J. London Math. Soc. (2) 87 (2013) 622–640 C2013 London Mathematical Society doi:10.1112/jlms/jds059 Flat singularity theory

T. C. Wall We use the term ‘flat singularity theory ’ for the study of singularities of plane curves taking account of tangential singularities in the sense of points of inflexion. A first definition of flat equivalence was given in [4], together with a classification of singularities of low codimension. In this paper, we first discuss several definitions of equivalence taking account of flatness. Our preferred definition is to classify the germ of the curve relative to its tangent cone. We give a brief discussion of classification of germs relative to a fixed germ, with formulae for tangent spaces to orbits and for their codimensions in terms of invariants. We then apply these results to the somewhat different case of flat equivalence relations. We also offer a formal definition of ‘flat equisingularity ’ by numerical, essentially topological invariants. The bulk of the paper is devoted to obtaining explicit classifications of ADE type singularities up to flat equivalence. Throughout the paper we treat in parallel two models, according as a curve is given (e) by an equation or (p) by a parametrisation. We suppress most of the details for the (e) case, which can be deduced from results of Arnol’d [1]; in a final section we present the calculations for the (p) case