The one-dimensional line scheme of a certain family of quantum P3P3s

A quantum P3P3 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 P3P3 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 P3P3. We find that, as a closed subscheme of P5P5, the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a P3P3, four planar elliptic curves and two nonsingular conics.