popovvl@mi.ras.ru

Submission: 2015, Apr 19

According to the classical Matsumura's theorem, every algebraic variety endowed with a nontrivial rational action of a connected affine algebraic group is birationally isomorphic to a product of another algebraic variety and a projective space of positive dimension. We show that the classical proof of this theorem actually works only in characteristic zero and give a characteristic free proof of it. To this end we prove and use a characterization of connected linear algebraic groups that have the property that every rational action of this group on an irreducible algebraic variety is birationally equivalent to its regular action on an affine algebraic variety.

2010 Mathematics Subject Classification: 20Gxx, 13N15, 14R10

Keywords and Phrases: algebraic variety, affine algebraic group

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