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 +1

Let $f:X\to {\mathbb{P}}^1$ be a non-isotrivial family of semi-stable curves of genus $g\ge 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Ja…