On the classification of products of Hilbert schemes of points over a surface

Abstract

Let $S$ be a smooth projective surface over $\mathbb{C}$ and $S^{[n]}$ be the Hilbert scheme of $n$ points over $S$, for any positive integer $n$. Let ${\bf a}=(n_1,\ldots,n_r)$ and ${\bf b}=(m_1,\ldots,m_s)$ be two distinct partitions of any positive integer $n$. We prove that, under certain conditions, the $2n$-dimensional schemes $S^{[{\bf a}]}=S^{[n_1]}\times \cdots \times S^{[n_r]}$ and $S^{[{\bf b}]}=S^{[m_1]}\times \cdots \times S^{[m_s]}$ are not isomorphic, using invariants like Betti numbers, Hodge numbers and Euler characteristics of the individual factors. We provide a complete classification of such product spaces for K3 surfaces using its inherent symplectic structure. Consequently, we obtain a complete classification for products of generalised Kummer varieties over any abelian surface.