Rational Components of Hilbert Schemes

Abstract

The Gröbner stratum of a monomial ideal i is an affine variety that parameterizes the family of all ideals having i as initial ideal (with respect to a fixed term ordering). The Gröbner 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 Pn. Using properties of the Gröbner strata we prove some sufficient conditions for the rationality of components of Hilbnp(z). We show for instance that all the smooth, irreducible components in Hilbnp(z) (or in its support) and the Reeves and Stillman component HRS 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.