The point variety of quantum polynomial rings

TL;DR

This paper characterizes the structure of the reduced point variety of quantum polynomial algebras, revealing it as a union of linear subspaces and providing detailed descriptions of its components and configurations.

Contribution

It offers a new geometric description of point varieties in quantum polynomial rings, including combinatorial classifications in low dimensions.

Findings

Reduced point variety is a union of linear subspaces

Irreducible components are explicitly described

Configurations are classified combinatorially in small dimensions

Abstract

We show that the reduced point variety of a quantum polynomial algebra is the union of specific linear subspaces in $\mathbb{P}^n$, we describe its irreducible components and give a combinatorial description of the possible configurations in small dimensions.