TITLE: Simplest example of jumping of cohomology of structure sheaf in smooth families?
QUESTION [20 upvotes]: Using Hodge theory (and the ill-defined Lefschetz principle), one can show that in characteristic 0, given a proper smooth family $X \rightarrow B$, the cohomology groups of the structure sheaf of the fibers are locally constant (as a function on $B$).  I'm aware of the existence of a number of counterexamples to the corresponding statement in positive characteristic.  Given the success of this question, I want to ask:  

What are the simplest examples of a proper smooth family exhibiting jumping of some cohomology group of the structure sheaf?

By the above discussion, such an example will necessarily be in positive characteristic.
By "simplest", I mean by one of the following measures.
(best) An example whose proof is as elementary as possible, and ideally short. 
An example with a simple conceptual underpinning.  (Well, the best answer would do well by both of the first two measures.)
A known example that is simple to state, but may have a complicated proof. (Ideally there should be a reference.)
An expected, folklore, or conjectured example.

REPLY [5 votes]: By the way, for Enriques this more than just an example - it reflects their classification: the moduli space of Enriques surfaces is connected in any characteristic. All such surfaces arise as desingularizations of quotients $Y/G$, where Y is a (possibly singular) complete intersection of $3$ quadrics in $\mathbb{P}^5$, and $G$ is a finite flat group scheme of length $2$.
In characteristic $p\neq2$, all Enriques surfaces satisfy $h^{01}=h^{10}=0$, and their moduli space is irreducible. Also, over an algebraically closed field of characteristic $p\neq2$, the only possibility for $G$ is $\mathbb{Z}/2\mathbb{Z}$, which is isomorphic to $\mu_2$.
In characteristic $p=2$, we have $h^{01}=0$, $h^{10}=1$ if the surface arises as a quotient by $G=\mu_2$ ("classical"), we have $h^{10}=h^{01}=1$ if $G=\alpha_2$ ("supersingular"), and finally, we have $h^{01}=1, h^{10}=0$ if $G=\mathbb{Z}/2\mathbb{Z}$ ("singular"). The moduli space is connected, but has two irreducible components: one corresponds to $\mu_2$-quotients, the other to $\mathbb{Z}/2\mathbb{Z}$-quotients, and their intersection to $\alpha_2$-quotients. 
Thus, the Hodge numbers reflect the position in the moduli space, and a jumping corresponds to a change in type.