Unobstructedness and dimension of families of codimension 3 ACM algebras

Abstract

The goal of this paper is to study irreducible families of codimension 3, Cohen-Macaulay quotients A of a polynomial ring R=k[x_0,x_1,...,x_n]; mainly, we study families of graded Cohen-Macaulay quotients A of codimension 1 on a codimension 2 Cohen-Macaulay algebra B defined by a regular section of (S^2K_B*)_t, the 2. symmetric power of the dual of canonical modul of B in degree t. We give lower bounds for the dimension of the irreducible components of the Hilbert scheme which contains Proj(A). The components are generically smooth and the bounds are sharp if t >> 0 and n=4 and 5. We also deal with a particular type of codimension 3, Cohen-Macaulay quotients A of R; concretely we restrict our attention to codimension 3 arithmetically Cohen-Macaulay subschemes X of P^n defined by the submaximal minors of a symmetric homogeneous matrix. We prove that such schemes are glicci and we give lower bounds for the dimension of the corresponding component of the Hilbert scheme. In the last part of the paper, we collect some questions/problems which naturally arise in our context.