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 
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/
@article{IM2_1972_6_2_a2,
     author = {V. L. Popov},
     title = {On the stability of the action of an algebraic group on an algebraic variety},
     journal = {Izvestiya. Mathematics},
     pages = {367--379},
     year = {1972},
     volume = {6},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_2_a2/}
}
TY  - JOUR
AU  - V. L. Popov
TI  - On the stability of the action of an algebraic group on an algebraic variety
JO  - Izvestiya. Mathematics
PY  - 1972
SP  - 367
EP  - 379
VL  - 6
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IM2_1972_6_2_a2/
LA  - en
ID  - IM2_1972_6_2_a2
ER  - 
%0 Journal Article
%A V. L. Popov
%T On the stability of the action of an algebraic group on an algebraic variety
%J Izvestiya. Mathematics
%D 1972
%P 367-379
%V 6
%N 2
%U http://geodesic.mathdoc.fr/item/IM2_1972_6_2_a2/
%G en
%F IM2_1972_6_2_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Popov V. L., “Kriterii stabilnosti deistviya poluprostoi gruppy na faktorialnom mnogoobrazii”, Izv. AN SSSR. Ser. matem., 34 (1970), 523–531 | MR

[2] Leng S., Algebra, Mir, M., 1968

[3] Shafarevich I. R., “Osnovy algebraicheskoi geometrii”, Uspekhi matem. nauk, XXIV:6(150) (1969), 3–184 | MR

[4] Serr Zh.-P., Algebraicheskie gruppy i polya klassov, Mir, M., 1968

[5] Rosenlicht M., “Some rationality questions on algebraic groups”, Ann. Mat., 43 (1957), 25–50 | DOI | MR | Zbl

[6] Seshadri C. S., “Some results on the quotient space by an algebraic group of automorphisms”, Math. Ann., 149 (1963), 286–301 | DOI | MR | Zbl

[7] Nagata M., “Invariants of a group in an affine ring”, J. Math. Kyoto Univ., 1963–1964, no. 3, 369–377 | MR

[8] Nagata M., “Invariants under semi-reductive action”, J. Math. Kyoto Univ., 5:2 (1966), 171–176 | MR | Zbl

[9] Nagata M., “Note on orbit spaces”, Osaka Math. J., 14 (1962), 21–31 | MR | Zbl

[10] Rosenlicht M., “On quotient varieties and the affine embedding of certain homogeneous spaces”, Trans. Amer. Math. Soc., 101 (1961), 211–223 | DOI | MR | Zbl

[11] Bialynicki–Birula A., “On homogeneous affine spaces of linear algebraic groups”, Am. J. Math., 85 (1963), 572 | MR