Newton-Okounkov bodies and Picard numbers on surfaces

We study the shapes of all Newton-Okounkov bodies Δᵥ(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 Δᵥ(D). The upper bounds are expressed in terms of Picard numbers and they are birationally invariant, as they do not depend on the model S̃ 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.