Example of a separated variety which is not quasi-projective

Question

I am following Kempf's book on algebraic varieties. A variety there is defined to be a space with functions having a finite affine open cover.

Call a variety $X$ separated if its diagonal $$\mathrm{diag} (X)=\{ (x,x)\mid x\in X\}$$ is closed in $X\times X$. In Lemma 3.3.2, Kempf proves that quasi-projective varieties are separated. My question is: does there exist an example of a separated variety which is non-quasi-projective?

PS: This is my first time posting a question here. Please feel free to suggest ways to make it better.

Answer 1

Yes, there are many examples of proper (thus separated) varieties which are not projective and therefore not quasi-projective. Here's a list of some examples:

• Hironaka's example.
• Nagata's existence theorem for nonprojective complete varieties associates to every function field of dimension 2 or more a complete nonprojective variety (some conditions on the base field are necessary in dimension 2).
• Hartshorne exercise II.7.13: Let $C=V(y^2z=x^3+x^2z)\subset\Bbb P^2_k$ be the nodal cubic for $k$ an algebraically closed field of characteristic not 2, which has smooth locus isomorphic to $\Bbb G_m$. Glue $C\times(\Bbb P^2\setminus \{0\})$ and $C\times(\Bbb P^2\setminus \{\infty\})$ along $C\times(\Bbb P^2\setminus \{0,\infty\})$ by the automorphism given by sending $(p,u)\mapsto (up,u)$. This has zero picard group and therefore cannot be projective.
• Hartshorne exercise III.5.9 produces an infinitesimal extension of $\Bbb P^2_k$ over a field of characteristic zero which isn't projective.
• Do complete nonprojective varieties arise "in nature"? on MO contains more discussion.