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If $X$ is a smooth complete non-projective variety of dimension at least three, let $U$ be a maximal (with respect to inclusion) open quasi-projective subset of $X$. Letting $Z=X-U$, it follows that the codimension of $Z$ in $X$ is at least 2. You can see this by first using Nagata's theorem and normalizing to get a dominant birational map $f: X' \rightarrow X$ where $X'$ is a projective normal variety and $f$ is an isomorphism on $U$. Then $f^{-1}$ defines an open embedding into $X'$ off of the exceptional locus $Z$, which by normality and Zariski's Main Theorem must be of codimension at least 2. It is also known (by using maximality and Seshadri's criterion) that $Z$ cannot contain an isolated point. See for a reference.

In Hironaka's example of a non-projective 3-fold (the one where he blows up two curves in different orders and glues), there are two maximal quasi-projective opens, the compliment of each being a smooth rational curve and hence of codimension exactly 2. Here are 3 questions.

1) Does anyone know of an example of a smooth complete non-projective variety where the compliment of a maximal quasi-projective open $Z$ is of codimension greater than 2?

2) An example where $Z$ is reducible?

3) An example where $Z$ is singular?

I am not sure how difficult these questions are, but I know very few examples of smooth complete non-projective varieties in general. Any interesting examples are welcomed.

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Note that nowadays the theory of toric varieties is an easy way to get a lot of examples of non-projective proper smooth varieties. – Torsten Ekedahl Mar 4 '11 at 8:06
Let $X$ be a proper non-projective variety. For 2), take the union of $X$ with another proper variety (say intersecting at one point). For 3), take $X\times \mathrm{Spec}k[t]/(t^2)$, or identify two rational points of an affine open subset of $X$. – Qing Liu Mar 4 '11 at 8:49
By variety I mean irreducible, reduced, and separated. Also, all the example of toric varieties I know don't satisfy 1),2), or 3). – Parsa Mar 4 '11 at 15:14
Sorry, I misread the question and mixed $X$ and $Z$ ! – Qing Liu Mar 4 '11 at 16:24

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