Two-periodic elliptic helices: classification and geometry
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912706382004224 |
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| author | Chan, Daniel Nyman, Adam |
| author_facet | Chan, Daniel Nyman, Adam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two-periodic elliptic helices: classification and geometry Chan, Daniel Nyman, Adam Algebraic Geometry Rings and Algebras Primary 14A22, Secondary 16S38 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. |
| title | Two-periodic elliptic helices: classification and geometry |
| topic | Algebraic Geometry Rings and Algebras Primary 14A22, Secondary 16S38 |
| url | https://arxiv.org/abs/2511.09825 |