Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Group completions via Hilbert schemes

Author: Michel Brion
Journal: J. Algebraic Geom. 12 (2003), 605-626
Published electronically: April 15, 2003
MathSciNet review: 1993758
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Abstract | References | Additional Information

Abstract: Let $X$ be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of $X$ belongs to a unique irreducible component ${\mathcal H}_X$ of the Hilbert scheme of $X\times X$. Moreover, ${\mathcal H}_X$ is isomorphic to the “wonderful completion” of the connected automorphism group of $X$; in particular, ${\mathcal H}_X$ is non-singular. We describe explicitly the degenerations of the diagonal in $X\times X$, that is, the points of ${\mathcal H}_X$; these subschemes of $X\times X$ are reduced and Cohen-Macaulay.

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Additional Information

Michel Brion
Affiliation: Université de Grenoble I, Département de Mathématiques, Institut Fourier, UMR 5582 du CNRS, 38402 Saint-Martin d’Hères Cedex, France
MR Author ID: 41725

Received by editor(s): November 10, 2000
Published electronically: April 15, 2003