On the minimal number of singular fibers in Lefschetz fibrations over the torus

TL;DR

This paper investigates the minimal number of singular fibers in genus-g Lefschetz fibrations over the torus, establishing lower bounds and exact values for specific genera.

Contribution

It proves that the minimal number of singular fibers is at least 3 and determines exact or near-exact values for various genera.

Findings

N(g,1) ≥ 3 for all g

N(g,1) ∈ {3,4} for g ≥ 3 and g=4

N(2,1) = 7

Abstract

We show that the minimal number of singular fibers $N(g,1)$ in a genus-$g$ Lefschetz fibration over the torus is at least $3$. As an application, we show that $N(g, 1) \in \{ 3, 4\}$ for $g\ge 5$, $N(g, 1) \in \{3, 4,5 \}$ for $g= 3, 4$ and $N(2,1) = 7$.