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.