Finding the inflexion points of a cubic in $ \mathbb{P}^{2}_{\mathbb{C}}.$

Question

For what values of $ m $ is the cubic $$ F = x_{0}^{3} + x_{1}^{2} + x_{2}^{3} + mx_{0}x_{1}x_{2} = 0 $$ in $ \mathbb{P}^{2}_{\mathbb{C}} $ nonsingular? Find its inflexion points.

I know that $$ \text{Sing}(F) = \Big\lbrace F = \frac{\partial F}{\partial x_{0}} = \frac{\partial F}{\partial x_{1}} = \frac{\partial F}{\partial x_{2}} = 0 \Big\rbrace $$

Using this information, I have that $$ \text{Sing}(F) = \lbrace (x_{0}:x_{1}:x_{2}) \in \mathbb{P}^{2}_{\mathbb{C}} \;|\; x_{0} = x_{1} = x_{2} \rbrace. $$ That is, $ \text{Sing}(F) = (1:1:1). $ Furthermore, at the singular point of $ F, m = -3. $

I am unsure about how to get the inflexion points. Do I need to compute the determinant of the Hessian matrix where $ m = -3$?

EDIT: Unless I've made an error, the determinant of the Hessian matrix yields $$ 216x_{0}x_{1}x_{2} - 6m^{2}x_{0}^{3} - 6m^{2}x^{3}_{2} + 2m^{3}x_{0}x_{1}x_{2} -6m^{2}x^{3}_{1} = 0 $$

I'm not sure how to proceed from here.

Answer 1

You need to solve simultaneously the original equation times the Hessian.

Adding $6m^2$ times the original equation to the Hessian gives $$(216+8m^3)x_0x_1x_2=0.$$ Unless $m^3=-27$ then $x_0x_1x_2=0$ so one of the variables vanishes. If $x_0=0$ then $x_1^3+x_2^2=0$ so you get three inflection points $(0:1:-\zeta)$ where $\zeta^3=1$. Overall then, you do get nine inflection points.