A Riemann–Roch theorem in tropical geometry

By Andreas Gathmann, Michael Kerber. 2006

197 citations recorded by the source corpus.

Recently, Baker and Norine have proven a Riemann–Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropical curves.

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

View Paper: https://www.semanticscholar.org/paper/3d781b871f656a1912415cb0bed639d82c608791

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Citations in the corpus

Linear series on metrized complexes of algebraic curves (2012, methodology): “The theory of divisors and linear equivalence on metrized complexes of curves generalizes both the classical theory for algebraic curves and the corresponding theory for metric graphs and tropical curves found in [26,36].”
A tropical proof of the Brill-Noether Theorem (2010, background): “Such metric graphs are examples of abstract tropical curves, in the sense of [13].”
Algebraic and tropical curves: comparing their moduli spaces (2011, methodology): “A proof of the RiemannRoch formula for a definition of rank extending the one used in [15] is given in [1].”
Brill-Noether varieties of k-gonal curves (2016, methodology): “out algebraic curves from the tropical Brill-Noether theory of the graph g;k;‘, i.e. of the loci Wr d p g;k;‘q, which can be dened in terms of the Baker-Norine rank of divisors on metric graphs (see [GK08] or [MZ08]). Our [P ] develops the tools 8 N. PFLUEGER necessary for this analysis, applicable ”
CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM (2013, background): “This can be viewed as the space of tropical rational functions on Γ ; see [ GK08 , MZ08 ].”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “Gathmann and Kerber [GK08] showed that one can deduce Theorem 4.”
The logarithmic Picard group and its tropicalization (2018, result): “ tropical Jacobian of a ﬁxed graph (not yet metrized) was only a ﬁnite set, but subdivision of the graph suggests the presence of a ﬁner geometric structure. This was explained by Gathmann and Kerber [GK08], who extended Baker and Norine’s results to metric graphs, and Amini and Caporaso added a ver”
Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem (2014, methodology): “This invariant satisfies the following Riemann–Roch theorem with respect to the canonical divisor K0 = ∑ v∈0(deg(v)− 2) v: Tropical Riemann–Roch Theorem [Baker and Norine 2007; Gathmann and Kerber 2008; Mikhalkin and Zharkov 2008].”
Tropical curves, their Jacobians and Theta functions (2006, background): “In particular, [GK06] (where this observation appeared first) formally deduces Theorem 7.”
Tropical hyperelliptic curves (2011, result): “.13 that G is therefore hyperelliptic. §2.3. Divisors on metric graphs and tropical curves. The basic theory of divisors on tropical curves that follows is due to [BN07] in the non-metric case and to [GK08], [MZ07] in the metric case. Let Γ be a metric graph. Then the divisor group Div(Γ) is the gro”