Geodesic
On ramified covers of the projective plane II: Generalizing Segre’s theory
Journal of the European Mathematical Society, Tome 14 (2012) no. 3, pp. 971-996.

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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. 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 PN and E to be the image of the double curve of a P3-model of X. In the classical Segre theory, a plane curve B is a branch curve of a smooth surface in P3 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 PN 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 PN.
DOI : 10.4171/jems/324
Classification : 14-XX, 00-XX
Keywords: branch curves, adjoint curves, ramified covers 
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Michael Friedman; Rebecca Lehman; Maxim Leyenson; Mina Teicher. On ramified covers of the projective plane II: Generalizing Segre’s theory. Journal of the European Mathematical Society, Tome 14 (2012) no. 3, pp. 971-996. doi : 10.4171/jems/324. http://geodesic.mathdoc.fr/articles/10.4171/jems/324/

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