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 <a href="http://mathoverflow.net/questions/68421/simplest-examples-of-nonisomorphic-complex-algebraic-varieties-with-isomorphic-an">this question</a>, 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.