Journal of the Mathematical Society of Japan

On the fundamental groups of non-generic $\mathbb{R}$-join-type curves, II

Christophe EYRAL and Mutsuo OKA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the fundamental groups of (the complements of) plane complex curves defined by equations of the form $f(y)=g(x)$, where $f$ and $g$ are polynomials with real coefficients and real roots (so-called $\mathbb{R}$-join-type curves). For generic (respectively, semi-generic) such polynomials, the groups in question are already considered in [6] (respectively, in [3]). In the present paper, we compute the fundamental groups of $\mathbb{R}$-join-type curves under a simple arithmetic condition on the multiplicities of the roots of $f$ and $g$ without assuming any (semi-)genericity condition.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 241-262.

Dates
First available in Project Euclid: 18 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730025

Digital Object Identifier
doi:10.2969/jmsj/06910241

Mathematical Reviews number (MathSciNet)
MR3597554

Zentralblatt MATH identifier
1368.14041

Subjects
Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]
Secondary: 14H20: Singularities, local rings [See also 13Hxx, 14B05] 14H45: Special curves and curves of low genus 14H50: Plane and space curves

Keywords
plane curves fundamental group bifurcation graph monodromy Zariski–van Kampen's pencil method

Citation

EYRAL, Christophe; OKA, Mutsuo. On the fundamental groups of non-generic $\mathbb{R}$-join-type curves, II. J. Math. Soc. Japan 69 (2017), no. 1, 241--262. doi:10.2969/jmsj/06910241. https://projecteuclid.org/euclid.jmsj/1484730025


Export citation

References

  • C. Eyral and M. Oka, Fundamental groups of join-type sextics via dessins d'enfants, Proc. Lond. Math. Soc. (3), 107 (2013), 76–120.
  • C. Eyral and M. Oka, Fundamental groups of join-type curves\textemdash achievements and perspectives, Proc. Japan Acad. Ser. A Math. Sci., 90 (2014), 43–47.
  • C. Eyral and M. Oka, On the fundamental groups of non-generic $\mathbb{R}$-join-type curves, In: Bridging algebra, geometry, and topology, Springer Proc. Math. Stat., 96, Springer, Cham, 2014, pp. 137–157.
  • C. Eyral and M. Oka, Classification of the fundamental groups of join-type curves of degree seven, J. Math. Soc. Japan, 67 (2015), 663–698.
  • E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math., 55 (1933), 255–260.
  • M. Oka, On the fundamental group of the complement of certain plane curves, J. Math. Soc. Japan, 30 (1978), 579–597.
  • M. Oka, Two transforms of plane curves and their fundamental groups, J. Math. Sci. Univ. Tokyo, 3 (1996), 399–443.
  • M. Oka, A survey on Alexander polynomials of plane curves, In: Singularités Franco–Japonaises, Sémin. Congr., 10, Soc. Math. France, Paris, 2005, pp.,209–232.
  • R. Thom, L'équivalence d'une fonction différentiable et d'un polynôme, Topology, 3 (1965), suppl. 2, 297–307.
  • 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.