TY - JOUR

T1 - On ramified covers of the projective plane I

T2 - Interpreting SEGRE's theory (with an appendix by Eugenii Shustin)

AU - Friedman, Michael

AU - Leyenson, Maxim

AU - Shustin, Eugenii

N1 - Funding Information:
This work is partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation.

PY - 2011/5

Y1 - 2011/5

N2 - We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

AB - We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

KW - Branched covering

KW - Zariski pairs

KW - adjoint curves

KW - singular phase curves

UR - http://www.scopus.com/inward/record.url?scp=79958033798&partnerID=8YFLogxK

U2 - 10.1142/S0129167X11006945

DO - 10.1142/S0129167X11006945

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AN - SCOPUS:79958033798

SN - 0129-167X

VL - 22

SP - 619

EP - 653

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 5

ER -