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

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 $N(2,1)=7$.

arXiv comment10 pages, 1 figure. An error in an earlier version is corrected