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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
\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
\jour Math. USSR-Izv.
\yr 1972
\vol 6
\issue 4
\pages 743--758

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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
    8. V. L. Popov, “Syzygies in the theory of invariants”, Math. USSR-Izv., 22:3 (1984), 507–585  mathnet  crossref  mathscinet  zmath
    9. Par Thierry Levasseur, “Dimension injective d'anneaux d'operateurs differentiels”, Communications in Algebra, 12:20 (1984), 2457  crossref
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    11. V. L. Popov, “Contractions of the actions of reductive algebraic groups”, Math. USSR-Sb., 58:2 (1987), 311–335  mathnet  crossref  mathscinet  zmath
    12. È. B. Vinberg, “Complexity of action of reductive groups”, Funct. Anal. Appl., 20:1 (1986), 1–11  mathnet  crossref  mathscinet  zmath  isi
    13. D. I. Panyushev, “Orbits of maximal dimension of solvable subgroups of reductive linear groups, and reduction for $U$-invariants”, Math. USSR-Sb., 60:2 (1988), 365–375  mathnet  crossref  mathscinet  zmath
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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
    19. Hanspeter Kraft, “Closures of conjugacy classes in g2”, Journal of Algebra, 126:2 (1989), 454  crossref
    20. D. I. Panyushev, “The canonical module of a quasihomogeneous normal affine $SL_2$-variety”, Math. USSR-Sb., 73:2 (1992), 569–578  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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
    24. I. V. Arzhantsev, “Contractions of affine spherical varieties”, Sb. Math., 190:7 (1999), 937–954  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. Dmitri I. Panyushev, “Actions of ‘nilpotent tori’ on G-varieties”, Indagationes Mathematicae, 10:4 (1999), 565  crossref
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    30. Jens Bender, Klaus Bongartz, “Minimal singularities in orbit closures of matrix pencils”, Linear Algebra and its Applications, 365 (2003), 13  crossref
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    32. P. Bravi, S. Cupit-Foutou, “Equivariant deformations of the affine multicone over a flag variety”, Advances in Mathematics, 217:6 (2008), 2800  crossref
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    35. D. I. Panyushev, “Actions of the Derived Group of a Maximal Unipotent Subgroup on G-Varieties”, Internat Math Res Notices, 2009  crossref  isi
    36. Kiumars Kaveh, A. G. Khovanskii, “Newton polytopes for horospherical spaces”, Mosc. Math. J., 11:2 (2011), 265–283  mathnet  mathscinet
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