2 Fixed mathematics according to a comment

## 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 $1 \to L \to \tilde{J} tilde{J}^{0} \to A \to 0,$ 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.

1

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

## 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 reduction $\tilde{J}$ fits into an exact sequence $1 \to L \to \tilde{J} \to A \to 0,$ 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.