On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles

TL;DR

This paper establishes bounds on the slope of hyperelliptic Lefschetz fibrations, revealing a key link between the slope and separating vanishing cycles, and improves bounds on their ratio for higher genus cases.

Contribution

It provides sharp bounds for the slope of hyperelliptic Lefschetz fibrations and clarifies the relationship with separating vanishing cycles, including improved bounds on their ratio.

Findings

Slope exceeds 4-4/g iff separating vanishing cycles are present.

Bounds on the ratio s/n are improved for genus g>1.

s <= n for g > 5, refining previous estimates.

Abstract

In this article we find an upper and lower bound for the slope of genus g hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate the strong connection, in general, between the slope of hyperelliptic genus g Lefschetz fibrations and the number of separating vanishing cycles. Specifically, we show that the slope is greater than 4-4/g if and only if the fibration contains separating vanishing cycles. We also improve the existing bound on s/n, the ratio of number of separating vanishing cycles to the number of non-separating vanishing cycles, for hyperelliptic Lefschetz fibrations of genus g>1. In particular we show that s<=n for such fibrations when g>5.