# 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
How to Cite This Entry:
Quasi-projective scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Quasi-projective_scheme&oldid=38802
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article