Rational components of Hilbert schemes

TL;DR

This paper investigates the rationality of components within Hilbert schemes using properties of Gr"obner strata, establishing conditions under which these components are rational, including smooth, irreducible components and specific known components.

Contribution

It provides new sufficient conditions for the rationality of Hilbert scheme components, linking Gr"obner strata properties with the structure of Hilbert schemes.

Findings

All smooth, irreducible components in Hilbert schemes are rational.

The Reeves and Stillman component $H_{RS}$ is rational.

Conditions for rationality of Hilbert scheme components are established.

Abstract

The Gr\"obner stratum of a monomial ideal $\id{j}$ is an affine variety that parametrizes the family of all ideals having $\id{j}$ as initial ideal (with respect to a fixed term ordering). The Gr\"obner 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 $\PP^n$. Using properties of the Gr\"obner 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.