an ample line bundle $L$ is a K\"ahler class and, therefore, $q_X(c_1(L))>0$ (cf.\ \ref{quadraticform}). Conversely, if $X$ admits a line bundle $L$ with $c_1(L)\in{\cal C}_X$, then there exists a closed $(1,1)$-current $T$ representing $c_1(L)$ such that $T-\varepsilon\omega$ is positive for some $\varepsilon>0$. By a result of Ji-Shiffman \cite{Shiffman} and Bonavero \cite{Bonavero} this implies that $X$ is Moishezon and, since it is also K\"ahler, that it is projective.\hspace*{\fill}$\Box$ \begin{remark} --- {\it i)} The theorem only asserts that $X$ is projective, but not that any line bundle $L$ with positive square $q_X(c_1(L))$ is ample (this is not even true for surfaces). A criterion for the ampleness of a line bundle will be discussed in Sect.\ \ref{amplecone}. However, if ${\rm Pic}(X)\cong L\cdot {\mathbb Z}$ with $q_X(c_1(L))>0$, then either $L$ or $L\makebox[0mm]{}^{{\scriptstyle\vee}}$ is ample. {\it ii)} In the original approach to prove the theorem I tried to avoid the result of Bonavero and Ji-Shiffman, which in turn relies on Demailly's very complicated holomorphic Morse inequalities. The argument should use the relatively easy Corollary \ref{demcor} on projective deformations and some kind of semi-continuity argument. But there are still some technical problems. Also note that if one is willing to use the holomorphic Morse inequalities (or rather their singular version) then one can in fact see that Corollary \ref{demcor} works without the projectivity assumption. {\it iii)} The techniques above can also be used to prove a result for K\"ahler surfaces I was not aware of before and which I could not find in the literature. However, I believe that going through the classification of surfaces an easier proof, not using singular Morse inequalities, should exist. The result is: A compact K\"ahler surface is projective if and only if the dual of the K\"ahler cone, i.e.\ the elements with positive intersection with any K\"ahler class, contains an inner integral point. \end{remark} \section{Birational Manifolds}\label{biratman} This section deals with the relation between birational irreducible symplectic manifolds and non-separated points in the moduli space ${\mathfrak M}_\Gamma$ (cf.\ \ref{moduli}). Already in dimension two, i.e.\ for K3 surfaces, the moduli space of marked irreducible symplectic manifolds is not separated (i.e.\ non-Hausdorff). But there, two non-separated points always correspond to just one K3 surface with two different markings. The situation is more subtle in higher dimensions: One easily generalizes the `Main Lemma' of Burns and Rapoport \cite{BurnsRapoport}, the algebraic version of which is \cite{MM}, to the effect that the underlying manifolds $X$ and $X'$ of two non-separated points $(X,\varphi),(X',\varphi')\in{\mathfrak M}_\Gamma$ are birational (Theorem \ref{BRMain}). But contrary to the two-dimensional case, this does not imply that $X$ and $X'$ are isomorphic. In fact, it is a standard procedure to construct via certain birational transformations out of one irreducible symplectic manifold new ones (cf.\ \ref{exbybir}). It is the goal of this section to prove that in general two birational irreducible symplectic manifolds correspond to non-separated points in ${\mathfrak M}_\Gamma$ (Theorem \ref{birat}). As the method is completely algebraic the result is limited to the case of projective manifolds (but see Sect.\ \ref{remarks} for the relation between a conjectural Global Torelli Theorem and this result for non-projective manifolds). This part is a continuation of the predecessor \cite{Huybrechts} of this paper, where the result was proved under an additional assumption on the codimension of the exceptional locus. For some details of the proof we will refer to \cite{Huybrechts}. The result has two applications. The first, to be considered in this section, concerns the classification of known examples of irreducible symplectic manifolds (cf.\ Sect.\ \ref{examples}). Corollary \ref{onlytwo} shows that all known examples of irreducible symplectic manifolds are deformation equivalent to one of the two standard examples: to the Hilbert scheme of points on a K3 surface or to a generalized Kummer variety (see Sect.\ \ref{examples} for their definition). In particular, in any real dimension $4n>4$ we know exactly two compact differentiable manifolds admitting an irreducible hyperk\"ahler metric. (For $n=1$ one knows that there exists exactly one: the real manifold underlying a K3 surface.) In the light of this result the list of examples of irreducible symplectic manifolds seems rather short. It might be noteworthy that the general results of this paper, e.g.\ the description of the K\"ahler cone \ref{kaehlerconethm}, the projectivity criterion \ref{proj} or the surjectivity of the period map \ref{surjper}, are by no means trivial (or a direct consequence of the K3 surface theory) for the deformations of the two standard series ${\rm Hilb}^n(S)$ and ${\rm K}^{n+1}(A)$. In fact, I do not see how one could possibly simplify the proofs for these special cases. The second application concerns a conjectural `Global Torelli Theorem' for higher dimensional irreducible symplectic manifolds. As this problem is intimately related to questions about the period map we postpone the discussion until Sect.\ \ref{remarks}. Before going into the subject we say a few words about what `non-separated' means in practice. To this end we formulate the following (almost tautological): \begin{lemma}\label{lemsep}--- Let $X$ and $X'$ be irreducible symplectic manifolds, let ${\cal X}\to S$ and ${\cal X}'\to S$ be deformations of $X$ and $X'$, respectively, and let $V\subset S$ be an open (in the analytic topology) non-empty subset such that {\it i)} $S$ is one-dimensional and {\it ii)} $0\in\partial V$, and {\it iii)} ${\cal X}|_V\cong {\cal X}'|_V$ (compatible with the projections to $S$). Then there exist markings $\varphi$ and $\varphi'$ of $X$ and $X'$, respectively, such that $(X,\varphi),(X',\varphi')\in{\mathfrak M}_\Gamma$ are non-separated. Conversely, if $(X,\varphi),(X'\varphi')\in{\mathfrak M}_\Gamma$ are two non-separated points in ${\mathfrak M}_\Gamma$, then there exist deformations ${\cal X}\to S$ and ${\cal X}'\to S$ and $V\subset S$ satisfying {\it i)}, {\it ii)}, and {\it iii)} such that for $t\in V$ the natural isomorphisms $H^2(X,{\mathbb Z})\cong H^2({\cal X}_t,{\mathbb Z})\cong H^2({\cal X}'_t,{\mathbb Z})\cong H^2(X',{\mathbb Z})$ are compatible with $\varphi$ and $\varphi'$.\hspace*{\fill}$\Box$ \end{lemma} \begin{remark}--- The arguments in the proof of Proposition \ref{Weylprop} show that in part two of the lemma one can in fact arrange things such that $V=S\setminus\{0\}$. See Remark \ref{topoo}. \end{remark} Let us now come to a straightforward generalization of the `Main Lemma' in \cite{BurnsRapoport}. The proof follows closely Beauville's expos\'e in \cite{Periodes}. \begin{theorem}\label{BRMain}--- If $(X,\varphi), (X',\varphi')\in{\mathfrak M}_\Gamma$ are non-separated points in the moduli space of marked irreducible symplectic manifolds, then $X$ and $X'$ are birational. \end{theorem} {\em Proof}. Consider deformations ${\cal X}\to S$ and ${\cal X}'\to S$ of ${\cal X}_0=X$ and ${\cal X}'_0=X'$, respectively, and $V\subset S$ as in \ref{lemsep}. In particular, we have for $t\in V$ the canonical isomorphism $H^2(X,{\mathbb Z})\cong H^2({\cal X}_t,{\mathbb Z})\cong H^2({\cal X}'_t,{\mathbb Z})\cong H^2(X',{\mathbb Z})$ which is compatible with $\varphi$ and $\varphi'$. Pick a sequence $t_i\in V$ converging to $0\in S$ and consider the corresponding isomorphisms $f_i:{\cal X}_{t_i}\cong {\cal X}'_{t_i}$ and their graphs $\Gamma_i\subset {\cal X}_{t_i}\times {\cal X}'_{t_i}$. By \cite{KSI} there exist K\"ahler forms $\omega_t$ and $\omega'_t$ on the fibres ${\cal X}_t$, respectively ${\cal X}'_t$, for all $t\in S$ depending continuously (this is enough) on $t$. The volume of $\Gamma_{t_i}$ with respect to these K\"ahler forms, can be computed by $$\begin{array}{ccl} vol(\Gamma_i)&=&\displaystyle{\int_{{\cal X}_{t_i}}(\omega_{t_i}+f_i^*\omega'_{t_i})^{2n}}\\ &=&\displaystyle{\int_{{\cal X}_{t_i}}([\omega_{t_i}]+f_i^*[\omega'_{t_i}])^{2n}}\\ &=&\displaystyle{\int_X([\omega_{t_i}]+({\varphi'}^{-1}\circ\varphi)[\omega'_{t_i}])^{2n}}.\\ \end{array}$$ (Use $f_i^*=({\varphi'}^{-1}\circ \varphi)$ via the isomorphisms $H^2(X,{\mathbb Z})\cong H^2({\cal X}_t,{\mathbb Z})$ and $H^2(X',{\mathbb Z})\cong H^2({\cal X}'_t,{\mathbb Z})$.) Hence $vol(\Gamma_i)$ converges to $\int_X([\omega_{0}]+({\varphi'}^{-1}\circ\varphi)[\omega'_{0}])^{2n}<\infty$. By a result of Bishop \cite {Bishop} the boundedness of the volume of $\Gamma_{i}$ is enough to conclude the existence of a limit cycle $\Gamma_\infty\subset X\times X'$ with the same cohomological properties as the $\Gamma_i$'s. In particular, $[\Gamma_\infty]\in H^{4n}(X\times X',{\mathbb Z})$ satisfies $p_*[\Gamma_\infty]=[X]\in H^0(X,{\mathbb Z})$; $p'_*[\Gamma_\infty]=[X']\in H^0(X',{\mathbb Z})$; and $p'_*([\Gamma_\infty].p^*\alpha)=({\varphi'}^{-1}\circ\varphi)(\alpha)$ for all $\alpha\in H^2(X,{\mathbb Z})$. Here, $p$ and $p'$ denote the two projections from $X\times X'$. Splitting $\Gamma_\infty$ into its irreducible components and using the first two properties we have either \begin{itemize} \item[$\bullet$] $\Gamma_\infty=Z+\sum Y_i$, where $p:Z\to X$ and $p':Z\to X'$ are generically one-to-one, or \item[$\bullet$] $\Gamma_\infty=Z+Z'+\sum Y_i$, where $p:Z\to X$ and $p':Z'\to X'$ are generically one-to-one, but neither $p':Z\to X'$ nor $p:Z'\to X$ is generically finite. \end{itemize} In both cases, $p_*[Y_i]=0$ and $p'_*[Y_i]=0$. The second possibility can be excluded: If $\sigma$ and $\sigma'$ are non-trivial holomorphic two-forms on $X$ and $X'$ respectively, then $({\varphi'}^{-1}\circ\varphi)([\sigma])=\lambda[\sigma']$ for some $\lambda\ne0$. Indeed, since the period domain, i.e.\ the quadric $Q\subset \IP(\Gamma_\IC)$, is separated, ${\varphi'}^{-1}\circ\varphi$ is compatible with the natural Hodge structures on $H^2(X,{\mathbb Z})$ and $H^2(X',{\mathbb Z})$. If $\Gamma_\infty=Z+Z'+\sum Y_i$, then $$\begin{array}{ccl} 0&\ne&\displaystyle{\lambda\int_{X'}(\sigma'\bar\sigma')^n=\int_{X'}({\varphi'}^{-1}\circ\varphi)[\sigma]({\sigma'}^{n-1}{\bar\sigma'}{}^{n})}\\ &=&\displaystyle{\int_{X'}p'_*([\Gamma_\infty].p^*[\sigma])({\sigma'}^{n-1}{\bar\sigma'}{}^n)=\int_{X\times X'}[\Gamma_\infty].p^*[\sigma].{p'}^*[{\sigma'}^{n-1}{\bar\sigma'}{}^n]}\\ &=&\displaystyle{\int_Zp^*[\sigma].{p'}^*[{\sigma'}^{n-1}{\bar\sigma'}{}^n]+\int_{Z'}p^*[\sigma].{p'}^*[{\sigma'}{}^{n-1}{\bar\sigma'}{}^n]+\sum\int_{Y_i}p^*[\sigma].{p'}^*[{\sigma'}{}^{n-1}{\bar\sigma'}{}^n]}.\\ \end{array}$$ The first and third term vanish, because $p'(Z)$ and $p'(Y_i)$ are of dimension $<2n$, but ${\bar\sigma'}{}^n$ is a $(0,2n)$-form on $X'$. For simplicity assume $Z'$ smooth (otherwise pass to a desingularization). Since $Z'\to X'$ is birational, $H^0(Z',\Omega^2_{Z'})={p'}^*({\sigma'})|_{Z'}\cdot\IC$. On the other hand, $p^*(\sigma)|_{Z'}\in H^0(Z',\Omega_{Z'}^2)$. Since $p^*(\sigma)|_{Z'}$ is everywhere degenerate, but ${p'}^*(\sigma')|_{Z'}$ is at least generically non-degenerate, we must have $p^*(\sigma)|_{Z'}=0$. Thus, also the second term vanishes. Contradiction. Therefore, only the decomposition $\Gamma_\infty=Z+\sum Y_i$ can occur. Since $Z\to X$ and $Z\to X'$ are generically one-to-one, $X$ and $X'$ are birational.\hspace*{\fill}$\Box$ \bigskip Note that the birational correspondence between $X$ and $X'$ constructed this way does not, in general, induce ${\varphi'}^{-1}\circ \varphi$ on $H^2$. The theorem (or rather its proof) has various interesting consequences and, with the proof still in mind, the reader may wish to have a look at Corollary \ref{corBRMain} already at this point in the discussion. \bigskip Let us now come to the converse: Are two birational irreducible symplectic manifolds non-separated in their moduli space? Recall that for an appropriate choice of the markings the periods of two birational irreducible symplectic manifolds are equal (cf.\ \ref{hodgeunderbir}). Therefore, their period points in $\IP(\Gamma_\IC)$ coincide. Let us consider the following situation: \bigskip \refstepcounter{theorem}\label{assump}{\bf \thetheorem} --- {\it i)} $X$ and $X'$ are irreducible symplectic manifolds. {\it ii)} $X'$ is projective. {\it iii)} There exists a birational map $f:X - - \to X'$. \bigskip Let $U\subset X$ and $U'\subset X'$ be the maximal open subsets where $f$, respectively $f^{-1}$, are regular. Then $U\cong U'$ and ${\rm codim}(X\setminus U), {\rm codim}(X'\setminus U')\geq2$. This is a general fact for varieties with nef canonical divisor, but see \cite{Huybrechts}. Since $X$ is K\"ahler and birational to the projective manifold $X'$, also $X$ is projective (cf.\ \ref{Moish}). For the algebraic Picard groups we have canonical isomorphisms ${\rm Pic}(X)\cong{\rm Pic}(U)\cong{\rm Pic}(U')\cong{\rm Pic}(X')$. If $L\in{\rm Pic}(X)$ we denote by $L'$ the associated line bundle on $X'$, i.e.\ $L'$ is the line bundle on $X'$ such that $L|_U\cong L'|_{U'}$. In the sequel, $L'$ will usually be ample. If $L$ is ample as well, then the isomorphism $U\cong U'$ can be extended to an isomorphism of $X$ and $X'$ and there is nothing to show. The following proposition was proved in \cite{Huybrechts}. \begin{proposition}\label{biratdefo}--- Under the assumptions \ref{assump}, let $L'$ be ample and let $\pi:({\cal X},{\cal L})\to S$ be a deformation of $({\cal X}_0,{\cal L}_0)=(X,L)$ over a smooth one-dimensional base $S$. If for any $n\gg0$ the dimension $h^0({\cal X}_t,{\cal L}_t^n)$ is constant in a neighbourhood of $0\in S$, then, after shrinking $S$ if necessary, there exists a deformation $({\cal X}',{\cal L}')\to S$ of $({\cal X}'_0,{\cal L}'_0)=(X',L')$ and an $S$-birational correspondence between ${\cal X}$ and ${\cal X}'$ respecting ${\cal L}$ and ${\cal L}'$.\hspace*{\fill}$\Box$ \end{proposition} The idea of the proof is as follows: The linear system $|{\cal L}^m_t|$ ($m\gg0$) defines a rational map from ${\cal X}$ to $\IP_S((\pi_*{\cal L}^m)\makebox[0mm]{}^{{\scriptstyle\vee}})$ and we define ${\cal X}'$ as the closure of the image of this rational map. Then one shows that ${\cal X}'$ is $S$-flat with special fibre $X'$. The line bundle ${{\cal L}'}^m$ is the restriction of the relative ${\cal O}(1)$ on $\IP_S((\pi_*{\cal L}^m)\makebox[0mm]{}^{{\scriptstyle\vee}})$. For details we refer to \cite{Huybrechts}. Note that the neighbourhood where $h^0({\cal X}_t,{\cal L}_t^m)\equiv const$ might very well depend on $n$. The main technical problem that limited the results in \cite{Huybrechts} to the case that ${\rm codim}(X\setminus U),{\rm codim}(X'\setminus U')\geq3$ was the assumption $h^0({\cal X}_t,{\cal L}_t^n)\equiv const$, which is easy to establish under this additional assumption on the codimension. The projectivity criterion Theorem \ref{proj}, which was not yet available in \cite{Huybrechts}, can be used to arrange things such that the assumption on $h^0({\cal X}_t,{\cal L}_t)$ holds. \begin{theorem}\label{birat}--- Let $X$ and $X'$ be birational projective irreducible symplectic manifolds. Then there exist deformations ${\cal X}\to S$ and ${\cal X}'\to S$ of ${\cal X}_0=X$ and ${\cal X}'_0=X'$, respectively, such that \begin{itemize} \item[$\bullet$] $S$ is smooth and one-dimensional. \item[$\bullet$] There exists an $S$-isomorphism ${\cal X}|_{S\setminus\{0\}}\cong {\cal X}'|_{S\setminus\{0\}}$. \end{itemize} \end{theorem} {\em Proof}. By \ref{Todd} there exist constants $(a_i)$ and $(a'_i)$ ($i=1,\ldots,n$) such that for any line bundle $L$ on $X$ (resp.\ $L'$ on $X'$) the Hirzebruch-Riemann-Roch formula can be written as $$\chi(X,L)=\sum_{i=1}^n\frac{a_i}{(2i)!}q_X(c_1(L))^i\phantom{XX} {\rm resp.}\phantom{XX}\chi(X',{L'})=\sum_{i=1}^n\frac{a'_i}{(2i)!}q_{X'}(c_1(L'))^i.$$ Without loss of generality we may assume that in the lexicographic order $(a_i)\geq(a'_i)$, i.e.\ $\sum a_i t^i\geq\sum a_i't^i$ for $t\gg0$. If now $L'\in{\rm Pic}(X')$ is ample and $L$ is the corresponding line bundle on $X$, then $q_X(c_1(L))=q_{X'}(c_1(L'))>0$. For the equality see \ref{hodgeunderbir} and for the inequality \ref{quadraticform}. Hence, $\chi(X,L^m)\geq\chi(X',{L'}^m)$ for $m\gg0$. Let $({\cal X},{\cal L})\to S$ be a deformation of $(X,L)$ over a smooth and one-dimensional base $S$ such that $\rho({\cal X}_t)=1$ for general $t\in S$, e.g.\ take a curve $S\subset {\it Def}(X,L)$ not contained in any ${\it Def}(X,M)$ where $M$ is a line bundle linearly independent of $L$ (cf.\ \ref{deformationlb}, \ref{hypersurfaces}). By Theorem \ref{proj} the positivity of $q_{{\cal X}_t}(c_1({\cal L}_t))=q_X(c_1(L))$ implies that all fibres ${\cal X}_t$ are projective. Hence, for general $t\in S$ either ${\cal L}_t$ or its dual is ample. The non-vanishing of $H^0(X,L^m)=H^0(X',{L'}^m)$ for $m\gg0$ and the semi-continuity of $h^0({\cal X}_t,{\cal L}_t)$ excludes the latter possibility for $t$ close to $0\in S$. Thus, we can apply the Kodaira Vanishing Theorem to ${\cal L}_t$: For general $t\in S$ one has $h^0({\cal X}_t,{\cal L}_t^m)=\chi({\cal X}_t,{\cal L}_t^m) =\chi(X,L^m)$ for all $m>0$. Using semi-continuity this yields $h^0(X,L^m)\geq h^0({\cal X}_t,{\cal L}_t^m)=\chi(X,L^m)$ for $m>0$. On the other hand, $h^0(X',{L'}^m)=h^0(X,L^m)$ (use ${\rm codim}(X\setminus U),{\rm codim}(X'\setminus U')\geq2$) and $\chi(X',{L'}^m)=h^0(X',{L'}^m)$ (use $L'$ ample on $X'$). Therefore, $$\chi(X',{L'}^m)=h^0(X,L^m)\geq h^0({\cal X}_t,{\cal L}_t^m)=\chi(X,L^m).$$ Under the assumptions $(a_i)\geq(a'_i)$ and $q_X(c_1(L))= q_{X'}(c_1(L'))>0$ this implies $h^0(X,L^m)=h^0({\cal X}_t,{\cal L}_t^m)$ for $m\gg0$ . Hence, for $m\gg0$ the dimension $h^0({\cal X}_t,{\cal L}_t^m)$ is constant in a neighbourhood of $0\in S$. Therefore, Proposition \ref{biratdefo} can be applied and we find a deformation $({\cal X}',{\cal L}')\to S$ of $(X',L')$ which is $S$-birational to ${\cal X}$. The rest of the proof is as the one of the corresponding theorem in \cite{Huybrechts}: For $t\in S$ general and close to $0\in S$ the birational correspondence ${\cal X}_t- - \to {\cal X}'_t$ is a birational correspondence between two projective manifolds with Picard number $\rho({\cal X}_t)=\rho({\cal X}_t')=1$, which must be an isomorphism. Hence, ${\cal X}- - \to {\cal X}'$ is an isomorphism on the general fibre. Restricting to an open neighbourhood of $0\in S$ we can in fact achieve that ${\cal X}|_{S\setminus\{0\}}\cong{\cal X}'|_{S\setminus\{0\}}$ (cf.\ \cite{Huybrechts}).\hspace*{\fill}$\Box$ \bigskip Note that the theorem is known as well for non-projective irreducible symplectic manifolds if the birational correspondence is described by an elementary transformation \cite{Huybrechts}. More in the spirit of the Main Lemma of Burns and Rapoport (see \ref{BRMain}) Theorem \ref{birat} can equivalently be formulated as \bigskip {\parindent0mm {\bf Theorem \ref{birat}'}} --- {\it Let $X$ and $X'$ be birational projective irreducible symplectic manifolds. Then there exist two markings $\varphi:H^2(X,{\mathbb Z})\cong \Gamma$ and $\varphi':H^2(X',{\mathbb Z})\cong\Gamma$ such that $(X,\varphi),(X',\varphi')\in{\mathfrak M}_\Gamma$ are non-separated points.\hspace*{\fill}$\Box$} \bigskip The fact that $X$ and $X'$ can be realized as the special fibres of the same family has strong consequences, which are not at all obvious just from the fact that they are birational. We only mention: \begin{corollary}--- If $X$ and $X'$ are birational projective irreducible symplectic manifolds, then: \begin{itemize} \item[{\it i)}] $X$ and $X'$ are diffeomorphic. \item[{\it ii)}] For all $k$ the weight-$k$ Hodge structures of $X$ and $X'$ are isomorphic.\hspace*{\fill}$\Box$ \end{itemize} \end{corollary} It is interesting to compare {\it ii)} in the corollary with a recent result of Batyrev and Kontsevich. They show that {\it ii)} holds true for all birational smooth projective manifolds with trivial canonical bundle; in particular for irreducible symplectic but also for Calabi-Yau manifolds. Assertion {\it i)} is not expected to hold for the more general class of projective manifolds with trivial canonical bundle. \bigskip {\bf Applications} \bigskip In Sect.\ \ref{examples} we provided a list of the known examples of irreducible symplectic manifolds. In most of the cases the verification of the defining properties was reduced to either of the two standard examples ${\rm Hilb}^n(S)$ or ${\rm K}^{n+1}(A)$, where $S$ is a K3 surface and $A$ is a torus. In fact, modulo birational correspondence all examples in Sect.\ \ref{examples} are deformation equivalent to one of these two. Theorem \ref{birat} in particular says that also birational irreducible symplectic manifolds are deformation equivalent. In particular, we wish to mention \begin{corollary}--- If $S$ is a K3 surface and $v=(v_0,v_1,v_2)$ is a Mukai vector with $v_1$ primitive, then for any $v$-generic polarization $H$ the moduli space $M_H(v)$ of semistable sheaves with Mukai vector $v$ is an irreducible symplectic manifold that is deformation equivalent to ${\rm Hilb}^n(S)$, where $n=(v,v)+2$. (For the notation see Sect.\ \ref{examples}, \cite{OG} or \cite{HL}.)\hspace*{\fill}$\Box$ \end{corollary} More generally, we formulate \begin{corollary}\label{onlytwo}--- All examples of irreducible symplectic manifolds in Sect.\ \ref{examples} are deformation equivalent (and hence diffeomorphic) either to ${\rm Hilb}^n(S)$ or to ${\rm K}^{n+1}(A)$, where $S$ is a K3 surface and $A$ is a torus. \end{corollary} {\em Proof}. The only problem is the projectivity assumption in Theorem \ref{birat}. But since all known examples of birational correspondences between non-projective irreducible symplectic manifolds are described in terms of elementary transformations along projective bundles, the result follows from \cite{Huybrechts}, where we gave an easy proof of Theorem \ref{birat} for elementary transformations without assuming the projectivity of $X$ and $X'$.\hspace*{\fill}$\Box$ \section{An Analogue of the Weyl-Action}\label{weylaction} The main obstacle to generalize the techniques in the theory of K3 surfaces to higher dimensions, besides the missing Global Torelli Theorem, is the absence of a Weyl-group, i.e.\ the group of automorphisms of the second cohomology generated by reflections in hypersurfaces orthogonal to some $(-2)$-class. To a certain extent, the following proposition (and its corollary) is a good replacement for the fact that the Weyl-group acts transitively on the set of chambers (see also Remark \ref{Weyl}). It has immediate consequences such as the description of the K\"ahler cone of very general irreducible symplectic manifolds (see \ref{Kaehlerconeforperiod} and \ref{Kaehlerconeforperiod2}). The question will be further pursued in Sect.\ \ref{kaehlercone}. A posteriori the not very concrete `generality' assumption in \ref{Weylprop} can be made more specific (Theorem \ref{Delignesuggestion}). This was pointed out to me by Deligne. \begin{proposition}\label{Weylprop}--- Let $(X,\varphi)\in{\mathfrak M}_\Gamma$ be a marked irreducible symplectic manifold. Assume $\alpha\in{\cal C}_X$ is general, i.e.\ $\alpha$ is contained in the complement of countably many nowhere dense closed subsets. Then there exists a point $(X',\varphi')\in{\mathfrak M}_\Gamma$, which cannot be separated from $(X,\varphi)$ such that $({\varphi'}^{-1}\circ\varphi)(\alpha)\in H^2(X',{\mathbb R})$ is a K\"ahler class. \end{proposition} Together with Theorem \ref{BRMain} (or rather its proof) the proposition shows \begin{corollary}\label{corBRMain} --- Let $X$ be an irreducible symplectic manifold of dimension $2n$ and let $\alpha\in{\cal C}_X$ be general. Then there exists another irreducible symplectic manifold $X'$ together with an effective cycle $\Gamma:=Z+\sum Y_i\subset X\times X'$ of dimension $2n$ satisfying the following conditions: \begin{itemize} \item[{\it i)}] $Z$ defines a birational map $X- - \to X'$. \item[{\it ii)}] The projections $Y_i\to X$ and $Y_i\to X'$ have positive fibre dimension. \item[{\it iii)}] $[\Gamma]_*:H^2(X,{\mathbb Z})\to H^2(X',{\mathbb Z})$ defines an isomorphism of Hodge structures compatible with $q_X$ and $q_{X'}$. Moreover, $[\Gamma]_*\circ[\Gamma]_*$ is the identity on $H^2(X,{\mathbb Z})$, resp.\ $H^2(X',{\mathbb Z})$. \item[{\it iv)}] $[\Gamma]_*(\alpha)\in{\cal K}_{X'}$.\hspace*{\fill}$\Box$ \end{itemize} \end{corollary} {\it Proof of Proposition \ref{Weylprop}.} First, let $\alpha\in {\cal C}_X$ be completely arbitrary. Choose a K\"ahler class $\gamma\in{\cal K}_X\subset{\cal C}_X$ and let $\beta:=(1+\varepsilon)\alpha-\varepsilon\gamma\in{\cal C}_X$, where $0<\varepsilon \ll1$. Then, pick a sequence $\beta_i\in H^2(X,\IQ)$ converging to $\beta$ such that $\beta_i\not\in H^{1,1}(X)$ (or, equivalently, $\beta_i\not\in{\cal C}_X$). The associated hypersurfaces $S_{\beta_i}\subset {\it Def}(X)$ (for the notation see \ref{hypersurfaces}) converge to $S_\beta\subset {\it Def}(X)$. Since $S_\beta\ne\emptyset$, also $S_{\beta_i}\ne\emptyset$ for $i\gg0$. Let $U\subset {\cal C}_X$ be an open neighbourhood of $\alpha$ such that for all $\alpha'\in U$ the class $((1+\varepsilon)\alpha'-\beta)/\varepsilon$ is still K\"ahler. Such an open neighbourhood exists, because ${\cal K}_X\subset {\cal C}_X$ is open. For any $\alpha'\in U$ let ${\cal X}(\alpha')\to T(\alpha')$ be the deformation obtained by restricting the universal deformation to $T(\alpha'):={\it Def}(X)\cap{\cal P}^{-1}(\IP(\varphi(F(\alpha')_\IC)))$ (see also \ref{twistor2}). Since $\alpha',\beta\in{\cal C}_X$ and, hence, $q_X(\alpha',\beta)\ne0$, the curve $T(\alpha')$ is not contained in $S_\beta$ (cf.\ \ref{pairing}, \ref{hypersurfaces}). Therefore, $T(\alpha')\cap S_\beta$ is zero-dimensional and, moreover, non-empty, as the orign $0$ is contained in $T(\alpha')\cap S_\beta$. Hence, $T(\alpha')\cap S_{\beta_i}$ is zero-dimensional and non-empty for $i\gg0$ as well. Thus, we find points $t_i\in T(\alpha')\cap S_{\beta_i}$ approaching $0$. We need the following \bigskip {\it Claim} --- For general $\alpha'\in U$ the points $t_i\in T(\alpha')\cap S_{\beta_i}$ are general points of $S_{\beta_i}$ in the sense that $H^{1,1}({\cal X}(\alpha')_{t_i})_\IQ$ is spanned by $\beta_i$ or, equivalently, that $\rho({\cal X}(\alpha')_{t_i})=1$. \bigskip {\it Proof of the Claim.} Consider the map $\psi_i:U\to S_{\beta_i}$ that sends $\alpha'\in U$ to the intersection $\{t_i\}=T(\alpha')\cap S_{\beta_i}$, which, at least locally, consists of a single point. The map $\psi_i$ is constant along the orbits of the natural ${\mathbb R}^*$-action on $U\subset H^{1,1}(X)_{\mathbb R}$. Thus $d\psi_i:T_{\alpha'}U\to T_{t_i} S_{\beta_i}$ factorizes through $T_{\alpha'}U=H^{1,1}(X)_{\mathbb R}\to H^{1,1}(X)_{\mathbb R}/\alpha'\cdot{\mathbb R} \to T_{t_i} S_{\beta_i}\subset H^{1,1}({\cal X}(\alpha')_{t_i})$. Using the period map, which is holomorphic, it is easy to extend the map $\psi_i$ to a holomorphic map $\tilde\psi_i:\tilde U\to S_{\beta_i}$, where $\tilde U$ is an open subset of the complex vector space $H^{1,1}(X)$ with $\tilde U\cap H^{1,1}(X)_{\mathbb R}=U$. Again, $\tilde \psi_i$ is constant along the orbits of the natural $\IC^*$-action on $\tilde U\subset H^{1,1}(X)$ and, therefore, its tangent map factorizes as follows $T_{\alpha'}\tilde U\cong H^{1,1}(X)\to H^{1,1}(X)/\alpha'\cdot\IC\to T_{t_i}S_{\beta_i}\subset H^{1,1}({\cal X}(\alpha')_{t_i})$. Next, one proves that the map $\tilde\psi:\tilde U\to S_{\beta_i}$ is injective modulo the $\IC^*$-action. Indeed, if $H^{2,0}=\sigma\cdot\IC$, then $F(\alpha')_\IC=\sigma\cdot\IC\oplus\bar\sigma\cdot\IC\oplus\alpha'\cdot\IC$. Therefore, for linearly independent $\alpha',\alpha''\in\tilde U\subset H^{1,1}(X)$ the two planes $\IP(\varphi(F(\alpha')_\IC)),\IP(\varphi(F(\alpha'')_\IC))\subset \IP(\Gamma_\IC)$ intersect in $\IP(\varphi(\sigma\cdot\IC\oplus\bar\sigma\cdot\IC))$. The latter space is a projective line which meets the period domain $Q\subset \IP(\Gamma_\IC)$ in exactly two points, namely $\IP(\varphi(\sigma\cdot\IC\oplus\bar\sigma\cdot\IC))\cap Q=\{\varphi[\sigma],\varphi[\bar\sigma]\}$. Shrinking ${\it Def}(X)$ such that $\varphi[\bar\sigma]\not\in{\cal P}({\it Def}(X))$, one has in particular $\varphi[\bar\sigma]\not\in{\cal P}(S_{\beta_i})$ and, therefore, $T(\alpha')\cap S_{\beta_i}\ne T(\alpha'')\cap S_{\beta_i}$ for linearly independent $\alpha'$ and $\alpha''$. Thus $\tilde\psi:\tilde U\to S_{\beta_i}$ is injective modulo $\IC^*$. But if $\tilde\psi_i:\tilde U\to S_{\beta_i}$ is injective modulo the $\IC^*$-action, then its tangent map $d\tilde\psi_i:H^{1,1}(X)/\alpha'\cdot\IC\to T_{t_i}S_{\beta_i}$ is injective for general $\alpha'\in \tilde U$. In fact, the set where $d\tilde \psi_i$ fails to be injective is a complex-analytic set and, therefore, cannot contain open parts of $U$. Thus, even for general $\alpha'\in U$ the tangent map can be assumed to be injective. Since both spaces $H^{1,1}(X)/\alpha'\cdot\IC$ and $T_{t_i}S_{t_i}$ have dimension $h^{1,1}(X)-1$, the tangent map $d\tilde\psi_i$ at such a point must be bijective. In particular, ${\rm im}(d\tilde\psi_i)$ is not contained in any $T_{t_i}S_\delta\cap T_{t_i}S_{\beta_i}$ for any $(1,1)$-class $\delta$ on ${\cal X}(\alpha')_{t_i}$ that is linearly independent of $\beta_i$. Since $d\tilde\psi_i$ is $\IC$-linear and $H^{1,1}(X)_{\mathbb R}$ spans $H^{1,1}(X)$, this also shows that ${\rm im}(d\psi_i)$ is not contained in any such $T_{t_i}S_\delta\cap T_{t_i}S_{\beta_i}$ . Hence, the image of $\psi_i:U\to S_{\beta_i}$ is not contained in any hypersurface $S_\delta\cap S_{\beta_i}$ with $\delta$ linearly independent of $\beta_i$. As there are only countably many $\beta_i$'s and $\delta$'s to be considered, one can assume that for the general $\alpha'\in U$ the intersection $T(\alpha')\cap S_{\beta_i}$ is not contained in any $S_\delta$ for $\delta$ linearly independent of $\beta_i$. In other words, if $\{t_i\}=T(\alpha')\cap S_{\beta_i}$ with $\alpha'$ general, then $\rho({\cal X}(\alpha')_{t_i})=1$. This proves the claim. \bigskip Let us now replace $\alpha$ by a general (in the above sense) $\alpha'\in U$. Analogously, replace $\gamma=((1+\varepsilon)\alpha-\beta)/\varepsilon$ by the K\"ahler class $\gamma'=((1+\varepsilon)\alpha'-\beta)/\varepsilon$. In other words, from now on we are in the following situation: $\alpha\in {\cal C}_X$, $\gamma=((1+\varepsilon)\alpha-\beta)/\varepsilon\in{\cal K}_X$, and there are points $t_i\in T(\alpha)\cap S_{\beta_i}$ converging to $0$ such that $\rho({\cal X}(\alpha)_{t_i})=1$. Let ${\cal X}:={\cal X}(\alpha)\to T(\alpha)$ and denote by $\alpha_t$ a $(1,1)$-class on ${\cal X}_t$ that spans the orthogonal complement of $P({\cal X}_t):=(\sigma_t\cdot\IC+\bar\sigma_t\cdot \IC)_{\mathbb R}$ in $F(\alpha)\subset H^2(X,{\mathbb R})\cong H^2({\cal X}_t,{\mathbb R})$. We may choose $\alpha_t$ depending continuously on $t$ such that $\alpha_0=\alpha$. Next, let $\gamma_i:=((1+\varepsilon)\alpha_{t_i}-\beta)/\varepsilon$ which is considered as a class on $X$ or as a class on ${\cal X}_{t_i}$ via the isomorphism $H^2(X,{\mathbb R})\cong H^2({\cal X}_{t_i},{\mathbb R})$. Since the union of the K\"ahler cones ${\cal K}_{{\cal X}_t}$ in $\bigcup_t H^{1,1}({\cal X}_t)_{\mathbb R}$ is open, $\gamma_i$ is a K\"ahler class on ${\cal X}_{t_i}$ for $i\gg0$ (see also the arguments in the proof of Fujiki's Theorem \ref{fujiki}). On the other hand, for $i\gg0$ the class $\beta_i$ on ${\cal X}_{t_i}$ is of type $(1,1)$ and $q_X(\beta_i)>0$. Theorem \ref{proj} then asserts that ${\cal X}_{t_i}$ is projective. Since $H^{1,1}({\cal X}_{t_i})_\IQ=\beta_i\cdot\IQ$ and $\beta_i\in{\cal C}_{{\cal X}_{t_i}}$ for $i\gg0$, the class $\beta_i$ is ample on ${\cal X}_{t_i}$. Thus, $\alpha_{t_i}$ is contained in the segment $[\gamma_i,\beta_i]$ joining the two K\"ahler classes $\gamma_i$ and $\beta_i$ on ${\cal X}_{t_i}$. Since the K\"ahler cone is convex, $\alpha_{t_i}$ is a K\"ahler class on ${\cal X}_{t_i}$ for $i\gg0$. All we need from the preceeding discussion is the following statement: \begin{itemize}\item[] {\it If $X$ is an irreducible symplectic manifold and $\alpha\in{\cal C}_X$ is general, then there exists a point $t\in T(\alpha)$ such that $\alpha_t$ is K\"ahler on ${\cal X}_t={\cal X}(\alpha)_t$.} \end{itemize} Let us fix this point $t$. We denote the induced marking $H^2({\cal X}_t,{\mathbb Z})\cong H^2(X,{\mathbb Z})\rpfeil{5}{\varphi}\Gamma$ of ${\cal X}_t$ by $\varphi_t$. Then $\varphi(F(\alpha))=\varphi(P(X)\oplus\alpha\cdot{\mathbb R})=\varphi_t(P({\cal X}_t)\oplus\alpha_t\cdot{\mathbb R})=\varphi_t(F(\alpha_t))$. This way one identifies $T(\alpha)$ with an open subset of the base of the twistor space associated to $({\cal X}_t,\alpha_t)$. (Note that for an arbitrary $(1,1)$-class $\delta$ the space $T(\delta)$ is only locally defined but if $\delta$ is a K\"ahler class then $T(\delta)$ means the complete base of the twistor space, which is a $\IP^1$, see \ref{deformationhk},\ \ref{twistor2}.) Hence, there are two families ${\cal X}\to T(\alpha)$ and ${\cal X}'\to T(\alpha)$ over the same base $T(\alpha)$, where the latter is (an open subset of) the twistor space of ${\cal X}_{t}\cong{\cal X}'_t$. Both deformations are endowed with the natural markings $\varphi$ and $\varphi'$ such that ${\varphi_t'}^{-1}\circ\varphi_t$ is induced by the isomorphism ${\cal X}_t\cong{\cal X}'_t$. Moreover, $\alpha'_s:=({\varphi_s'}^{-1}\circ\varphi_s)(\alpha_s)$ is a K\"ahler class on ${\cal X}_s'$ for all $s\in T(\alpha)$. In particular, $\alpha_0'$ is a K\"ahler class on $X':={\cal X}_0'$. It remains to show that $(X',\varphi'):= ({\cal X}'_0,\varphi'_0)$ cannot be separated from $(X,\varphi)$. Consider the maximal open subset $V\subset T(\alpha)$ containing $t$ such that there exists an isomorphism ${\cal X}|_V\cong{\cal X}'|_V$ extending ${\cal X}_t\cong{\cal X}_t'$. (Use the Local Torelli Theorem \ref{periodmapdef} in order to show that $V$ is open.) Denote by $\overline V$ the closure of $V$ in $T(\alpha)$ and let $\partial V:=\overline V\setminus V$ be its boundary. Let $s\in \partial V$. Then there exists an effective cycle $\Gamma=Z+ \sum Y_i\subset {\cal X}_s\times{\cal X}'_s$ as in the proof of \ref{BRMain}. In particular, ${\cal X}_s\leftarrow Z\to{\cal X}_s'$ is a birational correspondence, the projections $Y_i\to{\cal X}_s$ and $Y_i\to{\cal X}_s'$ have positive fibre dimension, and $[\Gamma]_*={\varphi_s'}^{-1}\circ\varphi_s$. Let us assume that in addition ${\cal X}_s'$ neither contains non-trivial curves nor effective divisors, e.g.\ $H^{1,1}({\cal X}_s')_{\mathbb Z}=0$ (cf.\ \ref{quadraticform}). Then, ${\cal X}_s\cong Z\cong{\cal X}_s'$ and $[Y_i]_*: H^2({\cal X}_s,{\mathbb Z})\to H^2({\cal X}_s',{\mathbb Z})$ is trivial (see Lemma \ref{lem} below). Since $\alpha'_s=({\varphi_s'}^{-1}\circ\varphi)(\alpha_s)=[\Gamma]_*(\alpha)=[Z]_*( \alpha_s)$, this yields that $\alpha_s$ is a K\"ahler class on ${\cal X}_s$. Next we claim that in fact $\Gamma=Z$. Indeed, if we compute the volume with respect to the K\"ahler class $p_1^*\alpha_s+p_2^*\alpha'_s$, then $$\begin{array}{rcl} vol(\Gamma)&=&vol(Z)+\sum vol(Y_i)\\ &=&\displaystyle{\int_Z (p_1^*\alpha_s+p_2^*\alpha_s')^{2n}}+\sum vol(Y_i)\\ &=&\displaystyle{\int_{{\cal X}_s'}([Z]_*(\alpha_s)+\alpha_s')^{2n}}+\sum vol(Y_i)\\ &=&\displaystyle{\int_{{\cal X}_s'}([\Gamma]_*(\alpha_s)+\alpha_s')^{2n}}+\sum vol(Y_i)\\ &=&\displaystyle{\int_{{\cal X}_s'}(2\alpha_s')^{2n}}+\sum vol(Y_i)\\ \end{array}$$ On the other hand, if $s_i\in V$ converges to $s$, then the volume of the graph $\Gamma_i$ of the isomorphism ${\cal X}_{s_i}\cong{\cal X}_{s_i}'$ converges to $vol(\Gamma)$. But $vol(\Gamma_i)=\int_{\Gamma_i}(p_1^*\alpha_{s_i}+p_2^*\alpha'_{s_i})^{2n}=\int_{{\cal X}_{s_i}'}(2\alpha'_{s_i})^{2n}$. Hence, $\sum vol(Y_i)=0$, i.e.\ $\Gamma=Z$. But this would contradict the maximality of $V$. Thus, if $s\in\partial V$, then ${\cal X}'_s$ either contains non-trivial curves or divisors. By \ref{twistor2} the set of points $s\in T(\alpha)$ with this property is countable. Hence, $\partial V$ is countable. But then $\partial V$ could not separate two non-empty open subset $V$ and $T(\alpha)\setminus \overline V$. Hence, $\overline V=T(\alpha)$, which proves that $(X,\varphi)$ and $(X',\varphi')$ cannot be separated.\hspace*{\fill}$\Box$ \begin{remark}\label{topoo}--- In fact, we can modify ${\cal X}'$ appropriately such that it becomes isomorphic to ${\cal X}$ over $T(\alpha)\setminus\{0\}$. (However, it will not be a twistor space any longer.) Indeed, any countable closed subset in $T(\alpha)$ has an isolated point, but in the neighbourhood of an isolated point $\ne0$ one can replace ${\cal X}'$ by ${\cal X}$. \end{remark} \begin{remark}\label{Weyl}--- The transitivity of the action of the Weyl-group on the set of chambers of a K3 surface is equivalent to the following statement: If $X$ is a K3 surface and $\alpha\in{\cal C}_X$ is general in the sense that it is not orthogonal to any $(-2)$-curve, then there exists a cycle $\Gamma=\Delta+\sum C_i\times C_i\subset X\times X$, where $\Delta$ is the diagonal and the $C_i$'s are $(-2)$-curves, such that $[\Gamma]_*(\alpha)$ is a K\"ahler class. In this light, the above proposition is a weak generalization of the transitivity of the action of the Weyl-group. Unfortunately, it seems to be hard to specify the assumption on $\alpha$ to be `general'. In particular, one would like to replace it by an open condition. For a K3 surface this is granted by the fact that the union of all walls is closed (and hence its complement is open) in the positive cone. However, as we will see below (Corollary \ref{Delignehelps}), a class is `general' in the sense of the proposition if it is not orthogonal to any integral class. \end{remark} The following lemma is rather elementary. It was used in the previous proof and will come up again in Sect.\ \ref{kaehlercone}. \begin{lemma}\label{lem}--- Let $X$ and $X'$ be compact complex manifolds of dimension $m$ and let $Y\subset X\times X'$ be an $m$-dimensional irreducible subvariety such that the projection $p':Y\to X'$ has positive fibre dimension. If the induced homomorphism $[Y]_*:=p'_*([Y].p^*(\,.\,)): H^2(X,{\mathbb Z})\to H^2(X',{\mathbb Z})$ is non-zero, then $p'(Y)\subset X'$ is a divisor. In this case $[Y]_*(\alpha)=(\int_C\alpha)[p'(Y)]$, where $C={p'}^{-1}(x)$ is the general fibre curve of $p':Y\to p'(Y)$.\hspace*{\fill}$\Box$ \end{lemma} Before approaching the K\"ahler cone of an arbitrary irreducible symplectic manifold in Sect.\ \ref{kaehlercone} let us deduce here some immediate consequences of Proposition \ref{Weylprop}. \begin{corollary}\label{Kaehlerconeforperiod}--- Let $X$ be an irreducible symplectic manifold without effective divisors. Then for general $\alpha\in{\cal C}_X$ there exists a birational correspondence $Z\subset X\times X'$ between $X$ and another irreducible symplectic manifold $X'$ with $[Z]_*(\alpha)\in{\cal K}_{X'}$. If in addition $X$ contains no rational curves, then ${\cal C}_X={\cal K}_X$ \end{corollary} {\em Proof}. If $X$ does not contain any divisor, then by the lemma the maps $[Y_i]: H^2(X',{\mathbb Z})\to H^2(X,{\mathbb Z})$ are trivial, where $\Gamma=Z+\sum Y_i$ is as in Corollary \ref{corBRMain}. Hence, if $\alpha':=[\Gamma]_*(\alpha)$, which is a K\"ahler class by Corollary \ref{corBRMain}, then $\alpha=[\Gamma]_*(\alpha')= [Z]_*(\alpha')$ and, therefore, $[Z]_*(\alpha)=\alpha'\in{\cal K}_{X'}$. If $X$ does not contain any rational curves, then any birational correspondence extends to an isomorphism. Hence, $\alpha$ is K\"ahler.\hspace*{\fill}$\Box$ \bigskip From the last assertion and \ref{twistor2} one easily obtains also the following \begin{corollary}\label{kaehlerconecor}\label{Kaehlerconeforperiod2}--- If $H^{1,1}(X)_{\mathbb Z}=0$, i.e.\ there exists no non-trivial line bundle on $X$, then ${\cal C}_X={\cal K}_X$. In particular, if $X$ is a general irreducible symplectic manifold then ${\cal C}_X={\cal K}_X$. \end{corollary} {\em Proof}. The assumption $H^{1,1}(X)_{\mathbb Z}=0$ implies $H^{2n-1,2n-1}(X)_{\mathbb Z}=0$ by \ref{quadraticform}. Hence, $X$ contains neither curves nor effective divisors. Thus we can apply the previous corollary. For the second assertion recall that the deformations of $X$ that admit non-trivial line bundles form a countable union of hypersurfaces in ${\it Def}(X)$ (cf.\ \ref{deformationlb}).\hspace*{\fill}$\Box$ \bigskip Once the K\"ahler cone of a general irreducible symplectic manifold is described (Corollary \ref{kaehlerconecor}), Proposition \ref{Weylprop} can be sharpened to yield: \begin{corollary}\label{Delignehelps} --- Let $(X,\varphi)\in{\mathfrak M}_\Gamma$ be a marked irreducible symplectic manifold. Assume that $\alpha\in{\cal C}_X$ is not orthogonal to any $0\ne\beta\in H^{1,1}(X)_{\mathbb Z}$. Then there exists a point $(X',\varphi')\in{\mathfrak M}_\Gamma$ which cannot be separated from $(X,\varphi)$ such that $({\varphi'}^{-1}\circ\varphi)(\alpha)$ is in the K\"ahler cone of $X'$. \end{corollary} {\em Proof}. As in the proof of Proposition \ref{Weylprop} we consider the `twistor space' ${\cal X}(\alpha)\to T(\alpha)$. By the assumption on $\alpha$, the general fibre ${\cal X}_t:={\cal X}(\alpha)_t$ satisfies $H^{1,1}({\cal X}_t)_{\mathbb Z}=0$. By Corollary \ref{kaehlerconecor} this implies ${\cal K}_{{\cal X}_t}={\cal C}_{{\cal X}_t}$. If $\alpha_t$ is a $(1,1)$-class on ${\cal X}_t$ that spans the orthogonal complement of $P({\cal X}_t):=(\sigma_t\cdot\IC+\bar\sigma_t\cdot \IC)_{\mathbb R}$ in $F(\alpha)\subset H^2(X,{\mathbb R})\cong H^2({\cal X}_t,{\mathbb R})$ (as in the proof of \ref{Weylprop}), then the class $\alpha_t$ is K\"ahler on ${\cal X}_t$ for general $t$ close to $0$. Now proceed as in the final paragraph of the proof of \ref{Weylprop}\hspace*{\fill}$\Box$. \bigskip Once a point $(X',\varphi')\in {\mathfrak M}_\Gamma$ non-separated from $(X,\varphi)$ with $({\varphi'}^{-1}\circ\varphi)(\alpha)$ in the K\"ahler cone ${\cal K}_{X'}$ has been shown to exist, one easily proves \begin{theorem}\label{Delignesuggestion}--- Let ${\cal X}\to {\it Def}(X)$ be the universal deformation of an irreducible symplectic manifold $X$ and let $\alpha\in{\cal C}_X$ be a class not orthogonal to any $0\ne\beta\in H^{1,1}(X)_{\mathbb Z}$. Then there exists another irreducible symplectic manifold $X'$ and a smooth proper family ${\cal X}'\to {\it Def}(X)$ with $X'={\cal X}'_0$ such that over an open subset containing the complement of the union of all hypersurfaces $S_\beta$ with $\beta\in H^2(X,{\mathbb Z})$ both families ${\cal X}$ and ${\cal X}'$ are isomorphic and the induced isomorphism $H^2(X,{\mathbb R})\cong H^2(X',{\mathbb R})$ maps $\alpha$ to a K\"ahler class on $X'$. \end{theorem} {\em Proof}. Again, this is just a reformulation of the proof of \ref{Weylprop} with the extra input \ref{Delignehelps}, which in turn is based on \ref{Weylprop} in its original form. Fix a marking $\varphi$ of $X$ and take $(X',\varphi')$ as in \ref{Delignehelps}. Consider ${\it Def}(X')$ as an open subset of ${\mathfrak M}_\Gamma$. Using the period map, both spaces ${\it Def}(X)$ and ${\it Def}(X')$ can be identified. Thus, we have two marked families $({\cal X},\varphi)\to{\it Def}(X)$ and $({\cal X}',\varphi')\to{\it Def}(X)$, both universal for $X$ respectively $X'$. Moreover, for some point $t\in{\it Def}(X)$ there is an isomorphism $f:{\cal X}_t\cong{\cal X}'_t$ with $f^*={\varphi'}^{-1}\circ\varphi$. Let $V$ be the maximal open subset such that there exists an isomorphism ${\cal X}|_V\cong{\cal X}'|_V$ extending $f$. As we have seen in the proof of \ref{Weylprop}, for a point $s$ in the boundary $\partial V:=\overline V\setminus V$ the group $H^{1,1}({\cal X}_s)_{\mathbb Z}$ is non-trivial. Hence, $\partial V$ is contained in the union of all hypersurfaces $S_\beta$ with $\beta\in H^2(X,{\mathbb Z})$. Since a countable union of real codimension two subsets cannot separate two non-empty open subsets, one of the open sets $V$ or ${\it Def}(X)\setminus \overline V$ must be empty. Hence, $\overline V={\it Def}(X)$.\hspace*{\fill}$\Box$ \bigskip It seems likely that in the above theorem one can achieve that the open subset is in fact the complement of finitely many hypersurfaces $S_{\beta_i}$ with $\beta_i\in H^{1,1}(X)_{\mathbb Z}$, but at the moment I do not know how to prove this. \section{The Ample Cone}\label{amplecone} After having established a projectivity criterion for irreducible symplectic manifolds in Sect.\ \ref{projectivity}, we now strive for an understanding of the ample cone of a projective irreducible symplectic manifold. Recall the following well-known results: \bigskip \refstepcounter{theorem}\label{Nakai}{\bf \thetheorem} --- Let $X$ be a projective variety. A line bundle $L$ on $X$ is ample if and only if $\int_Zc_1^i(L)>0$ for any integral subscheme $Z\subset X$ of dimension $i\leq\dim(X)$ (cf.\ \cite{HartshorneAmple}). \refstepcounter{theorem}\label{ampleK3}{\bf \thetheorem} --- Let $X$ be a K3 surface. A line bundle $L$ on $X$ is ample if and only if $c_1(L)$ is contained in the positive cone and $\int_Cc_1(L)>0$ for all $(-2)$-curves (i.e.\ irreducible smooth and rational) $C\subset X$ (cf.\ \cite{BPV}, \cite{Periodes}). \bigskip The second statement, a special case of the Nakai-Moishezon criterion, says that on a two-dimensional irreducible symplectic manifold the ample cone is the integral part of the positive cone that is positive on all $(-2)$-curves. The main result of this section (Corollary \ref{ampleconethm}) is formulated in this spirit. However, the result is extremely weak compared to \ref{ampleK3} as it only says that an integral class in the positive cone that cannot be separated from the K\"ahler cone by any {\it integral} wall is ample. The problem one has to face in higher dimensions is that an analogue of $(-2)$-curves is not (yet?) available. \bigskip Recall, the following notation: ${\cal K}_X\subset H^{1,1}(X)_{\mathbb R}$ is the cone of all K\"ahler classes and the positive cone ${\cal C}_X$ is the component of $\{\alpha\in H^{1,1}(X)_{\mathbb R}|q_X(\alpha)>0\}$ that contains ${\cal K}_X$. We begin with a variant of \ref{Nakai} in the case of projective manifolds with trivial canonical bundle. The following proposition, which is a straightforward consequence of the Basepoint-Free Theorem \cite{CKM}, says that in order to check whether a line bundle is ample it suffices to test it on subvarieties which are either $X$ itself or of dimension one. \begin{proposition}--- Let $X$ be a projective manifold with $K_X\cong{\cal O}_X$. Then a line bundle $L$ on $X$ is ample if and only if $\int c_1(L)^{\dim(X)}>0$ and $\int_Cc_1(L)>0$ for all curves $C\subset X$. \end{proposition} {\em Proof}. Obviously, if $L$ is ample, then both inequalities hold. If $L$ is a line bundle satisfying both inequalities, then $L$ is nef and $L\otimes K\makebox[0mm]{}^{{\scriptstyle\vee}}_X\cong L$ is nef and big. Then, the Basepoint-Free Theorem shows that $L^m$ is globally generated for $m\gg0$. But any globally generated line bundle that is positive on all curves is ample (cf.\ \cite{HartshorneAmple}).\hspace*{\fill}$\Box$ \bigskip Applied to irreducible symplectic manifolds this yields: \begin{corollary}\label{NakaionHK}--- Let $X$ be a projective irreducible symplectic manifold of dimension $2n$ and let $L$ be a line bundle on $X$. Then \begin{itemize} \item[$\bullet$] $L$ is ample if and only if $c_1(L)\in{\cal C}_X$ and $\int_Cc_1(L)>0$ for all curves $C\subset X$. \item[$\bullet$] $L$ is in the closure of the ample cone if and only if $c_1(L)\in\overline{\cal C}_X$ and $L$ is nef. \end{itemize} \end{corollary} {\em Proof}. Observe that for the first assertion we do not need the projectivity of $X$, for $c_1(L)\in{\cal C}_X$ implies that $X$ is projective (Theorem \ref{proj}). Both assertions follow directly from the proposition: If $L$ is ample, then certainly $\int_Cc_1(L)>0$, $q_X(c_1(L))>0$, and $q_X(c_1(L),\alpha)>0$ for any K\"ahler class $\alpha$ (cf.\ \ref{posconedef}) and, therefore, $c_1(L)\in{\cal C}_X$. Conversely, if $c_1(L)\in{\cal C}_X$ then $\int_Xc_1^{2n}(L)>0$ by \ref{Todd}. For the second assertion replace $L$ by $L^m\otimes M$ with $M$ an ample line bundle and $m\gg0$.\hspace*{\fill}$\Box$ \bigskip Using the isomorphism $cL_{\sigma\bar\sigma}^{n-1}:H^{1,1}(X)_\IQ\cong H^{2n-1,2n-1}(X)_\IQ$ (cf.\ \ref{quadraticform}) Corollary \ref{NakaionHK} allows us to formulate also the following \begin{corollary}\label{ampleconethm}--- Let $X$ be an irreducible symplectic manifold and let $L$ be a line bundle on $X$. Then $L$ is ample if and only if $c_1(L)$ satisfies \begin{itemize} \item[{\it i)}] $q_X(c_1(L),\,.\,)$ is positive on ${\cal K}_X$. \item[{\it ii)}] If $M\in {\rm Pic}(X)$ such that $q_X(c_1(M),\,.\,)$ is positive on ${\cal K}_X$, then $q_X(c_1(M),c_1(L))>0$. \end{itemize} \end{corollary} {\em Proof}. If $L$ is ample, then $c_1(L)\in{\cal K}_X$ and, therefore, {\it i)} and {\it ii)} follow. If $L$ satisfies {\it i)} and {\it ii)}, then $c_1(L)\in{\cal C}_X$. Hence, $X$ is projective (Theorem \ref{proj}). Thus it remains to verify that $L$ is positive on all curves (\ref{NakaionHK}). If $C$ is a curve then $[C]\in H^{2n-1,2n-1}(X)_\IQ$ and therefore there exists an $\alpha\in H^{1,1}(X)_\IQ$ with $cL_{\sigma\bar\sigma}^{n-1}(\alpha)=[C]$. Since any K\"ahler class $[\omega]$ is positive on $C$, one has $q_X(\alpha,\omega)>0$. Hence, $q_X(\alpha,c_1(L))>0$ by assumption, for $\alpha$ is a rational class.\hspace*{\fill}$\Box$ \bigskip The statement seems rather weak, and indeed, it cannot be considered as a true generalization of the Nakai-Moishezon criterion for K3 surfaces. However, it is non-trivial. E.g., if $X$ is a projective irreducible symplectic manifold such that its Picard group is spanned by two line bundles $L_1$ and $L_2$ with $L_1$ ample and $L_2$ non-ample, then there exists a (${\mathbb Z}$-)linear combination of $L_1$ and $L_2$ that is negative (with respect to $q_X$) on $L_2$ and positive on the whole K\"ahler (and not only on the ample!) cone. Of course, if $h^{1,1}(X)=\rho(X)$, then the assertion of the corollary is void. Note that the description of the ample cone given in this section does not make use of Proposition \ref{Weylprop}, but I expect that Corollary \ref{ampleconethm} together with Proposition \ref{Weylprop} implies that the ample cone is finitely polyhedral. But at the moment I do not know how to prove this. \section{The K\"ahler Cone}\label{kaehlercone} Here we slightly generalize the result of the previous section and give a description of the K\"ahler cone. Again, the result is much weaker than the known ones for K3 surfaces and says that a class in the positive cone that cannot be separated by any integral wall from the K\"ahler cone is K\"ahler itself. It might be noteworthy that the description of the ample cone of a K3 surface is a rather easy consequence of the Nakai-Moishezon criterion, whereas the description of the K\"ahler cone of a K3 surface relies on the Global Torelli Theorem, which is in no form available in higher dimensions. The result can most powerfully be applied to the case where there are no or only few integral classes. In particular it generalizes Corollary \ref{kaehlerconecor} to the case that $X$ is projective and the Picard number is one (cf.\ \ref{nameless}). \begin{theorem}\label{kaehlerconethm}--- Let $X$ be an irreducible symplectic manifold. Then a class $\alpha$ is contained in the closure of the K\"ahler cone $\overline{{\cal K}}_X$ if and only if \begin{itemize} \item[{\it i)}] $\alpha\in\overline{{\cal C}}_X$. \item[{\it ii)}] If $q_X(c_1(M),\,.\,)$ is non-negative on the K\"ahler cone ${\cal K}_X$ for a line bundle $M\in{\rm Pic}(X)$, then $q_X(c_1(M),\alpha)\geq0$. \end{itemize} \end{theorem} {\em Proof}. The principal idea of the proof is modelled on Beauville's expos\'e in \cite{Periodes}. Arguments using the Weyl-group are replaced by Proposition \ref{Weylprop}. It is obvious that {\it i)} and {\it ii)} are necessary for $\alpha\in\overline{{\cal K}}_X$. For the converse, assume that $\alpha$ satisfies both conditions. We first consider $\alpha+\varepsilon\cdot\gamma$, where $0<\varepsilon\ll1$ and $\gamma$ is a general K\"ahler class. This class is contained in ${\cal C}_X$ and satisfies the strong inequality in {\it ii)} for all line bundles $M$ with $q_X(c_1(M),\,.\,)$ positive on ${\cal K}_X$. If we can show that for general $\gamma$ the class $\alpha+\varepsilon\gamma$ is a K\"ahler class, then $\alpha\in\overline{{\cal K}}_X$. Certainly, $\alpha+\varepsilon\gamma$ is general (in the sense of Proposition \ref{Weylprop}) if $\gamma$ is a general K\"ahler class. Thus, we only have to deal with the following situation: $\alpha$ is a general class in ${\cal C}_X$ such that $q_X(c_1(M),\alpha)>0$ whenever $q_X(c_1(M),\,.\,)$ is positive on ${\cal K}_X$, where $M$ is an arbitrary line bundle on $X$. In this situation we show that $\alpha$ is a K\"ahler class. Indeed, by Proposition \ref{Weylprop} and Corollary \ref{corBRMain} there exists another irreducible symplectic manifold $X'$ together with an effective cycle $\Gamma=Z+\sum Y_i\subset X\times X'$ satisfying {\it i)-iv)} of Corollary \ref{corBRMain}. In particular, there exists a K\"ahler class $\alpha'$ on $X'$ with $[\Gamma]_*(\alpha)=\alpha'$. The latter condition is equivalent to $[\Gamma]_*(\alpha')=\alpha$. Since $[\Gamma]_*$ respects the quadratic form, we have $q_X(\alpha)=q_{X'}(\alpha')$. By \ref{hodgeunderbir} any birational correspondence $Z$ respects the quadratic form as well. Hence, $q_{X'}(\alpha')=q_{X}([Z]_*(\alpha'))$. This yields $$\begin{array}{ccl} 0&=&q_X(\alpha)-q_X([Z]_*(\alpha'))\\ &=&q_X(\alpha+[Z]_*(\alpha'),\alpha-[Z]_*(\alpha'))\\ &=&q_X(\alpha+[Z]_*(\alpha'),\sum[Y_i]_*(\alpha')).\\ \end{array}$$ By Lemma \ref{lem} only those components $Y_i$ contribute for which $V_i:=p(Y_i)$ is a divisor. Moreover, in this case $[Y_i]_*(\alpha')=(\int_{C_i}\alpha')[V_i]$, where $C_i$ is the generic fibre of $p:Y_i\to V_i$. Since $\alpha'$ is K\"ahler, $\int_{C_i}\alpha'>0$. By \ref{Todd} a K\"ahler class is positive (with respect to $q$) on any effective divisor. Hence, any $q_X((\int_{C_i}\alpha')[V_i],\,.\,)$ is positive on ${\cal K}_X$. Using the assumption on $\alpha$, this implies $q_X(\sum(\int_{C_i}\alpha')[V_i],\alpha)>0$. Since $\alpha'$ is K\"ahler, one also has $q_X([Z]_*(\alpha'),[V_i])=q_{X'}(\alpha',[Z]_*[V_i])>0$ (cf.\ \ref{Todd}). Altogether, this yields that if $\sum[Y_i]_*(\alpha')\ne0$ then $0=q_X(\alpha+[Z]_*(\alpha'),\sum(\int_{C_i}\alpha')[V_i])>0$. Hence, $\sum[Y_i]_*(\alpha')=0$ and, therefore, $\alpha=[\Gamma]_*(\alpha')=[Z]_*(\alpha')$. If $Z$ does not define an isomorphism $X\cong X'$ then there exists a (rational) curve $C\subset X$ such that $\int_C\alpha=\int_C[Z]_*(\alpha')<0$, e.g.\ take a curve in the fibre of $Z\to X'$. On the other hand, there exists a rational class $\beta\in H^{1,1}(X)_\IQ$ such that $cL_{\sigma\bar\sigma}^{n-1}(\beta)= [C]$ (cf.\ \ref{quadraticform}). Since any K\"ahler class is positive on $C$, the linear form $q_X(\beta,\,.\,)$ is positive on ${\cal K}_X$. By the assumption this yields $\int_C\alpha>0$. Contradiction. Thus $X\cong Z\cong X'$ and, therefore, $\alpha\in{\cal K}_X$.\hspace*{\fill}$\Box$ \bigskip The following is another instance where the K\"ahler cone can completely be described in terms of the period. \begin{corollary}\label{nameless}--- Let $X$ be an irreducible symplectic manifold and assume that ${\rm Pic}(X)$ is spanned by a line bundle $L$ such that $q_X(c_1(L))\geq0$. Then ${\cal K}_X={\cal C}_X$, i.e.\ any class $\alpha\in{\cal C}_X$ is K\"ahler. \end{corollary} {\em Proof}. Of course, it suffices to prove that $\overline{{\cal C}}_X=\overline{{\cal K}}_X$. Thus we can apply the theorem. Without loss of generality we can assume that $q_X(c_1(L),\,.\,)$ is non-negative on $\overline{{\cal C}}_X$ (Hodge Index). Therefore, $\overline{{\cal C}}_X\subset\overline{{\cal K}}_X$.\hspace*{\fill}$\Box$ \section{Surjectivity of the Period Map}\label{periodmap} Recall, that for a lattice $\Gamma$ of index $(3,b-3)$ we defined the period domain $Q$ as the set $\{x\in\IP(\Gamma_\IC)|q_\Gamma(x)=0,~q_\Gamma(x+\bar x)>0\}$, which is an open set of a smooth quadric. Also recall, that the period map ${\cal P}:{\mathfrak M}_\Gamma\to \IP(\Gamma_\IC)$ takes values in $Q$. (cf.\ \ref{moduli}). In this section we present a proof of the following \begin{theorem}\label{surjper}--- Let ${\mathfrak M}_\Gamma^o$ be a non-empty connected component of the moduli space ${\mathfrak M}_\Gamma$ of marked irreducible symplectic manifolds. Then the period map $${\cal P}:{\mathfrak M}_\Gamma^o\to Q$$ is surjective. \end{theorem} The proof is, once again, modelled on Beauville's presentation in \cite{Periodes}. One proceeds in two steps. The first part of the proof consists of showing that all points of $Q$ are equivalent with respect to a certain equivalence relation defined below. This part is a word-by-word copy of the known arguments. The second part, where it is shown that the image of the period map is invariant under the equivalence relation, deviates from the standard proofs even for K3 surfaces. The description of the K\"ahler cone (Corollary \ref{Kaehlerconeforperiod}) turns out to be crucial for this part. Let us first recall the following lemma (cf.\ \cite{Periodes}): \begin{lemma}--- The map sending a point $x\in Q\subset\IP(\Gamma_\IC)$ to $P(x):=(x\cdot\IC\oplus\bar x\cdot\IC)\cap\Gamma_{\mathbb R}$ defines a natural isomorphism between $Q$ and the Grassmannian $\tilde Q$ of positive oriented planes in $\Gamma_{\mathbb R}$.\hspace*{\fill}$\Box$ \end{lemma} This lemma enables one to prove that the period domain $Q$ is connected. See \cite{Periodes} for the complete argument. \begin{definition}--- Two points $x,y\in Q$ are called equivalent if there exists a sequence $x=x_1,x_2,\ldots,x_k=y\in Q$ such that the subspaces $\langle P(x_i),P(x_{i+1})\rangle\subset\Gamma_{\mathbb R}$ are of dimension three and such that $q_\Gamma|_{\langle P(x_i),P(x_{i+1})\rangle}$ is positive definite. (A subspace of dimension three with this property is called a positive $3$-space.) \end{definition} One can easily show that the set of points equivalent to a fixed $x\in Q$ is open. According to \cite{Periodes} this together with the connectivity of the period domain $Q$ is enough to prove (again, for the complete argument see \cite{Periodes}): \begin{lemma}--- Any two points in $Q$ are equivalent.\hspace*{\fill}$\Box$ \end{lemma} That these results are valid in this generality, and hence applicable to higher dimensional irreducible symplectic manifolds, was also noticed by Verbitsky \cite{Verbitsky}. This lemma together with the following one immediately proves Theorem \ref{surjper}. \begin{lemma}--- If $x,y\in Q$ such that $\langle P(x),P(y)\rangle$ is a positive $3$-space, then $x\in{\cal P}({\mathfrak M}_\Gamma^o)$ if and only if $y\in{\cal P}({\mathfrak M}_\Gamma^o)$. \end{lemma} {\em Proof}. Assume $x={\cal P}((X,\varphi))$, i.e.\ $P(x)=\varphi(P(X))$, with $(X,\varphi)\in{\mathfrak M}_\Gamma^o$. I claim that one can deform $X$ slightly such that \begin{itemize} \item[{\it i)}] $\langle P(x),P(y)\rangle$ is still a positive $3$-space. \item[{\it ii)}] $\rho(X)=0$. \end{itemize} This is proved as follows: Identify ${\it Def}(X)$ with a small open neighbourhood of $x$ in $Q$ via the period map (Local Torelli Theorem). Then consider the countable union of hypersurfaces $T=\bigcup S_\alpha\subset {\it Def}(X)$ where $\alpha\in H^2(X,{\mathbb Z})$ (see \ref{hypersurfaces}). To achieve {\it i)} and {\it ii)} it suffices to find $t\in {\it Def}(X)\setminus T$ such that $\langle \varphi(P({\cal X}_t)),P(y)\rangle$ is a positive $3$-space. Of course, the positivity is harmless as long as $t$ is close to $0$ and $\dim\langle \varphi(P({\cal X}_t)),P(y)\rangle=3$. First, consider those $t$ for which $\langle \varphi(P({\cal X}_t)),P(y)\rangle=\langle P(x),P(y)\rangle$. They are parametrized by (an open subset of) of the non-degenerate quadric $Q\cap \langle P(x),P(y)\rangle_\IC$ (for a similar argument see \ref{twistor2}). Moving $t$ slightly in $Q\cap\langle P(x),P(y)\rangle_\IC$ we can assume that the orthogonal complement $k_0\cdot{\mathbb R}\subset\langle P(x), P(y)\rangle$ of $P(x)$ is not contained in $P(y)$. Now fix a basis $P(y)=\langle a,b\rangle$. Then to any $k$ in a neighbourhood of $k_0$ in $\Gamma_{\mathbb R}$ we associate $P_k:=\langle q(k,k)a-q(a,k)k,q(k,k)b-q(b,k)k\rangle$ -- the orthogonal complement of $k$ in $\langle k,P(y)\rangle$. Then $\langle P_k,P(y)\rangle$ is a positive $3$-space. Let $T'\subset {\it Def}(X)$ be the subset of those $t\in{\it Def}(X)$ such that $\varphi(P({\cal X}_t))=P_k$ for some $k$ in a neighbourhood of $k_0\in\Gamma_{\mathbb R}$. If $T'\not\subset T$ we are done. If $T'\subset T$, then there exists an $\alpha$ such that $T'\subset S_\alpha$ (note that $T'$ is locally irreducible). Since $T'$ certainly contains those points for which $\langle\varphi(P({\cal X}_t)),P(y)\rangle=\langle P(x),P(y)\rangle$, we in particular have (locally!) $Q\cap\langle P(x),P(y)\rangle_\IC\subset T'\subset S_\alpha$. Since $S_\alpha$ is the hyperplane section defined by $q(\alpha,\,.\,)$ and $Q\cap \langle P(x),P(y)\rangle_\IC$ is a non-degenerate quadric, this shows that $q(\alpha,\,.\,)$ vanishes on $\langle P(x),P(y)\rangle$. Moreover, since $T'\subset S_\alpha$, the spaces $P_k$ are all orthogonal to $\alpha$. Hence, $q(q(k,k)a-q(a,k)k,\alpha)=0$ and $q(q(k,k)b-q(b,k)k,\alpha)=0$ for all $k$ in a neighbourhood of $k_0$. Also $q(a,\alpha)=q(b,\alpha)=0$ and, therefore, $q(a,k)q(k,\alpha) =q(b,k)q(k,\alpha)=0$ for all $k$ in a neighbourhood of $k_0$. Contradiction. Corollary \ref{Kaehlerconeforperiod} then shows that for $X$ satisfying {\it i)} and {\it ii)} above one has ${\cal C}_X={\cal K}_X$. Thus, if $\beta$ spans the orthogonal complement of $P(x)$ in $\langle P(x),P(y)\rangle$, then, after replacing $\beta$ by $-\beta$ if necessary, $\alpha:=\varphi^{-1}(\beta)\in{\cal C}_X$ is a K\"ahler class. On the other hand, $y\in\IP(\varphi(F(\alpha)_\IC))\subset \IP(\Gamma_\IC)$. Since there exists the twistor space ${\cal X}(\alpha)\to T(\alpha)=\IP(\varphi(F(\alpha)_\IC))$ and $T(\alpha)\subset{\mathfrak M}_\Gamma^o$, this suffices to conclude that $y\in {\cal P}({\mathfrak M}_\Gamma^o)$.\hspace*{\fill}$\Box$ \bigskip Note that in the original proof for K3 surfaces the class $\beta$ is chosen to be rational and not orthogonal to any $(-2)$-curve. Therefore, modulo the action of the Weyl-group, it is positive on all $(-2)$-curves and, hence, ample by the Nakai-Moishezon criterion. In the approach above the generalization of the Nakai-Moishezon criterion as proved in Sect.\ \ref{amplecone} does not suffice to prove the surjectivity along this line. Also note, that the order of the arguments is different compared to the original proof for K3 surfaces. For K3 surfaces one first proves the surjectivity of the period map using the Nakai-Moishezon criterion and then applies it to derive a description of the K\"ahler cone. Whereas here, we first described the K\"ahler cone and then applied the result, which is, however, far less explicit than the known one for K3 surfaces, to prove the surjectivity of the period map. \section{Automorphisms}\label{auto} Not much is known about the automorphism group of an irreducible symplectic manifold of dimension greater than two. In this section we collect some results related to this question and add some remarks. We introduce the following notations: Let $X$ be an irreducible symplectic manifold. Then \begin{itemize} \item[--] $Aut(X)$ is the group of holomorphic automorphisms of $X$. \item[--] $Birat(X)$ is the group of birational automorphisms, i.e.\ of birational maps $X- - \to X$. \item[--] $A(X)\subset Aut(H^2(X,{\mathbb Z}))$ is the group of automorphisms of $H^2(X,{\mathbb Z})$ which are com\-pa\-tible with the Hodge structure and the quadratic form $q_X$ and map the K\"ahler cone to the K\"ahler cone. \item[--] $B(X)\subset Aut(H^2(X,{\mathbb Z}))$ is the group of automorphisms of $H^2(X,{\mathbb Z})$ which are compa\-tible with the Hodge structure and the quadratic form $q_X$. \item[--] $a:Aut(X)\to Aut(H^2(X,{\mathbb Z}))$ maps an automorphism $f$ to $f^*$. \item[--] $b:Birat(X)\to Aut(H^2(X,{\mathbb Z}))$ maps a birational map $f$ to $f^*$. \end{itemize} \begin{proposition}--- {\it i)} ${\rm im}(a)\subset A(X)$, {\it ii)} ${\rm im}(b)\subset B(X)$, {\it iii)} $b^{-1}(A(X))=Aut(X)\subset Birat(X)$, {\it iv)} $\ker(a)=\ker(b)$, and {\it v)} $\ker(a)$ is finite. \end{proposition} {\em Proof}. {\it i)} is trivial, {\it ii)} follows from Lemma \ref{hodgeunderbir}. In order to prove {\it iii)} one has to show that any birational map which maps a K\"ahler class to a K\"ahler class can be extended to an automorphism. This was proved in \cite{Fujiki3}. {\it iv)} is an easy consequence of {\it iii)}. To prove {\it v)} one evokes two standard facts: Firstly, the group of isometries of a compact Riemannian manifold is compact and, secondly, the Calabi-Yau metric with respect to a fixed K\"ahler class is unique. Hence, an automorphism acting trivially on the second cohomology leaves invariant the K\"ahler class and, hence, the Calabi-Yau metric, i.e.\ it is an isometry. Since the group $Aut(X)$ is discrete, this suffices to conclude that $\ker(a)$ is finite. \hspace*{\fill}$\Box$ \bigskip The natural inclusion $Aut(X)\subset Birat(X)$ is in general proper. Indeed, Beauville constructed in \cite{Beauville2} an example of a birational automorphism of ${\rm Hilb}^n(S)$, where $S$ is a special K3 surface, which does not extend to an automorphism (cf.\ \ref{bevdeb}). However, one has \begin{proposition}--- If $X$ is general, i.e.\ together with a marking $\varphi$ it is a general point in ${\mathfrak M}_\Gamma$, then $Aut(X)=Birat(X)$. \end{proposition} {\em Proof}. This stems from the fact that the general irreducible symplectic manifold $X$ does not contain any (rational) curves (cf.\ \ref{twistor2}). But a birational automorphism of a variety without rational curves extends to a holomorphic automorphism.\hspace*{\fill}$\Box$ \bigskip For K3 surfaces one certainly has $Aut(X)=Birat(X)$. Moreover, the Global Torelli Theorem for K3 surfaces in particular asserts that $a:Aut(X)\to A(X)$ is an isomorphism. Using this fact one also shows that $a$ is injective for the Hilbert scheme of a K3 surface \cite{Beauville2}. However, due to an example of Beauville \cite{Beauville2}, we know that $a$ is not injective in general (This time the counterexample is provided by a generalized Kummer variety). Since the graph of an automorphism in the kernel of $a$ deforms (at least infinitesimally) in all directions with $X$, we shall not even expect that $a$ is injective for general $X$. \bigskip {\bf Questions} --- {\it i)} Is $a$ surjective? {\it ii)} Is $A(X)=\{1\}$ for general $X$? \bigskip The fact that $a$ is not injective has the unpleasant consequence that the moduli space ${\mathfrak M}_\Gamma$ of marked irreducible symplectic manifolds is not fine (contrary to the K3 surface case). For a discussion of this and other questions related to the moduli space see the next section. \section{Further Remarks}\label{remarks} As mentioned in the introduction, the two problems I consider the most important in the theory are: \begin{itemize} \item[--] Is there a Global Torelli Theorem for irreducible symplectic manifolds in higher dimensions? \item[--] What are the possible deformation (or diffeomorphism) types of irreducible symplectic manifolds? \end{itemize} The second question alludes to the rather easy fact that any two K3 surfaces are deformation equivalent and, hence, diffeomorphic. From dimension four on we know exactly two different deformation (diffeomorphism) types of irreducible symplectic manifolds (cf.\ Sect.\ \ref{biratman}). But there is no obvious reason why there should be no other possibilities. Due to Debarre's counterexample \cite{Debarre} (cf.\ \ref{bevdeb}) we know that the the Global Torelli Theorem as formulated for K3 surfaces fails in higher dimensions. However, there seems to be no counterexample known to the following speculation which was first formulated by Mukai \cite{Mu2}. \bigskip {\bf Speculation}\refstepcounter{theorem}\label{Spec} {\bf \thetheorem} --- {\it Two irreducible symplectic manifolds with isomorphic periods are birational.} \bigskip The known proofs of the Global Torelli Theorem for K3 surfaces break down in higher dimensions. This is mainly due to a missing analogue of Kummer surfaces. Kummer surface are dense in the moduli spaces of K3 surfaces and whether a K3 surface is a Kummer surface can easily be read off its period. (Quartic hypersurfaces would be another such distinguished class of K3 surfaces \cite{Friedman}.) In higher dimensions we neither have a good class of manifolds which are dense in ${\mathfrak M}_\Gamma$ nor do we know a `typical' class of manifolds which could be recognized by its period. In this light, it would be interesting to find an answer to the following \bigskip {\bf Question} --- Let $X$ be an irreducible symplectic manifold with the period of a Hilbert scheme ${\rm Hilb}^n(S)$ of a K3 surface $S$. Is $X$ birational to ${\rm Hilb}^n(S)$? \bigskip Let us conclude with a few remarks on the moduli space ${\mathfrak M}_\Gamma$ of marked manifolds. By definition ${\mathfrak M}_\Gamma$ is the set $\{(X,\varphi)\}/\sim$. Here $X$ is an irreducible symplectic manifold and $\varphi:H^2(X,{\mathbb Z})\cong\Gamma$ is an isomorphism of lattices. Two marked manifolds $(X,\varphi)$ and $(X',\varphi')$ are equivalent if there exists an isomorphism $g:X\cong X'$ with $g^*=\pm({\varphi}^{-1}\circ\varphi')$. That this set can be given the structure of a smooth, although non-separated, manifold is a consequence of the unobstructedness of symplectic manifolds and the Local Torelli Theorem. The period map ${\cal P}:{\mathfrak M}_\Gamma\to Q\subset\IP(\Gamma_\IC)$ exhibits ${\mathfrak M}_\Gamma$ as a space \'etale over of the period domain $Q$. The above speculation together with \ref{BRMain} and \ref{surjper} is equivalent to \bigskip {\bf Speculation}\label{Spec'} {\bf \thetheorem'} --- {\it If ${\mathfrak M}_\Gamma\ne\emptyset$ then the fibre over a general point $x\in Q$ is exactly one point.} \bigskip Note also that once the first speculation is answered positively it can be used to generalize Theorem \ref{birat} to the effect that any two birational irreducible symplectic manifolds (projective or not) correspond to non-separated points in the moduli space. It is tempting to define another moduli space which does not distinguish between birational manifolds. Let ${\mathfrak N}_\Gamma$ be the set $\{(X,\varphi)\}/\approx$, where the $(X,\varphi)$ are as above and $(X,\varphi)\approx(X',\varphi')$ if and only if there exists a birational map $g:X- - \to X'$ with $g^*=\pm({\varphi}^{-1}\circ\varphi')$. Of course, there is a natural map ${\mathfrak M}_\Gamma\to{\mathfrak N}_\Gamma$. Theorem \ref{birat} proves that two points in the same fibre of this map, at least if they are projective, are contained in the same component of ${\mathfrak M}_\Gamma$. In fact, they are non-separated there. Thus, the general form of Theorem \ref{birat}, i.e.\ without the projectivity assumption, would prove that the number of components of ${\mathfrak M}_\Gamma$ and ${\mathfrak N}_\Gamma$ is the same. Can ${\mathfrak N}_\Gamma$ be endowed with the structure of a manifold? If a birational map can always be extended to birational maps between all nearby fibres in the Kuranishi family, then local patching would supply ${\mathfrak N}_\Gamma$ with the structure of a manifold. Of course, if the Global Torelli Theorem in the formulation of the first speculation above holds true, then this would be trivial. Also note that the period map ${\cal P}:{\mathfrak M}_\Gamma\to Q$ naturally factorizes through a map ${\mathfrak N}_\Gamma\to Q$. {\footnotesize

