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Ultrametric Spaces of Branches on Arborescent Singularities

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Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Abstract

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A ⋅ B their intersection number in the sense of Mumford. If L is a fixed branch, we define U L(A, B) = (L ⋅ A)(L ⋅ B)(A ⋅ B)−1 when A ≠ B and U L(A, A) = 0 otherwise. We generalize a theorem of Płoski concerning smooth germs of surfaces, by proving that whenever S is arborescent, then U L is an ultrametric on the set of branches of S different from L. We compute the maximum of U L, which gives an analog of a theorem of Teissier. We show that U L encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both S and L are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of S. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.

This paper is dedicated to Antonio Campillo and Arkadiusz Płoski.

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Acknowledgements

This research was partially supported by the French grant ANR-12-JS01-0002-01 SUSI and Labex CEMPI (ANR-11-LABX-0007-01), and also by the Spanish Projects MTM2016-80659-P, MTM2016-76868-C2-1-P. The third author is grateful to María Angelica Cueto, András Némethi and Dmitry Stepanov for inspiring conversations. We are also grateful to Nicholas Duchon for having sent us his thesis, and to Charles Favre, Mattias Jonsson, András Némethi, Walter Neumann and Matteo Ruggiero for their comments on a previous version of this paper.

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Authors and Affiliations

  1. Departamento de Matemáticas, Estadística e I.O. Sección de Matemáticas, Universidad de La Laguna, La Laguna, Tenerife, España

    Evelia R. García Barroso

  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain

    Pedro D. González Pérez

  3. Facultad de Ciencias Matemáticas, Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Madrid, España

    Pedro D. González Pérez

  4. Département de Mathématiques, Université de Lille, Villeneuve d’Ascq Cedex, France

    Patrick Popescu-Pampu

Authors
  1. Evelia R. García Barroso
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  2. Pedro D. González Pérez
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  3. Patrick Popescu-Pampu
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Correspondence to Patrick Popescu-Pampu .

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Editors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Departamento de Álgebra, Instituto de Matemáticas (IMUS), Universidad de Sevilla, Sevilla, Spain

    Luis Narváez Macarro

  3. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Sebastià Xambó-Descamps

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García Barroso, E.R., González Pérez, P.D., Popescu-Pampu, P. (2018). Ultrametric Spaces of Branches on Arborescent Singularities. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_3

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