Complex Surfaces and Coverings

This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally intersecting divisors. Applying this to linear arrangements in the complex projective plane, the chapter first blows up the projective plane at non-transverse intersection points, that is, at those points of the arrangement where more than two lines intersect. These points are called singular points of the arrangement. This gives rise to a complex surface and transversely intersecting divisors that contain the proper transforms of the original lines. The chapter also introduces the divisor class group, their intersection numbers, and the canonical divisor class. Finally, it describes the Chern numbers of a complex surface in order to define the proportionality deviation of a complex surface and to study its behavior with respect to finite covers.