Automorphisms of punctual Hilbert schemes and symmetric powers of varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914004184596480 |
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| author | Bansal, Ashima Sarkar, Supravat Vats, Shivam |
| author_facet | Bansal, Ashima Sarkar, Supravat Vats, Shivam |
| contents | We classify complex smooth projective surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal. This completely answers a question raised by Belmans, Oberdieck and Rennemo, and extends previous works by Boissi{è}re-Sarti, Hayashi, Sasaki, Girardet and Wang. We reduce this to studying the existence of non-natural automorphisms of symmetric powers. We study this question for higher dimensional varieties too, giving some sufficient conditions guaranteeing every automorphism of a symmetric power to be natural. As a corollary, we characterize smooth projective surfaces of Kodaira dimension $\geq 1$ whose punctual Hilbert scheme has a non-natural automorphism, this time not assuming the automorphism preserves the big diagonal. We also address the question, when a smooth projective variety is determined up to isomorphism by its punctual Hilbert scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18059 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphisms of punctual Hilbert schemes and symmetric powers of varieties Bansal, Ashima Sarkar, Supravat Vats, Shivam Algebraic Geometry 14J50 We classify complex smooth projective surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal. This completely answers a question raised by Belmans, Oberdieck and Rennemo, and extends previous works by Boissi{è}re-Sarti, Hayashi, Sasaki, Girardet and Wang. We reduce this to studying the existence of non-natural automorphisms of symmetric powers. We study this question for higher dimensional varieties too, giving some sufficient conditions guaranteeing every automorphism of a symmetric power to be natural. As a corollary, we characterize smooth projective surfaces of Kodaira dimension $\geq 1$ whose punctual Hilbert scheme has a non-natural automorphism, this time not assuming the automorphism preserves the big diagonal. We also address the question, when a smooth projective variety is determined up to isomorphism by its punctual Hilbert scheme. |
| title | Automorphisms of punctual Hilbert schemes and symmetric powers of varieties |
| topic | Algebraic Geometry 14J50 |
| url | https://arxiv.org/abs/2508.18059 |