TY - JOUR

T1 - On ramified covers of the projective plane II

T2 - Generalizing Segre's theory

AU - Friedman, M.

AU - Lehman, R.

AU - Leyenson, M.

AU - Teicher, M.

PY - 2012

Y1 - 2012

N2 - 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.

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 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.

UR - http://www.scopus.com/inward/record.url?scp=84860605988&partnerID=8YFLogxK

U2 - 10.4171/JEMS/324

DO - 10.4171/JEMS/324

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AN - SCOPUS:84860605988

SN - 1435-9855

VL - 14

SP - 971

EP - 996

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 3

ER -