Variation of Geometric Invariant Theory Quotients

TL;DR

This paper investigates how geometric invariant theory quotients change with different choices of ample line bundles, showing finiteness of equivalence classes and describing their variation as birational transformations.

Contribution

It proves the finiteness of equivalence classes of ample line bundles producing quotients and describes how quotients vary across chambers as birational transformations.

Findings

Finitely many equivalence classes of line bundles produce projective quotients.

Chambers in a convex cone correspond to these classes.

Crossing walls between chambers induces birational transformations.

Abstract

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing 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 fundamental questions naturally 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.