Surface classification and local and global fundamentals groups, I

Given a smooth complex surface S, and a compact connected global normalcrossings divisor $D = \cup_i D_i$, we consider the local fundamental group,i.e., the fundamental group Gamma of T-D, where T is a good tubularneighbourhood of D. One has a surjection of Gamma onto the fundamental group of D, and the kernel$\sK$ is normally generated by geometric loops $\ga_i$ around the curve $D_i$.Among the main results, which are strong generalizations of a well knowntheorem of Mumford, is the nontriviality of $\ga_i$ in the local fundamentalgroup, provided all the curves $D_i$ of genus zero have selfintersection <= -2.(in particular this holds if the canonical divisor is nef on D), and under thetechnical assumption that the dual graph of D is a tree.