{"id":2170,"date":"2016-02-15T01:00:01","date_gmt":"2016-02-15T01:00:01","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2170"},"modified":"2018-08-15T08:49:19","modified_gmt":"2018-08-15T08:49:19","slug":"27-lines-on-a-cubic-surface","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/02\/15\/27-lines-on-a-cubic-surface\/","title":{"rendered":"27 Lines on a Cubic Surface"},"content":{"rendered":"
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\"27

27 Lines on a Cubic Surface – Greg Egan<\/p><\/div>\n<\/div>\n

This animation by Greg Egan<\/a> shows 27 lines on a surface defined by cubic equations: the Clebsch surface<\/a>. It illustrates a remarkable fact: any smooth cubic surface contains 27 lines.<\/p>\n

In its most symmetrical presentation, the Clebsch surface is defined by the equations<\/p>\n

$$ \\begin{array}{c} x_0+x_1+x_2+x_3+x_4 = 0 \\\\
\nx_0^3+x_1^3+x_2^3+x_3^3+x_4^3 = 0 \\end{array} $$<\/p>\n

where \\(x_0, \\dots, x_4\\) are complex variables. These two equations pick out a subset \\(S \\subset \\mathbb{C}^5\\) with complex dimension 3. Note that if \\((x_0, \\dots, x_4) \\in \\mathbb{C}^5 \\) is a solution, so is any multiple \\((cx_0, \\dots , cx_4) \\). We may thus projectivize<\/a> \\(S\\), treating any solution as ‘the same’ as any multiple of that solution. The result is a submanifold \\(X\\) of the complex projective space<\/a> \\(\\mathbb{C}\\mathrm{P}^4\\). This has complex dimension 2, and it is a smooth algebraic variety<\/a>, so we call it a smooth complex surface<\/b>.<\/p>\n

We can also eliminate the variable $x_0$ in the above equations, and define the Clebsch surface starting from the single homogeneous cubic equation<\/p>\n

$$ x_1^3+x_2^3+x_3^3+x_4^3 = (x_1+x_2+x_3+x_4)^3 $$<\/p>\n

in variables \\(x_1, \\dots, x_4\\). Projectivizing this, we get a smooth complex surface in \\(\\mathbb{C}\\mathrm{P}^3\\). Since it is defined by a homogeneous cubic equation, it is called a smooth cubic surface<\/a><\/b>.<\/p>\n

The Clebsch surface illustrates a remarkable result in algebraic geometry, the Cayley–Salmon theorem<\/a><\/b>, which states that any smooth cubic surface contains 27 straight lines. No matter how one goes about it, the proof of this theorem is nontrivial. For a very helpful introduction to the main ideas, see:<\/p>\n

\u2022 Jack Huizenga, Algebraic geometry: why are there exactly 27 straight lines on a smooth cubic surface?<\/a>, Quora<\/i>.<\/p>\n

For the Clebsch surface, the 27 lines are easiest to describe as 2-dimensional subspaces of \\(\\mathbb{C}^5\\). We first define<\/p>\n

$$ \\zeta = e^{2 \\pi i \/ 5}. $$<\/p>\n

The 27 2-dimensional subspaces are:<\/p>\n

\u2022 The 15 subspaces obtained from the one containing \\((1,-1,0,0,0)\\) and \\((0,0,1,-1,0)\\) by permuting the 5 coordinates in an arbitrary way.<\/p>\n

\u2022 The 12 subspaces obtained from the one containing \\( (1, \\zeta, \\zeta^2 , \\zeta^3, \\zeta^4) \\) and \\( (1, \\zeta^{-1}, \\zeta^{-2} , \\zeta^{-3}, \\zeta^{-4}) \\) by permuting the 5 coordinates in an arbitrary way.<\/p>\n

It is easy to check that all these subspaces lie in \\(S \\subset \\mathbb{C}^5\\).<\/p>\n

