Computing the equivariant Brauer group

TL;DR

This paper develops methods to compute the Brauer group of quotient stacks formed by smooth projective rational varieties with finite abelian group actions, providing explicit calculations for certain surfaces.

Contribution

It introduces effective computational techniques for the Brauer group of quotient stacks, especially for G-minimal del Pezzo surfaces, using Galois cohomology residues and fixed locus geometry.

Findings

Computed Br([X/G]) for all G-minimal del Pezzo surfaces

Developed residue-based methods in Galois cohomology

Provided explicit examples in dimensions 2 and 3

Abstract

Let X be a smooth projective rational variety carrying a regular action of a finite abelian group G. We give examples of effective computation of the Brauer group of the quotient stack [X/G] in dimensions 2 and 3 using residues in Galois cohomology and the geometry of fixed loci. In particular, we compute Br([X/G]) for all G-minimal del Pezzo surfaces.