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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."

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Certainly it does not suffice to take the schematic closure if $S$ is nonreduced. For instance, let $S$ be $\text{Spec}\ k[x,y]/\langle x^2, xy \rangle$. Let $A$ be $E \times_{\text{Spec} k} S$, where $E$ is an elliptic curve over $k$ with specified zero point $z$. Let $p \in S$ be the closed point with maximal ideal $\langle x,y\rangle$. Let $U$ be the open complement in $A$ of the closed point $(z,p)$. Let $\zeta:S\to A$ be the zero section with image $\{z\}\times S$. Let $D$ be the intersection of $U$ with the image Cartier divisor $\zeta(S)$. Then $D'$ is the underlying reduced scheme of $\zeta(S)$, and this is not a Cartier divisor in $A$, nor is it flat over $S$.

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  • $\begingroup$ Thank you very much! I find this an extremely instructive example to keep in mind when thinking about schematic image. As for the proof in "Neron models" where this is coming from, they must then be using their previous reduction to a regular Noetherian base at this point. I don't understand though how they manage to get the $S$-flatness of $W^{g - 1}$ (or that it is a divisor). $\endgroup$
    – Question Mark
    Commented Jan 25, 2015 at 5:52
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    $\begingroup$ @QuestionMark: The theta-divisor is a red herring, and can be completely ignored for the purpose of building the canonical principal polarization of the relative Pic$^0$ of such relative curves. (If you don't want projective assumptions, you have to use a bit more descent technique.) One just uses an fppf-descent generalization of the same method which is used for curves over a field in the absence of a rational point; this is explained very cleanly in Chapter 6 of Mumford's GIT book (proof of Prop. 6.9 in section 6.1, and so on). $\endgroup$
    – user74230
    Commented Jan 25, 2015 at 6:23

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