Kummer surfaces

In 1864 Ernst Eduard Kummer gave the following real one-dimensional family of surfaces of degree four (quartics):

$X(4,mu) = {(3-mu^2)(x^2+y^2+z^2-mu^2w^2)^2-(3mu^2-1)pqrs = 0}, mu real, p=w-z-sqrt(2)x, q=w-z+sqrt(2)x, r=w+z+sqrt(2)y, s=w+z-sqrt(2)y$

Most of this surfaces admit $µ(4)=16$ ordinary nodes, which can of course be complex. By varying µ one gets the following table:

`µ`	type	nodes		picture
`µ`	type	real	complex	picture
$0<=$ `µ`²<1/3	Kummersurface	4	12	4 real points
`µ`²=1/3	double sphere	/	/	28K
$1/3<$ `µ`²<1	Kummersurface	4	12	34K
`µ`²=1	Steiner roman surface	/	/	29K
$1<$ `µ`²<3	Kummersurface	16	0	52K
`µ`²=3	four planes	/	/	35K
$3<$ `µ`²	Kummersurface	16	0	52K
`µ`²=oo	Kummersurface	16	0	52K

All surfaces are invariant under the symmetry group of the tetrahedron S₄. Moreover, the surface X_4,oo is invariant under the symmetry group of the cube. In the modern classification of surfaces these quartics belong to the family of Kummer surfaces. Kummer surfaces are K3 surfaces containing 16 rational, disjoint (-2)-curves.

Stephan Endraß

Last update: Thursday, 06-Feb-2003 08:52:25 CET

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