  Smooth complex polarized varieties $(X,L)$ with a vector subspace $V
\subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that the
locus ${\Cal D}(X,V)$ of singular elements of $|V|$ has codimension equal to
$\dim(X)-i$, $i=3,4,5$, the last case under the additional assumption that $X$
has Picard number one. In fact it is proven that this codimension cannot be
$\dim(X)-4$ while it is $\dim(X)-3$ if and only if $(X,L)$ is a scroll over a
smooth curve. When the codimension is $\dim(X)-5$ and the Picard number is one
only the Pl\"ucker embedding of the Grassmannian of lines in $\Bbb P^4$ or one
of its hyperplane sections appear. One of the main ingredients is the
computation of the top Chern class of the first jet bundle of scrolls and
hyperquadric fibrations. Further consequences of these computations are also
provided.
