Barth Sextic

The Barth-sextic is a Sextic Surface in complex three-dimensional projective space having the maximum possible number of Ordinary Double Points (65). It is given by the implicit equation

$$4(\phi^2 x^2 - y^2)(\phi^2 y^2 - z^2)(\phi^2 z^2 - x^2)-(1+2\phi)(x^2 + y^2 + z^2 - w^2)^2w^2 = 0.$$

where $\phi$ is the Golden Mean, and $w$ is a parameter (Endraß, Nordstrand), taken as $w=1$ in the above plot. The Barth sextic is invariant under the Icosahedral Group. Under the map

$$(x,y,z,w)\to(x^2,y^2,z^2,w^2),$$

the surface is the eightfold cover of the Cayley Cubic (Endraß).

See also Algebraic Surface, Barth Decic, Cayley Cubic, Ordinary Double Point, Sextic Surface

References

Barth, W. ``Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron.'' J. Alg. Geom. 5, 173-186, 1996.

Endraß, S. ``Flächen mit vielen Doppelpunkten.'' DMV-Mitteilungen 4, 17-20, 4/1995.

Endraß, S. ``Barth's Sextic.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/Ebarthsextic.shtml.

Nordstrand, T. ``Barth Sextic.'' http://www.uib.no/people/nfytn/sexttxt.htm.