Varieties of small degree with respect to codimension and ample vector bundles

Abstract

Let $\mathscr{E}$ be an ample vector bundle on a projective manifold X, with a section vanishing on a smooth subvariety Z of the expected dimension, and let H be an ample line bundle on X inducing a very ample ample line bundle H_Z on Z. Triplets (X, $\mathscr{E}$, H) as above are classified assuming that Z, embedded by |H_Z|, is a variety of small degree with respect to codimension.