Birational Splitting and Algebraic Group Actions

According to the classical theorem, every 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 Ps 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. 1. Throughout this note k stands for an algebraically closed field of arbitrary characteristic which serves as domain of definition for each of the algebraic varieties considered below. Each algebraic variety is identified with its set of k-rational points. We use freely the standard notation and conventions of [PV 94], [Sp 98] and refer to [Ro 56], [Ro 61], [Ro 63], [PV 94], [Po 13] regarding the definitions and basic properties of rational and regular (morphic) actions of algebraic groups on algebraic varieties. Given a rational action of such a group G on an irreducible algebraic variety X, we denote by πG,X : X 99K X -G a rational quotient of this action; the latter means that X -G and πG,X are respectively an irreducible variety and a dominant rational map such that π G,X(k(X -G)) = k(X). 2. Up to a change of notation and terminology, the following statement appeared in classical paper [Ma 63, Thm. 1]: Theorem 1. Assume that a connected linear algebraic group G acts rationally and nontrivially on an irreducible algebraic variety X, and let B be a Borel subgroup of G. Then X is birationally isomorphic to P×X -B, where πB,X : X 99K X -B is a rational quotient of the natural rational action of B on X and 0 < s 6 dimB. In [Ma 63] no restriction on char k is imposed, but actually the brief argument given there in support of Theorem 1 works only if char k = 0. We reproduce it below in order to pinpoint where the restriction char k = 0 is implicitly used. Argument from [Ma 63] supporting Theorem 1. Since G is generated by its Borel subgroups and since all Borel subgroups are conjugate to each other, ∗ Supported by grants RFFI 14-01-00160, NX–2998.2014.1, and the granting program Contemporary Problems of Theoretical Mathematics of the Mathematics Branch of the Russian Academy of Sciences.