Families of low dimensional determinantal schemes

Abstract

A scheme X \subset \PP^{n} of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t x t minors of a homogeneous t x (t+c-1) matrix (f_{ij}). Given integers a_0 \le a_1 \le ...\le a_{t+c-2} and b_1 \le ...\le b_t, we denote by W_s(b;a) \subset Hilb(\PP^{n}) the stratum of standard determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_j-b_i and Hilb(\PP^{n}) is the Hilbert scheme (if n-c > 0, resp. the postulation Hilbert scheme if n-c = 0). Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of W_s(b;a) in Hilb(\PP^{n}) and we show that Hilb(\PP^{n}) is generically smooth along W_s(b;a) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of dim W_s(b;a) appearing in [26].