Okounkov bodies for ample line bundles

TL;DR

This paper constructs specific flags of subvarieties on complex projective varieties to demonstrate that the associated Okounkov bodies of ample line bundles are rational polytopes, advancing understanding in algebraic geometry.

Contribution

It introduces a method to construct admissible flags ensuring the Okounkov body of an ample line bundle is a rational polytope, providing new geometric insights.

Findings

Okounkov bodies are rational polytopes for certain flags

Construction of admissible flags on complex projective varieties

Enhancement of geometric understanding of line bundles

Abstract

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a nonsingular complex projective variety $X$. We construct an admissable flag $X_0 \subseteq X_1 \subseteq...\subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope.