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The attempted classification of regular algebras of global dimension four,so-called quantum $\mathbb P^3$s, has been a driving force for modern researchin noncommutative algebra. Inspired by the work of Artin, Tate, and Van denBergh, geometric methods via schemes of $d$-linear modules have been developedby various researchers to further their classification. In this work, wecompute the line scheme of a certain family of algebras whose generic member isa candidate for a generic quadratic quantum $\mathbb P^3$. We find that, viewedas a closed subscheme of $\mathbb P^5$, the generic member has aone-dimensional line scheme consisting of eight curves: one nonplanar ellipticcurve in a $\mathbb P^3$, one nonplanar rational curve with a unique singularpoint, two planar elliptic curves, and two subschemes, each consisting of theunion of a nonsingular conic and a line.

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