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Quantum Projective Planes Finite over their Centers

For a $3$-dimensional quantum polynomial algebra $A=\mathcal{A}(E,σ)$, Artin-Tate-Van den Bergh showed that $A$ is finite over its center if and only if $|σ|<\infty$. 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 $|ν^*σ^3|<\infty$ where $ν$ 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.