"Maximal families of Gorenstein algebras"
Author(s): Kleppe JO
Source: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 358 (7): 3133-3167 2006
Abstract: The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring R. Let GradAlg(H)(R) be the scheme parametrizing graded quotients of R with Hilbert function H. We prove there is a close relationship between the irreducible components of GradAlgH( R), whose general member is a Gorenstein codimension (c + 1) quotient, and the irreducible components of GradAlgH(')(R), whose general member B is a codimension c Cohen-Macaulay algebra of Hilbert function H-' related to H. If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of B, this relationship actually determines a well-defined injective mapping from such "Cohen-Macaulay" components of GradAlgH(')(R) to "Gorenstein" components of GradAlg(H)(R), in which generically smooth components correspond. Moreover the dimension of the "Gorenstein" components is computed in terms of the dimension of the corresponding "Cohen-Macaulay" component and a sum of two invariants of B. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.