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).

**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.

© 1996-9

1999-05-25