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.Comment: 24 page
