Mordell-Weil lattices and toric decompositions of plane curves

Abstract

We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve $C$ with the Mordell--Weil rank of certain isotrivial families of jacobians over ${\mathbf{P}}^2$ of discriminant $C$. In the second part we introduce a height pairing on the $(2,3,6)$ quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial $t^2-t+1$, but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.

References in corpus (2)

• Oka's conjecture on irreducible plane sextics
• Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves

Cited by in corpus (5)

• Representations of divisors on hyperelliptic curves, Gröbner bases and plane curves with quasi-toric relations
• Deformations of hypersurfaces with non-constant Alexander polynomial
• Curves with rational families of quasi-toric relations
• A remark on a Nagell-Lutz type statement for the Jacobian of a curve of genus 2 and a (2, 3, 6) quasi-torus decomposition of a sextic with 9 cusps
• On the Abel-Jacobi map of an elliptic surface and the topology of cubic-line arrangements