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arXiv · 2507.19163

Fano schemes of sub-maximal elementary symmetric functions

Alexandru Chirvasitu
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Abstract

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.

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Alexandru Chirvasitu. 2025-08-04. Fano schemes of sub-maximal elementary symmetric functions. https://arxiv.org/abs/2507.19163

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