The Principle Describing Possible Combinations of Singularities in Deformations of a Fixed Singularity

This is a review of my recent work. What kinds of combinations of singularities can appear in small deformation fibers of a fixed singularity? We consider this problem for hypersurface singularities on complex analytic spaces of dimension 2. Recall first that a connected Dynkin graph of type A, D or E corresponds to a surface singularity called a rational double point. (Durfee [3].) When the fixed singularity is a rational double point corresponding to a Dynkin graph Γ0, the answer to our problem is well-known. (1) Any small deformation fiber has only rational double points as singularities. (2) A combination of rational double points corresponding to a Dynkin graph Γ appears on a small deformation fiber if and only if Γ is a subgraph of Γ0. (The type of each component of Γ corresponds to the type of the singularity on a small deformation fiber Y and the number of components of each type corresponds to the number of singularities of each type on Y .)