# Hesse configuration

A Hesse configuration is a set $P$ of nine non-collinear points in the projective plane^{} 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 primitive^{} third root of unity. For such $K$ the projective automorphism group $\mathrm{PGL}(3,K)$ acts transitively on all possible Hesse configurations.

The configuration $P$ with its intersection structure of 12 lines is isomorphic^{} to the affine space^{} $A={\mathbb{F}}^{2}$ where $\mathbb{F}$ is a field with three elements.

The group $\mathrm{\Gamma}\subset \mathrm{PGL}(3,K)$ of all symmetries^{} that map $P$ onto itself has order 216 and it is isomorphic to the group of affine transformations^{} of $A$ that have determinant^{} 1. The stabilizer^{} in $\mathrm{\Gamma}$ of any of the 12 lines through $P$ is a cyclic subgroup of order three and $\mathrm{\Gamma}$ is generated by these subgroups^{}.

The symmetry group $\mathrm{\Gamma}$ is isomorphic to $G(K)/Z(K)$ where $G(K)\subset \mathrm{GL}(3,K)$ is a group of order 648 generated by reflections^{} of order three and $Z(K)$ is its cyclic center of order three. The reflection group $G(\u2102)$ is called the Hesse group which appears as ${G}_{25}$ in the classification of finite complex reflection groups by Shephard and Todd.

If $K$ is algebraically closed^{} and the characteristic^{} of $K$ is not 2 or 3 then the nine inflection points of an elliptic curve^{} $E$ 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 |