On cluster categories of weighted projective lines with at most three weights

Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$ the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can be realized as a generalized cluster category of quiver with potential. In this note, under the assumption that $\mathbb{X}$ has at most three weights or is of tubular type, we prove that if the generalized cluster category $\mathcal{C}_{(Q,W)}$ of a Jacobi-finite non-degenerate quiver with potential $(Q,W)$ shares a $2$-CY tilted algebra with $\mathcal{C}_\mathbb{X}$, then $\mathcal{C}_{(Q,W)}$ is triangle equivalent to $\mathcal{C}_\mathbb{X}$. As a byproduct, a $2$-CY tilted algebra of $\mathcal{C}_\mathbb{X}$ is determined by its quiver provided that $\mathbb{X}$ has at most three weights. To this end, for any weighted projective line $\mathbb{X}$ with at most three weights, we also obtain a realization of $\mathcal{C}_\mathbb{X}$ via Buan-Iyama-Reiten-Scott's construction of $2$-CY categories arising from preprojective algebras.

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