Birational Isomorphisms between Twisted Group Actions
Journal of Lie Theory, Volume 16 (2006) no. 4, pp. 791-802
\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 $m \ge 0$.
DOI:
10.5802/jolt.432
Classification:
14L30, 14E07, 16K20
Keywords: Group action, algebraic group, no-name lemma, birational isomorphism, central simple algebra, Galois cohomology
Keywords: Group action, algebraic group, no-name lemma, birational isomorphism, central simple algebra, Galois cohomology
@article{JOLT_2006_16_4_a7,
author = {Z. Reichstein and A. Vistoli},
title = {Birational {Isomorphisms} between {Twisted} {Group} {Actions}},
journal = {Journal of Lie Theory},
pages = {791--802},
year = {2006},
volume = {16},
number = {4},
doi = {10.5802/jolt.432},
zbl = {1109.14033},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.432/}
}
Z. Reichstein; A. Vistoli. Birational Isomorphisms between Twisted Group Actions. Journal of Lie Theory, Volume 16 (2006) no. 4, pp. 791-802. doi: 10.5802/jolt.432
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