Fibers of cyclic covering fibrations of a ruled surface

摘要

We give an algorithm to classify singular fibers of finite cyclic coveringfibrations of a ruled surface by using singularity diagrams. As the firstapplication, we classify all fibers of 3-cyclic covering fibrations of genus 4of a ruled surface and show that the signature of a complex surface with thisfibration is non-positive by computing the local signature for any fiber. Asthe second application, we classify all fibers of hyperelliptic fibrations ofgenus 3 into 12 types according to the Horikawa index. We also prove thatfinite cyclic covering fibrations of a ruled surface have no multiple fibers ifthe degree of the covering is greater than 3.