The Classification of Singular Points of Algebraic Curves

Abstract

Introduction.To each singular point P of an algebraic curve C there is associated the local ring of C at P and the completion of this local ring, called the complete local ring at P. One approach to the problem of classification of singular points is to consider as equivalent all singular points whose complete local rings are isomorphic.The purpose of this paper is to obtain a scheme for the classification of the complete local rings of those singular points that are cusps or "one-branch" singularities.This paper will be divided into five sections.In the first we show how the problem of classification of complete local rings can be translated into a problem involving the regular operation of an algebraic group on an abstract variety.The theory of algebraic groups and algebraic groups as transformation groups on a variety has been well developed in papers by Borel [1], Rosenlicht [5], and Weil [7].Using this theory, in the second section we prove a theorem that provides the required solution.The third section records the construction of canonical forms for a nontrivial example.In the fourth section we list complete sets of canonical forms for certain simple classes of rings.In the final section we discuss the effect of the field characteristic and the problem of extending this approach to "multiple-branch" singularities.1.The construction of the algebraic model.By a cusp we mean a singular point P of the curve C whose local ring oP is contained in one and only one valuation ring R of the function field Q(C), where Q is the universal domain.The complete local ring o* at such a point is described by the following theorem.Theorem 1.The complete local ring of a curve C at a cusp P is isomorphic to a proper complete subring of the formal power series ringQ.{J_tJ]which contains Q. and a conductor of of in fi [[i]].Proof.If M is the maximal ideal of the valuation ring R, it can be shown there exists an integer n so that M"cz oP.(See Theorem 2 in [6].)It follows that the topology induced on oP as a subset of R is its own topology as a local ring.Since