On the Arakelov inequality in positive characteristic

摘要

In this note, we generalize the Arakelov inequality in positive characteristic for non-isotrivial semistable families of curves of which are liftable to (resp. W(k)). As a consequence, we give an analogue of Beauville's conjecture in positive characteristic: there are at least 5 singular fibers for non-isotrivial semistable families of curves of over which are liftable to W(k).