The one-dimensional nonreduced line scheme of two families of quadratic quantum $\mathbb{P}^{\bf 3}$s

The classification of quantum $\mathbb{P}^2$s was completed by M. Artin et al. decades ago, but the classification of quadratic algebras that are viewed as quantum $\mathbb{P}^3$s is still an open problem. Based on work of M. Van den Bergh, it is believed that a ``generic'' quadratic quantum $\mathbb{P}^3$ should have a finite point scheme and a one-dimensional line scheme. Two families of quadratic quantum $\mathbb{P}^3$s with these geometric properties are presented herein, where each family member has a line scheme that is either a union of lines or is a union of a line, a conic and a curve. Moreover, we prove that, under certain conditions, if $A$ is a quadratic quantum $\mathbb{P}^3$ that contains a subalgebra $B$ that is a quadratic quantum $\mathbb{P}^2$, then the point scheme of $B$ embeds in the line scheme of $A$.