On families of complex curves over $\mathbb{P}^{1}$ with two singular fibers

Abstract

Let $f\colon S \to \mathbb{P}^{1}$ be a family of genus $g \geq 2$ curves with two singular fibers $F_{1}$ and $F_{2}$. We show that $F_{1} = {F_{2}}^{*}$ and $F_{2} = {F_{1}}^{*}$ are dual to each other, $S$ is a ruled surface, the geometric genera of the singular fibers are equal to the irregularity of the surface, and the virtual Mordell--Weil rank of $f$ is zero. We prove also that $c_{1}^{2}(S) \leq -2$ if $g = 2$, and $c_{1}^{2}(S) \leq -4$ if $g > 2$. As an application, we will classify all such fibrations of genus $g = 2$.