My question is a reference request for the following fact: if $k$ is a field and $X$ a proper smooth surface over $k$, then $X \rightarrow \mathrm{Spec}\, k$ is projective. Where is this wellknown fact proved (in the stated generality)?

6$\begingroup$ Theorem 1.2.8 in Badescu's book "Algebraic Surfaces" proves this theorem of ZariskiGoldman over any algebraically closed field, and the general case reduces to that via "norm of line bundles" or other reasons. $\endgroup$– user74230Feb 25, 2015 at 6:40

2$\begingroup$ Several classical references are given on II, §4, p. 105 of Hartshorne's book. $\endgroup$– Damian RösslerFeb 25, 2015 at 11:14
1 Answer
Quoting from a very nice paper by Stefan Schroeer we have: "The criterion of Zariski [3, Cor. 4, p. 328] tells us that a normal surface $Z$ is projective if and only if the set of points $z \in Z$ whose local ring $\mathcal{O}_{Z,z}$ is not $\mathbf{Q}$factorial allows an affine open neighborhood." The reference [3] is the following:
S. Kleiman: Toward a numerical theory of ampleness. Annals of Math. 84 (1966), 293–344.
The paper of Stefan is here.