This is a question that has come to mind as I have been studying Vakil’s notes on algebraic geometry.
I am studying this in my own time out of interest and afterwards wish to look at other books (specifically Silverman’s GTM book on Elliptic Curves).
One thing I know about Silverman’s book, and many other similar ones, is that they cover AG from a “classical” perspective - no schemes in sight in proving Bezout’s theorem and so forth as far as I know.
To what extent would my knowledge from Vakil’s notes be applicable to this context? Is there any sense in which much of the general theory seen in, e.g. the arithmetic of elliptic curves would be subsumed by what I will have learned while studying schemes?
I have found similar questions through searches, asking which elementary problems schemes can solve. This question is not quite that, as rather I am wondering how the theory of schemes helps simplify or illuminate classical geometry that already has classical proofs at the undergraduate level.
I studied “classical” AG books several years ago at university, so while I remember enough to motivate schemes, I can’t remember enough specifics to note specifically how they can be used here.