Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves. (English) Zbl 1326.14090
Summary: We establish a correspondence between the rank of Mordell-Weil group of the complex elliptic threefold associated with a plane curve \(\mathcal{C} \subset \mathbb{P}^2(\mathbb C)\) with equation \(F=0\), certain roots of the Alexander polynomial associated with the fundamental group \(\pi_1(\mathbb{P}^2(\mathbb C)\smallsetminus\mathcal{C})\) and the polynomial solutions for the functional equation of type
\[
h_1^pF_1+h_2^qF_2+h_3^rF_3=0
\]
where \(F=F_1F_2F_3\). This correspondence is obtained for curves in a certain class which includes the curves having introduced here \(\delta\)-essential singularities and in particular for all curves with ADE singularities.{
} As a consequence we find a linear bound for the degree of the Alexander polynomial in terms of the degree of \(\mathcal{C}\) for curves with \(\delta\)-essential singularities and in particular arbitrary ADE singularities.
MSC:
14J30 | \(3\)-folds |
14H50 | Plane and space curves |
14H20 | Singularities of curves, local rings |
14F35 | Homotopy theory and fundamental groups in algebraic geometry |
11G05 | Elliptic curves over global fields |