2009/03/19 by Michael Friedman, Michaël Friedman, Friedman, Michael +2
Engineering · Mathematics · #14E20 (Secondary) #14H45 #14H50 #14H51 #14J99 (Primary) #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.AG #msc:14E20 #msc:14H45 #msc:14H50 #msc:14H51 #msc:14J99
paper · pdf · doi:10.48550/arxiv.0903.3359
Minor changes. A subsection on the dimension of the component (in the variety of nodal cuspidal plane curves) which consists of branch curves of smooth surfaces in P^3 was added
openalex publication_date 2009/03/19 · arxiv created 2010/07/31 · arxiv updated 2010/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve 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 thegeography_ of branch curves among all nodal-cuspidal curves? Second: what is thegeometry_ of branch curves? In other words, how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic curve 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 two questions, mentioning both simple and some non-trivial results. We continue with study of classical work of Beniamino Segre which gives a complete answer to the second question in the case when S is a smooth surface in a three-dimensional projective space. We give an interpretation of Segre's work in terms of a study of Picard group of 0-cycles on a singular plane curve B. We also review examples of small degree. The Appendix written by Eugenii Shustin shows the existence of many new Zariski pairs of plane curves. We hope to continue this paper with a generalization to the case of smooth surfaces in a projective space of any dimension.