My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} \rightarrow X$ be its normalization. We have established the surjectivity of the natural pullback map $\pi^*: \mathrm{Pic}^0_{X/K} \rightarrow \mathrm{Pic}^0_{\widetilde{X}/K}$ of $K$-group schemes (the target being an abelian variety) and we would like to investigate the kernel $T := \mathrm{Ker}(\pi^*)$. In particular, we would like to show that $T$ is a torus. My question is: why is $T$ a torus?

The argument given for this in loc. cit. seems to be that $T$ being a torus somehow follows from an exact sequence of the form $$1 \rightarrow K^* \rightarrow \prod_{i = 1}^r K^* \rightarrow \prod_{j = 1}^N K^* \rightarrow \mathrm{Pic}(X) \rightarrow \mathrm{Pic}(\widetilde{X}) \rightarrow 1. $$ I understand the proof of this sequence, but this sequence seems to be entirely about abelian groups (at least with the level of justification given in the cited proof), so I fail to understand how it manages to imply the structure of $T$ as an algebraic group.

I would be very grateful if someone could spell out the argument or provide any clarifying comments.

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    $\begingroup$ What is written there is not at all a rigorous proof. But it contains nearly all of the key ideas. One has to think carefully about the functorial meaning of the entire calculation in terms of ${\rm{R}}^i(f_{\ast})$'s for the structure map $f$ between fppf sites to turn it into a rigorous justification (and if $K$ isn't algebraically closed then one has to use the hard fact that the non-smooth points are $K$-etale to get Weil-restricted GL$_1$'s attached to non-smooth points for describing the relevant tori over the ground field). The proof of Prop. 9 gives ample inspiration (read ahead!). $\endgroup$
    – user74230
    Jan 16 '15 at 4:00
  • $\begingroup$ Would you mind writing out (an outline of) the details as an answer? That would be extremely helpful. I've read the proof of Prop. 9 but that is more of the same: start out by pretending to work with Zariski sheaves, then move to other sites with no justification (or mentioning to which ones: big etale or small etale?); as a bonus put in some cryptic references to Serre's book (written in pre-EGA terminology). Here are the points of Ex. 8 that seem most obscure to me: (1) If we are to carry out the argument in the fppf site, then why is (*) exact (how to deal with arbitrary base change)? ... $\endgroup$ Jan 16 '15 at 6:35
  • $\begingroup$ ... (2) How is $\pi_* \mathbb{G}_m$ representable on the fppf site ($\pi$ is only finite, not locally free); same for the quotient? (I guess your helpful remark about the residue fields at non-smooth points is important here; on this subject, one should refer to III.2.7 of Freitag-Kiehl for a proof of the related remark preceding Thm. 7). (3) If one works in the fppf site, why is $R^1f_* \mathscr{Q} = 0$ (again how to deal with arbitrary base change)? $\endgroup$ Jan 16 '15 at 6:41
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    $\begingroup$ @QuestionMark: There are definitely other references for this, as well. If memory serves (and I often get these things wrong), Altman-Kleiman prove this in their paper on compactifications of Picard schemes of semistable curves. $\endgroup$ Jan 16 '15 at 12:34
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    $\begingroup$ @user74230: I have already put in a number of hours into this (hence the decision to ask on MO), and I don't feel that it is right that the reader should be expected to redo some of the proofs. Of course, the struggle is useful in the end (but also very time-consuming). $\endgroup$ Jan 16 '15 at 16:48

One may consult Corollary 12.5 in Oda, Seshadri "Compactifications of the generalized Jacobian variety" for a seemingly more complete argument.


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