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Special Fibres and Critical Locus for a Pencil of Plane Curve Singularities

Published online by Cambridge University Press:  04 December 2007

F. Delgado  and
H. Maugendre
F. Delgado
Affiliation:
Universidad de Valladolid, Dpto. Algebra, Geometría y Topología, 47005 Valladolid, Spain. e-mail: [email protected]
H. Maugendre
Affiliation:
Institut Fourier, Université de Grenoble I, BP 74, F -38402 Saint-Martin d′Hères, France. e-mail: [email protected]
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Abstract

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We will analyze the relationships between the special fibres of a pencil Λ of plane curve singularities and the Jacobian curve J of Λ (defined by the zero locus of the Jacobian determinant for any fixed basis ϕϕ′ ∈ Λ). From the results, we find decompositions of J (and of any special fibre of the pencil) in terms of the minimal resolution of Λ. Using these decompositions and the topological type of any generic pair of curves of Λ, we obtain some topological information about J. More precise decompositions for J can be deduced from the minimal embedded resolution of any pair of fibres (not necessarily generic) or from the minimal embedded resolution of all the special fibres.

Keywords

critical locusJacobian curvepencilsplane curve singularitiesspecial fibre.
Type
Research Article
Information
Compositio Mathematica , Volume 136 , Issue 1 , March 2003 , pp. 69 - 87
Copyright
© 2003 Kluwer Academic Publishers