Barth's sextic

We only know one surface of degree six (sextic) admitting

µ(6)=65

ordinary nodes. It is the surface

X₆ constructed by W. Barth in 1994:

$X6 = {4(t^2x^2-y^2)(t^2y^2-z^2)(t^2z^2-x^2) -(1+2t)(x^2+y^2+z^2-w^2)^2w^2 = 0}, t = (1+sqrt(5))/2$

[150x150 pixel image of X6] 38K [150x150 pixel transparent image of X6] 43K

X₆ is invariant under the symmetry group of the icosahedron A₅. Its equation was derived as a linear combination of two A₅-invariant polynomials of degree six. Moreover, under the mapping

$(x:y:z:w)\|--->(x$ ²:y²:z²:w²)

X₆ is the eightfold cover of a Cayley-cubic. The following computer generated animation shows the rotation of X₆ about a line connecting opposite vertices of the icosahedron.

It is the short version of the following FLI-animation:

Animation of

X₆ (512x512 pixel, 288 frames, FLI)

It can be played using xanim on most unix machines.

Stephan Endraß

Last update: Thursday, 06-Feb-2003 08:52:33 CET

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