Equisingular deformations of plane curves in arbitrary characteristic

Abstract

The purpose of this paper is to develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniversal deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial (respectively, weakly equisingular) deformations, that is, deformations which become trivial (respectively, equisingular) after a finite and dominant base change. The semiuniversal base space for weakly equisingular deformations is, in general, not smooth but becomes smooth after a finite and purely inseparable base extension. The proof of this fact is more complicated and we introduce new constructions which may have further applications in the theory of singularities in positive characteristic.