TITLE: How non-projective can a complete variety be?
QUESTION [12 upvotes]: Given a positive integer $n$, can you give an example of a connected smooth proper $\mathbb{C}$-scheme such that any $n$ closed points are contained in a common affine open, but there is $n+1$ closed points that are not contained in a common affine open? What is the minimum dimension of such an example (probably larger than 2, because every smooth proper surface is projective)?

REPLY [10 votes]: For every positive integer $n$, Farnik has constructed a smooth complete variety over an algebraically closed field with $n=\mathrm{sup}\{$$m$: every $m$ points of in $X$ are contained in a common affine open subset of $X\}$. The construction suggests that the varieties are threefolds so the minimum dimension appears to be $3$.
M. Farnik, On strengthening of the Kleiman-Chevalley criterion, Proceedings of the AMS 141 no 11 (2013) pp 4005-4013, doi:10.1090/S0002-9939-2013-11695-3. Link at AMS site.