The Jacobian algebra of a graded Gorenstein singularity (Q1102335)

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 \(\dim(A)=1\).