On fundamental groups related to degeneratale surfaces: Conjectures and examples: 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 × Cg, where Cg is a curve of genus g, π1(ℂℙ2- B) is a quotient of an Artin group associated to the degeneration.