Poincaré duality for singular tropical hypersurfaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908766326226944 |
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| author | Dentan, Samuel |
| author_facet | Dentan, Samuel |
| contents | We establish a partial extension of the Poincaré duality theorem of Jell-Rau-Shaw to tropical hypersurfaces arising from non-primitive triangulations. We introduce a notion of level of primitivity for triangulations of lattice polytopes and show that tropical hypersurfaces satisfy a partial form of Poincaré duality determined by this level. This notion of primitivity is defined modulo a fixed integral domain and is weaker than the classical notion of primitivity. Moreover, we obtain a generalization of complete Poincaré duality over this integral domain for tropical hypersurfaces whose underlying triangulations are primitive modulo the integral domain. As a corollary, we show that any tropical hypersurface obtained by patchworking from a triangualtion of a simple lattice polytope satisfies complete Poincaré duality over the field of rational numbers, providing a converse to a theorem of Aksnes. Throughout, we allow triangulations that are not necessarily convex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24548 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poincaré duality for singular tropical hypersurfaces Dentan, Samuel Algebraic Geometry We establish a partial extension of the Poincaré duality theorem of Jell-Rau-Shaw to tropical hypersurfaces arising from non-primitive triangulations. We introduce a notion of level of primitivity for triangulations of lattice polytopes and show that tropical hypersurfaces satisfy a partial form of Poincaré duality determined by this level. This notion of primitivity is defined modulo a fixed integral domain and is weaker than the classical notion of primitivity. Moreover, we obtain a generalization of complete Poincaré duality over this integral domain for tropical hypersurfaces whose underlying triangulations are primitive modulo the integral domain. As a corollary, we show that any tropical hypersurface obtained by patchworking from a triangualtion of a simple lattice polytope satisfies complete Poincaré duality over the field of rational numbers, providing a converse to a theorem of Aksnes. Throughout, we allow triangulations that are not necessarily convex. |
| title | Poincaré duality for singular tropical hypersurfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.24548 |