Projective bundle

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a projective bundle is a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> whose fibers are <a href="/facts/Projective_space/7MSpA9cM">projective spaces</a>.
By definition, a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> X over a <a href="/facts/Noetherian_scheme/amyCoMDf">Noetherian scheme</a> S is a Pn-bundle if it is locally a projective n-space; i.e., 
 
 
 
 X
 
 ×
 
 S
 
 
 U
 ≃
 
 
 P
 
 
 U
 
 
 n
 
 
 
 
 {\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}}
 
 and transition automorphisms are linear. Over a regular scheme S such as a <a href="/facts/Smooth_variety/Y76o7b9T">smooth variety</a>, every projective bundle is of the form 
 
 
 
 
 P
 
 (
 E
 )
 
 
 {\displaystyle \mathbb {P} (E)}
 
 for some vector bundle (<a href="/facts/Locally_free_sheaf/h2NoUN7e">locally free sheaf</a>) E.

Projective bundle open-in-new

Projective bundle