On numerical Newton-Okounkov bodies and the existence of Minkowski bases

TL;DR

This paper introduces the concept of numerical Newton-Okounkov bodies for boundary classes of the big cone and demonstrates that varieties with rational polyhedral global Okounkov bodies admit Minkowski bases incorporating these bodies, linking their dimension to the numerical Kodaira dimension.

Contribution

It defines numerical Newton-Okounkov bodies for boundary classes and proves the existence of Minkowski bases under rational polyhedral global Okounkov bodies.

Findings

Numerical Newton-Okounkov bodies are well-defined for boundary classes.

Varieties with rational polyhedral global Okounkov bodies admit Minkowski bases including these bodies.

The dimension of the numerical Newton-Okounkov body equals the numerical Kodaira dimension.

Abstract

Towards the boundary of the big cone, Newton-Okounkov bodies do not vary continuously and in fact the body of a boundary class is not well defined. Using the global Okounkov body one can nonetheless define a numerical invariant, the numerical Newton-Okounkov body. We show that if a normal projective variety has a rational polyhedral global Okounkov body, it admits a Minkowski basis provided one includes numerical Newton-Okounkov bodies above non-big classes. Under the same assumption, we also show that the dimension of the numerical Newton-Okounkov body is the numerical Kodaira dimension.