On ramified covers of the projective plane I: Interpreting SEGRE's theory (with an appendix by Eugenii Shustin)

Michael Friedman*, Maxim Leyenson, Eugenii Shustin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, 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. The main question that arises is with respect to 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. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. 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. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

Original languageEnglish
Pages (from-to)619-653
Number of pages35
JournalInternational Journal of Mathematics
Volume22
Issue number5
DOIs
StatePublished - May 2011

Keywords

  • Branched covering
  • Zariski pairs
  • adjoint curves
  • singular phase curves

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