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

TL;DR

This paper investigates the fundamental groups of certain real algebraic curves called -join-type curves, focusing on those with outer singularities, extending previous work on curves with only inner singularities.

Contribution

It analyzes the fundamental groups of -join-type curves with outer singularities, providing new insights beyond the case of only inner singularities.

Findings

Fundamental groups of specific -join-type curves with outer singularities are characterized.

The paper extends existing results to non-generic cases with outer singularities.

New methods for computing fundamental groups in these cases are introduced.

Abstract

An \emph{$\mathbb{R}$-join-type curve} is a curve in $\mathbb{C}^2$ defined by an equation of the form \begin{equation*} a\cdot\prod_{j=1}^\ell (y-\beta_j)^{\nu_j} = b\cdot\prod_{i=1}^m (x-\alpha_i)^{\lambda_i}, \end{equation*} where the coefficients $a$, $b$, $\alpha_i$ and $\beta_j$ are \emph{real} numbers. For generic values of $a$ and $b$, the singular locus of the curve consists of the points $(\alpha_i,\beta_j)$ with $\lambda_i,\nu_j\geq 2$ (so-called \emph{inner} singularities). In the non-generic case, the inner singularities are not the only ones: the curve may also have \emph{`outer'} singularities. The fundamental groups of (the complements of) curves having only inner singularities are considered in \cite{O}. In the present paper, we investigate the fundamental groups of a special class of curves possessing outer singularities.