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

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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

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

First available in Project Euclid: 18 January 2017

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Zentralblatt MATH identifier

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

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


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.

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