Ideals generated by submaximal minors

TL;DR

This paper studies families of arithmetically Gorenstein schemes defined by submaximal minors of matrices, proving irreducibility, smoothness, and computing their dimensions under certain conditions.

Contribution

It establishes that these families form irreducible components of the Hilbert scheme and computes their dimensions, advancing understanding of their geometric properties.

Findings

The closure of W(b;a) is an irreducible component of the Hilbert scheme.

Hilb^{p(x)}(P^n) is generically smooth along W(b;a).

The dimension of W(b;a) is explicitly computed in terms of matrix degrees.

Abstract

The goal of this paper is to study irreducible families W(b;a) of codimension 4, arithmetically Gorenstein schemes X of P^n defined by the submaximal minors of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is an irreducible component of Hilb^{p(x)}(P^n), we show that Hilb^{p(x)}(P^n) is generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms of a_j and b_i. To achieve these results we first prove that X is determined by a regular section of the twisted conormal sheaf I_Y/I^2_Y(s) where s=deg(det(A)) and Y is a codimension 2, arithmetically Cohen-Macaulay scheme of P^n defined by the maximal minors of the matrix obtained deleting a suitable row of A.