On the biregular geometry of the Fulton-MacPherson compactification

Abstract

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 $n\ne 2$ or $\dim (X)\ge 2$, 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 $g(C)\ne 1$ we classify the dominant morphisms $C[n]\to C[r]$, 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 $n\ne 2$, while $Aut(C[2])\cong S_2⋉(Aut(C)\times Aut(C))$. 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 $g(C_i)\ge 2$. Finally, using the techniques developed to deal with Fulton-MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces ${\overline{M}}_{0,n}({\mathbb{P}}^N,d)$.

30 pages

References in corpus (4)

• Configuration spaces of C and CP^1: some analytic properties
• Moduli spaces of weighted pointed stable rational curves via GIT
• Cones of effective divisors on the blown-up P^3 in general lines
• The automorphism group of $M_{0,n}^{\text{trop}}$ and ${\overline{M}}_{0,n}^{\text{trop}}$

Cited by in corpus (4)

• On the birational geometry of spaces of complete forms I: collineations and quadrics
• Birational geometry of moduli spaces of configurations of points on the line
• When is ${\overline{M}}_{0,n}({\mathbb{P}}^1,1)$ a Mori dream space?
• On the rigidity of moduli of weighted pointed stable curves