Mordell-Weil lattices and toric decompositions of plane curves

TL;DR

This paper explores the relationship between Mordell-Weil lattices, Alexander polynomials, and toric decompositions of plane curves, introducing a height pairing to distinguish Zariski pairs of curves with similar invariants.

Contribution

It extends previous results linking Alexander polynomials with Mordell-Weil ranks and introduces a height pairing on quasi-toric decompositions to differentiate Zariski pairs.

Findings

Constructed degree 12 curves with 30 cusps and Alexander polynomial t^2 - t + 1

Demonstrated the height pairing distinguishes Zariski pairs of curves

Connected Mordell-Weil lattices with plane curve decompositions

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.