Editing Kummer surface (section)

== Level 2 structure ==

=== Kummer's 16<sub>6</sub> configuration ===
There are several crucial points which relate the geometric, algebraic, and combinatorial aspects of the configuration of the nodes of the kummer quartic:

* Any symmetric odd theta divisor on <math>Jac(C)</math> is given by the set points <math>\{q-w|q\in C\}</math>, where w is a Weierstrass point on <math>C</math>. This theta divisor contains six 2-torsion points: <math>w'-w</math> such that <math>w'</math> is a Weierstrass point.
* Two odd theta divisors given by Weierstrass points <math>w,w'</math> intersect at <math>0</math> and at <math>w-w'</math>.
* The translation of the Jacobian by a two torsion point is an isomorphism of the Jacobian as an algebraic surface, which maps the set of 2-torsion points to itself.
* In the complete linear system <math>|2\Theta_C|</math> on <math>Jac(C)</math>, any odd theta divisor is mapped to a conic, which is the intersection of the Kummer quartic with a plane. Moreover, this complete linear system is invariant under shifts by 2-torsion points.

Hence we have a configuration of <math>16</math> conics in <math>\mathbb{P}^3</math>; where each contains 6 nodes, and such that the intersection of each two is along 2 nodes. This configuration is called the <math>16_6</math> configuration or the [[Kummer configuration]].

=== Weil pairing ===
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. The Weil pairing is given in this case by
<math>\langle p_1-p_2,p_3-p_4\rangle=\#\{p_1,p_2\}\cap\{p_3,p_4\}</math>. One can recover a lot of the group theoretic invariants of the group <math>Sp_4(2)</math> via the geometry of the <math>16_6</math> configuration.

=== Group theory, algebra and geometry ===
Below is a list of group theoretic invariants and their geometric incarnation in the 16<sub>6</sub> configuration.
 
* [[Polar line]]s
* [[Apolar complex]]es
* [[Klein configuration]]
* [[Fundamental quadric]]s
* [[Fundamental tetrahedra]]
* [[Rosenhain tetrad]]s
* [[Adolph Göpel 1812-1847]]s