Classification of Simple Plane Curve Singularities and Their Auslander-reiten Quiver

This thesis is largely a consolidation of parts of lectures given by Yoshino, Y. at Tokyo Metropolitan University in 1987 (cf. [17]). We aim to investigate the Cohen-Macaulay modules of simple plane curve singularities. We consider such simple plane curve singularities algebraically as quotient rings R = k[[x, y]]/(f) of the formal power series ring in two variables over an algebraically closed field k. In fact, the study of Cohen-Macaulay modules over such rings reduces to the study of torsion-free R-modules. We prove a classification theorem of one-dimensional rings R by the power series f , as well as an easy extension of this theorem to higher dimensions. We label the resultant classes of R by the classical Dynkin types An, Dn, E6, E7, E8 (cf. [6]). For the case of An in dimension one, we construct the quiver of morphisms between indecomposable torsion-free modules. We will make reference to an extra structure element of this quiver, which indicates the existence of Auslander-Reiten sequences of the torsion-free modules. A complete list of all such Auslander-Reiten quivers will be provided in appendix A.