Cox rings over nonclosed fields

TL;DR

This paper extends the concept of Cox rings and Cox sheaves to varieties over nonclosed fields, exploring their existence, classification, and applications in arithmetic geometry.

Contribution

It introduces a compatible definition of Cox rings over nonclosed fields and analyzes their relation to torsors and arithmetic applications.

Findings

Established a definition of Cox rings over nonclosed fields

Analyzed existence and classification of Cox rings in this setting

Connected Cox rings to torsors and demonstrated arithmetic applications

Abstract

We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.