Quasi-projective scheme

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A locally closed sub-scheme of a projective space $ \mathbf{P}^{n} $. In other words, a quasi-projective scheme is an open sub-scheme of a projective scheme. A scheme $ X $ over a field is quasi-projective if and only if there exists on $ X $ an invertible ample sheaf. A generalization of the notion of a quasi-projective scheme is that of a quasi-projective morphism, that is, a morphism of schemes that is the composition of an open imbedding and a projective morphism. A scheme that is both quasi-projective and complete is projective.


[a1] R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 10 & 103. MR0463157 Zbl 0367.14001
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Quasi-projective scheme. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article