On the biregular geometry of the Fulton-MacPherson compactification

Let $X[n]$ be the Fulton-MacPherson compactification of the configuration space of $n$ ordered points on a smooth projective variety $X$. We prove that if either $n\neq 2$ or $\dim(X)\geq 2$, then the connected component of the identity of $Aut(X[n])$ is isomorphic to the connected component of the identity of $Aut(X)$. When $X = C$ is a curve of genus $g(C)\neq 1$ we classify the dominant morphisms $C[n]\rightarrow C[r]$, and thanks to this we manage to compute the whole automorphism group of $C[n]$, namely $Aut(C[n])\cong S_n\times Aut(C)$ for any $n\neq 2$, while $Aut(C[2])\cong S_2\ltimes (Aut(C)\times Aut(C))$. Furthermore, we extend these results on the automorphisms to the case where $X = C_1\times ... \times C_r$ is a product of curves of genus $g(C_i)\geq 2$. Finally, using the techniques developed to deal with Fulton-MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces $\overline{M}_{0,n}(\mathbb{P}^N,d)$.