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

  • Evelia R. García Barroso
  • Pedro D. González Pérez
  • Patrick Popescu-PampuEmail author
Original Paper


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


Branch Characteristic exponent Contact Eggers-Wall tree Newton-Puiseux series Plane curve singularities Semivaluation Splice diagram Rooted tree Valuation Valuative tree 

Mathematics Subject Classification

14B05 (primary) 32S25 



This research was partially supported by the French grants ANR-12-JS01-0002-125-01 SUSI, ANR-17-CE40-0023-02 LISA and Labex CEMPI (ANR-11-LABX-0007-01), and also by the Spanish grants MTM2016-80659-P, MTM2016-76868-C2-1-P. The authors are grateful to Manuel González Villa and Adrien Poteaux for their interest in this work, which led to the clarification of several parts of it.


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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estadística e I.O. Sección de MatemáticasUniversidad de La LagunaSan Cristóbal de La LagunaSpain
  2. 2.Departmento de Álgebra, Geometría y Topología, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  3. 3.Département de MathsUniversité LilleVilleneuve d’Ascq CedexFrance

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