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Every non-singular complete surface is projective. On the other hand, there are non-projective complete surfaces (see e.g. Excercise II.7.13 of Hartshorne) - and there are such examples where the surface is also normal (see e.g. this ). All the examples I have seen of complete normal non-projective surfaces are non-rational. Hence the question: are there (complete) rational non-projective normal surfaces?

Edit: I just saw a previous question which asked for examples of normal non-projective varieties. So I guess this is a sub-question of that one.

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    $\begingroup$ There exists non-projective toric varieties, which are of course rational. $\endgroup$
    – J.C. Ottem
    Commented Jan 5, 2011 at 15:32
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    $\begingroup$ ...but all complete toric surfaces are projective. $\endgroup$
    – Dave Anderson
    Commented Jan 5, 2011 at 18:11

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Nagata constructs a normal complete rational surface in the paper Existence theorems for nonprojective complete algebraic varieties (see Section 4). His construction uses a blow-up of the plane in 12 points in special position.

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