On the Fundamental Groups of Non-generic $\mathbb{R}$-Join-Type Curves

Abstract

An $\mathbb{R}$ -join-type curve is a curve in $\mathbb{C}^{2}$ defined by an equation of the form $a \cdot \prod _{j=1}^{\ell}(y -\beta _{j})^{\nu _{j}} = b \cdot \prod _{i=1}^{m}(x -\alpha _{i})^{\lambda _{i}},$ where the coefficients a, b, α _i and β _j are real numbers. For generic values of a and b, the singular locus of the curve consists of the points (α _i , β _j ) with λ _i , ν _j ≥ 2 (so-called inner singularities). In the non-generic case, the inner singularities are not the only ones: the curve may also have “outer” singularities. The fundamental groups of (the complements of) curves having only inner singularities are considered in Oka (J Math Soc Jpn 30:579–597, 1978). In the present paper, we investigate the fundamental groups of a special class of curves possessing outer singularities.

Dedicated to S. Papadima and A. Dimca for their 60th birthday