Projectivization

In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space [math]\displaystyle{ {\mathbb P}(V) }[/math], whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of [math]\displaystyle{ {\mathbb P}(V) }[/math] 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 [math]\displaystyle{ {\mathbb P}(V) }[/math] is the quotient of the open set V\{0} of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of [math]\displaystyle{ {\mathbb P}(V) }[/math] in the sense of algebraic geometry is one less than the dimension of the vector space V.
• Projectivization is functorial with respect to injective linear maps: if

[math]\displaystyle{ f: V\to W }[/math]

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

[math]\displaystyle{ \mathbb{P}(f): \mathbb{P}(V)\to \mathbb{P}(W). }[/math]

In particular, the general linear group GL(V) acts on the projective space [math]\displaystyle{ {\mathbb P}(V) }[/math] by automorphisms.

Projective completion

A related procedure embeds a vector space V over a field K into the projective space [math]\displaystyle{ {\mathbb P}(V\oplus K) }[/math] of the same dimension. To every vector v of V, it associates the line spanned by the vector (v, 1) of V ⊕ K.

Generalization

In algebraic geometry, there is a procedure that associates a projective variety 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 [math]\displaystyle{ {\mathbb P}(V). }[/math] This Proj construction gives rise to a contravariant functor from the category of graded commutative rings and surjective graded maps to the category of projective schemes.