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

We compute the automorphism group of the dual complex ${\mathsf{T}}_{d,n}$ of the boundary divisor in the Kontsevich moduli space ${\bar{\mathcal{M}}}_{0,n}({\mathbb{P}}^r,d)$. When $d\ge 2$, we find that $\mathrm{A}\mathrm{u}\mathrm{t}({\mathsf{T}}_{d,n})\cong {\mathbb{S}}_n$, while $\mathrm{A}\mathrm{u}\mathrm{t}({\mathsf{T}}_{1,n})\cong {\mathbb{S}}_{n+1}$ for all $n\ge 4$. 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 $X[n]$.