Cusp with two isolated-point components
= Cusp with two isolated-point components
The algebraic set in $\mathbb A^3$ defined by
$$
xy=0,
\qquad y^2-z^3+xz=0,
\qquad x(x+y+2z+1)=0
$$
is 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.