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

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.