Torus invariant divisors

TL;DR

This paper characterizes invariant divisors on normal varieties with torus actions using polyhedral divisors and divisorial fans, providing explicit descriptions of divisor classes, canonical divisors, and global sections.

Contribution

It introduces a comprehensive framework for describing invariant divisors on such varieties via divisorial fans, extending the understanding of their divisor class groups and line bundle sections.

Findings

Invariant divisors correspond to faces of divisorial fans.

Divisor class groups are explicitly described.

Global sections relate to weight polytopes and the curve Y.

Abstract

Using the language of polyhedral divisors and divisorial fans 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.