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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.

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    $\begingroup$ 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). $\endgroup$ – Aaron Landesman Mar 25 at 23:40

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