Connections between the geometry of a projective variety and of an ample section

Let X be a smooth complex projective variety and let Z be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dimZ=dimX−r. We show, with some examples, that in general the Kleiman-Mori cones NE(Z) and NE(X) are different. We then give a necessary and sufficient condition for an extremal ray in NE(X) to be also extremal in NE(Z). We apply this result to the case r=1 and Z a Fano manifold of high index; in particular we classify all X with an ample divisor which is a Mukai manifold of dimension ≥4. In the last section we prove a general result in case Z is a minimal variety with 0≤κ(Z)<dimZ.