Projective bundle explained

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.,

X x _{S} U \simeq P ^{n} _{U} and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form P (E) for some vector bundle (locally free sheaf) E.

The projective bundle of a vector bundle

Every vector bundle over a variety X gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an obstruction in the cohomology group H²(X,O*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable open cover. On triple overlaps, any lift of these transition functions satisfies the cocycle condition up to an invertible function. The collection of these functions forms a 2-cocycle which vanishes in H²(X,O*) only if the projective bundle is the projectivization of a vector bundle. In particular, if X is a compact Riemann surface then H²(X,O*)=0, and so this obstruction vanishes.

G _{1(E)} of 1-planes in E.

The projective bundle P(E) of a vector bundle E is characterized by the universal property that says:

Given a morphism f: T → X, to factorize f through the projection map is to specify a line subbundle of f^*E.For example, taking f to be p, one gets the line subbundle O(-1) of p^*E, called the tautological line bundle on P(E). Moreover, this O(-1) is a universal bundle in the sense that when a line bundle L gives a factorization f = p ∘ g, L is the pullback of O(-1) along g. See also Cone#O(1) for a more explicit construction of O(-1).

On P(E), there is a natural exact sequence (called the tautological exact sequence):

0 \to l{O} _{P(E)} ( -1 ) \to p ^{*} E \to Q \to 0 where Q is called the tautological quotient-bundle