Symmetric real cubic surfaces and their lines

\[ \newcommand{\Cb}{\mathbb{C}} \newcommand{\Pb}{\mathbb{P}} \newcommand{\Rb}{\mathbb{R}} \]

I am lucky enough to have many friends, and lots of them do very wonderful mathematics. Two of these fine people, Sidhanth Raman and Thomas Brazelton, recently put up a really cool paper¹ in which they show, among other things, that the monodromy group of lines on symmetric cubic surfaces is the Klein 4-group. I’m especially interested in enumerative phenomena like this because of the close relationship to solving equations (in particular, Farb–Wolfson² proved that the resolvent degree of finding a line on a generic smooth cubic surface is at most 3). Jordan³ showed that the connected 27 lines cover has Galois group given by the Weyl group of \(E_6\)—in particular, Sidhanth and Tommy showed that restricting to symmetric surfaces renders a very difficult problem into a solvable one. Even better: they gave an explicit formula in radicals for the lines!

With all this in mind, I coded up a Shader that demonstrates the underlying phenomena:

TipControls

Use WASD for motion (hold space to boost) and arrow (↑←↓→) keys to turn the camera. Holding shift with the WASD keys moves us through the parameter space.

The parameter space of symmetric smooth cubics surfaces is \(2\)-dimensional—indeed, it is a discriminant complement in \(\Pb^2\). Any such surface can be given as the zero set of a form \[ A \sum_{i} X_i^3 + B \sum_{i,j} X_i^2 X_j + C \sum_{i,j,k} X_i X_j X_k \] in the four variables \(X_0, X_1, X_2, X_3\), and is uniquely determined up to scaling. By visually identifying \(\Rb\Pb^2\) with its double cover \(S^2\), we can move through the space of all such cubics using rotation matrices. This is shown in our “mini-map” in the bottom left corner:

• Surfaces in the orange regions are smooth with 27 real lines
• Surfaces in the purple and black regions are smooth, with only 3 real lines
  • In the purple, the real surface has two connected components
  • In the black, the real surface is connected
• Surfaces along the white curves are not smooth

More soon!

Footnotes

1. Thomas Brazelton and Sidhanth Raman, Monodromy in the space of symmetric cubic surfaces with a line, arXiv, 2025.

2. Benson Farb and Jesse Wolfson, Resolvent degree, Hilbert’s 13th Problem and geometry, L’Enseignement Mathématique 65 (2019), no. 3, 303–376.

3. Camille Jordan, Traité des substitutions et des équations algébriques, Gauthier Villars, 1870.