Noncommutative linear systems and noncommutative elliptic curves

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 $\mathsf{C}$ over a quadratic $\mathbb{Z}$-indexed algebra $A$. We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from $\mathsf{Proj}\mathrm{End}(\underline{\mathcal{L}})$ to $\mathsf{Proj}A$. 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 $B:=\mathrm{End}(\underline{\mathcal{L}})$. 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 $\mathsf{Proj}B$ 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.

37 pages