Toric varieties modulo reflections

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 fun...

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Bibliographic Details
Main Authors Crowley, Colin, Gong, Tao, Simpson, Connor
Format Journal Article
LanguageEnglish
Published 18.10.2024
Subjects
Mathematics - Algebraic Geometry
Mathematics - Combinatorics
Mathematics - Commutative Algebra
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Summary: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 the second author, and of Gui-Hu-Liu.
DOI:10.48550/arxiv.2410.14653