Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings

TL;DR

This paper introduces a new generalized coordinate ring associated with a degenerate Sklyanin algebra, revealing its geometric properties and its relation as a factor of the algebra, extending classical results.

Contribution

It defines the point parameter ring B for degenerate Sklyanin algebras and establishes its generation in degree one, linking it to the algebra's structure.

Findings

Ring B is generated in degree one

B is a factor of the degenerate Sklyanin algebra

The geometry of these algebras is surprisingly rich

Abstract

In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin-Tate-van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.