The geometries of Jordan nets and Jordan webs

Abstract

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space {mathbb{S}}^n of symmetric ntimes n matrices. We study the geometries of Jordan nets and webs: we classify the congruence orbits of Jordan nets (resp. webs) in {mathbb{S}}^n for nle 7 (resp. nle 5), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in mathbb{S}^n for nle 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 mathbb {S}^7 and Jordan webs in mathbb {S}^n for n=4,5.