  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.
