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 (
) (not both zero) and consists of the points in the plane whose homogeneous coordinates
Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity.
There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are 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.