mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a quasi-projective variety in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

is a

locally closed subset In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied: * E is the intersection of an open set and a closed set in X. * For each point x\in E ...

of a

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...

, i.e., the intersection inside some

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

of a Zariski-open and a

Zariski-closed In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is no ...

subset. A similar definition is used in

scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...

, where a ''quasi-projective scheme'' is a locally closed subscheme of some

Relationship to affine varieties

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

is a Zariski-open subset of a

, and since any closed affine subset

U

can be expressed as an intersection of the

projective completion In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

\bar

and the affine space embedded in the projective space, this implies that any

affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...

is quasiprojective. There are

locally closed In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied: * E is the intersection of an open set and a closed set in X. * For each point x\in E ...

subsets of projective space that are not affine, so that quasi-projective is more general than affine. Taking the complement of a single point in projective space of dimension at least 2 gives a non-affine quasi-projective variety. This is also an example of a quasi-projective variety that is neither affine nor projective.

Examples

Since quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as ''varieties''. Varieties isomorphic to affine algebraic varieties as quasi-projective varieties are called

affine varieties In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...

; similarly for projective varieties. For example, the complement of a point in the affine line, i.e.,

X=\mathbb^1 \setminus \

, 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 mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...

in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called quasi-affine. Quasi-projective varieties are ''locally affine'' in the same sense that a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

is locally Euclidean : every point of a quasi-projective variety has a neighborhood which is an affine variety. This yields a basis of affine sets for the

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

on a quasi-projective variety.

Citations

References

* {{refend Algebraic varieties

Relationship to affine varieties

Examples

See also

Citations

References