Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 The effect of points fattening in dimension three
- 2 Some remarks on surface moduli and determinants
- 3 Valuation spaces and multiplier ideals on singular varieties
- 4 Line arrangements modeling curves of high degree: Equations, syzygies, and secants
- 5 Rationally connected manifolds and semipositivity of the Ricci curvature
- 6 Subcanonical graded rings which are not Cohen–Macaulay
- 7 Threefold divisorial contractions to singularities of cE type
- 8 Special prime Fano fourfolds of degree 10 and index 2
- 9 Configuration spaces of complex and real spheres
- 10 Twenty points in ℙ3
- 11 The Betti table of a high-degree curve is asymptotically pure
- 12 Partial positivity: Geometry and cohomology of q-ample line bundles
- 13 Generic vanishing fails for singular varieties and in characteristic p > 0
- 14 Deformations of elliptic Calabi–Yau manifolds
- 15 Derived equivalence and non-vanishing loci II
- 16 The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces
- 17 Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions
- 18 Gaussian maps and generic vanishing I: Subvarieties of abelian varieties
- 19 Torsion points on cohomology support loci: From D-modules to Simpson's theorem
- 20 Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O'Grady
- References
6 - Subcanonical graded rings which are not Cohen–Macaulay
Published online by Cambridge University Press: 05 January 2015
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 The effect of points fattening in dimension three
- 2 Some remarks on surface moduli and determinants
- 3 Valuation spaces and multiplier ideals on singular varieties
- 4 Line arrangements modeling curves of high degree: Equations, syzygies, and secants
- 5 Rationally connected manifolds and semipositivity of the Ricci curvature
- 6 Subcanonical graded rings which are not Cohen–Macaulay
- 7 Threefold divisorial contractions to singularities of cE type
- 8 Special prime Fano fourfolds of degree 10 and index 2
- 9 Configuration spaces of complex and real spheres
- 10 Twenty points in ℙ3
- 11 The Betti table of a high-degree curve is asymptotically pure
- 12 Partial positivity: Geometry and cohomology of q-ample line bundles
- 13 Generic vanishing fails for singular varieties and in characteristic p > 0
- 14 Deformations of elliptic Calabi–Yau manifolds
- 15 Derived equivalence and non-vanishing loci II
- 16 The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces
- 17 Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions
- 18 Gaussian maps and generic vanishing I: Subvarieties of abelian varieties
- 19 Torsion points on cohomology support loci: From D-modules to Simpson's theorem
- 20 Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O'Grady
- References
Summary
Abstract
We answer a question by Jonathan Wahl, giving examples of regular surfaces (so that the canonical ring is Gorenstein) with the following properties:
(1) the canonical divisor KS ≡ rL is a positive multiple of an ample divisor L;
(2) the graded ring R := R(X, L) associated to L is not Cohen-Macaulay.
In the Appendix, Wahl shows how these examples lead to the existence of Cohen-Macaulay singularities with KX ℚ-Cartier which are not ℚ-Gorenstein, since their index one cover is not Cohen-Macaulay.
Dedicated to Rob Lazarsfeld on the occasion of his 60th birthday
1 Introduction
The situation that we consider in this paper is the following: L is an ample divisor on a complex projective manifold X of complex dimension n, and we assume that L is subcanonical, i.e., there exists an integer h such that we have the linear equivalence KX ≡ hL, where h ≠ 0. There are then two cases: h < 0 and X is a Fano manifold, or h > 0 and X is a manifold with ample canonical divisor (in particular X is of general type). Assume that X is a Fano manifold and that −KX = rL, with r > 0: then, by Kodaira vanishing,
Hj(mL) := Hj(OX(mL)) = 0, ∀m ∈ ℤ,∀ 1 ≤ j ≤ n − 1.
For m < 0 this follows from Kodaira vanishing (and holds for j ≥ 1), while for m ≥ 0 Serre duality gives hj(mL) = hn − j(K − mL) = hn−j((−r −m)L) = 0. At the other extreme, if KX is ample and KX ≡ rL (thus r > 0), by the same argument we get vanishing outside of the interval
0 ≤ m ≤ r.
- Type
- Chapter
- Information
- Recent Advances in Algebraic GeometryA Volume in Honor of Rob Lazarsfeld’s 60th Birthday, pp. 92 - 101Publisher: Cambridge University PressPrint publication year: 2015
References
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