Dynkin Graphs, Gabriélov Graphs and Triangle Singularities

We consider fourteen kinds of two-dimensional triangle hypersurface singularities, and consider what kinds of combinations of rational double points can appear on small deformation fibers of these triangle singularities. We show that possible combinations can be described by Gabriélov graphs and Dynkin graphs. 1. Review of results by Russian mathematicians In this article we assume that every variety is defined over the complex field C. First I explain some results by Arnold and Gabriélov briefly. In [1] Arnold has introduced an invariant m called modality or modules number, and has given a long classification list of hypersurface singularities. Modality m is a non-negative integer. Though we find singularities of any dimension in Arnold’s list, we consider singularities of dimension two in particular. His class of singularities with m = 0 coincides with the class of rational double points. It is well known that each rational double point corresponds to a connected Dynkin graph of type A, D or E in the theory of Lie algebras. (Durfee [3].) The class with m = 1 consists of three subclasses. (λ is a parameter.) 1. Three simple elliptic singularities: J10, X9, P8 2. Cusp singularities Tp, q, r. ( 1 p + 1 q + 1 r < 1 ) : x + y + z +λxyz = 0 (λ 6= 0). 3. fourteen triangle singularities (These fourteen are also called exceptional singularities.) E12 Z11 Q10 W12 S11 U 12 E13 Z12 Q11 W13 S12 E14 Z13 Q12 E12 : x 7 + y + z + λxy = 0 W12 : x 5 + y + z + λxy = 0 U12 : x 4 + y + z + λxyz = 0. (As for the other defining polynomials see Arnold [1].) His list continues in the case m ≥ 2, but we do not refer further. We go on to Gabriélov’s results. (Gabriélov [4].) Let f (x, y, z) = 0 be one of defining polynomials of fourteen hypersurface triangle singularities. It defines a singularity at the origin. We consider the Milnor fiber, i.e.,