Toric varieties modulo reflections
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915758186954752 |
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| author | Crowley, Colin Gong, Tao Simpson, Connor |
| author_facet | Crowley, Colin Gong, Tao Simpson, Connor |
| contents | Let $W$ be a finite group generated by reflections of a lattice $M$. If a lattice polytope $P \subset M \otimes_{\mathbb Z}\mathbb R$ is preserved by $W$, then we show that the quotient of the projective toric variety $X_P$ by $W$ is isomorphic to the toric variety $X_{P \cap D}$, where $D$ is a fundamental domain for the action of $W$. This answers a question of Horiguchi-Masuda-Shareshian-Song, and recovers results of Blume, of Song, of the second author, and of Gui-Hu-Liu. We also study quotients of real toric varieties, proving that $X_P^{\mathbb R} / W$ is contractible when $P$ is a permutohedron. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14653 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric varieties modulo reflections Crowley, Colin Gong, Tao Simpson, Connor Combinatorics Commutative Algebra Algebraic Geometry 14M25, 13F65, 57S12 Let $W$ be a finite group generated by reflections of a lattice $M$. If a lattice polytope $P \subset M \otimes_{\mathbb Z}\mathbb R$ is preserved by $W$, then we show that the quotient of the projective toric variety $X_P$ by $W$ is isomorphic to the toric variety $X_{P \cap D}$, where $D$ is a fundamental domain for the action of $W$. This answers a question of Horiguchi-Masuda-Shareshian-Song, and recovers results of Blume, of Song, of the second author, and of Gui-Hu-Liu. We also study quotients of real toric varieties, proving that $X_P^{\mathbb R} / W$ is contractible when $P$ is a permutohedron. |
| title | Toric varieties modulo reflections |
| topic | Combinatorics Commutative Algebra Algebraic Geometry 14M25, 13F65, 57S12 |
| url | https://arxiv.org/abs/2410.14653 |