11 - Ideals and clusters

Summary

In Chapter 8 we took a geometric approach to the combinatorics associated to a curve singularity, studying functions on the resolution tree. In this chapter we give a more algebraic presentation. This gives interesting information about the set of ideals in the local ring O₀: = O_{x, y} of O. We obtain a relation between these ideals and ‘clusters’ of infinitely near points, which can be formulated as a Galois correspondence between these.

This has two applications. One is a procedure (Enriques' ‘unloading algorithm’) leading from a numerical definition of an ideal to the effective numerical parameters defining it. The other is a lead in to the study of integral closures of ideals: we establish the surprisingly close connection between integrally closed ideals and exceptional cycles.

We briefly address the question of determinacy, that is, finding for each reduced f ∈ O₀ the least integer n such that the terms of degree n in the power series expansion of f are sufficient to determine the equisingularity type of the curve C_f.

In the final section we briefly discuss properties of plane curve singularities from the viewpoint of the local ring O_C, which is that taken in modern algebraic geometry.

Blowing up ideals

We study ideals I in the ring O₀, (which can be identified with ℂ{x, y}) of germs at O of holomorphic functions on the plane T₀. We begin by showing how I gives rise to an ideal in the local ring of the surface obtained by blowing up at a point, and establishing some basic results relating I to these blown up ideals.