Singularities of Cubic Surface
Cubic surfaces are implicit surfaces in projective three space with terms which are degree three or less. These surfaces can have several different types of singularity from the basic conical singularity A_{1} to those containing a cross-caps and a line of nodal points.
Projective space
Cubic surfaces are defined in projective space which is a lot like normal 3D space, but which some extra structure at infinity. Points in projective 3 space are specified by 4 number, written (x:y:z:w), with an equivalence relation so that all non zero multiples of a point are considered equivalent. Hence, (ax: ay: az: aw)=(x:y:z:w) for all not zero a. In particular if w is not zero then (x:y:z:w)=(x/w:y/w:z/w:1) and dropping the final number projects this to normal R^{3} so (x:y:z:w) is projected to (x/w,y/w,z/w) allowing the surface to be view in three dimensions.
One way to think of the space is the set of lines through the origin in R^{4}. If (x, y, z, w) is a points on such a line then so is (a x, a y, a z, a w) and the equivalence relation follows. Alternatively the space can be though of as the three sphere S^{3} with antipodal points identified.
For most of the surfaces below we have used a projection so that the points (1:0:0:0), (0:1:0:0), (0:0:1:0), (0:0:0:1) map to the vertices of a tetrahedron (1,1,1), (-1,-1,1), (1,-1,-1), (-1,1,-1). This is a particularly useful projection as the singularities of the expressions given by Cayley tend to lie at these four points.
Singularities
The type of singularity is determined by a classification due to Arnol'd, and are specified by their normal forms. In the complex case there is just one normal form for each, but in the real case changing the signs of the terms produce different images. Some of the alternate forms are shown, for many singularities one form is a single point, which is not shown.
The Cubic Surfaces
The following list of singularities follows Cayley. Schlafli was first to classify the various types. We use the modern notation for singularities following Bruce and Wall. Holzer and Labs distinguish between the different forms of the real singularities giving 45 types with rational double points.
Capitals X, Y, Z, W are used for coordinates in projective space and x, y, z for coordinates in R^{3}.
Class I: No singularities
There are a number of well known non singular cubics.
Clebsch surface
This surface contain the maximum possible 27 real lines.
X^{3}+Y^{3}+Z^{3}+W^{3}-(X+Y+Z+W)^{3}=0
One projection is 16 x^{3} + 16 y^{3} - 31 z^{3} + 24 x^{2} z - 48 x^{2} y - 48 x y^{2} +24 y^{2} z - 54 √3 z^{2} - 72 z
Fermat Cubic
X^{3}+Y^{3}+Z^{3}+W^{3}=0
Ref: Wikipedia
Class II: one A_{1} singularity
In Cayley's notation: C2
W (a X^{2} + b Y^{2} + c Z^{2} + f X Y+ g X Z+h Y Z)+2 k X Y Z
Class III: one A_{2} singularity
In Cayley's notation: B3
2W(X+Y+Z)(l X+m Y+n Z) + 2 k X Y Z
Some nice examples include
x z + y^{2}(x+y+z)
By Bruce Hunt
Class IV: two A_{1} singularities
In Cayley's notation: 2 C2
W X Z+Y^{2}(k Z+l W)+a X^{3} +b X^{2} Y+c X Y^{2} + d Y^{3}
Class V: one A_{3} singularity
In Cayley's notation: B4
W X Z +(X+Z)(Y^{2}-a X^{2}-b Z^{2})
(x^{2} + y^{2}) w + 2 x (z^{2} - 2 x^{2} - 4 y^{2}); w=1-y
Number KM26 in Holzer and Labs
Class VI: one A_{1} and and one A_{2} singularities
In Cayley's notation: B3+C2
W X Z+Y^{2} Z+a X^{3} +b X^{2} Y+c X Y^{2} + d Y^{3}
Class VII: one A_{4} singularity
In Cayley's notation: B5
W X Z +Y^{2} Z +Y X^{2}-Z^{3}
Class VIII: three A_{1} singularities
In Cayley's notation: 3 C2
Y^{3}+Y^{2}(X+Z+W)+4 a X Z W
Class IX: two A_{2} singularities
In Cayley's notation: 2 B3
W X Z+a X^{3}+ b X^{2} Y + c X Y^{2} + d Y^{3}
Class X: one A_{1} and one A_{3} singularities
In Cayley's notation: B4+C2
W X Z+(X+Z)(Y^{2}-X^{2})
Class XI: one A_{5} singularity
In Cayley's notation: B6
W X Z + Y^{2} Z+X^{3}- Z^{3}
Class XII: one D_{4} singularity
In Cayley's notation: U6
W(X+Y+Z)^{2}+X Y Z
Two nice examples of the surface show the different real forms of the singularity.
