Unobstructedness and dimension of families of Gorenstein algebras

Abstract

The goal of this paper is to develop tools to study maximal families of Gorenstein quotients A of a polynomial ring R. We prove a very general Theorem on deformations of the homogeneous coordinate ring of a scheme Proj(A) which is defined as the degeneracy locus of a regular section of the dual of some sheaf M^~ of rank r supported on say an arithmetically Cohen-Macaulay subscheme Proj(B) of Proj(R). Under certain conditions (notably; M maximally Cohen-Macaulay and the top exterior power of M^~ a twist of the canonical sheaf), then A is Gorenstein, and under additional assumptions, we show the unobstructedness of A and we give an explicit formula the dimension of any maximal family of Gorenstein quotients of R with fixed Hilbert function obtained by a regular section as above. The theorem also applies to Artinian quotients A. The case where M itself is a twist of the canonical module (r=1) was studied in a previous paper, while this paper concentrates on other low rank cases, notably r=2 and 3. In these cases regular sections of the first Koszul homology module and of normal sheaves to licci schemes (of say codimension 2) lead to Gorenstein quotients (of e.g. codimension 4) whose parameter spaces we examine. Our main applications are for Gorenstein quotients of codimension 4 of R since our assumptions are almost always satisfied in this case. Special attention are paid to arithmetically Gorenstein curves in P^5.