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 $g\ge 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}\ge 5$ if $k=\mathbb{C}$ and $g\ge 5$; we also prove that $s_{nc}\ge 4$ if ${\rm char}~k>0$andtherelativeJacobianof$f$ is non-smooth.