Abstract. We study ramified covers of the projective plane P 2. Given a smooth surface S in P n and a generic enough projection P n → P 2, we get a cover π: S → P 2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise: First, What is the geography of branch curves among all cuspidal-nodal curves? And second, what is the geometry of branch curves; i.e., how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e., form a special 0-cycle on the plane. We start with reviewing what is known about the answers to these questions, both simple and some non-trivial results. Secondly, the classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in P 3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. We also review examples of small degree. In addition, the Appendix written by E. Shustin shows the existence of new Zariski pairs.

Abstract. We investigate the problem of existence of degenerations of surfaces in P 3 with ordinary singularities into plane arrangements in general position. Introduction. In the article we investigate degenerations of surfaces in P 3 with ordinary singularities. To begin, consider the classical prototype of this situation, namely, degenerations of plane algebraic curves. As is known, any smooth projective curve can be projected to P 2 onto a nodal curve C — a curve with ordinary