Dimension of families of determinantal schemes.

Jan O. Kleppe and Rosa M. Mir'o-Roig

A scheme X of P^n+c of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t x (t+c-1) matrix and X is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers a₀, a₁,..., a_t+c-2 and b₁,...,b_t we denote by W(b;a) of Hilb^p(P^n+c) (resp. W_s(b;a)) the locus of good (resp. standard) determinantal schemes X of P^n+c of codimension c defined by the maximal minors of a t x (t+c-1) matrix (f_ij), i=1,...,t and j=0,...,t+c-2, with f_ij in k[x₀, x₁,..., x_n+c] is a homogeneous polynomial of degree a_j-b_i.

In this paper we address the following three fundamental problems : To determine (1) the dimension of W(b;a) (resp. W_s(b;a)) in terms of a_j and b_i, (2) whether the closure of W(b;a) is an irreducible component of Hilb^p(P^n+c), and (3) when Hilb^p( P^n+c) is generically smooth along W(b;a). Concerning question (1) we give an upper bound for the dimension of W(b;a) (resp. W_s(b;a)) which works for all integers a₀, a₁,..., a_t+c-2 and b₁,...,b_t, and we conjecture that this bound is sharp. The conjecture is proved for 1 < c < 6, and for c > 5 under some restriction on a₀, a₁,..., a_t+c-2 and b₁,...,b_t. For questions (2) and (3) we have an affirmative answer for 1 < c < 5 and n > 1, and for c > 4 under certain numerical assumptions.