Newton-Okounkov bodies and Picard numbers on surfaces
Julio José Moyano-Fernández, Matthias Nickel, Joaquim Roé
TL;DR
The paper investigates the shapes of Newton--Okounkov bodies $\Delta_v(D)$ for big divisors $D$ on surfaces across all rank-2 valuations $v$, establishing a birationally invariant upper bound $2\rho_D(S)+2$ on the number of vertices and showing sharpness when $\rho_D(S)$ is large, with $\rho_D(S)$ being the Picard number relative to $D$. It extends vertex-count results to big and nef divisors with $D$-positive flags, introducing $mv(D)$ (and $mv_D$) to quantify achievable vertex counts and proving that all counts $3\le k\le mv(D)$ occur. The work also analyzes $D$-orthogonal flags and reveals greater flexibility (via $mv^{\mathrm{Null}}(D)$) and noninvariance phenomena, followed by an infinitesimal-flag analysis that distinguishes Picard number $1$ surfaces (where all $\Delta_v(D)$ have at most four vertices) from higher Picard numbers, where five or more vertices can occur. Together, these results connect convex-geometric properties of Newton--Okounkov bodies with birational geometry and Picard data, suggesting that Newton--Okounkov bodies encode the Picard number of the surface.
Abstract
We study the shapes of all Newton-Okounkov bodies $Δ_{v}(D)$ 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 $Δ_{v}(D)$. 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.