In this paper, we are concerned with the relation between the ordinarity of surfaces of general type and the failure of the BMY inequality in positive characteristic. We consider semistable fibrations $\pi:S \longrightarrow C$ where $S$ is a smooth projective surface and $C$ is a smooth projective curve. Using the exact sequence relating the locally exact differential forms on $S$, $C$, and $S/C$, we prove an inequality relating $c_1^2$ and $c_2$ for ordinary surfaces which admit generically ordinary semistable fibrations. This inequality differs from the BMY inequality by a correcting term which vanishes if the fibration is ordinary.
Categories: math.AG