The Classification of 3-Calabi-Yau algebras with 3 generators and 3 quadratic relations

Abstract: Let $k$ be an algebraically closed field of characteristic not 2 or 3, $V$ a 3-dimensional vector space over $k$, $R$ a 3-dimensional subspace of $V \otimes V$, and $TV/(R)$ the quotient of the tensor algebra on $V$ by the ideal generated by $R$. Raf Bocklandt proved that if $TV/(R)$ is 3-Calabi-Yau, then it is isomorphic to $J({\sf{w}})$, the "Jacobian algebra" of some ${\sf{w}} \in V^{\otimes 3}$. This paper classifies the ${\sf{w}}\in V^{\otimes 3}$ such that $J({\sf{w}})$ is 3-Calabi-Yau. The classification depends on how ${\sf{w}}$ transforms under the action of the symmetric group $S_3$ on $V^{\otimes 3}$ and on the nature of the subscheme ${\overline{{\sf{w}}}=0} \subseteq \mathbb{P}^2$ where $\overline{{\sf{w}}}$ denotes the image of ${\sf{w}}$ in the symmetric algebra $SV$. Surprisingly, as ${\sf{w}}$ ranges over $V^{\otimes 3}-{0}$, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau $J({\sf{w}})$'s.