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
- The ultimate classical reference : R. W. H. T. Hudson Kummer's Quartic Surface ISBN 0521397901. Available online at http://www.hti.umich.edu:80/cgi/b/broker20/broker20?verb=Display&protocol=CGM&ver=1.0&identifier=oai:lib.umich.edu:ABR1780.0001.001 (this is the main source of the second part of this article)
- Igor Dolgachev's online notes on classical algebraic geometry (this is the main source of the first part of this article)