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](https://mathoverflow.net/questions/88798/hodge-numbers-in-a-family)).


I haven't been able to find such an example referenced in Deligne-lllusie.



Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump
**Update:** The details of this construction are now available in my blog post with Sean Cotner on [Thuses](https://thuses.com/algebraic-geometry/the-torsion-component-of-the-picard-scheme/).


I was recently interested in exactly the same question. But I failed to find any reference where this issue is discussed. So I came up with an example (hopefully correct) of such a family on myself. Presumably, this example is well-known to experts, I followed the footnotes of Grothendieck (page 10) in his paper ["On the de Rham cohomology of algebraic varieties"](http://www.numdam.org/article/PMIHES_1966__29__95_0.pdf) and Serre's example of a projective smooth variety with a failure of Hodge symmetry. Here is an idea of the construction.


Fix a base $S:=\operatorname{Spec} \bar{\mathbf{F}}\_p[T]\_{(T)}$. Then there is a smooth affine group scheme $G' \to S$ such that its generic fibre is isomorphic to $\mathbf G\_m$ and its special fibre is isomorphic to $\mathbf G\_a$ (it is an example [at the end of SGA 3, XVI, paragraph 3](https://webusers.imj-prg.fr/%7Epatrick.polo/SGA3/Expo16.pdf)). Since $G$ is smooth of relative pure dimension $1$, we know that the Frobenius homomorphism $Fr\_{G/S}: G' \to G'^{(p)}$ is finite flat of degree $p$. Consider its kernel $G:=\operatorname{Ker} (Fr\_{G/S})$ -- a finite flat $S$-group scheme of order $p$. We know by the construction that its generic fibre $G\_{\eta}$ is isomorphic to $\mu\_p$ and the special fibre $G\_{s}$ is isomorphic to $\alpha\_p$.


Now the idea is to mimic a construction of Serre (Propostion 16 in Serre's lecture [here](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/mexico.pdf)). We want to find a complete intersection $S$-scheme $X \subset \mathbf P^m\_S$ of relative dimension at least $3$, such that $X$ admits a free $G$-action and such that $Y:=X/G$ is smooth. Then $Y$ will be an example of a family, where hodge numbers jump. Observe that it is projective as a quotient of projective $S$-scheme by a finite flat $S$-group scheme.


Assume for a moment that we have such $X$ and denote the quotient morphism by $h: X \to Y$. Then it is rather easy to show (using Lefschetz hyperplane section for Picard groups, which is explained in detail in [SGA 2, Exp. XII](https://arxiv.org/pdf/math/0511279.pdf)) that $\mathbf{Pic}^{\tau}\_{X/S}$ is a trivial group scheme over $S$. A general descent type argument guarantees that
$$
\mathbf{Pic}^{\tau}\_{Y/S}=\operatorname{Ker} (h^\*: \mathbf{Pic}^{\tau}\_{Y/S} \to \mathbf{Pic}^{\tau}\_{X/S})=\operatorname{Hom}\_{S-gp}(G, \mathbf G\_{m,S})=G^D.
$$


So, we come up with an example of a smooth projective $S$-scheme such that the torsion component of its Picard functor is isomorphic to $\mathbf Z/p =\mu\_p^{D}$ on a generic fibre and isomorphic to $\alpha\_p=\alpha\_p^D$ on a special fibre.


Recall that the Torsion component of the Picard scheme is open in the Picard scheme for any locally projective flat family with geometrically integral fibres (as explained in FGA Explained) and the Lie algebra of the Picard scheme (or equivalently of its torsion component) is given by the first cohomology group of the structure sheaf. Thus
$$
\mathrm{H}^1(Y\_{\eta}, \mathcal O\_{Y\_{\eta}})=\operatorname{Lie}(\mathbf{Pic}\_{Y\_{\eta}/k(\eta)})=\operatorname{Lie}(\mathbf{Pic}^{\tau}\_{Y\_{\eta}/k(\eta)})=\operatorname{Lie}(\mathbf Z/p)=0
$$
and


$$
\mathrm{H}^1(Y\_s, \mathcal O\_{Y\_s})=\operatorname{Lie}(\mathbf{Pic}\_{Y\_{s}/k(s)})=\operatorname{Lie}(\mathbf{Pic}^{\tau}\_{Y\_{s}/k(s)})=\operatorname{Lie}(\alpha\_p)=\bar{\mathbf{F}}\_p.
$$
So, indeed, $h^{0,1}$ jumps in this family.


The last thing that we should address is the construction of such $X$. The idea is to adapt the proof of Serre's [Proposition 15](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/mexico.pdf) for any Noetherian local ring. But before doing this we need to find at least one action of $G$ on some projective space such that a closed locus of non-free action has codimension at least $3$ in the special fibre.


To construct such an action we consider a regular representation of $G$. Namely $G$ acts on $\mathcal O\_G(G)$ and this action is "algebraic", meaning that it defines an action on an affine scheme $V:=\operatorname{Spec\_S}(Sym^{\bullet}\mathcal O\_G(G)^{\vee})$. This induces an action of $G$ on the projective space $\mathbf P\_S(V)$. And I claim that the special fibre has exactly one point, where the action is not free (in a schematic sense). This point corresponds to a line of constants in a space of regular functions of $G$. Once we veriefied this we see that this action satisfies the condition of Serre's Lemma on special fibre for any $p\geq 5$. If $p\leq 3$ then one can take $\mathcal O\_G(G) \otimes \mathcal O\_G(G)$ insead and this will work.


Finally, there is a way to generalize Proposition 16 from Serre's paper in our context. But It will take too much space here. The basic idea is that you do the same things on a special fibre and then lift hyperplanes in an arbitrary way over the whole $S$. This is possible since $S$ is local. I wrote notes for myself a week ago. I can show it to you, if you write me a personal message, but I don't want to make them public (they are not polished at all).


The last thing to say here is that if you want a smooth projective family of **finite type** $\bar{\mathbf F}\_p$-schemes, where hodge numbers jump, you need to use standard spreading out techniques to spread this family over an open subscheme of $\mathbf A^1\_{\bar{\mathbf F}\_p}$.


