On the invariants of base changes of pencils of curves, II

In this part of the series, we shall investigate Deligne-Mumford semistable reductions from the point of view of numerical invariants. As an application, we obtain two numerical criterions for a base change to be stabilizing, and for a fibration to be isotrivial. We also obtain a canonical class inequality for any fibration. Some other applications are presented. Most of the results of this paper have arithmetical analogues. This paper will appear in Math. Z.