Braid Groups, Algebraic Surfaces and Fundamental Groups of Complements of Branch Curves

An overview of the braid group techniques in the theory of algebraic surfaces from Zariski to the latest results is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, namely B̃n which plays an important role in the description of fundamental groups of complements of branch curves. It turns out that all such groups are “almost solvable” B̃n-groups. Finally, the possible applications to study moduli spaces of surfaces of general type are described and new examples of positive signature spin surfaces whose fundamental groups can be computed using the above algorithm (Galois cover of Hirzebruch surfaces) are presented. 0. Introduction. This manuscript is based on our talk in Santa Cruz, July 1995. It presents the applications of the braid group technique to the study of algebraic surfaces and curves in general and to the moduli space of surfaces and the topology of complements of curves in particular. These techniques started with Enriques, Zariski and Van Kampen in the 30’s (see [VK], [Z]) and were revived by Moishezon in the late 70’s (see, e.g., [Mo1]). The manuscript includes a survey on the topology of complements of branch curve starting with Zariski’s results, as well as new results (related to a quotient B̃n of the braid group) and an open question on the topic. The manuscript is divided as follows: