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

摘要

An \emph{$\mathbb{R}$-join-type curve} is a curve in $\mathbb{C}^2$ definedby 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 ofthe curve consists of the points $(\alpha_i,\beta_j)$ with $\lambda_i,\nu_j\geq2$ (so-called \emph{inner} singularities). In the non-generic case, the innersingularities are not the only ones: the curve may also have \emph{`outer'}singularities. The fundamental groups of (the complements of) curves havingonly inner singularities are considered in \cite{O}. In the present paper, weinvestigate the fundamental groups of a special class of curves possessingouter singularities.