Nefness of adjoint bundles for ample vector bundles of corank 3 (Q2862566)

Let \(X\) be a smooth complex projective variety, and \(\mathcal E\) an ample vector bundle of rank \(\dim X -3 \geq 2\) on \(X\). In the paper under review, pairs \((X,\mathcal E)\) as above such that the adjoint bundle \(K_X + \det \mathcal E\) is not nef are studied. The main theorem describes the possible structures of these pairs, in the spirit of adjunction theory.NEWLINENEWLINEIn the last part of the paper two applications of the structure theorem are presented: the first is about pairs as above such that \(\mathcal E\) admits a section vanishing on a smooth variety \(Z\) of the expected dimension, whose adjoint bundle \(K_Z+ (\dim Z -3)H_Z\) is not nef, where \(H_Z\) is the restriction of an ample line bundle on \(X\). The second application is about describing classical scrolls which are not adjunction theoretic scrolls for some values of the dimensions of the scroll and of the base.