Past exam of the geometry course of the University of Oxford / 2009 / B / 5 / ii / Solution

The second-derivative matrix isIts determinant is $54[(4+{\lambda }^3)xyz-{\lambda }^2(x^3+y^3+z^3)]$. Hence the Hessian curve has equationIf $\lambda =0$, this is the union $xyz=0$ of three lines and is singular. If $\lambda \ne 0$, divide by ${\lambda }^2$ and write it as a Hesse cubic with parameter $\mu =-({\lambda }^3+4)/(3{\lambda }^2)$. The preceding singularity criterion gives ${\mu }^3=-1$, equivalentlyThus the complete condition isThe other values beyond the source's first family are zero and the three cube roots of eight.