Graded Ehrhart theory and toric geometry
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916919255236608 |
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| author | Cavey, Ian |
| author_facet | Cavey, Ian |
| contents | We give two new constructions of the harmonic algebra of a lattice polytope $P$, a bigraded algebra whose character is the $q$-Ehrhart series of $P$ defined by Reiner and Rhoades. First, we show that the harmonic algebra is the associated graded algebra of the semigroup algebra of $P$ with respect to a certain natural filtration, clarifying it's relationship with the more classical semigroup algebra. We then give a geometric interpretation of the harmonic algebra as a quotient of the ring of global sections of a certain family of line bundles on the blowup of the toric variety associated to $P$ at a generic point. Using this connection to toric geometry we resolve one the main conjectures of Reiner and Rhoades by showing that the harmonic algebra is not finitely generated in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graded Ehrhart theory and toric geometry Cavey, Ian Combinatorics Algebraic Geometry 05E14 (Primary) 05E40, 14M25 (Secondary) We give two new constructions of the harmonic algebra of a lattice polytope $P$, a bigraded algebra whose character is the $q$-Ehrhart series of $P$ defined by Reiner and Rhoades. First, we show that the harmonic algebra is the associated graded algebra of the semigroup algebra of $P$ with respect to a certain natural filtration, clarifying it's relationship with the more classical semigroup algebra. We then give a geometric interpretation of the harmonic algebra as a quotient of the ring of global sections of a certain family of line bundles on the blowup of the toric variety associated to $P$ at a generic point. Using this connection to toric geometry we resolve one the main conjectures of Reiner and Rhoades by showing that the harmonic algebra is not finitely generated in general. |
| title | Graded Ehrhart theory and toric geometry |
| topic | Combinatorics Algebraic Geometry 05E14 (Primary) 05E40, 14M25 (Secondary) |
| url | https://arxiv.org/abs/2508.19176 |