Morphisms to noncommutative projective lines

Chan, Douglas; Nyman, Adam

doi:10.1090/proc/15386

Proc. Amer. Math. Soc.

2021

DOI: 10.1090/proc/15386

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Morphisms to noncommutative projective lines

Douglas Chan¹,

Adam Nyman²

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2024

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Noncommutative linear systems and noncommutative elliptic curves

Chan,

Nyman

2024

Trans. Amer. Math. Soc.

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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.

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Noncommutative linear systems and noncommutative elliptic curves

Chan,

Nyman

2024

Trans. Amer. Math. Soc.

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Morphisms to noncommutative projective lines

Cited by 1 publication

References 14 publications

Noncommutative linear systems and noncommutative elliptic curves

Noncommutative linear systems and noncommutative elliptic curves

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