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

• Volume: 014, Issue: 3, page 971-996
• ISSN: 1435-9855

top

## Abstract

top
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 ${ℙ}^{N}$.

## How to cite

top

Friedman, Michael, et al. "On ramified covers of the projective plane II: Generalizing Segre’s theory." Journal of the European Mathematical Society 014.3 (2012): 971-996. <http://eudml.org/doc/277424>.

@article{Friedman2012,
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 $\mathbb \{P\}^N$.},
author = {Friedman, Michael, Lehman, Rebecca, Leyenson, Maxim, Teicher, Mina},
journal = {Journal of the European Mathematical Society},
keywords = {branch curves; adjoint curves; ramified covers; branch curves; adjoint curves; ramified covers},
language = {eng},
number = {3},
pages = {971-996},
publisher = {European Mathematical Society Publishing House},
title = {On ramified covers of the projective plane II: Generalizing Segre’s theory},
url = {http://eudml.org/doc/277424},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Friedman, Michael
AU - Lehman, Rebecca
AU - Leyenson, Maxim
AU - Teicher, Mina
TI - On ramified covers of the projective plane II: Generalizing Segre’s theory
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 3
SP - 971
EP - 996
AB - 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 $\mathbb {P}^N$.
LA - eng
KW - branch curves; adjoint curves; ramified covers; branch curves; adjoint curves; ramified covers
UR - http://eudml.org/doc/277424
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.