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.
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References
Arakelov, S.J.: Families of algebraic curves with fixed degeneracies. Math. USSR-Izv. 5, 1277–1302 (1971)
Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (1984)
Beauville, A.: Le nombre minimum de fibers singulieres d’un courbe stable sur \(\mathbb{P}^1\). Asterisque 86, 97–108 (1981)
Edge, W.L.: A pencil of four-nodal plane sextics. Math. Proc. Camb. Philos. Soc. 89, 413–421 (1981)
González-Aguilera, V., Rodríguez, R.E.: A pencil in \(\bar{\cal{M}}_6\) with three points at the boundary. Geom. Dedic. 42, 255–265 (1992)
Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann. 268, 159–171 (1984)
Parshin, A.: Algebraic curves over functions fields. Math. USSR-Izv. 2, 1145–1170 (1968)
Tan, S.-L.: The minimal number of singular fibers of a semi stable curve over \(\mathbb{P}^{1}\). J. Algebr. Geom. 4, 591–596 (1995)
Tan, S.-L., Tu, Y., Zamora, A.G.: On complex surfaces with \(5\) or \(6\) semistable singular fibers over \(\mathbb{P}^1\). Math. Z. 249, 427–438 (2005)
Vojta, P.: Diophantine Inequalities and Arakelov Theory. Appendix to Introduction to Arakelov Theory by S. Lang. Springer, Berlin (1988)
Zamora, A.G.: Some remarks on the Wiman-Edge Pencil. Proc. Edinb. Math. Soc. (in press)
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This paper was partially supported by CONACyT Grant 257079. The first author was supported by Conacyt Doctoral Scholarships. The second author was partially supported by Conacyt Grant 265621 for an Academic Sabbatical year.
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Castañeda-Salazar, M., Zamora, A.G. Semistable fibrations over \(\mathbb {P}^1\) with five singular fibers. Bol. Soc. Mat. Mex. 25, 13–19 (2019). https://doi.org/10.1007/s40590-017-0185-3
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DOI: https://doi.org/10.1007/s40590-017-0185-3