Newton--Okounkov bodies of curve classes

TL;DR

This paper develops a new theory for Newton--Okounkov bodies associated with curve classes, establishing their volume properties and proposing conjectures on their relations, with some cases proven.

Contribution

It introduces a novel definition of Newton--Okounkov bodies for curve classes and explores their volume properties and conjectural relations.

Findings

Volume of Newton--Okounkov body is a volume-type function of the curve

Conjectured relations between Newton--Okounkov bodies are supported in certain cases

New framework extends Newton--Okounkov body theory to curves

Abstract

The purpose of the paper is to initiate the development of the theory of Newton Okounkov bodies of curve classes. Our denition is based on making a fundamental property of NewtonOkounkov bodies hold also in the curve case: the volume of the NewtonOkounkov body of a curve is a volume-type function of the original curve. This construction allows us to conjecture a new relation between NewtonOkounkov bodies, we prove it in certain cases.