Rational Double Point

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 ($A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $D_4$, $D_5$, $E_6$ and $\tilde E_6$).

The eight types of rational double points (the $\tilde E_6$ type being the one excluded) can occur in only 20 combinations on a Cubic Surface (of which Fischer 1986 gives 19): $A_1$, $2A_1$, $3A_1$, $4A_1$, $A_2$, $(A_2, A_1)$, $2A_2$, $(2A_2, A_1)$, $3A_2$, $A_3$, $(A_3, A_1)$, $(A_3, 2A_1)$, $A_4$, ($A_4, A_1$), $A_5$, $(A_5, A_1)$, $D_4$, $D_5$, and $E_6$ (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 $\tilde E_6$. 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.