Riemann-Roch and Abel-Jacobi theory on a finite graph
Abstract
It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.
Cite
@article{arxiv.math/0608360,
title = {Riemann-Roch and Abel-Jacobi theory on a finite graph},
author = {Matthew Baker and Serguei Norine},
journal= {arXiv preprint arXiv:math/0608360},
year = {2007}
}
Comments
35 pages. v3: Several minor changes made, mostly fixing typographical errors. This is the final version, to appear in Adv. Math