In algebraic geometry, a Kummer quartic surface, first studied by Kummer (1864), is an irreducible algebraic surface of degree 4 in with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian 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 surface closely related to Kummer surfaces include Weddle surfaces, Wave surfaces, and tetrahedroids.
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“Just under the surface I shall be, all together at first, then separate and drift, through all the earth and perhaps in the end through a cliff into the sea, something of me.”
—Samuel Beckett (1906–1989)