The valuative tree is the projective limit of Eggers-Wall trees

Abstract

Consider a germ C of reduced curve on a smooth germ S of complex analytic surface. Assume that C contains a smooth branch L. Using the Newton-Puiseux series of C relative to any coordinate system (x, y) on S such that L is the y-axis, one may define the Eggers-Wall tree of ΘL (C) relative to L. Its ends are labeled by the branches of C and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically ΘL (C) into Favre and Jonsson’s valuative tree P (ν) of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural functions on ΘL (C) as pullbacks of other naturally defined functions on P (ν). As a consequence, we generalize the well-known inversion theorem for one branch: if L' is a second smooth branch of C, then the valuative embeddings of the Eggers-Wall trees ΘL' (C) and ΘL (C) identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space P (ν) is the projective limit of Eggers-Wall trees over all choices of curves C. As a supplementary result, we explain how to pass from ΘL (C) to an associated splice diagram.