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.