Discriminant loci of varieties with smooth normalization

Abstract

Let X be a smooth complex projective n-fold endowed with an ample and spanned line bundle (L). Under the assumption that Γ(L) defines a generically one-to-one map we describe the singular set of the general element in the main component of the discriminant locus of |L|. This description is used to show that (X:,L) is covered by linear 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.

Keywords:

• Ample and spanned line bundle
• generically one-to-one morphism
• discriminant locus
• nondegenerate quadratic singularity
• defect
• adjunction theory