The one-dimensional line scheme of a family of quadratic quantum P3s

The attempted classification of regular algebras of global dimension four, so-called quantum P3s, has been a driving force for modern research in noncommutative algebra. Inspired by the work of Artin, Tate, and Van den Bergh, geometric methods via schemes of d-linear modules have been developed by various researchers to further their classification. In this work, we compute the line scheme of a certain family of algebras whose defining relations involve a scalar α such that almost all of the algebras in the family are considered candidates for a generic quadratic quantum P3. We find that, for almost all α, the algebras have a one-dimensional line scheme consisting of eight curves. Viewing the line scheme as a closed subscheme of P5 via the Plücker embedding, it is the union of one nonplanar elliptic curve in a P3, one nonplanar rational curve with a unique singular point, two planar elliptic curves, and two subschemes, each consisting of the union of a nonsingular conic and a line.