Togliatti surface

In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by Eugenio G. Togliatti (1940). Arnaud Beauville (1980) proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal.

The surface with $w=1$ (real points, bounded by a sphere with radius=6).

3D model of same surface as above ($w=1$) bounded by the cube $[-10,10]^{3}$

See also

• Barth surface
• Endrass surface
• Sarti surface
• List of algebraic surfaces

References

• Beauville, Arnaud (1980), "Sur le nombre maximum de points doubles d'une surface dans $\mathbf {P} ^{3}(\mu (5)=31)$", Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 (PDF) (in French), Alphen aan den Rijn—Germantown, Md.: Sijthoff & Noordhoff, pp. 207–215, MR 0605342.
• Togliatti, Eugenio G. (1940), "Una notevole superficie di 5^o ordine con soli punti doppi isolati", Beiblatt (Festschrift Rudolf Fueter) (PDF), Vierteljschr. Naturforsch. Ges. Zürich (in Italian), vol. 85, pp. 127–132, MR 0004492.

External links

• Endraß, Stephan (2003). "Togliatti surfaces".
• Weisstein, Eric W. "Togliatti surface". MathWorld.