New components of Hilbert schemes of points and 2-step ideals
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918081677230080 |
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| author | Giovenzana, Franco Giovenzana, Luca Graffeo, Michele Lella, Paolo |
| author_facet | Giovenzana, Franco Giovenzana, Luca Graffeo, Michele Lella, Paolo |
| contents | This paper presents new examples of elementary and non-elementary irreducible components of the Hilbert scheme of points and its nested variants. The results are achieved via a careful analysis of the deformations of a class of finite colength ideals that are introduced in this paper and referred to as 2-step ideals. The most notable reducibility results pertain to the 4-nested Hilbert scheme of points on a smooth surface, the reducibility of $\text{Hilb}^{3,7}\mathbb{A}^4$, and a method to detect a large number of generically reduced elementary components. To demonstrate the feasibility of this approach, we provide an explicit description of 215 new generically reduced elementary components in dimensions 4, 5 and 6. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New components of Hilbert schemes of points and 2-step ideals Giovenzana, Franco Giovenzana, Luca Graffeo, Michele Lella, Paolo Algebraic Geometry Commutative Algebra 14C05 (Primary) 13D02, 13C05 (Secondary) This paper presents new examples of elementary and non-elementary irreducible components of the Hilbert scheme of points and its nested variants. The results are achieved via a careful analysis of the deformations of a class of finite colength ideals that are introduced in this paper and referred to as 2-step ideals. The most notable reducibility results pertain to the 4-nested Hilbert scheme of points on a smooth surface, the reducibility of $\text{Hilb}^{3,7}\mathbb{A}^4$, and a method to detect a large number of generically reduced elementary components. To demonstrate the feasibility of this approach, we provide an explicit description of 215 new generically reduced elementary components in dimensions 4, 5 and 6. |
| title | New components of Hilbert schemes of points and 2-step ideals |
| topic | Algebraic Geometry Commutative Algebra 14C05 (Primary) 13D02, 13C05 (Secondary) |
| url | https://arxiv.org/abs/2507.02789 |