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.
KeywordsLine Bundle Prime Divisor Global Section Torus Action Cartier Divisor
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- [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