A new counterexample to Nguyen's conjecture on surface fibration

Abstract

Suppose $f:S\rightarrow\mathbb{P}^1$ is a surface fibration of genus $g$ with $3$ singular fibers. If two of the singular fibers are semistable, Nguyen conjectured that $f$ does not exist for $g\ge2$. However, a counterexample for $g=2$ was discovered by Gong-Lu-Tan. Note that such kind of surface fibrations admit strong arithmetic properties but are rare in fact, and as such the counterexamples are important. In this paper, we construct a new one for $g=2$.