Osaka Journal of Mathematics

On families of complex curves over $\mathbb{P}^{1}$ with two singular fibers

Cheng Gong, Jun Lu, and Sheng-Li Tan

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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 $g = 2$.

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Osaka J. Math., Volume 53, Number 1 (2016), 83-101.

First available in Project Euclid: 19 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D06: Fibrations, degenerations 14C21: Pencils, nets, webs [See also 53A60] 14H10: Families, moduli (algebraic)


Gong, Cheng; Lu, Jun; Tan, Sheng-Li. On families of complex curves over $\mathbb{P}^{1}$ with two singular fibers. Osaka J. Math. 53 (2016), no. 1, 83--101.

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