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.