The n-Category Café

I started reading this paper because I wanted to finally understand the famous “27 lines on a cubic surface” and how they’re related to the smallest nontrivial representations of $\mathrm{E}_6$, which are 27-dimensional. Actually $\mathrm{E}_6$ has two nontrivial representations of this dimension, which are not isomorphic: one is the exceptional Jordan algebra, and one is its dual! The exceptional Jordan algebra consists of $3 \times 3$ self-adjoint octonionic matrices, so it has dimension

$$8+8+8+3 = 27$$

The determinant of such a matrix turns out to be well-defined despite the noncommutativity and nonassociativity of the octonions. $\mathrm{E}_6$ is the group of linear transformations of the exceptional Jordan algebra that preserves the determinant.

As you might expect, all this stuff going on in dimension 27 is just the tip of an iceberg—and Manivel explores quite a large chunk of that iceberg. But today let me just touch on the tip.

For starters, the Cayley–Salmon theorem says that every smooth cubic surface in $\mathbb{C}\mathrm{P}^3$ has exactly 27 lines on it.

(This is apparently one of those cases where mathematicians shared credit with the meal that inspired their work, like the Fermi–Pasta–Ulam problem.)

I can’t visualize those 27 lines in general. But Clebsch gave an example of a smooth real cubic surface where all the lines actually lie in $\mathbb{R}\mathrm{P}^3$, so you can see them. This is called the Clebsch surface, or Klein’s icosahedral cubic surface, because Klein also worked on it. It looks like this:

and the 27 lines look like this:

Please click on the pictures to see who created them! Here’s a model of it — one of those nice old plaster models you see in old universities:

This model is in Göttingen, photographed by Oliver Zauzig.

I would enjoy diving down the rabbit hole here and learning everything about this particular cubic surface, but I’ll resist for now! I’ll just say a few things:

First, the Clebsch surface can be described very nicely as a surface in $\mathbb{R}\mathrm{P}^4$ using the homogeneous equations

$$x_0+x_1+x_2+x_3+x_4 = 0$$ $$x_0^3+x_1^3+x_2^3+x_3^3+x_4^3 = 0$$

but then you can eliminate one variable and think of it as a surface in $\mathbb{R}\mathrm{P}^3$ given by the equation

$$x_1^3+x_2^3+x_3^3+x_4^3 = (x_1+x_2+x_3+x_4)^3$$

Second, the lines are actually defined over the golden field $\mathbb{Q}[\sqrt{5}]$. This may have something to do with why it’s called ‘Klein’s icosahedral cubic surface’ — I’ll avoid looking into that right now, but it may eventually be important, because there are nice relations between some exceptional Lie algebras and the golden field.

Third, you can see some points where three lines intersect: these are called Eckardt points and there are 10 of them.

Anyway, you may be wondering why there are 27 lines on a smooth cubic surface. The best argument I’ve seen so far, in terms of maximum friendliness, minimum jargon, and maximum total insight conveyed, is here:

• Jack Huizenga, Algebraic Geometry: Why are there exactly 27 straight lines on a smooth cubic surface?, answer on Quora.

I can’t say I fully understand it, since it’s fairly involved, but I still recommend it to anyone who knows a reasonable amount of algebraic geometry.

Anyway, I don’t think one needs to fully understand this to start wondering what $\mathrm{E}_6$ has to do with it. Here’s some of what Manivel has to say:

The configuration of the 27 lines on a smooth cubic surface in $\mathbb{C}\mathrm{P}^3$ has been thoroughly investigated by the classical algebraic geometers. It has been known for a long time that the automorphism group of this configuration can be identified with the Weyl group of the root system of type $\mathrm{E}_6$, of order 51,840. Moreover, the minimal representation $J$ of the simply connected complex Lie group of type $\mathrm{E}_6$ has dimension 27.

Here the letter $J$ means ‘exceptional Jordan algebra’.

This is a minuscule representation, meaning that the weight spaces are lines and that the Weyl group $W(\mathrm{E}_6)$ acts transitively on the weights. In fact one can recover the lines configuration of the cubic surface by defining two weights to be incident if they are not orthogonal with respect to the unique (up to scale) invariant scalar product. Conversely, one can recover the action of the Lie group $\mathrm{E}_6$ on $J$ from the line configuration.

I hope to come back to this and keep digging deeper. Right now I don’t even understand how the 27 weight spaces in $J$, which are 1-dimensional subspaces in a 27-dimensional space, are connected to 27 lines in some surface. But there are some other things in Manivel’s paper that I understand and like a lot.