Minkowski decomposition of Okounkov bodies on surfaces

TL;DR

This paper demonstrates that on certain smooth projective surfaces, Okounkov bodies of big divisors can be decomposed into Minkowski sums of simple geometric shapes, revealing a structured geometric property.

Contribution

It proves a Minkowski decomposition theorem for Okounkov bodies on surfaces with rational polyhedral pseudo-effective cones, linking them to nef divisors.

Findings

Okounkov bodies decompose as Minkowski sums of simplices and segments

Decomposition applies to big divisors with respect to general flags

Results hold on surfaces with rational polyhedral pseudo-effective cones

Abstract

We prove that the Okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo-effective cone is rational polyhedral decomposes as the Minkowski sum of finitely many simplices and line segments arising as Okounkov bodies of nef divisors.