# What do we lose if we only consider quasi-projective varieties?

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:

http://math.stanford.edu/~vakil/0506-216/216class4344.pdf

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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