Rings of Singularities

This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of positive integers $(p_1,p_2,p_3)$ a two-dimensional isolated graded singularity by an elementary procedure that works over any field $k$ (with no assumptions on characteristic, algebraic closedness or cardinality). This assignment starts from the triangle singularity $x_1^{p_1}+x_2^{p_2}+x_3^{p_3}$ and when applied to the Platonic (or Dynkin) triples, it produces the famous list of A-D-E-singularities. As another particular case, the procedure yields Arnold's famous strange duality list consisting of the 14 exceptional unimodular singularities (and an infinite sequence of further singularities not treated here in detail). As we are going to show, weighted projective lines and various triangulated categories attached to them play a key role in the study of the triangle and associated singularities.