Projective bundle

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.

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 ${\displaystyle G_1(E)}$ 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):

${\displaystyle 0\to {\mathcal{O}}_{\mathbf{P}(E)}(-1)\to p^{\ast }E\to Q\to 0}$

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:

${\displaystyle g:\mathbf{P}(E)\stackrel{\sim }{\to }\mathbf{P}(E\otimes L)}$

such that ${\displaystyle g^{\ast }(\mathcal{O}(-1))\simeq \mathcal{O}(-1)\otimes p^{\ast }L.}$ (In fact, one gets g by the universal property applied to the line bundle on the right.)