Discriminant loci of ample and spanned line bundles

Let (X, L, V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V subset of H-0(X, L) spans L. The discriminant locus D(X, V) subset of vertical bar V vertical bar is the algebraic subset of singular elements of vertical bar V vertical bar. We study the components of D(X, V) in connection with the jumping sets of (X, V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X, L, V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided. (c) 2007 Elsevier B.V. All rights reserved.