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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


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

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


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.''

Nordstrand, T. ``Barth Sextic.''

© 1996-9 Eric W. Weisstein