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

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. Miró-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.