Introduction
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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
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> 
> 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
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[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](http://arxiv.org/abs/0812.0371).



Relating the toric rank of a semistable curve and the first Betti number of its reduction graph
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).


