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 L _ ≔ ( L i ) i ∈ Z \underline {\mathcal {L}} ≔(\mathcal {L}_{i})_{i \in \mathbb {Z}} in an abelian category C \mathsf {C} over a quadratic Z \mathbb {Z} -indexed algebra A A . We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from Proj E n d ( L _ ) End(\underline {\mathcal {L}}) to Proj A 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 A is the quadratic part of B ≔ E n d ( L _ ) B≔End(\underline {\mathcal {L}}) . In this case, we identify B B as the quotient of the Koszul algebra A A by a normal family of regular elements of degree 3, and show that Proj B 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.