# 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 P$^n$-bundle if it is locally a projective n-space; i.e., $X\times _{S}U\simeq \mathbb {P} _{U}^{n}$ and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form $\mathbb {P} (E)$ 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 H$^2$(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 H$^2$(X,O*) only if the projective bundle is the projectivization of a vector bundle. In particular, if X is a compact Riemann surface then H$^2$(X,O*)=0, and so this obstruction vanishes.
The projective bundle of a vector bundle E is the same thing as the Grassmann bundle $G_{1}(E)$ 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):
    $0\to {\mathcal {O}}_{\mathbf {P} (E)}(-1)\to p^{*}E\to Q\to 0$
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:
    $g:\mathbf {P} (E){\overset {\sim }{\to }}\mathbf {P} (E\otimes L)$
such that $g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {O}}(-1)\otimes p^{*}L.$ (In fact, one gets g by the universal property applied to the line bundle on the right.)

## Examples

Many non-trivial examples of projective bundles can be found using fibrations over $\mathbb {P} ^{1}$ such as Lefschetz fibrations. For example, an elliptic K3 surface $X$ is a K3 surface with a fibration
   $\pi :X\to \mathbb {P} ^{1}$
such that the fibers $E_{p}$ for $p\in \mathbb {P} ^{1}$ are generically elliptic curves. Because every elliptic curve is a genus 1 curve with a distinguished point, there exists a global section of the fibration. Because of this global section, there exists a model of $X$ giving a morphism to the projective bundle
   $X\to \mathbb {P} ({\mathcal {O}}_{\mathbb {P} ^{1}}(4)\oplus {\mathcal {O}}_{\mathbb {P} ^{1}}(6)\oplus {\mathcal {O}}_{\mathbb {P} ^{1}})$
defined by the Weierstrass equation
   $y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}$
where $x,y,z$ represent the local coordinates of ${\mathcal {O}}_{\mathbb {P} ^{1}}(4),{\mathcal {O}}_{\mathbb {P} ^{1}}(6),{\mathcal {O}}_{\mathbb {P} ^{1}}$, respectively, and the coefficients
   $a_{i}\in H^{0}(\mathbb {P} ^{1},{\mathcal {O}}_{\mathbb {P} ^{1}}(2i))$
are sections of sheaves on $\mathbb {P} ^{1}$. Note this equation is well-defined because each term in the Weierstrass equation has total degree $12$ (meaning the degree of the coefficient plus the degree of the monomial. For example, ${\text{deg}}(a_{1}xyz)=2+(4+6+0)=12$).

## Cohomology ring and Chow group

Let X be a complex smooth projective variety and E a complex vector bundle of rank r on it. Let p: P(E) → X be the projective bundle of E. Then the cohomology ring H$^*$(P(E)) is an algebra over H$^*$(X) through the pullback p$^*$. Then the first Chern class ζ = c$_1$(O(1)) generates H$^*$(P(E)) with the relation
    $\zeta ^{r}+c_{1}(E)\zeta ^{r-1}+\cdots +c_{r}(E)=0$
where c$_i$(E) is the i-th Chern class of E. One interesting feature of this description is that one can define Chern classes as the coefficients in the relation; this is the approach taken by Grothendieck.
Over fields other than the complex field, the same description remains true with Chow ring in place of cohomology ring (still assuming X is smooth). In particular, for Chow groups, there is the direct sum decomposition
    $A_{k}(\mathbf {P} (E))=\bigoplus _{i=0}^{r-1}\zeta ^{i}A_{k-r+1+i}(X).$
As it turned out, this decomposition remains valid even if X is not smooth nor projective. In contrast, A$_k$(E) = A$_{k-r}$(X), via the Gysin homomorphism, morally because that the fibers of E, the vector spaces, are contractible.