Timeline for Existence of regular hypersurface sections
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 17, 2019 at 16:00 | comment | added | Laurent Moret-Bailly | @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. | |
| Apr 17, 2019 at 3:11 | comment | added | user127776 | @Laurent Moret-Bailly Could you please point me to a reference that contains the statement you've written. I searched for Rumely's theorem didn't find anything with the exactly same formulation. | |
| Apr 17, 2019 at 2:53 | comment | added | user127776 | @Laurent Moret-Bailly Thanks this was very helpful. | |
| Apr 15, 2019 at 9:56 | comment | added | Laurent Moret-Bailly | 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!) | |
| Apr 15, 2019 at 3:15 | comment | added | user127776 | 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. | |
| Apr 15, 2019 at 2:45 | comment | added | Piotr Achinger | Is the very ample line bundle fixed or is it allowed to vary? | |
| Apr 15, 2019 at 1:53 | history | asked | user127776 | CC BY-SA 4.0 |