Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the fibers are ordinary curves, the Hasse-Witt map $h\colon F^{*}(R^{n}f_{*}(O_{X}))\to R^{n}f_{*}(O_{X})$ is an isomorphism, where $F$ is the Frobenius morphism of $S$ (this result is due to Katz, *Slope filtrations of F-crystals,* in Asterisque '63).

Now, are there cases where the fibers are not ordinary curves but Hasse-Witt map is an isomorphism? Generally, are there reasonable conditions (of course, I mean besides ordinariness!) under which the Hasse-Witt map becomes an isomorphism?

Cohomologie cristalline, un survol, Expositiones Math. (1998). – ACL Dec 14 '11 at 7:05