On the Abel-Jacobi map of an elliptic surface and the topology of cubic-line arrangements

TL;DR

This paper explores the Abel-Jacobi map of elliptic surfaces, providing a detailed description of rational points in specific cases and applying these insights to the topology of cubic-line arrangements.

Contribution

It offers a new description of the Abel-Jacobi map for elliptic surfaces with rank one rational points and refines results on Zariski pairs in cubic-line arrangements.

Findings

Explicit description of $P_D$ for elliptic surfaces with rank one.

Refined understanding of the topology of cubic-line arrangements.

Improved classification of Zariski pairs.

Abstract

Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. We denote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonically associate a $C(C)$-rational point $P_D$. In this note, we give a description of $P_D$ of $E_S$, when the rank of the group of $C(C)$-rational points is one. We apply our description to refine our result on a Zariski pair for a cubic-line arrangement.