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January 2016 On families of complex curves over $\mathbb{P}^{1}$ with two singular fibers
Cheng Gong, Jun Lu, Sheng-Li Tan
Osaka J. Math. 53(1): 83-101 (January 2016).

Abstract

Let $f\colon S \to \mathbb{P}^{1}$ be a family of genus $g \geq 2$ curves with two singular fibers $F_{1}$ and $F_{2}$. We show that $F_{1} = {F_{2}}^{*}$ and $F_{2} = {F_{1}}^{*}$ are dual to each other, $S$ is a ruled surface, the geometric genera of the singular fibers are equal to the irregularity of the surface, and the virtual Mordell--Weil rank of $f$ is zero. We prove also that $c_{1}^{2}(S) \leq -2$ if $g = 2$, and $c_{1}^{2}(S) \leq -4$ if $g > 2$. As an application, we will classify all such fibrations of genus $g = 2$.

Citation

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Cheng Gong. Jun Lu. Sheng-Li Tan. "On families of complex curves over $\mathbb{P}^{1}$ with two singular fibers." Osaka J. Math. 53 (1) 83 - 101, January 2016.

Information

Published: January 2016
First available in Project Euclid: 19 February 2016

zbMATH: 1332.14018
MathSciNet: MR3466827

Subjects:
Primary: 14C21 , 14D06 , 14H10

Rights: Copyright © 2016 Osaka University and Osaka City University, Departments of Mathematics

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Vol.53 • No. 1 • January 2016
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