Blowup subalgebras of the Sklyanin algebra

TL;DR

This paper investigates specific degree-3 generated subalgebras within the Sklyanin algebra, revealing their geometric interpretation as blowups of the Sklyanin projective plane at up to seven points and establishing their algebraic properties.

Contribution

It characterizes certain graded subalgebras of the Sklyanin algebra as maximal orders corresponding to geometric blowups, linking algebraic and geometric structures.

Findings

Rings generated by degree-3 elements have good algebraic properties.

These rings correspond to blowups of the Sklyanin P^2 at ≤7 points.

Such subrings are exactly the maximal orders in the 3-Veronese quotient of S.

Abstract

We describe some interesting graded rings which are generated by degree-3 elements inside the Sklyanin algebra S, and prove that they have many good properties. Geometrically, these rings R correspond to blowups of the Sklyanin P^2 at 7 or fewer points. We show that the rings R are exactly those degree-3-generated subrings of S which are maximal orders in the quotient ring of the 3-Veronese of S.