Quintic Surface

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.

See also Algebraic Surface, Dervish, Kiss Surface, Ordinary Double Point

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 $5^o$ 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 $\Bbb{P}^4$ with $130$ Nodes.'' Topology 32, 857-864, 1993.