Noncommutative linear systems and noncommutative elliptic curves

TL;DR

This paper develops a noncommutative analogue of linear systems called helices, constructs examples on elliptic curves, and shows how they embed noncommutative elliptic curves into noncommutative projective planes.

Contribution

It introduces the concept of helices in abelian categories over quadratic algebras and relates them to noncommutative elliptic curves, generalizing classical elliptic helices.

Findings

Helices induce morphisms between noncommutative spaces.

Constructed examples of helices on elliptic curves.

Identified noncommutative elliptic curves as quotients of Koszul algebras.

Abstract

In this paper we introduce a noncommutative analogue of the notion of linear system, which we call a helix $\underline{\mathcal{L}} := (\mathcal{L}_{i})_{i \in \mathbb{Z}}$ in an abelian category ${\sf C}$ over a quadratic $\mathbb{Z}$-indexed algebra $A$. We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from ${\sf Proj }\operatorname{End}(\underline{\mathcal{L}})$ to ${\sf Proj }A$. We construct examples of helices of vector bundles on elliptic curves generalizing the elliptic helices of line bundles constructed by Bondal-Polishchuk, where $A$ is the quadratic part of $B:= \operatorname{End}(\underline{\mathcal{L}})$. In this case, we identify $B$ as the quotient of the Koszul algebra $A$ by a normal family of regular elements of degree 3, and show that ${\sf Proj }B$ is a noncommutative elliptic curve in the sense of Polishchuk. One interprets this as embedding the noncommutative elliptic curve as a cubic divisor in some noncommutative projective plane, hence generalizing some well-known results of Artin-Tate-Van den Bergh.