algebraic geometry; T-variety; line bundle; Cox ring; Okounkov body
In this dissertation, we focus on the description of equivariant line bundles on complexity-one T-varieties and two applications thereof.
Using the language of polyhedral divisors and divisorial fans developed by Klaus Altmann, Jürgen Hausen and Hendrik Süß, we describe equivariant line bundles in terms of so-called Cartier support functions on the underlying divisorial fan S. Furthermore, we give a precise description of their global sections and provide a vanishing result for cohomology groups of nef line bundles on certain complete rational complexity-one T-varieties. These results are then applied in two different ways.
First, given a Mori dream space TV(S) with free divisor class group we construct a polyhedral divisor on the projective line which corresponds to the Cox ring of TV(S). This polyhedral divisor not only allows for a detailed study of torus orbits and deformations but, in special cases, also for a downgrade to another polyhedral divisor previously constructed with different means by Klaus Altmann and Jarek Wisniewski in the same setting.
The second application lies within the realm of Okounkov bodies. We present a construction of two types of invariant flags and use these to compute Okounkov bodies of rational projective complexity-one T-varieties. In particular, we show that these are rational polytopes. Moreover, using results of Dave Anderson and Nathan Ilten, we exhibit explicit links to degenerations and T-deformations. Finally, we prove that the global Okounkov body of a rational projective complexity-one T-variety with respect to these two types of flags is rational polyhedral. This generalizes an analogous result previously obtained by Jose Gonzalez for projectivized rank two toric vector bundles over smooth projective toric varieties.
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