Geometric invariant theory and flips

TL;DR

This paper explores how geometric invariant theory quotients change with linearization choices, showing they are related by flips, and connects this to the minimal model program, providing explicit descriptions and applications to moduli problems.

Contribution

It demonstrates that GIT quotients are related by flips in good cases, linking GIT, Mori theory, and the minimal model program with explicit flip descriptions.

Findings

GIT quotients related by flips in certain cases

Explicit descriptions of flips via blow-ups and blow-downs

Application to various moduli problems and chamber structures

Abstract

We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and explain the relationship with the minimal model programme. Moreover, we express the flip as the blow-up and blow-down of specific ideal sheaves, leading, under certain hypotheses, to a quite explicit description of the flip. We apply these ideas to various familiar moduli problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos- Wentworth, and the author. Along the way we display a chamber structure, following Duistermaat-Heckman, on the space of all linearizations. We also give a new, easy proof of the Bialynicki-Birula decomposition theorem.