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

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 -