Geometry of bisections of elliptic surfaces and Zariski $$N$$ N -plets for conic arrangements

Abstract

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.

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

“…Among the switching classes in Figure 2, we have proved in Lemma 4.3 that the cases (5, 7), (5, 10) cannot appear. 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.…”

“…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).) For the remaining cases (6, 0), (6,4), (6,6), (…”

“…(We could have done the same for (6,14), (6,16), (6,20).) For the remaining cases (6, 0), (6,4), (6,6), (…”

“…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.…”

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

“…Also, in constructing plane curves which can be candidates for Zariski pairs, the first and the second authors introduced a new method by using the geometry of sections and multisections of an elliptic surface ( [5,6,17]). In [3,5,6], with the methods (b) and (d), they gave some examples for Zariski N -plet for arrangements of curves with low degrees.…”