Fano schemes of sub-maximal elementary symmetric functions

2025/07/25 by Chirvasitu, Alexandru · 1 citation
#05E05 #12J20 #13A50 #13F30 #14J45 #14M15 #14N05 #20C15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2507.19163

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⋅ 3⋯ (2d-1) pairings on a 2d-element set. This, in particular, confirming a conjecture of Ambartsoumian, Auel and Jebelli to the effect that (over ℝ) all isolated points are recoverable via integral star transforms with appropriate symbols.

Citations

• Symmetric star transforms and the algebraic geometry of their dual differential operators
• Inversion and Symmetries of the Star Transform

Cited by

• Symmetric star transforms and the algebraic geometry of their dual differential operators

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