9705

alg-geom/9705015

en

https://arxiv.org/abs/alg-geom/9705015

alg-geom/9705015

Giorgio Ottaviani

Lucia Fania and Giorgio Ottaviani

Boundedness for codimension two submanifolds of quadrics

AMS-LaTeX v2e, 22 pages, to the memory of Fernando Serrano

Arrondo, Sols and De Cataldo proved that there are only finitely many families of codimension two subvarieties not of general type in the smooth quadric of dimension $n+2$ for $n\ge 2 $, $n\neq 4$. In this paper we drop the assumption $n\neq 4$ from the previous result (obviously the assumption $n\ge 2$ cannot be removed).

alg-geom

\section{introduction} Ellingsrud and Peskine proved in \cite{ep} that smooth surfaces in $\Pin{4}$ not of general type have bounded degree. In \cite{boss1} this result has been estended to any non general type codimension two submanifolds in $\Pin{n+2}$, of dimension $n \geq 2$. In the same spirit Arrondo, Sols and De Cataldo proved in \cite{as}, \cite{deC} the following result \begin{theo}(Arrondo, Sols, De Cataldo) Let $X=X_n \subset Q_{n+2}$ be a smooth variety not of general type of dimension $n$ embedded in the smooth quadric $Q_{n+2}$ of dimension $n+2$. Let $n \geq 2$, $n \neq 4$. Then deg(X) is bounded. \end{theo} More precisely in \cite{as} it is proved the case n=2 while in \cite{deC} it is proved the case n=3 and it is observed that the case $n \geq 5$ follows by an inequality on Chern Classes along the lines of \cite{sc}. The aim of this paper is to drop the assumption $n \neq 4$ from the previous theorem. In fact we show the following \begin{theo} Let $X=X_4 \subset Q_6$ be a smooth 4-dimensional variety not of general type. Then deg(X) is bounded. \end{theo} As it is well known, see Prop. \brref{Hilbertcomp}, the theorem implies that there are only finitely many families of codimension two subvarieties not of general type in $Q_6$. The paper is structured as follows. In section 2. we fix our notation and give preliminary results that will be needed later on in the paper. The sections 3. and 4. are devoted to bounding the degree of a non general type 4-fold $X \subset Q_6$. In the last section we consider the problem of boundedness of non general type 4-folds in $\Pin{7}$. Among the log-special type 4-fold in $\Pin{7}$ (i.e., the image of the adjunction mapping has dimension less than 4) there are still two hard cases to be considered, namely the quadric bundles over surfaces and the scrolls over threefolds. We show that quadric bundles over surfaces have bounded degree with the only exception of those which lie in a 5-fold of degree 8. The same technique can be applied to a manifold $X$ of dimension $n+1$ embedded in $\Pin{2n+1}$ which is a quadric bundle over a surface. One can prove that there exists a function F(n) such that $deg(X) \leq F(n)$ or $X$ is contained in a variety of dimension $n+2$ and degree $[\frac{n(n+1)(4n^2+4n+1)}{6(4n-1)}]$, where $[x]$ is the greatest integer less than or equal to x. \section{Notations and Preliminaries} \subsection{NOTATION} \label{notation} Throughout this article, unless otherwise specified, $X$ denotes a smooth connected projective 4-fold defined over the complex field {\bf C}, which is contained in $Q_6$. Its structure sheaf is denoted by ${\cal O}_X$. For any coherent sheaf $\Im$ on $X$, $h^i(\Im )$ is the complex dimension of $H^i(S,\Im)$ and $\chi=\chi({\cal O}_X)=\sum_i(-1)^ih^i({\cal O}_X).$ The following notation is used:\\ X, smooth 4-fold in $Q_6$;\\ H class of hyperplane section of X, H = $\restrict{{\cal O}_{\Pin{7}}(1)}{X}$;\\ K class of canonical bundle of X;\\ $X^3$ generic 3-fold section of X;\\ S generic surface section of X;\\ C generic curve section of X;\\ g genus of C;\\ $c_i$ = Chern classes of X;\\ N the normal bundle of X in $Q_6, N_{X/Q_{6}}$.\\ Using the self intersection formula for the embedding of X $\subset Q_6$ \begin{equation} \label{selfint} c_2(N)=\frac{1}{2}dH^2 \end{equation} we get the following formulae for $KH^3, K^2H^2, K^3H, K^4$ as function of d, g, $\chi({\cal O}_X)$, $\chi({\cal O}_{{X}^3})$, $\chi(\cal{O}_S)$. \begin{eqnarray} \label{khcubo} K\cdot H^3 = 2g-2-3d \end{eqnarray} \begin{eqnarray} \label{kquadrohquadro} K^2\cdot H^2 = 6\chi({\cal O}_S)-12g+12+\frac{13}{2}d+\frac{1}{4}d^2 \end{eqnarray} \begin{eqnarray} \label{kcuboh} K^3\cdot H &=& -24\chi({\cal O}_{{X}^3})-48\chi({\cal O}_S) +48g-48+3d-3d^2+\\ & & d(g-1)\nonumber\end{eqnarray} \begin{eqnarray} \label{kquarta} K^4 &=& 120\chi({\cal O}_X)+216\chi({\cal O}_{X^{3}})+ \chi({\cal O}_S)\frac{9d+472}{2}+ \\ & &\frac{5d^3+1098d^2-16d(45g+434)-6144(g-1)}{48}. \nonumber \end{eqnarray} Note that \brref{khcubo} follows from the adjuction formula.\\ To prove \brref{kquadrohquadro} we reason as follows. From the long exact sequence \begin{equation} \label{tangX} 0\longrightarrow T_X \longrightarrow T_{Q_{6}|X} \longrightarrow N \longrightarrow 0 \end{equation} we get that $$c_2(N) = 16H^2+6H\cdot K+K^2-c_2$$ Such equality along with the self intersection formula \brref{selfint} gives \begin{equation} \label{cdue} c_2=(16-\frac{1}{2}d)H^2+6H\cdot K+K^2 \end{equation} Hence by dotting \brref{cdue} with $H^2$ we have \begin{equation} \label{cduehquadro} c_2\cdot H^2=(16-\frac{1}{2}d)H^4+6H^3\cdot K+H^2\cdot K^2 \end{equation} On the other hand \begin{equation} \label{cduehduefrom} c_2\cdot H^2=12\chi({\cal O}_S)-K^2\cdot H^2-(12g-12-11d). \end{equation} In fact using the following exact sequences $$0\longrightarrow T_{X^{3}} \longrightarrow T_{X|{X^{3}}} \longrightarrow H_{X^{3}} \longrightarrow 0$$ $$0\longrightarrow T_{S}\longrightarrow T_{X^{3}|S} \longrightarrow H_S \longrightarrow 0$$ we get that \begin{equation} \label{cdueh} c_2\cdot H=c_2(X^3)-K_{X^{3}}\cdot H_{X^{3}} \end{equation} Now \brref{cduehduefrom} will follow if we dot \brref{cdueh} with H and if we use the following facts: \begin{eqnarray} \label{cdueXtre} c_2(X^3)\cdot H_{X^{3}}&=&c_2(S)-K_S\cdot H_S=12\chi({\cal O}_S)-K_S^2-\\ & &(2g-2-d)\nonumber \end{eqnarray} and \begin{equation} \label{ksdue} K_S^2 = K^2\cdot H^2+8g-8-8d \end{equation} Combining \brref{cduehquadro} and \brref{cduehduefrom} we get \brref{kquadrohquadro}. In order to prove \brref{kcuboh} we do the following. By dotting \brref{cdue} with H$\cdot$ K we get \begin{equation} \label{cduehk} c_2\cdot H\cdot K=(16-\frac{1}{2}d)K\cdot H^3+6K^2\cdot H^2+K^3\cdot H \end{equation} On the other hand if we dot \brref{cdueh} with K and we use the fact that $c_2(X^3)\cdot K_{X^3}=-24\chi({\cal O}_{X^{3}})$ we have \begin{equation} \label{cduehkfrom} c_2\cdot H\cdot K=-24\chi({\cal O}_{X^{3}})-12\chi({\cal O}_S)+8g-8-6d \end{equation} Now \brref{kcuboh} is gotten by putting together \brref{cduehk} and \brref{cduehkfrom}. We now prove \brref{kquarta}. From the sequence \brref{tangX} we get that \begin{gather} \label{hcubo} 24H^3=c_3+c_2\cdot (6H+K)-\frac{1}{2}dK\cdot H^2\\ \label{hquarta} 22H^4=c_4+c_3\cdot (6H+K)-\frac{1}{2}dH^2\cdot c_2 \end{gather} Thus \begin{eqnarray} \label{ctre} c_3=(3d-72)H^3-(52-d)H^2\cdot K-12H\cdot K^2-K^3 \end{eqnarray} \begin{eqnarray} \label{cquattro} c_4&=&454d-26d^2+\frac{1}{4}d^3+(384-12d)H^3\cdot K+\\ & &\mbox{}(124-\frac{3}{2}d)H^2\cdot K^2+18H\cdot K^3+K^4\nonumber \end{eqnarray} By Riemann Roch theorem we know that \begin{equation} \label{RR} -720\chi({\cal O}_X)=K^4-4K^2\cdot c_2-3c_2^2+K\cdot c_3+c_4 \end{equation} Combining \brref{cdue}, \brref{RR}, \brref{cquattro}, \brref{ctre}, \brref{kcuboh}, \brref{kquadrohquadro} and \brref{khcubo} we get \brref{kquarta}. \begin{theo} (\cite{ccd}, Theorem 5.1) \label{ccdbound} Let C be an irreducible reduced curve of arithmetic genus g and degree d in the projective space $\Pin{n+1}$. Assume that C is not contained on any surface of degree $<$ s, with $d > \frac{2s}{n-1}\prod_{i=1}^{n-1} \sqrt[n-i]{n!s}$. Then \begin{eqnarray*} g-1 &\leq &\frac{d(d-1)}{2s}+\frac{d(s-2n+1)}{2(n-1)}+\frac{(d+s-1)(s-1)}{2s}+\\ & &\frac{(d-1)(n-2)(s+n-2)}{2s(n-1)}+\frac{(s-1)^2}{2(n-1)} \end{eqnarray*} \end{theo} \begin{prop} (\cite{as}, Proposition 6.3) \label{genusinQ3} Let C be a smooth curve of degree d and genus g in $Q_3$ that is not contained in any surface in $Q_3$ of degree strictly less than 2k. Then $$g-1 \leq \frac{d^2}{2k}+\frac{1}{2}(k-4)d$$ \end{prop} \begin{theo} (Castelnuovo bound \cite{har}) \label{harrisbound} Let V be an irreducible nondegenerate variety of dimension k and degree d in $\Pin{n}$. Put \begin{xalignat}{2} M=\left[\frac{d-1}{n-k}\right] &\quad and \quad \varepsilon = d-1-M(n-k) \notag \end{xalignat} where $[x]$ is the greatest integer less than or equal to x. Then \begin{eqnarray*} p_g(V)= h^{0}(\tilde{V}, \Omega^{k}) \leq \binom{M}{k+1}(n-k)+\binom{M}{k}\varepsilon \end{eqnarray*} where $\widetilde{V}$ is a resolution of V (i.e., $\widetilde{V}$ is a smooth variety mapping holomorphically and birationally to V) \end{theo} \begin{prop} \label{Hilbertpoli} Let X be a smooth 4-fold in $Q_6$. Then \begin{eqnarray*} \chi({\cal O}_X(t)) &=&\frac{1}{24}dt^4+\frac{1}{12}(2-2g+3d)t^3+ \frac{1}{24}(12\chi({\cal O}_S)-12g+12+11d)t^2\\ & &\mbox{}+\frac{1}{12}(12\chi({\cal O}_{X^{3}})+6\chi({\cal O}_S)-4g+4+3d)t+ \chi({\cal O}_X) \end{eqnarray*} \end{prop} \begin{pf} By the Riemann-Roch theorem we have $$\chi({\cal O}_X(t))= \frac{1}{24}H^4t^4+\frac{1}{12}(KH^3)t^3+ \frac{1}{24}(H^2\cdot K^2+c_2\cdot H^2)t^2- \frac{1}{24}c_2\cdot H\cdot Kt+\chi({\cal O}_X),$$ where $c_i = c_i(T_X)$. We now use \brref{cduehduefrom} and \brref{cduehkfrom} to get our claim. \end{pf} \begin{prop} \label{pgS} Let $S\subset Q_4$ be a surface of degree d contained in an irreducible threefold of degree $\sigma$, with $\sigma$ minimal. Then $$p_g(S)=h^2({\cal O}_S)\leq \frac{d^3}{24{\sigma}^2}+ \frac{d^2(\sigma-4)}{8\sigma}+ \frac{d(2{\sigma}^2-12\sigma+23)}{12}.$$ \end{prop} \begin{pf} Let C be the generic curve section of S. By \cite{as}, Proposition 6.4 for $d>>0$ we have $$g-1 \leq \frac{d^2}{4\sigma}+\frac{1}{2}(\sigma-3)d.$$ We let $$G(t) = \chi({\cal O}_{Q_{4}}(t)) =\left(\begin{array}{c}t+5\\5\end{array}\right)- \left(\begin{array}{c}t+3\\5\end{array}\right)$$ and $$\widetilde{F}(t) = G(t)-G(t-\sigma)-G(t-\frac{d}{2\sigma})+ G(t-\sigma-\frac{d}{2\sigma}).$$ We set $$P(t) = dt+\left[-\frac{d^2}{4\sigma}+ \frac{1}{2}(3-\sigma)d \right]$$ and {$$\qquad{\text F(t)=}\left\{\aligned \widetilde{F}(t) \ \ \ \ \ if \ \ t \leq -1\\ \widetilde{F}(0)-1 \ \ \ if \ \ t=0\\0 \ \ \ if \ \ t\geq1 \endaligned \right.$$} We have the following exact sequence $$H^1({\cal O}_C(t))\longrightarrow H^2({\cal O}_S(t-1)) \longrightarrow H^2({\cal O}_S(t)) \longrightarrow 0$$ from which it follows that {$${-h^2({\cal O}_S(t))+ h^2({\cal O}_{S}(t-1))\leq h^1({\cal O}_C(t))=} \left\{\aligned -\chi({\cal O}_C(t)) \leq -P(t) \ \ \ \ \ if \ \ t \leq -1\\ -\chi({\cal O}_C)+1 \leq -P(0)+1 \ \ \ if \ \ t=0\endaligned\right\}$$}\\ = F(t-1)-F(t), for t$\leq$ 0.\\ The same holds for t$\geq$ 1 since $h^1({\cal O}_C(t))=0$ and {$$\qquad{\text F(t-1)-F(t)=}\left\{\aligned \geq 0 \ \ \ \ \ if \ \ t=1\\ 0 \ \ \ if \ \ t \geq 2\endaligned \right.$$} From this it follows that F(t)-$h^2({\cal O}_S(t))$ is non increasing and since for t going to infinity it goes to zero it follows that $h^2({\cal O}_S(t)) \leq$ F(t) for all t. Thus evaluating it in t=0 we have $$F(0)=\frac{d^3}{24{\sigma}^2}+ \frac{d^2(\sigma-4)}{8\sigma}+ \frac{d(2{\sigma}^2-12\sigma+23)}{12}$$ and hence our claim. Note also that {$$\qquad{F(t)-F(t-1)=} \left\{\aligned dt+[-\frac{d^2}{4\sigma}+ \frac{1}{2}(3-\sigma)d] \ \ \ \ \ if \ \ t \leq -1\\ [-\frac{d^2}{4\sigma}+\frac{1}{2}(3-\sigma)d]-1 \ \ \ if \ \ t=0\endaligned\right.$$}\\ On passing note that if d is a multiple of 2$\sigma$ then $\widetilde{F}(t)$ is the Hilbert polynomial of the complete intersection in $Q_4$ of hypersurfaces in $\Pin{5}$ of degree $\sigma$ and $\frac{d}{2\sigma}$ while F(t) corresponds to $h^2({\cal O}_{V_{\sigma,\frac{d}{2\sigma}}}(t))$. \end{pf} \begin{prop}(\cite{deC}) \label{deC} Let $X^3 \subset V_{\sigma} \subset Q_5$. Then $$-\chi({\cal O}_{X{^3}}) \geq \frac{1}{192{\sigma}^3}d^4+l.t. in \sqrt d.$$ \end{prop} \begin{pf} For the proof see (\cite{deC}, Theorem 3.1.). It has to be noted that in there the coefficient of $d^4$ should be $\frac{1}{192{\sigma}^3}$. \end{pf} \begin{prop} \label{Hilbertcomp} For any fixed integer $d_0$ there are only finitely many irreducible components of the Hilbert scheme of 4-folds in $Q_6$ that contain 4-folds with $ d \leq d_0$. \end{prop} \begin{pf} By Harris' Castelnuovo bound, for $d \leq d_0$, there are finitely many possible values for g, $p_g(S), p_g(X^3), p_g(X)$ and hence for $\chi({\cal O}_S)$, $\chi({\cal O}_{X^{3}})$, $\chi({\cal O}_X)$ since $h^2({\cal O}_X) \leq p_g(S)$ and $h^1({\cal O}_S)=h^1({\cal O}_{X^{3}})=h^1({\cal O}_X)=0$. Thus there are only finitely many possibilities for the Hilbert polynomial \begin{eqnarray*} \chi({\cal O}_X(t))&=&\frac{1}{24}dt^4+\frac{1}{12}(2-2g+3d)t^3+ \frac{1}{24}(12\chi({\cal O}_S)-12g+12+11d)t^2\\ & &\mbox{}+\frac{1}{12}(12\chi({\cal O}_{X^{3}})+6\chi({\cal O}_S)-4g+4+3d)t+ \chi({\cal O}_X) \end{eqnarray*} \end{pf} \section{4-Folds on a Hypersurface of Fixed Degree} Let X be a 4-fold of degree d in $Q_6$ contained in an integral hypersurface $V_{\sigma} \in |{\cal O}_{Q_{6}}(\sigma)|.$ \begin{theo} \label{polin} Let $X \subset V_{\sigma} \subset Q_6$ be as above. There is a polynomial $P_{\sigma}(t)$ of degree 10 in $\sqrt d$ with positive leading coefficient, such that $$\chi({\cal O}_X) \geq P_{\sigma}(\sqrt d).$$ \end{theo} \begin{pf} Look at the following three exact sequences $$0 \longrightarrow {\cal O}_{\Pin{7}}(t-2) \longrightarrow {\cal O}_{\Pin{7}}(t) \longrightarrow {\cal O}_{Q_6}(t) \longrightarrow 0$$ $$0 \longrightarrow {\cal O}_{Q_6}(t-\sigma) \longrightarrow {\cal O}_{Q_6}(t) \longrightarrow {\cal O}_V(t) \longrightarrow 0$$ $$0 \longrightarrow {\cal I}_{X,V}(t) \longrightarrow {\cal O}_V(t) \longrightarrow {\cal O}_X(t) \longrightarrow 0$$ We use the first one to compute $\chi({\cal O}_{Q_6}(t))$ and the second one to compute \begin{eqnarray*} \label{chiOVt} \chi({\cal O}_{V}(t))&=&\frac{{\sigma}}{60}t^5+ \frac{(6{\sigma}-{\sigma}^2)}{24}t^4+ \frac{\sigma({\sigma}^2-9\sigma+26)}{18}t^3- \frac{\sigma({\sigma}^3-12{\sigma}^2+52\sigma-96)}{24}t^2\\ & &+\frac{\sigma(3{\sigma}^4-45{\sigma}^3+260{\sigma}^2-720\sigma+949)}{180}t\\ & &-\frac{\sigma({\sigma}^5-18{\sigma}^4+130{\sigma}^3-480{\sigma}^2+949\sigma-942)}{360}. \end{eqnarray*} Now use Prop.\brref{Hilbertpoli}, $\mu:=\mu_{\sigma}=\frac{1}{2}d^2+\sigma(\sigma-3)d-2\sigma(g-1)$ and the third exact sequence to compute \begin{eqnarray*} \label{chiideal} \chi({\cal I}_{X,V}(t))&=&\frac{{\sigma}}{60}t^5+ \frac{1}{24}((6-{\sigma}){\sigma}-d)t^4+\\ & &[\frac{3d^2+6d\sigma(\sigma-6)-2(3\mu-2{\sigma}^2({\sigma}^2-9\sigma+26))}{72\sigma}]t^3-\\ & &[\frac{12\chi({\cal O}_S)\sigma-2d^2+d\sigma(19-4\sigma)+4\mu+{\sigma}^2({\sigma}^3-12{\sigma}^2+52\sigma-96)}{24\sigma}]t^2-\\ & & [\frac{180\chi({\cal O}_S)\sigma+360\chi({\cal O}_{X^3})\sigma-15d^2+30d\sigma(4-\sigma)+30\mu}{360\sigma}\\ & &-\frac{2{\sigma}^2(3{\sigma}^4-45{\sigma}^3+260{\sigma}^2-720\sigma+949)}{360\sigma}]t-\\ & &\frac{\sigma({\sigma}^5-18{\sigma}^4+130{\sigma}^3-480{\sigma}^2+949\sigma-942)}{360}-\chi({\cal O}_X)\\ &:=& Q(t)-\chi({\cal O}_X) \end{eqnarray*} Thus $\chi({\cal O}_X) = Q(t)-\chi({\cal I}_{X,V}(t))$.\\ Define \begin{eqnarray*} t_1:= {\text {min}}\left \{ t\in {\Bbb N} | \delta:=2\sigma t-d > 0,\frac{{\delta}^2}{2}-\mu-\delta\sigma(\sigma-3)>0\right\} \end{eqnarray*} Then \begin{eqnarray*} \frac{d}{2\sigma}\leq t_1\leq \frac{d}{2\sigma}+ \frac{\sqrt {2d}}{2}+\sigma \end{eqnarray*} By plugging $t_1$ we get that \begin{eqnarray*} Q(t_1)&\geq& \frac{1}{60\cdot 2^5{\sigma}^4}d^5- \frac{1}{24\cdot 2^4{\sigma}^4}d^5- \frac{1}{24\cdot 2^3{\sigma}^4}d^5-\\ & &\frac{1}{2^3{\sigma}^2}d^2 \chi({\cal O}_S)- \frac{d}{2\sigma} \chi({\cal O}_{X^3})+ \textstyle{l.t. in \sqrt {d}} \end{eqnarray*} We now use Prop. \brref{deC} and Prop. \brref{pgS} in what above to get \begin{eqnarray*} Q(t_1)\geq \frac{d^5}{{\sigma}^4}(\frac{1}{60\cdot 2^5})+ \textstyle{l.t. in \sqrt {d}} \end{eqnarray*} and thus \begin{eqnarray} \chi({\cal O}_X) &\geq& Q(t_1)-\chi({\cal I}_{X,V}(t_1))\\ &\geq& -\chi({\cal I}_{X,V}(t_1))+ \frac{d^5}{{\sigma}^4}(\frac{1}{60\cdot 2^5})+ \textstyle{l.t. in \sqrt {d}}\nonumber \end{eqnarray} Moreover \begin{eqnarray*} -\chi({\cal I}_{X,V}(t_1)) &\geq& -h^0({\cal I}_{X,V}(t_1))- h^2({\cal I}_{X,V}(t_1))-h^4({\cal I}_{X,V}(t_1)) \end{eqnarray*} It will be enough to bound from above $h^2({\cal I}_{X,V}(t_1))$ since $h^0({\cal I}_{X,V}(t_1))$ and $h^4({\cal I}_{X,V}(t_1))$ have been bounded in (\cite {deC}, Lemma 3.3.). In order to bound $h^2({\cal I}_{X,V}(t_1))$ we consider the following exact sequences: $$0 \longrightarrow {\cal O}_{Q_5}(-\sigma) \longrightarrow {\cal I}_{{X^3},{Q_5}} \longrightarrow {\cal I}_{{X^3},{V^4}} \longrightarrow 0$$ $$0 \longrightarrow {\cal O}_{\Pin{6}}(-2) \longrightarrow {\cal I}_{{X^3},{\Pin{6}}} \longrightarrow {\cal I}_{{X^3},{Q_5}} \longrightarrow 0$$ By \cite{bm} \begin{xalignat}{2} H^1({\cal I}_{{X^3},{\Pin{6}}}(t)) = 0 &\quad for \quad t\geq4d-7 \notag \end{xalignat} The latter along with the above sequences give \begin{xalignat}{2} H^1({\cal I}_{{X^3},{V^4}}(t)) = 0 &\quad for \quad t\geq4d-7 \notag \end{xalignat} From $$0 \longrightarrow {\cal I}_{X,V}(k-1) \longrightarrow {\cal I}_{X,V}(k) \longrightarrow {\cal I}_{{X^3},{V^4}}(k) \longrightarrow 0$$ it follows that \begin{xalignat}{2} H^2({\cal I}_{X,V}(t)) = 0 &\quad for \quad t\geq 4d-8 \notag \end{xalignat} Moreover by (\cite{deC}, Lemma 3.3) we have that \begin{eqnarray*} h^2({\cal I}_{X,V}(t_1)) \leq \sum_{k=t_1+1}^{4d-7} h^1({\cal I}_{{X^3},{V^4}}(k)) \leq (4d-7)A d^{\frac{7}{2}}+... \leq 4Ad^{\frac{9}{2}}+... \end{eqnarray*} Hence \begin{eqnarray*} \chi({\cal O}_X) \geq Q(t_1)-\chi({\cal I}_{X,V}(t_1)) \geq Ad^5+ \mbox{...+ l.t. in d} \end{eqnarray*} which gives our claim. \end{pf} \begin{cor} \label{bound d} Let $X \subset V_{\sigma} \subset Q_6$ be as above. Assume that X is not of general type. Then there exists $d_0$ such that deg(X) $\leq d_0$. \end{cor} \begin{pf} Since X is not of general type we have $h^0(K_X(-1))$ = 0 from which it follows that $p_g(X) \leq p_g(X^3)$. This along with Harris bound give \begin{eqnarray} \label{chileq} \chi({\cal O}_{X})&=&1+h^2({\cal O}_X)-h^3({\cal O}_X)+ p_g(X) \leq 1+h^2({\cal O}_X)\\ & &+p_g(X^3) \leq 1+p_g(S)+p_g(X^3)\leq \frac{1}{216}d^4+ \mbox{l.t. in d}\nonumber \end{eqnarray} On the other hand by \brref{polin} we obtain that \begin{eqnarray} \label{chigeq} \chi({\cal O}_X)\geq \frac{1}{1920{\sigma}^4}d^5+ &\mbox{l.t. in d} \end{eqnarray} The boundedness of d will now follow from \brref{chileq} and \brref{chigeq}. Hence our claim. \end{pf} \section{Boundedness} \begin{prop} \label{chisless} Let X be a smooth 4-fold in $Q_6$. Denote $\chi({\cal O}_S)$, $\chi({\cal O}_{X^3})$, $\chi({\cal O}_X)$ by s, x and v respectively. Then \begin{itemize} \item[a)] $s \leq \frac{2}{3}\frac{(g-1)^2}{d}+\frac{5}{3}(g-1)-\frac{1}{24}d^2+\frac{5}{12}d.$ \item[b)] $-24x(2g-2-3d) \leq 36s^2+3s(d^2-22d-16(g-1))+\frac{1}{16}(d^4-92d^3+4d^2(12g+193)+32d(1-g)(g+8)+768(g-1)^2).$ \end{itemize} If X is not of general type then \begin{itemize} \item[c)] $v \leq 2s-x$ \end{itemize} \end{prop} \begin{pf} By the generalized Hodge index theorem we know that \begin{gather} \label{hodgeuno} (K^2\cdot H^2)H^4 \leq (K\cdot H^3)^2\\ \label{hodgedue} (K\cdot H^3)(K^3\cdot H) \leq (K^2\cdot H^2)^2. \end{gather} We observe that \brref{khcubo}, \brref{kquadrohquadro} and \brref{hodgeuno} give a) while \brref{khcubo}, \brref{kquadrohquadro}, \brref{kcuboh} and \brref{hodgedue} give b). In order to prove c) we use the following exact sequence \begin{equation} \label{Ksequence} 0\longrightarrow K_{X}(-1)\longrightarrow K_X\longrightarrow K_{X^3}(-1)\longrightarrow 0 \end{equation} Since X is not of general type we have $h^0(K_X(-1))$ = 0 and thus $h^0(K_X) \leq h^0(K_{X^3}(-1)) \leq h^0(K_{X^3}).$ By the Lefschetz theorem it follows that $h^2({\cal O}_X) \leq h^2(\cal{O}_S)$. Moreover being $h^1({\cal O}_X)$ = 0 it follows that $\chi({\cal O}_X) \leq \chi({\cal O}_S)+p_g(X^3) = 2\chi({\cal O}_S)-\chi({\cal O}_{X^3}).$ Hence our claim. \end{pf} \begin{prop} \label{chisgreater} Let X be a smooth 4-fold in $Q_6$. Then \begin{itemize} \item[a)] $24\chi({\cal O}_S) \geq d^2-2d-24(g-1).$ \item[b)] $240\chi({\cal O}_{X^3}) \leq 120\chi({\cal O}_X)+\frac{1}{2}(280-9d)\chi({\cal O}_S)- \frac{1}{48}(d^3-6d^2+16d(12g-13)-960(g-1)).$ \end{itemize} \end{prop} \begin{pf} Since N(-1) is globally generated, the Segre classes satisfy: \begin{xalignat}{2} s_2(N(-1)) \cdot H^2 \geq 0, &\quad s_4(N(-1)) \geq 0 \notag \end{xalignat} Recall that\\ $s_2 = c_1^2-c_2$\\ $s_4 = c_1^4+c_2^2-3c_1^2$.\\ Moreover\\ $c_1(N(-1)) = K+4H\\ c_2(N(-1)) = (\frac{1}{2}d-5)H^2 - H\cdot K$. Hence by \brref{khcubo}, \brref{kquadrohquadro} \begin{eqnarray} \label{stwo} 0&\leq&s_2(N(-1))\cdot H^2=K^2 \cdot H^2+16d+9K\cdot H^3-(\frac{1}{2}d-5)H^4\nonumber \\ &= &6\chi({\cal O}_S)+6(g-1)+\frac{1}{2}d-\frac{1}{4}d^2 \end{eqnarray} \begin{eqnarray} \label{sfour} 0&\leq&s_4(N(-1))=(434-13d)K\cdot H^3+(136-\frac{3}{2}d)K^2\cdot H^2\\ & &+19K^3\cdot H+K^4+521d-29d^2+\frac{1}{4}d^3=-240\chi({\cal O}_{X^3})\nonumber \\ & &+120\chi({\cal O}_X)+\frac{280-9d}{2}\chi({\cal O}_S)-\frac{d^3-6d^2+16d(12g-13)}{48}\nonumber\\ & &-20(g-1)\nonumber \end{eqnarray} \end{pf} \begin{theo} \label{comphilb} There are only finitely many irreducible components of the Hilbert scheme of smooth 4-folds in $Q_6$ that are not of general type. \end{theo} \begin{pf} Let X be a smooth 4-fold in $Q_6$ that is not of general type. By Prop.\brref{Hilbertcomp} it is enough to bound d = degX. We will do so by considering separately the cases 2g-2-3d $\leq$ 0 and 2g-2-3d $>$ 0. Assume that 2g-2-3d $\leq$ 0, i.e. g-1 $\leq \frac{3}{2}d$. Using Prop. \brref{chisless} along with a) in Prop. \brref{chisgreater} we get $$0 \leq -\frac{1}{2}d^2+6d$$ Hence d is bounded in this case. Assume now that 2g-2-3d $>$ 0. Using b) in Prop. \brref{chisgreater} along with c) and b) in Prop. \brref{chisless} we get \begin{eqnarray*} 0 &\leq & \frac{540}{2g-2-3d}s^2 + \frac{117d^2-6d(3g+707)+80(g-1)}{2(2g-2-3d)}s +\frac{d^4}{2g-2-3d}\\ & &+\frac{-d^3(g+2078)+6d^2(229g+2842)+304d(1-g)(3g+23)}{24(2g-2-3d)}\\ & &+\frac{76(g-1)^2}{2g-2-3d} \end{eqnarray*} Solving the above inequality with respect to s we see that either \begin{eqnarray} \label{sgreater} s \geq \frac{b+\sqrt {L}}{2160} \geq \frac{b}{2160} \end{eqnarray} or \begin{eqnarray} \label{sless} s \leq \frac{b-\sqrt {L}}{2160} \end{eqnarray} where $b = -117d^2+6d(3g+707)-80(g-1)$ and \begin{eqnarray*} L &=& 5049d^4-36d^3(107g+6793)+36d^2(9g^2-8978g+328809)+\\ & &960d(g-1)(339g+1915)-6560000(g-1)^2 \end{eqnarray*} If \brref{sgreater} holds then combining it with Prop. \brref{chisless} we get \begin{equation} \label{dquadro} 0 \leq \frac{d^2}{80}- \frac{d(3g+557)}{360}+ \frac{2(g-1)^2}{3d)}- \frac{46}{27}(g-1). \end{equation} If \brref{sless} holds then such inequality along with a) in Prop. \brref{chisgreater} gives \begin{eqnarray} \label{dquarta} 0&\leq& \frac{7}{864}d^4-\frac{d^3(5g+2203)}{6480}+ \frac{d^2(595-263g)}{3240}+\frac{7}{3}(g-1)^2\\ & &+\frac{d(1-g)(87g-5749)}{1620)}.\nonumber \end{eqnarray} Fix a positive integer k and let d $> 2k^2$. Assume that X does not lie on any hypersurface of $Q_6$ of degree strictly less than 2k. Then by Prop. \brref{genusinQ3} the genus of a general curve section of X satisfies \begin{equation} \label{gless} g-1 \leq \frac{d^2}{2k} + \frac{1}{2}(k-4)d \end{equation} Rewriting \brref{dquadro} in the following way $$0 \leq (g-1)\left[ \frac{2}{3d}(g-1)-\frac{d}{120}+\frac{46}{27} \right ]- \frac{7}{4}d + \frac{d^2}{80}$$ and using \brref{gless} we get \begin{eqnarray} \label{glessk} (g-1) \leq \frac{3k}{2(k-40)}d + &\mbox {l.t. in d}. \end{eqnarray} In the case \brref{dquarta} a similar reasoning yields \begin{eqnarray} \label{glessd} (g-1) \leq \frac{21}{2}d + &\mbox{ l.t. in d}. \end{eqnarray} The following inequality, gotten by combining a) in Prop. \brref{chisless} and a) in Prop. \brref{chisgreater} will be needed: \begin{equation} \label{eqcong} 0 \leq -\frac{1}{12}d^2 + \frac{1}{2}d + (g-1)\left[\frac{2}{3d}(g-1)+\frac{8}{3}\right ] \end{equation} Plugging \brref{glessk} and \brref{gless} in \brref{eqcong} gives \begin{eqnarray} \label{i} 0 \leq d^2(\frac{1}{2(k-40)}-\frac{1}{12}) + &\mbox{l.t. in d}. \end{eqnarray} Similarly, plugging \brref{glessd} and \brref{gless} in \brref{eqcong} gives \begin{eqnarray} \label{ii} 0 \leq d^2(\frac{7}{2k}-\frac{1}{12}) + &\mbox{l.t. in d}. \end{eqnarray} The coefficient of $d^2$, both in \brref{i} and \brref{ii} is negative for k=47. Hence d is bounded from above if X is not in a hypersurface of degree strictly less than 2 $\cdot$ 47. If X is not of general type and is contained in a hypersurface of degree less than or equal to 2$\cdot$ 47 then by Corollary \brref{bound d} there exists $d_0$ such that deg(X) $\leq d_0$. Hence the theorem is proved. \end{pf} \section{Quadric bundles over surfaces in $\Pin{7}$} Throughout this section $X$ will denote a smooth 4-fold of degree d in $\Pin{7}$ which is a quadric bundle over a surface. We will show that either its degree d is bounded or $X$ is contained in a $5$-fold of degree $8$. For 4-folds in $\Pin{7}$ by the selfintersection formula we have: $$c_3(N_{X|\Pin{7}}) = dH^3$$ \subsection{} \label{ciXinP7} From the exact sequence \begin{eqnarray*} \label{tangXinP7} 0\longrightarrow T_X \longrightarrow T_{\Pin{7}|X} \longrightarrow N_{X|\Pin{7}} \longrightarrow 0 \end{eqnarray*} we get that \begin{eqnarray*} c_3(X) &=& (56-d)H^3-28H^2\cdot c_1(X)+ 8H\cdot(c_1^2(X)-c_2(X))-c_1^3(X)+\\ & &2c_1(X)c_2(X)\end{eqnarray*} \begin{eqnarray*} c_4(X) &=& 70d-c_1(X)dH^3-c_2(X)(28H^2-8c_1(X)H+c_1^2(X)-c_2(X))\\ & &-c_3(X)(8H-c_1(X))\end{eqnarray*} \subsection{ Definition} A 4-fold $X$ is called a geometric quadric bundle if there exists a morphism $p:X \longrightarrow B$ onto a normal surface B such that every fibre $p^{-1}(b)$ is isomorphic to a quadric. A 4-fold X is a quadric bundle in the adjuction theoretic sense if there exists a morphism $p:X \longrightarrow B$ onto a normal surface B and an ample Cartier divisor L on B such $p^{*}L=K+2H$.\\ The following proposition relates the two notions. \begin{prop} \label{...} Let X be a quadric bundle in the adjuction theoretic sense. Then X is a geometric quadric bundle. Moreover the base B is smooth. \end{prop} \begin{pf} By (\cite {beso1}, Theorem 2.3) we know that p is equidimensional, being dim X=4. Moreover by (\cite {bes}, Theorem 8.2) the base B is smooth. \end{pf} We fix our notation which follows closely the one in \cite{boss2}. \subsection{Notations} \label{notazione quadric fib} Let $p:X \longrightarrow B$ be a geometric quadric bundle in $\Pin{7}$. We have a natural morphism $f:B \longrightarrow Gr(\Pin{3},\Pin{7})$. Let S be a generic surface section of X. Then $p:S \longrightarrow B$ is finite 2:1. Let 2R $\subset B$ be the ramification divisor of $p:S \longrightarrow B$. We set $p_{*}{{\cal O}_{X}}(1)$:= E, a rank 4 vector bundle over B. We have E = $f^{*}(U^{\vee})$, where U is the universal bundle of $Gr(\Pin{3},\Pin{7})$, in particular det E = $f^{*}(\stackrel{2}{\wedge}U^{\vee})$ is ample. Note that W:= P(E) is a $\Pin{3}$-bundle in the natural incidence variety $\Pin{7}\times Gr(\Pin{3},\Pin{7})$ whose projection $\pi$ into $\Pin{7}$ is the hypersurface V given by the union of all the 3-planes containing the quadrics of X. Moreover $\pi^{-1}(X) = \widetilde{X}$ is smooth and isomorphic to X. We denote the natural projection of W onto B also by p and by H the divisor on W corresponding to ${\cal O}_{W}(1)$. Hence $\widetilde{X} = 2H-p^{*}L$ for some divisor L on B. The divisor D $\subset B$ corresponding to points whose fibres are singular quadrics, is called the discriminant divisor. Moreover D = $c_1(E)-c_1(L\otimes E^{\vee}) = 2c_1(E)-4L.$ In fact $\widetilde{X}$ determines a section of $S^{2}E\otimes L^{\vee}$, hence a morphism $\phi: L\otimes E^{\vee}\longrightarrow E$. D is given by the equation det$\phi$ = 0. Thus our claim.\\ In order to bound d we need several preliminaries computations. \begin{prop} \label{ciW} \begin{eqnarray*} c_1(W)&=&4H-p^{*}c_1(E)+p^{*}c_1(B)\\ c_2(W)&=&6H^2+H\cdot[4p^{*}c_1(B)-3p^{*}c_1(E)]+ p^{*}c_2(B)-p^{*}c_1(E)\cdot p^{*}c_1(B)\\ & &+p^{*}c_2(E)\\ c_3(W)&=&4H^3+H^2\cdot[6p^{*}c_1(B)-3p^{*}c_1(E)]+ H\cdot[2p^{*}c_2(E)+4p^{*}c_2(B)\\ & &-3p^{*}c_1(E)\cdot p^{*}c_1(B)] \\ c_4(W)&=&4H^3\cdot p^{*}c_1(B)+H^2\cdot[6p^{*}c_2(B)-3p^{*}c_1(E)\cdot p^{*}c_1(B)] \end{eqnarray*} $H^4-H^3\cdot p^{*}c_1(E)+H^2\cdot p^{*}c_2(E)=0$ \end{prop} \begin{pf} Consider the sequence \begin{eqnarray*} \label{tangW} 0\longrightarrow {\cal O}_{W} \longrightarrow p^{*}E^{\vee} \otimes {\cal O}_{W}(1) \longrightarrow T_W \longrightarrow p^{*}T_B \longrightarrow 0 \end{eqnarray*} The Chern polynomial of $p^{*}E^{\vee} \otimes {\cal O}_{W}(1)$ is $1+c_1(p^{*}E^{\vee} \otimes {\cal O}_{W}(1))t+ c_2(p^{*}E^{\vee} \otimes {\cal O}_{W}(1))t^2+ c_3(p^{*}E^{\vee} \otimes {\cal O}_{W}(1))t^3 = 1+[4H-p^{*}c_1(E)]t+[6H^2-3p^{*}c_1(E)\cdot H+ p^{*}c_2(E)]t^2+[4H^3-3p^{*}c_1(E)\cdot H^2+2 p^{*}c_2(E)\cdot H]t^3.$ On the other hand $ch(T_W) = ch(p^{*}E^{\vee} \otimes {\cal O}_{W}(1)) \cdot ch(p^{*}T_B)$ hence we get that $1+c_1(W)t+c_2(W)t^2+c_3(W)t^3+c_4(W)t^4+c_5(W)t^5=\{1+ [4H-p^{*}c_1(E)]t+[6H^2-3p^{*}c_1(E)\cdot H+ p^{*}c_2(E)]t^2+[4H^3-3p^{*}c_1(E)\cdot H^2+2 p^{*}c_2(E)\cdot H]t^3\}\cdot \{1+p^{*}c_1(B)t+p^{*}c_2(B)t^2\}$. Expanding the right hand side we get the first four equations. The last one is the Wu-Chern equation on W = P(E), that is, $c_4(p^{*}E^{\vee} \otimes {\cal O}_{W}(1))=0.$ \end{pf} \begin{lemma} \label{c1E} $c_1(E) = 2R-\frac{D}{2}, \quad L = R-\frac{D}{2}$. \end{lemma} \begin{pf} We have $K_S=p^{*}(K_B+R)$, hence by the adjunction formula $K_X=-2H+p^{*}R+p^{*}K_B$. From Prop. \brref{ciW} $K_W=-4H+p^{*}c_1(E)-p^{*}c_1(B)$. Putting this together with the adjunction formula $K_X=\restrict{K_W}{X}+2H-p^{*}L$ gives $-p^{*}L+p^{*}c_1(E) = p^{*}R$, that is $c_1(E)=L+R$. Substituting this in $c_1(E)=\frac{D+4L}{2}$ we get $L = R-\frac{D}{2}$ and hence $c_1(E) = 2R-\frac{D}{2}$. \end{pf} \begin{prop} \label{ciX} \begin{eqnarray*} c_1(X) &=& 2H-p^{*}K_B-p^{*}R\\ c_2(X) &=& 2H^2+H\cdot[-p^{*}c_1(E)+2p^{*}c_1(B)]+p^{*}{R^2}- p^{*}c_1(E)\cdot p^{*}R\\ & &-p^{*}c_1(B)\cdot p^{*}R+ p^{*}c_2(B)+p^{*}c_2(E)\\ c_3(X) &=& H^2\cdot [2p^{*}c_1(B)+p^{*}c_1(E)- 2p^{*}R]+ H\cdot[2p^{*}c_2(B)-\\ & &p^{*}c_1(B)\cdot p^{*}c_1(E)-2p^{*}{R^2}+3p^{*}R\cdot p^{*}c_1(E)-p^{*}c_1^2(E)]\\ c_4(X) &=& H^3\cdot[-2p^{*}c_1(E)+4p^{*}R]+H^2\cdot[2p^{*}c_2(B)+p^{*}c_1(B)\cdot p^{*}c_1(E)\\ & &+6p^{*}{R^2}+3p^{*}c_1^2(E)-9p^{*}R\cdot p^{*}c_1(E)- 2p^{*}R\cdot p^{*}c_1(B)] \end{eqnarray*} \end{prop} \begin{pf} The following sequence \begin{eqnarray*} 0\longrightarrow T_X \longrightarrow T_{W|X} \longrightarrow {\cal O}(2H+R-c_1(E)) \longrightarrow 0 \end{eqnarray*} along with Prop. \brref{ciW} gives the proof. \end{pf} \begin{lemma} Let Z, $Z'$ be arbitrary divisors on B. Then \begin{itemize} \item [i)] $H^2\cdot p^{*}Z\cdot p^{*}Z' = 2Z\cdot Z'$ \item[ii)] $H^3\cdot p^{*}Z = (c_1(E)+R)\cdot Z = (3R-\frac{D}{2})\cdot Z$ \end{itemize} \label{intersecform} \end{lemma} \begin{pf} i) follows from the fact that the fibres of X over B are quadrics. As for ii) note that $H^3\cdot p^{*}Z$ is equal to the intersection product in W $$ H^3\cdot p^{*}Z\cdot (2H+p^{*}R-p^{*}c_1(E)).$$ Intersecting the Wu-Chern equation with $p^{*}Z$ we get that $$ H^4\cdot p^{*}Z-H^3\cdot p^{*}Z\cdot p^{*}c_1(E) =0.$$ Hence $H^3\cdot p^{*}Z\cdot (2H+p^{*}R-p^{*}c_1(E))= 2H^4\cdot p^{*}Z+ H^3\cdot p^{*}Z\cdot p^{*}R - H^3\cdot p^{*}Z\cdot p^{*}c_1(E) =2H^3\cdot p^{*}Z\cdot p^{*}c_1(E)+H^3\cdot p^{*}R\cdot p^{*}Z- H^3\cdot p^{*}Z\cdot p^{*}c_1(E) =(c_1(E)+R)\cdot Z = (3R-\frac{D}{2})\cdot Z$ \end{pf} \begin{prop} \label{deg,c2E} The surface f(B) in $Gr(\Pin{3},\Pin{7})$ has bidegree $(\delta, c_2(E))$ where \begin{xalignat}{2} \delta = degV = \frac{d-R\cdot c_1(E)+c_1^2(E)}{2}, &\quad c_2(E)= \frac{-d+R\cdot c_1(E)+c_1^2(E)}{2}\notag \end{xalignat} \end{prop} \begin{pf} We intersect the Wu-Chern equation in Prop. \brref{ciW} with H and we get $$\delta -H^4\cdot c_1(E)+H^3\cdot c_2(E) = 0.$$ Now cut the equation $X = 2H+p^{*}R-p^{*}c_1(E)$ with $H^4$ and we obtain $$d = 2\delta+H^3\cdot p^{*}R\cdot p^{*}c_1(E)- H^3\cdot p^{*}c_1^2(E).$$ From these two equalities we get our claim. \end{pf} \begin{prop} \label{ciXconsostituzioni} \begin{eqnarray*} c_1(X) &=& 2H+p^{*}c_1(B)-p^{*}R\\ c_2(X) &=& 2H^2+H\cdot[2p^{*}c_1(B)-2p^{*}R+\frac{1}{2}p^{*}D] -p^{*}{R^2}+p^{*}c_2(E)-\\ & & p^{*}c_1(B)\cdot p^{*}R+p^{*}c_2(B)+\frac{1}{2}p^{*}D\cdot p^{*}R\\ c_3(X) &=& H^2\cdot[-\frac{1}{2}p^{*}D+ 2p^{*}c_1(B)]+ H\cdot[\frac{1}{2}p^{*}c_1(B)\cdot p^{*}D-\\ & &2p^{*}c_1(B)\cdot p^{*}R+2p^{*}c_2(B)-\frac{1}{4}p^{*}{D^2} +\frac{1}{2}p^{*}D\cdot p^{*}R]\\ c_4(X) &=& H^3\cdot p^{*}D+H^2\cdot[-\frac{1}{2}p^{*}c_1(B)\cdot p^{*}D +2p^{*}c_2(B)+\frac{3}{4}p^{*}{D^2}-\frac{3}{2}p^{*}D\cdot p^{*}R] \end{eqnarray*} \end{prop} \begin{pf} Using \brref{c1E} - \brref{deg,c2E} and easy computations give the formulas for $c_2(X), c_3(X), c_4(X)$. \end{pf} \begin{prop} \label{system} Set $$P(d)=\frac{1}{9d^3-50d^2-10949d+169120}, \quad x = K_B^2, \quad y = D\cdot R.$$ The following hold:\\ \begin{eqnarray*} v=R^2&=&-\frac{1}{2}P(d)[-192864x+2842{d}^{3}+332024d-70224y- \\ & &53900{d}^{2}-15y{d}^{2}-49{d}^{4}+4974yd-3y{d}^{3}+9016dx]\\ z=D^2&=&P(d)[-882{d}^{4}+54y{d}^{3}+54684{d}^{3}-870y{d}^{2}-1100736{d}^{2}+\\ & &190512dx-39792yd+7112448d-3035648x+709632y]\\ f=c_2(B)&=&\frac {1}{16}P(d)[9{d}^{5}-328{d}^{4}+ 144{d}^{3}x+3y{d}^{3}+3036{d}^{3}-173y{d}^{2}-\\ & &37416{d}^{2}+712{d}^{2}x+2608yd-102048dx+728896d+\\ & &675584x-8576y]\\ u=K_B\cdot R&=&-\frac{1}{8}P(d)[-175392dx+1722{d}^{4}+696696{d}^{2} -22686yd-\\ & &52332{d}^{3}-11y{d}^{3}-21{d}^{5}+879y{d}^{2}+3864{d}^{2}x+1983744x-\\ & &3415104d+190784y]\\ t=K_B\cdot D&=&-\frac{1}{4}P(d)[-63{d}^{5}+5292{d}^{4} -164556{d}^{3}-33y{d}^{3}+ 2237760{d}^{2}+\\ & &13608{d}^{2}x+2703y{d}^{2}-11176704d-71880yd-516208dx+\\ & &624384y+4770304x] \end{eqnarray*} \end{prop} \begin{pf} We get a linear system of five equations in the unknowns f, t, u, v, z with coefficients rational functions of d. In fact from \brref{ciXinP7} we have $$c_3(X) = (56-d)H^3-28H^2\cdot c_1(X)+8H\cdot(c_1^2(X) -c_2(X))-c_1^3(X)+2c_1(X)c_2(X).$$ Substituting the values of $c_1(X), c_2(X), c_3(X)$ of Proposition \brref{ciXconsostituzioni} we get $(d-16){H}^{3}+\left(-12p^*R+\frac{3p^*D}{2}+ 14p^*c_1(B)\right)\cdot {H}^{2}+ (-6{p^*c_1(B)}^{2}-10{p^*R}^{2}+ 6p^*c_2(B)+4p^*c_2(E)-\frac {{p^*D}^{2}}{4}+6p^*c_1(B) \cdot p^*R-\frac{p^*c_1(B) \cdot p^*D}{2}+\frac{7p^*R\cdot p^*D}{2})\cdot H =0$ Cutting respectively with H, $p^{*}R$, $p^{*}K_B$, $p^{*}D$ we get four equations. For instance if we cut with $p^{*}D$ we obtain: $$(d-16)H^3\cdot p^{*}D +H^2\cdot \left (-12p^*R\cdot p^*(D)+\frac{1}{2}3p^*D\cdot p^*D +14p^*c_1(B)\cdot p^*D\right )=0$$ Using now lemma \brref{c1E} and lemma \brref{intersecform} we get $$(d-16)(3R-\frac{D}{2})\cdot D-24 R\cdot D+3D\cdot D+ 28c_1(B)\cdot D=0$$ and simplifying $$-72R\cdot D+11{D}^{2}+3dR\cdot D-\frac{{dD}^{2}}{2} -28K_B\cdot D=0$$ The fifth equation is gotten by substituting the values of Proposition \brref{ciXconsostituzioni} in the second formula of \brref{ciXinP7}: \begin{eqnarray*} c_4(X) &=& 70d-c_1(X)dH^3-c_2(X)(28H^2-8c_1(X)H+c_1^2(X)-c_2(X))\\ & &-c_3(X)(8H-c_1(X))\end{eqnarray*} Solving such system we get the claim. \end{pf} \begin{prop} \label{c2E,g} \begin{eqnarray*} c_2(E)&=&\frac{1}{4}P(d)[\left(41160d-360640\right )x+\left (-95{d}^{2}+5005d-69440\right)y\\ & &-165{d}^{4}+10390{d}^{3}-205070{d}^{2}+1225840d]\\ g-1&=&\frac{1}{8}P(d)[\left(1008{d}^{2}-49112d+566720\right)x +(223{d}^{2}-9857d+\\ & &109120)y+393{d}^{4}-21032{d}^{3}+353440{d}^{2}-1781360d] \end{eqnarray*} \end{prop} \begin{pf} Using Prop. \brref{deg,c2E} and Lemma \brref{c1E} we get $$c_2(E)=\frac{1}{2}\left(6R^2-\frac{5D\cdot R}{2}+\frac{D^2}{4}-d\right)$$ Moreover from the adjuction formula and Lemma \brref{intersecform} ii) we obtain \begin{eqnarray*} g-1&=&\frac{1}{2}d+\frac{1}{2}H^3(p^{*}K_B+p^{*}R)= \frac{1}{2}d+\frac{1}{2}(p^{*}K_B+p^{*}R)(3R-\frac{D}{2})\\ &=&\frac{1}{2}d+\frac{3}{2}K_B\cdot R-\frac{1}{4}K_B\cdot D+\frac{3}{2}R^2-\frac{1}{4}D\cdot R. \end{eqnarray*} Substituting the values of Prop. \brref{system} and simplifying we get the assertions. \end{pf} We need a Roth type result. \begin{prop} Let X be a codimension $3$ integral subvariety of $\Pin{n}$ of degree $d$. If the generic section $C=X\cap \Pin{4}$ with a linear $\Pin{4}$ is contained in a surface $S_{\sigma}\subset\Pin{4}$ of degree $\sigma$ with $\sigma^2 \leq d$ then X itself is contained in a codimension $2$ subvariety $V_{\sigma}\subset\Pin{n}$ of degree $\sigma$. \label{roth} \end{prop} \begin{pf} We first check that the generic section $S=X\cap \Pin{5}$ with a linear $\Pin{5}$ is contained in a 3-fold $Y_{\sigma}\subset\Pin{5}$ of degree $\sigma$. By the assumptions, the surface $S_{\sigma}$ is unique. On the contrary, suppose there are two such surfaces $S'_{\sigma}$ and $S''_{\sigma}$. There exists a linear projection $\pi$ from a point $p\in\Pin{4}$ on a hyperplane $\Pin{3}$ such that $\pi (S'_{\sigma})$ and $\pi (S''_{\sigma})$ are two irreducible distinct surfaces of degree $\sigma$ containing $\pi (C)$ against the Bezout theorem. We get a rational map from $(\Pin{5}) ^{\vee}$ in the Hilbert scheme of degree $\sigma$ surfaces of $\Pin{4}$. It follows that the closure of all surfaces $S_{\sigma}$ is the 3-fold $Y_{\sigma}$ we looked for. By iterating this process we get the thesis. \end{pf} \begin{prop} Let X be a quadric bundle in $\Pin{7}$. Then $d\leq 2963$ or $X$ is contained in a $5$-fold of degree $8$. \label{bound} \end{prop} \begin{pf} We consider the possible values of x and y compatible with the following three inequalities: $D\cdot R \geq 0$, \quad\mbox{(Lemma 4.15 in \cite{boss2})}\\ $c_2(E) \geq 0$, \quad\mbox{(Proposition \brref{deg,c2E})}\\ $c_1(E)\cdot D \geq 0$, \quad\mbox{($c_1$(E) is ample, see Notations \brref{notazione quadric fib})}.\\ Using Lemma \brref{intersecform}, Prop. \brref{deg,c2E}, Prop. \brref{system}, Prop. \brref{c2E,g} and easy computations we have\\ $y \geq 0\\ \left (41160d-360640\right )x +\left (-95{d}^{2}+5005d-69440\right )y-165{d}^{4}+10390{d}^{3}-205070{d}^{2}\\ +1225840d \geq 0\\ -\left (190512d-3035648\right )x - \left (18{d}^{3}-670{d}^{2}+4004d +33152\right )y+882{d}^{4}-54684{d}^{3}\\ +1100736{d}^{2}-7112448d \geq 0$\\ These inequalities bound the inside of a triangle whose vertices are: $$A_d= \left (\frac {d\left ( 33{d}^{3}-2078{d}^{2}+41014d-245168\right )}{8232d-72128},0\right )$$ $$B_d=\left (\frac {9d\left ({d}^{3}-62{d}^{2}+1248d-8064 \right )}{1944d-30976},0\right )$$ $$C_d=\left (\frac {\left (33{d}^{3}-2192{d}^{2}+46900d-316064 \right )d}{8232d-131712},\frac {2d\left (69d-872\right ) }{21d-336}\right )$$ The minimum and the maximum of g-1 considered as a function in x and y with d fixed (see Prop. \brref{c2E,g}) have to be assumed in one of the vertices. Substituting the coordinates of $A_d, B_d, C_d$ in the expression of g-1 we get respectively $g-1=\frac{(33{d}^{2}-293d-552)d}{588d-5152}$, $\frac{(63{d}^{2}-1320d+5192)d}{972d-15488}$, $\frac{(33{d}^{2}-407d-1008)d}{588d-9408}.$ Hence we have in our case \begin{eqnarray} \label{gcompreso} {\frac {\left (33{d}^{2}-293d-552\right )d}{588d-5152}} \leq g-1 \leq{ \frac {\left (63{d}^{2}-1320d+5192\right )d}{972d-15488}} \end{eqnarray} We now distinguish two cases. Suppose first that the curve section C is not contained in any surface of degree 8. Then from Theorem \brref{ccdbound} and \brref{gcompreso} we have $${\frac {\left (33{d}^{2}-293d-552\right )d}{588d-5152}} \leq \frac{d^2}{18}+\frac{5}{3}d+\frac{347}{18}$$ that is $3d^3-8881d^2-29706d+893872 \leq 0$, which implies $d \leq 2963$. If otherwise then C is contained in a octic, then from Proposition \brref{roth} it follows that also $X$ is contained in a octic. \end{pf} \begin{rem} We like to remark that a similar reasoning gives an analogous result for manifolds $X$ of dimension $n+1$ and degree $d$ embedded in $\Pin{2n+1}$ which is a quadric bundle over a surface. More precisely we have the following \end{rem} \begin{prop} Let $X$ be a manifold of dimension $n+1$ and degree d embedded in $\Pin{2n+1}$ which is a quadric bundle over a surface. Then there exists a function F(n) such that $d \leq F(n)$ or $X$ is contained in a variety of dimension $n+2$ and degree $[\frac{n(n+1)(4n^2+4n+1)}{6(4n-1)}]$, where $[x]$ is the greatest integer less than or equal to x. \label{Bound} \end{prop} \begin{pf} Similarly as in proposition \brref{bound} we get that $$Ad^2+O(d) \leq g-1 \leq Bd^2+O(d)$$ where $A=\frac{3(4n-1)}{n(n+1)(4n^2+4n+1)}$ and $B=\frac{3(2n+1)}{(n+1)(2n^3+2n^2+2n+3)}$. We now use Theorem \brref{ccdbound} and Proposition \brref{roth} to see that there exists a function F(n) such that $d \leq F(n)$ or $X$ is contained in a variety of dimension $n+2$ and degree $[\frac{n(n+1)(4n^2+4n+1)}{6(4n-1)}]$, where $[x]$ is the greatest integer less than or equal to x. \end{pf}

