Fano varieties with finitely generated semigroups in the Okounkov body construction

Shin-Yao Jow National Tsing Hua University

mathscidoc:1803.01001

Mathematical Research Letters, 24, (2), 421-426, 2017.3
The Okounkov body is a construction which, to an effective divisor $D$ on an $n$-dimensional algebraic variety $X$, associates a convex body $\Delta(D)$ in the $n$-dimensional Euclidean space $\RR^n$. It may be seen as a generalization of the moment polytope of an ample divisor on a toric variety, and it encodes rich numerical information about the divisor $D$. When constructing the Okounkov body, an intermediate product is a lattice semigroup $\Gamma(D)\subset \NN^{n+1}$, which we will call the Okounkov semigroup. Recently it was discovered that finite generation of the Okounkov semigroup has interesting geometric implication for $X$ regarding toric degenerations and integrable systems, however the finite generation condition is difficult to establish except for some special varieties $X$. In this article, we show that smooth projective Fano varieties of coindex${}\le 2$ have finitely generated Okounkov semigroups, providing the first family of nontrivial higher dimensional examples that are not coming from representation theory. Our result also gives a partial answer to a question of Anderson, K\"uronya, and Lozovanu.
Okounkov body, finitely generated lattice semigroup, Fano variety, del Pezzo variety, toric degeneration, integrable system
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@inproceedings{shin-yao2017fano,
  title={Fano varieties with finitely generated semigroups in the Okounkov body construction},
  author={Shin-Yao Jow},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180307010923824425969},
  booktitle={Mathematical Research Letters},
  volume={24},
  number={2},
  pages={421-426},
  year={2017},
}
Shin-Yao Jow. Fano varieties with finitely generated semigroups in the Okounkov body construction. 2017. Vol. 24. In Mathematical Research Letters. pp.421-426. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180307010923824425969.
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