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

Abstract: We show that the projection morphism $X^{[3,4]} \longrightarrow X<sup>{[3]}$ 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 $X<sup>{[3,4]}$ has canonical Gorenstein singularities. As a corollary, we specify the singularities of several nested Hilbert schemes.