Convex bodies appearing as Okounkov bodies of divisors

By Alex Kuronya, Victor Lozovanu, C. Maclean. 2010

77 citations recorded by the source corpus.

Abstract Based on the work of Okounkov (Okounkov, 1996 [15] , 2003 [16] ), Lazarsfeld and Mustaţă (2009) [13] and Kaveh and Khovanskii (preprint) [10] have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then give a complete characterisation of the set of convex bodies that arise as Okounkov bodies of R -divisors on smooth projective surfaces. Such Okounkov bodies are always polygons, satisfying certain combinatorial criteria. Finally, we construct two examples of non-polyhedral Okounkov bodies. In the first one, the variety we deal with is Fano and the line bundle is ample. In the second one, we find a Mori dream space variety such that under small perturbations of the flag the Okounkov body remains non-polyhedral.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3df6f42c2799d054f26f6cb88b7a211c9e058c1b

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Citations in the corpus

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory (2009, background): “ton polytope and their deﬁnition, in general, is not related with the group action. It is unlikely that one can completely understand the shape of a Newton-Okounkov body in the general situation (see [Kuronya-Lozovanu-Maxlean10] for some results in this direction). In [Kaveh-Khovanskii10a], we retur”
Crystal bases and Newton–Okounkov bodies (2011, background): “ers). The valuations of the form vY• constitute a large class of valuations on C(X) and one often is interested in Newton-Okounkov bodies with respect to such valuations (see for example [LaMu08] and [KLM12]). It is a natural question whether the highest term valuation vw0 can be realized as a valua”
Vanishing sequences and Okounkov bodies (2013, methodology): “As in [17], we can use [6] to obtain the existence of a smooth quartic surface S ⊆ P3 without (−2)-curve (and hence such that every effective divisor is nef) and of two smooth irreducible curves C, C ′ ⊂ S such that C, C ′ and OS(1) generate the Néron-Severi group of S.”
Okounkov bodies and toric degenerations (2010, background): “S-0902967, and also by the Clay Mathematics Institute as a Liftoﬀ Fellow. 1 2 DAVE ANDERSON generated. In fact, the numbers appearing as volumes or coordinates of Okounkov bodies can be quite general [KLM]; they certainly may be irrational. One situation, however, is easy. When X is a smooth toric v”