Multiplicities of classical varieties

The j-multiplicity plays an important role in the intersection theory of Stuckrad-Vogel cycles, while recent developments confirm the connections between the epsilon-multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the j-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the j-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the j- and epsilon-multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.