  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 C^2 or in CP^2. 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.
