Components of Hilbert Schemes of low degree and smoothable algebras
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912756736720896 |
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| author | Gałązka, Maciej Keneshlou, Hanieh Šivic, Klemen |
| author_facet | Gałązka, Maciej Keneshlou, Hanieh Šivic, Klemen |
| contents | In this article, we describe the irreducible components of the Hilbert scheme of $d$ points on $\mathbb{A}^n$ for $d=9,10$. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a specific Hilbert function. We further prove that any finite local algebra of degrees $9,10$ and the socle dimension $2$ is smoothable. As the main consequence, we establish the equality of the cactus Grassmann and the secant Grassmann variety in the corresponding cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Components of Hilbert Schemes of low degree and smoothable algebras Gałązka, Maciej Keneshlou, Hanieh Šivic, Klemen Algebraic Geometry 14C05, 15A27, 13E10, 14N07 In this article, we describe the irreducible components of the Hilbert scheme of $d$ points on $\mathbb{A}^n$ for $d=9,10$. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a specific Hilbert function. We further prove that any finite local algebra of degrees $9,10$ and the socle dimension $2$ is smoothable. As the main consequence, we establish the equality of the cactus Grassmann and the secant Grassmann variety in the corresponding cases. |
| title | Components of Hilbert Schemes of low degree and smoothable algebras |
| topic | Algebraic Geometry 14C05, 15A27, 13E10, 14N07 |
| url | https://arxiv.org/abs/2512.09428 |