  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
$V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset
|V|$ is the algebraic subset of singular elements of $|V|$. We study the
components of $\Cal D(X,V)$ 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.
