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

Michael Friedman; Rebecca Lehman; Maxim Leyenson; Mina Teicher

Journal of the European Mathematical Society (2012)

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

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topFriedman, 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 -

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