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On Lipschitz normally embedded complex surface germs

Part of: Local theory General commutative ring theory Singularities

Published online by Cambridge University Press:  27 May 2022

André Belotto da Silva ,
Lorenzo Fantini  and
Anne Pichon
André Belotto da Silva
Affiliation:
IMJ-PRG, CNRS 7586, Université de Paris, Institut de Mathématiques de Jussieu Paris Rive Gauche, 75013 Paris, France andre.belotto@imj-prg.fr
Lorenzo Fantini
Affiliation:
Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique and CNRS, Institut Polytechnique de Paris, 91120 Palaiseau, France lorenzo.fantini@polytechnique.edu
Anne Pichon
Affiliation:
Aix Marseille Univ, CNRS, I2M, Marseille, France anne.pichon@univ-amu.fr
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Abstract

We undertake a systematic study of Lipschitz normally embedded normal complex surface germs. We prove, in particular, that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz normally embedded surface singularity.

Keywords

complex surface singularitiesLipschitz geometryLipschitz normal embeddingspolar varietiesdiscriminant varietiesvaluation spaces

MSC classification

Primary: 32S25: Surface and hypersurface singularities
Secondary: 13A18: Valuations and their generalizations 14B05: Singularities

Information

Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 3 , March 2022 , pp. 623 - 653
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

This work is dedicated to Norbert A'Campo

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