Advertisement

Mathematische Zeitschrift

, Volume 255, Issue 4, pp 793–812 | Cite as

Simple immersions of wonderful varieties

  • Guido Pezzini
Article

Abstract

Let G be a semisimple connected linear algebraic group over \({\mathbb{C}}\) , and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample.

Mathematics Subject Classification (2000)

14L30 14M17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahiezer D.N. (1983) Equivariant completions of homogeneous algebraic varieties by homogeneous divisors. Ann. Global Anal. Geom. 1(1): 49–78CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bialynicki-Birula A. (1973) Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bialynicki-Birula A. (1976) Some properties of the decomposition of algebraic varieties determined by actions of a torus. Bull. Acad. Sci. Séri. Sci. Math. Astronom. Phys. 24, 667–674MathSciNetGoogle Scholar
  4. 4.
    Brion M. (1989) Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J. 58(2): 397–424CrossRefMathSciNetGoogle Scholar
  5. 5.
    Brion M. (1989) On spherical varieties of rank one. CMS Conf. Proc. 10, 31–41MathSciNetGoogle Scholar
  6. 6.
    Brion M. (1990) Vers une généralisation des espaces symétriques. J. Algebra 134(1): 115–143CrossRefMathSciNetGoogle Scholar
  7. 7.
    Brion M. (1993) Variétés sphériques et théorie de Mori. Duke Math. J. 72, 369–404CrossRefMathSciNetGoogle Scholar
  8. 8.
    Brion, M.: Variétés sphériques. http://www-fourier.ujf-grenoble.fr/~mbrion/spheriques.psGoogle Scholar
  9. 9.
    De Concini, C., Procesi, C.: Complete symmetric varieties. Invariant theory (Montecatini, 1982), pp. 1–44. Lecture Notes in Math., vol. 996. Springer, Berlin Heidelberg New York (1983)Google Scholar
  10. 10.
    Knop, F., Kraft, H., Luna, D., Vust, T.: Local Properties of Algebraic Group Actions. In: Kraft, H., Slodowy, P., Springer T.: (eds.) Algebraische Transformationsgruppen und Invariantentheorie, pp. 63–76 DMV-Seminar 13. Birkhäuser Verlag (Basel-Boston) (1989)Google Scholar
  11. 11.
    Huckleberry A., Snow D. (1982) Almost-homogeneous Kähler manifolds with hypersurface orbits. Osaka J. Math. 19, 763–786MathSciNetGoogle Scholar
  12. 12.
    Knop F. (1996) Automorphisms, root systems, and compactifications of homogeneous varieties. J. Am. Math. Soc. 9(1): 153–174CrossRefMathSciNetGoogle Scholar
  13. 13.
    Luna D. (1996) Toute variété magnifique est sphérique. Transform. Groups 1(3): 249–258CrossRefMathSciNetGoogle Scholar
  14. 14.
    Luna, D.: Grosses cellules pour les variétés sphériques. Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, pp. 267–280 Cambridge University Press, Cambridge (1997)Google Scholar
  15. 15.
    Luna D. (2001) Variétés sphériques de type A. Inst. Hautes Études Sci. Publ. Math. 94, 161–226MathSciNetGoogle Scholar
  16. 16.
    Luna D. (2002) Sur les plongements de Demazure. J. Algebra 258, 205–215CrossRefMathSciNetGoogle Scholar
  17. 17.
    Wasserman B. (1996) Wonderful varieties of rank two. Transform. Groups 1(4): 375–403CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institut FourierUniversité Joseph FourierSaint-Martin d’HèresFrance
  2. 2.Dipartimento di MatematicaUniversità La SapienzaRomaItaly

Personalised recommendations