Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves (1008.2018v2)

Abstract: We show that the degree of the Alexander polynomial of an irreducible plane algebraic curve with nodes and cusps as the only singularities does not exceed ${5 \over 3}d-2$ where $d$ is the degree of the curve. We also show that the Alexander polynomial $\Delta_C(t)$ of an irreducible curve $C={F=0}\subset \mathbb P^2$ whose singularities are nodes and cusps is non-trivial if and only if there exist homogeneous polynomials $f$, $g$, and $h$ such that $f^3+g^2+Fh^6=0$. This is obtained as a consequence of the correspondence, described here, between Alexander polynomials and ranks of Mordell-Weil groups of certain threefolds over function fields. All results also are extended to the case of reducible curves and Alexander polynomials $\Delta_{C,\epsilon}(t)$ corresponding to surjections $\epsilon: \pi_1(\mathbb P^2\setminus C_0 \cup C) \rightarrow \mathbb Z$, where $C_0$ is a line at infinity. In addition, we provide a detailed description of the collection of relations of $F$ as above in terms of the multiplicities of the roots of $\Delta_{C,\epsilon}(t)$. This generalization is made in the context of a larger class of singularities i.e. those which lead to rational orbifolds of elliptic type.