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 Proj $End(\underline {\mathcal {L}})$ to 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 $B≔End(\underline {\mathcal {L}})$. 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 Proj $B$ is a noncommutative elliptic curve in the sense of Polishchuk [J. Geom. Phys. 50 (2004), pp. 162–187]. 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.