The birational geometry of GIT quotients
Ruadhaí Dervan, Rémi Reboulet
TL;DR
The paper addresses the limitation of classical VGIT, which yields only finitely many birational models of a GIT quotient, by constructing a universal object $V^G\underline{X}$ that parameterises all quotients over all birational models of a projective variety $X$ with a $G$-action. Using Zariski--Riemann spaces and projective limits, the authors prove that $V^G\underline{X}$ is canonically isomorphic to the Zariski--Riemann space of any GIT quotient and thus encodes the full birational geometry of GIT quotients. They establish universality (cofinality) and analyze the infinitesimally stable locus in the analytified limit, showing it forms an open dense subset of the compactified universal VGIT space. This work provides a birationally complete framework for GIT quotients, linking VGIT with birational models and suggesting a canonical compactification of stability data across all quotients. The results illuminate how quotients transform under birational changes of the ambient space and offer a new lens on moduli problems via a universal, though non-schematic, limit object.
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.