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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.

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  • $\begingroup$ 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. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 30, 2018 at 18:46
  • $\begingroup$ @R.vanDobbendeBruyn The family $\pi$ is question is not smooth. $\endgroup$
    – Jana
    Commented Apr 30, 2018 at 20:11
  • $\begingroup$ Possibly relevant: arxiv preprint, especially Corollary 5.6 $\endgroup$
    – Axel Stäbler
    Commented May 2, 2018 at 13:18

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