The Geometries of Jordan nets and Jordan webs
Arthur Bik and Henrik Eisenmann
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Submission date: 12. Jan. 2022
MSC-Numbers: 17C50, 14M15, 14L30, 65K10
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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.