Combinatorial patchworking: Back from tropical geometry

Brugallé, Erwan; López de Medrano, Lucía; Rau, Johannes

doi:10.1090/tran/9054

Combinatorial patchworking: Back from tropical geometry
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by Erwan Brugallé, Lucía López de Medrano and Johannes Rau;

Trans. Amer. Math. Soc. 377 (2024), 6793-6826

DOI: https://doi.org/10.1090/tran/9054

Published electronically: July 29, 2024

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Abstract:

We show that, once translated to the dual setting of convex triangulations of lattice polytopes, results and methods from Arnal, Renaudineau, and Shaw [Ann. H. Lebesgue 4 (2021), pp. 1347–1387], Renaudineau and Shaw [Ann. Sci. Éc. Norm. Supér. (4) 56 (2023), pp. 945–980], Jell, Rau, and Shaw [Épijournal Géom. Algébrique 2 (2018), p. 27] and Reanaudineau, Rau, and Shaw [Real phase structures on tropical manifolds and patchworks in higher codimension, in preparation] extend to non-convex triangulations. So, while the translation of Viro’s patchworking method to the setting of tropical hypersurfaces has inspired several tremendous developments over the last two decades, we return to the original polytope setting in order to generalize and simplify some results regarding the topology of $T$-submanifolds of real toric varieties.

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Combinatorial patchworking: Back from tropical geometry
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Abstract: