Fundamental group of a class of rational cuspidal curves
Let \(C\) be a rational cuspidal curve of degree \(d\) in \(\mathbb{P}^2_\mathbb{C}\) having exactly three singular points one of which has multiplicity \(d-2\). According to the work of \textit{H. Flenner} and \textit{M. Zaidenberg} [Manuscr. Math. 89, No. 4, 439-459 (1996; Zbl 0868.14014)], these curves are classified as \(C=C_{d,a,b}\), where \((d,a,b) \in\mathbb{Z}^3\), \(d>3\), \(a\geq b>0\), \(a+b =d-2\). In this paper, the authors give an explicit finite presentation of \(\pi_1 (\mathbb{P}^2-C_{d,a,b})\) by generators and relations, which expecially claims that \(\pi_1 (\mathbb{P}^2- C_{d,a,b})\) depends only on \(d\) and \(\text{GCD} (2a+1, 2b+1)\). -- In the corollaries, the authors examine the triples \((d,a,b)\) for which the corresponding fundamental groups are abelian, or admit quotients of finite or infinite triangle groups.
scientific article; zbMATH DE number 1038291