A vanishing theorem for finitely supported ideals in regular local rings
A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.
On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups
Let $(R, \mathfrak{m}, \Bbbk)$ be a Noetherian three-dimensional Cohen-Macaulay analytically unramified ring and $I$ an $\mathfrak{m}$-primary $R$-ideal. Write $X = \mathrm{Proj}\left(\oplus_{n \in \mathbb{N}} \overline{I^n}t^n\right)$. We prove some consequences of the vanishing of $\mathrm{H}^2(X, \mathscr{O}_X)$, whose length equals the the constant term $\bar e_3(I)$ of the normal Hilbert polynomial of $I$. Firstly, $X$ is Cohen-Macaulay. Secondly, if the extended Rees ring $A := \oplus_{n \in \mathbb{Z}} \overline{I^n}t^n$ is not Cohen-Macaulay, and either $R$ is equicharacteristic or $\overline{I} = \mathfrak{m}$, then $\bar e_2(I) - \mathrm{length}_R\left(\frac{\overline{I^2}}{I\overline{I}}\right) \geq 3$; this estimate is proved using Boij-S\"oderberg theory of coherent sheaves on $\mathbb{P}^2_\Bbbk$. The two results above are related to a conjecture of S. Itoh (J. Algebra, 1992). Thirdly, $\mathrm{H}^2_E(X, I^m\mathscr{O}_X) = 0$ for all integers $m$, where $E$ is the exceptional divisor in $X$. Finally, if additionally $R$ is regular and $X$ is pseudo-rational, then the adjoint ideals $\widetilde{I^n}, n \geq 1$ satisfy $\widetilde{I^n} = I\widetilde{I^{n-1}}$ for all $n \geq 3$. The last two results are related to conjectures of J. Lipman (Math. Res. Lett., 1994).
ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS IN TWO-DIMENSIONAL REGULAR LOCAL RINGS
Introduction. In the study of the singularity of a complex plane curve germ (C, 0), given locally, say, by f(x, y) = 0, important roles are played by the adjoint curves (those having a local equation g = 0 such that g restricts on C to an element in the conductor ideal of the one-dimensional local ring OC,0 of germs of holomorphic functions), and by the polar curves (those given locally by an equation of the form afx + bfy + cf = 0 with a, b, c ∈ C). In this paper, we introduce corresponding complete ideals C and P in the convergent power series ring C{x, y}, and indeed in any two-dimensional regular local ring. The classical theory of base points of linear systems of curves on smooth surfaces, exposed at length in [4, book 4] and summarized in [16, Chaps. 1 and 2], inspired Zariski to create the theory of complete (=integrally closed) ideals in twodimensional regular local rings [15, pp. 199–201]. Over the past half century, this theory has been further developed, by Zariski himself and others [17, Appendix 5], [6, Chaps. II and V], [7], [5], [12], [8]. A complete ideal p in C{x, y} encodes local “base conditions” to be satisfied by certain complete linear systems of curves C through the origin; the adjoint ideal Cp (defined in (2.1)) encodes the conditions which are then satisfied by the adjoints of a generic such C, and the polar ideal Pp (defined in (5.1)) does the same for curves whose multiplicities at the infinitely near base points of the linear system are at least as big as those of the generic polar of C. These “polar multiplicities” were worked out in [4, pp. 374–381] (and for an extensive modern treatment of local polars of plane curves, see [1], [2], [3]). They are just the integers which make up the point basis of P, i.e., the si prescribed in Theorem (5.2) below in terms of the point basis {ri} of p, which {ri} may also be thought of as the sequence