Projectivization
{{Short description|Mathematics procedure}}
{{Refimprove|date=August 2024}}
In mathematics, projectivization is a procedure which associates with a non-zero vector space {{math | V}} a projective space {{math | P(V)}}, whose elements are one-dimensional subspaces of {{math | V}}. More generally, any subset {{math | S}} of {{math | V}} closed under scalar multiplication defines a subset of {{math | P(V)}} formed by the lines contained in {{math | S}} and is called the projectivization of {{math | S}}.{{Cite web |last=Weisstein |first=Eric W. |title=Projectivization |url=https://mathworld.wolfram.com/Projectivization.html |access-date=2024-08-27 |website=mathworld.wolfram.com |language=en}}
Properties
- Projectivization is a special case of the factorization by a group action: the projective space {{math | P(V)}} is the quotient of the open set {{math|V \ {{mset|0}}}} of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of {{math | P(V)}} in the sense of algebraic geometry is one less than the dimension of the vector space {{math | V}}.
- Projectivization is functorial with respect to injective linear maps: if
::
: is a linear map with trivial kernel then {{math | f}} defines an algebraic map of the corresponding projective spaces,
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: In particular, the general linear group GL(V) acts on the projective space {{math | P(V)}} by automorphisms.
Projective completion
A related procedure embeds a vector space {{math |V}} over a field {{math |K}} into the projective space {{math | P(V ⊕ K)}} of the same dimension. To every vector {{math |v}} of {{math |V}}, it associates the line spanned by the vector {{math|(v, 1)}} of {{math|V ⊕ K}}.
Generalization
{{main|Projective bundle}}
In algebraic geometry, there is a procedure that associates a projective variety {{math |Proj S}} with a graded commutative algebra {{math |S}} (under some technical restrictions on {{math |S}}). If {{math |S}} is the algebra of polynomials on a vector space {{math |V}} then {{math |Proj S}} is {{math |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.