= Unique smooth flex of a cuspidal cubic
{title2=$p=[0:1:0],\quad L=\{z=0\}$}
Over a <field> of <characteristic> different from two and three, let $D=\{y^2z=x^3\}\subset\mathbb P^2$. Its homogeneous <Hessian matrix> has <determinant> $24xy^2$. Its common zeros with $D$ are the cusp $[0:0:1]$ and the smooth <point> $p=[0:1:0]$. The <Hessian criterion for a flex> therefore leaves exactly one smooth <flex>.
At $p$, the <projective tangent line> is $L=\{z=0\}$. Restricting the cubic equation to $L$ gives $x^3=0$, so its <intersection multiplicity> at $p$ is three. Consequently a triple <projective line> and a <cuspidal cubic> meeting only at a smooth <point> are <projectively equivalent> to $(3L,D)$ under these <field> hypotheses.
The <characteristic> restriction matters. In <characteristic> three, every smooth <point> can be written $[a:1:a^3]$ and has <projective tangent line> $z=a^3y$. Substitution gives $(x-ay)^3=0$, so every smooth <point> has triple contact. The unique-flex conclusion then fails.
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