Upper bounds on the slope of certain fibered surfaces

TL;DR

This paper derives upper bounds for the slope of specific fibered surfaces, including cyclic coverings of elliptic and hyperelliptic fibrations, extending and providing new proofs for existing bounds.

Contribution

It establishes slope equalities and new upper bounds for cyclic covering fibrations of elliptic, bielliptic, and hyperelliptic surfaces, enhancing understanding of their geometric properties.

Findings

Upper bounds for cyclic covering fibrations of elliptic surfaces.

A new proof of Xiao's upper bound for hyperelliptic fibrations.

Slope equality for certain fibered surfaces.

Abstract

We establish the slope equality and give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface including bielliptic fibrations of genus greater than 5. We also give an upper bound of the slope for triple cyclic covering fibrations of a ruled surface and hyperelliptic fibrations, which provides a new proof of Xiao's upper bound.