Semistable fibrations over P1 with five singular fibers

Let X be a non-singular, projective surface and f : X → P 1 a non-isotrivial, semistable fibration defined over 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 bira...

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Bibliographic Details
Published in:Boletín de la Sociedad Matemática Mexicana 2019-03, Vol.25 (1), p.13-19
Main Authors: Castañeda-Salazar, M., Zamora, A. G.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Original Article
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Summary:Let X be a non-singular, projective surface and f : X → P 1 a non-isotrivial, semistable fibration defined over 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 ≤ 17 . The proof is based on a “vertical”version of Miyaoka’s inequality and positivity properties of the relative canonical divisor.
ISSN:1405-213X
2296-4495
DOI:10.1007/s40590-017-0185-3