  Let E be a rank two vector bundle on a scheme X. The following three
structures are shown to be equivalent : a) A primitive quadratic map q: E -->
L, with values in an invertible module L. b) A double covering f: Y --> X
endowed with an invertible module F on Y, plus an isomorphism from the direct
image of F to E. c) An effective Cartier divisor on the projective space P(E),
of degree two over X. The passages from one of these points of view to another
are carefully settled in their greatest generality: we only need that 2 is
invertible on X. We then restrict to the projective space P_n, and we prove
that for any double covering Y of P_n, the homomorphism on Picard groups it
induces is an isomorphism if n ? 3; we finally apply this result to quadratic
forms on rank two vector bundles on P_n.
