I have a bit general question about a proof /reduction technique in algebraic geometry which encountered to me quite often:
Sometimes if one has a morphism $f:X \to Y$ of schemes one makes wlog the assumption that $X$ is reduced, so $X = X_{red}$. Why and im which cases it can be done? Some "famous" examples?
The situation is usually when you have a closed subset $T$ of another scheme $X$ and want to give it a scheme structure. There are many possible structures one may take, but there is a unique choice which is reduced. Meaning, there is a unique reduced closed subscheme $Z \subset X$ whose underlying topological space is equal to that of $T$. You can look here at [Stacks, tag 01IZ] for more information on this, including a universal property.