Ample and spanned vector bundles of sectional genera three (Q757508)

Pairs (X,E) consisting of a complex projective manifold X of dimension \(\geq 2\) and of an ample vector bundle E of rank \(\geq 2\) on X are classified under the assumptions that E is spanned by its global sections and that either \(g(X,\det (E))=3\) or \(g(P,H)=3\), where \(P={\mathbb{P}}_ X(E)\) and H stands for the tautological line bundle of E.