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](http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/on_non_proj.pdf) ). 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](https://mathoverflow.net/questions/3624/nonprojective-surface)  which asked for examples of normal non-projective varieties. So I guess this is a sub-question of that one.



Are there non-projective normal surfaces which are rational?
Nagata constructs a normal complete rational surface in the paper [Existence theorems for nonprojective complete algebraic varieties](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-2/issue-4A/Existence-theorems-for-nonprojective-complete-algebraic-varieties/10.1215/ijm/1255454111.full) (see Section 4). His construction uses a blow-up of the plane in 12 points in special position.


