# Does the local Bertini theorem in mixed characteristic imply the global Bertini theorem

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, projective morphism with $X$ a regular scheme. Is it true that a general hypersurface of $H_X$ is a regular scheme, flat over $\mbox{Spec}(R)$?

If I understand correctly, this is true "locally" by a result of Flenner proven in Die Sätze von Bertini für lokale Ringe.

• Note that this type of question is much easier if you replace '$X$ a regular scheme' by '$\pi$ a smooth morphism', because the latter is well-behaved in families. The statement for $\pi$ smooth follows from the case over $\operatorname{Spec} k$, because the smooth locus is open. – R. van Dobben de Bruyn Apr 30 '18 at 18:46
• @R.vanDobbendeBruyn The family $\pi$ is question is not smooth. – Jana Apr 30 '18 at 20:11
• Possibly relevant: arxiv preprint, especially Corollary 5.6 – Axel Stäbler May 2 '18 at 13:18