Class XIII: two A_{1} and one A_{2} singularities
In Cayley's notation: B3+2 C2
W X Z+Y^{2}(X+Y+Z)
Class XIV: one A_{1} and one A_{4} singularities
In Cayley's notation: B5+C2
W X Z+Y^{2} Z+Y X^{2}
Class XV: 1 D_{5} singularity
In Cayley's notation: U7
W X^{2}+X Z^{2}+Y^{2} Z
Class XVI: four A_{1} singularities
In Cayley's notation: 4 C2
W(X Y+X Z+Y Z)+X Y Z
Cayley's Cubic
Cayley's expression X Y Z + X Y W + X Y Z + Y Z W=0 has nodes at the four points (1:0:0:0), (0:1:0:0), (0:0:1:0) and (0:0:0:1) we can apply a rotation so that these points map to four verticies of cube center at (0:0:0:1), that is the points (1:1:1:1), (-1:-1:1:1), (1:-1:-1:1), (1:-1:-1:1), ()-1:1:-1:1). Projecting these points to R^{3} via (x:y:z:w) → (x/w,y/w,z/w) gives vertices on a cube center origin. Plotting the resulting expression
X Y Z + X Y W + X Z W + Y Z W;
X = x + y + z + w;
Y = -x - y + z + w;
Z = x -y -z + w;
W = -x + y - z + w;
w = 1;
gives a surface where the nodal points are clearly visable.
S6
4-x^{2} + z x^{2} + y^{2} z - 2 z^{3} + 3 z^{2} - z;
A different projection of the same surface. From University of Turin
Class XVII: one A_{1} and two A_{2} singularities
In Cayley's notation: 2 B3 + C2
W X Z+X Y^{2}+Y^{3}
Class XVIII: two A_{1} and one A_{3} singularities
In Cayley's notation: B4 +2 C2
W X Z+(X+Z)Y^{2}
Class XIX: one A_{1} and one A_{5} singularity
In Cayley's notation: B6 + C2
W X Z + Y^{2} Z + X^{3}
Class XX: one E_{6} singularity
In Cayley's notation: U8
W X^{2} + X Z^{2} + Y^{3}
Class XXI: three A_{2} singularities
In Cayley's notation: 3 B3
W X Z+Y^{3}
Cayley 1
p_{1}=2 x^{3}-6 x y^{2};
p_{3}=x^{2}+y^{2};
a_{1} = -1; a_{2} = -1; a_{3}=-3 a_{1};
a_{4} = 1; a_{5}=a_{2}^{2}/(3 a_{1});
a_{6}=-a_{2}; a_{7}=a_{1};
Class XXII: two cross-caps and a nodal line
In Cayley's notation: S(1,1)
W X^{2} + Z Y^{2}
Class XXIII: nodal line
In Cayley's notation: S(1,1)
X(W X+Y Z)+Y^{3}
References
- Bruce, J. W.; Wall, C. T. C. (1979), "On the classification of cubic surfaces", The Journal of the London Mathematical Society. Second Series 19 (2): 245–256, doi:10.1112/jlms/s2-19.2.245, MR533323, ISSN 0024-6107 - Has details on the modern classification.
- Cayley, Arthur (1869), "A Memoir on Cubic Surfaces", Philosophical Transactions of the Royal Society of London (The Royal Society) 159: 231–326, ISSN 0080-4614 - The classification this page is based on.
- Holzer, Stephan; Labs, Oliver, "Illustrating the Classification of Real Cubic Surfaces" PDF - Visulise the 45 topological types for real cubic surfaces.
- modelli di superfici La collezione dei modelli di superfici del Dipartimento di Matematica dell'Università di Torino. (The Models of the Department of Mathematics University of Turin). - A collection of 30 models originally done in plaster, cover many of the surfaces seen here.
- H. Knörrer and T. Miller. Topologische Typen reeller kubischer Flächen. Mathematische Zeitschrift, 195, 1987. - Classification of the 45 real types.
- Cubic surface home page - A website dedicated to cubic surfaces. Has images of each type.
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