Geometry and Arithmetic

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pp: 75–89

DOI: 10.4171/119-1/5

Mordell–Weil groups and Zariski triples

José Ignacio Cogolludo-Agustín[1] and Remke Kloosterman[2]

(1) Universidad de Zaragoza, Spain
(2) Humboldt-Universität zu Berlin, Germany

We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant $j$-invariant 0 and discriminant curve $C_{12,m}$. Finally we consider a general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space.

Keywords: Zariski pairs and triples, Alexander polynomial, cuspidal curves, Mordell–Weil groups