Endraß Octic

Endraß surfaces are a pair of Octic Surfaces which have 168 Ordinary Double Points. This is the maximum number known to exist for an Octic Surface, although the rigorous upper bound is 174. The equations of the surfaces $X_8^\pm$ are

$64(x^2-w^2)(y^2-w^2)[(x+y)^2-2w^2][(x-y)^2-2w^2]-\{-4(1\pm\sqrt{2}\,)(x^2+y^2)^2$
$+[8(2\pm\sqrt{2}\,)z^2+2(2\pm 7\sqrt{2}\,)w^2](x^2+y^2)$
$-16z^4+8(1\mp 2\sqrt{2}\,)z^2w^2-(1+12\sqrt{2}\,)w^4\}^2=0,$

where $w$ is a parameter taken as $w=1$ in the above plots. All Ordinary Double Points of $X_8^+$ are real, while 24 of those in $X_8^-$ are complex. The surfaces were discovered in a 5-D family of octics with 112 nodes, and are invariant under the Group $D_8\otimes Z_2$.

See also Octic Surface

References

Endraß, S. ``Octics with 168 Nodes.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/Eendrassoctic.shtml.

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

Endraß, S. ``A Proctive Surface of Degree Eight with 168 Nodes.'' J. Algebraic Geom. 6, 325-334, 1997.