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

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 $X[n]$.