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.

On the period of Li, Pertusi, and Zhao’s symplectic variety

“…Conjecture 1 has been recently disproved by the authors by providing a ten-dimensional family of zerodimensional schemes in Hilb 12 A 3 with odd-dimensional tangent space [13,Corollary 3.2]. This argument provides counterexamples to the parity conjecture for any (r, d ) with r 1 and d 12. In Section 1 we propose a variant of the counterexamples given in [13] showing that, paying the price of increasing the rank one has counterexamples of smaller length.…”

“…For instance, when dim X = 2, the nested Hilbert scheme is irreducible for r 2 and reducible in general for r 5. The cases r = 3, 4 are not yet understood, [37,12].…”