Slopes of Fibered Surfaces with a Finite Cyclic Automorphism

TL;DR

This paper investigates the slopes of finite cyclic covering fibrations on fibered surfaces, establishing bounds and equalities, and analyzing the distribution of global signature contributions.

Contribution

It provides the optimal lower bound, slope equality, and upper bound for slopes of finite cyclic covering fibrations on ruled surfaces, with insights into signature concentration.

Findings

Established the best possible lower bound for slopes.

Derived the slope equality for certain fibrations.

Provided an upper bound for slopes on ruled surfaces.

Abstract

We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.