Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Moduli of affine schemes with reductive group action

Authors: Valery Alexeev and Michel Brion
Journal: J. Algebraic Geom. 14 (2005), 83-117
Published electronically: July 6, 2004
MathSciNet review: 2092127
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Abstract | References | Additional Information

Abstract: For a connected reductive group $G$ and a finite-dimensional $G$-module $V$, we study the invariant Hilbert scheme that parameterizes closed $G$-stable subschemes of $V$ affording a fixed, multiplicity-finite representation of $G$ in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus $T$ of $G$, together with an open $T$-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine $G$-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of $G$. We show that $V$ contains only finitely many multiplicity-free $G$-subvarieties, up to the action of the centralizer of $G$ in $\operatorname{GL}(V)$. As a consequence, there are only finitely many isomorphism classes of affine $G$-varieties affording a prescribed multiplicity-free representation in their coordinate ring.

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Additional Information

Valery Alexeev
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Michel Brion
Affiliation: Institut Fourier, B. P. 74, 38402 Saint-Martin d’Hères Cedex, France

Received by editor(s): January 31, 2003
Published electronically: July 6, 2004
Additional Notes: The first author was partially supported by NSF grant 0101280. Part of this work was done during the second author’s stay at the University of Georgia in January, 2003

American Mathematical Society