Flexes of a Hesse cubic
= Flexes of a Hesse cubic
{title2=$x^3+y^3+c z^3-dxyz=0$}
For $c\ne0$ and $d^3\ne27c$ in characteristic zero, the displayed <plane cubic> is smooth. Its Hessian determinant is $-6d^2(x^3+y^3+cz^3)+(216c-2d^3)xyz$. On the cubic this becomes $8(27c-d^3)xyz$. Hence all nine <plane cubic flexes> lie on the coordinate axes: $(1:-\zeta:0)$, $(-\sqrt[3]c\,\zeta:0:1)$ and $(0:-\sqrt[3]c\,\zeta:1)$ for $\zeta^3=1$.