Singular Fibers and Kodaira Dimensions

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 $κ(X)\leq κ(F)+1$. If $κ(X)=κ(F)+1$, then $s>\frac{4}m+2$. If $κ(X)\geq 0$, then $s\geq\frac{4}m+2$. In particular, if $m=1$, $s=6$ and $κ(X)=0$, then the family $f$ is Teichmüller.

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