Rational Components of Hilbert Schemes

Abstract

The Groebner stratum of a monomial ideal $\mathfrak{j}$ is an affine variety that parameterizes the family of all ideals having $\mathfrak{j}$ as initial ideal (with respect to a fixed term ordering). The Groebner strata can be equipped in a natural way with a structure of homogeneous variety and are in a close connection with Hilbert schemes of subschemes in the projective space ${\mathbf{P}}^n$. Using properties of the Groebner strata we prove some sufficient conditions for the rationality of components of $\mathcal{H}{\text{ilb}}_{p(z)}^n$. We show for instance that all the smooth, irreducible components in $\mathcal{H}{\text{ilb}}_{p(z)}^n$ (or in its support) and the Reeves and Stillman component $H_{RS}$ are rational. We also obtain sufficient conditions for isomorphisms between strata corresponding to pairs of ideals defining a same subscheme, that can strongly improve an explicit computation of their equations.