Hesse pencil [SEP] named after
Hesse pencil
In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation
Each curve in the family is determined by a pair of parameter values (formula_2) (not both zero) and consists of the points in the plane whose homogeneous coordinates formula_3 satisfy the equation for those parameters. Multiplying both formula_4 and formula_5 by the same scalar does not change the curve, so there
equivalent in pairs giving only nine points. The family of cubics through these nine points forms the Hesse pencil. More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points.
The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the