Multiplicities of discriminants

We compute the multiplicity of the discriminant of a line bundle £ over a nonsingular varietyS at a given sectionX, in terms of the Chern classes of £ and of the cotangent bundle ofS, and the Segre classes of the jacobian scheme ofX inS. ForS a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities ofX _red. Also, under certain hypotheses, we provide formulas for the “higher discriminants” that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be “small”.