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Semistable fibrations over \(\mathbb {P}^1\) with five singular fibers. (English) Zbl 1444.14031

Summary: 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.

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14J25 Special surfaces
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References:

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