Work

The birational geometry of GIT quotients

Year: 2025

Type: article

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.

Source: Bulletin of the London Mathematical Society

DOI: 10.1112/blms.70072

Authors: Ruadhaí Dervan, Rémi Reboulet

Institutions: University of Glasgow, Lyon 1 Université, Institut Camille Jordan

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,930

Funders: Royal Society, Knut och Alice Wallenbergs Stiftelse