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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 A1 to those containing a cross-caps and a line of nodal points.

Caylay's Cubics

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 R3 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 R4. 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 S3 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.

TypeNormal formCurve in 2DSurface in 3D
A1 x2+y2+z2 A1 singularity 2D A1 singularity cone A1 singularity point
A2 x3+y2+z2 A2 singularity 2D A2 singularity -- A2 singularity +-
A3 x4+y2+z2 A3 singularity 2D A3 singularity -- A3 singularity +-
A4 x5+y2+z2 A4 singularity 2D A4 singularity -- A4 singularity +-
A5 x6+y2+z2 A5 singularity 2D A5 singularity -- A5 singularity +-
D4 x3+x y2+z2 D4 singularity 2D D4 singularity -- D4 singularity +-
D5 x4+x y2+z2 D5 singularity 2D D5 singularity -- D5 singularity +-
E6 x3+y4+z2 E6 singularity 2D E6 singularity -- E6 singularity +-
6 x3+y3+z3
 +k x y z
E6~ singularity E6~ singularity +

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 R3.

Class I: No singularities

There are a number of well known non singular cubics.

Clebsch surface

Diagonal Surface of Clebsch

This surface contain the maximum possible 27 real lines.

X3+Y3+Z3+W3-(X+Y+Z+W)3=0

One projection is 16 x3 + 16 y3 - 31 z3 + 24 x2 z - 48 x2 y - 48 x y2 +24 y2 z - 54 √3 z2 - 72 z

Fermat Cubic

Fermat Cublic

X3+Y3+Z3+W3=0

Ref: Wikipedia

Class II: one A1 singularity

In Cayley's notation: C2

Cayley Cublic II

W (a X2 + b Y2 + c Z2 + f X Y+ g X Z+h Y Z)+2 k X Y Z

Class III: one A2 singularity

In Cayley's notation: B3

Cayley Cublic III

2W(X+Y+Z)(l X+m Y+n Z) + 2 k X Y Z

Some nice examples include

Hunt A2

x z + y2(x+y+z)

By Bruce Hunt

 

Cayley 3

S11

x2+y2+z3+3.2 (x3-3 x y2)

From University of Turin

Class IV: two A1 singularities

In Cayley's notation: 2 C2

Cayley Cublic IV

W X Z+Y2(k Z+l W)+a X3 +b X2 Y+c X Y2 + d Y3

Class V: one A3 singularity

In Cayley's notation: B4

Cayley Cublic V

W X Z +(X+Z)(Y2-a X2-b Z2)

 

Cubic KM26

(x2 + y2) w + 2 x (z2 - 2 x2 - 4 y2); w=1-y

Number KM26 in Holzer and Labs

Class VI: one A1 and and one A2 singularities

In Cayley's notation: B3+C2

Cayley Cublic VI

W X Z+Y2 Z+a X3 +b X2 Y+c X Y2 + d Y3

Class VII: one A4 singularity

In Cayley's notation: B5

Cayley Cublic VII

W X Z +Y2 Z +Y X2-Z3

Class VIII: three A1 singularities

In Cayley's notation: 3 C2

Cayley Cublic VIII

Y3+Y2(X+Z+W)+4 a X Z W

Class IX: two A2 singularities

In Cayley's notation: 2 B3

Cayley Cublic IX

W X Z+a X3+ b X2 Y + c X Y2 + d Y3

Class X: one A1 and one A3 singularities

In Cayley's notation: B4+C2

Cayley Cublic X

W X Z+(X+Z)(Y2-X2)

Class XI: one A5 singularity

In Cayley's notation: B6

Cayley Cublic XI

W X Z + Y2 Z+X3- Z3

Class XII: one D4 singularity

In Cayley's notation: U6

Cayley Cublic XII

W(X+Y+Z)2+X Y Z

Two nice examples of the surface show the different real forms of the singularity.

S16

S16

z2 ( z + 4) + y (y - √3) x) ( y + √3 x)

From University of Turin

 

S17

S17

z2 ( z + 4) + y (x2 + y2)

From University of Turin

Class XIII: two A1 and one A2 singularities

In Cayley's notation: B3+2 C2

Cayley Cublic XIII

W X Z+Y2(X+Y+Z)

Class XIV: one A1 and one A4 singularities

In Cayley's notation: B5+C2

Cayley Cublic XIV

W X Z+Y2 Z+Y X2

Class XV: 1 D5 singularity

In Cayley's notation: U7

Cayley Cublic XV

W X2+X Z2+Y2 Z

Class XVI: four A1 singularities

In Cayley's notation: 4 C2

Cayley Cublic XVI

W(X Y+X Z+Y Z)+X Y Z

Cayley's Cubic

Cayley's Cublic
4 (x2+y2+z2) + 16 x y z - 1;

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 R3 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

S6

4-x2 + z x2 + y2 z - 2 z3 + 3 z2 - z;

A different projection of the same surface. From University of Turin

Class XVII: one A1 and two A2 singularities

In Cayley's notation: 2 B3 + C2

Cayley Cublic XVII

W X Z+X Y2+Y3

Class XVIII: two A1 and one A3 singularities

In Cayley's notation: B4 +2 C2

Cayley Cublic XVIII

W X Z+(X+Z)Y2

Class XIX: one A1 and one A5 singularity

In Cayley's notation: B6 + C2

Cayley Cublic XIX

W X Z + Y2 Z + X3

Class XX: one E6 singularity

In Cayley's notation: U8

Cayley Cublic XX

W X2 + X Z2 + Y3

Class XXI: three A2 singularities

In Cayley's notation: 3 B3

Cayley Cublic XXI

W X Z+Y3

Cayley 1

Cayley 1
p1 + (a2z+a3)p3 + a4 z3 + a5z2 + a6 z + a7;
p1=2 x3-6 x y2;
p3=x2+y2;
a1 = -1; a2 = -1; a3=-3 a1;
a4 = 1; a5=a22/(3 a1);
a6=-a2; a7=a1;

Class XXII: two cross-caps and a nodal line

In Cayley's notation: S(1,1)

Cayley Cublic XXII

W X2 + Z Y2

Class XXIII: nodal line

In Cayley's notation: S(1,1)

Cayley Cublic XXIII

X(W X+Y Z)+Y3

References

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