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.

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.

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 of $B$). I'm aware of the existence of a number of counterexamples. 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.

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.