On the classification of products of Hilbert schemes of points over a surface
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| Format: | Preprint |
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2026
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| _version_ | 1866918425124667392 |
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| author | Dey, Arijit Mukherjee, Arijit Pahari, Anubhab |
| author_facet | Dey, Arijit Mukherjee, Arijit Pahari, Anubhab |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01374 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the classification of products of Hilbert schemes of points over a surface Dey, Arijit Mukherjee, Arijit Pahari, Anubhab Algebraic Geometry 14C05, 14C30, 05A17, 14Q10, 14Q15, 14J28, 53D05 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. |
| title | On the classification of products of Hilbert schemes of points over a surface |
| topic | Algebraic Geometry 14C05, 14C30, 05A17, 14Q10, 14Q15, 14J28, 53D05 |
| url | https://arxiv.org/abs/2604.01374 |