Local positivity of linear series on surfaces

TL;DR

This paper explores the local positivity of line bundles on surfaces using Newton-Okounkov polygons, providing geometric characterizations of ample and nef bundles and describing Seshadri constants through convex geometry.

Contribution

It introduces a convex geometric approach to understanding local positivity of line bundles on surfaces, including characterizations and descriptions of Seshadri constants.

Findings

Characterization of ample and nef line bundles via Newton-Okounkov bodies

Description of Seshadri constants using infinitesimal Newton-Okounkov polygons

Reproof of known results on Seshadri constants on surfaces

Abstract

We study asymptotic invariants of linear series on surfaces with the help of Newton-Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton-Okounkov bodies, treating the infinitesimal case as well. One of the main results is a description of moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri constants on surfaces.