Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes (Q402675)

Let \(R\) be a polynomial ring over a field, occasionally (but not usually) of characteristic zero. Let \(M\) be a finitely generated graded \(R\)-module. Let \(\mathcal A\) be a homogeneous presentation matrix for \(M\), and assume that the ideal \(I_t (\mathcal A)\) of maximal minors of \(\mathcal A\) has maximal codimension in \(R\). Let \(X = \text{Proj}(R/I_t(\mathcal A))\). Assuming that \(X\) is smooth in a sufficiently large open subset, \(\dim X \geq 1\), and a weak technical condition, the author shows that the local graded deformation functor of \(M\) is isomorphic to the local Hilbert (scheme) functor at \(X \subset \text{Proj}(R)\). The author shows that the Hilbert scheme is smooth at \((X)\), and obtains an explicit formula for the dimension of its local ring. As a corollary he proves a conjecture of his with Miró-Roig 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 of the Hilbert scheme, and he proves another conjecture of the same authors about the dimension of this component if \(\dim X \geq 1\). Finally, he obtains results about the vanishing of the normal sheaf of \(X\) in \(\text{Proj}(R)\), and shows how his results can be extended from \(R\) to any Cohen-Macaulay quotient of \(R\).