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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
  • 56 Downloads

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 (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.

Keywords

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 

Notes

Acknowledgements

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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