On the additivity of Newton-Okounkov bodies

TL;DR

This paper investigates the additivity property of Newton-Okounkov bodies on two-dimensional subcones of the ample cone, establishing conditions under which they behave additively and deriving related inequalities.

Contribution

It proves the additivity of Newton-Okounkov bodies on certain cones and discusses optimality and applications to intersection number inequalities.

Findings

Additivity holds on two-dimensional subcones of the ample cone.

A necessary condition for additivity is identified.

An inequality between intersection numbers of nef line bundles is derived.

Abstract

We study the additivity of Newton-Okounkov bodies. Our main result states that on two-dimensional subcones of the ample cone the Newto-Okounkov body associated to an appropriate flag acts additively. We prove this by induction relying on the slice formula for Newton-Okounkov bodies. Moreover, we discuss a necessary condition for the additivity showing that our result is optimal in general situations. As an application, we deduce an inequality between intersection numbers of nef line bundles.