The birational geometry of GIT quotients

TL;DR

This paper constructs a comprehensive space parametrizing all GIT quotients of a variety's birational models, capturing the full birational geometry of GIT quotients and providing a compactification of the stable orbit set.

Contribution

It introduces a natural, simple construction that encapsulates all possible GIT quotients across birational models, extending beyond previous limitations.

Findings

Constructs a space parametrising all GIT quotients of birational models.

Captures the entire birational geometry of GIT quotients.

Provides a compactification of the birational stable orbit set.

Abstract

Geometric Invariant Theory (GIT) produces quotients of algebraic varieties by reductive groups. If the variety is projective, this quotient depends on a choice of polarisation; by work of Dolgachev-Hu and Thaddeus, it is known that two quotients of the same variety using different polarisations are related by birational transformations. Only finitely many birational varieties arise in this way: variation of GIT fails to capture the entirety of the birational geometry of GIT quotients. We construct a space parametrising all possible GIT quotients of all birational models of the variety in a simple and natural way, which captures the entirety of the birational geometry of GIT quotients in a precise sense. It yields in particular a compactification of a birational analogue of the set of stable orbits of the variety.