On the automorphisms of Hassett's moduli spaces

Abstract

Let ${\overline{\mathcal{M}}}_{g,A[n]}$ be the moduli stack parametrizing weighted stable curves, and let ${\overline{M}}_{g,A[n]}$ be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of ${\mathcal{M}}_{g,n}$ and $M_{g,n}$ respectively, by assigning rational weights $A=(a_1,...,a_n)$, $0<a_i\le 1$ to the markings. In particular, the classical Deligne-Mumford compactification arises for $a_1=...=a_n=1$. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of ${\overline{M}}_{0,n}$ developed by M. Kapranov, while in higher genus they may be related to the LMMP on ${\overline{M}}_{g,n}$. We compute the automorphism groups of most of the Hassett's spaces appearing in the Kapranov's blow-up construction. Furthermore, if $g\ge 1$ we compute the automorphism groups of all Hassett's spaces. In particular, we prove that if $g\ge 1$ and $2g-2+n\ge 3$ then the automorphism groups of both ${\overline{\mathcal{M}}}_{g,A[n]}$ and ${\overline{M}}_{g,A[n]}$ are isomorphic to a subgroup of $S_n$ whose elements are permutations preserving the weight data in a suitable sense.

Exposition improved. Fixed a gap, pointed out by J. Hwang, in the factorization theorems

References in corpus (3)

• The automorphism group of ${\overline{M}}_{g,n}$
• Configuration spaces of C and CP^1: some analytic properties
• On some fibrations of ${\overline{M}}_{0,n}$

Cited by in corpus (1)

• On the rigidity of moduli of curves in arbitrary characteristic