Singularities of the nested Hilbert scheme of points of length 3, 4

We show that the projection morphism $X^{[3,4]} \longrightarrow X^{[3]}$ is flat even if it has reducible fiber. After showing the rational singularities of the fiber of residual morphism $\textrm{res}_{3,4} :X^{[3,4]} \longrightarrow X$, we conclude that $X^{[3,4]}$ has canonical Gorenstein singularities. As a corollary, we specify the singularities of several nested Hilbert schemes.