There are nine possible types of Isolated Singularities on a Cubic Surface, eight of them rational double points. Each type of Isolated Singularity has an associated normal form and Coxeter-Dynkin Diagram (, , , , , , , and ).
The eight types of rational double points (the type being the one excluded) can occur in only 20 combinations on a Cubic Surface (of which Fischer 1986 gives 19): , , , , , , , , , , , , , (), , , , , and (Looijenga 1978, Bruce and Wall 1979, Fischer 1986).
In particular, on a Cubic Surface, precisely those configurations of rational double points occur for which the disjoint union of the Coxeter-Dynkin Diagram is a Subgraph of the Coxeter-Dynkin Diagram . Also, a surface specializes to a more complicated one precisely when its graph is contained in the graph of the other one (Fischer 1986).
See also Coxeter-Dynkin Diagram, Cubic Surface, Isolated Singularity
References
Bruce, J. and Wall, C. T. C. ``On the Classification of Cubic Surfaces.'' J. London Math. Soc. 19, 245-256, 1979.
Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums.
Braunschweig, Germany: Vieweg, p. 13, 1986.
Fischer, G. (Ed.). Plates 14-31 in
Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 17-31, 1986.
Looijenga, E. ``On the Semi-Universal Deformation of a Simple Elliptic Hypersurface Singularity. Part II: The
Discriminant.'' Topology 17, 23-40, 1978.
Rodenberg, C. ``Modelle von Flächen dritter Ordnung.'' In Mathematische Abhandlungen aus dem
Verlage Mathematischer Modelle von Martin Schilling. Halle a. S., 1904.