A quintic surface is an Algebraic Surface of degree 5. Togliatti (1940, 1949) showed that quintic surfaces having 31 Ordinary Double Points exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points was the maximum possible, and quintic surfaces having 31 Ordinary Double Points are therefore sometimes called Togliatti Surfaces. van Straten (1993) subsequently constructed a 3-D family of solutions and in 1994, Barth derived the example known as the Dervish.

**References**

Beauville, A. ``Surfaces algébriques complexes.'' *Astérisque* **54**, 1-172, 1978.

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

Hunt, B. ``Algebraic Surfaces.'' http://www.mathematik.uni-kl.de/~wwwagag/Galerie.html.

Togliatti, E. G. ``Una notevole superficie de ordine con soli punti doppi isolati.'' *Vierteljschr. Naturforsch. Ges. Zürich*
**85**, 127-132, 1940.

Togliatti, E. ``Sulle superficie monoidi col massimo numero di punti doppi.'' *Ann. Mat. Pura Appl.* **30**, 201-209, 1949.

van Straten, D. ``A Quintic Hypersurface in with Nodes.'' *Topology* **32**, 857-864, 1993.

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1999-05-25