Rational nef and anti-nef polytopes are not uniform

TL;DR

This paper provides counterexamples demonstrating that rational nef and anti-nef polytopes lack uniformity even in simple surface pairs, but shows conditions under which nef polytopes are uniform.

Contribution

It answers a question by Chen-Han by providing explicit examples and establishes uniformity of nef polytopes under bounded Cartier indices.

Findings

Rational nef and anti-nef polytopes are not always uniform.

Counterexamples exist even for klt surface pairs.

Nef polytopes are uniform when Cartier indices are bounded.

Abstract

We give two examples which show that rational nef and anti-nef polytopes are not uniform even for klt surface pairs, answering a question of Chen-Han. We also show that rational nef polytopes are uniform when the Cartier indices are uniformly bounded.