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

We show that the projection morphism $X^{[3,4]}\text{lra}{X}^{[3]}$ is flat even if it has reducible fiber. After analyzing blow-up constructions related to $X^{[3,4]}$, we conclude that $X^{[3,4]}$ has canonical Gorenstein singularities. As a corollary, we specify the singularities of several nested Hilbert schemes.