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

A quantum ${\mathbb{P}}^3$ is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum ${\mathbb P}^3$ exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. In this article, we compute the line scheme of a family of algebras whose generic member is a candidate for a generic quadratic quantum ${\mathbb{P}}^3$. We find that, as a closed subscheme of ${\mathbb{P}}^5$, the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a ${\mathbb{P}}^3$, four planar elliptic curves and two nonsingular conics.