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Recently I posted an "announcement" on arxiv where I said something to the effect of "this is the first example we know where contracting (a tree of) rational curves from a non-singular algebraic surface (over $\mathbb{C}$) leads to a (normal) non-algebraic surface." Now that I am writing up the actual paper, I thought that may be I should broaden my knowledge base! So I ask: is there some example of this sort already known?

Remarks:

  1. Grauert constructed (in this article: mathscinet link, springerlink) such non-algebraic surfaces by blowing down curves of genus $\geq 2$ (in Section 4.8, Example d) and remarked that he did not know if it is possible (to construct non-algebraic normal surfaces) from blowing down tori.

  2. An example of Nagata described in a previous question of mine shows that it is indeed possible with tori.

  3. In the second paragraph of the first page of this article (mathscinet link) Artin mentions an example of Hironaka that shows that "in general there are no numerical criteria equivalent with (algebraic) contractibility of a given curve." Does anyone know what is this example?

  4. Happy New Year!

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Check out Schröer's papers arxiv.org/abs/math/9911004 and arxiv.org/abs/math/9911119. –  Angelo Dec 30 '12 at 21:32
    
@Angelo: Thanks! But I knew of these (perhaps should have mentioned them in the question) and they do not contain (and as far as I can see, do not shed any light on the construction of) any such examples. –  auniket Dec 30 '12 at 21:41
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Algebraic spaces are "algebraic". It says so right in the name! –  Jason Starr Dec 31 '12 at 11:31
    
@Jason: OK, I admit it wasn't such a good choice of words :) Would you prefer if I change "something non-algebraic" to "non-algebraic surfaces"? –  auniket Dec 31 '12 at 13:33
    
@auniket: Of course I understand what you mean, so my comment is joking. I think "non-schematic" is a bit more precise than "non-algebraic". –  Jason Starr Dec 31 '12 at 15:55

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