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arXiv · 1409.6593

On the minimal number of singular fibers with non-compact Jacobians for families of curves over $\mathbb P^1$

Xin Lu·Sheng-Li Tan·Wan-Yuan Xu·Kang Zuo
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Abstract

Let $f:X \to \mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $g\geq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}\geq 5$ if $k=\mathbb C$ and $g\geq 5$; we also prove that $s_{nc}\geq 4$ if ${\rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth.

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Xin Lu, Sheng-Li Tan, Wan-Yuan Xu, Kang Zuo. 2014-09-23. On the minimal number of singular fibers with non-compact Jacobians for families of curves over $\mathbb P^1$. https://arxiv.org/abs/1409.6593

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