Irreducibility and singularities of some nested Hilbert schemes

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 $k>22$.

20 pages, comments welcome