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Introduction

Let $k$ be a local field. Let $C$ be the spectrum of $\mathcal{O}_{k}$. Let $X/k$ be a smooth projective curve with a semistable model $\mathcal{X}/C$.

Let $J$ be the Jacobian of $X$. The identity component of the reduction $\tilde{J}$ fits into an exact sequence \[ 1 \to L \to \tilde{J}^{0} \to A \to 0, \] where $L$ is a linear group, and $A$ is an abelian variety. Furthermore $L$ can be decomposed in a torus $T$ and unipotent group $U$. (Since we have a semistable model, $U = 0$.) Let $t$ denote the rank of this torus $T$.

We can also associate a reduction graph $G$ to $X$, see e.g., [Liu, §10.4], but as Liu points out there, there are different ways of defining a reduction graph. To this graph we can associate its first Betti number, which is the rank of the first homology group. (Observe that $\beta(G)$, equals $\#E - \#V + 1$. I.e., the number of edges left after removing a spanning tree.)

Claim

In [Zha, lem 5.2.2] it is suggested that $t$ equals the $\beta(G)$.

I am looking for a proof of this claim. The first result that comes to mind is [Liu, prp 10.1.51(c)] but this is not exactly the result I am looking for. If we let $u$ denote the rank of $U$ (see above), then this proposition says $\beta(G) \le t + u$. Since $u = 0$ in our case, this does give an inequality in one direction.

What makes the question more difficult is that (as far as I see) Zhang does not specify which definition of reduction graph he uses. I guess he uses the same definition as [Yam, 1.10], since that article is very much related to the results of [Zha]. This definition is pretty much the same as the one in [Liu], although I think [Yam] allows loops in the reduction graph, for self-intersections of irreducible components. The problem, of course, is that allowing or disallowing loops has quite an impact on the rank of the first homology group. Allowing loops raises $\beta(G)$, and therefore the above inequality ($\beta(G) \le t$) is no longer necessarily true.

Question

Since I am only taking my first steps into the theory of reduction graphs I do not know whether the ramblings above or the question that follows makes any sense. M question essentially boils down to:

Under which conditions are $t$ and $\beta(G)$ equal, and why?


References

[Liu] Liu, Qing. Algebraic geometry and arithmetic curves. Translated from the French by Reinie Erné. Oxford Graduate Texts in Mathematics, 6. Oxford Science Publications. Oxford University Press, Oxford, 2002. xvi+576 pp.

[Yam] Kazuhiko Yamaki. Graph invariants and the height of the Gross-Schoen cycle. 2009. url: http://jairo.nii.ac.jp/0019/00096332/en.

[Zha] Shou-Wu Zhang. “Gross–Schoen Cycles and Dualising Sheaves”. arXiv:math/0812.0371.

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2 Answers 2

up vote 5 down vote accepted

Your "reduction" $\widetilde{J}$ is really the identity component of the reduction. Also, I think you should assume $\mathcal{X}$ is regular (as may be arranged). The big theorem that is relevant here is due to Raynaud (see 9.5/4 in "Neron Models"): the relative identity component of the Neron model is the separated open subgroup scheme ${\rm{Pic}}^0_{\mathcal{X}/C}$ inside the algebraic space ${\rm{Pic}}_{\mathcal{X}/C}$, so passing to special fibers gives that $\widetilde{J}$ is isomorphic to ${\rm{Pic}}^0_{\mathcal{X}_{\kappa}/\kappa}$, where $\kappa$ is the residue field of $k$.

So now the problem has nothing to do with Neron models and is entirely about how to compute the toric part of ${\rm{Pic}}^0_{X/K}$ for a connected semistable curve $X$ over an algebraically closed field $K$. We claim that the character group of the torus is canonically isomorphic to the integral homology of the (connected) graph associated to this semistable curve (so the rank $t$ of the torus is always the degree-1 Betti number of the graph). This is nicely explained (modulo a few details) in 9.2/8 of "Neron Models" using some exact sequences for the etale topology; the definition of the graph that is used there is given in the paragraph preceding 9.2/8 (using loops for "self-crossing" irreducible components).

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Thank you very much for your answer. I indeed forgot to write down that $\tilde{J}$ should be the identity component. (I do have it that way in my notes, and I will edit my question accordingly.) I will see to lay my hands on "Neron Models". I want to thank you very much for you remark about the canonical isomorphism between the character group and the homology. Because that was actually the result I was looking for, but I did not want to make this question too complicated. I would accept this answer twice if I could (-; –  jmc Dec 1 '12 at 10:54

I never studied this, but here is I guess the natural approach:

Suppost that the reduction is just a cycle of smooth curves $C_1, \ldots, C_k$ with $C_i$ intersecting $C_{(i+1)\mod k}$ in one point $p_i = q_{i+1\mod k}$ with $p_i\neq q_i$.

How do we construct a line bundle on this cycle? Such a bundle $L$ gives us by pullback a collection of line bundles $L_i$ on $C_i$, hence a (surjective) map $\tilde J\to A:= \prod Pic^0(C_i)$. What is the kernel? Suppose $L$ is trivial on every $C_i$ and choose a trivialization. This gives us isomorphisms $L_{p_i} \to L_{q_i}$. The composition $L_{p_1} \to L_{q_2} \to L_{p_2} \to \ldots \to L_{q_2} \to L_{p_1}$ is an automorphism of a $1$-dimensional vector space, i.e., an element $t$ of a $1$-dimensional torus $\mathbb{G}_m$. This is all the data we need.

In case there are more cycles, we will get such a $t$ for every cycle, hence the dimension of the toric part of $\tilde J$ will be equal to the number of cycles in the graph.

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Thank you very much for your answer. It gave me some new ideas to think about. However, it is not clear to me where you use semistability. And what are your assumptions on the definition of the reduction graph? –  jmc Dec 1 '12 at 10:51
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Dear Johan, Piotr's proof sketch is exactly the intuition behind the "exact sequences for the etale topology" referred to in xuhan's answer. As for where he uses semistability, he is using it in his description of the special fibre as a union of curves with normal crossings. Regards, Matthew –  Emerton Dec 1 '12 at 12:24
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@Johan: Consider the case over a field $K$ in which a $C$ has one "self-crossing", at some $x \in C(K)$, so normalization $C' \rightarrow C$ is an isomorphism over $C - x$, with $x$-fiber a disjoint union of points $x_1$ and $x_2$ in $C'(K)$. Using the etale-local structure of $(C,x)$, a line bundle $L$ on $C$ "is" a line bundle on $C'$ equipped with an identification $L(x_1) \simeq L(x_2)$. This relativizes over any base scheme and so provides a $\mathbf{G}_m$, corresponding to a "loop" at $C$ in the graph. If $x$ isn't $K$-rational it is at least $K$-etale, and one gets non-split tori. –  user29283 Dec 1 '12 at 18:34
    
@Emerton, @xuhan, thanks both of you for explaining even more. I will try to get more feeling for the reduction in case of semistability. But I do not have experience with étale topology, and also not with normal crossings. Anyway, thanks again for the help. –  jmc Dec 2 '12 at 20:12

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