We are looking at a triple of correlations that relate three variables:

In my previous post, we saw these pictures of triples of correlations:

The yellow shape is the space of possible correlations .

- On the left, we intersect with the plane (i.e. we set correlation to zero).
- On the right, the plane is (i.e. the three correlations sum to one).

The intersection of the yellow shape with the blue plane is all correlations where these extra linear conditions hold. The intersection seems to look different in the two pictures. How can we describe the difference? What are the possible shapes of intersections that can occur?

These questions were studied in the paper Nets of Conics by C.T.C. Wall from 1977. In the paper, Wall gives an algebraic classification, with 26 possibilities for what the intersection can look like:

Where do our two examples fit into Wall’s classification?

To see this, we look at the points on the boundary of the yellow shape. Such a point gives a matrix

with determinant equal to zero. To get Wall’s pictures, the boundary is extended **all** points where the determinant vanishes, ignoring the requirement that come from correlations (e.g. we now get points with coordinates outside of the range -1 to 1). In our first example, we get the extended picture:

or, from a different angle

Wall’s pictures plot the intersection of the yellow shape with the blue plane, shading in the correlations. For our first example, we get a shaded circle:

All of Wall’s intersections have cubic (degree three) equations, but here we have a circle, which is degree two. We recover the cubic if we apply a change of coordinates:

In terms of equations, the full cubic polynomial is

which factors as .

Table 1 of Wall’s paper tells us we are in type D*. Then, from Wall’s Figure 3 above, we see that our first example is sub-type D*c:

We follow the same recipe to find our second example in Wall’s classification. First, we extend the picture of the yellow shape and the blue plane:

Already we start to see some differences between the two examples. The intersection of the yellow shape with the blue plane gives the picture

The intersection is the cubic curve (or elliptic curve) with equation . This polynomial can’t be factored, and has no singular points. This means it is in Wall’s type A, which has four sub-types. Wall’s Figure 2, above, shows that it is sub-type Ab or Ac, since it has a shaded region. We distinguish Ab from Ac using what Wall calls a “preferred point”.

To find the preferred point, we apply a change of coordinates to convert the cubic to “canonical form” , where is a cubic (e.g. following the instructions here). In this example, we apply the change of coordinates

.

to and then set . This transforms the cubic to . In the new coordinates, the cubic is:

Going back to Wall’s pictures, we find that we’re in sub-type Ab:

In his paper, Wall says this classification is “intrinsically interesting, and involves some pleasant geometry”.

What do you think?