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 surface == | == Geometry of the Kummer surface == | ||
=== | === Singular quartic surfaces and the double plane model === | ||
Let <math>K\subset\mathbb{P}^2 </math> be a quartic surface, and let <math>p</math> be a singular point of this surface. Then the projection from <math>p</math> on | |||
<math>\mathbb{P}^2</math> is a double cover. The ramification locus of the double cover is a plane curve <math>C</math> of degree 6, and all the nodes of <math>K</math> which are not <math>p</math> map to nodes of <math>C</math>. The maximal number possible number of nodes on a sextic curve is obtained when the curve is a a union of <math>6</math> lines, in which case we have 15 nodes. | |||
=== | === Kummers's quartic surfaces and kummer varieties of Jacobians === | ||
=== The quadric line complex === | |||
=== The | |||
== Geometry and combinatorics of the level structure == | == Geometry and combinatorics of the level structure == | ||
Revision as of 16:40, 5 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. Then the projection from on is a double cover. 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 . 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.
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
- 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)