= Hesse pencil
{c}
{title2=$x^3+y^3+z^3+3\lambda xyz=0$}
{wiki}
This pencil consists of projective cubic curves generated by $x^3+y^3+z^3$ and $xyz$. The displayed finite parameter gives a nonsingular cubic exactly when $\lambda^3\ne-1$; the parameter at infinity gives the triangle $xyz=0$. Multiplying the three gradient equations proves the finite singularity criterion. The <Hessian curve> of a member has equation $\lambda^2(x^3+y^3+z^3)-(\lambda^3+4)xyz=0$, so the Hessian stays in the same pencil.
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