Irreducibility and singularities of some nested Hilbert schemes

TL;DR

This paper investigates the geometric properties of nested Hilbert schemes of points on surfaces, establishing their irreducibility, singularity types, and computing their Picard groups, while also providing examples of reducibility.

Contribution

It proves irreducibility and klt singularities of certain nested Hilbert schemes, computes their Picard groups under specific conditions, and identifies cases of reducibility for larger nested schemes.

Findings

$S^{[n,n+1,n+2]}$ is an irreducible local complete intersection with klt singularities.

Computed the Picard group of $S^{[n,n+1,n+2]}$ when $h^1(S,\mathcal{O}_S)=0$.

Established reducibility for nested Hilbert schemes with more than 22 points.

Abstract

Let $S$ be a smooth projective surface over $\mathbb{C}$. We study the local and global geometry of the nested Hilbert scheme of points $S^{[n,n+1,n+2]}$. In particular, we show that $S^{[n,n+1,n+2]}$ is an irreducible local complete intersection with klt singularities. In addition, we compute the Picard group of $S^{[n,n+1,n+2]}$ when $h^1(S,\mathcal{O}_S) = 0$. From the irreducibility of $S^{[n,n+1,n+2]}$, we deduce irreducibility for four other infinite families of nested Hilbert schemes. We give the first explicit example of a reducible nested Hilbert scheme, which allows us to show that $S^{[n_1,\dots,n_k]}$ is reducible for $k > 22$.