Torus invariant divisors

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.Comment: 16 pages; 5 pictures; small changes in the layout, further typos remove

Flexible affine cones and flexible coverings

“…By [36, Section 4] (or [31]), every invariant Cartier divisor arises in this way. We have D h ∼ D h if and only if h P − h P is affine linear for every P, i.e.…”

The Cox ring of a complexity-one horospherical variety

“…To do this, we apply [20,Prop. 3.13] to determine the prime T-divisors on X that are described by some extremal rays of tails of elements of S(E ) or by some vertices of non-trivial slices.…”

Families of invariant divisors on rational complexity-one T -varieties

“…We now recall the construction of T -varieties from p-divisors and divisorial fans, see [AH06] and [AHS08], as well as recalling the description of invariant Cartier divisors on complexity-one T -varieties [PS11]. For an introduction to and a survey of the theory of T -varieties, see [AIP + 12].…”

“…Suppose now that |S P | is convex for all P . We now recall the description of invariant Cartier divisors on X(S) from [PS11].…”

“…If K ∈ T-CaDiv ′ (X s ), we can assume (after possible modification with an invariant principal divisor) that it is of the form stated in [PS11,Theorem 3.19]. Coupled with [PS11, Proposition 3.16], we have that K = D h , with h ∈ CaSF ′ (S (s) ) defined as follows:…”