The special virtue of the Clebsch surface is that all 27 lines are visible even when we restrict attention to real<\/em> solutions, obtaining an ordinary 2-dimensional real<\/em> surface that we can draw, rather than a complex surface.<\/p>\n

To obtain the above picture, Greg Egan first considered the space of real solutions of<\/p>\n

$$ x_1^3+x_2^3+x_3^3+x_4^3 = (x_1+x_2+x_3+x_4)^3 .$$<\/p>\n

He then projectivized this space of solutions, obtaining a surface in the real projective space<\/a> \\(\\mathbb{R}\\mathrm{P}^3\\). To draw the surface in \\(\\mathbb{R}^3\\) he then picked a certain plane in \\(\\mathbb{R}\\mathrm{P}^3\\) to be the ‘plane at infinity’. Following Bruce Hunt, he chose<\/p>\n

$$ x_0 + x_1 + x_2 + 4x_3 = 0. $$<\/p>\n

The resulting surface in \\(\\mathbb{R}^3\\) has the equation<\/p>\n

$$ \\begin{array}{cc} 81 (x^3 + y^3 + z^3) \\; – \\;
\n189 (x^2 y + x^2 z + x y^2 + x z^2 + y^2 z + y z^2) \\; + \\\\
\n54 xyz \\; + \\; 126(xy + xz + yz) \\; – \\; 9(x^2 + y^2 + z^2) \\; – \\; 9(x + y + z) \\; + \\; 1 \\; \\; = \\; \\; 0. \\end{array} $$<\/p>\n

This surface omits some points in the Clebsch cubic, which have become ‘points at infinity’.<\/p>\n

The picture above is a rotating view of this surface, the 27 lines on this surface, and 7 of the 10 Eckardt points<\/b>: points where 3 lines meet. The remaining 3 Eckardt points are at infinity.<\/p>\n

He drew the lines in a way that makes them easy to count: there are 3 lines of each of 9 colors, with each triple having an order-3 rotational symmetry. The three Eckardt points at infinity can also be easily described in terms of this image: they are the 3 points at infinity where parallel red, green and blue lines meet.<\/p>\n

Here are some still views of the Clebsch cubic:<\/p>\n

\n
\"Clebsch

Clebsch Cubic (Still View) – Greg Egan<\/p><\/div>\n<\/div>\n

\n
\"Clebsch

Clebsch Cubic (Symmetrical Still View) – Greg Egan<\/p><\/div>\n<\/div>\n

\n
\"Clebsch

Clebsch Cubic (Still View Showing Hole) – Greg Egan<\/p><\/div>\n<\/div>\n

The last view shows a hole in the Clebsch cubic that is invisible in the other views!<\/p>\n

Bruce Hunt has given a detailed explanation of the Clebsch cubic surface, invaluable for anyone wanting to understand this and also the beautiful combinatorics surrounding the 27 lines on any smooth cubic:<\/p>\n

\u2022 Bruce Hunt, The Geometry of Some Special Arithmetic Quotients<\/i>, Chapter 4: The 27 lines on a cubic surface<\/a>, Springer Lecture Notes in Mathematics 1637, Springer, Berlin, 1996.<\/p>\n

The 27 lines on the cubic are also connected to the 27-dimensional exceptional Jordan algebra<\/a>, which (along with its dual) provides the smallest nontrivial representation of the exceptional Lie group E6<\/sub><\/a>. This is part of a much larger story discussed here:<\/p>\n

\u2022 Laurent Manivel, Configurations of lines and models of Lie algebras,<\/a> Journal of Algebra<\/i> 304<\/b> (2006), 457\u2013486.<\/p>\n

For more on the Clebsch surface see this other post:<\/p>\n

\u2022 Clebsch surface<\/a>.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment <\/a>here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This animation by Greg Egan<\/a> shows 27 lines on a surface defined by cubic equations: the Clebsch surface<\/a>. It illustrates a remarkable fact: any smooth cubic surface contains 27 lines.<\/p>\n

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