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

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 113; no. 4; pp. 4051 - 4105
• Main Authors: García Barroso, Evelia R., González Pérez, Pedro D., Popescu-Pampu, Patrick
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2019; Springer Nature B.V; Springer
• Subjects: Applications of Mathematics; Coordinates; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Multiplication; Original Paper; Theoretical; Trees; Plane curve singularities; Valuation; Splice diagram; Eggers-Wall tree; Contact; Valuative tree; Semivaluation; Newton-Puiseux series; 32S25; 14B05 (primary); Branch; Characteristic exponent; Rooted tree
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-019-00646-z

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.