Surface classification and local and global fundamental groups

Given a smooth complex surface S, and a compact connected global normal crossing divisor D = boolean OR(i) D-i, we consider the local fundamental group pi(1)(T \ D), where T is a good tubular neighbourhood of D. One has an exact sequence 1 -> K -> Gamma := pi(1)(T - D) -> Pi := pi(1)(D) -> 1, and the kernel K is normally generated by geometric loops gamma(i) around the curve D-i. Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of gamma(i) in Gamma = pi(1)(T - D), provided all the curves D-i of genus zero have self-intersection D-i(2) <= -2 (in particular this holds if the canonical divisor K-S is nef on D), and under the technical assumption that the dual graph of D is a tree.