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

By Matthew Baker, Sergey Norin. 2006

512 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/92170231069fd144805ba9c356ee002db123381b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Brill-Noether theory for curves of a fixed gonality (2017, background): “This is an immediate consequence of the tropical Abel-Jacobi theorem [10].”
Metric properties of the tropical Abel–Jacobi map (2009, background): “In order to work with tropical curves and their Jacobians from this point of view, one needs to be able to pass freely between different models for Γ , so it is desirable to set up a general theory of Jacobians for weighted graphs which corresponds to the usual notion of J (G) from [1, 3] when all e”
Lifting harmonic morphisms II: tropical curves and metrized complexes (2014, background): “, [Amini 2013; Baker and Norine 2007] for the definition and basic properties of reduced divisors).”
Algebraic and tropical curves: comparing their moduli spaces (2011, background): “Norine (in [4]) for combinatorial graphs; see also the work on G.”
Linear series on metrized complexes of algebraic curves (2012, methodology): “Our proof follows and extends the original arguments of [8] in the proof of Riemann–Roch theorem for graphs.”
A tropical proof of the Brill-Noether Theorem (2010, background): “Our starting points are the theory of ranks of divisors on graphs, as developed by Baker and Norine in their groundbreaking paper [4], and Baker’s Specialization Lemma [2], which says that the dimension of the complete linear system of a divisor on a smooth curve over a discretely valued field is le”
Tropical independence II: The maximal rank conjecture for quadrics (2015, background): “Seminal works in the development of this theory include [Baker and Norine 2007; Baker 2008; Amini and Baker 2015].”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, methodology): “3 from the Baker-Norine theorem using a clever approximation argument, whereas Mikhalkin and Zharkov [MZ08] generalized the method of proof from [BN07] to the metric graph setting.”
Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem (2014, methodology): “In other words, it has rank at least r in the sense of [Baker and Norine 2007].”