Can we recover a reduced finite type $k$-scheme from its set of closed points, if $k$ is not algebraically closed?

Question

Let $k$ be a field, which is not algebraically closed. For a $k$-scheme, let us denote by $X_0$ the set of all closed points of $X$. Let $X$ be a reduced finite type $k$-scheme. Can we recover $X$ from $X_0$? If $X$ is a smooth projective $k$-scheme, can we recover $X$ form $X_0$?