Irreducibility of Some Nested Hilbert Schemes

TL;DR

This paper proves the irreducibility of various nested Hilbert schemes of points on smooth projective surfaces, extending understanding of their geometric structure.

Contribution

It establishes the irreducibility of several classes of nested Hilbert schemes, which was previously unknown.

Findings

Proves irreducibility of $S^{[n,m]}$ and related schemes.

Extends irreducibility results to schemes with more nested levels.

Provides new insights into the geometry of nested Hilbert schemes.

Abstract

Let $S$ be a smooth projective surface over $\mathbb{C}$. Let $S^{[n_1,\dots,n_k]}$ denote the nested Hilbert scheme which parametrizes zero-dimensional subschemes $\xi_{n_1} \subset \ldots \subset \xi_{n_k}$ where $\xi_i$ is a closed subscheme of $S$ of length $i$. We show that $S^{[n,m]}$, $S^{[n,m,m+1]}$, $S^{[n,n+1,m]}$, $S^{[n,n+1,m,m+1]}$, $S^{[n,n+2,m]}$ and $S^{[n,n+2,m,m+1]}$ are irreducible.