 Quasi-projective variety

A quasi-projective variety is an irreducible variety admitting a locally closed embedding in projective space. Thus it can be written as an open subset of a projective variety. This includes both affine varieties and projective varieties.

An open subset of an affine variety is quasi-projective. A blowup of an algebraic variety along a closed center is also quasi-projective when the original algebraic variety is quasi-projective: its morphism to that base is projective, and a projective morphism over a quasi-projective base has a quasi-projective source. This hypothesis on the base is needed; a blowup of an arbitrary algebraic variety need not provide a quasi-projective example.

The complement of a point in the projective line is the affine line, while the projective line itself remains a projective example. The definition organizes open pieces of projective geometry without requiring all their regular functions to come from one global affine chart.

In algebraic geometry, a **quasi-projective variety** is a type of algebraic variety that can be viewed as an open subset of a projective variety.