Projective bundle

In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.

By definition, a scheme X over a Noetherian scheme S is a Pⁿ-bundle if it is locally a projective n-space; i.e., ${\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}}$ and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form ${\displaystyle \mathbb {P} (E)}$ for some vector bundle (locally free sheaf) E.^[1]