Mordell-Weil groups and Zariski triples

TL;DR

This paper constructs a Zariski triple of degree 12 curves with identical cusp counts but distinct Alexander polynomials, and explores related elliptic threefolds and deformation spaces.

Contribution

It demonstrates the existence of Zariski triples with specific properties and provides explicit generators for associated elliptic threefolds, advancing understanding of curve and threefold classifications.

Findings

Existence of three irreducible degree 12 curves with same cusps, different Alexander polynomials

Explicit generators for elliptic threefolds with constant j-invariant 0

Dimension calculations for equisingular deformation spaces of base-changed curves

Abstract

We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant $j$-invariant 0 and discriminant curve $C_{12,m}$. Finally we consider general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space.