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 Izv. Akad. Nauk SSSR Ser. Mat., 1972, Volume 36, Issue 2, Pages 371–385 (Mi izv2300)

On the stability of the action of an algebraic group on an algebraic variety

V. L. Popov

Abstract: 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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English version:
Mathematics of the USSR-Izvestiya, 1972, 6:2, 367–379

Bibliographic databases:

UDC: 519.4
MSC: Primary 20G15, 20G99, 14M15; Secondary 14M05, 10C30

Citation: V. L. Popov, “On the stability of the action of an algebraic group on an algebraic variety”, Izv. Akad. Nauk SSSR Ser. Mat., 36:2 (1972), 371–385; Math. USSR-Izv., 6:2 (1972), 367–379

Citation in format AMSBIB
\Bibitem{Pop72} \by V.~L.~Popov \paper On the stability of the action of an algebraic group on an algebraic variety \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1972 \vol 36 \issue 2 \pages 371--385 \mathnet{http://mi.mathnet.ru/izv2300} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=301028} \zmath{https://zbmath.org/?q=an:0232.14018} \transl \jour Math. USSR-Izv. \yr 1972 \vol 6 \issue 2 \pages 367--379 \crossref{https://doi.org/10.1070/IM1972v006n02ABEH001877} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. È. B. Vinberg, V. L. Popov, “On a class of quasihomogeneous affine varieties”, Math. USSR-Izv., 6:4 (1972), 743–758
2. V. L. Popov, “Quasihomogeneous affine algebraic varieties of the group $SL(2)$”, Math. USSR-Izv., 7:4 (1973), 793–831
3. V. L. Popov, “Classification of affine algebraic surfaces that are quasihomogeneous with respect to an algebraic group”, Math. USSR-Izv., 7:5 (1973), 1039–1056
4. V. L. Popov, “Classification of three-dimensional affine algebraic varieties that are quasi-homogeneous with respect to an algebraic group”, Math. USSR-Izv., 9:3 (1975), 535–576
5. Frank Grosshans, “Localization and invariant theory”, Advances in Mathematics, 21:1 (1976), 50
6. F. A. Bogomolov, “Holomorphic tensors and vector bundles on projective varieties”, Math. USSR-Izv., 13:3 (1979), 499–555
7. A. G. Élashvili, “Orbits of maximum dimension for borel subgroups of semisimple linear Lie groups”, Funct. Anal. Appl., 21:1 (1987), 84–86
8. V. L. Popov, “Closed orbits of Borel subgroups”, Math. USSR-Sb., 63:2 (1989), 375–392
9. A. B. Anisimov, “On stability of diagonal actions and tensor invariants”, Sb. Math., 203:4 (2012), 500–513
10. Mitsuyasu Hashimoto, “Good filtrations and strong F-regularity of the ring of -invariants”, Journal of Algebra, 370 (2012), 198
11. Mitsuyasu Hashimoto, “Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants”, Communications in Algebra, 43:4 (2015), 1524
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