Geodesic
On the stability of the action of an algebraic group on an algebraic variety
Izvestiya. Mathematics , Tome 6 (1972) no. 2, pp. 367-379

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We prove the following fact: if a connected algebraic group having no rational characters acts regularly on a normal irreducible algebraic variety $X$ with periodic divisor class group $ClX$, then for the orbit $O_x$ of a point $x\in X$ in general position to be closed, it is sufficient that $O_x$ be an affine variety; moreover, if $X$ is affine, this condition is also sufficient. 
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     author = {V. L. Popov},
     title = {On the stability of the action of an algebraic group on an algebraic variety},
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     number = {2},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_2_a2/}
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V. L. Popov. On the stability of the action of an algebraic group on an algebraic variety. Izvestiya. Mathematics , Tome 6 (1972) no. 2, pp. 367-379. http://geodesic.mathdoc.fr/item/IM2_1972_6_2_a2/