When does the image of a morphism of schemes support scheme structure?

Let $$Y$$ be a qcqs scheme, $$f:X\rightarrow Y$$ be a quasi-compact morphism locally of finite presentation. Are there any conditions on $$f$$ which

• do not force $$f(X)$$ be open or closed but force it to be locally closed;
• do not force $$f(X)$$ to be locally closed or homeomorphic to the underlying topological space of $$X$$ but force it to be homeomorphic to the underlying topological space of some scheme?

In principle, I do not mind adding some geometric hypotheses (say $$X$$ is of finite type over an algebraically closed field of characteristic 0) if that makes life simpler.

• I'm not certain if this is what you're looking for, but if f is a map of finite type schemes over a noetherian base, then there is a valuative criterion for when $f(X) \rightarrow Y$ is a locally closed immersion. See, Chapter 1, Corollary 2.13 of Shinichi Mochizuki. Foundations of p-adic Teichmuller Theory, volume 11. American Mathematical Soc., 2014. Essentially it says that the map will be an immersion if and only if for all DVRs R with Spec R mapping to Y such that both the special and generic points factor through f(X) then you can extend the map from Spec R to f(X). – Aaron Landesman Mar 25 at 23:40