Abstract
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.
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Notes
A convex piecewise-linear function means a function which is locally written as the maximum of finitely many affine linear functions. It has finitely many slopes if and only if it can be written as the maximum of finitely many affine linear functions.
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Acknowledgements
I am deeply grateful to my advisor Hiroshi Iritani for his advice. This paper would not have been possible without his guidance. Special thanks go to Yuji Odaka for his helpful advice. In particular, he suggested to study the Riemann-Roch inequality for surfaces with trivial canonical class. I would like to thank Dustin Cartwright for his valuable comments. He informed me of his definition of \(h^{0}(X,D)\) and suggested to study the relationship between \(h^{0}(X,D)\) and \(\dim H^{0}(X,\mathcal {O}(D))\), and also to study the integrality of \(\frac{1}{n!}D^{n}\). Theorems 44 and 47 were then obtained.
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This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 18J21511.
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Sumi, K. Tropical theta functions and Riemann–Roch inequality for tropical Abelian surfaces. Math. Z. 297, 1329–1351 (2021). https://doi.org/10.1007/s00209-020-02559-9
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DOI: https://doi.org/10.1007/s00209-020-02559-9