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 l
ast 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 $Bbb P^4$ 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.
Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a sub
manifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.
