= Unique smooth flex of a cuspidal cubic
{title2=$p=[0:1:0],\quad L=\{z=0\}$}
For $D=\{y^2z=x^3\}$, the determinant of the homogeneous <Hessian matrix> is $24xy^2$. Its intersections with $D$ are the cusp $[0:0:1]$ and the smooth point $[0:1:0]$. The <Hessian criterion for a flex> therefore gives a unique smooth <flex>, with tangent $z=0$ and contact order three. Consequently every pair consisting of a triple line and a cuspidal cubic meeting only at a smooth point is projectively equivalent to $(3\{z=0\},D)$.
Back to article page
