On the minimal number of singular fibers with non-compact Jacobians for   families of curves over $\mathbb P^1$
  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. 
Xin Lu, Sheng-Li Tan, Wan-Yuan Xu, Kang Zuo
math.AG