On the discriminant of spanned line bundles

Let X be an irreducible smooth complex projective variety of dimension n, L a nontrivial spanned line bundle on X, V \subseteq H^0(X,L) a vector subspace od global sections spanning L with dim(V)=N+1. The discriminant variety D(X,V) parameterizes the singular elements of |V|. By Bertini's theorem one can write dim(D(X,V))=N-1-k with k a non-negative integer. An upper bound on k in terms of N, n, and the dimension of the fibers of the map defined by V is established. Moreover, triplets (X,L,V) for which k is near the maximum are classified.