MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
There exists non-projective toric varieties, which are of course rational. – J.C. Ottem Jan 5 '11 at 15:32
...but all complete toric surfaces are projective. – Dave Anderson Jan 5 '11 at 18:11
up vote 5 down vote accepted

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.

share|cite|improve this answer
Thanks! I will check it out. – auniket Jan 7 '11 at 9:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.