Multiplicities of Classical Varieties

TL;DR

This paper explores the relationships between various multiplicities of classical varieties, providing formulas and computations for specific cases like rational normal scrolls and determinantal varieties, linking algebraic geometry with random matrix theory.

Contribution

It establishes a relationship between $j$-multiplicity and fiber cone degree, and computes these multiplicities for rational normal scrolls and determinantal varieties using standard monomial theory.

Findings

Computed $j$-multiplicity for rational normal scrolls.

Derived formulas for $j$- and $\epsilon$-multiplicities of determinantal varieties.

Connected multiplicity calculations to integrals in random matrix theory.

Abstract

The $j$-multiplicity plays an important role in the intersection theory of St\"uckrad-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.