Low dimensional discriminant loci and scrolls

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 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 Plucker embedding of the Grassmannian of lines in 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.