9705

alg-geom/9705006

en

https://arxiv.org/abs/alg-geom/9705006

alg-geom/9705006

Carlos Simpson

Carlos Simpson

Mixed twistor structures

LaTeX

The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety. We give some examples of generalizations of classical results for variations of mixed Hodge structure, to the twistor setting. This supports a ``meta-theorem'' (which we state but don't prove) that one can everywhere replace the word ``Hodge'' by the word ``twistor''. We show that the jet spaces of hyperk\"ahler or more generally hypercomplex manifolds have natural mixed twistor structures which determine the hypercomplex structure in a formal neighborhood of a point.

alg-geom

\section*{Mixed twistor structures} Carlos Simpson \newline CNRS, UMR 5580, Universit\'e Paul Sabatier\newline 31062 Toulouse CEDEX, France \bigskip The purpose of this paper is to introduce the notion of {\em mixed twistor structure} as a generalization of the notion of mixed Hodge structure. Recall that a mixed Hodge structure is a vector space $V$ with three filtrations $F$, $F'$ and $W$ (the first two decreasing, the last increasing) such that the two filtrations $F$ and $F'$ induce $i$-opposed filtrations on $Gr^W_i(V)$. (Generally $(V,W)$ is required to have a real structure and $F'$ is the complex conjugate of $F$ but that is not very relevant for us here.) Given a MHS $(V,W,F,F')$ we can form the {\em Rees bundle} $E:=\xi (V,F,F')$ over ${\bf P} ^1$ \cite{NAHT} \cite{SantaCruz}. In brief this is obtained from the trivial bundle $V\times {\bf P} ^1$ by using $F$ to make an elementary transformation over $0$ and $F'$ to make an elementary transformation over $\infty$. The bundle $E$ is graded by strict subbundles which we denote $W_{i}E$ and the condition of opposedness of the filtrations is equivalent to the condition that $Gr^W_i(E)$ be semistable of slope $i$ on ${\bf P} ^1$ (in other words a direct sum of copies of ${\cal O} _{{\bf P} ^1}(i)$). The notion of mixed twistor structure is simply obtained by abstracting this situation: an MTS is a pair consisting of a bundle $E$ over ${\bf P} ^1$ and a filtration by strict subbundles $W_iE$ such that the $Gr^W_i(E)$ are semistable of slope $i$. In the construction starting with a mixed Hodge structure the resulting $(E,W)$ has an action of ${\bf G}_m$ covering the standard action on ${\bf P} ^1$ and in fact the mixed Hodge structures are simply the ${\bf G}_m$-equivariant mixed twistor structures. Thus, in some sense, the passage from ``Hodge'' to ``Twistor'' is simply forgetting to have an action of ${\bf G}_m$. This principle occured already, in a primitive way, in the passage from systems of Hodge bundles (cf \cite{YMTU}) to Higgs bundles in \cite{CVHS}. We will give some generalizations of basic classical results for mixed Hodge structures, to the mixed twistor setting. The process of making these generalizations is relatively direct although some work must be done to develop the appropriate notion of {\em variation of mixed twistor structure}. The overall idea is that we have the following \noindent {\bf Meta-theorem} {\em If the words ``mixed Hodge structure'' (resp. ``variation of mixed Hodge structure'') are replaced by the words ``mixed twistor structure'' (resp. ``variation of mixed twistor structure'') in the hypotheses and conclusions of any theorem in Hodge theory, then one obtains a true statement. The proof of the new statement will be analogous to the proof of the old statement.} We don't prove this meta-theorem but support it with several examples using the basic theorems of mixed Hodge theory. The utility of the notion of mixed twistor structure, and of the above meta-theorem, is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety, where up until now the theory of weights has only been available for variations of Hodge structure. This phrase needs the further explanation that a harmonic bundle (cf \cite{Hitchin1} \cite{Corlette} \cite{HBLS}) yields a variation of pure twistor structure (i.e. of mixed twistor structure with only one nonzero weight-graded quotient) and in fact, up to choosing the weight a variation of pure twistor structure is essentially the same thing as a harmonic bundle. In particular, any irreducible representation of $\pi _1(X,x)$ for a compact K\"ahler manifold $X$, underlies a variation of pure twistor structure unique up to shift of weight (i.e. tensorization with constant pure rank one twistor structures). This is explained in Lemma \ref{harmonic} below. Apart from the above meta-theorem as a way of obtaining new statements, a possible area where we obtain a new type of object is the following: over the moduli space $M_B$ of representations of $\pi _1(X)$ for a compact K\"ahler variety $X$, we may have several families of mixed twistor structures, for example cohomology of open or singular subvarieties of $X$ with coefficients in the VMTS corresponding to $\rho \in M_B$, or some types of rational homotopy invariants relative to the representation $\rho$ (such as the relative Malcev completion or its analogues for higher homotopy). These natural families of MTS give classifying maps $M_B\rightarrow {\cal MTS}$ to the moduli stack of mixed twistor structures (the maps will not usually be algebraic but should be real analytic, for example). It might be interesting to study these classifying maps. The terminology ``twistor'' comes from the particular case of weight one twistor structures with a certain kind of real structure ({\em antipodal}). These are equivalent to the twistor bundles over ${\bf P} ^1$ of quaternionic vector spaces, see \cite{HKLR} for a nice exposition. Deligne in \cite{DeligneLetter} originally explained to me how to associate a quaternionic vector space to a weight one real Hodge structure and how to interpret the twistor space for Hitchin's hyperk\"ahler structure in terms of moduli of $\lambda$-connections, a notion very closely related to the Hodge filtration on the de Rham complex. This eventually led to a study of the formal neighborhood of the twistor lines in the twistor space, to an interpretation in terms of mixed Hodge structures (for the case of the moduli space of representations), and consequently to the present definition. In \S 7 below we discuss the relation between mixed twistor structures and the formal power series for hyperk\"ahler (or more precisely any hypercomplex) structures at a point. The case of the moduli space of representations of $\pi _1$ of a smooth projective variety is taken up briefly in \S 8. Here is an outline of the paper. {\em \S 1---Mixed twistor structures:} definition of mixed twistor structures, and the relation with mixed Hodge structures. The theorem that the category of MTS is abelian. As an aside at the end we point out that this theorem works with ${\bf P} ^1$ replaced by any projective variety as base (this remark is not used later, though). {\em \S 2---Real structures:} two different kinds of real structures for a mixed twistor structure: circular and antipodal. The weight one pure antipodal real twistor structures are the same as the quaternionic vector spaces. {\em \S 3---Variations of mixed twistor structure:} definition of ${\cal C} ^{\infty}$ families of mixed twistor structure, variations of mixed twistor structure, polarizations, real structures and a holomorphic interpretation as a pair of $\lambda$-connections on the two standard affine lines in ${\bf P} ^1$. {\em \S 4---Cohomology of smooth compact K\"{a}hler manifolds with VMTS coefficients:} we treat the cohomology of smooth compact K\"ahler manifolds, first with variations of pure twistor structure as coefficients, then with mixed coefficients. We give a holomorphic interpretation using $\lambda$-connections. {\em \S 5---Cohomology of open and singular varieties:} generalization of the basic theorems about cohomology of open and singular varieties to the mixed twistor case. This involves a notion of {\em mixed twistor complex}. A somewhat new point here involves the notion of {\em patching} complexes of sheaves on open sets of ${\bf P} ^1$; this replaces certain arguments involving the real structure in the theory of mixed Hodge complexes. {\em \S 6---Nilpotent orbits and the limiting mixed twistor structure:} we give a conjectural version of the nilpotent orbit theorem relating the degeneration of a harmonic bundle on a punctured disc to a limiting mixed twistor structure. {\em \S 7---Jet bundles of hypercomplex manifolds:} description of the formal germ of a hypercomplex structure in the neighborhood of a point by a mixed twistor structure on the jet space at the point. This comes from looking at the jet bundle of the twistor space along a twistor line. Following the philosophy described above of looking at classifying maps, we define the {\em Gauss map} from a hypercomplex manifold into the moduli stack for mixed twistor structures on the jet spaces. {\em \S 8---The moduli space of representations:} this is a discussion similar to that of the previous section, for moduli spaces of representations of fundamental groups of smooth projective varieties. Using the family of moduli spaces $M_{Del}$ of \cite{SantaCruz} as a replacement for the twistor space, we are able to treat singular points: we get a mixed twistor structure on the generalization of the jet space at a singular point, and this determines the formal neighborhood of a twistor line. {\em Notations:} we work in the analytic category of complex analytic spaces (usually smooth complex varieties) and when talking about projective varieties we mean the associated analytic varieties with the usual topology. \numero{Mixed twistor structures} Fix the projective line ${\bf P} ^1$ with points $0,1,\infty$, with the standard line bundle ${\cal O} _{{\bf P} ^1}(1)$. A {\em twistor structure} is a vector bundle $E$ on ${\bf P} ^1$. The {\em underlying vector space} is $E_1$, the fiber of the bundle $E$ over $1\in {\bf P} ^1$. We say that a twistor structure $E$ is {\em pure of weight $w$} if $E$ is a semistable vector bundle of slope $w$, which is equivalent to saying that $E$ is a direct sum of copies of ${\cal O} _{{\bf P} ^1}(w)$. A {\em mixed twistor structure} is a twistor structure $E$ filtered by an increasing sequence of strict subbundles $W_iE$ such that for all $i$, $Gr^W_i(E)= W_iE / W_{i-1}E$ is pure of weight $i$. The filtration $W_{\cdot}E$ is called the {\em weight filtration}. A mixed twistor structure is said to be {\em pure} if the associated graded is nontrivial in only one degree. \subnumero{Relation with mixed Hodge structures} As described in \cite{NAHT} and (\cite{SantaCruz} \S 5), if $V$ is a complex vector space with two decreasing filtrations $F$ and $F'$, then we obtain a vector bundle $\xi (V; F, F')$ on ${\bf P} ^1$. This is the {\em twistor structure associated to the pair of filtrations $F$ and $F'$}. The two filtrations define a pure Hodge structure of weight $w$ if and only if $\xi (V; F, F')$ is a pure twistor structure of weight $w$. Suppose $V$ is a vector space with three filtrations $W_{\cdot}$ (increasing) and $F$ and $F'$ (decreasing). Then putting $$ W_i \xi (V; F,F'):= \xi (W_iV; F, F') $$ we obtain a filtration by strict subbundles. The associated-graded $Gr ^W_i(\xi (V; F,F')$ is pure of weight $w$ if and only if $F$ and $F'$ induce a pure Hodge structure of weight $w$ on $Gr ^W_i(V)$. The condition in the previous paragraph is what we call a {\em complex mixed Hodge structure} (a definition which should have been made long ago). Note that a real mixed Hodge structure is simply a real vector space $V_{{\bf R}}$ with filtration $W_{\cdot}$ and complex filtration $F$ of $V_{{\bf C}}$ such that $W, F, \overline{F}$ form a complex mixed Hodge structure on $V_{{\bf C}}$. Similarly if $V,W,F,F'$ is a complex MHS then $V\oplus \overline{V}$ has a structure of real MHS (using the real structure which interchanges $V$ and $\overline{V}$). We have the following characterization. \begin{lemma} If $V_{{\bf R}}$ is a real vector space with increasing real filtration $W$, and with complex filtration (decreasing) $F$ of $V_{{\bf C}}$, then these data define a real mixed Hodge structure if and only if $\xi (V_{{\bf C}}; F, \overline{F})$ with the induced filtration $W_{\cdot}\xi (V_{{\bf C}}; F, \overline{F})$ is a mixed twistor structure. \end{lemma} \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} The group ${\bf G}_m$ acts on ${\bf P} ^1$ by translation. Denote the morphism of this action by $$ \mu : {\bf G}_m \times {\bf P} ^1 \rightarrow {\bf P} ^1. $$ A {\em ${\bf G}_m$-equivariant mixed twistor structure} is a mixed twistor structure $(E, W)$ together with an isomorphism of filtered coherent sheaves on ${\bf G}_m \times {\bf P} ^1$ $$ \rho : \mu ^{\ast}(E,W) \cong p_2^{\ast}(E,W) $$ such that the two resulting morphisms on ${\bf G}_m \times {\bf G}_m \times {\bf P} ^1$ $$ (\mu \circ \mu )^{\ast}(E,W) \cong p_3^{\ast}(E, W) $$ are equal. If we view $E$ as a filtered vector bundle over ${\bf P} ^1$ this just means that the action of ${\bf G}_m$ is lifted to an action on the total space of the bundle preserving the filtration. A {\em morphism} of equivariant mixed twistor structures is a morphism of filtered bundles compatible with the action. \begin{proposition} The category of ${\bf G}_m$-equivariant mixed twistor structures is naturally equivalent to the category of complex mixed Hodge structures (see below). \end{proposition} {\em Proof:} As was seen in \cite{NAHT} and \cite{SantaCruz} a bundle over ${\bf P} ^1$ equivariant for the action of ${\bf G}_m$ is the same thing as a vector space $V=E_1$ together with two decreasing filtrations $F$ and $F'$: the bundle $E$ is recovered as $\xi (V, F, F')$. Restriction to the fiber $V= E_1$ over $1\in {\bf P} ^1$ induces an inclusion-preserving bijection between the set of strict subsheaves of $V$ preserved by ${\bf G}_m$, and the set of subspaces of $V$. To see this, the restriction to the fiber may be viewed as the composition of restriction to ${\bf G}_m\subset {\bf P} ^1$ then restriction to the fiber. The second arrow is obviously a bijection between subsheaves of $E|_{{\bf G}_m}$ preserved by ${\bf G}_m$ and subspaces of $E_1$ (using the action of ${\bf G}_m$). Note by the way that any equivariant subsheaf of $E|_{{\bf G}_m}$ is automatically strict. Recall that a strict subsheaf of a vector bundle over a curve is determined by its restriction to any Zariski open set; this implies that restriction is a bijection between all strict subsheaves of $E$ and all strict subsheaves of $E|_{{\bf G}_m}$ (and this bijection preserves the subset of those which are equivariant). In view of this bijection, we obtain a bijection between the set of increasing filtrations of $E$ by strict subsheaves preserved by ${\bf G}_m$, and the set of increasing filtrations of $V$ by subspaces. In view of this and the first paragraph of the proof, an equivariant vector bundle $E$ over ${\bf P} ^1$ with increasing filtration by strict subsheaves $W_nE$ is the same thing as a vector space $V=E_1$ together with an increasing filtration $W_nV$ and two decreasing filtrations $F$ and $F'$. The construction in the reverse sense is the Rees bundle construction $E = \xi (V, F, F')$. Finally note that $$ Gr ^{WE}_n(E) = \xi ( Gr ^{WV}_n(V), F_{Gr}, F'_{Gr}) $$ where $F_{Gr}$ and $F'_{Gr}$ are the filtrations induced on the associated-graded (the Rees-bundle construction is compatible with taking strict subquotients, as can be verified for each filtration $F$ and $F'$ separately using splittings). The two filtrations $F_{Gr}$ and $F'_{Gr}$ are $n$-opposed if and only if the Rees bundle is semistable of slope $n$. Thus the data $(V,W,F, F')$ defines a mixed Hodge structure if and only if $(E,W)$ is a mixed twistor structure. We leave to the reader to verify full faithfulness of this correspondence for morphisms. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} We don't actually use the above proposition anywhere but it is crucial for understanding the analogy between all of the results which we give below, and the corresponding results for mixed Hodge structures. A somewhat pertinent remark is that our proofs for mixed twistor structures are all ${\bf G}_m$-equivariant in case the input-data is ${\bf G}_m$-equivariant. Thus our proofs give proofs of the corresponding statements for mixed Hodge structures. One hesitates to say that these constitute ``new'' proofs since they are really just recopying the old proofs for mixed Hodge structures into this new language. \subnumero{The abelian category of mixed twistor structures} The first basic theorem in the theory of mixed Hodge structures is the fact that the category of MHS is abelian. This generalizes to mixed twistor structures and the proof is essentially the same. \begin{lemma} \label{abelian} {\rm (cf \cite{Hodge2}, 2.3.5(i))} The category of mixed twistor structures is abelian. \end{lemma} {\em Proof:} Suppose $f: (E, W) \rightarrow (E', W')$ is a morphism of mixed twistor structures. {\bf Step 1}\,\, Define the {\em cokernel} of $f$ to be $(E'', W'')$ where $E''$ is the cokernel of $f$ considered as a map of coherent sheaves, and $W''$ is the filtration of $E''$ induced by $W'$. For the moment, this is just a coherent sheaf filtered by subsheaves. Let $A \subset E$ denote the kernel subsheaf of $f$. We work by induction on the size of the interval where the two weight filtrations $W$ and $W'$ are supported. If this interval has size $1$ then the cokernel is just the cokernel of a map of semistable sheaves of the same slope, so the cokernel is also semistable. Now consider an interval size of at least two. Let $Gr ^W_i(E)$ and $Gr ^{W'}_i(E')$ be the highest nonzero pair of terms (in other words $i$ is the smallest integer such that $W_i=E$ and $W'_i = E'$). Then we have an exact sequence $$ Gr ^W_i(E) \rightarrow Gr ^{W'}_i(E') \rightarrow Gr ^{W''}_i(E'') \rightarrow 0. $$ In particular, $Gr ^{W''}_i(E'')$ is semistable of slope $i$. Let $V_{i-1} \subset E$ be the inverse image of $W'_{i-1}$. We have an exact sequence $$ 0 \rightarrow W_{i-1} \rightarrow V_{i-1} \rightarrow B\rightarrow 0 $$ where $B$ is the kernel of the map $$ Gr ^W_i(E) \rightarrow Gr ^{W'}_i(E') . $$ Since this is a map of semistable sheaves of slope $i$, the kernel $B$ is semistable of slope $i$. Thus $V_{i-1}$ is a mixed twistor structure (with the weight filtration induced by that of $E$). By the induction hypothesis, the cokernel of the morphism $W_{i-1}\rightarrow W'_{i-1}$ of mixed twistor structures concentrated in a smaller interval (the $i$-th graded pieces are zero this time), is again a mixed twistor structure concentrated in the same interval. In particular, $W'_{i-1} /f(W_{i-1})$ cannot support a morphism from a semistable sheaf of weight $\geq i$. Thus the morphism $$ B = V_{i-1} /W_{i-1} \rightarrow W'_{i-1} /f(W_{i-1}) $$ is zero. Thus $$ f(V_{i-1}) = f(W_{i-1}). $$ Note that $$ f(V_i) = f(E) \cap W'_{i-1} = \ker (W'_{i-1} \rightarrow W''_{i-1}). $$ Therefore we get an exact sequence $$ W_{i-1} \rightarrow W'_{i-1} \rightarrow W'' _{i-1} \rightarrow 0. $$ Furthermore the weight filtration on $W'' _{i-1}$ is that induced by the weight filtration of $W'_{i-1}$. Hence $W'' _{i-1}$ is the cokernel of the map $W_{i-1} \rightarrow W'_{i-1}$, and by our induction hypothesis this cokernel is a mixed twistor structure. Finally, $E''$ is an extension of $Gr ^{W''}_i(E'')$ (which we have seen to be pure of weight $i$ at the start) by $W'' _{i-1}$, thus $E''$ with its weight filtration $W''$ is a mixed twistor structure. Along the way, we have proved that $E''$ is a bundle, for $Gr ^{W''}_i(E'')$ is a bundle (since it is a cokernel of a map of semistable bundles of the same slope) and $W'' _{i-1}$ is a bundle by induction. {\bf Step 2}\, \, Let $A \subset E$ be the kernel of $f$ considered as a morphism of coherent sheaves, and put $W_i A := A \cap W_i$. Note that $A$ is a strict subbundle of $E$ and $W_iA$ are strict subbundles of $A$. We would like to show that $(A, W_{\cdot }A)$ is a mixed twistor structure, i.e. to show that $Gr ^{WA}_i(A)$ is semistable of slope $i$. (Actually the following proof is just the dual of the proof of step 1.) Let $i$ denote the smallest number where either $W_i$ or $W'_i$ is nonzero. Thus $W_i$ and $W'_i$ are both semistable bundles of slope $i$. Let $V_i := f^{-1}(W'_i)\subset E$. We have that $V_i/W_i$ is the kernel of the morphism of mixed twistor structures $E/W_i \rightarrow E'/W'_i$. By induction (on the size of the interval where the weight graded quotients of $E$ and $E'$ live, just as in Step 1) we can suppose that $V_i/W_i$ with its filtration induced by $W_j/W_i$ is a mixed twistor structure, substructure of $E/W_i$. In particular the weights of $V_i/W_i$ are $> i$, which implies that any morphism of $V_i / W_i $ to a semistable bundle of slope $\leq i$ is zero. Thus the morphism $V_i/W_i \rightarrow W'_i / f(W_i)$ is zero. This implies that $$ V_i = W_i + A, $$ and from this formula the sequence $$ 0\rightarrow A / W_{i}A \rightarrow E / W_{i} \rightarrow E' / W' _{i} \rightarrow 0 $$ is exact. Note also that $A/W_iA = V_i /W_i$. For $j \geq i$ we have $W_jA / W_iA = A / W_iA \cap W_j /W_i$. In particular the filtration induced on $A/W_iA$ by the filtration $W_iA$ is the same as the kernel filtration for the above exact sequence. By induction on the size of the range where the weight filtration lives, we have that $A/W_iA$ is a mixed twistor structure. On the other hand, $W_iA $ is the kernel of the map $W_i \rightarrow W'_i$ of semistable bundles of slope $i$, so $W_iA $ is semistable of slope $i$. Note of course that $W_{i-1}A=0$. Thus for any $j$ we have that $Gr ^{WA}_j(A)$ is semistable of slope $j$ and $A$ is a mixed twistor structure. {\bf Step 3}\,\, We have to show that the image is equal to the coimage. In view of what we have seen with cokernels and kernels, this comes down to showing that if $f$ is an isomorphism on coherent sheaves $E \cong E'$ then $f(W_i)= W'_i$. Again, suppose that this is true for pairs of MTS with weight filtrations concentrated in any smaller range. Let $W_i$ and $W'_i$ be the lowest nonzero pair of levels in the weight filtration. Above we established in this case the formula $$ f^{-1}(W'_i ) = W_i + {\rm ker}(f) $$ but by hypothesis ${\rm ker}(f)=0$ here so $f^{-1}(W'_i )= W_i$. Since $f$ is an isomorphism, $f(W_i)=W'_i$. Now we can look at the isomorphism $E/W_i \cong E' /W'_i$ and by our inductive hypothesis the weight filtrations there are the same. Thus $f$ induces for all $j$ an isomorphism $Gr ^W_j(E)\cong Gr ^{W'}_j(E')$, and this implies that $f(W_j)=W'_j$. We have now completed all of the verifications necessary to show that our category is abelian (cf \cite{Hodge2}). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Remark:} The reader may generalize the other parts (ii)--(v) of Theorem 2.3.5 of \cite{Hodge2}. Note that the correct generalization of part (v) is that the functor ``fiber over $0\in {\bf P} ^1$'' is exact, and this may be further generalized to the statement that the functor ``fiber over $t\in {\bf P} ^1$'' is exact for any point $t$ (in the Hodge case all points are equivalent to $0$, $1$ or $\infty$ by the ${\bf G}_m$ action; the point $\infty$ corresponds to the complex conjugate filtration, so nothing is new in the Hodge case but in the twistor case all points are different). \subnumero{Aside: some generalizations of the above theorem} The above theorem that we get an abelian category depends only on some fairly standard results about semistability. These results hold in much greater generality than just semistable bundles on ${\bf P} ^1$. We could make the following definition. If $X$ is a projective scheme then a {\em mixed sheaf of pure dimension $d$ on $X$} is a sheaf $E$ of pure dimension $d$ with a filtration $W_P$ by subsheaves indexed by polynomials $P\in {\bf Q} [x]$ (increasing for the ordering of polynomials by values at large $x$) such that there are only a finite number of jumps in the filtration and such that $Gr _P^W(E)$ is a semistable sheaf of pure dimension $d$ with normalized Hilbert polynomial $P$ \cite{Moduli}. By the same proof as above, the category of mixed sheaves of pure dimension $d$ is abelian. When $d=dim (X)$ these become torsion-free sheaves. We can generalize further if anybody is interested! Let $\Lambda$ be a sheaf of rings of differential operators such as considered in \cite{Moduli}. Then a {\em mixed $\Lambda$-module} is a $\Lambda$-module $E$ of some pure dimension $d$ with a filtration by sub-$\Lambda$-modules $W_P$ indexed by rational polynomials $P$, with a finite number of jumps, such that $Gr ^W_P(E)$ is a semistable $\Lambda$-module with normalized Hilbert polynomial $P$. There are probably other variants, for example with parabolic structure, etc. \subnumero{Moduli of mixed twistor structures} There is an obvious notion of algebraic family of mixed twistor structures: if $S$ is a scheme then a family of MTS over $S$ is a bundle $E$ on $S\times {\bf P} ^1$ provided with a filtration by strict subbundles $W_iE$ such that in the fiber over each point $s\in S$ this gives a MTS. If we associate to each scheme $S$ the category of families of MTS over $S$, we obtain a stack ${\cal MTS}^{\rm any}$. The superscript denotes the fact that we include any morphisms of MTS, in particular it is not a stack of groupoids. The associated stack of groupoids which we denote by ${\cal MTS}$ is an algebraic stack in the sense of Artin. It is locally of finite type as we shall see below: the pieces corresponding to fixing the dimensions of the weight-graded pieces, are of finite type. There is a natural action of ${\bf G}_m$ and the fixed points (in the stack-theoretic sense) are exactly the ${\bf G}_m$-equivariant mixed twistor structures, i.e. complex mixed Hodge structures. We will attempt briefly to give some idea of what ${\cal MTS}$ looks like. A {\em framed $MTS$} is a mixed twistor structure $(E,W)$ provided with isomorphisms $$ \beta _n: Gr ^W_n(E) \cong {\cal O} _{{\bf P} ^1}(n)^{b_n}. $$ This is equivalent to the data of frames for the vector spaces $Gr ^W_n (E_1)$. A {\em framed family of MTS} over a base scheme $S$ is just a MTS over $S$ provided with isomorphisms $\beta _n$ on ${\bf P} ^1 \times S$ as above. Any automorphism of a framed MTS (or family of framed MTS) fixing the framing, is the identity. To prove this, suppose that $f$ is such an automorphism. Then $f - 1$ induces zero on the associated graded pieces, hence it is zero (because kernel and cokernel commute with taking $Gr ^W$---but this is easy to prove along the lines of the above theorem anyway). We claim that the functor which to any scheme $S$ associates the set of isomorphism classes of framed families of MTS over $S$, is representable by a scheme $Fr {\cal MTS}$ locally of finite type (of finite type if we fix the dimensions of the graded pieces i.e. the $b_n$ above). Let $Fr {\cal MTS}(b_{\cdot})$ denote the part corresponding to MTS where $Gr ^W_n(E)$ has rank $b_n$. The group $$ GL(b_{\cdot}):= \prod _{n} GL(b_n) $$ acts on $Fr {\cal MTS}(b_{\cdot})$ with stack-theoretic quotient ${\cal MTS}(b_{\cdot})$ (the open and closed substack of ${\cal MTS}$ with given dimensions of graded pieces). The stack ${\cal MTS}$ is the disjoint union of the ${\cal MTS}(b_{\cdot})$ so it is locally of finite type. We now prove the representability claim of the previous paragraph. Write $b_{\cdot } = (b_0, \ldots , b_k)$ with the rest being zero. We may proceed by induction on $k$, the case $k=0$ being obvious (the representing scheme is just a point). Thus we may assume that $Y=Fr {\cal MTS}(0, b_1, \ldots , b_k)$ exists. There is a universal family of framed MTS $(E', W')$ on $Y\times {\bf P} ^1$. Suppose $S$ is a scheme and $(E_S, W)$ is a family of framed MTS on $S$ with dimensions $b_0, \ldots , b_n$. Then $E_S/W_1E_S$ is a family of framed MTS with dimensions $0, b_1, \ldots , b_k$ so we obtain a morphism $p:S\rightarrow Y$ and $p^{\ast}(E',W')=(E_S/W_1,W)$ with framings. The data of $(E_S, W)$ is equivalent to the data of $(E_S/W_1,W)$ together with an extension of the bundle $E_S/W_1$ by ${\cal O} _{{\bf P} ^1\times S}(0)^{b_0}$. Thus we have a lifting of $p$ to a morphism from $S$ into the relative $Ext^1$-bundle (which we show to be a bundle two paragraphs below) $$ Ext ^1_{{\bf P} ^1\times Y/Y}(E', {\cal O} _{{\bf P} ^1\times Y}^{b_0})\rightarrow Y. $$ This lifting is unique, and the pullback of the universal extension gives $(E_S, W)$. Thus $Fr{\cal MTS}(b_0, \ldots , b_k)$ is the relative $Ext^1$-bundle refered to above. To sum up the previous paragraph, there are schemes $Fr{\cal MTS}(b_j, \ldots , b_k)$ and universal families of framed MTS $(E^{\rm univ}(b_j, \ldots , b_k), W)$ which are filtered bundles over $$ Fr{\cal MTS}(b_j, \ldots , b_k)\times {\bf P} ^1. $$ There are natural morphisms $$ Fr{\cal MTS}(b_j, \ldots , b_k)\rightarrow Fr{\cal MTS}(b_{j+1}, \ldots , b_k) $$ such that the upper scheme is naturally identified with the relative $Ext^1$-bundle classifying extensions in the ${\bf P} ^1$-direction of $E^{\rm univ}(b_{j+1}, \ldots , b_k)$ by ${\cal O} _{{\bf P} ^1} (j)^{b_j}$. We need to point out that the relative $Ext^1$ referred to above are indeed bundles over the base $Y$. This is by semicontinuity theory using the fact that for any point the corresponding $Ext^0$ is zero. Indeed the fiber over any point $y\in Y$ of the bundle $E'=E^{\rm univ}(b_{j+1}, \ldots , b_k)$ (over $\{ y\} \times {\bf P} ^1$) is an extension of bundles which are stable of slopes $\geq j+1$ so it decomposes into factors of degree $\geq j+1$. Thus there are no homomorphisms to ${\cal O} _{{\bf P} ^1} (j)^{b_j}$ so the $Ext^0=Hom$ is zero. Since we work on ${\bf P} ^1$ there are no other terms and semicontinuity implies that the family of $Ext^1$ spaces all have the same dimension and fit together into a vector bundle which we have referred to as $Ext ^1_{{\bf P} ^1\times Y/Y}(E', {\cal O} _{{\bf P} ^1\times Y}^{b_0})$. From this description, $Fr{\cal MTS}(b_j, \ldots , b_k)$ is obtained as a sequence of vector bundles, eventually over a point. Thus the underlying topological space is contractible. Hence the homotopy type of ${\cal MTS}(b_j, \ldots , b_k)^{\rm top}$ is the same as that of $BGL(b_j, \ldots , b_k)^{\rm top}$. We give a formula for the dimension of the stack ${\cal MTS}(b_j, \ldots , b_k)$ (recall that this means roughly speaking the dimension of the space of orbits minus the dimension of the stabilizer group). The dimension is equal to $$ dim (Fr{\cal MTS}(b_j, \ldots , b_k))- (b_j^2 + \ldots + b_k^2) $$ the latter term being the dimension of $GL(b_j, \ldots , b_k)$. For $dim(Fr{\cal MTS}(b_j, \ldots , b_k))$ we use the above expression as a series of fiber bundles. Note that the dimension of $$ Ext ^1({\cal O} _{{\bf P} ^1}(i), {\cal O} _{{\bf P} ^1}(j))= H^1({\bf P} ^1, {\cal O} _{{\bf P} ^1}(j-i)) $$ is $i-j-1$ when $i>j$, by Riemann-Roch. Thus we have $$ dim (Fr{\cal MTS}(b_j, \ldots , b_k))= dim (Fr{\cal MTS}(b_{j+1}, \ldots , b_k)) + \sum _{i=j+1}^{k} (i-j-1)b_ib_j. $$ Thus $$ dim (Fr{\cal MTS}(b_j, \ldots , b_k))=\sum _{j\leq u < i \leq k} (i-u-1)b_ib_j $$ and the term $dim \, GL(b_j,\ldots , b_k)$ fits in nicely to give $$ dim ({\cal MTS}(b_{\cdot}))=\sum _{u \leq i } (i-u-1)b_ib_j . $$ Note in particular that for most well spaced-out sequences $b_{\cdot}$ the dimension is positive, and in particular the family of orbits is nontrivial. This means that there are ``moduli'' of mixed twistor structures. {\em Questions:} Define a natural $GL(b_{\cdot})$-linearized line bundle on $Fr{\cal MTS}(b_{\cdot})$. What are the stable or semistable points for the action, and what does the GIT quotient space look like? Are the mixed twistor structures which come up in nature (e.g. the mixed Hodge structures) semistable? \numero{Real structures} To complete the circle of definitions and comparisons, we define some notions of {\em real twistor structure} and {\em real mixed twistor structure}. The situation is more complicated than for mixed Hodge structures, because there are many possible antiholomorphic involutions of ${\bf P} ^1$. We will isolate for our purposes two examples. Let $\sigma _{{\bf P} ^1}$ denote the antipodal involution of ${\bf P} ^1$ (it is antilinear, interchanges $0$ and $\infty$ and interchanges $1$ and $-1$). An {\em antipodal real twistor structure} is a bundle $E$ on ${\bf P} ^1$ with antilinear involution $\sigma$ lying over $\sigma _{{\bf P} ^1}$. An {\em antipodal real mixed twistor structure} is an antipodal real twistor structure with filtration by strict subbundles preserved by $\sigma$. In considering polarizations in the next section, we will make use of the following sheaf-theoretic point of view. If $E$ is a vector bundle on ${\bf P} ^1$ considered as a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules then put $$ \sigma ^{\ast}(E) (U):= E(\sigma _{{\bf P} ^1}U). $$ The structure of ${\cal O} _{{\bf P} ^1}(U)$-module is defined by setting, for $e\in E(\sigma _{{\bf P} ^1}U)$ and $a\in {\cal O}_{{\bf P} ^1}(U)$, $a\cdot e := \overline{\sigma _{\ast}(a)}e$. Note that $\sigma ^{\ast}(E)$ is again a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules. However the functor $\sigma ^{\ast}$ is antilinear: the induced map on ${\bf C}$-vector spaces $$ Hom (E, E')\rightarrow Hom (\sigma ^{\ast}(E),\sigma ^{\ast}(E')) $$ is antilinear. Also there is an induced map $$ H^i(E) \rightarrow H^i(\sigma ^{\ast}(E)), $$ again antilinear. Finally the functor $\sigma ^{\ast}$ is an involution (its square is naturally equal to the identity). An involution of $E$ lying over $\sigma _{{\bf P} ^1}$ (the notion defined in the previous paragraph, which concerns the total space of the bundle) can be expressed in sheaf-theoretic language as a morphism $$ f: E \rightarrow \sigma ^{\ast}(E) $$ such that $\sigma ^{\ast}(f)\circ f = 1_E$. Let $\tau _{{\bf P} ^1}$ denote the antiholomorphic involution which preserves the unit circle. A {\em circular real twistor structure} is a bundle $E$ on ${\bf P} ^1$ with antilinear involution $\tau$ lying over $\tau _{{\bf P} ^1}$. A {\em circular real mixed twistor structure} is a circular real twistor structure with filtration by strict subbundles preserved by $\tau$. Note that a sheaf-theoretic discussion similar to that which we have given for $\sigma$, exists for $\tau$ also. The difference between these two notions is that $\sigma _{{\bf P} ^1}$ has no fixed points, whereas $1$ is a fixed point of $\tau _{{\bf P} ^1}$. Thus if $E$ has a circular real structure then $E_1$ is a real vector space, whereas this is not necessarily the case for an antipodal real structure. We now investigate more closely what these structures mean in the pure case. There are two different situations depending on the parity of the weight. Suppose $E$ is an antipodal (resp. circular) real twistor structure which is pure of weight $0$. Then $H^0({\bf P} ^1, E)$ is a complex vector space with antilinear involution induced by $\sigma$ (resp. $\tau$), or in other words it is a real vector space. Note that $$ E = H^0({\bf P} ^1, E)\otimes _{{\bf C} } {\cal O} _{{\bf P} ^1} $$ preserving the involutions. Conversely any real vector space gives rise to an antipodal (resp. circular) real twistor structure which is pure of weight $0$ and these constructions are inverses. In the circular case note that the isomorphism $$ H^0({\bf P} ^1, E) \cong E_1 $$ is compatible with the involution $\tau$. Suppose $E$ is an antipodal real twistor structure which is pure of weight $1$. Then we obtain a structure of quaternionic vector space. Conversely a quaternionic vector space gives an antipodal real twistor structure pure of weight $1$, and these constructions are inverses. These are both well described in \cite{HKLR}. This is the example which lends the name ``twistor''. We start by describing this second direction which is just the classical ``twistor space'' construction of Penrose (cf \cite{Penrose} \cite{Hitchin2} \cite{HKLR}). A quaternionic vector space is a real vector space $V$ with operations $I$, $J$ and $K$ satisfying the relations $$ I^2=J^2=K^2 = -1, \;\;\; IJ=K \;\; \mbox{etc.} $$ For any triple of reals $(x,y,z)$ with $x^2+ y^2 + z^2= 1$ we can define a complex structure $xI + yJ + zK$ on $V$. Combined with the standard complex structure on $S^2= {\bf P} ^1$ this serves to define an almost complex structure on the bundle $E=V\times {\bf P} ^1$. This almost complex structure is integrable, as can be checked by a direct calculation, or else by noting that the bundle $L:={\cal O} _{{\bf P} ^1}(1) \oplus {\cal O} _{{\bf P} ^1}(1)$ has an antipodal real involution wherefrom the invariant sections give a trivialization of the real bundle $L\cong {\bf P} ^1 \times {\bf H}$ such that the points $0$, $1$ and $i$ in ${\bf P} ^1$ give a quaternionic triple of complex structures $I$, $J$ and $K$ on ${\bf H}$ whose resulting twistor almost complex structure recovers $L$ (this is a calculation which needs to be done and for which we refer to \cite{HKLR}). As any quaternionic $V$ is a direct sum of copies of ${\bf H}$ we obtain the integrability of the twistor bundle $E$ (which is a direct sum of copies of $L$), as well as the fact that the twistor bundle is pure of weight $1$ with antipodal real structure. We now describe how an antipodal real twistor structure $E$ pure of weight $1$ comes from a quaternionic vector space. The underlying real vector space $A$ is the space of sections of $E$ which are preserved by the involution $\sigma$. Note that $\sigma$ induces an antilinear involution on $H^0({\bf P}^1, E)$ and the fixed points form a real subspace $$ A= H^0({\bf P}^1, E)^{\sigma } $$ whose dimension is half the complex dimension of $H^0({\bf P}^1, E)$. In turn, the complex dimension of $H^0({\bf P}^1, E)$ is twice the complex rank of $E$ (by purity of weight $1$) so $$ dim _{{\bf R}}(A)= rk (E). $$ This suggests that the evaluation morphism $A\rightarrow E_p$ for any point $p\in {\bf P} ^1$, should be an isomorphism. We prove this: if $e\in E_p$ then $\sigma (e)\in E_{\sigma p}$ and by purity of weight $1$ there is a unique section $f: {\bf P} ^1 \rightarrow E$ such that $f(p)=e$ and $f(\sigma p)= \sigma (e)$. Uniqueness implies that $\sigma ^{\ast}(f)=f$. Uniqueness also gives injectivity of the morphism $A\rightarrow E_p$ and the above construction gives surjectivity. Now for every point $p\in {\bf P} ^1$ we obtain a complex structure $J_p$ on $A$ by pulling back the complex structure from $E_p$. The fact that $e\mapsto \sigma (e)$ is antilinear means that $J_{\sigma p}=-J_p$. Let $I=J_0$, $J=J_1$ and $K=J_{i}$. We claim that these provide a quaternionic triple and that for any other point $q$ the complex structure $J_q$ is that obtained from $(I,J,K)$ by the twistor space construction described above. To prove this claim let $L$ be the standard rank two pure twistor structure of weight $1$ with antipodal involution, corresponding to the quaternionic vector space ${\bf H}$. The bundle $\underline{Hom} (L, E)=L^{\ast}\otimes E$ is pure of weight $0$ and has an antipodal involution $\sigma$, thus it is the same as a vector space with real structure. The real sections are the morphisms $L\rightarrow E$ compatible with the antipodal involutions; let $Hom (L,E)^{\sigma }$ denote this space of real sections. It has real dimension equal to the complex dimension of $Hom (L,E)$ which in turn is the rank of $L^{\ast}\otimes E$ or $2r(E)$. If we fix a section $u$ of $L$ which is preserved by $\sigma$ then evaluation at $u$ gives a morphism $$ \epsilon : Hom (L,E)^{\sigma } \rightarrow A. $$ We claim that this morphism is injective. In fact if a $\sigma$-invariant map $f:L\rightarrow E$ sends $u$ to zero then the bundle $ker(f)\subset L$ is nontrivial, but it is a $\sigma$-invariant bundle, pure of weight $1$ so by the above discussion it corresponds to a nonzero subspace of $H^0({\bf P} ^1, L)^{\sigma}={\bf H}$ which is preserved by all of the complex structures---hence it must be all of ${\bf H}$ so $f=0$. Now injectivity of the map $\epsilon$ implies surjectivity since the real dimensions of the two sides coincide. Thus any section $a\in A$ is in the image of a morphism $L\rightarrow E$. The commutation conditions for the various complex structures on $L$ imply the same conditions for those complex structures of $E$ as applied to $a$. This proves our claim that $(I,J,K)$ form a quaternionic triple. The same proof shows that the other $J_q$ are as they are supposed to be. It is obvious that $(E, \sigma )$ comes from this quaternionic structure by the twistor construction. There is a canonical antipodal and circular real twistor structure of weight two which corresponds to the Tate Hodge structure. We call it the ``Tate twistor'' (!), denoted $T(1)$. Tensorisation and dual leads to Tate twistors $T(n)$ for all $n$. By tensoring with these, any pure antipodal or circular real twistor structure is equivalent to one of weight $0$ or weight $1$, so the above discussions apply: we obtain either a real vector space or a quaternionic one. Note, in passing, the fact that a complex vector space with quaternionic structure must have even dimension. This carries over the Hodge-theory fact that a real Hodge structure of odd weight must have even dimension (on the other hand, there is no such restriction for even weight). The line bundle ${\cal O} (1)$ pure of weight $1$ (not to be confused with $T(1)$ which is of weight two) has a natural circular real structure. On the dual ${\cal O} (-1)$ this is described as follows: over a point $P=[a:b] \in {\bf P} ^1$ the fiber ${\cal O} (-1)_P$ is just the line of $(x,y)$ proportional to $(a,b)$. Thus the total space of the bundle ${\cal O} (-1)$ is just ${\bf A}^2$ (blown up at the origin). We define the antilinear map $$ \tau : {\bf A}^2 \rightarrow {\bf A}^2 $$ by $$ \tau (x,y) = (\overline{y}, \overline{x}). $$ This extends to the blow-up at the origin as an antilinear involution covering the involution $\tau _{{\bf P} ^1}$, thus giving a circular real structure on ${\cal O} (-1)$. The dual is a circular real structure on ${\cal O} (1)$. By tensoring with a power of one of these, any pure twistor structure with circular real structure becomes the same as one of weight zero, so it is just a real vector space. This equivalence is compatible with taking the fiber $E_1$ over $1$ (which has a real structure since $1$ is a fixed point of $\tau _{{\bf P} ^1}$). Thus a circular real structure of any weight is just a real structure on the underlying vector space. \subnumero{Real structures in the Hodge case} Let $H$ be the group of conformal automorphisms of ${\bf P} ^1$ which preserve the set $\{ 0, \infty\}$ and which act on this set trivially when the orientation is preserved, nontrivially when the orientation is changed. The connected component of $H$ is just ${\bf G}_m$ (the group of holomorphic automorphisms fixing $0$ and $\infty$). There are exactly two components, and the other component is equal to ${\bf G}_m \cdot \sigma_{{\bf P} ^1}$ or equally well ${\bf G}_m \cdot \tau _{{\bf P} ^1}$. In particular $H$ may be expressed as the group of automorphisms generated by ${\bf G}_m$ and either one of $\sigma _{{\bf P} ^1}$ or $\tau _{{\bf P} ^1}$. \begin{proposition} An $H$-equivariant mixed twistor structure $(E,W)$ is the same thing as an ${\bf R}$-mixed Hodge structure. \end{proposition} {\em Proof:} Left to the reader. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} This proposition says that mixed twistor structures which are both ${\bf G}_m$-equivariant and have either an antipodal or circular real structure, are real mixed Hodge structures. Since the groups generated by ${\bf G}_m$ and either of $\sigma _{{\bf P} ^1}$ or $\tau _{{\bf P} ^1}$ are the same, we don't see the difference between antipodal and circular real structures in the ${\bf G}_m$-equivariant (Hodge) situation. In the present note we shall not consider the problem of real twistor structures any further (except one small place at the end when we discuss twistor spaces for hypercomplex structures). The reader may imagine how to incorporate real structures into all of the various statements we shall make: essentially all constructions are equivariant for $\sigma$ or $\tau$ if these involutions are given for the input data. \numero{Variations of mixed twistor structure} The problem of dealing with ${\cal C} ^{\infty}$ families of twistor structures can be solved in several different ways. The first would be to note that the category of twistor structures has generators ${\cal O} _{{\bf P}^1}(w)$ and that the morphism spaces $Hom ({\cal O} _{{\bf P} ^1}(w), {\cal O} _{{\bf P} ^1}(w'))= {\bf C} [\lambda , \mu ]_{w'-w}$ are easily described as the spaces of homogeneous polynomials (similarly for the extension groups). One can then ``tensor'' this category with any ring or sheaf of rings, in particular with ${\cal C} ^{\infty}_X$ for a manifold $X$. Another more concrete approach would be to say that a ${\cal C} ^{\infty}$ bundle of twistor structures over a manifold $X$ is a vector bundle $E$ on $X\times {\bf P} ^1$ provided with a complex structure operator $\overline{\partial} _{{\bf P} ^1}$ in the ${\bf P} ^1$-direction. For any $x\in X$ the restriction $E|_{\{ x\} \times {\bf P} ^1}$ becomes a holomorphic vector bundle on ${\bf P} ^1$, algebraic by GAGA. We will sometimes make use of this interpretation. Finally, our main interpretation is sheaf-theoretic: if $X$ is a ${\cal C} ^{\infty}$ manifold, let ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$ be the sheaf of functions on $X\times {\bf P} ^1$ (in the usual topology) which are ${\cal C}^{\infty}$ and which are holomorphic in the ${\bf P} ^1$-direction (in other words the functions annihilated by the operator $\overline{\partial} _{{\bf P} ^1}$ refered to above). A {\em ${\cal C} ^{\infty}$ family of twistor structures on $X$} is now simply a locally free sheaf ${\cal E}$ of ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-modules on $X\times {\bf P} ^1$. We often refer to this simply as a ``bundle'', or sometimes a ``${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-module'', on $X\times {\bf P} ^1$. A {\em strict filtration} of such a bundle is a filtration of ${\cal E}$ by subsheaves such that locally on $X\times {\bf P} ^1$ the filtration comes from a decomposition of ${\cal E}$ into a direct sum of locally free ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-modules. If $W_{\cdot}$ is a strict filtration then the $Gr ^W_i({\cal E} )$ are again locally free ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-modules. If $V$ is a ${\cal C} ^{\infty}$ bundle on $X$ provided with filtrations $F$ and $F'$ then by doing the Rees bundle construction in a ${\cal C} ^{\infty}$ family we obtain a locally free ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-module $\xi (V, F, F')$ on $X\times {\bf P} ^1$. The above constructions are functorial for ${\cal C} ^{\infty}$ morphisms $X'\rightarrow X$, notably for inclusions of points $x\hookrightarrow X$. Thus the above definitions restrict to the definitions from the beginning, over each point $x\in X$. A {\em ${\cal C} ^{\infty}$ family of mixed twistor structures over $X$} is a ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-module ${\cal E}$ with strict filtration $W_{\cdot}$ such that the $Gr ^W_i({\cal E} )$ are pure of weight $i$ (which means that for each $x\in X$ the corresponding holomorphic bundle on $\{ x\} \times {\bf P} ^1$ is pure of weight $i$). Suppose $X$ is a complex manifold. The cotangent bundle $T^{\ast}_{{\bf C}}(X)$, which we often confuse with its sheaf ${\cal A} ^{1}_X$ of sections, has a Hodge structure of weight $1$ and consequently a twistor structure pure of weight $1$. This twistor structure is a ${\cal C} ^{\infty}_X {\cal O} _{{\bf P} ^1}$-module which is isomorphic to $T^{\ast}_{{\bf C}}(X) \otimes _{{\bf C}} {\cal O} _{{\bf P} ^1}(1)$. To be precise, we have a decomposition $$ {\cal A} ^{1}_X = {\cal A} ^{1,0}_X \oplus {\cal A} ^{0,1}_X, $$ and the twistor structure which is $\xi ({\cal A} ^{1}_X, F, F')$ decomposes as $$ \xi ({\cal A} ^{1}_X, F, F')= \xi ({\cal A} ^{1,0}_X, F, F')\oplus \xi ({\cal A} ^{0,1}_X, F, F'), $$ where now the filtrations on the pieces ${\cal A} ^{1,0}_X$ and ${\cal A} ^{0,1}_X$ are trivial (but shifted differently). These trivial filtrations induce isomorphisms $$ \xi ({\cal A} ^{1,0}_X, F, F')\cong {\cal A} ^{1,0}_X\otimes _{{\bf C}} {\cal O} _{{\bf P}^1}(1),\;\;\; \xi ({\cal A} ^{0,1}_X, F, F')\cong {\cal A} ^{0,1}_X\otimes _{{\bf C}} {\cal O} _{{\bf P}^1}(1) $$ and their sum is the isomorphism $$ \phi :\xi ({\cal A} ^{1}_X, F, F')\cong {\cal A} ^{1}_X\otimes _{{\bf C}} {\cal O} _{{\bf P}^1}(1) $$ in question. There is a natural inclusion of bundles $$ \iota : {\cal A} ^{1}_X\otimes _{{\bf C}} {\cal O} _{{\bf P} ^1}\hookrightarrow \xi ({\cal A} ^{1}_X, F, F') $$ and this decomposes as inclusions on each of the pieces. If we fix the standard sections $\lambda$ and $\mu$ of of ${\cal O} (1)$ which vanish respectively at $0$ and $\infty$, we can write $$ \phi \iota (\alpha ^{1,0} + \alpha ^{0,1})= \lambda \alpha ^{1,0} + \mu \alpha ^{0,1}. $$ The inclusion $\iota$ fixes notation for $\xi ({\cal A} ^{1}_X, F, F')$ and the above formula fixes notation for the isomorphism $\phi$. We denote for short $\xi ({\cal A} ^{1}_X, F, F')$ by $\xi {\cal A} ^1_X$. The same construction applies to the whole exterior algebra ${\cal A} ^{\cdot}_X$ to give a graded ${\cal C} ^{\infty}_X{\cal O} _{{\bf P} ^1}$-module $\xi{\cal A} ^{\cdot}_X$ on $X\times {\bf P} ^1$. The $\xi{\cal A} ^i_X$ have twistor structures of weight $i$. There is a natural differential operator $$ {\bf d}: \xi{\cal A} ^0_X \rightarrow \xi{\cal A} ^1_X $$ which is a morphism of twistor structures. It is just the composition of the usual exterior derivative $d$ followed by $\iota$, so it can be written $$ {\bf d} = \lambda \partial + \mu \overline{\partial} $$ where again $\lambda$ and $\mu$ are the sections of ${\cal O} _{{\bf P} ^1}(1)$ which vanish respectively at $0$ and $\infty$. The differential extends to the comples $\xi{\cal A} ^i_X$ have twistor structures of weight $i$ and we obtain a complex $\xi{\cal A} ^{\cdot}_X$ with differential ${\bf d}$. We generalize the definition of \cite{SteenbrinkZucker} to the twistor case. A {\em variation of mixed twistor structure} \footnote{This is the twistor analogue of the complex variations of mixed Hodge structure; one could also talk about antipodal or circular real variations of mixed twistor structure which would be analogues of ${\bf R}$-VMHS, this is left to the reader.} is a ${\cal C} ^{\infty}$ family of mixed twistor structures $({\cal E} , W_{\cdot})$ on $X$ (i.e. a ${\cal C} ^{\infty}_X{\cal O} _{{\bf P} ^1}$-module ${\cal E}$ with strict filtration $W_{\cdot}$ on $X\times {\bf P} ^1$) together with an operator $$ D: {\cal E} \rightarrow {\cal E} \otimes _{{\cal C} ^{\infty}_X{\cal O} _{{\bf P} ^1}}\xi {\cal A} ^1_X $$ respecting the weight filtration, such that the Leibniz rule $$ D(ae) = {\bf d}(a) e + aD(e) $$ is satisfied, and such that $D^2=0$. A {\em pure variation of twistor structure} is the same as above but with the associated graded of the weight filtration concentrated in one degree. Suppose given a variation of mixed twistor structure. For any $\lambda \in {\bf A}^1$ we obtain an {\em underlying $\lambda$-connection} by taking the fiber over $\lambda \in {\bf P} ^1$. Note that the fiber of $\xi {\cal A} ^1_X$ over $\lambda \in {\bf P} ^1$ is naturally identified with ${\cal A} ^1_X$ and via this identification, ${\bf d}$ corresponds to $\lambda \partial + \overline{\partial}$. If $({\cal E} , D)$ is a variation of twistor structure then the bundle ${\cal E} _{\lambda}:= {\cal E} |_{X\times \{ \lambda \} }$ is a ${\cal C} ^{\infty}$ bundle on $X$ with operator $D_{\lambda}$ having symbol $\lambda \partial + \overline{\partial}$ and square zero. If we decompose according to Hodge type $$ D_{\lambda}= D^{1,0}_{\lambda} + D^{0,1}_{\lambda} $$ then $D^{0,1}_{\lambda}$ provides an integrable holomorphic structure for ${\cal E} _{\lambda}$ and $D^{1,0}_{\lambda}$ becomes a holomorphic $\lambda$-connection (i.e. operator satisfying Leibniz for $\lambda d$ and having square zero---cf \cite{SantaCruz}; this definition was made by Deligne in \cite{DeligneLetter}). In particular for $\lambda = 0$ we obtain an underlying Higgs bundle and for $\lambda = 1$ we obtain an underlying flat bundle. The $\lambda$-connections come in a holomorphic family indexed by ${\bf A}^1$, as will be explained in further detail below. On a slightly different note, for every point $x\in X$ we obtain a mixed twistor structure $({\cal E}, W_{\cdot})_x$. This gives a map $X\rightarrow {\cal MTS}$ to the moduli stack of mixed twistor structures; note however that it is only a ${\cal C}^{\infty}$ map and not holomorphic. We call this the {\em classifying map} for the variation of mixed twistor structure $({\cal E}, W_{\cdot})$. \subnumero{Polarizations} We define a notion of {\em polarization} for pure variations of twistor structures. This will be used as a characterization of those variations which correspond to harmonic bundles, cf Lemma \ref{harmonic} below. As we will not use the notion of polarization anywhere else, the reader may prefer to skip directly to Lemma \ref{harmonic} and take as the definition of polarizable variation, one which corresponds to a harmonic bundle by the construction of \ref{harmonic}. The {\em raison d'etre} of the definition we give below of polarization is just to emphasize the analogy with variations of Hodge structure. Suppose $({\cal E} , {\bf d})$ is a pure variation of twistor structure of weight $w$ on $X$. We obtain the locally free sheaf of ${\cal C} ^{\infty}_{X}{\cal O} _{{\bf P} ^1}$-modules $\sigma ^{\ast}({\cal E} )$ on $X\times {\bf P} ^1$ as follows: if $U\subset X\times {\bf P} ^1$ is an open set then $(1\times \sigma )(U)$ is an open subset of $X\times {\bf P} ^1$ (and all open subsets are obtained this way). We set $$ \sigma ^{\ast}({\cal E} )((1\times \sigma )(U)):={\cal E} (U). $$ We give this a structure of ${\cal C} ^{\infty}_{X}{\cal O} _{{\bf P} ^1}$-module as follows: if $e\in {\cal E} (U)$ and $$ a\in {\cal C} ^{\infty}_{X}{\cal O} _{{\bf P} ^1}((1\times \sigma )(U)) $$ then $a$ times $e$ is defined to be equal to $$ \overline{(1\times \sigma )^{\ast}(a)} e\in {\cal E} (U). $$ The complex conjugate is required in order to obtain a section $\overline{(1\times \sigma )^{\ast}(a)}$ which is holomorphic in the ${\bf P} ^1$-direction. This has the effect that the functor ${\cal E} \mapsto \sigma ^{\ast}({\cal E} )$ is a ${\bf C}$-antilinear functor. We give $\sigma ^{\ast}({\cal E} )$ a structure of variation of twistor structure on $\overline{X}$. For this, we define an operator $$ \sigma ^{\ast}(D): \sigma ^{\ast}({\cal E} ) \rightarrow \sigma ^{\ast}({\cal E} ) \otimes _{{\cal C} ^{\infty}_{X}{\cal O} _{{\bf P} ^1}}\xi{\cal A} ^1_X $$ by using a morphism $$ \sigma ^{\ast}(\xi{\cal A} ^1_X)\rightarrow \xi{\cal A} ^1_X. $$ This morphism is defined as follows. Sections of $\xi{\cal A} ^1_X$ may be written in the form $\alpha ' \lambda + \alpha '' \mu$ where $\alpha '$ and $\alpha ''$ are functions on $X\times {\bf P} ^1$ taking values respectively in the $1,0$ and $0,1$ forms on $X$, and holomorphically varying in the ${\bf P} ^1$-direction. Furthermore $\alpha '$ is allowed to have one pole at the zero of $\lambda$ (i.e. along $X\times \{ 0\}$) and $\alpha '' $ is allowed to have one pole at the zero of $\mu$ (i.e. along $X\times \{ \infty \}$). Noting that $\sigma ^{\ast}(\xi{\cal A} ^1_X)(U) = \xi {\cal A} ^1_X ((1\times \sigma )(U))$, we define the morphism $$ \psi :\xi {\cal A} ^1_X ((1\times \sigma )(U))\rightarrow \xi {\cal A} ^1_X(U) $$ by $$ \psi (\alpha ' \lambda + \alpha '' \mu ):= \overline{\sigma ^{\ast}(\alpha '' )}\lambda + \overline{\sigma ^{\ast}(\alpha ' )}\mu . $$ This expression still satisfies the conditios described above concerning the allowable poles of the coefficients, since $\sigma$ interchanges $0$ and $\infty$. Using this morphism we obtain the operator $\sigma ^{\ast}(D)$, making $(\sigma ^{\ast}({\cal E} ), \sigma ^{\ast}(D))$ into a variation of twistor structure. We note in passing that there isn't really any other way to define the antipodal conjugate $(\sigma ^{\ast}({\cal E} ), \sigma ^{\ast}(D))$; this is a contrast with the situation in Hodge theory where one doesn't really see any inner logic requiring the various changes of signs. (In a certain sense we have transfered the question to the simple fact of choosing to work with the antipodal involution, which turns out to contain all of the necessary sign changes.) A {\em polarization} of $({\cal E} , {\bf d})$ is a bilinear pairing $$ P: {\cal E} \otimes \sigma ^{\ast} ({\cal E} ) \rightarrow T(w) $$ which is a morphism of variation of twistor structures, and which is {\em positive hermitian}. This last notion is defined as follows: the form $P$ induces a morphism of trivial bundles $$ {\cal E} (-w) \otimes \sigma ^{\ast}({\cal E} (-w)) \rightarrow T(w) \otimes {\cal O} (-w) \otimes {\cal O} (-w) \cong {\cal O} , $$ hence a morphism on the corresponding vector spaces; but $H^0(\sigma ^{\ast}({\cal E} (-w)) )\cong H^0({\cal E} (-w))$ by an antilinear isomorphism, and we say that $P$ is {\em hermitian} (resp. {\em positive hermitian}) if the resulting antilinear form on the vector space $H^0({\cal E} (-w))$ is hermitian (resp. positive hermitian). Note that the hermitian condition can be expressed in sheaf-theoretic terms without refering to $H^0$ but I don't see a nice way to do this for the positivity condition. We say that the variation $({\cal E} , {\bf d})$ is {\em polarizable} if there exists a polarization. We say that a variation of mixed twistor structure $({\cal E} , W_{\cdot}, {\bf d})$ is {\em graded-polarizable} if the associated-graded pieces $Gr ^W_i({\cal E} )$ are polarizable as pure variations of twistor structure. \begin{lemma} \label{harmonic} If $V$ is a flat bundle with pluriharmonic metric then $V$ underlies a polarizable pure variation of twistor structure. In particular, if $X$ is a compact K\"{a}hler manifold then any irreducible representation of $\pi _1(X)$ corresponds to a flat bundle underlying a pure variation of twistor structure. This variation is unique up to change of weight (by tensoring with a one dimensional twistor structure). Any polarizable variation of pure twistor structure is a direct sum of ones obtained from irreducible representations by this construction. \end{lemma} {\em Proof:} Suppose ${\cal E}$ is a pure variation of twistor structure of weight zero (the other weights are treated by tensoring with ${\cal O} _{{\bf P} ^1}(w)$). Let $p: X\times {\bf P} ^1\rightarrow X$ denote the first projection. Then $E:=p_{\ast}({\cal E} )$ is a locally free ${\cal C}^{\infty}_X$-module on $X$ of rank equal to the rank of ${\cal E}$, and we have $$ {\cal E} = p^{-1}(E)\otimes _{p^{-1}{\cal C}^{\infty}_X} {\cal C}^{\infty}_X{\cal O} _{{\bf P} ^1}. $$ Similarly $p_{\ast} \xi {\cal A} ^0_X = {\cal A} ^0_X$. However, $$ p_{\ast} \xi {\cal A} ^1_X \cong {\cal A} ^1_X \otimes _{{\bf C}} {\bf C} \langle \lambda , \mu \rangle $$ is a bundle of rank twice that of ${\cal A} ^1_X$, since $\xi {\cal A} ^1_X \cong {\cal A} ^1_X \otimes {\cal O} _{{\bf P} ^1} (1)$ as described previously. We have $$ p_{\ast}({\bf d}) = \lambda \partial + \mu \overline{\partial} , $$ and $p_{\ast}(D)$ is an operator $$ p_{\ast}(D): E\rightarrow E \otimes _{{\cal C}^{\infty}_X} {\cal A} ^1_X \otimes _{{\bf C}} {\bf C} \langle \lambda , \mu \rangle $$ whose symbol is $\lambda \partial + \mu \overline{\partial}$. We can write $$ p_{\ast}(D) = \lambda D' + \mu D'' $$ where $D'$ and $D''$ are operators from $E$ to $E\otimes _{{\cal C}^{\infty}_X} {\cal A} ^1_X$. These operators are uniquely determined by the above equation and furthermore it follows that $D'$ has symbol $\partial$ and $D''$ has symbol $\overline{\partial}$. The condition $D^2=0$ implies the integrability conditions $(D')^2=0$, $(D'' ) ^2=0$ and $D'D'' + D'' D' = 0$. Conversely suppose $(E,D', D'')$ is a bundle with operators $D'$ and $D''$ with symbols $\partial$ and $\overline{\partial}$ respectively and satisfying the above integrability conditions. Then setting $$ {\cal E} := p^{-1}(E)\otimes _{p^{-1}{\cal C}^{\infty}_X} {\cal C}^{\infty}_X{\cal O} _{{\bf P} ^1}, $$ we get a ${\cal C} ^{\infty}$ family of pure twistor structures of weight zero. Putting $D:= \lambda D' + \mu D''$ considered as an operator from ${\cal E}$ to ${\cal E} \otimes _{{\cal C}^{\infty}_X{\cal O} _{{\bf P} ^1}} \xi {\cal A} ^1_X$ via the previously-mentionned isomorphism $\xi {\cal A} ^1_X\cong {\cal A} ^1_X \otimes _{{\bf C} } {\cal O} _{{\bf P} ^1}(1)$ (and via consideration of $\lambda $ and $\mu$ as sections of ${\cal O} _{{\bf P} ^1}(1)$), we obtain a variation of pure twistor structure $({\cal E} , D)$. These constructions establish a one to one correspondence between pure variations of twistor structure $({\cal E} , D)$ of weight zero, and triples $(E,D', D'')$ as in the definition of harmonic bundle \cite{HBLS}. We claim that the variation $({\cal E} , D)$ is polarizable if and only if the operators $D'$ and $D''$ are related by a metric (which is thus a harmonic metric) $K$ on $E$ according to the definitions in \cite{HBLS}. The first thing to note is that under the above correspondence (assuming ${\cal E}$ is of weight zero) we have that $p_{\ast}(\sigma ^{\ast}{\cal E} )= \overline{E}$ is the complex conjugate ${\cal C} ^{\infty}$ bundle. We have to calculate $p_{\ast}(\sigma ^{\ast} D)$. Note that we can decompose $D'$ and $D''$ according to Hodge type of forms and write $$ p_{\ast}(D) = \lambda D'_{1,0} + \lambda D' _{0,1} + \mu D''_{1,0} + \mu D'' _{0,1} . $$ If we write this in a form ready to apply the morphism $\psi$ used above to define $\sigma ^{\ast}(D)$ it becomes $$ p_{\ast}(D) = \lambda (D'_{1,0} + \frac{\mu }{\lambda}D''_{1,0} ) + \mu ( \frac{\lambda}{\mu} D' _{0,1} + D'' _{0,1} ). $$ Now $p_{\ast}(\sigma ^{\ast} D)$ is obtained by applying $\sigma ^{\ast}$ to the ${\cal E}$-coefficients, and by applying the operation $\psi$ to the form coefficients and $\lambda $ and $\mu$. We write this as $$ p_{\ast}(\sigma ^{\ast} D)= \lambda \cdot \overline{\sigma ^{\ast} ( \frac{\lambda}{\mu} D' _{0,1} + D'' _{0,1} )} + \mu \cdot \overline{\sigma ^{\ast} (D'_{1,0} + \frac{\mu }{\lambda}D''_{1,0} )}. $$ The pieces $D'_{1,0}$ and so on are invariant under $\sigma ^{\ast}$ because they are constant in the ${\bf P} ^1$-direction. However, note that $$ \overline{\sigma ^{\ast}(\frac{\lambda}{\mu})} =-\frac{\mu}{\lambda} $$ since the antipodal involution is written $t\mapsto -\overline{t}^{-1}$ in terms of the coordinate $t$ on ${\bf A}^1$. Thus we have $$ p_{\ast}(\sigma ^{\ast} D)= \lambda \cdot ( -\frac{\mu}{\lambda} \overline{D}' _{0,1} + \overline{D}'' _{0,1} ) + \mu \cdot (\overline{D}'_{1,0} - \frac{\lambda }{\mu}\overline{D}''_{1,0} ) $$ $$ = \lambda (\overline{D}'' _{0,1}- \overline{D}''_{1,0}) + \mu (\overline{D}'_{1,0}-\overline{D}' _{0,1}). $$ In other words the decomposition $$ p_{\ast}(\sigma ^{\ast} D)= \lambda (\sigma ^{\ast}D)' + \mu (\sigma ^{\ast} D)'' $$ is given by $$ (\sigma ^{\ast}D)' = \overline{D}'' _{0,1}- \overline{D}''_{1,0} $$ and $$ (\sigma ^{\ast}D)'' = \overline{D}'_{1,0}-\overline{D}' _{0,1}. $$ Thus the triple associated to $(\sigma ^{\ast}{\cal E} , \sigma ^{\ast}D)$ is $(\overline{E}, \overline{D}'' _{0,1}- \overline{D}''_{1,0}, \overline{D}'' _{0,1}- \overline{D}''_{1,0})$. A polarization $P$ of ${\cal E}$ corresponds to a morphism $$ K:= p_{\ast}(P): E\otimes \overline{E} \rightarrow {\bf C} $$ which, by hypothesis, is a positive definite hermitian form on $E$. The morphism $K$ intertwines the pair of operators $(D', D'')$ on $E$ with the pair of operators $(overline{D}'' _{0,1}- \overline{D}''_{1,0}, \overline{D}'' _{0,1}- \overline{D}''_{1,0})$ on $\overline{E}$---which exactly says that $K$ is a harmonic metric for the triple $(E,D', D'')$. Conversely by following the above formulas in the other direction, a harmonic metric leads to a polarization which is a morphism of mixed twistor structures. This completes the proof of the claimed correspondence between polarizations and harmonic metrics. This claim implies the lemma, modulo the remark that we can pass from variations $({\cal E} , D)$ pure of weight $w$ to variations pure of weight zero (and back again) by tensorization by the constant rank one twistor structure ${\cal O} _{{\bf P} ^1}(-w)$ (resp. ${\cal O} _{{\bf P} ^1}(w)$). Note that these twistor structures admit polarizations, so tensoring with them preserves the polarization condition. