On the topology of arrangements of a cubic and its inflectional tangents

Abstract

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 .

Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines

“…For the other cases, we can find explicit combinations of minimal vectors ±u ij ∈ E * 7 giving the desired switching classes, for example: Next, for the case |I| = 6, among the switching classes in Figure 3, (6,14), (6,16), (6,20) cannot appear due to Corollary 4.2. Also we can easily check that the switching classes (6, 10) 2 , (6, 10) 3 , (6, 12) 1 , (6, 12) 2 contain an induced sub switching class equivalent to (5,7), hence these cannot appear either due to Corollary 4.1. (We could have done the same for (6,14), (6,16), (6,20).)…”

“…However, as the number of irreducible components increase, these invariants become more increasingly complex, and it becomes difficult to manage the data they provide. One approach in resolving this problem is to consider all sub-arrangements of the curves as done in [6], [5]. In this paper, we take another step further in this direction and apply the comcept of two-graphs and switching classes from graph theory (see Section 2 for definitions) in order to clarify the data of the splitting invariants, which enables us to find a new Zariski 5-ple of degree 9 and a 9-ple of degree 10.…”

Galois covers of graphs and embedded topology of plane curves

“…In [19], he partially distinguished the embedded topology of Artal arrangements. In this section, we define Artal arrangements of type (p 1 , p 2 , p 3 ) for three partitions p i of an integer d ≥ 3, which is a generalization of Artal arrangements defined in [6] and [19]. (i) For a partition p = (e 1 , .…”

A note on the topology of arrangements for a smooth plane quartic and its bitangent lines

“…The first is the connected number introduced by T. Shirane in [16], which will be the key tool in distinguishing the Zariski pair that is claimed to exist in Theorem 0.1 (i). Another is the method considered and refined in [4], where the analysis of sub-arrangements effectively distinguishes arrangements with many irreducible components. This method distinguishes the Zariski triple that is claimed to exist in Theorem 0.1 (ii).…”