Enriques diagrams and adjacency of planar curve singularities

We study adjacency of equisingularity types of planar 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 show an example of singular curve of type D′ that can be deformed to a curve of type D without D′ being adjacent to D. This suggests that analytical rather than topological equivalence should be considered when studying adjacency of singularity types. Introduction A class of reduced (germs of) planar curve singularities D is said to be adjacent to the class D when every member of the class D can be deformed into a member of the class D by an arbitrarily small deformation. If this can be done with a linear deformation, then we say that D is linearly adjacent to D. It is well known that equisingularity and topological equivalence of reduced germs of curves on smooth surfaces are equivalent, and that analytical equivalence implies topological equivalence (see for instance [18], [19] or [2]). We shall focus on the equisingularity (or topological equivalence) classes, and we will call them simply types. The Enriques diagrams introduced by Enriques in [5, IV.I] represent the types: two reduced curves are equisingular at O if and only if their associated Enriques diagrams are isomorphic (see [2, 3.9]). In [1] Arnold classified critical points of functions with modality at most two; this implies the classification of types (of planar curve singularities) with multiplicity at most four. He also described some adjacencies between them, introducing the so-called series of types A, D, E and J . The construction of series was generalized by Siersma in [16] using a kind of Enriques diagrams; in particular, he classified types of multiplicity at most five. As we shall see below, all adjacencies within one series are linear. Both authors were partially supported by CAICYT PB98-1185, Generalitat de Catalunya 98-SGR-00024 and “EAGER”, European Union contract HPRN-CT-2000-00099.