{"id":2813,"date":"2016-09-01T01:00:46","date_gmt":"2016-09-01T01:00:46","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2813"},"modified":"2016-11-02T16:08:48","modified_gmt":"2016-11-02T16:08:48","slug":"kummers-quartic-surface","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/09\/01\/kummers-quartic-surface\/","title":{"rendered":"Kummer Quartic"},"content":{"rendered":"
\n
\"Kummer<\/a>

Kummer Quartic Surface – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

A quartic surface<\/b> is one defined by a polynomial equation of degree 4. 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

The Kummer surfaces<\/a> are the quartic nodal surfaces with the largest possible number of ordinary double points, namely 16. In the above picture, Abdelaziz Nait Merzouk<\/a> has drawn the real points of a Kummer surface.<\/p>\n

There are many different Kummer surfaces, but they all have the same topology and they can all be constructed in the same way. Start with a smooth complex curve<\/a> of genus 2<\/a>, say \\(C\\):<\/p>\n

\n
\"Genus<\/a>

Genus Two Curve – Oleg Alexandrov<\/p><\/div>\n<\/div>\n

Then, form its Jacobian variety<\/a> \\(J(C)\\). This is the space of isomorphism classes of holomorphic line bundles<\/a> over \\(C\\) with vanishing first Chern class<\/a>. As a topological space, this Jacobian is always product of 4 copies of a circle, but it is equipped with the structure of a complex surface in a way that depends on \\(C\\). It is an abelian group, thanks to our ability to tensor line bundles, so it has an automorphism \\(x \\mapsto -x\\) called the Kummer involution<\/b>. The quotient of the Jacobian by this involution is a Kummer surface<\/b>. The ordinary double points come from the points in \\(J(C)\\) with \\(x = -x\\); there are \\(16 = 2^4\\) of these in a product of 4 copies of the circle group.<\/p>\n

Here is a ‘cut’ view of the same Kummer surface. Again, only the real points are shown:<\/p>\n

\n
\"Cut<\/a>

Cut Kummer’s Quartic Surface – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

For an introduction to Kummer surfaces, see:<\/p>\n

• Dylan Wilson, Introducing K3 surfaces: Kummer surfaces<\/a>, 2014.<\/p>\n

Explicit equations for the Kummer surfaces can be found here:<\/p>\n

• Kummer surface<\/a>, Wolfram MathWorld<\/i>.<\/p>\n

Abdelaziz Nait Merzouk created these pictures of a Kummer surface and made them available on Google+<\/a> under a Creative Commons Attribution-ShareAlike 3.0 Unported<\/a> license.\ufeff Oleg Alexandrov made the picture of a complex curve of genus two and put it in the public domain on Wikicommons<\/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 here<\/a> and let me know!<\/p>\n

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

A quartic surface<\/b> is one defined by a polynomial equation of degree 4. An ordinary double point<\/b> is a point where a surface looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$. The Kummer surfaces<\/a> are the quartic surfaces with the largest possible number of ordinary double points, namely 16. This picture by Abdelaziz Nait Merzouk<\/a> shows the real points of a Kummer surface.<\/p>\n

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