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Birational isomorphisms between twisted group actions

Zinovy Reichstein·Angelo Vistoli
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Abstract

Let X be an algebraic variety with a generically free action of a connected algebraic group G. Given an automorphism u of G, we will denote by X^u the same variety X with the G-action given by twisted by u. V. L. Popov asked if X and X^u are always G-equivariantly birationally isomorphic. We construct examples to show that this is not the case in general, but we prove that X and X^u are always stably birationally isomorphic, i.e., birationally isomorphic after taking products with an affine space with trivial G-action.

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Zinovy Reichstein, Angelo Vistoli. 2005-04-14. Birational isomorphisms between twisted group actions. https://arxiv.org/abs/math/0504306

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