Theo de Jong

An Algorithm for Computing the Integral Closure

In this article we describe an algorithm for computing the integral closure of a reduced Noetherian ring, provided this integral closure is finitely generated as a module.

The Normalization: a new Algorithm, Implementation and Comparisons

We present a new algorithm for computing the normalization \(\bar{R}\) of a reduced affine ring R, together with some remarks on efficiency based on our experience with an implementation of this algorithm in SINGULAR (cf. [2]).

Plane Curve Singularities

This chapter is devoted to the study of plane curve singularities. First of all, from the existence of the normalization, and the fact that one-dimensional normal singularities are smooth, which are results proved in the previous chapter, it follows that one can parameterize curve singularities. Thus, if R is the local ring of an irreducible plane curve singularity, we have an injection R ⊂ ℂ{t}. From the Finiteness of Normalization Theorem 1.5.19 it follows without much difficulty that the codimension of R in ℂ{t} (as ℂ-vector space) is finite. This codimension is called the i-invariant. Here we prove the converse of this statement, that is, given a subring R of ℂ {t} of finite codimension, generated by two elements, say x(t) and y(t), then R the local ring of an irreducible plane curve singularity, say given by f(x,y) = 0. In order to obtain information on the order of f, we introduce the intersection multiplicity of two (irreducible) plane curve singularities (C, o) and (D, o). This number can be calculated as follows. Consider the normalization ℂ{t} of O c,o , and let (D, o) be given by g = 0. Via the inclusion O C,o ∈ℂ{t}, we can consider at g as an element of ℂ{t}. Then the intersection multiplicity is given by the vanishing order of g. From this it easily follows that if R = ℂ{x(t),y(t)}, and f ∈ ℂ{x,y} is irreducible with f(x(t),y(t)) = 0, then the order of f in y is equal to ord t (x(t)), and similar for x. As an application we prove that if we have a convergent power series f ∈ ℂ {x,y}, which is irreducible when considered as a formal power series, then it is already irreducible as a convergent power series.

Classification of Simple Hypersurface Singularities

This chapter is devoted to the study of hypersurface singularities. There are various equivalence relations one can consider on the set of all functions f ∈ ℂ {x 1,... ,x n }. The two most important ones are right equivalence and contact equivalence. Two functions f and g are called right equivalent if there exists an automorphism φ of ℂ {x 1,... ,x n } such that φ(f) = g. They are called contact equivalent if there exists an automorphism φ and a unit u in ℂ{x 1,...,x n} such that φ(f) = u{g. Until now we considered contact equivalence in this book (without using this name), but in this chapter we will mainly consider right equivalence, because in this case the theorems are somewhat easier to formulate and to prove. (One does not have to worry about the unit.) The aim of this chapter is to give a beginning of the classification of functions up to right equivalence. The first thing is to consider Finite Determinacy Theorems. Let m be the maximal ideal of ℂ{x 1,... ,x n }. A function / is called (right) k-determined if for all elements g with f - g ∈ m k +1 the functions / and g are right equivalent. In particular, a k-determined f is right equivalent to a polynomial of degree at most k. A function is called finitely determined, if there exists a k ∈ℕ such that / is k-determined. It is maybe a little surprising that all functions which have an isolated singularity are finitely determined. The proof, however, is not difficult at all: it is a direct application of Newton’s Lemma, see 3.3.34. This result is not strong enough for our purposes, however. We want to find a small k such that / is k-determined.

Affine Algebraic Geometry

The purpose of this chapter is to introduce the reader to the basic definitions, ideas and notations of affine algebraic geometry. Loosely speaking, an affine algebraic set is the zero set of finitely many polynomials in an affine space K n , where K is an algebraically closed field.1 Some of the results of this chapter are true and proved over arbitrary algebraically closed fields. This is the reason why we do not consider only the case K = ℂ as we will do from Chapter 3 onwards. Admittedly, our main interest in this book is the behavior of such zero sets in the neighborhood of a point. We will discuss and study the algebraic case first, however, simply because the proofs of the statements in this case usually are easier. This is because one does not have to bother one-selves with convergence questions. Having grasped the main geometric and algebraic ideas in the affine case, the reader is hopefully ready to tackle the local case in Chapter 3. A second reason for doing the affine case, too, is that we need to do some algebraic geometry, in order to classify hypersurface singularities, see Chapter 9.

The Principle of Conservation of Number

The aim of this chapter is to study the behavior of certain invariants of singularities in a family. The idea is that the invariants should be constant in the following sense: if in the family a singular point splits into several singular points by varying the parameter, then the sum of the invariants of those singular points should be equal to the invariant of the point we started with.

Further Development of Analytic Geometry

In this chapter we will continue with the basics of the theory of germs of analytic spaces (and of affine varieties). Probably the most important invariant of (irreducible) germs of analytic spaces is the dimension. There are various ways to introduce it. We decided to give, in fact, four definitions of dimension. We introduce this many of them, so that we have a flexible development of the theory.

Basics of Analytic Geometry

In this chapter we start studying local analytic geometry, that is, the zero sets of analytic functions in a (small) neighborhood of a point. To see why we want to do so, we look at the affine hypersurface in ℂ2 defined by f(x,y) = y 2-x 2(x + 1) = 0. In a small neighborhood U of (0,0) we see two “parts”, or components of f(x,y) = 0.

Deformations of Singularities

In the final chapter of this book, we study deformations of germs of complex spaces. The ultimate goal is to prove the existence of a semi-universal deformation of (X, o), in case it has an isolated singularity. As the proof of this theorem is quite involved, we will first treat some special cases. So, in Section 10.1, we will consider isolated hypersurface singularities (X, o) given by f = 0.