On ramified covers of the projective plane II: Generalizing Segre's theory

M. Friedman, R. Lehman, M. Leyenson, M. Teicher

Research output: Contribution to journalArticlepeer-review

Abstract

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in ℙ 3. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, B and E, we give a necessary and sufficient condition for B to be the branch curve of a surface X in ℙ N and E to be the image of the double curve of a ℙ 3-model of X. In the classical Segre theory, a plane curve B is a branch curve of a smooth surface in ℙ 3 iff its 0-cycle of singularities is special with respect to a linear system of plane curves of particular degree. Here we prove that B is a branch curve of a surface in ℙ N iff (part of) the cycle of singularities of the union of B and E is special with respect to the linear system of plane curves of a particular low degree. In particular, given just a curve B, we provide some necessary conditions for B to be a branch curve of a smooth surface in ℙ N.

Original languageEnglish
Pages (from-to)971-996
Number of pages26
JournalJournal of the European Mathematical Society
Volume14
Issue number3
DOIs
StatePublished - 2012
Externally publishedYes

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