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Remark:} This lemma is the fundamental reason for the utility of the notion of twistor structure: it applies to any irreducible representation of the fundamental group, and not just certain ones (the variations of Hodge structure). \subnumero{Description in terms of $t$-connections on $X$ and $\overline{X}$} On $X\times {\bf A}^1 \subset X \times {\bf P} ^1$ we have an isomorphism $$ \Phi : {\cal A} ^i_X |_{X\times {\bf A}^1 } \cong A^i (X\times {\bf A}^1/ {\bf A}^1). $$ Via this isomorphism we have $$ \Phi \circ {\bf d} = \overline{\partial} + t \partial $$ where $t$ is the coordinate on ${\bf P} ^1$. Using this isomorphism we can identify an operator $D$ with an operator $$ D_0 : {\cal E} \rightarrow {\cal E} \otimes A^1(X\times {\bf A}^1/ {\bf A}^1) $$ satisfying the Leibniz rule $$ D_0 (ae) = aD_0(e) + (\overline{\partial} a + t \partial a) e. $$ Decompose according to type as $D_0 = D_0^{1,0}+ D_0^{0,1}$. Furthermore we can add the operator $\partial $ in the ${\bf A} ^1$-direction to obtain an operator $$ D_0^{0,1} + \overline{\partial} : {\cal E} \rightarrow A^{0,1}(X\times {\bf A}^1, {\cal E} ). $$ This operator satisfies Leibniz' rule for the symbol $\partial_{X\times {\bf A}^1}$ and it is integrable (since $D_0^2= 0$, $\overline{\partial} ^2=0$ and $[\overline{\partial} , D_0]=0$). Thus this operator defines a holomorphic structure for the bundle ${\cal E}|_{X\times {\bf A}^1}$. Denote this holomorphic bundle---or more precisely its sheaf of holomorphic sections---by ${\cal F}$. The remaining part $D^{1,0}_0$ commutes with the holomorphic structure and satisfies Leibniz' rule for the symbol $t\partial$. In other words this operator leads to a morphism of holomorphic sheaves $$ \nabla : {\cal F} \rightarrow {\cal F} \otimes _{{\cal O} _{X\times {\bf A}^1}} \Omega ^1_{X\times {\bf A}^1/{\bf A}^1} $$ satisfying the Leibniz rule $\nabla (af)= a\nabla (f) + d(a)\nabla (f)$ for sections $a$ of ${\cal O} _{X\times {\bf A}^1}$ and $f$ of ${\cal F}$. Furthermore $\nabla ^2=0$. Thus $({\cal F} , \nabla )$ is a {\em $t$-connection on $X\times {\bf A}^1/{\bf A}^1$}. We obtain by symmetry a similar description on the other standard affine open set ${\bf A}^1\subset {\bf P} ^1$ whose coordinate is $t^{-1}$, but with $X$ replaced by $\overline{X}$. We are forced to replace $X$ by $\overline{X}$ because it is the $1,0$ part of the connection on $X$ whose symbol doesn't degenerate. This yields a $t^{-1}$-connection $({\cal F} ', \nabla ')$ on $\overline{X}\times {\bf A}^1/{\bf A}^1$. In order to relate these two descriptions, in other words to understand the glueing between the objects $({\cal F} , \nabla )$ and $({\cal F} ', \nabla ')$ we need a different trivialization of ${\cal A} ^i_X$ over $X\times {\bf G}_m \subset X \times {\bf P} ^1$, $$ \Psi : {\cal A} ^1_X |_{X\times {\bf G}_m} \cong A^1(X\times {\bf G}_m / {\bf G}_m ) $$ such that $$ \Psi \circ {\bf d} = d = \overline{\partial} + \partial . $$ Via this trivialization, ${\cal E} |_{X\times {\bf G}_m}$ becomes a vector bundle with a flat connection relative to the base $X\times {\bf G}_m$ and a commuting operator $\overline{\partial}$ in the horizontal direction. The sheaf of sections annihilated by the connection and the $\overline{\partial}$-operator is a locally free sheaf of $p_1^{-1}({\cal O} _{{\bf G}_m})$-modules on $X^{\rm top}\times {\bf G}_m$, in other words it is a holomorphic family of local systems which we denote ${\cal L} = \{ L_t\} _{t\in {\bf G}_m}$ on $X^{\rm top}$. (If we choose a basepoint $x\in X$ then this is the same as a holomorphic family of representations of $\pi _1(X,x)$ modulo holomorphic changes of basis.) The relationship between $({\cal F} , \nabla )|_{X\times {\bf G}_m}$ and ${\cal L}$ is that for any $t\in {\bf G}_m$, $t^{-1}\nabla$ is a holomorphic flat connection and $L_t$ is its local system of flat sections. By symmetry a similar interpretation holds for $({\cal F} ', \nabla ')$. \numero{Cohomology of smooth compact K\"{a}hler manifolds with VMTS coefficients} Classically, the cohomology of a smooth compact K\"{a}hler manifold with constant coefficients was the first object to be provided with a pure Hodge structure. Applying our meta-theorem to this statement doesn't yield anything other than trading in the Hodge structure for a twistor structure, since the notion of Hodge structure doesn't appear in the hypothesis. One of the main starting points for modern Hodge theory was Deligne's observation that the K\"ahler identities also work with coefficients in a variation of Hodge structure \cite{DeligneLetSerre}. This leads to the theorem that the $k$-th cohomology of a smooth compact K\"ahler manifold with coefficients in a pure variation of Hodge structure of weight $n$, carries a pure Hodge structure of weight $n+k$. This was generalized slightly in \cite{SteenbrinkZucker} to the case of coefficients in a variation of mixed Hodge structure, giving a mixed Hodge structure on the cohomology. We generalize to twistor structures in this section. \begin{theorem} \label{coho1} {\rm (\cite{DeligneLetSerre}, \cite{SteenbrinkZucker})} Suppose $({\cal E} , W_{\cdot} , {\bf d})$ is a $Gr$-polarizable variation of mixed twistor structure on a compact K\"ahler manifold $X$. Then $H^k(X, {\cal E} _1)$ (the cohomology with coefficients in the underlying flat bundle) carries a natural mixed twistor structure. If the variation is pure of weight $n$ then the $k$-th cohomology is pure of weight $n+k$. \end{theorem} \subnumero{Pure coefficients} First we treat the case of pure coefficients. Suppose that $X$ is a smooth compact K\"{a}hler variety and $({\cal E} ,{\bf d})$ is a polarizable variation of pure twistor structure of weight $n$ on $X$. Let $L_1$ denote the underlying flat bundle (it is the fiber over $1$ of the associated family of flat bundles $\{ L_t\}$). We will construct a pure twistor structure of weight $n+k$ on the $k$-th cohomology of $X$ with coefficients in $L_1$. In this case note that there is a harmonic bundle $(E, D', D'')$ and $$ {\cal E}= p_1^{\ast} E \otimes p_2^{\ast}{\cal O} _{{\bf P} ^1}(n), $$ with ${\bf d}= \lambda D' + \mu D'' $. Here the pullbacks $p_1^{\ast}$ and $p_2^{\ast}$ take ${\cal C} ^{\infty}_X$-modules (resp. ${\cal O} _{{\bf P} ^1}$-modules) to ${\bf C} ^{\infty}_X{\cal O} _{{\bf P} ^1}$-modules on $X\times {\bf P} ^1$. Let $\xi {\cal A} ^{\cdot}({\cal E} )$ be the twistor complex of forms with coefficients in ${\cal E}$, a complex of ${\cal C} ^{\infty}_X{\cal O} _{{\bf P} ^1}$-modules on $X\times {\bf P} ^1$ with operator ${\bf d}$. We have $$ \xi {\cal A} ^i({\cal E} ) \cong p_1^{\ast}(A^i(E)) \otimes p_2^{\ast}{\cal O} _{{\bf P} ^1}(n+i). $$ Let $H^i(E) \subset A^i(E)$ denote the subspace of harmonic $i$-forms with coefficients in $E$. We have for any $(a,b)\neq (0,0)\in {\bf C} ^2$ isomorphisms $$ H^i(E) \stackrel{\cong}{\rightarrow} H^i(A^{\cdot}(E), aD' + bD''). $$ This is shown in \cite{HBLS} for $(a,b) = (0,1)$ and $(1,1)$, but the same proof (which is just a direct generalization of the standard utilization of the K\"ahler identities) works in general---one defines the laplacian for $aD' + bD''$ and shows that it is proportional to the laplacian for $D''$. We get back to Deligne's original idea about extending the K\"ahler identities. In particular, the above morphism induces an isomorphism $$ H^i(E) \otimes {\cal O} _{{\bf P} ^1}(n+i) \stackrel{\cong}{\rightarrow} M:= R^ip_{2,\ast}(\xi {\cal A} ^{\cdot}({\cal E} ), {\bf d}). $$ This proves that the higher direct image is a bundle $M$ which is pure of weight $n+i$. We now give a second, analytic construction of $M$, which will be useful later on. However, we refer to the above differential-geometric point of view to prove purity of the weight quotients of our structure. Let ${\cal F}$, ${\cal L}$ and ${\cal F} '$ be the triple associated to ${\cal E}$ consisting of a $t$-connection $({\cal F} , \nabla )$ on $X\times {\bf A}^1$, a family of local systems ${\cal L} = \{ L_t\}$ and a $t^{-1}$-connection $({\cal F} ', \nabla ')$ on the other $\overline{X}\times {\bf A}^1$. Recall that $L_t$ is the flat bundle over $X^{\rm top}$ associated to the holomorphic flat connection $({\cal F} _t, t^{-1} \nabla _t)$ on $X$, and it is also the flat bundle associated to the connection $({\cal F}' _{t^{-1}}, t \nabla _{t^{-1}})$ on $\overline{X}$. Let $\xi \Omega ^{\cdot}_{X}({\cal F} )$ denote the Rees bundle complex of the complex of relative differentials with coefficients in ${\cal F}$, with differential given by the $t$-connection $\nabla$. More precisely we have $$ \xi \Omega ^{\cdot}_{X}({\cal F} )= \xi (\Omega ^{\cdot}_X, F) \otimes _{{\cal O} _{X\times {\bf P} ^1}}{\cal F} , $$ where the filtration $F$ is the usual Hodge filtration (i.e. the ``stupid filtration''). {\em Remark:} there shouldn't be too much confusion between our use of the symbol $\xi$ here for the Rees bundle construction with one filtration, and its use elsewhere for the Rees bundle construction with two filtrations: it depends on whether we are working over the affine line (as is presently the case) or over ${\bf P} ^1$ (as was the case in the differential-geometric construction above). For $t\neq 0$ the hypercohomology of $\xi \Omega ^{\cdot}_{X}({\cal F} )|_{X\times \{ t\} }$ calculates the cohomology of the local system $L_t$. Thus over ${\bf G}_m$ we have $$ {\bf R}^kp_{2,\ast}(\xi \Omega ^{\cdot}_{X}({\cal F} ) |_{X\times {\bf G}_m} \cong {\bf R}^kp_{2,\ast}{\cal L} . $$ Let $\xi \Omega ^{\cdot}_{\overline{X}}({\cal F} ')$ denote the Rees bundle complex of the complex of relative differentials with coefficients in ${\cal F}'$, with differential given by the $t^{-1}$-connection $\nabla '$. Note that this is now taking place over the affine line neighborhood of $\infty$. Again, over ${\bf G}_m$ we have $$ {\bf R}^kp_{2,\ast}(\xi \Omega ^{\cdot}_{\overline{X}} ({\cal F} ') |_{\overline{X}\times {\bf G}_m} \cong {\bf R}^kp_{2,\ast}{\cal L} . $$ We can use these two isomorphisms to glue together ${\bf R}^kp_{2,\ast}(\xi \Omega ^{\cdot}_{X}({\cal F} )$ and ${\bf R}^kp_{2,\ast}(\xi \Omega ^{\cdot}_{\overline{X}} ({\cal F} ')$ to obtain a bundle $M$ over ${\bf P} ^1$. This bundle is the same as that constructed previously; in particular it is pure of weight $n+k$. \subnumero{Mixed coefficients} For this section suppose that $X$ is a smooth compact K\"{a}hler variety and $(E,W,{\bf d})$ is a graded-polarizable variation of mixed twistor structure on $X$. Let $L_1$ denote the underlying flat bundle (it is the fiber over $1$ of the associated family of flat bundles $\{ L_t\}$). We will construct a mixed twistor structure on the $k$-th cohomology of $X$ with coefficients in $L_1$ (fix $k$ for the rest of this section). Let $\xi {\cal A} ^i_X(E)$ be the twistor complex of forms with coefficients in $E$. It is a complex of locally free ${\cal C} ^{\infty}_X{\cal O} _{{\bf P} ^1}$-modules on $X\times {\bf P} ^1$ with operator ${\bf d}$. Define the {\em pre-weight filtration} as $$ W^{\rm pre}_n \xi {\cal A}^i_X(E):= \xi {\cal A}^i_X(W_nE). $$ Let $M$ denote the $k$-th cohomology sheaf on ${\bf P} ^1$ of the filtered complex $$ {\cal M} ^{\cdot} := p_{2,\ast} (\xi {\cal A}^i_X(E), W^{\rm pre}, {\bf d}). $$ Note that $M$ has a filtration which we denote $W^{\rm pre}M$ and we define $$ W_mM := W^{\rm pre}_{m-k}M . $$ We claim that $(M,W)$ is a mixed twistor structure. Given the previous result in the pure case, this is essentially the same as Lemma \ref{degen} below but we give the direct argument here as a warmup. The spectral sequence for the cohomology of the filtered complex ${\cal M} ^{\cdot}$ starts with $$ E^{p,q}_1 = H^{p+q} (Gr ^{W^{\rm pre}}_{-p}(\xi {\cal A}^{\cdot}_X(E)) ) $$ $$ = H^{p+q} (\xi {\cal A} ^{\cdot}_X (Gr ^W_{-p}(E))), $$ which is a pure twistor structure of weight $p+q-p = q$ by the above result for the pure polarized variations of twistor structure $Gr ^W_{-p}(E)$. The differential $d_1:E^{p,q}_1 \rightarrow E^{p+1,q}_1$ is a morphism of twistor structures of the same weight, so the cohomology of the differential $d_1$ are pure twistor structures $E^{p,q}_2$ of weight $q$. Now the differential $d_2: E^{p,q}_2 \rightarrow E^{p+2, q-1}_2$ must vanish since it is a morphism from a semistable bundle of slope $q$ to a semistable bundle of slope $q-1$. Thus $E^{p,q}_3=E^{p,q}_2$ and arguing by induction, $E^{p,q}_2=E^{p,q}_{\infty}$ are pure twistor structures of weight $q$. But $$ E^{p,q}_{\infty} = Gr ^{W^{\rm pre}}_{-p}H^{p+q}(\xi {\cal A}^{\cdot}_X(E)), $$ in particular $$ Gr ^W_m (M)= Gr ^{W^{\rm pre}}_{m-k}H^{k}(\xi {\cal A}^{\cdot}_X(E)) = E^{k-m, m}_{\infty}. $$ This shows that $Gr ^W_m (M)$ is pure of weight $m$ as desired. One can do an analytic construction of $M$ parallel to the analytic construction in the pure case. This would essentially be repeating what we will say below concerning patching, so for now it is left to the reader. We have now proved Theorem \ref{coho1}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Cohomology of open and singular varieties} We generalize \cite{Hodge2}. For now we stay with the easy situation of coefficients which extend across a compactification, and don't attack the problem of coefficients which are a local system only defined on the open variety---since even in the case of variations of Hodge structure, this is a much harder problem. \begin{theorem} \label{TwistorII} {\rm (cf \cite{Hodge2}, 3.2.5)} Suppose $X$ is a compact K\"{a}hler manifold with a divisor $D\subset X$ with normal crossings. Let $U= X-D$. Suppose $({\cal E} , W_{\cdot}, D)$ is a variation of mixed twistor structure on $X$ with underlying flat bundle $V$. Then the cohomology $H^i(U, V|_U)$ carries a natural mixed twistor structure, which can be described as the higher direct image of the de Rham complex of $({\cal E} , W_{\cdot}, D)|_U$ via the projection $U\times {\bf P} ^1 \rightarrow {\bf P} ^1$. This mixed twistor structure is functorial in $U$ (independent of the compactification). \end{theorem} We adopt the following definition: if $X$ is any complex analytic space, then a {\em variation of mixed twistor structure on $X$} is a functorial assignment, for every smooth complex manifold $M$ mapping to $X$, of a variation of mixed twistor structure on $M$. To be precise about this, it means that for every morphism $f: M\rightarrow X$ from a smooth complex manifold we have a VMTS ${\bf E}(f) = ({\cal E} (f), W_{\cdot}(f), {\bf d}(f))$ on $M$; and whenever $a: M\rightarrow M'$ and $f':M'\rightarrow X$ with $f=f'a$ then we have an isomorphism $\epsilon (a): a^{\ast}({\bf E}(f'))\cong {\bf E}(f)$, such that the cocycle condition holds: if $M\stackrel{a}{\rightarrow} M'\stackrel{b}{\rightarrow} M''$ and $f'': M'' \rightarrow X$ with $f'=f''b$ and $f=f''ba$ then the composition $$ a^{\ast} b^{\ast}({\bf E}(f'')) \stackrel{a^{\ast}\epsilon (b)}{\rightarrow} a^{\ast}({\bf E}(f'))\stackrel{\epsilon (a)}{\rightarrow} {\bf E}(f) $$ is equal to $\epsilon (ba)$. We obtain in particular the same type of functorial collection of flat bundles on smooth manifolds mapping to $X$, which gives a flat bundle over any simplicial resolution of singularities as in \cite{Hodge3}. The fundamental group of the topological realization of such a simplicial resolution is the same as that of $X$ so this collection of flat bundles comes from a local system on $X$. We call this the {\em underlying flat bundle} of the variation ${\bf E}$. \begin{theorem} \label{TwistorIII} {\rm (cf \cite{Hodge3} and \cite{SteenbrinkZucker})} If $X$ is a complex projective variety (possibly singular) with a Zariski open subset $U\subset X$ and if $({\cal E} , W_{\cdot}, D)$ is a variation of mixed twistor structure on $X$ with underlying flat bundle $V$, then the cohomology $H^i(U, V|_U)$ carries a natural mixed twistor structure which is functorial in $U$. \end{theorem} {\em Remark:} The same yoga of weights as in \cite{Hodge3}, Theor\`eme 8.2.4, depending on the openness and singularity of the variety, holds here. \subnumero{Mixed twistor complexes} For the proofs of theorems \ref{TwistorII} and \ref{TwistorIII} (which we do together) we proceed exactly as in \cite{Hodge2} and \cite{Hodge3}. In fact we look at what is really going on in \cite{Hodge3} and plug in the stuff from \cite{Hodge2}. Start with the following definitions. A {\em mixed twistor complex} is a filtered complex $(M^{\cdot}, W^{\rm pre}_{\cdot})$ of sheaves of ${\cal O} _{{\bf P} ^1}$-modules on ${\bf P} ^1$ such that $$ \underline{H}^i(Gr ^{W^{\rm pre}}_n(M^{\cdot})) $$ (the cohomology sheaf) is a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules of finite rank, pure of weight $n+i$. This shift of weights is the reason we call this the {\em pre-weight filtration} and use the superfix $W^{\rm pre}$. \begin{lemma} \label{degen} {\rm (\cite{Hodge3} Scholie 8.1.9)} Suppose $(M^{\cdot}, W^{\rm pre}_{\cdot})$ is a mixed twistor complex. Then the spectral sequence for a filtered complex which calculates the cohomology sheaves $\underline{H}^i(M^{\cdot})$ degenerates at $E_3$ (i.e. $d_r=0$ for $r\geq 3$). The cohomology sheaves are locally free sheaves of ${\cal O} _{{\bf P} ^1}$-modules and the filtration induced by the pre-weight filtration, when shifted to $$ W_n\underline{H}^i(M^{\cdot}):= W^{\rm pre}_{n-i}\underline{H}^i(M^{\cdot}) $$ is the weight filtration for a mixed twistor structure on $\underline{H}^i(M^{\cdot})$. \end{lemma} {\em Proof:} The spectral sequence in question starts with $$ E^{p,q}_1 = Gr ^{W^{\rm pre}}_{-p}(M^{p+q}), $$ with differential $d_1: E_1^{p,q}\rightarrow E_1^{p,q+1}$ being the associated-graded of the differential of $M^{\cdot}$. The term $E^{p,q}_2$ is the cohomology of the differential $d_1$ and has differential $d_2: E^{p,q}_2\rightarrow E^{p+1,q}_2$. The cohomology of $d_1$ is just the cohomology sheaf which appears in the definition of mixed twistor complex above (with $i=p+q$ and $n=-p$); thus, by definition $E^{p,q}_2$ is a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules, pure of weight $q$. The differential $d_2$ is a morphism of semistable bundles of the same slope $q$, so $E^{p,q}_3$, i.e. the cohomology of $d_2$, is again a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules, pure of weight $q$. Now for $r\geq 3$ we have $d_r: E^{p,q}_r\rightarrow E^{p+r-1, q+2-r}$ which is a morphism between bundles which are, argueing inductively on $r$, pure of weights $q$ and $q+2-r < q$ respectively; thus $d_r=0$ and the spectral sequence degenerates. The limit of the spectral sequence is $$ E^{p,q}_3 = Gr ^{W^{\rm pre}}_{-p}(\underline{H}^{p+q}(M^{\cdot})), $$ and the fact that the associated graded pieces are locally free implies that $\underline{H}^i(M^{\cdot})$ is locally free and its induced filtration is by strict subbundles. The above $E^{p,q}_3$ being pure of weight $q = (p+q) + (-p)$ we can rewrite as saying that $W^{\rm pre}_{n-i}\underline{H}^i(M^{\cdot})$ is pure of weight $i+ (n-i)=n$. This shows that the weight filtration as shifted in the statement of the lemma gives a mixed twistor structure. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} Our mixed twistor complexes will generally come from a more global situation. Suppose $Y$ is a topological space mapping to ${\bf P} ^1$ with map denoted $p$. A {\em mixed twistor complex on $Y/{\bf P} ^1$} is a filtered complex of sheaves $(N^{\cdot}, W^{\rm pre})$ of $p^{-1}({\cal O} _{{\bf P} ^1})$-modules on $Y$ such that $$ {\bf R}^i p_{\ast} (Gr ^{W^{\rm pre}}_n(N^{\cdot})) $$ is a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules, pure of weight $n+i$. In this case, let ${\rm Go}(N^{\cdot})$ be the canonical Godement resolution (or any other canonical flasque resolution), commuting with subquotients. Set $$ M^{\cdot} = p_{\ast}{\rm Go}(N^{\cdot}) $$ with filtration $W^{\rm pre}M = p_{\ast}(W^{\rm pre}N^{\cdot})$. The direct images of these resolutions calculate the higher direct images of the complexes in question. Since $Gr ^{{\rm Go}(W^{\em pre})}_n({\rm Go} (N^{\cdot})= {\rm Go} (Gr ^{W^{\rm pre}}_n(N^{\cdot})$ we have $$ \underline{H}^i(Gr^{W^{\rm pre}M}_n(M^{\cdot})) = {\bf R}^i p_{\ast} (Gr ^{W^{\rm pre}}_n(N^{\cdot})) $$ which is, by hypothesis, pure of weight $n+i$. Thus $(M^{\cdot}, W^{\rm pre}M)$ is a mixed twistor complex on ${\bf P} ^1$. Of course in the case $Y={\bf P} ^1$ we recover the usual notion of mixed twistor complex (noting however that in the passage from $N$ to $M$ in this case, the complex will be replaced by its Godement resolution). \subnumero{Patching} The complexes constructed in \cite{Hodge2} for calculating the cohomology of open varieties are closely tied to the holomorphic structure of $X$. This construction will only work over the standard ${\bf A}^1$---neighborhood of $0$. A similar construction will work for the holomorphic structure $\overline{X}$ for the other standard neighborhood ${\bf A}^1$ of $\infty$. We need to patch these two together, and here is where the relationship with the Leray spectral sequence (pointed out by N. Katz, according to \cite{Hodge2}) comes in. Over ${\bf G}_m$ this allows us to relate either of the previous two constructions with a topological construction for flat bundles. We return to the trichotomy $({\cal F} , \nabla )$---$\{ L_t\}$---$({\cal F} ', \nabla ')$ that was discussed in \S 3. For each of these regimes we obtain a complex of sheaves on the appropriate open subset of ${\bf P} ^1$. We then need to patch them together to obtain a mixed twistor complex. This is the analogue in the present situation of the two components, holomorphic and topological, in Deligne's definition of mixed Hodge complex \cite{Hodge3} (our third component $({\cal F} ', \nabla ')$ being, in that case, the complex conjugate of the first component $({\cal F} , \nabla )$, because Deligne considered real objects rather than complex ones as we do here). So we discuss in general how to patch together complexes of sheaves to obtain mixed twistor complexes. The basic situation we will treat is the following one: suppose we have filtered complexes $M^{\cdot}$ and $N^{\cdot}$ of sheaves of ${\cal O}$-modules (with the pre-weight filtrations denoted $W^{\rm pre}M$ etc.), respectively over the standard neighborhoods ${\bf A}^1$ of $0$ and $\infty$ in ${\bf P} ^1$. Suppose we have a filtered complex $P^{\cdot}$ of sheaves of ${\cal O}$-modules on ${\bf G}_m$ (the intersection of the two affine lines). Finally suppose that we have filtered quasiisomorphisms $$ M^{\cdot}|_{{\bf G}_m} \leftarrow P^{\cdot} \rightarrow N^{\cdot} |_{{\bf G}_m} . $$ We construct a filtered complex of sheaves of ${\cal O}$-modules on ${\bf P} ^1$ denoted $Patch (M\leftarrow P \rightarrow N)$ (or just $Patch$ for short) in the following way. \newline (1)\, Let $i$ denote one of the three inclusions ${\bf A}^1\hookrightarrow {\bf P} ^1$ (twice) or ${\bf G}_m \hookrightarrow {\bf P} ^1$. Let $Ri_{\rm ex}$ denote a fixed functorial choice of right derived functor (a real functor from complexes to complexes, not just a functor on the derived category) of some extension functor for the inclusion $i$. For example this could be $i_{\ast} \circ {\rm Go}$, the composition of direct image with the Godement resolution. We require that there be fixed a functorial quasiisomorphism $$ {\cal F} ^{\cdot}\rightarrow i^{\ast} Ri_{\rm ex}({\cal F} ^{\cdot}) $$ (which is the case notably in our example). \newline (2)\, Let $M^{\cdot} _{\rm ex}:= Ri_{\rm ex}M^{\cdot}$ be this extension functor applied to the filtered complex $M^{\cdot}$ (resp. $P^{\cdot}_{\rm ex}:=Ri_{\rm ex}P^{\cdot}$, $N^{\cdot}_{\rm ex} :=Ri_{\rm ex}N^{\cdot}$ with $i$ being the appropriate inclusions). They are again filtered complexes of sheaves of ${\cal O} $-modules. \newline (3)\, In general if $f:A^{\cdot} \rightarrow B^{\cdot}$ is a map of filtered complexes (or complexes of sheaves) let $Cone (A\rightarrow B)$ denote the filtered complex (or complex of sheaves) defined as $$ Cone(A\rightarrow B)^k := A^{k+1} \oplus B^k $$ with differential equal to $d_A + d_B + f$. \newline (4)\, Now note that we have a morphism of filtered complexes of sheaves of ${\cal O}$-modules $$ P^{\cdot}_{\rm ex} \rightarrow M^{\cdot}_{\rm ex}\oplus N^{\cdot}_{\rm ex} $$ (put in a minus sign in one of the factors). We define $$ Patch( M \leftarrow P \rightarrow N):= Cone (P^{\cdot}_{\rm ex} \rightarrow M^{\cdot}_{\rm ex}\oplus N^{\cdot}_{\rm ex}). $$ We claim that $$ \underline{H}^i(Gr^{W^{\rm pre}}_n Patch( M \leftarrow P \rightarrow N)) $$ is the sheaf of ${\cal O}$-modules on ${\bf P} ^1$ obtained by glueing together $\underline{H}^i(Gr^{W^{\rm pre}}_nM^{\cdot})$ over the first neighborhood ${\bf A}^1$, with $\underline{H}^i(Gr^{W^{\rm pre}}_nN^{\cdot})$ over the second neighborhood ${\bf A}^1$, via the isomorphisms of cohomology sheaves induced by the filtered quasiisomorphisms $$ M^{\cdot}|_{{\bf G}_m} \leftarrow P^{\cdot} \rightarrow N^{\cdot} |_{{\bf G}_m} . $$ Suppose $U$ is a connected open set contained in the first affine neighborhood. Then $U\cap {\bf A}^1$ (intersection with the other affine neighborhood) is equal to $U\cap {\bf G}_m$ and $P|_{U\cap {\bf G}_m}\rightarrow N|_{U\cap {\bf G}_m}$ is a filtered quasiisomorphism. Thus $$ P^{\cdot}_{\rm ex}|_U\rightarrow N^{\cdot}_{\rm ex}|_U $$ is a filtered quasiisomorphism. In general if $A \rightarrow B$ is a filtered quasiisomorphism of complexes of sheaves and $A\rightarrow C$ is any morphism then $B \rightarrow Cone (A \rightarrow B\oplus C)$ is a filtered quasiisomorphism. In our case this says that the composition $$ M^{\cdot} |_U \rightarrow M^{\cdot}_{\rm ex} |_U \rightarrow Patch (M \leftarrow P \rightarrow N) |_U $$ is a filtered quasiisomorphism (the first arrow is the filtered quasiisomorphism coming from our assumption in (2) above). Hence the cohomology sheaf of the associated graded of $Patch$ restricted to $U$ is naturally isomorphic to that of $M$. Similarly if $U'$ is a neighborhood contained in the second ${\bf A}^1$ then the cohomology sheaf is naturally isomorphic to that of $N$. Finally, if $U$ is contained in ${\bf G}_m$ then the composition of these two isomorphisms is equal to that which comes from the diagram $M\leftarrow P\rightarrow N$ (the minus sign we put in before garanties that we are not missing a minus sign here!). Thus in order to construct a mixed twistor complex $Patch$ we just need to have all of the above data with the property that the cohomology sheaves of the associated-graded for the pre-weight filtration, when glued together, become pure of the right weights. We can make a similar construction in any more complicated situation of a chain of quasiisomorphisms. For example the actual situation we will need to consider is when we have the sequence of filtered quasiisomorphisms of filtered complexes over ${\bf G}_m$, $$ M |_{{\bf G}_m} \leftarrow P \rightarrow Q \leftarrow R \rightarrow N|_{{\bf G}_m} . $$ This can be replaced by the sequence $$ M |_{{\bf G}_m} \leftarrow Cone '(P \oplus R \rightarrow Q) \rightarrow N|_{{\bf G}_m} $$ where $Cone '$ is the cone but shifted in such a way that it is normalized for the first variable (the cone we defined above was normalized for the second variable). Now we can directly apply the previous discussion and define $$ Patch (M,P,Q,R,N):= Patch (M \leftarrow Cone '(P \oplus R \rightarrow Q) \rightarrow N). $$ One amusing point to notice is that in this case there are two stray minus signs which cancel out. This is probably quite lucky if one wants to look at the real situation, where a single minus sign might be very painful. Finally, the whole thing goes through equally well in the relative situation. If $p:Y\rightarrow {\bf P} ^1$ is a morphism of topological spaces and if we have filtered complexes of sheaves $M$ and $N$ on the two $p^{-1}({\bf A}^1)$ and a filtered complex of sheaves $P$ on $p^{-1}({\bf G}_m )$ with filtered quasiisomorphisms $$ M|_{p^{-1}({\bf G}_m )} \leftarrow P \rightarrow N|_{p^{-1}({\bf G}_m )} $$ then we obtain a filtered complex $Patch (M \leftarrow P\rightarrow N)$ of sheaves on $Y\times {\bf P} ^1$, which has the effect of patching together the cohomology sheaves of the complexes $M$ and $N$ along $P$. We can then take the direct image down to ${\bf P} ^1$ as was described previously. On the other hand, we could also take the direct images of $M$, $N$ and $P$ down to ${\bf P} ^1$ and then patch them together. The answers in these two cases are not quite the same but are quasiisomorphic. For simplicity we choose the route of first taking the direct image then patching. Thus, to be concrete, we obtain a filtered complex of sheaves $$ Patch ( R^ip_{\ast}(M,P,N)):= Patch \left( R^ip_{\ast}(M)\leftarrow R^ip_{\ast}(P) \rightarrow R^ip_{\ast}(N)\right) $$ on ${\bf P} ^1$, patching together the higher direct images (the higher direct images being calculated by canonical flasque resolutions, for example). There is a spectral sequence $$ R^i p_{\ast}Gr ^{W^{\rm pre}}_n(M) \Rightarrow Gr ^{W^{\rm pre}}_nR^ip_{\ast}M $$ and the same for $P$ and $N$, each of these being a spectral sequence of sheaves over the appropriate open subsets of ${\bf P} ^1$. Note that filtered quasiisomorphisms induce isomorphisms of spectral sequences. Thus we may patch together the spectral sequences to obtain a spectral sequence (of sheaves on ${\bf P} ^1$) which we denote $$ Patch (R^i p_{\ast}Gr ^{W^{\rm pre}}_n(M,P,N)) \Rightarrow Gr ^{W^{\rm pre}}_nPatch ( R^ip_{\ast}(M,P,N)). $$ The terminology on the right was defined above and similarly the notation on the left is defined as $$ Patch (R^i p_{\ast}Gr ^{W^{\rm pre}}_n(M,P,N)):= $$ $$ Patch \left( R^i p_{\ast}Gr ^{W^{\rm pre}}_n(M)\leftarrow R^i p_{\ast}Gr ^{W^{\rm pre}}_n(P)\rightarrow R^i p_{\ast}Gr ^{W^{\rm pre}}_n(N) \right) . $$ If the beginning term of the spectral sequence is a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules pure of weight $n+i$ then the spectral sequence degenerates after the next term, and the answer is again pure of weight $n+i$ (cf the argument of Lemma \ref{degen}). Thus in order to obtain a mixed twistor complex by patching, it suffices that the resulting patch of the higher direct images of the associated graded, $$ Patch \, \left( R^i p_{\ast}Gr ^{W^{\rm pre}}_n(M), R^i p_{\ast}Gr ^{W^{\rm pre}}_n(P),R^i p_{\ast}Gr ^{W^{\rm pre}}_n(N) \right) $$ be a locally free sheaf of ${\cal O} _{{\bf P} ^1}$-modules, pure of weight $n+i$. A similar statement works in the situation of patching $M,P,Q,R,N$ (which is the case we will use). \subnumero{Logarithmic complexes for mixed twistor structures} Suppose now that $Z$ is a smooth variety with a normal crossings divisor $D$, and that $U= Z-D$. Suppose $E$ is a variation of mixed twistor structure on $Z$, which we will now look at in terms of the three weighted objects ${\cal F}$ (with $t$-connection $\nabla$), $L_t$ for $t\in {\bf G}_m$, and ${\cal G}$ which was denoted ${\cal F} '$ previously. These lie respectively over ${\bf A}^1$, ${\bf G}_m$ and the other ${\bf A}^1$. We obtain the following complexes on subsets of the topological space $Y:= Z^{\rm top}\times {\bf P} ^1$. First, $M^{\cdot}$ is the complex over ${\bf A}^1$ of holomorphic logarithmic differentials with coefficients in ${\cal F}$, $$ M^i = \xi \Omega ^i_Z(\log D) \otimes _{{\cal O}_{X\times {\bf A} ^1} } {\cal F} $$ with differential coming from the $t$-connection $\nabla$ and weight filtration combining the weight filtration of \cite{Hodge2} with the filtration of ${\cal F}$. Here as below, the symbol $\xi$ refers to the operation described \cite{NAHT}, \cite{SantaCruz}, with respect to the Hodge filtration of $\Omega ^i_X(\log D)$ (which in this case is the ``stupid'' filtration). Second, $N^{\cdot}$ is the same thing for ${\cal G}$ on $\overline{Z}$ transported to $Y$ via the isomorphism $\overline{Z}^{\rm top} \cong Y$. Finally, we define several filtered complexes $P$, $Q$ and $R$ on the inverse image of ${\bf G}_m$. Recall that over $t\in {\bf G}_m$ we have the local system $L_t$ which is the local system of flat sections of $({\cal F} , t^{-1} \nabla )$ and similarly for ${\cal G}$. Let ${\cal A}^{\cdot}_U$ denote the sheaf of ${\cal C} ^{\infty}$ differential forms on $U$ and let $j: U\rightarrow Z$ denote the inclusion. We have filtered quasiisomorphisms (cf \cite{Hodge2} p. 33, the map called $\beta$ plus a discussion analogous to that of the top of page 33) $$ (\xi \Omega ^{\cdot}_Z(\log D) \otimes _{{\cal O}_{X\times {\bf G}_m }} {\cal F} , \tau ) \rightarrow (j_{\ast}\xi \Omega ^{\cdot}_U\otimes _{{\cal O}_{X\times {\bf G}_m }} {\cal F}, \tau ) $$ $$ \rightarrow (\xi {\cal A}^{\cdot}_U\otimes _{{\cal O} _{X\times {\bf G}_m }}{\cal F} , \tau ) = (\xi {\cal A}^{\cdot} _U \otimes _{p^{-1}{\cal O} _{{\bf G}_m }} {\cal L} , \tau ). $$ At the end note that the family ${\cal L}$ is considered as a locally free sheaf of $p^{-1}{\cal O} _{{\bf G}_m}$-modules on $Y\times _{{\bf P} ^1} {\bf G}_m$. Here the filtrations $\tau$ are the ``intelligent'' truncations of the complexes of differentials, tensored with the weight filtrations of ${\cal F}$ or ${\cal L}$. On the other hand, we have (again see \cite{Hodge2} p. 33, the map called $\alpha$) the filtered quasiisomorphism over ${\bf G}_m$, $$ (\xi \Omega ^{\cdot}_Z(\log D) \otimes _{{\cal O}_{X\times {\bf G}_m}} {\cal F} , W^{\rm pre} ) \leftarrow (\xi \Omega ^{\cdot}_Z(\log D) \otimes _{{\cal O}_{X\times {\bf G}_m}} {\cal F} , \tau ) . $$ In view of this we put $$ P^{\cdot} := (\xi \Omega ^{\cdot}_Z(\log D) \otimes _{{\cal O}_{X\times {\bf G}_m}} {\cal F} , \tau ) $$ over ${\bf G}_m$ and $$ Q^{\cdot} := (\xi {\cal A}^{\cdot} _U \otimes _{p^{-1}{\cal O} _{{\bf G}_m}} {\cal L} , \tau ) $$ again over ${\bf G}_m$. Note that $Q$ now makes no reference to the complex structure so it is the same with respect to $Z$ or $\overline{Z}$. Let $R$ be the same thing as $P$ but on $\overline{Z}$ and using ${\cal G}$. We obtain the diagram of filtered quasiisomorphisms of complexes on $Y\times _{{\bf P} ^1} {\bf G}_m$, $$ M |_{{\bf G}_m} \leftarrow P \rightarrow Q \leftarrow R \rightarrow N|_{{\bf G}_m} . $$ Put $$ MTC(E):= Patch (Rp_{\ast}M,Rp_{\ast}P,Rp_{\ast}Q,Rp_{\ast}R,Rp_{\ast}N) $$ with these and the previous notations. We claim that this is a mixed twistor complex. This claim implies that the cohomology sheaves of $MTC(E)$ have natural mixed twistor structures. The fiber over $1$ is the same as the hypercohomology on $Z^{\rm top}$ of the complex $Q$, which is just the cohomology of $U$ with coefficients in the local system $L_1 |_U$. This will therefore prove Theorem \ref{TwistorII}. \subnumero{Proof of claim} An {\em extension} of filtered complexes of sheaves is a short exact sequence of complexes inducing short exact sequences on all levels of the filtrations. This induces an extension of the associated-graded complexes. An extension of VMTS $$ 0 \rightarrow E ' \rightarrow E \rightarrow E'' \rightarrow 0 $$ induces an extension of filtered complexes $$ 0 \rightarrow MTC(E ') \rightarrow MTC(E) \rightarrow MTC(E'') \rightarrow 0, $$ hence an extension $$ 0 \rightarrow Gr _n^{W^{\rm pre}}MTC(E ') \rightarrow Gr _n^{W^{\rm pre}} MTC(E) \rightarrow Gr _n^{W^{\rm pre}}MTC(E'') \rightarrow 0. $$ From this we get a long exact sequence of cohomology sheaves. Suppose we know that $$ \underline{H}^iGr _n^{W^{\rm pre}}MTC(E ') \;\; \mbox{and} \;\; \underline{H}^iGr _n^{W^{\rm pre}}MTC(E '') $$ are pure of weight $n+i$, and suppose we know also that the connecting maps in the long exact sequence are zero. Then we can conclude that $\underline{H}^iGr _n^{W^{\rm pre}}MTC(E )$ are pure of weight $n+i$. This last phrase may be restated as saying that if we know the connecting maps in the long exact sequence of cohomology of associated graded pieces are zero, then $MTC(E ')$ and $MTC(E '')$ being mixed twistor complexes implies that $MTC(E )$ is a mixed twistor complex. We prove that for any VMTS $E$, $MTC(E)$ is a mixed twistor complex, by induction on the size of the interval containing the weight-graded pieces of $E$. If the size of this interval is $1$ then $E$ is pure and we will treat this case below. Suppose that $E$ is in an interval of size $n$ and that we know the result for any VMTS in an interval of size $<n$. Let $W_iE$ be the lowest nonzero piece of the weight filtration. Apply the preceeding discussion to the extension of VMTS $$ 0\rightarrow W_iE \rightarrow E \rightarrow E/W_iE \rightarrow 0. $$ By our inductive hypothesis $MTC(W_iE)$ and $MTC(E/W_iE)$ are mixed twistor complexes. In order to be able to conclude that $MTC(E)$ is a mixed twistor complex, we just have to know that the connecting maps in the long exact sequence calculating $H^jGr ^{W^{\rm pre}}_nMTC(E)$, are zero. But these connecting maps are obtained by a construction of the form, take an element with coefficients in $E'' := E/W_iE$ and lift it to an element with coefficients in $E$, then take the coboundary which will have coefficients in $E' := W_iE$. The preweight of our element (which lies in some complex of forms or Godement resolution) is the sum of the weight of the coefficient in $E$ plus the weights of the other stuff. But the weight of the coefficient in $E$ is strictly decreased by the operation described above, since the weights of $E'$ are strictly lower than the weights of $E''$. Thus, this operation is zero on the associated graded for the preweight filtration, in other words the connecting map in question is zero. This completes the proof of the claim modulo the case where $E$ is pure, which we now treat. It suffices to consider the case where $E$ is pure of weight zero. By the discussion of the previous subsection (and with the same notations as there), it suffices to check that the patching of $R^ip_{\ast}Gr^{W^{\rm pre}}_n({\bf x} )$ for ${\bf x} = M,P,Q,R,N$, be pure of weight $n+i$. Let $D^{(k)}$ denote the disjoint union of the intersections of $k$ smooth components of $D$. We assume that the irreducible components of $D$ are smooth so that the sheaves denoted $\varepsilon$ in \cite{Hodge2} are trivial. The {\em residue map} gives an isomorphism $$ res: Gr^{W^{\rm pre}}_n(M^{i}) \cong \xi \Omega ^{i-n}_{D^{(n)}}\otimes {\cal F} |_{D^{(n)}} \otimes {\cal O} _{{\bf A}^1}(n\cdot 0) $$ with differential induced by the $t$-connection $\nabla$. The term ${\cal O} _{{\bf A}^1}(n\cdot 0)$ (which means the sheaf of functions with poles of order $n$ at the origin) comes from the fact that the residue map contracts out $n$ things of the form $dz_i /z_i$, which provide twists due to the construction $\xi$. Over $t\neq 0$ (i.e. over ${\bf G}_m \subset {\bf P} ^1$) this complex is a resolution of the local system $L_t |_{D^{(n)}}$. Thus we have $$ R^ip_{\ast}Gr^{W^{\rm pre}}_n(M^{\cdot}) = R^{i-n}p_{\ast}(\xi \Omega ^{\cdot}_{D^{(n)}}\otimes {\cal F} |_{D^{(n)}}) \otimes {\cal O} _{{\bf A}^1}(n\cdot 0), $$ and this restricts (via the natural isomorphism) to $R^{i-n}p_{\ast}({\cal L} |_{D^{(n)}})$ over ${\bf G}_m$. The direct images of the associated-graded pieces of the complexes $P,Q,R$ all give the same answer $R^{i-n}p_{\ast}({\cal L} |_{D^{(n)}})$. The same holds for $R^{i-n}p_{\ast}Gr^{W^{\rm pre}}_n(N^{\cdot})$ and again this restricts to $R^{i-n}p_{\ast}({\cal L} |_{D^{(n)}})$ (which doesn't depend on the complex structure of $D^{(n)}$). The patching of these direct images of associated-graded pieces for $M,P,Q,R,N$ yields the filtered coherent sheaf constructed in Theorem \ref{coho1}, for our VMTS $E$ pulled back to $D^{(n)}$, tensored by ${\cal O} _{{\bf P} ^1}(2n)$ because of term ${\cal O} _{{\bf A}^1}(n\cdot 0)$ and the similar term in the neighborhood of $\infty$. By the result of \S 4 (recall that here we are treating the case where $E$ is pure of weight $0$) this patching is pure of weight $(i-n)+ 2n =n+i$. This completes the proof of the claim (and hence the proof of Theorem \ref{TwistorII}). \subnumero{The simplicial situation} In order to prove Theorem \ref{TwistorIII} we need to consider a simplicial situation. Any simplicial scheme $X_{\cdot}$ (of finite type) can be replaced (via the method of \cite{Hodge3}) by a simplicial collection of projective smooth schemes $Z_{\cdot}$ containing normal crossing divisors $D_{\cdot}$ such that the maps are compatible with the normal crossing divisors. The topological type of the original simplicial scheme is recovered by taking the topological type of the simplicial open subschemes $U_{\cdot} = X_{\cdot} - D_{\cdot}$ (complement of the divisor with normal crossings). For each $k$ we have constructed a mixed twistor complex $(M^{\cdot}_k, W^{\rm pre})$ calculating the cohomology of $U_k$ using the normal crossings compactification $(X_k, D_k)$. This is functorial (in a contravariant way): the face maps give morphisms of mixed twistor complexes $$ (M^{\cdot}_k, W^{\rm pre}) \rightarrow (M^{\cdot}_{k+1}, W^{\rm pre}). $$ In other words, we get a {\em cosimplicial mixed twistor complex}. The cohomology of the simplicial scheme is calculated by taking the associated double complex (with new differential the alternating sum of the face maps) and then taking its associated single complex. We just have to provide this with a structure of mixed twistor complex. This is of course exactly what is explained in \cite{Hodge3}. We briefly sketch the argument. We obtain a total complex $$ N^j = \bigoplus _{i+k=j}M^i_k, $$ and we define its preweight filtration by $$ W^{\rm pre}_nN^j := \bigoplus _{i+k=j} W^{\rm pre}_{n+k}M^i_k. $$ The two differentials are compatible with this new weight filtration and in fact the differential $M^i_k \rightarrow M^i_{k+1}$ induces the zero map on the associated-graded for the weight filtration. In particular, the double complex $Gr^{W^{\rm pre}}_n(M^{\cdot}_{\cdot})$ has one of its differentials vanishing; the other differential is just the differential in each mixed twistor complex $M^{\cdot}_k$. Thus $$ H^j(Gr^{W^{\rm pre}}_n(M^{\cdot}_{\cdot}))= \bigoplus _k H^{j-k}(Gr^{W^{\rm pre}}_{n+k}(M^{\cdot}_k)), $$ and this is pure of weight $n+j$ since each of the $M^{\cdot}_k$ is a mixed twistor complex. This shows that the total complex is again a mixed twistor complex. The proof that the resulting mixed twistor structure on the cohomology is independant of the choice of desingularization and normal crossing compactification, (and, what is pretty much the same thing, functoriality) is exactly the same as in \cite{Hodge2} \cite{Hodge3}. This completes the proof of Theorem \ref{TwistorIII}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} We have also obtained the simplicial version: \begin{theorem} \label{TwistorSimpl} Suppose $X_{\cdot}$ is a simplicial scheme with each $X_k$ projective over $Spec ({\bf C} )$, and suppose $U_{\cdot} \subset X_{\cdot}$ is a simplicial open subscheme. Suppose $(E, W, {\bf d})$ is a graded-polarizable variation of mixed twistor structure on $X_{\cdot}$ (i.e. a functorial association of VMTS for every smooth scheme mapping to $X_k$, with isomorphisms of compatibility for the simplicial maps). Then there is a mixed twistor structure on $H^i(U_{\cdot}, L)$ where $L$ is the flat bundle associated to $(E_1, {\bf d}_1)$. This mixed twistor structure is functorial for maps of VMTS and also for maps of simplicial schemes $U_{\cdot}$ (from the cohomology of $E$ on the target to the cohomology of the pullback of $E$ on the domain). \end{theorem} \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Remark:} For now, we have not made any study of VMTS on open varieties, so we are forced to assume the existence of extensions to projective varieties $X_k$ and furthermore that these extensions should be organized in a simplicial way. Eventually these restrictions should be gotten rid of (possibly in exchange for a condition of admissibility of the VMTS on the open varieties). \numero{Nilpotent orbits and the limiting mixed twistor structure} Suppose $U=X-P$ is a compact curve with one point removed (actually any number of points is ok but we work near one) and suppose $E$ is a harmonic bundle on $U$. We suppose that the monodromy of $E$ at $P$ is unipotent, and that the flat sections have sub-polynomial growth. This last condition means that, in the language of \cite{HBNC}, the associated filtered local system has trivial filtration. Let ${\cal E}$ be the associated variation of twistor structure considered as a bundle over $U\times {\bf P} ^1$, normalized to have weight zero say. Let $({\cal F} _U, {\cal L} , {\cal G} _U)$ be the triple of a $\lambda$-connection ${\cal F}_U$, a family of local systems ${\cal L}$ on $U$, and a $\lambda ^{-1}$-connection ${\cal G} _U$ on $\overline{Z}$. Let ${\cal F}$ be the canonical extension of ${\cal F}_U$ to a logarithmic $\lambda$-connection on $Z$. Let ${\cal G}$ be the canonical extension of ${\cal G}_U$ to a logarithmic $\lambda ^{-1}$-connection on $\overline{Z}$. These extensions have nilpotent transformations in the fiber over $P$, hence they have monodromy weight filtrations (cf \cite{Schmid} for example). The dimensions of the associate-graded of the weight filtrations are the same for all $\lambda $ (cf \cite{HBNC}). This implies that the weight filtrations are filtrations of the bundles ${\cal F} _P$ (defined to be the restriction to ${\bf A}^1 = \{ P\} \times {\bf A}^1$) and ${\cal G} _P$ (similarly, the restriction to the other copy of $\{ P\} \times {\bf A}^1$), by strict subbundles. Fix a tangent vector $\eta$ at $P$, fix a point $P_0$ nearby $P$, and fix a path going from $P_0$ to $P$ arriving by the direction $\eta$. For any logarithmic flat connection on a neighborhood of $P$ there is an isomorphism with a fixed connection with constant coefficients on a trivial bundle, and this isomorphism determines an isomorphism between the fiber over $P$ and the fiber over $P_0$. The isomorphism of fibers is well defined depending on the choice of tangent vector \cite{DeligneRegSing}. Furthermore this isomorphism depends analytically on any parameters, and takes the monodromy operator on the fiber over $P_0$ to the exponential of the residue of the connection over $P$. In particular it preserves the weight filtrations. Applying this to our situation, note that the family ${\cal L} _{P_0}$ of fibers of the $L_t$ over $P_0$ has a family of weight filtrations. The above isomorphisms give an isomorphism of filtered bundles $$ {\cal L} _{P_0} \cong {\cal F} _P $$ over ${\bf G}_m$. Similarly we have an isomorphism of filtered bundles $$ {\cal L} _{P_0} \cong {\cal G} _P $$ over ${\bf G}_m$ and we can use these to glue together the filtered bundles ${\cal F}_P$ and ${\cal G}_P$ to give a filtered bundle $(M, W)$ on ${\bf P} ^1$. Recall that $T(1)$ denotes the Tate twistor, of rank one and weight two. It is isomorphic to the tangent bundle of ${\bf P} ^1$ (and this is probably the right way to say canonically what it is). \begin{conjecture} \label{LimitingMTS} {\rm (\cite{Schmid})} The filtered bundle defined above is a mixed twistor structure. (We call it the {\em limiting mixed twistor structure of $E$}). Furthermore the logarithms of monodromy operators glue together to give a morphism of mixed twistor structures $$ N: M \rightarrow M\otimes _{{\cal O} _{{\bf P} ^1}}T(1) $$ such that the weight filtration is deduced from $N$ in the now standard way. \end{conjecture} Once this is proved, we would then like to generalize the nilpotent orbit theorem \cite{Schmid}, saying that the asymptotic behavior of our variation of twistor structure is well approximated by the behavior of a standard variation (``nilpotent orbit'') deduced directly from the limiting mixed twistor structure. All of this should then be generalized to higher dimensions, where it might help us to get a hold on the behavior of harmonic bundles near normal crossings divisors. \numero{Jet-bundles of hypercomplex manifolds} Let $U$ be a smooth hyperk\"ahler or more generally hypercomplex manifold. \footnote{ The definition of {\em hypercomplex manifold} is essentially the that of {\em hyperk\"ahler manifold} (cf \cite{HKLR}) minus the condition of existence of a metric. The original reference for hypercomplex manifolds seems to be Boyer \cite{Boyer}---see also the preprint of Kaledin \cite{Kaledin}. } The product $T:= U \times {\bf P} ^1$ has a complex structure called the {\em twistor space} of the quaternionic structure of $U$ \cite{Hitchin1} \cite{Hitchin2} \cite{HKLR} \cite{Kaledin}. Basically an almost-hypercomplex structure is just a ${\cal C} ^{\infty}$ family of actions of the quaternions on the tangent spaces of $U$, which gives an almost-complex structure on $T$. Integrability of this complex structure is exactly the condition of $U$ being hypercomplex (see Theorem 1 of \cite{Kaledin}, or \cite{HKLR} for the hyperk\"ahler case). The projection $p:T\rightarrow {\bf P} ^1$ and the horizontal sections for the product structure (called {\em twistor lines} \cite{HKLR}) $\zeta : {\bf P} ^1\rightarrow T$ are holomorphic maps. Fix a twistor line $\zeta$ corresponding to a point $u\in U$. The normal bundle $N_{\zeta /T}$---a holomorphic vector bundle on ${\bf P} ^1$ of rank $r$ equal to $\frac{1}{2}dim _{{\bf R}}(U)$---is the twistor-bundle of the tangent space $T(U)_u$ with its quaternionic structure. In particular, $N_{\zeta /T}$ is a pure twistor structure of weight $1$, i.e. it is isomorphic to a direct sum of $r$ copies of ${\cal O} _{{\bf P} ^1}(1)$. Let $J^n_{\zeta /T}$ denote the $n$-th relative jet bundle of $T$ over ${\bf P} ^1$ along the section $\zeta$. It may be described as follows. Let $I_{\zeta/ T}$ denote the ideal sheaf of the section $\zeta$. The coherent sheaf $Q_{n, \zeta /T} := {\cal O} _T / I_{\zeta / T} ^{n+1}$ on $T$ is flat and proper over ${\bf P} ^1$. Its direct image $p _{\ast}(Q_{n, \zeta /T} )$ is a vector bundle on ${\bf P} ^1$ and the jet bundle is the dual of this vector bundle. We have inclusions $$ J^0 _{\zeta /T} \subset \ldots \subset J^n_{\zeta /T} $$ and the quotients are naturally symmetric powers of the normal bundle: $$ J^m_{\zeta /T} / J^{m-1}_{\zeta /T} \cong Sym ^m(N_{\zeta /T}). $$ In particular, these quotients are pure twistor structures of weight $m$. Thus if we put $W_m(J^n_{\zeta /T}) := J^m_{\zeta /T}$ we obtain a mixed twistor structure. The fiber over $1$ is the jet space $J^n_u$ of $U$ at $u$ (for the complex structure corresponding to $1\in {\bf P} ^1$---this depends on how we normalize the twistor space). We conclude the phrase ``the jet spaces of a hypercomplex manifold have natural mixed twistor structures''. Taking the direct limit or union of the $J^n_u$ we get an ind-object $J^{\infty}_u$ in the category of mixed twistor structures. The weight quotients are finite dimensional, and in fact the finite $J^n_u$ are recovered as the pieces of the weight filtration of $J^{\infty}_u$. By abuse of notation we will say that $J^{\infty}_u$ is a mixed twistor structure which determines in particular the mixed twistor structures $J^n_u$. The dual of the jet space $J^n_u$ is $Q_{n,u}:={\cal O} _{U,u} /{\bf m}_u^{n+1}$. This also has a mixed twistor structure, dual of the above. The algebra structure of $Q_{n,u}$ comes from a coalgebra structure of $J^{\infty}_u$ (coalgebra in the category of ind-mixed twistor structures). The mixed twistor structure $J^{\infty}_u$ together with its coalgebra structure determines the algebras $Q_{n,u}$. In fact the twistor structure gives the coherent sheaves of algebras denoted ${\cal F}$ above, and taking the $Spec$ at each stage and formally taking the direct limit we obtain the formal completion of $T$ along the section $\zeta$. Thus ``the mixed twistor structure on the jet space $J^{\infty}_u$ of $U$ at $u$ determines the formal completion of the twistor space $T$''. Finally the twistor space $T$ has an antilinear involution $\sigma _T$ covering the antipodal involution $\sigma _{{\bf P} ^1}$. In particular the mixed twistor structure $J^{\infty}_u$ has an antipodal real structure and, via the previous paragraph, from this real structure we recover the involution $\sigma _T$ on the formal completion along the section $\zeta$. Nearby the section $\zeta$ the only other sections invariant under the involution $\sigma$ are the twistor lines (this local uniqueness of solutions of an elliptic equation may be verified in the linearization which is the normal bundle $N_{\zeta /T}$). Hence the antipodal real structure of $J^{\infty}_u$ determines the product structure of the formal completion of $T$. Therefore it determines the hypercomplex structure of $U$ in the formal neighborhood of $u$. {\em Question:} How can we include the data of the hyperk\"ahler metric in this point of view? We obtain a dictionary between formal germs of integrable quaternionic structure $(u, \hat{U})$ and mixed twistor structures $J^{\infty}$ with coalgebra structure and antipodal real structure (these last two being compatible) such that the associated graded of the weight filtration is the symmetric algebra. This gives a coordinate-free description of the power series of a quaternionic structure. \subnumero{The ``Gauss map''} Suppose $U$ is a hypercomplex manifold of real dimension $4d$, and fix a positive integer $n$. Then for each $u\in U$ we have constructed a mixed twistor structure on $J^n_u(U)$. The dimensions of the weight-graded quotients are $(a_0, a_1, \ldots , a_n)$ with $a_i$ being the rank of $Sym ^i ({\bf C} ^{2d})$. We obtain a morphism to the moduli stack of mixed twistor structures $$ U\rightarrow {\cal MTS}(a_0,\ldots , a_n). $$ We can think of this as a ``Gauss map'' for the hypercomplex structure. Actually this can be lifted a little bit more, using the coalgebra structure on the jet bundles. We have an isomorphism of complex vector spaces (using the complex structure $j$, say) $$ Gr ^W_m(J^n_u)\cong Sym ^{m}(T(U)_u). $$ In particular, if we choose a basis for $T(U)_u$ then this gives a basis for each of the $Gr ^W_m(J^n_u)$, in other words a framing. Thus, over the frame bundle of the tangent bundle we obtain a lifting of our map to $Fr{\cal MTS}(a_0,\ldots , a_n)$. We can be a little bit more precise about this. Note that the associated-graded of the tensor product of two filtered vector spaces, is naturally isomorphic to the tensor product of the associated-graded vector spaces. In this way, a coalgebra structure on a filtered vector space $J$ (which we now assume infinite dimensional, no longer fixing $n$) gives a coalgebra structure on $Gr^W(J)$. If $T$ is a vector space then let $Fr{\cal MTS}Cog(Sym^{\cdot}T)$ denote the moduli stack of pairs $(J, \beta )$ where $J$ is a mixed twistor structure with coalgebra structure and $\beta : Gr ^W(J_1)\cong Sym^{\cdot}T$ is an isomorphism of coalgebras. But this moduli stack is in fact a projective limit of schemes, because of the rigidity property of the remark following Lemma \ref{abelian}. Note also that the coalgebra structure, if it exists, is uniquely determined by $\beta$ again by the same rigidity property. Thus letting $a_i$ denote the dimension of $Sym ^i(T)$ we have $$ Fr{\cal MTS}Cog(Sym^{\cdot}T) \subset \lim _{\leftarrow , n} Fr{\cal MTS} (a_0,\ldots , a_n). $$ Finally, $GL(T)\cong GL(a_1,{\bf C} )$ acts on $Fr{\cal MTS}Cog(Sym^{\cdot}T)$ and our {\em Gauss map} is a real analytic map $$ \Phi : U\rightarrow Fr{\cal MTS}Cog(Sym^{\cdot}T)/GL(T). $$ The associated $GL(T)$-bundle on $U$ is the tangent bundle (for the complex structure over $1\in {\bf P} ^1$). {\em Problem:} what are the differential equations satisfied by $\Phi$ corresponding to the fact that it parametrizes the family of jet spaces of a manifold $U$? \numero{The moduli space of representations} Suppose $X$ is a connected smooth projective variety with basepoint $x$. Let $G=GL(n)$ (or any other reductive group) and let $M(X,G)$ denote the moduli space of representations of $\pi _1(X, x)$ in $G$ up to conjugacy \cite{Lubotsky-Magid}. The set of smooth points of $M(X,G)$ has a hyperk\"ahler structure \cite{Hitchin1} \cite{Hitchin2} \cite{SantaCruz}. Thus we can apply the above remarks to any smooth open set $U\subset M (X, G)$. As remarked in (\cite{SantaCruz} \S 3), any smooth stratum for a canonical stratification of $M(X,G)$ also inherits a hyperk\"ahler structure so we could also apply the previous remarks there. We find that the jet spaces $J^n_{\rho}(M(X,G))$ at any smooth point $\rho$ have natural mixed twistor structures, and the same for jet spaces of smooth strata. This can be generalized to singular points and points of the representation spaces as follows. In general if $Z$ is a scheme and $z\in Z$, denote by $Q_{n,z}(Z)$ the algebra ${\cal O} _{Z,z}/{\bf m}_z^{n+1}$, and let $J^n_z(Z)$ denote the ${\bf C}$-linear dual coalgebra. For our pointed connected smooth projective variety $(X,x)$ as above, let $R(X,G,x)$ denote the scheme of representations of $\pi _1(X,x)$ in $G$ (before dividing out by conjugation). Note that $M(X,G)$ is the geometric invariant theory categorical quotient of $R(X,G,x)$ by the action of $G$. Deligne's construction as described in \cite{NAHT} \cite{SantaCruz} gives an analogue of the twistor space for $R(X,G,x)$, namely a complex analytic space $R_{Del}(X,G,x)$ over ${\bf P} ^ 1$ with ``twistor lines'' corresponding to the semisimple representations of $\pi _1(X,x)$ in $G$. The group $G$ acts and the categorical quotient denoted $M_{Del}(X,G)$ gives, over the smooth points, the twistor space for $M(X,G)$. In Deligne's interpretation, $M_{Del}(X,G)$ constitutes a twistor space for the ``singular hyperk\"ahler structure" on $M(X,G)$ (in particular there is an antipodal involution $\sigma$). Suppose $\eta : {\bf P} ^1 \rightarrow R_{Del}(X,G,x)$ is a twistor line corresponding to a semisimple representation $\rho$. By Goldman-Millson theory \cite{Goldman-Millson} (cf \cite{SantaCruz} and \cite{Moduli} for the present application) the scheme $R_{Del}(X,G,x)$ is analytically formally trivial along the section $\eta$ and furthermore the singularities in the transversal direction are quadratic. In particular we can form the family of jet bundles $J^n_{\eta} (R_{Del}(X,G,x))$ which is a bundle of coalgebras over ${\bf P} ^1$ (and which we denote simply by $J^n$ for short, below). This bundle is dual to the bundle of algebras $Q_{n, \zeta}(R_{Del}(X,G,x))$. \begin{theorem} \label{JetBdlRepSpace} The jet bundles $J^n_{\eta} (R_{Del}(X,G,x))$ have natural mixed twistor structures. \end{theorem} {\em Proof:} The group $G$ acts. Let $G\eta$ denote the orbit (fiber-by-fiber) of the section $\eta$. Let $W_kQ_{n, \zeta}(R_{Del}(X,G,x))$ be the $k$-th power of the ideal of $G\eta \cap Spec ( Q_{n, \zeta}(R_{Del}(X,G,x)))$ in $Q_{n, \zeta}(R_{Del}(X,G,x))$. This is the weight filtration. By Luna's etale slice theorem the space $R_{Del}(X,G,x)$ has a product structure formally around the section $\eta$, $$ R_{Del}(X,G,x)^{\wedge} = G\eta ^{\wedge} \times Z^{\wedge} . $$ (to be precise this holds locally in the etale topology over the base ${\bf P} ^1$). We will use a different filtration to study this weight filtration. Define the {\em tangent cone filtration} $V_k$ of $Q_{n,u} (U)$ to be the filtration by the powers of the maximal ideal. To fix notations, let $$ Q_{\infty ,u}(U) = \lim _{\leftarrow , n} Q_{n ,u}(U) $$ and put $$ V_{-k}Q_{\infty ,u}(U):= {\bf m}_u ^kQ_{\infty ,u}(U). $$ The terminology comes from the fact that $Spec (Gr ^V(Q_{\infty,u}(U))$ is the tangent cone of $U$ at $u$. Note of course that $$ Q_{n ,u}(U) = Q_{\infty ,u}(U)/V_{-n-1}Q_{\infty ,u}(U). $$ The dual of $V$ is a filtration which we also denote by $V_k$ (indexed in the positive range of values of $k$ this time) of the jet space $J^{\infty}_u(U)$. The above discussion works in our family (since the families are locally trivial over ${\bf P} ^1$) to give filtrations $V$ of $Q_{\infty , \eta} (R_{Del}(X,G,x))$ and $J^{\infty}_{\eta}(R_{Del}(X,G,x))$. If $U$ is a product, $U= U_1 \times U_2$ then we have $$ Gr ^V(Q_{\infty , u}(U))= Gr ^V(Q_{\infty , u}(U_1))\otimes Gr ^V(Q_{\infty , u}(U_2)), $$ in other words, the tangent cone is the product of the two tangent cones. This is proved by interpreting the tangent cone as the scheme cut out by the initial forms of the equations. Now suppose that $U$ is a product as above and let $W$ be the filtration by powers of the ideal of the factor $U_1$. The filtration induced by $W$ on $Gr ^V(Q_{\infty , u}(U))$ is again the filtration by powers of the ideal of the first factor. This is the same as the filtration induced by the grading of the second factor only, in the above tensor product structure. By the lemma of two filtrations (\cite{Hodge2}) we have $$ Gr ^W_m Gr ^V_k (Q_{\infty , u}(U))= Gr ^V_k Gr ^W_m (Q_{\infty , u}(U)). $$ The same holds in the situation of our family over ${\bf P}^1$. Thus in order to prove that $Q_{\infty , \eta}(R_{Del}(X,G,x))$ is a mixed twistor structure, it suffices to prove that $$ Gr ^V_{\cdot}Q_{\infty , \eta}(R_{Del}(X,G,x)) $$ is a mixed twistor structure. Note that $$ Gr ^V_{k}Q_{\infty , \eta}(R_{Del}(X,G,x))= (J^k/J^{k-1})^{\ast}, $$ and in particular $$ Gr ^V_{1}Q_{\infty , \eta}(R_{Del}(X,G,x))= (J^1/J^{0})^{\ast} = N^{\ast} $$ where $N$ denotes the normal bundle (isomorphic to $J^1/J^0$). The algebra structure of $Gr^V_{\cdot}$ gives a morphism $$ Sym ^{k}(N^{\ast}) \rightarrow (J^k/J^{k-1})^{\ast} \rightarrow 0 $$ which is a surjection because $V$ is by definition the filtration by powers of the maximal ideal. By Goldman and Millson-theory the family of algebras $Q_{\infty , \eta}(R_{Del}(X,G,x))$ is quadratic, hence isomorphic to the tangent cone and in particular the tangent cone is quadratic. This means that the kernel of the above surjection comes from the quadratic relations, in other words we have an exact sequence $$ P\otimes Sym ^{k-2} (N^{\ast}) \rightarrow Sym ^{k}(N^{\ast}) \rightarrow (J^k/J^{k-1})^{\ast} \rightarrow 0 $$ where $P \subset Sym ^2(N^{\ast})$ is the module of quadratic relations, that is the kernel of the map $Sym ^2(N^{\ast}) \rightarrow J^2 / J^1$. We claim that the filtration induced by $W$ on $(J^k/J^{k-1})^{\ast}$ is the image of the filtration of $Sym ^{k}(N^{\ast})$ induced by $W$ on $N^{\ast}= (J^1/J^0)^{\ast}$. This is a general fact coming from the product situation $U=U_1\times U_2$ with our filtrations $V$ and $W$ as above. Write $U_i= Spec (A_i)$ and let ${\bf m}_i\subset A_i$ be the maximal ideal of the origin. Then $U_1\times U_2 = Spec (A)$ with $A=A_1\otimes A_2$, $W$ is the filtration by powers of ${\bf m} _2A$ and $V$ is the filtration by powers of ${\bf m}_1 A+ {\bf m}_2A$. The statement which needs to be proved $(\ast )$ is that $$ ({\bf m} ^1A + {\bf m} ^2A) ^{\otimes p} \otimes ({\bf m} _2A)^{\otimes q} \rightarrow \frac{({\bf m} _2A)^q \cap ({\bf m} _1 A+ {\bf m} _2A)^{p+q}}{({\bf m} _1 A+ {\bf m} _2A)^{p+q+1}} $$ is surjective. Choose a splittings $A_i\cong Gr (A_i)$ which we think of as direct sum decompositions $A_i \cong \bigoplus A_i^k$ compatible with the filtrations by powers of the maximal ideal. The product structure has an upper diagonal form for this decomposition, with the product in $Gr(A_i)$ along the diagonal. We get a decomposition $A= \bigoplus A^{j,k}$ with $A^{j,k}= A_1^j\otimes A_2^k$. The decomposition of $A$ into pieces of the form $\bigoplus _{j+k=n}A^{j,k}$ corresponds to a splitting $A\cong Gr(A)$. In particular, $$ ({\bf m} _1 A+ {\bf m} _2A)^{n} = \bigoplus _{j+k\geq n}A^{j,k}. $$ Similarly $$ ({\bf m} _2A)^q= \bigoplus _{k\geq q}A^{j,k}. $$ From this we get $$ ({\bf m} _2A)^q \cap ({\bf m} _1 A+ {\bf m} _2A)^{p+q} = \bigoplus _{j+k \geq n, k\geq q}A^{j,k} . $$ The product on the associated graded induces a surjective morphism $$ (A^{1,0})^{\otimes j}\otimes (A^{0,1})^{\otimes k} \rightarrow A^{j,k}, $$ and the product morphism from the left side into $A$ is equal to this morphism plus pieces which go into $\bigoplus _{j'+k'\geq j+k+1}A^{j',k'}$. This implies the statement $(\ast )$ which we are trying to prove. Thus in the exact sequence of the previous paragraph, the filtration induced on $(J^k/J^{k-1})^{\ast}$ by the weight filtration of the middle term, is equal to the weight filtration. The morphism $P\otimes Sym ^{k-2} (N^{\ast}) \rightarrow Sym ^{k}(N^{\ast})$ is compatible with the filtrations induced by $W$. We claim that $N$ and $P$ are mixed twistor structures. Thus the morphism is a morphism of mixed twistor structures, and the cokernel is a mixed twistor structure. This cokernel is equal to $(J^k/J^{k-1})^{\ast}$ with its filtration $W$. Thus $Gr ^V_{k}Q_{\infty , \eta}(R_{Del}(X,G,x))$ is a mixed twistor structure so, as explained above with the lemma of two filtrations, we are done. We now prove that $N$ and $P$ are mixed twistor structures. First of all, as in \cite{Goldman-Millson} \cite{Moduli}, $N$ is calculated as the first cohomology of the complex $$ \ker (\xi {\cal A}^0_X(End (E)\rightarrow End (E_x))\rightarrow \xi {\cal A} ^1_X(End (E))\rightarrow \xi {\cal A} ^2_X(End(E)). $$ Thus there is an exact sequence $$ 0\rightarrow H^0(End (E))\rightarrow End (E_x) \rightarrow N \rightarrow H^1(X\times {\bf P} ^1 / {\bf P} ^1, End (E))\rightarrow 0 $$ and the image of $End (E_x)$ in $N$ is the tangent space of the orbit, that is to say that it is $W_0$. From \ref{coho1} the cohomology $H^i(X\times {\bf P} ^1 / {\bf P} ^1, End (E))$ is a pure twistor structure of weight $i$ (note that $End(E)$ is pure of weight zero). From $i=0$ and the fact that $End (E_x)$ is pure of weight zero we find that $W_0$ is pure of weight zero. From $i=1$ we get that $H^1(End(E))$, which is the piece $W_1/W_0$, is pure of weight $1$. This proves that $N$ is a mixed twistor structure. Next, we have a cup-product $N \otimes N \rightarrow H^2(End (E))$ which is actually symmetric (the antisymmetry on odd-degree classes cancels the antisymmetry of the Lie bracket). This gives a morphism $Sym ^2(N)\rightarrow H^2(End(E))$. The part $W_1Sym ^2(N)$ goes to zero since, by the group action, there are no obstructions in the direction of the group orbit. It follows that the cokernel and the image are pure of weight $2$. The module of relations $P$ is the image of the dual morphism $$ H^2(End(E))^{\ast} \rightarrow Sym ^2(N^{\ast}), $$ so $P$ is a pure twistor structure of weight $-2$. This completes the proof that the jet spaces of the representation space have natural mixed twistor structures. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Application to the relative completion:} Taking some sort of Tannakian dual of the completions of the representation spaces $R(X,G,x)$ where $G$ runs through all groups, at points corresponding to all representations associated to a given semisimple representation $\rho$, should give back the {\em relative Malcev completion} of $\pi _1(X,x)$ at the representation $\rho$. One should be able to use this to obtain a mixed twistor structure on the relative Malcev completion. Alternatively, a direct generalization of Hain's technique \cite{Hain} should also work. One would like to check, in fact, that the two constructions agree. As you can tell from the tense of this paragraph, I haven't looked into this at all! {\em Application to the moduli space:} The group $G$ doesn't act on the formal local ring of $R(X,G,x)$ at a point $\rho$ (semisimple representation), however we do obtain an infinitesimal action of the Lie algebra ${\bf g}$. Assume that $G$ is connected. Then the formal local ring of $M(X,G)$ at $\rho$ is the subring of ${\bf g}$-invariants in the formal local ring of $R(X,G,x)$ at $\rho$. This latter has a mixed twistor structure (it is the dual of $J^{\infty}$). The action of ${\bf g}$ preserves the mixed twistor structure so the subring of invariants is a mixed twistor structure. Again taking the dual, $J^{\infty}_{\rho}(M(X,G))$ has a mixed twistor structure. Note that if $\rho$ is not a smooth point, the weight filtration is not at all the same as the easy filtration, and $J^1$ might very well start of with pieces of relatively high weight (essentially because the lower-degree terms don't survive as invariants).

