A new look at the Kummer surface

W. L. Edge
1967 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Rummer's surface has the base surface F of a certain net 31 of quadrics in [5] for a non-singular model. All the quadrics of 31 have a common self-polar simplex 2, and 31 can, in a double-infinity of ways, be based on a quadric 12i and two quadrics that 12i reciprocates into each other. F is invariant under harmonic inversions in the vertices and opposite bounding primes of 2 and ( §2) contains 32 lines. In §3 it is shown, conversely, that those quadrics for which a given simplex is self-polar
more » ... nd which contain a line of general position constitute a net of this kind. Each quadric of 31 is shown, in §5, to contain 32 tangent planes of F. This is confirmed by another argument in §6, where it is explained how this fact establishes an involution of pairs of conjugate directions at any point of F; this involves a system of asymptotic curves. That these are intersections of F with quadrics for which, too, 2 is self-polar is seen in §7; that they correspond to the traditional asymptotic curves on K will be proved in §12. After describing a less-specialized surface <£ in §8, the rudiments of the geometry on F are alluded to in §9; §10 introduces the Weddle surface W as a projection of F and avails itself of the classical birational correspondence between W and K. The association between the 15 quadric line cones through F and the 15 separations of the nodes of K into complementary octads is explained in §11. The details of the mapping of curves, of lower orders, on F and K into each other are discussed in § §12-15. § §16-17 end the paper with a brief note on tetrahedroids. 1. The following pages concern matters related to the geometry of the surface F, in projective space [5], common to the three quadrics where the summations over i run from 0 to 5 and no two of the six coefficients a t are equal. It is sometimes convenient to speak of the quadric 0 K = 0 merely as "the quadric 12 K ." The first appearance of these three equations is in Klein's early paper (11) on linear and quadratic complexes. He uses his mapping of the lines of [3] by the points of 12 0 , so that the lines of a general quadratic complex are mapped on the threefold Q 0 = Gi = 0. The equations (1.1) are all on p. 223 of (11),
doi:10.4153/cjm-1967-087-5 fatcat:fuqnuvao3zctrbh4yj7lmuumwu