Fundamental group of plane curves and related invariants

Abstract

The article under review contains a study of the topology of a pair (P2,C), where C is an algebraic curve in the complex projective plane. The basic problem is to find invariants which are sensitive enough to distinguish many pairs, and for which there is an algorithm for checking this. The homology of the complement is certainly computable in this sense, but it is too coarse to be really useful. The fundamental group of the complement, by contrast, is very sensitive. The article reviews the Zariski-van Kampen method for finding a presentation for it. However, it is not clear whether the isomorphism problem for this class of groups is solvable. The article surveys many other invariants, such as the Alexander polynomial and characteristic varieties, which are more computable. This last set of invariants was introduced, in this context, by A. S. Libgober [in Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), 215–254, Kluwer Acad. Publ., Dordrecht, 2001