## Graduation Semester and Year

2019

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Michaela Vancliff

## Abstract

In the 1980s, M. Artin, J. Tate, and M. Van den Bergh applied geometric techniques to noncommutative algebras. Their work introduced algebraic concepts called point modules and line modules and an associated geometric concept, which was later called the point scheme. In 2002, Shelton and Vancliff defined the concept of line scheme and developed a method for computing the line scheme of any quadratic algebra that satisfies certain conditions. Artin, Tate, and Van den Bergh were able to classify so-called quantum P²s by their point scheme; quantum P³s however are much more challenging and involve computing the line scheme. In this thesis, we compute the point variety and line scheme of a certain quadratic algebra that is an iterated Ore extension of a polynomial ring on two variables. This algebra was provided in a paper authored by Stephenson and Vancliff as a counter example to a prior open problem. We find the closed points of the point scheme and compute the line scheme. Our work shows that the line scheme consists of three components: two of those components have increased multiplicity and direct us to a certain subalgebra of the algebra. We also consider an algebra known as the Exotic Elliptic algebra. We investigate whether or not there is a relationship between central elements of the algebra and the line scheme of the algebra. In particular, we examine right ideals corresponding to (right) line modules for central elements.

## Keywords

Line scheme, Nonreduced, Noncommutative, Algebraic geometry, Vancliff, Artin, Quantum

## Disciplines

Mathematics | Physical Sciences and Mathematics

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Mastriania, Anthony IV, "Some Quadratic Regular Algebras on Four Generators Wtih A 1-Dimensional Nonreduced Line Scheme" (2019). *Mathematics Dissertations*. 214.

https://mavmatrix.uta.edu/math_dissertations/214

## Comments

Degree granted by The University of Texas at Arlington