Line-scheme structure for the most generic 4-dimensional quadratic AS regular algebras

Establish that the line scheme of the most generic quadratic Artin–Schelter regular algebra of dimension four with four generators and six relations is the union of two spatial elliptic curves and four planar elliptic curves.

Background

The authors survey known geometric invariants of quadratic AS regular algebras, where line schemes play a key role in classification. For 4-dimensional cases with four generators and six relations, the most generic situation is conjectured to exhibit a specific decomposition of the line scheme.

This conjecture is cited from prior work and remains an open description of the geometry for the most generic such algebras, motivating comparisons with the line schemes computed in this paper.