Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers

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

“…One of our new feature of this article concerns (I): Construction of plane curves via geometry and arithmetic of sections for certain rational elliptic surfaces. This basic idea can be found in [21] by the first autor and in [5] by the first and second authors. In this article, however, we make use of the arithmetic of sections more intensively than previous papers.…”

“…For (II), in order to distinguish (P 2 , B 1 ) and (P 2 , B 2 ), our tool is Galois covers branched along B i developed in [5,21]. Note that there are various other tools, for example, the fundamental group π 1 (P 2 \ B i , * ), braid monodromy and Alexander invariants.…”

“…We explain our object more concretely. The first and second authors have studied Zariski pairs (or N -ples) for arrangements of curves whose irreducible components are of low degree, less than or equal to 4, via geometry of sections and bisections of rational elliptic surfaces ( [5,21]). In this article, we also continue to study such objects.…”

“…We keep our notation in Combinatorics 1. Our construction is similar to that in [21]. We first note that L i := f Q,zo (s Pi ) (i = 1, 2, 3) are lines connecting the node of E and p i .…”

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.…”