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Journal of Lie Theory 16 (2006), No. 4, 791--802
Copyright Heldermann Verlag 2006

Birational Isomorphisms between Twisted Group Actions

Zinovy Reichstein
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Angelo Vistoli
Dipartimento di Matematica, UniversitÓ di Bologna, Piazza di Porta S. Donato 5, 40137 Bologna, Italy


\def\A{{\mathbb A}} Let $X$ be an algebraic variety with a generically free action of a connected algebraic group $G$. Given an automorphism $\phi \colon G\to G$, we will denote by $X^{\phi}$ the same variety $X$ with the $G$-action given by $g \colon x\to\phi(g) \cdot x$. We construct examples of $G$-varieties $X$ such that $X$ and $X^{\phi}$ are not $G$-equivariantly isomorphic. The problem of whether or not such examples can exist in the case where $X$ is a vector space with a generically free linear action, remains open. On the other hand, we prove that $X$ and $X^{\phi}$ are always stably birationally isomorphic, i.e., $X \times {\A}^m$ and $X^{\phi} \times {\A}^m$ are $G$-equivariantly birationally isomorphic for a suitable $m \ge 0$.

Keywords: Group action, algebraic group, no-name lemma, birational isomorphism, central simple algebra, Galois cohomology.

MSC: 14L30, 14E07, 16K20

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