Newton-Okounkov polygons with a small number of vertices and Picard number

TL;DR

This paper investigates the structure of Newton-Okounkov polygons on algebraic surfaces, revealing how their vertices relate to the Picard number and the presence of negative curves, with specific results for elliptic K3 surfaces.

Contribution

It characterizes the maximum vertices of Newton-Okounkov polygons in relation to Picard number and negative curves on surfaces, especially elliptic K3 surfaces.

Findings

mv(S) = 4 iff (S)  2 and no negative curves

mv(S) = 5 iff (S) = 2 and negative curves present

mv(S)  3 does not determine (S) for (S)  3 or more

Abstract

Newton-Okounkov bodies serve as a bridge between algebraic geometry and convex geometry, enabling the application of combinatorial and geometric methods to the study of linear systems on algebraic varieties. This paper contributes to understanding the algebro-geometric information encoded in the collection of all Newton-Okounkov bodies on a given surface, focusing on Newton-Okounkov polygons with few vertices and on elliptic K3 surfaces. Let S be an algebraic surface and mv(S) be the maximum number of vertices of the Newton-Okounkov bodies of S. We prove that mv(S) = 4 if and only if its Picard number \rho(S) is at least 2 and S contains no negative irreducible curve. Additionally, if S contains a negative curve, then \rho(S) = 2 if and only if mv(S) = 5. Furthermore, we provide an example involving two elliptic K3 surfaces to demonstrate that when \rho(S) \geq 3, mv(S) no longer determines the Picard number \rho(S).