On fundamental groups related to the Hirzebruch surface F 1

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ2 or in ℂℙ2. In this article, we show that these groups, for the Hirzebruch surface F 1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces.