If $X$ is a $k$-scheme of finite type with $k$ a field, there is a surjective map $\gamma$ from the set of $\overline{k}$-rational points of $X \times_{\mathrm{Spec}(k)}\mathrm{Spec}(\overline{k})$, with $\overline{k}$ an algebraic closure of $k$, to the set of closed points of $X$. Each closed point of $X$ corresponds to a $\mathrm{Gal}(\overline{k}/k)$-orbit of $\overline{k}$-rational points.
If I remember well, this is not true for general $k$-schemes. What are some ineteresting examples where the map $\gamma$ is not surjective ?