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

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 ${\mathbb{P}}^3$. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$, we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in ${\mathbb{P}}^N$ and $E$ to be the image of the double curve of a ${\mathbb{P}}^3$-model of $X$.
In the classical Segre theory, a plane curve $B$ is a branch curve of a smooth surface in ${\mathbb{P}}^3$ iff its 0-cycle of singularities is special with respect to a linear system of plane curves of particular degree. Here we prove that $B$ is a branch curve of a surface in ${\mathbb{P}}^N$ 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 ${\mathbb{P}}^N$.