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.  
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 Kummer surface ==
== Geometry of the Kummer surface ==

Revision as of 10:55, 8 March 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 Kummer surface

Singular quartic surfaces and the double plane model

Let be a quartic surface, and let be a singular point of this surface. Identifying the lines in thorugh the point with , we get a double cover from the blow up of at to ; this double cover is given by sending , and any line in the tangent cone of in to itself. The ramification locus of the double cover is a plane curve of degree 6, and all the nodes of which are not map to nodes of .

By the genus degree formula, the maximal number possible number of nodes on a sextic curve is obtained when the curve is a 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 simple project from another node). A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below.

Since 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. This double cover may be mapped to <mathbb>\mathbb{P}^3</math>, under a map which blows down the doubel cover of the special conic, and is an isomorphism elsewhere (w.o. proof).

Kummers's quartic surfaces and kummer varieties of Jacobians

The quadric line complex

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)