TITLE: Are there non-projective normal surfaces which are rational?
QUESTION [9 upvotes]: 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.

REPLY [5 votes]: 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.