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On Sextic Curves with Big Milnor Number

  • Enrique Artal Bartolo
  • Jorge Carmona Ruber
  • José Ignacio Cogolludo Agustín
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

In this work we present an exhaustive description, up to projective isomorphism, of all irreducible sextic curves in ℙ2 having a singular point of type ,\( \mathbb{A}_n ,n \geqslant 15 \) n ≥ 15, only rational singularities and global Milnor number at least 18. Moreover, we develop a method for an explicit construction of sextic curves with at least eight — possibly infinitely near — double points. This method allows us to express such sextic curves in terms of arrangements of curves with lower degrees and it provides a geometric picture of possible deformations. Because of the large number of cases, we have chosen to carry out only a few to give some insights into the general situation.

2000 Mathematics Subject Classification

Primary 14HlO, 14H30, 14D06j Secondary 14Q05, 32S20, 32S50, 14H52 

Key words and phrases

Equisingular family sextic curves deformation fundamental group 

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References

  1. [1]
    E. Artal Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), no. 2, 223–247.MathSciNetzbMATHGoogle Scholar
  2. [2]
    E. Artal Bartolo, J. Carmona, J.I. Cogolludo, and H. Tokunaga, On curves with singular points in special position, J. Knot Theory Ramifications 10 (2001), no. 4, 547–578.MathSciNetCrossRefGoogle Scholar
  3. [3]
    A. I. Degtyarëv, Alexander polynomial of a curve of degree six, J. Knot Theory Ramifications 3 (1994), no. 4, 439–454.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992.CrossRefGoogle Scholar
  5. [5]
    K. Kodaira, On the structure of compact complex analytic surfaces. II, Amer. J. Math. 88 (1966), 682–721.Google Scholar
  6. [6]
    A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), no. 4, 833–851.Google Scholar
  7. [7]
    I. Luengo, On the existence of complete families of projective plane curves, which are obstructed, J. London Math. Soc. (2) 36 (1987), no. 1, 33–43.Google Scholar
  8. [8]
    S. Yu. Orevkov and E. I. Shustin, Flexible - algebraically unrealizable curves: rehabilitation of Hilbert-Rohn-Gudkov approach, Preprint, 2000.Google Scholar
  9. [9]
    D.T. Pho, Classification of singularities on torus curves of type (2, 3), to appear in Kodai Math. J., 2001.Google Scholar
  10. [10]
    D.T. Pho and M. Oka, Fundamental group of sextics of torus type, this Volume.Google Scholar
  11. [11]
    T. Shioda, On the Mordell-Weil lattices,Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240.MathSciNetzbMATHGoogle Scholar
  12. [12]
    H. Tokunaga, Some examples of Zariski pairs arising from certain elliptic K3 surfaces. II. Degtyarev’s conjecture, Math. Z. 230 (1999), no. 2, 389–400.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    J.-G. Yang, Sextic curves with simple singularities,Tohoku Math. J. (2) 48 (1996), no. 2, 203–227.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    H. Yoshihara, On plane rational curves, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 4, 152–155.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929), 305–328.Google Scholar
  16. [16]
    O. Zariski, On the irregularity of cyclic multiple planes, Ann. Math. 32 (1931), 445–489.Google Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Enrique Artal Bartolo
    • 1
  • Jorge Carmona Ruber
    • 2
  • José Ignacio Cogolludo Agustín
    • 3
  1. 1.Departamento de MatemáticasUniversidad de ZaragozaZaragozaSpain
  2. 2.Departamento de Sistemas informáticos y programaciónUniversidad ComplutenseMadridSpain
  3. 3.Departamento de ÁlgebraUniversidad ComplutenseMadridSpain

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