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 well-known fact proved (in the stated generality)?



A proper smooth surface is projective
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](http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/on_non_proj.pdf).


