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arXiv · 2207.05562

Riemann-Roch for Toric Rank Functions

Dalton Bidleman
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Abstract

In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann--Roch holds for trees and polygons. We also prove an asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large degree). Finally, we also provide magma code and computational evidence that Riemann--Roch holds for the toric rank function.

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Dalton Bidleman. 2022-07-09. Riemann-Roch for Toric Rank Functions. https://arxiv.org/abs/2207.05562

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