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What do we lose if we only consider quasi-projective varieties? What are merits of considering varieties which are not quasi-projective?

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Seems a reasonable question to me, so I upvoted. – paul garrett Dec 30 '12 at 21:05
This is a good question. But it could be a little more precise: what do you mean by varieties ? Are they separated or do you even restrict to proper varieties ? – user18119 Dec 30 '12 at 22:32
@QiL They are varieties in the sense of Serre, i.e. they are separated and not necessarily proper. – Makoto Kato Dec 30 '12 at 22:53
Separated algebraic varieties are open subvarieties of proper algebraic varieties by a theorem of Nagata. So I think the real question would be why to consider proper varieties which are not necessarily projective. – user18119 Dec 30 '12 at 23:02
@MakotoKato: Not to nitpick, but what is interesting, is not who voted to close, but the reason why. Anyhow, I think this is a good question. – Fredrik Meyer Dec 31 '12 at 15:55

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:

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"we can glue two disjoint varieties along isomorphic closed subvarieties and get a variety" Could you explain the motivation for this construction? – Makoto Kato Jan 1 '13 at 20:48

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.

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