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Kolibri Surface

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kolibri surface

The kolibri surface is the quartic surface

y^2=x^3+x^2z^2,

named by Hauser. Its German name means "hummingbird." Its singular points form the z-axis.

For a fixed nonzero value z=c, the section is the nodal cubic y^2=x^2(x+c^2), whose two tangent lines at the origin of that section are y=+/-cx. At z=0, the section becomes the semicubical parabola y^2=x^3. Thus the type of the singularity changes at the origin, even though the set of singular points is a straight line. Faber and Hauser (2010, pp. 386-390) analyze its resolution of singularities.

See also

Quartic Surface, Resolution of Singularities, Semicubical Parabola

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References

Faber, E. and Hauser, H. "Today's Menu: Geometry and Resolution of Singular Algebraic Surfaces." Bull. Amer. Math. Soc. 47, 373-417, 2010. https://doi.org/10.1090/S0273-0979-10-01295-4.Hauser, H. "Gallery of Singular Algebraic Surfaces: Kolibri." https://homepage.univie.ac.at/herwig.hauser/gallery.html.

Cite this as:

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

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