Projectivization

Source: Wikipedia, the free encyclopedia.

In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P(V), whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of P(V) formed by the lines contained in S and is called the projectivization of S.[1]

Properties

is a linear map with trivial kernel then f defines an algebraic map of the corresponding projective spaces,
In particular, the general linear group GL(V) acts on the projective space P(V) by automorphisms.

Projective completion

A related procedure embeds a vector space V over a field K into the projective space P(VK) of the same dimension. To every vector v of V, it associates the line spanned by the vector (v, 1) of VK.

Generalization

In

contravariant functor from the category of graded commutative rings and surjective graded maps to the category of projective schemes
.

References

  1. ^ Weisstein, Eric W. "Projectivization". mathworld.wolfram.com. Retrieved 2024-08-27.