FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 1, PAGES 125-133

**Varieties birationally isomorphic to affine
$G$-varieties**

A. V. Petukhov

Abstract

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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 $\$\; \backslash bar\; X\; \$$ 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 $\$\; \backslash bar\; X\; \$$.

Location: http://mech.math.msu.su/~fpm/eng/k09/k091/k09109h.htm

Last modified: December 2, 2009