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The number of singular fibers in hyperelliptic Lefschetz fibrations. (English) Zbl 1464.57030

The paper under review concerns the study of the lower bounds for the minimal number of singular fibres of genus-\(g\) Lefschetz fibrations over the two-sphere.
Let \(M_g\) be the minimal number of singular fibres in all genus-\(g\) hyperelliptic Lefschetz fibrations over the two-sphere, with total space a complex surface (considered as a four-dimensional manifold) and let \(N_g\) be the minimal number of singular fibres in all genus-\(g\) hyperelliptic Lefschetz fibrations over the two-sphere.
The author estimates \(N_g\) for \(4\leq g \leq 10\). A more exhaustive estimate is given for \(M_g\), proving the following: for \(g\) even, if \(g\geq 4\) then \(M_g=2g+4\); for \(g\) odd, if \(g\geq 7\) then \(M_g\geq 2g+6\).

MSC:

57K40 General topology of 4-manifolds
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
20F38 Other groups related to topology or analysis
57K43 Symplectic structures in 4 dimensions
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:

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