Unobstructedness and Dimension of Families of Standard Determinantal Ideals

Abstract

Let ℙ^r be the r-dimensional projective space over a field K and fix a numerical polynomial $p\left( t \right) = \sum\nolimits_{i = 0}^r {a_i \left( {_i^{t + r} } \right) \in \mathbb{Q}\left[ t \right]}$ with $a_i \in \mathbb{Z}$ for all i. We consider the contravariant functor

$$ \underline {Hilb} _{p\left( t \right)}^r :\left( {{{Sch} \mathord{\left/ {\vphantom {{Sch} K}} \right. \kern-\nulldelimiterspace} K}} \right) \to \left( {Sets} \right) $$

defined by $\underline {Hilb} _{p\left( t \right)}^r \left( S \right): = \left\{ {flat families} \right. \mathcal{X} \subset \mathbb{P}^r \times S$ of closed subschemes of ℙ^r parameterized by S with fibers having Hilbert polynomial p(t).