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: = Ox, 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 Cf.

In the final section we briefly discuss properties of plane curve singularities from the viewpoint of the local ring OC, 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.