In this question I ask whether ambient spaces descend to models of varieties.


Let $k\subset K$ be a non-trivial extension of algebraically closed fields, e.g., $\overline{\mathbb Q}\subset \mathbb C$.


Let $X$ be a projective variety over $K$ which can be defined over $k$ (as an abstract scheme).


Assume that $X$ can be embedded in $\mathbb P^n\_K$, i.e., there is a closed immersion from $X$ into $\mathbb P^n\_K$. Is there a model $X\_0$ of $X$ over $k$ such that $X\_0$ can be embedded in $\mathbb P^n\_k$?


What if $k$ is of characteristic zero?



Minimal projective space containing projective variety independent of base field
Write $X=Y\times\_k K$ for some variety $Y$ over $k$. The closed immersion $i: Y\times\_k K\to\mathbb P^n\_K$ is defined by using finitely many coefficients in $K$, so it is defined over a finitely generated $k$-algebra $A$: 
$$i\_S: Y\times\_k S\to \mathbb P^n\_S=\mathbb P^n\_k \times\_k S $$
where $S$ is the affine variety associated to $A$, and $i\_S$ is an $S$-morphism. 
Take a closed point $s\in S$ and you will have a closed immersion 
$$ Y=Y\times \{ s \} \to \mathbb P^n\_k \times \{ s\}=\mathbb P^n\_k.$$


