Timeline for Varieties whose residue fields do not generate the algebraic closure of the ground field
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 3, 2023 at 7:35 | history | became hot network question | |||
| Jul 3, 2023 at 6:24 | comment | added | Laurent Moret-Bailly | I agree with Sándor. It even suffices to treat the case of $\mathbb{A}^1$ and use a nonconstant morphism $U\to \mathbb{A}^1$ where $U\subset X$ is an open affine. | |
| Jul 3, 2023 at 5:26 | vote | accept | stupid_question_bot | ||
| Jul 3, 2023 at 3:20 | comment | added | Sándor Kovács | I may be misunderstanding the question, but if $X={\rm Spec} K[x_1,\dots,x_n]$, then surely $X(L)\neq X(\overline K)$ if $L\neq \overline K$ and otherwise take an affine cover of $X$ and use Noether normalization. | |
| Jul 3, 2023 at 3:18 | comment | added | Arno Fehm | I assume you mean $K\subseteq L\subseteq \bar{K}$? | |
| Jul 3, 2023 at 3:17 | answer | added | Arno Fehm | timeline score: 5 | |
| Jul 3, 2023 at 2:20 | comment | added | stupid_question_bot | @Gro-Tsen thanks, added this condition. Also added the condition that $K$ be a number field, since that’s the case I’m most interested in. | |
| Jul 3, 2023 at 2:17 | history | edited | stupid_question_bot | CC BY-SA 4.0 |
added 21 characters in body
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| Jul 2, 2023 at 23:43 | comment | added | Gro-Tsen | You want to add some non-triviality condition ($\dim X>0$ I guess?) to avoid the obvious counterexample $X = \operatorname{Spec} K$. | |
| Jul 2, 2023 at 23:34 | history | asked | stupid_question_bot | CC BY-SA 4.0 |