In this paper, we introduce splitting numbers of subvarieties in a smooth complex variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. In particular cases, we show that splitting numbers enable us to distinguish topologies of complex plane curves even if fundamental groups of complements of plane curves are isomorphic. Consequently, we prove that there are π 1 -equivalent Zariski kplets for any integer k ≥ 2.
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 .
Abstract.e splitting number of a plane irreducible curve for a Galois cover is e ective to distinguish the embedded topology of plane curves. In this paper, we de ne the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ , where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total in ectional tangents.
The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the splitting graph. By using the splitting graph, we classify the embedded topology of plane curves consisting of one smooth curve and non-concurrent three lines, called Artal arrangements.
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