Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes

TL;DR

This paper establishes a connection between deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes, proving smoothness, dimension formulas, and confirming conjectures related to determinantal loci.

Contribution

It proves the isomorphism between local graded deformation functors and Hilbert scheme functors at determinantal schemes, confirming conjectures on smoothness and dimension of determinantal loci.

Findings

Hilbert scheme is smooth at determinantal schemes under certain conditions.

Explicit formula for the dimension of the Hilbert scheme at these schemes.

Confirmation of conjectures on the structure and dimension of determinantal loci.

Abstract

Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_t(\cA) generated by the maximal minors of a homogeneous presentation matrix, \cA, of M has maximal codimension in R). Suppose X:=Proj(R/I_t(\cA)) is smooth in a sufficiently large open subset and dim X > 0. Then we prove that the local graded deformation functor of M is isomorphic to the local Hilbert (scheme) functor at X \subset Proj(R) under a week assumption which holds if dim X > 1. Under this assumptions we get that the Hilbert scheme is smooth at (X), and we give an explicit formula for the dimension of its local ring. As a corollary we prove a conjecture of R. M. Mir\'o-Roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component V of the Hilbert scheme. Also their conjecture on the dimension of V is proved for dim X > 0. The cohomology H^i_{*}({\cN}_X) of the normal sheaf of X in Proj(R) is shown to vanish for 0 < i < dim X-1. Finally the mentioned results, slightly adapted, remain true replacing R by any Cohen-Macaulay quotient of a polynomial ring.