On the biregular geometry of Fulton-MacPherson configuration spaces

Alex Massarenti

21 apr 2016 -- 12:30

Aula Tricerri, DiMaI, Università di Firenze

Abstract.

The Fulton-MacPherson conguration space $X[n]$ is a natural compactification of the configuration space of n ordered points on a smooth projective variety $X$. The Kontsevich moduli space $\overline{M}_{0,n}(P^N,d)$ parametrizing stable maps from n-pointed rational curves to a projective space is another widely studied algebraic variety and plays a central role in algebraic geometry, string theory and Gromov-Witten theory. These two spaces are related by an isomorphism $P^1[n] = \overline{M}_{0,n}(P^1,d)$. Furthermore, the Fulton-MacPherson conguration space $C[n]$ of n ordered points in a smooth projective curve $C$ is closely related to the Deligne-Mumford compactification $\overline{M}_{g,n}$ of the moduli space of smooth curves of genus g with n-marked points. Indeed, $\overline{M}_{0,n}$ is a GIT quotient of $P^1[n]$, and if $g(C)\geq 3$ then $C[n]$ appears as the general fiber of the forgetful morphism $ \overline{M}_{g,n} \to \overline{M}_{g}$. We will prove that if either $n \not= 2$ 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 non-elliptic curve we will classify the dominant morphisms $C[n] \to C[r]$ and thanks to this we will manage to compute the whole automorphism group of $C[n]$, namely $ Aut(C[n]) = S_{n} \times  Aut(C) $ for any $n \not= 2$. Finally, using the techniques developed to deal with the Fulton-MacPherson conguration spaces, we will study the automorphism groups of some Kontsevich moduli spaces $\overline{M}_{0,n}(P^N,d)$ .