24 - Gaffney's work on equisingularity

Summary

Abstract

A survey of equisingularity theory focussed on Terry Gaffney's work.

The article begins with an account of the early history of equisingularity. Next I develop notation, particularly for polar varieties; recall the theory of integral closures of ideals, show how Gaffney generalised this to integral closures of modules, and list a variety of applications he has made.

The invariants available are classical and Buchsbaum-Rim multiplicities of modules, polar multiplicities and Segre numbers of ideals, and generalisations to modules. Some of the main theorems are of the form: the constancy of certain numerical invariants of a family imply equisingularity of the family (usually in the form of Whitney triviality). Many of the proofs use results showing that constancy of some invariants implies an integral dependence relation. One notable paper gives a sufficient condition for topological triviality of families of maps.

Introduction

The classification of singularities of plane curves was achieved in 1932 by Brauner [2], Burau [6], [7] and Zariski [60]: it yields an easily stated, necessary and sufficient condition for topological equivalence, which clearly does not imply analytic equivalence. Probably the simplest example is the case of 4 concurrent lines xy(x + y)(x + ty) = 0 with t an invariant of analytic, but not of topological equivalence.

This situation presents the problem of creating a theory of equivalence of families of objects (e.g. algebraic varieties or morphisms) which will say when the members of the family are essentially the same.