On rational additive group actions

TL;DR

This paper characterizes rational additive group actions on algebraic varieties over characteristic zero fields using integrability of velocity vector fields, extending classical regular action correspondences and exploring applications to invariants and toric varieties.

Contribution

It extends the classical correspondence between regular additive group actions and locally nilpotent derivations to rational actions via integrability conditions.

Findings

Complete characterization of rational additive group actions on algebraic varieties.

Analysis of the rational Makar-Limanov invariant for affine varieties.

Description of rational homogeneous additive actions on toric varieties.

Abstract

We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical correspondence between regular actions of the additive group on affine algebraic varieties and the so-called locally nilpotent derivations of their coordinate rings. This leads in particular to a complete characterization of regular additive group actions on semi-affine varieties in terms of their associated vector fields. Among other applications, we review properties of the rational counter-part of the Makar-Limanov invariant for affine varieties and describe the structure of rational homogeneous additive group actions on toric varieties.