# On the number of vertices of Newton–Okounkov polygons

### Joaquim Roé

Universitat Autònoma de Barcelona, Bellaterra, Spain### Tomasz Szemberg

Pedagogical University Cracow, Poland

## 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 Küronya, Lozovanu and Maclean.

## Cite this article

Joaquim Roé, Tomasz Szemberg, On the number of vertices of Newton–Okounkov polygons. Rev. Mat. Iberoam. 38 (2022), no. 5, pp. 1383–1397

DOI 10.4171/RMI/1369