Discriminant loci of varieties with smooth normalization

Let X be a smooth complex projective n-fold endowed with an ample and spanned line bundle L. Under the assumption that |L| defines a generically one-to-one map we describe the singular set and the general element in the main component of the discriminant locus of |L|. The description is used to show that (X,L) is covered by lienar P^k's, where k+1 stands for the codimension of the main component. We also give some applications relating k to the spectral value of (X,L) and discuss some examples.