Elliptic surfaces of rank one and the topology of cubic-line arrangements
type: Item
instance of: scholarly article
title: Elliptic surfaces of rank one and the topology of cubic-line arrangements
zbMATH Open document ID: 1453.14091
author: Shinzo Bannai
author: Hiro-O Tokunaga
published in: Journal of Number Theory
publication date: 2021-01-25T00:00:00Z
review text: In the paper under review, the author studies the geometry of elliptic surfaces of rank one and its applications towards the existence of Zariski pairs. Let \(\phi: S \rightarrow C\) be an elliptic surface over a smooth projective curve such that \(\phi\) is relatively minimal, there exists a section \(O: C \rightarrow S\), and there exists at least one degenerate fiber. We denote the set of sections of \(\phi: S \rightarrow C\) by \(\mathrm{MW}(S)\), the Mordell-Weil group, which is non-empty. Let \(E_{S}\) denote the generic fiber of \(\phi\) and we denote by \(E_{S}(\mathbb{C}(C))\) the set of \(\mathbb{C}(C)\) -rational points of \(E_{S}\) and thus \((E_{S},O)\) is an elliptic curve defined over \(\mathbb{C}(C)\) and \(E_{S}(\mathbb{C}(C))\) has the structure of a finitely generated abelian group. We say that an elliptic surface \(\phi:S \rightarrow C\) is an elliptic surface of rank one if \(\mathrm{rk} \,E_{S}(\mathbb{C}(C)) = 1\). Let \(D\) be a divisor on \(S\), by restricting \(D\) to \(E_{S}\) we have a divisor \(\mathfrak{d}\) in \(E_{S}\) defined over \(\mathbb{C}(C)\). If we apply Abel's theorem to \(\mathfrak{d}\), we have \(P_{D} \in E_{S}(\mathbb{C}(C))\) and thus the corresponding section \(s(D) \in\mathrm{ MW}(S)\). In the paper under review the authors study \(n\)-divisiblity of \(P_{D}\) in the setting when \(E_{S}(\mathbb{C}(C)) = \mathbb{Z}P_{o} \oplus E_{S}(\mathbb{C}(C))_{\mathrm{tor}}\) for some \(P_{o} \in E_{S}(\mathbb{C}(C))\) and then they apply this result to consider the embedded topology of reducible curves. The first main result can be formulated as follows. Theorem A. Suppose that \(\mathrm{rk} \, E_{S}(\mathbb{C}(C))=1\). Let \(n\) be an integer such that \(P_{D} = nP_{o} + P_{\tau}\), \(P_{\tau} \in E_{S}(\mathbb{C}(C))_{\mathrm{tor}}\). Then we have \[n^{2} = - \frac{\phi_{o}(D) \cdot \phi_{o}(D)}{\langle P_{o},P_{o}\rangle}, \quad \quad n = - \frac{\phi_{o}(D) \cdot \phi(P_{o})}{\langle P_{o}, P_{o} \rangle},\] where \(\cdot\) a…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 7300429
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: Mordell-Weil rank one
zbMATH Keywords: multi-sections
zbMATH Keywords: topology of reducible plane curves
MaRDI profile type: Publication
full work available at URL: j.jnt.2020.06.005
OpenAlex ID: W3042581360
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DOI: 10.1016/J.JNT.2020.06.005
description: scientific article