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 ...

, projectivization is a procedure which associates with a non-zero

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

''V'' a

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 ...

(V)

, whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multiplication defines a subset of

(V)

formed by the lines contained in ''S'' and is called the projectivization of ''S''.

Properties

* Projectivization is a special case of the factorization by a group action: the projective space

(V)

is the quotient of the open set ''V''\ of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

(V)

in the sense of

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 one less than the dimension of the vector space ''V''. * Projectivization is functorial with respect to

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

linear maps: if ::

f: V\to W

: is a linear map with trivial kernel then ''f'' defines an algebraic map of the corresponding projective spaces, ::

\mathbb(f): \mathbb(V)\to \mathbb(W).

: In particular, the general linear group GL(''V'') acts on the projective space

(V)

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...

Projective completion

A related procedure embeds a vector space ''V'' over a field ''K'' into the projective space

(V\oplus K)

of the same dimension. To every vector ''v'' of ''V'', it associates the line spanned by the vector of .

Generalization

, there is a procedure that associates 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 wi ...

Proj ''S'' with a graded commutative algebra ''S'' (under some technical restrictions on ''S''). If ''S'' is the algebra of polynomials on a vector space ''V'' then Proj ''S'' is

{\mathbb P}(V).

This

Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not funct ...

gives rise to a

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

from the category of graded commutative rings and surjective graded maps to the category of projective schemes. Projective geometry Linear algebra