  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.
