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

TL;DR

This paper investigates the geometric properties of nested Hilbert schemes of points, establishing flatness and canonical Gorenstein singularities for specific cases, and clarifying the nature of their singularities.

Contribution

It demonstrates flatness of the projection from nested to ordinary Hilbert schemes and characterizes their singularities as canonical Gorenstein, providing new insights into their structure.

Findings

Projection $X^{[3,4]} o X^{[3]}$ is flat despite reducible fibers

$X^{[3,4]}$ has canonical Gorenstein singularities

Clarifies singularities of several nested Hilbert schemes

Abstract

We show that the projection morphism $X^{[3,4]} \lra X^{[3]}$ 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.