The Jacobian algebra of a graded Gorenstein singularity
type: Item
instance of: scholarly article
title: The Jacobian algebra of a graded Gorenstein singularity
zbMATH Open document ID: 0644.14001
DOI: 10.1215/S0012-7094-87-05540-2
author: Jonathan Wahl
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: This paper was motivated by the discovery of \textit{J. Damon} that for certain 2-dimensional graded complete intersections A, the dual of the module \(T^ 1\) of first order deformations is cyclic. Here this is proved for all graded Gorenstein isolated 2-dimensional singularities: indeed, the dual is isomorphic to the Jacobian algebra. As the latter is much easier to compute, this leads to a new method of calculation of \(T^ 1\), and this is applied to simple elliptic singularities, triangle singularities and canonical cones. In this case, this leads to a new map \(\phi: \wedge^ 2 H^ 0(C,K_ C)\to H^ 0(C,K_ C^{\otimes 3})\) which is never surjective if C is a smooth divisor on a K3 surface but is surjective if C is a complete intersection (with a few exceptions in codimensions \(\leq 3\)). It is also shown that for A Gorenstein of dimension 3, \(T^ 1\) is selfdual, and an appendix treats the case \(\dim(A)=1\).
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 32C37
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 32S05
zbMATH DE Number: 4049759
zbMATH Keywords: dual of first order deformations
zbMATH Keywords: graded Gorenstein isolated 2- dimensional singularities
zbMATH Keywords: Jacobian algebra
zbMATH Keywords: elliptic singularities
zbMATH Keywords: triangle singularities
zbMATH Keywords: canonical cones
MaRDI profile type: Publication
cites work: Q3347989
cites work: Q3973653
cites work: Topological triviality of versal unfoldings of complete intersections
cites work: Divisorenklassengruppen quasihomogener Singularitäten.
cites work: Dualität in der lokalen Kohomologie isolierter Singularitäten
cites work: Q4748277
cites work: The Milnor number and deformations of complex curve singularities
cites work: The dimension of smoothing components
cites work: Über den Derivationenmodul und das Jacobi-Ideal von Kurvensingularitäten
cites work: Brill-Noether-Petri without degenerations
cites work: Q4765389
cites work: Q4765390
cites work: Some remarks on isolated singularity and their application to algebraic manifolds
cites work: Q4058771
cites work: Normal surface singularities with \(\mathbb{C}^*\) action
cites work: Q4172153
cites work: Q3661709
cites work: Equisingular deformations of normal surface singularities. I
cites work: Equations defining rational singularities
cites work: Smoothings of normal surface singularities
cites work: Q3714188
cites work: Q3317211
cites work: Deformations of quasi-homogeneous surface singularities
cites work: Deformation von Gorenstein-Singularitäten der Kodimension 3
cites work: Some remarks concerning Demazure’s construction of normal graded rings
cites work: Existence of \(L^ 2-\)integrable holomorphic forms and lower estimates of T//\(v^ 1\).
cites work: Sections hyperplanes des surfaces K3
cites work: Q3222301
cites work: Q3478539
cites work: Derivations of negative weight and non-smoothability of certain singularities
full work available at URL: s0012-7094-87-05540-2
OpenAlex ID: W2026746483
Recommended article: When the Jacobian algebra of a curve singularity is Gorenstein
Recommended article: Q4946480
Recommended article: Singularity categories of Gorenstein monomial algebras
Recommended article: Gorenstein differential graded algebras
Recommended article: Gorenstein Graded Algebras and the Evaluation Map
Recommended article: Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities
Recommended article: Q3717380
Recommended article: Graded Lie algebras and quasihomogeneous complete intersection singularities
Recommended article: The Jordan type of graded Artinian Gorenstein algebras
Recommended article: Gröbner deformations and F‐singularities
description: scientific article; zbMATH DE number 4049759