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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Ultrametric properties for valuation spaces of normal surface singularities
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by Evelia R. García Barroso, Pedro D. González Pérez, Patrick Popescu-Pampu and Matteo Ruggiero PDF
Trans. Amer. Math. Soc. 372 (2019), 8423-8475 Request permission

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
  • Email: ergarcia@ull.es
  • Pedro D. González Pérez
  • Affiliation: Departamento Álgebra, Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain
  • Email: pgonzalez@mat.ucm.es
  • Patrick Popescu-Pampu
  • Affiliation: Laboratoire Paul Painlevé, Université de Lille, 59655 Villeneuve-d’Ascq, France
  • MR Author ID: 695298
  • Email: patrick.popescu-pampu@univ-lille.fr
  • Matteo Ruggiero
  • Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot - Université de Paris, 75013 Paris, France
  • MR Author ID: 974797
  • Email: matteo.ruggiero@imj-prg.fr
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8423-8475
  • MSC (2010): Primary 14B05; Secondary 14J17, 32S25
  • DOI: https://doi.org/10.1090/tran/7854
  • MathSciNet review: 4029701