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

TL;DR

The paper investigates Newton-Okounkov bodies on smooth projective surfaces, focusing on the maximal number of vertices mv(S) of these polygons and its relation to the Picard number $\rho(S)$. It proves sharp characterizations for mv(S)=3,4,5: mv(S)=3 occurs exactly when $\rho(S)=1$, mv(S)=4 occurs when $\rho(S)>1$ and there are no negative irreducible curves, and mv(S)=5 occurs when a negative curve is present and $\rho(S)=2$. Extending to elliptic K3 surfaces, the authors show that mv(S) does not determine $\rho(S)$ for $\rho(S)\ge 3$ by constructing a counterexample with two elliptic K3 surfaces having distinct Picard numbers but equal maximal vertex counts. The results connect Newton-Okounkov polygons with lattice-theoretic data from the Néron-Severi group and the Mordell-Weil lattice, illustrating both the discriminating power and the limitations of mv(S) as a Picard-number invariant. The work also situates these findings in the broader conjectural landscape about infinitesimal flags and Picard data, highlighting directions for future refinement of the Newton-Okounkov–Picard correspondence.

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 ρ(S) is at least 2 and S contains no negative irreducible curve. Additionally, if S contains a negative curve, then ρ(S) = 2 if and only if mv(S) = 5. Furthermore, we provide an example involving two elliptic K3 surfaces to demonstrate that when ρ(S) \geq 3, mv(S) no longer determines the Picard number ρ(S).

Figures (1)

• Figure 1: $A_n$ ($n$ vertices)

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: roe2022number Theorem 5.5
• Example 1
• Example 2
• Definition 2.1
• Theorem 3.1
• proof
• Corollary 3.2
• proof