Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves

TL;DR

This paper establishes bounds on the Alexander polynomial degree of plane curves with nodes and cusps, links it to Mordell-Weil groups of elliptic threefolds, and extends results to reducible curves and more general singularities.

Contribution

It introduces a novel connection between Alexander polynomials of plane curves and Mordell-Weil groups of elliptic threefolds, providing bounds and characterizations for these polynomials.

Findings

Bound on Alexander polynomial degree: ${5 ackslash 3}d-2$ for degree d curves.

Non-trivial Alexander polynomial characterized by specific polynomial relations.

Extension of results to reducible curves and broader singularity classes.

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.