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 by t minors of a homogeneous t by (t+c-1) matrix (fij). Given integers a0 a1 ... at+c-2 and b1 ... bt, we denote by Ws(b;a) the stratum of GradAlg(H) 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).