Nine different types of singularities are possible on a cubic surface, according to [Wikipedia.](https://en.wikipedia.org/wiki/Cubic_surface) How exactly is the "type" of singularity defined? I know that the number corresponding to the singularity is the number of degrees of freedom removed, but how can we say that the two different surfaces with a $D\_4$ singularity have the same type of singularity while the surface with an $A\_4$ singularity has another type? What properties can tell the difference between the two singularities?



How do we define the type of a singularity on a cubic surface?
All the singularities involved in this classification are Rational Double Points. These singularities are taut, in other words, their analytic type is uniquely determined by the configuration of curves in their minimal resolution.


Such a resolution is a finite set of (-2)-curves whose dual diagram is a Dynkin diagram of the same type of the singularity.


In particular, since the Dynkin diagram of type $A\_4$ is non-isomorphic to the Dynkin diagram of tipe $D\_4$, this allows you to distinguish the two singularities.


