Families of determinantal schemes

TL;DR

This paper investigates the dimension and smoothness of families of determinantal schemes in projective space, extending previous results by relaxing numerical assumptions and analyzing their position within the Hilbert scheme.

Contribution

It extends prior work by determining the dimension and irreducibility of determinantal scheme families under weaker conditions.

Findings

Determined the dimension of W(b;a) under relaxed assumptions.

Established conditions for W(b;a) to be a generically smooth irreducible component.

Extended the understanding of determinantal schemes in the Hilbert scheme context.

Abstract

Given integers a_0 \le a_1 \le ... \le a_{t+c-2} and b_1 \le ... \le b_t, we denote by W(b;a) \subset Hilb^p(\PP^{n}) the locus of good determinantal schemes X \subset \PP^{n} of codimension c defined by the maximal minors of a t x (t+c-1) homogeneous matrix with entries homogeneous polynomials of degree a_j-b_i. The goal of this short note is to extend and complete the results given by the authors in [10] and determine under weakened numerical assumptions the dimension of W(b;a), as well as whether the closure of W(b;a) is a generically smooth irreducible component of the Hilbert scheme Hilb^p(\PP^{n}).