Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and people go to reasonable lengths to include these examples all over the place, so they're easy to find. However, Hartshorne does say that singular complete surfaces are not all projective. Is there a simple example? A complete normal surface that is not projective? Is there some "least singular" possible such surface? I suspect that normality is too much to hope for, but I can't quite phrase why I think this, so is every normal complete surface projective?
[Math] Nonprojective Surface
ag.algebraic-geometryalgebraic-surfacesexamples
Best Answer
There is a construction of a proper normal non-projective surface here .
There is an example given by Nagata in his paper "Existence theorems for nonprojective complete algebraic varieties" in the Illinois Journal, but I don't know where this is available on the web.
Over a finite field complete + normal implies projective for surfaces.