JournalsjemsVol. 14, No. 3pp. 971–996

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

  • Mina Teicher

    Bar-Ilan University, Ramat-Gan, Israel
  • Michael Friedman

    Bar-Ilan University, Ramat-Gan, Israel
  • Rebecca Lehman

    Massachusetts Institute of Technology, Cambridge, USA
  • Maxim Leyenson

    Bar-Ilan University, Ramat-Gan, Israel
On ramified covers of the projective plane II: Generalizing Segre’s theory cover
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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 P3\mathbb P^3. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, BB and EE, we give a necessary and sufficient condition for BB to be the branch curve of a surface XX in PN\mathbb P^N and EE to be the image of the double curve of a P3\mathbb P^3-model of XX.
In the classical Segre theory, a plane curve BB is a branch curve of a smooth surface in P3\mathbb 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 BB is a branch curve of a surface in PN\mathbb P^N iff (part of) the cycle of singularities of the union of BB and EE is special with respect to the linear system of plane curves of a particular low degree. In particular, given just a curve BB, we provide some necessary conditions for BB to be a branch curve of a smooth surface in PN\mathbb P^N.

Cite this article

Mina Teicher, Michael Friedman, Rebecca Lehman, Maxim Leyenson, On ramified covers of the projective plane II: Generalizing Segre’s theory. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 971–996

DOI 10.4171/JEMS/324