## 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 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}.

Original language | English |
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Pages (from-to) | 971-996 |

Number of pages | 26 |

Journal | Journal of the European Mathematical Society |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |