Permutohedral complexes and rational curves with cyclic action

TL;DR

This paper introduces a new moduli space of rational curves with cyclic automorphisms, revealing a combinatorial structure linked to permutohedral complexes and complex reflection groups, extending previous toric cases.

Contribution

It defines a novel moduli space with boundary stratification encoded by a permutohedral complex, generalizing known cases to non-toric settings with cyclic symmetry.

Findings

Boundary strata correspond to a permutohedral complex

The structure encodes a complex reflection group action

Extends previous toric moduli space results

Abstract

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.