Unobstructedness and dimension of families of Gorenstein algebras
Journal article, Peer reviewed
Postprint version of article originally published in collectanea mathematica. original article can be found at u r l: http://www.collectanea.ub.edu/index.php/ collectanea/article/view/4110
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Original versionKleppe, J.O. (2007). Unobstructedness and dimension of families of Gorenstein algebras. Collectanea Mathematica, 58 (2), 199-238 http://www.collectanea.ub.edu/index.php/Collectanea/article/view/4110
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 some sheaf M~ of rank r supported on an arithmetically Cohen-Macaulay subscheme Proj B of Proj R. Under certain conditions (notably; M maximally Cohen-Macaulay and the r-th exterior power of M~ a twist of the canonical sheaf), then A is Gorenstein, and, under additional assumptions, we show the unobstructedness of A (also for an Artinian A) and we compute the dimension of maximal families of Gorenstein quotients with fixed Hilbert function obtained via such a regular section. 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 satisfied in this case. Special attention are paid to arithmetically Gorenstein curves in P^5.