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 $K$.



Varieties whose residue fields do not generate the algebraic closure of the ground field
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.


