Toric varieties modulo reflections
Colin Crowley, Tao Gong, Connor Simpson

TL;DR
This paper investigates the quotient of toric varieties by reflection groups, showing it can be described via intersections with fundamental domains, and studies real toric variety quotients, notably proving contractibility for permutohedra.
Contribution
It provides a new description of quotients of toric varieties under reflection group actions and extends understanding to real toric varieties, answering open questions.
Findings
Quotient of $X_P$ by $W$ is isomorphic to $X_{P igcap D}$.
The quotient $X_P^{ ext{R}} / W$ is contractible for permutohedra.
Recovers several known results in the literature.
Abstract
Let be a finite group generated by reflections of a lattice . If a lattice polytope is preserved by , then we show that the quotient of the projective toric variety by is isomorphic to the toric variety , where is a fundamental domain for the action of . 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 is contractible when is a permutohedron.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
