Newton-Okounkov bodies and Picard numbers on surfaces

TL;DR

This paper investigates the shapes of Newton-Okounkov bodies on surfaces, establishing bounds related to Picard numbers and proposing a conjecture linking these bodies to the Picard number of the surface.

Contribution

It provides upper bounds and exact characterizations of Newton-Okounkov bodies on surfaces, connecting their geometry to Picard numbers and proving the conjecture for Picard number 1.

Findings

Upper bounds for vertices of Newton-Okounkov bodies in terms of Picard numbers

Exact determination of Newton-Okounkov bodies for certain cases

Proof that Newton-Okounkov bodies determine Picard number 1 surfaces

Abstract

We study the shapes of all Newton-Okounkov bodies $\Delta_{v}(D)$ of a given big divisor $D$ on a surface $S$ with respect to all rank 2 valuations $v$ of $K(S)$. We obtain upper bounds for, and in many cases we determine exactly, the possible numbers of vertices of the bodies $\Delta_{v}(D)$. The upper bounds are expressed in terms of Picard numbers and they are birationally invariant, as they do not depend on the model $\tilde{S}$ where the valuation $v$ becomes a flag valuation. We also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor $D$ determines the Picard number of $S$, and prove that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.