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

Let $X$ be a smooth complex projective variety and let $Z = (s = 0)$ be a smooth submanifold which is the zero locus of a section of an ample vector bundle $mathcal E$ of rank $r$ with $dim Z = dim X - r$. We show with some examples that in general the Kleiman--Mori cones $overline{{ m NE}(Z)}$ and $overline{{ m NE}(X)}$ are different. We then give a necessary and sufficient condition for an extremal ray in $overline{{ m NE}(X)}$ to be also extremal in $overline{{ m NE}(Z)}$. 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 $geq 4$. In the last section we prove a general result in case $Z$ is a minimal variety with $0 leq kappa (Z) < dim Z$.