Arithmetic toric varieties

TL;DR

This paper investigates the classification of toric varieties over a field using Galois cohomology, providing methods to compute cohomology groups and classify forms of projective spaces and surfaces.

Contribution

It introduces a cohomological approach to classify k-forms of toric varieties, especially for cyclic Galois groups, and links cohomology to the class group presentation.

Findings

Computed Galois cohomology for cyclic Galois groups.

Classified k-forms of projective spaces.

Studied k-forms of surfaces.

Abstract

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.