Generic p-rank for semi-stable fibrations

We show that over a positive characteristic field, the semi-positivity theorem for a semi-stable fibration of a proper smooth surface to a proper smooth curve partially depends on the p-rank of the generic fiber of the fibration. With this result, we can prove that in the moduli space of proper smooth curves over a number field, a certain 1-dimensional point has a reduction of positive p-rank at almost all places. Also we construct a counterexample for Parshin\u27s conjecture concerning the Miyaoka-Yau inequality over a field of positive characteristic