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

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.

20 pages, 9 figures