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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 E^~_6).

The eight types of rational double points (the E^~_6 type being the one excluded) can occur in only 20 combinations on a cubic surface (of which Fischer 1986a 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 1986a).

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 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 1986a).


See also

Coxeter-Dynkin Diagram, Cubic Surface, Double Point, Isolated Singularity, Ordinary Double Point

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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.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, p. 13, 1986a.Fischer, G. (Ed.). Plates 14-31 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 17-31, 1986b.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.

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Rational Double Point

Cite this as:

Weisstein, Eric W. "Rational Double Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RationalDoublePoint.html

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