Quasi-projective scheme

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.

References

[a1]	R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 10 & 103. MR0463157 Zbl 0367.14001