Journal of the Mathematical Society of Japan

Classification of the fundamental groups of join-type curves of degree seven

Christophe EYRAL and Mutsuo OKA

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We compute the fundamental groups $\pi_1(\mathbb{P}^2\setminus C)$ for all complex curves $C$ of degree $7$ defined by an equation of the form

$$\prod_{j=1}^\ell (Y-\beta_j Z)^{\nu_j} = c\cdot\prod_{i=1}^m (X-\alpha_i Z)^{\lambda_i},$$

where $\sum_{j=1}^\ell \nu_j=\sum_{i=1}^m \lambda_i$ is the degree of the curve, $c\in\mathbb{R}\setminus \{0\}$, and $\beta_1,\ldots,\beta_\ell$ (respectively $\alpha_1,\ldots,\alpha_m$) mutually distinct real numbers.

Article information

J. Math. Soc. Japan, Volume 67, Number 2 (2015), 663-698.

First available in Project Euclid: 21 April 2015

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Mathematical Reviews number (MathSciNet)

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 Zariski–van Kampen pencil method monodromy


EYRAL, Christophe; OKA, Mutsuo. Classification of the fundamental groups of join-type curves of degree seven. J. Math. Soc. Japan 67 (2015), no. 2, 663--698. doi:10.2969/jmsj/06720663.

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