Newton-Okounkov bodies and Picard numbers on 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.

25 pages. Revised version: the proof of Theorem 4.6 (Theorem C) has been rewritten to overcome a gap in (former) Lemma 4.4. Exposition has been improved throughout

Cited by in corpus (1)

• On the valuative Nagata conjecture