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Variation of geometric invariant theory quotients

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Abstract

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI, USA

    Igor V. Dolgachev

  2. Department of Mathematics, University of Texas at Arlington, 76019, Arlington, TX, USA

    Yi Hu

Authors
  1. Igor V. Dolgachev
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  2. Yi Hu
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Additional information

Research supported in part by a NSF grant.

Research supported in part by NSF grant DMS 9401695.

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Dolgachev, I.V., Hu, Y. Variation of geometric invariant theory quotients. Publications Mathématiques de L’Institut des Hautes Scientifiques 87, 5–51 (1998). https://doi.org/10.1007/BF02698859

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  • DOI: https://doi.org/10.1007/BF02698859

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