The geometries of Jordan nets and Jordan webs.

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space S 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 S 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 S 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 S 7 and Jordan webs in S n for n = 4 , 5 .