# Do complete non-projective varieties arise “in nature”?

I'm aware of the existence of complete (abstract) algebraic varieties that are not projective but, probably due to my ignorance, I have the impression that they arise only as very particular examples constructed just with the purpose of finding such an example. My question (perhaps a bit vague) is:

Are there exemples in the literature in which complete non-projective varieties appear without "being expected" from the beginning or without just being the goal of the construction or proof?

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There is lots of non-projective complete toric varieties... –  Piotr Achinger Nov 5 '12 at 2:05
I think there is an example of a projective variety $X$ with an action of a finite group $G$ such that the quotient $X/G$ is not projective. Would this fit your question? –  J.C. Ottem Nov 5 '12 at 2:59
@J.C. Ottem: There is certainly no such example; for a finite group acting on a quasi-projective variety the quotient is also quasi-projective. What you are perhaps thinking of is that there is an action of a finite group on a non quasi-projective variety such that the quotient (which always exists as an algebraic space) is not a variety. –  ulrich Nov 5 '12 at 4:23
@Dima Pasechnik: If a quotient of a projective variety exists as a quasi-projective variety then the quotient is obviously projective (being the image of a projective variety). –  ulrich Nov 5 '12 at 6:13
@JC Ottern: ... or you might be thinking of a non-liftable (from char. $p>0$ to $0$) variety. There is an example of Serre involving a quotient construction. –  Damian Rössler Nov 5 '12 at 7:35

Assume $X$ is a projective threefold with $n$ ordinary double points. Then $X$ has $2^n$ small resolutions of singularities. Usually, almost of all of them are non-projective.
@Piotr Aichinger: a birational morphism between two regular surfaces is a sequence of blow-ups but this is not true in higher dimensions. In general, there is the "strong factorisation conjecture", to the effect that there are morphisms $X'\to X$, $X'\to Y$, which are sequences of blow-ups along regular centers, and such that as a birational map (not morphism) $X\to Y$ is equal to the inverse of $X'\to X$ composed with $X'\to Y$. See the papers by Wlodarczyk, Abramovich et al. for this and the "weak factorisation conjecture". –  Damian Rössler Nov 5 '12 at 7:47