TITLE: Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump
QUESTION [8 upvotes]: Let $\mathbb F$ be an algebraic closure of the field of order $p$. Let $S=\textrm{Spec}(\mathbb F[[z]])$ with special point $s$ and generic point $\eta$. I'm looking for an example of a smooth projective morphism: $X\rightarrow S$
and integers $i,j$ so that $$h^i(X_{\eta},\Omega^j_{X_\eta})<h^i(X_{s},\Omega^j_{X_s}).$$
Note that this can't happen in characteristic $0$ (see e.g. Hodge numbers in a family).
I haven't been able to find such an example referenced in Deligne-lllusie.

REPLY [6 votes]: This is a comment on gdb's gorgeous answer, for people like me who aren't so comfortable with the Picard functor. Let $X$ be a space with a free action by one of the groups schemes $\mathbb{Z}/p$, $\mu_p$ or $\alpha_p$ and let $X \to Y$ be the quotient map. Assume also that $H^0(X, \mathcal{O}) = k$ (our ground field). 
Here is a concrete check that $H^1(Y, \mathcal{O}) \to H^1(X, \mathcal{O})$ is injective in the $\mu_p$ case, but has kernel for $\alpha_p$ and $\mathbb{Z}/p$.
The $\mu_p$ case A $\mu_p$ action on a ring is equivalent to a $(\mathbb{Z}/p)$-grading on that ring, and this passes to localization, so a $\mu_p$ action on $X$ is a $(\mathbb{Z}/p)$-grading $\mathcal{O}_X = \bigoplus \mathcal{O}^j_X$ of the sheaf of rings $\mathcal{O}_X$. The ringed space $Y$ is the same topological space with structure sheaf $\mathcal{O}_Y = \mathcal{O}_X^0$. So $H^1(Y, \mathcal{O}_Y)$ is a direct summand of $H^1(X, \mathcal{O}_X) = \bigoplus H^1(X, \mathcal{O}_X^j)$.
The $\alpha_p$ case An $\alpha_p$ action on a ring is a $p$-nilpotent derivation, and again this passes to localizations, so an $\alpha_p$ action on $X$ is a derivation $D : \mathcal{O}_X \to \mathcal{O}_X$ with $D^p=0$. We have $\mathcal{O}_Y \cong \mathrm{Ker}(D)$. Choose an affine cover $V_i = \mathrm{Spec}(B_i)$ of $Y$ and let $U_i =\mathrm{Spec}(A_i)$ be the preimage cover of $X$. Using that the $\alpha_p$ action is free, for each $i$, choose $x_i \in A_i$ with $D x_i = 1$. Then $D(x_i-x_j)=0$, so $x_i-x_j$ descends to $V_i \cap V_j$ and is a Cech cocycle in $H^1(Y, \mathcal{O})$ which is a coboundary in $H^1(X, \mathcal{O})$. I claim that it is not a coboundary in $H^1(Y, \mathcal{O})$. If it were, so we had $x_i - x_j = f_i - f_j$ with $f_i \in B_i$, then $x_i - f_i$ would glue to a global section $s$ of $\mathcal{O}_X$ with $Ds=1$, contradicting that the only global sections of $\mathcal{O}_X$ are constants.
The $\mathbb{Z}/p$ case (used in Serre's original paper, though not in gdb's answer): A $\mathbb{Z}/p$-action on a ring is an automorphism $\phi$ with $\phi^p = \mathrm{Id}$. So we have such an automorphism of $\mathcal{O}_X$. Put $\Delta(a) = \phi(a) - a$, so $\Delta^p=0$. Again, the freeness of the action lets us find $x_i \in A_i$ with $\Delta(x_i)=1$ (in other words, write $A_i$ as an Artin-Schreier cover $A_i = B_i[x_i]/(x_i^p-x_i-y_i)$ for some $y_i \in B_i$, normalized so that $\phi(x_i) = x_i +1$).  Then $\Delta(x_i-x_j) = 0$ so $x_i-x_j$ descends to a cocycle on $Y$, and the argument goes just as in the previous case (replacing $Ds=1$ with $\Delta(s)=1$).