Fano schemes of sub-maximal elementary symmetric functions

TL;DR

This paper characterizes the rational points on the Fano scheme of subspaces contained in the zero locus of a specific elementary symmetric polynomial, confirming a conjecture about their recoverability over the reals.

Contribution

It describes the structure of rational points on a Fano scheme associated with elementary symmetric functions and confirms a conjecture regarding their recoverability over the reals.

Findings

Isolated points exist only when the dimension is twice the subspace dimension.

These points correspond to pairings on a set of size 2d.

All isolated points are recoverable via integral star transforms over the reals.

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.