Fano schemes of sub-maximal elementary symmetric functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908478380965888 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective $(d-1)$-spaces contained in the zero locus of $E_{\dim V-1}$. Isolated points exist precisely for $\dim V=2d$, in which case they are in bijection with the $1\cdot 3\cdots (2d-1)$ pairings on a $2d$-element set. This, in particular, confirming a conjecture of Ambartsoumian, Auel and Jebelli to the effect that (over $\mathbb{R}$) all isolated points are recoverable via integral star transforms with appropriate symbols. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fano schemes of sub-maximal elementary symmetric functions Chirvasitu, Alexandru Algebraic Geometry Combinatorics Rings and Algebras Representation Theory 14J45, 05E05, 14M15, 20C15, 13A50, 13F30, 12J20, 14N05 Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective $(d-1)$-spaces contained in the zero locus of $E_{\dim V-1}$. Isolated points exist precisely for $\dim V=2d$, in which case they are in bijection with the $1\cdot 3\cdots (2d-1)$ pairings on a $2d$-element set. This, in particular, confirming a conjecture of Ambartsoumian, Auel and Jebelli to the effect that (over $\mathbb{R}$) all isolated points are recoverable via integral star transforms with appropriate symbols. |
| title | Fano schemes of sub-maximal elementary symmetric functions |
| topic | Algebraic Geometry Combinatorics Rings and Algebras Representation Theory 14J45, 05E05, 14M15, 20C15, 13A50, 13F30, 12J20, 14N05 |
| url | https://arxiv.org/abs/2507.19163 |