Newton-Okounkov bodies and Picard numbers on surfaces

Abstract

We study the shapes of all Newton-Okounkov bodies ∆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 ∆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 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.