Abstract
Let \(\mathscr {L} \rightarrow X\) be an ample line bundle over a complex normal projective variety X. We construct a flag \(X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X\) of subvarieties for which the associated Okounkov body for \(\mathscr {L}\) is a rational simplex. In the case when X is a homogeneous surface, and the pseudoeffective cone of X is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone for a generic choice of a flag of subvarieties. Finally, we provide an application to the asymptotic study of group representations.
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Acknowledgments
The author would like to thank Jacopo Gandini and Joachim Hilgert for interesting discussions on Okounkov bodies, as well as Shin-Yao Jow and David Schmitz for helpful comments on an earlier version of this manuscript. Finally, I would like the anonymous referee for useful remarks that helped improve the exposition.
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H. Seppänen supported by the DFG Priority Programme 1388 “Representation Theory”.
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Seppänen, H. Okounkov bodies for ample line bundles with applications to multiplicities for group representations. Beitr Algebra Geom 57, 735–749 (2016). https://doi.org/10.1007/s13366-016-0295-5
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DOI: https://doi.org/10.1007/s13366-016-0295-5