9705

alg-geom/9705018

en

https://arxiv.org/abs/alg-geom/9705018

alg-geom/9705018

Biran Paul

Paul Biran

Constructing new ample divisors out of old ones

22 pages, written in Amslatex. Several correction made

We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the ample cone gives rise to an algorithm for constructing new ample divisors. We then propose a conjecture relating continued fractions approximations and Seshadri-like constants of line bundles over rational surfaces. By applying our algorithm recursively we verify our conjecture in many cases and obtain new asymptotic estimates on these constants. Finally, we explain the intuition behind the gluing theorem in terms of symplectic geometry and propose generalizations.

alg-geom

\section{Introduction} \setcounter{thm}{0} \renewcommand{\thethm \label{sect-Intro} The main objective of this paper is to propose a method for constructing new ample divisors on rational surfaces by gluing two given ones. Recall that a divisor $D$ on an algebraic variety $X$ is ample if the corresponding line bundle $\calO_X(D)$ is ample, and is called {\em nef (numerically effective)} if there exists an ample divisor $A$ such that $A+kD$ is ample for every $k>0$. We refer the reader to \cite{Dem,Ha-Ample} for excellent expositions on various aspects of the theory of ample and nef line bundles. Of fundamental importance is the determination of those classes in $\mbox{Pic}(X)$ which are ample. Although this problem has a very simple solution for smooth curves, already in dimension two the problem becomes much harder. It turns out the even for relatively simple surfaces, such as rational, the complete answer is not known. Several conjectures in this direction exist, however at the present time only estimates on the {\em ample cone} -- the cone generated by the ample classes in $\mbox{Pic}(X)$ -- are known. For example, let $d,m>0$ and consider the divisor class $$D=\pi^* \calO_{\CPTU}(d) - m\sum_{j=1}^N E_j$$ on the blow-up $\pi:V_N \rightarrow \CPTU$ of $\CPTU$ at $N\geq 9$ generic points. Nagata conjectured in~\cite{Nag} that {\em $D$ is ample iff $D\cdot D >0$}, but was able to prove it only for $N's$ which are squares. In~\cite{Xu-Curves} Xu proved that {\em $D$ is ample provided that $\frac{m}{d}< \frac{\sqrt{N-1}}{N}$}. By making a more detailed analysis of the case $m=1$, Xu proved in \cite{Xu-Divisors} that {\em when $d\geq 3$ the divisor class $D=\pi^* \calO_{\CPTU}(d) - \sum_{j=1}^N E_j$ is ample iff $D \cdot D > 0$} (see also K\"{u}chle~\cite{Ku} for a generalization for arbitrary surfaces and~\cite{Ang} for an analogous result for $\CC P^3$). Closely related is the problem of computing {\em Seshadri constants} of ample line bundles, which measure their local positivity. The Seshadri constant $\calE(\calL,p)$ of the line bundle $\calL$ at the point $p\in X$ is defined to be {\em the supremum of all those $\eps\geq 0$ for which the $\RR$-divisor class $\pi^*\calL -\eps E$ is nef on the blow-up $\pi:\wtldX_p \rightarrow X$ of $X$ at the point $p$ with exceptional divisor $E$}. Seshadri constant has been studied much by Demailly (\cite{Dem}), Ein, K\"{u}chle, Lazarsfeld (\cite{EL},\cite{EKL},\cite{Laz}), and Xu (\cite{Xu-Ample}). A considerable part of these works is devoted to computations and estimates from below on the values of these constants. The present paper is largely motivated by the problem of computing Seshadri constants and the determination of the ample cone of rational surfaces. Our main results provide an algorithmic method for constructing new ample divisors out of the knowledge of ample divisors on simpler rational surfaces. By applying the algorithm recursively we obtain in Section~\ref{sect-Asymptotics} new estimates on Seshadri-like constants and detect new ample divisors. We then propose in Section~\ref{sect-Conjecture} a conjecture naturally arising from our method which relates {\em continued fractions expansions of $\sqrt{N}$} with the ample cone of $\CPTU$ blown-up at $N$ points. Finally we interpret in Section~\ref{sect-Symplectic} our main results in the language of Symplectic Geometry and explain the intuition behind them. Our main tool is Shustin's version of the Viro method for gluing curves with singularities. \section{Main results} \setcounter{thm}{0} \renewcommand{\thethm Our main results deal with {\em simple rational surfaces} $S$, which by definition are blow-ups $\Th:S\rightarrow \CPTU$ of $\CPTU$ at $n$ distinct points $p_1, \ldots, p_n \in \CPTU$.\footnote{Note that we regard $\CPTU$ itself as a simple rational surface too (this corresponds to $n=0$).} We denote by $E^S_i=\Th^{-1}(p_i) \quad i=1,\ldots,n$ the {\em standard exceptional divisors} of the blow-up and write $\Sig^S$ for the union $\cup_{i=1}^n E^S_i$. Finally, we write $L^S$ for be a divisor on $S$, obtained by pulling back via $\Th$ a projective line in $\CPTU$ which does not pass through any of the points $p_1,\ldots,p_n$. A vector $(d;\al_1,\ldots,\al_k) \in \ZZ_+ \times \ZZ^k_{\mbox{\tiny $\geq 0$}}$ is called ample (resp. nef) if there exists a simple rational surface $V$, on which the divisor $dL^V-\sum_{j=1}^k \al_j E^V_j$ is ample (resp. nef). \ \\ Our first result is the following gluing theorem: \begin{thm} \label{thm-glue1} Let $(d;m_1, \ldots m_n, m)$ be an ample (resp. nef) vector and $(m; \al_1, \ldots, \al_k) \in \ZZ_+^{k+1}$ a nef vector. Then $v=(d;m_1,\ldots,m_n,\al_1,\ldots, \al_k)$ is ample (resp. nef). Moreover, $v$ can be realized by an ample (resp. nef) divisor on a very general rational surface. \end{thm} By a {\em very general} choice of points $q_1,\ldots,q_r$ in an algebraic variety $X$ we mean that $(q_1,\ldots ,q_r)$ is allowed to vary in a subset of the configuration space $\calC_r(X)=\{(x_1, \ldots, x_r)\in X^r \mid x_i \neq x_j \}$ whose complement is contained in a countable union of proper subvarieties of $\calC_r(X)$. By a {\em very general rational surface} we mean one which is obtained by blowing-up points $q_1,\ldots,q_r \in \CPTU$ which may be chosen to be very general. We shall actually prove a stronger result which allows us to keep the blown-up points corresponding to the first ample vector fixed, thus giving information also on ample divisors on non-generic rational surfaces. The precise statement is: \begin{thm} \label{thm-glue2} Let $D$ be a divisor on a simple rational surface $S$. Suppose that there exists a point $p\in S\setminus (\Sig^S \cup Supp\,D)$ and $m>0$ such that $\pi_p^* D-mE$ is ample on the blow-up $\pi_p:\wtldS_p\rightarrow S$ of $S$ at $p$ with exceptional divisor $E$. Let $(m;\al_1, \ldots, \al_k) \in \ZZ_+^{k+1}$ be a nef vector. Then for a very general choice of points $q_1, \ldots, q_k \in S\setminus(\Sig^S\cup Supp\,D)$ the divisor $$\pi^*D -\sum_{j=1}^k \al_j E_j$$ is ample on the blow-up $\pi:\wtldS\rightarrow S$ of $S$ at $q_1, \ldots, q_k$ with exceptional divisors $E_j=\pi^{-1}(q_j)$. \end{thm} Proofs of Theorems~\ref{thm-glue1} and~\ref{thm-glue2} appear in Section~\ref{sect-Gluing}. Theorem~\ref{thm-glue1} in combination with the action of the Cremona group on the ample cone give rise to an algorithmic procedure for detecting new ample classes in the Picard group of rational surfaces. The algorithm will be explained in Section~\ref{subsect-Alg}. \subsection{Applications to Seshadri constants} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-Applic} Given an ample line bundle $\calL \rightarrow S$ on a surface, and a vector $w=(w_1,\ldots, w_N)$ of positive numbers we define the {\em $w$-weighted remainder} of $\calL$ at the $N$ distinct points $p_1,\ldots,p_N \in S$ to be the quantity $$\calR^w(\calL, p_1,\ldots,p_N)=\frac{1}{\calL \cdot \calL} \inf_{0\leq\eps\in \RR} \left\{ \calL_{\eps} \cdot \calL_{\eps} \bigg| \calL_{\eps} = \pi^*\calL -\eps\sum_{j=1}^N w_j E_j \quad \mbox{is nef} \right\},$$ where $\pi:\wtldS\rightarrow S$ is the blow-up of $S$ at the points $p_1,\ldots,p_N$ with exceptional divisors $E_i=\pi^{-1}(p_i)$. It is obvious that $0 \leq \calR^w < 1$. Note that $\calR^w$ remains invariant under rescalings of $\calL$ and of $w$, namely $\calR^{aw}(b\calL, p_1, \ldots p_N)=\calR^{w}(\calL, p_1,\ldots p_N)$ for every $a,b>0$. It is convenient to define also a more global invariant, namely $$\calR^w_N (\calL)= \inf \left\{ \calR^{w}(\calL, p_1,\ldots p_N) \mid p_1, \ldots p_N \in S \quad \mbox{are distinct points} \right\}.$$ Restricting to the case of homogeneous weights we obtain the {\em homogeneous remainders} $$\calR(\calL, p_1, \ldots, p_N)= \calR^{w_h}(\calL, p_1, \ldots, p_N), \qquad \calR_N(\calL) = \calR^{w_h}(\calL),$$ where $w_h=(1, \ldots, 1)$. The constants $\calR^w(\calL,p_1, \ldots,p_N)$ are obvious generalizations of the Seshadri constants $\calE(\calL,p)$ from section~\ref{sect-Intro} (see also~\cite{Xu-Ample} for similar Seshadri-like constants). Several theorems and conjectures related to the ample cone can be neatly formulated using the constants $\calR_N$. For example, Nagata's conjecture from Section~\ref{sect-Intro} can be reformulated as ``{\em \ $\calR_N(\Oone)=0$ when $N \geq 9$ \ }''. Similarly, Xu's result from Section~\ref{sect-Intro} asserts that $\calR_N(\Oone) \leq \frac{1}{N}$. In Section~\ref{subsect-prfs1} we shall prove the following asymptotic result: \begin{thm} \label{thm-asymp1} \begin{enumerate} \item[1)] For $N=a^2l^2+2l$, $a,l\in \NN$ \ \ $\calR_N(\Oone)\leq \frac{1}{(a^2l+1)^2}$. \item[2)] For $N=a^2l^2-2l$, $a,l,\in \NN$ \ \ $\calR_N(\Oone)\leq \frac{1}{(a^2l-1)^2}$. \item[3)] If $N=a^2l^2+l$ with $1<l\in \NN$ and suppose that $l > \frac{a}{2^{k-1}}$, where $k$ is the maximal non-negative integer for which $a\equiv 0 \mod 2^k$. Then $\calR_N(\Oone)\leq\frac{1}{(2a^2l+1)^2}$. \end{enumerate} \end{thm} In Section~\ref{sect-Conjecture} we shall view this result in a more general context by proposing a conjecture which bounds $\calR_N(\Oone)$ in terms of continued fractions approximations of $\sqrt{N}$. Our methods also yield, as a corollary, the following generalization of a theorem of Xu~\cite{Xu-Divisors} and K\"{u}chle~\cite{Ku}: \begin{cor} \label{cor-coef2} Let $d>0$. The divisor $D=\pi^* \calO_{\CPTU}(d)-2\sum_{j=1}^N E_j$ on the blow-up of $\CPTU$ at $N$ very general points is nef iff $D\cdot D\geq 0$. \end{cor} The proof appears in Section~\ref{subsect-prfs1}. Let us conclude this section with the following, somewhat amusing, corollary of Theorem~\ref{thm-glue1}. \begin{cor} \label{cor-Nag} If Nagata's conjecture holds for $N_1$ and $N_2$ then it holds also for $N_1N_2$. \end{cor} The proof is given in Section~\ref{subsect-prfs2}. \section{Gluing curves on rational surfaces} \setcounter{thm}{0} \renewcommand{\thethm \label{sect-Gluing} We shall derive Theorem~\ref{thm-glue1} as a corollary from Theorem~\ref{thm-glue2}. The proof of Theorem~\ref{thm-glue2} is based on a technique for "gluing" singular curves, which was developed by Shustin in~\cite{Sh1}. This method generalizes Viro's method (see~\cite{Vir}) for gluing curves to singular cases. Suppose that $C_1, \ldots, C_n$ are plane curves with Newton Polygons $\Delta_1, \ldots, \Delta_n$ which have mutually disjoint interiors and match together to a bigger polygon $\Delta = \Delta_1 \cup \ldots \cup \Delta_n$. Shustin's method allows, under some transversality conditions on the equisingular strarta corresponding to $C_1, \ldots, C_n$, to construct a new curve $C$ with Newton polygon $\Delta$ and with singular points "inherited" from $C_1,\ldots,C_n$. We refer the reader to~\cite{Sh1} for a detailed presentation of the general method and to~\cite{Sh2} for interesting applications in other directions. Here, we shall make use only of a tip of the power of this method, by applying it to two curves with disjoint Newton polygons. The application of Shustin's technique to our problem is summed up in the following proposition which will be the main ingredient in the proof of Theorem~\ref{thm-glue2}. Most of the proof presented below is essentially an adjustment of the arguments appearing in the proof of Theorem 3.1 of~\cite{Sh1} to our specific situation. \begin{prop} \label{prop-glue} Let $D$ be an effective divisor on a simple rational surface $S$, and $p\in S \setminus\Sig^S$ a point with $\mbox{\em mult}_p D=m>0$. Let $C$ be an effective divisor on another simple rational surface $V$, lying in the linear system $|m' L^V-\sum_{j=1}^k \al_j E^V_j|$. Suppose that $D,C$ satisfy the following conditions: \begin{enumerate} \item[1)] $0<m'<m$. \item[2)] $H^1(\wtldS_p, \calO_{\wtldS_p}(\pi^*_p D-mE))=0$, where $\pi_p:\wtldS_p\rightarrow S$ is the blow-up of $S$ at the point $p$ with exceptional divisor $E$. \item[3)] $H^1(V,\calO_V(C))=0$. \item[4)] Each of $C,D$ does not have any of the standard exceptional divisors $E^V_i, E^S_j$ as one of its components. \item[5)] $D$ is an irreducible curve. \end{enumerate} Then, there exist $k$ distinct points $q_1, \ldots q_k \in S \setminus (\Sig^S \cup Supp\, D)$ and a curve $\wtld{D}$ on the blow-up $\pi:\wtldS \rightarrow S$ of $S$ at $q_1,\ldots,q_k$ with exceptional divisors $E_j=\pi^{-1}(q_j)$, which has the following properties: \begin{enumerate} \item[1)] $\wtld{D}\in |\pi^*D-\sum_{j=1}^k \al_j E_j|$. \item[2)] The curve $\wtld{D}$ is irreducible. \end{enumerate} \end{prop} {\bf Proof. } The idea of the proof is basically the following. By passing to the underlying projective planes of $\wtldS_p$ and $V$ we obtain from $D$ and $C$ two singular curves $C_1$ and $C_2$ and a point, still denoted by $p$, such that $\mbox{mult}_p C_1 > \deg C_2$. This inequality implies that the Newton polygons of $C_1$ and $C_2$ with respect to an affine chart centered at $p$ are disjoint. The next step is to construct two deformations $C_{1,t}$ and $C_{2,t}$ of $C_1$ and $C_2$ which are equisingular for $t>0$ and such that each of them contains a deformations of the union of the singular points of $C_1$ and $C_2$ except of the one at the point $p$ which might disappear. These two deformations are then glued using the {\em Viro polynomial}. Shustin's method requires the Newton polygons of each of $C_{1,t}$ and $C_{2,t}$ to be contained in the union, say $\Delta$, of the ones of $C_1$ and $C_2$. In order to construct deformations which satisfy this, one has to prove roughly speaking that the equisingular strata of $C_1$ and $C_2$ intersect transversally the space of curves with Newton polygons $\Delta$. This is precisely what the conditions of vanishing of the $H^1$'s is needed for. Let us give now the precise details of the proof. Suppose that $S$ is obtained by blowing-up $\Th_{S}:S \rightarrow \CPTU$ at $p_1, \ldots, p_N \in \CPTU$ and that $V$ is obtained by blowing-up $\Th_{V}:V \rightarrow \CPTU$ at $q_1^0, \ldots, q_k^0 \in \CPTU$. Put $p_0=\Th_{S}(p), \quad C_1=\Th_{S}(D)\subset \CPTU$ and $C_2=\Th_{V}(C)\subset \CPTU$. Assuming that $D\in|dL^S-\sum_{i=1}^n m_i E^S_i|$ we see that: \begin{itemize} \item $C_1$ is a plane curve of degree $d$ and has singularities of orders $m_1, \ldots, m_n$ at the points $p_1, \ldots, p_n$ and a singular point of order $m$ at $p_0$. \item $C_2$ is a plane curve of degree $m'$ and has singularities of orders $\al_1,\ldots,\al_k$ at the points $q_1^0, \ldots, q_k^0$. \end{itemize} In view of what we have to prove there is no loss of generality in assuming that $q_j^0 \not\in C_1$ for every $1\leq j\leq k$. Choose an affine chart $\CC^2 \subset \CPTU$ with coordinates $(x,y)$ such that $p_0=(0,0)$ and such that $p_1,\ldots, p_n, q_1^0, \ldots, q_k^0 \in (\CC^*)^2 \subset \CC^2 \subset \CPTU$. Let $F_1(x,y),F_2(x,y)$ be polynomials of degrees $d$ and $m'$ respectively, such that $C_1 \cap \CC^2 = \{ F_1=0\}$ and $C_2\cap \CC^2 = \{ F_2=0 \}$. Set $$\Delta_1=\{(i,j) \in \ZZ^2_{\geq 0} | m\leq i+j \leq d\}, \qquad \Delta_2 = \{(i,j) \in \ZZ^2_{\geq 0} | 0\leq i+j \leq m' \},$$ and put $\Delta = \Delta_1 \cup \Delta_2$. With these notations, we may write $$F_1(x,y)=\sum_{(i,j)\in \Delta_1} a_{ij}x^i y^j, \qquad F_2(x,y)=\sum_{(i,j)\in \Delta_2} a_{ij}x^i y^j.$$ Let $\overline{\Delta}_1\supset \Delta_1$ and $\overline{\Delta}_2 \supset \Delta_2$ be two slightly larger triangles with disjoint interiors. More precisely, let $\dlt>0$ be a small enough number such that $m'+2\dlt < d-2\dlt$ and set $$ \overline{\Delta}_1=\{(i,j) \in \RR^2_{\geq 0} | m-\dlt \leq i+j \leq d\}, \qquad \overline{\Delta}_1=\{(i,j) \in \RR^2_{\geq 0} | 0\leq i+j \leq m'+\dlt \}. $$ Next, choose a strictly convex continuous piecewise linear function $\nu:\RR^2 \rightarrow \RR$ such that the restrictions of $\nu$ to each of $\overline{\Delta}_1,\overline{\Delta}_2$ coincides with some linear function $\ell_1,\ell_2:\RR^2 \rightarrow \RR$ with $\ell_1 \neq \ell_2$. We shall define the curve $\wtld{D}$ as the zero locus of a polynomial lying in the following family: $$F_t(x,y)=\sum_{(i,j) \in \Delta} A_{ij}(t)x^i y^j t^{\nu(i,j)} \qquad t>0,$$ with $\lim_{t\to 0}A_{ij}(t)=a_{ij}$. The polynomial $F_t$ is called the {\em Viro polynomial}. More precisely, we claim that by a correct choice of of the coefficients $A_{ij}(t)$, and of a homogeneous change of coordinates $(x,y)\rightarrow T_t(x,y)$, the curve $D_t=\{F_t(T_t(x,y))=0\}$ will have the following properties for $t>0$ small enough: \begin{enumerate} \item[1)] $\mbox{mult}_{p_i} D_t = m_i$ for every $1\leq i \leq n$. \item[2)] There exists $k$ points ${q_1}_t, \ldots, {q_k}_t$ depending smoothly on $t>0$ such that ${q_j}_t \neq p_i$ for every $i,j$ and $\mbox{mult}_{{q_j}_t} D_t = \al_j$ for every $1\leq j \leq k$. \item[3)] $D_t$ is an irreducible curve of degree $d$. \end{enumerate} If we manage to prove this then the statement of the proposition will immediately follow. Indeed, let $t_0>0$ be small enough such that properties 1-3 above hold. Consider $\overline{D}_{t_0} \subset \CPTU$, the closure of $D_{t_0}$ in $\CPTU$, and let $\wtld{D}_{t_0}$ be the proper transform of $\overline{D}_{t_0}$ in $\wtldS$, the blow-up of $\CPTU$ at $p_1,\ldots,p_n, {q_1}_{t_0}, \ldots ,{q_k}_{t_0}$. Clearly $\wtld{D}_t \in |\pi^*D -\sum_{j=1}^k \al_j E_j|$, where $\pi:\wtldS \rightarrow S$ denotes the blow-up of $S$ at $q_1={q_1}_{t_0}, \ldots ,{q_k}_{t_0}$. Let us prove the existence of the coefficients $A_{ij}(t)$ having the claimed properties. For this end, set $\nu_1=\nu-\ell_1, \quad \nu_2=\nu-\ell_2$. Note that since $\nu$ is strictly convex and $\ell_1\neq \ell_2$ by construction, we must have ${\nu_1}_{|_{\overline{\Delta}_2}} > 0, \quad {\nu_2}_{|_{\overline{\Delta}_1}}>0$. Consider the following deformations of $F_1(x,y), F_2(x,y)$: \\ $$F_{1,t}(x,y) = \sum_{(i,j) \in \Delta} A_{ij}(t)x^i y^j t^{\nu_1(i,j)}, \eqno(1)$$ $$F_{2,t}(x,y) = \sum_{(i,j) \in \Delta} A_{ij}(t)x^i y^j t^{\nu_2(i,j)}. \eqno(2)$$ An easy computation gives:\footnote{Here we use the convention that $t^0 \equiv 1$ and so the families $F_{1,t},F_{2,t}$ extend smoothly to $t\geq 0$.} $$F_{1,t}(x,y) = F_1(x,y)+ \sum_{(i,j)\in\Delta_2} A_{ij}(t)x^i y^j t^{\nu_1(i,j)} + \sum_{(i,j)\in\Delta_1} (A_{ij}(t)-a_{ij})x^i y^j, \eqno(3)$$ $$F_{2,t}(x,y) = F_2(x,y)+ \sum_{(i,j)\in\Delta_1} A_{ij}(t)x^i y^j t^{\nu_2(i,j)} + \sum_{(i,j)\in\Delta_2} (A_{ij}(t)-a_{ij})x^i y^j, \eqno(4)$$ and $$F_t(x,y)=t^{c^{1}_0}F_{1,t} (t^{c^1_1}x, t^{c^1_2}y) = t^{c^2_0}F_{2,t} (t^{c^2_1}x, t^{c^2_2}y), \eqno(5)$$ where $c_i^1, c_j^2$ are the coefficients of the linear functions $\ell_1,\ell_2$, namely $$\ell_1(i,j)=c^1_0+c^1_1 i + c^1_2 j, \qquad \ell_2(i,j)=c^2_0+c^2_1 i + c^2_2 j.$$ Since $A_{ij}(t) \ \begin{Sb} \longrightarrow \\ t \to 0 \end{Sb} a_{ij}$ we see from (3) above that $F_{1,t} \rightarrow F_1$ as $t\rightarrow 0$. As $F_1(x,y)$ is assumed to be irreducible and $F_{1,t}$ is a deformation of $F_1$ (of the same degree) we see that $F_{1,t}$ is irreducible for $t>0$ small enough. In view of (5) we conclude that $F_t(x,y)$ is irreducible for $t>0$ small too. From (5) above we also see that the curve $\{ F_t(x,y)=0 \}$ will have the same (topological) types of singularities as each of the curves $\{F_{1,t}(x,y)=0\}, \quad \{F_{2,t}(x,y)=0 \}$. Put $$T_t (x,y)=(t^{-c^1_1}x, t^{-c^1_2}y), \qquad T'_t(x,y)=(t^{-c^2_1}x, t^{-c^2_2}y)$$ and write $$D_t=\{F_t(\,T_t(x,y)\,)=0\}, \quad {q_j}_t=T^{-1}_t\circ T'_t (q_j^0) \mbox{\ \ for every $1\leq j \leq k$}.$$ Clearly, the maps $(x,y)\rightarrow T_t(x,y)$ and $(x,y)\rightarrow T'_t(x,y)$ extend to a family of biholomorphisms of $\CPTU$ depending smoothly on $t>0$. Note that from the definition of the points ${q_j}_t$ it easily follows that for a generic choice of $t>0$ the points ${q_j}_t$ will be distinct from the $p_i$'s. In particular there exit arbitrarily small values $t>0$ for which the points ${q_j}_t$ will not collide with the $p_i$'s. Putting $D_t=\{F_t(\,T_t(x,y)\,)=0\}$, the problem is reduced to proving the following \\ {\bf Lemma.} There exists a smooth deformation $\{ A_{ij}(t) \}_{0<t<\eps}$ of the coefficients $a_{ij},\quad (i,j)\in\Delta$ with the following properties: \begin{enumerate} \item[1)] $\lim_{t\to 0}A_{ij} = a_{ij}$. \item[2)] The curve $\{F_{1,t}(x,y)=0\}$ passes through $p_1, \ldots, p_n$ with multiplicities $m_1, \ldots, m_n$. \item[3)] The curve $\{F_{1,t}(x,y)=0 \}$ passes through $q_1^0, \ldots, q_k^0$ with multiplicities $\al_1, \ldots ,\al_k$. \end{enumerate} \noindent {\em Proof of the Lemma}. Let $\calP(\Delta_1), \calP(\Delta_2)$ be the spaces of polynomials in the variables $(x,y)$ with Newton diagrams contained in $\Delta_1,\Delta_2$, respectively. For every point $q\in \CC^2$, we denote by $J^{(r)}_q$ the space of $r$ jets of holomorphic functions at the point $q$, viewed as a vector space and write $j^{(r)}_q(F) \in J^{(r)}_q$ the $r$'th jet of $F$ at the point $q$. Consider the linear maps $$R_1:\calP(\Delta_1) \rightarrow \bigoplus_{i=1}^n J^{(m_i)}_{p_i}, \qquad R_2:\calP(\Delta_2) \rightarrow \bigoplus_{j=1}^k J^{(\al_j)}_{q_j^0}$$ defined by $$R_1(F) = \left(j^{(m_1)}_{p_1}(F), \ldots, j^{(m_n)}_{p_n}(F)\right), \qquad R_2(F) = \left(j^{(\al_1)}_{q_1^0}(F), \ldots, j^{(\al_k)}_{q_k^0}(F)\right).$$ We claim that they are both surjective. To see this let us denote for every $q\in \CPTU$ by $\calJ_q$ the ideal sheaf corresponding to the point $q$. Consider the ideal sheaf $\calJ_{X_1}=\prod_{i=1}^n \calJ^{m_i}_{p_i} \cdot \calJ^m_p$ on $\CPTU$, and let $X_1\subset \CPTU$ be the zero-dimensional subscheme defined by $\calJ_{X_1}$, with structure sheaf $\calO_{X_1}=\calO_{\CPTU}/\calJ_{X_1}$. Tensoring the structural exact sequence of $X_1$ by $\calO_{\CPTU}(d)$ we obtain the following exact sequence $$0\rightarrow \calJ_{X_1}(d) \rightarrow \calO_{\CPTU}(d)\rightarrow\calO_{X_1}(d)\rightarrow 0,$$ where for any sheaf $\cal{F}$ we denote $\calF(d)=\calF \otimes \calO_{\CPTU}(d)$. Passing to cohomologies we obtain: \[ \begin{CD} 0 @>>> H^0(\calJ_{X_1}(d)) @>>> H^0(\calO_{\CPTU}(d)) @>{R_{X_1}}>> H^0(\calO_{X_1}(d)) @>>> H^1(\calJ_{X_1}(d)) @>>> \ldots \end{CD} \] where the map ${R_{X_1}}$ is induced by the restriction $\calR_{X_1}: \calO_{\CPTU} \rightarrow \calO_{X_1}$. Since $H^1(\CPTU,\calJ_{X_1}(d)) \cong H^1(\wtldS_p, \calO_{\wtldS_p}(D-mE))$ and the latter vanishes by assumption we see that the map $R_{X_1}$ is surjective. The choice of the affine chart $\CC^2 \subset \CPTU$ induces an isomorphism $i_1:\calP(d) \rightarrow H^0(\calO_{\CPTU}(d))$, where $\calP(d)$ denotes the space of polynomials in $(x,y)$ of degree not more than $d$. Similarly, we obtain an isomorphism $i_1':\oplus_{i=1}^n J^{(m_i)}_{p_i} \oplus J^{(m)}_{p_0} \rightarrow H^0(\calO_{X_1})$. Denoting by $$\wtld{R}_1 : \calP(d) \rightarrow \bigoplus_{i=1}^n J^{(m_i)}_{p_i} \bigoplus J^{(m)}_{p_0}$$ the linear map $$\wtld{R}_1(F) = \left(j^{(m_1)}_{p_1}(F), \ldots, j^{(m_n)}_{p_n}(F), j^{(m)}_p(F)\right),$$ we obtain the following commutative diagram: \[ \begin{CD} \calP(d) @>{\wtld{R}_1}>> \bigoplus_{i=1}^n J^{(m_i)}_{p_i} \bigoplus J^{(m)}_{p_0} \\ @V{i_1}VV @VV{i_1'}V \\ H^0(\calO_{\CPTU}(d)) @>{R_{X_1}}>> H^0(\calO_{X_1}(d)) \end{CD} \] As $R_{X_1}$ is surjective so is $\wtld{R}_1$. But $\wtld{R}_1^{-1}(\oplus_{i=1}^n J^{(m_i)}_{p_i})= \calP(\Delta_1) \subset \calP(d)$ and ${\wtld{R}_1}|_{\calP(\Delta_1)} = R_1$. This implies that $R_1$ is indeed surjective. The case of $R_2$ is easier. Replacing $d$ by $m'$, $X_1$ by the subscheme $X_2\subset \CPTU$ defined by the ideal sheaf $\calJ_{X_2}=\prod_{j=1}^k \calJ^{\al_j}_{q^0_j}$, and $R_{X_1}$ by the the restriction map $R_{X_2}$, we obtain the commutative diagram: \[ \begin{CD} \calP(\Delta_2) @>{R_2}>> \bigoplus_{j=1}^k J^{(\al_i)}_{q^0_i} \\ @V{i_2}VV @VV{i_2'}V \\ H^0(\calO_{\CPTU}(m')) @>{R_{X_2}}>> H^0(\calO_{X_2}(m')) \end{CD} \] where $i_2$ and $i_2'$ are obvious isomorphisms induced by the choice of the affine chart $\CC^2 \subset \CPTU$. The vanishing of $H^1(V,\calO_{V}(C)) \cong H^1(\CPTU,\calJ_{X_2}(m'))$ implies, as before, the surjectivity of $R_{X_2}$ and consequently that of $R_2$. To conclude the proof of the lemma, consider the smooth family of linear maps $$R^{(t)}:\calP(\Delta) \rightarrow \bigoplus_{i=1}^n J^{(m_i)}_{p_i} \;\;\bigoplus \;\; \bigoplus_{j=1}^k J^{(\al_j)}_{q^0_j},$$ defined by:\footnote{As before, using the convention that $t^0 \equiv 1$ the family $R^{(t)}$ extends also for $t=0$.} $$R^{(t)}\big(\sum_{(i,j)\in \Delta} A_{ij} x^i y^j\big) = \left(R_1\big(\sum_{(i,j) \in \Delta} A_{ij} x^i y^j t^{\nu_1(i,j)}\big), \quad R_2\big(\sum_{(i,j) \in \Delta} A_{ij} x^i y^j t^{\nu_2(i,j)}\big) \right).$$ Substituting $t=0$ we have, under the direct sum decomposition $\calP(\Delta)=\calP(\Delta_1) \oplus \calP(\Delta_2)$, that $R^{(0)}=R_1 \oplus R_2$, hence $R^{(0)}$ is surjective. Since the family $R^{(t)}$ depends smoothly on $t$ we conclude that $R^{(t)}$ remains surjective for $t>0$ small enough. By the (linear) implicit function theorem there exists a smooth deformation $\{A_{ij}(t)\}_{0\leq t \leq \eps}$ of $a_{ij}$, such that $R^{(t)}(\sum_{(i,j)\in \Delta} A_{ij}(t)x^i y^j)=0$. This means that $F_{1,t}(x,y)$ vanishes to order $m_i$ at $p_i$ for every $1\leq i \leq n$ and $F_{2,t}(x,y)$ vanishes to order $\al_j$ at $q_j^0$ for every $1\leq j \leq k$. This concludes the proof of the lemma and thus of the whole proposition. \ \hfill \rule{.75em}{.75em} \subsection{Passing from specific points to very general} \setcounter{thm}{0} \renewcommand{\thethm In what follows we shall detect several useful ample (resp. nef) vectors $(d;\al_1, \ldots,\al_k)$ by choosing $k$ points $q_1,\ldots,q_k\in \CPTU$ to lie in a very specific convenient position which is not generic. The following lemma shows that this vectors remain ample (resp. nef) also for a very general choice of the points $q_1,\ldots,q_k$. \begin{lem} \label{lem-very_general} Let $F$ be a divisor on a simple rational surface $S$, and $q^{(0)}_1, \ldots q^{(0)}_k \in S \setminus (\Sig^S \cup Supp\,F)$ distinct points. Let $\pi_0:\wtldS_0\rightarrow S$ be the blow-up of $S$ at $q^0_1,\ldots q^0_k$ with exceptional divisors $E^0_i=\pi^{-1}_0 (q^0_i) \, i=1,\ldots,k$. Suppose that for some $f_1,\ldots,f_k \geq 0$ the divisor $\pi^*_0F -\sum_{j=1}^k f_j E^0_j$ is ample (resp. nef). Then, for a very general choice of points $q_1,\ldots,q_k \in S \setminus (\Sig^S\cup Supp\,F)$ the divisor $$\pi^*F-\sum_{j=1}^k f_j E_j$$ is ample (resp. nef) on the blow-up $\pi:\wtldS \rightarrow S$ of $S$ at $q_1,\ldots, q_k$ with exceptional divisors $E_j=\pi^{-1}(q_j) \quad j=1,\ldots,k$. \end{lem} {\bf Proof. } The idea of the proof is very simple. Since $\wtld{F}_0=\pi^*_0F -\sum_{j=1}^k f_j E^0_j$ is assumed to be nef on the blow-up of $S$ at $q^0_1,\ldots q^0_k$, all the divisor classes which intersect $\wtld{F}_0$ negatively do not admit any effective representatives. Now, the point is that if a divisor class on the blow-up of $S$ at specific points has no effective representatives then the same will continue to hold also on the blow-up at generic points. The lemma now follows because $\mbox{Pic}(\wtldS)$ is countable. Let us give now the precise details. We prove the lemma for the ``nef'' case, the ``ample'' being very similar. Consider the following subset of $\mbox{Pic}(S)\times \ZZ_{\geq 0}^k$: $$\calB=\left\{(A,a_1,\ldots a_k) \in \mbox{Pic}(S)\times \ZZ_{\geq 0}^k \bigg| F\cdot A - \sum_{j=1}^k f_j a_j < 0 \right\}.$$ Notice that $\calB$ is a countable set. We claim that for every $B=(A,a_1,\ldots a_k)\in \calB$ there exists a non-empty Zariski-open subset $\calU_B \subset \calC_k(S\setminus (\Sig^S \cup Supp\,F))$ such that for every $(q_1, \ldots, q_k) \in \calU_B$, the surface $\wtldS$ obtained by blowing-up $S$,\ $\pi:\wtldS \rightarrow S$, at $q_1, \ldots, q_k$ does not admit any effective divisor in the class $\pi^*[A]-\sum_{j=1}^k a_j [E_j]$. Once this is proved, we take $\calV=\cap_{B\in \calB} \calU_B$. Obviously $\calV$ is a very general subset of $\calC_k(S\setminus (\Sig^S \cup Supp\,F))$ having the needed properties. Let us prove the existence of the Zariski-open sets $\calU_B$ claimed above. For this end put $\calC=\calC_k(S\setminus (\Sig^S \cup Supp\,F)), \quad X=\calC \times S$, and denote by $pr:X \rightarrow \calC$ the obvious projection. Consider the subvarieties $Y_j\subset X, \quad j=1,\ldots,k$, defined by $$Y_j=\{ ((x_1, \ldots, x_k),x)\mid x=x_j \}.$$ The $Y_j$'s are smooth disjoint subvarieties of $X$ each of which is mapped by $pr$ isomorphically onto $\calC$. Let $\Th: \wtld{X} \rightarrow X$ be the blow-up of $X$ along $Y=\cup_{j=1}^k Y_j$ and write $\wtld{E}_j=\Th^{-1}(Y_j)$ for the exceptional divisors. Given $B=(A,a_1,\ldots a_k)\in \calB$, we denote by $\calL$ the line bundle $$\calL=\calO_{\wtldX}(\wtld{A}-\sum_{j=1}^k a_j \wtld{E}_j) \in \mbox{Pic}(\wtldX),$$ where $\wtld{A}\in \mbox{Div}(\wtldX)$ is the divisor $\Th^*(\calC \times A)$. Finally, for every $\undl{q} \in \calC$ we write $\calL_{\undl{q}}$ for the restriction of $\calL$ to the surface $\wtldS_{\undl{q}} = \Th^{-1} pr^{-1}(\undl{q})$. Let $\undl{q}=(q_1, \ldots, q_k)\in \calC$. It is easy to see that the map $\pi_{\undl{q}}:\wtld{S}_{\undl{q}} \rightarrow S$ defined by the composition $\wtldS_{\undl{q}} \stackrel{\Th}{\longrightarrow} X \stackrel{pr_{_S}}{\longrightarrow} S$ is just the blow-up of $S$ at $q_1, \ldots, q_k$, and that $$\calL_{\undl{q}}=\calO_{\wtld{S}_{\undl{q}}}(\pi^*_{\undl{q}} A-\sum_{j=1}^k a_j E_j).$$ Now, for $\undl{q}^0=(q_1^0, \ldots, q_k^0)$ we know that $\dim H^0(\wtld{S}_{\undl{q}^0},\calL_{\undl{q}^0})=0$ because $\pi_0^*F-\sum_{j=1}^k f_j E_j$ is nef. It follows from the semicontinuity theorem (see~\cite{Ha-AG}) that there exits a Zariski-open neighborhood of $\undl{q}^0$, \ $\calU_B \subset \calC$ such that for every $\undl{q} \in \calU_B$, \ $H^0(\wtld{S}_{\undl{q}},\calL_{\undl{q}})=0$. \ \hfill \rule{.75em}{.75em} \subsection{Proof of the gluing Theorem} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-prf_glue_thm} Now we are in position to prove Theorem~\ref{thm-glue2}. {\bf Proof. } We divide the proof into three steps. In the first step we prove that the resulting divisor $\wtldD=\pi^*D-\sum_{j=1}^k \al_j E_j$ is nef provided that $D_p=\pi_p^*D-mE$ and $v=(m;\al_1,\ldots, \al_k)$ are nef. In the second step we prove that the theorem holds under the assumption that both $D_p$ and $v$ are ample. Finally, in the third step we prove the theorem in its full generality by reducing the problem to the first two steps. {\em Step 1.} Assuming that $D_p=\pi_p^*D-mE$ and $v=(m;\al_1,\ldots,\al_k)$ are nef we prove that $\wtldD=\pi^*D-\sum_{j=1}^k E_j$ is nef. We claim that there exists $N_0>0$ and a divisor $A$ on $S$ such that for every $N>0$ there exists a very general subset $G_N\subset \calC_k (S \setminus (\Sig^S\cup Supp\, D))$ such that for every $(q_1,\ldots, q_k) \in G_N$ the divisor $$D_N=\pi^*A+(N+N_0)\pi^*D - N \sum_{j=1}^k \al_j E_j$$ is nef on the blow-up $\pi:\wtldS \rightarrow S$ of $S$ at $q_1, \ldots, q_k$. Once this is proved step 1 of the proof will be concluded as follows: put $G=\cap_{N=1}^{\infty} G_N$. Clearly $G \subset \calC_k (S \setminus (\Sig^S\cup Supp\, D))$ is a very general subset. Let $(q_1, \ldots, q_k) \in G$ and let $C \subset \wtldS$ be a curve. Since $D_N$ is nef we have $$0\leq D_N \cdot C =A\cdot C +(N+N_0)D\cdot C -N\sum_{j=1}^k C\cdot E_j.$$ Dividing by $N$ and letting $N\rightarrow \infty$ we obtain that $$\wtldD\cdot C = (\pi^*D-\sum_{j=1}^k \al_j E_j)\cdot C \geq 0.$$ As $v$ is nef, we have $m^2\geq \sum_{j=1}^k \al_j^2$ and so $$\wtldD\cdot \wtldD = D \cdot D - \sum_{j=1}^k \al_j^2 \geq D\cdot D-m^2 = D_p \cdot D_p \geq 0,$$ the latter inequality following from the nefness of $D_p$. Thus $\wtldD$ is nef. Let us prove now the existence of $N_0,A,G_N$ claimed above. For the divisor $A$ we choose any divisor on $S$ such that $\pi_p^*A-E$ is ample on $\wtldS_p$. The nefness of $v$ means by definition that there exit $k$ distinct points $q^0_1, \ldots, q^0_k \in \CPTU$ such that the divisor $mL^V-\sum_{j=1}^k \al_j E^V_j$ is nef on $V$ -- the blow-up of $\CPTU$ at $q^0_1, \ldots, q^0_k$. For $N_0>0$ we choose any integer for which $B=N_0 L^V - \sum_{j=1}^k E^V_j$ is ample on $V$. For every $N>0$ define $L_N'\in \mbox{Div}(\wtldS_p),\,\, L_N''\in \mbox{Div}(V)$ to be: $$L_N'=(\pi_p^*A-E)+(N+N_0)D_p= \pi_p^*A+(N+N_0)\pi_p^*D-(mN+mN_0+1)E,$$ $$L_N''=B+N(mL^V-\sum_{j=1}^k \al_jE_J^V)= (Nm+ N_0)L^V-N\sum_{j=1}^k E^V_j.$$ It easily follows from our assumptions on $D_p$ and on $v$ that $L_N',L_N''$ are ample for every $N>0$. Choose an intger $r_N>0$ for which the following two conditions are satisfied: \begin{enumerate} \item[1)] $r_N L_N'$ and $r_N L_N''$ are very ample. \item[2)] $H^1(\wtldS_p, \calO_{\wtldS_p}(r_N L_N'))=0$, and $H^1(V, \calO_V(r_N L_N''))=0$. \end{enumerate} Choose irreducible curves $\wtldC_N' \in |r_N L_N'|$ and $\wtldC_N'' \in |r_N L_N''|$ and put $C_N'=\pi_p(\wtldC_N') \subset S$. By Proposition~\ref{prop-glue} there exist $q_1, \ldots, q_k \in S\setminus (\Sig^S \cup C_N')$ such that the surface $\wtldS$ obtained by blowing-up $\pi:\wtldS \rightarrow S$ at $q_1, \ldots, q_k$ admits an irreducible curve $C_N$ in the linear system $$C_N\in \left| r_N \bigg(\pi^*A+(N+N_0)\pi^*D - N \sum_{j=1}^k \al_j E_j\bigg) \right| = \left| r_N D_N \right|.$$ Noting that $$D_N\cdot D_N \geq N^2(D \cdot D - \sum_{j=1}^k \al_j^2) \geq N^2(D\cdot D -m^2)=N^2 D_p\cdot D_p \geq 0,$$ we conclude that $D_N$ intersects every curve non-negatively and so it is is nef on $\wtldS$. By Lemma~\ref{lem-very_general} we may assume that $(q_1, \ldots,q_k)$ vary in some very general subset $G_N\subset \calC (S \setminus (\Sig^S\cup Supp\, D))$. This completes the proof of step 1. {\em Step 2.} Assuming $D_p$ and $v$ are both ample we prove that $\wtldD=\pi^*D-\sum_{j=1}^k \al_j E_j$ is ample. Here it is more convenient to work with $\QQ$-divisors. First note that step 1 remains true if we take $m$ and $\al_j$ to be rational numbers. It follows from Seshadri's criterion for ampleness (see\cite{Ha-Ample}) that there exists a positive rational number $\eps$ such that $\pi^*D-(1+\eps)mE$ is ample. Since $( (1+\eps)m; (1+\eps)\al_1, \ldots, (1+\eps)\al_k )$ is ample too we have from step 1 that $\wtldD_{\eps}=\pi^*D-(1+\eps)\sum_{j=1}^k \al_j E_j$ is nef. Let us prove that $\wtldD$ is ample by applying Nakai-Moishezon criterion. Indeed, let $\wtldC \subset \wtldS$ be a curve. If $\wtldC=E_j$ is one of the standard exceptional divisors then $\wtldD \cdot \wtldC = \al_j > 0$ (recall that $\al_j >0$ because we assume that $v$ is ample). Otherwise, let $C=\pi(\wtldC)$. If $\wtldC$ does not pass through any of the exceptional divisors then $\wtldD\cdot \wtldC= D\cdot C >0$ because $D$ itself is ample for $D_p$ is. Suppose now that there exits a $j_0$ such that $\wtldC\cdot E_{j_0}>0$. In this case $\wtldD \cdot \wtld C = D\cdot C - \sum_{j=1}^k \al_j \wtldC\cdot E_j > \wtldD_{\eps} \cdot \wtldC \geq 0$. Finally note that $\wtldD\cdot \wtldD > \wtldD_{\eps}\cdot \wtldD_{\eps} \geq 0$. {\em Step 3.} Consider the general case. The case of $D_p$ nef has been treated in step 1 so we may assume that $D_p$ is ample and $v$ is nef. Similarly to step 2 we choose a positive rational number $\eps$ such that both $\pi_p^*D-(1+\eps)E$ and $((1+\eps)m; \al_1, \ldots, \al_k)$ are ample. By step 2 we have that $\pi_p^*D-\sum_{j=1}^k E_j$ is ample. \ \hfill \rule{.75em}{.75em} \ \\ Theorem~\ref{thm-glue1} follows immediately from Theorem~\ref{thm-glue2} by taking $D=dL^S-\sum_{i=1}^n m_i E^S_i$ for a suitable simple rational surface $S$. \ \hfill \rule{.75em}{.75em} \section{Asymptotics on the remainders of $\Oone$} \setcounter{thm}{0} \renewcommand{\thethm \label{sect-Asymptotics} In order to obtain estimates on $\calR_N(\Oone)$ we shall extensively use Theorem~\ref{thm-glue1} in combination with the {\em Cremona action}. The point is, that the Cremona group acts on the set of ample (resp. nef) vectors. Let us briefly summerize the needed facts about the Cremona action. We refer the reader to~\cite{Do-Or} for more details. \subsection{The Cremona action on the ample cone} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-Cremona} Denote by $(H_k,\langle \, , \, \rangle) \quad (k\geq 3)$ the hyperbolic lattice $H_k=\ZZ l \oplus\ZZ e_1 \oplus \ldots \oplus \ZZ e_k$ with the bilinear form $\langle\, ,\, \rangle$ defined by $\langle l,l\rangle=1, \quad \langle l, e_j\rangle=0, \quad \langle e_i,e_j\rangle=-\dlt_{ij}$. Consider the subgroup $Cr_k \subset Aut(H_k,\langle \, , \,\rangle)$, generated by: \begin{enumerate} \item[1)] The symmetric group $S_k \hookrightarrow Aut(H_k,\langle\, ,\,\rangle)$ acting on the last $k$ components. \item[2)] The reflection $R_{123}:(H_k,\langle \, ,\,\rangle) \rightarrow (H_k,\langle \, ,\,\rangle)$ defined by $R_{123}(\eta)=\eta + \langle \eta, r_{123}\rangle r_{123}$, where $r_{123}=l-e_1-e_2-e_3$. \end{enumerate} The group $Cr_k$ is called the {\em Cremona group}. It is easily seen that the reflections $R_{ijk}(\eta)=\eta + \langle \eta, r_{ijk}\rangle r_{ijk}$, where $r_{ijk}=l-e_i-e_j-e_k$, belong to $Cr_k$. Let us mention one more useful transformation which we denote by $SR$. The transformation $SR$ takes a vector $v=(d;m_1, \ldots, m_k) \in H_k$ and sorts it. In other words $SR(v)=(d;m_{\tau(1)}, \ldots, m_{\tau(k)})$, where $\tau$ is a permutation of $\{1,\ldots, k\}$ for which $m_{\tau(1)} \geq \ldots \geq m_{\tau(k)}$. It is obvious that for every vector $v\in H_k$ there exists $\sig \in Cr_k$ such that $SR(v)=\sig(v)$. Given a simple rational surface obtained by blowing up $\Th:V\rightarrow \CPTU$ of $p_1,\ldots,p_n \in \CPTU$, there is an isomorphism of lattices $m_{\Th}:(\mbox{Pic}(V),\,\cdot \,) \rightarrow (H_k,\langle \, ,\, \rangle)$, where $\cdot$ stands for the intersection form on $\mbox{Pic}(V)$. The isomorphism $m_{\Th}$ sends $L^V$ to $l$ and $E^V_i$ to $e_i$. To deduce that $Cr_k$ acts on the set of ample (resp. nef) vectors we need the following lemma which essentially appears in~\cite{Do-Or}. \begin{lem} \label{lem-marking} Let $V$ be a simple rational surface obtained by blowing-up $\Th:V\rightarrow \CPTU$ points $p_1,\ldots,p_n \in \CPTU$ in general position. Then for every $\sig \in Cr_k$ there exists a simple rational surface $V_{\sig}$ obtained by blowing-up $\Th_{\sig} : V_{\sig} \rightarrow \CPTU$ points $q_1,\ldots q_k$ in general position and a biholomorphism $f_{\sig}:V_{\sig} \rightarrow V$ making the following diagram commutative: \[ \begin{CD} \mbox{\em Pic}(V) @>{f^*_{\sig}}>> \mbox{\em Pic}(V_{\sig}) \\ @V{m_{\Th}}VV @V{m_{\Th_{\sig}}}VV \\ H_k @>{\sig}>> H_k \end{CD} \] \end{lem} Combining this with Lemma~\ref{lem-very_general} we immediately obtain the following \begin{lem} \label{lem-Cremona} When $k\geq 3$ the group $Cr_k$ acts on the set of ample (resp. nef) vectors viewed as a subset of $H_k$. \end{lem} \noindent {\bf Remark.} From Lemma~\ref{lem-very_general} it follows that there exists (at least) one simple rational surface $S$, obtained by blowing-up $k$ distinct points in $\CPTU$, $\Th:\wtldS\rightarrow \CPTU$ such that $\calL \in \mbox{Pic}(S)$ is ample (resp. nef) iff $m_{\Th}(\calL)$ is ample (resp. nef). Hence, the set of ample (resp. nef) vectors is closed under addition and multiplication by positive (resp. non-negative) integers. Henceforth we shall denote by $\calK_k \subset H_k \otimes \RR$ (resp. $\calKbar_k$) the cone generated by all ample (resp. nef) vectors. \subsection{An algorithmic procedure for detecting ample classes} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-Alg} Given two vectors $v_1=(d;m_1, \ldots, m_n) \in H_n$ and $v_2=(\dlt; \al_1, \ldots, \al_k) \in H_k$ with $\dlt=m_i$ for some $1\leq i\leq n$, define a new vector $v_1 \dzi v_2 \in H_{n+k-1}$ by setting $$v_1 \dzi v_2=(d; m_1,\ldots, m_{i-1}, \al_1, \ldots, \al_k, m_{i+1}, \ldots, m_n).$$ Theorem~\ref{thm-glue1} asserts that if $v_1$ is ample (resp. nef) and $v_2$ is nef, then $v_1 \dzi v_2$ is ample (resp. nef). Given a vector $v_0 \in H_N$ the ampleness of which we want to prove we try to find a decomposition $v_0=v_1 \#_{_{i_1}} u_1$ where $u_1 \in H_{k_1}$ is known to be nef and $v_1\in H_{n_1},\,\,(k_1+n_1 - 1 = N)$. If $v_1$ turns to be ample then we are done in view of Theorem~\ref{thm-glue1}. To check the ampleness of $v_1$ we first "simplify" it by applying to it Cremona transformations. For example, we may try, using Cremona transformations to reduce the degree of $v_1$ (by the degree of $v=(d;\mu_1, \ldots, \mu_k)$ we mean $\deg\,v=d$). Let $v_1'$ be a simpler vector in the same orbit of $v_1$ under the action of $Cr_{n_1}$ (e.g. $v_1'$ having minimal degree in the orbit, or having some other convenient feature). By Lemma~\ref{lem-Cremona} $v_1$ is ample iff $v_1'$ is. Now we apply the whole process to $v_1'$ and so on. In this way we obtain a sequence of vectors $v_1,u_1,v_1', \ldots, v_r,u_r,v_r'$ where $v_j'$ is a Cremona simplification of $v_j \in H_{n_j}$, $u_j \in H_{k_j}$ is a nef vector and $v_j'=v_{j+1} \#_{_{i_{j+1}}} u_{j+1}$ for some $i_{j+1}$. Note that at each stage the number of points decreases, namely $n_{j+1}<n_j$ provided that $k_j > 1$. The process ends successfully as soon as we are able to prove that $v_r$ is ample for some $r$. We remark that if one of the $v_j$ turns out not to be ample then process fails to give any information because the converse of Theorem~\ref{thm-glue1} is not true. However, we may attempt to find other decomposition sequences $v_1,u_1,v_1', \ldots,v_r,u_r,v_r'$ (see Section~\ref{sect-Remarks}). The same procedure can be applied for proving nefness of a vector $v_0$ by requiring that $v_r$ is nef instead of ample. In the next subsection we shall apply this process in order to prove Theorem~\ref{thm-asymp1} and Corollary~\ref{cor-coef2}. In order to make the preceding procedure applicable we must first endow ourselves with an initial large enough collection of ample and nef vectors which will play the role of the $u_j$'s and of $v_r$. To simplify notations let us agree that $(d;\al_1^{\times r_1}, \ldots \al_k^{\times r_k})$ stands for $$(d;\underbrace{\al_1, \ldots, \al_1}_{\mbox{\tiny $r_1$ times}}, \ldots \ldots ,\underbrace{\al_k, \ldots, \al_k}_{\mbox{\tiny $r_k$ times}}) \in H_N,$$ where $N=\sum_{j=1}^k r_j$. The next lemma provides a modest initial collection of ample and nef vectors which is sufficient for our purposes. \begin{lem} \label{lem-vectors} The following vectors are nef (resp. ample) on a very general rational surface: \begin{enumerate} \item[1)] $(d; 1^{\times r})$, where $d^2 \geq r$ (resp. $d^2 > r$). \item[2)] $(d; m_1, m_2, 1^{\times r})$ where $d\geq m_1+m_2$ and $d^2 \geq m_1^2+m_2^2+r$. \end{enumerate} \end{lem} \noindent {\bf Remark.} The ``ample'' case of statement 1 above has been proved by Xu in~\cite{Xu-Divisors} and by K\"{u}chle in~\cite{Ku}. Below however, we present an alternative proof suggested by Ilya Tyomkin. {\bf Proof. } Notice first that in view of Lemma~\ref{lem-very_general} it is enough to prove that the above vectors are nef (resp. ample) on a specific simple rational surface.\\ \ \\ 1) Consider first the case $d^2>r$. In~\cite{Nag} (consult also~\cite{Sh-Ty}) Nagata proved that if $N$ is a square, then for generic points $p_1, \ldots, p_N\in \CPTU$ and for every irreducible curve $C\subset \CPTU$ the following strict inequality holds: $$\mbox{deg}(C) > \frac{\sum_{j=1}^N \mbox{mult}_{p_j}(C)}{\sqrt{N}}. \eqno(1)$$ Let $V_r$ be the blow-up of $\CPTU$ at $r$ generic points and denote by $\Th:\wtld{V}_r \rightarrow V_r$ the blow-up of $V_r$ at $d^2-r$ generic points. Thus $\wtld{V}_r$ is the blow-up of $\CPTU$ at $N=d^2$ generic points and it follows from inequality (1) that the divisor $\wtld{D}=\Th^*(dL^{V_r}-\sum_{j=1}^r E^{V_r}_j) - \sum_{j=r+1}^{d^2} E_j$ intersects every curve positively. This immediately implies that $D=dL^{V_r}-\sum_{j=1}^r E^{V_r}_j$ intersects any curve in $V_r$ positively. As $D\cdot D > 0$ the statement follows from Nakai-Moishezon criterion (see~\cite{Ha-Ample}). The proof for the nef case ($d^2 \geq r$) is much easier. Indeed, let $C\subset \CPTU$ be an irreducible curve of degree $d$, and let $p_1, \ldots, p_r$ be distinct points on $C$ at which $C$ is smooth. Let $V$ be the blow-up of $\CPTU$ at $p_1, \ldots, p_r$ and let $D$ be the proper transform of $C$ in $V$, $D\in | d L^V-\sum_{j=1}^r E_j|$. As $D$ is an irreducible curve of non-negative self intersection the vector $(d; 1^{\times r})$ corresponding to the divisor class of $D$ is nef on $V$. \ \\ \ \\ 2) Set $D=dL - m_1 E_1 - m_2 E_2$. Consider the linear system $|D|$ on $V_2$ -- the blow-up of $\CPTU$ at 2 points. As $D=m_1(L-E_1) + m_2(L-E_2) + (d-m_1-m_2)L$ it is easy to see that $|D|$ is not empty and has no base-points, hence by Bertini theorem there exists an irreducible (smooth) curve $C\in |D|$. Choose $r$ distinct points $p_1, \ldots, p_r \in C \setminus (E_1 \cup E_2)$ and let$\wtld{V}$ be the blow-up of $V$ at $p_1, \ldots, p_r$. Finally denote by $\wtld{C}$ be the proper transform of $C$ in $\wtld{V}$. We have $\wtld{C} \in |dL- m_1E_1 -m_2E_2 - \sum_{j=3}^{r+2} E_j|$. As $\wtld{C}$ is irreducible and $\wtld{C} \cdot \wtld{C} \geq 0$, the vector $(d;m_1,m_2,1^{\times r})$ is nef. \ \hfill \rule{.75em}{.75em} \ \\ \\ {\bf Remark.} Note that the cones $\calK_n$ (resp. $\calKbar_n$) can be explicitly computed when $n<9$ (see~\cite{Dmz},~\cite{F-M}), and so can be joined to the initial collection of ample and nef vectors to be applied in the framework of the process mentioned above. \subsection{Proofs of Theorem~\ref{thm-asymp1} and Corollary~\ref{cor-coef2}} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-prfs1} We start with the proof of Theorem~\ref{thm-asymp1}. {\bf Proof. } 1) Let $N=a^2l^2+2l$ and $v_0=(a^2l+1;a^{\times N})$. As $\langle v_0, v_0 \rangle = 1$, nefness of $v_0$ will give the needed estimate for $\calR_N(\Oone)$. The decomposition $N=(al-1)^2 + n$, where $n=2al+2l-1$, leads us to $v_0=v_1 \#_{_1} u_1$ where $$v_1=\Big( a^2l+1; a(al-1), a^{\times n} \Big) \in H_{n+1}, \quad u_1=a\Big(al-1; 1^{\times (al-1)^2}\Big)\in H_{(al-1)^2}.$$ By Lemma~\ref{lem-vectors} $u_1$ is nef, hence in view of Theorem~\ref{thm-glue1} we are reduced to proving that $v_1$ is nef. This turns out to be easy by using Cremona transformations. Indeed let $$v_1'=R_{1,n-1,n} \circ R_{1,n-3,n-2} \circ \ldots \circ R_{123} (v_1),$$ where $R_{ijk}\in Cr_{n+1}$ are defined in Section~\ref{subsect-Cremona}. A straight forward computation shows that $v_1'=(a+l; l-1, 1^{\times n-1},a)$. This vector is nef by Lemma~\ref{lem-vectors}, and therefore $v_1$ too. \\ \noindent 2) Let $N=a^2l^2-2l$ and $v_0=(a^2l-1; a^{\times N})$. Again $\lbr v_0,v_0\rbr = 1$, hence in order to prove the needed estimate on $\calR_N(\Oone)$ we have to prove that $v_0$ is nef. Using the decomposition $N=(al-2)^2+n_1$, where $n_1=4al-2l-4$, we note that $v_0=v_1 \#_{_1} u_1$ where $$v_1=\Big( a^2l-1; a(al-2), a^{\times n_1} \Big) \in H_{n_1+1}, \quad u_1=a\Big( al-2; 1^{\times (al-2)^2}\Big) \in H_{(al-2)^2}.$$ By Lemma~\ref{lem-vectors} $u_1$ is nef. We are thus reduced to proving nefness of $v_1$. By applying similar Cremona transformations as in 1, we obtain that $v_1'=(a^2l-2al+l+1; (al-l-2)(a-1), (a-1)^{\times n_1})$ lies in the same orbit as $v_1$. Let us apply now the same algorithm again on $v_1'$. For this, consider the decomposition $v_1'=v_2\#_{_1} u_2$, where $$v_2=\Big(a^2 l-2al+l+1; (al-l-1)(a-1), (a-1)^{\times 2al-1}\Big) \in H_{2al-1},$$ $$u_2=(a-1)\Big( al-l-1; al-l-2, 1^{\times 2al-2l-2}\Big).$$ By Lemma~\ref{lem-vectors} $u_2$ is nef, thus we are reduced to proving that $v_2$ is nef. Using similar Cremona transformations as in 1 we obtain that $v_2'=\big(a+l-1; l-1, 1^{\times 2al-2}, a-1 \big)$ lies in the same orbit as $v_2$. But by Lemma~\ref{lem-vectors} $v_2'$ is nef. \\ 3) Let $N=a^2 l^2+l$ and suppose that $a=2^k b$ with $k\geq 0$ and $b$ odd. The assumption appearing in the statement of the Theorem is that $l>2b$. Note that we may assume that $l$ is odd, since when $l$ is even we have $N=(2a)^2(\frac{l}{2})^2+2\frac{l}{2}$ and this is already covered in 1 above. In order to prove the needed estimate on $\calR_N(\Oone)$ we have to show that the vector $v=\big(2a^2 l+1; 2a^{\times (a^2l^2+l)}\big)$ is nef. Let us prove a slightly stronger statement, namely: {\em Claim.} The vector $v_0=\big(2a^2 l+1; 2a^{\times (a^2l^2+l)}, 1\big)$ is nef. We argue by induction on $k$. Consider first the case $k=0$. We have $v_0=w\#_{_1}u$, where $$w=\Big(2a^2l+1; 2a(al-1), 2a^{\times 2al+l-1},1\Big), \quad \mbox{and} \quad u=2a\Big(al-1; 1^{\times (al-1)^2}\Big).$$ The latter being nef, we are reduced to proving nefness of $w$. Applying suitable Cremona transformation to $w$, we obtain the vector $$w'=\Big(\frac{l+1}{2}+a; \frac{l-1}{2}-a, 1^{\times 2al+l}\Big).$$ Since $l>2b=2a$ we have that $\frac{l-1}{2}-a\geq 0$ and so $w'$ is nef by Lemma~\ref{lem-vectors}. This completes the basis of the induction. Let us turn now to the case $k>0$. We have $$v_0=\Big(\,\big(v_1\#_{_3}u_1\big)\#_{_2}u_1\Big)\#_{_1}u_1,$$ where $$v_1=\Big(2a^2l+1;(a^2l)^{\times 3},2a^{\times (\frac{a}{2}l)^2+l}, 1\Big), \quad u_1=2a\Big(\frac{a}{2} l; 1^{\times (\frac{a}{2}l)^2}\Big).$$ Again, $u_1$ is nef. As for $v_1$, it lies in the same orbit under the Cremona action as the vector $v_1'=\big(a^2l+2; 2a^{\times (\frac{a}{2}l)^2+l}, 1^{\times 4}\big)$. Consider now the decomposition $v_1'=v_2 \# (2; 1^{\times 4})$, where $v_2=\big(a^2l+2; 2a^{\times (\frac{a}{2}l)^2+l},2\big)$ and $\#$ stands for gluing at the last coordinate of $v_2$. As $(2; 1^{\times 4})$ is nef, it is enough to prove that $v_2$ is nef. Putting $c=\frac{a}{2}=2^{k-1}b$, we have that $$v_2=2(2c^2l+1; 2c^{\times c^2 l^2+l},1).$$ By the induction hypothesis $v_2$ is nef. This completes the proof of the claim. The Theorem now follows easily. \ \hfill \rule{.75em}{.75em} \ \\ Let us turn now to the proof of Corollary~\ref{cor-coef2}. {\bf Proof. } Let $D=\pi^*\calO_{\CPTU}(d)-2\sum_{j=1}^N E_j$ and suppose that $D \cdot D \geq 0$. {\em Step 1.} Consider first the case $N=k^2+k$ for some $k$. By Theorem~\ref{thm-asymp1}-3 $$\calR_N(\Oone)\leq \frac{1}{(2k+1)^2}.$$ Since $D\cdot D =1$, this implies that $D$ is nef. {\em Step 2.} Consider the general case. The condition $D\cdot D \geq 0$ reads $d^2\geq 4N$. We may assume that $d$ is odd, for the case of $d$ even is precisely the contents of Xu's theorem from Section~\ref{sect-Intro} (see~\cite{Xu-Divisors}. Writing $d=2k+1$, the condition $d^2\geq 4N$ gives $k^2+k >N$. By step 1, $\pi^*\calO_{\CPTU}(d)-2\sum_{j=1}^{k^2+k}E_j$ is nef, hence also $\pi^*\calO_{\CPTU}(d)-2\sum_{j=1}^N E_j$. \ \hfill \rule{.75em}{.75em} \ \\ \\ {\bf Remark.} More careful considerations, in the spirit of the proof of Theorem~\ref{thm-asymp1} actually show that {\em when $d>5$, the divisor $D=\pi^*\calO_{\CPTU}(d)-2\sum_{j=1}^N E_j$ is ample iff $D\cdot D>0$.} To prove this one has to sharpen first the second statement of Lemma~\ref{lem-vectors} and prove that $(d; m_1, m_2, 1^{\times r})$ is ample when $d > m_1+m_2$ and $d^2 > m_1^2+m_2^2+r$. This can be done by similar, though more delicate, arguments to those used to prove nefness of these vectors. Then, using the ``ample+nef $\Rightarrow$ ample'' case of Theorem~\ref{thm-glue1} one deduces as in the proof of Theorem~\ref{thm-asymp1} that the divisor $D$ is ample for $N=k^2+k$, when $k>2$. The case of general $N$ can be easily reduced to $N=k^2+k$ as in the preceding proof. \subsection{Proof of Corollary~\ref{cor-Nag}} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-prfs2} {\bf Proof. } Let $N=N_1 N_2$. Nagata's conjecture for $N$ is equivalent to the nefness of vector $v=(d; m^{\times N})$ for every $d,m>0$ which satisfy $d^2-Nm^2>0$. Let $d,m$ be two such numbers. Choose a positive rational number $x$ such that $d^2 > x^2 N_2 > m^2 N$. The assumption of Nagata's conjecture for $N_1$ and $N_2$ implies that the vectors $u=(x;m^{N_1}) \in \QQ^{N_1+1}$ and $w=(d; x^{\times N_2}) \in \QQ^{N_2+1}$ are nef. We have $v=\big( \ldots ((w \#_{_{N_2}}u) \#_{_{N_2-1}}u) \ldots \big)\#_{_{1}}u$. Observing that Theorem~\ref{thm-glue2} remains valid also for vectors of rational numbers, we conclude that $v$ is also nef. \ \hfill \rule{.75em}{.75em} \section{A conjecture relating continued fractions and remainders of a line bundle} \setcounter{thm}{0} \renewcommand{\thethm \label{sect-Conjecture} The goal of this section is to propose a conjecture concerning estimates on the values of the homogeneous remainders of $\Oone$, defined in Section~\ref{subsect-Applic}. It turns out that all the cases appearing in the statement of Theorem~\ref{thm-asymp1} are particular cases of this conjecture. Let us first recall some relevant facts from classical number theory. Given a square-free number $N$, consider the following Diophantine equation in the unknowns $d,m$ $$d^2-Nm^2=1.$$ This equation had been attached-to the name {\em Pell's equation} in the ancient literature and has been extensively studied by many mathematicians in the 17'th and 18'th centuries including Leonard Euler (see~\cite{Niv,Ir-Ro, VndP}). the classical result about the solutions of this equation is that all solutions come from {\em continued fractions expansions} of $\sqrt{N}$. Let us write $\lbr a_0, a_1, \ldots, a_n \rbr$ for the continued fractions expansion \[ a_0+ \cfrac{1}{a_1+\cfrac{1}{\ddots +\cfrac{1}{a_{n-1} +\cfrac{1}{a_n}}}} \] \noindent Similarly, we denote by $\lbr a_0, a_1, \ldots \rbr$ an infinite continued fractions expansion. It is not hard to see that the continued fractions expansion of $\sqrt{N}$ must be of the following special periodic form $$\sqrt{N}=\lbr a_0,a_1,\ldots, a_{n-1}, 2a_0, a_1,\ldots, a_{n-1}, 2a_0, \ldots \rbr,$$ hence we shall write from now on $\sqrt{N}=\lbr a_0,\overline{a_1,\ldots,a_{n-1}, 2a_0} \rbr$ where the bar denotes the periodic part. Moreover, it turns out that $a_i=a_{n-i}$ for every $1\leq i \leq n-1$, (i.e. $(a_1,\ldots,a_{n-1})$ is a palindrome). Define a rational number $\frac{d}{m}$ as follows: if $n$ is even put $$\frac{d}{m}=\lbr a_0, a_1,\ldots, a_{n-1} \rbr,$$ while for $n$ odd $$\frac{d}{m}=\lbr a_0, a_1,\ldots, a_{n-1}, 2a_0, a_1,\ldots, a_{n-1}\rbr.$$ It is well known that $(d,m)$ provides the minimal solution of Pell's equation, called the {\em fundamental solution}. Moreover, any other solution of Pell's equation is obtained in a similar manner -- by truncating the infinite continued fraction of $\sqrt{N}$ one term before the end of one of its periods. More precisely, $(d,m)$ solves Pell's equation iff $$\frac{d}{m}=\lbr a_0,\overline{a_1,\ldots, a_{n-1}, 2a_0}^{\, \times r}, a_1, \ldots, a_{n-1}\rbr , \,\,\, \mbox{where $r$ is odd if $n$ is odd}.$$ This formula means that the periodic part \ $a_1, \ldots a_{n-1}, 2a_0$ \ should be taken $r$ times and then once more without the last member $2a_0$. The number $r$ is allowed to be any non-negative integer in case $n$ is even, and $r$ must be odd if $n$ is odd. \ \\ Our conjecture is the following \begin{cnj} \label{cnj-CF} Let $N>9$ be a square-free number and let $\frac{d}{m}$ be the fundamental solution of the corresponding Pell's equation. Then:\\ {\em 1)} The vector $(d;m^{\times N})$ is nef. \\ {\em 2)} $\calR_N(\Oone)\leq \frac{1}{d^2}$. \end{cnj} Our conjecture is much weaker than Nagata's, on the other hand it seems more accessible. Indeed, our methods provide a proof for the conjecture in the following cases: \\ \ \\ 1. {\em Consider the case that $\sqrt{N}$ has a 2-periodic continued fractions expansion $\sqrt{N}=\lbr a_0,\overline{a_1,2a_0} \rbr$.} It is easy to see that this is the case iff $a_1|2a_0$ and $N=a_0^2 + 2\frac{a_0}{a_1}$. The solution of Pell's equation is $d=1_0a_1+1,\; m=a_1$. \begin{enumerate} \item[A)] {\em Suppose that $a_1|a_0$}. Putting $a=a_1$ and $l=\frac{a_0}{a_1}$ we get $N=a^2l^2+2l$, and so by Theorem~\ref{thm-asymp1} our conjecture holds in this case. \item[B)] {\em Suppose that $a_1 \nmid a_0$ and $2^{k}a_0 > a_1^2$, where $k$ is the maximal integer for which $2^k | a_1$}. Since $a_1|2a_0$, $a_1$ must be even. Putting $a=\frac{a_1}{2}$ and $l=\frac{2a_0}{a_1}$ we obtain $N=a^2l^2+l$ and $l > \frac{a}{2^{k-2}}$. By Theorem~\ref{thm-asymp1} our conjecture holds. \end{enumerate} \ \\ 2. {\em Consider $N$'s of the form $N=a^2l^2-2l$}. It is not hard to see that $d=a^2l-1,m=a$ satisfy Pell's equation $d^2-Nm^2=1$. By Theorem~\ref{thm-asymp1} we have $\calR_N(\Oone)\leq \frac{1}{d^2}$. Therefore, if $(d',m')$ is the fundamental solution of Pell's equation then $\calR_N(\Oone)\leq\frac{1}{d^2} \leq \frac{1}{{d'}^2}$, and so the conjecture holds. Note that in this case the expansion of $\sqrt{N}$ will usually be longer than 2 (example: $\sqrt{14}= \lbr 3,\overline{1,2,1,6} \rbr$).\footnote{It is not hard to see that if $N=a^2l^2-2l$ then $\sqrt{N}$ has 2-periodic expansion with minus signs.} \\ \ \\ 3. Let us mention two other examples which do not fall into the above categories. \begin{enumerate} \item[A)] $N=19$. In this case $\sqrt{19}=\lbr 4,\overline{2,1,3,1,2,8} \rbr$. The fundamental solution is $d=170, m=39$. Thus, our conjecture suggests that $\calR_{19}(\Oone) \leq \frac{1}{{170}^2}$. \item [B)] $N=22$. In this case $\sqrt{22}=\lbr 4,\overline{1,2,4,2,1,8}\rbr$. The fundamental solution is $d=197, m=42$. Thus, our conjecture suggests that $\calR_{22}\leq \frac{1}{{197}^2}$. \end{enumerate} Let us prove that the conjecture indeed holds in the cases 3.A and 3.B. We start with $N=19$. By Lemma~\ref{lem-vectors} the vector $u=39(2; 1^{\times 4})=(78; 39^{\times 4})$ is nef. We have $$(170; 39^{\times 19})= (\,(\,(170;78^{\times 3}, 39^{\times 7}) \#_{_{3}} u) \#_{_{2}} u) \#_{_{1}}u.$$ Thus, we are reduced to proving that $v=(170;78^{\times 3}, 39^{\times 7})$ is nef. To do this we apply the following Cremona transformations successively: \begin{enumerate} \item[1)] Replace $v$ by $R_{123} (v)$. \item[2)] Sort the vector $v$, that is, replace $v$ by $SR(v)$, where the transformation $SR$ is the one defined in~\ref{subsect-Cremona}. \end{enumerate} Applying this process enough times we finally arrive to the vector $(1; 0^{\times 10})$ which is nef. The case $N=22$ is similar. Here we use the decomposition $$(197; 42^{\times 22})= (\,(\,(\,(197;84^{\times 4}, 42^{\times 6}) \#_{_{4}} u) \#_{_{3}} u) \#_{_{2}} u) \#_{_{1}}u,$$ with $u=42(2; 1^{\times 4})=(84; 42^{\times 4}).$ Applying the preceding process successively to $(197;84^{\times 4}, 42^{\times 6})$ we obtain again the vector $(1; 0^{\times 10})$ which is nef. \section{The limits of the algorithm} \setcounter{thm}{0} \renewcommand{\thethm \label{sect-Remarks} In its present version, the algorithm described in Section~\ref{subsect-Alg} has the disadvantage that it does not tell which decomposition $v'_j=v_{j+1}\# u_{j+1}$ one should choose at each stage in order the whole process to end successfully. We would like to emphasize that this decision is sometimes crucial as the following example shows: \\ Let $v_0=(10; 3^{\times 11}) \in H_{11}$. if one tries to decompose $v_0$ as $v_0=(10;3^{\times 2}, 9) \# (9;3^{\times 9})$ the process will fail to give any information on $v_0$. The reason is that although $(9;3^{\times 9})$ is nef $(10;3^{\times 2}, 9)$ is not, and so we cannot apply the gluing theorem. However, the decomposition $v_0=(10;3^{\times 7}, 6) \# (6; 3^4)$ will eventually lead to a successful ending of the algorithm, thus proving that $v_0$ is nef. It would be useful of course to find a rule for choosing the ``optimal'' decomposition at each stage. Finally, let us mention one simple example for which it seems that the algorithm fails to give information always. Consider the vector $v_0=(19,6^{\times 10})$ which by Nagata's conjecture should be ample. However, it seems that the vector $v_0$ is {\em indecomposable} in the sense that it is impossible to find even nef vectors $v_1 \in H_{n_1}, u_1\in H_{k_1}$ with $n_1,k_1 < 10$, such that $v_0'=v_1\# v_2$ lies in the same orbit as $v_0$ under the Cremona action. It would be interesting to find the precise conditions for an ample (resp. nef) vector $v$ to be indecomposable. \section{Symplectic interpretations} \setcounter{thm}{0} \renewcommand{\thethm \label{sect-Symplectic} The purpose of this section is to explain the intuition which give rise to the gluing Theorem~\ref{thm-glue2}. Interestingly enough this comes from symplectic geometry. Symplectic geometry is the branch of geometry dealing with the structure of symplectic manifolds which are by definition pairs, $(M,\Om)$, consisting of a smooth manifold $M$ and a non-degenerated closed differential 2-form $\Om$. The reader is referred to~\cite{A-G} and~\cite{M-S} for the foundations. Due to developments in this field of research in the last decade, many analogies has been discovered between symplectic and complex manifolds. These become especially striking in dimension 4, where symplectic 4-manifolds play the role of complex surfaces. In several cases it turned out that algebro-geometric considerations, remain true when properly translated into the symplectic category, and so gave rise to new theorems in the symplectic framework. This principle is reflected very well in the classification of rational and ruled symplectic manifolds of Lalonde and McDuff, in the symplectic packing theorems of McDuff and Polterovich, in Ruan's symplectization of the extremal rays theory etc. In this paper we have, in some sense, reversed this direction of reasoning. Our main theorem is in fact an algebro-geometric translation of a very simple symplectic fact arising from the theory of symplectic packing. We refer the reader to~\cite{M-P} for an excellent exposition on the symplectic packing problem. Recall from~\cite{M-P} that a {\em symplectic packing} of $(M,\Om)$ by $N$ balls of radii $\lam_1,\ldots, \lam_N$ is a symplectic embedding $$\phipack,$$ where $B(\lam_j)$ stands for the standard Euclidean closed ball of radius $\lam_j$ of the same dimension as $M$, endowed with its standard symplectic structure $\omstd=dx_1\wedge dy_1 + \ldots + dx_n \wedge dy_n$. It was discovered by McDuff that every symplectic packing gives rise to a symplectic form $\wtldOm$ on the blow-up $\Th:\wtldM \rightarrow M$ of $M$ at the points $p_1=\vphi_1(0), \ldots p_N=\vphi_N(0)$. This form lies in the cohomology class $$[\wtldOm]=[\Th^* \Om]-\pi \sum_{j=1}^N \lam_j^2 e_j, \eqno(1)$$ where $e_j$ denotes the Poincar\'{e} dual to the homology class of the exceptional divisor $E_j$ of the blow-up. This procedure is called {\em symplectic blowing-up}. Conversely, given a symplectic form $\wtldOm$ on $\wtldM$ which is non-degenerated on the exceptional divisors $E_j$ and with cohomology class as in (1) above, one can perform {\em symplectic blowing-down} at the exceptional divisors and obtain a symplectic form $\Om$ on $M$ and a symplectic packing $\phipack$. Consider the symplectic manifold $(\CC P^2, \sig)$ where $\sig$ is the Fubini-Studi K\"{a}hler form normalized such that the area of a projective line is $\pi$. Its cohomology class is $\pi l$, where $l\in H^2(\CPTU, \ZZ)$ is the standard positive generator. Call a vector of positive numbers $(d;m_1,\ldots, m_k)$ {\em symplectic} if the cohomology class $$d \Th_V^*l - \sum_{j=1}^k m_j e_j$$ can be represented by a symplectic form $\wtld{\om}$ on some blow-up $\Th_V:V \rightarrow \CC P^2$ of $\CC P^2$ at some $k$ distinct points, in such a way that $\wtld{\om}$ is non-degenerated on the exceptional divisors. Now, let $M$ be a complex surface and $\Th_p:\wtldM_p \rightarrow M$ its blow-up at the point $p\in M$ with exceptional divisor $E$. Denote by $e$ the Poincar\'{e} dual to the homology class of $E$. \begin{prop} \label{prop-Symplectic} Let $a \in H^2(M)$ and suppose that there exists a positive number $m$ such that the cohomology class $\Th_p^*a -me\in H^2(\wtldM_p)$ can be represented by a symplectic form whose restriction to $E$ is non-degenerated.\footnote{This means that $E$ is a symplectic submanifold with respect to this form.} Then, for every symplectic vector $(m; \al_1, \ldots, \al_k)$, the cohomology class $\Th^*a-\sum_{j=1}^k \al_j e_j$ on the blow-up $\Th:\wtldM \rightarrow M$ at some $k$ points can be represented by a symplectic form. \end{prop} The proof is based on the following very simple observation. If $\Th_p^*a -me$ has a symplectic representative $\wtldOm$, then by symplectic blowing down one obtains a symplectic form $\Om$ on $M$ and an embedding $\vphi$ of a standard 4-dimensional ball of radius $\sqrt{\frac{m}{\pi}}$ into $(M, \Om)$. The same argument with slight modifications, applied to the vector $(m; \al_1, \ldots, \al_k)$, implies that the standard ball of radius $\sqrt{\frac{m}{\pi}}$ admits a symplectic packing, say $\phi$, by $k$ balls of radii $\sqrt{\frac{\al_1}{\pi}}, \ldots, \sqrt{\frac{\al_k}{\pi}}$. Composing these two embeddings we conclude that $(M, \Om)$ admits a symplectic packing $\vphi \circ \phi$ of $k$ balls of radii $\sqrt{\frac{\al_1}{\pi}}, \ldots, \sqrt{\frac{\al_k}{\pi}}$. The proposition follows now from symplectic blowing-up. For completeness, here are the precise arguments of the proof. {\bf Proof. } Let $\wtldOm$ be a symplectic form on $\wtldM_p$ lying in the cohomology class $\Th_p^*a -me$ and suppose that the restriction of $\wtldOm$ to $E$ is non-degenerated. Applying symplectic blowing-down to $\wtldOm$ we obtain a symplectic form $\Om$ on $M$ lying in the cohomology class \ $a$ \ and a symplectic embedding $\vphi:B(\sqrt{\frac{m}{\pi}}) \rightarrow (M, \Om)$. Let $\wtld{\om}$ be a symplectic form on the blow-up $\pi:V \rightarrow \CPTU$ of $\CPTU$ lying in the cohomology class $m\Th_V^*l-\sum_{j=1}^k \al_j e_j$ and whose restriction to the exceptional divisors $E_j$ is non-degenerated. Blowing-down symplectically we obtain a symplectic form $\om$ on $\CPTU$ lying in the cohomology class \ $ m l$ \ and a symplectic packing $\psi:\coprod_{j=1}^k B(\sqrt{\frac{\al_j}{\pi}}) \rightarrow (\CPTU, \om)$. Since any two cohomologous symplectic forms on $\CPTU$ are symplectomorphic we may assume that $\om=\frac{m}{\pi}\sig$. It can be proved by the methods of~\cite{M-P} that there exists a symplectic submanifold (with respect to $\om$) $L\subset M$, homologous to a projective line, which is disjoint from $\mbox{Image}\;\psi$. It is well known that $(\CPTU \setminus L, \frac{m}{\pi}\sig) \approx B(\sqrt{\frac{m}{\pi}})$. We thus obtain a symplectic packing $\phi:\coprod_{j=1}^k B(\sqrt{\frac{\al_j}{\pi}}) \rightarrow B(\sqrt{\frac{m}{\pi}})$. The composition $\vphi\circ\phi$ is a symplectic packing of $(M,\Om)$ by $k$ balls of radii $\sqrt{\frac{\al_1}{\pi}}, \ldots, \sqrt{\frac{\al_k}{\pi}}$. Blowing-up symplectically with respect to this embedding yields a symplectic form on the blow-up $\Th: \wtldM \rightarrow M$ of $M$ at $k$ points, which lies in the cohomology class $\Th^* a - \sum_{j=1}^k \al_je_j$. \ \hfill \rule{.75em}{.75em} \ \\ Let us try to translate Proposition~\ref{prop-Symplectic} to the language of algebraic geometry. Keeping in mind that in the symplectic category the role of K\"{a}hler forms is played by symplectic forms and the role of complex submanifolds by symplectic submanifolds, the K\"{a}hlerian translation should read: {\em ``If the cohomology class $\Th_p^*a-me$ has a K\"{a}hler representative than for every K\"{a}hler vector $(m; \al_1, \ldots, \al_k)$ the cohomology class $\Th^*a - \sum_{j=1}^k \al_j e_j$ has a K\"{a}hler representative too''.} \\ Here, we call a vector $(m; \al_1, \ldots, \al_k)$ K\"{a}hler if the cohomology class $$m \Th_V^*l - \sum_{j=1}^k \al_j e_j$$ can be represented by a K\"{a}hler form $\wtld{\om}$ on some simple rational surface $\Th_V:V \rightarrow \CC P^2$ obtained by blowing-up $\CC P^2$ at some $k$ distinct points. Due to Lefschetz theorem on $(1,1)$ classes and Kodaira's embedding theorem it follows that on a complex manifold there is a bijection -- via Poincar\'{e} duality -- between the set of homology classes of ample $\QQ$-divisors and the set of rational cohomology classes which can be represented by K\"{a}hler forms. Poincar\'{e} dualizing the ``K\"{a}hlerian translation'' we are naturally led to the following: {\em \ \ ``Let $D$ be a divisor on $M$ such that $\Th_p^*D-mE$ is ample. Then for every ample vector $(m; \al_1, \ldots, \al_k)$ the divisor $\Th^*D -\sum_{j=1}^k \al_j E_j$ is ample too''.} \ \ This is precisely the contents of Theorem~\ref{thm-glue2} for the case that $M$ is a simple rational surface. The technical machinery which made the whole translation rigorous is Shustin's curve gluing technique which we used in Section~\ref{sect-Gluing}. We would like to emphasize that the same ``symplectic reasoning'' suggests that if we replace the surface $S$ in the statement of Theorem~\ref{thm-glue2} by any projective surface, the Theorem should remain true. Similar symplectic arguments suggest that an appropriate version of Theorem~\ref{thm-glue2} should hold also for higher dimensions than 2. It would be of course interesting to know whether Theorem~\ref{thm-glue2} continues to hold for smooth algebraic surfaces over an arbitrary algebraically closed field. We leave these discussions to another opportunity. \subsection{Symplectic meaning of the remainders $\calR_N(\calL)$} \setcounter{thm}{0} \renewcommand{\thethm \label{subsect-Symplectic_meaning} In section~\ref{subsect-Applic} we have defined the homogeneous remainders $\calR_N(\calL)$ of an ample line bundle $\calL$ over a surface. The definition naturally extends to $n$-dimensional smooth varieties $X$ in the following obvious way. Given $p_1, \ldots, p_N \in X$, set $$\calR(\calL, p_1,\ldots,p_N)=\frac{1}{\calL ^n} \inf_{0\leq\eps\in \RR} \left\{ \calL_{\eps}^n \bigg| \calL_{\eps} = \pi^*\calL -\eps\sum_{j=1}^N E_j \quad \mbox{is nef} \right\},$$ where $\pi:\wtldX\rightarrow X$ is the blow-up of $X$ at the points $p_1,\ldots,p_N$ with exceptional divisors $E_i=\pi^{-1}(p_i)$. To get a more global invariant, define $$\calR_N (\calL)= \inf \left\{ \calR(\calL, p_1,\ldots p_N) \mid p_1, \ldots p_N \in X \quad \mbox{are distinct points} \right\}.$$ Let us explain now the symplectic meaning of these constants. Let $(M, \Om)$ be a symplectic manifold. Following McDuff and Polterovich define the following quantity $$ v_N(M,\Om) = \sup_{\vphi, \,\lam} \frac{\mbox{Vol}(\mbox{Image\, }\vphi)} {\mbox{Vol}(M,\Om)},$$ where $(\vphi, \lam)$ passes over all the possible symplectic packings $\vphi$ of $(M,\Om)$ with $N$ equal balls of varying radius $\lam$. The volume of the manifolds is defined to be $\mbox{Vol}(M,\Om)=\int_M \frac{1}{n!} \Om^{\wedge n}$. The constants $v_N(M, \Om)$ admit values between $0$ and $1$ and measure the maximal part of the volume of $(M, \Om)$ which can be filled by symplectic packing with $N$ equal balls. When $v_N=1$ we say that there exists a {\em full packing} of $(M,\Om)$ by $N$ equal balls, while in the case $v_N<1$ we say that there exists a {\em packing obstruction}. In view of the preceding discussion it is easy to see that the homogeneous remainders $\calR_N(\calL)$ of an ample line bundle over a complex manifold $M$, play the algebro-geometric role of the quantity $1-v_N(M, \Om)$, where $\Om$ is a K\"{a}hler form representing the first Chern class of $\calL, \; c_1(\calL)$. In fact, it is not hard to prove that the following inequality holds $$1-v_N(M,\Om) \leq \calR_N(\calL). \eqno(2)$$ Note that there are cases in which one always has equality in (2). For example, it follows from the work of McDuff and Polterovich (see~\cite{M-P}) that this is the case for $\CPTU$ when $N < 9$. The point is that the symplectic cone and the K\"{a}hler cone of del Pezzo surfaces coincide. Note that in (real) dimension 4, more is known about the constants $v_N$ than about $\calR_N$ (see~\cite{Bi}). It would be interesting to know whether there exist cases in which a strict inequality occurs in (2). \\ Let us conclude by pointing out another interesting approach to bounding Seshadri constants via symplectic packing, due to Lazarsfeld (see~\cite{Laz}). The idea is that given a K\"{a}hler form $\Om$ on a complex manifold and a symplectic packing $\vphi$ which is also holomorphic, the symplectic blow-up of $\Om$ associated to $\vphi$ will be K\"{a}hler. This situation happens when the associated K\"{a}hler metric on the image of $\vphi$ is flat. Applying this to the case of a principally polarized abelian variety, Lazarsfeld obtains non-trivial estimates on Seshadri constatnts of the corresponding ample divisor. \subsection*{Acknowledgments} I am extremely grateful to Prof. Eugenii Shustin for explaining me his approach to Viro method with singularities and for numerous enlightening discussions from which I learned a good deal. I would like to thank Prof. Leonid Polterovich for his interest in this work and for encouraging and interesting me to work on this project. Special thanks are due to Prof. Joseph Bernstein for the encouragement and for drawing my attention to interesting points which I was not aware of. I have also benefited from discussions with Michael Thaddeus who gave me the reference to~\cite{Do-Or} and with Ilya Tyomkin who shared with me his insights on the problems discussed in this paper. I wish to thank these people.

