Morphisms to noncommutative projective lines

TL;DR

This paper establishes conditions for morphisms from noncommutative spaces to noncommutative projective lines and constructs specific degree two covers of these lines by noncommutative elliptic curves.

Contribution

It generalizes classical projective morphisms to the noncommutative setting and constructs explicit examples of noncommutative elliptic curve covers.

Findings

Provided sufficient conditions for morphisms to noncommutative projective lines.

Constructed degree two covers of noncommutative projective lines by elliptic curves.

Extended classical geometric concepts to noncommutative algebraic geometry.

Abstract

Let $k$ be a field, let ${\sf C}$ be a $k$-linear abelian category, let $\underline{\mathcal{L}}:=\{\mathcal{L}_{i}\}_{i \in \mathbb{Z}}$ be a sequence of objects in ${\sf C}$, 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 morphism from the noncommutative space ${\sf Proj }B_{\underline{\mathcal{L}}}$ to a noncommutative projective line in the sense of \cite{abstractp1}, 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 by a noncommutative elliptic curve in the sense of Polishchuk.