Some new Zariski pairs of sextic curves
Two plane curves \(C_1\) and \(C_2\) of the same degree are said to be a Zariski pair if they have the same combinatorial data, but \(({\mathbb P}^2 ,C_1)\) and \(({\mathbb P}^2,C_2)\) are not homeomorphic (see e.g. \textit{E. Artal-Bartolo} and \textit{H. Tokunaga} [Proc. Lond. Math. Soc., III. Ser. 80, No.1, 127--144 (2000; Zbl 1031.14012)]). Let \(C\) be a reduced sextic curve with simple singularities only and let \(X\) be the \(K3\) surface obtained from the double cover branched over \(C\). It is known that the orthogonal complement in \(H^2(X,{\mathbb Z})\) of the sublattice generated by all irreducible components of the inverse image of \(C\) in \(X\) is a topological invariant of the pair \(({\mathbb P}^2,C)\) (see \textit{I. Shimada} [Mich. Math. J. 59, No. 3, 621--665 (2010; Zbl 1230.14041)]). By \textit{I. Shimada}'s theorem and Nikulin's lattice theory, a weaker topological invariant of \(({\mathbb P}^2,C)\) is the discriminant group of the primitive hull of this sublattice (see e.g. [Adv. Geom. 8, No. 2, 205--225 (2008; Zbl 1153.14017)]). In this paper, the authors show that there are Zariski pairs of sextic curves which cannot be detected by the above invariants and that the discriminant group of the primitive hull of the sublattice generated by the \(-2\) curves arising from the simple singularities is a topological invariant of \(({\mathbb P}^2,C)\); using this invariant, they find twelve Zariski new pairs: each of them consists of a conic and a quartic component. Among these pairs, one has Milnor number \(19\) and the other eleven are its perturbation.
scientific article; zbMATH DE number 6110244
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