The Stacks project

Remark 44.2.7. Let $f : X \to S$ be a morphism of schemes. The assumption of Proposition 44.2.6 and hence the conclusion holds in each of the following cases:

  1. $X$ is quasi-affine,

  2. $f$ is quasi-affine,

  3. $f$ is quasi-projective,

  4. $f$ is locally projective,

  5. there exists an ample invertible sheaf on $X$,

  6. there exists an $f$-ample invertible sheaf on $X$, and

  7. there exists an $f$-very ample invertible sheaf on $X$.

Namely, in each of these cases, every finite set of points of a fibre $X_ s$ is contained in a quasi-compact open $U$ of $X$ which comes with an ample invertible sheaf, is isomorphic to an open of an affine scheme, or is isomorphic to an open of $\text{Proj}$ of a graded ring (in each case this follows by unwinding the definitions). Thus the existence of suitable affine opens by Properties, Lemma 28.29.5.

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