Tohoku Mathematical Journal

Quadratic vanishing cycles, reduction curves and reduction of the monodromy group of plane curve singularities

Norbert A'Campo

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Abstract

The geometric local monodromy of a plane curve singularity is a diffeomorphism of a compact oriented surface with non empty boundary. The monodromy diffeomorphism is a product of right Dehn twists, where the number of factors is equal to the rank of the first homology of the surface. The core curves of the Dehn twists are quadratic vanishing cycles of the singularity. Moreover, the monodromy diffeomorphism decomposes along reduction curves into pieces, which are invariant, such that the restriction of the monodromy on each piece is isotopic to a diffeomorphism of finite order. In this paper we determine the mutual positions of the core curves of the Dehn twists, which appear in the decomposition of the monodromy, together with the positions of the reduction curves of the monodromy.

Article information

Source
Tohoku Math. J. (2), Volume 53, Number 4 (2001), 533-552.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113247799

Digital Object Identifier
doi:10.2748/tmj/1113247799

Mathematical Reviews number (MathSciNet)
MR1862217

Zentralblatt MATH identifier
1065.14031

Subjects
Primary: 14H20: Singularities, local rings [See also 13Hxx, 14B05]
Secondary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14H50: Plane and space curves 32S40: Monodromy; relations with differential equations and D-modules 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45]

Citation

A'Campo, Norbert. Quadratic vanishing cycles, reduction curves and reduction of the monodromy group of plane curve singularities. Tohoku Math. J. (2) 53 (2001), no. 4, 533--552. doi:10.2748/tmj/1113247799. https://projecteuclid.org/euclid.tmj/1113247799


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References

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