  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 the dual of some sheaf
M^~ of rank r supported on say an arithmetically Cohen-Macaulay subscheme
Proj(B) of Proj(R). Under certain conditions (notably; M maximally
Cohen-Macaulay and the top exterior power of M^~ a twist of the canonical
sheaf), then A is Gorenstein, and under additional assumptions, we show the
unobstructedness of A and we give an explicit formula the dimension of any
maximal family of Gorenstein quotients of R with fixed Hilbert function
obtained by a regular section as above. The theorem also applies to Artinian
quotients A.
  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 almost always satisfied in this case. Special attention are
paid to arithmetically Gorenstein curves in P^5.
