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

  1. González Pérez, Pedro D.
  2. García Barroso, Evelia R.
  3. Popescu-Pampu, Patrick
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

  2. 2 Universidad Complutense de Madrid
    info

    Universidad Complutense de Madrid

    Madrid, España

    ROR 02p0gd045

Journal:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

ISSN: 1578-7303,  1579-1505

Year of publication: 2019

Volume: 113

Issue: 4

Pages: 4051-4105

Type: Article

DOI: 10.1007/S13398-019-00646-Z GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

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 treeΘL(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 ΘL(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 ΘL(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 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(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 ΘL(C) to an associated splice diagram.

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