Hostname: page-component-77f85d65b8-5ngxj Total loading time: 0 Render date: 2026-03-27T01:39:15.444Z Has data issue: false hasContentIssue false
  • English
  • Français

On Lipschitz normally embedded complex surface germs

Part of: Singularities Local theory General commutative ring theory

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work is dedicated to Norbert A'Campo

References

Artin, M., On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136.CrossRefGoogle Scholar
Belotto da Silva, A., Fantini, L. and Pichon, A., Inner geometry of complex surfaces: a valuative approach, Geom. Topol. 26 (2022), 163–219.10.2140/gt.2022.26.163CrossRefGoogle Scholar
Birbrair, L. and Mendes, R., Arc criterion of normal embedding, in Singularities and foliations. Geometry, topology and applications, Springer Proceedings in Mathematics and Statistics, vol. 222 (Springer, Cham, 2018), 549–553.CrossRefGoogle Scholar
Birbrair, L., Neumann, W. D. and Pichon, A., The thick-thin decomposition and the Bilipschitz classification of normal surface singularities, Acta Math. 212 (2014), 199–256.10.1007/s11511-014-0111-8CrossRefGoogle Scholar
Bondil, R., Discriminant of a generic projection of a minimal normal surface singularity, C. R. Math. Acad. Sci. Paris 337 (2003), 195–200.10.1016/S1631-073X(03)00260-7CrossRefGoogle Scholar
Bondil, R., General elements of an m-primary ideal on a normal surface singularity, in Singularités Franco-Japonaises, Séminaires & Congrès, vol. 10 (Société Mathématique de France, Paris, 2005), 11–20.Google Scholar
Bondil, R., Fine polar invariants of minimal singularities of surfaces, J. Singul. 14 (2016), 91–112.Google Scholar
Brieskorn, E. and Knörrer, H., Plane algebraic curves (Birkhäuser, Basel, 1986); translated from the German by John Stillwell.CrossRefGoogle Scholar
Fantini, L., Normalized Berkovich spaces and surface singularities, Trans. Amer. Math. Soc. 370 (2018), 7815–7859.CrossRefGoogle Scholar
Fernandes, A., Topological equivalence of complex curves and bi-Lipschitz homeomorphisms, Michigan Math. J. 51 (2003), 593–606.10.1307/mmj/1070919562CrossRefGoogle Scholar
Fernandes, A. and Sampaio, J. E., Tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. Appendix by Anne Pichon and Walter D. Neumann, Int. Math. Res. Not. IMRN 2019 (2019), 4880–4897.CrossRefGoogle Scholar
Fernández de Bobadilla, J., Heinze, S., Pereira, M. P. and Sampaio, J. E., Moderately discontinuous homology, Commun. Pure Appl. Math. (2021), doi:10.1002/cpa.22013.CrossRefGoogle Scholar
García Barroso, E. R., González Pérez, P. D. and Popescu-Pampu, P., The valuative tree is the projective limit of Eggers–Wall trees, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), 4051–4105.CrossRefGoogle Scholar
Grauert, H., Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368.10.1007/BF01441136CrossRefGoogle Scholar
Gonzalez-Sprinberg, G. and Lejeune-Jalabert, M., Families of smooth curves on surface singularities and wedges, Ann. Polon. Math. 67 (1997), 179–190.10.4064/ap-67-2-179-190CrossRefGoogle Scholar
Hironaka, H., On Nash blowing-up, in Arithmetic and geometry, Vol. II, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, Mass., 1983), 103–111.10.1007/978-1-4757-9286-7_6CrossRefGoogle Scholar
Kerner, D., Pedersen, H. M. and Ruas, M. A. S., Lipschitz normal embeddings in the space of matrices, Math. Z. 290 (2018), 485–507.10.1007/s00209-017-2027-4CrossRefGoogle Scholar
Kollár, J., Toward moduli of singular varieties, Compos. Math. 56 (1985), 369–398.Google Scholar
Laufer, H. B., Normal two-dimensional singularities, Annals of Mathematics Studies, vol. 71 (Princeton University Press; University of Tokyo Press, Princeton, NJ; Tokyo, 1971).Google Scholar
Laufer, H. B., On rational singularities, Amer. J. Math. 94 (1972), 597–608.10.2307/2374639CrossRefGoogle Scholar
