On the biregular geometry of the Fulton-MacPherson compactification

TL;DR

This paper investigates the automorphism groups of Fulton-MacPherson compactifications and related moduli spaces, establishing their structure and classifying morphisms for certain algebraic varieties, especially curves of genus not equal to one.

Contribution

It provides a detailed description of the automorphism groups of Fulton-MacPherson spaces and classifies morphisms for curves, extending results to products of curves and related moduli spaces.

Findings

Automorphism group of $X[n]$ is isomorphic to that of $X$ under certain conditions.

Automorphism group of $C[n]$ is $S_n imes Aut(C)$ for $n eq 2$.

Classified dominant morphisms $C[n] ightarrow C[r]$ for curves of genus not equal to one.

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 $n\neq 2$ 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 $\overline{M}_{0,n}(\mathbb{P}^N,d)$.