Simple curve singularities

Journal article, 2015

In this paper we classify simple parametrisations of complex curve singularities of arbitrary embedding dimension. Simple means that all neighbouring singularities fall in finitely many equivalence classes. We take the neighbouring singularities to be the ones occurring in the versal deformation of the parametrisation. This leads to a smaller list than that obtained by looking at the neighbours in the space of multi-germs with a fixed number of branches. Our simple parametrisations are the same as the complex version of the fully simple singularities of Zhitomirskii, who classified real plane and space curve singularities. The list of simple parametrisations of plane curves is the A-D-E list. Also for space curves the list coincides with the lists of simple curves of Giusti and Frühbis-Krüger, in the sense of deformations of the curve. For higher embedding dimension no classification of simple curves is available, but we conjecture that even there the list is exactly that of curves with simple parametrisations.