The Geometries of Jordan nets and Jordan webs

Arthur Bik and Henrik Eisenmann

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Submission date: 12. Jan. 2022
Pages: 41
Bibtex
MSC-Numbers: 17C50, 14M15, 14L30, 65K10
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Abstract:
A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space ?ⁿ of symmetric n × n matrices. We study the geometries of Jordan nets and webs: we classify the congruence-orbits of Jordan nets (resp. webs) in ?ⁿ for n ≤ 7 (resp. n ≤ 5), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in ?ⁿ for n ≤ 5, these obstructions show that our list of degenerations is complete. For n = 6, the existence of one degeneration is still undetermined.

To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions, and then used it to compute the degenerations between Jordan nets in ?⁷ and Jordan webs in ?ⁿ for n = 4,5.