I want to get into some of the big classification problems in algebraic geometry, but have a very broad question. Ultimately we would like to classify all varieties over some field up to isomorphism, and this is done via moduli theory. I am studying the moduli of elliptic curves at the moment. From the moduli space we construct we are able to obtain information about the relationship between different varieties based on the geometry of the moduli space. For example varieties in the same irreducible component can somehow be deformed to eachother.
On the other hand it seems like the completion of this program is hopelessly difficult for general varieties. Instead it is common to attempt to classify such objects only up to birational equivalence. This is the goal of the minimal model program.
But it seems that these two aren't really running in parallel in the sense that the latter doesn't appear to be a direct stepping stone for the former. I guess what I was hoping for is that classififying up to birational equivalence would somehow be a big step towards classification up to isomorphism. But when we classify curves up to isomorphism we appeal to the genus. But for birational equivalence we just normalize and apply Chow's lemma.
I know this is a broad question, but is there some geometry of the moduli space that would tell us when two varieties are birational equivalent, such as being in the same connected component or some such simple test? Or even more broadly, would the completion of the minimal model program give us any insight into what the moduli space for some family of varieties would look like?
I left this as a soft question since I probably don't even know enough about the subject to ask it in a precise manner. But hopefully some experts understand what I am trying to get at.