Singular fibers and Kodaira dimensions

Abstract

Let \(f:\,X \rightarrow \mathbb {P}^1\) 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 Teichmuller.