A note on complex plane curve singularities up to diffeomorphism and their rigidity

TL;DR

This paper demonstrates that smooth diffeomorphisms between plane curve germs with singularities imply complex isomorphism or conjugation, providing a constructive approach via Taylor series and extending results to non-irreducible hypersurfaces.

Contribution

It establishes that smooth equivalences between plane curve germs with singularities lead to complex isomorphisms or conjugates, with a constructive proof using Taylor series, and extends to certain non-irreducible hypersurfaces.

Findings

Smooth diffeomorphism implies complex isomorphism or conjugation for plane curve germs.

Constructive proof via Taylor series of the diffeomorphism.

Extension of results to non-irreducible hypersurfaces with specific singularities.

Abstract

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component of zero-dimensional isosingular locus. Moreover, we provide a general overview of the different classifications of plane curve singularities.