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After I learnt about projective schemes over a field, I tried to figure out if for a given projective scheme there is some "natural" closed immersion into a projective space. It turns out unless you impose some restrictions on Picard rank, nothing like this exists. Yet somehow, most of the geometric properties of a projective variety do not really depend on the choice of a closed immersion. One can also characterize complete varieties pretty intrinsically.

Therefore, the question: does there exist a characterization of projective morphisms of schemes which does not involve the word "exists"?

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  • $\begingroup$ This would imply to choose a canonical ample sheaf inside the Picard group. and it doesn't seem feasible, in my opinion. $\endgroup$
    – Leo Alonso
    Commented Apr 3, 2019 at 8:53
  • $\begingroup$ @LeoAlonso I am not entirely sure this would imply that we have a canonical choice of ample sheaf. There may be some non-trivial condition equivalent to existence of a very ample line bundle but which does not itself give the choice of such a bundle (though I agree that such scenario is somewhat unlikely). $\endgroup$
    – user137767
    Commented Apr 3, 2019 at 9:11

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The following may be relevant. Let $k$ be an algebraically closed field. A variety is an integral separated scheme of finite type over $k$. Benoist has shown that a normal variety is quasi-projective iff every finite subset is contained in an affine open subvariety. If you have reasonable definitions, properness+quasi-projectiveness is equivalent to projectiveness.

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