The One-Dimensional Line Scheme of a Family of Quadratic Quantum $\mathbb P^3$s

Abstract

The attempted classification of regular algebras of global dimension four, so-called quantum $\mathbb P^3$s, 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 generic member is a candidate for a generic quadratic quantum $\mathbb P^3$. We find that, viewed as a closed subscheme of $\mathbb P^5$, the generic member has a one-dimensional line scheme consisting of eight curves: one nonplanar elliptic curve in a $\mathbb P^3$, 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.