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?