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A combinatorial smoothness criterion for spherical varieties

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Abstract

We suggest a combinatorial criterion for the smoothness of an arbitrary spherical variety using the classification of multiplicity-free spaces, generalizing an earlier result of Camus for spherical varieties of type A.

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References

  1. Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox Rings, Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge (2015)

  2. Avdeev R.: Normalizers of solvable spherical subgroups. Math. Notes 94(1–2), 20–31 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bravi P., Luna D.: An introduction to wonderful varieties with many examples of type F4. J. Algebra 329, 4–51 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Batyrev V., Moreau A.: The arc space of horospherical varieties and motivic integration. Compos. Math. 149(8), 1327–1352 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bravi P., Pezzini G.: Wonderful subgroups of reductive groups and spherical systems. J. Algebra 409, 101–147 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Benson C., Ratcliff G.: A classification of multiplicity free actions. J. Algebra 181(1), 152–186 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brion, M.: Variétés spheriques. http://www-fourier.ujf-grenoble.fr/~mbrion/spheriques.pdf

  8. Brion M.: Représentations exceptionnelles des groupes semi-simples. Ann. Sci. Ec. Norm. Sup. 18(2), 345–387 (1985)

    MATH  MathSciNet  Google Scholar 

  9. Brion M.: Vers une généralisation des espaces symétriques. J. Algebra 134(1), 115–143 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brion M.: Sur la géométrie des variétés sphériques. Comment. Math. Helv. 66(2), 237–262 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brion M.: The total coordinate ring of a wonderful variety. J. Algebra 313(1), 61–99 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Camus, R.: Variétés sphériques affines lisses. Thèse de doctorat, Université Joseph Fourier (2001)

  13. Cupit-Foutou, S.: Wonderful varieties: a geometrical realization. arXiv:0907.2852v4

  14. Gagliardi G.: The Cox ring of a spherical embedding. J. Algebra 397, 548–569 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kac V.G.: Some remarks on nilpotent orbits. J. Algebra 64(1), 190–213 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  16. Knop, F.: The Luna-Vust theory of spherical embeddings. In: Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) (Madras), Manoj Prakashan, pp. 225–249 (1991)

  17. Knop F.: Automorphisms, root systems, and compactifications of homogeneous varieties. J. Am. Math. Soc. 9(1), 153–174 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Knop F.: Some remarks on multiplicity free spaces. Representation theory and algebraic geometry. In: Broer, A., Sabidussi, G. (eds.) NATO ASI Series C, vol. 514, pp. 301–317. Kluwer, Dortrecht (1998)

    Google Scholar 

  19. Knop F., Van Steirteghem B.: Classification of smooth affine spherical varieties. Transform. Groups 11(3), 495–516 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Leahy A.S.: A classification of multiplicity free representations. J. Lie Theory 8(2), 367–391 (1998)

    MATH  MathSciNet  Google Scholar 

  21. Losev I.V.: Uniqueness property for spherical homogeneous spaces. Duke Math. J. 147(2), 315–343 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Luna, D.: Slices étales. Sur les groupes algébriques, Soc. Math. France, Paris. Bull. Soc. Math. France, Paris, Mémoire, vol. 33, pp. 81–105 (1973)

  23. Luna, D.: Variétés sphériques de type A. Publ. Math. Inst. Hautes Études Sci., no. 94, pp. 161–226 (2001)

  24. Luna D., Vust Th.: Plongements d’espaces homogènes. Comment. Math. Helv. 58(2), 186–245 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pasquier B.: Variétés horosphériques de Fano. Bull. Soc. Math. Fr. 136(2), 195–225 (2008)

    MATH  MathSciNet  Google Scholar 

  26. Pauer F.: Glatte Einbettungen von G/U. Math. Ann. 262(3), 421–429 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  27. Timashev, D.A.: Homogeneous spaces and equivariant embeddings. In: Encyclopaedia of Mathematical Sciences, Invariant Theory and Algebraic Transformation Groups, 8, vol. 138. Springer, Heidelberg (2011)

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Correspondence to Giuliano Gagliardi.

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Gagliardi, G. A combinatorial smoothness criterion for spherical varieties. manuscripta math. 146, 445–461 (2015). https://doi.org/10.1007/s00229-014-0713-7

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  • DOI: https://doi.org/10.1007/s00229-014-0713-7

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