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 () (not both zero) and consists of the points in the plane whose homogeneous coordinates satisfy the equation for those parameters. Multiplying both and by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either or (but not both) to be set to zero.