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 It(A) generated by the maximal minors of a homogeneous presentation matrix, A, of M has maximal codimension in R). Suppose X:=Proj(R/It(A)) is smooth in a sufficiently large open subset and dimX⩾1. Then we prove that the local graded deformation functor of M is isomorphic to the local Hilbert (scheme) functor at X⊂Proj(R) under a weak assumption which holds if dimX⩾2. Under this assumption 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 dimX⩾1. The cohomology H⁎i(NX) of the normal sheaf of X in Proj(R) is shown to vanish for 1⩽i⩽dimX−2. Finally the mentioned results, slightly adapted, remain true replacing R by any Cohen–Macaulay quotient of a polynomial ring.