Let $f:,X to mathbb{P}^1$ be a non-isotrivial semi-stable family of varieties of dimension $m$ over $mathbb{P}^1$ with $s$ singular fibers. Assume that the smooth fibers $F$ are minimal, i.e., their canonical line bundles are semiample. Then $kappa(X)leq kappa(F)+1$. If $kappa(X)=kappa(F)+1$, then $s>frac{4}m+2$. If $kappa(X)geq 0$, then $sgeqfrac{4}m+2$. In particular, if $m=1$, $s=6$ and $kappa(X)=0$, then the family $f$ is Teichmuller.
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