The Barth Sextic

At least since Kummer's work (1860s) on 16-nodal quartics algebraic geometers ask the question how many isolated singularities a surface of degree d in projective 3-space can have. The answer is only known up to degree 6: Let us call this maximum number μ(d), then: μ(1)=0, μ(2)=1, μ(3)=4, μ(4)=16, μ(5)=31, μ(6)=65. See Oliver's Ph.D. thesis and publications for a lot more on surfaces with many singularities, e.g.: μ(7)≥99 (2004, O. Labs), μ(7)≤104 (1983, Varchenko).

In 1996, W. Barth showed μ(6)≥65 by constructing a surface of degree six (a sextic) with 65 nodes. His equation is very simple: B65 = P - α Q^2 where α is a suitably chosen constant, Q is the unit sphere and P is a product of all six planes which are orthogonal to the diagonals of a regular icosahedron.

The animation starts by visualizing that the pencil xy-(z+ε)(z-ε) specializes to the cone (a node) for ε=0. Analoguously, each intersection of Q with the double lines of P is a node of B65. Choosing an α related to the golden section yields a sextic with 65 nodes (15 of which are at infinity).

This animation was made by Hans-Christian and Oliver using surfex.

This was #6 on our geometric animations advent calender:
http://www.calendar.algebraicsurface.net/

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