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Calypso Surface

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calypso surface

The calypso surface is the cubic surface

x^2+y^2z=z^2,

named by Hauser. Its only singular point is the origin. Its section by the plane y=0 consists of the two lines x=+/-z, whereas its section by y=1 is the hyperbola (z-1/2)^2-x^2=1/4.

The map (x,y,z)|->(xz,y,z) takes the calypso surface onto the calyx surface. Conversely, the substitution x=Xz in the equation of the calyx surface, followed by removal of the factor z^2, gives the calypso equation. This is a step in an explicit resolution of singularities by algebraic blow-ups (Hauser 2007, p. 33).

See also

Algebraic Blow-Up, Calyx Surface, Cubic Surface, Resolution of Singularities

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References

Hauser, H. "Desingularization of Ideals and Varieties." In Singularity Theory: Proceedings of the 2005 Marseille Singularity School and Conference (Ed. D. Chéniot, N. Dutertre, C. Murolo, D. Trotman, and A. Pichon). Singapore: World Scientific, pp. 33-69, 2007. https://doi.org/10.1142/9789812707499_0002.Hauser, H. "Gallery of Singular Algebraic Surfaces: Calypso." https://homepage.univie.ac.at/herwig.hauser/gallery.html.

Cite this as:

Weisstein, Eric W. "Calypso Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CalypsoSurface.html

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