Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise dense (and perhaps $S$-quasi-compact) open subscheme $U \subset A$ and an effective relative (to $S$) Cartier divisor $D \subset U$. One takes the schematic image $D'$ of $D \rightarrow A$. Is the closed subscheme $D' \subset A$ a relative (to $S$) effective Cartier divisor? I.e., is $D'$ flat over $S$ and locally on $A$ cut out by a single nonzero divisor?

Something like this comes up in the construction of a $\Theta$-divisor for a proper smooth $S$-curve of genus $g \ge 2$. Namely, my question is inspired by the desire to understand the sentence "Furthermore, $W^{g - 1}$ is an effective relative Cartier divisor on $P$, usually denoted by $\Theta_\sigma$." on p. 261 of "Neron models."