CHAPTER 1 - EXAMPLES OF ISOLATED SINGULAR POINTS

Summary

Loosely speaking, an analytic germ (X, x) in C^N is called a complete intersection if the minimal number of equations by which it can be defined equals its codimension in C^N. Although such germs will be our principal object of study, we must realize that quite often analytic germs are not given to us as the common zero set of a specific set of equations. In such cases it is unreasonable to expect these germs to be complete intersections. We illustrate this by describing several constructions of singular germs, most of which fail to yield complete intersections in general. Some of the germs which happen to be complete intersections, will reappear when we make a beginning of the classification in chapter 7. Another goal of this chapter is to make the reader acquainted with, several interesting examples to which the theory we are going to develop may be applied. We do not always provide full proofs of the properties attributed to these singularities. The reader shouldn't feel uneasy about this, for in such cases we will not make any use of them.

Hypersurface singularities

(1. 1) Let X be an analytic set in an open ∪ ⊂ Cⁿ⁺¹ and let x ∈ X. The ideal 1_{X, x} of holomorphic functions at x vanishing on X is principal and nonzero if and only if each irreducible component of the germ (X, x) is of dimension n (see for instance Whitney (1972), Ch. 2, Thm's 10 C, D).