Irreducibility of Some Nested Hilbert Schemes

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 ${\xi }_{n_1}\subset \dots \subset {\xi }_{n_k}$ 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.