Journal of the Mathematical Society of Japan

Irreducible plane sextics with large fundamental groups

Alex DEGTYAREV

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Abstract

We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.

Article information

Source
J. Math. Soc. Japan, Volume 61, Number 4 (2009), 1131-1169.

Dates
First available in Project Euclid: 6 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1257520503

Digital Object Identifier
doi:10.2969/jmsj/06141131

Mathematical Reviews number (MathSciNet)
MR2588507

Zentralblatt MATH identifier
1183.14040

Subjects
Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]
Secondary: 14H45: Special curves and curves of low genus

Keywords
plane sextic torus type fundamental group trigonal curve

Citation

DEGTYAREV, Alex. Irreducible plane sextics with large fundamental groups. J. Math. Soc. Japan 61 (2009), no. 4, 1131--1169. doi:10.2969/jmsj/06141131. https://projecteuclid.org/euclid.jmsj/1257520503


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References

  • V. I. Arnol'd, A. N. Varchenko and S. M. Guseĭn-Zade, Singularities of differentiable maps, I, The classification of critical points, caustics and wave fronts, Nauka, Moscow, 1982 (Russian); English translation: Monographs in Mathematics, 82, Birkhäuser Boston Inc., Boston, MA, 1985.
  • J. I. Cogolludo, Fundamental group for some cuspidal curves, Bull. London Math. Soc., 31 (1999), 136–142.
  • A. Degtyarev, Isotopy classification of complex plane projective curves of degree 5, Algebra i Analis, 1 (1989), 78–101 (Russian); English translation in Leningrad Math. J., 1 (1990), 881–904.
  • A. Degtyarev, Alexander polynomial of a curve of degree six, J. Knot Theory Ramifications, 3 (1994), 439–454.
  • A. Degtyarev, Quintics in $\mbi{C}\mathrm{p}^{2}$ with nonabelian fundamental group, Algebra i Analis, 11 (1999), 130–151; (Russian); English translation in Leningrad Math. J., 11 (2000), 809–826.
  • A. Degtyarev, On deformations of singular plane sextics, J. Algebraic Geom., 17 (2008), 101–135.
  • A. Degtyarev, Oka's conjecture on irreducible plane sextics, J. London Math. Soc., 78 (2008), 329–351.
  • A. Degtyarev, Oka's conjecture on irreducible plane sextics. II, J. Knot Theory Ramifications., to appear.
  • A. Degtyarev, Zariski $k$-plets via dessins d'enfants, Comment. Math. Helv., 84 (2009), 639–671.
  • A. Degtyarev, On irreducible sextics with non-abelian fundamental group, Proceedings of Niigata–Toyama Conferences 2007, Adv. Stud. Pure Math., arXiv:0711.3070, to appear.
  • A. Degtyarev, Fundamental groups of symmetric sextics, J. Math. Kyoto Univ., 48 (2008), 765–792.
  • C. Eyral and M. Oka, Fundamental groups of the complements of certain plane non-tame torus sextics, Topology Appl., 153 (2006), 1705–1721.
  • E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math., 55 (1933), 255–260.
  • W. Magnus, Braid groups: A survey, Lecture Notes in Math., 372, Springer, 1974, pp. 463–487.
  • W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Presentations of groups in terms of generators and relations, Second revised edition, Dover Publications, Inc., New York, 1976.
  • A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J., 49 (1982), 833–851.
  • A. Libgober, Alexander modules of plane algebraic curves, Contemporary Math., 20 (1983), 231–247.
  • V. V. Nikulin, Integer quadratic forms and some of their geometrical applications, Izv. Akad. Nauk SSSR, Ser. Mat., 43 (1979), 111–177 (Russian); English translation in Math. USSR–Izv., 14 (1980), 103–167.
  • M. V. Nori, Zariski conjecture and related problems, Ann. Sci. École Norm. Sup., 4 série, 16 (1983), 305–344.
  • M. Oka, Alexander polynomial of sextics, J. Knot Theory Ramifications, 12 (2003), 619–636.
  • M. Oka, Zariski pairs on sextics, I, Vietnam J. Math., 33 (2005), Special Issue, 81–92.
  • M. Oka and D. T. Pho, Classification of sextics of torus type, Tokyo J. Math., 25 (2002), 399–433.
  • M. Oka and D. T. Pho, Fundamental group of sextics of torus type, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp. 151–180.
  • A. Özgüner, Classical Zariski pairs with nodes, M.Sc. thesis, Bilkent University, 2007.
  • H. Tokunaga, Irreducible plane curves with the Albanese dimension 2, Proc. Amer. Math. Soc., 127 (1999), 1935–1940.
  • H. Tokunaga, A note on triple covers of $\mbi{P}^{2}$, to appear.
  • 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.
  • O. Zariski, The topological discriminant group of a Riemann surface of genus $p$, Amer. J. Math., 59 (1937), 335–358.
  • J.-G. Yang, Sextic curves with simple singularities, Tohoku Math. J. (2), 48 (1996), 203–227.