Are there non-projective normal surfaces which are rational? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:47:42Z http://mathoverflow.net/feeds/question/51180 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51180/are-there-non-projective-normal-surfaces-which-are-rational Are there non-projective normal surfaces which are rational? auniket 2011-01-05T06:26:43Z 2011-01-06T20:51:27Z <p>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. <a href="http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/on_non_proj.pdf" rel="nofollow"> this </a>). 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?</p> <p>Edit: I just saw <a href="http://mathoverflow.net/questions/3624/nonprojective-surface" rel="nofollow"> a previous question </a> which asked for examples of normal non-projective varieties. So I guess this is a sub-question of that one.</p> http://mathoverflow.net/questions/51180/are-there-non-projective-normal-surfaces-which-are-rational/51341#51341 Answer by J.C. Ottem for Are there non-projective normal surfaces which are rational? J.C. Ottem 2011-01-06T20:51:27Z 2011-01-06T20:51:27Z <p>Nagata constructs a normal complete rational surface in the paper <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.ijm/1255454111&amp;page=record" rel="nofollow">Existence theorems for nonprojective complete algebraic varieties</a> (see Section 4). His construction uses a blow-up of the plane in 12 points in special position.</p>