The geometry of representations of 3-dimensional Sklyanin algebras

The representation scheme ${\mathtt{r}\mathtt{e}\mathtt{p}}_{n}A$ of the 3-dimensional Sklyanin algebra $A$ associated to a plane elliptic curve and n-torsion point contains singularities over the augmentation ideal $\mathfrak{m}$. We investigate the semi-stable representations of the noncommutative blow-up algebra $B=A \oplus \mathfrak{m}t \oplus\mathfrak{m}^2 t^2 \oplus …$ to obtain a partial resolution of the central singularity ${\tt proj} Z(B) \rightarrow {\tt spec} Z(A)$ such that the remaining singularities in the exceptional fiber determine an elliptic curve and are all of type $$\mathbb{C}$ \times $\mathbb{C}$^2/$\mathbb{Z}$_n$.