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 (${\displaystyle \lambda ,\mu }$) (not both zero) and consists of the points in the plane whose homogeneous coordinates ${\displaystyle (x,y,z)}$ satisfy the equation for those parameters. Multiplying both ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ 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 ${\displaystyle \lambda }$ or ${\displaystyle \mu }$ (but not both) to be set to zero.

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