Two-periodic elliptic helices: Classification and geometry

Abstract

Let $k$ denote an algebraically closed field of characteristic zero, and let $X$ denote a smooth elliptic curve over $k$. In this paper, motivated by our earlier work in 2021, we think of two-periodic elliptic helices as noncommutative analogues of degree two line bundles over $X$. We classify and study two-periodic elliptic helices in order to generalize the theory of double covers of ${\mathbb{P}}^1$ by $X$ to the noncommutative setting. This leads to the following problem: given an integer $d>2$ and a real number $\theta \in \mathbb{Q}+\mathbb{Q}\sqrt{d^2-4}$, classify elliptic helices inducing double covers of ${\mathbb{P}}_d^1$ by ${\mathsf{C}}^{\theta }$, where ${\mathbb{P}}_d^1$ is Piontkovski’s noncommutative projective line and ${\mathsf{C}}^{\theta }$ is Polishchuk’s noncommutative elliptic curve. We find examples of $d$ and $\theta$ such that there is essentially one numerical class of elliptic helices and examples of $d$ and $\theta$ such that there are several distinct numerical classes of elliptic helices, in contrast to the commutative situation.