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
reichst@math.ubc.ca
Angelo Vistoli
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40137 Bologna, Italy
vistoli@dm.unibo.it
[Abstract-pdf]
\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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