Additive group actions on affine T-varieties of complexity one in arbitrary characteristic

TL;DR

This paper classifies additive group actions on affine T-varieties of complexity one over arbitrary characteristic fields, extending previous results to positive characteristic and introducing new derivations related to Frobenius maps.

Contribution

It generalizes the classification of additive group actions to positive characteristic fields and introduces rationally homogeneous locally finite iterative higher derivations.

Findings

Complete description of additive group actions in the toric case.

Classification of normalized additive group actions on affine T-varieties.

Introduction of rationally homogeneous locally finite iterative higher derivations.

Abstract

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero. With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine T-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these additive group actions in the toric situation.