Okounkov bodies on projectivizations of rank two toric vector bundles

TL;DR

This paper explicitly describes the global Okounkov body of projectivizations of rank two toric vector bundles, showing it is a rational polyhedral cone and that these spaces are Mori dream spaces.

Contribution

It provides an explicit description of the Okounkov body for rank two toric vector bundles using Klyachko filtrations, establishing their rational polyhedral nature and Mori dream space status.

Findings

Okounkov body is a rational polyhedral cone

P(E) is a Mori dream space

Explicit description via Klyachko filtrations

Abstract

The global Okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety. In the case of a rank two toric vector bundle E on a smooth projective toric variety, we use its Klyachko filtrations to give an explicit description of the global Okounkov body of P(E). In particular, we show that this is a rational polyhedral cone and that P(E) is a Mori dream space.