SOME QUADRATIC QUANTUM P³s WITH A LINEAR ONE-DIMENSIONAL LINE SCHEME

It is believed that quadratic Artin-Shelter regular (AS-regular) algebras of global dimension four (sometimes called quadratic quantum P3s can be classified using a geometry similar to that developed in the 1980’s by Artin, Tate, and Van den Bergh. Their geometry involved studying a scheme (later called the point scheme) that parametrizes the point modules over a graded algebra. The notion of line scheme (which parametrizes line modules) was introduced later by Shelton and Vancliff. It is known that “generic” quadratic quantum P3s have a finite point scheme and one-dimensional line scheme. A family of algebras with these properties is presented herein where each member has a line scheme that is a union of lines. Moreover, we prove that if a quadratic quantum P3, denoted A, is an Ore extension of a quadratic quantum P2, denoted B, then the point variety of B is embedded in the line variety of A. Indeed, this result is generalized to prove that, under certain conditions, if A is a quadratic quantum P3 that contains a subalgebra isomorphic to a quadratic quantum P2, then the point variety of the subalgebra is embedded in the line variety of A.