Projective bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
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*). In particular, if X is a compact Riemann surface, the obstruction vanishes i.e. H2(X,O*)=0.
The projective bundle of a vector bundle E is the same thing as the Grassmann bundle of 1-planes in E.
The projective bundle P(E) of a vector bundle E is characterized by the universal property that says:
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).
Let E ⊂ F be vector bundles (locally free sheaves of finite rank) on X and G = F/E. Let q: P(F) → X be the projection. Then the natural map O(-1) → q*F → q*G is a global section of the sheaf hom Hom(O(-1), q*G) = q*G ⊗ O(1). Moreover, this natural map vanishes at a point exactly when the point is a line in a fiber of E; in other words, the zero-locus of this section is P(E).
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