Ample vector bundles with sections vanishing along conic fibrations over curves (Q1127842)

Let \({\mathcal E}\) be an ample vector bundle of rank \(r\geq 2\) on a complex projective manifold \(X\) of dimension \(n=r+2\) having a section whose zero locus is a smooth surface \(Z\). Pairs \((X,{\mathcal E})\) as above are classified under the assumption that \((Z, L_Z)\) is a conic fibration over a smooth curve for some ample line bundle \(L\) on \(X\).