Cuspidal cubic

The projective algebraic curve $D=\{y^{2}z=x^3\}\subset {\mathbb{P}}^2$ is an irreducible cubic with a cusp at $[0:0:1]$. In the chart $z=1$, its equation is $y^2=x^3$, with parametrization $(x,y)=(t^2,t^3)$. Its affine version $y^2=z^3$, used below, differs only by renaming a coordinate. At the point $[0:1:0]$, the chart $y=1$ gives $z=x^3$, which is smooth and exhibits its unique smooth flex of a cuspidal cubic.

For $D=\{y^{2}z=x^3\}$, the determinant of the homogeneous Hessian matrix is $24x{y}^2$. 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 $(3\{z=0\},D)$.

The algebraic set in ${\mathbb{A}}^3$ defined byis the disjoint union of the cuspidal cubic $V(x,y^2-z^3)$ and the two reduced points $(-1,0,0)$ and $(1,0,-1)$. Its radical vanishing ideal is the intersection of the three corresponding prime ideals.