Kummer Surface

The Kummer surfaces are a family of Quartic Surfaces given by the algebraic equation

$$(x^2+y^2+z^2-\mu^2w^2)^2-\lambda pqrs=0,$$ (1)

where

$$\lambda\equiv {3\mu^2-1\over 3-\mu^2},$$ (2)

$p$, $q$, $r$, and $s$ are the Tetrahedral Coordinates

$$p = w-z-\sqrt{2}\,x$$ (3)

$$q = w-z+\sqrt{2}\,x$$ (4)

$$r = w+z+\sqrt{2}\,y$$ (5)

$$s = w+z-\sqrt{2}\,y,$$ (6)

and $w$ is a parameter which, in the above plots, is set to $w=1$. The above plots correspond to $\mu^2=1/3$

$$(3x^2+3y^2+3z^2+1)^2=0,$$ (7)

(double sphere), 2/3, 1

$$x^4-2x^2y^2+y^4+4x^2z+4y^2z+4x^2z^2+4y^2z^2=0$$ (8)

(Roman Surface), $\sqrt{2}$, $\sqrt{3}$

$$[(z-1)^2-2x^2][y^2-(z+1)^2]=0$$ (9)

(four planes), 2, and 5. The case $0\leq\mu^2\leq 1/3$ corresponds to four real points.

The following table gives the number of Ordinary Double Points for various ranges of $\mu^2$, corresponding to the preceding illustrations.

$0\leq\mu^2\leq {\textstyle{1\over 3}}$	4	12
$\mu^2={\textstyle{1\over 3}}$
${\textstyle{1\over 3}}\leq\mu^2<1$	4	12
$\mu^2=1$
$1<\mu^2<3$	16	0
$\mu^2=3$
$\mu^2>3$	16	0

The Kummer surfaces can be represented parametrically by hyperelliptic Theta Functions. Most of the Kummer surfaces admit 16 Ordinary Double Points, the maximum possible for a Quartic Surface. A special case of a Kummer surface is the Tetrahedroid.

Nordstrand gives the implicit equations as

$$x^4+y^4+z^4-x^2-y^2-z^2-x^2y^2-x^2z^2-y^2z^2+1 = 0$$ (10)

or

$$x^4+y^4+z^4+a(x^2+y^2+z^2)+b(x^2y^2+x^2z^2+y^2z^2)+cxyz-1=0.$$ (11)

See also Quartic Surface, Roman Surface, Tetrahedroid

References

Endraß, S. ``Flächen mit vielen Doppelpunkten.'' DMV-Mitteilungen 4, 17-20, Apr. 1995.

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

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 14-19, 1986.

Fischer, G. (Ed.). Plates 34-37 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 33-37, 1986.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 183, 1994.

Hudson, R. Kummer's Quartic Surface. Cambridge, England: Cambridge University Press, 1990.

Kummer, E. ``Über die Flächen vierten Grades mit sechszehn singulären Punkten.'' Ges. Werke 2, 418-432.

Kummer, E. ``Über Strahlensysteme, deren Brennflächen Flächen vierten Grades mit sechszehn singulären Punkten sind.'' Ges. Werke 2, 418-432.

Nordstrand, T. ``Kummer's Surface.'' http://www.uib.no/people/nfytn/kummtxt.htm.