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)?

  • 6
    $\begingroup$ 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. $\endgroup$
    – user74230
    Feb 25, 2015 at 6:40
  • 2
    $\begingroup$ Several classical references are given on II, §4, p. 105 of Hartshorne's book. $\endgroup$ Feb 25, 2015 at 11:14

1 Answer 1


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.


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