  In this paper we 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 semiuniveral 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, those 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. For the proof of this fact we introduce some
constructions which may have further applications in the theory of
singularities in positive characteristic.
