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

• Published in: arXiv.org
• Main Authors: García Barroso, Evelia R, González Pérez, Pedro D, Popescu-Pampu, Patrick
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.07.2018
• Subjects: Coordinates; Mathematical analysis; Mathematics - Algebraic Geometry; Multiplication; Theorems; Trees
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1807.02841

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 {\em 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 prove an inversion theorem generalizing the well-known Abhyankar-Zariski inversion theorem concerning 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.