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

Shinzo Bannai; Hiro-o Tokunaga

paper

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

arXiv:1802.06661

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.

16 pages