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Transactions of the American Mathematical Society

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Ultrametric properties for valuation spaces of normal surface singularities

Authors: Evelia R. García Barroso, Pedro D. González Pérez, Patrick Popescu-Pampu and Matteo Ruggiero
Journal: Trans. Amer. Math. Soc. 372 (2019), 8423-8475
MSC (2010): Primary 14B05; Secondary 14J17, 32S25
Published electronically: August 1, 2019
MathSciNet review: 4029701
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Abstract: Let $ L$ be a fixed branch - that is, an irreducible germ of curve - on a normal surface singularity $ X$. If $ A,B$ are two other branches, define $ u_{L}(A,B) := \dfrac {(L \cdot A) \> (L \cdot B)}{A \cdot B}$, where $ A \cdot B$ denotes the intersection number of $ A$ and $ B$. Call $ X$ arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of Płoski by proving that whenever $ X$ is arborescent, the function $ u_{L}$ is an ultrametric on the set of branches on $ X$ different from $ L$. In the present paper we prove that, conversely, if $ u_{L}$ is an ultrametric, then $ X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $ X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $ u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $ L$ to be an arbitrary semivaluation on $ X$ and by defining $ u_{L}$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $ X$ is arborescent, and without any restriction on $ X$ we exhibit special subspaces of the space of semivaluations in restriction to which $ u_{L}$ is still an ultrametric.

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Additional Information

Evelia R. García Barroso
Affiliation: Departamento Matemáticas, Estadística e Investigación Operativa, Universidad de La Laguna, 38271 La Laguna Islas Canarias, Spain

Pedro D. González Pérez
Affiliation: Departamento Álgebra, Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain

Patrick Popescu-Pampu
Affiliation: Laboratoire Paul Painlevé, Université de Lille, 59655 Villeneuve-d’Ascq, France

Matteo Ruggiero
Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot - Université de Paris, 75013 Paris, France

Keywords: Arborescent singularity, B-divisor, birational geometry, block, brick, cut-vertex, cyclic element, intersection number, normal surface singularity, semivaluation, tree, ultrametric, valuation
Received by editor(s): December 5, 2018
Received by editor(s) in revised form: March 12, 2019
Published electronically: August 1, 2019
Additional Notes: This research was partially supported by the French grants ANR-12-JS01-0002-01 SUSI, ANR-17-CE40-0023-02 LISA, ANR-17-CE40-0002-01 Fatou, and Labex CEMPI (ANR-11-LABX-0007-01), and also by the Spanish Projects MTM2016-80659-P and MTM2016-76868-C2-1-P
Article copyright: © Copyright 2019 American Mathematical Society