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

TL;DR

The work analyzes the flatness and singularities of nested Hilbert schemes, focusing on the first projection $\pi_1: X^{[3,4]} \to X^{[3]}$. By combining étale local models, projective-embedding techniques, and blow-up strategies, the authors prove that $\pi_1$ is flat and that $X^{[3,4]}$ has canonical Gorenstein singularities, with explicit fiber decompositions in the non-curvilinear case and corollaries for related nested schemes. The approach yields an alternative, projective-geometry-based proof of flatness via Hilbert polynomials, and extends the singularity analysis to $X^{[1,2,3]}$, $X^{[2,3,4]}$, $X^{[1,2,3,4]}$, and related spaces, establishing rational and canonical singularities in these cases. The paper also poses open questions about normality, Cohen–Macaulayness, and rationality of higher-length nested Hilbert schemes and the singularities of the universal families $\mathcal{Z}_4$.

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.

Theorems & Definitions (16)

• Proposition 1.1
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• proof
• Remark 2.3
• proof : Proof 1 of proposition \ref{['main flat']}
• Definition 3.1: cf. BS88 and catanese90
• proof : Proof 2 of proposition \ref{['main flat']}
• Proposition 4.1