Towards the Classification of Semistable Fibrations Having Exactly Five Singular Fibers

Abstract

Let X be a nonsingular complex projective surface. Given a semistable non isotrivial fibration \(f: X \rightarrow \mathbb {P}^{1}\) with general non-hyperelliptic fiber of genus \(g\ge 4\), we show that, if the number of singular fibers is 5, then \(g\le 11\), thus improving the previously known bound \(g\le 17\). Furthermore, we show that, for each possible genus, the general fiber has gonality at most 5. The corresponding fibrations are described as the resolution of concrete pencils of curves on minimal rational surfaces.