Plane Curve Singularities via Divides

Norbert A’Campo () and Pablo Portilla Cuadrado ()
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Norbert A’Campo: University of Basel
Pablo Portilla Cuadrado: Universidad Politécnica de Madrid

Chapter Chapter 1 in Handbook of Geometry and Topology of Singularities VIII, 2026, pp 1-84 from Springer

Abstract: Abstract Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism is explicitly given as a product of Dehn twists. Complex isolated plane curve singularities provide a classical fibered link, the Milnor fibration, with its Milnor monodromy, monodromy group, and vanishing cycles. This surveys puts together much of the work done on divides and their role in the topology of isolated plane curve singularities. We review two complementary approaches for constructing divides: one via embedded resolution techniques and controlled real deformations, and another via Chebyshev polynomials, which yield explicit real morsifications. A combinatorial description of the Milnor fiber is developed, leading to an explicit factorization of the geometric monodromy as a product of right-handed Dehn twists. We further explore the structure of reduction curves that arise from the Nielsen description of quasi-finite mapping classes and from iterated cabling operations on divides. The interplay between the geometric and integral homological monodromies is analyzed, with special attention to symmetries induced by complex conjugation and strong invertibility phenomena. In particular, the integral homological monodromy for isolated plane curve singularities can be computed effectively. In contrast, for complex hypersurface singularities in higher dimensions no method of computation of the integral homology monodromy is known. Connections with mapping class groups, contact and symplectic geometry, and Lefschetz fibrations are also discussed. We conclude by outlining several open problems and conjectures related to the characterization of divides among fibered links, the presentation of geometric monodromy groups, and the existence of symplectic fillings compatible with the natural fibration structures.

Date: 2026
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DOI: 10.1007/978-3-031-99571-2_1

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