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Let $K$ be a number field, $\overline{K}$ an algebraic closure, and $X$ be a positive dimensional finite type $K$-scheme.

Could there exist a proper subfield $L\subset\overline{K}$ such that the natural inclusion $$X(L)\hookrightarrow X(\overline{K})$$ is surjective?

If so, what sorts of conditions can we put on $X$ to ensure that this can't happen? I'm also curious about the answer for more general fields $K$.

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    $\begingroup$ You want to add some non-triviality condition ($\dim X>0$ I guess?) to avoid the obvious counterexample $X = \operatorname{Spec} K$. $\endgroup$
    – Gro-Tsen
    Commented Jul 2, 2023 at 23:43
  • $\begingroup$ @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. $\endgroup$
    – stupid_question_bot
    Commented Jul 3, 2023 at 2:20
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    $\begingroup$ I assume you mean $K\subseteq L\subseteq \bar{K}$? $\endgroup$
    – Arno Fehm
    Commented Jul 3, 2023 at 3:18
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    $\begingroup$ 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. $\endgroup$
    – Sándor Kovács
    Commented Jul 3, 2023 at 3:20
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    $\begingroup$ 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. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jul 3, 2023 at 6:24

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No this never happens: Without loss of generality, $X$ is affine. The case $X=\mathbb{A}^n_K$ is obvious, the general case then follows from Noether normalization and going up.

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  • $\begingroup$ Posted independently by @SándorKovács a few minutes later. Also, of course, it's obvious for $\mathbb A_K^n$ as long as $n$ is positive, as @‍Gro-Tsen mentioned. $\endgroup$
    – LSpice
    Commented Jul 3, 2023 at 3:37
  • $\begingroup$ Ah, of course. Nice! $\endgroup$
    – stupid_question_bot
    Commented Jul 3, 2023 at 5:26

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