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
DOI: https://doi.org/10.1090/S1056-3911-04-00377-7
Published electronically: July 6, 2004
MathSciNet review: 2092127
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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
Email: valery@math.uga.edu

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

DOI: https://doi.org/10.1090/S1056-3911-04-00377-7
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