{"id":2833,"date":"2016-09-15T01:00:55","date_gmt":"2016-09-15T01:00:55","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2833"},"modified":"2016-10-30T05:06:14","modified_gmt":"2016-10-30T05:06:14","slug":"togliatti-quintic-surface","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/09\/15\/togliatti-quintic-surface\/","title":{"rendered":"Togliatti Quintic"},"content":{"rendered":"
\n
\"Togliatti<\/a>

Togliatti Quintic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

A quintic surface<\/b> is one defined by a polynomial equation of degree 5. 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

A Togliatti surface<\/a> is a quintic nodal surface with the largest possible number of ordinary double points, namely 31. In the above picture, Abdelaziz Nait Merzouk<\/a> has drawn the real points of a Togliatti surface.<\/p>\n

This surface is described by a homogeneous quintic equation in four variables, say \\(w,x,y,z\\), which is then intersected with the hyperplane \\(w = 1\\). Here is a version rotated around the \\(wz\\) plane before intersecting with the hyperplane \\(w = 1\\):<\/p>\n

\n
\"Rotated<\/a>

Rotated Togliatti Quintic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

This version is sometimes called the ‘dervish’, due to its resemblance to a whirling dervish.<\/p>\n

The first example of a Togliatti quintic was constructed in 1940:<\/p>\n

• Eugenio G. Togliatti, Una notevole superficie di 5° ordine con soli punti doppi isolati<\/a>, Vierteljschr. Naturforsch. Ges. Z\u00fcrich<\/i> 85<\/b> (1940), 127–132.<\/p>\n

In 1980, Beauville proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal:<\/p>\n

• Arnaud Beauville, Sur le nombre maximum de points doubles d’une surface dans \\(\\mathrm{P}^3\\) (\u03bc(5) = 31)<\/a>, Journ\u00e9es de G\u00e9ometrie Alg\u00e9brique d’Angers, Juillet 1979\/Algebraic Geometry, Angers, 1979<\/i>, Alphen aan den Rijn\u2014Germantown, Md.: Sijthoff & Noordhoff, 1980, pp. 207–215.<\/p>\n

Abdelaziz Nait Merzouk created these pictures of a Kummer surface and made them available on Google+<\/a> and made them available under a Creative Commons Attribution-ShareAlike 3.0 Unported<\/a> license.\ufeff <\/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 here<\/a> and let me know!<\/p>\n

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

A quintic surface<\/b> is one defined by a polynomial equation of degree 5. 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 \\(x^2 + y^2 = z^2\\). A Togliatti surface<\/a> is a quintic nodal surface with the largest possible number of ordinary double points, namely 31. Here Abdelaziz Nait Merzouk<\/a> has drawn the real points of a Togliatti surface.<\/p>\n

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