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

单词	Syzygetic pencil
释义	Syzygetic pencil 英语百科 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 ( $\lambda,\mu$ ) (not both zero) and consists of the points in the plane whose homogeneous coordinates $(x,y,z)$ satisfy the equation for those parameters. Multiplying both $\lambda$ and $\mu$ 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 $\lambda$ or $\mu$ (but not both) to be set to zero.
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Syzygetic pencil

Hesse pencil