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

(1) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) |

(9) |

(10) | |||

(11) | |||

(12) |

(Nordstrand).

The dervish is invariant under the Group and contains exactly 15 lines. Five of these are the intersection of the surface with a -invariant cone containing 16 nodes, five are the intersection of the surface with a -invariant plane containing 10 nodes, and the last five are the intersection of the surface with a second -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.

© 1996-9

1999-05-24