Let $X$ be a projective variety over an imperfect (hence infinite and char(k)=p>0) field $k$. If the local rings of $X$ are all regular, then can we say that a general hyperplane section $H$ is also regular? If it helps, you can assume any combination of the following hypothesis on $k$:
$k$ contains a perfect (infinite) subfield $k\_0$ or $k$ is a $F$-finite field or $k$ is a differentially finite over an (infinite) perfect subfield $k\_0$ i.e., $\Omega\_{k/k\_0}$ is a finite dimensional vector space over $k\_0$, or perhaps even the most geometric scenario that $k$ is a finite dimensional function field over an algebraically closed field $k\_0$. 


These type of results are very useful for studying varieties over imperfect fields; in particular studying families in positive characteristic over algebraically closed fields. 


Note that there is a theorem of Seidenberg which says that if $X$ is a normal projective variety of dimension at least $2$ over an infinite field $k$, then a general hyperplane section $H$ of $X$ is irreducible and normal. 



Bertini's type theorems over imperfect fields
I needed to know the answer to this myself, so here is a good reference:



> 
> Hubert Flenner, Liam O’Carroll, and Wolfgang Vogel, *Joins and
>  intersections,* Springer Monographs in Mathematics, Springer-Verlag,
>  Berlin, 1999. MR [1724388](https://mathscinet.ams.org/mathscinet-getitem?mr=1724388) DOI [10.1007/978-3-662-03817-8](https://doi.org/10.1007/978-3-662-03817-8)
> 
> 
> 


The relevant results are the following:


**Theorem 3.4.10.** *Let $X$ be a variety over an infinite field $K$, and let $D$ be a Cartier divisor on $X$. Assume that $\Gamma \subseteq \lvert D \rvert$ is a finite-dimensional linear system that is not composed with a pencil and satisfies $\operatorname{codim} \operatorname{Bs}(\Gamma) \ge 2$. Then, a generic member of $\Gamma$ is irreducible.*


**Corollary 3.4.14.** *Let $X \subseteq \mathbf{P}^n\_K$ be a projective scheme over an infinite field $K$ which is regular (resp. normal, reduced, satisfies $R\_k$). Then, for a generic hyperplane $H \subseteq \mathbf{P}^n\_K$, the intersection $X \cap H$ has the same property.*


