On the number of vertices of Newton--Okounkov polygons

TL;DR

This paper investigates the geometric properties of Newton--Okounkov polygons on smooth surfaces, revealing their connection to intersection theory and Picard numbers, and characterizing which polygons can arise as such bodies.

Contribution

It provides a detailed geometric description of Newton--Okounkov polygons for ample divisors on smooth surfaces and links their shape to intersection numbers and Picard numbers.

Findings

Sides of polygons relate to irreducible curves and intersection numbers.

Characterization of possible k-gons as Newton--Okounkov bodies.

Connection between polygon shape and surface Picard number.

Abstract

The Newton--Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton--Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D. As a consequence of our description we determine the numbers k such that D admits some k-gon as a Newton--Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of K\"uronya, Lozovanu and Maclean.