Quasiprojective variety

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In mathematics, a quasiprojective variety in algebraic geometry is, in non-intrinsic terms, a Zariski-open subset of a projective variety, or equivalently the intersection of a Zariski closed and a Zariski open set. These sets are also referred to as locally-closed.

For example, affine space is a Zariski-open subset of projective space, and this implies that any affine variety is quasiprojective. There are locally closed subsets of projective space that are not affine, so that quasiprojective is more general than affine. Taking the complement of a single point in projective space of dimension at least 2 gives a non-affine quasiprojective variety. There exist quasiprojective varieties that are neither affine nor projective, such as the complement of a point in projective space of dimension 2, and the previous example.

Since quasiprojective varieties generalize both affine and projective varieties, they are sometimes referred to simply as varieties. Varieties isomorphic to affine algebraic varieties as quasiprojective varieties are called affine varieties; similarly for projective varieties. For example, the complement of a point in the affine line is isomorphic to the zero set of the polynomial xy − 1 in the affine plane. As an affine set the complement is not closed since any polynomial zero on the complement must be zero on the affine line. For another example, the complement of any conic in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called quasi-affine.

Short of the generality of schemes, quasiprojective varieties are in some sense the most general notion of a variety that one would wish to study. Every quasiprojective variety defines a scheme, but the class of schemes is much more general. Quasiprojective varieties have the advantage of an intrinsic definition, not depending on a particular embedding into an ambient space.

Quasiprojective 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 open variety. This yields a basis of affine open sets for the Zariski topology on a quasiprojective variety.