Some beautiful algebraic surfaces

copyright Bruce Hunt 1995

Compiled and presented by Bruce Hunt

All pictures here were formed with the help of vort -- very ordinary ray tracing, a public domain package with which one can draw surfaces. All Images here are now public domain, but I would appreciate it if anyone using an image for professional purposes would credit the source. If not mentioned otherwise, all equations below were derived by myself, and without exception all images were created by myself.
To start with, a quadric
The source file is Quadric.scn in ~hunt/DRAWING/TDOT. Of course this is not particularly interesting. Let us now pass to

Cubic surfaces.

Cubic surfaces
a) Cayley cubic : Cayley.scn, Cayleyblue.scn, Cayleywithtri.scn, Cayleypent.scn , all of which are in the directory ~hunt/DRAWING/TDOT.
It is well known that there is a unique cubic surface which has four ordinary double points, usually called the Cayley cubic after its discoverer, Arthur Cayley . A picture of it is:
For the explicit equation and more information, click here.
A Tritangent of a cubic surface is a plane which intersects the cubic surface in the union of three lines (instead of a irreducible cubic curve), and the three intersection points of the three lines are then tangent points, hence the name. A picture with tritangents is the following:
We remark that this cubic surface has symmetry group S_4, the symmetric group on four letters. This symmetry group permutes the four double points.
Passing now onto smooth cubic surfaces, there is a unique such surface which has S_5, the symmetric group on 5 letters, as symmetry group. It is named the Clebsch cubic after its discoverer. A picture is
Once again, we can draw some of the tritangents (of which there are exactly 45 on a smooth cubic surface; there are also exactly 27 lines which are entirely contained in a smooth cubic surface).
For the equation of the Clebsch cubic, click here. Some other resources: there is a movie showing how this cubic surface looks by rotating it on the vertical axis, Clebsch.*.mix in the directory mentioned above, and also a movie Clebschdef.*.mix in the subdirectory .../movies, which shows how the Clebsch cubic can be deformed into the Cayley cubic.
For a lot of information on cubic surfaces, including a discussion of the equation for the 27 lines, see Chapter 4 of the book The geometry of some special arithmetic quotients of bounded symmetric domains.

Quartic surfaces

There are several special kinds of quartic surfaces, the Kummer surfaces, named after W. Kummer, the symmetroids , which are the zero locus of the determinant of a symmetric 4 x 4 matrix of linear forms, and the desmic surfaces, so-called because of their relation to desmic tetrahedra. For the Kummer surfaces details can be found in the book Kummer surfaces by Hudson, while for the others details can be found in the book Quartic surfaces by Jessop. For some general results, see Appendix B.5 of the book The geometry of some special arithmetic quotients of bounded symmetric domains.
A typical example of a symmetroid is the Hessian of a cubic surface. A picture of the Hessian of the Clebsch cubic is as follows:
In this image a mirror is behind the surface, with a light in one of the branches. Another intersting case is the Hessian of the Cayley cubic. Here we have the following image:
It is interesting to note that while a general symmetroid and also a general hessian of a smooth cubic surface has 10 ordinary double points, the hessian of the Cayley cubic above has 14 . The additional four come from the four ordinary double points of the Cayley surface. For a more detailed discussion of this quartic, along with its equation, click here.
As far as the desmic surfaces are concenrned, these are more special than the symmetroids: there is just a one-dimensional family of them, and they have 12 ordinary double points, which are the vertices of three tetrahedra in three-space. These tetrahedra are so-called desmic tetrahedra because they are in perspective with one another. A typical picture is the following :

A movie which shows how this desmic surface can be obtained from the four planes :
by deformation is contained in the directory ~hunt/DRAWING/TDOT/movies, as desmic.*.mix.
The most famous quartic surfaces are the Kummer surfaces. These are actually special cases of symmetroids, for which there are six addition ordinary double points, so 16 such altogether. A typical picture is the following (which however is more symmetric than will generically be the case):
Other Kummer surfaces are the following: 

                                .
In fact, we can find a one-dimensional family of the Kummer surfaces which degenerate into four planes, as was the case for the desmic surfaces (this can be done for any quartic, since four planes is just a degenerate quartic).

ART

If we use vort to make a picture, we are free to choose materials and colors. Here is an example of what it looks like if we make a model of the Clebsch cubic above out of glass. For this to be more interesting, we include a torus which is reflected in the glass. The torus is in red. In fact, we are looking at the cubic surface from above, and the torus is above, reflecting in the surface. Note that the reflection is not just a torus, but a pretzel. The reason is that the top of the surface is an inflection point.
One can also view a movie of this, as it rotates. The reflection is beautiful to watch. Source: art : ``movie -d .1 Clebschart.*.mix'' im Verzeichnis TDOT.

Quintics

Isn't this a beautiful quintic?
There is a unique quintic with 30 double points and symmetry group the icosahedral group, this is it. The equation was provided by W. Barth and D. v. Straten. Thanks.
The maximal number of ordinary double points a quintic surface can have is 31, and the existence of such surfaces was proven by Togliatti. However, he did not give the equation of a single such surface. The equations for such surfaces were derived by D. v. Straten, and then W. Barth found one will a Z_5-symmetry. This surface is depicted here. To see more of the double points we have included two mirrors into the image, so one can view from the bottom as well as from the back side.
Finally there is the follwing beautiful object, which does not have so many ordinary double points, but is a special space section of the invariant quintic under the Weyl group of the exceptional Lie group E_6. More about this quintic fourfold can be found in Chapter 6 of The geometry of some special arithmetic quotients of bounded symmetric domains, or a more self-contained version in A gem of the modular universe.

A Sextic

The following sextic was also found by W. Barth. It has 65 ordinary double points, the maximal possible number.
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