Noncommutative linear systems and noncommutative elliptic curves.

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. [ABSTRACT FROM AUTHOR]