MORPHISMS TO NONCOMMUTATIVE PROJECTIVE LINES

Let k be a field, let C be a k-linear abelian category, let (L) under bar := {L-i}(i is an element of Z) be a sequence of objects in C, and let B-(L) under bar be the associated orbit algebra. We describe sufficient conditions on (L) under bar such that there is a canonical functor from the noncommutative space ProjB((L) under bar )to a noncommutative projective line in the sense of Nyman [J. Noncommut. Geom. 13 (2019), pp. 517-552], 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 (see Dmitri Piontkovski [J. Algebra 319 (2008), pp. 3280-3290]) by a noncommutative elliptic curve in the sense of Polishchuk [J. Geom. Phys. 50 (2004), pp. 162-187].