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

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 Θ L ( C ) of 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 Θ L ( C ) into Favre and Jonsson’s valuative tree P ( V ) of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural functions on Θ L ( C ) as pullbacks of other naturally defined functions on P ( V ) . 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 Θ L ′ ( C ) and Θ L ( C ) identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space P ( V ) is the projective limit of Eggers-Wall trees over all choices of curves C . As a supplementary result, we explain how to pass from Θ L ( C ) to an associated splice diagram.