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?