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This might be a bit of a broad question, or maybe even questions.

Recently I have learned about the connection between algebraic geometry and graph theory, via the dual graph of a curve. I have also seen people call it the reduction graph.

I find it really fascinating that on both sides there is a notion of divisors, and for example the statement of Riemann-Roch is identical (up to notation). Further, both sides seem to have genera, Picard groups and other notions that I only knew of in algebraic geometry before. I even saw a kind of Riemann-Hurwitz formula for graphs.

In short I would like to know if there exists an article or book that introduces this connection for someone who knows both fields at graduate level, but close to nothing about the connection between them.

Specifically I would like to have answers to these questions (an answer or a reference is perfect):

  • What is the general/natural setting of this connection? I.e., what are necessary and/or sufficient conditions on the curve to have a "nice connection"? (Where "nice connection" is purposely vague.)

Assuming that our curve satisfies the above conditions:

  • What invariants carry over? I have read that the genus of the curve is also the genus of the dual graph. How about the Picard group and/or other notions that we have on both sides of the connection?
  • Is this connection functorial for a suitable category of suitable curves?

I realize that these questions are not very specific (of which the fact that this question does not contain any LaTeX markup might be a witness), but I hope that someone can give me pointers to introductory material. All articles that Google supplies to me seem to assume that the reader is already familiar with quite a few facts about this connection between curves and dual graphs.

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Reduction graphs arise naturally in the reduction theory of a curve. More precisely, let $X$ be a curve over a number field $K$. (It's more natural to consider a local field $K$.) Let $O_K$ be the ring of integers of $K$. Then, if $\mathcal{X}$ is a model of $X$ over $O_K$, you can consider the "reduction" of $\mathcal{X}$ at a finite place $v$ of $O_K$, i.e, the geometric fibre of $\mathcal X\to $ Spec $O_K$ over $v$. The "reduction" behaviour of your curve $X$ at the place $v$ can be tautologically read off its "reduction graph". See Chapter 10.1.4 of Liu's book for more details. –  Ariyan Javanpeykar Oct 27 '12 at 17:31
    
@Ariyan Thanks for the reference. I inderstood the definition, but I wondered if there were any 'natural' conditions that one usually puts on the curve $X/K$, or the model $\mathcal{X}/O_{K}$. Finiteness conditions (which one?) or properness (probably not for the model), regularness, etc... But, I will first take a look at [§10, Liu]. –  jmc Oct 28 '12 at 21:27
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@Johan. I posted a long comment as an answer below. The curves we consider are in general smooth projective and geometrically connected over a number field (or local field) $K$. The theory becomes nice if the genus of $X$ is positive. There are different "natural" models that one can consider for $X$ over $O_K$. All of these give different reduction graphs. –  Ariyan Javanpeykar Oct 29 '12 at 8:11

1 Answer 1

I'm not sure if this should be posted as an answer, but it became too long to post as a comment.

Let $X$ be a smooth projective geometrically connected curve of positive genus over a number field $K$. (The condition on the genus will be used below.) Let me explain some elements of the theory of models for $X$ over $O_K$.

Firstly, there exists a model of $X$ over $O_K$. In fact, there is a closed immersion $X\to\mathbf{P}^n_{K}$. The Zariski closure of $X$ in $\mathbf{P}^n_{O_K}$ via $X\to \mathbf{P}^n_K \subset \mathbf{P}^n_{O_K}$ gives a model for $X$ over $\mathcal{O}_K$. It is a projective model. It is irreducible and reduced as a scheme. Normalizing this scheme gives a normal (projective) model. Now, you can use "resolution of singularities", e.g., Lipman's theorem, to obtain a regular model for $X$ over $O_K$. Then, subsequently contracting all the $-1$-curves (also called exceptional curves) on this regular model you will obtain a regular projective model $\mathcal{X}_{min}$ for $X$ over $O_K$ which is "minimal". We call $\mathcal{X}_{min}$ the minimal regular model of $X$ over $O_K$. It is this minimality condition which is quite natural (for curves of positive genus).

Before continuing, let me give some references for the above paragraph. You can find them all in Liu's book.

For basic facts on the fibres of a model for $X$ over $O_K$ see Chapter 8.3.1.

Desingularization of a normal model is explained in Chapter 8.3.4. For example, a precise statement of Lipman's theorem can be found in Theorem 8.3.44.

Exceptional divisors on a model are defined in Definition 9.3.1.

There are two notions of minimality (Definition 9.3.12) which are shown to coincide in Corollary 9.3.24 (when the generic fibre has positive genus).

Finally, the existence of the minimal regular model is obtained in Therem 9.3.21.

To summarize, the minimal regular model exists and is unique. Thus, it is "natural" to look at the dual reduction graphs of this model.

The (geometric) fibres of $\mathcal{X}_{min}$ are, in general, very complicated. Of course, by Proposition 8.3.11, almost all of them are smooth. But, the geometric fibre over a "bad" place will be a singular curve with "complicated" singularities.

This brings us to semi-stable reduction. In fact, if your reduction isn't a smooth curve, you could hope for the next best thing: semi-stability.

A curve over an algebraically closed field is semi-stable if it is connected, reduced and has only ordinary double singularities. The model $\mathcal{X}_{min}$ doesn't have semi-stable geometric fibres in general, but a deep theorem of Grothendieck and Mumford states that there exists a finite field extension $L/K$ such that the minimal regular model of $X_L$ over $O_L$ is semi-stable, i.e., its geometric fibres are semi-stable. Considering the reduction graph of this model is also very natural; see Definition 10.3.17 in Liu's book.

(Caution: just because the singularities of the geometric fibres have "easy" singularities, doesn't mean the configuration of the irreducible fibres is easy. In fact, determining the configuration is a difficult problem in arithmetic geometry. For example, the semi-stable reduction of the modular curve $X_0(n)$ (for all $n$) has only been achieved recently by Jared Weinstein; see http://arxiv.org/abs/1010.4241 .)

There is also the notion of stability. This is stronger than semi-stability. The stable reduction of a curve is also "natural" to consider.

I'll finish with a quick note on elliptic curves.

Let $E/K$ be an elliptic curve. Then you can consider the minimal regular model and, for some suitable $L/K$, the semi-stable reduction of $E_L$ over $O_L$.

It is also natural to ask whether $E$ has a model over $O_K$ which extends the group structure and the smoothness property of $E/K$. Such a model doesn't exist in general if we demand properness. If we drop the properness condition, then such a model exists. (The finiteness conditions being as usual: the model is of finite type and separated over the base scheme Spec $O_K$.) You can then ask for a model which extends this group structure in the "best possible" way. This brings us to Neron models. Such a model for $E$ over $O_K$ always exists. In fact, you can show that the smooth locus $\mathcal{E}_{min}^{sm}$ of the minimal regular model $\mathcal E_{min}$ of $E$ over $O_K$ is the Neron model of $E$ over $O_K$; see Theorem 10.2 for a discussion of this beautiful theory.

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@Ariyan Thank you very much for this 'comment'! I already new about the Néron model for elliptic curves, but you really helped me with the minimal regular model. I hope to get enough feeling for these configurations to be able to understand why for example the genus of the reduction graph is the same as the genus of the curve. –  jmc Oct 29 '12 at 11:56

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