Convex bodies appearing as Okounkov bodies of divisors

TL;DR

This paper investigates the set of convex bodies that can be realized as Okounkov bodies of divisors on smooth projective varieties, providing a countability result, a complete characterization for surfaces, and examples of non-polyhedral bodies.

Contribution

It characterizes the set of Okounkov bodies for divisors, showing they are countable, describes all such bodies on surfaces as polygons, and presents examples of non-polyhedral bodies.

Findings

The set of Okounkov bodies on smooth projective varieties is countable.

On surfaces, Okounkov bodies are polygons with specific properties.

Examples of non-polyhedral Okounkov bodies are constructed.

Abstract

Based on the work of Okounkov (\cite{Ok96}, \cite{Ok03}), Lazarsfeld and Musta\c t\u a (\cite{LM08}) and Kaveh and Khovanskii (\cite{KK08}) 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.