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Let's say a morphism $f:X\to Y$ is compactifiable if it admits a factorization $f = pj$ with $j:X\to P$ an open immersion and $p:P\to Y$ proper.

In SGA 4 Exp. XVII, Deligne says that Nagata proved that any morphism of separated integral northerian schemes is compactifiable but that he didn't understand the proof.

My questions:

  • Where can I find a proof of Nagata's theorem?
  • What about the complex analytic setting?
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    $\begingroup$ I just noticed that your question was "Where can I find a proof that any algebraic morphism of quasi-projective varieties is compactifiable?". This is pretty much obvious; just embed $X$ into $\mathbb P^N$, and take the closure of $X$ in $\mathbb P^N \times Y$. $\endgroup$ – Angelo Apr 24 '10 at 13:55
  • $\begingroup$ Obvious indeed. Changed the question. $\endgroup$ – AFK Apr 24 '10 at 14:13
  • $\begingroup$ Nagata's theorem doesn't need the "integral" hypothesis (even though Nagata's original formulation may have had it, possibly due to his use of earlier style of alg. geometry; I don't remember). The references mentioned below avoid it. $\endgroup$ – BCnrd Apr 24 '10 at 16:34
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Brian Conrad has written up a proof of Nagata's theorem, starting from notes of Deligne: http://math.stanford.edu/~conrad/papers/nagatafinal.pdf. About the analytic case, I have no idea.

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For the algebraic case, see also Lutkebohmert: On compactifications of schemes.

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The following article of Conrad, Lieblich and Olsson settles the case of algebraic spaces.

http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5008v1.pdf

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    $\begingroup$ Rydh has a proof by a different approach (see his webpage, or C-L-O Introduction), and there is yet a 3rd approach in progress by Temkin & Temkin. $\endgroup$ – BCnrd Apr 24 '10 at 16:31

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