Two-periodic elliptic helices: classification and geometry

TL;DR

The paper develops a noncommutative analogue of degree-two line bundles on an elliptic curve by classifying two-periodic elliptic helices in ${\rm Coh}(X)$ and linking them to double covers of noncommutative projective lines $\mathbb{P}^1_d$ by noncommutative elliptic curves ${\sf C}^{\theta}$. It establishes a precise classification criterion based on $d=\dim\operatorname{Hom}(\mathcal{E}_{-1},\mathcal{E}_0)$, proves an irrational negative limit slope $\theta$ for $d>2$, and shows ampleness of these helices, yielding noncommutative homogeneous coordinate rings $B_{\underline{\mathcal{E}}}$ for ${\sf C}^{\theta}$. A robust framework of $\mathbb{Z}$-algebras and an Eilenberg–Watts theorem for $\mathbb{Z}$-algebras is developed to compare autoequivalences and construct double covers, with explicit results on the autoequivalences of the nc projective line and the behavior of covers under helix operations. The work provides a foundation for understanding how noncommutative elliptic curves map onto nc projective lines, including criteria for when multiple numerical classes of helices arise and when unique covers exist. Overall, it advances noncommutative algebraic geometry by integrating helix theory, ampleness, and double-cover constructions in a coherent, category-theoretic framework.

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 $θ\in \mathbb{Q}+\mathbb{Q}\sqrt{d^2-4}$, classify elliptic helices inducing double covers of $\mathbb{P}^{1}_{d}$ by ${\sf C}^θ$, where $\mathbb{P}^{1}_{d}$ is Piontkovski's noncommutative projective line and ${\sf C}^θ$ is Polischuk's noncommutative elliptic curve. We find examples of $d$ and $θ$ such that there is essentially one numerical class of elliptic helices and examples of $d$ and $θ$ such that there are several distinct numerical classes of elliptic helices, in contrast to the commutative situation.

Theorems & Definitions (63)

• Proposition 1.1
• Theorem 1.2
• Proposition 1.5
• Proposition 1.6
• Remark 2.4
• Theorem 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof