In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve inflectional tangent lines.
A k-Artal arrangement is a reducible algebraic curve composed of a smooth cubic and k inflectional tangents. By studying the topological properties of their subarrangements, we prove that for k = 3, 4, 5, 6, there exist Zariski pairs of k-Artal arrangements. These Zariki pairs can be distinguished in a geometric way by the number of collinear triples in the set of singular points contained in the cubic.1.1. Subarrangements. We here reformulate our idea in [3] more precisely. Let B o be a (possibly empty) reduced plane curve B o . We define Curve Bo red to be the set of the reduced plane curves of the form B o + B, where B is a reduced curve with no common component with B o . Let B = B 1 + · · · + B r denote the irreducible decomposition of B. For a subset I of the power set 2 {1,...,r} of {1, . . . , r}, which does not contain the empty set ∅, we define the sub set Sub I (B o , B) of Curve Bo red by: Sub I (B o , B) := B o + i∈I B i I ∈ I .
In this article, we prove that two versal Galois covers for S 4 and A 5 introduced in [17], [18] and [19] are birationally distinct to each other. As a corollary, we obtain two non-conjugate embeddings of S 4 and A 5 into Cr 2 (C).
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