Laufer, H. B., On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257–1295.CrossRefGoogle Scholar
Lê, D. T., Geometry of complex surface singularities, in Singularities—Sapporo 1998, Advanced Studies in Pure Mathematics, vol. 29 (Kinokuniya, Tokyo, 2000), 163–180.Google Scholar
Misev, F. and Pichon, A., Lipschitz normal embedding among superisolated singularities, Int. Math. Res. Not. IMRN 2021 (2021), 13546–13569.CrossRefGoogle Scholar
Mostowski, T., Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.) 243 (1985), 46.Google Scholar
Mostowski, T., Tangent cones and Lipschitz stratifications, in Singularities (Warsaw, 1985), Banach Center Publications, vol. 20 (PWN, Warsaw, 1988), 303–322.Google Scholar
Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Études Sci. 9 (1961), 5–22.CrossRefGoogle Scholar
Némethi, A., Five lectures on normal surface singularities, in Low dimensional topology (Eger, 1996/Budapest, 1998), Bolyai Society Mathematical Studies, vol. 8 (János Bolyai Mathematical Society, Budapest, 1999), 269–351, with the assistance of Ágnes Szilárd and Sándor Kovács.Google Scholar
Neumann, W. D., A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–344.10.1090/S0002-9947-1981-0632532-8CrossRefGoogle Scholar
Neumann, W. D. and Pichon, A., Complex analytic realization of links, in Intelligence of low dimensional topology 2006, Knots Everything, vol. 40 (World Scientific, Hackensack, NJ, 2007), 231–238.10.1142/9789812770967_0029CrossRefGoogle Scholar
Neumann, W. D. and Pichon, A., Lipschitz geometry of complex curves, J. Singul. 10 (2014), 225–234.Google Scholar
Neumann, W. D., Pedersen, H. M. and Pichon, A., A characterization of Lipschitz normally embedded surface singularities, J. Lond. Math. Soc. (2) 101 (2020), 612–640.10.1112/jlms.12279CrossRefGoogle Scholar
Neumann, W. D., Pedersen, H. M. and Pichon, A., Minimal surface singularities are Lipschitz normally embedded, J. Lond. Math. Soc. (2) 101 (2020), 641–658.10.1112/jlms.12280CrossRefGoogle Scholar
Parusiński, A., Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier (Grenoble) 38 (1988), 189–213.CrossRefGoogle Scholar
Parusiński, A., Lipschitz stratification of subanalytic sets, Ann. Sci. Éc. Norm. Sup. (4) 27 (1994), 661–696.CrossRefGoogle Scholar
Pham, F. and Teissier, B., Fractions lipschitziennes d'une algèbre analytique complexe et saturation de Zariski. Prépublications Ecole Polytechnique (1969), No. M17.0669.Google Scholar
Pichon, A., An introduction to Lipschitz geometry of complex singularities, in Introduction to Lipschitz geometry of singularities, Lecture Notes in Mathematics, vol. 2280 (Springer, Cham, 2020), 167–216.CrossRefGoogle Scholar
Popescu-Pampu, P., On the invariance of the semigroup of a quasi-ordinary surface singularity, C. R. Math. Acad. Sci. Paris 334 (2002), 1001–1106.CrossRefGoogle Scholar
Popescu-Pampu, P., On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity, Duke Math. J. 124 (2004), 67–104.10.1215/S0012-7094-04-12413-3CrossRefGoogle Scholar
Remmert, R., Local theory of complex spaces, in Several complex variables, VII, Encyclopaedia of Mathematical Sciences, vol. 74 (Springer, Berlin, 1994), 7–96.CrossRefGoogle Scholar
Spivakovsky, M., Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. of Math. (2) 131 (1990), 411–491.10.2307/1971467CrossRefGoogle Scholar
Teissier, B., Résolution simultanée: II - Résolution simultanée et cycles évanescents. Séminaire sur les singularités des surfaces, 1976–1977. Talk:9.Google Scholar
Teissier, B., Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, in Algebraic geometry (La Rábida, 1981), Lecture Notes in Mathematics, vol. 961 (Springer, Berlin, 1982), 314–491.CrossRefGoogle Scholar
Valette, G., Lipschitz triangulations, Illinois J. Math. 49 (2005), 953–979.CrossRefGoogle Scholar
Zariski, O., The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689.10.2307/1968949CrossRefGoogle Scholar