Plane Curve Singularities

Abstract

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.