Enriques Diagrams and Adjacency of Planar Curve Singularities

We study adjacency of equisingularity types of planar complex curve singularities in terms of their Enriques diagrams. The goal is, given two equisingularity types, to determine whether one of them is adjacent to the other. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also obtain new examples of exceptional deformations, i.e., singular curves of type mathcal{D}' that can be deformed to a curve of type mathcal{D} without mathcal{D}' being adjacent to mathcal{D}.