# 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.