2019/11/21 by Joaquím Roé, Roé, Joaquim, Tomasz Szemberg +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1911.09708
openalex publication_date 2019/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.