9705

alg-geom/9705028

en

https://arxiv.org/abs/alg-geom/9705028

alg-geom/9705028

David R. Morrison

David R. Morrison

Through the Looking Glass

latex2e, 22 pages with 1 figure

Mirror Symmetry III (D. H. Phong, L. Vinet, and S.-T. Yau, eds.), American Mathematical Society and International Press, 1999, pp. 263-277

DUK-M-97-04

It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure that the Calabi-Yau condition be preserved.) There has been speculation that all Calabi-Yau threefolds could be linked in this way, and considerable evidence has been amassed in this direction. We propose here a natural way to relate this construction to the string-theoretic phenomenon known as ``mirror symmetry.'' We formulate a conjecture which in principle could predict mirror partners for all Calabi-Yau threefolds, provided that all were indeed linked by the degeneration/resolution process. The conjecture produces new mirrors from old, and so requires some initial mirror manifold construction---such as Greene-Plesser orbifolding---as a starting point. (Lecture given at the CIRM conference, Trento, June 1994, and at the Workshop on Complex Geometry and Mirror Symmetry, Montr\'eal, March 1995.)

alg-geom

\section{Introduction} \label{sec:1} One of the most intriguing conjectures in complex analytic geometry is the spe\-cu\-la\-tion---essentially due to Herb Clemens, with refinements by Miles Reid and others---that the moduli spaces of all nonsingular compact complex threefolds with trivial canonical bundle might possibly be connected into a single family. Clemens introduced\footnote{The construction is hinted at in \cite{clmnone, clmntwo}, and given in detail in \cite{friedman}; the consequences for connecting moduli spaces are spelled out in \cite{reid} and \cite{friedmansurvey}, and the local geometry of the transition was analyzed in \cite{cd}.} a transition process among the moduli spaces of these various threefolds in which the complex structure on one such threefold is allowed to acquire singularities, and those singularities are then resolved by means of a bimeromorphic map from a second such threefold. If the threefolds involved have compatible K\"ahler metrics (which therefore determine ``Calabi--Yau structures''), then the second step in this process can be viewed as a sort of inverse to allowing the K\"ahler metric on the nonsingular model to degenerate to some kind of ``K\"ahler metric with singularities,'' the singularities being concentrated on the exceptional set of the resolution map. It is natural to wonder how this process might be related to the string-theoretic phenomenon known as ``mirror symmetry'' \cite{dixon, lvw, cls, gp}, which predicts that the moduli space of complex structures on one Calabi--Yau threefold $X$ should in many cases be locally isomorphic to the space of (complexified) K\"ahler structures on a ``mirror partner'' $Y$ of $X$, and {\it vice versa}. Viewed through the ``looking glass'' of mirror symmetry, the two ways of approaching the singular space in Clemens' transition appear complementary---one involves a specialization of complex structure parameters, the other involves a specialization of K\"ahler parameters---and indeed their r\^oles should be reversed when passing to mirror partners. The thesis of this lecture is that the whenever such a transition exists, it ought to enable us to predict the mirror partner of one of the Calabi--Yau manifolds involved in the transition from a knowledge of the mirror partner of the other. We will explain this idea in general terms, give some evidence and examples, and then formulate a specific conjecture which implements it. After this lecture was delivered, there were a number of new developments in string theory related to this construction; we describe those briefly at the end. \section{Extremal Transitions and Mirror Symmetry }\label{sec:2} Use of the term ``Calabi--Yau'' varies; our conventions are as follows. Let $X$ be a compact, connected, oriented manifold of dimension $2n$. A {\it Calabi--Yau metric}\/ on $X$ is a Riemannian metric whose (global) holonomy is a subgroup of $\SU(n)$. A {\it Calabi--Yau structure}\/ on $X$ is a choice of complex structure with trivial canonical bundle together with a K\"ahler metric; each Calabi--Yau structure determines a unique Calabi--Yau metric according to Yau's solution \cite{Yau} of the Calabi conjecture \cite{Calabi}. Finally, $X$ is a {\it Calabi--Yau manifold}\/ if it admits Calabi--Yau structures (and hence Calabi--Yau metrics). Let $X$ be a Calabi--Yau manifold. The simplest way to specify a family of Calabi--Yau structures on $X$ is by means of a proper smooth holomorphic map $\pi:{\cal X}\to S$ satisfying $\omega_{{\cal X}/S}={\cal O}_S$ whose fibers ${\cal X}_s:=\pi^{-1}(s)$ are diffeomorphic to $X$, together with a fixed K\"ahler metric on the total space ${\cal X}$, or more generally, a family of K\"ahler metrics depending on parameters. We can allow the complex structures to acquire singularities by enlarging the family to $\overline\pi:\overline{\cal X}\to\overline{S}$, where $S\subset\overline{S}$ is an open subset whose complement is closed, and where $\overline\pi$ is still assumed to be a proper holomorphic map such that $\omega_{\smash{\overline{\cal X}/\overline S}}= {\cal O}_{\smash{\overline S}}$, but is no longer assumed to be smooth. The fibers $\overline{\cal X}_{\sigma}:=\overline\pi^{-1}(\sigma)$ for $\sigma\in \widehat{S}:=\overline{S}-S$ may then be complex analytic spaces with singularities; of course, such spaces do not have K\"ahler metrics {\it per se}. Suppose that the singularities of the fibers $\overline{\cal X}_{\sigma}$ can be simultaneously resolved for $\sigma\in \widehat{S}$ by manifolds with Calabi--Yau structures.\footnote{A familiar instance of this is the small resolution of a collection of ordinary double points \cite{Atiyah, Hirz, TianYau}; note that we are demanding that a K\"ahler metric exist on the resolved space, which puts restrictions on the global configuration of the singularities.} That is, suppose that there exists a manifold $\widehat X$, a family of complex structures $\widehat\pi:\widehat{\cal X}\to \widehat{S}$ on $\widehat{X}$ such that $\omega_{\smash{\widehat{\cal X}/\widehat{S}}}={\cal O}_{\smash{\widehat{S}}}$ whose total space $\widehat{\cal X}$ is a K\"ahler manifold, and a proper bimeromorphic map $f:\widehat{\cal X}\to \overline{\cal X}_{\smash{\widehat{S}}} :=\overline\pi^{-1}(\widehat{S})$ which commutes with projection to $\widehat{S}$. (In general, it will be necessary to shrink $\overline{S}$ before this simultaneous resolution is possible---if it is possible at all.) If in addition, the bimeromorphic map $f_{\sigma}:\widehat{\cal X}_{\sigma}\to\overline{\cal X}_{\sigma}$ is an {\it extremal contraction}\/ in the sense of Mori theory (which means that for every algebraic curve $C\subset \widehat{\cal X}_{\sigma}$, we have $\dim f(C)=0$ if and only if the class of $C$ lies in some fixed face ${\cal F}$ of the Mori cone of $\widehat{\cal X}_{\sigma}$), then we say that $X$ and $\widehat{X}$ are related by an {\it extremal transition}. We can also describe an extremal transition ``in reverse,'' starting with $\widehat{X}$. In this version of the definition, we should begin with a family $\widehat{\cal X}\to \widehat{S}$ of complex structures on $\widehat{X}$ whose total space is a K\"ahler manifold, fix a birational extremal contraction $\widehat{\cal X}_\sigma\to \overline{\cal X}_\sigma$ (which is determined simply by the choice of an appropriate face ${\cal F}$ of the Mori cone), and ask for a smoothing of the contracted spaces. That is, we look for a space $\overline{S}$ into which $\widehat{S}$ can be embedded as a closed subset, and a family $\overline{\cal X}\to \overline{S}$ such that $\overline{\cal X}_s$ is smooth for $s\in \overline{S}-\widehat{S}$ (and diffeomorphic to $X$) whereas $\overline{\cal X}_{\smash{\widehat{S}}}$ is the same family of singular spaces as before. If this smoothing exists, then $X$ and $\widehat{X}$ are related by an extremal transition as above. This ``reverse'' description can actually be thought of as varying the K\"ahler parameters, rather than the complex structure parameters. Consider a family of K\"ahler metrics on the total space $\widehat{X}$ whose cohomology class approaches a point on the dual ${\cal F}^\perp$ of the face ${\cal F}$. Metrically, the curves $C$ lying in that face will approach zero area in such a process, and so the space with metric included ``approaches'' the contracted space $\overline{\cal X}$. In string theory, the choice of K\"ahler class serves as an important parameter in the theory. In fact, a more natural parameter for string theory is a complexification of the K\"ahler class, varying throughout an open subset of the {\it complexified K\"ahler moduli space}\/ ${\cal M}_{\text{K\"ah}}(\widehat{\cal X}_\sigma):= {\cal K}_{\Bbb C}(\widehat{\cal X}_\sigma)/\Aut(\widehat{\cal X}_\sigma)$, where \begin{equation} {\cal K}_{\Bbb C}(\widehat{\cal X}_\sigma):= \{B+i\omega\in H^2(\widehat{X},{\Bbb C}/{\Bbb Z})\ |\ \text{$\omega$ is a K\"ahler class on $\widehat{\cal X}_\sigma$}\} \end{equation} is the {\it complexified K\"ahler cone}, and $\Aut(\widehat{\cal X}_\sigma)$ is the group of holomorphic automorphisms. The face ${\cal F}$ of the Mori cone corresponding to an extremal contraction determines a boundary ``wall'' ${\cal F}^{\perp}$ of the complexified K\"ahler cone, and---provided that the singularities of $\overline{{\cal X}}_\sigma$ are sufficiently mild---the intersection \begin{equation} {\cal F}^\perp\cap\overline{{\cal K}_{\Bbb C}(\widehat{\cal X}_\sigma)} \end{equation} will coincide with $\overline{{\cal K}_{\Bbb C}({\cal X}_s)}$ for generic nearby $s\in S$. If the automorphism groups are reasonably well-behaved, there is then a natural inclusion \begin{equation}\label{inclusion} \overline{{\cal M}_{\text{K\"ah}}({\cal X}_s)}\subset \overline{{\cal M}_{\text{K\"ah}}(\widehat{\cal X}_\sigma)} \end{equation} (using the partial compactifications described in \cite{compact}). {\it Mirror symmetry}\/ refers to a phenomenon in string theory in which certain pairs of Calabi--Yau manifolds produce isomorphic physical theories, in such a way that the complexified K\"ahler moduli space of the first Calabi--Yau manifold is mapped to the ordinary (complex structure) moduli space of the second and {\it vice versa}, establishing local isomorphisms between those moduli spaces. Our basic proposal is that the mirror of an extremal transition should be another such transition, from a smooth space $Y$ specializing to a singular space $\overline{Y}$ (using the complex structure) which has a resolution of singularities $\widehat{Y}$. In this mirror version, {\it the mirror partner of $X$ should be $\widehat{Y}$, while the mirror partner of $\widehat{X}$ should be $Y$}. Under the mirror map, the inclusion \begin{equation}\label{inclX} \widehat{S}\subset\overline S \end{equation} between compactified parameter spaces for complex structures on $\widehat{X}$ and $X$ should locally map to the inclusion \begin{equation}\label{inclY} \overline{{\cal M}_{\text{K\"ah}}({\cal Y}_t)}\subset \overline{{\cal M}_{\text{K\"ah}}(\widehat{\cal Y}_\tau)} \end{equation} (and similarly for complex structures on $\widehat{Y}$ and $Y$ mapping to K\"ahler structures on $X$ and $\widehat{X}$). In particular, if we know the mirror partner of $X$ (resp.~$\widehat{X}$), and if we can predict where in the moduli space the mirror of the extremal transition should occur, then we should be able to determine the mirror partner of $\widehat{X}$ (resp.~$X$). We will make this proposal more precise in section \ref{sec:4}. \section{Evidence and Examples}\label{sec:3} \subsection{ Hodge numbers of conifold transitions}\label{subsec:31} In addition to the formal analogy between K\"ahler and complex structures which led to our ``looking glass'' interpretation of the mirror of an extremal transition, we were motivated by a computation of the effect on cohomology of the simplest type of extremal transition which has been known since the work of Clemens. (See \cite{werner, schoen, WvG} for general versions of this computation). Suppose that the complex dimension of our Calabi--Yau manifolds is three, and suppose that all singularities of $\overline{\cal X}_v$ are ordinary double points. (In this case, we will follow the conventions of the physics literature \cite{ghone} and refer to the extremal transition as a {\it conifold transition.}\footnote{We are assuming that the physical model does not have ``discrete torsion'' at the ordinary double points; otherwise, the picture is somewhat different \cite{VW, stabs}.}) Generally, we might expect that acquiring a double point places one condition on moduli. However, the double points may fail to impose independent conditions. Clemens' computation relates the failure of double points to impose independent conditions on the moduli of $X$ to the relative Picard number of the small resolution $\widehat{X}$, in the following way: if there are $\delta$ double points which only impose $\sigma:=\delta-\rho$ conditions on moduli then \medskip \noindent (i) $H^3(\widehat{X})\subset H^3(X)$ is a subspace of codimension $2\sigma$ (the inclusion comes from the identification of $H^3(\widehat{X})$ with the space of invariant cycles), and \medskip \noindent (ii) $\oplus H^{2k}(X) \subset \oplus H^{2k}(\widehat{X})$ is a subspace of codimension $2\rho$ (the inclusion is induced by pullbacks of divisors). \medskip \noindent In other words, each failure to impose a condition on moduli leads to an extra class in $H^2(\widehat{X})$. One of the properties of mirror symmetry for threefolds is that $H^3$ and $\oplus H^{2k}$ are exchanged when passing to a mirror partner. Thus, if mirror partners $\widehat{Y}$ and $Y$ are known for $X$ and $\widehat{X}$, respectively, then by using these mirror isomorphisms of cohomology we find that \medskip \noindent (i) $\oplus H^{2k}(Y) \subset \oplus H^{2k}(\widehat{Y})$ has codimension $2\sigma$, and \medskip \noindent (ii) $H^3(\widehat{Y})\subset H^3(Y)$ has codimension $2\rho$. \medskip \noindent It then seems very natural to conjecture that $Y$ and $\widehat{Y}$ should be related by a conifold transition in the opposite direction: it should be possible to allow the complex structure on $Y$ to acquire $\delta$ ordinary double points which this time impose only $\rho$ conditions on moduli, such that the singular spaces obtained can be resolved by $\widehat{Y}$.\footnote{Recent results \cite{KMP} indicate that this conjecture can only hold for generic moduli, if we insist on using only conifold transitions rather than the more general extremal transitions.} This would be a special case of our general principle relating mirror symmetry to extremal transitions. \subsection{Extremal transitions among toric hypersurfaces}\label{subsec:32} One context in which certain extremal transitions have a very explicit realization is the case of Calabi--Yau hypersurfaces in toric varieties. An {\it $n{+}1$-dimensional toric variety}\/ $V$ is a $T$-equivariant compactification of the algebraic torus $T:=({\Bbb C}^*)^{n+1}$ (often specified by the combinatorial data encoded in a {\it fan}). To describe a hypersurface within such a variety we need a polynomial---the defining equation---whose constituent monomials can be thought of as ${\Bbb C}^*$-valued characters $\chi:T\to{\Bbb C}^*$. Batyrev \cite{batyrev} has given an elegant condition which characterizes when such a hypersurface will be Calabi--Yau. The condition is stated in terms of the {\it Newton polyhedron}\/ of the polynomial, which is the polyhedron spanned by the monomials appearing in the equation. That is, if the polynomial is $\sum_{a\in M} c_a \chi_a$ where $M$ is the lattice of characters on $T$, then the Newton polyhedron is the convex hull \begin{equation} {\cal P}:=\Hull(\{a\ |\ c_a\ne0\})\subset M\otimes{\Bbb R}. \end{equation} Batyrev's criterion says that the generic such hypersurface is Calabi--Yau\footnote{We must use a slight generalization of the term ``Calabi--Yau'' here, in which Gorenstein canonical singularities are allowed on these hypersurface. However, when $n\le3$ it is always possible to choose the fan in such a way that the generic such hypersurface is nonsingular, and we can then speak of Calabi--Yau {\it manifolds}\/ in this context.} if ${\cal P}$ is {\it reflexive}, which means that (1) ${\cal P}$ is the convex hull of the lattice points it contains, (2) there is a unique lattice point $a_0$ in the interior of ${\cal P}$, and (3) the polar polyhedron (with respect to $a_0$) defined by \begin{equation} {\cal P}^\circ:=\{x\in N\otimes{\Bbb R}\ |\ \langle x,a\rangle - \langle x,a_0 \rangle \ge -1\ \hbox{for all}\ a\in{\cal P}\} \end{equation} has its vertices at lattice points of the dual lattice $N:=\Hom(M,{\Bbb Z})$. Conversely, given a reflexive polyhedron ${\cal P}$, there is an associated family of Calabi--Yau hypersurfaces with defining equation \begin{equation}\label{poly} f_{\cal P}=\sum_{a\in{\cal P}\cap M}c_a\chi_a \end{equation} embedded in a toric variety $V_{\cal P}$ determined by some fan whose one-dimensional cones are the rays ${\Bbb R}\vec{v}$ for $\vec{v}\in {\cal P}^\circ\cap N$. (The toric variety $V_{\cal P}$ is not uniquely specified by this, since we need to choose the fan, not just the one-dimensional cones; however, $\Pic(V_{\cal P})$ is independent of the choice of fan.) There is a simple class of {\it toric extremal transitions}\/ which can be described in these terms.\footnote{This construction was independently noted by Berglund, Katz and Klemm \cite{bkk}.} Suppose we have two reflexive polyhedra ${\cal Q}\subset{\cal P}$. Since all monomials appearing in $f_{\cal Q}$ also appear in $f_{\cal P}$, we can regard the hypersurfaces $X_{\cal Q}\subset V_{\cal Q}$ associated to ${\cal Q}$ as being limits of the hypersurfaces $X_{\cal P}\subset V_{\cal P}$ associated to ${\cal P}$. In fact, they will have worse than generic singularities when embedded in the toric variety $V_{\cal P}$, but those singularities can be improved, or in some cases (including the case $n\le3$) fully resolved, by further triangulation. In fact, the unique interior point $a_0$ of ${\cal Q}$ must also be the unique interior point of ${\cal P}$, which implies that \begin{equation} {\cal P}^\circ \subset {\cal Q}^\circ. \end{equation} Starting with a triangulation of ${\cal P}^\circ$, we will be able to further triangulate by including vertices of ${\cal Q}^\circ$. This construction provides an extremal transition as defined above, when the spaces involved are nonsingular (as is the case when $n\le3$): the family of complex structures acquires canonical singularities when certain coefficients in the defining equation are set to zero, and those singularities can be simultaneously resolved by maps which are extremal contractions. More precisely, the complex structures on the Calabi--Yau manifold $X_{\cal P}$ have a natural parameter space $S$ which is an open subset of \begin{equation}\label{barS} \overline{S}:={\Bbb C}^{{\cal P}\cap M} \gitquot \Aut(V_{\cal P})\times{\Bbb C}^*. \end{equation} (The notation indicates a quotient in the sense of Geometric Invariant Theory.) One of the subsets of $\overline{S}$ along which the Calabi--Yau spaces approach a singular space $\overline{X}$ is the set \begin{equation}\label{hatS} \widehat{S}:={\Bbb C}^{{\cal Q}\cap M} \gitquot \Aut(V_{\cal Q})\times{\Bbb C}^* \end{equation} in which all coefficients $c_a$ in eq.~\eqref{poly} with $a\not\in{\cal Q}$ have been set to zero. This is where our extremal transition is located. Refining a triangulation of ${\cal P}^\circ$ to a triangulation of ${\cal Q}^\circ$ produces the resolution of singularities $\widehat{X_{\cal Q}}\to\overline{X}$. In addition to giving a criterion for when a toric hypersurface is Calabi--Yau, Batyrev made a simple and beautiful conjecture concerning a possible mirror partner for any such hypersurface. To find the mirror family for $X_{\cal P}$, Batyrev's conjecture states that we should simply use the polar polyhedron ${\cal P}^\circ$ to determine a new family of hypersurfaces---let's call it $\widehat{Y}_{{\cal P}^\circ}$. Batyrev's construction exhibits a perfect compatibility with toric extremal transitions: since ${\cal P}^\circ \subset {\cal Q}^\circ$, we {\it automatically}\/ get an extremal transition between $Y_{{\cal Q}^\circ}$ and $\widehat{Y}_{{\cal P}^\circ}$. Moreover, Batyrev's formula \cite{batyrev} for the Hodge numbers of these hypersurfaces shows that---as in the case of our conifold transition conjecture---mirror symmetry is compatible with extremal transitions in the manner stated. In fact, using a description of the K\"ahler cones of $Y_{{\cal Q}^\circ}$ and $\widehat{Y}_{{\cal P}^\circ}$ in terms of ${\cal Q}$ and ${\cal P}$, together with an extension of Batyrev's conjecture known as the ``monomial-divisor mirror map'' \cite{mondiv}, one can verify that the inclusion $\widehat{S}\subset\overline{S}$ between the spaces from eqs.~\eqref{hatS} and \eqref{barS} is locally isomorphic to the inclusion\footnote{Actually, we are only working with a subset of the K\"ahler moduli space here corresponding to ``toric'' divisors---divisors which arise by restriction from a divisor on the ambient toric variety. Similarly, the parameter spaces $\overline{S}$ and $\widehat{S}$ only capture those complex structure parameters which preserve the property that the Calabi--Yau space can be embedded as a hypersurface in a toric variety.} \begin{equation} \overline{{\cal M}_{\text{K\"ah}}(Y_{{\cal Q}^\circ})}\subset \overline{{\cal M}_{\text{K\"ah}}(\widehat{Y}_{{\cal P}^\circ})} \end{equation} between complexified K\"ahler moduli spaces, with the corresponding face ${\cal F}^\perp$ of the K\"ahler cone determined by the inclusion ${\cal Q}\subset{\cal P}$. \subsection{An explicit example}\label{subsec:33} The relationship between mirror symmetry and extremal transitions can be seen very clearly in an explicit example worked out some years ago by the author in collaboration with P.~Candelas, X. de~la~Ossa, A.~Font and S.~Katz \cite{cdfkm}.\footnote{We shall use notation compatible with \cite{Small, MP}, where further details about this example were analyzed.} The transition is most easily described from the ``reverse'' perspective, beginning with $\widehat{Y}$ and varying the K\"ahler class to eventually produce $Y$. We begin with the weighted projective space $\wp$, and let $\pi: \widehat{\p}\to\wp$ be the blowup of the singular locus of $\wp$. Then $\widehat{\p}$ is a smooth fourfold containing smooth anti-canonical divisors $\widehat{Y}$, which are Calabi--Yau threefolds. The induced blowdown $\pi:\widehat{Y}\to\overline{Y}$ is an extremal contraction. Moreover, the Calabi--Yau space $\overline{Y}$ (which is defined by a homogeneous polynomial of weighted degree $8$, and has canonical singularities) can be smoothed. This is most easily seen by using the linear system ${\cal O}(2)$ to map $\wp$ to $\p^5$, where the image is a quadric hypersurface of rank 3. The image of $\overline{Y}$ under this mapping is the intersection of this singular quadric hypersurface with a smooth hypersurface of degree $4$. This has a smoothing to a space $Y$, the intersection of smooth hypersurfaces of degrees $2$ and $4$. To describe $\widehat{\p}$ in standard toric geometry language, we begin with the description of the weighted projective space $\wp$, which is determined by the following lattice points in $N={\Bbb Z}^4$: \begin{equation*} \begin{array}{ll} \multicolumn{2}{c}{v_1=(-1,-2,-2,-2),}\\ v_2=(1,0,0,0),&v_4=(0,0,1,0),\\ v_3=(0,1,0,0),&v_5=(0,0,0,1). \end{array} \end{equation*} The blowup is described by including the additional lattice point \begin{equation*} v_6=\frac{v_1+v_2}2=(0,-1,-1,-1). \end{equation*} Each lattice point $v_i$ determines a toric divisor $D_i$, whose classes generate $\Pic(\widehat{\p})$. If we choose a basis for the linear relations among the $v_i$'s, say \begin{align*} v_3+v_4+v_5+v_6 & =0\\ v_1+v_2-2v_6 & =0 \end{align*} then there is a corresponding basis $\eta_1$, $\eta_2$ of $\Pic(\widehat{\p})$ for which \begin{align*} [D_3]=[D_4]=[D_5]&=\eta_1\\ [D_1]=[D_2]&=\eta_2\\ [D_6]&=\eta_1-2\eta_2. \end{align*} It turns out that $\eta_1$ and $\eta_2$ generate the K\"ahler cone of $\widehat{\p}$. We can write an arbitrary K\"ahler class in the form $a_1\eta_1+a_2\eta_2$, and use $a_1$, $a_2$ as coordinates on the K\"ahler cone; the natural coordinates to use on the complexified K\"ahler cone are then $\exp(2\pi i\,a_1)$, $\exp(2\pi i\, a_2)$. The K\"ahler cone has two faces given by $\{a_1=0\}$ and $\{a_2=0\}$. The first face corresponds to a pencil of K3 surfaces, and has no associated extremal transition. It is the second face $\{a_2=0\}$ which is associated to our extremal transition. The class $\eta_1$ which spans that face contains $2D_1+D_6, D_1+D_2+D_6,2D_2+D_6,D_3, D_4, D_5$ as representatives, and the corresponding linear system maps $\widehat{\p}$ to $\wp$, shrinking the exceptional divisor of the blowup map to zero size. As remarked above, the hypersurfaces $\overline{Y}$ in $\wp$ can then be smoothed (after reembedding $\wp$ as a rank 3 quadric in $\p^5$); this gives a complete description of the extremal transition. Candidate mirror partners are known for both $\widehat{Y}$ and $Y$ \cite{gp, lt}, so it is natural to look for a connection between these---a mirror image of the extremal transition mentioned above. This was also found in \cite{cdfkm}. The candidate mirror partner of $\widehat{Y}$ is the desingularization of an anti-canonical hypersurface in $\wp/G$, where $G$ is the image in $\Aut(\wp)$ of \begin{equation*} \widetilde{G}=\{ \vec{\lambda} \in({\Bbb C}^*)^5\ |\ \lambda_1^8=\lambda_2^8=\lambda_3^4=\lambda_4^4=\lambda_5^4= \lambda_1\lambda_2\lambda_3\lambda_4\lambda_5=1\}. \end{equation*} The complex structure moduli space of the mirror manifold $X$ can be described in terms of some analogous parameters. Rather than using a redundant description in terms of toric divisors, we can this time use a redundant description in terms of monomials. To describe the mirror of $\widehat{Y}$, in fact, we should consider the family of polynomials in the homogeneous coordinates $x_1$, \dots, $x_5$ \begin{equation} c_0 x_1x_2x_3x_4x_5 + c_1x_1^8 + c_2x_2^8 + c_3x_3^4 + c_4x_4^4 + c_5x_5^4 + c_6 x_1^4x_2^4. \end{equation} The connection with the divisors on the mirror becomes more apparent if we divide the polynomial by $x_1x_2x_3x_4x_5$ and rewrite in terms of the basis of the torus $T=({\Bbb C}^*)^4$ defined by \begin{align*} t_1&=x_1^{-1}x_2^{7}x_3^{-1}x_4^{-1}x_5^{-1}\\ t_2&=x_1^{-1}x_2^{-1}x_3^{3}x_4^{-1}x_5^{-1}\\ t_3&=x_1^{-1}x_2^{-1}x_3^{-1}x_4^{3}x_5^{-1}\\ t_4&=x_1^{-1}x_2^{-1}x_3^{-1}x_4^{-1}x_5^{3}. \end{align*} In this basis, the defining polynomial of $X$ becomes \begin{equation} c_0+c_1t_1^{-1}t_2^{-2}t_3^{-2}t_4^{-2} +c_2t_1+c_3t_2+c_4t_3+c_5t_4+c_6t_2^{-1}t_3^{-1}t_4^{-1} \end{equation} and the exponents on the monomials correspond to the $v_i$'s of the mirror. The coordinates on the complex structure moduli space which are analogous to the $\exp(2\pi i\,a_j)$'s are then \begin{equation} q_1:=c_3c_4c_5c_6/c_0^4;\qquad q_2:=c_1c_2/c_6^2. \end{equation} (These are local coordinates on the parameter space $\overline{S}={\Bbb C}^{7} \gitquot T\times{\Bbb C}^*$.) \iffigs \begin{figure}[t] \begin{center} \begin{picture}(3,4.3)(0,0) \put(.15,.2){\epsfxsize=3in\epsfbox{moduli.eps}} \put(2.8,3.8){$(q_1^{\vphantom1},q_2^{\vphantom1})$} \put(2.8,1.7){$(q_1^{\vphantom1}q_2^{1\smash/2},q_2^{-1\smash/2})$} \put(-.7,.2){$(q_1^{-1}q_2^{-1\smash/2},q_2^{-1\smash/2})$} \put(-.3,3.82){$(q_1^{-1},q_2^{\vphantom1})$} \put(1.3,2.8){$q_2=1/4$} \put(2.1,3.3){$\Delta_0$} \end{picture} \end{center} \caption{The moduli space for Calabi--Yau hypersurfaces in $\wp/G$.}\label{fig:1} \end{figure} \fi The moduli space is illustrated in figure \ref{fig:1}. The figure displays four coordinate charts, which cover the entire moduli space: the coordinates in each chart are indicated near the point which is the center of the coordinate chart. (The dotted lines indicate the approximate division into ``phase regions,'' described as cones in the variables ${\frac1{2\pi i}}\log(q_j)$: cf.~\cite{MP}. Note that our figure is rotated by $90^\circ$ with respect to figure 5 of \cite{cdfkm}.) The chart in the upper right corner with coordinates $(q_1,q_2)$ is centered at the so-called ``large complex structure limit'' point. The ``discriminant locus'' where these hypersurfaces become singular has five components: the principal component, labeled $\Delta_0$ in the figure, is the curve defined by \begin{equation} \Delta_0=\left\{ q_2={\frac14}\left(1-{\frac1{256}q_1}\right)^2 \right\}; \end{equation} the other components are described by $\{q_1=0\}$, $\{q_2=0\}$, $\{q_1^{-1}=0\}$, and $\{q_2=1/4\}$.\footnote{Note that in \cite{cdfkm}, the principal component was called $C_{\text{con}}$, and the other components were called $D_{(1,0)}$, $C_\infty$, $C_0$, and $C_1$, respectively.} (There is also some monodromy around the ``orbifold locus'' $\{q_2^{-1\smash/2}=0\}$.) Along the locus where $q_2=1/4$, we found in \cite{cdfkm} an extremal transition, as follows. (Notice that this is not a special case of example \ref{subsec:32}, since we are not simply setting coefficients to zero.) When $q_2=1/4$, choose square roots $\sqrt{c_1}$ and $\sqrt{c_2}$ which are related by requiring that $2\sqrt{c_1}\sqrt{c_2}=c_6$. Let $\Gamma$ be the image in $\Aut(\p^5)$ of \begin{equation*} \widetilde{\Gamma}:=\{ \vec{\mu} \in({\Bbb C}^*)^6\ |\ \mu_0^2=\mu_1^4=\mu_2^4=\mu_3^2=\mu_4^2=\mu_5^2 =\mu_1\mu_2=\mu_0\mu_3\mu_4\mu_5=1\}. \end{equation*} Then we can define a rational map $\wp/G\to \p^5/\Gamma$ by \begin{align*} y_0&=\sqrt{c_1}\,x_1^4+\sqrt{c_2}\,x_2^4 &y_3&=x_3^2\\ y_1&=x_1\sqrt{x_3x_4x_5} &y_4&=x_4^2\\ y_2&=x_2\sqrt{x_3x_4x_5} &y_5&=x_5^2 \end{align*} (using the same value of $\sqrt{x_3x_4x_5}$ in both $y_1$ and $y_2$). The image of $\wp/G$ satisfies the equation \begin{equation} \sqrt{c_1}\,y_1^4+\sqrt{c_2}\,y_2^4=y_0y_3y_4y_5, \end{equation} while the image of the Calabi--Yau hypersurface additionally satisfies the equation \begin{equation} y_0^2+c_0y_1y_2+c_3y_3^2+c_4y_4^2+c_5y_5^2. \end{equation} It can be easily checked that the hypersurface in $\wp/G$ is mapped {\it birationally}\/ to the complete intersection in $\p^5/\Gamma$, which is the candidate mirror partner for the complete intersection in $\p^5$ of bidegree $(2,4)$! \section{The Location of the Extremal Transition}\label{sec:4} In order to make the principle formulated in section \ref{sec:2} more precise, we need to recall the conjectural correspondence between boundary points of complex structure moduli spaces, and possible mirror partners of a given Calabi--Yau manifold $X$ \cite{AGM, compact}. If we compactify the complex structure moduli space in such a way that the boundary is a divisor with normal crossings, then candidates for ``large complex structure limit points'' can be identified by the properties of the monodromy transformations around boundary divisors. If the moduli space has dimension $r$, then any such candidate point $P$ should lie at the intersection of $r$ boundary divisors $D_i$ whose monodromy transformations $T_1$, \dots, $T_r$ define a ``monodromy weight filtration'' which is opposite to the Hodge filtration \cite{deligne, compact}. The conjecture is that any such point will have an associated mirror partner $\widehat{Y}$, together with a choice\footnote{This choice may look a bit unnatural, but it is needed to get reasonable coordinates; the independence from choices would follow from the ``cone conjecture.'' See \cite{compact} for a discussion of this issue.} of a simplicial rational polyhedral cone $\Pi$ contained in the closure of the K\"ahler cone of $\widehat{Y}$, in such a way that under the mirror map $\mu$, the complement of the $D_i$'s in a neighborhood of $P$ is mapped to an open subset in the closure $\overline{\cal D}_\Pi$ of the space \begin{equation} {\cal D}_\Pi:= \{\beta\in H^2(\widehat{Y},{\Bbb C}/{\Bbb Z})\ |\ \Im(\beta)\in\Pi\}. \end{equation} In the strongest form of the conjecture, one asserts that $\Pi$ can be chosen so that it is generated by a basis $e^1$, \dots, $e^r$ of $H^2(\widehat{Y},{\Bbb Z})/(\text{torsion})$. In this case, if we write a general element of $H^2(\widehat{Y},{\Bbb C})$ in the form $\sum t_je^j$ and let $w_j:=\exp(2\pi i\,t_j)$, we can describe ${\cal D}_\Pi$ in coordinates as \begin{equation} {\cal D}_\Pi=\{(w_1,\dots,w_r)\ |\ 0<|w_j|<1\ \forall j\}. \end{equation} This space has a partial compactification \begin{equation} {\cal D}_\Pi^-:=\{(w_1,\dots,w_r)\ |\ 0\le|w_j|<1\ \forall j\}, \end{equation} and $P$ maps to the {\it distinguished limit point}\/ $\mu(P)=(0,\dots,0)\in{\cal D}_\Pi^-$. If we choose $\Pi$ so that it shares with the K\"ahler cone a face associated to an extremal transition (say the face spanned by the first $k$ vectors), then the extremal transition na\"{\i}vely would be expected at $t_{k+1}=\dots=t_r=0$, i.e., at $w_{k+1}=\dots=w_r=1$. The location of the extremal transition may, however, be modified by quantum effects in the complexified K\"ahler moduli space. We expect it to occur at a place where the conformal field theory has a singularity, and this is measured by poles in the correlation functions of the quantum theory. The locus $w_{k+1}=\dots=w_r=1$ will meet the boundary stratum in ${\cal D}_\Pi^-$ defined by $w_1=w_2=\dots=w_{k}=0$, and the ``large radius'' approximation to the quantum field theory should be good near that boundary stratum. In fact, we should be able use the behavior of correlation functions along that boundary stratum to predict the location of the extremal transition in the complex structure moduli space. To do so, we will need to use the ``flat coordinates'' $z_1$, \dots, $z_r$ which are intrinsically associated to the large complex structure limit point, since the mirror map $\mu$ has the property that $\mu^*(w_j)=z_j$. Recall how the flat coordinates are defined: the monodromy properties of the periods near the large complex structure limit point $P$ guarantee that if $q_1$, \dots, $q_r$ are local coordinates such that the boundary divisors intersecting at $P$ are given by $D_j=\{q_j=0\}$, then there are periods integrals $\varpi_j=\int_{\gamma_j}\Omega$ of the holomorphic $n$-form $\Omega$ with the property that $\varpi_0$ is single-valued near $P$, while \begin{equation} \varpi_j=\frac{\varpi_0}{2\pi i}\log(q_j)+ \text{single-valued function}. \end{equation} The flat coordinates are then given by $z_j=\exp(2\pi i{\varpi_j}/{\varpi_0})$; they are uniquely determined up to multiplication by constants.\footnote{For a discussion of how those constants should be fixed, see \cite{predictions}.} To learn what we should expect concerning the location of the extremal transition, let us again consider the example from section \ref{subsec:33}. In that example, we should study the boundary curve $B$ (in the complex structure moduli space of $X$) defined by $z_1=0$, or equivalently by $q_1=0$. The period integrals of $X$ satisfy certain Gelfand-Kapranov-Zelevinsky hypergeometric differential equations \cite{batyrev:vmhs}; when restricted to the locus $q_1=0$, there is a single such equation, which can be read off of the formula $q_2=c_1c_2/c_6^2$ as being \begin{equation} \left((q_2\frac{d}{dq_2})(q_2\frac{d}{dq_2}) - q_2(-2q_2\frac{d}{dq_2})(-2q_2\frac{d}{dq_2}-1)\right)\varpi(0,q_2)=0. \end{equation} This has a general solution near $q_2=0$ given by \cite{Small} \begin{equation} \varpi(0,q_2) =C_1+C_2\log\left(\frac{2q_2}{1-2q_2+\sqrt{1-4q_2}}\right), \end{equation} choosing the branch of the square root which is near $1$ when $q_2$ is near $0$. It follow that the flat coordinate along $B$ is given by \begin{equation} z_2|_B=\frac{2q_2}{1-2q_2+\sqrt{1-4q_2}}. \end{equation} Note that (as can be seen in figure \ref{fig:1}), $B$ meets two other components of the discriminant locus, at $q_2=1/4$ and at $q_2\to\infty$ (the latter being the intersection with the ``orbifold locus''). When $q_2=1/4$, we have $z_2=1$ whereas when $q_2\to\infty$ we have $z_2=-1$. Thus, there will be poles in correlation functions precisely at $z_2=\pm1$. More generally, we should expect that poles in correlation functions could occur at several distinct values of $|z_r|$. (This is known to happen in other examples \cite{Small, gmv}.) The large radius approximation can only be trusted for values of $|w_r|$ less than the minimum value at which a pole occurs, so we shall expect that any extremal transition whose occurrence is predicted by mirror symmetry will occur at a pole where the value of $|z_r|=|\mu^*(w_r)|$ is minimal. As the present example shows, such a pole need not be unique. Returning to the general case, we formulate the following conjecture concerning the location of the extremal transition, which we hope is not too far off the mark. We assume that a mirror pair $(X,\widehat{Y})$ is somehow known, corresponding to the large complex structure limit point $P$ (for $X$) and the subcone $\Pi$ of the K\"ahler cone of $\widehat{Y}$. Let ${\cal F}^\perp=\Pi\cap\Span\{e^1,\dots,e^k\}$ be a face of $\Pi$. \begin{conjecture} There exist a compactification $\overline{\cal M}$ of the complex structure moduli space of $X$ containing $P=D_1\cap\dots\cap D_r$, together with components $\Delta_{k+1}$, \dots, $\Delta_r$ of the boundary of $\overline{\cal M}$ along which $X$ acquires canonical singularities, and a stratification \begin{equation} \Delta_{k+1}\cup\dots\cup\Delta_r=\coprod_{\sigma<{\cal F}} \Delta_\sigma \end{equation} of the union of those components, indexed by subcones of the dual face ${\cal F}$ of the Mori cone, such that the intersection of the stratum $\Delta_\sigma$ with \begin{equation} B_\sigma:=D_{\sigma(1)}\cap\dots\cap D_{\sigma(s)}=\{z_{\sigma(1)}=\dots=z_{\sigma(s)}\} \end{equation} lies at the minimum possible distance from the origin, among locations of poles of correlation functions on $B_\sigma$. Furthermore, the singular Calabi--Yau space $\overline{X}_{\Delta_{\cal F}}$ has a Calabi--Yau desingularization $\widehat{X}\to\overline{X}_{\Delta_{\cal F}}$ if and only if the contracted space $\overline{Y}_{\cal F}$ has a Calabi--Yau smoothing $Y$. In this case, the Calabi--Yau manifolds $\widehat{X}$ and $Y$ should be mirror partners. \end{conjecture} We have limited our discussion to neighborhoods of large complex structure limit points, which are the mirrors of K\"ahler cones (of various birational models of $\widehat{Y}$). It is frequently possible to analytically continue the complexified K\"ahler moduli space beyond these K\"ahler cones \cite{beyond}, but we don't know good criteria for deciding about the existence of extremal transitions (on the ``K\"ahler moduli'' side) in such regions of the moduli space. Extremal transitions in such regions, if they exist, would evade detection in the sort of analysis given here. \section{Recent developments}\label{subsec:recent} In the physics literature, conifold transitions were first observed in a process known as ``splitting'' which related various families of complete intersection Calabi--Yau threefolds \cite{cdls}. Considerable effort was expended in showing that all then-known examples of Calabi--Yau threefolds could be connected into a single web \cite{ghone, ghtwo, cgh}, and it was also observed that these connections occurred at finite distance in the moduli space (with respect to the natural ``Zamolodchikov'' metric on that space) \cite{cghfinite}. However, a physical mechanism implementing these transitions was unknown. Some months after this lecture was delivered, a new mechanism was proposed in the physics literature for realizing an extremal transition as a physical process in type II string theory \cite{strom, gms}. The mechanism in its original form only applies to conifold transitions, but there are now indications \cite{KMP, MVII, Wittennewest, morsei, GMS, IMS} that similar mechanisms will enable all extremal transitions to be realized in physics. Motivated by this, the subpolyhedron construction described in section \ref{subsec:32} was subsequently used \cite{CGGK, ACJM} to show that all Calabi--Yau hypersurfaces in weighted projective spaces can be linked by extremal transitions. In another direction---perhaps closer in spirit to the original approach of Clemens and Reid---Kontsevich \cite{kontsevich} has made a fascinating construction involving Lagrangian analytic cones in an infinite-dimensional space, and has conjectured a mirror symmetry relationship in terms of these cones which would involve {\it all}\/ symplectic complex threefolds with trivial canonical bundle (even those for which the symplectic structure does not arise from a K\"ahler structure). Finally, there was been a recent geometric reformulation of the basic mirror symmetry property in physics \cite{SYZ} (see also \cite{underlying, GW}), in terms of fibrations of a Calabi--Yau manifold by special Lagrangian tori. It is an important and challenging problem to understand how such fibrations behave under an extremal transition. Such an understanding could ultimately lead to a proof of the conjecture in section \ref{sec:4} using the new ``geometric'' definition of mirror symmetry. \subsection*{ Acknowledgments}\label{subsec:ack} I am grateful to Mark Gross for insisting that the analysis given here should not be limited to the case of small resolutions, and for discussions on other points. I would also like to thank Paul Aspinwall, Bob Friedman, Sheldon Katz, Ronen Plesser, Miles Reid, Pelham Wilson, Edward Witten and especially Brian Greene for useful discussions. \def\leavevmode\hbox to3em{\hrulefill}\thinspace{\leavevmode\hbox to3em{\hrulefill}\thinspace}

9705

alg-geom/9705002

en

https://arxiv.org/abs/alg-geom/9705002

alg-geom/9705002

Tom Bridgeland

Tom Bridgeland

Fourier-Mukai transforms for elliptic surfaces

Many minor changes, published in J. reine angew. math. vol. 498

We compute a large number of moduli spaces of stable bundles on a general algebraic elliptic surface using a new class of Fourier-Mukai type transforms.

