Constructing Lefschetz Fibrations with Arbitrary Slope

TL;DR

The paper demonstrates the existence of genus-$g$ Lefschetz fibrations over the two-sphere with any slope in the interval (2,8), for sufficiently large genus.

Contribution

It constructs Lefschetz fibrations with arbitrary slopes in (2,8), expanding the known range of such fibrations.

Findings

Existence of genus-$g$ Lefschetz fibrations with slope $r$ for any $r$ in (2,8)

Construction works for sufficiently large genus-$g$

Fills a gap in the classification of Lefschetz fibrations

Abstract

We prove that for any rational number $r\in (2,8)$, there exists a genus-$g$ Lefschetz fibration over the two-sphere with large enough genus-$g$ having the slope is $r$.