The geometry of representations of 3-dimensional Sklyanin algebras

TL;DR

This paper explores the geometric structure of representations of 3D Sklyanin algebras, revealing how certain singularities relate to elliptic curves and specific quotient singularities through a noncommutative blow-up process.

Contribution

It introduces a partial resolution of singularities in the representation scheme of 3D Sklyanin algebras using semi-stable representations of a noncommutative blow-up algebra.

Findings

Singularities over the augmentation ideal are characterized.

A partial resolution relates the scheme to an elliptic curve.

Remaining singularities are of type  imes \u001C^2/_n.

Abstract

The representation scheme ${\tt rep}_n A$ 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 $\mathbb{C} \times \mathbb{C}^2/\mathbb{Z}_n$.