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## Birational splitting and algebraic group actions

European Journal of Mathematics. 2016. Vol. 2. No. 1 . P. 283-290.

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 the projective space of a positive dimension. 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.