Geometry of weak-bitangent lines to quartic curves and sections on certain rational elliptic surfaces (Q6545070)

It is well known that a smooth qurtic curve has twenty-eight bitangent lines. In this article, the author introduces the notion of weak-bitangent line for a reduced qurtic curve as a generalization of bitangent lines.\N\NLet \(\mathscr{Q}\) be a reduced qurtic curve. A line \(L\) is said to be a weak-bitangent line if for any \(p \in \mathscr{Q} \cap L\), the intersection multiplicity of \(\mathscr{Q}\) and \(L\) at \(p\) is even.\N\NIn this article, the author studies weak-bitangent lines of a reduced qurtic curve \(\mathscr{Q}\) in the case when \(\mathscr{Q}\) satisfies the following condition \((\dag)\):\N\N\((\dag)\) \(\mathscr{Q}\) is irreducibel or is the union of smooth conics \(\mathscr{G}_1 + \mathscr{G}_2\), where \(\mathscr{G}_1\) and \(\mathscr{G}_2\) meet transversely.\N\NIt can be constructed a rational elliptic surface from the above \(\mathscr{Q}\) and a smooth point \(z_0\) on \(\mathscr{Q}\). Let \(S_{\mathscr{Q}}\) be the minimal resolution of the double cover of the projective plane \(\mathbb{P}^2\) ( over the complex number field) branched along \(\mathscr{Q}\). The pencil of lines pasing through \(z_0\) induces a pencil of genus \(1\) curves \(\Lambda_{z_0}\) on \(S_{\mathscr{Q}}\), which has a unique base point of multiplicity \(2\). Resolving the indeteminancy for the rational map induced by \(\Lambda_{z_0}\), we obtain an elliptic fibration \(\varphi_{\mathscr{Q}, z_{0}} : S_{\mathscr{Q}, z_{0}} \to \mathbb{P}^1\) with a section \(O\) arising from \(z_0\). This is a rational elliptic surface. The canonical map from \(S_{\mathscr{Q},z_0} \) to \(\mathbb{P}^2\) is denoted by \(\tilde{f}_{\mathscr{Q},z_0} : S_{\mathscr{Q},z_0} \to \mathbb{P}^2\).\N\NA section \(s\) of \(S_{\mathscr{Q}, z_0}\) is said to be a \textit{line-section} if \(\tilde{f}_{\mathscr{Q},z_0}(s)\) is a line in \(\mathbb{P}^2\). A rational point \(P\) of the generic fiber \(E_{\mathscr{Q},z_0}\) of \(\varphi_{\mathscr{Q},z_0}\) is said to be a \textit{line-point} if the correspoding section \(s_P\) is a line-section.\N\NA weak-bitangent line gives rise to two line-sections of \(S_{\mathscr{Q},z_0}\) and vice-versa, if \(\mathscr{Q}\) and \(z_0\) satisfy \((\dag)\) and the following condition \((\ddag)\):\N\N\((\ddag)\) The tangent line at \(z_0\) meets \(\mathscr{Q}\) at two distinct points other than \(z_0\).\N\NThen the pull-back of a weak-bitangent line \(L\) contains two sections \(S^{+}_{L}\) and \(S^{-}_{L}\) of \(S_{\mathscr{Q}, z_0}\). In particular, a weak-bitangent line gives rise to two rational points \(P_{s^{+}_{L}}\) and \(P_{s^{-}_{L}} = [-1]P_{s^{+}_{L}}\).\N\NUnder these settings, the author showed the following theorem.\N\N{Theorem 1.3.} Let \(\mathscr{Q}\) be a reduced qurtic curve satisfying \((\dag)\) and let \(z_0\) be a smooth point on \(\mathscr{Q}\) satisfying \((\ddag)\). For three distinct weak-bitagent lines \(L_1, L_2\) and \(L_3\), let \(P_i\) (\(i=1,2,3)\) be line-points such that \(L_i = \tilde{f}_{\mathscr{Q},z_0}(s_{P_i})\). If \(P_4 = P_1 \dot{+} P_2 \dot{+} P_3\) (as the addition on the elliptic curve \(E_{\mathscr{Q},z_0}\)) is a line-point, then all intersection points of \(\mathscr{Q}\) and \(L_1+L_2+L_3+L_4\) (the sum as divisors) lie on a conic, where \(L_4\) is the line \(\tilde{f}_{\mathscr{Q}, z_0}(s_{P_4})\).\N\NIn the poof of the theorem, the author uses Mumford representations in order to describe divisor classes on elliptic curves.\N\NFurthermore, the author gives classification of weak-bitangent lines of singular qurtic curves satisfying \((\dag)\) by using a result of [\textit{K. Oguiso} and \textit{T. Shioda}, Comment. Math. Univ. St. Pauli 40, No. 1, 83--99; Appendix 1: 96--97; Appendix 2: 97--98 (1991; Zbl 0757.14011)]. As a result, the author gives new proofs for some clasical results on singular qurtic curves and their bitangent lines.