Classifying Quadratic Quantum P^2s by using Graded Skew Clifford Algebras

TL;DR

This paper classifies quadratic regular algebras of global dimension three, linking them to graded skew Clifford algebras and describing their construction via twists and extensions, with a focus on specific point schemes.

Contribution

It establishes a connection between quadratic regular algebras and graded skew Clifford algebras, providing a classification based on point schemes and algebraic constructions.

Findings

Most such algebras are twists of graded skew Clifford algebras or their Ore extensions.

Classified all quadratic regular algebras with nodal or cuspidal cubic point schemes.

Connected algebraic structures to geometric properties of cubic curves.

Abstract

We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P^2.