9 - Bi-Lipschitz equivalence, integral closure and invariants

Summary

Abstract

In this note we relate several different ways of looking at an infinitesimal notion of bi-Lipschitz equisingularity. In the case of curves we show how invariants related to these notions can be used to tell if a family is bi-Lipschitz equisingular.

Introduction

The study of bi-Lipschitz equisingularity was started by Zariski [22], Pham and Teissier [18], [19], and was further developed by Lipman [13], Mostowski [14], [15], Parusinski [16], [17], Birbrair [2] and others.

In this note, we begin the study of bi-Lipschitz equisingularity from the perspective of our previous work on Whitney equisingularity. In the approach of that work, the study of the equisingularity condition is developed along two avenues. One direction is the study of the appropriate closure notion on modules and applying it to the Jacobian module of a singularity [4]. The other direction is through the study of analytic invariants which control the particular stratification condition [5]. The interaction between the two approaches is useful in understanding each approach.

In section two of this paper, we work on the first approach, looking to the hypersurface case, and to similar constructions for motivation. The construction which seems most promising, defining the saturation of I via the blow-up of X by I, is the analogue of a construction used to study the weak sub-integral closure of an ideal in [11], written with Marie Vitulli.

In section three, we apply the theory of analytic invariants to describe when a family of space curves is bi-Lipschitz equisingular.