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Izv. Akad. Nauk SSSR Ser. Mat., 1972, Volume 36, Issue 4, Pages 749–764 (Mi izv2332)  

This article is cited in 50 scientific papers (total in 52 papers)

On a class of quasihomogeneous affine varieties

È. B. Vinberg, V. L. Popov


Abstract: We give a classification of irreducible affine algebraic varieties which are quasihomogeneous with respect to a regular action by a connected linear group of automorphisms and are such that the isotropy subgroup of a point in general position contains a maximal unipotent subgroup of the group of transformations. We find criteria for the normality and factoriality of such varieties. We compute the divisor class group and give a complete description of the orbits in such varieties.

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English version:
Mathematics of the USSR-Izvestiya, 1972, 6:4, 743–758

Bibliographic databases:

UDC: 519.4
MSC: Primary 20G05, 14M15; Secondary 14M05, 14M20
Received: 18.10.1971

Citation: È. B. Vinberg, V. L. Popov, “On a class of quasihomogeneous affine varieties”, Izv. Akad. Nauk SSSR Ser. Mat., 36:4 (1972), 749–764; Math. USSR-Izv., 6:4 (1972), 743–758

Citation in format AMSBIB
\Bibitem{VinPop72}
\by \`E.~B.~Vinberg, V.~L.~Popov
\paper On a~class of quasihomogeneous affine varieties
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1972
\vol 36
\issue 4
\pages 749--764
\mathnet{http://mi.mathnet.ru/izv2332}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=313260}
\zmath{https://zbmath.org/?q=an:0248.14014}
\transl
\jour Math. USSR-Izv.
\yr 1972
\vol 6
\issue 4
\pages 743--758
\crossref{https://doi.org/10.1070/IM1972v006n04ABEH001898}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. L. Popov, “Quasihomogeneous affine algebraic varieties of the group $SL(2)$”, Math. USSR-Izv., 7:4 (1973), 793–831  mathnet  crossref  mathscinet  zmath
    2. 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  mathnet  crossref  mathscinet  zmath
    3. V. L. Popov, “Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles”, Math. USSR-Izv., 8:2 (1974), 301–327  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    5. D. N. Akhiezer, “Dense orbits with two ends”, Math. USSR-Izv., 11:2 (1977), 293–307  mathnet  crossref  mathscinet  zmath
    6. V. I. Danilov, “The geometry of toric varieties”, Russian Math. Surveys, 33:2 (1978), 97–154  mathnet  crossref  mathscinet  zmath
    7. Franz Pauer, “Normale Einbettungen vonG/U”, Math Ann, 257:3 (1981), 371  crossref  mathscinet  zmath  isi
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    16. D. I. Panyushev, “The structure of the canonical module and the Gorenstein property for some quasihomogeneous varieties”, Math. USSR-Sb., 65:1 (1990), 81–95  mathnet  crossref  mathscinet  zmath
    17. T Levasseur, S.P Smith, “Primitive ideals and nilpotent orbits in type G2”, Journal of Algebra, 114:1 (1988), 81  crossref
    18. T Levasseur, S.P Smith, J.T Stafford, “The minimal nilpotent orbit, the Joseph ideal, and differential operators”, Journal of Algebra, 116:2 (1988), 480  crossref
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    21. Klaus Bongartz, “Minimal singularities for representations of Dynkin quivers”, Comment Math Helv, 69:1 (1994), 575  crossref  mathscinet  zmath  isi
    22. D. V. Alekseevskii, V. O. Bugaenko, G. I. Olshanskii, V. L. Popov, O. V. Schwarzman, “Érnest Borisovich Vinberg (on his 60th birthday)”, Russian Math. Surveys, 52:6 (1997), 1335–1343  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    23. I. V. Arzhantsev, “On $\operatorname{SL}_2$-actions of complexity one”, Izv. Math., 61:4 (1997), 685–698  mathnet  crossref  crossref  mathscinet  zmath  isi
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