Polyhedral divisors and torus actions of complexity one over arbitrary fields

TL;DR

This paper extends the combinatorial description of affine torus-varieties of complexity one to arbitrary fields, using polyhedral divisors and Galois descent, and explores associated multigraded algebras over Dedekind domains.

Contribution

It generalizes the polyhedral divisor presentation of affine $ ext{T}$-varieties of complexity one to arbitrary fields and introduces Galois invariant polyhedral divisors for non-split tori.

Findings

Polyhedral divisor presentation holds over any field.

Describes how associated algebras change under scalar extension.

Introduces Galois invariant polyhedral divisors for non-split tori.

Abstract

We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{G}$-varieties of complexity one over a field, where $\mathbf{G}$ is a (not-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.