I have recently started to learn algebraic geometry and this question has been bugging me.
An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is a zero set of a collection of polynomials in projective space.
Affine space has the property that a group action is available on it while projective space has the property that it is compact.
So one of the two definitions comes with something algebraic in addition while the other comes with something topological.
But it is not clear to me what the advantages of either are over the other. Compactness always makes things easier so it would not surprise me if it would make things easier in algebraic geometry, too. But exactly how it does is not clear to me.
I guess it will depend on which branch of algebraic geometry, too.
So let's assume, just for the purpose of this question, that I want to study singularities of varieties.
Please could someone explain to me, perhaps even provide a simple example, what the advantages of one type of variety is over the other? How to use the group action or the compactness to classify singularities?
I think once I see examples it will all become very clear.