On ramified covers of the projective plane I: Segre's theory and classification in small degrees, with Appendix by Eugenii Shustin

TL;DR

This paper explores the classification and properties of ramified covers of the projective plane, focusing on the geometry and geography of branch curves, classical results, and new examples of Zariski pairs, with potential for generalization.

Contribution

It provides a detailed analysis of classical and new results on branch curves, interprets Segre's work in modern terms, and introduces new Zariski pairs, advancing understanding of ramified covers.

Findings

Segre's classical work fully characterizes branch curves for surfaces in P^3.

New Zariski pairs of plane curves are constructed, showing diverse topological types.

Conditions for a curve to be a branch curve are clarified, such as cusps lying on a conic.

Abstract

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 the_geography_ of branch curves among all nodal-cuspidal curves? Second: what is the_geometry_ 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.