Maximal families of Gorenstein algebras

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring . Let be the scheme parametrizing graded quotients of with Hilbert function . We prove there is a close relationship between the irreducible components of , whose general member is a Gorenstein codimension quotient, and the irreducible components of , whose general member is a codimension Cohen-Macaulay algebra of Hilbert function related to . If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of , this relationship actually determines a well-defined injective mapping from such ``Cohen-Macaulay' components of to ``Gorenstein' components of , 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 . 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.