alg-geom

\section{Introduction} In recent years moduli spaces of stable bundles on projective surfaces have been extensively studied. Although various general results are known, few of these spaces have been described explicitly. One important development was S. Mukai's discovery [11] of a transform which allowed him to compute some of the moduli spaces on Abelian varieties [14]. This technique has recently been extended to cover K3 surfaces [4], [13], [17]. Here we introduce similar transforms for elliptic surfaces. Much is already known about moduli of rank 2 bundles on elliptic surfaces thanks to work by R. Friedman [7], [8], amongst others. This research has led to several important results, including the smooth classification of elliptic surfaces. In this paper we use Mukai's techniques to study bundles of higher rank. \subsection{} Let $X\lRa{\pi} C$ be a relatively minimal algebraic elliptic surface over ${\mathbb C}$. Given a sheaf $E$ on $X$ we write its Chern class as a triple $$(r(E),c_1(E),c_2(E))\in{\mathbb Z}\times\operatorname{NS}(X)\times{\mathbb Z}.$$ Here $\operatorname{NS}(X)$ is the N\'eron-Severi group of $X$, i.e. the subgroup of $H^2(X,{\mathbb Z})$ generated by the Chern classes of line bundles on $X$. We denote the element of $\operatorname{NS}(X)$ corresponding to a fibre of $\pi$ by $f$. Let $(r,\Lambda,k)$ be a triple as above, and assume that $r>1$ is coprime to $\Lambda\cdot f$. Then, as Friedman observed, there exist polarizations of $X$ with respect to which a torsion-free sheaf $E$ of Chern class $(r,\Lambda,k)$ is stable iff the restriction of $E$ to the general fibre of $\pi$ is stable. For these polarizations, and sheaves of this Chern class, the notions of Gieseker stability, $\mu$-semi-stability and $\mu$-stability all coincide. Taking such a polarization, we define ${\mathcal M}={\mathcal M}_X(r,\Lambda,k)$ to be the moduli space of stable torsion-free sheaves on $X$ of Chern class $(r,\Lambda,k)$. An argument of Friedman's shows that the projective scheme ${\mathcal M}$ is smooth of dimension $\dim(\Pic^{\circ}(X))+2t$, where \begin{equation} \label{first} 2t=2rk-(r-1)\Lambda^2-(r^2-1)\chi({\mathscr O}_X). \end{equation} If $t<0$ then ${\mathcal M}$ is empty, so we shall assume that $t$ is non-negative. Our main result (Theorem 1.1 below) states that ${\mathcal M}$ is irreducible and birationally equivalent to $\Pic^{\circ}(Y)\times\operatorname{Hilb}^t(Y)$, where $Y$ is another elliptic surface over $C$. \subsection{} Let us define $\lambda_X$ to be the highest common factor of the fibre degrees of sheaves on $X$. Equivalently, $\lambda_X$ is the smallest positive integer such that $X\lRa{\pi} C$ has a holomorphic $\lambda_X$-multisection. Given a pair of integers $a>0$ and $b$ such that $a\lambda_X$ is coprime to $b$, we define an elliptic surface $J_X(a,b)$ over $C$, whose fibre over a point $p\in C$ is canonically identified with (a component of) the moduli space of rank $a$, degree $b$, stable sheaves on the fibre $X_p$. To do this, take a polarization of $X$ of fibre degree coprime to $b$, and let ${\mathcal M}(X/C)\to C$ denote the relative moduli space of stable pure-dimension 1 sheaves on the fibres of $\pi$ (see [18]). Then define $J_X(a,b)$ to be the union of those components of ${\mathcal M}(X/C)$ which contain a rank $a$, degree $b$ vector bundle supported on a non-singular fibre of $\pi$. The details of this construction, and the proof that $J_X(a,b)$ is an elliptic surface, are given in section 4. The essential point is that each component of the moduli of stable sheaves on an elliptic curve is again an elliptic curve, so that $J_X(a,b)$ has a natural elliptic fibration structure. \begin{thm} \label{moduli} The moduli space ${\mathcal M}={\mathcal M}_X(r,\Lambda,k)$ is a smooth (non-empty) projective variety and is birationally equivalent to $$\Pic^{\circ}(J_X(a,b))\times\operatorname{Hilb}^t(J_X(a,b)),$$ where $(a,b)$ is the unique pair of integers satisfying $br-a(\Lambda\cdot f)=1$ and $0<a<r$. Furthermore, if $r>at$ the birational equivalence extends to give an isomorphism of varieties. \end{thm} This generalises a result of Friedman who looked at the rank 2 moduli. In that case one always has $a=1$, and the surface $J_X(a,b)$ is the relative Picard scheme $J^b(X)$ of [7]. \subsection{} To prove Theorem \ref{moduli} we use a relative version of the Fourier-Mukai transform. Let ${D}(S)$ denote the bounded derived category of coherent sheaves on a Noetherian scheme $S$. In general, given two smooth varieties $X$ and $Y$ and an object ${\mathcal P}$ of ${D}(X\times Y)$, one defines a functor $\Phi^{\mathcal P}_{Y\to X}:{D}(Y)\longrightarrow {D}(X)$, by the formula $$\Phi^{\mathcal P}_{Y\to X}(-)=\mathbf R\pi_{X,*}({\mathcal P}\mathbin{\buildrel{\mathbf L}\over{\tensor}}\pi^*_Y(-)),$$ where $\pi_X$ and $\pi_Y$ are the projections from $X\times Y$ to $X$ and $Y$ respectively. In some very special circumstances this functor is an equivalence of categories, and one then has a useful tool for studying sheaves on $X$ and $Y$. Examples include the original Fourier-Mukai transform [11] (where $X$ is an Abelian variety and $Y$ is its dual) and the generalised transforms of [4] and [17] (where $X$ and $Y$ are isogenous K3 surfaces). \begin{thm} \label{biggy} Given integers $a>0$ and $b$ with $a\lambda_X$ coprime to $b$, let $Y=J_X(a,b)$. Then there exist tautological sheaves on $X\times Y$, supported on $X\times_C Y$, and for each such sheaf ${\mathcal P}$, the functor $\Phi^{{\mathcal P}}_{Y\to X}:{D}(Y)\longrightarrow {D}(X)$ is an equivalence of categories. \end{thm} Although this result involves derived categories in an essential way, we shall see that the functors $\Phi^{{\mathcal P}}_{Y\to X}$ often take sheaves to sheaves, and thus yield concrete results about moduli of vector bundles. Indeed, once the basic properties of these functors are known, the proof of Theorem \ref{moduli} is relatively simple, and it seems likely that the same functors will prove useful for solving other moduli problems on elliptic surfaces. One might also expect similar results for higher-dimensional elliptic fibrations. \begin{nota} All schemes will be Noetherian ${\mathbb C}$-schemes and all morphisms will be morphisms over ${\mathbb C}$. By a sheaf on a scheme $X$ we mean a coherent ${\mathscr O}_X$-module. A point of $X$ will mean a closed point. ${D}(X)$ denotes the bounded derived category of sheaves on $X$. We refer to [9] for properties of ${D}(X)$. Given an object $E$ of ${D}(X)$ let $\mathscr H^i(E)$ denote the $i$th cohomology sheaf of $E$. We say that $E$ is a sheaf if $\mathscr H^i(E)=0$ when $i\neq 0$. The derived dual, $\mathbf R\operatorname{{\mathcal H}om}(E,{\mathscr O}_X)$, is denoted $E^{\vee}$, and $E[n]$ denotes the object $E$ shifted to the left by $n$ places. By a variety we mean an integral, separated scheme of finite type over ${\mathbb C}$. The canonical bundle of a smooth variety $X$ is written $\omega_X$, and for objects $E$ and $F$ of ${D}(X)$ we define $$\chi(E,F)=\sum(-1)^i\dim_{\,{\mathbb C}}\operatorname{Hom}_{{D}(X)}(E,F[i]),$$ and write $\operatorname{ch}(E)$ for the Chern character of $E$. This defines a map $$\operatorname{ch}:{D}(X)\longrightarrow H^{2*}(X,{\mathbb C}).$$ We denote the image (an Abelian group) by $\operatorname{ch}(X)$. We refer to [18] for the definitions of pure-dimension sheaves and stable sheaves on a projective scheme $X$. They depend on the polarization chosen for $X$. \end{nota} \begin{ack} This work forms part of my PhD project, which is funded by the EPSRC. I would like to thank my supervisor, Antony Maciocia, for teaching me about Fourier-Mukai transforms, and for providing a great deal of help and encouragement. The original idea of looking for relative transforms on elliptic surfaces was his. \end{ack} \section{General Properties of $\Phi^{{\mathcal P}}_{Y\to X}$} Throughout this section $X$ and $Y$ are smooth varieties and ${\mathcal P}$ is an object of ${D}(X\times Y)$. We shall be mainly interested in the case when ${\mathcal P}$ is a sheaf on $X\times Y$, flat over both factors. Let $\Phi$ denote the corresponding functor $\Phi_{Y\to X}^{{\mathcal P}}:{D}(Y)\to{D}(X)$ defined in the introduction. We state various properties of $\Phi$ which we shall need. These all appear in some form in Mukai's original papers on Abelian varieties and K3 surfaces [11], [13], [14]. The general results have been worked out by A. I. Bondal and D. O. Orlov [6], [17], and A. Maciocia [10]. \subsection{} First we define WIT (weak index theorem) sheaves. Given an object $E$ of ${D}(Y)$ put $$\Phi^i(E)=\mathscr H^i(\Phi(E)).$$ A sheaf $E$ on $Y$ is said to be $\Phi$-WIT$_i$ if $\Phi^j(E)=0$ for all $j\neq i$, or equivalently if $\Phi(E)[i]$ is a sheaf. We say $E$ is $\Phi$-WIT if it is $\Phi$-WIT$_i$ for some $i$, and in this case we often write $\Hat E$ for $\Phi^i(E)$, and refer to $\Hat E$ as the {\it transform} of $E$. \subsection{} By Grothendieck's Riemann-Roch theorem there is a group homomorphism $\operatorname{ch}(\Phi)$ making the following diagram commute $$ \begin{array}{ccc} {D}(Y) &\lRa{\Phi} &{D}(X) \\ \scriptstyle{\operatorname{ch}}\big\downarrow &&\scriptstyle{\operatorname{ch}}\big\downarrow \\ \operatorname{ch}(Y) &\lRa{\operatorname{ch}(\Phi)} &\operatorname{ch}(X) \end{array} $$ It is given by $$\operatorname{ch}(\Phi)(y)=\pi_{X,*}(p\cdot\pi_Y^*y),$$ where $p=\operatorname{ch}({\mathcal P})\cdot\pi_Y^*(\operatorname{td}_Y)$ and $\operatorname{td}_Y$ is the Todd class of $Y$. \subsection{} Define $$\mathcal Q=({\mathcal P}^{\vee}\tensor\pi_X^*\omega_X)[\dim X+\dim Y-\dim {\mathcal P}],$$ and put $\Psi=\Phi^{\mathcal Q}_{X\to Y}$. It is a simple consequence of Grothendieck-Verdier duality (see e.g. [6], Lemma 1.2) that $\Psi[\dim {\mathcal P}-\dim Y]$ is a left adjoint of $\Phi$. If $\Phi$ is fully faithful one has an isomorphism of functors $$\Psi\circ\Phi\cong \operatorname{Id}_{{D}(Y)}[\dim Y-\dim{\mathcal P}].$$ The reason for the apparently strange choice of shift in the definition of $\mathcal Q$ is to give it a good chance of being a sheaf. For example when ${\mathcal P}$ is a vector bundle, so is $\mathcal Q$. \subsection{} Assume that ${\mathcal P}$ is a sheaf on $X\times Y$, flat over both factors. Then $\Phi$ preserves families of sheaves. In detail, let $S$ be a scheme and $\mathcal E$ an $S$-flat sheaf on $Y\times S$. Then $$U=\{s\in S:\mathcal E_s\mbox{ is } \Phi\mbox{-WIT}_i\}$$ is the set of points of an open subscheme of $S$. Furthermore there exists a $U$-flat sheaf $\mathcal F$ on $X\times U$, such that for all $s\in U$, $\mathcal F_s=\Phi^i(\mathcal E_s)$. The proof of this result is identical to that of [14], Theorem 1.6. \subsection{} Suppose now that $\Phi$ is fully faithful, and take sheaves $A$ and $B$ on $Y$, with $A$ $\Phi$-WIT$_a$ and $B$ $\Phi$-WIT$_b$. Then for all $i$, one has $$\operatorname{Hom}_{{D}(Y)}(A,B[i])=\operatorname{Hom}_{{D}(X)}(\Hat A[-a],\Hat B[i-b]).$$ Rewriting this gives the identity \begin{equation} \label{percy} \operatorname{Ext}_Y^i(A,B)=\operatorname{Ext}_X^{i+a-b}(\Hat A,\Hat B), \end{equation} which is referred to as the Parseval theorem. As a special case, note that if $A$ is simple then so is $\Hat A$. \subsection{} Assume that $X$ and $Y$ have the same dimension. Recall that a $Y$-flat sheaf ${\mathcal P}$ on $X\times Y$ is said to be {\it strongly simple} over $Y$ if ${\mathcal P}_y$ is simple for all $y\in Y$, and if for any pair $y_1,y_2$ of distinct points of $Y$ and any integer $i$ one has $\operatorname{Ext}^i_X({\mathcal P}_{y_1},{\mathcal P}_{y_2})=0$. \begin{thm} \label{equivalence} Let ${\mathcal P}$ be a $Y$-flat sheaf on $X\times Y$. Then $\Phi$ is fully faithful iff ${\mathcal P}$ is strongly simple over $Y$. If ${\mathcal P}$ is flat over $X$ and $Y$ then $\Phi$ is an equivalence iff ${\mathcal P}$ is strongly simple over both factors. \qed \end{thm} The main idea behind the proof was given by Mukai ([13], Theorem 4.9) and the result has appeared in various forms since then. The most general statement is due to Bondal and Orlov ([6], Theorem 1.1). Following Orlov we shall make essential use of the following lemma. It is an immediate consequence of [5], Lemma 3.1. \begin{lemma} \label{useful} Suppose $\Phi$ is fully faithful. Then $\Phi$ is an equivalence iff for any object $E$ of ${D}(X)$, $\Psi(E)\cong 0$ implies $E\cong 0$. \qed \end{lemma} \section{Fourier-Mukai Transforms for Elliptic Curves} Here we illustrate the results of the last section by considering the case when $X$ and $Y$ are elliptic curves. \subsection{} Let $X$ be an elliptic curve. Given a sheaf $E$ on $X$ we write its Chern class as a pair of integers $(r(E),d(E))$. Let $a$ and $b$ be coprime integers with $a>0$ and let $Y$ be the moduli space of stable bundles on $X$ of Chern class $(a,b)$. In fact it is a consequence of the work of Atiyah ([2], Theorem 7) that $Y$ is isomorphic to $X$; we preserve the distinction for clarity. Let ${\mathcal P}$ be a tautological bundle on $X\times Y$, and put $$\Phi=\Phi^{{\mathcal P}}_{Y\to X},\qquad \Psi=\Phi^{{\mathcal P}^{\vee}}_{X\to Y}.$$ As we noted in 2.3, $\Psi[1]$ is a left adjoint of $\Phi$. \begin{prop} \label{yawn} The functor $\Phi$ is an equivalence. \end{prop} \begin{pf} First note that ${\mathcal P}$ is strongly simple over $Y$, since for any pair $P_1,P_2$ of non-isomorphic stable bundles on $X$ with the same Chern class, Serre duality gives $$\operatorname{Ext}^1_X(P_2,P_1)=\operatorname{Hom}_X(P_1,P_2)^{\vee}=0.$$ It follows from Theorem \ref{equivalence} that $\Phi$ is fully faithful, so $\Psi\circ\Phi\cong \operatorname{Id}_{{D}(Y)}[-1]$. Next observe that the group homomorphism $\operatorname{ch}(\Phi)$ must be an isomorphism, since $$\operatorname{ch}(X)\cong\operatorname{ch}(Y)={\mathbb Z}\oplus{\mathbb Z},$$ and $\operatorname{ch}(\Psi)\circ\operatorname{ch}(\Phi)=-\operatorname{Id}_{\operatorname{ch}(Y)}$. To complete the proof use Lemma \ref{useful}. Suppose $E$ is an object of ${D}(X)$ such that $\Psi(E)\cong 0$. Consider the hypercohomology spectral sequence $$E^{p,q}_2=\Psi^p(\mathscr H^q(E))\implies \Psi^{p+q}(E)=0.$$ Since $E^{p,q}_2=0$ unless $p=0$ or 1, the spectral sequence degenerates at the $r=2$ level. It follows that $\Psi(\mathscr H^q(E))=0$ for all $q$. But then, since $\operatorname{ch}(\Psi)$ is an isomorphism, one has that $\mathscr H^q(E)=0$ for all $q$, so $E\cong 0$. \qed \end{pf} The group homomorphism $\operatorname{ch}(\Phi)$ is invertible and takes $(0,1)$ to $(a,b)$, so must be given by some matrix $$\mat{c}{a}{d}{b}$$ where $c$ and $d$ are integers satisfying $bc-ad=\pm 1$. Then $\operatorname{ch}(\Psi)$, which is the inverse of $-\operatorname{ch}(\Phi)$, takes $(0,1)$ to $\pm (a,-c)$. Since $\Psi$ is given by a sheaf on $X\times Y$, we must take the positive sign, so that in fact $bc-ad=1$. This condition does not define $c$ and $d$ uniquely: we may replace them by $c+na$ and $d+nb$ for any integer $n$. This corresponds to twisting ${\mathcal P}$ by the pull-back of a line bundle of degree $n$ on $Y$. By varying $n$ we obtain all possible values of $c$ and $d$. \begin{thm} \label{ellcur} Let $X$ be an elliptic curve and take an element $$A=\mat{c}{a}{d}{b}\in\mbox{SL}_2({\mathbb Z}),$$ such that $a>0$. Then there exist vector bundles on $X\times X$ which are strongly simple over both factors, and which restrict to give bundles of Chern class $(a,c)$ on the first factor and $(a,b)$ on the second. For any such bundle ${\mathcal P}$, the resulting functor $\Phi=\Phi^{{\mathcal P}}_{X\to X}$ is an equivalence, and satisfies $$\col{r(\Phi E)}{d(\Phi E)}=\mat{c}{a}{d}{b}\col{r(E)}{d(E)},$$ for all objects $E$ of ${D}(X)$. \qed \end{thm} For the usual Fourier-Mukai transform ${\mathcal F}$ on $X$ (see [11]) one has $$A=\mat{0}{1}{-1}{0}.$$ The equivalences we have found are not essentially new, since one can check that they can all be obtained from composites of the functors ${\mathcal F}$ and $L\tensor(-)$ for line bundles $L$ on $X$. Later, however, we shall try to apply the transforms on each fibre of an elliptic surface, and this will only be possible for certain choices of $A$. \subsection{} We conclude by showing that the transforms take simple sheaves to simple sheaves. \begin{prop} \label{stable} Let $X$ be an elliptic curve and ${\mathcal P}$ a bundle on $X\times X$, strongly simple over one factor. Then $\Phi=\Phi^{{\mathcal P}}_{X\to X}$ is an equivalence. Furthermore any simple sheaf $E$ on $X$ is $\Phi$-WIT and the transform $\Hat E$ is a simple sheaf. \end{prop} \begin{pf} The argument of Proposition \ref{yawn} shows that $\Phi$ is an equivalence, so defining $\Psi$ as above, there is an isomorphism $$\Psi\circ\Phi\cong \operatorname{Id}_{{D}(X)}[-1],$$ and hence, for any sheaf $E$ on $X$, a spectral sequence $$E^{p,q}_2=\Psi^p(\Phi^q(E))\implies\left\{\begin{array}{ll} E &\mbox{ if $p+q=1$} \\ 0 &\mbox{ otherwise.} \end{array} \right. $$ Now $E^{p,q}_2=0$ unless $0\leq p,q\leq 1$, so this gives a short exact sequence $$0\longrightarrow \Psi^1(\Phi^0(E))\longrightarrow E\longrightarrow \Psi^0(\Phi^1(E))\longrightarrow 0,$$ together with the information that $\Phi^0(E)$ is $\Psi$-WIT$_1$ and $\Phi^1(E)$ is $\Psi$-WIT$_0$. The Parseval theorem then implies that $$\operatorname{Ext}^1_X(\Psi^0(\Phi^1(E)),\Psi^1(\Phi^0(E)))=0,$$ so $E$ is given by a trivial extension. If $E$ is simple it follows that one of the two sheaves $\Psi^1(\Phi^0(E))$ or $\Psi^0(\Phi^1(E))$ is zero, and $E$ is $\Phi$-WIT. The transform $\Hat E$ is then simple, as we noted in 2.5. \qed \end{pf} \begin{remark} A straightforward application of Serre duality shows that a simple sheaf on $X$ is either a stable vector bundle or the skyscraper sheaf of a point of $X$. In particular, a vector bundle on $X$ is simple iff it is stable. \end{remark} \section{The Elliptic Surfaces $J_X(a,b)$} In this section we introduce the elliptic surfaces $J_X(a,b)$ mentioned in the introduction. By an elliptic surface we shall mean a smooth variety $X$ of dimension 2 together with a smooth curve $C$ and a relatively minimal morphism $\pi:X\to C$ whose general fibre is an elliptic curve. We often abuse notation and refer to $X$ as an elliptic surface, or an elliptic surface over $C$, and take the morphism $\pi$ as given. \subsection{} Let $X\lRa{\pi} C$ be an elliptic surface. Recall ([3], V.12.3) that the canonical bundle of $X$ takes the form $$\omega_X=\pi^*(L)\tensor {\mathscr O}_X(\sum{(m_i-1)f_i)},$$ where $L$ is a line bundle on $C$ and $m_1f_1,\cdots,m_kf_k$ are the multiple fibres of $\pi$. This formula depends on the assumption that $\pi$ is relatively minimal. We denote the algebraic equivalence class of a fibre of $\pi$ by $f$, and for any object $E$ of ${D}(X)$ define the {\it fibre degree} of $E$ to be $$d(E)=c_1(E)\cdot f.$$ Note that the restriction of a sheaf $E$ on $X$ to a general fibre of $\pi$ has Chern class $(r(E),d(E))$. We say that a sheaf $E$ on $X$ is a {\it fibre sheaf} if $r(E)=d(E)=0$, or equivalently if the support of $E$ is contained in the union of finitely many fibres of $\pi$. In this case $E\tensor\omega_X$ has the same Chern class as $E$, and for any other sheaf $F$ on $X$, \begin{equation} \label{serre} \chi(E,F)=\chi(F,E). \end{equation} Let $\lambda_X$ denote the highest common factor of the fibre degrees of sheaves on $X$. Equivalently $\lambda_X$ is the smallest positive integer such that there is a divisor $\sigma$ on $X$ with $\sigma\cdot f=\lambda_X$. Note that, by Riemann-Roch, given a divisor of positive fibre degree, we can add a large multiple of $f$ and obtain an effective divisor of the same fibre degree. \subsection{} Let $X\lRa{\pi} C$ be an elliptic surface and fix integers $a>0$ and $b$, with $a\lambda_X$ coprime to $b$. Take a polarization of $X$ of fibre degree coprime to $b$. By the results of [18], there exists a relative moduli scheme ${\mathcal M}(X/C)\to C$, whose points represent stable pure-dimension 1 sheaves on fibres of $\pi$. \begin{dfn} Let $J_X(a,b)$ be the union of those components of ${\mathcal M}(X/C)$ which contain a point representing a rank $a$, degree $b$ vector bundle on a non-singular fibre of $\pi$. Let $\Hat{\pi}$ denote the natural map $\Hat{\pi}:J_X(a,b)\longrightarrow C.$ \end{dfn} First note that the coprimality assumptions we have made imply that $Y=J_X(a,b)$ is a fine moduli scheme. Thus $Y$ is a projective scheme whose points all represent strictly stable sheaves. Furthermore, by an argument of Mukai ([13], Theorem A.6), there is a tautological sheaf ${\mathcal P}$ on $X\times_C Y$, such that for each point $y\in Y$, the stable sheaf corresponding to $y$ is given by ${\mathcal P}_y$, the restriction of ${\mathcal P}$ to $X_{\Hat{\pi}(y)}\times\{y\}$. Let $U$ be the set of points $p\in C$ such that the fibre $X_p$ is non-singular. The fibre of $\Hat{\pi}$ over a point $p\in U$ is the moduli space of rank $a$, degree $b$ stable sheaves on $X_p$, which, as we noted in section 3, is isomorphic to $X_p$. Thus $\Hat{\pi}$ is an elliptic fibration. Clearly $\Hat{\pi}$ is dominant, hence surjective, so there is some component of $Y$ which contains sheaves supported on every fibre of $\pi$. Any other component of $Y$ must contain a sheaf supported on a non-singular fibre, but the fibre of $\Hat{\pi}$ over every point of $U$ is connected. It follows that $Y$ is connected. \smallskip Now let ${\mathcal M}(X)$ denote the moduli space of stable pure-dimension 1 sheaves on $X$, and let $Z$ be the union of those components of ${\mathcal M}(X)$ which contain a rank $a$, degree $b$ sheaf supported on a non-singular fibre of $\pi$. Thus points of $Z$ correspond to strictly stable sheaves of Chern class $(0,af,-b)$. There is a natural `extension by zero' morphism $i:Y\longrightarrow Z$, which maps a point $y\in Y$ representing the stable sheaf ${\mathcal P}_y$ on the fibre $X_{\Hat{\pi}(y)}$, to the point $z\in Z$ representing the stable sheaf on $X$ obtained by extending ${\mathcal P}_y$ by zero. This morphism $i$ induces a bijection on points, since every stable sheaf of Chern class $(0,af,-b)$ is supported on some fibre of $\pi$ (the support of a stable sheaf must be connected). I claim that $Z$ is a non-singular projective surface; it will follow from this that $i$ is an isomorphism and that $Y$ is an elliptic surface over $C$. \smallskip Given a point $y\in Y$ we shall identify the sheaf ${\mathcal P}_y$ with its extension by zero on $X$. If $y\in Y$ is such that ${\mathcal P}_y$ is supported on a non-singular fibre of $\pi$, then \begin{equation} \label{phew} {\mathcal P}_y={\mathcal P}_y\tensor \omega_X, \end{equation} because the restriction of $\omega_X$ to any non-singular fibre of $\pi$ is trivial. By EGA III.7.7.8 the dimension of the space $$\operatorname{Hom}_X({\mathcal P}_y,{\mathcal P}_y\tensor \omega_X)$$ is upper semi-continuous on $Y$, so for all $y\in Y$ there is a non-zero morphism ${\mathcal P}_y\to{\mathcal P}_y\tensor\omega_X$. But both these sheaves are stable with the same Chern class, so they are isomorphic and \ref{phew} holds for all $y\in Y$. The Riemann-Roch formula gives $$\chi({\mathcal P}_y,{\mathcal P}_y)=-(af)^2=0,$$ so the Zariski tangent space to $Z$ at a point $i(y)$, which is given by $$\operatorname{Ext}^1_X({\mathcal P}_y,{\mathcal P}_y)$$ (see e.g. [19]), always has dimension 2. Now $Y$ fibres over $C$ with elliptic fibres, so has dimension at least 2, and it then follows that $Z$ is a non-singular projective surface as claimed. \smallskip Extending our tautological sheaf ${\mathcal P}$ by zero, we obtain a sheaf on $X\times Y$ which we shall also denote by ${\mathcal P}$, such that for each point $y\in Y$, ${\mathcal P}_y$ is a stable sheaf of Chern class $(0,af,-b)$ on $X$. For any two distinct points $y_1$, $y_2$ of $Y$, Serre duality implies that $$\operatorname{Ext}_X^2({\mathcal P}_{y_1},{\mathcal P}_{y_2})=\operatorname{Hom}_X({\mathcal P}_{y_2},{\mathcal P}_{y_1})^{\vee}=0,$$ and since $\chi({\mathcal P}_{y_1},{\mathcal P}_{y_2})=0$, this is enough to show that ${\mathcal P}$ is strongly simple over $Y$. By Theorem \ref{equivalence}, the functor $\Phi=\Phi^{{\mathcal P}}_{Y\to X}$ is fully faithful. \begin{prop} \label{nearly} The scheme $Y=J_X(a,b)$ is an elliptic surface over $C$. Furthermore, the sheaf ${\mathcal P}$ is strongly simple over $Y$, so $\Phi=\Phi^{{\mathcal P}}_{Y\to X}$ is fully faithful. \end{prop} \begin{pf} It only remains to show that $Y$ is relatively minimal over $C$. Suppose not, i.e. that there exists a $(-1)$-curve $D$ contained in a fibre of $\Hat{\pi}$. Then $\mathcal K_Y\cdot D<0$, so $$\chi({\mathscr O}_D,{\mathscr O}_Y)=\chi(\omega_Y|_D)\neq\chi({\mathscr O}_D)=\chi({\mathscr O}_Y,{\mathscr O}_D).$$ Since $\Phi$ is fully faithful this implies that $\chi(E,F)\neq\chi(F,E),$ where $E=\Phi({\mathscr O}_D)$ and $F=\Phi({\mathscr O}_Y)$. But for each $i$, $\mathscr H^i(E)$ is a fibre sheaf (because ${\mathscr O}_D$ is), so this contradicts \ref{serre}. \qed \end{pf} \begin{remark} \label{tiup} Suppose we use two different polarizations of $X$ to define elliptic surfaces $J_X(a,b)$ and $J'_X(a,b)$ over $C$. Then, since the stability of a sheaf on a smooth curve does not depend on a choice of polarization, the two spaces will be isomorphic over the open subset $U$ considered above, and hence birational. Since both are relatively minimal over $C$, [3], Proposition III.8.4 implies that they are isomorphic as elliptic surfaces over $C$. \end{remark} \subsection{} In the next section we show that the functor $\Phi$ is an equivalence. For now, let us note that the restriction of $\Phi$ to a non-singular fibre of $\pi$ yields one of the transforms considered in section 3. Indeed, if $p\in C$ and $i_p:X_p\hookrightarrow X$ and $j_p:Y_p\hookrightarrow Y$ are the inclusion of the non-singular fibres $X_p$ and $Y_p$, then a simple base-change (see [6], Lemma 1.3) gives an isomorphism of functors $$\L i_p^*\circ\Phi\cong\Phi_p\circ\L j_p^*.$$ Here $\Phi_p$ is the functor $\Phi_{Y_p\to X_p}^{{\mathcal P}_p}$ and ${\mathcal P}_p$, the restriction of ${\mathcal P}$ to $X_p\times Y_p$, is a tautological bundle parameterising stable bundles on $X_p$ of rank $a$ and degree $b$. Thus $\Phi_p$ coincides with one of the transforms of Theorem \ref{ellcur}, and in particular there is a matrix $$\mat{c}{a}{d}{b}\in\mbox{SL}_2({\mathbb Z}),$$ such that for all objects $E$ of ${D}(Y)$, \begin{equation} \label{blag} \col{r(\Phi E)}{d(\Phi E)}=\mat{c}{a}{d}{b}\col{r(E)}{d(E)}. \end{equation} Furthermore, Proposition \ref{stable} gives \begin{lemma} \label{whoop} Let $E$ be a $\Phi$-WIT sheaf on $Y$ whose restriction to the general fibre of $\Hat{\pi}$ is simple. Then the restriction of $\Hat E$ to the general fibre of $\pi$ is also simple. \qed \end{lemma} \section{Fourier-Mukai Transforms for Elliptic Surfaces} Here we prove Theorem \ref{biggy}. Let $X\lRa{\pi} C$ be an elliptic surface, fix integers $a>0$ and $b$, with $a\lambda_X$ coprime to $b$, and let $Y$ denote the elliptic surface $\Hat{\pi}:J_X(a,b)\longrightarrow C$ defined in the last section. Fix a tautological sheaf on $X\times_C Y$, and extend by zero to obtain a sheaf ${\mathcal P}$ on $X\times Y$. Let $\mathcal Q$ be the object $({\mathcal P}^{\vee}\tensor\pi_X^*\omega_X)[1]$ of ${D}(X\times Y)$, and define functors $$\Phi=\Phi^{{\mathcal P}}_{Y\to X},\qquad\Psi=\Phi^{\mathcal Q}_{X\to Y}.$$ As we noted in 2.3, $\Psi[1]$ is a left adjoint of $\Phi$. \begin{lemma} \label{tech} The object $\mathcal Q$ is a sheaf on $X\times Y$. Moreover, ${\mathcal P}$ and $\mathcal Q$ are both flat over $X$ and $Y$. \end{lemma} \begin{pf} For each point $(x,y)\in X\times Y$, consider the commutative diagram $$ \begin{array}{ccc} \operatorname{Spec}{\mathbb C} & \lRa{j_x} & X \\ \scriptstyle{j_y}\big\downarrow && \scriptstyle{i_y}\big\downarrow \\ Y & \lRa{i_x} & X\times Y \end{array} $$ where $j_x$ and $i_x$ are the inclusions of $\{x\}$ in $X$ and $\{x\}\times Y$ in $X\times Y$ respectively. Similarly for $j_y$ and $i_y$. First note that by an argument of Mukai ([12], p. 105), any pure-dimension 1 sheaf on any surface has a locally-free resolution of length 2. This implies that for all $y\in Y$, $({\mathcal P}_y)^{\vee}[1]$ is a sheaf on $X$. But $$\L i_y^*({\mathcal P}^{\vee}[1])=(\L i_y^*({\mathcal P}))^{\vee}[1]=({\mathcal P}_y)^{\vee}[1],$$ so the corresponding hypercohomology spectral sequence implies that ${\mathcal P}^{\vee}[1]$ is a sheaf on $X\times Y$, flat over $Y$ (see [6], Proposition 1.5). To show that ${\mathcal P}$ is flat over $X$ consider the spectral sequence $$E_{p,q}^2=\L_p j_y^*(\L_q i_x^*({\mathcal P}))\implies\L_{(p+q)}j_x^*({\mathcal P}_y).$$ Here $\L_p f^*(E)$ denotes the $(-p)$th cohomology sheaf of $\L f^*(E)$. Since ${\mathcal P}_y$ has a two-step resolution, the right-hand side is non-zero only if $p+q=0$ or 1, so one concludes that $$\L_1 j_y^*(\L_1 i_x^*({\mathcal P}))=0,$$ for all $y\in Y$. This implies that $\L_1 i_x^*({\mathcal P})$ is locally free on $Y$. But for any $x\in X$ one can find $y\in Y$ such that $(x,y)$ does not lie in the support of ${\mathcal P}$, so $\L_1 i_x^*({\mathcal P})=0$ for all $x\in X$, and ${\mathcal P}$ is flat over $X$. Finally, the isomorphism $$\L i_x^*({\mathcal P}^{\vee}[1])\cong({\mathcal P}_x)^{\vee}[1],$$ implies that both sides are sheaves on $Y$, so ${\mathcal P}^{\vee}[1]$ is flat over $X$. \qed \end{pf} The next lemma shows that the relationship between $X$ and $Y$ is entirely symmetrical. \begin{lemma} \label{crafty} There exists an integer $c$ such that $X\cong J_Y(a,c).$ \end{lemma} \begin{pf} If $X_p$ is a non-singular fibre of $\pi$ then the restriction of ${\mathcal P}$ to $X_p\times Y_p$ is a tautological bundle parameterising stable bundles on $X_p$. By the results of section 3, this bundle is strongly simple over both factors, so for any point $x\in X$ lying on a non-singular fibre of $\pi$, the sheaf ${\mathcal P}_x$ is a stable sheaf on $Y$. Let its Chern class be $(0,af,-c)$. I claim that $c$ is coprime to $a\lambda_Y$, so that $J_Y(a,c)$ is well-defined. Assuming this for the moment, note that as in Remark \ref{tiup}, the two elliptic surfaces $X$ and $J_Y(a,c)$ over $C$ are isomorphic away from the singular fibres, so are isomorphic. Since the object $\mathcal Q_x$ of ${D}(Y)$ has Chern class $(0,af,c)$, to prove the claim it will be enough to exhibit an object $E$ of ${D}(Y)$ such that $\chi(\mathcal Q_x,E)=1$. But this is possible by \ref{blag}, since the result of section 2.3 implies that $$\chi(\mathcal Q_x,E)=-\chi({\mathbb C}_x,\Phi E)=-r(\Phi E),$$ for any object $E$ of ${D}(Y)$. \qed \end{pf} \smallskip We can now prove Theorem \ref{biggy}. By Proposition \ref{nearly}, $\Phi$ is fully faithful, so $\Psi\circ\Phi\cong\operatorname{Id}_{{D}(Y)}[-1]$. It follows that $\operatorname{ch}(\Psi)\circ\operatorname{ch}(\Phi)=-\operatorname{Id}_{\operatorname{ch}(Y)},$ and $\operatorname{ch}(\Phi)$ embeds $\operatorname{ch}(Y)$ as a direct summand of $\operatorname{ch}(X)$. Applying Lemma \ref{crafty}, we can repeat the argument and obtain $\operatorname{ch}(X)$ as a direct summand of $\operatorname{ch}(Y)$. This shows that $\operatorname{ch}(\Phi)$ is an isomorphism. We complete the proof by applying Lemma \ref{useful}. Suppose $E$ is an object of ${D}(X)$ such that $\Psi(E)\cong 0$. Consider the hypercohomology spectral sequence $$E^{p,q}_2=\Psi^p(\mathscr H^q(E))\implies\Psi^{p+q}(E)=0.$$ Since $\mathcal Q$ is supported on $X\times_C Y$ and is flat over $X$ and $Y$, one has that $E^{p,q}_2=0$ unless $p=0$ or 1. It follows that the spectral sequence degenerates at the $r=2$ level, so $\Psi(\mathscr H^q(E))=0$ for all $q$. But since $\operatorname{ch}(\Psi)$ is an isomorphism, and a non-zero sheaf has non-zero Chern character, this implies that $\mathscr H^q(E)=0$ for all $q$, so $E\cong 0$. \smallskip We summarise our results in the following theorem. \begin{thm} \label{super} Let $X\lRa{\pi} C$ be an elliptic surface and take an element $$\mat{c}{a}{d}{b}\in\mbox{SL}_2({\mathbb Z}),$$ such that $\lambda_X$ divides $d$ and $a>0$. Let $Y$ be the elliptic surface $J_X(a,b)$ over $C$. Then there exist sheaves ${\mathcal P}$ on $X\times Y$, flat and strongly simple over both factors such that for any point $(x,y)\in X\times Y$, ${\mathcal P}_y$ has Chern class $(0,af,-b)$ on $X$ and ${\mathcal P}_x$ has Chern class $(0,af,-c)$ on $Y$. For any such sheaf ${\mathcal P}$, the resulting functor $\Phi=\Phi^{{\mathcal P}}_{Y\to X}$ is an equivalence and satisfies \begin{equation} \label{num} \col{r(\Phi E)}{d(\Phi E)}=\mat{c}{a}{d}{b}\col{r(E)}{d(E)}, \end{equation} for all objects $E$ of ${D}(Y)$. \end{thm} \begin{pf} Take a tautological sheaf ${\mathcal P}$ on $X\times Y$ and put $\Phi=\Phi^{{\mathcal P}}_{Y\to X}$. As we showed above, $\Phi$ is an equivalence and there exist integers $c$ and $d$ such that \ref{num} holds. Now $\lambda_X$ divides $d(\Phi E)$ for any object $E$ of ${D}(Y)$, so $\lambda_X$ divides $\lambda_Y$ and $d$. By symmetry $\lambda_X=\lambda_Y$. As in section 3, $c$ and $d$ are not uniquely defined: we can replace them by $c+n\lambda_Xa$ and $d+n\lambda_Xb$ by twisting ${\mathcal P}$ by the pull-back of a line bundle of fibre degree $n\lambda_X$ on $Y$. \qed \end{pf} \begin{remark} \label{stability} As a corollary of the proof of Theorem \ref{super}, note that we can always choose ${\mathcal P}$ so that for some polarization of $X$, ${\mathcal P}_y$ is stable for all $y\in Y$. By Lemma \ref{crafty}, we could also view $X$ as a moduli space of sheaves on $Y$, and take ${\mathcal P}$ such that for some polarization of $Y$, ${\mathcal P}_x^{\vee}[1]$ is stable for all $x\in X$. \end{remark} \section{Properties of the Transforms} Let $X\lRa{\pi} C$ be an elliptic surface, fix an element $$\mat{c}{a}{d}{b}\in\mbox{SL}_2({\mathbb Z}),$$ with $a>0$ and $\lambda_X$ dividing $d$, let $Y$ be the elliptic surface $J_X(a,b)$ and take a sheaf ${\mathcal P}$ on $X\times Y$ as in Theorem \ref{super}. As in the last section, define the sheaf $\mathcal Q=({\mathcal P}^{\vee}\tensor\pi_X^*\omega_X)[1],$ and the functors $\Phi$ and $\Psi$. Since $\Phi$ is an equivalence, one has isomorphisms \begin{equation} \label{star} \Psi\circ\Phi\cong \operatorname{Id}_{{D}(Y)}[-1],\qquad\Phi\circ\Psi\cong \operatorname{Id}_{{D}(X)}[-1]. \end{equation} In this section we give some properties of the transforms which will be useful in section 7. Note that, because of the symmetry of the situation, for each result we give here, there will be another result obtained by exchanging $\Phi$ and $\Psi$, and $X$ and $Y$. \subsection{} The functor $\Phi$ is left exact, because ${\mathcal P}$ is flat over $Y$. Thus given a short exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0,$$ one obtains a long exact sequence \begin{eqnarray} \label{long} 0&\longrightarrow&\Phi^0(A)\longrightarrow\Phi^0(B)\longrightarrow\Phi^0(C) \\ &\longrightarrow&\Phi^1(A)\longrightarrow\Phi^1(B)\longrightarrow\Phi^1(C)\lra0. \nonumber \end{eqnarray} \subsection{} The isomorphisms \ref{star} imply that if $E$ is a $\Psi$-WIT$_i$ sheaf on $X$ ($i=0,1$), then $\Hat E$ is a $\Phi$-WIT$_{1-i}$ sheaf on $Y$. More generally, for any sheaf $E$ on $X$ there is a spectral sequence $$E^{p,q}_2=\Phi^p(\Psi^q(E))\implies\left\{\begin{array}{ll} E &\mbox{ if $p+q=1$} \\ 0 &\mbox{ otherwise.} \end{array} \right. $$ Since $E^{p,q}_2=0$ unless $0\leq p,q\leq 1$, this yields a short exact sequence $$0\longrightarrow \Phi^1(\Psi^0(E))\longrightarrow E\longrightarrow \Phi^0(\Psi^1(E))\longrightarrow 0,$$ together with the information that $\Psi^0(E)$ is $\Phi$-WIT$_1$ and $\Psi^1(E)$ is $\Phi$-WIT$_0$. Note that \ref{star} then implies that $\Phi^1(\Psi^0(E))$ is $\Psi$-WIT$_0$ and $\Phi^0(\Psi^1(E))$ is $\Psi$-WIT$_1$. \begin{lemma} \label{specseq} For any sheaf $E$ on $X$ there is a unique short exact sequence $$0\longrightarrow A\longrightarrow E\longrightarrow B\longrightarrow 0,$$ such that $A$ is $\Psi$-WIT$_0$ and $B$ is $\Psi$-WIT$_1$. \end{lemma} \begin{pf} For the uniqueness, suppose there is another such sequence $$0\longrightarrow A'\longrightarrow E\longrightarrow B'\longrightarrow 0.$$ Then, since $A'$ is $\Psi$-WIT$_0$ and $B$ is $\Psi$-WIT$_1$, the Parseval theorem implies that there is no non-zero map $A'\to B$, so the inclusion of $A'$ in $E$ factors through $A$. By symmetry $A=A'$. \qed \end{pf} \subsection{} Given a torsion-free sheaf $E$ on $X$, put $\mu(E)=d(E)/r(E)$. \begin{lemma} \label{chern} Let $E$ be a torsion-free sheaf on $X$. If $E$ is $\Psi$-WIT$_0$ then $\mu(E)\geq b/a$. Similarly if $E$ is $\Psi$-WIT$_1$ then $\mu(E)\leq b/a$. \end{lemma} \begin{pf} If $E$ is $\Psi$-WIT$_1$ then $\Psi(E)[1]$ is a sheaf so $r(\Psi E)\leq 0$. Similarly, if $E$ is $\Psi$-WIT$_0$ then $r(\Psi E)\geq 0$. Since $\Psi[1]$ is the inverse of $\Phi$, one has $$\col{r(\Psi E)}{d(\Psi E)}=\mat{-b}{a}{d}{-c}\col{r(E)}{d(E)},$$ for any object $E$ of ${D}(X)$. The result follows. \qed \end{pf} A similar argument gives \begin{lemma} \label{fibre} Let $T$ be a $\Psi$-WIT$_1$ torsion sheaf on $X$. Then $T$ is a fibre sheaf. \qed \end{lemma} Combining Lemma \ref{chern} with Lemma \ref{specseq} we obtain \begin{lemma} \label{wit} Let $E$ be a torsion-free sheaf on $X$ such that the restriction of $E$ to the general fibre of $\pi$ is stable. Suppose $\mu(E)<b/a$. Then $E$ is $\Psi$-WIT$_1$. \end{lemma} \begin{pf} Consider the short exact sequence of Lemma \ref{specseq}. If $A$ is non-zero, it is torsion-free and one has $\mu(A)\geq b/a>\mu(E)$. Restricting to the general fibre of $\pi$ this gives a contradiction. Hence $A=0$ and $E$ is $\Psi$-WIT$_1$. \qed \end{pf} \subsection{} The final result we shall need is \begin{lemma} \label{new} A sheaf $F$ on $Y$ is $\Phi$-WIT$_0$ iff $$\operatorname{Hom}_Y(F,\mathcal Q_x)=0\qquad\forall x\in X.$$ \end{lemma} \begin{pf} First note that $\mathcal Q_x=\Psi({\mathbb C}_x)$ is $\Phi$-WIT$_1$. If $F$ is $\Phi$-WIT$_0$, then the Parseval theorem implies that there are no non-zero maps $F\to\mathcal Q_x$. Conversely, if $F$ is not $\Phi$-WIT$_0$, then by the argument of Lemma \ref{specseq}, there is a surjection $F\to B$ with $B$ a $\Phi$-WIT$_1$ sheaf. Applying the Parseval theorem again gives $$\operatorname{Hom}_Y(B,\mathcal Q_x)=\operatorname{Hom}_X(\Hat B,{\mathbb C}_x).$$ Since $\Hat B$ is non-zero, there exists an $x\in X$ and a non-zero map $B\to \mathcal Q_x$, hence a non-zero map $F\to \mathcal Q_x$. \qed \end{pf} \section{Application to Moduli of Stable Sheaves} In this section we use the relative transforms we have developed to prove Theorem \ref{moduli}. \subsection{} Let $X\lRa{\pi} C$ be an elliptic surface and fix a triple $$(r,\Lambda,k)\in\mathbb N\times\operatorname{NS}(X)\times{\mathbb Z},$$ such that $r$ is coprime to $d=\Lambda\cdot f$. The proof of the following result is entirely analagous to the rank $2$ case ([7], Theorem I.3.3) so we omit it (see also [15], Proposition I.1.6). \begin{prop} There exist polarizations of $X$ with respect to which a torsion-free sheaf $E$ on $X$ with Chern class $(r,\Lambda,k)$ is $\mu$-stable whenever it is $\mu$-semi-stable, and this is the case iff the restriction of $E$ to all but finitely many fibres of $\pi$ is stable. \qed \end{prop} Taking such a polarization, define ${\mathcal M}={\mathcal M}_X(r,\Lambda,k)$ to be the (fine) moduli space of stable torsion-free sheaves on $X$ of Chern class $(r,\Lambda,k)$. We identify the closed points of ${\mathcal M}$ with the stable sheaves which they represent. As in the rank $2$ case ([7], Lemma III.3.6), one shows that for any $E\in{\mathcal M}$, $$\operatorname{Ext}^2_X(E,E)=H^2(X,{\mathscr O}_X).$$ It then follows from the general results of [1] that ${\mathcal M}$, if non-empty, is smooth of dimension $\dim (\Pic^{\circ}(X))+2t$, where $$2t=2rk-(r-1)\Lambda^2-(r^2-1)\chi({\mathscr O}_X),$$ and that if $t<0$ then ${\mathcal M}$ is empty. In what follows we take $r>1$ and assume that $t$ is non-negative. Let $a$ and $b$ be the unique pair of integers satisfying $br-ad=1$, with $0<a<r$. Let $\Hat{\pi}:Y\to C$ be the elliptic surface $J_X(a,b)$ and put $$\mathscr N={\mathcal M}_Y(1,0,t)=\Pic^{\circ}(Y)\times\operatorname{Hilb}^t(Y).$$ We shall prove Theorem \ref{moduli} by showing that ${\mathcal M}$ is birationally equivalent to $\mathscr N$. \subsection{} Let ${\mathcal P}$ be a sheaf on $X\times Y$ as in Theorem \ref{super}, with matrix $$\mat{r}{a}{d}{b},$$ and define equivalences of categories $\Phi$ and $\Psi$ as in section 6. As we noted in Remark \ref{stability}, we can assume that we have chosen ${\mathcal P}$, and a polarization of $Y$, so that $\mathcal Q_x$ is a stable sheaf for all $x\in X$. Take a sheaf $E$ on $X$ of Chern class $(r,\Lambda,k)$. The formula given in the proof of Lemma \ref{chern} shows that $\Psi(E)$ has rank 1 and fibre degree 0. Twisting ${\mathcal P}$ by the pull-back of a line bundle on $Y$ we can assume that $c_1(\Psi E)=0$, and the formula \ref{percy}, together with Riemann-Roch then implies that $\Psi(E)$ has Chern class $(1,0,t)$. Note that, by Lemma \ref{wit}, any element $E$ of ${\mathcal M}$ is $\Psi$-WIT$_1$. Define $$\mathscr U=\{E\in{\mathcal M}:\Hat E\mbox{ is torsion-free}\}.$$ By the result of section 2.4, $\mathscr U$ is an open subscheme of ${\mathcal M}$. Also define the open subscheme $$\mathscr V=\{F\in \mathscr N:F\mbox{ is }\Phi\mbox{-WIT}_0\}.$$ \begin{lemma} The transform $\Phi$ gives an isomorphism between the schemes $\mathscr U$ and $\mathscr V$. \end{lemma} \begin{pf} For any point $E\in \mathscr U$, $E$ is $\Psi$-WIT$_1$ and $\Hat E\in \mathscr V$. Suppose now that $F\in \mathscr V$ and put $E=\Hat{F}$. Claim that $E\in \mathscr U$. By Lemma \ref{whoop} the restriction of $E$ to the general fibre of $X$ is stable, so it is only neccesary to check that $E$ is torsion-free. Suppose $E$ has a torsion subsheaf $T$. Then since $E$ is $\Psi$-WIT$_1$, the long exact sequence \ref{long} implies that $T$ is $\Psi$-WIT$_1$ also, hence, by Lemma \ref{fibre}, a fibre sheaf. Applying $\Psi$ gives a sequence $$0\longrightarrow\Psi^0(E/T)\lRa{f} \Hat{T}\longrightarrow F\longrightarrow\Psi^1(E/T)\longrightarrow 0.$$ Since $F$ is torsion-free and $\Hat T$ is a fibre sheaf, $f$ must be an isomorphism. But, by the result of 6.2, $\Psi^0(E/T)$ is $\Phi$-WIT$_1$, and $\Hat{T}$ is $\Phi$-WIT$_0$. It follows that both sheaves are zero, so $T=0$ and $E$ is torsion-free. The proof of the lemma is completed by appealing to the general result quoted in section 2.4. \qed \end{pf} Clearly, we need to show that $\mathscr U$ and $\mathscr V$ are non-empty. Take $F\in\mathscr N$. Then $F=L\tensor\mathcal I_Z$, with $L\in\Pic^{\circ}(Y)$ and $Z$ a zero-dimensional subscheme of $Y$ of length $t$. By Lemma \ref{new}, $F$ is $\Phi$-WIT$_0$ precisely when there is no non-zero map $F\to\mathcal Q_x$ for any $x\in X$. Since $\mathcal Q_x$ is supported on the fibre $Y_{\pi(x)}$ of $\Hat{\pi}$, any map $F\to\mathcal Q_x$ factors via $F|_{Y_{\pi(x)}}$, and hence via a stable, pure dimension 1 sheaf on $Y$ of Chern class $(0,f,s)$, where $s$ is the number of points of $Z$ lying on the fibre $Y_{\pi(x)}$. Now $\mathcal Q_x$ has Chern class $(0,af,r)$, and is stable, so if $s<r/a$, any map $F\to\mathcal Q_x$ is zero. This argument, and the fact that $r>a$, gives the following results. \begin{lemma} Let $F=L\tensor\mathcal I_Z$, with $L\in\Pic^{\circ}(Y)$ and $Z$ a set of $t$ points $\{y_1,\cdots,y_n\}$ lying on distinct fibres of $\pi:Y\to C$. Then $F$ is an element of $\mathscr V$. \qed \end{lemma} \begin{lemma} If $r>at$ then $\mathscr V=\mathscr N$. \qed \end{lemma} \begin{remark} Applying $\Phi$ to the short exact sequence $$0\longrightarrow F\longrightarrow L\longrightarrow {\mathscr O}_Z\longrightarrow 0,$$ gives a sequence $$0\longrightarrow \Hat F\longrightarrow {\Hat L}\longrightarrow \oplus{\mathcal P}_{y_i}\longrightarrow 0.$$ Let us assume for simplicity that $X$ is simply-connected. Then we see that an open subset of ${\mathcal M}$ is obtained from the fixed bundle $\Hat{{\mathscr O}_Y}$ by taking $t$ distinct non-singular fibres $\{f_1,\cdots,f_t\}$ of $\pi$ and stable bundles $P_i$ of rank $a$ and degree $b$ on $f_i$, and taking the kernel of the unique morphism $$\Hat{{\mathscr O}_Y}\longrightarrow\oplus P_i.$$ Furthermore, when $X$ is nodal, the proof of [7], Proposition III.3.11 shows that $\Hat{{\mathscr O}_Y}$ is the unique sheaf (up to twists) on $X$ whose restriction to every reduction of a fibre of $\pi$ is stable. In the rank $2$ case, this corresponds to Friedman's method of constructing bundles using elementary modifications ([7], section III.3). \end{remark} \subsection{} To complete the proof of Theorem 1.1 we must show that ${\mathcal M}$ is irreducible, i.e. that ${\mathcal M}$ has only one connected component. Let us suppose, for contradiction, that there is a connected component $\mathscr W$ of ${\mathcal M}$ which does not meet $\mathscr U$. Let $E$ be a point of $\mathscr W$. Then $E$ is $\Psi$-WIT$_1$, and the transform $\Hat E$ is a sheaf of Chern class $(1,0,t)$ on $Y$, with a non-zero torsion sheaf. By the argument of Lemma \ref{whoop}, the restriction of $\Hat E$ to the general fibre of $\Hat{\pi}$ is simple, hence stable. \begin{lemma} Let $n\geq 1$ be an integer. Then for a general zero-dimensional subscheme $Z\in\operatorname{Hilb}^{rn}(Y)$, there is a unique morphism $\Hat E\to{\mathscr O}_Z$. Furthermore, for general $Z$, this morphism surjects and the kernel $K$ is $\Phi$-WIT$_0$. The transform $\Hat K$ is then an element of the moduli space $$\tilde{{\mathcal M}}={\mathcal M}_X(r,\Lambda-(rna)f,k+rnb-rna(\Lambda\cdot f)).$$ \end{lemma} \begin{pf} We may suppose that $Z$ consists of $rn$ points lying on distinct non-singular fibres $f_1,\cdots ,f_{rn}$ of $\Hat{\pi}$. We can also suppose that $\Hat E$ is locally-free at each of the points of $Z$. Then there is a unique morphism $\Hat E\to{\mathscr O}_Z$ and this map surjects, giving an exact sequence $$0\longrightarrow K\longrightarrow \Hat E\longrightarrow {\mathscr O}_Z\longrightarrow 0.$$ By Lemma \ref{new}, to prove that $K$ is $\Phi$-WIT$_0$, we must show that there are no non-zero morphisms $K\to\mathcal Q_x$ for any $x\in X$. We only need to check this when $\mathcal Q_x$ is supported on one of the fibres $f_1,\cdots ,f_{rn}$ since the restrictions of $\Hat E$ and $K$ to any other fibre are identical, and $\Hat E$ is $\Phi$-WIT$_0$. But we can always take $Z$ so that the restriction of $\Hat E$ to each of the fibres $f_i$ is a degree 0 line bundle. This will be enough since $\mathcal Q_x$ is stable of degree $-r$. \qed \end{pf} Twisting by ${\mathscr O}_X(anf)$ gives an isomorphism between the spaces $\tilde{{\mathcal M}}$ and ${\mathcal M}_X(r,\Lambda,k+n)$, so a general theorem of Gieseker-Li and O'Grady (see e.g. [16]) implies that for large enough $n$, $\tilde{{\mathcal M}}$ is irreducible. It follows from the results of 7.2 that the general element of $\tilde{{\mathcal M}}$ has torsion-free transform. Now the construction of the lemma gives a rational map $$\theta:\mathscr W\times\operatorname{Hilb}^{rn}(Y)\dashrightarrow\tilde{{\mathcal M}},$$ and since all points in the image of $\theta$ have non-torsion-free transforms, $\theta$ cannot be dominant. But we shall show below that the general fibre of $\theta$ is zero-dimensional. Since $\theta$ is a map between two varieties of the same dimension, this will give a contradiction. Take an element of $\tilde{{\mathcal M}}$, and let $K$ be its transform. We must show that there are only finitely many pairs $$(E,Z)\in\mathscr W\times\operatorname{Hilb}^{rn}(Y),$$ such that $Z$ consists of $rn$ distinct points at which $\Hat E$ is locally free, and $K=\Hat E\tensor\mathcal I_Z$. Given such a pair, note that $Z$ does not meet the support of the torsion subsheaf of $\Hat E$, so the torsion subsheaves $T$ of $\Hat E$ and $\Hat E\tensor\mathcal I_Z=K$ are equal. Thus $Z$ is a subset of the finite set of points at which $K/T$ is not locally-free. This implies that the number of possible choices of $Z$ is finite. Finally, if we have two pairs $(E_1,Z)$, and $(E_2,Z)$ then $E_1=E_2$, because there is only one extension of $K$ by ${\mathscr O}_Z$ which is locally-free at each of the points of $Z$. This completes the proof. \subsection{} We conclude with the following simple corollary of Theorem \ref{moduli}. \begin{cor} Given integers $a>0$ and $b$ with $a\lambda_X$ coprime to $b$, and a positive integer $t$, there are polarizations of $X$ such that a component of the moduli space of torsion-free stable sheaves on $X$ is isomorphic to $$\Pic^{\circ}(J_X(a,b))\times\operatorname{Hilb}^t(J_X(a,b)).$$ \end{cor} \begin{pf} Take $r>at$ such that $br$ is congruent to 1 modulo $a\lambda_X$. Then there exists a divisor $\Lambda$ on $X$ such that $d=\Lambda\cdot f$ is coprime to $r$ and $br-ad=1$. Adding multiples of $f$ to $\Lambda$ if neccesary, it is easy to check that one can choose $k$ such that \ref{first} holds. Applying Theorem \ref{moduli} then gives the result. \qed \end{pf} \section*{References} \small {EGA} {\it A. Grothendieck, J. Dieudonn{\'e},} El{\'e}ments de g{\'e}om{\'e}trie alg{\'e}brique, Publ. Math. I.H.E.S. [1] {\it V.I. Artamkin,} On deformations of sheaves, Math. USSR-Izv. {\bf 32} (1989), 663-668. [2] {\it M.F. Atiyah,} Vector bundles over an elliptic curve, Proc. London Math. Soc. {\bf 7} (1957), 414-452. [3] {\it W. Barth, C. Peters, A. Van de Ven,} Compact Complex Surfaces, Ergebnisse Math. Grenzgeb. (3), vol. 4, Springer-Verlag, 1984. [4] {\it C. Bartocci, U. Bruzzo, D. Hern{\'a}ndez-Ruip{\'e}rez,} A Fourier-Mukai transform for stable bundles on K3 surfaces, J. reine angew. Math. {\bf 486} (1997), 1-16. [5] {\it A.I. Bondal,} Representations of associative algebras and coherent sheaves, Math. USSR Izv. {\bf 34} (1990), 23-42. [6] {\it A.I. Bondal, D.O. Orlov,} Semiorthogonal decomposition for algebraic varieties, Preprint alg-geom 9506012. [7] {\it R. Friedman,} Vector bundles and SO(3)-invariants for elliptic surfaces, J. Amer. Math. Soc. {\bf 8} (1995), 29-139. [8] {\it R. Friedman, J.W. Morgan,} Smooth four-manifolds and complex surfaces, Ergebnisse Math. Grenzgeb. (3), vol. 27, Springer-Verlag, 1994. [9] {\it R. Hartshorne,} Residues and duality, Lect. Notes Math. {\bf 20}, Springer-Verlag, 1966. [10] {\it A. Maciocia,} Generalized Fourier-Mukai transforms, J. reine angew. Math. {\bf 480} (1996), 197-211. [11] {\it S. Mukai,} Duality between $D(X)$ and $D(\Hat X)$ with its application to Picard sheaves, Nagoya Math. J. {\bf 81} (1981), 153-175. [12] {\it S. Mukai,} Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. math. {\bf 77} (1984), 101-116. [13] {\it S. Mukai,} On the moduli space of bundles on K3 surfaces I, in: Vector Bundles on Algebraic Varieties, M.F. Atiyah et al., Oxford University Press (1987), 341-413. [14] {\it S. Mukai,} Fourier functor and its application to the moduli of bundles on an abelian variety, Adv. Pure Math. {\bf 10} (1987), 515-550. [15] {\it K.G. O'Grady,} The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, Preprint alg-geom 9510001. [16] {\it K.G. O'Grady,} Moduli of vector bundles on surfaces, Preprint alg-geom 9609015. [17] {\it D.O. Orlov,} Equivalences of derived categories and K3 surfaces, Preprint alg-geom 9606006. [18] {\it C.T. Simpson,} Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. {\bf 79} (1994) 47-129. [19] {\it Y.T. Siu, G. Trautmann,} Deformations of coherent analytic sheaves with compact supports, Memoirs of Amer. Math. Soc. {\bf 29} (1981), no. 238. \bigskip Department of Mathematics and Statistics, The University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK. email: [email protected] \end{document}

9512

alg-geom/9512008

en

https://arxiv.org/abs/alg-geom/9512008

alg-geom/9512008

Peter Schenzel

Applications of Koszul homology to numbers of generators and syzygies

To appear in J. Pure Appl. Algebra, LaTeX2e

Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of projective varieties. There are also investigations about the shifts and the dimension of Betti numbers.

