Riemann–Roch theory for weighted graphs and tropical curves

By Omid Amini, Lucia Caporaso. 2011

66 citations recorded by the source corpus.

Abstract We define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices; we prove that the Riemann–Roch theorem holds in both cases. We extend Baker’s Specialization Lemma to weighted graphs.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9ea07ec9d8655eb07354668ea7225591879e750e

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In Algebra and Number Theory

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Citations in the corpus

Tropical hyperelliptic curves (2011, background): “ ∼ E′ }. Deﬁnition 2.3. A metric graph is hyperelliptic if it has a divisor of degree 2 and rank 1. Let us extend the above deﬁnition to abstract tropical curves. The deﬁnition we will give is due to [ABC11]. First, we recall: Deﬁnition 2.4. An abstract tropical curve is a triple (G,w,l), where (G,l”
Lifting harmonic morphisms II: tropical curves and metrized complexes (2014, background): “According to the specialization inequality [Baker 2008; Amini and Caporaso 2013; Amini and Baker 2014]), for any divisor D in Div(X) one has”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, methodology): “OF LINEAR SERIES FROM THE TROPICAL POINT OF VIEW 15 Figure 7. Two metric graphs of genus 3, the rst of which is hyperelliptic, and the second of which is not. (All edges have length 1.) x2. Following [AC13], we dene a new metric graph # by attaching !(x) loops of arbitrary positive length at each po”
Gonality of algebraic curves and graphs (2012, methodology): “l, the free abelian group generated by the points of Γ (viewed as a metric space). The weightless case has been carefully studied in [GK08], for example; the general case has been recently treated in [AC11], to which we refer for the deﬁnition of the rank rΓ(D) of any D ∈ Div(Γ) and its basic proper”
Algebraic and tropical curves: comparing their moduli spaces (2011, methodology): “ful tool to relate the combinatorial with the algebro-geometric divisor theory, can be reﬁned to be applicable in the classical algebro-geometric set-up (see [7]) and extended to weighted graphs (see [1]). A related problem is the extension of the Riemann-Roch theorem, and of other classical theorem”
Linear series on metrized complexes of algebraic curves (2012, methodology): “This loss of information was partially remedied in [6] by looking at vertex-weighted graphs, where the weight attached to a vertex is the genus of the corresponding irreducible component of the special fiber.”