Families of artinian and low dimensional determinantal rings

TL;DR

This paper studies families of artinian and low-dimensional determinantal rings, extending dimension and smoothness results, and explores their implications for the structure of Hilbert schemes and Gorenstein liaison classes.

Contribution

It extends previous results to artinian and low-dimensional determinantal rings, analyzing their deformation properties and implications for Hilbert scheme components.

Findings

GradAlg(H) is generically smooth along determinantal strata under certain conditions.

General elements in these families are glicci, linking to Gorenstein liaison theory.

Ghost terms in resolutions can disappear or persist under deformation.

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 by t minors of a homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a}) the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are homogeneous of degrees a_j-b_i. In this paper we extend previous results on the dimension and codimension of W_s(\underline{b};\underline{a}) in GradAlg(H) to {\it artinian determinantal rings}, and we show that GradAlg(H) is generically smooth along W_s(\underline{b};\underline{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).