Proper polyhedral divisors and torus actions over arbitrary fields

TL;DR

This paper develops a combinatorial framework using proper polyhedral divisors with Galois actions to describe normal affine varieties with torus actions over any field, extending previous theories.

Contribution

It introduces a new algebro-geometric combinatorial description for torus actions over arbitrary fields using proper polyhedral divisors with Galois semilinear actions.

Findings

Provides a classification of affine varieties with torus actions over arbitrary fields.

Extends the combinatorial description to non-algebraically closed fields.

Connects polyhedral divisors with Galois actions to geometric properties.

Abstract

We provide a algebro-geometric combinatorial description of geometrically integral geometrically normal affine varieties endowed with an effective action of an algebraic torus over arbitrary fields. This description is achieved in terms of proper polyhedral divisors endowed with a Galois semilinear action.