Birational splitting and algebraic group actions

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 ${\mathbf{P}}^s$ 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.

Acknowledgements added

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