Let $X\to \operatorname{Spec } K$ be a projective integral variety. Let $L|K$ be a field extension and consider an $L$ point $x\colon \operatorname{Spec} L\to X$ in $X$ (morphism over $K$).
Under which conditions on the extension $L|K$, the morphism $x \colon \operatorname{Spec} L\to X$ is projective or quasi-projective?
If $x$ is a closed point (i.e. $L|K$ is finite) then clearly the morphism is projective. But what about the other cases? For instance assume that the closure $\overline{\{x\}}$ is a proper subvariety of $X$, then I suspect that the morphism $x\colon \operatorname{Spec} L\to X$ must have some special properties.