alg-geom

\section{Introduction} Let $X \subset \mathbb P^n_K$ be an algebraic variety, $K$ an algebraically closed field. Let $\mathcal J_X$ denote the ideal sheaf of $X.$ Then $\mathcal J_X$ admits a finite minimal resolution \begin{displaymath}\mathcal F_{\bullet} : 0 \to \mathcal F_s \to \ldots \to \mathcal F_i \to \ldots \to \mathcal F_1 \to \mathcal J_X \to 0, \end{displaymath}where $\mathcal F_i \simeq \oplus_{j\in \mathbb Z} \mathcal O^{i_j}(-j).$ Here only finitely many $i_j$ are non-zero. The resolution $\mathcal F_{\bullet}$ reflects several geometric and arithmetic properties of $X.$ For instance, the length $s$ of $\mathcal F_{\bullet}$ satisfies $s \geq \codim X.$ The equality characterizes when $X$ is arithmetically Cohen-Macaulay. On the other hand $$ \reg M := \max \{ j - i \; | \; j \in \mathbb Z \mbox{ and } i_j \neq 0 \} $$ is called the Castelnuovo-Mumford regularity of $X.$ It is determined by the vanishing of the cohomology $H^i(X, \mathcal J_X(n)),$ see e.g., \cite{dM}. The rank of $\mathcal F_1$ is determined by the minimal number of generators $\mu (I_X)$ of $I_X,$ the saturated ideal of $X$ in $R = K[x_0,\ldots,x_n].$ There is a classical result by P.~Dubreil, see \cite{pD}, that for $X$ a set of points in the projective plane $\mathbb P^2_K$ it follows that $$ \mu (I_X) \leq a(I_X) + 1, $$ where $a(I_X)$ denotes the smallest degree of a hypersurface that contains $X.$ There is an extension of this result to the case of $X$ arithmetically Cohen-Macaulay of codimension two, see \cite{DGM} or \cite{pS1}. More recently H.~Martin and J.~Migliore extended Dubreil's Theorem to $X$ a locally Cohen-Macaulay scheme, see \cite[Theorem 2.5]{MM}. One of the main points of the present paper is an extension of their result to an arbitrary scheme $X \subset \mathbb P^n_K.$ In fact it turns out that $$ \mu (I_X) \leq a(I_X) + 1 + C(I_X) $$ for a certain correcting term $C(I_X),$ see \ref{4.1} and \ref{4.3}. The integer $C(I_X)$ is determined by the dimensions of the Koszul homology modules of $H^i_{\ast}(X, \mathcal O_X)$ with respect to a certain system of linear forms. These kinds of invariants have been considered by M.~Green in his fundamental paper \cite{mG}. In fact we develop a complete local analogue of these modules for any finitely generated graded $R$-module $M.$ These invariants $$ H_i(\underline l ; H_{\mathfrak m}^j(M)),\quad i, j \in \mathbb Z, $$ $\underline l = l_1, \ldots, l_r, $ a system of generic linear forms, are graded $R$-modules of finite length. We call them Green modules of $M$ with respect to $\underline l.$ As indicated in Green's paper, see \cite{mG}, its graded components play an important r\^ole in getting information about the minimal free resolution of $M$ over $R.$ Under additional assumptions on $X$ resp. $M$ there are explicit geometric interpretations of the Green modules. So e.g. in the case of $X$ an arithmetically Cohen-Macaulay scheme of codimension three it follows that $\mu (I_X) \leq a(I_X) + 1 + \deg X.$ On the other hand the Green modules are intimately related to the structure of the minimal free resolution of $\mathcal F_{\bullet}$ of $M.$ In fact they are the ingredients of a spectral sequence for computing $\Tor^R_i(K, M), i \in \mathbb Z.$ That is, they describe in a certain sense the Betti numbers and their shifts. This is worked out in more detail in Section 5, where it is shown that $$ \reg X = \max \{j - i \; | \; j \geq \codim X \mbox{ and } i_j \neq 0 \}, $$ see \ref{5.2}. That is, the regularity of $X$ is completely determined by the tail of $\mathcal F_{\bullet}.$ More precisely, under certain additional assumptions there is an explicit computation of $i_j$ in terms of the graded components of the Green modules, see \ref{5.5} for the precise statement. It turns out that this is a generalization of M.~Green's duality theorem, see \cite[Section 2]{mG}. On the other hand Theorem 5.5 is a far reaching generalization of P.~Rao's observation on how much the resolution of the Hartshorne-Rao module $M(C)$ of a curve $C \subset \mathbb P^3_K$ determines the resolution of $\mathcal J_C$ at the tail, see \cite[(2.5)]{pR}. One application of this type concerns the resolution of certain curves $C \subset \mathbb P^n_K$ of arithmetic genus $g_a(C) = 0,$ see \ref{5.7}. While the applications of our results are motivated by geometric questions we formulate and prove them in terms of graded modules over $R$ and its local cohomology modules $H_{\mathfrak m}^j(M).$ To this end we fix a few homolo\-gical preliminaries in Section 2. These concern Koszul homology, local cohomology, and some basic facts about spectral sequences. The spectral sequence related to a double complex is in several variations one of the basic tools of our investigations. In Section 3 we summarize the details about the Green modules. The most important result is \ref{3.4}. It proves the finite length of $H_i(\underline l ; H_{\mathfrak m}^j(M))$ for a generic system of linear forms, i.e., for almost all $t \in \mathbb Z$ it follows that $$ H_i(\underline l ; H_{\mathfrak m}^j(M))_{i+t} = 0 \, \mbox{ for all } i, j \in \mathbb Z. $$ Section 4 is devoted to the estimates of the number of generators of $I_X,$ i.e., to the desired variations of Dubreil's theorem. The final Section 5 concerns the relation of the syzygies of the modules of `deficiencies' $H_{\mathfrak m}^j(M)$ to those of $M.$ In particular it yields the new characterization of $\reg M$ resp. the generalization of Green's duality theorem. The author is grateful for the referee's careful reading of the manuscript. \section{Koszul homology and local cohomology} First fix a few notation and conventions. Let $A = \oplus _{n \geq 0} A_n$ denote a graded Noetherian ring such that $A_0 = K$ is an infinite field and $A = K[A_1].$ Then $A$ is an epimorphic image of the polynomial ring $R = K[x_1, \ldots, x_r]$ in the variables $x_1, \ldots, x_r, \, r = \dim_K A_1.$ Let $M = \oplus_{n \in \mathbb Z} M_n$ denote a graded $A$-module. For $k \in \mathbb Z$ let $M(k)$ denote the module $M$ with the grading given by $[M(k)]_n = M_{k+n}, \, n \in \mathbb Z.$ Mostly we consider a graded $A$-module as an module over $R.$ For more details about graded modules and rings see \cite[1.5]{BH}. Let $X^{\bullet}, Y^{\bullet}$ denote two complexes of graded $A$-modules. Let $Z^{\bullet}$ denote the single complex associated to the double complex $X^{\bullet} \otimes_A Y^{\bullet}.$ Then there are the following two spectral sequences \begin{displaymath} \begin{array}{clllcll} E_2^{ij} & = & H^i(X^{\bullet} \otimes_A H^j(Y^{\bullet})) & \Rightarrow & E^{i+j} & = & H^{i+j}(Z^{\bullet}) \quad \mbox{and } \\ `E_2^{ij} & = & H^i(H^j(X^{\bullet}) \otimes_A Y^{\bullet}) & \Rightarrow & `E^{i+j} & = & H^{i+j}(Z^{\bullet}). \end{array} \end{displaymath} See e.g. \cite[Appendix A3]{dE} or \cite[5.6]{cW} for an introduction and the basic results concerning spectral sequences. Here we remark that by the definitions all the homomorphisms are homogeneous of degree zero. Let $\underline f = f_1, \ldots, f_s$ denote a system of homogeneous elements. Then $K_{\bullet}(\underline f ; A)$ denotes the Koszul complex with respect to $\underline f.$ Fix the following definitions \begin{displaymath} \begin{array}{ll} K_{\bullet}(\underline f ; M) := K_{\bullet}(\underline f ; A) \otimes_A M, \quad & H_i(\underline f ; M) := H_i(K_{\bullet}(\underline f ; M)), \\ K^{\bullet}(\underline f ; M) := \Hom_A(K_{\bullet}(\underline f ; A), M), \quad & H^i(\underline f ; M) := H^i(K^{\bullet}(\underline f ; M)), \end{array} \end{displaymath} where $i \in \mathbb Z,$ see e.g. \cite[1.6]{BH} or \cite[Section 17]{dE}. Note that all the modules resp. complexes are graded. The homomorphims are homogeneous of degree zero. In particular let $\underline{\mathfrak m} = x_1, \ldots, x_r$ denote a generating set of $\mathfrak m,$ the ideal generated by all forms of positive degree. Then $K_{\bullet}(\underline x ; R)$ provides a finite free resolution of $K,$ the residue field. Therefore \begin{displaymath} H_i( \mathfrak m; M) \simeq \Tor_i^R(K, M) \mbox{ and } H^i( \mathfrak m; M) \simeq \Ext_R^i(K, M), i \in \mathbb Z. \end{displaymath} In the following split $\mathfrak m$ into two subsets $\underline x$ and $\underline y.$ Then we compare their Koszul homologies. \begin{lemma} \label{2.1} Put $\underline x = x_1, \ldots, x_s$ and $\underline y = x_{s+1}, \ldots, x_r$ for an integer $1 \leq s < r.$ Then $$ \dim _K H_n(\mathfrak m; M) \leq \sum_{i=\max\{0,n-(r-s)\}}^{\min\{s,n\}} \dim _K H_i(\underline x; H_{n-i}(\underline y;M)), \quad n \in \mathbb N, $$ for any finitely generated graded $R$-module $M.$ \end{lemma} \begin{proof} First note that $$ K_{\bullet}( \mathfrak m;M) \simeq K_{\bullet}(\underline x ; R) \otimes_R K_{\bullet}(\underline y; M) $$ as follows by view of the construction of the Koszul complex. But then there is the following spectral sequence $$ E^2_{ij} = H_i(\underline x; H_j(\underline y; M)) \Rightarrow E_{i+j} = H_{i+j}(\mathfrak m ;M). $$ Moreover note that all the $E^2_{ij}$-terms are finite dimensional $K$-vector spaces. This follows because all of them are annihilated by $\mathfrak m.$ The subsequent terms $E^n_{ij}$ are subquotients of $E^2_{ij}.$ So they are also finite dimensional and $$ \dim_K E^n_{ij}\, \leq \, \dim_K E^2_{ij} \quad \mbox{for all } n \geq 2. $$ Now for large $n$ one has $E^{\infty}_{ij} = E^n_{ij}.$ Furthermore it is known that $$ E_{i+j} = H_{i+j}(\mathfrak m;M) $$ admits a finite filtration whose quotients are $E^{\infty}_{i,n-i},\, i = 0,1, \ldots, n.$ This implies that $$ \dim_K H_n(\mathfrak m;M) = \sum_{i=0}^n \dim_K E^{\infty}_{i,n-i}. $$ So the claim follows because of the above estimates. \end{proof} For further investigations the case of $n = 1$ is of a particular interest. To this end formulate it as a separate Corollary. \begin{corollary} \label{2.2} Let $\underline x, \underline y,$ and $M$ as in \ref{2.1}. Then $$ \dim_K H_1(\mathfrak m; M) \leq \dim_K H_0(\underline x ;H_1(\underline y; M)) + \dim_K H_1(\underline x; H_0(\underline y; M)). $$ \end{corollary} The general idea behind \ref{2.1} is a bound of the Betti numbers $$ b_n(M) := \dim _K \Tor_n^R(K, M). $$ For $n = 1$ and $M = R/I, I$ a homogeneous ideal of $R,$ this yields a bound for the minimal number of generators $\mu(I)$ of $I,$ see \ref{4.1}. For several investigations we need the local cohomology modules $H_{\mathfrak m}^i(M), i \in \mathbb Z, $ of $M$ with respect to $\mathfrak m.$ To this end denote by $K^{\bullet}_f(A)$ the complex $0 \to A \to A_f \to 0,$ where $f$ denotes a homogeneous element and $A_f$ is the localization with respect to $f.$ The middle homomorphism denotes the canonical map into the localization. For $\underline{\mathfrak m} = x_1. \ldots, x_r$ define $$ K^{\bullet} := \otimes_{i=1}^r K^{\bullet}_{x_i} \quad \mbox{and } \quad K^{\bullet}(M) := K^{\bullet} \otimes_A M $$ the \v Cech complex of $A$ and $M.$ Then there are canonical isomorphisms $$ H_{\mathfrak m}^i(M) \simeq H^i(K^{\bullet} \otimes_A M)\, \mbox{ for all } i \in \mathbb Z. $$ For the details of this facts see e.g. \cite{rH} or \cite[3.5]{BH}. \section{The Green modules} Let $M$ denote a finitely generated graded $R$-module. In this section we introduce certain invariants related to the local cohomology and the Koszul homology of $M.$ To this end we need the notion of a generic system of elements. \begin{definition} \label{3.1} A system of linear elements $\underline l = l_1, \ldots,l_s$ is said to be a generic linear system of elements with respect to $M$ provided $$ l_i \not\in \mathfrak p \mbox{ for all } \mathfrak p \in (\Ass_R(M/(l_1, \ldots, l_{i-1})M) \setminus \{\mathfrak m\}. $$ Here $\mathfrak m$ denotes the ideal generated by the variables $x_1, \ldots, x_r$ in $R.$ \end{definition} Note that $\underline l$ is a generic system of linear elements if and only if the following quotients $$ ((l_1,\ldots,l_{i-1})M :_M l_i)/(l_1,\ldots,l_{i-1})M, \, i = 1,\ldots,s, $$ are graded $R$-modules of finite length. This observation is helpful in order to check whether a given $\underline l$ is a generic linear system. The most important property of a general linear system is related to a certain finiteness property of $H_i(\underline l ; H_{\mathfrak m}^j(M))$ which we shall prove in this section. In order to do that we need another auxiliary statement. \begin{lemma} \label{3.2} Let $\underline l = l_1, \ldots, l_s$ denote a generic linear system with respect to $M.$ Then $H^i(\underline l ; M)$ is an $R$-module of finite length in the following two cases: \begin{itemize} \item[(a)] $i < s,$ \item[(b)] for all $i \in \mathbb Z,$ provided $s \geq \dim_R M.$ \end{itemize} \end{lemma} \begin{proof} First prove the claim in (b). To this end note that $$ (\underline l, \Ann_R M) H^i(\underline l; M) = 0 \, \mbox{ for all }\, i \in \mathbb Z. $$ Therefore $\Supp _R H^i(\underline l; M) \subseteq \Supp_R M/\underline l M.$ Since $s\geq \dim_R M$ it follows that $\Supp _R M\underline l M \subseteq V(\mathfrak m).$ Recall that $\underline l$ is generically chosen. This proves (b) since $H^i(\underline l; M)$ is a finitely generated $R$-module. The statement in (a) will be shown by an induction on $d := \dim_RM.$ First note that the case $d = 0$ is covered by the claim proved in (b). So let $d > 0.$ First suppose that $\depth_R M > 0.$ Then $l = l_1$ is an $M$-regular element. The short exact sequence $$ 0 \to M(-1) \to M \to M/lM \to 0 $$ induces short exact sequences $$ 0 \to H^i(\underline l;M) \to H^i(\underline l; M/lM) \to H^{i+1}(\underline l; M)(-1) \to 0 $$ for all $i \in \mathbb Z.$ Note that $l H^i(\underline l; M) = 0.$ Hence the induced maps on the Koszul cohomology are trivial. Now put $\underline l' = l_2, \ldots, l_s.$ Then $$ H^i(\underline l;M/lM) \simeq H^i(\underline l';M/lM) \oplus H^{i-1}(\underline l';M/lM)(-1), \quad i \in \mathbb Z, $$ see \cite[1.6]{BH}. Note that $l$ acts trivially by multiplication on $M/lM,$ Now by induction hypothesis $H^i(\underline l'; M/lM)$ is of finite length for all $i < s-1.$ Therefore $H^i(\underline l; M/lM)$ is of finite length for all $i < s-1.$ Whence the above short exact sequence proves the claim. Finally let $\depth_R M = 0.$ Then $N := \cup_{n \geq 1} (0 :_M l^n)$ is an $R$-module of finite length as follows by the definition of $\underline l.$ Then $\depth_R M/N > 0. $ By the first part of the inductive step and $\dim _R M/N = d$ the claim is true for $M/N.$ Note that $\underline l$ forms a generic system of linear forms with respect to $M/N$ as easily seen by a localization argument with respect to non-maximal prime ideals. By (b) the claim is true for $N$ and all $i \geq 0$ since $\dim_R N = 0.$ So the final statement for $M$ follows from the induced long exact Koszul cohomology sequence derived from $0 \to N \to M \to M/N \to 0.$ \end{proof} In the following consider the functor $\Hom_K(\Box, K) = {\Box}^{\vee}$ on the category of graded $R$-modules. By the graded version of the Local Duality Theorem, see \cite[3.6.19]{BH}, it turns out that there is a natural graded isomorphism of degree zero $$ H_{\mathfrak m}^i(M) \simeq (\Ext_R^{r-i}(M, R(-r))^{\vee}, \quad i \in \mathbb Z. $$ Put $K^i_M := \Ext_R^{r-i}(M, R(-r)), \, i \in \mathbb Z.$ Then $K^i_M = 0$ for $i<0$ and $i > \dim_R M := d.$ In particular $K_M := K_M^d$ is called the canonical module of $M.$ These modules are studied in a systematic way in \cite[\S 3]{pS2}. Here we mention only that $$ \dim_R K^i_M \leq i,\, \mbox{ for } 0 \leq i < d,\, \mbox{ and }\, \dim_R K_M = d, $$ see \cite[3.1.1]{pS2} for the details. For the next results we need another definition of genericity. It is related to the modules of `deficiency' $K^i_M.$ \begin{definition} \label{3.3} A generic system of linear elements $\underline l = l_1, \ldots, l_s$ is called a strongly generic linear system of elements with respect to $M$ provided it is a generic linear system of elements for all $K^i_M, \, i = 0, \ldots,d.$ \end{definition} Because $K$ is an infinite field it is clear that strongly generic linear systems of elements with respect to $M$ always exist. Their construction is just an application of prime avoidance arguments. \begin{theorem} \label{3.4} Suppose that $\underline l = l_1, \ldots,l_s$ is a strongly generic linear system of elements with respect to $M.$ Then $H_i(\underline l; H_{\mathfrak m}^j(M))$ is a graded $R$-module of finite length in the following two cases: \begin{itemize} \item[(a)] $i < s,$ and \item[(b)] for all $i \in \mathbb Z$ provided $s \geq j.$ \end{itemize} \end{theorem} \begin{proof} First observe that there are canonical isomorphisms $$ H_i(\underline l; N^{\vee}) \simeq (H^i(\underline l;N))^{\vee}, \quad i \in \mathbb Z. $$ This follows because of the isomorphism of complexes $$ K_{\bullet}(\underline l;R) \otimes_R (N)^{\vee} \simeq (\Hom_R (K_{\bullet}(\underline l;R), N))^{\vee}, $$ which is well known. Here $N$ denotes an arbitrary graded $R$-module. Put $N = K^i_M.$ Then it follows that $$ H_i(\underline l; H_{\mathfrak m}^j(M)) \simeq (H^i(\underline l; K^j_M))^{\vee} \, \mbox{ for all } \, i, j \in \mathbb Z. $$ By \ref{3.2} it is known that $H^i(\underline l; K^j_M)$ is an $R$-module of finite length for $i < s$ resp. for all $i \in \mathbb Z$ provided $s \geq \dim_R K^j_M.$ Because of $\dim_R K^j_M \leq j$ this finishes the proof. \end{proof} In his paper \cite{mG} M.~Green considered the following situation. Let $\mathcal F$ denote a coherent sheaf on $X,$ a compact complex manifold. Then he considered the vector spaces $\mathcal K^i_{p,q}(X, \mathcal F).$ Let $i \geq 1.$ Then it is easy to see, see \cite{mG}, that $$ \mathcal K^i_{p,q}(X, \mathcal F) \simeq H_p(\mathfrak m; H_{\mathfrak m}^{i+1}(M))_{p+q}, $$ where $M$ denotes the associated graded module to $\mathcal F.$ So in an obvious way the Koszul homology modules $ H_p(\mathfrak m; H_{\mathfrak m}^{i+1}(M))$ are graded analogues of the invariants introduced by M.~Green. As an application of \ref{3.4} it turns out that $\mathcal K^i_{p,q}(X, \mathcal F) = 0$ for all $q \ll 0$ resp. for all $q \gg 0$ provided $\underline l$ is strongly generically chosen. For the numerical influence of these finitely many nonvanishing $\mathcal K^i_{p,q}(X, \mathcal F)$ on free resolutions see the results in Section 5. The most important feature of $H_p(\mathfrak m; H_{\mathfrak m}^{i+1}(M))$ is that it is one of the ingredients of a spectral sequence. In the following let $M$ denote a finitely generated graded $R$-module. Choose $\underline l = l_1,\ldots,l_s, \, s \geq \dim_R M,$ a generic linear system of elements with respect to $M.$ Then consider the following complexes $K^{\bullet},$ the \v Cech complex, $K_{\bullet}(\underline l; M),$ the Koszul complex of $M$ with respect to $\underline l,$ and $C^{\bullet} := K^{\bullet} \otimes_R K_{\bullet}(\underline l; M).$ Then there is the following spectral sequence $$ H_{\mathfrak m}^i(H_j(\underline l ; M)) \Rightarrow H_{j-i}(C^{\bullet}). $$ Because of the choice of $\underline l$ it turns out that $H_j(\underline l;M) \simeq H^{s-j}(\underline l;M)(-s)$ are $R$-modules of finite length for all $j \in \mathbb Z,$ see \ref{3.2}. Because of the basic properties of local cohomology it yields that $$ H_{\mathfrak m}^i(H_j(\underline l;M)) = \begin{cases} H_j(\underline l;M) & \mbox{ for } i = 0 \\ 0 & \mbox{ for } i \not= 0. \end{cases} $$ Therefore the spectral sequence degenerates partially to the isomorphisms $H_j(C^{\bullet}) \simeq H_j(\underline l ; M)$ for all $j \in \mathbb Z.$ The second spectral sequence for the corresponding double complex is $H_j(\underline l; H_{\mathfrak m}^i(M)) \Rightarrow H_{j-i}(C^{\bullet}).$ Putting this together it proves the following \begin{lemma} \label{3.5} Let $M$ be a finitely generated graded $R$-module. Let $\underline l = l_1,\ldots,l_s, \, s \geq \dim_R M$ be a generic linear system with respect to $M.$ Then there is a spectral sequence $$ E_2^{-j,i} = H_j(\underline l; H_{\mathfrak m}^i(M)) \Rightarrow E^{-j+i} = H_{j-i}(\underline l; M), $$ where all the derived homomorphisms are homogeneous of degree zero. \end{lemma} In the more special situation of a strongly generic linear system with respect to $M$ not only $H_{j-i}(\underline l;M)$ but also $H_j(\underline l; H_{\mathfrak m}^i(M))$ are modules of finite length for all $i, j \in \mathbb Z,$ see \ref{3.4}. Therefore there is an estimate for the length of $H_{j-i}(\underline l;M).$ \begin{corollary} \label{3.6} Suppose that $\underline l = l_1,\ldots,l_s, \, s \geq \dim_R M,$ denotes a strongly generic linear system of elements with respect to $M.$ Then $$ L_A(H_n(\underline l;M)) \leq \sum_{i=0}^{\min\{s-n,d\}} L_A(H_{n+i}(\underline l;H_{\mathfrak m}^i(M)), $$ for all $n \in \mathbb Z,$ where $d = \dim_RM.$ \end{corollary} \begin{proof} First note that $H_{\mathfrak m}^i(M) = 0$ for all $i > d$ resp. $H_j(\underline l;M) = 0$ for all $j > s.$ Then the estimate follows by the same line of reasoning as in the proof of \ref{2.1}. \end{proof} The spectral sequence in \ref{3.5} has several more applications in Section 4 and Section 5. Here we want to add just two simple consequences. They are helpful also in different situations. \begin{corollary} \label{3.7} Let $M$ be a finitely generated graded $R$-module. Suppose that $\underline l = l_1,\ldots,l_s, \, s \geq \dim_R M$ denotes a generic linear sytem with respect to $M.$ Then \begin{itemize} \item[(a)] $H_{s-t}(\underline l; M) \simeq H_s(\underline l; H_{\mathfrak m}^t(M)),$ \, where $t = \depth_RM,$ and \item[(b)] $H_{i-d}(\underline l;M) \simeq H_i(\underline l; H_{\mathfrak m}^d(M)),$ \, for all $i \in \mathbb Z,$ provided $M$ is a $d$-dimensional Cohen-Macaulay module. \end{itemize} \end{corollary} \begin{proof} In order to prove (a) consider the spectral sequence in \ref{3.5}. Take the terms $E_2^{-j,i}$ with $j-i = s-t.$ Then $$ E_2^{-j,i} = \begin{cases} 0 & \mbox{ for }\, j > s \mbox{ or } i < t \text{ and } \\ H_s(\underline l; H_{\mathfrak m}^t(M)) & \mbox{ for }\, j = s \mbox{ and } i = t. \end{cases} $$ But this means that the spectral sequence degenerates partially to the desired isomorphism. The claim in (b) follows by a similar argument since $H_{\mathfrak m}^i(M) = 0$ for all $i \not= d$ in the case of a Cohen-Macaulay module $M.$ \end{proof} For an extension of the results of this section to the situation of a finitely generated module over a local ring, see \cite{pS3}. \section{Bounds on the number of generators} For homogeneous ideals $I \subset R = K[x_1,\ldots,x_r], r \geq 2,$ such that $I$ is a perfect ideal of codimension two it is known that $$ \mu (I) \leq a(I) + 1, $$ where $a(I) = \min \{ n \in Z \; | \; I_n \not= 0\},$ the initial degree of $I.$ Note that $a(I)$ is equal to the minimal degree of a non-zero form contained in $I.$ This estimate is a generalization of a corresponding bound given by P.~Dubreil in the case of $r = 3,$ see \cite{pD}. For the proof see e.g. \cite{DGM} resp. \cite{pS1}. An approach related to Hilbert functions is developed in \cite{DGM}, while \cite{pS1} contains a proof based on the Hilbert-Burch Theorem. In the following put $\underline x = x_1, x_2,\, \underline y = x_3,\ldots,x_r, \, r \geq 3,$ where $x_1,\ldots,x_r$ denotes a set of generators of $\mathfrak m.$ In a certain sense the following result is a generalization of Dubreil's Theorem. \begin{theorem} \label{4.1} Let $I \subset R$ denote a homogeneous ideal of codimension at least two. Then $$ \mu (I) \leq a(I) + 1 + \mu(H_1(\underline y; R/I)), $$ where $\underline y$ is chosen generically with respect to $R/I.$ \end{theorem} \begin{proof} First put $S = R/\underline y R$ and $J = I S.$ Then we obtain the bound $$ \mu(I) \leq \mu(J) + \dim_K H_0(\underline x; H_1(\underline y; R/I)), $$ as follows by \ref{2.2}. By the generic choice of $\underline y$ it is known that $a(I) = a(J).$ Now $J$ is a perfect ideal of codimension two in $S.$ Therefore $\mu (J) \leq a(J) + 1.$ Finally the dimension of the vector space $H_0(\underline x; H_1(\underline y; R/I))$ coincides with the number of generators of $H_1(\underline y; R/I).$ \end{proof} In fact \ref{4.1} is a generalization of J.~Migliore's result, see \cite[Corollary 3.3]{jM}, in the case of the defining ideal $\mathcal J_C$ of a curve $C \subset \mathbb P^3_K.$ Here we extend his result to an arbitrary projective scheme. \begin{corollary} \label{4.2} Let $I \subset R$ denote a homogeneous ideal with $\codim I \geq 2.$ Put $t = \depth R/I.$ Then $$ \mu(I) \leq a(I) + 1 + \mu(H_{t+1}(\underline y; H_{\mathfrak m}^t(R/I))), $$ where $\underline y$ is chosen strongly generic with respect to $R/I.$ \end{corollary} \begin{proof} By \ref{3.7} there is the following isomorphism $$ H_1(\underline y; R/I) \simeq H_{t+1}(\underline y; H_{\mathfrak m}^t(R/I)). $$ Therefore the claim is a consequence of \ref{4.1}. \end{proof} In the situation of $I$ the saturated defining ideal of curve $C \subset \mathbb P^3_K$ it follows that $t = 1.$ Therefore $H_2(l_1,l_2; H_{\mathfrak m}^1(R/I))$ is just the submodule of $H^1_{\ast}(\mathcal J_C)$ annihilated by $l_1, l_2,$ see \cite[Corollary 3.3]{jM}. Besides of its vanishing it is known that Koszul homology is difficult to handle. So for the rest of this section there are several approaches in order to estimate the term $\mu(H_1(\underline y; R/I))$ in \ref{4.1}. \begin{corollary} \label{4.3} Let $I \subset R$ denote a homogeneous ideal of codimension at least two with $d = \dim R/I.$ Then $$ \mu(I) \leq a(I) + 1 + \sum_{i=0}^d L_R(H_{i+1}(\underline y; H_{\mathfrak m}^i(R/I))). $$ Moreover, suppose that $H_{\mathfrak m}^i(R/I)$ are graded $R$-modules of finite length for $ i = 0,1,\ldots,d-1.$ Then $$ \mu(I) \leq a(I) + 1 + \sum_{i=0}^{d-1} \binom{r-2}{i+1} L_R(H_{\mathfrak m}^i(R/I)) + L_R(H_{d+1}(\underline y; H_{\mathfrak m}^d(R/I))). $$ Here $\underline y $ is chosen strongly generic with respect to $R/I.$ \end{corollary} \begin{proof} Under the additional assumption that $\underline y $ is a strongly generic system of linear forms with respect to $R/I$ it follows that $H_{i+1}(\underline y; H_{\mathfrak m}^i(R/I))$ are graded $R$-modules of finite length, see \ref{3.4}. Then the spectral sequence in \ref{3.5} provides the following estimate $$ L_R(H_1(\underline y; R/I)) \leq \sum_{i=0}^d L_R(H_{i+1}(\underline y; H_{\mathfrak m}^i(R/I))). $$ By virtue of \ref{4.1} this proves the first part of the claim. Under the additional assumption of the finite length of $H_{\mathfrak m}^i(R/I)$ for $ i = 0,1,\ldots,d-1,$ it is easy to see that $$ L_R(H_{i+1}(\underline y; H_{\mathfrak m}^i(R/I))) \leq \binom{r-2}{i+1} L_R(H_{\mathfrak m}^i(R/I)), \quad i = 0,\ldots,d-1. $$ To this end consider the definition of the Koszul homology. Therefore the second bound follows. \end{proof} Note that \ref{4.3} was shown by H.~Martin and J.~Migliore, see \cite[Theorem 2.5]{MM}, under the additional assumption that $\Proj R/I$ is equidimensional and a Cohen-Macaulay scheme. Of particular interest is the case of $\codim I = 2.$ In this situation the term $H_{d+1}(\underline y; H_{\mathfrak m}^d(R/I))$ does not occur since $d + 1 = r - 1 > r -2,$ the number of elements of $\underline y.$ In the following let $\sigma (N)$ denote the socle dimension of $N.$ That means $\sigma (N) = \dim_K \Hom_R(R/\mathfrak m, N)$ for an arbitrary $R$-module $N.$ \begin{corollary} \label{4.4} Let $I \subset R$ denote a perfect homogeneous ideal of codimension at least three. Then $$ \mu(I) \leq a(I) + 1 + \sigma (H^{d+1}(\underline y; K_{R/I})), $$ where $\underline y$ is chosen strongly generic with respect to $R/I.$ Here $K_{R/I}$ denotes the canonical module of $R/I.$ \end{corollary} \begin{proof} Because $R/I$ is a Cohen-Macaulay ring we have to estimate $\mu(H_{d+1}(\underline y; H_{\mathfrak m}^d(R/I), d = \dim R/I,$ see \ref{4.2}. But now $$ H_{d+1}(\underline y; H_{\mathfrak m}^d(R/I)) \simeq (H^{d+1}(\underline y; K_{R/I}))^{\vee}. $$ Therefore the dimension of $R/\mathfrak m \otimes_R H_{d+1}(\underline y; H_{\mathfrak m}^d(R/I))$ is equal to the socle dimension of $H^{d+1}(\underline y; K_{R/I})$ as easily seen. \end{proof} Of a particular interest is the case of a Gorenstein ideal of codimension three. In this situation it follows: \begin{corollary} \label{4.5} Let $I \subset R, \underline y$ be as in \ref{4.4}. Suppose that $R/I$ is a Gorenstein ring and $\codim I = 3.$ Then $\mu(I) \leq 2 a(I) + 1.$ \end{corollary} \begin{proof} Because $R/I$ is a Gorenstein ring it is known that $R/I \simeq K_{R/I}.$ Put $\underline z = x_3,\ldots,x_{r-1}, \, y = x_r.$ Now define $S = R/\underline z R, \, J = IS.$ Then $H^{d+1}(\underline y; K_{R/I}) \simeq S/(J, yS).$ But now the socle dimension of $S/(J, yS)$ is equal to the type of $S/(J, yS),$ or what is the same, to the minimal number of generators of $L$ minus one, $\mu(L) - 1,$ where $T = S/yS$ and $L = J T.$ Recall that $L$ is a perfect ideal of codimenssion two in $T.$ But then $\mu (L) \leq a(L) +1$ by Dubreil's Theorem. Finally $a(I) = a(L)$ since $\underline y$ is chosen generically. Therefore by \ref{4.4} the claim is shown to be true. \end{proof} Note that this result follows also by the Buchsbaum-Eisenbud structure theorem for Gorenstein ideals of codimension three. For the details see \cite{pS1}. A further result including the degree is the following: \begin{corollary} \label{4.6} Let $I \subset R, \, \underline y$ be as in \ref{4.4}. Suppose $\codim I = 3.$ Then $$ \mu(I) \leq a(I) + 1 + e(R/I), $$ where $e(R/I)$ denotes the multiplicity of $R/I.$ \end{corollary} \begin{proof} First note that by 4.4 it is obviously true that $$ \sigma(H^{d+1}(\underline y; K_{R/I})) \leq L_R(H^{d+1}(\underline y; K_{R/I})) \leq L_R(K_{R/I}/\underline z K_{R/I}). $$ Here let $\underline y$ be generated by $y_1,\ldots,y_{d+1}$ and $\underline z = y_1,\ldots,y_d.$ Then $\underline z$ forms a system of parameters for $K_{R/I}$ and $R/I$ as well. Furthermore $L_R(K_{R/I}/\underline y K_{R/I}) \leq L_R(K_{R/I}/\underline z K_{R/I}).$ Because $R/I$ is a Cohen-Macaulay ring, $K_{R/I}$ is a Cohen-Macaulay module and therefore $$ L_R(K_{R/I}/\underline z K_{R/I}) = e(\underline z; K_{R/I}) = e(\underline z; R/I). $$ Because of the generic choice of the linear elements in $\underline z$ this completes the proof. \end{proof} The bound in \ref{4.6} is rather rough. It would be of some interest to find a common generalization of \ref{4.5} and \ref{4.6}. \section{Koszul homology and syzygies} As before let $R = K[x_1,\ldots,x_r]$ denote the polynomial ring in $r$ variables. For a graded $R$-module $M$ define $$ a(M) = \min \{n \in \mathbb Z \; | \; M_n \not= 0 \} \text{ and } e(M) = \max \{n \in \mathbb Z \; | \; M_n \not= 0 \}. $$ It is well known that $e(H_{\mathfrak m}^i(M)) < \infty$ for all $i \in \mathbb Z.$ \begin{definition} \label{5.1} The Castelnuovo-Mumford regularity $\reg M$ of $M$ is defined by $$ \reg M = \max \{ e(H_{\mathfrak m}^i(M)) + i \; | \; i \in \mathbb Z\}. $$ Note that $e(0) = - \infty.$ \end{definition} It is a well known fact that $$ \reg M = \max\{ e(\Tor_i^R(K, M)) -i \; | \; 0 \leq i \leq r\}. $$ So $\reg M$ yields a bound on the maximal degree in a minimal generating set of the syzygy modules of $M.$ It reflects the structure of the minimal free resolution $F_{\bullet}$ of $M$ over $R,$ where $$ F_{\bullet} : 0 \to F_s \to \ldots \to F_i \to \ldots \to F_0 \to M \to 0, $$ with $F_i \simeq \oplus_{j \in \mathbb Z} R^{i_j}(-j)$ and $i_j = \dim_K \Tor_i^R(K, M)_j.$ Suppose that $M$ is a Cohen-Macaulay module. Then $\reg M = e(\Tor_c^R(K, M)) -c,$ where $c = r - \dim_R M$ denotes the codimension of $M.$ This follows easily since $\Hom_R(F_{\bullet}, R(-r))$ gives a minimal free resolution of $K_M = \Ext_R^c(M, R(-r)),$ the canonical module of $M.$ On the other hand it was observed by P.~Rao, see \cite[(2.5)]{pR}, that in the case of $I$ the defining ideal of a curve $C \subset \mathbb P^3_K$ the Hartshorne-Rao module $M(C) \simeq H_{\mathfrak m}^1(R/I)$ gives certain information on the tail of the minimal free resolution of $R/I.$ In the following we shall generalize both of these observations. Firstly we describe $\reg M$ in terms of the $\Tor$s in a certain range. Secondly we shall clarify how the minimal free resolutions of $H_{\mathfrak m}^i(M),$ the `modules of deficiency', determine the minimal free resolution of $M.$ Both considerations turn out by a careful study of the spectral sequence given in \ref{3.5}. \begin{theorem} \label{5.2} Let $M$ denote a finitely generated graded $R$-module. Let $s \in \mathbb N.$ then the following two integers coincide \begin{itemize} \item[(a)] $\max \{e(H_{\mathfrak m}^i(M)) + i \; | \; 0 \leq i \leq s \}$ and \item[(b)] $\max \{e(\Tor_j^R(K, M)) - j \; | \; r - s \leq j \leq r\}. $ \end{itemize} In particular for $s = \dim_R M$ it follows that $$ \reg M = \max \{e(\Tor_j^R(K, M)) - j \; | \; c \leq j \leq r\}, $$ where $c = r - \dim_R M$ denotes the codimension of $M.$ \end{theorem} Before proving 5.2 we separate two partial results as Lemmas. They concern results in this direction which seem to be of some independent interest. \begin{lemma} \label{5.3} Suppose that $H_s(\mathfrak m; M)_{s+t} \not= 0$ for a certain $t \in \mathbb Z$ and $r - i \leq s \leq r.$ Then there exists an $j \in \mathbb Z$ such that $0 \leq j \leq i$ and $H_{\mathfrak m}^j(M)_{t-j} \not= 0.$ \end{lemma} \begin{proof} Assume the contrary, i.e., $H_{\mathfrak m}^j(M)_{t-j} = 0$ for all $0 \leq j \leq i.$ Then consider the spectral sequence $$ [E_2^{-s-j,j}]_{t+s} = H_{s+j}(\mathfrak m; H_{\mathfrak m}^i(M))_{t+s} \Rightarrow [E^{-s}]_{t+s} = H_s(\mathfrak m;M)_{t+s} $$ as defined in \ref{4.5}. Recall that all the homomorphisms are homogeneous of degree zero. Now the corresponding $E_2$-term is a subquotient of $$ [\oplus H_{\mathfrak m}^j(M)^{\binom {r}{s+j}}(-s-j)]_{t+s}. $$ Let $j \leq i.$ Then this vectorspace is zero by the assumption about the local cohomology. Let $j > i.$ Then $s + j > s + i \geq r$ and $\binom{r}{s+j} = 0.$ Therefore the corresponding $E_2$-term $[E_2^{-s-j,j}]_{t+s}$ is zero for all $j \in \mathbb Z.$ But then also all the subsequent stages are zero, i.e., $[E_{\infty}^{-s-j,j}]_{t+s} = 0$ for all $j \in \mathbb Z.$ Therefore $[E^{-s}]_{t+s} = H_s(\mathfrak m;M)_{t+s} = 0,$ contradicting the assumption. \end{proof} The second partial result shows that a certain non-vanishing of $H_{\mathfrak m}^i(M))$ yields the existence of a minimal generator of a higher syzygy module. \begin{lemma} \label{5.4} Suppose that there are integers $s, b$ such that the following conditions are satisfied: \begin{itemize} \item[(a)] $H_{\mathfrak m}^i(M)_{b+1-i} = 0$ for all $i < s$ and \item[(b)] $H_r(\mathfrak m; H_{\mathfrak m}^s(M))_{b+r-s} \not= 0$ \end{itemize} Then it follows that $H_{r-s}(\mathfrak m;M)_{b+r-s} \not= 0.$ \end{lemma} Note that the condition (b) in \ref{5.4} means that $H_{\mathfrak m}^s(M)$ possesses a socle generator in degree $b - s.$ Recall that $r$ denotes the number of generators of $\mathfrak m.$ \begin{proof} As above we consider the spectral sequence $$ E_2^{-r,s} = H_r(\mathfrak m; H_{\mathfrak m}^s(M)) \Rightarrow E^{-r+s} = H_{r-s}(\mathfrak m;M) $$ in degree $b+r-s.$ The subsequent stages of $[E_2^{-r,s}]_{b+r-s}$ are derived by the cohomology of the following sequence $$ [E_n^{-r-n,s+n-1}]_{b+r-s} \to [E_n^{-r,s}]_{b+r-s} \to [E_n^{-r+n,s-n+1}]_{b+r-s} $$ for $n \geq 2.$ But now $[E_n^{-r-n,s+n-1}]_{b+r-s}$ resp. $[E_n^{-r+n,s-n+1}]_{b+r-s}$ are subquotients of $$ H_{r+n}(\mathfrak m; H_{\mathfrak m}^{s+n-1}(M))_{b+r-s} = 0 \text{ resp. } H_{r-n}(\mathfrak m; H_{\mathfrak m}^{s-n+1}(M))_{b+r-s} = 0. $$ For the second module recall that it is a subquotient of $$ [\oplus H_{\mathfrak m}^{s-n+1}(M)^{\binom{r}{r-n}}(-r+n)]_{b+r-s} = 0, \quad n \geq 2. $$ Therefore $[E_2^{-r,s}]_{b+r-s} = [E_{\infty}^{-r,s}]_{b+r-s} \not= 0$ and $$ [E^{-r+s}]_{b+r-s} \simeq H_{r-s}(\mathfrak m; M)_{b+r-s} \not= 0 $$ as follows by the filtration with the corresponding $E_{\infty}$-terms. \end{proof} \begin{proof} {\it (Theorem 5.2).} First of all let us introduce two abbreviations. Put $a := \max \{ e(\Tor_j^R(K, M)) -j \; | \; r-s \leq j \leq r\}.$ Then by \ref{5.3} it follows that $a \leq b,$ where $b:= \max \{ e(H_{\mathfrak m}^i(M)) + i \; | \; 0 \leq i \leq s\}.$ On the other hand choose $j$ an integer $0 \leq j \leq s$ such that $b = e(H_{\mathfrak m}^j(M)) + j.$ Then $H_{\mathfrak m}^j(M)_{b-j} \not= 0,$ $ H_{\mathfrak m}^j(M)_{c-j} = 0$ for all $c > b,$ and $H_{\mathfrak m}^i(M)_{b+1-i} = 0 $ for all $i < j.$ Recall that this means that $H_{\mathfrak m}^j(M)$ has a socle generator in degree $b - s.$ Therefore Lemma \ref{5.4} applies and $\Tor^R_{r-j}(K, M)_{b+r-j} \not= 0.$ In other words, $b \leq a,$ as required. \end{proof} An easy byproduct of our investigations is the above mentioned fact that $$ \reg M = e(\Tor_c^R(K, M)) - c, \, c = r - \dim M, $$ provided $M$ is a Cohen-Macaulay module. \begin{theorem} \label{5.5} Let $M$ be a finitely generated graded $R$-module with $d = \dim _R M.$ Suppose there is an integer $j \in \mathbb Z$ such that for all $q \in \mathbb Z$ either \begin{itemize} \item[(a)] $H_{\mathfrak m}^q(M)_{j-q} = 0$ or \item[(b)] $H_{\mathfrak m}^p(M)_{j+1-q} = 0$ for all $p < q $ and $H_{\mathfrak m}^p(M)_{j-1-q} = 0$ for all $p >q.$ \end{itemize} Then for $s \in \mathbb Z$ it follows that \begin{itemize} \item[(1)] $\Tor_s^R(K, M)_{s+j} \simeq \oplus_{i=0}^{r-s} \Tor_{s+i}^R(K, H_{\mathfrak m}^i(M))_{s+j}$ provided $s > c,$ and \item[(2)] $\Tor_s^R(K, M)_{s+j} \simeq \oplus_{i=0}^{d-1} \Tor_{s+i}^R(K, H_{\mathfrak m}^i(M))_{s+j} \oplus \Tor_{c-s}^R(K, K_M)^{\vee}_{r-s-j},$ provided $s \leq c,$ \end{itemize} where $K_M = \Ext_R^c(M, R(-r)), \, c = \codim M,$ denotes the canonical mo\-dule of $M.$ \end{theorem} \begin{proof} As above consider the spectral sequence $$ E_2^{-s-i,i} = H_{s+i}(\mathfrak m; H_{\mathfrak m}^i(M)) \Rightarrow E^{-s} = H_{s}(\mathfrak m;M) $$ in degree $s+j,$ see \ref{3.5}. Firstly we claim that $[E_2^{-s-i,i}]_{s+j} \simeq [E_{\infty}^{-s-i,i}]_{s+j}$ for all $s \in \mathbb Z.$ Because $[E_2^{-s-i,i}]_{s+j}$ is a subquotient of $$ [\oplus H_{\mathfrak m}^i(M)^{\binom{r}{s+i}}(-s-i)]_{s+j} $$ The claim is true provided $H_{\mathfrak m}^i(M)_{j-i} = 0.$ Suppose that $H_{\mathfrak m}^i(M)_{j-i} \not= 0.$ In order to prove the claim in this case too note that $[E_{n+1}^{-s-i,i}]_{s+j}$ is the cohomology at $$ [E_n^{-s-i-n,i+n-1}]_{s+j} \to [E_n^{-s-i,i}]_{s+j} \to [E_2^{-s-i+n,i-n+1}]_{s+j}. $$ Then the module at the left resp. the right is a subquotient of $$ H_{s+i+n}(\mathfrak m; H_{\mathfrak m}^{i+n-1}(M))_{s+j} \text{ resp. } H_{s+i-n}(\mathfrak m; H_{\mathfrak m}^{i-n+1}(M))_{s+j}. $$ Therefore both of them vanish. But this means that the $E_2$-term coincides with the corresponding $E_{\infty}$-term. So the target of the spectral sequence $H_s(\mathfrak m;M)_{s+j}$ admitts a finite filtration whose quotients are $H_{s+i}(\mathfrak m; H_{\mathfrak m}^i(M))_{s+j}.$ Because all of these modules are finite dimensional vectorspaces it follows that $$ H_s(\mathfrak m;M)_{s+j} \simeq \oplus_{i=0}^{r-s} H_{s+i}(\mathfrak m; H_{\mathfrak m}^i(M))_{s+j} $$ for all $s \in \mathbb Z.$ In the case of $s > c$ it is known that $r - s < d.$ Hence the first part of the claim is shown to be true. In the remaining case $s \leq c$ the summation is taken from $i = 0,\ldots,d.$ Therefore we have to interprete the summand $H_{s+d}(\mathfrak m; H_{\mathfrak m}^d(M))_{s+j}.$ By the Local Duality Theorem $H_{\mathfrak m}^d(M) \simeq (K_M)^{\vee}.$ Therefore there are the following isomorphisms $$ H_{s+d}(\mathfrak m; (K_M)^{\vee})_{s+j} \simeq (H^{s+d}(\mathfrak m; K_M)^{\vee})_{s+j} \simeq H_{r-d-s}(\mathfrak m; K_M)^{\vee}_{r-s-j}, $$ which proves the second part of the claim. \end{proof} As an application of \ref{5.5} we derive M.~Green's duality theorem \cite[Section 2]{mG}, see also \cite[Theorem 1.2]{NP} for a similar approach of the original statement. \begin{corollary} \label{5.6} Suppose there exists an integer $j \in \mathbb Z$ such that $$H_{\mathfrak m}^q(M)_{j-q} = H_{\mathfrak m}^q(M)_{j+1-q} = 0 $$ for all $q < \dim_R M.$ Then $$ \Tor_s^R(K, M)_{s+j} \simeq \Tor_{c-s}^R(K, K_M)^{\vee}_{r-s-j}, $$ for all $s \in \mathbb Z,$ where $c = \codim M.$ \end{corollary} \begin{proof} It follows that the assumptions of Theorem 5.5 are satisfied for $j$ because of $H_{\mathfrak m}^p(M)_{j-1-p} = 0$ for all $p > \dim M.$ Therefore the isomorphism is a consequence of (1) and (2) in 5.5. To this end recall that $$ \Tor^R_{s+i}(K, H_{\mathfrak m}^i(M))_{s+j} \simeq H_{s+i}(\mathfrak m; H_{\mathfrak m}^i(M))_{s+j} = 0, $$ as follows by the vanishing of $H_{\mathfrak m}^i(M)_{s+j}$ for all $j \in \mathbb Z.$ \end{proof} M.~Green's duality theorem in \ref{5.6} relates the Betti numbers of $M$ to those of $K_M.$ Because of the strong vanishing assumptions in \ref{5.6} very often it does not give strong information about Betti numbers. Often it says just the vanishing which follows also by different arguments, e.g., the regularity of $M.$ Theorem \ref{5.5} is more subtle. In a certain sense it is an extension of P.~R.~Rao's argument, see \cite[(2.5)]{pR}. We shall illustrate its usefulness by the following example. \begin{example} \label{5.7} Let $C \subset \mathbb P^n_K$ denote a reduced integral non-degenerate curve over an algebraically closed field $K.$ Suppose that $C$ is non-singular and of genus $g(C) = 0.$ Let $A = R/I$ denote its coordinate ring, i.e., $R = K[x_0,\ldots,x_n]$ and $I$ its homogeneous defining ideal. Then $$ \Tor_s^R(K, R/I)_{s+j} \simeq \Tor^R_{s+1}(K, H_{\mathfrak m}^1(R/I))_{s+j} $$ for all $s \geq 1$ and all $j \geq 3.$ To this end recall that $A$ is a two-dimensional domain. Moreover it is well-known that $H_{\mathfrak m}^q(R/I) = 0$ for all $q \leq 0$ and $q > 2.$ Furthermore it is easy to see that $H_{\mathfrak m}^1(R/I)_{j-1} = 0$ for all $j \leq 1.$ Moreover $H_{\mathfrak m}^2(R/I)_{j-1-2} = 0$ for all $j \geq 3$ as follows because of $g(C) = 0.$ That is for $j \geq 3$ one might apply \ref{5.5}. In order to conclude we have to show that $\Tor^R_{c-s}(K, K_{R/I})_{r-s-j} = 0$ for $j \geq 3.$ To this end note that $$ H_{c-s}(\mathfrak m; K_{R/I})^{\vee}_{r-s-j} \simeq H_{s+2}(\mathfrak m; H_{\mathfrak m}^2(R/I))_{s+j} $$ as is shown in the proof of \ref{5.5}. But this vanishes for $j \geq 2$ as is easily seen. \end{example}

9512

alg-geom/9512011

en

https://arxiv.org/abs/alg-geom/9512011

alg-geom/9512011

Vd

Valeri V.Dolotin

On Invariant Theory

6 pages, LaTeX. Insignificant cosmetic changes

Here we develop a technique of computing the invariants of $n-$ary forms and systems of forms using the discriminants of corresponding multilinear forms built of their partial derivatives, which should be cosidered as analogues of classical discriminants and resultants for binary forms.

alg-geom

\section{Introduction} Let $f(y_0,y_1,...,y_n)=\displaystyle \sum^{}_{\alpha_1\le{...}\le{\alpha_d}}c_{\alpha_1{...}{\alpha_{d}}% }y_{\alpha_{1}} {...}y_{\alpha_{d}}$ be an $n+1-$ary form of degree $d$, which is a polynomial of $y$ of homogeneous degree $d$. \begin{Definition} {\rm The value of coefficients $(c_{\alpha_1...\alpha_d})$ is called {\it discriminantal} if for this value the system of equations \begin{equation}\label{E1.1} \frac{\partial{f}}{\partial{y_{\alpha}}}=0,\quad \alpha=0,...,n \end{equation} has a solution in ${\bf P}_n$, i.e. a nonzero one.} \end{Definition} If the set of all discriminantal values of $c_{\alpha_1...\alpha_d}$ is an algebraic manifold of codimension 1 in the space of coefficients, then it is called {\it the discriminant} of $f$, denoted by $D(f)$. In particular case when $f$ is a $d-$linear form, its coefficients may be viewed as elements of a $d-$dimensional matrix $(a_{i_1...i_d})$. In this paper we show how the invariants of $n-$ary forms can be produced from the discriminants of multilinear forms (determinants of multidimensional matricies), which should be considered as the generalization of the operation of taking classical hessians and resultants. The algorithm of computation of discriminants of multilinear forms is considered in paper [1]. \noindent {\bf Acknoledgment} I want to thank Professor I.M.Gelfand for numerous critical discussions, which, in particular, helped me to turn aside from the "naive" definition of a "skew defferential" of a "skew form" $F=\sum_{i_1<\dots<i_n}f_{i_1\dots i_n}dx_{i_1}\dots dx_{i_n}$ as $$d_{\varepsilon}F=\sum_{i_1<\dots<i_k<\dots< i_{n+1}}\varepsilon^{i_k}\frac{\partial f_{i_1\dots{\widehat i_k}\dots i_{n+1}}}{\partial x_{i_k}}dx_{i_1}\dots dx_{i_{n+1}}$$ \section{Hyperpolarisation and Hyperhessians} \subsection{Basic example. Binary forms} {\bf Example.} Let $f=c_{30}x^3+c_{21}x^2y+c_{12}xy^2+c_{03}y^3$ be a homogeneouse polinomial of 3-rd degree. Take its full polarisation form, which is a 3-linear form $(a_{i_1i_2i_3})$ with coefficients $$ a_{111}=\frac{\partial^3{f}}{\partial{x^3}}% $$ $$ a_{112}=a_{121}=a_{211}=\frac{\partial^3{f}}{\partial{x^2}\partial{y}}% $$ $$ a_{122}=a_{212}=a_{221}=\frac{\partial^3{f}}{\partial{x}\partial^2{y}}% $$ $$ a_{222}=\frac{\partial^3{f}}{\partial{y^3}}% $$ \begin{Proposition} The discriminant of the full polarisation form $(a_{i_1i_2i_3})$ is equal to the discriminant of $f$. \end{Proposition} \noindent Now let $f=c_{k0}x^k+c_{{k-1},1}x^{k-1}y^1+...+c_{0k}y^k$ be a homogeneous polynomial of degree $k$. Let $(a_{i_1...i_k})$ be a $k-$linear form (${% \underbrace{2\times{...}\times{2}}_{\mbox{\scriptsize{k times}}}}$ matrix) with coefficients $$ a_{i_1...i_k}=\frac{\partial^{i+j}f}{\partial{x^i}\partial{y^j}}% $$ where $j$ is the number of $i_k$ which are equal to 1, $j$ is the number of $% i_k$ which are equal to 2. \begin{Proposition} The discriminant of the full polarisation form $(a_{i_1...i_k})$ is a product of $GL_2$ invariants of binary form $f$, and in particular is divisible by the discriminant of $f$. \end{Proposition} \subsection{General Setting} Throughout the paper let $V$ be a linear vector space of dimension $n+1$ and let ${\bf P}_{n}:={\bf P}(V)$ denote its projectivisation. Let $f(x_0,x_1,\dots,x_n)$ be a homogeneous polynomial of degree $k$, or a global section of ${\cal O}_{{\bf P}_{n}}(k)$. For a sequence of positive integers ${K}:=(k_1,...,k_d)$ we have a $d$-linear form with coefficients \begin{equation} \label{E1.2} a_{{I}_1...{I}_d}=\frac{\partial ^{|{I}_1+...+ {I}_d|}f}{\partial {\bf {x}}^{{I}_1+...+{I}_d}} \end{equation} \noindent where ${I}_m\in \{(i_0,...,i_n)|\quad i_0+...+i_n=k_m \}$ are multiidecies. For a multiindex ${I}=(i_0,...,i_n)$ we denote $\partial {X^I}:= \partial {x_0^{i_0}}...\partial {x_n^{i_n}}$. Note that $|{I}_1+...+{I}_d| =k_1+...+k_d$. Each $a_{{I}_1...{I}_d}$ is again a homogeneous polynomial of $(x_0,...,x_n)$ but of degree $k-(k_1+...+k_d)$. \begin{Definition} {\rm We will call the form $(a_{{I}_1...{I}_d})$ defined by (1.2) {\it ${K}$-polarisation form} (or just a {\it hyperpolarisation})and denote it as $P^{(k_1...k_d)}(f)$.} \end{Definition} So ${K}$-polarisation is a map from the space ${\cal O}(k)$ of homogeneous polynomials of degree $k$ to the space of $d-$linear forms with coefficients in ${\cal O}(k-|{K}|)$ which we will denote by ${\cal O}(k-|{K}|)\otimes {S^{K}T^*(V)}$ (where we use the notation $S^{K}U :=S^{k_1}U\otimes\dots\otimes S^{k_d}U$): $${\cal O}(k)\to {\cal O}(k-|{K}|)\otimes {S^{K}T^*(V)}$$ $$f\mapsto P^{(k_1\dots k_d)}(f)$$ \noindent {\bf Example} For $k_1=\dots =k_d=1$ the corresponding hyperpolarisation form is the form of usual $d$-th polarisation of $f$. \begin{Definition} {\rm The discriminant of the form $(a_{{I}_1...{I}_d})=P^{(k_1\dots k_d)}(f)$ is called $(k_1,...,k_d)-${\it hessian} (or just {\it hyperhessian}) of $f$ and is denoted as} $$ {\cal H}^{(k_1...k_d)}(f):=D(a_{{I}_1...{I}_d })= D(P^{(k_1...k_d)}(f))$$ \end{Definition} \noindent {\bf Example} The usual hessian corresponds to the case when $d=2$ and $k_1=k_2=1$. \begin{Proposition} Let $k_1+...+k_d=k$. Then the ${K}$-hessian ${\cal H}^{(k_1...k_d)}(f)=D(a_{{\bf I}_1...{I}_d})$ is a product of $GL_n$ invariants of $n$-ary form $f$. \end{Proposition} So, provided we know how to compute discriminants of $d-$linear forms (see paper [1] for an outline of the algorithm), each partition $k_1,...,k_d$ of an integer number $k=k_1+...+k_d$ gives us a set of invariants of $n-$ary form $f$ of degree $k$. In Section 4 we will give an example of use of hyperhessians. \section{Hyperjacobians and Hyperresultants} \subsection{Basic Example} Let $f_1=c^{\prime}_{20}x^2+c^{\prime}_{11}xy+c^{\prime}_{02}y^2$ and $f_2=c''_{20}x^2+c''_{11}xy+c''_{02}y^2$ be a pair of polynomials. Take a 3-linear form $(a_{i_1i_2j})$ (with $2\times 2\times 2$ matrix of coefficients), such that $$ a_{11j}=\frac{\partial^2{f_j}}{\partial{x^2}} $$ $$ a_{12j}=a_{21j}=\frac{\partial^2{f_j}}{\partial{x}\partial{y}} $$ $$ a_{22j}=\frac{\partial^2{f_j}}{\partial{y^2}} $$ i.e. $a_{i_1i_2j}=(P^{(1,1)}(f_j))_{i_1i_2}$. \begin{Proposition} The discriminant of the form $(a_{i_1i_2j})$ is equal to the resultant of $f_1$ and $f_2$ $$D(a_{i_1i_2j})=Res(f_1,f_2)$$ \end{Proposition} Now let $f_1=c'_{k0}x^k+c'_{{k-1},1}x^{k-1}y^1+...+c'_{0k}y^k$ and $f_2=c''_{k0}x^k+c''_{{k-1},1}x^{k-1}y^1+...+c''_{0k}y^k$ be a pair of homogeneous polynomials of degree $k$. Let $(a_{i_1...i_kj})$ be a $k+1$-linear form with ${\underbrace{2\times{...}\times{2}}_{\mbox{\scriptsize{k+1 times}}}}$ matrix of coefficients $$ a_{i_1...i_kj}= \frac{\partial^{j_1+j_2}f_j}{\partial{x^{j_1}}\partial{y^{j_2}}} $$ where $j_1$ is the number of $i_k$ which are equal to 1, $j_2$ is the number of $i_k$ which are equal to 2. \begin{Proposition} The discriminant of the form $(a_{i_1...i_kj})$ is a product of $GL_2$ invariants of a pair of binary forms $f_1,f_2$, and in particular is divisible by the resultant $Res(f_1,f_2)$. \end{Proposition} \subsection{General Setting} Let ${M}:=(m_1,...,m_{d_1})$ be a sequence of positive integers and let $F^{M}(k):=\{f_{I}\}_{{I}\le{M}}$ be a system of homogeneous polynomials of degree $k$ ennumerated by the indecies ${I}=(i_1,\dots,i_{d_1})$, or a global section of ${\cal O}(k)^{M}:={\cal O}(k)^{\oplus m_1}\otimes\dots\otimes {\cal O}(k)^{\oplus m_{d_1}}$. Let ${K}:=(k_1,...,k_{d_2})$. For each $f_{I}$ take its $K$-polarisation form $P^{(k_1...k_d)}(f_{I})$ as described in Section 1.2. Then we get a $(d_1+d_2)$-linear form $(a_{i_1\dots i_{d_1}{J}_1...{J}_{d_2}})$ such that \begin{equation} \label{E2.3} a_{i_1\dots i_{d_1}{J}_1...{J}_{d_2}}= (P^{(k_1...k_d)}(f_{i_1\dots i_{d_1}}))_{{J}_1\dots{J}_{d_2}} \end{equation} where as in Section 1.2 ${J}_i\in \{(j_0,...,j_n)|\quad j_0+...+j_n=k_i\}$ are multiidecies. Each $a_{i_1\dots i_{d_1}{J}_1...{J}_{d_2}}$ is a homogeneous polynomial of $(x_0,\dots,x_n)$ of degree $k-|K|$. \begin{Definition} {\rm The form defined by (2.3) is called $K$-{\it Jacobi form} of the section $(f_{i_1\dots i_{d_1}})_{I\le M}$, considered as a map of ${\bf P}_n$ to the space of $n_1\times\dots\times n_{d_1}$ tensors (or a tensor field on ${\bf P}_n$) and is denoted by} $$J^K(F^M)=\frac{{\rm D}^K((f_I)_{I\le M})}{{\rm D}(x_1,\dots,x_n)^{|K|}}$$ {\rm The discriminant of $K$-Jacobi form is called $K$-{\it Jacobian} of $(f_{i_1\dots i_{d_1}})_{I\le M}$ (or just a {\it hyperjacobian}).} \end{Definition} So taking the $K$-Jacobi form of tensor fields $F^M\in{\cal O}(k)^{M}$ gives us a map $$J^K:{\cal O}(k)^{M}\to{\cal O}(k-|K|)^{M}\otimes S^{K}T^*(V)$$ \noindent {\bf Example} For $F^M=(f_1,\dots,f_m)$ and $K=(1)$ the hyperpolarisation form $P^{(1)}(f_i)$ is the differential $df_i$ and the corresponding $K$-Jacobi form is the usual Jacobi matrix of the map ${\bf P}_n\to{\bf P}_m$. \subsection{Hyperresultant} Let $F^M$ be a section of ${\cal O}(k)^{\oplus m}$, i.e. is a map ${\bf P}_n\to {\bf P}_m$ of degree $k$, or a system $(f_1,\dots,f_m)$ of $m$ homogeneous polynomials of degree $k$. For $K=\underbrace{(1,\dots,1)}_{k \ {\rm times}}$ the hyperpolarisation form $P^K(f_i)$ is the form with constant coefficients. For the corresponding $K$-Jacobian we will reserve a special name. \begin{Definition} {\rm The $\underbrace{(1,\dots,1)}_{k\ {\rm times}}$-Jacobian of a system $(f_1,\dots,f_m)$ of polynomials of degree $k$ is called {\it hyperresultant} of this system and is denoted by $$R_m(f_1,...,f_m):=D((a_{i_1\dots i_kj})_ { i_1,\dots,i_k\le n \ j\le m })=D(P^{(1,\dots,1)}(f_j)_{j\le m})$$}. \end{Definition} \begin{Proposition} For $m=2$ the hyperresultant $R_2(f_1,f_2)$ is divisible by the usual resultant $R(f_1,f_2)$. \end{Proposition} \noindent {\bf Example} Let $f_1=a_{20}x^2+a_{11}xy+a_{02}y^2, \ f_2= b_{20}x^2+b_{11}xy+b_{02}y^2$ and $f_3= c_{20}x^2+c_{11}xy+c_{02}y^2$. If we write $f_i(x,y)$ as functions of a nonhomogeneous variable $x^{\prime}:= \frac{x}{y}$ then the Wronskian $$ W(f_1,f_2,f_3)=\left|{ \matrix{ \frac{\partial^2{f_1}}{\partial{x'^2}} & \frac{\partial{f_1}}{\partial{x'}} & f_1 \cr \frac{\partial^2{f_2}}{\partial{x'^2}} & \frac{\partial{f_2}}{\partial{x'}} & f_2 \cr \frac{\partial^2{f_3}}{\partial{x'^2}} & \frac{\partial{f_3}}{\partial{x'}} & f_3 \cr }}\right| $$ is equal to 0 iff $f_1,f_2$ and $f_3$ are linearly dependent. On the other hand for each $f_i$ we take the $2\times 2$ matrix $P^{(1,1)}(f_i)$ and make from these matricies a $2\times 2\times 3$ form $R_3(f_1,f_2,f_3)$ according to (2.3). Then for 3-resultant of $f_1, f_2$ and $f_3$ (the discriminant of $J^{(1,1)}(f_1,f_2,f_3)$) there is an equality $$R_3(f_1,f_2,f_3)=W(f_1,f_2,f_3)^2$$ \subsection{Jacobi Sequence} Let $K=(1)$. Then the corresponding 1-Jacobi map defines a sequence $${\cal O}(k)\textmap{J^{(1)}}{\cal O}(k-1)\otimes T^*(V)\textmap{J^{(1)}}\dots \textmap{J^{(1)}}{\cal O}(0)\otimes T^*(V)^{\otimes k}$$ Now let us extend the Jacobi map from homogeneous polynomials to the space ${\cal O}$ of analytic (or for some purposes just differentiable) functions on an $n$-dimensional local chart $X$ and denote $T^*(X)\cong{\cal O}\otimes T^*_x(X)$ (for any given $x\in X$) the space of analytic sections of cotangent bundle of $X$. This gives us an infinite sequence: $${\cal O}\textmap{J^{(1)}}T^*(X)\textmap{J^{(1)}}\dots \textmap{J^{(1)}}T^*(X)^{\otimes k}\textmap{J^{(1)}}\dots$$ which we will call a {\it Jacobi sequence}. \section{Gramm Complexes} \subsection{Combinatorial Remark} Let $d>0$ be an integer and $k_1\ge\dots\ge k_p$ be its partition $k_1+\dots+k_p=d$. The group $S_d/S_{k1}\times\dots\times S_{k_p}$ has a free representation on the set of words of length $d$ of $p$ letters. There is a noncanonical but, nevertheless, natural lexicographic ordering on this representation space, which induces an oredering on the group. For $\sigma\in S_d/S_{k1}\times\dots\times S_{k_p}$ let ${\rm ord}(\sigma)$ denote the ordinal number of the group element $\sigma$ with respect to this ordering. For an integer $d$ let $\varepsilon$ be the primitive $d!$-th root of unity $\varepsilon^{d!}=1$. \noindent {\bf Notation } For an $n$-dimensional vector space $V$ let $V^{\otimes d}_{k}\subset V^{\otimes d}$ denote the subspace of $d$-vectors with the condition of $\varepsilon^k$-{\it skew symmetry} of their coordinates: $$a_{\sigma(i_1\dots i_d)}=\varepsilon^{k\ \scriptsize{ord}(\sigma)}a_{i_1\dots i_d}$$ Here each set of indices $i_1,\dots,i_d$ is arranged as: $$i_1=\dots=i_{k_1}$$ $$i_{k_1+1}=\dots=i_{k_1+k_2}$$ $$\dots$$ $$i_{k_1+\dots+k_{d-1}+1}=\dots=i_{d}$$ with $k_1\ge\dots\ge k_d$, and $\varepsilon^{k\frac{d!}{k_1!\dots k_d!}}=1$. Then such a set may be considered as the initial (in lexicographic ordering) element of representation space of the group $S_d/S_{k1}\times\dots\times S_{k_p}\ni\sigma$ with the corresponding well defined ordinal function ${\rm ord}(\sigma)$ on it. \begin{Proposition} $$V^{\otimes d}=\bigoplus_{k=1}^{k=d!}V^{\otimes d}_{k}$$ \end{Proposition} \noindent {\bf Example} Let $\dim V=3$, $d=3$. Then $$V^{\otimes 3}=V^{\otimes 3}_0\oplus V^{\otimes 3}_1\oplus V^{\otimes 3}_5\oplus V^{\otimes 3}_2\oplus V^{\otimes 3}_4\oplus V^{\otimes 3}_3.$$ Here $V^{\otimes 3}_0$ is symmetric part of dimension 10 with nonzero components with idices sets $(111),(222),(333),$ $(112),(113),(223),(221),(331),(332),(123)$, $V^{\otimes 3}_1$ and $V^{\otimes 3}_5$ are parts of dimension 1 with index set $(123)$, $V^{\otimes 3}_2$ and $V^{\otimes 3}_4$ are parts of dimension $7$ with index set $(112),(113),(223),(221),(331),(332),(123)$, $V^{\otimes 3}_3$ is antisymmetric part of dimension $1$ with index set $(123)$. An example of the generic element of $V^{\otimes 3}_2$ is $$ a_{112}e_1\otimes e_1\otimes e_2+ \varepsilon^2a_{112}e_1\otimes e_2\otimes e_1+ \varepsilon^4a_{112}e_2\otimes e_1\otimes e_1+$$ $$ a_{113}e_1\otimes e_1\otimes e_3+ \varepsilon^2a_{113}e_1\otimes e_3\otimes e_1+ \varepsilon^4a_{113}e_3\otimes e_1\otimes e_1+$$ $$ a_{223}e_2\otimes e_2\otimes e_3+ \varepsilon^2a_{223}e_2\otimes e_3\otimes e_2+ \varepsilon^4a_{223}e_3\otimes e_2\otimes e_2+$$ $$ a_{221}e_2\otimes e_2\otimes e_1+ \varepsilon^2a_{221}e_2\otimes e_1\otimes e_2+ \varepsilon^4a_{221}e_1\otimes e_2\otimes e_2+$$ $$ a_{331}e_3\otimes e_3\otimes e_1+ \varepsilon^2a_{331}e_3\otimes e_1\otimes e_3+ \varepsilon^4a_{331}e_1\otimes e_3\otimes e_3+$$ $$ a_{332}e_3\otimes e_3\otimes e_2+ \varepsilon^2a_{332}e_3\otimes e_2\otimes e_3+ \varepsilon^4a_{332}e_2\otimes e_3\otimes e_3+$$ $$ a_{123}e_1\otimes e_2\otimes e_3+ \varepsilon^2a_{123}e_1\otimes e_3\otimes e_2+ \varepsilon^4a_{123}e_2\otimes e_1\otimes e_3+ \varepsilon^0a_{123}e_2\otimes e_3\otimes e_1+ \varepsilon^2a_{123}e_3\otimes e_1\otimes e_2+ \varepsilon^4a_{123}e_3\otimes e_2\otimes e_1 $$ where $\varepsilon$ is the primitive $3!$-th root of unity. {\bf Notation} For a given basis in $V$ denote by $p_k$ the projection map of $V^{\otimes d}$ onto $\varepsilon^k$-skew symmetric component $V^{\otimes d}_k$ for $k\le d!$: $$p_k:V^{\otimes d}\to V^{\otimes d}_k.$$ \subsection{Gramm Forms} Let $F\in U^{*\otimes d}$ be a $d$-linear form. Then for each $\{u_1,\dots,u_m\}\in\underbrace{U\times\dots\times U}_{d\ {\rm times}}$ we have a set of expressions $$<u_{i_1},\dots,u_{i_d}>:=F(u_{i_1},\dots,u_{i_d}),\quad i_1,\dots,i_d=1,...,m.$$ Let $D(<u>)$ be the discriminant of the $\underbrace{m\times\dots\times m}_{d\ {\rm times}}$ form with coefficients $<u_{i_1},\dots,u_{i_d}>$. Then $D(<u>)^{m/d{\rm deg}D(<u>)}$ is an expression of $u_1,...,u_m$ which has homogeneous degree $1$ with respect to each of $u_i$. \begin{Definition} {\rm The function $$G^d:\{u_1,\dots,u_m\}\mapsto D(<u>)^{\frac{m}{d{\rm deg}D(<u>)}}$$ is called the $d$-{\it Gramm form}.} \end{Definition} \noindent {\bf Example} Let $d=2$. Then $(<u_{i_1},u_{i_2}>)_{1\le i_1,i_2\le m}$ is the usual Gramm matrix of the $m$-tuple of vectors $u_1,\dots,u_m$. \begin{Proposition} Let $g\in GL(m)$. Then $G^d(<g(u)>)=\mid g\mid G^d(<u>)$. \end{Proposition} \noindent {\bf Proof} This is the consequence of the fact that the discriminant of $\underbrace{m\times\dots\times m}_{d\ {\rm times}}$ form is a $GL(m)$ invariant. \begin{Proposition} If $u_1,\dots,u_m$ are linearly dependent then $G^d(<u>)\equiv 0$. \end{Proposition} In particular, if the number of vectors $u_i$ is less then the "dimensionality" of the form $m<d$ then $G^d(u_1,\dots,u_m)\equiv 0$. Let $X$ be a manifold, and let $F_x$ be a section of $T^*(X)^{\otimes d}$, i.e. a field of $d-$linear forms on $T(X)^{\times m}$. Then the map $G^d(F_x):u_1,\dots,u_m\mapsto G^d(u_1,\dots,u_m;F_x)$ gives us a measure of integration on any $d$-submanifold of $X$. Let $\mu(X)$ denote the space of analytic measures on $X$. Thus taking a Gramm form defines a map: $$T^*(X)^{\otimes d}\to \mu(X)$$ $$F(\bullet)\mapsto G^d(\bullet;F)$$ \subsection{Gramm Complex} According to Section 3.1 the $\underbrace{m\times\dots\times m}_{d\ {\rm times}}$ form $(<u_{i_1},\dots,u_{i_d}>)$ may be decomposed into $\varepsilon^k$-skew symmetric parts $(<u_{i_1},\dots,u_{i_d}>)_k$: $$\begin{array}{ccc} u_1\times\dots\times u_m&\textmap{}&(<u_{i_1,\dots,u_{i_d}}>)\\ &&\downarrow^{p_k}\\ &&(<u_{i_1,\dots,u_{i_d}}>)_k \end{array}\quad {\rm for}\ k=1,...,d!$$ \begin{Definition} Functions $G^d_{k}: u_1,\dots,u_m\mapsto D((F(u_{i_1},\dots,u_{i_d}))_k)$ are called $k$-{\it skew Gramm forms}. \end{Definition} If $X$ is a manifold and $F$ is a field of $d-$linear forms, i.e. a section of $T^*(X)^{\otimes d}$, then each $G^d_k,\quad k=1,\dots,d!$ gives us a measure on $X$. Thus taking a $k$-skew Gramm forms defines a set of maps: $$\begin{array}{ccc} T^*(X)^{\otimes d}&\textmap{G^d_1}&\mu(X)\\ &\textmap{G^d_2}&\mu(X)\\ &\dots&\\ &\textmap{G^d_{d!}}&\mu(X)\\ \end{array}$$ Each image $G^d_k(T^*(X)^{\otimes d})$ is a linear subspace $\mu^d_k(X)$ in the space $\mu(X)$. For a fixed $k$ the map $G^{\bullet}_k$ applied to the Jacobi sequence: $$\begin{array}{ccccccccccc} {\cal O}&\textmap{J^{(1)}}&T^*(X)&\textmap{J^{(1)}}&\dots& \textmap{J^{(1)}}&T^*(X)^{\otimes k}&\textmap{J^{(1)}}& T^*(X)^{\otimes k+1}&\textmap{J^{(1)}}&\dots\\ \downarrow^{G^0_k}&&\downarrow^{G^1_k}&&\dots&&\downarrow^{G^k_k}&& \downarrow^{G^{k+1}_k}&&\dots\\ 0&&\dots&&0&&\mu^k_k(X)&&\mu^{k+1}_k(X)&\dots\\ \end{array}$$ gives us an infinite sequence $$\begin{array}{ccccccccccccccc} 0&\textmap{\delta_k}&\dots&\textmap{\delta_k}&0& \textmap{\delta_k}&\mu^k_k(X)&\textmap{\delta_k}&\mu^{k+1}_k(X) &\textmap{\delta_k}&\dots&\textmap{\delta_k} &\mu^n_k(X)&\textmap{\delta_k}\dots \end{array}$$ \begin{Proposition} For integer $1\le k< d!$ the operator $\delta^k$ has a final order: $$\delta^d\equiv 0.$$ \end{Proposition} \begin{Definition} We call the operator $\delta_k$ $k-${\it differential} and the sequence () $k-${\it Gramm complex}. \end{Definition} \noindent {\bf Examples} \noindent 1. For $k=d!$ (or, equivalently, $k=0$) the 0-defferential corresponds to symmetric part of Gramm form and $\delta_k$ has infinite order. \noindent 2. For $k=d!/2$ the $k$-Gramm complex is just the de Rham complex of our manifold $X$. So the set of $k$-Gramm complexes may be considered as a deformation connecting de Rham complex with differential $\delta^2\equiv 0$ with the complex (Example 1.) with differential of infinite order. \section{Applications of Hyperjacobians} Let $f=c_{40}x^4+a_{31}x^3y+c_{22}x^2y^2+c_{13}x^1y^3+c_{04}y^4$ be a binary form of degree 4. It is known (see [2]) that the ring of $GL_2$ invariants of $f$ is generated by the polynomials: 1) Hankel determinant $$ {\cal {H}}:={\frac{1}{8}}\left|{ \matrix{ 24c_{40} & 6c_{31} & 4c_{22} \cr 6c_{31} & 4c_{22} & 6c_{13} \cr 4c_{22} & 6c_{13} & 24c_{04} }}\right| $$ 2) apolara $$ {\cal {A}}:=c_{22}^2+3c_{31}c_{13}+12c_{40}c_{04} $$ Denote also by ${\cal {D}}$ another invariant from this ring - the discriminant of $f$.\\ On the other hand we can take a (1,1,1)-hessian of $f$, denoted by $f^{111}:=H_{111}(f)$. $f^{111}$ is again a polynomial in $(x,y)$ of degree 4. \begin{Proposition} $$D(f^{111})=2^{36}3^6{\cal{D}}{\cal{H}}^{6}$$ $$R(f,f^{111})=2^{24}3^{12}{\cal{D}}^2{\cal{A}}^4$$ \end{Proposition} So as soon as we know how to compute the hyperhessian $H_{111}(f)$ (i.e. the 3-dimensional determinants, see [1] for the outline of an algorithm) we can get the generators of the ring of invariants by taking the classical discriminants and resultant of $f$ and $f^{111}$. The experiments with polynomials and their systems of higher degree and of more variables are left to the reader.

9512

alg-geom/9512015

en

https://arxiv.org/abs/alg-geom/9512015

alg-geom/9512015

Dan Abrmovich

Dan Abramovich

Lang maps and Harris's conjecture - a note in search for content

LaTeX2e in LaTeX 2.09 compatibility mode. 5 pages in \large font, with no figures, author-supplied HTML file available at http://math.bu.edu/INDIVIDUAL/abrmovic/langmap/langmap.html

The Lang map, namely the universal dominant rational map to a variety of general type, is constructed and briefly discussed in relation with arithmetic conjectures of Harris, Lang and Manin. Existence of the Lang map follows from the additivity of Kodaira dimension, but the fine structure depends on conjectures on birational classification of algebraic varieties. Serious applications of the Lang map are still being searched.

alg-geom

\section{Introduction} We work over fields of characteristic 0. Let $X$ be a variety of general type defined over a number field $K$. A well known conjecture of S. Lang \cite{langbul} states that the set of rational points $X(K)$ is not Zariski - dense in $X$. As noted in \cite{a}, this implies that if $X$ is a variety which only {\em dominates} a variety of general type then $X(K)$ is still not dense in $X$. J. Harris proposed a way to quantify this situation \cite{harris}: define the {\em Lang dimension} of a variety to be the maximal dimension of a variety of general type which it dominates. Harris conjectured in particular that if the Lang dimension is 0 then for {\em some} number field $L\supset K$ we have that the set of $L$ rational points $X(L)$ is dense in $X$. The full statement of Harris's conjecture will be given below (Conjecture \ref{harcon}). The purpose of this note is to provide a geometric context for Harris's conjecture, by showing the existence of a universal dominant map to a variety of general type, which we call {\em the Lang map.} \section{The Lang map} \begin{th} Let $X$ be an irreducible variety ofver a field $k$, $char(k)=0$. There exists a variety of general type $L(X)$ and a dominant rational map $L:X \das L(X)$, defined over $k$, satisfying the following universal property: Given a field $K \supset k$ and a dominant rational map $f:X_K\das Z$ defined over $K$, where $Z$ is of general type, then there exists a unique dominant rational map $L(f): L(X)\das Z$ such that $L(f)\circ L = f$. \end{th} \begin{dfn} The universal dominant rational map $L:X\das L(X) $ is called {\em the Lang map}\footnote{I believe this name is appropriate since (1) $L$ is closely related to Lang's conjecture, and (2) the construction resembles in many ways the constructions of Albanese, trace etc. in Lang's book \cite{langAV}. I'm also surprised that such a definition has not been made before. (Is that true?)}. The dimension $\dim L(X)$ is called {\em the Lang dimension} of $X$. \end{dfn} \begin{lem}\label{two-maps} Assume $f_i:X_K\das Z_i$ are dominant rational maps with irreducible general fiber, where $Z_i$ are varieties of general type, $i=1,2$. Then there exists a variety of general type $Z$ over $K$ and dominant rational maps $f:X\das Z$ and $g_i: Z \das Z_i$ such that $g_i f = f_i$. \end{lem} {\bf Proof}. Let $Z = Im(f_1 \times f_2:X \das Z_1\times Z_2),$ and let $f:X\das Z$ be the induced map. The map $g_i:Z\das Z_i$ is dominant and has irreducible general fiber. We claim that $Z$ is a variety of general type. By Viehweg's additivity theorem (\cite{viehweg1}, Satz III), it suffices to show that the generic fiber of $g_1$ is of general type. This follows since the fibers of $g_1$ sweep $Z_{2}$. (Specifically, let $d$ be the dimension of the generic fiber of $g_2$. Choose a general codimension-$d$ plane section $H\subset Z_1$, then $g_1^{-1}H\rar Z_2$ is generically finite and dominant, therefore $g_1^{-1}H$ is of general type, therefore the generic fiber of $g_1^{-1}H\rar Z_1$ is of general type.)\qed \begin{lem} Given a field extension $K\supset k$, let $l_K$ be the maximal dimension of a variety of general type $Y/K$ such that there exists a dominant rational map $L_K: X_K\das Y$ with irreducible general fiber. Let $l=\max_{K\supset k}l_K$, and let $K\supset k$ be an extension such that $l_K=l$. Then any such map $L_K$ is the Lang map of $X_K$. \end{lem} {\bf Proof.} Given an extension $E\supset K$ let $f_2:X_{E} \das Z_2$ be a dominant rational map. By the lemma above with $L_K=f_1$ there exists a variety $Z$ and a dominant rational map with irreducible general fiber $f:X_E\das Z$ dominating both $Y_E$ and $Z_2$. By maximality $\dim Y= \dim Z_2$ and since the general fibers of $Z\das Y_E$ are irreducible, $Z\das Y_E$ is birational. The map $g_2 \circ g_1^{-1}:Y_E\das Z_2$ gives the required dominant rational map. \qed {\bf Proof of the theorem.} Using Stein factorization we may restrict attention to maps with irreducible general fibers. As above, let $l=\max_{K\supset k}l_K$, and let $K\supset k$ be an extension such that $l_K=l$. We need to show that $L_K$ can be descended to $k$. First, we may assume that $K$ is finitely generated over $k$, since both $Y$ and $L_K$ require only finitely many coefficients in their defining equations. Next, we descend $L_K$ to an algebraic extension of $k$. Choose a model $B$ for $K$, and a model $\Y\rar B$ for $Y$. We have a dominant rational map $X_B\das \Y$ over $B$. There exists a point $p\in B$ with $[k(p):k]$ finite, such that $\Y_p$ is a variety of general type of dimension $l$ and such that the rational map $X_p\das \Y_p$ exists. The lemma above shows that $(\Y_p)_K$ is birational to $Y$. Alternatively, this step follows since by theorems of Maehara (see \cite{moriwaki}) and Kobayashi - Ochiai (see \cite{d-m}) the set of rational maps to varieties of general type $X\das Z$ is discrete, therefore each $f:X_K \das Z$ is birationally equivalent to a map defined over a finite extension of $k$. We may therefore replace $K$ by an algebraic Galois extension of $k$, which we still call $K$. Let $Gal(K/k)=\{\sigma_1,\ldots,\sigma_m\}$. For any $1\leq i\leq m$ we have a rational map $(f_1\times\sigma_i\circ f_1): X\das Y\times Y^{\sigma_i}$. Applying lemma \ref{two-maps} we obtain a birational map $Y\das Y^{\sigma}.$ There are open sets $U_i\subset Y^{\sigma_i}$ over which these maps are regular isomorphisms, giving rise to descent data for $U_1$ to $k$. \qed Is there a way to describe the fibers of the Lang map $X\das L(X)$? A first approximation is provided by the following: \begin{prp} The generic fiber of the Lang map has Lang dimension 0. \end{prp} {\bf Proof. } Let $\eta\in L(X)$ be the generic point and let $X_\eta \das L(X_\eta)$ be the lang map of the generic fiber. Let $M\rar L(X)$ be a model of $L(X_\eta)$. By definition, the generic fiber $M_\eta=L(X_\eta)$ of $M$ is of general type, therefore by Viehweg's additivity theorem $M$ is of general type, and by definition $M$ is birational to $L(X)$.\qed \begin{ques}\label{open} Is there an open set in $X$ where the Lang map is defined and the fibers have Lang dimension 0? \end{ques} We will see that the answer is yes, if one assumes the following inspiring conjecture of higher dimensional classification theory: \begin{conj}[see Conjecture 1.24 of \cite{flab}]\label{uni/kod} \begin{enumerate} \item Let $X$ be a variety in characteristic 0. Then either $X$ is uniruled, or ${\operatorname{Kod}}(X) \geq 0$. \item If ${\operatorname{Kod}}(X) \geq 0$ then there is an open set in $X$ where the fibers of the Iitaka fibration have Kodaira dimension 0. \end{enumerate} \end{conj} This conjecture allows us to ``construct'' the Lang map ``from above'': \begin{prp}\label{constr} Assume that conjecture \ref{uni/kod} holds true. Then there is a finite sequence of dominant rational maps $$X\das X_1\das\cdots \das X_n = L(X)$$ where each map $X_i\das X_{i+1}$ is either an MRC fibration (see \cite{kmm}, 2.7) or an Iitaka fibration. In particular, the answer to question \ref{open} is ``yes''. \end{prp} {\bf The proof} is obvious. We remark that since \ref{uni/kod} is known when the fibers have dimension $\leq 2$. In particular, \ref{constr} is known unconditionally when $\dim X\leq 3$. \section{Harris's conjecture} As mentioned above, we define {\em the Lang dimension} of a variety $X$ to be $\dim L(X)$, and Lang's conjecture implies that if $K$ is a number field, and if $X/K$ has positive Lang dimension, then $X(K)$ is not Zariski - dense in $X$. In \cite{harris}, J. Harris proposed a complementary statement: \begin{conj}[Harris's conjecture, weak form]\label{harwk} Let $X$ be a variety of Lang dimension 0 defined over a number field $K$. Then for some finite extension $E\supset K$ the set of $E$-rational points $X(E)$ is Zariski dense in $X$. \end{conj} It is illuminating to consider the motivating case of an elliptic surface of positive rank. Let $\pi_0:X_0\rar \bfp^1$ be a pencil of cubics through 9 rational points in $\bfp^2$. By choosing the base points in general position we can guarantee that the pencil has 12 irreducible singular fibers which are nodal rational curves. The Mordell - Weil group of $\pi_0$ has rank 8. The relative dualizing sheaf $\omega_{\pi_0}= \co_{X_0}(F_0)$ where $F_0$ is a fiber. Let $f:\bfp^1\rar\bfp^1$ be a map of degree at least 3. Let $\pi:X\rar\bfp^1$ be the pull-back of $X_0$ along $f$. Then $\omega_\pi = \co_X(3F)$, therefore $\omega_X = \co_X(F)$ and $X$ has Kodaira dimension 1. The Iitaka fibration is simply $\pi$. The elliptic surface $X$ still has a Mordell - Weil group of rank 8 of sections. By applying these sections to rational points on $\bfp^1$ we see that the set of rational points $X(\bfq)$ is dense in $X$. It is not hard to modify this example to obtain a varying family of elliptic surfaces which has a dense collection of sections. Let $B$ be a curve and let $g: B\times \bfp^1 \rar \bfp^1$ be a family of rational functions on $\bfp^1$ which varies in moduli (such families exists as soon as the degree is at least 3). Let $Y$ be the pullback of $X$ to $B\times \bfp^1$. Then $p: Y\rar B$ is a family of elliptic surfaces, of variation $Var(p)=1$, and relative kodaira dimension 1. By composing sections of $E$ with $g$ and arbitrary rational maps $B\rar \bfp^1$, we see that $p$ has a dense collection of sections. Harris's weak conjecture for elliptic surfaces is attributed to Manin. In case of surfaces over $\bfp^1_\bfq$, it has been related to the conjecture of Birch and Swinnerton-Dyer: let $\pi:X\rar \bfp^1$ be an elliptic surface defined over $\bfq$. In \cite{manduchi}, E. Manduchi shows that under certain assumptions on the behavior of the $j$ function, the set of points in $\bfp^1(\bfq)$ where the fiber has root number $-1$ is dense (in the classical topology). According to the conjecture of Birch and Swinnerton-Dyer, the root number gives the parity of the Mordell-Weil rank. It is likely that some of Manduchi's conditions (at least condition (1) in Theorem 1 of \cite{manduchi}) can be relaxed once one passes to a number field. What can be said in case $0<\dim L(X) <\dim X$? In \cite{harris2}, Harris proposed the following definition: \begin{dfn} The {\em diophantine dimension,} $\operatorname{Ddim}(X)$ is defined as follows: $$\begin{array}{cccc} \operatorname{Ddim}(X) :=& \min & \max & \dim(\overline{U(E)})\\[-2mm] & _{\emptyset \neq U\subset X\mbox{ open }} & _{[E:K]<\infty} & \end{array}$$ \end{dfn} Harris proceeded to propose the following: \begin{conj}[Harris's conjecture]\label{harcon} For any variety $X$ over a number field, $$ \operatorname{Ddim}(X) + \dim L(X) = \dim X.$$ \end{conj} I do not know whether or not Harris himself believes this conjecture. This does not really matter. What is appealing in this conjecture, apart from it's ``tightness'', is that any evidence, either for or against it, is likely to be of much interest. For lack of any better results, we just note that proposition \ref{constr} directly implies the following: \begin{prp} Assuming \ref{uni/kod}, Lang's conjecture together with the weak form of Harris's conjecture \ref{harwk} implies Harris's conjecture \ref{harcon}. \end{prp} {\sc Acknowledgements } \small I would like to thank D. Bertrand, J. Harris, J. Koll\'ar, K. Matsuki, D. Rohrlich and J. F. Voloch for discussions related to this note.