  Given a smooth complex surface S, and a compact connected global normal
crossings 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 tubular
neighbourhood 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 known
theorem of Mumford, is the nontriviality of $\ga_i$ in the local fundamental
group, 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 the
technical assumption that the dual graph of D is a tree.
