Projective Geometry Associated to some Quadratic, Regular Algebras of Global Dimension Four
Abstract
The attempted classification of regular algebras of global dimension four, so-called quantum P³s, has been a driving force for modern research in noncommutative algebra. Inspired by the work of Artin, Tate, and Van den Bergh, geometric methods via schemes of d-linear modules have been developed by various researchers to further their classification. In this thesis, we compute and analyze the line scheme of two families of algebras -- for both families, almost every algebra can be considered a candidate for a generic quadratic quantum P³. For the first family of algebras, we find that, viewed as a closed subscheme of P⁵, the generic member has a one-dimensional line scheme consisting of eight irreducible curves: one nonplanar elliptic curve in a P³, one nonplanar rational curve with a unique singular point, two planar elliptic curves, and two subschemes, each consisting of the union of a nonsingular conic and a line. For the second family of algebras, we find that, viewed as a closed subscheme of P⁵, the generic member has a one-dimensional line scheme consisting of seven irreducible curves: three nonplanar elliptic curves in a P³ and four planar elliptic curves. Additionally, regarding the first family of algebras, we relate distinguished points of the line scheme to distinguished elements in the algebras. In particular, we explore a connection between certain right ideals of the algebras and how they intersect with a particular family of normalizing sequences.