On a Projective Bundle Formula

For a vector bundle E → X with X a smooth variety over a field we prove a formula for the proper pushforward π∗ : A ∗ P(E ) → A∗X, where π : P(E ) → X denotes the associated projective bundle of lines in E . Our formula is relative in the sense that it depends on the rank of E while being independent of the dimension of X. As applications, we apply the formula to compute the pushforward to X of the Chern class of a fibration Y → X, whose fibers are degree d projective hypersurfaces with d arbitrary, and then derive a ‘relative Bézout-type formula’ for the intersection of dim(P(E )) divisors in P(E ) in terms of invariants of X.