Jan O. Kleppe
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 GradAlg^H(R) whose general member is a Gorenstein codimension (c+1) quotient, and the irreducible components of 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 GradAlg^{H'}(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.