On the automorphisms of Hassett's moduli spaces

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 $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ 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 $g\geq 1$ we compute the automorphism groups of all Hassett's spaces. In particular, we prove that if $g\geq 1$ and $2g-2+n\geq 3$ 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.