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Riemann-Roch and Abel-Jacobi theory on a finite graph

Matthew Baker·Serguei Norine
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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.

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Matthew Baker, Serguei Norine. 2007-07-09. Riemann-Roch and Abel-Jacobi theory on a finite graph. https://arxiv.org/abs/math/0608360

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