Contributions to the Problem of equisingularity

The general problem which we propose in these lectures is the following: given an irreducible subvariety W of the singular locus of an algebraic (or algebroid) variety V and given a simple point Q of W, give a precise meaning to the following intuitive statement: “

the singularity which

V

has at the point

Q is ‘not worse’ than (or is ‘of the same type’ as) the singularity which V has at the general point of W”. We briefly phrase this statement as follows: “V

is equisingular along

W,

at

Q.” It is understood that we require the solution to consist not merely of some plausible definition and some reasonable consequences, but above all of a body of

criteria

of various nature (algebro-geometric, topological and differentio-geometric) and of the proofs of

equivalence

of these various criteria.

In this lecture we give, in the first place, a complete solution of this problem in the special case of cod

V

W

= 1. We also treat a special type of equisingularity which we call

equisaturation

; we are led to this concept by our algebraic theory of

saturation and saturated local rings.

For both of these topics we need a thorough analysis of some old and new aspects of the concept of

equivalent

singularities of plane algebroid curves. This analysis is developed in Sections 1–6. In the last section we discuss connections with the differentiogeometric conditions A and B of Whitney-Thom.