In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subscheme of some projective space.
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Relationship to affine varieties
An affine space is a Zariski-open subset of a projective space, and since any closed affine subset
Examples
Since quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as varieties. Varieties isomorphic to affine algebraic varieties as quasi-projective varieties are called affine varieties; similarly for projective varieties. For example, the complement of a point in the affine line, i.e.
Quasi-projective varieties are locally affine in the sense that a manifold is locally Euclidean — every point of a quasiprojective variety has a neighborhood given by an affine variety. This yields a basis of affine sets for the Zariski topology on a quasi-projective variety.