Perhaps a basic question... how does one study affine algebraic geometry via projective geometry? For example, suppose I have two affine varieties which I want to prove are isomorphic, would it help to look at the projective closures (assuming I don't know of any other method for proving they are isomorphic)? How does one go back to the affine case from the projective closure (or "projectivization")? Sorry if this sounds confusing.

Thanks,
Morton