Quasiprojective

Quasiprojective, or quasi-projective, refers to a class of algebraic varieties (or schemes) over a field that can be realized as an open subset of a projective variety. Concretely, a k-variety X is quasiprojective if there exists a projective k-variety Y and an open immersion X → Y. Equivalently, X can be viewed as a locally closed subvariety of some projective space P^n_k. This places quasiprojective varieties between affine and projective varieties: every affine variety is quasiprojective, and projective varieties are quasiprojective.

Properties and equivalences: If X is quasiprojective, any open subset U ⊂ X is quasiprojective, and any closed subvariety of X is quasiprojective. Quasiprojective varieties are of finite type over k and are separated. Many standard constructions preserve quasiprojectivity, such as taking open subschemes and closed subvarieties of a quasiprojective variety. However, not every finite-type k-scheme is quasiprojective, and some moduli spaces are not.

Examples: The projective space P^n_k is projective and hence quasiprojective. The affine space A^n_k embeds as an open subset of P^n_k, so it is quasiprojective. A^n_k minus a hyperplane or minus a finite set of points is also quasiprojective. In general, any locally closed subset of P^n_k is quasiprojective.

Context and utility: Quasiprojectivity is a convenient hypothesis in algebraic geometry because it allows the use of projective techniques on open subsets, including the existence of ample line bundles and the ability to embed into projective space after suitable restriction. It thus broadens the scope of methods beyond strictly projective varieties while retaining many of their powerful tools.