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

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.