Mordell-Weil groups and Zariski triples

Abstract

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 general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space.

References in corpus (3)

• Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
• Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
• Cuspidal plane curves, syzygies and a bound on the MW-rank

Cited by in corpus (3)

• On some conjectures about free and nearly free divisors
• Kummer covers and braid monodromy
• On the Abel-Jacobi map of an elliptic surface and the topology of cubic-line arrangements