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

Abstract

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.