Barth Decic

$\begin{figure}\begin{center}\BoxedEPSF{BarthDecic.epsf}\end{center}\end{figure}$

The Barth decic is a Decic Surface in complex three-dimensional projective space having the maximum possible number of Ordinary Double Points (345). It is given by the implicit equation

$8(x^2-\phi^4 y^2) (y^2-\phi^4 z^2) (z^2-\phi^4 x^2)(x^4+y^4+z^4-2 x^2 y^2-2 x^2 z^2-2 y^2 z^2)$

$+(3+5\phi) (x^2+y^2+z^2-w^2)^2 [x^2+y^2+z^2-(2-\phi) w^2]^2 w^2=0,$

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

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 Decic.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/Ebarthdecic.shtml.

Nordstrand, T. ``Batch Decic.'' http://www.uib.no/people/nfytn/bdectxt.htm.

$8(x^2-\phi^4 y^2) (y^2-\phi^4 z^2) (z^2-\phi^4 x^2)(x^4+y^4+z^4-2 x^2 y^2-2 x^2 z^2-2 y^2 z^2)$
$+(3+5\phi) (x^2+y^2+z^2-w^2)^2 [x^2+y^2+z^2-(2-\phi) w^2]^2 w^2=0,$