quasiprojectivity

Quasi-projectivity (often written quasi-projective or quasiprojective) is a property in algebraic geometry describing a class of varieties and, more generally, schemes over a field. A k-variety X is called quasi-projective if there exists an embedding of X into projective space P^n_k as a locally closed subscheme. Equivalently, X is an open subset of a projective k-variety. This characterization places X between affine and projective varieties: every affine variety is quasi-projective, and every projective variety is quasi-projective (as a closed subset of some projective space).

Basic properties and examples: Quasi-projective varieties are of finite type over k and are separated. Open subsets of quasi-projective varieties are again quasi-projective, and products of quasi-projective varieties are quasi-projective. Examples include affine space A^n_k, projective space P^n_k, and any open subset of a projective variety. Any closed subvariety of a projective space is quasi-projective because it is locally closed.

Limitations and scope: Not every finite type k-scheme is quasi-projective; there exist schemes of finite type over k that are not quasi-projective. The notion also extends to quasi-projective morphisms: a morphism X -> S is quasi-projective if X can be embedded as a locally closed subscheme of a projective bundle over S. Quasi-projectivity provides a useful balance between flexibility and structure, enabling the use of projective techniques while accommodating common open or embedded subsets.