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A Riemann-Roch theorem in tropical geometry. (English) Zbl 1187.14066
Recently, M. Baker and S. Norine [Adv. Math. 215, 766–788 (2007; Zbl 1124.05049)] have proven a Riemann–Roch theorem for graphs. The authors extend this result to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropical curves. Their proof relies on the result of Baker and Norine; a different (independent) proof of the Riemann–Roch theorem for tropical curves is given by G. Mikhalkin and I. Zharkov [Contemp. Math. 465, 203–230 (2008; Zbl 1152.14028)].

MSC:
14T05 Tropical geometry (MSC2010)
14H55 Riemann surfaces; Weierstrass points; gap sequences
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C38 Paths and cycles
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References:
[1] Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. (to appear, 2007), preprint math.CO/0608360 · Zbl 1124.05049
[2] Gathmann, A., Markwig, H.: Kontsevich’s formula and the WDVV equations in tropical geometry. Preprint math.AG/0509628 · Zbl 1131.14057
[3] Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and Theta functions. preprint math. AG/0612267 · Zbl 1152.14028
[4] Zhang S. (1993). Admissible pairing on a curve. Invent. Math. 112: 171–193 · Zbl 0795.14015
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