Kummer surface

In algebraic geometry, a Kummer quartic surface, first studied by Ernst Kummer (1864), is an irreducible nodal surface of degree 4 in

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.

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.

be a quartic surface with an ordinary double point p, near which K looks like a quadratic cone.

; this double cover is given by sending q ≠ p ↦

The ramification locus of the double cover is a plane curve C of degree 6, and all the nodes of K which are not p map to nodes of C. By the genus degree formula, the maximal possible number of nodes on a sextic curve is obtained when the curve is a union of

lines, in which case we have 15 nodes.

Hence the maximal number of nodes on a quartic is 16, and in this case they are all simple nodes (to show that

is a simple node, the tangent cone to this point is mapped to a conic under the double cover.

This conic is in fact tangent to the six lines (w.o proof).

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.

, under a map which blows down the double cover of the special conic, and is an isomorphism elsewhere (w.o.

is a hyperelliptic curve the map from the symmetric product

, is the blow down of the graph of the hyperelliptic involution to the canonical divisor class.

This double cover is the one which already appeared above: The 6 lines are the images of the odd symmetric theta divisors on

, and each of the six lines is naturally isomorphic to the dual canonical system

via the identification of theta divisors and translates of the curve

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

are Weierstrass points (which are the odd theta characteristics in this in genus 2).

Hence the branch points of the canonical map

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

maps to the complete linear system

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

There are several crucial points which relate the geometric, algebraic, and combinatorial aspects of the configuration of the nodes of the kummer quartic: Hence we have a configuration of

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 166 configuration.

This article incorporates material from the Citizendium article "Kummer surface", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.