Cubic Equations Of Rational Triple Points

The main purpose of the paper is to give one cubic denning equation for each rational triple singularity of dimension two. There are 9 types of rational triple singularities according to Artin's classification [1]. Artin proved also that these singularities can be embedded into C~4, and later Tyurian [9] gave explicitly 3 defining equations for each singularity. On the other hand, Tyurian proved also in [10] that a rational triple point is determined uniquely by their dual gragh. So isomorphically, there are 9 rational triple points. It is well known that any surface singularity can be birationally embedded into C~3. So a surface singular point can be defined by one equation in three variables which determines uniquely the singularity by normalization. Theoretically, a rational triple point can be defined by a cubic equation. On the other hand, any singular point defined by a cubic equation can be resolved explicitly by Tan's method [5]. Therefore, it is very interesting to find the cubic defining equations for 9 rational triple points. Tan has found the cubic equations explicitly for 4 rational triple points . In this paper, we will give the cubic equations for the remaining 5 rational triple points [2].