Ample vector bundles and intrinsic quadric fibrations over irrational curves

Let E be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X. This work is part of the following problem: to study and classify the pair (X, E) assuming the existence of a regular section s ∈Γ (X, E) whose zero locus is a special subvariety of X . In [2] and [11], the case of Z quadric fibration, respectively of diimension 2 or more, over a smooth curve is discussed under the further hypothesis that the quadric fibration structure is induced on Z by an ample line bundle L on X . Here the same situation is considered, and classification is given assuming the base curve to be irrational, in the more general case that the quadric fibration structure of Z is intrinsic, i.e. not a priori induced by a polarization of X .