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 $n$. Under suitable hypotheses, we show that the products of Hilbert schemes $S^{[{\bf a}]}$ and $S^{[{\bf b}]}$ associated to two distinct partitions ${\bf a}$ and ${\bf b}$ of $n$ are non-isomorphic, using Betti numbers, Hodge numbers and Euler characteristics of the individual factors. The classifications obtained using Betti numbers and Euler characteristics extend over $\overline{\F}_q$ as well, where $\F_q$ is the finite field with $q$ elements. We also provide a complete classification of such product spaces for K3 surfaces using their inherent symplectic structure.