Components of the nested Hilbert scheme of few points

TL;DR

This paper investigates the structure and existence of elementary components in the nested Hilbert scheme of points on smooth and singular varieties, providing new bounds and methods for constructing non-reduced components.

Contribution

It introduces a new lower bound for non-smoothable nestings of fat points and develops a systematic method to construct generically non-reduced elementary components.

Findings

Established a lower bound for non-smoothable fat point nestings in higher dimensions.

Developed a systematic construction method for non-reduced elementary components.

Proved the existence of elementary components in Hilbert schemes on singular hypersurfaces with high multiplicity.

Abstract

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points on a smooth $n$-fold, for $n\geqslant 4$. Moreover, we implement a systematic method to build generically non-reduced elementary components. We also investigate the problem of irreducibility of the Hilbert scheme of points on a singular hypersurface of $\mathbb A^3$. Explicitly, we show that the Hilbert scheme of points on a hypersurface of $\mathbb{A}^3$ having a singularity of multiplicity at least 5 admits elementary components.