Towards the classification of semistable fibrations having exactly five singular fibers

Abstract: Let X be a non singular projective surface. Given a semistable non isotrivial fibration f over a smooth rational curve with general fiber non hyperelliptic of genus g bigger than 3, we show that if the number s of singular fibers is 5, then the genus is less or equal to 11, thus improving the previously known bound (g lesser or equal to 17). Furthermore we show that for each possible genus the general fiber has gonality at most 5 and we describe the corresponding fibrations as the resolution of concrete pencils of curves on minimal rational surfaces.