Global Okounkov bodies for Bott-Samelson varieties

TL;DR

This paper proves that the global Okounkov bodies of Bott-Samelson and related Mori dream spaces are rational polyhedral, enabling combinatorial analysis of line bundle sections over Schubert and flag varieties.

Contribution

It establishes the rational polyhedrality of global Okounkov bodies for a broad class of Mori dream spaces, including Bott-Samelson and flag varieties.

Findings

Okounkov bodies of Schubert varieties are rational polyhedral

Global Okounkov body of flag variety G/B is rational polyhedral

Asymptotic dimensions of weight spaces relate to lattice point counting in polytopes

Abstract

We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. In fact, we prove more generally that this holds for any Mori dream space which admits a flag of Mori dream spaces satisfying a certain regularity condition. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety $G/B$ is rational polyhedral. As an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the counting of lattice points in polytopes.