Segre Numbers and Hypersurface Singularities

TL;DR

This paper generalizes the concept of multiplicity of ideals to Segre numbers and applies these to study hypersurface singularities and equisingularity conditions.

Contribution

It introduces Segre numbers as a new invariant and extends key theorems involving multiplicity to this broader context.

Findings

Generalization of multiplicity to Segre numbers

Extension of the Rees-B"oger theorem

Application to equisingularity conditions

Abstract

We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-B\"oger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of various equisingularity conditions, such as Verdier's condition W, and conditions $A_f$ and $W_f$.