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Introduction to Plane Curve Singularities. Toric Resolution Tower and Puiseux Pairs

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Arrangements, Local Systems and Singularities

Part of the book series: Progress in Mathematics ((PM,volume 283))

Abstract

We give an elementary introduction of the toric modification, using an irreducible plane curve germ. We explain also the relation between the tower of the toric modifications which gives a resolution of the curve and the Puiseux pairs.

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References

  1. N. A’Campo. La fonction zeta d’une monodromie. Comm. Math. Helv. 50:539–580, 1975.

    MathSciNet  Google Scholar 

  2. N. A’Campo and M. Oka. Geometry of plane curves via Tchirnhausen resolution tower. Osaka J. Math. 33:1003–1033, 1996.

    MATH  MathSciNet  Google Scholar 

  3. P. Griffiths and J. Harris. Principles of algebraic geometry Wiley Classics Library. John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.

    MATH  Google Scholar 

  4. R. C. Gunning and H. Rossi. Analytic functions of several complex variables. Prentice-Hall Inc., Englewood Cliffs, N.J., 1965.

    MATH  Google Scholar 

  5. A. G. Kouchnirenko. Polyédres de Newton et nombres de Milnor. Invent. Math., 32:1–31, 1976.

    Article  MathSciNet  Google Scholar 

  6. S. Lang. Algebra. Addison-Wesley, Reading, Mass., second edition, 1984.

    MATH  Google Scholar 

  7. J. Milnor. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, N.J., 1965.

    MATH  Google Scholar 

  8. J. Milnor. Singular Points of Complex Hypersurface, volume 61 of Annals Math. Studies. Princeton Univ. Press, 1968.

    Google Scholar 

  9. M. Oka. On the resolution of the hypersurface singularities. In Complex analytic singularities, volume 8 of Adv. Stud. Pure Math., pages 405–436. North-Holland, Amsterdam, 1987.

    Google Scholar 

  10. M. Oka. Geometry of plane curves via toroidal resolution. In Algebraic geometry and singularities (La Rábida, 1991), pages 95–121. Birkhäuser, Basel, 1996.

    Google Scholar 

  11. M. Oka. Non-degenerate complete intersection singularity. Hermann, Paris, 1997.

    MATH  Google Scholar 

  12. L. D. Tráng and M. Oka. On resolution complexity of plane curves. Kodai Math. J., 18(1):1–36, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. N. Varchenko. Zeta-function of monodromy and Newton’s diagram. Invent. Math., 37(3):253–262, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. T. C. Wall. Singular points of plane curves, volume 63 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2004.

    Google Scholar 

  15. J. A. Wolf. Differentiable fibre spaces and mappings compatible with Riemannian metrics. Michigan Math. J., 11:65–70, 1964.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Tokyo University of Science, Wakamiyacho 26, Shinjuku-ku, Tokyo, Japan

    Mutsuo Oka

Authors
  1. Mutsuo Oka
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, Case 7012, 2 place Jussieu, 75251, Paris Cedex 05, France

    Fouad El Zein

  2. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, P.O. Box 11-236, Beirut, Lebanon

    Fouad El Zein

  3. Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    Alexandru I. Suciu

  4. Mathematics Department, Galatasaray University, Ciragan Cad. No 36 Besiktas, 34357, Istanbul, Turkey

    Meral Tosun  & A. Muhammed Uludağ  & 

  5. Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA

    Sergey Yuzvinsky

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© 2009 Birkhäuser Verlag Basel/Switzerland

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Oka, M. (2009). Introduction to Plane Curve Singularities. Toric Resolution Tower and Puiseux Pairs. In: El Zein, F., Suciu, A.I., Tosun, M., Uludağ, A.M., Yuzvinsky, S. (eds) Arrangements, Local Systems and Singularities. Progress in Mathematics, vol 283. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0209-9_9

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