Projectivization

<h2 id="properties">Properties</h2>
<ul><li>Projectivization is a special case of the <a href="/facts/Quotient_space_(topology)/muE0qQVC">factorization</a> by a <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a>: the projective space P(V) 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 <a href="/facts/Dimension_(mathematics)/0ktmUyac">dimension</a> of P(V) in the sense of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> is one less than the dimension of the vector space V.</li>
<li>Projectivization is <a href="/facts/Functor/l2u3rIBE">functorial</a> with respect to <a href="/facts/Injective/5ES6tqf5">injective</a> linear maps: if</li></ul>

f
        :
        V
        →
        W
      
    
    {\displaystyle f:V\to W}

is a linear map with trivial <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">kernel</a> then f defines an algebraic map of the corresponding projective spaces,

P
        
        (
        f
        )
        :
        
          P
        
        (
        V
        )
        →
        
          P
        
        (
        W
        )
        .
      
    
    {\displaystyle \mathbf {P} (f):\mathbf {P} (V)\to \mathbf {P} (W).}

In particular, the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> GL(V) acts on the projective space P(V) by <a href="/facts/Automorphism/WnyxNFkQ">automorphisms</a>.
<h2 id="projective-completion">Projective completion</h2>
A related procedure embeds a vector space V over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K into the projective space P(V ⊕ K) of the same dimension. To every vector v of V, it associates the line spanned by the vector (v, 1) of V ⊕ K.

<h2 id="generalization">Generalization</h2>
Main article: <a href="/facts/Projective_bundle/W7jjGmlW">Projective bundle</a>
In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, there is a procedure that associates a <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a> Proj S with a <a href="/facts/Graded_algebra/AIUjkSaP">graded commutative algebra</a> S (under some technical restrictions on S). If S is the <a href="/facts/Ring_of_polynomial_functions/q99CvhPj">algebra of polynomials on a vector space</a> V then Proj S is P(V). This <a href="/facts/Proj_construction/gcvPmjpU">Proj construction</a> gives rise to a <a href="/facts/Contravariant_functor/l2u3rIBE">contravariant functor</a> from the category of graded commutative rings and surjective graded maps to the category of projective <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a>.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Projectivization of a vector space: projective geometry definition vs algebraic geometry definition". Mathematics Stack Exchange. Retrieved 2024-08-22. <a href="https://math.stackexchange.com/questions/2298714/projectivization-of-a-vector-space-projective-geometry-definition-vs-algebraic" target="_blank">https://math.stackexchange.com/questions/2298714/projectivization-of-a-vector-space-projective-geometry-definition-vs-algebraic</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weisstein, Eric W. "Projectivization". mathworld.wolfram.com. Retrieved 2024-08-27. <a href="https://mathworld.wolfram.com/Projectivization.html" target="_blank">https://mathworld.wolfram.com/Projectivization.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Projectivization open-in-new

Projectivization