On the number of singular fibers of a semistable fibration: Further Consequences of Tan's Inequality

Abstract. Let f: S − → P 1 be a semistable and non isotrivial fibration, we prove that if S is nonruled and g ≥ 3 then the number s of singular fibers is at least 6. Furthermore, if S is ruled and g is large enough with respect to some topological invariant of S, then we also obtain s ≥ 6. These result are refinements of Theorems by A. Beauville and S-L. Tan. 1.