Rational components of Hilbert schemes

Abstract

The Gröbner stratum of a monomial ideal $\\mathfrak{j}$ is an affine variety that parametrizes the family of all ideals having $\\mathfrak{j}$ as initial ideal (with respect to a fixed term ordering). The Gröbner strata can be equipped in a natural way of a structure of homogeneous variety and are in a close connection with Hilbert schemes of subvarieties in the projective space $\\mathbb{P}^n$. Using properties of the Gröbner strata we prove some sufficient conditions for the rationality of components of $\\hilb_{p(z)}^n$. We show for instance that all the smooth, irreducible components in $\\hilb_{p(z)}^n$ (or in its support) and the Reeves and Stillman component $H_RS$ are rational.