{"id":2791,"date":"2016-08-15T01:00:49","date_gmt":"2016-08-15T01:00:49","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2791"},"modified":"2016-08-31T10:00:09","modified_gmt":"2016-08-31T10:00:09","slug":"cayleys-nodal-cubic-surface","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/08\/15\/cayleys-nodal-cubic-surface\/","title":{"rendered":"Cayley’s Nodal Cubic Surface"},"content":{"rendered":"
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\"Cayley's<\/a>

Cayley’s Nodal Cubic Surface – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

A cubic surface<\/b> is one defined by a polynomial equation of degree 3. A nodal surface<\/b> is one whose only singularities are ordinary double points<\/b>: that is, points where it looks like the origin of the cone in 3-dimensional space defined by<\/p>\n

$$ x^2 + y^2 = z^2 .$$<\/p>\n

Cayley’s nodal cubic surface<\/a><\/b>, drawn above by Abdelaziz Nait Merzouk<\/a>, is the cubic surface with the largest possible number of ordinary double points, namely 4. In fact, every cubic with 4 ordinary double points is isomorphic to this one.<\/p>\n

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\"Cayley's<\/a>

Cayley’s Nodal Cubic Surface (Cut Version) – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

Cayley’s nodal cubic surface is described by this equation<\/p>\n

$$ wxy + wxz + wyz + xyz = 0 $$<\/p>\n

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<\/a> \\(S\\), treating any solution as ‘the same’ as any multiple of that solution. The result is an algebraic variety<\/a> \\(X\\) in the complex projective space<\/a> \\(\\mathbb{C}\\mathrm{P}^3\\). This variety has complex dimension 2, so it is called a complex surface<\/b>. 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\\). <\/p>\n

Sitting inside \\(\\mathbb{R}\\mathrm{P}^3\\) we in turn have many copies of ordinary 3-dimensional space, \\(\\mathbb{R}^3\\). The pictures above show the portion of Cayley’s nodal cubic surface living in one of these copies. <\/p>\n

The simple double points in Cayley’s nodal cubic occur where three of the coordinates \\(w,x,y,z\\) are zero. The hyperplane \\(w + x + y + z = 1\\) determines a copy of \\(\\mathbb{C}^3\\) inside \\(\\mathbb{C}\\mathrm{P}^3\\), and taking all four coordinates to be real gives a copy of \\(\\mathbb{R}^3\\) in which these double points lie at the vertices of a regular tetrahedron. Indeed, the symmetry group of Cayley’s nodal cubic is \\(\\mathrm{S}_4\\), the symmetry group of a tetrahedron.<\/p>\n

Puzzle 1.<\/b> There are 9 lines on Cayley’s nodal cubic surface. 6 of these lines contain the edges of the tetrahedron described above. What are the other 3 lines?<\/p>\n

Some interesting properties of Cayley’s nodal cubic surface are discussed here:<\/p>\n

• Bruce Hunt, Nice modular varieties<\/a>, Experimental Mathematics<\/i> 9<\/b> (2000), 613–622.<\/p>\n

In particular, he explains how it is a compactification of a quotient of a ball, and a moduli space for certain abelian fourfolds.<\/p>\n

Puzzle 2.<\/b> Show that after a change of variables, Cayley’s nodal cubic surface can also be described by the equation<\/p>\n

$$ w^3 + x^3 + y^3 + z^3 = (w + x + y + z)^3 $$<\/p>\n

Abdelaziz Nait Merzouk created the pictures above and made them available on Google+<\/a> under a Creative Commons Attribution-ShareAlike 3.0 Unported<\/a> license.\ufeff<\/p>\n

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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 here<\/a> and let me know!<\/p>\n

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

A cubic surface<\/b> is one defined by a polynomial equation of degree 3. Cayley’s nodal cubic surface<\/a>, drawn above by Abdelaziz Nait Merzouk<\/a>, is the cubic surface with the largest possible number of ordinary double points<\/b> and no other singularities: 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 4 ordinary double points, shown here at the vertices of a regular tetrahedron.<\/p>\n

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