Kummer surface

In algebraic geometry, a Kummer quartic surface, first studied by Kummer (1864), is an irreducible nodal surface of degree 4 in ${\displaystyle \mathbb {P} ^{3}}$ 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.

Let ${\displaystyle K\subset \mathbb {P} ^{3}}$ be a quartic surface with an ordinary double point p, near which K looks like a quadratic cone. 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. Identifying the lines in ${\displaystyle \mathbb {P} ^{3}}$ through the point p with ${\displaystyle \mathbb {P} ^{2}}$, we get a double cover from the blow up of K at p to ${\displaystyle \mathbb {P} ^{2}}$; this double cover is given by sending q ≠ p ↦ ${\displaystyle \scriptstyle {\overline {pq}}}$, and any line in the tangent cone of p in K to itself. 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.

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