Israel Journal of Mathematics

, Volume 182, Issue 1, pp 481–504 | Cite as

Torus invariant divisors

  • Lars PetersenEmail author
  • Hendrik Süss


Using the language of Altmann, Hausen and Süß, we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture, X is given by a divisorial fan on a smooth projective curve Y. Cartier divisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one-dimensional faces of it. Furthermore, we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(D h ) will be determined by a subset of a weight polytope associated to h, and global sections of specific line bundles on the underlying curve Y.


Line Bundle Prime Divisor Global Section Torus Action Cartier Divisor 
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  1. [AH06]
    K. Altmann and J. Hausen, Polyhedral divisors and algebraic torus actions, Mathematische Annalen 334 (2006), 557–607.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [AHS08]
    K. Altmann, J. Hausen and H. Süß, Gluing affine torus actions via divisorial fans, Transformation Groups 13 (2008), 215–242.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [Dem01]
    J. -P. Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect, Notes, Abdus Salam Int. Cent. Theoret. Phys. 6, Trieste, 2001, pp. 1–148.Google Scholar
  4. [FZ03]
    H. Flenner and M. Zaidenberg, Normal affine surfaces with ℂ*-actions, Osaka Journal of Mathematics 40 (2003), 981–1009.zbMATHMathSciNetGoogle Scholar
  5. [KKMB73]
    G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Mathematics 339, Springer-Verlag, New York, 1973.zbMATHGoogle Scholar
  6. [Lie08]
    A. Liendo, Affine T-varieties of complexity one and locally nilpotent derivations, Trasformation Group 15 (2010), 389–425.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [OW77]
    P. Orlik and P. Wagreich, Algebraic surfaces with k*-action, Acta Mathematica 138 (1977), 43–81.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Tim97]
    D. A. Timashev, Classification of G-varieties of complexity 1 Math. USSR-Izv., 61 (1997), 363–397.zbMATHMathSciNetGoogle Scholar
  9. [Tim00]
    D. A. Timashev, Cartier divisors and geometry of normal G-varieties, Transformation Groups 5 (2000), 181–204.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Institut für Mathematik und InformatikFreie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, LS Algebra und GeometrieBrandenburgische Technische Universität CottbusCottbusGermany

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