  In this paper, we proved that, for a semi-stable fibration of a proper smooth
surface to a proper smooth curve over a filed of positive characteristic, if
the p-rank of the generic fiber is 0, then the base change of the fibration by
a sufficiently many iterative Frobenius morphism of the base curve violates the
semi-positivity theorem. As an application, we suggest a statement on a
distribution of p-ranks of reductions for a certain non-closed point in the
moduli space over a number field.
