Morphisms to noncommutative projective lines

Abstract

Let k be a field, let C be a k-linear abelian category, let L := {L i } i∈Z be a sequence of objects in C, and let B L be the associated orbit algebra. We describe sufficient conditions on L such that there is a canonical morphism from the noncommutative space ProjB L to a noncommutative projective line in the sense of [6], generalizing the usual construction of a map from a scheme X to P 1 defined by an invertible sheaf L generated by two global sections. We then apply our results to construct, for every natural number d > 2, a degree two cover of Piontkovski's dth noncommutative projective line [7] by a noncommutative elliptic curve in the sense of Polishchuk [9].