# Existence of regular hypersurface sections

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!

• Is the very ample line bundle fixed or is it allowed to vary? – Piotr Achinger Apr 15 at 2:45
• 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. – user127776 Apr 15 at 3:15
• 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!) – Laurent Moret-Bailly Apr 15 at 9:56
• @Laurent Moret-Bailly Thanks this was very helpful. – user127776 Apr 17 at 2:53
• @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. – Laurent Moret-Bailly Apr 17 at 16:00