Abstract
Let a linear algebraic group G act on an algebraic variety X. Classification of all these actions, in particular birational classification, is of great interest. A complete classification related to Galois cohomologies of the group G was established. Another important question is reducibility, in some sense, of this action to an action of G on an affine variety. It has been shown that if the stabilizer of a typical point under the action of a reductive group G on a variety X is reductive, then X is birationally isomorphic to an affine variety \( \bar X \) with stable action of G. In this paper, I show that if a typical orbit of the action of G is quasiaffine, then the variety X is birationally isomorphic to an affine variety \( \bar X \).
Similar content being viewed by others
References
V. Popov, “Sections in invariant theory,” in: The Sophus Lie Memorial Conf. Proc., Oslo (1992).
Z. Reichstein and N. Vonessen, “Stable affine models for algebraic group actions,” J. Lie Theory, 14, No. 2, 563–568 (2004).
R. W. Richardson, Jr., “Deformations of Lie subgroups and the variation of isotropy subgroups,” Acta Math., 129, 35–73 (1972).
E. B. Vinberg and V. L. Popov, “Theory of invariants,” in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Vol. 55, VINITI, Moscow (1989), pp. 137–290.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 15, No. 1, pp. 125–133, 2009.
Rights and permissions
About this article
Cite this article
Petukhov, A.V. Varieties birationally isomorphic to affine G-varieties. J Math Sci 166, 773–778 (2010). https://doi.org/10.1007/s10958-010-9893-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-9893-1