Dervish

A Quintic Surface having the maximum possible number of Ordinary Double Points (31), which was constructed by W. Barth in 1994 (Endraß). The implicit equation of the surface is

$64(x-w)[x^4-4x^3w-10x^2y^2-4x^2w^2$
$+ 16xw^3-20xy^2w+5y^4+16w^4-20y^2w^2]$
$-5\sqrt{5-\sqrt{5}}\,(2z-\sqrt{5-\sqrt{5}}\,w)[4(x^2+y^2+z^2)+(1+3\sqrt{5}\,)w^2]^2,\quad$	(1)

where $w$ is a parameter (Endraß). The surface can also be described by the equation

$$aF+q=0,$$ (2)

where

$$F=h_1 h_2 h_3 h_4 h_5,$$ (3)

$$h_1 = x-z$$ (4)

$$h_2 = \cos\left({2\pi\over 5}\right)x-\sin\left({2\pi\over 5}\right)y-z$$ (5)

$$h_3 = \cos\left({4\pi\over 5}\right)x-\sin\left({4\pi\over 5}\right)y-z$$ (6)

$$h_4 = \cos\left({6\pi\over 5}\right)x-\sin\left({6\pi\over 5}\right)y-z$$ (7)

$$h_5 = \cos\left({8\pi\over 5}\right)x-\sin\left({8\pi\over 5}\right)y-z$$ (8)

$$q=(1-cz)(x^2+y^2-1+rz^2)^2,$$ (9)

and

$$r = {\textstyle{1\over 4}}(1+\sqrt{5}\,)$$ (10)

$$a = -{8\over 5}\left({1+{1\over\sqrt{5}}}\right)\sqrt{5-\sqrt{5}}$$ (11)

$$c = {\textstyle{1\over 2}}\sqrt{5-\sqrt{5}}\,$$ (12)

(Nordstrand).

The dervish is invariant under the Group $D_5$ and contains exactly 15 lines. Five of these are the intersection of the surface with a $D_5$-invariant cone containing 16 nodes, five are the intersection of the surface with a $D_5$-invariant plane containing 10 nodes, and the last five are the intersection of the surface with a second $D_5$-invariant plane containing no nodes (Endraß).

References

Endraß, S. ``Togliatti Surfaces.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/Etogliatti.shtml.

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

Endraß, S. Symmetrische Fläche mit vielen gewöhnlichen Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996.

Nordstrand, T. ``Dervish.'' http://www.uib.no/people/nfytn/dervtxt.htm.