Rational components of Hilbert schemes

By Paolo Lella, Margherita Roggero. 2009

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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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d9286c2f5c3cd395991135397d0b12d0f49e0e7e