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≠2 or dim⁡(X)≥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)≠1 we classify the dominant morphisms C[n]→C[r], and thanks to this we manage to compute the whole automorphism group of C[n], namely Aut(C[n])≅Sn×Aut(C) for any n≠2, while Aut(C[2])≅S2⋉(Aut(C)×Aut(C)). Furthermore, we extend these results on the automorphisms to the case where X=C1×&×Cr is a product of curves of genus g(Ci)≥2. Finally, using the techniques developed to deal with Fulton–MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces M‾0,n(PN,d).