Newton–Okounkov bodies sprouting on the valuative tree

© 2016, Springer-Verlag Italia. Given a smooth projective algebraic surface X, a point O∈ X and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely nearO, in the sense that there is a sequence of blowups X′→ X, mapping the smooth, irreducible rational curve E⊂ X′ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case X= P2.