Singular Points of Reducible Sextic Curves

There are 106 individual types of singular points for reducible complex sextic curves. 1. Introduction Extensive studies of simple singularities of complex sextic curves have been made by Urabe [1, 2] and Yang [3]; see also [4]. Singular points of sextic curves of torus type have been classified by Oka and Pho [5, 6]. The authors have classified all individual types of singular points for irreducible real and complex sextic curves in a previous paper [7]. In this paper, we will determine the individual types of singular points for reducible complex sextic curves. There are 106 types. The proof is as elementary as possible and relies heavily on Puiseux expansions, which are computed by using Maple when necessary. Definition of the Equivalence Relation. Let us begin with the definition of the equivalence relation on singular points of algebraic curves. The equivalence relation is that two singular points are equivalent if and only if their probranches have the same exponents of contact. The term “probranch” was used by Wall in [8] to refer to the distinct Puiseux expansions of an algebraic curve at a point. In the same book, Wall defines the exponent of contact between two probranches to be the smallest exponent such that the corresponding terms in the two Puiseux expansions have unequal coefficients. Given an algebraic curve with a singular point at the origin, let us now describe how to associate a tree diagram to this singular point once we have the Puiseux expansions. Each time at least one probranch separates, record the smallest exponent where that happens. Place all such exponents in a row at the top. For each exponent in the top row, there corresponds a column of vertices. Each Puiseux expansion corresponds to exactly one vertex in that column, and those expansions with the same coefficients up to that exponent correspond to the same vertex. We start with one vertex on the left corresponding to the power zero. Line segments are drawn connecting the vertices from left to right, where each polygonal path from left to right corresponds to Puiseux expansions having the same set of coefficients up to a given exponent. The diagram stops at the first exponent where each vertex in that column corresponds to exactly one Puiseux expansion. This tree diagram uniquely specifies the singularity type (up to permutations of vertices within columns) provided that no tangent line at the origin is vertical. It follows from [8, Lemma？？4.1.1] that the diagram we assign to a singular point is invariant under a linear change of coordinates. The diagram just codifies in