Rendiconti del Seminario Matematico della Università di Padova

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Volume 126, 2011, pp. 11–45
DOI: 10.4171/RSMUP/126-2

Rational Components of Hilbert Schemes

Paolo Lella[1] and Margherita Roggero[2]

(1) Università degli Studi di Torino, Italy
(2) Università degli Studi di Torino, Italy

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.

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Lella Paolo, Roggero Margherita: Rational Components of Hilbert Schemes. Rend. Sem. Mat. Univ. Padova 126 (2011), 11-45. doi: 10.4171/RSMUP/126-2