Deformation and Unobstructedness of Determinantal Schemes

Abstract

Let $Hilb ^{p(t)}(P^n)$ be the Hilbert scheme of closed subschemes of $P^n$ with Hilbert polynomial $p(t) \in Q[t]$, and let $W:= \overline{W(\underline{b};\underline{a};r)}$ be the closure of the locus in $Hilb ^{p(t)}(P^n)$ of determinantal schemes defined by the vanishing of the $(t-r+1)\times (t - r+1)$ minors of some matrix $\mathcal A$ of size $t\times (t+c-1)$ with $ij$-enty a homogeneous form of degree $a_j-b_i$ and with $r$ satisfying $\max\{1,2-c\} \le r < t$. $W$ is an irreducible algebraic set. First of all, we compute an upper $r$-independent bound for the dimension of $W$ in terms of $a_j$ and $b_i$ which is sharp for $r=1$. In the linear case ($a_j = 1, b_i=0$) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all $r$. Then, we study to what extent $W$ is a generically smooth component of $Hilb ^{p(t)}(P^n)$. Under some weak numerical assumptions on the integers $a_j$ and $b_i$ (or under some depth conditions) we conjecture and often prove that $W$ is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of $(X)\in W$ and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp. $r$). These results extend previous results on standard determinantal schemes to determinantal schemes; i.e. previous results of the authors on $W(\underline{b};\underline{a};1)$ to $W$ with $1\le r < t$ and $c\ge 2-r$. Finally, deformations of exterior powers of the cokernel of the map determined by $\mathcal A$ are studied and proven to be given as deformations of $X \subset P^n$ if $\dim X \ge 3$. The work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section.