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 work in \cite{CN}, 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 $\mathbb{P}^{1}_{d}$ by ${\sf C}^{\theta}$, where $\mathbb{P}^{1}_{d}$ is Piontkovski's noncommutative projective line and ${\sf C}^{\theta}$ is Polischuk'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.