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

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

Edit: Thanks for the replies. Being new to AG let me try and rephrase my quandary: It seems the projective setting is the most convenient to study AG but if I want to study properties of affine varieties, how does one use results of projective varieties in the affine case? I know this sounds vague but it is a fundamental doubt I have.