The semi-invariant ring as the Cox ring of a GIT quotient

2024/04/18 by Gwyn Bellamy, Alastair Craw, Bellamy, Gwyn +3
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2404.12225

openalex publication_date 2024/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study GIT quotients X_θ=V / / _θG whose linearisation map defines an isomorphism between the group of characters of G and the Picard group of X_θ modulo torsion. Our main result establishes that the Cox ring of X_θ is isomorphic to the semi-invariant ring of the θ-stable locus in V. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of n-points in the affine plane.

Related

• The Cox ring of an embedded variety
• Cox rings of minimal resolutions of surface quotient singularities
• Cox rings and symplectic quotient singularities with torus action
• P-Divisors of Cox Rings
• Cox rings of some symplectic resolutions of quotient singularities