Editing Kummer surface (section)

=== The double plane and Kummer varieties of Jacobians ===
Starting from a smooth curve <math>C</math> of genus 2, we may identify the Jacobian <math>Jac(C)</math>
with <math>Pic^2(C)</math> under the map <math>x\mapsto x+K_C</math>. We now observe two facts: Since <math>C</math> is a [[hyperelliptic curve]] the map from the symmetric product
<math>Sym^2 C</math> to <math>Pic^2 C</math>, defined by <math>\{p,q\}\mapsto p+q</math>, is the blow down of the graph of the hyperelliptic  involution to the [[canonical divisor]] class. Moreover, the canonical map <math>C\to|K_C|^*</math> is a double cover. Hence we get a double cover <math>Kum(C)\to Sym^2|K_C|^*</math>.

This double cover is the one which already appeared above: The 6 lines are the images of the odd symmetric [[theta divisors]] on <math>Jac(C)</math>, while the conic is the image of the blown-up 0. The conic is isomorphic to the canonical system via the isomorphism <math>T_0 Jac(C)\cong |K_C|^*</math>, and each of the six lines is naturally isomorphic to the dual canonical system <math>|K_C|^*</math> via the identification of theta divisors and translates of the curve <math>C</math>. There is a 1-1 correspondence between pairs of odd symmetric theta divisors and 2-torsion points on the Jacobian given by the fact that <math>(\Theta+w_1)\cap(\Theta+w_2)=\{w_1-w_2,0\}</math>, where <math>w_1,w_2</math> are Weierstrass points  (which are the odd theta characteristics in this in genus 2). Hence the branch points of the canonical map <math>C\mapsto |K_C|^*</math> appear on each of these copies of the canonical system as the intersection points of the lines and the tangency points of the lines and the conic.

Finally, since we know that every Kummer quartic is a Kummer variety of a Jacobian of a hyperelliptic curve, we show how to reconstruct Kummer quartic surface directly from the Jacobian of a genus 2 curve: The Jacobian of <math>C</math> maps to the complete [[linear system]] <math>|O_{Jac(C)}(2\Theta_C)|\cong\mathbb{P}^{2^2-1}</math> (see the article on [[Abelian variety|Abelian varieties]]). This map factors through the Kummer variety as a degree 4 map which has 16 nodes at the images of the 2-torsion points on <math>Jac(C)</math>.