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Smooth complex polarized varieties $(X,L)$ with a vector subspace $V\subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that thelocus ${\Cal D}(X,V)$ 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 asmooth curve. When the codimension is $\dim(X)-5$ and the Picard number is oneonly the Pl\"ucker embedding of the Grassmannian of lines in $\Bbb P^4$ or oneof its hyperplane sections appear. One of the main ingredients is thecomputation of the top Chern class of the first jet bundle of scrolls andhyperquadric fibrations. Further consequences of these computations are alsoprovided.

Low dimensional discriminant loci and scrolls