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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.2012-12-30
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    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 ?2012-12-30
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    @QiL They are varieties in the sense of Serre, i.e. they are separated and not necessarily proper.2012-12-30
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    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.2012-12-30
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    @QiL "So I think the real question would be why to consider proper varieties which are not necessarily projective." I would like to know the reason why.2012-12-30
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    I'm curious to know who voted to close.2012-12-31
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    @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.2012-12-31
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    **I have removed off-topic comments.**2013-01-02
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    I noticed that someone serially upvoted for my questions and answers including this one. While I appreciate them, I would like to point out that serial upvotes are automatically reversed by the system.2013-11-27

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