# A proper smooth surface is projective

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)?

• Theorem 1.2.8 in Badescu's book "Algebraic Surfaces" proves this theorem of Zariski--Goldman over any algebraically closed field, and the general case reduces to that via "norm of line bundles" or other reasons. Feb 25, 2015 at 6:40
• Several classical references are given on II, §4, p. 105 of Hartshorne's book. Feb 25, 2015 at 11:14

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: