Fibers of Cyclic Covering Fibrations of a Ruled Surface

TL;DR

This paper presents an algorithm for classifying singular fibers in cyclic covering fibrations of ruled surfaces, applies it to specific cases, and analyzes the signature and fiber types, revealing structural properties and constraints.

Contribution

It introduces a novel classification algorithm for singular fibers using singularity diagrams and applies it to classify fibers in specific cyclic coverings and hyperelliptic fibrations.

Findings

Classified all fibers of 3-cyclic coverings of genus 4

Determined the non-positivity of the surface signature

Proved no multiple fibers exist for coverings of degree greater than 3

Abstract

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