 Hesse pencil

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.