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 × 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 smooth variety, every projective bundle is of the form P ( E ) {\displaystyle \mathbb {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. The projective bundle of a vector bundle E is the same thing as the Grassmann bundle G 1 ( E ) {\displaystyle 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 p: P(E) → X 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 → O P ( E ) ( − 1 ) → p ∗ E → Q → 0 {\displaystyle 0\to {\mathcal {O}}_{\mathbf {P} (E)}(-1)\to p^{*}E\to Q\to 0}
where Q is called the tautological quotient-bundle. 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 E; in other words, the zero-locus of this section is P(E). A particularly useful instance of this construction is when F is the direct sum E ⊕ 1 of E and the trivial line bundle (i.e., the structure sheaf). Then P(E) is a hyperplane in P(E ⊕ 1), called the hyperplane at infinity, and the complement of P(E) can be identified with E. In this way, P(E ⊕ 1) is referred to as the projective completion (or "compactification") of E. The projective bundle P(E) is stable under twisting E by a line bundle; precisely, given a line bundle L, there is the natural isomorphism:
g : P ( E ) → ∼ P ( E ⊗ L ) {\displaystyle g:\mathbf {P} (E){\overset {\sim }{\to }}\mathbf {P} (E\otimes L)}
such that g ∗ ( O ( − 1 ) ) ≃ O ( − 1 ) ⊗ p ∗ L . {\displaystyle g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {O}}(-1)\otimes p^{*}L.} (In fact, one gets g by the universal property applied to the line bundle on the right.)
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