Kummers quartic surface

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Any projective line through p then meets K with multiplicity two at p , and will therefore meet the quartic K in just two other points. A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below.

chongranrelewalt.mlaic geometry - Kummer's quartic surface and the Dirac operator - MathOverflow

This conic is in fact tangent to the six lines w. Conversely, given a configuration of a conic and six lines which tangent to it in the plane, we may define the double cover of the plane ramified over the union of these 6 lines. There are several crucial points which relate the geometric, algebraic, and combinatorial aspects of the configuration of the nodes of the kummer quartic:. The 2-torsion points on an Abelian variety admit a symplectic bilinear form called the Weil pairing.

In the case of Jacobians of curves of genus two, every nontrivial 2-torsion point is uniquely expressed as a difference between two of the six Weierstrass points of the curve. Below is a list of group theoretic invariants and their geometric incarnation in the 16 6 configuration. This article incorporates material from the Citizendium article " Kummer surface ", which is licensed under the Creative Commons Attribution-ShareAlike 3.

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Cambridge, England: University Press, Kummer, E. Berlin: Springer-Verlag, pp. Nordstrand, T.

Kummer Surface

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