Families of artinian and low dimensional determinantal rings.

Abstract

Let GradAlg(H) be the scheme parameterizing graded quotients of R = k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert scheme of P^n if we restrict to quotients of positive dimension, see definition below). A graded quotient A = R/I of codimension c is called standard determinantal if the ideal I can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (f_ij). Given integers a_0 ≤ a_1 ≤ ... ≤ a_{t+c−2} and b_1≤...≤b_t, we denote by W_s(b;a)⊂ GradAlg(H) the stratum of determinantal rings where f_ij ∈ R are homogeneous of degree a_j−b_i. In this paper we extend previous results on the dimension and codimension of W_s(b;a) in GradAlg(H) to artinian determinantal rings, and we show that GradAlg(H) is generically smooth along W_s(b;a) under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of P^n is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of GradAlg(H).