4
$\begingroup$

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 http://www.ams.org/journals/proc/2005-133-09/S0002-9939-05-07859-7/home.html 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.

$\endgroup$
4
  • 1
    $\begingroup$ Note that nowadays the theory of toric varieties is an easy way to get a lot of examples of non-projective proper smooth varieties. $\endgroup$ Mar 4, 2011 at 8:06
  • $\begingroup$ 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 $X$. $\endgroup$
    – Qing Liu
    Mar 4, 2011 at 8:49
  • $\begingroup$ By variety I mean irreducible, reduced, and separated. Also, all the example of toric varieties I know don't satisfy 1),2), or 3). $\endgroup$
    – Parsa
    Mar 4, 2011 at 15:14
  • $\begingroup$ Sorry, I misread the question and mixed $X$ and $Z$ ! $\endgroup$
    – Qing Liu
    Mar 4, 2011 at 16:24

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.