A survey on Zariski pairs

Geometry of bisections of elliptic surfaces and Zariski N-plets II

“…By combining the two invariants, we have the following table where the top row indicates the value of ♯ (Φ 1 Q,C (i) ) −1 (1) and the left most column indicates the splitting type. 0 1 2 (0, 4) Q + C 3 + C 4 -Q + C 1 + C 2 (1, 3) Q + C 3 + C 5 --(2, 2) Q + C 3 + C 6 Q + C 1 + C 3 -Now it is immediate that (Q + C (1) , Q + C (2) , Q + C (3) , Q + C (4) , Q + C (5) ) forms a Zariski 5-plet. Hence under the assumption that C 1 , .…”

“…In this article, we continue to study the topology of reducible plane curves via the geometry of elliptic surface as in [3,6,18,20]. As for terminologies about Zariski pairs or Zariski N -plets, we refer to [2].…”

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

“…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 . Plane curves given by f Q,zo (s [2]Pi ) (i = 1, 2, 3) are contact conics by a similar argument to that in [21, p. 633]. Suppose that C i meets L j transversely for any i, j. Then…”

“…The first condition can be replaced by the combinatorics (or the combinatorial type) of B i . For the precise definition of the combinatorics, see [2] (It can also be found in [21]). Since the combinatorics is easier to treat with, we always consider that of B i .…”

“…Since the '90's there have been a lot of results on Zariski pairs by many mathematicians via various methods (see the reference [2], for example). As we noted in [2], there are two main ingredients in the study of Zariski pairs. Namely, (I) To find reduced curves B 1 , B 2 with the same combinatorics so that B 1 , B 2 have certain different features, and (II) To prove (P 2 , B 1 ) is not homeomorphic to (P 2 , B 2 ) based on the different feature as above.…”