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A Quick Trip into Local Singularities of Complex Curves and Surfaces

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Introduction to Lipschitz Geometry of Singularities

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2280))

Abstract

In these notes we give a summary of some of the properties of curve and surface singularities needed in the study of Lipschitz geometry of singular varieties. In particular, we describe normalization and resolution processes, and we introduce the concepts of polar curves and exceptional tangents for surfaces.

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References

  1. J.M. Aroca, H. Hironaka, J.L. Vicente, Desingularization Theorems, vol. 30. Memorias de Matemática del Instituto “Jorge Juan” [Mathematical Memoirs of the Jorge Juan Institute] (Consejo Superior de Investigaciones Científicas, Madrid, 1977)

    Google Scholar 

  2. M. Artin, On isolated rational singularities of surfaces. Amer. J. Math. 88, 129–136 (1966)

    Article  MathSciNet  Google Scholar 

  3. R.-O. Buchweitz, G.-M. Greuel, The Milnor number and deformations of complex curve singularities. Invent. Math. 58(3), 241–281 (1980)

    Article  MathSciNet  Google Scholar 

  4. C. Brücker, G.-M. Greuel, Deformationen isolierter Kurvensingularitäten mit eingebetteten Komponenten. Manuscripta Math. 70(1), 93–114 (1990)

    Article  MathSciNet  Google Scholar 

  5. E. Brieskorn, H. Knörrer, Plane algebraic curves, in Modern Birkhäuser Classics (Birkhäuser/Springer Basel AG, Basel, 1986). Translated from the German original by John Stillwell, [2012] reprint of the 1986 edition

    Google Scholar 

  6. R. Bondil, D.T. Lê, Résolution des singularités de surfaces par éclatements normalisés (multiplicité, multiplicité polaire, et singularités minimales), in Trends in Singularities. Trends in Mathematics (Birkhäuser, Basel, 2002), pp. 31–81

    Google Scholar 

  7. L. Birbrair, W.D. Neumann, A. Pichon, The thick-thin decomposition and the bilipschitz classification of normal surface singularities. Acta Math. 212(2), 199–256 (2014)

    Article  MathSciNet  Google Scholar 

  8. W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, in Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4 (Springer, Berlin, 1984)

    Google Scholar 

  9. S.D. Cutkosky, Resolution of singularities, in Graduate Studies in Mathematics, vol. 63 (American Mathematical Society, Providence, 2004)

    Google Scholar 

  10. T. de Jong, G. Pfister, Local analytic geometry, in Advanced Lectures in Mathematics (Friedr. Vieweg & Sohn, Braunschweig, 2000). Basic theory and applications

    Google Scholar 

  11. G.-M. Greuel, C. Lossen, E. Shustin, Introduction to Singularities and Deformations. Springer Monographs in Mathematics (Springer, Berlin, 2007)

    Google Scholar 

  12. G.-M. Greuel, Equisingular and equinormalizable deformations of isolated non-normal singularities. Methods Appl. Anal. 24(2), 215–276 (2017)

    MathSciNet  MATH  Google Scholar 

  13. R. Hartshorne, Algebraic Geometry (Springer, New York, 1977). Graduate Texts in Mathematics, No. 52

    Google Scholar 

  14. H. Hauser, J. Lipman, F. Oort, A. Quirós (eds.), Resolution of Singularities, vol. 181. Progress in Mathematics (Birkhäuser Verlag, Basel, 2000). A research textbook in tribute to Oscar Zariski, Papers from the Working Week held in Obergurgl, September 7–14, 1997

    Google Scholar 

  15. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203 (1964); ibid. (2), 79, 205–326 (1964)

    Google Scholar 

  16. K.-H. Kiyek, J.L. Vicente, Resolution of curve and surface singularities, in Characteristic Zero, Algebra and Applications, vol. 4 (Kluwer Academic Publishers, Dordrecht, 2004), xxii+483 pp.

