Simple curve singularities

TL;DR

This paper classifies simple parametrizations of complex curve singularities, showing they correspond to well-known lists like A-D-E for plane curves and extending to space curves, with conjectures for higher dimensions.

Contribution

It provides a classification of simple parametrizations of complex curve singularities, linking them to known singularity lists and proposing conjectures for higher dimensions.

Findings

The simple parametrizations of plane curves form the A-D-E list.

For space curves, the list matches known classifications by Giusti and Frühbis-Krüger.

Conjecture: simple parametrizations in higher dimensions are exactly the simple curves.

Abstract

We classify simple parametrisations of complex curve singularities. 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 a fixed space of multi-germs. Our simple parametrisations are the same as 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 Fr\"uhbis-Kr\"uger, 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.