Quasiprojective variety

In mathematics, a quasiprojective 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.

For example, affine space is a Zariski-open subset of projective space, and since any closed affine subset $U$ can be expressed as an intersection of the projective completion and the affine space embedded in the projective space, 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. This is also an example of quasiprojective varieties that are neither affine nor projective.

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, i.e. $X=mathbb\{A\}^1-0$, is isomorphic to the zero set of the polynomial $xy-1$ in the affine plane. As an affine set X 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.

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

References

* Igor R. Shafarevich, "Basic Algebraic Geometry 1", Springer-Verlag 1999: Chapter 1 Section 4.