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arXiv · 2301.09227

The birational geometry of GIT quotients

Ruadhaí Dervan·Rémi Reboulet
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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.

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Ruadhaí Dervan, Rémi Reboulet. 2025-03-29. The birational geometry of GIT quotients. https://arxiv.org/abs/2301.09227

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