Singular Fibers and Kodaira Dimensions

Abstract

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

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