Birational splitting and algebraic group actions

TL;DR

The paper extends the classical birational splitting theorem for algebraic varieties with group actions to positive characteristic fields, providing a characteristic-free proof and characterizing certain algebraic groups.

Contribution

It offers a characteristic-free proof of the birational splitting theorem and characterizes algebraic groups with specific rational action properties.

Findings

Classical theorem holds only in characteristic 0.

A new characteristic-free proof is provided.

Characterization of algebraic groups with birational action properties.

Abstract

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and ${\bf P}^s$ with positive $s$. We show that the classical proof of this theorem actually works only in characteristic $0$ and we give a characteristic free proof of it. To this end we prove and use a characterization of connected linear algebraic groups $G$ with the property that every rational action of $G$ on an irreducible algebraic variety is birationally equivalent to a regular action of $G$ on an affine algebraic variety.