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Archiv der Mathematik

, Volume 108, Issue 1, pp 17–27 | Cite as

The Cox ring of a complexity-one horospherical variety

  • Kevin Langlois
  • Ronan TerpereauEmail author
Open Access
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Abstract

Cox rings are intrinsic objects naturally generalizing homogeneous coordinate rings of projective spaces. A complexity-one horospherical variety is a normal variety equipped with a reductive group action whose general orbit is horospherical and of codimension one. In this note, we provide a presentation by generators and relations for the Cox rings of complete rational complexity-one horospherical varieties.

Keywords

Action of algebraic groups Luna–Vust theory 

Mathematics Subject Classification

14L30 14M27 14M25 

References

  1. 1.
    I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings, 144, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2015.Google Scholar
  2. 2.
    Altmann K., Hausen J.: Polyhedral divisors and algebraic torus actions. Math. Ann. 334, 557–607 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Altmann K., Hausen J., Süss H.: Gluing affine torus actions via divisorial fans. Transform. Groups 13, 215–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Altmann K., Kiritchenko V., Petersen L.: Merging divisorial with colored fans. Michigan Math. J. 64, 3–38 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Brion, Lectures on the geometry of flag varieties, In: Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, 33–85.Google Scholar
  6. 6.
    Brion M.: The total coordinate ring of a wonderful variety. J. Algebra 313, 61–99 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cox D. A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4, 17–50 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.Google Scholar
  9. 9.
    Gagliardi G.: The Cox ring of a spherical embedding. J. Algebra 397, 548–569 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Herppich E.: On Fano varieties with torus action of complexity 1. Proc. Edinb. Math. Soc. (2) 57, 737–753 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. Hausen and E. Herppich, Factorially graded rings of complexity one, Torsors, étale homotopy and applications to rational points, 414–428, London Math. Soc. Lecture Note Ser, 405, Cambridge Univ. Press, Cambridge, 2013.Google Scholar
  12. 12.
    Hausen J., Süss H.: The Cox ring of an algebraic variety with torus action. Adv. Math. 225, 977–1012 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Knop F.: Weylgruppe und Momentabbildung. Invent. Math. 99, 1–23 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    F. Knop, The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 225–249, Manoj Prakashan, Madras, 1991.Google Scholar
  15. 15.
    Kempf G. R., Ramanathan A.: Multicones over Schubert varieties. Invent. Math. 87, 353–363 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    K. Langlois, C. Pech, and M. Raibaut, Stringy invariants for horospherical varieties of complexity one, arXiv:1511.03852.
  17. 17.
    Langlois K., Terpereau R.: On the geometry of normal horospherical G-varieties of complexity one. J. Lie Theory 26, 49–78 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Luna D., Vust T.: Plongements d’espaces homogènes. Comment. Math. Helv. 58, 186–245 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    McKernan J.: Mori dream spaces. Jpn. J. Math. 5, 127–151 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Petersen L., Süss H.: Torus invariant divisors. Israel J. Math. 182, 481–504 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sumihiro H.: Equivariant completion. J. Math. Kyoto Univ. 14, 1–28 (1974)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Süss H.: Equivariant Fano threefolds with 2-torus action: a picture book. Doc. Math. 19, 905–940 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Timashev D. A.: Equivariant Classification of G-manifolds of complexity 1. Izv. Ross. Akad. Nauk Ser. Mat. 61, 127–162 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    D. A. Timashev, Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, 138, Invariant Theory and Algebraic Transformation Groups, 8, Springer, Heidelberg, 2011.Google Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematisches Institut, Heinrich Heine UniversitätDüsseldorfGermany
  2. 2.Max Planck Institut für MathematikBonnGermany

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