Semistable fibrations over $$\mathbb {P}^1$$ P 1 with five singular fibers

TLDR: In this paper, the authors deduce necessary conditions for the number s of singular fibers being 5, provided that the genus of the fibration is greater than or equal to 2 and is at least 6 if the surface is not birationally ruled.

Abstract: Let X be a non-singular, projective surface and $$f: X\rightarrow \mathbb {P}^1$$ a non-isotrivial, semistable fibration defined over $$\mathbb {C}$$ . It is known that the number s of singular fibers must be at least 5, provided that the genus of the fibration is greater than or equal to 2 and is at least 6 if the surface is not birationally ruled. In this paper, we deduce necessary conditions for the number s of singular fibers being 5. Concretely, we prove that if $$s=5$$ , then the condition $$(K_X+F)^2=0$$ holds unless S is rational and $$g\le 17$$ . The proof is based on a “vertical”version of Miyaoka’s inequality and positivity properties of the relative canonical divisor.