Maximal families of Gorenstein algebras

Abstract:

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring $R$. Let $\operatorname {GradAlg}^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$. We prove there is a close relationship between the irreducible components of $\operatorname {GradAlg}^H(R)$, whose general member is a Gorenstein codimension $(c+1)$ quotient, and the irreducible components of $\operatorname {GradAlg}^{H’}(R)$, 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 $\operatorname {GradAlg}^{H’}(R)$ to “Gorenstein” components of $\operatorname {GradAlg}^{H}(R)$, 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.