Kummer surface: Difference between revisions

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In [[algebraic geometry]] Kummer's quartic surface is an [[irreducible]] [[algebraic surface]] over a field <math>K</math> of characteristic different then 2, which is a hypersurface of degree 4 in <math>\mathbb{P}^3</math>  with 16 [[singularities]]; the maximal possible number of singularities of a quartic surface. It is a remarkable fact that 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 <math>x\mapsto-x</math>. 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.
== Geometry of the surface ==
== Geometry of the surface ==



Revision as of 18:57, 28 February 2007

In algebraic geometry Kummer's quartic surface is an irreducible algebraic surface over a field of characteristic different then 2, which is a hypersurface of degree 4 in with 16 singularities; the maximal possible number of singularities of a quartic surface. It is a remarkable fact that 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 . 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.

Geometry of the surface

The quadric line complex

Kummer's quartic surface

Kummers's quartic surfaces as kummer varieties of Jacobians

The double plane model

Kummer varieties of Jacobians as Kummer quartics

Geometry and combinatorics of the level structure

Polar lines

Apolar complexes

Klien's configuration

Kummer's configurations

fundamental quadrics

fundamental tetrahedra

Rosenheim tetrads

Gopel tetrads

References

  • Igor Dolgachev's online notes on classical algebraic geometry (this is the main source of the first part of this article)