Elliptic surfaces of rank one and the topology of cubic-line arrangements
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 \(\cdot\) and \(\langle, \rangle\) mean the intersection and height pairing, respectively, and \(\phi, \phi_{o}\) are homomorphisms described in Section \(2\) therein. Since properties of \(P_{D} \in E_{S}(\mathbb{C}(C))\) play important roles in order to study the existence/non-existence of dihedral covers of \(\mathbb{P}^{2}\), one can use them in order to obtain an observation for the embedded topology of plane curves which arise from \(D\). Consider the following combinatorics. Let \(E\) be a nodal cubic and \(L_{i}\) with \(i \in \{0,1,2,3\}\) be four lines as below, and we put \(\mathcal{B} = E + \sum_{i=0}^{e}L_{i}\): i) \(L_{0}\) is a transversal line to \(E\) and we put \(E\cap L_{0} = \{p_{1}, p_{2},p_{3}\}\), ii) \(L_{i}\) is a line through \(p_{i}\) and tangent to \(E\) at a point \(q_{i}\) distinct from \(p_{i}\), for each \(i \in \{1,2,3\}\), iii) \(L_{1},L_{2},L_{3}\) are not concurrent and we put \(L_{i}\cap L_{j} = \{r_{k}\}\) for \(\{i,j,k\} = \{1,2,3\}\). For \(\mathcal{B}\) with the above combinatorics we call it Type I (Type II, respectively) if \(q_{1},q_{2},q_{3}\) are collinear (not collinear, respectively). Theorem B. Let \((\mathcal{B}^{1}, \mathcal{B}^{2})\) be a pair of plane curve with the above combinatorics such that their Types are distinct. Then both of the fundamental groups \(\pi_{1}(\mathbb{P}^{2} \setminus \mathcal{B}^{j}, *)\) with \(j \in \{1,2\}\) are non-abelian and there exists no homomorphism between \((\mathbb{P}^{2}, \mathcal{B}^{1})\) and \((\mathbb{P}^{2},\mathcal{B}^{2})\).
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