Morphisms to noncommutative projective lines

Abstract:

Let $k$ be a field, let ${\mathsf {C}}$ be a $k$-linear abelian category, let $\underline {\mathcal {L}}\colonequals \{\mathcal {L}_{i}\}_{i \in \mathbb {Z}}$ be a sequence of objects in ${\mathsf {C}}$, and let $B_{\underline {\mathcal {L}}}$ be the associated orbit algebra. We describe sufficient conditions on $\underline {\mathcal {L}}$ such that there is a canonical functor from the noncommutative space ${\mathsf {Proj }}B_{\underline {\mathcal {L}}}$ 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 $\mathbb {P}^{1}$ defined by an invertible sheaf $\mathcal {L}$ 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 $d$th 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].