Arbitrarily large jumps in the de Rham and Hodge cohomology of families in characteristic $p$

We construct smooth projective families of algebraic varieties in characteristic $p$ such that the dimensions of the de Rham and Hodge cohomology groups of the fibers can be made to jump by an arbitrarily large amount. To do this, we first construct an example using the classifying stack of a finite flat group scheme which degenerates $\mathbb{Z}\mathrm{/}{p}^2\mathbb{Z}$ to ${\alpha }_p\oplus {\alpha }_p$. Along the way, we give a self-contained exposition of the construction of Godeaux–Serre varieties.