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

Abstract

Let $E$ 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 $(X,E,H)$ as above are classified assuming that $Z$, embedded by $\left|H_Z\right|$, is a variety of small degree with respect to codimension.