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

TL;DR

This paper classifies when certain product schemes of Hilbert schemes of points over surfaces are isomorphic, using invariants like Betti, Hodge numbers, and Euler characteristics, with complete results for K3 surfaces and abelian surfaces.

Contribution

It provides a classification of product schemes of Hilbert schemes of points over surfaces, especially for K3 and abelian surfaces, based on their invariants.

Findings

Product schemes are not isomorphic under certain conditions.

Complete classification for K3 surface products.

Complete classification for generalized Kummer varieties.

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.