# Quasi-projective scheme

From Encyclopedia of Mathematics

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