Geodesic
Varieties birationally isomorphic to affine $G$-varieties
Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 1, pp. 125-133.

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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 $\overline X$ 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 $\overline X$. 
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A. V. Petukhov. Varieties birationally isomorphic to affine $G$-varieties. Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 1, pp. 125-133. http://geodesic.mathdoc.fr/item/FPM_2009_15_1_a8/

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