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 . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, and , we give a necessary and sufficient condition for to be the branch curve of a surface in and to be the image of the double curve of a -model of .
In the classical Segre theory, a plane curve is a branch curve of a smooth surface in iff its 0-cycle of singularities is special with respect to a linear system of plane curves of particular degree. Here we prove that is a branch curve of a surface in iff (part of) the cycle of singularities of the union of and is special with respect to the linear system of plane curves of a particular low degree. In particular, given just a curve , we provide some necessary conditions for to be a branch curve of a smooth surface in .
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. 971–996DOI 10.4171/JEMS/324