On fundamental groups related to degeneratale surfaces: Conjectures and examples

We argue that for a smooth surface S, considered as a ramified cover over ℂℙ², branched over a nodal-cuspidal curve B ⊂ ℂℙ², one could use the structure of the fundamental group of the complement of the branch curve π₂(ℂℙ²- B) to understand other properties of the surface and its degeneration and vice-versa. In this paper, we look at embedded-degeneratable surfaces - a class of surfaces admitting a planar degeneration with a few combinatorial conditions imposed on its degeneration. We close a conjecture of Teicher on the virtual solvability of π₁ (ℂℙ²- B) for these surfaces and present two new conjectures on the structure of this group, regarding non-embedded-degeneratable surfaces. We prove two theorems supporting our conjectures, and show that for ℂℙ¹ × C_g, where C_g is a curve of genus g, π₁(ℂℙ²- B) is a quotient of an Artin group associated to the degeneration.