I've been learning about non-projective complete varieties and I am trying to get a handle on how crazy they can get. 


$\textbf{Question:}$ Let $V$ be a complete threefold over $\mathbb{C}$. Given distinct points $P,Q$ on $V$, does there exist a complete surface $S\subset V$ which contains neither $P$ nor $Q$?


I would prefer it if $V$ were smooth, though I'd be surprised if that were relevant. I would also be interested in examples/proofs over fields of finite characteristic.


Thanks!



Non-projective smooth complete threefolds with a pair of points intersecting every surface
I do not know the answer to the question. However, let me point out that the answer is negative if you look for surfaces avoiding more points. More precisely, there exists a smooth complete threefold that does not satisfy Sándor's property $NC\_{10}$: it contains 10 points that cannot be simultaneously avoided by a divisor.


In [Fujino - Payne, Smooth complete toric threefolds with no nontrivial nef line bundles], the authors construct varieties as indicated in the title. Their best example $X$ has Picard rank $5$, and they show that it is optimal.
 Now, the property that $X$ has no nontrivial nef line bundle is equivalent to the fact that the cone of effective curves is the whole vector space $N\_1(X)$ of $1$-cycles up to numerical equivalence. Since this vector space is $5$-dimensional, it will be possible to find $10$ integral curves $C\_1,\dots,C\_{10}$ on $X$ spanning this vector space as a cone (see [Davis, Theory of positive linear dependance, Theorem 6.7]). Choose a point $P\_i$ in $C\_i$. I claim that no surface in $X$ avoids all the $P\_i$. Indeed, such a surface $S$ would intersect non-negatively the $C\_i$, hence any curve. This is obviously impossible: working in an affine neighbourhood of a point of $S$, one constructs a curve intersecting $S$ positively.


Note that in Fujino and Payne's precise example, it should be possible to find less than $10$ integral curves spanning $N\_1(X)$ as a cone, but certainly not less than $6$ curves.


Finally, toric varieties are never going to provide counter-examples to your original question, as any two points in a toric variety have a common affine neighbourhood.


