Families of low dimensional determinantal schemes

TL;DR

This paper studies families of low-dimensional determinantal schemes, determining their codimension in the Hilbert scheme and analyzing smoothness properties, with specific results for zero and one-dimensional cases.

Contribution

It characterizes the codimension and smoothness of families of determinantal schemes, providing new insights especially for zero-dimensional cases and counterexamples to previous conjectures.

Findings

Determines codimension of determinantal scheme families in Hilbert schemes.

Shows Hilbert schemes are generically smooth along these families under certain conditions.

Provides a counterexample to a previous conjecture on the dimension of these families.

Abstract

A scheme X \subset \PP^{n} of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t x t minors of a homogeneous t x (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(b;a) \subset Hilb(\PP^{n}) the stratum of standard determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_j-b_i and Hilb(\PP^{n}) is the Hilbert scheme (if n-c > 0, resp. the postulation Hilbert scheme if n-c = 0). Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of W_s(b;a) in Hilb(\PP^{n}) and we show that Hilb(\PP^{n}) is generically smooth along W_s(b;a) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of dim W_s(b;a) appearing in [26].