Kummer surface

In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb{P}^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution x ↦ −x. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces. Other surfaces closely related to Kummer surfaces include Weddle surfaces, wave surfaces, and tetrahedroids.

In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb{P}^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution x ↦ −x. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces. Other surfaces closely related to Kummer surfaces include Weddle surfaces, wave surfaces, and tetrahedroids. This double cover is the one which already appeared above: The 6 lines are the images of the odd symmetric theta divisors on Jac(C) , while the conic is the image of the blown-up 0. The conic is isomorphic to the canonical system via the isomorphism T_0 Jac(C)\cong |K_C|^* , and each of the six lines is naturally isomorphic to the dual canonical system |K_C|^* via the identification of theta divisors and translates of the curve C . 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 (\Theta+w_1)\cap(\Theta+w_2)=\{w_1-w_2,0\} , where w_1,w_2 are Weierstrass points (which are the odd theta characteristics in this in genus 2). Hence the branch points of the canonical map C\mapsto |K_C|^* 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 C maps to the complete line complex |O_{Jac(C)}(2\Theta_C)|\cong\mathbb{P}^{2^2-1} (see the article on 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 Jac(C) . \langle p_1-p_2,p_3-p_4\rangle=\#\{p_1,p_2\}\cap\{p_3,p_4\} . One can recover a lot of the group theoretic invariants of the group Sp_4(2) via the geometry of the 16_6 configuration.

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