Toric varieties modulo reflections

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 $W$ 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.