The Geometries of Jordan nets and Jordan webs

TL;DR

This paper classifies the geometric configurations of Jordan nets and webs embedded in symmetric matrices, analyzing their degenerations and obstructions up to certain dimensions, and introduces an algorithm to explore these degenerations.

Contribution

It provides a complete classification of Jordan nets and webs in low dimensions, and develops an algorithm to determine degenerations between their orbits.

Findings

Classified congruence-orbits of Jordan nets and webs in specified dimensions.

Identified degenerations and obstructions, with completeness for certain cases.

Developed and verified an algorithm to compute degenerations between orbits.

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 $n\times n$ 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 $n\leq 7$ (resp. $n\leq 5$), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $\mathbb{S}^n$ for $n\leq5$, 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 $n=4,5$.