mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a quasi-projective variety in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

is a locally closed subset of a

projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

, 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 defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

subset. A similar definition is used in

scheme theory In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different s ...

, where a ''quasi-projective scheme'' is a locally closed

subscheme This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

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 relat ...

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 is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the de ...

\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 is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...

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 (see

Morphism of algebraic varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regu ...

) are called

affine varieties In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...

; 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 (when one assumes that the base field be

algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...

or at least infinite) since any proper closed subset of

\mathbb A^1

is finite. More generally, the variety

\mathbb A^n\setminus\

, with

f\in k_1,\ldots,x_n /math>, is isomorphic to the hypersurface in \mathbb A^given by the equation x_f-1=0 ., Chapter I, Lemma 4.2 For another example, the complement of any

conic A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, thou ...

in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called quasi-affine. Quasi-projective varieties (like their generalization, schemes) 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 defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

on a quasi-projective variety.

Citations

References

* {{refend Algebraic varieties