Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular codimension 1 subvariety?

I think the full version of the Bertini theorem is not known and requires assumption of some conjectures including abc conjecture (At least over $\mathbb{Z}$). I was wondering whether the existence of just one such a hypersurface is known or not. Thanks!

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    $\begingroup$ Is the very ample line bundle fixed or is it allowed to vary? $\endgroup$ – Piotr Achinger Apr 15 at 2:45
  • $\begingroup$ I'm not sure about your question but I assume you are asking whether $X$ is embedded in some $\mathbb{P}^n$ for some fixed embedding or you can embed it differently. If that's the case then yes it is allowed to vary. $\endgroup$ – user127776 Apr 15 at 3:15
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    $\begingroup$ If $X$ is smooth over $O_K$, you can at least find a finite extension $L$ of $K$ such that a smooth hyperplane section exists after base change to $O_L$. This follows from Rumely's existence theorem (and of course, it does not answer the question!) $\endgroup$ – Laurent Moret-Bailly Apr 15 at 9:56
  • $\begingroup$ @Laurent Moret-Bailly Thanks this was very helpful. $\endgroup$ – user127776 Apr 17 at 2:53
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    $\begingroup$ @user127776 Fix an embedding of $X$ in $\mathbb{P}^n_{O_K}$ and consider the open subscheme $U$ in the dual projective space consisting of hyperplanes meeting $X$ transversally. Then $U$ is surjective over $\mathrm{Spec}(O_K)$, with geometrically irreducible fibers. Rumely's theorem then says that $U(O_L)\neq\emptyset$ for some finite extension $L$ of $K$. See for instance Theorem 1.7 in numdam.org/item/?id=ASENS_1989_4_22_2_161_0. $\endgroup$ – Laurent Moret-Bailly Apr 17 at 16:00

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