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.

allpoints, then the inverse image of a general hyperplane is regular. I think this condition is satisfied for closed embeddings. $\endgroup$ – Takumi Murayama Oct 5 '17 at 21:54