Let X be a smooth complex projective surface, and let \(\overline{S}\) be the branch curve of a generic projection of \(X\) onto the projective plane. The fundamental group of the complement of \(\overline{S}\) is a projective deformation invariant. The main result of this paper is the computation of a presentation of this group for the Hirzebruch surface \({\mathbb F}_2\) embedded by the linear system \(|2E_0+2C|\): here C is a fiber and \(E_0\) is the zero section.
An important tool for the computation is the regeneration from M. Friedman and M. Teicher [Pure Appl. Math. Q. 4, No. 2, 383–425 (2008; Zbl 1168.14022)].