= Solution
For $V=Z(f_1,\ldots,f_r)$, $T_{V,p}$ is the common kernel of $df_i(p)$; $\dim V$ is the transcendence degree of its <function> field, and $p$ is smooth when $\dim T_{V,p}=\dim V$. A nonzero maximal Jacobian minor defines a nonempty open smooth locus (generic rank gives nonemptiness).
A line $ux+vy+wz=0$ corresponds to $[u:v:w]\in(\mathbb P^2)^*$. Lines through $P=[a:b:c]$ satisfy $au+bv+cw=0$, a projective line. For a <smooth curve> $F=0$, the dual morphism is $P\mapsto[F_x(P):F_y(P):F_z(P)]$.
For $F=x^3+y^3+z^3$, eliminating $x,y,z$ from $[u:v:w]=[x^2:y^2:z^2]$ gives
$$
u^6+v^6+w^6-2u^3v^3-2u^3w^3-2v^3w^3=0.
$$
The dual sextic is singular: its nine cusps are the tangent lines at the nine flexes of the cubic. Thus points of $X$ mapping to singularities are precisely inflection points, where the tangent has higher contact.
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