Hesse configuration

A Hesse configuration is a set P of nine non-collinear points in the projective planeMathworldPlanetmath over a field K such that any line through two points of P contains exactly three points of P. Then there are 12 such lines through P. A Hesse configuration exists if and only if the field K contains a primitivePlanetmathPlanetmath third root of unity. For such K the projective automorphism group PGL(3,K) acts transitively on all possible Hesse configurations.

The configuration P with its intersection structure of 12 lines is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to the affine spacePlanetmathPlanetmath A=𝔽2 where 𝔽 is a field with three elements.

The group ΓPGL(3,K) of all symmetriesMathworldPlanetmathPlanetmath that map P onto itself has order 216 and it is isomorphic to the group of affine transformationsMathworldPlanetmath of A that have determinantMathworldPlanetmath 1. The stabilizerMathworldPlanetmath in Γ of any of the 12 lines through P is a cyclic subgroup of order three and Γ is generated by these subgroupsMathworldPlanetmathPlanetmath.

The symmetry group Γ is isomorphic to G(K)/Z(K) where G(K)GL(3,K) is a group of order 648 generated by reflectionsMathworldPlanetmathPlanetmath of order three and Z(K) is its cyclic center of order three. The reflection group G() is called the Hesse group which appears as G25 in the classification of finite complex reflection groups by Shephard and Todd.

If K is algebraically closedMathworldPlanetmath and the characteristicPlanetmathPlanetmathPlanetmath of K is not 2 or 3 then the nine inflection points of an elliptic curveMathworldPlanetmath E over K form a Hesse configuration.

Title Hesse configuration
Canonical name HesseConfiguration
Date of creation 2013-03-22 14:04:04
Last modified on 2013-03-22 14:04:04
Owner debosberg (3620)
Last modified by debosberg (3620)
Numerical id 8
Author debosberg (3620)
Entry type Definition
Classification msc 51A05
Classification msc 51A45
Classification msc 51E20
Related topic ProjectiveSpace
Related topic AffineSpace
Related topic EllipticCurve