# 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

## 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 $\mathbb P^N$.

## 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