Projectivization

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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]

Contents

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 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 P(V). 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.

References

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