What do we lose if we only consider quasi-projective varieties? What are merits of considering varieties which are not quasi-projective?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
It's nice that, with the intrinsic definition of variety, we can glue two disjoint varieties along isomorphic closed subvarieties and get a variety (or at least a scheme). We can't do this if we insist on only studying quasi-projective varieties:
There are natural constructions in algebraic geometry which lead to a priori non-quasi-projective varieties. One important example, and I am pretty sure this is one of the motivations of Weil to define "abstract algebraic varieties", is the algebraic (i.e. not Abel-Jacobi) construction of Jacobians of algebraic curves. The construction of Weil in the 40's gives a proper algebraic variety. Of course now it is known that abelian varieties are projective. But it is important to know that some "natural" algebraic varieties are not directly constructed as projective varieties.
Jacobian varieties are moduli spaces (of divisors on curves). There are other moduli spaces (e.g. that of smooth or stable curves of given genus) which are naturally separated or proper varieties. It is only after they are constructed, and if we are happy, that we can prove they are quasi-projective.
There is other reason to consider proper or separated algebraic varieties because we have valuative criterion to decide whether a variety is separated or proper. It is much harder to show the quasi-projectivity. Sometimes the properness of a projective variety is enough for what we need.