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Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. Wedenote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonicallyassociate 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. Weapply our description to refine our result on a Zariski pair for a cubic-linearrangement.

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