Irreducibility of some nested Hilbert schemes

Abstract

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