Classifying torsors of tori with Brauer groups

TL;DR

This paper introduces a framework using Mackey functors to classify torsors of algebraic tori via Brauer groups, extending previous results to a broader class of retract rational tori.

Contribution

It generalizes Blunk's classification of tori associated with del Pezzo surfaces to all retract rational tori using Mackey functors and Brauer groups.

Findings

Provides a unified classification framework for torsors of algebraic tori.

Extends previous classifications to a larger class of tori.

Uses Mackey functors to relate torsors to Brauer groups.

Abstract

Using Mackey functors, we provide a general framework for classifying torsors of algebraic tori in terms of Brauer groups of finite field extensions of the base field. This generalizes Blunk's description of the tori associated to del Pezzo surfaces of degree 6 to all retract rational tori, essentially the largest class for which this is possible.