  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.
