Barth Decic

A decic surface is one defined by a polynomial equation of degree 10. The Barth decic, drawn above by Abdelaziz Nait Merzouk, is currently the decic surface with the largest known number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by

$$x^2 + y^2 = z^2 .$$

It has 345 ordinary double points, while the best known upper bound for a decic surface that’s smooth except for such singularities is 360.

The Barth decic is defined by this homogeneous polynomial equation of degree 10 in four variables $w,x,y,z$:

$$\begin{array}{c} 8 (x^2 – \Phi^4 y^2) (y^2 – \Phi^4 z^2) (z^2 – \Phi^4 x^2) \left( x^4 + y^4 + z^4 – 2 x^2 y^2 – 2 x^2 z^2 – 2 y^2 z^2\right) \\ + (3 + 5 \Phi) w^2 \left( x^2 + y^2 + z^2 – w^2 \right)^2 \left( x^2 + y^2 + z^2 – (2-\Phi)^2 w^2)\right) = 0 \end{array}$$

where

$$\Phi = \frac{\sqrt{5} + 1}{2}$$

is the golden ratio. This equation determines a subset $S \subset \mathbb{C}^4$ with complex dimension 2. Note that if $(w, x, y, z \in \mathbb{C}^4$ is a solution, so is any multiple $(cw, cx, cy, cz)$. We may thus projectivize $S$, treating any solution as ‘the same’ as any multiple of that solution. The result is an algebraic variety $X$ in the complex projective space $\mathbb{C}\mathrm{P}^3$. This variety has complex dimension 2, so it is called a complex surface. To obtain an ordinary real 2-dimensional surface we may take its intersection with a copy of $\mathbb{R}\mathrm{P}^3$ in $\mathbb{C}\mathrm{P}^3$.

Sitting inside $\mathbb{R}\mathrm{P}^3$ we in turn have many copies of ordinary 3-dimensional space, $\mathbb{R}^3$. The picture above shows the portion of the Barth decic living in one of these copies. Concretely, this consists of real solutions of the above equation where $w = 1$.

But we also have ‘points at infinity’. If you march off in either of two opposite directions in $\mathbb{R}^3$, you will approach one of these points at infinity. The points at infinity form a projective plane, that is, a copy of $\mathbb{R}\mathrm{P}^2$. Concretely, these points at infinity are the points in $\mathbb{R}\mathrm{P}^2$ coming from points $(x,y,z,w) \in \mathbb{R}^4$ with $w = 0$.

The Barth decic has 345 ordinary double points. However, 45 of these are points at infinity, so they are not visible in the above picture, or this one:

We can bring the double points at infinity into view by rotating $\mathbb{R}\mathrm{P}^3$ slightly. If we slice the resulting surface to see it better, we obtain a picture like this:

We can also compress $\mathbb{R}^3$ into a ball, so that the points at infinity lie on the surface of this ball. More precisely, the surface of this ball is a 2-sphere, a double cover of $\mathbb{R}P^2$, so any antipodal pair of points in this 2-sphere correspond to the same point at infinity.

This gives the following view of the Barth decic:

You can see visually that the compressed Barth decic meets the 2-sphere in 10 great circles. To see this mathematically, we can take the equation for the Barth decic and set $w = 0$:

$$\begin{array}{c} (x^2 – \Phi^4 y^2) (y^2 – \Phi^4 z^2) (z^2 – \Phi^4 x^2) \left( x^4 + y^4 + z^4 – 2 x^2 y^2 – 2 x^2 z^2 – 2 y^2 z^2\right) = 0 \end{array}$$

This factors into 10 linear functions:

$$(x – \Phi^2 y)(x + \Phi^2 y)(y – \Phi^2 z)(y + \Phi^2 z)(z – \Phi^2 x)(z + \Phi^2 x)(x-y-z)(x+y-z)(x-y+z)(x+y+z) = 0$$

Each of these defines a plane in $\mathbb{R}^3$ whose intersection with the unit 2-sphere is one of the 10 great circles. These 10 great circles are orthogonal to the lines going through the opposite corners of a regular dodecahedron:

You can see 5 double points in each face of the dodecahedron and 1 at the midpoint of each edge, for a total of $5 \times 12 + 30 = 90$. However, antipodal points on the sphere count as the same point at infinity, so we get a total of $90/2 = 45$ double points at infinity.

For more related pictures see:

• Abdelaziz Nait Merzouk, Barth decic and dodecahedron.

It is worth comparing the Barth sextic:

• Barth sextic.

The group of rotation and reflection symmetries of an icosahedron, $\mathrm{A}_5 \times \mathbb{Z}/2$, acts as symmetries of both the Barth sextic and the Barth decic. Barth introduced these surfaces here:

• Wolf Barth, Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, Journal of Algebraic Geometry 5 (1994), 173–186.

Ordinary double points are also known as nodes. In 1984, Miyaoka proved that a decic surface in $\mathbb{C}\mathrm{P}^3$ with only rational double points can have at most 360 such points:

• Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159–171.

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