Automorphisms of the boundary complex of $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$

TL;DR

This paper determines the automorphism groups of the boundary complex in certain moduli spaces of stable maps, revealing symmetry structures that differ based on the degree of the maps.

Contribution

It explicitly computes the automorphism groups of the boundary complex in Kontsevich moduli spaces, showing a distinction between degree 1 and higher degrees.

Findings

For d ≥ 2, automorphism group is the symmetric group on n elements.

For d = 1, automorphism group is the symmetric group on n+1 elements.

The boundary complex for d=1 relates to the Fulton–MacPherson compactification, with non-extendable automorphisms.

Abstract

We compute the automorphism group of the dual complex $\mathsf{T}_{d, n}$ of the boundary divisor in the Kontsevich moduli space $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$. When $d \geq 2$, we find that $\mathrm{Aut}(\mathsf{T}_{d, n}) \cong \mathbb{S}_{n}$, while $\mathrm{Aut}(\mathsf{T}_{1, n}) \cong \mathbb{S}_{n + 1}$ for all $n \geq 4$. The complex $\mathsf{T}_{1, n}$ is also the dual complex of the boundary divisor in the Fulton--MacPherson compactification of the configuration space of $n$ points on $X$, if $X$ is any smooth, proper, and connected algebraic variety over $\mathbb{C}$. Following work of Massarenti, this implies that $\mathsf{T}_{1, n}$ admits automorphisms which in general do not extend to $X[n]$.