    Google Scholar 

  17. J. Kollár, Lectures on resolution of singularities, in Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, 2007)

    MATH  Google Scholar 

  18. H.B. Laufer, Normal Two-Dimensional Singularities (Princeton University Press, Princeton; University of Tokyo Press, Tokyo, 1971). Annals of Mathematics Studies, No. 71

    Google Scholar 

  19. D.T. Lê, B. Teissier, Limites d’espaces tangents en géométrie analytique. Comment. Math. Helv. 63(4), 540–578 (1988)

    MathSciNet  MATH  Google Scholar 

  20. D.T. Lê, C. Weber, Résoudre est un jeu d’enfants. Rev. Semin. Iberoam. Mat. Singul. Tordesillas 3(1), 3–23 (2000)

    MathSciNet  Google Scholar 

  21. J. Milnor, Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, No. 61 (Princeton University Press, Princeton; University of Tokyo Press, Tokyo, 1968)

    Google Scholar 

  22. R. Narasimhan, Introduction to the Theory of Analytic Spaces. Lecture Notes in Mathematics, No. 25 (Springer, Berlin, 1966)

    Google Scholar 

  23. F. Pham, B. Teissier, Lipschitz fractions of a complex analytic algebra and Zariski saturation, in Introduction to Lipschitz Geometry of Singularities: Lecture Notes of the International School on Singularity Theory and Lipschitz Geometry, Cuernavaca, June 2018, ed. by W. Neumann, A. Pichon. Lecture Notes in Mathematics, vol. 2280 (Springer, Cham, 2020), pp. 309–337. https://doi.org/10.1007/978-3-030-61807-0_10

  24. P. Popescu-Pampu, Introduction to Jung’s method of resolution of singularities, in Topology of Algebraic Varieties and Singularities. Contemporary Mathematics, vol. 538 (American Mathematical Society, Providence, 2011), pp. 401–432

    Google Scholar 

  25. J. Snoussi, Limites d’espaces tangents à une surface normale. Comment. Math. Helv. 76(1), 61–88 (2001)

    Article  MathSciNet  Google Scholar 

  26. J. Snoussi, The Nash modification and hyperplane sections on surfaces. Bull. Braz. Math. Soc. (N.S.) 36(3), 309–317 (2005)

    Google Scholar 

  27. M. Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations. Ann. Math. (2) 131(3), 411–491 (1990)

    Google Scholar 

  28. B. Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, in Algebraic Geometry (La Rábida, 1981), vol. 961. Lecture Notes in Mathematics (Springer, Berlin, 1982), pp. 314–491

    Google Scholar 

  29. C.T.C. Wall, Singular Points of Plane Curves. London Mathematical Society Student Texts, vol. 63 (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  30. H. Whitney, Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, 1965), pp. 205–244

    Google Scholar 

  31. O. Zariski, The reduction of the singularities of an algebraic surface. Ann. Math. (2) 40, 639–689 (1939)

    Google Scholar 

  32. O. Zariski, P. Samuel, Commutative Algebra, vol. II (Springer, New York, 1975). Reprint of the 1960 edition, Graduate Texts in Mathematics, vol. 29

    Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, Mexico

    Jawad Snoussi

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  1. Jawad Snoussi
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Correspondence to Jawad Snoussi .

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Editors and Affiliations

  1. Department of Mathematics, Barnard College, Columbia University, New York, NY, USA

    Walter Neumann

  2. Aix Marseille Univ CNRS, Centrale Marseille, I2M, Marseille, France

    Anne Pichon

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Snoussi, J. (2020). A Quick Trip into Local Singularities of Complex Curves and Surfaces. In: Neumann, W., Pichon, A. (eds) Introduction to Lipschitz Geometry of Singularities . Lecture Notes in Mathematics, vol 2280. Springer, Cham. https://doi.org/10.1007/978-3-030-61807-0_2

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