Branch curves of some cubic surfaces

(This documentation will be improved)

• The first script visualizes the [real locus of the] branch curve of (a generic projection of) a smooth cubic surface to a plane. This famous curve is also called "Zariski sextic". It has six cuspidal points, all of them on a conic; but, paradoxically, the same number of parameters (moduli) as any plane sextic with six cusps, without any conditions on a conic.

For the geometric introduction, please see a separate file Geometric introduction

This curve has a large fundamental group of the complement (PSL(2,Z)); (this was an intriguing discovery of Oscar Zariski); much larger than in the case of a plane sextic with six cusps not on a conic (in which this is abelian of order six).

• The second script demonstrates the branch curve of (a generic projection of) a cubic surface with a double line.

(There are two natural notions of a branch curve in this case; the first definition, (which can given by taking the surface normalization, for example), gives a curve isomorphic to the classical deltoid curve; we called this curve "the pure ramification curve" in this Arxiv text.

The second definition (which can be given via the support of the sheaf of relative differentials; or, classically, via the intersection of the surface with the polar surface) is the union of the pure ramification curve with the image of the double curve (double line, in this case).

(One can prove that the image of a double curve is always tangent to the pure ramification curve at some smooth points on the pure ramification curve.)

The Macaulay code intended for finding with a nicely positioned cubic surface with a double line defined over the reals, and with all cusps defined over the reals will be posted separately. (Moreover, we get everything defined over the integers Z).

Sample output

For the smooth cubic surface:

For the cubic with a double line:

References

The scripts are written in the surf programming language

Attributions

Maxim Leyenson,

License

This code is distributed under the BSD 4-Clause "Original" or "Old" License, see the LICENSE file. (That is, mention my name somewhere when you are forking the code. I am looking for a job!)

Synopsis

Usage

For the case of a smooth cubic surface (Zariski sextic):

      $ cd zariski-sextic.2d/sign-minus 
      $ surf-alggeo-nox curve.minus.surf

(creates a bunch of images)

      $ cd zariski-sextic.2d/sign-plus 
      $ surf-alggeo-nox curve.plus.surf

For the cubic surface with a double line (pure branch curve is a Deltoid):

      $ cd ramification-for-a-cubic-surface-with-double-line/surf
      $ surf-alggeo-nox plot-ramification-curve.surf

or

      $ cd ramification-for-a-cubic-surface-with-double-line/matplotlib
      $ python3 plot-full-ramification.py

to see the image of the double line, too. (There is a quirk in the Debian's version of surf which I am trying to bypass with mathplotlib)

Requirements

• surf

Installing surf

In the Ubuntu derivatives,

   # apt install -y  surf-alggeo-nox