Families of artinian and low dimensional determinantal rings

Abstract

Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x0,…,xn] with Hilbert function H (it is a subscheme of the Hilbert scheme of Pn 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 (fij). Given integers a0≤a1≤...≤at+c−2 and b1≤...≤bt, we denote by Ws(b_;a_)⊂GradAlg(H) the stratum of determinantal rings where fij∈R are homogeneous of degrees aj−bi.In this paper we extend previous results on the dimension and codimension of Ws(b_;a_) in GradAlg(H) to artinian determinantal rings, and we show that GradAlg(H) is generically smooth along Ws(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 Pn 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).