I've just begun to learn algebraic geometry, and to my understanding Grothendieck's revolutionary idea was to define a generalization for algebraic varieties (affine, projective, quasi-affine & quasi-projective) that eliminated restrictions imposed by working over algebraically closed fields, by instead working over general commutative rings. This lead to modern algebraic geometry and scheme theory.
I've got some questions regarding things that I've not yet grasped.
- Why are there 3 types of varieties and what is the relationship between them?
- If affine, quasi-affine and projective varieties are all special cases of quasi-projective varieties, then why not just use them in the first place?
- If schemes are a generalization of algebraic varieties, then how are affine, projective, quasi-affine & quasi-projective varieties defined in terms of schemes?