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Splitting of certain singular fibers of genus two. (English) Zbl 1097.14010

This very instructive paper provides a detailed construction of two holomorphic families of compact genus \(2\) Riemann surfaces over the disk previously introduced by the author [in: Topology and Teichmüller spaces, Proc. 37th Taniguchi symp. Katinkulta, Finland, 1995, 123–148 (1996; Zbl 0921.57006)].
The salient feature of the examples is the behaviour of their singular fibers under small real one-parameter perturbations. More precisely, in each family, the single singular fiber splits into several simpler singular fibers that, in fact, do not split any further. Concrete equations are given for the families and their deformations, and the types of the new singular fibers together with their corresponding vanishing cycles are explicitely determined. As a nice consequence, the monodromy of the original singular fiber is decomposed into a product of Dehn twists, thus giving a relation in the genus \(2\) mapping class group.

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
57M50 General geometric structures on low-dimensional manifolds
55R55 Fiberings with singularities in algebraic topology

Citations:

Zbl 0921.57006
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