Cuspidal cubic

The affine plane curve $y^2=z^3$ is an irreducible cuspidal cubic. Away from the origin its tangent line is $2ydy-3z^{2}dz=0$; at the origin both partial derivatives vanish, so its Zariski tangent space is the whole ambient plane and the origin is singular.

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.