Quantum Projective Planes Finite over their Centers

TL;DR

This paper characterizes when 3-dimensional quantum polynomial algebras are finite over their centers, linking this property to the existence of fat point modules and automorphism conditions, with implications for noncommutative algebraic geometry.

Contribution

It establishes a precise equivalence between the finiteness over the center, the existence of fat point modules, and automorphism order conditions for quantum projective planes.

Findings

A quantum polynomial algebra has a fat point module iff its associated quantum projective plane is finite over its center.

If the second Hessian of E is zero, then the algebra has no fat point module.

Finiteness over the center is characterized by the order of a specific automorphism related to the Nakayama automorphism.

Abstract

For a $3$-dimensional quantum polynomial algebra $A=\mathcal{A}(E,\sigma)$, Artin-Tate-Van den Bergh showed that $A$ is finite over its center if and only if $|\sigma|<\infty$. Moreover, Artin showed that if $A$ is finite over its center and $E\neq \mathbb{P}^2$, then $A$ has a fat point module, which plays an important role in noncommutative algebraic geometry, however the converse is not true in general. In this paper, we will show that, if $E\neq \mathbb{P}^2$, then $A$ has a fat point module if and only if the quantum projective plane $\mathsf{Proj}_{{\rm nc}} A$ is finite over its center in the sense of this paper if and only if $|\nu^*\sigma^3|<\infty$ where $\nu$ is the Nakayama automorphism of $A$.In particular, we will show that if the second Hessian of $E$ is zero, then $A$ has no fat point module.