Twisted forms of toric varieties

TL;DR

This paper studies the classification of forms of toric varieties over arbitrary fields, introducing new cohomological maps that generalize classical results like the Brauer class and Severi-Brauer varieties.

Contribution

It defines an injective cohomological map from forms of toric varieties to a non-abelian second cohomology set, extending classical classifications to more general isomorphisms.

Findings

Established a cohomological classification framework for toric variety forms.

Generalized the Brauer class to a broader setting involving arbitrary isomorphisms.

Connected forms of toric varieties to forms of separable algebras, extending known correspondences.

Abstract

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M.~Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebras