Okounkov bodies of finitely generated divisors

TL;DR

This paper proves that the Okounkov body of a big divisor with a finitely generated section ring is a rational simplex, and extends this result to all big divisors on surfaces, with additional conditions ensuring finite generation of the semigroup.

Contribution

It establishes the rational simplex structure of Okounkov bodies for finitely generated divisors and extends results to all big divisors on surfaces under certain conditions.

Findings

Okounkov body of finitely generated divisor is a rational simplex

On surfaces, all big divisors have rational simplex Okounkov bodies

Under certain conditions, the associated semigroup is finitely generated

Abstract

We show that the Okounkov body of a big divisor with finitely generated section ring is a rational simplex, for an appropriate choice of flag; furthermore, when the ambient variety is a surface, the same holds for every big divisor. Under somewhat more restrictive hypotheses, we also show that the corresponding semigroup is finitely generated.