Multiplicities of classical varieties. Issue 4 (5th March 2015) - Jeffries, Jack; Montaño, Jonathan; Varbaro, Matteo
Abstract : Thej ‐multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between theϵ ‐multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between thej ‐multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute thej ‐multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute thej ‐ andϵ ‐multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.