Singularities on Cubic Surfaces

Rodenberg's Classification of Singularities

Many of the models in Series VII illustrate the different possible configurations of singularities on cubics. This classification of cubics by singularity configuration does not distinguish surfaces which are projectively equivalent (i.e. related by a linear change of coordinates), because there would be impractically many models if so. (Note that models VII 2, 3, 4, 5, 6 are all projectively equivalent. This is initially surprising, given how different the models look.

Here we will consider two singularities to be "the same" if it is possible to apply a biholomorphic local change of coordinates (satisfying nice smoothness conditions, with its inverse satisfying these too) to transform from one to the other (and back again). Rodenberg described two surfaces as having the same "shape" if they can be continuously deformed into each other without changing the "order" of any singularity, or passing the surface through singularities.

This description can be made more concrete using modern singularity theory. The details of the classification of these singularities are quite technical. For more information on the types of singularites present in our models, namely $A_{n}, D_{n}, E_{6}$, see Miles Reid's wonderful account of surfaces and singularities [1]. In short, an $A_{n}$ singularity is like one given by $x^2 + y^2 + z ^{n+1} = 0$ at the origin, $D_{n}$ corresponds to $x^2 + y^{2} z + z^{n−1} = 0$ at the origin, and $E_{6}$ corresponds to $x^2 + y^3 + z^4 = 0$ at the origin.

There are around 20 possible configurations of singularities for cubic surfaces, illustrated in Series VII. The Clebsch cubic, Model VII 1, has no singularities, and models VII 2 to 19 represent the various configurations of finitely many singularities, some projectively equivalent to each other. VII 20, 21 and 22 have infinitely many singularities, all of which make up a curve. It turns out that any cubic with more than four singularites must have a curve consisting of singularities. 

 

The Models

The theory behind this classification by singularity type is quite complicated, but the aesthetic beauty of the models and their range of shapes is self-evident. Recall that model VII 1 (the Clebsch Diagonal Surface), is a cubic with no singularities and 27 lines. Also the ruled cubics VII 20, 21, 22 have infinitely many double points, and infinitely many lines. Now we shall explore the other types of singularities and number of lines for the other models in Rodenberg's classification:

 

VII 2 (large & small), 3, 5, 6 - four real ordinary $A_{1}$ double points, and 9 lines. (VII 4 is missing from our collection)

 
 

VII 7 - Three $A_{1}$ double points, and 12 lines. 

 

VII 8 - The "outside" of model VII 7, turned upside down.

 

VII 9 - Three $A_{2}$ double points, and 3 lines. 

 

VII 10 - One $A_{2}$ double point, and 15 lines.

 

VII 11 - One $A_{2}$ double point, and 15 lines. This is a different configuration to VII 10.

 

VII 12 - One $A_{3}$ and two $A_{1}$ double points, and 5 lines. 

 

VII 13 - One $A_{3}$ double point, and two lines.

 

VII 14 - One $A_{4}$ and one $A_{1}$ double point, and 4 lines.

 

VII 15 - One $A_{5}$ and one $A_{1}$ double point, and 2 lines.

 

VII 16 - One $D_{4}$ double point, and 6 lines.

 

VII 17 - One $D_{4}$ double point, and 6 lines. This is a different configuration to VII 16.

 

VII 18 - One $D_{5}$ double point, and 3 lines. 

 

VII 19 - One $E_{6}$ double point, and 1 line. 

 

References

[1] Chapters on Algebraic Surfaces, Miles Reid (1996), 80-109 (Chapter 4), arXiv:alg-geom/9602006v1