Small Lefschetz Fibrations on Simply-Connected $4$-Manifolds

TL;DR

This paper constructs explicit small Lefschetz fibrations on simply-connected 4-manifolds, providing bounds on the minimal number of singular fibers and exploring their properties across different genera.

Contribution

It introduces explicit constructions of nonhyperelliptic and hyperelliptic genus 4 Lefschetz fibrations on exotic simply-connected 4-manifolds, and analyzes minimal singular fiber counts.

Findings

Constructed genus 4 Lefschetz fibrations on exotic 4-manifolds.

Established upper bounds for minimal singular fibers.

Proved minimal number is 18 for hyperelliptic genus 3 fibrations.

Abstract

We consider simply-connected $4$-manifolds admitting Lefschetz fibrations over the $2$-sphere. We explicitly construct nonhyperelliptic and hyperelliptic Lefschetz fibrations of genus $4$ on simply-connected $4$-manifolds which are exotic symplectic $4$-manifolds in the homeomorphism classes of $\mathbb{C}P^{2}\#8\overline{\mathbb{C} P^{2}}$ and $\mathbb{C} P^{2}\#9\overline{\mathbb{C} P^{2}}$, respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to $18$ for $g=3$ when such fibrations are hyperelliptic. Moreover, we discuss these numbers for higher genera.